id	sid	tid	token	lemma	pos
ma-125	1	1	2023	2023	NUM
ma-125	1	2	ada	ada	PROPN
ma-125	1	3	academica	academica	PROPN
ma-125	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-125	1	5	.	.	PUNCT
ma-125	2	1	j.	j.	PROPN
ma-125	2	2	math	math	PROPN
ma-125	2	3	.	.	PUNCT
ma-125	3	1	anal	anal	ADJ
ma-125	3	2	.	.	PUNCT
ma-125	4	1	3	3	NUM
ma-125	4	2	(	(	PUNCT
ma-125	4	3	2023	2023	NUM
ma-125	4	4	)	)	PUNCT
ma-125	4	5	11doi	11doi	NUM
ma-125	4	6	:	:	PUNCT
ma-125	4	7	10.28924	10.28924	NUM
ma-125	4	8	/	/	SYM
ma-125	4	9	ada	ada	PROPN
ma-125	4	10	/	/	SYM
ma-125	4	11	ma.3.11	ma.3.11	ADJ
ma-125	4	12	woven	weave	VERB
ma-125	4	13	k	k	PROPN
ma-125	4	14	−	−	PROPN
ma-125	4	15	g−fusion	g−fusion	NOUN
ma-125	4	16	frames	frame	NOUN
ma-125	4	17	in	in	ADP
ma-125	4	18	hilbert	hilbert	PROPN
ma-125	4	19	c∗−modules	c∗−modules	PROPN
ma-125	4	20	fakhr	fakhr	PROPN
ma-125	4	21	-	-	PROPN
ma-125	4	22	dine	dine	NOUN
ma-125	4	23	nhari1	nhari1	PROPN
ma-125	4	24	,	,	PUNCT
ma-125	4	25	mohamed	mohamed	PROPN
ma-125	4	26	rossafi2,∗	rossafi2,∗	PROPN
ma-125	4	27	1laboratory	1laboratory	NUM
ma-125	4	28	analysis	analysis	NOUN
ma-125	4	29	,	,	PUNCT
ma-125	4	30	geometry	geometry	NOUN
ma-125	4	31	and	and	CCONJ
ma-125	4	32	applications	application	NOUN
ma-125	4	33	department	department	NOUN
ma-125	4	34	of	of	ADP
ma-125	4	35	mathematics	mathematic	NOUN
ma-125	4	36	,	,	PUNCT
ma-125	4	37	faculty	faculty	NOUN
ma-125	4	38	of	of	ADP
ma-125	4	39	sciences	science	NOUN
ma-125	4	40	,	,	PUNCT
ma-125	4	41	university	university	NOUN
ma-125	4	42	of	of	ADP
ma-125	4	43	ibn	ibn	PROPN
ma-125	4	44	tofail	tofail	NOUN
ma-125	4	45	,	,	PUNCT
ma-125	4	46	p.	p.	PROPN
ma-125	4	47	o.	o.	PROPN
ma-125	4	48	box	box	PROPN
ma-125	4	49	133	133	NUM
ma-125	4	50	kenitra	kenitra	PROPN
ma-125	4	51	,	,	PUNCT
ma-125	4	52	morocco	morocco	PROPN
ma-125	4	53	nharidoc@gmail.com	nharidoc@gmail.com	PUNCT
ma-125	5	1	2lasma	2lasma	NUM
ma-125	5	2	laboratory	laboratory	NOUN
ma-125	5	3	,	,	PUNCT
ma-125	5	4	department	department	NOUN
ma-125	5	5	of	of	ADP
ma-125	5	6	mathematics	mathematic	NOUN
ma-125	5	7	,	,	PUNCT
ma-125	5	8	faculty	faculty	NOUN
ma-125	5	9	of	of	ADP
ma-125	5	10	sciences	sciences	PROPN
ma-125	5	11	dhar	dhar	PROPN
ma-125	5	12	el	el	PROPN
ma-125	5	13	mahraz	mahraz	PROPN
ma-125	5	14	,	,	PUNCT
ma-125	5	15	university	university	NOUN
ma-125	5	16	sidi	sidi	NOUN
ma-125	5	17	mohamed	mohamed	PROPN
ma-125	5	18	ben	ben	PROPN
ma-125	5	19	abdellah	abdellah	PROPN
ma-125	5	20	,	,	PUNCT
ma-125	5	21	p.	p.	PROPN
ma-125	5	22	o.	o.	PROPN
ma-125	5	23	box	box	PROPN
ma-125	5	24	1796	1796	NUM
ma-125	5	25	fez	fez	PROPN
ma-125	5	26	atlas	atlas	PROPN
ma-125	5	27	,	,	PUNCT
ma-125	5	28	morocco	morocco	PROPN
ma-125	5	29	rossafimohamed@gmail.com	rossafimohamed@gmail.com	X
ma-125	6	1	∗correspondence	∗correspondence	NOUN
ma-125	6	2	:	:	PUNCT
ma-125	6	3	rossafimohamed@gmail.com	rossafimohamed@gmail.com	X
ma-125	7	1	abstract	abstract	ADJ
ma-125	7	2	.	.	PUNCT
ma-125	8	1	in	in	ADP
ma-125	8	2	this	this	DET
ma-125	8	3	paper	paper	NOUN
ma-125	8	4	,	,	PUNCT
ma-125	8	5	we	we	PRON
ma-125	8	6	introduced	introduce	VERB
ma-125	8	7	the	the	DET
ma-125	8	8	notion	notion	NOUN
ma-125	8	9	of	of	ADP
ma-125	8	10	woven	woven	ADJ
ma-125	9	1	k	k	PROPN
ma-125	9	2	−	−	PROPN
ma-125	9	3	g−fusion	g−fusion	NOUN
ma-125	9	4	frames	frame	NOUN
ma-125	9	5	in	in	ADP
ma-125	9	6	hilbert	hilbert	PROPN
ma-125	9	7	c∗−modules	c∗−modules	PROPN
ma-125	9	8	.	.	PUNCT
ma-125	10	1	we	we	PRON
ma-125	10	2	present	present	VERB
ma-125	10	3	necessary	necessary	ADJ
ma-125	10	4	and	and	CCONJ
ma-125	10	5	sufficient	sufficient	ADJ
ma-125	10	6	conditions	condition	NOUN
ma-125	10	7	for	for	ADP
ma-125	10	8	these	these	DET
ma-125	10	9	woven	weave	VERB
ma-125	10	10	and	and	CCONJ
ma-125	10	11	also	also	ADV
ma-125	10	12	constructthem	constructthem	NOUN
ma-125	10	13	by	by	ADP
ma-125	10	14	linear	linear	PROPN
ma-125	10	15	bounded	bound	VERB
ma-125	10	16	operator	operator	NOUN
ma-125	10	17	.	.	PUNCT
ma-125	11	1	finally	finally	ADV
ma-125	11	2	we	we	PRON
ma-125	11	3	study	study	VERB
ma-125	11	4	perturbation	perturbation	NOUN
ma-125	11	5	of	of	ADP
ma-125	11	6	weaving	weave	VERB
ma-125	11	7	k	k	PROPN
ma-125	11	8	−	−	PROPN
ma-125	11	9	g−fusion	g−fusion	NOUN
ma-125	11	10	frames	frame	NOUN
ma-125	11	11	.	.	PUNCT
ma-125	12	1	1	1	X
ma-125	12	2	.	.	X
ma-125	12	3	introduction	introduction	NOUN
ma-125	12	4	basis	basis	NOUN
ma-125	12	5	is	be	AUX
ma-125	12	6	one	one	NUM
ma-125	12	7	of	of	ADP
ma-125	12	8	the	the	DET
ma-125	12	9	most	most	ADV
ma-125	12	10	important	important	ADJ
ma-125	12	11	concepts	concept	NOUN
ma-125	12	12	in	in	ADP
ma-125	12	13	vector	vector	NOUN
ma-125	12	14	spaces	space	NOUN
ma-125	12	15	study	study	NOUN
ma-125	12	16	.	.	PUNCT
ma-125	13	1	however	however	ADV
ma-125	13	2	,	,	PUNCT
ma-125	13	3	frames	frame	NOUN
ma-125	13	4	generaliseorthonormal	generaliseorthonormal	ADJ
ma-125	13	5	bases	basis	NOUN
ma-125	13	6	and	and	CCONJ
ma-125	13	7	were	be	AUX
ma-125	13	8	introduced	introduce	VERB
ma-125	13	9	by	by	ADP
ma-125	13	10	duffin	duffin	PROPN
ma-125	13	11	and	and	CCONJ
ma-125	13	12	schaefer	schaefer	PROPN
ma-125	13	13	[	[	X
ma-125	13	14	3	3	NUM
ma-125	13	15	]	]	PUNCT
ma-125	13	16	in	in	ADP
ma-125	13	17	1952	1952	NUM
ma-125	13	18	to	to	PART
ma-125	13	19	analyse	analyse	VERB
ma-125	13	20	some	some	DET
ma-125	13	21	deepproblems	deepproblem	NOUN
ma-125	13	22	in	in	ADP
ma-125	13	23	nonharmonic	nonharmonic	ADJ
ma-125	13	24	fourier	fourier	NOUN
ma-125	13	25	series	series	NOUN
ma-125	13	26	by	by	ADP
ma-125	13	27	abstracting	abstract	VERB
ma-125	13	28	the	the	DET
ma-125	13	29	fundamental	fundamental	ADJ
