id	sid	tid	token	lemma	pos
ma-58	1	1	2022	2022	NUM
ma-58	1	2	ada	ada	PROPN
ma-58	1	3	academica	academica	PROPN
ma-58	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-58	1	5	.	.	PUNCT
ma-58	2	1	j.	j.	PROPN
ma-58	2	2	math	math	PROPN
ma-58	2	3	.	.	PUNCT
ma-58	3	1	anal	anal	ADJ
ma-58	3	2	.	.	PUNCT
ma-58	3	3	2	2	NUM
ma-58	3	4	(	(	PUNCT
ma-58	3	5	2022	2022	NUM
ma-58	3	6	)	)	PUNCT
ma-58	3	7	4doi	4doi	NOUN
ma-58	3	8	:	:	PUNCT
ma-58	3	9	10.28924	10.28924	NUM
ma-58	3	10	/	/	SYM
ma-58	3	11	ada	ada	PROPN
ma-58	3	12	/	/	SYM
ma-58	3	13	ma.2.4	ma.2.4	PROPN
ma-58	3	14	∗-k	∗-k	NOUN
ma-58	3	15	-	-	PUNCT
ma-58	3	16	operator	operator	NOUN
ma-58	3	17	frame	frame	NOUN
ma-58	3	18	for	for	ADP
ma-58	3	19	hom∗a(x	hom∗a(x	NOUN
ma-58	3	20	)	)	PUNCT
ma-58	3	21	mohamed	mohamed	PROPN
ma-58	3	22	rossafi1,∗	rossafi1,∗	PROPN
ma-58	3	23	,	,	PUNCT
ma-58	3	24	roumaissae	roumaissae	PROPN
ma-58	3	25	el	el	PROPN
ma-58	3	26	jazzar2	jazzar2	PROPN
ma-58	3	27	and	and	CCONJ
ma-58	3	28	ali	ali	PROPN
ma-58	3	29	kacha2	kacha2	PROPN
ma-58	3	30	1lasma	1lasma	NUM
ma-58	3	31	laboratory	laboratory	NOUN
ma-58	3	32	department	department	NOUN
ma-58	3	33	of	of	ADP
ma-58	3	34	mathematics	mathematic	NOUN
ma-58	3	35	,	,	PUNCT
ma-58	3	36	faculty	faculty	NOUN
ma-58	3	37	of	of	ADP
ma-58	3	38	sciences	sciences	PROPN
ma-58	3	39	dhar	dhar	PROPN
ma-58	3	40	el	el	PROPN
ma-58	3	41	mahraz	mahraz	PROPN
ma-58	3	42	,	,	PUNCT
ma-58	3	43	university	university	NOUN
ma-58	3	44	sidi	sidi	NOUN
ma-58	3	45	mohamed	mohamed	PROPN
ma-58	3	46	ben	ben	PROPN
ma-58	3	47	abdellah	abdellah	PROPN
ma-58	3	48	,	,	PUNCT
ma-58	4	1	b.	b.	PROPN
ma-58	4	2	p.	p.	NOUN
ma-58	4	3	1796	1796	NUM
ma-58	5	1	fes	fes	PROPN
ma-58	5	2	atlas	atlas	PROPN
ma-58	5	3	,	,	PUNCT
ma-58	5	4	morocco	morocco	PROPN
ma-58	5	5	mohamed.rossafi@usmba.ac.ma	mohamed.rossafi@usmba.ac.ma	NUM
ma-58	5	6	2laboratory	2laboratory	NUM
ma-58	5	7	of	of	ADP
ma-58	5	8	partial	partial	ADJ
ma-58	5	9	differential	differential	NOUN
ma-58	5	10	equations	equation	NOUN
ma-58	5	11	,	,	PUNCT
ma-58	5	12	spectral	spectral	ADJ
ma-58	5	13	algebra	algebra	NOUN
ma-58	5	14	and	and	CCONJ
ma-58	5	15	geometry	geometry	NOUN
ma-58	5	16	department	department	NOUN
ma-58	5	17	of	of	ADP
ma-58	5	18	mathematics	mathematic	NOUN
ma-58	5	19	,	,	PUNCT
ma-58	5	20	faculty	faculty	NOUN
ma-58	5	21	of	of	ADP
ma-58	5	22	sciences	science	NOUN
ma-58	5	23	,	,	PUNCT
ma-58	5	24	university	university	NOUN
ma-58	5	25	ibn	ibn	PROPN
ma-58	5	26	tofail	tofail	NOUN
ma-58	5	27	,	,	PUNCT
ma-58	5	28	kenitra	kenitra	PROPN
ma-58	5	29	,	,	PUNCT
ma-58	5	30	morocco	morocco	PROPN
ma-58	5	31	roumaissae.eljazzar@uit.ac.ma	roumaissae.eljazzar@uit.ac.ma	PROPN
ma-58	5	32	,	,	PUNCT
ma-58	5	33	ali.kacha@yahoo.fr	ali.kacha@yahoo.fr	PUNCT
ma-58	5	34	∗correspondence	∗correspondence	NOUN
ma-58	5	35	:	:	PUNCT
ma-58	5	36	rossafimohamed@gmail.com	rossafimohamed@gmail.com	X
ma-58	5	37	abstract	abstract	ADJ
ma-58	5	38	.	.	PUNCT
ma-58	6	1	in	in	ADP
ma-58	6	2	this	this	DET
ma-58	6	3	work	work	NOUN
ma-58	6	4	,	,	PUNCT
ma-58	6	5	we	we	PRON
ma-58	6	6	introduce	introduce	VERB
ma-58	6	7	the	the	DET
ma-58	6	8	concept	concept	NOUN
ma-58	6	9	of	of	ADP
ma-58	6	10	∗-k	∗-k	NOUN
ma-58	6	11	-	-	PUNCT
ma-58	6	12	operator	operator	NOUN
ma-58	6	13	frames	frame	NOUN
ma-58	6	14	in	in	ADP
ma-58	6	15	hilbert	hilbert	PROPN
ma-58	6	16	pro	pro	PROPN
ma-58	6	17	-	-	NOUN
ma-58	6	18	c∗-modules	c∗-module	NOUN
ma-58	6	19	,	,	PUNCT
ma-58	6	20	which	which	PRON
ma-58	6	21	is	be	AUX
ma-58	6	22	a	a	DET
ma-58	6	23	generalization	generalization	NOUN
ma-58	6	24	of	of	ADP
ma-58	6	25	k	k	ADJ
ma-58	6	26	-	-	NOUN
ma-58	6	27	operator	operator	NOUN
ma-58	6	28	frame	frame	NOUN
ma-58	6	29	.	.	PUNCT
ma-58	7	1	we	we	PRON
ma-58	7	2	present	present	VERB
ma-58	7	3	the	the	DET
ma-58	7	4	analysis	analysis	NOUN
ma-58	7	5	operator	operator	NOUN
ma-58	7	6	,	,	PUNCT
ma-58	7	7	the	the	DET
ma-58	7	8	synthesisoperator	synthesisoperator	NOUN
ma-58	7	9	and	and	CCONJ
ma-58	7	10	the	the	DET
ma-58	7	11	frame	frame	NOUN
ma-58	7	12	operator	operator	NOUN
ma-58	7	13	.	.	PUNCT
ma-58	8	1	we	we	PRON
ma-58	8	2	also	also	ADV
ma-58	8	3	give	give	VERB
ma-58	8	4	some	some	DET
ma-58	8	5	properties	property	NOUN
ma-58	8	6	and	and	CCONJ
ma-58	8	7	we	we	PRON
ma-58	8	8	study	study	VERB
ma-58	8	9	the	the	DET
ma-58	8	10	tensor	tensor	NOUN
ma-58	8	11	product	product	NOUN
ma-58	8	12	of	of	ADP
ma-58	8	13	∗-k	∗-k	NOUN
ma-58	8	14	-	-	PUNCT
ma-58	8	15	operator	operator	NOUN
ma-58	8	16	frame	frame	NOUN
ma-58	8	17	for	for	ADP
ma-58	8	18	hilbert	hilbert	PROPN
ma-58	8	19	pro	pro	PROPN
ma-58	8	20	-	-	NOUN
ma-58	8	21	c∗-modules	c∗-module	NOUN
ma-58	8	22	.	.	NOUN
ma-58	9	1	1	1	X
ma-58	9	2	.	.	X
ma-58	9	3	introduction	introduction	NOUN
ma-58	9	4	duffin	duffin	PROPN
ma-58	9	5	and	and	CCONJ
ma-58	9	6	schaeffer	schaeffer	PROPN
ma-58	9	7	introduced	introduce	VERB
ma-58	9	8	the	the	DET
ma-58	9	9	notion	notion	NOUN
ma-58	9	10	of	of	ADP
ma-58	9	11	frame	frame	NOUN
ma-58	9	12	in	in	ADP
ma-58	9	13	nonharmonic	nonharmonic	ADJ
ma-58	9	14	fourier	fourier	NOUN
ma-58	9	15	analysis	analysis	NOUN
ma-58	9	16	in	in	ADP
ma-58	9	17	1952	1952	NUM
ma-58	10	1	[	[	X
ma-58	10	2	3].in	3].in	NUM
ma-58	10	3	1986	1986	NUM
ma-58	10	4	the	the	DET
ma-58	10	5	work	work	NOUN
ma-58	10	6	of	of	ADP
ma-58	10	7	duffin	duffin	PROPN
ma-58	10	8	and	and	CCONJ
ma-58	10	9	schaeffer	schaeffer	PROPN
ma-58	10	10	were	be	AUX
ma-58	10	11	reintroduced	reintroduce	VERB
ma-58	10	12	and	and	CCONJ
ma-58	10	13	developed	develop	VERB
ma-58	10	14	by	by	ADP
ma-58	10	15	grossman	grossman	PROPN
ma-58	10	16	andmeyer	andmeyer	PROPN
ma-58	11	1	[	[	X
ma-58	11	2	7	7	NUM
ma-58	11	3	]	]	PUNCT
ma-58	11	4	.	.	PUNCT
ma-58	12	1	the	the	DET
ma-58	12	2	concept	concept	NOUN
ma-58	12	3	of	of	ADP
ma-58	12	4	frame	frame	NOUN
ma-58	12	5	on	on	ADP
ma-58	12	6	hilbert	hilbert	NOUN
ma-58	12	7	space	space	NOUN
ma-58	12	8	has	have	AUX
ma-58	12	9	already	already	ADV
ma-58	12	10	been	be	AUX
ma-58	12	11	successfully	successfully	ADV
ma-58	12	12	extended	extend	VERB
ma-58	12	13	to	to	ADP
ma-58	12	14	proc∗-algebras	proc∗-algebras	PROPN
ma-58	12	15	and	and	CCONJ
ma-58	12	16	hilbert	hilbert	NOUN
ma-58	12	17	modules	module	NOUN
ma-58	12	18	.	.	PUNCT
ma-58	13	1	many	many	ADJ
ma-58	13	2	properties	property	NOUN
ma-58	13	3	of	of	ADP
ma-58	13	4	frames	frame	NOUN
ma-58	13	5	in	in	ADP
ma-58	13	6	hilbert	hilbert	PROPN
ma-58	13	7	c∗-modules	c∗-modules	PROPN
ma-58	13	8	are	be	AUX
ma-58	13	9	valid	valid	ADJ
ma-58	13	10	forframes	forframe	NOUN
ma-58	13	11	of	of	ADP
ma-58	13	12	multipliers	multiplier	NOUN
ma-58	13	13	in	in	ADP
ma-58	13	14	hilbert	hilbert	NOUN
ma-58	13	15	modules	module	NOUN
ma-58	13	16	over	over	ADP
ma-58	13	17	pro	pro	ADJ
ma-58	13	18	-	-	ADJ
ma-58	13	19	c∗-algebras	c∗-algebra	NOUN
ma-58	13	20	[	[	X
ma-58	13	21	9].operator	9].operator	NUM
ma-58	13	22	frames	frame	NOUN
ma-58	13	23	for	for	ADP
ma-58	13	24	b(h	b(h	NOUN
ma-58	13	25	)	)	PUNCT
ma-58	13	26	is	be	AUX
ma-58	13	27	a	a	DET
ma-58	13	28	new	new	ADJ
ma-58	13	29	notion	notion	NOUN
ma-58	13	30	of	of	ADP
ma-58	13	31	frames	frame	NOUN
ma-58	13	32	that	that	PRON
ma-58	13	33	li	li	PROPN
ma-58	13	34	and	and	CCONJ
ma-58	13	35	cio	cio	PROPN
ma-58	13	36	introduced	introduce	VERB
ma-58	13	37	in	in	ADP
ma-58	13	38	[	[	X
ma-58	13	39	11	11	NUM
ma-58	13	40	]	]	PUNCT
ma-58	13	41	andgeneralized	andgeneralize	VERB
ma-58	13	42	by	by	ADP
ma-58	13	43	rossafi	rossafi	NOUN
ma-58	13	44	in	in	ADP
ma-58	13	45	[	[	X
ma-58	13	46	16	16	NUM
ma-58	13	47	]	]	PUNCT
ma-58	13	48	.	.	PUNCT
ma-58	14	1	in	in	ADP
ma-58	14	2	this	this	DET
ma-58	14	3	work	work	NOUN
ma-58	14	4	we	we	PRON
ma-58	14	5	introduce	introduce	VERB
ma-58	14	6	the	the	DET
ma-58	14	7	notion	notion	NOUN
ma-58	14	8	of	of	ADP
ma-58	14	9	∗-k	∗-k	ADJ
ma-58	14	10	-	-	PUNCT
ma-58	14	11	operator	operator	NOUN
ma-58	14	12	frame	frame	NOUN
ma-58	14	13	for	for	ADP
ma-58	14	14	thespace	thespace	NOUN
ma-58	14	15	hom∗a(x	hom∗a(x	NOUN
ma-58	14	16	)	)	PUNCT
ma-58	14	17	of	of	ADP
ma-58	14	18	all	all	DET
ma-58	14	19	adjointable	adjointable	ADJ
ma-58	14	20	operators	operator	NOUN
ma-58	14	21	on	on	ADP
ma-58	14	22	a	a	DET
ma-58	14	23	hilbert	hilbert	NOUN
ma-58	14	24	pro	pro	NOUN
ma-58	14	25	-	-	NOUN
ma-58	14	26	c∗-module	c∗-module	NOUN
ma-58	14	27	for	for	SCONJ
ma-58	14	28	x	x	SYM
ma-58	14	29	.this	.this	PRON
ma-58	14	30	paper	paper	NOUN
ma-58	14	31	is	be	AUX
ma-58	14	32	divided	divide	VERB
ma-58	14	33	into	into	ADP
ma-58	14	34	three	three	NUM
ma-58	14	35	sections	section	NOUN
ma-58	14	36	.	.	PUNCT
ma-58	15	1	in	in	ADP
ma-58	15	2	section	section	NOUN
ma-58	15	3	2	2	NUM
ma-58	15	4	we	we	PRON
ma-58	15	5	recall	recall	VERB
ma-58	15	6	some	some	DET
ma-58	15	7	fundamental	fundamental	ADJ
ma-58	15	8	definitionsand	definitionsand	NOUN
ma-58	15	9	notations	notation	NOUN
ma-58	15	10	of	of	ADP
ma-58	15	11	hilbert	hilbert	PROPN
ma-58	15	12	pro	pro	PROPN
ma-58	15	13	-	-	NOUN
ma-58	15	14	c∗-modules	c∗-module	NOUN
ma-58	15	15	.	.	PUNCT
ma-58	16	1	in	in	ADP
ma-58	16	2	section	section	NOUN
ma-58	16	3	3	3	NUM
ma-58	16	4	we	we	PRON
ma-58	16	5	introduce	introduce	VERB
ma-58	16	6	the	the	DET
ma-58	16	7	∗-k	∗-k	NOUN
ma-58	16	8	-	-	PUNCT
ma-58	16	9	operator	operator	NOUN
ma-58	16	10	frame	frame	NOUN
ma-58	16	11	andwe	andwe	NOUN
ma-58	16	12	give	give	VERB
ma-58	16	13	some	some	PRON
ma-58	16	14	of	of	ADP
ma-58	16	15	its	its	PRON
ma-58	16	16	properties	property	NOUN
ma-58	16	17	.	.	PUNCT
ma-58	17	1	lastly	lastly	ADV
ma-58	17	2	we	we	PRON
ma-58	17	3	investigate	investigate	VERB
ma-58	17	4	tensor	tensor	NOUN
ma-58	17	5	product	product	NOUN
ma-58	17	6	of	of	ADP
ma-58	17	7	hilbert	hilbert	PROPN
ma-58	17	8	pro	pro	PROPN
ma-58	17	9	-	-	NOUN
ma-58	17	10	c∗-modules	c∗-module	NOUN
ma-58	17	11	,	,	PUNCT
ma-58	17	12	weshow	weshow	ADP
ma-58	17	13	that	that	SCONJ
ma-58	17	14	tensor	tensor	NOUN
ma-58	17	15	product	product	NOUN
ma-58	17	16	of	of	ADP
ma-58	17	17	∗-k	∗-k	NOUN
ma-58	17	18	-	-	PUNCT
ma-58	17	19	operator	operator	NOUN
ma-58	17	20	frames	frame	NOUN
ma-58	17	21	for	for	ADP
ma-58	17	22	hilbert	hilbert	PROPN
ma-58	17	23	pro	pro	PROPN
ma-58	17	24	-	-	NOUN
ma-58	17	25	c∗-modules	c∗-module	NOUN
ma-58	17	26	x	x	NOUN
ma-58	17	27	and	and	CCONJ
ma-58	17	28	y	y	PROPN
ma-58	17	29	,	,	PUNCT
ma-58	17	30	present	present	VERB
ma-58	17	31	an	an	DET
ma-58	17	32	∗-k	∗-k	ADJ
ma-58	17	33	-	-	PUNCT
ma-58	17	34	operator	operator	NOUN
ma-58	17	35	frames	frame	NOUN
ma-58	17	36	for	for	ADP
ma-58	17	37	x	x	PROPN
ma-58	17	38	⊗	⊗	PROPN
ma-58	17	39	y	y	PROPN
ma-58	17	40	,	,	PUNCT
ma-58	17	41	and	and	CCONJ
ma-58	17	42	tensor	tensor	NOUN
ma-58	17	43	product	product	NOUN
ma-58	17	44	of	of	ADP
ma-58	17	45	their	their	PRON
ma-58	17	46	frame	frame	NOUN
ma-58	17	47	operators	operator	NOUN
ma-58	17	48	is	be	AUX
ma-58	17	49	the	the	DET
ma-58	17	50	frame	frame	NOUN
ma-58	17	51	operatorof	operatorof	NOUN
ma-58	17	52	their	their	PRON
ma-58	17	53	tensor	tensor	NOUN
ma-58	17	54	product	product	NOUN
ma-58	17	55	of	of	ADP
ma-58	17	56	∗-k	∗-k	NOUN
ma-58	17	57	-	-	PUNCT
ma-58	17	58	operator	operator	NOUN
ma-58	17	59	frames	frame	NOUN
ma-58	17	60	.	.	PUNCT
ma-58	18	1	received	receive	VERB
ma-58	18	2	:	:	PUNCT
ma-58	18	3	18	18	NUM
ma-58	18	4	nov	nov	PROPN
ma-58	18	5	2021	2021	NUM
ma-58	18	6	.	.	PUNCT
ma-58	19	1	key	key	ADJ
ma-58	19	2	words	word	NOUN
ma-58	19	3	and	and	CCONJ
ma-58	19	4	phrases	phrase	NOUN
ma-58	19	5	.	.	PUNCT
ma-58	20	1	frame	frame	NOUN
ma-58	20	2	;	;	PUNCT
ma-58	20	3	∗-k	∗-k	NOUN
ma-58	20	4	-	-	PUNCT
ma-58	20	5	operator	operator	NOUN
ma-58	20	6	frame	frame	NOUN
ma-58	20	7	;	;	PUNCT
ma-58	20	8	k	k	X
ma-58	20	9	-	-	PUNCT
ma-58	20	10	operator	operator	NOUN
ma-58	20	11	frame	frame	NOUN
ma-58	20	12	pro	pro	ADJ
ma-58	20	13	-	-	ADJ
ma-58	20	14	c∗-algebra	c∗-algebra	NOUN
ma-58	20	15	;	;	PUNCT
ma-58	20	16	hilbert	hilbert	ADJ
ma-58	20	17	pro	pro	PROPN
ma-58	20	18	-	-	NOUN
ma-58	20	19	c∗-modules	c∗-module	NOUN
ma-58	20	20	;	;	PUNCT
ma-58	20	21	tensorproduct	tensorproduct	NOUN
ma-58	20	22	.	.	PUNCT
ma-58	21	1	1	1	NUM
ma-58	21	2	https://adac.ee	https://adac.ee	PROPN
ma-58	21	3	https://doi.org/10.28924/ada/ma.2.4	https://doi.org/10.28924/ada/ma.2.4	PROPN
ma-58	21	4	eur	eur	NOUN
ma-58	21	5	.	.	PUNCT
ma-58	22	1	j.	j.	PROPN
ma-58	22	2	math	math	PROPN
ma-58	22	3	.	.	PUNCT
ma-58	23	1	anal	anal	PROPN
ma-58	23	2	.	.	PUNCT
ma-58	24	1	10.28924	10.28924	NUM
ma-58	24	2	/	/	SYM
ma-58	24	3	ada	ada	PROPN
ma-58	24	4	/	/	SYM
ma-58	24	5	ma.2.4	ma.2.4	PROPN
ma-58	24	6	22	22	NUM
ma-58	24	7	.	.	PUNCT
ma-58	25	1	preliminaries	preliminary	NOUN
ma-58	25	2	the	the	DET
ma-58	25	3	basic	basic	ADJ
ma-58	25	4	information	information	NOUN
ma-58	25	5	about	about	ADP
ma-58	25	6	pro	pro	ADJ
ma-58	25	7	-	-	ADJ
ma-58	25	8	c∗-algebras	c∗-algebra	NOUN
ma-58	25	9	can	can	AUX
ma-58	25	10	be	be	AUX
ma-58	25	11	found	find	VERB
ma-58	25	12	in	in	ADP
ma-58	25	13	the	the	DET
ma-58	25	14	works	work	NOUN
ma-58	26	1	[	[	X
ma-58	26	2	4–6,8	4–6,8	NOUN
ma-58	26	3	,	,	PUNCT
ma-58	26	4	12,14,15	12,14,15	NUM
ma-58	26	5	]	]	PUNCT
ma-58	26	6	.	.	PUNCT
ma-58	27	1	c∗-algebra	c∗-algebra	X
ma-58	27	2	whose	whose	DET
ma-58	27	3	topology	topology	NOUN
ma-58	27	4	is	be	AUX
ma-58	27	5	induced	induce	VERB
ma-58	27	6	by	by	ADP
ma-58	27	7	a	a	DET
ma-58	27	8	family	family	NOUN
ma-58	27	9	of	of	ADP
ma-58	27	10	continuous	continuous	ADJ
ma-58	27	11	c∗-seminorms	c∗-seminorm	NOUN
ma-58	27	12	instead	instead	ADV
ma-58	27	13	of	of	ADP
ma-58	27	14	a	a	DET
ma-58	27	15	c∗-norm	c∗-norm	NOUN
ma-58	27	16	is	be	AUX
ma-58	27	17	called	call	VERB
ma-58	27	18	pro	pro	ADJ
ma-58	27	19	-	-	ADJ
ma-58	27	20	c∗-algebra	c∗-algebra	NOUN
ma-58	27	21	.	.	PUNCT
ma-58	27	22	hilbert	hilbert	PROPN
ma-58	27	23	pro	pro	PROPN
ma-58	27	24	-	-	NOUN
ma-58	27	25	c∗-modules	c∗-module	NOUN
ma-58	27	26	are	be	AUX
ma-58	27	27	generalizations	generalization	NOUN
ma-58	27	28	of	of	ADP
ma-58	27	29	hilbert	hilbert	PROPN
ma-58	27	30	spacesby	spacesby	ADJ
ma-58	27	31	allowing	allow	VERB
ma-58	27	32	the	the	DET
ma-58	27	33	inner	inner	ADJ
ma-58	27	34	product	product	NOUN
ma-58	27	35	to	to	PART
ma-58	27	36	take	take	VERB
ma-58	27	37	values	value	NOUN
ma-58	27	38	in	in	ADP
ma-58	27	39	a	a	DET
ma-58	27	40	pro	pro	ADJ
ma-58	27	41	-	-	ADJ
ma-58	27	42	c∗-algebra	c∗-algebra	NOUN
ma-58	27	43	rather	rather	ADV
ma-58	27	44	than	than	ADP
ma-58	27	45	in	in	ADP
ma-58	27	46	the	the	DET
ma-58	27	47	field	field	NOUN
ma-58	27	48	of	of	ADP
ma-58	27	49	complexnumbers.pro	complexnumbers.pro	NOUN
ma-58	27	50	-	-	PUNCT
ma-58	27	51	c∗-algebra	c∗-algebra	NOUN
ma-58	27	52	is	be	AUX
ma-58	27	53	defined	define	VERB
ma-58	27	54	as	as	ADP
ma-58	27	55	a	a	DET
ma-58	27	56	complete	complete	ADJ
ma-58	27	57	hausdorff	hausdorff	NOUN
ma-58	27	58	complex	complex	ADJ
ma-58	27	59	topological	topological	ADJ
ma-58	27	60	∗-algebra	∗-algebra	NOUN
ma-58	27	61	a	a	DET
ma-58	27	62	whosetopology	whosetopology	NOUN
ma-58	27	63	is	be	AUX
ma-58	27	64	determined	determine	VERB
ma-58	27	65	by	by	ADP
ma-58	27	66	its	its	PRON
ma-58	27	67	continuous	continuous	ADJ
ma-58	27	68	c∗-seminorms	c∗-seminorm	NOUN
ma-58	27	69	in	in	ADP
ma-58	27	70	the	the	DET
ma-58	27	71	sens	sen	NOUN
ma-58	27	72	that	that	PRON
ma-58	27	73	a	a	DET
ma-58	27	74	net	net	ADJ
ma-58	27	75	{	{	PUNCT
ma-58	27	76	aα	aα	NOUN
ma-58	27	77	}	}	PUNCT
ma-58	27	78	converges	converge	NOUN
ma-58	27	79	to	to	ADP
ma-58	27	80	0if	0if	ADJ
ma-58	27	81	and	and	CCONJ
ma-58	27	82	only	only	ADV
ma-58	27	83	if	if	SCONJ
ma-58	27	84	p(aα	p(aα	NOUN
ma-58	27	85	)	)	PUNCT
ma-58	27	86	converges	converge	VERB
ma-58	27	87	to	to	ADP
ma-58	27	88	0	0	NUM
ma-58	27	89	for	for	ADP
ma-58	27	90	all	all	DET
ma-58	27	91	continuous	continuous	ADJ
ma-58	27	92	c∗-seminorm	c∗-seminorm	NOUN
ma-58	27	93	p	p	NOUN
ma-58	27	94	on	on	ADP
ma-58	27	95	a	a	DET
ma-58	27	96	[	[	X
ma-58	27	97	8,10,15	8,10,15	NUM
ma-58	27	98	]	]	PUNCT
ma-58	27	99	,	,	PUNCT
ma-58	27	100	and	and	CCONJ
ma-58	27	101	we	we	PRON
ma-58	27	102	have:1	have:1	VERB
ma-58	27	103	)	)	PUNCT
ma-58	27	104	p(ab	p(ab	PROPN
ma-58	27	105	)	)	PUNCT
ma-58	27	106	≤	≤	PROPN
