id	sid	tid	token	lemma	pos
ejpam-5263	1	1	european	european	PROPN
ejpam-5263	1	2	journal	journal	PROPN
ejpam-5263	1	3	of	of	ADP
ejpam-5263	1	4	pure	pure	ADJ
ejpam-5263	1	5	and	and	CCONJ
ejpam-5263	1	6	applied	apply	VERB
ejpam-5263	1	7	mathematics	mathematic	NOUN
ejpam-5263	1	8	vol	vol	NOUN
ejpam-5263	1	9	.	.	PROPN
ejpam-5263	2	1	17	17	NUM
ejpam-5263	2	2	,	,	PUNCT
ejpam-5263	2	3	no	no	INTJ
ejpam-5263	2	4	.	.	NOUN
ejpam-5263	2	5	3	3	NUM
ejpam-5263	2	6	,	,	PUNCT
ejpam-5263	2	7	2024	2024	NUM
ejpam-5263	2	8	,	,	PUNCT
ejpam-5263	2	9	1937	1937	NUM
ejpam-5263	2	10	-	-	SYM
ejpam-5263	2	11	1947	1947	NUM
ejpam-5263	2	12	issn	issn	PROPN
ejpam-5263	2	13	1307	1307	NUM
ejpam-5263	2	14	-	-	SYM
ejpam-5263	2	15	5543	5543	NUM
ejpam-5263	2	16	–	–	PUNCT
ejpam-5263	2	17	ejpam.com	ejpam.com	X
ejpam-5263	2	18	published	publish	VERB
ejpam-5263	2	19	by	by	ADP
ejpam-5263	2	20	new	new	PROPN
ejpam-5263	2	21	york	york	PROPN
ejpam-5263	2	22	business	business	PROPN
ejpam-5263	2	23	global	global	PROPN
ejpam-5263	2	24	on	on	ADP
ejpam-5263	2	25	inner	inner	ADJ
ejpam-5263	2	26	products	product	NOUN
ejpam-5263	2	27	derived	derive	VERB
ejpam-5263	2	28	from	from	ADP
ejpam-5263	2	29	the	the	DET
ejpam-5263	2	30	standard	standard	ADJ
ejpam-5263	2	31	n	n	CCONJ
ejpam-5263	2	32	-	-	PUNCT
ejpam-5263	2	33	inner	inner	ADJ
ejpam-5263	2	34	product	product	NOUN
ejpam-5263	2	35	on	on	ADP
ejpam-5263	2	36	an	an	DET
ejpam-5263	2	37	inner	inner	ADJ
ejpam-5263	2	38	product	product	NOUN
ejpam-5263	2	39	space	space	NOUN
ejpam-5263	2	40	adam	adam	PROPN
ejpam-5263	2	41	adam1∗	adam1∗	PROPN
ejpam-5263	2	42	,	,	PUNCT
ejpam-5263	2	43	steven	steven	PROPN
ejpam-5263	2	44	rante2	rante2	PROPN
ejpam-5263	2	45	,	,	PUNCT
ejpam-5263	2	46	hendra	hendra	PROPN
ejpam-5263	2	47	gunawan2	gunawan2	PROPN
ejpam-5263	2	48	1	1	NUM
ejpam-5263	2	49	department	department	NOUN
ejpam-5263	2	50	of	of	ADP
ejpam-5263	2	51	mathematics	mathematics	PROPN
ejpam-5263	2	52	,	,	PUNCT
ejpam-5263	2	53	institut	institut	PROPN
ejpam-5263	2	54	teknologi	teknologi	PROPN
ejpam-5263	2	55	kalimantan	kalimantan	PROPN
ejpam-5263	2	56	,	,	PUNCT
ejpam-5263	2	57	balikpapan	balikpapan	PROPN
ejpam-5263	2	58	76127	76127	NUM
ejpam-5263	2	59	,	,	PUNCT
ejpam-5263	2	60	indonesia	indonesia	PROPN
ejpam-5263	2	61	2	2	NUM
ejpam-5263	2	62	analysis	analysis	NOUN
ejpam-5263	2	63	and	and	CCONJ
ejpam-5263	2	64	geometry	geometry	NOUN
ejpam-5263	2	65	group	group	NOUN
ejpam-5263	2	66	,	,	PUNCT
ejpam-5263	2	67	faculty	faculty	NOUN
ejpam-5263	2	68	of	of	ADP
ejpam-5263	2	69	mathematics	mathematic	NOUN
ejpam-5263	2	70	and	and	CCONJ
ejpam-5263	2	71	natural	natural	ADJ
ejpam-5263	2	72	sciences	science	NOUN
ejpam-5263	2	73	,	,	PUNCT
ejpam-5263	2	74	bandung	bandung	PROPN
ejpam-5263	2	75	institute	institute	PROPN
ejpam-5263	2	76	of	of	ADP
ejpam-5263	2	77	technology	technology	PROPN
ejpam-5263	2	78	,	,	PUNCT
ejpam-5263	2	79	bandung	bandung	PROPN
ejpam-5263	2	80	40132	40132	NUM
ejpam-5263	2	81	,	,	PUNCT
ejpam-5263	2	82	indonesia	indonesia	PROPN
ejpam-5263	2	83	abstract	abstract	NOUN
ejpam-5263	2	84	.	.	PUNCT
ejpam-5263	3	1	in	in	ADP
ejpam-5263	3	2	this	this	DET
ejpam-5263	3	3	paper	paper	NOUN
ejpam-5263	3	4	,	,	PUNCT
ejpam-5263	3	5	we	we	PRON
ejpam-5263	3	6	study	study	VERB
ejpam-5263	3	7	relations	relation	NOUN
ejpam-5263	3	8	between	between	ADP
ejpam-5263	3	9	inner	inner	ADJ
ejpam-5263	3	10	products	product	NOUN
ejpam-5263	3	11	derived	derive	VERB
ejpam-5263	3	12	from	from	ADP
ejpam-5263	3	13	the	the	DET
ejpam-5263	3	14	standard	standard	ADJ
ejpam-5263	3	15	n	n	CCONJ
ejpam-5263	3	16	-	-	PUNCT
ejpam-5263	3	17	inner	inner	ADJ
ejpam-5263	3	18	product	product	NOUN
ejpam-5263	3	19	defined	define	VERB
ejpam-5263	3	20	on	on	ADP
ejpam-5263	3	21	an	an	DET
ejpam-5263	3	22	inner	inner	ADJ
ejpam-5263	3	23	product	product	NOUN
ejpam-5263	3	24	space	space	NOUN
ejpam-5263	3	25	.	.	PUNCT
ejpam-5263	4	1	in	in	ADP
ejpam-5263	4	2	particular	particular	ADJ
ejpam-5263	4	3	,	,	PUNCT
ejpam-5263	4	4	we	we	PRON
ejpam-5263	4	5	are	be	AUX
ejpam-5263	4	6	interested	interested	ADJ
ejpam-5263	4	7	in	in	ADP
ejpam-5263	4	8	knowing	know	VERB
ejpam-5263	4	9	when	when	SCONJ
ejpam-5263	4	10	orthogonality	orthogonality	NOUN
ejpam-5263	4	11	with	with	ADP
ejpam-5263	4	12	respect	respect	NOUN
ejpam-5263	4	13	to	to	ADP
ejpam-5263	4	14	the	the	DET
ejpam-5263	4	15	original	original	ADJ
ejpam-5263	4	16	inner	inner	ADJ
ejpam-5263	4	17	product	product	NOUN
ejpam-5263	4	18	is	be	AUX
ejpam-5263	4	19	preserved	preserve	VERB
ejpam-5263	4	20	by	by	ADP
ejpam-5263	4	21	the	the	DET
ejpam-5263	4	22	derived	derive	VERB
ejpam-5263	4	23	inner	inner	ADJ
ejpam-5263	4	24	product	product	NOUN
ejpam-5263	4	25	.	.	PUNCT
ejpam-5263	5	1	2020	2020	NUM
ejpam-5263	5	2	mathematics	mathematic	NOUN
ejpam-5263	5	3	subject	subject	NOUN
ejpam-5263	5	4	classifications	classification	NOUN
ejpam-5263	5	5	:	:	PUNCT
ejpam-5263	5	6	46c50	46c50	NUM
ejpam-5263	5	7	,	,	PUNCT
ejpam-5263	5	8	46b20	46b20	NUM
ejpam-5263	5	9	,	,	PUNCT
ejpam-5263	5	10	46c05	46c05	NUM
ejpam-5263	5	11	key	key	ADJ
ejpam-5263	5	12	words	word	NOUN
ejpam-5263	5	13	and	and	CCONJ
ejpam-5263	5	14	phrases	phrase	NOUN
ejpam-5263	5	15	:	:	PUNCT
ejpam-5263	5	16	inner	inner	ADJ
ejpam-5263	5	17	products	product	NOUN
ejpam-5263	5	18	,	,	PUNCT
ejpam-5263	5	19	n	n	CCONJ
ejpam-5263	5	20	-	-	PUNCT
ejpam-5263	5	21	inner	inner	ADJ
ejpam-5263	5	22	products	product	NOUN
ejpam-5263	5	23	,	,	PUNCT
ejpam-5263	5	24	orthogonality	orthogonality	NOUN
ejpam-5263	5	25	,	,	PUNCT
ejpam-5263	5	26	orthogonal	orthogonal	ADJ
ejpam-5263	5	27	set	set	NOUN
ejpam-5263	5	28	,	,	PUNCT
ejpam-5263	5	29	orthonormal	orthonormal	ADJ
ejpam-5263	5	30	basis	basis	NOUN
ejpam-5263	5	31	.	.	PUNCT
ejpam-5263	6	1	1	1	X
ejpam-5263	6	2	.	.	X
ejpam-5263	6	3	introduction	introduction	NOUN
ejpam-5263	6	4	let	let	VERB
ejpam-5263	6	5	x	x	PRON
ejpam-5263	6	6	be	be	AUX
ejpam-5263	6	7	a	a	DET
ejpam-5263	6	8	real	real	ADJ
ejpam-5263	6	9	vector	vector	NOUN
ejpam-5263	6	10	space	space	NOUN
ejpam-5263	6	11	of	of	ADP
ejpam-5263	6	12	dimension	dimension	NOUN
ejpam-5263	6	13	d	d	PROPN
ejpam-5263	6	14	≥	≥	NOUN
ejpam-5263	6	15	n	n	CCONJ
ejpam-5263	6	16	and	and	CCONJ
ejpam-5263	6	17	let	let	VERB
ejpam-5263	6	18	⟨	⟨	NOUN
ejpam-5263	6	19	·	·	PUNCT
ejpam-5263	6	20	,	,	PUNCT
ejpam-5263	6	21	·	·	PUNCT
ejpam-5263	6	22	|	|	ADV
ejpam-5263	6	23	·	·	PUNCT
ejpam-5263	6	24	,	,	PUNCT
ejpam-5263	6	25	.	.	PUNCT
ejpam-5263	6	26	.	.	PUNCT
ejpam-5263	7	1	.	.	PUNCT
ejpam-5263	8	1	,	,	PUNCT
ejpam-5263	8	2	·	·	PUNCT
ejpam-5263	8	3	⟩	⟩	NOUN
ejpam-5263	8	4	:	:	PUNCT
ejpam-5263	8	5	xn+1	xn+1	NUM
ejpam-5263	8	6	→	→	SYM
ejpam-5263	8	7	r	r	NOUN
ejpam-5263	8	8	be	be	AUX
ejpam-5263	8	9	a	a	DET
ejpam-5263	8	10	function	function	NOUN
ejpam-5263	8	11	such	such	ADJ
ejpam-5263	8	12	that	that	PRON
ejpam-5263	8	13	for	for	ADP
ejpam-5263	8	14	every	every	DET
ejpam-5263	8	15	x0	x0	PROPN
ejpam-5263	8	16	,	,	PUNCT
ejpam-5263	8	17	x1	x1	PROPN
ejpam-5263	8	18	,	,	PUNCT
ejpam-5263	8	19	.	.	PUNCT
ejpam-5263	8	20	.	.	PUNCT
ejpam-5263	9	1	.	.	PUNCT
ejpam-5263	10	1	,	,	PUNCT
ejpam-5263	10	2	xn	xn	PROPN
ejpam-5263	10	3	,	,	PUNCT
ejpam-5263	10	4	xn+1	xn+1	PROPN
ejpam-5263	10	5	∈	∈	PROPN
ejpam-5263	10	6	x	x	X
ejpam-5263	10	7	and	and	CCONJ
ejpam-5263	10	8	α	α	NOUN
ejpam-5263	10	9	∈	∈	NOUN
ejpam-5263	11	1	r	r	NOUN
ejpam-5263	11	2	we	we	PRON
ejpam-5263	11	3	have	have	VERB
ejpam-5263	11	4	(	(	PUNCT
ejpam-5263	11	5	i1	i1	NOUN
ejpam-5263	11	6	)	)	PUNCT
ejpam-5263	12	1	⟨x1	⟨x1	PROPN
ejpam-5263	12	2	,	,	PUNCT
ejpam-5263	12	3	x1|x2	x1|x2	PRON
ejpam-5263	12	4	,	,	PUNCT
ejpam-5263	12	5	.	.	PUNCT
ejpam-5263	12	6	.	.	PUNCT
ejpam-5263	12	7	.	.	PUNCT
ejpam-5263	13	1	,	,	PUNCT
ejpam-5263	13	2	xn⟩	xn⟩	PROPN
ejpam-5263	13	3	≥	≥	X
ejpam-5263	13	4	0	0	NUM
ejpam-5263	13	5	and	and	CCONJ
ejpam-5263	13	6	⟨x1	⟨x1	NOUN
ejpam-5263	13	7	,	,	PUNCT
ejpam-5263	13	8	x1|x2	x1|x2	PRON
ejpam-5263	13	9	,	,	PUNCT
ejpam-5263	13	10	.	.	PUNCT
ejpam-5263	13	11	.	.	PUNCT
ejpam-5263	14	1	.	.	PUNCT
ejpam-5263	15	1	,	,	PUNCT
ejpam-5263	15	2	xn⟩	xn⟩	PROPN
ejpam-5263	16	1	=	=	SYM
ejpam-5263	16	2	0	0	PUNCT
ejpam-5263	17	1	if	if	SCONJ
ejpam-5263	17	2	and	and	CCONJ
ejpam-5263	17	3	only	only	ADV
ejpam-5263	17	4	if	if	SCONJ
ejpam-5263	17	5	x1	x1	PROPN
ejpam-5263	17	6	,	,	PUNCT
ejpam-5263	17	7	x2	x2	PROPN
ejpam-5263	17	8	,	,	PUNCT
ejpam-5263	17	9	.	.	PUNCT
ejpam-5263	17	10	.	.	PUNCT
ejpam-5263	18	1	.	.	PUNCT
ejpam-5263	19	1	,	,	PUNCT
ejpam-5263	19	2	xn	xn	PROPN
ejpam-5263	19	3	are	be	AUX
ejpam-5263	19	4	linearly	linearly	ADV
ejpam-5263	19	5	dependent	dependent	ADJ
ejpam-5263	19	6	;	;	PUNCT
ejpam-5263	19	7	(	(	PUNCT
ejpam-5263	19	8	i2	i2	NOUN
ejpam-5263	19	9	)	)	PUNCT
ejpam-5263	19	10	⟨x1	⟨x1	PROPN
ejpam-5263	19	11	,	,	PUNCT
ejpam-5263	19	12	x1|x2	x1|x2	PRON
ejpam-5263	19	13	,	,	PUNCT
ejpam-5263	19	14	.	.	PUNCT
ejpam-5263	19	15	.	.	PUNCT
ejpam-5263	20	1	.	.	PUNCT
ejpam-5263	21	1	,	,	PUNCT
ejpam-5263	21	2	xn⟩	xn⟩	PROPN
ejpam-5263	22	1	=	=	X
ejpam-5263	22	2	⟨xi1	⟨xi1	PROPN
ejpam-5263	22	3	,	,	PUNCT
ejpam-5263	22	4	xi1	xi1	PROPN
ejpam-5263	22	5	|xi2	|xi2	PROPN
ejpam-5263	22	6	,	,	PUNCT
ejpam-5263	22	7	.	.	PUNCT
ejpam-5263	22	8	.	.	PUNCT
ejpam-5263	22	9	.	.	PUNCT
ejpam-5263	23	1	,	,	PUNCT
ejpam-5263	23	2	xin⟩	xin⟩	PROPN
ejpam-5263	23	3	for	for	ADP
ejpam-5263	23	4	any	any	DET
ejpam-5263	23	5	permutation	permutation	NOUN
ejpam-5263	23	6	;	;	PUNCT
ejpam-5263	23	7	{	{	PUNCT
ejpam-5263	23	8	i1	i1	PROPN
ejpam-5263	23	9	,	,	PUNCT
ejpam-5263	23	10	i2	i2	PROPN
ejpam-5263	23	11	,	,	PUNCT
ejpam-5263	23	12	.	.	PUNCT
ejpam-5263	23	13	.	.	PUNCT
ejpam-5263	24	1	.	.	PUNCT
ejpam-5263	25	1	,	,	PUNCT
ejpam-5263	25	2	in	in	ADP
ejpam-5263	25	3	}	}	PUNCT
ejpam-5263	25	4	of	of	ADP
ejpam-5263	25	5	(	(	PUNCT
ejpam-5263	25	6	1	1	NUM
ejpam-5263	25	7	,	,	PUNCT
ejpam-5263	25	8	.	.	PUNCT
ejpam-5263	25	9	.	.	PUNCT
ejpam-5263	26	1	.	.	PUNCT
ejpam-5263	26	2	,	,	PUNCT
ejpam-5263	27	1	n	n	CCONJ
ejpam-5263	27	2	)	)	PUNCT
ejpam-5263	28	1	;	;	PUNCT
ejpam-5263	28	2	(	(	PUNCT
ejpam-5263	28	3	i3	i3	NOUN
ejpam-5263	28	4	)	)	PUNCT
ejpam-5263	28	5	⟨x0	⟨x0	NOUN
ejpam-5263	28	6	,	,	PUNCT
ejpam-5263	28	7	x1|x2	x1|x2	PROPN
ejpam-5263	28	8	,	,	PUNCT
ejpam-5263	28	9	.	.	PUNCT
ejpam-5263	28	10	.	.	PUNCT
ejpam-5263	28	11	.	.	PUNCT
ejpam-5263	29	1	,	,	PUNCT
ejpam-5263	29	2	xn⟩	xn⟩	PROPN
ejpam-5263	29	3	=	=	SYM
ejpam-5263	29	4	⟨x1	⟨x1	NOUN
ejpam-5263	29	5	,	,	PUNCT
ejpam-5263	29	6	x0|x2	x0|x2	PROPN
ejpam-5263	29	7	,	,	PUNCT
ejpam-5263	29	8	.	.	PUNCT
ejpam-5263	29	9	.	.	PUNCT
ejpam-5263	29	10	.	.	PUNCT
ejpam-5263	30	1	,	,	PUNCT
ejpam-5263	30	2	xn⟩	xn⟩	PROPN
ejpam-5263	30	3	;	;	PUNCT
ejpam-5263	30	4	(	(	PUNCT
ejpam-5263	30	5	i4	i4	PROPN
ejpam-5263	30	6	)	)	PUNCT
ejpam-5263	30	7	⟨αx0	⟨αx0	NOUN
ejpam-5263	30	8	,	,	PUNCT
ejpam-5263	30	9	x1|x2	x1|x2	PROPN
ejpam-5263	30	10	,	,	PUNCT
ejpam-5263	30	11	.	.	PUNCT
ejpam-5263	30	12	.	.	PUNCT
ejpam-5263	31	1	.	.	PUNCT
ejpam-5263	32	1	,	,	PUNCT
ejpam-5263	32	2	xn⟩	xn⟩	PROPN
ejpam-5263	32	3	=	=	SYM
ejpam-5263	32	4	α⟨x0	α⟨x0	NOUN
ejpam-5263	32	5	,	,	PUNCT
ejpam-5263	32	6	x1|x2	x1|x2	ADV
ejpam-5263	32	7	,	,	PUNCT
ejpam-5263	32	8	.	.	PUNCT
ejpam-5263	32	9	.	.	PUNCT
ejpam-5263	32	10	.	.	PUNCT
ejpam-5263	33	1	,	,	PUNCT
ejpam-5263	33	2	xn⟩	xn⟩	PROPN
ejpam-5263	33	3	;	;	PUNCT
ejpam-5263	33	4	(	(	PUNCT
ejpam-5263	33	5	i5	i5	NOUN
ejpam-5263	33	6	)	)	PUNCT
ejpam-5263	34	1	⟨x0	⟨x0	NOUN
ejpam-5263	34	2	+	+	NOUN
ejpam-5263	34	3	xn+1	xn+1	NUM
ejpam-5263	34	4	,	,	PUNCT
ejpam-5263	34	5	x1|x2	x1|x2	ADV
ejpam-5263	34	6	,	,	PUNCT
ejpam-5263	34	7	.	.	PUNCT
ejpam-5263	34	8	.	.	PUNCT
ejpam-5263	35	1	.	.	PUNCT
ejpam-5263	36	1	,	,	PUNCT
ejpam-5263	36	2	xn⟩	xn⟩	PROPN
ejpam-5263	37	1	=	=	SYM
ejpam-5263	37	2	⟨x0	⟨x0	PROPN
ejpam-5263	37	3	,	,	PUNCT
ejpam-5263	37	4	x1|x2	x1|x2	PROPN
ejpam-5263	37	5	,	,	PUNCT
ejpam-5263	37	6	.	.	PUNCT
ejpam-5263	37	7	.	.	PUNCT
ejpam-5263	37	8	.	.	PUNCT
ejpam-5263	38	1	,	,	PUNCT
ejpam-5263	38	2	xn⟩+	xn⟩+	PROPN
ejpam-5263	38	3	⟨xn+1	⟨xn+1	PROPN
ejpam-5263	38	4	,	,	PUNCT
ejpam-5263	38	5	x1|x2	x1|x2	PROPN
ejpam-5263	38	6	,	,	PUNCT
ejpam-5263	38	7	.	.	PUNCT
ejpam-5263	38	8	.	.	PUNCT
ejpam-5263	39	1	.	.	PUNCT
ejpam-5263	40	1	,	,	PUNCT
ejpam-5263	40	2	xn⟩.	xn⟩.	PROPN
ejpam-5263	40	3	∗corresponding	∗corresponde	VERB
ejpam-5263	40	4	author	author	NOUN
ejpam-5263	40	5	.	.	PUNCT
ejpam-5263	41	1	doi	doi	NOUN
ejpam-5263	41	2	:	:	PUNCT
ejpam-5263	41	3	https://doi.org/10.29020/nybg.ejpam.v17i3.5263	https://doi.org/10.29020/nybg.ejpam.v17i3.5263	PROPN
ejpam-5263	41	4	email	email	NOUN
ejpam-5263	41	5	addresses	address	NOUN
ejpam-5263	41	6	:	:	PUNCT
ejpam-5263	41	7	adam@lecturer.itk.ac.id	adam@lecturer.itk.ac.id	PROPN
ejpam-5263	41	8	(	(	PUNCT
ejpam-5263	41	9	adam	adam	PROPN
ejpam-5263	41	10	adam	adam	PROPN
ejpam-5263	41	11	)	)	PUNCT
ejpam-5263	41	12	,	,	PUNCT
ejpam-5263	41	13	stevengrante@gmail.com	stevengrante@gmail.com	PROPN
ejpam-5263	41	14	(	(	PUNCT
ejpam-5263	41	15	steven	steven	PROPN
ejpam-5263	41	16	rante	rante	PROPN
ejpam-5263	41	17	)	)	PUNCT
ejpam-5263	41	18	,	,	PUNCT
ejpam-5263	41	19	hgunawan@math.itb.ac.id	hgunawan@math.itb.ac.id	PROPN
ejpam-5263	41	20	(	(	PUNCT
ejpam-5263	41	21	hendra	hendra	PROPN
ejpam-5263	41	22	gunawan	gunawan	PROPN
ejpam-5263	41	23	)	)	PUNCT
ejpam-5263	41	24	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5263	41	25	1937	1937	NUM
ejpam-5263	42	1	©	©	PROPN
ejpam-5263	42	2	2024	2024	NUM
ejpam-5263	42	3	ejpam	ejpam	NOUN
ejpam-5263	42	4	all	all	DET
ejpam-5263	42	5	rights	right	NOUN
ejpam-5263	42	6	reserved	reserve	VERB
ejpam-5263	42	7	.	.	PUNCT
ejpam-5263	43	1	a.	a.	PROPN
ejpam-5263	43	2	adam	adam	PROPN
ejpam-5263	43	3	,	,	PUNCT
ejpam-5263	43	4	s.	s.	PROPN
ejpam-5263	43	5	rante	rante	PROPN
ejpam-5263	43	6	,	,	PUNCT
ejpam-5263	43	7	h.	h.	PROPN
ejpam-5263	43	8	gunawan	gunawan	PROPN
ejpam-5263	43	9	/	/	PUNCT
ejpam-5263	43	10	eur	eur	PROPN
ejpam-5263	43	11	.	.	PUNCT
ejpam-5263	44	1	j.	j.	PROPN
ejpam-5263	44	2	pure	pure	PROPN
ejpam-5263	44	3	appl	appl	PROPN
ejpam-5263	44	4	.	.	PROPN
ejpam-5263	44	5	math	math	PROPN
ejpam-5263	44	6	,	,	PUNCT
ejpam-5263	44	7	17	17	NUM
ejpam-5263	44	8	(	(	PUNCT
ejpam-5263	44	9	3	3	NUM
ejpam-5263	44	10	)	)	PUNCT
ejpam-5263	44	11	(	(	PUNCT
ejpam-5263	44	12	2024	2024	NUM
ejpam-5263	44	13	)	)	PUNCT
ejpam-5263	44	14	,	,	PUNCT
ejpam-5263	44	15	1937	1937	NUM
ejpam-5263	44	16	-	-	SYM
ejpam-5263	44	17	1947	1947	NUM
ejpam-5263	44	18	1938	1938	NUM
ejpam-5263	44	19	the	the	DET
ejpam-5263	44	20	function	function	NOUN
ejpam-5263	44	21	⟨	⟨	VERB
ejpam-5263	44	22	·	·	PUNCT
ejpam-5263	44	23	,	,	PUNCT
ejpam-5263	44	24	·	·	PUNCT
ejpam-5263	44	25	|	|	ADV
ejpam-5263	44	26	·	·	PUNCT
ejpam-5263	44	27	,	,	PUNCT
ejpam-5263	44	28	.	.	PUNCT
ejpam-5263	44	29	.	.	PUNCT
ejpam-5263	44	30	.	.	PUNCT
ejpam-5263	45	1	,	,	PUNCT
ejpam-5263	45	2	·	·	PUNCT
ejpam-5263	45	3	⟩	⟩	NOUN
ejpam-5263	45	4	is	be	AUX
ejpam-5263	45	5	called	call	VERB
ejpam-5263	45	6	an	an	DET
ejpam-5263	45	7	n	n	CCONJ
ejpam-5263	45	8	-	-	PUNCT
ejpam-5263	45	9	inner	inner	ADJ
ejpam-5263	45	10	product	product	NOUN
ejpam-5263	45	11	introduced	introduce	VERB
ejpam-5263	45	12	by	by	ADP
ejpam-5263	45	13	misiak	misiak	NOUN
ejpam-5263	45	14	in	in	ADP
ejpam-5263	45	15	1989	1989	NUM
ejpam-5263	45	16	[	[	X
ejpam-5263	45	17	14	14	NUM
ejpam-5263	45	18	]	]	PUNCT
ejpam-5263	45	19	.	.	PUNCT
ejpam-5263	46	1	here	here	ADV
ejpam-5263	46	2	,	,	PUNCT
ejpam-5263	46	3	the	the	DET
ejpam-5263	46	4	pair	pair	NOUN
ejpam-5263	46	5	(	(	PUNCT
ejpam-5263	46	6	x	x	NOUN
ejpam-5263	46	7	,	,	PUNCT
ejpam-5263	46	8	⟨	⟨	NOUN
ejpam-5263	46	9	·	·	SYM
ejpam-5263	46	10	,	,	PUNCT
ejpam-5263	46	11	·	·	PUNCT
ejpam-5263	46	12	|	|	ADV
ejpam-5263	46	13	·	·	PUNCT
ejpam-5263	46	14	,	,	PUNCT
ejpam-5263	46	15	.	.	PUNCT
ejpam-5263	46	16	.	.	PUNCT
ejpam-5263	47	1	.	.	PUNCT
ejpam-5263	48	1	,	,	PUNCT
ejpam-5263	48	2	·	·	PUNCT
ejpam-5263	48	3	⟩	⟩	NOUN
ejpam-5263	48	4	)	)	PUNCT
ejpam-5263	48	5	is	be	AUX
ejpam-5263	48	6	called	call	VERB
ejpam-5263	48	7	an	an	DET
ejpam-5263	48	8	n	n	CCONJ
ejpam-5263	48	9	-	-	PUNCT
ejpam-5263	48	10	inner	inner	ADJ
ejpam-5263	48	11	product	product	NOUN
ejpam-5263	48	12	space	space	NOUN
ejpam-5263	48	13	.	.	PUNCT
ejpam-5263	49	1	if	if	SCONJ
ejpam-5263	49	2	(	(	PUNCT
ejpam-5263	49	3	x	x	NOUN
ejpam-5263	49	4	,	,	PUNCT
ejpam-5263	49	5	⟨	⟨	NOUN
ejpam-5263	49	6	·	·	SYM
ejpam-5263	49	7	,	,	PUNCT
ejpam-5263	49	8	·	·	PUNCT
ejpam-5263	49	9	⟩	⟩	NOUN
ejpam-5263	49	10	)	)	PUNCT
ejpam-5263	49	11	is	be	AUX
ejpam-5263	49	12	a	a	DET
ejpam-5263	49	13	real	real	ADJ
ejpam-5263	49	14	inner	inner	ADJ
ejpam-5263	49	15	product	product	NOUN
ejpam-5263	49	16	space	space	NOUN
ejpam-5263	49	17	of	of	ADP
ejpam-5263	49	18	dimension	dimension	NOUN
ejpam-5263	49	19	d	d	PROPN
ejpam-5263	49	20	≥	≥	NUM
ejpam-5263	49	21	n	n	CCONJ
ejpam-5263	49	22	,	,	PUNCT
ejpam-5263	49	23	we	we	PRON
ejpam-5263	49	24	can	can	AUX
ejpam-5263	49	25	define	define	VERB
ejpam-5263	49	26	the	the	DET
ejpam-5263	49	27	standard	standard	ADJ
ejpam-5263	49	28	n	n	CCONJ
ejpam-5263	49	29	-	-	PUNCT
ejpam-5263	49	30	inner	inner	ADJ
ejpam-5263	49	31	product	product	NOUN
ejpam-5263	49	32	by	by	ADP
ejpam-5263	49	33	⟨x0	⟨x0	PROPN
ejpam-5263	49	34	,	,	PUNCT
ejpam-5263	49	35	x1|x2	x1|x2	ADV
ejpam-5263	49	36	,	,	PUNCT
ejpam-5263	49	37	.	.	PUNCT
ejpam-5263	49	38	.	.	PUNCT
ejpam-5263	50	1	.	.	PUNCT
ejpam-5263	51	1	,	,	PUNCT
ejpam-5263	51	2	xn⟩	xn⟩	PROPN
ejpam-5263	52	1	=	=	SYM
ejpam-5263	52	2	∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣	PROPN
ejpam-5263	53	1	⟨x0	⟨x0	PROPN
ejpam-5263	53	2	,	,	PUNCT
ejpam-5263	53	3	x1⟩	x1⟩	PUNCT
ejpam-5263	54	1	⟨x0	⟨x0	NOUN
ejpam-5263	54	2	,	,	PUNCT
ejpam-5263	54	3	x2⟩	x2⟩	PUNCT
ejpam-5263	54	4	.	.	PUNCT
ejpam-5263	54	5	.	.	PUNCT
ejpam-5263	54	6	.	.	PUNCT
ejpam-5263	55	1	⟨x0	⟨x0	NOUN
ejpam-5263	55	2	,	,	PUNCT
ejpam-5263	55	3	xn⟩	xn⟩	PROPN
ejpam-5263	55	4	⟨x2	⟨x2	PROPN
ejpam-5263	55	5	,	,	PUNCT
ejpam-5263	55	6	x1⟩	x1⟩	PROPN
ejpam-5263	55	7	⟨x2	⟨x2	PROPN
ejpam-5263	55	8	,	,	PUNCT
ejpam-5263	55	9	x2⟩	x2⟩	PUNCT
ejpam-5263	55	10	.	.	PUNCT
ejpam-5263	55	11	.	.	PUNCT
ejpam-5263	55	12	.	.	PUNCT
ejpam-5263	56	1	⟨x2	⟨x2	PROPN
ejpam-5263	56	2	,	,	PUNCT
ejpam-5263	56	3	xn⟩	xn⟩	PROPN
ejpam-5263	56	4	...	...	PUNCT
ejpam-5263	56	5	...	...	PUNCT
ejpam-5263	56	6	.	.	PUNCT
ejpam-5263	56	7	.	.	PUNCT
ejpam-5263	56	8	.	.	PUNCT
ejpam-5263	56	9	...	...	PUNCT
ejpam-5263	57	1	⟨xn	⟨xn	NOUN
ejpam-5263	57	2	,	,	PUNCT
ejpam-5263	57	3	x1⟩	x1⟩	PUNCT
ejpam-5263	57	4	⟨xn	⟨xn	NOUN
ejpam-5263	57	5	,	,	PUNCT
ejpam-5263	57	6	x2⟩	x2⟩	PUNCT
ejpam-5263	57	7	.	.	PUNCT
ejpam-5263	57	8	.	.	PUNCT
ejpam-5263	57	9	.	.	PUNCT
ejpam-5263	58	1	⟨xn	⟨xn	NOUN
ejpam-5263	58	2	,	,	PUNCT
ejpam-5263	58	3	xn⟩	xn⟩	PROPN
ejpam-5263	58	4	∣∣∣∣∣∣∣∣∣.	∣∣∣∣∣∣∣∣∣.	PROPN
ejpam-5263	58	5	so	so	ADV
ejpam-5263	58	6	,	,	PUNCT
ejpam-5263	58	7	a	a	DET
ejpam-5263	58	8	real	real	ADJ
ejpam-5263	58	9	inner	inner	ADJ
ejpam-5263	58	10	product	product	NOUN
ejpam-5263	58	11	space	space	NOUN
ejpam-5263	58	12	of	of	ADP
ejpam-5263	58	13	dimension	dimension	NOUN
ejpam-5263	58	14	d	d	PROPN
ejpam-5263	58	15	≥	≥	NOUN
ejpam-5263	58	16	n	n	CCONJ
ejpam-5263	58	17	with	with	ADP
ejpam-5263	58	18	the	the	DET
ejpam-5263	58	19	n	n	CCONJ
ejpam-5263	58	20	-	-	PUNCT
ejpam-5263	58	21	inner	inner	ADJ
ejpam-5263	58	22	product	product	NOUN
ejpam-5263	58	23	defined	define	VERB
ejpam-5263	58	24	above	above	ADV
ejpam-5263	58	25	is	be	AUX
ejpam-5263	58	26	an	an	DET
ejpam-5263	58	27	example	example	NOUN
ejpam-5263	58	28	of	of	ADP
ejpam-5263	58	29	an	an	DET
ejpam-5263	58	30	n	n	CCONJ
ejpam-5263	58	31	-	-	PUNCT
ejpam-5263	58	32	inner	inner	ADJ
ejpam-5263	58	33	product	product	NOUN
ejpam-5263	58	34	space	space	NOUN
ejpam-5263	58	35	,	,	PUNCT
ejpam-5263	58	36	which	which	PRON
ejpam-5263	58	37	we	we	PRON
ejpam-5263	58	38	can	can	AUX
ejpam-5263	58	39	call	call	VERB
ejpam-5263	58	40	a	a	DET
ejpam-5263	58	41	standard	standard	ADJ
ejpam-5263	58	42	n	n	CCONJ
ejpam-5263	58	43	-	-	PUNCT
ejpam-5263	58	44	inner	inner	ADJ
ejpam-5263	58	45	product	product	NOUN
ejpam-5263	58	46	space	space	NOUN
ejpam-5263	58	47	.	.	PUNCT
ejpam-5263	59	1	next	next	ADJ
ejpam-5263	59	2	,	,	PUNCT
ejpam-5263	59	3	from	from	ADP
ejpam-5263	59	4	n	n	CCONJ
ejpam-5263	59	5	-	-	PUNCT
ejpam-5263	59	6	inner	inner	ADJ
ejpam-5263	59	7	product	product	NOUN
ejpam-5263	59	8	space	space	NOUN
ejpam-5263	59	9	we	we	PRON
ejpam-5263	59	10	can	can	AUX
ejpam-5263	59	11	derive	derive	VERB
ejpam-5263	59	12	n	n	CCONJ
ejpam-5263	59	13	-	-	PUNCT
ejpam-5263	59	14	norm	norm	NOUN
ejpam-5263	59	15	,	,	PUNCT
ejpam-5263	59	16	defined	define	VERB
ejpam-5263	59	17	by	by	ADP
ejpam-5263	59	18	∥x1	∥x1	NOUN
ejpam-5263	59	19	,	,	PUNCT
ejpam-5263	59	20	x2	x2	PROPN
ejpam-5263	59	21	,	,	PUNCT
ejpam-5263	59	22	.	.	PUNCT
ejpam-5263	59	23	.	.	PUNCT
ejpam-5263	60	1	.	.	PUNCT
ejpam-5263	61	1	,	,	PUNCT
ejpam-5263	61	2	xn∥	xn∥	PROPN
ejpam-5263	61	3	=	=	SYM
ejpam-5263	61	4	⟨x1	⟨x1	NOUN
ejpam-5263	61	5	,	,	PUNCT
ejpam-5263	61	6	x1|x2	x1|x2	PRON
ejpam-5263	61	7	,	,	PUNCT
ejpam-5263	61	8	.	.	PUNCT
ejpam-5263	61	9	.	.	PUNCT
ejpam-5263	61	10	.	.	PUNCT
ejpam-5263	62	1	,	,	PUNCT
ejpam-5263	62	2	xn⟩	xn⟩	PROPN
ejpam-5263	62	3	1	1	NUM
ejpam-5263	62	4	2	2	NUM
ejpam-5263	62	5	.	.	PUNCT
ejpam-5263	63	1	furthermore	furthermore	ADV
ejpam-5263	63	2	,	,	PUNCT
ejpam-5263	63	3	an	an	DET
ejpam-5263	63	4	n	n	NOUN
ejpam-5263	63	5	-	-	PUNCT
ejpam-5263	63	6	norm	norm	NOUN
ejpam-5263	63	7	on	on	ADP
ejpam-5263	63	8	x	x	X
ejpam-5263	63	9	is	be	AUX
ejpam-5263	63	10	a	a	DET
ejpam-5263	63	11	function	function	NOUN
ejpam-5263	63	12	∥	∥	NUM
ejpam-5263	63	13	·	·	PUNCT
ejpam-5263	63	14	,	,	PUNCT
ejpam-5263	63	15	.	.	PUNCT
