id	sid	tid	token	lemma	pos
ma-199	1	1	2024	2024	NUM
ma-199	1	2	ada	ada	PROPN
ma-199	1	3	academica	academica	PROPN
ma-199	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-199	1	5	.	.	PUNCT
ma-199	2	1	j.	j.	PROPN
ma-199	2	2	math	math	PROPN
ma-199	2	3	.	.	PUNCT
ma-199	3	1	anal	anal	ADJ
ma-199	3	2	.	.	PUNCT
ma-199	4	1	4	4	NUM
ma-199	4	2	(	(	PUNCT
ma-199	4	3	2024	2024	NUM
ma-199	4	4	)	)	PUNCT
ma-199	5	1	1doi	1doi	NUM
ma-199	5	2	:	:	PUNCT
ma-199	5	3	10.28924	10.28924	NUM
ma-199	5	4	/	/	SYM
ma-199	5	5	ada	ada	PROPN
ma-199	5	6	/	/	SYM
ma-199	5	7	ma.4.1	ma.4.1	PROPN
ma-199	5	8	frame	frame	NOUN
ma-199	5	9	operators	operator	NOUN
ma-199	5	10	for	for	ADP
ma-199	5	11	frames	frame	NOUN
ma-199	5	12	in	in	ADP
ma-199	5	13	krein	krein	PROPN
ma-199	5	14	spaces	space	NOUN
ma-199	5	15	shah	shah	PROPN
ma-199	5	16	jahan1	jahan1	PROPN
ma-199	5	17	,	,	PUNCT
ma-199	5	18	p.	p.	PROPN
ma-199	5	19	sam	sam	PROPN
ma-199	6	1	johnson2,∗	johnson2,∗	PROPN
ma-199	6	2	1department	1department	NUM
ma-199	6	3	of	of	ADP
ma-199	6	4	mathematics	mathematic	NOUN
ma-199	6	5	,	,	PUNCT
ma-199	6	6	central	central	ADJ
ma-199	6	7	university	university	PROPN
ma-199	6	8	of	of	ADP
ma-199	6	9	haryana	haryana	PROPN
ma-199	6	10	,	,	PUNCT
ma-199	6	11	haryana	haryana	PROPN
ma-199	6	12	123029	123029	NUM
ma-199	6	13	,	,	PUNCT
ma-199	7	1	india	india	PROPN
ma-199	7	2	shahjahan@cuh.ac.in	shahjahan@cuh.ac.in	PROPN
ma-199	8	1	2department	2department	NUM
ma-199	8	2	of	of	ADP
ma-199	8	3	mathematical	mathematical	ADJ
ma-199	8	4	and	and	CCONJ
ma-199	8	5	computational	computational	ADJ
ma-199	8	6	sciences	science	NOUN
ma-199	8	7	,	,	PUNCT
ma-199	8	8	national	national	PROPN
ma-199	8	9	institute	institute	PROPN
ma-199	8	10	of	of	ADP
ma-199	8	11	technology	technology	PROPN
ma-199	8	12	karnataka	karnataka	PROPN
ma-199	8	13	,	,	PUNCT
ma-199	8	14	surathkal	surathkal	PROPN
ma-199	8	15	575025	575025	NUM
ma-199	8	16	,	,	PUNCT
ma-199	8	17	india	india	PROPN
ma-199	8	18	sam@nitk.edu.in	sam@nitk.edu.in	PROPN
ma-199	8	19	∗correspondence	∗correspondence	NOUN
ma-199	8	20	:	:	PUNCT
ma-199	8	21	sam@nitk.edu.in	sam@nitk.edu.in	X
ma-199	8	22	abstract	abstract	ADJ
ma-199	8	23	.	.	PUNCT
ma-199	9	1	in	in	ADP
ma-199	9	2	recent	recent	ADJ
ma-199	9	3	years	year	NOUN
ma-199	9	4	,	,	PUNCT
ma-199	9	5	frames	frame	NOUN
ma-199	9	6	in	in	ADP
ma-199	9	7	krein	krein	ADJ
ma-199	9	8	spaces	space	NOUN
ma-199	9	9	and	and	CCONJ
ma-199	9	10	several	several	ADJ
ma-199	9	11	generalizations	generalization	NOUN
ma-199	9	12	have	have	AUX
ma-199	9	13	been	be	AUX
ma-199	9	14	extensivelystudied	extensivelystudie	VERB
ma-199	9	15	.	.	PUNCT
ma-199	10	1	in	in	ADP
ma-199	10	2	this	this	DET
ma-199	10	3	paper	paper	NOUN
ma-199	10	4	,	,	PUNCT
ma-199	10	5	we	we	PRON
ma-199	10	6	propose	propose	VERB
ma-199	10	7	an	an	DET
ma-199	10	8	alternative	alternative	ADJ
ma-199	10	9	way	way	NOUN
ma-199	10	10	of	of	ADP
ma-199	10	11	looking	look	VERB
ma-199	10	12	at	at	ADP
ma-199	10	13	the	the	DET
ma-199	10	14	notion	notion	NOUN
ma-199	10	15	of	of	ADP
ma-199	10	16	frames	frame	NOUN
ma-199	10	17	in	in	ADP
ma-199	10	18	kreinspaces	kreinspace	NOUN
ma-199	10	19	and	and	CCONJ
ma-199	10	20	give	give	VERB
ma-199	10	21	a	a	DET
ma-199	10	22	necessary	necessary	ADJ
ma-199	10	23	and	and	CCONJ
ma-199	10	24	sufficient	sufficient	ADJ
ma-199	10	25	condition	condition	NOUN
ma-199	10	26	for	for	ADP
ma-199	10	27	a	a	DET
ma-199	10	28	sequence	sequence	NOUN
ma-199	10	29	in	in	ADP
ma-199	10	30	a	a	DET
ma-199	10	31	krein	krein	ADJ
ma-199	10	32	space	space	NOUN
ma-199	10	33	to	to	PART
ma-199	10	34	be	be	AUX
ma-199	10	35	a	a	DET
ma-199	10	36	besselsequence	besselsequence	NOUN
ma-199	10	37	.	.	PUNCT
ma-199	11	1	we	we	PRON
ma-199	11	2	observe	observe	VERB
ma-199	11	3	that	that	SCONJ
ma-199	11	4	a	a	DET
ma-199	11	5	subsequence	subsequence	NOUN
ma-199	11	6	of	of	ADP
ma-199	11	7	a	a	DET
ma-199	11	8	frame	frame	NOUN
ma-199	11	9	in	in	ADP
ma-199	11	10	a	a	DET
ma-199	11	11	krein	krein	NOUN
ma-199	11	12	space	space	NOUN
ma-199	11	13	need	need	AUX
ma-199	11	14	not	not	PART
ma-199	11	15	be	be	AUX
ma-199	11	16	a	a	DET
ma-199	11	17	frame	frame	NOUN
ma-199	11	18	.	.	PUNCT
ma-199	12	1	also	also	ADV
ma-199	12	2	,	,	PUNCT
ma-199	12	3	two	two	NUM
ma-199	12	4	complementary	complementary	ADJ
ma-199	12	5	subsequences	subsequence	NOUN
ma-199	12	6	are	be	AUX
ma-199	12	7	considered	consider	VERB
ma-199	12	8	in	in	ADP
ma-199	12	9	which	which	PRON
ma-199	12	10	one	one	NUM
ma-199	12	11	of	of	ADP
ma-199	12	12	them	they	PRON
ma-199	12	13	is	be	AUX
ma-199	12	14	a	a	DET
ma-199	12	15	frame	frame	NOUN
ma-199	12	16	for	for	ADP
ma-199	12	17	a	a	DET
ma-199	12	18	krein	krein	NOUN
ma-199	12	19	space.we	space.we	PRON
ma-199	12	20	obtain	obtain	VERB
ma-199	12	21	necessary	necessary	ADJ
ma-199	12	22	and	and	CCONJ
ma-199	12	23	sufficient	sufficient	ADJ
ma-199	12	24	conditions	condition	NOUN
ma-199	12	25	under	under	ADP
ma-199	12	26	which	which	PRON
ma-199	12	27	the	the	DET
ma-199	12	28	other	other	ADJ
ma-199	12	29	one	one	NOUN
ma-199	12	30	is	be	AUX
ma-199	12	31	also	also	ADV
ma-199	12	32	a	a	DET
ma-199	12	33	frame	frame	NOUN
ma-199	12	34	for	for	ADP
ma-199	12	35	the	the	DET
ma-199	12	36	kreinspace	kreinspace	NOUN
ma-199	12	37	.	.	PUNCT
ma-199	13	1	1	1	X
ma-199	13	2	.	.	X
ma-199	13	3	introduction	introduction	NOUN
ma-199	13	4	hilbert	hilbert	PROPN
ma-199	13	5	space	space	NOUN
ma-199	13	6	frames	frame	NOUN
ma-199	13	7	were	be	AUX
ma-199	13	8	originally	originally	ADV
ma-199	13	9	introduced	introduce	VERB
ma-199	13	10	by	by	ADP
ma-199	13	11	duffin	duffin	PROPN
ma-199	13	12	and	and	CCONJ
ma-199	13	13	schaeffer	schaeffer	VERB
ma-199	13	14	[	[	X
ma-199	13	15	7	7	NUM
ma-199	13	16	]	]	PUNCT
ma-199	13	17	to	to	PART
ma-199	13	18	deal	deal	VERB
ma-199	13	19	with	with	ADP
ma-199	13	20	someproblems	someproblem	NOUN
ma-199	13	21	in	in	ADP
ma-199	13	22	non	non	ADJ
ma-199	13	23	-	-	ADJ
ma-199	13	24	harmonic	harmonic	ADJ
ma-199	13	25	fourier	fourier	NOUN
ma-199	13	26	analysis	analysis	NOUN
ma-199	13	27	.	.	PUNCT
ma-199	14	1	the	the	DET
ma-199	14	2	linear	linear	ADJ
ma-199	14	3	independence	independence	NOUN
ma-199	14	4	property	property	NOUN
ma-199	14	5	for	for	ADP
ma-199	14	6	a	a	DET
ma-199	14	7	(	(	PUNCT
ma-199	14	8	hamel	hamel	PROPN
ma-199	14	9	)	)	PUNCT
ma-199	14	10	basis	basis	NOUN
ma-199	14	11	,	,	PUNCT
ma-199	14	12	which	which	PRON
ma-199	14	13	allows	allow	VERB
ma-199	14	14	every	every	DET
ma-199	14	15	vector	vector	NOUN
ma-199	14	16	to	to	PART
ma-199	14	17	be	be	AUX
ma-199	14	18	uniquely	uniquely	ADV
ma-199	14	19	represented	represent	VERB
ma-199	14	20	as	as	ADP
ma-199	14	21	a	a	DET
ma-199	14	22	linear	linear	ADJ
ma-199	14	23	combination	combination	NOUN
ma-199	14	24	is	be	AUX
ma-199	14	25	very	very	ADV
ma-199	14	26	restrictive	restrictive	ADJ
ma-199	14	27	forpractical	forpractical	ADJ
ma-199	14	28	problems	problem	NOUN
ma-199	14	29	.	.	PUNCT
ma-199	15	1	frames	frame	NOUN
ma-199	15	2	allow	allow	VERB
ma-199	15	3	each	each	DET
ma-199	15	4	element	element	NOUN
ma-199	15	5	in	in	ADP
ma-199	15	6	the	the	DET
ma-199	15	7	space	space	NOUN
ma-199	15	8	to	to	PART
ma-199	15	9	be	be	AUX
ma-199	15	10	written	write	VERB
ma-199	15	11	as	as	ADP
ma-199	15	12	a	a	DET
ma-199	15	13	linear	linear	NOUN
ma-199	15	14	combinationof	combinationof	NOUN
ma-199	15	15	the	the	DET
ma-199	15	16	elements	element	NOUN
ma-199	15	17	in	in	ADP
ma-199	15	18	the	the	DET
ma-199	15	19	frame	frame	NOUN
ma-199	15	20	,	,	PUNCT
ma-199	15	21	but	but	CCONJ
ma-199	15	22	linear	linear	ADJ
ma-199	15	23	independence	independence	NOUN
ma-199	15	24	is	be	AUX
ma-199	15	25	not	not	PART
ma-199	15	26	required	require	VERB
ma-199	15	27	.	.	PUNCT
ma-199	16	1	frames	frame	NOUN
ma-199	16	2	can	can	AUX
ma-199	16	3	be	be	AUX
ma-199	16	4	viewed	view	VERB
ma-199	16	5	asredundant	asredundant	ADJ
ma-199	16	6	bases	basis	NOUN
ma-199	16	7	which	which	PRON
ma-199	16	8	are	be	AUX
ma-199	16	9	generalization	generalization	NOUN
ma-199	16	10	of	of	ADP
ma-199	16	11	riesz	riesz	PROPN
ma-199	16	12	bases	basis	NOUN
ma-199	16	13	.	.	PUNCT
ma-199	17	1	this	this	DET
ma-199	17	2	redundancy	redundancy	NOUN
ma-199	17	3	property	property	NOUN
ma-199	17	4	sometimes	sometimes	ADV
ma-199	17	5	isextremely	isextremely	ADV
ma-199	17	6	important	important	ADJ
ma-199	17	7	in	in	ADP
ma-199	17	8	applications	application	NOUN
ma-199	17	9	such	such	ADJ
ma-199	17	10	as	as	ADP
ma-199	17	11	sampling	sample	VERB
ma-199	17	12	theory	theory	NOUN
ma-199	17	13	[	[	X
ma-199	17	14	9	9	NUM
ma-199	17	15	]	]	PUNCT
ma-199	17	16	,	,	PUNCT
ma-199	17	17	filter	filter	NOUN
ma-199	17	18	banks	bank	NOUN
ma-199	17	19	[	[	X
ma-199	17	20	3	3	NUM
ma-199	17	21	]	]	PUNCT
ma-199	17	22	,	,	PUNCT
ma-199	17	23	signal	signal	NOUN
ma-199	17	24	and	and	CCONJ
ma-199	17	25	imageprocessing	imageprocesse	VERB
ma-199	17	26	[	[	X
ma-199	17	27	6	6	NUM
ma-199	17	28	]	]	PUNCT
ma-199	17	29	and	and	CCONJ
ma-199	17	30	so	so	ADV
ma-199	17	31	on	on	ADV
ma-199	17	32	.	.	PUNCT
ma-199	18	1	definition	definition	NOUN
ma-199	18	2	1.1	1.1	NUM
ma-199	18	3	.	.	PUNCT
ma-199	19	1	[	[	X
ma-199	19	2	4	4	X
ma-199	19	3	]	]	PUNCT
ma-199	19	4	let	let	VERB
ma-199	19	5	h	h	PRON
ma-199	19	6	be	be	AUX
ma-199	19	7	a	a	DET
ma-199	19	8	hilbert	hilbert	NOUN
ma-199	19	9	space	space	NOUN
ma-199	20	1	and	and	CCONJ
ma-199	20	2	i	i	PRON
ma-199	20	3	be	be	VERB
ma-199	20	4	a	a	DET
ma-199	20	5	countable	countable	ADJ
ma-199	20	6	index	index	NOUN
ma-199	20	7	set	set	NOUN
ma-199	20	8	.	.	PUNCT
ma-199	21	1	a	a	DET
ma-199	21	2	collection	collection	NOUN
ma-199	21	3	{	{	PUNCT
ma-199	21	4	fn}n∈i	fn}n∈i	NOUN
ma-199	21	5	in	in	ADP
ma-199	21	6	a	a	DET
ma-199	21	7	hilbert	hilbert	NOUN
ma-199	21	8	space	space	NOUN
ma-199	21	9	h	h	NOUN
ma-199	21	10	is	be	AUX
ma-199	21	11	said	say	VERB
ma-199	21	12	to	to	PART
ma-199	21	13	be	be	AUX
ma-199	21	14	a	a	DET
ma-199	21	15	frame	frame	NOUN
ma-199	21	16	for	for	ADP
ma-199	21	17	h	h	NOUN
ma-199	21	18	if	if	SCONJ
ma-199	21	19	there	there	PRON
ma-199	21	20	exist	exist	VERB
ma-199	21	21	a	a	DET
ma-199	21	22	,	,	PUNCT
ma-199	21	23	b	b	X
ma-199	21	24	>	>	X
ma-199	21	25	0	0	NUM
ma-199	21	26	such	such	ADJ
ma-199	21	27	that	that	DET
ma-199	21	28	a‖f	a‖f	NOUN
ma-199	21	29	‖2	‖2	NOUN
ma-199	21	30	≤	≤	NUM
ma-199	21	31	∑	∑	PUNCT
ma-199	21	32	n∈i	n∈i	PROPN
ma-199	21	33	|〈f	|〈f	NOUN
ma-199	21	34	,	,	PUNCT
ma-199	21	35	fn〉|2	fn〉|2	PROPN
ma-199	21	36	≤	≤	PUNCT
ma-199	21	37	b‖f	b‖f	ADJ
ma-199	21	38	‖2	‖2	NOUN
ma-199	21	39	,	,	PUNCT
ma-199	21	40	∀f	∀f	PROPN
ma-199	21	41	∈	∈	PROPN
ma-199	21	42	h.	h.	NOUN
ma-199	21	43	received	receive	VERB
ma-199	21	44	:	:	PUNCT
ma-199	21	45	19	19	NUM
ma-199	21	46	nov	nov	PROPN
ma-199	21	47	2023.2020	2023.2020	NUM
ma-199	21	48	mathematics	mathematics	PROPN
ma-199	21	49	subject	subject	ADJ
ma-199	21	50	classification	classification	NOUN
ma-199	21	51	.	.	PUNCT
ma-199	22	1	42c15	42c15	NUM
ma-199	22	2	,	,	PUNCT
ma-199	22	3	46c05	46c05	NUM
ma-199	22	4	,	,	PUNCT
ma-199	22	5	46c20	46c20	NUM
ma-199	22	6	.	.	PUNCT
ma-199	23	1	key	key	ADJ
ma-199	23	2	words	word	NOUN
ma-199	23	3	and	and	CCONJ
ma-199	23	4	phrases	phrase	NOUN
ma-199	23	5	.	.	PUNCT
ma-199	24	1	krein	krein	ADJ
ma-199	24	2	space	space	NOUN
ma-199	24	3	;	;	PUNCT
ma-199	24	4	bessel	bessel	ADJ
ma-199	24	5	sequence	sequence	NOUN
ma-199	24	6	;	;	PUNCT
ma-199	24	7	frame	frame	NOUN
ma-199	24	8	sequence	sequence	NOUN
ma-199	24	9	;	;	PUNCT
ma-199	24	10	frame	frame	NOUN
ma-199	24	11	operator.1	operator.1	PROPN
ma-199	25	1	https://adac.ee	https://adac.ee	PROPN
ma-199	25	2	https://doi.org/10.28924/ada/ma.4.1	https://doi.org/10.28924/ada/ma.4.1	PROPN
ma-199	25	3	https://orcid.org/0000-0002-5966-9185	https://orcid.org/0000-0002-5966-9185	PROPN
ma-199	25	4	https://orcid.org/0000-0003-3461-5380	https://orcid.org/0000-0003-3461-5380	NOUN
ma-199	25	5	eur	eur	PROPN
ma-199	25	6	.	.	PUNCT
ma-199	26	1	j.	j.	PROPN
ma-199	26	2	math	math	PROPN
ma-199	26	3	.	.	PUNCT
ma-199	27	1	anal	anal	PROPN
ma-199	27	2	.	.	PUNCT
ma-199	28	1	10.28924	10.28924	NUM
ma-199	28	2	/	/	SYM
ma-199	28	3	ada	ada	PROPN
ma-199	28	4	/	/	SYM
ma-199	28	5	ma.4.1	ma.4.1	PROPN
ma-199	28	6	2we	2we	NOUN
ma-199	28	7	now	now	ADV
ma-199	28	8	look	look	VERB
ma-199	28	9	at	at	ADP
ma-199	28	10	the	the	DET
ma-199	28	11	definition	definition	NOUN
ma-199	28	12	of	of	ADP
ma-199	28	13	frame	frame	NOUN
ma-199	28	14	which	which	PRON
ma-199	28	15	is	be	AUX
ma-199	28	16	equivalent	equivalent	ADJ
ma-199	28	17	to	to	PART
ma-199	28	18	perceive	perceive	VERB
ma-199	28	19	as	as	SCONJ
ma-199	28	20	the	the	DET
ma-199	28	21	map	map	NOUN
ma-199	28	22	h	h	NOUN
ma-199	28	23	3	3	NUM
ma-199	28	24	f	f	SYM
ma-199	28	25	7→	7→	NUM
ma-199	28	26	∑	∑	PROPN
ma-199	28	27	n∈i	n∈i	PROPN
ma-199	28	28	〈	〈	PROPN
ma-199	28	29	f	f	X
ma-199	28	30	,	,	PUNCT
ma-199	28	31	fn〉fn	fn〉fn	PROPN
ma-199	28	32	∈	∈	PROPN
ma-199	28	33	h	h	NOUN
ma-199	28	34	(	(	PUNCT
ma-199	28	35	1	1	X
ma-199	28	36	)	)	PUNCT
ma-199	28	37	which	which	PRON
ma-199	28	38	is	be	AUX
ma-199	28	39	a	a	DET
ma-199	28	40	well	well	ADV
ma-199	28	41	-	-	PUNCT
ma-199	28	42	defined	define	VERB
ma-199	28	43	bounded	bound	VERB
ma-199	28	44	positive	positive	ADJ
ma-199	28	45	invertible	invertible	ADJ
ma-199	28	46	operator.the	operator.the	DET
ma-199	28	47	bounded	bounded	ADJ
ma-199	28	48	linear	linear	ADJ
ma-199	28	49	operator	operator	NOUN
ma-199	28	50	s	s	PART
ma-199	28	51	:	:	PUNCT
ma-199	28	52	h	h	NOUN
ma-199	28	53	−→	−→	NOUN
ma-199	28	54	h	h	NOUN
ma-199	28	55	defined	define	VERB
ma-199	28	56	by	by	ADP
ma-199	28	57	sf	sf	NOUN
ma-199	28	58	=	=	PUNCT
ma-199	28	59	∑	∑	PUNCT
ma-199	28	60	n∈i	n∈i	PROPN
ma-199	28	61	〈	〈	PROPN
ma-199	28	62	f	f	PROPN
ma-199	28	63	,	,	PUNCT
ma-199	28	64	fn〉fn	fn〉fn	PROPN
ma-199	28	65	,	,	PUNCT
ma-199	28	66	f	f	PROPN
ma-199	28	67	∈	∈	PROPN
ma-199	28	68	h	h	NOUN
ma-199	28	69	,	,	PUNCT
ma-199	28	70	is	be	AUX
ma-199	28	71	known	know	VERB
ma-199	28	72	as	as	ADP
ma-199	28	73	the	the	DET
ma-199	28	74	frame	frame	NOUN
ma-199	28	75	operator	operator	NOUN
ma-199	28	76	associated	associate	VERB
ma-199	28	77	to	to	ADP
ma-199	28	78	the	the	DET
ma-199	28	79	frame	frame	NOUN
ma-199	28	80	{	{	PUNCT
ma-199	28	81	fn}n∈i	fn}n∈i	NOUN
ma-199	28	82	.	.	PUNCT
ma-199	29	1	this	this	DET
ma-199	29	2	operator	operator	NOUN
ma-199	29	3	s	s	VERB
ma-199	29	4	is	be	AUX
ma-199	29	5	bounded	bounded	ADJ
ma-199	29	6	invert	invert	NOUN
ma-199	29	7	-	-	PUNCT
ma-199	29	8	ible	ible	ADJ
ma-199	29	9	,	,	PUNCT
ma-199	29	10	positive	positive	ADJ
ma-199	29	11	and	and	CCONJ
ma-199	29	12	self	self	NOUN
ma-199	29	13	adjoint	adjoint	NOUN
ma-199	29	14	.	.	PUNCT
ma-199	30	1	it	it	PRON
ma-199	30	2	allows	allow	VERB
ma-199	30	3	to	to	PART
ma-199	30	4	reconstruct	reconstruct	VERB
ma-199	30	5	each	each	DET
ma-199	30	6	vector	vector	NOUN
ma-199	30	7	in	in	ADP
ma-199	30	8	terms	term	NOUN
ma-199	30	9	of	of	ADP
ma-199	30	10	the	the	DET
ma-199	30	11	sequence	sequence	NOUN
ma-199	30	12	{	{	PUNCT
ma-199	30	13	fn}n∈ias	fn}n∈ias	PROPN
ma-199	30	14	follows	follow	VERB
ma-199	30	15	:	:	PUNCT
ma-199	30	16	f	f	X
ma-199	30	17	=	=	SYM
ma-199	30	18	∑	∑	PROPN
ma-199	30	19	n∈i	n∈i	PROPN
ma-199	30	20	〈	〈	PROPN
ma-199	30	21	f	f	X
ma-199	30	22	,	,	PUNCT
ma-199	30	23	s−1fn〉fn	s−1fn〉fn	PUNCT
ma-199	31	1	=	=	PUNCT
ma-199	31	2	∑	∑	PUNCT
ma-199	31	3	n∈i	n∈i	PROPN
ma-199	31	4	〈	〈	PROPN
ma-199	31	5	f	f	PROPN
ma-199	31	6	,	,	PUNCT
ma-199	31	7	fn〉s−1fn	fn〉s−1fn	PROPN
ma-199	31	8	.	.	PUNCT
ma-199	32	1	(	(	PUNCT
ma-199	32	2	2	2	X
ma-199	32	3	)	)	PUNCT
ma-199	32	4	the	the	DET
ma-199	32	5	formula	formula	NOUN
ma-199	32	6	(	(	PUNCT
ma-199	32	7	2	2	X
ma-199	32	8	)	)	PUNCT
ma-199	32	9	is	be	AUX
ma-199	32	10	known	know	VERB
ma-199	32	11	as	as	ADP
ma-199	32	12	reconstruction	reconstruction	NOUN
ma-199	32	13	formula	formula	NOUN
ma-199	32	14	associated	associate	VERB
ma-199	32	15	to	to	ADP
ma-199	32	16	{	{	PUNCT
ma-199	32	17	fn}n∈i	fn}n∈i	X
ma-199	32	18	and	and	CCONJ
ma-199	32	19	if	if	SCONJ
ma-199	32	20	s	s	VERB
ma-199	32	21	=	=	VERB
ma-199	32	22	i	i	PROPN
ma-199	32	23	,	,	PUNCT
ma-199	32	24	then	then	ADV
ma-199	32	25	thereconstruction	thereconstruction	NOUN
ma-199	32	26	formula	formula	NOUN
ma-199	32	27	resembles	resemble	VERB
ma-199	32	28	the	the	DET
ma-199	32	29	fourier	fourier	ADJ
ma-199	32	30	series	series	NOUN
ma-199	32	31	of	of	ADP
ma-199	32	32	f	f	PROPN
ma-199	32	33	associated	associate	VERB
ma-199	32	34	with	with	ADP
ma-199	32	35	the	the	DET
ma-199	32	36	orthonormal	orthonormal	ADJ
ma-199	32	37	sequence	sequence	NOUN
ma-199	32	38	{	{	PUNCT
ma-199	32	39	fn}n∈i	fn}n∈i	NOUN
ma-199	32	40	.the	.the	DET
ma-199	32	41	concept	concept	NOUN
ma-199	32	42	of	of	ADP
ma-199	32	43	indefinite	indefinite	ADJ
ma-199	32	44	inner	inner	ADJ
ma-199	32	45	product	product	NOUN
ma-199	32	46	was	be	AUX
ma-199	32	47	first	first	ADV
ma-199	32	48	found	find	VERB
ma-199	32	49	in	in	ADP
ma-199	32	50	a	a	DET
ma-199	32	51	paper	paper	NOUN
ma-199	32	52	on	on	ADP
ma-199	32	53	quantum	quantum	ADJ
ma-199	32	54	field	field	NOUN
ma-199	32	55	theory	theory	NOUN
ma-199	32	56	bydirac	bydirac	NOUN
ma-199	32	57	in	in	ADP
ma-199	32	58	1942	1942	NUM
ma-199	33	1	[	[	X
ma-199	33	2	5	5	NUM
ma-199	33	3	]	]	PUNCT
ma-199	33	4	.	.	PUNCT
ma-199	34	1	pontrjagin	pontrjagin	NOUN
ma-199	34	2	gave	give	VERB
ma-199	34	3	the	the	DET
ma-199	34	4	mathematical	mathematical	ADJ
ma-199	34	5	interpretation	interpretation	NOUN
ma-199	34	6	of	of	ADP
ma-199	34	7	indefinite	indefinite	ADJ
ma-199	34	8	inner	inner	ADJ
ma-199	34	9	product.giribet	product.giribet	PROPN
ma-199	34	10	et	et	NOUN
ma-199	34	11	al	al	PROPN
ma-199	34	12	.	.	PROPN
ma-199	34	13	have	have	AUX
ma-199	34	14	introduced	introduce	VERB
ma-199	34	15	and	and	CCONJ
ma-199	34	16	studied	study	VERB
ma-199	34	17	frames	frame	NOUN
ma-199	34	18	for	for	ADP
ma-199	34	19	krein	krein	ADJ
ma-199	34	20	spaces	space	NOUN
ma-199	34	21	[	[	X
ma-199	34	22	8	8	NUM
ma-199	34	23	]	]	PUNCT
ma-199	34	24	.	.	PUNCT
ma-199	35	1	motivated	motivate	VERB
ma-199	35	2	by	by	ADP
ma-199	35	3	the	the	DET
ma-199	35	4	equivalentdefinition	equivalentdefinition	NOUN
ma-199	35	5	of	of	ADP
ma-199	35	6	frame	frame	NOUN
ma-199	35	7	as	as	SCONJ
ma-199	35	8	given	give	VERB
ma-199	35	9	in	in	ADP
ma-199	35	10	(	(	PUNCT
ma-199	35	11	1	1	NUM
ma-199	35	12	)	)	PUNCT
ma-199	35	13	,	,	PUNCT
