id	sid	tid	token	lemma	pos
iajs-2952	1	1	ihjpas	ihjpas	PROPN
iajs-2952	1	2	.	.	PUNCT
iajs-2952	2	1	36(1)2023	36(1)2023	NUM
iajs-2952	2	2	311	311	NUM
iajs-2952	2	3	this	this	DET
iajs-2952	2	4	work	work	NOUN
iajs-2952	2	5	is	be	AUX
iajs-2952	2	6	licensed	license	VERB
iajs-2952	2	7	under	under	ADP
iajs-2952	2	8	a	a	DET
iajs-2952	2	9	creative	creative	ADJ
iajs-2952	2	10	commons	common	NOUN
iajs-2952	2	11	attribution	attribution	NOUN
iajs-2952	2	12	4.0	4.0	NUM
iajs-2952	2	13	international	international	ADJ
iajs-2952	2	14	license	license	NOUN
iajs-2952	2	15	the	the	DET
iajs-2952	2	16	completion	completion	NOUN
iajs-2952	2	17	of	of	ADP
iajs-2952	2	18	generalized	generalized	ADJ
iajs-2952	2	19	2	2	NUM
iajs-2952	2	20	-	-	PUNCT
iajs-2952	2	21	inner	inner	ADJ
iajs-2952	2	22	product	product	NOUN
iajs-2952	2	23	spaces	space	VERB
iajs-2952	2	24	abstract	abstract	ADV
iajs-2952	2	25	a	a	DET
iajs-2952	2	26	complete	complete	ADJ
iajs-2952	2	27	metric	metric	ADJ
iajs-2952	2	28	space	space	NOUN
iajs-2952	2	29	is	be	AUX
iajs-2952	2	30	a	a	DET
iajs-2952	2	31	well	well	ADV
iajs-2952	2	32	-	-	PUNCT
iajs-2952	2	33	known	know	VERB
iajs-2952	2	34	concept	concept	NOUN
iajs-2952	2	35	.	.	PUNCT
iajs-2952	3	1	kreyszig	kreyszig	PROPN
iajs-2952	3	2	shows	show	VERB
iajs-2952	3	3	that	that	SCONJ
iajs-2952	3	4	every	every	DET
iajs-2952	3	5	non	non	ADJ
iajs-2952	3	6	-	-	ADJ
iajs-2952	3	7	complete	complete	ADJ
iajs-2952	3	8	metric	metric	ADJ
iajs-2952	3	9	space	space	NOUN
iajs-2952	3	10	𝑊	𝑊	PROPN
iajs-2952	3	11	can	can	AUX
iajs-2952	3	12	be	be	AUX
iajs-2952	3	13	developed	develop	VERB
iajs-2952	3	14	into	into	ADP
iajs-2952	3	15	a	a	DET
iajs-2952	3	16	complete	complete	ADJ
iajs-2952	3	17	metric	metric	ADJ
iajs-2952	3	18	space	space	NOUN
iajs-2952	3	19	ŵ	ŵ	PROPN
iajs-2952	3	20	,	,	PUNCT
iajs-2952	3	21	referred	refer	VERB
iajs-2952	3	22	to	to	ADP
iajs-2952	3	23	as	as	ADP
iajs-2952	3	24	completion	completion	NOUN
iajs-2952	3	25	of	of	ADP
iajs-2952	3	26	𝑊.	𝑊.	PROPN
iajs-2952	3	27	we	we	PRON
iajs-2952	3	28	use	use	VERB
iajs-2952	3	29	the	the	DET
iajs-2952	3	30	b	b	NOUN
iajs-2952	3	31	-	-	PUNCT
iajs-2952	3	32	cauchy	cauchy	ADJ
iajs-2952	3	33	sequence	sequence	NOUN
iajs-2952	3	34	to	to	PART
iajs-2952	3	35	form	form	VERB
iajs-2952	3	36	�	�	PROPN
iajs-2952	3	37	̂	̂	PROPN
iajs-2952	3	38	�	�	NOUN
iajs-2952	3	39	which	which	PRON
iajs-2952	3	40	“	"	PUNCT
iajs-2952	3	41	is	be	AUX
iajs-2952	3	42	the	the	DET
iajs-2952	3	43	set	set	NOUN
iajs-2952	3	44	of	of	ADP
iajs-2952	3	45	all	all	DET
iajs-2952	3	46	b	b	NOUN
iajs-2952	3	47	-	-	PUNCT
iajs-2952	3	48	cauchy	cauchy	ADJ
iajs-2952	3	49	sequences	sequence	NOUN
iajs-2952	3	50	equivalence	equivalence	NOUN
iajs-2952	3	51	classes	class	NOUN
iajs-2952	3	52	”	"	PUNCT
iajs-2952	3	53	.	.	PUNCT
iajs-2952	4	1	after	after	ADP
iajs-2952	4	2	that	that	PRON
iajs-2952	4	3	,	,	PUNCT
iajs-2952	4	4	we	we	PRON
iajs-2952	4	5	prove	prove	VERB
iajs-2952	4	6	�	�	PROPN
iajs-2952	4	7	̂	̂	SYM
iajs-2952	4	8	�	�	PROPN
iajs-2952	4	9	to	to	PART
iajs-2952	4	10	be	be	AUX
iajs-2952	4	11	a	a	DET
iajs-2952	4	12	2	2	NUM
iajs-2952	4	13	-	-	PUNCT
iajs-2952	4	14	normed	normed	ADJ
iajs-2952	4	15	space	space	NOUN
iajs-2952	4	16	.	.	PUNCT
iajs-2952	5	1	then	then	ADV
iajs-2952	5	2	,	,	PUNCT
iajs-2952	5	3	we	we	PRON
iajs-2952	5	4	construct	construct	VERB
iajs-2952	5	5	an	an	DET
iajs-2952	5	6	isometric	isometric	NOUN
iajs-2952	5	7	by	by	ADP
iajs-2952	5	8	defining	define	VERB
iajs-2952	5	9	the	the	DET
iajs-2952	5	10	function	function	NOUN
iajs-2952	5	11	from	from	ADP
iajs-2952	5	12	w	w	PROPN
iajs-2952	5	13	to	to	ADP
iajs-2952	5	14	ŵ0	ŵ0	PROPN
iajs-2952	5	15	;	;	PUNCT
iajs-2952	5	16	thus	thus	ADV
iajs-2952	5	17	ŵ0	ŵ0	VERB
iajs-2952	5	18	and	and	CCONJ
iajs-2952	5	19	w	w	PROPN
iajs-2952	5	20	are	be	AUX
iajs-2952	5	21	isometric	isometric	ADJ
iajs-2952	5	22	,	,	PUNCT
iajs-2952	5	23	where	where	SCONJ
iajs-2952	5	24	ŵ0	ŵ0	ADV
iajs-2952	5	25	is	be	AUX
iajs-2952	5	26	the	the	DET
iajs-2952	5	27	subset	subset	NOUN
iajs-2952	5	28	of	of	ADP
iajs-2952	5	29	ŵ	ŵ	PROPN
iajs-2952	5	30	composed	compose	VERB
iajs-2952	5	31	of	of	ADP
iajs-2952	5	32	the	the	DET
iajs-2952	5	33	equivalence	equivalence	NOUN
iajs-2952	5	34	classes	class	NOUN
iajs-2952	5	35	that	that	PRON
iajs-2952	5	36	contains	contain	VERB
iajs-2952	5	37	constant	constant	ADJ
iajs-2952	5	38	b	b	X
iajs-2952	5	39	-	-	PUNCT
iajs-2952	5	40	cauchy	cauchy	ADJ
iajs-2952	5	41	sequences	sequence	NOUN
iajs-2952	5	42	.	.	PUNCT
iajs-2952	6	1	finally	finally	ADV
iajs-2952	6	2	,	,	PUNCT
iajs-2952	6	3	we	we	PRON
iajs-2952	6	4	prove	prove	VERB
iajs-2952	6	5	that	that	SCONJ
iajs-2952	6	6	�	�	PROPN
iajs-2952	6	7	̂	̂	VERB
iajs-2952	6	8	�	�	NOUN
iajs-2952	6	9	0	0	NUM
iajs-2952	6	10	is	be	AUX
iajs-2952	6	11	dense	dense	ADJ
iajs-2952	6	12	in	in	ADP
iajs-2952	6	13	�	�	PROPN
iajs-2952	6	14	̂	̂	SYM
iajs-2952	6	15	�	�	PROPN
iajs-2952	6	16	,	,	PUNCT
iajs-2952	6	17	�	�	PROPN
iajs-2952	6	18	̂	̂	VERB
iajs-2952	6	19	�	�	NOUN
iajs-2952	6	20	is	be	AUX
iajs-2952	6	21	complete	complete	ADJ
iajs-2952	6	22	and	and	CCONJ
iajs-2952	6	23	the	the	DET
iajs-2952	6	24	uniqueness	uniqueness	NOUN
iajs-2952	6	25	of	of	ADP
iajs-2952	6	26	�	�	PROPN
iajs-2952	6	27	̂	̂	VERB
iajs-2952	6	28	�	�	NOUN
iajs-2952	6	29	is	be	AUX
iajs-2952	6	30	up	up	ADP
iajs-2952	6	31	to	to	ADP
iajs-2952	6	32	isometrics	isometric	NOUN
iajs-2952	6	33	.	.	PUNCT
iajs-2952	7	1	keywords	keyword	NOUN
iajs-2952	7	2	:	:	PUNCT
iajs-2952	7	3	b	b	X
iajs-2952	7	4	-	-	PUNCT
iajs-2952	7	5	cauchy	cauchy	ADJ
iajs-2952	7	6	sequence	sequence	NOUN
iajs-2952	7	7	,	,	PUNCT
iajs-2952	7	8	equivalent	equivalent	ADJ
iajs-2952	7	9	class	class	NOUN
iajs-2952	7	10	,	,	PUNCT
iajs-2952	7	11	metric	metric	ADJ
iajs-2952	7	12	space	space	NOUN
iajs-2952	7	13	,	,	PUNCT
iajs-2952	7	14	completion	completion	NOUN
iajs-2952	7	15	generalized	generalize	VERB
iajs-2952	7	16	2	2	NUM
iajs-2952	7	17	-	-	PUNCT
iajs-2952	7	18	inner	inner	ADJ
iajs-2952	7	19	product	product	NOUN
iajs-2952	7	20	space	space	NOUN
iajs-2952	7	21	.	.	PUNCT
iajs-2952	8	1	1	1	X
iajs-2952	8	2	.	.	X
iajs-2952	8	3	introduction	introduction	NOUN
iajs-2952	8	4	cho	cho	PROPN
iajs-2952	8	5	and	and	CCONJ
iajs-2952	8	6	freese	freese	PROPN
iajs-2952	8	7	[	[	X
iajs-2952	8	8	3	3	NUM
iajs-2952	8	9	-	-	SYM
iajs-2952	8	10	4	4	NUM
iajs-2952	8	11	]	]	PUNCT
iajs-2952	8	12	introduced	introduce	VERB
iajs-2952	8	13	2	2	NUM
iajs-2952	8	14	-	-	PUNCT
iajs-2952	8	15	normed	norme	VERB
iajs-2952	8	16	space	space	NOUN
iajs-2952	8	17	by	by	ADP
iajs-2952	8	18	:	:	PUNCT
iajs-2952	8	19	let	let	VERB
iajs-2952	8	20	w	w	NOUN
iajs-2952	8	21	be	be	AUX
iajs-2952	8	22	a	a	DET
iajs-2952	8	23	real	real	ADJ
iajs-2952	8	24	linear	linear	ADJ
iajs-2952	8	25	space	space	NOUN
iajs-2952	8	26	with	with	ADP
iajs-2952	8	27	a	a	DET
iajs-2952	8	28	dimension	dimension	NOUN
iajs-2952	8	29	greater	great	ADJ
iajs-2952	8	30	than	than	ADP
iajs-2952	8	31	1	1	NUM
iajs-2952	8	32	.	.	PUNCT
iajs-2952	8	33	suppose	suppose	VERB
iajs-2952	8	34	that	that	SCONJ
iajs-2952	8	35	‖	‖	PROPN
iajs-2952	8	36	,	,	PUNCT
iajs-2952	8	37	‖	‖	ADJ
iajs-2952	8	38	is	be	AUX
iajs-2952	8	39	a	a	DET
iajs-2952	8	40	real	real	ADV
iajs-2952	8	41	-	-	PUNCT
iajs-2952	8	42	valued	value	VERB
iajs-2952	8	43	function	function	NOUN
iajs-2952	8	44	on	on	ADP
iajs-2952	8	45	w	w	PROPN
iajs-2952	8	46	×	×	PROPN
iajs-2952	8	47	w	w	NOUN
iajs-2952	8	48	for	for	ADP
iajs-2952	8	49	all	all	DET
iajs-2952	8	50	w	w	PROPN
iajs-2952	8	51	,	,	PUNCT
iajs-2952	8	52	y	y	PROPN
iajs-2952	8	53	,	,	PUNCT
iajs-2952	8	54	z	z	NOUN
iajs-2952	8	55	in	in	ADP
iajs-2952	8	56	w	w	PROPN
iajs-2952	8	57	and	and	CCONJ
iajs-2952	8	58	α	α	NOUN
iajs-2952	8	59	∈	∈	NOUN
iajs-2952	8	60	ℝ	ℝ	NOUN
iajs-2952	8	61	satisfying	satisfy	VERB
iajs-2952	8	62	the	the	DET
iajs-2952	8	63	following	follow	VERB
iajs-2952	8	64	requirements	requirement	NOUN
iajs-2952	8	65	:	:	PUNCT
iajs-2952	8	66	1	1	NUM
iajs-2952	8	67	.	.	NUM
iajs-2952	8	68	‖w	‖w	NOUN
iajs-2952	8	69	,	,	PUNCT
iajs-2952	8	70	y‖	y‖	PROPN
iajs-2952	8	71	=	=	PUNCT
iajs-2952	8	72	0	0	PUNCT
iajs-2952	9	1	if	if	SCONJ
iajs-2952	9	2	and	and	CCONJ
iajs-2952	9	3	only	only	ADV
iajs-2952	9	4	if	if	SCONJ
iajs-2952	9	5	w	w	PROPN
iajs-2952	9	6	and	and	CCONJ
iajs-2952	9	7	y	y	PROPN
iajs-2952	9	8	are	be	AUX
iajs-2952	9	9	linearly	linearly	ADV
iajs-2952	9	10	dependent	dependent	ADJ
iajs-2952	9	11	.	.	PUNCT
iajs-2952	10	1	2	2	X
iajs-2952	10	2	.	.	NUM
iajs-2952	10	3	‖w	‖w	NOUN
iajs-2952	10	4	,	,	PUNCT
iajs-2952	10	5	y‖	y‖	PROPN
iajs-2952	10	6	=	=	SYM
iajs-2952	11	1	‖y	‖y	PROPN
iajs-2952	11	2	,	,	PUNCT
iajs-2952	11	3	w‖	w‖	PROPN
iajs-2952	11	4	3	3	NUM
iajs-2952	11	5	.	.	PUNCT
iajs-2952	12	1	‖αw	‖αw	NOUN
iajs-2952	12	2	,	,	PUNCT
iajs-2952	12	3	y‖	y‖	NOUN
iajs-2952	12	4	=	=	SYM
iajs-2952	12	5	|α|‖w	|α|‖w	PROPN
iajs-2952	12	6	,	,	PUNCT
iajs-2952	12	7	y‖	y‖	NOUN
iajs-2952	12	8	4	4	NUM
iajs-2952	12	9	.	.	NUM
iajs-2952	12	10	‖w	‖w	PROPN
iajs-2952	13	1	+	+	PUNCT
iajs-2952	13	2	y	y	NOUN
iajs-2952	13	3	,	,	PUNCT
iajs-2952	13	4	z‖	z‖	VERB
iajs-2952	13	5	≤	≤	NUM
iajs-2952	13	6	‖w	‖w	NOUN
iajs-2952	13	7	,	,	PUNCT
iajs-2952	13	8	z‖	z‖	NOUN
iajs-2952	13	9	+	+	CCONJ
iajs-2952	13	10	‖y	‖y	ADJ
iajs-2952	13	11	,	,	PUNCT
iajs-2952	13	12	z‖	z‖	NOUN
iajs-2952	13	13	then	then	ADV
iajs-2952	13	14	‖	‖	PROPN
iajs-2952	13	15	,	,	PUNCT
iajs-2952	13	16	‖	‖	PROPN
iajs-2952	13	17	is	be	AUX
iajs-2952	13	18	called	call	VERB
iajs-2952	13	19	a	a	DET
iajs-2952	13	20	2	2	NUM
iajs-2952	13	21	-	-	PUNCT
iajs-2952	13	22	norm	norm	NOUN
iajs-2952	13	23	on	on	ADP
iajs-2952	13	24	𝑊	𝑊	PROPN
iajs-2952	13	25	and	and	CCONJ
iajs-2952	13	26	the	the	DET
iajs-2952	13	27	pair	pair	NOUN
iajs-2952	13	28	(	(	PUNCT
iajs-2952	13	29	𝑊	𝑊	PROPN
iajs-2952	13	30	,	,	PUNCT
iajs-2952	13	31	‖	‖	ADJ
iajs-2952	13	32	,	,	PUNCT
iajs-2952	13	33	‖	‖	NUM
iajs-2952	13	34	)	)	PUNCT
iajs-2952	13	35	is	be	AUX
iajs-2952	13	36	called	call	VERB
iajs-2952	13	37	a	a	DET
iajs-2952	13	38	linear	linear	ADJ
iajs-2952	13	39	2	2	NUM
iajs-2952	13	40	-	-	PUNCT
iajs-2952	13	41	normed	norme	VERB
iajs-2952	13	42	space	space	NOUN
iajs-2952	13	43	or	or	CCONJ
iajs-2952	13	44	2	2	NUM
iajs-2952	13	45	-	-	PUNCT
iajs-2952	13	46	normed	norme	VERB
iajs-2952	13	47	space	space	NOUN
iajs-2952	13	48	.	.	PUNCT
iajs-2952	14	1	for	for	ADP
iajs-2952	14	2	more	more	ADJ
iajs-2952	14	3	details	detail	NOUN
iajs-2952	14	4	,	,	PUNCT
iajs-2952	14	5	see	see	VERB
iajs-2952	14	6	[	[	X
iajs-2952	14	7	11	11	NUM
iajs-2952	14	8	-	-	SYM
iajs-2952	14	9	12	12	NUM
iajs-2952	14	10	]	]	PUNCT
iajs-2952	14	11	doi.org/10.30526/36.1.2952	doi.org/10.30526/36.1.2952	NOUN
iajs-2952	14	12	article	article	NOUN
iajs-2952	14	13	history	history	NOUN
iajs-2952	14	14	:	:	PUNCT
iajs-2952	14	15	received	receive	VERB
iajs-2952	14	16	14	14	NUM
iajs-2952	14	17	augest	aug	ADJ
iajs-2952	14	18	2022	2022	NUM
iajs-2952	14	19	,	,	PUNCT
iajs-2952	14	20	accepted	accept	VERB
iajs-2952	14	21	11	11	NUM
iajs-2952	14	22	september	september	PROPN
iajs-2952	14	23	2022	2022	NUM
iajs-2952	14	24	,	,	PUNCT
iajs-2952	14	25	published	publish	VERB
iajs-2952	14	26	in	in	ADP
iajs-2952	14	27	january	january	PROPN
iajs-2952	14	28	2023	2023	NUM
iajs-2952	14	29	.	.	PUNCT
iajs-2952	15	1	ibn	ibn	PROPN
iajs-2952	15	2	al	al	PROPN
iajs-2952	15	3	-	-	PUNCT
iajs-2952	15	4	haitham	haitham	PROPN
iajs-2952	15	5	journal	journal	PROPN
iajs-2952	15	6	for	for	ADP
iajs-2952	15	7	pure	pure	ADJ
iajs-2952	15	8	and	and	CCONJ
iajs-2952	15	9	applied	applied	ADJ
iajs-2952	15	10	sciences	sciences	PROPN
iajs-2952	15	11	journal	journal	PROPN
iajs-2952	15	12	homepage	homepage	NOUN
iajs-2952	15	13	:	:	PUNCT
iajs-2952	15	14	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	PROPN
iajs-2952	15	15	safa	safa	PROPN
iajs-2952	15	16	l.	l.	PROPN
iajs-2952	15	17	hamad	hamad	PROPN
iajs-2952	15	18	department	department	PROPN
iajs-2952	15	19	of	of	ADP
iajs-2952	15	20	mathematics	mathematics	PROPN
iajs-2952	15	21	,	,	PUNCT
iajs-2952	15	22	college	college	NOUN
iajs-2952	15	23	of	of	ADP
iajs-2952	15	24	sciences	science	NOUN
iajs-2952	15	25	,	,	PUNCT
iajs-2952	15	26	university	university	NOUN
iajs-2952	15	27	of	of	ADP
iajs-2952	15	28	baghdadiraq	baghdadiraq	PROPN
iajs-2952	15	29	.	.	PUNCT
iajs-2952	16	1	safalafta2019@gmail.com	safalafta2019@gmail.com	PROPN
iajs-2952	16	2	zeana	zeana	PROPN
iajs-2952	16	3	z.	z.	PROPN
iajs-2952	16	4	jamil	jamil	PROPN
iajs-2952	16	5	department	department	PROPN
iajs-2952	16	6	of	of	ADP
iajs-2952	16	7	mathematics	mathematics	PROPN
iajs-2952	16	8	,	,	PUNCT
iajs-2952	16	9	college	college	NOUN
iajs-2952	16	10	of	of	ADP
iajs-2952	16	11	sciences	science	NOUN
iajs-2952	16	12	,	,	PUNCT
iajs-2952	16	13	university	university	NOUN
iajs-2952	16	14	of	of	ADP
iajs-2952	16	15	baghdadiraq	baghdadiraq	PROPN
iajs-2952	16	16	.	.	PUNCT
iajs-2952	17	1	zina.z@sc.uobaghdad.edu.iq	zina.z@sc.uobaghdad.edu.iq	PROPN
iajs-2952	17	2	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
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iajs-2952	18	2	mailto:safalafta2019@gmail.com	mailto:safalafta2019@gmail.com	PROPN
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iajs-2952	18	4	ihjpas	ihjpas	PROPN
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iajs-2952	19	6	[	[	X
iajs-2952	19	7	10	10	NUM
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iajs-2952	19	11	generalized	generalized	ADJ
iajs-2952	19	12	2	2	NUM
iajs-2952	19	13	-	-	PUNCT
iajs-2952	19	14	inner	inner	ADJ
iajs-2952	19	15	product	product	NOUN
iajs-2952	19	16	space	space	NOUN
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iajs-2952	19	26	generalized	generalized	ADJ
iajs-2952	19	27	2	2	NUM
iajs-2952	19	28	-	-	PUNCT
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iajs-2952	19	30	product	product	NOUN
iajs-2952	19	31	space	space	NOUN
iajs-2952	19	32	if	if	SCONJ
iajs-2952	19	33	there	there	PRON
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iajs-2952	19	35	a	a	DET
iajs-2952	19	36	complex	complex	ADJ
iajs-2952	19	37	valued	value	VERB
iajs-2952	19	38	function	function	NOUN
iajs-2952	19	39	〈	〈	PROPN
iajs-2952	19	40	(	(	PUNCT
iajs-2952	19	41	,	,	PUNCT
iajs-2952	19	42	)	)	PUNCT
iajs-2952	19	43	,	,	PUNCT
iajs-2952	19	44	(	(	PUNCT
iajs-2952	19	45	,	,	PUNCT
iajs-2952	19	46	)	)	PUNCT
iajs-2952	19	47	〉	〉	NOUN
iajs-2952	19	48	on	on	ADP