ejpam-5263	63	16	.	.	PUNCT
ejpam-5263	64	1	.	.	PUNCT
ejpam-5263	65	1	,	,	PUNCT
ejpam-5263	66	1	·	·	PUNCT
ejpam-5263	66	2	∥	∥	X
ejpam-5263	66	3	:	:	PUNCT
ejpam-5263	66	4	xn	xn	PUNCT
ejpam-5263	66	5	→	→	PUNCT
ejpam-5263	66	6	r	r	NOUN
ejpam-5263	66	7	such	such	ADJ
ejpam-5263	66	8	that	that	PRON
ejpam-5263	66	9	for	for	ADP
ejpam-5263	66	10	every	every	DET
ejpam-5263	66	11	x0	x0	PROPN
ejpam-5263	66	12	,	,	PUNCT
ejpam-5263	66	13	x1	x1	PROPN
ejpam-5263	66	14	,	,	PUNCT
ejpam-5263	66	15	.	.	PUNCT
ejpam-5263	66	16	.	.	PUNCT
ejpam-5263	67	1	.	.	PUNCT
ejpam-5263	68	1	,	,	PUNCT
ejpam-5263	68	2	xn	xn	PROPN
ejpam-5263	68	3	,	,	PUNCT
ejpam-5263	68	4	xn+1	xn+1	PROPN
ejpam-5263	68	5	∈	∈	PROPN
ejpam-5263	68	6	x	x	X
ejpam-5263	68	7	and	and	CCONJ
ejpam-5263	68	8	α	α	NOUN
ejpam-5263	68	9	∈	∈	PROPN
ejpam-5263	68	10	r	r	NOUN
ejpam-5263	68	11	,	,	PUNCT
ejpam-5263	68	12	the	the	DET
ejpam-5263	68	13	function	function	NOUN
ejpam-5263	68	14	satisfying	satisfy	VERB
ejpam-5263	68	15	the	the	DET
ejpam-5263	68	16	following	follow	VERB
ejpam-5263	68	17	properties	property	NOUN
ejpam-5263	68	18	:	:	PUNCT
ejpam-5263	68	19	(	(	PUNCT
ejpam-5263	68	20	n1	n1	NOUN
ejpam-5263	68	21	)	)	PUNCT
ejpam-5263	68	22	∥x1	∥x1	NOUN
ejpam-5263	68	23	,	,	PUNCT
ejpam-5263	68	24	x2	x2	PROPN
ejpam-5263	68	25	,	,	PUNCT
ejpam-5263	68	26	.	.	PUNCT
ejpam-5263	68	27	.	.	PUNCT
ejpam-5263	68	28	.	.	PUNCT
ejpam-5263	69	1	,	,	PUNCT
ejpam-5263	69	2	xn∥	xn∥	PROPN
ejpam-5263	69	3	≥	≥	PROPN
ejpam-5263	69	4	0	0	NUM
ejpam-5263	69	5	and	and	CCONJ
ejpam-5263	69	6	∥x1	∥x1	NOUN
ejpam-5263	69	7	,	,	PUNCT
ejpam-5263	69	8	x2	x2	PROPN
ejpam-5263	69	9	,	,	PUNCT
ejpam-5263	69	10	.	.	PUNCT
ejpam-5263	69	11	.	.	PUNCT
ejpam-5263	69	12	.	.	PUNCT
ejpam-5263	70	1	,	,	PUNCT
ejpam-5263	70	2	xn∥	xn∥	PROPN
ejpam-5263	71	1	=	=	PUNCT
ejpam-5263	71	2	0	0	PUNCT
ejpam-5263	72	1	if	if	SCONJ
ejpam-5263	72	2	and	and	CCONJ
ejpam-5263	72	3	only	only	ADV
ejpam-5263	72	4	if	if	SCONJ
ejpam-5263	72	5	x1	x1	PROPN
ejpam-5263	72	6	,	,	PUNCT
ejpam-5263	72	7	x2	x2	PROPN
ejpam-5263	72	8	,	,	PUNCT
ejpam-5263	72	9	.	.	PUNCT
ejpam-5263	72	10	.	.	PUNCT
ejpam-5263	73	1	.	.	PUNCT
ejpam-5263	74	1	,	,	PUNCT
ejpam-5263	74	2	xn	xn	PROPN
ejpam-5263	74	3	are	be	AUX
ejpam-5263	74	4	linearly	linearly	ADV
ejpam-5263	74	5	dependent	dependent	ADJ
ejpam-5263	74	6	;	;	PUNCT
ejpam-5263	74	7	(	(	PUNCT
ejpam-5263	74	8	n2	n2	ADJ
ejpam-5263	74	9	)	)	PUNCT
ejpam-5263	74	10	∥x1	∥x1	NOUN
ejpam-5263	74	11	,	,	PUNCT
ejpam-5263	74	12	x2	x2	PROPN
ejpam-5263	74	13	,	,	PUNCT
ejpam-5263	74	14	.	.	PUNCT
ejpam-5263	74	15	.	.	PUNCT
ejpam-5263	75	1	.	.	PUNCT
ejpam-5263	76	1	,	,	PUNCT
ejpam-5263	76	2	xn∥	xn∥	PROPN
ejpam-5263	76	3	is	be	AUX
ejpam-5263	76	4	invariant	invariant	ADJ
ejpam-5263	76	5	under	under	ADP
ejpam-5263	76	6	permutation	permutation	NOUN
ejpam-5263	76	7	;	;	PUNCT
ejpam-5263	76	8	(	(	PUNCT
ejpam-5263	76	9	n3	n3	NOUN
ejpam-5263	76	10	)	)	PUNCT
ejpam-5263	76	11	∥αx1	∥αx1	PROPN
ejpam-5263	76	12	,	,	PUNCT
ejpam-5263	76	13	x2	x2	PROPN
ejpam-5263	76	14	,	,	PUNCT
ejpam-5263	76	15	.	.	PUNCT
ejpam-5263	76	16	.	.	PUNCT
ejpam-5263	77	1	.	.	PUNCT
ejpam-5263	78	1	,	,	PUNCT
ejpam-5263	78	2	xn∥	xn∥	PROPN
ejpam-5263	78	3	=	=	SYM
ejpam-5263	78	4	|α|	|α|	NUM
ejpam-5263	78	5	∥x1	∥x1	NOUN
ejpam-5263	78	6	,	,	PUNCT
ejpam-5263	78	7	x2	x2	PROPN
ejpam-5263	78	8	,	,	PUNCT
ejpam-5263	78	9	.	.	PUNCT
ejpam-5263	78	10	.	.	PUNCT
ejpam-5263	79	1	.	.	PUNCT
ejpam-5263	80	1	,	,	PUNCT
ejpam-5263	80	2	xn∥	xn∥	PROPN
ejpam-5263	80	3	;	;	PUNCT
ejpam-5263	80	4	(	(	PUNCT
ejpam-5263	80	5	n4	n4	PROPN
ejpam-5263	80	6	)	)	PUNCT
ejpam-5263	80	7	∥x0	∥x0	NOUN
ejpam-5263	81	1	+	+	CCONJ
ejpam-5263	82	1	x1	x1	ADJ
ejpam-5263	82	2	,	,	PUNCT
ejpam-5263	82	3	x2	x2	PROPN
ejpam-5263	82	4	,	,	PUNCT
ejpam-5263	82	5	.	.	PUNCT
ejpam-5263	82	6	.	.	PUNCT
ejpam-5263	82	7	.	.	PUNCT
ejpam-5263	83	1	,	,	PUNCT
ejpam-5263	83	2	xn∥	xn∥	PROPN
ejpam-5263	83	3	≤	≤	PROPN
ejpam-5263	83	4	∥x0	∥x0	NOUN
ejpam-5263	83	5	,	,	PUNCT
ejpam-5263	83	6	x2	x2	PROPN
ejpam-5263	83	7	,	,	PUNCT
ejpam-5263	83	8	.	.	PUNCT
ejpam-5263	83	9	.	.	PUNCT
ejpam-5263	83	10	.	.	PUNCT
ejpam-5263	84	1	,	,	PUNCT
ejpam-5263	84	2	xn∥+	xn∥+	PUNCT
ejpam-5263	84	3	∥x1	∥x1	PROPN
ejpam-5263	84	4	,	,	PUNCT
ejpam-5263	84	5	x2	x2	PROPN
ejpam-5263	84	6	,	,	PUNCT
ejpam-5263	84	7	.	.	PUNCT
ejpam-5263	84	8	.	.	PUNCT
ejpam-5263	84	9	.	.	PUNCT
ejpam-5263	85	1	,	,	PUNCT
ejpam-5263	85	2	xn∥.	xn∥.	PROPN
ejpam-5263	85	3	geometrically	geometrically	ADV
ejpam-5263	85	4	,	,	PUNCT
ejpam-5263	85	5	∥x1	∥x1	NOUN
ejpam-5263	85	6	,	,	PUNCT
ejpam-5263	85	7	x2	x2	PROPN
ejpam-5263	85	8	,	,	PUNCT
ejpam-5263	85	9	.	.	PUNCT
ejpam-5263	85	10	.	.	PUNCT
ejpam-5263	85	11	.	.	PUNCT
ejpam-5263	86	1	,	,	PUNCT
ejpam-5263	86	2	xn∥	xn∥	PROPN
ejpam-5263	86	3	represents	represent	VERB
ejpam-5263	86	4	the	the	DET
ejpam-5263	86	5	generalized	generalized	ADJ
ejpam-5263	86	6	volume	volume	NOUN
ejpam-5263	86	7	of	of	ADP
ejpam-5263	86	8	an	an	DET
ejpam-5263	86	9	n	n	ADV
ejpam-5263	86	10	-	-	PUNCT
ejpam-5263	86	11	dimensional	dimensional	ADJ
ejpam-5263	86	12	parallelepiped	parallelepipe	VERB
ejpam-5263	86	13	spanned	span	VERB
ejpam-5263	86	14	by	by	ADP
ejpam-5263	86	15	x1	x1	PROPN
ejpam-5263	86	16	,	,	PUNCT
ejpam-5263	86	17	x2	x2	PROPN
ejpam-5263	86	18	,	,	PUNCT
ejpam-5263	86	19	.	.	PUNCT
ejpam-5263	86	20	.	.	PUNCT
ejpam-5263	87	1	.	.	PUNCT
ejpam-5263	88	1	,	,	PUNCT
ejpam-5263	88	2	xn	xn	X
ejpam-5263	88	3	.	.	PUNCT
ejpam-5263	89	1	then	then	ADV
ejpam-5263	89	2	,	,	PUNCT
ejpam-5263	89	3	⟨x0	⟨x0	PROPN
ejpam-5263	89	4	,	,	PUNCT
ejpam-5263	89	5	x1|x2	x1|x2	PROPN
ejpam-5263	89	6	,	,	PUNCT
ejpam-5263	89	7	.	.	PUNCT
ejpam-5263	89	8	.	.	PUNCT
ejpam-5263	90	1	.	.	PUNCT
ejpam-5263	91	1	,	,	PUNCT
ejpam-5263	91	2	xn⟩	xn⟩	PROPN
ejpam-5263	91	3	∥x0	∥x0	PROPN
ejpam-5263	91	4	,	,	PUNCT
ejpam-5263	91	5	x2	x2	PROPN
ejpam-5263	91	6	,	,	PUNCT
ejpam-5263	91	7	.	.	PUNCT
ejpam-5263	91	8	.	.	PUNCT
ejpam-5263	91	9	.	.	PUNCT
ejpam-5263	92	1	,	,	PUNCT
ejpam-5263	92	2	xn∥	xn∥	PROPN
ejpam-5263	92	3	∥x1	∥x1	PROPN
ejpam-5263	92	4	,	,	PUNCT
ejpam-5263	92	5	x2	x2	PROPN
ejpam-5263	92	6	,	,	PUNCT
ejpam-5263	92	7	.	.	PUNCT
ejpam-5263	92	8	.	.	PUNCT
ejpam-5263	92	9	.	.	PUNCT
ejpam-5263	93	1	,	,	PUNCT
ejpam-5263	93	2	xn∥	xn∥	PROPN
ejpam-5263	93	3	is	be	AUX
ejpam-5263	93	4	the	the	DET
ejpam-5263	93	5	cosine	cosine	NOUN
ejpam-5263	93	6	of	of	ADP
ejpam-5263	93	7	the	the	DET
ejpam-5263	93	8	angle	angle	NOUN
ejpam-5263	93	9	between	between	ADP
ejpam-5263	93	10	two	two	NUM
ejpam-5263	93	11	parallelepipeds	parallelepiped	NOUN
ejpam-5263	93	12	spanned	span	VERB
ejpam-5263	93	13	by	by	ADP
ejpam-5263	93	14	x0	x0	PROPN
ejpam-5263	93	15	,	,	PUNCT
ejpam-5263	93	16	x2	x2	PROPN
ejpam-5263	93	17	,	,	PUNCT
ejpam-5263	93	18	.	.	PUNCT
ejpam-5263	93	19	.	.	PUNCT
ejpam-5263	94	1	.	.	PUNCT
ejpam-5263	95	1	,	,	PUNCT
ejpam-5263	95	2	xn	xn	PROPN
ejpam-5263	95	3	and	and	CCONJ
ejpam-5263	95	4	x1	x1	PROPN
ejpam-5263	95	5	,	,	PUNCT
ejpam-5263	95	6	x2	x2	PROPN
ejpam-5263	95	7	,	,	PUNCT
ejpam-5263	95	8	.	.	PUNCT
ejpam-5263	95	9	.	.	PUNCT
ejpam-5263	96	1	.	.	PUNCT
ejpam-5263	97	1	,	,	PUNCT
ejpam-5263	97	2	xn	xn	X
ejpam-5263	97	3	.	.	PUNCT
ejpam-5263	97	4	see	see	VERB
ejpam-5263	98	1	[	[	X
ejpam-5263	98	2	9	9	NUM
ejpam-5263	98	3	,	,	PUNCT
ejpam-5263	98	4	10	10	NUM
ejpam-5263	98	5	,	,	PUNCT
ejpam-5263	98	6	15	15	NUM
ejpam-5263	98	7	]	]	PUNCT
ejpam-5263	98	8	for	for	ADP
ejpam-5263	98	9	more	more	ADJ
ejpam-5263	98	10	properties	property	NOUN
ejpam-5263	98	11	of	of	ADP
ejpam-5263	98	12	n	n	CCONJ
ejpam-5263	98	13	-	-	PUNCT
ejpam-5263	98	14	inner	inner	ADJ
ejpam-5263	98	15	products	product	NOUN
ejpam-5263	98	16	.	.	PUNCT
ejpam-5263	99	1	the	the	DET
ejpam-5263	99	2	related	relate	VERB
ejpam-5263	99	3	results	result	NOUN
ejpam-5263	99	4	may	may	AUX
ejpam-5263	99	5	also	also	ADV
ejpam-5263	99	6	be	be	AUX
ejpam-5263	99	7	found	find	VERB
ejpam-5263	99	8	in	in	ADP
ejpam-5263	99	9	[	[	X
ejpam-5263	99	10	3–6	3–6	NUM
ejpam-5263	99	11	,	,	PUNCT
ejpam-5263	99	12	13	13	NUM
ejpam-5263	99	13	,	,	PUNCT
ejpam-5263	99	14	16	16	NUM
ejpam-5263	99	15	,	,	PUNCT
ejpam-5263	99	16	17	17	NUM
ejpam-5263	99	17	]	]	PUNCT
ejpam-5263	99	18	.	.	PUNCT
ejpam-5263	100	1	historically	historically	ADV
ejpam-5263	100	2	,	,	PUNCT
ejpam-5263	100	3	numerous	numerous	ADJ
ejpam-5263	100	4	authors	author	NOUN
ejpam-5263	100	5	have	have	AUX
ejpam-5263	100	6	introduced	introduce	VERB
ejpam-5263	100	7	and	and	CCONJ
ejpam-5263	100	8	developed	develop	VERB
ejpam-5263	100	9	several	several	ADJ
ejpam-5263	100	10	concepts	concept	NOUN
ejpam-5263	100	11	of	of	ADP
ejpam-5263	100	12	orthogonality	orthogonality	NOUN
ejpam-5263	100	13	in	in	ADP
ejpam-5263	100	14	2	2	NUM
ejpam-5263	100	15	-	-	PUNCT
ejpam-5263	100	16	normed	norme	VERB
ejpam-5263	100	17	spaces	space	NOUN
ejpam-5263	100	18	and	and	CCONJ
ejpam-5263	100	19	2	2	NUM
ejpam-5263	100	20	-	-	PUNCT
ejpam-5263	100	21	inner	inner	ADJ
ejpam-5263	100	22	product	product	NOUN
ejpam-5263	100	23	spaces	space	VERB
ejpam-5263	100	24	[	[	X
ejpam-5263	100	25	1	1	NUM
ejpam-5263	100	26	,	,	PUNCT
ejpam-5263	100	27	2	2	NUM
ejpam-5263	100	28	,	,	PUNCT
ejpam-5263	100	29	7	7	NUM
ejpam-5263	100	30	,	,	PUNCT
ejpam-5263	100	31	11	11	NUM
ejpam-5263	100	32	,	,	PUNCT
ejpam-5263	100	33	12	12	NUM
ejpam-5263	100	34	,	,	PUNCT
ejpam-5263	100	35	15	15	NUM
ejpam-5263	100	36	]	]	PUNCT
ejpam-5263	100	37	.	.	PUNCT
ejpam-5263	101	1	just	just	ADV
ejpam-5263	101	2	as	as	SCONJ
ejpam-5263	101	3	the	the	DET
ejpam-5263	101	4	concepts	concept	NOUN
ejpam-5263	101	5	of	of	ADP
ejpam-5263	101	6	orthogonality	orthogonality	NOUN
ejpam-5263	101	7	in	in	ADP
ejpam-5263	101	8	normed	normed	ADJ
ejpam-5263	101	9	spaces	space	NOUN
ejpam-5263	101	10	draw	draw	VERB
ejpam-5263	101	11	inspiration	inspiration	NOUN
ejpam-5263	101	12	from	from	ADP
ejpam-5263	101	13	those	those	PRON
ejpam-5263	101	14	in	in	ADP
ejpam-5263	101	15	inner	inner	ADJ
ejpam-5263	101	16	product	product	NOUN
ejpam-5263	101	17	spaces	space	NOUN
ejpam-5263	101	18	,	,	PUNCT
ejpam-5263	101	19	the	the	DET
ejpam-5263	101	20	notions	notion	NOUN
ejpam-5263	101	21	of	of	ADP
ejpam-5263	101	22	orthogonality	orthogonality	NOUN
ejpam-5263	101	23	in	in	ADP
ejpam-5263	101	24	2	2	NUM
ejpam-5263	101	25	-	-	PUNCT
ejpam-5263	101	26	normed	norme	VERB
ejpam-5263	101	27	spaces	space	NOUN
ejpam-5263	101	28	are	be	AUX
ejpam-5263	101	29	similarly	similarly	ADV
ejpam-5263	101	30	linked	link	VERB
ejpam-5263	101	31	to	to	ADP
ejpam-5263	101	32	those	those	PRON
ejpam-5263	101	33	in	in	ADP
ejpam-5263	101	34	2	2	NUM
ejpam-5263	101	35	-	-	PUNCT
ejpam-5263	101	36	inner	inner	ADJ
ejpam-5263	101	37	product	product	NOUN
ejpam-5263	101	38	spaces	space	VERB
ejpam-5263	101	39	.	.	PUNCT
ejpam-5263	102	1	in	in	ADP
ejpam-5263	102	2	[	[	X
ejpam-5263	102	3	11	11	NUM
ejpam-5263	102	4	]	]	PUNCT
ejpam-5263	102	5	,	,	PUNCT
ejpam-5263	102	6	it	it	PRON
ejpam-5263	102	7	is	be	AUX
ejpam-5263	102	8	shown	show	VERB
ejpam-5263	102	9	that	that	SCONJ
ejpam-5263	102	10	the	the	DET
ejpam-5263	102	11	standard	standard	ADJ
ejpam-5263	102	12	definition	definition	NOUN
ejpam-5263	102	13	of	of	ADP
ejpam-5263	102	14	orthogonality	orthogonality	NOUN
ejpam-5263	102	15	in	in	ADP
ejpam-5263	102	16	a	a	DET
ejpam-5263	102	17	2	2	NUM
ejpam-5263	102	18	-	-	PUNCT
ejpam-5263	102	19	inner	inner	ADJ
ejpam-5263	102	20	product	product	NOUN
ejpam-5263	102	21	space	space	NOUN
ejpam-5263	102	22	(	(	PUNCT
ejpam-5263	102	23	x	x	NOUN
ejpam-5263	102	24	,	,	PUNCT
ejpam-5263	102	25	⟨	⟨	NOUN
ejpam-5263	102	26	·	·	SYM
ejpam-5263	102	27	,	,	PUNCT
ejpam-5263	102	28	·	·	PUNCT
ejpam-5263	102	29	|·⟩	|·⟩	X
ejpam-5263	102	30	)	)	PUNCT
ejpam-5263	102	31	with	with	ADP
ejpam-5263	102	32	dim(x	dim(x	PROPN
ejpam-5263	102	33	)	)	PUNCT
ejpam-5263	102	34	≥	≥	NOUN
ejpam-5263	102	35	3	3	NUM
ejpam-5263	102	36	,	,	PUNCT
ejpam-5263	102	37	is	be	AUX
ejpam-5263	102	38	as	as	SCONJ
ejpam-5263	102	39	follows	follow	VERB
ejpam-5263	102	40	:	:	PUNCT
ejpam-5263	102	41	a.	a.	NOUN
ejpam-5263	102	42	adam	adam	PROPN
ejpam-5263	102	43	,	,	PUNCT
ejpam-5263	102	44	s.	s.	PROPN
ejpam-5263	102	45	rante	rante	PROPN
ejpam-5263	102	46	,	,	PUNCT
ejpam-5263	102	47	h.	h.	PROPN
ejpam-5263	102	48	gunawan	gunawan	PROPN
ejpam-5263	102	49	/	/	PUNCT
ejpam-5263	102	50	eur	eur	PROPN
ejpam-5263	102	51	.	.	PUNCT
ejpam-5263	103	1	j.	j.	PROPN
ejpam-5263	103	2	pure	pure	PROPN
ejpam-5263	103	3	appl	appl	PROPN
ejpam-5263	103	4	.	.	PROPN
ejpam-5263	103	5	math	math	PROPN
ejpam-5263	103	6	,	,	PUNCT
ejpam-5263	103	7	17	17	NUM
ejpam-5263	103	8	(	(	PUNCT
ejpam-5263	103	9	3	3	NUM
ejpam-5263	103	10	)	)	PUNCT
ejpam-5263	103	11	(	(	PUNCT
ejpam-5263	103	12	2024	2024	NUM
ejpam-5263	103	13	)	)	PUNCT
ejpam-5263	103	14	,	,	PUNCT
ejpam-5263	103	15	1937	1937	NUM
ejpam-5263	103	16	-	-	SYM
ejpam-5263	103	17	1947	1947	NUM
ejpam-5263	103	18	1939	1939	NUM
ejpam-5263	103	19	definition	definition	NOUN
ejpam-5263	103	20	1	1	NUM
ejpam-5263	103	21	(	(	PUNCT
ejpam-5263	103	22	g	g	NOUN
ejpam-5263	103	23	-	-	PUNCT
ejpam-5263	103	24	orthogonality	orthogonality	NOUN
ejpam-5263	103	25	in	in	ADP
ejpam-5263	103	26	2	2	NUM
ejpam-5263	103	27	-	-	PUNCT
ejpam-5263	103	28	inner	inner	ADJ
ejpam-5263	103	29	product	product	NOUN
ejpam-5263	103	30	spaces	space	VERB
ejpam-5263	103	31	)	)	PUNCT
ejpam-5263	103	32	.	.	PUNCT
ejpam-5263	104	1	let	let	AUX
ejpam-5263	104	2	(	(	PUNCT
ejpam-5263	104	3	x	x	NOUN
ejpam-5263	104	4	,	,	PUNCT
ejpam-5263	104	5	⟨	⟨	NOUN
ejpam-5263	104	6	·	·	SYM
ejpam-5263	104	7	,	,	PUNCT
ejpam-5263	104	8	·	·	PUNCT
ejpam-5263	104	9	|·⟩	|·⟩	X
ejpam-5263	104	10	)	)	PUNCT
ejpam-5263	104	11	be	be	VERB
ejpam-5263	104	12	a	a	DET
ejpam-5263	104	13	2	2	NUM
ejpam-5263	104	14	-	-	PUNCT
ejpam-5263	104	15	inner	inner	ADJ
ejpam-5263	104	16	product	product	NOUN
ejpam-5263	104	17	spaces	space	VERB
ejpam-5263	104	18	.	.	PUNCT
ejpam-5263	105	1	x1	x1	PROPN
ejpam-5263	105	2	is	be	AUX
ejpam-5263	105	3	g	g	NOUN
ejpam-5263	105	4	-	-	PUNCT
ejpam-5263	105	5	orthogonal	orthogonal	ADJ
ejpam-5263	105	6	to	to	ADP
ejpam-5263	105	7	x2	x2	PRON
ejpam-5263	105	8	if	if	SCONJ
ejpam-5263	106	1	and	and	CCONJ
ejpam-5263	106	2	only	only	ADV
ejpam-5263	106	3	if	if	SCONJ
ejpam-5263	106	4	there	there	PRON
ejpam-5263	106	5	exists	exist	VERB
ejpam-5263	106	6	a	a	DET
ejpam-5263	106	7	subspace	subspace	NOUN
ejpam-5263	106	8	v	v	ADP
ejpam-5263	106	9	of	of	ADP
ejpam-5263	106	10	x	x	PUNCT
ejpam-5263	106	11	with	with	ADP
ejpam-5263	106	12	codim(v	codim(v	ADJ
ejpam-5263	106	13	)	)	PUNCT
ejpam-5263	106	14	=	=	PUNCT
ejpam-5263	107	1	1	1	NUM
ejpam-5263	107	2	such	such	ADJ
ejpam-5263	107	3	that	that	SCONJ
ejpam-5263	107	4	⟨x1	⟨x1	NOUN
ejpam-5263	107	5	,	,	PUNCT
ejpam-5263	107	6	x2|x⟩	x2|x⟩	PUNCT
ejpam-5263	108	1	=	=	SYM
ejpam-5263	108	2	0	0	PROPN
ejpam-5263	108	3	for	for	ADP
ejpam-5263	108	4	all	all	DET
ejpam-5263	108	5	x	x	SYM
ejpam-5263	108	6	∈	∈	NOUN
ejpam-5263	108	7	v	v	NOUN
ejpam-5263	108	8	(	(	PUNCT
ejpam-5263	108	9	denoted	denote	VERB
ejpam-5263	108	10	by	by	ADP
ejpam-5263	108	11	x1⊥gx2	x1⊥gx2	PROPN
ejpam-5263	108	12	)	)	PUNCT
ejpam-5263	108	13	.	.	PUNCT
ejpam-5263	109	1	we	we	PRON
ejpam-5263	109	2	can	can	AUX
ejpam-5263	109	3	say	say	VERB
ejpam-5263	109	4	this	this	DET
ejpam-5263	109	5	definition	definition	NOUN
ejpam-5263	109	6	is	be	AUX
ejpam-5263	109	7	standard	standard	ADJ
ejpam-5263	109	8	because	because	SCONJ
ejpam-5263	109	9	when	when	SCONJ
ejpam-5263	109	10	x	x	PRON
ejpam-5263	109	11	is	be	AUX
ejpam-5263	109	12	a	a	DET
ejpam-5263	109	13	standard	standard	ADJ
ejpam-5263	109	14	2	2	NUM
ejpam-5263	109	15	-	-	PUNCT
ejpam-5263	109	16	inner	inner	ADJ
ejpam-5263	109	17	product	product	NOUN
ejpam-5263	109	18	space	space	NOUN
ejpam-5263	109	19	,	,	PUNCT
ejpam-5263	109	20	we	we	PRON
ejpam-5263	109	21	have	have	VERB
ejpam-5263	109	22	x1⊥x2	x1⊥x2	PROPN
ejpam-5263	110	1	if	if	SCONJ
ejpam-5263	110	2	and	and	CCONJ
ejpam-5263	110	3	only	only	ADV
ejpam-5263	110	4	if	if	SCONJ
ejpam-5263	110	5	x1⊥gx2	x1⊥gx2	PROPN
ejpam-5263	110	6	.	.	PUNCT
ejpam-5263	111	1	the	the	DET
ejpam-5263	111	2	definition	definition	NOUN
ejpam-5263	111	3	of	of	ADP
ejpam-5263	111	4	g	g	NOUN
ejpam-5263	111	5	-	-	PUNCT
ejpam-5263	111	6	orthogonality	orthogonality	NOUN
ejpam-5263	111	7	provided	provide	VERB
ejpam-5263	111	8	above	above	ADV
ejpam-5263	111	9	represents	represent	VERB
ejpam-5263	111	10	an	an	DET
ejpam-5263	111	11	enhancement	enhancement	NOUN
ejpam-5263	111	12	over	over	ADP
ejpam-5263	111	13	the	the	DET
ejpam-5263	111	14	definition	definition	NOUN
ejpam-5263	111	15	proposed	propose	VERB
ejpam-5263	111	16	by	by	ADP
ejpam-5263	111	17	cho	cho	PROPN
ejpam-5263	111	18	and	and	CCONJ
ejpam-5263	111	19	kim	kim	PROPN
ejpam-5263	112	1	[	[	X
ejpam-5263	112	2	2	2	NUM
ejpam-5263	112	3	]	]	PUNCT
ejpam-5263	112	4	and	and	CCONJ
ejpam-5263	112	5	godini	godini	X
ejpam-5263	112	6	[	[	X
ejpam-5263	112	7	7	7	X
ejpam-5263	112	8	]	]	PUNCT
ejpam-5263	112	9	as	as	SCONJ
ejpam-5263	112	10	demonstrated	demonstrate	VERB
ejpam-5263	112	11	in	in	ADP
ejpam-5263	112	12	[	[	X
ejpam-5263	112	13	11	11	NUM
ejpam-5263	112	14	]	]	PUNCT
ejpam-5263	112	15	.	.	PUNCT
ejpam-5263	113	1	on	on	ADP
ejpam-5263	113	2	the	the	DET
ejpam-5263	113	3	other	other	ADJ
ejpam-5263	113	4	hand	hand	NOUN
ejpam-5263	113	5	,	,	PUNCT
ejpam-5263	113	6	cho	cho	PROPN
ejpam-5263	113	7	and	and	CCONJ
ejpam-5263	113	8	kim	kim	PROPN
ejpam-5263	113	9	’s	’s	PART
ejpam-5263	113	10	concept	concept	NOUN
ejpam-5263	113	11	of	of	ADP
ejpam-5263	113	12	orthogonality	orthogonality	NOUN
ejpam-5263	113	13	and	and	CCONJ
ejpam-5263	113	14	godini	godini	PROPN
ejpam-5263	113	15	can	can	AUX
ejpam-5263	113	16	be	be	AUX
ejpam-5263	113	17	seen	see	VERB
ejpam-5263	113	18	as	as	ADP
ejpam-5263	113	19	a	a	DET
ejpam-5263	113	20	development	development	NOUN
ejpam-5263	113	21	of	of	ADP
ejpam-5263	113	22	khan	khan	PROPN
ejpam-5263	113	23	and	and	CCONJ
ejpam-5263	113	24	siddiqui	siddiqui	PROPN
ejpam-5263	113	25	’s	’s	PART
ejpam-5263	113	26	concept	concept	NOUN
ejpam-5263	113	27	of	of	ADP
ejpam-5263	113	28	orthogonality	orthogonality	NOUN
ejpam-5263	113	29	[	[	X
ejpam-5263	113	30	12	12	NUM
ejpam-5263	113	31	]	]	PUNCT
ejpam-5263	113	32	.	.	PUNCT
ejpam-5263	114	1	furthermore	furthermore	ADV
ejpam-5263	114	2	,	,	PUNCT
ejpam-5263	114	3	one	one	PRON
ejpam-5263	114	4	can	can	AUX
ejpam-5263	114	5	define	define	VERB
ejpam-5263	114	6	the	the	DET
ejpam-5263	114	7	notion	notion	NOUN
ejpam-5263	114	8	of	of	ADP
ejpam-5263	114	9	g	g	NOUN
ejpam-5263	114	10	-	-	PUNCT
ejpam-5263	114	11	orthogonality	orthogonality	NOUN
ejpam-5263	114	12	in	in	ADP
ejpam-5263	114	13	n	n	CCONJ
ejpam-5263	114	14	-	-	PUNCT
ejpam-5263	114	15	inner	inner	ADJ
ejpam-5263	114	16	product	product	NOUN
ejpam-5263	114	17	spaces	space	NOUN
ejpam-5263	114	18	as	as	SCONJ
ejpam-5263	114	19	follows	follow	VERB
ejpam-5263	114	20	:	:	PUNCT
ejpam-5263	114	21	definition	definition	NOUN
ejpam-5263	114	22	2	2	NUM
ejpam-5263	114	23	(	(	PUNCT
ejpam-5263	114	24	g	g	NOUN
ejpam-5263	114	25	-	-	PUNCT
ejpam-5263	114	26	orthogonality	orthogonality	NOUN
ejpam-5263	114	27	in	in	ADP
ejpam-5263	114	28	n	n	CCONJ
ejpam-5263	114	29	-	-	PUNCT
ejpam-5263	114	30	inner	inner	ADJ
ejpam-5263	114	31	product	product	NOUN
ejpam-5263	114	32	spaces	space	VERB
ejpam-5263	114	33	)	)	PUNCT
ejpam-5263	114	34	.	.	PUNCT
ejpam-5263	115	1	let	let	VERB
ejpam-5263	115	2	(	(	PUNCT
ejpam-5263	115	3	x	x	NOUN
ejpam-5263	115	4	,	,	PUNCT
ejpam-5263	115	5	⟨	⟨	NOUN
ejpam-5263	115	6	·	·	SYM
ejpam-5263	115	7	,	,	PUNCT
ejpam-5263	115	8	·	·	PUNCT
ejpam-5263	115	9	|	|	ADV
ejpam-5263	115	10	·	·	PUNCT
ejpam-5263	115	11	,	,	PUNCT
ejpam-5263	115	12	.	.	PUNCT
ejpam-5263	115	13	.	.	PUNCT
ejpam-5263	115	14	.	.	PUNCT
ejpam-5263	116	1	,	,	PUNCT
ejpam-5263	116	2	·	·	PUNCT
ejpam-5263	116	3	⟩	⟩	NOUN
ejpam-5263	116	4	)	)	PUNCT
ejpam-5263	116	5	be	be	VERB
ejpam-5263	116	6	an	an	DET
ejpam-5263	116	7	ninner	ninner	NOUN
ejpam-5263	116	8	product	product	NOUN
ejpam-5263	116	9	spaces	space	VERB
ejpam-5263	116	10	with	with	ADP
ejpam-5263	116	11	dim(x	dim(x	PROPN
ejpam-5263	116	12	)	)	PUNCT
ejpam-5263	116	13	≥	≥	NOUN
ejpam-5263	116	14	n	n	NOUN
ejpam-5263	117	1	+	+	NUM
ejpam-5263	117	2	1	1	NUM
ejpam-5263	117	3	.	.	X
ejpam-5263	117	4	x1	x1	PROPN
ejpam-5263	117	5	is	be	AUX
ejpam-5263	117	6	g	g	NOUN
ejpam-5263	117	7	-	-	PUNCT
ejpam-5263	117	8	orthogonal	orthogonal	ADJ
ejpam-5263	117	9	to	to	ADP
ejpam-5263	117	10	x2	x2	PRON
ejpam-5263	117	11	if	if	SCONJ
ejpam-5263	118	1	and	and	CCONJ
ejpam-5263	118	2	only	only	ADV
ejpam-5263	118	3	if	if	SCONJ
ejpam-5263	118	4	there	there	PRON
ejpam-5263	118	5	exists	exist	VERB
ejpam-5263	118	6	a	a	DET
ejpam-5263	118	7	subspace	subspace	NOUN
ejpam-5263	118	8	v	v	ADP
ejpam-5263	118	9	of	of	ADP
ejpam-5263	118	10	x	x	PUNCT
ejpam-5263	118	11	with	with	ADP
ejpam-5263	118	12	codim(v	codim(v	ADJ
ejpam-5263	118	13	)	)	PUNCT
ejpam-5263	118	14	=	=	PUNCT
ejpam-5263	119	1	1	1	NUM
ejpam-5263	119	2	such	such	ADJ
ejpam-5263	119	3	that	that	SCONJ
ejpam-5263	119	4	⟨x1	⟨x1	NOUN
ejpam-5263	119	5	,	,	PUNCT
ejpam-5263	119	6	x2|x3	x2|x3	NUM
ejpam-5263	119	7	,	,	PUNCT
ejpam-5263	119	8	.	.	PUNCT
ejpam-5263	119	9	.	.	PUNCT
ejpam-5263	119	10	.	.	PUNCT
ejpam-5263	120	1	,	,	PUNCT
ejpam-5263	120	2	xn+1⟩	xn+1⟩	X
ejpam-5263	120	3	=	=	SYM
ejpam-5263	120	4	0	0	NUM
ejpam-5263	120	5	for	for	ADP
ejpam-5263	120	6	all	all	DET
ejpam-5263	120	7	x3	x3	ADJ
ejpam-5263	120	8	,	,	PUNCT
ejpam-5263	120	9	.	.	PUNCT
ejpam-5263	120	10	.	.	PUNCT
ejpam-5263	121	1	.	.	PUNCT
ejpam-5263	122	1	,	,	PUNCT
ejpam-5263	122	2	xn+1	xn+1	PROPN
ejpam-5263	122	3	∈	∈	PROPN
ejpam-5263	122	4	v	v	NOUN
ejpam-5263	122	5	(	(	PUNCT