ma-199	35	14	in	in	ADP
ma-199	35	15	this	this	DET
ma-199	35	16	paper	paper	NOUN
ma-199	35	17	,	,	PUNCT
ma-199	35	18	we	we	PRON
ma-199	35	19	propose	propose	VERB
ma-199	35	20	an	an	DET
ma-199	35	21	alternative	alternative	ADJ
ma-199	35	22	way	way	NOUN
ma-199	35	23	of	of	ADP
ma-199	35	24	looking	look	VERB
ma-199	35	25	at	at	ADP
ma-199	35	26	thenotion	thenotion	NOUN
ma-199	35	27	of	of	ADP
ma-199	35	28	frames	frame	NOUN
ma-199	35	29	in	in	ADP
ma-199	35	30	krein	krein	ADJ
ma-199	35	31	spaces	space	NOUN
ma-199	35	32	by	by	ADP
ma-199	35	33	decomposing	decompose	VERB
ma-199	35	34	the	the	DET
ma-199	35	35	index	index	NOUN
ma-199	35	36	set	set	VERB
ma-199	35	37	i	i	PRON
ma-199	35	38	in	in	ADP
ma-199	35	39	a	a	DET
ma-199	35	40	natural	natural	ADJ
ma-199	35	41	way	way	NOUN
ma-199	35	42	and	and	CCONJ
ma-199	35	43	obtain	obtain	VERB
ma-199	35	44	somenew	somenew	ADJ
ma-199	35	45	results	result	NOUN
ma-199	35	46	on	on	ADP
ma-199	35	47	frames	frame	NOUN
ma-199	35	48	sequences.the	sequences.the	DET
ma-199	35	49	paper	paper	NOUN
ma-199	35	50	is	be	AUX
ma-199	35	51	organized	organize	VERB
ma-199	35	52	as	as	SCONJ
ma-199	35	53	follows	follow	VERB
ma-199	35	54	.	.	PUNCT
ma-199	36	1	standard	standard	ADJ
ma-199	36	2	definition	definition	NOUN
ma-199	36	3	of	of	ADP
ma-199	36	4	krein	krein	ADJ
ma-199	36	5	space	space	NOUN
ma-199	36	6	is	be	AUX
ma-199	36	7	given	give	VERB
ma-199	36	8	in	in	ADP
ma-199	36	9	section	section	NOUN
ma-199	36	10	2along	2along	NUM
ma-199	36	11	with	with	ADP
ma-199	36	12	some	some	DET
ma-199	36	13	notations	notation	NOUN
ma-199	36	14	and	and	CCONJ
ma-199	36	15	examples	example	NOUN
ma-199	36	16	which	which	PRON
ma-199	36	17	will	will	AUX
ma-199	36	18	be	be	AUX
ma-199	36	19	used	use	VERB
ma-199	36	20	in	in	ADP
ma-199	36	21	the	the	DET
ma-199	36	22	sequel	sequel	NOUN
ma-199	36	23	.	.	PUNCT
ma-199	37	1	in	in	ADP
ma-199	37	2	section	section	NOUN
ma-199	37	3	3	3	NUM
ma-199	37	4	,	,	PUNCT
ma-199	37	5	we	we	PRON
ma-199	37	6	definethe	definethe	VERB
ma-199	37	7	concept	concept	NOUN
ma-199	37	8	of	of	ADP
ma-199	37	9	bessel	bessel	ADJ
ma-199	37	10	sequence	sequence	NOUN
ma-199	37	11	in	in	ADP
ma-199	37	12	krein	krein	ADJ
ma-199	37	13	spaces	space	NOUN
ma-199	37	14	and	and	CCONJ
ma-199	37	15	give	give	VERB
ma-199	37	16	a	a	DET
ma-199	37	17	necessary	necessary	ADJ
ma-199	37	18	and	and	CCONJ
ma-199	37	19	sufficient	sufficient	ADJ
ma-199	37	20	condition	condition	NOUN
ma-199	37	21	fora	fora	ADJ
ma-199	37	22	sequence	sequence	NOUN
ma-199	37	23	to	to	PART
ma-199	37	24	be	be	AUX
ma-199	37	25	a	a	DET
ma-199	37	26	bessel	bessel	ADJ
ma-199	37	27	sequence	sequence	NOUN
ma-199	37	28	in	in	ADP
ma-199	37	29	krein	krein	ADJ
ma-199	37	30	spaces	space	NOUN
ma-199	37	31	.	.	PUNCT
ma-199	38	1	in	in	ADP
ma-199	38	2	section	section	NOUN
ma-199	38	3	4	4	NUM
ma-199	38	4	,	,	PUNCT
ma-199	38	5	we	we	PRON
ma-199	38	6	give	give	VERB
ma-199	38	7	the	the	DET
ma-199	38	8	definition	definition	NOUN
ma-199	38	9	of	of	ADP
ma-199	38	10	framefor	framefor	ADP
ma-199	38	11	krein	krein	NOUN
ma-199	38	12	space	space	NOUN
ma-199	38	13	and	and	CCONJ
ma-199	38	14	study	study	NOUN
ma-199	38	15	operators	operator	NOUN
ma-199	38	16	associated	associate	VERB
ma-199	38	17	to	to	ADP
ma-199	38	18	the	the	DET
ma-199	38	19	frame	frame	NOUN
ma-199	38	20	.	.	PUNCT
ma-199	39	1	in	in	ADP
ma-199	39	2	the	the	DET
ma-199	39	3	last	last	ADJ
ma-199	39	4	section	section	NOUN
ma-199	39	5	,	,	PUNCT
ma-199	39	6	we	we	PRON
ma-199	39	7	study	study	VERB
ma-199	39	8	framesequences	framesequence	NOUN
ma-199	39	9	in	in	ADP
ma-199	39	10	krein	krein	ADJ
ma-199	39	11	spaces	space	NOUN
ma-199	39	12	.	.	PUNCT
ma-199	40	1	in	in	ADP
ma-199	40	2	general	general	ADJ
ma-199	40	3	,	,	PUNCT
ma-199	40	4	if	if	SCONJ
ma-199	40	5	{	{	PUNCT
ma-199	40	6	fn}n∈i	fn}n∈i	NOUN
ma-199	40	7	is	be	AUX
ma-199	40	8	a	a	DET
ma-199	40	9	frame	frame	NOUN
ma-199	40	10	in	in	ADP
ma-199	40	11	a	a	DET
ma-199	40	12	krein	krein	ADJ
ma-199	40	13	space	space	NOUN
ma-199	40	14	and	and	CCONJ
ma-199	40	15	{	{	PUNCT
ma-199	40	16	nk	nk	NOUN
ma-199	40	17	}	}	PUNCT
ma-199	40	18	is	be	AUX
ma-199	40	19	any	any	DET
ma-199	40	20	infiniteincreasing	infiniteincrease	VERB
ma-199	40	21	sequence	sequence	NOUN
ma-199	40	22	in	in	ADP
ma-199	40	23	i	i	PRON
ma-199	40	24	,	,	PUNCT
ma-199	40	25	then	then	ADV
ma-199	40	26	{	{	PUNCT
ma-199	40	27	fnk	fnk	NOUN
ma-199	40	28	}	}	PUNCT
ma-199	40	29	need	need	AUX
ma-199	40	30	not	not	PART
ma-199	40	31	be	be	AUX
ma-199	40	32	a	a	DET
ma-199	40	33	frame	frame	NOUN
ma-199	40	34	sequence	sequence	NOUN
ma-199	40	35	.	.	PUNCT
ma-199	41	1	we	we	PRON
ma-199	41	2	provide	provide	VERB
ma-199	41	3	some	some	DET
ma-199	41	4	sufficientconditions	sufficientcondition	NOUN
ma-199	41	5	under	under	ADP
ma-199	41	6	which	which	PRON
ma-199	41	7	subsequences	subsequence	VERB
ma-199	41	8	become	become	VERB
ma-199	41	9	frame	frame	NOUN
ma-199	41	10	sequence	sequence	NOUN
ma-199	41	11	for	for	ADP
ma-199	41	12	the	the	DET
ma-199	41	13	krein	krein	ADJ
ma-199	41	14	space	space	NOUN
ma-199	41	15	.	.	PUNCT
ma-199	42	1	2	2	X
ma-199	42	2	.	.	X
ma-199	42	3	preliminaries	preliminary	NOUN
ma-199	42	4	let	let	VERB
ma-199	42	5	k	k	PRON
ma-199	42	6	be	be	AUX
ma-199	42	7	a	a	DET
ma-199	42	8	complex	complex	ADJ
ma-199	42	9	vector	vector	NOUN
ma-199	42	10	space	space	NOUN
ma-199	42	11	with	with	ADP
ma-199	42	12	a	a	DET
ma-199	42	13	hermitian	hermitian	ADJ
ma-199	42	14	sesquilinear	sesquilinear	NOUN
ma-199	42	15	form	form	NOUN
ma-199	42	16	defined	define	VERB
ma-199	42	17	on	on	ADP
ma-199	42	18	it	it	PRON
ma-199	42	19	.	.	PUNCT
ma-199	43	1	then	then	ADV
ma-199	43	2	wecall	wecall	PROPN
ma-199	43	3	(	(	PUNCT
ma-199	43	4	k	k	NOUN
ma-199	43	5	,	,	PUNCT
ma-199	43	6	[	[	X
ma-199	43	7	.	.	PUNCT
ma-199	43	8	,	,	PUNCT
ma-199	43	9	.	.	PUNCT
ma-199	44	1	]	]	X
ma-199	44	2	)	)	PUNCT
ma-199	44	3	an	an	DET
ma-199	44	4	inner	inner	ADJ
ma-199	44	5	product	product	NOUN
ma-199	44	6	space	space	NOUN
ma-199	44	7	.	.	PUNCT
ma-199	45	1	an	an	DET
ma-199	45	2	element	element	NOUN
ma-199	45	3	x	x	SYM
ma-199	45	4	∈	∈	PROPN
ma-199	45	5	k	k	PROPN
ma-199	45	6	is	be	AUX
ma-199	45	7	called	call	VERB
ma-199	45	8	neutral	neutral	ADJ
ma-199	45	9	,	,	PUNCT
ma-199	45	10	positive	positive	ADJ
ma-199	45	11	,	,	PUNCT
ma-199	45	12	or	or	CCONJ
ma-199	45	13	negativeif	negativeif	NOUN
ma-199	46	1	[	[	X
ma-199	46	2	x	x	X
ma-199	46	3	,	,	PUNCT
ma-199	46	4	x	x	X
ma-199	46	5	]	]	X
ma-199	47	1	=	=	SYM
ma-199	47	2	0	0	NUM
ma-199	47	3	,	,	PUNCT
ma-199	47	4	[	[	X
ma-199	47	5	x	x	X
ma-199	47	6	,	,	PUNCT
ma-199	47	7	x	x	X
ma-199	47	8	]	]	PUNCT
ma-199	47	9	>	>	X
ma-199	47	10	0	0	NUM
ma-199	47	11	,	,	PUNCT
ma-199	47	12	or	or	CCONJ
ma-199	47	13	[	[	X
ma-199	47	14	x	x	X
ma-199	47	15	,	,	PUNCT
ma-199	47	16	x	x	X
ma-199	47	17	]	]	PUNCT
ma-199	47	18	<	<	X
ma-199	47	19	0	0	NUM
ma-199	47	20	respectively	respectively	ADV
ma-199	47	21	.	.	PUNCT
ma-199	48	1	if	if	SCONJ
ma-199	48	2	k	k	PROPN
ma-199	48	3	contains	contain	VERB
ma-199	48	4	positive	positive	ADJ
ma-199	48	5	as	as	ADV
ma-199	48	6	well	well	ADV
ma-199	48	7	as	as	ADP
ma-199	48	8	negativeelements	negativeelement	NOUN
ma-199	48	9	,	,	PUNCT
ma-199	48	10	then	then	ADV
ma-199	48	11	it	it	PRON
ma-199	48	12	is	be	AUX
ma-199	48	13	called	call	VERB
ma-199	48	14	an	an	DET
ma-199	48	15	indefinite	indefinite	ADJ
ma-199	48	16	inner	inner	ADJ
ma-199	48	17	product	product	NOUN
ma-199	48	18	space	space	NOUN
ma-199	48	19	,	,	PUNCT
ma-199	48	20	otherwise	otherwise	ADV
ma-199	48	21	it	it	PRON
ma-199	48	22	is	be	AUX
ma-199	48	23	called	call	VERB
ma-199	48	24	a	a	DET
ma-199	48	25	semi	semi	ADJ
ma-199	48	26	-	-	ADJ
ma-199	48	27	definiteinner	definiteinner	ADJ
ma-199	48	28	product	product	NOUN
ma-199	48	29	space	space	NOUN
ma-199	48	30	.	.	PUNCT
ma-199	49	1	we	we	PRON
ma-199	49	2	refer	refer	VERB
ma-199	49	3	[	[	X
ma-199	49	4	1	1	NUM
ma-199	49	5	,	,	PUNCT
ma-199	49	6	2	2	NUM
ma-199	49	7	]	]	PUNCT
ma-199	49	8	for	for	ADP
ma-199	49	9	basics	basic	NOUN
ma-199	49	10	on	on	ADP
ma-199	49	11	indefinite	indefinite	ADJ
ma-199	49	12	inner	inner	ADJ
ma-199	49	13	product	product	NOUN
ma-199	49	14	spaces	space	VERB
ma-199	49	15	.	.	PUNCT
ma-199	50	1	https://doi.org/10.28924/ada/ma.4.1	https://doi.org/10.28924/ada/ma.4.1	NUM
ma-199	50	2	eur	eur	PROPN
ma-199	50	3	.	.	PUNCT
ma-199	51	1	j.	j.	PROPN
ma-199	51	2	math	math	PROPN
ma-199	51	3	.	.	PUNCT
ma-199	52	1	anal	anal	PROPN
ma-199	52	2	.	.	PUNCT
ma-199	53	1	10.28924	10.28924	NUM
ma-199	53	2	/	/	SYM
ma-199	53	3	ada	ada	PROPN
ma-199	53	4	/	/	SYM
ma-199	53	5	ma.4.1	ma.4.1	PROPN
ma-199	53	6	3an	3an	PROPN
ma-199	53	7	indefinite	indefinite	ADJ
ma-199	53	8	inner	inner	ADJ
ma-199	53	9	product	product	NOUN
ma-199	53	10	space	space	NOUN
ma-199	53	11	(	(	PUNCT
ma-199	53	12	k	k	NOUN
ma-199	53	13	,	,	PUNCT
ma-199	53	14	[	[	X
ma-199	53	15	.	.	PUNCT
ma-199	53	16	,	,	PUNCT
ma-199	53	17	.	.	PUNCT
ma-199	54	1	]	]	PUNCT
ma-199	54	2	)	)	PUNCT
ma-199	54	3	is	be	AUX
ma-199	54	4	decomposable	decomposable	ADJ
ma-199	54	5	if	if	SCONJ
ma-199	54	6	it	it	PRON
ma-199	54	7	can	can	AUX
ma-199	54	8	be	be	AUX
ma-199	54	9	written	write	VERB
ma-199	54	10	as	as	ADP
ma-199	54	11	an	an	DET
ma-199	54	12	orthogonaldirect	orthogonaldirect	ADJ
ma-199	54	13	sum	sum	NOUN
ma-199	54	14	of	of	ADP
ma-199	54	15	a	a	DET
ma-199	54	16	neutral	neutral	ADJ
ma-199	54	17	subspace	subspace	NOUN
ma-199	54	18	k0	k0	PROPN
ma-199	54	19	,	,	PUNCT
ma-199	54	20	a	a	DET
ma-199	54	21	positive	positive	ADJ
ma-199	54	22	definite	definite	ADJ
ma-199	54	23	subspace	subspace	NOUN
ma-199	54	24	k+	k+	NOUN
ma-199	54	25	and	and	CCONJ
ma-199	54	26	a	a	DET
ma-199	54	27	negative	negative	ADJ
ma-199	54	28	definitesubspace	definitesubspace	NOUN
ma-199	54	29	k−	k−	PROPN
ma-199	54	30	:	:	PUNCT
ma-199	55	1	k	k	X
ma-199	55	2	=	=	PUNCT
ma-199	55	3	k0[+̇]k+[+̇]k−.	k0[+̇]k+[+̇]k−.	PROPN
ma-199	55	4	(	(	PUNCT
ma-199	55	5	3	3	NUM
ma-199	55	6	)	)	PUNCT
ma-199	55	7	then	then	ADV
ma-199	55	8	(	(	PUNCT
ma-199	55	9	3	3	X
ma-199	55	10	)	)	PUNCT
ma-199	55	11	is	be	AUX
ma-199	55	12	known	know	VERB
ma-199	55	13	as	as	ADP
ma-199	55	14	a	a	DET
ma-199	55	15	fundamental	fundamental	ADJ
ma-199	55	16	decomposition	decomposition	NOUN
ma-199	55	17	of	of	ADP
ma-199	55	18	k.an	k.an	PROPN
ma-199	55	19	indefinite	indefinite	ADJ
ma-199	55	20	inner	inner	ADJ
ma-199	55	21	product	product	NOUN
ma-199	55	22	space	space	NOUN
ma-199	55	23	(	(	PUNCT
ma-199	55	24	k	k	NOUN
ma-199	55	25	,	,	PUNCT
ma-199	55	26	[	[	X
ma-199	55	27	.	.	PUNCT
ma-199	55	28	,	,	PUNCT
ma-199	55	29	.	.	PUNCT
ma-199	56	1	]	]	PUNCT
ma-199	56	2	)	)	PUNCT
ma-199	56	3	is	be	AUX
ma-199	56	4	a	a	DET
ma-199	56	5	krein	krein	ADJ
ma-199	56	6	space	space	NOUN
ma-199	56	7	if	if	SCONJ
ma-199	56	8	it	it	PRON
ma-199	56	9	can	can	AUX
ma-199	56	10	be	be	AUX
ma-199	56	11	written	write	VERB
ma-199	56	12	as	as	ADP
ma-199	56	13	an	an	DET
ma-199	56	14	orthogonaldirect	orthogonaldirect	ADJ
ma-199	56	15	sum	sum	NOUN
ma-199	56	16	of	of	ADP
ma-199	56	17	a	a	DET
ma-199	56	18	positive	positive	ADJ
ma-199	56	19	definite	definite	ADJ
ma-199	56	20	subspace	subspace	NOUN
ma-199	56	21	k+	k+	NOUN
ma-199	56	22	and	and	CCONJ
ma-199	56	23	a	a	DET
ma-199	56	24	negative	negative	ADJ
ma-199	56	25	definite	definite	ADJ
ma-199	56	26	subspace	subspace	NOUN
ma-199	56	27	k−	k−	PROPN
ma-199	56	28	such	such	ADJ
ma-199	56	29	that	that	PRON
ma-199	56	30	(	(	PUNCT
ma-199	56	31	k+	k+	X
ma-199	56	32	,	,	PUNCT
ma-199	56	33	[	[	X
ma-199	56	34	.	.	PUNCT
ma-199	56	35	,	,	PUNCT
ma-199	56	36	.	.	PUNCT
ma-199	57	1	]	]	PUNCT
ma-199	57	2	)	)	PUNCT
ma-199	58	1	and	and	CCONJ
ma-199	58	2	(	(	PUNCT
ma-199	58	3	k−,−	k−,−	NOUN
ma-199	58	4	[	[	X
ma-199	58	5	.	.	PUNCT
ma-199	58	6	,	,	PUNCT
ma-199	58	7	.	.	PUNCT
ma-199	59	1	]	]	PUNCT
ma-199	59	2	)	)	PUNCT
ma-199	59	3	are	be	AUX
ma-199	59	4	hilbert	hilbert	NOUN
ma-199	59	5	spaces	space	NOUN
ma-199	59	6	.	.	PUNCT
ma-199	60	1	let	let	VERB
ma-199	60	2	a	a	DET
ma-199	60	3	fundamental	fundamental	ADJ
ma-199	60	4	decomposition	decomposition	NOUN
ma-199	60	5	of	of	ADP
ma-199	60	6	a	a	DET
ma-199	60	7	krein	krein	NOUN
ma-199	60	8	space	space	NOUN
ma-199	60	9	k	k	PROPN
ma-199	60	10	be	be	AUX
ma-199	60	11	given	give	VERB
ma-199	60	12	by	by	ADP
ma-199	60	13	k	k	PROPN
ma-199	60	14	=	=	PUNCT
ma-199	60	15	k+[+̇]k−	k+[+̇]k−	X
ma-199	60	16	(	(	PUNCT
ma-199	60	17	4	4	NUM
ma-199	60	18	)	)	PUNCT
ma-199	60	19	and	and	CCONJ
ma-199	60	20	p±	p±	AUX
ma-199	60	21	be	be	AUX
ma-199	60	22	the	the	DET
ma-199	60	23	orthogonal	orthogonal	ADJ
ma-199	60	24	projections	projection	NOUN
ma-199	60	25	onto	onto	ADP
ma-199	60	26	k±.	k±.	PROPN
ma-199	60	27	the	the	DET
ma-199	60	28	linear	linear	PROPN
ma-199	60	29	map	map	NOUN
ma-199	60	30	j	j	PROPN
ma-199	61	1	=	=	PRON
ma-199	61	2	p+	p+	PROPN
ma-199	61	3	−	−	PROPN
ma-199	61	4	p−	p−	NOUN
ma-199	61	5	is	be	AUX
ma-199	61	6	called	call	VERB
ma-199	61	7	the	the	DET
ma-199	61	8	fundamental	fundamental	ADJ
ma-199	61	9	symmetry	symmetry	NOUN
ma-199	61	10	corresponding	correspond	VERB
ma-199	61	11	to	to	ADP
ma-199	61	12	(	(	PUNCT
ma-199	61	13	4	4	NUM
ma-199	61	14	)	)	PUNCT
ma-199	61	15	.	.	PUNCT
ma-199	62	1	then	then	ADV
ma-199	62	2	(	(	PUNCT
ma-199	62	3	f	f	X
ma-199	62	4	,	,	PUNCT
ma-199	62	5	g)j	g)j	NOUN
ma-199	62	6	=	=	PUNCT
ma-199	63	1	[	[	X
ma-199	63	2	jf	jf	INTJ
ma-199	63	3	,	,	PUNCT
ma-199	63	4	g	g	NOUN
ma-199	63	5	]	]	X
ma-199	63	6	is	be	AUX
ma-199	63	7	a	a	DET
ma-199	63	8	positive	positive	ADJ
ma-199	63	9	definite	definite	ADJ
ma-199	63	10	inner	inner	ADJ
ma-199	63	11	product	product	NOUN
ma-199	63	12	on	on	ADP
ma-199	63	13	k	k	PROPN
ma-199	63	14	,	,	PUNCT
ma-199	63	15	called	call	VERB
ma-199	63	16	j	j	NOUN
ma-199	63	17	-	-	ADJ
ma-199	63	18	inner	inner	ADJ
ma-199	63	19	product	product	NOUN
ma-199	63	20	corresponding	correspond	VERB
ma-199	63	21	to	to	ADP
ma-199	63	22	the	the	DET
ma-199	63	23	fundamentaldecomposition	fundamentaldecomposition	NOUN
ma-199	63	24	(	(	PUNCT
ma-199	63	25	4	4	NUM
ma-199	63	26	)	)	PUNCT
ma-199	63	27	.	.	PUNCT
ma-199	64	1	we	we	PRON
ma-199	64	2	can	can	AUX
ma-199	64	3	write	write	VERB
ma-199	64	4	(	(	PUNCT
ma-199	64	5	f	f	PROPN
ma-199	64	6	,	,	PUNCT
ma-199	64	7	f	f	PROPN
ma-199	64	8	)	)	PUNCT
ma-199	64	9	j	j	PROPN
ma-199	65	1	=	=	PUNCT
ma-199	66	1	[	[	X
ma-199	66	2	jf	jf	PROPN
ma-199	66	3	,	,	PUNCT
ma-199	66	4	f	f	X
ma-199	66	5	]	]	PUNCT
ma-199	67	1	=	=	PUNCT
ma-199	68	1	[	[	X
ma-199	68	2	(	(	PUNCT
ma-199	68	3	2p	2p	NUM
ma-199	68	4	+	+	CCONJ
ma-199	68	5	−	−	NOUN
ma-199	68	6	i)f	i)f	NOUN
ma-199	68	7	,	,	PUNCT
ma-199	68	8	f	f	X
ma-199	68	9	]	]	X
ma-199	68	10	=	=	SYM
ma-199	68	11	2[p+f	2[p+f	NUM
ma-199	68	12	,	,	PUNCT
ma-199	68	13	p+f	p+f	X
ma-199	68	14	]	]	PUNCT
ma-199	68	15	−	−	PROPN
ma-199	69	1	[	[	X
ma-199	69	2	f	f	X
ma-199	69	3	,	,	PUNCT
ma-199	69	4	f	f	X
ma-199	69	5	]	]	PUNCT
ma-199	69	6	.	.	PUNCT
ma-199	70	1	(	(	PUNCT
ma-199	70	2	5	5	X
ma-199	70	3	)	)	PUNCT
ma-199	70	4	the	the	DET
ma-199	70	5	corresponding	correspond	VERB
ma-199	70	6	norm	norm	NOUN
ma-199	70	7	(	(	PUNCT
ma-199	70	8	called	call	VERB
ma-199	70	9	j	j	NOUN
ma-199	70	10	-	-	NOUN
ma-199	70	11	norm	norm	NOUN
ma-199	70	12	)	)	PUNCT
ma-199	70	13	is	be	AUX
ma-199	70	14	denoted	denote	VERB
ma-199	70	15	by	by	ADP
ma-199	70	16	‖f	‖f	PRON
ma-199	70	17	‖j	‖j	PUNCT
ma-199	71	1	=	=	SYM
ma-199	71	2	(	(	PUNCT
ma-199	71	3	f	f	PROPN
ma-199	71	4	,	,	PUNCT
ma-199	71	5	f	f	PROPN
ma-199	71	6	)	)	PUNCT
ma-199	71	7	1	1	NUM
ma-199	71	8	2	2	NUM
ma-199	71	9	j	j	NOUN
ma-199	71	10	=	=	SYM
ma-199	72	1	[	[	X
ma-199	72	2	jf	jf	PROPN
ma-199	72	3	,	,	PUNCT
ma-199	72	4	f	f	X
ma-199	72	5	]	]	PUNCT
ma-199	72	6	1	1	NUM
ma-199	72	7	2	2	NUM
ma-199	72	8	.	.	PUNCT
ma-199	72	9	example	example	NOUN
ma-199	72	10	2.1	2.1	NUM
ma-199	72	11	.	.	PUNCT
ma-199	73	1	consider	consider	VERB
ma-199	73	2	k	k	NOUN
ma-199	73	3	=	=	PUNCT
ma-199	73	4	`	`	PUNCT
ma-199	73	5	2(n	2(n	NUM
ma-199	73	6	)	)	PUNCT
ma-199	73	7	,	,	PUNCT
ma-199	73	8	the	the	DET
ma-199	73	9	linear	linear	ADJ
ma-199	73	10	space	space	NOUN
ma-199	73	11	of	of	ADP
ma-199	73	12	square	square	ADJ
ma-199	73	13	-	-	PUNCT
ma-199	73	14	summable	summable	ADJ
ma-199	73	15	sequences	sequence	NOUN
ma-199	73	16	,	,	PUNCT
ma-199	73	17	with	with	ADP
ma-199	73	18	[	[	X
ma-199	73	19	f	f	X
ma-199	73	20	,	,	PUNCT
ma-199	73	21	g	g	NOUN
ma-199	73	22	]	]	X
ma-199	73	23	=	=	SYM
ma-199	73	24	∞∑	∞∑	NUM
ma-199	73	25	n=1	n=1	PROPN
ma-199	73	26	(	(	PUNCT
ma-199	73	27	−1)nfngn	−1)nfngn	VERB
ma-199	73	28	for	for	ADP
ma-199	73	29	f	f	PROPN
ma-199	73	30	=	=	SYM
ma-199	73	31	(	(	PUNCT
ma-199	73	32	fn	fn	NOUN
ma-199	73	33	)	)	PUNCT
ma-199	73	34	∞	∞	PROPN
ma-199	73	35	n=1	n=1	PROPN
ma-199	73	36	,	,	PUNCT
ma-199	73	37	g	g	NOUN
ma-199	73	38	=	=	SYM
ma-199	73	39	(	(	PUNCT
ma-199	73	40	gn	gn	PROPN
ma-199	73	41	)	)	PUNCT
ma-199	73	42	∞	∞	PROPN
ma-199	73	43	n=1	n=1	PROPN
ma-199	73	44	∈	∈	PROPN
ma-199	73	45	k.	k.	PROPN
ma-199	73	46	let	let	VERB
ma-199	73	47	k+	k+	NOUN
ma-199	73	48	=	=	PRON
ma-199	73	49	{	{	PUNCT
ma-199	73	50	(	(	PUNCT
ma-199	73	51	fn	fn	NOUN
ma-199	73	52	)	)	PUNCT
ma-199	73	53	∞	∞	PROPN
ma-199	73	54	n=1	n=1	PROPN
ma-199	73	55	:	:	PUNCT
ma-199	73	56	fn	fn	NOUN
ma-199	73	57	=	=	NOUN
ma-199	73	58	0	0	PUNCT
ma-199	74	1	if	if	SCONJ
ma-199	74	2	n	n	NOUN
ma-199	74	3	is	be	AUX
ma-199	74	4	odd	odd	ADJ
ma-199	74	5	}	}	PUNCT
ma-199	74	6	and	and	CCONJ
ma-199	74	7	k−	k−	PROPN
ma-199	75	1	=	=	SYM
ma-199	75	2	{	{	PUNCT
ma-199	75	3	(	(	PUNCT
ma-199	75	4	fn	fn	NOUN
ma-199	75	5	)	)	PUNCT
ma-199	75	6	∞	∞	PROPN
ma-199	75	7	n=1	n=1	PROPN
ma-199	75	8	:	:	PUNCT
ma-199	75	9	fn	fn	NOUN
ma-199	75	10	=	=	NOUN
ma-199	75	11	0	0	PUNCT
ma-199	76	1	if	if	SCONJ
ma-199	76	2	n	n	NOUN
ma-199	76	3	is	be	AUX
ma-199	76	4	even	even	ADV
ma-199	76	5	}	}	PUNCT
ma-199	76	6	.	.	PUNCT
ma-199	77	1	then	then	ADV
ma-199	77	2	k	k	PROPN
ma-199	77	3	=	=	PUNCT
ma-199	77	4	k+[+̇]k−	k+[+̇]k−	NOUN
ma-199	77	5	,	,	PUNCT
ma-199	77	6	where	where	SCONJ
ma-199	77	7	k+	k+	NOUN
ma-199	77	8	and	and	CCONJ
ma-199	77	9	k−	k−	PROPN
ma-199	77	10	are	be	AUX
ma-199	77	11	complete	complete	ADJ
ma-199	77	12	with	with	ADP
ma-199	77	13	respect	respect	NOUN
ma-199	77	14	to	to	ADP
ma-199	77	15	the	the	DET
ma-199	77	16	induced	induced	ADJ
ma-199	77	17	norm	norm	NOUN
ma-199	77	18	and	and	CCONJ
ma-199	77	19	hence	hence	ADV
