id	sid	tid	token	lemma	pos
easat-9645	1	1	edelweiss	edelweiss	PROPN
easat-9645	1	2	applied	apply	VERB
easat-9645	1	3	science	science	NOUN
easat-9645	1	4	and	and	CCONJ
easat-9645	1	5	technology	technology	NOUN
easat-9645	1	6	issn	issn	PROPN
easat-9645	1	7	:	:	PUNCT
easat-9645	1	8	2576	2576	NUM
easat-9645	1	9	-	-	SYM
easat-9645	1	10	8484	8484	NUM
easat-9645	1	11	vol	vol	NOUN
easat-9645	1	12	.	.	PROPN
easat-9645	2	1	9	9	NUM
easat-9645	2	2	,	,	PUNCT
easat-9645	2	3	no	no	INTJ
easat-9645	2	4	.	.	NOUN
easat-9645	2	5	8	8	NUM
easat-9645	2	6	,	,	PUNCT
easat-9645	2	7	1498	1498	NUM
easat-9645	2	8	-	-	SYM
easat-9645	2	9	1523	1523	NUM
easat-9645	2	10	2025	2025	NUM
easat-9645	2	11	publisher	publisher	NOUN
easat-9645	2	12	:	:	PUNCT
easat-9645	2	13	learning	learn	VERB
easat-9645	2	14	gate	gate	NOUN
easat-9645	2	15	doi	doi	PROPN
easat-9645	2	16	:	:	PUNCT
easat-9645	2	17	10.55214/2576	10.55214/2576	NUM
easat-9645	2	18	-	-	SYM
easat-9645	2	19	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	3	1	©	©	PROPN
easat-9645	3	2	2025	2025	NUM
easat-9645	3	3	by	by	ADP
easat-9645	3	4	the	the	DET
easat-9645	3	5	authors	author	NOUN
easat-9645	3	6	;	;	PUNCT
easat-9645	3	7	licensee	licensee	PROPN
easat-9645	3	8	learning	learning	NOUN
easat-9645	3	9	gate	gate	NOUN
easat-9645	3	10	©	©	PROPN
easat-9645	3	11	2025	2025	NUM
easat-9645	3	12	by	by	ADP
easat-9645	3	13	the	the	DET
easat-9645	3	14	authors	author	NOUN
easat-9645	3	15	;	;	PUNCT
easat-9645	3	16	licensee	licensee	PROPN
easat-9645	3	17	learning	learning	NOUN
easat-9645	3	18	gate	gate	NOUN
easat-9645	3	19	history	history	NOUN
easat-9645	3	20	:	:	PUNCT
easat-9645	3	21	received	receive	VERB
easat-9645	3	22	:	:	PUNCT
easat-9645	3	23	20	20	NUM
easat-9645	3	24	june	june	PROPN
easat-9645	3	25	2025	2025	NUM
easat-9645	3	26	;	;	PUNCT
easat-9645	3	27	revised	revise	VERB
easat-9645	3	28	:	:	PUNCT
easat-9645	3	29	8	8	NUM
easat-9645	3	30	august	august	PROPN
easat-9645	3	31	2025	2025	NUM
easat-9645	3	32	;	;	PUNCT
easat-9645	3	33	accepted	accept	VERB
easat-9645	3	34	:	:	PUNCT
easat-9645	3	35	12	12	NUM
easat-9645	3	36	august	august	PROPN
easat-9645	3	37	2025	2025	NUM
easat-9645	3	38	;	;	PUNCT
easat-9645	3	39	published	publish	VERB
easat-9645	3	40	:	:	PUNCT
easat-9645	4	1	25	25	NUM
easat-9645	4	2	august	august	PROPN
easat-9645	4	3	2025	2025	NUM
easat-9645	4	4	*	*	PUNCT
easat-9645	4	5	correspondence	correspondence	NOUN
easat-9645	4	6	:	:	PUNCT
easat-9645	4	7	mannan.iu31@uttarauniversity.edu.bd	mannan.iu31@uttarauniversity.edu.bd	PROPN
easat-9645	4	8	hilbert	hilbert	PROPN
easat-9645	4	9	and	and	CCONJ
easat-9645	4	10	inner	inner	ADJ
easat-9645	4	11	product	product	NOUN
easat-9645	4	12	spaces	space	VERB
easat-9645	4	13	:	:	PUNCT
easat-9645	5	1	theory	theory	NOUN
easat-9645	5	2	,	,	PUNCT
easat-9645	5	3	visualization	visualization	NOUN
easat-9645	5	4	,	,	PUNCT
easat-9645	5	5	and	and	CCONJ
easat-9645	5	6	applications	application	NOUN
easat-9645	5	7	in	in	ADP
easat-9645	5	8	machine	machine	NOUN
easat-9645	5	9	learning	learn	VERB
easat-9645	5	10	md	md	PROPN
easat-9645	5	11	.	.	PUNCT
easat-9645	6	1	abdul	abdul	PROPN
easat-9645	6	2	mannan1	mannan1	PROPN
easat-9645	6	3	*	*	PROPN
easat-9645	6	4	,	,	PUNCT
easat-9645	6	5	mohammad	mohammad	PROPN
easat-9645	6	6	alauddin2	alauddin2	PROPN
easat-9645	6	7	,	,	PUNCT
easat-9645	6	8	md	md	PROPN
easat-9645	6	9	nazmul	nazmul	PROPN
easat-9645	6	10	hasan3	hasan3	PROPN
easat-9645	6	11	,	,	PUNCT
easat-9645	6	12	provakar	provakar	PROPN
easat-9645	6	13	ghose4	ghose4	PROPN
easat-9645	6	14	,	,	PUNCT
easat-9645	6	15	md	md	PROPN
easat-9645	6	16	.	.	PROPN
easat-9645	6	17	amzad	amzad	PROPN
easat-9645	6	18	hossain5	hossain5	PROPN
easat-9645	6	19	,	,	PUNCT
easat-9645	6	20	md	md	PROPN
easat-9645	6	21	.	.	PROPN
easat-9645	6	22	shafikul	shafikul	PROPN
easat-9645	6	23	islam6	islam6	PROPN
easat-9645	6	24	,	,	PUNCT
easat-9645	6	25	md	md	PROPN
easat-9645	6	26	.	.	PROPN
easat-9645	6	27	shafiul	shafiul	PROPN
easat-9645	7	1	alam	alam	PROPN
easat-9645	7	2	chowdhury7	chowdhury7	PROPN
easat-9645	7	3	,	,	PUNCT
easat-9645	7	4	bilkish	bilkish	ADJ
easat-9645	7	5	akter8	akter8	NOUN
easat-9645	7	6	,	,	PUNCT
easat-9645	7	7	sahib	sahib	PROPN
easat-9645	7	8	jada	jada	PROPN
easat-9645	7	9	eyakub	eyakub	PROPN
easat-9645	7	10	khan9	khan9	PROPN
easat-9645	7	11	1department	1department	NUM
easat-9645	7	12	of	of	ADP
easat-9645	7	13	mathematics	mathematics	PROPN
easat-9645	7	14	,	,	PUNCT
easat-9645	7	15	uttara	uttara	PROPN
easat-9645	7	16	university	university	PROPN
easat-9645	7	17	,	,	PUNCT
easat-9645	7	18	dhaka	dhaka	PROPN
easat-9645	7	19	,	,	PUNCT
easat-9645	7	20	bangladesh	bangladesh	PROPN
easat-9645	7	21	;	;	PUNCT
easat-9645	7	22	mannan.iu31@uttarauniversity.edu.bd	mannan.iu31@uttarauniversity.edu.bd	PROPN
easat-9645	7	23	(	(	PUNCT
easat-9645	7	24	m.a.m	m.a.m	ADJ
easat-9645	7	25	.	.	PUNCT
easat-9645	7	26	)	)	PUNCT
easat-9645	7	27	.	.	PUNCT
easat-9645	8	1	2,6,7department	2,6,7department	NUM
easat-9645	8	2	of	of	ADP
easat-9645	8	3	computer	computer	NOUN
easat-9645	8	4	science	science	NOUN
easat-9645	8	5	and	and	CCONJ
easat-9645	8	6	engineering	engineering	NOUN
easat-9645	8	7	,	,	PUNCT
easat-9645	8	8	uttara	uttara	PROPN
easat-9645	8	9	university	university	PROPN
easat-9645	8	10	,	,	PUNCT
easat-9645	8	11	dhaka	dhaka	PROPN
easat-9645	8	12	,	,	PUNCT
easat-9645	8	13	bangladesh	bangladesh	PROPN
easat-9645	8	14	;	;	PUNCT
easat-9645	8	15	alauddinsorwer@gmail.com	alauddinsorwer@gmail.com	X
easat-9645	8	16	(	(	PUNCT
easat-9645	8	17	m.a	m.a	NOUN
easat-9645	8	18	.	.	PROPN
easat-9645	8	19	)	)	PUNCT
easat-9645	9	1	shafik.cse@gmail.com	shafik.cse@gmail.com	PUNCT
easat-9645	10	1	(	(	PUNCT
easat-9645	10	2	m.s.i	m.s.i	NOUN
easat-9645	10	3	.	.	PUNCT
easat-9645	10	4	)	)	PUNCT
easat-9645	11	1	shafiul.cse@uttarauniversity.edu.bd	shafiul.cse@uttarauniversity.edu.bd	VERB
easat-9645	11	2	(	(	PUNCT
easat-9645	11	3	m.s.a.c	m.s.a.c	PROPN
easat-9645	11	4	.	.	PUNCT
easat-9645	11	5	)	)	PUNCT
easat-9645	11	6	.	.	PUNCT
easat-9645	12	1	3pompea	3pompea	NUM
easat-9645	12	2	college	college	NOUN
easat-9645	12	3	of	of	ADP
easat-9645	12	4	business	business	NOUN
easat-9645	12	5	,	,	PUNCT
easat-9645	12	6	university	university	NOUN
easat-9645	12	7	of	of	ADP
easat-9645	12	8	new	new	PROPN
easat-9645	12	9	haven	haven	NOUN
easat-9645	12	10	,	,	PUNCT
easat-9645	12	11	west	west	PROPN
easat-9645	12	12	haven	haven	PROPN
easat-9645	12	13	,	,	PUNCT
easat-9645	12	14	ct	ct	PROPN
easat-9645	12	15	,	,	PUNCT
easat-9645	12	16	usa	usa	PROPN
easat-9645	12	17	:	:	PUNCT
easat-9645	12	18	mhasa9@unh.newhaven.edu	mhasa9@unh.newhaven.edu	PROPN
easat-9645	12	19	(	(	PUNCT
easat-9645	12	20	m.n.h	m.n.h	ADJ
easat-9645	12	21	.	.	PUNCT
easat-9645	12	22	)	)	PUNCT
easat-9645	12	23	.	.	PUNCT
easat-9645	13	1	4pompea	4pompea	NUM
easat-9645	13	2	college	college	NOUN
easat-9645	13	3	of	of	ADP
easat-9645	13	4	business	business	NOUN
easat-9645	13	5	,	,	PUNCT
easat-9645	13	6	university	university	NOUN
easat-9645	13	7	of	of	ADP
easat-9645	13	8	new	new	PROPN
easat-9645	13	9	haven	haven	NOUN
easat-9645	13	10	,	,	PUNCT
easat-9645	13	11	west	west	PROPN
easat-9645	13	12	haven	haven	PROPN
easat-9645	13	13	,	,	PUNCT
easat-9645	13	14	ct	ct	PROPN
easat-9645	13	15	,	,	PUNCT
easat-9645	13	16	usa	usa	PROPN
easat-9645	13	17	:	:	PUNCT
easat-9645	13	18	pghos1@unh.newhaven.edu	pghos1@unh.newhaven.edu	PROPN
easat-9645	13	19	(	(	PUNCT
easat-9645	13	20	p.g	p.g	PROPN
easat-9645	13	21	.	.	PROPN
easat-9645	13	22	)	)	PUNCT
easat-9645	13	23	.	.	PUNCT
easat-9645	14	1	5department	5department	NUM
easat-9645	14	2	of	of	ADP
easat-9645	14	3	education	education	NOUN
easat-9645	14	4	,	,	PUNCT
easat-9645	14	5	uttara	uttara	PROPN
easat-9645	14	6	university	university	PROPN
easat-9645	14	7	,	,	PUNCT
easat-9645	14	8	dhaka	dhaka	PROPN
easat-9645	14	9	,	,	PUNCT
easat-9645	14	10	bangladesh	bangladesh	PROPN
easat-9645	14	11	:	:	PUNCT
easat-9645	14	12	amzad_uuedu@uttarauniversity.edu.bd	amzad_uuedu@uttarauniversity.edu.bd	PROPN
easat-9645	14	13	(	(	PUNCT
easat-9645	14	14	m.a.h	m.a.h	PROPN
easat-9645	14	15	.	.	PUNCT
easat-9645	14	16	)	)	PUNCT
easat-9645	14	17	.	.	PUNCT
easat-9645	15	1	8department	8department	NUM
easat-9645	15	2	of	of	ADP
easat-9645	15	3	economics	economic	NOUN
easat-9645	15	4	,	,	PUNCT
easat-9645	15	5	govt	govt	NOUN
easat-9645	15	6	.	.	PUNCT
easat-9645	16	1	titumir	titumir	PROPN
easat-9645	16	2	college	college	PROPN
easat-9645	16	3	,	,	PUNCT
easat-9645	16	4	dhaka	dhaka	PROPN
easat-9645	16	5	,	,	PUNCT
easat-9645	16	6	bangladesh	bangladesh	PROPN
easat-9645	16	7	:	:	PUNCT
easat-9645	16	8	bilkishakter2035@gmail.com	bilkishakter2035@gmail.com	X
easat-9645	16	9	(	(	PUNCT
easat-9645	16	10	b.a	b.a	PROPN
easat-9645	16	11	.	.	PROPN
easat-9645	16	12	)	)	PUNCT
easat-9645	16	13	.	.	PUNCT
easat-9645	17	1	9biman	9biman	NUM
easat-9645	17	2	bangladesh	bangladesh	PROPN
easat-9645	17	3	airlines	airline	NOUN
easat-9645	17	4	,	,	PUNCT
easat-9645	17	5	kurmitola	kurmitola	PROPN
easat-9645	17	6	,	,	PUNCT
easat-9645	17	7	dhaka	dhaka	PROPN
easat-9645	17	8	,	,	PUNCT
easat-9645	17	9	bangladesh	bangladesh	PROPN
easat-9645	17	10	:	:	PUNCT
easat-9645	17	11	sk2xenon@gmail.com	sk2xenon@gmail.com	X
easat-9645	17	12	(	(	PUNCT
easat-9645	17	13	s.j	s.j	PROPN
easat-9645	17	14	.	.	PROPN
easat-9645	17	15	)	)	PUNCT
easat-9645	17	16	.	.	PUNCT
easat-9645	18	1	abstract	abstract	ADV
easat-9645	18	2	:	:	PUNCT
easat-9645	18	3	this	this	DET
easat-9645	18	4	study	study	NOUN
easat-9645	18	5	investigates	investigate	VERB
easat-9645	18	6	the	the	DET
easat-9645	18	7	mathematical	mathematical	ADJ
easat-9645	18	8	structure	structure	NOUN
easat-9645	18	9	of	of	ADP
easat-9645	18	10	hilbert	hilbert	PROPN
easat-9645	18	11	spaces	space	NOUN
easat-9645	18	12	,	,	PUNCT
easat-9645	18	13	defined	define	VERB
easat-9645	18	14	as	as	ADP
easat-9645	18	15	complete	complete	ADJ
easat-9645	18	16	inner	inner	ADJ
easat-9645	18	17	product	product	NOUN
easat-9645	18	18	spaces	space	NOUN
easat-9645	18	19	,	,	PUNCT
easat-9645	18	20	and	and	CCONJ
easat-9645	18	21	their	their	PRON
easat-9645	18	22	significance	significance	NOUN
easat-9645	18	23	in	in	ADP
easat-9645	18	24	both	both	CCONJ
easat-9645	18	25	theoretical	theoretical	ADJ
easat-9645	18	26	and	and	CCONJ
easat-9645	18	27	applied	applied	ADJ
easat-9645	18	28	contexts	contexts	NOUN
easat-9645	18	29	.	.	PUNCT
easat-9645	19	1	we	we	PRON
easat-9645	19	2	begin	begin	VERB
easat-9645	19	3	by	by	ADP
easat-9645	19	4	exploring	explore	VERB
easat-9645	19	5	their	their	PRON
easat-9645	19	6	foundational	foundational	ADJ
easat-9645	19	7	properties	property	NOUN
easat-9645	19	8	,	,	PUNCT
easat-9645	19	9	including	include	VERB
easat-9645	19	10	inner	inner	ADJ
easat-9645	19	11	products	product	NOUN
easat-9645	19	12	,	,	PUNCT
easat-9645	19	13	orthogonality	orthogonality	NOUN
easat-9645	19	14	,	,	PUNCT
easat-9645	19	15	and	and	CCONJ
easat-9645	19	16	completeness	completeness	NOUN
easat-9645	19	17	,	,	PUNCT
easat-9645	19	18	which	which	PRON
easat-9645	19	19	extend	extend	VERB
easat-9645	19	20	euclidean	euclidean	ADJ
easat-9645	19	21	geometric	geometric	ADJ
easat-9645	19	22	concepts	concept	NOUN
easat-9645	19	23	to	to	PART
easat-9645	19	24	infinite	infinite	VERB
easat-9645	19	25	-	-	PUNCT
easat-9645	19	26	dimensional	dimensional	ADJ
easat-9645	19	27	settings	setting	NOUN
easat-9645	19	28	.	.	PUNCT
easat-9645	20	1	key	key	ADJ
easat-9645	20	2	mathematical	mathematical	ADJ
easat-9645	20	3	tools	tool	NOUN
easat-9645	20	4	,	,	PUNCT
easat-9645	20	5	including	include	VERB
easat-9645	20	6	the	the	DET
easat-9645	20	7	cauchy	cauchy	PROPN
easat-9645	20	8	–	–	PUNCT
easat-9645	20	9	schwarz	schwarz	NOUN
easat-9645	20	10	inequality	inequality	NOUN
easat-9645	20	11	,	,	PUNCT
easat-9645	20	12	triangle	triangle	NOUN
easat-9645	20	13	inequality	inequality	NOUN
easat-9645	20	14	,	,	PUNCT
easat-9645	20	15	polarization	polarization	NOUN
easat-9645	20	16	identity	identity	NOUN
easat-9645	20	17	,	,	PUNCT
easat-9645	20	18	and	and	CCONJ
easat-9645	20	19	apollonius	apollonius	NOUN
easat-9645	20	20	identity	identity	NOUN
easat-9645	20	21	,	,	PUNCT
easat-9645	20	22	are	be	AUX
easat-9645	20	23	analyzed	analyze	VERB
easat-9645	20	24	to	to	PART
easat-9645	20	25	highlight	highlight	VERB
easat-9645	20	26	the	the	DET
easat-9645	20	27	analytical	analytical	ADJ
easat-9645	20	28	framework	framework	NOUN
easat-9645	20	29	of	of	ADP
easat-9645	20	30	hilbert	hilbert	PROPN
easat-9645	20	31	spaces	space	NOUN
easat-9645	20	32	and	and	CCONJ
easat-9645	20	33	their	their	PRON
easat-9645	20	34	relationship	relationship	NOUN
easat-9645	20	35	to	to	ADP
easat-9645	20	36	normed	normed	ADJ
easat-9645	20	37	spaces	space	NOUN
easat-9645	20	38	and	and	CCONJ
easat-9645	20	39	banach	banach	NOUN
easat-9645	20	40	spaces	space	VERB
easat-9645	20	41	.	.	PUNCT
easat-9645	21	1	then	then	ADV
easat-9645	21	2	we	we	PRON
easat-9645	21	3	examine	examine	VERB
easat-9645	21	4	practical	practical	ADJ
easat-9645	21	5	applications	application	NOUN
easat-9645	21	6	in	in	ADP
easat-9645	21	7	quantum	quantum	ADJ
easat-9645	21	8	mechanics	mechanic	NOUN
easat-9645	21	9	,	,	PUNCT
easat-9645	21	10	signal	signal	ADJ
easat-9645	21	11	processing	processing	NOUN
easat-9645	21	12	,	,	PUNCT
easat-9645	21	13	and	and	CCONJ
easat-9645	21	14	machine	machine	NOUN
easat-9645	21	15	learning	learning	NOUN
easat-9645	21	16	,	,	PUNCT
easat-9645	21	17	where	where	SCONJ
easat-9645	21	18	the	the	DET
easat-9645	21	19	inner	inner	ADJ
easat-9645	21	20	product	product	NOUN
easat-9645	21	21	structure	structure	NOUN
easat-9645	21	22	enables	enable	VERB
easat-9645	21	23	techniques	technique	NOUN
easat-9645	21	24	like	like	ADP
easat-9645	21	25	kernel	kernel	PROPN
easat-9645	21	26	methods	method	NOUN
easat-9645	21	27	,	,	PUNCT
easat-9645	21	28	support	support	VERB
easat-9645	21	29	vector	vector	NOUN
easat-9645	21	30	machines	machine	NOUN
easat-9645	21	31	,	,	PUNCT
easat-9645	21	32	and	and	CCONJ
easat-9645	21	33	principal	principal	ADJ
easat-9645	21	34	component	component	NOUN
easat-9645	21	35	analysis	analysis	NOUN
easat-9645	21	36	.	.	PUNCT
easat-9645	22	1	we	we	PRON
easat-9645	22	2	provide	provide	VERB
easat-9645	22	3	matlab	matlab	PROPN
easat-9645	22	4	-	-	PUNCT
easat-9645	22	5	based	base	VERB
easat-9645	22	6	visualizations	visualization	NOUN
easat-9645	22	7	are	be	AUX
easat-9645	22	8	provided	provide	VERB
easat-9645	22	9	,	,	PUNCT
easat-9645	22	10	illustrating	illustrate	VERB
easat-9645	22	11	concepts	concept	NOUN
easat-9645	22	12	such	such	ADJ
easat-9645	22	13	as	as	ADP
easat-9645	22	14	projections	projection	NOUN
easat-9645	22	15	and	and	CCONJ
easat-9645	22	16	orthonormal	orthonormal	ADJ
easat-9645	22	17	expansions	expansion	NOUN
easat-9645	22	18	in	in	ADP
easat-9645	22	19	computational	computational	ADJ
easat-9645	22	20	contexts	contexts	NOUN
easat-9645	22	21	.	.	PUNCT
easat-9645	23	1	this	this	DET
easat-9645	23	2	work	work	NOUN
easat-9645	23	3	integrates	integrate	VERB
easat-9645	23	4	rigorous	rigorous	ADJ
easat-9645	23	5	mathematical	mathematical	ADJ
easat-9645	23	6	analysis	analysis	NOUN
easat-9645	23	7	with	with	ADP
easat-9645	23	8	practical	practical	ADJ
easat-9645	23	9	demonstrations	demonstration	NOUN
easat-9645	23	10	,	,	PUNCT
easat-9645	23	11	offering	offer	VERB
easat-9645	23	12	valuable	valuable	ADJ
easat-9645	23	13	insights	insight	NOUN
easat-9645	23	14	for	for	ADP
easat-9645	23	15	students	student	NOUN
easat-9645	23	16	and	and	CCONJ
easat-9645	23	17	researchers	researcher	NOUN
easat-9645	23	18	in	in	ADP
easat-9645	23	19	mathematics	mathematic	NOUN
easat-9645	23	20	and	and	CCONJ
easat-9645	23	21	data	datum	NOUN
easat-9645	23	22	science	science	NOUN
easat-9645	23	23	.	.	PUNCT
easat-9645	24	1	keywords	keyword	NOUN
easat-9645	24	2	:	:	PUNCT
easat-9645	24	3	hilbert	hilbert	NOUN
easat-9645	24	4	spaces	space	NOUN
easat-9645	24	5	,	,	PUNCT
easat-9645	24	6	inner	inner	ADJ
easat-9645	24	7	product	product	NOUN
easat-9645	24	8	spaces	space	NOUN
easat-9645	24	9	,	,	PUNCT
easat-9645	24	10	matlab	matlab	PROPN
easat-9645	24	11	visualization	visualization	NOUN
easat-9645	24	12	,	,	PUNCT
easat-9645	24	13	normed	normed	PROPN
easat-9645	24	14	spaces	space	NOUN
easat-9645	24	15	,	,	PUNCT
easat-9645	24	16	principal	principal	ADJ
easat-9645	24	17	component	component	NOUN
easat-9645	24	18	analysis	analysis	NOUN
easat-9645	24	19	(	(	PUNCT
easat-9645	24	20	pca	pca	NOUN
easat-9645	24	21	)	)	PUNCT
easat-9645	24	22	,	,	PUNCT
easat-9645	24	23	reproducing	reproduce	VERB
easat-9645	24	24	kernel	kernel	PROPN
easat-9645	24	25	hilbert	hilbert	PROPN
easat-9645	24	26	space	space	NOUN
easat-9645	24	27	(	(	PUNCT
easat-9645	24	28	rkhs	rkhs	PROPN
easat-9645	24	29	)	)	PUNCT
easat-9645	24	30	,	,	PUNCT
easat-9645	24	31	support	support	VERB
easat-9645	24	32	vector	vector	NOUN
easat-9645	24	33	machines	machine	NOUN
easat-9645	24	34	(	(	PUNCT
easat-9645	24	35	svm	svm	PROPN
easat-9645	24	36	)	)	PUNCT
easat-9645	24	37	.	.	PUNCT
easat-9645	25	1	1	1	X
easat-9645	25	2	.	.	X
easat-9645	25	3	introduction	introduction	NOUN
easat-9645	25	4	this	this	DET
easat-9645	25	5	paper	paper	NOUN
easat-9645	25	6	investigates	investigate	VERB
easat-9645	25	7	the	the	DET
easat-9645	25	8	structure	structure	NOUN
easat-9645	25	9	and	and	CCONJ
easat-9645	25	10	significance	significance	NOUN
easat-9645	25	11	of	of	ADP
easat-9645	25	12	hilbert	hilbert	NOUN
easat-9645	25	13	spaces	space	NOUN
easat-9645	25	14	,	,	PUNCT
easat-9645	25	15	which	which	PRON
easat-9645	25	16	are	be	AUX
easat-9645	25	17	complete	complete	ADJ
easat-9645	25	18	inner	inner	ADJ
easat-9645	25	19	product	product	NOUN
easat-9645	25	20	spaces	space	NOUN
easat-9645	25	21	.	.	PUNCT
easat-9645	26	1	addressing	address	VERB
easat-9645	26	2	the	the	DET
easat-9645	26	3	fundamental	fundamental	ADJ
easat-9645	26	4	geometric	geometric	ADJ
easat-9645	26	5	concepts	concept	NOUN
easat-9645	26	6	such	such	ADJ
easat-9645	26	7	as	as	ADP
easat-9645	26	8	distance	distance	NOUN
easat-9645	26	9	,	,	PUNCT
easat-9645	26	10	angle	angle	NOUN
easat-9645	26	11	,	,	PUNCT
easat-9645	26	12	and	and	CCONJ
easat-9645	26	13	orthogonality	orthogonality	NOUN
easat-9645	26	14	,	,	PUNCT
easat-9645	26	15	familiar	familiar	ADJ
easat-9645	26	16	in	in	ADP
easat-9645	26	17	finite	finite	ADJ
easat-9645	26	18	dimensional	dimensional	ADJ
easat-9645	26	19	euclidean	euclidean	ADJ
easat-9645	26	20	spaces	space	NOUN
easat-9645	26	21	,	,	PUNCT
easat-9645	26	22	be	be	AUX
easat-9645	26	23	extended	extend	VERB
easat-9645	26	24	to	to	PART
easat-9645	26	25	infinite	infinite	VERB
easat-9645	26	26	dimensional	dimensional	ADJ
easat-9645	26	27	settings	setting	NOUN
easat-9645	26	28	.	.	PUNCT
easat-9645	27	1	hilbert	hilbert	PROPN
easat-9645	27	2	spaces	space	NOUN
easat-9645	27	3	,	,	PUNCT
easat-9645	27	4	defined	define	VERB
easat-9645	27	5	as	as	ADP
easat-9645	27	6	complete	complete	ADJ
easat-9645	27	7	inner	inner	ADJ
easat-9645	27	8	product	product	NOUN
easat-9645	27	9	spaces	space	NOUN
easat-9645	27	10	,	,	PUNCT
easat-9645	27	11	provide	provide	VERB
easat-9645	27	12	a	a	DET
easat-9645	27	13	robust	robust	ADJ
easat-9645	27	14	framework	framework	NOUN
easat-9645	27	15	for	for	ADP
easat-9645	27	16	this	this	DET
easat-9645	27	17	generalization	generalization	NOUN
easat-9645	27	18	,	,	PUNCT
easat-9645	27	19	blending	blend	VERB
easat-9645	27	20	algebraic	algebraic	ADJ
easat-9645	27	21	rigor	rigor	NOUN
easat-9645	27	22	with	with	ADP
easat-9645	27	23	geometric	geometric	ADJ
easat-9645	27	24	intuition	intuition	NOUN
easat-9645	27	25	[	[	X
easat-9645	27	26	1	1	NUM
easat-9645	27	27	,	,	PUNCT
easat-9645	27	28	2	2	NUM
easat-9645	27	29	]	]	PUNCT
easat-9645	27	30	.	.	PUNCT
easat-9645	28	1	their	their	PRON
easat-9645	28	2	completeness	completeness	NOUN
easat-9645	28	3	,	,	PUNCT
easat-9645	28	4	ensuring	ensure	VERB
easat-9645	28	5	that	that	SCONJ
easat-9645	28	6	every	every	DET
easat-9645	28	7	cauchy	cauchy	ADJ
easat-9645	28	8	sequence	sequence	NOUN
easat-9645	28	9	converges	converge	VERB
easat-9645	28	10	within	within	ADP
easat-9645	28	11	the	the	DET
easat-9645	28	12	space	space	NOUN
easat-9645	28	13	,	,	PUNCT
easat-9645	28	14	makes	make	VERB
easat-9645	28	15	them	they	PRON
easat-9645	28	16	a	a	DET
easat-9645	28	17	cornerstone	cornerstone	NOUN
easat-9645	28	18	of	of	ADP
easat-9645	28	19	both	both	DET
easat-9645	28	20	theoretical	theoretical	ADJ
easat-9645	28	21	mathematics	mathematic	NOUN
easat-9645	28	22	and	and	CCONJ
easat-9645	28	23	applied	applied	ADJ
easat-9645	28	24	sciences	science	NOUN
easat-9645	28	25	.	.	PUNCT
easat-9645	29	1	this	this	DET
easat-9645	29	2	study	study	NOUN
easat-9645	29	3	aims	aim	VERB
easat-9645	29	4	to	to	PART
easat-9645	29	5	elucidate	elucidate	VERB
easat-9645	29	6	the	the	DET
easat-9645	29	7	properties	property	NOUN
easat-9645	29	8	of	of	ADP
easat-9645	29	9	hilbert	hilbert	PROPN
easat-9645	29	10	spaces	space	NOUN
easat-9645	29	11	,	,	PUNCT
easat-9645	29	12	demonstrate	demonstrate	VERB
easat-9645	29	13	their	their	PRON
easat-9645	29	14	analytical	analytical	ADJ
easat-9645	29	15	power	power	NOUN
easat-9645	29	16	through	through	ADP
easat-9645	29	17	key	key	ADJ
easat-9645	29	18	mathematical	mathematical	ADJ
easat-9645	29	19	tools	tool	NOUN
easat-9645	29	20	,	,	PUNCT
easat-9645	29	21	and	and	CCONJ
easat-9645	29	22	showcase	showcase	VERB
easat-9645	29	23	their	their	PRON
easat-9645	29	24	applications	application	NOUN
easat-9645	29	25	in	in	ADP
easat-9645	29	26	fields	field	NOUN
easat-9645	29	27	such	such	ADJ
easat-9645	29	28	as	as	ADP
easat-9645	29	29	quantum	quantum	ADJ
easat-9645	29	30	mechanics	mechanic	NOUN
easat-9645	29	31	,	,	PUNCT
easat-9645	29	32	signal	signal	NOUN
easat-9645	29	33	processing	processing	NOUN
easat-9645	29	34	,	,	PUNCT
easat-9645	29	35	and	and	CCONJ
easat-9645	29	36	machine	machine	NOUN
easat-9645	29	37	learning	learning	NOUN
easat-9645	29	38	.	.	PUNCT
easat-9645	30	1	by	by	ADP
easat-9645	30	2	integrating	integrate	VERB
easat-9645	30	3	rigorous	rigorous	ADJ
easat-9645	30	4	theory	theory	NOUN
easat-9645	30	5	with	with	ADP
easat-9645	30	6	computational	computational	ADJ
easat-9645	30	7	tools	tool	NOUN
easat-9645	30	8	like	like	ADP
easat-9645	30	9	matlab	matlab	PROPN
easat-9645	30	10	based	base	VERB
easat-9645	30	11	visualizations	visualization	NOUN
easat-9645	30	12	of	of	ADP
easat-9645	30	13	mathematics	mathematic	NOUN
easat-9645	30	14	and	and	CCONJ
easat-9645	30	15	data	datum	NOUN
easat-9645	30	16	science	science	NOUN
easat-9645	30	17	.	.	PUNCT
easat-9645	31	1	https://orcid.org/0000-0002-8730-1783	https://orcid.org/0000-0002-8730-1783	VERB
easat-9645	32	1	https://orcid.org/0009-0004-5563-4497	https://orcid.org/0009-0004-5563-4497	ADJ
easat-9645	32	2	https://orcid.org/0009-0007-2555-6047	https://orcid.org/0009-0007-2555-6047	PROPN
easat-9645	32	3	https://orcid.org/0000-0002-9829-2560	https://orcid.org/0000-0002-9829-2560	PROPN
easat-9645	32	4	https://orcid.org/0000-0003-3115-7820	https://orcid.org/0000-0003-3115-7820	PROPN
easat-9645	32	5	https://orcid.org/0000-0002-3019-8260	https://orcid.org/0000-0002-3019-8260	PROPN
easat-9645	32	6	https://orcid.org/0009-0007-7327-6090	https://orcid.org/0009-0007-7327-6090	PROPN
easat-9645	32	7	https://orcid.org/0009-0003-5256-4422	https://orcid.org/0009-0003-5256-4422	PROPN
easat-9645	32	8	1499	1499	NUM
easat-9645	32	9	edelweiss	edelweiss	PROPN
easat-9645	32	10	applied	apply	VERB
easat-9645	32	11	science	science	NOUN
easat-9645	32	12	and	and	CCONJ
easat-9645	32	13	technology	technology	NOUN
easat-9645	32	14	issn	issn	PROPN
easat-9645	32	15	:	:	PUNCT
easat-9645	32	16	2576	2576	NUM
easat-9645	32	17	-	-	SYM
easat-9645	32	18	8484	8484	NUM
easat-9645	32	19	vol	vol	NOUN
easat-9645	32	20	.	.	PROPN
easat-9645	33	1	9	9	NUM
easat-9645	33	2	,	,	PUNCT
easat-9645	33	3	no	no	INTJ
easat-9645	33	4	.	.	NOUN
easat-9645	33	5	8	8	NUM
easat-9645	33	6	:	:	SYM
easat-9645	33	7	1498	1498	NUM
easat-9645	33	8	-	-	SYM
easat-9645	33	9	1523	1523	NUM
easat-9645	33	10	,	,	PUNCT
easat-9645	33	11	2025	2025	NUM
easat-9645	33	12	doi	doi	NOUN
easat-9645	33	13	:	:	PUNCT
easat-9645	33	14	10.55214/2576	10.55214/2576	NUM
easat-9645	33	15	-	-	SYM
easat-9645	33	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	33	17	©	©	PROPN
easat-9645	33	18	2025	2025	NUM
easat-9645	33	19	by	by	ADP
easat-9645	33	20	the	the	DET
easat-9645	33	21	authors	author	NOUN
easat-9645	33	22	;	;	PUNCT
easat-9645	33	23	licensee	licensee	PROPN
easat-9645	33	24	learning	learning	NOUN
easat-9645	33	25	gate	gate	VERB
easat-9645	33	26	1.1	1.1	NUM
easat-9645	33	27	.	.	PUNCT
easat-9645	34	1	historical	historical	ADJ
easat-9645	34	2	background	background	NOUN
easat-9645	34	3	and	and	CCONJ
easat-9645	34	4	literature	literature	NOUN
easat-9645	34	5	review	review	VERB
easat-9645	34	6	the	the	DET
easat-9645	34	7	concept	concept	NOUN
easat-9645	34	8	of	of	ADP
easat-9645	34	9	hilbert	hilbert	NOUN
easat-9645	34	10	spaces	space	NOUN
easat-9645	34	11	emerged	emerge	VERB
easat-9645	34	12	from	from	ADP
easat-9645	34	13	the	the	DET
easat-9645	34	14	need	need	NOUN
easat-9645	34	15	to	to	PART
easat-9645	34	16	generalize	generalize	VERB
easat-9645	34	17	euclidean	euclidean	ADJ
easat-9645	34	18	geometry	geometry	NOUN
easat-9645	34	19	to	to	ADP
easat-9645	34	20	abstract	abstract	ADJ
easat-9645	34	21	vector	vector	NOUN
easat-9645	34	22	spaces	space	NOUN
easat-9645	34	23	,	,	PUNCT
easat-9645	34	24	including	include	VERB
easat-9645	34	25	those	those	PRON
easat-9645	34	26	with	with	ADP
easat-9645	34	27	infinite	infinite	ADJ
easat-9645	34	28	dimensions	dimension	NOUN
easat-9645	34	29	.	.	PUNCT
easat-9645	35	1	at	at	ADP
easat-9645	35	2	the	the	DET
easat-9645	35	3	core	core	NOUN
easat-9645	35	4	of	of	ADP
easat-9645	35	5	this	this	DET
easat-9645	35	6	generalization	generalization	NOUN
easat-9645	35	7	is	be	AUX
easat-9645	35	8	the	the	DET
easat-9645	35	9	inner	inner	ADJ
easat-9645	35	10	product	product	NOUN
easat-9645	35	11	,	,	PUNCT
easat-9645	35	12	a	a	DET
easat-9645	35	13	bilinear	bilinear	NOUN
easat-9645	35	14	form	form	NOUN
easat-9645	35	15	that	that	PRON
easat-9645	35	16	formalizes	formalize	VERB
easat-9645	35	17	notions	notion	NOUN
easat-9645	35	18	of	of	ADP
easat-9645	35	19	length	length	NOUN
easat-9645	35	20	,	,	PUNCT
easat-9645	35	21	angle	angle	NOUN
easat-9645	35	22	,	,	PUNCT
easat-9645	35	23	and	and	CCONJ
easat-9645	35	24	orthogonality	orthogonality	NOUN
easat-9645	35	25	in	in	ADP
easat-9645	35	26	a	a	DET
easat-9645	35	27	way	way	NOUN
easat-9645	35	28	that	that	PRON
easat-9645	35	29	extends	extend	VERB
easat-9645	35	30	naturally	naturally	ADV
easat-9645	35	31	from	from	ADP
easat-9645	35	32	finite	finite	NOUN
easat-9645	35	33	to	to	ADP
easat-9645	35	34	infinite	infinite	ADJ
easat-9645	35	35	dimensions	dimension	NOUN
easat-9645	35	36	.	.	PUNCT
easat-9645	36	1	a	a	DET
easat-9645	36	2	hilbert	hilbert	NOUN
easat-9645	36	3	space	space	NOUN
easat-9645	36	4	is	be	AUX
easat-9645	36	5	an	an	DET
easat-9645	36	6	inner	inner	ADJ
easat-9645	36	7	product	product	NOUN
easat-9645	36	8	space	space	NOUN
easat-9645	36	9	that	that	PRON
easat-9645	36	10	is	be	AUX
easat-9645	36	11	complete	complete	ADJ
easat-9645	36	12	with	with	ADP
easat-9645	36	13	respect	respect	NOUN
easat-9645	36	14	to	to	ADP
easat-9645	36	15	the	the	DET
easat-9645	36	16	norm	norm	NOUN
easat-9645	36	17	induced	induce	VERB
easat-9645	36	18	by	by	ADP
easat-9645	36	19	the	the	DET
easat-9645	36	20	inner	inner	ADJ
easat-9645	36	21	product	product	NOUN
easat-9645	36	22	,	,	PUNCT
easat-9645	36	23	ensuring	ensure	VERB
easat-9645	36	24	convergence	convergence	NOUN
easat-9645	36	25	of	of	ADP
easat-9645	36	26	cauchy	cauchy	PROPN
easat-9645	36	27	sequences	sequence	NOUN
easat-9645	36	28	a	a	DET
easat-9645	36	29	property	property	NOUN
easat-9645	36	30	critical	critical	ADJ
easat-9645	36	31	for	for	ADP
easat-9645	36	32	theoretical	theoretical	ADJ
easat-9645	36	33	and	and	CCONJ
easat-9645	36	34	applied	applied	ADJ
easat-9645	36	35	contexts	contexts	NOUN
easat-9645	36	36	.	.	PUNCT
easat-9645	37	1	this	this	DET
easat-9645	37	2	completeness	completeness	NOUN
easat-9645	37	3	distinguishes	distinguish	VERB
easat-9645	37	4	hilbert	hilbert	NOUN
easat-9645	37	5	spaces	space	NOUN
easat-9645	37	6	from	from	ADP
easat-9645	37	7	general	general	ADJ
easat-9645	37	8	normed	normed	PROPN
easat-9645	37	9	spaces	space	NOUN
easat-9645	37	10	,	,	PUNCT
easat-9645	37	11	preserving	preserve	VERB
easat-9645	37	12	geometric	geometric	ADJ
easat-9645	37	13	properties	property	NOUN
easat-9645	37	14	that	that	PRON
easat-9645	37	15	enable	enable	VERB
easat-9645	37	16	powerful	powerful	ADJ
easat-9645	37	17	analytical	analytical	ADJ
easat-9645	37	18	techniques	technique	NOUN
easat-9645	37	19	.	.	PUNCT
easat-9645	38	1	foundational	foundational	ADJ
easat-9645	38	2	work	work	NOUN
easat-9645	38	3	introducing	introduce	VERB
easat-9645	38	4	sampling	sample	VERB
easat-9645	38	5	techniques	technique	NOUN
easat-9645	38	6	in	in	ADP
easat-9645	38	7	discrete	discrete	ADJ
easat-9645	38	8	reproducing	reproduce	VERB
easat-9645	38	9	kernel	kernel	PROPN
easat-9645	38	10	hilbert	hilbert	PROPN
easat-9645	38	11	spaces	space	NOUN
easat-9645	38	12	(	(	PUNCT
easat-9645	38	13	rkhs	rkh	NOUN
easat-9645	38	14	)	)	PUNCT
easat-9645	39	1	[	[	X
easat-9645	39	2	3	3	NUM
easat-9645	39	3	,	,	PUNCT
easat-9645	39	4	4	4	NUM
easat-9645	39	5	]	]	PUNCT
easat-9645	39	6	.	.	PUNCT
easat-9645	40	1	the	the	DET
easat-9645	40	2	development	development	NOUN
easat-9645	40	3	of	of	ADP
easat-9645	40	4	hilbert	hilbert	PROPN
easat-9645	40	5	spaces	space	NOUN
easat-9645	40	6	contributions	contribution	NOUN
easat-9645	40	7	by	by	ADP
easat-9645	40	8	mathematicians	mathematician	NOUN
easat-9645	40	9	like	like	ADP
easat-9645	40	10	schmidt	schmidt	NOUN
easat-9645	40	11	[	[	X
easat-9645	40	12	5	5	NUM
easat-9645	40	13	]	]	PUNCT
easat-9645	40	14	refined	refine	VERB
easat-9645	40	15	the	the	DET
easat-9645	40	16	geometric	geometric	ADJ
easat-9645	40	17	interpretation	interpretation	NOUN
easat-9645	40	18	and	and	CCONJ
easat-9645	40	19	terminology	terminology	NOUN
easat-9645	40	20	,	,	PUNCT
easat-9645	40	21	laying	lay	VERB
easat-9645	40	22	the	the	DET
easat-9645	40	23	groundwork	groundwork	NOUN
easat-9645	40	24	for	for	ADP
easat-9645	40	25	modern	modern	ADJ
easat-9645	40	26	functional	functional	ADJ
easat-9645	40	27	analysis	analysis	NOUN
easat-9645	40	28	.	.	PUNCT
easat-9645	41	1	fundamental	fundamental	ADJ
easat-9645	41	2	tools	tool	NOUN
easat-9645	41	3	,	,	PUNCT
easat-9645	41	4	such	such	ADJ
easat-9645	41	5	as	as	ADP
easat-9645	41	6	the	the	DET
easat-9645	41	7	cauchy	cauchy	PROPN
easat-9645	41	8	–	–	PUNCT
easat-9645	41	9	schwarz	schwarz	NOUN
easat-9645	41	10	inequality	inequality	NOUN
easat-9645	41	11	and	and	CCONJ
easat-9645	41	12	bunyakovsky	bunyakovsky	ADJ
easat-9645	41	13	inequality	inequality	NOUN
easat-9645	41	14	[	[	X
easat-9645	41	15	6	6	NUM
easat-9645	41	16	]	]	PUNCT
easat-9645	41	17	triangle	triangle	NOUN
easat-9645	41	18	inequality	inequality	NOUN
easat-9645	41	19	[	[	X
easat-9645	41	20	7	7	NUM
easat-9645	41	21	]	]	PUNCT
easat-9645	41	22	and	and	CCONJ
easat-9645	41	23	polarization	polarization	NOUN
easat-9645	41	24	identity	identity	NOUN
easat-9645	41	25	[	[	X
easat-9645	41	26	8	8	NUM
easat-9645	41	27	]	]	PUNCT
easat-9645	41	28	form	form	NOUN
easat-9645	41	29	the	the	DET
easat-9645	41	30	analytical	analytical	ADJ
easat-9645	41	31	backbone	backbone	NOUN
easat-9645	41	32	of	of	ADP
easat-9645	41	33	hilbert	hilbert	PROPN
easat-9645	41	34	space	space	NOUN
easat-9645	41	35	theory	theory	NOUN
easat-9645	41	36	.	.	PUNCT
easat-9645	42	1	these	these	DET
easat-9645	42	2	results	result	VERB
easat-9645	42	3	not	not	PART
easat-9645	42	4	only	only	ADV
easat-9645	42	5	provide	provide	VERB
easat-9645	42	6	deep	deep	ADJ
easat-9645	42	7	insights	insight	NOUN
easat-9645	42	8	into	into	ADP
easat-9645	42	9	the	the	DET
easat-9645	42	10	structure	structure	NOUN
easat-9645	42	11	of	of	ADP
easat-9645	42	12	these	these	DET
easat-9645	42	13	spaces	space	NOUN
easat-9645	42	14	but	but	CCONJ
easat-9645	42	15	also	also	ADV
easat-9645	42	16	facilitate	facilitate	VERB
easat-9645	42	17	practical	practical	ADJ
easat-9645	42	18	computations	computation	NOUN
easat-9645	42	19	in	in	ADP
easat-9645	42	20	diverse	diverse	ADJ
easat-9645	42	21	scientific	scientific	ADJ
easat-9645	42	22	domains	domain	NOUN
easat-9645	42	23	.	.	PUNCT
easat-9645	43	1	hilbert	hilbert	PROPN
easat-9645	43	2	spaces	space	NOUN
easat-9645	43	3	have	have	AUX
easat-9645	43	4	become	become	VERB
easat-9645	43	5	indispensable	indispensable	ADJ
easat-9645	43	6	in	in	ADP
easat-9645	43	7	fields	field	NOUN
easat-9645	43	8	such	such	ADJ
easat-9645	43	9	as	as	ADP
easat-9645	43	10	functional	functional	ADJ
easat-9645	43	11	analysis	analysis	NOUN
easat-9645	43	12	,	,	PUNCT
easat-9645	43	13	quantum	quantum	NOUN
easat-9645	43	14	mechanics	mechanic	NOUN
easat-9645	43	15	,	,	PUNCT
easat-9645	43	16	and	and	CCONJ
easat-9645	43	17	partial	partial	ADJ
easat-9645	43	18	differential	differential	ADJ
easat-9645	43	19	equations	equation	NOUN
easat-9645	43	20	[	[	X
easat-9645	43	21	9	9	NUM
easat-9645	43	22	]	]	PUNCT
easat-9645	43	23	.	.	PUNCT
easat-9645	44	1	the	the	DET
easat-9645	44	2	role	role	NOUN
easat-9645	44	3	of	of	ADP
easat-9645	44	4	hilbert	hilbert	NOUN
easat-9645	44	5	spaces	space	NOUN
easat-9645	44	6	in	in	ADP
easat-9645	44	7	quantum	quantum	ADJ
easat-9645	44	8	mechanics	mechanic	NOUN
easat-9645	44	9	,	,	PUNCT
easat-9645	44	10	focusing	focus	VERB
easat-9645	44	11	on	on	ADP
easat-9645	44	12	their	their	PRON
easat-9645	44	13	mathematical	mathematical	ADJ
easat-9645	44	14	structure	structure	NOUN
easat-9645	44	15	and	and	CCONJ
easat-9645	44	16	applications	application	NOUN
easat-9645	44	17	[	[	X
easat-9645	44	18	10	10	NUM
easat-9645	44	19	]	]	PUNCT
easat-9645	44	20	.	.	PUNCT
easat-9645	44	21	in	in	ADP
easat-9645	44	22	quantum	quantum	ADJ
easat-9645	44	23	mechanics	mechanic	NOUN
easat-9645	44	24	,	,	PUNCT
easat-9645	44	25	for	for	ADP
easat-9645	44	26	instance	instance	NOUN
easat-9645	44	27	,	,	PUNCT
easat-9645	44	28	the	the	DET
easat-9645	44	29	state	state	NOUN
easat-9645	44	30	space	space	NOUN
easat-9645	44	31	of	of	ADP
easat-9645	44	32	a	a	DET
easat-9645	44	33	quantum	quantum	ADJ
easat-9645	44	34	system	system	NOUN
easat-9645	44	35	is	be	AUX
easat-9645	44	36	modeled	model	VERB
easat-9645	44	37	as	as	ADP
easat-9645	44	38	a	a	DET
easat-9645	44	39	hilbert	hilbert	NOUN
easat-9645	44	40	space	space	NOUN
easat-9645	44	41	,	,	PUNCT
easat-9645	44	42	where	where	SCONJ
easat-9645	44	43	the	the	DET
easat-9645	44	44	inner	inner	ADJ
easat-9645	44	45	product	product	NOUN
easat-9645	44	46	governs	govern	VERB
easat-9645	44	47	probabilistic	probabilistic	ADJ
easat-9645	44	48	interpretations	interpretation	NOUN
easat-9645	44	49	of	of	ADP
easat-9645	44	50	wave	wave	NOUN
easat-9645	44	51	functions	function	NOUN
easat-9645	44	52	.	.	PUNCT
easat-9645	45	1	similarly	similarly	ADV
easat-9645	45	2	,	,	PUNCT
easat-9645	45	3	in	in	ADP
easat-9645	45	4	signal	signal	ADJ
easat-9645	45	5	processing	processing	NOUN
easat-9645	45	6	,	,	PUNCT
easat-9645	45	7	hilbert	hilbert	NOUN
easat-9645	45	8	spaces	space	NOUN
easat-9645	45	9	underpin	underpin	VERB
easat-9645	45	10	techniques	technique	NOUN
easat-9645	45	11	for	for	ADP
easat-9645	45	12	analyzing	analyze	VERB
easat-9645	45	13	and	and	CCONJ
easat-9645	45	14	reconstructing	reconstruct	VERB
easat-9645	45	15	signals	signal	NOUN
easat-9645	45	16	.	.	PUNCT
easat-9645	46	1	more	more	ADV
easat-9645	46	2	recently	recently	ADV
easat-9645	46	3	,	,	PUNCT
easat-9645	46	4	the	the	DET
easat-9645	46	5	relevance	relevance	NOUN
easat-9645	46	6	of	of	ADP
easat-9645	46	7	hilbert	hilbert	NOUN
easat-9645	46	8	spaces	space	NOUN
easat-9645	46	9	has	have	AUX
easat-9645	46	10	surged	surge	VERB
easat-9645	46	11	in	in	ADP
easat-9645	46	12	machine	machine	NOUN
easat-9645	46	13	learning	learning	NOUN
easat-9645	46	14	,	,	PUNCT
easat-9645	46	15	where	where	SCONJ
easat-9645	46	16	the	the	DET
easat-9645	46	17	inner	inner	ADJ
easat-9645	46	18	product	product	NOUN
easat-9645	46	19	structure	structure	NOUN
easat-9645	46	20	supports	support	VERB
easat-9645	46	21	methods	method	NOUN
easat-9645	46	22	like	like	ADP
easat-9645	46	23	support	support	NOUN
easat-9645	46	24	vector	vector	NOUN
easat-9645	46	25	machines	machine	NOUN
easat-9645	46	26	(	(	PUNCT
easat-9645	46	27	svms	svms	NOUN
easat-9645	46	28	)	)	PUNCT
easat-9645	46	29	,	,	PUNCT
easat-9645	46	30	principal	principal	ADJ
easat-9645	46	31	component	component	NOUN
easat-9645	46	32	analysis	analysis	NOUN
easat-9645	46	33	(	(	PUNCT
easat-9645	46	34	pca	pca	NOUN
easat-9645	46	35	)	)	PUNCT
easat-9645	46	36	,	,	PUNCT
easat-9645	46	37	and	and	CCONJ
easat-9645	46	38	kernel	kernel	PROPN
easat-9645	46	39	methods	method	NOUN
easat-9645	46	40	[	[	PUNCT
easat-9645	46	41	11	11	NUM
easat-9645	46	42	-	-	SYM
easat-9645	46	43	13	13	NUM
easat-9645	46	44	]	]	PUNCT
easat-9645	46	45	.	.	PUNCT
easat-9645	47	1	these	these	DET
easat-9645	47	2	techniques	technique	NOUN
easat-9645	47	3	leverage	leverage	VERB
easat-9645	47	4	the	the	DET
easat-9645	47	5	geometry	geometry	NOUN
easat-9645	47	6	of	of	ADP
easat-9645	47	7	high	high	ADJ
easat-9645	47	8	or	or	CCONJ
easat-9645	47	9	infinite	infinite	ADJ
easat-9645	47	10	dimensional	dimensional	ADJ
easat-9645	47	11	hilbert	hilbert	NOUN
easat-9645	47	12	spaces	space	NOUN
easat-9645	47	13	to	to	PART
easat-9645	47	14	address	address	VERB
easat-9645	47	15	complex	complex	ADJ
easat-9645	47	16	data	datum	NOUN
easat-9645	47	17	analysis	analysis	NOUN
easat-9645	47	18	tasks	task	NOUN
easat-9645	47	19	,	,	PUNCT
easat-9645	47	20	demonstrating	demonstrate	VERB
easat-9645	47	21	the	the	DET
easat-9645	47	22	practical	practical	ADJ
easat-9645	47	23	impact	impact	NOUN
easat-9645	47	24	of	of	ADP
easat-9645	47	25	abstract	abstract	ADJ
easat-9645	47	26	mathematical	mathematical	ADJ
easat-9645	47	27	theory	theory	NOUN
easat-9645	47	28	[	[	X
easat-9645	47	29	8	8	NUM
easat-9645	47	30	,	,	PUNCT
easat-9645	47	31	14	14	NUM
easat-9645	47	32	]	]	PUNCT
easat-9645	47	33	.	.	PUNCT
easat-9645	48	1	this	this	DET
easat-9645	48	2	paper	paper	NOUN
easat-9645	48	3	explores	explore	VERB
easat-9645	48	4	the	the	DET
easat-9645	48	5	theoretical	theoretical	ADJ
easat-9645	48	6	foundations	foundation	NOUN
easat-9645	48	7	of	of	ADP
easat-9645	48	8	hilbert	hilbert	NOUN
easat-9645	48	9	spaces	space	NOUN
easat-9645	48	10	and	and	CCONJ
easat-9645	48	11	their	their	PRON
easat-9645	48	12	inner	inner	ADJ
easat-9645	48	13	product	product	NOUN
easat-9645	48	14	structure	structure	NOUN
easat-9645	48	15	,	,	PUNCT
easat-9645	48	16	supported	support	VERB
easat-9645	48	17	by	by	ADP
easat-9645	48	18	matlab	matlab	PROPN
easat-9645	48	19	based	base	VERB
easat-9645	48	20	visualizations	visualization	NOUN
easat-9645	48	21	to	to	PART
easat-9645	48	22	illustrate	illustrate	VERB
easat-9645	48	23	abstract	abstract	ADJ
easat-9645	48	24	concepts	concept	NOUN
easat-9645	48	25	.	.	PUNCT
easat-9645	49	1	by	by	ADP
easat-9645	49	2	examining	examine	VERB
easat-9645	49	3	applications	application	NOUN
easat-9645	49	4	in	in	ADP
easat-9645	49	5	both	both	CCONJ
easat-9645	49	6	traditional	traditional	ADJ
easat-9645	49	7	and	and	CCONJ
easat-9645	49	8	modern	modern	ADJ
easat-9645	49	9	contexts	context	NOUN
easat-9645	49	10	,	,	PUNCT
easat-9645	49	11	we	we	PRON
easat-9645	49	12	highlight	highlight	VERB
easat-9645	49	13	the	the	DET
easat-9645	49	14	versatility	versatility	NOUN
easat-9645	49	15	of	of	ADP
easat-9645	49	16	hilbert	hilbert	PROPN
easat-9645	49	17	spaces	space	NOUN
easat-9645	49	18	in	in	ADP
easat-9645	49	19	bridging	bridge	VERB
easat-9645	49	20	pure	pure	ADJ
easat-9645	49	21	mathematics	mathematic	NOUN
easat-9645	49	22	and	and	CCONJ
easat-9645	49	23	computational	computational	ADJ
easat-9645	49	24	practice	practice	NOUN
easat-9645	49	25	.	.	PUNCT
easat-9645	50	1	our	our	PRON
easat-9645	50	2	work	work	NOUN
easat-9645	50	3	aims	aim	VERB
easat-9645	50	4	to	to	PART
easat-9645	50	5	provide	provide	VERB
easat-9645	50	6	a	a	DET
easat-9645	50	7	unified	unified	ADJ
easat-9645	50	8	perspective	perspective	NOUN
easat-9645	50	9	,	,	PUNCT
easat-9645	50	10	making	make	VERB
easat-9645	50	11	the	the	DET
easat-9645	50	12	theory	theory	NOUN
easat-9645	50	13	accessible	accessible	ADJ
easat-9645	50	14	while	while	SCONJ
easat-9645	50	15	emphasizing	emphasize	VERB
easat-9645	50	16	its	its	PRON
easat-9645	50	17	relevance	relevance	NOUN
easat-9645	50	18	to	to	ADP
easat-9645	50	19	contemporary	contemporary	ADJ
easat-9645	50	20	challenges	challenge	NOUN
easat-9645	50	21	in	in	ADP
easat-9645	50	22	science	science	NOUN
easat-9645	50	23	and	and	CCONJ
easat-9645	50	24	technology	technology	NOUN
easat-9645	50	25	[	[	X
easat-9645	50	26	15	15	NUM
easat-9645	50	27	,	,	PUNCT
easat-9645	50	28	16	16	NUM
easat-9645	50	29	]	]	PUNCT
easat-9645	50	30	.	.	PUNCT
easat-9645	51	1	1.2	1.2	NUM
easat-9645	51	2	.	.	PUNCT
easat-9645	51	3	preliminaries	preliminary	NOUN
easat-9645	51	4	:	:	PUNCT
easat-9645	51	5	normed	normed	ADJ
easat-9645	51	6	and	and	CCONJ
easat-9645	51	7	inner	inner	ADJ
easat-9645	51	8	product	product	NOUN
easat-9645	51	9	spaces	space	VERB
easat-9645	51	10	this	this	DET
easat-9645	51	11	subsection	subsection	NOUN
easat-9645	51	12	introduces	introduce	VERB
easat-9645	51	13	the	the	DET
easat-9645	51	14	foundational	foundational	ADJ
easat-9645	51	15	concepts	concept	NOUN
easat-9645	51	16	and	and	CCONJ
easat-9645	51	17	definitions	definition	NOUN
easat-9645	51	18	necessary	necessary	ADJ
easat-9645	51	19	for	for	ADP
easat-9645	51	20	understanding	understand	VERB
easat-9645	51	21	hilbert	hilbert	NOUN
easat-9645	51	22	spaces	space	NOUN
easat-9645	51	23	.	.	PUNCT
easat-9645	52	1	a	a	DET
easat-9645	52	2	normed	normed	ADJ
easat-9645	52	3	space	space	NOUN
easat-9645	52	4	is	be	AUX
easat-9645	52	5	a	a	DET
easat-9645	52	6	vector	vector	NOUN
easat-9645	52	7	space	space	NOUN
easat-9645	52	8	equipped	equip	VERB
easat-9645	52	9	with	with	ADP
easat-9645	52	10	a	a	DET
easat-9645	52	11	norm	norm	NOUN
easat-9645	52	12	,	,	PUNCT
easat-9645	52	13	which	which	PRON
easat-9645	52	14	induces	induce	VERB
easat-9645	52	15	a	a	DET
easat-9645	52	16	metric	metric	NOUN
easat-9645	52	17	to	to	PART
easat-9645	52	18	measure	measure	VERB
easat-9645	52	19	distance	distance	NOUN
easat-9645	52	20	.	.	PUNCT
easat-9645	53	1	an	an	DET
easat-9645	53	2	inner	inner	ADJ
easat-9645	53	3	product	product	NOUN
easat-9645	53	4	space	space	NOUN
easat-9645	53	5	is	be	AUX
easat-9645	53	6	a	a	DET
easat-9645	53	7	normed	normed	ADJ
easat-9645	53	8	space	space	NOUN
easat-9645	53	9	with	with	ADP
easat-9645	53	10	an	an	DET
easat-9645	53	11	inner	inner	ADJ
easat-9645	53	12	product	product	NOUN
easat-9645	53	13	,	,	PUNCT
easat-9645	53	14	a	a	DET
easat-9645	53	15	bilinear	bilinear	NOUN
easat-9645	53	16	form	form	NOUN
easat-9645	53	17	that	that	PRON
easat-9645	53	18	defines	define	VERB
easat-9645	53	19	geometric	geometric	ADJ
easat-9645	53	20	properties	property	NOUN
easat-9645	53	21	like	like	ADP
easat-9645	53	22	orthogonality	orthogonality	NOUN
easat-9645	53	23	and	and	CCONJ
easat-9645	53	24	angle	angle	NOUN
easat-9645	53	25	.	.	PUNCT
easat-9645	54	1	a	a	DET
easat-9645	54	2	hilbert	hilbert	NOUN
easat-9645	54	3	space	space	NOUN
easat-9645	54	4	is	be	AUX
easat-9645	54	5	an	an	DET
easat-9645	54	6	inner	inner	ADJ
easat-9645	54	7	product	product	NOUN
easat-9645	54	8	space	space	NOUN
easat-9645	54	9	that	that	PRON
easat-9645	54	10	is	be	AUX
easat-9645	54	11	complete	complete	ADJ
easat-9645	54	12	with	with	ADP
easat-9645	54	13	respect	respect	NOUN
easat-9645	54	14	to	to	ADP
easat-9645	54	15	the	the	DET
easat-9645	54	16	norm	norm	NOUN
easat-9645	54	17	induced	induce	VERB
easat-9645	54	18	by	by	ADP
easat-9645	54	19	the	the	DET
easat-9645	54	20	inner	inner	ADJ
easat-9645	54	21	product	product	NOUN
easat-9645	54	22	,	,	PUNCT
easat-9645	54	23	meaning	mean	VERB
easat-9645	54	24	every	every	DET
easat-9645	54	25	cauchy	cauchy	ADJ
easat-9645	54	26	sequence	sequence	NOUN
easat-9645	54	27	converges	converge	VERB
easat-9645	54	28	to	to	ADP
easat-9645	54	29	a	a	DET
easat-9645	54	30	point	point	NOUN
easat-9645	54	31	in	in	ADP
easat-9645	54	32	the	the	DET
easat-9645	54	33	space	space	NOUN
easat-9645	54	34	.	.	PUNCT
easat-9645	55	1	normed	normed	PROPN
easat-9645	55	2	spaces	space	VERB
easat-9645	55	3	:	:	PUNCT
easat-9645	55	4	a	a	DET
easat-9645	55	5	normed	normed	ADJ
easat-9645	55	6	on	on	ADP
easat-9645	55	7	x	x	SYM
easat-9645	55	8	is	be	AUX
easat-9645	55	9	a	a	DET
easat-9645	55	10	real	real	ADJ
easat-9645	55	11	function	function	NOUN
easat-9645	56	1	‖•‖	‖•‖	PROPN
easat-9645	56	2	:	:	PUNCT
easat-9645	56	3	𝑋	𝑋	PROPN
easat-9645	56	4	→	→	SYM
easat-9645	56	5	𝑅defined	𝑅defined	PROPN
easat-9645	56	6	on	on	ADP
easat-9645	56	7	x	x	SYM
easat-9645	56	8	such	such	ADJ
easat-9645	56	9	that	that	PRON
easat-9645	56	10	for	for	ADP
easat-9645	56	11	any	any	DET
easat-9645	56	12	𝑥	𝑥	NOUN
easat-9645	56	13	,	,	PUNCT
easat-9645	56	14	𝑦	𝑦	NOUN
easat-9645	56	15	∈	∈	NOUN
easat-9645	56	16	𝑋	𝑋	NOUN
easat-9645	56	17	and	and	CCONJ
easat-9645	56	18	for	for	ADP
easat-9645	56	19	all	all	PRON
easat-9645	56	20	𝜆	𝜆	PRON
easat-9645	56	21	∈	∈	PROPN
easat-9645	56	22	𝐾.	𝐾.	PROPN
easat-9645	56	23	i.	i.	NOUN
easat-9645	56	24	‖𝑥‖	‖𝑥‖	PROPN
easat-9645	56	25	≥	≥	NUM
easat-9645	56	26	0	0	NUM
easat-9645	56	27	,	,	PUNCT
easat-9645	56	28	ii	ii	NOUN
easat-9645	56	29	.	.	PUNCT
easat-9645	57	1	‖𝑥‖	‖𝑥‖	PROPN
easat-9645	57	2	=	=	PUNCT
easat-9645	58	1	0𝑖𝑓𝑎𝑛𝑑𝑜𝑛𝑙𝑦𝑖𝑓𝑥	0𝑖𝑓𝑎𝑛𝑑𝑜𝑛𝑙𝑦𝑖𝑓𝑥	PUNCT
easat-9645	58	2	=	=	SYM
easat-9645	58	3	0	0	NUM
easat-9645	58	4	,	,	PUNCT
easat-9645	58	5	iii	iii	X
easat-9645	58	6	.	.	NOUN
easat-9645	58	7	xx	xx	NUM
easat-9645	58	8			NOUN
easat-9645	58	9	=	=	SYM
easat-9645	58	10	iv	iv	X
easat-9645	58	11	.	.	PUNCT
easat-9645	58	12	‖𝑥	‖𝑥	PROPN
easat-9645	59	1	+	+	CCONJ
easat-9645	59	2	𝑦‖	𝑦‖	PROPN
easat-9645	59	3	≤	≤	PROPN
easat-9645	59	4	‖𝑥‖	‖𝑥‖	PROPN
easat-9645	59	5	+	+	CCONJ
easat-9645	59	6	‖𝑦‖(𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦	‖𝑦‖(𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒𝑖𝑛𝑒𝑞𝑢𝑎𝑙𝑖𝑡𝑦	NOUN
easat-9645	59	7	)	)	PUNCT
easat-9645	59	8	a	a	DET
easat-9645	59	9	norm	norm	NOUN
easat-9645	59	10	on	on	ADP
easat-9645	59	11	x	x	PUNCT
easat-9645	59	12	defines	define	VERB
easat-9645	59	13	a	a	DET
easat-9645	59	14	metric	metric	ADJ
easat-9645	59	15	d	d	NOUN
easat-9645	59	16	on	on	ADP
easat-9645	59	17	x	x	PUNCT
easat-9645	59	18	which	which	PRON
easat-9645	59	19	is	be	AUX
easat-9645	59	20	given	give	VERB
easat-9645	59	21	by	by	ADP
easat-9645	59	22	𝑑(𝑥	𝑑(𝑥	PROPN
easat-9645	59	23	,	,	PUNCT
easat-9645	59	24	𝑦	𝑦	X
easat-9645	59	25	)	)	PUNCT
easat-9645	59	26	=	=	SYM
easat-9645	59	27	‖𝑥	‖𝑥	PROPN
easat-9645	59	28	−	−	PROPN
easat-9645	59	29	𝑦‖	𝑦‖	PROPN
easat-9645	59	30	;	;	PUNCT
easat-9645	59	31	𝑥	𝑥	X
easat-9645	59	32	,	,	PUNCT
easat-9645	59	33	𝑦	𝑦	NOUN
easat-9645	59	34	∈	∈	NOUN
easat-9645	60	1	𝑋and	𝑋and	PROPN
easat-9645	60	2	is	be	AUX
easat-9645	60	3	called	call	VERB
easat-9645	60	4	the	the	DET
easat-9645	60	5	metric	metric	NOUN
easat-9645	60	6	induced	induce	VERB
easat-9645	60	7	by	by	ADP
easat-9645	60	8	the	the	DET
easat-9645	60	9	norm	norm	NOUN
easat-9645	60	10	.	.	PUNCT
easat-9645	61	1	the	the	DET
easat-9645	61	2	normed	normed	PROPN
easat-9645	61	3	space	space	NOUN
easat-9645	61	4	is	be	AUX
easat-9645	61	5	denoted	denote	VERB
easat-9645	61	6	by	by	ADP
easat-9645	61	7	(	(	PUNCT
easat-9645	61	8	)	)	PUNCT
easat-9645	61	9	•,x	•,x	VERB
easat-9645	61	10	or	or	CCONJ
easat-9645	61	11	simply	simply	ADV
easat-9645	61	12	by	by	ADP
easat-9645	61	13	x	x	SYM
easat-9645	61	14	inner	inner	ADJ
easat-9645	61	15	product	product	NOUN
easat-9645	61	16	spaces	space	VERB
easat-9645	61	17	:	:	PUNCT
easat-9645	61	18	let	let	VERB
easat-9645	61	19	x	x	PRON
easat-9645	61	20	be	be	AUX
easat-9645	61	21	a	a	DET
easat-9645	61	22	vector	vector	NOUN
easat-9645	61	23	space	space	NOUN
easat-9645	61	24	over	over	ADP
easat-9645	61	25	the	the	DET
easat-9645	61	26	field	field	NOUN
easat-9645	61	27	k	k	PROPN
easat-9645	61	28	of	of	ADP
easat-9645	61	29	real	real	ADJ
easat-9645	61	30	or	or	CCONJ
easat-9645	61	31	complex	complex	ADJ
easat-9645	61	32	.	.	PUNCT
easat-9645	62	1	then	then	ADV
easat-9645	62	2	a	a	DET
easat-9645	62	3	mapping	mapping	NOUN
easat-9645	62	4	⟨.	⟨.	NOUN
easat-9645	62	5	,	,	PUNCT
easat-9645	62	6	.	.	PUNCT
easat-9645	63	1	⟩	⟩	NOUN
easat-9645	63	2	:	:	PUNCT
easat-9645	64	1	𝑋	𝑋	PROPN
easat-9645	64	2	×	×	NOUN
easat-9645	64	3	𝑋	𝑋	PROPN
easat-9645	64	4	→	→	SYM
easat-9645	64	5	𝐾	𝐾	PROPN
easat-9645	64	6	is	be	AUX
easat-9645	64	7	called	call	VERB
easat-9645	64	8	an	an	DET
easat-9645	64	9	inner	inner	ADJ
easat-9645	64	10	product	product	NOUN
easat-9645	64	11	of	of	ADP
easat-9645	64	12	any	any	DET
easat-9645	64	13	𝑥	𝑥	NOUN
easat-9645	64	14	,	,	PUNCT
easat-9645	64	15	𝑦	𝑦	NOUN
easat-9645	64	16	∈	∈	NOUN
easat-9645	64	17	𝑋	𝑋	NOUN
easat-9645	64	18	and	and	CCONJ
easat-9645	64	19	for	for	ADP
easat-9645	64	20	all	all	DET
easat-9645	64	21	𝛼	𝛼	PRON
easat-9645	64	22	∈	∈	PROPN
easat-9645	64	23	𝐾	𝐾	PROPN
easat-9645	64	24	,	,	PUNCT
easat-9645	64	25	which	which	PRON
easat-9645	64	26	satisfies	satisfy	VERB
easat-9645	64	27	the	the	DET
easat-9645	64	28	following	follow	VERB
easat-9645	64	29	conditions	condition	NOUN
easat-9645	64	30	:	:	PUNCT
easat-9645	64	31	1500	1500	NUM
easat-9645	64	32	edelweiss	edelweiss	PROPN
easat-9645	64	33	applied	apply	VERB
easat-9645	64	34	science	science	NOUN