ejpam-5263	122	6	denoted	denote	VERB
ejpam-5263	122	7	by	by	ADP
ejpam-5263	122	8	x1⊥gx2	x1⊥gx2	PROPN
ejpam-5263	122	9	)	)	PUNCT
ejpam-5263	122	10	.	.	PUNCT
ejpam-5263	123	1	with	with	ADP
ejpam-5263	123	2	this	this	DET
ejpam-5263	123	3	definition	definition	NOUN
ejpam-5263	123	4	,	,	PUNCT
ejpam-5263	123	5	in	in	ADP
ejpam-5263	123	6	a	a	DET
ejpam-5263	123	7	standard	standard	ADJ
ejpam-5263	123	8	n	n	CCONJ
ejpam-5263	123	9	-	-	PUNCT
ejpam-5263	123	10	inner	inner	ADJ
ejpam-5263	123	11	product	product	NOUN
ejpam-5263	123	12	space	space	NOUN
ejpam-5263	123	13	,	,	PUNCT
ejpam-5263	123	14	g	g	NOUN
ejpam-5263	123	15	-	-	PUNCT
ejpam-5263	123	16	orthogonality	orthogonality	NOUN
ejpam-5263	123	17	is	be	AUX
ejpam-5263	123	18	also	also	ADV
ejpam-5263	123	19	equivalent	equivalent	ADJ
ejpam-5263	123	20	to	to	ADP
ejpam-5263	123	21	the	the	DET
ejpam-5263	123	22	usual	usual	ADJ
ejpam-5263	123	23	orthogonality	orthogonality	NOUN
ejpam-5263	123	24	(	(	PUNCT
ejpam-5263	123	25	with	with	ADP
ejpam-5263	123	26	respect	respect	NOUN
ejpam-5263	123	27	to	to	ADP
ejpam-5263	123	28	the	the	DET
ejpam-5263	123	29	inner	inner	ADJ
ejpam-5263	123	30	product	product	NOUN
ejpam-5263	123	31	)	)	PUNCT
ejpam-5263	123	32	.	.	PUNCT
ejpam-5263	124	1	in	in	ADP
ejpam-5263	124	2	other	other	ADJ
ejpam-5263	124	3	words	word	NOUN
ejpam-5263	124	4	,	,	PUNCT
ejpam-5263	124	5	x1⊥x2	x1⊥x2	PROPN
ejpam-5263	125	1	if	if	SCONJ
ejpam-5263	125	2	and	and	CCONJ
ejpam-5263	125	3	only	only	ADV
ejpam-5263	125	4	if	if	SCONJ
ejpam-5263	125	5	x1⊥gx2	x1⊥gx2	PROPN
ejpam-5263	125	6	.	.	PUNCT
ejpam-5263	126	1	we	we	PRON
ejpam-5263	126	2	can	can	AUX
ejpam-5263	126	3	see	see	VERB
ejpam-5263	126	4	that	that	SCONJ
ejpam-5263	126	5	the	the	DET
ejpam-5263	126	6	definition	definition	NOUN
ejpam-5263	126	7	requires	require	VERB
ejpam-5263	126	8	the	the	DET
ejpam-5263	126	9	assumption	assumption	NOUN
ejpam-5263	126	10	that	that	SCONJ
ejpam-5263	126	11	the	the	DET
ejpam-5263	126	12	dimension	dimension	NOUN
ejpam-5263	126	13	of	of	ADP
ejpam-5263	126	14	x	x	PUNCT
ejpam-5263	126	15	is	be	AUX
ejpam-5263	126	16	greater	great	ADJ
ejpam-5263	126	17	than	than	ADP
ejpam-5263	126	18	n.	n.	NOUN
ejpam-5263	126	19	in	in	ADP
ejpam-5263	126	20	[	[	PUNCT
ejpam-5263	126	21	11	11	NUM
ejpam-5263	126	22	]	]	PUNCT
ejpam-5263	126	23	,	,	PUNCT
ejpam-5263	126	24	it	it	PRON
ejpam-5263	126	25	is	be	AUX
ejpam-5263	126	26	shown	show	VERB
ejpam-5263	126	27	that	that	SCONJ
ejpam-5263	126	28	if	if	SCONJ
ejpam-5263	126	29	we	we	PRON
ejpam-5263	126	30	define	define	VERB
ejpam-5263	126	31	g	g	NOUN
ejpam-5263	126	32	-	-	PUNCT
ejpam-5263	126	33	orthogonality	orthogonality	NOUN
ejpam-5263	126	34	in	in	ADP
ejpam-5263	126	35	the	the	DET
ejpam-5263	126	36	standard	standard	ADJ
ejpam-5263	126	37	2	2	NUM
ejpam-5263	126	38	-	-	PUNCT
ejpam-5263	126	39	inner	inner	ADJ
ejpam-5263	126	40	product	product	NOUN
ejpam-5263	126	41	space	space	NOUN
ejpam-5263	126	42	x	x	PUNCT
ejpam-5263	126	43	of	of	ADP
ejpam-5263	126	44	dimension	dimension	NOUN
ejpam-5263	126	45	2	2	NUM
ejpam-5263	126	46	,	,	PUNCT
ejpam-5263	126	47	any	any	DET
ejpam-5263	126	48	pair	pair	NOUN
ejpam-5263	126	49	of	of	ADP
ejpam-5263	126	50	linearly	linearly	ADV
ejpam-5263	126	51	independent	independent	ADJ
ejpam-5263	126	52	vectors	vector	NOUN
ejpam-5263	126	53	becomes	become	VERB
ejpam-5263	126	54	g	g	NOUN
ejpam-5263	126	55	-	-	PUNCT
ejpam-5263	126	56	orthogonal	orthogonal	NOUN
ejpam-5263	126	57	.	.	PUNCT
ejpam-5263	127	1	similarly	similarly	ADV
ejpam-5263	127	2	,	,	PUNCT
ejpam-5263	127	3	if	if	SCONJ
ejpam-5263	127	4	we	we	PRON
ejpam-5263	127	5	define	define	VERB
ejpam-5263	127	6	g	g	NOUN
ejpam-5263	127	7	-	-	PUNCT
ejpam-5263	127	8	orthogonality	orthogonality	NOUN
ejpam-5263	127	9	for	for	ADP
ejpam-5263	127	10	a	a	DET
ejpam-5263	127	11	standard	standard	ADJ
ejpam-5263	127	12	n	n	CCONJ
ejpam-5263	127	13	-	-	PUNCT
ejpam-5263	127	14	inner	inner	ADJ
ejpam-5263	127	15	product	product	NOUN
ejpam-5263	127	16	space	space	NOUN
ejpam-5263	127	17	x	x	PUNCT
ejpam-5263	127	18	of	of	ADP
ejpam-5263	127	19	dimension	dimension	NOUN
ejpam-5263	127	20	n	n	CCONJ
ejpam-5263	127	21	,	,	PUNCT
ejpam-5263	127	22	any	any	DET
ejpam-5263	127	23	pair	pair	NOUN
ejpam-5263	127	24	of	of	ADP
ejpam-5263	127	25	linearly	linearly	ADV
ejpam-5263	127	26	independent	independent	ADJ
ejpam-5263	127	27	vectors	vector	NOUN
ejpam-5263	127	28	also	also	ADV
ejpam-5263	127	29	becomes	become	VERB
ejpam-5263	127	30	g	g	NOUN
ejpam-5263	127	31	-	-	PUNCT
ejpam-5263	127	32	orthogonal	orthogonal	NOUN
ejpam-5263	127	33	.	.	PUNCT
ejpam-5263	128	1	indeed	indeed	ADV
ejpam-5263	128	2	,	,	PUNCT
ejpam-5263	128	3	this	this	DET
ejpam-5263	128	4	fact	fact	NOUN
ejpam-5263	128	5	is	be	AUX
ejpam-5263	128	6	undesirable	undesirable	ADJ
ejpam-5263	128	7	.	.	PUNCT
ejpam-5263	129	1	it	it	PRON
ejpam-5263	129	2	is	be	AUX
ejpam-5263	129	3	necessary	necessary	ADJ
ejpam-5263	129	4	to	to	PART
ejpam-5263	129	5	adopt	adopt	VERB
ejpam-5263	129	6	a	a	DET
ejpam-5263	129	7	different	different	ADJ
ejpam-5263	129	8	approach	approach	NOUN
ejpam-5263	129	9	to	to	PART
ejpam-5263	129	10	establish	establish	VERB
ejpam-5263	129	11	orthogonality	orthogonality	NOUN
ejpam-5263	129	12	in	in	ADP
ejpam-5263	129	13	n	n	CCONJ
ejpam-5263	129	14	-	-	PUNCT
ejpam-5263	129	15	inner	inner	ADJ
ejpam-5263	129	16	product	product	NOUN
ejpam-5263	129	17	spaces	space	NOUN
ejpam-5263	129	18	of	of	ADP
ejpam-5263	129	19	dimension	dimension	NOUN
ejpam-5263	129	20	n	n	CCONJ
ejpam-5263	129	21	in	in	ADP
ejpam-5263	129	22	a	a	DET
ejpam-5263	129	23	general	general	ADJ
ejpam-5263	129	24	sense	sense	NOUN
ejpam-5263	129	25	.	.	PUNCT
ejpam-5263	130	1	meanwhile	meanwhile	ADV
ejpam-5263	130	2	,	,	PUNCT
ejpam-5263	130	3	note	note	VERB
ejpam-5263	130	4	that	that	SCONJ
ejpam-5263	130	5	if	if	SCONJ
ejpam-5263	130	6	(	(	PUNCT
ejpam-5263	130	7	x	x	NOUN
ejpam-5263	130	8	,	,	PUNCT
ejpam-5263	130	9	⟨	⟨	NOUN
ejpam-5263	130	10	·	·	SYM
ejpam-5263	130	11	,	,	PUNCT
ejpam-5263	130	12	·	·	PUNCT
ejpam-5263	130	13	|	|	ADV
ejpam-5263	130	14	·	·	PUNCT
ejpam-5263	130	15	,	,	PUNCT
ejpam-5263	130	16	.	.	PUNCT
ejpam-5263	130	17	.	.	PUNCT
ejpam-5263	131	1	.	.	PUNCT
ejpam-5263	132	1	,	,	PUNCT
ejpam-5263	132	2	·	·	PUNCT
ejpam-5263	132	3	⟩	⟩	NOUN
ejpam-5263	132	4	)	)	PUNCT
ejpam-5263	132	5	is	be	AUX
ejpam-5263	132	6	an	an	DET
ejpam-5263	132	7	arbitrary	arbitrary	ADJ
ejpam-5263	132	8	n	n	CCONJ
ejpam-5263	132	9	-	-	PUNCT
ejpam-5263	132	10	inner	inner	ADJ
ejpam-5263	132	11	product	product	NOUN
ejpam-5263	132	12	space	space	NOUN
ejpam-5263	132	13	and	and	CCONJ
ejpam-5263	132	14	a	a	DET
ejpam-5263	132	15	=	=	NOUN
ejpam-5263	132	16	{	{	PUNCT
ejpam-5263	132	17	a1	a1	PROPN
ejpam-5263	132	18	,	,	PUNCT
ejpam-5263	132	19	a2	a2	PROPN
ejpam-5263	132	20	,	,	PUNCT
ejpam-5263	132	21	.	.	PUNCT
ejpam-5263	132	22	.	.	PUNCT
ejpam-5263	133	1	.	.	PUNCT
ejpam-5263	134	1	,	,	PUNCT
ejpam-5263	134	2	an	an	PRON
ejpam-5263	134	3	}	}	PUNCT
ejpam-5263	134	4	is	be	AUX
ejpam-5263	134	5	a	a	DET
ejpam-5263	134	6	set	set	NOUN
ejpam-5263	134	7	of	of	ADP
ejpam-5263	134	8	n	n	CCONJ
ejpam-5263	134	9	linearly	linearly	ADV
ejpam-5263	134	10	independent	independent	ADJ
ejpam-5263	134	11	vectors	vector	NOUN
ejpam-5263	134	12	in	in	ADP
ejpam-5263	134	13	x	x	NOUN
ejpam-5263	134	14	,	,	PUNCT
ejpam-5263	134	15	then	then	ADV
ejpam-5263	134	16	one	one	PRON
ejpam-5263	134	17	may	may	AUX
ejpam-5263	134	18	observe	observe	VERB
ejpam-5263	134	19	that	that	PRON
ejpam-5263	134	20	⟨x	⟨x	VERB
ejpam-5263	134	21	,	,	PUNCT
ejpam-5263	134	22	y⟩a	y⟩a	NUM
ejpam-5263	134	23	:	:	PUNCT
ejpam-5263	134	24	=	=	SYM
ejpam-5263	134	25	∑	∑	PUNCT
ejpam-5263	134	26	{	{	PUNCT
ejpam-5263	134	27	i2,	i2,	NOUN
ejpam-5263	134	28	...	...	PUNCT
ejpam-5263	134	29	in}⊂{1,2,	in}⊂{1,2,	NOUN
ejpam-5263	134	30	...	...	PUNCT
ejpam-5263	134	31	,n	,n	NOUN
ejpam-5263	134	32	}	}	PUNCT
ejpam-5263	134	33	⟨x	⟨x	VERB
ejpam-5263	134	34	,	,	PUNCT
ejpam-5263	134	35	y|ai2	y|ai2	NOUN
ejpam-5263	134	36	,	,	PUNCT
ejpam-5263	134	37	.	.	PUNCT
ejpam-5263	134	38	.	.	PUNCT
ejpam-5263	135	1	.	.	PUNCT
ejpam-5263	136	1	,	,	PUNCT
ejpam-5263	136	2	ain⟩	ain⟩	PROPN
ejpam-5263	136	3	defines	define	VERB
ejpam-5263	136	4	an	an	DET
ejpam-5263	136	5	inner	inner	ADJ
ejpam-5263	136	6	product	product	NOUN
ejpam-5263	136	7	on	on	ADP
ejpam-5263	136	8	x.	x.	NOUN
ejpam-5263	136	9	there	there	PRON
ejpam-5263	136	10	are	be	VERB
ejpam-5263	136	11	n	n	PRON
ejpam-5263	136	12	terms	term	NOUN
ejpam-5263	136	13	in	in	ADP
ejpam-5263	136	14	the	the	DET
ejpam-5263	136	15	above	above	ADJ
ejpam-5263	136	16	sum	sum	NOUN
ejpam-5263	136	17	,	,	PUNCT
ejpam-5263	136	18	as	as	SCONJ
ejpam-5263	136	19	there	there	PRON
ejpam-5263	136	20	are	be	VERB
ejpam-5263	136	21	n	n	DET
ejpam-5263	136	22	subsets	subset	NOUN
ejpam-5263	136	23	of	of	ADP
ejpam-5263	136	24	{	{	PUNCT
ejpam-5263	136	25	1	1	NUM
ejpam-5263	136	26	,	,	PUNCT
ejpam-5263	136	27	2	2	NUM
ejpam-5263	136	28	,	,	PUNCT
ejpam-5263	136	29	.	.	PUNCT
ejpam-5263	136	30	.	.	PUNCT
ejpam-5263	137	1	.	.	PUNCT
ejpam-5263	138	1	,	,	PUNCT
ejpam-5263	138	2	n	n	CCONJ
ejpam-5263	138	3	}	}	PUNCT
ejpam-5263	138	4	consisting	consist	VERB
ejpam-5263	138	5	of	of	ADP
ejpam-5263	138	6	n−	n−	NOUN
ejpam-5263	138	7	1	1	NUM
ejpam-5263	138	8	elements	element	NOUN
ejpam-5263	138	9	.	.	PUNCT
ejpam-5263	139	1	if	if	SCONJ
ejpam-5263	139	2	dimx	dimx	NOUN
ejpam-5263	139	3	=	=	SYM
ejpam-5263	139	4	d	d	X
ejpam-5263	139	5	<	<	X
ejpam-5263	139	6	∞	∞	PROPN
ejpam-5263	139	7	,	,	PUNCT
ejpam-5263	139	8	we	we	PRON
ejpam-5263	139	9	can	can	AUX
ejpam-5263	139	10	also	also	ADV
ejpam-5263	139	11	define	define	VERB
ejpam-5263	139	12	an	an	DET
ejpam-5263	139	13	inner	inner	ADJ
ejpam-5263	139	14	product	product	NOUN
ejpam-5263	139	15	by	by	ADP
ejpam-5263	139	16	the	the	DET
ejpam-5263	139	17	above	above	ADJ
ejpam-5263	139	18	formula	formula	NOUN
ejpam-5263	139	19	using	use	VERB
ejpam-5263	139	20	a	a	DET
ejpam-5263	139	21	set	set	NOUN
ejpam-5263	139	22	of	of	ADP
ejpam-5263	139	23	d	d	X
ejpam-5263	139	24	linearly	linearly	ADV
ejpam-5263	139	25	independent	independent	ADJ
ejpam-5263	139	26	vectors	vector	NOUN
ejpam-5263	139	27	in	in	ADP
ejpam-5263	139	28	x	x	PROPN
ejpam-5263	139	29	(	(	PUNCT
ejpam-5263	139	30	see	see	VERB
ejpam-5263	139	31	[	[	X
ejpam-5263	139	32	8	8	NUM
ejpam-5263	139	33	]	]	NUM
ejpam-5263	139	34	)	)	PUNCT
ejpam-5263	139	35	.	.	PUNCT
ejpam-5263	140	1	thus	thus	ADV
ejpam-5263	140	2	,	,	PUNCT
ejpam-5263	140	3	starting	start	VERB
ejpam-5263	140	4	from	from	ADP
ejpam-5263	140	5	an	an	DET
ejpam-5263	140	6	inner	inner	ADJ
ejpam-5263	140	7	product	product	NOUN
ejpam-5263	140	8	,	,	PUNCT
ejpam-5263	140	9	we	we	PRON
ejpam-5263	140	10	can	can	AUX
ejpam-5263	140	11	define	define	VERB
ejpam-5263	140	12	the	the	DET
ejpam-5263	140	13	standard	standard	ADJ
ejpam-5263	140	14	n	n	CCONJ
ejpam-5263	140	15	-	-	PUNCT
ejpam-5263	140	16	inner	inner	ADJ
ejpam-5263	140	17	product	product	NOUN
ejpam-5263	140	18	,	,	PUNCT
ejpam-5263	140	19	and	and	CCONJ
ejpam-5263	140	20	then	then	ADV
ejpam-5263	140	21	from	from	ADP
ejpam-5263	140	22	the	the	DET
ejpam-5263	140	23	n	n	CCONJ
ejpam-5263	140	24	-	-	PUNCT
ejpam-5263	140	25	inner	inner	ADJ
ejpam-5263	140	26	product	product	NOUN
ejpam-5263	140	27	,	,	PUNCT
ejpam-5263	140	28	we	we	PRON
ejpam-5263	140	29	can	can	AUX
ejpam-5263	140	30	derive	derive	VERB
ejpam-5263	140	31	a	a	DET
ejpam-5263	140	32	new	new	ADJ
ejpam-5263	140	33	inner	inner	ADJ
ejpam-5263	140	34	product	product	NOUN
ejpam-5263	140	35	.	.	PUNCT
ejpam-5263	141	1	it	it	PRON
ejpam-5263	141	2	is	be	AUX
ejpam-5263	141	3	then	then	ADV
ejpam-5263	141	4	interesting	interesting	ADJ
ejpam-5263	141	5	to	to	PART
ejpam-5263	141	6	investigate	investigate	VERB
ejpam-5263	141	7	how	how	SCONJ
ejpam-5263	141	8	the	the	DET
ejpam-5263	141	9	new	new	ADJ
ejpam-5263	141	10	inner	inner	ADJ
ejpam-5263	141	11	product	product	NOUN
ejpam-5263	141	12	derived	derive	VERB
ejpam-5263	141	13	from	from	ADP
ejpam-5263	141	14	standard	standard	ADJ
ejpam-5263	141	15	n	n	CCONJ
ejpam-5263	141	16	-	-	PUNCT
ejpam-5263	141	17	inner	inner	ADJ
ejpam-5263	141	18	product	product	NOUN
ejpam-5263	141	19	relates	relate	VERB
ejpam-5263	141	20	to	to	ADP
ejpam-5263	141	21	the	the	DET
ejpam-5263	141	22	original	original	ADJ
ejpam-5263	141	23	inner	inner	ADJ
ejpam-5263	141	24	product	product	NOUN
ejpam-5263	141	25	on	on	ADP
ejpam-5263	141	26	(	(	PUNCT
ejpam-5263	141	27	x	x	X
ejpam-5263	141	28	,	,	PUNCT
ejpam-5263	141	29	⟨	⟨	NOUN
ejpam-5263	141	30	·	·	SYM
ejpam-5263	141	31	,	,	PUNCT
ejpam-5263	141	32	·	·	PUNCT
ejpam-5263	141	33	⟩	⟩	NOUN
ejpam-5263	141	34	)	)	PUNCT
ejpam-5263	141	35	.	.	PUNCT
ejpam-5263	142	1	in	in	ADP
ejpam-5263	142	2	particular	particular	ADJ
ejpam-5263	142	3	,	,	PUNCT
ejpam-5263	142	4	we	we	PRON
ejpam-5263	142	5	would	would	AUX
ejpam-5263	142	6	like	like	VERB
ejpam-5263	142	7	to	to	PART
ejpam-5263	142	8	know	know	VERB
ejpam-5263	142	9	whether	whether	SCONJ
ejpam-5263	142	10	a.	a.	NOUN
ejpam-5263	142	11	adam	adam	PROPN
ejpam-5263	142	12	,	,	PUNCT
ejpam-5263	142	13	s.	s.	PROPN
ejpam-5263	142	14	rante	rante	PROPN
ejpam-5263	142	15	,	,	PUNCT
ejpam-5263	142	16	h.	h.	PROPN
ejpam-5263	142	17	gunawan	gunawan	PROPN
ejpam-5263	142	18	/	/	PUNCT
ejpam-5263	142	19	eur	eur	PROPN
ejpam-5263	142	20	.	.	PUNCT
ejpam-5263	143	1	j.	j.	PROPN
ejpam-5263	143	2	pure	pure	PROPN
ejpam-5263	143	3	appl	appl	PROPN
ejpam-5263	143	4	.	.	PROPN
ejpam-5263	143	5	math	math	PROPN
ejpam-5263	143	6	,	,	PUNCT
ejpam-5263	143	7	17	17	NUM
ejpam-5263	143	8	(	(	PUNCT
ejpam-5263	143	9	3	3	NUM
ejpam-5263	143	10	)	)	PUNCT
ejpam-5263	143	11	(	(	PUNCT
ejpam-5263	143	12	2024	2024	NUM
ejpam-5263	143	13	)	)	PUNCT
ejpam-5263	143	14	,	,	PUNCT
ejpam-5263	143	15	1937	1937	NUM
ejpam-5263	143	16	-	-	SYM
ejpam-5263	143	17	1947	1947	NUM
ejpam-5263	143	18	1940	1940	NUM
ejpam-5263	143	19	or	or	CCONJ
ejpam-5263	143	20	not	not	PART
ejpam-5263	143	21	the	the	DET
ejpam-5263	143	22	new	new	ADJ
ejpam-5263	143	23	inner	inner	ADJ
ejpam-5263	143	24	product	product	NOUN
ejpam-5263	143	25	preserves	preserve	VERB
ejpam-5263	143	26	orthogonality	orthogonality	NOUN
ejpam-5263	143	27	.	.	PUNCT
ejpam-5263	144	1	this	this	PRON
ejpam-5263	144	2	generally	generally	ADV
ejpam-5263	144	3	depends	depend	VERB
ejpam-5263	144	4	on	on	ADP
ejpam-5263	144	5	the	the	DET
ejpam-5263	144	6	set	set	NOUN
ejpam-5263	144	7	a	a	PRON
ejpam-5263	144	8	that	that	SCONJ
ejpam-5263	144	9	we	we	PRON
ejpam-5263	144	10	choose	choose	VERB
ejpam-5263	144	11	in	in	ADP
ejpam-5263	144	12	the	the	DET
ejpam-5263	144	13	definition	definition	NOUN
ejpam-5263	144	14	of	of	ADP
ejpam-5263	144	15	the	the	DET
ejpam-5263	144	16	new	new	ADJ
ejpam-5263	144	17	inner	inner	ADJ
ejpam-5263	144	18	product	product	NOUN
ejpam-5263	144	19	.	.	PUNCT
ejpam-5263	145	1	in	in	ADP
ejpam-5263	145	2	the	the	DET
ejpam-5263	145	3	next	next	ADJ
ejpam-5263	145	4	sections	section	NOUN
ejpam-5263	145	5	,	,	PUNCT
ejpam-5263	145	6	we	we	PRON
ejpam-5263	145	7	present	present	VERB
ejpam-5263	145	8	necessary	necessary	ADJ
ejpam-5263	145	9	and	and	CCONJ
ejpam-5263	145	10	sufficient	sufficient	ADJ
ejpam-5263	145	11	conditions	condition	NOUN
ejpam-5263	145	12	for	for	SCONJ
ejpam-5263	145	13	the	the	DET
ejpam-5263	145	14	set	set	NOUN
ejpam-5263	145	15	a	a	PRON
ejpam-5263	145	16	to	to	PART
ejpam-5263	145	17	give	give	VERB
ejpam-5263	145	18	the	the	DET
ejpam-5263	145	19	positive	positive	ADJ
ejpam-5263	145	20	answer	answer	NOUN
ejpam-5263	145	21	.	.	PUNCT
ejpam-5263	146	1	with	with	ADP
ejpam-5263	146	2	this	this	DET
ejpam-5263	146	3	approach	approach	NOUN
ejpam-5263	146	4	,	,	PUNCT
ejpam-5263	146	5	we	we	PRON
ejpam-5263	146	6	have	have	VERB
ejpam-5263	146	7	an	an	DET
ejpam-5263	146	8	alternative	alternative	ADJ
ejpam-5263	146	9	way	way	NOUN
ejpam-5263	146	10	to	to	PART
ejpam-5263	146	11	establish	establish	VERB
ejpam-5263	146	12	the	the	DET
ejpam-5263	146	13	orthogonality	orthogonality	NOUN
ejpam-5263	146	14	of	of	ADP
ejpam-5263	146	15	two	two	NUM
ejpam-5263	146	16	vectors	vector	NOUN
ejpam-5263	146	17	in	in	ADP
ejpam-5263	146	18	arbitrary	arbitrary	ADJ
ejpam-5263	146	19	n	n	CCONJ
ejpam-5263	146	20	-	-	PUNCT
ejpam-5263	146	21	inner	inner	ADJ
ejpam-5263	146	22	product	product	NOUN
ejpam-5263	146	23	spaces	space	VERB
ejpam-5263	146	24	because	because	SCONJ
ejpam-5263	146	25	we	we	PRON
ejpam-5263	146	26	can	can	AUX
ejpam-5263	146	27	define	define	VERB
ejpam-5263	146	28	the	the	DET
ejpam-5263	146	29	inner	inner	ADJ
ejpam-5263	146	30	product	product	NOUN
ejpam-5263	146	31	on	on	ADP
ejpam-5263	146	32	n	n	CCONJ
ejpam-5263	146	33	-	-	PUNCT
ejpam-5263	146	34	inner	inner	ADJ
ejpam-5263	146	35	product	product	NOUN
ejpam-5263	146	36	spaces	space	VERB
ejpam-5263	146	37	.	.	PUNCT
ejpam-5263	147	1	2	2	X
ejpam-5263	147	2	.	.	X
ejpam-5263	147	3	the	the	DET
ejpam-5263	147	4	n	n	ADV
ejpam-5263	147	5	-	-	PUNCT
ejpam-5263	147	6	dimensional	dimensional	ADJ
ejpam-5263	147	7	case	case	NOUN
ejpam-5263	147	8	let	let	VERB
ejpam-5263	147	9	(	(	PUNCT
ejpam-5263	147	10	x	x	NOUN
ejpam-5263	147	11	,	,	PUNCT
ejpam-5263	147	12	⟨	⟨	NOUN
ejpam-5263	147	13	·	·	SYM
ejpam-5263	147	14	,	,	PUNCT
ejpam-5263	147	15	·	·	PUNCT
ejpam-5263	147	16	|	|	ADV
ejpam-5263	147	17	·	·	PUNCT
ejpam-5263	147	18	,	,	PUNCT
ejpam-5263	147	19	.	.	PUNCT
ejpam-5263	147	20	.	.	PUNCT
ejpam-5263	147	21	.	.	PUNCT
ejpam-5263	148	1	,	,	PUNCT
ejpam-5263	148	2	·	·	PUNCT
ejpam-5263	148	3	⟩	⟩	NOUN
ejpam-5263	148	4	)	)	PUNCT
ejpam-5263	148	5	be	be	VERB
ejpam-5263	148	6	a	a	DET
ejpam-5263	148	7	standard	standard	ADJ
ejpam-5263	148	8	n	n	CCONJ
ejpam-5263	148	9	-	-	PUNCT
ejpam-5263	148	10	inner	inner	ADJ
ejpam-5263	148	11	product	product	NOUN
ejpam-5263	148	12	space	space	NOUN
ejpam-5263	148	13	.	.	PUNCT
ejpam-5263	149	1	as	as	SCONJ
ejpam-5263	149	2	indicated	indicate	VERB
ejpam-5263	149	3	in	in	ADP
ejpam-5263	149	4	[	[	X
ejpam-5263	149	5	10	10	NUM
ejpam-5263	149	6	]	]	PUNCT
ejpam-5263	149	7	and	and	CCONJ
ejpam-5263	149	8	[	[	X
ejpam-5263	149	9	11	11	NUM
ejpam-5263	149	10	]	]	PUNCT
ejpam-5263	149	11	,	,	PUNCT
ejpam-5263	149	12	the	the	DET
ejpam-5263	149	13	n	n	CCONJ
ejpam-5263	149	14	-	-	PUNCT
ejpam-5263	149	15	dimensional	dimensional	ADJ
ejpam-5263	149	16	case	case	NOUN
ejpam-5263	149	17	is	be	AUX
ejpam-5263	149	18	special	special	ADJ
ejpam-5263	149	19	.	.	PUNCT
ejpam-5263	150	1	so	so	ADV
ejpam-5263	150	2	we	we	PRON
ejpam-5263	150	3	shall	shall	AUX
ejpam-5263	150	4	first	first	ADV
ejpam-5263	150	5	pay	pay	VERB
ejpam-5263	150	6	attention	attention	NOUN
ejpam-5263	150	7	to	to	ADP
ejpam-5263	150	8	the	the	DET
ejpam-5263	150	9	case	case	NOUN
ejpam-5263	150	10	where	where	SCONJ
ejpam-5263	150	11	dimx	dimx	NOUN
ejpam-5263	150	12	=	=	PUNCT
ejpam-5263	150	13	n.	n.	NOUN
ejpam-5263	150	14	our	our	PRON
ejpam-5263	150	15	results	result	NOUN
ejpam-5263	150	16	are	be	AUX
ejpam-5263	150	17	the	the	DET
ejpam-5263	150	18	following	follow	VERB
ejpam-5263	150	19	theorems	theorem	NOUN
ejpam-5263	150	20	.	.	PUNCT
ejpam-5263	151	1	theorem	theorem	NOUN
ejpam-5263	151	2	1	1	NUM
ejpam-5263	151	3	.	.	PUNCT
ejpam-5263	152	1	let	let	VERB
ejpam-5263	152	2	a	a	DET
ejpam-5263	152	3	=	=	PUNCT
ejpam-5263	152	4	{	{	PUNCT
ejpam-5263	152	5	a1	a1	PROPN
ejpam-5263	152	6	,	,	PUNCT
ejpam-5263	152	7	a2	a2	PROPN
ejpam-5263	152	8	,	,	PUNCT
ejpam-5263	152	9	.	.	PUNCT
ejpam-5263	152	10	.	.	PUNCT
ejpam-5263	153	1	.	.	PUNCT
ejpam-5263	154	1	,	,	PUNCT
ejpam-5263	154	2	an	an	PRON
ejpam-5263	154	3	}	}	PUNCT
ejpam-5263	154	4	⊂	⊂	PROPN
ejpam-5263	154	5	x	x	PUNCT
ejpam-5263	154	6	be	be	AUX
ejpam-5263	154	7	an	an	DET
ejpam-5263	154	8	orthogonal	orthogonal	ADJ
ejpam-5263	154	9	set	set	VERB
ejpam-5263	154	10	with	with	ADP
ejpam-5263	154	11	∥ai∥	∥ai∥	NOUN
ejpam-5263	155	1	=	=	SYM
ejpam-5263	155	2	α	α	X
ejpam-5263	155	3	>	>	X
ejpam-5263	155	4	0	0	PUNCT
ejpam-5263	156	1	for	for	ADP
ejpam-5263	156	2	all	all	DET
ejpam-5263	156	3	i	i	PRON
ejpam-5263	156	4	=	=	NOUN
ejpam-5263	156	5	1	1	NUM
ejpam-5263	156	6	,	,	PUNCT
ejpam-5263	156	7	2	2	NUM
ejpam-5263	156	8	,	,	PUNCT
ejpam-5263	156	9	.	.	PUNCT
ejpam-5263	156	10	.	.	PUNCT
ejpam-5263	156	11	.	.	PUNCT
ejpam-5263	157	1	,	,	PUNCT
ejpam-5263	157	2	n.	n.	PROPN
ejpam-5263	157	3	then	then	ADV
ejpam-5263	157	4	⟨x	⟨x	VERB
ejpam-5263	157	5	,	,	PUNCT
ejpam-5263	157	6	y⟩a	y⟩a	NUM
ejpam-5263	157	7	=	=	SYM
ejpam-5263	157	8	0	0	PUNCT
ejpam-5263	158	1	if	if	SCONJ
ejpam-5263	158	2	and	and	CCONJ
ejpam-5263	158	3	only	only	ADV
ejpam-5263	158	4	if	if	SCONJ
ejpam-5263	158	5	⟨x	⟨x	VERB
ejpam-5263	158	6	,	,	PUNCT
ejpam-5263	158	7	y⟩	y⟩	NOUN
ejpam-5263	158	8	=	=	NOUN
ejpam-5263	158	9	0	0	NUM
ejpam-5263	158	10	for	for	ADP
ejpam-5263	158	11	all	all	DET
ejpam-5263	158	12	x	x	NOUN
ejpam-5263	158	13	,	,	PUNCT
ejpam-5263	158	14	y	y	PROPN
ejpam-5263	158	15	∈	∈	PROPN
ejpam-5263	158	16	x.	x.	NOUN
ejpam-5263	158	17	proof	proof	NOUN
ejpam-5263	158	18	.	.	PUNCT
ejpam-5263	159	1	for	for	ADP
ejpam-5263	159	2	any	any	DET
ejpam-5263	159	3	subset	subset	NOUN
ejpam-5263	159	4	{	{	PUNCT
ejpam-5263	159	5	i2	i2	PROPN
ejpam-5263	159	6	,	,	PUNCT
ejpam-5263	159	7	.	.	PUNCT
ejpam-5263	159	8	.	.	PUNCT
ejpam-5263	159	9	.	.	PUNCT
ejpam-5263	160	1	,	,	PUNCT
ejpam-5263	160	2	in	in	ADP
ejpam-5263	160	3	}	}	PUNCT
ejpam-5263	160	4	⊂	⊂	PROPN
ejpam-5263	160	5	{	{	PUNCT
ejpam-5263	160	6	1	1	NUM
ejpam-5263	160	7	,	,	PUNCT
ejpam-5263	160	8	2	2	NUM
ejpam-5263	160	9	,	,	PUNCT
ejpam-5263	160	10	.	.	PUNCT
ejpam-5263	160	11	.	.	PUNCT
ejpam-5263	160	12	.	.	PUNCT
ejpam-5263	160	13	,	,	PUNCT
ejpam-5263	160	14	n	n	CCONJ
ejpam-5263	160	15	}	}	PUNCT
ejpam-5263	160	16	,	,	PUNCT
ejpam-5263	160	17	we	we	PRON