ma-199	77	20	k	k	PROPN
ma-199	77	21	is	be	AUX
ma-199	77	22	a	a	DET
ma-199	77	23	krein	krein	ADJ
ma-199	77	24	space	space	NOUN
ma-199	77	25	.	.	PUNCT
ma-199	78	1	theorem	theorem	VERB
ma-199	78	2	2.1	2.1	NUM
ma-199	78	3	.	.	PUNCT
ma-199	79	1	[	[	X
ma-199	79	2	2	2	X
ma-199	79	3	]	]	PUNCT
ma-199	79	4	let	let	VERB
ma-199	79	5	k	k	PRON
ma-199	79	6	be	be	AUX
ma-199	79	7	a	a	DET
ma-199	79	8	krein	krein	ADJ
ma-199	79	9	space	space	NOUN
ma-199	79	10	.	.	PUNCT
ma-199	80	1	then	then	ADV
ma-199	80	2	the	the	DET
ma-199	80	3	following	follow	VERB
ma-199	80	4	are	be	AUX
ma-199	80	5	equivalent:(1	equivalent:(1	PROPN
ma-199	80	6	)	)	PUNCT
ma-199	80	7	there	there	PRON
ma-199	80	8	exists	exist	VERB
ma-199	80	9	a	a	DET
ma-199	80	10	fundamental	fundamental	ADJ
ma-199	80	11	decomposition	decomposition	NOUN
ma-199	80	12	of	of	ADP
ma-199	80	13	k.(2	k.(2	NUM
ma-199	80	14	)	)	PUNCT
ma-199	80	15	there	there	PRON
ma-199	80	16	exists	exist	VERB
ma-199	80	17	a	a	DET
ma-199	80	18	maximal	maximal	ADJ
ma-199	80	19	uniformly	uniformly	ADV
ma-199	80	20	positive	positive	ADJ
ma-199	80	21	ortho	ortho	NOUN
ma-199	80	22	-	-	PUNCT
ma-199	80	23	complemented	complement	VERB
ma-199	80	24	subspace.(3	subspace.(3	NOUN
ma-199	80	25	)	)	PUNCT
ma-199	80	26	there	there	PRON
ma-199	80	27	exists	exist	VERB
ma-199	80	28	a	a	DET
ma-199	80	29	maximal	maximal	ADJ
ma-199	80	30	uniformly	uniformly	ADV
ma-199	80	31	negative	negative	ADJ
ma-199	80	32	ortho	ortho	ADV
ma-199	80	33	-	-	PUNCT
ma-199	80	34	complemented	complement	VERB
ma-199	80	35	subspace.(4	subspace.(4	NOUN
ma-199	80	36	)	)	PUNCT
ma-199	80	37	there	there	PRON
ma-199	80	38	exists	exist	VERB
ma-199	80	39	a	a	DET
ma-199	80	40	mapping	mapping	NOUN
ma-199	80	41	j	j	PROPN
ma-199	80	42	in	in	ADP
ma-199	80	43	k	k	PROPN
ma-199	81	1	such	such	ADJ
ma-199	81	2	that	that	PRON
ma-199	81	3	j	j	PROPN
ma-199	81	4	=	=	PUNCT
ma-199	81	5	j∗	j∗	PROPN
ma-199	81	6	=	=	PUNCT
ma-199	81	7	j−1	j−1	PROPN
ma-199	81	8	.	.	PROPN
ma-199	82	1	https://doi.org/10.28924/ada/ma.4.1	https://doi.org/10.28924/ada/ma.4.1	NUM
ma-199	82	2	eur	eur	PROPN
ma-199	82	3	.	.	PUNCT
ma-199	83	1	j.	j.	PROPN
ma-199	83	2	math	math	PROPN
ma-199	83	3	.	.	PUNCT
ma-199	84	1	anal	anal	PROPN
ma-199	84	2	.	.	PUNCT
ma-199	85	1	10.28924	10.28924	NUM
ma-199	85	2	/	/	SYM
ma-199	85	3	ada	ada	PROPN
ma-199	85	4	/	/	SYM
ma-199	85	5	ma.4.1	ma.4.1	PROPN
ma-199	85	6	4different	4different	NUM
ma-199	85	7	fundamental	fundamental	ADJ
ma-199	85	8	decompositions	decomposition	NOUN
ma-199	85	9	induce	induce	VERB
ma-199	85	10	different	different	ADJ
ma-199	85	11	j	j	NOUN
ma-199	85	12	-	-	PUNCT
ma-199	85	13	norms	norm	NOUN
ma-199	85	14	.	.	PUNCT
ma-199	86	1	hence	hence	ADV
ma-199	86	2	various	various	ADJ
ma-199	86	3	norms	norm	NOUN
ma-199	86	4	can	can	AUX
ma-199	86	5	bedefined	bedefine	VERB
ma-199	86	6	on	on	ADP
ma-199	86	7	a	a	DET
ma-199	86	8	krein	krein	ADJ
ma-199	86	9	space	space	NOUN
ma-199	86	10	by	by	ADP
ma-199	86	11	choosing	choose	VERB
ma-199	86	12	different	different	ADJ
ma-199	86	13	underlying	underlie	VERB
ma-199	86	14	fundamental	fundamental	ADJ
ma-199	86	15	decompositions	decomposition	NOUN
ma-199	86	16	.	.	PUNCT
ma-199	87	1	example	example	NOUN
ma-199	87	2	2.2	2.2	NUM
ma-199	87	3	.	.	PUNCT
ma-199	88	1	let	let	VERB
ma-199	88	2	k	k	PRON
ma-199	88	3	be	be	AUX
ma-199	88	4	a	a	DET
ma-199	88	5	two	two	NUM
ma-199	88	6	-	-	PUNCT
ma-199	88	7	dimensional	dimensional	ADJ
ma-199	88	8	vector	vector	NOUN
ma-199	88	9	space	space	NOUN
ma-199	88	10	with	with	ADP
ma-199	88	11	basis	basis	NOUN
ma-199	88	12	{	{	PUNCT
ma-199	88	13	e1	e1	PROPN
ma-199	88	14	,	,	PUNCT
ma-199	88	15	e2	e2	PROPN
ma-199	88	16	}	}	PUNCT
ma-199	88	17	and	and	CCONJ
ma-199	88	18	an	an	DET
ma-199	88	19	indefinite	indefinite	ADJ
ma-199	88	20	inner	inner	ADJ
ma-199	88	21	product	product	NOUN
ma-199	88	22	defined	define	VERB
ma-199	88	23	by	by	ADP
ma-199	88	24	[	[	PUNCT
ma-199	88	25	e1	e1	NOUN
ma-199	88	26	,	,	PUNCT
ma-199	88	27	e1	e1	NOUN
ma-199	88	28	]	]	PUNCT
ma-199	88	29	=	=	SYM
ma-199	88	30	1	1	NUM
ma-199	88	31	,	,	PUNCT
ma-199	88	32	[	[	X
ma-199	88	33	e2	e2	NOUN
ma-199	88	34	,	,	PUNCT
ma-199	88	35	e2	e2	PROPN
ma-199	88	36	]	]	PUNCT
ma-199	88	37	=	=	SYM
ma-199	88	38	−1	−1	NOUN
ma-199	88	39	and	and	CCONJ
ma-199	88	40	[	[	X
ma-199	88	41	e1	e1	NOUN
ma-199	88	42	,	,	PUNCT
ma-199	88	43	e2	e2	X
ma-199	88	44	]	]	PUNCT
ma-199	89	1	=	=	SYM
ma-199	89	2	0	0	X
ma-199	89	3	.	.	PUNCT
ma-199	90	1	if	if	SCONJ
ma-199	90	2	we	we	PRON
ma-199	90	3	take	take	VERB
ma-199	90	4	y	y	NOUN
ma-199	90	5	=	=	PUNCT
ma-199	90	6	span{e1	span{e1	NOUN
ma-199	90	7	}	}	PUNCT
ma-199	90	8	,	,	PUNCT
ma-199	90	9	then	then	ADV
ma-199	90	10	it	it	PRON
ma-199	90	11	is	be	AUX
ma-199	90	12	a	a	DET
ma-199	90	13	maximal	maximal	ADJ
ma-199	90	14	uniformly	uniformly	ADV
ma-199	90	15	positive	positive	ADJ
ma-199	90	16	definite	definite	ADJ
ma-199	90	17	subspace	subspace	NOUN
ma-199	90	18	and	and	CCONJ
ma-199	90	19	hence	hence	ADV
ma-199	90	20	there	there	PRON
ma-199	90	21	exists	exist	VERB
ma-199	90	22	a	a	DET
ma-199	90	23	fundamental	fundamental	ADJ
ma-199	90	24	decomposition	decomposition	NOUN
ma-199	90	25	of	of	ADP
ma-199	90	26	k	k	PROPN
ma-199	90	27	with	with	ADP
ma-199	90	28	k+	k+	NOUN
ma-199	90	29	=	=	SYM
ma-199	90	30	y	y	PROPN
ma-199	90	31	and	and	CCONJ
ma-199	90	32	k−	k−	PROPN
ma-199	90	33	=	=	SYM
ma-199	90	34	span{e2	span{e2	NOUN
ma-199	90	35	}	}	PUNCT
ma-199	90	36	.	.	PUNCT
ma-199	91	1	choosing	choose	VERB
ma-199	91	2	k+n	k+n	PROPN
ma-199	91	3	=	=	SYM
ma-199	91	4	span{(n	span{(n	ADJ
ma-199	91	5	,	,	PUNCT
ma-199	91	6	1	1	NUM
ma-199	91	7	)	)	PUNCT
ma-199	91	8	}	}	PUNCT
ma-199	91	9	and	and	CCONJ
ma-199	91	10	k−n	k−n	PROPN
ma-199	91	11	=	=	SYM
ma-199	91	12	span{(1	span{(1	PROPN
ma-199	91	13	,	,	PUNCT
ma-199	91	14	n	n	CCONJ
ma-199	91	15	)	)	PUNCT
ma-199	91	16	}	}	PUNCT
ma-199	91	17	where	where	SCONJ
ma-199	91	18	n	n	X
ma-199	91	19	>	>	X
ma-199	91	20	1	1	NUM
ma-199	91	21	,	,	PUNCT
ma-199	91	22	we	we	PRON
ma-199	91	23	get	get	VERB
ma-199	91	24	several	several	ADJ
ma-199	91	25	fundamental	fundamental	ADJ
ma-199	91	26	decompositions	decomposition	NOUN
ma-199	91	27	.	.	PUNCT
ma-199	92	1	the	the	DET
ma-199	92	2	corresponding	corresponding	ADJ
ma-199	92	3	fundamental	fundamental	ADJ
ma-199	92	4	symmetries	symmetry	NOUN
ma-199	92	5	jn	jn	PROPN
ma-199	92	6	are	be	AUX
ma-199	92	7	given	give	VERB
ma-199	92	8	by	by	ADP
ma-199	92	9	jn	jn	PROPN
ma-199	92	10	=	=	PROPN
ma-199	92	11	(	(	PUNCT
ma-199	92	12	n2	n2	ADJ
ma-199	92	13	+	+	PROPN
ma-199	92	14	1	1	NUM
ma-199	92	15	n2−1	n2−1	NOUN
ma-199	92	16	−2n	−2n	PROPN
ma-199	92	17	n2−1	n2−1	NOUN
ma-199	92	18	2n	2n	NUM
ma-199	92	19	n2−1	n2−1	ADP
ma-199	92	20	−(n2	−(n2	ADV
ma-199	92	21	+	+	NOUN
ma-199	92	22	1	1	NUM
ma-199	92	23	)	)	PUNCT
ma-199	92	24	n2−1	n2−1	NOUN
ma-199	92	25	)	)	PUNCT
ma-199	92	26	.	.	PUNCT
ma-199	93	1	here	here	ADV
ma-199	93	2	we	we	PRON
ma-199	93	3	can	can	AUX
ma-199	93	4	see	see	VERB
ma-199	93	5	that	that	SCONJ
ma-199	93	6	the	the	DET
ma-199	93	7	fundamental	fundamental	ADJ
ma-199	93	8	symmetries	symmetry	NOUN
ma-199	93	9	jn	jn	PROPN
ma-199	93	10	satisfy	satisfy	VERB
ma-199	93	11	j2n	j2n	VERB
ma-199	94	1	=	=	PUNCT
ma-199	95	1	in	in	ADV
ma-199	95	2	,	,	PUNCT
ma-199	95	3	[	[	X
ma-199	95	4	jnf	jnf	X
ma-199	95	5	,	,	PUNCT
ma-199	95	6	g	g	NOUN
ma-199	95	7	]	]	X
ma-199	95	8	=	=	PUNCT
ma-199	96	1	[	[	X
ma-199	96	2	f	f	X
ma-199	96	3	,	,	PUNCT
ma-199	96	4	jng	jng	PROPN
ma-199	96	5	]	]	PUNCT
ma-199	96	6	and	and	CCONJ
ma-199	96	7	[	[	X
ma-199	96	8	jnf	jnf	X
ma-199	96	9	,	,	PUNCT
ma-199	96	10	jng	jng	PROPN
ma-199	96	11	]	]	X
ma-199	96	12	=	=	PUNCT
ma-199	97	1	[	[	X
ma-199	97	2	f	f	X
ma-199	97	3	,	,	PUNCT
ma-199	97	4	g	g	NOUN
ma-199	97	5	]	]	PUNCT
ma-199	97	6	for	for	ADP
ma-199	97	7	all	all	DET
ma-199	97	8	f	f	PROPN
ma-199	97	9	,	,	PUNCT
ma-199	97	10	g	g	PROPN
ma-199	97	11	∈	∈	PROPN
ma-199	97	12	k.	k.	PROPN
ma-199	98	1	3	3	X
ma-199	98	2	.	.	PUNCT
ma-199	98	3	frame	frame	NOUN
ma-199	98	4	operator	operator	NOUN
ma-199	98	5	for	for	ADP
ma-199	98	6	frames	frame	NOUN
ma-199	98	7	in	in	ADP
ma-199	98	8	krein	krein	ADJ
ma-199	98	9	spaces	space	NOUN
ma-199	98	10	let	let	VERB
ma-199	98	11	k	k	PRON
ma-199	98	12	be	be	AUX
ma-199	98	13	a	a	DET
ma-199	98	14	krein	krein	ADJ
ma-199	98	15	space	space	NOUN
ma-199	98	16	and	and	CCONJ
ma-199	98	17	let	let	VERB
ma-199	98	18	{	{	PUNCT
ma-199	98	19	fn}n∈i	fn}n∈i	VERB
ma-199	98	20	be	be	AUX
ma-199	98	21	a	a	DET
ma-199	98	22	sequence	sequence	NOUN
ma-199	98	23	in	in	ADP
ma-199	98	24	k.	k.	PROPN
ma-199	98	25	in	in	ADP
ma-199	98	26	relation	relation	NOUN
ma-199	98	27	to	to	ADP
ma-199	98	28	the	the	DET
ma-199	98	29	sequence	sequence	NOUN
ma-199	98	30	{	{	PUNCT
ma-199	98	31	fn}n∈i	fn}n∈i	X
ma-199	98	32	,	,	PUNCT
ma-199	98	33	the	the	DET
ma-199	98	34	index	index	NOUN
ma-199	98	35	set	set	VERB
ma-199	98	36	i	i	PRON
ma-199	98	37	is	be	AUX
ma-199	98	38	decomposed	decompose	VERB
ma-199	98	39	as	as	ADP
ma-199	98	40	i+	i+	NOUN
ma-199	98	41	=	=	NOUN
ma-199	98	42	{	{	PUNCT
ma-199	98	43	n	n	NOUN
ma-199	98	44	∈	∈	NOUN
ma-199	99	1	i	i	PRON
ma-199	99	2	:	:	PUNCT
ma-199	100	1	[	[	X
ma-199	100	2	fn	fn	X
ma-199	100	3	,	,	PUNCT
ma-199	100	4	fn	fn	NOUN
ma-199	100	5	]	]	X
ma-199	100	6	≥	≥	NOUN
ma-199	100	7	0	0	NUM
ma-199	100	8	}	}	PUNCT
ma-199	100	9	and	and	CCONJ
ma-199	100	10	i−	i−	PROPN
ma-199	100	11	=	=	SYM
ma-199	100	12	{	{	PUNCT
ma-199	100	13	n	n	NOUN
ma-199	100	14	∈	∈	NOUN
ma-199	101	1	i	i	PRON
ma-199	101	2	:	:	PUNCT
ma-199	102	1	[	[	X
ma-199	102	2	fn	fn	X
ma-199	102	3	,	,	PUNCT
ma-199	102	4	fn	fn	ADP
ma-199	102	5	]	]	X
ma-199	102	6	<	<	X
ma-199	102	7	0	0	NUM
ma-199	102	8	}	}	PUNCT
ma-199	102	9	.	.	PUNCT
ma-199	103	1	it	it	PRON
ma-199	103	2	iseasy	iseasy	ADJ
ma-199	103	3	to	to	PART
ma-199	103	4	observe	observe	VERB
ma-199	103	5	that	that	SCONJ
ma-199	103	6	`	`	PUNCT
ma-199	103	7	2(i	2(i	NUM
ma-199	103	8	)	)	PUNCT
ma-199	103	9	is	be	AUX
ma-199	103	10	the	the	DET
ma-199	103	11	orthogonal	orthogonal	ADJ
ma-199	103	12	direct	direct	ADJ
ma-199	103	13	sum	sum	NOUN
ma-199	103	14	of	of	ADP
ma-199	103	15	`	`	PUNCT
ma-199	103	16	2(i+	2(i+	NUM
ma-199	103	17	)	)	PUNCT
ma-199	103	18	and	and	CCONJ
ma-199	103	19	`	`	PUNCT
ma-199	103	20	2(i−	2(i−	NUM
ma-199	103	21	)	)	PUNCT
ma-199	103	22	.	.	PUNCT
ma-199	104	1	definition	definition	NOUN
ma-199	104	2	3.1	3.1	NUM
ma-199	104	3	.	.	PUNCT
ma-199	105	1	a	a	DET
ma-199	105	2	sequence	sequence	NOUN
ma-199	105	3	{	{	PUNCT
ma-199	105	4	fn}n∈i	fn}n∈i	NOUN
ma-199	105	5	in	in	ADP
ma-199	105	6	a	a	DET
ma-199	105	7	krein	krein	ADJ
ma-199	105	8	space	space	NOUN
ma-199	105	9	k	k	PROPN
ma-199	105	10	is	be	AUX
ma-199	105	11	called	call	VERB
ma-199	105	12	a	a	DET
ma-199	105	13	bessel	bessel	ADJ
ma-199	105	14	sequence	sequence	NOUN
ma-199	105	15	if	if	SCONJ
ma-199	105	16	there	there	PRON
ma-199	105	17	exists	exist	VERB
ma-199	105	18	a	a	DET
ma-199	105	19	constant	constant	ADJ
ma-199	105	20	b	b	NOUN
ma-199	105	21	>	>	X
ma-199	105	22	0	0	NUM
ma-199	105	23	such	such	ADJ
ma-199	105	24	that	that	SCONJ
ma-199	105	25	∑	∑	PROPN
ma-199	105	26	n∈i	n∈i	PROPN
ma-199	105	27	|[fn	|[fn	PROPN
ma-199	105	28	,	,	PUNCT
ma-199	105	29	f	f	X
ma-199	105	30	]	]	PUNCT
ma-199	105	31	|2	|2	NUM
ma-199	105	32	≤	≤	NOUN
ma-199	105	33	b‖f	b‖f	ADJ
ma-199	105	34	‖2	‖2	NOUN
ma-199	105	35	,	,	PUNCT
ma-199	105	36	for	for	ADP
ma-199	105	37	all	all	DET
ma-199	105	38	f	f	PROPN
ma-199	105	39	∈	∈	PROPN
ma-199	105	40	k.	k.	PROPN
ma-199	105	41	(	(	PUNCT
ma-199	105	42	6	6	NUM
ma-199	105	43	)	)	PUNCT
ma-199	105	44	the	the	DET
ma-199	105	45	constant	constant	ADJ
ma-199	105	46	b	b	NOUN
ma-199	105	47	in	in	ADP
ma-199	105	48	the	the	DET
ma-199	105	49	inequality	inequality	NOUN
ma-199	105	50	(	(	PUNCT
ma-199	105	51	6	6	NUM
ma-199	105	52	)	)	PUNCT
ma-199	105	53	is	be	AUX
ma-199	105	54	called	call	VERB
ma-199	105	55	a	a	DET
ma-199	105	56	bessel	bessel	NOUN
ma-199	105	57	bound	bind	VERB
ma-199	105	58	for	for	ADP
ma-199	105	59	{	{	PUNCT
ma-199	105	60	fn}n∈i	fn}n∈i	X
ma-199	105	61	.	.	PUNCT
ma-199	105	62	theorem	theorem	VERB
ma-199	105	63	3.1	3.1	NUM
ma-199	105	64	.	.	PUNCT
ma-199	106	1	let	let	VERB
ma-199	106	2	{	{	PUNCT
ma-199	106	3	fn}n∈i	fn}n∈i	VERB
ma-199	106	4	be	be	AUX
ma-199	106	5	a	a	DET
ma-199	106	6	sequence	sequence	NOUN
ma-199	106	7	in	in	ADP
ma-199	106	8	a	a	DET
ma-199	106	9	krein	krein	NOUN
ma-199	106	10	space	space	NOUN
ma-199	107	1	k.	k.	PROPN
ma-199	107	2	then	then	ADV
ma-199	107	3	{	{	PUNCT
ma-199	107	4	fn}n∈i	fn}n∈i	NOUN
ma-199	107	5	is	be	AUX
ma-199	107	6	a	a	DET
ma-199	107	7	bessel	bessel	ADJ
ma-199	107	8	sequence	sequence	NOUN
ma-199	107	9	with	with	ADP
ma-199	107	10	a	a	DET
ma-199	107	11	bessel	bessel	NOUN
ma-199	107	12	bound	bind	VERB
ma-199	107	13	b	b	NOUN
ma-199	108	1	if	if	SCONJ
ma-199	109	1	and	and	CCONJ
ma-199	109	2	only	only	ADV
ma-199	109	3	if	if	SCONJ
ma-199	109	4	the	the	DET
ma-199	109	5	operators	operator	NOUN
ma-199	109	6	t+	t+	PUNCT
ma-199	109	7	:	:	PUNCT
ma-199	109	8	`	`	PUNCT
ma-199	109	9	2(i+	2(i+	X
ma-199	109	10	)	)	PUNCT
ma-199	109	11	−→	−→	ADJ
ma-199	109	12	k+	k+	NOUN
ma-199	109	13	defined	define	VERB
ma-199	109	14	by	by	ADP
ma-199	109	15	t+{cn}n∈i+	t+{cn}n∈i+	PROPN
ma-199	109	16	=	=	PROPN
ma-199	109	17	∑	∑	PROPN
ma-199	109	18	n∈i+	n∈i+	X
ma-199	109	19	cnfn	cnfn	PROPN
ma-199	109	20	and	and	CCONJ
ma-199	109	21	t−	t−	PROPN
ma-199	109	22	:	:	PUNCT
ma-199	109	23	`	`	PUNCT
ma-199	109	24	2(i−	2(i−	NUM
ma-199	109	25	)	)	PUNCT
ma-199	109	26	−→	−→	PROPN
ma-199	109	27	k−	k−	PROPN
ma-199	109	28	defined	define	VERB
ma-199	109	29	by	by	ADP
ma-199	109	30	t−{cn}n∈i−	t−{cn}n∈i−	NOUN
ma-199	109	31	=	=	PUNCT
ma-199	109	32	∑	∑	ADP
ma-199	109	33	n∈i−	n∈i−	PROPN
ma-199	109	34	cnfn	cnfn	NOUN
ma-199	109	35	are	be	AUX
ma-199	109	36	well	well	ADV
ma-199	109	37	defined	define	VERB
ma-199	109	38	bounded	bounded	ADJ
ma-199	109	39	operators	operator	NOUN
ma-199	109	40	and	and	CCONJ
ma-199	109	41	‖t‖	‖t‖	PROPN
ma-199	109	42	≤	≤	PROPN
ma-199	109	43	√	√	NUM
ma-199	109	44	b	b	NOUN
ma-199	109	45	,	,	PUNCT
ma-199	109	46	where	where	SCONJ
ma-199	109	47	t	t	NOUN
ma-199	109	48	=	=	SYM
ma-199	109	49	t+	t+	PUNCT
ma-199	109	50	+	+	NUM
ma-199	109	51	t−.	t−.	NOUN
ma-199	109	52	proof	proof	NOUN
ma-199	109	53	.	.	PUNCT
ma-199	110	1	suppose	suppose	VERB
ma-199	110	2	first	first	ADV
ma-199	110	3	that	that	SCONJ
ma-199	110	4	{	{	PUNCT
ma-199	110	5	fn}n∈i	fn}n∈i	NOUN
ma-199	110	6	is	be	AUX
ma-199	110	7	a	a	DET
ma-199	110	8	bessel	bessel	ADJ
ma-199	110	9	sequence	sequence	NOUN
ma-199	110	10	with	with	ADP
ma-199	110	11	a	a	DET
ma-199	110	12	bessel	bessel	NOUN
ma-199	110	13	bound	bind	VERB
ma-199	110	14	b.	b.	PROPN
ma-199	110	15	let	let	VERB
ma-199	110	16	{	{	PUNCT
ma-199	110	17	cn}n∈i+	cn}n∈i+	VERB
ma-199	110	18	∈	∈	PROPN
ma-199	110	19	`	`	PUNCT
ma-199	110	20	2(i+)and	2(i+)and	NUM
ma-199	110	21	{	{	PUNCT
ma-199	110	22	cn}n∈i−	cn}n∈i−	X
ma-199	110	23	∈	∈	PROPN
ma-199	110	24	`	`	PUNCT
ma-199	110	25	2(i−	2(i−	NUM
ma-199	110	26	)	)	PUNCT
ma-199	110	27	.	.	PUNCT
ma-199	111	1	let	let	VERB
ma-199	111	2	`	`	PUNCT
ma-199	111	3	,	,	PUNCT
ma-199	111	4	m	m	VERB
ma-199	111	5	∈	∈	NOUN
ma-199	111	6	i−	i−	ADJ
ma-199	111	7	such	such	ADJ
ma-199	111	8	that	that	SCONJ
ma-199	111	9	`	`	PUNCT
ma-199	111	10	>	>	X
ma-199	111	11	m.	m.	NOUN
ma-199	111	12	then∥∥∥∥∥∑̀	then∥∥∥∥∥∑̀	PROPN
ma-199	111	13	n=1	n=1	PROPN
ma-199	111	14	cnfn	cnfn	PROPN
ma-199	111	15	−	−	PROPN
ma-199	111	16	m∑	m∑	VERB
ma-199	111	17	n=1	n=1	PROPN
ma-199	111	18	cnfn	cnfn	NOUN
ma-199	111	19	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-199	111	20	=	=	SYM
ma-199	111	21	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-199	111	22	∑̀	∑̀	NOUN
ma-199	111	23	n	n	CCONJ
ma-199	111	24	=	=	NOUN
ma-199	111	25	m+1	m+1	NUM
ma-199	111	26	cnfn	cnfn	NOUN
ma-199	111	27	∥∥∥∥∥	∥∥∥∥∥	NOUN
ma-199	111	28	=	=	SYM
ma-199	111	29	sup	sup	NOUN
ma-199	111	30	‖g‖=1	‖g‖=1	PROPN
ma-199	111	31	|	|	NOUN
ma-199	111	32	[	[	PUNCT
ma-199	111	33	∑̀	∑̀	NOUN
ma-199	111	34	n	n	CCONJ
ma-199	111	35	=	=	NOUN
ma-199	111	36	m+1	m+1	NUM
ma-199	111	37	cnfn	cnfn	NOUN
ma-199	111	38	,	,	PUNCT
ma-199	111	39	g]|	g]|	VERB
ma-199	111	40	≤	≤	NUM
ma-199	111	41	sup	sup	NOUN
ma-199	111	42	‖g‖=1	‖g‖=1	PROPN
ma-199	111	43	∑̀	∑̀	NOUN
ma-199	111	44	n	n	CCONJ
ma-199	111	45	=	=	NOUN
ma-199	111	46	m+1	m+1	NUM
ma-199	111	47	|[cnfn	|[cnfn	PROPN
ma-199	111	48	,	,	PUNCT
ma-199	111	49	g]|	g]|	NOUN
ma-199	111	50	https://doi.org/10.28924/ada/ma.4.1	https://doi.org/10.28924/ada/ma.4.1	PRON
ma-199	111	51	eur	eur	NOUN
ma-199	111	52	.	.	PUNCT
ma-199	112	1	j.	j.	PROPN
ma-199	112	2	math	math	PROPN
ma-199	112	3	.	.	PUNCT
ma-199	113	1	anal	anal	PROPN
ma-199	113	2	.	.	PUNCT
ma-199	114	1	10.28924	10.28924	NUM
ma-199	114	2	/	/	SYM
ma-199	114	3	ada	ada	PROPN
ma-199	114	4	/	/	SYM
ma-199	114	5	ma.4.1	ma.4.1	PROPN
ma-199	114	6	5	5	NUM
ma-199	114	7	≤	≤	NUM
ma-199	114	8	(	(	PUNCT
ma-199	114	9	∑̀	∑̀	NOUN
ma-199	114	10	n	n	CCONJ
ma-199	114	11	=	=	SYM
ma-199	114	12	m+1	m+1	NOUN
ma-199	114	13	|cn|2	|cn|2	PUNCT
ma-199	114	14	)	)	PUNCT
ma-199	114	15	1	1	NUM
ma-199	114	16	2	2	NUM
ma-199	114	17	sup	sup	NOUN
ma-199	114	18	‖g‖=1	‖g‖=1	PROPN
ma-199	114	19	(	(	PUNCT
ma-199	114	20	∑̀	∑̀	NOUN
ma-199	114	21	n	n	CCONJ
ma-199	114	22	=	=	NOUN
ma-199	114	23	m+1	m+1	NUM
ma-199	114	24	|[fn	|[fn	NOUN
ma-199	114	25	,	,	PUNCT
ma-199	114	26	g]|2	g]|2	NOUN
ma-199	114	27	)	)	PUNCT
ma-199	114	28	1	1	NUM
ma-199	114	29	2	2	NUM
ma-199	114	30	≤	≤	NUM
ma-199	114	31	√	√	NUM
ma-199	114	32	b	b	NOUN
ma-199	114	33	(	(	PUNCT
ma-199	114	34	∑̀	∑̀	NOUN
ma-199	114	35	n	n	CCONJ
ma-199	114	36	=	=	SYM
ma-199	114	37	m+1	m+1	NOUN
ma-199	114	38	|cn|2	|cn|2	PUNCT
ma-199	114	39	)	)	PUNCT
ma-199	114	40	1	1	NUM
ma-199	114	41	2	2	NUM
ma-199	114	42	.	.	PUNCT
ma-199	115	1	since	since	SCONJ
ma-199	115	2	{	{	PUNCT
ma-199	115	3	cn}n∈i−	cn}n∈i−	X
ma-199	115	4	∈	∈	PROPN
ma-199	115	5	`	`	PUNCT
ma-199	115	6	2(i−	2(i−	NUM
ma-199	115	7	)	)	PUNCT
ma-199	115	8	,	,	PUNCT
ma-199	115	9	{	{	PUNCT
ma-199	115	10	∑`n=1	∑`n=1	PUNCT
ma-199	115	11	|cn|2	|cn|2	PUNCT
ma-199	115	12	}	}	PUNCT
ma-199	115	13	is	be	AUX
ma-199	115	14	a	a	DET
ma-199	115	15	cauchy	cauchy	ADJ
ma-199	115	16	sequence	sequence	NOUN
ma-199	115	17	in	in	ADP
ma-199	115	18	c.	c.	PROPN
ma-199	115	19	the	the	DET
ma-199	115	20	above	above	ADJ