easat-9645	64	35	and	and	CCONJ
easat-9645	64	36	technology	technology	NOUN
easat-9645	64	37	issn	issn	PROPN
easat-9645	64	38	:	:	PUNCT
easat-9645	64	39	2576	2576	NUM
easat-9645	64	40	-	-	SYM
easat-9645	64	41	8484	8484	NUM
easat-9645	64	42	vol	vol	NOUN
easat-9645	64	43	.	.	PROPN
easat-9645	65	1	9	9	NUM
easat-9645	65	2	,	,	PUNCT
easat-9645	65	3	no	no	INTJ
easat-9645	65	4	.	.	NOUN
easat-9645	65	5	8	8	NUM
easat-9645	65	6	:	:	SYM
easat-9645	65	7	1498	1498	NUM
easat-9645	65	8	-	-	SYM
easat-9645	65	9	1523	1523	NUM
easat-9645	65	10	,	,	PUNCT
easat-9645	65	11	2025	2025	NUM
easat-9645	65	12	doi	doi	NOUN
easat-9645	65	13	:	:	PUNCT
easat-9645	65	14	10.55214/2576	10.55214/2576	NUM
easat-9645	65	15	-	-	SYM
easat-9645	65	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	65	17	©	©	PROPN
easat-9645	65	18	2025	2025	NUM
easat-9645	65	19	by	by	ADP
easat-9645	65	20	the	the	DET
easat-9645	65	21	authors	author	NOUN
easat-9645	65	22	;	;	PUNCT
easat-9645	66	1	licensee	licensee	PROPN
easat-9645	66	2	learning	learning	PROPN
easat-9645	66	3	gate	gate	PROPN
easat-9645	66	4	i.	i.	PROPN
easat-9645	66	5	〈	〈	PROPN
easat-9645	66	6	𝑥	𝑥	PROPN
easat-9645	66	7	,	,	PUNCT
easat-9645	66	8	𝑥	𝑥	NOUN
easat-9645	66	9	〉	〉	X
easat-9645	66	10	≥	≥	NOUN
easat-9645	66	11	0	0	NUM
easat-9645	66	12	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
easat-9645	66	13	〈	〈	PROPN
easat-9645	66	14	𝑥	𝑥	PROPN
easat-9645	66	15	,	,	PUNCT
easat-9645	66	16	𝑥	𝑥	NOUN
easat-9645	66	17	〉	〉	NOUN
easat-9645	66	18	=	=	SYM
easat-9645	66	19	0	0	NUM
easat-9645	66	20	⇔	⇔	X
easat-9645	66	21	x	x	X
easat-9645	66	22	=	=	SYM
easat-9645	66	23	0	0	NUM
easat-9645	66	24	ii	ii	PROPN
easat-9645	66	25	.	.	PUNCT
easat-9645	67	1	〈	〈	PROPN
easat-9645	67	2	𝑥	𝑥	PROPN
easat-9645	67	3	,	,	PUNCT
easat-9645	67	4	𝑦〉̅̅	𝑦〉̅̅	PROPN
easat-9645	67	5	̅̅	̅̅	PROPN
easat-9645	67	6	̅̅	̅̅	PROPN
easat-9645	67	7	̅	̅	PROPN
easat-9645	67	8	=	=	PUNCT
easat-9645	67	9	〈	〈	NOUN
easat-9645	67	10	𝑦	𝑦	NOUN
easat-9645	67	11	,	,	PUNCT
easat-9645	67	12	𝑥	𝑥	PRON
easat-9645	67	13	〉	〉	X
easat-9645	67	14	iii	iii	NOUN
easat-9645	67	15	.	.	PUNCT
easat-9645	68	1	〈	〈	PROPN
easat-9645	68	2	𝛼𝑥	𝛼𝑥	PROPN
easat-9645	68	3	,	,	PUNCT
easat-9645	68	4	𝑦	𝑦	NOUN
easat-9645	68	5	〉	〉	NOUN
easat-9645	68	6	=	=	SYM
easat-9645	68	7	𝛼〈𝑥	𝛼〈𝑥	PROPN
easat-9645	68	8	,	,	PUNCT
easat-9645	68	9	𝑦	𝑦	NUM
easat-9645	68	10	〉	〉	NOUN
easat-9645	68	11	iv	iv	NUM
easat-9645	68	12	.	.	PUNCT
easat-9645	69	1	〈𝑥	〈𝑥	ADP
easat-9645	69	2	+	+	NUM
easat-9645	69	3	𝑦	𝑦	X
easat-9645	69	4	,	,	PUNCT
easat-9645	69	5	𝑧	𝑧	PRON
easat-9645	69	6	〉	〉	NOUN
easat-9645	69	7	=	=	PUNCT
easat-9645	69	8	〈	〈	PROPN
easat-9645	69	9	𝑥	𝑥	PROPN
easat-9645	69	10	,	,	PUNCT
easat-9645	69	11	𝑧	𝑧	PRON
easat-9645	69	12	〉	〉	NOUN
easat-9645	69	13	+	+	X
easat-9645	69	14	〈	〈	PROPN
easat-9645	69	15	𝑦	𝑦	NOUN
easat-9645	69	16	,	,	PUNCT
easat-9645	69	17	𝑧	𝑧	PRON
easat-9645	69	18	〉	〉	NOUN
easat-9645	69	19	∀	∀	X
easat-9645	69	20	𝑥	𝑥	NOUN
easat-9645	69	21	,	,	PUNCT
easat-9645	69	22	𝑦	𝑦	NOUN
easat-9645	69	23	,	,	PUNCT
easat-9645	69	24	𝑧	𝑧	DET
easat-9645	69	25	∈	∈	NOUN
easat-9645	69	26	𝑋	𝑋	NOUN
easat-9645	69	27	the	the	DET
easat-9645	69	28	vector	vector	NOUN
easat-9645	69	29	space	space	NOUN
easat-9645	69	30	x	x	PUNCT
easat-9645	69	31	together	together	ADV
easat-9645	69	32	with	with	ADP
easat-9645	69	33	inner	inner	ADJ
easat-9645	69	34	product	product	NOUN
easat-9645	69	35	⟨.	⟨.	NOUN
easat-9645	69	36	,	,	PUNCT
easat-9645	69	37	.	.	PUNCT
easat-9645	70	1	⟩	⟩	NOUN
easat-9645	70	2	is	be	AUX
easat-9645	70	3	called	call	VERB
easat-9645	70	4	an	an	DET
easat-9645	70	5	inner	inner	ADJ
easat-9645	70	6	product	product	NOUN
easat-9645	70	7	or	or	CCONJ
easat-9645	70	8	prehilbert	prehilbert	ADJ
easat-9645	70	9	space	space	NOUN
easat-9645	70	10	.	.	PUNCT
easat-9645	71	1	the	the	DET
easat-9645	71	2	inner	inner	ADJ
easat-9645	71	3	product	product	NOUN
easat-9645	71	4	space	space	NOUN
easat-9645	71	5	is	be	AUX
easat-9645	71	6	denoted	denote	VERB
easat-9645	71	7	by	by	ADP
easat-9645	71	8	(	(	PUNCT
easat-9645	71	9	𝑋	𝑋	PROPN
easat-9645	71	10	,	,	PUNCT
easat-9645	71	11	⟨.	⟨.	PROPN
easat-9645	71	12	,	,	PUNCT
easat-9645	71	13	.	.	PUNCT
easat-9645	72	1	⟩	⟩	NOUN
easat-9645	72	2	)	)	PUNCT
easat-9645	72	3	the	the	DET
easat-9645	72	4	space	space	NOUN
easat-9645	72	5	𝑙(𝑝	𝑙(𝑝	NOUN
easat-9645	72	6	):	):	PUNCT
easat-9645	72	7	let	let	VERB
easat-9645	72	8	𝑝𝑘	𝑝𝑘	PART
easat-9645	72	9	be	be	AUX
easat-9645	72	10	+	+	PROPN
easat-9645	72	11	𝑣𝑒	𝑣𝑒	X
easat-9645	72	12	𝑜𝑓	𝑜𝑓	ADP
easat-9645	72	13	𝑅	𝑅	PROPN
easat-9645	72	14	that	that	PRON
easat-9645	72	15	is	be	AUX
easat-9645	72	16	bounded	bound	VERB
easat-9645	72	17	above	above	ADV
easat-9645	72	18	,	,	PUNCT
easat-9645	72	19	s.t	s.t	PROPN
easat-9645	72	20	.	.	PROPN
easat-9645	72	21	0	0	PUNCT
easat-9645	73	1	<	<	X
easat-9645	73	2	𝑝𝑘	𝑝𝑘	PRON
easat-9645	73	3	≤	≤	NUM
easat-9645	73	4	sup	sup	NOUN
easat-9645	73	5	𝑝𝑘	𝑝𝑘	NOUN
easat-9645	73	6	=	=	PUNCT
easat-9645	73	7	𝐻	𝐻	NOUN
easat-9645	73	8	<	<	X
easat-9645	73	9	∞	∞	PROPN
easat-9645	73	10	.	.	PUNCT
easat-9645	74	1	then	then	ADV
easat-9645	74	2	the	the	DET
easat-9645	74	3	space	space	NOUN
easat-9645	74	4	𝑙(𝑝	𝑙(𝑝	NOUN
easat-9645	74	5	)	)	PUNCT
easat-9645	74	6	is	be	AUX
easat-9645	74	7	defined	define	VERB
easat-9645	74	8	as	as	ADP
easat-9645	74	9	𝑙(𝑝	𝑙(𝑝	NOUN
easat-9645	74	10	)	)	PUNCT
easat-9645	74	11	=	=	PRON
easat-9645	75	1	{	{	PUNCT
easat-9645	75	2	𝑥	𝑥	X
easat-9645	75	3	=	=	SYM
easat-9645	75	4	(	(	PUNCT
easat-9645	75	5	𝑥𝑘	𝑥𝑘	NOUN
easat-9645	75	6	):	):	PUNCT
easat-9645	75	7	∑	∑	PUNCT
easat-9645	75	8	|𝑥𝑘|𝑝𝑘	|𝑥𝑘|𝑝𝑘	VERB
easat-9645	75	9	<	<	X
easat-9645	75	10	∞𝑘	∞𝑘	NOUN
easat-9645	75	11	}	}	PUNCT
easat-9645	75	12	.	.	PUNCT
easat-9645	76	1	also	also	ADV
easat-9645	76	2	𝑑(𝑥	𝑑(𝑥	PROPN
easat-9645	76	3	,	,	PUNCT
easat-9645	76	4	𝑦	𝑦	NOUN
easat-9645	76	5	)	)	PUNCT
easat-9645	76	6	=	=	SYM
easat-9645	76	7	(	(	PUNCT
easat-9645	76	8	∑	∑	PROPN
easat-9645	76	9	|𝑥𝑘	|𝑥𝑘	PRON
easat-9645	76	10	−	−	PROPN
easat-9645	76	11	𝑦𝑘|𝑝𝑘	𝑦𝑘|𝑝𝑘	NOUN
easat-9645	76	12	𝑘	𝑘	X
easat-9645	76	13	)	)	PUNCT
easat-9645	76	14	1	1	NUM
easat-9645	76	15	𝑀	𝑀	PROPN
easat-9645	76	16	the	the	DET
easat-9645	76	17	space	space	NOUN
easat-9645	76	18	𝑙𝑝	𝑙𝑝	NOUN
easat-9645	76	19	:	:	PUNCT
easat-9645	76	20	let	let	VERB
easat-9645	76	21	𝑝𝑘	𝑝𝑘	VERB
easat-9645	76	22	=	=	SYM
easat-9645	76	23	𝑝	𝑝	PROPN
easat-9645	76	24	be	be	AUX
easat-9645	76	25	+	+	ADJ
easat-9645	76	26	𝑣𝑒	𝑣𝑒	X
easat-9645	76	27	𝑜𝑓	𝑜𝑓	ADP
easat-9645	76	28	𝑅	𝑅	PROPN
easat-9645	76	29	that	that	PRON
easat-9645	76	30	is	be	AUX
easat-9645	76	31	bounded	bound	VERB
easat-9645	76	32	above	above	ADV
easat-9645	76	33	,	,	PUNCT
easat-9645	76	34	s.t	s.t	PROPN
easat-9645	76	35	.	.	PROPN
easat-9645	76	36	1	1	NUM
easat-9645	76	37	≤	≤	NOUN
easat-9645	77	1	p	p	X
easat-9645	77	2	<	<	X
easat-9645	77	3	∞	∞	PROPN
easat-9645	77	4	.	.	PUNCT
easat-9645	78	1	then	then	ADV
easat-9645	78	2	the	the	DET
easat-9645	78	3	space	space	NOUN
easat-9645	78	4	𝑙𝑝	𝑙𝑝	NOUN
easat-9645	78	5	is	be	AUX
easat-9645	78	6	defined	define	VERB
easat-9645	78	7	as	as	ADP
easat-9645	78	8	𝑙𝑝	𝑙𝑝	NOUN
easat-9645	78	9	=	=	PUNCT
easat-9645	78	10	{	{	PUNCT
easat-9645	78	11	𝑥	𝑥	X
easat-9645	78	12	=	=	SYM
easat-9645	78	13	(	(	PUNCT
easat-9645	78	14	𝑥𝑘	𝑥𝑘	PROPN
easat-9645	78	15	):	):	PUNCT
easat-9645	78	16	∑	∑	PUNCT
easat-9645	78	17	|𝑥𝑘|𝑝	|𝑥𝑘|𝑝	VERB
easat-9645	78	18	<	<	X
easat-9645	78	19	∞𝑘	∞𝑘	NOUN
easat-9645	78	20	}	}	PUNCT
easat-9645	78	21	.	.	PUNCT
easat-9645	79	1	also	also	ADV
easat-9645	79	2	𝑑(𝑥	𝑑(𝑥	PROPN
easat-9645	79	3	,	,	PUNCT
easat-9645	79	4	𝑦	𝑦	NOUN
easat-9645	79	5	)	)	PUNCT
easat-9645	79	6	=	=	SYM
easat-9645	79	7	(	(	PUNCT
easat-9645	79	8	∑	∑	PROPN
easat-9645	79	9	|𝑥𝑘	|𝑥𝑘	PRON
easat-9645	79	10	−	−	PROPN
easat-9645	79	11	𝑦𝑘|𝑝	𝑦𝑘|𝑝	VERB
easat-9645	79	12	𝑘	𝑘	X
easat-9645	79	13	)	)	PUNCT
easat-9645	79	14	1	1	NUM
easat-9645	79	15	𝑝	𝑝	NOUN
easat-9645	79	16	the	the	DET
easat-9645	79	17	space	space	NOUN
easat-9645	79	18	𝑙∞(𝑝	𝑙∞(𝑝	NOUN
easat-9645	79	19	):	):	PUNCT
easat-9645	79	20	let	let	VERB
easat-9645	79	21	𝑝𝑘	𝑝𝑘	PART
easat-9645	79	22	be	be	AUX
easat-9645	79	23	+	+	PROPN
easat-9645	79	24	𝑣𝑒	𝑣𝑒	X
easat-9645	79	25	𝑜𝑓	𝑜𝑓	ADP
easat-9645	79	26	𝑅	𝑅	PROPN
easat-9645	79	27	that	that	PRON
easat-9645	79	28	is	be	AUX
easat-9645	79	29	bounded	bound	VERB
easat-9645	79	30	above	above	ADV
easat-9645	79	31	,	,	PUNCT
easat-9645	79	32	s.t	s.t	PROPN
easat-9645	79	33	.	.	PROPN
easat-9645	79	34	0	0	PUNCT
easat-9645	80	1	<	<	X
easat-9645	80	2	𝑝𝑘	𝑝𝑘	PRON
easat-9645	80	3	≤	≤	NUM
easat-9645	80	4	sup	sup	NOUN
easat-9645	80	5	𝑝𝑘	𝑝𝑘	NOUN
easat-9645	80	6	=	=	PUNCT
easat-9645	80	7	𝐻	𝐻	NOUN
easat-9645	80	8	<	<	X
easat-9645	80	9	∞	∞	PROPN
easat-9645	80	10	.	.	PUNCT
easat-9645	81	1	then	then	ADV
easat-9645	81	2	the	the	DET
easat-9645	81	3	space	space	NOUN
easat-9645	81	4	𝑙(𝑝	𝑙(𝑝	NOUN
easat-9645	81	5	)	)	PUNCT
easat-9645	81	6	is	be	AUX
easat-9645	81	7	defined	define	VERB
easat-9645	81	8	as	as	ADP
easat-9645	81	9	𝑙∞(𝑝	𝑙∞(𝑝	NOUN
easat-9645	81	10	)	)	PUNCT
easat-9645	82	1	=	=	PRON
easat-9645	82	2	{	{	PUNCT
easat-9645	82	3	𝑥	𝑥	X
easat-9645	82	4	=	=	SYM
easat-9645	82	5	(	(	PUNCT
easat-9645	82	6	𝑥𝑘	𝑥𝑘	PROPN
easat-9645	82	7	):	):	PUNCT
easat-9645	82	8	sup	sup	NOUN
easat-9645	82	9	𝑘	𝑘	ADP
easat-9645	82	10	|𝑥𝑘|𝑝𝑘	|𝑥𝑘|𝑝𝑘	NOUN
easat-9645	82	11	<	<	X
easat-9645	82	12	∞	∞	NUM
easat-9645	82	13	}	}	PUNCT
easat-9645	82	14	.	.	PUNCT
easat-9645	83	1	also	also	ADV
easat-9645	83	2	𝑑(𝑥	𝑑(𝑥	PROPN
easat-9645	83	3	,	,	PUNCT
easat-9645	83	4	𝑦	𝑦	X
easat-9645	83	5	)	)	PUNCT
easat-9645	83	6	=	=	SYM
easat-9645	83	7	sup	sup	NOUN
easat-9645	83	8	𝑘	𝑘	PROPN
easat-9645	84	1	|𝑥𝑘	|𝑥𝑘	NUM
easat-9645	84	2	−	−	PROPN
easat-9645	85	1	𝑦𝑘|	𝑦𝑘|	PROPN
easat-9645	85	2	𝑝𝑘	𝑝𝑘	PROPN
easat-9645	85	3	𝑀	𝑀	PROPN
easat-9645	85	4	the	the	DET
easat-9645	85	5	space	space	NOUN
easat-9645	85	6	𝑙∞	𝑙∞	PROPN
easat-9645	85	7	𝑜𝑟	𝑜𝑟	ADP
easat-9645	85	8	𝑙∞	𝑙∞	PROPN
easat-9645	85	9	:	:	PUNCT
easat-9645	85	10	let	let	VERB
easat-9645	85	11	𝑝𝑘	𝑝𝑘	VERB
easat-9645	85	12	=	=	SYM
easat-9645	85	13	𝑝	𝑝	PROPN
easat-9645	85	14	be	be	AUX
easat-9645	85	15	+	+	ADJ
easat-9645	85	16	𝑣𝑒	𝑣𝑒	X
easat-9645	85	17	𝑜𝑓	𝑜𝑓	ADP
easat-9645	85	18	𝑅	𝑅	PROPN
easat-9645	85	19	that	that	PRON
easat-9645	85	20	is	be	AUX
easat-9645	85	21	bounded	bound	VERB
easat-9645	85	22	above	above	ADV
easat-9645	85	23	,	,	PUNCT
easat-9645	85	24	s.t	s.t	PROPN
easat-9645	85	25	.	.	PROPN
easat-9645	85	26	1	1	NUM
easat-9645	85	27	≤	≤	NOUN
easat-9645	86	1	p	p	X
easat-9645	86	2	<	<	X
easat-9645	86	3	∞	∞	PROPN
easat-9645	86	4	.	.	PUNCT
easat-9645	87	1	then	then	ADV
easat-9645	87	2	the	the	DET
easat-9645	87	3	space	space	NOUN
easat-9645	87	4	𝑙∞	𝑙∞	PROPN
easat-9645	87	5	is	be	AUX
easat-9645	87	6	defined	define	VERB
easat-9645	87	7	as	as	ADP
easat-9645	87	8	𝑙∞	𝑙∞	PROPN
easat-9645	87	9	=	=	SYM
easat-9645	87	10	{	{	PUNCT
easat-9645	87	11	𝑥	𝑥	X
easat-9645	87	12	=	=	SYM
easat-9645	87	13	(	(	PUNCT
easat-9645	87	14	𝑥𝑘	𝑥𝑘	PROPN
easat-9645	87	15	):	):	PUNCT
easat-9645	87	16	sup	sup	NOUN
easat-9645	87	17	𝑘	𝑘	INTJ
easat-9645	87	18	|𝑥𝑘|𝑝	|𝑥𝑘|𝑝	NOUN
easat-9645	87	19	<	<	X
easat-9645	87	20	∞	∞	NUM
easat-9645	87	21	}	}	PUNCT
easat-9645	87	22	.	.	PUNCT
easat-9645	88	1	also	also	ADV
easat-9645	88	2	𝑑(𝑥	𝑑(𝑥	PROPN
easat-9645	88	3	,	,	PUNCT
easat-9645	88	4	𝑦	𝑦	X
easat-9645	88	5	)	)	PUNCT
easat-9645	88	6	=	=	SYM
easat-9645	88	7	sup	sup	NOUN
easat-9645	88	8	𝑘	𝑘	PROPN
easat-9645	89	1	|𝑥𝑘	|𝑥𝑘	NUM
easat-9645	89	2	−	−	PROPN
easat-9645	90	1	𝑦𝑘|	𝑦𝑘|	PROPN
easat-9645	90	2	the	the	DET
easat-9645	90	3	space	space	NOUN
easat-9645	90	4	𝑐(𝑝	𝑐(𝑝	ADV
easat-9645	90	5	):	):	PUNCT
easat-9645	90	6	let	let	VERB
easat-9645	90	7	𝑝𝑘	𝑝𝑘	PART
easat-9645	90	8	be	be	AUX
easat-9645	90	9	+	+	PROPN
easat-9645	90	10	𝑣𝑒	𝑣𝑒	X
easat-9645	90	11	𝑜𝑓	𝑜𝑓	ADP
easat-9645	90	12	𝑅	𝑅	PROPN
easat-9645	90	13	that	that	PRON
easat-9645	90	14	is	be	AUX
easat-9645	90	15	bounded	bound	VERB
easat-9645	90	16	above	above	ADV
easat-9645	90	17	,	,	PUNCT
easat-9645	90	18	s.t	s.t	PROPN
easat-9645	90	19	.	.	PROPN
easat-9645	90	20	0	0	PUNCT
easat-9645	91	1	<	<	X
easat-9645	91	2	𝑝𝑘	𝑝𝑘	PRON
easat-9645	91	3	≤	≤	NUM
easat-9645	91	4	sup	sup	NOUN
easat-9645	91	5	𝑝𝑘	𝑝𝑘	NOUN
easat-9645	91	6	=	=	PUNCT
easat-9645	91	7	𝐻	𝐻	NOUN
easat-9645	91	8	<	<	X
easat-9645	91	9	∞	∞	PROPN
easat-9645	91	10	.	.	PUNCT
easat-9645	92	1	then	then	ADV
easat-9645	92	2	the	the	DET
easat-9645	92	3	space	space	NOUN
easat-9645	92	4	𝑐(𝑝	𝑐(𝑝	ADV
easat-9645	92	5	)	)	PUNCT
easat-9645	92	6	is	be	AUX
easat-9645	92	7	defined	define	VERB
easat-9645	92	8	as	as	ADP
easat-9645	92	9	𝑙(𝑝	𝑙(𝑝	NOUN
easat-9645	92	10	)	)	PUNCT
easat-9645	92	11	=	=	PRON
easat-9645	93	1	{	{	PUNCT
easat-9645	93	2	𝑥	𝑥	X
easat-9645	93	3	=	=	SYM
easat-9645	93	4	(	(	PUNCT
easat-9645	93	5	𝑥𝑘	𝑥𝑘	PROPN
easat-9645	93	6	):	):	PUNCT
easat-9645	93	7	|𝑥𝑘	|𝑥𝑘	NUM
easat-9645	93	8	−	−	PROPN
easat-9645	93	9	𝑙|𝑝𝑘	𝑙|𝑝𝑘	VERB
easat-9645	93	10	<	<	X
easat-9645	93	11	∞	∞	NUM
easat-9645	93	12	∀	∀	X
easat-9645	93	13	𝑙	𝑙	PRON
easat-9645	93	14	∈	∈	PROPN
easat-9645	93	15	𝐶	𝐶	PROPN
easat-9645	93	16	}	}	PUNCT
easat-9645	93	17	.	.	PUNCT
easat-9645	94	1	also	also	ADV
easat-9645	94	2	𝑑(𝑥	𝑑(𝑥	PROPN
easat-9645	94	3	,	,	PUNCT
easat-9645	94	4	𝑦	𝑦	X
easat-9645	94	5	)	)	PUNCT
easat-9645	94	6	=	=	SYM
easat-9645	94	7	sup	sup	NOUN
easat-9645	94	8	𝑘	𝑘	PROPN
easat-9645	95	1	|𝑥𝑘	|𝑥𝑘	NUM
easat-9645	95	2	−	−	PROPN
easat-9645	96	1	𝑦𝑘|	𝑦𝑘|	PROPN
easat-9645	96	2	𝑝𝑘	𝑝𝑘	PROPN
easat-9645	96	3	𝑀	𝑀	PROPN
easat-9645	96	4	the	the	DET
easat-9645	96	5	space	space	NOUN
easat-9645	96	6	𝑐	𝑐	NOUN
easat-9645	96	7	:	:	PUNCT
easat-9645	96	8	let	let	VERB
easat-9645	96	9	𝑝𝑘	𝑝𝑘	VERB
easat-9645	96	10	=	=	SYM
easat-9645	96	11	𝑝	𝑝	PROPN
easat-9645	96	12	be	be	AUX
easat-9645	96	13	+	+	ADJ
easat-9645	96	14	𝑣𝑒	𝑣𝑒	X
easat-9645	96	15	𝑜𝑓	𝑜𝑓	ADP
easat-9645	96	16	𝑅	𝑅	PROPN
easat-9645	96	17	that	that	PRON
easat-9645	96	18	is	be	AUX
easat-9645	96	19	bounded	bound	VERB
easat-9645	96	20	above	above	ADV
easat-9645	96	21	,	,	PUNCT
easat-9645	96	22	s.t	s.t	PROPN
easat-9645	96	23	.	.	PROPN
easat-9645	96	24	1	1	NUM
easat-9645	96	25	≤	≤	NOUN
easat-9645	97	1	p	p	X
easat-9645	97	2	<	<	X
easat-9645	97	3	∞	∞	PROPN
easat-9645	97	4	.	.	PUNCT
easat-9645	98	1	then	then	ADV
easat-9645	98	2	the	the	DET
easat-9645	98	3	space	space	NOUN
easat-9645	98	4	𝑐	𝑐	PROPN
easat-9645	98	5	is	be	AUX
easat-9645	98	6	defined	define	VERB
easat-9645	98	7	as	as	ADP
easat-9645	98	8	𝑐	𝑐	PROPN
easat-9645	98	9	=	=	PUNCT
easat-9645	98	10	{	{	PUNCT
easat-9645	98	11	𝑥	𝑥	X
easat-9645	98	12	=	=	SYM
easat-9645	98	13	(	(	PUNCT
easat-9645	98	14	𝑥𝑘	𝑥𝑘	PROPN
easat-9645	98	15	):	):	PUNCT
easat-9645	98	16	|𝑥𝑘	|𝑥𝑘	X
easat-9645	98	17	−	−	PROPN
easat-9645	98	18	𝑙|𝑝	𝑙|𝑝	VERB
easat-9645	98	19	<	<	X
easat-9645	98	20	∞	∞	NUM
easat-9645	98	21	}	}	PUNCT
easat-9645	98	22	.	.	PUNCT
easat-9645	99	1	also	also	ADV
easat-9645	99	2	,	,	PUNCT
easat-9645	99	3	𝑑(𝑥	𝑑(𝑥	PROPN
easat-9645	99	4	,	,	PUNCT
easat-9645	99	5	𝑦	𝑦	X
easat-9645	99	6	)	)	PUNCT
easat-9645	99	7	=	=	SYM
easat-9645	99	8	sup	sup	NOUN
easat-9645	99	9	𝑘	𝑘	PROPN
easat-9645	99	10	|𝑥𝑘	|𝑥𝑘	NUM
easat-9645	99	11	−	−	PROPN
easat-9645	99	12	𝑦𝑘|	𝑦𝑘|	PROPN
easat-9645	99	13	the	the	DET
easat-9645	99	14	space	space	NOUN
easat-9645	99	15	𝑐0(𝑝	𝑐0(𝑝	NOUN
easat-9645	99	16	):	):	PUNCT
easat-9645	99	17	let	let	VERB
easat-9645	99	18	𝑝𝑘	𝑝𝑘	PART
easat-9645	99	19	be	be	AUX
easat-9645	99	20	+	+	PROPN
easat-9645	99	21	𝑣𝑒	𝑣𝑒	X
easat-9645	99	22	𝑜𝑓	𝑜𝑓	ADP
easat-9645	99	23	𝑅	𝑅	PROPN
easat-9645	99	24	that	that	PRON
easat-9645	99	25	is	be	AUX
easat-9645	99	26	bounded	bound	VERB
easat-9645	99	27	above	above	ADV
easat-9645	99	28	,	,	PUNCT
easat-9645	99	29	s.t	s.t	PROPN
easat-9645	99	30	.	.	PROPN
easat-9645	99	31	0	0	PUNCT
easat-9645	100	1	<	<	X
easat-9645	100	2	𝑝𝑘	𝑝𝑘	PRON
easat-9645	100	3	≤	≤	NUM
easat-9645	100	4	sup	sup	NOUN
easat-9645	100	5	𝑝𝑘	𝑝𝑘	NOUN
easat-9645	100	6	=	=	PUNCT
easat-9645	100	7	𝐻	𝐻	NOUN
easat-9645	100	8	<	<	X
easat-9645	100	9	∞	∞	PROPN
easat-9645	100	10	.	.	PUNCT
easat-9645	101	1	then	then	ADV
easat-9645	101	2	the	the	DET
easat-9645	101	3	space	space	NOUN
easat-9645	101	4	𝑐(𝑝	𝑐(𝑝	ADV
easat-9645	101	5	)	)	PUNCT
easat-9645	101	6	is	be	AUX
easat-9645	101	7	defined	define	VERB
easat-9645	101	8	as	as	ADP
easat-9645	101	9	𝑙(𝑝	𝑙(𝑝	NOUN
easat-9645	101	10	)	)	PUNCT
easat-9645	101	11	=	=	PRON
easat-9645	102	1	{	{	PUNCT
easat-9645	102	2	𝑥	𝑥	X
easat-9645	102	3	=	=	SYM
easat-9645	102	4	(	(	PUNCT
easat-9645	102	5	𝑥𝑘	𝑥𝑘	NOUN
easat-9645	102	6	):	):	PUNCT
easat-9645	102	7	|𝑥𝑘|𝑝𝑘	|𝑥𝑘|𝑝𝑘	X
easat-9645	102	8	<	<	X
easat-9645	102	9	∞	∞	NUM
easat-9645	102	10	}	}	PUNCT
easat-9645	102	11	.	.	PUNCT
easat-9645	103	1	also	also	ADV
easat-9645	103	2	,	,	PUNCT
easat-9645	103	3	𝑑(𝑥	𝑑(𝑥	PROPN
easat-9645	103	4	,	,	PUNCT
easat-9645	103	5	𝑦	𝑦	X
easat-9645	103	6	)	)	PUNCT
easat-9645	103	7	=	=	SYM
easat-9645	103	8	sup	sup	NOUN
easat-9645	103	9	𝑘	𝑘	PROPN
easat-9645	104	1	|𝑥𝑘	|𝑥𝑘	NUM
easat-9645	104	2	−	−	PROPN
easat-9645	105	1	𝑦𝑘|	𝑦𝑘|	PROPN
easat-9645	105	2	𝑝𝑘	𝑝𝑘	PROPN
easat-9645	105	3	𝑀	𝑀	PROPN
easat-9645	105	4	the	the	DET
easat-9645	105	5	space	space	NOUN
easat-9645	105	6	𝑐0	𝑐0	NOUN
easat-9645	105	7	:	:	PUNCT
easat-9645	105	8	let	let	VERB
easat-9645	105	9	𝑝𝑘	𝑝𝑘	VERB
easat-9645	105	10	=	=	SYM
easat-9645	105	11	𝑝	𝑝	PROPN
easat-9645	105	12	be	be	AUX
easat-9645	105	13	+	+	ADJ
easat-9645	105	14	𝑣𝑒	𝑣𝑒	X
easat-9645	105	15	𝑜𝑓	𝑜𝑓	ADP
easat-9645	105	16	𝑅	𝑅	PROPN
easat-9645	105	17	that	that	PRON
easat-9645	105	18	is	be	AUX
easat-9645	105	19	bounded	bound	VERB
easat-9645	105	20	above	above	ADV
easat-9645	105	21	,	,	PUNCT
easat-9645	105	22	s.t	s.t	PROPN
easat-9645	105	23	.	.	PROPN
easat-9645	105	24	1	1	NUM
easat-9645	105	25	≤	≤	NOUN
easat-9645	106	1	p	p	X
easat-9645	106	2	<	<	X
easat-9645	106	3	∞	∞	PROPN
easat-9645	106	4	.	.	PUNCT
easat-9645	107	1	then	then	ADV
easat-9645	107	2	the	the	DET
easat-9645	107	3	space	space	NOUN
easat-9645	107	4	𝑐0	𝑐0	NOUN
easat-9645	107	5	is	be	AUX
easat-9645	107	6	defined	define	VERB
easat-9645	107	7	as	as	ADP
easat-9645	107	8	𝑐0	𝑐0	NOUN
easat-9645	107	9	=	=	SYM
easat-9645	107	10	{	{	PUNCT
easat-9645	107	11	𝑥	𝑥	X
easat-9645	107	12	=	=	SYM
easat-9645	107	13	(	(	PUNCT
easat-9645	107	14	𝑥𝑘	𝑥𝑘	NOUN
easat-9645	107	15	):	):	PUNCT
easat-9645	107	16	|𝑥𝑘|𝑝	|𝑥𝑘|𝑝	ADP
easat-9645	107	17	<	<	X
easat-9645	107	18	∞	∞	NUM
easat-9645	107	19	}	}	PUNCT
easat-9645	107	20	.	.	PUNCT
easat-9645	108	1	also	also	ADV
easat-9645	108	2	,	,	PUNCT
easat-9645	108	3	𝑑(𝑥	𝑑(𝑥	PROPN
easat-9645	108	4	,	,	PUNCT
easat-9645	108	5	𝑦	𝑦	X
easat-9645	108	6	)	)	PUNCT
easat-9645	108	7	=	=	SYM
easat-9645	108	8	sup	sup	PROPN
easat-9645	108	9	𝑘	𝑘	PROPN
easat-9645	108	10	|𝑥𝑘	|𝑥𝑘	NUM
easat-9645	108	11	−	−	PROPN
easat-9645	108	12	𝑦𝑘|	𝑦𝑘|	PROPN
easat-9645	108	13	,	,	PUNCT
easat-9645	108	14	1.3	1.3	NUM
easat-9645	108	15	.	.	PUNCT
easat-9645	108	16	arrangement	arrangement	NOUN
easat-9645	108	17	of	of	ADP
easat-9645	108	18	the	the	DET
easat-9645	108	19	article	article	NOUN
easat-9645	108	20	the	the	DET
easat-9645	108	21	structure	structure	NOUN
easat-9645	108	22	of	of	ADP
easat-9645	108	23	this	this	DET
easat-9645	108	24	paper	paper	NOUN
easat-9645	108	25	is	be	AUX
easat-9645	108	26	organized	organize	VERB
easat-9645	108	27	as	as	SCONJ
easat-9645	108	28	follows	follow	VERB
easat-9645	108	29	.	.	PUNCT
easat-9645	109	1	section	section	NOUN
easat-9645	109	2	2	2	NUM
easat-9645	109	3	presents	present	VERB
easat-9645	109	4	the	the	DET
easat-9645	109	5	parallelogram	parallelogram	NOUN
easat-9645	109	6	law	law	NOUN
easat-9645	109	7	and	and	CCONJ
easat-9645	109	8	examines	examine	VERB
easat-9645	109	9	its	its	PRON
easat-9645	109	10	importance	importance	NOUN
easat-9645	109	11	in	in	ADP
easat-9645	109	12	identifying	identify	VERB
easat-9645	109	13	hilbert	hilbert	NOUN
easat-9645	109	14	spaces	space	NOUN
easat-9645	109	15	.	.	PUNCT
easat-9645	110	1	section	section	NOUN
easat-9645	110	2	3	3	NUM
easat-9645	110	3	focuses	focus	VERB
easat-9645	110	4	on	on	ADP
easat-9645	110	5	the	the	DET
easat-9645	110	6	development	development	NOUN
easat-9645	110	7	and	and	CCONJ
easat-9645	110	8	formulation	formulation	NOUN
easat-9645	110	9	of	of	ADP
easat-9645	110	10	hilbert	hilbert	PROPN
easat-9645	110	11	spaces	space	NOUN
easat-9645	110	12	.	.	PUNCT
easat-9645	111	1	section	section	NOUN
easat-9645	111	2	4	4	NUM
easat-9645	111	3	highlights	highlight	NOUN
easat-9645	111	4	general	general	ADJ
easat-9645	111	5	applications	application	NOUN
easat-9645	111	6	of	of	ADP
easat-9645	111	7	hilbert	hilbert	PROPN
easat-9645	111	8	spaces	space	NOUN
easat-9645	111	9	.	.	PUNCT
easat-9645	112	1	section	section	NOUN
easat-9645	112	2	5	5	NUM
easat-9645	112	3	the	the	DET
easat-9645	112	4	discussion	discussion	NOUN
easat-9645	112	5	to	to	ADP
easat-9645	112	6	their	their	PRON
easat-9645	112	7	use	use	NOUN
easat-9645	112	8	in	in	ADP
easat-9645	112	9	machine	machine	NOUN
easat-9645	112	10	learning	learning	NOUN
easat-9645	112	11	.	.	PUNCT
easat-9645	113	1	section	section	NOUN
easat-9645	113	2	6	6	NUM
easat-9645	113	3	explores	explore	NOUN
easat-9645	113	4	applications	application	NOUN
easat-9645	113	5	of	of	ADP
easat-9645	113	6	inner	inner	ADJ
easat-9645	113	7	product	product	NOUN
easat-9645	113	8	spaces	space	NOUN
easat-9645	113	9	,	,	PUNCT
easat-9645	113	10	followed	follow	VERB
easat-9645	113	11	by	by	ADP
easat-9645	113	12	section	section	NOUN
easat-9645	113	13	7	7	NUM
easat-9645	113	14	,	,	PUNCT
easat-9645	113	15	which	which	PRON
easat-9645	113	16	considers	consider	VERB
easat-9645	113	17	their	their	PRON
easat-9645	113	18	role	role	NOUN
easat-9645	113	19	in	in	ADP
easat-9645	113	20	machine	machine	NOUN
easat-9645	113	21	learning	learning	NOUN
easat-9645	113	22	.	.	PUNCT
easat-9645	114	1	section	section	NOUN
easat-9645	114	2	8	8	NUM
easat-9645	114	3	integrates	integrate	VERB
easat-9645	114	4	the	the	DET
easat-9645	114	5	applications	application	NOUN
easat-9645	114	6	of	of	ADP
easat-9645	114	7	both	both	CCONJ
easat-9645	114	8	inner	inner	ADJ
easat-9645	114	9	product	product	NOUN
easat-9645	114	10	and	and	CCONJ
easat-9645	114	11	hilbert	hilbert	NOUN
easat-9645	114	12	spaces	space	NOUN
easat-9645	114	13	in	in	ADP
easat-9645	114	14	machine	machine	NOUN
easat-9645	114	15	learning	learn	VERB
easat-9645	114	16	contexts	contexts	NOUN
easat-9645	114	17	.	.	PUNCT
easat-9645	115	1	section	section	NOUN
easat-9645	115	2	9	9	NUM
easat-9645	115	3	provides	provide	VERB
easat-9645	115	4	a	a	DET
easat-9645	115	5	conceptual	conceptual	ADJ
easat-9645	115	6	perspective	perspective	NOUN
easat-9645	115	7	on	on	ADP
easat-9645	115	8	human	human	ADJ
easat-9645	115	9	reasoning	reasoning	NOUN
easat-9645	115	10	in	in	ADP
easat-9645	115	11	machine	machine	NOUN
easat-9645	115	12	learning	learn	VERB
easat-9645	115	13	through	through	ADP
easat-9645	115	14	the	the	DET
easat-9645	115	15	lens	lens	NOUN
easat-9645	115	16	of	of	ADP
easat-9645	115	17	these	these	DET
easat-9645	115	18	spaces	space	NOUN
easat-9645	115	19	.	.	PUNCT
easat-9645	116	1	finally	finally	ADV
easat-9645	116	2	,	,	PUNCT
easat-9645	116	3	section	section	NOUN
easat-9645	116	4	10	10	NUM
easat-9645	116	5	offers	offer	VERB
easat-9645	116	6	concluding	conclude	VERB
easat-9645	116	7	remarks	remark	NOUN
easat-9645	116	8	.	.	PUNCT
easat-9645	117	1	2	2	X
easat-9645	117	2	.	.	X
easat-9645	117	3	parallelogram	parallelogram	NOUN
easat-9645	117	4	law	law	NOUN
easat-9645	117	5	and	and	CCONJ
easat-9645	117	6	characterization	characterization	NOUN
easat-9645	117	7	of	of	ADP
easat-9645	117	8	hilbert	hilbert	NOUN
easat-9645	117	9	space	space	NOUN
easat-9645	117	10	in	in	ADP
easat-9645	117	11	classical	classical	ADJ
easat-9645	117	12	geometry	geometry	NOUN
easat-9645	117	13	,	,	PUNCT
easat-9645	117	14	the	the	DET
easat-9645	117	15	parallelogram	parallelogram	NOUN
easat-9645	117	16	law	law	NOUN
easat-9645	117	17	asserts	assert	VERB
easat-9645	117	18	that	that	SCONJ
easat-9645	117	19	the	the	DET
easat-9645	117	20	total	total	NOUN
easat-9645	117	21	of	of	ADP
easat-9645	117	22	the	the	DET
easat-9645	117	23	squares	square	NOUN
easat-9645	117	24	of	of	ADP
easat-9645	117	25	a	a	DET
easat-9645	117	26	parallelogram	parallelogram	NOUN
easat-9645	117	27	’s	’s	PART
easat-9645	117	28	diagonals	diagonal	NOUN
easat-9645	117	29	is	be	AUX
easat-9645	117	30	equal	equal	ADJ
easat-9645	117	31	to	to	ADP
easat-9645	117	32	the	the	DET
easat-9645	117	33	sum	sum	NOUN
easat-9645	117	34	of	of	ADP
easat-9645	117	35	the	the	DET
easat-9645	117	36	squares	square	NOUN
easat-9645	117	37	of	of	ADP
easat-9645	117	38	its	its	PRON
easat-9645	117	39	four	four	NUM
easat-9645	117	40	sides	side	NOUN
easat-9645	117	41	.	.	PUNCT
easat-9645	118	1	this	this	DET
easat-9645	118	2	concept	concept	NOUN
easat-9645	118	3	has	have	VERB
easat-9645	118	4	a	a	DET
easat-9645	118	5	significant	significant	ADJ
easat-9645	118	6	counterpart	counterpart	NOUN
easat-9645	118	7	in	in	ADP
easat-9645	118	8	functional	functional	ADJ
easat-9645	118	9	analysis	analysis	NOUN
easat-9645	118	10	.	.	PUNCT
easat-9645	119	1	in	in	ADP
easat-9645	119	2	particular	particular	ADJ
easat-9645	119	3	,	,	PUNCT
easat-9645	119	4	within	within	ADP
easat-9645	119	5	normed	normed	ADJ
easat-9645	119	6	vector	vector	NOUN
easat-9645	119	7	spaces	space	NOUN
easat-9645	119	8	,	,	PUNCT
easat-9645	119	9	an	an	DET
easat-9645	119	10	analogous	analogous	ADJ
easat-9645	119	11	identity	identity	NOUN
easat-9645	119	12	holds	hold	VERB
easat-9645	119	13	in	in	ADP
easat-9645	119	14	inner	inner	ADJ
easat-9645	119	15	product	product	NOUN
easat-9645	119	16	spaces	space	NOUN
easat-9645	119	17	.	.	PUNCT
easat-9645	120	1	this	this	DET
easat-9645	120	2	identity	identity	NOUN
easat-9645	120	3	is	be	AUX
easat-9645	120	4	universally	universally	ADV
easat-9645	120	5	valid	valid	ADJ
easat-9645	120	6	in	in	ADP
easat-9645	120	7	inner	inner	ADJ
easat-9645	120	8	product	product	NOUN
easat-9645	120	9	spaces	space	NOUN
easat-9645	120	10	and	and	CCONJ
easat-9645	120	11	plays	play	VERB
easat-9645	120	12	a	a	DET
easat-9645	120	13	vital	vital	ADJ
easat-9645	120	14	role	role	NOUN
easat-9645	120	15	in	in	ADP
easat-9645	120	16	identifying	identify	VERB
easat-9645	120	17	when	when	SCONJ
easat-9645	120	18	a	a	DET
easat-9645	120	19	normed	normed	ADJ
easat-9645	120	20	space	space	NOUN
easat-9645	120	21	originates	originate	NOUN
easat-9645	120	22	from	from	ADP
easat-9645	120	23	an	an	DET
easat-9645	120	24	inner	inner	ADJ
easat-9645	120	25	product	product	NOUN
easat-9645	120	26	.	.	PUNCT
easat-9645	121	1	it	it	PRON
easat-9645	121	2	leads	lead	VERB
easat-9645	121	3	to	to	ADP
easat-9645	121	4	a	a	DET
easat-9645	121	5	significant	significant	ADJ
easat-9645	121	6	result	result	NOUN
easat-9645	121	7	that	that	SCONJ
easat-9645	121	8	a	a	DET
easat-9645	121	9	normed	normed	ADJ
easat-9645	121	10	linear	linear	ADJ
easat-9645	121	11	space	space	NOUN
easat-9645	121	12	is	be	AUX
easat-9645	121	13	a	a	DET
easat-9645	121	14	hilbert	hilbert	NOUN
easat-9645	121	15	space	space	NOUN
easat-9645	121	16	if	if	SCONJ
easat-9645	122	1	and	and	CCONJ
easat-9645	122	2	only	only	ADV
easat-9645	122	3	if	if	SCONJ
easat-9645	122	4	its	its	PRON
easat-9645	122	5	norm	norm	NOUN
easat-9645	122	6	satisfies	satisfy	VERB
easat-9645	122	7	the	the	DET
easat-9645	122	8	parallelogram	parallelogram	NOUN
easat-9645	122	9	law	law	NOUN
easat-9645	122	10	.	.	PUNCT
easat-9645	123	1	if	if	SCONJ
easat-9645	123	2	a	a	DET
easat-9645	123	3	normed	normed	ADJ
easat-9645	123	4	space	space	NOUN
easat-9645	123	5	satisfies	satisfie	NOUN
easat-9645	123	6	this	this	DET
easat-9645	123	7	condition	condition	NOUN
easat-9645	123	8	,	,	PUNCT
easat-9645	123	9	it	it	PRON
easat-9645	123	10	is	be	AUX
easat-9645	123	11	possible	possible	ADJ
easat-9645	123	12	to	to	PART
easat-9645	123	13	define	define	VERB
easat-9645	123	14	an	an	DET
easat-9645	123	15	inner	inner	ADJ
easat-9645	123	16	product	product	NOUN
easat-9645	123	17	that	that	PRON
easat-9645	123	18	induces	induce	VERB
easat-9645	123	19	the	the	DET
easat-9645	123	20	norm	norm	NOUN
easat-9645	123	21	,	,	PUNCT
easat-9645	123	22	effectively	effectively	ADV
easat-9645	123	23	transforming	transform	VERB
easat-9645	123	24	the	the	DET
easat-9645	123	25	space	space	NOUN
easat-9645	123	26	into	into	ADP
easat-9645	123	27	an	an	DET
easat-9645	123	28	inner	inner	ADJ
easat-9645	123	29	product	product	NOUN
easat-9645	123	30	space	space	NOUN
easat-9645	123	31	.	.	PUNCT
easat-9645	124	1	if	if	SCONJ
easat-9645	124	2	,	,	PUNCT
easat-9645	124	3	in	in	ADP
easat-9645	124	4	addition	addition	NOUN
easat-9645	124	5	,	,	PUNCT
easat-9645	124	6	the	the	DET
easat-9645	124	7	space	space	NOUN
easat-9645	124	8	is	be	AUX
easat-9645	124	9	complete	complete	ADJ
easat-9645	124	10	under	under	ADP
easat-9645	124	11	this	this	DET
easat-9645	124	12	1501	1501	NUM
easat-9645	124	13	edelweiss	edelweiss	PROPN
easat-9645	124	14	applied	apply	VERB
easat-9645	124	15	science	science	NOUN
easat-9645	124	16	and	and	CCONJ
easat-9645	124	17	technology	technology	NOUN
easat-9645	124	18	issn	issn	PROPN
easat-9645	124	19	:	:	PUNCT
easat-9645	124	20	2576	2576	NUM
easat-9645	124	21	-	-	SYM
easat-9645	124	22	8484	8484	NUM
easat-9645	124	23	vol	vol	NOUN
easat-9645	124	24	.	.	PROPN
easat-9645	125	1	9	9	NUM
easat-9645	125	2	,	,	PUNCT
easat-9645	125	3	no	no	INTJ
easat-9645	125	4	.	.	NOUN
easat-9645	125	5	8	8	NUM
easat-9645	125	6	:	:	SYM
easat-9645	125	7	1498	1498	NUM
easat-9645	125	8	-	-	SYM
easat-9645	125	9	1523	1523	NUM
easat-9645	125	10	,	,	PUNCT
easat-9645	125	11	2025	2025	NUM
easat-9645	125	12	doi	doi	NOUN
easat-9645	125	13	:	:	PUNCT
easat-9645	125	14	10.55214/2576	10.55214/2576	NUM
easat-9645	125	15	-	-	SYM
easat-9645	125	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	125	17	©	©	PROPN
easat-9645	125	18	2025	2025	NUM
easat-9645	125	19	by	by	ADP
easat-9645	125	20	the	the	DET
easat-9645	125	21	authors	author	NOUN
easat-9645	125	22	;	;	PUNCT
easat-9645	125	23	licensee	licensee	PROPN
easat-9645	125	24	learning	learn	VERB
easat-9645	125	25	gate	gate	PROPN
easat-9645	125	26	norm	norm	NOUN
easat-9645	125	27	,	,	PUNCT
easat-9645	125	28	it	it	PRON
easat-9645	125	29	qualifies	qualify	VERB
easat-9645	125	30	as	as	ADP
easat-9645	125	31	a	a	DET
easat-9645	125	32	hilbert	hilbert	NOUN
easat-9645	125	33	space	space	NOUN
easat-9645	125	34	.	.	PUNCT
easat-9645	126	1	thus	thus	ADV
easat-9645	126	2	,	,	PUNCT
easat-9645	126	3	the	the	DET
easat-9645	126	4	parallelogram	parallelogram	NOUN
easat-9645	126	5	law	law	NOUN
easat-9645	126	6	not	not	PART
easat-9645	126	7	only	only	ADV
easat-9645	126	8	captures	capture	VERB
easat-9645	126	9	a	a	DET
easat-9645	126	10	fundamental	fundamental	ADJ
easat-9645	126	11	geometric	geometric	ADJ
easat-9645	126	12	principle	principle	NOUN
easat-9645	126	13	but	but	CCONJ
easat-9645	126	14	also	also	ADV
easat-9645	126	15	serves	serve	VERB
easat-9645	126	16	as	as	ADP
easat-9645	126	17	a	a	DET
easat-9645	126	18	powerful	powerful	ADJ
easat-9645	126	19	analytical	analytical	ADJ
easat-9645	126	20	tool	tool	NOUN
easat-9645	126	21	for	for	ADP
easat-9645	126	22	recognizing	recognize	VERB
easat-9645	126	23	hilbert	hilbert	NOUN
easat-9645	126	24	spaces	space	NOUN
easat-9645	126	25	within	within	ADP
easat-9645	126	26	the	the	DET
easat-9645	126	27	wider	wide	ADJ
easat-9645	126	28	context	context	NOUN
easat-9645	126	29	of	of	ADP
easat-9645	126	30	normed	normed	ADJ
easat-9645	126	31	vector	vector	NOUN
easat-9645	126	32	spaces	space	NOUN
easat-9645	126	33	.	.	PUNCT
easat-9645	127	1	2.1	2.1	NUM
easat-9645	127	2	.	.	PUNCT
easat-9645	127	3	theorem	theorem	NOUN
easat-9645	127	4	(	(	PUNCT
easat-9645	127	5	parallelogram	parallelogram	NOUN
easat-9645	127	6	law	law	NOUN
easat-9645	127	7	)	)	PUNCT
easat-9645	127	8	for	for	ADP
easat-9645	127	9	any	any	DET
easat-9645	127	10	two	two	NUM
easat-9645	127	11	elements	element	NOUN
easat-9645	127	12	x	x	PUNCT
easat-9645	127	13	and	and	CCONJ
easat-9645	127	14	y	y	PROPN
easat-9645	127	15	belonging	belong	VERB
easat-9645	127	16	to	to	ADP
easat-9645	127	17	an	an	DET
easat-9645	127	18	inner	inner	ADJ
easat-9645	127	19	product	product	NOUN
easat-9645	127	20	space	space	NOUN
easat-9645	127	21	x	x	NOUN
easat-9645	127	22	,	,	PUNCT
easat-9645	127	23	then	then	ADV
easat-9645	127	24	‖𝑥	‖𝑥	PUNCT
easat-9645	127	25	+	+	CCONJ
easat-9645	127	26	𝑦‖2	𝑦‖2	X
easat-9645	127	27	+	+	NOUN
easat-9645	127	28	‖𝑥	‖𝑥	NOUN
easat-9645	127	29	−	−	NOUN
easat-9645	127	30	𝑦‖2	𝑦‖2	X
easat-9645	127	31	=	=	SYM
easat-9645	127	32	2‖𝑥‖2	2‖𝑥‖2	PROPN
easat-9645	127	33	+	+	CCONJ
easat-9645	127	34	2‖𝑦‖2	2‖𝑦‖2	NUM
easat-9645	127	35	figure	figure	NOUN
easat-9645	127	36	1	1	NUM
easat-9645	127	37	.	.	PUNCT
easat-9645	127	38	matlab	matlab	PROPN
easat-9645	127	39	visualization	visualization	NOUN
easat-9645	127	40	of	of	ADP
easat-9645	127	41	the	the	DET
easat-9645	127	42	parallelogram	parallelogram	NOUN
easat-9645	127	43	law	law	NOUN
easat-9645	127	44	.	.	PUNCT
easat-9645	128	1	the	the	DET
easat-9645	128	2	figure	figure	NOUN
easat-9645	128	3	1	1	NUM
easat-9645	128	4	,	,	PUNCT
easat-9645	128	5	provides	provide	VERB
easat-9645	128	6	a	a	DET
easat-9645	128	7	geometric	geometric	ADJ
easat-9645	128	8	illustration	illustration	NOUN
easat-9645	128	9	of	of	ADP
easat-9645	128	10	the	the	DET
easat-9645	128	11	parallelogram	parallelogram	NOUN
easat-9645	128	12	law	law	NOUN
easat-9645	128	13	in	in	ADP
easat-9645	128	14	a	a	DET
easat-9645	128	15	2d	2d	NUM
easat-9645	128	16	inner	inner	ADJ
easat-9645	128	17	product	product	NOUN
easat-9645	128	18	space	space	NOUN
easat-9645	128	19	.	.	PUNCT
easat-9645	129	1	two	two	NUM
easat-9645	129	2	vectors	vector	NOUN
easat-9645	129	3	x	x	SYM
easat-9645	129	4	(	(	PUNCT
easat-9645	129	5	red	red	ADJ
easat-9645	129	6	)	)	PUNCT
easat-9645	129	7	and	and	CCONJ
easat-9645	129	8	y	y	PROPN
easat-9645	129	9	(	(	PUNCT
easat-9645	129	10	blue	blue	ADJ
easat-9645	129	11	)	)	PUNCT
easat-9645	129	12	originate	originate	VERB
easat-9645	129	13	from	from	ADP
easat-9645	129	14	the	the	DET
easat-9645	129	15	origin	origin	NOUN
easat-9645	129	16	and	and	CCONJ
easat-9645	129	17	form	form	VERB
easat-9645	129	18	the	the	DET
easat-9645	129	19	adjacent	adjacent	ADJ
easat-9645	129	20	sides	side	NOUN
easat-9645	129	21	of	of	ADP
easat-9645	129	22	a	a	DET
easat-9645	129	23	parallelogram	parallelogram	NOUN
easat-9645	129	24	.	.	PUNCT
easat-9645	130	1	their	their	PRON
easat-9645	130	2	vector	vector	NOUN
easat-9645	130	3	sum	sum	NOUN
easat-9645	130	4	𝑥	𝑥	PROPN
easat-9645	130	5	+	+	NUM
easat-9645	130	6	𝑦	𝑦	SYM
easat-9645	130	7	(	(	PUNCT
easat-9645	130	8	green	green	ADJ
easat-9645	130	9	)	)	PUNCT
easat-9645	130	10	and	and	CCONJ
easat-9645	130	11	difference	difference	NOUN
easat-9645	130	12	𝑥	𝑥	DET
easat-9645	130	13	−	−	PROPN
easat-9645	130	14	𝑦	𝑦	SYM
easat-9645	130	15	(	(	PUNCT
easat-9645	130	16	magenta	magenta	NOUN
easat-9645	130	17	)	)	PUNCT
easat-9645	130	18	serve	serve	VERB
easat-9645	130	19	as	as	ADP
easat-9645	130	20	the	the	DET
easat-9645	130	21	diagonals	diagonal	NOUN
easat-9645	130	22	.	.	PUNCT
easat-9645	131	1	the	the	DET
easat-9645	131	2	shaded	shaded	ADJ
easat-9645	131	3	parallelogram	parallelogram	NOUN
easat-9645	131	4	visually	visually	ADV
easat-9645	131	5	demonstrates	demonstrate	VERB
easat-9645	131	6	that	that	SCONJ
easat-9645	131	7	the	the	DET
easat-9645	131	8	sum	sum	NOUN
easat-9645	131	9	of	of	ADP
easat-9645	131	10	the	the	DET
easat-9645	131	11	squared	square	VERB
easat-9645	131	12	lengths	length	NOUN
easat-9645	131	13	‖𝑥	‖𝑥	PROPN
easat-9645	131	14	+	+	CCONJ
easat-9645	131	15	𝑦‖2	𝑦‖2	X
easat-9645	131	16	+	+	NOUN
easat-9645	131	17	‖𝑥	‖𝑥	NOUN
easat-9645	131	18	−	−	NOUN
easat-9645	131	19	𝑦‖2	𝑦‖2	PROPN
easat-9645	131	20	=	=	SYM
easat-9645	131	21	2‖𝑥‖2	2‖𝑥‖2	PROPN
easat-9645	131	22	+	+	CCONJ
easat-9645	131	23	2‖𝑦‖2,confirming	2‖𝑦‖2,confirme	VERB
easat-9645	131	24	the	the	DET
easat-9645	131	25	parallelogram	parallelogram	NOUN
easat-9645	131	26	law	law	NOUN
easat-9645	131	27	in	in	ADP
easat-9645	131	28	inner	inner	ADJ
easat-9645	131	29	product	product	NOUN
easat-9645	131	30	spaces	space	NOUN
easat-9645	131	31	.	.	PUNCT
easat-9645	132	1	2.2	2.2	NUM
easat-9645	132	2	.	.	PUNCT
easat-9645	132	3	theorem	theorem	NOUN
easat-9645	132	4	prove	prove	VERB
easat-9645	132	5	that	that	SCONJ
easat-9645	132	6	the	the	DET
easat-9645	132	7	space	space	NOUN
easat-9645	132	8	𝑙𝑝	𝑙𝑝	NOUN
easat-9645	132	9	is	be	AUX
easat-9645	132	10	not	not	PART
easat-9645	132	11	a	a	DET
easat-9645	132	12	hilbert	hilbert	NOUN
easat-9645	132	13	space	space	NOUN
easat-9645	132	14	for	for	ADP
easat-9645	132	15	𝑝	𝑝	PROPN
easat-9645	132	16	≠	≠	PROPN
easat-9645	132	17	2	2	NUM
easat-9645	132	18	2.2.1	2.2.1	NUM
easat-9645	132	19	.	.	PUNCT
easat-9645	133	1	geometrical	geometrical	ADJ
easat-9645	133	2	explanation	explanation	NOUN
easat-9645	133	3	why	why	SCONJ
easat-9645	133	4	the	the	DET
easat-9645	133	5	space	space	NOUN
easat-9645	133	6	𝑙𝑝	𝑙𝑝	NOUN
easat-9645	133	7	is	be	AUX
easat-9645	133	8	not	not	PART
easat-9645	133	9	a	a	DET
easat-9645	133	10	hilbert	hilbert	NOUN
easat-9645	133	11	space	space	NOUN
easat-9645	133	12	for	for	ADP
easat-9645	133	13	𝑝	𝑝	PROPN
easat-9645	133	14	≠	≠	PROPN
easat-9645	133	15	2	2	NUM
easat-9645	133	16	.	.	PUNCT
easat-9645	133	17	to	to	PART
easat-9645	133	18	geometrically	geometrically	ADV
easat-9645	133	19	understand	understand	VERB
easat-9645	133	20	and	and	CCONJ
easat-9645	133	21	why	why	SCONJ
easat-9645	133	22	the	the	DET
easat-9645	133	23	space	space	NOUN
easat-9645	133	24	𝑙𝑝	𝑙𝑝	NOUN
easat-9645	133	25	is	be	AUX
easat-9645	133	26	not	not	PART
easat-9645	133	27	a	a	DET
easat-9645	133	28	hilbert	hilbert	NOUN
easat-9645	133	29	space	space	NOUN
easat-9645	133	30	unless	unless	SCONJ
easat-9645	133	31	𝑝	𝑝	NOUN
easat-9645	133	32	=	=	SYM
easat-9645	133	33	2	2	NUM
easat-9645	133	34	,	,	PUNCT
easat-9645	133	35	we	we	PRON
easat-9645	133	36	must	must	AUX
easat-9645	133	37	look	look	VERB
easat-9645	133	38	at	at	ADP
easat-9645	133	39	the	the	DET
easat-9645	133	40	shape	shape	NOUN
easat-9645	133	41	of	of	ADP
easat-9645	133	42	unit	unit	NOUN
easat-9645	133	43	balls	ball	NOUN
easat-9645	133	44	and	and	CCONJ
easat-9645	133	45	the	the	DET
easat-9645	133	46	notion	notion	NOUN
easat-9645	133	47	of	of	ADP
easat-9645	133	48	angles	angle	NOUN
easat-9645	133	49	in	in	ADP
easat-9645	133	50	these	these	DET
easat-9645	133	51	spaces	space	NOUN
easat-9645	133	52	.	.	PUNCT
easat-9645	134	1	1502	1502	NUM
easat-9645	134	2	edelweiss	edelweiss	PROPN
easat-9645	134	3	applied	apply	VERB
easat-9645	134	4	science	science	NOUN
easat-9645	134	5	and	and	CCONJ
easat-9645	134	6	technology	technology	NOUN
easat-9645	134	7	issn	issn	PROPN
easat-9645	134	8	:	:	PUNCT
easat-9645	134	9	2576	2576	NUM
easat-9645	134	10	-	-	SYM
easat-9645	134	11	8484	8484	NUM
easat-9645	134	12	vol	vol	NOUN
easat-9645	134	13	.	.	PROPN
easat-9645	135	1	9	9	NUM
easat-9645	135	2	,	,	PUNCT
easat-9645	135	3	no	no	INTJ
easat-9645	135	4	.	.	NOUN
easat-9645	135	5	8	8	NUM
easat-9645	135	6	:	:	SYM
easat-9645	135	7	1498	1498	NUM
easat-9645	135	8	-	-	SYM
easat-9645	135	9	1523	1523	NUM
easat-9645	135	10	,	,	PUNCT
easat-9645	135	11	2025	2025	NUM
easat-9645	135	12	doi	doi	NOUN
easat-9645	135	13	:	:	PUNCT
easat-9645	135	14	10.55214/2576	10.55214/2576	NUM
easat-9645	135	15	-	-	SYM
easat-9645	135	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	135	17	©	©	PROPN
easat-9645	135	18	2025	2025	NUM
easat-9645	135	19	by	by	ADP
easat-9645	135	20	the	the	DET
easat-9645	135	21	authors	author	NOUN
easat-9645	135	22	;	;	PUNCT
easat-9645	135	23	licensee	licensee	PROPN
easat-9645	135	24	learning	learning	PROPN
easat-9645	135	25	gate	gate	PROPN
easat-9645	135	26	i.	i.	PROPN
easat-9645	135	27	hilbert	hilbert	PROPN
easat-9645	135	28	spaces	space	NOUN
easat-9645	135	29	and	and	CCONJ
easat-9645	135	30	inner	inner	ADJ
easat-9645	135	31	product	product	NOUN
easat-9645	135	32	geometry	geometry	NOUN
easat-9645	135	33	in	in	ADP
easat-9645	135	34	a	a	DET
easat-9645	135	35	hilbert	hilbert	PROPN
easat-9645	135	36	space	space	PROPN
easat-9645	135	37	i.	i.	PROPN
easat-9645	135	38	e.	e.	PROPN
easat-9645	135	39	𝑙2	𝑙2	PROPN
easat-9645	135	40	,	,	PUNCT
easat-9645	135	41	geometry	geometry	NOUN
easat-9645	135	42	behaves	behave	VERB
easat-9645	135	43	similarly	similarly	ADV
easat-9645	135	44	to	to	ADP
easat-9645	135	45	euclidean	euclidean	ADJ
easat-9645	135	46	space	space	NOUN
easat-9645	135	47	:	:	PUNCT
easat-9645	135	48	a	a	X
easat-9645	135	49	)	)	PUNCT
easat-9645	135	50	a	a	DET
easat-9645	135	51	well	well	ADV
easat-9645	135	52	-	-	PUNCT
easat-9645	135	53	defined	define	VERB
easat-9645	135	54	inner	inner	ADJ
easat-9645	135	55	product	product	NOUN
easat-9645	135	56	spaces	space	NOUN
easat-9645	135	57	,	,	PUNCT
easat-9645	135	58	b	b	X
easat-9645	135	59	)	)	PUNCT
easat-9645	135	60	define	define	VERB
easat-9645	135	61	angles	angle	NOUN
easat-9645	135	62	and	and	CCONJ
easat-9645	135	63	orthogonality	orthogonality	NOUN
easat-9645	135	64	,	,	PUNCT
easat-9645	135	65	c	c	NOUN
easat-9645	135	66	)	)	PUNCT
easat-9645	135	67	the	the	DET
easat-9645	135	68	unit	unit	NOUN
easat-9645	135	69	ball	ball	NOUN
easat-9645	135	70	(	(	PUNCT
easat-9645	135	71	set	set	NOUN
easat-9645	135	72	of	of	ADP
easat-9645	135	73	points	point	NOUN
easat-9645	135	74	with	with	ADP
easat-9645	135	75	norm	norm	NOUN
easat-9645	135	76	≤	≤	NUM
easat-9645	135	77	1	1	NUM
easat-9645	135	78	)	)	PUNCT
easat-9645	135	79	is	be	AUX
easat-9645	135	80	round	round	ADJ
easat-9645	135	81	—	—	PUNCT
easat-9645	135	82	a	a	DET
easat-9645	135	83	perfect	perfect	ADJ
easat-9645	135	84	circle	circle	NOUN
easat-9645	135	85	(	(	PUNCT
easat-9645	135	86	in	in	ADP
easat-9645	135	87	2d	2d	NUM
easat-9645	135	88	)	)	PUNCT
easat-9645	135	89	or	or	CCONJ
easat-9645	135	90	sphere	sphere	NOUN
easat-9645	135	91	(	(	PUNCT
easat-9645	135	92	in	in	ADP
easat-9645	135	93	higher	high	ADJ
easat-9645	135	94	dimensions	dimension	NOUN
easat-9645	135	95	)	)	PUNCT
easat-9645	135	96	,	,	PUNCT
easat-9645	136	1	d	d	X
easat-9645	136	2	)	)	PUNCT
easat-9645	136	3	the	the	DET
easat-9645	136	4	parallelogram	parallelogram	NOUN
easat-9645	136	5	law	law	NOUN
easat-9645	136	6	holds	hold	VERB
easat-9645	136	7	:	:	PUNCT
easat-9645	136	8	it	it	PRON
easat-9645	136	9	characterizes	characterize	VERB
easat-9645	136	10	how	how	SCONJ
easat-9645	136	11	vector	vector	NOUN
easat-9645	136	12	lengths	length	NOUN
easat-9645	136	13	and	and	CCONJ
easat-9645	136	14	angles	angle	NOUN
easat-9645	136	15	interact	interact	VERB
easat-9645	136	16	.	.	PUNCT
easat-9645	137	1	ii	ii	PROPN
easat-9645	137	2	.	.	PUNCT
easat-9645	138	1	𝑙2	𝑙2	PROPN
easat-9645	138	2	spaces	space	NOUN
easat-9645	138	3	and	and	CCONJ
easat-9645	138	4	geometry	geometry	NOUN
easat-9645	138	5	of	of	ADP
easat-9645	138	6	unit	unit	NOUN
easat-9645	138	7	balls	ball	NOUN
easat-9645	138	8	let	let	VERB
easat-9645	138	9	us	we	PRON
easat-9645	138	10	consider	consider	VERB
easat-9645	138	11	the	the	DET
easat-9645	138	12	unit	unit	NOUN
easat-9645	138	13	ball	ball	NOUN
easat-9645	138	14	in	in	ADP
easat-9645	138	15	r2	r2	PROPN
easat-9645	138	16	under	under	ADP
easat-9645	138	17	different	different	ADJ
easat-9645	138	18	𝑙𝑝	𝑙𝑝	NOUN
easat-9645	138	19	norms	norm	NOUN
easat-9645	138	20	:	:	PUNCT
easat-9645	138	21	p	p	NOUN
easat-9645	138	22	unit	unit	NOUN
easat-9645	138	23	ball	ball	NOUN
easat-9645	138	24	shape	shape	NOUN
easat-9645	138	25	geometry	geometry	NOUN
easat-9645	138	26	behavior	behavior	NOUN
easat-9645	138	27	p=1	p=1	NOUN
easat-9645	138	28	diamond	diamond	NOUN
easat-9645	138	29	-	-	PUNCT
easat-9645	138	30	shaped	shape	VERB
easat-9645	138	31	corners	corner	NOUN
easat-9645	138	32	,	,	PUNCT
easat-9645	138	33	no	no	DET
easat-9645	138	34	smoothness	smoothness	ADJ
easat-9645	138	35	p=2	p=2	X
easat-9645	138	36	perfect	perfect	ADJ
easat-9645	138	37	circle	circle	NOUN
easat-9645	138	38	euclidean	euclidean	NOUN
easat-9645	138	39	,	,	PUNCT
easat-9645	138	40	inner	inner	ADJ
easat-9645	138	41	product	product	NOUN
easat-9645	138	42	exists	exist	VERB
easat-9645	138	43	𝑃	𝑃	NOUN
easat-9645	138	44	=	=	PUNCT
easat-9645	138	45	∞	∞	NUM
easat-9645	138	46	square	square	ADJ
easat-9645	138	47	flat	flat	ADJ
easat-9645	138	48	sides	side	NOUN
easat-9645	138	49	,	,	PUNCT
easat-9645	138	50	angles	angle	VERB
easat-9645	138	51	not	not	PART
easat-9645	138	52	well	well	ADV
easat-9645	138	53	-	-	PUNCT
easat-9645	138	54	defined	define	VERB
easat-9645	138	55	𝑝	𝑝	PROPN
easat-9645	138	56	≠	≠	PROPN
easat-9645	138	57	2	2	NUM
easat-9645	138	58	smooth	smooth	ADJ
easat-9645	138	59	but	but	CCONJ
easat-9645	138	60	not	not	PART
easat-9645	138	61	circular	circular	ADJ
easat-9645	138	62	geometry	geometry	NOUN
easat-9645	138	63	is	be	AUX
easat-9645	138	64	distorted	distort	VERB
easat-9645	138	65	iii	iii	NOUN
easat-9645	138	66	.	.	PUNCT
easat-9645	139	1	why	why	SCONJ
easat-9645	139	2	this	this	DET
easat-9645	139	3	matters	matter	NOUN
easat-9645	139	4	for	for	ADP
easat-9645	139	5	hilbert	hilbert	NOUN
easat-9645	139	6	spaces	space	VERB
easat-9645	139	7	a	a	DET
easat-9645	139	8	hilbert	hilbert	NOUN
easat-9645	139	9	space	space	NOUN
easat-9645	139	10	is	be	AUX
easat-9645	139	11	more	more	ADJ
easat-9645	139	12	than	than	ADP
easat-9645	139	13	just	just	ADV
easat-9645	139	14	a	a	DET
easat-9645	139	15	complete	complete	ADJ
easat-9645	139	16	normed	normed	ADJ
easat-9645	139	17	space	space	NOUN
easat-9645	139	18	such	such	ADJ
easat-9645	139	19	that	that	DET
easat-9645	139	20	project	project	NOUN
easat-9645	139	21	one	one	NUM
easat-9645	139	22	vector	vector	NOUN
easat-9645	139	23	onto	onto	ADP
easat-9645	139	24	another	another	PRON
easat-9645	139	25	,	,	PUNCT
easat-9645	139	26	and	and	CCONJ
easat-9645	139	27	angles	angle	NOUN
easat-9645	139	28	make	make	VERB
easat-9645	139	29	sense	sense	NOUN
easat-9645	139	30	.	.	PUNCT
easat-9645	140	1	for	for	ADP
easat-9645	140	2	𝑝	𝑝	PROPN
easat-9645	140	3	≠	≠	PROPN
easat-9645	140	4	2	2	NUM
easat-9645	140	5	,	,	PUNCT
easat-9645	140	6	these	these	DET
easat-9645	140	7	geometric	geometric	ADJ
easat-9645	140	8	tools	tool	NOUN
easat-9645	140	9	break	break	VERB
easat-9645	140	10	down	down	ADP
easat-9645	140	11	:	:	PUNCT
easat-9645	140	12	a	a	X
easat-9645	140	13	)	)	PUNCT
easat-9645	140	14	no	no	DET
easat-9645	140	15	proper	proper	ADJ
easat-9645	140	16	projection	projection	NOUN
easat-9645	140	17	theorem	theorem	NOUN
easat-9645	140	18	,	,	PUNCT
easat-9645	140	19	b	b	NOUN
easat-9645	140	20	)	)	PUNCT
easat-9645	140	21	no	no	DET
easat-9645	140	22	orthogonal	orthogonal	ADJ
easat-9645	140	23	decomposition	decomposition	NOUN
easat-9645	140	24	,	,	PUNCT
easat-9645	140	25	c	c	NOUN
easat-9645	140	26	)	)	PUNCT
easat-9645	140	27	no	no	DET
easat-9645	140	28	true	true	ADJ
easat-9645	140	29	"	"	PUNCT
easat-9645	140	30	angles	angle	NOUN
easat-9645	140	31	"	"	PUNCT
easat-9645	140	32	between	between	ADP
easat-9645	140	33	vectors	vector	NOUN
easat-9645	140	34	.	.	PUNCT
easat-9645	141	1	the	the	DET
easat-9645	141	2	geometry	geometry	NOUN
easat-9645	141	3	of	of	ADP
easat-9645	141	4	𝑙𝑝	𝑙𝑝	PROPN
easat-9645	141	5	space	space	NOUN
easat-9645	141	6	for	for	ADP
easat-9645	141	7	𝑝	𝑝	PROPN
easat-9645	141	8	≠	≠	PROPN
easat-9645	141	9	2	2	NUM
easat-9645	141	10	is	be	AUX
easat-9645	141	11	not	not	PART
easat-9645	141	12	euclidean	euclidean	ADJ
easat-9645	141	13	:	:	PUNCT
easat-9645	141	14	the	the	DET
easat-9645	141	15	unit	unit	NOUN
easat-9645	141	16	balls	ball	NOUN
easat-9645	141	17	are	be	AUX
easat-9645	141	18	not	not	PART
easat-9645	141	19	round	round	ADJ
easat-9645	141	20	,	,	PUNCT
easat-9645	141	21	the	the	DET
easat-9645	141	22	norm	norm	NOUN
easat-9645	141	23	does	do	AUX
easat-9645	141	24	not	not	PART
easat-9645	141	25	ips	ip	NOUN
easat-9645	141	26	,	,	PUNCT
easat-9645	141	27	and	and	CCONJ
easat-9645	141	28	𝑙𝑝	𝑙𝑝	PROPN
easat-9645	141	29	fails	fail	VERB
easat-9645	141	30	to	to	PART
easat-9645	141	31	be	be	AUX
easat-9645	141	32	a	a	DET
easat-9645	141	33	hilbert	hilbert	NOUN
easat-9645	141	34	space	space	NOUN
easat-9645	141	35	unless	unless	SCONJ
easat-9645	141	36	p=2	p=2	PROPN
easat-9645	141	37	.	.	PUNCT
easat-9645	142	1	proof	proof	NOUN
easat-9645	142	2	:	:	PUNCT
easat-9645	142	3	let	let	VERB
easat-9645	142	4	us	we	PRON
easat-9645	142	5	take	take	VERB
easat-9645	142	6	𝑥	𝑥	X
easat-9645	142	7	=	=	SYM
easat-9645	142	8	(	(	PUNCT
easat-9645	142	9	1,1,0,0	1,1,0,0	NUM
easat-9645	142	10	,	,	PUNCT
easat-9645	142	11	.	.	PUNCT
easat-9645	142	12	.	.	PUNCT
easat-9645	142	13	.	.	PUNCT
easat-9645	142	14	)	)	PUNCT
easat-9645	143	1	∈	∈	PROPN
easat-9645	143	2	𝑙𝑝𝑎𝑛𝑑	𝑙𝑝𝑎𝑛𝑑	NOUN
easat-9645	144	1	𝑥	𝑥	NOUN
easat-9645	144	2	=	=	SYM
easat-9645	144	3	(	(	PUNCT
easat-9645	144	4	1	1	NUM
easat-9645	144	5	,	,	PUNCT
easat-9645	144	6	−1,0,0	−1,0,0	NOUN
easat-9645	144	7	,	,	PUNCT
easat-9645	144	8	.	.	PUNCT
easat-9645	144	9	.	.	PUNCT
easat-9645	144	10	.	.	PUNCT
easat-9645	144	11	)	)	PUNCT
easat-9645	145	1	∈	∈	PROPN
easat-9645	146	1	𝑙𝑝.	𝑙𝑝.	CCONJ
easat-9645	146	2	then	then	ADV
easat-9645	146	3	𝑥	𝑥	X
easat-9645	146	4	+	+	PUNCT
easat-9645	146	5	𝑦	𝑦	SYM
easat-9645	146	6	=	=	SYM
easat-9645	146	7	(	(	PUNCT
easat-9645	146	8	2,0,0	2,0,0	NUM
easat-9645	146	9	,	,	PUNCT
easat-9645	146	10	.	.	PUNCT
easat-9645	146	11	.	.	PUNCT
easat-9645	146	12	.	.	PUNCT
easat-9645	146	13	)	)	PUNCT
easat-9645	147	1	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-9645	148	1	𝑥	𝑥	PRON
easat-9645	148	2	−	−	PROPN
easat-9645	148	3	𝑦	𝑦	SYM
easat-9645	148	4	=	=	SYM
easat-9645	148	5	(	(	PUNCT
easat-9645	148	6	0,2,0,0	0,2,0,0	NOUN
easat-9645	148	7	,	,	PUNCT
easat-9645	148	8	.	.	PUNCT
easat-9645	148	9	.	.	PUNCT
easat-9645	148	10	.	.	PUNCT
easat-9645	148	11	)	)	PUNCT
easat-9645	148	12	.	.	PUNCT
easat-9645	149	1	we	we	PRON
easat-9645	149	2	have	have	VERB