ejpam-5263	160	18	have	have	AUX
ejpam-5263	160	19	⟨x	⟨x	VERB
ejpam-5263	160	20	,	,	PUNCT
ejpam-5263	160	21	y|ai2	y|ai2	NOUN
ejpam-5263	160	22	,	,	PUNCT
ejpam-5263	160	23	ai3	ai3	NOUN
ejpam-5263	160	24	,	,	PUNCT
ejpam-5263	160	25	.	.	PUNCT
ejpam-5263	160	26	.	.	PUNCT
ejpam-5263	161	1	.	.	PUNCT
ejpam-5263	162	1	,	,	PUNCT
ejpam-5263	162	2	ain⟩	ain⟩	X
ejpam-5263	163	1	=	=	SYM
ejpam-5263	163	2	∣∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣∣	PROPN
ejpam-5263	163	3	⟨x	⟨x	VERB
ejpam-5263	163	4	,	,	PUNCT
ejpam-5263	163	5	y⟩	y⟩	NOUN
ejpam-5263	163	6	⟨x	⟨x	VERB
ejpam-5263	163	7	,	,	PUNCT
ejpam-5263	163	8	ai2⟩	ai2⟩	PROPN
ejpam-5263	163	9	.	.	PUNCT
ejpam-5263	163	10	.	.	PUNCT
ejpam-5263	163	11	.	.	PUNCT
ejpam-5263	164	1	⟨x	⟨x	VERB
ejpam-5263	164	2	,	,	PUNCT
ejpam-5263	164	3	ain⟩	ain⟩	PROPN
ejpam-5263	164	4	⟨ai2	⟨ai2	PROPN
ejpam-5263	164	5	,	,	PUNCT
ejpam-5263	164	6	y⟩	y⟩	NOUN
ejpam-5263	164	7	⟨ai2	⟨ai2	PROPN
ejpam-5263	164	8	,	,	PUNCT
ejpam-5263	164	9	ai2⟩	ai2⟩	PROPN
ejpam-5263	164	10	.	.	PUNCT
ejpam-5263	164	11	.	.	PUNCT
ejpam-5263	164	12	.	.	PUNCT
ejpam-5263	165	1	⟨ai2	⟨ai2	PROPN
ejpam-5263	165	2	,	,	PUNCT
ejpam-5263	165	3	ain⟩	ain⟩	PROPN
ejpam-5263	165	4	...	...	PUNCT
ejpam-5263	165	5	...	...	PUNCT
ejpam-5263	165	6	.	.	PUNCT
ejpam-5263	165	7	.	.	PUNCT
ejpam-5263	165	8	.	.	PUNCT
ejpam-5263	166	1	...	...	PUNCT
ejpam-5263	167	1	⟨ain	⟨ain	PROPN
ejpam-5263	167	2	,	,	PUNCT
ejpam-5263	167	3	y⟩	y⟩	NOUN
ejpam-5263	167	4	⟨ain	⟨ain	PROPN
ejpam-5263	167	5	,	,	PUNCT
ejpam-5263	167	6	ai2⟩	ai2⟩	PROPN
ejpam-5263	167	7	.	.	PUNCT
ejpam-5263	167	8	.	.	PUNCT
ejpam-5263	167	9	.	.	PUNCT
ejpam-5263	168	1	⟨ain	⟨ain	PROPN
ejpam-5263	168	2	,	,	PUNCT
ejpam-5263	168	3	ain⟩	ain⟩	NOUN
ejpam-5263	168	4	∣∣∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣∣∣	PROPN
ejpam-5263	168	5	n×n	n×n	PROPN
ejpam-5263	168	6	=	=	PUNCT
ejpam-5263	168	7	⟨x	⟨x	VERB
ejpam-5263	168	8	,	,	PUNCT
ejpam-5263	168	9	y⟩	y⟩	NOUN
ejpam-5263	168	10	∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣	PROPN
ejpam-5263	168	11	∥ai2∥	∥ai2∥	VERB
ejpam-5263	168	12	2	2	NUM
ejpam-5263	168	13	0	0	NUM
ejpam-5263	168	14	.	.	PUNCT
ejpam-5263	168	15	.	.	PUNCT
ejpam-5263	169	1	.	.	PUNCT
ejpam-5263	170	1	0	0	NUM
ejpam-5263	170	2	0	0	NUM
ejpam-5263	170	3	∥ai3∥	∥ai3∥	X
ejpam-5263	170	4	2	2	NUM
ejpam-5263	170	5	.	.	PUNCT
ejpam-5263	170	6	.	.	PUNCT
ejpam-5263	170	7	.	.	PUNCT
ejpam-5263	171	1	0	0	NUM
ejpam-5263	171	2	...	...	PUNCT
ejpam-5263	171	3	...	...	PUNCT
ejpam-5263	171	4	.	.	PUNCT
ejpam-5263	171	5	.	.	PUNCT
ejpam-5263	172	1	.	.	PUNCT
ejpam-5263	173	1	...	...	PUNCT
ejpam-5263	174	1	0	0	NUM
ejpam-5263	174	2	0	0	NUM
ejpam-5263	174	3	.	.	PUNCT
ejpam-5263	174	4	.	.	PUNCT
ejpam-5263	174	5	.	.	PUNCT
ejpam-5263	175	1	∥ain∥	∥ain∥	X
ejpam-5263	175	2	2	2	NUM
ejpam-5263	175	3	∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣	PROPN
ejpam-5263	175	4	(	(	PUNCT
ejpam-5263	175	5	n−1)×(n−1	n−1)×(n−1	PROPN
ejpam-5263	175	6	)	)	PUNCT
ejpam-5263	175	7	−	−	PROPN
ejpam-5263	175	8	⟨x	⟨x	VERB
ejpam-5263	175	9	,	,	PUNCT
ejpam-5263	175	10	ai2⟩	ai2⟩	PROPN
ejpam-5263	175	11	∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣	PROPN
ejpam-5263	175	12	⟨ai2	⟨ai2	PROPN
ejpam-5263	175	13	,	,	PUNCT
ejpam-5263	175	14	y⟩	y⟩	NOUN
ejpam-5263	175	15	0	0	NUM
ejpam-5263	175	16	.	.	PUNCT
ejpam-5263	175	17	.	.	PUNCT
ejpam-5263	176	1	.	.	PUNCT
ejpam-5263	177	1	0	0	PUNCT
ejpam-5263	178	1	⟨ai3	⟨ai3	NOUN
ejpam-5263	178	2	,	,	PUNCT
ejpam-5263	178	3	y⟩	y⟩	NOUN
ejpam-5263	178	4	∥ai3∥	∥ai3∥	X
ejpam-5263	178	5	2	2	X
ejpam-5263	178	6	.	.	PUNCT
ejpam-5263	178	7	.	.	PUNCT
ejpam-5263	178	8	.	.	PUNCT
ejpam-5263	179	1	0	0	NUM
ejpam-5263	179	2	...	...	PUNCT
ejpam-5263	179	3	...	...	PUNCT
ejpam-5263	179	4	.	.	PUNCT
ejpam-5263	179	5	.	.	PUNCT
ejpam-5263	179	6	.	.	PUNCT
ejpam-5263	180	1	...	...	PUNCT
ejpam-5263	181	1	⟨ain	⟨ain	NOUN
ejpam-5263	181	2	,	,	PUNCT
ejpam-5263	181	3	y⟩	y⟩	NOUN
ejpam-5263	181	4	0	0	NUM
ejpam-5263	181	5	.	.	PUNCT
ejpam-5263	181	6	.	.	PUNCT
ejpam-5263	181	7	.	.	PUNCT
ejpam-5263	182	1	∥ain∥	∥ain∥	X
ejpam-5263	182	2	2	2	NUM
ejpam-5263	182	3	∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣	PROPN
ejpam-5263	182	4	(	(	PUNCT
ejpam-5263	182	5	n−1)×(n−1	n−1)×(n−1	PROPN
ejpam-5263	182	6	)	)	PUNCT
ejpam-5263	182	7	+	+	CCONJ
ejpam-5263	182	8	·	·	PUNCT
ejpam-5263	182	9	·	·	PUNCT
ejpam-5263	182	10	·	·	PUNCT
ejpam-5263	183	1	+	+	NUM
ejpam-5263	183	2	(	(	PUNCT
ejpam-5263	183	3	−1)n−1⟨x	−1)n−1⟨x	NOUN
ejpam-5263	183	4	,	,	PUNCT
ejpam-5263	183	5	ain⟩	ain⟩	PROPN
ejpam-5263	183	6	∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣	PROPN
ejpam-5263	183	7	⟨ai2	⟨ai2	PROPN
ejpam-5263	183	8	,	,	PUNCT
ejpam-5263	183	9	y⟩	y⟩	NOUN
ejpam-5263	183	10	∥ai2∥	∥ai2∥	PROPN
ejpam-5263	183	11	2	2	NUM
ejpam-5263	183	12	.	.	PUNCT
ejpam-5263	183	13	.	.	PUNCT
ejpam-5263	184	1	.	.	PUNCT
ejpam-5263	185	1	0	0	NUM
ejpam-5263	185	2	...	...	PUNCT
ejpam-5263	185	3	...	...	PUNCT
ejpam-5263	185	4	.	.	PUNCT
ejpam-5263	185	5	.	.	PUNCT
ejpam-5263	186	1	.	.	PUNCT
ejpam-5263	187	1	...	...	PUNCT
ejpam-5263	188	1	⟨ain−1	⟨ain−1	PROPN
ejpam-5263	188	2	,	,	PUNCT
ejpam-5263	188	3	y⟩	y⟩	NOUN
ejpam-5263	188	4	0	0	NUM
ejpam-5263	188	5	.	.	PUNCT
ejpam-5263	188	6	.	.	PUNCT
ejpam-5263	189	1	.	.	PUNCT
ejpam-5263	190	1	∥∥ain−1	∥∥ain−1	NUM
ejpam-5263	191	1	∥∥2	∥∥2	PROPN
ejpam-5263	191	2	⟨ain	⟨ain	PROPN
ejpam-5263	191	3	,	,	PUNCT
ejpam-5263	191	4	y⟩	y⟩	NOUN
ejpam-5263	191	5	0	0	NUM
ejpam-5263	191	6	.	.	PUNCT
ejpam-5263	191	7	.	.	PUNCT
ejpam-5263	191	8	.	.	PUNCT
ejpam-5263	192	1	0	0	NUM
ejpam-5263	192	2	∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣	PROPN
ejpam-5263	192	3	(	(	PUNCT
ejpam-5263	192	4	n−1)×(n−1	n−1)×(n−1	PROPN
ejpam-5263	192	5	)	)	PUNCT
ejpam-5263	192	6	=	=	PUNCT
ejpam-5263	192	7	⟨x	⟨x	VERB
ejpam-5263	192	8	,	,	PUNCT
ejpam-5263	192	9	y⟩	y⟩	NOUN
ejpam-5263	192	10	n∏	n∏	PROPN
ejpam-5263	192	11	j=2	j=2	PROPN
ejpam-5263	192	12	∥∥aij∥∥2	∥∥aij∥∥2	PROPN
ejpam-5263	192	13	−	−	PROPN
ejpam-5263	192	14	⟨x	⟨x	VERB
ejpam-5263	192	15	,	,	PUNCT
ejpam-5263	192	16	ai2⟩⟨ai2	ai2⟩⟨ai2	INTJ
ejpam-5263	192	17	,	,	PUNCT
ejpam-5263	193	1	y⟩	y⟩	NOUN
ejpam-5263	193	2	n∏	n∏	PROPN
ejpam-5263	193	3	j=3	j=3	PROPN
ejpam-5263	193	4	∥∥aij∥∥2	∥∥aij∥∥2	PUNCT
ejpam-5263	194	1	−	−	PROPN
ejpam-5263	194	2	·	·	PUNCT
ejpam-5263	194	3	·	·	PUNCT
ejpam-5263	194	4	·	·	PUNCT
ejpam-5263	195	1	−	−	PROPN
ejpam-5263	195	2	⟨x	⟨x	NUM
ejpam-5263	195	3	,	,	PUNCT
ejpam-5263	195	4	ain⟩⟨ain	ain⟩⟨ain	PUNCT
ejpam-5263	195	5	,	,	PUNCT
ejpam-5263	195	6	y⟩	y⟩	NOUN
ejpam-5263	196	1	n−1∏	n−1∏	PROPN
ejpam-5263	196	2	j=2	j=2	PROPN
ejpam-5263	196	3	∥∥aij∥∥2	∥∥aij∥∥2	PUNCT
ejpam-5263	197	1	=	=	PRON
ejpam-5263	197	2	[	[	PUNCT
ejpam-5263	197	3	⟨x	⟨x	NUM
ejpam-5263	197	4	,	,	PUNCT
ejpam-5263	197	5	y⟩	y⟩	NOUN
ejpam-5263	197	6	−	−	PROPN
ejpam-5263	197	7	⟨x	⟨x	VERB
ejpam-5263	197	8	,	,	PUNCT
ejpam-5263	197	9	ai2⟩⟨ai2	ai2⟩⟨ai2	INTJ
ejpam-5263	197	10	,	,	PUNCT
ejpam-5263	198	1	y⟩	y⟩	NOUN
ejpam-5263	198	2	∥ai2∥	∥ai2∥	PROPN
ejpam-5263	199	1	2	2	NUM
ejpam-5263	199	2	−	−	NOUN
ejpam-5263	199	3	⟨x	⟨x	NUM
ejpam-5263	199	4	,	,	PUNCT
ejpam-5263	199	5	ai3⟩⟨ai3	ai3⟩⟨ai3	PROPN
ejpam-5263	199	6	,	,	PUNCT
ejpam-5263	199	7	y⟩	y⟩	NOUN
ejpam-5263	199	8	∥ai3∥	∥ai3∥	VERB
ejpam-5263	199	9	2	2	X
ejpam-5263	199	10	−	−	NOUN
ejpam-5263	199	11	·	·	PUNCT
ejpam-5263	199	12	·	·	PUNCT
ejpam-5263	199	13	·	·	PUNCT
ejpam-5263	199	14	−	−	PROPN
ejpam-5263	199	15	⟨x	⟨x	NUM
ejpam-5263	199	16	,	,	PUNCT
ejpam-5263	199	17	ain⟩⟨ain	ain⟩⟨ain	PUNCT
ejpam-5263	199	18	,	,	PUNCT
ejpam-5263	200	1	y⟩	y⟩	NOUN
ejpam-5263	200	2	∥ain∥	∥ain∥	X
ejpam-5263	200	3	2	2	X
ejpam-5263	200	4	]	]	PUNCT
ejpam-5263	200	5	n∏	n∏	PROPN
ejpam-5263	200	6	j=2	j=2	PROPN
ejpam-5263	200	7	∥∥aij∥∥2	∥∥aij∥∥2	PUNCT
ejpam-5263	200	8	.	.	PUNCT
ejpam-5263	201	1	a.	a.	PROPN
ejpam-5263	201	2	adam	adam	PROPN
ejpam-5263	201	3	,	,	PUNCT
ejpam-5263	201	4	s.	s.	PROPN
ejpam-5263	201	5	rante	rante	PROPN
ejpam-5263	201	6	,	,	PUNCT
ejpam-5263	201	7	h.	h.	PROPN
ejpam-5263	201	8	gunawan	gunawan	PROPN
ejpam-5263	201	9	/	/	PUNCT
ejpam-5263	201	10	eur	eur	PROPN
ejpam-5263	201	11	.	.	PUNCT
ejpam-5263	202	1	j.	j.	PROPN
ejpam-5263	202	2	pure	pure	PROPN
ejpam-5263	202	3	appl	appl	PROPN
ejpam-5263	202	4	.	.	PROPN
ejpam-5263	202	5	math	math	PROPN
ejpam-5263	202	6	,	,	PUNCT
ejpam-5263	202	7	17	17	NUM
ejpam-5263	202	8	(	(	PUNCT
ejpam-5263	202	9	3	3	NUM
ejpam-5263	202	10	)	)	PUNCT
ejpam-5263	202	11	(	(	PUNCT
ejpam-5263	202	12	2024	2024	NUM
ejpam-5263	202	13	)	)	PUNCT
ejpam-5263	202	14	,	,	PUNCT
ejpam-5263	202	15	1937	1937	NUM
ejpam-5263	202	16	-	-	SYM
ejpam-5263	202	17	1947	1947	NUM
ejpam-5263	202	18	1941	1941	NUM
ejpam-5263	202	19	let	let	VERB
ejpam-5263	202	20	{	{	PUNCT
ejpam-5263	202	21	1	1	NUM
ejpam-5263	202	22	,	,	PUNCT
ejpam-5263	202	23	2	2	NUM
ejpam-5263	202	24	,	,	PUNCT
ejpam-5263	202	25	.	.	PUNCT
ejpam-5263	202	26	.	.	PUNCT
ejpam-5263	202	27	.	.	PUNCT
ejpam-5263	203	1	n}\{i2	n}\{i2	NOUN
ejpam-5263	203	2	,	,	PUNCT
ejpam-5263	203	3	.	.	PUNCT
ejpam-5263	203	4	.	.	PUNCT
ejpam-5263	203	5	.	.	PUNCT
ejpam-5263	204	1	in	in	ADP
ejpam-5263	204	2	}	}	PUNCT
ejpam-5263	204	3	=	=	SYM
ejpam-5263	204	4	{	{	PUNCT
ejpam-5263	204	5	i1	i1	NOUN
ejpam-5263	204	6	}	}	PUNCT
ejpam-5263	204	7	.	.	PUNCT
ejpam-5263	205	1	by	by	ADP
ejpam-5263	205	2	parseval	parseval	NOUN
ejpam-5263	205	3	’s	’s	PART
ejpam-5263	205	4	identity	identity	NOUN
ejpam-5263	205	5	,	,	PUNCT
ejpam-5263	205	6	we	we	PRON
ejpam-5263	205	7	have	have	AUX
ejpam-5263	205	8	⟨x	⟨x	VERB
ejpam-5263	205	9	,	,	PUNCT
ejpam-5263	205	10	y⟩	y⟩	NOUN
ejpam-5263	206	1	=	=	PUNCT
ejpam-5263	206	2	n∑	n∑	PROPN
ejpam-5263	206	3	j=1	j=1	PROPN
ejpam-5263	206	4	⟨x	⟨x	VERB
ejpam-5263	206	5	,	,	PUNCT
ejpam-5263	206	6	aij	aij	PROPN
ejpam-5263	206	7	⟩⟨aij	⟩⟨aij	NOUN
ejpam-5263	206	8	,	,	PUNCT
ejpam-5263	206	9	y⟩∥∥aij∥∥2	y⟩∥∥aij∥∥2	PROPN
ejpam-5263	206	10	(	(	PUNCT
ejpam-5263	206	11	since	since	SCONJ
ejpam-5263	206	12	{	{	PUNCT
ejpam-5263	206	13	i1	i1	PROPN
ejpam-5263	206	14	,	,	PUNCT
ejpam-5263	206	15	i2	i2	PROPN
ejpam-5263	206	16	,	,	PUNCT
ejpam-5263	206	17	.	.	PUNCT
ejpam-5263	206	18	.	.	PUNCT
ejpam-5263	207	1	.	.	PUNCT
ejpam-5263	208	1	,	,	PUNCT
ejpam-5263	208	2	in	in	ADP
ejpam-5263	208	3	}	}	PUNCT
ejpam-5263	208	4	=	=	PUNCT
ejpam-5263	208	5	{	{	PUNCT
ejpam-5263	208	6	1	1	NUM
ejpam-5263	208	7	,	,	PUNCT
ejpam-5263	208	8	2	2	NUM
ejpam-5263	208	9	,	,	PUNCT
ejpam-5263	208	10	.	.	PUNCT
ejpam-5263	208	11	.	.	PUNCT
ejpam-5263	208	12	.	.	PUNCT
ejpam-5263	208	13	,	,	PUNCT
ejpam-5263	208	14	n	n	CCONJ
ejpam-5263	208	15	}	}	PUNCT
ejpam-5263	208	16	as	as	ADP
ejpam-5263	208	17	sets	set	NOUN
ejpam-5263	208	18	)	)	PUNCT
ejpam-5263	208	19	.	.	PUNCT
ejpam-5263	209	1	hence	hence	ADV
ejpam-5263	209	2	it	it	PRON
ejpam-5263	209	3	follows	follow	VERB
ejpam-5263	209	4	that	that	SCONJ
ejpam-5263	209	5	⟨x	⟨x	VERB
ejpam-5263	209	6	,	,	PUNCT
ejpam-5263	209	7	y|ai2	y|ai2	NOUN
ejpam-5263	209	8	,	,	PUNCT
ejpam-5263	209	9	ai3	ai3	NOUN
ejpam-5263	209	10	,	,	PUNCT
ejpam-5263	209	11	.	.	PUNCT
ejpam-5263	209	12	.	.	PUNCT
ejpam-5263	210	1	.	.	PUNCT
ejpam-5263	211	1	,	,	PUNCT
ejpam-5263	211	2	ain⟩	ain⟩	X
ejpam-5263	211	3	=	=	PUNCT
ejpam-5263	211	4	⟨x	⟨x	VERB
ejpam-5263	211	5	,	,	PUNCT
ejpam-5263	211	6	ai1⟩⟨ai1	ai1⟩⟨ai1	ADJ
ejpam-5263	211	7	,	,	PUNCT
ejpam-5263	211	8	y⟩	y⟩	NOUN
ejpam-5263	211	9	∥ai1∥	∥ai1∥	VERB
ejpam-5263	211	10	2	2	NUM
ejpam-5263	211	11	n∏	n∏	PROPN
ejpam-5263	211	12	j=2	j=2	PROPN
ejpam-5263	211	13	∥∥aij∥∥2	∥∥aij∥∥2	PUNCT
ejpam-5263	211	14	=	=	PUNCT
ejpam-5263	211	15	⟨x	⟨x	VERB
ejpam-5263	211	16	,	,	PUNCT
ejpam-5263	211	17	ai1⟩⟨ai1	ai1⟩⟨ai1	ADJ
ejpam-5263	211	18	,	,	PUNCT
ejpam-5263	211	19	y⟩	y⟩	NOUN
ejpam-5263	211	20	∥ai1∥	∥ai1∥	ADP
ejpam-5263	211	21	4	4	NUM
ejpam-5263	211	22	n∏	n∏	PROPN
ejpam-5263	211	23	i=1	i=1	PROPN
ejpam-5263	211	24	∥ai∥2	∥ai∥2	PROPN
ejpam-5263	211	25	.	.	PUNCT
ejpam-5263	212	1	summing	sum	VERB
ejpam-5263	212	2	the	the	DET
ejpam-5263	212	3	above	above	ADJ
ejpam-5263	212	4	expressions	expression	NOUN
ejpam-5263	212	5	for	for	ADP
ejpam-5263	212	6	i1	i1	PROPN
ejpam-5263	212	7	=	=	PUNCT
ejpam-5263	212	8	1	1	NUM
ejpam-5263	212	9	,	,	PUNCT
ejpam-5263	212	10	2	2	NUM
ejpam-5263	212	11	,	,	PUNCT
ejpam-5263	212	12	.	.	PUNCT
ejpam-5263	212	13	.	.	PUNCT
ejpam-5263	213	1	.	.	PUNCT
ejpam-5263	214	1	,	,	PUNCT
ejpam-5263	214	2	n	n	CCONJ
ejpam-5263	214	3	,	,	PUNCT
ejpam-5263	214	4	we	we	PRON
ejpam-5263	214	5	get	get	AUX
ejpam-5263	214	6	⟨x	⟨x	VERB
ejpam-5263	214	7	,	,	PUNCT
ejpam-5263	214	8	y⟩a	y⟩a	NUM
ejpam-5263	214	9	=	=	SYM
ejpam-5263	214	10	∑	∑	PUNCT
ejpam-5263	214	11	{	{	PUNCT
ejpam-5263	214	12	i2,	i2,	NOUN
ejpam-5263	214	13	...	...	PUNCT
ejpam-5263	214	14	in}⊂{1,2,	in}⊂{1,2,	NOUN
ejpam-5263	214	15	...	...	PUNCT
ejpam-5263	214	16	,n	,n	NOUN
ejpam-5263	214	17	}	}	PUNCT
ejpam-5263	214	18	⟨x	⟨x	VERB
ejpam-5263	214	19	,	,	PUNCT
ejpam-5263	214	20	y|ai2	y|ai2	NOUN
ejpam-5263	214	21	,	,	PUNCT
ejpam-5263	214	22	.	.	PUNCT
ejpam-5263	214	23	.	.	PUNCT
ejpam-5263	215	1	.	.	PUNCT
ejpam-5263	216	1	,	,	PUNCT
ejpam-5263	216	2	ain⟩	ain⟩	X
ejpam-5263	216	3	=	=	X
ejpam-5263	217	1	[	[	PUNCT
ejpam-5263	217	2	⟨x	⟨x	VERB
ejpam-5263	217	3	,	,	PUNCT
ejpam-5263	217	4	a1⟩⟨a1	a1⟩⟨a1	PROPN
ejpam-5263	217	5	,	,	PUNCT
ejpam-5263	217	6	y⟩	y⟩	NOUN
ejpam-5263	217	7	∥a1∥4	∥a1∥4	PROPN
ejpam-5263	217	8	+	+	CCONJ
ejpam-5263	217	9	⟨x	⟨x	ADJ
ejpam-5263	217	10	,	,	PUNCT
ejpam-5263	217	11	a2⟩⟨a2	a2⟩⟨a2	NOUN
ejpam-5263	217	12	,	,	PUNCT
ejpam-5263	217	13	y⟩	y⟩	NOUN
ejpam-5263	217	14	∥a2∥4	∥a2∥4	ADV
ejpam-5263	217	15	+	+	CCONJ
ejpam-5263	217	16	·	·	PUNCT
ejpam-5263	217	17	·	·	PUNCT
ejpam-5263	217	18	·	·	PUNCT
ejpam-5263	217	19	+	+	CCONJ
ejpam-5263	217	20	⟨x	⟨x	VERB
ejpam-5263	217	21	,	,	PUNCT
ejpam-5263	217	22	an⟩⟨an	an⟩⟨an	PROPN
ejpam-5263	217	23	,	,	PUNCT
ejpam-5263	217	24	y⟩	y⟩	NOUN
ejpam-5263	217	25	∥an∥4	∥an∥4	X
ejpam-5263	217	26	]	]	PUNCT
ejpam-5263	217	27	n∏	n∏	PROPN
ejpam-5263	217	28	i=1	i=1	PROPN
ejpam-5263	217	29	∥ai∥2	∥ai∥2	PROPN
ejpam-5263	217	30	.	.	PUNCT
ejpam-5263	218	1	however	however	ADV
ejpam-5263	218	2	,	,	PUNCT
ejpam-5263	218	3	we	we	PRON
ejpam-5263	218	4	are	be	AUX
ejpam-5263	218	5	assuming	assume	VERB
ejpam-5263	218	6	that	that	SCONJ
ejpam-5263	218	7	∥ai∥	∥ai∥	NOUN
ejpam-5263	218	8	=	=	SYM
ejpam-5263	218	9	α	α	PROPN
ejpam-5263	218	10	for	for	ADP
ejpam-5263	218	11	all	all	DET
ejpam-5263	218	12	i	i	PRON
ejpam-5263	218	13	=	=	NOUN
ejpam-5263	218	14	1	1	NUM
ejpam-5263	218	15	,	,	PUNCT
ejpam-5263	218	16	2	2	NUM
ejpam-5263	218	17	,	,	PUNCT
ejpam-5263	218	18	.	.	PUNCT
ejpam-5263	218	19	.	.	PUNCT
ejpam-5263	219	1	.	.	PUNCT
ejpam-5263	220	1	,	,	PUNCT
ejpam-5263	221	1	n	n	CCONJ
ejpam-5263	222	1	and	and	CCONJ
ejpam-5263	222	2	so	so	ADV
ejpam-5263	222	3	we	we	PRON
ejpam-5263	222	4	obtain	obtain	VERB
ejpam-5263	222	5	⟨x	⟨x	VERB
ejpam-5263	222	6	,	,	PUNCT
ejpam-5263	222	7	y⟩a	y⟩a	NUM
ejpam-5263	222	8	=	=	SYM
ejpam-5263	222	9	[	[	PUNCT
ejpam-5263	222	10	⟨x	⟨x	VERB
ejpam-5263	222	11	,	,	PUNCT
ejpam-5263	222	12	a1⟩⟨a1	a1⟩⟨a1	ADJ
ejpam-5263	222	13	,	,	PUNCT
ejpam-5263	222	14	y⟩	y⟩	NOUN
ejpam-5263	222	15	α4	α4	NOUN
ejpam-5263	222	16	+	+	CCONJ
ejpam-5263	222	17	⟨x	⟨x	NUM
ejpam-5263	222	18	,	,	PUNCT
ejpam-5263	222	19	a2⟩⟨a2	a2⟩⟨a2	NOUN
ejpam-5263	222	20	,	,	PUNCT
ejpam-5263	222	21	y⟩	y⟩	NOUN
ejpam-5263	222	22	α4	α4	NOUN
ejpam-5263	222	23	+	+	CCONJ
ejpam-5263	222	24	·	·	PUNCT
ejpam-5263	222	25	·	·	PUNCT
ejpam-5263	222	26	·	·	PUNCT
ejpam-5263	223	1	+	+	CCONJ
ejpam-5263	223	2	⟨x	⟨x	VERB
ejpam-5263	223	3	,	,	PUNCT
ejpam-5263	223	4	an⟩⟨an	an⟩⟨an	PROPN
ejpam-5263	223	5	,	,	PUNCT
ejpam-5263	223	6	y⟩	y⟩	NOUN
ejpam-5263	223	7	α4	α4	NOUN
ejpam-5263	223	8	]	]	PUNCT
ejpam-5263	223	9	n∏	n∏	PROPN
ejpam-5263	223	10	i=1	i=1	PROPN
ejpam-5263	223	11	α2	α2	PROPN
ejpam-5263	223	12	=	=	PUNCT
ejpam-5263	223	13	⟨x	⟨x	VERB
ejpam-5263	223	14	,	,	PUNCT
ejpam-5263	223	15	y⟩	y⟩	NOUN
ejpam-5263	223	16	α2	α2	PROPN
ejpam-5263	223	17	α2n	α2n	PROPN
ejpam-5263	224	1	=	=	SYM
ejpam-5263	224	2	α2(n−1)⟨x	α2(n−1)⟨x	PROPN
ejpam-5263	224	3	,	,	PUNCT
ejpam-5263	224	4	y⟩.	y⟩.	NUM
ejpam-5263	224	5	since	since	SCONJ
ejpam-5263	224	6	α	α	PRON
ejpam-5263	224	7	̸=	̸=	PROPN
ejpam-5263	224	8	0	0	NUM
ejpam-5263	224	9	,	,	PUNCT
ejpam-5263	224	10	we	we	PRON
ejpam-5263	224	11	conclude	conclude	VERB
ejpam-5263	224	12	that	that	PRON
ejpam-5263	224	13	⟨x	⟨x	VERB
ejpam-5263	224	14	,	,	PUNCT
ejpam-5263	224	15	y⟩a	y⟩a	X
ejpam-5263	224	16	=	=	SYM
ejpam-5263	224	17	0	0	PUNCT
ejpam-5263	225	1	if	if	SCONJ
ejpam-5263	225	2	and	and	CCONJ
ejpam-5263	225	3	only	only	ADV
ejpam-5263	225	4	if	if	SCONJ
ejpam-5263	225	5	⟨x	⟨x	VERB
ejpam-5263	225	6	,	,	PUNCT
ejpam-5263	225	7	y⟩	y⟩	NOUN
ejpam-5263	225	8	=	=	NOUN
ejpam-5263	225	9	0	0	PROPN
ejpam-5263	225	10	,	,	PUNCT
ejpam-5263	225	11	which	which	PRON
ejpam-5263	225	12	proves	prove	VERB
ejpam-5263	225	13	the	the	DET
ejpam-5263	225	14	theorem	theorem	PROPN
ejpam-5263	225	15	.	.	PUNCT
ejpam-5263	225	16	theorem	theorem	NOUN
ejpam-5263	225	17	2	2	NUM
ejpam-5263	225	18	.	.	PUNCT
ejpam-5263	226	1	let	let	VERB
ejpam-5263	226	2	a	a	DET
ejpam-5263	226	3	=	=	PUNCT
ejpam-5263	226	4	{	{	PUNCT
ejpam-5263	226	5	b1	b1	NOUN
ejpam-5263	226	6	,	,	PUNCT
ejpam-5263	226	7	b2	b2	NOUN
ejpam-5263	226	8	,	,	PUNCT
ejpam-5263	226	9	.	.	PUNCT
ejpam-5263	226	10	.	.	PUNCT
ejpam-5263	227	1	.	.	PUNCT
ejpam-5263	228	1	,	,	PUNCT
ejpam-5263	228	2	bn	bn	AUX
ejpam-5263	228	3	}	}	PUNCT
ejpam-5263	228	4	be	be	AUX
ejpam-5263	228	5	a	a	DET
ejpam-5263	228	6	set	set	NOUN
ejpam-5263	228	7	of	of	ADP
ejpam-5263	228	8	n	n	CCONJ
ejpam-5263	228	9	linearly	linearly	ADV
ejpam-5263	228	10	independent	independent	ADJ
ejpam-5263	228	11	vectors	vector	NOUN
ejpam-5263	228	12	in	in	ADP
ejpam-5263	228	13	x.	x.	PROPN
ejpam-5263	228	14	then	then	ADV
ejpam-5263	228	15	⟨x	⟨x	VERB
ejpam-5263	228	16	,	,	PUNCT
ejpam-5263	228	17	y⟩a	y⟩a	NUM
ejpam-5263	228	18	=	=	PUNCT
ejpam-5263	228	19	⟨x	⟨x	VERB
ejpam-5263	228	20	,	,	PUNCT
ejpam-5263	228	21	y⟩	y⟩	NOUN
ejpam-5263	229	1	if	if	SCONJ
ejpam-5263	229	2	and	and	CCONJ
ejpam-5263	229	3	only	only	ADV
ejpam-5263	229	4	if	if	SCONJ
ejpam-5263	229	5	a	a	PRON
ejpam-5263	229	6	is	be	AUX
ejpam-5263	229	7	an	an	DET
ejpam-5263	229	8	orthonormal	orthonormal	ADJ
ejpam-5263	229	9	basis	basis	NOUN
ejpam-5263	229	10	for	for	ADP
ejpam-5263	229	11	x.	x.	NOUN
ejpam-5263	229	12	proof	proof	NOUN
ejpam-5263	229	13	.	.	PUNCT
ejpam-5263	230	1	the	the	DET
ejpam-5263	230	2	sufficient	sufficient	ADJ
ejpam-5263	230	3	part	part	NOUN
ejpam-5263	230	4	follows	follow	VERB
ejpam-5263	230	5	immediately	immediately	ADV
ejpam-5263	230	6	from	from	ADP
ejpam-5263	230	7	the	the	DET
ejpam-5263	230	8	previous	previous	ADJ
ejpam-5263	230	9	theorem	theorem	NOUN
ejpam-5263	230	10	.	.	PROPN
ejpam-5263	231	1	for	for	ADP
ejpam-5263	231	2	the	the	DET
ejpam-5263	231	3	necessary	necessary	ADJ
ejpam-5263	231	4	part	part	NOUN
ejpam-5263	231	5	,	,	PUNCT
ejpam-5263	231	6	suppose	suppose	VERB
ejpam-5263	231	7	that	that	SCONJ
ejpam-5263	231	8	⟨x	⟨x	VERB
ejpam-5263	231	9	,	,	PUNCT
ejpam-5263	231	10	y⟩a	y⟩a	X
ejpam-5263	231	11	=	=	PUNCT
ejpam-5263	231	12	⟨x	⟨x	VERB
ejpam-5263	231	13	,	,	PUNCT
ejpam-5263	231	14	y⟩	y⟩	NOUN
ejpam-5263	231	15	for	for	SCONJ
ejpam-5263	231	16	all	all	DET
ejpam-5263	231	17	x	x	NOUN
ejpam-5263	231	18	,	,	PUNCT
ejpam-5263	231	19	y	y	PROPN
ejpam-5263	231	20	∈	∈	PROPN
ejpam-5263	231	21	x.	x.	VERB
ejpam-5263	231	22	to	to	PART
ejpam-5263	231	23	prove	prove	VERB
ejpam-5263	231	24	that	that	SCONJ
ejpam-5263	231	25	a	a	PRON
ejpam-5263	231	26	is	be	AUX
ejpam-5263	231	27	an	an	DET
ejpam-5263	231	28	orthonormal	orthonormal	ADJ
ejpam-5263	231	29	basis	basis	NOUN
ejpam-5263	231	30	for	for	ADP
ejpam-5263	231	31	x	x	X
ejpam-5263	231	32	,	,	PUNCT
ejpam-5263	231	33	let	let	VERB
ejpam-5263	231	34	us	we	PRON
ejpam-5263	231	35	first	first	ADJ
ejpam-5263	231	36	compute	compute	NOUN
ejpam-5263	231	37	(	(	PUNCT
ejpam-5263	231	38	bi	bi	NOUN
ejpam-5263	231	39	,	,	PUNCT
ejpam-5263	231	40	bj	bj	NOUN
ejpam-5263	231	41	)	)	PUNCT
ejpam-5263	231	42	for	for	ADP
ejpam-5263	231	43	i	i	PROPN
ejpam-5263	231	44	̸=	̸=	PROPN
ejpam-5263	231	45	j.	j.	PROPN
ejpam-5263	231	46	we	we	PRON
ejpam-5263	231	47	have	have	AUX
ejpam-5263	231	48	⟨bi	⟨bi	NOUN
ejpam-5263	231	49	,	,	PUNCT
ejpam-5263	231	50	bj⟩	bj⟩	X
ejpam-5263	231	51	=	=	SYM
ejpam-5263	231	52	⟨bi	⟨bi	NOUN
ejpam-5263	231	53	,	,	PUNCT
ejpam-5263	231	54	bj⟩a	bj⟩a	X
ejpam-5263	231	55	=	=	SYM
ejpam-5263	231	56	∑	∑	PUNCT
ejpam-5263	231	57	{	{	PUNCT
ejpam-5263	231	58	i2,	i2,	NOUN
ejpam-5263	231	59	...	...	PUNCT
ejpam-5263	231	60	in}⊂{1,2,	in}⊂{1,2,	NOUN
ejpam-5263	231	61	...	...	PUNCT
ejpam-5263	231	62	,n	,n	NOUN
ejpam-5263	231	63	}	}	PUNCT
ejpam-5263	231	64	⟨bi	⟨bi	NUM
ejpam-5263	231	65	,	,	PUNCT
ejpam-5263	231	66	bj	bj	ADP
ejpam-5263	231	67	|bi2	|bi2	PROPN
ejpam-5263	231	68	,	,	PUNCT
ejpam-5263	231	69	.	.	PUNCT
ejpam-5263	231	70	.	.	PUNCT
ejpam-5263	232	1	.	.	PUNCT
ejpam-5263	233	1	,	,	PUNCT
ejpam-5263	233	2	bin⟩.	bin⟩.	NOUN
ejpam-5263	233	3	for	for	ADP
ejpam-5263	233	4	any	any	DET
ejpam-5263	233	5	{	{	PUNCT
ejpam-5263	233	6	i2	i2	NOUN
ejpam-5263	233	7	,	,	PUNCT
ejpam-5263	233	8	.	.	PUNCT
ejpam-5263	233	9	.	.	PUNCT
ejpam-5263	234	1	.	.	PUNCT
ejpam-5263	235	1	,	,	PUNCT
ejpam-5263	235	2	in	in	ADP
ejpam-5263	235	3	}	}	PUNCT
ejpam-5263	235	4	⊂	⊂	PROPN
ejpam-5263	235	5	{	{	PUNCT
ejpam-5263	235	6	1	1	NUM
ejpam-5263	235	7	,	,	PUNCT
ejpam-5263	235	8	2	2	NUM
ejpam-5263	235	9	,	,	PUNCT
ejpam-5263	235	10	.	.	PUNCT
ejpam-5263	235	11	.	.	PUNCT
ejpam-5263	235	12	.	.	PUNCT
ejpam-5263	235	13	,	,	PUNCT
ejpam-5263	235	14	n	n	CCONJ
ejpam-5263	235	15	}	}	PUNCT
ejpam-5263	235	16	,	,	PUNCT