iajs-2952	19	49	w2	w2	PROPN
iajs-2952	19	50	×	×	PROPN
iajs-2952	19	51	w2	w2	NOUN
iajs-2952	19	52	such	such	ADJ
iajs-2952	19	53	that	that	SCONJ
iajs-2952	19	54	a	a	DET
iajs-2952	19	55	,	,	PUNCT
iajs-2952	19	56	b	b	NOUN
iajs-2952	19	57	,	,	PUNCT
iajs-2952	19	58	c	c	NOUN
iajs-2952	19	59	,	,	PUNCT
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iajs-2952	19	63	,	,	PUNCT
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iajs-2952	19	65	α	α	PRON
iajs-2952	19	66	∈	∈	PROPN
iajs-2952	19	67	c	c	X
iajs-2952	19	68	,	,	PUNCT
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iajs-2952	19	70	the	the	DET
iajs-2952	19	71	following	following	NOUN
iajs-2952	19	72	:	:	PUNCT
iajs-2952	20	1	1	1	X
iajs-2952	20	2	.	.	X
iajs-2952	21	1	〈	〈	PROPN
iajs-2952	21	2	(	(	PUNCT
iajs-2952	21	3	a	a	DET
iajs-2952	21	4	,	,	PUNCT
iajs-2952	21	5	b	b	NOUN
iajs-2952	21	6	)	)	PUNCT
iajs-2952	21	7	,	,	PUNCT
iajs-2952	21	8	(	(	PUNCT
iajs-2952	21	9	c	c	X
iajs-2952	21	10	,	,	PUNCT
iajs-2952	21	11	d	d	NOUN
iajs-2952	21	12	)	)	PUNCT
iajs-2952	21	13	〉	〉	NOUN
iajs-2952	21	14	=	=	SYM
iajs-2952	21	15	〈	〈	PROPN
iajs-2952	21	16	(	(	PUNCT
iajs-2952	21	17	c	c	NOUN
iajs-2952	21	18	,	,	PUNCT
iajs-2952	21	19	d	d	NOUN
iajs-2952	21	20	)	)	PUNCT
iajs-2952	21	21	,	,	PUNCT
iajs-2952	21	22	(	(	PUNCT
iajs-2952	21	23	a	a	PRON
iajs-2952	21	24	,	,	PUNCT
iajs-2952	21	25	b)〉̅̅	b)〉̅̅	ADJ
iajs-2952	21	26	̅̅	̅̅	PROPN
iajs-2952	21	27	̅̅	̅̅	PROPN
iajs-2952	21	28	̅̅	̅̅	PROPN
iajs-2952	21	29	̅̅	̅̅	PROPN
iajs-2952	21	30	̅̅	̅̅	PROPN
iajs-2952	21	31	̅̅	̅̅	PROPN
iajs-2952	21	32	̅̅	̅̅	PROPN
iajs-2952	21	33	̅	̅	PROPN
iajs-2952	21	34	2	2	NUM
iajs-2952	21	35	.	.	PUNCT
iajs-2952	22	1	if	if	SCONJ
iajs-2952	22	2	a	a	PRON
iajs-2952	22	3	and	and	CCONJ
iajs-2952	22	4	b	b	NOUN
iajs-2952	22	5	are	be	AUX
iajs-2952	22	6	linear	linear	ADJ
iajs-2952	22	7	independent	independent	ADJ
iajs-2952	22	8	in	in	ADP
iajs-2952	22	9	w	w	PROPN
iajs-2952	22	10	,	,	PUNCT
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iajs-2952	22	12	〈	〈	PROPN
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iajs-2952	22	14	a	a	DET
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iajs-2952	22	17	)	)	PUNCT
iajs-2952	22	18	,	,	PUNCT
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iajs-2952	22	20	c	c	X
iajs-2952	22	21	,	,	PUNCT
iajs-2952	22	22	d	d	NOUN
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iajs-2952	22	26	0	0	NUM
iajs-2952	22	27	.	.	NOUN
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iajs-2952	24	1	〈	〈	PROPN
iajs-2952	24	2	(	(	PUNCT
iajs-2952	24	3	a	a	DET
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iajs-2952	24	9	c	c	X
iajs-2952	24	10	,	,	PUNCT
iajs-2952	24	11	d	d	NOUN
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iajs-2952	24	16	,	,	PUNCT
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iajs-2952	24	21	c	c	X
iajs-2952	24	22	,	,	PUNCT
iajs-2952	24	23	d	d	NOUN
iajs-2952	24	24	)	)	PUNCT
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iajs-2952	24	26	.	.	PUNCT
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iajs-2952	25	2	.	.	X
iajs-2952	26	1	〈	〈	PROPN
iajs-2952	26	2	(	(	PUNCT
iajs-2952	26	3	αa	αa	PROPN
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iajs-2952	26	6	,	,	PUNCT
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iajs-2952	26	11	c	c	X
iajs-2952	26	12	,	,	PUNCT
iajs-2952	26	13	d	d	NOUN
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iajs-2952	26	18	,	,	PUNCT
iajs-2952	26	19	b	b	NOUN
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iajs-2952	26	22	(	(	PUNCT
iajs-2952	26	23	c	c	X
iajs-2952	26	24	,	,	PUNCT
iajs-2952	26	25	d	d	NOUN
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iajs-2952	26	27	〉	〉	NOUN
iajs-2952	26	28	+	+	X
iajs-2952	26	29	〈	〈	PROPN
iajs-2952	26	30	(	(	PUNCT
iajs-2952	26	31	e	e	NOUN
iajs-2952	26	32	,	,	PUNCT
iajs-2952	26	33	b	b	NOUN
iajs-2952	26	34	)	)	PUNCT
iajs-2952	26	35	,	,	PUNCT
iajs-2952	26	36	(	(	PUNCT
iajs-2952	26	37	c	c	X
iajs-2952	26	38	,	,	PUNCT
iajs-2952	26	39	d	d	NOUN
iajs-2952	26	40	)	)	PUNCT
iajs-2952	26	41	〉	〉	NOUN
iajs-2952	26	42	.	.	PUNCT
iajs-2952	27	1	for	for	ADP
iajs-2952	27	2	more	more	ADJ
iajs-2952	27	3	details	detail	NOUN
iajs-2952	27	4	,	,	PUNCT
iajs-2952	27	5	see	see	VERB
iajs-2952	27	6	[	[	X
iajs-2952	27	7	8][2][5	8][2][5	NUM
iajs-2952	27	8	]	]	PUNCT
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iajs-2952	28	1	ghafoor	ghafoor	PROPN
iajs-2952	29	1	[	[	X
iajs-2952	29	2	6	6	NUM
iajs-2952	29	3	]	]	PUNCT
iajs-2952	29	4	,	,	PUNCT
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iajs-2952	29	6	that	that	SCONJ
iajs-2952	29	7	the	the	DET
iajs-2952	29	8	generalized	generalized	ADJ
iajs-2952	29	9	2	2	NUM
iajs-2952	29	10	-	-	PUNCT
iajs-2952	29	11	inner	inner	ADJ
iajs-2952	29	12	product	product	NOUN
iajs-2952	29	13	space	space	NOUN
iajs-2952	29	14	is	be	AUX
iajs-2952	29	15	a	a	DET
iajs-2952	29	16	2	2	NUM
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iajs-2952	29	19	space	space	NOUN
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iajs-2952	29	22	,	,	PUNCT
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iajs-2952	29	29	y	y	PROPN
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iajs-2952	29	31	,	,	PUNCT
iajs-2952	29	32	(	(	PUNCT
iajs-2952	29	33	w	w	PROPN
iajs-2952	29	34	,	,	PUNCT
iajs-2952	29	35	y	y	NOUN
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iajs-2952	29	37	〉	〉	NOUN
iajs-2952	29	38	1	1	NUM
iajs-2952	29	39	2	2	NUM
iajs-2952	29	40	.	.	PUNCT
iajs-2952	30	1	after	after	ADP
iajs-2952	30	2	many	many	ADJ
iajs-2952	30	3	failures	failure	NOUN
iajs-2952	30	4	to	to	PART
iajs-2952	30	5	define	define	VERB
iajs-2952	30	6	orthogonal	orthogonal	ADJ
iajs-2952	30	7	vectors	vector	NOUN
iajs-2952	30	8	in	in	ADP
iajs-2952	30	9	2	2	NUM
iajs-2952	30	10	-	-	PUNCT
iajs-2952	30	11	normed	normed	ADJ
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iajs-2952	30	20	,	,	PUNCT
iajs-2952	30	21	[	[	X
iajs-2952	30	22	10	10	NUM
iajs-2952	30	23	]	]	SYM
iajs-2952	30	24	defined	define	VERB
iajs-2952	30	25	orthogonal	orthogonal	ADJ
iajs-2952	30	26	vectors	vector	NOUN
iajs-2952	30	27	in	in	ADP
iajs-2952	30	28	a	a	DET
iajs-2952	30	29	2	2	NUM
iajs-2952	30	30	-	-	PUNCT
iajs-2952	30	31	normed	norme	VERB
iajs-2952	30	32	space	space	NOUN
iajs-2952	30	33	by	by	ADP
iajs-2952	30	34	restriction	restriction	NOUN
iajs-2952	30	35	space	space	NOUN
iajs-2952	30	36	w	w	PROPN
iajs-2952	30	37	×	×	PROPN
iajs-2952	30	38	w	w	NOUN
iajs-2952	30	39	to	to	ADP
iajs-2952	30	40	w	w	PROPN
iajs-2952	30	41	×	×	PROPN
iajs-2952	30	42	〈	〈	PROPN
iajs-2952	30	43	b	b	SYM
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iajs-2952	30	47	<	<	X
iajs-2952	30	48	b	b	X
iajs-2952	30	49	>	>	X
iajs-2952	30	50	is	be	AUX
iajs-2952	30	51	a	a	DET
iajs-2952	30	52	non	non	ADJ
iajs-2952	30	53	-	-	ADJ
iajs-2952	30	54	zero	zero	ADJ
iajs-2952	30	55	subspace	subspace	NOUN
iajs-2952	30	56	in	in	ADP
iajs-2952	30	57	w.	w.	PROPN
iajs-2952	30	58	thus	thus	ADV
iajs-2952	30	59	,	,	PUNCT
iajs-2952	30	60	the	the	DET
iajs-2952	30	61	domain	domain	NOUN
iajs-2952	30	62	of	of	ADP
iajs-2952	30	63	the	the	DET
iajs-2952	30	64	generalized	generalized	ADJ
iajs-2952	30	65	2	2	NUM
iajs-2952	30	66	-	-	PUNCT
iajs-2952	30	67	inner	inner	ADJ
iajs-2952	30	68	product	product	NOUN
iajs-2952	30	69	restriction	restriction	NOUN
iajs-2952	30	70	with	with	ADP
iajs-2952	30	71	space	space	NOUN
iajs-2952	30	72	w2	w2	NOUN
iajs-2952	30	73	×	×	PROPN
iajs-2952	30	74	〈	〈	NOUN
iajs-2952	30	75	b〉2	b〉2	NOUN
iajs-2952	30	76	.	.	PUNCT
iajs-2952	31	1	it	it	PRON
iajs-2952	31	2	is	be	AUX
iajs-2952	31	3	well	well	ADV
iajs-2952	31	4	-	-	PUNCT
iajs-2952	31	5	known	know	VERB
iajs-2952	31	6	that	that	SCONJ
iajs-2952	31	7	there	there	PRON
iajs-2952	31	8	are	be	VERB
iajs-2952	31	9	incomplete	incomplete	ADJ
iajs-2952	31	10	metric	metric	ADJ
iajs-2952	31	11	spaces	space	NOUN
iajs-2952	31	12	.	.	PUNCT
iajs-2952	32	1	kreyszig	kreyszig	PROPN
iajs-2952	32	2	,	,	PUNCT
iajs-2952	32	3	in	in	ADP
iajs-2952	32	4	1978	1978	NUM
iajs-2952	32	5	[	[	X
iajs-2952	32	6	7	7	NUM
iajs-2952	32	7	]	]	PUNCT
iajs-2952	32	8	discussed	discuss	VERB
iajs-2952	32	9	the	the	DET
iajs-2952	32	10	strategy	strategy	NOUN
iajs-2952	32	11	for	for	ADP
iajs-2952	32	12	completing	complete	VERB
iajs-2952	32	13	every	every	DET
iajs-2952	32	14	metric	metric	ADJ
iajs-2952	32	15	space	space	NOUN
iajs-2952	32	16	by	by	ADP
iajs-2952	32	17	defining	define	VERB
iajs-2952	32	18	an	an	DET
iajs-2952	32	19	equivalent	equivalent	ADJ
iajs-2952	32	20	relation	relation	NOUN
iajs-2952	32	21	on	on	ADP
iajs-2952	32	22	cauchy	cauchy	ADJ
iajs-2952	32	23	sequences	sequence	NOUN
iajs-2952	32	24	and	and	CCONJ
iajs-2952	32	25	the	the	DET
iajs-2952	32	26	metric	metric	NOUN
iajs-2952	32	27	on	on	ADP
iajs-2952	32	28	it	it	PRON
iajs-2952	32	29	.	.	PUNCT
iajs-2952	33	1	in	in	ADP
iajs-2952	33	2	2001	2001	NUM
iajs-2952	33	3	,	,	PUNCT
iajs-2952	33	4	cho	cho	PROPN
iajs-2952	33	5	and	and	CCONJ
iajs-2952	33	6	freese	freese	PROPN
iajs-2952	34	1	[	[	X
iajs-2952	34	2	3	3	NUM
iajs-2952	34	3	]	]	PUNCT
iajs-2952	34	4	used	use	VERB
iajs-2952	34	5	the	the	DET
iajs-2952	34	6	same	same	ADJ
iajs-2952	34	7	strategy	strategy	NOUN
iajs-2952	34	8	for	for	ADP
iajs-2952	34	9	the	the	DET
iajs-2952	34	10	completion	completion	NOUN
iajs-2952	34	11	of	of	ADP
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iajs-2952	34	13	2normed	2normed	NUM
iajs-2952	34	14	space	space	NOUN
iajs-2952	34	15	,	,	PUNCT
iajs-2952	34	16	but	but	CCONJ
iajs-2952	34	17	some	some	DET
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iajs-2952	34	20	when	when	SCONJ
iajs-2952	34	21	defining	define	VERB
iajs-2952	34	22	a	a	DET
iajs-2952	34	23	metric	metric	NOUN
iajs-2952	34	24	on	on	ADP
iajs-2952	34	25	it	it	PRON
iajs-2952	34	26	,	,	PUNCT
iajs-2952	34	27	and	and	CCONJ
iajs-2952	34	28	then	then	ADV
iajs-2952	34	29	,	,	PUNCT
iajs-2952	34	30	he	he	PRON
iajs-2952	34	31	had	have	VERB
iajs-2952	34	32	to	to	PART
iajs-2952	34	33	give	give	VERB
iajs-2952	34	34	another	another	DET
iajs-2952	34	35	condition	condition	NOUN
iajs-2952	34	36	on	on	ADP
iajs-2952	34	37	the	the	DET
iajs-2952	34	38	space	space	NOUN
iajs-2952	34	39	.	.	PUNCT
iajs-2952	35	1	despite	despite	SCONJ
iajs-2952	35	2	that	that	PRON
iajs-2952	35	3	all	all	DET
iajs-2952	35	4	generalized	generalize	VERB
iajs-2952	35	5	2	2	NUM
iajs-2952	35	6	-	-	PUNCT
iajs-2952	35	7	inner	inner	ADJ
iajs-2952	35	8	product	product	NOUN
iajs-2952	35	9	space	space	NOUN
iajs-2952	35	10	is	be	AUX
iajs-2952	35	11	2	2	NUM
iajs-2952	35	12	-	-	PUNCT
iajs-2952	35	13	normed	normed	ADJ
iajs-2952	35	14	space	space	NOUN
iajs-2952	35	15	,	,	PUNCT
iajs-2952	35	16	but	but	CCONJ
iajs-2952	35	17	in	in	ADP
iajs-2952	35	18	this	this	DET
iajs-2952	35	19	paper	paper	NOUN
iajs-2952	35	20	,	,	PUNCT
iajs-2952	35	21	we	we	PRON
iajs-2952	35	22	give	give	VERB
iajs-2952	35	23	a	a	DET
iajs-2952	35	24	completion	completion	NOUN
iajs-2952	35	25	of	of	ADP
iajs-2952	35	26	the	the	DET
iajs-2952	35	27	generalized	generalized	ADJ
iajs-2952	35	28	2	2	NUM
iajs-2952	35	29	-	-	PUNCT
iajs-2952	35	30	inner	inner	ADJ
iajs-2952	35	31	product	product	NOUN
iajs-2952	35	32	space	space	NOUN
iajs-2952	35	33	without	without	ADP
iajs-2952	35	34	need	need	NOUN
iajs-2952	35	35	any	any	DET
iajs-2952	35	36	condition	condition	NOUN
iajs-2952	35	37	using	use	VERB
iajs-2952	35	38	the	the	DET
iajs-2952	35	39	b	b	NOUN
iajs-2952	35	40	-	-	PUNCT
iajs-2952	35	41	cauchy	cauchy	ADJ
iajs-2952	35	42	sequences	sequence	NOUN
iajs-2952	35	43	.	.	PUNCT
iajs-2952	36	1	this	this	DET
iajs-2952	36	2	paper	paper	NOUN
iajs-2952	36	3	includes	include	VERB
iajs-2952	36	4	two	two	NUM
iajs-2952	36	5	sections	section	NOUN
iajs-2952	36	6	.	.	PUNCT
iajs-2952	37	1	the	the	DET
iajs-2952	37	2	first	first	ADJ
iajs-2952	37	3	section	section	NOUN
iajs-2952	37	4	discusses	discuss	VERB
iajs-2952	37	5	some	some	PRON
iajs-2952	37	6	of	of	ADP
iajs-2952	37	7	the	the	DET
iajs-2952	37	8	properties	property	NOUN
iajs-2952	37	9	of	of	ADP
iajs-2952	37	10	the	the	DET
iajs-2952	37	11	bcauchy	bcauchy	ADJ
iajs-2952	37	12	sequence	sequence	NOUN
iajs-2952	37	13	in	in	ADP
iajs-2952	37	14	a	a	DET
iajs-2952	37	15	generalized	generalized	ADJ
iajs-2952	37	16	2	2	NUM
iajs-2952	37	17	-	-	PUNCT
iajs-2952	37	18	inner	inner	ADJ
iajs-2952	37	19	product	product	NOUN
iajs-2952	37	20	space	space	NOUN
iajs-2952	37	21	.	.	PUNCT
iajs-2952	38	1	the	the	DET
iajs-2952	38	2	second	second	ADJ
iajs-2952	38	3	section	section	NOUN
iajs-2952	38	4	proves	prove	VERB
iajs-2952	38	5	the	the	DET
iajs-2952	38	6	completion	completion	NOUN
iajs-2952	38	7	of	of	ADP
iajs-2952	38	8	the	the	DET
iajs-2952	38	9	generalized	generalized	ADJ
iajs-2952	38	10	2	2	NUM
iajs-2952	38	11	-	-	PUNCT
iajs-2952	38	12	inner	inner	ADJ
iajs-2952	38	13	product	product	NOUN
iajs-2952	38	14	.	.	PUNCT
iajs-2952	39	1	we	we	PRON
iajs-2952	39	2	will	will	AUX
iajs-2952	39	3	abbreviate	abbreviate	VERB
iajs-2952	39	4	‖w	‖w	NOUN
iajs-2952	39	5	,	,	PUNCT
iajs-2952	39	6	b‖by	b‖by	PROPN
iajs-2952	39	7	‖w‖b	‖w‖b	NOUN
iajs-2952	39	8	in	in	ADP
iajs-2952	39	9	the	the	DET
iajs-2952	39	10	sequel	sequel	NOUN
iajs-2952	39	11	.	.	PUNCT
iajs-2952	40	1	2	2	X
iajs-2952	40	2	.	.	X
iajs-2952	40	3	b	b	X
iajs-2952	40	4	-	-	PUNCT
iajs-2952	40	5	cauchy	cauchy	ADJ
iajs-2952	40	6	sequences	sequence	NOUN
iajs-2952	40	7	.	.	PUNCT
iajs-2952	41	1	this	this	DET
iajs-2952	41	2	section	section	NOUN
iajs-2952	41	3	discusses	discuss	VERB
iajs-2952	41	4	some	some	PRON
iajs-2952	41	5	of	of	ADP
iajs-2952	41	6	the	the	DET
iajs-2952	41	7	properties	property	NOUN
iajs-2952	41	8	of	of	ADP
iajs-2952	41	9	the	the	DET
iajs-2952	41	10	b	b	PROPN
iajs-2952	41	11	-	-	PUNCT
iajs-2952	41	12	cauchy	cauchy	ADJ
iajs-2952	41	13	sequence	sequence	NOUN
iajs-2952	41	14	in	in	ADP
iajs-2952	41	15	a	a	DET
iajs-2952	41	16	generalized	generalized	ADJ
iajs-2952	41	17	2inner	2inner	NUM
iajs-2952	41	18	product	product	NOUN
iajs-2952	41	19	space	space	NOUN
iajs-2952	41	20	.	.	PUNCT
iajs-2952	42	1	mazaheri	mazaheri	NOUN
iajs-2952	42	2	and	and	CCONJ
iajs-2952	42	3	kazemi	kazemi	VERB
iajs-2952	42	4	in	in	ADP
iajs-2952	42	5	[	[	X
iajs-2952	42	6	9	9	NUM
iajs-2952	42	7	]	]	PUNCT
iajs-2952	42	8	introduce	introduce	VERB
iajs-2952	42	9	a	a	DET
iajs-2952	42	10	b	b	NOUN