ma-58	27	107	p(a)p(b)2	p(a)p(b)2	NOUN
ma-58	27	108	)	)	PUNCT
ma-58	27	109	p(a∗a	p(a∗a	NOUN
ma-58	27	110	)	)	PUNCT
ma-58	28	1	=	=	PUNCT
ma-58	28	2	p(a)2for	p(a)2for	VERB
ma-58	28	3	all	all	DET
ma-58	28	4	a	a	DET
ma-58	28	5	,	,	PUNCT
ma-58	28	6	b	b	X
ma-58	28	7	∈	∈	PROPN
ma-58	28	8	aif	aif	PROPN
ma-58	28	9	the	the	DET
ma-58	28	10	topology	topology	NOUN
ma-58	28	11	of	of	ADP
ma-58	28	12	pro	pro	ADJ
ma-58	28	13	-	-	ADJ
ma-58	28	14	c∗-algebra	c∗-algebra	NOUN
ma-58	28	15	is	be	AUX
ma-58	28	16	determined	determine	VERB
ma-58	28	17	by	by	ADP
ma-58	28	18	only	only	ADV
ma-58	28	19	countably	countably	ADV
ma-58	28	20	many	many	ADJ
ma-58	28	21	c∗-seminorms	c∗-seminorm	NOUN
ma-58	28	22	,	,	PUNCT
ma-58	28	23	then	then	ADV
ma-58	28	24	it	it	PRON
ma-58	28	25	iscalled	iscalle	VERB
ma-58	28	26	a	a	DET
ma-58	28	27	σ-c∗-algebra.we	σ-c∗-algebra.we	NOUN
ma-58	28	28	denote	denote	NOUN
ma-58	28	29	by	by	ADP
ma-58	28	30	sp(a	sp(a	NOUN
ma-58	28	31	)	)	PUNCT
ma-58	28	32	the	the	DET
ma-58	28	33	spectrum	spectrum	NOUN
ma-58	28	34	of	of	ADP
ma-58	28	35	a	a	DET
ma-58	28	36	such	such	ADJ
ma-58	28	37	that	that	PRON
ma-58	28	38	:	:	PUNCT
ma-58	28	39	sp(a	sp(a	X
ma-58	28	40	)	)	PUNCT
ma-58	28	41	=	=	SYM
ma-58	29	1	{	{	PUNCT
ma-58	29	2	λ	λ	X
ma-58	29	3	∈	∈	NOUN
ma-58	29	4	c	c	NOUN
ma-58	29	5	:	:	PUNCT
ma-58	29	6	λ1a	λ1a	NOUN
ma-58	29	7	−	−	PROPN
ma-58	29	8	a	a	PRON
ma-58	29	9	is	be	AUX
ma-58	29	10	not	not	PART
ma-58	29	11	invertible	invertible	ADJ
ma-58	29	12	}	}	PUNCT
ma-58	29	13	forall	forall	VERB
ma-58	29	14	a	a	DET
ma-58	29	15	∈	∈	NOUN
ma-58	29	16	a.	a.	NOUN
ma-58	29	17	where	where	SCONJ
ma-58	29	18	a	a	PRON
ma-58	29	19	is	be	AUX
ma-58	29	20	unital	unital	ADJ
ma-58	29	21	pro	pro	ADJ
ma-58	29	22	-	-	ADJ
ma-58	29	23	c∗-algebra	c∗-algebra	NOUN
ma-58	29	24	with	with	ADP
ma-58	29	25	unite	unite	VERB
ma-58	29	26	1a.the	1a.the	DET
ma-58	29	27	set	set	NOUN
ma-58	29	28	of	of	ADP
ma-58	29	29	all	all	DET
ma-58	29	30	continuous	continuous	ADJ
ma-58	29	31	c∗-seminorms	c∗-seminorm	NOUN
ma-58	29	32	on	on	ADP
ma-58	29	33	a	a	PRON
ma-58	29	34	is	be	AUX
ma-58	29	35	denoted	denote	VERB
ma-58	29	36	by	by	ADP
ma-58	29	37	s(a	s(a	PROPN
ma-58	29	38	)	)	PUNCT
ma-58	29	39	.	.	PUNCT
ma-58	30	1	if	if	SCONJ
ma-58	30	2	a+	a+	PRON
ma-58	30	3	denotes	denote	VERB
ma-58	30	4	the	the	DET
ma-58	30	5	set	set	NOUN
ma-58	30	6	of	of	ADP
ma-58	30	7	allpositive	allpositive	ADJ
ma-58	30	8	elements	element	NOUN
ma-58	30	9	of	of	ADP
ma-58	30	10	a	a	PRON
ma-58	30	11	,	,	PUNCT
ma-58	30	12	then	then	ADV
ma-58	30	13	a+	a+	PUNCT
ma-58	30	14	is	be	AUX
ma-58	30	15	a	a	DET
ma-58	30	16	closed	closed	ADJ
ma-58	30	17	convex	convex	NOUN
ma-58	30	18	c∗-seminorms	c∗-seminorm	NOUN
ma-58	30	19	on	on	ADP
ma-58	30	20	a.	a.	NOUN
ma-58	30	21	example	example	NOUN
ma-58	30	22	2.1	2.1	NUM
ma-58	30	23	.	.	PUNCT
ma-58	31	1	every	every	DET
ma-58	31	2	c∗-algebra	c∗-algebra	PROPN
ma-58	31	3	is	be	AUX
ma-58	31	4	a	a	DET
ma-58	31	5	pro	pro	ADJ
ma-58	31	6	-	-	ADJ
ma-58	31	7	c∗-algebra	c∗-algebra	ADJ
ma-58	31	8	.	.	PUNCT
ma-58	31	9	proposition	proposition	NOUN
ma-58	31	10	2.2	2.2	NUM
ma-58	31	11	.	.	PUNCT
ma-58	32	1	[	[	X
ma-58	32	2	8	8	NUM
ma-58	32	3	]	]	PUNCT
ma-58	32	4	let	let	VERB
ma-58	32	5	a	a	PRON
ma-58	32	6	be	be	AUX
ma-58	32	7	a	a	DET
ma-58	32	8	unital	unital	ADJ
ma-58	32	9	pro	pro	ADJ
ma-58	32	10	-	-	NOUN
ma-58	32	11	c∗-algebra	c∗-algebra	NOUN
ma-58	32	12	with	with	ADP
ma-58	32	13	an	an	DET
ma-58	32	14	identity	identity	NOUN
ma-58	32	15	1a	1a	NOUN
ma-58	32	16	.	.	PUNCT
ma-58	33	1	then	then	ADV
ma-58	33	2	for	for	ADP
ma-58	33	3	any	any	DET
ma-58	33	4	p	p	PROPN
ma-58	33	5	∈	∈	PROPN
ma-58	33	6	s(a	s(a	PROPN
ma-58	33	7	)	)	PUNCT
ma-58	33	8	,	,	PUNCT
ma-58	33	9	we	we	PRON
ma-58	33	10	have:(1	have:(1	NOUN
ma-58	33	11	)	)	PUNCT
ma-58	33	12	p(a	p(a	NOUN
ma-58	33	13	)	)	PUNCT
ma-58	33	14	=	=	SYM
ma-58	33	15	p(a∗	p(a∗	NOUN
ma-58	33	16	)	)	PUNCT
ma-58	33	17	for	for	ADP
ma-58	33	18	all	all	DET
ma-58	33	19	a	a	DET
ma-58	33	20	∈	∈	PROPN
ma-58	33	21	a(2	a(2	PROPN
ma-58	33	22	)	)	PUNCT
ma-58	33	23	p	p	NOUN
ma-58	33	24	(	(	PUNCT
ma-58	33	25	1a	1a	X
ma-58	33	26	)	)	PUNCT
ma-58	33	27	=	=	SYM
ma-58	33	28	1(3	1(3	X
ma-58	33	29	)	)	PUNCT
ma-58	33	30	if	if	SCONJ
ma-58	33	31	a	a	PRON
ma-58	33	32	,	,	PUNCT
ma-58	33	33	b	b	NOUN
ma-58	33	34	∈	∈	PROPN
ma-58	33	35	a+	a+	PUNCT
ma-58	33	36	and	and	CCONJ
ma-58	33	37	a	a	DET
ma-58	33	38	≤	≤	NUM
ma-58	33	39	b	b	NOUN
ma-58	33	40	,	,	PUNCT
ma-58	33	41	then	then	ADV
ma-58	33	42	p(a	p(a	PROPN
ma-58	33	43	)	)	PUNCT
ma-58	33	44	≤	≤	NOUN
ma-58	33	45	p(b)(4	p(b)(4	PROPN
ma-58	33	46	)	)	PUNCT
ma-58	33	47	if	if	SCONJ
ma-58	33	48	1a	1a	NOUN
ma-58	33	49	≤	≤	NUM
ma-58	33	50	b	b	PROPN
ma-58	33	51	,	,	PUNCT
ma-58	33	52	then	then	ADV
ma-58	33	53	b	b	NOUN
ma-58	33	54	is	be	AUX
ma-58	33	55	invertible	invertible	ADJ
ma-58	33	56	and	and	CCONJ
ma-58	33	57	b−1	b−1	PROPN
ma-58	33	58	≤	≤	NUM
ma-58	33	59	1a(5	1a(5	NUM
ma-58	33	60	)	)	PUNCT
ma-58	33	61	if	if	SCONJ
ma-58	33	62	a	a	PRON
ma-58	33	63	,	,	PUNCT
ma-58	33	64	b	b	X
ma-58	33	65	∈	∈	PROPN
ma-58	33	66	a+	a+	PUNCT
ma-58	33	67	are	be	AUX
ma-58	33	68	invertible	invertible	ADJ
ma-58	33	69	and	and	CCONJ
ma-58	33	70	0	0	NUM
ma-58	33	71	≤	≤	NOUN
ma-58	33	72	a	a	DET
ma-58	33	73	≤	≤	NUM
ma-58	33	74	b	b	NOUN
ma-58	33	75	,	,	PUNCT
ma-58	33	76	then	then	ADV
ma-58	33	77	0	0	NUM
ma-58	33	78	≤	≤	NUM
ma-58	33	79	b−1	b−1	PROPN
ma-58	33	80	≤	≤	NUM
ma-58	33	81	a−1(6	a−1(6	NOUN
ma-58	33	82	)	)	PUNCT
ma-58	33	83	if	if	SCONJ
ma-58	33	84	a	a	DET
ma-58	33	85	,	,	PUNCT
ma-58	33	86	b	b	NOUN
ma-58	33	87	,	,	PUNCT
ma-58	33	88	c	c	PROPN
ma-58	33	89	∈	∈	PROPN
ma-58	33	90	a	a	PRON
ma-58	33	91	and	and	CCONJ
ma-58	33	92	a	a	DET
ma-58	33	93	≤	≤	PROPN
ma-58	33	94	b	b	NOUN
ma-58	33	95	then	then	ADV
ma-58	33	96	c∗ac	c∗ac	NOUN
ma-58	33	97	≤	≤	PROPN
ma-58	33	98	c∗bc(7	c∗bc(7	PROPN
ma-58	33	99	)	)	PUNCT
ma-58	33	100	if	if	SCONJ
ma-58	33	101	a	a	PRON
ma-58	33	102	,	,	PUNCT
ma-58	33	103	b	b	NOUN
ma-58	33	104	∈	∈	PROPN
ma-58	33	105	a+	a+	PUNCT
ma-58	33	106	and	and	CCONJ
ma-58	33	107	a2	a2	PROPN
ma-58	33	108	≤	≤	NUM
ma-58	33	109	b2	b2	NOUN
ma-58	33	110	,	,	PUNCT
ma-58	33	111	then	then	ADV
ma-58	33	112	0	0	NUM
ma-58	33	113	≤	≤	NOUN
ma-58	33	114	a	a	DET
ma-58	33	115	≤	≤	NUM
ma-58	33	116	b	b	NOUN
ma-58	33	117	definition	definition	NOUN
ma-58	33	118	2.3	2.3	NUM
ma-58	33	119	.	.	PUNCT
ma-58	34	1	[	[	X
ma-58	34	2	15	15	NUM
ma-58	34	3	]	]	X
ma-58	34	4	a	a	DET
ma-58	34	5	pre	pre	ADJ
ma-58	34	6	-	-	ADJ
ma-58	34	7	hilbert	hilbert	ADJ
ma-58	34	8	module	module	NOUN
ma-58	34	9	over	over	ADP
ma-58	34	10	pro	pro	ADJ
ma-58	34	11	-	-	ADJ
ma-58	34	12	c∗-algebra	c∗-algebra	ADJ
ma-58	34	13	a	a	PRON
ma-58	34	14	,	,	PUNCT
ma-58	34	15	is	be	AUX
ma-58	34	16	a	a	DET
ma-58	34	17	complex	complex	ADJ
ma-58	34	18	vector	vector	NOUN
ma-58	34	19	space	space	NOUN
ma-58	34	20	ewhich	ewhich	NOUN
ma-58	34	21	is	be	AUX
ma-58	34	22	also	also	ADV
ma-58	34	23	a	a	DET
ma-58	34	24	left	left	ADJ
ma-58	34	25	a	a	DET
ma-58	34	26	-	-	PUNCT
ma-58	34	27	module	module	NOUN
ma-58	34	28	compatible	compatible	ADJ
ma-58	34	29	with	with	ADP
ma-58	34	30	the	the	DET
ma-58	34	31	complex	complex	ADJ
ma-58	34	32	algebra	algebra	NOUN
ma-58	34	33	structure	structure	NOUN
ma-58	34	34	,	,	PUNCT
ma-58	34	35	equipped	equip	VERB
ma-58	34	36	with	with	ADP
ma-58	34	37	an	an	DET
ma-58	34	38	a	a	ADV
ma-58	34	39	-	-	PUNCT
ma-58	34	40	valued	value	VERB
ma-58	34	41	inner	inner	ADJ
ma-58	34	42	product	product	NOUN
ma-58	34	43	〈	〈	PROPN
ma-58	34	44	.	.	PROPN
ma-58	34	45	,	,	PUNCT
ma-58	34	46	.	.	PUNCT
ma-58	35	1	〉	〉	PROPN
ma-58	35	2	e×e	e×e	PROPN
ma-58	35	3	→	→	PUNCT
ma-58	35	4	a	a	PRON
ma-58	35	5	which	which	PRON
ma-58	35	6	is	be	AUX
ma-58	35	7	c	c	NOUN
ma-58	35	8	-	-	PUNCT
ma-58	35	9	and	and	CCONJ
ma-58	35	10	a	a	DET
ma-58	35	11	-	-	PUNCT
ma-58	35	12	linear	linear	NOUN
ma-58	35	13	in	in	ADP
ma-58	35	14	its	its	PRON
ma-58	35	15	first	first	ADJ
ma-58	35	16	variable	variable	NOUN
ma-58	35	17	and	and	CCONJ
ma-58	35	18	satisfiesthe	satisfiesthe	ADJ
ma-58	35	19	following	follow	VERB
ma-58	35	20	conditions:1	conditions:1	NOUN
ma-58	35	21	)	)	PUNCT
ma-58	35	22	〈	〈	PROPN
ma-58	35	23	ξ	ξ	X
ma-58	35	24	,	,	PUNCT
ma-58	35	25	η〉∗	η〉∗	NOUN
ma-58	35	26	=	=	SYM
ma-58	35	27	〈	〈	PROPN
ma-58	35	28	η	η	PROPN
ma-58	35	29	,	,	PUNCT
ma-58	35	30	ξ	ξ	PROPN
ma-58	35	31	〉	〉	NOUN
ma-58	35	32	for	for	ADP
ma-58	35	33	every	every	DET
ma-58	35	34	ξ	ξ	PROPN
ma-58	35	35	,	,	PUNCT
ma-58	35	36	η	η	PROPN
ma-58	35	37	∈	∈	PROPN
ma-58	35	38	e2	e2	PROPN
ma-58	35	39	)	)	PUNCT
ma-58	35	40	〈	〈	PROPN
ma-58	35	41	ξ	ξ	PROPN
ma-58	35	42	,	,	PUNCT
ma-58	35	43	ξ	ξ	PROPN
ma-58	35	44	〉	〉	NUM
ma-58	35	45	≥	≥	NOUN
ma-58	35	46	0	0	NUM
ma-58	35	47	for	for	ADP
ma-58	35	48	every	every	DET
ma-58	35	49	ξ	ξ	PROPN
ma-58	35	50	∈	∈	PROPN
ma-58	35	51	e	e	ADP
ma-58	35	52	https://doi.org/10.28924/ada/ma.2.4	https://doi.org/10.28924/ada/ma.2.4	PROPN
ma-58	35	53	eur	eur	NOUN
ma-58	35	54	.	.	PUNCT
ma-58	36	1	j.	j.	PROPN
ma-58	36	2	math	math	PROPN
ma-58	36	3	.	.	PUNCT
ma-58	37	1	anal	anal	PROPN
ma-58	37	2	.	.	PUNCT
ma-58	38	1	10.28924	10.28924	NUM
ma-58	38	2	/	/	SYM
ma-58	38	3	ada	ada	PROPN
ma-58	38	4	/	/	SYM
ma-58	38	5	ma.2.4	ma.2.4	PROPN
ma-58	38	6	33	33	NUM
ma-58	38	7	)	)	PUNCT
ma-58	38	8	〈	〈	PROPN
ma-58	38	9	ξ	ξ	X
ma-58	38	10	,	,	PUNCT
ma-58	38	11	ξ	ξ	NOUN
ma-58	38	12	〉	〉	NOUN
ma-58	38	13	=	=	SYM
ma-58	38	14	0	0	PUNCT
ma-58	39	1	if	if	SCONJ
ma-58	39	2	and	and	CCONJ
ma-58	39	3	only	only	ADV
ma-58	39	4	if	if	SCONJ
ma-58	39	5	ξ	ξ	X
ma-58	39	6	=	=	PUNCT
ma-58	39	7	0for	0for	ADP
ma-58	39	8	every	every	DET
ma-58	39	9	ξ	ξ	PROPN
ma-58	39	10	,	,	PUNCT
ma-58	39	11	η	η	PROPN
ma-58	39	12	∈	∈	PROPN
ma-58	39	13	e.	e.	PROPN
ma-58	39	14	we	we	PRON
ma-58	39	15	say	say	VERB
ma-58	39	16	e	e	NOUN
ma-58	39	17	is	be	AUX
ma-58	39	18	a	a	DET
ma-58	39	19	hilbert	hilbert	NOUN
ma-58	39	20	a	a	DET
ma-58	39	21	-	-	PUNCT
ma-58	39	22	module	module	NOUN
ma-58	39	23	(	(	PUNCT
ma-58	39	24	or	or	CCONJ
ma-58	39	25	hilbert	hilbert	NOUN
ma-58	39	26	pro	pro	NOUN
ma-58	39	27	-	-	NOUN
ma-58	39	28	c∗-module	c∗-module	NOUN
ma-58	39	29	over	over	ADP
ma-58	39	30	a	a	PRON
ma-58	39	31	)	)	PUNCT
ma-58	39	32	.	.	PUNCT
ma-58	40	1	if	if	SCONJ
ma-58	40	2	e	e	PRON
ma-58	40	3	iscomplete	iscomplete	VERB
ma-58	40	4	with	with	ADP
ma-58	40	5	respect	respect	NOUN
ma-58	40	6	to	to	ADP
ma-58	40	7	the	the	DET
ma-58	40	8	topology	topology	NOUN
ma-58	40	9	determined	determine	VERB
ma-58	40	10	by	by	ADP
ma-58	40	11	the	the	DET
ma-58	40	12	family	family	NOUN
ma-58	40	13	of	of	ADP
ma-58	40	14	seminorms	seminorm	NOUN
ma-58	40	15	p̄e(ξ	p̄e(ξ	NOUN
ma-58	40	16	)	)	PUNCT
ma-58	40	17	=	=	SYM
ma-58	40	18	√	√	NUM
ma-58	40	19	p(〈ξ	p(〈ξ	NOUN
ma-58	40	20	,	,	PUNCT
ma-58	40	21	ξ	ξ	PROPN
ma-58	40	22	〉	〉	NOUN
ma-58	40	23	)	)	PUNCT
ma-58	40	24	ξ	ξ	PROPN
ma-58	40	25	∈	∈	PROPN
ma-58	40	26	e	e	NOUN
ma-58	40	27	,	,	PUNCT
ma-58	40	28	p	p	PROPN
ma-58	40	29	∈	∈	PROPN
ma-58	40	30	s(a	s(a	PROPN
ma-58	40	31	)	)	PUNCT
ma-58	40	32	let	let	VERB
ma-58	40	33	a	a	PRON
ma-58	40	34	be	be	AUX
ma-58	40	35	a	a	DET
ma-58	40	36	pro	pro	ADJ
ma-58	40	37	-	-	ADJ
ma-58	40	38	c∗-algebra	c∗-algebra	ADJ
ma-58	40	39	and	and	CCONJ
ma-58	40	40	let	let	VERB
ma-58	40	41	x	x	PRON
ma-58	40	42	and	and	CCONJ
ma-58	40	43	y	y	PROPN
ma-58	40	44	be	be	AUX
ma-58	40	45	hilbert	hilbert	PROPN
ma-58	40	46	a	a	DET
ma-58	40	47	-	-	PUNCT
ma-58	40	48	modules	module	NOUN
ma-58	40	49	and	and	CCONJ
ma-58	40	50	assume	assume	VERB
ma-58	40	51	that	that	SCONJ
ma-58	40	52	i	i	PRON
ma-58	40	53	and	and	CCONJ
ma-58	40	54	j	j	PROPN
ma-58	40	55	becountable	becountable	ADJ
ma-58	40	56	index	index	NOUN
ma-58	40	57	sets	set	NOUN
ma-58	40	58	.	.	PUNCT
ma-58	41	1	a	a	DET
ma-58	41	2	bounded	bounded	ADJ
ma-58	41	3	a	a	DET
ma-58	41	4	-	-	PUNCT
ma-58	41	5	module	module	NOUN
ma-58	41	6	map	map	NOUN
ma-58	41	7	from	from	ADP
ma-58	41	8	x	x	PUNCT
ma-58	41	9	to	to	ADP
ma-58	41	10	y	y	PROPN
ma-58	41	11	is	be	AUX
ma-58	41	12	called	call	VERB
ma-58	41	13	an	an	DET
ma-58	41	14	operators	operator	NOUN
ma-58	41	15	from	from	ADP
ma-58	41	16	x	x	PUNCT
ma-58	41	17	to	to	ADP
ma-58	41	18	y	y	PROPN
ma-58	41	19	.we	.we	PUNCT
ma-58	41	20	denote	denote	VERB
ma-58	41	21	the	the	DET
ma-58	41	22	set	set	NOUN
ma-58	41	23	of	of	ADP
ma-58	41	24	all	all	DET
ma-58	41	25	operator	operator	NOUN
ma-58	41	26	from	from	ADP
ma-58	41	27	x	x	PUNCT
ma-58	41	28	to	to	ADP
ma-58	41	29	y	y	PROPN
ma-58	41	30	by	by	ADP
ma-58	41	31	homa(x	homa(x	PROPN
ma-58	41	32	,	,	PUNCT
ma-58	41	33	y	y	PROPN
ma-58	41	34	)	)	PUNCT
ma-58	41	35	.	.	PUNCT
ma-58	42	1	definition	definition	NOUN
ma-58	42	2	2.4	2.4	NUM
ma-58	42	3	.	.	PUNCT
ma-58	43	1	[	[	X
ma-58	43	2	1	1	X
ma-58	43	3	]	]	PUNCT
ma-58	43	4	an	an	DET
ma-58	43	5	a	a	DET
ma-58	43	6	-	-	PUNCT
ma-58	43	7	module	module	NOUN
ma-58	43	8	map	map	NOUN
ma-58	43	9	t	t	NOUN
ma-58	43	10	:	:	PUNCT
ma-58	43	11	x	x	PUNCT
ma-58	43	12	−→	−→	NOUN
ma-58	43	13	y	y	PROPN
ma-58	43	14	is	be	AUX
ma-58	43	15	adjointable	adjointable	ADJ
ma-58	43	16	if	if	SCONJ
ma-58	43	17	there	there	PRON
ma-58	43	18	is	be	VERB
ma-58	43	19	a	a	DET
ma-58	43	20	map	map	NOUN
ma-58	43	21	t	t	NOUN
ma-58	43	22	∗	∗	NOUN
ma-58	43	23	:	:	PUNCT
ma-58	44	1	y	y	PROPN
ma-58	44	2	−→	−→	NOUN
ma-58	44	3	xsuch	xsuch	PROPN
ma-58	44	4	that	that	PRON
ma-58	44	5	〈	〈	PROPN
ma-58	44	6	tξ	tξ	AUX
ma-58	44	7	,	,	PUNCT
ma-58	44	8	η	η	PROPN
ma-58	44	9	〉	〉	NOUN
ma-58	44	10	=	=	SYM
ma-58	44	11	〈	〈	PROPN
ma-58	44	12	ξ	ξ	PROPN
ma-58	44	13	,	,	PUNCT
ma-58	44	14	t	t	PROPN
ma-58	44	15	∗η	∗η	PROPN
ma-58	44	16	〉	〉	PROPN
ma-58	44	17	for	for	ADP
ma-58	44	18	all	all	DET
ma-58	44	19	ξ	ξ	X
ma-58	44	20	∈	∈	PROPN
ma-58	44	21	x	x	X
ma-58	44	22	,	,	PUNCT
ma-58	44	23	η	η	PROPN
ma-58	44	24	∈	∈	PROPN
ma-58	44	25	y	y	PROPN
ma-58	44	26	,	,	PUNCT
ma-58	44	27	and	and	CCONJ
ma-58	44	28	is	be	AUX
ma-58	44	29	called	call	VERB
ma-58	44	30	bounded	bounded	ADJ
ma-58	44	31	if	if	SCONJ
ma-58	44	32	for	for	ADP
ma-58	44	33	all	all	PRON
ma-58	44	34	p	p	PRON
ma-58	44	35	∈	∈	PROPN
ma-58	44	36	s(a	s(a	PROPN
ma-58	44	37	)	)	PUNCT
ma-58	44	38	,	,	PUNCT
ma-58	44	39	thereis	thereis	PROPN
ma-58	44	40	mp	mp	PROPN
ma-58	44	41	>	>	X
ma-58	44	42	0	0	NUM
ma-58	44	43	such	such	ADJ
ma-58	44	44	that	that	DET
ma-58	44	45	p̄y(tξ	p̄y(tξ	NOUN
ma-58	44	46	)	)	PUNCT
ma-58	44	47	≤	≤	NUM
ma-58	44	48	mpp̄x	mpp̄x	NOUN
ma-58	44	49	(	(	PUNCT
ma-58	44	50	ξ	ξ	NOUN
ma-58	44	51	)	)	PUNCT
ma-58	44	52	for	for	ADP
ma-58	44	53	all	all	DET
ma-58	44	54	ξ	ξ	X
ma-58	44	55	∈	∈	PROPN
ma-58	44	56	x	x	SYM
ma-58	44	57	.we	.we	PUNCT
ma-58	44	58	denote	denote	VERB
ma-58	44	59	by	by	ADP
ma-58	44	60	hom∗a(x	hom∗a(x	NOUN
ma-58	44	61	,	,	PUNCT
ma-58	44	62	y	y	PROPN
ma-58	44	63	)	)	PUNCT
ma-58	44	64	,	,	PUNCT
ma-58	44	65	the	the	DET
ma-58	44	66	set	set	NOUN
ma-58	44	67	of	of	ADP
ma-58	44	68	all	all	DET
ma-58	44	69	adjointable	adjointable	ADJ
ma-58	44	70	operator	operator	NOUN
ma-58	44	71	from	from	ADP
ma-58	44	72	x	x	PUNCT
ma-58	44	73	to	to	ADP
ma-58	44	74	y	y	PROPN
ma-58	44	75	and	and	CCONJ
ma-58	44	76	hom∗a(x	hom∗a(x	NOUN
ma-58	44	77	)	)	PUNCT
ma-58	45	1	=	=	SYM
ma-58	45	2	hom∗a(x	hom∗a(x	NOUN
ma-58	45	3	,	,	PUNCT
ma-58	45	4	x	x	X
ma-58	45	5	)	)	PUNCT
ma-58	45	6	definition	definition	NOUN
ma-58	45	7	2.5	2.5	NUM
ma-58	45	8	.	.	PUNCT
ma-58	46	1	[	[	X
ma-58	46	2	1	1	X
ma-58	46	3	]	]	PUNCT
ma-58	46	4	let	let	VERB
ma-58	46	5	a	a	PRON
ma-58	46	6	be	be	AUX
ma-58	46	7	a	a	DET
ma-58	46	8	pro	pro	ADJ
ma-58	46	9	-	-	ADJ
ma-58	46	10	c∗-algebra	c∗-algebra	ADJ
ma-58	46	11	and	and	CCONJ
ma-58	46	12	x	x	SYM
ma-58	46	13	,	,	PUNCT
ma-58	46	14	y	y	PROPN
ma-58	46	15	be	be	VERB
ma-58	46	16	two	two	NUM
ma-58	46	17	hilbert	hilbert	NOUN
ma-58	46	18	a	a	NOUN
ma-58	46	19	-	-	PUNCT
ma-58	46	20	modules	module	NOUN
ma-58	46	21	.	.	PUNCT
ma-58	47	1	the	the	DET
ma-58	47	2	operator	operator	NOUN
ma-58	47	3	t	t	NOUN
ma-58	47	4	:	:	PUNCT
ma-58	47	5	x	x	X
ma-58	47	6	→	→	SYM
ma-58	47	7	y	y	PROPN
ma-58	47	8	is	be	AUX
ma-58	47	9	called	call	VERB
ma-58	47	10	uniformly	uniformly	ADV
ma-58	47	11	bounded	bound	VERB
ma-58	47	12	below	below	ADV