ejpam-5263	235	17	observe	observe	VERB
ejpam-5263	235	18	that	that	SCONJ
ejpam-5263	235	19	bi	bi	PROPN
ejpam-5263	235	20	∈	∈	PROPN
ejpam-5263	235	21	{	{	PUNCT
ejpam-5263	235	22	bi2	bi2	INTJ
ejpam-5263	235	23	,	,	PUNCT
ejpam-5263	235	24	.	.	PUNCT
ejpam-5263	235	25	.	.	PUNCT
ejpam-5263	236	1	.	.	PUNCT
ejpam-5263	237	1	,	,	PUNCT
ejpam-5263	237	2	bin	bin	NOUN
ejpam-5263	237	3	}	}	PUNCT
ejpam-5263	237	4	or	or	CCONJ
ejpam-5263	237	5	bj	bj	ADP
ejpam-5263	237	6	∈	∈	PROPN
ejpam-5263	237	7	{	{	PUNCT
ejpam-5263	237	8	bi2	bi2	NOUN
ejpam-5263	237	9	,	,	PUNCT
ejpam-5263	237	10	.	.	PUNCT
ejpam-5263	237	11	.	.	PUNCT
ejpam-5263	237	12	.	.	PUNCT
ejpam-5263	238	1	,	,	PUNCT
ejpam-5263	238	2	bin	bin	NOUN
ejpam-5263	238	3	}	}	PUNCT
ejpam-5263	238	4	,	,	PUNCT
ejpam-5263	238	5	because	because	SCONJ
ejpam-5263	238	6	{	{	PUNCT
ejpam-5263	238	7	bi2	bi2	INTJ
ejpam-5263	238	8	,	,	PUNCT
ejpam-5263	238	9	.	.	PUNCT
ejpam-5263	238	10	.	.	PUNCT
ejpam-5263	238	11	.	.	PUNCT
ejpam-5263	239	1	,	,	PUNCT
ejpam-5263	239	2	bin	bin	NOUN
ejpam-5263	239	3	}	}	PUNCT
ejpam-5263	239	4	consists	consist	VERB
ejpam-5263	239	5	of	of	ADP
ejpam-5263	239	6	n−1	n−1	ADJ
ejpam-5263	239	7	elements	element	NOUN
ejpam-5263	239	8	of	of	ADP
ejpam-5263	239	9	a.	a.	NOUN
ejpam-5263	239	10	consequently	consequently	ADV
ejpam-5263	239	11	,	,	PUNCT
ejpam-5263	239	12	⟨bi	⟨bi	X
ejpam-5263	239	13	,	,	PUNCT
ejpam-5263	239	14	bj	bj	ADP
ejpam-5263	239	15	|bi2	|bi2	PROPN
ejpam-5263	239	16	,	,	PUNCT
ejpam-5263	239	17	.	.	PUNCT
ejpam-5263	239	18	.	.	PUNCT
ejpam-5263	240	1	.	.	PUNCT
ejpam-5263	241	1	,	,	PUNCT
ejpam-5263	241	2	bin⟩	bin⟩	VERB
ejpam-5263	241	3	=	=	PUNCT
ejpam-5263	241	4	0	0	NUM
ejpam-5263	241	5	,	,	PUNCT
ejpam-5263	241	6	because	because	SCONJ
ejpam-5263	241	7	two	two	NUM
ejpam-5263	241	8	rows	row	NOUN
ejpam-5263	241	9	or	or	CCONJ
ejpam-5263	241	10	two	two	NUM
ejpam-5263	241	11	columns	column	NOUN
ejpam-5263	241	12	in	in	ADP
ejpam-5263	241	13	the	the	DET
ejpam-5263	241	14	determinant	determinant	NOUN
ejpam-5263	241	15	will	will	AUX
ejpam-5263	241	16	be	be	AUX
ejpam-5263	241	17	identical	identical	ADJ
ejpam-5263	241	18	.	.	PUNCT
ejpam-5263	242	1	since	since	SCONJ
ejpam-5263	242	2	this	this	PRON
ejpam-5263	242	3	is	be	AUX
ejpam-5263	242	4	true	true	ADJ
ejpam-5263	242	5	for	for	ADP
ejpam-5263	242	6	any	any	DET
ejpam-5263	242	7	{	{	PUNCT
ejpam-5263	242	8	i2	i2	NOUN
ejpam-5263	242	9	,	,	PUNCT
ejpam-5263	242	10	.	.	PUNCT
ejpam-5263	242	11	.	.	PUNCT
ejpam-5263	243	1	.	.	PUNCT
ejpam-5263	244	1	,	,	PUNCT
ejpam-5263	244	2	in	in	ADP
ejpam-5263	244	3	}	}	PUNCT
ejpam-5263	244	4	⊂	⊂	PROPN
ejpam-5263	244	5	{	{	PUNCT
ejpam-5263	244	6	1	1	NUM
ejpam-5263	244	7	,	,	PUNCT
ejpam-5263	244	8	2	2	NUM
ejpam-5263	244	9	,	,	PUNCT
ejpam-5263	244	10	.	.	PUNCT
ejpam-5263	244	11	.	.	PUNCT
ejpam-5263	244	12	.	.	PUNCT
ejpam-5263	244	13	,	,	PUNCT
ejpam-5263	244	14	n	n	CCONJ
ejpam-5263	244	15	}	}	PUNCT
ejpam-5263	244	16	,	,	PUNCT
ejpam-5263	244	17	we	we	PRON
ejpam-5263	244	18	conclude	conclude	VERB
ejpam-5263	244	19	that	that	PRON
ejpam-5263	244	20	⟨bi	⟨bi	NOUN
ejpam-5263	244	21	,	,	PUNCT
ejpam-5263	244	22	bj⟩	bj⟩	PUNCT
ejpam-5263	244	23	=	=	PUNCT
ejpam-5263	244	24	∑	∑	PUNCT
ejpam-5263	244	25	{	{	PUNCT
ejpam-5263	244	26	i2,	i2,	NOUN
ejpam-5263	244	27	...	...	PUNCT
ejpam-5263	244	28	in}⊂{1,2,	in}⊂{1,2,	NOUN
ejpam-5263	244	29	...	...	PUNCT
ejpam-5263	244	30	,n	,n	NOUN
ejpam-5263	244	31	}	}	PUNCT
ejpam-5263	244	32	⟨bi	⟨bi	NUM
ejpam-5263	244	33	,	,	PUNCT
ejpam-5263	244	34	bj	bj	ADP
ejpam-5263	244	35	|bi2	|bi2	PROPN
ejpam-5263	244	36	,	,	PUNCT
ejpam-5263	244	37	.	.	PUNCT
ejpam-5263	244	38	.	.	PUNCT
ejpam-5263	245	1	.	.	PUNCT
ejpam-5263	246	1	,	,	PUNCT
ejpam-5263	246	2	bin⟩	bin⟩	VERB
ejpam-5263	246	3	=	=	PUNCT
ejpam-5263	247	1	0	0	X
ejpam-5263	247	2	.	.	PUNCT
ejpam-5263	248	1	let	let	VERB
ejpam-5263	248	2	us	we	PRON
ejpam-5263	248	3	now	now	ADV
ejpam-5263	248	4	compute	compute	VERB
ejpam-5263	248	5	⟨bi	⟨bi	PRON
ejpam-5263	248	6	,	,	PUNCT
ejpam-5263	248	7	bi⟩	bi⟩	NUM
ejpam-5263	248	8	for	for	ADP
ejpam-5263	248	9	i	i	PRON
ejpam-5263	248	10	=	=	NOUN
ejpam-5263	248	11	1	1	NUM
ejpam-5263	248	12	,	,	PUNCT
ejpam-5263	248	13	2	2	NUM
ejpam-5263	248	14	,	,	PUNCT
ejpam-5263	248	15	.	.	PUNCT
ejpam-5263	248	16	.	.	PUNCT
ejpam-5263	249	1	.	.	PUNCT
ejpam-5263	250	1	,	,	PUNCT
ejpam-5263	250	2	n.	n.	PROPN
ejpam-5263	250	3	notice	notice	VERB
ejpam-5263	250	4	that	that	SCONJ
ejpam-5263	250	5	if	if	SCONJ
ejpam-5263	250	6	bi	bi	PROPN
ejpam-5263	250	7	∈	∈	PROPN
ejpam-5263	250	8	{	{	PUNCT
ejpam-5263	250	9	bi2	bi2	INTJ
ejpam-5263	250	10	,	,	PUNCT
ejpam-5263	250	11	.	.	PUNCT
ejpam-5263	250	12	.	.	PUNCT
ejpam-5263	250	13	.	.	PUNCT
ejpam-5263	250	14	,	,	PUNCT
ejpam-5263	250	15	bin	bin	NOUN
ejpam-5263	250	16	}	}	PUNCT
ejpam-5263	250	17	,	,	PUNCT
ejpam-5263	250	18	then	then	ADV
ejpam-5263	250	19	we	we	PRON
ejpam-5263	250	20	have	have	VERB
ejpam-5263	250	21	⟨bi	⟨bi	NOUN
ejpam-5263	250	22	,	,	PUNCT
ejpam-5263	250	23	bi|bi2	bi|bi2	X
ejpam-5263	250	24	,	,	PUNCT
ejpam-5263	250	25	.	.	PUNCT
ejpam-5263	250	26	.	.	PUNCT
ejpam-5263	251	1	.	.	PUNCT
ejpam-5263	252	1	,	,	PUNCT
ejpam-5263	252	2	bin⟩	bin⟩	VERB
ejpam-5263	252	3	=	=	PUNCT
ejpam-5263	253	1	0	0	X
ejpam-5263	253	2	.	.	PUNCT
ejpam-5263	254	1	meanwhile	meanwhile	ADV
ejpam-5263	254	2	,	,	PUNCT
ejpam-5263	254	3	if	if	SCONJ
ejpam-5263	254	4	bi	bi	PROPN
ejpam-5263	254	5	̸∈	̸∈	PROPN
ejpam-5263	254	6	{	{	PUNCT
ejpam-5263	254	7	bi2	bi2	PROPN
ejpam-5263	254	8	,	,	PUNCT
ejpam-5263	254	9	.	.	PUNCT
ejpam-5263	254	10	.	.	PUNCT
ejpam-5263	254	11	.	.	PUNCT
ejpam-5263	255	1	,	,	PUNCT
ejpam-5263	255	2	bin	bin	NOUN
ejpam-5263	255	3	}	}	PUNCT
ejpam-5263	255	4	then	then	ADV
ejpam-5263	255	5	—	—	PUNCT
ejpam-5263	255	6	by	by	ADP
ejpam-5263	255	7	the	the	DET
ejpam-5263	255	8	properties	property	NOUN
ejpam-5263	255	9	of	of	ADP
ejpam-5263	255	10	the	the	DET
ejpam-5263	255	11	standard	standard	PROPN
ejpam-5263	255	12	n−product	n−product	PROPN
ejpam-5263	255	13	—	—	PUNCT
ejpam-5263	255	14	we	we	PRON
ejpam-5263	255	15	have	have	VERB
ejpam-5263	255	16	a.	a.	NOUN
ejpam-5263	255	17	adam	adam	PROPN
ejpam-5263	255	18	,	,	PUNCT
ejpam-5263	255	19	s.	s.	PROPN
ejpam-5263	255	20	rante	rante	PROPN
ejpam-5263	255	21	,	,	PUNCT
ejpam-5263	255	22	h.	h.	PROPN
ejpam-5263	255	23	gunawan	gunawan	PROPN
ejpam-5263	255	24	/	/	PUNCT
ejpam-5263	255	25	eur	eur	PROPN
ejpam-5263	255	26	.	.	PUNCT
ejpam-5263	256	1	j.	j.	PROPN
ejpam-5263	256	2	pure	pure	PROPN
ejpam-5263	256	3	appl	appl	PROPN
ejpam-5263	256	4	.	.	PROPN
ejpam-5263	256	5	math	math	PROPN
ejpam-5263	256	6	,	,	PUNCT
ejpam-5263	256	7	17	17	NUM
ejpam-5263	256	8	(	(	PUNCT
ejpam-5263	256	9	3	3	NUM
ejpam-5263	256	10	)	)	PUNCT
ejpam-5263	256	11	(	(	PUNCT
ejpam-5263	256	12	2024	2024	NUM
ejpam-5263	256	13	)	)	PUNCT
ejpam-5263	256	14	,	,	PUNCT
ejpam-5263	256	15	1937	1937	NUM
ejpam-5263	256	16	-	-	SYM
ejpam-5263	256	17	1947	1947	NUM
ejpam-5263	256	18	1942	1942	NUM
ejpam-5263	256	19	⟨bi	⟨bi	NOUN
ejpam-5263	256	20	,	,	PUNCT
ejpam-5263	256	21	bi|bi2	bi|bi2	X
ejpam-5263	256	22	,	,	PUNCT
ejpam-5263	256	23	.	.	PUNCT
ejpam-5263	256	24	.	.	PUNCT
ejpam-5263	256	25	.	.	PUNCT
ejpam-5263	257	1	,	,	PUNCT
ejpam-5263	257	2	bin⟩	bin⟩	VERB
ejpam-5263	257	3	=	=	PUNCT
ejpam-5263	257	4	⟨b1	⟨b1	PROPN
ejpam-5263	257	5	,	,	PUNCT
ejpam-5263	257	6	b1|b2	b1|b2	NOUN
ejpam-5263	257	7	,	,	PUNCT
ejpam-5263	257	8	.	.	PUNCT
ejpam-5263	257	9	.	.	PUNCT
ejpam-5263	258	1	.	.	PUNCT
ejpam-5263	259	1	,	,	PUNCT
ejpam-5263	259	2	bn⟩	bn⟩	NOUN
ejpam-5263	259	3	=	=	SYM
ejpam-5263	259	4	∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣	PROPN
ejpam-5263	259	5	⟨b1	⟨b1	PROPN
ejpam-5263	259	6	,	,	PUNCT
ejpam-5263	259	7	b1⟩	b1⟩	PUNCT
ejpam-5263	260	1	⟨b1	⟨b1	PROPN
ejpam-5263	260	2	,	,	PUNCT
ejpam-5263	260	3	b2⟩	b2⟩	NOUN
ejpam-5263	260	4	.	.	PUNCT
ejpam-5263	260	5	.	.	PUNCT
ejpam-5263	260	6	.	.	PUNCT
ejpam-5263	261	1	⟨b1	⟨b1	NOUN
ejpam-5263	261	2	,	,	PUNCT
ejpam-5263	261	3	bn⟩	bn⟩	NOUN
ejpam-5263	261	4	⟨b2	⟨b2	PROPN
ejpam-5263	261	5	,	,	PUNCT
ejpam-5263	261	6	b1⟩	b1⟩	PUNCT
ejpam-5263	262	1	⟨b2	⟨b2	PROPN
ejpam-5263	262	2	,	,	PUNCT
ejpam-5263	262	3	b2⟩	b2⟩	NOUN
ejpam-5263	262	4	.	.	PUNCT
ejpam-5263	262	5	.	.	PUNCT
ejpam-5263	262	6	.	.	PUNCT
ejpam-5263	263	1	⟨b2	⟨b2	NOUN
ejpam-5263	263	2	,	,	PUNCT
ejpam-5263	263	3	bn⟩	bn⟩	NOUN
ejpam-5263	263	4	...	...	PUNCT
ejpam-5263	263	5	...	...	PUNCT
ejpam-5263	263	6	.	.	PUNCT
ejpam-5263	263	7	.	.	PUNCT
ejpam-5263	263	8	.	.	PUNCT
ejpam-5263	263	9	...	...	PUNCT
ejpam-5263	264	1	⟨bn	⟨bn	NUM
ejpam-5263	264	2	,	,	PUNCT
ejpam-5263	264	3	b1⟩	b1⟩	NOUN
ejpam-5263	264	4	⟨bn	⟨bn	PROPN
ejpam-5263	264	5	,	,	PUNCT
ejpam-5263	264	6	b2⟩	b2⟩	X
ejpam-5263	264	7	.	.	PUNCT
ejpam-5263	264	8	.	.	PUNCT
ejpam-5263	264	9	.	.	PUNCT
ejpam-5263	265	1	⟨bn	⟨bn	PROPN
ejpam-5263	265	2	,	,	PUNCT
ejpam-5263	265	3	bn⟩	bn⟩	PUNCT
ejpam-5263	265	4	∣∣∣∣∣∣∣∣∣	∣∣∣∣∣∣∣∣∣	PUNCT
ejpam-5263	266	1	=	=	SYM
ejpam-5263	266	2	n∏	n∏	PROPN
ejpam-5263	266	3	j=1	j=1	NOUN
ejpam-5263	266	4	∥bj∥2	∥bj∥2	PROPN
ejpam-5263	266	5	.	.	PUNCT
ejpam-5263	267	1	hence	hence	ADV
ejpam-5263	267	2	,	,	PUNCT
ejpam-5263	267	3	we	we	PRON
ejpam-5263	267	4	obtain	obtain	VERB
ejpam-5263	267	5	⟨bi	⟨bi	NOUN
ejpam-5263	267	6	,	,	PUNCT
ejpam-5263	267	7	bi⟩a	bi⟩a	PROPN
ejpam-5263	267	8	=	=	PUNCT
ejpam-5263	267	9	∑	∑	PUNCT
ejpam-5263	267	10	{	{	PUNCT
ejpam-5263	267	11	i2,	i2,	NOUN
ejpam-5263	267	12	...	...	PUNCT
ejpam-5263	267	13	in}⊂{1,2,	in}⊂{1,2,	NOUN
ejpam-5263	267	14	...	...	PUNCT
ejpam-5263	267	15	,n	,n	NOUN
ejpam-5263	267	16	}	}	PUNCT
ejpam-5263	267	17	⟨bi	⟨bi	NUM
ejpam-5263	267	18	,	,	PUNCT
ejpam-5263	267	19	bi|bi2	bi|bi2	X
ejpam-5263	267	20	,	,	PUNCT
ejpam-5263	267	21	.	.	PUNCT
ejpam-5263	267	22	.	.	PUNCT
ejpam-5263	267	23	.	.	PUNCT
ejpam-5263	268	1	,	,	PUNCT
ejpam-5263	268	2	bin⟩	bin⟩	VERB
ejpam-5263	268	3	=	=	PUNCT
ejpam-5263	268	4	n∏	n∏	NOUN
ejpam-5263	268	5	j=1	j=1	NOUN
ejpam-5263	268	6	∥bj∥2	∥bj∥2	PROPN
ejpam-5263	268	7	.	.	PUNCT
ejpam-5263	269	1	by	by	ADP
ejpam-5263	269	2	our	our	PRON
ejpam-5263	269	3	hypothesis	hypothesis	NOUN
ejpam-5263	269	4	,	,	PUNCT
ejpam-5263	269	5	∥bi∥2	∥bi∥2	PUNCT
ejpam-5263	269	6	=	=	SYM
ejpam-5263	269	7	∥bi∥2a	∥bi∥2a	PROPN
ejpam-5263	269	8	=	=	SYM
ejpam-5263	269	9	n∏	n∏	PROPN
ejpam-5263	269	10	j=1	j=1	NOUN
ejpam-5263	270	1	∥bj∥2	∥bj∥2	PROPN
ejpam-5263	270	2	.	.	PUNCT
ejpam-5263	271	1	this	this	PRON
ejpam-5263	271	2	holds	hold	VERB
ejpam-5263	271	3	only	only	ADV
ejpam-5263	271	4	if	if	SCONJ
ejpam-5263	271	5	∥bi∥	∥bi∥	NUM
ejpam-5263	271	6	=	=	SYM
ejpam-5263	271	7	1	1	NUM
ejpam-5263	271	8	for	for	ADP
ejpam-5263	271	9	all	all	DET
ejpam-5263	271	10	i	i	PRON
ejpam-5263	271	11	=	=	NOUN
ejpam-5263	271	12	1	1	NUM
ejpam-5263	271	13	,	,	PUNCT
ejpam-5263	271	14	2	2	NUM
ejpam-5263	271	15	,	,	PUNCT
ejpam-5263	271	16	.	.	PUNCT
ejpam-5263	271	17	.	.	PUNCT
ejpam-5263	272	1	.	.	PUNCT
ejpam-5263	273	1	,	,	PUNCT
ejpam-5263	273	2	n.	n.	NOUN
ejpam-5263	273	3	to	to	PART
ejpam-5263	273	4	sum	sum	VERB
ejpam-5263	273	5	up	up	ADP
ejpam-5263	273	6	,	,	PUNCT
ejpam-5263	273	7	we	we	PRON
ejpam-5263	273	8	have	have	AUX
ejpam-5263	273	9	proved	prove	VERB
ejpam-5263	273	10	that	that	SCONJ
ejpam-5263	273	11	⟨x	⟨x	VERB
ejpam-5263	273	12	,	,	PUNCT
ejpam-5263	273	13	y⟩a	y⟩a	X
ejpam-5263	273	14	=	=	PUNCT
ejpam-5263	273	15	⟨x	⟨x	VERB
ejpam-5263	273	16	,	,	PUNCT
ejpam-5263	273	17	y⟩	y⟩	NOUN
ejpam-5263	273	18	if	if	SCONJ
ejpam-5263	274	1	and	and	CCONJ
ejpam-5263	274	2	only	only	ADV
ejpam-5263	274	3	if	if	SCONJ
ejpam-5263	274	4	a	a	PRON
ejpam-5263	274	5	is	be	AUX
ejpam-5263	274	6	an	an	DET
ejpam-5263	274	7	orthonormal	orthonormal	ADJ
ejpam-5263	274	8	basis	basis	NOUN
ejpam-5263	274	9	for	for	ADP
ejpam-5263	274	10	x.	x.	NOUN
ejpam-5263	274	11	3	3	X
ejpam-5263	274	12	.	.	PUNCT
ejpam-5263	275	1	the	the	DET
ejpam-5263	275	2	higher	high	ADJ
ejpam-5263	275	3	dimensional	dimensional	ADJ
ejpam-5263	275	4	case	case	NOUN
ejpam-5263	275	5	let	let	VERB
ejpam-5263	275	6	us	we	PRON
ejpam-5263	275	7	now	now	ADV
ejpam-5263	275	8	consider	consider	VERB
ejpam-5263	275	9	the	the	DET
ejpam-5263	275	10	case	case	NOUN
ejpam-5263	275	11	where	where	SCONJ
ejpam-5263	275	12	n+1	n+1	PROPN
ejpam-5263	275	13	≤	≤	PUNCT
ejpam-5263	275	14	d	d	NOUN
ejpam-5263	275	15	=	=	SYM
ejpam-5263	275	16	dim	dim	NOUN
ejpam-5263	275	17	x	x	PUNCT
ejpam-5263	275	18	<	<	X
ejpam-5263	275	19	∞.	∞.	PROPN
ejpam-5263	275	20	let	let	VERB
ejpam-5263	275	21	a	a	DET
ejpam-5263	275	22	:	:	PUNCT
ejpam-5263	275	23	=	=	NUM
ejpam-5263	275	24	{	{	PUNCT
ejpam-5263	275	25	a1	a1	PROPN
ejpam-5263	275	26	,	,	PUNCT
ejpam-5263	275	27	a2	a2	PROPN
ejpam-5263	275	28	,	,	PUNCT
ejpam-5263	275	29	.	.	PUNCT
ejpam-5263	275	30	.	.	PUNCT
ejpam-5263	276	1	.	.	PUNCT
ejpam-5263	277	1	,	,	PUNCT
ejpam-5263	277	2	ad	ad	NOUN
ejpam-5263	277	3	}	}	PUNCT
ejpam-5263	277	4	be	be	AUX
ejpam-5263	277	5	a	a	DET
ejpam-5263	277	6	set	set	NOUN
ejpam-5263	277	7	of	of	ADP
ejpam-5263	277	8	linearly	linearly	ADV
ejpam-5263	277	9	independent	independent	ADJ
ejpam-5263	277	10	vectors	vector	NOUN
ejpam-5263	277	11	in	in	ADP
ejpam-5263	277	12	x.	x.	PROPN
ejpam-5263	277	13	(	(	PUNCT
ejpam-5263	277	14	what	what	PRON
ejpam-5263	277	15	happens	happen	VERB
ejpam-5263	277	16	if	if	SCONJ
ejpam-5263	277	17	we	we	PRON
ejpam-5263	277	18	use	use	VERB
ejpam-5263	277	19	only	only	ADV
ejpam-5263	277	20	n	n	ADP
ejpam-5263	277	21	vectors	vector	NOUN
ejpam-5263	277	22	will	will	AUX
ejpam-5263	277	23	be	be	AUX
ejpam-5263	277	24	discussed	discuss	VERB
ejpam-5263	277	25	later	later	ADV
ejpam-5263	277	26	,	,	PUNCT
ejpam-5263	277	27	together	together	ADV
ejpam-5263	277	28	with	with	ADP
ejpam-5263	277	29	the	the	DET
ejpam-5263	277	30	case	case	NOUN
ejpam-5263	277	31	where	where	SCONJ
ejpam-5263	277	32	d	d	PROPN
ejpam-5263	277	33	=	=	SYM
ejpam-5263	277	34	∞.	∞.	PROPN
ejpam-5263	277	35	)	)	PUNCT
ejpam-5263	277	36	we	we	PRON
ejpam-5263	277	37	define	define	VERB
ejpam-5263	277	38	the	the	DET
ejpam-5263	277	39	following	follow	VERB
ejpam-5263	277	40	inner	inner	ADJ
ejpam-5263	277	41	product	product	NOUN
ejpam-5263	277	42	on	on	ADP
ejpam-5263	277	43	x	x	NOUN
ejpam-5263	277	44	:	:	PUNCT
ejpam-5263	277	45	⟨x	⟨x	VERB
ejpam-5263	277	46	,	,	PUNCT
ejpam-5263	277	47	y⟩a	y⟩a	NUM
ejpam-5263	277	48	:	:	PUNCT
ejpam-5263	277	49	=	=	SYM
ejpam-5263	277	50	∑	∑	PUNCT
ejpam-5263	277	51	{	{	PUNCT
ejpam-5263	277	52	i2,	i2,	NOUN
ejpam-5263	277	53	...	...	PUNCT
ejpam-5263	277	54	in}⊂{1,2,	in}⊂{1,2,	NOUN
ejpam-5263	277	55	...	...	PUNCT
ejpam-5263	277	56	,n	,n	NOUN
ejpam-5263	277	57	}	}	PUNCT
ejpam-5263	277	58	⟨x	⟨x	VERB
ejpam-5263	277	59	,	,	PUNCT
ejpam-5263	277	60	y|ai2	y|ai2	NOUN
ejpam-5263	277	61	,	,	PUNCT
ejpam-5263	277	62	.	.	PUNCT
ejpam-5263	277	63	.	.	PUNCT
ejpam-5263	278	1	.	.	PUNCT
ejpam-5263	279	1	,	,	PUNCT
ejpam-5263	279	2	ain⟩.	ain⟩.	PROPN
ejpam-5263	279	3	note	note	VERB
ejpam-5263	279	4	that	that	SCONJ
ejpam-5263	279	5	there	there	PRON
ejpam-5263	279	6	are	be	VERB
ejpam-5263	279	7	(	(	PUNCT
ejpam-5263	279	8	d	d	X
ejpam-5263	279	9	n−	n−	NOUN
ejpam-5263	279	10	1	1	NUM
ejpam-5263	279	11	)	)	PUNCT
ejpam-5263	279	12	terms	term	NOUN
ejpam-5263	279	13	in	in	ADP
ejpam-5263	279	14	the	the	DET
ejpam-5263	279	15	above	above	ADJ
ejpam-5263	279	16	sum	sum	NOUN
ejpam-5263	279	17	.	.	PUNCT
ejpam-5263	280	1	analogous	analogous	ADJ
ejpam-5263	280	2	to	to	PART
ejpam-5263	280	3	theorem	theorem	VERB
ejpam-5263	280	4	2.1	2.1	NUM
ejpam-5263	280	5	,	,	PUNCT
ejpam-5263	280	6	we	we	PRON
ejpam-5263	280	7	have	have	VERB
ejpam-5263	280	8	the	the	DET
ejpam-5263	280	9	following	follow	VERB
ejpam-5263	280	10	theorem	theorem	VERB
ejpam-5263	280	11	.	.	PUNCT
ejpam-5263	281	1	theorem	theorem	NOUN
ejpam-5263	281	2	3	3	X
ejpam-5263	281	3	.	.	PUNCT
ejpam-5263	282	1	let	let	VERB
ejpam-5263	282	2	a	a	PRON
ejpam-5263	282	3	=	=	PUNCT
ejpam-5263	282	4	{	{	PUNCT
ejpam-5263	282	5	a1	a1	PROPN
ejpam-5263	282	6	,	,	PUNCT
ejpam-5263	282	7	a2	a2	PROPN
ejpam-5263	282	8	,	,	PUNCT
ejpam-5263	282	9	.	.	PUNCT
ejpam-5263	282	10	.	.	PUNCT
ejpam-5263	283	1	.	.	PUNCT
ejpam-5263	284	1	,	,	PUNCT
ejpam-5263	284	2	ad	ad	NOUN
ejpam-5263	284	3	}	}	PUNCT
ejpam-5263	284	4	⊂	⊂	PROPN
ejpam-5263	284	5	x	x	PUNCT
ejpam-5263	284	6	be	be	AUX
ejpam-5263	284	7	an	an	DET
ejpam-5263	284	8	orthogonal	orthogonal	ADJ
ejpam-5263	284	9	set	set	VERB
ejpam-5263	284	10	with	with	ADP
ejpam-5263	284	11	∥ai∥	∥ai∥	NOUN
ejpam-5263	285	1	=	=	SYM
ejpam-5263	285	2	α	α	X
ejpam-5263	285	3	>	>	X
ejpam-5263	285	4	0	0	PUNCT
ejpam-5263	286	1	for	for	ADP
ejpam-5263	286	2	all	all	DET
ejpam-5263	286	3	i	i	PRON
ejpam-5263	286	4	=	=	NOUN
ejpam-5263	286	5	1	1	NUM
ejpam-5263	286	6	,	,	PUNCT
ejpam-5263	286	7	2	2	NUM
ejpam-5263	286	8	,	,	PUNCT
ejpam-5263	286	9	.	.	PUNCT
ejpam-5263	286	10	.	.	PUNCT
ejpam-5263	286	11	.	.	PUNCT
ejpam-5263	287	1	,	,	PUNCT
ejpam-5263	287	2	d.	d.	PROPN
ejpam-5263	287	3	then	then	ADV
ejpam-5263	287	4	⟨x	⟨x	VERB
ejpam-5263	287	5	,	,	PUNCT
ejpam-5263	287	6	y⟩a	y⟩a	NUM
ejpam-5263	287	7	=	=	SYM
ejpam-5263	287	8	0	0	PUNCT
ejpam-5263	288	1	if	if	SCONJ
ejpam-5263	288	2	and	and	CCONJ
ejpam-5263	288	3	only	only	ADV
ejpam-5263	288	4	if	if	SCONJ
ejpam-5263	288	5	⟨x	⟨x	VERB
ejpam-5263	288	6	,	,	PUNCT
ejpam-5263	288	7	y⟩	y⟩	NOUN
ejpam-5263	288	8	=	=	NOUN
ejpam-5263	288	9	0	0	NUM
ejpam-5263	288	10	for	for	ADP
ejpam-5263	288	11	all	all	DET
ejpam-5263	288	12	x	x	NOUN
ejpam-5263	288	13	,	,	PUNCT
ejpam-5263	288	14	y	y	PROPN
ejpam-5263	288	15	∈	∈	PROPN
ejpam-5263	288	16	x.	x.	NOUN
ejpam-5263	288	17	proof	proof	NOUN
ejpam-5263	288	18	.	.	PUNCT
ejpam-5263	289	1	let	let	VERB
ejpam-5263	289	2	i	i	PRON
ejpam-5263	289	3	d	d	NOUN
ejpam-5263	289	4	:	:	PUNCT
ejpam-5263	289	5	=	=	SYM
ejpam-5263	289	6	{	{	PUNCT
ejpam-5263	289	7	1	1	NUM
ejpam-5263	289	8	,	,	PUNCT
ejpam-5263	289	9	2	2	NUM
ejpam-5263	289	10	,	,	PUNCT
ejpam-5263	289	11	.	.	PUNCT
ejpam-5263	289	12	.	.	PUNCT
ejpam-5263	290	1	.	.	PUNCT
ejpam-5263	291	1	,	,	PUNCT
ejpam-5263	291	2	d	d	X
ejpam-5263	291	3	}	}	PUNCT
ejpam-5263	291	4	.	.	PUNCT
ejpam-5263	292	1	for	for	ADP
ejpam-5263	292	2	any	any	DET
ejpam-5263	292	3	i0	i0	PROPN
ejpam-5263	292	4	:	:	PUNCT
ejpam-5263	292	5	=	=	SYM
ejpam-5263	292	6	{	{	PUNCT
ejpam-5263	292	7	i2	i2	PROPN
ejpam-5263	292	8	,	,	PUNCT
ejpam-5263	292	9	.	.	PUNCT
ejpam-5263	292	10	.	.	PUNCT
ejpam-5263	292	11	.	.	PUNCT
ejpam-5263	293	1	,	,	PUNCT
ejpam-5263	293	2	in	in	ADP
ejpam-5263	293	3	}	}	PUNCT
ejpam-5263	293	4	⊂	⊂	PROPN
ejpam-5263	293	5	i	i	PROPN
ejpam-5263	293	6	d	d	PROPN
ejpam-5263	293	7	,	,	PUNCT
ejpam-5263	293	8	let	let	VERB
ejpam-5263	293	9	i1	i1	PROPN
ejpam-5263	293	10	:	:	PUNCT
ejpam-5263	294	1	=	=	SYM
ejpam-5263	294	2	i	i	PROPN
ejpam-5263	294	3	d	d	PROPN
ejpam-5263	294	4	\	\	PROPN
ejpam-5263	294	5	i0	i0	PROPN
ejpam-5263	294	6	.	.	PUNCT
ejpam-5263	295	1	then	then	ADV
ejpam-5263	295	2	,	,	PUNCT
ejpam-5263	295	3	we	we	PRON
ejpam-5263	295	4	have	have	AUX
ejpam-5263	295	5	⟨x	⟨x	VERB
ejpam-5263	295	6	,	,	PUNCT
ejpam-5263	295	7	y|ai2	y|ai2	NOUN
ejpam-5263	295	8	,	,	PUNCT
ejpam-5263	295	9	.	.	PUNCT
ejpam-5263	295	10	.	.	PUNCT
ejpam-5263	295	11	.	.	PUNCT
ejpam-5263	296	1	ain⟩	ain⟩	X
ejpam-5263	297	1	=	=	SYM
ejpam-5263	297	2	∑	∑	PUNCT
ejpam-5263	297	3	i∈i1	i∈i1	PROPN
ejpam-5263	297	4	⟨x	⟨x	VERB
ejpam-5263	297	5	,	,	PUNCT
ejpam-5263	297	6	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	297	7	,	,	PUNCT
ejpam-5263	297	8	y⟩	y⟩	NOUN
ejpam-5263	297	9	∥ai∥2	∥ai∥2	VERB
ejpam-5263	298	1	∏	∏	NUM
ejpam-5263	298	2	i∈i0	i∈i0	NOUN
ejpam-5263	298	3	∥ai∥2	∥ai∥2	VERB
ejpam-5263	299	1	=	=	X
ejpam-5263	299	2	[	[	PUNCT
ejpam-5263	299	3	⟨x	⟨x	NUM
ejpam-5263	299	4	,	,	PUNCT
ejpam-5263	299	5	y⟩	y⟩	NOUN
ejpam-5263	299	6	−	−	NOUN
ejpam-5263	299	7	∑	∑	PROPN
ejpam-5263	299	8	i∈i0	i∈i0	NOUN
ejpam-5263	299	9	⟨x	⟨x	VERB
ejpam-5263	299	10	,	,	PUNCT
ejpam-5263	299	11	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	299	12	,	,	PUNCT
ejpam-5263	299	13	y⟩	y⟩	NOUN
ejpam-5263	299	14	∥ai∥2	∥ai∥2	VERB
ejpam-5263	299	15	]	]	PUNCT
ejpam-5263	300	1	∏	∏	NUM
ejpam-5263	300	2	i∈i0	i∈i0	NOUN
ejpam-5263	300	3	∥ai∥2	∥ai∥2	VERB
ejpam-5263	300	4	.	.	PUNCT
ejpam-5263	301	1	a.	a.	PROPN
ejpam-5263	301	2	adam	adam	PROPN
ejpam-5263	301	3	,	,	PUNCT
ejpam-5263	301	4	s.	s.	PROPN
ejpam-5263	301	5	rante	rante	PROPN
ejpam-5263	301	6	,	,	PUNCT
ejpam-5263	301	7	h.	h.	PROPN
ejpam-5263	301	8	gunawan	gunawan	PROPN
ejpam-5263	301	9	/	/	PUNCT
ejpam-5263	301	10	eur	eur	PROPN
ejpam-5263	301	11	.	.	PUNCT
ejpam-5263	302	1	j.	j.	PROPN
ejpam-5263	302	2	pure	pure	PROPN
ejpam-5263	302	3	appl	appl	PROPN
ejpam-5263	302	4	.	.	PROPN
ejpam-5263	302	5	math	math	PROPN
ejpam-5263	302	6	,	,	PUNCT
ejpam-5263	302	7	17	17	NUM
ejpam-5263	302	8	(	(	PUNCT
ejpam-5263	302	9	3	3	NUM
ejpam-5263	302	10	)	)	PUNCT
ejpam-5263	302	11	(	(	PUNCT
ejpam-5263	302	12	2024	2024	NUM
ejpam-5263	302	13	)	)	PUNCT
ejpam-5263	302	14	,	,	PUNCT
ejpam-5263	302	15	1937	1937	NUM
ejpam-5263	302	16	-	-	SYM
ejpam-5263	302	17	1947	1947	NUM
ejpam-5263	302	18	1943	1943	NUM
ejpam-5263	302	19	summing	sum	VERB
ejpam-5263	302	20	over	over	ADP
ejpam-5263	302	21	all	all	DET
ejpam-5263	302	22	subsets	subset	NOUN
ejpam-5263	302	23	i0	i0	PROPN
ejpam-5263	302	24	⊂	⊂	PROPN
ejpam-5263	303	1	i	i	PROPN
ejpam-5263	303	2	d	d	PROPN
ejpam-5263	303	3	and	and	CCONJ
ejpam-5263	303	4	using	use	VERB
ejpam-5263	303	5	the	the	DET
ejpam-5263	303	6	assumption	assumption	NOUN
ejpam-5263	303	7	that	that	SCONJ
ejpam-5263	303	8	∥ai∥	∥ai∥	VERB
ejpam-5263	303	9	=	=	PUNCT
ejpam-5263	303	10	α	α	PROPN
ejpam-5263	303	11	for	for	ADP
ejpam-5263	303	12	all	all	DET
ejpam-5263	303	13	i	i	PRON
ejpam-5263	303	14	=	=	NOUN
ejpam-5263	303	15	1	1	NUM
ejpam-5263	303	16	,	,	PUNCT