iajs-2952	42	11	-	-	PUNCT
iajs-2952	42	12	cauchy	cauchy	ADJ
iajs-2952	42	13	sequence	sequence	NOUN
iajs-2952	42	14	concept	concept	NOUN
iajs-2952	42	15	as	as	SCONJ
iajs-2952	42	16	follows	follow	VERB
iajs-2952	42	17	definition	definition	NOUN
iajs-2952	42	18	(	(	PUNCT
iajs-2952	42	19	2.1)[9	2.1)[9	NOUN
iajs-2952	42	20	]	]	X
iajs-2952	42	21	:	:	PUNCT
iajs-2952	42	22	let	let	VERB
iajs-2952	42	23	𝑊	𝑊	PRON
iajs-2952	42	24	be	be	AUX
iajs-2952	42	25	a	a	DET
iajs-2952	42	26	generalized	generalized	ADJ
iajs-2952	42	27	2	2	NUM
iajs-2952	42	28	-	-	PUNCT
iajs-2952	42	29	inner	inner	ADJ
iajs-2952	42	30	product	product	NOUN
iajs-2952	42	31	space	space	NOUN
iajs-2952	42	32	,	,	PUNCT
iajs-2952	42	33	0	0	NUM
iajs-2952	42	34	≠	≠	PROPN
iajs-2952	42	35	𝑏	𝑏	PRON
iajs-2952	42	36	∈	∈	PROPN
iajs-2952	42	37	𝑊	𝑊	PROPN
iajs-2952	42	38	,	,	PUNCT
iajs-2952	42	39	{	{	PUNCT
iajs-2952	42	40	𝑤𝑛	𝑤𝑛	NOUN
iajs-2952	42	41	}	}	PUNCT
iajs-2952	42	42	be	be	AUX
iajs-2952	42	43	a	a	DET
iajs-2952	42	44	sequence	sequence	NOUN
iajs-2952	42	45	in	in	ADP
iajs-2952	42	46	𝑊	𝑊	PROPN
iajs-2952	42	47	,	,	PUNCT
iajs-2952	42	48	then	then	ADV
iajs-2952	42	49	,	,	PUNCT
iajs-2952	42	50	it	it	PRON
iajs-2952	42	51	is	be	AUX
iajs-2952	42	52	called	call	VERB
iajs-2952	42	53	a	a	DET
iajs-2952	42	54	b	b	NOUN
iajs-2952	42	55	-	-	PUNCT
iajs-2952	42	56	cauchy	cauchy	ADJ
iajs-2952	42	57	sequence	sequence	NOUN
iajs-2952	42	58	if	if	SCONJ
iajs-2952	42	59	lim	lim	PROPN
iajs-2952	42	60	𝑛,𝑚→∞	𝑛,𝑚→∞	PROPN
iajs-2952	42	61	‖𝑤𝑛	‖𝑤𝑛	VERB
iajs-2952	42	62	−	−	PROPN
iajs-2952	42	63	𝑤𝑚	𝑤𝑚	NOUN
iajs-2952	42	64	,	,	PUNCT
iajs-2952	42	65	𝑏‖	𝑏‖	X
iajs-2952	43	1	=	=	SYM
iajs-2952	43	2	0	0	X
iajs-2952	43	3	.	.	PUNCT
iajs-2952	44	1	the	the	DET
iajs-2952	44	2	following	follow	VERB
iajs-2952	44	3	definition	definition	NOUN
iajs-2952	44	4	has	have	AUX
iajs-2952	44	5	been	be	AUX
iajs-2952	44	6	devised	devise	VERB
iajs-2952	44	7	from	from	ADP
iajs-2952	44	8	[	[	X
iajs-2952	44	9	9	9	NUM
iajs-2952	44	10	]	]	PUNCT
iajs-2952	44	11	definition	definition	NOUN
iajs-2952	44	12	(	(	PUNCT
iajs-2952	44	13	2.2	2.2	NUM
iajs-2952	44	14	):	):	PUNCT
iajs-2952	44	15	an	an	DET
iajs-2952	44	16	open	open	ADJ
iajs-2952	44	17	ball	ball	NOUN
iajs-2952	44	18	of	of	ADP
iajs-2952	44	19	radius	radius	NOUN
iajs-2952	44	20	𝑟	𝑟	NOUN
iajs-2952	44	21	and	and	CCONJ
iajs-2952	44	22	centered	center	VERB
iajs-2952	44	23	at	at	ADP
iajs-2952	44	24	y	y	PROPN
iajs-2952	44	25	in	in	ADP
iajs-2952	44	26	a	a	DET
iajs-2952	44	27	generalized	generalized	ADJ
iajs-2952	44	28	2	2	NUM
iajs-2952	44	29	-	-	PUNCT
iajs-2952	44	30	inner	inner	ADJ
iajs-2952	44	31	product	product	NOUN
iajs-2952	44	32	space	space	NOUN
iajs-2952	44	33	w	w	NOUN
iajs-2952	44	34	is	be	AUX
iajs-2952	44	35	defined	define	VERB
iajs-2952	44	36	as	as	ADP
iajs-2952	44	37	:	:	PUNCT
iajs-2952	44	38	br(y	br(y	NUM
iajs-2952	44	39	)	)	PUNCT
iajs-2952	44	40	≔	≔	NOUN
iajs-2952	44	41	{	{	PUNCT
iajs-2952	44	42	w	w	NOUN
iajs-2952	44	43	∈	∈	PROPN
iajs-2952	44	44	w	w	PROPN
iajs-2952	44	45	:	:	PUNCT
iajs-2952	44	46	‖w	‖w	NOUN
iajs-2952	44	47	−	−	PROPN
iajs-2952	45	1	y‖b	y‖b	NOUN
iajs-2952	45	2	<	<	X
iajs-2952	45	3	r	r	NOUN
iajs-2952	45	4	}	}	PUNCT
iajs-2952	45	5	.	.	PUNCT
iajs-2952	46	1	the	the	DET
iajs-2952	46	2	following	follow	VERB
iajs-2952	46	3	proposition	proposition	NOUN
iajs-2952	46	4	is	be	AUX
iajs-2952	46	5	a	a	DET
iajs-2952	46	6	characterization	characterization	NOUN
iajs-2952	46	7	of	of	ADP
iajs-2952	46	8	b	b	NOUN
iajs-2952	46	9	-	-	PUNCT
iajs-2952	46	10	cauchy	cauchy	ADJ
iajs-2952	46	11	sequences	sequence	NOUN
iajs-2952	46	12	in	in	ADP
iajs-2952	46	13	a	a	DET
iajs-2952	46	14	generalized	generalized	ADJ
iajs-2952	46	15	2inner	2inner	NUM
iajs-2952	46	16	product	product	NOUN
iajs-2952	46	17	space	space	NOUN
iajs-2952	46	18	.	.	PUNCT
iajs-2952	47	1	but	but	CCONJ
iajs-2952	47	2	first	first	ADV
iajs-2952	47	3	,	,	PUNCT
iajs-2952	47	4	we	we	PRON
iajs-2952	47	5	define	define	VERB
iajs-2952	47	6	a	a	DET
iajs-2952	47	7	neighborhood	neighborhood	NOUN
iajs-2952	47	8	of	of	ADP
iajs-2952	47	9	0	0	NUM
iajs-2952	47	10	.	.	PUNCT
iajs-2952	48	1	definition	definition	NOUN
iajs-2952	48	2	(	(	PUNCT
iajs-2952	48	3	2.3	2.3	NUM
iajs-2952	48	4	):	):	PUNCT
iajs-2952	48	5	if	if	SCONJ
iajs-2952	48	6	w	w	NOUN
iajs-2952	48	7	is	be	AUX
iajs-2952	48	8	a	a	DET
iajs-2952	48	9	point	point	NOUN
iajs-2952	48	10	in	in	ADP
iajs-2952	48	11	a	a	DET
iajs-2952	48	12	generalized	generalized	ADJ
iajs-2952	48	13	2	2	NUM
iajs-2952	48	14	-	-	PUNCT
iajs-2952	48	15	inner	inner	ADJ
iajs-2952	48	16	product	product	NOUN
iajs-2952	48	17	space	space	NOUN
iajs-2952	48	18	w	w	PROPN
iajs-2952	48	19	,	,	PUNCT
iajs-2952	48	20	then	then	ADV
iajs-2952	48	21	a	a	DET
iajs-2952	48	22	neighborhood	neighborhood	NOUN
iajs-2952	48	23	u	u	NOUN
iajs-2952	48	24	of	of	ADP
iajs-2952	48	25	w	w	PROPN
iajs-2952	48	26	is	be	AUX
iajs-2952	48	27	a	a	DET
iajs-2952	48	28	set	set	NOUN
iajs-2952	48	29	containing	contain	VERB
iajs-2952	48	30	br(w	br(w	PUNCT
iajs-2952	48	31	)	)	PUNCT
iajs-2952	48	32	for	for	ADP
iajs-2952	48	33	some	some	DET
iajs-2952	48	34	r	r	NOUN
iajs-2952	48	35	>	>	X
iajs-2952	48	36	0	0	NUM
iajs-2952	48	37	,	,	PUNCT
iajs-2952	48	38	i.e	i.e	PRON
iajs-2952	48	39	,	,	PUNCT
iajs-2952	48	40	there	there	PRON
iajs-2952	48	41	exists	exist	VERB
iajs-2952	48	42	r	r	NOUN
iajs-2952	48	43	>	>	X
iajs-2952	48	44	0	0	NUM
iajs-2952	48	45	,	,	PUNCT
iajs-2952	48	46	such	such	ADJ
iajs-2952	48	47	that	that	SCONJ
iajs-2952	48	48	w	w	PROPN
iajs-2952	48	49	∈	∈	PROPN
iajs-2952	48	50	br(w	br(w	PUNCT
iajs-2952	48	51	)	)	PUNCT
iajs-2952	49	1	⊂	⊂	PROPN
iajs-2952	49	2	u.	u.	PROPN
iajs-2952	49	3	ihjpas	ihjpas	PROPN
iajs-2952	49	4	.	.	PUNCT
iajs-2952	50	1	36(1)2023	36(1)2023	NUM
iajs-2952	50	2	313	313	NUM
iajs-2952	50	3	proposition	proposition	NOUN
iajs-2952	50	4	(	(	PUNCT
iajs-2952	50	5	2.4	2.4	NUM
iajs-2952	50	6	):	):	PUNCT
iajs-2952	50	7	let	let	VERB
iajs-2952	50	8	𝑊	𝑊	PRON
iajs-2952	50	9	be	be	AUX
iajs-2952	50	10	a	a	DET
iajs-2952	50	11	generalized	generalized	ADJ
iajs-2952	50	12	2	2	NUM
iajs-2952	50	13	-	-	PUNCT
iajs-2952	50	14	inner	inner	ADJ
iajs-2952	50	15	product	product	NOUN
iajs-2952	50	16	space	space	NOUN
iajs-2952	50	17	,	,	PUNCT
iajs-2952	50	18	{	{	PUNCT
iajs-2952	50	19	wn	wn	NOUN
iajs-2952	50	20	}	}	PUNCT
iajs-2952	50	21	is	be	AUX
iajs-2952	50	22	a	a	DET
iajs-2952	50	23	b	b	PROPN
iajs-2952	50	24	-	-	PUNCT
iajs-2952	50	25	cauchy	cauchy	ADJ
iajs-2952	50	26	sequence	sequence	NOUN
iajs-2952	50	27	in	in	ADP
iajs-2952	50	28	w	w	NOUN
iajs-2952	50	29	if	if	SCONJ
iajs-2952	51	1	and	and	CCONJ
iajs-2952	51	2	only	only	ADV
iajs-2952	51	3	if	if	SCONJ
iajs-2952	51	4	for	for	ADP
iajs-2952	51	5	any	any	DET
iajs-2952	51	6	neighborhood	neighborhood	NOUN
iajs-2952	51	7	u	u	NOUN
iajs-2952	51	8	of	of	ADP
iajs-2952	51	9	0	0	NUM
iajs-2952	51	10	;	;	PUNCT
iajs-2952	51	11	there	there	PRON
iajs-2952	51	12	is	be	VERB
iajs-2952	51	13	an	an	DET
iajs-2952	51	14	integer	integer	NOUN
iajs-2952	51	15	m(u	m(u	PROPN
iajs-2952	51	16	)	)	PUNCT
iajs-2952	51	17	such	such	ADJ
iajs-2952	51	18	that	that	PRON
iajs-2952	51	19	for	for	ADP
iajs-2952	51	20	all	all	DET
iajs-2952	51	21	n	n	CCONJ
iajs-2952	51	22	,	,	PUNCT
iajs-2952	51	23	m	m	VERB
iajs-2952	51	24	≥	≥	NOUN
iajs-2952	51	25	m(u	m(u	NOUN
iajs-2952	51	26	)	)	PUNCT
iajs-2952	51	27	implies	imply	VERB
iajs-2952	51	28	that	that	SCONJ
iajs-2952	51	29	wn	wn	PROPN
iajs-2952	51	30	−	−	PROPN
iajs-2952	51	31	wm	wm	PROPN
iajs-2952	51	32	∈	∈	PROPN
iajs-2952	51	33	u.	u.	NOUN
iajs-2952	51	34	proof	proof	NOUN
iajs-2952	51	35	:	:	PUNCT
iajs-2952	51	36	let	let	VERB
iajs-2952	51	37	u	u	PRON
iajs-2952	51	38	be	be	AUX
iajs-2952	51	39	a	a	DET
iajs-2952	51	40	neighborhood	neighborhood	NOUN
iajs-2952	51	41	of	of	ADP
iajs-2952	51	42	0	0	NUM
iajs-2952	51	43	,	,	PUNCT
iajs-2952	51	44	then	then	ADV
iajs-2952	51	45	there	there	PRON
iajs-2952	51	46	exists	exist	VERB
iajs-2952	51	47	ε	ε	PROPN
iajs-2952	51	48	>	>	X
iajs-2952	51	49	0	0	NUM
iajs-2952	51	50	such	such	ADJ
iajs-2952	51	51	that	that	PRON
iajs-2952	51	52	b0(ε	b0(ε	NOUN
iajs-2952	51	53	)	)	PUNCT
iajs-2952	51	54	⊆	⊆	NUM
iajs-2952	51	55	u.	u.	NOUN
iajs-2952	51	56	since	since	SCONJ
iajs-2952	51	57	{	{	PUNCT
iajs-2952	51	58	wn	wn	PROPN
iajs-2952	51	59	}	}	PUNCT
iajs-2952	51	60	is	be	AUX
iajs-2952	51	61	a	a	DET
iajs-2952	51	62	b	b	PROPN
iajs-2952	51	63	-	-	PUNCT
iajs-2952	51	64	cauchy	cauchy	ADJ
iajs-2952	51	65	sequence	sequence	NOUN
iajs-2952	51	66	in	in	ADP
iajs-2952	51	67	w	w	PROPN
iajs-2952	51	68	,	,	PUNCT
iajs-2952	51	69	thus	thus	ADV
iajs-2952	51	70	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-2952	51	71	𝑛,𝑚→∞	𝑛,𝑚→∞	NOUN
iajs-2952	51	72	‖𝑤𝑛	‖𝑤𝑛	PUNCT
iajs-2952	51	73	−	−	PROPN
iajs-2952	51	74	𝑤𝑚‖𝑏	𝑤𝑚‖𝑏	PROPN
iajs-2952	51	75	=	=	PUNCT
iajs-2952	51	76	0	0	PUNCT
iajs-2952	51	77	.	.	PUNCT
iajs-2952	52	1	it	it	PRON
iajs-2952	52	2	implies	imply	VERB
iajs-2952	52	3	that	that	SCONJ
iajs-2952	52	4	there	there	PRON
iajs-2952	52	5	exists	exist	VERB
iajs-2952	52	6	m(ε	m(ε	NUM
iajs-2952	52	7	)	)	PUNCT
iajs-2952	52	8	>	>	X
iajs-2952	52	9	0	0	NUM
iajs-2952	52	10	such	such	ADJ
iajs-2952	52	11	that	that	PRON
iajs-2952	52	12	‖wn	‖wn	NUM
iajs-2952	52	13	−	−	PROPN
iajs-2952	52	14	wm‖b	wm‖b	NOUN
iajs-2952	52	15	<	<	X
iajs-2952	52	16	ε	ε	PROPN
iajs-2952	52	17	;	;	PUNCT
iajs-2952	52	18	n	n	CCONJ
iajs-2952	52	19	,	,	PUNCT
iajs-2952	52	20	m	m	VERB
iajs-2952	52	21	≥	≥	NOUN
iajs-2952	52	22	m(ε	m(ε	NUM
iajs-2952	52	23	)	)	PUNCT
iajs-2952	52	24	.	.	PUNCT
iajs-2952	53	1	then	then	ADV
iajs-2952	53	2	,	,	PUNCT
iajs-2952	53	3	wn	wn	PROPN
iajs-2952	53	4	−	−	PROPN
iajs-2952	53	5	wm	wm	PROPN
iajs-2952	53	6	∈	∈	PROPN
iajs-2952	53	7	b0(ε	b0(ε	PROPN
iajs-2952	53	8	)	)	PUNCT
iajs-2952	53	9	⊆	⊆	NUM
iajs-2952	53	10	u.	u.	NOUN
iajs-2952	53	11	conversely	conversely	ADV
iajs-2952	53	12	,	,	PUNCT
iajs-2952	53	13	let	let	VERB
iajs-2952	53	14	{	{	PUNCT
iajs-2952	53	15	wn	wn	AUX
iajs-2952	53	16	}	}	PUNCT
iajs-2952	53	17	be	be	AUX
iajs-2952	53	18	a	a	DET
iajs-2952	53	19	sequence	sequence	NOUN
iajs-2952	53	20	in	in	ADP
iajs-2952	53	21	w	w	ADP
iajs-2952	53	22	such	such	ADJ
iajs-2952	53	23	that	that	PRON
iajs-2952	53	24	for	for	ADP
iajs-2952	53	25	every	every	DET
iajs-2952	53	26	neighborhood	neighborhood	NOUN
iajs-2952	53	27	u	u	NOUN
iajs-2952	53	28	of	of	ADP
iajs-2952	53	29	0	0	NUM
iajs-2952	53	30	there	there	PRON
iajs-2952	53	31	is	be	VERB
iajs-2952	53	32	an	an	DET
iajs-2952	53	33	integer	integer	NOUN
iajs-2952	53	34	m(u	m(u	PROPN
iajs-2952	53	35	)	)	PUNCT
iajs-2952	53	36	>	>	X
iajs-2952	53	37	0	0	NUM
iajs-2952	53	38	;	;	PUNCT
iajs-2952	53	39	wn	wn	PROPN
iajs-2952	53	40	−	−	PROPN
iajs-2952	53	41	wm	wm	PROPN
iajs-2952	53	42	∈	∈	PROPN
iajs-2952	53	43	u	u	PROPN
iajs-2952	53	44	where	where	SCONJ
iajs-2952	53	45	n	n	CCONJ
iajs-2952	53	46	,	,	PUNCT
iajs-2952	53	47	m	m	VERB
iajs-2952	53	48	≥	≥	NOUN
iajs-2952	53	49	m(u	m(u	NUM
iajs-2952	53	50	)	)	PUNCT
iajs-2952	53	51	.	.	PUNCT
iajs-2952	54	1	then	then	ADV
iajs-2952	54	2	,	,	PUNCT
iajs-2952	54	3	there	there	PRON
iajs-2952	54	4	exists	exist	VERB
iajs-2952	54	5	δ(u	δ(u	NOUN
iajs-2952	54	6	)	)	PUNCT
iajs-2952	54	7	>	>	X
iajs-2952	54	8	0	0	NUM
iajs-2952	55	1	such	such	ADJ
iajs-2952	55	2	that	that	PRON
iajs-2952	55	3	‖wn	‖wn	NUM
iajs-2952	55	4	−	−	PROPN
iajs-2952	55	5	wm‖b	wm‖b	PROPN
iajs-2952	55	6	<	<	X
iajs-2952	55	7	δ	δ	PROPN
iajs-2952	55	8	,	,	PUNCT
iajs-2952	55	9	where	where	SCONJ
iajs-2952	55	10	n	n	CCONJ
iajs-2952	55	11	,	,	PUNCT
iajs-2952	55	12	m	m	VERB
iajs-2952	55	13	≥	≥	NOUN
iajs-2952	55	14	m(u	m(u	NUM
iajs-2952	55	15	)	)	PUNCT
iajs-2952	55	16	.	.	PUNCT
iajs-2952	56	1	it	it	PRON
iajs-2952	56	2	implies	imply	VERB
iajs-2952	56	3	that	that	SCONJ
iajs-2952	56	4	for	for	ADP
iajs-2952	56	5	all	all	DET
iajs-2952	56	6	ε	ε	PROPN
iajs-2952	56	7	>	>	X
iajs-2952	56	8	0	0	PROPN
iajs-2952	56	9	,	,	PUNCT
iajs-2952	56	10	there	there	PRON
iajs-2952	56	11	exists	exist	VERB
iajs-2952	56	12	m(ε	m(ε	NUM
iajs-2952	56	13	)	)	PUNCT
iajs-2952	56	14	such	such	ADJ
iajs-2952	56	15	that	that	PRON
iajs-2952	56	16	‖wn	‖wn	NUM
iajs-2952	56	17	−	−	PROPN
iajs-2952	56	18	wm‖b	wm‖b	NOUN
iajs-2952	56	19	<	<	X
iajs-2952	56	20	ε	ε	PROPN
iajs-2952	56	21	for	for	ADP
iajs-2952	56	22	n	n	CCONJ
iajs-2952	56	23	,	,	PUNCT
iajs-2952	56	24	m	m	VERB
iajs-2952	56	25	≥	≥	NOUN
iajs-2952	56	26	m(ε	m(ε	NUM
iajs-2952	56	27	)	)	PUNCT
iajs-2952	56	28	,	,	PUNCT
iajs-2952	56	29	then	then	ADV
iajs-2952	56	30	,	,	PUNCT
iajs-2952	56	31	𝑙𝑖𝑚	𝑙𝑖𝑚	VERB
iajs-2952	56	32	𝑛,𝑚→∞	𝑛,𝑚→∞	NOUN
iajs-2952	56	33	‖𝑤𝑛	‖𝑤𝑛	VERB
iajs-2952	56	34	−	−	PROPN
iajs-2952	56	35	𝑤𝑚‖𝑏	𝑤𝑚‖𝑏	PROPN
iajs-2952	56	36	=	=	SYM
iajs-2952	56	37	0	0	NUM
iajs-2952	56	38	.	.	PUNCT
iajs-2952	57	1	thus	thus	ADV
iajs-2952	57	2	,	,	PUNCT
iajs-2952	57	3	{	{	PUNCT
iajs-2952	57	4	wn	wn	PROPN
iajs-2952	57	5	}	}	PUNCT
iajs-2952	57	6	is	be	AUX
iajs-2952	57	7	a	a	DET
iajs-2952	57	8	b	b	PROPN
iajs-2952	57	9	-	-	PUNCT
iajs-2952	57	10	cauchy	cauchy	ADJ
iajs-2952	57	11	sequence	sequence	NOUN
iajs-2952	57	12	in	in	ADP
iajs-2952	57	13	w.∎	w.∎	NOUN
iajs-2952	57	14	3	3	NUM
iajs-2952	57	15	.	.	PUNCT
iajs-2952	57	16	completion	completion	NOUN
iajs-2952	57	17	of	of	ADP
iajs-2952	57	18	the	the	DET
iajs-2952	57	19	generalized	generalized	ADJ
iajs-2952	57	20	2	2	NUM
iajs-2952	57	21	-	-	PUNCT
iajs-2952	57	22	inner	inner	ADJ
iajs-2952	57	23	product	product	NOUN
iajs-2952	57	24	spaces	space	VERB
iajs-2952	57	25	.	.	PUNCT
iajs-2952	58	1	kreyszig	kreyszig	PROPN
iajs-2952	59	1	[	[	X
iajs-2952	59	2	1	1	NUM
iajs-2952	59	3	]	]	PUNCT
iajs-2952	59	4	states	state	VERB
iajs-2952	59	5	few	few	ADJ
iajs-2952	59	6	steps	step	NOUN
iajs-2952	59	7	to	to	PART
iajs-2952	59	8	prove	prove	VERB
iajs-2952	59	9	that	that	SCONJ
iajs-2952	59	10	an	an	DET
iajs-2952	59	11	arbitrary	arbitrary	ADJ
iajs-2952	59	12	incomplete	incomplete	ADJ
iajs-2952	59	13	metric	metric	ADJ
iajs-2952	59	14	space	space	NOUN
iajs-2952	59	15	can	can	AUX
iajs-2952	59	16	be	be	AUX
iajs-2952	59	17	completed	complete	VERB
iajs-2952	59	18	.	.	PUNCT
iajs-2952	60	1	in	in	ADP
iajs-2952	60	2	this	this	DET
iajs-2952	60	3	section	section	NOUN
iajs-2952	60	4	,	,	PUNCT
iajs-2952	60	5	we	we	PRON
iajs-2952	60	6	follow	follow	VERB
iajs-2952	60	7	kreyszig	kreyszig	PROPN
iajs-2952	60	8	strategy	strategy	NOUN
iajs-2952	60	9	to	to	PART