ma-58	47	13	,	,	PUNCT
ma-58	47	14	if	if	SCONJ
ma-58	47	15	there	there	PRON
ma-58	47	16	exists	exist	VERB
ma-58	47	17	c	c	NOUN
ma-58	47	18	>	>	X
ma-58	47	19	0	0	NUM
ma-58	47	20	such	such	ADJ
ma-58	47	21	that	that	PRON
ma-58	47	22	for	for	ADP
ma-58	47	23	each	each	DET
ma-58	47	24	p	p	PROPN
ma-58	47	25	∈	∈	PROPN
ma-58	47	26	s(a	s(a	PROPN
ma-58	47	27	)	)	PUNCT
ma-58	47	28	,	,	PUNCT
ma-58	47	29	p̄y(tξ	p̄y(tξ	NOUN
ma-58	47	30	)	)	PUNCT
ma-58	47	31	6	6	NUM
ma-58	47	32	cp̄x	cp̄x	NOUN
ma-58	47	33	(	(	PUNCT
ma-58	47	34	ξ	ξ	NOUN
ma-58	47	35	)	)	PUNCT
ma-58	47	36	,	,	PUNCT
ma-58	47	37	for	for	ADP
ma-58	47	38	all	all	DET
ma-58	47	39	ξ	ξ	X
ma-58	47	40	∈	∈	NOUN
ma-58	47	41	x	x	X
ma-58	47	42	and	and	CCONJ
ma-58	47	43	is	be	AUX
ma-58	47	44	called	call	VERB
ma-58	47	45	uniformly	uniformly	ADV
ma-58	47	46	bounded	bound	VERB
ma-58	47	47	above	above	ADV
ma-58	47	48	if	if	SCONJ
ma-58	47	49	there	there	PRON
ma-58	47	50	exists	exist	VERB
ma-58	47	51	c′	c′	ADV
ma-58	47	52	>	>	X
ma-58	47	53	0	0	NUM
ma-58	47	54	such	such	ADJ
ma-58	47	55	that	that	PRON
ma-58	47	56	for	for	ADP
ma-58	47	57	each	each	DET
ma-58	47	58	p	p	PROPN
ma-58	47	59	∈	∈	PROPN
ma-58	47	60	s(a	s(a	PROPN
ma-58	47	61	)	)	PUNCT
ma-58	47	62	,	,	PUNCT
ma-58	47	63	p̄y(tξ	p̄y(tξ	NOUN
ma-58	47	64	)	)	PUNCT
ma-58	47	65	>	>	X
ma-58	47	66	c′p̄x	c′p̄x	PROPN
ma-58	47	67	(	(	PUNCT
ma-58	47	68	ξ	ξ	NOUN
ma-58	47	69	)	)	PUNCT
ma-58	47	70	,	,	PUNCT
ma-58	47	71	for	for	ADP
ma-58	47	72	all	all	DET
ma-58	47	73	ξ	ξ	X
ma-58	47	74	∈	∈	NOUN
ma-58	47	75	x	x	PUNCT
ma-58	47	76	‖t‖∞	‖t‖∞	PROPN
ma-58	47	77	=	=	SYM
ma-58	47	78	inf{m	inf{m	PROPN
ma-58	47	79	:	:	PUNCT
ma-58	47	80	m	m	VERB
ma-58	47	81	is	be	AUX
ma-58	47	82	an	an	DET
ma-58	47	83	upper	upper	ADJ
ma-58	47	84	bound	bind	VERB
ma-58	47	85	for	for	ADP
ma-58	47	86	t	t	PROPN
ma-58	47	87	}	}	PUNCT
ma-58	47	88	p̂y(t	p̂y(t	PROPN
ma-58	47	89	)	)	PUNCT
ma-58	48	1	=	=	SYM
ma-58	48	2	sup	sup	X
ma-58	48	3	{	{	PUNCT
ma-58	48	4	p̄y(t	p̄y(t	PROPN
ma-58	48	5	(	(	PUNCT
ma-58	48	6	x	x	NOUN
ma-58	48	7	)	)	PUNCT
ma-58	48	8	)	)	PUNCT
ma-58	48	9	:	:	PUNCT
ma-58	49	1	ξ	ξ	X
ma-58	49	2	∈	∈	PROPN
ma-58	49	3	x	x	X
ma-58	49	4	,	,	PUNCT
ma-58	49	5	p̄x	p̄x	NOUN
ma-58	49	6	(	(	PUNCT
ma-58	49	7	ξ	ξ	NOUN
ma-58	49	8	)	)	PUNCT
ma-58	49	9	6	6	NUM
ma-58	49	10	1}it	1}it	NOUN
ma-58	49	11	’s	’s	NOUN
ma-58	49	12	clear	clear	ADJ
ma-58	49	13	to	to	PART
ma-58	49	14	see	see	VERB
ma-58	49	15	that	that	PRON
ma-58	49	16	,	,	PUNCT
ma-58	49	17	p̂(t	p̂(t	PUNCT
ma-58	49	18	)	)	PUNCT
ma-58	49	19	6	6	NUM
ma-58	49	20	‖t‖∞	‖t‖∞	NOUN
ma-58	49	21	for	for	ADP
ma-58	49	22	all	all	DET
ma-58	49	23	p	p	PRON
ma-58	49	24	∈	∈	PROPN
ma-58	49	25	s(a	s(a	PROPN
ma-58	49	26	)	)	PUNCT
ma-58	49	27	.	.	PUNCT
ma-58	50	1	proposition	proposition	NOUN
ma-58	50	2	2.6	2.6	NUM
ma-58	50	3	.	.	PUNCT
ma-58	51	1	[	[	X
ma-58	51	2	2	2	NUM
ma-58	51	3	]	]	PUNCT
ma-58	51	4	.	.	PUNCT
ma-58	52	1	let	let	VERB
ma-58	52	2	x	x	PRON
ma-58	52	3	be	be	AUX
ma-58	52	4	a	a	DET
ma-58	52	5	hilbert	hilbert	NOUN
ma-58	52	6	module	module	NOUN
ma-58	52	7	over	over	ADP
ma-58	52	8	pro	pro	ADJ
ma-58	52	9	-	-	ADJ
ma-58	52	10	c∗-algebra	c∗-algebra	ADJ
ma-58	52	11	a	a	NOUN
ma-58	52	12	and	and	CCONJ
ma-58	52	13	t	t	PROPN
ma-58	52	14	be	be	AUX
ma-58	52	15	an	an	DET
ma-58	52	16	invertible	invertible	ADJ
ma-58	52	17	element	element	NOUN
ma-58	52	18	in	in	ADP
ma-58	52	19	hom∗a(x	hom∗a(x	NOUN
ma-58	52	20	)	)	PUNCT
ma-58	52	21	such	such	ADJ
ma-58	52	22	that	that	SCONJ
ma-58	52	23	both	both	PRON
ma-58	52	24	are	be	AUX
ma-58	52	25	uniformly	uniformly	ADV
ma-58	52	26	bounded	bound	VERB
ma-58	52	27	.	.	PUNCT
ma-58	53	1	then	then	ADV
ma-58	53	2	for	for	ADP
ma-58	53	3	each	each	DET
ma-58	53	4	ξ	ξ	X
ma-58	53	5	∈	∈	PROPN
ma-58	53	6	x	x	X
ma-58	53	7	,	,	PUNCT
ma-58	53	8	∥∥t−1∥∥−2∞	∥∥t−1∥∥−2∞	X
ma-58	53	9	〈	〈	PROPN
ma-58	53	10	ξ	ξ	PROPN
ma-58	53	11	,	,	PUNCT
ma-58	53	12	ξ	ξ	PROPN
ma-58	53	13	〉	〉	NOUN
ma-58	53	14	≤	≤	NOUN
ma-58	53	15	〈	〈	PROPN
ma-58	53	16	tξ	tξ	AUX
ma-58	53	17	,	,	PUNCT
ma-58	53	18	tξ	tξ	VERB
ma-58	53	19	〉	〉	NOUN
ma-58	53	20	≤	≤	NOUN
ma-58	53	21	‖t‖2∞〈ξ	‖t‖2∞〈ξ	NUM
ma-58	53	22	,	,	PUNCT
ma-58	53	23	ξ	ξ	PROPN
ma-58	53	24	〉	〉	NUM
ma-58	53	25	.	.	NOUN
ma-58	54	1	3	3	NUM
ma-58	54	2	.	.	X
ma-58	54	3	∗-k	∗-k	NOUN
ma-58	54	4	-	-	PUNCT
ma-58	54	5	operator	operator	NOUN
ma-58	54	6	frame	frame	NOUN
ma-58	54	7	for	for	ADP
ma-58	54	8	hom∗a(x	hom∗a(x	NOUN
ma-58	54	9	)	)	PUNCT
ma-58	55	1	we	we	PRON
ma-58	55	2	begin	begin	VERB
ma-58	55	3	this	this	DET
ma-58	55	4	section	section	NOUN
ma-58	55	5	with	with	ADP
ma-58	55	6	the	the	DET
ma-58	55	7	definition	definition	NOUN
ma-58	55	8	of	of	ADP
ma-58	55	9	a	a	DET
ma-58	55	10	k	k	ADJ
ma-58	55	11	-	-	PUNCT
ma-58	55	12	operator	operator	NOUN
ma-58	55	13	frame	frame	NOUN
ma-58	55	14	.	.	PUNCT
ma-58	56	1	definition	definition	NOUN
ma-58	56	2	3.1	3.1	NUM
ma-58	56	3	.	.	PUNCT
ma-58	57	1	let	let	VERB
ma-58	57	2	{	{	PUNCT
ma-58	57	3	ti}i∈i	ti}i∈i	NOUN
ma-58	57	4	be	be	AUX
ma-58	57	5	a	a	DET
ma-58	57	6	family	family	NOUN
ma-58	57	7	of	of	ADP
ma-58	57	8	adjointable	adjointable	PROPN
ma-58	57	9	operators	operator	NOUN
ma-58	57	10	on	on	ADP
ma-58	57	11	a	a	DET
ma-58	57	12	hilbert	hilbert	NOUN
ma-58	57	13	a	a	DET
ma-58	57	14	-	-	PUNCT
ma-58	57	15	module	module	NOUN
ma-58	57	16	x	x	PUNCT
ma-58	57	17	over	over	ADP
ma-58	57	18	aunital	aunital	ADJ
ma-58	57	19	pro	pro	ADJ
ma-58	57	20	-	-	ADJ
ma-58	57	21	c∗-algebra	c∗-algebra	ADJ
ma-58	57	22	,	,	PUNCT
ma-58	57	23	and	and	CCONJ
ma-58	57	24	let	let	VERB
ma-58	57	25	k	k	PROPN
ma-58	57	26	∈	∈	PROPN
ma-58	57	27	hom∗a(x	hom∗a(x	NOUN
ma-58	57	28	)	)	PUNCT
ma-58	57	29	.	.	PUNCT
ma-58	58	1	{	{	PUNCT
ma-58	58	2	ti}i∈i	ti}i∈i	NOUN
ma-58	58	3	is	be	AUX
ma-58	58	4	called	call	VERB
ma-58	58	5	a	a	DET
ma-58	58	6	k	k	ADJ
ma-58	58	7	-	-	PUNCT
ma-58	58	8	operator	operator	NOUN
ma-58	58	9	frame	frame	NOUN
ma-58	58	10	for	for	ADP
ma-58	58	11	hom∗a(x	hom∗a(x	NOUN
ma-58	58	12	)	)	PUNCT
ma-58	58	13	,	,	PUNCT
ma-58	58	14	if	if	SCONJ
ma-58	58	15	there	there	PRON
ma-58	58	16	exist	exist	VERB
ma-58	58	17	two	two	NUM
ma-58	58	18	positive	positive	ADJ
ma-58	58	19	constants	constant	NOUN
ma-58	58	20	a	a	PRON
ma-58	58	21	,	,	PUNCT
ma-58	58	22	b	b	X
ma-58	58	23	>	>	X
ma-58	58	24	0	0	NUM
ma-58	58	25	such	such	ADJ
ma-58	58	26	that	that	SCONJ
ma-58	58	27	a〈k∗ξ	a〈k∗ξ	NOUN
ma-58	58	28	,	,	PUNCT
ma-58	58	29	k∗ξ	k∗ξ	PROPN
ma-58	58	30	〉	〉	NOUN
ma-58	58	31	≤	≤	NOUN
ma-58	58	32	∑	∑	PUNCT
ma-58	58	33	i∈i	i∈i	ADJ
ma-58	58	34	〈	〈	PROPN
ma-58	58	35	tiξ	tiξ	PROPN
ma-58	58	36	,	,	PUNCT
ma-58	58	37	tiξ	tiξ	PROPN
ma-58	58	38	〉	〉	PROPN
ma-58	58	39	≤	≤	PROPN
ma-58	58	40	b〈ξ	b〈ξ	PUNCT
ma-58	58	41	,	,	PUNCT
ma-58	58	42	ξ〉,∀ξ	ξ〉,∀ξ	VERB
ma-58	58	43	∈	∈	PROPN
ma-58	58	44	x	x	X
ma-58	58	45	.	.	PUNCT
ma-58	59	1	(	(	PUNCT
ma-58	59	2	3.1	3.1	NUM
ma-58	59	3	)	)	PUNCT
ma-58	59	4	https://doi.org/10.28924/ada/ma.2.4	https://doi.org/10.28924/ada/ma.2.4	NUM
ma-58	59	5	eur	eur	NOUN
ma-58	59	6	.	.	PUNCT
ma-58	60	1	j.	j.	PROPN
ma-58	60	2	math	math	PROPN
ma-58	60	3	.	.	PUNCT
ma-58	61	1	anal	anal	PROPN
ma-58	61	2	.	.	PUNCT
ma-58	62	1	10.28924	10.28924	NUM
ma-58	62	2	/	/	SYM
ma-58	62	3	ada	ada	PROPN
ma-58	62	4	/	/	SYM
ma-58	62	5	ma.2.4	ma.2.4	PROPN
ma-58	62	6	4the	4the	NUM
ma-58	62	7	numbers	number	NOUN
ma-58	62	8	a	a	PRON
ma-58	62	9	and	and	CCONJ
ma-58	62	10	b	b	NOUN
ma-58	62	11	are	be	AUX
ma-58	62	12	called	call	VERB
ma-58	62	13	lower	low	ADJ
ma-58	62	14	and	and	CCONJ
ma-58	62	15	upper	upper	ADJ
ma-58	62	16	bound	bind	VERB
ma-58	62	17	of	of	ADP
ma-58	62	18	the	the	DET
ma-58	62	19	k	k	ADJ
ma-58	62	20	-	-	NOUN
ma-58	62	21	operator	operator	NOUN
ma-58	62	22	frame	frame	NOUN
ma-58	62	23	,	,	PUNCT
ma-58	62	24	respectively	respectively	ADV
ma-58	62	25	.	.	PUNCT
ma-58	63	1	if	if	SCONJ
ma-58	63	2	a〈k∗ξ	a〈k∗ξ	NOUN
ma-58	63	3	,	,	PUNCT
ma-58	63	4	k∗ξ	k∗ξ	PROPN
ma-58	63	5	〉	〉	NOUN
ma-58	63	6	=	=	SYM
ma-58	63	7	∑	∑	ADP
ma-58	63	8	i∈i	i∈i	ADJ
ma-58	63	9	〈	〈	PROPN
ma-58	63	10	tiξ	tiξ	PROPN
ma-58	63	11	,	,	PUNCT
ma-58	63	12	tiξ	tiξ	PROPN
ma-58	63	13	〉	〉	PROPN
ma-58	64	1	,	,	PUNCT
ma-58	64	2	the	the	DET
ma-58	64	3	k	k	NOUN
ma-58	64	4	-	-	PUNCT
ma-58	64	5	operator	operator	NOUN
ma-58	64	6	frame	frame	NOUN
ma-58	64	7	is	be	AUX
ma-58	64	8	an	an	DET
ma-58	64	9	a	a	DET
ma-58	64	10	-	-	PUNCT
ma-58	64	11	tight	tight	NOUN
ma-58	64	12	.	.	PUNCT
ma-58	65	1	if	if	SCONJ
ma-58	65	2	a	a	DET
ma-58	65	3	=	=	NOUN
ma-58	65	4	1	1	NUM
ma-58	65	5	,	,	PUNCT
ma-58	65	6	it	it	PRON
ma-58	65	7	is	be	AUX
ma-58	65	8	called	call	VERB
ma-58	65	9	a	a	DET
ma-58	65	10	normalized	normalize	VERB
ma-58	65	11	tight	tight	ADJ
ma-58	65	12	k	k	ADJ
ma-58	65	13	-	-	NOUN
ma-58	65	14	operator	operator	NOUN
ma-58	65	15	frame	frame	NOUN
ma-58	65	16	or	or	CCONJ
ma-58	65	17	aparseval	aparseval	NOUN
ma-58	65	18	k	k	ADJ
ma-58	65	19	-	-	PUNCT
ma-58	65	20	operator	operator	NOUN
ma-58	65	21	frame	frame	NOUN
ma-58	65	22	.	.	PUNCT
ma-58	66	1	we	we	PRON
ma-58	66	2	will	will	AUX
ma-58	66	3	now	now	ADV
ma-58	66	4	move	move	VERB
ma-58	66	5	to	to	PART
ma-58	66	6	define	define	VERB
ma-58	66	7	the	the	DET
ma-58	66	8	∗-k	∗-k	ADJ
ma-58	66	9	-	-	PUNCT
ma-58	66	10	operator	operator	NOUN
ma-58	66	11	frame	frame	NOUN
ma-58	66	12	for	for	ADP
ma-58	66	13	hom∗a(x	hom∗a(x	NOUN
ma-58	66	14	)	)	PUNCT
ma-58	66	15	.	.	PUNCT
ma-58	67	1	definition	definition	NOUN
ma-58	67	2	3.2	3.2	NUM
ma-58	67	3	.	.	PUNCT
ma-58	68	1	let	let	VERB
ma-58	68	2	{	{	PUNCT
ma-58	68	3	ti}i∈i	ti}i∈i	NOUN
ma-58	68	4	be	be	AUX
ma-58	68	5	a	a	DET
ma-58	68	6	family	family	NOUN
ma-58	68	7	of	of	ADP
ma-58	68	8	adjointable	adjointable	PROPN
ma-58	68	9	operators	operator	NOUN
ma-58	68	10	on	on	ADP
ma-58	68	11	a	a	DET
ma-58	68	12	hilbert	hilbert	NOUN
ma-58	68	13	a	a	DET
ma-58	68	14	-	-	PUNCT
ma-58	68	15	module	module	NOUN
ma-58	68	16	x	x	SYM
ma-58	68	17	overa	overa	NOUN
ma-58	68	18	unital	unital	ADJ
ma-58	68	19	pro	pro	ADJ
ma-58	68	20	-	-	ADJ
ma-58	68	21	c∗-algebra	c∗-algebra	ADJ
ma-58	68	22	,	,	PUNCT
ma-58	68	23	and	and	CCONJ
ma-58	68	24	let	let	VERB
ma-58	68	25	k	k	PROPN
ma-58	68	26	∈	∈	PROPN
ma-58	68	27	hom∗a(x	hom∗a(x	NOUN
ma-58	68	28	)	)	PUNCT
ma-58	68	29	.	.	PUNCT
ma-58	69	1	{	{	PUNCT
ma-58	69	2	ti}i∈i	ti}i∈i	NOUN
ma-58	69	3	is	be	AUX
ma-58	69	4	called	call	VERB
ma-58	69	5	a	a	DET
ma-58	69	6	∗-k	∗-k	ADJ
ma-58	69	7	-	-	PUNCT
ma-58	69	8	operator	operator	NOUN
ma-58	69	9	frame	frame	NOUN
ma-58	69	10	for	for	ADP
ma-58	69	11	hom∗a(h	hom∗a(h	NOUN
ma-58	69	12	)	)	PUNCT
ma-58	69	13	,	,	PUNCT
ma-58	69	14	if	if	SCONJ
ma-58	69	15	there	there	PRON
ma-58	69	16	exists	exist	VERB
ma-58	69	17	two	two	NUM
ma-58	69	18	nonzero	nonzero	PROPN
ma-58	69	19	elements	element	NOUN
ma-58	69	20	a	a	PRON
ma-58	69	21	and	and	CCONJ
ma-58	69	22	b	b	NOUN
ma-58	69	23	in	in	ADP
ma-58	69	24	a	a	DET
ma-58	69	25	such	such	ADJ
ma-58	69	26	that	that	DET
ma-58	69	27	a〈k∗ξ	a〈k∗ξ	NOUN
ma-58	69	28	,	,	PUNCT
ma-58	69	29	k∗ξ〉a∗	k∗ξ〉a∗	PROPN
ma-58	69	30	≤	≤	NOUN
ma-58	69	31	∑	∑	PUNCT
ma-58	69	32	i∈i	i∈i	ADJ
ma-58	69	33	〈	〈	PROPN
ma-58	69	34	tiξ	tiξ	PROPN
ma-58	69	35	,	,	PUNCT
ma-58	69	36	tiξ	tiξ	PROPN
ma-58	69	37	〉	〉	PROPN
ma-58	69	38	≤	≤	PROPN
ma-58	69	39	b〈ξ	b〈ξ	PUNCT
ma-58	69	40	,	,	PUNCT
ma-58	69	41	ξ〉b∗,∀ξ	ξ〉b∗,∀ξ	VERB
ma-58	69	42	∈	∈	NOUN
ma-58	69	43	x	x	X
ma-58	69	44	.	.	PUNCT
ma-58	70	1	(	(	PUNCT
ma-58	70	2	3.2	3.2	NUM
ma-58	70	3	)	)	PUNCT
ma-58	70	4	the	the	DET
ma-58	70	5	elements	element	NOUN
ma-58	70	6	a	a	PRON
ma-58	70	7	and	and	CCONJ
ma-58	70	8	b	b	NOUN
ma-58	70	9	are	be	AUX
ma-58	70	10	called	call	VERB
ma-58	70	11	lower	low	ADJ
ma-58	70	12	and	and	CCONJ
ma-58	70	13	upper	upper	ADJ
ma-58	70	14	bounds	bound	NOUN
ma-58	70	15	of	of	ADP
ma-58	70	16	the	the	DET
ma-58	70	17	∗-k	∗-k	ADJ
ma-58	70	18	-	-	PUNCT
ma-58	70	19	operator	operator	NOUN
ma-58	70	20	frame	frame	NOUN
ma-58	70	21	,	,	PUNCT
ma-58	70	22	respectively.if	respectively.if	PROPN
ma-58	70	23	a〈k∗ξ	a〈k∗ξ	NOUN
ma-58	70	24	,	,	PUNCT
ma-58	70	25	k∗ξ〉∗	k∗ξ〉∗	PROPN
ma-58	70	26	=	=	SYM
ma-58	71	1	∑	∑	PROPN
ma-58	71	2	i∈i	i∈i	ADJ
ma-58	71	3	〈	〈	PROPN
ma-58	71	4	tiξ	tiξ	PROPN
ma-58	71	5	,	,	PUNCT
ma-58	71	6	tiξ	tiξ	PROPN
ma-58	71	7	〉	〉	PROPN
ma-58	71	8	,	,	PUNCT
ma-58	71	9	the	the	DET
ma-58	71	10	∗-k	∗-k	NOUN
ma-58	71	11	-	-	PUNCT
ma-58	71	12	operator	operator	NOUN
ma-58	71	13	frame	frame	NOUN
ma-58	71	14	is	be	AUX
ma-58	71	15	an	an	DET
ma-58	71	16	a	a	DET
ma-58	71	17	-	-	PUNCT
ma-58	71	18	tight	tight	NOUN
ma-58	71	19	.	.	PUNCT
ma-58	72	1	if	if	SCONJ
ma-58	72	2	a	a	DET
ma-58	72	3	=	=	NOUN
ma-58	72	4	1	1	NUM
ma-58	72	5	,	,	PUNCT
ma-58	72	6	it	it	PRON
ma-58	72	7	is	be	AUX
ma-58	72	8	called	call	VERB
ma-58	72	9	a	a	DET
ma-58	72	10	normalized	normalize	VERB
ma-58	72	11	tight	tight	ADJ
ma-58	72	12	∗-k	∗-k	NOUN
ma-58	72	13	-	-	PUNCT
ma-58	72	14	operator	operator	NOUN
ma-58	72	15	frameor	frameor	NOUN
ma-58	72	16	a	a	DET
ma-58	72	17	parseval	parseval	NOUN
ma-58	72	18	∗-k	∗-k	NOUN
ma-58	72	19	-	-	PUNCT
ma-58	72	20	operator	operator	NOUN
ma-58	72	21	frame	frame	NOUN
ma-58	72	22	.	.	PUNCT
ma-58	73	1	example	example	NOUN
ma-58	73	2	3.3	3.3	NUM
ma-58	73	3	.	.	PUNCT
ma-58	74	1	let	let	VERB
ma-58	74	2	l∞	l∞	NOUN
ma-58	74	3	be	be	AUX
ma-58	74	4	the	the	DET
ma-58	74	5	set	set	NOUN
ma-58	74	6	of	of	ADP
ma-58	74	7	all	all	DET
ma-58	74	8	bounded	bounded	ADJ
ma-58	74	9	complex	complex	ADV
ma-58	74	10	-	-	PUNCT
ma-58	74	11	valued	value	VERB
ma-58	74	12	sequences	sequence	NOUN
ma-58	74	13	.	.	PUNCT
ma-58	75	1	for	for	ADP
ma-58	75	2	any	any	DET
ma-58	75	3	u	u	NOUN
ma-58	75	4	=	=	PUNCT
ma-58	75	5	{	{	PUNCT
ma-58	75	6	uj}j∈n	uj}j∈n	PROPN
ma-58	75	7	,	,	PUNCT
ma-58	75	8	v	v	NOUN
ma-58	75	9	=	=	SYM
ma-58	75	10	{	{	PUNCT
ma-58	75	11	vj}j∈n	vj}j∈n	ADP
ma-58	75	12	∈	∈	PROPN
ma-58	75	13	l∞	l∞	NOUN
ma-58	75	14	,	,	PUNCT
ma-58	75	15	we	we	PRON
ma-58	75	16	define	define	VERB
ma-58	75	17	uv	uv	NOUN
ma-58	75	18	=	=	PUNCT
ma-58	75	19	{	{	PUNCT
ma-58	75	20	ujvj}j∈n	ujvj}j∈n	PROPN
ma-58	75	21	,	,	PUNCT
ma-58	75	22	u∗	u∗	NOUN
ma-58	75	23	=	=	SYM
ma-58	75	24	{	{	PUNCT
ma-58	75	25	ūj}j∈n	ūj}j∈n	NOUN
ma-58	75	26	,	,	PUNCT
ma-58	75	27	‖u‖	‖u‖	PROPN
ma-58	75	28	=	=	PUNCT
ma-58	75	29	sup	sup	NOUN
ma-58	75	30	j∈n	j∈n	NOUN
ma-58	75	31	|uj	|uj	PROPN
ma-58	76	1	|	|	ADV
ma-58	76	2	.	.	PUNCT
ma-58	77	1	then	then	ADV
ma-58	77	2	a	a	PRON
ma-58	77	3	=	=	X
ma-58	77	4	{	{	PUNCT
ma-58	77	5	l∞	l∞	NOUN
ma-58	77	6	,	,	PUNCT
ma-58	77	7	‖.‖	‖.‖	NOUN
ma-58	77	8	}	}	PUNCT
ma-58	77	9	is	be	AUX
ma-58	77	10	a	a	DET
ma-58	77	11	c∗-algebra	c∗-algebra	PROPN
ma-58	77	12	.	.	PUNCT
ma-58	78	1	then	then	ADV
ma-58	78	2	a	a	PRON
ma-58	78	3	is	be	AUX
ma-58	78	4	pro-c∗-algebra.let	pro-c∗-algebra.let	NOUN
ma-58	78	5	x	x	PUNCT
ma-58	79	1	=	=	SYM
ma-58	79	2	c0	c0	NOUN
ma-58	79	3	be	be	VERB
ma-58	79	4	the	the	DET
ma-58	79	5	set	set	NOUN
ma-58	79	6	of	of	ADP
ma-58	79	7	all	all	DET
ma-58	79	8	null	null	ADJ
ma-58	79	9	sequences	sequence	NOUN
ma-58	79	10	.	.	PUNCT
ma-58	80	1	for	for	ADP
ma-58	80	2	any	any	DET
ma-58	80	3	u	u	NOUN
ma-58	80	4	,	,	PUNCT
ma-58	80	5	v	v	NOUN
ma-58	80	6	∈	∈	NOUN
ma-58	80	7	x	x	X
ma-58	80	8	we	we	PRON
ma-58	80	9	define	define	VERB
ma-58	80	10	〈	〈	PROPN
ma-58	80	11	u	u	NOUN
ma-58	80	12	,	,	PUNCT
ma-58	80	13	v	v	NOUN
ma-58	80	14	〉	〉	NOUN
ma-58	80	15	=	=	PUNCT
ma-58	80	16	uv∗	uv∗	ADJ
ma-58	80	17	=	=	X
ma-58	80	18	{	{	PUNCT
ma-58	80	19	uj	uj	PROPN
ma-58	80	20	ūj}j∈n	ūj}j∈n	PROPN
ma-58	80	21	.	.	PUNCT
ma-58	81	1	therefore	therefore	ADV
ma-58	81	2	x	x	X
ma-58	81	3	is	be	AUX
ma-58	81	4	a	a	DET
ma-58	81	5	hilbert	hilbert	NOUN
ma-58	81	6	a-module.define	a-module.define	NOUN
ma-58	81	7	fj	fj	PROPN
ma-58	82	1	=	=	PUNCT
ma-58	82	2	{	{	PUNCT
ma-58	82	3	f	f	X
ma-58	82	4	ji	ji	X
ma-58	82	5	}	}	PUNCT
ma-58	82	6	i∈n∗	i∈n∗	VERB
ma-58	82	7	by	by	ADP
ma-58	82	8	f	f	PROPN
ma-58	82	9	ji	ji	PROPN
ma-58	82	10	=	=	NOUN
ma-58	82	11	1	1	NUM
ma-58	82	12	2	2	NUM
ma-58	83	1	+	+	SYM
ma-58	83	2	1	1	NUM
ma-58	83	3	i	i	PRON
ma-58	83	4	if	if	SCONJ
ma-58	83	5	i	i	PRON
ma-58	83	6	=	=	SYM
ma-58	83	7	j	j	PROPN
ma-58	83	8	and	and	CCONJ
ma-58	83	9	f	f	PROPN
ma-58	83	10	ji	ji	PROPN
ma-58	84	1	=	=	NOUN
ma-58	84	2	0	0	PUNCT
ma-58	85	1	if	if	SCONJ