ejpam-5263	303	17	2	2	NUM
ejpam-5263	303	18	,	,	PUNCT
ejpam-5263	303	19	.	.	PUNCT
ejpam-5263	303	20	.	.	PUNCT
ejpam-5263	304	1	.	.	PUNCT
ejpam-5263	305	1	n	n	CCONJ
ejpam-5263	305	2	,	,	PUNCT
ejpam-5263	305	3	we	we	PRON
ejpam-5263	305	4	obtain	obtain	VERB
ejpam-5263	305	5	⟨x	⟨x	VERB
ejpam-5263	305	6	,	,	PUNCT
ejpam-5263	305	7	y⟩a	y⟩a	NUM
ejpam-5263	305	8	=	=	SYM
ejpam-5263	305	9	∑	∑	PROPN
ejpam-5263	305	10	i0⊂id	i0⊂id	PROPN
ejpam-5263	305	11	[	[	PUNCT
ejpam-5263	305	12	⟨x	⟨x	NUM
ejpam-5263	305	13	,	,	PUNCT
ejpam-5263	305	14	y⟩	y⟩	NOUN
ejpam-5263	305	15	−	−	PROPN
ejpam-5263	305	16	⟨x	⟨x	NUM
ejpam-5263	305	17	,	,	PUNCT
ejpam-5263	305	18	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	305	19	,	,	PUNCT
ejpam-5263	305	20	y⟩	y⟩	NOUN
ejpam-5263	305	21	∥ai∥2	∥ai∥2	VERB
ejpam-5263	305	22	]	]	PUNCT
ejpam-5263	306	1	α2(n−1	α2(n−1	X
ejpam-5263	306	2	)	)	PUNCT
ejpam-5263	306	3	=	=	PUNCT
ejpam-5263	307	1	∑	∑	PUNCT
ejpam-5263	307	2	i0⊂id	i0⊂id	PROPN
ejpam-5263	307	3	α2(n−1)⟨x	α2(n−1)⟨x	NUM
ejpam-5263	307	4	,	,	PUNCT
ejpam-5263	307	5	y⟩	y⟩	NOUN
ejpam-5263	307	6	−	−	PROPN
ejpam-5263	308	1	∑	∑	PROPN
ejpam-5263	308	2	i0⊂id	i0⊂id	PROPN
ejpam-5263	308	3	∑	∑	VERB
ejpam-5263	308	4	i∈i0	i∈i0	VERB
ejpam-5263	308	5	α2(n−1	α2(n−1	PROPN
ejpam-5263	308	6	)	)	PUNCT
ejpam-5263	308	7	⟨x	⟨x	NUM
ejpam-5263	308	8	,	,	PUNCT
ejpam-5263	308	9	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	308	10	,	,	PUNCT
ejpam-5263	308	11	y⟩	y⟩	NOUN
ejpam-5263	308	12	α2	α2	PROPN
ejpam-5263	308	13	.	.	PUNCT
ejpam-5263	309	1	the	the	DET
ejpam-5263	309	2	first	first	ADJ
ejpam-5263	309	3	sum	sum	NOUN
ejpam-5263	309	4	on	on	ADP
ejpam-5263	309	5	the	the	DET
ejpam-5263	309	6	right	right	ADJ
ejpam-5263	309	7	hand	hand	NOUN
ejpam-5263	309	8	side	side	NOUN
ejpam-5263	309	9	is	be	AUX
ejpam-5263	309	10	equal	equal	ADJ
ejpam-5263	309	11	to	to	ADP
ejpam-5263	309	12	(	(	PUNCT
ejpam-5263	309	13	d	d	NUM
ejpam-5263	309	14	n−	n−	NOUN
ejpam-5263	309	15	1	1	NUM
ejpam-5263	309	16	)	)	PUNCT
ejpam-5263	309	17	α2(n−1)⟨x	α2(n−1)⟨x	NUM
ejpam-5263	309	18	,	,	PUNCT
ejpam-5263	309	19	y⟩.	y⟩.	NUM
ejpam-5263	309	20	for	for	ADP
ejpam-5263	309	21	the	the	DET
ejpam-5263	309	22	second	second	ADJ
ejpam-5263	309	23	sum	sum	NOUN
ejpam-5263	309	24	,	,	PUNCT
ejpam-5263	309	25	we	we	PRON
ejpam-5263	309	26	observe	observe	VERB
ejpam-5263	309	27	that	that	SCONJ
ejpam-5263	309	28	each	each	DET
ejpam-5263	309	29	expression	expression	NOUN
ejpam-5263	309	30	⟨x	⟨x	NUM
ejpam-5263	309	31	,	,	PUNCT
ejpam-5263	309	32	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	309	33	,	,	PUNCT
ejpam-5263	309	34	y⟩	y⟩	NOUN
ejpam-5263	309	35	occurs	occur	VERB
ejpam-5263	309	36	precisely	precisely	ADV
ejpam-5263	309	37	(	(	PUNCT
ejpam-5263	309	38	d−	d−	PROPN
ejpam-5263	309	39	1	1	NUM
ejpam-5263	309	40	n−	n−	NOUN
ejpam-5263	309	41	2	2	NUM
ejpam-5263	309	42	)	)	PUNCT
ejpam-5263	309	43	times	time	NOUN
ejpam-5263	309	44	for	for	ADP
ejpam-5263	309	45	all	all	PRON
ejpam-5263	309	46	i	i	PRON
ejpam-5263	309	47	∈	∈	PROPN
ejpam-5263	310	1	i	i	PRON
ejpam-5263	310	2	d.	d.	PROPN
ejpam-5263	310	3	hence	hence	ADV
ejpam-5263	310	4	,	,	PUNCT
ejpam-5263	310	5	by	by	ADP
ejpam-5263	310	6	parseval	parseval	NOUN
ejpam-5263	310	7	’s	’s	PART
ejpam-5263	310	8	identity	identity	NOUN
ejpam-5263	310	9	,	,	PUNCT
ejpam-5263	310	10	the	the	DET
ejpam-5263	310	11	second	second	ADJ
ejpam-5263	310	12	sum	sum	NOUN
ejpam-5263	310	13	is	be	AUX
ejpam-5263	310	14	equal	equal	ADJ
ejpam-5263	310	15	(	(	PUNCT
ejpam-5263	310	16	d−	d−	PROPN
ejpam-5263	310	17	1	1	NUM
ejpam-5263	310	18	n−	n−	NOUN
ejpam-5263	310	19	2	2	NUM
ejpam-5263	310	20	)	)	PUNCT
ejpam-5263	310	21	α2(n−1)⟨x	α2(n−1)⟨x	NUM
ejpam-5263	310	22	,	,	PUNCT
ejpam-5263	310	23	y⟩.	y⟩.	PRON
ejpam-5263	310	24	therefore	therefore	ADV
ejpam-5263	310	25	we	we	PRON
ejpam-5263	310	26	get	get	AUX
ejpam-5263	310	27	⟨x	⟨x	VERB
ejpam-5263	310	28	,	,	PUNCT
ejpam-5263	310	29	y⟩a	y⟩a	X
ejpam-5263	310	30	=	=	PUNCT
ejpam-5263	311	1	[	[	X
ejpam-5263	311	2	(	(	PUNCT
ejpam-5263	311	3	d	d	NUM
ejpam-5263	311	4	n−	n−	NOUN
ejpam-5263	311	5	1	1	NUM
ejpam-5263	311	6	)	)	PUNCT
ejpam-5263	311	7	−	−	PROPN
ejpam-5263	311	8	(	(	PUNCT
ejpam-5263	311	9	d−	d−	PROPN
ejpam-5263	311	10	1	1	NUM
ejpam-5263	311	11	n−	n−	NOUN
ejpam-5263	311	12	2	2	NUM
ejpam-5263	311	13	)	)	PUNCT
ejpam-5263	311	14	]	]	PUNCT
ejpam-5263	311	15	α2(n−1)⟨x	α2(n−1)⟨x	NUM
ejpam-5263	311	16	,	,	PUNCT
ejpam-5263	311	17	y⟩	y⟩	NOUN
ejpam-5263	311	18	=	=	PUNCT
ejpam-5263	311	19	(	(	PUNCT
ejpam-5263	311	20	d−	d−	PROPN
ejpam-5263	311	21	1	1	NUM
ejpam-5263	311	22	n−	n−	NOUN
ejpam-5263	311	23	1	1	NUM
ejpam-5263	311	24	)	)	PUNCT
ejpam-5263	311	25	α2(n−1)⟨x	α2(n−1)⟨x	NUM
ejpam-5263	311	26	,	,	PUNCT
ejpam-5263	311	27	y⟩	y⟩	NOUN
ejpam-5263	311	28	,	,	PUNCT
ejpam-5263	311	29	which	which	PRON
ejpam-5263	311	30	gives	give	VERB
ejpam-5263	311	31	us	we	PRON
ejpam-5263	311	32	the	the	DET
ejpam-5263	311	33	desired	desire	VERB
ejpam-5263	311	34	conclusion	conclusion	NOUN
ejpam-5263	311	35	.	.	PUNCT
ejpam-5263	312	1	corollary	corollary	ADJ
ejpam-5263	312	2	1	1	NUM
ejpam-5263	312	3	.	.	PUNCT
ejpam-5263	313	1	if	if	SCONJ
ejpam-5263	313	2	a	a	PRON
ejpam-5263	313	3	=	=	PUNCT
ejpam-5263	313	4	{	{	PUNCT
ejpam-5263	313	5	a1	a1	PROPN
ejpam-5263	313	6	,	,	PUNCT
ejpam-5263	313	7	a2	a2	PROPN
ejpam-5263	313	8	,	,	PUNCT
ejpam-5263	313	9	.	.	PUNCT
ejpam-5263	313	10	.	.	PUNCT
ejpam-5263	313	11	.	.	PUNCT
ejpam-5263	314	1	,	,	PUNCT
ejpam-5263	314	2	ad	ad	NOUN
ejpam-5263	314	3	}	}	PUNCT
ejpam-5263	314	4	is	be	AUX
ejpam-5263	314	5	an	an	DET
ejpam-5263	314	6	orthonormal	orthonormal	ADJ
ejpam-5263	314	7	basis	basis	NOUN
ejpam-5263	314	8	for	for	ADP
ejpam-5263	314	9	x	x	X
ejpam-5263	314	10	,	,	PUNCT
ejpam-5263	314	11	then	then	ADV
ejpam-5263	314	12	⟨x	⟨x	VERB
ejpam-5263	314	13	,	,	PUNCT
ejpam-5263	314	14	y⟩a	y⟩a	NUM
ejpam-5263	314	15	=(	=(	NOUN
ejpam-5263	314	16	d−	d−	PROPN
ejpam-5263	314	17	1	1	NUM
ejpam-5263	314	18	n−	n−	PROPN
ejpam-5263	314	19	1	1	NUM
ejpam-5263	314	20	)	)	PUNCT
ejpam-5263	314	21	⟨x	⟨x	VERB
ejpam-5263	314	22	,	,	PUNCT
ejpam-5263	314	23	y⟩	y⟩	NOUN
ejpam-5263	314	24	for	for	ADP
ejpam-5263	314	25	all	all	DET
ejpam-5263	314	26	x	x	NOUN
ejpam-5263	314	27	,	,	PUNCT
ejpam-5263	314	28	y	y	PROPN
ejpam-5263	314	29	∈	∈	PROPN
ejpam-5263	314	30	x.	x.	NOUN
ejpam-5263	314	31	remark	remark	VERB
ejpam-5263	314	32	1	1	NUM
ejpam-5263	314	33	.	.	PUNCT
ejpam-5263	315	1	the	the	DET
ejpam-5263	315	2	converse	converse	NOUN
ejpam-5263	315	3	of	of	ADP
ejpam-5263	315	4	the	the	DET
ejpam-5263	315	5	above	above	ADJ
ejpam-5263	315	6	corollary	corollary	NOUN
ejpam-5263	315	7	does	do	AUX
ejpam-5263	315	8	not	not	PART
ejpam-5263	315	9	hold	hold	VERB
ejpam-5263	315	10	.	.	PUNCT
ejpam-5263	316	1	to	to	PART
ejpam-5263	316	2	give	give	VERB
ejpam-5263	316	3	an	an	DET
ejpam-5263	316	4	example	example	NOUN
ejpam-5263	316	5	,	,	PUNCT
ejpam-5263	316	6	let	let	VERB
ejpam-5263	316	7	d	d	NOUN
ejpam-5263	316	8	=	=	NOUN
ejpam-5263	316	9	dimx	dimx	NOUN
ejpam-5263	316	10	=	=	SYM
ejpam-5263	316	11	3	3	NUM
ejpam-5263	316	12	and	and	CCONJ
ejpam-5263	316	13	n	n	NOUN
ejpam-5263	316	14	=	=	SYM
ejpam-5263	316	15	2	2	X
ejpam-5263	316	16	.	.	PUNCT
ejpam-5263	316	17	let	let	VERB
ejpam-5263	316	18	a	a	DET
ejpam-5263	316	19	=	=	PUNCT
ejpam-5263	316	20	{	{	PUNCT
ejpam-5263	316	21	a1	a1	PROPN
ejpam-5263	316	22	,	,	PUNCT
ejpam-5263	316	23	a2	a2	PROPN
ejpam-5263	316	24	,	,	PUNCT
ejpam-5263	316	25	a3	a3	NOUN
ejpam-5263	316	26	}	}	PUNCT
ejpam-5263	316	27	be	be	VERB
ejpam-5263	316	28	linearly	linearly	ADV
ejpam-5263	316	29	independent	independent	ADJ
ejpam-5263	316	30	set	set	NOUN
ejpam-5263	316	31	in	in	ADP
ejpam-5263	316	32	x.	x.	NOUN
ejpam-5263	316	33	suppose	suppose	VERB
ejpam-5263	316	34	that	that	SCONJ
ejpam-5263	316	35	⟨x	⟨x	VERB
ejpam-5263	316	36	,	,	PUNCT
ejpam-5263	316	37	y⟩a	y⟩a	PROPN
ejpam-5263	316	38	=	=	SYM
ejpam-5263	316	39	2⟨x	2⟨x	PROPN
ejpam-5263	316	40	,	,	PUNCT
ejpam-5263	316	41	y⟩	y⟩	NOUN
ejpam-5263	316	42	for	for	ADP
ejpam-5263	316	43	all	all	DET
ejpam-5263	316	44	x	x	NOUN
ejpam-5263	316	45	,	,	PUNCT
ejpam-5263	316	46	y	y	PROPN
ejpam-5263	316	47	∈	∈	PROPN
ejpam-5263	316	48	x.	x.	NOUN
ejpam-5263	317	1	we	we	PRON
ejpam-5263	317	2	would	would	AUX
ejpam-5263	317	3	like	like	VERB
ejpam-5263	317	4	to	to	PART
ejpam-5263	317	5	check	check	VERB
ejpam-5263	317	6	whether	whether	SCONJ
ejpam-5263	317	7	we	we	PRON
ejpam-5263	317	8	have	have	VERB
ejpam-5263	317	9	∥ai∥	∥ai∥	NOUN
ejpam-5263	318	1	=	=	SYM
ejpam-5263	318	2	1	1	NUM
ejpam-5263	318	3	for	for	ADP
ejpam-5263	318	4	i	i	PRON
ejpam-5263	318	5	=	=	NOUN
ejpam-5263	318	6	1	1	NUM
ejpam-5263	318	7	,	,	PUNCT
ejpam-5263	318	8	2	2	NUM
ejpam-5263	318	9	,	,	PUNCT
ejpam-5263	318	10	3	3	NUM
ejpam-5263	318	11	and	and	CCONJ
ejpam-5263	318	12	⟨ai	⟨ai	NOUN
ejpam-5263	318	13	,	,	PUNCT
ejpam-5263	318	14	aj⟩	aj⟩	X
ejpam-5263	318	15	=	=	SYM
ejpam-5263	318	16	0	0	NUM
ejpam-5263	318	17	for	for	ADP
ejpam-5263	318	18	i	i	PRON
ejpam-5263	318	19	̸=	̸=	PROPN
ejpam-5263	318	20	j.	j.	PROPN
ejpam-5263	318	21	notice	notice	VERB
ejpam-5263	318	22	that	that	SCONJ
ejpam-5263	318	23	⟨x	⟨x	VERB
ejpam-5263	318	24	,	,	PUNCT
ejpam-5263	318	25	y⟩a	y⟩a	NUM
ejpam-5263	318	26	=	=	SYM
ejpam-5263	318	27	3∑	3∑	NUM
ejpam-5263	318	28	i=1	i=1	PRON
ejpam-5263	318	29	⟨x	⟨x	NUM
ejpam-5263	318	30	,	,	PUNCT
ejpam-5263	318	31	y|ai⟩	y|ai⟩	NOUN
ejpam-5263	318	32	=	=	NUM
ejpam-5263	318	33	3∑	3∑	NUM
ejpam-5263	318	34	i=1	i=1	PRON
ejpam-5263	318	35	∣∣∣∣	∣∣∣∣	NOUN
ejpam-5263	318	36	⟨x	⟨x	VERB
ejpam-5263	318	37	,	,	PUNCT
ejpam-5263	318	38	y⟩	y⟩	NOUN
ejpam-5263	318	39	⟨x	⟨x	NUM
ejpam-5263	318	40	,	,	PUNCT
ejpam-5263	318	41	ai⟩	ai⟩	PROPN
ejpam-5263	318	42	⟨ai	⟨ai	NOUN
ejpam-5263	318	43	,	,	PUNCT
ejpam-5263	318	44	y⟩	y⟩	NOUN
ejpam-5263	318	45	⟨ai	⟨ai	NOUN
ejpam-5263	318	46	,	,	PUNCT
ejpam-5263	318	47	ai⟩	ai⟩	PROPN
ejpam-5263	318	48	∣∣∣∣	∣∣∣∣	VERB
ejpam-5263	318	49	for	for	ADP
ejpam-5263	318	50	all	all	DET
ejpam-5263	318	51	x	x	NOUN
ejpam-5263	318	52	,	,	PUNCT
ejpam-5263	318	53	y	y	PROPN
ejpam-5263	318	54	∈	∈	PROPN
ejpam-5263	318	55	x.	x.	NOUN
ejpam-5263	318	56	from	from	ADP
ejpam-5263	318	57	the	the	DET
ejpam-5263	318	58	hypothesis	hypothesis	NOUN
ejpam-5263	318	59	,	,	PUNCT
ejpam-5263	318	60	we	we	PRON
ejpam-5263	318	61	have	have	VERB
ejpam-5263	318	62	⟨ai	⟨ai	NOUN
ejpam-5263	318	63	,	,	PUNCT
ejpam-5263	318	64	aj⟩a	aj⟩a	PROPN
ejpam-5263	319	1	=	=	SYM
ejpam-5263	319	2	2⟨ai	2⟨ai	NUM
ejpam-5263	319	3	,	,	PUNCT
ejpam-5263	319	4	aj⟩	aj⟩	VERB
ejpam-5263	319	5	for	for	ADP
ejpam-5263	319	6	i	i	PRON
ejpam-5263	319	7	,	,	PUNCT
ejpam-5263	319	8	j	j	PROPN
ejpam-5263	319	9	=	=	SYM
ejpam-5263	319	10	1	1	NUM
ejpam-5263	319	11	,	,	PUNCT
ejpam-5263	319	12	2	2	NUM
ejpam-5263	319	13	,	,	PUNCT
ejpam-5263	319	14	3	3	NUM
ejpam-5263	319	15	,	,	PUNCT
ejpam-5263	319	16	which	which	PRON
ejpam-5263	319	17	may	may	AUX
ejpam-5263	319	18	be	be	AUX
ejpam-5263	319	19	rewritten	rewrite	VERB
ejpam-5263	319	20	as	as	ADP
ejpam-5263	319	21	∥a1∥2	∥a1∥2	ADV
ejpam-5263	319	22	∥a2∥2	∥a2∥2	NUM
ejpam-5263	319	23	−	−	PUNCT
ejpam-5263	319	24	⟨a1	⟨a1	PROPN
ejpam-5263	319	25	,	,	PUNCT
ejpam-5263	319	26	a2⟩2	a2⟩2	PROPN
ejpam-5263	319	27	+	+	CCONJ
ejpam-5263	319	28	∥a1∥2	∥a1∥2	PROPN
ejpam-5263	319	29	∥a3∥2	∥a3∥2	NOUN
ejpam-5263	319	30	−	−	PROPN
ejpam-5263	320	1	⟨a1	⟨a1	PROPN
ejpam-5263	320	2	,	,	PUNCT
ejpam-5263	320	3	a3⟩2	a3⟩2	PROPN
ejpam-5263	320	4	=	=	PUNCT
ejpam-5263	320	5	2	2	NUM
ejpam-5263	320	6	∥a1∥2	∥a1∥2	ADV
ejpam-5263	320	7	,	,	PUNCT
ejpam-5263	320	8	∥a1∥2	∥a1∥2	ADV
ejpam-5263	320	9	∥a2∥2	∥a2∥2	NUM
ejpam-5263	320	10	−	−	PUNCT
ejpam-5263	320	11	⟨a1	⟨a1	PROPN
ejpam-5263	320	12	,	,	PUNCT
ejpam-5263	320	13	a2⟩2	a2⟩2	PROPN
ejpam-5263	320	14	+	+	CCONJ
ejpam-5263	320	15	∥a2∥2	∥a2∥2	NUM
ejpam-5263	320	16	∥a3∥2	∥a3∥2	NOUN
ejpam-5263	320	17	−	−	PROPN
ejpam-5263	320	18	⟨a2	⟨a2	PROPN
ejpam-5263	320	19	,	,	PUNCT
ejpam-5263	320	20	a3⟩2	a3⟩2	PROPN
ejpam-5263	320	21	=	=	NOUN
ejpam-5263	320	22	2	2	NUM
ejpam-5263	320	23	∥a2∥2	∥a2∥2	NUM
ejpam-5263	320	24	,	,	PUNCT
ejpam-5263	320	25	∥a1∥2	∥a1∥2	PROPN
ejpam-5263	320	26	∥a3∥2	∥a3∥2	PROPN
ejpam-5263	320	27	−	−	PROPN
ejpam-5263	320	28	⟨a1	⟨a1	PROPN
ejpam-5263	320	29	,	,	PUNCT
ejpam-5263	320	30	a3⟩2	a3⟩2	PROPN
ejpam-5263	320	31	+	+	CCONJ
ejpam-5263	320	32	∥a2∥2	∥a2∥2	NUM
ejpam-5263	320	33	∥a3∥2	∥a3∥2	PROPN
ejpam-5263	320	34	−	−	PROPN
ejpam-5263	321	1	⟨a2	⟨a2	PROPN
ejpam-5263	321	2	,	,	PUNCT
ejpam-5263	321	3	a3⟩2	a3⟩2	PROPN
ejpam-5263	321	4	=	=	PROPN
ejpam-5263	321	5	2	2	NUM
ejpam-5263	321	6	∥a3∥2	∥a3∥2	PROPN
ejpam-5263	321	7	,	,	PUNCT
ejpam-5263	321	8	⟨a1	⟨a1	PROPN
ejpam-5263	321	9	,	,	PUNCT
ejpam-5263	321	10	a2⟩	a2⟩	PUNCT
ejpam-5263	321	11	∥a3∥2	∥a3∥2	PROPN
ejpam-5263	321	12	−	−	PROPN
ejpam-5263	321	13	⟨a1	⟨a1	PROPN
ejpam-5263	321	14	,	,	PUNCT
ejpam-5263	321	15	a3⟩⟨a2	a3⟩⟨a2	NOUN
ejpam-5263	321	16	,	,	PUNCT
ejpam-5263	321	17	a3⟩	a3⟩	NOUN
ejpam-5263	321	18	=	=	SYM
ejpam-5263	321	19	2⟨a1	2⟨a1	NUM
ejpam-5263	321	20	,	,	PUNCT
ejpam-5263	321	21	a2⟩	a2⟩	NUM
ejpam-5263	321	22	,	,	PUNCT
ejpam-5263	321	23	⟨a1	⟨a1	PROPN
ejpam-5263	321	24	,	,	PUNCT
ejpam-5263	321	25	a3⟩	a3⟩	NOUN
ejpam-5263	322	1	∥a2∥2	∥a2∥2	NUM
ejpam-5263	322	2	−	−	PROPN
ejpam-5263	322	3	⟨a1	⟨a1	PROPN
ejpam-5263	322	4	,	,	PUNCT
ejpam-5263	322	5	a2⟩⟨a2	a2⟩⟨a2	NOUN
ejpam-5263	322	6	,	,	PUNCT
ejpam-5263	322	7	a3⟩	a3⟩	NOUN
ejpam-5263	322	8	=	=	SYM
ejpam-5263	322	9	2⟨a1	2⟨a1	NUM
ejpam-5263	322	10	,	,	PUNCT
ejpam-5263	322	11	a3⟩	a3⟩	NOUN
ejpam-5263	322	12	,	,	PUNCT
ejpam-5263	322	13	⟨a2	⟨a2	NOUN
ejpam-5263	322	14	,	,	PUNCT
ejpam-5263	322	15	a3⟩	a3⟩	NOUN
ejpam-5263	322	16	∥a1∥2	∥a1∥2	ADV
ejpam-5263	322	17	−	−	PROPN
ejpam-5263	323	1	⟨a1	⟨a1	PROPN
ejpam-5263	323	2	,	,	PUNCT
ejpam-5263	323	3	a2⟩⟨a1	a2⟩⟨a1	PROPN
ejpam-5263	323	4	,	,	PUNCT
ejpam-5263	323	5	a3⟩	a3⟩	NOUN
ejpam-5263	323	6	=	=	SYM
ejpam-5263	323	7	2⟨a2	2⟨a2	NUM
ejpam-5263	323	8	,	,	PUNCT
ejpam-5263	323	9	a3⟩.	a3⟩.	ADJ
ejpam-5263	323	10	a.	a.	NOUN
ejpam-5263	323	11	adam	adam	PROPN
ejpam-5263	323	12	,	,	PUNCT
ejpam-5263	323	13	s.	s.	PROPN
ejpam-5263	323	14	rante	rante	PROPN
ejpam-5263	323	15	,	,	PUNCT
ejpam-5263	323	16	h.	h.	PROPN
ejpam-5263	323	17	gunawan	gunawan	PROPN
ejpam-5263	323	18	/	/	PUNCT
ejpam-5263	323	19	eur	eur	PROPN
ejpam-5263	323	20	.	.	PUNCT
ejpam-5263	324	1	j.	j.	PROPN
ejpam-5263	324	2	pure	pure	PROPN
ejpam-5263	324	3	appl	appl	PROPN
ejpam-5263	324	4	.	.	PROPN
ejpam-5263	324	5	math	math	PROPN
ejpam-5263	324	6	,	,	PUNCT
ejpam-5263	324	7	17	17	NUM
ejpam-5263	324	8	(	(	PUNCT
ejpam-5263	324	9	3	3	NUM
ejpam-5263	324	10	)	)	PUNCT
ejpam-5263	324	11	(	(	PUNCT
ejpam-5263	324	12	2024	2024	NUM
ejpam-5263	324	13	)	)	PUNCT
ejpam-5263	324	14	,	,	PUNCT
ejpam-5263	324	15	1937	1937	NUM
ejpam-5263	324	16	-	-	SYM
ejpam-5263	324	17	1947	1947	NUM
ejpam-5263	324	18	1944	1944	NUM
ejpam-5263	324	19	let	let	VERB
ejpam-5263	324	20	a	a	DET
ejpam-5263	324	21	:	:	PUNCT
ejpam-5263	324	22	=	=	SYM
ejpam-5263	324	23	∥a1∥	∥a1∥	PROPN
ejpam-5263	324	24	,	,	PUNCT
ejpam-5263	324	25	b	b	X
ejpam-5263	324	26	:	:	PUNCT
ejpam-5263	324	27	=	=	SYM
ejpam-5263	324	28	∥a2∥	∥a2∥	PROPN
ejpam-5263	324	29	,	,	PUNCT
ejpam-5263	324	30	c	c	NOUN
ejpam-5263	324	31	:	:	PUNCT
ejpam-5263	325	1	=	=	SYM
ejpam-5263	325	2	∥a3∥	∥a3∥	ADJ
ejpam-5263	325	3	,	,	PUNCT
ejpam-5263	325	4	d	d	X
ejpam-5263	325	5	:	:	PUNCT
ejpam-5263	325	6	=	=	SYM
ejpam-5263	325	7	⟨a1	⟨a1	PROPN
ejpam-5263	325	8	,	,	PUNCT
ejpam-5263	325	9	a2⟩	a2⟩	NUM
ejpam-5263	325	10	,	,	PUNCT
ejpam-5263	325	11	e	e	X
ejpam-5263	325	12	:	:	PUNCT
ejpam-5263	325	13	=	=	SYM
ejpam-5263	325	14	⟨a1	⟨a1	PROPN
ejpam-5263	325	15	,	,	PUNCT
ejpam-5263	325	16	a3⟩	a3⟩	NOUN
ejpam-5263	325	17	,	,	PUNCT
ejpam-5263	325	18	f	f	X
ejpam-5263	325	19	:	:	PUNCT
ejpam-5263	325	20	=	=	SYM
ejpam-5263	325	21	⟨a2	⟨a2	NOUN
ejpam-5263	325	22	,	,	PUNCT
ejpam-5263	325	23	a3⟩.	a3⟩.	ADJ
ejpam-5263	325	24	then	then	ADV
ejpam-5263	325	25	a2b2	a2b2	VERB
ejpam-5263	325	26	−d2	−d2	NOUN
ejpam-5263	325	27	+	+	ADJ
ejpam-5263	325	28	a2c2	a2c2	INTJ
ejpam-5263	325	29	−	−	PROPN
ejpam-5263	325	30	e2	e2	NOUN
ejpam-5263	325	31	=	=	SYM
ejpam-5263	325	32	2a2	2a2	NUM
ejpam-5263	325	33	,	,	PUNCT
ejpam-5263	325	34	a2b2	a2b2	PUNCT
ejpam-5263	326	1	−d2	−d2	VERB
ejpam-5263	326	2	+	+	ADJ
ejpam-5263	326	3	b2c2	b2c2	X
ejpam-5263	326	4	−	−	NOUN
ejpam-5263	326	5	f	f	NOUN
ejpam-5263	326	6	2	2	NUM
ejpam-5263	326	7	=	=	SYM
ejpam-5263	326	8	2b2	2b2	NUM
ejpam-5263	326	9	,	,	PUNCT
ejpam-5263	326	10	a2c2	a2c2	ADP
ejpam-5263	326	11	−	−	PROPN
ejpam-5263	326	12	e2	e2	NOUN
ejpam-5263	326	13	+	+	PROPN
ejpam-5263	326	14	b2c2	b2c2	PROPN
ejpam-5263	326	15	−	−	NOUN
ejpam-5263	326	16	f	f	NOUN
ejpam-5263	326	17	2	2	NUM
ejpam-5263	326	18	=	=	SYM
ejpam-5263	326	19	2c2	2c2	NUM
ejpam-5263	326	20	,	,	PUNCT
ejpam-5263	326	21	dc2	dc2	PROPN
ejpam-5263	326	22	−	−	PROPN
ejpam-5263	326	23	ef	ef	PROPN
ejpam-5263	326	24	=	=	NOUN
ejpam-5263	326	25	2d	2d	PROPN
ejpam-5263	326	26	,	,	PUNCT
ejpam-5263	326	27	eb2	eb2	PROPN
ejpam-5263	326	28	−df	−df	NOUN
ejpam-5263	326	29	=	=	SYM
ejpam-5263	326	30	2e	2e	NUM
ejpam-5263	326	31	,	,	PUNCT
ejpam-5263	326	32	fa2	fa2	PROPN
ejpam-5263	326	33	−de	−de	X
ejpam-5263	326	34	=	=	SYM
ejpam-5263	326	35	2f	2f	NOUN
ejpam-5263	326	36	.	.	PUNCT
ejpam-5263	327	1	observe	observe	VERB
ejpam-5263	327	2	that	that	SCONJ
ejpam-5263	327	3	a	a	DET
ejpam-5263	327	4	=	=	SYM
ejpam-5263	327	5	b	b	NOUN
ejpam-5263	327	6	=	=	SYM
ejpam-5263	327	7	c	c	NOUN
ejpam-5263	327	8	=	=	SYM
ejpam-5263	327	9	1	1	NUM
ejpam-5263	327	10	,	,	PUNCT
ejpam-5263	327	11	d	d	NOUN
ejpam-5263	327	12	=	=	SYM
ejpam-5263	327	13	e	e	NOUN
ejpam-5263	327	14	=	=	PUNCT
ejpam-5263	327	15	f	f	PROPN
ejpam-5263	327	16	=	=	SYM
ejpam-5263	327	17	0	0	NUM
ejpam-5263	327	18	satisfy	satisfy	VERB
ejpam-5263	327	19	the	the	DET
ejpam-5263	327	20	above	above	ADJ
ejpam-5263	327	21	equations	equation	NOUN
ejpam-5263	327	22	simultaneously	simultaneously	ADV
ejpam-5263	327	23	.	.	PUNCT
ejpam-5263	328	1	we	we	PRON
ejpam-5263	328	2	shall	shall	AUX
ejpam-5263	328	3	see	see	VERB
ejpam-5263	328	4	that	that	SCONJ
ejpam-5263	328	5	there	there	PRON
ejpam-5263	328	6	are	be	VERB
ejpam-5263	328	7	other	other	ADJ
ejpam-5263	328	8	possible	possible	ADJ
ejpam-5263	328	9	solutions	solution	NOUN
ejpam-5263	328	10	with	with	ADP
ejpam-5263	328	11	d	d	PROPN
ejpam-5263	328	12	,	,	PUNCT
ejpam-5263	328	13	e	e	NOUN
ejpam-5263	328	14	,	,	PUNCT
ejpam-5263	328	15	f	f	PROPN
ejpam-5263	328	16	̸=	̸=	PROPN
ejpam-5263	328	17	0	0	NUM
ejpam-5263	328	18	.	.	PUNCT
ejpam-5263	329	1	multiplying	multiply	VERB
ejpam-5263	329	2	both	both	DET
ejpam-5263	329	3	sides	side	NOUN
ejpam-5263	329	4	in	in	ADP
ejpam-5263	329	5	the	the	DET
ejpam-5263	329	6	last	last	ADJ
ejpam-5263	329	7	three	three	NUM
ejpam-5263	329	8	equations	equation	NOUN
ejpam-5263	329	9	by	by	ADP
ejpam-5263	329	10	d	d	PROPN
ejpam-5263	329	11	,	,	PUNCT
ejpam-5263	329	12	e	e	NOUN
ejpam-5263	329	13	,	,	PUNCT
ejpam-5263	329	14	and	and	CCONJ
ejpam-5263	329	15	f	f	PROPN
ejpam-5263	329	16	(	(	PUNCT
ejpam-5263	329	17	respectively	respectively	ADV
ejpam-5263	329	18	)	)	PUNCT
ejpam-5263	329	19	and	and	CCONJ
ejpam-5263	329	20	rearranging	rearrange	VERB
ejpam-5263	329	21	the	the	DET
ejpam-5263	329	22	terms	term	NOUN
ejpam-5263	329	23	,	,	PUNCT
ejpam-5263	329	24	we	we	PRON
ejpam-5263	329	25	obtain	obtain	VERB
ejpam-5263	329	26	d2	d2	PROPN
ejpam-5263	329	27	+	+	CCONJ
ejpam-5263	329	28	e2	e2	PROPN
ejpam-5263	329	29	=	=	PUNCT
ejpam-5263	329	30	a2b2	a2b2	PROPN
ejpam-5263	330	1	+	+	ADJ
ejpam-5263	330	2	a2c2	a2c2	INTJ
ejpam-5263	330	3	−	−	PROPN
ejpam-5263	330	4	2a2	2a2	NUM
ejpam-5263	330	5	,	,	PUNCT
ejpam-5263	330	6	(	(	PUNCT
ejpam-5263	330	7	1	1	X
ejpam-5263	330	8	)	)	PUNCT
ejpam-5263	330	9	d2	d2	NOUN
ejpam-5263	330	10	+	+	CCONJ
ejpam-5263	330	11	f	f	PROPN
ejpam-5263	330	12	2	2	NUM
ejpam-5263	330	13	=	=	SYM
ejpam-5263	331	1	a2b2	a2b2	PROPN
ejpam-5263	331	2	+	+	ADJ
ejpam-5263	331	3	b2c2	b2c2	X
ejpam-5263	331	4	−	−	PROPN
ejpam-5263	331	5	2b2	2b2	NUM
ejpam-5263	331	6	,	,	PUNCT
ejpam-5263	331	7	(	(	PUNCT
ejpam-5263	331	8	2	2	NUM
ejpam-5263	331	9	)	)	PUNCT
ejpam-5263	331	10	e2	e2	NOUN
ejpam-5263	331	11	+	+	CCONJ
ejpam-5263	331	12	f	f	PROPN
ejpam-5263	331	13	2	2	NUM
ejpam-5263	331	14	=	=	SYM
ejpam-5263	331	15	a2c2	a2c2	X
ejpam-5263	332	1	+	+	NOUN
ejpam-5263	332	2	b2c2	b2c2	PROPN
ejpam-5263	332	3	−	−	NOUN
ejpam-5263	332	4	2c2	2c2	NUM
ejpam-5263	332	5	,	,	PUNCT
ejpam-5263	332	6	(	(	PUNCT
ejpam-5263	332	7	3	3	X
ejpam-5263	332	8	)	)	PUNCT
ejpam-5263	332	9	c2d2	c2d2	X
ejpam-5263	333	1	−	−	NOUN
ejpam-5263	333	2	2d2	2d2	NUM
ejpam-5263	333	3	=	=	SYM
ejpam-5263	333	4	def	def	PROPN
ejpam-5263	333	5	,	,	PUNCT
ejpam-5263	333	6	(	(	PUNCT
ejpam-5263	333	7	4	4	NUM
ejpam-5263	333	8	)	)	PUNCT
ejpam-5263	333	9	b2e2	b2e2	PROPN
ejpam-5263	333	10	−	−	PROPN
ejpam-5263	333	11	2e2	2e2	NUM
ejpam-5263	333	12	=	=	SYM
ejpam-5263	333	13	def	def	PROPN
ejpam-5263	333	14	,	,	PUNCT
ejpam-5263	333	15	(	(	PUNCT
ejpam-5263	333	16	5	5	X
ejpam-5263	333	17	)	)	PUNCT
ejpam-5263	333	18	a2f	a2f	NOUN
ejpam-5263	333	19	2	2	NUM
ejpam-5263	333	20	−	−	NOUN
ejpam-5263	333	21	2f	2f	NUM
ejpam-5263	333	22	2	2	NUM
ejpam-5263	333	23	=	=	SYM
ejpam-5263	333	24	def	def	ADJ
ejpam-5263	333	25	.	.	PUNCT
ejpam-5263	334	1	(	(	PUNCT
ejpam-5263	334	2	6	6	NUM
ejpam-5263	334	3	)	)	PUNCT
ejpam-5263	334	4	from	from	ADP
ejpam-5263	334	5	(	(	PUNCT
ejpam-5263	334	6	1	1	NUM
ejpam-5263	334	7	)	)	PUNCT
ejpam-5263	334	8	,	,	PUNCT
ejpam-5263	334	9	(	(	PUNCT
ejpam-5263	334	10	2	2	NUM
ejpam-5263	334	11	)	)	PUNCT
ejpam-5263	334	12	,	,	PUNCT
ejpam-5263	334	13	and	and	CCONJ
ejpam-5263	334	14	(	(	PUNCT
ejpam-5263	334	15	3	3	NUM