ma-199	115	21	calculation	calculation	NOUN
ma-199	115	22	showsthat	showsthat	PROPN
ma-199	115	23	{	{	PUNCT
ma-199	115	24	∑`n=1	∑`n=1	PROPN
ma-199	115	25	cnfn}`∈i−	cnfn}`∈i−	PROPN
ma-199	115	26	is	be	AUX
ma-199	115	27	a	a	DET
ma-199	115	28	cauchy	cauchy	ADJ
ma-199	115	29	sequence	sequence	NOUN
ma-199	115	30	in	in	ADP
ma-199	115	31	k−	k−	PROPN
ma-199	115	32	,	,	PUNCT
ma-199	115	33	and	and	CCONJ
ma-199	115	34	so	so	ADV
ma-199	115	35	it	it	PRON
ma-199	115	36	is	be	AUX
ma-199	115	37	convergent	convergent	ADJ
ma-199	115	38	.	.	PUNCT
ma-199	116	1	hence	hence	ADV
ma-199	116	2	t−	t−	PROPN
ma-199	116	3	is	be	AUX
ma-199	116	4	welldefined	welldefine	VERB
ma-199	116	5	.	.	PUNCT
ma-199	117	1	with	with	ADP
ma-199	117	2	similar	similar	ADJ
ma-199	117	3	arguments	argument	NOUN
ma-199	117	4	and	and	CCONJ
ma-199	117	5	by	by	ADP
ma-199	117	6	considering	consider	VERB
ma-199	117	7	`	`	PUNCT
ma-199	117	8	,	,	PUNCT
ma-199	117	9	m	m	VERB
ma-199	117	10	∈	∈	ADJ
ma-199	117	11	i+	i+	NOUN
ma-199	117	12	with	with	ADP
ma-199	117	13	`	`	PUNCT
ma-199	117	14	>	>	X
ma-199	117	15	m	m	VERB
ma-199	117	16	one	one	PRON
ma-199	117	17	may	may	AUX
ma-199	117	18	prove	prove	VERB
ma-199	117	19	that	that	SCONJ
ma-199	117	20	{	{	PUNCT
ma-199	117	21	∑	∑	PUNCT
ma-199	117	22	`	`	PUNCT
ma-199	117	23	n=1	n=1	PROPN
ma-199	117	24	cnfn}`∈i+	cnfn}`∈i+	NOUN
ma-199	117	25	is	be	AUX
ma-199	117	26	a	a	DET
ma-199	117	27	cauchy	cauchy	ADJ
ma-199	117	28	sequence	sequence	NOUN
ma-199	117	29	in	in	ADP
ma-199	117	30	k+	k+	NOUN
ma-199	118	1	and	and	CCONJ
ma-199	118	2	so	so	ADV
ma-199	118	3	it	it	PRON
ma-199	118	4	is	be	AUX
ma-199	118	5	convergent	convergent	ADJ
ma-199	118	6	.	.	PUNCT
ma-199	119	1	thus	thus	ADV
ma-199	119	2	t−	t−	PROPN
ma-199	119	3	and	and	CCONJ
ma-199	119	4	t+	t+	NOUN
ma-199	119	5	arewell	arewell	PROPN
ma-199	119	6	defined	define	VERB
ma-199	119	7	and	and	CCONJ
ma-199	119	8	boundedness	boundedness	NOUN
ma-199	119	9	follows	follow	VERB
ma-199	119	10	from	from	ADP
ma-199	119	11	the	the	DET
ma-199	119	12	above	above	ADJ
ma-199	119	13	calculation	calculation	NOUN
ma-199	119	14	.	.	PUNCT
ma-199	120	1	clearly	clearly	ADV
ma-199	120	2	t−	t−	PROPN
ma-199	120	3	and	and	CCONJ
ma-199	120	4	t+	t+	NOUN
ma-199	120	5	are	be	AUX
ma-199	120	6	linear.conversely	linear.conversely	ADV
ma-199	120	7	,	,	PUNCT
ma-199	120	8	suppose	suppose	VERB
ma-199	120	9	that	that	SCONJ
ma-199	120	10	t−	t−	PROPN
ma-199	120	11	and	and	CCONJ
ma-199	120	12	t+	t+	NOUN
ma-199	120	13	are	be	AUX
ma-199	120	14	well	well	ADV
ma-199	120	15	defined	define	VERB
ma-199	120	16	bounded	bounded	ADJ
ma-199	120	17	linear	linear	PROPN
ma-199	120	18	operators	operator	NOUN
ma-199	120	19	with	with	ADP
ma-199	120	20	their	their	PRON
ma-199	120	21	adjoints	adjoint	NOUN
ma-199	120	22	t−∗	t−∗	NOUN
ma-199	120	23	:	:	PUNCT
ma-199	120	24	k−	k−	PROPN
ma-199	120	25	−→	−→	ADJ
ma-199	120	26	`	`	PUNCT
ma-199	120	27	2(i−	2(i−	NUM
ma-199	120	28	)	)	PUNCT
ma-199	120	29	and	and	CCONJ
ma-199	120	30	t+∗	t+∗	ADV
ma-199	120	31	:	:	PUNCT
ma-199	120	32	k+	k+	PUNCT
ma-199	120	33	−→	−→	ADJ
ma-199	120	34	`	`	PUNCT
ma-199	120	35	2(i+	2(i+	X
ma-199	120	36	)	)	PUNCT
ma-199	120	37	defined	define	VERB
ma-199	120	38	by	by	ADP
ma-199	120	39	t−∗f	t−∗f	PROPN
ma-199	120	40	=	=	SYM
ma-199	120	41	{	{	PUNCT
ma-199	120	42	−[f	−[f	PROPN
ma-199	120	43	,	,	PUNCT
ma-199	120	44	fn]}n∈i−	fn]}n∈i−	NUM
ma-199	120	45	and	and	CCONJ
ma-199	120	46	t+∗f	t+∗f	NOUN
ma-199	120	47	=	=	PUNCT
ma-199	120	48	{	{	PUNCT
ma-199	121	1	[	[	X
ma-199	121	2	f	f	X
ma-199	121	3	,	,	PUNCT
ma-199	121	4	fn]}n∈i+	fn]}n∈i+	PROPN
ma-199	121	5	respectively	respectively	ADV
ma-199	121	6	.	.	PUNCT
ma-199	122	1	since	since	SCONJ
ma-199	122	2	the	the	DET
ma-199	122	3	adjoint	adjoint	NOUN
ma-199	122	4	of	of	ADP
ma-199	122	5	a	a	DET
ma-199	122	6	bounded	bound	VERB
ma-199	122	7	operator	operator	NOUN
ma-199	122	8	is	be	AUX
ma-199	122	9	bounded	bound	VERB
ma-199	122	10	,	,	PUNCT
ma-199	122	11	‖t−∗‖	‖t−∗‖	NOUN
ma-199	122	12	=	=	SYM
ma-199	122	13	‖t−‖	‖t−‖	PROPN
ma-199	122	14	and	and	CCONJ
ma-199	122	15	‖t+∗‖	‖t+∗‖	NOUN
ma-199	123	1	=	=	NOUN
ma-199	123	2	‖t+‖.	‖t+‖.	NOUN
ma-199	123	3	also	also	ADV
ma-199	123	4	we	we	PRON
ma-199	123	5	have	have	VERB
ma-199	123	6	‖t+∗f	‖t+∗f	NOUN
ma-199	123	7	‖2	‖2	NOUN
ma-199	123	8	≤	≤	X
ma-199	123	9	‖t+‖2‖f	‖t+‖2‖f	PROPN
ma-199	123	10	‖2	‖2	NOUN
ma-199	123	11	,	,	PUNCT
ma-199	123	12	for	for	ADP
ma-199	123	13	all	all	DET
ma-199	123	14	f	f	PROPN
ma-199	123	15	∈	∈	PROPN
ma-199	123	16	k+	k+	NOUN
ma-199	123	17	and	and	CCONJ
ma-199	123	18	‖t−∗f	‖t−∗f	DET
ma-199	123	19	‖2	‖2	NOUN
ma-199	123	20	≤	≤	NUM
ma-199	123	21	‖t−‖2‖f	‖t−‖2‖f	PROPN
ma-199	123	22	‖2	‖2	NOUN
ma-199	123	23	,	,	PUNCT
ma-199	123	24	for	for	ADP
ma-199	123	25	all	all	DET
ma-199	123	26	f	f	PROPN
ma-199	123	27	∈	∈	PROPN
ma-199	123	28	k−.so	k−.so	NOUN
ma-199	123	29	‖t	‖t	NOUN
ma-199	123	30	∗f	∗f	PROPN
ma-199	123	31	‖2	‖2	NOUN
ma-199	123	32	≤	≤	NOUN
ma-199	124	1	‖t‖2‖f	‖t‖2‖f	DET
ma-199	124	2	‖2	‖2	NOUN
ma-199	124	3	,	,	PUNCT
ma-199	124	4	for	for	ADP
ma-199	124	5	all	all	DET
ma-199	124	6	f	f	PROPN
ma-199	124	7	∈	∈	PROPN
ma-199	124	8	k.	k.	PROPN
ma-199	124	9	hence	hence	ADV
ma-199	124	10	{	{	PUNCT
ma-199	124	11	fn}n∈i	fn}n∈i	NOUN
ma-199	124	12	is	be	AUX
ma-199	124	13	a	a	DET
ma-199	124	14	bessel	bessel	ADJ
ma-199	124	15	sequence	sequence	NOUN
ma-199	124	16	.	.	PUNCT
ma-199	125	1	�	�	PROPN
ma-199	125	2	corollary	corollary	ADJ
ma-199	125	3	3.2	3.2	NUM
ma-199	125	4	.	.	PUNCT
ma-199	126	1	let	let	VERB
ma-199	126	2	{	{	PUNCT
ma-199	126	3	fn}n∈i	fn}n∈i	VERB
ma-199	126	4	be	be	AUX
ma-199	126	5	a	a	DET
ma-199	126	6	sequence	sequence	NOUN
ma-199	126	7	in	in	ADP
ma-199	126	8	a	a	DET
ma-199	126	9	krein	krein	ADJ
ma-199	126	10	space	space	NOUN
ma-199	126	11	k	k	NOUN
ma-199	126	12	such	such	ADJ
ma-199	126	13	that	that	SCONJ
ma-199	126	14	both	both	DET
ma-199	126	15	∑	∑	PROPN
ma-199	126	16	n∈i+	n∈i+	PROPN
ma-199	126	17	cnfn	cnfn	PROPN
ma-199	126	18	and∑	and∑	PART
ma-199	126	19	n∈i−	n∈i−	PROPN
ma-199	126	20	cnfn	cnfn	NOUN
ma-199	126	21	are	be	AUX
ma-199	126	22	convergent	convergent	ADJ
ma-199	126	23	for	for	ADP
ma-199	126	24	all	all	PRON
ma-199	126	25	{	{	PUNCT
ma-199	126	26	cn}n∈i−	cn}n∈i−	X
ma-199	126	27	∈	∈	PROPN
ma-199	126	28	`	`	PUNCT
ma-199	126	29	2(i−	2(i−	NUM
ma-199	126	30	)	)	PUNCT
ma-199	126	31	and	and	CCONJ
ma-199	126	32	{	{	PUNCT
ma-199	126	33	cn}n∈i+	cn}n∈i+	PROPN
ma-199	126	34	∈	∈	PROPN
ma-199	126	35	`	`	PUNCT
ma-199	126	36	2(i+	2(i+	NUM
ma-199	126	37	)	)	PUNCT
ma-199	126	38	.	.	PUNCT
ma-199	127	1	then	then	ADV
ma-199	127	2	{	{	PUNCT
ma-199	127	3	fn}n∈i	fn}n∈i	NOUN
ma-199	127	4	is	be	AUX
ma-199	127	5	a	a	DET
ma-199	127	6	bessel	bessel	ADJ
ma-199	127	7	sequence	sequence	NOUN
ma-199	127	8	.	.	PUNCT
ma-199	128	1	the	the	DET
ma-199	128	2	condition	condition	NOUN
ma-199	128	3	(	(	PUNCT
ma-199	128	4	6	6	NUM
ma-199	128	5	)	)	PUNCT
ma-199	128	6	remains	remain	VERB
ma-199	128	7	unchanged	unchanged	ADJ
ma-199	128	8	regardless	regardless	ADV
ma-199	128	9	how	how	SCONJ
ma-199	128	10	the	the	DET
ma-199	128	11	elements	element	NOUN
ma-199	128	12	of	of	ADP
ma-199	128	13	{	{	PUNCT
ma-199	128	14	fn}n∈i	fn}n∈i	NOUN
ma-199	128	15	are	be	AUX
ma-199	128	16	numbered	number	VERB
ma-199	128	17	.	.	PUNCT
ma-199	129	1	corollary	corollary	ADJ
ma-199	129	2	3.3	3.3	NUM
ma-199	129	3	.	.	PUNCT
ma-199	130	1	let	let	VERB
ma-199	130	2	{	{	PUNCT
ma-199	130	3	fn}n∈i	fn}n∈i	VERB
ma-199	130	4	be	be	AUX
ma-199	130	5	a	a	DET
ma-199	130	6	bessel	bessel	ADJ
ma-199	130	7	sequence	sequence	NOUN
ma-199	130	8	in	in	ADP
ma-199	130	9	a	a	DET
ma-199	130	10	krein	krein	NOUN
ma-199	130	11	space	space	NOUN
ma-199	131	1	k.	k.	PROPN
ma-199	132	1	then	then	ADV
ma-199	132	2	∑	∑	PROPN
ma-199	132	3	n∈i+	n∈i+	PROPN
ma-199	132	4	cnfn	cnfn	PROPN
ma-199	132	5	and∑	and∑	PART
ma-199	132	6	n∈i−	n∈i−	PROPN
ma-199	132	7	cnfn	cnfn	NOUN
ma-199	132	8	converge	converge	VERB
ma-199	132	9	unconditionally	unconditionally	ADV
ma-199	132	10	for	for	ADP
ma-199	132	11	all	all	DET
ma-199	132	12	{	{	PUNCT
ma-199	132	13	cn}n∈i+	cn}n∈i+	PROPN
ma-199	132	14	∈	∈	PROPN
ma-199	132	15	`	`	PUNCT
ma-199	132	16	2(i+	2(i+	NUM
ma-199	132	17	)	)	PUNCT
ma-199	132	18	and	and	CCONJ
ma-199	132	19	{	{	PUNCT
ma-199	132	20	cn}n∈i−	cn}n∈i−	X
ma-199	132	21	∈	∈	PROPN
ma-199	132	22	`	`	PUNCT
ma-199	132	23	2(i−	2(i−	NUM
ma-199	132	24	)	)	PUNCT
ma-199	132	25	respectively	respectively	ADV
ma-199	132	26	.	.	PUNCT
ma-199	133	1	the	the	DET
ma-199	133	2	following	follow	VERB
ma-199	133	3	example	example	NOUN
ma-199	133	4	illustrates	illustrate	VERB
ma-199	133	5	that	that	SCONJ
ma-199	133	6	how	how	SCONJ
ma-199	133	7	the	the	DET
ma-199	133	8	norm	norm	NOUN
ma-199	133	9	of	of	ADP
ma-199	133	10	a	a	DET
ma-199	133	11	single	single	ADJ
ma-199	133	12	element	element	NOUN
ma-199	133	13	actually	actually	ADV
ma-199	133	14	depends	depend	VERB
ma-199	133	15	uponthe	uponthe	DET
ma-199	133	16	choice	choice	NOUN
ma-199	133	17	of	of	ADP
ma-199	133	18	fundamental	fundamental	ADJ
ma-199	133	19	decomposition	decomposition	NOUN
ma-199	133	20	.	.	PUNCT
ma-199	134	1	if	if	SCONJ
ma-199	134	2	a	a	DET
ma-199	134	3	frame	frame	NOUN
ma-199	134	4	is	be	AUX
ma-199	134	5	defined	define	VERB
ma-199	134	6	relative	relative	ADJ
ma-199	134	7	to	to	ADP
ma-199	134	8	fundamental	fundamental	ADJ
ma-199	134	9	decom	decom	NOUN
ma-199	134	10	-	-	PUNCT
ma-199	134	11	position	position	NOUN
ma-199	134	12	,	,	PUNCT
ma-199	134	13	then	then	ADV
ma-199	134	14	frame	frame	NOUN
ma-199	134	15	bounds	bound	NOUN
ma-199	134	16	will	will	AUX
ma-199	134	17	vary	vary	VERB
ma-199	134	18	arbitrarily	arbitrarily	ADV
ma-199	134	19	when	when	SCONJ
ma-199	134	20	difference	difference	NOUN
ma-199	134	21	fundamental	fundamental	ADJ
ma-199	134	22	decompositions	decomposition	NOUN
ma-199	134	23	areconsidered	areconsidere	VERB
ma-199	134	24	.	.	PUNCT
ma-199	134	25	example	example	NOUN
ma-199	134	26	3.2	3.2	NUM
ma-199	134	27	.	.	PUNCT
ma-199	135	1	consider	consider	VERB
ma-199	135	2	the	the	DET
ma-199	135	3	two	two	NUM
ma-199	135	4	dimensional	dimensional	ADJ
ma-199	135	5	minkowski	minkowski	ADJ
ma-199	135	6	space	space	NOUN
ma-199	135	7	k	k	NOUN
ma-199	135	8	=	=	PUNCT
ma-199	135	9	r2	r2	PROPN
ma-199	135	10	with	with	ADP
ma-199	135	11	the	the	DET
ma-199	135	12	inner	inner	ADJ
ma-199	135	13	product	product	NOUN
ma-199	135	14	[	[	X
ma-199	135	15	f	f	X
ma-199	135	16	,	,	PUNCT
ma-199	135	17	g	g	NOUN
ma-199	135	18	]	]	X
ma-199	135	19	=	=	SYM
ma-199	135	20	f1g1−f2g2	f1g1−f2g2	NOUN
ma-199	135	21	where	where	SCONJ
ma-199	135	22	f	f	PROPN
ma-199	135	23	=	=	PRON
ma-199	135	24	(	(	PUNCT
ma-199	135	25	f1	f1	PROPN
ma-199	135	26	,	,	PUNCT
ma-199	135	27	f2	f2	PROPN
ma-199	135	28	)	)	PUNCT
ma-199	135	29	,	,	PUNCT
ma-199	135	30	g	g	PROPN
ma-199	135	31	=	=	SYM
ma-199	135	32	(	(	PUNCT
ma-199	135	33	g1	g1	PROPN
ma-199	135	34	,	,	PUNCT
ma-199	135	35	g2	g2	PROPN
ma-199	135	36	)	)	PUNCT
ma-199	135	37	∈	∈	PROPN
ma-199	135	38	r2	r2	PROPN
ma-199	135	39	.	.	PUNCT
ma-199	136	1	consider	consider	VERB
ma-199	136	2	the	the	DET
ma-199	136	3	fundamental	fundamental	ADJ
ma-199	136	4	decompositions	decomposition	NOUN
ma-199	136	5	with	with	ADP
ma-199	136	6	k+n	k+n	PROPN
ma-199	136	7	=	=	SYM
ma-199	136	8	span{(n+1n	span{(n+1n	PROPN
ma-199	136	9	,	,	PUNCT
ma-199	136	10	n−1	n−1	PROPN
ma-199	136	11	n	n	NOUN
ma-199	136	12	)	)	PUNCT
ma-199	136	13	}	}	PUNCT
ma-199	136	14	and	and	CCONJ
ma-199	136	15	k−n	k−n	PROPN
ma-199	136	16	=	=	SYM
ma-199	136	17	span{(n−1n	span{(n−1n	PROPN
ma-199	136	18	,	,	PUNCT
ma-199	136	19	n+1	n+1	PROPN
ma-199	136	20	n	n	PROPN
ma-199	136	21	)	)	PUNCT
ma-199	136	22	}	}	PUNCT
ma-199	137	1	where	where	SCONJ
ma-199	137	2	n	n	X
ma-199	137	3	>	>	X
ma-199	137	4	1	1	X
ma-199	137	5	.	.	PUNCT
ma-199	138	1	then	then	ADV
ma-199	138	2	we	we	PRON
ma-199	138	3	get	get	VERB
ma-199	138	4	‖f	‖f	PRON
ma-199	138	5	‖2jn	‖2jn	NOUN
ma-199	139	1	=	=	SYM
ma-199	139	2	1	1	NUM
ma-199	139	3	4	4	NUM
ma-199	139	4	[	[	X
ma-199	139	5	(	(	PUNCT
ma-199	139	6	2n	2n	ADJ
ma-199	139	7	+	+	CCONJ
ma-199	139	8	2	2	NUM
ma-199	139	9	/	/	SYM
ma-199	139	10	n)(f	n)(f	NOUN
ma-199	139	11	21	21	NUM
ma-199	139	12	+	+	CCONJ
ma-199	139	13	f	f	PROPN
ma-199	139	14	2	2	NUM
ma-199	139	15	2	2	NUM
ma-199	139	16	)	)	PUNCT
ma-199	140	1	+	+	CCONJ
ma-199	140	2	4f1f2(1	4f1f2(1	NUM
ma-199	140	3	/	/	SYM
ma-199	140	4	n	n	CCONJ
ma-199	140	5	−	−	NOUN
ma-199	140	6	n	n	CCONJ
ma-199	140	7	)	)	PUNCT
ma-199	140	8	]	]	PUNCT
ma-199	140	9	.	.	PUNCT
ma-199	141	1	let	let	VERB
ma-199	141	2	f	f	PROPN
ma-199	141	3	=	=	SYM
ma-199	141	4	(	(	PUNCT
ma-199	141	5	1	1	NUM
ma-199	141	6	,	,	PUNCT
ma-199	141	7	1	1	NUM
ma-199	141	8	)	)	PUNCT
ma-199	141	9	and	and	CCONJ
ma-199	141	10	g	g	NOUN
ma-199	141	11	=	=	SYM
ma-199	141	12	(	(	PUNCT
ma-199	141	13	1	1	NUM
ma-199	141	14	,	,	PUNCT
ma-199	141	15	0	0	NUM
ma-199	141	16	)	)	PUNCT
ma-199	141	17	.	.	PUNCT
ma-199	142	1	then	then	ADV
ma-199	142	2	‖f	‖f	PRON
ma-199	142	3	‖2jn	‖2jn	NOUN
ma-199	143	1	=	=	SYM
ma-199	143	2	2	2	NUM
ma-199	143	3	n	n	NUM
ma-199	143	4	and	and	CCONJ
ma-199	143	5	‖g‖2jn	‖g‖2jn	X
ma-199	143	6	=	=	NOUN
ma-199	143	7	1	1	NUM
ma-199	143	8	2(n	2(n	NUM
ma-199	143	9	+	+	CCONJ
ma-199	143	10	1	1	NUM
ma-199	143	11	n	n	NUM
ma-199	143	12	)	)	PUNCT
ma-199	143	13	.	.	PUNCT
ma-199	144	1	https://doi.org/10.28924/ada/ma.4.1	https://doi.org/10.28924/ada/ma.4.1	NUM
ma-199	144	2	eur	eur	PROPN
ma-199	144	3	.	.	PUNCT
ma-199	145	1	j.	j.	PROPN
ma-199	145	2	math	math	PROPN
ma-199	145	3	.	.	PUNCT
ma-199	146	1	anal	anal	PROPN
ma-199	146	2	.	.	PUNCT
ma-199	147	1	10.28924	10.28924	NUM
ma-199	147	2	/	/	SYM
ma-199	147	3	ada	ada	PROPN
ma-199	147	4	/	/	SYM
ma-199	147	5	ma.4.1	ma.4.1	PROPN
ma-199	147	6	6let	6let	PROPN
ma-199	147	7	{	{	PUNCT
ma-199	147	8	fn}n∈i	fn}n∈i	PUNCT
ma-199	147	9	be	be	AUX
ma-199	147	10	a	a	DET
ma-199	147	11	bessel	bessel	ADJ
ma-199	147	12	sequence	sequence	NOUN
ma-199	147	13	in	in	ADP
ma-199	147	14	k.	k.	PROPN
ma-199	147	15	then	then	ADV
ma-199	147	16	both	both	DET
ma-199	147	17	{	{	PUNCT
ma-199	147	18	fn}n∈i+	fn}n∈i+	PROPN
ma-199	147	19	⊂	⊂	PROPN
ma-199	147	20	k+	k+	PROPN
ma-199	147	21	and	and	CCONJ
ma-199	147	22	{	{	PUNCT
ma-199	147	23	fn}n∈i−	fn}n∈i−	ADJ
ma-199	147	24	⊂	⊂	PROPN
ma-199	147	25	k−	k−	PROPN
ma-199	147	26	arebessel	arebessel	PROPN
ma-199	147	27	sequences	sequence	NOUN
ma-199	147	28	.	.	PUNCT
ma-199	148	1	define	define	VERB
ma-199	148	2	t+	t+	NOUN
ma-199	148	3	:	:	PUNCT
ma-199	148	4	`	`	PUNCT
ma-199	148	5	2(i+	2(i+	X
ma-199	148	6	)	)	PUNCT
ma-199	148	7	−→	−→	ADJ
ma-199	148	8	k+	k+	NOUN
ma-199	148	9	and	and	CCONJ
ma-199	148	10	t−	t−	PROPN
ma-199	148	11	:	:	PUNCT
ma-199	148	12	`	`	PUNCT
ma-199	148	13	2(i−	2(i−	NUM
ma-199	148	14	)	)	PUNCT
ma-199	148	15	−→	−→	NOUN
ma-199	148	16	k−	k−	NOUN
ma-199	148	17	by	by	ADP
ma-199	148	18	t+{cn	t+{cn	NOUN
ma-199	148	19	}	}	PUNCT
ma-199	148	20	=	=	SYM
ma-199	148	21	∑	∑	PROPN
ma-199	148	22	n∈i+	n∈i+	X
ma-199	148	23	cnfn	cnfn	PROPN
ma-199	148	24	and	and	CCONJ
ma-199	148	25	t−{cn	t−{cn	NOUN
ma-199	148	26	}	}	PUNCT
ma-199	148	27	=	=	PUNCT
ma-199	148	28	∑	∑	PROPN
ma-199	148	29	n∈i−	n∈i−	PROPN
ma-199	148	30	cnfn	cnfn	NOUN
ma-199	148	31	respectively	respectively	ADV
ma-199	148	32	.	.	PUNCT
ma-199	149	1	then	then	ADV
ma-199	149	2	t+	t+	PUNCT
ma-199	149	3	and	and	CCONJ
ma-199	149	4	t−	t−	PROPN
ma-199	149	5	are	be	AUX
ma-199	149	6	both	both	PRON
ma-199	149	7	bounded	bound	VERB
ma-199	149	8	linear	linear	PROPN
ma-199	149	9	operators	operator	NOUN
ma-199	149	10	and	and	CCONJ
ma-199	149	11	are	be	AUX
ma-199	149	12	called	call	VERB
ma-199	149	13	synthesis	synthesis	NOUN
ma-199	149	14	opera	opera	NOUN
ma-199	149	15	-	-	PUNCT
ma-199	149	16	tors	tor	NOUN
ma-199	149	17	.	.	PUNCT
ma-199	150	1	define	define	VERB
ma-199	150	2	t+∗	t+∗	NOUN
ma-199	150	3	:	:	PUNCT
ma-199	150	4	k+	k+	X
ma-199	150	5	−→	−→	ADJ
ma-199	150	6	`	`	PUNCT
ma-199	150	7	2(i+	2(i+	NUM
ma-199	150	8	)	)	PUNCT
ma-199	150	9	and	and	CCONJ
ma-199	150	10	t−∗	t−∗	NOUN
ma-199	150	11	:	:	PUNCT
ma-199	150	12	k−	k−	PROPN
ma-199	150	13	−→	−→	ADJ
ma-199	150	14	`	`	PUNCT
ma-199	150	15	2(i−	2(i−	NUM
ma-199	150	16	)	)	PUNCT
ma-199	150	17	by	by	ADP
ma-199	150	18	t+∗f	t+∗f	NOUN
ma-199	150	19	=	=	SYM
ma-199	150	20	{	{	PUNCT
ma-199	150	21	[	[	X
ma-199	150	22	f	f	X
ma-199	150	23	,	,	PUNCT
ma-199	150	24	fn]}n∈i+	fn]}n∈i+	PROPN
ma-199	150	25	and	and	CCONJ
ma-199	150	26	t−∗f	t−∗f	PROPN
ma-199	150	27	=	=	SYM
ma-199	150	28	{	{	PUNCT
ma-199	150	29	[	[	X
ma-199	150	30	f	f	X
ma-199	150	31	,	,	PUNCT
ma-199	150	32	fn]}n∈i−are	fn]}n∈i−are	PROPN
ma-199	150	33	called	call	VERB
ma-199	150	34	analysis	analysis	NOUN
ma-199	150	35	operators	operator	NOUN
ma-199	150	36	.	.	PUNCT
ma-199	151	1	thus	thus	ADV
ma-199	151	2	s+	s+	ADV
ma-199	151	3	=	=	SYM
ma-199	151	4	t+t+∗	t+t+∗	PROPN
ma-199	151	5	:	:	PUNCT
ma-199	151	6	k+	k+	X
ma-199	151	7	−→	−→	ADJ
ma-199	151	8	k+	k+	NOUN
ma-199	151	9	given	give	VERB
ma-199	151	10	by	by	ADP
ma-199	151	11	s+f	s+f	PROPN
ma-199	151	12	=	=	SYM
ma-199	151	13	t+t+∗f	t+t+∗f	PROPN
ma-199	151	14	=	=	NOUN
ma-199	151	15	∑	∑	X
ma-199	151	16	n∈i+	n∈i+	ADP
ma-199	151	17	[	[	X
ma-199	151	18	f	f	X
ma-199	151	19	,	,	PUNCT
ma-199	151	20	fn]fn	fn]fn	ADJ
ma-199	151	21	and	and	CCONJ
ma-199	151	22	s−	s−	PROPN
ma-199	151	23	=	=	SYM
ma-199	151	24	t−t−∗	t−t−∗	NOUN
ma-199	151	25	:	:	PUNCT
ma-199	151	26	k−	k−	PROPN
ma-199	151	27	−→	−→	PROPN
ma-199	151	28	k−	k−	PROPN
ma-199	151	29	given	give	VERB
ma-199	151	30	by	by	ADP
ma-199	151	31	s−f	s−f	NOUN
ma-199	151	32	=	=	SYM
ma-199	151	33	t−t−∗f	t−t−∗f	PROPN
ma-199	151	34	=	=	PUNCT
ma-199	151	35	∑	∑	PUNCT
ma-199	151	36	n∈i−	n∈i−	PROPN
ma-199	151	37	[	[	X
ma-199	151	38	f	f	X
ma-199	151	39	,	,	PUNCT
ma-199	151	40	fn]fn.therefore	fn]fn.therefore	ADP
ma-199	151	41	the	the	DET
ma-199	151	42	frame	frame	NOUN
ma-199	151	43	operator	operator	NOUN
ma-199	151	44	s	s	PART
ma-199	151	45	=	=	PUNCT
ma-199	151	46	s+	s+	ADV
ma-199	151	47	+	+	CCONJ
ma-199	151	48	s−	s−	PROPN
ma-199	151	49	:	:	PUNCT
ma-199	152	1	k	k	X
ma-199	152	2	−→	−→	NOUN
ma-199	152	3	k	k	PROPN
ma-199	152	4	is	be	AUX
ma-199	152	5	defined	define	VERB
ma-199	152	6	by	by	ADP
ma-199	152	7	sf	sf	PROPN
ma-199	152	8	=	=	PUNCT
ma-199	152	9	(	(	PUNCT
ma-199	152	10	s+	s+	X
ma-199	152	11	+	+	X
ma-199	152	12	s−)f	s−)f	X
ma-199	152	13	=	=	SYM