ma-58	85	2	i	i	PRON
ma-58	85	3	6=	6=	VERB
ma-58	85	4	j	j	PROPN
ma-58	85	5	∀j	∀j	PROPN
ma-58	85	6	∈	∈	PROPN
ma-58	85	7	n∗.now	n∗.now	NOUN
ma-58	85	8	define	define	VERB
ma-58	85	9	the	the	DET
ma-58	85	10	adjointable	adjointable	NOUN
ma-58	85	11	operator	operator	NOUN
ma-58	85	12	tj	tj	NOUN
ma-58	85	13	:	:	PUNCT
ma-58	85	14	x	x	X
ma-58	85	15	→	→	SYM
ma-58	85	16	x	x	SYM
ma-58	85	17	,	,	PUNCT
ma-58	85	18	tj{(ξi)i	tj{(ξi)i	NOUN
ma-58	85	19	}	}	PUNCT
ma-58	85	20	=	=	PUNCT
ma-58	86	1	(	(	PUNCT
ma-58	86	2	ξi	ξi	NOUN
ma-58	86	3	f	f	PROPN
ma-58	86	4	j	j	PROPN
ma-58	87	1	i	i	INTJ
ma-58	87	2	)	)	PUNCT
ma-58	88	1	i	i	PRON
ma-58	88	2	.then	.then	VERB
ma-58	88	3	for	for	ADP
ma-58	88	4	every	every	DET
ma-58	88	5	x	x	SYM
ma-58	88	6	∈	∈	PROPN
ma-58	88	7	x	x	INTJ
ma-58	88	8	we	we	PRON
ma-58	88	9	have∑	have∑	VERB
ma-58	88	10	j∈n	j∈n	NOUN
ma-58	88	11	〈	〈	PROPN
ma-58	88	12	tjξ	tjξ	NOUN
ma-58	88	13	,	,	PUNCT
ma-58	88	14	tjξ	tjξ	NOUN
ma-58	88	15	〉	〉	NUM
ma-58	88	16	=	=	PUNCT
ma-58	88	17	{	{	PUNCT
ma-58	88	18	1	1	NUM
ma-58	88	19	2	2	NUM
ma-58	88	20	+	+	SYM
ma-58	88	21	1	1	NUM
ma-58	88	22	i	i	NOUN
ma-58	88	23	}	}	PUNCT
ma-58	88	24	i∈n∗〈ξ	i∈n∗〈ξ	PROPN
ma-58	88	25	,	,	PUNCT
ma-58	88	26	ξ	ξ	PROPN
ma-58	88	27	〉	〉	NOUN
ma-58	88	28	{	{	PUNCT
ma-58	88	29	1	1	NUM
ma-58	88	30	2	2	NUM
ma-58	88	31	+	+	SYM
ma-58	88	32	1	1	NUM
ma-58	88	33	i	i	NOUN
ma-58	88	34	}	}	PUNCT
ma-58	88	35	i∈n∗	i∈n∗	NOUN
ma-58	88	36	.	.	PUNCT
ma-58	89	1	so	so	ADV
ma-58	89	2	{	{	PUNCT
ma-58	89	3	tj}j	tj}j	PROPN
ma-58	89	4	is	be	AUX
ma-58	89	5	a	a	DET
ma-58	89	6	{	{	PUNCT
ma-58	89	7	12	12	NUM
ma-58	89	8	+	+	NUM
ma-58	89	9	1	1	NUM
ma-58	89	10	i	i	NOUN
ma-58	89	11	}	}	PUNCT
ma-58	89	12	i∈n∗-tight	i∈n∗-tight	PROPN
ma-58	89	13	∗-operator	∗-operator	NOUN
ma-58	89	14	frame.let	frame.let	X
ma-58	89	15	k	k	PROPN
ma-58	89	16	:	:	PUNCT
ma-58	89	17	h	h	NOUN
ma-58	89	18	→	→	SYM
ma-58	89	19	h	h	PRON
ma-58	89	20	defined	define	VERB
ma-58	89	21	by	by	ADP
ma-58	89	22	kξ	kξ	NOUN
ma-58	89	23	=	=	PROPN
ma-58	89	24	{	{	PUNCT
ma-58	89	25	ξii	ξii	ADV
ma-58	89	26	}	}	PUNCT
ma-58	89	27	i∈n∗	i∈n∗	NOUN
ma-58	89	28	.then	.then	X
ma-58	89	29	for	for	ADP
ma-58	89	30	every	every	DET
ma-58	89	31	ξ	ξ	PROPN
ma-58	89	32	∈	∈	PROPN
ma-58	89	33	x	x	INTJ
ma-58	89	34	we	we	PRON
ma-58	89	35	have	have	VERB
ma-58	89	36	〈	〈	PROPN
ma-58	89	37	k∗ξ	k∗ξ	PROPN
ma-58	89	38	,	,	PUNCT
ma-58	89	39	k∗ξ	k∗ξ	PROPN
ma-58	89	40	〉	〉	NOUN
ma-58	89	41	≤	≤	NUM
ma-58	89	42	∑	∑	ADP
ma-58	89	43	j∈n	j∈n	NOUN
ma-58	89	44	〈	〈	PROPN
ma-58	89	45	tjξ	tjξ	NOUN
ma-58	89	46	,	,	PUNCT
ma-58	89	47	tjξ	tjξ	NOUN
ma-58	89	48	〉	〉	NUM
ma-58	89	49	=	=	PUNCT
ma-58	89	50	{	{	PUNCT
ma-58	89	51	1	1	NUM
ma-58	89	52	2	2	NUM
ma-58	89	53	+	+	SYM
ma-58	89	54	1	1	NUM
ma-58	89	55	i	i	NOUN
ma-58	89	56	}	}	PUNCT
ma-58	89	57	i∈n∗〈ξ	i∈n∗〈ξ	PROPN
ma-58	89	58	,	,	PUNCT
ma-58	89	59	ξ	ξ	PROPN
ma-58	89	60	〉	〉	NOUN
ma-58	89	61	{	{	PUNCT
ma-58	89	62	1	1	NUM
ma-58	89	63	2	2	NUM
ma-58	89	64	+	+	SYM
ma-58	89	65	1	1	NUM
ma-58	89	66	i	i	NOUN
ma-58	89	67	}	}	PUNCT
ma-58	89	68	i∈n∗	i∈n∗	PROPN
ma-58	89	69	.	.	PUNCT
ma-58	90	1	https://doi.org/10.28924/ada/ma.2.4	https://doi.org/10.28924/ada/ma.2.4	NUM
ma-58	90	2	eur	eur	NOUN
ma-58	90	3	.	.	PUNCT
ma-58	91	1	j.	j.	PROPN
ma-58	91	2	math	math	PROPN
ma-58	91	3	.	.	PUNCT
ma-58	92	1	anal	anal	PROPN
ma-58	92	2	.	.	PUNCT
ma-58	93	1	10.28924	10.28924	NUM
ma-58	93	2	/	/	SYM
ma-58	93	3	ada	ada	PROPN
ma-58	93	4	/	/	SYM
ma-58	93	5	ma.2.4	ma.2.4	PROPN
ma-58	93	6	5this	5this	NUM
ma-58	93	7	shows	show	VERB
ma-58	93	8	that	that	SCONJ
ma-58	93	9	{	{	PUNCT
ma-58	93	10	tj}j∈n	tj}j∈n	ADP
ma-58	93	11	is	be	AUX
ma-58	93	12	an	an	DET
ma-58	93	13	∗-k	∗-k	ADJ
ma-58	93	14	-	-	PUNCT
ma-58	93	15	operator	operator	NOUN
ma-58	93	16	frame	frame	NOUN
ma-58	93	17	with	with	ADP
ma-58	93	18	bounds	bound	NOUN
ma-58	93	19	1	1	NUM
ma-58	93	20	,	,	PUNCT
ma-58	93	21	{	{	PUNCT
ma-58	93	22	12	12	NUM
ma-58	93	23	+	+	NUM
ma-58	93	24	1	1	NUM
ma-58	93	25	i	i	NOUN
ma-58	93	26	}	}	PUNCT
ma-58	93	27	i∈n∗	i∈n∗	PROPN
ma-58	93	28	.	.	PUNCT
ma-58	94	1	remark	remark	VERB
ma-58	94	2	3.4	3.4	NUM
ma-58	94	3	.	.	PUNCT
ma-58	95	1	(	(	PUNCT
ma-58	95	2	1	1	X
ma-58	95	3	)	)	PUNCT
ma-58	95	4	every	every	DET
ma-58	95	5	∗-operator	∗-operator	NOUN
ma-58	95	6	frame	frame	NOUN
ma-58	95	7	for	for	ADP
ma-58	95	8	hom∗a(x	hom∗a(x	NOUN
ma-58	95	9	)	)	PUNCT
ma-58	95	10	is	be	AUX
ma-58	95	11	an	an	DET
ma-58	95	12	∗-k	∗-k	ADJ
ma-58	95	13	-	-	PUNCT
ma-58	95	14	operator	operator	NOUN
ma-58	95	15	frame	frame	NOUN
ma-58	95	16	,	,	PUNCT
ma-58	95	17	for	for	ADP
ma-58	95	18	any	any	DET
ma-58	95	19	k	k	PROPN
ma-58	95	20	∈	∈	PROPN
ma-58	95	21	hom∗a(x	hom∗a(x	NOUN
ma-58	95	22	):	):	PUNCT
ma-58	95	23	k	k	PROPN
ma-58	95	24	6=	6=	ADP
ma-58	95	25	0.(2	0.(2	NUM
ma-58	95	26	)	)	PUNCT
ma-58	95	27	if	if	SCONJ
ma-58	95	28	k	k	PROPN
ma-58	95	29	∈	∈	PROPN
ma-58	95	30	hom∗a(x	hom∗a(x	NOUN
ma-58	95	31	)	)	PUNCT
ma-58	95	32	is	be	AUX
ma-58	95	33	a	a	DET
ma-58	95	34	surjective	surjective	ADJ
ma-58	95	35	operator	operator	NOUN
ma-58	95	36	,	,	PUNCT
ma-58	95	37	then	then	ADV
ma-58	95	38	every	every	DET
ma-58	95	39	∗-k	∗-k	ADJ
ma-58	95	40	-	-	PUNCT
ma-58	95	41	operator	operator	NOUN
ma-58	95	42	frame	frame	NOUN
ma-58	95	43	for	for	ADP
ma-58	95	44	hom∗a(x	hom∗a(x	NOUN
ma-58	95	45	)	)	PUNCT
ma-58	95	46	isan	isan	ADJ
ma-58	95	47	∗-operator	∗-operator	NOUN
ma-58	95	48	frame	frame	NOUN
ma-58	95	49	.	.	PUNCT
ma-58	96	1	example	example	NOUN
ma-58	96	2	3.5	3.5	NUM
ma-58	96	3	.	.	PUNCT
ma-58	97	1	let	let	VERB
ma-58	97	2	x	x	PRON
ma-58	97	3	be	be	AUX
ma-58	97	4	a	a	DET
ma-58	97	5	finitely	finitely	ADV
ma-58	97	6	or	or	CCONJ
ma-58	97	7	countably	countably	ADV
ma-58	97	8	generated	generate	VERB
ma-58	97	9	hilbert	hilbert	PROPN
ma-58	97	10	a	a	DET
ma-58	97	11	-	-	PUNCT
ma-58	97	12	module	module	NOUN
ma-58	97	13	.	.	PUNCT
ma-58	98	1	hom∗a(x	hom∗a(x	NOUN
ma-58	98	2	)	)	PUNCT
ma-58	98	3	.	.	PUNCT
ma-58	99	1	let	let	VERB
ma-58	99	2	k	k	PROPN
ma-58	99	3	∈	∈	PROPN
ma-58	99	4	hom∗a(x	hom∗a(x	NOUN
ma-58	99	5	)	)	PUNCT
ma-58	99	6	an	an	DET
ma-58	99	7	invertible	invertible	ADJ
ma-58	99	8	element	element	NOUN
ma-58	99	9	such	such	ADJ
ma-58	99	10	that	that	SCONJ
ma-58	99	11	both	both	PRON
ma-58	99	12	are	be	AUX
ma-58	99	13	uniformly	uniformly	ADV
ma-58	99	14	bounded	bound	VERB
ma-58	99	15	and	and	CCONJ
ma-58	99	16	k	k	X
ma-58	99	17	6=	6=	PROPN
ma-58	99	18	0	0	X
ma-58	99	19	.	.	PUNCT
ma-58	100	1	let	let	VERB
ma-58	100	2	{	{	PUNCT
ma-58	100	3	ti}i∈i	ti}i∈i	NOUN
ma-58	100	4	be	be	AUX
ma-58	100	5	an	an	DET
ma-58	100	6	∗-operator	∗-operator	NOUN
ma-58	100	7	frame	frame	NOUN
ma-58	100	8	for	for	ADP
ma-58	100	9	x	x	PUNCT
ma-58	100	10	with	with	ADP
ma-58	100	11	bounds	bound	NOUN
ma-58	100	12	a	a	PRON
ma-58	100	13	and	and	CCONJ
ma-58	100	14	b	b	NOUN
ma-58	100	15	,	,	PUNCT
ma-58	100	16	respectively	respectively	ADV
ma-58	100	17	.	.	PUNCT
ma-58	101	1	we	we	PRON
ma-58	101	2	have	have	VERB
ma-58	101	3	a〈ξ	a〈ξ	NOUN
ma-58	101	4	,	,	PUNCT
ma-58	101	5	ξ〉a∗	ξ〉a∗	PROPN
ma-58	101	6	≤	≤	PROPN
ma-58	101	7	∑	∑	PUNCT
ma-58	101	8	i∈i	i∈i	ADJ
ma-58	101	9	〈	〈	PROPN
ma-58	101	10	tiξ	tiξ	PROPN
ma-58	101	11	,	,	PUNCT
ma-58	101	12	tiξ	tiξ	PROPN
ma-58	101	13	〉	〉	PROPN
ma-58	101	14	≤	≤	PROPN
ma-58	101	15	b〈ξ	b〈ξ	PUNCT
ma-58	101	16	,	,	PUNCT
ma-58	101	17	ξ〉b∗,∀ξ	ξ〉b∗,∀ξ	VERB
ma-58	101	18	∈	∈	NOUN
ma-58	101	19	x	x	X
ma-58	101	20	.	.	PUNCT
ma-58	102	1	or	or	CCONJ
ma-58	102	2	〈	〈	PROPN
ma-58	102	3	k∗ξ	k∗ξ	PROPN
ma-58	102	4	,	,	PUNCT
ma-58	102	5	k∗ξ	k∗ξ	PROPN
ma-58	102	6	〉	〉	NOUN
ma-58	102	7	≤	≤	NUM
ma-58	102	8	‖k‖2∞〈ξ	‖k‖2∞〈ξ	ADV
ma-58	102	9	,	,	PUNCT
ma-58	102	10	ξ〉,∀ξ	ξ〉,∀ξ	VERB
ma-58	102	11	∈	∈	PROPN
ma-58	102	12	x	x	X
ma-58	102	13	.then	.then	X
ma-58	102	14	‖k‖−1∞	‖k‖−1∞	ADJ
ma-58	102	15	a〈k∗ξ	a〈k∗ξ	NOUN
ma-58	102	16	,	,	PUNCT
ma-58	102	17	k∗ξ〉(‖k‖−1∞	k∗ξ〉(‖k‖−1∞	PROPN
ma-58	102	18	a)∗	a)∗	PROPN
ma-58	102	19	≤	≤	NOUN
ma-58	102	20	∑	∑	PUNCT
ma-58	102	21	i∈i	i∈i	ADJ
ma-58	102	22	〈	〈	PROPN
ma-58	102	23	tiξ	tiξ	PROPN
ma-58	102	24	,	,	PUNCT
ma-58	102	25	tiξ	tiξ	PROPN
ma-58	102	26	〉	〉	PROPN
ma-58	102	27	≤	≤	PROPN
ma-58	102	28	b〈ξ	b〈ξ	PUNCT
ma-58	102	29	,	,	PUNCT
ma-58	102	30	ξ〉b∗,∀ξ	ξ〉b∗,∀ξ	VERB
ma-58	102	31	∈	∈	NOUN
ma-58	102	32	x	x	X
ma-58	102	33	.	.	PUNCT
ma-58	103	1	so	so	ADV
ma-58	103	2	{	{	PUNCT
ma-58	103	3	ti}i∈i	ti}i∈i	NOUN
ma-58	103	4	is	be	AUX
ma-58	103	5	∗-k	∗-k	NOUN
ma-58	103	6	-	-	PUNCT
ma-58	103	7	operator	operator	NOUN
ma-58	103	8	frame	frame	NOUN
ma-58	103	9	for	for	ADP
ma-58	103	10	x	x	PUNCT
ma-58	103	11	with	with	ADP
ma-58	103	12	bounds	bound	NOUN
ma-58	103	13	‖k‖−1∞	‖k‖−1∞	NUM
ma-58	103	14	a	a	PRON
ma-58	103	15	and	and	CCONJ
ma-58	103	16	b	b	NOUN
ma-58	103	17	,	,	PUNCT
ma-58	103	18	respectively	respectively	ADV
ma-58	103	19	.	.	PUNCT
ma-58	104	1	in	in	ADP
ma-58	104	2	what	what	PRON
ma-58	104	3	follows	follow	VERB
ma-58	104	4	,	,	PUNCT
ma-58	104	5	we	we	PRON
ma-58	104	6	introduce	introduce	VERB
ma-58	104	7	the	the	DET
ma-58	104	8	analysis	analysis	NOUN
ma-58	104	9	,	,	PUNCT
ma-58	104	10	the	the	DET
ma-58	104	11	synthesis	synthesis	NOUN
ma-58	104	12	and	and	CCONJ
ma-58	104	13	the	the	DET
ma-58	104	14	frame	frame	NOUN
ma-58	104	15	operator	operator	NOUN
ma-58	104	16	.	.	PUNCT
ma-58	105	1	we	we	PRON
ma-58	105	2	alsoestablish	alsoestablish	VERB
ma-58	105	3	some	some	DET
ma-58	105	4	properties.let	properties.let	NOUN
ma-58	105	5	{	{	PUNCT
ma-58	105	6	ti}i∈i	ti}i∈i	X
ma-58	105	7	be	be	AUX
ma-58	105	8	an	an	DET
ma-58	105	9	∗-k	∗-k	ADJ
ma-58	105	10	-	-	PUNCT
ma-58	105	11	operator	operator	NOUN
ma-58	105	12	frame	frame	NOUN
ma-58	105	13	for	for	ADP
ma-58	105	14	hom∗a(x	hom∗a(x	NOUN
ma-58	105	15	)	)	PUNCT
ma-58	105	16	.	.	PUNCT
ma-58	106	1	define	define	VERB
ma-58	106	2	an	an	DET
ma-58	106	3	operator	operator	NOUN
ma-58	106	4	r	r	NOUN
ma-58	106	5	:	:	PUNCT
ma-58	106	6	x	x	X
ma-58	106	7	→	→	SYM
ma-58	106	8	l2(x	l2(x	PROPN
ma-58	106	9	)	)	PUNCT
ma-58	106	10	by	by	ADP
ma-58	106	11	rξ	rξ	X
ma-58	106	12	=	=	SYM
ma-58	106	13	{	{	PUNCT
ma-58	106	14	tiξ}i∈i	tiξ}i∈i	PROPN
ma-58	106	15	,	,	PUNCT
ma-58	106	16	∀ξ	∀ξ	X
ma-58	106	17	∈	∈	NOUN
ma-58	106	18	x	x	X
ma-58	106	19	,	,	PUNCT
ma-58	106	20	then	then	ADV
ma-58	106	21	r	r	NOUN
ma-58	106	22	is	be	AUX
ma-58	106	23	called	call	VERB
ma-58	106	24	the	the	DET
ma-58	106	25	analysis	analysis	NOUN
ma-58	106	26	operator	operator	NOUN
ma-58	106	27	.	.	PUNCT
ma-58	107	1	the	the	DET
ma-58	107	2	adjoint	adjoint	NOUN
ma-58	107	3	of	of	ADP
ma-58	107	4	the	the	DET
ma-58	107	5	analysis	analysis	NOUN
ma-58	107	6	operator	operator	NOUN
ma-58	107	7	r	r	NOUN
ma-58	107	8	,	,	PUNCT
ma-58	107	9	r∗	r∗	VERB
ma-58	107	10	:	:	PUNCT
ma-58	107	11	l2(x	l2(x	PROPN
ma-58	107	12	)	)	PUNCT
ma-58	107	13	→	→	SYM
ma-58	107	14	x	x	X
ma-58	107	15	is	be	AUX
ma-58	107	16	given	give	VERB
ma-58	107	17	by	by	ADP
ma-58	107	18	r∗({ξi}i	r∗({ξi}i	PROPN
ma-58	107	19	)	)	PUNCT
ma-58	107	20	=	=	PUNCT
ma-58	108	1	∑	∑	PUNCT
ma-58	108	2	i∈i	i∈i	PROPN
ma-58	108	3	t	t	PROPN
ma-58	108	4	∗	∗	NOUN
ma-58	109	1	i	i	PRON
ma-58	109	2	ξi	ξi	VERB
ma-58	109	3	,	,	PUNCT
ma-58	109	4	∀{ξi}i	∀{ξi}i	NOUN
ma-58	109	5	∈	∈	NOUN
ma-58	109	6	l2(x	l2(x	PROPN
ma-58	109	7	)	)	PUNCT
ma-58	109	8	.	.	PUNCT
ma-58	110	1	the	the	DET
ma-58	110	2	operator	operator	NOUN
ma-58	110	3	r∗	r∗	NOUN
ma-58	110	4	is	be	AUX
ma-58	110	5	calledthe	calledthe	ADJ
ma-58	110	6	synthesis	synthesis	NOUN
ma-58	110	7	operator	operator	NOUN
ma-58	110	8	.	.	PUNCT
ma-58	111	1	by	by	ADP
ma-58	111	2	composing	compose	VERB
ma-58	111	3	r	r	NOUN
ma-58	111	4	and	and	CCONJ
ma-58	111	5	r∗	r∗	PROPN
ma-58	111	6	,	,	PUNCT
ma-58	111	7	the	the	DET
ma-58	111	8	frame	frame	NOUN
ma-58	111	9	operator	operator	NOUN
ma-58	111	10	s	s	PART
ma-58	111	11	:	:	PUNCT
ma-58	111	12	x	x	SYM
ma-58	111	13	→	→	PUNCT
ma-58	111	14	x	x	X
ma-58	111	15	is	be	AUX
ma-58	111	16	given	give	VERB
ma-58	111	17	by	by	ADP
ma-58	111	18	sξ	sξ	PROPN
ma-58	111	19	=	=	SYM
ma-58	111	20	r∗rξ	r∗rξ	NOUN
ma-58	111	21	=	=	PUNCT
ma-58	111	22	∑	∑	PROPN
ma-58	111	23	i∈i	i∈i	PROPN
ma-58	111	24	t	t	PROPN
ma-58	111	25	∗	∗	NOUN
ma-58	111	26	i	i	PRON
ma-58	111	27	tiξ.note	tiξ.note	VERB
ma-58	111	28	that	that	SCONJ
ma-58	111	29	s	s	AUX
ma-58	111	30	need	need	AUX
ma-58	111	31	not	not	PART
ma-58	111	32	be	be	AUX
ma-58	111	33	invertible	invertible	ADJ
ma-58	111	34	in	in	ADP
ma-58	111	35	general	general	ADJ
ma-58	111	36	.	.	PUNCT
ma-58	112	1	but	but	CCONJ
ma-58	112	2	under	under	ADP
ma-58	112	3	some	some	DET
ma-58	112	4	condition	condition	NOUN
ma-58	112	5	s	s	VERB
ma-58	112	6	will	will	AUX
ma-58	112	7	be	be	AUX
ma-58	112	8	invertible	invertible	ADJ
ma-58	112	9	.	.	PUNCT
ma-58	113	1	theorem	theorem	VERB
ma-58	113	2	3.6	3.6	NUM
ma-58	113	3	.	.	PUNCT
ma-58	114	1	let	let	VERB
ma-58	114	2	k	k	PRON
ma-58	114	3	be	be	AUX
ma-58	114	4	a	a	DET
ma-58	114	5	surjective	surjective	ADJ
ma-58	114	6	operators	operator	NOUN
ma-58	114	7	in	in	ADP
ma-58	114	8	hom∗a(x	hom∗a(x	NOUN
ma-58	114	9	)	)	PUNCT
ma-58	114	10	.	.	PUNCT
ma-58	115	1	if	if	SCONJ
ma-58	115	2	{	{	PUNCT
ma-58	115	3	ti}i∈i	ti}i∈i	NOUN
ma-58	115	4	is	be	AUX
ma-58	115	5	an	an	DET
ma-58	115	6	∗-k	∗-k	ADJ
ma-58	115	7	-	-	PUNCT
ma-58	115	8	operator	operator	NOUN
ma-58	115	9	frame	frame	NOUN
ma-58	115	10	for	for	ADP
ma-58	115	11	hom∗a(x	hom∗a(x	NOUN
ma-58	115	12	)	)	PUNCT
ma-58	115	13	,	,	PUNCT
ma-58	115	14	then	then	ADV
ma-58	115	15	the	the	DET
ma-58	115	16	frame	frame	NOUN
ma-58	115	17	operator	operator	NOUN
ma-58	115	18	s	s	VERB
ma-58	115	19	is	be	AUX
ma-58	115	20	positive	positive	ADJ
ma-58	115	21	,	,	PUNCT
ma-58	115	22	invertible	invertible	ADJ
ma-58	115	23	and	and	CCONJ
ma-58	115	24	adjointable	adjointable	ADJ
ma-58	115	25	.	.	PUNCT
ma-58	116	1	in	in	ADP
ma-58	116	2	addition	addition	NOUN
ma-58	116	3	we	we	PRON
ma-58	116	4	have	have	VERB
ma-58	116	5	the	the	DET
ma-58	116	6	reconstruction	reconstruction	NOUN
ma-58	116	7	formula	formula	NOUN
ma-58	116	8	,	,	PUNCT
ma-58	116	9	ξ	ξ	X
ma-58	116	10	=	=	PUNCT
ma-58	116	11	∑	∑	PROPN
ma-58	116	12	i∈i	i∈i	PROPN
ma-58	116	13	t	t	PROPN
ma-58	116	14	∗	∗	NOUN
ma-58	117	1	i	i	PRON
ma-58	117	2	tis	tis	PROPN
ma-58	117	3	−1ξ	−1ξ	PROPN
ma-58	117	4	,	,	PUNCT
ma-58	117	5	∀ξ	∀ξ	X
ma-58	117	6	∈	∈	NOUN
ma-58	117	7	x	x	X
ma-58	117	8	.	.	PUNCT
ma-58	118	1	proof	proof	NOUN
ma-58	118	2	.	.	PUNCT
ma-58	119	1	we	we	PRON
ma-58	119	2	start	start	VERB
ma-58	119	3	by	by	ADP
ma-58	119	4	showing	show	VERB
ma-58	119	5	that	that	PRON
ma-58	119	6	,	,	PUNCT
ma-58	119	7	s	s	VERB
ma-58	119	8	is	be	AUX
ma-58	119	9	a	a	DET
ma-58	119	10	self	self	NOUN
ma-58	119	11	-	-	PUNCT
ma-58	119	12	adjoint	adjoint	NOUN
ma-58	119	13	operator	operator	NOUN
ma-58	119	14	.	.	PUNCT
ma-58	120	1	by	by	ADP
ma-58	120	2	definition	definition	NOUN
ma-58	120	3	we	we	PRON
ma-58	120	4	have	have	VERB
ma-58	120	5	∀ξ	∀ξ	NOUN
ma-58	120	6	,	,	PUNCT
ma-58	120	7	η	η	PROPN
ma-58	120	8	∈	∈	PROPN
ma-58	120	9	h	h	NOUN
ma-58	120	10	〈	〈	PROPN
ma-58	120	11	sξ	sξ	PROPN
ma-58	120	12	,	,	PUNCT
ma-58	120	13	η	η	PROPN
ma-58	120	14	〉	〉	NOUN
ma-58	120	15	=	=	SYM
ma-58	120	16	〈	〈	SYM
ma-58	120	17	∑	∑	PROPN
ma-58	120	18	i∈i	i∈i	ADJ
ma-58	120	19	t	t	PROPN
ma-58	120	20	∗i	∗i	PROPN
ma-58	120	21	tiξ	tiξ	PROPN
ma-58	120	22	,	,	PUNCT
ma-58	120	23	η	η	NOUN
ma-58	120	24	〉	〉	NOUN
ma-58	120	25	=	=	SYM
ma-58	120	26	∑	∑	ADP
ma-58	120	27	i∈i	i∈i	ADJ
ma-58	120	28	〈	〈	PROPN
ma-58	120	29	t	t	PROPN
ma-58	120	30	∗i	∗i	PROPN
ma-58	120	31	tiξ	tiξ	PROPN
ma-58	120	32	,	,	PUNCT
ma-58	120	33	η	η	NOUN
ma-58	120	34	〉	〉	PROPN
ma-58	120	35	=	=	SYM
ma-58	120	36	∑	∑	PROPN
ma-58	120	37	i∈i	i∈i	ADJ
ma-58	120	38	〈	〈	PROPN
ma-58	120	39	ξ	ξ	PROPN
ma-58	120	40	,	,	PUNCT
ma-58	120	41	t	t	PROPN
ma-58	120	42	∗i	∗i	PROPN
ma-58	120	43	tiη	tiη	PROPN
ma-58	120	44	〉	〉	PROPN
ma-58	120	45	https://doi.org/10.28924/ada/ma.2.4	https://doi.org/10.28924/ada/ma.2.4	PROPN
ma-58	120	46	eur	eur	NOUN
ma-58	120	47	.	.	PUNCT
ma-58	121	1	j.	j.	PROPN
ma-58	121	2	math	math	PROPN
ma-58	121	3	.	.	PUNCT
ma-58	122	1	anal	anal	PROPN
ma-58	122	2	.	.	PUNCT
ma-58	123	1	10.28924	10.28924	NUM
ma-58	123	2	/	/	SYM
ma-58	123	3	ada	ada	PROPN
ma-58	123	4	/	/	SYM
ma-58	123	5	ma.2.4	ma.2.4	PROPN