easat-9645	149	3	‖𝑥‖	‖𝑥‖	PROPN
easat-9645	149	4	=	=	PUNCT
easat-9645	149	5	(	(	PUNCT
easat-9645	149	6	∑|𝑥𝑘|𝑝	∑|𝑥𝑘|𝑝	NUM
easat-9645	149	7	𝑛	𝑛	PRON
easat-9645	149	8	𝑘=1	𝑘=1	NOUN
easat-9645	149	9	)	)	PUNCT
easat-9645	149	10	1	1	NUM
easat-9645	149	11	𝑝	𝑝	NOUN
easat-9645	149	12	=	=	SYM
easat-9645	149	13	(	(	PUNCT
easat-9645	149	14	|1|𝑝	|1|𝑝	NOUN
easat-9645	149	15	+	+	X
easat-9645	149	16	|1|𝑝	|1|𝑝	ADJ
easat-9645	149	17	+	+	CCONJ
easat-9645	149	18	0	0	NUM
easat-9645	150	1	+	+	CCONJ
easat-9645	150	2	0	0	NUM
easat-9645	151	1	+	+	NOUN
easat-9645	151	2	.	.	PUNCT
easat-9645	151	3	.	.	PUNCT
easat-9645	151	4	.	.	PUNCT
easat-9645	152	1	+0	+0	ADP
easat-9645	152	2	)	)	PUNCT
easat-9645	152	3	1	1	NUM
easat-9645	152	4	𝑝	𝑝	NOUN
easat-9645	152	5	=	=	SYM
easat-9645	152	6	2	2	NUM
easat-9645	152	7	𝑎𝑛𝑑	𝑎𝑛𝑑	NOUN
easat-9645	152	8	‖𝑦‖	‖𝑦‖	PROPN
easat-9645	152	9	=	=	SYM
easat-9645	152	10	(	(	PUNCT
easat-9645	152	11	|1|𝑝	|1|𝑝	NOUN
easat-9645	152	12	+	+	NOUN
easat-9645	152	13	|−1|𝑝	|−1|𝑝	NOUN
easat-9645	152	14	+	+	CCONJ
easat-9645	152	15	0	0	NUM
easat-9645	153	1	+	+	NUM
easat-9645	153	2	0	0	NUM
easat-9645	154	1	+	+	NOUN
easat-9645	154	2	.	.	PUNCT
easat-9645	154	3	.	.	PUNCT
easat-9645	154	4	.	.	PUNCT
easat-9645	155	1	+0	+0	ADP
easat-9645	155	2	)	)	PUNCT
easat-9645	155	3	1	1	NUM
easat-9645	155	4	𝑝	𝑝	NOUN
easat-9645	155	5	=	=	SYM
easat-9645	155	6	2	2	NUM
easat-9645	155	7	also	also	ADV
easat-9645	155	8	‖𝑥	‖𝑥	PROPN
easat-9645	156	1	+	+	CCONJ
easat-9645	156	2	𝑦‖	𝑦‖	PROPN
easat-9645	156	3	=	=	SYM
easat-9645	156	4	(	(	PUNCT
easat-9645	156	5	|2|𝑝	|2|𝑝	X
easat-9645	156	6	+	+	X
easat-9645	156	7	0	0	NUM
easat-9645	156	8	+	+	NOUN
easat-9645	156	9	0	0	NUM
easat-9645	156	10	+	+	NOUN
easat-9645	156	11	.	.	PUNCT
easat-9645	156	12	.	.	PUNCT
easat-9645	156	13	.	.	PUNCT
easat-9645	157	1	+0	+0	ADP
easat-9645	157	2	)	)	PUNCT
easat-9645	157	3	1	1	NUM
easat-9645	157	4	𝑝	𝑝	NOUN
easat-9645	157	5	=	=	SYM
easat-9645	157	6	2	2	NUM
easat-9645	157	7	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-9645	157	8	‖𝑥	‖𝑥	PROPN
easat-9645	157	9	−	−	NOUN
easat-9645	157	10	𝑦‖	𝑦‖	PROPN
easat-9645	157	11	=	=	SYM
easat-9645	157	12	(	(	PUNCT
easat-9645	157	13	0	0	NUM
easat-9645	157	14	+	+	CCONJ
easat-9645	157	15	|2|𝑝	|2|𝑝	ADJ
easat-9645	157	16	+	+	X
easat-9645	157	17	0	0	NUM
easat-9645	157	18	+	+	CCONJ
easat-9645	157	19	0	0	NUM
easat-9645	157	20	+	+	NOUN
easat-9645	157	21	.	.	PUNCT
easat-9645	157	22	.	.	PUNCT
easat-9645	157	23	.	.	PUNCT
easat-9645	158	1	+0	+0	ADP
easat-9645	158	2	)	)	PUNCT
easat-9645	158	3	1	1	NUM
easat-9645	158	4	𝑝	𝑝	NOUN
easat-9645	158	5	=	=	SYM
easat-9645	158	6	2	2	NUM
easat-9645	158	7	so	so	SCONJ
easat-9645	158	8	that	that	SCONJ
easat-9645	158	9	‖𝑥	‖𝑥	NOUN
easat-9645	159	1	+	+	CCONJ
easat-9645	159	2	𝑦‖2	𝑦‖2	X
easat-9645	159	3	+	+	NOUN
easat-9645	159	4	‖𝑥	‖𝑥	NOUN
easat-9645	159	5	−	−	NOUN
easat-9645	159	6	𝑦‖2	𝑦‖2	X
easat-9645	159	7	=	=	NOUN
easat-9645	159	8	22	22	NUM
easat-9645	159	9	+	+	CCONJ
easat-9645	159	10	22	22	NUM
easat-9645	159	11	=	=	SYM
easat-9645	159	12	8	8	NUM
easat-9645	159	13	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-9645	159	14	2‖𝑥‖2	2‖𝑥‖2	PROPN
easat-9645	159	15	+	+	CCONJ
easat-9645	159	16	2‖𝑦‖2	2‖𝑦‖2	NUM
easat-9645	159	17	=	=	SYM
easat-9645	159	18	2	2	NUM
easat-9645	159	19	(	(	PUNCT
easat-9645	159	20	2	2	NUM
easat-9645	159	21	2	2	NUM
easat-9645	159	22	𝑝	𝑝	NOUN
easat-9645	159	23	+	+	CCONJ
easat-9645	159	24	2	2	NUM
easat-9645	159	25	2	2	NUM
easat-9645	159	26	𝑝	𝑝	NOUN
easat-9645	159	27	)	)	PUNCT
easat-9645	159	28	if	if	SCONJ
easat-9645	159	29	p=2	p=2	PROPN
easat-9645	159	30	,	,	PUNCT
easat-9645	159	31	then	then	ADV
easat-9645	159	32	2‖𝑥‖2	2‖𝑥‖2	PROPN
easat-9645	159	33	+	+	CCONJ
easat-9645	159	34	2‖𝑦‖2	2‖𝑦‖2	NUM
easat-9645	159	35	=	=	SYM
easat-9645	159	36	2(2	2(2	NUM
easat-9645	159	37	+	+	CCONJ
easat-9645	159	38	2	2	X
easat-9645	159	39	)	)	PUNCT
easat-9645	159	40	=	=	SYM
easat-9645	159	41	8	8	NUM
easat-9645	159	42	thus	thus	ADV
easat-9645	159	43	,	,	PUNCT
easat-9645	159	44	for	for	ADP
easat-9645	159	45	p=2	p=2	PROPN
easat-9645	159	46	,	,	PUNCT
easat-9645	159	47	the	the	DET
easat-9645	159	48	parallelogram	parallelogram	NOUN
easat-9645	159	49	law	law	NOUN
easat-9645	159	50	‖𝑥	‖𝑥	PROPN
easat-9645	159	51	+	+	CCONJ
easat-9645	159	52	𝑦‖2	𝑦‖2	X
easat-9645	159	53	+	+	NOUN
easat-9645	159	54	‖𝑥	‖𝑥	NOUN
easat-9645	159	55	−	−	NOUN
easat-9645	159	56	𝑦‖2	𝑦‖2	X
easat-9645	159	57	=	=	SYM
easat-9645	159	58	2‖𝑥‖2	2‖𝑥‖2	PROPN
easat-9645	159	59	+	+	CCONJ
easat-9645	159	60	2‖𝑦‖2	2‖𝑦‖2	NOUN
easat-9645	159	61	is	be	AUX
easat-9645	159	62	satisfied	satisfied	ADJ
easat-9645	159	63	.	.	PUNCT
easat-9645	160	1	this	this	PRON
easat-9645	160	2	implies	imply	VERB
easat-9645	160	3	that	that	SCONJ
easat-9645	160	4	the	the	DET
easat-9645	160	5	space	space	NOUN
easat-9645	160	6	𝑙2	𝑙2	PROPN
easat-9645	160	7	is	be	AUX
easat-9645	160	8	a	a	DET
easat-9645	160	9	hilbert	hilbert	NOUN
easat-9645	160	10	space	space	NOUN
easat-9645	160	11	.	.	PUNCT
easat-9645	161	1	when	when	SCONJ
easat-9645	161	2	𝑝	𝑝	PRON
easat-9645	161	3	≠	≠	PROPN
easat-9645	161	4	2	2	NUM
easat-9645	161	5	,	,	PUNCT
easat-9645	161	6	the	the	DET
easat-9645	161	7	parallelogram	parallelogram	NOUN
easat-9645	161	8	law	law	NOUN
easat-9645	161	9	does	do	AUX
easat-9645	161	10	not	not	PART
easat-9645	161	11	hold	hold	VERB
easat-9645	161	12	.	.	PUNCT
easat-9645	162	1	consequently	consequently	ADV
easat-9645	162	2	,	,	PUNCT
easat-9645	162	3	the	the	DET
easat-9645	162	4	space	space	NOUN
easat-9645	162	5	𝑙𝑝	𝑙𝑝	NOUN
easat-9645	162	6	is	be	AUX
easat-9645	162	7	not	not	PART
easat-9645	162	8	a	a	DET
easat-9645	162	9	hilbert	hilbert	NOUN
easat-9645	162	10	space	space	NOUN
easat-9645	162	11	for	for	ADP
easat-9645	162	12	values	value	NOUN
easat-9645	162	13	of	of	ADP
easat-9645	162	14	𝑝	𝑝	NOUN
easat-9645	162	15	other	other	ADJ
easat-9645	162	16	than	than	ADP
easat-9645	162	17	2	2	NUM
easat-9645	162	18	.	.	PUNCT
easat-9645	162	19	to	to	PART
easat-9645	162	20	visualize	visualize	VERB
easat-9645	162	21	the	the	DET
easat-9645	162	22	failure	failure	NOUN
easat-9645	162	23	of	of	ADP
easat-9645	162	24	the	the	DET
easat-9645	162	25	parallelogram	parallelogram	NOUN
easat-9645	162	26	law	law	NOUN
easat-9645	162	27	in	in	ADP
easat-9645	162	28	matlab	matlab	PROPN
easat-9645	162	29	for	for	ADP
easat-9645	162	30	𝑙𝑝	𝑙𝑝	PROPN
easat-9645	162	31	spaces	space	NOUN
easat-9645	162	32	when	when	SCONJ
easat-9645	162	33	𝑝	𝑝	PROPN
easat-9645	162	34	≠	≠	PROPN
easat-9645	162	35	2	2	NUM
easat-9645	162	36	,	,	PUNCT
easat-9645	162	37	we	we	PRON
easat-9645	162	38	can	can	AUX
easat-9645	162	39	create	create	VERB
easat-9645	162	40	a	a	DET
easat-9645	162	41	2d	2d	NUM
easat-9645	162	42	plot	plot	NOUN
easat-9645	162	43	showing	show	VERB
easat-9645	162	44	how	how	SCONJ
easat-9645	162	45	the	the	DET
easat-9645	162	46	quantity	quantity	NOUN
easat-9645	162	47	𝐷(𝑝	𝐷(𝑝	NOUN
easat-9645	162	48	)	)	PUNCT
easat-9645	162	49	=	=	SYM
easat-9645	163	1	‖𝑥	‖𝑥	NOUN
easat-9645	164	1	+	+	CCONJ
easat-9645	164	2	𝑦‖𝑝	𝑦‖𝑝	ADP
easat-9645	164	3	2	2	NUM
easat-9645	164	4	+	+	NUM
easat-9645	164	5	‖𝑥	‖𝑥	NOUN
easat-9645	164	6	−	−	NOUN
easat-9645	164	7	𝑦‖𝑝	𝑦‖𝑝	ADP
easat-9645	164	8	2	2	NUM
easat-9645	164	9	,	,	PUNCT
easat-9645	164	10	varies	vary	VERB
easat-9645	164	11	with	with	ADP
easat-9645	164	12	p	p	NOUN
easat-9645	164	13	and	and	CCONJ
easat-9645	164	14	compare	compare	VERB
easat-9645	164	15	it	it	PRON
easat-9645	164	16	to	to	ADP
easat-9645	164	17	the	the	DET
easat-9645	164	18	value	value	NOUN
easat-9645	164	19	4	4	NUM
easat-9645	164	20	(	(	PUNCT
easat-9645	164	21	which	which	PRON
easat-9645	164	22	is	be	AUX
easat-9645	164	23	required	require	VERB
easat-9645	164	24	by	by	ADP
easat-9645	164	25	the	the	DET
easat-9645	164	26	parallelogram	parallelogram	PROPN
easat-9645	164	27	law	law	NOUN
easat-9645	164	28	)	)	PUNCT
easat-9645	164	29	.	.	PUNCT
easat-9645	165	1	𝐹𝑜𝑟	𝐹𝑜𝑟	PROPN
easat-9645	165	2	𝑝	𝑝	NOUN
easat-9645	165	3	=	=	SYM
easat-9645	165	4	2	2	NUM
easat-9645	165	5	,	,	PUNCT
easat-9645	165	6	𝐷(𝑝	𝐷(𝑝	X
easat-9645	165	7	)	)	PUNCT
easat-9645	165	8	=	=	SYM
easat-9645	165	9	4	4	NUM
easat-9645	165	10	;	;	PUNCT
easat-9645	165	11	𝑓𝑜𝑟	𝑓𝑜𝑟	X
easat-9645	165	12	𝑝	𝑝	PRON
easat-9645	165	13	≠	≠	PROPN
easat-9645	165	14	2	2	NUM
easat-9645	165	15	,	,	PUNCT
easat-9645	165	16	𝐷(𝑝	𝐷(𝑝	X
easat-9645	165	17	)	)	PUNCT
easat-9645	165	18	≠	≠	PROPN
easat-9645	165	19	4	4	NUM
easat-9645	165	20	.	.	X
easat-9645	165	21	1503	1503	NUM
easat-9645	165	22	edelweiss	edelweiss	PROPN
easat-9645	165	23	applied	apply	VERB
easat-9645	165	24	science	science	NOUN
easat-9645	165	25	and	and	CCONJ
easat-9645	165	26	technology	technology	NOUN
easat-9645	165	27	issn	issn	PROPN
easat-9645	165	28	:	:	PUNCT
easat-9645	165	29	2576	2576	NUM
easat-9645	165	30	-	-	SYM
easat-9645	165	31	8484	8484	NUM
easat-9645	165	32	vol	vol	NOUN
easat-9645	165	33	.	.	PROPN
easat-9645	166	1	9	9	NUM
easat-9645	166	2	,	,	PUNCT
easat-9645	166	3	no	no	INTJ
easat-9645	166	4	.	.	NOUN
easat-9645	166	5	8	8	NUM
easat-9645	166	6	:	:	SYM
easat-9645	166	7	1498	1498	NUM
easat-9645	166	8	-	-	SYM
easat-9645	166	9	1523	1523	NUM
easat-9645	166	10	,	,	PUNCT
easat-9645	166	11	2025	2025	NUM
easat-9645	166	12	doi	doi	NOUN
easat-9645	166	13	:	:	PUNCT
easat-9645	166	14	10.55214/2576	10.55214/2576	NUM
easat-9645	166	15	-	-	SYM
easat-9645	166	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	166	17	©	©	PROPN
easat-9645	166	18	2025	2025	NUM
easat-9645	166	19	by	by	ADP
easat-9645	166	20	the	the	DET
easat-9645	166	21	authors	author	NOUN
easat-9645	166	22	;	;	PUNCT
easat-9645	166	23	licensee	licensee	PROPN
easat-9645	166	24	learning	learn	VERB
easat-9645	166	25	gate	gate	NOUN
easat-9645	166	26	figure	figure	NOUN
easat-9645	166	27	2	2	NUM
easat-9645	166	28	.	.	PUNCT
easat-9645	166	29	matlab	matlab	PROPN
easat-9645	166	30	visualization	visualization	NOUN
easat-9645	166	31	of	of	ADP
easat-9645	166	32	the	the	DET
easat-9645	166	33	deviation	deviation	NOUN
easat-9645	166	34	from	from	ADP
easat-9645	166	35	parallelogram	parallelogram	NOUN
easat-9645	166	36	law	law	NOUN
easat-9645	166	37	in	in	ADP
easat-9645	166	38	𝑙𝑝	𝑙𝑝	PROPN
easat-9645	166	39	norms	norm	NOUN
easat-9645	166	40	.	.	PUNCT
easat-9645	167	1	this	this	DET
easat-9645	167	2	figure	figure	NOUN
easat-9645	167	3	2	2	NUM
easat-9645	167	4	,	,	PUNCT
easat-9645	167	5	shows	show	VERB
easat-9645	167	6	that	that	SCONJ
easat-9645	167	7	𝑙𝑝	𝑙𝑝	PROPN
easat-9645	167	8	spaces	space	NOUN
easat-9645	167	9	deviate	deviate	VERB
easat-9645	167	10	most	most	ADV
easat-9645	167	11	from	from	ADP
easat-9645	167	12	the	the	DET
easat-9645	167	13	parallelogram	parallelogram	NOUN
easat-9645	167	14	law	law	NOUN
easat-9645	167	15	when	when	SCONJ
easat-9645	167	16	p	p	NOUN
easat-9645	167	17	is	be	AUX
easat-9645	167	18	far	far	ADV
easat-9645	167	19	from	from	ADP
easat-9645	167	20	2	2	NUM
easat-9645	167	21	,	,	PUNCT
easat-9645	167	22	and	and	CCONJ
easat-9645	167	23	the	the	DET
easat-9645	167	24	deviation	deviation	NOUN
easat-9645	167	25	gradually	gradually	ADV
easat-9645	167	26	decreases	decrease	VERB
easat-9645	167	27	as	as	ADP
easat-9645	167	28	p	p	NOUN
easat-9645	167	29	approaches	approach	NOUN
easat-9645	167	30	2	2	NUM
easat-9645	167	31	,	,	PUNCT
easat-9645	167	32	highlighting	highlight	VERB
easat-9645	167	33	that	that	SCONJ
easat-9645	167	34	only	only	ADV
easat-9645	167	35	when	when	SCONJ
easat-9645	167	36	𝑝	𝑝	NOUN
easat-9645	167	37	=	=	SYM
easat-9645	167	38	2	2	NUM
easat-9645	167	39	does	do	AUX
easat-9645	167	40	the	the	DET
easat-9645	167	41	norm	norm	NOUN
easat-9645	167	42	come	come	VERB
easat-9645	167	43	from	from	ADP
easat-9645	167	44	an	an	DET
easat-9645	167	45	inner	inner	ADJ
easat-9645	167	46	product	product	NOUN
easat-9645	167	47	.	.	PUNCT
easat-9645	168	1	2.3	2.3	NUM
easat-9645	168	2	.	.	PUNCT
easat-9645	168	3	theorem	theorem	NOUN
easat-9645	168	4	(	(	PUNCT
easat-9645	168	5	cauchy	cauchy	PROPN
easat-9645	168	6	schwartz	schwartz	PROPN
easat-9645	168	7	inequality	inequality	PROPN
easat-9645	168	8	)	)	PUNCT
easat-9645	168	9	if	if	SCONJ
easat-9645	168	10	𝑋	𝑋	PROPN
easat-9645	168	11	is	be	AUX
easat-9645	168	12	an	an	DET
easat-9645	168	13	inner	inner	ADJ
easat-9645	168	14	product	product	NOUN
easat-9645	168	15	space	space	NOUN
easat-9645	168	16	and	and	CCONJ
easat-9645	168	17	𝑥	𝑥	NOUN
easat-9645	168	18	,	,	PUNCT
easat-9645	168	19	𝑦	𝑦	NOUN
easat-9645	168	20	∈	∈	NOUN
easat-9645	168	21	x	x	NOUN
easat-9645	168	22	,	,	PUNCT
easat-9645	168	23	then	then	ADV
easat-9645	168	24	|	|	ADV
easat-9645	168	25	<	<	X
easat-9645	168	26	𝑥	𝑥	PROPN
easat-9645	168	27	,	,	PUNCT
easat-9645	168	28	y	y	PROPN
easat-9645	168	29	>	>	X
easat-9645	168	30	|	|	ADJ
easat-9645	168	31	≤	≤	NUM
easat-9645	168	32	‖𝑥‖‖𝑦‖.	‖𝑥‖‖𝑦‖.	NUM
easat-9645	168	33	2.3.1	2.3.1	NUM
easat-9645	168	34	.	.	PUNCT
easat-9645	169	1	geometrical	geometrical	ADJ
easat-9645	169	2	interpretation	interpretation	NOUN
easat-9645	169	3	of	of	ADP
easat-9645	169	4	the	the	DET
easat-9645	169	5	cauchy	cauchy	PROPN
easat-9645	169	6	–	–	PUNCT
easat-9645	169	7	schwarz	schwarz	PROPN
easat-9645	169	8	inequality	inequality	NOUN
easat-9645	169	9	the	the	DET
easat-9645	169	10	cauchy	cauchy	PROPN
easat-9645	169	11	–	–	PUNCT
easat-9645	169	12	schwarz	schwarz	PROPN
easat-9645	169	13	inequality	inequality	NOUN
easat-9645	169	14	provides	provide	VERB
easat-9645	169	15	a	a	DET
easat-9645	169	16	geometric	geometric	ADJ
easat-9645	169	17	bound	bind	VERB
easat-9645	169	18	on	on	ADP
easat-9645	169	19	the	the	DET
easat-9645	169	20	inner	inner	ADJ
easat-9645	169	21	product	product	NOUN
easat-9645	169	22	of	of	ADP
easat-9645	169	23	two	two	NUM
easat-9645	169	24	vectors	vector	NOUN
easat-9645	169	25	.	.	PUNCT
easat-9645	170	1	in	in	ADP
easat-9645	170	2	euclidean	euclidean	ADJ
easat-9645	170	3	space	space	NOUN
easat-9645	170	4	,	,	PUNCT
easat-9645	170	5	the	the	DET
easat-9645	170	6	inner	inner	ADJ
easat-9645	170	7	product	product	NOUN
easat-9645	170	8	<	<	X
easat-9645	170	9	𝑥	𝑥	PROPN
easat-9645	170	10	,	,	PUNCT
easat-9645	170	11	y	y	PROPN
easat-9645	170	12	>	>	X
easat-9645	170	13	can	can	AUX
easat-9645	170	14	be	be	AUX
easat-9645	170	15	written	write	VERB
easat-9645	170	16	as	as	ADP
easat-9645	170	17	:	:	PUNCT
easat-9645	170	18	<	<	X
easat-9645	170	19	𝑥	𝑥	X
easat-9645	170	20	,	,	PUNCT
easat-9645	170	21	y	y	PROPN
easat-9645	170	22	>	>	X
easat-9645	170	23	=	=	PUNCT
easat-9645	170	24	‖𝑥‖‖𝑦‖.	‖𝑥‖‖𝑦‖.	NOUN
easat-9645	170	25	𝑐𝑜𝑠𝜃.	𝑐𝑜𝑠𝜃.	NOUN
easat-9645	170	26	using	use	VERB
easat-9645	170	27	this	this	PRON
easat-9645	170	28	,	,	PUNCT
easat-9645	170	29	the	the	DET
easat-9645	170	30	cauchy	cauchy	PROPN
easat-9645	170	31	–	–	PUNCT
easat-9645	170	32	schwarz	schwarz	PROPN
easat-9645	170	33	inequality	inequality	NOUN
easat-9645	170	34	becomes	become	VERB
easat-9645	170	35	:	:	PUNCT
easat-9645	171	1	|	|	ADV
easat-9645	171	2	<	<	X
easat-9645	171	3	𝑥	𝑥	PROPN
easat-9645	171	4	,	,	PUNCT
easat-9645	171	5	y	y	PROPN
easat-9645	171	6	>	>	X
easat-9645	171	7	|	|	NOUN
easat-9645	171	8	≤	≤	NUM
easat-9645	171	9	‖𝑥‖.	‖𝑥‖.	PUNCT
easat-9645	171	10	‖𝑦‖.	‖𝑦‖.	PUNCT
easat-9645	171	11	𝑐𝑜𝑠𝜃	𝑐𝑜𝑠𝜃	X
easat-9645	171	12	≤	≤	ADJ
easat-9645	171	13	‖𝑥‖.	‖𝑥‖.	ADP
easat-9645	171	14	‖𝑦‖.	‖𝑦‖.	PUNCT
easat-9645	171	15	since	since	SCONJ
easat-9645	171	16	∣	∣	PROPN
easat-9645	171	17	𝑐𝑜𝑠	𝑐𝑜𝑠	NOUN
easat-9645	171	18	𝜃	𝜃	X
easat-9645	171	19	∣≤	∣≤	NUM
easat-9645	171	20	1for	1for	NUM
easat-9645	171	21	all	all	DET
easat-9645	171	22	real	real	ADJ
easat-9645	171	23	angles	angle	NOUN
easat-9645	171	24	θ	θ	PROPN
easat-9645	171	25	.	.	PUNCT
easat-9645	172	1	equality	equality	NOUN
easat-9645	172	2	occurs	occur	VERB
easat-9645	172	3	when	when	SCONJ
easat-9645	172	4	𝜃	𝜃	X
easat-9645	172	5	=	=	SYM
easat-9645	172	6	0	0	NUM
easat-9645	172	7	𝑜𝑟	𝑜𝑟	PRON
easat-9645	172	8	𝜋	𝜋	NOUN
easat-9645	172	9	,	,	PUNCT
easat-9645	172	10	meaning	mean	VERB
easat-9645	172	11	𝑥	𝑥	PROPN
easat-9645	172	12	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-9645	172	13	𝑦	𝑦	DET
easat-9645	172	14	point	point	NOUN
easat-9645	172	15	in	in	ADP
easat-9645	172	16	the	the	DET
easat-9645	172	17	same	same	ADJ
easat-9645	172	18	or	or	CCONJ
easat-9645	172	19	opposite	opposite	ADJ
easat-9645	172	20	directions	direction	NOUN
easat-9645	172	21	and	and	CCONJ
easat-9645	172	22	are	be	AUX
easat-9645	172	23	thus	thus	ADV
easat-9645	172	24	linearly	linearly	ADV
easat-9645	172	25	dependent	dependent	ADJ
easat-9645	172	26	.	.	PUNCT
easat-9645	173	1	in	in	ADP
easat-9645	173	2	geometric	geometric	ADJ
easat-9645	173	3	terms	term	NOUN
easat-9645	173	4	,	,	PUNCT
easat-9645	173	5	this	this	DET
easat-9645	173	6	inequality	inequality	NOUN
easat-9645	173	7	implies	imply	VERB
easat-9645	173	8	that	that	SCONJ
easat-9645	173	9	the	the	DET
easat-9645	173	10	projection	projection	NOUN
easat-9645	173	11	of	of	ADP
easat-9645	173	12	one	one	NUM
easat-9645	173	13	vector	vector	NOUN
easat-9645	173	14	onto	onto	ADP
easat-9645	173	15	another	another	PRON
easat-9645	173	16	can	can	AUX
easat-9645	173	17	not	not	PART
easat-9645	173	18	be	be	AUX
easat-9645	173	19	longer	long	ADJ
easat-9645	173	20	than	than	ADP
easat-9645	173	21	the	the	DET
easat-9645	173	22	product	product	NOUN
easat-9645	173	23	of	of	ADP
easat-9645	173	24	their	their	PRON
easat-9645	173	25	lengths	length	NOUN
easat-9645	173	26	.	.	PUNCT
easat-9645	174	1	it	it	PRON
easat-9645	174	2	ensures	ensure	VERB
easat-9645	174	3	that	that	SCONJ
easat-9645	174	4	the	the	DET
easat-9645	174	5	cosine	cosine	NOUN
easat-9645	174	6	of	of	ADP
easat-9645	174	7	the	the	DET
easat-9645	174	8	angle	angle	NOUN
easat-9645	174	9	between	between	ADP
easat-9645	174	10	two	two	NUM
easat-9645	174	11	vectors	vector	NOUN
easat-9645	174	12	always	always	ADV
easat-9645	174	13	lies	lie	VERB
easat-9645	174	14	between	between	ADP
easat-9645	174	15	−1	−1	NOUN
easat-9645	174	16	and	and	CCONJ
easat-9645	174	17	1	1	NUM
easat-9645	174	18	,	,	PUNCT
easat-9645	174	19	preserving	preserve	VERB
easat-9645	174	20	the	the	DET
easat-9645	174	21	familiar	familiar	ADJ
easat-9645	174	22	geometric	geometric	ADJ
easat-9645	174	23	structure	structure	NOUN
easat-9645	174	24	even	even	ADV
easat-9645	174	25	in	in	ADP
easat-9645	174	26	abstract	abstract	ADJ
easat-9645	174	27	inner	inner	ADJ
easat-9645	174	28	product	product	NOUN
easat-9645	174	29	spaces	space	VERB
easat-9645	174	30	.	.	PUNCT
easat-9645	175	1	2.3.2	2.3.2	X
easat-9645	175	2	.	.	X
easat-9645	175	3	matlab	matlab	PROPN
easat-9645	175	4	visualization	visualization	NOUN
easat-9645	175	5	of	of	ADP
easat-9645	175	6	the	the	DET
easat-9645	175	7	cauchy	cauchy	PROPN
easat-9645	175	8	–	–	PUNCT
easat-9645	175	9	schwarz	schwarz	PROPN
easat-9645	175	10	inequality	inequality	NOUN
easat-9645	175	11	the	the	DET
easat-9645	175	12	cauchy	cauchy	PROPN
easat-9645	175	13	–	–	PUNCT
easat-9645	175	14	schwarz	schwarz	PROPN
easat-9645	175	15	inequality	inequality	NOUN
easat-9645	175	16	is	be	AUX
easat-9645	175	17	a	a	DET
easat-9645	175	18	key	key	ADJ
easat-9645	175	19	result	result	NOUN
easat-9645	175	20	stating	state	VERB
easat-9645	175	21	that	that	SCONJ
easat-9645	175	22	the	the	DET
easat-9645	175	23	absolute	absolute	ADJ
easat-9645	175	24	value	value	NOUN
easat-9645	175	25	of	of	ADP
easat-9645	175	26	the	the	DET
easat-9645	175	27	inner	inner	ADJ
easat-9645	175	28	product	product	NOUN
easat-9645	175	29	of	of	ADP
easat-9645	175	30	two	two	NUM
easat-9645	175	31	vectors	vector	NOUN
easat-9645	175	32	does	do	AUX
easat-9645	175	33	not	not	PART
easat-9645	175	34	exceed	exceed	VERB
easat-9645	175	35	the	the	DET
easat-9645	175	36	product	product	NOUN
easat-9645	175	37	of	of	ADP
easat-9645	175	38	their	their	PRON
easat-9645	175	39	norms	norm	NOUN
easat-9645	175	40	.	.	PUNCT
easat-9645	176	1	it	it	PRON
easat-9645	176	2	is	be	AUX
easat-9645	176	3	crucial	crucial	ADJ
easat-9645	176	4	for	for	ADP
easat-9645	176	5	proving	prove	VERB
easat-9645	176	6	many	many	ADJ
easat-9645	176	7	other	other	ADJ
easat-9645	176	8	results	result	NOUN
easat-9645	176	9	in	in	ADP
easat-9645	176	10	1504	1504	NUM
easat-9645	176	11	edelweiss	edelweiss	PROPN
easat-9645	176	12	applied	apply	VERB
easat-9645	176	13	science	science	NOUN
easat-9645	176	14	and	and	CCONJ
easat-9645	176	15	technology	technology	NOUN
easat-9645	176	16	issn	issn	PROPN
easat-9645	176	17	:	:	PUNCT
easat-9645	176	18	2576	2576	NUM
easat-9645	176	19	-	-	SYM
easat-9645	176	20	8484	8484	NUM
easat-9645	176	21	vol	vol	NOUN
easat-9645	176	22	.	.	PROPN
easat-9645	177	1	9	9	NUM
easat-9645	177	2	,	,	PUNCT
easat-9645	177	3	no	no	INTJ
easat-9645	177	4	.	.	NOUN
easat-9645	177	5	8	8	NUM
easat-9645	177	6	:	:	SYM
easat-9645	177	7	1498	1498	NUM
easat-9645	177	8	-	-	SYM
easat-9645	177	9	1523	1523	NUM
easat-9645	177	10	,	,	PUNCT
easat-9645	177	11	2025	2025	NUM
easat-9645	177	12	doi	doi	NOUN
easat-9645	177	13	:	:	PUNCT
easat-9645	177	14	10.55214/2576	10.55214/2576	NUM
easat-9645	177	15	-	-	SYM
easat-9645	177	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	177	17	©	©	PROPN
easat-9645	177	18	2025	2025	NUM
easat-9645	177	19	by	by	ADP
easat-9645	177	20	the	the	DET
easat-9645	177	21	authors	author	NOUN
easat-9645	177	22	;	;	PUNCT
easat-9645	177	23	licensee	licensee	PROPN
easat-9645	177	24	learning	learn	VERB
easat-9645	177	25	gate	gate	VERB
easat-9645	177	26	functional	functional	ADJ
easat-9645	177	27	analysis	analysis	NOUN
easat-9645	177	28	,	,	PUNCT
easat-9645	177	29	hilbert	hilbert	NOUN
easat-9645	177	30	spaces	space	NOUN
easat-9645	177	31	,	,	PUNCT
easat-9645	177	32	and	and	CCONJ
easat-9645	177	33	vector	vector	NOUN
easat-9645	177	34	geometry	geometry	NOUN
easat-9645	177	35	.	.	PUNCT
easat-9645	178	1	here	here	ADV
easat-9645	178	2	,	,	PUNCT
easat-9645	178	3	plots	plot	VERB
easat-9645	178	4	two	two	NUM
easat-9645	178	5	vectors	vector	NOUN
easat-9645	178	6	𝑥	𝑥	X
easat-9645	178	7	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
easat-9645	178	8	𝑦	𝑦	NOUN
easat-9645	178	9	,	,	PUNCT
easat-9645	178	10	the	the	DET
easat-9645	178	11	projection	projection	NOUN
easat-9645	178	12	of	of	ADP
easat-9645	178	13	𝑥	𝑥	PROPN
easat-9645	178	14	𝑜𝑛𝑡𝑜	𝑜𝑛𝑡𝑜	NOUN
easat-9645	178	15	𝑦	𝑦	NOUN
easat-9645	178	16	and	and	CCONJ
easat-9645	178	17	computes	compute	VERB
easat-9645	178	18	both	both	DET
easat-9645	178	19	sides	side	NOUN
easat-9645	178	20	of	of	ADP
easat-9645	178	21	the	the	DET
easat-9645	178	22	cauchy	cauchy	PROPN
easat-9645	178	23	–	–	PUNCT
easat-9645	178	24	schwarz	schwarz	PROPN
easat-9645	178	25	inequality	inequality	NOUN
easat-9645	178	26	.	.	PUNCT
easat-9645	179	1	figure	figure	NOUN
easat-9645	179	2	3	3	NUM
easat-9645	179	3	.	.	PUNCT
easat-9645	179	4	matlab	matlab	PROPN
easat-9645	179	5	visualization	visualization	NOUN
easat-9645	179	6	of	of	ADP
easat-9645	179	7	the	the	DET
easat-9645	179	8	cauchy	cauchy	PROPN
easat-9645	179	9	–	–	PUNCT
easat-9645	179	10	schwarz	schwarz	PROPN
easat-9645	179	11	inequality	inequality	NOUN
easat-9645	179	12	.	.	PUNCT
easat-9645	180	1	the	the	DET
easat-9645	180	2	figure	figure	NOUN
easat-9645	180	3	3	3	NUM
easat-9645	180	4	,	,	PUNCT
easat-9645	180	5	illustrates	illustrate	VERB
easat-9645	180	6	the	the	DET
easat-9645	180	7	cauchy	cauchy	PROPN
easat-9645	180	8	-	-	PUNCT
easat-9645	180	9	schwarz	schwarz	PROPN
easat-9645	180	10	inequality	inequality	NOUN
easat-9645	180	11	and	and	CCONJ
easat-9645	180	12	the	the	DET
easat-9645	180	13	projection	projection	NOUN
easat-9645	180	14	of	of	ADP
easat-9645	180	15	vector	vector	NOUN
easat-9645	180	16	x	x	PUNCT
easat-9645	180	17	onto	onto	ADP
easat-9645	180	18	vector	vector	NOUN
easat-9645	180	19	y	y	PROPN
easat-9645	180	20	in	in	ADP
easat-9645	180	21	a	a	DET
easat-9645	180	22	2d	2d	NUM
easat-9645	180	23	space	space	NOUN
easat-9645	180	24	.	.	PUNCT
easat-9645	181	1	the	the	DET
easat-9645	181	2	red	red	ADJ
easat-9645	181	3	vector	vector	NOUN
easat-9645	181	4	x	x	PUNCT
easat-9645	181	5	extends	extend	VERB
easat-9645	181	6	from	from	ADP
easat-9645	181	7	(	(	PUNCT
easat-9645	181	8	0,0	0,0	NOUN
easat-9645	181	9	)	)	PUNCT
easat-9645	181	10	to	to	ADP
easat-9645	181	11	approximately	approximately	ADV
easat-9645	181	12	(	(	PUNCT
easat-9645	181	13	2.5	2.5	NUM
easat-9645	181	14	,	,	PUNCT
easat-9645	181	15	1	1	NUM
easat-9645	181	16	)	)	PUNCT
easat-9645	181	17	,	,	PUNCT
easat-9645	181	18	and	and	CCONJ
easat-9645	181	19	the	the	DET
easat-9645	181	20	blue	blue	ADJ
easat-9645	181	21	vector	vector	NOUN
easat-9645	181	22	y	y	PROPN
easat-9645	181	23	extends	extend	VERB
easat-9645	181	24	to	to	ADP
easat-9645	181	25	(	(	PUNCT
easat-9645	181	26	2	2	NUM
easat-9645	181	27	,	,	PUNCT
easat-9645	181	28	2	2	NUM
easat-9645	181	29	)	)	PUNCT
easat-9645	181	30	.	.	PUNCT
easat-9645	182	1	the	the	DET
easat-9645	182	2	green	green	PROPN
easat-9645	182	3	vector	vector	PROPN
easat-9645	182	4	shows	show	VERB
easat-9645	182	5	the	the	DET
easat-9645	182	6	projection	projection	NOUN
easat-9645	182	7	of	of	ADP
easat-9645	182	8	x	x	PUNCT
easat-9645	182	9	onto	onto	ADP
easat-9645	182	10	y.	y.	NOUN
easat-9645	182	11	the	the	DET
easat-9645	182	12	inequality	inequality	NOUN
easat-9645	182	13	|	|	ADV
easat-9645	182	14	<	<	X
easat-9645	182	15	𝑥	𝑥	PROPN
easat-9645	182	16	,	,	PUNCT
easat-9645	182	17	y	y	PROPN
easat-9645	182	18	>	>	X
easat-9645	183	1	|	|	PROPN
easat-9645	183	2	≈	≈	PROPN
easat-9645	183	3	8.00	8.00	NUM
easat-9645	183	4	𝑡𝑜	𝑡𝑜	NOUN
easat-9645	183	5	8.94	8.94	NUM
easat-9645	183	6	is	be	AUX
easat-9645	183	7	close	close	ADJ
easat-9645	183	8	to	to	ADP
easat-9645	183	9	‖𝑥‖.	‖𝑥‖.	X
easat-9645	183	10	‖𝑦‖.	‖𝑦‖.	PRON
easat-9645	183	11	2.4	2.4	NUM
easat-9645	183	12	.	.	PUNCT
easat-9645	184	1	theorem	theorem	VERB
easat-9645	184	2	the	the	DET
easat-9645	184	3	space	space	NOUN
easat-9645	184	4	c	c	NOUN
easat-9645	185	1	[	[	X
easat-9645	185	2	a	a	X
easat-9645	185	3	,	,	PUNCT
easat-9645	185	4	b	b	NOUN
easat-9645	185	5	]	]	X
easat-9645	185	6	is	be	AUX
easat-9645	185	7	not	not	PART
easat-9645	185	8	an	an	DET
easat-9645	185	9	inner	inner	ADJ
easat-9645	185	10	product	product	NOUN
easat-9645	185	11	space	space	NOUN
easat-9645	185	12	,	,	PUNCT
easat-9645	185	13	hence	hence	ADV
easat-9645	185	14	not	not	PART
easat-9645	185	15	a	a	DET
easat-9645	185	16	hilbert	hilbert	NOUN
easat-9645	185	17	space	space	NOUN
easat-9645	185	18	.	.	PUNCT
easat-9645	186	1	2.4.1	2.4.1	X
easat-9645	186	2	.	.	PUNCT
easat-9645	187	1	geometrical	geometrical	ADJ
easat-9645	187	2	explanation	explanation	NOUN
easat-9645	187	3	why	why	SCONJ
easat-9645	187	4	c	c	AUX
easat-9645	188	1	[	[	X
easat-9645	188	2	a	a	X
easat-9645	188	3	,	,	PUNCT
easat-9645	188	4	b	b	NOUN
easat-9645	188	5	]	]	X
easat-9645	188	6	is	be	AUX
easat-9645	188	7	not	not	PART
easat-9645	188	8	a	a	DET
easat-9645	188	9	hilbert	hilbert	NOUN
easat-9645	188	10	space	space	NOUN
easat-9645	188	11	?	?	PUNCT
easat-9645	189	1	here	here	ADV
easat-9645	189	2	,	,	PUNCT
easat-9645	189	3	c	c	X
easat-9645	190	1	[	[	X
easat-9645	190	2	a	a	X
easat-9645	190	3	,	,	PUNCT
easat-9645	190	4	b	b	NOUN
easat-9645	190	5	]	]	X
easat-9645	190	6	is	be	AUX
easat-9645	190	7	not	not	PART
easat-9645	190	8	a	a	DET
easat-9645	190	9	hilbert	hilbert	NOUN
easat-9645	190	10	space	space	NOUN
easat-9645	190	11	because	because	SCONJ
easat-9645	190	12	its	its	PRON
easat-9645	190	13	norm	norm	NOUN
easat-9645	190	14	is	be	AUX
easat-9645	190	15	not	not	PART
easat-9645	190	16	induced	induce	VERB
easat-9645	190	17	by	by	ADP
easat-9645	190	18	an	an	DET
easat-9645	190	19	inner	inner	ADJ
easat-9645	190	20	product	product	NOUN
easat-9645	190	21	,	,	PUNCT
easat-9645	190	22	and	and	CCONJ
easat-9645	190	23	it	it	PRON
easat-9645	190	24	is	be	AUX
easat-9645	190	25	not	not	PART
easat-9645	190	26	complete	complete	ADJ
easat-9645	190	27	under	under	ADP
easat-9645	190	28	the	the	DET
easat-9645	190	29	𝐿2	𝐿2	PROPN
easat-9645	190	30	norm	norm	NOUN
easat-9645	190	31	,	,	PUNCT
easat-9645	190	32	which	which	PRON
easat-9645	190	33	is	be	AUX
easat-9645	190	34	essential	essential	ADJ
easat-9645	190	35	for	for	ADP
easat-9645	190	36	hilbert	hilbert	NOUN
easat-9645	190	37	spaces	space	NOUN
easat-9645	190	38	.	.	PUNCT
easat-9645	191	1	i.	i.	PROPN
easat-9645	191	2	geometry	geometry	PROPN
easat-9645	191	3	in	in	ADP
easat-9645	191	4	hilbert	hilbert	PROPN
easat-9645	191	5	spaces	space	NOUN
easat-9645	191	6	(	(	PUNCT
easat-9645	191	7	i.	i.	PROPN
easat-9645	191	8	e.	e.	PROPN
easat-9645	191	9	l2[a	l2[a	PROPN
easat-9645	191	10	,	,	PUNCT
easat-9645	191	11	b	b	NOUN
easat-9645	191	12	]	]	X
easat-9645	191	13	)	)	PUNCT
easat-9645	191	14	in	in	ADP
easat-9645	191	15	hilbert	hilbert	PROPN
easat-9645	191	16	spaces	space	VERB
easat-9645	191	17	:	:	PUNCT
easat-9645	191	18	a	a	X
easat-9645	191	19	)	)	PUNCT
easat-9645	191	20	the	the	DET
easat-9645	191	21	norm	norm	NOUN
easat-9645	191	22	comes	come	VERB
easat-9645	191	23	from	from	ADP
easat-9645	191	24	an	an	DET
easat-9645	191	25	inner	inner	ADJ
easat-9645	191	26	product	product	NOUN
easat-9645	191	27	,	,	PUNCT
easat-9645	191	28	b	b	X
easat-9645	191	29	)	)	PUNCT
easat-9645	191	30	define	define	VERB
easat-9645	191	31	angles	angle	NOUN
easat-9645	191	32	and	and	CCONJ
easat-9645	191	33	orthogonality	orthogonality	NOUN
easat-9645	191	34	,	,	PUNCT
easat-9645	191	35	c	c	NOUN
easat-9645	191	36	)	)	PUNCT
easat-9645	191	37	there	there	PRON
easat-9645	191	38	is	be	VERB
easat-9645	191	39	a	a	DET
easat-9645	191	40	clear	clear	ADJ
easat-9645	191	41	pythagorean	pythagorean	PROPN
easat-9645	191	42	geometry	geometry	NOUN
easat-9645	191	43	:	:	PUNCT
easat-9645	191	44	‖𝑥	‖𝑥	NOUN
easat-9645	191	45	+	+	CCONJ
easat-9645	191	46	𝑦‖2	𝑦‖2	X
easat-9645	191	47	=	=	NOUN
easat-9645	191	48	‖𝑥‖2	‖𝑥‖2	NOUN
easat-9645	191	49	+	+	CCONJ
easat-9645	191	50	‖𝑦‖2	‖𝑦‖2	NOUN
easat-9645	191	51	𝑖𝑓	𝑖𝑓	ADP
easat-9645	191	52	𝑥	𝑥	NOUN
easat-9645	191	53	⊥	⊥	PROPN
easat-9645	191	54	𝑦	𝑦	PROPN
easat-9645	191	55	,	,	PUNCT
easat-9645	191	56	d	d	NOUN
easat-9645	191	57	)	)	PUNCT
easat-9645	191	58	a	a	DET
easat-9645	191	59	project	project	NOUN
easat-9645	191	60	of	of	ADP
easat-9645	191	61	a	a	DET
easat-9645	191	62	function	function	NOUN
easat-9645	191	63	onto	onto	ADP
easat-9645	191	64	a	a	DET
easat-9645	191	65	subspace	subspace	NOUN
easat-9645	191	66	(	(	PUNCT
easat-9645	191	67	like	like	ADP
easat-9645	191	68	fourier	fourier	PROPN
easat-9645	191	69	series	series	PROPN
easat-9645	191	70	projection	projection	PROPN
easat-9645	191	71	)	)	PUNCT
easat-9645	191	72	,	,	PUNCT
easat-9645	191	73	e	e	X
easat-9645	191	74	)	)	PUNCT
easat-9645	191	75	the	the	DET
easat-9645	191	76	geometry	geometry	NOUN
easat-9645	191	77	behaves	behave	VERB
easat-9645	191	78	like	like	ADP
easat-9645	191	79	euclidean	euclidean	ADJ
easat-9645	191	80	space	space	NOUN
easat-9645	191	81	,	,	PUNCT
easat-9645	191	82	but	but	CCONJ
easat-9645	191	83	infinite	infinite	ADJ
easat-9645	191	84	-	-	PUNCT
easat-9645	191	85	dimensional	dimensional	ADJ
easat-9645	191	86	.	.	PUNCT
easat-9645	192	1	1505	1505	NUM
easat-9645	192	2	edelweiss	edelweiss	PROPN
easat-9645	192	3	applied	apply	VERB
easat-9645	192	4	science	science	NOUN
easat-9645	192	5	and	and	CCONJ
easat-9645	192	6	technology	technology	NOUN
easat-9645	192	7	issn	issn	PROPN
easat-9645	192	8	:	:	PUNCT
easat-9645	192	9	2576	2576	NUM
easat-9645	192	10	-	-	SYM
easat-9645	192	11	8484	8484	NUM
easat-9645	192	12	vol	vol	NOUN
easat-9645	192	13	.	.	PROPN
easat-9645	193	1	9	9	NUM
easat-9645	193	2	,	,	PUNCT
easat-9645	193	3	no	no	INTJ
easat-9645	193	4	.	.	NOUN
easat-9645	193	5	8	8	NUM
easat-9645	193	6	:	:	SYM
easat-9645	193	7	1498	1498	NUM
easat-9645	193	8	-	-	SYM
easat-9645	193	9	1523	1523	NUM
easat-9645	193	10	,	,	PUNCT
easat-9645	193	11	2025	2025	NUM
easat-9645	193	12	doi	doi	NOUN
easat-9645	193	13	:	:	PUNCT
easat-9645	193	14	10.55214/2576	10.55214/2576	NUM
easat-9645	193	15	-	-	SYM
easat-9645	193	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	193	17	©	©	PROPN
easat-9645	193	18	2025	2025	NUM
easat-9645	193	19	by	by	ADP
easat-9645	193	20	the	the	DET
easat-9645	193	21	authors	author	NOUN
easat-9645	193	22	;	;	PUNCT
easat-9645	193	23	licensee	licensee	PROPN
easat-9645	193	24	learning	learning	PROPN
easat-9645	193	25	gate	gate	PROPN
easat-9645	193	26	ii	ii	PROPN
easat-9645	193	27	.	.	PUNCT
easat-9645	193	28	geometry	geometry	NOUN
easat-9645	193	29	in	in	ADP
easat-9645	193	30	c	c	PROPN
easat-9645	193	31	[	[	X
easat-9645	193	32	a	a	X
easat-9645	193	33	,	,	PUNCT
easat-9645	193	34	b	b	NOUN
easat-9645	193	35	]	]	X
easat-9645	194	1	the	the	DET
easat-9645	194	2	space	space	NOUN
easat-9645	194	3	c	c	NOUN
easat-9645	195	1	[	[	X
easat-9645	195	2	a	a	X
easat-9645	195	3	,	,	PUNCT
easat-9645	195	4	b	b	NOUN
easat-9645	195	5	]	]	X
easat-9645	195	6	,	,	PUNCT
easat-9645	195	7	with	with	ADP
easat-9645	195	8	the	the	DET
easat-9645	195	9	supremum	supremum	ADJ
easat-9645	195	10	norm	norm	NOUN
easat-9645	195	11	:	:	PUNCT
easat-9645	195	12	‖𝑓‖∞	‖𝑓‖∞	NOUN
easat-9645	195	13	=	=	PUNCT
easat-9645	195	14	𝑠𝑢𝑝	𝑠𝑢𝑝	NOUN
easat-9645	195	15	𝑡∈[𝑎,𝑏	𝑡∈[𝑎,𝑏	PROPN
easat-9645	195	16	]	]	PUNCT
easat-9645	196	1	|𝑓(𝑡)|	|𝑓(𝑡)|	PROPN
easat-9645	196	2	iii	iii	PROPN
easat-9645	196	3	.	.	PUNCT
easat-9645	196	4	geometrical	geometrical	ADJ
easat-9645	196	5	difference	difference	NOUN
easat-9645	196	6	via	via	ADP
easat-9645	196	7	unit	unit	NOUN
easat-9645	196	8	balls	ball	NOUN
easat-9645	196	9	in	in	ADP
easat-9645	196	10	2d	2d	NUM
easat-9645	196	11	geometry	geometry	NOUN
easat-9645	196	12	,	,	PUNCT
easat-9645	196	13	you	you	PRON
easat-9645	196	14	can	can	AUX
easat-9645	196	15	visualize	visualize	VERB
easat-9645	196	16	normed	normed	ADJ
easat-9645	196	17	spaces	space	NOUN
easat-9645	196	18	by	by	ADP
easat-9645	196	19	looking	look	VERB
easat-9645	196	20	at	at	ADP
easat-9645	196	21	unit	unit	NOUN
easat-9645	196	22	balls	ball	NOUN
easat-9645	196	23	(	(	PUNCT
easat-9645	196	24	the	the	DET
easat-9645	196	25	set	set	NOUN
easat-9645	196	26	of	of	ADP
easat-9645	196	27	vectors	vector	NOUN
easat-9645	196	28	/	/	SYM
easat-9645	196	29	functions	function	NOUN
easat-9645	196	30	with	with	ADP
easat-9645	196	31	norm	norm	NOUN
easat-9645	196	32	1	1	NUM
easat-9645	196	33	)	)	PUNCT
easat-9645	196	34	.	.	PUNCT
easat-9645	197	1	space	space	NOUN
easat-9645	197	2	norm	norm	PROPN
easat-9645	197	3	unit	unit	NOUN
easat-9645	197	4	ball	ball	NOUN
easat-9645	197	5	shape	shape	NOUN
easat-9645	197	6	geometry	geometry	NOUN
easat-9645	197	7	type	type	NOUN
easat-9645	197	8	𝑙2	𝑙2	PROPN
easat-9645	197	9	2	2	NUM
easat-9645	197	10	‖𝑥‖2	‖𝑥‖2	NOUN
easat-9645	197	11	circle	circle	NOUN
easat-9645	197	12	inner	inner	ADJ
easat-9645	197	13	product	product	NOUN
easat-9645	197	14	space	space	NOUN
easat-9645	197	15	𝑙∞	𝑙∞	PROPN
easat-9645	197	16	2	2	NUM
easat-9645	197	17	‖𝑥‖∞	‖𝑥‖∞	PROPN
easat-9645	197	18	square	square	ADJ
easat-9645	197	19	not	not	PART
easat-9645	197	20	inner	inner	ADJ
easat-9645	197	21	product	product	NOUN
easat-9645	197	22	c	c	NOUN
easat-9645	198	1	[	[	X
easat-9645	198	2	a	a	X
easat-9645	198	3	,	,	PUNCT
easat-9645	198	4	b	b	NOUN
easat-9645	198	5	]	]	X
easat-9645	198	6	‖𝑓‖∞	‖𝑓‖∞	PRON
easat-9645	198	7	infinite	infinite	ADJ
easat-9645	198	8	-	-	PUNCT
easat-9645	198	9	dimensional	dimensional	ADJ
easat-9645	198	10	cube	cube	NOUN
easat-9645	198	11	-	-	PUNCT
easat-9645	198	12	like	like	ADJ
easat-9645	198	13	no	no	DET
easat-9645	198	14	angles	angle	NOUN
easat-9645	198	15	,	,	PUNCT
easat-9645	198	16	no	no	DET
easat-9645	198	17	projection	projection	NOUN
easat-9645	198	18	in	in	ADP
easat-9645	198	19	c	c	PROPN
easat-9645	199	1	[	[	X
easat-9645	199	2	a	a	X
easat-9645	199	3	,	,	PUNCT
easat-9645	199	4	b	b	NOUN
easat-9645	199	5	]	]	X
easat-9645	199	6	,	,	PUNCT
easat-9645	199	7	the	the	DET
easat-9645	199	8	geometry	geometry	NOUN
easat-9645	199	9	is	be	AUX
easat-9645	199	10	dominated	dominate	VERB
easat-9645	199	11	by	by	ADP
easat-9645	199	12	uniform	uniform	ADJ
easat-9645	199	13	height	height	NOUN
easat-9645	199	14	across	across	ADP
easat-9645	199	15	the	the	DET
easat-9645	199	16	interval	interval	NOUN
easat-9645	199	17	,	,	PUNCT
easat-9645	199	18	not	not	PART
easat-9645	199	19	by	by	ADP
easat-9645	199	20	averaging	average	VERB
easat-9645	199	21	as	as	ADP
easat-9645	199	22	in	in	ADP
easat-9645	199	23	l2	l2	NOUN
easat-9645	199	24	.	.	PUNCT
easat-9645	200	1	this	this	PRON
easat-9645	200	2	makes	make	VERB
easat-9645	200	3	the	the	DET
easat-9645	200	4	"	"	PUNCT
easat-9645	200	5	roundness	roundness	NOUN
easat-9645	200	6	"	"	PUNCT
easat-9645	200	7	(	(	PUNCT
easat-9645	200	8	essential	essential	ADJ
easat-9645	200	9	for	for	ADP
easat-9645	200	10	inner	inner	ADJ
easat-9645	200	11	products	product	NOUN
easat-9645	200	12	)	)	PUNCT
easat-9645	200	13	absent	absent	ADJ
easat-9645	200	14	.	.	PUNCT
easat-9645	201	1	the	the	DET
easat-9645	201	2	space	space	NOUN
easat-9645	201	3	c	c	NOUN
easat-9645	202	1	[	[	X
easat-9645	202	2	a	a	X
easat-9645	202	3	,	,	PUNCT
easat-9645	202	4	b	b	NOUN
easat-9645	202	5	]	]	PUNCT
easat-9645	202	6	lacks	lack	VERB
easat-9645	202	7	the	the	DET
easat-9645	202	8	euclidean	euclidean	ADJ
easat-9645	202	9	-	-	PUNCT
easat-9645	202	10	like	like	ADJ
easat-9645	202	11	geometry	geometry	NOUN
easat-9645	202	12	of	of	ADP
easat-9645	202	13	hilbert	hilbert	PROPN
easat-9645	202	14	spaces	space	NOUN
easat-9645	202	15	:	:	PUNCT
easat-9645	202	16	the	the	DET
easat-9645	202	17	unit	unit	NOUN
easat-9645	202	18	ball	ball	NOUN
easat-9645	202	19	is	be	AUX
easat-9645	202	20	not	not	PART
easat-9645	202	21	round	round	ADJ
easat-9645	202	22	,	,	PUNCT
easat-9645	202	23	there	there	PRON
easat-9645	202	24	is	be	VERB
easat-9645	202	25	no	no	DET
easat-9645	202	26	inner	inner	ADJ
easat-9645	202	27	product	product	NOUN
easat-9645	202	28	,	,	PUNCT
easat-9645	202	29	so	so	CCONJ
easat-9645	202	30	no	no	DET
easat-9645	202	31	angles	angle	NOUN
easat-9645	202	32	,	,	PUNCT
easat-9645	202	33	no	no	DET
easat-9645	202	34	orthogonality	orthogonality	NOUN
easat-9645	202	35	,	,	PUNCT
easat-9645	202	36	and	and	CCONJ
easat-9645	202	37	no	no	DET
easat-9645	202	38	projections	projection	NOUN
easat-9645	202	39	and	and	CCONJ
easat-9645	202	40	even	even	ADV
easat-9645	202	41	under	under	ADP
easat-9645	202	42	the	the	DET
easat-9645	202	43	l2	l2	NOUN
easat-9645	202	44	inner	inner	ADJ
easat-9645	202	45	product	product	NOUN
easat-9645	202	46	,	,	PUNCT
easat-9645	202	47	it	it	PRON
easat-9645	202	48	's	be	AUX
easat-9645	202	49	not	not	PART
easat-9645	202	50	complete	complete	ADJ
easat-9645	202	51	,	,	PUNCT
easat-9645	202	52	so	so	SCONJ
easat-9645	202	53	you	you	PRON
easat-9645	202	54	ca	can	AUX
easat-9645	202	55	n't	not	PART
easat-9645	202	56	use	use	VERB
easat-9645	202	57	hilbert	hilbert	NOUN
easat-9645	202	58	space	space	NOUN
easat-9645	202	59	geometry	geometry	NOUN
easat-9645	202	60	reliably	reliably	ADV
easat-9645	202	61	.	.	PUNCT
easat-9645	203	1	figure	figure	VERB
easat-9645	203	2	4	4	NUM
easat-9645	203	3	.	.	PUNCT
easat-9645	203	4	matlab	matlab	PROPN
easat-9645	203	5	visualization	visualization	NOUN
easat-9645	203	6	for	for	ADP
easat-9645	203	7	the	the	DET
easat-9645	203	8	space	space	NOUN
easat-9645	203	9	c	c	NOUN
easat-9645	204	1	[	[	X
easat-9645	204	2	a	a	X
easat-9645	204	3	,	,	PUNCT
easat-9645	204	4	b	b	NOUN
easat-9645	204	5	]	]	X
easat-9645	204	6	.	.	PUNCT
easat-9645	205	1	the	the	DET
easat-9645	205	2	figure	figure	NOUN
easat-9645	205	3	4	4	NUM
easat-9645	205	4	,	,	PUNCT
easat-9645	205	5	illustrates	illustrate	VERB
easat-9645	205	6	the	the	DET
easat-9645	205	7	pointwise	pointwise	ADJ
easat-9645	205	8	convergence	convergence	NOUN
easat-9645	205	9	of	of	ADP
easat-9645	205	10	the	the	DET
easat-9645	205	11	sequence	sequence	NOUN
easat-9645	205	12	𝑓𝑛(𝑥	𝑓𝑛(𝑥	NOUN
easat-9645	205	13	)	)	PUNCT
easat-9645	206	1	=	=	PRON
easat-9645	206	2	𝑥𝑛	𝑥𝑛	AUX
easat-9645	206	3	𝑎𝑟𝑒	𝑎𝑟𝑒	NOUN
easat-9645	206	4	𝑖𝑛	𝑖𝑛	NOUN
easat-9645	206	5	𝐶[0,1	𝐶[0,1	ADP
easat-9645	206	6	]	]	PUNCT
easat-9645	206	7	.	.	PUNCT
easat-9645	207	1	it	it	PRON
easat-9645	207	2	shows	show	VERB
easat-9645	207	3	curves	curve	NOUN
easat-9645	207	4	for	for	ADP
easat-9645	207	5	𝑥1(𝑏𝑙𝑢𝑒	𝑥1(𝑏𝑙𝑢𝑒	NUM
easat-9645	207	6	)	)	PUNCT
easat-9645	207	7	,	,	PUNCT
easat-9645	207	8	𝑥2(𝑐𝑦𝑎𝑛	𝑥2(𝑐𝑦𝑎𝑛	NUM
easat-9645	207	9	)	)	PUNCT
easat-9645	207	10	,	,	PUNCT
easat-9645	207	11	𝑥5	𝑥5	PROPN
easat-9645	207	12	(	(	PUNCT
easat-9645	207	13	𝑔𝑟𝑒𝑒𝑛	𝑔𝑟𝑒𝑒𝑛	PROPN
easat-9645	207	14	)	)	PUNCT
easat-9645	207	15	,	,	PUNCT
easat-9645	207	16	𝑥10(𝑦𝑒𝑙𝑙𝑜𝑤	𝑥10(𝑦𝑒𝑙𝑙𝑜𝑤	NUM
easat-9645	207	17	)	)	PUNCT
easat-9645	207	18	,	,	PUNCT
easat-9645	207	19	𝑥50(𝑜𝑟𝑎𝑛𝑔𝑒	𝑥50(𝑜𝑟𝑎𝑛𝑔𝑒	NOUN
easat-9645	207	20	)	)	PUNCT
easat-9645	207	21	,	,	PUNCT
easat-9645	207	22	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-9645	207	23	𝑥100	𝑥100	PROPN
easat-9645	207	24	(	(	PUNCT
easat-9645	207	25	𝑟𝑒𝑑	𝑟𝑒𝑑	NOUN
easat-9645	207	26	)	)	PUNCT
easat-9645	207	27	,	,	PUNCT
easat-9645	207	28	with	with	ADP
easat-9645	207	29	the	the	DET
easat-9645	207	30	pointwise	pointwise	NOUN
easat-9645	207	31	limit	limit	NOUN
easat-9645	207	32	(	(	PUNCT
easat-9645	207	33	black	black	ADJ
easat-9645	207	34	dashed	dash	VERB
easat-9645	207	35	line	line	NOUN
easat-9645	207	36	)	)	PUNCT
easat-9645	207	37	approaching	approach	VERB
easat-9645	207	38	0	0	NUM
easat-9645	207	39	for	for	ADP
easat-9645	207	40	𝑥	𝑥	PRON
easat-9645	207	41	∈	∈	PROPN
easat-9645	207	42	[	[	X
easat-9645	207	43	0,1	0,1	NUM
easat-9645	207	44	)	)	PUNCT
easat-9645	207	45	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-9645	207	46	1	1	NUM
easat-9645	207	47	𝑎𝑡	𝑎𝑡	ADP
easat-9645	207	48	𝑥	𝑥	NOUN
easat-9645	207	49	=	=	SYM
easat-9645	207	50	1	1	NUM
easat-9645	207	51	.	.	PUNCT
easat-9645	207	52	1506	1506	NUM
easat-9645	207	53	edelweiss	edelweiss	PROPN
easat-9645	207	54	applied	apply	VERB
easat-9645	207	55	science	science	NOUN
easat-9645	207	56	and	and	CCONJ
easat-9645	207	57	technology	technology	NOUN
easat-9645	207	58	issn	issn	PROPN
easat-9645	207	59	:	:	PUNCT
easat-9645	207	60	2576	2576	NUM
easat-9645	207	61	-	-	SYM
easat-9645	207	62	8484	8484	NUM
easat-9645	207	63	vol	vol	NOUN
easat-9645	207	64	.	.	PROPN
easat-9645	208	1	9	9	NUM
easat-9645	208	2	,	,	PUNCT
easat-9645	208	3	no	no	INTJ
easat-9645	208	4	.	.	NOUN
easat-9645	208	5	8	8	NUM
easat-9645	208	6	:	:	SYM
easat-9645	208	7	1498	1498	NUM
easat-9645	208	8	-	-	SYM
easat-9645	208	9	1523	1523	NUM
easat-9645	208	10	,	,	PUNCT
easat-9645	208	11	2025	2025	NUM
easat-9645	208	12	doi	doi	NOUN
easat-9645	208	13	:	:	PUNCT
easat-9645	208	14	10.55214/2576	10.55214/2576	NUM
easat-9645	208	15	-	-	SYM
easat-9645	208	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	208	17	©	©	PROPN
easat-9645	208	18	2025	2025	NUM
easat-9645	208	19	by	by	ADP
easat-9645	208	20	the	the	DET
easat-9645	208	21	authors	author	NOUN
easat-9645	208	22	;	;	PUNCT
easat-9645	208	23	licensee	licensee	PROPN
easat-9645	208	24	learning	learning	NOUN
easat-9645	208	25	gate	gate	NOUN
easat-9645	208	26	3	3	NUM
easat-9645	208	27	.	.	NOUN
easat-9645	208	28	result	result	NOUN
easat-9645	208	29	and	and	CCONJ
easat-9645	208	30	discussion	discussion	NOUN
easat-9645	208	31	hilbert	hilbert	NOUN
easat-9645	208	32	spaces	space	NOUN
easat-9645	208	33	,	,	PUNCT
easat-9645	208	34	as	as	ADP
easat-9645	208	35	complete	complete	ADJ
easat-9645	208	36	inner	inner	ADJ
easat-9645	208	37	product	product	NOUN
easat-9645	208	38	spaces	space	NOUN
easat-9645	208	39	,	,	PUNCT
easat-9645	208	40	extend	extend	VERB
easat-9645	208	41	euclidean	euclidean	ADJ
easat-9645	208	42	geometry	geometry	NOUN
easat-9645	208	43	to	to	PART
easat-9645	208	44	infinite	infinite	VERB
easat-9645	208	45	dimensions	dimension	NOUN
easat-9645	208	46	while	while	SCONJ
easat-9645	208	47	preserving	preserve	VERB
easat-9645	208	48	key	key	ADJ
easat-9645	208	49	properties	property	NOUN
easat-9645	208	50	like	like	ADP
easat-9645	208	51	orthogonality	orthogonality	NOUN
easat-9645	208	52	and	and	CCONJ
easat-9645	208	53	completeness	completeness	NOUN
easat-9645	208	54	.	.	PUNCT
easat-9645	209	1	they	they	PRON
easat-9645	209	2	underpin	underpin	VERB
easat-9645	209	3	important	important	ADJ
easat-9645	209	4	theorems	theorem	NOUN
easat-9645	209	5	and	and	CCONJ
easat-9645	209	6	enable	enable	VERB
easat-9645	209	7	machine	machine	NOUN
easat-9645	209	8	learning	learning	NOUN
easat-9645	209	9	methods	method	NOUN
easat-9645	209	10	such	such	ADJ
easat-9645	209	11	as	as	ADP
easat-9645	209	12	svms	svms	NOUN
easat-9645	209	13	and	and	CCONJ
easat-9645	209	14	kernel	kernel	PROPN
easat-9645	209	15	pca	pca	PROPN
easat-9645	209	16	through	through	ADP
easat-9645	209	17	reproducing	reproduce	VERB
easat-9645	209	18	kernel	kernel	PROPN
easat-9645	209	19	hilbert	hilbert	PROPN
easat-9645	209	20	spaces	space	VERB
easat-9645	209	21	.	.	PUNCT
easat-9645	210	1	matlab	matlab	PROPN
easat-9645	210	2	visualizations	visualization	NOUN
easat-9645	210	3	help	help	VERB
easat-9645	210	4	illustrate	illustrate	VERB
easat-9645	210	5	these	these	DET
easat-9645	210	6	geometric	geometric	ADJ
easat-9645	210	7	concepts	concept	NOUN
easat-9645	210	8	in	in	ADP
easat-9645	210	9	data	data	NOUN
easat-9645	210	10	analysis	analysis	NOUN
easat-9645	210	11	and	and	CCONJ
easat-9645	210	12	learning	learning	NOUN
easat-9645	210	13	.	.	PUNCT
easat-9645	211	1	3	3	NUM
easat-9645	211	2	.	.	NOUN
easat-9645	211	3	1	1	NUM
easat-9645	211	4	.	.	X
easat-9645	211	5	theorem	theorem	NOUN
easat-9645	211	6	prove	prove	VERB
easat-9645	211	7	that	that	SCONJ
easat-9645	211	8	the	the	DET
easat-9645	211	9	euclidean	euclidean	ADJ
easat-9645	211	10	space	space	NOUN
easat-9645	211	11	ℝ𝑛	ℝ𝑛	X
easat-9645	211	12	is	be	AUX
easat-9645	211	13	a	a	DET
easat-9645	211	14	hilbert	hilbert	NOUN
easat-9645	211	15	space	space	NOUN
easat-9645	211	16	.	.	PUNCT
easat-9645	212	1	proof	proof	NOUN
easat-9645	212	2	:	:	PUNCT
easat-9645	212	3	the	the	DET
easat-9645	212	4	space	space	NOUN
easat-9645	212	5	rn	rn	PROPN
easat-9645	212	6	is	be	AUX
easat-9645	212	7	a	a	DET
easat-9645	212	8	hilbert	hilbert	NOUN
easat-9645	212	9	space	space	NOUN
easat-9645	212	10	with	with	ADP
easat-9645	212	11	inner	inner	ADJ
easat-9645	212	12	product	product	NOUN
easat-9645	212	13	defined	define	VERB
easat-9645	212	14	by	by	ADP
easat-9645	212	15	〈	〈	PROPN
easat-9645	212	16	𝑎	𝑎	NOUN
easat-9645	212	17	,	,	PUNCT
easat-9645	212	18	𝑏	𝑏	DET
easat-9645	212	19	〉	〉	NOUN
easat-9645	212	20	=	=	SYM
easat-9645	212	21	𝑎1𝑏1	𝑎1𝑏1	X
easat-9645	212	22	+	+	X
easat-9645	212	23	𝑎2𝑏2	𝑎2𝑏2	PUNCT
easat-9645	212	24	+	+	CCONJ
easat-9645	212	25	.	.	PUNCT
easat-9645	212	26	.	.	PUNCT
easat-9645	212	27	.	.	PUNCT
easat-9645	213	1	+	+	PUNCT
easat-9645	213	2	𝑎𝑛𝑏𝑛	𝑎𝑛𝑏𝑛	NOUN
easat-9645	213	3	(	(	PUNCT
easat-9645	213	4	1	1	NUM
easat-9645	213	5	)	)	PUNCT
easat-9645	213	6	where	where	SCONJ
easat-9645	213	7	𝑎	𝑎	NOUN
easat-9645	213	8	=	=	SYM
easat-9645	213	9	(	(	PUNCT
easat-9645	213	10	𝑎𝑖	𝑎𝑖	NOUN
easat-9645	213	11	)	)	PUNCT
easat-9645	213	12	=	=	SYM
easat-9645	213	13	(	(	PUNCT
easat-9645	213	14	𝑎1	𝑎1	PROPN
easat-9645	213	15	,	,	PUNCT
easat-9645	213	16	𝑎2	𝑎2	PROPN
easat-9645	213	17	,	,	PUNCT
easat-9645	213	18	.	.	PUNCT
easat-9645	213	19	.	.	PUNCT
easat-9645	213	20	.	.	PUNCT
easat-9645	214	1	,	,	PUNCT
easat-9645	214	2	𝑎𝑛	𝑎𝑛	PROPN
easat-9645	214	3	)	)	PUNCT
easat-9645	214	4	∈	∈	PROPN
easat-9645	214	5	rn	rn	PROPN
easat-9645	214	6	and	and	CCONJ
easat-9645	214	7	𝑏	𝑏	NOUN
easat-9645	214	8	=	=	PUNCT
easat-9645	214	9	(	(	PUNCT
easat-9645	214	10	𝑏𝑖	𝑏𝑖	NOUN
easat-9645	214	11	)	)	PUNCT
easat-9645	214	12	=	=	SYM
easat-9645	214	13	(	(	PUNCT
easat-9645	214	14	𝑏1	𝑏1	NOUN
easat-9645	214	15	,	,	PUNCT
easat-9645	214	16	𝑏2	𝑏2	PROPN
easat-9645	214	17	,	,	PUNCT
easat-9645	214	18	.	.	PUNCT
easat-9645	214	19	.	.	PUNCT
easat-9645	215	1	.	.	PUNCT
easat-9645	216	1	,	,	PUNCT
easat-9645	216	2	𝑏𝑛	𝑏𝑛	NOUN
easat-9645	216	3	)	)	PUNCT
easat-9645	216	4	∈rn	∈rn	NOUN
easat-9645	216	5	in	in	ADP
easat-9645	216	6	fact	fact	NOUN
easat-9645	216	7	,	,	PUNCT
easat-9645	216	8	from	from	ADP
easat-9645	216	9	(	(	PUNCT
easat-9645	216	10	1	1	X
easat-9645	216	11	)	)	PUNCT
easat-9645	216	12	we	we	PRON
easat-9645	216	13	obtain,‖𝑎‖	obtain,‖𝑎‖	PUNCT
easat-9645	217	1	=	=	SYM
easat-9645	217	2	⟨𝑎	⟨𝑎	NOUN
easat-9645	217	3	,	,	PUNCT
easat-9645	217	4	𝑎⟩	𝑎⟩	X
easat-9645	217	5	1	1	NUM
easat-9645	217	6	2	2	X
easat-9645	217	7	=	=	SYM
easat-9645	217	8	⟨𝑎1	⟨𝑎1	PROPN
easat-9645	217	9	2	2	NUM
easat-9645	217	10	+	+	NOUN
easat-9645	217	11	𝑎2	𝑎2	NOUN
easat-9645	217	12	2	2	NUM
easat-9645	217	13	+	+	PROPN
easat-9645	217	14	.	.	PUNCT
easat-9645	217	15	.	.	PUNCT
easat-9645	217	16	.	.	PUNCT
easat-9645	218	1	+	+	ADV
easat-9645	218	2	𝑎𝑛	𝑎𝑛	PROPN
easat-9645	218	3	2⟩	2⟩	NUM
easat-9645	218	4	1	1	NUM
easat-9645	218	5	2	2	NUM
easat-9645	218	6	and	and	CCONJ
easat-9645	218	7	𝑑(𝑎	𝑑(𝑎	NOUN
easat-9645	218	8	,	,	PUNCT
easat-9645	218	9	𝑏	𝑏	NOUN
easat-9645	218	10	)	)	PUNCT
easat-9645	218	11	=	=	VERB
easat-9645	218	12	‖𝑎	‖𝑎	NOUN
easat-9645	218	13	−	−	PROPN
easat-9645	218	14	𝑏‖	𝑏‖	X
easat-9645	219	1	=	=	PUNCT
easat-9645	220	1	⟨𝑎	⟨𝑎	NOUN
easat-9645	221	1	−	−	PROPN
easat-9645	221	2	𝑏	𝑏	NOUN
easat-9645	221	3	,	,	PUNCT
easat-9645	221	4	𝑎	𝑎	PRON
easat-9645	221	5	−	−	NOUN
easat-9645	221	6	𝑏⟩	𝑏⟩	NOUN
easat-9645	221	7	1	1	NUM
easat-9645	221	8	2	2	NUM
easat-9645	221	9	=	=	SYM
easat-9645	221	10	⟨(𝑎1	⟨(𝑎1	PROPN
easat-9645	221	11	−	−	PROPN
easat-9645	221	12	𝑏1)2	𝑏1)2	NUM
easat-9645	222	1	+	+	CCONJ
easat-9645	222	2	(	(	PUNCT
easat-9645	222	3	𝑎2	𝑎2	NOUN
easat-9645	222	4	−	−	PROPN
easat-9645	222	5	𝑏2)2	𝑏2)2	ADV
easat-9645	222	6	+	+	PROPN
easat-9645	222	7	.	.	PUNCT
easat-9645	222	8	.	.	PUNCT
easat-9645	222	9	.	.	PUNCT
easat-9645	223	1	+	+	ADV
easat-9645	223	2	(	(	PUNCT
easat-9645	223	3	𝑎𝑛	𝑎𝑛	PROPN
easat-9645	223	4	−	−	PROPN
easat-9645	223	5	𝑏𝑛)2⟩	𝑏𝑛)2⟩	NUM
easat-9645	223	6	1	1	NUM
easat-9645	223	7	2	2	NUM
easat-9645	223	8	firstly	firstly	ADV