iajs-2952	60	10	prove	prove	VERB
iajs-2952	60	11	the	the	DET
iajs-2952	60	12	completion	completion	NOUN
iajs-2952	60	13	of	of	ADP
iajs-2952	60	14	the	the	DET
iajs-2952	60	15	generalized	generalized	ADJ
iajs-2952	60	16	2	2	NUM
iajs-2952	60	17	-	-	PUNCT
iajs-2952	60	18	inner	inner	ADJ
iajs-2952	60	19	product	product	NOUN
iajs-2952	60	20	space	space	NOUN
iajs-2952	60	21	:	:	PUNCT
iajs-2952	61	1	step1	step1	PROPN
iajs-2952	61	2	:	:	PUNCT
iajs-2952	61	3	forming	form	VERB
iajs-2952	61	4	�	�	PROPN
iajs-2952	61	5	̂	̂	SYM
iajs-2952	61	6	�	�	NOUN
iajs-2952	61	7	is	be	AUX
iajs-2952	61	8	the	the	DET
iajs-2952	61	9	set	set	NOUN
iajs-2952	61	10	of	of	ADP
iajs-2952	61	11	all	all	DET
iajs-2952	61	12	b	b	NOUN
iajs-2952	61	13	-	-	PUNCT
iajs-2952	61	14	cauchy	cauchy	ADJ
iajs-2952	61	15	sequence	sequence	NOUN
iajs-2952	61	16	equivalence	equivalence	NOUN
iajs-2952	61	17	classes	class	NOUN
iajs-2952	61	18	.	.	PUNCT
iajs-2952	62	1	definition	definition	NOUN
iajs-2952	62	2	(	(	PUNCT
iajs-2952	62	3	3.1	3.1	NUM
iajs-2952	62	4	):	):	PUNCT
iajs-2952	62	5	two	two	NUM
iajs-2952	62	6	b	b	X
iajs-2952	62	7	-	-	PUNCT
iajs-2952	62	8	cauchy	cauchy	ADJ
iajs-2952	62	9	sequences	sequence	NOUN
iajs-2952	62	10	{	{	PUNCT
iajs-2952	62	11	wn	wn	NOUN
iajs-2952	62	12	}	}	PUNCT
iajs-2952	62	13	and	and	CCONJ
iajs-2952	62	14	{	{	PUNCT
iajs-2952	62	15	yn	yn	NOUN
iajs-2952	62	16	}	}	PUNCT
iajs-2952	62	17	in	in	ADP
iajs-2952	62	18	a	a	DET
iajs-2952	62	19	generalized	generalized	ADJ
iajs-2952	62	20	2	2	NUM
iajs-2952	62	21	-	-	PUNCT
iajs-2952	62	22	inner	inner	ADJ
iajs-2952	62	23	product	product	NOUN
iajs-2952	62	24	space	space	NOUN
iajs-2952	62	25	w	w	AUX
iajs-2952	62	26	have	have	VERB
iajs-2952	62	27	a	a	DET
iajs-2952	62	28	relation	relation	NOUN
iajs-2952	62	29	denoted	denote	VERB
iajs-2952	62	30	by	by	ADP
iajs-2952	62	31	{	{	PUNCT
iajs-2952	62	32	wn}~{yn	wn}~{yn	NUM
iajs-2952	62	33	}	}	PUNCT
iajs-2952	62	34	,	,	PUNCT
iajs-2952	62	35	if	if	SCONJ
iajs-2952	62	36	for	for	ADP
iajs-2952	62	37	every	every	DET
iajs-2952	62	38	neighborhood	neighborhood	NOUN
iajs-2952	62	39	u	u	NOUN
iajs-2952	62	40	of	of	ADP
iajs-2952	62	41	0	0	NUM
iajs-2952	62	42	there	there	PRON
iajs-2952	62	43	is	be	VERB
iajs-2952	62	44	an	an	DET
iajs-2952	62	45	integer	integer	NOUN
iajs-2952	62	46	m(u	m(u	PROPN
iajs-2952	62	47	)	)	PUNCT
iajs-2952	63	1	such	such	ADJ
iajs-2952	63	2	that	that	SCONJ
iajs-2952	63	3	n	n	NUM
iajs-2952	63	4	≥	≥	NOUN
iajs-2952	63	5	m(u	m(u	PROPN
iajs-2952	63	6	)	)	PUNCT
iajs-2952	63	7	implies	imply	VERB
iajs-2952	63	8	that	that	SCONJ
iajs-2952	63	9	wn	wn	PROPN
iajs-2952	63	10	−	−	PROPN
iajs-2952	63	11	yn	yn	PROPN
iajs-2952	63	12	∈	∈	PROPN
iajs-2952	63	13	u.	u.	VERB
iajs-2952	63	14	it	it	PRON
iajs-2952	63	15	is	be	AUX
iajs-2952	63	16	clear	clear	ADJ
iajs-2952	63	17	that	that	SCONJ
iajs-2952	63	18	~	~	PUNCT
iajs-2952	63	19	is	be	AUX
iajs-2952	63	20	a	a	DET
iajs-2952	63	21	reflexive	reflexive	ADJ
iajs-2952	63	22	and	and	CCONJ
iajs-2952	63	23	symmetric	symmetric	ADJ
iajs-2952	63	24	relation	relation	NOUN
iajs-2952	63	25	.	.	PUNCT
iajs-2952	64	1	the	the	DET
iajs-2952	64	2	following	follow	VERB
iajs-2952	64	3	proposition	proposition	NOUN
iajs-2952	64	4	shows	show	VERB
iajs-2952	64	5	that	that	SCONJ
iajs-2952	64	6	this	this	DET
iajs-2952	64	7	relation	relation	NOUN
iajs-2952	64	8	is	be	AUX
iajs-2952	64	9	equivalent	equivalent	ADJ
iajs-2952	64	10	.	.	PUNCT
iajs-2952	65	1	proposition	proposition	NOUN
iajs-2952	65	2	(	(	PUNCT
iajs-2952	65	3	3.2	3.2	NUM
iajs-2952	65	4	):	):	PUNCT
iajs-2952	65	5	the	the	DET
iajs-2952	65	6	relation	relation	NOUN
iajs-2952	65	7	~	~	PUNCT
iajs-2952	65	8	on	on	ADP
iajs-2952	65	9	the	the	DET
iajs-2952	65	10	set	set	NOUN
iajs-2952	65	11	of	of	ADP
iajs-2952	65	12	b	b	PROPN
iajs-2952	65	13	-	-	PUNCT
iajs-2952	65	14	cauchy	cauchy	ADJ
iajs-2952	65	15	sequences	sequence	NOUN
iajs-2952	65	16	in	in	ADP
iajs-2952	65	17	w	w	PROPN
iajs-2952	65	18	is	be	AUX
iajs-2952	65	19	an	an	DET
iajs-2952	65	20	equivalent	equivalent	ADJ
iajs-2952	65	21	relation	relation	NOUN
iajs-2952	65	22	on	on	ADP
iajs-2952	65	23	w.	w.	PROPN
iajs-2952	65	24	proof	proof	NOUN
iajs-2952	65	25	:	:	PUNCT
iajs-2952	65	26	let	let	VERB
iajs-2952	65	27	{	{	PUNCT
iajs-2952	65	28	wn}~{yn	wn}~{yn	X
iajs-2952	65	29	}	}	PUNCT
iajs-2952	65	30	and	and	CCONJ
iajs-2952	65	31	{	{	PUNCT
iajs-2952	65	32	yn}~{zn	yn}~{zn	NOUN
iajs-2952	65	33	}	}	PUNCT
iajs-2952	65	34	.	.	PUNCT
iajs-2952	66	1	let	let	VERB
iajs-2952	66	2	,	,	PUNCT
iajs-2952	66	3	u	u	NOUN
iajs-2952	66	4	is	be	AUX
iajs-2952	66	5	an	an	DET
iajs-2952	66	6	arbitrary	arbitrary	ADJ
iajs-2952	66	7	neighborhood	neighborhood	NOUN
iajs-2952	66	8	of	of	ADP
iajs-2952	66	9	0	0	NUM
iajs-2952	66	10	.	.	PUNCT
iajs-2952	67	1	there	there	PRON
iajs-2952	67	2	exists	exist	VERB
iajs-2952	67	3	a	a	DET
iajs-2952	67	4	neighborhood	neighborhood	NOUN
iajs-2952	67	5	v	v	NOUN
iajs-2952	67	6	of	of	ADP
iajs-2952	67	7	0	0	NUM
iajs-2952	67	8	such	such	ADJ
iajs-2952	67	9	that	that	PRON
iajs-2952	67	10	v	v	NOUN
iajs-2952	67	11	+	+	X
iajs-2952	67	12	v	v	ADP
iajs-2952	67	13	⊂	⊂	X
iajs-2952	67	14	u.	u.	NOUN
iajs-2952	67	15	by	by	ADP
iajs-2952	67	16	definition	definition	NOUN
iajs-2952	67	17	(	(	PUNCT
iajs-2952	67	18	2.1	2.1	NUM
iajs-2952	67	19	)	)	PUNCT
iajs-2952	67	20	and	and	CCONJ
iajs-2952	67	21	for	for	ADP
iajs-2952	67	22	this	this	DET
iajs-2952	67	23	v	v	NOUN
iajs-2952	67	24	,	,	PUNCT
iajs-2952	67	25	there	there	PRON
iajs-2952	67	26	exists	exist	VERB
iajs-2952	67	27	an	an	DET
iajs-2952	67	28	integer	integer	NOUN
iajs-2952	67	29	m	m	VERB
iajs-2952	67	30	such	such	ADJ
iajs-2952	67	31	that	that	SCONJ
iajs-2952	67	32	wn	wn	PROPN
iajs-2952	67	33	−	−	PROPN
iajs-2952	67	34	yn	yn	PROPN
iajs-2952	67	35	,	,	PUNCT
iajs-2952	67	36	yn	yn	PROPN
iajs-2952	67	37	−	−	PROPN
iajs-2952	67	38	zn	zn	PROPN
iajs-2952	67	39	∈	∈	PROPN
iajs-2952	67	40	v	v	NOUN
iajs-2952	67	41	for	for	ADP
iajs-2952	67	42	n	n	PRON
iajs-2952	67	43	≥	≥	NOUN
iajs-2952	67	44	m.	m.	NOUN
iajs-2952	67	45	hence	hence	ADV
iajs-2952	67	46	,	,	PUNCT
iajs-2952	67	47	wn	wn	PROPN
iajs-2952	67	48	−	−	PROPN
iajs-2952	67	49	zn	zn	PROPN
iajs-2952	67	50	=	=	SYM
iajs-2952	67	51	(	(	PUNCT
iajs-2952	67	52	wn	wn	INTJ
iajs-2952	67	53	−	−	PROPN
iajs-2952	67	54	yn	yn	PROPN
iajs-2952	67	55	)	)	PUNCT
iajs-2952	68	1	+	+	CCONJ
iajs-2952	68	2	(	(	PUNCT
iajs-2952	68	3	yn	yn	INTJ
iajs-2952	68	4	−	−	PROPN
iajs-2952	68	5	zn	zn	X
iajs-2952	68	6	)	)	PUNCT
iajs-2952	68	7	is	be	AUX
iajs-2952	68	8	an	an	DET
iajs-2952	68	9	element	element	NOUN
iajs-2952	68	10	of	of	ADP
iajs-2952	68	11	u	u	NOUN
iajs-2952	68	12	for	for	ADP
iajs-2952	68	13	n	n	NUM
iajs-2952	68	14	≥	≥	NOUN
iajs-2952	68	15	m.	m.	NOUN
iajs-2952	68	16	therefore	therefore	ADV
iajs-2952	68	17	,	,	PUNCT
iajs-2952	68	18	{	{	PUNCT
iajs-2952	68	19	wn}~{zn}.∎	wn}~{zn}.∎	PRON
iajs-2952	68	20	define	define	VERB
iajs-2952	68	21	ŵ	ŵ	PROPN
iajs-2952	68	22	:	:	PUNCT
iajs-2952	68	23	=	=	X
iajs-2952	68	24	{	{	PUNCT
iajs-2952	68	25	ŵ	ŵ	X
iajs-2952	68	26	:	:	PUNCT
iajs-2952	68	27	ŵ	ŵ	X
iajs-2952	68	28	is	be	AUX
iajs-2952	68	29	equivalent	equivalent	ADJ
iajs-2952	68	30	class	class	NOUN
iajs-2952	68	31	of	of	ADP
iajs-2952	68	32	b	b	PROPN
iajs-2952	68	33	-	-	PUNCT
iajs-2952	68	34	cauchy	cauchy	ADJ
iajs-2952	68	35	sequences	sequence	NOUN
iajs-2952	68	36	}	}	PUNCT
iajs-2952	68	37	.	.	PUNCT
iajs-2952	69	1	step	step	NOUN
iajs-2952	69	2	2	2	NUM
iajs-2952	69	3	:	:	PUNCT
iajs-2952	69	4	proof	proof	NOUN
iajs-2952	69	5	�	�	PROPN
iajs-2952	69	6	̂	̂	SYM
iajs-2952	69	7	�	�	NOUN
iajs-2952	69	8	vector	vector	NOUN
iajs-2952	69	9	space	space	NOUN
iajs-2952	69	10	.	.	PUNCT
iajs-2952	70	1	let	let	VERB
iajs-2952	70	2	ŵ,ŷ	ŵ,ŷ	ADJ
iajs-2952	70	3	in	in	ADP
iajs-2952	70	4	ŵ.	ŵ.	NOUN
iajs-2952	70	5	define	define	VERB
iajs-2952	70	6	the	the	DET
iajs-2952	70	7	terms	term	NOUN
iajs-2952	70	8	addition	addition	NOUN
iajs-2952	70	9	and	and	CCONJ
iajs-2952	70	10	scalar	scalar	ADJ
iajs-2952	70	11	multiplication	multiplication	NOUN
iajs-2952	70	12	.	.	PUNCT
iajs-2952	71	1	on	on	ADP
iajs-2952	71	2	ŵ	ŵ	PROPN
iajs-2952	71	3	where	where	SCONJ
iajs-2952	71	4	{	{	PUNCT
iajs-2952	71	5	wn	wn	PROPN
iajs-2952	71	6	}	}	PUNCT
iajs-2952	71	7	∈	∈	PROPN
iajs-2952	71	8	ŵ	ŵ	PROPN
iajs-2952	71	9	and	and	CCONJ
iajs-2952	71	10	{	{	PUNCT
iajs-2952	71	11	yn	yn	PROPN
iajs-2952	71	12	}	}	PUNCT
iajs-2952	71	13	∈	∈	PROPN
iajs-2952	71	14	ŷ	ŷ	NUM
iajs-2952	71	15	,	,	PUNCT
iajs-2952	71	16	as	as	SCONJ
iajs-2952	71	17	shown	show	VERB
iajs-2952	71	18	below	below	ADP
iajs-2952	71	19	:	:	PUNCT
iajs-2952	71	20	•	•	NUM
iajs-2952	71	21	�	�	PROPN
iajs-2952	71	22	̂	̂	NUM
iajs-2952	71	23	�	�	PROPN
iajs-2952	71	24	+	+	SYM
iajs-2952	71	25	�	�	PROPN
iajs-2952	71	26	̂	̂	SYM
iajs-2952	71	27	�	�	NOUN
iajs-2952	71	28	=	=	PUNCT
iajs-2952	71	29	{	{	PUNCT
iajs-2952	71	30	𝑤𝑛	𝑤𝑛	NOUN
iajs-2952	71	31	+	+	CCONJ
iajs-2952	71	32	𝑦𝑛	𝑦𝑛	NOUN
iajs-2952	71	33	}	}	PUNCT
iajs-2952	71	34	•	•	NUM
iajs-2952	71	35	𝛼ŵ	𝛼ŵ	PROPN
iajs-2952	71	36	=	=	SYM
iajs-2952	71	37	{	{	PUNCT
iajs-2952	71	38	𝛼wn	𝛼wn	NOUN
iajs-2952	71	39	}	}	PUNCT
iajs-2952	71	40	ihjpas	ihjpas	PROPN
iajs-2952	71	41	.	.	PUNCT
iajs-2952	72	1	36(1)2023	36(1)2023	NUM
iajs-2952	72	2	314	314	NUM
iajs-2952	72	3	the	the	DET
iajs-2952	72	4	following	follow	VERB
iajs-2952	72	5	proposition	proposition	NOUN
iajs-2952	72	6	explains	explain	VERB
iajs-2952	72	7	that	that	SCONJ
iajs-2952	72	8	the	the	DET
iajs-2952	72	9	two	two	NUM
iajs-2952	72	10	operations	operation	NOUN
iajs-2952	72	11	defined	define	VERB
iajs-2952	72	12	on	on	ADP
iajs-2952	72	13	ŵ	ŵ	PROPN
iajs-2952	72	14	are	be	AUX
iajs-2952	72	15	well	well	ADV
iajs-2952	72	16	-	-	PUNCT
iajs-2952	72	17	defined	define	VERB
iajs-2952	72	18	because	because	SCONJ
iajs-2952	72	19	they	they	PRON
iajs-2952	72	20	are	be	AUX
iajs-2952	72	21	unaffected	unaffected	ADJ
iajs-2952	72	22	by	by	ADP
iajs-2952	72	23	the	the	DET
iajs-2952	72	24	elements	element	NOUN
iajs-2952	72	25	chosen	choose	VERB
iajs-2952	72	26	from	from	ADP
iajs-2952	72	27	{	{	PUNCT
iajs-2952	72	28	ŵn	ŵn	NOUN
iajs-2952	72	29	}	}	PUNCT
iajs-2952	72	30	and	and	CCONJ
iajs-2952	72	31	{	{	PUNCT
iajs-2952	72	32	ŷn	ŷn	PROPN
iajs-2952	72	33	}	}	PUNCT
iajs-2952	72	34	.	.	PUNCT
iajs-2952	73	1	but	but	CCONJ
iajs-2952	73	2	first	first	ADV
iajs-2952	73	3	,	,	PUNCT
iajs-2952	73	4	we	we	PRON
iajs-2952	73	5	need	need	VERB
iajs-2952	73	6	the	the	DET
iajs-2952	73	7	following	follow	VERB
iajs-2952	73	8	proposition	proposition	NOUN
iajs-2952	73	9	:	:	PUNCT
iajs-2952	73	10	proposition	proposition	NOUN
iajs-2952	73	11	(	(	PUNCT
iajs-2952	73	12	3.3	3.3	NUM
iajs-2952	73	13	):	):	PUNCT
iajs-2952	73	14	a	a	DET
iajs-2952	73	15	b	b	X
iajs-2952	73	16	-	-	PUNCT
iajs-2952	73	17	cauchy	cauchy	ADJ
iajs-2952	73	18	sequence	sequence	NOUN
iajs-2952	73	19	{	{	PUNCT
iajs-2952	73	20	wn	wn	NOUN
iajs-2952	73	21	}	}	PUNCT
iajs-2952	73	22	is	be	AUX
iajs-2952	73	23	equivalent	equivalent	ADJ
iajs-2952	73	24	to	to	ADP
iajs-2952	73	25	{	{	PUNCT
iajs-2952	73	26	an	an	PRON
iajs-2952	73	27	}	}	PUNCT
iajs-2952	73	28	in	in	ADP
iajs-2952	73	29	a	a	DET
iajs-2952	73	30	generalized	generalized	ADJ
iajs-2952	73	31	2	2	NUM
iajs-2952	73	32	-	-	PUNCT
iajs-2952	73	33	inner	inner	ADJ
iajs-2952	73	34	product	product	NOUN
iajs-2952	73	35	space	space	NOUN
iajs-2952	73	36	w	w	NOUN
iajs-2952	74	1	if	if	SCONJ
iajs-2952	75	1	and	and	CCONJ
iajs-2952	75	2	only	only	ADV
iajs-2952	75	3	if	if	SCONJ
iajs-2952	75	4	lim	lim	PROPN
iajs-2952	75	5	n→∞	n→∞	X
iajs-2952	75	6	‖wn	‖wn	NUM
iajs-2952	75	7	−	−	PROPN
iajs-2952	75	8	an‖b	an‖b	PROPN
iajs-2952	75	9	=	=	NOUN
iajs-2952	75	10	0	0	X
iajs-2952	75	11	.	.	PUNCT
iajs-2952	76	1	proof	proof	NOUN
iajs-2952	76	2	:	:	PUNCT
iajs-2952	76	3	let	let	VERB
iajs-2952	76	4	u	u	PRON
iajs-2952	76	5	be	be	AUX
iajs-2952	76	6	a	a	DET
iajs-2952	76	7	neighborhood	neighborhood	NOUN
iajs-2952	76	8	of	of	ADP
iajs-2952	76	9	zero	zero	NUM
iajs-2952	76	10	,	,	PUNCT
iajs-2952	76	11	then	then	ADV
iajs-2952	76	12	,	,	PUNCT
iajs-2952	76	13	there	there	PRON
iajs-2952	76	14	exists	exist	VERB
iajs-2952	76	15	m(u	m(u	PROPN
iajs-2952	76	16	)	)	PUNCT
iajs-2952	76	17	>	>	X
iajs-2952	76	18	0	0	NUM
iajs-2952	76	19	,	,	PUNCT
iajs-2952	76	20	such	such	ADJ
iajs-2952	76	21	that	that	SCONJ
iajs-2952	76	22	wn	wn	PROPN
iajs-2952	76	23	−	−	PROPN
iajs-2952	76	24	an	an	DET
iajs-2952	76	25	∈	∈	PROPN
iajs-2952	76	26	u	u	NOUN
iajs-2952	76	27	for	for	ADP
iajs-2952	76	28	n	n	PRON
iajs-2952	76	29	≥	≥	NOUN
iajs-2952	76	30	m(u	m(u	NUM
iajs-2952	76	31	)	)	PUNCT
iajs-2952	76	32	.	.	PUNCT
iajs-2952	77	1	hence	hence	ADV
iajs-2952	77	2	,	,	PUNCT
iajs-2952	77	3	for	for	ADP
iajs-2952	77	4	all	all	DET
iajs-2952	77	5	neighborhood	neighborhood	NOUN
iajs-2952	77	6	u	u	NOUN
iajs-2952	77	7	of	of	ADP
iajs-2952	77	8	0	0	NUM
iajs-2952	77	9	,	,	PUNCT
iajs-2952	77	10	there	there	PRON
iajs-2952	77	11	exists	exist	VERB
iajs-2952	77	12	ε	ε	PROPN
iajs-2952	77	13	=	=	SYM
iajs-2952	77	14	ε(u	ε(u	PROPN
iajs-2952	77	15	)	)	PUNCT
iajs-2952	77	16	>	>	X
iajs-2952	77	17	0	0	NUM
iajs-2952	78	1	such	such	ADJ
iajs-2952	78	2	that	that	PRON
iajs-2952	78	3	‖wn	‖wn	NUM
iajs-2952	78	4	−	−	PROPN
iajs-2952	78	5	an‖b	an‖b	ADP
iajs-2952	78	6	<	<	X
iajs-2952	78	7	ε	ε	PROPN
iajs-2952	78	8	;	;	PUNCT
iajs-2952	78	9	n	n	PRON
iajs-2952	78	10	≥	≥	NOUN
iajs-2952	78	11	m(u	m(u	NUM
iajs-2952	78	12	)	)	PUNCT
iajs-2952	78	13	.	.	PUNCT
iajs-2952	79	1	then	then	ADV
iajs-2952	79	2	,	,	PUNCT
iajs-2952	79	3	for	for	ADP
iajs-2952	79	4	every	every	DET
iajs-2952	79	5	δ	δ	PROPN
iajs-2952	79	6	>	>	X
iajs-2952	79	7	0	0	PROPN
iajs-2952	79	8	,	,	PUNCT
iajs-2952	79	9	there	there	PRON
iajs-2952	79	10	exists	exist	VERB
iajs-2952	79	11	m(δ	m(δ	PROPN
iajs-2952	79	12	)	)	PUNCT
iajs-2952	79	13	>	>	X
iajs-2952	79	14	0	0	NUM
iajs-2952	80	1	such	such	ADJ
iajs-2952	80	2	that	that	PRON
iajs-2952	80	3	‖wn	‖wn	NUM
iajs-2952	80	4	−	−	PROPN
iajs-2952	80	5	an‖b	an‖b	ADP
iajs-2952	80	6	<	<	X
iajs-2952	80	7	δ	δ	PROPN
iajs-2952	80	8	for	for	ADP
iajs-2952	80	9	all	all	DET
iajs-2952	80	10	n	n	PRON
iajs-2952	80	11	≥	≥	NOUN
iajs-2952	80	12	m(δ	m(δ	PROPN
iajs-2952	80	13	)	)	PUNCT
iajs-2952	80	14	,	,	PUNCT
iajs-2952	80	15	therefore	therefore	ADV
iajs-2952	80	16	,	,	PUNCT
iajs-2952	80	17	lim	lim	PROPN
iajs-2952	80	18	n→∞	n→∞	X
iajs-2952	80	19	‖wn	‖wn	NUM