ma-58	123	6	6	6	NUM
ma-58	123	7	=	=	SYM
ma-58	123	8	〈	〈	PROPN
ma-58	123	9	ξ	ξ	PROPN
ma-58	123	10	,	,	PUNCT
ma-58	123	11	∑	∑	PUNCT
ma-58	123	12	i∈i	i∈i	ADJ
ma-58	123	13	t	t	NOUN
ma-58	123	14	∗i	∗i	PROPN
ma-58	123	15	tiη	tiη	NOUN
ma-58	123	16	〉	〉	NOUN
ma-58	123	17	=	=	SYM
ma-58	123	18	〈	〈	PROPN
ma-58	123	19	ξ	ξ	PROPN
ma-58	123	20	,	,	PUNCT
ma-58	123	21	sη	sη	PROPN
ma-58	123	22	〉	〉	PROPN
ma-58	123	23	.	.	PUNCT
ma-58	124	1	then	then	ADV
ma-58	124	2	s	s	VERB
ma-58	124	3	is	be	AUX
ma-58	124	4	a	a	DET
ma-58	124	5	selfadjoint.the	selfadjoint.the	DET
ma-58	124	6	operator	operator	NOUN
ma-58	124	7	s	s	VERB
ma-58	124	8	is	be	AUX
ma-58	124	9	clearly	clearly	ADV
ma-58	124	10	positive.by	positive.by	PROPN
ma-58	124	11	(	(	PUNCT
ma-58	124	12	2	2	NUM
ma-58	124	13	)	)	PUNCT
ma-58	124	14	in	in	ADP
ma-58	124	15	remark	remark	NOUN
ma-58	124	16	3.4	3.4	NUM
ma-58	124	17	{	{	PUNCT
ma-58	124	18	ti}i∈i	ti}i∈i	X
ma-58	124	19	is	be	AUX
ma-58	124	20	an	an	DET
ma-58	124	21	∗-operator	∗-operator	NOUN
ma-58	124	22	frame	frame	NOUN
ma-58	124	23	for	for	ADP
ma-58	124	24	hom∗a(x	hom∗a(x	NOUN
ma-58	124	25	)	)	PUNCT
ma-58	125	1	.the	.the	DET
ma-58	125	2	definition	definition	NOUN
ma-58	125	3	of	of	ADP
ma-58	125	4	an	an	DET
ma-58	125	5	∗-operator	∗-operator	NOUN
ma-58	125	6	gives	give	VERB
ma-58	125	7	a1〈ξ	a1〈ξ	NOUN
ma-58	125	8	,	,	PUNCT
ma-58	125	9	ξ〉a∗1	ξ〉a∗1	PROPN
ma-58	125	10	≤	≤	NOUN
ma-58	125	11	∑	∑	PUNCT
ma-58	125	12	i∈i	i∈i	ADJ
ma-58	125	13	〈	〈	PROPN
ma-58	125	14	tiξ	tiξ	PROPN
ma-58	125	15	,	,	PUNCT
ma-58	125	16	tiξ	tiξ	PROPN
ma-58	125	17	〉	〉	PROPN
ma-58	125	18	≤	≤	NUM
ma-58	125	19	b〈ξ	b〈ξ	PUNCT
ma-58	125	20	,	,	PUNCT
ma-58	125	21	ξ〉b∗.	ξ〉b∗.	NUM
ma-58	125	22	thus	thus	ADV
ma-58	125	23	by	by	ADP
ma-58	125	24	the	the	DET
ma-58	125	25	definition	definition	NOUN
ma-58	125	26	of	of	ADP
ma-58	125	27	norm	norm	NOUN
ma-58	125	28	in	in	ADP
ma-58	125	29	l2(x	l2(x	PROPN
ma-58	125	30	)	)	PUNCT
ma-58	126	1	p̄x	p̄x	NOUN
ma-58	126	2	(	(	PUNCT
ma-58	126	3	rξ)2	rξ)2	PROPN
ma-58	126	4	=	=	PUNCT
ma-58	126	5	p̄x	p̄x	NOUN
ma-58	126	6	(	(	PUNCT
ma-58	126	7	∑	∑	ADP
ma-58	126	8	i∈i	i∈i	ADJ
ma-58	126	9	〈	〈	PROPN
ma-58	126	10	tiξ	tiξ	PROPN
ma-58	126	11	,	,	PUNCT
ma-58	126	12	tiξ	tiξ	PROPN
ma-58	126	13	〉	〉	PROPN
ma-58	126	14	)	)	PUNCT
ma-58	126	15	≤	≤	NOUN
ma-58	126	16	p̄x	p̄x	NOUN
ma-58	126	17	(	(	PUNCT
ma-58	126	18	b)2p(〈ξ	b)2p(〈ξ	X
ma-58	126	19	,	,	PUNCT
ma-58	126	20	ξ〉),∀ξ	ξ〉),∀ξ	NOUN
ma-58	126	21	∈	∈	NOUN
ma-58	126	22	x	x	X
ma-58	126	23	.	.	PUNCT
ma-58	127	1	(	(	PUNCT
ma-58	127	2	3.3	3.3	NUM
ma-58	127	3	)	)	PUNCT
ma-58	127	4	therefore	therefore	ADV
ma-58	127	5	r	r	NOUN
ma-58	127	6	is	be	AUX
ma-58	127	7	well	well	ADV
ma-58	127	8	defined	define	VERB
ma-58	127	9	and	and	CCONJ
ma-58	127	10	p̄x	p̄x	NOUN
ma-58	127	11	(	(	PUNCT
ma-58	127	12	r	r	NOUN
ma-58	127	13	)	)	PUNCT
ma-58	127	14	≤	≤	NOUN
ma-58	127	15	p̄x	p̄x	NOUN
ma-58	127	16	(	(	PUNCT
ma-58	127	17	b	b	NOUN
ma-58	127	18	)	)	PUNCT
ma-58	127	19	.	.	PUNCT
ma-58	128	1	it	it	PRON
ma-58	128	2	’s	’	VERB
ma-58	128	3	clear	clear	ADJ
ma-58	128	4	that	that	SCONJ
ma-58	128	5	r	r	NOUN
ma-58	128	6	is	be	AUX
ma-58	128	7	a	a	DET
ma-58	128	8	linear	linear	ADJ
ma-58	128	9	a	a	DET
ma-58	128	10	-	-	PUNCT
ma-58	128	11	module	module	NOUN
ma-58	128	12	map	map	NOUN
ma-58	128	13	.	.	PUNCT
ma-58	129	1	wewill	wewill	PROPN
ma-58	129	2	then	then	ADV
ma-58	129	3	show	show	VERB
ma-58	129	4	that	that	SCONJ
ma-58	129	5	the	the	DET
ma-58	129	6	range	range	NOUN
ma-58	129	7	of	of	ADP
ma-58	129	8	r	r	NOUN
ma-58	129	9	is	be	AUX
ma-58	129	10	closed	close	VERB
ma-58	129	11	.	.	PUNCT
ma-58	130	1	let	let	VERB
ma-58	130	2	{	{	PUNCT
ma-58	130	3	rξn}n∈n	rξn}n∈n	ADV
ma-58	130	4	be	be	AUX
ma-58	130	5	a	a	DET
ma-58	130	6	sequence	sequence	NOUN
ma-58	130	7	in	in	ADP
ma-58	130	8	the	the	DET
ma-58	130	9	range	range	NOUN
ma-58	130	10	of	of	ADP
ma-58	130	11	r	r	NOUN
ma-58	130	12	suchthat	suchthat	PROPN
ma-58	130	13	limn→∞rξn	limn→∞rξn	PUNCT
ma-58	130	14	=	=	SYM
ma-58	130	15	η	η	PROPN
ma-58	130	16	.	.	PROPN
ma-58	130	17	for	for	ADP
ma-58	130	18	n	n	CCONJ
ma-58	130	19	,	,	PUNCT
ma-58	130	20	m	m	PROPN
ma-58	130	21	∈	∈	PROPN
ma-58	130	22	n	n	CCONJ
ma-58	130	23	,	,	PUNCT
ma-58	130	24	we	we	PRON
ma-58	130	25	have	have	VERB
ma-58	130	26	p(a〈ξn	p(a〈ξn	NOUN
ma-58	131	1	−	−	PROPN
ma-58	131	2	ξm	ξm	PROPN
ma-58	131	3	,	,	PUNCT
ma-58	131	4	ξn	ξn	ADP
ma-58	131	5	−	−	PROPN
ma-58	131	6	ξm〉a∗	ξm〉a∗	PROPN
ma-58	131	7	)	)	PUNCT
ma-58	131	8	≤	≤	NOUN
ma-58	132	1	p(〈r(ξn	p(〈r(ξn	ADJ
ma-58	132	2	−	−	NOUN
ma-58	132	3	ξm	ξm	NOUN
ma-58	132	4	)	)	PUNCT
ma-58	132	5	,	,	PUNCT
ma-58	132	6	r(ξn	r(ξn	PROPN
ma-58	132	7	−	−	PROPN
ma-58	132	8	ξm	ξm	PROPN
ma-58	132	9	)	)	PUNCT
ma-58	132	10	〉	〉	PROPN
ma-58	132	11	)	)	PUNCT
ma-58	133	1	=	=	SYM
ma-58	133	2	p̄x	p̄x	NOUN
ma-58	133	3	(	(	PUNCT
ma-58	133	4	r(ξn	r(ξn	PROPN
ma-58	133	5	−	−	PROPN
ma-58	133	6	ξm))2	ξm))2	NOUN
ma-58	133	7	.	.	PUNCT
ma-58	134	1	seeing	see	VERB
ma-58	134	2	that	that	SCONJ
ma-58	134	3	{	{	PUNCT
ma-58	134	4	rξn}n∈n	rξn}n∈n	PRON
ma-58	134	5	is	be	AUX
ma-58	134	6	cauchy	cauchy	ADJ
ma-58	134	7	sequence	sequence	NOUN
ma-58	134	8	in	in	ADP
ma-58	134	9	x	x	SYM
ma-58	134	10	,	,	PUNCT
ma-58	134	11	then	then	ADV
ma-58	134	12	p(a〈ξn	p(a〈ξn	PROPN
ma-58	134	13	−	−	PROPN
ma-58	134	14	ξm	ξm	PROPN
ma-58	134	15	,	,	PUNCT
ma-58	134	16	ξn	ξn	PROPN
ma-58	134	17	−	−	PROPN
ma-58	134	18	ξm〉a∗)→	ξm〉a∗)→	PROPN
ma-58	134	19	0	0	NUM
ma-58	134	20	,	,	PUNCT
ma-58	134	21	as	as	ADP
ma-58	134	22	n	n	X
ma-58	134	23	,	,	PUNCT
ma-58	134	24	m	m	VERB
ma-58	134	25	→∞.note	→∞.note	ADJ
ma-58	134	26	that	that	SCONJ
ma-58	134	27	for	for	ADP
ma-58	134	28	n	n	CCONJ
ma-58	134	29	,	,	PUNCT
ma-58	134	30	m	m	PROPN
ma-58	134	31	∈	∈	PROPN
ma-58	134	32	n	n	CCONJ
ma-58	134	33	,	,	PUNCT
ma-58	134	34	p(〈ξn	p(〈ξn	PROPN
ma-58	134	35	−	−	PROPN
ma-58	134	36	ξm	ξm	PROPN
ma-58	134	37	,	,	PUNCT
ma-58	134	38	ξn	ξn	PROPN
ma-58	134	39	−	−	PROPN
ma-58	134	40	ξm	ξm	NOUN
ma-58	134	41	〉	〉	NOUN
ma-58	134	42	)	)	PUNCT
ma-58	134	43	=	=	PUNCT
ma-58	135	1	p(a−1a〈ξn	p(a−1a〈ξn	NOUN
ma-58	135	2	−	−	PROPN
ma-58	136	1	ξm	ξm	PROPN
ma-58	136	2	,	,	PUNCT
ma-58	136	3	ξn	ξn	NOUN
ma-58	136	4	−	−	PROPN
ma-58	136	5	ξm〉a∗(a∗)−1	ξm〉a∗(a∗)−1	NOUN
ma-58	136	6	)	)	PUNCT
ma-58	136	7	≤	≤	NUM
ma-58	136	8	p(a−1)2p(a〈ξn	p(a−1)2p(a〈ξn	NOUN
ma-58	136	9	−	−	PROPN
ma-58	137	1	ξm	ξm	PROPN
ma-58	137	2	,	,	PUNCT
ma-58	137	3	ξn	ξn	VERB
ma-58	137	4	−	−	PROPN
ma-58	137	5	ξm〉a∗	ξm〉a∗	PROPN
ma-58	137	6	)	)	PUNCT
ma-58	137	7	.	.	PUNCT
ma-58	138	1	thus	thus	ADV
ma-58	138	2	the	the	DET
ma-58	138	3	sequence	sequence	NOUN
ma-58	138	4	{	{	PUNCT
ma-58	138	5	ξn}n∈n	ξn}n∈n	PROPN
ma-58	138	6	is	be	AUX
ma-58	138	7	cauchy	cauchy	ADJ
ma-58	138	8	and	and	CCONJ
ma-58	138	9	hence	hence	ADV
ma-58	138	10	there	there	PRON
ma-58	138	11	exists	exist	VERB
ma-58	138	12	ξ	ξ	PROPN
ma-58	138	13	∈	∈	PROPN
ma-58	138	14	x	x	PUNCT
ma-58	138	15	such	such	ADJ
ma-58	138	16	that	that	SCONJ
ma-58	138	17	ξn	ξn	PROPN
ma-58	138	18	→	→	SYM
ma-58	138	19	ξ	ξ	PROPN
ma-58	138	20	as	as	ADP
ma-58	138	21	n	n	NOUN
ma-58	138	22	→∞.again	→∞.again	PUNCT
ma-58	138	23	by	by	ADP
ma-58	138	24	(	(	PUNCT
ma-58	138	25	3.3	3.3	NUM
ma-58	138	26	)	)	PUNCT
ma-58	138	27	,	,	PUNCT
ma-58	138	28	we	we	PRON
ma-58	138	29	have	have	VERB
ma-58	138	30	p̄x	p̄x	NOUN
ma-58	138	31	(	(	PUNCT
ma-58	138	32	r(ξn	r(ξn	PROPN
ma-58	138	33	−	−	PROPN
ma-58	138	34	ξm))2	ξm))2	SYM
ma-58	138	35	≤	≤	ADJ
ma-58	138	36	p̄x	p̄x	NOUN
ma-58	138	37	(	(	PUNCT
ma-58	138	38	b)2p(〈ξn	b)2p(〈ξn	PROPN
ma-58	138	39	−	−	PROPN
ma-58	138	40	ξ	ξ	PROPN
ma-58	138	41	,	,	PUNCT
ma-58	138	42	ξn	ξn	INTJ
ma-58	138	43	−	−	PROPN
ma-58	138	44	ξ	ξ	PROPN
ma-58	138	45	〉	〉	NOUN
ma-58	138	46	)	)	PUNCT
ma-58	138	47	.	.	PUNCT
ma-58	139	1	thus	thus	ADV
ma-58	139	2	p(rξn	p(rξn	PRON
ma-58	139	3	−	−	PROPN
ma-58	139	4	rξ	rξ	NOUN
ma-58	139	5	)	)	PUNCT
ma-58	139	6	→	→	SYM
ma-58	139	7	0	0	NUM
ma-58	139	8	as	as	ADP
ma-58	139	9	n	n	PROPN
ma-58	139	10	→	→	SYM
ma-58	139	11	∞	∞	PROPN
ma-58	139	12	implies	imply	VERB
ma-58	139	13	that	that	SCONJ
ma-58	139	14	rξ	rξ	PROPN
ma-58	139	15	=	=	SYM
ma-58	139	16	η	η	PROPN
ma-58	139	17	.	.	PUNCT
ma-58	140	1	it	it	PRON
ma-58	140	2	is	be	AUX
ma-58	140	3	therefore	therefore	ADV
ma-58	140	4	concluded	conclude	VERB
ma-58	140	5	that	that	SCONJ
ma-58	140	6	therange	therange	NOUN
ma-58	140	7	of	of	ADP
ma-58	140	8	r	r	NOUN
ma-58	140	9	is	be	AUX
ma-58	140	10	closed	closed	ADJ
ma-58	140	11	.	.	PUNCT
ma-58	141	1	we	we	PRON
ma-58	141	2	now	now	ADV
ma-58	141	3	show	show	VERB
ma-58	141	4	that	that	SCONJ
ma-58	141	5	r	r	NOUN
ma-58	141	6	is	be	AUX
ma-58	141	7	injective	injective	ADJ
ma-58	141	8	.	.	PUNCT
ma-58	142	1	let	let	VERB
ma-58	142	2	ξ	ξ	X
ma-58	142	3	∈	∈	PROPN
ma-58	142	4	x	x	X
ma-58	142	5	and	and	CCONJ
ma-58	142	6	rξ	rξ	X
ma-58	142	7	=	=	SYM
ma-58	142	8	0	0	X
ma-58	142	9	.	.	PUNCT
ma-58	143	1	note	note	VERB
ma-58	143	2	that	that	SCONJ
ma-58	143	3	a〈ξ	a〈ξ	NOUN
ma-58	143	4	,	,	PUNCT
ma-58	143	5	ξ〉a∗	ξ〉a∗	PROPN
ma-58	143	6	≤	≤	NUM
ma-58	143	7	〈	〈	PROPN
ma-58	143	8	rξ	rξ	PROPN
ma-58	143	9	,	,	PUNCT
ma-58	143	10	rξ	rξ	PROPN
ma-58	143	11	〉	〉	NOUN
ma-58	143	12	then	then	ADV
ma-58	143	13	〈	〈	PROPN
ma-58	143	14	ξ	ξ	PROPN
ma-58	143	15	,	,	PUNCT
ma-58	143	16	ξ	ξ	NOUN
ma-58	143	17	〉	〉	NOUN
ma-58	143	18	=	=	SYM
ma-58	143	19	0	0	PUNCT
ma-58	144	1	so	so	ADV
ma-58	144	2	ξ	ξ	X
ma-58	144	3	=	=	SYM
ma-58	144	4	0	0	NUM
ma-58	144	5	i.e.	i.e.	X
ma-58	144	6	r	r	NOUN
ma-58	144	7	is	be	AUX
ma-58	144	8	injective.for	injective.for	ADP
ma-58	144	9	ξ	ξ	PROPN
ma-58	144	10	∈	∈	PROPN
ma-58	144	11	x	x	X
ma-58	144	12	and	and	CCONJ
ma-58	144	13	{	{	PUNCT
ma-58	144	14	ξi}i∈i	ξi}i∈i	INTJ
ma-58	144	15	∈	∈	PROPN
ma-58	144	16	l2(x	l2(x	PROPN
ma-58	144	17	)	)	PUNCT
ma-58	144	18	we	we	PRON
ma-58	144	19	have	have	VERB
ma-58	144	20	〈	〈	PROPN
ma-58	144	21	rξ	rξ	PROPN
ma-58	144	22	,	,	PUNCT
ma-58	144	23	{	{	PUNCT
ma-58	144	24	ξi}i∈i	ξi}i∈i	NOUN
ma-58	144	25	〉	〉	NOUN
ma-58	144	26	=	=	SYM
ma-58	144	27	〈	〈	PROPN
ma-58	144	28	{	{	PUNCT
ma-58	144	29	tiξ}i∈i	tiξ}i∈i	PROPN
ma-58	144	30	,	,	PUNCT
ma-58	144	31	{	{	PUNCT
ma-58	144	32	ξi}i∈i	ξi}i∈i	NOUN
ma-58	144	33	〉	〉	NOUN
ma-58	144	34	=	=	SYM
ma-58	144	35	∑	∑	ADP
ma-58	144	36	i∈i	i∈i	ADJ
ma-58	144	37	〈	〈	PROPN
ma-58	144	38	tiξ	tiξ	PROPN
ma-58	144	39	,	,	PUNCT
ma-58	144	40	ξi	ξi	NOUN
ma-58	144	41	〉	〉	NUM
ma-58	144	42	=	=	PUNCT
ma-58	144	43	∑	∑	ADP
ma-58	144	44	i∈i	i∈i	ADJ
ma-58	144	45	〈	〈	PROPN
ma-58	144	46	ξ	ξ	PROPN
ma-58	144	47	,	,	PUNCT
ma-58	144	48	t	t	PROPN
ma-58	145	1	∗i	∗i	PROPN
ma-58	145	2	ξi	ξi	PROPN
ma-58	145	3	〉	〉	NUM
ma-58	145	4	=	=	SYM
ma-58	145	5	〈	〈	PROPN
ma-58	145	6	ξ	ξ	X
ma-58	145	7	,	,	PUNCT
ma-58	145	8	∑	∑	ADP
ma-58	145	9	i∈i	i∈i	ADJ
ma-58	145	10	t	t	PROPN
ma-58	145	11	∗i	∗i	PROPN
ma-58	145	12	ξi	ξi	PROPN
ma-58	145	13	〉	〉	PROPN
ma-58	145	14	.	.	PUNCT
ma-58	146	1	then	then	ADV
ma-58	146	2	r∗({ξi}i∈i	r∗({ξi}i∈i	VERB
ma-58	146	3	)	)	PUNCT
ma-58	147	1	=	=	PUNCT
ma-58	147	2	∑	∑	PUNCT
ma-58	147	3	i∈i	i∈i	PROPN
ma-58	147	4	t	t	PROPN
ma-58	147	5	∗	∗	NOUN
ma-58	148	1	i	i	PRON
ma-58	148	2	ξi	ξi	VERB
ma-58	148	3	.	.	PUNCT
ma-58	149	1	since	since	SCONJ
ma-58	149	2	r	r	NOUN
ma-58	149	3	is	be	AUX
ma-58	149	4	injective	injective	ADJ
ma-58	149	5	,	,	PUNCT
ma-58	149	6	then	then	ADV
ma-58	149	7	the	the	DET
ma-58	149	8	operator	operator	NOUN
ma-58	149	9	r∗	r∗	NOUN
ma-58	149	10	has	have	AUX
ma-58	149	11	closed	close	VERB
ma-58	149	12	range	range	NOUN
ma-58	149	13	and	and	CCONJ
ma-58	149	14	x	x	NOUN
ma-58	149	15	=	=	SYM
ma-58	149	16	range(r∗	range(r∗	PROPN
ma-58	149	17	)	)	PUNCT
ma-58	149	18	,	,	PUNCT
ma-58	149	19	therefore	therefore	ADV
ma-58	149	20	s	s	VERB
ma-58	149	21	=	=	PUNCT
ma-58	149	22	r∗r	r∗r	ADJ
ma-58	149	23	is	be	AUX
ma-58	149	24	invertible	invertible	ADJ
ma-58	149	25	�	�	PROPN
ma-58	149	26	https://doi.org/10.28924/ada/ma.2.4	https://doi.org/10.28924/ada/ma.2.4	PROPN
ma-58	149	27	eur	eur	NOUN
ma-58	149	28	.	.	PUNCT
ma-58	150	1	j.	j.	PROPN
ma-58	150	2	math	math	PROPN
ma-58	150	3	.	.	PUNCT
ma-58	151	1	anal	anal	PROPN
ma-58	151	2	.	.	PUNCT
ma-58	152	1	10.28924	10.28924	NUM
ma-58	152	2	/	/	SYM
ma-58	152	3	ada	ada	PROPN
ma-58	152	4	/	/	SYM
ma-58	152	5	ma.2.4	ma.2.4	PROPN
ma-58	152	6	7let	7let	PROPN
ma-58	152	7	k	k	PROPN
ma-58	152	8	∈	∈	PROPN
ma-58	152	9	hom∗a(x	hom∗a(x	NOUN
ma-58	152	10	)	)	PUNCT
ma-58	152	11	,	,	PUNCT
ma-58	152	12	in	in	ADP
ma-58	152	13	the	the	DET
ma-58	152	14	following	follow	VERB
ma-58	152	15	theorem	theorem	NOUN
ma-58	152	16	we	we	PRON
ma-58	152	17	constructed	construct	VERB
ma-58	152	18	an	an	DET
ma-58	152	19	∗-k	∗-k	ADJ
ma-58	152	20	-	-	PUNCT
ma-58	152	21	operator	operator	NOUN
ma-58	152	22	frame	frame	NOUN
ma-58	152	23	by	by	ADP
ma-58	152	24	using	use	VERB
ma-58	152	25	an	an	DET
ma-58	152	26	∗-operator	∗-operator	NOUN
ma-58	152	27	frame	frame	NOUN
ma-58	152	28	.	.	PUNCT
ma-58	153	1	theorem	theorem	VERB
ma-58	153	2	3.7	3.7	NUM
ma-58	153	3	.	.	PUNCT
ma-58	154	1	let	let	AUX
ma-58	154	2	{	{	PUNCT
ma-58	154	3	ti}i∈i	ti}i∈i	NOUN
ma-58	154	4	be	be	AUX
ma-58	154	5	an	an	DET
ma-58	154	6	∗-k	∗-k	ADJ
ma-58	154	7	-	-	PUNCT
ma-58	154	8	operator	operator	NOUN
ma-58	154	9	frame	frame	NOUN
ma-58	154	10	in	in	ADP
ma-58	154	11	x	x	PUNCT
ma-58	154	12	with	with	ADP
ma-58	154	13	bounds	bound	NOUN
ma-58	154	14	a	a	PRON
ma-58	154	15	,	,	PUNCT
ma-58	154	16	b	b	NOUN
ma-58	154	17	and	and	CCONJ
ma-58	154	18	k	k	PROPN
ma-58	154	19	∈	∈	PROPN
ma-58	154	20	hom∗a(x	hom∗a(x	NOUN
ma-58	154	21	)	)	PUNCT
ma-58	154	22	be	be	AUX
ma-58	154	23	an	an	DET
ma-58	154	24	invertible	invertible	ADJ
ma-58	154	25	element	element	NOUN
ma-58	154	26	such	such	ADJ
ma-58	154	27	that	that	SCONJ
ma-58	154	28	both	both	PRON
ma-58	154	29	are	be	AUX
ma-58	154	30	uniformly	uniformly	ADV
ma-58	154	31	bounded	bound	VERB
ma-58	154	32	.	.	PUNCT
ma-58	155	1	then	then	ADV
ma-58	155	2	{	{	PUNCT
ma-58	155	3	tik}i∈i	tik}i∈i	ADV
ma-58	155	4	is	be	AUX
ma-58	155	5	an	an	DET
ma-58	155	6	∗-k∗-operator	∗-k∗-operator	NOUN
ma-58	155	7	frame	frame	NOUN
ma-58	155	8	in	in	ADP
ma-58	155	9	x	x	PUNCT
ma-58	155	10	with	with	ADP
ma-58	155	11	bounds	bound	NOUN
ma-58	155	12	a	a	DET
ma-58	155	13	,	,	PUNCT
ma-58	155	14	‖k‖∞b	‖k‖∞b	PROPN
ma-58	155	15	.	.	PUNCT
ma-58	156	1	the	the	DET
ma-58	156	2	frame	frame	NOUN
ma-58	156	3	operator	operator	NOUN
ma-58	156	4	of	of	ADP
ma-58	156	5	{	{	PUNCT
ma-58	156	6	tik}i∈i	tik}i∈i	ADV
ma-58	156	7	is	be	AUX
ma-58	156	8	s′	s′	ADJ
ma-58	156	9	=	=	PUNCT
ma-58	156	10	k∗sk	k∗sk	PROPN
ma-58	156	11	,	,	PUNCT
ma-58	156	12	where	where	SCONJ
ma-58	156	13	s	s	NOUN
ma-58	156	14	is	be	AUX
ma-58	156	15	the	the	DET
ma-58	156	16	frame	frame	NOUN
ma-58	156	17	operator	operator	NOUN
ma-58	156	18	of	of	ADP
ma-58	156	19	{	{	PUNCT
ma-58	156	20	ti}i∈i	ti}i∈i	NOUN
ma-58	156	21	.	.	PUNCT
ma-58	157	1	proof	proof	NOUN
ma-58	157	2	.	.	PUNCT
ma-58	158	1	from	from	ADP
ma-58	158	2	a〈ξ	a〈ξ	NOUN
ma-58	158	3	,	,	PUNCT
ma-58	158	4	ξ〉a∗	ξ〉a∗	PROPN
ma-58	158	5	≤	≤	PROPN
ma-58	158	6	∑	∑	PUNCT
ma-58	158	7	i∈i	i∈i	ADJ
ma-58	158	8	〈	〈	PROPN
ma-58	158	9	tiξ	tiξ	PROPN
ma-58	158	10	,	,	PUNCT
ma-58	158	11	tiξ	tiξ	PROPN
ma-58	158	12	〉	〉	PROPN
ma-58	158	13	≤	≤	PROPN
ma-58	158	14	b〈ξ	b〈ξ	PUNCT
ma-58	158	15	,	,	PUNCT
ma-58	158	16	ξ〉b∗,∀ξ	ξ〉b∗,∀ξ	VERB
ma-58	158	17	∈	∈	NOUN
ma-58	158	18	x	x	X
ma-58	158	19	.	.	PUNCT
ma-58	159	1	we	we	PRON
ma-58	159	2	get	get	VERB
ma-58	159	3	for	for	ADP
ma-58	159	4	all	all	DET
ma-58	159	5	ξ	ξ	X
ma-58	159	6	∈	∈	PROPN
ma-58	159	7	x	x	X
ma-58	159	8	,	,	PUNCT
ma-58	159	9	a〈kξ	a〈kξ	PROPN
ma-58	159	10	,	,	PUNCT
ma-58	159	11	kξ〉a∗	kξ〉a∗	PROPN
ma-58	159	12	≤	≤	PROPN
ma-58	159	13	∑	∑	PUNCT
ma-58	159	14	i∈i	i∈i	ADJ
ma-58	159	15	〈	〈	PROPN
ma-58	159	16	tikξ	tikξ	NOUN
ma-58	159	17	,	,	PUNCT
ma-58	159	18	tikξ	tikξ	VERB
ma-58	159	19	〉	〉	PROPN
ma-58	159	20	≤	≤	NUM
ma-58	159	21	b〈kξ	b〈kξ	PROPN
ma-58	159	22	,	,	PUNCT
ma-58	159	23	kξ〉b∗	kξ〉b∗	PROPN
ma-58	159	24	≤	≤	X
ma-58	159	25	‖k‖∞b〈ξ	‖k‖∞b〈ξ	ADJ
ma-58	159	26	,	,	PUNCT
ma-58	159	27	ξ〉(‖k‖∞b)∗.	ξ〉(‖k‖∞b)∗.	PROPN
ma-58	159	28	then	then	ADV
ma-58	159	29	{	{	PUNCT
ma-58	159	30	tik}i∈i	tik}i∈i	ADV