ma-199	152	14	∑	∑	PUNCT
ma-199	152	15	n∈i+	n∈i+	X
ma-199	152	16	[	[	X
ma-199	152	17	f	f	X
ma-199	152	18	,	,	PUNCT
ma-199	152	19	fn]fn	fn]fn	ADJ
ma-199	152	20	+	+	CCONJ
ma-199	152	21	∑	∑	PROPN
ma-199	152	22	n∈i−	n∈i−	PROPN
ma-199	152	23	[	[	X
ma-199	152	24	f	f	X
ma-199	152	25	,	,	PUNCT
ma-199	152	26	fn]fn	fn]fn	ADJ
ma-199	152	27	.	.	PUNCT
ma-199	153	1	esmeral	esmeral	PROPN
ma-199	153	2	et	et	PROPN
ma-199	153	3	al	al	PROPN
ma-199	153	4	.	.	PUNCT
ma-199	154	1	[	[	X
ma-199	154	2	5	5	NUM
ma-199	154	3	]	]	PUNCT
ma-199	154	4	have	have	AUX
ma-199	154	5	given	give	VERB
ma-199	154	6	a	a	DET
ma-199	154	7	defintion	defintion	NOUN
ma-199	154	8	of	of	ADP
ma-199	154	9	frame	frame	NOUN
ma-199	154	10	which	which	PRON
ma-199	154	11	involves	involve	VERB
ma-199	154	12	fundamental	fundamental	ADJ
ma-199	154	13	symmetry	symmetry	NOUN
ma-199	154	14	of	of	ADP
ma-199	154	15	thekrein	thekrein	ADJ
ma-199	154	16	space	space	NOUN
ma-199	154	17	k.	k.	PROPN
ma-199	154	18	as	as	SCONJ
ma-199	154	19	shown	show	VERB
ma-199	154	20	in	in	ADP
ma-199	154	21	the	the	DET
ma-199	154	22	example	example	NOUN
ma-199	154	23	3.2	3.2	NUM
ma-199	154	24	,	,	PUNCT
ma-199	154	25	for	for	ADP
ma-199	154	26	sufficiently	sufficiently	ADV
ma-199	154	27	large	large	ADJ
ma-199	154	28	values	value	NOUN
ma-199	154	29	on	on	ADP
ma-199	154	30	n	n	CCONJ
ma-199	154	31	,	,	PUNCT
ma-199	154	32	j	j	PROPN
ma-199	154	33	-	-	PUNCT
ma-199	154	34	norms	norm	NOUN
ma-199	154	35	of	of	ADP
ma-199	154	36	elementsof	elementsof	NOUN
ma-199	154	37	k	k	PROPN
ma-199	154	38	can	can	AUX
ma-199	154	39	be	be	AUX
ma-199	154	40	too	too	ADV
ma-199	154	41	small	small	ADJ
ma-199	154	42	or	or	CCONJ
ma-199	154	43	too	too	ADV
ma-199	154	44	large	large	ADJ
ma-199	154	45	.	.	PUNCT
ma-199	155	1	thus	thus	ADV
ma-199	155	2	we	we	PRON
ma-199	155	3	propose	propose	VERB
ma-199	155	4	the	the	DET
ma-199	155	5	following	following	ADJ
ma-199	155	6	definition	definition	NOUN
ma-199	155	7	for	for	ADP
ma-199	155	8	frame	frame	NOUN
ma-199	155	9	in	in	ADP
ma-199	155	10	kreinspaces	kreinspace	NOUN
ma-199	155	11	.	.	PUNCT
ma-199	156	1	definition	definition	NOUN
ma-199	156	2	3.3	3.3	NUM
ma-199	156	3	.	.	PUNCT
ma-199	157	1	let	let	VERB
ma-199	157	2	{	{	PUNCT
ma-199	157	3	fn}n∈i	fn}n∈i	VERB
ma-199	157	4	be	be	AUX
ma-199	157	5	a	a	DET
ma-199	157	6	bessel	bessel	ADJ
ma-199	157	7	sequence	sequence	NOUN
ma-199	157	8	in	in	ADP
ma-199	157	9	k.	k.	PROPN
ma-199	158	1	the	the	DET
ma-199	158	2	sequence	sequence	NOUN
ma-199	158	3	{	{	PUNCT
ma-199	158	4	fn}n∈i	fn}n∈i	NOUN
ma-199	158	5	}	}	PUNCT
ma-199	158	6	is	be	AUX
ma-199	158	7	said	say	VERB
ma-199	158	8	to	to	PART
ma-199	158	9	be	be	AUX
ma-199	158	10	a	a	DET
ma-199	158	11	frame	frame	NOUN
ma-199	158	12	if	if	SCONJ
ma-199	158	13	the	the	DET
ma-199	158	14	frame	frame	NOUN
ma-199	158	15	operator	operator	NOUN
ma-199	158	16	s	s	PART
ma-199	158	17	=	=	PUNCT
ma-199	158	18	s+	s+	ADV
ma-199	158	19	+	+	CCONJ
ma-199	158	20	s−	s−	PROPN
ma-199	158	21	:	:	PUNCT
ma-199	159	1	k	k	PROPN
ma-199	159	2	−→	−→	PROPN
ma-199	159	3	k	k	PROPN
ma-199	159	4	defined	define	VERB
ma-199	159	5	by	by	ADP
ma-199	159	6	sf	sf	PROPN
ma-199	159	7	=	=	PUNCT
ma-199	159	8	(	(	PUNCT
ma-199	159	9	s+	s+	X
ma-199	159	10	+	+	X
ma-199	159	11	s−)f	s−)f	X
ma-199	159	12	=	=	SYM
ma-199	159	13	∑	∑	PUNCT
ma-199	159	14	n∈i+	n∈i+	X
ma-199	159	15	[	[	X
ma-199	159	16	f	f	X
ma-199	159	17	,	,	PUNCT
ma-199	159	18	fn]fn	fn]fn	ADJ
ma-199	159	19	+	+	CCONJ
ma-199	159	20	∑	∑	PROPN
ma-199	159	21	n∈i−	n∈i−	PROPN
ma-199	159	22	[	[	X
ma-199	159	23	f	f	X
ma-199	159	24	,	,	PUNCT
ma-199	159	25	fn]fn	fn]fn	PROPN
ma-199	159	26	is	be	AUX
ma-199	159	27	a	a	DET
ma-199	159	28	bounded	bounded	ADJ
ma-199	159	29	positive	positive	ADJ
ma-199	159	30	invertible	invertible	ADJ
ma-199	159	31	operator	operator	NOUN
ma-199	159	32	.	.	PUNCT
ma-199	160	1	the	the	DET
ma-199	160	2	following	follow	VERB
ma-199	160	3	is	be	AUX
ma-199	160	4	the	the	DET
ma-199	160	5	frame	frame	NOUN
ma-199	160	6	decomposition	decomposition	NOUN
ma-199	160	7	theorem	theorem	NOUN
ma-199	160	8	for	for	ADP
ma-199	160	9	the	the	DET
ma-199	160	10	krein	krein	PROPN
ma-199	160	11	space	space	NOUN
ma-199	160	12	k	k	PROPN
ma-199	160	13	,	,	PUNCT
ma-199	160	14	which	which	PRON
ma-199	160	15	states	state	VERB
ma-199	160	16	that	that	SCONJ
ma-199	160	17	if	if	SCONJ
ma-199	160	18	{	{	PUNCT
ma-199	160	19	fn}n∈i	fn}n∈i	NOUN
ma-199	160	20	is	be	AUX
ma-199	160	21	a	a	DET
ma-199	160	22	frame	frame	NOUN
ma-199	160	23	for	for	ADP
ma-199	160	24	a	a	DET
ma-199	160	25	krein	krein	ADJ
ma-199	160	26	space	space	NOUN
ma-199	160	27	k	k	NOUN
ma-199	160	28	,	,	PUNCT
ma-199	160	29	then	then	ADV
ma-199	160	30	every	every	DET
ma-199	160	31	element	element	NOUN
ma-199	160	32	can	can	AUX
ma-199	160	33	be	be	AUX
ma-199	160	34	written	write	VERB
ma-199	160	35	as	as	ADP
ma-199	160	36	a	a	DET
ma-199	160	37	linear	linear	ADJ
ma-199	160	38	combinationof	combinationof	NOUN
ma-199	160	39	frame	frame	NOUN
ma-199	160	40	elements	element	NOUN
ma-199	160	41	.	.	PUNCT
ma-199	161	1	theorem	theorem	VERB
ma-199	161	2	3.4	3.4	NUM
ma-199	161	3	.	.	PUNCT
ma-199	162	1	let	let	VERB
ma-199	162	2	{	{	PUNCT
ma-199	162	3	fn}n∈i	fn}n∈i	VERB
ma-199	162	4	be	be	AUX
ma-199	162	5	a	a	DET
ma-199	162	6	frame	frame	NOUN
ma-199	162	7	for	for	ADP
ma-199	162	8	a	a	DET
ma-199	162	9	krein	krein	ADJ
ma-199	162	10	space	space	NOUN
ma-199	162	11	k	k	PROPN
ma-199	162	12	with	with	ADP
ma-199	162	13	the	the	DET
ma-199	162	14	frame	frame	NOUN
ma-199	162	15	operator	operator	NOUN
ma-199	162	16	s.	s.	PROPN
ma-199	162	17	then	then	ADV
ma-199	162	18	f	f	PROPN
ma-199	163	1	=	=	PUNCT
ma-199	163	2	∑	∑	PROPN
ma-199	163	3	n∈i+	n∈i+	X
ma-199	164	1	[	[	X
ma-199	164	2	f	f	X
ma-199	164	3	,	,	PUNCT
ma-199	164	4	s+	s+	ADV
ma-199	164	5	−1	−1	NOUN
ma-199	164	6	fn]fn	fn]fn	ADJ
ma-199	164	7	+	+	CCONJ
ma-199	164	8	∑	∑	PROPN
ma-199	164	9	n∈i−	n∈i−	PROPN
ma-199	164	10	[	[	X
ma-199	164	11	f	f	X
ma-199	164	12	,	,	PUNCT
ma-199	164	13	s−	s−	PROPN
ma-199	164	14	−1	−1	NOUN
ma-199	164	15	fn]fn	fn]fn	ADJ
ma-199	164	16	,	,	PUNCT
ma-199	164	17	for	for	ADP
ma-199	164	18	all	all	DET
ma-199	164	19	f	f	PROPN
ma-199	164	20	∈	∈	PROPN
ma-199	164	21	k	k	PROPN
ma-199	164	22	,	,	PUNCT
ma-199	164	23	(	(	PUNCT
ma-199	164	24	7	7	NUM
ma-199	164	25	)	)	PUNCT
ma-199	164	26	and	and	CCONJ
ma-199	164	27	both	both	DET
ma-199	164	28	the	the	DET
ma-199	164	29	series	series	NOUN
ma-199	164	30	converges	converge	VERB
ma-199	164	31	unconditionally	unconditionally	ADV
ma-199	164	32	for	for	ADP
ma-199	164	33	all	all	DET
ma-199	164	34	f	f	PROPN
ma-199	164	35	∈	∈	PROPN
ma-199	164	36	k.	k.	PROPN
ma-199	165	1	https://doi.org/10.28924/ada/ma.4.1	https://doi.org/10.28924/ada/ma.4.1	PROPN
ma-199	165	2	eur	eur	PROPN
ma-199	165	3	.	.	PUNCT
ma-199	166	1	j.	j.	PROPN
ma-199	166	2	math	math	PROPN
ma-199	166	3	.	.	PUNCT
ma-199	167	1	anal	anal	PROPN
ma-199	167	2	.	.	PUNCT
ma-199	168	1	10.28924	10.28924	NUM
ma-199	168	2	/	/	SYM
ma-199	168	3	ada	ada	PROPN
ma-199	168	4	/	/	SYM
ma-199	168	5	ma.4.1	ma.4.1	PROPN
ma-199	168	6	7	7	NUM
ma-199	168	7	proof	proof	NOUN
ma-199	168	8	.	.	PUNCT
ma-199	169	1	since	since	SCONJ
ma-199	169	2	the	the	DET
ma-199	169	3	operator	operator	NOUN
ma-199	169	4	s	s	VERB
ma-199	169	5	is	be	AUX
ma-199	169	6	self	self	NOUN
ma-199	169	7	adjoint	adjoint	NOUN
ma-199	169	8	and	and	CCONJ
ma-199	169	9	invertible	invertible	ADJ
ma-199	169	10	,	,	PUNCT
ma-199	169	11	we	we	PRON
ma-199	169	12	have	have	VERB
ma-199	169	13	f	f	NOUN
ma-199	169	14	=	=	SYM
ma-199	169	15	ss−1f	ss−1f	PROPN
ma-199	170	1	=	=	SYM
ma-199	170	2	∑	∑	PUNCT
ma-199	170	3	n∈i	n∈i	PROPN
ma-199	170	4	[	[	X
ma-199	170	5	s−1f	s−1f	NOUN
ma-199	170	6	,	,	PUNCT
ma-199	170	7	fn]fn	fn]fn	ADJ
ma-199	170	8	=	=	PUNCT
ma-199	170	9	∑	∑	PUNCT
ma-199	170	10	n∈i+	n∈i+	X
ma-199	171	1	[	[	X
ma-199	171	2	s+	s+	ADP
ma-199	171	3	−1	−1	NOUN
ma-199	171	4	f	f	PROPN
ma-199	171	5	,	,	PUNCT
ma-199	171	6	fn]fn	fn]fn	ADJ
ma-199	171	7	+	+	CCONJ
ma-199	171	8	∑	∑	PROPN
ma-199	171	9	n∈i−	n∈i−	PROPN
ma-199	171	10	[	[	X
ma-199	171	11	s−	s−	PROPN
ma-199	171	12	−1	−1	NOUN
ma-199	171	13	f	f	PROPN
ma-199	171	14	,	,	PUNCT
ma-199	171	15	fn]fn	fn]fn	ADJ
ma-199	171	16	=	=	PUNCT
ma-199	171	17	∑	∑	PUNCT
ma-199	171	18	n∈i+	n∈i+	X
ma-199	171	19	[	[	X
ma-199	171	20	f	f	X
ma-199	171	21	,	,	PUNCT
ma-199	171	22	s+	s+	ADV
ma-199	171	23	−1	−1	NOUN
ma-199	171	24	fn]fn	fn]fn	ADJ
ma-199	171	25	+	+	CCONJ
ma-199	171	26	∑	∑	PROPN
ma-199	171	27	n∈i−	n∈i−	PROPN
ma-199	171	28	[	[	X
ma-199	171	29	f	f	X
ma-199	171	30	,	,	PUNCT
ma-199	171	31	s−	s−	PROPN
ma-199	171	32	−1	−1	ADV
ma-199	171	33	fn]fn	fn]fn	ADJ
ma-199	171	34	since	since	SCONJ
ma-199	171	35	{	{	PUNCT
ma-199	171	36	fn}n∈i	fn}n∈i	NOUN
ma-199	171	37	is	be	AUX
ma-199	171	38	a	a	DET
ma-199	171	39	frame	frame	NOUN
ma-199	171	40	with	with	ADP
ma-199	171	41	[	[	X
ma-199	171	42	f	f	X
ma-199	171	43	,	,	PUNCT
ma-199	171	44	s+−1fn	s+−1fn	ADP
ma-199	171	45	]	]	X
ma-199	171	46	∈	∈	PROPN
ma-199	171	47	`	`	PUNCT
ma-199	171	48	2(i+	2(i+	NUM
ma-199	171	49	)	)	PUNCT
ma-199	171	50	and	and	CCONJ
ma-199	171	51	[	[	X
ma-199	171	52	f	f	X
ma-199	171	53	,	,	PUNCT
ma-199	171	54	s−−1fn]fn	s−−1fn]fn	VERB
ma-199	171	55	∈	∈	PROPN
ma-199	171	56	`	`	PUNCT
ma-199	171	57	2(i−	2(i−	PROPN
ma-199	171	58	)	)	PUNCT
ma-199	171	59	,	,	PUNCT
ma-199	171	60	the	the	DET
ma-199	171	61	unconditionalconvergence	unconditionalconvergence	NOUN
ma-199	171	62	follows	follow	VERB
ma-199	171	63	from	from	ADP
ma-199	171	64	corollary	corollary	ADJ
ma-199	171	65	3.3	3.3	NUM
ma-199	171	66	.	.	PUNCT
ma-199	172	1	�	�	PROPN
ma-199	172	2	4	4	NUM
ma-199	172	3	.	.	PUNCT
ma-199	172	4	frame	frame	NOUN
ma-199	172	5	sequences	sequence	NOUN
ma-199	172	6	in	in	ADP
ma-199	172	7	krein	krein	ADJ
ma-199	172	8	spaces	space	NOUN
ma-199	172	9	we	we	PRON
ma-199	172	10	begin	begin	VERB
ma-199	172	11	this	this	DET
ma-199	172	12	section	section	NOUN
ma-199	172	13	with	with	ADP
ma-199	172	14	the	the	DET
ma-199	172	15	following	follow	VERB
ma-199	172	16	definitions	definition	NOUN
ma-199	172	17	for	for	ADP
ma-199	172	18	frame	frame	NOUN
ma-199	172	19	sequence	sequence	NOUN
ma-199	172	20	definition	definition	NOUN
ma-199	172	21	4.1	4.1	NUM
ma-199	172	22	.	.	PUNCT
ma-199	173	1	let	let	VERB
ma-199	173	2	k	k	PRON
ma-199	173	3	be	be	AUX
ma-199	173	4	a	a	DET
ma-199	173	5	krein	krein	ADJ
ma-199	173	6	space	space	NOUN
ma-199	173	7	.	.	PUNCT
ma-199	174	1	a	a	DET
ma-199	174	2	sequence	sequence	NOUN
ma-199	174	3	{	{	PUNCT
ma-199	174	4	fn}n∈i	fn}n∈i	NOUN
ma-199	174	5	∈	∈	NOUN
ma-199	174	6	k	k	PROPN
ma-199	174	7	is	be	AUX
ma-199	174	8	called	call	VERB
ma-199	174	9	a(a	a(a	PROPN
ma-199	174	10	)	)	PUNCT
ma-199	174	11	frame	frame	NOUN
ma-199	174	12	sequence	sequence	NOUN
ma-199	174	13	if	if	SCONJ
ma-199	174	14	it	it	PRON
ma-199	174	15	is	be	AUX
ma-199	174	16	a	a	DET
ma-199	174	17	frame	frame	NOUN
ma-199	174	18	for	for	ADP
ma-199	174	19	[	[	X
ma-199	174	20	fn	fn	X
ma-199	174	21	]	]	X
ma-199	174	22	=	=	SYM
ma-199	174	23	span{fn	span{fn	NOUN
ma-199	174	24	:	:	PUNCT
ma-199	174	25	n	n	PRON
ma-199	174	26	∈	∈	PROPN
ma-199	174	27	i}.(b	i}.(b	PROPN
ma-199	174	28	)	)	PUNCT
ma-199	174	29	exact	exact	ADJ
ma-199	174	30	if	if	SCONJ
ma-199	174	31	removal	removal	NOUN
ma-199	174	32	of	of	ADP
ma-199	174	33	an	an	DET
ma-199	174	34	arbitrary	arbitrary	ADJ
ma-199	174	35	fn	fn	NOUN
ma-199	174	36	render	render	VERB
ma-199	174	37	the	the	DET
ma-199	174	38	collection	collection	NOUN
ma-199	174	39	{	{	PUNCT
ma-199	174	40	fn	fn	NOUN
ma-199	174	41	}	}	PUNCT
ma-199	174	42	no	no	ADV
ma-199	174	43	longer	long	ADV
ma-199	174	44	a	a	DET
ma-199	174	45	frame	frame	NOUN
ma-199	174	46	for	for	ADP
ma-199	174	47	the	the	DET
ma-199	174	48	krein	krein	PROPN
ma-199	174	49	space	space	NOUN
ma-199	174	50	k.(c	k.(c	NOUN
ma-199	174	51	)	)	PUNCT
ma-199	174	52	near	near	ADP
ma-199	174	53	exact	exact	ADJ
ma-199	174	54	if	if	SCONJ
ma-199	174	55	it	it	PRON
ma-199	174	56	can	can	AUX
ma-199	174	57	be	be	AUX
ma-199	174	58	made	make	VERB
ma-199	174	59	exact	exact	ADJ
ma-199	174	60	by	by	ADP
ma-199	174	61	removing	remove	VERB
ma-199	174	62	finitely	finitely	ADV
ma-199	174	63	many	many	ADJ
ma-199	174	64	elements	element	NOUN
ma-199	174	65	from	from	ADP
ma-199	174	66	it	it	PRON
ma-199	174	67	.	.	PUNCT
ma-199	175	1	example	example	NOUN
ma-199	175	2	4.2	4.2	NUM
ma-199	175	3	.	.	PUNCT
ma-199	176	1	let	let	AUX
ma-199	176	2	{	{	PUNCT
ma-199	176	3	fn}n∈i	fn}n∈i	VERB
ma-199	176	4	be	be	AUX
ma-199	176	5	a	a	DET
ma-199	176	6	sequence	sequence	NOUN
ma-199	176	7	of	of	ADP
ma-199	176	8	unit	unit	NOUN
ma-199	176	9	orthonormal	orthonormal	ADJ
ma-199	176	10	vectors	vector	NOUN
ma-199	176	11	in	in	ADP
ma-199	176	12	krein	krein	NOUN
ma-199	176	13	space	space	NOUN
ma-199	176	14	k	k	PROPN
ma-199	176	15	and	and	CCONJ
ma-199	176	16	{	{	PUNCT
ma-199	176	17	nk	nk	NOUN
ma-199	176	18	}	}	PUNCT
ma-199	176	19	be	be	AUX
ma-199	176	20	any	any	DET
ma-199	176	21	infinite	infinite	ADJ
ma-199	176	22	increasing	increase	VERB
ma-199	176	23	subset	subset	NOUN
ma-199	176	24	of	of	ADP
ma-199	176	25	i.	i.	PROPN
ma-199	176	26	then	then	ADV
ma-199	176	27	{	{	PUNCT
ma-199	176	28	fnk	fnk	NOUN
ma-199	176	29	}	}	PUNCT
ma-199	176	30	is	be	AUX
ma-199	176	31	a	a	DET
ma-199	176	32	frame	frame	NOUN
ma-199	176	33	sequence	sequence	NOUN
ma-199	176	34	.	.	PUNCT
ma-199	177	1	let	let	VERB
ma-199	177	2	{	{	PUNCT
ma-199	177	3	fn}n∈i	fn}n∈i	VERB
ma-199	177	4	be	be	AUX
ma-199	177	5	a	a	DET
ma-199	177	6	frame	frame	NOUN
ma-199	177	7	for	for	ADP
ma-199	177	8	a	a	DET
ma-199	177	9	krein	krein	ADJ
ma-199	177	10	space	space	NOUN
ma-199	177	11	k	k	PROPN
ma-199	177	12	and	and	CCONJ
ma-199	177	13	{	{	PUNCT
ma-199	177	14	nk	nk	NOUN
ma-199	177	15	}	}	PUNCT
ma-199	177	16	be	be	AUX
ma-199	177	17	any	any	DET
ma-199	177	18	infinite	infinite	ADJ
ma-199	177	19	increasing	increase	VERB
ma-199	177	20	sequence	sequence	NOUN
ma-199	177	21	in	in	ADP
ma-199	177	22	i	i	PRON
ma-199	177	23	.then	.then	PUNCT
ma-199	177	24	{	{	PUNCT
ma-199	177	25	fnk	fnk	NOUN
ma-199	177	26	}	}	PUNCT
ma-199	177	27	need	need	AUX
ma-199	177	28	not	not	PART
ma-199	177	29	be	be	AUX
ma-199	177	30	a	a	DET
ma-199	177	31	frame	frame	NOUN
ma-199	177	32	sequence	sequence	NOUN
ma-199	177	33	.	.	PUNCT
ma-199	178	1	example	example	NOUN
ma-199	178	2	4.3	4.3	NUM
ma-199	178	3	.	.	PUNCT
ma-199	179	1	let	let	VERB
ma-199	179	2	{	{	PUNCT
ma-199	179	3	fn}n∈i	fn}n∈i	VERB
ma-199	179	4	be	be	AUX
ma-199	179	5	a	a	DET
ma-199	179	6	sequence	sequence	NOUN
ma-199	179	7	of	of	ADP
ma-199	179	8	unit	unit	NOUN
ma-199	179	9	orthonormal	orthonormal	ADJ
ma-199	179	10	vectors	vector	NOUN
ma-199	179	11	in	in	ADP
ma-199	179	12	a	a	DET
ma-199	179	13	krein	krein	NOUN
ma-199	179	14	space	space	NOUN
ma-199	179	15	k.	k.	PROPN
ma-199	179	16	define	define	VERB
ma-199	179	17	a	a	DET
ma-199	179	18	sequence	sequence	NOUN
ma-199	179	19	{	{	PUNCT
ma-199	179	20	hn}n∈i	hn}n∈i	NOUN
ma-199	179	21	∈	∈	PROPN
ma-199	179	22	k	k	X
ma-199	179	23	by	by	ADP
ma-199	179	24	hn	hn	PROPN
ma-199	179	25	=	=	PROPN
ma-199	179	26	1√	1√	PROPN
ma-199	179	27	n	n	CCONJ
ma-199	179	28	fn	fn	NOUN
ma-199	179	29	,	,	PUNCT
ma-199	179	30	n	n	PROPN
ma-199	179	31	∈	∈	PROPN
ma-199	179	32	i.	i.	NOUN
ma-199	179	33	let	let	VERB
ma-199	179	34	nk	nk	PROPN
ma-199	179	35	=	=	PROPN
ma-199	179	36	nk−1	nk−1	PROPN
ma-199	180	1	+	+	SYM
ma-199	180	2	(	(	PUNCT
ma-199	180	3	k	k	NOUN
ma-199	180	4	−	−	PROPN
ma-199	180	5	1	1	NUM
ma-199	180	6	)	)	PUNCT
ma-199	180	7	,	,	PUNCT
ma-199	181	1	k	k	PROPN
ma-199	181	2	∈	∈	PROPN
ma-199	182	1	i	i	PRON
ma-199	182	2	and	and	CCONJ
ma-199	182	3	n0	n0	NOUN
ma-199	182	4	=	=	PUNCT
ma-199	182	5	±1	±1	PROPN
ma-199	182	6	.	.	PUNCT
ma-199	183	1	then	then	ADV
ma-199	183	2	{	{	PUNCT
ma-199	183	3	nk	nk	PROPN
ma-199	183	4	}	}	PUNCT
ma-199	183	5	is	be	AUX
ma-199	183	6	an	an	DET
ma-199	183	7	infinite	infinite	ADJ
ma-199	183	8	increasing	increase	VERB
ma-199	183	9	sequence	sequence	NOUN
ma-199	183	10	in	in	ADP
ma-199	183	11	i	i	PRON
ma-199	183	12	.	.	PUNCT
ma-199	184	1	define	define	VERB
ma-199	184	2	another	another	DET
ma-199	184	3	sequence	sequence	NOUN
ma-199	184	4	{	{	PUNCT
ma-199	184	5	gn}n∈i	gn}n∈i	NOUN
ma-199	184	6	∈	∈	X
ma-199	184	7	k	k	X
ma-199	184	8	by	by	ADP
ma-199	184	9	g1	g1	PROPN
ma-199	184	10	=	=	SYM
ma-199	184	11	h1	h1	PROPN
ma-199	184	12	,	,	PUNCT
ma-199	184	13	gnk	gnk	NOUN
ma-199	184	14	=	=	PUNCT
ma-199	184	15	gnk+1	gnk+1	NOUN
ma-199	184	16	=	=	SYM
ma-199	184	17	gnk+2	gnk+2	NOUN
ma-199	184	18	=	=	SYM
ma-199	184	19	·	·	PUNCT
ma-199	184	20	·	·	PUNCT
ma-199	184	21	·	·	PUNCT
ma-199	185	1	=	=	PRON
ma-199	185	2	gnk+1	gnk+1	VERB
ma-199	185	3	−	−	PROPN
ma-199	185	4	1	1	NUM
ma-199	185	5	=	=	SYM
ma-199	185	6	hk	hk	PROPN
ma-199	185	7	,	,	PUNCT
ma-199	185	8	k	k	X
ma-199	185	9	≥	≥	NUM
ma-199	185	10	2	2	NUM
ma-199	185	11	.	.	PUNCT
ma-199	186	1	then	then	ADV
ma-199	186	2	{	{	PUNCT
ma-199	186	3	gn}n∈i	gn}n∈i	NOUN
ma-199	186	4	is	be	AUX
ma-199	186	5	a	a	DET
ma-199	186	6	tight	tight	ADJ
ma-199	186	7	frame	frame	NOUN
ma-199	186	8	for	for	ADP
ma-199	186	9	k.	k.	PROPN
ma-199	186	10	but	but	CCONJ
ma-199	186	11	note	note	VERB
ma-199	186	12	that	that	SCONJ
ma-199	186	13	{	{	PUNCT
ma-199	186	14	gnk	gnk	NOUN
ma-199	186	15	}	}	PUNCT
ma-199	186	16	=	=	SYM
ma-199	186	17	{	{	PUNCT
ma-199	186	18	hk	hk	NOUN
ma-199	186	19	}	}	PUNCT
ma-199	186	20	is	be	AUX
ma-199	186	21	not	not	PART
ma-199	186	22	a	a	DET
ma-199	186	23	frame	frame	NOUN
ma-199	186	24	sequence	sequence	NOUN
ma-199	186	25	.	.	PUNCT
ma-199	187	1	the	the	DET
ma-199	187	2	following	follow	VERB
ma-199	187	3	theorem	theorem	NOUN
ma-199	187	4	gives	give	VERB
ma-199	187	5	a	a	DET
ma-199	187	6	necessary	necessary	ADJ
ma-199	187	7	and	and	CCONJ
ma-199	187	8	sufficient	sufficient	ADJ
ma-199	187	9	condition	condition	NOUN
ma-199	187	10	for	for	ADP
ma-199	187	11	a	a	DET
ma-199	187	12	existence	existence	NOUN
ma-199	187	13	of	of	ADP
ma-199	187	14	a	a	DET
ma-199	187	15	subsequenceto	subsequenceto	NOUN
ma-199	187	16	be	be	AUX
ma-199	187	17	a	a	DET
ma-199	187	18	frame	frame	NOUN
ma-199	187	19	for	for	SCONJ
ma-199	187	20	a	a	DET
ma-199	187	21	krein	krein	NOUN
ma-199	187	22	space	space	NOUN
ma-199	187	23	k.	k.	PROPN
ma-199	187	24	theorem	theorem	VERB
ma-199	187	25	4.1	4.1	NUM
ma-199	187	26	.	.	PUNCT