ejpam-5263	334	16	)	)	PUNCT
ejpam-5263	334	17	,	,	PUNCT
ejpam-5263	334	18	we	we	PRON
ejpam-5263	334	19	get	get	VERB
ejpam-5263	334	20	d2	d2	NOUN
ejpam-5263	334	21	=	=	SYM
ejpam-5263	334	22	a2b2	a2b2	PROPN
ejpam-5263	334	23	−a2	−a2	NOUN
ejpam-5263	334	24	−b2	−b2	PROPN
ejpam-5263	334	25	+	+	CCONJ
ejpam-5263	334	26	c2	c2	PROPN
ejpam-5263	334	27	,	,	PUNCT
ejpam-5263	334	28	(	(	PUNCT
ejpam-5263	334	29	7	7	X
ejpam-5263	334	30	)	)	PUNCT
ejpam-5263	334	31	e2	e2	NOUN
ejpam-5263	334	32	=	=	SYM
ejpam-5263	334	33	a2c2	a2c2	PROPN
ejpam-5263	334	34	−a2	−a2	NOUN
ejpam-5263	334	35	+	+	PROPN
ejpam-5263	334	36	b2	b2	NOUN
ejpam-5263	334	37	−	−	PROPN
ejpam-5263	334	38	c2	c2	PROPN
ejpam-5263	334	39	,	,	PUNCT
ejpam-5263	334	40	(	(	PUNCT
ejpam-5263	334	41	8)	8)	NUM
ejpam-5263	334	42	f	f	NOUN
ejpam-5263	334	43	2	2	NUM
ejpam-5263	334	44	=	=	SYM
ejpam-5263	334	45	b2c2	b2c2	X
ejpam-5263	334	46	+	+	NOUN
ejpam-5263	334	47	a2	a2	PROPN
ejpam-5263	334	48	−b2	−b2	ADJ
ejpam-5263	334	49	−	−	PROPN
ejpam-5263	334	50	c2	c2	PROPN
ejpam-5263	334	51	.	.	PUNCT
ejpam-5263	335	1	(	(	PUNCT
ejpam-5263	335	2	9	9	NUM
ejpam-5263	335	3	)	)	PUNCT
ejpam-5263	335	4	from	from	ADP
ejpam-5263	335	5	(	(	PUNCT
ejpam-5263	335	6	4	4	NUM
ejpam-5263	335	7	)	)	PUNCT
ejpam-5263	335	8	,	,	PUNCT
ejpam-5263	335	9	(	(	PUNCT
ejpam-5263	335	10	5	5	NUM
ejpam-5263	335	11	)	)	PUNCT
ejpam-5263	335	12	,	,	PUNCT
ejpam-5263	335	13	and	and	CCONJ
ejpam-5263	335	14	(	(	PUNCT
ejpam-5263	335	15	6	6	NUM
ejpam-5263	335	16	)	)	PUNCT
ejpam-5263	335	17	,	,	PUNCT
ejpam-5263	335	18	we	we	PRON
ejpam-5263	335	19	get	get	VERB
ejpam-5263	335	20	(	(	PUNCT
ejpam-5263	335	21	c2	c2	PROPN
ejpam-5263	335	22	−	−	PROPN
ejpam-5263	335	23	2)d2	2)d2	PROPN
ejpam-5263	335	24	=	=	SYM
ejpam-5263	335	25	(	(	PUNCT
ejpam-5263	335	26	b2	b2	NOUN
ejpam-5263	335	27	−	−	PROPN
ejpam-5263	335	28	2)e2	2)e2	PROPN
ejpam-5263	335	29	=	=	SYM
ejpam-5263	335	30	(	(	PUNCT
ejpam-5263	335	31	a2	a2	PROPN
ejpam-5263	335	32	−	−	PROPN
ejpam-5263	335	33	2)f	2)f	NOUN
ejpam-5263	335	34	2	2	NUM
ejpam-5263	335	35	=	=	SYM
ejpam-5263	335	36	def	def	ADJ
ejpam-5263	335	37	.	.	PUNCT
ejpam-5263	336	1	(	(	PUNCT
ejpam-5263	336	2	10	10	NUM
ejpam-5263	336	3	)	)	PUNCT
ejpam-5263	336	4	substituting	substitute	VERB
ejpam-5263	336	5	(	(	PUNCT
ejpam-5263	336	6	7	7	NUM
ejpam-5263	336	7	)	)	PUNCT
ejpam-5263	336	8	,	,	PUNCT
ejpam-5263	336	9	(	(	PUNCT
ejpam-5263	336	10	8)	8)	NUM
ejpam-5263	336	11	,	,	PUNCT
ejpam-5263	336	12	and	and	CCONJ
ejpam-5263	336	13	(	(	PUNCT
ejpam-5263	336	14	9	9	NUM
ejpam-5263	336	15	)	)	PUNCT
ejpam-5263	336	16	into	into	ADP
ejpam-5263	336	17	(	(	PUNCT
ejpam-5263	336	18	10	10	NUM
ejpam-5263	336	19	)	)	PUNCT
ejpam-5263	336	20	,	,	PUNCT
ejpam-5263	336	21	we	we	PRON
ejpam-5263	336	22	obtain	obtain	VERB
ejpam-5263	336	23	(	(	PUNCT
ejpam-5263	336	24	b2	b2	NOUN
ejpam-5263	336	25	−	−	PROPN
ejpam-5263	336	26	c2)(a2	c2)(a2	NOUN
ejpam-5263	337	1	+	+	NOUN
ejpam-5263	337	2	b2	b2	NOUN
ejpam-5263	337	3	+	+	CCONJ
ejpam-5263	337	4	c2	c2	PROPN
ejpam-5263	337	5	−	−	PROPN
ejpam-5263	337	6	4	4	NUM
ejpam-5263	337	7	)	)	PUNCT
ejpam-5263	337	8	=	=	SYM
ejpam-5263	337	9	0	0	NUM
ejpam-5263	337	10	,	,	PUNCT
ejpam-5263	337	11	(	(	PUNCT
ejpam-5263	337	12	a2	a2	PROPN
ejpam-5263	337	13	−	−	PROPN
ejpam-5263	337	14	c2)(a2	c2)(a2	PUNCT
ejpam-5263	338	1	+	+	NOUN
ejpam-5263	338	2	b2	b2	NOUN
ejpam-5263	338	3	+	+	CCONJ
ejpam-5263	338	4	c2	c2	PROPN
ejpam-5263	338	5	−	−	PROPN
ejpam-5263	338	6	4	4	NUM
ejpam-5263	338	7	)	)	PUNCT
ejpam-5263	338	8	=	=	SYM
ejpam-5263	338	9	0	0	NUM
ejpam-5263	338	10	,	,	PUNCT
ejpam-5263	338	11	(	(	PUNCT
ejpam-5263	338	12	a2	a2	PROPN
ejpam-5263	338	13	−b2)(a2	−b2)(a2	PROPN
ejpam-5263	338	14	+	+	NOUN
ejpam-5263	338	15	b2	b2	NOUN
ejpam-5263	338	16	+	+	CCONJ
ejpam-5263	338	17	c2	c2	PROPN
ejpam-5263	338	18	−	−	PROPN
ejpam-5263	338	19	4	4	NUM
ejpam-5263	338	20	)	)	PUNCT
ejpam-5263	338	21	=	=	SYM
ejpam-5263	338	22	0	0	X
ejpam-5263	338	23	.	.	PUNCT
ejpam-5263	339	1	now	now	ADV
ejpam-5263	339	2	one	one	PRON
ejpam-5263	339	3	may	may	AUX
ejpam-5263	339	4	check	check	VERB
ejpam-5263	339	5	that	that	PRON
ejpam-5263	339	6	a	a	DET
ejpam-5263	339	7	=	=	SYM
ejpam-5263	339	8	b	b	NOUN
ejpam-5263	339	9	=	=	SYM
ejpam-5263	339	10	c	c	NOUN
ejpam-5263	339	11	=	=	SYM
ejpam-5263	339	12	4	4	NUM
ejpam-5263	339	13	3	3	NUM
ejpam-5263	339	14	,	,	PUNCT
ejpam-5263	340	1	d	d	PROPN
ejpam-5263	340	2	+	+	NUM
ejpam-5263	341	1	e	e	NOUN
ejpam-5263	341	2	+	+	CCONJ
ejpam-5263	341	3	f	f	NOUN
ejpam-5263	341	4	=	=	SYM
ejpam-5263	341	5	−2	−2	PROPN
ejpam-5263	341	6	3	3	NUM
ejpam-5263	341	7	satisfy	satisfy	NOUN
ejpam-5263	341	8	the	the	DET
ejpam-5263	341	9	above	above	ADJ
ejpam-5263	341	10	equations	equation	NOUN
ejpam-5263	341	11	simultaneously	simultaneously	ADV
ejpam-5263	341	12	.	.	PUNCT
ejpam-5263	342	1	this	this	PRON
ejpam-5263	342	2	tells	tell	VERB
ejpam-5263	342	3	us	we	PRON
ejpam-5263	342	4	that	that	SCONJ
ejpam-5263	342	5	a	a	PRON
ejpam-5263	342	6	is	be	AUX
ejpam-5263	342	7	not	not	PART
ejpam-5263	342	8	necessarily	necessarily	ADV
ejpam-5263	342	9	an	an	DET
ejpam-5263	342	10	orthonormal	orthonormal	ADJ
ejpam-5263	342	11	basis	basis	NOUN
ejpam-5263	342	12	.	.	PUNCT
ejpam-5263	343	1	a.	a.	NOUN
ejpam-5263	343	2	adam	adam	PROPN
ejpam-5263	343	3	,	,	PUNCT
ejpam-5263	343	4	s.	s.	PROPN
ejpam-5263	343	5	rante	rante	PROPN
ejpam-5263	343	6	,	,	PUNCT
ejpam-5263	343	7	h.	h.	PROPN
ejpam-5263	343	8	gunawan	gunawan	PROPN
ejpam-5263	343	9	/	/	PUNCT
ejpam-5263	343	10	eur	eur	PROPN
ejpam-5263	343	11	.	.	PUNCT
ejpam-5263	344	1	j.	j.	PROPN
ejpam-5263	344	2	pure	pure	PROPN
ejpam-5263	344	3	appl	appl	PROPN
ejpam-5263	344	4	.	.	PROPN
ejpam-5263	344	5	math	math	PROPN
ejpam-5263	344	6	,	,	PUNCT
ejpam-5263	344	7	17	17	NUM
ejpam-5263	344	8	(	(	PUNCT
ejpam-5263	344	9	3	3	NUM
ejpam-5263	344	10	)	)	PUNCT
ejpam-5263	344	11	(	(	PUNCT
ejpam-5263	344	12	2024	2024	NUM
ejpam-5263	344	13	)	)	PUNCT
ejpam-5263	344	14	,	,	PUNCT
ejpam-5263	344	15	1937	1937	NUM
ejpam-5263	344	16	-	-	SYM
ejpam-5263	344	17	1947	1947	NUM
ejpam-5263	344	18	1945	1945	NUM
ejpam-5263	344	19	we	we	PRON
ejpam-5263	344	20	now	now	ADV
ejpam-5263	344	21	come	come	VERB
ejpam-5263	344	22	to	to	ADP
ejpam-5263	344	23	the	the	DET
ejpam-5263	344	24	case	case	NOUN
ejpam-5263	344	25	where	where	SCONJ
ejpam-5263	344	26	d	d	NOUN
ejpam-5263	344	27	=	=	SYM
ejpam-5263	344	28	dim	dim	ADJ
ejpam-5263	344	29	x	x	NOUN
ejpam-5263	344	30	=	=	SYM
ejpam-5263	344	31	∞.	∞.	NUM
ejpam-5263	344	32	we	we	PRON
ejpam-5263	344	33	assume	assume	VERB
ejpam-5263	344	34	that	that	SCONJ
ejpam-5263	344	35	x	x	PRON
ejpam-5263	344	36	is	be	AUX
ejpam-5263	344	37	separable	separable	ADJ
ejpam-5263	344	38	and	and	CCONJ
ejpam-5263	344	39	b	b	NOUN
ejpam-5263	344	40	:	:	PUNCT
ejpam-5263	344	41	=	=	X
ejpam-5263	344	42	{	{	PUNCT
ejpam-5263	344	43	ai	ai	INTJ
ejpam-5263	344	44	:	:	PUNCT
ejpam-5263	344	45	i	i	NOUN
ejpam-5263	344	46	=	=	NOUN
ejpam-5263	344	47	1	1	NUM
ejpam-5263	344	48	,	,	PUNCT
ejpam-5263	344	49	2	2	NUM
ejpam-5263	344	50	,	,	PUNCT
ejpam-5263	344	51	3	3	NUM
ejpam-5263	344	52	,	,	PUNCT
ejpam-5263	344	53	.	.	PUNCT
ejpam-5263	344	54	.	.	PUNCT
ejpam-5263	344	55	.	.	PUNCT
ejpam-5263	345	1	}	}	PUNCT
ejpam-5263	345	2	is	be	AUX
ejpam-5263	345	3	an	an	DET
ejpam-5263	345	4	orthogonal	orthogonal	ADJ
ejpam-5263	345	5	basis	basis	NOUN
ejpam-5263	345	6	for	for	ADP
ejpam-5263	345	7	x.	x.	NOUN
ejpam-5263	345	8	thus	thus	ADV
ejpam-5263	345	9	for	for	ADP
ejpam-5263	345	10	all	all	DET
ejpam-5263	345	11	x	x	NOUN
ejpam-5263	345	12	,	,	PUNCT
ejpam-5263	345	13	y	y	PROPN
ejpam-5263	345	14	∈	∈	PROPN
ejpam-5263	345	15	x	x	X
ejpam-5263	345	16	,	,	PUNCT
ejpam-5263	345	17	we	we	PRON
ejpam-5263	345	18	have	have	VERB
ejpam-5263	345	19	parseval	parseval	NOUN
ejpam-5263	345	20	’s	’s	PART
ejpam-5263	345	21	identity	identity	NOUN
ejpam-5263	345	22	that	that	PRON
ejpam-5263	345	23	∞∑	∞∑	NUM
ejpam-5263	345	24	i=1	i=1	PRON
ejpam-5263	345	25	⟨x	⟨x	ADJ
ejpam-5263	345	26	,	,	PUNCT
ejpam-5263	345	27	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	345	28	,	,	PUNCT
ejpam-5263	345	29	y⟩	y⟩	NOUN
ejpam-5263	345	30	∥ai∥2	∥ai∥2	VERB
ejpam-5263	345	31	will	will	AUX
ejpam-5263	345	32	converge	converge	VERB
ejpam-5263	345	33	to	to	ADP
ejpam-5263	345	34	⟨x	⟨x	NUM
ejpam-5263	345	35	,	,	PUNCT
ejpam-5263	345	36	y⟩.	y⟩.	PRON
ejpam-5263	345	37	next	next	ADJ
ejpam-5263	345	38	,	,	PUNCT
ejpam-5263	345	39	let	let	VERB
ejpam-5263	345	40	a	a	DET
ejpam-5263	345	41	:	:	PUNCT
ejpam-5263	345	42	=	=	NUM
ejpam-5263	345	43	{	{	PUNCT
ejpam-5263	345	44	a1	a1	PROPN
ejpam-5263	345	45	,	,	PUNCT
ejpam-5263	345	46	a2	a2	PROPN
ejpam-5263	345	47	,	,	PUNCT
ejpam-5263	345	48	.	.	PUNCT
ejpam-5263	345	49	.	.	PUNCT
ejpam-5263	346	1	.	.	PUNCT
ejpam-5263	347	1	,	,	PUNCT
ejpam-5263	347	2	an	an	X
ejpam-5263	347	3	}	}	PUNCT
ejpam-5263	347	4	,	,	PUNCT
ejpam-5263	347	5	where	where	SCONJ
ejpam-5263	347	6	the	the	DET
ejpam-5263	347	7	vectors	vector	NOUN
ejpam-5263	347	8	ai	ai	VERB
ejpam-5263	347	9	’s	’s	ADV
ejpam-5263	347	10	are	be	AUX
ejpam-5263	347	11	the	the	DET
ejpam-5263	347	12	first	first	ADJ
ejpam-5263	347	13	n	n	NOUN
ejpam-5263	347	14	vectors	vector	NOUN
ejpam-5263	347	15	in	in	ADP
ejpam-5263	347	16	b.	b.	PROPN
ejpam-5263	347	17	we	we	PRON
ejpam-5263	347	18	define	define	VERB
ejpam-5263	347	19	⟨x	⟨x	VERB
ejpam-5263	347	20	,	,	PUNCT
ejpam-5263	347	21	y⟩a	y⟩a	NUM
ejpam-5263	347	22	:	:	PUNCT
ejpam-5263	347	23	=	=	SYM
ejpam-5263	347	24	∑	∑	PUNCT
ejpam-5263	347	25	{	{	PUNCT
ejpam-5263	347	26	i2,	i2,	X
ejpam-5263	347	27	...	...	PUNCT
ejpam-5263	347	28	,in}⊂{1,2,	,in}⊂{1,2,	NUM
ejpam-5263	347	29	...	...	PUNCT
ejpam-5263	347	30	,n	,n	NOUN
ejpam-5263	347	31	}	}	PUNCT
ejpam-5263	347	32	⟨x	⟨x	VERB
ejpam-5263	347	33	,	,	PUNCT
ejpam-5263	347	34	y|ai2	y|ai2	NOUN
ejpam-5263	347	35	,	,	PUNCT
ejpam-5263	347	36	.	.	PUNCT
ejpam-5263	347	37	.	.	PUNCT
ejpam-5263	348	1	.	.	PUNCT
ejpam-5263	349	1	,	,	PUNCT
ejpam-5263	349	2	ain⟩	ain⟩	VERB
ejpam-5263	349	3	for	for	ADP
ejpam-5263	349	4	all	all	DET
ejpam-5263	349	5	x	x	NOUN
ejpam-5263	349	6	,	,	PUNCT
ejpam-5263	349	7	y	y	PROPN
ejpam-5263	349	8	∈	∈	PROPN
ejpam-5263	349	9	x.	x.	NOUN
ejpam-5263	350	1	then	then	ADV
ejpam-5263	350	2	we	we	PRON
ejpam-5263	350	3	have	have	VERB
ejpam-5263	350	4	the	the	DET
ejpam-5263	350	5	following	follow	VERB
ejpam-5263	350	6	theorem	theorem	VERB
ejpam-5263	350	7	.	.	PUNCT
ejpam-5263	350	8	theorem	theorem	NOUN
ejpam-5263	350	9	4	4	NUM
ejpam-5263	350	10	.	.	PUNCT
ejpam-5263	350	11	for	for	ADP
ejpam-5263	350	12	all	all	DET
ejpam-5263	350	13	x	x	NOUN
ejpam-5263	350	14	,	,	PUNCT
ejpam-5263	350	15	y	y	PROPN
ejpam-5263	350	16	∈	∈	PROPN
ejpam-5263	351	1	x	x	X
ejpam-5263	351	2	,	,	PUNCT
ejpam-5263	351	3	we	we	PRON
ejpam-5263	351	4	have	have	AUX
ejpam-5263	351	5	⟨x	⟨x	VERB
ejpam-5263	351	6	,	,	PUNCT
ejpam-5263	351	7	y⟩a	y⟩a	X
ejpam-5263	351	8	=	=	SYM
ejpam-5263	351	9	[	[	PUNCT
ejpam-5263	351	10	n∑	n∑	NOUN
ejpam-5263	351	11	i=1	i=1	PROPN
ejpam-5263	351	12	⟨x	⟨x	NUM
ejpam-5263	351	13	,	,	PUNCT
ejpam-5263	351	14	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	351	15	,	,	PUNCT
ejpam-5263	351	16	y⟩	y⟩	NOUN
ejpam-5263	351	17	∥ai∥4	∥ai∥4	PUNCT
ejpam-5263	352	1	+	+	NUM
ejpam-5263	352	2	n	n	CCONJ
ejpam-5263	352	3	∞∑	∞∑	NUM
ejpam-5263	352	4	i	i	NOUN
ejpam-5263	352	5	=	=	NOUN
ejpam-5263	352	6	n+1	n+1	PRON
ejpam-5263	352	7	⟨x	⟨x	NUM
ejpam-5263	352	8	,	,	PUNCT
ejpam-5263	352	9	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	352	10	,	,	PUNCT
ejpam-5263	352	11	y⟩	y⟩	NOUN
ejpam-5263	352	12	∥ai∥2	∥ai∥2	VERB
ejpam-5263	352	13	]	]	PUNCT
ejpam-5263	353	1	n∏	n∏	NOUN
ejpam-5263	353	2	j=1	j=1	NOUN
ejpam-5263	353	3	∥aj∥2	∥aj∥2	PUNCT
ejpam-5263	353	4	.	.	PUNCT
ejpam-5263	354	1	in	in	ADP
ejpam-5263	354	2	particular	particular	ADJ
ejpam-5263	354	3	,	,	PUNCT
ejpam-5263	354	4	if	if	SCONJ
ejpam-5263	354	5	∥ai∥	∥ai∥	PRON
ejpam-5263	354	6	=	=	SYM
ejpam-5263	354	7	α	α	NOUN
ejpam-5263	354	8	for	for	ADP
ejpam-5263	354	9	i	i	PRON
ejpam-5263	354	10	=	=	NOUN
ejpam-5263	354	11	1	1	NUM
ejpam-5263	354	12	,	,	PUNCT
ejpam-5263	354	13	2	2	NUM
ejpam-5263	354	14	,	,	PUNCT
ejpam-5263	354	15	.	.	PUNCT
ejpam-5263	354	16	.	.	PUNCT
ejpam-5263	354	17	.	.	PUNCT
ejpam-5263	355	1	,	,	PUNCT
ejpam-5263	355	2	n	n	CCONJ
ejpam-5263	355	3	then	then	ADV
ejpam-5263	355	4	⟨x	⟨x	VERB
ejpam-5263	355	5	,	,	PUNCT
ejpam-5263	355	6	y⟩a	y⟩a	NUM
ejpam-5263	355	7	=	=	SYM
ejpam-5263	355	8	α2(n−1	α2(n−1	NUM
ejpam-5263	355	9	)	)	PUNCT
ejpam-5263	356	1	[	[	PUNCT
ejpam-5263	356	2	n∑	n∑	NOUN
ejpam-5263	356	3	i=1	i=1	PROPN
ejpam-5263	356	4	⟨x	⟨x	NUM
ejpam-5263	356	5	,	,	PUNCT
ejpam-5263	356	6	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	356	7	,	,	PUNCT
ejpam-5263	356	8	y⟩	y⟩	NOUN
ejpam-5263	356	9	α2	α2	PROPN
ejpam-5263	356	10	+	+	CCONJ
ejpam-5263	356	11	n	n	PROPN
ejpam-5263	356	12	∞∑	∞∑	NUM
ejpam-5263	356	13	i	i	NOUN
ejpam-5263	356	14	=	=	NOUN
ejpam-5263	356	15	n+1	n+1	PRON
ejpam-5263	356	16	⟨x	⟨x	NUM
ejpam-5263	356	17	,	,	PUNCT
ejpam-5263	356	18	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	356	19	,	,	PUNCT
ejpam-5263	356	20	y⟩	y⟩	NOUN
ejpam-5263	356	21	∥ai∥2	∥ai∥2	VERB
ejpam-5263	356	22	]	]	PUNCT
ejpam-5263	356	23	.	.	PUNCT
ejpam-5263	357	1	proof	proof	NOUN
ejpam-5263	357	2	.	.	PUNCT
ejpam-5263	358	1	the	the	DET
ejpam-5263	358	2	proof	proof	NOUN
ejpam-5263	358	3	is	be	AUX
ejpam-5263	358	4	similar	similar	ADJ
ejpam-5263	358	5	to	to	ADP
ejpam-5263	358	6	the	the	DET
ejpam-5263	358	7	proof	proof	NOUN
ejpam-5263	358	8	of	of	ADP
ejpam-5263	358	9	theorem	theorem	ADJ
ejpam-5263	358	10	2.1	2.1	NUM
ejpam-5263	358	11	but	but	CCONJ
ejpam-5263	358	12	this	this	DET
ejpam-5263	358	13	time	time	NOUN
ejpam-5263	358	14	we	we	PRON
ejpam-5263	358	15	have	have	AUX
ejpam-5263	358	16	⟨x	⟨x	VERB
ejpam-5263	358	17	,	,	PUNCT
ejpam-5263	358	18	y|ai2	y|ai2	NOUN
ejpam-5263	358	19	,	,	PUNCT
ejpam-5263	358	20	.	.	PUNCT
ejpam-5263	358	21	.	.	PUNCT
ejpam-5263	359	1	.	.	PUNCT
ejpam-5263	360	1	,	,	PUNCT
ejpam-5263	360	2	ain⟩	ain⟩	X
ejpam-5263	360	3	=	=	X
ejpam-5263	361	1	[	[	PUNCT
ejpam-5263	361	2	⟨x	⟨x	VERB
ejpam-5263	361	3	,	,	PUNCT
ejpam-5263	361	4	ai1⟩⟨ai1	ai1⟩⟨ai1	ADJ
ejpam-5263	361	5	,	,	PUNCT
ejpam-5263	361	6	y⟩	y⟩	NOUN
ejpam-5263	361	7	∥ai1∥	∥ai1∥	ADP
ejpam-5263	361	8	4	4	NUM
ejpam-5263	361	9	+	+	SYM
ejpam-5263	361	10	1	1	NUM
ejpam-5263	361	11	∥ai1∥	∥ai1∥	ADP
ejpam-5263	361	12	2	2	NUM
ejpam-5263	361	13	∞∑	∞∑	NUM
ejpam-5263	361	14	i	i	PRON
ejpam-5263	361	15	=	=	NOUN
ejpam-5263	361	16	n+1	n+1	PRON
ejpam-5263	361	17	⟨x	⟨x	NUM
ejpam-5263	361	18	,	,	PUNCT
ejpam-5263	361	19	ai1⟩⟨ai1	ai1⟩⟨ai1	ADJ
ejpam-5263	361	20	,	,	PUNCT
ejpam-5263	361	21	y⟩	y⟩	NOUN
ejpam-5263	361	22	∥ai∥2	∥ai∥2	VERB
ejpam-5263	361	23	]	]	PUNCT
ejpam-5263	362	1	n∏	n∏	NOUN
ejpam-5263	362	2	j=1	j=1	NOUN
ejpam-5263	362	3	∥aj∥2	∥aj∥2	X
ejpam-5263	362	4	,	,	PUNCT
ejpam-5263	362	5	where	where	SCONJ
ejpam-5263	362	6	{	{	PUNCT
ejpam-5263	362	7	i1	i1	NOUN
ejpam-5263	362	8	}	}	PUNCT
ejpam-5263	362	9	=	=	PUNCT
ejpam-5263	362	10	{	{	PUNCT
ejpam-5263	362	11	1	1	NUM
ejpam-5263	362	12	,	,	PUNCT
ejpam-5263	362	13	2	2	NUM
ejpam-5263	362	14	,	,	PUNCT
ejpam-5263	362	15	.	.	PUNCT
ejpam-5263	362	16	.	.	PUNCT
ejpam-5263	362	17	.	.	PUNCT
ejpam-5263	362	18	,	,	PUNCT
ejpam-5263	362	19	n	n	CCONJ
ejpam-5263	362	20	}	}	PUNCT
ejpam-5263	362	21	\	\	NOUN
ejpam-5263	362	22	{	{	PUNCT
ejpam-5263	362	23	i2	i2	PROPN
ejpam-5263	362	24	,	,	PUNCT
ejpam-5263	362	25	.	.	PUNCT
ejpam-5263	362	26	.	.	PUNCT
ejpam-5263	362	27	.	.	PUNCT
ejpam-5263	363	1	,	,	PUNCT
ejpam-5263	363	2	in	in	ADP
ejpam-5263	363	3	}	}	PUNCT
ejpam-5263	363	4	.	.	PUNCT
ejpam-5263	364	1	summing	sum	VERB
ejpam-5263	364	2	all	all	DET
ejpam-5263	364	3	these	these	DET
ejpam-5263	364	4	expression	expression	NOUN
ejpam-5263	364	5	for	for	ADP
ejpam-5263	364	6	all	all	DET
ejpam-5263	364	7	subsets	subset	NOUN
ejpam-5263	364	8	{	{	PUNCT
ejpam-5263	364	9	i2	i2	PROPN
ejpam-5263	364	10	,	,	PUNCT
ejpam-5263	364	11	.	.	PUNCT
ejpam-5263	364	12	.	.	PUNCT
ejpam-5263	365	1	.	.	PUNCT
ejpam-5263	366	1	,	,	PUNCT
ejpam-5263	366	2	in	in	ADP
ejpam-5263	366	3	}	}	PUNCT
ejpam-5263	366	4	⊂	⊂	PROPN
ejpam-5263	366	5	{	{	PUNCT
ejpam-5263	366	6	1	1	NUM
ejpam-5263	366	7	,	,	PUNCT
ejpam-5263	366	8	2	2	NUM
ejpam-5263	366	9	,	,	PUNCT
ejpam-5263	366	10	.	.	PUNCT
ejpam-5263	366	11	.	.	PUNCT
ejpam-5263	366	12	.	.	PUNCT
ejpam-5263	366	13	,	,	PUNCT
ejpam-5263	366	14	n	n	CCONJ
ejpam-5263	366	15	}	}	PUNCT
ejpam-5263	366	16	,	,	PUNCT
ejpam-5263	366	17	we	we	PRON
ejpam-5263	366	18	obtain	obtain	VERB
ejpam-5263	366	19	⟨x	⟨x	VERB
ejpam-5263	366	20	,	,	PUNCT
ejpam-5263	366	21	y⟩a	y⟩a	NUM
ejpam-5263	366	22	=	=	SYM
ejpam-5263	367	1	[	[	PUNCT
ejpam-5263	367	2	n∑	n∑	NOUN
ejpam-5263	367	3	i=1	i=1	PROPN
ejpam-5263	367	4	⟨x	⟨x	NUM
ejpam-5263	367	5	,	,	PUNCT
ejpam-5263	367	6	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	367	7	,	,	PUNCT
ejpam-5263	367	8	y⟩	y⟩	NOUN
ejpam-5263	367	9	∥ai∥4	∥ai∥4	PUNCT
ejpam-5263	368	1	+	+	CCONJ
ejpam-5263	368	2	n∑	n∑	X
ejpam-5263	368	3	i=1	i=1	NUM
ejpam-5263	368	4	1	1	NUM
ejpam-5263	368	5	∥ai∥2	∥ai∥2	VERB
ejpam-5263	368	6	∞∑	∞∑	NUM
ejpam-5263	368	7	i	i	PRON
ejpam-5263	368	8	=	=	NOUN
ejpam-5263	368	9	n+1	n+1	PRON
ejpam-5263	368	10	⟨x	⟨x	NUM
ejpam-5263	368	11	,	,	PUNCT
ejpam-5263	368	12	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	368	13	,	,	PUNCT
ejpam-5263	368	14	y⟩	y⟩	NOUN
ejpam-5263	368	15	∥ai∥2	∥ai∥2	VERB
ejpam-5263	368	16	]	]	PUNCT
ejpam-5263	369	1	n∏	n∏	NOUN
ejpam-5263	369	2	j=1	j=1	NOUN
ejpam-5263	369	3	∥aj∥2	∥aj∥2	PUNCT
ejpam-5263	369	4	.	.	PUNCT
ejpam-5263	370	1	in	in	ADP
ejpam-5263	370	2	particular	particular	ADJ
ejpam-5263	370	3	,	,	PUNCT
ejpam-5263	370	4	if	if	SCONJ
ejpam-5263	370	5	∥ai∥	∥ai∥	PRON
ejpam-5263	370	6	=	=	SYM
ejpam-5263	370	7	α	α	NOUN
ejpam-5263	370	8	for	for	ADP
ejpam-5263	370	9	i	i	PRON
ejpam-5263	370	10	=	=	NOUN
ejpam-5263	370	11	1	1	NUM
ejpam-5263	370	12	,	,	PUNCT
ejpam-5263	370	13	2	2	NUM
ejpam-5263	370	14	,	,	PUNCT
ejpam-5263	370	15	.	.	PUNCT
ejpam-5263	370	16	.	.	PUNCT
ejpam-5263	370	17	.	.	PUNCT
ejpam-5263	371	1	,	,	PUNCT
ejpam-5263	371	2	n	n	CCONJ
ejpam-5263	371	3	,	,	PUNCT
ejpam-5263	371	4	then	then	ADV
ejpam-5263	371	5	we	we	PRON
ejpam-5263	371	6	have	have	AUX
ejpam-5263	371	7	⟨x	⟨x	VERB
ejpam-5263	371	8	,	,	PUNCT
ejpam-5263	371	9	y⟩a	y⟩a	NUM
ejpam-5263	371	10	=	=	SYM
ejpam-5263	371	11	α2(n−1	α2(n−1	NUM
ejpam-5263	371	12	)	)	PUNCT
ejpam-5263	372	1	[	[	PUNCT
ejpam-5263	372	2	n∑	n∑	NOUN
ejpam-5263	372	3	i=1	i=1	PROPN
ejpam-5263	372	4	⟨x	⟨x	NUM
ejpam-5263	372	5	,	,	PUNCT
ejpam-5263	372	6	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	372	7	,	,	PUNCT
ejpam-5263	372	8	y⟩	y⟩	NOUN
ejpam-5263	372	9	α2	α2	PROPN
ejpam-5263	372	10	+	+	CCONJ
ejpam-5263	372	11	n	n	PROPN
ejpam-5263	372	12	∞∑	∞∑	NUM
ejpam-5263	372	13	i	i	NOUN
ejpam-5263	372	14	=	=	NOUN
ejpam-5263	372	15	n+1	n+1	PRON
ejpam-5263	372	16	⟨x	⟨x	NUM
ejpam-5263	372	17	,	,	PUNCT
ejpam-5263	372	18	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	372	19	,	,	PUNCT
ejpam-5263	372	20	y⟩	y⟩	NOUN
ejpam-5263	372	21	∥ai∥2	∥ai∥2	VERB
ejpam-5263	372	22	]	]	PUNCT
ejpam-5263	372	23	as	as	SCONJ
ejpam-5263	372	24	claimed	claim	VERB
ejpam-5263	372	25	.	.	PUNCT
ejpam-5263	373	1	remark	remark	PROPN
ejpam-5263	373	2	2	2	NUM
ejpam-5263	373	3	.	.	PUNCT
ejpam-5263	373	4	note	note	VERB
ejpam-5263	373	5	that	that	SCONJ
ejpam-5263	373	6	if	if	SCONJ
ejpam-5263	373	7	∥ai∥	∥ai∥	ADP
ejpam-5263	373	8	=	=	SYM
ejpam-5263	373	9	1	1	NUM
ejpam-5263	373	10	for	for	ADP
ejpam-5263	373	11	i	i	PRON
ejpam-5263	373	12	=	=	NOUN
ejpam-5263	373	13	1	1	NUM
ejpam-5263	373	14	,	,	PUNCT
ejpam-5263	373	15	2	2	NUM
ejpam-5263	373	16	,	,	PUNCT
ejpam-5263	373	17	3	3	NUM
ejpam-5263	373	18	,	,	PUNCT
ejpam-5263	373	19	.	.	PUNCT
ejpam-5263	373	20	.	.	PUNCT
ejpam-5263	374	1	.	.	PUNCT
ejpam-5263	375	1	(	(	PUNCT
ejpam-5263	375	2	that	that	PRON
ejpam-5263	375	3	is	is	ADV
ejpam-5263	375	4	,	,	PUNCT
ejpam-5263	375	5	b	b	PRON
ejpam-5263	375	6	is	be	AUX
ejpam-5263	375	7	an	an	DET
ejpam-5263	375	8	orthonormal	orthonormal	ADJ
ejpam-5263	375	9	basis	basis	NOUN
ejpam-5263	375	10	for	for	ADP
ejpam-5263	375	11	x	x	NOUN
ejpam-5263	375	12	)	)	PUNCT
ejpam-5263	375	13	,	,	PUNCT
ejpam-5263	375	14	then	then	ADV
ejpam-5263	375	15	the	the	DET
ejpam-5263	375	16	conclusion	conclusion	NOUN
ejpam-5263	375	17	in	in	ADP
ejpam-5263	375	18	the	the	DET
ejpam-5263	375	19	above	above	ADJ
ejpam-5263	375	20	theorem	theorem	NOUN
ejpam-5263	375	21	tells	tell	VERB
ejpam-5263	375	22	us	we	PRON
ejpam-5263	375	23	that	that	PRON
ejpam-5263	375	24	⟨x	⟨x	VERB
ejpam-5263	375	25	,	,	PUNCT
ejpam-5263	376	1	y⟩a	y⟩a	PROPN
ejpam-5263	376	2	=	=	SYM
ejpam-5263	376	3	n∑	n∑	PROPN
ejpam-5263	376	4	i=1	i=1	PROPN
ejpam-5263	376	5	⟨x	⟨x	NUM
ejpam-5263	376	6	,	,	PUNCT
ejpam-5263	376	7	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	376	8	,	,	PUNCT
ejpam-5263	376	9	y⟩+	y⟩+	PROPN
ejpam-5263	376	10	n	n	ADP
ejpam-5263	376	11	∞∑	∞∑	NUM
ejpam-5263	376	12	i	i	PROPN
ejpam-5263	376	13	=	=	NOUN
ejpam-5263	376	14	n+1	n+1	PRON
ejpam-5263	376	15	⟨x	⟨x	NUM
ejpam-5263	376	16	,	,	PUNCT
ejpam-5263	376	17	ai⟩⟨ai	ai⟩⟨ai	PROPN
ejpam-5263	376	18	,	,	PUNCT
ejpam-5263	376	19	y⟩	y⟩	NOUN
ejpam-5263	376	20	for	for	ADP
ejpam-5263	376	21	all	all	DET
ejpam-5263	376	22	x	x	NOUN
ejpam-5263	377	1	,	,	PUNCT
ejpam-5263	377	2	y	y	PROPN
ejpam-5263	377	3	∈	∈	PROPN
ejpam-5263	377	4	x.	x.	NOUN
ejpam-5263	377	5	references	reference	VERB
ejpam-5263	377	6	1946	1946	NUM
ejpam-5263	377	7	corollary	corollary	ADJ
ejpam-5263	377	8	2	2	NUM
ejpam-5263	377	9	.	.	PUNCT
ejpam-5263	377	10	suppose	suppose	VERB
ejpam-5263	377	11	that	that	SCONJ