ma-58	159	31	is	be	AUX
ma-58	159	32	an	an	DET
ma-58	159	33	∗-k∗-operator	∗-k∗-operator	NOUN
ma-58	159	34	frame	frame	NOUN
ma-58	159	35	in	in	ADP
ma-58	159	36	x	x	PUNCT
ma-58	159	37	with	with	ADP
ma-58	159	38	bounds	bound	NOUN
ma-58	159	39	a	a	PRON
ma-58	159	40	,	,	PUNCT
ma-58	159	41	‖k‖∞b.by	‖k‖∞b.by	PROPN
ma-58	159	42	definition	definition	NOUN
ma-58	159	43	of	of	ADP
ma-58	159	44	s	s	PROPN
ma-58	159	45	,	,	PUNCT
ma-58	159	46	we	we	PRON
ma-58	159	47	have	have	VERB
ma-58	159	48	skξ	skξ	VERB
ma-58	159	49	=	=	SYM
ma-58	159	50	∑	∑	PUNCT
ma-58	159	51	i∈i	i∈i	PROPN
ma-58	159	52	t	t	PROPN
ma-58	159	53	∗	∗	NOUN
ma-58	159	54	i	i	PRON
ma-58	159	55	tikξ	tikξ	VERB
ma-58	159	56	.	.	PUNCT
ma-58	160	1	then	then	ADV
ma-58	160	2	k∗sk	k∗sk	PROPN
ma-58	160	3	=	=	SYM
ma-58	160	4	k∗	k∗	PROPN
ma-58	160	5	∑	∑	PUNCT
ma-58	160	6	i∈i	i∈i	ADJ
ma-58	160	7	t	t	PROPN
ma-58	160	8	∗i	∗i	NOUN
ma-58	160	9	tikξ	tikξ	NOUN
ma-58	161	1	=	=	PUNCT
ma-58	161	2	∑	∑	ADP
ma-58	161	3	i∈i	i∈i	ADJ
ma-58	161	4	k∗t	k∗t	PROPN
ma-58	161	5	∗i	∗i	NOUN
ma-58	161	6	tikξ	tikξ	NOUN
ma-58	161	7	.	.	PUNCT
ma-58	162	1	hence	hence	ADV
ma-58	162	2	s′	s′	ADJ
ma-58	162	3	=	=	PUNCT
ma-58	162	4	k∗sk	k∗sk	PROPN
ma-58	162	5	.	.	PUNCT
ma-58	162	6	�	�	PROPN
ma-58	162	7	corollary	corollary	ADJ
ma-58	162	8	3.8	3.8	NUM
ma-58	162	9	.	.	PUNCT
ma-58	163	1	let	let	VERB
ma-58	163	2	k	k	PROPN
ma-58	163	3	∈	∈	PROPN
ma-58	163	4	hom∗a(x	hom∗a(x	NOUN
ma-58	163	5	)	)	PUNCT
ma-58	163	6	and	and	CCONJ
ma-58	163	7	{	{	PUNCT
ma-58	163	8	ti}i∈i	ti}i∈i	NOUN
ma-58	163	9	be	be	AUX
ma-58	163	10	an	an	DET
ma-58	163	11	∗-operator	∗-operator	NOUN
ma-58	163	12	frame	frame	NOUN
ma-58	163	13	.	.	PUNCT
ma-58	164	1	then	then	ADV
ma-58	164	2	{	{	PUNCT
ma-58	164	3	tis−1k}i∈i	tis−1k}i∈i	PROPN
ma-58	164	4	is	be	AUX
ma-58	164	5	an	an	DET
ma-58	164	6	∗-k∗-operator	∗-k∗-operator	NOUN
ma-58	164	7	frame	frame	NOUN
ma-58	164	8	,	,	PUNCT
ma-58	164	9	where	where	SCONJ
ma-58	164	10	s	s	NOUN
ma-58	164	11	is	be	AUX
ma-58	164	12	the	the	DET
ma-58	164	13	frame	frame	NOUN
ma-58	164	14	operator	operator	NOUN
ma-58	164	15	of	of	ADP
ma-58	164	16	{	{	PUNCT
ma-58	164	17	ti}i∈i	ti}i∈i	NOUN
ma-58	164	18	.	.	PUNCT
ma-58	165	1	proof	proof	NOUN
ma-58	165	2	.	.	PUNCT
ma-58	166	1	result	result	NOUN
ma-58	166	2	of	of	ADP
ma-58	166	3	the	the	DET
ma-58	166	4	theorem	theorem	NOUN
ma-58	166	5	3.7	3.7	NUM
ma-58	166	6	for	for	ADP
ma-58	166	7	the	the	DET
ma-58	166	8	∗-operator	∗-operator	NOUN
ma-58	166	9	frame	frame	NOUN
ma-58	166	10	{	{	PUNCT
ma-58	166	11	tis−1}i∈i	tis−1}i∈i	PROPN
ma-58	166	12	.	.	PUNCT
ma-58	167	1	�	�	PROPN
ma-58	167	2	4	4	NUM
ma-58	167	3	.	.	PUNCT
ma-58	167	4	tensor	tensor	NOUN
ma-58	167	5	product	product	NOUN
ma-58	167	6	we	we	PRON
ma-58	167	7	denote	denote	VERB
ma-58	167	8	by	by	ADP
ma-58	167	9	a⊗b	a⊗b	PROPN
ma-58	167	10	,	,	PUNCT
ma-58	167	11	the	the	DET
ma-58	167	12	minimal	minimal	ADJ
ma-58	167	13	or	or	CCONJ
ma-58	167	14	injective	injective	ADJ
ma-58	167	15	tensor	tensor	NOUN
ma-58	167	16	product	product	NOUN
ma-58	167	17	of	of	ADP
ma-58	167	18	the	the	DET
ma-58	167	19	pro	pro	ADJ
ma-58	167	20	-	-	ADJ
ma-58	167	21	c∗-algebras	c∗-algebra	NOUN
ma-58	167	22	a	a	PRON
ma-58	167	23	and	and	CCONJ
ma-58	167	24	b	b	NOUN
ma-58	167	25	,	,	PUNCT
ma-58	167	26	itis	itis	VERB
ma-58	167	27	the	the	DET
ma-58	167	28	completion	completion	NOUN
ma-58	167	29	of	of	ADP
ma-58	167	30	the	the	DET
ma-58	167	31	algebraic	algebraic	ADJ
ma-58	167	32	tensor	tensor	NOUN
ma-58	167	33	product	product	NOUN
ma-58	167	34	a⊗alg	a⊗alg	PROPN
ma-58	168	1	b	b	PROPN
ma-58	168	2	with	with	ADP
ma-58	168	3	respect	respect	NOUN
ma-58	168	4	to	to	ADP
ma-58	168	5	the	the	DET
ma-58	168	6	topology	topology	NOUN
ma-58	168	7	determinedby	determinedby	VERB
ma-58	168	8	a	a	DET
ma-58	168	9	family	family	NOUN
ma-58	168	10	of	of	ADP
ma-58	168	11	c∗-seminorms	c∗-seminorm	NOUN
ma-58	168	12	.	.	PUNCT
ma-58	168	13	suppose	suppose	VERB
ma-58	168	14	that	that	SCONJ
ma-58	168	15	x	x	PRON
ma-58	168	16	is	be	AUX
ma-58	168	17	a	a	DET
ma-58	168	18	hilbert	hilbert	NOUN
ma-58	168	19	module	module	NOUN
ma-58	168	20	over	over	ADP
ma-58	168	21	a	a	DET
ma-58	168	22	pro	pro	ADJ
ma-58	168	23	-	-	ADJ
ma-58	168	24	c∗-algebra	c∗-algebra	ADJ
ma-58	168	25	a	a	PRON
ma-58	168	26	and	and	CCONJ
ma-58	168	27	y	y	PROPN
ma-58	168	28	is	be	AUX
ma-58	168	29	a	a	DET
ma-58	168	30	hilbert	hilbert	NOUN
ma-58	168	31	module	module	NOUN
ma-58	168	32	over	over	ADP
ma-58	168	33	a	a	DET
ma-58	168	34	pro	pro	ADJ
ma-58	168	35	-	-	ADJ
ma-58	168	36	c∗-algebra	c∗-algebra	ADJ
ma-58	168	37	b.	b.	NOUN
ma-58	168	38	the	the	DET
ma-58	168	39	algebraic	algebraic	ADJ
ma-58	168	40	tensor	tensor	NOUN
ma-58	168	41	product	product	NOUN
ma-58	168	42	x	x	PUNCT
ma-58	168	43	⊗alg	⊗alg	PROPN
ma-58	168	44	y	y	PROPN
ma-58	168	45	of	of	ADP
ma-58	168	46	x	x	PUNCT
ma-58	168	47	and	and	CCONJ
ma-58	168	48	y	y	PROPN
ma-58	168	49	is	be	AUX
ma-58	168	50	a	a	DET
ma-58	168	51	pre	pre	ADJ
ma-58	168	52	-	-	ADJ
ma-58	168	53	hilbert	hilbert	ADJ
ma-58	168	54	a⊗	a⊗	NOUN
ma-58	168	55	b	b	NOUN
ma-58	168	56	-	-	PUNCT
ma-58	168	57	module	module	NOUN
ma-58	168	58	with	with	ADP
ma-58	168	59	the	the	DET
ma-58	168	60	action	action	NOUN
ma-58	168	61	of	of	ADP
ma-58	168	62	a⊗	a⊗	PROPN
ma-58	168	63	b	b	PROPN
ma-58	168	64	on	on	ADP
ma-58	168	65	x	x	SYM
ma-58	168	66	⊗alg	⊗alg	PROPN
ma-58	168	67	y	y	PROPN
ma-58	168	68	defined	define	VERB
ma-58	168	69	by	by	ADP
ma-58	168	70	(	(	PUNCT
ma-58	168	71	ξ	ξ	PROPN
ma-58	168	72	⊗	⊗	PROPN
ma-58	168	73	η)(a	η)(a	PROPN
ma-58	168	74	⊗	⊗	PROPN
ma-58	168	75	b	b	NOUN
ma-58	168	76	)	)	PUNCT
ma-58	168	77	=	=	SYM
ma-58	169	1	ξa	ξa	PROPN
ma-58	169	2	⊗	⊗	PROPN
ma-58	169	3	ηb	ηb	PROPN
ma-58	169	4	for	for	ADP
ma-58	169	5	all	all	DET
ma-58	169	6	ξ	ξ	X
ma-58	169	7	∈	∈	PROPN
ma-58	169	8	x	x	X
ma-58	169	9	,	,	PUNCT
ma-58	169	10	η	η	PROPN
ma-58	169	11	∈	∈	PROPN
ma-58	169	12	y	y	PROPN
ma-58	169	13	,	,	PUNCT
ma-58	169	14	a	a	DET
ma-58	169	15	∈	∈	PROPN
ma-58	169	16	a	a	PRON
ma-58	169	17	and	and	CCONJ
ma-58	169	18	b	b	PROPN
ma-58	169	19	∈	∈	PROPN
ma-58	169	20	b	b	PROPN
ma-58	169	21	and	and	CCONJ
ma-58	169	22	the	the	DET
ma-58	169	23	inner	inner	ADJ
ma-58	169	24	product	product	NOUN
ma-58	169	25	〈	〈	PROPN
ma-58	169	26	·	·	SYM
ma-58	169	27	,	,	PUNCT
ma-58	169	28	·	·	PUNCT
ma-58	169	29	〉	〉	NOUN
ma-58	169	30	:	:	PUNCT
ma-58	169	31	(	(	PUNCT
ma-58	169	32	x	x	X
ma-58	169	33	⊗alg	⊗alg	NOUN
ma-58	169	34	y)×	y)×	NOUN
ma-58	169	35	(	(	PUNCT
ma-58	169	36	x	x	SYM
ma-58	169	37	⊗alg	⊗alg	PROPN
ma-58	169	38	y)→	y)→	PROPN
ma-58	169	39	a⊗alg	a⊗alg	PROPN
ma-58	169	40	b.	b.	PROPN
ma-58	170	1	defined	define	VERB
ma-58	170	2	by	by	ADP
ma-58	170	3	〈	〈	PROPN
ma-58	170	4	ξ1	ξ1	PROPN
ma-58	170	5	⊗	⊗	PROPN
ma-58	170	6	η1	η1	NOUN
ma-58	170	7	,	,	PUNCT
ma-58	170	8	ξ2	ξ2	PROPN
ma-58	170	9	⊗	⊗	PROPN
ma-58	170	10	η2	η2	PROPN
ma-58	170	11	〉	〉	NOUN
ma-58	170	12	=	=	SYM
ma-58	170	13	〈	〈	PROPN
ma-58	170	14	ξ1	ξ1	NOUN
ma-58	170	15	,	,	PUNCT
ma-58	170	16	ξ2	ξ2	NOUN
ma-58	170	17	〉	〉	PROPN
ma-58	170	18	⊗	⊗	PROPN
ma-58	170	19	〈	〈	PROPN
ma-58	170	20	η1	η1	NOUN
ma-58	170	21	,	,	PUNCT
ma-58	170	22	η2〉and	η2〉and	NOUN
ma-58	170	23	we	we	PRON
ma-58	170	24	know	know	VERB
ma-58	170	25	that	that	PRON
ma-58	170	26	for	for	ADP
ma-58	170	27	z	z	NOUN
ma-58	170	28	=	=	SYM
ma-58	171	1	∑n	∑n	PROPN
ma-58	171	2	i=1	i=1	PROPN
ma-58	171	3	ξi⊗ηi	ξi⊗ηi	NOUN
ma-58	171	4	in	in	ADP
ma-58	171	5	x⊗algy	x⊗algy	NOUN
ma-58	171	6	we	we	PRON
ma-58	171	7	have	have	VERB
ma-58	171	8	〈	〈	PROPN
ma-58	171	9	z	z	PROPN
ma-58	171	10	,	,	PUNCT
ma-58	171	11	z〉a⊗b	z〉a⊗b	X
ma-58	171	12	=	=	PUNCT
ma-58	171	13	∑	∑	PUNCT
ma-58	172	1	i	i	PRON
ma-58	172	2	,	,	PUNCT
ma-58	172	3	j〈ξi	j〈ξi	PROPN
ma-58	172	4	,	,	PUNCT
ma-58	172	5	ξj〉a⊗〈ηi	ξj〉a⊗〈ηi	PROPN
ma-58	172	6	,	,	PUNCT
ma-58	172	7	ηj〉b	ηj〉b	PROPN
ma-58	172	8	≥	≥	NUM
ma-58	172	9	0and	0and	PROPN
ma-58	172	10	〈	〈	PROPN
ma-58	172	11	z	z	PROPN
ma-58	172	12	,	,	PUNCT
ma-58	172	13	z〉a⊗b	z〉a⊗b	X
ma-58	172	14	=	=	SYM
ma-58	172	15	0	0	PROPN
ma-58	172	16	iff	iff	PROPN
ma-58	172	17	z	z	PROPN
ma-58	172	18	=	=	SYM
ma-58	172	19	0	0	PROPN
ma-58	172	20	.	.	PUNCT
ma-58	173	1	https://doi.org/10.28924/ada/ma.2.4	https://doi.org/10.28924/ada/ma.2.4	NUM
ma-58	173	2	eur	eur	NOUN
ma-58	173	3	.	.	PUNCT
ma-58	174	1	j.	j.	PROPN
ma-58	174	2	math	math	PROPN
ma-58	174	3	.	.	PUNCT
ma-58	175	1	anal	anal	PROPN
ma-58	175	2	.	.	PUNCT
ma-58	176	1	10.28924	10.28924	NUM
ma-58	176	2	/	/	SYM
ma-58	176	3	ada	ada	PROPN
ma-58	176	4	/	/	SYM
ma-58	176	5	ma.2.4	ma.2.4	PROPN
ma-58	176	6	8the	8the	DET
ma-58	176	7	external	external	ADJ
ma-58	176	8	tensor	tensor	NOUN
ma-58	176	9	product	product	NOUN
ma-58	176	10	of	of	ADP
ma-58	176	11	x	x	PROPN
ma-58	176	12	and	and	CCONJ
ma-58	176	13	y	y	PROPN
ma-58	176	14	is	be	AUX
ma-58	176	15	the	the	DET
ma-58	176	16	hilbert	hilbert	NOUN
ma-58	176	17	module	module	NOUN
ma-58	176	18	x	x	PUNCT
ma-58	176	19	⊗y	⊗y	NOUN
ma-58	176	20	over	over	ADP
ma-58	176	21	a⊗b	a⊗b	PROPN
ma-58	176	22	obtained	obtain	VERB
ma-58	176	23	by	by	ADP
ma-58	176	24	thecompletion	thecompletion	NOUN
ma-58	176	25	of	of	ADP
ma-58	176	26	the	the	DET
ma-58	176	27	pre	pre	NOUN
ma-58	176	28	-	-	NOUN
ma-58	176	29	hilbert	hilbert	ADJ
ma-58	176	30	a⊗	a⊗	NOUN
ma-58	176	31	b	b	NOUN
ma-58	176	32	-	-	PUNCT
ma-58	176	33	module	module	NOUN
ma-58	176	34	x	x	SYM
ma-58	176	35	⊗alg	⊗alg	NOUN
ma-58	177	1	y	y	PROPN
ma-58	177	2	.if	.if	PUNCT
ma-58	178	1	p	p	PRON
ma-58	178	2	∈	∈	PROPN
ma-58	178	3	m(x	m(x	PROPN
ma-58	178	4	)	)	PUNCT
ma-58	178	5	and	and	CCONJ
ma-58	178	6	q	q	PROPN
ma-58	178	7	∈	∈	PROPN
ma-58	178	8	m(y	m(y	NOUN
ma-58	178	9	)	)	PUNCT
ma-58	179	1	then	then	ADV
ma-58	179	2	there	there	PRON
ma-58	179	3	is	be	VERB
ma-58	179	4	a	a	DET
ma-58	179	5	unique	unique	ADJ
ma-58	179	6	adjointable	adjointable	NOUN
ma-58	179	7	module	module	NOUN
ma-58	179	8	morphism	morphism	NOUN
ma-58	179	9	p	p	PROPN
ma-58	179	10	⊗	⊗	PROPN
ma-58	179	11	q	q	NOUN
ma-58	179	12	:	:	PUNCT
ma-58	179	13	a⊗b	a⊗b	X
ma-58	179	14	→	→	SYM
ma-58	179	15	x	x	SYM
ma-58	179	16	⊗y	⊗y	NOUN
ma-58	179	17	such	such	ADJ
ma-58	179	18	that	that	SCONJ
ma-58	179	19	(	(	PUNCT
ma-58	179	20	p	p	X
ma-58	179	21	⊗q)(a⊗	⊗q)(a⊗	PROPN
ma-58	179	22	b	b	NOUN
ma-58	179	23	)	)	PUNCT
ma-58	179	24	=	=	SYM
ma-58	180	1	p	p	X
ma-58	180	2	(	(	PUNCT
ma-58	180	3	a)⊗q(b	a)⊗q(b	PROPN
ma-58	180	4	)	)	PUNCT
ma-58	180	5	and	and	CCONJ
ma-58	180	6	(	(	PUNCT
ma-58	180	7	p	p	X
ma-58	180	8	⊗q)∗(a⊗	⊗q)∗(a⊗	PROPN
ma-58	180	9	b	b	PROPN
ma-58	180	10	)	)	PUNCT
ma-58	180	11	=	=	SYM
ma-58	181	1	p	p	PROPN
ma-58	181	2	∗(a)⊗q∗(b)for	∗(a)⊗q∗(b)for	ADP
ma-58	181	3	all	all	DET
ma-58	181	4	a	a	DET
ma-58	181	5	∈	∈	NOUN
ma-58	181	6	a	a	PRON
ma-58	181	7	and	and	CCONJ
ma-58	181	8	for	for	ADP
ma-58	181	9	all	all	DET
ma-58	181	10	b	b	NOUN
ma-58	181	11	∈	∈	ADP
ma-58	181	12	b	b	PROPN
ma-58	181	13	(	(	PUNCT
ma-58	181	14	see	see	VERB
ma-58	181	15	,	,	PUNCT
ma-58	181	16	for	for	ADP
ma-58	181	17	example	example	NOUN
ma-58	181	18	,	,	PUNCT
ma-58	181	19	cite	cite	VERB
ma-58	181	20	the	the	DET
ma-58	181	21	minimal	minimal	ADJ
ma-58	181	22	or	or	CCONJ
ma-58	181	23	injective	injective	ADJ
ma-58	181	24	tensor	tensor	NOUN
ma-58	181	25	product	product	NOUN
ma-58	181	26	ofthe	ofthe	PRON
ma-58	181	27	pro	pro	ADJ
ma-58	181	28	-	-	NOUN
ma-58	181	29	c∗-algebras	c∗-algebra	NOUN
ma-58	181	30	a	a	PRON
ma-58	181	31	and	and	CCONJ
ma-58	181	32	b	b	NOUN
ma-58	181	33	,	,	PUNCT
ma-58	181	34	denoted	denote	VERB
ma-58	181	35	by	by	ADP
ma-58	181	36	a⊗b	a⊗b	PROPN
ma-58	181	37	,	,	PUNCT
ma-58	181	38	is	be	AUX
ma-58	181	39	the	the	DET
ma-58	181	40	completion	completion	NOUN
ma-58	181	41	of	of	ADP
ma-58	181	42	the	the	DET
ma-58	181	43	algebraic	algebraic	ADJ
ma-58	181	44	tensor	tensor	NOUN
ma-58	181	45	product	product	NOUN
ma-58	181	46	a⊗alg	a⊗alg	PROPN
ma-58	181	47	b	b	PROPN
ma-58	181	48	with	with	ADP
ma-58	181	49	respect	respect	NOUN
ma-58	181	50	to	to	ADP
ma-58	181	51	the	the	DET
ma-58	181	52	topology	topology	NOUN
ma-58	181	53	determined	determine	VERB
ma-58	181	54	by	by	ADP
ma-58	181	55	a	a	DET
ma-58	181	56	family	family	NOUN
ma-58	181	57	of	of	ADP
ma-58	181	58	c∗-seminorms	c∗-seminorm	NOUN
ma-58	181	59	.	.	PUNCT
ma-58	182	1	suppose	suppose	VERB
ma-58	182	2	that	that	SCONJ
ma-58	182	3	xis	xis	PROPN
ma-58	182	4	a	a	DET
ma-58	182	5	hilbert	hilbert	NOUN
ma-58	182	6	module	module	NOUN
ma-58	182	7	over	over	ADP
ma-58	182	8	a	a	DET
ma-58	182	9	pro	pro	ADJ
ma-58	182	10	-	-	ADJ
ma-58	182	11	c∗-algebra	c∗-algebra	ADJ
ma-58	182	12	a	a	PRON
ma-58	182	13	and	and	CCONJ
ma-58	182	14	y	y	PROPN
ma-58	182	15	is	be	AUX
ma-58	182	16	a	a	DET
ma-58	182	17	hilbert	hilbert	NOUN
ma-58	182	18	module	module	NOUN
ma-58	182	19	over	over	ADP
ma-58	182	20	a	a	DET
ma-58	182	21	pro	pro	ADJ
ma-58	182	22	-	-	ADJ
ma-58	182	23	c∗-algebra	c∗-algebra	ADJ
ma-58	182	24	b.the	b.the	DET
ma-58	182	25	algebraic	algebraic	ADJ
ma-58	182	26	tensor	tensor	NOUN
ma-58	182	27	product	product	NOUN
ma-58	182	28	x	x	PUNCT
ma-58	182	29	⊗alg	⊗alg	PROPN
ma-58	182	30	y	y	PROPN
ma-58	182	31	of	of	ADP
ma-58	182	32	x	x	PUNCT
ma-58	182	33	and	and	CCONJ
ma-58	182	34	y	y	PROPN
ma-58	182	35	is	be	AUX
ma-58	182	36	a	a	DET
ma-58	182	37	pre	pre	ADJ
ma-58	182	38	-	-	ADJ
ma-58	182	39	hilbert	hilbert	ADJ
ma-58	182	40	a⊗b	a⊗b	NOUN
ma-58	182	41	-	-	PUNCT
ma-58	182	42	module	module	NOUN
ma-58	182	43	with	with	ADP
ma-58	182	44	the	the	DET
ma-58	182	45	actionof	actionof	NOUN
ma-58	182	46	a⊗	a⊗	PROPN
ma-58	182	47	b	b	PROPN
ma-58	182	48	on	on	ADP
ma-58	182	49	x	x	SYM
ma-58	182	50	⊗alg	⊗alg	PROPN
ma-58	182	51	y	y	PROPN
ma-58	182	52	defined	define	VERB
ma-58	182	53	by	by	ADP
ma-58	182	54	(	(	PUNCT
ma-58	182	55	ξ	ξ	PROPN
ma-58	182	56	⊗	⊗	PROPN
ma-58	182	57	η)(a	η)(a	PROPN
ma-58	182	58	⊗	⊗	PROPN
ma-58	182	59	b	b	NOUN
ma-58	182	60	)	)	PUNCT
ma-58	182	61	=	=	SYM
ma-58	183	1	ξa	ξa	PROPN
ma-58	183	2	⊗	⊗	PROPN
ma-58	183	3	ηb	ηb	PROPN
ma-58	183	4	for	for	ADP
ma-58	183	5	all	all	DET
ma-58	183	6	ξ	ξ	X
ma-58	183	7	∈	∈	PROPN
ma-58	183	8	x	x	X
ma-58	183	9	,	,	PUNCT
ma-58	183	10	η	η	PROPN
ma-58	183	11	∈	∈	PROPN
ma-58	183	12	y	y	PROPN
ma-58	183	13	,	,	PUNCT
ma-58	183	14	a	a	DET
ma-58	183	15	∈	∈	PROPN
ma-58	183	16	a	a	PRON
ma-58	183	17	and	and	CCONJ
ma-58	183	18	b	b	PROPN
ma-58	183	19	∈	∈	PROPN
ma-58	183	20	b	b	PROPN
ma-58	183	21	and	and	CCONJ
ma-58	183	22	the	the	DET
ma-58	183	23	inner	inner	ADJ
ma-58	183	24	product	product	NOUN
ma-58	183	25	〈	〈	PROPN
ma-58	183	26	·	·	SYM
ma-58	183	27	,	,	PUNCT
ma-58	183	28	·	·	PUNCT
ma-58	183	29	〉	〉	NOUN
ma-58	183	30	:	:	PUNCT
ma-58	183	31	(	(	PUNCT
ma-58	183	32	x	x	X
ma-58	183	33	⊗alg	⊗alg	NOUN
ma-58	183	34	y)×	y)×	NOUN
ma-58	183	35	(	(	PUNCT
ma-58	183	36	x	x	SYM
ma-58	183	37	⊗alg	⊗alg	PROPN
ma-58	183	38	y)→	y)→	PROPN
ma-58	183	39	a⊗alg	a⊗alg	PROPN
ma-58	183	40	b.	b.	PROPN
ma-58	184	1	defined	define	VERB
ma-58	184	2	by	by	ADP
ma-58	184	3	〈	〈	PROPN
ma-58	184	4	ξ1	ξ1	PROPN
ma-58	184	5	⊗	⊗	PROPN
ma-58	184	6	η1	η1	NOUN
ma-58	184	7	,	,	PUNCT
ma-58	184	8	ξ2	ξ2	PROPN
ma-58	184	9	⊗	⊗	PROPN
ma-58	184	10	η2	η2	PROPN
ma-58	184	11	〉	〉	NOUN
ma-58	184	12	=	=	SYM
ma-58	184	13	〈	〈	PROPN
ma-58	184	14	ξ1	ξ1	NOUN
ma-58	184	15	,	,	PUNCT
ma-58	184	16	ξ2	ξ2	NOUN
ma-58	184	17	〉	〉	PROPN
ma-58	184	18	⊗	⊗	PROPN
ma-58	184	19	〈	〈	PROPN
ma-58	184	20	η1	η1	NOUN
ma-58	184	21	,	,	PUNCT
ma-58	184	22	η2	η2	PROPN
ma-58	184	23	〉	〉	NOUN
ma-58	184	24	we	we	PRON
ma-58	184	25	also	also	ADV
ma-58	184	26	know	know	VERB
ma-58	184	27	that	that	SCONJ
ma-58	184	28	for	for	ADP
ma-58	184	29	z	z	NOUN
ma-58	185	1	=	=	SYM
ma-58	185	2	∑n	∑n	PROPN
ma-58	185	3	i=1	i=1	PROPN
ma-58	185	4	ξi⊗ηi	ξi⊗ηi	NOUN
ma-58	185	5	in	in	ADP
ma-58	185	6	x⊗algy	x⊗algy	NOUN
ma-58	185	7	we	we	PRON
ma-58	185	8	have	have	VERB
ma-58	185	9	〈	〈	PROPN
ma-58	185	10	z	z	PROPN
ma-58	185	11	,	,	PUNCT
ma-58	185	12	z〉a⊗b	z〉a⊗b	X
ma-58	185	13	=	=	PUNCT
ma-58	185	14	∑	∑	PUNCT
ma-58	186	1	i	i	PRON
ma-58	186	2	,	,	PUNCT
ma-58	186	3	j〈ξi	j〈ξi	PROPN
ma-58	186	4	,	,	PUNCT
ma-58	186	5	ξj〉a⊗〈ηi	ξj〉a⊗〈ηi	PROPN
ma-58	186	6	,	,	PUNCT
ma-58	186	7	ηj〉b	ηj〉b	PROPN
ma-58	186	8	≥	≥	NUM
ma-58	186	9	0and	0and	PROPN