ma-199	188	1	let	let	VERB
ma-199	188	2	{	{	PUNCT
ma-199	188	3	fn}n∈i	fn}n∈i	VERB
ma-199	188	4	be	be	AUX
ma-199	188	5	a	a	DET
ma-199	188	6	frame	frame	NOUN
ma-199	188	7	for	for	ADP
ma-199	188	8	a	a	DET
ma-199	188	9	krein	krein	ADJ
ma-199	188	10	space	space	NOUN
ma-199	188	11	k	k	PROPN
ma-199	188	12	and	and	CCONJ
ma-199	188	13	let	let	VERB
ma-199	188	14	{	{	PUNCT
ma-199	188	15	mk	mk	X
ma-199	188	16	}	}	PUNCT
ma-199	188	17	and	and	CCONJ
ma-199	188	18	{	{	PUNCT
ma-199	188	19	nk	nk	NOUN
ma-199	188	20	}	}	PUNCT
ma-199	188	21	be	be	AUX
ma-199	188	22	two	two	NUM
ma-199	188	23	infinite	infinite	ADJ
ma-199	188	24	increasing	increase	VERB
ma-199	188	25	sequences	sequence	NOUN
ma-199	188	26	in	in	ADP
ma-199	188	27	i	i	PRON
ma-199	188	28	with	with	ADP
ma-199	188	29	{	{	PUNCT
ma-199	188	30	m+k	m+k	NUM
ma-199	188	31	}	}	PUNCT
ma-199	188	32	∪{n	∪{n	PROPN
ma-199	189	1	+	+	CCONJ
ma-199	189	2	k	k	X
ma-199	189	3	}	}	PUNCT
ma-199	189	4	=	=	SYM
ma-199	189	5	i+	i+	NOUN
ma-199	189	6	and	and	CCONJ
ma-199	189	7	{	{	PUNCT
ma-199	189	8	m−k	m−k	NOUN
ma-199	189	9	}	}	PUNCT
ma-199	189	10	∪{n	∪{n	NOUN
ma-199	189	11	−	−	NOUN
ma-199	189	12	k	k	NOUN
ma-199	189	13	}	}	PUNCT
ma-199	189	14	=	=	SYM
ma-199	189	15	i−.	i−.	NOUN
ma-199	189	16	if	if	SCONJ
ma-199	189	17	{	{	PUNCT
ma-199	189	18	fmk}mk∈i	fmk}mk∈i	NOUN
ma-199	189	19	is	be	AUX
ma-199	189	20	a	a	DET
ma-199	189	21	frame	frame	NOUN
ma-199	189	22	,	,	PUNCT
ma-199	189	23	then	then	ADV
ma-199	189	24	{	{	PUNCT
ma-199	189	25	fnk}nk∈i	fnk}nk∈i	NOUN
ma-199	189	26	is	be	AUX
ma-199	189	27	a	a	DET
ma-199	189	28	frame	frame	NOUN
ma-199	189	29	if	if	SCONJ
ma-199	189	30	and	and	CCONJ
ma-199	189	31	only	only	ADV
ma-199	189	32	if	if	SCONJ
ma-199	189	33	there	there	PRON
ma-199	189	34	exists	exist	VERB
ma-199	189	35	a	a	DET
ma-199	189	36	bounded	bounded	ADJ
ma-199	189	37	linear	linear	ADJ
ma-199	189	38	operator	operator	NOUN
ma-199	189	39	t	t	NOUN
ma-199	189	40	:	:	PUNCT
ma-199	189	41	`	`	PUNCT
ma-199	189	42	2(i	2(i	NUM
ma-199	189	43	)	)	PUNCT
ma-199	189	44	−→	−→	NOUN
ma-199	189	45	`	`	PUNCT
ma-199	189	46	2(i	2(i	NUM
ma-199	189	47	)	)	PUNCT
ma-199	189	48	such	such	ADJ
ma-199	189	49	that	that	SCONJ
ma-199	189	50	t	t	NOUN
ma-199	189	51	=	=	SYM
ma-199	189	52	t++t−	t++t−	PROPN
ma-199	189	53	,	,	PUNCT
ma-199	189	54	where	where	SCONJ
ma-199	189	55	t+	t+	NOUN
ma-199	189	56	:	:	PUNCT
ma-199	189	57	`	`	PUNCT
ma-199	189	58	2(i+	2(i+	X
ma-199	189	59	)	)	PUNCT
ma-199	189	60	−→	−→	ADJ
ma-199	189	61	`	`	PUNCT
ma-199	189	62	2(i+	2(i+	X
ma-199	189	63	)	)	PUNCT
ma-199	189	64	defined	define	VERB
ma-199	189	65	by	by	ADP
ma-199	189	66	t+{[f	t+{[f	NOUN
ma-199	190	1	+	+	PROPN
ma-199	190	2	nk	nk	PROPN
ma-199	190	3	,	,	PUNCT
ma-199	190	4	f	f	PROPN
ma-199	191	1	+	+	ADJ
ma-199	191	2	]	]	X
ma-199	191	3	}	}	PUNCT
ma-199	191	4	=	=	SYM
ma-199	191	5	{	{	PUNCT
ma-199	191	6	[	[	X
ma-199	191	7	f	f	X
ma-199	191	8	+	+	PROPN
ma-199	191	9	mk	mk	PROPN
ma-199	191	10	,	,	PUNCT
ma-199	191	11	f	f	PROPN
ma-199	192	1	+	+	NOUN
ma-199	192	2	]	]	X
ma-199	192	3	}	}	PUNCT
ma-199	192	4	,	,	PUNCT
ma-199	192	5	f	f	PROPN
ma-199	192	6	+	+	CCONJ
ma-199	192	7	∈	∈	PROPN
ma-199	192	8	k+	k+	NOUN
ma-199	192	9	and	and	CCONJ
ma-199	192	10	t−	t−	PROPN
ma-199	192	11	:	:	PUNCT
ma-199	192	12	`	`	PUNCT
ma-199	192	13	2(i−	2(i−	NUM
ma-199	192	14	)	)	PUNCT
ma-199	193	1	−→	−→	NOUN
ma-199	193	2	`	`	PUNCT
ma-199	193	3	2(i−	2(i−	NUM
ma-199	193	4	)	)	PUNCT
ma-199	193	5	is	be	AUX
ma-199	193	6	defined	define	VERB
ma-199	193	7	by	by	ADP
ma-199	193	8	t−{[f	t−{[f	PROPN
ma-199	193	9	−nk	−nk	NOUN
ma-199	193	10	,	,	PUNCT
ma-199	193	11	f	f	PROPN
ma-199	194	1	−	−	NOUN
ma-199	194	2	]	]	X
ma-199	194	3	}	}	PUNCT
ma-199	194	4	=	=	SYM
ma-199	194	5	{	{	PUNCT
ma-199	195	1	[	[	X
ma-199	195	2	f	f	X
ma-199	195	3	−mk	−mk	PROPN
ma-199	195	4	,	,	PUNCT
ma-199	195	5	f	f	PROPN
ma-199	196	1	−	−	NOUN
ma-199	196	2	]	]	PUNCT
ma-199	196	3	}	}	PUNCT
ma-199	196	4	,	,	PUNCT
ma-199	196	5	f	f	PROPN
ma-199	196	6	−	−	PROPN
ma-199	196	7	∈	∈	PROPN
ma-199	196	8	k−.	k−.	PROPN
ma-199	196	9	https://doi.org/10.28924/ada/ma.4.1	https://doi.org/10.28924/ada/ma.4.1	ADP
ma-199	196	10	eur	eur	NOUN
ma-199	196	11	.	.	PUNCT
ma-199	197	1	j.	j.	PROPN
ma-199	197	2	math	math	PROPN
ma-199	197	3	.	.	PUNCT
ma-199	198	1	anal	anal	PROPN
ma-199	198	2	.	.	PUNCT
ma-199	199	1	10.28924	10.28924	NUM
ma-199	199	2	/	/	SYM
ma-199	199	3	ada	ada	PROPN
ma-199	199	4	/	/	SYM
ma-199	199	5	ma.4.1	ma.4.1	PROPN
ma-199	199	6	8	8	NUM
ma-199	199	7	proof	proof	NOUN
ma-199	199	8	.	.	PUNCT
ma-199	199	9	suppose	suppose	VERB
ma-199	199	10	that	that	SCONJ
ma-199	199	11	{	{	PUNCT
ma-199	199	12	fmk}mk∈i	fmk}mk∈i	NOUN
ma-199	199	13	is	be	AUX
ma-199	199	14	a	a	DET
ma-199	199	15	frame	frame	NOUN
ma-199	199	16	with	with	ADP
ma-199	199	17	lower	low	ADJ
ma-199	199	18	frame	frame	NOUN
ma-199	199	19	bounds	bound	VERB
ma-199	199	20	a	a	PRON
ma-199	199	21	and	and	CCONJ
ma-199	199	22	a′	a′	PROPN
ma-199	199	23	.	.	PUNCT
ma-199	200	1	then∑	then∑	PROPN
ma-199	200	2	mk∈i+	mk∈i+	PROPN
ma-199	200	3	|[fmk	|[fmk	PRON
ma-199	200	4	,	,	PUNCT
ma-199	200	5	f	f	PROPN
ma-199	201	1	+	+	NOUN
ma-199	201	2	]	]	X
ma-199	201	3	|2	|2	X
ma-199	201	4	=	=	PUNCT
ma-199	201	5	∑	∑	PUNCT
ma-199	201	6	nk∈i+	nk∈i+	X
ma-199	201	7	‖[t+{[f	‖[t+{[f	PUNCT
ma-199	202	1	+	+	PROPN
ma-199	202	2	nk	nk	PROPN
ma-199	202	3	,	,	PUNCT
ma-199	202	4	f	f	PROPN
ma-199	203	1	+	+	ADJ
ma-199	203	2	]	]	X
ma-199	203	3	}	}	PUNCT
ma-199	203	4	‖	‖	ADJ
ma-199	203	5	≤	≤	ADV
ma-199	203	6	‖t+‖	‖t+‖	NUM
ma-199	203	7	∑	∑	PUNCT
ma-199	203	8	nk∈i+	nk∈i+	X
ma-199	203	9	|[fnk	|[fnk	NOUN
ma-199	203	10	,	,	PUNCT
ma-199	203	11	f	f	PROPN
ma-199	203	12	+	+	NOUN
ma-199	203	13	]	]	X
ma-199	203	14	|2	|2	NUM
ma-199	203	15	.	.	PUNCT
ma-199	204	1	so	so	ADV
ma-199	204	2	,	,	PUNCT
ma-199	204	3	we	we	PRON
ma-199	204	4	have	have	VERB
ma-199	204	5	∑	∑	PROPN
ma-199	204	6	nk∈i+	nk∈i+	X
ma-199	204	7	|[fnk	|[fnk	NOUN
ma-199	204	8	,	,	PUNCT
ma-199	204	9	f	f	PROPN
ma-199	204	10	+	+	NOUN
ma-199	204	11	]	]	X
ma-199	204	12	|2	|2	X
ma-199	204	13	≥	≥	NOUN
ma-199	204	14	∑	∑	PUNCT
ma-199	204	15	mk∈i+	mk∈i+	PROPN
ma-199	204	16	|[fmk	|[fmk	PRON
ma-199	204	17	,	,	PUNCT
ma-199	204	18	f	f	PROPN
ma-199	205	1	+	+	ADJ
ma-199	205	2	]	]	X
ma-199	205	3	|2	|2	X
ma-199	205	4	‖t+‖	‖t+‖	NUM
ma-199	205	5	≥	≥	NOUN
ma-199	205	6	a	a	DET
ma-199	205	7	‖t‖‖f	‖t‖‖f	PROPN
ma-199	205	8	+	+	NOUN
ma-199	205	9	‖2	‖2	NOUN
ma-199	205	10	.	.	PUNCT
ma-199	206	1	similarly	similarly	ADV
ma-199	206	2	,	,	PUNCT
ma-199	206	3	we	we	PRON
ma-199	206	4	have	have	VERB
ma-199	206	5	∑	∑	ADV
ma-199	206	6	nk∈i−	nk∈i−	ADJ
ma-199	206	7	|[fnk	|[fnk	NOUN
ma-199	206	8	,	,	PUNCT
ma-199	206	9	f	f	PROPN
ma-199	206	10	−]|2	−]|2	PROPN
ma-199	206	11	≥	≥	PUNCT
ma-199	206	12	a	a	DET
ma-199	206	13	′	′	NUM
ma-199	206	14	‖t‖‖f	‖t‖‖f	PROPN
ma-199	206	15	−‖2	−‖2	NOUN
ma-199	206	16	.	.	PUNCT
ma-199	207	1	hence	hence	ADV
ma-199	207	2	{	{	PUNCT
ma-199	207	3	fnk}nk∈i	fnk}nk∈i	PROPN
ma-199	207	4	is	be	AUX
ma-199	207	5	a	a	DET
ma-199	207	6	frame	frame	NOUN
ma-199	207	7	for	for	ADP
ma-199	207	8	the	the	DET
ma-199	207	9	krein	krein	NOUN
ma-199	207	10	space	space	NOUN
ma-199	207	11	k.conversely	k.conversely	ADV
ma-199	207	12	,	,	PUNCT
ma-199	207	13	suppose	suppose	VERB
ma-199	207	14	that	that	SCONJ
ma-199	207	15	{	{	PUNCT
ma-199	207	16	fnk}nk∈i	fnk}nk∈i	NOUN
ma-199	207	17	is	be	AUX
ma-199	207	18	a	a	DET
ma-199	207	19	frame	frame	NOUN
ma-199	207	20	for	for	ADP
ma-199	207	21	the	the	DET
ma-199	207	22	krein	krein	PROPN
ma-199	207	23	space	space	NOUN
ma-199	208	1	k.	k.	PROPN
ma-199	208	2	then	then	ADV
ma-199	208	3	there	there	PRON
ma-199	208	4	exist	exist	VERB
ma-199	208	5	operators	operator	NOUN
ma-199	208	6	t1	t1	VERB
ma-199	208	7	+	+	CCONJ
ma-199	208	8	:	:	PUNCT
ma-199	208	9	`	`	PUNCT
ma-199	208	10	2(i+	2(i+	X
ma-199	208	11	)	)	PUNCT
ma-199	208	12	−→	−→	NOUN
ma-199	208	13	k1	k1	NOUN
ma-199	208	14	given	give	VERB
ma-199	208	15	by	by	ADP
ma-199	208	16	t1+{[f	t1+{[f	PROPN
ma-199	208	17	+	+	PROPN
ma-199	208	18	nk	nk	PROPN
ma-199	208	19	,	,	PUNCT
ma-199	208	20	f	f	PROPN
ma-199	209	1	+	+	ADJ
ma-199	209	2	]	]	X
ma-199	209	3	}	}	PUNCT
ma-199	209	4	−→	−→	ADJ
ma-199	209	5	f	f	NOUN
ma-199	209	6	+	+	X
ma-199	209	7	and	and	CCONJ
ma-199	209	8	t	t	X
ma-199	209	9	∗1	∗1	PROPN
ma-199	210	1	+	+	CCONJ
ma-199	210	2	:	:	PUNCT
ma-199	210	3	k1	k1	NOUN
ma-199	210	4	−→	−→	ADJ
ma-199	210	5	`	`	PUNCT
ma-199	210	6	2(i+	2(i+	NUM
ma-199	210	7	)	)	PUNCT
ma-199	210	8	given	give	VERB
ma-199	210	9	by	by	ADP
ma-199	210	10	t	t	NOUN
ma-199	210	11	∗1	∗1	PUNCT
ma-199	211	1	+	+	PUNCT
ma-199	211	2	f	f	NOUN
ma-199	211	3	+	+	X
ma-199	211	4	=	=	SYM
ma-199	211	5	{	{	PUNCT
ma-199	212	1	[	[	X
ma-199	212	2	f	f	X
ma-199	212	3	+	+	NOUN
ma-199	212	4	nk	nk	PROPN
ma-199	212	5	,	,	PUNCT
ma-199	212	6	f	f	PROPN
ma-199	213	1	+	+	ADJ
ma-199	213	2	]	]	X
ma-199	213	3	}	}	PUNCT
ma-199	213	4	and	and	CCONJ
ma-199	213	5	similarly	similarly	ADV
ma-199	213	6	there	there	PRON
ma-199	213	7	exist	exist	VERB
ma-199	213	8	operators	operator	NOUN
ma-199	213	9	t1−	t1−	NOUN
ma-199	213	10	:	:	PUNCT
ma-199	213	11	`	`	PUNCT
ma-199	213	12	2(i−	2(i−	NUM
ma-199	213	13	)	)	PUNCT
ma-199	213	14	−→	−→	PROPN
ma-199	213	15	k2	k2	NOUN
ma-199	213	16	given	give	VERB
ma-199	213	17	by	by	ADP
ma-199	213	18	t1−{[f	t1−{[f	NOUN
ma-199	213	19	−nk	−nk	NOUN
ma-199	213	20	,	,	PUNCT
ma-199	213	21	f	f	PROPN
ma-199	214	1	−	−	NOUN
ma-199	214	2	]	]	X
ma-199	214	3	}	}	PUNCT
ma-199	214	4	=	=	SYM
ma-199	214	5	f	f	NOUN
ma-199	214	6	−and	−and	NOUN
ma-199	214	7	t−1	t−1	PROPN
ma-199	214	8	∗	∗	NOUN
ma-199	214	9	:	:	PUNCT
ma-199	214	10	k2	k2	ADJ
ma-199	214	11	−→	−→	NOUN
ma-199	214	12	`	`	PUNCT
ma-199	214	13	2(i−	2(i−	NUM
ma-199	214	14	)	)	PUNCT
ma-199	214	15	given	give	VERB
ma-199	214	16	by	by	ADP
ma-199	214	17	t−1	t−1	PROPN
ma-199	214	18	∗	∗	NOUN
ma-199	214	19	f	f	NOUN
ma-199	214	20	−	−	PROPN
ma-199	215	1	=	=	PUNCT
ma-199	215	2	{	{	PUNCT
ma-199	216	1	[	[	X
ma-199	216	2	f	f	X
ma-199	216	3	−nk	−nk	NOUN
ma-199	216	4	,	,	PUNCT
ma-199	216	5	f	f	PROPN
ma-199	216	6	−	−	NOUN
ma-199	216	7	]	]	PUNCT
ma-199	216	8	}	}	PUNCT
ma-199	216	9	.	.	PUNCT
ma-199	217	1	also	also	ADV
ma-199	217	2	,	,	PUNCT
ma-199	217	3	since	since	SCONJ
ma-199	217	4	{	{	PUNCT
ma-199	217	5	fmk}mk∈i	fmk}mk∈i	NOUN
ma-199	217	6	is	be	AUX
ma-199	217	7	a	a	DET
ma-199	217	8	frame	frame	NOUN
ma-199	217	9	forthe	forthe	DET
ma-199	217	10	krein	krein	PROPN
ma-199	217	11	space	space	PROPN
ma-199	217	12	k	k	PROPN
ma-199	217	13	,	,	PUNCT
ma-199	217	14	there	there	PRON
ma-199	217	15	exist	exist	VERB
ma-199	217	16	operators	operator	NOUN
ma-199	217	17	t+2	t+2	NUM
ma-199	217	18	:	:	PUNCT
ma-199	217	19	`	`	PUNCT
ma-199	217	20	2(i+	2(i+	X
ma-199	217	21	)	)	PUNCT
ma-199	217	22	−→	−→	NOUN
ma-199	217	23	k1	k1	NOUN
ma-199	217	24	given	give	VERB
ma-199	217	25	by	by	ADP
ma-199	217	26	t2+{[f	t2+{[f	VERB
ma-199	217	27	+	+	PROPN
ma-199	217	28	mk	mk	NOUN
ma-199	217	29	,	,	PUNCT
ma-199	217	30	f	f	PROPN
ma-199	218	1	+	+	ADJ
ma-199	218	2	]	]	X
ma-199	218	3	}	}	PUNCT
ma-199	218	4	=	=	SYM
ma-199	218	5	f	f	PROPN
ma-199	219	1	+	+	CCONJ
ma-199	219	2	and	and	CCONJ
ma-199	219	3	t+2	t+2	NUM
ma-199	219	4	∗	∗	NOUN
ma-199	219	5	:	:	PUNCT
ma-199	219	6	k1	k1	VERB
ma-199	219	7	−→	−→	ADJ
ma-199	219	8	`	`	PUNCT
ma-199	219	9	2(i+	2(i+	NUM
ma-199	219	10	)	)	PUNCT
ma-199	219	11	given	give	VERB
ma-199	219	12	by	by	ADP
ma-199	219	13	t+2	t+2	X
ma-199	219	14	∗f	∗f	NOUN
ma-199	219	15	+	+	PUNCT
ma-199	219	16	=	=	SYM
ma-199	219	17	{	{	PUNCT
ma-199	220	1	[	[	X
ma-199	220	2	f	f	X
ma-199	220	3	+	+	PROPN
ma-199	220	4	mk	mk	PROPN
ma-199	220	5	,	,	PUNCT
ma-199	220	6	f	f	PROPN
ma-199	221	1	+	+	NOUN
ma-199	221	2	]	]	X
ma-199	221	3	}	}	PUNCT
ma-199	221	4	.similarly	.similarly	ADV
ma-199	221	5	,	,	PUNCT
ma-199	221	6	there	there	PRON
ma-199	221	7	exist	exist	VERB
ma-199	221	8	operators	operator	NOUN
ma-199	222	1	t2−	t2−	NOUN
ma-199	222	2	:	:	PUNCT
ma-199	222	3	`	`	PUNCT
ma-199	222	4	2(i−	2(i−	NUM
ma-199	222	5	)	)	PUNCT
ma-199	222	6	−→	−→	PROPN
ma-199	222	7	k2	k2	NOUN
ma-199	222	8	given	give	VERB
ma-199	222	9	by	by	ADP
ma-199	222	10	t2−{[f	t2−{[f	PROPN
ma-199	222	11	−mk	−mk	PROPN
ma-199	222	12	,	,	PUNCT
ma-199	222	13	f	f	PROPN
ma-199	223	1	−	−	NOUN
ma-199	223	2	]	]	X
ma-199	223	3	}	}	PUNCT
ma-199	223	4	=	=	SYM
ma-199	223	5	f	f	PROPN
ma-199	223	6	−	−	NOUN
ma-199	223	7	and	and	CCONJ
ma-199	223	8	t−2	t−2	PROPN
ma-199	223	9	∗	∗	NOUN
ma-199	223	10	:	:	PUNCT
ma-199	223	11	k2	k2	ADJ
ma-199	223	12	−→	−→	NOUN
ma-199	223	13	`	`	PUNCT
ma-199	223	14	2(i−	2(i−	NUM
ma-199	223	15	)	)	PUNCT
ma-199	223	16	given	give	VERB
ma-199	223	17	by	by	ADP
ma-199	223	18	t−2	t−2	PROPN
ma-199	223	19	∗f	∗f	NOUN
ma-199	223	20	−	−	PROPN
ma-199	224	1	=	=	SYM
ma-199	224	2	{	{	PUNCT
ma-199	225	1	[	[	X
ma-199	225	2	f	f	X
ma-199	225	3	−mk	−mk	PROPN
ma-199	225	4	,	,	PUNCT
ma-199	225	5	f	f	PROPN
ma-199	226	1	−	−	NOUN
ma-199	226	2	]	]	PUNCT
ma-199	226	3	}	}	PUNCT
ma-199	226	4	.	.	PUNCT
ma-199	227	1	then	then	ADV
ma-199	227	2	t+	t+	PUNCT
ma-199	227	3	=	=	SYM
ma-199	227	4	t+2	t+2	NUM
ma-199	227	5	∗	∗	NOUN
ma-199	227	6	t1	t1	NOUN
ma-199	227	7	+	+	CCONJ
ma-199	227	8	:	:	PUNCT
ma-199	227	9	`	`	PUNCT
ma-199	227	10	2(i+	2(i+	X
ma-199	227	11	)	)	PUNCT
ma-199	227	12	−→	−→	ADJ
ma-199	227	13	`	`	PUNCT
ma-199	227	14	2(i+	2(i+	X
ma-199	227	15	)	)	PUNCT
ma-199	227	16	is	be	AUX
ma-199	227	17	abounded	abound	VERB
ma-199	227	18	linear	linear	ADJ
ma-199	227	19	operator	operator	NOUN
ma-199	227	20	such	such	ADJ
ma-199	227	21	that	that	DET
ma-199	227	22	t+{[f	t+{[f	NOUN
ma-199	228	1	+	+	NOUN
ma-199	228	2	nk	nk	PROPN
ma-199	228	3	,	,	PUNCT
ma-199	228	4	f	f	PROPN
ma-199	229	1	+	+	ADJ
ma-199	229	2	]	]	X
ma-199	229	3	}	}	PUNCT
ma-199	229	4	=	=	SYM
ma-199	229	5	{	{	PUNCT
ma-199	229	6	[	[	X
ma-199	229	7	f	f	X
ma-199	229	8	+	+	PROPN
ma-199	229	9	mk	mk	PROPN
ma-199	229	10	,	,	PUNCT
ma-199	229	11	f	f	PROPN
ma-199	230	1	+	+	NOUN
ma-199	230	2	]	]	X
ma-199	230	3	}	}	PUNCT
ma-199	230	4	,	,	PUNCT
ma-199	230	5	f	f	PROPN
ma-199	230	6	+	+	CCONJ
ma-199	230	7	∈	∈	PROPN
ma-199	230	8	k+	k+	NOUN
ma-199	230	9	and	and	CCONJ
ma-199	230	10	t−	t−	PROPN
ma-199	230	11	=	=	PROPN
ma-199	230	12	t−2	t−2	PROPN
ma-199	230	13	∗	∗	NOUN
ma-199	230	14	t1	t1	NOUN
ma-199	230	15	−	−	PROPN
ma-199	230	16	:	:	PUNCT
ma-199	230	17	`	`	PUNCT
ma-199	230	18	2(i−	2(i−	NUM
ma-199	230	19	)	)	PUNCT
ma-199	230	20	−→	−→	NOUN
ma-199	230	21	`	`	PUNCT
ma-199	230	22	2(i−	2(i−	NUM
ma-199	230	23	)	)	PUNCT
ma-199	230	24	is	be	AUX
ma-199	230	25	a	a	DET
ma-199	230	26	bounded	bounded	ADJ
ma-199	230	27	linear	linear	ADJ
ma-199	230	28	operator	operator	NOUN
ma-199	230	29	such	such	ADJ
ma-199	230	30	that	that	DET
ma-199	230	31	t−{[f	t−{[f	NOUN
ma-199	230	32	−nk	−nk	NOUN
ma-199	230	33	,	,	PUNCT
ma-199	230	34	f	f	PROPN
ma-199	230	35	−	−	NOUN
ma-199	230	36	]	]	X
ma-199	230	37	}	}	PUNCT
ma-199	230	38	=	=	SYM
ma-199	230	39	{	{	PUNCT
ma-199	231	1	[	[	X
ma-199	231	2	f	f	X
ma-199	231	3	−mk	−mk	PROPN
ma-199	231	4	,	,	PUNCT
ma-199	231	5	f	f	PROPN
ma-199	232	1	−	−	NOUN
ma-199	232	2	]	]	PUNCT
ma-199	232	3	}	}	PUNCT
ma-199	232	4	,	,	PUNCT
ma-199	232	5	f	f	PROPN
ma-199	232	6	−	−	PROPN
ma-199	232	7	∈	∈	PROPN
ma-199	232	8	k−.	k−.	PROPN
ma-199	232	9	�	�	PROPN
ma-199	232	10	next	next	ADV
ma-199	232	11	,	,	PUNCT
ma-199	232	12	we	we	PRON
ma-199	232	13	give	give	VERB
ma-199	232	14	a	a	DET
ma-199	232	15	sufficient	sufficient	ADJ
ma-199	232	16	condition	condition	NOUN
ma-199	232	17	for	for	ADP
ma-199	232	18	two	two	NUM
ma-199	232	19	subsequences	subsequence	NOUN
ma-199	232	20	of	of	ADP
ma-199	232	21	a	a	DET
ma-199	232	22	frame	frame	NOUN
ma-199	232	23	for	for	SCONJ
ma-199	232	24	k	k	PROPN
ma-199	232	25	to	to	PART
ma-199	232	26	be	be	AUX
ma-199	232	27	a	a	DET
ma-199	232	28	frame	frame	NOUN
ma-199	232	29	sequence	sequence	NOUN
ma-199	232	30	.	.	PUNCT
ma-199	233	1	theorem	theorem	VERB
ma-199	233	2	4.2	4.2	NUM
ma-199	233	3	.	.	PUNCT
ma-199	234	1	let	let	VERB
ma-199	234	2	{	{	PUNCT
ma-199	234	3	fn}n∈i	fn}n∈i	VERB
ma-199	234	4	be	be	AUX
ma-199	234	5	a	a	DET
ma-199	234	6	frame	frame	NOUN
ma-199	234	7	for	for	ADP
ma-199	234	8	a	a	DET
ma-199	234	9	krein	krein	NOUN
ma-199	234	10	space	space	NOUN
ma-199	234	11	k.	k.	PROPN
ma-199	235	1	let	let	VERB
ma-199	235	2	{	{	PUNCT
ma-199	235	3	mk	mk	X
ma-199	235	4	}	}	PUNCT
ma-199	235	5	and	and	CCONJ
ma-199	235	6	{	{	PUNCT
ma-199	235	7	nk	nk	NOUN
ma-199	235	8	}	}	PUNCT
ma-199	235	9	be	be	AUX
ma-199	235	10	two	two	NUM
ma-199	235	11	infinite	infinite	ADJ
ma-199	235	12	increasing	increase	VERB
ma-199	235	13	sequences	sequence	NOUN
ma-199	235	14	in	in	ADP
ma-199	235	15	i	i	PRON
ma-199	235	16	with	with	ADP
ma-199	235	17	{	{	PUNCT
ma-199	235	18	m+k	m+k	NUM
ma-199	235	19	}	}	PUNCT
ma-199	235	20	∪	∪	NOUN
ma-199	235	21	{	{	PUNCT
ma-199	235	22	n	n	NOUN
ma-199	235	23	+	+	CCONJ
ma-199	235	24	k	k	NOUN
ma-199	235	25	}	}	PUNCT
ma-199	235	26	=	=	SYM
ma-199	236	1	i+	i+	NOUN
ma-199	236	2	and	and	CCONJ
ma-199	236	3	{	{	PUNCT
ma-199	236	4	m−k	m−k	NOUN
ma-199	236	5	}	}	PUNCT
ma-199	236	6	∪	∪	VERB
ma-199	236	7	{	{	PUNCT
ma-199	236	8	n	n	NOUN
ma-199	236	9	−	−	PROPN
ma-199	236	10	k	k	NOUN
ma-199	236	11	}	}	PUNCT
ma-199	236	12	=	=	SYM
ma-199	236	13	i−.	i−.	NOUN
ma-199	236	14	let	let	VERB
ma-199	236	15	k1	k1	NOUN
ma-199	236	16	=	=	PUNCT
ma-199	237	1	[	[	X
ma-199	237	2	f	f	X
ma-199	237	3	+	+	NOUN
ma-199	237	4	mk	mk	X
ma-199	237	5	]	]	PUNCT
ma-199	237	6	∩	∩	PROPN
ma-199	238	1	[	[	X
ma-199	238	2	f	f	X
ma-199	238	3	+	+	NUM
ma-199	238	4	nk	nk	PROPN
ma-199	238	5	]	]	X
ma-199	238	6	.	.	PUNCT
ma-199	239	1	if	if	SCONJ
ma-199	239	2	k1	k1	PROPN
ma-199	239	3	is	be	AUX
ma-199	239	4	a	a	DET
ma-199	239	5	finite	finite	ADJ
ma-199	239	6	dimensional	dimensional	ADJ
ma-199	239	7	space	space	NOUN
ma-199	239	8	,	,	PUNCT
ma-199	239	9	then	then	ADV
ma-199	239	10	{	{	PUNCT
ma-199	239	11	f	f	PROPN
ma-199	239	12	+	+	PROPN
ma-199	239	13	mk	mk	X
ma-199	239	14	}	}	PUNCT
ma-199	239	15	and	and	CCONJ
ma-199	239	16	{	{	PUNCT
ma-199	239	17	f	f	PROPN
ma-199	239	18	+	+	PROPN
ma-199	239	19	nk	nk	PROPN
ma-199	239	20	}	}	PUNCT
ma-199	239	21	are	be	AUX
ma-199	239	22	frame	frame	NOUN
ma-199	239	23	sequences	sequence	NOUN
ma-199	239	24	for	for	ADP
ma-199	239	25	k+	k+	NOUN
ma-199	239	26	.	.	PUNCT
ma-199	240	1	further	far	ADV
ma-199	240	2	,	,	PUNCT
ma-199	240	3	if	if	SCONJ
ma-199	240	4	k2	k2	ADJ