ma-125	13	30	notion	notion	NOUN
ma-125	13	31	of	of	ADP
ma-125	13	32	gabor	gabor	PROPN
ma-125	14	1	[	[	X
ma-125	14	2	5	5	NUM
ma-125	14	3	]	]	PUNCT
ma-125	14	4	for	for	ADP
ma-125	14	5	signalprocessing	signalprocesse	VERB
ma-125	14	6	.	.	PUNCT
ma-125	15	1	in	in	ADP
ma-125	15	2	2000	2000	NUM
ma-125	15	3	,	,	PUNCT
ma-125	15	4	frank	frank	PROPN
ma-125	15	5	-	-	PUNCT
ma-125	15	6	larson	larson	PROPN
ma-125	15	7	[	[	X
ma-125	15	8	4	4	NUM
ma-125	15	9	]	]	PUNCT
ma-125	15	10	introduced	introduce	VERB
ma-125	15	11	the	the	DET
ma-125	15	12	concept	concept	NOUN
ma-125	15	13	of	of	ADP
ma-125	15	14	frames	frame	NOUN
ma-125	15	15	in	in	ADP
ma-125	15	16	hilbet	hilbet	PROPN
ma-125	15	17	c∗−modulesas	c∗−modulesas	PROPN
ma-125	15	18	a	a	DET
ma-125	15	19	generalization	generalization	NOUN
ma-125	15	20	of	of	ADP
ma-125	15	21	frames	frame	NOUN
ma-125	15	22	in	in	ADP
ma-125	15	23	hilbert	hilbert	PROPN
ma-125	15	24	spaces	space	NOUN
ma-125	15	25	.	.	PUNCT
ma-125	16	1	the	the	DET
ma-125	16	2	basic	basic	ADJ
ma-125	16	3	idea	idea	NOUN
ma-125	16	4	was	be	AUX
ma-125	16	5	to	to	PART
ma-125	16	6	consider	consider	VERB
ma-125	16	7	modules	module	NOUN
ma-125	16	8	over	over	ADP
ma-125	16	9	c∗−algebras	c∗−algebra	NOUN
ma-125	16	10	of	of	ADP
ma-125	16	11	linear	linear	ADJ
ma-125	16	12	spaces	space	NOUN
ma-125	16	13	and	and	CCONJ
ma-125	16	14	to	to	PART
ma-125	16	15	allow	allow	VERB
ma-125	16	16	the	the	DET
ma-125	16	17	inner	inner	ADJ
ma-125	16	18	product	product	NOUN
ma-125	16	19	to	to	PART
ma-125	16	20	take	take	VERB
ma-125	16	21	values	value	NOUN
ma-125	16	22	in	in	ADP
ma-125	16	23	the	the	DET
ma-125	16	24	c∗−algebras	c∗−algebra	NOUN
ma-125	16	25	[	[	SYM
ma-125	16	26	6].many	6].many	NUM
ma-125	16	27	generalizations	generalization	NOUN
ma-125	16	28	of	of	ADP
ma-125	16	29	the	the	DET
ma-125	16	30	concept	concept	NOUN
ma-125	16	31	of	of	ADP
ma-125	16	32	frame	frame	NOUN
ma-125	16	33	have	have	AUX
ma-125	16	34	been	be	AUX
ma-125	16	35	defined	define	VERB
ma-125	16	36	in	in	ADP
ma-125	16	37	hilbert	hilbert	PROPN
ma-125	16	38	c∗-modules	c∗-module	NOUN
ma-125	16	39	[	[	X
ma-125	16	40	7,9,11–16].throughout	7,9,11–16].throughout	ADP
ma-125	16	41	this	this	DET
ma-125	16	42	paper	paper	NOUN
ma-125	16	43	,	,	PUNCT
ma-125	16	44	h	h	PROPN
ma-125	16	45	is	be	AUX
ma-125	16	46	considered	consider	VERB
ma-125	16	47	to	to	PART
ma-125	16	48	be	be	AUX
ma-125	16	49	a	a	DET
ma-125	16	50	countably	countably	ADV
ma-125	16	51	generated	generate	VERB
ma-125	16	52	hilbert	hilbert	PROPN
ma-125	16	53	c∗−module	c∗−module	PROPN
ma-125	16	54	.	.	PUNCT
ma-125	17	1	let	let	VERB
ma-125	17	2	{	{	PUNCT
ma-125	17	3	hj}j∈j	hj}j∈j	ADP
ma-125	17	4	are	be	AUX
ma-125	17	5	the	the	DET
ma-125	17	6	collection	collection	NOUN
ma-125	17	7	of	of	ADP
ma-125	17	8	hilbert	hilbert	PROPN
ma-125	17	9	c∗−module	c∗−module	PROPN
ma-125	17	10	and	and	CCONJ
ma-125	17	11	{	{	PUNCT
ma-125	17	12	wj}j∈j	wj}j∈j	X
ma-125	17	13	is	be	AUX
ma-125	17	14	a	a	DET
ma-125	17	15	collection	collection	NOUN
ma-125	17	16	of	of	ADP
ma-125	17	17	closed	closed	ADJ
ma-125	17	18	orthogonallycomplemented	orthogonallycomplemente	VERB
ma-125	17	19	submodules	submodule	NOUN
ma-125	17	20	of	of	ADP
ma-125	17	21	h	h	NOUN
ma-125	17	22	,	,	PUNCT
ma-125	17	23	where	where	SCONJ
ma-125	17	24	j	j	PROPN
ma-125	17	25	be	be	AUX
ma-125	17	26	finite	finite	ADJ
ma-125	17	27	or	or	CCONJ
ma-125	17	28	countable	countable	ADJ
ma-125	17	29	index	index	NOUN
ma-125	17	30	set	set	NOUN
ma-125	17	31	.	.	PUNCT
ma-125	18	1	end∗a(h	end∗a(h	PROPN
ma-125	18	2	,	,	PUNCT
ma-125	18	3	hj	hj	PROPN
ma-125	18	4	)	)	PUNCT
ma-125	18	5	is	be	AUX
ma-125	18	6	a	a	DET
ma-125	18	7	setof	setof	NOUN
ma-125	18	8	all	all	DET
ma-125	18	9	adjointable	adjointable	ADJ
ma-125	18	10	operator	operator	NOUN
ma-125	18	11	from	from	ADP
ma-125	18	12	h	h	NOUN
ma-125	18	13	to	to	PART
ma-125	18	14	hj	hj	PROPN
ma-125	18	15	.	.	PUNCT
ma-125	19	1	in	in	ADP
ma-125	19	2	particular	particular	ADJ
ma-125	19	3	end∗a(h	end∗a(h	NOUN
ma-125	19	4	)	)	PUNCT
ma-125	19	5	denote	denote	VERB
ma-125	19	6	the	the	DET
ma-125	19	7	set	set	NOUN
ma-125	19	8	of	of	ADP
ma-125	19	9	all	all	DET
ma-125	19	10	adjointableoperators	adjointableoperator	NOUN
ma-125	19	11	on	on	ADP
ma-125	19	12	h.	h.	PROPN
ma-125	19	13	pwj	pwj	PROPN
ma-125	19	14	denote	denote	VERB
ma-125	19	15	the	the	DET
ma-125	19	16	orthogonal	orthogonal	ADJ
ma-125	19	17	projection	projection	NOUN
ma-125	19	18	onto	onto	ADP
ma-125	19	19	the	the	DET
ma-125	19	20	closed	closed	ADJ
ma-125	19	21	submodule	submodule	NOUN
ma-125	19	22	orthogonally	orthogonally	ADV
ma-125	19	23	received	receive	VERB
ma-125	19	24	:	:	PUNCT
ma-125	19	25	31	31	NUM
ma-125	19	26	jul	jul	PROPN
ma-125	19	27	2022.2010	2022.2010	NUM
ma-125	19	28	mathematics	mathematic	NOUN
ma-125	19	29	subject	subject	ADJ
ma-125	19	30	classification	classification	NOUN
ma-125	19	31	.	.	PUNCT
ma-125	20	1	primary	primary	ADJ
ma-125	20	2	41a58	41a58	NOUN
ma-125	20	3	;	;	PUNCT
ma-125	20	4	secondary	secondary	ADJ
ma-125	20	5	42c15	42c15	NUM
ma-125	20	6	.	.	PUNCT
ma-125	21	1	key	key	ADJ
ma-125	21	2	words	word	NOUN
ma-125	21	3	and	and	CCONJ
ma-125	21	4	phrases	phrase	NOUN
ma-125	21	5	.	.	PUNCT
ma-125	22	1	fusion	fusion	NOUN
ma-125	22	2	frames	frame	NOUN
ma-125	22	3	;	;	PUNCT
ma-125	22	4	k	k	PROPN
ma-125	22	5	−	−	PROPN
ma-125	22	6	g−fusion	g−fusion	NOUN
ma-125	22	7	frames	frame	NOUN
ma-125	22	8	;	;	PUNCT
ma-125	22	9	woven	woven	ADJ
ma-125	22	10	k	k	PROPN
ma-125	22	11	−	−	NOUN
ma-125	22	12	g−fusion	g−fusion	NOUN
ma-125	22	13	frames	frame	NOUN
ma-125	22	14	;	;	PUNCT
ma-125	22	15	c∗-algebra	c∗-algebra	NOUN
ma-125	22	16	;	;	PUNCT