ma-58	186	10	〈	〈	PROPN
ma-58	186	11	z	z	PROPN
ma-58	186	12	,	,	PUNCT
ma-58	186	13	z〉a⊗b	z〉a⊗b	X
ma-58	186	14	=	=	SYM
ma-58	186	15	0	0	PROPN
ma-58	187	1	iff	iff	PROPN
ma-58	187	2	z	z	PROPN
ma-58	187	3	=	=	PUNCT
ma-58	188	1	0.the	0.the	DET
ma-58	188	2	external	external	ADJ
ma-58	188	3	tensor	tensor	NOUN
ma-58	188	4	product	product	NOUN
ma-58	188	5	of	of	ADP
ma-58	188	6	x	x	PROPN
ma-58	188	7	and	and	CCONJ
ma-58	188	8	y	y	PROPN
ma-58	188	9	is	be	AUX
ma-58	188	10	the	the	DET
ma-58	188	11	hilbert	hilbert	NOUN
ma-58	188	12	module	module	NOUN
ma-58	188	13	x	x	PUNCT
ma-58	188	14	⊗y	⊗y	NOUN
ma-58	188	15	over	over	ADP
ma-58	188	16	a⊗b	a⊗b	PROPN
ma-58	188	17	obtained	obtain	VERB
ma-58	188	18	by	by	ADP
ma-58	188	19	thecompletion	thecompletion	NOUN
ma-58	188	20	of	of	ADP
ma-58	188	21	the	the	DET
ma-58	188	22	pre	pre	NOUN
ma-58	188	23	-	-	NOUN
ma-58	188	24	hilbert	hilbert	ADJ
ma-58	188	25	a⊗	a⊗	NOUN
ma-58	188	26	b	b	NOUN
ma-58	188	27	-	-	PUNCT
ma-58	188	28	module	module	NOUN
ma-58	188	29	x	x	SYM
ma-58	188	30	⊗alg	⊗alg	NOUN
ma-58	188	31	y	y	PROPN
ma-58	188	32	.if	.if	PUNCT
ma-58	189	1	p	p	PRON
ma-58	189	2	∈	∈	PROPN
ma-58	189	3	m(x	m(x	PROPN
ma-58	189	4	)	)	PUNCT
ma-58	189	5	and	and	CCONJ
ma-58	189	6	q	q	PROPN
ma-58	189	7	∈	∈	PROPN
ma-58	189	8	m(y	m(y	NOUN
ma-58	189	9	)	)	PUNCT
ma-58	190	1	then	then	ADV
ma-58	190	2	there	there	PRON
ma-58	190	3	is	be	VERB
ma-58	190	4	a	a	DET
ma-58	190	5	unique	unique	ADJ
ma-58	190	6	adjointable	adjointable	NOUN
ma-58	190	7	module	module	NOUN
ma-58	190	8	morphism	morphism	NOUN
ma-58	190	9	p	p	PROPN
ma-58	190	10	⊗	⊗	PROPN
ma-58	190	11	q	q	NOUN
ma-58	190	12	:	:	PUNCT
ma-58	190	13	a⊗b	a⊗b	X
ma-58	190	14	→	→	SYM
ma-58	190	15	x	x	SYM
ma-58	190	16	⊗y	⊗y	NOUN
ma-58	190	17	such	such	ADJ
ma-58	190	18	that	that	SCONJ
ma-58	190	19	(	(	PUNCT
ma-58	190	20	p	p	X
ma-58	190	21	⊗q)(a⊗	⊗q)(a⊗	PROPN
ma-58	190	22	b	b	NOUN
ma-58	190	23	)	)	PUNCT
ma-58	190	24	=	=	SYM
ma-58	191	1	p	p	X
ma-58	191	2	(	(	PUNCT
ma-58	191	3	a)⊗q(b	a)⊗q(b	PROPN
ma-58	191	4	)	)	PUNCT
ma-58	191	5	and	and	CCONJ
ma-58	191	6	(	(	PUNCT
ma-58	191	7	p	p	X
ma-58	191	8	⊗q)∗(a⊗	⊗q)∗(a⊗	PROPN
ma-58	191	9	b	b	PROPN
ma-58	191	10	)	)	PUNCT
ma-58	191	11	=	=	SYM
ma-58	192	1	p	p	PROPN
ma-58	192	2	∗(a)⊗q∗(b)for	∗(a)⊗q∗(b)for	ADP
ma-58	192	3	all	all	DET
ma-58	192	4	a	a	DET
ma-58	192	5	∈	∈	NOUN
ma-58	192	6	a	a	PRON
ma-58	192	7	and	and	CCONJ
ma-58	192	8	for	for	ADP
ma-58	192	9	all	all	DET
ma-58	192	10	b	b	NOUN
ma-58	192	11	∈	∈	ADP
ma-58	192	12	b	b	PROPN
ma-58	192	13	(	(	PUNCT
ma-58	192	14	see	see	VERB
ma-58	192	15	,	,	PUNCT
ma-58	192	16	for	for	ADP
ma-58	192	17	example	example	NOUN
ma-58	192	18	,	,	PUNCT
ma-58	192	19	[	[	X
ma-58	192	20	9])let	9])let	X
ma-58	192	21	i	i	PRON
ma-58	192	22	and	and	CCONJ
ma-58	192	23	j	j	PROPN
ma-58	192	24	be	be	AUX
ma-58	192	25	countable	countable	ADJ
ma-58	192	26	index	index	NOUN
ma-58	192	27	sets	set	NOUN
ma-58	192	28	.	.	PUNCT
ma-58	193	1	theorem	theorem	VERB
ma-58	193	2	4.1	4.1	NUM
ma-58	193	3	.	.	PUNCT
ma-58	194	1	let	let	VERB
ma-58	194	2	x	x	PRON
ma-58	194	3	and	and	CCONJ
ma-58	194	4	y	y	PROPN
ma-58	194	5	be	be	VERB
ma-58	194	6	two	two	NUM
ma-58	194	7	hilbert	hilbert	NOUN
ma-58	194	8	pro	pro	ADJ
ma-58	194	9	-	-	NOUN
ma-58	194	10	c∗-modules	c∗-module	NOUN
ma-58	194	11	over	over	ADP
ma-58	194	12	unitary	unitary	ADJ
ma-58	194	13	pro	pro	ADJ
ma-58	194	14	-	-	ADJ
ma-58	194	15	c∗-algebras	c∗-algebra	NOUN
ma-58	194	16	a	a	PRON
ma-58	194	17	and	and	CCONJ
ma-58	194	18	b	b	NOUN
ma-58	194	19	,	,	PUNCT
ma-58	194	20	respectively	respectively	ADV
ma-58	194	21	.	.	PUNCT
ma-58	195	1	let	let	AUX
ma-58	195	2	{	{	PUNCT
ma-58	195	3	ti}i∈i	ti}i∈i	NOUN
ma-58	195	4	⊂	⊂	PROPN
ma-58	195	5	hom∗a(x	hom∗a(x	NOUN
ma-58	195	6	)	)	PUNCT
ma-58	195	7	be	be	AUX
ma-58	195	8	an	an	DET
ma-58	195	9	∗-k	∗-k	ADJ
ma-58	195	10	-	-	PUNCT
ma-58	195	11	operator	operator	NOUN
ma-58	195	12	frame	frame	NOUN
ma-58	195	13	for	for	ADP
ma-58	195	14	x	x	PUNCT
ma-58	195	15	with	with	ADP
ma-58	195	16	bounds	bound	NOUN
ma-58	195	17	a	a	DET
ma-58	195	18	and	and	CCONJ
ma-58	195	19	b	b	NOUN
ma-58	195	20	and	and	CCONJ
ma-58	195	21	frame	frame	NOUN
ma-58	195	22	operators	operator	NOUN
ma-58	195	23	st	st	PROPN
ma-58	195	24	and	and	CCONJ
ma-58	195	25	{	{	PUNCT
ma-58	195	26	pj}j∈j	pj}j∈j	X
ma-58	195	27	⊂	⊂	X
ma-58	195	28	hom∗b(y	hom∗b(y	PROPN
ma-58	195	29	)	)	PUNCT
ma-58	195	30	be	be	VERB
ma-58	195	31	an	an	DET
ma-58	195	32	∗-l	∗-l	ADJ
ma-58	195	33	-	-	PUNCT
ma-58	195	34	operator	operator	NOUN
ma-58	195	35	frame	frame	NOUN
ma-58	195	36	for	for	ADP
ma-58	195	37	k	k	PROPN
ma-58	195	38	with	with	ADP
ma-58	195	39	bounds	bound	NOUN
ma-58	195	40	c	c	PROPN
ma-58	195	41	and	and	CCONJ
ma-58	195	42	d	d	NOUN
ma-58	195	43	and	and	CCONJ
ma-58	195	44	frame	frame	NOUN
ma-58	195	45	operators	operator	NOUN
ma-58	195	46	sl	sl	VERB
ma-58	195	47	.	.	PUNCT
ma-58	196	1	then	then	ADV
ma-58	196	2	{	{	PUNCT
ma-58	196	3	ti	ti	PROPN
ma-58	196	4	⊗	⊗	PROPN
ma-58	196	5	lj}i∈i	lj}i∈i	NOUN
ma-58	196	6	,	,	PUNCT
ma-58	196	7	j∈j	j∈j	NOUN
ma-58	196	8	is	be	AUX
ma-58	196	9	an	an	DET
ma-58	196	10	∗-k⊗l	∗-k⊗l	NOUN
ma-58	196	11	-	-	PUNCT
ma-58	196	12	operator	operator	NOUN
ma-58	196	13	frame	frame	NOUN
ma-58	196	14	for	for	ADP
ma-58	196	15	hibert	hibert	NOUN
ma-58	196	16	a⊗	a⊗	PROPN
ma-58	196	17	b	b	NOUN
ma-58	196	18	-	-	PUNCT
ma-58	196	19	module	module	NOUN
ma-58	196	20	x	x	PUNCT
ma-58	196	21	⊗	⊗	PROPN
ma-58	196	22	y	y	PROPN
ma-58	196	23	with	with	ADP
ma-58	196	24	frame	frame	NOUN
ma-58	196	25	operator	operator	NOUN
ma-58	196	26	st	st	PROPN
ma-58	196	27	⊗	⊗	PROPN
ma-58	196	28	sp	sp	ADP
ma-58	196	29	and	and	CCONJ
ma-58	196	30	bounds	bound	VERB
ma-58	196	31	a⊗	a⊗	PROPN
ma-58	196	32	c	c	PROPN
ma-58	196	33	and	and	CCONJ
ma-58	196	34	b	b	PROPN
ma-58	196	35	⊗d	⊗d	NOUN
ma-58	196	36	.	.	PUNCT
ma-58	197	1	proof	proof	NOUN
ma-58	197	2	.	.	PUNCT
ma-58	198	1	the	the	DET
ma-58	198	2	defintion	defintion	NOUN
ma-58	198	3	of	of	ADP
ma-58	198	4	∗-k	∗-k	NOUN
ma-58	198	5	-	-	PUNCT
ma-58	198	6	operator	operator	NOUN
ma-58	198	7	frame	frame	NOUN
ma-58	198	8	{	{	PUNCT
ma-58	198	9	ti}i∈i	ti}i∈i	SYM
ma-58	198	10	and	and	CCONJ
ma-58	198	11	∗-l	∗-l	ADJ
ma-58	198	12	-	-	PUNCT
ma-58	198	13	operator	operator	NOUN
ma-58	198	14	frame	frame	NOUN
ma-58	198	15	{	{	PUNCT
ma-58	198	16	pj}j∈j	pj}j∈j	PRON
ma-58	198	17	gives	give	VERB
ma-58	198	18	a〈k∗ξ	a〈k∗ξ	NOUN
ma-58	198	19	,	,	PUNCT
ma-58	198	20	k∗ξ〉aa∗	k∗ξ〉aa∗	PROPN
ma-58	198	21	≤	≤	PROPN
ma-58	198	22	∑	∑	PUNCT
ma-58	198	23	i∈i	i∈i	ADJ
ma-58	198	24	〈	〈	PROPN
ma-58	198	25	tiξ	tiξ	PROPN
ma-58	198	26	,	,	PUNCT
ma-58	198	27	tiξ〉a	tiξ〉a	NOUN
ma-58	198	28	≤	≤	NUM
ma-58	198	29	b〈ξ	b〈ξ	NUM
ma-58	198	30	,	,	PUNCT
ma-58	198	31	ξ〉ab∗,∀ξ	ξ〉ab∗,∀ξ	PROPN
ma-58	198	32	∈	∈	PROPN
ma-58	198	33	x	x	X
ma-58	198	34	.	.	PUNCT
ma-58	199	1	c〈l∗η	c〈l∗η	PROPN
ma-58	199	2	,	,	PUNCT
ma-58	199	3	l∗η〉bc∗	l∗η〉bc∗	NOUN
ma-58	199	4	≤	≤	NOUN
ma-58	199	5	∑	∑	PUNCT
ma-58	199	6	j∈j	j∈j	NOUN
ma-58	199	7	〈	〈	PROPN
ma-58	199	8	pjη	pjη	PROPN
ma-58	199	9	,	,	PUNCT
ma-58	199	10	pjη〉b	pjη〉b	VERB
ma-58	199	11	≤	≤	ADJ
ma-58	199	12	d〈η	d〈η	NOUN
ma-58	199	13	,	,	PUNCT
ma-58	199	14	η〉bd∗,∀η	η〉bd∗,∀η	PROPN
ma-58	199	15	∈	∈	PROPN
ma-58	199	16	y.	y.	PROPN
ma-58	199	17	https://doi.org/10.28924/ada/ma.2.4	https://doi.org/10.28924/ada/ma.2.4	PROPN
ma-58	199	18	eur	eur	NOUN
ma-58	199	19	.	.	PUNCT
ma-58	200	1	j.	j.	PROPN
ma-58	200	2	math	math	PROPN
ma-58	200	3	.	.	PUNCT
ma-58	201	1	anal	anal	PROPN
ma-58	201	2	.	.	PUNCT
ma-58	202	1	10.28924	10.28924	NUM
ma-58	202	2	/	/	SYM
ma-58	202	3	ada	ada	PROPN
ma-58	202	4	/	/	SYM
ma-58	202	5	ma.2.4	ma.2.4	PROPN
ma-58	202	6	9therefore	9therefore	NUM
ma-58	202	7	(	(	PUNCT
ma-58	202	8	a〈k∗ξ	a〈k∗ξ	NOUN
ma-58	202	9	,	,	PUNCT
ma-58	202	10	k∗ξ〉aa∗)⊗	k∗ξ〉aa∗)⊗	NOUN
ma-58	202	11	(	(	PUNCT
ma-58	202	12	c〈l∗η	c〈l∗η	PROPN
ma-58	202	13	,	,	PUNCT
ma-58	202	14	l∗η〉bc∗	l∗η〉bc∗	NOUN
ma-58	202	15	)	)	PUNCT
ma-58	202	16	≤	≤	NOUN
ma-58	202	17	∑	∑	PUNCT
ma-58	202	18	i∈i	i∈i	ADJ
ma-58	202	19	〈	〈	PROPN
ma-58	202	20	tiξ	tiξ	PROPN
ma-58	202	21	,	,	PUNCT
ma-58	202	22	tiξ〉a	tiξ〉a	NOUN
ma-58	202	23	⊗	⊗	PROPN
ma-58	202	24	∑	∑	PROPN
ma-58	202	25	j∈j	j∈j	PROPN
ma-58	202	26	〈	〈	PROPN
ma-58	202	27	pjη	pjη	PROPN
ma-58	202	28	,	,	PUNCT
ma-58	202	29	pjη〉b	pjη〉b	NOUN
ma-58	202	30	≤	≤	NUM
ma-58	202	31	(	(	PUNCT
ma-58	202	32	b〈ξ	b〈ξ	NUM
ma-58	202	33	,	,	PUNCT
ma-58	202	34	ξ〉ab∗)⊗	ξ〉ab∗)⊗	NOUN
ma-58	202	35	(	(	PUNCT
ma-58	202	36	d〈η	d〈η	NOUN
ma-58	202	37	,	,	PUNCT
ma-58	202	38	η〉bd∗),∀ξ	η〉bd∗),∀ξ	NOUN
ma-58	202	39	∈	∈	PROPN
ma-58	202	40	x	x	X
ma-58	202	41	,	,	PUNCT
ma-58	202	42	∀η	∀η	PROPN
ma-58	202	43	∈	∈	PROPN
ma-58	202	44	y.then	y.then	X
ma-58	202	45	(	(	PUNCT
ma-58	202	46	a⊗	a⊗	PROPN
ma-58	202	47	c)(〈k∗ξ	c)(〈k∗ξ	PROPN
ma-58	202	48	,	,	PUNCT
ma-58	202	49	k∗ξ〉a	k∗ξ〉a	VERB
ma-58	202	50	⊗	⊗	PROPN
ma-58	202	51	〈	〈	PROPN
ma-58	202	52	l∗η	l∗η	PROPN
ma-58	202	53	,	,	PUNCT
ma-58	202	54	l∗η〉b)(a∗	l∗η〉b)(a∗	ADJ
ma-58	202	55	⊗	⊗	PROPN
ma-58	202	56	c∗	c∗	PROPN
ma-58	202	57	)	)	PUNCT
ma-58	202	58	≤	≤	NOUN
ma-58	202	59	∑	∑	PUNCT
ma-58	202	60	i∈i	i∈i	ADJ
ma-58	202	61	,	,	PUNCT
ma-58	202	62	j∈j	j∈j	NOUN
ma-58	202	63	〈	〈	PROPN
ma-58	202	64	tiξ	tiξ	PROPN
ma-58	202	65	,	,	PUNCT
ma-58	202	66	tiξ〉a	tiξ〉a	NOUN
ma-58	202	67	⊗	⊗	PROPN
ma-58	202	68	〈	〈	PROPN
ma-58	202	69	pjη	pjη	PROPN
ma-58	202	70	,	,	PUNCT
ma-58	202	71	pjη〉b	pjη〉b	NOUN
ma-58	202	72	≤	≤	NUM
ma-58	202	73	(	(	PUNCT
ma-58	202	74	b	b	NOUN
ma-58	202	75	⊗d)(〈ξ	⊗d)(〈ξ	ADV
ma-58	202	76	,	,	PUNCT
ma-58	202	77	ξ〉a	ξ〉a	VERB
ma-58	202	78	⊗	⊗	PROPN
ma-58	202	79	〈	〈	PROPN
ma-58	202	80	η	η	PROPN
ma-58	202	81	,	,	PUNCT
ma-58	202	82	η〉b)(b∗	η〉b)(b∗	PROPN
ma-58	202	83	⊗d∗),∀ξ	⊗d∗),∀ξ	VERB
ma-58	202	84	∈	∈	PROPN
ma-58	202	85	x	x	X
ma-58	202	86	,	,	PUNCT
ma-58	202	87	∀η	∀η	X
ma-58	202	88	∈	∈	PROPN
ma-58	202	89	y.consequently	y.consequently	ADV
ma-58	202	90	we	we	PRON
ma-58	202	91	have	have	VERB
ma-58	202	92	(	(	PUNCT
ma-58	202	93	a⊗	a⊗	NOUN
ma-58	202	94	c)〈k∗ξ	c)〈k∗ξ	PROPN
ma-58	202	95	⊗	⊗	PROPN
ma-58	202	96	l∗η	l∗η	PROPN
ma-58	202	97	,	,	PUNCT
ma-58	202	98	k∗ξ	k∗ξ	PROPN
ma-58	202	99	⊗	⊗	PROPN
ma-58	202	100	l∗η〉a⊗b(a⊗	l∗η〉a⊗b(a⊗	NOUN
ma-58	202	101	c)∗	c)∗	PROPN
ma-58	202	102	≤	≤	NOUN
ma-58	202	103	∑	∑	PUNCT
ma-58	202	104	i∈i	i∈i	ADJ
ma-58	202	105	,	,	PUNCT
ma-58	202	106	j∈j	j∈j	PROPN
ma-58	202	107	〈	〈	PROPN
ma-58	202	108	tiξ	tiξ	PROPN
ma-58	202	109	⊗	⊗	PROPN
ma-58	202	110	pjη	pjη	PROPN
ma-58	202	111	,	,	PUNCT
ma-58	202	112	tiξ	tiξ	PROPN
ma-58	202	113	⊗	⊗	PROPN
ma-58	202	114	pjη〉a⊗b	pjη〉a⊗b	PROPN
ma-58	202	115	≤	≤	PROPN
ma-58	202	116	(	(	PUNCT
ma-58	202	117	b	b	X
ma-58	202	118	⊗d)〈ξ	⊗d)〈ξ	PROPN
ma-58	202	119	⊗	⊗	PROPN
ma-58	202	120	η	η	PROPN
ma-58	202	121	,	,	PUNCT
ma-58	202	122	ξ	ξ	PROPN
ma-58	202	123	⊗	⊗	PROPN
ma-58	202	124	η〉a⊗b(b	η〉a⊗b(b	PROPN
ma-58	202	125	⊗d)∗,∀ξ	⊗d)∗,∀ξ	PROPN
ma-58	202	126	∈	∈	PROPN
ma-58	202	127	x	x	X
ma-58	202	128	,	,	PUNCT
ma-58	202	129	∀η	∀η	X
ma-58	202	130	∈	∈	PROPN
ma-58	202	131	y.	y.	NOUN
ma-58	202	132	then	then	ADV
ma-58	202	133	for	for	ADP
ma-58	202	134	all	all	DET
ma-58	202	135	ξ	ξ	PROPN
ma-58	202	136	⊗	⊗	PROPN
ma-58	202	137	η	η	PROPN
ma-58	202	138	in	in	ADP
ma-58	202	139	x	x	PROPN
ma-58	202	140	⊗	⊗	PROPN
ma-58	202	141	y	y	NOUN
ma-58	202	142	we	we	PRON
ma-58	202	143	have	have	VERB
ma-58	202	144	(	(	PUNCT
ma-58	202	145	a⊗	a⊗	NOUN
ma-58	202	146	c)〈(k	c)〈(k	PROPN
ma-58	202	147	⊗	⊗	PROPN
ma-58	202	148	l)∗(ξ	l)∗(ξ	PROPN
ma-58	202	149	⊗	⊗	PROPN
ma-58	202	150	η	η	PROPN
ma-58	202	151	)	)	PUNCT
ma-58	202	152	,	,	PUNCT
ma-58	202	153	(	(	PUNCT
ma-58	202	154	k	k	PROPN
ma-58	202	155	⊗	⊗	PROPN
ma-58	202	156	l)∗(ξ	l)∗(ξ	PROPN
ma-58	202	157	⊗	⊗	PROPN
ma-58	202	158	η)〉a⊗b(a⊗	η)〉a⊗b(a⊗	NOUN
ma-58	202	159	c)∗	c)∗	PROPN
ma-58	202	160	≤	≤	PROPN
ma-58	202	161	∑	∑	PUNCT
ma-58	202	162	i∈i	i∈i	ADJ
ma-58	202	163	,	,	PUNCT
ma-58	202	164	j∈j	j∈j	NOUN
ma-58	202	165	〈	〈	PROPN
ma-58	202	166	(	(	PUNCT
ma-58	202	167	ti	ti	PROPN
ma-58	202	168	⊗	⊗	PROPN
ma-58	202	169	pj)(ξ	pj)(ξ	PROPN
ma-58	202	170	⊗	⊗	PROPN
ma-58	202	171	η	η	PROPN
ma-58	202	172	)	)	PUNCT
ma-58	202	173	,	,	PUNCT
ma-58	202	174	(	(	PUNCT
ma-58	202	175	ti	ti	PROPN
ma-58	202	176	⊗	⊗	PROPN
ma-58	202	177	pj)(ξ	pj)(ξ	PROPN
ma-58	202	178	⊗	⊗	PROPN
ma-58	202	179	η)〉a⊗b	η)〉a⊗b	PROPN
ma-58	202	180	≤	≤	PROPN
ma-58	202	181	(	(	PUNCT
ma-58	202	182	b	b	X
ma-58	202	183	⊗d)〈ξ	⊗d)〈ξ	PROPN
ma-58	202	184	⊗	⊗	PROPN
ma-58	202	185	η	η	PROPN
ma-58	202	186	,	,	PUNCT
ma-58	202	187	ξ	ξ	PROPN
ma-58	202	188	⊗	⊗	PROPN
ma-58	202	189	η〉a⊗b(b	η〉a⊗b(b	PROPN
ma-58	202	190	⊗d)∗.	⊗d)∗.	NOUN
ma-58	202	191	the	the	DET
ma-58	202	192	last	last	ADJ
ma-58	202	193	inequality	inequality	NOUN
ma-58	202	194	is	be	AUX
ma-58	202	195	true	true	ADJ
ma-58	202	196	for	for	ADP
ma-58	202	197	every	every	DET
ma-58	202	198	finite	finite	ADJ
ma-58	202	199	sum	sum	NOUN
ma-58	202	200	of	of	ADP
ma-58	202	201	elements	element	NOUN
ma-58	202	202	in	in	ADP
ma-58	202	203	x	x	PUNCT
ma-58	202	204	⊗alg	⊗alg	PROPN
ma-58	202	205	y	y	PROPN
ma-58	202	206	and	and	CCONJ
ma-58	202	207	then	then	ADV
ma-58	202	208	it	it	PRON
ma-58	202	209	’s	’	VERB
ma-58	202	210	true	true	ADJ
ma-58	202	211	for	for	ADP
ma-58	202	212	all	all	DET
ma-58	202	213	z	z	NOUN
ma-58	202	214	∈	∈	NOUN
ma-58	202	215	x	x	NOUN
ma-58	202	216	⊗k	⊗k	ADJ
ma-58	202	217	.	.	PUNCT
ma-58	203	1	it	it	PRON
ma-58	203	2	shows	show	VERB
ma-58	203	3	that	that	SCONJ
ma-58	203	4	{	{	PUNCT
ma-58	203	5	ti	ti	PROPN
ma-58	203	6	⊗	⊗	PROPN
ma-58	203	7	pj}i∈i	pj}i∈i	PROPN
ma-58	203	8	,	,	PUNCT
ma-58	203	9	j∈j	j∈j	NOUN
ma-58	203	10	is	be	AUX
ma-58	203	11	an	an	DET
ma-58	203	12	∗-k	∗-k	ADJ
ma-58	203	13	⊗	⊗	ADJ
ma-58	203	14	l	l	NOUN
ma-58	203	15	-	-	NOUN
ma-58	203	16	operator	operator	NOUN
ma-58	203	17	frame	frame	NOUN
ma-58	203	18	for	for	ADP
ma-58	203	19	hilbert	hilbert	PROPN
ma-58	203	20	a⊗b	a⊗b	PROPN
ma-58	203	21	-	-	PUNCT
ma-58	203	22	module	module	NOUN
ma-58	203	23	x	x	PUNCT
ma-58	203	24	⊗	⊗	PROPN
ma-58	203	25	y	y	PROPN
ma-58	203	26	with	with	ADP
ma-58	203	27	lower	low	ADJ
ma-58	203	28	and	and	CCONJ
ma-58	203	29	upper	upper	ADJ
ma-58	203	30	bounds	bound	NOUN
ma-58	203	31	a⊗	a⊗	PROPN
ma-58	203	32	c	c	PROPN
ma-58	203	33	and	and	CCONJ
ma-58	203	34	b	b	PROPN
ma-58	203	35	⊗d	⊗d	PROPN
ma-58	203	36	,	,	PUNCT
ma-58	203	37	respectively.by	respectively.by	PROPN
ma-58	203	38	the	the	DET
ma-58	203	39	definition	definition	NOUN
ma-58	203	40	of	of	ADP
ma-58	203	41	frame	frame	NOUN
ma-58	203	42	operator	operator	NOUN
ma-58	203	43	st	st	PROPN
ma-58	203	44	and	and	CCONJ
ma-58	203	45	sp	sp	ADP
ma-58	203	46	we	we	PRON
ma-58	203	47	have	have	VERB
ma-58	203	48	st	st	PROPN
ma-58	203	49	ξ	ξ	PROPN
ma-58	203	50	=	=	PUNCT
ma-58	203	51	∑	∑	PROPN
ma-58	203	52	i∈i	i∈i	ADJ
ma-58	203	53	t	t	PROPN
ma-58	203	54	∗i	∗i	PROPN
ma-58	203	55	tiξ	tiξ	PROPN
ma-58	203	56	,	,	PUNCT
ma-58	203	57	∀ξ	∀ξ	NOUN
ma-58	203	58	∈	∈	PROPN
ma-58	203	59	x	x	X
ma-58	203	60	.	.	PUNCT
ma-58	204	1	spη	spη	PROPN
ma-58	204	2	=	=	PUNCT
ma-58	204	3	∑	∑	PUNCT
ma-58	204	4	j∈j	j∈j	NOUN
ma-58	204	5	p	p	PROPN
ma-58	204	6	∗j	∗j	PROPN
ma-58	204	7	pjη	pjη	PROPN
ma-58	204	8	,	,	PUNCT
ma-58	204	9	∀η	∀η	X
ma-58	204	10	∈	∈	PROPN
ma-58	204	11	y.	y.	PROPN
ma-58	204	12	therefore	therefore	ADV
ma-58	204	13	(	(	PUNCT
ma-58	204	14	st	st	PROPN
ma-58	204	15	⊗	⊗	PROPN
ma-58	204	16	sp	sp	PROPN
ma-58	204	17	)	)	PUNCT
ma-58	204	18	(	(	PUNCT
ma-58	204	19	ξ	ξ	PROPN
ma-58	204	20	⊗	⊗	PROPN
ma-58	204	21	η	η	PROPN
ma-58	204	22	)	)	PUNCT
ma-58	204	23	=	=	SYM
ma-58	204	24	st	st	PROPN
ma-58	204	25	ξ	ξ	PROPN
ma-58	204	26	⊗	⊗	PROPN
ma-58	204	27	spη	spη	PROPN
ma-58	205	1	=	=	PUNCT
ma-58	205	2	∑	∑	PUNCT
ma-58	205	3	i∈i	i∈i	ADJ
ma-58	205	4	t	t	PROPN
ma-58	205	5	∗i	∗i	PROPN
ma-58	205	6	tiξ	tiξ	PROPN
ma-58	205	7	⊗	⊗	PROPN
ma-58	205	8	∑	∑	PROPN
ma-58	205	9	j∈j	j∈j	PROPN
ma-58	205	10	p	p	PROPN
ma-58	205	11	∗j	∗j	PROPN
ma-58	205	12	pjη	pjη	NOUN
ma-58	205	13	=	=	PUNCT
ma-58	205	14	∑	∑	PUNCT
ma-58	205	15	i∈i	i∈i	ADJ
ma-58	205	16	,	,	PUNCT
ma-58	205	17	j∈j	j∈j	NOUN
ma-58	205	18	t	t	PROPN
ma-58	205	19	∗i	∗i	PROPN
ma-58	205	20	tiξ	tiξ	PROPN
ma-58	205	21	⊗	⊗	PROPN
ma-58	205	22	p	p	PROPN