iajs-2952	80	20	−	−	PROPN
iajs-2952	80	21	an‖b	an‖b	PROPN
iajs-2952	80	22	=	=	NOUN
iajs-2952	80	23	0	0	PROPN
iajs-2952	80	24	for	for	ADP
iajs-2952	80	25	n	n	PRON
iajs-2952	80	26	≥	≥	NOUN
iajs-2952	80	27	m(δ	m(δ	PROPN
iajs-2952	80	28	)	)	PUNCT
iajs-2952	80	29	.	.	PUNCT
iajs-2952	81	1	conversely	conversely	ADV
iajs-2952	81	2	,	,	PUNCT
iajs-2952	81	3	let	let	VERB
iajs-2952	81	4	{	{	PUNCT
iajs-2952	81	5	wn	wn	NOUN
iajs-2952	81	6	}	}	PUNCT
iajs-2952	81	7	,	,	PUNCT
iajs-2952	81	8	{	{	PUNCT
iajs-2952	81	9	an	an	PRON
iajs-2952	81	10	}	}	PUNCT
iajs-2952	81	11	be	be	AUX
iajs-2952	81	12	b	b	NOUN
iajs-2952	81	13	-	-	PUNCT
iajs-2952	81	14	cauchy	cauchy	ADJ
iajs-2952	81	15	sequences	sequence	NOUN
iajs-2952	81	16	in	in	ADP
iajs-2952	81	17	w	w	ADP
iajs-2952	81	18	such	such	ADJ
iajs-2952	81	19	that	that	PRON
iajs-2952	81	20	for	for	ADP
iajs-2952	81	21	every	every	DET
iajs-2952	81	22	neighborhood	neighborhood	NOUN
iajs-2952	81	23	u	u	NOUN
iajs-2952	81	24	of	of	ADP
iajs-2952	81	25	0	0	NUM
iajs-2952	81	26	,	,	PUNCT
iajs-2952	81	27	there	there	PRON
iajs-2952	81	28	exists	exist	VERB
iajs-2952	81	29	ε	ε	PROPN
iajs-2952	81	30	>	>	X
iajs-2952	81	31	0	0	NUM
iajs-2952	81	32	such	such	ADJ
iajs-2952	81	33	that	that	PRON
iajs-2952	81	34	bε(0	bε(0	NOUN
iajs-2952	81	35	)	)	PUNCT
iajs-2952	82	1	⊂	⊂	PROPN
iajs-2952	82	2	u.	u.	PROPN
iajs-2952	82	3	by	by	ADP
iajs-2952	82	4	our	our	PRON
iajs-2952	82	5	hypothesis	hypothesis	NOUN
iajs-2952	82	6	lim	lim	PROPN
iajs-2952	82	7	n→∞	n→∞	X
iajs-2952	83	1	‖wn	‖wn	NUM
iajs-2952	83	2	−	−	PROPN
iajs-2952	83	3	an‖b	an‖b	PROPN
iajs-2952	83	4	=	=	PROPN
iajs-2952	83	5	0	0	NUM
iajs-2952	83	6	,	,	PUNCT
iajs-2952	83	7	then	then	ADV
iajs-2952	83	8	there	there	PRON
iajs-2952	83	9	exists	exist	VERB
iajs-2952	83	10	m(ε	m(ε	NUM
iajs-2952	83	11	)	)	PUNCT
iajs-2952	83	12	>	>	X
iajs-2952	83	13	0	0	NUM
iajs-2952	84	1	such	such	ADJ
iajs-2952	84	2	that	that	PRON
iajs-2952	84	3	‖wn	‖wn	NUM
iajs-2952	84	4	−	−	PROPN
iajs-2952	84	5	an‖b	an‖b	ADP
iajs-2952	84	6	<	<	X
iajs-2952	84	7	ε	ε	PROPN
iajs-2952	84	8	for	for	ADP
iajs-2952	84	9	n	n	PROPN
iajs-2952	84	10	≥	≥	NOUN
iajs-2952	84	11	m(ε	m(ε	NUM
iajs-2952	84	12	)	)	PUNCT
iajs-2952	84	13	.	.	PUNCT
iajs-2952	85	1	hence	hence	ADV
iajs-2952	85	2	,	,	PUNCT
iajs-2952	85	3	wn	wn	PROPN
iajs-2952	85	4	−	−	PROPN
iajs-2952	85	5	an	an	DET
iajs-2952	85	6	∈	∈	PROPN
iajs-2952	85	7	bε(0	bε(0	PROPN
iajs-2952	85	8	)	)	PUNCT
iajs-2952	85	9	⊂	⊂	PROPN
iajs-2952	85	10	u	u	NOUN
iajs-2952	85	11	for	for	ADP
iajs-2952	85	12	n	n	PROPN
iajs-2952	85	13	≥	≥	NOUN
iajs-2952	85	14	m(ε	m(ε	NUM
iajs-2952	85	15	)	)	PUNCT
iajs-2952	85	16	,	,	PUNCT
iajs-2952	85	17	then	then	ADV
iajs-2952	85	18	{	{	PUNCT
iajs-2952	85	19	wn}~{an}.∎	wn}~{an}.∎	NOUN
iajs-2952	85	20	proposition	proposition	NOUN
iajs-2952	85	21	(	(	PUNCT
iajs-2952	85	22	3.4	3.4	NUM
iajs-2952	85	23	):	):	PUNCT
iajs-2952	85	24	if	if	SCONJ
iajs-2952	85	25	{	{	PUNCT
iajs-2952	85	26	an	an	NOUN
iajs-2952	85	27	}	}	PUNCT
iajs-2952	85	28	and	and	CCONJ
iajs-2952	85	29	{	{	PUNCT
iajs-2952	85	30	bn	bn	X
iajs-2952	85	31	}	}	PUNCT
iajs-2952	85	32	are	be	AUX
iajs-2952	85	33	equivalent	equivalent	ADJ
iajs-2952	85	34	to	to	ADP
iajs-2952	85	35	{	{	PUNCT
iajs-2952	85	36	wn	wn	PROPN
iajs-2952	85	37	}	}	PUNCT
iajs-2952	85	38	and	and	CCONJ
iajs-2952	85	39	{	{	PUNCT
iajs-2952	85	40	yn	yn	NOUN
iajs-2952	85	41	}	}	PUNCT
iajs-2952	85	42	in	in	ADP
iajs-2952	85	43	a	a	DET
iajs-2952	85	44	generalized	generalized	ADJ
iajs-2952	85	45	2	2	NUM
iajs-2952	85	46	-	-	PUNCT
iajs-2952	85	47	inner	inner	ADJ
iajs-2952	85	48	product	product	NOUN
iajs-2952	85	49	space	space	NOUN
iajs-2952	85	50	w.	w.	PROPN
iajs-2952	85	51	then	then	ADV
iajs-2952	85	52	,	,	PUNCT
iajs-2952	85	53	{	{	PUNCT
iajs-2952	85	54	an	an	DET
iajs-2952	85	55	+	+	NOUN
iajs-2952	85	56	bn	bn	ADJ
iajs-2952	85	57	}	}	PUNCT
iajs-2952	85	58	is	be	AUX
iajs-2952	85	59	equivalent	equivalent	ADJ
iajs-2952	85	60	to	to	ADP
iajs-2952	85	61	{	{	PUNCT
iajs-2952	85	62	wn	wn	PROPN
iajs-2952	85	63	+	+	PROPN
iajs-2952	85	64	yn	yn	PROPN
iajs-2952	85	65	}	}	PUNCT
iajs-2952	85	66	and	and	CCONJ
iajs-2952	85	67	{	{	PUNCT
iajs-2952	85	68	αan	αan	NOUN
iajs-2952	85	69	}	}	PUNCT
iajs-2952	85	70	is	be	AUX
iajs-2952	85	71	equivalent	equivalent	ADJ
iajs-2952	85	72	to	to	ADP
iajs-2952	85	73	{	{	PUNCT
iajs-2952	85	74	αwn	αwn	NOUN
iajs-2952	85	75	}	}	PUNCT
iajs-2952	85	76	.	.	PUNCT
iajs-2952	86	1	moreover	moreover	ADV
iajs-2952	86	2	,	,	PUNCT
iajs-2952	86	3	ŵ	ŵ	X
iajs-2952	86	4	is	be	AUX
iajs-2952	86	5	a	a	DET
iajs-2952	86	6	linear	linear	ADJ
iajs-2952	86	7	space	space	NOUN
iajs-2952	86	8	.	.	PUNCT
iajs-2952	87	1	proof	proof	NOUN
iajs-2952	87	2	:	:	PUNCT
iajs-2952	87	3	since	since	SCONJ
iajs-2952	87	4	{	{	PUNCT
iajs-2952	87	5	an}~{wn	an}~{wn	NOUN
iajs-2952	87	6	}	}	PUNCT
iajs-2952	87	7	and	and	CCONJ
iajs-2952	87	8	{	{	PUNCT
iajs-2952	87	9	bn}~{yn	bn}~{yn	NOUN
iajs-2952	87	10	}	}	PUNCT
iajs-2952	87	11	thus	thus	ADV
iajs-2952	87	12	we	we	PRON
iajs-2952	87	13	get	get	VERB
iajs-2952	87	14	‖(wn	‖(wn	NUM
iajs-2952	87	15	+	+	X
iajs-2952	87	16	yn	yn	NOUN
iajs-2952	87	17	)	)	PUNCT
iajs-2952	87	18	−	−	PROPN
iajs-2952	87	19	(	(	PUNCT
iajs-2952	87	20	an	an	DET
iajs-2952	87	21	+	+	PUNCT
iajs-2952	87	22	bn)‖b	bn)‖b	NOUN
iajs-2952	87	23	≤	≤	NOUN
iajs-2952	87	24	‖wn	‖wn	NUM
iajs-2952	87	25	−	−	NOUN
iajs-2952	87	26	an‖b	an‖b	PROPN
iajs-2952	87	27	+	+	PROPN
iajs-2952	88	1	‖yn	‖yn	PROPN
iajs-2952	89	1	−	−	PROPN
iajs-2952	89	2	bn‖b	bn‖b	PROPN
iajs-2952	89	3	then	then	ADV
iajs-2952	89	4	lim	lim	PROPN
iajs-2952	89	5	𝑛,𝑚→∞	𝑛,𝑚→∞	PROPN
iajs-2952	89	6	‖(wn	‖(wn	VERB
iajs-2952	89	7	+	+	CCONJ
iajs-2952	89	8	yn	yn	NOUN
iajs-2952	89	9	)	)	PUNCT
iajs-2952	89	10	−	−	PROPN
iajs-2952	90	1	(	(	PUNCT
iajs-2952	90	2	an	an	DET
iajs-2952	90	3	+	+	X
iajs-2952	90	4	bn)‖b	bn)‖b	NOUN
iajs-2952	90	5	=	=	SYM
iajs-2952	90	6	0	0	NUM
iajs-2952	90	7	…	…	PUNCT
iajs-2952	90	8	(	(	PUNCT
iajs-2952	90	9	1	1	X
iajs-2952	90	10	)	)	PUNCT
iajs-2952	90	11	on	on	ADP
iajs-2952	90	12	the	the	DET
iajs-2952	90	13	other	other	ADJ
iajs-2952	90	14	hand	hand	NOUN
iajs-2952	90	15	,	,	PUNCT
iajs-2952	90	16	lim	lim	PROPN
iajs-2952	90	17	𝑛→∞	𝑛→∞	NUM
iajs-2952	90	18	‖αwn	‖αwn	PROPN
iajs-2952	90	19	−	−	X
iajs-2952	90	20	αan‖b	αan‖b	SYM
iajs-2952	90	21	=	=	SYM
iajs-2952	90	22	0	0	NUM
iajs-2952	90	23	...	...	PUNCT
iajs-2952	91	1	(	(	PUNCT
iajs-2952	91	2	2	2	X
iajs-2952	91	3	)	)	PUNCT
iajs-2952	91	4	it	it	PRON
iajs-2952	91	5	implies	imply	VERB
iajs-2952	91	6	that	that	SCONJ
iajs-2952	91	7	{	{	PUNCT
iajs-2952	91	8	an	an	DET
iajs-2952	91	9	+	+	X
iajs-2952	91	10	bn}~{wn	bn}~{wn	NOUN
iajs-2952	91	11	+	+	CCONJ
iajs-2952	91	12	yn	yn	NOUN
iajs-2952	91	13	}	}	PUNCT
iajs-2952	91	14	and	and	CCONJ
iajs-2952	91	15	{	{	PUNCT
iajs-2952	91	16	αan}~{αwn	αan}~{αwn	NOUN
iajs-2952	91	17	}	}	PUNCT
iajs-2952	91	18	.	.	PUNCT
iajs-2952	92	1	therefore	therefore	ADV
iajs-2952	92	2	,	,	PUNCT
iajs-2952	92	3	from	from	ADP
iajs-2952	92	4	(	(	PUNCT
iajs-2952	92	5	1	1	NUM
iajs-2952	92	6	)	)	PUNCT
iajs-2952	92	7	,	,	PUNCT
iajs-2952	92	8	(	(	PUNCT
iajs-2952	92	9	2	2	X
iajs-2952	92	10	)	)	PUNCT
iajs-2952	92	11	and	and	CCONJ
iajs-2952	92	12	proposition	proposition	NOUN
iajs-2952	92	13	(	(	PUNCT
iajs-2952	92	14	2.3	2.3	NUM
iajs-2952	92	15	)	)	PUNCT
iajs-2952	92	16	,	,	PUNCT
iajs-2952	92	17	ŵ	ŵ	X
iajs-2952	92	18	is	be	AUX
iajs-2952	92	19	a	a	DET
iajs-2952	92	20	linear	linear	NOUN
iajs-2952	92	21	space.∎	space.∎	X
iajs-2952	92	22	step3	step3	PROPN
iajs-2952	92	23	:	:	PUNCT
iajs-2952	92	24	proof	proof	NOUN
iajs-2952	92	25	�	�	PROPN
iajs-2952	92	26	̂	̂	VERB
iajs-2952	92	27	�	�	NOUN
iajs-2952	92	28	is	be	AUX
iajs-2952	92	29	a	a	DET
iajs-2952	92	30	2	2	NUM
iajs-2952	92	31	-	-	PUNCT
iajs-2952	92	32	normed	normed	ADJ
iajs-2952	92	33	space	space	NOUN
iajs-2952	92	34	.	.	PUNCT
iajs-2952	93	1	we	we	PRON
iajs-2952	93	2	will	will	AUX
iajs-2952	93	3	define	define	VERB
iajs-2952	93	4	a	a	DET
iajs-2952	93	5	2	2	NUM
iajs-2952	93	6	-	-	PUNCT
iajs-2952	93	7	norm	norm	NOUN
iajs-2952	93	8	function	function	NOUN
iajs-2952	93	9	on	on	ADP
iajs-2952	93	10	the	the	DET
iajs-2952	93	11	space	space	NOUN
iajs-2952	93	12	ŵ.	ŵ.	NOUN
iajs-2952	93	13	as	as	ADP
iajs-2952	93	14	:	:	PUNCT
iajs-2952	93	15	‖.	‖.	PROPN
iajs-2952	93	16	‖b̂	‖b̂	PROPN
iajs-2952	93	17	:	:	PUNCT
iajs-2952	93	18	ŵ	ŵ	X
iajs-2952	93	19	×	×	NOUN
iajs-2952	93	20	<	<	X
iajs-2952	93	21	b̂	b̂	NOUN
iajs-2952	93	22	>	>	PUNCT
iajs-2952	93	23	→	→	PUNCT
iajs-2952	93	24	r+	r+	PRON
iajs-2952	93	25	is	be	AUX
iajs-2952	93	26	defined	define	VERB
iajs-2952	93	27	as	as	ADP
iajs-2952	93	28	:	:	PUNCT
iajs-2952	93	29	‖ŵ	‖ŵ	PROPN
iajs-2952	93	30	−	−	PROPN
iajs-2952	93	31	ŷ‖b̂	ŷ‖b̂	NOUN
iajs-2952	94	1	=	=	PROPN
iajs-2952	94	2	lim	lim	PROPN
iajs-2952	94	3	𝑛→∞	𝑛→∞	NUM
iajs-2952	94	4	‖wn	‖wn	NUM
iajs-2952	94	5	−	−	PROPN
iajs-2952	94	6	yn‖b	yn‖b	PROPN
iajs-2952	94	7	…	…	PUNCT
iajs-2952	94	8	(	(	PUNCT
iajs-2952	94	9	3	3	X
iajs-2952	94	10	)	)	PUNCT
iajs-2952	94	11	where	where	SCONJ
iajs-2952	94	12	{	{	PUNCT
iajs-2952	94	13	wn	wn	PROPN
iajs-2952	94	14	}	}	PUNCT
iajs-2952	94	15	∈	∈	PROPN
iajs-2952	94	16	ŵ	ŵ	PROPN
iajs-2952	94	17	,	,	PUNCT
iajs-2952	94	18	{	{	PUNCT
iajs-2952	94	19	yn	yn	PROPN
iajs-2952	94	20	}	}	PUNCT
iajs-2952	94	21	∈	∈	PROPN
iajs-2952	94	22	ŷ.	ŷ.	NOUN
iajs-2952	95	1	the	the	DET
iajs-2952	95	2	function	function	NOUN
iajs-2952	95	3	is	be	AUX
iajs-2952	95	4	well	well	ADV
iajs-2952	95	5	-	-	PUNCT
iajs-2952	95	6	defined	define	VERB
iajs-2952	95	7	as	as	SCONJ
iajs-2952	95	8	follows	follow	VERB
iajs-2952	95	9	:	:	PUNCT
iajs-2952	95	10	proposition	proposition	NOUN
iajs-2952	95	11	(	(	PUNCT
iajs-2952	95	12	3.5	3.5	NUM
iajs-2952	95	13	):	):	PUNCT
iajs-2952	95	14	if	if	SCONJ
iajs-2952	95	15	w	w	NOUN
iajs-2952	95	16	is	be	AUX
iajs-2952	95	17	a	a	DET
iajs-2952	95	18	generalized	generalized	ADJ
iajs-2952	95	19	2	2	NUM
iajs-2952	95	20	-	-	PUNCT
iajs-2952	95	21	inner	inner	ADJ
iajs-2952	95	22	product	product	NOUN
iajs-2952	95	23	space	space	NOUN
iajs-2952	95	24	,	,	PUNCT
iajs-2952	95	25	then	then	ADV
iajs-2952	95	26	for	for	ADP
iajs-2952	95	27	any	any	DET
iajs-2952	95	28	two	two	NUM
iajs-2952	95	29	b	b	X
iajs-2952	95	30	-	-	PUNCT
iajs-2952	95	31	cauchy	cauchy	ADJ
iajs-2952	95	32	sequences	sequence	NOUN
iajs-2952	95	33	{	{	PUNCT
iajs-2952	95	34	wn	wn	NOUN
iajs-2952	95	35	}	}	PUNCT
iajs-2952	95	36	and	and	CCONJ
iajs-2952	95	37	{	{	PUNCT
iajs-2952	95	38	yn	yn	NOUN
iajs-2952	95	39	}	}	PUNCT
iajs-2952	95	40	in	in	ADP
iajs-2952	95	41	w	w	PROPN
iajs-2952	95	42	:	:	PUNCT
iajs-2952	95	43	ihjpas	ihjpas	PROPN
iajs-2952	95	44	.	.	PUNCT
iajs-2952	96	1	36(1)2023	36(1)2023	NUM
iajs-2952	96	2	315	315	NUM
iajs-2952	96	3	1	1	NUM
iajs-2952	96	4	.	.	PUNCT
iajs-2952	97	1	lim	lim	PROPN
iajs-2952	97	2	𝑛→∞	𝑛→∞	NUM
iajs-2952	97	3	‖wn	‖wn	NUM
iajs-2952	97	4	−	−	PROPN
iajs-2952	97	5	yn‖b	yn‖b	PROPN
iajs-2952	97	6	exists	exist	VERB
iajs-2952	97	7	.	.	PUNCT
iajs-2952	98	1	2	2	X
iajs-2952	98	2	.	.	X
iajs-2952	98	3	for	for	ADP
iajs-2952	98	4	pairs	pair	NOUN
iajs-2952	98	5	of	of	ADP
iajs-2952	98	6	equivalent	equivalent	ADJ
iajs-2952	98	7	b	b	NOUN
iajs-2952	98	8	-	-	PUNCT
iajs-2952	98	9	cauchy	cauchy	ADJ
iajs-2952	98	10	sequences	sequence	NOUN
iajs-2952	98	11	{	{	PUNCT
iajs-2952	98	12	an}~{wn	an}~{wn	NOUN
iajs-2952	98	13	}	}	PUNCT
iajs-2952	98	14	and	and	CCONJ
iajs-2952	98	15	{	{	PUNCT
iajs-2952	98	16	bn}~{yn	bn}~{yn	NOUN
iajs-2952	98	17	}	}	PUNCT
iajs-2952	98	18	,	,	PUNCT
iajs-2952	98	19	lim	lim	PROPN
iajs-2952	98	20	𝑛→∞	𝑛→∞	NUM
iajs-2952	98	21	‖wn	‖wn	NUM
iajs-2952	98	22	−	−	PROPN
iajs-2952	98	23	yn‖b	yn‖b	PROPN
iajs-2952	98	24	=	=	SYM
iajs-2952	98	25	lim	lim	PROPN
iajs-2952	98	26	𝑛→∞	𝑛→∞	NUM
iajs-2952	98	27	‖an	‖an	PROPN
iajs-2952	98	28	−	−	PROPN
iajs-2952	98	29	bn‖b	bn‖b	PROPN
iajs-2952	98	30	.	.	PUNCT
iajs-2952	99	1	proof	proof	NOUN
iajs-2952	99	2	:	:	PUNCT
iajs-2952	100	1	1	1	X
iajs-2952	100	2	.	.	X
iajs-2952	101	1	let	let	VERB
iajs-2952	101	2	{	{	PUNCT
iajs-2952	101	3	wn	wn	PROPN
iajs-2952	101	4	}	}	PUNCT
iajs-2952	101	5	∈	∈	PROPN
iajs-2952	101	6	ŵ	ŵ	PROPN
iajs-2952	101	7	,	,	PUNCT
iajs-2952	101	8	{	{	PUNCT
iajs-2952	101	9	yn	yn	PROPN
iajs-2952	101	10	}	}	PUNCT
iajs-2952	101	11	∈	∈	PROPN
iajs-2952	101	12	ŷ	ŷ	AUX
iajs-2952	101	13	be	be	AUX
iajs-2952	101	14	any	any	DET
iajs-2952	101	15	two	two	NUM
iajs-2952	101	16	b	b	X
iajs-2952	101	17	-	-	PUNCT
iajs-2952	101	18	cauchy	cauchy	ADJ
iajs-2952	101	19	sequences	sequence	NOUN
iajs-2952	101	20	,	,	PUNCT
iajs-2952	101	21	then	then	ADV
iajs-2952	101	22	‖wn	‖wn	NUM
iajs-2952	101	23	−	−	PROPN
iajs-2952	102	1	yn‖b	yn‖b	PROPN
iajs-2952	102	2	≤	≤	NOUN
iajs-2952	102	3	‖wn	‖wn	NUM
iajs-2952	102	4	−	−	NOUN
iajs-2952	102	5	wm‖b	wm‖b	NOUN
iajs-2952	102	6	+	+	CCONJ
iajs-2952	102	7	‖wm	‖wm	NUM
iajs-2952	102	8	−	−	PROPN
iajs-2952	102	9	ym‖b	ym‖b	NOUN
iajs-2952	102	10	+	+	CCONJ
iajs-2952	102	11	‖ym	‖ym	NUM
iajs-2952	102	12	−	−	PROPN
iajs-2952	102	13	yn‖b	yn‖b	PROPN
iajs-2952	102	14	hence	hence	ADV
iajs-2952	102	15	,	,	PUNCT
iajs-2952	102	16	‖wn	‖wn	NUM
iajs-2952	102	17	−	−	PROPN
iajs-2952	102	18	yn‖b	yn‖b	PROPN
iajs-2952	102	19	−	−	PROPN
iajs-2952	102	20	‖wm	‖wm	NUM
iajs-2952	102	21	−	−	NUM
iajs-2952	102	22	ym‖b	ym‖b	PROPN
iajs-2952	102	23	≤	≤	NOUN
iajs-2952	102	24	‖wn	‖wn	NUM
iajs-2952	102	25	−	−	NOUN
iajs-2952	102	26	wm‖b	wm‖b	NOUN
iajs-2952	102	27	+	+	CCONJ
iajs-2952	102	28	‖ym	‖ym	NUM
iajs-2952	102	29	−	−	PROPN
iajs-2952	102	30	yn‖b	yn‖b	PROPN
iajs-2952	102	31	on	on	ADP
iajs-2952	102	32	the	the	DET
iajs-2952	102	33	other	other	ADJ
iajs-2952	102	34	hand	hand	NOUN
iajs-2952	102	35	,	,	PUNCT
iajs-2952	102	36	if	if	SCONJ
iajs-2952	102	37	we	we	PRON
iajs-2952	102	38	change	change	VERB
iajs-2952	102	39	m	m	VERB
iajs-2952	102	40	by	by	ADP
iajs-2952	102	41	n	n	CCONJ
iajs-2952	102	42	,	,	PUNCT
iajs-2952	102	43	‖wm	‖wm	NUM
iajs-2952	102	44	−	−	PROPN
iajs-2952	102	45	ym‖b	ym‖b	PROPN
iajs-2952	102	46	−	−	PROPN
iajs-2952	102	47	‖wn	‖wn	NUM
iajs-2952	102	48	−	−	PROPN
iajs-2952	102	49	yn‖b	yn‖b	PROPN
iajs-2952	102	50	≤	≤	PROPN
iajs-2952	102	51	‖wm	‖wm	NUM
iajs-2952	102	52	−	−	PROPN
iajs-2952	102	53	wn‖b	wn‖b	PROPN
iajs-2952	102	54	+	+	PROPN
iajs-2952	102	55	‖yn	‖yn	PROPN
iajs-2952	102	56	−	−	PROPN
iajs-2952	102	57	ym‖b	ym‖b	NOUN
iajs-2952	102	58	it	it	PRON
iajs-2952	102	59	implies	imply	VERB