easat-9645	223	9	,	,	PUNCT
easat-9645	223	10	we	we	PRON
easat-9645	223	11	will	will	AUX
easat-9645	223	12	prove	prove	VERB
easat-9645	223	13	that	that	SCONJ
easat-9645	223	14	the	the	DET
easat-9645	223	15	euclidean	euclidean	ADJ
easat-9645	223	16	space	space	NOUN
easat-9645	223	17	ℝ𝑛	ℝ𝑛	PROPN
easat-9645	223	18	is	be	AUX
easat-9645	223	19	complete	complete	ADJ
easat-9645	223	20	.	.	PUNCT
easat-9645	224	1	we	we	PRON
easat-9645	224	2	know	know	VERB
easat-9645	224	3	that	that	SCONJ
easat-9645	224	4	the	the	DET
easat-9645	224	5	metric	metric	NOUN
easat-9645	224	6	of	of	ADP
easat-9645	224	7	ℝ𝑛	ℝ𝑛	PROPN
easat-9645	224	8	is	be	AUX
easat-9645	224	9	𝑑(𝑎	𝑑(𝑎	PROPN
easat-9645	224	10	,	,	PUNCT
easat-9645	224	11	𝑏	𝑏	NOUN
easat-9645	224	12	)	)	PUNCT
easat-9645	224	13	=	=	SYM
easat-9645	224	14	(	(	PUNCT
easat-9645	224	15	∑	∑	PUNCT
easat-9645	224	16	(	(	PUNCT
easat-9645	224	17	𝑎𝑖	𝑎𝑖	ADP
easat-9645	224	18	−	−	PROPN
easat-9645	224	19	𝑏𝑖)2𝑛	𝑏𝑖)2𝑛	NOUN
easat-9645	224	20	𝑖=1	𝑖=1	PUNCT
easat-9645	224	21	)	)	PUNCT
easat-9645	224	22	1	1	NUM
easat-9645	224	23	2	2	NUM
easat-9645	224	24	where	where	SCONJ
easat-9645	224	25	𝑎𝑖	𝑎𝑖	ADV
easat-9645	224	26	=	=	NOUN
easat-9645	224	27	𝑎1	𝑎1	PROPN
easat-9645	224	28	,	,	PUNCT
easat-9645	224	29	𝑎2	𝑎2	PROPN
easat-9645	224	30	,	,	PUNCT
easat-9645	224	31	…	…	PUNCT
easat-9645	224	32	…	…	PUNCT
easat-9645	224	33	.	.	PUNCT
easat-9645	224	34	.	.	PUNCT
easat-9645	225	1	,	,	PUNCT
easat-9645	225	2	𝑎𝑛	𝑎𝑛	PRON
easat-9645	225	3	and	and	CCONJ
easat-9645	225	4	𝑏𝑖	𝑏𝑖	ADP
easat-9645	225	5	=	=	PUNCT
easat-9645	225	6	𝑏1	𝑏1	NOUN
easat-9645	225	7	,	,	PUNCT
easat-9645	225	8	𝑏2	𝑏2	NOUN
easat-9645	225	9	…	…	PUNCT
easat-9645	225	10	…	…	PUNCT
easat-9645	225	11	…	…	PUNCT
easat-9645	225	12	,	,	PUNCT
easat-9645	225	13	𝑏𝑛	𝑏𝑛	NOUN
easat-9645	225	14	,	,	PUNCT
easat-9645	225	15	∀	∀	NOUN
easat-9645	225	16	𝑎𝑖	𝑎𝑖	ADJ
easat-9645	225	17	,	,	PUNCT
easat-9645	225	18	𝑏𝑖	𝑏𝑖	ADP
easat-9645	225	19	∈	∈	NOUN
easat-9645	225	20	ℝ	ℝ	PROPN
easat-9645	225	21	let	let	VERB
easat-9645	225	22	<	<	X
easat-9645	225	23	𝑎𝑛	𝑎𝑛	AUX
easat-9645	225	24	>	>	X
easat-9645	225	25	be	be	AUX
easat-9645	225	26	a	a	DET
easat-9645	225	27	cauchy	cauchy	ADJ
easat-9645	225	28	sequence	sequence	NOUN
easat-9645	225	29	in	in	ADP
easat-9645	225	30	ℝ𝑛.	ℝ𝑛.	PROPN
easat-9645	225	31	for	for	ADP
easat-9645	225	32	every	every	DET
easat-9645	225	33	𝜖	𝜖	PROPN
easat-9645	225	34	>	>	X
easat-9645	225	35	0	0	NUM
easat-9645	225	36	∃	∃	PROPN
easat-9645	225	37	𝑚	𝑚	PROPN
easat-9645	225	38	,	,	PUNCT
easat-9645	225	39	𝑟	𝑟	X
easat-9645	225	40	∈	∈	PROPN
easat-9645	225	41	ℕ	ℕ	PROPN
easat-9645	225	42	s.t	s.t	PROPN
easat-9645	225	43	.	.	PUNCT
easat-9645	225	44	𝑑(𝑎𝑚	𝑑(𝑎𝑚	PROPN
easat-9645	225	45	,	,	PUNCT
easat-9645	225	46	𝑎𝑟	𝑎𝑟	NOUN
easat-9645	225	47	)	)	PUNCT
easat-9645	225	48	=	=	SYM
easat-9645	226	1	(	(	PUNCT
easat-9645	226	2	∑	∑	PUNCT
easat-9645	226	3	(	(	PUNCT
easat-9645	226	4	𝑎𝑖	𝑎𝑖	X
easat-9645	226	5	(	(	PUNCT
easat-9645	226	6	𝑚	𝑚	NOUN
easat-9645	226	7	)	)	PUNCT
easat-9645	226	8	−	−	PROPN
easat-9645	226	9	𝑎𝑖	𝑎𝑖	INTJ
easat-9645	226	10	(	(	PUNCT
easat-9645	226	11	𝑟	𝑟	NOUN
easat-9645	226	12	)	)	PUNCT
easat-9645	226	13	)	)	PUNCT
easat-9645	226	14	2	2	NUM
easat-9645	226	15	𝑛	𝑛	PRON
easat-9645	226	16	𝑖=1	𝑖=1	PROPN
easat-9645	226	17	)	)	PUNCT
easat-9645	226	18	1	1	NUM
easat-9645	226	19	2	2	NUM
easat-9645	226	20	<	<	X
easat-9645	226	21	𝜖	𝜖	X
easat-9645	226	22	∀	∀	X
easat-9645	226	23	𝑚	𝑚	NOUN
easat-9645	226	24	,	,	PUNCT
easat-9645	226	25	𝑟	𝑟	X
easat-9645	226	26	∈	∈	PROPN
easat-9645	226	27	ℕ	ℕ	PROPN
easat-9645	226	28	(	(	PUNCT
easat-9645	226	29	2	2	NUM
easat-9645	226	30	)	)	PUNCT
easat-9645	226	31	both	both	DET
easat-9645	226	32	sides	side	NOUN
easat-9645	226	33	squaring	square	VERB
easat-9645	226	34	in(𝑖	in(𝑖	NOUN
easat-9645	226	35	)	)	PUNCT
easat-9645	226	36	,	,	PUNCT
easat-9645	226	37	then	then	ADV
easat-9645	226	38	we	we	PRON
easat-9645	226	39	get	get	VERB
easat-9645	226	40	(	(	PUNCT
easat-9645	226	41	𝑎𝑖	𝑎𝑖	X
easat-9645	226	42	(	(	PUNCT
easat-9645	226	43	𝑚	𝑚	NOUN
easat-9645	226	44	)	)	PUNCT
easat-9645	226	45	−	−	PROPN
easat-9645	226	46	𝑎𝑖	𝑎𝑖	INTJ
easat-9645	226	47	(	(	PUNCT
easat-9645	226	48	𝑟	𝑟	NOUN
easat-9645	226	49	)	)	PUNCT
easat-9645	226	50	)	)	PUNCT
easat-9645	227	1	2	2	NUM
easat-9645	227	2	<	<	X
easat-9645	227	3	𝜖2	𝜖2	X
easat-9645	227	4	∀	∀	X
easat-9645	227	5	𝑖	𝑖	NOUN
easat-9645	227	6	=	=	NOUN
easat-9645	227	7	1,2,3	1,2,3	NUM
easat-9645	227	8	,	,	PUNCT
easat-9645	227	9	…	…	PUNCT
easat-9645	227	10	…	…	PUNCT
easat-9645	227	11	…	…	PUNCT
easat-9645	227	12	,	,	PUNCT
easat-9645	227	13	𝑛	𝑛	DET
easat-9645	227	14	⇒	⇒	NOUN
easat-9645	227	15	|𝑎𝑖	|𝑎𝑖	NUM
easat-9645	227	16	(	(	PUNCT
easat-9645	227	17	𝑚	𝑚	NOUN
easat-9645	227	18	)	)	PUNCT
easat-9645	227	19	−	−	PROPN
easat-9645	227	20	𝑎𝑖	𝑎𝑖	INTJ
easat-9645	227	21	(	(	PUNCT
easat-9645	227	22	𝑟	𝑟	NOUN
easat-9645	227	23	)	)	PUNCT
easat-9645	228	1	|	|	ADV
easat-9645	228	2	<	<	X
easat-9645	228	3	𝜖	𝜖	X
easat-9645	228	4	fixed	fix	VERB
easat-9645	228	5	𝑖	𝑖	SYM
easat-9645	228	6	,	,	PUNCT
easat-9645	228	7	(	(	PUNCT
easat-9645	228	8	1	1	NUM
easat-9645	228	9	≤	≤	NUM
easat-9645	228	10	𝑖	𝑖	SYM
easat-9645	228	11	≤	≤	NOUN
easat-9645	228	12	𝑛	𝑛	NOUN
easat-9645	228	13	)	)	PUNCT
easat-9645	228	14	,	,	PUNCT
easat-9645	228	15	and	and	CCONJ
easat-9645	228	16	<	<	X
easat-9645	228	17	𝑎𝑖	𝑎𝑖	PRON
easat-9645	228	18	1	1	NUM
easat-9645	228	19	,	,	PUNCT
easat-9645	228	20	𝑎𝑖	𝑎𝑖	ADP
easat-9645	228	21	2	2	NUM
easat-9645	228	22	…	…	PUNCT
easat-9645	228	23	…	…	PUNCT
easat-9645	228	24	…	…	PUNCT
easat-9645	228	25	…	…	PUNCT
easat-9645	228	26	>	>	X
easat-9645	228	27	is	be	AUX
easat-9645	228	28	a	a	DET
easat-9645	228	29	cauchy	cauchy	ADJ
easat-9645	228	30	sequence	sequence	NOUN
easat-9645	228	31	of	of	ADP
easat-9645	228	32	ℝ.	ℝ.	PROPN
easat-9645	228	33	so	so	SCONJ
easat-9645	228	34	it	it	PRON
easat-9645	228	35	converges	converge	VERB
easat-9645	228	36	i.e.	i.e.	ADV
easat-9645	228	37	𝑎𝑖	𝑎𝑖	ADP
easat-9645	228	38	𝑚	𝑚	NOUN
easat-9645	228	39	→	→	SYM
easat-9645	228	40	𝑎	𝑎	NOUN
easat-9645	228	41	as	as	ADP
easat-9645	228	42	𝑚	𝑚	PROPN
easat-9645	228	43	→	→	SYM
easat-9645	229	1	∞.	∞.	PROPN
easat-9645	229	2	using	use	VERB
easat-9645	229	3	this	this	PRON
easat-9645	229	4	n	n	NUM
easat-9645	229	5	limits	limit	NOUN
easat-9645	229	6	,	,	PUNCT
easat-9645	229	7	we	we	PRON
easat-9645	229	8	define	define	VERB
easat-9645	229	9	𝑎	𝑎	X
easat-9645	229	10	=	=	PUNCT
easat-9645	229	11	(	(	PUNCT
easat-9645	229	12	𝑎1	𝑎1	INTJ
easat-9645	229	13	,	,	PUNCT
easat-9645	229	14	𝑎2	𝑎2	PROPN
easat-9645	229	15	,	,	PUNCT
easat-9645	229	16	…	…	PUNCT
easat-9645	229	17	…	…	PUNCT
easat-9645	229	18	.	.	PUNCT
easat-9645	229	19	.	.	PUNCT
easat-9645	230	1	,	,	PUNCT
easat-9645	230	2	𝑎𝑛	𝑎𝑛	PROPN
easat-9645	230	3	)	)	PUNCT
easat-9645	230	4	.	.	PUNCT
easat-9645	231	1	clearly	clearly	ADV
easat-9645	231	2	𝑎	𝑎	PRON
easat-9645	231	3	∈	∈	ADJ
easat-9645	231	4	ℝ𝑛	ℝ𝑛	NOUN
easat-9645	231	5	,	,	PUNCT
easat-9645	231	6	from(2	from(2	PROPN
easat-9645	231	7	)	)	PUNCT
easat-9645	231	8	,	,	PUNCT
easat-9645	231	9	𝑎𝑟	𝑎𝑟	X
easat-9645	231	10	→	→	SYM
easat-9645	231	11	𝑎	𝑎	X
easat-9645	231	12	𝑎𝑠	𝑎𝑠	ADP
easat-9645	231	13	𝑟	𝑟	NOUN
easat-9645	231	14	→	→	SYM
easat-9645	231	15	∞	∞	PROPN
easat-9645	231	16	,	,	PUNCT
easat-9645	231	17	then	then	ADV
easat-9645	231	18	we	we	PRON
easat-9645	231	19	have	have	VERB
easat-9645	231	20	𝑑(𝑎𝑚	𝑑(𝑎𝑚	NOUN
easat-9645	231	21	,	,	PUNCT
easat-9645	231	22	𝑎	𝑎	X
easat-9645	231	23	)	)	PUNCT
easat-9645	231	24	≤	≤	X
easat-9645	231	25	𝜖	𝜖	PROPN
easat-9645	231	26	,	,	PUNCT
easat-9645	231	27	𝑚	𝑚	X
easat-9645	231	28	>	>	X
easat-9645	231	29	𝑁	𝑁	PROPN
easat-9645	231	30	so	so	ADV
easat-9645	231	31	,	,	PUNCT
easat-9645	231	32	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
easat-9645	231	33	𝑚→∞	𝑚→∞	X
easat-9645	231	34	𝑎𝑚	𝑎𝑚	NOUN
easat-9645	231	35	=	=	PUNCT
easat-9645	231	36	𝑎.	𝑎.	NOUN
easat-9645	231	37	hence	hence	ADV
easat-9645	231	38	ℝ𝑛	ℝ𝑛	ADV
easat-9645	231	39	is	be	AUX
easat-9645	231	40	complete	complete	ADJ
easat-9645	231	41	.	.	PUNCT
easat-9645	232	1	if	if	SCONJ
easat-9645	232	2	n=3	n=3	NOUN
easat-9645	232	3	,	,	PUNCT
easat-9645	232	4	then	then	ADV
easat-9645	232	5	(	(	PUNCT
easat-9645	232	6	1	1	X
easat-9645	232	7	)	)	PUNCT
easat-9645	232	8	gives	give	VERB
easat-9645	232	9	⟨𝑎	⟨𝑎	NOUN
easat-9645	232	10	,	,	PUNCT
easat-9645	232	11	𝑏⟩	𝑏⟩	PUNCT
easat-9645	233	1	=	=	PUNCT
easat-9645	233	2	𝑎.	𝑎.	NOUN
easat-9645	234	1	𝑏	𝑏	NOUN
easat-9645	234	2	=	=	SYM
easat-9645	234	3	𝑎1𝑏1	𝑎1𝑏1	PROPN
easat-9645	234	4	+	+	CCONJ
easat-9645	234	5	𝑎2𝑏2	𝑎2𝑏2	PUNCT
easat-9645	234	6	+	+	CCONJ
easat-9645	234	7	𝑎3𝑏3	𝑎3𝑏3	X
easat-9645	234	8	of	of	ADP
easat-9645	234	9	𝑎	𝑎	NOUN
easat-9645	234	10	=	=	PUNCT
easat-9645	234	11	(	(	PUNCT
easat-9645	234	12	𝑎1	𝑎1	PROPN
easat-9645	234	13	,	,	PUNCT
easat-9645	234	14	𝑎2	𝑎2	PROPN
easat-9645	234	15	,	,	PUNCT
easat-9645	234	16	𝑎3	𝑎3	PROPN
easat-9645	234	17	)	)	PUNCT
easat-9645	234	18	∈	∈	PROPN
easat-9645	234	19	𝑅	𝑅	PROPN
easat-9645	234	20	and	and	CCONJ
easat-9645	234	21	𝑏	𝑏	NOUN
easat-9645	234	22	=	=	PUNCT
easat-9645	234	23	(	(	PUNCT
easat-9645	234	24	𝑏1	𝑏1	NOUN
easat-9645	234	25	,	,	PUNCT
easat-9645	234	26	𝑏2	𝑏2	NOUN
easat-9645	234	27	,	,	PUNCT
easat-9645	234	28	𝑏3	𝑏3	PROPN
easat-9645	234	29	)	)	PUNCT
easat-9645	234	30	∈	∈	PROPN
easat-9645	234	31	𝑅	𝑅	PROPN
easat-9645	234	32	and	and	CCONJ
easat-9645	234	33	the	the	DET
easat-9645	234	34	orthogonality	orthogonality	NOUN
easat-9645	234	35	⟨𝑎	⟨𝑎	NUM
easat-9645	234	36	,	,	PUNCT
easat-9645	234	37	𝑏⟩	𝑏⟩	PUNCT
easat-9645	234	38	=	=	PUNCT
easat-9645	234	39	𝑎.	𝑎.	NOUN
easat-9645	235	1	𝑏	𝑏	NOUN
easat-9645	235	2	=	=	NOUN
easat-9645	235	3	0	0	NOUN
easat-9645	235	4	this	this	DET
easat-9645	235	5	concept	concept	NOUN
easat-9645	235	6	is	be	AUX
easat-9645	235	7	consistent	consistent	ADJ
easat-9645	235	8	with	with	ADP
easat-9645	235	9	the	the	DET
easat-9645	235	10	fundamental	fundamental	ADJ
easat-9645	235	11	idea	idea	NOUN
easat-9645	235	12	of	of	ADP
easat-9645	235	13	perpendicularity	perpendicularity	NOUN
easat-9645	235	14	,	,	PUNCT
easat-9645	235	15	meaning	mean	VERB
easat-9645	235	16	that	that	SCONJ
easat-9645	235	17	two	two	NUM
easat-9645	235	18	vectors	vector	NOUN
easat-9645	235	19	are	be	AUX
easat-9645	235	20	orthogonal	orthogonal	ADJ
easat-9645	235	21	if	if	SCONJ
easat-9645	235	22	their	their	PRON
easat-9645	235	23	inner	inner	ADJ
easat-9645	235	24	product	product	NOUN
easat-9645	235	25	is	be	AUX
easat-9645	235	26	zero	zero	NUM
easat-9645	235	27	.	.	PUNCT
easat-9645	236	1	therefore	therefore	ADV
easat-9645	236	2	,	,	PUNCT
easat-9645	236	3	the	the	DET
easat-9645	236	4	euclidean	euclidean	ADJ
easat-9645	236	5	space	space	NOUN
easat-9645	236	6	ℝ𝑛	ℝ𝑛	X
easat-9645	236	7	is	be	AUX
easat-9645	236	8	a	a	DET
easat-9645	236	9	hilbert	hilbert	NOUN
easat-9645	236	10	space	space	NOUN
easat-9645	236	11	.	.	PUNCT
easat-9645	237	1	1507	1507	NUM
easat-9645	237	2	edelweiss	edelweiss	PROPN
easat-9645	237	3	applied	apply	VERB
easat-9645	237	4	science	science	NOUN
easat-9645	237	5	and	and	CCONJ
easat-9645	237	6	technology	technology	NOUN
easat-9645	237	7	issn	issn	PROPN
easat-9645	237	8	:	:	PUNCT
easat-9645	237	9	2576	2576	NUM
easat-9645	237	10	-	-	SYM
easat-9645	237	11	8484	8484	NUM
easat-9645	237	12	vol	vol	NOUN
easat-9645	237	13	.	.	PROPN
easat-9645	238	1	9	9	NUM
easat-9645	238	2	,	,	PUNCT
easat-9645	238	3	no	no	INTJ
easat-9645	238	4	.	.	NOUN
easat-9645	238	5	8	8	NUM
easat-9645	238	6	:	:	SYM
easat-9645	238	7	1498	1498	NUM
easat-9645	238	8	-	-	SYM
easat-9645	238	9	1523	1523	NUM
easat-9645	238	10	,	,	PUNCT
easat-9645	238	11	2025	2025	NUM
easat-9645	238	12	doi	doi	NOUN
easat-9645	238	13	:	:	PUNCT
easat-9645	238	14	10.55214/2576	10.55214/2576	NUM
easat-9645	238	15	-	-	SYM
easat-9645	238	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	238	17	©	©	PROPN
easat-9645	238	18	2025	2025	NUM
easat-9645	238	19	by	by	ADP
easat-9645	238	20	the	the	DET
easat-9645	238	21	authors	author	NOUN
easat-9645	238	22	;	;	PUNCT
easat-9645	238	23	licensee	licensee	PROPN
easat-9645	238	24	learning	learn	VERB
easat-9645	238	25	gate	gate	PROPN
easat-9645	238	26	figure	figure	NOUN
easat-9645	238	27	5	5	NUM
easat-9645	238	28	.	.	PUNCT
easat-9645	238	29	matlab	matlab	PROPN
easat-9645	238	30	visualization	visualization	NOUN
easat-9645	238	31	for	for	ADP
easat-9645	238	32	the	the	DET
easat-9645	238	33	euclidean	euclidean	ADJ
easat-9645	238	34	space	space	NOUN
easat-9645	238	35	in	in	ADP
easat-9645	238	36	r2	r2	PROPN
easat-9645	238	37	.	.	PUNCT
easat-9645	239	1	the	the	DET
easat-9645	239	2	figure	figure	NOUN
easat-9645	239	3	5	5	NUM
easat-9645	239	4	,	,	PUNCT
easat-9645	239	5	shows	show	VERB
easat-9645	239	6	vectors	vector	NOUN
easat-9645	239	7	in	in	ADP
easat-9645	239	8	r2	r2	PROPN
easat-9645	239	9	with	with	ADP
easat-9645	239	10	an	an	DET
easat-9645	239	11	angle	angle	NOUN
easat-9645	239	12	of	of	ADP
easat-9645	239	13	45∘.	45∘.	NUM
easat-9645	239	14	the	the	DET
easat-9645	239	15	red	red	ADJ
easat-9645	239	16	vector	vector	NOUN
easat-9645	239	17	x	x	PUNCT
easat-9645	239	18	extends	extend	VERB
easat-9645	239	19	from	from	ADP
easat-9645	239	20	(	(	PUNCT
easat-9645	239	21	0	0	NUM
easat-9645	239	22	,	,	PUNCT
easat-9645	239	23	0	0	NUM
easat-9645	239	24	)	)	PUNCT
easat-9645	239	25	to	to	PART
easat-9645	239	26	approximately	approximately	ADV
easat-9645	239	27	(	(	PUNCT
easat-9645	239	28	3	3	NUM
easat-9645	239	29	,	,	PUNCT
easat-9645	239	30	1	1	NUM
easat-9645	239	31	)	)	PUNCT
easat-9645	239	32	,	,	PUNCT
easat-9645	239	33	the	the	DET
easat-9645	239	34	blue	blue	ADJ
easat-9645	239	35	vector	vector	NOUN
easat-9645	239	36	y	y	PROPN
easat-9645	239	37	extends	extend	VERB
easat-9645	239	38	to	to	ADP
easat-9645	239	39	about	about	ADP
easat-9645	239	40	(	(	PUNCT
easat-9645	239	41	2	2	NUM
easat-9645	239	42	,	,	PUNCT
easat-9645	239	43	2	2	NUM
easat-9645	239	44	)	)	PUNCT
easat-9645	239	45	,	,	PUNCT
easat-9645	239	46	and	and	CCONJ
easat-9645	239	47	the	the	DET
easat-9645	239	48	green	green	ADJ
easat-9645	239	49	vector	vector	NOUN
easat-9645	239	50	represents	represent	VERB
easat-9645	239	51	the	the	DET
easat-9645	239	52	projection	projection	NOUN
easat-9645	239	53	of	of	ADP
easat-9645	239	54	x	x	PUNCT
easat-9645	239	55	onto	onto	ADP
easat-9645	239	56	y	y	PROPN
easat-9645	239	57	,	,	PUNCT
easat-9645	239	58	ending	end	VERB
easat-9645	239	59	at	at	ADP
easat-9645	239	60	around	around	ADV
easat-9645	239	61	(	(	PUNCT
easat-9645	239	62	1.5	1.5	NUM
easat-9645	239	63	,	,	PUNCT
easat-9645	239	64	1.5	1.5	NUM
easat-9645	239	65	)	)	PUNCT
easat-9645	239	66	.	.	PUNCT
easat-9645	240	1	example	example	NOUN
easat-9645	240	2	(	(	PUNCT
easat-9645	240	3	orthogonality	orthogonality	NOUN
easat-9645	240	4	in	in	ADP
easat-9645	240	5	rn	rn	PROPN
easat-9645	240	6	):	):	PUNCT
easat-9645	240	7	in	in	ADP
easat-9645	240	8	r2	r2	PROPN
easat-9645	240	9	,	,	PUNCT
easat-9645	240	10	let	let	VERB
easat-9645	240	11	𝑥	𝑥	X
easat-9645	240	12	=	=	SYM
easat-9645	240	13	(	(	PUNCT
easat-9645	240	14	1,0	1,0	NUM
easat-9645	240	15	)	)	PUNCT
easat-9645	240	16	,	,	PUNCT
easat-9645	240	17	𝑦	𝑦	NOUN
easat-9645	240	18	=	=	SYM
easat-9645	240	19	(	(	PUNCT
easat-9645	240	20	0,1	0,1	NUM
easat-9645	240	21	)	)	PUNCT
easat-9645	240	22	.	.	PUNCT
easat-9645	241	1	then	then	ADV
easat-9645	241	2	⟨𝑥	⟨𝑥	NOUN
easat-9645	241	3	,	,	PUNCT
easat-9645	241	4	𝑦⟩	𝑦⟩	NOUN
easat-9645	241	5	=	=	PUNCT
easat-9645	241	6	1	1	NUM
easat-9645	241	7	⋅	⋅	X
easat-9645	241	8	0	0	NUM
easat-9645	242	1	+	+	CCONJ
easat-9645	242	2	0	0	NUM
easat-9645	242	3	⋅	⋅	NUM
easat-9645	242	4	1	1	NUM
easat-9645	242	5	=	=	SYM
easat-9645	242	6	0	0	NUM
easat-9645	242	7	,	,	PUNCT
easat-9645	242	8	so	so	ADV
easat-9645	242	9	x	x	X
easat-9645	242	10	and	and	CCONJ
easat-9645	242	11	y	y	PROPN
easat-9645	242	12	are	be	AUX
easat-9645	242	13	orthogonal	orthogonal	ADJ
easat-9645	242	14	,	,	PUNCT
easat-9645	242	15	representing	represent	VERB
easat-9645	242	16	perpendicular	perpendicular	ADJ
easat-9645	242	17	vectors	vector	NOUN
easat-9645	242	18	.	.	PUNCT
easat-9645	243	1	example	example	NOUN
easat-9645	243	2	(	(	PUNCT
easat-9645	243	3	completeness	completeness	NOUN
easat-9645	243	4	in	in	ADP
easat-9645	243	5	rn	rn	PROPN
easat-9645	243	6	):	):	PUNCT
easat-9645	243	7	consider	consider	VERB
easat-9645	243	8	a	a	DET
easat-9645	243	9	cauchy	cauchy	ADJ
easat-9645	243	10	sequence	sequence	NOUN
easat-9645	243	11	(	(	PUNCT
easat-9645	243	12	𝑥𝑘	𝑥𝑘	NOUN
easat-9645	243	13	)	)	PUNCT
easat-9645	243	14	𝑖𝑛	𝑖𝑛	NOUN
easat-9645	243	15	𝑅2	𝑅2	NOUN
easat-9645	243	16	,	,	PUNCT
easat-9645	243	17	where	where	SCONJ
easat-9645	243	18	𝑥𝑘	𝑥𝑘	X
easat-9645	243	19	=	=	PUNCT
easat-9645	243	20	(	(	PUNCT
easat-9645	243	21	1	1	NUM
easat-9645	243	22	−	−	PROPN
easat-9645	243	23	1	1	NUM
easat-9645	243	24	𝑘	𝑘	NOUN
easat-9645	243	25	,	,	PUNCT
easat-9645	243	26	1	1	NUM
easat-9645	243	27	𝑘	𝑘	NOUN
easat-9645	243	28	)	)	PUNCT
easat-9645	243	29	.	.	PUNCT
easat-9645	244	1	𝐹𝑜𝑟	𝐹𝑜𝑟	DET
easat-9645	244	2	𝑘	𝑘	X
easat-9645	244	3	>	>	X
easat-9645	244	4	𝑚	𝑚	NOUN
easat-9645	244	5	,	,	PUNCT
easat-9645	244	6	𝑡he	𝑡he	X
easat-9645	244	7	distance	distance	NOUN
easat-9645	244	8	is	be	AUX
easat-9645	244	9	‖𝑥𝑘	‖𝑥𝑘	PROPN
easat-9645	244	10	−	−	NOUN
easat-9645	244	11	𝑥𝑚‖	𝑥𝑚‖	PROPN
easat-9645	244	12	=	=	SYM
easat-9645	244	13	√	√	PROPN
easat-9645	244	14	(	(	PUNCT
easat-9645	244	15	1	1	NUM
easat-9645	244	16	𝑘	𝑘	PRON
easat-9645	244	17	−	−	NOUN
easat-9645	244	18	1	1	NUM
easat-9645	244	19	𝑚	𝑚	NOUN
easat-9645	244	20	)	)	PUNCT
easat-9645	244	21	2	2	NUM
easat-9645	244	22	+	+	CCONJ
easat-9645	244	23	(	(	PUNCT
easat-9645	244	24	1	1	NUM
easat-9645	244	25	𝑚	𝑚	PART
easat-9645	244	26	−	−	PROPN
easat-9645	244	27	1	1	NUM
easat-9645	244	28	𝑘	𝑘	NOUN
easat-9645	244	29	)	)	PUNCT
easat-9645	244	30	2	2	NUM
easat-9645	244	31	,	,	PUNCT
easat-9645	244	32	which	which	PRON
easat-9645	244	33	approaches	approach	VERB
easat-9645	244	34	0	0	NUM
easat-9645	244	35	as	as	ADP
easat-9645	244	36	𝑚	𝑚	PROPN
easat-9645	244	37	,	,	PUNCT
easat-9645	244	38	𝑘	𝑘	X
easat-9645	244	39	→	→	SYM
easat-9645	244	40	∞	∞	NUM
easat-9645	244	41	.	.	PUNCT
easat-9645	245	1	the	the	DET
easat-9645	245	2	sequence	sequence	NOUN
easat-9645	245	3	converges	converge	VERB
easat-9645	245	4	to	to	ADP
easat-9645	245	5	(	(	PUNCT
easat-9645	245	6	1,0	1,0	NUM
easat-9645	245	7	)	)	PUNCT
easat-9645	245	8	∈	∈	NOUN
easat-9645	245	9	𝑅2	𝑅2	NOUN
easat-9645	245	10	,	,	PUNCT
easat-9645	245	11	confirming	confirm	VERB
easat-9645	245	12	completeness	completeness	NOUN
easat-9645	245	13	.	.	PUNCT
easat-9645	246	1	3.2	3.2	NUM
easat-9645	246	2	.	.	PUNCT
easat-9645	246	3	theorem	theorem	NOUN
easat-9645	246	4	prove	prove	VERB
easat-9645	246	5	that	that	SCONJ
easat-9645	246	6	the	the	DET
easat-9645	246	7	unitary	unitary	ADJ
easat-9645	246	8	space	space	NOUN
easat-9645	246	9	𝐶𝑛	𝐶𝑛	PROPN
easat-9645	246	10	is	be	AUX
easat-9645	246	11	a	a	DET
easat-9645	246	12	hilbert	hilbert	NOUN
easat-9645	246	13	space	space	NOUN
easat-9645	246	14	.	.	PUNCT
easat-9645	247	1	proof	proof	NOUN
easat-9645	247	2	:	:	PUNCT
easat-9645	247	3	the	the	DET
easat-9645	247	4	space	space	NOUN
easat-9645	247	5	cn	cn	PROPN
easat-9645	247	6	is	be	AUX
easat-9645	247	7	a	a	DET
easat-9645	247	8	hilbert	hilbert	NOUN
easat-9645	247	9	space	space	NOUN
easat-9645	247	10	with	with	ADP
easat-9645	247	11	inner	inner	ADJ
easat-9645	247	12	product	product	NOUN
easat-9645	247	13	defined	define	VERB
easat-9645	247	14	by	by	ADP
easat-9645	247	15	〈	〈	PROPN
easat-9645	247	16	𝑎	𝑎	NOUN
easat-9645	247	17	,	,	PUNCT
easat-9645	247	18	𝑏	𝑏	DET
easat-9645	247	19	〉	〉	NOUN
easat-9645	247	20	=	=	SYM
easat-9645	247	21	𝑎1𝑏1	𝑎1𝑏1	PROPN
easat-9645	247	22	̅̅̅	̅̅̅	PROPN
easat-9645	247	23	+	+	PROPN
easat-9645	247	24	𝑎2𝑏2	𝑎2𝑏2	PROPN
easat-9645	247	25	̅̅	̅̅	NOUN
easat-9645	247	26	̅	̅	NOUN
easat-9645	247	27	+	+	X
easat-9645	247	28	.	.	PUNCT
easat-9645	247	29	.	.	PUNCT
easat-9645	247	30	.	.	PUNCT
easat-9645	248	1	+	+	ADJ
easat-9645	248	2	𝑎𝑛𝑏𝑛	𝑎𝑛𝑏𝑛	PROPN
easat-9645	248	3	̅̅	̅̅	PROPN
easat-9645	248	4	̅	̅	NOUN
easat-9645	248	5	(	(	PUNCT
easat-9645	248	6	3	3	NUM
easat-9645	248	7	)	)	PUNCT
easat-9645	248	8	where	where	SCONJ
easat-9645	248	9	𝑎	𝑎	NOUN
easat-9645	248	10	=	=	SYM
easat-9645	248	11	(	(	PUNCT
easat-9645	248	12	𝑎𝑖	𝑎𝑖	NOUN
easat-9645	248	13	)	)	PUNCT
easat-9645	248	14	=	=	SYM
easat-9645	248	15	(	(	PUNCT
easat-9645	248	16	𝑎1	𝑎1	PROPN
easat-9645	248	17	,	,	PUNCT
easat-9645	248	18	𝑎2	𝑎2	PROPN
easat-9645	248	19	,	,	PUNCT
easat-9645	248	20	.	.	PUNCT
easat-9645	248	21	.	.	PUNCT
easat-9645	249	1	.	.	PUNCT
easat-9645	250	1	,	,	PUNCT
easat-9645	250	2	𝑎𝑛	𝑎𝑛	PROPN
easat-9645	250	3	)	)	PUNCT
easat-9645	250	4	∈	∈	PROPN
easat-9645	250	5	cn	cn	PROPN
easat-9645	250	6	and	and	CCONJ
easat-9645	250	7	𝑏	𝑏	NOUN
easat-9645	250	8	=	=	PUNCT
easat-9645	250	9	(	(	PUNCT
easat-9645	250	10	𝑏𝑖	𝑏𝑖	NOUN
easat-9645	250	11	)	)	PUNCT
easat-9645	250	12	=	=	SYM
easat-9645	250	13	(	(	PUNCT
easat-9645	250	14	𝑏1	𝑏1	NOUN
easat-9645	250	15	,	,	PUNCT
easat-9645	250	16	𝑏2	𝑏2	PROPN
easat-9645	250	17	,	,	PUNCT
easat-9645	250	18	.	.	PUNCT
easat-9645	250	19	.	.	PUNCT
easat-9645	251	1	.	.	PUNCT
easat-9645	252	1	,	,	PUNCT
easat-9645	252	2	𝑏𝑛	𝑏𝑛	NOUN
easat-9645	252	3	)	)	PUNCT
easat-9645	252	4	∈cn	∈cn	NOUN
easat-9645	252	5	in	in	ADP
easat-9645	252	6	fact	fact	NOUN
easat-9645	252	7	,	,	PUNCT
easat-9645	252	8	from	from	ADP
easat-9645	252	9	(	(	PUNCT
easat-9645	252	10	3	3	X
easat-9645	252	11	)	)	PUNCT
easat-9645	252	12	we	we	PRON
easat-9645	252	13	obtain	obtain	VERB
easat-9645	252	14	,	,	PUNCT
easat-9645	252	15	‖𝑎‖	‖𝑎‖	NOUN
easat-9645	253	1	=	=	SYM
easat-9645	253	2	⟨𝑎	⟨𝑎	NOUN
easat-9645	253	3	,	,	PUNCT
easat-9645	253	4	𝑎⟩	𝑎⟩	X
easat-9645	253	5	1	1	NUM
easat-9645	253	6	2	2	NUM
easat-9645	253	7	=	=	SYM
easat-9645	253	8	(	(	PUNCT
easat-9645	253	9	𝑎1𝑏1	𝑎1𝑏1	X
easat-9645	253	10	+	+	X
easat-9645	253	11	𝑎2𝑏2	𝑎2𝑏2	X
easat-9645	253	12	+	+	NOUN
easat-9645	253	13	.	.	PUNCT
easat-9645	253	14	.	.	PUNCT
easat-9645	253	15	.	.	PUNCT
easat-9645	254	1	+	+	NOUN
easat-9645	254	2	𝑎𝑛𝑏𝑛	𝑎𝑛𝑏𝑛	NOUN
easat-9645	254	3	)	)	PUNCT
easat-9645	254	4	1	1	NUM
easat-9645	254	5	2	2	NUM
easat-9645	254	6	=	=	SYM
easat-9645	254	7	(	(	PUNCT
easat-9645	254	8	|𝑎|1	|𝑎|1	PROPN
easat-9645	254	9	2	2	NUM
easat-9645	254	10	+	+	CCONJ
easat-9645	254	11	|𝑎2|2	|𝑎2|2	PROPN
easat-9645	254	12	+	+	PROPN
easat-9645	254	13	.	.	PUNCT
easat-9645	254	14	.	.	PUNCT
easat-9645	254	15	.	.	PUNCT
easat-9645	255	1	+	+	X
easat-9645	255	2	|𝑎𝑛|2	|𝑎𝑛|2	NOUN
easat-9645	255	3	)	)	PUNCT
easat-9645	255	4	1	1	NUM
easat-9645	255	5	2	2	NUM
easat-9645	255	6	and	and	CCONJ
easat-9645	255	7	from	from	ADP
easat-9645	255	8	this	this	PRON
easat-9645	255	9	the	the	DET
easat-9645	255	10	unitary	unitary	ADJ
easat-9645	255	11	metric	metric	NOUN
easat-9645	255	12	defined	define	VERB
easat-9645	255	13	by	by	ADP
easat-9645	255	14	𝑑(𝑎	𝑑(𝑎	PROPN
easat-9645	255	15	,	,	PUNCT
easat-9645	255	16	𝑏	𝑏	NOUN
easat-9645	255	17	)	)	PUNCT
easat-9645	255	18	=	=	VERB
easat-9645	256	1	‖𝑎	‖𝑎	NOUN
easat-9645	256	2	−	−	PROPN
easat-9645	256	3	𝑏‖	𝑏‖	X
easat-9645	257	1	=	=	PUNCT
easat-9645	257	2	⟨𝑎	⟨𝑎	NOUN
easat-9645	258	1	−	−	PROPN
easat-9645	258	2	𝑏	𝑏	NOUN
easat-9645	258	3	,	,	PUNCT
easat-9645	258	4	𝑎	𝑎	PRON
easat-9645	258	5	−	−	NOUN
easat-9645	258	6	𝑏⟩	𝑏⟩	NOUN
easat-9645	258	7	1	1	NUM
easat-9645	258	8	2	2	NUM
easat-9645	258	9	=	=	NOUN
easat-9645	258	10	⟨|𝑎1	⟨|𝑎1	NOUN
easat-9645	258	11	−	−	PROPN
easat-9645	258	12	𝑏1|2	𝑏1|2	PROPN
easat-9645	258	13	+	+	CCONJ
easat-9645	258	14	|𝑎2	|𝑎2	NOUN
easat-9645	258	15	−	−	PROPN
easat-9645	258	16	𝑏2|2	𝑏2|2	PROPN
easat-9645	258	17	+	+	PROPN
easat-9645	258	18	.	.	PUNCT
easat-9645	258	19	.	.	PUNCT
easat-9645	258	20	.	.	PUNCT
easat-9645	259	1	+	+	ADV
easat-9645	259	2	|𝑎𝑛	|𝑎𝑛	X
easat-9645	259	3	−	−	NUM
easat-9645	259	4	𝑏𝑛|2⟩	𝑏𝑛|2⟩	NUM
easat-9645	259	5	1	1	NUM
easat-9645	259	6	2	2	NUM
easat-9645	259	7	1508	1508	NUM
easat-9645	259	8	edelweiss	edelweiss	PROPN
easat-9645	259	9	applied	apply	VERB
easat-9645	259	10	science	science	NOUN
easat-9645	259	11	and	and	CCONJ
easat-9645	259	12	technology	technology	NOUN
easat-9645	259	13	issn	issn	PROPN
easat-9645	259	14	:	:	PUNCT
easat-9645	259	15	2576	2576	NUM
easat-9645	259	16	-	-	SYM
easat-9645	259	17	8484	8484	NUM
easat-9645	259	18	vol	vol	NOUN
easat-9645	259	19	.	.	PROPN
easat-9645	260	1	9	9	NUM
easat-9645	260	2	,	,	PUNCT
easat-9645	260	3	no	no	INTJ
easat-9645	260	4	.	.	NOUN
easat-9645	260	5	8	8	NUM
easat-9645	260	6	:	:	SYM
easat-9645	260	7	1498	1498	NUM
easat-9645	260	8	-	-	SYM
easat-9645	260	9	1523	1523	NUM
easat-9645	260	10	,	,	PUNCT
easat-9645	260	11	2025	2025	NUM
easat-9645	260	12	doi	doi	NOUN
easat-9645	260	13	:	:	PUNCT
easat-9645	260	14	10.55214/2576	10.55214/2576	NUM
easat-9645	260	15	-	-	SYM
easat-9645	260	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	260	17	©	©	PROPN
easat-9645	260	18	2025	2025	NUM
easat-9645	260	19	by	by	ADP
easat-9645	260	20	the	the	DET
easat-9645	260	21	authors	author	NOUN
easat-9645	260	22	;	;	PUNCT
easat-9645	260	23	licensee	licensee	PROPN
easat-9645	260	24	learning	learning	NOUN
easat-9645	260	25	gate	gate	NOUN
easat-9645	260	26	firstly	firstly	ADV
easat-9645	260	27	,	,	PUNCT
easat-9645	260	28	we	we	PRON
easat-9645	260	29	will	will	AUX
easat-9645	260	30	prove	prove	VERB
easat-9645	260	31	that	that	SCONJ
easat-9645	260	32	the	the	DET
easat-9645	260	33	unitary	unitary	ADJ
easat-9645	260	34	space	space	NOUN
easat-9645	260	35	𝐶𝑛	𝐶𝑛	PROPN
easat-9645	260	36	is	be	AUX
easat-9645	260	37	complete	complete	ADJ
easat-9645	260	38	.	.	PUNCT
easat-9645	261	1	we	we	PRON
easat-9645	261	2	know	know	VERB
easat-9645	261	3	that	that	SCONJ
easat-9645	261	4	the	the	DET
easat-9645	261	5	metric	metric	NOUN
easat-9645	261	6	in	in	ADP
easat-9645	261	7	𝐶𝑛	𝐶𝑛	PROPN
easat-9645	261	8	is	be	AUX
easat-9645	261	9	𝑑(𝑎	𝑑(𝑎	PROPN
easat-9645	261	10	,	,	PUNCT
easat-9645	261	11	𝑏	𝑏	NOUN
easat-9645	261	12	)	)	PUNCT
easat-9645	261	13	=	=	SYM
easat-9645	261	14	(	(	PUNCT
easat-9645	261	15	∑	∑	PROPN
easat-9645	261	16	|𝑎𝑖	|𝑎𝑖	X
easat-9645	261	17	−	−	PROPN
easat-9645	261	18	𝑏𝑖|2𝑛	𝑏𝑖|2𝑛	NOUN
easat-9645	261	19	𝑖=1	𝑖=1	PROPN
easat-9645	261	20	)	)	PUNCT
easat-9645	261	21	1	1	NUM
easat-9645	261	22	2	2	NUM
easat-9645	261	23	where	where	SCONJ
easat-9645	261	24	𝑎𝑖	𝑎𝑖	ADV
easat-9645	261	25	=	=	NOUN
easat-9645	261	26	𝑎1	𝑎1	PROPN
easat-9645	261	27	,	,	PUNCT
easat-9645	261	28	𝑎2	𝑎2	PROPN
easat-9645	261	29	,	,	PUNCT
easat-9645	261	30	…	…	PUNCT
easat-9645	261	31	…	…	PUNCT
easat-9645	261	32	.	.	PUNCT
easat-9645	261	33	.	.	PUNCT
easat-9645	262	1	,	,	PUNCT
easat-9645	262	2	𝑎𝑛	𝑎𝑛	PRON
easat-9645	262	3	and	and	CCONJ
easat-9645	262	4	𝑏𝑖	𝑏𝑖	ADP
easat-9645	262	5	=	=	PUNCT
easat-9645	262	6	𝑏1	𝑏1	NOUN
easat-9645	262	7	,	,	PUNCT
easat-9645	262	8	𝑏2	𝑏2	NOUN
easat-9645	262	9	…	…	PUNCT
easat-9645	262	10	…	…	PUNCT
easat-9645	262	11	…	…	PUNCT
easat-9645	262	12	,	,	PUNCT
easat-9645	262	13	𝑏𝑛	𝑏𝑛	ADP
easat-9645	262	14	let	let	VERB
easat-9645	262	15	<	<	X
easat-9645	262	16	𝑥𝑛	𝑥𝑛	AUX
easat-9645	262	17	>	>	X
easat-9645	262	18	be	be	AUX
easat-9645	262	19	a	a	DET
easat-9645	262	20	cauchy	cauchy	ADJ
easat-9645	262	21	sequence	sequence	NOUN
easat-9645	262	22	in	in	ADP
easat-9645	262	23	𝐶𝑛.	𝐶𝑛.	PROPN
easat-9645	262	24	for	for	ADP
easat-9645	262	25	every	every	DET
easat-9645	262	26	𝜖	𝜖	PROPN
easat-9645	262	27	>	>	X
easat-9645	262	28	0	0	NUM
easat-9645	262	29	∃	∃	PROPN
easat-9645	262	30	𝑚	𝑚	PROPN
easat-9645	262	31	,	,	PUNCT
easat-9645	262	32	𝑛	𝑛	PRON
easat-9645	262	33	∈	∈	PROPN
easat-9645	262	34	ℕ	ℕ	PROPN
easat-9645	262	35	s.	s.	PROPN
easat-9645	262	36	t.	t.	PROPN
easat-9645	262	37	𝑑(𝑎𝑚	𝑑(𝑎𝑚	PROPN
easat-9645	262	38	,	,	PUNCT
easat-9645	262	39	𝑎𝑛	𝑎𝑛	PROPN
easat-9645	262	40	)	)	PUNCT
easat-9645	262	41	=	=	SYM
easat-9645	263	1	(	(	PUNCT
easat-9645	263	2	∑	∑	PUNCT
easat-9645	263	3	(	(	PUNCT
easat-9645	263	4	𝑎𝑖	𝑎𝑖	X
easat-9645	263	5	(	(	PUNCT
easat-9645	263	6	𝑚	𝑚	NOUN
easat-9645	263	7	)	)	PUNCT
easat-9645	263	8	−	−	PROPN
easat-9645	263	9	𝑎𝑖	𝑎𝑖	INTJ
easat-9645	263	10	(	(	PUNCT
easat-9645	263	11	𝑛	𝑛	NOUN
easat-9645	263	12	)	)	PUNCT
easat-9645	263	13	)	)	PUNCT
easat-9645	263	14	2	2	NUM
easat-9645	263	15	𝑛	𝑛	PRON
easat-9645	263	16	𝑖=1	𝑖=1	PROPN
easat-9645	263	17	)	)	PUNCT
easat-9645	263	18	1	1	NUM
easat-9645	263	19	2	2	NUM
easat-9645	263	20	<	<	X
easat-9645	263	21	𝜖	𝜖	X
easat-9645	263	22	∀	∀	X
easat-9645	263	23	𝑚	𝑚	NOUN
easat-9645	263	24	,	,	PUNCT
easat-9645	263	25	𝑛	𝑛	PRON
easat-9645	263	26	∈	∈	PROPN
easat-9645	263	27	ℕ	ℕ	PROPN
easat-9645	263	28	(	(	PUNCT
easat-9645	263	29	4	4	NOUN
easat-9645	263	30	)	)	PUNCT
easat-9645	263	31	both	both	DET
easat-9645	263	32	sides	side	NOUN
easat-9645	263	33	squaring	square	VERB
easat-9645	263	34	in(4	in(4	PROPN
easat-9645	263	35	)	)	PUNCT
easat-9645	263	36	,	,	PUNCT
easat-9645	263	37	then	then	ADV
easat-9645	263	38	we	we	PRON
easat-9645	263	39	get	get	VERB
easat-9645	263	40	(	(	PUNCT
easat-9645	263	41	𝑎𝑖	𝑎𝑖	X
easat-9645	263	42	(	(	PUNCT
easat-9645	263	43	𝑚	𝑚	NOUN
easat-9645	263	44	)	)	PUNCT
easat-9645	263	45	−	−	PROPN
easat-9645	263	46	𝑎𝑖	𝑎𝑖	INTJ
easat-9645	263	47	(	(	PUNCT
easat-9645	263	48	𝑛	𝑛	NOUN
easat-9645	263	49	)	)	PUNCT
easat-9645	263	50	)	)	PUNCT
easat-9645	264	1	2	2	NUM
easat-9645	264	2	<	<	X
easat-9645	264	3	𝜖2	𝜖2	X
easat-9645	264	4	∀	∀	X
easat-9645	264	5	𝑖	𝑖	NOUN
easat-9645	264	6	=	=	NOUN
easat-9645	264	7	1,2,3	1,2,3	NUM
easat-9645	264	8	,	,	PUNCT
easat-9645	264	9	…	…	PUNCT
easat-9645	264	10	…	…	PUNCT
easat-9645	264	11	…	…	PUNCT
easat-9645	264	12	,	,	PUNCT
easat-9645	264	13	𝑛	𝑛	DET
easat-9645	264	14	⇒	⇒	NOUN
easat-9645	264	15	|𝑎𝑖	|𝑎𝑖	NUM
easat-9645	264	16	(	(	PUNCT
easat-9645	264	17	𝑚	𝑚	NOUN
easat-9645	264	18	)	)	PUNCT
easat-9645	264	19	−	−	PROPN
easat-9645	264	20	𝑎𝑖	𝑎𝑖	INTJ
easat-9645	264	21	(	(	PUNCT
easat-9645	264	22	𝑛	𝑛	NOUN
easat-9645	264	23	)	)	PUNCT
easat-9645	264	24	|	|	ADV
easat-9645	264	25	<	<	X
easat-9645	264	26	𝜖	𝜖	X
easat-9645	264	27	fixed	fix	VERB
easat-9645	264	28	𝑖	𝑖	SYM
easat-9645	264	29	,	,	PUNCT
easat-9645	264	30	(	(	PUNCT
easat-9645	264	31	1	1	NUM
easat-9645	264	32	≤	≤	NUM
easat-9645	264	33	𝑖	𝑖	SYM
easat-9645	264	34	≤	≤	NOUN
easat-9645	264	35	𝑛	𝑛	NOUN
easat-9645	264	36	)	)	PUNCT
easat-9645	264	37	,	,	PUNCT
easat-9645	264	38	the	the	DET
easat-9645	264	39	sequence	sequence	NOUN
easat-9645	264	40	<	<	X
easat-9645	264	41	𝑥𝑖	𝑥𝑖	PROPN
easat-9645	264	42	1	1	NUM
easat-9645	264	43	,	,	PUNCT
easat-9645	264	44	𝑥𝑖	𝑥𝑖	ADV
easat-9645	264	45	2	2	NUM
easat-9645	264	46	…	…	PUNCT
easat-9645	264	47	…	…	PUNCT
easat-9645	264	48	…	…	PUNCT
easat-9645	264	49	…	…	PUNCT
easat-9645	264	50	>	>	X
easat-9645	264	51	is	be	AUX
easat-9645	264	52	a	a	DET
easat-9645	264	53	cauchy	cauchy	ADJ
easat-9645	264	54	sequence	sequence	NOUN
easat-9645	264	55	of	of	ADP
easat-9645	264	56	𝐶.	𝐶.	PROPN
easat-9645	264	57	so	so	SCONJ
easat-9645	264	58	it	it	PRON
easat-9645	264	59	converges	converge	VERB
easat-9645	264	60	i.e.	i.e.	ADV
easat-9645	264	61	𝑎𝑖	𝑎𝑖	ADP
easat-9645	264	62	𝑚	𝑚	NOUN
easat-9645	264	63	→	→	SYM
easat-9645	264	64	𝑎	𝑎	NOUN
easat-9645	264	65	as	as	ADP
easat-9645	264	66	𝑚	𝑚	PROPN
easat-9645	264	67	→	→	SYM
easat-9645	265	1	∞.	∞.	PROPN
easat-9645	265	2	using	use	VERB
easat-9645	265	3	this	this	PRON
easat-9645	265	4	n	n	NUM
easat-9645	265	5	limits	limit	NOUN
easat-9645	265	6	,	,	PUNCT
easat-9645	265	7	we	we	PRON
easat-9645	265	8	define	define	VERB
easat-9645	265	9	𝑎	𝑎	X
easat-9645	265	10	=	=	PUNCT
easat-9645	265	11	(	(	PUNCT
easat-9645	265	12	𝑎1	𝑎1	INTJ
easat-9645	265	13	,	,	PUNCT
easat-9645	265	14	𝑎2	𝑎2	PROPN
easat-9645	265	15	,	,	PUNCT
easat-9645	265	16	…	…	PUNCT
easat-9645	265	17	…	…	PUNCT
easat-9645	265	18	.	.	PUNCT
easat-9645	265	19	.	.	PUNCT
easat-9645	266	1	,	,	PUNCT
easat-9645	266	2	𝑎𝑛	𝑎𝑛	PROPN
easat-9645	266	3	)	)	PUNCT
easat-9645	266	4	.	.	PUNCT
easat-9645	267	1	clearly	clearly	ADV
easat-9645	267	2	𝑎	𝑎	X
easat-9645	267	3	∈	∈	PROPN
easat-9645	267	4	𝐶𝑛	𝐶𝑛	PROPN
easat-9645	267	5	,	,	PUNCT
easat-9645	267	6	from(2	from(2	PROPN
easat-9645	267	7	)	)	PUNCT
easat-9645	267	8	,	,	PUNCT
easat-9645	267	9	𝑎𝑛	𝑎𝑛	PRON
easat-9645	267	10	→	→	SYM
easat-9645	267	11	𝑎	𝑎	X
easat-9645	267	12	𝑎𝑠	𝑎𝑠	NOUN
easat-9645	267	13	𝑛	𝑛	PROPN
easat-9645	267	14	→	→	SYM
easat-9645	267	15	∞	∞	PROPN
easat-9645	267	16	,	,	PUNCT
easat-9645	267	17	then	then	ADV
easat-9645	267	18	we	we	PRON
easat-9645	267	19	have	have	VERB
easat-9645	267	20	𝑑(𝑎𝑚	𝑑(𝑎𝑚	NOUN
easat-9645	267	21	,	,	PUNCT
easat-9645	267	22	𝑎	𝑎	X
easat-9645	267	23	)	)	PUNCT
easat-9645	267	24	≤	≤	X
easat-9645	267	25	𝜖	𝜖	PROPN
easat-9645	267	26	,	,	PUNCT
easat-9645	267	27	𝑚	𝑚	X
easat-9645	267	28	>	>	X
easat-9645	267	29	𝑁	𝑁	PROPN
easat-9645	267	30	so	so	ADV
easat-9645	267	31	,	,	PUNCT
easat-9645	267	32	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
easat-9645	267	33	𝑚→∞	𝑚→∞	X
easat-9645	267	34	𝑎𝑚	𝑎𝑚	NOUN
easat-9645	267	35	=	=	PUNCT
easat-9645	267	36	𝑎.	𝑎.	NOUN
easat-9645	267	37	hence	hence	ADV
easat-9645	267	38	the	the	DET
easat-9645	267	39	unitary	unitary	ADJ
easat-9645	267	40	space	space	NOUN
easat-9645	267	41	𝐶𝑛	𝐶𝑛	PROPN
easat-9645	267	42	is	be	AUX
easat-9645	267	43	complete	complete	ADJ
easat-9645	267	44	if	if	SCONJ
easat-9645	267	45	n=3	n=3	NOUN
easat-9645	267	46	,	,	PUNCT
easat-9645	267	47	then	then	ADV
easat-9645	267	48	(	(	PUNCT
easat-9645	267	49	3	3	X
easat-9645	267	50	)	)	PUNCT
easat-9645	267	51	gives	give	VERB
easat-9645	267	52	⟨𝑎	⟨𝑎	NOUN
easat-9645	267	53	,	,	PUNCT
easat-9645	267	54	𝑏⟩	𝑏⟩	PUNCT
easat-9645	267	55	=	=	PUNCT
easat-9645	267	56	𝑎.	𝑎.	NOUN
easat-9645	268	1	𝑏	𝑏	NOUN
easat-9645	268	2	=	=	SYM
easat-9645	268	3	𝑎1𝑏1	𝑎1𝑏1	PROPN
easat-9645	268	4	+	+	CCONJ
easat-9645	268	5	𝑎2𝑏2	𝑎2𝑏2	PUNCT
easat-9645	268	6	+	+	CCONJ
easat-9645	268	7	𝑎3𝑏3	𝑎3𝑏3	X
easat-9645	268	8	of	of	ADP
easat-9645	268	9	𝑎	𝑎	NOUN
easat-9645	268	10	=	=	PUNCT
easat-9645	268	11	(	(	PUNCT
easat-9645	268	12	𝑎1	𝑎1	PROPN
easat-9645	268	13	,	,	PUNCT
easat-9645	268	14	𝑎2	𝑎2	PROPN
easat-9645	268	15	,	,	PUNCT
easat-9645	268	16	𝑎3	𝑎3	PROPN
easat-9645	268	17	)	)	PUNCT
easat-9645	268	18	∈	∈	PROPN
easat-9645	268	19	𝐶	𝐶	PROPN
easat-9645	268	20	and	and	CCONJ
easat-9645	268	21	𝑏	𝑏	NOUN
easat-9645	268	22	=	=	PUNCT
easat-9645	268	23	(	(	PUNCT
easat-9645	268	24	𝑏1	𝑏1	NOUN
easat-9645	268	25	,	,	PUNCT
easat-9645	268	26	𝑏2	𝑏2	NOUN
easat-9645	268	27	,	,	PUNCT
easat-9645	268	28	𝑏3	𝑏3	PROPN
easat-9645	268	29	)	)	PUNCT
easat-9645	268	30	∈	∈	PROPN
easat-9645	268	31	𝐶	𝐶	PROPN
easat-9645	268	32	and	and	CCONJ
easat-9645	268	33	the	the	DET
easat-9645	268	34	orthogonality	orthogonality	NOUN
easat-9645	268	35	⟨𝑎	⟨𝑎	NUM
easat-9645	268	36	,	,	PUNCT
easat-9645	268	37	𝑏⟩	𝑏⟩	PUNCT
easat-9645	268	38	=	=	PUNCT
easat-9645	268	39	𝑎.	𝑎.	NOUN
easat-9645	269	1	𝑏	𝑏	NOUN
easat-9645	269	2	=	=	NOUN
easat-9645	269	3	0	0	PROPN
easat-9645	269	4	.	.	PUNCT
easat-9645	270	1	this	this	DET
easat-9645	270	2	concept	concept	NOUN
easat-9645	270	3	is	be	AUX
easat-9645	270	4	consistent	consistent	ADJ
easat-9645	270	5	with	with	ADP
easat-9645	270	6	the	the	DET
easat-9645	270	7	fundamental	fundamental	ADJ
easat-9645	270	8	idea	idea	NOUN
easat-9645	270	9	of	of	ADP
easat-9645	270	10	orthogonality	orthogonality	NOUN
easat-9645	270	11	,	,	PUNCT
easat-9645	270	12	meaning	mean	VERB
easat-9645	270	13	that	that	SCONJ
easat-9645	270	14	two	two	NUM
easat-9645	270	15	vectors	vector	NOUN
easat-9645	270	16	are	be	AUX
easat-9645	270	17	orthogonal	orthogonal	ADJ
easat-9645	270	18	if	if	SCONJ
easat-9645	270	19	their	their	PRON
easat-9645	270	20	inner	inner	ADJ
easat-9645	270	21	product	product	NOUN
easat-9645	270	22	is	be	AUX
easat-9645	270	23	zero	zero	NUM
easat-9645	270	24	.	.	PUNCT
easat-9645	271	1	therefore	therefore	ADV
easat-9645	271	2	,	,	PUNCT
easat-9645	271	3	the	the	DET
easat-9645	271	4	unitary	unitary	ADJ
easat-9645	271	5	space	space	NOUN
easat-9645	271	6	𝐶𝑛	𝐶𝑛	PROPN
easat-9645	271	7	is	be	AUX
easat-9645	271	8	a	a	DET
easat-9645	271	9	hilbert	hilbert	NOUN
easat-9645	271	10	space	space	NOUN
easat-9645	271	11	.	.	PUNCT
easat-9645	272	1	figure	figure	NOUN
easat-9645	272	2	6	6	NUM
easat-9645	272	3	.	.	PUNCT
easat-9645	272	4	matlab	matlab	PROPN
easat-9645	272	5	visualization	visualization	NOUN
easat-9645	272	6	for	for	ADP
easat-9645	272	7	the	the	DET
easat-9645	272	8	unitary	unitary	ADJ
easat-9645	272	9	space	space	NOUN
easat-9645	272	10	in	in	ADP
easat-9645	272	11	c2	c2	PROPN
easat-9645	272	12	.	.	PUNCT
easat-9645	273	1	1509	1509	NUM
easat-9645	273	2	edelweiss	edelweiss	PROPN
easat-9645	273	3	applied	apply	VERB
easat-9645	273	4	science	science	NOUN
easat-9645	273	5	and	and	CCONJ
easat-9645	273	6	technology	technology	NOUN
easat-9645	273	7	issn	issn	PROPN
easat-9645	273	8	:	:	PUNCT
easat-9645	273	9	2576	2576	NUM
easat-9645	273	10	-	-	SYM
easat-9645	273	11	8484	8484	NUM
easat-9645	273	12	vol	vol	NOUN
easat-9645	273	13	.	.	PROPN
easat-9645	274	1	9	9	NUM
easat-9645	274	2	,	,	PUNCT
easat-9645	274	3	no	no	INTJ
easat-9645	274	4	.	.	NOUN
easat-9645	274	5	8	8	NUM
easat-9645	274	6	:	:	SYM
easat-9645	274	7	1498	1498	NUM
easat-9645	274	8	-	-	SYM
easat-9645	274	9	1523	1523	NUM
easat-9645	274	10	,	,	PUNCT
easat-9645	274	11	2025	2025	NUM
easat-9645	274	12	doi	doi	NOUN
easat-9645	274	13	:	:	PUNCT
easat-9645	274	14	10.55214/2576	10.55214/2576	NUM
easat-9645	274	15	-	-	SYM
easat-9645	274	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	274	17	©	©	PROPN
easat-9645	274	18	2025	2025	NUM
easat-9645	274	19	by	by	ADP
easat-9645	274	20	the	the	DET
easat-9645	274	21	authors	author	NOUN
easat-9645	274	22	;	;	PUNCT
easat-9645	274	23	licensee	licensee	PROPN
easat-9645	274	24	learning	learning	NOUN
easat-9645	274	25	gate	gate	VERB
easat-9645	274	26	the	the	DET
easat-9645	274	27	figure	figure	NOUN
easat-9645	274	28	6	6	NUM
easat-9645	274	29	,	,	PUNCT
easat-9645	274	30	depicts	depict	VERB
easat-9645	274	31	complex	complex	ADJ
easat-9645	274	32	vectors	vector	NOUN
easat-9645	274	33	in	in	ADP
easat-9645	274	34	c2	c2	PROPN
easat-9645	274	35	with	with	ADP
easat-9645	274	36	an	an	DET
easat-9645	274	37	angle	angle	NOUN
easat-9645	274	38	of	of	ADP
easat-9645	274	39	68.180	68.180	NUM
easat-9645	274	40	.	.	PUNCT
easat-9645	275	1	the	the	DET
easat-9645	275	2	red	red	ADJ
easat-9645	275	3	solid	solid	ADJ
easat-9645	275	4	line	line	NOUN
easat-9645	275	5	represents	represent	VERB
easat-9645	275	6	the	the	DET
easat-9645	275	7	real	real	ADJ
easat-9645	275	8	part	part	NOUN
easat-9645	275	9	of	of	ADP
easat-9645	275	10	z	z	NOUN
easat-9645	275	11	(	(	PUNCT
easat-9645	275	12	re(z	re(z	NOUN
easat-9645	275	13	)	)	PUNCT
easat-9645	275	14	)	)	PUNCT
easat-9645	275	15	,	,	PUNCT
easat-9645	275	16	the	the	DET
easat-9645	275	17	blue	blue	ADJ
easat-9645	275	18	dashed	dash	VERB
easat-9645	275	19	line	line	NOUN
easat-9645	275	20	represents	represent	VERB
easat-9645	275	21	the	the	DET
easat-9645	275	22	real	real	ADJ
easat-9645	275	23	part	part	NOUN
easat-9645	275	24	of	of	ADP
easat-9645	275	25	w	w	PROPN
easat-9645	275	26	(	(	PUNCT
easat-9645	275	27	re(w	re(w	ADJ
easat-9645	275	28	)	)	PUNCT
easat-9645	275	29	)	)	PUNCT
easat-9645	275	30	,	,	PUNCT
easat-9645	275	31	the	the	DET
easat-9645	275	32	red	red	ADJ
easat-9645	275	33	dashed	dash	VERB
easat-9645	275	34	line	line	NOUN
easat-9645	275	35	represents	represent	VERB
easat-9645	275	36	the	the	DET
easat-9645	275	37	imaginary	imaginary	ADJ
easat-9645	275	38	part	part	NOUN
easat-9645	275	39	of	of	ADP
easat-9645	275	40	z	z	PROPN
easat-9645	275	41	(	(	PUNCT
easat-9645	275	42	im(z	im(z	NOUN
easat-9645	275	43	)	)	PUNCT
easat-9645	275	44	)	)	PUNCT
easat-9645	275	45	,	,	PUNCT
easat-9645	275	46	and	and	CCONJ
easat-9645	275	47	the	the	DET
easat-9645	275	48	blue	blue	ADJ
easat-9645	275	49	solid	solid	ADJ
easat-9645	275	50	line	line	NOUN
easat-9645	275	51	represents	represent	VERB
easat-9645	275	52	the	the	DET
easat-9645	275	53	imaginary	imaginary	ADJ
easat-9645	275	54	part	part	NOUN
easat-9645	275	55	of	of	ADP
easat-9645	275	56	w	w	PROPN
easat-9645	275	57	(	(	PUNCT
easat-9645	275	58	im(w	im(w	X
easat-9645	275	59	)	)	PUNCT
easat-9645	275	60	)	)	PUNCT
easat-9645	275	61	,	,	PUNCT
easat-9645	275	62	showing	show	VERB
easat-9645	275	63	their	their	PRON
easat-9645	275	64	components	component	NOUN
easat-9645	275	65	in	in	ADP
easat-9645	275	66	a	a	DET
easat-9645	275	67	2d	2d	NUM
easat-9645	275	68	plane	plane	NOUN
easat-9645	275	69	.	.	PUNCT
easat-9645	276	1	example	example	NOUN
easat-9645	276	2	(	(	PUNCT
easat-9645	276	3	orthogonality	orthogonality	NOUN
easat-9645	276	4	in	in	ADP
easat-9645	276	5	𝐶𝑛	𝐶𝑛	PROPN
easat-9645	276	6	):	):	PUNCT
easat-9645	276	7	in	in	ADV
easat-9645	276	8	𝐶2	𝐶2	ADV
easat-9645	276	9	,	,	PUNCT
easat-9645	276	10	let	let	VERB
easat-9645	276	11	𝑥	𝑥	VERB
easat-9645	276	12	=	=	SYM
easat-9645	276	13	(	(	PUNCT
easat-9645	276	14	1	1	NUM
easat-9645	276	15	,	,	PUNCT
easat-9645	276	16	𝑖	𝑖	ADJ
easat-9645	276	17	)	)	PUNCT
easat-9645	276	18	,	,	PUNCT
easat-9645	276	19	𝑦	𝑦	NOUN
easat-9645	276	20	=	=	SYM
easat-9645	276	21	(	(	PUNCT
easat-9645	276	22	𝑖	𝑖	NOUN
easat-9645	276	23	,	,	PUNCT
easat-9645	276	24	−1	−1	NOUN
easat-9645	276	25	)	)	PUNCT
easat-9645	276	26	.	.	PUNCT
easat-9645	277	1	then	then	ADV
easat-9645	277	2	⟨𝑥	⟨𝑥	NOUN
easat-9645	277	3	,	,	PUNCT
easat-9645	277	4	𝑦⟩	𝑦⟩	NOUN
easat-9645	277	5	=	=	SYM
easat-9645	277	6	1	1	X
easat-9645	277	7	·	·	PUNCT
easat-9645	277	8	(	(	PUNCT
easat-9645	277	9	−𝑖	−𝑖	PROPN
easat-9645	277	10	)	)	PUNCT
easat-9645	277	11	+	+	PUNCT
easat-9645	277	12	𝑖	𝑖	SYM
easat-9645	277	13	·	·	PUNCT
easat-9645	277	14	1	1	NUM
easat-9645	277	15	=	=	SYM
easat-9645	277	16	0	0	NUM
easat-9645	277	17	.	.	PUNCT
easat-9645	278	1	so	so	ADV
easat-9645	278	2	𝑥	𝑥	X
easat-9645	278	3	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-9645	278	4	(	(	PUNCT
easat-9645	278	5	𝑖	𝑖	SYM
easat-9645	278	6	,	,	PUNCT
easat-9645	278	7	1	1	NUM
easat-9645	278	8	)	)	PUNCT
easat-9645	278	9	are	be	AUX
easat-9645	278	10	orthogonal	orthogonal	ADJ
easat-9645	278	11	.	.	PUNCT
easat-9645	279	1	3.3	3.3	NUM
easat-9645	279	2	.	.	PUNCT
easat-9645	279	3	theorem	theorem	NOUN
easat-9645	279	4	prove	prove	VERB
easat-9645	279	5	that	that	SCONJ
easat-9645	279	6	the	the	DET
easat-9645	279	7	space	space	NOUN
easat-9645	279	8	𝑙2	𝑙2	PROPN
easat-9645	279	9	is	be	AUX
easat-9645	279	10	a	a	DET
easat-9645	279	11	hilbert	hilbert	NOUN
easat-9645	279	12	space	space	NOUN
easat-9645	279	13	.	.	PUNCT
easat-9645	280	1	proof	proof	NOUN
easat-9645	280	2	:	:	PUNCT
easat-9645	280	3	the	the	DET
easat-9645	280	4	space	space	NOUN
easat-9645	280	5	𝑙2	𝑙2	PROPN
easat-9645	280	6	is	be	AUX
easat-9645	280	7	a	a	DET
easat-9645	280	8	hilbert	hilbert	NOUN
easat-9645	280	9	space	space	NOUN
easat-9645	280	10	with	with	ADP
easat-9645	280	11	inner	inner	ADJ
easat-9645	280	12	product	product	NOUN
easat-9645	280	13	defined	define	VERB
easat-9645	280	14	by	by	ADP
easat-9645	280	15	〈	〈	PROPN
easat-9645	280	16	𝑎	𝑎	NOUN
easat-9645	280	17	,	,	PUNCT
easat-9645	280	18	𝑏	𝑏	DET
easat-9645	280	19	〉	〉	NOUN
easat-9645	280	20	=	=	SYM
easat-9645	280	21	𝑎1𝑏1	𝑎1𝑏1	PROPN
easat-9645	280	22	̅̅̅	̅̅̅	PROPN
easat-9645	280	23	+	+	PROPN
easat-9645	280	24	𝑎2𝑏2	𝑎2𝑏2	PROPN
easat-9645	280	25	̅̅	̅̅	NOUN
easat-9645	280	26	̅	̅	NOUN
easat-9645	280	27	+	+	X
easat-9645	280	28	.	.	PUNCT
easat-9645	280	29	.	.	PUNCT
easat-9645	280	30	.	.	PUNCT
easat-9645	281	1	+	+	ADJ
easat-9645	281	2	𝑎𝑛𝑏𝑛	𝑎𝑛𝑏𝑛	PROPN
easat-9645	281	3	̅̅	̅̅	PROPN
easat-9645	281	4	̅	̅	NOUN
easat-9645	281	5	(	(	PUNCT
easat-9645	281	6	5	5	NUM
easat-9645	281	7	)	)	PUNCT
easat-9645	281	8	where	where	SCONJ
easat-9645	281	9	𝑎	𝑎	NOUN
easat-9645	281	10	=	=	SYM
easat-9645	281	11	(	(	PUNCT
easat-9645	281	12	𝑎𝑖	𝑎𝑖	NOUN
easat-9645	281	13	)	)	PUNCT
easat-9645	281	14	=	=	SYM
easat-9645	281	15	(	(	PUNCT
easat-9645	281	16	𝑎1	𝑎1	PROPN
easat-9645	281	17	,	,	PUNCT
easat-9645	281	18	𝑎2	𝑎2	PROPN
easat-9645	281	19	,	,	PUNCT
easat-9645	281	20	.	.	PUNCT
easat-9645	281	21	.	.	PUNCT
easat-9645	282	1	.	.	PUNCT
easat-9645	283	1	,	,	PUNCT
easat-9645	283	2	𝑎𝑛	𝑎𝑛	PROPN
easat-9645	283	3	)	)	PUNCT
easat-9645	283	4	∈	∈	PROPN
easat-9645	283	5	𝑙2	𝑙2	PROPN
easat-9645	283	6	and	and	CCONJ
easat-9645	283	7	𝑏	𝑏	NOUN
easat-9645	283	8	=	=	PUNCT
easat-9645	283	9	(	(	PUNCT
easat-9645	283	10	𝑏𝑖	𝑏𝑖	NOUN
easat-9645	283	11	)	)	PUNCT
easat-9645	283	12	=	=	SYM
easat-9645	283	13	(	(	PUNCT
easat-9645	283	14	𝑏1	𝑏1	NOUN
easat-9645	283	15	,	,	PUNCT
easat-9645	283	16	𝑏2	𝑏2	PROPN
easat-9645	283	17	,	,	PUNCT
easat-9645	283	18	.	.	PUNCT
easat-9645	283	19	.	.	PUNCT
easat-9645	284	1	.	.	PUNCT
easat-9645	285	1	,	,	PUNCT
easat-9645	285	2	𝑏𝑛	𝑏𝑛	NOUN
easat-9645	285	3	)	)	PUNCT
easat-9645	285	4	∈	∈	PROPN
easat-9645	285	5	𝑙2	𝑙2	PROPN
easat-9645	285	6	in	in	ADP
easat-9645	285	7	fact	fact	NOUN
easat-9645	285	8	,	,	PUNCT
easat-9645	285	9	from	from	ADP
easat-9645	285	10	(	(	PUNCT
easat-9645	285	11	3	3	X
easat-9645	285	12	)	)	PUNCT
easat-9645	285	13	we	we	PRON
easat-9645	285	14	obtain	obtain	VERB
easat-9645	285	15	,	,	PUNCT
easat-9645	285	16	‖𝑎‖	‖𝑎‖	NOUN
easat-9645	285	17	=	=	SYM
easat-9645	286	1	⟨𝑎	⟨𝑎	NOUN
easat-9645	286	2	,	,	PUNCT
easat-9645	286	3	𝑎⟩	𝑎⟩	X
easat-9645	286	4	1	1	NUM
easat-9645	286	5	2	2	NUM
easat-9645	286	6	=	=	SYM
easat-9645	286	7	(	(	PUNCT
easat-9645	286	8	𝑎1𝑏1	𝑎1𝑏1	X
easat-9645	286	9	+	+	X
easat-9645	286	10	𝑎2𝑏2	𝑎2𝑏2	X
easat-9645	286	11	+	+	NOUN
easat-9645	286	12	.	.	PUNCT
easat-9645	286	13	.	.	PUNCT
easat-9645	286	14	.	.	PUNCT
easat-9645	287	1	+	+	NOUN
easat-9645	287	2	𝑎𝑛𝑏𝑛	𝑎𝑛𝑏𝑛	NOUN
easat-9645	287	3	)	)	PUNCT
easat-9645	287	4	1	1	NUM
easat-9645	287	5	2	2	NUM
easat-9645	287	6	=	=	SYM
easat-9645	287	7	(	(	PUNCT
easat-9645	287	8	|𝑎|1	|𝑎|1	PROPN
easat-9645	287	9	2	2	NUM
easat-9645	287	10	+	+	CCONJ
easat-9645	287	11	|𝑎2|2	|𝑎2|2	PROPN
easat-9645	287	12	+	+	PROPN
easat-9645	287	13	.	.	PUNCT
easat-9645	287	14	.	.	PUNCT
easat-9645	287	15	.	.	PUNCT
easat-9645	288	1	+	+	X
easat-9645	288	2	|𝑎𝑛|2	|𝑎𝑛|2	NOUN
easat-9645	288	3	)	)	PUNCT
easat-9645	288	4	1	1	NUM
easat-9645	288	5	2	2	NUM
easat-9645	288	6	and	and	CCONJ
easat-9645	288	7	from	from	ADP
easat-9645	288	8	this	this	PRON
easat-9645	288	9	the	the	DET
easat-9645	288	10	unitary	unitary	ADJ
easat-9645	288	11	metric	metric	NOUN
easat-9645	288	12	defined	define	VERB
easat-9645	288	13	by	by	ADP
easat-9645	288	14	𝑑(𝑎	𝑑(𝑎	PROPN
easat-9645	288	15	,	,	PUNCT
easat-9645	288	16	𝑏	𝑏	NOUN
easat-9645	288	17	)	)	PUNCT
easat-9645	288	18	=	=	VERB
easat-9645	289	1	‖𝑎	‖𝑎	NOUN
easat-9645	289	2	−	−	PROPN
easat-9645	289	3	𝑏‖	𝑏‖	X
easat-9645	290	1	=	=	PUNCT
easat-9645	290	2	⟨𝑎	⟨𝑎	NOUN
easat-9645	291	1	−	−	PROPN
easat-9645	291	2	𝑏	𝑏	NOUN
easat-9645	291	3	,	,	PUNCT
easat-9645	291	4	𝑎	𝑎	PRON
easat-9645	291	5	−	−	NOUN
easat-9645	291	6	𝑏⟩	𝑏⟩	NOUN
easat-9645	291	7	1	1	NUM
easat-9645	291	8	2	2	NUM
easat-9645	291	9	=	=	NOUN
easat-9645	291	10	⟨|𝑎1	⟨|𝑎1	NOUN
easat-9645	291	11	−	−	PROPN
easat-9645	291	12	𝑏1|2	𝑏1|2	PROPN
easat-9645	291	13	+	+	CCONJ
easat-9645	291	14	|𝑎2	|𝑎2	NOUN
easat-9645	291	15	−	−	PROPN
easat-9645	291	16	𝑏2|2	𝑏2|2	PROPN
easat-9645	291	17	+	+	PROPN
easat-9645	291	18	.	.	PUNCT
easat-9645	291	19	.	.	PUNCT
easat-9645	291	20	.	.	PUNCT
easat-9645	292	1	+	+	ADV
easat-9645	292	2	|𝑎𝑛	|𝑎𝑛	X
easat-9645	292	3	−	−	NUM
easat-9645	292	4	𝑏𝑛|2⟩	𝑏𝑛|2⟩	NUM
easat-9645	292	5	1	1	NUM
easat-9645	292	6	2	2	NUM
easat-9645	292	7	firstly	firstly	ADV
easat-9645	292	8	,	,	PUNCT
easat-9645	292	9	we	we	PRON
easat-9645	292	10	will	will	AUX
easat-9645	292	11	prove	prove	VERB
easat-9645	292	12	that	that	SCONJ
easat-9645	292	13	the	the	DET
easat-9645	292	14	unitary	unitary	ADJ
easat-9645	292	15	space	space	NOUN
easat-9645	292	16	𝑙2	𝑙2	NOUN
easat-9645	292	17	is	be	AUX
easat-9645	292	18	complete	complete	ADJ
easat-9645	292	19	.	.	PUNCT
easat-9645	293	1	we	we	PRON
easat-9645	293	2	know	know	VERB
easat-9645	293	3	that	that	SCONJ
easat-9645	293	4	the	the	DET
easat-9645	293	5	metric	metric	NOUN
easat-9645	293	6	in	in	ADP
easat-9645	293	7	𝐶𝑛	𝐶𝑛	PROPN
easat-9645	293	8	is	be	AUX
easat-9645	293	9	𝑑(𝑎	𝑑(𝑎	PROPN
easat-9645	293	10	,	,	PUNCT
easat-9645	293	11	𝑏	𝑏	NOUN
easat-9645	293	12	)	)	PUNCT
easat-9645	293	13	=	=	SYM
easat-9645	293	14	(	(	PUNCT
easat-9645	293	15	∑	∑	PROPN
easat-9645	293	16	|𝑎𝑖	|𝑎𝑖	X
easat-9645	293	17	−	−	PROPN
easat-9645	293	18	𝑏𝑖|2𝑛	𝑏𝑖|2𝑛	NOUN
easat-9645	293	19	𝑖=1	𝑖=1	PROPN
easat-9645	293	20	)	)	PUNCT
easat-9645	293	21	1	1	NUM
easat-9645	293	22	2	2	NUM
easat-9645	293	23	where	where	SCONJ
easat-9645	293	24	𝑎𝑖	𝑎𝑖	ADV
easat-9645	293	25	=	=	NOUN
easat-9645	293	26	𝑎1	𝑎1	PROPN
easat-9645	293	27	,	,	PUNCT
easat-9645	293	28	𝑎2	𝑎2	PROPN
easat-9645	293	29	,	,	PUNCT
easat-9645	293	30	…	…	PUNCT
easat-9645	293	31	…	…	PUNCT
easat-9645	293	32	.	.	PUNCT
easat-9645	293	33	.	.	PUNCT
easat-9645	294	1	,	,	PUNCT
easat-9645	294	2	𝑎𝑛	𝑎𝑛	PRON
easat-9645	294	3	and	and	CCONJ
easat-9645	294	4	𝑏𝑖	𝑏𝑖	ADP
easat-9645	294	5	=	=	PUNCT
easat-9645	294	6	𝑏1	𝑏1	NOUN
easat-9645	294	7	,	,	PUNCT
easat-9645	294	8	𝑏2	𝑏2	NOUN
easat-9645	294	9	…	…	PUNCT
easat-9645	294	10	…	…	PUNCT
easat-9645	294	11	…	…	PUNCT
easat-9645	294	12	,	,	PUNCT
easat-9645	294	13	𝑏𝑛	𝑏𝑛	ADP
easat-9645	294	14	let	let	VERB
easat-9645	294	15	<	<	X
easat-9645	294	16	𝑥𝑛	𝑥𝑛	AUX
easat-9645	294	17	>	>	X
easat-9645	294	18	be	be	AUX
easat-9645	294	19	a	a	DET
easat-9645	294	20	cauchy	cauchy	ADJ
easat-9645	294	21	sequence	sequence	NOUN
easat-9645	294	22	in	in	ADP
easat-9645	294	23	𝑙2	𝑙2	PROPN
easat-9645	294	24	.	.	PUNCT
easat-9645	295	1	for	for	ADP
easat-9645	295	2	every	every	PRON
easat-9645	295	3	𝜖	𝜖	PROPN
easat-9645	295	4	>	>	X
easat-9645	295	5	0	0	NUM
easat-9645	295	6	∃	∃	PROPN
easat-9645	295	7	𝑚	𝑚	PROPN
easat-9645	295	8	,	,	PUNCT
easat-9645	295	9	𝑛	𝑛	PRON
easat-9645	295	10	∈	∈	PROPN
easat-9645	295	11	ℕ	ℕ	PROPN
easat-9645	295	12	s.	s.	PROPN
easat-9645	295	13	t.	t.	PROPN
easat-9645	295	14	𝑑(𝑎𝑚	𝑑(𝑎𝑚	PROPN
easat-9645	295	15	,	,	PUNCT
easat-9645	295	16	𝑎𝑛	𝑎𝑛	PROPN