ma-58	205	23	∗j	∗j	PROPN
ma-58	205	24	pjη	pjη	NOUN
ma-58	205	25	=	=	PUNCT
ma-58	205	26	∑	∑	PUNCT
ma-58	205	27	i∈i	i∈i	ADJ
ma-58	205	28	,	,	PUNCT
ma-58	205	29	j∈j	j∈j	NOUN
ma-58	205	30	(	(	PUNCT
ma-58	205	31	t	t	PROPN
ma-58	205	32	∗i	∗i	PROPN
ma-58	205	33	⊗	⊗	PROPN
ma-58	205	34	p	p	PROPN
ma-58	205	35	∗j	∗j	PROPN
ma-58	205	36	)	)	PUNCT
ma-58	205	37	(	(	PUNCT
ma-58	205	38	tiξ	tiξ	PROPN
ma-58	205	39	⊗	⊗	PROPN
ma-58	205	40	pjη	pjη	PROPN
ma-58	205	41	)	)	PUNCT
ma-58	205	42	https://doi.org/10.28924/ada/ma.2.4	https://doi.org/10.28924/ada/ma.2.4	PROPN
ma-58	205	43	eur	eur	NOUN
ma-58	205	44	.	.	PUNCT
ma-58	206	1	j.	j.	PROPN
ma-58	206	2	math	math	PROPN
ma-58	206	3	.	.	PUNCT
ma-58	207	1	anal	anal	PROPN
ma-58	207	2	.	.	PUNCT
ma-58	208	1	10.28924	10.28924	NUM
ma-58	208	2	/	/	SYM
ma-58	208	3	ada	ada	PROPN
ma-58	208	4	/	/	SYM
ma-58	208	5	ma.2.4	ma.2.4	PROPN
ma-58	208	6	10	10	NUM
ma-58	208	7	=	=	NOUN
ma-58	208	8	∑	∑	PUNCT
ma-58	208	9	i∈i	i∈i	ADJ
ma-58	208	10	,	,	PUNCT
ma-58	208	11	j∈j	j∈j	NOUN
ma-58	208	12	(	(	PUNCT
ma-58	208	13	t	t	PROPN
ma-58	208	14	∗i	∗i	PROPN
ma-58	208	15	⊗	⊗	PROPN
ma-58	208	16	p	p	PROPN
ma-58	208	17	∗j	∗j	PROPN
ma-58	208	18	)	)	PUNCT
ma-58	208	19	(	(	PUNCT
ma-58	208	20	ti	ti	PROPN
ma-58	208	21	⊗	⊗	PROPN
ma-58	208	22	pj)(ξ	pj)(ξ	PROPN
ma-58	208	23	⊗	⊗	PROPN
ma-58	208	24	η	η	PROPN
ma-58	208	25	)	)	PUNCT
ma-58	208	26	=	=	PUNCT
ma-58	209	1	∑	∑	PUNCT
ma-58	209	2	i∈i	i∈i	ADJ
ma-58	209	3	,	,	PUNCT
ma-58	209	4	j∈j	j∈j	NOUN
ma-58	209	5	(	(	PUNCT
ma-58	209	6	ti	ti	NOUN
ma-58	209	7	⊗	⊗	PROPN
ma-58	209	8	pj)∗(ti	pj)∗(ti	PROPN
ma-58	209	9	⊗	⊗	PROPN
ma-58	209	10	pj)(ξ	pj)(ξ	PROPN
ma-58	209	11	⊗	⊗	PROPN
ma-58	209	12	η	η	PROPN
ma-58	209	13	)	)	PUNCT
ma-58	209	14	.	.	PUNCT
ma-58	210	1	then	then	ADV
ma-58	210	2	by	by	ADP
ma-58	210	3	the	the	DET
ma-58	210	4	uniqueness	uniqueness	NOUN
ma-58	210	5	of	of	ADP
ma-58	210	6	frame	frame	NOUN
ma-58	210	7	operator	operator	NOUN
ma-58	210	8	,	,	PUNCT
ma-58	210	9	the	the	DET
ma-58	210	10	last	last	ADJ
ma-58	210	11	expression	expression	NOUN
ma-58	210	12	is	be	AUX
ma-58	210	13	equal	equal	ADJ
ma-58	210	14	to	to	ADP
ma-58	210	15	st⊗p	st⊗p	VERB
ma-58	210	16	(	(	PUNCT
ma-58	210	17	ξ⊗η	ξ⊗η	NOUN
ma-58	210	18	)	)	PUNCT
ma-58	210	19	.	.	PUNCT
ma-58	211	1	consequentlywe	consequentlywe	AUX
ma-58	211	2	have	have	VERB
ma-58	211	3	(	(	PUNCT
ma-58	211	4	st	st	PROPN
ma-58	211	5	⊗sp	⊗sp	NOUN
ma-58	211	6	)	)	PUNCT
ma-58	211	7	(	(	PUNCT
ma-58	211	8	ξ⊗η	ξ⊗η	X
ma-58	211	9	)	)	PUNCT
ma-58	212	1	=	=	PUNCT
ma-58	212	2	st⊗p	st⊗p	PROPN
ma-58	212	3	(	(	PUNCT
ma-58	212	4	ξ⊗η	ξ⊗η	NOUN
ma-58	212	5	)	)	PUNCT
ma-58	212	6	.	.	PUNCT
ma-58	213	1	the	the	DET
ma-58	213	2	last	last	ADJ
ma-58	213	3	equality	equality	NOUN
ma-58	213	4	is	be	AUX
ma-58	213	5	true	true	ADJ
ma-58	213	6	for	for	ADP
ma-58	213	7	every	every	DET
ma-58	213	8	finite	finite	ADJ
ma-58	213	9	sum	sum	NOUN
ma-58	213	10	of	of	ADP
ma-58	213	11	elementsin	elementsin	PROPN
ma-58	213	12	x	x	SYM
ma-58	213	13	⊗alg	⊗alg	PROPN
ma-58	213	14	y	y	PROPN
ma-58	214	1	and	and	CCONJ
ma-58	214	2	then	then	ADV
ma-58	214	3	it	it	PRON
ma-58	214	4	’s	’	VERB
ma-58	214	5	true	true	ADJ
ma-58	214	6	for	for	ADP
ma-58	214	7	all	all	DET
ma-58	214	8	z	z	NOUN
ma-58	214	9	∈	∈	PROPN
ma-58	214	10	x	x	PUNCT
ma-58	214	11	⊗	⊗	PROPN
ma-58	214	12	y	y	PROPN
ma-58	214	13	.	.	PUNCT
ma-58	215	1	it	it	PRON
ma-58	215	2	follows	follow	VERB
ma-58	215	3	that	that	SCONJ
ma-58	215	4	(	(	PUNCT
ma-58	215	5	st	st	PROPN
ma-58	215	6	⊗	⊗	PROPN
ma-58	215	7	sp	sp	PROPN
ma-58	215	8	)	)	PUNCT
ma-58	215	9	(	(	PUNCT
ma-58	215	10	z	z	NOUN
ma-58	215	11	)	)	PUNCT
ma-58	215	12	=	=	SYM
ma-58	215	13	st⊗p	st⊗p	NOUN
ma-58	215	14	(	(	PUNCT
ma-58	215	15	z	z	NOUN
ma-58	215	16	)	)	PUNCT
ma-58	215	17	.	.	PUNCT
ma-58	216	1	thus	thus	ADV
ma-58	216	2	st⊗p	st⊗p	VERB
ma-58	216	3	=	=	SYM
ma-58	216	4	st	st	PROPN
ma-58	216	5	⊗	⊗	PROPN
ma-58	216	6	sp	sp	PROPN
ma-58	216	7	.	.	PUNCT
ma-58	217	1	�	�	PROPN
ma-58	217	2	references	reference	NOUN
ma-58	217	3	[	[	X
ma-58	217	4	1	1	NUM
ma-58	217	5	]	]	X
ma-58	217	6	n.	n.	PROPN
ma-58	217	7	haddadzadeh	haddadzadeh	PROPN
ma-58	217	8	,	,	PUNCT
ma-58	217	9	g	g	NOUN
ma-58	217	10	-	-	PUNCT
ma-58	217	11	frames	frame	NOUN
ma-58	217	12	in	in	ADP
ma-58	217	13	hilbert	hilbert	NOUN
ma-58	217	14	modules	module	NOUN
ma-58	217	15	over	over	ADP
ma-58	217	16	pro	pro	ADJ
ma-58	217	17	-	-	ADJ
ma-58	217	18	c*-algebras	c*-algebras	ADJ
ma-58	217	19	,	,	PUNCT
ma-58	217	20	int	int	NOUN
ma-58	217	21	.	.	PUNCT
ma-58	218	1	j.	j.	PROPN
ma-58	218	2	ind	ind	PROPN
ma-58	218	3	.	.	PUNCT
ma-58	219	1	math	math	NOUN
ma-58	219	2	.	.	PUNCT
ma-58	220	1	9(4	9(4	NUM
ma-58	220	2	)	)	PUNCT
ma-58	220	3	(	(	PUNCT
ma-58	220	4	2017	2017	NUM
ma-58	220	5	)	)	PUNCT
ma-58	220	6	259	259	NUM
ma-58	220	7	-	-	SYM
ma-58	220	8	267.[2	267.[2	NUM
ma-58	220	9	]	]	PUNCT
ma-58	220	10	m.	m.	NOUN
ma-58	220	11	azhini	azhini	PROPN
ma-58	220	12	and	and	CCONJ
ma-58	220	13	n.	n.	PROPN
ma-58	220	14	haddadzadeh	haddadzadeh	PROPN
ma-58	220	15	,	,	PUNCT
ma-58	220	16	fusion	fusion	NOUN
ma-58	220	17	frames	frame	NOUN
ma-58	220	18	in	in	ADP
ma-58	220	19	hilbert	hilbert	NOUN
ma-58	220	20	modules	module	NOUN
ma-58	220	21	over	over	ADP
ma-58	220	22	pro	pro	ADJ
ma-58	220	23	-	-	ADJ
ma-58	220	24	c∗-algebras	c∗-algebra	NOUN
ma-58	220	25	,	,	PUNCT
ma-58	220	26	int	int	NOUN
ma-58	220	27	.	.	PUNCT
ma-58	221	1	j.	j.	PROPN
ma-58	221	2	ind	ind	PROPN
ma-58	221	3	.	.	PUNCT
ma-58	222	1	math	math	NOUN
ma-58	222	2	.	.	PUNCT
ma-58	223	1	5(2)(2013	5(2)(2013	NUM
ma-58	223	2	)	)	PUNCT
ma-58	223	3	article	article	NOUN
ma-58	223	4	i	i	NOUN
ma-58	223	5	d	d	PROPN
ma-58	223	6	ijim-00211.[3	ijim-00211.[3	PROPN
ma-58	223	7	]	]	X
ma-58	223	8	r.	r.	PROPN
ma-58	223	9	j.	j.	PROPN
ma-58	223	10	duffin	duffin	PROPN
ma-58	223	11	and	and	CCONJ
ma-58	223	12	a.	a.	PROPN
ma-58	223	13	c.	c.	PROPN
ma-58	223	14	schaeffer	schaeffer	PROPN
ma-58	223	15	,	,	PUNCT
ma-58	223	16	a	a	DET
ma-58	223	17	class	class	NOUN
ma-58	223	18	of	of	ADP
ma-58	223	19	nonharmonic	nonharmonic	ADJ
ma-58	223	20	fourier	fourier	NOUN
ma-58	223	21	series	series	NOUN
ma-58	223	22	,	,	PUNCT
ma-58	223	23	trans	trans	PROPN
ma-58	223	24	.	.	PROPN
ma-58	224	1	amer	amer	PROPN
ma-58	224	2	.	.	PUNCT
ma-58	224	3	math	math	PROPN
ma-58	224	4	.	.	PUNCT
ma-58	225	1	soc	soc	PROPN
ma-58	225	2	.	.	PUNCT
ma-58	226	1	72	72	NUM
ma-58	226	2	(	(	PUNCT
ma-58	226	3	1952	1952	NUM
ma-58	226	4	)	)	PUNCT
ma-58	226	5	341	341	NUM
ma-58	226	6	-	-	SYM
ma-58	226	7	366.[4	366.[4	NUM
ma-58	226	8	]	]	PUNCT
ma-58	226	9	m.	m.	NOUN
ma-58	226	10	fragoulopoulou	fragoulopoulou	PROPN
ma-58	226	11	,	,	PUNCT
ma-58	226	12	an	an	DET
ma-58	226	13	introduction	introduction	NOUN
ma-58	226	14	to	to	ADP
ma-58	226	15	the	the	DET
ma-58	226	16	representation	representation	NOUN
ma-58	226	17	theory	theory	NOUN
ma-58	226	18	of	of	ADP
ma-58	226	19	topological	topological	ADJ
ma-58	226	20	∗-algebras	∗-algebra	NOUN
ma-58	226	21	,	,	PUNCT
ma-58	226	22	schriftenreihe	schriftenreihe	NOUN
ma-58	226	23	,	,	PUNCT
ma-58	226	24	univ.münster	univ.münster	NOUN
ma-58	226	25	,	,	PUNCT
ma-58	226	26	48	48	NUM
ma-58	226	27	(	(	PUNCT
ma-58	226	28	1988	1988	NUM
ma-58	226	29	)	)	PUNCT
ma-58	226	30	1	1	NUM
ma-58	226	31	-	-	SYM
ma-58	226	32	81.[5	81.[5	NUM
ma-58	226	33	]	]	PUNCT
ma-58	226	34	m.	m.	NOUN
ma-58	226	35	fragoulopoulou	fragoulopoulou	PROPN
ma-58	226	36	,	,	PUNCT
ma-58	226	37	tensor	tensor	NOUN
ma-58	226	38	products	product	NOUN
ma-58	226	39	of	of	ADP
ma-58	226	40	enveloping	envelop	VERB
ma-58	226	41	locally	locally	ADV
ma-58	226	42	c∗-algebras	c∗-algebra	NOUN
ma-58	226	43	,	,	PUNCT
ma-58	226	44	schriftenreihe	schriftenreihe	NOUN
ma-58	226	45	,	,	PUNCT
ma-58	226	46	univ	univ	PROPN
ma-58	226	47	.	.	PUNCT
ma-58	227	1	münster	münster	NOUN
ma-58	227	2	(	(	PUNCT
ma-58	227	3	1997	1997	NUM
ma-58	227	4	)	)	PUNCT
ma-58	227	5	1	1	NUM
ma-58	227	6	-	-	SYM
ma-58	227	7	81.[6	81.[6	NUM
ma-58	227	8	]	]	PUNCT
ma-58	227	9	m.	m.	NOUN
ma-58	227	10	fragoulopoulou	fragoulopoulou	PROPN
ma-58	227	11	,	,	PUNCT
ma-58	227	12	topological	topological	ADJ
ma-58	227	13	algebras	algebra	NOUN
ma-58	227	14	with	with	ADP
ma-58	227	15	involution	involution	NOUN
ma-58	227	16	,	,	PUNCT
ma-58	227	17	north	north	PROPN
ma-58	227	18	holland	holland	PROPN
ma-58	227	19	,	,	PUNCT
ma-58	227	20	amsterdam	amsterdam	PROPN
ma-58	227	21	,	,	PUNCT
ma-58	227	22	2005.[7	2005.[7	NUM
ma-58	227	23	]	]	X
ma-58	227	24	a.	a.	NOUN
ma-58	227	25	grossman	grossman	PROPN
ma-58	227	26	,	,	PUNCT
ma-58	227	27	and	and	CCONJ
ma-58	227	28	y.	y.	PROPN
ma-58	227	29	meyer	meyer	PROPN
ma-58	227	30	,	,	PUNCT
ma-58	227	31	painless	painless	ADJ
ma-58	227	32	nonorthogonal	nonorthogonal	ADJ
ma-58	227	33	expansions	expansion	NOUN
ma-58	227	34	,	,	PUNCT
ma-58	227	35	j.	j.	PROPN
ma-58	227	36	math	math	PROPN
ma-58	227	37	.	.	PUNCT
ma-58	228	1	phys	phy	NOUN
ma-58	228	2	.	.	PUNCT
ma-58	229	1	27	27	NUM
ma-58	229	2	(	(	PUNCT
ma-58	229	3	1986	1986	NUM
ma-58	229	4	)	)	PUNCT
ma-58	229	5	1271	1271	NUM
ma-58	229	6	-	-	SYM
ma-58	229	7	1283.[8	1283.[8	NUM
ma-58	229	8	]	]	PUNCT
ma-58	229	9	a.	a.	NOUN
ma-58	229	10	inoue	inoue	PROPN
ma-58	229	11	,	,	PUNCT
ma-58	229	12	locally	locally	ADV
ma-58	229	13	c∗-algebra	c∗-algebra	PROPN
ma-58	229	14	,	,	PUNCT
ma-58	229	15	mem	mem	PROPN
ma-58	229	16	.	.	PUNCT
ma-58	229	17	fac	fac	PROPN
ma-58	229	18	.	.	PUNCT
ma-58	230	1	sci	sci	PROPN
ma-58	230	2	.	.	PUNCT
ma-58	230	3	kyushu	kyushu	PROPN
ma-58	230	4	univ	univ	PROPN
ma-58	230	5	.	.	PUNCT
ma-58	231	1	ser	ser	PROPN
ma-58	231	2	.	.	PUNCT
ma-58	232	1	a	a	DET
ma-58	232	2	,	,	PUNCT
ma-58	232	3	math	math	NOUN
ma-58	232	4	.	.	PUNCT
ma-58	233	1	25(2	25(2	NUM
ma-58	233	2	)	)	PUNCT
ma-58	233	3	(	(	PUNCT
ma-58	233	4	1972	1972	NUM
ma-58	233	5	)	)	PUNCT
ma-58	233	6	197	197	NUM
ma-58	233	7	-	-	SYM
ma-58	233	8	235.[9	235.[9	NUM
ma-58	233	9	]	]	PUNCT
ma-58	233	10	m.	m.	NOUN
ma-58	233	11	joita	joita	PROPN
ma-58	233	12	,	,	PUNCT
ma-58	233	13	on	on	ADP
ma-58	233	14	frames	frame	NOUN
ma-58	233	15	in	in	ADP
ma-58	233	16	hilbert	hilbert	NOUN
ma-58	233	17	modules	module	NOUN
ma-58	233	18	over	over	ADP
ma-58	233	19	pro	pro	ADJ
ma-58	233	20	-	-	ADJ
ma-58	233	21	c∗-algebras	c∗-algebra	NOUN
ma-58	233	22	,	,	PUNCT
ma-58	233	23	topol	topol	NOUN
ma-58	233	24	.	.	PUNCT
ma-58	234	1	appl	appl	PROPN
ma-58	234	2	.	.	PUNCT
ma-58	235	1	156	156	NUM
ma-58	235	2	(	(	PUNCT
ma-58	235	3	2008	2008	NUM
ma-58	235	4	)	)	PUNCT
ma-58	235	5	83	83	NUM
ma-58	235	6	-	-	SYM
ma-58	235	7	92.[10	92.[10	NUM
ma-58	235	8	]	]	PUNCT
ma-58	235	9	e.	e.	PROPN
ma-58	235	10	c.	c.	PROPN
ma-58	235	11	lance	lance	PROPN
ma-58	235	12	,	,	PUNCT
ma-58	235	13	hilbert	hilbert	PROPN
ma-58	235	14	c∗-modules	c∗-modules	PROPN
ma-58	235	15	,	,	PUNCT
ma-58	235	16	a	a	DET
ma-58	235	17	toolkit	toolkit	NOUN
ma-58	235	18	for	for	ADP
ma-58	235	19	operator	operator	NOUN
ma-58	235	20	algebraists	algebraist	NOUN
ma-58	235	21	,	,	PUNCT
ma-58	235	22	london	london	PROPN
ma-58	235	23	math	math	NOUN
ma-58	235	24	.	.	PUNCT
ma-58	236	1	soc	soc	PROPN
ma-58	236	2	.	.	PUNCT
ma-58	237	1	lecture	lecture	NOUN
ma-58	237	2	note	note	NOUN
ma-58	237	3	series	series	PROPN
ma-58	237	4	210.cambridge	210.cambridge	PROPN
ma-58	237	5	univ	univ	PROPN
ma-58	237	6	.	.	PUNCT
ma-58	238	1	press	press	PROPN
ma-58	238	2	,	,	PUNCT
ma-58	238	3	cambridge	cambridge	PROPN
ma-58	238	4	,	,	PUNCT
ma-58	238	5	1995.[11	1995.[11	NUM
ma-58	238	6	]	]	PUNCT
ma-58	238	7	c.	c.	PROPN
ma-58	238	8	y.	y.	PROPN
ma-58	238	9	li	li	PROPN
ma-58	238	10	and	and	CCONJ
ma-58	238	11	h.	h.	PROPN
ma-58	238	12	x.	x.	PROPN
ma-58	238	13	cao	cao	PROPN
ma-58	238	14	,	,	PUNCT
ma-58	238	15	operator	operator	NOUN
ma-58	238	16	frames	frame	NOUN
ma-58	238	17	for	for	ADP
ma-58	238	18	b(h	b(h	NOUN
ma-58	238	19	)	)	PUNCT
ma-58	238	20	,	,	PUNCT
ma-58	238	21	wavelet	wavelet	NOUN
ma-58	238	22	analysis	analysis	NOUN
ma-58	238	23	and	and	CCONJ
ma-58	238	24	applications	application	NOUN
ma-58	238	25	.	.	PUNCT
ma-58	239	1	birkhäuser	birkhäuser	PROPN
ma-58	239	2	basel	basel	PROPN
ma-58	239	3	,	,	PUNCT
ma-58	239	4	2006.67	2006.67	NUM
ma-58	239	5	-	-	SYM
ma-58	239	6	82.[12	82.[12	NUM
ma-58	239	7	]	]	X
ma-58	239	8	a.	a.	NOUN
ma-58	239	9	mallios	mallios	NOUN
ma-58	239	10	,	,	PUNCT
ma-58	239	11	topological	topological	ADJ
ma-58	239	12	algebras	algebra	NOUN
ma-58	239	13	:	:	PUNCT
ma-58	239	14	selected	select	VERB
ma-58	239	15	topics	topic	NOUN
ma-58	239	16	,	,	PUNCT
ma-58	239	17	north	north	NOUN
ma-58	239	18	holland	holland	PROPN
ma-58	239	19	,	,	PUNCT
ma-58	239	20	amsterdam	amsterdam	PROPN
ma-58	239	21	,	,	PUNCT
ma-58	239	22	1986.[13	1986.[13	PROPN
ma-58	239	23	]	]	PUNCT
ma-58	239	24	m.	m.	NOUN
ma-58	239	25	naroei	naroei	PROPN
ma-58	239	26	and	and	CCONJ
ma-58	239	27	a.	a.	NOUN
ma-58	239	28	nazari	nazari	PROPN
ma-58	239	29	,	,	PUNCT
ma-58	239	30	some	some	DET
ma-58	239	31	properties	property	NOUN
ma-58	239	32	of	of	ADP
ma-58	239	33	–	–	PUNCT
ma-58	239	34	frames	frame	NOUN
ma-58	239	35	in	in	ADP
ma-58	239	36	hilbert	hilbert	NOUN
ma-58	239	37	modules	module	NOUN
ma-58	239	38	over	over	ADP
ma-58	239	39	pro	pro	ADJ
ma-58	239	40	-	-	ADJ
ma-58	239	41	c∗-algebras	c∗-algebra	NOUN
ma-58	239	42	,	,	PUNCT
ma-58	239	43	sahand	sahand	NOUN
ma-58	239	44	commun.math	commun.math	PROPN
ma-58	239	45	.	.	PUNCT
ma-58	240	1	anal	anal	PROPN
ma-58	240	2	.	.	PUNCT
ma-58	241	1	16	16	NUM
ma-58	241	2	(	(	PUNCT
ma-58	241	3	2019	2019	NUM
ma-58	241	4	)	)	PUNCT
ma-58	241	5	105–117.[14	105–117.[14	PROPN
ma-58	241	6	]	]	X
ma-58	241	7	n.	n.	PROPN
ma-58	241	8	c.	c.	PROPN
ma-58	241	9	phillips	phillips	PROPN
ma-58	241	10	,	,	PUNCT
ma-58	241	11	inverse	inverse	NOUN
ma-58	241	12	limits	limit	NOUN
ma-58	241	13	of	of	ADP
ma-58	241	14	c*-algebras	c*-algebra	NOUN
ma-58	241	15	,	,	PUNCT
ma-58	241	16	j.	j.	PROPN
ma-58	241	17	oper	oper	PROPN
ma-58	241	18	.	.	PROPN
ma-58	241	19	theory	theory	NOUN
ma-58	241	20	.	.	PUNCT
ma-58	242	1	19	19	NUM
ma-58	242	2	(	(	PUNCT
ma-58	242	3	1988	1988	NUM
ma-58	242	4	)	)	PUNCT
ma-58	242	5	159	159	NUM
ma-58	242	6	-	-	SYM
ma-58	242	7	195.[15	195.[15	NUM
ma-58	242	8	]	]	X
ma-58	242	9	n.	n.	PROPN
ma-58	242	10	c.	c.	PROPN
ma-58	242	11	phillips	phillips	PROPN
ma-58	242	12	,	,	PUNCT
ma-58	242	13	representable	representable	ADJ
ma-58	242	14	k	k	NOUN
ma-58	242	15	-	-	NOUN
ma-58	242	16	theory	theory	NOUN
ma-58	242	17	for	for	ADP
ma-58	242	18	σ	σ	PROPN
ma-58	242	19	-c∗-algebras	-c∗-algebras	PROPN
ma-58	242	20	,	,	PUNCT
ma-58	242	21	k	k	NOUN
ma-58	242	22	-	-	NOUN
ma-58	242	23	theory	theory	NOUN
ma-58	242	24	,	,	PUNCT
ma-58	242	25	3	3	NUM
ma-58	242	26	(	(	PUNCT
ma-58	242	27	1989	1989	NUM
ma-58	242	28	)	)	PUNCT
ma-58	242	29	441	441	NUM
ma-58	242	30	-	-	SYM
ma-58	242	31	478.[16	478.[16	NUM
ma-58	242	32	]	]	PUNCT
ma-58	242	33	m.	m.	NOUN
ma-58	242	34	rossafi	rossafi	NOUN
ma-58	242	35	and	and	CCONJ
ma-58	242	36	s.	s.	PROPN
ma-58	242	37	kabbaj	kabbaj	PROPN
ma-58	242	38	,	,	PUNCT
ma-58	242	39	operator	operator	NOUN
ma-58	242	40	frame	frame	NOUN
ma-58	242	41	for	for	ADP
ma-58	242	42	end∗a(h	end∗a(h	NOUN
ma-58	242	43	)	)	PUNCT
ma-58	242	44	,	,	PUNCT
ma-58	242	45	j.	j.	PROPN
ma-58	242	46	linear	linear	PROPN
ma-58	242	47	topol	topol	PROPN
ma-58	242	48	.	.	PUNCT
ma-58	243	1	algebra	algebra	NOUN
ma-58	243	2	.	.	PUNCT
ma-58	244	1	8	8	NUM
ma-58	244	2	(	(	PUNCT
ma-58	244	3	2019	2019	NUM
ma-58	244	4	)	)	PUNCT
ma-58	244	5	85	85	NUM
ma-58	244	6	-	-	SYM
ma-58	244	7	95	95	NUM
ma-58	244	8	.	.	PUNCT
ma-58	245	1	https://doi.org/10.28924/ada/ma.2.4	https://doi.org/10.28924/ada/ma.2.4	NUM
ma-58	245	2	1	1	NUM
ma-58	245	3	.	.	PUNCT
ma-58	245	4	introduction	introduction	NOUN
ma-58	245	5	2	2	NUM
ma-58	245	6	.	.	PUNCT
ma-58	245	7	preliminaries	preliminary	NOUN
ma-58	245	8	3	3	NUM
ma-58	245	9	.	.	PUNCT
ma-58	246	1	-k	-k	PUNCT
ma-58	246	2	-	-	PUNCT
ma-58	246	3	operator	operator	NOUN
ma-58	246	4	frame	frame	NOUN
ma-58	246	5	for	for	ADP
ma-58	246	6	homa(x	homa(x	PROPN
ma-58	246	7	)	)	PUNCT
ma-58	246	8	4	4	NUM
ma-58	246	9	.	.	NOUN
ma-58	246	10	tensor	tensor	NOUN
ma-58	246	11	product	product	NOUN
ma-58	246	12	references	reference	NOUN