ma-125	22	17	hilbert	hilbert	PROPN
ma-125	22	18	c∗-modules	c∗-modules	PROPN
ma-125	22	19	.	.	PROPN
ma-125	23	1	1	1	NUM
ma-125	23	2	https://adac.ee	https://adac.ee	PROPN
ma-125	23	3	https://doi.org/10.28924/ada/ma.3.11	https://doi.org/10.28924/ada/ma.3.11	PROPN
ma-125	23	4	eur	eur	NOUN
ma-125	23	5	.	.	PUNCT
ma-125	24	1	j.	j.	PROPN
ma-125	24	2	math	math	PROPN
ma-125	24	3	.	.	PUNCT
ma-125	25	1	anal	anal	PROPN
ma-125	25	2	.	.	PUNCT
ma-125	26	1	10.28924	10.28924	NUM
ma-125	26	2	/	/	SYM
ma-125	26	3	ada	ada	PROPN
ma-125	26	4	/	/	SYM
ma-125	26	5	ma.3.11	ma.3.11	ADJ
ma-125	26	6	2complemented	2complemented	NUM
ma-125	26	7	wj	wj	PROPN
ma-125	26	8	of	of	ADP
ma-125	26	9	h.	h.	PROPN
ma-125	26	10	define	define	VERB
ma-125	26	11	the	the	DET
ma-125	26	12	module	module	NOUN
ma-125	26	13	l2({hj}j∈j	l2({hj}j∈j	NOUN
ma-125	26	14	)	)	PUNCT
ma-125	27	1	=	=	PRON
ma-125	27	2	{	{	PUNCT
ma-125	27	3	{	{	PUNCT
ma-125	27	4	fj}j∈j	fj}j∈j	PROPN
ma-125	27	5	:	:	PUNCT
ma-125	27	6	fj	fj	PROPN
ma-125	27	7	∈	∈	PROPN
ma-125	27	8	hj	hj	PROPN
ma-125	27	9	,	,	PUNCT
ma-125	27	10	‖	‖	PROPN
ma-125	27	11	∑	∑	PROPN
ma-125	27	12	j∈j	j∈j	NOUN
ma-125	27	13	〈	〈	PROPN
ma-125	27	14	fj	fj	PROPN
ma-125	27	15	,	,	PUNCT
ma-125	27	16	fj〉‖	fj〉‖	PROPN
ma-125	27	17	<	<	X
ma-125	27	18	∞	∞	NUM
ma-125	27	19	}	}	PUNCT
ma-125	27	20	with	with	ADP
ma-125	27	21	a−valued	a−value	VERB
ma-125	27	22	inner	inner	ADJ
ma-125	27	23	product	product	NOUN
ma-125	27	24	〈	〈	PROPN
ma-125	27	25	f	f	PROPN
ma-125	27	26	,	,	PUNCT
ma-125	27	27	g	g	PROPN
ma-125	27	28	〉	〉	NUM
ma-125	27	29	=	=	SYM
ma-125	27	30	∑	∑	PUNCT
ma-125	27	31	j∈j〈fj	j∈j〈fj	PROPN
ma-125	27	32	,	,	PUNCT
ma-125	27	33	gj	gj	PROPN
ma-125	27	34	〉	〉	PROPN
ma-125	27	35	,	,	PUNCT
ma-125	27	36	where	where	SCONJ
ma-125	27	37	f	f	AUX
ma-125	27	38	=	=	PRON
ma-125	27	39	{	{	PUNCT
ma-125	27	40	fj}j∈j	fj}j∈j	NOUN
ma-125	27	41	and	and	CCONJ
ma-125	27	42	g	g	PROPN
ma-125	27	43	=	=	SYM
ma-125	27	44	{	{	PUNCT
ma-125	27	45	gj}j∈j	gj}j∈j	ADJ
ma-125	27	46	,	,	PUNCT
ma-125	27	47	clearly	clearly	ADV
ma-125	27	48	l2({hj}j∈j	l2({hj}j∈j	ADJ
ma-125	27	49	)	)	PUNCT
ma-125	27	50	is	be	AUX
ma-125	27	51	a	a	DET
ma-125	27	52	hilbert	hilbert	PROPN
ma-125	27	53	a−module	a−module	PROPN
ma-125	27	54	.	.	NOUN
ma-125	27	55	definition	definition	NOUN
ma-125	27	56	1.1	1.1	NUM
ma-125	27	57	.	.	PUNCT
ma-125	28	1	[	[	X
ma-125	28	2	8	8	NUM
ma-125	28	3	]	]	PUNCT
ma-125	28	4	let	let	VERB
ma-125	28	5	a	a	PRON
ma-125	28	6	be	be	AUX
ma-125	28	7	a	a	DET
ma-125	28	8	unital	unital	ADJ
ma-125	28	9	c∗-algebra	c∗-algebra	NOUN
ma-125	28	10	and	and	CCONJ
ma-125	28	11	h	h	NOUN
ma-125	28	12	be	be	VERB
ma-125	28	13	a	a	DET
ma-125	28	14	left	left	ADJ
ma-125	28	15	a	a	DET
ma-125	28	16	-	-	PUNCT
ma-125	28	17	module	module	NOUN
ma-125	28	18	,	,	PUNCT
ma-125	28	19	such	such	ADJ
ma-125	28	20	that	that	SCONJ
ma-125	28	21	the	the	DET
ma-125	28	22	linearstructures	linearstructure	NOUN
ma-125	28	23	of	of	ADP
ma-125	28	24	a	a	PRON
ma-125	28	25	and	and	CCONJ
ma-125	28	26	h	h	NOUN
ma-125	28	27	are	be	AUX
ma-125	28	28	compatible	compatible	ADJ
ma-125	28	29	.	.	PUNCT
ma-125	29	1	h	h	NOUN
ma-125	29	2	is	be	AUX
ma-125	29	3	a	a	DET
ma-125	29	4	pre	pre	NOUN
ma-125	29	5	-	-	ADJ
ma-125	29	6	hilbert	hilbert	ADJ
ma-125	29	7	a	a	NOUN
ma-125	29	8	-	-	PUNCT
ma-125	29	9	module	module	NOUN
ma-125	29	10	if	if	SCONJ
ma-125	29	11	h	h	NOUN
ma-125	29	12	is	be	AUX
ma-125	29	13	equipped	equip	VERB
ma-125	29	14	with	with	ADP
ma-125	29	15	an	an	DET
ma-125	29	16	a	a	ADV
ma-125	29	17	-	-	PUNCT
ma-125	29	18	valued	value	VERB
ma-125	29	19	inner	inner	ADJ
ma-125	29	20	product	product	NOUN
ma-125	29	21	〈	〈	PROPN
ma-125	29	22	.	.	PROPN
ma-125	29	23	,	,	PUNCT
ma-125	29	24	.	.	PUNCT
ma-125	30	1	〉	〉	NOUN
ma-125	30	2	:	:	PUNCT
ma-125	31	1	h×h	h×h	PROPN
ma-125	31	2	→	→	SYM
ma-125	31	3	a	a	X
ma-125	31	4	,	,	PUNCT
ma-125	31	5	such	such	ADJ
ma-125	31	6	that	that	PRON
ma-125	31	7	is	be	AUX
ma-125	31	8	sesquilinear	sesquilinear	ADJ
ma-125	31	9	,	,	PUNCT
ma-125	31	10	positive	positive	ADJ
ma-125	31	11	definite	definite	ADJ
ma-125	31	12	and	and	CCONJ
ma-125	31	13	respectsthe	respectsthe	NOUN
ma-125	31	14	module	module	NOUN
ma-125	31	15	action	action	NOUN
ma-125	31	16	.	.	PUNCT
ma-125	32	1	in	in	ADP
ma-125	32	2	the	the	DET
ma-125	32	3	other	other	ADJ
ma-125	32	4	words,(i	words,(i	NOUN
ma-125	32	5	)	)	PUNCT
ma-125	33	1	〈	〈	PROPN
ma-125	33	2	f	f	PROPN
ma-125	33	3	,	,	PUNCT
ma-125	33	4	f	f	PROPN
ma-125	33	5	〉	〉	PROPN
ma-125	33	6	≥	≥	NOUN
ma-125	33	7	0	0	NUM
ma-125	33	8	for	for	ADP
ma-125	33	9	all	all	DET
ma-125	33	10	f	f	PROPN
ma-125	33	11	∈	∈	PROPN
ma-125	33	12	h	h	NOUN
ma-125	33	13	and	and	CCONJ
ma-125	33	14	〈	〈	PROPN
ma-125	33	15	f	f	PROPN
ma-125	33	16	,	,	PUNCT
ma-125	33	17	f	f	PROPN
ma-125	33	18	〉	〉	PROPN
ma-125	33	19	=	=	NOUN
ma-125	33	20	0	0	PUNCT
ma-125	34	1	if	if	SCONJ
ma-125	34	2	and	and	CCONJ
ma-125	34	3	only	only	ADV
ma-125	34	4	if	if	SCONJ
ma-125	34	5	f	f	PROPN
ma-125	34	6	=	=	SYM
ma-125	34	7	0.(ii	0.(ii	PROPN
ma-125	34	8	)	)	PUNCT
ma-125	34	9	〈	〈	PROPN
ma-125	34	10	af	af	PROPN
ma-125	34	11	+	+	SYM
ma-125	34	12	g	g	PROPN
ma-125	34	13	,	,	PUNCT
ma-125	34	14	h	h	NOUN
ma-125	34	15	〉	〉	NOUN
ma-125	34	16	=	=	SYM
ma-125	34	17	a〈f	a〈f	X
ma-125	34	18	,	,	PUNCT
ma-125	34	19	h〉+	h〉+	PROPN
ma-125	34	20	〈	〈	PROPN
ma-125	34	21	g	g	NOUN
ma-125	34	22	,	,	PUNCT
ma-125	34	23	h	h	NOUN
ma-125	34	24	〉	〉	NUM
ma-125	34	25	for	for	ADP
ma-125	34	26	all	all	DET
ma-125	34	27	a	a	DET
ma-125	34	28	∈	∈	PROPN
ma-125	34	29	a	a	PRON
ma-125	34	30	and	and	CCONJ
ma-125	34	31	f	f	PROPN
ma-125	34	32	,	,	PUNCT
ma-125	34	33	g	g	PROPN
ma-125	34	34	,	,	PUNCT