iajs-2952	102	60	that	that	SCONJ
iajs-2952	102	61	|‖wn	|‖wn	NOUN
iajs-2952	103	1	−	−	PROPN
iajs-2952	103	2	yn‖b	yn‖b	PROPN
iajs-2952	104	1	−	−	PROPN
iajs-2952	104	2	‖wm	‖wm	NUM
iajs-2952	104	3	−	−	PROPN
iajs-2952	104	4	ym‖b|	ym‖b|	PROPN
iajs-2952	105	1	≤	≤	NOUN
iajs-2952	105	2	‖wn	‖wn	NUM
iajs-2952	105	3	−	−	NOUN
iajs-2952	105	4	wm‖b	wm‖b	NOUN
iajs-2952	105	5	+	+	CCONJ
iajs-2952	106	1	‖yn	‖yn	NUM
iajs-2952	106	2	−	−	PROPN
iajs-2952	106	3	ym‖b	ym‖b	NOUN
iajs-2952	106	4	…	…	PUNCT
iajs-2952	106	5	(	(	PUNCT
iajs-2952	106	6	4	4	NUM
iajs-2952	106	7	)	)	PUNCT
iajs-2952	106	8	thus	thus	ADV
iajs-2952	106	9	,	,	PUNCT
iajs-2952	106	10	by	by	ADP
iajs-2952	106	11	taking	take	VERB
iajs-2952	106	12	n	n	PRON
iajs-2952	106	13	,	,	PUNCT
iajs-2952	106	14	m	m	PROPN
iajs-2952	106	15	→	→	SYM
iajs-2952	106	16	∞	∞	NUM
iajs-2952	106	17	and	and	CCONJ
iajs-2952	106	18	definition	definition	NOUN
iajs-2952	106	19	(	(	PUNCT
iajs-2952	106	20	1.1	1.1	NUM
iajs-2952	106	21	)	)	PUNCT
iajs-2952	106	22	,	,	PUNCT
iajs-2952	106	23	it	it	PRON
iajs-2952	106	24	follows	follow	VERB
iajs-2952	106	25	that	that	SCONJ
iajs-2952	106	26	lim	lim	PROPN
iajs-2952	106	27	𝑛→∞	𝑛→∞	NUM
iajs-2952	106	28	|‖wn	|‖wn	PUNCT
iajs-2952	106	29	−	−	PROPN
iajs-2952	106	30	yn‖b	yn‖b	NUM
iajs-2952	106	31	−	−	PROPN
iajs-2952	106	32	‖wm	‖wm	NUM
iajs-2952	107	1	−	−	NOUN
iajs-2952	107	2	ym‖b|	ym‖b|	PROPN
iajs-2952	107	3	=	=	PUNCT
iajs-2952	107	4	0	0	PUNCT
iajs-2952	108	1	then	then	ADV
iajs-2952	108	2	,	,	PUNCT
iajs-2952	108	3	{	{	PUNCT
iajs-2952	108	4	‖wn	‖wn	NUM
iajs-2952	108	5	−	−	PROPN
iajs-2952	108	6	yn‖b	yn‖b	PROPN
iajs-2952	108	7	}	}	PUNCT
iajs-2952	108	8	is	be	AUX
iajs-2952	108	9	a	a	DET
iajs-2952	108	10	cauchy	cauchy	ADJ
iajs-2952	108	11	sequence	sequence	NOUN
iajs-2952	108	12	in	in	ADP
iajs-2952	108	13	r.	r.	PROPN
iajs-2952	108	14	but	but	CCONJ
iajs-2952	108	15	,	,	PUNCT
iajs-2952	108	16	r	r	NOUN
iajs-2952	108	17	is	be	AUX
iajs-2952	108	18	complete	complete	ADJ
iajs-2952	108	19	,	,	PUNCT
iajs-2952	108	20	thus	thus	ADV
iajs-2952	108	21	lim	lim	NOUN
iajs-2952	108	22	𝑛→∞	𝑛→∞	NUM
iajs-2952	108	23	‖wn	‖wn	NUM
iajs-2952	108	24	−	−	PROPN
iajs-2952	108	25	yn‖b	yn‖b	PROPN
iajs-2952	108	26	exists	exist	VERB
iajs-2952	108	27	.	.	PUNCT
iajs-2952	109	1	2	2	X
iajs-2952	109	2	.	.	X
iajs-2952	109	3	let	let	VERB
iajs-2952	109	4	{	{	PUNCT
iajs-2952	109	5	an}~{wn	an}~{wn	VERB
iajs-2952	109	6	}	}	PUNCT
iajs-2952	109	7	and	and	CCONJ
iajs-2952	109	8	{	{	PUNCT
iajs-2952	109	9	bn}~{yn	bn}~{yn	NOUN
iajs-2952	109	10	}	}	PUNCT
iajs-2952	109	11	.	.	PUNCT
iajs-2952	110	1	by	by	ADP
iajs-2952	110	2	the	the	DET
iajs-2952	110	3	same	same	ADJ
iajs-2952	110	4	argument	argument	NOUN
iajs-2952	110	5	of	of	ADP
iajs-2952	110	6	(	(	PUNCT
iajs-2952	110	7	4	4	NUM
iajs-2952	110	8	)	)	PUNCT
iajs-2952	110	9	,	,	PUNCT
iajs-2952	110	10	it	it	PRON
iajs-2952	110	11	implies	imply	VERB
iajs-2952	110	12	that	that	SCONJ
iajs-2952	110	13	|‖wn	|‖wn	NOUN
iajs-2952	111	1	−	−	PROPN
iajs-2952	111	2	yn‖b	yn‖b	PROPN
iajs-2952	112	1	−	−	PROPN
iajs-2952	113	1	‖an	‖an	NUM
iajs-2952	113	2	−	−	PUNCT
iajs-2952	113	3	bn‖b|	bn‖b|	PROPN
iajs-2952	113	4	≤	≤	PROPN
iajs-2952	114	1	‖wn	‖wn	NUM
iajs-2952	114	2	−	−	NOUN
iajs-2952	114	3	an‖b	an‖b	PROPN
iajs-2952	114	4	+	+	PROPN
iajs-2952	115	1	‖yn	‖yn	NUM
iajs-2952	115	2	−	−	PROPN
iajs-2952	115	3	bn‖b	bn‖b	PROPN
iajs-2952	115	4	by	by	ADP
iajs-2952	115	5	taking	take	VERB
iajs-2952	115	6	n	n	PRON
iajs-2952	115	7	,	,	PUNCT
iajs-2952	115	8	m	m	PROPN
iajs-2952	115	9	→	→	SYM
iajs-2952	115	10	∞	∞	NUM
iajs-2952	115	11	and	and	CCONJ
iajs-2952	115	12	proposition	proposition	NOUN
iajs-2952	115	13	(	(	PUNCT
iajs-2952	115	14	2.3	2.3	NUM
iajs-2952	115	15	)	)	PUNCT
iajs-2952	115	16	,	,	PUNCT
iajs-2952	115	17	we	we	PRON
iajs-2952	115	18	get	get	VERB
iajs-2952	115	19	lim	lim	PROPN
iajs-2952	115	20	𝑛→∞	𝑛→∞	NUM
iajs-2952	115	21	‖wn	‖wn	NUM
iajs-2952	115	22	−	−	PROPN
iajs-2952	116	1	yn‖b	yn‖b	PROPN
iajs-2952	117	1	=	=	SYM
iajs-2952	118	1	lim	lim	PROPN
iajs-2952	118	2	𝑛→∞	𝑛→∞	NUM
iajs-2952	118	3	‖an	‖an	PROPN
iajs-2952	118	4	−	−	PROPN
iajs-2952	118	5	bn‖b	bn‖b	PROPN
iajs-2952	118	6	.∎	.∎	PUNCT
iajs-2952	119	1	from	from	ADP
iajs-2952	119	2	equation	equation	NOUN
iajs-2952	119	3	(	(	PUNCT
iajs-2952	119	4	3	3	NUM
iajs-2952	119	5	)	)	PUNCT
iajs-2952	119	6	and	and	CCONJ
iajs-2952	119	7	proposition	proposition	NOUN
iajs-2952	119	8	(	(	PUNCT
iajs-2952	119	9	2.3	2.3	NUM
iajs-2952	119	10	)	)	PUNCT
iajs-2952	119	11	,	,	PUNCT
iajs-2952	119	12	the	the	DET
iajs-2952	119	13	conditions	condition	NOUN
iajs-2952	119	14	(	(	PUNCT
iajs-2952	119	15	1	1	NUM
iajs-2952	119	16	-	-	SYM
iajs-2952	119	17	3	3	NUM
iajs-2952	119	18	)	)	PUNCT
iajs-2952	119	19	of	of	ADP
iajs-2952	119	20	a	a	DET
iajs-2952	119	21	2	2	NUM
iajs-2952	119	22	-	-	PUNCT
iajs-2952	119	23	normed	norme	VERB
iajs-2952	119	24	space	space	NOUN
iajs-2952	119	25	are	be	AUX
iajs-2952	119	26	done	do	VERB
iajs-2952	119	27	.	.	PUNCT
iajs-2952	120	1	proposition	proposition	NOUN
iajs-2952	120	2	(	(	PUNCT
iajs-2952	120	3	3.6	3.6	NUM
iajs-2952	120	4	):	):	PUNCT
iajs-2952	120	5	(	(	PUNCT
iajs-2952	120	6	ŵ	ŵ	X
iajs-2952	120	7	,	,	PUNCT
iajs-2952	120	8	‖.	‖.	PROPN
iajs-2952	120	9	‖b̂	‖b̂	PROPN
iajs-2952	120	10	)	)	PUNCT
iajs-2952	120	11	is	be	AUX
iajs-2952	120	12	a	a	DET
iajs-2952	120	13	2	2	NUM
iajs-2952	120	14	-	-	PUNCT
iajs-2952	120	15	normed	norme	VERB
iajs-2952	120	16	space	space	NOUN
iajs-2952	120	17	.	.	PUNCT
iajs-2952	121	1	proof	proof	NOUN
iajs-2952	121	2	:	:	PUNCT
iajs-2952	121	3	since	since	SCONJ
iajs-2952	121	4	‖ŵ	‖ŵ	PROPN
iajs-2952	121	5	−	−	PROPN
iajs-2952	121	6	ẑ‖b̂	ẑ‖b̂	PROPN
iajs-2952	121	7	=	=	SYM
iajs-2952	121	8	lim	lim	PROPN
iajs-2952	121	9	𝑛→∞	𝑛→∞	NUM
iajs-2952	121	10	‖wn	‖wn	PROPN
iajs-2952	121	11	−	−	PROPN
iajs-2952	121	12	zn‖	zn‖	PROPN
iajs-2952	121	13	b	b	PROPN
iajs-2952	121	14	≤	≤	PROPN
iajs-2952	121	15	lim	lim	NOUN
iajs-2952	121	16	𝑛→∞	𝑛→∞	NUM
iajs-2952	121	17	‖wn	‖wn	NUM
iajs-2952	121	18	−	−	PROPN
iajs-2952	121	19	yn‖b	yn‖b	PROPN
iajs-2952	122	1	+	+	CCONJ
iajs-2952	122	2	lim	lim	PROPN
iajs-2952	122	3	𝑛→∞	𝑛→∞	NUM
iajs-2952	122	4	‖yn	‖yn	PROPN
iajs-2952	122	5	−	−	PROPN
iajs-2952	122	6	zn‖b	zn‖b	PROPN
iajs-2952	122	7	=	=	SYM
iajs-2952	122	8	‖ŵ	‖ŵ	PROPN
iajs-2952	122	9	−	−	PROPN
iajs-2952	122	10	ŷ‖b̂	ŷ‖b̂	NOUN
iajs-2952	122	11	+	+	CCONJ
iajs-2952	122	12	‖ŷ	‖ŷ	PROPN
iajs-2952	122	13	−	−	PROPN
iajs-2952	122	14	ẑ‖b̂	ẑ‖b̂	PROPN
iajs-2952	122	15	,	,	PUNCT
iajs-2952	122	16	then	then	ADV
iajs-2952	122	17	(	(	PUNCT
iajs-2952	122	18	ŵ	ŵ	X
iajs-2952	122	19	,	,	PUNCT
iajs-2952	122	20	‖	‖	ADJ
iajs-2952	122	21	,	,	PUNCT
iajs-2952	122	22	‖b̂	‖b̂	PROPN
iajs-2952	122	23	)	)	PUNCT
iajs-2952	122	24	is	be	AUX
iajs-2952	122	25	a	a	DET
iajs-2952	122	26	2	2	NUM
iajs-2952	122	27	-	-	PUNCT
iajs-2952	122	28	norm.∎	norm.∎	NOUN
iajs-2952	122	29	step4	step4	NOUN
iajs-2952	122	30	:	:	PUNCT
iajs-2952	122	31	construction	construction	NOUN
iajs-2952	122	32	of	of	ADP
iajs-2952	122	33	an	an	DET
iajs-2952	122	34	isometric	isometric	ADJ
iajs-2952	122	35	𝑇	𝑇	PROPN
iajs-2952	122	36	:	:	PUNCT
iajs-2952	122	37	𝑊	𝑊	PROPN
iajs-2952	122	38	→	→	SYM
iajs-2952	122	39	�	�	PROPN
iajs-2952	122	40	̂	̂	VERB
iajs-2952	122	41	�	�	PROPN
iajs-2952	122	42	0	0	NUM
iajs-2952	122	43	⊂	⊂	PROPN
iajs-2952	122	44	�	�	PROPN
iajs-2952	122	45	̂	̂	NOUN
iajs-2952	122	46	�	�	PROPN
iajs-2952	122	47	.	.	PUNCT
iajs-2952	123	1	let	let	AUX
iajs-2952	123	2	ŵ0	ŵ0	VERB
iajs-2952	123	3	be	be	AUX
iajs-2952	123	4	the	the	DET
iajs-2952	123	5	subset	subset	NOUN
iajs-2952	123	6	of	of	ADP
iajs-2952	123	7	ŵ	ŵ	PROPN
iajs-2952	123	8	composed	compose	VERB
iajs-2952	123	9	of	of	ADP
iajs-2952	123	10	the	the	DET
iajs-2952	123	11	equivalence	equivalence	NOUN
iajs-2952	123	12	classes	class	NOUN
iajs-2952	123	13	containing	contain	VERB
iajs-2952	123	14	constant	constant	ADJ
iajs-2952	123	15	b	b	PROPN
iajs-2952	123	16	-	-	PUNCT
iajs-2952	123	17	cauchy	cauchy	ADJ
iajs-2952	123	18	sequences	sequence	NOUN
iajs-2952	123	19	.	.	PUNCT
iajs-2952	124	1	define	define	VERB
iajs-2952	124	2	a	a	DET
iajs-2952	124	3	function	function	NOUN
iajs-2952	124	4	t	t	NOUN
iajs-2952	124	5	:	:	PUNCT
iajs-2952	124	6	w	w	PROPN
iajs-2952	124	7	→	→	SYM
iajs-2952	124	8	ŵ0	ŵ0	PROPN
iajs-2952	124	9	⊂	⊂	X
iajs-2952	124	10	ŵ	ŵ	X
iajs-2952	124	11	by	by	ADP
iajs-2952	124	12	t(w	t(w	NOUN
iajs-2952	124	13	)	)	PUNCT
iajs-2952	124	14	=	=	SYM
iajs-2952	124	15	ŵ	ŵ	X
iajs-2952	124	16	=	=	SYM
iajs-2952	124	17	(	(	PUNCT
iajs-2952	124	18	w	w	PROPN
iajs-2952	124	19	,	,	PUNCT
iajs-2952	124	20	w	w	PROPN
iajs-2952	124	21	,	,	PUNCT
iajs-2952	124	22	…	…	PUNCT
iajs-2952	124	23	)	)	PUNCT
iajs-2952	124	24	.	.	PUNCT
iajs-2952	125	1	it	it	PRON
iajs-2952	125	2	is	be	AUX
iajs-2952	125	3	clearly	clearly	ADV
iajs-2952	125	4	that	that	SCONJ
iajs-2952	125	5	t	t	PROPN
iajs-2952	125	6	is	be	AUX
iajs-2952	125	7	a	a	DET
iajs-2952	125	8	welldefined	welldefine	VERB
iajs-2952	125	9	onto	onto	ADP
iajs-2952	125	10	and	and	CCONJ
iajs-2952	125	11	one	one	NUM
iajs-2952	125	12	to	to	ADP
iajs-2952	125	13	one	one	NUM
iajs-2952	125	14	.	.	PUNCT
iajs-2952	126	1	in	in	ADP
iajs-2952	126	2	fact	fact	NOUN
iajs-2952	126	3	,	,	PUNCT
iajs-2952	126	4	‖tw	‖tw	PROPN
iajs-2952	126	5	−	−	PROPN
iajs-2952	126	6	ty‖b	ty‖b	PROPN
iajs-2952	126	7	=	=	PUNCT
iajs-2952	126	8	‖ŵ	‖ŵ	PROPN
iajs-2952	126	9	−	−	PROPN
iajs-2952	126	10	ŷ‖b̂	ŷ‖b̂	NOUN
iajs-2952	127	1	=	=	SYM
iajs-2952	127	2	lim	lim	PROPN
iajs-2952	127	3	𝑛→∞	𝑛→∞	NUM
iajs-2952	127	4	‖w	‖w	X
iajs-2952	127	5	−	−	PROPN
iajs-2952	127	6	y‖b	y‖b	NOUN
iajs-2952	127	7	=	=	SYM
iajs-2952	127	8	‖w	‖w	NOUN
iajs-2952	127	9	−	−	PROPN
iajs-2952	127	10	y‖b	y‖b	NOUN
iajs-2952	127	11	,	,	PUNCT
iajs-2952	127	12	thus	thus	ADV
iajs-2952	127	13	,	,	PUNCT
iajs-2952	127	14	ŵ0	ŵ0	PROPN
iajs-2952	127	15	and	and	CCONJ
iajs-2952	127	16	w	w	PROPN
iajs-2952	127	17	are	be	AUX
iajs-2952	127	18	isometric	isometric	ADJ
iajs-2952	127	19	.	.	PUNCT
iajs-2952	128	1	ihjpas	ihjpas	PROPN
iajs-2952	128	2	.	.	PUNCT
iajs-2952	129	1	36(1)2023	36(1)2023	NUM
iajs-2952	129	2	316	316	NUM
iajs-2952	129	3	step	step	NOUN
iajs-2952	129	4	5	5	NUM
iajs-2952	129	5	:	:	PUNCT
iajs-2952	129	6	proof	proof	NOUN
iajs-2952	129	7	�	�	PROPN
iajs-2952	129	8	̂	̂	VERB
iajs-2952	129	9	�	�	NOUN
iajs-2952	129	10	0	0	NUM
iajs-2952	129	11	is	be	AUX
iajs-2952	129	12	dense	dense	ADJ
iajs-2952	129	13	in	in	ADP
iajs-2952	129	14	�	�	PROPN
iajs-2952	129	15	̂	̂	NOUN
iajs-2952	129	16	�	�	PROPN
iajs-2952	129	17	.	.	PUNCT
iajs-2952	130	1	proposition	proposition	NOUN
iajs-2952	130	2	(	(	PUNCT
iajs-2952	130	3	3.7	3.7	NUM
iajs-2952	130	4	):	):	PUNCT
iajs-2952	130	5	if	if	SCONJ
iajs-2952	130	6	w	w	NOUN
iajs-2952	130	7	is	be	AUX
iajs-2952	130	8	a	a	DET
iajs-2952	130	9	generalized	generalized	ADJ
iajs-2952	130	10	2	2	NUM
iajs-2952	130	11	-	-	PUNCT
iajs-2952	130	12	inner	inner	ADJ
iajs-2952	130	13	product	product	NOUN
iajs-2952	130	14	space	space	NOUN
iajs-2952	130	15	,	,	PUNCT
iajs-2952	130	16	then	then	ADV
iajs-2952	130	17	,	,	PUNCT
iajs-2952	130	18	ŵ0	ŵ0	X
iajs-2952	130	19	is	be	AUX
iajs-2952	130	20	dense	dense	ADJ
iajs-2952	130	21	in	in	ADP
iajs-2952	130	22	ŵ.	ŵ.	ADJ
iajs-2952	130	23	proof	proof	NOUN
iajs-2952	130	24	:	:	PUNCT
iajs-2952	130	25	let	let	VERB
iajs-2952	130	26	ŵ	ŵ	X
iajs-2952	130	27	∈	∈	PROPN
iajs-2952	130	28	ŵ	ŵ	X
iajs-2952	131	1	−	−	PROPN
iajs-2952	131	2	ŵ0	ŵ0	NOUN
iajs-2952	131	3	,	,	PUNCT
iajs-2952	131	4	then	then	ADV
iajs-2952	131	5	,	,	PUNCT
iajs-2952	131	6	there	there	PRON
iajs-2952	131	7	exists	exist	VERB
iajs-2952	131	8	a	a	DET
iajs-2952	131	9	b	b	NOUN
iajs-2952	131	10	-	-	PUNCT
iajs-2952	131	11	cauchy	cauchy	ADJ
iajs-2952	131	12	sequence	sequence	NOUN
iajs-2952	131	13	{	{	PUNCT
iajs-2952	131	14	wn	wn	PROPN
iajs-2952	131	15	}	}	PUNCT
iajs-2952	131	16	∈	∈	PROPN
iajs-2952	131	17	ŵ	ŵ	X
iajs-2952	131	18	where	where	SCONJ
iajs-2952	131	19	{	{	PUNCT
iajs-2952	131	20	wn	wn	NOUN
iajs-2952	131	21	}	}	PUNCT
iajs-2952	131	22	=	=	SYM
iajs-2952	131	23	{	{	PUNCT
iajs-2952	131	24	w1	w1	NOUN
iajs-2952	131	25	,	,	PUNCT
iajs-2952	131	26	w2	w2	NOUN
iajs-2952	131	27	,	,	PUNCT
iajs-2952	131	28	…	…	PUNCT
iajs-2952	131	29	}	}	PUNCT
iajs-2952	131	30	.	.	PUNCT
iajs-2952	132	1	define	define	VERB
iajs-2952	132	2	ŵm	ŵm	NUM
iajs-2952	132	3	=	=	SYM
iajs-2952	132	4	{	{	PUNCT
iajs-2952	132	5	wm	wm	PROPN
iajs-2952	132	6	,	,	PUNCT
iajs-2952	132	7	wm	wm	PROPN
iajs-2952	132	8	,	,	PUNCT
iajs-2952	132	9	…	…	PUNCT
iajs-2952	132	10	}	}	PUNCT
iajs-2952	132	11	for	for	ADP
iajs-2952	132	12	all	all	DET
iajs-2952	132	13	m	m	NOUN
iajs-2952	132	14	∈	∈	NOUN
iajs-2952	132	15	n	n	CCONJ
iajs-2952	132	16	,	,	PUNCT
iajs-2952	132	17	thus	thus	ADV
iajs-2952	132	18	�	�	PROPN
iajs-2952	132	19	̂	̂	SYM
iajs-2952	132	20	�	�	PROPN
iajs-2952	132	21	m	m	NOUN
iajs-2952	132	22	∈	∈	PROPN
iajs-2952	132	23	x̂0	x̂0	PROPN
iajs-2952	132	24	.	.	PUNCT
iajs-2952	133	1	hence	hence	ADV
iajs-2952	133	2	,	,	PUNCT
iajs-2952	133	3	by	by	ADP
iajs-2952	133	4	definition	definition	NOUN
iajs-2952	133	5	(	(	PUNCT
iajs-2952	133	6	1.1	1.1	NUM
iajs-2952	133	7	)	)	PUNCT
iajs-2952	133	8	‖ŵm	‖ŵm	NOUN
iajs-2952	133	9	−	−	NOUN
iajs-2952	133	10	ŵ‖b̂	ŵ‖b̂	NOUN
iajs-2952	134	1	=	=	PROPN
iajs-2952	134	2	lim	lim	PROPN
iajs-2952	134	3	𝑛,𝑚→∞	𝑛,𝑚→∞	PROPN
iajs-2952	134	4	‖wn	‖wn	NUM
iajs-2952	134	5	−	−	NUM
iajs-2952	134	6	wm‖b	wm‖b	NOUN
iajs-2952	134	7	=	=	SYM
iajs-2952	134	8	0	0	NUM
iajs-2952	134	9	.	.	PUNCT
iajs-2952	135	1	then	then	ADV
iajs-2952	135	2	,	,	PUNCT
iajs-2952	135	3	ŵ0	ŵ0	PROPN
iajs-2952	135	4	is	be	AUX
iajs-2952	135	5	dense	dense	ADJ
iajs-2952	135	6	in	in	ADP
iajs-2952	135	7	ŵ.∎	ŵ.∎	PROPN
iajs-2952	135	8	step	step	VERB
iajs-2952	135	9	6	6	NUM
iajs-2952	135	10	:	:	PUNCT
iajs-2952	135	11	proof	proof	NOUN
iajs-2952	135	12	completeness	completeness	NOUN
iajs-2952	135	13	of	of	ADP
iajs-2952	135	14	�	�	PROPN
iajs-2952	135	15	̂	̂	PROPN
iajs-2952	135	16	�	�	PROPN
iajs-2952	135	17	.	.	PUNCT
iajs-2952	136	1	theorem	theorem	NOUN
iajs-2952	136	2	(	(	PUNCT
iajs-2952	136	3	3.8	3.8	NUM
iajs-2952	136	4	):	):	PUNCT
iajs-2952	136	5	if	if	SCONJ
iajs-2952	136	6	w	w	NOUN
iajs-2952	136	7	is	be	AUX
iajs-2952	136	8	a	a	DET
iajs-2952	136	9	generalized	generalized	ADJ
iajs-2952	136	10	2	2	NUM
iajs-2952	136	11	-	-	PUNCT
iajs-2952	136	12	inner	inner	ADJ
iajs-2952	136	13	product	product	NOUN
iajs-2952	136	14	space	space	NOUN