easat-9645	295	17	)	)	PUNCT
easat-9645	295	18	=	=	SYM
easat-9645	295	19	(	(	PUNCT
easat-9645	295	20	∑	∑	PUNCT
easat-9645	295	21	(	(	PUNCT
easat-9645	295	22	𝑎𝑖	𝑎𝑖	X
easat-9645	295	23	(	(	PUNCT
easat-9645	295	24	𝑚	𝑚	NOUN
easat-9645	295	25	)	)	PUNCT
easat-9645	295	26	−	−	PROPN
easat-9645	295	27	𝑎𝑖	𝑎𝑖	INTJ
easat-9645	295	28	(	(	PUNCT
easat-9645	295	29	𝑛	𝑛	NOUN
easat-9645	295	30	)	)	PUNCT
easat-9645	295	31	)	)	PUNCT
easat-9645	296	1	2	2	NUM
easat-9645	296	2	𝑛	𝑛	PRON
easat-9645	296	3	𝑖=1	𝑖=1	PROPN
easat-9645	296	4	)	)	PUNCT
easat-9645	296	5	1	1	NUM
easat-9645	296	6	2	2	NUM
easat-9645	296	7	<	<	X
easat-9645	296	8	𝜖	𝜖	X
easat-9645	296	9	∀	∀	X
easat-9645	296	10	𝑚	𝑚	NOUN
easat-9645	296	11	,	,	PUNCT
easat-9645	296	12	𝑛	𝑛	PRON
easat-9645	296	13	∈	∈	PROPN
easat-9645	296	14	ℕ	ℕ	PROPN
easat-9645	296	15	(	(	PUNCT
easat-9645	296	16	6	6	NUM
easat-9645	296	17	)	)	PUNCT
easat-9645	296	18	both	both	DET
easat-9645	296	19	sides	side	NOUN
easat-9645	296	20	squaring	square	VERB
easat-9645	296	21	in(4	in(4	PROPN
easat-9645	296	22	)	)	PUNCT
easat-9645	296	23	,	,	PUNCT
easat-9645	296	24	then	then	ADV
easat-9645	296	25	we	we	PRON
easat-9645	296	26	get	get	VERB
easat-9645	296	27	(	(	PUNCT
easat-9645	296	28	𝑎𝑖	𝑎𝑖	X
easat-9645	296	29	(	(	PUNCT
easat-9645	296	30	𝑚	𝑚	NOUN
easat-9645	296	31	)	)	PUNCT
easat-9645	296	32	−	−	PROPN
easat-9645	296	33	𝑎𝑖	𝑎𝑖	INTJ
easat-9645	296	34	(	(	PUNCT
easat-9645	296	35	𝑛	𝑛	NOUN
easat-9645	296	36	)	)	PUNCT
easat-9645	296	37	)	)	PUNCT
easat-9645	296	38	2	2	NUM
easat-9645	296	39	<	<	X
easat-9645	296	40	𝜖2	𝜖2	X
easat-9645	296	41	∀	∀	X
easat-9645	296	42	𝑖	𝑖	NOUN
easat-9645	296	43	=	=	NOUN
easat-9645	296	44	1,2,3	1,2,3	NUM
easat-9645	296	45	,	,	PUNCT
easat-9645	296	46	…	…	PUNCT
easat-9645	296	47	…	…	PUNCT
easat-9645	296	48	…	…	PUNCT
easat-9645	296	49	,	,	PUNCT
easat-9645	296	50	𝑛	𝑛	DET
easat-9645	296	51	⇒	⇒	NOUN
easat-9645	296	52	|𝑎𝑖	|𝑎𝑖	NUM
easat-9645	296	53	(	(	PUNCT
easat-9645	296	54	𝑚	𝑚	NOUN
easat-9645	296	55	)	)	PUNCT
easat-9645	296	56	−	−	PROPN
easat-9645	296	57	𝑎𝑖	𝑎𝑖	INTJ
easat-9645	296	58	(	(	PUNCT
easat-9645	296	59	𝑛	𝑛	NOUN
easat-9645	296	60	)	)	PUNCT
easat-9645	296	61	|	|	ADV
easat-9645	296	62	<	<	X
easat-9645	296	63	𝜖	𝜖	X
easat-9645	296	64	fixed	fix	VERB
easat-9645	296	65	𝑖	𝑖	SYM
easat-9645	296	66	,	,	PUNCT
easat-9645	296	67	(	(	PUNCT
easat-9645	296	68	1	1	NUM
easat-9645	296	69	≤	≤	NUM
easat-9645	296	70	𝑖	𝑖	SYM
easat-9645	296	71	≤	≤	NOUN
easat-9645	296	72	𝑛	𝑛	NOUN
easat-9645	296	73	)	)	PUNCT
easat-9645	296	74	,	,	PUNCT
easat-9645	296	75	the	the	DET
easat-9645	296	76	sequence	sequence	NOUN
easat-9645	296	77	<	<	X
easat-9645	296	78	𝑥𝑖	𝑥𝑖	PROPN
easat-9645	296	79	1	1	NUM
easat-9645	296	80	,	,	PUNCT
easat-9645	296	81	𝑥𝑖	𝑥𝑖	ADV
easat-9645	296	82	2	2	NUM
easat-9645	296	83	…	…	PUNCT
easat-9645	296	84	…	…	PUNCT
easat-9645	296	85	…	…	PUNCT
easat-9645	296	86	…	…	PUNCT
easat-9645	296	87	>	>	X
easat-9645	296	88	is	be	AUX
easat-9645	296	89	a	a	DET
easat-9645	296	90	cauchy	cauchy	ADJ
easat-9645	296	91	sequence	sequence	NOUN
easat-9645	296	92	of	of	ADP
easat-9645	296	93	𝑅.	𝑅.	NOUN
easat-9645	296	94	so	so	SCONJ
easat-9645	296	95	it	it	PRON
easat-9645	296	96	converges	converge	VERB
easat-9645	296	97	i.e.	i.e.	ADV
easat-9645	296	98	𝑎𝑖	𝑎𝑖	ADP
easat-9645	296	99	𝑚	𝑚	NOUN
easat-9645	296	100	→	→	SYM
easat-9645	296	101	𝑎	𝑎	NOUN
easat-9645	296	102	as	as	ADP
easat-9645	296	103	𝑚	𝑚	PROPN
easat-9645	296	104	→	→	SYM
easat-9645	296	105	∞.	∞.	PROPN
easat-9645	296	106	using	use	VERB
easat-9645	296	107	this	this	PRON
easat-9645	296	108	n	n	NUM
easat-9645	296	109	limits	limit	NOUN
easat-9645	296	110	,	,	PUNCT
easat-9645	296	111	we	we	PRON
easat-9645	296	112	define	define	VERB
easat-9645	296	113	𝑎	𝑎	X
easat-9645	296	114	=	=	PUNCT
easat-9645	296	115	(	(	PUNCT
easat-9645	296	116	𝑎1	𝑎1	INTJ
easat-9645	296	117	,	,	PUNCT
easat-9645	296	118	𝑎2	𝑎2	PROPN
easat-9645	296	119	,	,	PUNCT
easat-9645	296	120	…	…	PUNCT
easat-9645	296	121	…	…	PUNCT
easat-9645	296	122	.	.	PUNCT
easat-9645	296	123	.	.	PUNCT
easat-9645	297	1	,	,	PUNCT
easat-9645	297	2	𝑎𝑛	𝑎𝑛	PROPN
easat-9645	297	3	)	)	PUNCT
easat-9645	297	4	.	.	PUNCT
easat-9645	298	1	clearly	clearly	ADV
easat-9645	298	2	𝑎	𝑎	PROPN
easat-9645	298	3	∈	∈	PROPN
easat-9645	298	4	𝑙2	𝑙2	NOUN
easat-9645	298	5	,	,	PUNCT
easat-9645	298	6	from(2	from(2	PROPN
easat-9645	298	7	)	)	PUNCT
easat-9645	298	8	,	,	PUNCT
easat-9645	298	9	𝑎𝑛	𝑎𝑛	PRON
easat-9645	298	10	→	→	SYM
easat-9645	298	11	𝑎	𝑎	X
easat-9645	298	12	𝑎𝑠	𝑎𝑠	NOUN
easat-9645	298	13	𝑛	𝑛	PROPN
easat-9645	298	14	→	→	SYM
easat-9645	298	15	∞	∞	PROPN
easat-9645	298	16	,	,	PUNCT
easat-9645	298	17	then	then	ADV
easat-9645	298	18	we	we	PRON
easat-9645	298	19	have	have	VERB
easat-9645	298	20	𝑑(𝑎𝑚	𝑑(𝑎𝑚	NOUN
easat-9645	298	21	,	,	PUNCT
easat-9645	298	22	𝑎	𝑎	X
easat-9645	298	23	)	)	PUNCT
easat-9645	298	24	≤	≤	X
easat-9645	298	25	𝜖	𝜖	PROPN
easat-9645	298	26	,	,	PUNCT
easat-9645	298	27	𝑚	𝑚	X
easat-9645	298	28	>	>	X
easat-9645	298	29	𝑁	𝑁	PROPN
easat-9645	298	30	so	so	ADV
easat-9645	298	31	,	,	PUNCT
easat-9645	298	32	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
easat-9645	298	33	𝑚→∞	𝑚→∞	X
easat-9645	298	34	𝑎𝑚	𝑎𝑚	NOUN
easat-9645	298	35	=	=	PUNCT
easat-9645	298	36	𝑎.	𝑎.	NOUN
easat-9645	298	37	hence	hence	ADV
easat-9645	298	38	the	the	DET
easat-9645	298	39	unitary	unitary	ADJ
easat-9645	298	40	space	space	NOUN
easat-9645	298	41	𝑙2	𝑙2	NOUN
easat-9645	298	42	is	be	AUX
easat-9645	298	43	complete	complete	ADJ
easat-9645	298	44	if	if	SCONJ
easat-9645	298	45	n=3	n=3	NOUN
easat-9645	298	46	,	,	PUNCT
easat-9645	298	47	then	then	ADV
easat-9645	298	48	(	(	PUNCT
easat-9645	298	49	3	3	X
easat-9645	298	50	)	)	PUNCT
easat-9645	298	51	gives	give	VERB
easat-9645	298	52	⟨𝑎	⟨𝑎	NOUN
easat-9645	298	53	,	,	PUNCT
easat-9645	298	54	𝑏⟩	𝑏⟩	PUNCT
easat-9645	299	1	=	=	PUNCT
easat-9645	299	2	𝑎.	𝑎.	NOUN
easat-9645	300	1	𝑏	𝑏	NOUN
easat-9645	300	2	=	=	SYM
easat-9645	300	3	𝑎1𝑏1	𝑎1𝑏1	PROPN
easat-9645	300	4	+	+	CCONJ
easat-9645	300	5	𝑎2𝑏2	𝑎2𝑏2	PUNCT
easat-9645	300	6	+	+	CCONJ
easat-9645	300	7	𝑎3𝑏3	𝑎3𝑏3	X
easat-9645	300	8	of	of	ADP
easat-9645	300	9	𝑎	𝑎	NOUN
easat-9645	300	10	=	=	PUNCT
easat-9645	300	11	(	(	PUNCT
easat-9645	300	12	𝑎1	𝑎1	PROPN
easat-9645	300	13	,	,	PUNCT
easat-9645	300	14	𝑎2	𝑎2	PROPN
easat-9645	300	15	,	,	PUNCT
easat-9645	300	16	𝑎3	𝑎3	PROPN
easat-9645	300	17	)	)	PUNCT
easat-9645	300	18	∈	∈	PROPN
easat-9645	300	19	𝑅	𝑅	PROPN
easat-9645	300	20	and	and	CCONJ
easat-9645	300	21	𝑏	𝑏	NOUN
easat-9645	300	22	=	=	PUNCT
easat-9645	300	23	(	(	PUNCT
easat-9645	300	24	𝑏1	𝑏1	NOUN
easat-9645	300	25	,	,	PUNCT
easat-9645	300	26	𝑏2	𝑏2	NOUN
easat-9645	300	27	,	,	PUNCT
easat-9645	300	28	𝑏3	𝑏3	PROPN
easat-9645	300	29	)	)	PUNCT
easat-9645	300	30	∈	∈	PROPN
easat-9645	300	31	𝑅	𝑅	PROPN
easat-9645	300	32	and	and	CCONJ
easat-9645	300	33	the	the	DET
easat-9645	300	34	orthogonality	orthogonality	NOUN
easat-9645	300	35	⟨𝑎	⟨𝑎	NUM
easat-9645	300	36	,	,	PUNCT
easat-9645	300	37	𝑏⟩	𝑏⟩	PUNCT
easat-9645	300	38	=	=	PUNCT
easat-9645	300	39	𝑎.	𝑎.	NOUN
easat-9645	301	1	𝑏	𝑏	NOUN
easat-9645	301	2	=	=	NOUN
easat-9645	301	3	0	0	PROPN
easat-9645	301	4	.	.	PUNCT
easat-9645	302	1	this	this	DET
easat-9645	302	2	concept	concept	NOUN
easat-9645	302	3	is	be	AUX
easat-9645	302	4	consistent	consistent	ADJ
easat-9645	302	5	with	with	ADP
easat-9645	302	6	the	the	DET
easat-9645	302	7	fundamental	fundamental	ADJ
easat-9645	302	8	idea	idea	NOUN
easat-9645	302	9	of	of	ADP
easat-9645	302	10	orthogonality	orthogonality	NOUN
easat-9645	302	11	,	,	PUNCT
easat-9645	302	12	meaning	mean	VERB
easat-9645	302	13	that	that	SCONJ
easat-9645	302	14	two	two	NUM
easat-9645	302	15	vectors	vector	NOUN
easat-9645	302	16	are	be	AUX
easat-9645	302	17	orthogonal	orthogonal	ADJ
easat-9645	302	18	if	if	SCONJ
easat-9645	302	19	their	their	PRON
easat-9645	302	20	inner	inner	ADJ
easat-9645	302	21	product	product	NOUN
easat-9645	302	22	is	be	AUX
easat-9645	302	23	zero	zero	NUM
easat-9645	302	24	.	.	PUNCT
easat-9645	303	1	therefore	therefore	ADV
easat-9645	303	2	,	,	PUNCT
easat-9645	303	3	the	the	DET
easat-9645	303	4	space	space	NOUN
easat-9645	303	5	𝑙2	𝑙2	PROPN
easat-9645	303	6	is	be	AUX
easat-9645	303	7	a	a	DET
easat-9645	303	8	hilbert	hilbert	NOUN
easat-9645	303	9	space	space	NOUN
easat-9645	303	10	.	.	PUNCT
easat-9645	304	1	1510	1510	NUM
easat-9645	304	2	edelweiss	edelweiss	PROPN
easat-9645	304	3	applied	apply	VERB
easat-9645	304	4	science	science	NOUN
easat-9645	304	5	and	and	CCONJ
easat-9645	304	6	technology	technology	NOUN
easat-9645	304	7	issn	issn	PROPN
easat-9645	304	8	:	:	PUNCT
easat-9645	304	9	2576	2576	NUM
easat-9645	304	10	-	-	SYM
easat-9645	304	11	8484	8484	NUM
easat-9645	304	12	vol	vol	NOUN
easat-9645	304	13	.	.	PROPN
easat-9645	305	1	9	9	NUM
easat-9645	305	2	,	,	PUNCT
easat-9645	305	3	no	no	INTJ
easat-9645	305	4	.	.	NOUN
easat-9645	305	5	8	8	NUM
easat-9645	305	6	:	:	SYM
easat-9645	305	7	1498	1498	NUM
easat-9645	305	8	-	-	SYM
easat-9645	305	9	1523	1523	NUM
easat-9645	305	10	,	,	PUNCT
easat-9645	305	11	2025	2025	NUM
easat-9645	305	12	doi	doi	NOUN
easat-9645	305	13	:	:	PUNCT
easat-9645	305	14	10.55214/2576	10.55214/2576	NUM
easat-9645	305	15	-	-	SYM
easat-9645	305	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	305	17	©	©	PROPN
easat-9645	305	18	2025	2025	NUM
easat-9645	305	19	by	by	ADP
easat-9645	305	20	the	the	DET
easat-9645	305	21	authors	author	NOUN
easat-9645	305	22	;	;	PUNCT
easat-9645	305	23	licensee	licensee	PROPN
easat-9645	305	24	learning	learn	VERB
easat-9645	305	25	gate	gate	NOUN
easat-9645	305	26	figure	figure	NOUN
easat-9645	305	27	7	7	NUM
easat-9645	305	28	.	.	PUNCT
easat-9645	305	29	matlab	matlab	PROPN
easat-9645	305	30	visualization	visualization	NOUN
easat-9645	305	31	of	of	ADP
easat-9645	305	32	𝑙2	𝑙2	NOUN
easat-9645	305	33	truncations	truncation	NOUN
easat-9645	305	34	the	the	DET
easat-9645	305	35	figure	figure	NOUN
easat-9645	305	36	7	7	NUM
easat-9645	305	37	,	,	PUNCT
easat-9645	305	38	shows	show	VERB
easat-9645	305	39	that	that	SCONJ
easat-9645	305	40	𝑙2geometrically	𝑙2geometrically	ADV
easat-9645	305	41	by	by	ADP
easat-9645	305	42	projecting	project	VERB
easat-9645	305	43	infinite	infinite	ADJ
easat-9645	305	44	-	-	PUNCT
easat-9645	305	45	dimensional	dimensional	ADJ
easat-9645	305	46	concepts	concept	NOUN
easat-9645	305	47	onto	onto	ADP
easat-9645	305	48	2d	2d	NUM
easat-9645	305	49	or	or	CCONJ
easat-9645	305	50	3d	3d	NUM
easat-9645	305	51	subspaces	subspace	NOUN
easat-9645	305	52	of	of	ADP
easat-9645	305	53	rn	rn	PROPN
easat-9645	305	54	.	.	PUNCT
easat-9645	306	1	the	the	DET
easat-9645	306	2	inner	inner	ADJ
easat-9645	306	3	product	product	NOUN
easat-9645	306	4	,	,	PUNCT
easat-9645	306	5	norm	norm	NOUN
easat-9645	306	6	,	,	PUNCT
easat-9645	306	7	and	and	CCONJ
easat-9645	306	8	orthogonality	orthogonality	NOUN
easat-9645	306	9	behave	behave	VERB
easat-9645	306	10	analogously	analogously	ADV
easat-9645	306	11	to	to	ADP
easat-9645	306	12	how	how	SCONJ
easat-9645	306	13	they	they	PRON
easat-9645	306	14	do	do	AUX
easat-9645	306	15	in	in	ADP
easat-9645	306	16	euclidean	euclidean	ADJ
easat-9645	306	17	space	space	NOUN
easat-9645	306	18	.	.	PUNCT
easat-9645	307	1	this	this	DET
easat-9645	307	2	visualization	visualization	NOUN
easat-9645	307	3	preserves	preserve	VERB
easat-9645	307	4	the	the	DET
easat-9645	307	5	core	core	ADJ
easat-9645	307	6	hilbert	hilbert	PROPN
easat-9645	307	7	space	space	NOUN
easat-9645	307	8	geometry	geometry	NOUN
easat-9645	307	9	:	:	PUNCT
easat-9645	307	10	norm	norm	NOUN
easat-9645	307	11	,	,	PUNCT
easat-9645	307	12	angle	angle	NOUN
easat-9645	307	13	,	,	PUNCT
easat-9645	307	14	projection	projection	NOUN
easat-9645	307	15	,	,	PUNCT
easat-9645	307	16	and	and	CCONJ
easat-9645	307	17	orthogonality	orthogonality	NOUN
easat-9645	307	18	.	.	PUNCT
easat-9645	308	1	also	also	ADV
easat-9645	308	2	,	,	PUNCT
easat-9645	308	3	this	this	DET
easat-9645	308	4	matlab	matlab	PROPN
easat-9645	308	5	code	code	NOUN
easat-9645	308	6	visualizes	visualize	VERB
easat-9645	308	7	three	three	NUM
easat-9645	308	8	orthogonal	orthogonal	ADJ
easat-9645	308	9	sequences	sequence	NOUN
easat-9645	308	10	(	(	PUNCT
easat-9645	308	11	finite	finite	NOUN
easat-9645	308	12	truncations	truncation	NOUN
easat-9645	308	13	)	)	PUNCT
easat-9645	308	14	as	as	ADP
easat-9645	308	15	vectors	vector	NOUN
easat-9645	308	16	in	in	ADP
easat-9645	308	17	r3	r3	PROPN
easat-9645	308	18	.	.	PUNCT
easat-9645	309	1	example	example	NOUN
easat-9645	309	2	(	(	PUNCT
easat-9645	309	3	orthogonality	orthogonality	NOUN
easat-9645	309	4	in	in	ADP
easat-9645	309	5	𝑙2	𝑙2	PROPN
easat-9645	309	6	):	):	PUNCT
easat-9645	309	7	let	let	VERB
easat-9645	309	8	𝑥	𝑥	VERB
easat-9645	309	9	=	=	SYM
easat-9645	309	10	(	(	PUNCT
easat-9645	309	11	1	1	NUM
easat-9645	309	12	,	,	PUNCT
easat-9645	309	13	0	0	NUM
easat-9645	309	14	,	,	PUNCT
easat-9645	309	15	0	0	NUM
easat-9645	309	16	,	,	PUNCT
easat-9645	309	17	.	.	PUNCT
easat-9645	309	18	.	.	PUNCT
easat-9645	309	19	.	.	PUNCT
easat-9645	310	1	)	)	PUNCT
easat-9645	310	2	,	,	PUNCT
easat-9645	310	3	𝑦	𝑦	NOUN
easat-9645	310	4	=	=	SYM
easat-9645	310	5	(	(	PUNCT
easat-9645	310	6	0	0	NUM
easat-9645	310	7	,	,	PUNCT
easat-9645	310	8	1	1	NUM
easat-9645	310	9	,	,	PUNCT
easat-9645	310	10	0	0	NUM
easat-9645	310	11	,	,	PUNCT
easat-9645	310	12	.	.	PUNCT
easat-9645	310	13	.	.	PUNCT
easat-9645	310	14	.	.	PUNCT
easat-9645	310	15	)	)	PUNCT
easat-9645	310	16	.	.	PUNCT
easat-9645	311	1	then	then	ADV
easat-9645	311	2	⟨𝑥	⟨𝑥	NOUN
easat-9645	311	3	,	,	PUNCT
easat-9645	311	4	𝑦⟩	𝑦⟩	NOUN
easat-9645	311	5	=	=	PUNCT
easat-9645	311	6	1	1	NUM
easat-9645	311	7	·	·	SYM
easat-9645	311	8	0	0	PUNCT
easat-9645	312	1	+	+	CCONJ
easat-9645	312	2	0	0	NUM
easat-9645	312	3	·	·	SYM
easat-9645	312	4	1	1	NUM
easat-9645	313	1	+	+	SYM
easat-9645	313	2	0	0	NUM
easat-9645	313	3	=	=	SYM
easat-9645	313	4	0	0	NUM
easat-9645	313	5	,	,	PUNCT
easat-9645	313	6	so	so	ADV
easat-9645	313	7	x	x	X
easat-9645	313	8	and	and	CCONJ
easat-9645	313	9	y	y	PROPN
easat-9645	313	10	are	be	AUX
easat-9645	313	11	orthogonal	orthogonal	ADJ
easat-9645	313	12	.	.	PUNCT
easat-9645	313	13	example	example	NOUN
easat-9645	313	14	(	(	PUNCT
easat-9645	313	15	completeness	completeness	NOUN
easat-9645	313	16	in	in	ADP
easat-9645	313	17	𝑙2	𝑙2	PROPN
easat-9645	313	18	):	):	PUNCT
easat-9645	313	19	consider	consider	VERB
easat-9645	313	20	𝑥𝑘	𝑥𝑘	X
easat-9645	313	21	=	=	PUNCT
easat-9645	313	22	(	(	PUNCT
easat-9645	313	23	1	1	NUM
easat-9645	313	24	𝑘	𝑘	INTJ
easat-9645	313	25	,	,	PUNCT
easat-9645	313	26	1	1	NUM
easat-9645	313	27	𝑘2	𝑘2	PROPN
easat-9645	313	28	,	,	PUNCT
easat-9645	313	29	.	.	PUNCT
easat-9645	313	30	.	.	PUNCT
easat-9645	314	1	.	.	PUNCT
easat-9645	315	1	1	1	NUM
easat-9645	315	2	𝑘𝑛	𝑘𝑛	INTJ
easat-9645	315	3	,	,	PUNCT
easat-9645	315	4	0	0	NUM
easat-9645	315	5	,	,	PUNCT
easat-9645	315	6	.	.	PUNCT
easat-9645	315	7	.	.	PUNCT
easat-9645	315	8	.	.	PUNCT
easat-9645	315	9	)	)	PUNCT
easat-9645	315	10	.	.	PUNCT
easat-9645	316	1	compute	compute	VERB
easat-9645	316	2	‖𝑥𝑘	‖𝑥𝑘	NUM
easat-9645	316	3	−	−	PROPN
easat-9645	316	4	𝑥𝑚‖2	𝑥𝑚‖2	ADV
easat-9645	316	5	=	=	SYM
easat-9645	316	6	∑	∑	PUNCT
easat-9645	316	7	|	|	ADV
easat-9645	316	8	1	1	NUM
easat-9645	316	9	𝑘𝑛	𝑘𝑛	NOUN
easat-9645	316	10	−	−	PROPN
easat-9645	316	11	1	1	NUM
easat-9645	316	12	𝑚𝑛|	𝑚𝑛|	PROPN
easat-9645	316	13	2	2	NUM
easat-9645	316	14	𝑚𝑖𝑛(𝑘,𝑚	𝑚𝑖𝑛(𝑘,𝑚	NOUN
easat-9645	316	15	)	)	PUNCT
easat-9645	316	16	𝑛=1	𝑛=1	NOUN
easat-9645	317	1	+	+	CCONJ
easat-9645	317	2	∑	∑	PROPN
easat-9645	317	3	|	|	ADV
easat-9645	317	4	1	1	NUM
easat-9645	317	5	𝑘𝑛|	𝑘𝑛|	NOUN
easat-9645	317	6	2	2	NUM
easat-9645	317	7	𝑚𝑎𝑥(𝑘,𝑚	𝑚𝑎𝑥(𝑘,𝑚	NOUN
easat-9645	317	8	)	)	PUNCT
easat-9645	317	9	𝑛=min(𝑘,𝑚)+1	𝑛=min(𝑘,𝑚)+1	NOUN
easat-9645	317	10	,	,	PUNCT
easat-9645	317	11	which	which	PRON
easat-9645	317	12	approaches	approach	VERB
easat-9645	317	13	0	0	NUM
easat-9645	317	14	.	.	PUNCT
easat-9645	318	1	the	the	DET
easat-9645	318	2	limit	limit	NOUN
easat-9645	318	3	(	(	PUNCT
easat-9645	318	4	0	0	NUM
easat-9645	318	5	,	,	PUNCT
easat-9645	318	6	0	0	NUM
easat-9645	318	7	,	,	PUNCT
easat-9645	318	8	.	.	PUNCT
easat-9645	318	9	.	.	PUNCT
easat-9645	318	10	.	.	PUNCT
easat-9645	318	11	)	)	PUNCT
easat-9645	319	1	∈	∈	PROPN
easat-9645	319	2	𝑙2	𝑙2	PROPN
easat-9645	319	3	,	,	PUNCT
easat-9645	319	4	confirming	confirm	VERB
easat-9645	319	5	completeness	completeness	NOUN
easat-9645	319	6	3.4	3.4	NUM
easat-9645	319	7	.	.	PUNCT
easat-9645	320	1	theorem	theorem	NOUN
easat-9645	320	2	(	(	PUNCT
easat-9645	320	3	polarization	polarization	NOUN
easat-9645	320	4	identity	identity	NOUN
easat-9645	320	5	and	and	CCONJ
easat-9645	320	6	hilbert	hilbert	NOUN
easat-9645	320	7	space	space	NOUN
easat-9645	320	8	)	)	PUNCT
easat-9645	320	9	let	let	VERB
easat-9645	320	10	b	b	X
easat-9645	320	11	be	be	AUX
easat-9645	320	12	a	a	DET
easat-9645	320	13	complex	complex	ADJ
easat-9645	320	14	banach	banach	NOUN
easat-9645	320	15	space	space	NOUN
easat-9645	320	16	,	,	PUNCT
easat-9645	320	17	and	and	CCONJ
easat-9645	320	18	suppose	suppose	VERB
easat-9645	320	19	the	the	DET
easat-9645	320	20	norm	norm	NOUN
easat-9645	320	21	∥⋅∥	∥⋅∥	PROPN
easat-9645	320	22	on	on	ADP
easat-9645	320	23	b	b	NUM
easat-9645	320	24	satisfies	satisfie	NOUN
easat-9645	320	25	the	the	DET
easat-9645	320	26	parallelogram	parallelogram	NOUN
easat-9645	320	27	law	law	NOUN
easat-9645	320	28	.	.	PUNCT
easat-9645	321	1	define	define	VERB
easat-9645	321	2	a	a	DET
easat-9645	321	3	map	map	NOUN
easat-9645	321	4	⟨⋅,⋅⟩	⟨⋅,⋅⟩	PUNCT
easat-9645	322	1	on	on	ADP
easat-9645	322	2	𝐵	𝐵	NOUN
easat-9645	322	3	×	×	NOUN
easat-9645	322	4	𝐵	𝐵	NOUN
easat-9645	322	5	by	by	ADP
easat-9645	322	6	4⟨𝑝	4⟨𝑝	NUM
easat-9645	322	7	,	,	PUNCT
easat-9645	322	8	𝑞⟩	𝑞⟩	AUX
easat-9645	322	9	:	:	PUNCT
easat-9645	322	10	=	=	X
easat-9645	322	11	∥	∥	X
easat-9645	322	12	𝑝	𝑝	NOUN
easat-9645	323	1	+	+	CCONJ
easat-9645	323	2	𝑞	𝑞	X
easat-9645	323	3	∥2−∥	∥2−∥	PROPN
easat-9645	323	4	𝑝	𝑝	PROPN
easat-9645	323	5	−	−	PROPN
easat-9645	323	6	𝑞	𝑞	X
easat-9645	323	7	∥2	∥2	X
easat-9645	323	8	+	+	NOUN
easat-9645	323	9	𝑖	𝑖	PUNCT
easat-9645	323	10	∥	∥	PROPN
easat-9645	323	11	𝑝	𝑝	PROPN
easat-9645	324	1	+	+	NUM
easat-9645	324	2	𝑖𝑞	𝑖𝑞	PROPN
easat-9645	324	3	∥2−	∥2−	PROPN
easat-9645	324	4	𝑖	𝑖	PUNCT
easat-9645	324	5	∥	∥	PROPN
easat-9645	324	6	𝑝	𝑝	PROPN
easat-9645	324	7	−	−	PROPN
easat-9645	324	8	𝑖𝑞	𝑖𝑞	NOUN
easat-9645	324	9	∥2	∥2	NOUN
easat-9645	324	10	.	.	PUNCT
easat-9645	325	1	then	then	ADV
easat-9645	325	2	⟨⋅,⋅⟩	⟨⋅,⋅⟩	PROPN
easat-9645	325	3	is	be	AUX
easat-9645	325	4	an	an	DET
easat-9645	325	5	inner	inner	ADJ
easat-9645	325	6	product	product	NOUN
easat-9645	325	7	,	,	PUNCT
easat-9645	325	8	and	and	CCONJ
easat-9645	325	9	hence	hence	ADV
easat-9645	325	10	b	b	NOUN
easat-9645	325	11	,	,	PUNCT
easat-9645	325	12	being	be	AUX
easat-9645	325	13	complete	complete	ADJ
easat-9645	325	14	,	,	PUNCT
easat-9645	325	15	is	be	AUX
easat-9645	325	16	a	a	DET
easat-9645	325	17	hilbert	hilbert	NOUN
easat-9645	325	18	space	space	NOUN
easat-9645	325	19	.	.	PUNCT
easat-9645	326	1	proof	proof	NOUN
easat-9645	326	2	:	:	PUNCT
easat-9645	326	3	to	to	PART
easat-9645	326	4	show	show	VERB
easat-9645	326	5	that	that	SCONJ
easat-9645	326	6	b	b	NOUN
easat-9645	326	7	is	be	AUX
easat-9645	326	8	a	a	DET
easat-9645	326	9	hilbert	hilbert	NOUN
easat-9645	326	10	space	space	NOUN
easat-9645	326	11	,	,	PUNCT
easat-9645	326	12	then	then	ADV
easat-9645	326	13	the	the	DET
easat-9645	326	14	map	map	NOUN
easat-9645	326	15	⟨⋅,⋅⟩	⟨⋅,⋅⟩	PUNCT
easat-9645	326	16	satisfies	satisfy	VERB
easat-9645	326	17	the	the	DET
easat-9645	326	18	axioms	axiom	NOUN
easat-9645	326	19	of	of	ADP
easat-9645	326	20	an	an	DET
easat-9645	326	21	inner	inner	ADJ
easat-9645	326	22	product	product	NOUN
easat-9645	326	23	on	on	ADP
easat-9645	326	24	a	a	DET
easat-9645	326	25	complex	complex	ADJ
easat-9645	326	26	vector	vector	NOUN
easat-9645	326	27	space	space	NOUN
easat-9645	326	28	.	.	PUNCT
easat-9645	327	1	1511	1511	NUM
easat-9645	327	2	edelweiss	edelweiss	PROPN
easat-9645	327	3	applied	apply	VERB
easat-9645	327	4	science	science	NOUN
easat-9645	327	5	and	and	CCONJ
easat-9645	327	6	technology	technology	NOUN
easat-9645	327	7	issn	issn	PROPN
easat-9645	327	8	:	:	PUNCT
easat-9645	327	9	2576	2576	NUM
easat-9645	327	10	-	-	SYM
easat-9645	327	11	8484	8484	NUM
easat-9645	327	12	vol	vol	NOUN
easat-9645	327	13	.	.	PROPN
easat-9645	328	1	9	9	NUM
easat-9645	328	2	,	,	PUNCT
easat-9645	328	3	no	no	INTJ
easat-9645	328	4	.	.	NOUN
easat-9645	328	5	8	8	NUM
easat-9645	328	6	:	:	SYM
easat-9645	328	7	1498	1498	NUM
easat-9645	328	8	-	-	SYM
easat-9645	328	9	1523	1523	NUM
easat-9645	328	10	,	,	PUNCT
easat-9645	328	11	2025	2025	NUM
easat-9645	328	12	doi	doi	NOUN
easat-9645	328	13	:	:	PUNCT
easat-9645	328	14	10.55214/2576	10.55214/2576	NUM
easat-9645	328	15	-	-	SYM
easat-9645	328	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	328	17	©	©	PROPN
easat-9645	328	18	2025	2025	NUM
easat-9645	328	19	by	by	ADP
easat-9645	328	20	the	the	DET
easat-9645	328	21	authors	author	NOUN
easat-9645	328	22	;	;	PUNCT
easat-9645	328	23	licensee	licensee	PROPN
easat-9645	328	24	learning	learning	NOUN
easat-9645	328	25	gate	gate	PROPN
easat-9645	328	26	3.4.1	3.4.1	PROPN
easat-9645	328	27	.	.	PUNCT
easat-9645	329	1	positivity	positivity	NOUN
easat-9645	329	2	and	and	CCONJ
easat-9645	329	3	definiteness	definiteness	NOUN
easat-9645	329	4	set	set	VERB
easat-9645	329	5	𝑞	𝑞	PROPN
easat-9645	329	6	=	=	NOUN
easat-9645	329	7	𝑝.	𝑝.	PROPN
easat-9645	329	8	then	then	ADV
easat-9645	329	9	4⟨𝑝	4⟨𝑝	NUM
easat-9645	329	10	,	,	PUNCT
easat-9645	329	11	𝑝⟩	𝑝⟩	PUNCT
easat-9645	330	1	=	=	NOUN
easat-9645	330	2	∥	∥	X
easat-9645	330	3	𝑝	𝑝	NOUN
easat-9645	330	4	+	+	CCONJ
easat-9645	330	5	𝑞	𝑞	X
easat-9645	330	6	∥2−∥	∥2−∥	PROPN
easat-9645	330	7	𝑝	𝑝	PROPN
easat-9645	330	8	−	−	PROPN
easat-9645	330	9	𝑞	𝑞	X
easat-9645	330	10	∥2	∥2	X
easat-9645	330	11	+	+	NOUN
easat-9645	330	12	𝑖	𝑖	PUNCT
easat-9645	330	13	∥	∥	PROPN
easat-9645	330	14	𝑝	𝑝	PROPN
easat-9645	331	1	+	+	NUM
easat-9645	331	2	𝑖𝑞	𝑖𝑞	PROPN
easat-9645	331	3	∥2−	∥2−	PROPN
easat-9645	331	4	𝑖	𝑖	PUNCT
easat-9645	331	5	∥	∥	PROPN
easat-9645	331	6	𝑝	𝑝	PROPN
easat-9645	331	7	−	−	PROPN
easat-9645	331	8	𝑖𝑞	𝑖𝑞	PROPN
easat-9645	331	9	∥2	∥2	NOUN
easat-9645	332	1	=	=	NOUN
easat-9645	332	2	∥	∥	X
easat-9645	332	3	2𝑝	2𝑝	NOUN
easat-9645	332	4	∥2−	∥2−	PROPN
easat-9645	332	5	0	0	PUNCT
easat-9645	333	1	+	+	NUM
easat-9645	333	2	𝑖	𝑖	SYM
easat-9645	333	3	∥	∥	PUNCT
easat-9645	333	4	𝑝(1	𝑝(1	NOUN
easat-9645	333	5	+	+	NUM
easat-9645	333	6	𝑖	𝑖	X
easat-9645	333	7	)	)	PUNCT
easat-9645	333	8	∥2−	∥2−	PROPN
easat-9645	333	9	𝑖	𝑖	SYM
easat-9645	333	10	∥	∥	PUNCT
easat-9645	333	11	𝑝(1	𝑝(1	NOUN
easat-9645	333	12	−	−	NUM
easat-9645	333	13	𝑖	𝑖	NOUN
easat-9645	333	14	)	)	PUNCT
easat-9645	333	15	∥2	∥2	NOUN
easat-9645	333	16	=	=	SYM
easat-9645	334	1	4	4	NUM
easat-9645	334	2	∥	∥	SYM
easat-9645	334	3	𝑝	𝑝	ADP
easat-9645	334	4	∥2	∥2	NOUN
easat-9645	334	5	+	+	CCONJ
easat-9645	334	6	𝑖	𝑖	NOUN
easat-9645	334	7	∣	∣	ADJ
easat-9645	334	8	1	1	NUM
easat-9645	334	9	+	+	CCONJ
easat-9645	334	10	𝑖	𝑖	SYM
easat-9645	334	11	∣2∥	∣2∥	PROPN
easat-9645	334	12	𝑝	𝑝	PROPN
easat-9645	334	13	∥2−	∥2−	PROPN
easat-9645	334	14	𝑖	𝑖	PROPN
easat-9645	334	15	∣	∣	NOUN
easat-9645	334	16	1	1	NUM
easat-9645	334	17	−	−	NOUN
easat-9645	334	18	𝑖	𝑖	SYM
easat-9645	334	19	∣2∥	∣2∥	PROPN
easat-9645	334	20	𝑝	𝑝	PROPN
easat-9645	334	21	∥2	∥2	NOUN
easat-9645	334	22	=	=	SYM
easat-9645	334	23	4	4	NUM
easat-9645	334	24	∥	∥	SYM
easat-9645	334	25	𝑝	𝑝	PRON
easat-9645	334	26	∥2	∥2	NOUN
easat-9645	334	27	+	+	NUM
easat-9645	334	28	2𝑖	2𝑖	NOUN
easat-9645	334	29	∥	∥	PUNCT
easat-9645	334	30	𝑝	𝑝	PROPN
easat-9645	334	31	∥2−	∥2−	PROPN
easat-9645	334	32	2𝑖	2𝑖	NOUN
easat-9645	334	33	∥	∥	PUNCT
easat-9645	334	34	𝑝	𝑝	NOUN
easat-9645	334	35	∥2=	∥2=	NOUN
easat-9645	334	36	4	4	NUM
easat-9645	334	37	∥	∥	SYM
easat-9645	334	38	𝑝	𝑝	PRON
easat-9645	334	39	∥2	∥2	NOUN
easat-9645	334	40	.	.	PUNCT
easat-9645	335	1	thus	thus	ADV
easat-9645	335	2	,	,	PUNCT
easat-9645	335	3	⟨𝑝	⟨𝑝	NOUN
easat-9645	335	4	,	,	PUNCT
easat-9645	335	5	𝑝⟩	𝑝⟩	PUNCT
easat-9645	336	1	=	=	NOUN
easat-9645	336	2	∥	∥	X
easat-9645	336	3	𝑝	𝑝	NOUN
easat-9645	336	4	∥2≥	∥2≥	PROPN
easat-9645	336	5	0	0	NUM
easat-9645	336	6	and	and	CCONJ
easat-9645	336	7	⟨𝑝	⟨𝑝	NOUN
easat-9645	336	8	,	,	PUNCT
easat-9645	336	9	𝑝⟩	𝑝⟩	PUNCT
easat-9645	336	10	=	=	SYM
easat-9645	336	11	0	0	NUM
easat-9645	336	12	implies	imply	VERB
easat-9645	336	13	∥	∥	NUM
easat-9645	336	14	𝑝	𝑝	PROPN
easat-9645	336	15	∥=	∥=	NOUN
easat-9645	336	16	0	0	NUM
easat-9645	336	17	,	,	PUNCT
easat-9645	336	18	hence	hence	ADV
easat-9645	336	19	𝑝	𝑝	NOUN
easat-9645	336	20	=	=	SYM
easat-9645	336	21	0	0	NUM
easat-9645	336	22	.	.	PUNCT
easat-9645	337	1	3.4.2	3.4.2	NUM
easat-9645	337	2	.	.	PUNCT
easat-9645	337	3	conjugate	conjugate	ADJ
easat-9645	337	4	symmetry	symmetry	NOUN
easat-9645	337	5	take	take	VERB
easat-9645	337	6	the	the	DET
easat-9645	337	7	complex	complex	ADJ
easat-9645	337	8	conjugate	conjugate	NOUN
easat-9645	337	9	of	of	ADP
easat-9645	337	10	the	the	DET
easat-9645	337	11	polarization	polarization	NOUN
easat-9645	337	12	identity	identity	NOUN
easat-9645	337	13	:	:	PUNCT
easat-9645	337	14	4〈𝑝	4〈𝑝	NUM
easat-9645	337	15	,	,	PUNCT
easat-9645	337	16	𝑞〉̅̅	𝑞〉̅̅	ADJ
easat-9645	337	17	̅̅	̅̅	PROPN
easat-9645	337	18	̅̅	̅̅	PROPN
easat-9645	337	19	̅	̅	NOUN
easat-9645	337	20	=	=	SYM
easat-9645	337	21	∥	∥	X
easat-9645	337	22	𝑝	𝑝	NOUN
easat-9645	338	1	+	+	CCONJ
easat-9645	338	2	𝑞	𝑞	X
easat-9645	338	3	∥2−∥	∥2−∥	PROPN
easat-9645	338	4	𝑝	𝑝	PROPN
easat-9645	338	5	−	−	PROPN
easat-9645	338	6	𝑞	𝑞	X
easat-9645	338	7	∥2−	∥2−	PROPN
easat-9645	338	8	𝑖	𝑖	PRON
easat-9645	338	9	∥	∥	PROPN
easat-9645	338	10	𝑝	𝑝	PROPN
easat-9645	339	1	+	+	CCONJ
easat-9645	339	2	𝑖𝑞	𝑖𝑞	NUM
easat-9645	340	1	∥2	∥2	NOUN
easat-9645	340	2	+	+	NOUN
easat-9645	340	3	𝑖	𝑖	PUNCT
easat-9645	340	4	∥	∥	PROPN
easat-9645	340	5	𝑝	𝑝	PROPN
easat-9645	340	6	−	−	PROPN
easat-9645	340	7	𝑖𝑞	𝑖𝑞	PROPN
easat-9645	340	8	∥2	∥2	NOUN
easat-9645	340	9	by	by	ADP
easat-9645	340	10	symmetry	symmetry	NOUN
easat-9645	340	11	and	and	CCONJ
easat-9645	340	12	norm	norm	NOUN
easat-9645	340	13	properties	property	NOUN
easat-9645	340	14	:	:	PUNCT
easat-9645	340	15	4〈𝑝	4〈𝑝	NUM
easat-9645	340	16	,	,	PUNCT
easat-9645	340	17	𝑞〉̅̅	𝑞〉̅̅	ADJ
easat-9645	340	18	̅̅	̅̅	PROPN
easat-9645	340	19	̅̅	̅̅	PROPN
easat-9645	340	20	̅	̅	NOUN
easat-9645	340	21	=	=	SYM
easat-9645	340	22	4⟨𝑝	4⟨𝑝	NUM
easat-9645	340	23	,	,	PUNCT
easat-9645	340	24	𝑝⟩	𝑝⟩	PUNCT
easat-9645	340	25	⇒	⇒	PROPN
easat-9645	340	26	〈	〈	PROPN
easat-9645	340	27	𝑝	𝑝	PROPN
easat-9645	340	28	,	,	PUNCT
easat-9645	340	29	𝑞〉̅̅	𝑞〉̅̅	ADJ
easat-9645	340	30	̅̅	̅̅	PROPN
easat-9645	340	31	̅̅	̅̅	PROPN
easat-9645	340	32	̅	̅	PROPN
easat-9645	340	33	=	=	SYM
easat-9645	340	34	⟨𝑝	⟨𝑝	NOUN
easat-9645	340	35	,	,	PUNCT
easat-9645	340	36	𝑝⟩	𝑝⟩	PUNCT
easat-9645	341	1	3.4.3	3.4.3	X
easat-9645	341	2	.	.	PUNCT
easat-9645	342	1	linearity	linearity	NOUN
easat-9645	342	2	in	in	ADP
easat-9645	342	3	the	the	DET
easat-9645	342	4	first	first	ADJ
easat-9645	342	5	argument	argument	NOUN
easat-9645	342	6	let	let	VERB
easat-9645	342	7	𝑝	𝑝	NOUN
easat-9645	342	8	,	,	PUNCT
easat-9645	342	9	𝑞	𝑞	X
easat-9645	342	10	,	,	PUNCT
easat-9645	342	11	𝑟	𝑟	X
easat-9645	342	12	∈	∈	PROPN
easat-9645	342	13	𝐵.	𝐵.	NOUN
easat-9645	342	14	we	we	PRON
easat-9645	342	15	have	have	VERB
easat-9645	342	16	,	,	PUNCT
easat-9645	342	17	⟨𝑝	⟨𝑝	VERB
easat-9645	342	18	+	+	CCONJ
easat-9645	342	19	𝑞	𝑞	PROPN
easat-9645	342	20	,	,	PUNCT
easat-9645	342	21	𝑟⟩	𝑟⟩	X
easat-9645	342	22	=	=	SYM
easat-9645	342	23	⟨𝑝	⟨𝑝	NOUN
easat-9645	342	24	,	,	PUNCT
easat-9645	342	25	𝑟⟩	𝑟⟩	X
easat-9645	342	26	+	+	SYM
easat-9645	342	27	⟨𝑞	⟨𝑞	ADJ
easat-9645	342	28	,	,	PUNCT
easat-9645	342	29	𝑟⟩.	𝑟⟩.	VERB
easat-9645	342	30	then	then	ADV
easat-9645	342	31	we	we	PRON
easat-9645	342	32	applying	apply	VERB
easat-9645	342	33	the	the	DET
easat-9645	342	34	polarization	polarization	NOUN
easat-9645	342	35	identity	identity	NOUN
easat-9645	342	36	to	to	ADP
easat-9645	342	37	𝑝	𝑝	NOUN
easat-9645	342	38	+	+	CCONJ
easat-9645	342	39	𝑞	𝑞	PROPN
easat-9645	342	40	and	and	CCONJ
easat-9645	342	41	𝑟	𝑟	NOUN
easat-9645	342	42	,	,	PUNCT
easat-9645	342	43	expanding	expand	VERB
easat-9645	342	44	each	each	DET
easat-9645	342	45	term	term	NOUN
easat-9645	342	46	,	,	PUNCT
easat-9645	342	47	and	and	CCONJ
easat-9645	342	48	using	use	VERB
easat-9645	342	49	the	the	DET
easat-9645	342	50	parallelogram	parallelogram	NOUN
easat-9645	342	51	law	law	NOUN
easat-9645	342	52	repeatedly	repeatedly	ADV
easat-9645	342	53	s.t	s.t	PROPN
easat-9645	342	54	.	.	PUNCT
easat-9645	342	55	∥	∥	PUNCT
easat-9645	342	56	𝑝	𝑝	PROPN
easat-9645	343	1	+	+	NUM
easat-9645	343	2	𝑞	𝑞	X
easat-9645	343	3	+	+	X
easat-9645	343	4	𝑟	𝑟	NUM
easat-9645	343	5	∥2−∥	∥2−∥	PROPN
easat-9645	343	6	𝑝	𝑝	PROPN
easat-9645	343	7	+	+	CCONJ
easat-9645	343	8	𝑞	𝑞	X
easat-9645	343	9	−	−	PROPN
easat-9645	343	10	𝑟	𝑟	NOUN
easat-9645	343	11	∥2=	∥2=	NOUN
easat-9645	343	12	(	(	PUNCT
easat-9645	343	13	∥	∥	PROPN
easat-9645	343	14	𝑝	𝑝	NOUN
easat-9645	344	1	+	+	NUM
easat-9645	344	2	𝑟	𝑟	NOUN
easat-9645	344	3	∥2−∥	∥2−∥	PROPN
easat-9645	344	4	𝑝	𝑝	PROPN
easat-9645	344	5	−	−	NOUN
easat-9645	344	6	𝑟	𝑟	NOUN
easat-9645	344	7	∥2	∥2	X
easat-9645	344	8	)	)	PUNCT
easat-9645	345	1	+	+	CCONJ
easat-9645	345	2	(	(	PUNCT
easat-9645	345	3	∥	∥	X
easat-9645	345	4	𝑞	𝑞	X
easat-9645	345	5	+	+	NOUN
easat-9645	345	6	𝑟	𝑟	NOUN
easat-9645	345	7	∥2−∥	∥2−∥	NOUN
easat-9645	345	8	𝑞	𝑞	PART
easat-9645	345	9	−	−	PROPN
easat-9645	345	10	𝑟	𝑟	NOUN
easat-9645	345	11	∥2	∥2	X
easat-9645	345	12	)	)	PUNCT
easat-9645	345	13	similarly	similarly	ADV
easat-9645	345	14	for	for	ADP
easat-9645	345	15	the	the	DET
easat-9645	345	16	imaginary	imaginary	ADJ
easat-9645	345	17	parts	part	NOUN
easat-9645	345	18	,	,	PUNCT
easat-9645	345	19	⟨𝑝	⟨𝑝	NOUN
easat-9645	345	20	+	+	CCONJ
easat-9645	345	21	𝑞	𝑞	PROPN
easat-9645	345	22	,	,	PUNCT
easat-9645	345	23	𝑟⟩	𝑟⟩	X
easat-9645	345	24	=	=	SYM
easat-9645	345	25	⟨𝑝	⟨𝑝	NOUN
easat-9645	345	26	,	,	PUNCT
easat-9645	345	27	𝑟⟩	𝑟⟩	X
easat-9645	345	28	+	+	SYM
easat-9645	345	29	⟨𝑞	⟨𝑞	ADJ
easat-9645	345	30	,	,	PUNCT
easat-9645	345	31	𝑟⟩	𝑟⟩	X
easat-9645	345	32	3.4.4	3.4.4	NUM
easat-9645	345	33	.	.	PUNCT
easat-9645	346	1	homogeneity	homogeneity	NOUN
easat-9645	346	2	in	in	ADP
easat-9645	346	3	the	the	DET
easat-9645	346	4	first	first	ADJ
easat-9645	346	5	argument	argument	NOUN
easat-9645	346	6	we	we	PRON
easat-9645	346	7	verify	verify	VERB
easat-9645	346	8	for	for	ADP
easat-9645	346	9	scalar	scalar	ADJ
easat-9645	346	10	𝛼	𝛼	PRON
easat-9645	346	11	∈	∈	PROPN
easat-9645	346	12	𝐶	𝐶	PROPN
easat-9645	346	13	,	,	PUNCT
easat-9645	346	14	•	•	NOUN
easat-9645	347	1	𝐶𝑎𝑠𝑒	𝐶𝑎𝑠𝑒	PROPN
easat-9645	347	2	𝛼	𝛼	NOUN
easat-9645	347	3	=	=	PUNCT
easat-9645	347	4	𝑛	𝑛	PRON
easat-9645	347	5	∈	∈	PROPN
easat-9645	347	6	𝑁	𝑁	PROPN
easat-9645	347	7	:	:	PUNCT
easat-9645	347	8	𝑈𝑠𝑒	𝑈𝑠𝑒	PROPN
easat-9645	347	9	𝑖𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛	𝑖𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛	NOUN
easat-9645	347	10	𝑜𝑛	𝑜𝑛	PROPN
easat-9645	347	11	𝑛.	𝑛.	NOUN
easat-9645	347	12	•	•	ADP
easat-9645	348	1	𝐶𝑎𝑠𝑒	𝐶𝑎𝑠𝑒	PROPN
easat-9645	348	2	𝛼	𝛼	NOUN
easat-9645	348	3	=	=	NOUN
easat-9645	348	4	−1	−1	NOUN
easat-9645	348	5	:	:	PUNCT
easat-9645	349	1	𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒	𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑒	ADJ
easat-9645	349	2	−	−	PROPN
easat-9645	349	3	𝑝	𝑝	NOUN
easat-9645	349	4	𝑖𝑛	𝑖𝑛	NOUN
easat-9645	349	5	𝑡ℎ𝑒	𝑡ℎ𝑒	PROPN
easat-9645	349	6	𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦	𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦	NOUN
easat-9645	349	7	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
easat-9645	349	8	𝑠ℎ𝑜𝑤	𝑠ℎ𝑜𝑤	VERB
easat-9645	349	9	⟨−𝑝	⟨−𝑝	PROPN
easat-9645	349	10	,	,	PUNCT
easat-9645	349	11	𝑞⟩	𝑞⟩	VERB
easat-9645	349	12	=	=	PUNCT
easat-9645	349	13	−⟨𝑝	−⟨𝑝	NOUN
easat-9645	349	14	,	,	PUNCT
easat-9645	349	15	𝑞⟩	𝑞⟩	VERB
easat-9645	349	16	•	•	PUNCT
easat-9645	349	17	𝐶𝑎𝑠𝑒	𝐶𝑎𝑠𝑒	PROPN
easat-9645	349	18	𝛼	𝛼	NOUN
easat-9645	349	19	∈	∈	NOUN
easat-9645	349	20	𝑄	𝑄	NOUN
easat-9645	349	21	:	:	PUNCT
easat-9645	349	22	𝑊𝑟𝑖𝑡𝑒	𝑊𝑟𝑖𝑡𝑒	PROPN
easat-9645	349	23	𝛼	𝛼	NOUN
easat-9645	349	24	=	=	SYM
easat-9645	349	25	𝑠	𝑠	PROPN
easat-9645	349	26	𝑡	𝑡	PROPN
easat-9645	349	27	,	,	PUNCT
easat-9645	349	28	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-9645	349	29	𝑣𝑒𝑟𝑖𝑓𝑦	𝑣𝑒𝑟𝑖𝑓𝑦	NOUN
easat-9645	349	30	𝑢𝑠𝑖𝑛𝑔	𝑢𝑠𝑖𝑛𝑔	PROPN
easat-9645	349	31	𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑡𝑦	𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑡𝑦	PROPN
easat-9645	349	32	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
easat-9645	349	33	𝑠𝑐𝑎𝑙𝑎𝑟	𝑠𝑐𝑎𝑙𝑎𝑟	VERB
easat-9645	349	34	𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛.	𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛.	NOUN
easat-9645	349	35	•	•	ADP
easat-9645	350	1	𝐶𝑎𝑠𝑒	𝐶𝑎𝑠𝑒	PROPN
easat-9645	350	2	𝛼	𝛼	NOUN
easat-9645	350	3	=	=	SYM
easat-9645	350	4	𝑖	𝑖	SYM
easat-9645	350	5	:	:	PUNCT
easat-9645	350	6	𝑅𝑒𝑝𝑙𝑎𝑐𝑒	𝑅𝑒𝑝𝑙𝑎𝑐𝑒	PROPN
easat-9645	350	7	𝑝	𝑝	PROPN
easat-9645	350	8	𝑤𝑖𝑡ℎ	𝑤𝑖𝑡ℎ	NOUN
easat-9645	350	9	𝑖𝑝	𝑖𝑝	INTJ
easat-9645	350	10	𝑖𝑛	𝑖𝑛	PROPN
easat-9645	350	11	𝑡ℎ𝑒	𝑡ℎ𝑒	PROPN
easat-9645	350	12	𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦	𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦	PROPN
easat-9645	350	13	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-9645	350	14	𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦	𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦	NOUN
easat-9645	350	15	:	:	PUNCT
easat-9645	351	1	⟨𝑖𝑝	⟨𝑖𝑝	PROPN
easat-9645	351	2	,	,	PUNCT
easat-9645	351	3	𝑞⟩	𝑞⟩	VERB
easat-9645	351	4	=	=	SYM
easat-9645	351	5	𝑖⟨𝑝	𝑖⟨𝑝	PROPN
easat-9645	351	6	,	,	PUNCT
easat-9645	351	7	𝑞⟩	𝑞⟩	VERB
easat-9645	351	8	•	•	PUNCT
easat-9645	352	1	𝐶𝑎𝑠𝑒	𝐶𝑎𝑠𝑒	PROPN
easat-9645	352	2	𝛼	𝛼	NOUN
easat-9645	352	3	=	=	X
easat-9645	352	4	𝑎	𝑎	PROPN
easat-9645	352	5	+	+	NUM
easat-9645	352	6	𝑖𝑏	𝑖𝑏	INTJ
easat-9645	352	7	∈	∈	PROPN
easat-9645	352	8	𝐶	𝐶	PROPN
easat-9645	352	9	:	:	PUNCT
easat-9645	352	10	𝑈𝑠𝑒	𝑈𝑠𝑒	PROPN
easat-9645	352	11	𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑡𝑦	𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑡𝑦	PROPN
easat-9645	352	12	:	:	PUNCT
easat-9645	352	13	⟨𝛼𝑝	⟨𝛼𝑝	NOUN
easat-9645	352	14	,	,	PUNCT
easat-9645	352	15	𝑞⟩	𝑞⟩	VERB
easat-9645	352	16	=	=	PUNCT
easat-9645	352	17	⟨𝑎𝑝	⟨𝑎𝑝	PROPN
easat-9645	353	1	+	+	CCONJ
easat-9645	353	2	𝑖𝑏𝑝	𝑖𝑏𝑝	ADJ
easat-9645	353	3	,	,	PUNCT
easat-9645	353	4	𝑞⟩	𝑞⟩	VERB
easat-9645	353	5	=	=	SYM
easat-9645	354	1	𝑎⟨𝑝	𝑎⟨𝑝	PROPN
easat-9645	354	2	,	,	PUNCT
easat-9645	354	3	𝑞⟩	𝑞⟩	VERB
easat-9645	355	1	+	+	CCONJ
easat-9645	355	2	𝑖𝑏⟨𝑝	𝑖𝑏⟨𝑝	PROPN
easat-9645	355	3	,	,	PUNCT
easat-9645	355	4	𝑞⟩	𝑞⟩	VERB
easat-9645	355	5	=	=	SYM
easat-9645	356	1	𝛼⟨𝑝	𝛼⟨𝑝	PROPN
easat-9645	356	2	,	,	PUNCT
easat-9645	356	3	𝑞⟩	𝑞⟩	VERB
easat-9645	356	4	⇒	⇒	PROPN
easat-9645	356	5	⟨𝛼𝑝	⟨𝛼𝑝	PROPN
easat-9645	356	6	,	,	PUNCT
easat-9645	356	7	𝑞⟩	𝑞⟩	VERB
easat-9645	356	8	=	=	SYM
easat-9645	357	1	𝛼⟨𝑝	𝛼⟨𝑝	PROPN
easat-9645	357	2	,	,	PUNCT
easat-9645	357	3	𝑞.	𝑞.	VERB
easat-9645	357	4	the	the	DET
easat-9645	357	5	map	map	NOUN
easat-9645	357	6	⟨⋅,⋅⟩	⟨⋅,⋅⟩	PUNCT
easat-9645	357	7	satisfies	satisfie	NOUN
easat-9645	357	8	are	be	AUX
easat-9645	357	9	all	all	DET
easat-9645	357	10	the	the	DET
easat-9645	357	11	properties	property	NOUN
easat-9645	357	12	of	of	ADP
easat-9645	357	13	an	an	DET
easat-9645	357	14	inner	inner	ADJ
easat-9645	357	15	product	product	NOUN
easat-9645	357	16	.	.	PUNCT
easat-9645	358	1	since	since	SCONJ
easat-9645	358	2	b	b	PROPN
easat-9645	358	3	is	be	AUX
easat-9645	358	4	already	already	ADV
easat-9645	358	5	a	a	DET
easat-9645	358	6	banach	banach	NOUN
easat-9645	358	7	space	space	NOUN
easat-9645	358	8	(	(	PUNCT
easat-9645	358	9	i.e.	i.e.	X
easat-9645	358	10	,	,	PUNCT
easat-9645	358	11	complete	complete	ADJ
easat-9645	358	12	)	)	PUNCT
easat-9645	358	13	,	,	PUNCT
easat-9645	358	14	thus	thus	ADV
easat-9645	358	15	,	,	PUNCT
easat-9645	358	16	b	b	X
easat-9645	358	17	satisfied	satisfy	VERB
easat-9645	358	18	all	all	DET
easat-9645	358	19	the	the	DET
easat-9645	358	20	conditions	condition	NOUN
easat-9645	358	21	of	of	ADP
easat-9645	358	22	an	an	DET
easat-9645	358	23	inner	inner	ADJ
easat-9645	358	24	product	product	NOUN
easat-9645	358	25	space	space	NOUN
easat-9645	358	26	.	.	PUNCT
easat-9645	359	1	therefore	therefore	ADV
easat-9645	359	2	𝐵	𝐵	PROPN
easat-9645	359	3	is	be	AUX
easat-9645	359	4	an	an	DET
easat-9645	359	5	inner	inner	ADJ
easat-9645	359	6	product	product	NOUN
easat-9645	359	7	space	space	NOUN
easat-9645	359	8	and	and	CCONJ
easat-9645	359	9	hence	hence	ADV
easat-9645	359	10	𝐵	𝐵	PROPN
easat-9645	359	11	is	be	AUX
easat-9645	359	12	a	a	DET
easat-9645	359	13	hilbert	hilbert	NOUN
easat-9645	359	14	space	space	NOUN
easat-9645	359	15	.	.	PUNCT
easat-9645	360	1	3.4.5	3.4.5	X
easat-9645	360	2	.	.	PUNCT
easat-9645	360	3	interpretation	interpretation	NOUN
easat-9645	360	4	•	•	ADP
easat-9645	361	1	the	the	DET
easat-9645	361	2	polarization	polarization	NOUN
easat-9645	361	3	identity	identity	NOUN
easat-9645	361	4	extracts	extract	NOUN
easat-9645	361	5	full	full	ADJ
easat-9645	361	6	geometric	geometric	ADJ
easat-9645	361	7	information	information	NOUN
easat-9645	361	8	(	(	PUNCT
easat-9645	361	9	lengths	length	NOUN
easat-9645	361	10	and	and	CCONJ
easat-9645	361	11	angles	angle	NOUN
easat-9645	361	12	)	)	PUNCT
easat-9645	361	13	from	from	ADP
easat-9645	361	14	norms	norm	NOUN
easat-9645	361	15	.	.	PUNCT
easat-9645	362	1	•	•	NUM
easat-9645	362	2	by	by	ADP
easat-9645	362	3	measuring	measure	VERB
easat-9645	362	4	lengths	length	NOUN
easat-9645	362	5	of	of	ADP
easat-9645	362	6	carefully	carefully	ADV
easat-9645	362	7	combined	combine	VERB
easat-9645	362	8	vectors	vector	NOUN
easat-9645	362	9	,	,	PUNCT
easat-9645	362	10	it	it	PRON
easat-9645	362	11	reconstructs	reconstruct	VERB
easat-9645	362	12	the	the	DET
easat-9645	362	13	inner	inner	ADJ
easat-9645	362	14	product	product	NOUN
easat-9645	362	15	the	the	DET
easat-9645	362	16	fundamental	fundamental	ADJ
easat-9645	362	17	tool	tool	NOUN
easat-9645	362	18	for	for	ADP
easat-9645	362	19	angle	angle	NOUN
easat-9645	362	20	,	,	PUNCT
easat-9645	362	21	projection	projection	NOUN
easat-9645	362	22	,	,	PUNCT
easat-9645	362	23	and	and	CCONJ
easat-9645	362	24	orthogonality	orthogonality	NOUN
easat-9645	362	25	in	in	ADP
easat-9645	362	26	hilbert	hilbert	PROPN
easat-9645	362	27	spaces	space	NOUN
easat-9645	362	28	.	.	PUNCT
easat-9645	363	1	•	•	NUM
easat-9645	363	2	the	the	DET
easat-9645	363	3	matlab	matlab	PROPN
easat-9645	363	4	plot	plot	NOUN
easat-9645	363	5	shows	show	VERB
easat-9645	363	6	how	how	SCONJ
easat-9645	363	7	real	real	ADJ
easat-9645	363	8	and	and	CCONJ
easat-9645	363	9	imaginary	imaginary	ADJ
easat-9645	363	10	combinations	combination	NOUN
easat-9645	363	11	of	of	ADP
easat-9645	363	12	vectors	vector	NOUN
easat-9645	363	13	contribute	contribute	VERB
easat-9645	363	14	to	to	ADP
easat-9645	363	15	the	the	DET
easat-9645	363	16	full	full	ADJ
easat-9645	363	17	inner	inner	ADJ
easat-9645	363	18	product	product	NOUN
easat-9645	363	19	.	.	PUNCT
easat-9645	364	1	1512	1512	NUM
easat-9645	364	2	edelweiss	edelweiss	PROPN
easat-9645	364	3	applied	apply	VERB
easat-9645	364	4	science	science	NOUN
easat-9645	364	5	and	and	CCONJ
easat-9645	364	6	technology	technology	NOUN
easat-9645	364	7	issn	issn	PROPN
easat-9645	364	8	:	:	PUNCT
easat-9645	364	9	2576	2576	NUM
easat-9645	364	10	-	-	SYM
easat-9645	364	11	8484	8484	NUM
easat-9645	364	12	vol	vol	NOUN
easat-9645	364	13	.	.	PROPN
easat-9645	365	1	9	9	NUM
easat-9645	365	2	,	,	PUNCT
easat-9645	365	3	no	no	INTJ
easat-9645	365	4	.	.	NOUN
easat-9645	365	5	8	8	NUM
easat-9645	365	6	:	:	SYM
easat-9645	365	7	1498	1498	NUM
easat-9645	365	8	-	-	SYM
easat-9645	365	9	1523	1523	NUM
easat-9645	365	10	,	,	PUNCT
easat-9645	365	11	2025	2025	NUM
easat-9645	365	12	doi	doi	NOUN
easat-9645	365	13	:	:	PUNCT
easat-9645	365	14	10.55214/2576	10.55214/2576	NUM
easat-9645	365	15	-	-	SYM
easat-9645	365	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	365	17	©	©	PROPN
easat-9645	365	18	2025	2025	NUM
easat-9645	365	19	by	by	ADP
easat-9645	365	20	the	the	DET
easat-9645	365	21	authors	author	NOUN
easat-9645	365	22	;	;	PUNCT
easat-9645	365	23	licensee	licensee	PROPN
easat-9645	365	24	learning	learn	VERB
easat-9645	365	25	gate	gate	PROPN
easat-9645	365	26	figure	figure	NOUN
easat-9645	365	27	8	8	NUM
easat-9645	365	28	.	.	PUNCT
easat-9645	366	1	matlab	matlab	PROPN
easat-9645	366	2	visualization	visualization	NOUN
easat-9645	366	3	of	of	ADP
easat-9645	366	4	the	the	DET
easat-9645	366	5	polarization	polarization	NOUN
easat-9645	366	6	identity	identity	NOUN
easat-9645	366	7	.	.	PUNCT
easat-9645	367	1	the	the	DET
easat-9645	367	2	figure	figure	NOUN
easat-9645	367	3	8	8	NUM
easat-9645	367	4	,	,	PUNCT
easat-9645	367	5	visualizes	visualize	VERB
easat-9645	367	6	the	the	DET
easat-9645	367	7	polarization	polarization	NOUN
easat-9645	367	8	identity	identity	NOUN
easat-9645	367	9	in	in	ADP
easat-9645	367	10	𝐶2showing	𝐶2showing	NOUN
easat-9645	367	11	vectors	vector	NOUN
easat-9645	367	12	x	x	SYM
easat-9645	367	13	(	(	PUNCT
easat-9645	367	14	red	red	PROPN
easat-9645	367	15	)	)	PUNCT
easat-9645	367	16	,	,	PUNCT
easat-9645	367	17	y	y	PROPN
easat-9645	367	18	(	(	PUNCT
easat-9645	367	19	blue	blue	PROPN
easat-9645	367	20	)	)	PUNCT
easat-9645	367	21	,	,	PUNCT
easat-9645	367	22	x	x	X
easat-9645	368	1	+	+	CCONJ
easat-9645	368	2	y	y	PROPN
easat-9645	368	3	(	(	PUNCT
easat-9645	368	4	green	green	NOUN
easat-9645	368	5	)	)	PUNCT
easat-9645	368	6	,	,	PUNCT
easat-9645	368	7	x	x	X
easat-9645	368	8	+	+	NUM
easat-9645	368	9	iy	iy	PROPN
easat-9645	368	10	(	(	PUNCT
easat-9645	368	11	magenta	magenta	PROPN
easat-9645	368	12	)	)	PUNCT
easat-9645	368	13	,	,	PUNCT
easat-9645	368	14	and	and	CCONJ
easat-9645	368	15	x	x	X
easat-9645	368	16	−	−	NOUN
easat-9645	368	17	iy(cyan	iy(cyan	ADJ
easat-9645	368	18	)	)	PUNCT
easat-9645	368	19	originating	originate	VERB
easat-9645	368	20	from	from	ADP
easat-9645	368	21	the	the	DET
easat-9645	368	22	origin	origin	NOUN
easat-9645	368	23	in	in	ADP
easat-9645	368	24	a	a	DET
easat-9645	368	25	3d	3d	NUM
easat-9645	368	26	space	space	NOUN
easat-9645	368	27	with	with	ADP
easat-9645	368	28	axes	axis	NOUN
easat-9645	368	29	𝑅𝑒(𝑥1	𝑅𝑒(𝑥1	NOUN
easat-9645	368	30	)	)	PUNCT
easat-9645	368	31	,	,	PUNCT
easat-9645	368	32	𝑅𝑒(𝑥2	𝑅𝑒(𝑥2	NUM
easat-9645	368	33	)	)	PUNCT
easat-9645	368	34	,	,	PUNCT
easat-9645	368	35	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-9645	368	36	𝐼𝑚	𝐼𝑚	PROPN
easat-9645	368	37	(	(	PUNCT
easat-9645	368	38	𝑥	𝑥	NOUN
easat-9645	368	39	)	)	PUNCT
easat-9645	368	40	,	,	PUNCT
easat-9645	368	41	illustrating	illustrate	VERB
easat-9645	368	42	the	the	DET
easat-9645	368	43	relationship	relationship	NOUN
easat-9645	368	44	between	between	ADP
easat-9645	368	45	complex	complex	ADJ
easat-9645	368	46	vector	vector	NOUN
easat-9645	368	47	components	component	NOUN
easat-9645	368	48	.	.	PUNCT
easat-9645	369	1	3.5	3.5	NUM
easat-9645	369	2	.	.	PUNCT
easat-9645	369	3	discussion	discussion	NOUN
easat-9645	369	4	the	the	DET
easat-9645	369	5	preceding	precede	VERB
easat-9645	369	6	results	result	NOUN
easat-9645	369	7	confirm	confirm	VERB
easat-9645	369	8	that	that	SCONJ
easat-9645	369	9	ℝ𝑛	ℝ𝑛	PROPN
easat-9645	369	10	,	,	PUNCT
easat-9645	369	11	𝐶𝑛	𝐶𝑛	PROPN
easat-9645	369	12	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
easat-9645	369	13	𝑙2are	𝑙2are	PROPN
easat-9645	369	14	hilbert	hilbert	NOUN
easat-9645	369	15	spaces	space	NOUN
easat-9645	369	16	by	by	ADP
easat-9645	369	17	virtue	virtue	NOUN
easat-9645	369	18	of	of	ADP
easat-9645	369	19	possessing	possess	VERB
easat-9645	369	20	both	both	CCONJ
easat-9645	369	21	an	an	DET
easat-9645	369	22	inner	inner	ADJ
easat-9645	369	23	product	product	NOUN
easat-9645	369	24	structure	structure	NOUN
easat-9645	369	25	and	and	CCONJ
easat-9645	369	26	completeness	completeness	NOUN
easat-9645	369	27	.	.	PUNCT
easat-9645	370	1	these	these	DET
easat-9645	370	2	two	two	NUM
easat-9645	370	3	features	feature	NOUN
easat-9645	370	4	are	be	AUX
easat-9645	370	5	essential	essential	ADJ
easat-9645	370	6	in	in	ADP
easat-9645	370	7	extending	extend	VERB
easat-9645	370	8	geometric	geometric	ADJ
easat-9645	370	9	intuition	intuition	NOUN
easat-9645	370	10	from	from	ADP
easat-9645	370	11	euclidean	euclidean	ADJ
easat-9645	370	12	spaces	space	NOUN
easat-9645	370	13	to	to	ADP
easat-9645	370	14	more	more	ADJ
easat-9645	370	15	abstract	abstract	ADJ
easat-9645	370	16	settings	setting	NOUN
easat-9645	370	17	.	.	PUNCT
easat-9645	371	1	orthogonality	orthogonality	NOUN
easat-9645	371	2	allows	allow	VERB
easat-9645	371	3	for	for	ADP
easat-9645	371	4	decomposition	decomposition	NOUN
easat-9645	371	5	and	and	CCONJ
easat-9645	371	6	projection	projection	NOUN
easat-9645	371	7	of	of	ADP
easat-9645	371	8	elements	element	NOUN
easat-9645	371	9	a	a	DET
easat-9645	371	10	mechanism	mechanism	NOUN
easat-9645	371	11	at	at	ADP
easat-9645	371	12	the	the	DET
easat-9645	371	13	heart	heart	NOUN
easat-9645	371	14	of	of	ADP
easat-9645	371	15	many	many	ADJ
easat-9645	371	16	computational	computational	ADJ
easat-9645	371	17	techniques	technique	NOUN
easat-9645	371	18	such	such	ADJ
easat-9645	371	19	as	as	ADP
easat-9645	371	20	principal	principal	ADJ
easat-9645	371	21	component	component	NOUN
easat-9645	371	22	analysis	analysis	NOUN
easat-9645	371	23	(	(	PUNCT
easat-9645	371	24	pca	pca	NOUN
easat-9645	371	25	)	)	PUNCT
easat-9645	371	26	in	in	ADP
easat-9645	371	27	machine	machine	NOUN
easat-9645	371	28	learning	learning	NOUN
easat-9645	371	29	.	.	PUNCT
easat-9645	372	1	completeness	completeness	NOUN
easat-9645	372	2	guarantees	guarantee	VERB
easat-9645	372	3	that	that	SCONJ
easat-9645	372	4	limits	limit	NOUN
easat-9645	372	5	of	of	ADP
easat-9645	372	6	cauchy	cauchy	ADJ
easat-9645	372	7	sequences	sequence	NOUN
easat-9645	372	8	remain	remain	VERB
easat-9645	372	9	within	within	ADP
easat-9645	372	10	the	the	DET
easat-9645	372	11	space	space	NOUN
easat-9645	372	12	,	,	PUNCT
easat-9645	372	13	providing	provide	VERB
easat-9645	372	14	a	a	DET
easat-9645	372	15	foundation	foundation	NOUN
easat-9645	372	16	for	for	ADP
easat-9645	372	17	the	the	DET
easat-9645	372	18	convergence	convergence	NOUN
easat-9645	372	19	of	of	ADP
easat-9645	372	20	iterative	iterative	ADJ
easat-9645	372	21	numerical	numerical	ADJ
easat-9645	372	22	algorithms	algorithm	NOUN
easat-9645	372	23	.	.	PUNCT
easat-9645	373	1	together	together	ADV
easat-9645	373	2	,	,	PUNCT
easat-9645	373	3	these	these	DET
easat-9645	373	4	properties	property	NOUN
easat-9645	373	5	support	support	VERB
easat-9645	373	6	a	a	DET
easat-9645	373	7	broad	broad	ADJ
easat-9645	373	8	range	range	NOUN
easat-9645	373	9	of	of	ADP
easat-9645	373	10	applications	application	NOUN
easat-9645	373	11	,	,	PUNCT
easat-9645	373	12	from	from	ADP
easat-9645	373	13	the	the	DET
easat-9645	373	14	mathematical	mathematical	ADJ
easat-9645	373	15	formulation	formulation	NOUN
easat-9645	373	16	of	of	ADP
easat-9645	373	17	quantum	quantum	NOUN
easat-9645	373	18	systems	system	NOUN
easat-9645	373	19	to	to	PART
easat-9645	373	20	signal	signal	VERB
easat-9645	373	21	representation	representation	NOUN
easat-9645	373	22	and	and	CCONJ
easat-9645	373	23	processing	processing	NOUN
easat-9645	373	24	in	in	ADP
easat-9645	373	25	engineering	engineering	NOUN
easat-9645	373	26	.	.	PUNCT
easat-9645	374	1	4	4	X
easat-9645	374	2	.	.	X
easat-9645	374	3	applications	application	NOUN
easat-9645	374	4	of	of	ADP
easat-9645	374	5	hilbert	hilbert	PROPN
easat-9645	374	6	spaces	space	VERB
easat-9645	374	7	4.1	4.1	NUM
easat-9645	374	8	.	.	PUNCT
easat-9645	374	9	quantum	quantum	ADJ
easat-9645	374	10	mechanics	mechanic	NOUN
easat-9645	374	11	in	in	ADP
easat-9645	374	12	quantum	quantum	ADJ
easat-9645	374	13	mechanics	mechanic	NOUN
easat-9645	374	14	,	,	PUNCT
easat-9645	374	15	the	the	DET
easat-9645	374	16	state	state	NOUN
easat-9645	374	17	of	of	ADP
easat-9645	374	18	a	a	DET
easat-9645	374	19	physical	physical	ADJ
easat-9645	374	20	system	system	NOUN
easat-9645	374	21	is	be	AUX
easat-9645	374	22	described	describe	VERB
easat-9645	374	23	by	by	ADP
easat-9645	374	24	a	a	DET
easat-9645	374	25	vector	vector	NOUN
easat-9645	374	26	in	in	ADP
easat-9645	374	27	a	a	DET
easat-9645	374	28	complex	complex	ADJ
easat-9645	374	29	hilbert	hilbert	NOUN
easat-9645	374	30	space	space	NOUN
easat-9645	374	31	.	.	PUNCT
easat-9645	375	1	observable	observable	ADJ
easat-9645	375	2	physical	physical	ADJ
easat-9645	375	3	quantities	quantity	NOUN
easat-9645	375	4	correspond	correspond	VERB
easat-9645	375	5	to	to	ADP
easat-9645	375	6	self	self	NOUN
easat-9645	375	7	-	-	PUNCT
easat-9645	375	8	adjoint	adjoint	NOUN
easat-9645	375	9	(	(	PUNCT
easat-9645	375	10	hermitian	hermitian	PROPN
easat-9645	375	11	)	)	PUNCT
easat-9645	375	12	operators	operator	NOUN
easat-9645	375	13	acting	act	VERB
easat-9645	375	14	on	on	ADP
easat-9645	375	15	this	this	DET
easat-9645	375	16	space	space	NOUN
easat-9645	375	17	.	.	PUNCT
easat-9645	376	1	the	the	DET
easat-9645	376	2	inner	inner	ADJ
easat-9645	376	3	product	product	NOUN
easat-9645	376	4	on	on	ADP
easat-9645	376	5	the	the	DET
easat-9645	376	6	hilbert	hilbert	NOUN
easat-9645	376	7	space	space	NOUN
easat-9645	376	8	is	be	AUX
easat-9645	376	9	crucial	crucial	ADJ
easat-9645	376	10	for	for	ADP
easat-9645	376	11	interpreting	interpret	VERB
easat-9645	376	12	measurement	measurement	NOUN
easat-9645	376	13	results	result	NOUN
easat-9645	376	14	probabilistically	probabilistically	ADV