ma-125	34	35	h	h	NOUN
ma-125	34	36	∈	∈	PROPN
ma-125	34	37	h.(iii	h.(iii	VERB
ma-125	34	38	)	)	PUNCT
ma-125	34	39	〈	〈	PROPN
ma-125	34	40	f	f	PROPN
ma-125	34	41	,	,	PUNCT
ma-125	34	42	g	g	PROPN
ma-125	34	43	〉	〉	NOUN
ma-125	34	44	=	=	SYM
ma-125	35	1	〈	〈	PROPN
ma-125	35	2	g	g	NOUN
ma-125	35	3	,	,	PUNCT
ma-125	35	4	f	f	PROPN
ma-125	35	5	〉	〉	PROPN
ma-125	35	6	∗	∗	NOUN
ma-125	35	7	for	for	ADP
ma-125	35	8	all	all	DET
ma-125	35	9	f	f	PROPN
ma-125	35	10	,	,	PUNCT
ma-125	35	11	g	g	PROPN
ma-125	35	12	∈	∈	PROPN
ma-125	35	13	h.for	h.for	ADP
ma-125	35	14	f	f	PROPN
ma-125	35	15	∈	∈	PROPN
ma-125	35	16	h	h	NOUN
ma-125	35	17	,	,	PUNCT
ma-125	35	18	we	we	PRON
ma-125	35	19	define	define	VERB
ma-125	35	20	||f	||f	NOUN
ma-125	35	21	||	||	PUNCT
ma-125	36	1	=	=	PUNCT
ma-125	36	2	||〈f	||〈f	PROPN
ma-125	36	3	,	,	PUNCT
ma-125	36	4	f	f	PROPN
ma-125	36	5	〉	〉	PROPN
ma-125	36	6	||	||	PROPN
ma-125	36	7	1	1	NUM
ma-125	36	8	2	2	NUM
ma-125	36	9	.	.	PUNCT
ma-125	37	1	if	if	SCONJ
ma-125	37	2	h	h	NOUN
ma-125	37	3	is	be	AUX
ma-125	37	4	complete	complete	ADJ
ma-125	37	5	with	with	ADP
ma-125	37	6	||.||	||.||	NOUN
ma-125	37	7	,	,	PUNCT
ma-125	37	8	it	it	PRON
ma-125	37	9	is	be	AUX
ma-125	37	10	called	call	VERB
ma-125	37	11	a	a	DET
ma-125	37	12	hilbert	hilbert	NOUN
ma-125	37	13	a	a	DET
ma-125	37	14	-	-	PUNCT
ma-125	37	15	moduleor	moduleor	NOUN
ma-125	37	16	a	a	DET
ma-125	37	17	hilbert	hilbert	NOUN
ma-125	37	18	c∗-module	c∗-module	NOUN
ma-125	37	19	over	over	ADP
ma-125	37	20	a.	a.	NOUN
ma-125	37	21	for	for	ADP
ma-125	37	22	every	every	DET
ma-125	37	23	a	a	PRON
ma-125	37	24	in	in	ADP
ma-125	37	25	a	a	DET
ma-125	37	26	c∗-algebra	c∗-algebra	PROPN
ma-125	37	27	a	a	NOUN
ma-125	37	28	,	,	PUNCT
ma-125	37	29	we	we	PRON
ma-125	37	30	have	have	VERB
ma-125	37	31	|a|	|a|	NOUN
ma-125	37	32	=	=	SYM
ma-125	37	33	(	(	PUNCT
ma-125	37	34	a∗a	a∗a	X
ma-125	37	35	)	)	PUNCT
ma-125	37	36	1	1	NUM
ma-125	37	37	2	2	NUM
ma-125	37	38	and	and	CCONJ
ma-125	37	39	the	the	DET
ma-125	37	40	a	a	ADV
ma-125	37	41	-	-	PUNCT
ma-125	37	42	valued	value	VERB
ma-125	37	43	norm	norm	NOUN
ma-125	37	44	on	on	ADP
ma-125	37	45	h	h	NOUN
ma-125	37	46	is	be	AUX
ma-125	37	47	defined	define	VERB
ma-125	37	48	by	by	ADP
ma-125	37	49	|f	|f	PROPN
ma-125	38	1	|	|	NOUN
ma-125	38	2	=	=	SYM
ma-125	39	1	〈	〈	PROPN
ma-125	39	2	f	f	PROPN
ma-125	39	3	,	,	PUNCT
ma-125	39	4	f	f	PROPN
ma-125	39	5	〉	〉	PROPN
ma-125	39	6	1	1	NUM
ma-125	39	7	2	2	NUM
ma-125	39	8	for	for	ADP
ma-125	39	9	f	f	PROPN
ma-125	39	10	∈	∈	PROPN
ma-125	39	11	h.	h.	PROPN
ma-125	39	12	lemma	lemma	PROPN
ma-125	39	13	1.2	1.2	NUM
ma-125	39	14	.	.	PUNCT
ma-125	40	1	[	[	X
ma-125	40	2	10	10	NUM
ma-125	40	3	]	]	X
ma-125	40	4	let	let	AUX
ma-125	40	5	{	{	PUNCT
ma-125	40	6	wj}j∈j	wj}j∈j	ADP
ma-125	40	7	be	be	AUX
ma-125	40	8	a	a	DET
ma-125	40	9	sequence	sequence	NOUN
ma-125	40	10	of	of	ADP
ma-125	40	11	orthogonally	orthogonally	ADV
ma-125	40	12	complemented	complement	VERB
ma-125	40	13	closed	closed	ADJ
ma-125	40	14	submodules	submodule	NOUN
ma-125	40	15	of	of	ADP
ma-125	40	16	h	h	NOUN
ma-125	40	17	and	and	CCONJ
ma-125	40	18	t	t	PROPN
ma-125	40	19	∈	∈	PROPN
ma-125	40	20	end∗a(h	end∗a(h	NOUN
ma-125	40	21	)	)	PUNCT
ma-125	40	22	invertible	invertible	ADJ
ma-125	40	23	,	,	PUNCT
ma-125	40	24	if	if	SCONJ
ma-125	40	25	t	t	PROPN
ma-125	40	26	∗twj	∗twj	VERB
ma-125	40	27	⊂	⊂	PROPN
ma-125	40	28	wj	wj	PROPN
ma-125	40	29	for	for	ADP
ma-125	40	30	each	each	DET
ma-125	40	31	j	j	PROPN
ma-125	40	32	∈	∈	PROPN
ma-125	40	33	j	j	PROPN
ma-125	40	34	,	,	PUNCT
ma-125	40	35	then	then	ADV
ma-125	40	36	{	{	PUNCT
ma-125	40	37	twj}j∈j	twj}j∈j	NOUN
ma-125	40	38	is	be	AUX
ma-125	40	39	a	a	DET
ma-125	40	40	sequence	sequence	NOUN
ma-125	40	41	of	of	ADP
ma-125	40	42	orthogonally	orthogonally	ADV
ma-125	40	43	complemented	complement	VERB
ma-125	40	44	closed	closed	ADJ
ma-125	40	45	submodules	submodule	NOUN
ma-125	40	46	and	and	CCONJ
ma-125	40	47	pwj	pwj	VERB
ma-125	40	48	t	t	PROPN
ma-125	40	49	∗	∗	NOUN
ma-125	40	50	=	=	PUNCT
ma-125	41	1	pwj	pwj	PROPN
ma-125	41	2	t	t	PROPN
ma-125	41	3	∗ptwj	∗ptwj	NUM
ma-125	41	4	.	.	PUNCT
ma-125	42	1	lemma	lemma	PROPN
ma-125	42	2	1.3	1.3	NUM
ma-125	42	3	.	.	PUNCT
ma-125	43	1	[	[	X
ma-125	43	2	2	2	NUM
ma-125	43	3	]	]	PUNCT
ma-125	43	4	.	.	PUNCT
ma-125	44	1	let	let	VERB
ma-125	44	2	h	h	NOUN
ma-125	44	3	and	and	CCONJ
ma-125	44	4	k	k	PROPN
ma-125	44	5	two	two	NUM
ma-125	44	6	hilbert	hilbert	PROPN
ma-125	44	7	a	a	NOUN
ma-125	44	8	-	-	PUNCT
ma-125	44	9	modules	module	NOUN
ma-125	44	10	and	and	CCONJ
ma-125	44	11	t	t	NOUN
ma-125	44	12	∈	∈	PROPN
ma-125	44	13	end∗a(h	end∗a(h	NOUN
ma-125	44	14	,	,	PUNCT
ma-125	44	15	k	k	NOUN
ma-125	44	16	)	)	PUNCT
ma-125	44	17	.	.	PUNCT
ma-125	45	1	then	then	ADV
ma-125	45	2	the	the	DET
ma-125	45	3	following	follow	VERB
ma-125	45	4	statements	statement	NOUN
ma-125	45	5	are	be	AUX
ma-125	45	6	equivalent:(i	equivalent:(i	NOUN
ma-125	45	7	)	)	PUNCT
ma-125	46	1	t	t	PROPN
ma-125	46	2	is	be	AUX
ma-125	46	3	surjective.(ii	surjective.(ii	NOUN
ma-125	46	4	)	)	PUNCT
ma-125	47	1	t	t	PROPN
ma-125	47	2	∗	∗	NOUN
ma-125	47	3	is	be	AUX
ma-125	47	4	bounded	bound	VERB
ma-125	47	5	below	below	ADV
ma-125	47	6	with	with	ADP
ma-125	47	7	respect	respect	NOUN
ma-125	47	8	to	to	ADP
ma-125	47	9	norm	norm	NOUN
ma-125	47	10	,	,	PUNCT
ma-125	47	11	i.e.	i.e.	X
ma-125	47	12	,	,	PUNCT
ma-125	47	13	there	there	PRON
ma-125	47	14	is	be	VERB
ma-125	47	15	m	m	PROPN
ma-125	47	16	>	>	X
ma-125	47	17	0	0	NUM
ma-125	47	18	such	such	ADJ
ma-125	47	19	that	that	DET
ma-125	47	20	‖t	‖t	NOUN
ma-125	47	21	∗x‖	∗x‖	PROPN
ma-125	47	22	≥	≥	NOUN
ma-125	47	23	m‖x‖	m‖x‖	NOUN
ma-125	47	24	for	for	ADP