iajs-2952	136	15	,	,	PUNCT
iajs-2952	136	16	then	then	ADV
iajs-2952	136	17	ŵ	ŵ	PROPN
iajs-2952	136	18	is	be	AUX
iajs-2952	136	19	complete	complete	ADJ
iajs-2952	136	20	.	.	PUNCT
iajs-2952	137	1	proof	proof	NOUN
iajs-2952	137	2	:	:	PUNCT
iajs-2952	137	3	let	let	VERB
iajs-2952	137	4	{	{	PUNCT
iajs-2952	137	5	ŵn	ŵn	NOUN
iajs-2952	137	6	}	}	PUNCT
iajs-2952	137	7	be	be	AUX
iajs-2952	137	8	a	a	DET
iajs-2952	137	9	b	b	NOUN
iajs-2952	137	10	-	-	PUNCT
iajs-2952	137	11	cauchy	cauchy	ADJ
iajs-2952	137	12	sequence	sequence	NOUN
iajs-2952	137	13	in	in	ADP
iajs-2952	137	14	ŵ.	ŵ.	NOUN
iajs-2952	137	15	since	since	SCONJ
iajs-2952	137	16	ŵ0	ŵ0	PROPN
iajs-2952	137	17	is	be	AUX
iajs-2952	137	18	dense	dense	ADJ
iajs-2952	137	19	in	in	ADP
iajs-2952	137	20	ŵ	ŵ	PROPN
iajs-2952	137	21	,	,	PUNCT
iajs-2952	137	22	thus	thus	ADV
iajs-2952	137	23	there	there	PRON
iajs-2952	137	24	exists	exist	VERB
iajs-2952	137	25	{	{	PUNCT
iajs-2952	137	26	ẑn	ẑn	NOUN
iajs-2952	137	27	}	}	SYM
iajs-2952	137	28	∈	∈	NOUN
iajs-2952	137	29	ŵ0	ŵ0	NOUN
iajs-2952	137	30	such	such	ADJ
iajs-2952	137	31	that	that	SCONJ
iajs-2952	137	32	‖ŵn	‖ŵn	NOUN
iajs-2952	137	33	−	−	PROPN
iajs-2952	137	34	ẑn‖b̂	ẑn‖b̂	NOUN
iajs-2952	137	35	=	=	NOUN
iajs-2952	137	36	0	0	PROPN
iajs-2952	137	37	.	.	PUNCT
iajs-2952	138	1	but	but	CCONJ
iajs-2952	138	2	‖ẑn	‖ẑn	X
iajs-2952	139	1	−	−	PROPN
iajs-2952	139	2	ẑm‖b̂	ẑm‖b̂	PROPN
iajs-2952	139	3	≤	≤	NOUN
iajs-2952	139	4	‖ẑn	‖ẑn	ADV
iajs-2952	140	1	−	−	ADP
iajs-2952	140	2	�	�	PROPN
iajs-2952	140	3	̂	̂	NOUN
iajs-2952	140	4	�	�	NOUN
iajs-2952	140	5	n‖b̂	n‖b̂	NOUN
iajs-2952	140	6	+	+	NUM
iajs-2952	140	7	‖ŵn	‖ŵn	NOUN
iajs-2952	140	8	−	−	NUM
iajs-2952	140	9	ŵm‖b̂	ŵm‖b̂	PROPN
iajs-2952	140	10	+	+	CCONJ
iajs-2952	140	11	‖ŵm	‖ŵm	VERB
iajs-2952	140	12	−	−	PROPN
iajs-2952	140	13	ẑm‖b̂.	ẑm‖b̂.	PROPN
iajs-2952	140	14	then	then	ADV
iajs-2952	140	15	,	,	PUNCT
iajs-2952	140	16	by	by	ADP
iajs-2952	140	17	equation	equation	NOUN
iajs-2952	140	18	(	(	PUNCT
iajs-2952	140	19	3	3	NUM
iajs-2952	140	20	)	)	PUNCT
iajs-2952	140	21	and	and	CCONJ
iajs-2952	140	22	definition	definition	NOUN
iajs-2952	140	23	(	(	PUNCT
iajs-2952	140	24	1.1	1.1	NUM
iajs-2952	140	25	)	)	PUNCT
iajs-2952	140	26	and	and	CCONJ
iajs-2952	140	27	if	if	SCONJ
iajs-2952	140	28	we	we	PRON
iajs-2952	140	29	take	take	VERB
iajs-2952	140	30	n	n	PRON
iajs-2952	140	31	,	,	PUNCT
iajs-2952	140	32	m	m	PROPN
iajs-2952	140	33	→	→	SYM
iajs-2952	140	34	∞	∞	PROPN
iajs-2952	140	35	,	,	PUNCT
iajs-2952	140	36	we	we	PRON
iajs-2952	140	37	get	get	VERB
iajs-2952	140	38	lim	lim	PROPN
iajs-2952	140	39	𝑛,𝑚→∞	𝑛,𝑚→∞	PROPN
iajs-2952	140	40	‖ẑn	‖ẑn	ADV
iajs-2952	141	1	−	−	ADP
iajs-2952	141	2	ẑm‖b̂	ẑm‖b̂	NOUN
iajs-2952	141	3	=	=	SYM
iajs-2952	141	4	0	0	PROPN
iajs-2952	141	5	,	,	PUNCT
iajs-2952	141	6	it	it	PRON
iajs-2952	141	7	implies	imply	VERB
iajs-2952	141	8	that	that	SCONJ
iajs-2952	141	9	{	{	PUNCT
iajs-2952	141	10	ẑn	ẑn	X
iajs-2952	141	11	}	}	PUNCT
iajs-2952	141	12	is	be	AUX
iajs-2952	141	13	a	a	DET
iajs-2952	141	14	b	b	PROPN
iajs-2952	141	15	-	-	PUNCT
iajs-2952	141	16	cauchy	cauchy	ADJ
iajs-2952	141	17	sequence	sequence	NOUN
iajs-2952	141	18	in	in	ADP
iajs-2952	141	19	ŵ0	ŵ0	PROPN
iajs-2952	141	20	.	.	PUNCT
iajs-2952	142	1	but	but	CCONJ
iajs-2952	142	2	w	w	PROPN
iajs-2952	142	3	and	and	CCONJ
iajs-2952	142	4	ŵ	ŵ	PROPN
iajs-2952	142	5	are	be	AUX
iajs-2952	142	6	isometric	isometric	ADJ
iajs-2952	142	7	.	.	PUNCT
iajs-2952	143	1	thus	thus	ADV
iajs-2952	143	2	,	,	PUNCT
iajs-2952	143	3	there	there	PRON
iajs-2952	143	4	exists	exist	VERB
iajs-2952	143	5	a	a	DET
iajs-2952	143	6	b	b	NOUN
iajs-2952	143	7	-	-	PUNCT
iajs-2952	143	8	cauchy	cauchy	ADJ
iajs-2952	143	9	sequence	sequence	NOUN
iajs-2952	143	10	{	{	PUNCT
iajs-2952	143	11	zn	zn	NOUN
iajs-2952	143	12	}	}	PUNCT
iajs-2952	143	13	in	in	ADP
iajs-2952	143	14	w	w	ADP
iajs-2952	143	15	which	which	PRON
iajs-2952	143	16	is	be	AUX
iajs-2952	143	17	contained	contain	VERB
iajs-2952	143	18	in	in	ADP
iajs-2952	143	19	an	an	DET
iajs-2952	143	20	equivalent	equivalent	ADJ
iajs-2952	143	21	class	class	NOUN
iajs-2952	143	22	in	in	ADP
iajs-2952	143	23	ŵ	ŵ	PROPN
iajs-2952	143	24	,	,	PUNCT
iajs-2952	143	25	say	say	VERB
iajs-2952	143	26	ŵ.	ŵ.	PROPN
iajs-2952	143	27	note	note	VERB
iajs-2952	143	28	that	that	SCONJ
iajs-2952	143	29	,	,	PUNCT
iajs-2952	143	30	‖ŵn	‖ŵn	NOUN
iajs-2952	143	31	−	−	NUM
iajs-2952	143	32	ŵ‖b̂	ŵ‖b̂	NOUN
iajs-2952	143	33	≤	≤	NOUN
iajs-2952	143	34	‖ŵn	‖ŵn	NOUN
iajs-2952	143	35	−	−	PROPN
iajs-2952	143	36	ẑn‖b̂	ẑn‖b̂	NOUN
iajs-2952	143	37	+	+	CCONJ
iajs-2952	143	38	‖ẑn	‖ẑn	ADV
iajs-2952	143	39	−	−	ADP
iajs-2952	143	40	ŵ‖b̂	ŵ‖b̂	NOUN
iajs-2952	143	41	=	=	PUNCT
iajs-2952	143	42	‖ŵn	‖ŵn	NOUN
iajs-2952	143	43	−	−	PROPN
iajs-2952	143	44	ẑn‖b̂	ẑn‖b̂	NOUN
iajs-2952	143	45	+	+	CCONJ
iajs-2952	144	1	‖ẑn	‖ẑn	ADV
iajs-2952	144	2	−	−	PROPN
iajs-2952	144	3	ẑn‖b̂.	ẑn‖b̂.	PROPN
iajs-2952	144	4	thus	thus	ADV
iajs-2952	144	5	,	,	PUNCT
iajs-2952	144	6	lim	lim	PROPN
iajs-2952	144	7	𝑛→∞	𝑛→∞	NUM
iajs-2952	144	8	‖ŵn	‖ŵn	NOUN
iajs-2952	144	9	−	−	X
iajs-2952	144	10	ŵ‖b̂	ŵ‖b̂	NOUN
iajs-2952	144	11	=	=	PUNCT
iajs-2952	144	12	0	0	NUM
iajs-2952	144	13	.	.	PUNCT
iajs-2952	145	1	therefore	therefore	ADV
iajs-2952	145	2	,	,	PUNCT
iajs-2952	145	3	ŵ	ŵ	X
iajs-2952	145	4	is	be	AUX
iajs-2952	145	5	complete.∎	complete.∎	ADJ
iajs-2952	145	6	step7	step7	NOUN
iajs-2952	145	7	:	:	PUNCT
iajs-2952	145	8	proof	proof	ADJ
iajs-2952	145	9	uniqueness	uniqueness	NOUN
iajs-2952	145	10	of	of	ADP
iajs-2952	145	11	�	�	PROPN
iajs-2952	145	12	̂	̂	PROPN
iajs-2952	145	13	�	�	PROPN
iajs-2952	145	14	up	up	ADP
iajs-2952	145	15	to	to	ADP
iajs-2952	145	16	isometrics	isometric	NOUN
iajs-2952	145	17	.	.	PUNCT
iajs-2952	146	1	theorem	theorem	NOUN
iajs-2952	146	2	(	(	PUNCT
iajs-2952	146	3	3.9	3.9	NUM
iajs-2952	146	4	):	):	PUNCT
iajs-2952	146	5	the	the	DET
iajs-2952	146	6	space	space	NOUN
iajs-2952	146	7	ŵ	ŵ	X
iajs-2952	146	8	is	be	AUX
iajs-2952	146	9	unique	unique	ADJ
iajs-2952	146	10	up	up	ADP
iajs-2952	146	11	to	to	ADP
iajs-2952	146	12	isometrics	isometric	NOUN
iajs-2952	146	13	.	.	PUNCT
iajs-2952	147	1	proof	proof	NOUN
iajs-2952	147	2	:	:	PUNCT
iajs-2952	147	3	let	let	VERB
iajs-2952	147	4	ŷ	ŷ	X
iajs-2952	147	5	be	be	AUX
iajs-2952	147	6	another	another	DET
iajs-2952	147	7	completion	completion	NOUN
iajs-2952	147	8	to	to	ADP
iajs-2952	147	9	w	w	VERB
iajs-2952	147	10	with	with	ADP
iajs-2952	147	11	a	a	DET
iajs-2952	147	12	dense	dense	ADJ
iajs-2952	147	13	subset	subset	NOUN
iajs-2952	147	14	ŷ0	ŷ0	NOUN
iajs-2952	147	15	in	in	ADP
iajs-2952	147	16	ŷ.	ŷ.	PROPN
iajs-2952	147	17	then	then	ADV
iajs-2952	147	18	,	,	PUNCT
iajs-2952	147	19	there	there	PRON
iajs-2952	147	20	exists	exist	VERB
iajs-2952	147	21	s	s	NOUN
iajs-2952	147	22	:	:	PUNCT
iajs-2952	147	23	w	w	PROPN
iajs-2952	147	24	→	→	SYM
iajs-2952	147	25	ŷ0	ŷ0	NOUN
iajs-2952	147	26	is	be	AUX
iajs-2952	147	27	isometric	isometric	ADJ
iajs-2952	147	28	by	by	ADP
iajs-2952	147	29	step	step	NOUN
iajs-2952	147	30	4	4	NUM
iajs-2952	147	31	defined	define	VERB
iajs-2952	147	32	by	by	ADP
iajs-2952	147	33	s(w	s(w	NOUN
iajs-2952	147	34	)	)	PUNCT
iajs-2952	147	35	=	=	SYM
iajs-2952	148	1	ŷ	ŷ	NUM
iajs-2952	148	2	=	=	SYM
iajs-2952	148	3	(	(	PUNCT
iajs-2952	148	4	y	y	PROPN
iajs-2952	148	5	,	,	PUNCT
iajs-2952	148	6	y	y	PROPN
iajs-2952	148	7	,	,	PUNCT
iajs-2952	148	8	…	…	PUNCT
iajs-2952	148	9	)	)	PUNCT
iajs-2952	148	10	.	.	PUNCT
iajs-2952	149	1	we	we	PRON
iajs-2952	149	2	will	will	AUX
iajs-2952	149	3	define	define	VERB
iajs-2952	149	4	h	h	NOUN
iajs-2952	149	5	:	:	PUNCT
iajs-2952	149	6	ŵ0	ŵ0	PROPN
iajs-2952	149	7	→	→	PUNCT
iajs-2952	149	8	ŷ0	ŷ0	NOUN
iajs-2952	149	9	by	by	ADP
iajs-2952	149	10	h(ŵ	h(ŵ	PROPN
iajs-2952	149	11	)	)	PUNCT
iajs-2952	149	12	=	=	SYM
iajs-2952	149	13	st−1(ŵ	st−1(ŵ	PROPN
iajs-2952	149	14	)	)	PUNCT
iajs-2952	149	15	.	.	PUNCT
iajs-2952	150	1	it	it	PRON
iajs-2952	150	2	implis	impli	VERB
iajs-2952	150	3	that	that	PRON
iajs-2952	150	4	ŵ0	ŵ0	VERB
iajs-2952	150	5	isometric	isometric	ADJ
iajs-2952	150	6	to	to	ADP
iajs-2952	150	7	ŷ0	ŷ0	PROPN
iajs-2952	150	8	.	.	PUNCT
iajs-2952	151	1	for	for	ADP
iajs-2952	151	2	ŷ1,ŷ2	ŷ1,ŷ2	PROPN
iajs-2952	151	3	in	in	ADP
iajs-2952	151	4	ŷ	ŷ	NUM
iajs-2952	151	5	there	there	PRON
iajs-2952	151	6	exists	exist	VERB
iajs-2952	151	7	b	b	X
iajs-2952	151	8	-	-	PUNCT
iajs-2952	151	9	cauchy	cauchy	ADJ
iajs-2952	151	10	sequences	sequence	NOUN
iajs-2952	151	11	{	{	PUNCT
iajs-2952	151	12	ŷ1n	ŷ1n	NOUN
iajs-2952	151	13	}	}	PUNCT
iajs-2952	151	14	,	,	PUNCT
iajs-2952	151	15	{	{	PUNCT
iajs-2952	151	16	ŷ2n	ŷ2n	VERB
iajs-2952	151	17	}	}	PUNCT
iajs-2952	151	18	in	in	ADP
iajs-2952	151	19	ŷ0	ŷ0	PROPN
iajs-2952	151	20	such	such	ADJ
iajs-2952	151	21	that	that	DET
iajs-2952	151	22	ŷ1n	ŷ1n	PROPN
iajs-2952	151	23	→	→	SYM
iajs-2952	151	24	ŷ1	ŷ1	PROPN
iajs-2952	151	25	and	and	CCONJ
iajs-2952	151	26	ŷ2n	ŷ2n	PROPN
iajs-2952	151	27	→	→	SYM
iajs-2952	151	28	ŷ2	ŷ2	PROPN
iajs-2952	151	29	.	.	PUNCT
iajs-2952	152	1	thus	thus	ADV
iajs-2952	152	2	,	,	PUNCT
iajs-2952	152	3	by	by	ADP
iajs-2952	152	4	equation	equation	NOUN
iajs-2952	152	5	(	(	PUNCT
iajs-2952	152	6	4	4	NUM
iajs-2952	152	7	)	)	PUNCT
iajs-2952	152	8	|‖	|‖	NUM
iajs-2952	152	9	�	�	NOUN
iajs-2952	152	10	̂	̂	SYM
iajs-2952	152	11	�	�	NOUN
iajs-2952	152	12	1	1	NUM
iajs-2952	152	13	−	−	PROPN
iajs-2952	152	14	�	�	PROPN
iajs-2952	152	15	̂	̂	SYM
iajs-2952	152	16	�	�	PROPN
iajs-2952	152	17	2‖	2‖	NUM
iajs-2952	152	18	�	�	PROPN
iajs-2952	152	19	̂	̂	SYM
iajs-2952	152	20	�	�	PROPN
iajs-2952	152	21	−	−	PROPN
iajs-2952	152	22	‖	‖	PROPN
iajs-2952	152	23	�	�	PROPN
iajs-2952	152	24	̂	̂	VERB
iajs-2952	152	25	�	�	NOUN
iajs-2952	152	26	1𝑛	1𝑛	NOUN
iajs-2952	152	27	−	−	PROPN
iajs-2952	152	28	�	�	PROPN
iajs-2952	152	29	̂	̂	SYM
iajs-2952	152	30	�	�	PROPN
iajs-2952	152	31	2𝑛‖𝑏|	2𝑛‖𝑏|	NUM
iajs-2952	152	32	≤	≤	NUM
iajs-2952	152	33	‖	‖	PROPN
iajs-2952	152	34	�	�	PROPN
iajs-2952	152	35	̂	̂	VERB
iajs-2952	152	36	�	�	NOUN
iajs-2952	152	37	1	1	NUM
iajs-2952	152	38	−	−	PROPN
iajs-2952	152	39	�	�	PROPN
iajs-2952	152	40	̂	̂	NOUN
iajs-2952	152	41	�	�	NOUN
iajs-2952	152	42	1𝑛‖𝑏	1𝑛‖𝑏	NUM
iajs-2952	152	43	−	−	PROPN
iajs-2952	152	44	‖	‖	PROPN
iajs-2952	152	45	�	�	PROPN
iajs-2952	152	46	̂	̂	VERB
iajs-2952	152	47	�	�	NOUN
iajs-2952	152	48	2	2	NUM
iajs-2952	152	49	−	−	PROPN
iajs-2952	152	50	�	�	PROPN
iajs-2952	152	51	̂	̂	NOUN
iajs-2952	152	52	�	�	NOUN
iajs-2952	152	53	2𝑛‖𝑏	2𝑛‖𝑏	ADJ
iajs-2952	152	54	→	→	SYM
iajs-2952	152	55	0	0	NUM
iajs-2952	152	56	by	by	ADP
iajs-2952	152	57	taking	take	VERB
iajs-2952	152	58	n	n	PRON
iajs-2952	152	59	→	→	SYM
iajs-2952	152	60	∞	∞	PROPN
iajs-2952	152	61	‖	‖	PROPN
iajs-2952	152	62	�	�	PROPN
iajs-2952	152	63	̂	̂	VERB
iajs-2952	152	64	�	�	NOUN
iajs-2952	152	65	1	1	NUM
iajs-2952	153	1	−	−	PROPN
iajs-2952	153	2	�	�	PROPN
iajs-2952	153	3	̂	̂	SYM
iajs-2952	153	4	�	�	PROPN
iajs-2952	153	5	2‖	2‖	NUM
iajs-2952	153	6	�	�	PROPN
iajs-2952	153	7	̂	̂	SYM
iajs-2952	153	8	�	�	NOUN
iajs-2952	153	9	=	=	SYM
iajs-2952	153	10	lim	lim	PROPN
iajs-2952	153	11	𝑛→∞	𝑛→∞	NUM
iajs-2952	153	12	‖	‖	PROPN
iajs-2952	153	13	�	�	PROPN
iajs-2952	153	14	̂	̂	SYM
iajs-2952	153	15	�	�	NOUN
iajs-2952	153	16	1𝑛	1𝑛	NOUN
iajs-2952	153	17	−	−	PROPN
iajs-2952	153	18	�	�	PROPN
iajs-2952	153	19	̂	̂	NOUN
iajs-2952	153	20	�	�	NOUN
iajs-2952	153	21	2𝑛‖𝑏	2𝑛‖𝑏	ADJ
iajs-2952	153	22	(	(	PUNCT
iajs-2952	153	23	5	5	NUM
iajs-2952	153	24	)	)	PUNCT
iajs-2952	153	25	by	by	ADP
iajs-2952	153	26	the	the	DET
iajs-2952	153	27	same	same	ADJ
iajs-2952	153	28	argument	argument	NOUN
iajs-2952	153	29	‖	‖	PROPN
iajs-2952	153	30	�	�	PROPN
iajs-2952	153	31	̂	̂	VERB
iajs-2952	153	32	�	�	NOUN
iajs-2952	153	33	1	1	NUM
iajs-2952	153	34	−	−	PROPN
iajs-2952	153	35	�	�	PROPN
iajs-2952	153	36	̂	̂	SYM
iajs-2952	153	37	�	�	PROPN
iajs-2952	153	38	2‖	2‖	NUM
iajs-2952	153	39	�	�	PROPN
iajs-2952	153	40	̂	̂	SYM
iajs-2952	153	41	�	�	NOUN
iajs-2952	153	42	=	=	SYM
iajs-2952	153	43	lim	lim	PROPN
iajs-2952	153	44	𝑛→∞	𝑛→∞	NUM
iajs-2952	153	45	‖	‖	PROPN
iajs-2952	153	46	�	�	PROPN
iajs-2952	153	47	̂	̂	SYM
iajs-2952	153	48	�	�	NOUN
iajs-2952	153	49	1𝑛	1𝑛	NOUN
iajs-2952	153	50	−	−	PROPN
iajs-2952	153	51	�	�	PROPN
iajs-2952	153	52	̂	̂	NOUN
iajs-2952	153	53	�	�	NOUN
iajs-2952	153	54	2𝑛‖𝑏	2𝑛‖𝑏	ADJ
iajs-2952	153	55	(	(	PUNCT
iajs-2952	153	56	6	6	NUM
iajs-2952	153	57	)	)	PUNCT
iajs-2952	153	58	ihjpas	ihjpa	NOUN
iajs-2952	153	59	.	.	PUNCT
iajs-2952	154	1	36(1)2023	36(1)2023	NUM
iajs-2952	154	2	317	317	NUM
iajs-2952	154	3	thus	thus	ADV
iajs-2952	154	4	,	,	PUNCT
iajs-2952	154	5	by	by	ADP
iajs-2952	154	6	(	(	PUNCT
iajs-2952	154	7	5	5	NUM
iajs-2952	154	8	)	)	PUNCT
iajs-2952	154	9	and	and	CCONJ
iajs-2952	154	10	(	(	PUNCT
iajs-2952	154	11	6	6	X
iajs-2952	154	12	)	)	PUNCT
iajs-2952	154	13	we	we	PRON
iajs-2952	154	14	get	get	VERB
iajs-2952	154	15	‖	‖	ADJ
iajs-2952	154	16	�	�	PROPN
iajs-2952	154	17	̂	̂	VERB
iajs-2952	154	18	�	�	NOUN
iajs-2952	154	19	1	1	NUM
iajs-2952	154	20	−	−	PROPN
iajs-2952	154	21	�	�	PROPN
iajs-2952	154	22	̂	̂	SYM
iajs-2952	154	23	�	�	PROPN
iajs-2952	154	24	2‖	2‖	NUM
iajs-2952	154	25	�	�	PROPN
iajs-2952	154	26	̂	̂	SYM
iajs-2952	154	27	�	�	NOUN
iajs-2952	154	28	=	=	SYM
iajs-2952	154	29	lim	lim	PROPN
iajs-2952	154	30	𝑛→∞	𝑛→∞	NUM
iajs-2952	154	31	‖	‖	PROPN
iajs-2952	154	32	�	�	PROPN
iajs-2952	154	33	̂	̂	SYM
iajs-2952	154	34	�	�	NOUN
iajs-2952	154	35	1𝑛	1𝑛	NOUN
iajs-2952	154	36	−	−	PROPN
iajs-2952	154	37	�	�	PROPN
iajs-2952	154	38	̂	̂	NOUN
iajs-2952	154	39	�	�	NOUN
iajs-2952	154	40	2𝑛‖𝑏	2𝑛‖𝑏	PROPN
iajs-2952	154	41	=	=	SYM
iajs-2952	154	42	lim	lim	PROPN
iajs-2952	154	43	𝑛→∞	𝑛→∞	NUM
iajs-2952	154	44	‖	‖	PROPN
iajs-2952	154	45	�	�	PROPN
iajs-2952	154	46	̂	̂	SYM
iajs-2952	154	47	�	�	NOUN
iajs-2952	154	48	1𝑛	1𝑛	NOUN
iajs-2952	154	49	−	−	PROPN
iajs-2952	154	50	�	�	PROPN
iajs-2952	154	51	̂	̂	NOUN
iajs-2952	154	52	�	�	NOUN
iajs-2952	154	53	2𝑛‖𝑏	2𝑛‖𝑏	ADJ
iajs-2952	154	54	=	=	SYM
iajs-2952	154	55	‖	‖	PROPN
iajs-2952	154	56	�	�	PROPN
iajs-2952	154	57	̂	̂	VERB
iajs-2952	154	58	�	�	NOUN
iajs-2952	154	59	1	1	NUM
iajs-2952	154	60	−	−	PROPN
iajs-2952	154	61	�	�	PROPN
iajs-2952	154	62	̂	̂	SYM
iajs-2952	154	63	�	�	PROPN
iajs-2952	154	64	2‖	2‖	NUM
iajs-2952	154	65	�	�	PROPN
iajs-2952	154	66	̂	̂	SYM
iajs-2952	154	67	�	�	PROPN
iajs-2952	154	68	it	it	PRON
iajs-2952	154	69	implies	imply	VERB