easat-9645	376	15	:	:	PUNCT
easat-9645	376	16	the	the	DET
easat-9645	376	17	squared	square	VERB
easat-9645	376	18	magnitude	magnitude	NOUN
easat-9645	376	19	of	of	ADP
easat-9645	376	20	the	the	DET
easat-9645	376	21	inner	inner	ADJ
easat-9645	376	22	product	product	NOUN
easat-9645	376	23	between	between	ADP
easat-9645	376	24	two	two	NUM
easat-9645	376	25	state	state	NOUN
easat-9645	376	26	vectors	vector	NOUN
easat-9645	376	27	represents	represent	VERB
easat-9645	376	28	the	the	DET
easat-9645	376	29	1513	1513	NUM
easat-9645	376	30	edelweiss	edelweiss	PROPN
easat-9645	376	31	applied	apply	VERB
easat-9645	376	32	science	science	NOUN
easat-9645	376	33	and	and	CCONJ
easat-9645	376	34	technology	technology	NOUN
easat-9645	376	35	issn	issn	PROPN
easat-9645	376	36	:	:	PUNCT
easat-9645	376	37	2576	2576	NUM
easat-9645	376	38	-	-	SYM
easat-9645	376	39	8484	8484	NUM
easat-9645	376	40	vol	vol	NOUN
easat-9645	376	41	.	.	PROPN
easat-9645	377	1	9	9	NUM
easat-9645	377	2	,	,	PUNCT
easat-9645	377	3	no	no	INTJ
easat-9645	377	4	.	.	NOUN
easat-9645	377	5	8	8	NUM
easat-9645	377	6	:	:	SYM
easat-9645	377	7	1498	1498	NUM
easat-9645	377	8	-	-	SYM
easat-9645	377	9	1523	1523	NUM
easat-9645	377	10	,	,	PUNCT
easat-9645	377	11	2025	2025	NUM
easat-9645	377	12	doi	doi	NOUN
easat-9645	377	13	:	:	PUNCT
easat-9645	377	14	10.55214/2576	10.55214/2576	NUM
easat-9645	377	15	-	-	SYM
easat-9645	377	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	377	17	©	©	PROPN
easat-9645	377	18	2025	2025	NUM
easat-9645	377	19	by	by	ADP
easat-9645	377	20	the	the	DET
easat-9645	377	21	authors	author	NOUN
easat-9645	377	22	;	;	PUNCT
easat-9645	377	23	licensee	licensee	PROPN
easat-9645	377	24	learning	learning	NOUN
easat-9645	377	25	gate	gate	NOUN
easat-9645	377	26	probability	probability	NOUN
easat-9645	377	27	of	of	ADP
easat-9645	377	28	transitioning	transition	VERB
easat-9645	377	29	from	from	ADP
easat-9645	377	30	one	one	NUM
easat-9645	377	31	state	state	NOUN
easat-9645	377	32	to	to	ADP
easat-9645	377	33	another	another	PRON
easat-9645	377	34	.	.	PUNCT
easat-9645	378	1	this	this	DET
easat-9645	378	2	framework	framework	NOUN
easat-9645	378	3	offers	offer	VERB
easat-9645	378	4	a	a	DET
easat-9645	378	5	precise	precise	ADJ
easat-9645	378	6	mathematical	mathematical	ADJ
easat-9645	378	7	foundation	foundation	NOUN
easat-9645	378	8	for	for	ADP
easat-9645	378	9	the	the	DET
easat-9645	378	10	inherently	inherently	ADV
easat-9645	378	11	probabilistic	probabilistic	ADJ
easat-9645	378	12	behavior	behavior	NOUN
easat-9645	378	13	observed	observe	VERB
easat-9645	378	14	in	in	ADP
easat-9645	378	15	quantum	quantum	NOUN
easat-9645	378	16	systems	system	NOUN
easat-9645	378	17	.	.	PUNCT
easat-9645	379	1	4.2	4.2	NUM
easat-9645	379	2	.	.	PUNCT
easat-9645	379	3	signal	signal	NOUN
easat-9645	379	4	and	and	CCONJ
easat-9645	379	5	image	image	NOUN
easat-9645	379	6	processing	processing	NOUN
easat-9645	379	7	hilbert	hilbert	NOUN
easat-9645	379	8	spaces	space	NOUN
easat-9645	379	9	are	be	AUX
easat-9645	379	10	fundamental	fundamental	ADJ
easat-9645	379	11	in	in	ADP
easat-9645	379	12	fourier	fourier	ADJ
easat-9645	379	13	analysis	analysis	NOUN
easat-9645	379	14	and	and	CCONJ
easat-9645	379	15	the	the	DET
easat-9645	379	16	theory	theory	NOUN
easat-9645	379	17	of	of	ADP
easat-9645	379	18	wavelets	wavelet	NOUN
easat-9645	379	19	.	.	PUNCT
easat-9645	380	1	signals	signal	NOUN
easat-9645	380	2	are	be	AUX
easat-9645	380	3	often	often	ADV
easat-9645	380	4	modeled	model	VERB
easat-9645	380	5	as	as	ADP
easat-9645	380	6	elements	element	NOUN
easat-9645	380	7	of	of	ADP
easat-9645	380	8	l2	l2	NOUN
easat-9645	380	9	spaces	space	NOUN
easat-9645	380	10	,	,	PUNCT
easat-9645	380	11	and	and	CCONJ
easat-9645	380	12	decomposing	decompose	VERB
easat-9645	380	13	signals	signal	NOUN
easat-9645	380	14	into	into	ADP
easat-9645	380	15	orthonormal	orthonormal	ADJ
easat-9645	380	16	bases	basis	NOUN
easat-9645	380	17	(	(	PUNCT
easat-9645	380	18	e.g.	e.g.	ADV
easat-9645	380	19	,	,	PUNCT
easat-9645	380	20	sine	sine	ADJ
easat-9645	380	21	/	/	SYM
easat-9645	380	22	cosine	cosine	NOUN
easat-9645	380	23	in	in	ADP
easat-9645	380	24	fourier	fourier	NOUN
easat-9645	380	25	transforms	transform	NOUN
easat-9645	380	26	)	)	PUNCT
easat-9645	380	27	is	be	AUX
easat-9645	380	28	a	a	DET
easat-9645	380	29	core	core	NOUN
easat-9645	380	30	technique	technique	NOUN
easat-9645	380	31	.	.	PUNCT
easat-9645	381	1	4.3	4.3	NUM
easat-9645	381	2	.	.	PUNCT
easat-9645	381	3	machine	machine	NOUN
easat-9645	381	4	learning	learning	NOUN
easat-9645	381	5	and	and	CCONJ
easat-9645	381	6	data	datum	NOUN
easat-9645	381	7	science	science	NOUN
easat-9645	381	8	reproducing	reproduce	VERB
easat-9645	381	9	kernel	kernel	PROPN
easat-9645	381	10	hilbert	hilbert	PROPN
easat-9645	381	11	spaces	space	NOUN
easat-9645	381	12	(	(	PUNCT
easat-9645	381	13	rkhs	rkh	NOUN
easat-9645	381	14	)	)	PUNCT
easat-9645	381	15	are	be	AUX
easat-9645	381	16	used	use	VERB
easat-9645	381	17	in	in	ADP
easat-9645	381	18	support	support	NOUN
easat-9645	381	19	vector	vector	NOUN
easat-9645	381	20	machines	machine	NOUN
easat-9645	381	21	and	and	CCONJ
easat-9645	381	22	gaussian	gaussian	ADJ
easat-9645	381	23	processes	process	NOUN
easat-9645	381	24	.	.	PUNCT
easat-9645	382	1	these	these	DET
easat-9645	382	2	spaces	space	NOUN
easat-9645	382	3	enable	enable	VERB
easat-9645	382	4	kernel	kernel	NOUN
easat-9645	382	5	methods	method	NOUN
easat-9645	382	6	that	that	PRON
easat-9645	382	7	efficiently	efficiently	ADV
easat-9645	382	8	operate	operate	VERB
easat-9645	382	9	in	in	ADP
easat-9645	382	10	high	high	ADJ
easat-9645	382	11	-	-	PUNCT
easat-9645	382	12	dimensional	dimensional	ADJ
easat-9645	382	13	or	or	CCONJ
easat-9645	382	14	infinitedimensional	infinitedimensional	ADJ
easat-9645	382	15	feature	feature	NOUN
easat-9645	382	16	spaces	space	NOUN
easat-9645	382	17	.	.	PUNCT
easat-9645	383	1	4.4	4.4	NUM
easat-9645	383	2	.	.	PUNCT
easat-9645	384	1	numerical	numerical	ADJ
easat-9645	384	2	analysis	analysis	NOUN
easat-9645	384	3	and	and	CCONJ
easat-9645	384	4	pdes	pde	NOUN
easat-9645	384	5	hilbert	hilbert	NOUN
easat-9645	384	6	spaces	space	NOUN
easat-9645	384	7	provide	provide	VERB
easat-9645	384	8	the	the	DET
easat-9645	384	9	setting	setting	NOUN
easat-9645	384	10	for	for	ADP
easat-9645	384	11	the	the	DET
easat-9645	384	12	variational	variational	ADJ
easat-9645	384	13	formulation	formulation	NOUN
easat-9645	384	14	of	of	ADP
easat-9645	384	15	partial	partial	ADJ
easat-9645	384	16	differential	differential	ADJ
easat-9645	384	17	equations	equation	NOUN
easat-9645	384	18	.	.	PUNCT
easat-9645	385	1	the	the	DET
easat-9645	385	2	famous	famous	ADJ
easat-9645	385	3	lax	lax	PROPN
easat-9645	385	4	-	-	PUNCT
easat-9645	385	5	milgram	milgram	NOUN
easat-9645	385	6	theorem	theorem	NOUN
easat-9645	385	7	,	,	PUNCT
easat-9645	385	8	for	for	ADP
easat-9645	385	9	instance	instance	NOUN
easat-9645	385	10	,	,	PUNCT
easat-9645	385	11	guarantees	guarantee	VERB
easat-9645	385	12	the	the	DET
easat-9645	385	13	existence	existence	NOUN
easat-9645	385	14	and	and	CCONJ
easat-9645	385	15	uniqueness	uniqueness	NOUN
easat-9645	385	16	of	of	ADP
easat-9645	385	17	solutions	solution	NOUN
easat-9645	385	18	under	under	ADP
easat-9645	385	19	certain	certain	ADJ
easat-9645	385	20	conditions	condition	NOUN
easat-9645	385	21	in	in	ADP
easat-9645	385	22	a	a	DET
easat-9645	385	23	hilbert	hilbert	NOUN
easat-9645	385	24	space	space	NOUN
easat-9645	385	25	setting	setting	NOUN
easat-9645	385	26	.	.	PUNCT
easat-9645	386	1	5	5	X
easat-9645	386	2	.	.	X
easat-9645	386	3	applications	application	NOUN
easat-9645	386	4	of	of	ADP
easat-9645	386	5	hilbert	hilbert	PROPN
easat-9645	386	6	spaces	space	NOUN
easat-9645	386	7	in	in	ADP
easat-9645	386	8	machine	machine	NOUN
easat-9645	386	9	learning	learn	VERB
easat-9645	386	10	5.1	5.1	NUM
easat-9645	386	11	.	.	PUNCT
easat-9645	387	1	kernel	kernel	PROPN
easat-9645	387	2	methods	method	NOUN
easat-9645	387	3	and	and	CCONJ
easat-9645	387	4	reproducing	reproduce	VERB
easat-9645	387	5	kernel	kernel	PROPN
easat-9645	387	6	hilbert	hilbert	PROPN
easat-9645	387	7	spaces	space	NOUN
easat-9645	387	8	(	(	PUNCT
easat-9645	387	9	rkhs	rkh	NOUN
easat-9645	387	10	)	)	PUNCT
easat-9645	388	1	[	[	X
easat-9645	388	2	17	17	NUM
easat-9645	388	3	,	,	PUNCT
easat-9645	388	4	18	18	NUM
easat-9645	388	5	]	]	PUNCT
easat-9645	388	6	.	.	PUNCT
easat-9645	389	1	kernel	kernel	PROPN
easat-9645	389	2	methods	method	NOUN
easat-9645	389	3	implicitly	implicitly	ADV
easat-9645	389	4	map	map	VERB
easat-9645	389	5	data	datum	NOUN
easat-9645	389	6	into	into	ADP
easat-9645	389	7	high	high	ADJ
easat-9645	389	8	-	-	PUNCT
easat-9645	389	9	dimensional	dimensional	ADJ
easat-9645	389	10	hilbert	hilbert	NOUN
easat-9645	389	11	spaces	space	NOUN
easat-9645	389	12	without	without	ADP
easat-9645	389	13	computing	compute	VERB
easat-9645	389	14	the	the	DET
easat-9645	389	15	coordinates	coordinate	NOUN
easat-9645	389	16	directly	directly	ADV
easat-9645	389	17	.	.	PUNCT
easat-9645	390	1	this	this	PRON
easat-9645	390	2	allows	allow	VERB
easat-9645	390	3	for	for	ADP
easat-9645	390	4	:	:	PUNCT
easat-9645	390	5	non	non	ADJ
easat-9645	390	6	-	-	ADJ
easat-9645	390	7	linear	linear	ADJ
easat-9645	390	8	classification	classification	NOUN
easat-9645	390	9	using	use	VERB
easat-9645	390	10	linear	linear	ADJ
easat-9645	390	11	techniques	technique	NOUN
easat-9645	390	12	in	in	ADP
easat-9645	390	13	transformed	transform	VERB
easat-9645	390	14	spaces	space	NOUN
easat-9645	390	15	.	.	PUNCT
easat-9645	391	1	efficient	efficient	ADJ
easat-9645	391	2	implementation	implementation	NOUN
easat-9645	391	3	via	via	ADP
easat-9645	391	4	the	the	DET
easat-9645	391	5	kernel	kernel	PROPN
easat-9645	391	6	trick	trick	NOUN
easat-9645	391	7	:	:	PUNCT
easat-9645	392	1	k(x	k(x	PROPN
easat-9645	392	2	,	,	PUNCT
easat-9645	392	3	y	y	NOUN
easat-9645	392	4	)	)	PUNCT
easat-9645	392	5	=	=	PUNCT
easat-9645	392	6	⟨ϕ(x	⟨ϕ(x	NOUN
easat-9645	392	7	)	)	PUNCT
easat-9645	392	8	,	,	PUNCT
easat-9645	392	9	ϕ(y)⟩	ϕ(y)⟩	PART
easat-9645	392	10	figure	figure	VERB
easat-9645	392	11	9	9	NUM
easat-9645	392	12	.	.	PUNCT
easat-9645	392	13	matlab	matlab	PROPN
easat-9645	392	14	implementation	implementation	NOUN
easat-9645	392	15	of	of	ADP
easat-9645	392	16	the	the	DET
easat-9645	392	17	visualizing	visualize	VERB
easat-9645	392	18	a	a	DET
easat-9645	392	19	gaussian	gaussian	NOUN
easat-9645	392	20	(	(	PUNCT
easat-9645	392	21	rbf	rbf	PROPN
easat-9645	392	22	)	)	PUNCT
easat-9645	392	23	kernel	kernel	PROPN
easat-9645	392	24	and	and	CCONJ
easat-9645	392	25	feature	feature	NOUN
easat-9645	392	26	mapping	mapping	NOUN
easat-9645	392	27	.	.	PUNCT
easat-9645	393	1	1514	1514	NUM
easat-9645	393	2	edelweiss	edelweiss	PROPN
easat-9645	393	3	applied	apply	VERB
easat-9645	393	4	science	science	NOUN
easat-9645	393	5	and	and	CCONJ
easat-9645	393	6	technology	technology	NOUN
easat-9645	393	7	issn	issn	PROPN
easat-9645	393	8	:	:	PUNCT
easat-9645	393	9	2576	2576	NUM
easat-9645	393	10	-	-	SYM
easat-9645	393	11	8484	8484	NUM
easat-9645	393	12	vol	vol	NOUN
easat-9645	393	13	.	.	PROPN
easat-9645	394	1	9	9	NUM
easat-9645	394	2	,	,	PUNCT
easat-9645	394	3	no	no	INTJ
easat-9645	394	4	.	.	NOUN
easat-9645	394	5	8	8	NUM
easat-9645	394	6	:	:	SYM
easat-9645	394	7	1498	1498	NUM
easat-9645	394	8	-	-	SYM
easat-9645	394	9	1523	1523	NUM
easat-9645	394	10	,	,	PUNCT
easat-9645	394	11	2025	2025	NUM
easat-9645	394	12	doi	doi	NOUN
easat-9645	394	13	:	:	PUNCT
easat-9645	394	14	10.55214/2576	10.55214/2576	NUM
easat-9645	394	15	-	-	SYM
easat-9645	394	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	394	17	©	©	PROPN
easat-9645	394	18	2025	2025	NUM
easat-9645	394	19	by	by	ADP
easat-9645	394	20	the	the	DET
easat-9645	394	21	authors	author	NOUN
easat-9645	394	22	;	;	PUNCT
easat-9645	394	23	licensee	licensee	PROPN
easat-9645	394	24	learning	learning	NOUN
easat-9645	394	25	gate	gate	VERB
easat-9645	394	26	the	the	DET
easat-9645	394	27	figure	figure	NOUN
easat-9645	394	28	9	9	NUM
easat-9645	394	29	,	,	PUNCT
easat-9645	394	30	shows	show	VERB
easat-9645	394	31	a	a	DET
easat-9645	394	32	gaussian	gaussian	ADJ
easat-9645	394	33	rbf	rbf	PROPN
easat-9645	394	34	(	(	PUNCT
easat-9645	394	35	radial	radial	ADJ
easat-9645	394	36	basis	basis	NOUN
easat-9645	394	37	function	function	NOUN
easat-9645	394	38	)	)	PUNCT
easat-9645	394	39	kernel	kernel	PROPN
easat-9645	394	40	matrix	matrix	NOUN
easat-9645	394	41	,	,	PUNCT
easat-9645	394	42	visualized	visualize	VERB
easat-9645	394	43	as	as	ADP
easat-9645	394	44	a	a	DET
easat-9645	394	45	heatmap	heatmap	NOUN
easat-9645	394	46	.	.	PUNCT
easat-9645	395	1	the	the	DET
easat-9645	395	2	x	x	NOUN
easat-9645	395	3	and	and	CCONJ
easat-9645	395	4	y	y	PROPN
easat-9645	395	5	axes	axis	NOUN
easat-9645	395	6	range	range	VERB
easat-9645	395	7	from	from	ADP
easat-9645	395	8	-3	-3	INTJ
easat-9645	395	9	to	to	ADP
easat-9645	395	10	3	3	NUM
easat-9645	395	11	,	,	PUNCT
easat-9645	395	12	representing	represent	VERB
easat-9645	395	13	pairs	pair	NOUN
easat-9645	395	14	of	of	ADP
easat-9645	395	15	data	datum	NOUN
easat-9645	395	16	points	point	NOUN
easat-9645	395	17	.	.	PUNCT
easat-9645	396	1	the	the	DET
easat-9645	396	2	color	color	NOUN
easat-9645	396	3	gradient	gradient	NOUN
easat-9645	396	4	,	,	PUNCT
easat-9645	396	5	ranging	range	VERB
easat-9645	396	6	from	from	ADP
easat-9645	396	7	blue	blue	ADJ
easat-9645	396	8	(	(	PUNCT
easat-9645	396	9	0.1	0.1	NUM
easat-9645	396	10	)	)	PUNCT
easat-9645	396	11	𝑡𝑜	𝑡𝑜	PROPN
easat-9645	396	12	𝑟𝑒𝑑	𝑟𝑒𝑑	NOUN
easat-9645	396	13	(	(	PUNCT
easat-9645	396	14	1.0	1.0	NUM
easat-9645	396	15	)	)	PUNCT
easat-9645	396	16	,	,	PUNCT
easat-9645	396	17	indicates	indicate	VERB
easat-9645	396	18	the	the	DET
easat-9645	396	19	similarity	similarity	NOUN
easat-9645	396	20	between	between	ADP
easat-9645	396	21	these	these	DET
easat-9645	396	22	points	point	NOUN
easat-9645	396	23	,	,	PUNCT
easat-9645	396	24	with	with	ADP
easat-9645	396	25	warmer	warm	ADJ
easat-9645	396	26	colors	color	NOUN
easat-9645	396	27	showing	show	VERB
easat-9645	396	28	higher	high	ADJ
easat-9645	396	29	similarity	similarity	NOUN
easat-9645	396	30	.	.	PUNCT
easat-9645	397	1	the	the	DET
easat-9645	397	2	diagonal	diagonal	ADJ
easat-9645	397	3	line	line	NOUN
easat-9645	397	4	of	of	ADP
easat-9645	397	5	red	red	ADJ
easat-9645	397	6	indicates	indicate	VERB
easat-9645	397	7	maximum	maximum	ADJ
easat-9645	397	8	similarity	similarity	NOUN
easat-9645	397	9	where	where	SCONJ
easat-9645	397	10	points	point	NOUN
easat-9645	397	11	are	be	AUX
easat-9645	397	12	compared	compare	VERB
easat-9645	397	13	to	to	ADP
easat-9645	397	14	themselves	themselves	PRON
easat-9645	397	15	,	,	PUNCT
easat-9645	397	16	while	while	SCONJ
easat-9645	397	17	the	the	DET
easat-9645	397	18	similarity	similarity	NOUN
easat-9645	397	19	decreases	decrease	VERB
easat-9645	397	20	as	as	SCONJ
easat-9645	397	21	points	point	NOUN
easat-9645	397	22	move	move	VERB
easat-9645	397	23	away	away	ADV
easat-9645	397	24	from	from	ADP
easat-9645	397	25	the	the	DET
easat-9645	397	26	diagonal	diagonal	NOUN
easat-9645	397	27	.	.	PUNCT
easat-9645	398	1	5.2	5.2	NUM
easat-9645	398	2	.	.	PUNCT
easat-9645	399	1	support	support	NOUN
easat-9645	399	2	vector	vector	NOUN
easat-9645	399	3	machines	machine	NOUN
easat-9645	399	4	(	(	PUNCT
easat-9645	399	5	svms	svms	NOUN
easat-9645	399	6	)	)	PUNCT
easat-9645	399	7	svms	svms	NOUN
easat-9645	399	8	find	find	VERB
easat-9645	399	9	the	the	DET
easat-9645	399	10	optimal	optimal	ADJ
easat-9645	399	11	hyperplane	hyperplane	NOUN
easat-9645	399	12	in	in	ADP
easat-9645	399	13	a	a	DET
easat-9645	399	14	hilbert	hilbert	NOUN
easat-9645	399	15	space	space	NOUN
easat-9645	399	16	to	to	PART
easat-9645	399	17	separate	separate	VERB
easat-9645	399	18	data	datum	NOUN
easat-9645	399	19	.	.	PUNCT
easat-9645	400	1	the	the	DET
easat-9645	400	2	use	use	NOUN
easat-9645	400	3	of	of	ADP
easat-9645	400	4	kernels	kernel	NOUN
easat-9645	400	5	enables	enable	VERB
easat-9645	400	6	non	non	ADJ
easat-9645	400	7	-	-	ADJ
easat-9645	400	8	linear	linear	ADJ
easat-9645	400	9	boundaries	boundary	NOUN
easat-9645	400	10	in	in	ADP
easat-9645	400	11	the	the	DET
easat-9645	400	12	original	original	ADJ
easat-9645	400	13	space	space	NOUN
easat-9645	400	14	while	while	SCONJ
easat-9645	400	15	maintaining	maintain	VERB
easat-9645	400	16	computational	computational	ADJ
easat-9645	400	17	feasibility	feasibility	NOUN
easat-9645	400	18	.	.	PUNCT
easat-9645	401	1	svms	svms	NOUN
easat-9645	401	2	is	be	AUX
easat-9645	401	3	supervised	supervised	ADJ
easat-9645	401	4	algorithms	algorithm	NOUN
easat-9645	401	5	applied	apply	VERB
easat-9645	401	6	to	to	ADP
easat-9645	401	7	classification	classification	NOUN
easat-9645	401	8	and	and	CCONJ
easat-9645	401	9	regression	regression	NOUN
easat-9645	401	10	problems	problem	NOUN
easat-9645	401	11	.	.	PUNCT
easat-9645	402	1	they	they	PRON
easat-9645	402	2	identify	identify	VERB
easat-9645	402	3	the	the	DET
easat-9645	402	4	hyperplane	hyperplane	NOUN
easat-9645	402	5	that	that	PRON
easat-9645	402	6	maximizes	maximize	VERB
easat-9645	402	7	the	the	DET
easat-9645	402	8	margin	margin	NOUN
easat-9645	402	9	between	between	ADP
easat-9645	402	10	different	different	ADJ
easat-9645	402	11	classes	class	NOUN
easat-9645	402	12	in	in	ADP
easat-9645	402	13	a	a	DET
easat-9645	402	14	transformed	transform	VERB
easat-9645	402	15	feature	feature	NOUN
easat-9645	402	16	space	space	NOUN
easat-9645	402	17	.	.	PUNCT
easat-9645	403	1	for	for	ADP
easat-9645	403	2	datasets	dataset	NOUN
easat-9645	403	3	that	that	PRON
easat-9645	403	4	are	be	AUX
easat-9645	403	5	not	not	PART
easat-9645	403	6	linearly	linearly	ADV
easat-9645	403	7	separable	separable	ADJ
easat-9645	403	8	,	,	PUNCT
easat-9645	403	9	kernel	kernel	PROPN
easat-9645	403	10	functions	function	NOUN
easat-9645	403	11	are	be	AUX
easat-9645	403	12	used	use	VERB
easat-9645	403	13	to	to	PART
easat-9645	403	14	project	project	VERB
easat-9645	403	15	data	datum	NOUN
easat-9645	403	16	into	into	ADP
easat-9645	403	17	higher	high	ADJ
easat-9645	403	18	-	-	PUNCT
easat-9645	403	19	dimensional	dimensional	ADJ
easat-9645	403	20	spaces	space	NOUN
easat-9645	403	21	,	,	PUNCT
easat-9645	403	22	enabling	enable	VERB
easat-9645	403	23	the	the	DET
easat-9645	403	24	discovery	discovery	NOUN
easat-9645	403	25	of	of	ADP
easat-9645	403	26	a	a	DET
easat-9645	403	27	linear	linear	ADJ
easat-9645	403	28	boundary	boundary	NOUN
easat-9645	403	29	.	.	PUNCT
easat-9645	404	1	figure	figure	NOUN
easat-9645	404	2	10	10	NUM
easat-9645	404	3	.	.	PUNCT
easat-9645	405	1	matlab	matlab	PROPN
easat-9645	405	2	implementation	implementation	NOUN
easat-9645	405	3	:	:	PUNCT
easat-9645	405	4	visualizing	visualize	VERB
easat-9645	405	5	a	a	DET
easat-9645	405	6	svm	svm	ADJ
easat-9645	405	7	classification	classification	NOUN
easat-9645	405	8	.	.	PUNCT
easat-9645	406	1	the	the	DET
easat-9645	406	2	figure	figure	NOUN
easat-9645	406	3	10	10	NUM
easat-9645	406	4	,	,	PUNCT
easat-9645	406	5	illustrates	illustrate	VERB
easat-9645	406	6	svm	svm	PROPN
easat-9645	406	7	(	(	PUNCT
easat-9645	406	8	support	support	NOUN
easat-9645	406	9	vector	vector	NOUN
easat-9645	406	10	machine	machine	NOUN
easat-9645	406	11	)	)	PUNCT
easat-9645	406	12	classification	classification	NOUN
easat-9645	406	13	using	use	VERB
easat-9645	406	14	an	an	DET
easat-9645	406	15	rbf	rbf	PROPN
easat-9645	406	16	(	(	PUNCT
easat-9645	406	17	radial	radial	ADJ
easat-9645	406	18	basis	basis	NOUN
easat-9645	406	19	function	function	NOUN
easat-9645	406	20	)	)	PUNCT
easat-9645	406	21	kernel	kernel	NOUN
easat-9645	406	22	.	.	PUNCT
easat-9645	407	1	it	it	PRON
easat-9645	407	2	shows	show	VERB
easat-9645	407	3	two	two	NUM
easat-9645	407	4	classes	class	NOUN
easat-9645	407	5	:	:	PUNCT
easat-9645	407	6	class	class	NOUN
easat-9645	407	7	1	1	NUM
easat-9645	407	8	(	(	PUNCT
easat-9645	407	9	red	red	PROPN
easat-9645	407	10	′x′	′x′	PROPN
easat-9645	407	11	)	)	PUNCT
easat-9645	407	12	and	and	CCONJ
easat-9645	407	13	class	class	NOUN
easat-9645	407	14	−	−	NOUN
easat-9645	407	15	1	1	NUM
easat-9645	407	16	(	(	PUNCT
easat-9645	407	17	blue	blue	ADJ
easat-9645	407	18	′o′	′o′	PROPN
easat-9645	407	19	)	)	PUNCT
easat-9645	407	20	,	,	PUNCT
easat-9645	407	21	separated	separate	VERB
easat-9645	407	22	by	by	ADP
easat-9645	407	23	a	a	DET
easat-9645	407	24	nonlinear	nonlinear	ADJ
easat-9645	407	25	decision	decision	NOUN
easat-9645	407	26	boundary	boundary	ADJ
easat-9645	407	27	(	(	PUNCT
easat-9645	407	28	black	black	ADJ
easat-9645	407	29	curve	curve	NOUN
easat-9645	407	30	)	)	PUNCT
easat-9645	407	31	.	.	PUNCT
easat-9645	408	1	the	the	DET
easat-9645	408	2	dashed	dash	VERB
easat-9645	408	3	lines	line	NOUN
easat-9645	408	4	represent	represent	VERB
easat-9645	408	5	the	the	DET
easat-9645	408	6	margins	margin	NOUN
easat-9645	408	7	,	,	PUNCT
easat-9645	408	8	which	which	PRON
easat-9645	408	9	define	define	VERB
easat-9645	408	10	the	the	DET
easat-9645	408	11	boundary	boundary	NOUN
easat-9645	408	12	's	's	PART
easat-9645	408	13	width	width	NOUN
easat-9645	408	14	,	,	PUNCT
easat-9645	408	15	with	with	ADP
easat-9645	408	16	support	support	NOUN
easat-9645	408	17	vectors	vector	NOUN
easat-9645	408	18	(	(	PUNCT
easat-9645	408	19	data	data	PROPN
easat-9645	408	20	points	point	VERB
easat-9645	408	21	closest	close	ADJ
easat-9645	408	22	to	to	ADP
easat-9645	408	23	the	the	DET
easat-9645	408	24	boundary	boundary	NOUN
easat-9645	408	25	)	)	PUNCT
easat-9645	408	26	influencing	influence	VERB
easat-9645	408	27	its	its	PRON
easat-9645	408	28	position	position	NOUN
easat-9645	408	29	.	.	PUNCT
easat-9645	409	1	5.3	5.3	NUM
easat-9645	409	2	.	.	PUNCT
easat-9645	409	3	principal	principal	ADJ
easat-9645	409	4	component	component	NOUN
easat-9645	409	5	analysis	analysis	NOUN
easat-9645	409	6	(	(	PUNCT
easat-9645	409	7	pca	pca	NOUN
easat-9645	409	8	)	)	PUNCT
easat-9645	409	9	pca	pca	PROPN
easat-9645	409	10	is	be	AUX
easat-9645	409	11	a	a	DET
easat-9645	409	12	statistical	statistical	ADJ
easat-9645	409	13	technique	technique	NOUN
easat-9645	409	14	designed	design	VERB
easat-9645	409	15	to	to	PART
easat-9645	409	16	reduce	reduce	VERB
easat-9645	409	17	the	the	DET
easat-9645	409	18	dimensionality	dimensionality	NOUN
easat-9645	409	19	of	of	ADP
easat-9645	409	20	a	a	DET
easat-9645	409	21	dataset	dataset	NOUN
easat-9645	409	22	by	by	ADP
easat-9645	409	23	transforming	transform	VERB
easat-9645	409	24	it	it	PRON
easat-9645	409	25	into	into	ADP
easat-9645	409	26	a	a	DET
easat-9645	409	27	smaller	small	ADJ
easat-9645	409	28	set	set	NOUN
easat-9645	409	29	of	of	ADP
easat-9645	409	30	variables	variable	NOUN
easat-9645	409	31	while	while	SCONJ
easat-9645	409	32	retaining	retain	VERB
easat-9645	409	33	most	most	ADJ
easat-9645	409	34	of	of	ADP
easat-9645	409	35	the	the	DET
easat-9645	409	36	original	original	ADJ
easat-9645	409	37	information	information	NOUN
easat-9645	409	38	.	.	PUNCT
easat-9645	410	1	this	this	PRON
easat-9645	410	2	is	be	AUX
easat-9645	410	3	achieved	achieve	VERB
easat-9645	410	4	by	by	ADP
easat-9645	410	5	identifying	identify	VERB
easat-9645	410	6	new	new	ADJ
easat-9645	410	7	axes	axis	NOUN
easat-9645	410	8	,	,	PUNCT
easat-9645	410	9	known	know	VERB
easat-9645	410	10	as	as	ADP
easat-9645	410	11	principal	principal	ADJ
easat-9645	410	12	components	component	NOUN
easat-9645	410	13	,	,	PUNCT
easat-9645	410	14	which	which	PRON
easat-9645	410	15	are	be	AUX
easat-9645	410	16	linear	linear	ADJ
easat-9645	410	17	combinations	combination	NOUN
easat-9645	410	18	of	of	ADP
easat-9645	410	19	the	the	DET
easat-9645	410	20	original	original	ADJ
easat-9645	410	21	1515	1515	NUM
easat-9645	410	22	edelweiss	edelweiss	PROPN
easat-9645	410	23	applied	apply	VERB
easat-9645	410	24	science	science	NOUN
easat-9645	410	25	and	and	CCONJ
easat-9645	410	26	technology	technology	NOUN
easat-9645	410	27	issn	issn	PROPN
easat-9645	410	28	:	:	PUNCT
easat-9645	410	29	2576	2576	NUM
easat-9645	410	30	-	-	SYM
easat-9645	410	31	8484	8484	NUM
easat-9645	410	32	vol	vol	NOUN
easat-9645	410	33	.	.	PROPN
easat-9645	411	1	9	9	NUM
easat-9645	411	2	,	,	PUNCT
easat-9645	411	3	no	no	INTJ
easat-9645	411	4	.	.	NOUN
easat-9645	411	5	8	8	NUM
easat-9645	411	6	:	:	SYM
easat-9645	411	7	1498	1498	NUM
easat-9645	411	8	-	-	SYM
easat-9645	411	9	1523	1523	NUM
easat-9645	411	10	,	,	PUNCT
easat-9645	411	11	2025	2025	NUM
easat-9645	411	12	doi	doi	NOUN
easat-9645	411	13	:	:	PUNCT
easat-9645	411	14	10.55214/2576	10.55214/2576	NUM
easat-9645	411	15	-	-	SYM
easat-9645	411	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	411	17	©	©	PROPN
easat-9645	411	18	2025	2025	NUM
easat-9645	411	19	by	by	ADP
easat-9645	411	20	the	the	DET
easat-9645	411	21	authors	author	NOUN
easat-9645	411	22	;	;	PUNCT
easat-9645	411	23	licensee	licensee	PROPN
easat-9645	411	24	learning	learn	VERB
easat-9645	411	25	gate	gate	NOUN
easat-9645	411	26	features	feature	NOUN
easat-9645	411	27	.	.	PUNCT
easat-9645	412	1	these	these	DET
easat-9645	412	2	components	component	NOUN
easat-9645	412	3	are	be	AUX
easat-9645	412	4	ranked	rank	VERB
easat-9645	412	5	based	base	VERB
easat-9645	412	6	on	on	ADP
easat-9645	412	7	how	how	SCONJ
easat-9645	412	8	much	much	ADJ
easat-9645	412	9	of	of	ADP
easat-9645	412	10	the	the	DET
easat-9645	412	11	data	datum	NOUN
easat-9645	412	12	’s	’s	PART
easat-9645	412	13	variation	variation	NOUN
easat-9645	412	14	they	they	PRON
easat-9645	412	15	capture	capture	VERB
easat-9645	412	16	.	.	PUNCT
easat-9645	413	1	the	the	DET
easat-9645	413	2	first	first	ADJ
easat-9645	413	3	captures	capture	VERB
easat-9645	413	4	the	the	DET
easat-9645	413	5	greatest	great	ADJ
easat-9645	413	6	amount	amount	NOUN
easat-9645	413	7	of	of	ADP
easat-9645	413	8	variation	variation	NOUN
easat-9645	413	9	,	,	PUNCT
easat-9645	413	10	followed	follow	VERB
easat-9645	413	11	by	by	ADP
easat-9645	413	12	the	the	DET
easat-9645	413	13	second	second	ADJ
easat-9645	413	14	,	,	PUNCT
easat-9645	413	15	and	and	CCONJ
easat-9645	413	16	so	so	ADV
easat-9645	413	17	on	on	ADV
easat-9645	413	18	.	.	PUNCT
easat-9645	413	19	by	by	ADP
easat-9645	413	20	projecting	project	VERB
easat-9645	413	21	the	the	DET
easat-9645	413	22	data	datum	NOUN
easat-9645	413	23	onto	onto	ADP
easat-9645	413	24	these	these	DET
easat-9645	413	25	principal	principal	ADJ
easat-9645	413	26	components	component	NOUN
easat-9645	413	27	,	,	PUNCT
easat-9645	413	28	pca	pca	PROPN
easat-9645	413	29	helps	helps	AUX
easat-9645	413	30	simplify	simplify	VERB
easat-9645	413	31	complex	complex	ADJ
easat-9645	413	32	data	datum	NOUN
easat-9645	413	33	,	,	PUNCT
easat-9645	413	34	making	make	VERB
easat-9645	413	35	it	it	PRON
easat-9645	413	36	easier	easy	ADJ
easat-9645	413	37	to	to	PART
easat-9645	413	38	explore	explore	VERB
easat-9645	413	39	,	,	PUNCT
easat-9645	413	40	visualize	visualize	VERB
easat-9645	413	41	,	,	PUNCT
easat-9645	413	42	and	and	CCONJ
easat-9645	413	43	interpret	interpret	VERB
easat-9645	413	44	.	.	PUNCT
easat-9645	414	1	figure	figure	NOUN
easat-9645	414	2	11	11	NUM
easat-9645	414	3	.	.	PUNCT
easat-9645	415	1	matlab	matlab	PROPN
easat-9645	415	2	implementation	implementation	NOUN
easat-9645	415	3	:	:	PUNCT
easat-9645	415	4	visualizing	visualize	VERB
easat-9645	415	5	a	a	DET
easat-9645	415	6	pca	pca	NOUN
easat-9645	415	7	classification	classification	NOUN
easat-9645	415	8	with	with	ADP
easat-9645	415	9	quantum	quantum	ADJ
easat-9645	415	10	state	state	NOUN
easat-9645	415	11	vectors	vector	NOUN
easat-9645	415	12	as	as	ADP
easat-9645	415	13	unit	unit	NOUN
easat-9645	415	14	vectors	vector	NOUN
easat-9645	415	15	on	on	ADP
easat-9645	415	16	a	a	DET
easat-9645	415	17	3d	3d	PROPN
easat-9645	415	18	bloch	bloch	PROPN
easat-9645	415	19	sphere	sphere	PROPN
easat-9645	415	20	.	.	PUNCT
easat-9645	416	1	the	the	DET
easat-9645	416	2	figure11	figure11	ADJ
easat-9645	416	3	,	,	PUNCT
easat-9645	416	4	consists	consist	VERB
easat-9645	416	5	of	of	ADP
easat-9645	416	6	two	two	NUM
easat-9645	416	7	parts	part	NOUN
easat-9645	416	8	.	.	PUNCT
easat-9645	417	1	the	the	DET
easat-9645	417	2	left	left	ADJ
easat-9645	417	3	graph	graph	NOUN
easat-9645	417	4	,	,	PUNCT
easat-9645	417	5	"	"	PUNCT
easat-9645	417	6	pca	pca	NOUN
easat-9645	417	7	variance	variance	NOUN
easat-9645	417	8	explained	explain	VERB
easat-9645	417	9	,	,	PUNCT
easat-9645	417	10	"	"	PUNCT
easat-9645	417	11	shows	show	VERB
easat-9645	417	12	a	a	DET
easat-9645	417	13	plot	plot	NOUN
easat-9645	417	14	where	where	SCONJ
easat-9645	417	15	the	the	DET
easat-9645	417	16	y	y	NOUN
easat-9645	417	17	-	-	PUNCT
easat-9645	417	18	axis	axis	NOUN
easat-9645	417	19	represents	represent	VERB
easat-9645	417	20	the	the	DET
easat-9645	417	21	percentage	percentage	NOUN
easat-9645	417	22	of	of	ADP
easat-9645	417	23	variance	variance	NOUN
easat-9645	417	24	explained	explain	VERB
easat-9645	417	25	(	(	PUNCT
easat-9645	417	26	75	75	NUM
easat-9645	417	27	-	-	SYM
easat-9645	417	28	100	100	NUM
easat-9645	417	29	%	%	NOUN
easat-9645	417	30	)	)	PUNCT
easat-9645	417	31	and	and	CCONJ
easat-9645	417	32	the	the	DET
easat-9645	417	33	x	x	NOUN
easat-9645	417	34	-	-	NOUN
easat-9645	417	35	axis	axis	NOUN
easat-9645	417	36	represents	represent	VERB
easat-9645	417	37	the	the	DET
easat-9645	417	38	number	number	NOUN
easat-9645	417	39	of	of	ADP
easat-9645	417	40	principal	principal	ADJ
easat-9645	417	41	components	component	NOUN
easat-9645	417	42	(	(	PUNCT
easat-9645	417	43	1	1	NUM
easat-9645	417	44	to	to	ADP
easat-9645	417	45	2.8	2.8	NUM
easat-9645	417	46	)	)	PUNCT
easat-9645	417	47	.	.	PUNCT
easat-9645	418	1	the	the	DET
easat-9645	418	2	curve	curve	NOUN
easat-9645	418	3	rises	rise	VERB
easat-9645	418	4	steeply	steeply	ADV
easat-9645	418	5	,	,	PUNCT
easat-9645	418	6	indicating	indicate	VERB
easat-9645	418	7	that	that	SCONJ
easat-9645	418	8	most	most	ADJ
easat-9645	418	9	variance	variance	NOUN
easat-9645	418	10	is	be	AUX
easat-9645	418	11	captured	capture	VERB
easat-9645	418	12	with	with	ADP
easat-9645	418	13	a	a	DET
easat-9645	418	14	few	few	ADJ
easat-9645	418	15	components	component	NOUN
easat-9645	418	16	.	.	PUNCT
easat-9645	419	1	the	the	DET
easat-9645	419	2	right	right	ADJ
easat-9645	419	3	image	image	NOUN
easat-9645	419	4	,	,	PUNCT
easat-9645	419	5	"	"	PUNCT
easat-9645	419	6	quantum	quantum	ADJ
easat-9645	419	7	state	state	NOUN
easat-9645	419	8	vector	vector	NOUN
easat-9645	419	9	on	on	ADP
easat-9645	419	10	bloch	bloch	PROPN
easat-9645	419	11	sphere	sphere	ADV
easat-9645	419	12	,	,	PUNCT
easat-9645	419	13	"	"	PUNCT
easat-9645	419	14	depicts	depict	VERB
easat-9645	419	15	a	a	DET
easat-9645	419	16	3d	3d	NUM
easat-9645	419	17	sphere	sphere	NOUN
easat-9645	419	18	with	with	ADP
easat-9645	419	19	a	a	DET
easat-9645	419	20	red	red	ADJ
easat-9645	419	21	vector	vector	NOUN
easat-9645	419	22	pointing	pointing	NOUN
easat-9645	419	23	from	from	ADP
easat-9645	419	24	the	the	DET
easat-9645	419	25	origin	origin	NOUN
easat-9645	419	26	to	to	ADP
easat-9645	419	27	a	a	DET
easat-9645	419	28	point	point	NOUN
easat-9645	419	29	near	near	ADP
easat-9645	419	30	the	the	DET
easat-9645	419	31	top	top	NOUN
easat-9645	419	32	,	,	PUNCT
easat-9645	419	33	illustrating	illustrate	VERB
easat-9645	419	34	a	a	DET
easat-9645	419	35	quantum	quantum	ADJ
easat-9645	419	36	state	state	NOUN
easat-9645	419	37	vector	vector	NOUN
easat-9645	419	38	's	's	PART
easat-9645	419	39	position	position	NOUN
easat-9645	419	40	in	in	ADP
easat-9645	419	41	a	a	DET
easat-9645	419	42	bloch	bloch	PROPN
easat-9645	419	43	sphere	sphere	PROPN
easat-9645	419	44	representation	representation	PROPN
easat-9645	419	45	.	.	PUNCT
easat-9645	420	1	1516	1516	NUM
easat-9645	420	2	edelweiss	edelweiss	PROPN
easat-9645	420	3	applied	apply	VERB
easat-9645	420	4	science	science	NOUN
easat-9645	420	5	and	and	CCONJ
easat-9645	420	6	technology	technology	NOUN
easat-9645	420	7	issn	issn	PROPN
easat-9645	420	8	:	:	PUNCT
easat-9645	420	9	2576	2576	NUM
easat-9645	420	10	-	-	SYM
easat-9645	420	11	8484	8484	NUM
easat-9645	420	12	vol	vol	NOUN
easat-9645	420	13	.	.	PROPN
easat-9645	421	1	9	9	NUM
easat-9645	421	2	,	,	PUNCT
easat-9645	421	3	no	no	INTJ
easat-9645	421	4	.	.	NOUN
easat-9645	421	5	8	8	NUM
easat-9645	421	6	:	:	SYM
easat-9645	421	7	1498	1498	NUM
easat-9645	421	8	-	-	SYM
easat-9645	421	9	1523	1523	NUM
easat-9645	421	10	,	,	PUNCT
easat-9645	421	11	2025	2025	NUM
easat-9645	421	12	doi	doi	NOUN
easat-9645	421	13	:	:	PUNCT
easat-9645	421	14	10.55214/2576	10.55214/2576	NUM
easat-9645	421	15	-	-	SYM
easat-9645	421	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	421	17	©	©	PROPN
easat-9645	421	18	2025	2025	NUM
easat-9645	421	19	by	by	ADP
easat-9645	421	20	the	the	DET
easat-9645	421	21	authors	author	NOUN
easat-9645	421	22	;	;	PUNCT
easat-9645	421	23	licensee	licensee	PROPN
easat-9645	421	24	learning	learning	NOUN
easat-9645	421	25	gate	gate	NOUN
easat-9645	421	26	5.4	5.4	NUM
easat-9645	421	27	.	.	PUNCT
easat-9645	422	1	orthonormal	orthonormal	ADJ
easat-9645	422	2	basis	basis	NOUN
easat-9645	422	3	in	in	ADP
easat-9645	422	4	𝐿2	𝐿2	PROPN
easat-9645	422	5	(	(	PUNCT
easat-9645	422	6	fourier	fouri	ADJ
easat-9645	422	7	sine	sine	NOUN
easat-9645	422	8	and	and	CCONJ
easat-9645	422	9	cosine	cosine	NOUN
easat-9645	422	10	functions	function	NOUN
easat-9645	422	11	)	)	PUNCT
easat-9645	422	12	and	and	CCONJ
easat-9645	422	13	interactive	interactive	ADJ
easat-9645	422	14	function	function	NOUN
easat-9645	422	15	visualizations	visualization	NOUN
easat-9645	422	16	(	(	PUNCT
easat-9645	422	17	e.g.	e.g.	ADV
easat-9645	422	18	,	,	PUNCT
easat-9645	422	19	fourier	fourier	ADJ
easat-9645	422	20	basis	basis	NOUN
easat-9645	422	21	in	in	ADP
easat-9645	422	22	l²	l²	NOUN
easat-9645	422	23	space	space	NOUN
easat-9645	422	24	)	)	PUNCT
easat-9645	422	25	figure	figure	NOUN
easat-9645	422	26	12	12	NUM
easat-9645	422	27	.	.	PUNCT
easat-9645	423	1	matlab	matlab	PROPN
easat-9645	423	2	implementation	implementation	NOUN
easat-9645	423	3	:	:	PUNCT
easat-9645	423	4	this	this	PRON
easat-9645	423	5	visualizes	visualize	VERB
easat-9645	423	6	how	how	SCONJ
easat-9645	423	7	sine	sine	ADJ
easat-9645	423	8	and	and	CCONJ
easat-9645	423	9	cosine	cosine	NOUN
easat-9645	423	10	functions	function	NOUN
easat-9645	423	11	can	can	AUX
easat-9645	423	12	act	act	VERB
easat-9645	423	13	as	as	ADP
easat-9645	423	14	orthonormal	orthonormal	ADJ
easat-9645	423	15	bases	basis	NOUN
easat-9645	423	16	in	in	ADP
easat-9645	423	17	the	the	DET
easat-9645	423	18	𝐿2[0,2𝜋	𝐿2[0,2𝜋	PROPN
easat-9645	423	19	]	]	PUNCT
easat-9645	423	20	hilbert	hilbert	NOUN
easat-9645	423	21	space	space	NOUN
easat-9645	423	22	.	.	PUNCT
easat-9645	424	1	the	the	DET
easat-9645	424	2	figure	figure	NOUN
easat-9645	424	3	12	12	NUM
easat-9645	424	4	,	,	PUNCT
easat-9645	424	5	shows	show	VERB
easat-9645	424	6	orthonormal	orthonormal	ADJ
easat-9645	424	7	basis	basis	NOUN
easat-9645	424	8	functions	function	NOUN
easat-9645	424	9	in	in	ADP
easat-9645	424	10	𝐿2[0	𝐿2[0	PROPN
easat-9645	424	11	,	,	PUNCT
easat-9645	424	12	2𝜋]on	2𝜋]on	NUM
easat-9645	424	13	the	the	DET
easat-9645	424	14	left	left	NOUN
easat-9645	424	15	,	,	PUNCT
easat-9645	424	16	with	with	ADP
easat-9645	424	17	three	three	NUM
easat-9645	424	18	curves	curve	NOUN
easat-9645	424	19	:	:	PUNCT
easat-9645	424	20	sin(x	sin(x	PROPN
easat-9645	424	21	)	)	PUNCT
easat-9645	424	22	in	in	ADP
easat-9645	424	23	red	red	ADJ
easat-9645	424	24	,	,	PUNCT
easat-9645	424	25	cos(x	cos(x	PROPN
easat-9645	424	26	)	)	PUNCT
easat-9645	424	27	in	in	ADP
easat-9645	424	28	green	green	ADJ
easat-9645	424	29	,	,	PUNCT
easat-9645	424	30	and	and	CCONJ
easat-9645	424	31	sin(2x	sin(2x	VERB
easat-9645	424	32	)	)	PUNCT
easat-9645	424	33	in	in	ADP
easat-9645	424	34	blue	blue	ADJ
easat-9645	424	35	,	,	PUNCT
easat-9645	424	36	oscillating	oscillate	VERB
easat-9645	424	37	between	between	ADP
easat-9645	424	38	-1	-1	PUNCT
easat-9645	424	39	and	and	CCONJ
easat-9645	424	40	1	1	NUM
easat-9645	424	41	over	over	ADP
easat-9645	424	42	the	the	DET
easat-9645	424	43	interval	interval	NOUN
easat-9645	424	44	[	[	X
easat-9645	424	45	0	0	NUM
easat-9645	424	46	,	,	PUNCT
easat-9645	424	47	7	7	NUM
easat-9645	424	48	]	]	PUNCT
easat-9645	424	49	.	.	PUNCT
easat-9645	425	1	the	the	DET
easat-9645	425	2	right	right	ADJ
easat-9645	425	3	plot	plot	NOUN
easat-9645	425	4	zooms	zoom	NOUN
easat-9645	425	5	in	in	ADP
easat-9645	425	6	on	on	ADP
easat-9645	425	7	sin(x	sin(x	PROPN
easat-9645	425	8	)	)	PUNCT
easat-9645	425	9	,	,	PUNCT
easat-9645	425	10	displaying	display	VERB
easat-9645	425	11	its	its	PRON
easat-9645	425	12	standard	standard	ADJ
easat-9645	425	13	sinusoidal	sinusoidal	ADJ
easat-9645	425	14	wave	wave	NOUN
easat-9645	425	15	with	with	ADP
easat-9645	425	16	a	a	DET
easat-9645	425	17	single	single	ADJ
easat-9645	425	18	peak	peak	NOUN
easat-9645	425	19	and	and	CCONJ
easat-9645	425	20	trough	trough	NOUN
easat-9645	425	21	within	within	ADP
easat-9645	425	22	[	[	X
easat-9645	425	23	0	0	NUM
easat-9645	425	24	,	,	PUNCT
easat-9645	425	25	7	7	NUM
easat-9645	425	26	]	]	PUNCT
easat-9645	425	27	,	,	PUNCT
easat-9645	425	28	ranging	range	VERB
easat-9645	425	29	from	from	ADP
easat-9645	425	30	-1	-1	INTJ
easat-9645	425	31	to	to	ADP
easat-9645	425	32	1	1	NUM
easat-9645	425	33	.	.	NOUN
easat-9645	425	34	6	6	NUM
easat-9645	425	35	.	.	PUNCT
easat-9645	425	36	applications	application	NOUN
easat-9645	425	37	of	of	ADP
easat-9645	425	38	inner	inner	ADJ
easat-9645	425	39	product	product	NOUN
easat-9645	425	40	spaces	space	VERB
easat-9645	425	41	inner	inner	ADJ
easat-9645	425	42	product	product	NOUN
easat-9645	425	43	spaces	space	NOUN
easat-9645	425	44	play	play	VERB
easat-9645	425	45	a	a	DET
easat-9645	425	46	pivotal	pivotal	ADJ
easat-9645	425	47	role	role	NOUN
easat-9645	425	48	in	in	ADP
easat-9645	425	49	both	both	CCONJ
easat-9645	425	50	pure	pure	ADJ
easat-9645	425	51	and	and	CCONJ
easat-9645	425	52	applied	apply	VERB
easat-9645	425	53	mathematics	mathematic	NOUN
easat-9645	425	54	due	due	ADP
easat-9645	425	55	to	to	ADP
easat-9645	425	56	their	their	PRON
easat-9645	425	57	geometric	geometric	ADJ
easat-9645	425	58	structure	structure	NOUN
easat-9645	425	59	and	and	CCONJ
easat-9645	425	60	analytical	analytical	ADJ
easat-9645	425	61	properties	property	NOUN
easat-9645	425	62	.	.	PUNCT
easat-9645	426	1	their	their	PRON
easat-9645	426	2	applications	application	NOUN
easat-9645	426	3	span	span	VERB
easat-9645	426	4	across	across	ADP
easat-9645	426	5	various	various	ADJ
easat-9645	426	6	disciplines	discipline	NOUN
easat-9645	426	7	,	,	PUNCT
easat-9645	426	8	including	include	VERB
easat-9645	426	9	:	:	PUNCT
easat-9645	426	10	6.1	6.1	NUM
easat-9645	426	11	.	.	PUNCT
easat-9645	426	12	quantum	quantum	ADJ
easat-9645	426	13	mechanics	mechanic	NOUN
easat-9645	426	14	in	in	ADP
easat-9645	426	15	quantum	quantum	ADJ
easat-9645	426	16	theory	theory	NOUN
easat-9645	426	17	,	,	PUNCT
easat-9645	426	18	the	the	DET
easat-9645	426	19	state	state	NOUN
easat-9645	426	20	of	of	ADP
easat-9645	426	21	a	a	DET
easat-9645	426	22	physical	physical	ADJ
easat-9645	426	23	system	system	NOUN
easat-9645	426	24	is	be	AUX
easat-9645	426	25	represented	represent	VERB
easat-9645	426	26	by	by	ADP
easat-9645	426	27	a	a	DET
easat-9645	426	28	vector	vector	NOUN
easat-9645	426	29	in	in	ADP
easat-9645	426	30	a	a	DET
easat-9645	426	31	complex	complex	ADJ
easat-9645	426	32	hilbert	hilbert	NOUN
easat-9645	426	33	space	space	NOUN
easat-9645	426	34	(	(	PUNCT
easat-9645	426	35	a	a	DET
easat-9645	426	36	complete	complete	ADJ
easat-9645	426	37	inner	inner	ADJ
easat-9645	426	38	product	product	NOUN
easat-9645	426	39	space	space	NOUN
easat-9645	426	40	)	)	PUNCT
easat-9645	426	41	.	.	PUNCT
easat-9645	427	1	observables	observable	NOUN
easat-9645	427	2	such	such	ADJ
easat-9645	427	3	as	as	ADP
easat-9645	427	4	position	position	NOUN
easat-9645	427	5	and	and	CCONJ
easat-9645	427	6	momentum	momentum	NOUN
easat-9645	427	7	are	be	AUX
easat-9645	427	8	modeled	model	VERB
easat-9645	427	9	as	as	ADP
easat-9645	427	10	linear	linear	PROPN
easat-9645	427	11	operators	operator	NOUN
easat-9645	427	12	on	on	ADP
easat-9645	427	13	these	these	DET
easat-9645	427	14	spaces	space	NOUN
easat-9645	427	15	.	.	PUNCT
easat-9645	428	1	the	the	DET
easat-9645	428	2	inner	inner	ADJ
easat-9645	428	3	product	product	NOUN
easat-9645	428	4	allows	allow	VERB
easat-9645	428	5	the	the	DET
easat-9645	428	6	computation	computation	NOUN
easat-9645	428	7	of	of	ADP
easat-9645	428	8	probabilities	probability	NOUN
easat-9645	428	9	and	and	CCONJ
easat-9645	428	10	expectations	expectation	NOUN
easat-9645	428	11	,	,	PUNCT
easat-9645	428	12	forming	form	VERB
easat-9645	428	13	the	the	DET
easat-9645	428	14	core	core	NOUN
easat-9645	428	15	of	of	ADP
easat-9645	428	16	quantum	quantum	NOUN
easat-9645	428	17	measurement	measurement	NOUN
easat-9645	428	18	theory	theory	NOUN
easat-9645	428	19	.	.	PUNCT
easat-9645	429	1	6.2	6.2	NUM
easat-9645	429	2	.	.	PUNCT
easat-9645	429	3	signal	signal	NOUN
easat-9645	429	4	processing	processing	NOUN
easat-9645	429	5	and	and	CCONJ
easat-9645	429	6	fourier	fourier	NOUN
easat-9645	429	7	analysis	analysis	NOUN
easat-9645	429	8	inner	inner	ADJ
easat-9645	429	9	product	product	NOUN
easat-9645	429	10	spaces	space	NOUN
easat-9645	429	11	provide	provide	VERB
easat-9645	429	12	the	the	DET
easat-9645	429	13	foundation	foundation	NOUN
easat-9645	429	14	for	for	ADP
easat-9645	429	15	signal	signal	ADJ
easat-9645	429	16	decomposition	decomposition	NOUN
easat-9645	429	17	techniques	technique	NOUN
easat-9645	429	18	such	such	ADJ
easat-9645	429	19	as	as	ADP
easat-9645	429	20	the	the	DET
easat-9645	429	21	fourier	fourier	NOUN
easat-9645	429	22	transform	transform	NOUN
easat-9645	429	23	.	.	PUNCT
easat-9645	430	1	signals	signal	NOUN
easat-9645	430	2	can	can	AUX
easat-9645	430	3	be	be	AUX
easat-9645	430	4	represented	represent	VERB
easat-9645	430	5	as	as	ADP
easat-9645	430	6	sums	sum	NOUN
easat-9645	430	7	of	of	ADP
easat-9645	430	8	orthogonal	orthogonal	ADJ
easat-9645	430	9	basis	basis	NOUN
easat-9645	430	10	functions	function	NOUN
easat-9645	430	11	,	,	PUNCT
easat-9645	430	12	and	and	CCONJ
easat-9645	430	13	inner	inner	ADJ
easat-9645	430	14	products	product	NOUN
easat-9645	430	15	are	be	AUX
easat-9645	430	16	used	use	VERB
easat-9645	430	17	to	to	PART
easat-9645	430	18	compute	compute	VERB
easat-9645	430	19	the	the	DET
easat-9645	430	20	coefficients	coefficient	NOUN
easat-9645	430	21	in	in	ADP
easat-9645	430	22	these	these	DET
easat-9645	430	23	expansions	expansion	NOUN
easat-9645	430	24	.	.	PUNCT
easat-9645	431	1	this	this	PRON
easat-9645	431	2	is	be	AUX
easat-9645	431	3	critical	critical	ADJ
easat-9645	431	4	in	in	ADP
easat-9645	431	5	filtering	filter	VERB
easat-9645	431	6	,	,	PUNCT
easat-9645	431	7	compression	compression	NOUN
easat-9645	431	8	,	,	PUNCT
easat-9645	431	9	and	and	CCONJ
easat-9645	431	10	noise	noise	NOUN
easat-9645	431	11	reduction	reduction	NOUN
easat-9645	431	12	.	.	PUNCT
easat-9645	432	1	6.3	6.3	NUM
easat-9645	432	2	.	.	PUNCT
easat-9645	433	1	computer	computer	NOUN
easat-9645	433	2	graphics	graphic	NOUN
easat-9645	433	3	and	and	CCONJ
easat-9645	433	4	geometry	geometry	NOUN
easat-9645	433	5	in	in	ADP
easat-9645	433	6	3d	3d	PROPN
easat-9645	433	7	graphics	graphic	NOUN
easat-9645	433	8	,	,	PUNCT
easat-9645	433	9	the	the	DET
easat-9645	433	10	inner	inner	ADJ
easat-9645	433	11	product	product	NOUN
easat-9645	433	12	(	(	PUNCT
easat-9645	433	13	or	or	CCONJ
easat-9645	433	14	dot	dot	NOUN
easat-9645	433	15	product	product	NOUN
easat-9645	433	16	)	)	PUNCT
easat-9645	433	17	is	be	AUX
easat-9645	433	18	used	use	VERB
easat-9645	433	19	to	to	PART
easat-9645	433	20	determine	determine	VERB
easat-9645	433	21	angles	angle	NOUN
easat-9645	433	22	between	between	ADP
easat-9645	433	23	vectors	vector	NOUN
easat-9645	433	24	,	,	PUNCT
easat-9645	433	25	shading	shading	NOUN
easat-9645	433	26	,	,	PUNCT
easat-9645	433	27	and	and	CCONJ
easat-9645	433	28	projection	projection	NOUN
easat-9645	433	29	operations	operation	NOUN
easat-9645	433	30	.	.	PUNCT
easat-9645	434	1	this	this	PRON
easat-9645	434	2	underpins	underpin	VERB
easat-9645	434	3	rendering	render	VERB
easat-9645	434	4	techniques	technique	NOUN
easat-9645	434	5	and	and	CCONJ
easat-9645	434	6	physical	physical	ADJ
easat-9645	434	7	simulations	simulation	NOUN
easat-9645	434	8	.	.	PUNCT
easat-9645	435	1	1517	1517	NUM
easat-9645	435	2	edelweiss	edelweiss	PROPN
easat-9645	435	3	applied	apply	VERB
easat-9645	435	4	science	science	NOUN
easat-9645	435	5	and	and	CCONJ
easat-9645	435	6	technology	technology	NOUN
easat-9645	435	7	issn	issn	PROPN
easat-9645	435	8	:	:	PUNCT
easat-9645	435	9	2576	2576	NUM
easat-9645	435	10	-	-	SYM
easat-9645	435	11	8484	8484	NUM
easat-9645	435	12	vol	vol	NOUN
easat-9645	435	13	.	.	PROPN
easat-9645	436	1	9	9	NUM
easat-9645	436	2	,	,	PUNCT
easat-9645	436	3	no	no	INTJ
easat-9645	436	4	.	.	NOUN
easat-9645	436	5	8	8	NUM
easat-9645	436	6	:	:	SYM
easat-9645	436	7	1498	1498	NUM
easat-9645	436	8	-	-	SYM
easat-9645	436	9	1523	1523	NUM
easat-9645	436	10	,	,	PUNCT
easat-9645	436	11	2025	2025	NUM
easat-9645	436	12	doi	doi	NOUN
easat-9645	436	13	:	:	PUNCT
easat-9645	436	14	10.55214/2576	10.55214/2576	NUM
easat-9645	436	15	-	-	SYM
easat-9645	436	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	436	17	©	©	PROPN
easat-9645	436	18	2025	2025	NUM
easat-9645	436	19	by	by	ADP
easat-9645	436	20	the	the	DET
easat-9645	436	21	authors	author	NOUN
easat-9645	436	22	;	;	PUNCT
easat-9645	436	23	licensee	licensee	PROPN
easat-9645	436	24	learning	learning	NOUN
easat-9645	436	25	gate	gate	NOUN
easat-9645	436	26	6.4	6.4	NUM
easat-9645	436	27	.	.	PUNCT
easat-9645	437	1	statistics	statistic	NOUN
easat-9645	437	2	and	and	CCONJ
easat-9645	437	3	principal	principal	ADJ
easat-9645	437	4	component	component	NOUN
easat-9645	437	5	analysis	analysis	NOUN
easat-9645	437	6	(	(	PUNCT
easat-9645	437	7	pca	pca	NOUN
easat-9645	437	8	)	)	PUNCT
easat-9645	437	9	in	in	ADP
easat-9645	437	10	multivariate	multivariate	NOUN
easat-9645	437	11	statistics	statistic	NOUN
easat-9645	437	12	,	,	PUNCT
easat-9645	437	13	pca	pca	PROPN
easat-9645	437	14	uses	use	VERB
easat-9645	437	15	the	the	DET
easat-9645	437	16	inner	inner	ADJ
easat-9645	437	17	product	product	NOUN
easat-9645	437	18	to	to	PART
easat-9645	437	19	measure	measure	VERB
easat-9645	437	20	variance	variance	NOUN
easat-9645	437	21	and	and	CCONJ
easat-9645	437	22	correlation	correlation	NOUN
easat-9645	437	23	.	.	PUNCT
easat-9645	438	1	data	datum	NOUN
easat-9645	438	2	is	be	AUX
easat-9645	438	3	projected	project	VERB
easat-9645	438	4	onto	onto	ADP
easat-9645	438	5	orthogonal	orthogonal	ADJ
easat-9645	438	6	components	component	NOUN
easat-9645	438	7	(	(	PUNCT
easat-9645	438	8	eigenvectors	eigenvector	NOUN
easat-9645	438	9	of	of	ADP
easat-9645	438	10	the	the	DET
easat-9645	438	11	covariance	covariance	NOUN
easat-9645	438	12	matrix	matrix	NOUN
easat-9645	438	13	)	)	PUNCT
easat-9645	438	14	,	,	PUNCT
easat-9645	438	15	which	which	PRON
easat-9645	438	16	are	be	AUX
easat-9645	438	17	computed	compute	VERB
easat-9645	438	18	using	use	VERB
easat-9645	438	19	inner	inner	ADJ
easat-9645	438	20	product	product	NOUN
easat-9645	438	21	operations	operation	NOUN
easat-9645	438	22	.	.	PUNCT
easat-9645	439	1	6.5	6.5	NUM
easat-9645	439	2	.	.	PUNCT
easat-9645	440	1	functional	functional	ADJ
easat-9645	440	2	analysis	analysis	NOUN
easat-9645	440	3	and	and	CCONJ
easat-9645	440	4	pdes	pde	VERB
easat-9645	440	5	many	many	ADJ
easat-9645	440	6	boundary	boundary	ADJ
easat-9645	440	7	value	value	NOUN
easat-9645	440	8	problems	problem	NOUN
easat-9645	440	9	for	for	ADP
easat-9645	440	10	partial	partial	ADJ
easat-9645	440	11	differential	differential	ADJ
easat-9645	440	12	equations	equation	NOUN
easat-9645	440	13	are	be	AUX
easat-9645	440	14	solved	solve	VERB
easat-9645	440	15	within	within	ADP
easat-9645	440	16	inner	inner	ADJ
easat-9645	440	17	product	product	NOUN
easat-9645	440	18	spaces	space	NOUN
easat-9645	440	19	using	use	VERB
easat-9645	440	20	methods	method	NOUN
easat-9645	440	21	such	such	ADJ
easat-9645	440	22	as	as	ADP
easat-9645	440	23	the	the	DET
easat-9645	440	24	galerkin	galerkin	NOUN
easat-9645	440	25	or	or	CCONJ
easat-9645	440	26	ritz	ritz	PROPN
easat-9645	440	27	method	method	NOUN
easat-9645	440	28	.	.	PUNCT
easat-9645	441	1	these	these	PRON
easat-9645	441	2	rely	rely	VERB
easat-9645	441	3	on	on	ADP
easat-9645	441	4	projecting	project	VERB
easat-9645	441	5	infinitedimensional	infinitedimensional	ADJ
easat-9645	441	6	problems	problem	NOUN
easat-9645	441	7	into	into	ADP
easat-9645	441	8	finite	finite	ADJ
easat-9645	441	9	-	-	ADJ
easat-9645	441	10	dimensional	dimensional	ADJ
easat-9645	441	11	subspaces	subspace	NOUN
easat-9645	441	12	using	use	VERB
easat-9645	441	13	orthogonality	orthogonality	NOUN
easat-9645	441	14	conditions	condition	NOUN
easat-9645	441	15	.	.	PUNCT
easat-9645	442	1	6.6	6.6	NUM
easat-9645	442	2	.	.	PUNCT
easat-9645	442	3	signal	signal	ADJ
easat-9645	442	4	decomposition	decomposition	NOUN
easat-9645	442	5	using	use	VERB
easat-9645	442	6	orthonormal	orthonormal	ADJ
easat-9645	442	7	basis	basis	NOUN
easat-9645	442	8	(	(	PUNCT
easat-9645	442	9	fourier	fourier	NOUN
easat-9645	442	10	analysis	analysis	NOUN
easat-9645	442	11	)	)	PUNCT
easat-9645	442	12	signals	signal	NOUN
easat-9645	442	13	can	can	AUX
easat-9645	442	14	be	be	AUX
easat-9645	442	15	represented	represent	VERB
easat-9645	442	16	as	as	ADP
easat-9645	442	17	linear	linear	ADJ