ma-125	47	25	all	all	DET
ma-125	47	26	x	x	SYM
ma-125	47	27	∈	∈	PROPN
ma-125	47	28	k.(iii	k.(iii	PROPN
ma-125	47	29	)	)	PUNCT
ma-125	47	30	t	t	PROPN
ma-125	47	31	∗	∗	NOUN
ma-125	47	32	is	be	AUX
ma-125	47	33	bounded	bound	VERB
ma-125	47	34	below	below	ADV
ma-125	47	35	with	with	ADP
ma-125	47	36	respect	respect	NOUN
ma-125	47	37	to	to	ADP
ma-125	47	38	the	the	DET
ma-125	47	39	inner	inner	ADJ
ma-125	47	40	product	product	NOUN
ma-125	47	41	,	,	PUNCT
ma-125	47	42	i.e.	i.e.	X
ma-125	47	43	,	,	PUNCT
ma-125	47	44	there	there	PRON
ma-125	47	45	is	be	VERB
ma-125	47	46	m′	m′	NOUN
ma-125	47	47	>	>	X
ma-125	47	48	0	0	NUM
ma-125	47	49	such	such	ADJ
ma-125	47	50	that	that	SCONJ
ma-125	47	51	〈	〈	PROPN
ma-125	47	52	t	t	PROPN
ma-125	47	53	∗x	∗x	NOUN
ma-125	47	54	,	,	PUNCT
ma-125	47	55	t	t	PROPN
ma-125	47	56	∗x	∗x	PROPN
ma-125	47	57	〉	〉	PROPN
ma-125	47	58	≥	≥	NOUN
ma-125	47	59	m′〈x	m′〈x	NOUN
ma-125	47	60	,	,	PUNCT
ma-125	47	61	x	x	NOUN
ma-125	47	62	〉	〉	NOUN
ma-125	47	63	for	for	ADP
ma-125	47	64	all	all	DET
ma-125	47	65	x	x	SYM
ma-125	47	66	∈	∈	PROPN
ma-125	47	67	k.	k.	PROPN
ma-125	47	68	lemma	lemma	PROPN
ma-125	47	69	1.4	1.4	NUM
ma-125	47	70	.	.	PUNCT
ma-125	48	1	[	[	X
ma-125	48	2	1	1	NUM
ma-125	48	3	]	]	PUNCT
ma-125	48	4	.	.	PUNCT
ma-125	49	1	let	let	VERB
ma-125	49	2	u	u	PRON
ma-125	49	3	and	and	CCONJ
ma-125	49	4	h	h	PROPN
ma-125	49	5	two	two	NUM
ma-125	49	6	hilbert	hilbert	PROPN
ma-125	49	7	a	a	NOUN
ma-125	49	8	-	-	PUNCT
ma-125	49	9	modules	module	NOUN
ma-125	49	10	and	and	CCONJ
ma-125	49	11	t	t	NOUN
ma-125	49	12	∈	∈	PROPN
ma-125	49	13	end∗a(u	end∗a(u	NOUN
ma-125	49	14	,	,	PUNCT
ma-125	49	15	h	h	NOUN
ma-125	49	16	)	)	PUNCT
ma-125	49	17	.	.	PUNCT
ma-125	50	1	then:(i	then:(i	PROPN
ma-125	50	2	)	)	PUNCT
ma-125	51	1	if	if	SCONJ
ma-125	51	2	t	t	PROPN
ma-125	51	3	is	be	AUX
ma-125	51	4	injective	injective	ADJ
ma-125	51	5	and	and	CCONJ
ma-125	51	6	t	t	PROPN
ma-125	51	7	has	have	AUX
ma-125	51	8	closed	close	VERB
ma-125	51	9	range	range	NOUN
ma-125	51	10	,	,	PUNCT
ma-125	51	11	then	then	ADV
ma-125	51	12	the	the	DET
ma-125	51	13	adjointable	adjointable	NOUN
ma-125	51	14	map	map	NOUN
ma-125	51	15	t	t	PROPN
ma-125	51	16	∗t	∗t	PROPN
ma-125	51	17	is	be	AUX
ma-125	51	18	invertible	invertible	ADJ
ma-125	51	19	and	and	CCONJ
ma-125	51	20	‖(t	‖(t	PUNCT
ma-125	51	21	∗t	∗t	ADJ
ma-125	51	22	)	)	PUNCT
ma-125	51	23	−1‖−1	−1‖−1	NOUN
ma-125	51	24	≤	≤	NUM
ma-125	51	25	t	t	X
ma-125	51	26	∗t	∗t	PROPN
ma-125	51	27	≤	≤	PROPN
ma-125	51	28	‖t‖2	‖t‖2	NOUN
ma-125	51	29	.	.	PUNCT
ma-125	52	1	(	(	PUNCT
ma-125	52	2	ii	ii	NOUN
ma-125	52	3	)	)	PUNCT
ma-125	52	4	if	if	SCONJ
ma-125	52	5	t	t	PROPN
ma-125	52	6	is	be	AUX
ma-125	52	7	surjective	surjective	ADJ
ma-125	52	8	,	,	PUNCT
ma-125	52	9	then	then	ADV
ma-125	52	10	the	the	DET
ma-125	52	11	adjointable	adjointable	NOUN
ma-125	52	12	map	map	NOUN
ma-125	52	13	tt	tt	PROPN
ma-125	52	14	∗	∗	NOUN
ma-125	52	15	is	be	AUX
ma-125	52	16	invertible	invertible	ADJ
ma-125	52	17	and	and	CCONJ
ma-125	52	18	‖(tt	‖(tt	PROPN
ma-125	52	19	∗)−1‖−1	∗)−1‖−1	PROPN
ma-125	52	20	≤	≤	ADV
ma-125	52	21	tt	tt	PROPN
ma-125	52	22	∗	∗	VERB
ma-125	52	23	≤	≤	PROPN
ma-125	52	24	‖t‖2	‖t‖2	PROPN
ma-125	52	25	.	.	PUNCT
ma-125	53	1	definition	definition	NOUN
ma-125	53	2	1.5	1.5	NUM
ma-125	53	3	.	.	PUNCT
ma-125	54	1	[	[	X
ma-125	54	2	10	10	NUM
ma-125	54	3	]	]	X
ma-125	54	4	let	let	AUX
ma-125	54	5	{	{	PUNCT
ma-125	54	6	wi}i∈i	wi}i∈i	X
ma-125	54	7	be	be	AUX
ma-125	54	8	a	a	DET
ma-125	54	9	sequence	sequence	NOUN
ma-125	54	10	of	of	ADP
ma-125	54	11	closed	close	VERB
ma-125	54	12	orthogonally	orthogonally	ADV
ma-125	54	13	complemented	complement	VERB
ma-125	54	14	submodulesof	submodulesof	NOUN
ma-125	54	15	h	h	NOUN
ma-125	54	16	,	,	PUNCT
ma-125	54	17	{	{	PUNCT
ma-125	54	18	vi}i∈i	vi}i∈i	INTJ
ma-125	54	19	be	be	AUX
ma-125	54	20	a	a	DET
ma-125	54	21	familly	familly	NOUN
ma-125	54	22	of	of	ADP
ma-125	54	23	positive	positive	ADJ
ma-125	54	24	weights	weight	NOUN
ma-125	54	25	in	in	ADP
ma-125	54	26	a	a	PRON
ma-125	54	27	,	,	PUNCT
ma-125	54	28	i.e.	i.e.	X
ma-125	54	29	,	,	PUNCT
ma-125	54	30	each	each	DET
ma-125	54	31	vi	vi	PROPN
ma-125	54	32	is	be	AUX
ma-125	54	33	a	a	DET
ma-125	54	34	positive	positive	ADJ
ma-125	54	35	invertible	invertible	ADJ
ma-125	54	36	element	element	NOUN
ma-125	54	37	from	from	ADP
ma-125	54	38	https://doi.org/10.28924/ada/ma.3.11	https://doi.org/10.28924/ada/ma.3.11	PROPN
ma-125	54	39	eur	eur	PROPN
ma-125	54	40	.	.	PUNCT
ma-125	55	1	j.	j.	PROPN
ma-125	55	2	math	math	PROPN
ma-125	55	3	.	.	PUNCT
ma-125	56	1	anal	anal	PROPN
ma-125	56	2	.	.	PUNCT
ma-125	57	1	10.28924	10.28924	NUM
ma-125	57	2	/	/	SYM
ma-125	57	3	ada	ada	PROPN
ma-125	57	4	/	/	SYM
ma-125	57	5	ma.3.11	ma.3.11	ADJ
ma-125	57	6	3the	3the	DET
ma-125	57	7	center	center	NOUN
ma-125	57	8	of	of	ADP
ma-125	57	9	the	the	DET
ma-125	57	10	c∗−algebra	c∗−algebra	PROPN
ma-125	57	11	a	a	PRON
ma-125	57	12	and	and	CCONJ
ma-125	57	13	λi	λi	NOUN
ma-125	57	14	∈	∈	PROPN
ma-125	57	15	end∗a(h	end∗a(h	NOUN
ma-125	57	16	,	,	PUNCT
ma-125	57	17	hi	hi	INTJ
ma-125	57	18	)	)	PUNCT
ma-125	57	19	for	for	ADP
ma-125	57	20	all	all	PRON
ma-125	57	21	i	i	PRON
ma-125	57	22	∈	∈	VERB
ma-125	58	1	i	i	PRON
ma-125	58	2	.	.	PUNCT
ma-125	59	1	we	we	PRON
ma-125	59	2	say	say	VERB
ma-125	59	3	that	that	SCONJ
ma-125	59	4	λ	λ	PROPN
ma-125	59	5	=	=	PRON
ma-125	59	6	{	{	PUNCT
ma-125	59	7	wi	wi	PROPN
ma-125	59	8	,	,	PUNCT
ma-125	59	9	λi	λi	INTJ
ma-125	59	10	,	,	PUNCT
ma-125	59	11	vi}i∈iis	vi}i∈iis	VERB
ma-125	59	12	a	a	DET
ma-125	59	13	g−fusion	g−fusion	NOUN
ma-125	59	14	frame	frame	NOUN
ma-125	59	15	for	for	ADP
ma-125	59	16	h	h	NOUN