ejpam-5263	377	12	∥ai∥	∥ai∥	NOUN
ejpam-5263	377	13	=	=	PUNCT
ejpam-5263	377	14	α	α	PROPN
ejpam-5263	377	15	for	for	ADP
ejpam-5263	377	16	i	i	PRON
ejpam-5263	377	17	=	=	NOUN
ejpam-5263	377	18	1	1	NUM
ejpam-5263	377	19	,	,	PUNCT
ejpam-5263	377	20	2	2	NUM
ejpam-5263	377	21	,	,	PUNCT
ejpam-5263	377	22	.	.	PUNCT
ejpam-5263	377	23	.	.	PUNCT
ejpam-5263	378	1	.	.	PUNCT
ejpam-5263	379	1	,	,	PUNCT
ejpam-5263	379	2	n.	n.	NOUN
ejpam-5263	379	3	for	for	ADP
ejpam-5263	379	4	every	every	DET
ejpam-5263	379	5	x	x	NOUN
ejpam-5263	379	6	,	,	PUNCT
ejpam-5263	379	7	y	y	PROPN
ejpam-5263	379	8	∈	∈	PROPN
ejpam-5263	380	1	x	x	AUX
ejpam-5263	380	2	,	,	PUNCT
ejpam-5263	380	3	let	let	VERB
ejpam-5263	380	4	x	x	PRON
ejpam-5263	380	5	:	:	PUNCT
ejpam-5263	380	6	=	=	PUNCT
ejpam-5263	381	1	xa+x⊥a	xa+x⊥a	PROPN
ejpam-5263	381	2	and	and	CCONJ
ejpam-5263	381	3	y	y	PROPN
ejpam-5263	381	4	:	:	PUNCT
ejpam-5263	382	1	=	=	SYM
ejpam-5263	382	2	ya+y⊥a	ya+y⊥a	X
ejpam-5263	382	3	where	where	SCONJ
ejpam-5263	382	4	xa	xa	PROPN
ejpam-5263	382	5	and	and	CCONJ
ejpam-5263	382	6	ya	ya	PRON
ejpam-5263	382	7	are	be	AUX
ejpam-5263	382	8	the	the	DET
ejpam-5263	382	9	orthogonal	orthogonal	ADJ
ejpam-5263	382	10	projections	projection	NOUN
ejpam-5263	382	11	of	of	ADP
ejpam-5263	382	12	x	x	X
ejpam-5263	382	13	and	and	CCONJ
ejpam-5263	382	14	y	y	PROPN
ejpam-5263	382	15	on	on	ADP
ejpam-5263	382	16	span	span	NOUN
ejpam-5263	382	17	a	a	PRON
ejpam-5263	382	18	(	(	PUNCT
ejpam-5263	382	19	respectively	respectively	ADV
ejpam-5263	382	20	)	)	PUNCT
ejpam-5263	382	21	,	,	PUNCT
ejpam-5263	382	22	and	and	CCONJ
ejpam-5263	382	23	x⊥a	x⊥a	PROPN
ejpam-5263	382	24	and	and	CCONJ
ejpam-5263	382	25	y⊥a	y⊥a	PROPN
ejpam-5263	382	26	are	be	AUX
ejpam-5263	382	27	their	their	PRON
ejpam-5263	382	28	complements	complement	NOUN
ejpam-5263	382	29	(	(	PUNCT
ejpam-5263	382	30	respectively	respectively	ADV
ejpam-5263	382	31	)	)	PUNCT
ejpam-5263	382	32	.	.	PUNCT
ejpam-5263	383	1	if	if	SCONJ
ejpam-5263	383	2	⟨xa	⟨xa	NOUN
ejpam-5263	383	3	,	,	PUNCT
ejpam-5263	383	4	ya⟩a	ya⟩a	PROPN
ejpam-5263	383	5	=	=	SYM
ejpam-5263	383	6	0	0	NUM
ejpam-5263	383	7	and	and	CCONJ
ejpam-5263	383	8	⟨x⊥a	⟨x⊥a	PROPN
ejpam-5263	383	9	,	,	PUNCT
ejpam-5263	383	10	y⊥a⟩	y⊥a⟩	PROPN
ejpam-5263	383	11	=	=	SYM
ejpam-5263	383	12	0	0	NUM
ejpam-5263	383	13	,	,	PUNCT
ejpam-5263	383	14	then	then	ADV
ejpam-5263	383	15	⟨x	⟨x	VERB
ejpam-5263	383	16	,	,	PUNCT
ejpam-5263	383	17	y⟩	y⟩	NOUN
ejpam-5263	383	18	=	=	NOUN
ejpam-5263	383	19	0	0	X
ejpam-5263	383	20	.	.	PUNCT
ejpam-5263	384	1	conversely	conversely	ADV
ejpam-5263	384	2	,	,	PUNCT
ejpam-5263	384	3	if	if	SCONJ
ejpam-5263	384	4	⟨xa	⟨xa	NOUN
ejpam-5263	384	5	,	,	PUNCT
ejpam-5263	384	6	ya⟩	ya⟩	PUNCT
ejpam-5263	384	7	=	=	SYM
ejpam-5263	384	8	0	0	NUM
ejpam-5263	384	9	and	and	CCONJ
ejpam-5263	384	10	⟨x⊥a	⟨x⊥a	PROPN
ejpam-5263	384	11	,	,	PUNCT
ejpam-5263	384	12	y⊥a⟩	y⊥a⟩	PROPN
ejpam-5263	384	13	=	=	SYM
ejpam-5263	384	14	0	0	NUM
ejpam-5263	384	15	,	,	PUNCT
ejpam-5263	384	16	then	then	ADV
ejpam-5263	384	17	⟨x	⟨x	VERB
ejpam-5263	384	18	,	,	PUNCT
ejpam-5263	384	19	y⟩a	y⟩a	NUM
ejpam-5263	384	20	=	=	SYM
ejpam-5263	384	21	0	0	X
ejpam-5263	384	22	.	.	PUNCT
ejpam-5263	385	1	acknowledgements	acknowledgement	VERB
ejpam-5263	385	2	the	the	DET
ejpam-5263	385	3	first	first	ADJ
ejpam-5263	385	4	author	author	NOUN
ejpam-5263	385	5	is	be	AUX
ejpam-5263	385	6	supported	support	VERB
ejpam-5263	385	7	by	by	ADP
ejpam-5263	385	8	lppm	lppm	PROPN
ejpam-5263	385	9	institut	institut	PROPN
ejpam-5263	385	10	teknologi	teknologi	PROPN
ejpam-5263	385	11	kalimantan	kalimantan	PROPN
ejpam-5263	385	12	.	.	PUNCT
ejpam-5263	386	1	the	the	DET
ejpam-5263	386	2	third	third	ADJ
ejpam-5263	386	3	author	author	NOUN
ejpam-5263	386	4	is	be	AUX
ejpam-5263	386	5	supported	support	VERB
ejpam-5263	386	6	by	by	ADP
ejpam-5263	386	7	p3mi	p3mi	VERB
ejpam-5263	386	8	-	-	PUNCT
ejpam-5263	386	9	itb	itb	ADJ
ejpam-5263	386	10	program	program	NOUN
ejpam-5263	386	11	.	.	PUNCT
ejpam-5263	387	1	the	the	DET
ejpam-5263	387	2	authors	author	NOUN
ejpam-5263	387	3	express	express	VERB
ejpam-5263	387	4	their	their	PRON
ejpam-5263	387	5	gratitude	gratitude	NOUN
ejpam-5263	387	6	to	to	ADP
ejpam-5263	387	7	the	the	DET
ejpam-5263	387	8	editor	editor	NOUN
ejpam-5263	387	9	and	and	CCONJ
ejpam-5263	387	10	anonymous	anonymous	ADJ
ejpam-5263	387	11	reviewers	reviewer	NOUN
ejpam-5263	387	12	for	for	ADP
ejpam-5263	387	13	their	their	PRON
ejpam-5263	387	14	valuable	valuable	ADJ
ejpam-5263	387	15	comments	comment	NOUN
ejpam-5263	387	16	that	that	PRON
ejpam-5263	387	17	helped	helped	AUX
ejpam-5263	387	18	improve	improve	VERB
ejpam-5263	387	19	the	the	DET
ejpam-5263	387	20	quality	quality	NOUN
ejpam-5263	387	21	of	of	ADP
ejpam-5263	387	22	this	this	DET
ejpam-5263	387	23	work	work	NOUN
ejpam-5263	387	24	.	.	PUNCT
ejpam-5263	388	1	references	reference	NOUN
ejpam-5263	388	2	[	[	X
ejpam-5263	388	3	1	1	NUM
ejpam-5263	388	4	]	]	PUNCT
ejpam-5263	388	5	y	y	PROPN
ejpam-5263	388	6	j	j	PROPN
ejpam-5263	388	7	cho	cho	PROPN
ejpam-5263	388	8	,	,	PUNCT
ejpam-5263	388	9	c	c	PROPN
ejpam-5263	388	10	r	r	NOUN
ejpam-5263	388	11	diminnie	diminnie	NOUN
ejpam-5263	388	12	,	,	PUNCT
ejpam-5263	388	13	s	s	NOUN
ejpam-5263	388	14	gahler	gahler	NOUN
ejpam-5263	388	15	,	,	PUNCT
ejpam-5263	388	16	r	r	PROPN
ejpam-5263	388	17	w	w	PROPN
ejpam-5263	388	18	freese	freese	PROPN
ejpam-5263	388	19	,	,	PUNCT
ejpam-5263	388	20	and	and	CCONJ
ejpam-5263	388	21	e	e	NOUN
ejpam-5263	388	22	z	z	NOUN
ejpam-5263	388	23	andalafte	andalafte	PROPN
ejpam-5263	388	24	.	.	PUNCT
ejpam-5263	389	1	isosceles	isosceles	PROPN
ejpam-5263	389	2	orthogonal	orthogonal	ADJ
ejpam-5263	389	3	triple	triple	NOUN
ejpam-5263	389	4	in	in	ADP
ejpam-5263	389	5	linear	linear	ADJ
ejpam-5263	389	6	2	2	NUM
ejpam-5263	389	7	-	-	PUNCT
ejpam-5263	389	8	normed	norme	VERB
ejpam-5263	389	9	spaces	space	NOUN
ejpam-5263	389	10	.	.	PUNCT
ejpam-5263	390	1	math	math	NOUN
ejpam-5263	390	2	.	.	PUNCT
ejpam-5263	391	1	nachr	nachr	PROPN
ejpam-5263	391	2	.	.	PROPN
ejpam-5263	391	3	,	,	PUNCT
ejpam-5263	391	4	157:225–234	157:225–234	NUM
ejpam-5263	391	5	,	,	PUNCT
ejpam-5263	391	6	1992	1992	NUM
ejpam-5263	391	7	.	.	PUNCT
ejpam-5263	392	1	[	[	X
ejpam-5263	392	2	2	2	NUM
ejpam-5263	392	3	]	]	PUNCT
ejpam-5263	392	4	y	y	PROPN
ejpam-5263	392	5	j	j	PROPN
ejpam-5263	392	6	cho	cho	PROPN
ejpam-5263	392	7	and	and	CCONJ
ejpam-5263	392	8	s	s	PROPN
ejpam-5263	392	9	s	s	PROPN
ejpam-5263	392	10	kim	kim	PROPN
ejpam-5263	392	11	.	.	PUNCT
ejpam-5263	393	1	gateaux	gateaux	PROPN
ejpam-5263	393	2	derivatives	derivative	NOUN
ejpam-5263	393	3	and	and	CCONJ
ejpam-5263	393	4	2	2	NUM
ejpam-5263	393	5	-	-	PUNCT
ejpam-5263	393	6	inner	inner	ADJ
ejpam-5263	393	7	product	product	NOUN
ejpam-5263	393	8	spaces	space	NOUN
ejpam-5263	393	9	.	.	PUNCT
ejpam-5263	394	1	glas	glas	PROPN
ejpam-5263	394	2	.	.	PUNCT
ejpam-5263	394	3	mat	mat	PROPN
ejpam-5263	394	4	.	.	PUNCT
ejpam-5263	394	5	ser	ser	PROPN
ejpam-5263	394	6	.	.	PUNCT
ejpam-5263	394	7	iii	iii	PROPN
ejpam-5263	394	8	.	.	PROPN
ejpam-5263	394	9	,	,	PUNCT
ejpam-5263	394	10	27(47):197–203	27(47):197–203	NUM
ejpam-5263	394	11	,	,	PUNCT
ejpam-5263	394	12	1983	1983	NUM
ejpam-5263	394	13	.	.	PUNCT
ejpam-5263	395	1	[	[	X
ejpam-5263	395	2	3	3	X
ejpam-5263	395	3	]	]	X
ejpam-5263	395	4	p	p	NOUN
ejpam-5263	395	5	debnath	debnath	NOUN
ejpam-5263	395	6	and	and	CCONJ
ejpam-5263	395	7	m	m	PROPN
ejpam-5263	395	8	saha	saha	PROPN
ejpam-5263	395	9	.	.	PUNCT
ejpam-5263	396	1	categorization	categorization	NOUN
ejpam-5263	396	2	of	of	ADP
ejpam-5263	396	3	n	n	CCONJ
ejpam-5263	396	4	-	-	PUNCT
ejpam-5263	396	5	inner	inner	ADJ
ejpam-5263	396	6	product	product	NOUN
ejpam-5263	396	7	space	space	NOUN
ejpam-5263	396	8	.	.	PUNCT
ejpam-5263	397	1	asian	asian	ADJ
ejpam-5263	397	2	research	research	PROPN
ejpam-5263	397	3	j.	j.	PROPN
ejpam-5263	397	4	math	math	PROPN
ejpam-5263	397	5	,	,	PUNCT
ejpam-5263	397	6	11(4):1–10	11(4):1–10	NUM
ejpam-5263	397	7	,	,	PUNCT
ejpam-5263	397	8	2018	2018	NUM
ejpam-5263	397	9	.	.	PUNCT
ejpam-5263	398	1	[	[	X
ejpam-5263	398	2	4	4	X
ejpam-5263	398	3	]	]	SYM
ejpam-5263	398	4	c	c	NOUN
ejpam-5263	398	5	r	r	NOUN
ejpam-5263	398	6	diminnie	diminnie	NOUN
ejpam-5263	398	7	,	,	PUNCT
ejpam-5263	398	8	s	s	NOUN
ejpam-5263	398	9	gahler	gahler	NOUN
ejpam-5263	398	10	,	,	PUNCT
ejpam-5263	398	11	and	and	CCONJ
ejpam-5263	398	12	a	a	DET
ejpam-5263	398	13	white	white	ADJ
ejpam-5263	398	14	.	.	PUNCT
ejpam-5263	399	1	2	2	NUM
ejpam-5263	399	2	-	-	PUNCT
ejpam-5263	399	3	inner	inner	ADJ
ejpam-5263	399	4	product	product	NOUN
ejpam-5263	399	5	spaces	space	VERB
ejpam-5263	399	6	.	.	PUNCT
ejpam-5263	400	1	demonstratio	demonstratio	PROPN
ejpam-5263	400	2	math	math	PROPN
ejpam-5263	400	3	.	.	PUNCT
ejpam-5263	401	1	,	,	PUNCT
ejpam-5263	401	2	6:525–536	6:525–536	NOUN
ejpam-5263	401	3	,	,	PUNCT
ejpam-5263	401	4	1973	1973	NUM
ejpam-5263	401	5	.	.	PUNCT
ejpam-5263	402	1	[	[	X
ejpam-5263	402	2	5	5	NUM
ejpam-5263	402	3	]	]	SYM
ejpam-5263	402	4	c	c	NOUN
ejpam-5263	402	5	r	r	NOUN
ejpam-5263	402	6	diminnie	diminnie	NOUN
ejpam-5263	402	7	,	,	PUNCT
ejpam-5263	402	8	s	s	NOUN
ejpam-5263	402	9	gahler	gahler	NOUN
ejpam-5263	402	10	,	,	PUNCT
ejpam-5263	402	11	and	and	CCONJ
ejpam-5263	402	12	a	a	DET
ejpam-5263	402	13	white	white	ADJ
ejpam-5263	402	14	.	.	PUNCT
ejpam-5263	403	1	2	2	NUM
ejpam-5263	403	2	-	-	PUNCT
ejpam-5263	403	3	inner	inner	ADJ
ejpam-5263	403	4	product	product	NOUN
ejpam-5263	403	5	spaces	space	VERB
ejpam-5263	403	6	.	.	PUNCT
ejpam-5263	404	1	ii	ii	PROPN
ejpam-5263	404	2	.	.	PROPN
ejpam-5263	404	3	demonstratio	demonstratio	PROPN
ejpam-5263	404	4	math	math	PROPN
ejpam-5263	404	5	.	.	PUNCT
ejpam-5263	404	6	,	,	PUNCT
ejpam-5263	404	7	10:169–188	10:169–188	NUM
ejpam-5263	404	8	,	,	PUNCT
ejpam-5263	404	9	1977	1977	NUM
ejpam-5263	404	10	.	.	PUNCT
ejpam-5263	405	1	[	[	X
ejpam-5263	405	2	6	6	NUM
ejpam-5263	405	3	]	]	SYM
ejpam-5263	405	4	c	c	NOUN
ejpam-5263	405	5	r	r	NOUN
ejpam-5263	405	6	diminnie	diminnie	NOUN
ejpam-5263	405	7	and	and	CCONJ
ejpam-5263	405	8	a	a	DET
ejpam-5263	405	9	white	white	NOUN
ejpam-5263	405	10	.	.	PUNCT
ejpam-5263	406	1	a	a	DET
ejpam-5263	406	2	characterization	characterization	NOUN
ejpam-5263	406	3	of	of	ADP
ejpam-5263	406	4	2	2	NUM
ejpam-5263	406	5	-	-	PUNCT
ejpam-5263	406	6	inner	inner	ADJ
ejpam-5263	406	7	product	product	NOUN
ejpam-5263	406	8	spaces	space	VERB
ejpam-5263	406	9	.	.	PUNCT
ejpam-5263	407	1	math	math	NOUN
ejpam-5263	407	2	.	.	PUNCT
ejpam-5263	408	1	japon	japon	PROPN
ejpam-5263	408	2	.	.	PROPN
ejpam-5263	408	3	,	,	PUNCT
ejpam-5263	408	4	114:275–277	114:275–277	NUM
ejpam-5263	408	5	,	,	PUNCT
ejpam-5263	408	6	1983	1983	NUM
ejpam-5263	408	7	.	.	PUNCT
ejpam-5263	409	1	[	[	X
ejpam-5263	409	2	7	7	NUM
ejpam-5263	409	3	]	]	X
ejpam-5263	409	4	g	g	PROPN
ejpam-5263	409	5	godini	godini	PROPN
ejpam-5263	409	6	.	.	PUNCT
ejpam-5263	410	1	mr0743643	mr0743643	PROPN
ejpam-5263	410	2	(	(	PUNCT
ejpam-5263	410	3	85g:46028	85g:46028	NUM
ejpam-5263	410	4	)	)	PUNCT
ejpam-5263	410	5	.	.	PUNCT
ejpam-5263	411	1	mathematical	mathematical	ADJ
ejpam-5263	411	2	reviews	review	NOUN
ejpam-5263	411	3	on	on	ADP
ejpam-5263	411	4	the	the	DET
ejpam-5263	411	5	web	web	NOUN
ejpam-5263	411	6	,	,	PUNCT
ejpam-5263	411	7	mathscinet	mathscinet	NOUN
ejpam-5263	411	8	.	.	PUNCT
ejpam-5263	412	1	[	[	X
ejpam-5263	412	2	8	8	NUM
ejpam-5263	412	3	]	]	X
ejpam-5263	412	4	h	h	NOUN
ejpam-5263	412	5	gunawan	gunawan	X
ejpam-5263	412	6	.	.	PUNCT
ejpam-5263	413	1	inner	inner	ADJ
ejpam-5263	413	2	product	product	NOUN
ejpam-5263	413	3	on	on	ADP
ejpam-5263	413	4	n	n	CCONJ
ejpam-5263	413	5	-	-	PUNCT
ejpam-5263	413	6	inner	inner	ADJ
ejpam-5263	413	7	product	product	NOUN
ejpam-5263	413	8	spaces	space	VERB
ejpam-5263	413	9	.	.	PUNCT
ejpam-5263	414	1	soochow	soochow	PROPN
ejpam-5263	414	2	j.	j.	PROPN
ejpam-5263	414	3	math	math	PROPN
ejpam-5263	414	4	.	.	PUNCT
ejpam-5263	414	5	,	,	PUNCT
ejpam-5263	414	6	28(4):389	28(4):389	NUM
ejpam-5263	414	7	–	–	PUNCT
ejpam-5263	414	8	398	398	NUM
ejpam-5263	414	9	,	,	PUNCT
ejpam-5263	414	10	2002	2002	NUM
ejpam-5263	414	11	.	.	PUNCT
ejpam-5263	415	1	[	[	X
ejpam-5263	415	2	9	9	NUM
ejpam-5263	415	3	]	]	X
ejpam-5263	415	4	h	h	NOUN
ejpam-5263	415	5	gunawan	gunawan	PROPN
ejpam-5263	415	6	.	.	PUNCT
ejpam-5263	416	1	on	on	ADP
ejpam-5263	416	2	n	n	CCONJ
ejpam-5263	416	3	-	-	PUNCT
ejpam-5263	416	4	inner	inner	ADJ
ejpam-5263	416	5	products	product	NOUN
ejpam-5263	416	6	,	,	PUNCT
ejpam-5263	416	7	n	n	CCONJ
ejpam-5263	416	8	-	-	PUNCT
ejpam-5263	416	9	norms	norm	NOUN
ejpam-5263	416	10	,	,	PUNCT
ejpam-5263	416	11	and	and	CCONJ
ejpam-5263	416	12	the	the	DET
ejpam-5263	416	13	cauchy	cauchy	PROPN
ejpam-5263	416	14	-	-	PUNCT
ejpam-5263	416	15	schwartz	schwartz	PROPN
ejpam-5263	416	16	inequality	inequality	NOUN
ejpam-5263	416	17	.	.	PUNCT
ejpam-5263	417	1	sci	sci	PROPN
ejpam-5263	417	2	.	.	PROPN
ejpam-5263	417	3	japon	japon	PROPN
ejpam-5263	417	4	.	.	PUNCT
ejpam-5263	418	1	math	math	NOUN
ejpam-5263	418	2	.	.	PUNCT
ejpam-5263	418	3	,	,	PUNCT
ejpam-5263	419	1	55:53–60	55:53–60	PROPN
ejpam-5263	419	2	,	,	PUNCT
ejpam-5263	419	3	2002	2002	NUM
ejpam-5263	419	4	.	.	PUNCT
ejpam-5263	420	1	[	[	X
ejpam-5263	420	2	10	10	NUM
ejpam-5263	420	3	]	]	X
ejpam-5263	420	4	h	h	NOUN
ejpam-5263	420	5	gunawan	gunawan	X
ejpam-5263	420	6	.	.	PUNCT
ejpam-5263	421	1	n	n	CCONJ
ejpam-5263	421	2	-	-	PUNCT
ejpam-5263	421	3	inner	inner	ADJ
ejpam-5263	421	4	products	product	NOUN
ejpam-5263	421	5	,	,	PUNCT
ejpam-5263	421	6	n	n	CCONJ
ejpam-5263	421	7	-	-	PUNCT
ejpam-5263	421	8	norms	norm	NOUN
ejpam-5263	421	9	,	,	PUNCT
ejpam-5263	421	10	and	and	CCONJ
ejpam-5263	421	11	angles	angle	NOUN
ejpam-5263	421	12	between	between	ADP
ejpam-5263	421	13	two	two	NUM
ejpam-5263	421	14	subspaces	subspace	NOUN
ejpam-5263	421	15	.	.	PUNCT
ejpam-5263	422	1	in	in	ADP
ejpam-5263	422	2	h	h	PROPN
ejpam-5263	422	3	dutta	dutta	PROPN
ejpam-5263	422	4	,	,	PUNCT
ejpam-5263	422	5	m	m	VERB
ejpam-5263	422	6	ruzhansky	ruzhansky	ADJ
ejpam-5263	422	7	,	,	PUNCT
ejpam-5263	422	8	and	and	CCONJ
ejpam-5263	422	9	r	r	NOUN
ejpam-5263	422	10	p	p	PROPN
ejpam-5263	422	11	agarwal	agarwal	PROPN
ejpam-5263	422	12	,	,	PUNCT
ejpam-5263	422	13	editors	editor	NOUN
ejpam-5263	422	14	,	,	PUNCT
ejpam-5263	422	15	mathematical	mathematical	ADJ
ejpam-5263	422	16	analysis	analysis	NOUN
ejpam-5263	422	17	and	and	CCONJ
ejpam-5263	422	18	applications	application	NOUN
ejpam-5263	422	19	:	:	PUNCT
ejpam-5263	422	20	selected	select	VERB
ejpam-5263	422	21	topics	topic	NOUN
ejpam-5263	422	22	.	.	PUNCT
ejpam-5263	422	23	,	,	PUNCT
ejpam-5263	422	24	chapter	chapter	NOUN
ejpam-5263	422	25	13	13	NUM
ejpam-5263	422	26	.	.	PUNCT
ejpam-5263	423	1	john	john	PROPN
ejpam-5263	423	2	wiley	wiley	PROPN
ejpam-5263	423	3	&	&	CCONJ
ejpam-5263	423	4	sons	sons	PROPN
ejpam-5263	423	5	,	,	PUNCT
ejpam-5263	423	6	inc	inc	PROPN
ejpam-5263	423	7	.	.	PROPN
ejpam-5263	423	8	,	,	PUNCT
ejpam-5263	423	9	2018	2018	NUM
ejpam-5263	423	10	.	.	PUNCT
ejpam-5263	424	1	references	reference	NOUN
ejpam-5263	424	2	1947	1947	NUM
ejpam-5263	425	1	[	[	X
ejpam-5263	425	2	11	11	NUM
ejpam-5263	425	3	]	]	X
ejpam-5263	425	4	h	h	NOUN
ejpam-5263	425	5	gunawan	gunawan	X
ejpam-5263	425	6	,	,	PUNCT
ejpam-5263	425	7	m	m	NOUN
ejpam-5263	425	8	mashadi	mashadi	NOUN
ejpam-5263	425	9	,	,	PUNCT
ejpam-5263	425	10	s	s	PART
ejpam-5263	425	11	gemawati	gemawati	PROPN
ejpam-5263	425	12	,	,	PUNCT
ejpam-5263	425	13	n	n	PRON
ejpam-5263	425	14	supiamin	supiamin	NOUN
ejpam-5263	425	15	,	,	PUNCT
ejpam-5263	425	16	and	and	CCONJ
ejpam-5263	425	17	i	i	PRON
ejpam-5263	425	18	sihwaningrum	sihwaningrum	NOUN
ejpam-5263	425	19	.	.	PUNCT
ejpam-5263	426	1	orthogonality	orthogonality	NOUN
ejpam-5263	426	2	in	in	ADP
ejpam-5263	426	3	2	2	NUM
ejpam-5263	426	4	-	-	PUNCT
ejpam-5263	426	5	normed	norme	VERB
ejpam-5263	426	6	spaces	space	NOUN
ejpam-5263	426	7	revisited	revisit	VERB
ejpam-5263	426	8	.	.	PUNCT
ejpam-5263	427	1	publikacije	publikacije	NOUN
ejpam-5263	427	2	elektrotehnickog	elektrotehnickog	PROPN
ejpam-5263	427	3	fakulteta	fakulteta	ADJ
ejpam-5263	427	4	serija	serija	NOUN
ejpam-5263	427	5	:	:	PUNCT
ejpam-5263	427	6	matematika	matematika	PROPN
ejpam-5263	427	7	.	.	PROPN
ejpam-5263	427	8	,	,	PUNCT
ejpam-5263	427	9	17:76–83	17:76–83	NUM
ejpam-5263	427	10	,	,	PUNCT
ejpam-5263	427	11	2006	2006	NUM
ejpam-5263	427	12	.	.	PUNCT
ejpam-5263	428	1	[	[	X
ejpam-5263	428	2	12	12	NUM
ejpam-5263	428	3	]	]	PUNCT
ejpam-5263	428	4	a	a	DET
ejpam-5263	428	5	khan	khan	NOUN
ejpam-5263	428	6	and	and	CCONJ
ejpam-5263	428	7	a	a	DET
ejpam-5263	428	8	siddiqui	siddiqui	NOUN
ejpam-5263	428	9	.	.	PUNCT
ejpam-5263	429	1	b	b	X
ejpam-5263	429	2	-	-	PUNCT
ejpam-5263	429	3	orthogonality	orthogonality	NOUN
ejpam-5263	429	4	in	in	ADP
ejpam-5263	429	5	2	2	NUM
ejpam-5263	429	6	-	-	PUNCT
ejpam-5263	429	7	normed	norme	VERB
ejpam-5263	429	8	space	space	NOUN
ejpam-5263	429	9	.	.	PUNCT
ejpam-5263	430	1	bull	bull	NOUN
ejpam-5263	430	2	.	.	PUNCT
ejpam-5263	431	1	calcutta	calcutta	PROPN
ejpam-5263	431	2	math.soc	math.soc	PROPN
ejpam-5263	431	3	.	.	PROPN
ejpam-5263	431	4	,	,	PUNCT
ejpam-5263	431	5	74:216–222	74:216–222	PROPN
ejpam-5263	431	6	,	,	PUNCT
ejpam-5263	431	7	1982	1982	NUM
ejpam-5263	431	8	.	.	PUNCT
ejpam-5263	432	1	[	[	X
ejpam-5263	432	2	13	13	NUM
ejpam-5263	432	3	]	]	PUNCT
ejpam-5263	432	4	n	n	X
ejpam-5263	432	5	minculete	minculete	NOUN
ejpam-5263	432	6	and	and	CCONJ
ejpam-5263	432	7	r	r	NOUN
ejpam-5263	432	8	păltănea	păltănea	NOUN
ejpam-5263	432	9	.	.	PUNCT
ejpam-5263	433	1	weak	weak	ADJ
ejpam-5263	433	2	n	n	CCONJ
ejpam-5263	433	3	-	-	PUNCT
ejpam-5263	433	4	inner	inner	ADJ
ejpam-5263	433	5	product	product	NOUN
ejpam-5263	433	6	spaces	space	VERB
ejpam-5263	433	7	.	.	PUNCT
ejpam-5263	434	1	annals	annal	NOUN
ejpam-5263	434	2	of	of	ADP
ejpam-5263	434	3	functional	functional	ADJ
ejpam-5263	434	4	analysis	analysis	NOUN
ejpam-5263	434	5	.	.	PUNCT
ejpam-5263	434	6	,	,	PUNCT
ejpam-5263	434	7	12(22):1–22	12(22):1–22	NUM
ejpam-5263	434	8	,	,	PUNCT
ejpam-5263	434	9	2021	2021	NUM
ejpam-5263	434	10	.	.	PUNCT
ejpam-5263	435	1	[	[	X
ejpam-5263	435	2	14	14	NUM
ejpam-5263	435	3	]	]	X
ejpam-5263	435	4	a	a	DET
ejpam-5263	435	5	misiak	misiak	NOUN
ejpam-5263	435	6	.	.	PUNCT
ejpam-5263	436	1	n	n	CCONJ
ejpam-5263	436	2	-	-	PUNCT
ejpam-5263	436	3	inner	inner	ADJ
ejpam-5263	436	4	product	product	NOUN
ejpam-5263	436	5	spaces	space	VERB
ejpam-5263	436	6	.	.	PUNCT
ejpam-5263	437	1	math	math	NOUN
ejpam-5263	437	2	.	.	PUNCT
ejpam-5263	438	1	nachr	nachr	PROPN
ejpam-5263	438	2	.	.	PUNCT
ejpam-5263	438	3	,	,	PUNCT
ejpam-5263	438	4	140:299–319	140:299–319	NUM
ejpam-5263	438	5	,	,	PUNCT
ejpam-5263	438	6	1989	1989	NUM
ejpam-5263	438	7	.	.	PUNCT
ejpam-5263	439	1	[	[	X
ejpam-5263	439	2	15	15	NUM
ejpam-5263	439	3	]	]	X
ejpam-5263	439	4	a	a	DET
ejpam-5263	439	5	misiak	misiak	NOUN
ejpam-5263	439	6	.	.	PUNCT
ejpam-5263	439	7	orthogonality	orthogonality	NOUN
ejpam-5263	439	8	and	and	CCONJ
ejpam-5263	439	9	orthonormality	orthonormality	NOUN
ejpam-5263	439	10	in	in	ADP
ejpam-5263	439	11	n	n	CCONJ
ejpam-5263	439	12	-	-	PUNCT
ejpam-5263	439	13	inner	inner	ADJ
ejpam-5263	439	14	product	product	NOUN
ejpam-5263	439	15	spaces	space	VERB
ejpam-5263	439	16	.	.	PUNCT
ejpam-5263	440	1	math	math	NOUN
ejpam-5263	440	2	.	.	PUNCT
ejpam-5263	441	1	nachr	nachr	PROPN
ejpam-5263	441	2	.	.	PUNCT
ejpam-5263	441	3	,	,	PUNCT
ejpam-5263	441	4	143:249–261	143:249–261	NUM
ejpam-5263	441	5	,	,	PUNCT
ejpam-5263	441	6	1989	1989	NUM
ejpam-5263	441	7	.	.	PUNCT
ejpam-5263	442	1	[	[	X
ejpam-5263	442	2	16	16	NUM
ejpam-5263	442	3	]	]	PUNCT
ejpam-5263	442	4	a	a	DET
ejpam-5263	442	5	l	l	NOUN
ejpam-5263	442	6	soenjaya	soenjaya	NOUN
ejpam-5263	442	7	.	.	PUNCT
ejpam-5263	443	1	characterizations	characterization	NOUN
ejpam-5263	443	2	of	of	ADP
ejpam-5263	443	3	n	n	CCONJ
ejpam-5263	443	4	-	-	PUNCT
ejpam-5263	443	5	inner	inner	ADJ
ejpam-5263	443	6	product	product	NOUN
ejpam-5263	443	7	space	space	NOUN
ejpam-5263	443	8	.	.	PUNCT
ejpam-5263	444	1	int	int	NOUN
ejpam-5263	444	2	.	.	PUNCT
ejpam-5263	445	1	j.	j.	PROPN
ejpam-5263	445	2	pure	pure	PROPN
ejpam-5263	445	3	appl	appl	PROPN
ejpam-5263	445	4	.	.	PUNCT
ejpam-5263	445	5	math	math	PROPN
ejpam-5263	445	6	.	.	PUNCT
ejpam-5263	445	7	,	,	PUNCT
ejpam-5263	446	1	78(7):1011–1018	78(7):1011–1018	NOUN
ejpam-5263	446	2	,	,	PUNCT
ejpam-5263	446	3	2012	2012	NUM
ejpam-5263	446	4	.	.	PUNCT
ejpam-5263	447	1	[	[	X
ejpam-5263	447	2	17	17	NUM
ejpam-5263	447	3	]	]	X
ejpam-5263	447	4	k	k	PROPN
ejpam-5263	447	5	trencevski	trencevski	PROPN
ejpam-5263	447	6	and	and	CCONJ
ejpam-5263	447	7	rmalceski	rmalceski	ADJ
ejpam-5263	447	8	.	.	PUNCT
ejpam-5263	448	1	on	on	ADP
ejpam-5263	448	2	a	a	DET
ejpam-5263	448	3	generalized	generalized	ADJ
ejpam-5263	448	4	n	n	CCONJ
ejpam-5263	448	5	-	-	PUNCT
ejpam-5263	448	6	inner	inner	ADJ
ejpam-5263	448	7	product	product	NOUN
ejpam-5263	448	8	and	and	CCONJ
ejpam-5263	448	9	the	the	DET
ejpam-5263	448	10	corresponding	corresponding	ADJ
ejpam-5263	448	11	cauchy	cauchy	PROPN
ejpam-5263	448	12	-	-	PUNCT
ejpam-5263	448	13	schwarz	schwarz	PROPN
ejpam-5263	448	14	inequality	inequality	NOUN
ejpam-5263	448	15	.	.	PUNCT
ejpam-5263	449	1	j.	j.	PROPN
ejpam-5263	449	2	inequal	inequal	PROPN
ejpam-5263	449	3	.	.	PUNCT
ejpam-5263	450	1	pure	pure	ADJ
ejpam-5263	450	2	and	and	CCONJ
ejpam-5263	450	3	appl	appl	PROPN
ejpam-5263	450	4	.	.	PROPN
ejpam-5263	450	5	math	math	PROPN
ejpam-5263	450	6	.	.	PUNCT
ejpam-5263	451	1	,	,	PUNCT
ejpam-5263	451	2	7(2):1–10	7(2):1–10	NUM
ejpam-5263	451	3	,	,	PUNCT
ejpam-5263	451	4	2006	2006	NUM
ejpam-5263	451	5	.	.	PUNCT