easat-9645	442	18	combinations	combination	NOUN
easat-9645	442	19	of	of	ADP
easat-9645	442	20	orthonormal	orthonormal	ADJ
easat-9645	442	21	functions	function	NOUN
easat-9645	442	22	.	.	PUNCT
easat-9645	443	1	the	the	DET
easat-9645	443	2	coefficients	coefficient	NOUN
easat-9645	443	3	are	be	AUX
easat-9645	443	4	obtained	obtain	VERB
easat-9645	443	5	via	via	ADP
easat-9645	443	6	inner	inner	ADJ
easat-9645	443	7	products	product	NOUN
easat-9645	443	8	.	.	PUNCT
easat-9645	444	1	figure	figure	NOUN
easat-9645	444	2	13	13	NUM
easat-9645	444	3	.	.	PUNCT
easat-9645	445	1	matlab	matlab	PROPN
easat-9645	445	2	implementation	implementation	NOUN
easat-9645	445	3	:	:	PUNCT
easat-9645	445	4	visualizing	visualize	VERB
easat-9645	445	5	a	a	DET
easat-9645	445	6	signal	signal	ADJ
easat-9645	445	7	decomposition	decomposition	NOUN
easat-9645	445	8	using	use	VERB
easat-9645	445	9	inner	inner	ADJ
easat-9645	445	10	product	product	NOUN
easat-9645	445	11	.	.	PUNCT
easat-9645	446	1	the	the	DET
easat-9645	446	2	figure	figure	NOUN
easat-9645	446	3	13	13	NUM
easat-9645	446	4	,	,	PUNCT
easat-9645	446	5	illustrates	illustrate	VERB
easat-9645	446	6	signal	signal	ADJ
easat-9645	446	7	decomposition	decomposition	NOUN
easat-9645	446	8	using	use	VERB
easat-9645	446	9	the	the	DET
easat-9645	446	10	inner	inner	ADJ
easat-9645	446	11	product	product	NOUN
easat-9645	446	12	,	,	PUNCT
easat-9645	446	13	comparing	compare	VERB
easat-9645	446	14	an	an	DET
easat-9645	446	15	original	original	ADJ
easat-9645	446	16	signal	signal	NOUN
easat-9645	446	17	(	(	PUNCT
easat-9645	446	18	blue	blue	ADJ
easat-9645	446	19	)	)	PUNCT
easat-9645	446	20	and	and	CCONJ
easat-9645	446	21	its	its	PRON
easat-9645	446	22	reconstructed	reconstructed	ADJ
easat-9645	446	23	signal	signal	NOUN
easat-9645	446	24	(	(	PUNCT
easat-9645	446	25	red	red	ADJ
easat-9645	446	26	)	)	PUNCT
easat-9645	446	27	.	.	PUNCT
easat-9645	447	1	both	both	DET
easat-9645	447	2	signals	signal	NOUN
easat-9645	447	3	vary	vary	VERB
easat-9645	447	4	in	in	ADP
easat-9645	447	5	amplitude	amplitude	NOUN
easat-9645	447	6	from	from	ADP
easat-9645	447	7	-1.5	-1.5	PUNCT
easat-9645	447	8	to	to	ADP
easat-9645	447	9	1	1	NUM
easat-9645	447	10	over	over	ADP
easat-9645	447	11	time	time	NOUN
easat-9645	447	12	(	(	PUNCT
easat-9645	447	13	0	0	NUM
easat-9645	447	14	to	to	PART
easat-9645	447	15	7	7	NUM
easat-9645	447	16	)	)	PUNCT
easat-9645	447	17	,	,	PUNCT
easat-9645	447	18	showing	show	VERB
easat-9645	447	19	a	a	DET
easat-9645	447	20	similar	similar	ADJ
easat-9645	447	21	pattern	pattern	NOUN
easat-9645	447	22	with	with	ADP
easat-9645	447	23	peaks	peak	NOUN
easat-9645	447	24	and	and	CCONJ
easat-9645	447	25	troughs	troughs	NOUN
easat-9645	447	26	,	,	PUNCT
easat-9645	447	27	indicating	indicate	VERB
easat-9645	447	28	the	the	DET
easat-9645	447	29	reconstruction	reconstruction	NOUN
easat-9645	447	30	closely	closely	ADV
easat-9645	447	31	matches	match	VERB
easat-9645	447	32	the	the	DET
easat-9645	447	33	original	original	NOUN
easat-9645	447	34	.	.	PUNCT
easat-9645	448	1	7	7	X
easat-9645	448	2	.	.	X
easat-9645	448	3	applications	application	NOUN
easat-9645	448	4	of	of	ADP
easat-9645	448	5	inner	inner	ADJ
easat-9645	448	6	product	product	NOUN
easat-9645	448	7	spaces	space	NOUN
easat-9645	448	8	in	in	ADP
easat-9645	448	9	machine	machine	NOUN
easat-9645	448	10	learning	learn	VERB
easat-9645	448	11	inner	inner	ADJ
easat-9645	448	12	product	product	NOUN
easat-9645	448	13	spaces	space	NOUN
easat-9645	448	14	form	form	VERB
easat-9645	448	15	a	a	DET
easat-9645	448	16	mathematical	mathematical	ADJ
easat-9645	448	17	foundation	foundation	NOUN
easat-9645	448	18	for	for	ADP
easat-9645	448	19	many	many	ADJ
easat-9645	448	20	algorithms	algorithm	NOUN
easat-9645	448	21	in	in	ADP
easat-9645	448	22	machine	machine	NOUN
easat-9645	448	23	learning	learning	NOUN
easat-9645	448	24	,	,	PUNCT
easat-9645	448	25	offering	offer	VERB
easat-9645	448	26	a	a	DET
easat-9645	448	27	framework	framework	NOUN
easat-9645	448	28	to	to	PART
easat-9645	448	29	measure	measure	VERB
easat-9645	448	30	similarity	similarity	NOUN
easat-9645	448	31	,	,	PUNCT
easat-9645	448	32	define	define	VERB
easat-9645	448	33	geometry	geometry	NOUN
easat-9645	448	34	in	in	ADP
easat-9645	448	35	feature	feature	NOUN
easat-9645	448	36	spaces	space	NOUN
easat-9645	448	37	,	,	PUNCT
easat-9645	448	38	and	and	CCONJ
easat-9645	448	39	facilitate	facilitate	NOUN
easat-9645	448	40	learning	learn	VERB
easat-9645	448	41	in	in	ADP
easat-9645	448	42	high	high	ADJ
easat-9645	448	43	-	-	PUNCT
easat-9645	448	44	dimensional	dimensional	ADJ
easat-9645	448	45	settings	setting	NOUN
easat-9645	448	46	.	.	PUNCT
easat-9645	449	1	their	their	PRON
easat-9645	449	2	properties	property	NOUN
easat-9645	449	3	enable	enable	VERB
easat-9645	449	4	both	both	DET
easat-9645	449	5	theoretical	theoretical	ADJ
easat-9645	449	6	insights	insight	NOUN
easat-9645	449	7	and	and	CCONJ
easat-9645	449	8	practical	practical	ADJ
easat-9645	449	9	implementations	implementation	NOUN
easat-9645	449	10	in	in	ADP
easat-9645	449	11	various	various	ADJ
easat-9645	449	12	learning	learning	NOUN
easat-9645	449	13	paradigms	paradigm	NOUN
easat-9645	449	14	.	.	PUNCT
easat-9645	450	1	1518	1518	NUM
easat-9645	450	2	edelweiss	edelweiss	PROPN
easat-9645	450	3	applied	apply	VERB
easat-9645	450	4	science	science	NOUN
easat-9645	450	5	and	and	CCONJ
easat-9645	450	6	technology	technology	NOUN
easat-9645	450	7	issn	issn	PROPN
easat-9645	450	8	:	:	PUNCT
easat-9645	450	9	2576	2576	NUM
easat-9645	450	10	-	-	SYM
easat-9645	450	11	8484	8484	NUM
easat-9645	450	12	vol	vol	NOUN
easat-9645	450	13	.	.	PROPN
easat-9645	451	1	9	9	NUM
easat-9645	451	2	,	,	PUNCT
easat-9645	451	3	no	no	INTJ
easat-9645	451	4	.	.	NOUN
easat-9645	451	5	8	8	NUM
easat-9645	451	6	:	:	SYM
easat-9645	451	7	1498	1498	NUM
easat-9645	451	8	-	-	SYM
easat-9645	451	9	1523	1523	NUM
easat-9645	451	10	,	,	PUNCT
easat-9645	451	11	2025	2025	NUM
easat-9645	451	12	doi	doi	NOUN
easat-9645	451	13	:	:	PUNCT
easat-9645	451	14	10.55214/2576	10.55214/2576	NUM
easat-9645	451	15	-	-	SYM
easat-9645	451	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	451	17	©	©	PROPN
easat-9645	451	18	2025	2025	NUM
easat-9645	451	19	by	by	ADP
easat-9645	451	20	the	the	DET
easat-9645	451	21	authors	author	NOUN
easat-9645	451	22	;	;	PUNCT
easat-9645	451	23	licensee	licensee	PROPN
easat-9645	451	24	learning	learning	NOUN
easat-9645	451	25	gate	gate	NOUN
easat-9645	451	26	7	7	NUM
easat-9645	451	27	.	.	NOUN
easat-9645	451	28	1	1	NUM
easat-9645	451	29	.	.	X
easat-9645	451	30	similarity	similarity	NOUN
easat-9645	451	31	measures	measure	NOUN
easat-9645	451	32	and	and	CCONJ
easat-9645	451	33	kernel	kernel	NOUN
easat-9645	451	34	methods	method	NOUN
easat-9645	451	35	the	the	DET
easat-9645	451	36	inner	inner	ADJ
easat-9645	451	37	product	product	NOUN
easat-9645	451	38	is	be	AUX
easat-9645	451	39	frequently	frequently	ADV
easat-9645	451	40	used	use	VERB
easat-9645	451	41	to	to	PART
easat-9645	451	42	quantify	quantify	VERB
easat-9645	451	43	the	the	DET
easat-9645	451	44	similarity	similarity	NOUN
easat-9645	451	45	between	between	ADP
easat-9645	451	46	vectors	vector	NOUN
easat-9645	451	47	.	.	PUNCT
easat-9645	452	1	in	in	ADP
easat-9645	452	2	classification	classification	NOUN
easat-9645	452	3	and	and	CCONJ
easat-9645	452	4	clustering	clustering	NOUN
easat-9645	452	5	,	,	PUNCT
easat-9645	452	6	similarity	similarity	NOUN
easat-9645	452	7	functions	function	NOUN
easat-9645	452	8	guide	guide	VERB
easat-9645	452	9	decisions	decision	NOUN
easat-9645	452	10	about	about	ADP
easat-9645	452	11	groupings	grouping	NOUN
easat-9645	452	12	or	or	CCONJ
easat-9645	452	13	class	class	NOUN
easat-9645	452	14	membership	membership	NOUN
easat-9645	452	15	.	.	PUNCT
easat-9645	453	1	kernel	kernel	PROPN
easat-9645	453	2	methods	method	NOUN
easat-9645	453	3	extend	extend	VERB
easat-9645	453	4	this	this	DET
easat-9645	453	5	idea	idea	NOUN
easat-9645	453	6	by	by	ADP
easat-9645	453	7	computing	compute	VERB
easat-9645	453	8	inner	inner	ADJ
easat-9645	453	9	products	product	NOUN
easat-9645	453	10	in	in	ADP
easat-9645	453	11	transformed	transform	VERB
easat-9645	453	12	feature	feature	NOUN
easat-9645	453	13	spaces	space	NOUN
easat-9645	453	14	,	,	PUNCT
easat-9645	453	15	often	often	ADV
easat-9645	453	16	through	through	ADP
easat-9645	453	17	kernel	kernel	NOUN
easat-9645	453	18	functions	function	NOUN
easat-9645	453	19	.	.	PUNCT
easat-9645	454	1	this	this	PRON
easat-9645	454	2	is	be	AUX
easat-9645	454	3	central	central	ADJ
easat-9645	454	4	to	to	ADP
easat-9645	454	5	algorithms	algorithm	NOUN
easat-9645	454	6	like	like	ADP
easat-9645	454	7	the	the	DET
easat-9645	454	8	support	support	NOUN
easat-9645	454	9	vector	vector	NOUN
easat-9645	454	10	machine	machine	NOUN
easat-9645	454	11	(	(	PUNCT
easat-9645	454	12	svm	svm	PROPN
easat-9645	454	13	)	)	PUNCT
easat-9645	454	14	,	,	PUNCT
easat-9645	454	15	where	where	SCONJ
easat-9645	454	16	the	the	DET
easat-9645	454	17	decision	decision	NOUN
easat-9645	454	18	boundary	boundary	NOUN
easat-9645	454	19	is	be	AUX
easat-9645	454	20	constructed	construct	VERB
easat-9645	454	21	using	use	VERB
easat-9645	454	22	inner	inner	ADJ
easat-9645	454	23	products	product	NOUN
easat-9645	454	24	between	between	ADP
easat-9645	454	25	data	datum	NOUN
easat-9645	454	26	points	point	NOUN
easat-9645	454	27	in	in	ADP
easat-9645	454	28	a	a	PRON
easat-9645	454	29	high	high	ADV
easat-9645	454	30	-	-	PUNCT
easat-9645	454	31	dimensional	dimensional	ADJ
easat-9645	454	32	(	(	PUNCT
easat-9645	454	33	possibly	possibly	ADV
easat-9645	454	34	infinite	infinite	ADJ
easat-9645	454	35	-	-	PUNCT
easat-9645	454	36	dimensional	dimensional	ADJ
easat-9645	454	37	)	)	PUNCT
easat-9645	454	38	reproducing	reproduce	VERB
easat-9645	454	39	kernel	kernel	PROPN
easat-9645	454	40	hilbert	hilbert	PROPN
easat-9645	454	41	space	space	NOUN
easat-9645	454	42	(	(	PUNCT
easat-9645	454	43	rkhs	rkh	NOUN
easat-9645	454	44	)	)	PUNCT
easat-9645	454	45	.	.	PUNCT
easat-9645	455	1	7.2	7.2	NUM
easat-9645	455	2	.	.	PUNCT
easat-9645	455	3	principal	principal	ADJ
easat-9645	455	4	component	component	NOUN
easat-9645	455	5	analysis	analysis	NOUN
easat-9645	455	6	(	(	PUNCT
easat-9645	455	7	pca	pca	NOUN
easat-9645	455	8	)	)	PUNCT
easat-9645	455	9	pca	pca	PROPN
easat-9645	455	10	is	be	AUX
easat-9645	455	11	a	a	DET
easat-9645	455	12	dimensionality	dimensionality	NOUN
easat-9645	455	13	reduction	reduction	NOUN
easat-9645	455	14	technique	technique	NOUN
easat-9645	455	15	that	that	PRON
easat-9645	455	16	relies	rely	VERB
easat-9645	455	17	on	on	ADP
easat-9645	455	18	inner	inner	ADJ
easat-9645	455	19	products	product	NOUN
easat-9645	455	20	to	to	PART
easat-9645	455	21	compute	compute	VERB
easat-9645	455	22	the	the	DET
easat-9645	455	23	covariance	covariance	NOUN
easat-9645	455	24	matrix	matrix	NOUN
easat-9645	455	25	and	and	CCONJ
easat-9645	455	26	its	its	PRON
easat-9645	455	27	eigenvectors	eigenvector	NOUN
easat-9645	455	28	.	.	PUNCT
easat-9645	456	1	these	these	DET
easat-9645	456	2	eigenvectors	eigenvector	NOUN
easat-9645	456	3	form	form	VERB
easat-9645	456	4	an	an	DET
easat-9645	456	5	orthogonal	orthogonal	ADJ
easat-9645	456	6	basis	basis	NOUN
easat-9645	456	7	that	that	PRON
easat-9645	456	8	captures	capture	VERB
easat-9645	456	9	the	the	DET
easat-9645	456	10	directions	direction	NOUN
easat-9645	456	11	of	of	ADP
easat-9645	456	12	maximum	maximum	ADJ
easat-9645	456	13	variance	variance	NOUN
easat-9645	456	14	in	in	ADP
easat-9645	456	15	the	the	DET
easat-9645	456	16	data	datum	NOUN
easat-9645	456	17	.	.	PUNCT
easat-9645	457	1	by	by	ADP
easat-9645	457	2	projecting	project	VERB
easat-9645	457	3	data	datum	NOUN
easat-9645	457	4	onto	onto	ADP
easat-9645	457	5	these	these	DET
easat-9645	457	6	directions	direction	NOUN
easat-9645	457	7	using	use	VERB
easat-9645	457	8	inner	inner	ADJ
easat-9645	457	9	products	product	NOUN
easat-9645	457	10	,	,	PUNCT
easat-9645	457	11	pca	pca	NOUN
easat-9645	457	12	simplifies	simplifie	NOUN
easat-9645	457	13	the	the	DET
easat-9645	457	14	feature	feature	NOUN
easat-9645	457	15	space	space	NOUN
easat-9645	457	16	while	while	SCONJ
easat-9645	457	17	preserving	preserve	VERB
easat-9645	457	18	key	key	ADJ
easat-9645	457	19	information	information	NOUN
easat-9645	457	20	,	,	PUNCT
easat-9645	457	21	improving	improve	VERB
easat-9645	457	22	both	both	DET
easat-9645	457	23	interpretability	interpretability	NOUN
easat-9645	457	24	and	and	CCONJ
easat-9645	457	25	computational	computational	ADJ
easat-9645	457	26	efficiency	efficiency	NOUN
easat-9645	457	27	.	.	PUNCT
easat-9645	458	1	7.3	7.3	NUM
easat-9645	458	2	.	.	PUNCT
easat-9645	458	3	neural	neural	ADJ
easat-9645	458	4	networks	network	NOUN
easat-9645	458	5	and	and	CCONJ
easat-9645	458	6	optimization	optimization	NOUN
easat-9645	458	7	although	although	SCONJ
easat-9645	458	8	neural	neural	ADJ
easat-9645	458	9	networks	network	NOUN
easat-9645	458	10	are	be	AUX
easat-9645	458	11	nonlinear	nonlinear	ADJ
easat-9645	458	12	models	model	NOUN
easat-9645	458	13	,	,	PUNCT
easat-9645	458	14	inner	inner	ADJ
easat-9645	458	15	products	product	NOUN
easat-9645	458	16	still	still	ADV
easat-9645	458	17	appear	appear	VERB
easat-9645	458	18	in	in	ADP
easat-9645	458	19	the	the	DET
easat-9645	458	20	computation	computation	NOUN
easat-9645	458	21	of	of	ADP
easat-9645	458	22	neuron	neuron	NOUN
easat-9645	458	23	activations	activation	NOUN
easat-9645	458	24	,	,	PUNCT
easat-9645	458	25	especially	especially	ADV
easat-9645	458	26	in	in	ADP
easat-9645	458	27	fully	fully	ADV
easat-9645	458	28	connected	connected	ADJ
easat-9645	458	29	layers	layer	NOUN
easat-9645	458	30	.	.	PUNCT
easat-9645	459	1	each	each	DET
easat-9645	459	2	neuron	neuron	NOUN
easat-9645	459	3	computes	compute	VERB
easat-9645	459	4	a	a	DET
easat-9645	459	5	weighted	weight	VERB
easat-9645	459	6	inner	inner	ADJ
easat-9645	459	7	product	product	NOUN
easat-9645	459	8	between	between	ADP
easat-9645	459	9	the	the	DET
easat-9645	459	10	input	input	NOUN
easat-9645	459	11	vector	vector	NOUN
easat-9645	459	12	and	and	CCONJ
easat-9645	459	13	a	a	DET
easat-9645	459	14	weight	weight	NOUN
easat-9645	459	15	vector	vector	NOUN
easat-9645	459	16	,	,	PUNCT
easat-9645	459	17	followed	follow	VERB
easat-9645	459	18	by	by	ADP
easat-9645	459	19	a	a	DET
easat-9645	459	20	non	non	ADJ
easat-9645	459	21	-	-	ADJ
easat-9645	459	22	linear	linear	ADJ
easat-9645	459	23	activation	activation	NOUN
easat-9645	459	24	.	.	PUNCT
easat-9645	460	1	additionally	additionally	ADV
easat-9645	460	2	,	,	PUNCT
easat-9645	460	3	gradient	gradient	NOUN
easat-9645	460	4	-	-	PUNCT
easat-9645	460	5	based	base	VERB
easat-9645	460	6	optimization	optimization	NOUN
easat-9645	460	7	methods	method	NOUN
easat-9645	460	8	used	use	VERB
easat-9645	460	9	to	to	PART
easat-9645	460	10	train	train	VERB
easat-9645	460	11	networks	network	NOUN
easat-9645	460	12	rely	rely	VERB
easat-9645	460	13	on	on	ADP
easat-9645	460	14	inner	inner	ADJ
easat-9645	460	15	product	product	NOUN
easat-9645	460	16	-	-	PUNCT
easat-9645	460	17	based	base	VERB
easat-9645	460	18	notions	notion	NOUN
easat-9645	460	19	such	such	ADJ
easat-9645	460	20	as	as	ADP
easat-9645	460	21	gradient	gradient	ADJ
easat-9645	460	22	descent	descent	NOUN
easat-9645	460	23	directions	direction	NOUN
easat-9645	460	24	and	and	CCONJ
easat-9645	460	25	orthogonal	orthogonal	ADJ
easat-9645	460	26	projections	projection	NOUN
easat-9645	460	27	in	in	ADP
easat-9645	460	28	parameter	parameter	NOUN
easat-9645	460	29	space	space	NOUN
easat-9645	460	30	.	.	PUNCT
easat-9645	461	1	7.4	7.4	NUM
easat-9645	461	2	.	.	PUNCT
easat-9645	462	1	recommendation	recommendation	NOUN
easat-9645	462	2	systems	system	NOUN
easat-9645	462	3	matrix	matrix	NOUN
easat-9645	462	4	factorization	factorization	NOUN
easat-9645	462	5	techniques	technique	NOUN
easat-9645	462	6	for	for	ADP
easat-9645	462	7	collaborative	collaborative	ADJ
easat-9645	462	8	filtering	filtering	NOUN
easat-9645	462	9	in	in	ADP
easat-9645	462	10	recommendation	recommendation	NOUN
easat-9645	462	11	systems	system	NOUN
easat-9645	462	12	use	use	VERB
easat-9645	462	13	inner	inner	ADJ
easat-9645	462	14	products	product	NOUN
easat-9645	462	15	to	to	PART
easat-9645	462	16	model	model	VERB
easat-9645	462	17	user	user	NOUN
easat-9645	462	18	item	item	NOUN
easat-9645	462	19	interactions	interaction	NOUN
easat-9645	462	20	.	.	PUNCT
easat-9645	463	1	in	in	ADP
easat-9645	463	2	such	such	ADJ
easat-9645	463	3	systems	system	NOUN
easat-9645	463	4	,	,	PUNCT
easat-9645	463	5	the	the	DET
easat-9645	463	6	predicted	predict	VERB
easat-9645	463	7	rating	rating	NOUN
easat-9645	463	8	of	of	ADP
easat-9645	463	9	a	a	DET
easat-9645	463	10	user	user	NOUN
easat-9645	463	11	for	for	ADP
easat-9645	463	12	an	an	DET
easat-9645	463	13	item	item	NOUN
easat-9645	463	14	is	be	AUX
easat-9645	463	15	computed	compute	VERB
easat-9645	463	16	as	as	ADP
easat-9645	463	17	the	the	DET
easat-9645	463	18	inner	inner	ADJ
easat-9645	463	19	product	product	NOUN
easat-9645	463	20	of	of	ADP
easat-9645	463	21	the	the	DET
easat-9645	463	22	user	user	NOUN
easat-9645	463	23	’s	’s	PART
easat-9645	463	24	and	and	CCONJ
easat-9645	463	25	item	item	NOUN
easat-9645	463	26	’s	’s	PART
easat-9645	463	27	latent	latent	ADJ
easat-9645	463	28	feature	feature	NOUN
easat-9645	463	29	vectors	vector	NOUN
easat-9645	463	30	.	.	PUNCT
easat-9645	464	1	also	also	ADV
easat-9645	464	2	,	,	PUNCT
easat-9645	464	3	this	this	DET
easat-9645	464	4	manuscript	manuscript	NOUN
easat-9645	464	5	builds	build	VERB
easat-9645	464	6	on	on	ADP
easat-9645	464	7	our	our	PRON
easat-9645	464	8	earlier	early	ADJ
easat-9645	464	9	analytic	analytic	ADJ
easat-9645	464	10	work	work	NOUN
easat-9645	464	11	applying	apply	VERB
easat-9645	464	12	sylow	sylow	NOUN
easat-9645	464	13	’s	’s	PART
easat-9645	464	14	theorems	theorem	NOUN
easat-9645	464	15	to	to	ADP
easat-9645	464	16	small	small	ADJ
easat-9645	464	17	composite	composite	ADJ
easat-9645	464	18	orders	order	NOUN
easat-9645	464	19	[	[	X
easat-9645	464	20	19	19	NUM
easat-9645	464	21	,	,	PUNCT
easat-9645	464	22	20	20	NUM
easat-9645	464	23	]	]	PUNCT
easat-9645	464	24	.	.	PUNCT
easat-9645	465	1	8	8	X
easat-9645	465	2	.	.	PUNCT
easat-9645	465	3	applications	application	NOUN
easat-9645	465	4	of	of	ADP
easat-9645	465	5	inner	inner	ADJ
easat-9645	465	6	product	product	NOUN
easat-9645	465	7	and	and	CCONJ
easat-9645	465	8	hilbert	hilbert	NOUN
easat-9645	465	9	spaces	space	NOUN
easat-9645	465	10	in	in	ADP
easat-9645	465	11	machine	machine	NOUN
easat-9645	465	12	learning	learn	VERB
easat-9645	465	13	8.1	8.1	NUM
easat-9645	465	14	.	.	PUNCT
easat-9645	466	1	inner	inner	ADJ
easat-9645	466	2	product	product	NOUN
easat-9645	466	3	spaces	space	VERB
easat-9645	466	4	in	in	ADP
easat-9645	466	5	machine	machine	NOUN
easat-9645	466	6	learning	learn	VERB
easat-9645	466	7	an	an	DET
easat-9645	466	8	inner	inner	ADJ
easat-9645	466	9	product	product	NOUN
easat-9645	466	10	space	space	NOUN
easat-9645	466	11	provides	provide	VERB
easat-9645	466	12	a	a	DET
easat-9645	466	13	way	way	NOUN
easat-9645	466	14	to	to	PART
easat-9645	466	15	measure	measure	VERB
easat-9645	466	16	angles	angle	NOUN
easat-9645	466	17	and	and	CCONJ
easat-9645	466	18	distances	distance	NOUN
easat-9645	466	19	between	between	ADP
easat-9645	466	20	vectors	vector	NOUN
easat-9645	466	21	.	.	PUNCT
easat-9645	467	1	this	this	DET
easat-9645	467	2	structure	structure	NOUN
easat-9645	467	3	underlies	underlie	VERB
easat-9645	467	4	several	several	ADJ
easat-9645	467	5	machine	machine	NOUN
easat-9645	467	6	learning	learning	NOUN
easat-9645	467	7	concepts	concept	NOUN
easat-9645	467	8	,	,	PUNCT
easat-9645	467	9	especially	especially	ADV
easat-9645	467	10	where	where	SCONJ
easat-9645	467	11	similarity	similarity	NOUN
easat-9645	467	12	,	,	PUNCT
easat-9645	467	13	projection	projection	NOUN
easat-9645	467	14	,	,	PUNCT
easat-9645	467	15	and	and	CCONJ
easat-9645	467	16	orthogonality	orthogonality	NOUN
easat-9645	467	17	are	be	AUX
easat-9645	467	18	essential	essential	ADJ
easat-9645	467	19	.	.	PUNCT
easat-9645	468	1	also	also	ADV
easat-9645	468	2	,	,	PUNCT
easat-9645	468	3	the	the	DET
easat-9645	468	4	machine	machine	NOUN
easat-9645	468	5	learning	learn	VERB
easat-9645	468	6	concepts	concept	NOUN
easat-9645	468	7	are	be	AUX
easat-9645	468	8	like	like	ADP
easat-9645	468	9	similarity	similarity	NOUN
easat-9645	468	10	measures	measure	NOUN
easat-9645	468	11	,	,	PUNCT
easat-9645	468	12	pca	pca	PROPN
easat-9645	468	13	&	&	CCONJ
easat-9645	468	14	data	data	PROPN
easat-9645	468	15	compression	compression	PROPN
easat-9645	468	16	,	,	PUNCT
easat-9645	468	17	regression	regression	NOUN
easat-9645	468	18	&	&	CCONJ
easat-9645	468	19	optimization	optimization	NOUN
easat-9645	468	20	8.2	8.2	NUM
easat-9645	468	21	.	.	PUNCT
easat-9645	469	1	hilbert	hilbert	PROPN
easat-9645	469	2	spaces	space	NOUN
easat-9645	469	3	in	in	ADP
easat-9645	469	4	machine	machine	NOUN
easat-9645	469	5	learning	learn	VERB
easat-9645	469	6	a	a	DET
easat-9645	469	7	hilbert	hilbert	NOUN
easat-9645	469	8	space	space	NOUN
easat-9645	469	9	is	be	AUX
easat-9645	469	10	a	a	DET
easat-9645	469	11	complete	complete	ADJ
easat-9645	469	12	inner	inner	ADJ
easat-9645	469	13	product	product	NOUN
easat-9645	469	14	space	space	NOUN
easat-9645	469	15	,	,	PUNCT
easat-9645	469	16	allowing	allow	VERB
easat-9645	469	17	infinite	infinite	ADJ
easat-9645	469	18	-	-	PUNCT
easat-9645	469	19	dimensional	dimensional	ADJ
easat-9645	469	20	extensions	extension	NOUN
easat-9645	469	21	of	of	ADP
easat-9645	469	22	vector	vector	NOUN
easat-9645	469	23	space	space	NOUN
easat-9645	469	24	methods	method	NOUN
easat-9645	469	25	.	.	PUNCT
easat-9645	470	1	in	in	ADP
easat-9645	470	2	machine	machine	NOUN
easat-9645	470	3	learning	learning	NOUN
easat-9645	470	4	,	,	PUNCT
easat-9645	470	5	hilbert	hilbert	NOUN
easat-9645	470	6	spaces	space	NOUN
easat-9645	470	7	(	(	PUNCT
easat-9645	470	8	especially	especially	ADV
easat-9645	470	9	rkhs	rkh	NOUN
easat-9645	470	10	)	)	PUNCT
easat-9645	470	11	are	be	AUX
easat-9645	470	12	used	use	VERB
easat-9645	470	13	to	to	PART
easat-9645	470	14	generalize	generalize	VERB
easat-9645	470	15	linear	linear	ADJ
easat-9645	470	16	algorithms	algorithm	NOUN
easat-9645	470	17	to	to	ADP
easat-9645	470	18	nonlinear	nonlinear	ADJ
easat-9645	470	19	contexts	context	NOUN
easat-9645	470	20	through	through	ADP
easat-9645	470	21	kernel	kernel	NOUN
easat-9645	470	22	functions	function	NOUN
easat-9645	470	23	,	,	PUNCT
easat-9645	470	24	functional	functional	ADJ
easat-9645	470	25	regression	regression	NOUN
easat-9645	470	26	,	,	PUNCT
easat-9645	470	27	spectral	spectral	ADJ
easat-9645	470	28	learning	learning	NOUN
easat-9645	470	29	methods	method	NOUN
easat-9645	470	30	.	.	PUNCT
easat-9645	471	1	1519	1519	NUM
easat-9645	471	2	edelweiss	edelweiss	PROPN
easat-9645	471	3	applied	apply	VERB
easat-9645	471	4	science	science	NOUN
easat-9645	471	5	and	and	CCONJ
easat-9645	471	6	technology	technology	NOUN
easat-9645	471	7	issn	issn	PROPN
easat-9645	471	8	:	:	PUNCT
easat-9645	471	9	2576	2576	NUM
easat-9645	471	10	-	-	SYM
easat-9645	471	11	8484	8484	NUM
easat-9645	471	12	vol	vol	NOUN
easat-9645	471	13	.	.	PROPN
easat-9645	472	1	9	9	NUM
easat-9645	472	2	,	,	PUNCT
easat-9645	472	3	no	no	INTJ
easat-9645	472	4	.	.	NOUN
easat-9645	472	5	8	8	NUM
easat-9645	472	6	:	:	SYM
easat-9645	472	7	1498	1498	NUM
easat-9645	472	8	-	-	SYM
easat-9645	472	9	1523	1523	NUM
easat-9645	472	10	,	,	PUNCT
easat-9645	472	11	2025	2025	NUM
easat-9645	472	12	doi	doi	NOUN
easat-9645	472	13	:	:	PUNCT
easat-9645	472	14	10.55214/2576	10.55214/2576	NUM
easat-9645	472	15	-	-	SYM
easat-9645	472	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	472	17	©	©	PROPN
easat-9645	472	18	2025	2025	NUM
easat-9645	472	19	by	by	ADP
easat-9645	472	20	the	the	DET
easat-9645	472	21	authors	author	NOUN
easat-9645	472	22	;	;	PUNCT
easat-9645	472	23	licensee	licensee	PROPN
easat-9645	472	24	learning	learn	VERB
easat-9645	472	25	gate	gate	NOUN
easat-9645	472	26	table	table	NOUN
easat-9645	472	27	1	1	NUM
easat-9645	472	28	.	.	PUNCT
easat-9645	472	29	comparison	comparison	NOUN
easat-9645	472	30	in	in	ADP
easat-9645	472	31	inner	inner	ADJ
easat-9645	472	32	product	product	NOUN
easat-9645	472	33	and	and	CCONJ
easat-9645	472	34	hilbert	hilbert	NOUN
easat-9645	472	35	spaces	space	NOUN
easat-9645	472	36	in	in	ADP
easat-9645	472	37	machine	machine	NOUN
easat-9645	472	38	learning	learning	NOUN
easat-9645	472	39	.	.	PUNCT
easat-9645	473	1	feature	feature	VERB
easat-9645	473	2	inner	inner	ADJ
easat-9645	473	3	product	product	NOUN
easat-9645	473	4	space	space	NOUN
easat-9645	473	5	hilbert	hilbert	NOUN
easat-9645	473	6	space	space	NOUN
easat-9645	473	7	(	(	PUNCT
easat-9645	473	8	complete	complete	VERB
easat-9645	473	9	inner	inner	ADJ
easat-9645	473	10	product	product	NOUN
easat-9645	473	11	space	space	NOUN
easat-9645	473	12	)	)	PUNCT
easat-9645	473	13	dimensionality	dimensionality	NOUN
easat-9645	473	14	finite	finite	ADJ
easat-9645	473	15	-	-	ADJ
easat-9645	473	16	dimensional	dimensional	ADJ
easat-9645	473	17	can	can	AUX
easat-9645	473	18	be	be	AUX
easat-9645	473	19	infinite	infinite	ADJ
easat-9645	473	20	-	-	PUNCT
easat-9645	473	21	dimensional	dimensional	ADJ
easat-9645	473	22	completeness	completeness	NOUN
easat-9645	473	23	not	not	PART
easat-9645	473	24	necessarily	necessarily	ADV
easat-9645	473	25	complete	complete	ADJ
easat-9645	473	26	always	always	ADV
easat-9645	473	27	complete	complete	ADJ
easat-9645	473	28	use	use	NOUN
easat-9645	473	29	in	in	ADP
easat-9645	473	30	ml	ml	NOUN
easat-9645	473	31	feature	feature	NOUN
easat-9645	473	32	similarity	similarity	NOUN
easat-9645	473	33	,	,	PUNCT
easat-9645	473	34	projections	projection	NOUN
easat-9645	473	35	,	,	PUNCT
easat-9645	473	36	pca	pca	PROPN
easat-9645	473	37	,	,	PUNCT
easat-9645	473	38	regression	regression	NOUN
easat-9645	473	39	kernel	kernel	PROPN
easat-9645	473	40	methods	method	NOUN
easat-9645	473	41	,	,	PUNCT
easat-9645	473	42	gaussian	gaussian	NOUN
easat-9645	473	43	processes	process	NOUN
easat-9645	473	44	,	,	PUNCT
easat-9645	473	45	rkhs	rkh	NOUN
easat-9645	473	46	,	,	PUNCT
easat-9645	473	47	functional	functional	ADJ
easat-9645	473	48	learning	learn	VERB
easat-9645	473	49	examples	example	NOUN
easat-9645	473	50	in	in	ADP
easat-9645	473	51	ml	ml	NOUN
easat-9645	473	52	linear	linear	PROPN
easat-9645	473	53	regression	regression	NOUN
easat-9645	473	54	,	,	PUNCT
easat-9645	473	55	pca	pca	PROPN
easat-9645	473	56	,	,	PUNCT
easat-9645	473	57	cosine	cosine	NOUN
easat-9645	473	58	similarity	similarity	NOUN
easat-9645	473	59	,	,	PUNCT
easat-9645	473	60	k	k	NOUN
easat-9645	473	61	-	-	PUNCT
easat-9645	473	62	means	means	NOUN
easat-9645	473	63	svm	svm	PROPN
easat-9645	473	64	(	(	PUNCT
easat-9645	473	65	with	with	ADP
easat-9645	473	66	kernel	kernel	NOUN
easat-9645	473	67	)	)	PUNCT
easat-9645	473	68	,	,	PUNCT
easat-9645	473	69	gaussian	gaussian	NOUN
easat-9645	473	70	processes	process	NOUN
easat-9645	473	71	,	,	PUNCT
easat-9645	474	1	spectral	spectral	ADJ
easat-9645	474	2	learning	learning	NOUN
easat-9645	474	3	matlab	matlab	PROPN
easat-9645	474	4	tool	tool	NOUN
easat-9645	474	5	use	use	VERB
easat-9645	474	6	dot	dot	NOUN
easat-9645	474	7	,	,	PUNCT
easat-9645	474	8	eig	eig	NOUN
easat-9645	474	9	,	,	PUNCT
easat-9645	474	10	matrix	matrix	NOUN
easat-9645	474	11	algebra	algebra	NOUN
easat-9645	474	12	kernel	kernel	PROPN
easat-9645	474	13	computation	computation	NOUN
easat-9645	474	14	,	,	PUNCT
easat-9645	474	15	functional	functional	ADJ
easat-9645	474	16	inner	inner	ADJ
easat-9645	474	17	products	product	NOUN
easat-9645	474	18	mathematical	mathematical	ADJ
easat-9645	474	19	framework	framework	NOUN
easat-9645	474	20	euclidean	euclidean	PROPN
easat-9645	474	21	geometry	geometry	NOUN
easat-9645	474	22	functional	functional	ADJ
easat-9645	474	23	analysis	analysis	NOUN
easat-9645	474	24	figure	figure	NOUN
easat-9645	474	25	14	14	NUM
easat-9645	474	26	.	.	PUNCT
easat-9645	475	1	matlab	matlab	PROPN
easat-9645	475	2	simulation	simulation	PROPN
easat-9645	475	3	:	:	PUNCT
easat-9645	475	4	inner	inner	ADJ
easat-9645	475	5	product	product	NOUN
easat-9645	475	6	vs.	vs.	ADP
easat-9645	475	7	rkhs	rkh	NOUN
easat-9645	475	8	(	(	PUNCT
easat-9645	475	9	hilbert	hilbert	NOUN
easat-9645	475	10	space	space	NOUN
easat-9645	475	11	via	via	ADP
easat-9645	475	12	kernel	kernel	PROPN
easat-9645	475	13	)	)	PUNCT
easat-9645	475	14	.	.	PUNCT
easat-9645	476	1	this	this	DET
easat-9645	476	2	simulation	simulation	NOUN
easat-9645	476	3	compares	compare	VERB
easat-9645	476	4	linear	linear	ADJ
easat-9645	476	5	inner	inner	ADJ
easat-9645	476	6	product	product	NOUN
easat-9645	476	7	similarity	similarity	NOUN
easat-9645	476	8	and	and	CCONJ
easat-9645	476	9	kernel	kernel	NOUN
easat-9645	476	10	-	-	PUNCT
easat-9645	476	11	based	base	VERB
easat-9645	476	12	similarity	similarity	NOUN
easat-9645	476	13	(	(	PUNCT
easat-9645	476	14	in	in	ADP
easat-9645	476	15	rkhs	rkh	NOUN
easat-9645	476	16	)	)	PUNCT
easat-9645	476	17	.	.	PUNCT
easat-9645	477	1	the	the	DET
easat-9645	477	2	figure	figure	NOUN
easat-9645	477	3	14	14	NUM
easat-9645	477	4	,	,	PUNCT
easat-9645	477	5	compares	compare	VERB
easat-9645	477	6	the	the	DET
easat-9645	477	7	similarity	similarity	NOUN
easat-9645	477	8	of	of	ADP
easat-9645	477	9	inner	inner	ADJ
easat-9645	477	10	products	product	NOUN
easat-9645	477	11	in	in	ADP
easat-9645	477	12	euclidean	euclidean	NOUN
easat-9645	477	13	versus	versus	ADP
easat-9645	477	14	hilbert	hilbert	NOUN
easat-9645	477	15	space	space	NOUN
easat-9645	477	16	.	.	PUNCT
easat-9645	478	1	it	it	PRON
easat-9645	478	2	shows	show	VERB
easat-9645	478	3	a	a	DET
easat-9645	478	4	bar	bar	NOUN
easat-9645	478	5	chart	chart	NOUN
easat-9645	478	6	with	with	ADP
easat-9645	478	7	two	two	NUM
easat-9645	478	8	categories	category	NOUN
easat-9645	478	9	:	:	PUNCT
easat-9645	478	10	"	"	PUNCT
easat-9645	478	11	linear	linear	VERB
easat-9645	478	12	ip	ip	NOUN
easat-9645	478	13	"	"	PUNCT
easat-9645	478	14	and	and	CCONJ
easat-9645	478	15	"	"	PUNCT
easat-9645	478	16	kernel	kernel	NOUN
easat-9645	478	17	ip	ip	NOUN
easat-9645	478	18	(	(	PUNCT
easat-9645	478	19	rkhs	rkh	NOUN
easat-9645	478	20	)	)	PUNCT
easat-9645	478	21	"	"	PUNCT
easat-9645	478	22	.	.	PUNCT
easat-9645	479	1	the	the	DET
easat-9645	479	2	"	"	PUNCT
easat-9645	479	3	linear	linear	ADJ
easat-9645	479	4	ip	ip	NOUN
easat-9645	479	5	"	"	PUNCT
easat-9645	479	6	bar	bar	NOUN
easat-9645	479	7	reaches	reach	VERB
easat-9645	479	8	approximately	approximately	ADV
easat-9645	479	9	7	7	NUM
easat-9645	479	10	on	on	ADP
easat-9645	479	11	the	the	DET
easat-9645	479	12	similarity	similarity	NOUN
easat-9645	479	13	scale	scale	NOUN
easat-9645	479	14	,	,	PUNCT
easat-9645	479	15	while	while	SCONJ
easat-9645	479	16	the	the	DET
easat-9645	479	17	"	"	PUNCT
easat-9645	479	18	kernel	kernel	NOUN
easat-9645	479	19	ip	ip	NOUN
easat-9645	479	20	(	(	PUNCT
easat-9645	479	21	rkhs	rkh	NOUN
easat-9645	479	22	)	)	PUNCT
easat-9645	479	23	"	"	PUNCT
easat-9645	479	24	bar	bar	NOUN
easat-9645	479	25	is	be	AUX
easat-9645	479	26	much	much	ADV
easat-9645	479	27	lower	low	ADJ
easat-9645	479	28	,	,	PUNCT
easat-9645	479	29	around	around	ADP
easat-9645	479	30	1	1	NUM
easat-9645	479	31	,	,	PUNCT
easat-9645	479	32	indicating	indicate	VERB
easat-9645	479	33	a	a	DET
easat-9645	479	34	significant	significant	ADJ
easat-9645	479	35	difference	difference	NOUN
easat-9645	479	36	in	in	ADP
easat-9645	479	37	similarity	similarity	NOUN
easat-9645	479	38	measures	measure	NOUN
easat-9645	479	39	between	between	ADP
easat-9645	479	40	the	the	DET
easat-9645	479	41	two	two	NUM
easat-9645	479	42	methods	method	NOUN
easat-9645	479	43	.	.	PUNCT
easat-9645	480	1	1520	1520	NUM
easat-9645	480	2	edelweiss	edelweiss	PROPN
easat-9645	480	3	applied	apply	VERB
easat-9645	480	4	science	science	NOUN
easat-9645	480	5	and	and	CCONJ
easat-9645	480	6	technology	technology	NOUN
easat-9645	480	7	issn	issn	PROPN
easat-9645	480	8	:	:	PUNCT
easat-9645	480	9	2576	2576	NUM
easat-9645	480	10	-	-	SYM
easat-9645	480	11	8484	8484	NUM
easat-9645	480	12	vol	vol	NOUN
easat-9645	480	13	.	.	PROPN
easat-9645	481	1	9	9	NUM
easat-9645	481	2	,	,	PUNCT
easat-9645	481	3	no	no	INTJ
easat-9645	481	4	.	.	NOUN
easat-9645	481	5	8	8	NUM
easat-9645	481	6	:	:	SYM
easat-9645	481	7	1498	1498	NUM
easat-9645	481	8	-	-	SYM
easat-9645	481	9	1523	1523	NUM
easat-9645	481	10	,	,	PUNCT
easat-9645	481	11	2025	2025	NUM
easat-9645	481	12	doi	doi	NOUN
easat-9645	481	13	:	:	PUNCT
easat-9645	481	14	10.55214/2576	10.55214/2576	NUM
easat-9645	481	15	-	-	SYM
easat-9645	481	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	481	17	©	©	PROPN
easat-9645	481	18	2025	2025	NUM
easat-9645	481	19	by	by	ADP
easat-9645	481	20	the	the	DET
easat-9645	481	21	authors	author	NOUN
easat-9645	481	22	;	;	PUNCT
easat-9645	481	23	licensee	licensee	PROPN
easat-9645	481	24	learning	learn	VERB
easat-9645	481	25	gate	gate	NOUN
easat-9645	481	26	table	table	NOUN
easat-9645	481	27	2	2	NUM
easat-9645	481	28	.	.	PUNCT
easat-9645	482	1	the	the	DET
easat-9645	482	2	comparison	comparison	NOUN
easat-9645	482	3	of	of	ADP
easat-9645	482	4	inner	inner	ADJ
easat-9645	482	5	product	product	NOUN
easat-9645	482	6	spaces	space	NOUN
easat-9645	482	7	vs.	vs.	ADP
easat-9645	482	8	hilbert	hilbert	NOUN
easat-9645	482	9	spaces	space	NOUN
easat-9645	482	10	in	in	ADP
easat-9645	482	11	machine	machine	NOUN
easat-9645	482	12	learning	learning	NOUN
easat-9645	482	13	.	.	PUNCT
easat-9645	483	1	feature	feature	VERB
easat-9645	483	2	inner	inner	ADJ
easat-9645	483	3	product	product	NOUN
easat-9645	483	4	space	space	NOUN
easat-9645	483	5	(	(	PUNCT
easat-9645	483	6	e.g.	e.g.	ADV
easat-9645	483	7	,	,	PUNCT
easat-9645	483	8	linear	linear	ADJ
easat-9645	483	9	svm	svm	ADJ
easat-9645	483	10	)	)	PUNCT
easat-9645	483	11	hilbert	hilbert	NOUN
easat-9645	483	12	space	space	NOUN
easat-9645	483	13	(	(	PUNCT
easat-9645	483	14	e.g.	e.g.	ADV
easat-9645	483	15	,	,	PUNCT
easat-9645	483	16	kernel	kernel	PROPN
easat-9645	483	17	svm	svm	PROPN
easat-9645	483	18	with	with	ADP
easat-9645	483	19	rbf	rbf	PROPN
easat-9645	483	20	)	)	PUNCT
easat-9645	483	21	mathematical	mathematical	ADJ
easat-9645	483	22	foundation	foundation	NOUN
easat-9645	483	23	vector	vector	NOUN
easat-9645	483	24	space	space	NOUN
easat-9645	483	25	with	with	ADP
easat-9645	483	26	finite	finite	ADJ
easat-9645	483	27	-	-	ADJ
easat-9645	483	28	dimensional	dimensional	ADJ
easat-9645	483	29	inner	inner	ADJ
easat-9645	483	30	product	product	NOUN
easat-9645	483	31	complete	complete	VERB
easat-9645	483	32	inner	inner	ADJ
easat-9645	483	33	product	product	NOUN
easat-9645	483	34	space	space	NOUN
easat-9645	483	35	(	(	PUNCT
easat-9645	483	36	possibly	possibly	ADV
easat-9645	483	37	infinite	infinite	VERB
easat-9645	483	38	-	-	PUNCT
easat-9645	483	39	dim	dim	ADJ
easat-9645	483	40	)	)	PUNCT
easat-9645	483	41	key	key	ADJ
easat-9645	483	42	use	use	NOUN
easat-9645	483	43	in	in	ADP
easat-9645	483	44	ml	ml	NOUN
easat-9645	483	45	linear	linear	PROPN
easat-9645	483	46	classification	classification	NOUN
easat-9645	483	47	,	,	PUNCT
easat-9645	483	48	regression	regression	NOUN
easat-9645	483	49	,	,	PUNCT
easat-9645	483	50	pca	pca	PROPN
easat-9645	483	51	nonlinear	nonlinear	NOUN
easat-9645	483	52	classification	classification	NOUN
easat-9645	483	53	,	,	PUNCT
easat-9645	483	54	kernel	kernel	PROPN
easat-9645	483	55	methods	method	NOUN
easat-9645	483	56	decision	decision	NOUN
easat-9645	483	57	boundary	boundary	ADJ
easat-9645	483	58	shape	shape	NOUN
easat-9645	483	59	linear	linear	PROPN
easat-9645	483	60	(	(	PUNCT
easat-9645	483	61	hyperplane	hyperplane	NOUN
easat-9645	483	62	)	)	PUNCT
easat-9645	483	63	nonlinear	nonlinear	NOUN
easat-9645	483	64	(	(	PUNCT
easat-9645	483	65	curved	curved	ADJ
easat-9645	483	66	,	,	PUNCT
easat-9645	483	67	flexible	flexible	ADJ
easat-9645	483	68	)	)	PUNCT
easat-9645	483	69	flexibility	flexibility	NOUN
easat-9645	483	70	limited	limit	VERB
easat-9645	483	71	to	to	PART
easat-9645	483	72	linear	linear	VERB
easat-9645	483	73	separability	separability	NOUN
easat-9645	483	74	can	can	AUX
easat-9645	483	75	handle	handle	VERB
easat-9645	483	76	complex	complex	ADJ
easat-9645	483	77	,	,	PUNCT
easat-9645	483	78	nonlinear	nonlinear	ADJ
easat-9645	483	79	patterns	pattern	NOUN
easat-9645	483	80	similarity	similarity	NOUN
easat-9645	483	81	measure	measure	NOUN
easat-9645	483	82	euclidean	euclidean	NOUN
easat-9645	483	83	dot	dot	NOUN
easat-9645	483	84	product	product	NOUN
easat-9645	483	85	kernel	kernel	NOUN
easat-9645	483	86	-	-	PUNCT
easat-9645	483	87	induced	induce	VERB
easat-9645	483	88	inner	inner	ADJ
easat-9645	483	89	product	product	NOUN
easat-9645	483	90	(	(	PUNCT
easat-9645	483	91	e.g.	e.g.	ADV
easat-9645	483	92	,	,	PUNCT
easat-9645	483	93	gaussian	gaussian	ADJ
easat-9645	483	94	kernel	kernel	NOUN
easat-9645	483	95	)	)	PUNCT
easat-9645	483	96	svm	svm	ADJ
easat-9645	483	97	kernel	kernel	PROPN
easat-9645	483	98	function	function	PROPN
easat-9645	483	99	in	in	ADP
easat-9645	483	100	matlab	matlab	PROPN
easat-9645	483	101	'	'	PUNCT
easat-9645	483	102	linear	linear	NOUN
easat-9645	483	103	'	'	PUNCT
easat-9645	483	104	'	'	PUNCT
easat-9645	483	105	rbf	rbf	PROPN
easat-9645	483	106	'	'	PROPN
easat-9645	483	107	,	,	PUNCT
easat-9645	483	108	'	'	PUNCT
easat-9645	483	109	polynomial	polynomial	ADJ
easat-9645	483	110	'	'	PUNCT
easat-9645	483	111	,	,	PUNCT
easat-9645	483	112	or	or	CCONJ
easat-9645	483	113	custom	custom	NOUN
easat-9645	483	114	kernels	kernel	NOUN
easat-9645	483	115	performance	performance	NOUN
easat-9645	483	116	on	on	ADP
easat-9645	483	117	nonlinear	nonlinear	ADJ
easat-9645	483	118	data	datum	NOUN
easat-9645	483	119	poor	poor	ADJ
easat-9645	483	120	excellent	excellent	ADJ
easat-9645	483	121	computational	computational	ADJ
easat-9645	483	122	cost	cost	NOUN
easat-9645	483	123	low	low	ADV
easat-9645	483	124	higher	high	ADJ
easat-9645	483	125	(	(	PUNCT
easat-9645	483	126	due	due	ADP
easat-9645	483	127	to	to	ADP
easat-9645	483	128	kernel	kernel	PROPN
easat-9645	483	129	matrix	matrix	NOUN
easat-9645	483	130	computation	computation	NOUN
easat-9645	483	131	)	)	PUNCT
easat-9645	483	132	figure	figure	NOUN
easat-9645	483	133	15	15	NUM
easat-9645	483	134	.	.	PUNCT
easat-9645	484	1	matlab	matlab	PROPN
easat-9645	484	2	simulation	simulation	PROPN
easat-9645	484	3	:	:	PUNCT
easat-9645	484	4	linear	linear	PROPN
easat-9645	484	5	vs.	vs.	X
easat-9645	484	6	kernel	kernel	PROPN
easat-9645	484	7	svm	svm	PROPN
easat-9645	484	8	(	(	PUNCT
easat-9645	484	9	using	use	VERB
easat-9645	484	10	synthetic	synthetic	ADJ
easat-9645	484	11	data	datum	NOUN
easat-9645	484	12	)	)	PUNCT
easat-9645	484	13	.	.	PUNCT
easat-9645	485	1	we	we	PRON
easat-9645	485	2	generate	generate	VERB
easat-9645	485	3	nonlinear	nonlinear	ADJ
easat-9645	485	4	data	datum	NOUN
easat-9645	485	5	,	,	PUNCT
easat-9645	485	6	then	then	ADV
easat-9645	485	7	apply	apply	VERB
easat-9645	485	8	both	both	CCONJ
easat-9645	485	9	linear	linear	ADJ
easat-9645	485	10	and	and	CCONJ
easat-9645	485	11	kernel	kernel	PROPN
easat-9645	485	12	svms	svms	NOUN
easat-9645	485	13	to	to	PART
easat-9645	485	14	compare	compare	VERB
easat-9645	485	15	their	their	PRON
easat-9645	485	16	performance	performance	NOUN
easat-9645	485	17	—	—	PUNCT
easat-9645	485	18	demonstrating	demonstrate	VERB
easat-9645	485	19	the	the	DET
easat-9645	485	20	practical	practical	ADJ
easat-9645	485	21	contrast	contrast	NOUN
easat-9645	485	22	between	between	ADP
easat-9645	485	23	an	an	DET
easat-9645	485	24	inner	inner	ADJ
easat-9645	485	25	product	product	NOUN
easat-9645	485	26	space	space	NOUN
easat-9645	485	27	and	and	CCONJ
easat-9645	485	28	a	a	DET
easat-9645	485	29	hilbert	hilbert	NOUN
easat-9645	485	30	space	space	NOUN
easat-9645	485	31	.	.	PUNCT
easat-9645	486	1	the	the	DET
easat-9645	486	2	figure	figure	NOUN
easat-9645	486	3	15	15	NUM
easat-9645	486	4	,	,	PUNCT
easat-9645	486	5	shows	show	VERB
easat-9645	486	6	a	a	DET
easat-9645	486	7	bar	bar	NOUN
easat-9645	486	8	chart	chart	NOUN
easat-9645	486	9	comparing	compare	VERB
easat-9645	486	10	the	the	DET
easat-9645	486	11	accuracy	accuracy	NOUN
easat-9645	486	12	of	of	ADP
easat-9645	486	13	svm	svm	ADJ
easat-9645	486	14	classification	classification	NOUN
easat-9645	486	15	on	on	ADP
easat-9645	486	16	the	the	DET
easat-9645	486	17	iris	iris	NOUN
easat-9645	486	18	dataset	dataset	NOUN
easat-9645	486	19	using	use	VERB
easat-9645	486	20	linear	linear	ADJ
easat-9645	486	21	svm	svm	PROPN
easat-9645	486	22	and	and	CCONJ
easat-9645	486	23	kernel	kernel	PROPN
easat-9645	486	24	svm	svm	PROPN
easat-9645	486	25	.	.	PUNCT
easat-9645	487	1	both	both	DET
easat-9645	487	2	methods	method	NOUN
easat-9645	487	3	achieve	achieve	VERB
easat-9645	487	4	an	an	DET
easat-9645	487	5	accuracy	accuracy	NOUN
easat-9645	487	6	of	of	ADP
easat-9645	487	7	approximately	approximately	ADV
easat-9645	487	8	90	90	NUM
easat-9645	487	9	-	-	SYM
easat-9645	487	10	100	100	NUM
easat-9645	487	11	%	%	NOUN
easat-9645	487	12	,	,	PUNCT
easat-9645	487	13	with	with	ADP
easat-9645	487	14	no	no	DET
easat-9645	487	15	significant	significant	ADJ
easat-9645	487	16	difference	difference	NOUN
easat-9645	487	17	between	between	ADP
easat-9645	487	18	them	they	PRON
easat-9645	487	19	.	.	PUNCT
easat-9645	488	1	9	9	X
easat-9645	488	2	.	.	X
easat-9645	488	3	conceptual	conceptual	ADJ
easat-9645	488	4	overview	overview	NOUN
easat-9645	488	5	:	:	PUNCT
easat-9645	488	6	human	human	ADJ
easat-9645	488	7	thinking	thinking	NOUN
easat-9645	488	8	in	in	ADP
easat-9645	488	9	ml	ml	NOUN
easat-9645	488	10	with	with	ADP
easat-9645	488	11	inner	inner	ADJ
easat-9645	488	12	product	product	NOUN
easat-9645	488	13	and	and	CCONJ
easat-9645	488	14	hilbert	hilbert	NOUN
easat-9645	488	15	spaces	space	NOUN
easat-9645	488	16	we	we	PRON
easat-9645	488	17	visualize	visualize	VERB
easat-9645	488	18	the	the	DET
easat-9645	488	19	inner	inner	ADJ
easat-9645	488	20	product	product	NOUN
easat-9645	488	21	space	space	NOUN
easat-9645	488	22	as	as	SCONJ
easat-9645	488	23	the	the	DET
easat-9645	488	24	human	human	ADJ
easat-9645	488	25	thinking	thinking	NOUN
easat-9645	488	26	linearly	linearly	ADV
easat-9645	488	27	—	—	PUNCT
easat-9645	488	28	making	make	VERB
easat-9645	488	29	decisions	decision	NOUN
easat-9645	488	30	based	base	VERB
easat-9645	488	31	on	on	ADP
easat-9645	488	32	straight	straight	ADJ
easat-9645	488	33	-	-	PUNCT
easat-9645	488	34	line	line	NOUN
easat-9645	488	35	similarity	similarity	NOUN
easat-9645	488	36	or	or	CCONJ
easat-9645	488	37	dot	dot	NOUN
easat-9645	488	38	product	product	NOUN
easat-9645	488	39	intuition	intuition	NOUN
easat-9645	488	40	.	.	PUNCT
easat-9645	489	1	and	and	CCONJ
easat-9645	489	2	the	the	DET
easat-9645	489	3	hilbert	hilbert	NOUN
easat-9645	489	4	space	space	NOUN
easat-9645	489	5	(	(	PUNCT
easat-9645	489	6	via	via	ADP
easat-9645	489	7	rkhs	rkh	NOUN
easat-9645	489	8	)	)	PUNCT
easat-9645	489	9	as	as	ADP
easat-9645	489	10	the	the	DET
easat-9645	489	11	human	human	ADJ
easat-9645	489	12	1521	1521	NUM
easat-9645	489	13	edelweiss	edelweiss	PROPN
easat-9645	489	14	applied	apply	VERB
easat-9645	489	15	science	science	NOUN
easat-9645	489	16	and	and	CCONJ
easat-9645	489	17	technology	technology	NOUN
easat-9645	489	18	issn	issn	PROPN
easat-9645	489	19	:	:	PUNCT
easat-9645	489	20	2576	2576	NUM
easat-9645	489	21	-	-	SYM
easat-9645	489	22	8484	8484	NUM
easat-9645	489	23	vol	vol	NOUN
easat-9645	489	24	.	.	PROPN
easat-9645	490	1	9	9	NUM
easat-9645	490	2	,	,	PUNCT
easat-9645	490	3	no	no	INTJ
easat-9645	490	4	.	.	NOUN
easat-9645	490	5	8	8	NUM
easat-9645	490	6	:	:	SYM
easat-9645	490	7	1498	1498	NUM
easat-9645	490	8	-	-	SYM
easat-9645	490	9	1523	1523	NUM
easat-9645	490	10	,	,	PUNCT
easat-9645	490	11	2025	2025	NUM
easat-9645	490	12	doi	doi	NOUN
easat-9645	490	13	:	:	PUNCT
easat-9645	490	14	10.55214/2576	10.55214/2576	NUM
easat-9645	490	15	-	-	SYM
easat-9645	490	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	490	17	©	©	PROPN
easat-9645	490	18	2025	2025	NUM
easat-9645	490	19	by	by	ADP
easat-9645	490	20	the	the	DET
easat-9645	490	21	authors	author	NOUN
easat-9645	490	22	;	;	PUNCT
easat-9645	490	23	licensee	licensee	PROPN
easat-9645	490	24	learning	learn	VERB
easat-9645	490	25	gate	gate	NOUN
easat-9645	490	26	thinking	think	VERB
easat-9645	490	27	abstractly	abstractly	ADV
easat-9645	490	28	or	or	CCONJ
easat-9645	490	29	nonlinearly	nonlinearly	ADV
easat-9645	490	30	,	,	PUNCT
easat-9645	490	31	visualizing	visualize	VERB
easat-9645	490	32	patterns	pattern	NOUN
easat-9645	490	33	beyond	beyond	ADP
easat-9645	490	34	what	what	PRON
easat-9645	490	35	’s	’	VERB
easat-9645	490	36	visible	visible	ADJ
easat-9645	490	37	—	—	PUNCT
easat-9645	490	38	like	like	ADP
easat-9645	490	39	imagination	imagination	NOUN
easat-9645	490	40	mapping	mapping	NOUN
easat-9645	490	41	reality	reality	NOUN
easat-9645	490	42	to	to	ADP
easat-9645	490	43	a	a	DET
easat-9645	490	44	higher	higher	ADV
easat-9645	490	45	-	-	PUNCT
easat-9645	490	46	dimensional	dimensional	ADJ
easat-9645	490	47	space[21	space[21	NOUN
easat-9645	490	48	,	,	PUNCT
easat-9645	490	49	22	22	NUM
easat-9645	490	50	]	]	PUNCT
easat-9645	490	51	.	.	PUNCT
easat-9645	491	1	figure	figure	NOUN
easat-9645	491	2	16	16	NUM
easat-9645	491	3	.	.	PUNCT
easat-9645	492	1	human	human	ADJ
easat-9645	492	2	thinking	thinking	NOUN
easat-9645	492	3	in	in	ADP
easat-9645	492	4	ml	ml	NOUN
easat-9645	492	5	with	with	ADP
easat-9645	492	6	inner	inner	ADJ
easat-9645	492	7	product	product	NOUN
easat-9645	492	8	and	and	CCONJ
easat-9645	492	9	hilbert	hilbert	NOUN
easat-9645	492	10	spaces	space	NOUN
easat-9645	492	11	.	.	PUNCT
easat-9645	493	1	the	the	DET
easat-9645	493	2	figure	figure	NOUN
easat-9645	493	3	16	16	NUM
easat-9645	493	4	,	,	PUNCT
easat-9645	493	5	emphasizes	emphasize	VERB
easat-9645	493	6	how	how	SCONJ
easat-9645	493	7	human	human	ADJ
easat-9645	493	8	reasoning	reasoning	NOUN
easat-9645	493	9	evolves	evolve	VERB
easat-9645	493	10	:	:	PUNCT
easat-9645	493	11	•	•	ADV
easat-9645	493	12	in	in	ADP
easat-9645	493	13	the	the	DET
easat-9645	493	14	inner	inner	ADJ
easat-9645	493	15	product	product	NOUN
easat-9645	493	16	space	space	NOUN
easat-9645	493	17	,	,	PUNCT
easat-9645	493	18	the	the	DET
easat-9645	493	19	approach	approach	NOUN
easat-9645	493	20	is	be	AUX
easat-9645	493	21	direct	direct	ADJ
easat-9645	493	22	,	,	PUNCT
easat-9645	493	23	simple	simple	ADJ
easat-9645	493	24	,	,	PUNCT
easat-9645	493	25	and	and	CCONJ
easat-9645	493	26	linear	linear	NOUN
easat-9645	493	27	.	.	PUNCT
easat-9645	494	1	•	•	NOUN
easat-9645	494	2	in	in	ADP
easat-9645	494	3	the	the	DET
easat-9645	494	4	hilbert	hilbert	NOUN
easat-9645	494	5	space	space	NOUN
easat-9645	494	6	,	,	PUNCT
easat-9645	494	7	reasoning	reasoning	NOUN
easat-9645	494	8	becomes	become	VERB
easat-9645	494	9	abstract	abstract	ADJ
easat-9645	494	10	,	,	PUNCT
easat-9645	494	11	enabling	enable	VERB
easat-9645	494	12	the	the	DET
easat-9645	494	13	recognition	recognition	NOUN
easat-9645	494	14	of	of	ADP
easat-9645	494	15	nonlinear	nonlinear	ADJ
easat-9645	494	16	patterns	pattern	NOUN
easat-9645	494	17	—	—	PUNCT
easat-9645	494	18	similar	similar	ADJ
easat-9645	494	19	to	to	ADP
easat-9645	494	20	how	how	SCONJ
easat-9645	494	21	kernel	kernel	NOUN
easat-9645	494	22	methods	method	NOUN
easat-9645	494	23	work	work	VERB
easat-9645	494	24	in	in	ADP
easat-9645	494	25	machine	machine	NOUN
easat-9645	494	26	learning	learning	NOUN
easat-9645	494	27	.	.	PUNCT
easat-9645	495	1	10	10	NUM
easat-9645	495	2	.	.	PUNCT
easat-9645	495	3	limitations	limitation	NOUN
easat-9645	495	4	and	and	CCONJ
easat-9645	495	5	future	future	ADJ
easat-9645	495	6	work	work	NOUN
easat-9645	495	7	although	although	SCONJ
easat-9645	495	8	this	this	DET
easat-9645	495	9	study	study	NOUN
easat-9645	495	10	provides	provide	VERB
easat-9645	495	11	both	both	DET
easat-9645	495	12	conceptual	conceptual	ADJ
easat-9645	495	13	insight	insight	NOUN
easat-9645	495	14	and	and	CCONJ
easat-9645	495	15	computational	computational	ADJ
easat-9645	495	16	demonstrations	demonstration	NOUN
easat-9645	495	17	of	of	ADP
easat-9645	495	18	hilbert	hilbert	NOUN
easat-9645	495	19	and	and	CCONJ
easat-9645	495	20	inner	inner	ADJ
easat-9645	495	21	product	product	NOUN
easat-9645	495	22	spaces	space	VERB
easat-9645	495	23	with	with	ADP
easat-9645	495	24	matlab	matlab	PROPN
easat-9645	495	25	based	base	VERB
easat-9645	495	26	illustrations	illustration	NOUN
easat-9645	495	27	relevant	relevant	ADJ
easat-9645	495	28	to	to	ADP
easat-9645	495	29	machine	machine	NOUN
easat-9645	495	30	learning	learn	VERB
easat-9645	495	31	several	several	ADJ
easat-9645	495	32	limitations	limitation	NOUN
easat-9645	495	33	persist	persist	VERB
easat-9645	495	34	,	,	PUNCT
easat-9645	495	35	offering	offer	VERB
easat-9645	495	36	opportunities	opportunity	NOUN
easat-9645	495	37	for	for	ADP
easat-9645	495	38	further	further	ADJ
easat-9645	495	39	research	research	NOUN
easat-9645	495	40	and	and	CCONJ
easat-9645	495	41	refinement	refinement	NOUN
easat-9645	495	42	.	.	PUNCT
easat-9645	496	1	10.1	10.1	NUM
easat-9645	496	2	.	.	PUNCT
easat-9645	497	1	theoretical	theoretical	ADJ
easat-9645	497	2	limitations	limitation	NOUN
easat-9645	497	3	the	the	DET
easat-9645	497	4	present	present	ADJ
easat-9645	497	5	analysis	analysis	NOUN
easat-9645	497	6	is	be	AUX
easat-9645	497	7	limited	limit	VERB
easat-9645	497	8	to	to	ADP