ma-125	59	17	if	if	SCONJ
ma-125	60	1	and	and	CCONJ
ma-125	60	2	only	only	ADV
ma-125	60	3	if	if	SCONJ
ma-125	60	4	there	there	PRON
ma-125	60	5	exists	exist	VERB
ma-125	60	6	two	two	NUM
ma-125	60	7	constants	constant	NOUN
ma-125	60	8	0	0	PUNCT
ma-125	60	9	<	<	X
ma-125	60	10	a	a	DET
ma-125	60	11	≤	≤	NUM
ma-125	60	12	b	b	NOUN
ma-125	60	13	<	<	X
ma-125	60	14	∞	∞	NUM
ma-125	60	15	such	such	ADJ
ma-125	60	16	that	that	DET
ma-125	60	17	a〈x	a〈x	NOUN
ma-125	60	18	,	,	PUNCT
ma-125	60	19	x	x	X
ma-125	60	20	〉	〉	NOUN
ma-125	60	21	≤	≤	NOUN
ma-125	60	22	∑	∑	PUNCT
ma-125	60	23	i∈i	i∈i	ADV
ma-125	60	24	v2	v2	INTJ
ma-125	61	1	i	i	PRON
ma-125	61	2	〈	〈	PROPN
ma-125	61	3	λipwi	λipwi	PROPN
ma-125	61	4	x	x	X
ma-125	61	5	,	,	PUNCT
ma-125	61	6	λipwi	λipwi	PROPN
ma-125	61	7	x	x	PROPN
ma-125	61	8	〉	〉	PROPN
ma-125	61	9	≤	≤	NUM
ma-125	61	10	b〈x	b〈x	PUNCT
ma-125	61	11	,	,	PUNCT
ma-125	61	12	x	x	PROPN
ma-125	61	13	〉	〉	NOUN
ma-125	61	14	,	,	PUNCT
ma-125	61	15	∀x	∀x	X
ma-125	61	16	∈	∈	PROPN
ma-125	61	17	h.	h.	NOUN
ma-125	61	18	(	(	PUNCT
ma-125	61	19	1.1	1.1	NUM
ma-125	61	20	)	)	PUNCT
ma-125	61	21	the	the	DET
ma-125	61	22	constants	constant	NOUN
ma-125	61	23	a	a	PRON
ma-125	61	24	and	and	CCONJ
ma-125	61	25	b	b	NOUN
ma-125	61	26	are	be	AUX
ma-125	61	27	called	call	VERB
ma-125	61	28	the	the	DET
ma-125	61	29	lower	low	ADJ
ma-125	61	30	and	and	CCONJ
ma-125	61	31	upper	upper	ADJ
ma-125	61	32	bounds	bound	NOUN
ma-125	61	33	of	of	ADP
ma-125	61	34	g−fusion	g−fusion	NOUN
ma-125	61	35	frame	frame	NOUN
ma-125	61	36	,	,	PUNCT
ma-125	61	37	respectively	respectively	ADV
ma-125	61	38	.	.	PUNCT
ma-125	62	1	if	if	SCONJ
ma-125	62	2	a	a	DET
ma-125	62	3	=	=	SYM
ma-125	62	4	b	b	NOUN
ma-125	62	5	then	then	ADV
ma-125	62	6	λ	λ	PROPN
ma-125	62	7	is	be	AUX
ma-125	62	8	called	call	VERB
ma-125	62	9	tight	tight	ADJ
ma-125	62	10	g	g	NOUN
ma-125	62	11	-	-	PUNCT
ma-125	62	12	fusion	fusion	NOUN
ma-125	62	13	frame	frame	NOUN
ma-125	62	14	and	and	CCONJ
ma-125	62	15	if	if	SCONJ
ma-125	62	16	a	a	DET
ma-125	62	17	=	=	SYM
ma-125	62	18	b	b	NOUN
ma-125	62	19	=	=	SYM
ma-125	62	20	1	1	NUM
ma-125	62	21	then	then	ADV
ma-125	62	22	we	we	PRON
ma-125	62	23	say	say	VERB
ma-125	62	24	λ	λ	NOUN
ma-125	62	25	is	be	AUX
ma-125	62	26	a	a	DET
ma-125	62	27	parseval	parseval	NOUN
ma-125	62	28	g−fusionframe	g−fusionframe	NOUN
ma-125	62	29	.	.	PUNCT
ma-125	63	1	if	if	SCONJ
ma-125	63	2	λ	λ	PROPN
ma-125	63	3	satisfies	satisfy	VERB
ma-125	63	4	the	the	DET
ma-125	63	5	inequality∑	inequality∑	PROPN
ma-125	63	6	i∈i	i∈i	ADJ
ma-125	63	7	v2	v2	NOUN
ma-125	63	8	i	i	PRON
ma-125	63	9	〈	〈	PROPN
ma-125	63	10	λipwi	λipwi	PROPN
ma-125	63	11	x	x	X
ma-125	63	12	,	,	PUNCT
ma-125	63	13	λipwi	λipwi	PROPN
ma-125	63	14	x	x	PROPN
ma-125	63	15	〉	〉	PROPN
ma-125	63	16	≤	≤	NUM
ma-125	63	17	b〈x	b〈x	PUNCT
ma-125	63	18	,	,	PUNCT
ma-125	63	19	x	x	PROPN
ma-125	63	20	〉	〉	NOUN
ma-125	63	21	,	,	PUNCT
ma-125	63	22	∀x	∀x	X
ma-125	63	23	∈	∈	PROPN
ma-125	63	24	h.	h.	NOUN
ma-125	63	25	then	then	ADV
ma-125	63	26	it	it	PRON
ma-125	63	27	is	be	AUX
ma-125	63	28	called	call	VERB
ma-125	63	29	a	a	DET
ma-125	63	30	g−fusion	g−fusion	NOUN
ma-125	63	31	bessel	bessel	NOUN
ma-125	63	32	sequence	sequence	NOUN
ma-125	63	33	with	with	ADP
ma-125	63	34	bound	bind	VERB
ma-125	63	35	b	b	PROPN
ma-125	63	36	in	in	ADP
ma-125	63	37	h.	h.	PROPN
ma-125	63	38	definition	definition	NOUN
ma-125	63	39	1.6	1.6	NUM
ma-125	63	40	.	.	PUNCT
ma-125	64	1	[	[	X
ma-125	64	2	10]let	10]let	NUM
ma-125	64	3	λ	λ	NOUN
ma-125	64	4	=	=	SYM
ma-125	64	5	{	{	PUNCT
ma-125	64	6	wj	wj	PROPN
ma-125	64	7	,	,	PUNCT
ma-125	64	8	λj	λj	INTJ
ma-125	64	9	,	,	PUNCT
ma-125	64	10	vj}j∈j	vj}j∈j	PART
ma-125	64	11	be	be	AUX
ma-125	64	12	a	a	DET
ma-125	64	13	g−fusion	g−fusion	NOUN
ma-125	64	14	bessel	bessel	NOUN
ma-125	64	15	sequence	sequence	NOUN
ma-125	64	16	for	for	ADP
ma-125	64	17	h.	h.	PROPN
ma-125	64	18	then	then	ADV
ma-125	64	19	the	the	DET
ma-125	64	20	operator	operator	NOUN
ma-125	64	21	tλ	tλ	VERB
ma-125	64	22	:	:	PUNCT
ma-125	64	23	l2({hj}j∈j)→	l2({hj}j∈j)→	NOUN
ma-125	64	24	h	h	NOUN
ma-125	64	25	defined	define	VERB
ma-125	64	26	by	by	ADP
ma-125	64	27	tλ({fj}j∈j	tλ({fj}j∈j	ADP
ma-125	64	28	)	)	PUNCT
ma-125	64	29	=	=	PUNCT
ma-125	64	30	∑	∑	PUNCT
ma-125	64	31	j∈j	j∈j	NOUN
ma-125	64	32	vjpwj	vjpwj	NOUN
ma-125	64	33	λ∗j	λ∗j	PUNCT
ma-125	64	34	fj	fj	PROPN
ma-125	64	35	,	,	PUNCT
ma-125	64	36	∀{fj}j∈j	∀{fj}j∈j	NUM
ma-125	64	37	∈	∈	PROPN
ma-125	64	38	l2({hj}j∈j	l2({hj}j∈j	PROPN
ma-125	64	39	)	)	PUNCT
ma-125	64	40	.	.	PUNCT
ma-125	65	1	is	be	AUX
ma-125	65	2	called	call	VERB
ma-125	65	3	synthesis	synthesis	NOUN
ma-125	65	4	operator	operator	NOUN
ma-125	65	5	.	.	PUNCT
ma-125	66	1	we	we	PRON
ma-125	66	2	say	say	VERB
ma-125	66	3	the	the	DET
ma-125	66	4	adjoint	adjoint	PROPN
ma-125	66	5	uλ	uλ	ADP
ma-125	66	6	of	of	ADP
ma-125	66	7	the	the	DET
ma-125	66	8	synthesis	synthesis	NOUN
ma-125	66	9	operator	operator	NOUN
ma-125	66	10	the	the	DET
ma-125	66	11	analysis	analysis	NOUN
ma-125	66	12	operatorand	operatorand	NOUN
ma-125	67	1	it	it	PRON
ma-125	67	2	is	be	AUX
ma-125	67	3	defined	define	VERB
ma-125	67	4	by	by	ADP
ma-125	67	5	uλ	uλ	X
ma-125	67	6	:	:	PUNCT
ma-125	67	7	h	h	PROPN
ma-125	67	8	→	→	SYM
ma-125	67	9	l2({hj}j∈j	l2({hj}j∈j	NOUN
ma-125	67	10	)	)	PUNCT
ma-125	67	11	such	such	ADJ
ma-125	67	12	that	that	SCONJ
ma-125	67	13	uλ(f	uλ(f	PUNCT
ma-125	67	14	)	)	PUNCT
ma-125	68	1	=	=	SYM
ma-125	68	2	{	{	PUNCT
ma-125	68	3	vjλjpwj	vjλjpwj	PROPN
ma-125	68	4	(	(	PUNCT
ma-125	68	5	f	f	NOUN
ma-125	68	6	)	)	PUNCT
ma-125	68	7	}	}	PUNCT
ma-125	68	8	j∈j	j∈j	NOUN
ma-125	68	9	,	,	PUNCT
ma-125	68	10	∀f	∀f	PROPN
ma-125	68	11	∈	∈	PROPN