iajs-2952	154	70	that	that	SCONJ
iajs-2952	154	71	ŵ	ŵ	PROPN
iajs-2952	154	72	is	be	AUX
iajs-2952	154	73	isometric	isometric	ADJ
iajs-2952	154	74	to	to	ADP
iajs-2952	154	75	ŷ.∎	ŷ.∎	NOUN
iajs-2952	154	76	4	4	NUM
iajs-2952	154	77	.	.	PUNCT
iajs-2952	155	1	discussion	discussion	NOUN
iajs-2952	155	2	and	and	CCONJ
iajs-2952	155	3	conclusion	conclusion	NOUN
iajs-2952	155	4	a	a	DET
iajs-2952	155	5	complete	complete	ADJ
iajs-2952	155	6	metric	metric	ADJ
iajs-2952	155	7	space	space	NOUN
iajs-2952	155	8	is	be	AUX
iajs-2952	155	9	a	a	DET
iajs-2952	155	10	well	well	ADV
iajs-2952	155	11	-	-	PUNCT
iajs-2952	155	12	known	know	VERB
iajs-2952	155	13	concept	concept	NOUN
iajs-2952	155	14	.	.	PUNCT
iajs-2952	156	1	every	every	DET
iajs-2952	156	2	non	non	ADJ
iajs-2952	156	3	-	-	ADJ
iajs-2952	156	4	complete	complete	ADJ
iajs-2952	156	5	metric	metric	ADJ
iajs-2952	156	6	space	space	NOUN
iajs-2952	156	7	w	w	NOUN
iajs-2952	156	8	can	can	AUX
iajs-2952	156	9	be	be	AUX
iajs-2952	156	10	built	build	VERB
iajs-2952	156	11	into	into	ADP
iajs-2952	156	12	a	a	DET
iajs-2952	156	13	complete	complete	ADJ
iajs-2952	156	14	metric	metric	ADJ
iajs-2952	156	15	space	space	NOUN
iajs-2952	156	16	ŵ	ŵ	PROPN
iajs-2952	156	17	,	,	PUNCT
iajs-2952	156	18	which	which	PRON
iajs-2952	156	19	is	be	AUX
iajs-2952	156	20	known	know	VERB
iajs-2952	156	21	as	as	ADP
iajs-2952	156	22	a	a	DET
iajs-2952	156	23	completion	completion	NOUN
iajs-2952	156	24	of	of	ADP
iajs-2952	156	25	w.	w.	NOUN
iajs-2952	156	26	in	in	ADP
iajs-2952	156	27	this	this	DET
iajs-2952	156	28	paper	paper	NOUN
iajs-2952	156	29	,	,	PUNCT
iajs-2952	156	30	we	we	PRON
iajs-2952	156	31	construct	construct	VERB
iajs-2952	156	32	equivalent	equivalent	ADJ
iajs-2952	156	33	classes	class	NOUN
iajs-2952	156	34	of	of	ADP
iajs-2952	156	35	b	b	NOUN
iajs-2952	156	36	-	-	PUNCT
iajs-2952	156	37	cauchy	cauchy	ADJ
iajs-2952	156	38	sequences	sequence	NOUN
iajs-2952	156	39	to	to	PART
iajs-2952	156	40	complete	complete	VERB
iajs-2952	156	41	a	a	DET
iajs-2952	156	42	generalized	generalized	ADJ
iajs-2952	156	43	2	2	NUM
iajs-2952	156	44	-	-	PUNCT
iajs-2952	156	45	inner	inner	ADJ
iajs-2952	156	46	product	product	NOUN
iajs-2952	156	47	space	space	NOUN
iajs-2952	156	48	.	.	PUNCT
iajs-2952	157	1	references	reference	NOUN
iajs-2952	157	2	1.anshul	1.anshul	NUM
iajs-2952	157	3	,	,	PUNCT
iajs-2952	157	4	r.	r.	PROPN
iajs-2952	157	5	;	;	PUNCT
iajs-2952	157	6	ravinder	ravinder	PROPN
iajs-2952	157	7	,	,	PUNCT
iajs-2952	157	8	k.	k.	PROPN
iajs-2952	157	9	,	,	PUNCT
iajs-2952	157	10	s.	s.	PROPN
iajs-2952	157	11	,	,	PUNCT
iajs-2952	157	12	;	;	PUNCT
iajs-2952	157	13	sumit	sumit	PROPN
iajs-2952	157	14	,	,	PUNCT
iajs-2952	157	15	c.	c.	PROPN
iajs-2952	157	16	,	,	PUNCT
iajs-2952	157	17	stability	stability	NOUN
iajs-2952	157	18	of	of	ADP
iajs-2952	157	19	complex	complex	ADJ
iajs-2952	157	20	functional	functional	ADJ
iajs-2952	157	21	equations	equation	NOUN
iajs-2952	157	22	in	in	ADP
iajs-2952	157	23	2	2	NUM
iajs-2952	157	24	-	-	PUNCT
iajs-2952	157	25	banach	banach	NOUN
iajs-2952	157	26	spaces	space	NOUN
iajs-2952	157	27	.	.	PUNCT
iajs-2952	158	1	journal	journal	NOUN
iajs-2952	158	2	of	of	ADP
iajs-2952	158	3	mathematical	mathematical	ADJ
iajs-2952	158	4	physics	physics	NOUN
iajs-2952	158	5	,	,	PUNCT
iajs-2952	158	6	analysis	analysis	NOUN
iajs-2952	158	7	,	,	PUNCT
iajs-2952	158	8	geometry	geometry	NOUN
iajs-2952	158	9	.	.	PUNCT
iajs-2952	159	1	2021	2021	NUM
iajs-2952	159	2	,	,	PUNCT
iajs-2952	159	3	17	17	NUM
iajs-2952	159	4	,	,	PUNCT
iajs-2952	159	5	3	3	NUM
iajs-2952	159	6	,	,	PUNCT
iajs-2952	159	7	341–368	341–368	NUM
iajs-2952	159	8	.	.	PUNCT
iajs-2952	160	1	2.bahram	2.bahram	NUM
iajs-2952	160	2	,	,	PUNCT
iajs-2952	160	3	d.	d.	PROPN
iajs-2952	160	4	;	;	PUNCT
iajs-2952	161	1	mohammad	mohammad	PROPN
iajs-2952	161	2	,	,	PUNCT
iajs-2952	161	3	j.	j.	PROPN
iajs-2952	161	4	,	,	PUNCT
iajs-2952	161	5	atomic	atomic	ADJ
iajs-2952	161	6	systems	system	NOUN
iajs-2952	161	7	in	in	ADP
iajs-2952	161	8	2	2	NUM
iajs-2952	161	9	-	-	PUNCT
iajs-2952	161	10	inner	inner	ADJ
iajs-2952	161	11	product	product	NOUN
iajs-2952	161	12	spaces	space	NOUN
iajs-2952	161	13	,	,	PUNCT
iajs-2952	161	14	iranian	iranian	ADJ
iajs-2952	161	15	journal	journal	PROPN
iajs-2952	161	16	of	of	ADP
iajs-2952	161	17	mathematical	mathematical	ADJ
iajs-2952	161	18	sciences	sciences	PROPN
iajs-2952	161	19	and	and	CCONJ
iajs-2952	161	20	informatics	informatic	NOUN
iajs-2952	161	21	.	.	PUNCT
iajs-2952	161	22	2018	2018	NUM
iajs-2952	161	23	,	,	PUNCT
iajs-2952	161	24	13	13	NUM
iajs-2952	161	25	,	,	PUNCT
iajs-2952	161	26	1	1	NUM
iajs-2952	161	27	,	,	PUNCT
iajs-2952	161	28	103	103	NUM
iajs-2952	161	29	-	-	SYM
iajs-2952	161	30	110	110	NUM
iajs-2952	161	31	.	.	PUNCT
iajs-2952	162	1	3.cho	3.cho	NUM
iajs-2952	162	2	,	,	PUNCT
iajs-2952	162	3	y.	y.	PROPN
iajs-2952	162	4	,	,	PUNCT
iajs-2952	162	5	j	j	PROPN
iajs-2952	162	6	;	;	PUNCT
iajs-2952	162	7	freese	freese	PROPN
iajs-2952	162	8	,	,	PUNCT
iajs-2952	162	9	r.	r.	PROPN
iajs-2952	162	10	w.	w.	PROPN
iajs-2952	162	11	,	,	PUNCT
iajs-2952	162	12	geometry	geometry	NOUN
iajs-2952	162	13	of	of	ADP
iajs-2952	162	14	linear	linear	PROPN
iajs-2952	162	15	2	2	NUM
iajs-2952	162	16	-	-	PUNCT
iajs-2952	162	17	normed	norme	VERB
iajs-2952	162	18	spaces	space	NOUN
iajs-2952	162	19	,	,	PUNCT
iajs-2952	162	20	nova	nova	PROPN
iajs-2952	162	21	science	science	NOUN
iajs-2952	162	22	publishers	publisher	NOUN
iajs-2952	162	23	,	,	PUNCT
iajs-2952	162	24	new	new	PROPN
iajs-2952	162	25	york	york	PROPN
iajs-2952	162	26	,	,	PUNCT
iajs-2952	162	27	2001	2001	NUM
iajs-2952	162	28	.	.	PUNCT
iajs-2952	163	1	4.y	4.y	NUM
iajs-2952	163	2	.	.	PUNCT
iajs-2952	164	1	j.	j.	PROPN
iajs-2952	164	2	cho	cho	PROPN
iajs-2952	164	3	,	,	PUNCT
iajs-2952	164	4	m.	m.	NOUN
iajs-2952	164	5	matic	matic	PROPN
iajs-2952	164	6	,	,	PUNCT
iajs-2952	164	7	j.	j.	PROPN
iajs-2952	164	8	e.	e.	PROPN
iajs-2952	164	9	,	,	PUNCT
iajs-2952	164	10	pecaric	pecaric	ADJ
iajs-2952	164	11	,	,	PUNCT
iajs-2952	164	12	on	on	ADP
iajs-2952	164	13	gram	gram	PROPN
iajs-2952	164	14	’s	’s	PART
iajs-2952	164	15	deteminant	deteminant	NOUN
iajs-2952	164	16	in	in	ADP
iajs-2952	164	17	2	2	NUM
iajs-2952	164	18	-	-	PUNCT
iajs-2952	164	19	inner	inner	ADJ
iajs-2952	164	20	product	product	NOUN
iajs-2952	164	21	spaces	space	NOUN
iajs-2952	164	22	,	,	PUNCT
iajs-2952	164	23	j.	j.	PROPN
iajs-2952	164	24	korean	korean	PROPN
iajs-2952	164	25	math	math	PROPN
iajs-2952	164	26	.	.	PUNCT
iajs-2952	165	1	soc	soc	PROPN
iajs-2952	165	2	.	.	PROPN
iajs-2952	165	3	,	,	PUNCT
iajs-2952	165	4	2001	2001	NUM
iajs-2952	165	5	,	,	PUNCT
iajs-2952	165	6	38(4	38(4	NUM
iajs-2952	165	7	)	)	PUNCT
iajs-2952	165	8	,	,	PUNCT
iajs-2952	165	9	1125–1156	1125–1156	NUM
iajs-2952	165	10	.	.	PUNCT
iajs-2952	166	1	5.debnath	5.debnath	NUM
iajs-2952	166	2	,	,	PUNCT
iajs-2952	166	3	p.	p.	NOUN
iajs-2952	166	4	,	,	PUNCT
iajs-2952	166	5	saha	saha	PROPN
iajs-2952	166	6	,	,	PUNCT
iajs-2952	166	7	m.	m.	NOUN
iajs-2952	166	8	,	,	PUNCT
iajs-2952	166	9	categorization	categorization	NOUN
iajs-2952	166	10	of	of	ADP
iajs-2952	166	11	n	n	CCONJ
iajs-2952	166	12	-	-	PUNCT
iajs-2952	166	13	inner	inner	ADJ
iajs-2952	166	14	product	product	NOUN
iajs-2952	166	15	space	space	NOUN
iajs-2952	166	16	.	.	PUNCT
iajs-2952	167	1	asian	asian	ADJ
iajs-2952	167	2	res	re	NOUN
iajs-2952	167	3	.	.	PUNCT
iajs-2952	168	1	j.	j.	PROPN
iajs-2952	168	2	math	math	PROPN
iajs-2952	168	3	.	.	PUNCT
iajs-2952	168	4	2018,11(4	2018,11(4	PROPN
iajs-2952	168	5	)	)	PUNCT
iajs-2952	168	6	,	,	PUNCT
iajs-2952	168	7	1–10	1–10	NOUN
iajs-2952	168	8	.	.	PUNCT
iajs-2952	169	1	6.ghafoor	6.ghafoor	NUM
iajs-2952	169	2	,	,	PUNCT
iajs-2952	169	3	g.	g.	PROPN
iajs-2952	169	4	,	,	PUNCT
iajs-2952	169	5	r.	r.	PROPN
iajs-2952	169	6	;	;	PUNCT
iajs-2952	169	7	jamil	jamil	PROPN
iajs-2952	169	8	,	,	PUNCT
iajs-2952	169	9	z.	z.	PROPN
iajs-2952	169	10	,	,	PUNCT
iajs-2952	169	11	z.	z.	PROPN
iajs-2952	169	12	,	,	PUNCT
iajs-2952	169	13	study	study	NOUN
iajs-2952	169	14	of	of	ADP
iajs-2952	169	15	b	b	PROPN
iajs-2952	169	16	-	-	PUNCT
iajs-2952	169	17	hilbert	hilbert	NOUN
iajs-2952	169	18	spaces	space	NOUN
iajs-2952	169	19	and	and	CCONJ
iajs-2952	169	20	some	some	DET
iajs-2952	169	21	classes	class	NOUN
iajs-2952	169	22	of	of	ADP
iajs-2952	169	23	operators	operator	NOUN
iajs-2952	169	24	,	,	PUNCT
iajs-2952	169	25	university	university	NOUN
iajs-2952	169	26	of	of	ADP
iajs-2952	169	27	baghdad	baghdad	PROPN
iajs-2952	169	28	,	,	PUNCT
iajs-2952	169	29	baghdad	baghdad	PROPN
iajs-2952	169	30	,	,	PUNCT
iajs-2952	169	31	2018	2018	NUM
iajs-2952	169	32	,	,	PUNCT
iajs-2952	169	33	27	27	NUM
iajs-2952	169	34	-	-	SYM
iajs-2952	169	35	28	28	NUM
iajs-2952	169	36	.	.	PUNCT
iajs-2952	170	1	7.kreyszig	7.kreyszig	NUM
iajs-2952	170	2	,	,	PUNCT
iajs-2952	170	3	erwin	erwin	PROPN
iajs-2952	170	4	,	,	PUNCT
iajs-2952	170	5	introductory	introductory	ADJ
iajs-2952	170	6	functional	functional	ADJ
iajs-2952	170	7	analysis	analysis	NOUN
iajs-2952	170	8	with	with	ADP
iajs-2952	170	9	applications	application	NOUN
iajs-2952	170	10	,	,	PUNCT
iajs-2952	170	11	john	john	PROPN
iajs-2952	170	12	wiley	wiley	PROPN
iajs-2952	170	13	and	and	CCONJ
iajs-2952	170	14	sons	son	NOUN
iajs-2952	170	15	,	,	PUNCT
iajs-2952	170	16	new	new	PROPN
iajs-2952	170	17	york	york	PROPN
iajs-2952	170	18	,	,	PUNCT
iajs-2952	170	19	1978	1978	NUM
iajs-2952	170	20	.	.	PUNCT
iajs-2952	171	1	8.prasenjit	8.prasenjit	NUM
iajs-2952	171	2	,	,	PUNCT
iajs-2952	171	3	g.	g.	NOUN
iajs-2952	171	4	,	,	PUNCT
iajs-2952	171	5	frame	frame	NOUN
iajs-2952	171	6	operator	operator	NOUN
iajs-2952	171	7	for	for	ADP
iajs-2952	171	8	k	k	NOUN
iajs-2952	171	9	-	-	NOUN
iajs-2952	171	10	frame	frame	NOUN
iajs-2952	171	11	in	in	ADP
iajs-2952	171	12	2	2	NUM
iajs-2952	171	13	-	-	PUNCT
iajs-2952	171	14	inner	inner	ADJ
iajs-2952	171	15	product	product	NOUN
iajs-2952	171	16	space	space	NOUN
iajs-2952	171	17	,	,	PUNCT
iajs-2952	171	18	international	international	ADJ
iajs-2952	171	19	journal	journal	NOUN
iajs-2952	171	20	of	of	ADP
iajs-2952	171	21	mathematics	mathematics	NOUN
iajs-2952	171	22	trends	trend	NOUN
iajs-2952	171	23	and	and	CCONJ
iajs-2952	171	24	technology	technology	NOUN
iajs-2952	171	25	.	.	PUNCT
iajs-2952	172	1	2021	2021	NUM
iajs-2952	172	2	,	,	PUNCT
iajs-2952	172	3	67	67	NUM
iajs-2952	172	4	.	.	PUNCT
iajs-2952	173	1	9.mazaheri	9.mazaheri	NUM
iajs-2952	173	2	,	,	PUNCT
iajs-2952	173	3	h.	h.	PROPN
iajs-2952	173	4	;	;	PUNCT
iajs-2952	173	5	kazemi	kazemi	PROPN
iajs-2952	173	6	,	,	PUNCT
iajs-2952	173	7	r.	r.	PROPN
iajs-2952	173	8	,	,	PUNCT
iajs-2952	173	9	some	some	PRON
iajs-2952	173	10	results	result	VERB
iajs-2952	173	11	on	on	ADP
iajs-2952	173	12	2	2	NUM
iajs-2952	173	13	-	-	PUNCT
iajs-2952	173	14	inner	inner	ADJ
iajs-2952	173	15	product	product	NOUN
iajs-2952	173	16	spaces	space	NOUN
iajs-2952	173	17	,	,	PUNCT
iajs-2952	173	18	nove	nove	PROPN
iajs-2952	173	19	sad	sad	ADJ
iajs-2952	173	20	.	.	PUNCT
iajs-2952	174	1	j.math	j.math	PROPN
iajs-2952	174	2	.	.	PROPN
iajs-2952	175	1	2007	2007	NUM
iajs-2952	175	2	,	,	PUNCT
iajs-2952	175	3	37	37	NUM
iajs-2952	175	4	,	,	PUNCT
iajs-2952	175	5	35	35	NUM
iajs-2952	175	6	-	-	SYM
iajs-2952	175	7	40	40	NUM
iajs-2952	175	8	.	.	PUNCT
iajs-2952	176	1	10.riyas	10.riyas	NUM
iajs-2952	176	2	,	,	PUNCT
iajs-2952	176	3	p.	p.	NOUN
iajs-2952	176	4	,	,	PUNCT
iajs-2952	176	5	;	;	PUNCT
iajs-2952	176	6	ravindran	ravindran	NOUN
iajs-2952	176	7	,	,	PUNCT
iajs-2952	176	8	k.	k.	PROPN
iajs-2952	176	9	,	,	PUNCT
iajs-2952	176	10	t.	t.	PROPN
iajs-2952	176	11	,	,	PUNCT
iajs-2952	176	12	riesz	riesz	VERB
iajs-2952	176	13	theorems	theorem	NOUN
iajs-2952	176	14	and	and	CCONJ
iajs-2952	176	15	~adjoint	~adjoint	ADP
iajs-2952	176	16	operators	operator	NOUN
iajs-2952	176	17	on	on	ADP
iajs-2952	176	18	generalized	generalized	ADJ
iajs-2952	176	19	2	2	NUM
iajs-2952	176	20	-	-	PUNCT
iajs-2952	176	21	inner	inner	ADJ
iajs-2952	176	22	product	product	NOUN
iajs-2952	176	23	spaces	space	NOUN
iajs-2952	176	24	,	,	PUNCT
iajs-2952	176	25	global	global	ADJ
iajs-2952	176	26	journal	journal	NOUN
iajs-2952	176	27	mathematics	mathematic	NOUN
iajs-2952	176	28	,	,	PUNCT
iajs-2952	176	29	2015	2015	NUM
iajs-2952	176	30	,	,	PUNCT
iajs-2952	176	31	3	3	NUM
iajs-2952	176	32	,	,	PUNCT
iajs-2952	176	33	1	1	NUM
iajs-2952	176	34	,	,	PUNCT
iajs-2952	176	35	may	may	AUX
iajs-2952	176	36	18	18	NUM
iajs-2952	176	37	,	,	PUNCT
iajs-2952	176	38	244	244	NUM
iajs-2952	176	39	-	-	SYM
iajs-2952	176	40	254	254	NUM
iajs-2952	176	41	.	.	PUNCT
iajs-2952	177	1	11.sibel	11.sibel	PROPN
iajs-2952	177	2	,	,	PUNCT
iajs-2952	177	3	e.	e.	PROPN
iajs-2952	177	4	ideal	ideal	PROPN
iajs-2952	177	5	strong	strong	ADJ
iajs-2952	177	6	lacunary	lacunary	ADJ
iajs-2952	177	7	quasi	quasi	NOUN
iajs-2952	177	8	cauchy	cauchy	NOUN
iajs-2952	177	9	sequences	sequence	NOUN
iajs-2952	177	10	in	in	ADP
iajs-2952	177	11	2	2	NUM
iajs-2952	177	12	-	-	PUNCT
iajs-2952	177	13	normed	norme	VERB
iajs-2952	177	14	spaces	space	NOUN
iajs-2952	177	15	.	.	PUNCT
iajs-2952	178	1	aip	aip	PROPN
iajs-2952	178	2	conference	conference	NOUN
iajs-2952	178	3	proceedings	proceeding	NOUN
iajs-2952	178	4	2334	2334	NUM
iajs-2952	178	5	,	,	PUNCT
iajs-2952	178	6	040004	040004	NUM
iajs-2952	178	7	.2021	.2021	NOUN
iajs-2952	178	8	.	.	PUNCT
iajs-2952	179	1	12.vijayakumar	12.vijayakumar	NUM
iajs-2952	179	2	,	,	PUNCT
iajs-2952	179	3	s.	s.	PROPN
iajs-2952	179	4	;	;	PUNCT
iajs-2952	179	5	baskaran	baskaran	PROPN
iajs-2952	179	6	b.	b.	PROPN
iajs-2952	179	7	,	,	PUNCT
iajs-2952	179	8	a	a	DET
iajs-2952	179	9	characterization	characterization	NOUN
iajs-2952	179	10	of	of	ADP
iajs-2952	179	11	2	2	NUM
iajs-2952	179	12	-	-	PUNCT
iajs-2952	179	13	inner	inner	ADJ
iajs-2952	179	14	product	product	NOUN
iajs-2952	179	15	spaces	space	VERB
iajs-2952	179	16	.	.	PUNCT
iajs-2952	180	1	aip	aip	PROPN
iajs-2952	180	2	conference	conference	NOUN
iajs-2952	180	3	proceedings	proceeding	NOUN
iajs-2952	180	4	2282	2282	NUM
iajs-2952	180	5	,	,	PUNCT
iajs-2952	180	6	020040	020040	NUM
iajs-2952	180	7	.2020	.2020	VERB
iajs-2952	180	8	https://aip.scitation.org/author/vijayakumar%2c+s	https://aip.scitation.org/author/vijayakumar%2c+s	PROPN
iajs-2952	180	9	https://aip.scitation.org/author/baskaran%2c+b	https://aip.scitation.org/author/baskaran%2c+b	PROPN