easat-9645	497	9	separable	separable	ADJ
easat-9645	497	10	hilbert	hilbert	NOUN
easat-9645	497	11	spaces	space	NOUN
easat-9645	497	12	and	and	CCONJ
easat-9645	497	13	bounded	bound	VERB
easat-9645	497	14	linear	linear	PROPN
easat-9645	497	15	operators	operator	NOUN
easat-9645	497	16	.	.	PUNCT
easat-9645	498	1	advanced	advanced	ADJ
easat-9645	498	2	topics	topic	NOUN
easat-9645	498	3	such	such	ADJ
easat-9645	498	4	as	as	ADP
easat-9645	498	5	non	non	ADJ
easat-9645	498	6	-	-	ADJ
easat-9645	498	7	separable	separable	ADJ
easat-9645	498	8	hilbert	hilbert	NOUN
easat-9645	498	9	spaces	space	NOUN
easat-9645	498	10	,	,	PUNCT
easat-9645	498	11	unbounded	unbounded	ADJ
easat-9645	498	12	operators	operator	NOUN
easat-9645	498	13	,	,	PUNCT
easat-9645	498	14	and	and	CCONJ
easat-9645	498	15	spectral	spectral	ADJ
easat-9645	498	16	theory	theory	NOUN
easat-9645	498	17	were	be	AUX
easat-9645	498	18	not	not	PART
easat-9645	498	19	explored	explore	VERB
easat-9645	498	20	.	.	PUNCT
easat-9645	499	1	these	these	DET
easat-9645	499	2	areas	area	NOUN
easat-9645	499	3	,	,	PUNCT
easat-9645	499	4	however	however	ADV
easat-9645	499	5	,	,	PUNCT
easat-9645	499	6	are	be	AUX
easat-9645	499	7	crucial	crucial	ADJ
easat-9645	499	8	for	for	ADP
easat-9645	499	9	a	a	DET
easat-9645	499	10	deeper	deep	ADJ
easat-9645	499	11	understanding	understanding	NOUN
easat-9645	499	12	in	in	ADP
easat-9645	499	13	fields	field	NOUN
easat-9645	499	14	like	like	ADP
easat-9645	499	15	quantum	quantum	NOUN
easat-9645	499	16	mechanics	mechanic	NOUN
easat-9645	499	17	,	,	PUNCT
easat-9645	499	18	mathematical	mathematical	ADJ
easat-9645	499	19	physics	physics	NOUN
easat-9645	499	20	,	,	PUNCT
easat-9645	499	21	and	and	CCONJ
easat-9645	499	22	infinite	infinite	ADJ
easat-9645	499	23	-	-	PUNCT
easat-9645	499	24	dimensional	dimensional	ADJ
easat-9645	499	25	learning	learning	NOUN
easat-9645	499	26	theory	theory	NOUN
easat-9645	499	27	,	,	PUNCT
easat-9645	499	28	and	and	CCONJ
easat-9645	499	29	warrant	warrant	VERB
easat-9645	499	30	further	further	ADJ
easat-9645	499	31	investigation	investigation	NOUN
easat-9645	499	32	in	in	ADP
easat-9645	499	33	future	future	ADJ
easat-9645	499	34	work	work	NOUN
easat-9645	499	35	.	.	PUNCT
easat-9645	500	1	1522	1522	NUM
easat-9645	500	2	edelweiss	edelweiss	PROPN
easat-9645	500	3	applied	apply	VERB
easat-9645	500	4	science	science	NOUN
easat-9645	500	5	and	and	CCONJ
easat-9645	500	6	technology	technology	NOUN
easat-9645	500	7	issn	issn	PROPN
easat-9645	500	8	:	:	PUNCT
easat-9645	500	9	2576	2576	NUM
easat-9645	500	10	-	-	SYM
easat-9645	500	11	8484	8484	NUM
easat-9645	500	12	vol	vol	NOUN
easat-9645	500	13	.	.	PROPN
easat-9645	501	1	9	9	NUM
easat-9645	501	2	,	,	PUNCT
easat-9645	501	3	no	no	INTJ
easat-9645	501	4	.	.	NOUN
easat-9645	501	5	8	8	NUM
easat-9645	501	6	:	:	SYM
easat-9645	501	7	1498	1498	NUM
easat-9645	501	8	-	-	SYM
easat-9645	501	9	1523	1523	NUM
easat-9645	501	10	,	,	PUNCT
easat-9645	501	11	2025	2025	NUM
easat-9645	501	12	doi	doi	NOUN
easat-9645	501	13	:	:	PUNCT
easat-9645	501	14	10.55214/2576	10.55214/2576	NUM
easat-9645	501	15	-	-	SYM
easat-9645	501	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	501	17	©	©	PROPN
easat-9645	501	18	2025	2025	NUM
easat-9645	501	19	by	by	ADP
easat-9645	501	20	the	the	DET
easat-9645	501	21	authors	author	NOUN
easat-9645	501	22	;	;	PUNCT
easat-9645	501	23	licensee	licensee	PROPN
easat-9645	501	24	learning	learning	NOUN
easat-9645	501	25	gate	gate	NOUN
easat-9645	501	26	10.2	10.2	NUM
easat-9645	501	27	.	.	PUNCT
easat-9645	502	1	methodological	methodological	ADJ
easat-9645	502	2	and	and	CCONJ
easat-9645	502	3	modeling	model	VERB
easat-9645	502	4	gaps	gap	NOUN
easat-9645	502	5	this	this	DET
easat-9645	502	6	study	study	NOUN
easat-9645	502	7	focuses	focus	VERB
easat-9645	502	8	on	on	ADP
easat-9645	502	9	foundational	foundational	ADJ
easat-9645	502	10	concepts	concept	NOUN
easat-9645	502	11	and	and	CCONJ
easat-9645	502	12	techniques	technique	NOUN
easat-9645	502	13	and	and	CCONJ
easat-9645	502	14	does	do	AUX
easat-9645	502	15	not	not	PART
easat-9645	502	16	incorporate	incorporate	VERB
easat-9645	502	17	more	more	ADV
easat-9645	502	18	sophisticated	sophisticated	ADJ
easat-9645	502	19	machine	machine	NOUN
easat-9645	502	20	learning	learning	NOUN
easat-9645	502	21	models	model	NOUN
easat-9645	502	22	such	such	ADJ
easat-9645	502	23	as	as	ADP
easat-9645	502	24	deep	deep	ADJ
easat-9645	502	25	neural	neural	ADJ
easat-9645	502	26	networks	network	NOUN
easat-9645	502	27	,	,	PUNCT
easat-9645	502	28	support	support	VERB
easat-9645	502	29	vector	vector	NOUN
easat-9645	502	30	machines	machine	NOUN
easat-9645	502	31	with	with	ADP
easat-9645	502	32	kernel	kernel	PROPN
easat-9645	502	33	methods	method	NOUN
easat-9645	502	34	,	,	PUNCT
easat-9645	502	35	or	or	CCONJ
easat-9645	502	36	recent	recent	ADJ
easat-9645	502	37	developments	development	NOUN
easat-9645	502	38	in	in	ADP
easat-9645	502	39	deep	deep	ADJ
easat-9645	502	40	kernel	kernel	NOUN
easat-9645	502	41	learning	learning	NOUN
easat-9645	502	42	.	.	PUNCT
easat-9645	503	1	in	in	ADP
easat-9645	503	2	particular	particular	ADJ
easat-9645	503	3	,	,	PUNCT
easat-9645	503	4	the	the	DET
easat-9645	503	5	practical	practical	ADJ
easat-9645	503	6	utilization	utilization	NOUN
easat-9645	503	7	of	of	ADP
easat-9645	503	8	reproducing	reproduce	VERB
easat-9645	503	9	kernel	kernel	PROPN
easat-9645	503	10	hilbert	hilbert	PROPN
easat-9645	503	11	space	space	NOUN
easat-9645	503	12	(	(	PUNCT
easat-9645	503	13	rkhs	rkh	NOUN
easat-9645	503	14	)	)	PUNCT
easat-9645	503	15	frameworks	framework	NOUN
easat-9645	503	16	within	within	ADP
easat-9645	503	17	contemporary	contemporary	ADJ
easat-9645	503	18	deep	deep	ADJ
easat-9645	503	19	learning	learning	NOUN
easat-9645	503	20	models	model	NOUN
easat-9645	503	21	has	have	AUX
easat-9645	503	22	not	not	PART
easat-9645	503	23	been	be	AUX
easat-9645	503	24	addressed	address	VERB
easat-9645	503	25	.	.	PUNCT
easat-9645	504	1	bridging	bridge	VERB
easat-9645	504	2	this	this	DET
easat-9645	504	3	gap	gap	NOUN
easat-9645	504	4	could	could	AUX
easat-9645	504	5	be	be	AUX
easat-9645	504	6	a	a	DET
easat-9645	504	7	valuable	valuable	ADJ
easat-9645	504	8	direction	direction	NOUN
easat-9645	504	9	for	for	ADP
easat-9645	504	10	future	future	ADJ
easat-9645	504	11	research	research	NOUN
easat-9645	504	12	.	.	PUNCT
easat-9645	505	1	11	11	NUM
easat-9645	505	2	.	.	X
easat-9645	505	3	conclusion	conclusion	NOUN
easat-9645	505	4	this	this	DET
easat-9645	505	5	study	study	NOUN
easat-9645	505	6	demonstrates	demonstrate	VERB
easat-9645	505	7	that	that	SCONJ
easat-9645	505	8	hilbert	hilbert	NOUN
easat-9645	505	9	and	and	CCONJ
easat-9645	505	10	inner	inner	ADJ
easat-9645	505	11	product	product	NOUN
easat-9645	505	12	spaces	space	NOUN
easat-9645	505	13	generalize	generalize	VERB
easat-9645	505	14	the	the	DET
easat-9645	505	15	geometric	geometric	ADJ
easat-9645	505	16	and	and	CCONJ
easat-9645	505	17	algebraic	algebraic	ADJ
easat-9645	505	18	structure	structure	NOUN
easat-9645	505	19	of	of	ADP
easat-9645	505	20	euclidean	euclidean	ADJ
easat-9645	505	21	spaces	space	NOUN
easat-9645	505	22	to	to	PART
easat-9645	505	23	infinite	infinite	VERB
easat-9645	505	24	dimensional	dimensional	ADJ
easat-9645	505	25	settings	setting	NOUN
easat-9645	505	26	,	,	PUNCT
easat-9645	505	27	providing	provide	VERB
easat-9645	505	28	a	a	DET
easat-9645	505	29	rigorous	rigorous	ADJ
easat-9645	505	30	framework	framework	NOUN
easat-9645	505	31	for	for	ADP
easat-9645	505	32	projections	projection	NOUN
easat-9645	505	33	,	,	PUNCT
easat-9645	505	34	orthogonality	orthogonality	NOUN
easat-9645	505	35	,	,	PUNCT
easat-9645	505	36	and	and	CCONJ
easat-9645	505	37	completeness	completeness	NOUN
easat-9645	505	38	.	.	PUNCT
easat-9645	506	1	by	by	ADP
easat-9645	506	2	formally	formally	ADV
easat-9645	506	3	analyzing	analyze	VERB
easat-9645	506	4	ℝ𝑛	ℝ𝑛	PROPN
easat-9645	506	5	,	,	PUNCT
easat-9645	506	6	𝐶𝑛	𝐶𝑛	PROPN
easat-9645	506	7	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
easat-9645	506	8	𝑙2	𝑙2	PROPN
easat-9645	506	9	,	,	PUNCT
easat-9645	506	10	illustrating	illustrate	VERB
easat-9645	506	11	the	the	DET
easat-9645	506	12	parallelogram	parallelogram	NOUN
easat-9645	506	13	law	law	NOUN
easat-9645	506	14	,	,	PUNCT
easat-9645	506	15	and	and	CCONJ
easat-9645	506	16	applying	apply	VERB
easat-9645	506	17	these	these	DET
easat-9645	506	18	concepts	concept	NOUN
easat-9645	506	19	in	in	ADP
easat-9645	506	20	pca	pca	PROPN
easat-9645	506	21	,	,	PUNCT
easat-9645	506	22	svms	svms	NOUN
easat-9645	506	23	,	,	PUNCT
easat-9645	506	24	quantum	quantum	NOUN
easat-9645	506	25	mechanics	mechanic	NOUN
easat-9645	506	26	,	,	PUNCT
easat-9645	506	27	and	and	CCONJ
easat-9645	506	28	signal	signal	NOUN
easat-9645	506	29	processing	processing	NOUN
easat-9645	506	30	,	,	PUNCT
easat-9645	506	31	we	we	PRON
easat-9645	506	32	highlighted	highlight	VERB
easat-9645	506	33	the	the	DET
easat-9645	506	34	practical	practical	ADJ
easat-9645	506	35	and	and	CCONJ
easat-9645	506	36	theoretical	theoretical	ADJ
easat-9645	506	37	significance	significance	NOUN
easat-9645	506	38	of	of	ADP
easat-9645	506	39	hilbert	hilbert	NOUN
easat-9645	506	40	spaces	space	NOUN
easat-9645	506	41	.	.	PUNCT
easat-9645	507	1	while	while	SCONJ
easat-9645	507	2	the	the	DET
easat-9645	507	3	focus	focus	NOUN
easat-9645	507	4	was	be	AUX
easat-9645	507	5	on	on	ADP
easat-9645	507	6	separable	separable	ADJ
easat-9645	507	7	hilbert	hilbert	NOUN
easat-9645	507	8	spaces	space	NOUN
easat-9645	507	9	and	and	CCONJ
easat-9645	507	10	illustrative	illustrative	ADJ
easat-9645	507	11	computational	computational	ADJ
easat-9645	507	12	examples	example	NOUN
easat-9645	507	13	,	,	PUNCT
easat-9645	507	14	the	the	DET
easat-9645	507	15	work	work	NOUN
easat-9645	507	16	lays	lay	VERB
easat-9645	507	17	a	a	DET
easat-9645	507	18	foundation	foundation	NOUN
easat-9645	507	19	for	for	ADP
easat-9645	507	20	exploring	explore	VERB
easat-9645	507	21	advanced	advanced	ADJ
easat-9645	507	22	topics	topic	NOUN
easat-9645	507	23	such	such	ADJ
easat-9645	507	24	as	as	ADP
easat-9645	507	25	hilbert	hilbert	PROPN
easat-9645	507	26	schmidt	schmidt	PROPN
easat-9645	507	27	operators	operators	PROPN
easat-9645	507	28	,	,	PUNCT
easat-9645	507	29	rkhs	rkhs	NOUN
easat-9645	507	30	-	-	PUNCT
easat-9645	507	31	banach	banach	NOUN
easat-9645	507	32	duality	duality	NOUN
easat-9645	507	33	,	,	PUNCT
easat-9645	507	34	and	and	CCONJ
easat-9645	507	35	applications	application	NOUN
easat-9645	507	36	in	in	ADP
easat-9645	507	37	large	large	ADJ
easat-9645	507	38	-	-	PUNCT
easat-9645	507	39	scale	scale	NOUN
easat-9645	507	40	machine	machine	NOUN
easat-9645	507	41	learning	learning	NOUN
easat-9645	507	42	or	or	CCONJ
easat-9645	507	43	quantum	quantum	NOUN
easat-9645	507	44	computing	computing	NOUN
easat-9645	507	45	.	.	PUNCT
easat-9645	508	1	overall	overall	ADV
easat-9645	508	2	,	,	PUNCT
easat-9645	508	3	hilbert	hilbert	NOUN
easat-9645	508	4	spaces	space	NOUN
easat-9645	508	5	emerge	emerge	VERB
easat-9645	508	6	as	as	ADP
easat-9645	508	7	a	a	DET
easat-9645	508	8	central	central	ADJ
easat-9645	508	9	mathematical	mathematical	ADJ
easat-9645	508	10	framework	framework	NOUN
easat-9645	508	11	that	that	SCONJ
easat-9645	508	12	bridges	bridge	NOUN
easat-9645	508	13	theory	theory	NOUN
easat-9645	508	14	with	with	ADP
easat-9645	508	15	practical	practical	ADJ
easat-9645	508	16	algorithms	algorithm	NOUN
easat-9645	508	17	,	,	PUNCT
easat-9645	508	18	enabling	enable	VERB
easat-9645	508	19	robust	robust	ADJ
easat-9645	508	20	problemsolving	problemsolving	NOUN
easat-9645	508	21	across	across	ADP
easat-9645	508	22	mathematics	mathematic	NOUN
easat-9645	508	23	,	,	PUNCT
easat-9645	508	24	data	data	NOUN
easat-9645	508	25	science	science	NOUN
easat-9645	508	26	,	,	PUNCT
easat-9645	508	27	and	and	CCONJ
easat-9645	508	28	physics	physics	NOUN
easat-9645	508	29	.	.	PUNCT
easat-9645	509	1	transparency	transparency	NOUN
easat-9645	509	2	:	:	PUNCT
easat-9645	509	3	the	the	DET
easat-9645	509	4	authors	author	NOUN
easat-9645	509	5	confirm	confirm	VERB
easat-9645	509	6	that	that	SCONJ
easat-9645	509	7	the	the	DET
easat-9645	509	8	manuscript	manuscript	NOUN
easat-9645	509	9	is	be	AUX
easat-9645	509	10	an	an	DET
easat-9645	509	11	honest	honest	ADJ
easat-9645	509	12	,	,	PUNCT
easat-9645	509	13	accurate	accurate	ADJ
easat-9645	509	14	,	,	PUNCT
easat-9645	509	15	and	and	CCONJ
easat-9645	509	16	transparent	transparent	ADJ
easat-9645	509	17	account	account	NOUN
easat-9645	509	18	of	of	ADP
easat-9645	509	19	the	the	DET
easat-9645	509	20	study	study	NOUN
easat-9645	509	21	;	;	PUNCT
easat-9645	509	22	that	that	SCONJ
easat-9645	509	23	no	no	DET
easat-9645	509	24	vital	vital	ADJ
easat-9645	509	25	features	feature	NOUN
easat-9645	509	26	of	of	ADP
easat-9645	509	27	the	the	DET
easat-9645	509	28	study	study	NOUN
easat-9645	509	29	have	have	AUX
easat-9645	509	30	been	be	AUX
easat-9645	509	31	omitted	omit	VERB
easat-9645	509	32	;	;	PUNCT
easat-9645	509	33	and	and	CCONJ
easat-9645	509	34	that	that	SCONJ
easat-9645	509	35	any	any	DET
easat-9645	509	36	discrepancies	discrepancy	NOUN
easat-9645	509	37	from	from	ADP
easat-9645	509	38	the	the	DET
easat-9645	509	39	study	study	NOUN
easat-9645	509	40	as	as	SCONJ
easat-9645	509	41	planned	plan	VERB
easat-9645	509	42	have	have	AUX
easat-9645	509	43	been	be	AUX
easat-9645	509	44	explained	explain	VERB
easat-9645	509	45	.	.	PUNCT
easat-9645	510	1	this	this	DET
easat-9645	510	2	study	study	NOUN
easat-9645	510	3	followed	follow	VERB
easat-9645	510	4	all	all	DET
easat-9645	510	5	ethical	ethical	ADJ
easat-9645	510	6	practices	practice	NOUN
easat-9645	510	7	during	during	ADP
easat-9645	510	8	writing	writing	NOUN
easat-9645	510	9	.	.	PUNCT
easat-9645	511	1	acknowledgements	acknowledgement	NOUN
easat-9645	511	2	:	:	PUNCT
easat-9645	511	3	we	we	PRON
easat-9645	511	4	would	would	AUX
easat-9645	511	5	like	like	VERB
easat-9645	511	6	to	to	PART
easat-9645	511	7	thank	thank	VERB
easat-9645	511	8	my	my	PRON
easat-9645	511	9	respectable	respectable	ADJ
easat-9645	511	10	teacher	teacher	NOUN
easat-9645	511	11	prof	prof	NOUN
easat-9645	511	12	.	.	PUNCT
easat-9645	512	1	dr	dr	PROPN
easat-9645	512	2	.	.	PROPN
easat-9645	512	3	moqbul	moqbul	PROPN
easat-9645	512	4	hossain	hossain	PROPN
easat-9645	512	5	for	for	ADP
easat-9645	512	6	encouragement	encouragement	NOUN
easat-9645	512	7	and	and	CCONJ
easat-9645	512	8	valuable	valuable	ADJ
easat-9645	512	9	suggestions	suggestion	NOUN
easat-9645	512	10	.	.	PUNCT
easat-9645	513	1	copyright	copyright	NOUN
easat-9645	513	2	:	:	PUNCT
easat-9645	513	3	©	©	PROPN
easat-9645	513	4	2025	2025	NUM
easat-9645	513	5	by	by	ADP
easat-9645	513	6	the	the	DET
easat-9645	513	7	authors	author	NOUN
easat-9645	513	8	.	.	PUNCT
easat-9645	514	1	this	this	DET
easat-9645	514	2	open	open	ADJ
easat-9645	514	3	-	-	PUNCT
easat-9645	514	4	access	access	NOUN
easat-9645	514	5	article	article	NOUN
easat-9645	514	6	is	be	AUX
easat-9645	514	7	distributed	distribute	VERB
easat-9645	514	8	under	under	ADP
easat-9645	514	9	the	the	DET
easat-9645	514	10	terms	term	NOUN
easat-9645	514	11	and	and	CCONJ
easat-9645	514	12	conditions	condition	NOUN
easat-9645	514	13	of	of	ADP
easat-9645	514	14	the	the	DET
easat-9645	514	15	creative	creative	ADJ
easat-9645	514	16	commons	common	NOUN
easat-9645	514	17	attribution	attribution	NOUN
easat-9645	514	18	(	(	PUNCT
easat-9645	514	19	cc	cc	NOUN
easat-9645	514	20	by	by	ADP
easat-9645	514	21	)	)	PUNCT
easat-9645	514	22	license	license	NOUN
easat-9645	514	23	(	(	PUNCT
easat-9645	514	24	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-9645	514	25	)	)	PUNCT
easat-9645	514	26	.	.	PUNCT
easat-9645	515	1	references	reference	NOUN
easat-9645	515	2	[	[	X
easat-9645	515	3	1	1	NUM
easat-9645	515	4	]	]	PUNCT
easat-9645	515	5	p.	p.	NOUN
easat-9645	515	6	blanchard	blanchard	PROPN
easat-9645	515	7	and	and	CCONJ
easat-9645	515	8	e.	e.	PROPN
easat-9645	515	9	brüning	brüning	PROPN
easat-9645	515	10	,	,	PUNCT
easat-9645	515	11	"	"	PUNCT
easat-9645	515	12	inner	inner	ADJ
easat-9645	515	13	product	product	NOUN
easat-9645	515	14	spaces	space	NOUN
easat-9645	515	15	and	and	CCONJ
easat-9645	515	16	hilbert	hilbert	NOUN
easat-9645	515	17	spaces	space	NOUN
easat-9645	515	18	.	.	PUNCT
easat-9645	515	19	"	"	PUNCT
easat-9645	516	1	birkhäuser	birkhäuser	PROPN
easat-9645	516	2	,	,	PUNCT
easat-9645	516	3	boston	boston	PROPN
easat-9645	516	4	,	,	PUNCT
easat-9645	516	5	ma	ma	PROPN
easat-9645	516	6	:	:	PUNCT
easat-9645	516	7	springer	springer	NOUN
easat-9645	516	8	,	,	PUNCT
easat-9645	516	9	2015	2015	NUM
easat-9645	516	10	,	,	PUNCT
easat-9645	516	11	pp	pp	ADJ
easat-9645	516	12	.	.	PUNCT
easat-9645	517	1	213	213	NUM
easat-9645	517	2	-	-	SYM
easat-9645	517	3	225	225	NUM
easat-9645	517	4	.	.	PUNCT
easat-9645	518	1	https://doi.org/10.1007/978-1-4612-0049-9_14	https://doi.org/10.1007/978-1-4612-0049-9_14	NOUN
easat-9645	518	2	springerlink+1	springerlink+1	VERB
easat-9645	518	3	[	[	X
easat-9645	518	4	2	2	NUM
easat-9645	518	5	]	]	X
easat-9645	518	6	e.	e.	PROPN
easat-9645	518	7	provenzi	provenzi	PROPN
easat-9645	518	8	,	,	PUNCT
easat-9645	518	9	from	from	ADP
easat-9645	518	10	euclidean	euclidean	ADJ
easat-9645	518	11	to	to	ADP
easat-9645	518	12	hilbert	hilbert	PROPN
easat-9645	518	13	spaces	space	NOUN
easat-9645	518	14	:	:	PUNCT
easat-9645	518	15	introduction	introduction	NOUN
easat-9645	518	16	to	to	ADP
easat-9645	518	17	functional	functional	ADJ
easat-9645	518	18	analysis	analysis	NOUN
easat-9645	518	19	and	and	CCONJ
easat-9645	518	20	its	its	PRON
easat-9645	518	21	applications	application	NOUN
easat-9645	518	22	.	.	PUNCT
easat-9645	519	1	london	london	PROPN
easat-9645	519	2	,	,	PUNCT
easat-9645	519	3	uk	uk	PROPN
easat-9645	519	4	:	:	PUNCT
easat-9645	519	5	iste	iste	PROPN
easat-9645	519	6	ltd	ltd	PROPN
easat-9645	519	7	/	/	SYM
easat-9645	519	8	john	john	PROPN
easat-9645	519	9	wiley	wiley	PROPN
easat-9645	519	10	&	&	CCONJ
easat-9645	519	11	sons	son	NOUN
easat-9645	519	12	,	,	PUNCT
easat-9645	519	13	2021	2021	NUM
easat-9645	519	14	.	.	PUNCT
easat-9645	520	1	https://doi.org/10.1002/9781119851318	https://doi.org/10.1002/9781119851318	X
easat-9645	520	2	[	[	X
easat-9645	520	3	3	3	X
easat-9645	520	4	]	]	PUNCT
easat-9645	520	5	m.	m.	NOUN
easat-9645	520	6	foroutan	foroutan	PROPN
easat-9645	520	7	and	and	CCONJ
easat-9645	520	8	r.	r.	PROPN
easat-9645	520	9	asadi	asadi	PROPN
easat-9645	520	10	,	,	PUNCT
easat-9645	520	11	"	"	PUNCT
easat-9645	520	12	reproducing	reproduce	VERB
easat-9645	520	13	kernel	kernel	PROPN
easat-9645	520	14	hilbert	hilbert	PROPN
easat-9645	520	15	spaces	space	NOUN
easat-9645	520	16	via	via	ADP
easat-9645	520	17	sampling	sample	VERB
easat-9645	520	18	in	in	ADP
easat-9645	520	19	discrete	discrete	ADJ
easat-9645	520	20	spaces	space	NOUN
easat-9645	520	21	,	,	PUNCT
easat-9645	520	22	"	"	PUNCT
easat-9645	520	23	journal	journal	NOUN
easat-9645	520	24	of	of	ADP
easat-9645	520	25	analysis	analysis	NOUN
easat-9645	520	26	,	,	PUNCT
easat-9645	520	27	vol	vol	NOUN
easat-9645	520	28	.	.	PROPN
easat-9645	520	29	31	31	NUM
easat-9645	520	30	,	,	PUNCT
easat-9645	520	31	no	no	INTJ
easat-9645	520	32	.	.	NOUN
easat-9645	520	33	3	3	NUM
easat-9645	520	34	,	,	PUNCT
easat-9645	520	35	pp	pp	ADJ
easat-9645	520	36	.	.	PUNCT
easat-9645	520	37	1805	1805	NUM
easat-9645	520	38	-	-	SYM
easat-9645	520	39	1818	1818	NUM
easat-9645	520	40	,	,	PUNCT
easat-9645	520	41	2023	2023	NUM
easat-9645	520	42	.	.	PUNCT
easat-9645	520	43	https://doi.org/10.1007/s41478-022-00535-6	https://doi.org/10.1007/s41478-022-00535-6	NUM
easat-9645	521	1	[	[	X
easat-9645	521	2	4	4	X
easat-9645	521	3	]	]	PUNCT
easat-9645	521	4	j.	j.	PROPN
easat-9645	521	5	e.	e.	PROPN
easat-9645	521	6	gough	gough	PROPN
easat-9645	521	7	,	,	PUNCT
easat-9645	521	8	n.	n.	PROPN
easat-9645	521	9	h.	h.	PROPN
easat-9645	521	10	amini	amini	PROPN
easat-9645	521	11	,	,	PUNCT
easat-9645	521	12	and	and	CCONJ
easat-9645	521	13	h.	h.	PROPN
easat-9645	521	14	ding	ding	PROPN
easat-9645	521	15	,	,	PUNCT
easat-9645	521	16	"	"	PUNCT
easat-9645	521	17	reproducing	reproduce	VERB
easat-9645	521	18	kernel	kernel	PROPN
easat-9645	521	19	hilbert	hilbert	PROPN
easat-9645	521	20	space	space	NOUN
easat-9645	521	21	approach	approach	NOUN
easat-9645	521	22	to	to	ADP
easat-9645	521	23	non	non	ADJ
easat-9645	521	24	-	-	ADJ
easat-9645	521	25	markovian	markovian	ADJ
easat-9645	521	26	quantum	quantum	ADJ
easat-9645	521	27	stochastic	stochastic	NOUN
easat-9645	521	28	models	model	NOUN
easat-9645	521	29	,	,	PUNCT
easat-9645	521	30	"	"	PUNCT
easat-9645	521	31	journal	journal	NOUN
easat-9645	521	32	of	of	ADP
easat-9645	521	33	mathematical	mathematical	ADJ
easat-9645	521	34	physics	physics	NOUN
easat-9645	521	35	,	,	PUNCT
easat-9645	521	36	vol	vol	NOUN
easat-9645	521	37	.	.	PROPN
easat-9645	521	38	66	66	NUM
easat-9645	521	39	,	,	PUNCT
easat-9645	521	40	no	no	INTJ
easat-9645	521	41	.	.	NOUN
easat-9645	521	42	4	4	NUM
easat-9645	521	43	,	,	PUNCT
easat-9645	521	44	p.	p.	NOUN
easat-9645	521	45	042102	042102	NUM
easat-9645	521	46	,	,	PUNCT
easat-9645	521	47	2025	2025	NUM
easat-9645	521	48	.	.	PUNCT
easat-9645	522	1	[	[	X
easat-9645	522	2	5	5	X
easat-9645	522	3	]	]	PUNCT
easat-9645	522	4	e.	e.	PROPN
easat-9645	522	5	schmidt	schmidt	PROPN
easat-9645	522	6	,	,	PUNCT
easat-9645	522	7	"	"	PUNCT
easat-9645	522	8	zur	zur	NOUN
easat-9645	522	9	theorie	theorie	PROPN
easat-9645	522	10	der	der	PROPN
easat-9645	522	11	linearen	linearen	PROPN
easat-9645	522	12	und	und	VERB
easat-9645	522	13	nichtlinearen	nichtlinearen	PROPN
easat-9645	522	14	integralgleichungen	integralgleichungen	NOUN
easat-9645	522	15	,	,	PUNCT
easat-9645	522	16	"	"	PUNCT
easat-9645	522	17	mathematische	mathematische	NOUN
easat-9645	522	18	annalen	annalen	PROPN
easat-9645	522	19	,	,	PUNCT
easat-9645	522	20	vol	vol	NOUN
easat-9645	522	21	.	.	PROPN
easat-9645	523	1	63	63	NUM
easat-9645	523	2	,	,	PUNCT
easat-9645	523	3	no	no	INTJ
easat-9645	523	4	.	.	NOUN
easat-9645	523	5	4	4	NUM
easat-9645	523	6	,	,	PUNCT
easat-9645	523	7	pp	pp	ADJ
easat-9645	523	8	.	.	PUNCT
easat-9645	524	1	433	433	NUM
easat-9645	524	2	-	-	SYM
easat-9645	524	3	476	476	NUM
easat-9645	524	4	,	,	PUNCT
easat-9645	524	5	1907	1907	NUM
easat-9645	524	6	.	.	PUNCT
easat-9645	525	1	https://doi.org/10.1007/bf01449770	https://doi.org/10.1007/bf01449770	PUNCT
easat-9645	526	1	[	[	X
easat-9645	526	2	6	6	NUM
easat-9645	526	3	]	]	X
easat-9645	526	4	n.	n.	NOUN
easat-9645	526	5	minculete	minculete	NOUN
easat-9645	526	6	,	,	PUNCT
easat-9645	526	7	"	"	PUNCT
easat-9645	526	8	about	about	ADP
easat-9645	526	9	the	the	DET
easat-9645	526	10	cauchy	cauchy	NOUN
easat-9645	526	11	–	–	PUNCT
easat-9645	526	12	bunyakovsky	bunyakovsky	NOUN
easat-9645	526	13	–	–	PUNCT
easat-9645	526	14	schwarz	schwarz	PROPN
easat-9645	526	15	inequality	inequality	NOUN
easat-9645	526	16	for	for	ADP
easat-9645	526	17	hilbert	hilbert	NOUN
easat-9645	526	18	space	space	NOUN
easat-9645	526	19	operators	operator	NOUN
easat-9645	526	20	,	,	PUNCT
easat-9645	526	21	"	"	PUNCT
easat-9645	526	22	symmetry	symmetry	NOUN
easat-9645	526	23	,	,	PUNCT
easat-9645	526	24	vol	vol	NOUN
easat-9645	526	25	.	.	PROPN
easat-9645	526	26	13	13	NUM
easat-9645	526	27	,	,	PUNCT
easat-9645	526	28	no	no	INTJ
easat-9645	526	29	.	.	NOUN
easat-9645	526	30	2	2	NUM
easat-9645	526	31	,	,	PUNCT
easat-9645	526	32	p.	p.	NOUN
easat-9645	526	33	305	305	NUM
easat-9645	526	34	,	,	PUNCT
easat-9645	526	35	2021	2021	NUM
easat-9645	526	36	.	.	PUNCT
easat-9645	527	1	https://doi.org/10.3390/sym13020305	https://doi.org/10.3390/sym13020305	NOUN
easat-9645	528	1	[	[	X
easat-9645	528	2	7	7	NUM
easat-9645	528	3	]	]	X
easat-9645	528	4	d.	d.	PROPN
easat-9645	528	5	sain	sain	PROPN
easat-9645	528	6	,	,	PUNCT
easat-9645	528	7	"	"	PUNCT
easat-9645	528	8	on	on	ADP
easat-9645	528	9	a	a	DET
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easat-9645	528	11	of	of	ADP
easat-9645	528	12	a	a	DET
easat-9645	528	13	complementary	complementary	ADJ
easat-9645	528	14	triangle	triangle	NOUN
easat-9645	528	15	inequality	inequality	NOUN
easat-9645	528	16	in	in	ADP
easat-9645	528	17	hilbert	hilbert	NOUN
easat-9645	528	18	spaces	space	NOUN
easat-9645	528	19	and	and	CCONJ
easat-9645	528	20	banach	banach	NOUN
easat-9645	528	21	spaces	space	NOUN
easat-9645	528	22	,	,	PUNCT
easat-9645	528	23	"	"	PUNCT
easat-9645	528	24	indian	indian	ADJ
easat-9645	528	25	journal	journal	NOUN
easat-9645	528	26	of	of	ADP
easat-9645	528	27	pure	pure	ADJ
easat-9645	528	28	and	and	CCONJ
easat-9645	528	29	applied	applied	ADJ
easat-9645	528	30	mathematics	mathematic	NOUN
easat-9645	528	31	,	,	PUNCT
easat-9645	528	32	vol	vol	NOUN
easat-9645	528	33	.	.	PROPN
easat-9645	528	34	51	51	NUM
easat-9645	528	35	,	,	PUNCT
easat-9645	528	36	no	no	INTJ
easat-9645	528	37	.	.	NOUN
easat-9645	528	38	4	4	NUM
easat-9645	528	39	,	,	PUNCT
easat-9645	528	40	pp	pp	ADJ
easat-9645	528	41	.	.	PUNCT
easat-9645	528	42	1815	1815	NUM
easat-9645	528	43	-	-	SYM
easat-9645	528	44	1827	1827	NUM
easat-9645	528	45	,	,	PUNCT
easat-9645	528	46	2020	2020	NUM
easat-9645	528	47	.	.	PUNCT
easat-9645	529	1	https://doi.org/10.1007/s13226-0200498-1	https://doi.org/10.1007/s13226-0200498-1	PROPN
easat-9645	529	2	[	[	X
easat-9645	529	3	8	8	NUM
easat-9645	529	4	]	]	X
easat-9645	529	5	c.	c.	PROPN
easat-9645	529	6	bender	bender	PROPN
easat-9645	529	7	,	,	PUNCT
easat-9645	529	8	"	"	PUNCT
easat-9645	529	9	polarization	polarization	NOUN
easat-9645	529	10	identities	identity	NOUN
easat-9645	529	11	,	,	PUNCT
easat-9645	529	12	"	"	PUNCT
easat-9645	529	13	mathematics	mathematic	NOUN
easat-9645	529	14	,	,	PUNCT
easat-9645	529	15	vol	vol	NOUN
easat-9645	529	16	.	.	PROPN
easat-9645	529	17	11	11	NUM
easat-9645	529	18	,	,	PUNCT
easat-9645	529	19	no	no	INTJ
easat-9645	529	20	.	.	NOUN
easat-9645	529	21	3	3	NUM
easat-9645	529	22	,	,	PUNCT
easat-9645	529	23	p.	p.	NOUN
easat-9645	529	24	635	635	NUM
easat-9645	529	25	,	,	PUNCT
easat-9645	529	26	2023	2023	NUM
easat-9645	529	27	.	.	PUNCT
easat-9645	530	1	[	[	X
easat-9645	530	2	9	9	NUM
easat-9645	530	3	]	]	PUNCT
easat-9645	530	4	a.	a.	NOUN
easat-9645	530	5	j.	j.	PROPN
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easat-9645	530	7	,	,	PUNCT
easat-9645	530	8	quantum	quantum	NOUN
easat-9645	530	9	mechanics	mechanic	NOUN
easat-9645	530	10	:	:	PUNCT
easat-9645	530	11	a	a	DET
easat-9645	530	12	mathematical	mathematical	ADJ
easat-9645	530	13	introduction	introduction	NOUN
easat-9645	530	14	.	.	PUNCT
easat-9645	531	1	cambridge	cambridge	PROPN
easat-9645	531	2	,	,	PUNCT
easat-9645	531	3	uk	uk	PROPN
easat-9645	531	4	:	:	PUNCT
easat-9645	531	5	cambridge	cambridge	PROPN
easat-9645	531	6	university	university	PROPN
easat-9645	531	7	press	press	NOUN
easat-9645	531	8	,	,	PUNCT
easat-9645	531	9	2022	2022	NUM
easat-9645	531	10	.	.	PUNCT
easat-9645	532	1	https://doi.org/10.1017/9781009118026	https://doi.org/10.1017/9781009118026	X
easat-9645	533	1	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
easat-9645	533	2	https://doi.org/10.1007/978-1-4612-0049-9_14	https://doi.org/10.1007/978-1-4612-0049-9_14	VERB
easat-9645	533	3	https://doi.org/10.1002/9781119851318	https://doi.org/10.1002/9781119851318	ADP
easat-9645	533	4	https://doi.org/10.1007/s41478-022-00535-6	https://doi.org/10.1007/s41478-022-00535-6	NUM
easat-9645	533	5	https://doi.org/10.1007/bf01449770	https://doi.org/10.1007/bf01449770	NOUN
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easat-9645	533	7	https://doi.org/10.1007/s13226-020-0498-1	https://doi.org/10.1007/s13226-020-0498-1	PROPN
easat-9645	533	8	https://doi.org/10.1007/s13226-020-0498-1	https://doi.org/10.1007/s13226-020-0498-1	NUM
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easat-9645	533	10	1523	1523	NUM
easat-9645	533	11	edelweiss	edelweiss	PROPN
easat-9645	533	12	applied	apply	VERB
easat-9645	533	13	science	science	NOUN
easat-9645	533	14	and	and	CCONJ
easat-9645	533	15	technology	technology	NOUN
easat-9645	533	16	issn	issn	PROPN
easat-9645	533	17	:	:	PUNCT
easat-9645	533	18	2576	2576	NUM
easat-9645	533	19	-	-	SYM
easat-9645	533	20	8484	8484	NUM
easat-9645	533	21	vol	vol	NOUN
easat-9645	533	22	.	.	PROPN
easat-9645	534	1	9	9	NUM
easat-9645	534	2	,	,	PUNCT
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easat-9645	534	4	.	.	NOUN
easat-9645	534	5	8	8	NUM
easat-9645	534	6	:	:	SYM
easat-9645	534	7	1498	1498	NUM
easat-9645	534	8	-	-	SYM
easat-9645	534	9	1523	1523	NUM
easat-9645	534	10	,	,	PUNCT
easat-9645	534	11	2025	2025	NUM
easat-9645	534	12	doi	doi	NOUN
easat-9645	534	13	:	:	PUNCT
easat-9645	534	14	10.55214/2576	10.55214/2576	NUM
easat-9645	534	15	-	-	SYM
easat-9645	534	16	8484.v9i8.9645	8484.v9i8.9645	NUM
easat-9645	534	17	©	©	PROPN
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easat-9645	534	20	the	the	DET
easat-9645	534	21	authors	author	NOUN
easat-9645	534	22	;	;	PUNCT
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easat-9645	534	24	learning	learning	NOUN
easat-9645	534	25	gate	gate	NOUN
easat-9645	535	1	[	[	X
easat-9645	535	2	10	10	NUM
easat-9645	535	3	]	]	X
easat-9645	535	4	p.	p.	NOUN
easat-9645	535	5	vaidyanathan	vaidyanathan	PROPN
easat-9645	535	6	,	,	PUNCT
easat-9645	535	7	hilbert	hilbert	NOUN
easat-9645	535	8	spaces	space	NOUN
easat-9645	535	9	(	(	PUNCT
easat-9645	535	10	chap	chap	NOUN
easat-9645	535	11	.	.	PUNCT
easat-9645	535	12	 	 	SPACE
easat-9645	536	1	3	3	NUM
easat-9645	536	2	,	,	PUNCT
easat-9645	536	3	pp	pp	ADV
easat-9645	536	4	.	.	PUNCT
easat-9645	536	5	 	 	SPACE
easat-9645	537	1	71–110	71–110	PROPN
easat-9645	537	2	)	)	PUNCT
easat-9645	537	3	.	.	PUNCT
easat-9645	538	1	in	in	ADP
easat-9645	538	2	functional	functional	ADJ
easat-9645	538	3	analysis	analysis	NOUN
easat-9645	538	4	.	.	PUNCT
easat-9645	539	1	cambridge	cambridge	PROPN
easat-9645	539	2	,	,	PUNCT
easat-9645	539	3	uk	uk	PROPN
easat-9645	539	4	:	:	PUNCT
easat-9645	539	5	cambridge	cambridge	PROPN
easat-9645	539	6	university	university	PROPN
easat-9645	539	7	press	press	NOUN
easat-9645	539	8	,	,	PUNCT
easat-9645	539	9	2025	2025	NUM
easat-9645	539	10	.	.	PUNCT
easat-9645	540	1	https://doi.org/10.1017/9781009243926.004	https://doi.org/10.1017/9781009243926.004	PUNCT
easat-9645	540	2	[	[	X
easat-9645	540	3	11	11	NUM
easat-9645	540	4	]	]	PUNCT
easat-9645	540	5	t.	t.	PROPN
easat-9645	540	6	hofmann	hofmann	PROPN
easat-9645	540	7	,	,	PUNCT
easat-9645	540	8	b.	b.	PROPN
easat-9645	540	9	schölkopf	schölkopf	NOUN
easat-9645	540	10	,	,	PUNCT
easat-9645	540	11	and	and	CCONJ
easat-9645	540	12	a.	a.	PROPN
easat-9645	540	13	j.	j.	PROPN
easat-9645	540	14	smola	smola	PROPN
easat-9645	540	15	,	,	PUNCT
easat-9645	540	16	"	"	PUNCT
easat-9645	540	17	kernel	kernel	NOUN
easat-9645	540	18	methods	method	NOUN
easat-9645	540	19	in	in	ADP
easat-9645	540	20	machine	machine	NOUN
easat-9645	540	21	learning	learning	NOUN
easat-9645	540	22	,	,	PUNCT
easat-9645	540	23	"	"	PUNCT
easat-9645	540	24	annals	annals	NOUN
easat-9645	540	25	of	of	ADP
easat-9645	540	26	statistics	statistic	NOUN
easat-9645	540	27	,	,	PUNCT
easat-9645	540	28	vol	vol	NOUN
easat-9645	540	29	.	.	PROPN
easat-9645	540	30	36	36	NUM
easat-9645	540	31	,	,	PUNCT
easat-9645	540	32	no	no	INTJ
easat-9645	540	33	.	.	NOUN
easat-9645	540	34	3	3	NUM
easat-9645	540	35	,	,	PUNCT
easat-9645	540	36	pp	pp	ADJ
easat-9645	540	37	.	.	NOUN
easat-9645	540	38	1171–1220	1171–1220	NUM
easat-9645	540	39	,	,	PUNCT
easat-9645	540	40	2008	2008	NUM
easat-9645	540	41	.	.	PUNCT
easat-9645	541	1	https://doi.org/10.1214/009053607000000677	https://doi.org/10.1214/009053607000000677	X
easat-9645	542	1	[	[	X
easat-9645	542	2	12	12	NUM
easat-9645	542	3	]	]	PUNCT
easat-9645	542	4	a.	a.	NOUN
easat-9645	542	5	a.	a.	NOUN
easat-9645	542	6	amini	amini	PROPN
easat-9645	542	7	,	,	PUNCT
easat-9645	542	8	"	"	PUNCT
easat-9645	542	9	sampled	sample	VERB
easat-9645	542	10	forms	form	NOUN
easat-9645	542	11	of	of	ADP
easat-9645	542	12	functional	functional	ADJ
easat-9645	542	13	pca	pca	NOUN
easat-9645	542	14	in	in	ADP
easat-9645	542	15	reproducing	reproduce	VERB
easat-9645	542	16	kernel	kernel	PROPN
easat-9645	542	17	hilbert	hilbert	PROPN
easat-9645	542	18	spaces	space	NOUN
easat-9645	542	19	,	,	PUNCT
easat-9645	542	20	"	"	PUNCT
easat-9645	542	21	annals	annals	NOUN
easat-9645	542	22	of	of	ADP
easat-9645	542	23	statistics	statistic	NOUN
easat-9645	542	24	,	,	PUNCT
easat-9645	542	25	vol	vol	NOUN
easat-9645	542	26	.	.	PROPN
easat-9645	542	27	40	40	NUM
easat-9645	542	28	,	,	PUNCT
easat-9645	542	29	no	no	INTJ
easat-9645	542	30	.	.	NOUN
easat-9645	542	31	5	5	NUM
easat-9645	542	32	,	,	PUNCT
easat-9645	542	33	pp	pp	ADJ
easat-9645	542	34	.	.	PUNCT
easat-9645	542	35	2352–2388	2352–2388	NUM
easat-9645	542	36	,	,	PUNCT
easat-9645	542	37	2012	2012	NUM
easat-9645	542	38	.	.	PUNCT
easat-9645	543	1	https://doi.org/10.1214/12-aos1033	https://doi.org/10.1214/12-aos1033	VERB
easat-9645	543	2	[	[	SYM
easat-9645	543	3	13	13	NUM
easat-9645	543	4	]	]	PUNCT
easat-9645	543	5	j.	j.	PROPN
easat-9645	543	6	song	song	PROPN
easat-9645	543	7	and	and	CCONJ
easat-9645	543	8	b.	b.	PROPN
easat-9645	543	9	li	li	PROPN
easat-9645	543	10	,	,	PUNCT
easat-9645	543	11	"	"	PUNCT
easat-9645	543	12	nonlinear	nonlinear	ADJ
easat-9645	543	13	and	and	CCONJ
easat-9645	543	14	additive	additive	ADJ
easat-9645	543	15	principal	principal	ADJ
easat-9645	543	16	component	component	NOUN
easat-9645	543	17	analysis	analysis	NOUN
easat-9645	543	18	for	for	ADP
easat-9645	543	19	functional	functional	ADJ
easat-9645	543	20	data	datum	NOUN
easat-9645	543	21	,	,	PUNCT
easat-9645	543	22	"	"	PUNCT
easat-9645	543	23	journal	journal	NOUN
easat-9645	543	24	of	of	ADP
easat-9645	543	25	multivariate	multivariate	NOUN
easat-9645	543	26	analysis	analysis	NOUN
easat-9645	543	27	,	,	PUNCT
easat-9645	543	28	vol	vol	NOUN
easat-9645	543	29	.	.	PROPN
easat-9645	543	30	181	181	NUM
easat-9645	543	31	,	,	PUNCT
easat-9645	543	32	p.	p.	NOUN
easat-9645	543	33	104675	104675	NUM
easat-9645	543	34	,	,	PUNCT
easat-9645	543	35	2021	2021	NUM
easat-9645	543	36	.	.	PUNCT
easat-9645	544	1	[	[	X
easat-9645	544	2	14	14	NUM
easat-9645	544	3	]	]	X
easat-9645	544	4	a.	a.	PROPN
easat-9645	544	5	farooq	farooq	PROPN
easat-9645	544	6	,	,	PUNCT
easat-9645	544	7	c.	c.	PROPN
easat-9645	544	8	a.	a.	PROPN
easat-9645	544	9	galvis	galvis	PROPN
easat-9645	544	10	-	-	PUNCT
easat-9645	544	11	florez	florez	PROPN
easat-9645	544	12	,	,	PUNCT
easat-9645	544	13	and	and	CCONJ
easat-9645	544	14	s.	s.	PROPN
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easat-9645	544	16	,	,	PUNCT
easat-9645	544	17	"	"	PUNCT
easat-9645	544	18	quantum	quantum	NOUN
easat-9645	544	19	-	-	PUNCT
easat-9645	544	20	assisted	assist	VERB
easat-9645	544	21	hilbert	hilbert	NOUN
easat-9645	544	22	-	-	PUNCT
easat-9645	544	23	space	space	NOUN
easat-9645	544	24	gaussian	gaussian	ADJ
easat-9645	544	25	process	process	NOUN
easat-9645	544	26	regression	regression	NOUN
easat-9645	544	27	,	,	PUNCT
easat-9645	544	28	"	"	PUNCT
easat-9645	544	29	physical	physical	ADJ
easat-9645	544	30	review	review	NOUN
easat-9645	544	31	a	a	DET
easat-9645	544	32	,	,	PUNCT
easat-9645	544	33	vol	vol	NOUN
easat-9645	544	34	.	.	PROPN
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easat-9645	544	36	,	,	PUNCT
easat-9645	544	37	no	no	INTJ
easat-9645	544	38	.	.	NOUN
easat-9645	544	39	5	5	NUM
easat-9645	544	40	,	,	PUNCT
easat-9645	544	41	p.	p.	NOUN
easat-9645	544	42	052410	052410	NUM
easat-9645	544	43	,	,	PUNCT
easat-9645	544	44	2024	2024	NUM
easat-9645	544	45	.	.	PUNCT
easat-9645	545	1	https://doi.org/10.1103/physreva.109.052410	https://doi.org/10.1103/physreva.109.052410	VERB
easat-9645	545	2	[	[	X
easat-9645	545	3	15	15	NUM
easat-9645	545	4	]	]	X
easat-9645	545	5	r.	r.	PROPN
easat-9645	545	6	tuo	tuo	PROPN
easat-9645	545	7	,	,	PUNCT
easat-9645	545	8	s.	s.	PROPN
easat-9645	545	9	he	he	PRON
easat-9645	545	10	,	,	PUNCT
easat-9645	545	11	a.	a.	NOUN
easat-9645	545	12	pourhabib	pourhabib	NOUN
easat-9645	545	13	,	,	PUNCT
easat-9645	545	14	y.	y.	NOUN
easat-9645	545	15	ding	ding	PROPN
easat-9645	545	16	,	,	PUNCT
easat-9645	545	17	and	and	CCONJ
easat-9645	545	18	j.	j.	PROPN
easat-9645	545	19	z.	z.	PROPN
easat-9645	545	20	huang	huang	PROPN
easat-9645	545	21	,	,	PUNCT
easat-9645	545	22	"	"	PUNCT
easat-9645	545	23	a	a	DET
easat-9645	545	24	reproducing	reproduce	VERB
easat-9645	545	25	kernel	kernel	PROPN
easat-9645	545	26	hilbert	hilbert	PROPN
easat-9645	545	27	space	space	NOUN
easat-9645	545	28	approach	approach	NOUN
easat-9645	545	29	to	to	ADP
easat-9645	545	30	functional	functional	ADJ
easat-9645	545	31	calibration	calibration	NOUN
easat-9645	545	32	of	of	ADP
easat-9645	545	33	computer	computer	NOUN
easat-9645	545	34	models	model	NOUN
easat-9645	545	35	,	,	PUNCT
easat-9645	545	36	"	"	PUNCT
easat-9645	545	37	journal	journal	NOUN
easat-9645	545	38	of	of	ADP
easat-9645	545	39	the	the	DET
easat-9645	545	40	american	american	PROPN
easat-9645	545	41	statistical	statistical	PROPN
easat-9645	545	42	association	association	PROPN
easat-9645	545	43	,	,	PUNCT
easat-9645	545	44	vol	vol	NOUN
easat-9645	545	45	.	.	PROPN
easat-9645	545	46	118	118	NUM
easat-9645	545	47	,	,	PUNCT
easat-9645	545	48	no	no	INTJ
easat-9645	545	49	.	.	NOUN
easat-9645	545	50	542	542	NUM
easat-9645	545	51	,	,	PUNCT
easat-9645	545	52	pp	pp	ADJ
easat-9645	545	53	.	.	PUNCT
easat-9645	546	1	883	883	NUM
easat-9645	546	2	-	-	SYM
easat-9645	546	3	897	897	NUM
easat-9645	546	4	,	,	PUNCT
easat-9645	546	5	2023	2023	NUM
easat-9645	546	6	.	.	PUNCT
easat-9645	547	1	https://doi.org/10.1080/01621459.2021.1956938	https://doi.org/10.1080/01621459.2021.1956938	PROPN
easat-9645	548	1	[	[	X
easat-9645	548	2	16	16	NUM
easat-9645	548	3	]	]	PUNCT
easat-9645	548	4	s.	s.	PROPN
easat-9645	548	5	b.	b.	PROPN
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easat-9645	548	7	,	,	PUNCT
easat-9645	548	8	"	"	PUNCT
easat-9645	548	9	reproducing	reproduce	VERB
easat-9645	548	10	kernel	kernel	PROPN
easat-9645	548	11	hilbert	hilbert	PROPN
easat-9645	548	12	space	space	NOUN
easat-9645	548	13	method	method	NOUN
easat-9645	548	14	for	for	ADP
easat-9645	548	15	nonlinear	nonlinear	ADJ
easat-9645	548	16	boundary	boundary	ADJ
easat-9645	548	17	value	value	NOUN
easat-9645	548	18	problems	problem	NOUN
easat-9645	548	19	,	,	PUNCT
easat-9645	548	20	"	"	PUNCT
easat-9645	548	21	applied	apply	VERB
easat-9645	548	22	mathematical	mathematical	ADJ
easat-9645	548	23	modelling	modelling	NOUN
easat-9645	548	24	,	,	PUNCT
easat-9645	548	25	vol	vol	NOUN
easat-9645	548	26	.	.	PROPN
easat-9645	549	1	45	45	NUM
easat-9645	549	2	,	,	PUNCT
easat-9645	549	3	pp	pp	ADJ
easat-9645	550	1	.	.	PUNCT
easat-9645	550	2	1–18	1–18	NUM
easat-9645	550	3	,	,	PUNCT
easat-9645	550	4	2021	2021	NUM
easat-9645	550	5	.	.	PUNCT
easat-9645	551	1	[	[	X
easat-9645	551	2	17	17	NUM
easat-9645	551	3	]	]	X
easat-9645	551	4	b.	b.	PROPN
easat-9645	551	5	ghojogh	ghojogh	PROPN
easat-9645	551	6	,	,	PUNCT
easat-9645	551	7	a.	a.	NOUN
easat-9645	551	8	ghodsi	ghodsi	PROPN
easat-9645	551	9	,	,	PUNCT
easat-9645	551	10	f.	f.	PROPN
easat-9645	551	11	karray	karray	PROPN
easat-9645	551	12	,	,	PUNCT
easat-9645	551	13	and	and	CCONJ
easat-9645	551	14	m.	m.	NOUN
easat-9645	551	15	crowley	crowley	PROPN
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easat-9645	551	17	"	"	PUNCT
easat-9645	551	18	reproducing	reproduce	VERB
easat-9645	551	19	kernel	kernel	PROPN
easat-9645	551	20	hilbert	hilbert	PROPN
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easat-9645	551	26	,	,	PUNCT
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easat-9645	551	28	,	,	PUNCT
easat-9645	551	29	nystr\	nystr\	PROPN
easat-9645	551	30	"	"	PUNCT
easat-9645	551	31	om	om	PROPN
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easat-9645	551	33	,	,	PUNCT
easat-9645	551	34	and	and	CCONJ
easat-9645	551	35	use	use	NOUN
easat-9645	551	36	of	of	ADP
easat-9645	551	37	kernels	kernel	NOUN
easat-9645	551	38	in	in	ADP
easat-9645	551	39	machine	machine	NOUN
easat-9645	551	40	learning	learning	NOUN
easat-9645	551	41	:	:	PUNCT
easat-9645	551	42	tutorial	tutorial	NOUN
easat-9645	551	43	and	and	CCONJ
easat-9645	551	44	survey	survey	NOUN
easat-9645	551	45	,	,	PUNCT
easat-9645	551	46	"	"	PUNCT
easat-9645	551	47	arxiv	arxiv	PROPN
easat-9645	551	48	preprint	preprint	NOUN
easat-9645	551	49	arxiv:2106.08443	arxiv:2106.08443	NOUN
easat-9645	551	50	,	,	PUNCT
easat-9645	551	51	2021	2021	NUM
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easat-9645	552	2	[	[	X
easat-9645	552	3	18	18	NUM
easat-9645	552	4	]	]	PUNCT
easat-9645	552	5	c.	c.	PROPN
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easat-9645	552	7	,	,	PUNCT
easat-9645	552	8	e.	e.	PROPN
easat-9645	552	9	de	de	PROPN
easat-9645	552	10	vito	vito	PROPN
easat-9645	552	11	,	,	PUNCT
easat-9645	552	12	and	and	CCONJ
easat-9645	552	13	a.	a.	NOUN
easat-9645	552	14	toigo	toigo	PROPN
easat-9645	552	15	,	,	PUNCT
easat-9645	552	16	"	"	PUNCT
easat-9645	552	17	vector	vector	NOUN
easat-9645	552	18	valued	value	VERB
easat-9645	552	19	reproducing	reproduce	VERB
easat-9645	552	20	kernel	kernel	PROPN
easat-9645	552	21	hilbert	hilbert	PROPN
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easat-9645	552	23	of	of	ADP
easat-9645	552	24	integrable	integrable	ADJ
easat-9645	552	25	functions	function	NOUN
easat-9645	552	26	and	and	CCONJ
easat-9645	552	27	mercer	mercer	PROPN
easat-9645	552	28	theorem	theorem	PROPN
easat-9645	552	29	,	,	PUNCT
easat-9645	552	30	"	"	PUNCT
easat-9645	552	31	analysis	analysis	NOUN
easat-9645	552	32	and	and	CCONJ
easat-9645	552	33	applications	application	NOUN
easat-9645	552	34	,	,	PUNCT
easat-9645	552	35	vol	vol	NOUN
easat-9645	552	36	.	.	PROPN
easat-9645	553	1	4	4	NUM
easat-9645	553	2	,	,	PUNCT
easat-9645	553	3	no	no	INTJ
easat-9645	553	4	.	.	NOUN
easat-9645	553	5	04	04	NUM
easat-9645	553	6	,	,	PUNCT
easat-9645	553	7	pp	pp	ADJ
easat-9645	553	8	.	.	PUNCT
easat-9645	554	1	377	377	NUM
easat-9645	554	2	-	-	SYM
easat-9645	554	3	408	408	NUM
easat-9645	554	4	,	,	PUNCT
easat-9645	554	5	2006	2006	NUM
easat-9645	554	6	.	.	PUNCT
easat-9645	555	1	https://doi.org/10.1142/s0219530506000838	https://doi.org/10.1142/s0219530506000838	NUM
easat-9645	555	2	[	[	X
easat-9645	555	3	19	19	NUM
easat-9645	555	4	]	]	PUNCT
easat-9645	555	5	m.	m.	NOUN
easat-9645	555	6	a.	a.	NOUN
easat-9645	555	7	mannan	mannan	PROPN
easat-9645	555	8	,	,	PUNCT
easat-9645	555	9	m.	m.	NOUN
easat-9645	555	10	a.	a.	PROPN
easat-9645	555	11	ullah	ullah	PROPN
easat-9645	555	12	,	,	PUNCT
easat-9645	555	13	u.	u.	PROPN
easat-9645	555	14	k.	k.	PROPN
easat-9645	555	15	dey	dey	PROPN
easat-9645	555	16	,	,	PUNCT
easat-9645	555	17	and	and	CCONJ
easat-9645	555	18	m.	m.	NOUN
easat-9645	555	19	alauddin	alauddin	NOUN
easat-9645	555	20	,	,	PUNCT
easat-9645	555	21	"	"	PUNCT
easat-9645	555	22	a	a	DET
easat-9645	555	23	study	study	NOUN
easat-9645	555	24	on	on	ADP
easat-9645	555	25	sylow	sylow	NOUN
easat-9645	555	26	theorems	theorem	NOUN
easat-9645	555	27	for	for	ADP
easat-9645	555	28	finding	find	VERB
easat-9645	555	29	out	out	ADP
easat-9645	555	30	possible	possible	ADJ
easat-9645	555	31	subgroups	subgroup	NOUN
easat-9645	555	32	of	of	ADP
easat-9645	555	33	a	a	DET
easat-9645	555	34	group	group	NOUN
easat-9645	555	35	in	in	ADP
easat-9645	555	36	different	different	ADJ
easat-9645	555	37	types	type	NOUN
easat-9645	555	38	of	of	ADP
easat-9645	555	39	order	order	NOUN
easat-9645	555	40	,	,	PUNCT
easat-9645	555	41	"	"	PUNCT
easat-9645	555	42	mathematics	mathematic	NOUN
easat-9645	555	43	and	and	CCONJ
easat-9645	555	44	statistics	statistic	NOUN
easat-9645	555	45	,	,	PUNCT
easat-9645	555	46	vol	vol	NOUN
easat-9645	555	47	.	.	PROPN
easat-9645	555	48	10	10	NUM
easat-9645	555	49	,	,	PUNCT
easat-9645	555	50	pp	pp	ADJ
easat-9645	555	51	.	.	PUNCT
easat-9645	555	52	851	851	NUM
easat-9645	555	53	-	-	SYM
easat-9645	555	54	860	860	NUM
easat-9645	555	55	,	,	PUNCT
easat-9645	555	56	2022	2022	NUM
easat-9645	555	57	.	.	PUNCT
easat-9645	556	1	https://doi.org/10.13189/ms.2022.100416	https://doi.org/10.13189/ms.2022.100416	X
easat-9645	557	1	[	[	X
easat-9645	557	2	20	20	NUM
easat-9645	557	3	]	]	PUNCT
easat-9645	557	4	m.	m.	NOUN
easat-9645	557	5	a.	a.	NOUN
easat-9645	557	6	mannan	mannan	PROPN
easat-9645	557	7	,	,	PUNCT
easat-9645	557	8	h.	h.	PROPN
easat-9645	557	9	akter	akter	PROPN
easat-9645	557	10	,	,	PUNCT
easat-9645	557	11	and	and	CCONJ
easat-9645	557	12	m.	m.	PROPN
easat-9645	557	13	a.	a.	PROPN
easat-9645	557	14	ullah	ullah	PROPN
easat-9645	557	15	,	,	PUNCT
easat-9645	557	16	"	"	PUNCT
easat-9645	557	17	evaluate	evaluate	VERB
easat-9645	557	18	all	all	DET
easat-9645	557	19	the	the	DET
easat-9645	557	20	order	order	NOUN
easat-9645	557	21	of	of	ADP
easat-9645	557	22	every	every	DET
easat-9645	557	23	element	element	NOUN
easat-9645	557	24	in	in	ADP
easat-9645	557	25	the	the	DET
easat-9645	557	26	higher	high	ADJ
easat-9645	557	27	even	even	ADV
easat-9645	557	28	,	,	PUNCT
easat-9645	557	29	odd	odd	ADJ
easat-9645	557	30	,	,	PUNCT
easat-9645	557	31	and	and	CCONJ
easat-9645	557	32	prime	prime	ADJ
easat-9645	557	33	order	order	NOUN
easat-9645	557	34	of	of	ADP
easat-9645	557	35	group	group	NOUN
easat-9645	557	36	for	for	ADP
easat-9645	557	37	composition	composition	NOUN
easat-9645	557	38	,	,	PUNCT
easat-9645	557	39	"	"	PUNCT
easat-9645	557	40	science	science	NOUN
easat-9645	557	41	and	and	CCONJ
easat-9645	557	42	technology	technology	NOUN
easat-9645	557	43	indonesia	indonesia	PROPN
easat-9645	557	44	,	,	PUNCT
easat-9645	557	45	vol	vol	NOUN
easat-9645	557	46	.	.	PROPN
easat-9645	558	1	7	7	NUM
easat-9645	558	2	,	,	PUNCT
easat-9645	558	3	no	no	INTJ
easat-9645	558	4	.	.	NOUN
easat-9645	558	5	3	3	NUM
easat-9645	558	6	,	,	PUNCT
easat-9645	558	7	pp	pp	ADJ
easat-9645	558	8	.	.	PUNCT
easat-9645	559	1	333	333	NUM
easat-9645	559	2	-	-	SYM
easat-9645	559	3	343	343	NUM
easat-9645	559	4	,	,	PUNCT
easat-9645	559	5	2022	2022	NUM
easat-9645	559	6	.	.	PUNCT
easat-9645	560	1	https://doi.org/10.26554/sti.2022.7.3.333-343	https://doi.org/10.26554/sti.2022.7.3.333-343	PRON
easat-9645	561	1	[	[	X
easat-9645	561	2	21	21	NUM
easat-9645	561	3	]	]	X
easat-9645	561	4	b.	b.	PROPN
easat-9645	561	5	schölkopf	schölkopf	PROPN
easat-9645	561	6	,	,	PUNCT
easat-9645	561	7	a.	a.	NOUN
easat-9645	561	8	smola	smola	PROPN
easat-9645	561	9	,	,	PUNCT
easat-9645	561	10	and	and	CCONJ
easat-9645	561	11	k.-r	k.-r	PROPN
easat-9645	561	12	.	.	PUNCT
easat-9645	562	1	müller	müller	PROPN
easat-9645	562	2	,	,	PUNCT
easat-9645	562	3	"	"	PUNCT
easat-9645	562	4	nonlinear	nonlinear	ADJ
easat-9645	562	5	component	component	NOUN
easat-9645	562	6	analysis	analysis	NOUN
easat-9645	562	7	as	as	ADP
easat-9645	562	8	a	a	DET
easat-9645	562	9	kernel	kernel	PROPN
easat-9645	562	10	eigenvalue	eigenvalue	PROPN
easat-9645	562	11	problem	problem	NOUN
easat-9645	562	12	,	,	PUNCT
easat-9645	562	13	"	"	PUNCT
easat-9645	562	14	neural	neural	ADJ
easat-9645	562	15	computation	computation	NOUN
easat-9645	562	16	,	,	PUNCT
easat-9645	562	17	vol	vol	NOUN
easat-9645	562	18	.	.	PROPN
easat-9645	563	1	10	10	NUM
easat-9645	563	2	,	,	PUNCT
easat-9645	563	3	no	no	INTJ
easat-9645	563	4	.	.	NOUN
easat-9645	563	5	5	5	NUM
easat-9645	563	6	,	,	PUNCT
easat-9645	563	7	pp	pp	ADJ
easat-9645	563	8	.	.	PUNCT
easat-9645	564	1	1299	1299	NUM
easat-9645	564	2	-	-	SYM
easat-9645	564	3	1319	1319	NUM
easat-9645	564	4	,	,	PUNCT
easat-9645	564	5	1998	1998	NUM
easat-9645	564	6	.	.	PUNCT
easat-9645	565	1	https://doi.org/10.1162/089976698300017467	https://doi.org/10.1162/089976698300017467	X
easat-9645	566	1	[	[	X
easat-9645	566	2	22	22	NUM
easat-9645	566	3	]	]	X
easat-9645	566	4	s.	s.	PROPN
easat-9645	566	5	theodoridis	theodoridis	PROPN
easat-9645	566	6	and	and	CCONJ
easat-9645	566	7	k.	k.	PROPN
easat-9645	566	8	koutroumbas	koutroumbas	PROPN
easat-9645	566	9	,	,	PUNCT
easat-9645	566	10	pattern	pattern	NOUN
easat-9645	566	11	recognition	recognition	NOUN
easat-9645	566	12	,	,	PUNCT
easat-9645	566	13	4th	4th	ADJ
easat-9645	566	14	ed	ed	NOUN
easat-9645	566	15	.	.	PUNCT
easat-9645	567	1	boston	boston	PROPN
easat-9645	567	2	,	,	PUNCT
easat-9645	567	3	ma	ma	PROPN
easat-9645	567	4	,	,	PUNCT
easat-9645	567	5	usa	usa	PROPN
easat-9645	567	6	:	:	PUNCT
easat-9645	567	7	academic	academic	ADJ
easat-9645	567	8	press	press	NOUN
easat-9645	567	9	,	,	PUNCT
easat-9645	567	10	2009	2009	NUM
easat-9645	567	11	.	.	PUNCT
easat-9645	568	1	https://doi.org/10.1017/9781009243926.004	https://doi.org/10.1017/9781009243926.004	X
easat-9645	568	2	https://doi.org/10.1214/009053607000000677	https://doi.org/10.1214/009053607000000677	PRON
easat-9645	568	3	https://doi.org/10.1214/12-aos1033	https://doi.org/10.1214/12-aos1033	VERB
easat-9645	568	4	https://doi.org/10.1103/physreva.109.052410	https://doi.org/10.1103/physreva.109.052410	ADJ
easat-9645	568	5	https://doi.org/10.1080/01621459.2021.1956938	https://doi.org/10.1080/01621459.2021.1956938	PROPN
easat-9645	568	6	https://doi.org/10.48550/arxiv.2106.08443	https://doi.org/10.48550/arxiv.2106.08443	PROPN
easat-9645	568	7	https://doi.org/10.1142/s0219530506000838	https://doi.org/10.1142/s0219530506000838	NUM
easat-9645	568	8	https://doi.org/10.13189/ms.2022.100416	https://doi.org/10.13189/ms.2022.100416	NOUN
easat-9645	568	9	https://doi.org/10.26554/sti.2022.7.3.333-343	https://doi.org/10.26554/sti.2022.7.3.333-343	NOUN
easat-9645	568	10	https://doi.org/10.1162/089976698300017467	https://doi.org/10.1162/089976698300017467	NOUN