ma-125	68	12	h.	h.	NOUN
ma-125	68	13	the	the	DET
ma-125	68	14	operator	operator	NOUN
ma-125	68	15	sλ	sλ	NOUN
ma-125	68	16	:	:	PUNCT
ma-125	68	17	h	h	NOUN
ma-125	68	18	→	→	SYM
ma-125	68	19	h	h	PRON
ma-125	68	20	defined	define	VERB
ma-125	68	21	by	by	ADP
ma-125	68	22	sλf	sλf	NOUN
ma-125	68	23	=	=	SYM
ma-125	68	24	tλuλf	tλuλf	NOUN
ma-125	68	25	=	=	PUNCT
ma-125	68	26	∑	∑	PUNCT
ma-125	68	27	j∈j	j∈j	PROPN
ma-125	68	28	v2	v2	PROPN
ma-125	68	29	j	j	PROPN
ma-125	68	30	pwj	pwj	PROPN
ma-125	68	31	λ∗j	λ∗j	PUNCT
ma-125	68	32	λjpwj	λjpwj	PROPN
ma-125	68	33	(	(	PUNCT
ma-125	68	34	f	f	PROPN
ma-125	68	35	)	)	PUNCT
ma-125	68	36	,	,	PUNCT
ma-125	68	37	∀f	∀f	PROPN
ma-125	68	38	∈	∈	PROPN
ma-125	68	39	h.	h.	NOUN
ma-125	68	40	is	be	AUX
ma-125	68	41	called	call	VERB
ma-125	68	42	g−fusion	g−fusion	NOUN
ma-125	68	43	frame	frame	NOUN
ma-125	68	44	operator	operator	NOUN
ma-125	68	45	.	.	PUNCT
ma-125	69	1	it	it	PRON
ma-125	69	2	can	can	AUX
ma-125	69	3	be	be	AUX
ma-125	69	4	easily	easily	ADV
ma-125	69	5	verify	verify	VERB
ma-125	69	6	that	that	SCONJ
ma-125	69	7	〈	〈	PROPN
ma-125	69	8	sλf	sλf	PROPN
ma-125	69	9	,	,	PUNCT
ma-125	69	10	f	f	PROPN
ma-125	69	11	〉	〉	PROPN
ma-125	69	12	=	=	PUNCT
ma-125	70	1	∑	∑	PUNCT
ma-125	70	2	j∈j	j∈j	PROPN
ma-125	70	3	v2	v2	PROPN
ma-125	70	4	j	j	PROPN
ma-125	70	5	〈	〈	NOUN
ma-125	70	6	λjpwj	λjpwj	X
ma-125	70	7	(	(	PUNCT
ma-125	70	8	f	f	PROPN
ma-125	70	9	)	)	PUNCT
ma-125	70	10	,	,	PUNCT
ma-125	70	11	λjpwj	λjpwj	VERB
ma-125	70	12	(	(	PUNCT
ma-125	70	13	f	f	PROPN
ma-125	70	14	)	)	PUNCT
ma-125	70	15	〉	〉	PROPN
ma-125	70	16	,	,	PUNCT
ma-125	70	17	∀f	∀f	PROPN
ma-125	70	18	∈	∈	PROPN
ma-125	70	19	h.	h.	NOUN
ma-125	70	20	(	(	PUNCT
ma-125	70	21	1.2	1.2	NUM
ma-125	70	22	)	)	PUNCT
ma-125	70	23	furthermore	furthermore	ADV
ma-125	70	24	,	,	PUNCT
ma-125	70	25	if	if	SCONJ
ma-125	70	26	λ	λ	PROPN
ma-125	70	27	is	be	AUX
ma-125	70	28	a	a	DET
ma-125	70	29	g−fusion	g−fusion	NOUN
ma-125	70	30	frame	frame	NOUN
ma-125	70	31	with	with	ADP
ma-125	70	32	bounds	bound	NOUN
ma-125	70	33	a	a	PRON
ma-125	70	34	and	and	CCONJ
ma-125	70	35	b	b	NOUN
ma-125	70	36	,	,	PUNCT
ma-125	70	37	then	then	ADV
ma-125	70	38	a〈f	a〈f	PROPN
ma-125	70	39	,	,	PUNCT
ma-125	70	40	f	f	PROPN
ma-125	70	41	〉	〉	PROPN
ma-125	70	42	≤	≤	PUNCT
ma-125	71	1	〈	〈	PROPN
ma-125	71	2	sλf	sλf	VERB
ma-125	71	3	,	,	PUNCT
ma-125	71	4	f	f	PROPN
ma-125	71	5	〉	〉	PROPN
ma-125	71	6	≤	≤	PROPN
ma-125	71	7	b〈f	b〈f	PUNCT
ma-125	72	1	,	,	PUNCT
ma-125	72	2	f	f	PROPN
ma-125	72	3	〉	〉	PROPN
ma-125	72	4	,	,	PUNCT
ma-125	72	5	∀f	∀f	PROPN
ma-125	72	6	∈	∈	PROPN
ma-125	72	7	h.	h.	NOUN
ma-125	72	8	it	it	PRON
ma-125	72	9	easy	easy	ADJ
ma-125	72	10	to	to	PART
ma-125	72	11	see	see	VERB
ma-125	72	12	that	that	SCONJ
ma-125	72	13	the	the	DET
ma-125	72	14	operator	operator	NOUN
ma-125	72	15	sλ	sλ	NOUN
ma-125	72	16	is	be	AUX
ma-125	72	17	bounded	bound	VERB
ma-125	72	18	,	,	PUNCT
ma-125	72	19	self	self	NOUN
ma-125	72	20	-	-	PUNCT
ma-125	72	21	adjoint	adjoint	NOUN
ma-125	72	22	,	,	PUNCT
ma-125	72	23	positive	positive	ADJ
ma-125	72	24	,	,	PUNCT
ma-125	72	25	now	now	ADV
ma-125	72	26	we	we	PRON
ma-125	72	27	proof	proof	VERB
ma-125	72	28	the	the	DET
ma-125	72	29	inversibilityof	inversibilityof	ADJ
ma-125	72	30	sλ	sλ	NOUN
ma-125	72	31	.	.	PUNCT
ma-125	73	1	let	let	VERB
ma-125	73	2	f	f	PRON
ma-125	73	3	∈	∈	PROPN
ma-125	73	4	h	h	NOUN
ma-125	73	5	we	we	PRON
ma-125	73	6	have	have	VERB
ma-125	73	7	||uλ(f	||uλ(f	PROPN
ma-125	73	8	)	)	PUNCT
ma-125	73	9	||	||	PUNCT
ma-125	74	1	=	=	PUNCT
ma-125	74	2	||{vjλjpwj	||{vjλjpwj	PROPN
ma-125	74	3	(	(	PUNCT
ma-125	74	4	f	f	PROPN
ma-125	74	5	)	)	PUNCT
ma-125	74	6	}	}	PUNCT
ma-125	74	7	j∈i	j∈i	PROPN
ma-125	74	8	||	||	PUNCT
ma-125	75	1	=	=	PUNCT
ma-125	75	2	||	||	NOUN
ma-125	75	3	∑	∑	PUNCT
ma-125	75	4	j∈j	j∈j	NOUN
ma-125	75	5	v2	v2	PROPN
ma-125	75	6	j	j	PROPN
ma-125	75	7	〈	〈	NOUN
ma-125	75	8	λjpwj	λjpwj	X
ma-125	75	9	(	(	PUNCT
ma-125	75	10	f	f	PROPN
ma-125	75	11	)	)	PUNCT
ma-125	75	12	,	,	PUNCT
ma-125	75	13	λjpwj	λjpwj	VERB
ma-125	75	14	(	(	PUNCT
ma-125	75	15	f	f	PROPN
ma-125	75	16	)	)	PUNCT
ma-125	75	17	〉	〉	PROPN
ma-125	75	18	||	||	NOUN
ma-125	75	19	1	1	NUM
ma-125	75	20	2	2	NUM
ma-125	75	21	.	.	PUNCT
ma-125	76	1	since	since	SCONJ
ma-125	76	2	λ	λ	PROPN
ma-125	76	3	is	be	AUX
ma-125	76	4	g−fusion	g−fusion	NOUN
ma-125	76	5	frame	frame	NOUN
ma-125	76	6	then	then	ADV
ma-125	76	7	√	√	PROPN
ma-125	76	8	a||〈f	a||〈f	PROPN
ma-125	76	9	,	,	PUNCT
ma-125	77	1	f	f	PROPN
ma-125	77	2	〉	〉	PROPN
ma-125	77	3	||	||	NOUN
ma-125	77	4	1	1	NUM
ma-125	77	5	2	2	NUM
ma-125	77	6	≤	≤	NUM
ma-125	77	7	||uλf	||uλf	PROPN
ma-125	77	8	||.then	||.then	PUNCT
ma-125	78	1	√	√	PROPN
ma-125	78	2	a||f	a||f	PROPN
ma-125	78	3	||	||	NOUN
ma-125	79	1	≤	≤	NUM
ma-125	80	1	||uλf	||uλf	PROPN
ma-125	80	2	||	||	PROPN
ma-125	80	3	.	.	PUNCT
ma-125	81	1	https://doi.org/10.28924/ada/ma.3.11	https://doi.org/10.28924/ada/ma.3.11	PROPN
ma-125	81	2	eur	eur	PROPN
ma-125	81	3	.	.	PUNCT
ma-125	82	1	j.	j.	PROPN
ma-125	82	2	math	math	PROPN
ma-125	82	3	.	.	PUNCT
ma-125	83	1	anal	anal	PROPN
ma-125	83	2	.	.	PUNCT
ma-125	84	1	10.28924	10.28924	NUM
ma-125	84	2	/	/	SYM
ma-125	84	3	ada	ada	PROPN
ma-125	84	4	/	/	SYM
ma-125	84	5	ma.3.11	ma.3.11	PROPN
ma-125	85	1	4frome	4frome	NUM
ma-125	85	2	lemma	lemma	PROPN
ma-125	85	3	1.3	1.3	NUM
ma-125	85	4	,	,	PUNCT
ma-125	85	5	tλ	tλ	PART
ma-125	85	6	is	be	AUX
ma-125	85	7	surjective	surjective	ADJ
ma-125	85	8	and	and	CCONJ
ma-125	85	9	by	by	ADP
ma-125	85	10	lemma	lemma	PROPN
ma-125	85	11	1.4	1.4	NUM
ma-125	85	12	,	,	PUNCT
ma-125	85	13	tλuλ	tλuλ	PROPN
ma-125	85	14	=	=	SYM
ma-125	85	15	sλ	sλ	NOUN
ma-125	85	16	is	be	AUX
ma-125	85	17	invertible	invertible	ADJ