ma-199	240	5	=	=	PUNCT
ma-199	241	1	[	[	X
ma-199	241	2	f	f	X
ma-199	241	3	−mk	−mk	PROPN
ma-199	241	4	]	]	PUNCT
ma-199	241	5	∩	∩	NOUN
ma-199	241	6	[	[	X
ma-199	241	7	f	f	X
ma-199	241	8	−	−	PROPN
ma-199	241	9	nk	nk	PROPN
ma-199	241	10	]	]	PUNCT
ma-199	241	11	is	be	AUX
ma-199	241	12	finite	finite	ADJ
ma-199	241	13	dimensional	dimensional	ADJ
ma-199	241	14	,	,	PUNCT
ma-199	241	15	then	then	ADV
ma-199	241	16	{	{	PUNCT
ma-199	241	17	f	f	PROPN
ma-199	241	18	−mk	−mk	PROPN
ma-199	241	19	}	}	PUNCT
ma-199	241	20	and	and	CCONJ
ma-199	241	21	{	{	PUNCT
ma-199	241	22	f	f	NOUN
ma-199	241	23	−nk	−nk	PROPN
ma-199	241	24	}	}	PUNCT
ma-199	241	25	are	be	AUX
ma-199	241	26	frame	frame	NOUN
ma-199	241	27	sequences	sequence	NOUN
ma-199	241	28	for	for	ADP
ma-199	241	29	k−.	k−.	ADJ
ma-199	241	30	proof	proof	NOUN
ma-199	241	31	.	.	PUNCT
ma-199	242	1	let	let	VERB
ma-199	242	2	{	{	PUNCT
ma-199	242	3	`	`	PUNCT
ma-199	242	4	+	+	ADV
ma-199	242	5	k	k	VERB
ma-199	242	6	}	}	PUNCT
ma-199	242	7	be	be	AUX
ma-199	242	8	a	a	DET
ma-199	242	9	finite	finite	ADJ
ma-199	242	10	subsequence	subsequence	NOUN
ma-199	242	11	of	of	ADP
ma-199	242	12	{	{	PUNCT
ma-199	242	13	n+k	n+k	PROPN
ma-199	242	14	}	}	PUNCT
ma-199	242	15	such	such	ADJ
ma-199	242	16	that	that	SCONJ
ma-199	242	17	k1	k1	NOUN
ma-199	242	18	=	=	PUNCT
ma-199	243	1	[	[	X
ma-199	243	2	f`k	f`k	X
ma-199	243	3	]	]	X
ma-199	243	4	`	`	PUNCT
ma-199	243	5	k∈i+	k∈i+	PROPN
ma-199	243	6	.	.	PUNCT
ma-199	244	1	since	since	SCONJ
ma-199	244	2	k1	k1	PROPN
ma-199	244	3	is	be	AUX
ma-199	244	4	finitedimensional	finitedimensional	ADJ
ma-199	244	5	,	,	PUNCT
ma-199	244	6	{	{	PUNCT
ma-199	244	7	f	f	PROPN
ma-199	244	8	+	+	PROPN
ma-199	244	9	`	`	PUNCT
ma-199	244	10	k	k	NOUN
ma-199	244	11	}	}	PUNCT
ma-199	244	12	is	be	AUX
ma-199	244	13	a	a	DET
ma-199	244	14	frame	frame	NOUN
ma-199	244	15	for	for	ADP
ma-199	244	16	k1	k1	NOUN
ma-199	244	17	.	.	PUNCT
ma-199	245	1	let	let	VERB
ma-199	245	2	a′	a′	NOUN
ma-199	245	3	and	and	CCONJ
ma-199	245	4	b	b	PROPN
ma-199	245	5	′	′	NOUN
ma-199	245	6	be	be	AUX
ma-199	245	7	the	the	DET
ma-199	245	8	frame	frame	NOUN
ma-199	245	9	bounds	bound	NOUN
ma-199	245	10	for	for	ADP
ma-199	245	11	{	{	PUNCT
ma-199	245	12	f	f	PROPN
ma-199	245	13	+	+	PROPN
ma-199	245	14	`	`	PUNCT
ma-199	245	15	k	k	NOUN
ma-199	245	16	}	}	PUNCT
ma-199	245	17	.	.	PUNCT
ma-199	246	1	consider	consider	VERB
ma-199	246	2	{	{	PUNCT
ma-199	246	3	fnk}nk∈i+	fnk}nk∈i+	PROPN
ma-199	246	4	,	,	PUNCT
ma-199	246	5	let	let	VERB
ma-199	246	6	f	f	PRON
ma-199	246	7	+	+	PROPN
ma-199	246	8	∈	∈	PROPN
ma-199	246	9	{	{	PUNCT
ma-199	246	10	fnk}nk∈i+	fnk}nk∈i+	PROPN
ma-199	246	11	be	be	AUX
ma-199	246	12	any	any	DET
ma-199	246	13	element	element	NOUN
ma-199	246	14	.	.	PUNCT
ma-199	247	1	now	now	ADV
ma-199	247	2	,	,	PUNCT
ma-199	247	3	if	if	SCONJ
ma-199	247	4	f	f	PROPN
ma-199	247	5	+	+	PROPN
ma-199	247	6	[	[	X
ma-199	247	7	⊥]k1	⊥]k1	X
ma-199	247	8	,	,	PUNCT
ma-199	247	9	then∑	then∑	NOUN
ma-199	247	10	n∈i+	n∈i+	NOUN
ma-199	248	1	[	[	X
ma-199	248	2	f	f	X
ma-199	248	3	+	+	PROPN
ma-199	248	4	,	,	PUNCT
ma-199	248	5	fn	fn	NOUN
ma-199	248	6	]	]	X
ma-199	248	7	2	2	NUM
ma-199	248	8	=	=	PUNCT
ma-199	248	9	∑	∑	PUNCT
ma-199	248	10	nk∈i+	nk∈i+	X
ma-199	249	1	[	[	X
ma-199	249	2	f	f	X
ma-199	249	3	+	+	ADV
ma-199	249	4	,	,	PUNCT
ma-199	249	5	fnk	fnk	VERB
ma-199	249	6	]	]	PUNCT
ma-199	249	7	2	2	NUM
ma-199	249	8	≥	≥	NOUN
ma-199	249	9	a‖f	a‖f	NOUN
ma-199	249	10	+	+	NOUN
ma-199	249	11	‖2	‖2	NOUN
ma-199	249	12	.	.	PUNCT
ma-199	250	1	https://doi.org/10.28924/ada/ma.4.1	https://doi.org/10.28924/ada/ma.4.1	NUM
ma-199	250	2	eur	eur	PROPN
ma-199	250	3	.	.	PUNCT
ma-199	251	1	j.	j.	PROPN
ma-199	251	2	math	math	PROPN
ma-199	251	3	.	.	PUNCT
ma-199	252	1	anal	anal	PROPN
ma-199	252	2	.	.	PUNCT
ma-199	253	1	10.28924	10.28924	NUM
ma-199	253	2	/	/	SYM
ma-199	253	3	ada	ada	PROPN
ma-199	253	4	/	/	SYM
ma-199	253	5	ma.4.1	ma.4.1	PROPN
ma-199	253	6	9also	9also	NUM
ma-199	253	7	,	,	PUNCT
ma-199	253	8	if	if	SCONJ
ma-199	253	9	f	f	PROPN
ma-199	253	10	+	+	PROPN
ma-199	253	11	∈	∈	PROPN
ma-199	253	12	k1	k1	NOUN
ma-199	253	13	,	,	PUNCT
ma-199	253	14	then	then	ADV
ma-199	253	15	∑	∑	PUNCT
ma-199	253	16	nk∈i+	nk∈i+	X
ma-199	254	1	[	[	X
ma-199	254	2	f	f	X
ma-199	254	3	+	+	ADV
ma-199	254	4	,	,	PUNCT
ma-199	254	5	fnk	fnk	VERB
ma-199	254	6	]	]	PUNCT
ma-199	254	7	2	2	NUM
ma-199	254	8	≥	≥	NOUN
ma-199	254	9	∑	∑	PUNCT
ma-199	254	10	`	`	PUNCT
ma-199	254	11	k∈i+	k∈i+	X
ma-199	255	1	[	[	X
ma-199	255	2	f	f	X
ma-199	255	3	+	+	PROPN
ma-199	255	4	,	,	PUNCT
ma-199	255	5	flk	flk	PROPN
ma-199	255	6	]	]	PUNCT
ma-199	255	7	2	2	NUM
ma-199	255	8	≥	≥	NOUN
ma-199	255	9	a	a	DET
ma-199	255	10	′‖f	′‖f	PROPN
ma-199	255	11	+	+	NOUN
ma-199	255	12	‖2	‖2	NOUN
ma-199	255	13	.	.	PUNCT
ma-199	256	1	otherwise	otherwise	ADV
ma-199	256	2	,	,	PUNCT
ma-199	256	3	we	we	PRON
ma-199	256	4	have	have	VERB
ma-199	256	5	f	f	NOUN
ma-199	256	6	+	+	CCONJ
ma-199	257	1	=	=	PUNCT
ma-199	257	2	∑	∑	PUNCT
ma-199	257	3	αk	αk	ADP
ma-199	257	4	fnk	fnk	VERB
ma-199	257	5	=	=	PUNCT
ma-199	257	6	∑	∑	PUNCT
ma-199	257	7	αi	αi	INTJ
ma-199	257	8	fn	fn	NOUN
ma-199	258	1	+	+	CCONJ
ma-199	258	2	∑	∑	PROPN
ma-199	258	3	αj	αj	ADP
ma-199	258	4	fj	fj	PROPN
ma-199	258	5	,	,	PUNCT
ma-199	258	6	i	i	PRON
ma-199	258	7	∈	∈	PROPN
ma-199	258	8	{	{	PUNCT
ma-199	258	9	nk}\{`k	nk}\{`k	NOUN
ma-199	258	10	}	}	PUNCT
ma-199	258	11	,	,	PUNCT
ma-199	258	12	j	j	PROPN
ma-199	258	13	∈	∈	PROPN
ma-199	258	14	{	{	PUNCT
ma-199	258	15	`	`	PUNCT
ma-199	258	16	k	k	NOUN
ma-199	258	17	}	}	PUNCT
ma-199	258	18	=	=	SYM
ma-199	258	19	(	(	PUNCT
ma-199	258	20	f	f	PROPN
ma-199	258	21	+	+	NOUN
ma-199	258	22	)	)	PUNCT
ma-199	258	23	′	′	PUNCT
ma-199	259	1	+	+	CCONJ
ma-199	259	2	(	(	PUNCT
ma-199	259	3	f	f	X
ma-199	259	4	+	+	ADJ
ma-199	259	5	)	)	PUNCT
ma-199	259	6	′′	′′	PROPN
ma-199	259	7	,	,	PUNCT
ma-199	259	8	where	where	SCONJ
ma-199	259	9	(	(	PUNCT
ma-199	259	10	f	f	NOUN
ma-199	259	11	+	+	NOUN
ma-199	259	12	)	)	PUNCT
ma-199	259	13	′	′	PUNCT
ma-199	260	1	[	[	X
ma-199	260	2	⊥]k1	⊥]k1	X
ma-199	260	3	and	and	CCONJ
ma-199	260	4	(	(	PUNCT
ma-199	260	5	f	f	X
ma-199	260	6	+	+	ADJ
ma-199	260	7	)	)	PUNCT
ma-199	260	8	′′	′′	PROPN
ma-199	260	9	∈	∈	PROPN
ma-199	260	10	k1	k1	NOUN
ma-199	260	11	.	.	PUNCT
ma-199	261	1	thus	thus	ADV
ma-199	261	2	∑	∑	PUNCT
ma-199	261	3	nk∈i+	nk∈i+	X
ma-199	262	1	[	[	X
ma-199	262	2	f	f	X
ma-199	262	3	+	+	ADV
ma-199	262	4	,	,	PUNCT
ma-199	262	5	fnk	fnk	VERB
ma-199	262	6	]	]	PUNCT
ma-199	262	7	2	2	NUM
ma-199	262	8	=	=	SYM
ma-199	262	9	∑	∑	PUNCT
ma-199	263	1	[	[	X
ma-199	263	2	f	f	X
ma-199	263	3	+	+	ADJ
ma-199	263	4	,	,	PUNCT
ma-199	263	5	fn	fn	NOUN
ma-199	263	6	]	]	X
ma-199	263	7	2	2	NUM
ma-199	264	1	+	+	CCONJ
ma-199	264	2	∑	∑	PROPN
ma-199	265	1	[	[	X
ma-199	265	2	f	f	X
ma-199	265	3	+	+	PROPN
ma-199	265	4	,	,	PUNCT
ma-199	265	5	fj	fj	X
ma-199	265	6	]	]	PUNCT
ma-199	265	7	2	2	NUM
ma-199	265	8	,	,	PUNCT
ma-199	265	9	i	i	PRON
ma-199	265	10	∈	∈	PROPN
ma-199	265	11	{	{	PUNCT
ma-199	265	12	nk}\{`k	nk}\{`k	NOUN
ma-199	265	13	}	}	PUNCT
ma-199	265	14	,	,	PUNCT
ma-199	265	15	j	j	PROPN
ma-199	265	16	∈	∈	PROPN
ma-199	265	17	{	{	PUNCT
ma-199	265	18	`	`	PUNCT
ma-199	265	19	k	k	NOUN
ma-199	265	20	}	}	PUNCT
ma-199	265	21	=	=	PUNCT
ma-199	265	22	∑	∑	PUNCT
ma-199	266	1	[	[	X
ma-199	266	2	(	(	PUNCT
ma-199	266	3	f	f	NOUN
ma-199	266	4	+	+	NOUN
ma-199	266	5	)	)	PUNCT
ma-199	266	6	′	′	PUNCT
ma-199	267	1	+	+	CCONJ
ma-199	267	2	(	(	PUNCT
ma-199	267	3	f	f	X
ma-199	267	4	+	+	ADJ
ma-199	267	5	)	)	PUNCT
ma-199	267	6	′′	′′	PROPN
ma-199	267	7	,	,	PUNCT
ma-199	267	8	fn	fn	PROPN
ma-199	267	9	]	]	X
ma-199	267	10	2	2	NUM
ma-199	267	11	+	+	CCONJ
ma-199	267	12	∑	∑	PUNCT
ma-199	267	13	[	[	X
ma-199	267	14	(	(	PUNCT
ma-199	267	15	f	f	NOUN
ma-199	267	16	+	+	NOUN
ma-199	267	17	)	)	PUNCT
ma-199	267	18	′	′	PUNCT
ma-199	268	1	+	+	CCONJ
ma-199	268	2	(	(	PUNCT
ma-199	268	3	f	f	X
ma-199	268	4	+	+	ADJ
ma-199	268	5	)	)	PUNCT
ma-199	268	6	′′	′′	PROPN
ma-199	268	7	,	,	PUNCT
ma-199	268	8	fj	fj	PROPN
ma-199	268	9	]	]	PUNCT
ma-199	268	10	2	2	X
ma-199	268	11	=	=	SYM
ma-199	268	12	∑	∑	PUNCT
ma-199	268	13	[	[	X
ma-199	268	14	(	(	PUNCT
ma-199	268	15	f	f	NOUN
ma-199	268	16	+	+	NOUN
ma-199	268	17	)	)	PUNCT
ma-199	268	18	′	′	NOUN
ma-199	268	19	,	,	PUNCT
ma-199	268	20	fn	fn	X
ma-199	268	21	]	]	X
ma-199	268	22	2	2	NUM
ma-199	268	23	+	+	CCONJ
ma-199	268	24	∑	∑	PUNCT
ma-199	268	25	[	[	X
ma-199	268	26	(	(	PUNCT
ma-199	268	27	f	f	NOUN
ma-199	268	28	+	+	NOUN
ma-199	268	29	)	)	PUNCT
ma-199	268	30	′′	′′	PROPN
ma-199	268	31	,	,	PUNCT
ma-199	268	32	fj	fj	PROPN
ma-199	268	33	]	]	PUNCT
ma-199	268	34	2	2	NUM
ma-199	268	35	≥	≥	NOUN
ma-199	268	36	a‖(f	a‖(f	NOUN
ma-199	268	37	+	+	NOUN
ma-199	268	38	)	)	PUNCT
ma-199	268	39	′‖2	′‖2	PROPN
ma-199	268	40	+	+	NUM
ma-199	268	41	a′‖(f	a′‖(f	NOUN
ma-199	268	42	+	+	ADJ
ma-199	268	43	)	)	PUNCT
ma-199	268	44	′′‖	′′‖	PROPN
ma-199	268	45	≥	≥	PUNCT
ma-199	268	46	min	min	NOUN
ma-199	268	47	{	{	PUNCT
ma-199	268	48	a	a	DET
ma-199	268	49	2	2	NUM
ma-199	268	50	,	,	PUNCT
ma-199	268	51	a	a	DET
ma-199	268	52	′	′	NOUN
ma-199	268	53	2	2	NUM
ma-199	268	54	}	}	PUNCT
ma-199	268	55	‖f	‖f	PUNCT
ma-199	268	56	+	+	NOUN
ma-199	268	57	‖2	‖2	NOUN
ma-199	268	58	.	.	PUNCT
ma-199	269	1	hence	hence	ADV
ma-199	269	2	{	{	PUNCT
ma-199	269	3	fnk}nk∈i+	fnk}nk∈i+	PROPN
ma-199	269	4	is	be	AUX
ma-199	269	5	a	a	DET
ma-199	269	6	frame	frame	NOUN
ma-199	269	7	sequence	sequence	NOUN
ma-199	269	8	for	for	ADP
ma-199	269	9	k+	k+	NOUN
ma-199	269	10	.	.	PUNCT
ma-199	270	1	similarly	similarly	ADV
ma-199	270	2	we	we	PRON
ma-199	270	3	can	can	AUX
ma-199	270	4	show	show	VERB
ma-199	270	5	that	that	SCONJ
ma-199	270	6	{	{	PUNCT
ma-199	270	7	fmk}mk∈i+	fmk}mk∈i+	PROPN
ma-199	270	8	is	be	AUX
ma-199	270	9	a	a	DET
ma-199	270	10	framesequence	framesequence	NOUN
ma-199	270	11	for	for	ADP
ma-199	270	12	k+	k+	NOUN
ma-199	270	13	.	.	PUNCT
ma-199	271	1	also	also	ADV
ma-199	271	2	,	,	PUNCT
ma-199	271	3	in	in	ADP
ma-199	271	4	a	a	DET
ma-199	271	5	similar	similar	ADJ
ma-199	271	6	way	way	NOUN
ma-199	271	7	,	,	PUNCT
ma-199	271	8	one	one	PRON
ma-199	271	9	can	can	AUX
ma-199	271	10	prove	prove	VERB
ma-199	271	11	that	that	SCONJ
ma-199	271	12	{	{	PUNCT
ma-199	271	13	fnk}nk∈i−	fnk}nk∈i−	ADJ
ma-199	271	14	and	and	CCONJ
ma-199	271	15	{	{	PUNCT
ma-199	271	16	fmk}mk∈i−	fmk}mk∈i−	PROPN
ma-199	271	17	are	be	AUX
ma-199	271	18	framesequences	framesequence	NOUN
ma-199	271	19	for	for	ADP
ma-199	271	20	k−.	k−.	PROPN
ma-199	271	21	�	�	PROPN
ma-199	271	22	corollary	corollary	PROPN
ma-199	271	23	4.3	4.3	NUM
ma-199	271	24	.	.	PUNCT
ma-199	272	1	let	let	AUX
ma-199	272	2	{	{	PUNCT
ma-199	272	3	fn}n∈i	fn}n∈i	VERB
ma-199	272	4	be	be	AUX
ma-199	272	5	a	a	DET
ma-199	272	6	frame	frame	NOUN
ma-199	272	7	for	for	ADP
ma-199	272	8	a	a	DET
ma-199	272	9	krein	krein	NOUN
ma-199	272	10	space	space	NOUN
ma-199	272	11	k.	k.	PROPN
ma-199	272	12	let	let	VERB
ma-199	272	13	{	{	PUNCT
ma-199	272	14	mk	mk	X
ma-199	272	15	}	}	PUNCT
ma-199	272	16	and	and	CCONJ
ma-199	272	17	{	{	PUNCT
ma-199	272	18	nk	nk	NOUN
ma-199	272	19	}	}	PUNCT
ma-199	272	20	be	be	AUX
ma-199	272	21	two	two	NUM
ma-199	272	22	infinite	infinite	ADJ
ma-199	272	23	increasing	increase	VERB
ma-199	272	24	sequences	sequence	NOUN
ma-199	272	25	in	in	ADP
ma-199	272	26	i	i	PRON
ma-199	272	27	with	with	ADP
ma-199	272	28	{	{	PUNCT
ma-199	272	29	m+k	m+k	NUM
ma-199	272	30	}	}	PUNCT
ma-199	272	31	∪	∪	NOUN
ma-199	272	32	{	{	PUNCT
ma-199	272	33	n	n	NOUN
ma-199	272	34	+	+	CCONJ
ma-199	272	35	k	k	NOUN
ma-199	272	36	}	}	PUNCT
ma-199	272	37	=	=	SYM
ma-199	272	38	i+	i+	NOUN
ma-199	272	39	and	and	CCONJ
ma-199	272	40	{	{	PUNCT
ma-199	272	41	m−k	m−k	NOUN
ma-199	272	42	}	}	PUNCT
ma-199	272	43	∪	∪	VERB
ma-199	272	44	{	{	PUNCT
ma-199	272	45	n	n	NOUN
ma-199	272	46	−	−	PROPN
ma-199	272	47	k	k	NOUN
ma-199	272	48	}	}	PUNCT
ma-199	272	49	=	=	SYM
ma-199	272	50	i−.	i−.	PROPN
ma-199	272	51	let	let	VERB
ma-199	272	52	{	{	PUNCT
ma-199	272	53	fmk}mk∈i+	fmk}mk∈i+	PROPN
ma-199	272	54	and	and	CCONJ
ma-199	272	55	{	{	PUNCT
ma-199	272	56	fnk}nk∈i+	fnk}nk∈i+	PROPN
ma-199	272	57	be	be	AUX
ma-199	272	58	frames	frame	NOUN
ma-199	272	59	for	for	ADP
ma-199	272	60	[	[	X
ma-199	272	61	fmk	fmk	NOUN
ma-199	272	62	]	]	X
ma-199	272	63	mk∈i+	mk∈i+	PROPN
ma-199	272	64	and	and	CCONJ
ma-199	272	65	[	[	X
ma-199	272	66	fnk	fnk	X
ma-199	272	67	]	]	X
ma-199	272	68	nk∈i+	nk∈i+	X
ma-199	272	69	respectively	respectively	ADV
ma-199	272	70	and	and	CCONJ
ma-199	272	71	let	let	VERB
ma-199	272	72	{	{	PUNCT
ma-199	272	73	fmk}mk∈i−	fmk}mk∈i−	VERB
ma-199	272	74	and	and	CCONJ
ma-199	272	75	{	{	PUNCT
ma-199	272	76	fnk}nk∈i−	fnk}nk∈i−	ADJ
ma-199	272	77	be	be	NOUN
ma-199	272	78	frames	frame	NOUN
ma-199	272	79	for	for	ADP
ma-199	272	80	[	[	X
ma-199	272	81	fmk	fmk	NOUN
ma-199	272	82	]	]	PUNCT
ma-199	272	83	mk∈i−	mk∈i−	ADJ
ma-199	272	84	and	and	CCONJ
ma-199	272	85	[	[	X
ma-199	272	86	fnk	fnk	NOUN
ma-199	272	87	]	]	X
ma-199	272	88	nk∈i−	nk∈i−	ADJ
ma-199	272	89	respectively	respectively	ADV
ma-199	272	90	.	.	PUNCT
ma-199	273	1	if	if	SCONJ
ma-199	273	2	{	{	PUNCT
ma-199	273	3	g+i	g+i	PROPN
ma-199	273	4	}	}	PUNCT
ma-199	273	5	=	=	PUNCT
ma-199	273	6	{	{	PUNCT
ma-199	273	7	fmk}mk∈i+∪{fnk}nk∈i+	fmk}mk∈i+∪{fnk}nk∈i+	PROPN
ma-199	273	8	and	and	CCONJ
ma-199	273	9	{	{	PUNCT
ma-199	273	10	g−i	g−i	PROPN
ma-199	273	11	}	}	PUNCT
ma-199	273	12	=	=	PUNCT
ma-199	273	13	{	{	PUNCT
ma-199	273	14	fmk}mk∈i−	fmk}mk∈i−	ADV
ma-199	273	15	∪	∪	ADV
ma-199	273	16	{	{	PUNCT
ma-199	273	17	fnk}nk∈i−	fnk}nk∈i−	ADJ
ma-199	273	18	,	,	PUNCT
ma-199	273	19	then	then	ADV
ma-199	273	20	{	{	PUNCT
ma-199	273	21	g+i	g+i	PROPN
ma-199	273	22	}	}	PUNCT
ma-199	273	23	and	and	CCONJ
ma-199	273	24	{	{	PUNCT
ma-199	273	25	g−i	g−i	PROPN
ma-199	273	26	}	}	PUNCT
ma-199	273	27	are	be	AUX
ma-199	273	28	frame	frame	NOUN
ma-199	273	29	sequences	sequence	NOUN
ma-199	273	30	.	.	PUNCT
ma-199	274	1	proof	proof	NOUN
ma-199	274	2	.	.	PUNCT
ma-199	275	1	the	the	DET
ma-199	275	2	proof	proof	NOUN
ma-199	275	3	of	of	ADP
ma-199	275	4	the	the	DET
ma-199	275	5	corollary	corollary	NOUN
ma-199	275	6	follows	follow	VERB
ma-199	275	7	from	from	ADP
ma-199	275	8	the	the	DET
ma-199	275	9	theorem	theorem	ADJ
ma-199	275	10	4.2	4.2	NUM
ma-199	275	11	and	and	CCONJ
ma-199	275	12	the	the	DET
ma-199	275	13	fact	fact	NOUN
ma-199	275	14	that	that	SCONJ
ma-199	275	15	{	{	PUNCT
ma-199	275	16	fmk}mk∈i+	fmk}mk∈i+	PROPN
ma-199	275	17	and	and	CCONJ
ma-199	275	18	{	{	PUNCT
ma-199	275	19	fnk}nk∈i+	fnk}nk∈i+	PROPN
ma-199	275	20	are	be	AUX
ma-199	275	21	frames	frame	NOUN
ma-199	275	22	for	for	ADP
ma-199	275	23	the	the	DET
ma-199	275	24	[	[	X
ma-199	275	25	fmk	fmk	NOUN
ma-199	275	26	]	]	X
ma-199	275	27	mk∈i+	mk∈i+	PROPN
ma-199	275	28	and	and	CCONJ
ma-199	275	29	[	[	X
ma-199	275	30	fnk	fnk	X
ma-199	275	31	]	]	X
ma-199	275	32	nk∈i+	nk∈i+	X
ma-199	275	33	respectively	respectively	ADV
ma-199	275	34	.	.	PUNCT
ma-199	276	1	�	�	PROPN
ma-199	276	2	finally	finally	ADV
ma-199	276	3	,	,	PUNCT
ma-199	276	4	we	we	PRON
ma-199	276	5	give	give	VERB
ma-199	276	6	a	a	DET
ma-199	276	7	sufficient	sufficient	ADJ
ma-199	276	8	condition	condition	NOUN
ma-199	276	9	for	for	ADP
ma-199	276	10	the	the	DET
ma-199	276	11	exactness	exactness	NOUN
ma-199	276	12	of	of	ADP
ma-199	276	13	frames	frame	NOUN
ma-199	276	14	in	in	ADP
ma-199	276	15	a	a	DET
ma-199	276	16	krein	krein	NOUN
ma-199	276	17	space	space	NOUN
ma-199	276	18	k.	k.	PROPN
ma-199	276	19	theorem	theorem	VERB
ma-199	276	20	4.4	4.4	NUM
ma-199	276	21	.	.	PUNCT
ma-199	277	1	let	let	VERB
ma-199	277	2	{	{	PUNCT
ma-199	277	3	fn}n∈i	fn}n∈i	VERB
ma-199	277	4	be	be	AUX
ma-199	277	5	a	a	DET
ma-199	277	6	frame	frame	NOUN
ma-199	277	7	for	for	ADP
ma-199	277	8	a	a	DET
ma-199	277	9	krein	krein	ADJ
ma-199	277	10	space	space	NOUN
ma-199	277	11	(	(	PUNCT
ma-199	277	12	k	k	NOUN
ma-199	277	13	,	,	PUNCT
ma-199	277	14	[	[	X
ma-199	277	15	.	.	PUNCT
ma-199	277	16	,	,	PUNCT
ma-199	277	17	.	.	PUNCT
ma-199	278	1	]	]	PUNCT
ma-199	278	2	)	)	PUNCT
ma-199	279	1	with	with	ADP
ma-199	279	2	bounds	bound	NOUN
ma-199	279	3	a	a	PRON
ma-199	279	4	,	,	PUNCT
ma-199	279	5	a′	a′	PROPN
ma-199	279	6	and	and	CCONJ
ma-199	279	7	b	b	PROPN
ma-199	279	8	,	,	PUNCT
ma-199	279	9	b′	b′	NUM
ma-199	279	10	such	such	ADJ
ma-199	279	11	that	that	DET
ma-199	279	12	fn	fn	PROPN
ma-199	279	13	6=	6=	ADP
ma-199	279	14	0	0	NUM
ma-199	279	15	,	,	PUNCT
ma-199	279	16	for	for	ADP
ma-199	279	17	all	all	DET
ma-199	279	18	n	n	PRON
ma-199	279	19	∈	∈	PROPN
ma-199	279	20	i.	i.	NOUN
ma-199	279	21	if	if	SCONJ
ma-199	279	22	for	for	SCONJ
ma-199	279	23	every	every	DET
ma-199	279	24	infinite	infinite	ADJ
ma-199	279	25	increasing	increase	VERB
ma-199	279	26	sequence	sequence	NOUN
ma-199	279	27	{	{	PUNCT
ma-199	279	28	nk	nk	PROPN
ma-199	279	29	}	}	PUNCT
ma-199	279	30	∈	∈	PROPN
ma-199	279	31	i+	i+	NOUN
ma-199	279	32	and	and	CCONJ
ma-199	279	33	{	{	PUNCT
ma-199	279	34	mk	mk	PROPN
ma-199	279	35	}	}	PUNCT
ma-199	279	36	∈	∈	PROPN
ma-199	279	37	i−	i−	PROPN
ma-199	279	38	,	,	PUNCT
ma-199	279	39	{	{	PUNCT
ma-199	279	40	fnk}nk∈i+	fnk}nk∈i+	PROPN
ma-199	279	41	and	and	CCONJ
ma-199	279	42	{	{	PUNCT
ma-199	279	43	fmk}mk∈i−	fmk}mk∈i−	PROPN
ma-199	279	44	are	be	AUX
ma-199	279	45	frame	frame	NOUN
ma-199	279	46	sequences	sequence	NOUN
ma-199	279	47	with	with	ADP
ma-199	279	48	bounds	bound	NOUN
ma-199	279	49	a	a	DET
ma-199	279	50	,	,	PUNCT
ma-199	279	51	b	b	NOUN
ma-199	279	52	and	and	CCONJ
ma-199	279	53	a′	a′	NOUN
ma-199	279	54	,	,	PUNCT
ma-199	279	55	b′	b′	NUM
ma-199	279	56	respectively	respectively	ADV
ma-199	279	57	,	,	PUNCT
ma-199	279	58	then	then	ADV
ma-199	279	59	{	{	PUNCT
ma-199	279	60	fn}n∈i	fn}n∈i	NOUN
ma-199	279	61	is	be	AUX
ma-199	279	62	an	an	DET
ma-199	279	63	exact	exact	ADJ
ma-199	279	64	frame	frame	NOUN
ma-199	279	65	.	.	PUNCT
ma-199	280	1	https://doi.org/10.28924/ada/ma.4.1	https://doi.org/10.28924/ada/ma.4.1	ADP
ma-199	280	2	eur	eur	PROPN
ma-199	280	3	.	.	PUNCT
ma-199	281	1	j.	j.	PROPN
ma-199	281	2	math	math	PROPN
ma-199	281	3	.	.	PUNCT
ma-199	282	1	anal	anal	PROPN
ma-199	282	2	.	.	PUNCT
ma-199	283	1	10.28924	10.28924	NUM
ma-199	283	2	/	/	SYM
ma-199	283	3	ada	ada	PROPN
ma-199	283	4	/	/	SYM
ma-199	283	5	ma.4.1	ma.4.1	PROPN
ma-199	283	6	10	10	NUM
ma-199	283	7	proof	proof	NOUN
ma-199	283	8	.	.	PUNCT
ma-199	284	1	suppose	suppose	VERB
ma-199	284	2	on	on	ADP
ma-199	284	3	the	the	DET
ma-199	284	4	contrary	contrary	NOUN
ma-199	284	5	that	that	SCONJ
ma-199	284	6	{	{	PUNCT
ma-199	284	7	fn}n∈i	fn}n∈i	NOUN
ma-199	284	8	is	be	AUX
ma-199	284	9	not	not	PART
ma-199	284	10	an	an	DET
ma-199	284	11	exact	exact	ADJ
ma-199	284	12	frame	frame	NOUN
ma-199	284	13	.	.	PUNCT
ma-199	285	1	then	then	ADV
ma-199	285	2	,	,	PUNCT
ma-199	285	3	there	there	PRON
ma-199	285	4	existsm	existsm	VERB
ma-199	285	5	∈	∈	PROPN
ma-199	285	6	i	i	PRON
ma-199	285	7	such	such	ADJ
ma-199	285	8	that	that	SCONJ
ma-199	285	9	fm	fm	PROPN
ma-199	285	10	∈	∈	PROPN
ma-199	286	1	[	[	X
ma-199	286	2	fn	fn	X
ma-199	286	3	]	]	X
ma-199	286	4	,	,	PUNCT
ma-199	286	5	i	i	PROPN
ma-199	286	6	6=	6=	PROPN
ma-199	286	7	m.	m.	NOUN
ma-199	286	8	let	let	VERB
ma-199	286	9	{	{	PUNCT
ma-199	286	10	nk	nk	NOUN
ma-199	286	11	}	}	PUNCT
ma-199	286	12	be	be	AUX
ma-199	286	13	an	an	DET
ma-199	286	14	increasing	increase	VERB
ma-199	286	15	sequence	sequence	NOUN
ma-199	286	16	in	in	ADP
ma-199	286	17	i	i	PRON
ma-199	286	18	,	,	PUNCT
ma-199	286	19	given	give	VERB
ma-199	286	20	by	by	ADP
ma-199	286	21	nk	nk	PROPN
ma-199	286	22	=	=	PROPN
ma-199	286	23	k	k	PROPN
ma-199	286	24	,	,	PUNCT
ma-199	286	25	k	k	NOUN
ma-199	286	26	=	=	SYM
ma-199	286	27	1	1	NUM
ma-199	286	28	,	,	PUNCT
ma-199	286	29	2	2	NUM
ma-199	286	30	,	,	PUNCT
ma-199	286	31	3	3	NUM
ma-199	286	32	,	,	PUNCT
ma-199	286	33	.	.	PUNCT
ma-199	286	34	.	.	PUNCT
ma-199	286	35	.	.	PUNCT
ma-199	287	1	,	,	PUNCT
ma-199	287	2	m−1and	m−1and	PRON
ma-199	287	3	nk	nk	NOUN
ma-199	287	4	=	=	PROPN
ma-199	288	1	k	k	PROPN
ma-199	289	1	+	+	PROPN
ma-199	289	2	1	1	NUM
ma-199	289	3	,	,	PUNCT
ma-199	289	4	k	k	PROPN
ma-199	289	5	=	=	PUNCT
ma-199	289	6	m	m	PROPN
ma-199	289	7	,	,	PUNCT
ma-199	289	8	m	m	VERB
ma-199	289	9	+	+	NOUN
ma-199	289	10	1	1	NUM
ma-199	289	11	,	,	PUNCT
ma-199	289	12	.	.	PUNCT
ma-199	289	13	.	.	PUNCT
ma-199	289	14	.	.	PUNCT
ma-199	289	15	.	.	PUNCT
ma-199	290	1	since	since	SCONJ
ma-199	290	2	{	{	PUNCT
ma-199	290	3	fnk}nk∈i+	fnk}nk∈i+	PROPN
ma-199	290	4	is	be	AUX
ma-199	290	5	a	a	DET
ma-199	290	6	frame	frame	NOUN
ma-199	290	7	for	for	ADP
ma-199	290	8	k+	k+	NOUN
ma-199	290	9	and	and	CCONJ
ma-199	290	10	{	{	PUNCT
ma-199	290	11	fnk}nk∈i−	fnk}nk∈i−	ADJ
ma-199	290	12	is	be	AUX
ma-199	290	13	a	a	DET
ma-199	290	14	framefor	framefor	ADJ
ma-199	290	15	k−	k−	PROPN
ma-199	290	16	with	with	ADP
ma-199	290	17	bounds	bound	NOUN
ma-199	290	18	a	a	DET
ma-199	290	19	,	,	PUNCT
ma-199	290	20	b	b	NOUN
ma-199	290	21	and	and	CCONJ
ma-199	290	22	a′	a′	NOUN
ma-199	290	23	,	,	PUNCT
ma-199	290	24	b′	b′	NUM
ma-199	290	25	respectively	respectively	ADV
ma-199	290	26	,	,	PUNCT
ma-199	290	27	we	we	PRON
ma-199	290	28	have	have	VERB
ma-199	290	29	a‖f	a‖f	NOUN
ma-199	290	30	‖2	‖2	NOUN
ma-199	290	31	≤	≤	NUM
ma-199	290	32	∑	∑	PUNCT
ma-199	290	33	n	n	PROPN
ma-199	290	34	6	6	NUM
ma-199	290	35	=	=	NOUN
ma-199	290	36	m	m	NOUN
ma-199	290	37	n∈i+	n∈i+	NOUN
ma-199	290	38	[	[	X
ma-199	290	39	f	f	X
ma-199	290	40	,	,	PUNCT
ma-199	290	41	fn	fn	PROPN
ma-199	290	42	]	]	X
ma-199	290	43	2	2	NUM
ma-199	290	44	≤	≤	NOUN
ma-199	290	45	b‖f	b‖f	ADJ
ma-199	290	46	‖2	‖2	NOUN
ma-199	290	47	,	,	PUNCT
ma-199	290	48	for	for	SCONJ
ma-199	290	49	all	all	DET
ma-199	290	50	f	f	PROPN
ma-199	290	51	∈	∈	PROPN
ma-199	290	52	k+	k+	NOUN
ma-199	290	53	(	(	PUNCT
ma-199	290	54	8)	8)	NUM
ma-199	290	55	and	and	CCONJ
ma-199	290	56	a	a	DET
ma-199	290	57	′‖f	′‖f	NOUN
ma-199	290	58	‖2	‖2	NOUN
ma-199	290	59	≤	≤	NUM
ma-199	290	60	∑	∑	PUNCT
ma-199	290	61	n	n	PROPN
ma-199	290	62	6	6	NUM
ma-199	290	63	=	=	NOUN
ma-199	290	64	m	m	NOUN
ma-199	290	65	n∈i−	n∈i−	PROPN
ma-199	290	66	|[f	|[f	NOUN
ma-199	290	67	,	,	PUNCT
ma-199	290	68	fn]|2	fn]|2	VERB
ma-199	290	69	≤	≤	NUM
ma-199	290	70	b	b	NUM
ma-199	290	71	′‖f	′‖f	NOUN
ma-199	290	72	‖2	‖2	NOUN
ma-199	290	73	,	,	PUNCT
ma-199	290	74	for	for	ADP
ma-199	290	75	all	all	DET
ma-199	290	76	f	f	PROPN
ma-199	290	77	∈	∈	PROPN
ma-199	290	78	k−.	k−.	PROPN
ma-199	290	79	(	(	PUNCT
ma-199	290	80	9	9	NUM
ma-199	290	81	)	)	PUNCT
ma-199	290	82	since	since	SCONJ
ma-199	290	83	{	{	PUNCT
ma-199	290	84	fn}n∈i	fn}n∈i	NOUN
ma-199	290	85	is	be	AUX
ma-199	290	86	a	a	DET
ma-199	290	87	frame	frame	NOUN
ma-199	290	88	for	for	ADP
ma-199	290	89	the	the	DET
ma-199	290	90	krein	krein	NOUN
ma-199	290	91	space	space	NOUN
ma-199	290	92	(	(	PUNCT
ma-199	290	93	k	k	NOUN
ma-199	290	94	,	,	PUNCT
ma-199	290	95	[	[	X
ma-199	290	96	.	.	PUNCT
ma-199	290	97	,	,	PUNCT
ma-199	290	98	.	.	PUNCT
ma-199	291	1	]	]	PUNCT
ma-199	291	2	)	)	PUNCT
ma-199	291	3	,	,	PUNCT
ma-199	291	4	by	by	ADP
ma-199	291	5	(	(	PUNCT
ma-199	291	6	8)	8)	NUM
ma-199	291	7	,	,	PUNCT
ma-199	291	8	we	we	PRON
ma-199	291	9	have	have	VERB
ma-199	291	10	[	[	X
ma-199	291	11	f	f	X
ma-199	291	12	,	,	PUNCT
ma-199	291	13	fm	fm	PROPN
ma-199	291	14	]	]	X
ma-199	291	15	=	=	SYM
ma-199	291	16	0	0	NUM
ma-199	291	17	for	for	ADP
ma-199	291	18	all	all	DET
ma-199	291	19	f	f	PROPN
ma-199	291	20	∈	∈	PROPN
ma-199	291	21	k+	k+	PROPN
ma-199	291	22	.	.	PUNCT
ma-199	291	23	inparticular	inparticular	PROPN
ma-199	291	24	,	,	PUNCT
ma-199	291	25	[	[	X
ma-199	291	26	fm	fm	NOUN
ma-199	291	27	,	,	PUNCT
ma-199	291	28	fm	fm	NOUN
ma-199	291	29	]	]	X
ma-199	291	30	=	=	SYM
ma-199	291	31	0	0	X
ma-199	291	32	.	.	PUNCT
ma-199	292	1	this	this	PRON
ma-199	292	2	gives	give	VERB
ma-199	292	3	fm	fm	PROPN
ma-199	292	4	=	=	SYM
ma-199	292	5	0	0	X
ma-199	292	6	.	.	PUNCT
ma-199	293	1	also	also	ADV
ma-199	293	2	,	,	PUNCT
ma-199	293	3	by	by	ADP
ma-199	293	4	(	(	PUNCT
ma-199	293	5	8)	8)	NUM
ma-199	293	6	,	,	PUNCT
ma-199	293	7	|[f	|[f	NOUN
ma-199	293	8	−	−	PROPN
ma-199	293	9	,	,	PUNCT
ma-199	293	10	fm]|	fm]|	NOUN
ma-199	293	11	=	=	SYM
ma-199	293	12	0	0	NUM
ma-199	293	13	,	,	PUNCT
ma-199	293	14	for	for	ADP
ma-199	293	15	all	all	DET
ma-199	293	16	f	f	PROPN
ma-199	293	17	∈	∈	PROPN
ma-199	293	18	k−.	k−.	PROPN
ma-199	293	19	in	in	ADP
ma-199	293	20	particular	particular	ADJ
ma-199	293	21	|[fm	|[fm	NOUN
ma-199	293	22	,	,	PUNCT
ma-199	293	23	fm]|	fm]|	NOUN
ma-199	293	24	=	=	SYM
ma-199	293	25	0	0	PROPN
ma-199	293	26	.	.	PUNCT
ma-199	294	1	this	this	PRON
ma-199	294	2	implies	imply	VERB
ma-199	294	3	that	that	SCONJ
ma-199	294	4	fm	fm	PROPN
ma-199	294	5	=	=	SYM
ma-199	294	6	0	0	NUM
ma-199	294	7	which	which	PRON
ma-199	294	8	is	be	AUX
ma-199	294	9	a	a	DET
ma-199	294	10	contradiction	contradiction	NOUN
ma-199	294	11	.	.	PUNCT
ma-199	295	1	hence	hence	ADV
ma-199	295	2	{	{	PUNCT
ma-199	295	3	fnk}nk∈i	fnk}nk∈i	PROPN
ma-199	295	4	is	be	AUX
ma-199	295	5	an	an	DET
ma-199	295	6	exactframe	exactframe	NOUN
ma-199	295	7	.	.	PUNCT
ma-199	296	1	�	�	PROPN
ma-199	296	2	acknowledgementsthe	acknowledgementsthe	DET
ma-199	296	3	present	present	ADJ
ma-199	296	4	work	work	NOUN
ma-199	296	5	of	of	ADP
ma-199	296	6	the	the	DET
ma-199	296	7	first	first	ADJ
ma-199	296	8	author	author	NOUN
ma-199	296	9	is	be	AUX
ma-199	296	10	partially	partially	ADV
ma-199	296	11	supported	support	VERB
ma-199	296	12	by	by	ADP
ma-199	296	13	university	university	NOUN
ma-199	296	14	grants	grant	NOUN
ma-199	296	15	commission(ugc	commission(ugc	PROPN
ma-199	296	16	)	)	PUNCT
ma-199	296	17	,	,	PUNCT
ma-199	296	18	government	government	NOUN
ma-199	296	19	of	of	ADP
ma-199	296	20	india	india	PROPN
ma-199	296	21	.	.	PUNCT
ma-199	297	1	the	the	DET
ma-199	297	2	present	present	ADJ
ma-199	297	3	work	work	NOUN
ma-199	297	4	of	of	ADP
ma-199	297	5	the	the	DET
ma-199	297	6	second	second	ADJ
ma-199	297	7	author	author	NOUN
ma-199	297	8	is	be	AUX
ma-199	297	9	partially	partially	ADV
ma-199	297	10	supported	support	VERB
ma-199	297	11	byscience	byscience	NOUN
ma-199	297	12	and	and	CCONJ
ma-199	297	13	engineering	engineering	NOUN
ma-199	297	14	research	research	NOUN
ma-199	297	15	board	board	NOUN
ma-199	297	16	(	(	PUNCT
ma-199	297	17	serb	serb	NOUN
ma-199	297	18	)	)	PUNCT
ma-199	297	19	,	,	PUNCT
ma-199	297	20	government	government	NOUN
ma-199	297	21	of	of	ADP
ma-199	297	22	india	india	PROPN
ma-199	297	23	(	(	PUNCT
ma-199	297	24	reference	reference	NOUN
ma-199	297	25	number	number	NOUN
ma-199	297	26	:	:	PUNCT
ma-199	297	27	tar/2022/000219	tar/2022/000219	ADJ
ma-199	297	28	)	)	PUNCT
ma-199	297	29	.	.	PUNCT
ma-199	298	1	references	reference	NOUN
ma-199	298	2	[	[	X
ma-199	298	3	1	1	X
ma-199	298	4	]	]	PUNCT
ma-199	298	5	t.	t.	PROPN
ma-199	298	6	y.	y.	PROPN
ma-199	298	7	azizov	azizov	PROPN
ma-199	298	8	and	and	CCONJ
ma-199	298	9	i.	i.	PROPN
ma-199	298	10	s.	s.	PROPN
ma-199	298	11	iokhvidov	iokhvidov	PROPN
ma-199	298	12	,	,	PUNCT
ma-199	298	13	linear	linear	PROPN
ma-199	298	14	operators	operator	NOUN
ma-199	298	15	in	in	ADP
ma-199	298	16	spaces	space	NOUN
ma-199	298	17	with	with	ADP
ma-199	298	18	an	an	DET
ma-199	298	19	indefinite	indefinite	ADJ
ma-199	298	20	metric	metric	NOUN
ma-199	298	21	,	,	PUNCT
ma-199	298	22	john	john	PROPN
ma-199	298	23	wiley	wiley	PROPN
ma-199	298	24	&	&	CCONJ
ma-199	298	25	sons	son	NOUN
ma-199	298	26	,	,	PUNCT
ma-199	298	27	incorporated,1989.[2	incorporated,1989.[2	PROPN
ma-199	298	28	]	]	PUNCT
ma-199	298	29	j.	j.	PROPN
ma-199	298	30	bognár	bognár	PROPN
ma-199	298	31	,	,	PUNCT
ma-199	298	32	indefinite	indefinite	ADJ
ma-199	298	33	inner	inner	ADJ
ma-199	298	34	product	product	NOUN
ma-199	298	35	spaces	space	NOUN
ma-199	298	36	,	,	PUNCT
ma-199	298	37	vol	vol	NOUN
ma-199	298	38	.	.	PROPN
ma-199	298	39	78	78	NUM
ma-199	298	40	,	,	PUNCT
ma-199	298	41	springer	springer	NOUN
ma-199	298	42	science	science	PROPN
ma-199	298	43	&	&	CCONJ
ma-199	298	44	business	business	NOUN
ma-199	298	45	media	medium	NOUN
ma-199	298	46	,	,	PUNCT
ma-199	298	47	2012	2012	NUM
ma-199	298	48	.	.	PUNCT
ma-199	299	1	https://doi.org/	https://doi.org/	VERB
ma-199	299	2	10.1007/978	10.1007/978	NUM
ma-199	299	3	-	-	SYM
ma-199	299	4	3	3	NUM
ma-199	299	5	-	-	PUNCT
ma-199	299	6	642	642	NUM
ma-199	299	7	-	-	PUNCT
ma-199	299	8	65567	65567	NUM
ma-199	299	9	-	-	SYM
ma-199	299	10	8.[3	8.[3	NUM
ma-199	299	11	]	]	X
ma-199	299	12	p.	p.	NOUN
ma-199	299	13	g.	g.	PROPN
ma-199	299	14	casazza	casazza	PROPN
ma-199	299	15	and	and	CCONJ
ma-199	299	16	g.	g.	PROPN
ma-199	299	17	kutyniok	kutyniok	PROPN
ma-199	299	18	,	,	PUNCT
ma-199	299	19	finite	finite	PROPN
ma-199	299	20	frames	frame	NOUN
ma-199	299	21	:	:	PUNCT
ma-199	299	22	theory	theory	NOUN
ma-199	299	23	and	and	CCONJ
ma-199	299	24	applications	application	NOUN
ma-199	299	25	,	,	PUNCT
ma-199	299	26	springer	springer	NOUN
ma-199	299	27	science	science	PROPN
ma-199	299	28	&	&	CCONJ
ma-199	299	29	business	business	NOUN
ma-199	299	30	media	medium	NOUN
ma-199	299	31	,	,	PUNCT
ma-199	299	32	2012	2012	NUM
ma-199	299	33	.	.	PUNCT
ma-199	300	1	https://doi.org/10.1007/978-0-8176-8373-3.[4	https://doi.org/10.1007/978-0-8176-8373-3.[4	PROPN
ma-199	300	2	]	]	X
ma-199	300	3	i.	i.	PROPN
ma-199	300	4	daubechies	daubechies	PROPN
ma-199	300	5	,	,	PUNCT
ma-199	300	6	a.	a.	NOUN
ma-199	300	7	grossmann	grossmann	PROPN
ma-199	300	8	and	and	CCONJ
ma-199	300	9	y.	y.	PROPN
ma-199	300	10	meyer	meyer	PROPN
ma-199	300	11	,	,	PUNCT
ma-199	300	12	painless	painless	ADJ
ma-199	300	13	nonorthogonal	nonorthogonal	ADJ
ma-199	300	14	expansions	expansion	NOUN
ma-199	300	15	,	,	PUNCT
ma-199	300	16	j.	j.	PROPN
ma-199	300	17	math	math	PROPN
ma-199	300	18	.	.	PUNCT
ma-199	301	1	phys	phy	NOUN
ma-199	301	2	.	.	PUNCT
ma-199	302	1	27	27	NUM
ma-199	302	2	(	(	PUNCT
ma-199	302	3	1986	1986	NUM
ma-199	302	4	)	)	PUNCT
ma-199	302	5	1271	1271	NUM
ma-199	302	6	-	-	SYM
ma-199	302	7	1283	1283	NUM
ma-199	302	8	.	.	PUNCT
ma-199	303	1	https://doi.org/10.1063/1.527388.[5	https://doi.org/10.1063/1.527388.[5	X
ma-199	303	2	]	]	X
ma-199	303	3	p.	p.	NOUN
ma-199	303	4	a.	a.	PROPN
ma-199	303	5	m.	m.	PROPN
ma-199	303	6	dirac	dirac	PROPN
ma-199	303	7	,	,	PUNCT
ma-199	303	8	the	the	DET
ma-199	303	9	physical	physical	ADJ
ma-199	303	10	interpretation	interpretation	NOUN
ma-199	303	11	of	of	ADP
ma-199	303	12	the	the	DET
ma-199	303	13	quantum	quantum	NOUN
ma-199	303	14	dynamics	dynamic	NOUN
ma-199	303	15	,	,	PUNCT
ma-199	303	16	proc	proc	NOUN
ma-199	303	17	.	.	PUNCT
ma-199	304	1	r.	r.	PROPN
ma-199	304	2	soc	soc	PROPN
ma-199	304	3	.	.	PUNCT
ma-199	305	1	lond	lond	PROPN
ma-199	305	2	.	.	PUNCT
ma-199	306	1	ser	ser	PROPN
ma-199	306	2	.	.	PUNCT
ma-199	307	1	a	a	PRON
ma-199	307	2	,	,	PUNCT
ma-199	307	3	113	113	NUM
ma-199	307	4	(	(	PUNCT
ma-199	307	5	1927	1927	NUM
ma-199	307	6	)	)	PUNCT
ma-199	307	7	621	621	NUM
ma-199	307	8	-	-	SYM
ma-199	307	9	641	641	NUM
ma-199	307	10	.	.	PUNCT
ma-199	308	1	https://doi.org/10.1098/rspa.1927.0012.[6	https://doi.org/10.1098/rspa.1927.0012.[6	ADP
ma-199	308	2	]	]	X
ma-199	308	3	d.	d.	PROPN
ma-199	308	4	l.	l.	PROPN
ma-199	308	5	donoho	donoho	PROPN
ma-199	308	6	and	and	CCONJ
ma-199	308	7	m.	m.	PROPN
ma-199	308	8	elad	elad	PROPN
ma-199	308	9	,	,	PUNCT
ma-199	308	10	optimally	optimally	ADV
ma-199	308	11	sparse	sparse	VERB
ma-199	308	12	representation	representation	NOUN
ma-199	308	13	in	in	ADP
ma-199	308	14	general	general	ADJ
ma-199	308	15	(	(	PUNCT
ma-199	308	16	nonorthogonal	nonorthogonal	ADJ
ma-199	308	17	)	)	PUNCT
ma-199	308	18	dictionaries	dictionary	NOUN
ma-199	308	19	via	via	ADP
ma-199	308	20	`	`	PUNCT
ma-199	308	21	1	1	NUM
ma-199	308	22	mini	mini	NOUN
ma-199	308	23	-	-	NOUN
ma-199	308	24	mization	mization	NOUN
ma-199	308	25	,	,	PUNCT
ma-199	308	26	proc	proc	NOUN
ma-199	308	27	.	.	PUNCT
ma-199	309	1	nat	nat	PROPN
ma-199	309	2	.	.	PUNCT
ma-199	310	1	acad	acad	PROPN
ma-199	310	2	.	.	PUNCT
ma-199	311	1	sci	sci	PROPN
ma-199	311	2	.	.	PROPN
ma-199	311	3	100	100	NUM
ma-199	311	4	(	(	PUNCT
ma-199	311	5	2003	2003	NUM
ma-199	311	6	)	)	PUNCT
ma-199	311	7	2197	2197	NUM
ma-199	311	8	-	-	SYM
ma-199	311	9	2202	2202	NUM
ma-199	311	10	.	.	PUNCT
ma-199	312	1	https://doi.org/10.1073/pnas.0437847100.[7	https://doi.org/10.1073/pnas.0437847100.[7	PROPN
ma-199	312	2	]	]	X
ma-199	312	3	r.	r.	PROPN
ma-199	312	4	j.	j.	PROPN
ma-199	312	5	duffin	duffin	PROPN
ma-199	312	6	and	and	CCONJ
ma-199	312	7	a.	a.	PROPN
ma-199	312	8	c.	c.	PROPN
ma-199	312	9	schaeffer	schaeffer	PROPN
ma-199	312	10	,	,	PUNCT
ma-199	312	11	a	a	DET
ma-199	312	12	class	class	NOUN
ma-199	312	13	of	of	ADP
ma-199	312	14	nonharmonic	nonharmonic	ADJ
ma-199	312	15	fourier	fourier	NOUN
ma-199	312	16	series	series	NOUN
ma-199	312	17	,	,	PUNCT
ma-199	312	18	trans	trans	PROPN
ma-199	312	19	.	.	PROPN
ma-199	312	20	amer	amer	PROPN
ma-199	312	21	.	.	PUNCT
ma-199	312	22	math	math	PROPN
ma-199	312	23	.	.	PUNCT
ma-199	313	1	soc	soc	PROPN
ma-199	313	2	.	.	PUNCT
ma-199	314	1	72	72	NUM
ma-199	314	2	(	(	PUNCT
ma-199	314	3	1952	1952	NUM
ma-199	314	4	)	)	PUNCT
ma-199	314	5	341	341	NUM
ma-199	314	6	-	-	SYM
ma-199	314	7	366	366	NUM
ma-199	314	8	.	.	PUNCT
ma-199	315	1	https://doi.org/10.1090/s0002-9947-1952-0047179-6.[8	https://doi.org/10.1090/s0002-9947-1952-0047179-6.[8	PROPN
ma-199	315	2	]	]	PUNCT
ma-199	315	3	j.	j.	PROPN
ma-199	315	4	giribet	giribet	PROPN
ma-199	315	5	,	,	PUNCT
ma-199	315	6	a.	a.	PROPN
ma-199	315	7	maestripieri	maestripieri	PROPN
ma-199	315	8	,	,	PUNCT
ma-199	315	9	f.	f.	PROPN
ma-199	315	10	m.	m.	PROPN
ma-199	315	11	pería	pería	VERB
ma-199	315	12	and	and	CCONJ
ma-199	315	13	p.	p.	NOUN
ma-199	315	14	g.	g.	PROPN
ma-199	316	1	massey	massey	PROPN
ma-199	316	2	,	,	PUNCT
ma-199	316	3	on	on	ADP
ma-199	316	4	frames	frame	NOUN
ma-199	316	5	for	for	ADP
ma-199	316	6	krein	krein	ADJ
ma-199	316	7	spaces	space	NOUN
ma-199	316	8	,	,	PUNCT
ma-199	316	9	j.	j.	PROPN
ma-199	316	10	math	math	PROPN
ma-199	316	11	.	.	PUNCT
ma-199	317	1	anal	anal	PROPN
ma-199	317	2	.	.	PUNCT
ma-199	318	1	appl	appl	PROPN
ma-199	318	2	.	.	PROPN
ma-199	319	1	393	393	NUM
ma-199	319	2	(	(	PUNCT
ma-199	319	3	2012)122	2012)122	NOUN
ma-199	319	4	-	-	SYM
ma-199	319	5	137	137	NUM
ma-199	319	6	.	.	PUNCT
ma-199	320	1	https://doi.org/10.1016/j.jmaa.2012.03.040.[9	https://doi.org/10.1016/j.jmaa.2012.03.040.[9	PROPN
ma-199	320	2	]	]	PUNCT
ma-199	320	3	d.	d.	PROPN
ma-199	320	4	han	han	PROPN
ma-199	320	5	,	,	PUNCT
ma-199	320	6	k.	k.	PROPN
ma-199	320	7	kornelson	kornelson	PROPN
ma-199	320	8	,	,	PUNCT
ma-199	320	9	d.	d.	PROPN
ma-199	320	10	larson	larson	PROPN
ma-199	320	11	and	and	CCONJ
ma-199	320	12	e.	e.	PROPN
ma-199	320	13	weber	weber	PROPN
ma-199	320	14	,	,	PUNCT
ma-199	320	15	frames	frame	NOUN
ma-199	320	16	for	for	ADP
ma-199	320	17	undergraduates	undergraduate	NOUN
ma-199	320	18	,	,	PUNCT
ma-199	320	19	vol	vol	NOUN
ma-199	320	20	.	.	PROPN
ma-199	320	21	40	40	NUM
ma-199	320	22	.	.	PUNCT
ma-199	321	1	american	american	PROPN
ma-199	321	2	mathematical	mathematical	PROPN
ma-199	321	3	soc	soc	PROPN
ma-199	321	4	.	.	PUNCT
ma-199	322	1	,2007	,2007	PROPN
ma-199	322	2	.	.	PUNCT
ma-199	323	1	https://doi.org/10.28924/ada/ma.4.1	https://doi.org/10.28924/ada/ma.4.1	NUM
ma-199	323	2	https://doi.org/10.1007/978-3-642-65567-8	https://doi.org/10.1007/978-3-642-65567-8	PROPN
ma-199	323	3	https://doi.org/10.1007/978-3-642-65567-8	https://doi.org/10.1007/978-3-642-65567-8	PROPN
ma-199	323	4	https://doi.org/10.1007/978-0-8176-8373-3	https://doi.org/10.1007/978-0-8176-8373-3	NOUN
ma-199	323	5	https://doi.org/10.1063/1.527388	https://doi.org/10.1063/1.527388	PROPN
ma-199	323	6	https://doi.org/10.1098/rspa.1927.0012	https://doi.org/10.1098/rspa.1927.0012	PROPN
ma-199	323	7	https://doi.org/10.1073/pnas.0437847100	https://doi.org/10.1073/pnas.0437847100	VERB
ma-199	324	1	https://doi.org/10.1090/s0002-9947-1952-0047179-6	https://doi.org/10.1090/s0002-9947-1952-0047179-6	NOUN
ma-199	324	2	https://doi.org/10.1016/j.jmaa.2012.03.040	https://doi.org/10.1016/j.jmaa.2012.03.040	VERB
ma-199	324	3	1	1	NUM
ma-199	324	4	.	.	PUNCT
ma-199	324	5	introduction	introduction	NOUN
ma-199	324	6	2	2	NUM
ma-199	324	7	.	.	PUNCT
ma-199	324	8	preliminaries	preliminary	NOUN
ma-199	324	9	3	3	NUM
ma-199	324	10	.	.	PUNCT
ma-199	324	11	frame	frame	NOUN
ma-199	324	12	operator	operator	NOUN
ma-199	324	13	for	for	ADP
ma-199	324	14	frames	frame	NOUN
ma-199	324	15	in	in	ADP
ma-199	324	16	krein	krein	PROPN
ma-199	324	17	spaces	space	NOUN
ma-199	324	18	4	4	NUM
ma-199	324	19	.	.	PUNCT
ma-199	324	20	frame	frame	NOUN
ma-199	324	21	sequences	sequence	NOUN
ma-199	324	22	in	in	ADP
ma-199	324	23	krein	krein	ADJ
ma-199	324	24	spaces	space	NOUN
ma-199	324	25	references	reference	NOUN