ma-125	85	18	.	.	PUNCT
ma-125	86	1	we	we	PRON
ma-125	86	2	now	now	ADV
ma-125	86	3	,	,	PUNCT
ma-125	86	4	aih	aih	X
ma-125	86	5	≤	≤	NUM
ma-125	86	6	sλ	sλ	NOUN
ma-125	86	7	≤	≤	NUM
ma-125	86	8	bih	bih	ADV
ma-125	86	9	and	and	CCONJ
ma-125	86	10	this	this	PRON
ma-125	86	11	gives	give	VERB
ma-125	86	12	b−1ih	b−1ih	NOUN
ma-125	86	13	≤	≤	ADJ
ma-125	86	14	s−1	s−1	PROPN
ma-125	86	15	λ	λ	PROPN
ma-125	86	16	≤	≤	NOUN
ma-125	86	17	a−1ih	a−1ih	NOUN
ma-125	86	18	.	.	PUNCT
ma-125	87	1	2	2	X
ma-125	87	2	.	.	X
ma-125	87	3	woven	weave	VERB
ma-125	87	4	k	k	PROPN
ma-125	88	1	−	−	NOUN
ma-125	88	2	g−fusion	g−fusion	NOUN
ma-125	88	3	frames	frame	NOUN
ma-125	88	4	in	in	ADP
ma-125	88	5	hilbert	hilbert	PROPN
ma-125	88	6	c∗−modules	c∗−module	NOUN
ma-125	88	7	throughout	throughout	ADP
ma-125	88	8	this	this	DET
ma-125	88	9	paper	paper	NOUN
ma-125	88	10	,	,	PUNCT
ma-125	89	1	[	[	X
ma-125	89	2	m	m	X
ma-125	89	3	]	]	X
ma-125	89	4	=	=	X
ma-125	89	5	{	{	PUNCT
ma-125	89	6	1	1	NUM
ma-125	89	7	,	,	PUNCT
ma-125	89	8	2	2	NUM
ma-125	89	9	,	,	PUNCT
ma-125	89	10	...	...	PUNCT
ma-125	89	11	,	,	PUNCT
ma-125	89	12	m	m	VERB
ma-125	89	13	}	}	PUNCT
ma-125	89	14	for	for	ADP
ma-125	89	15	each	each	DET
ma-125	89	16	m	m	NOUN
ma-125	89	17	>	>	X
ma-125	89	18	1	1	NUM
ma-125	89	19	,	,	PUNCT
ma-125	89	20	{	{	PUNCT
ma-125	89	21	wi	wi	PROPN
ma-125	89	22	j}j∈j	j}j∈j	PROPN
ma-125	89	23	,	,	PUNCT
ma-125	89	24	i∈[m	i∈[m	PROPN
ma-125	89	25	]	]	PUNCT
ma-125	89	26	is	be	AUX
ma-125	89	27	a	a	DET
ma-125	89	28	collection	collection	NOUN
ma-125	89	29	of	of	ADP
ma-125	89	30	closedorthogonally	closedorthogonally	ADV
ma-125	89	31	complemented	complement	VERB
ma-125	89	32	submodules	submodule	NOUN
ma-125	89	33	of	of	ADP
ma-125	89	34	h	h	NOUN
ma-125	89	35	,	,	PUNCT
ma-125	89	36	{	{	PUNCT
ma-125	89	37	vi	vi	PROPN
ma-125	89	38	j}j∈j	j}j∈j	PROPN
ma-125	89	39	,	,	PUNCT
ma-125	89	40	i∈[m	i∈[m	PROPN
ma-125	89	41	]	]	PUNCT
ma-125	89	42	is	be	AUX
ma-125	89	43	a	a	DET
ma-125	89	44	family	family	NOUN
ma-125	89	45	of	of	ADP
ma-125	89	46	weights	weight	NOUN
ma-125	89	47	,	,	PUNCT
ma-125	89	48	k	k	PROPN
ma-125	89	49	∈	∈	PROPN
ma-125	89	50	end∗a(h)and	end∗a(h)and	PROPN
ma-125	89	51	{	{	PUNCT
ma-125	89	52	λi	λi	X
ma-125	89	53	j}j∈j	j}j∈j	X
ma-125	89	54	,	,	PUNCT
ma-125	89	55	i∈[m	i∈[m	NOUN
ma-125	89	56	]	]	X
ma-125	89	57	∈	∈	PROPN
ma-125	89	58	end∗a(h	end∗a(h	NOUN
ma-125	89	59	,	,	PUNCT
ma-125	89	60	hi	hi	PROPN
ma-125	89	61	j	j	NOUN
ma-125	89	62	)	)	PUNCT
ma-125	89	63	where	where	SCONJ
ma-125	89	64	hi	hi	INTJ
ma-125	89	65	j	j	PROPN
ma-125	89	66	are	be	AUX
ma-125	89	67	hilbert	hilbert	PROPN
ma-125	89	68	a−modules	a−module	NOUN
ma-125	89	69	.	.	PUNCT
ma-125	90	1	definition	definition	NOUN
ma-125	90	2	2.1	2.1	NUM
ma-125	90	3	.	.	PUNCT
ma-125	91	1	a	a	DET
ma-125	91	2	family	family	NOUN
ma-125	91	3	of	of	ADP
ma-125	91	4	g−fusion	g−fusion	NOUN
ma-125	91	5	frames	frame	NOUN
ma-125	91	6	{	{	PUNCT
ma-125	91	7	wi	wi	PROPN
ma-125	91	8	j	j	PROPN
ma-125	91	9	,	,	PUNCT
ma-125	91	10	λi	λi	PROPN
ma-125	91	11	j	j	PROPN
ma-125	91	12	,	,	PUNCT
ma-125	91	13	vi	vi	PROPN
ma-125	91	14	j}j∈j	j}j∈j	PROPN
ma-125	91	15	,	,	PUNCT
ma-125	91	16	i∈[m	i∈[m	PROPN
ma-125	91	17	]	]	PUNCT
ma-125	91	18	for	for	ADP
ma-125	91	19	h	h	NOUN
ma-125	91	20	is	be	AUX
ma-125	91	21	said	say	VERB
ma-125	91	22	to	to	PART
ma-125	91	23	be	be	AUX
ma-125	91	24	k−	k−	NOUN
ma-125	91	25	g−fusionwoven	g−fusionwoven	VERB
ma-125	91	26	if	if	SCONJ
ma-125	91	27	there	there	PRON
ma-125	91	28	exist	exist	VERB
ma-125	91	29	universal	universal	ADJ
ma-125	91	30	positive	positive	ADJ
ma-125	91	31	constants	constant	NOUN
ma-125	91	32	0	0	PUNCT
ma-125	91	33	<	<	X
ma-125	91	34	a	a	DET
ma-125	91	35	≤	≤	NUM
ma-125	91	36	b	b	NOUN
ma-125	91	37	such	such	ADJ
ma-125	91	38	that	that	DET
ma-125	91	39	for	for	ADP
ma-125	91	40	each	each	DET
ma-125	91	41	partition	partition	NOUN
ma-125	91	42	{	{	PUNCT
ma-125	91	43	σi}i∈[m]of	σi}i∈[m]of	PROPN
ma-125	91	44	j	j	PROPN
ma-125	91	45	,	,	PUNCT
ma-125	91	46	the	the	DET
ma-125	91	47	family	family	NOUN
ma-125	91	48	{	{	PUNCT
ma-125	91	49	wi	wi	PROPN
ma-125	91	50	j	j	PROPN
ma-125	91	51	,	,	PUNCT
ma-125	91	52	λi	λi	PROPN
ma-125	91	53	j	j	PROPN
ma-125	91	54	,	,	PUNCT
ma-125	91	55	vi	vi	PROPN
ma-125	91	56	j}j∈σi	j}j∈σi	NOUN
ma-125	91	57	,	,	PUNCT
ma-125	91	58	i∈[m	i∈[m	PROPN
ma-125	91	59	]	]	PUNCT
ma-125	91	60	is	be	AUX
ma-125	91	61	a	a	DET
ma-125	91	62	k	k	NOUN
ma-125	91	63	−	−	PROPN
ma-125	91	64	g−fusion	g−fusion	NOUN
ma-125	91	65	frame	frame	NOUN
ma-125	91	66	for	for	ADP
ma-125	91	67	h	h	NOUN
ma-125	91	68	with	with	ADP
ma-125	91	69	bounds	bound	NOUN
ma-125	91	70	a	a	PRON
ma-125	91	71	and	and	CCONJ
ma-125	91	72	b.	b.	PROPN
ma-125	92	1	in	in	ADP
ma-125	92	2	next	next	PROPN
ma-125	92	3	theorem	theorem	NOUN
ma-125	92	4	,	,	PUNCT
ma-125	92	5	we	we	PRON
ma-125	92	6	provide	provide	VERB
ma-125	92	7	a	a	DET
ma-125	92	8	necessary	necessary	ADJ
ma-125	92	9	and	and	CCONJ
ma-125	92	10	sufficient	sufficient	ADJ
ma-125	92	11	condition	condition	NOUN
ma-125	92	12	for	for	ADP
ma-125	92	13	weaving	weave	VERB
ma-125	92	14	k−g−fusion	k−g−fusion	PROPN
ma-125	92	15	frames	frame	NOUN
ma-125	92	16	.	.	PUNCT
ma-125	93	1	theorem	theorem	VERB
ma-125	93	2	2.2	2.2	NUM
ma-125	93	3	.	.	PUNCT
ma-125	94	1	assume	assume	VERB
ma-125	94	2	that	that	SCONJ
ma-125	94	3	{	{	PUNCT
ma-125	94	4	wj	wj	PROPN
ma-125	94	5	,	,	PUNCT
ma-125	94	6	λj	λj	INTJ
ma-125	94	7	,	,	PUNCT
ma-125	94	8	vj}j∈j	vj}j∈j	X
ma-125	94	9	and	and	CCONJ
ma-125	94	10	{	{	PUNCT
ma-125	94	11	vj	vj	INTJ
ma-125	94	12	,	,	PUNCT
ma-125	94	13	θj	θj	INTJ
ma-125	94	14	,	,	PUNCT
ma-125	94	15	µj}j∈j	µj}j∈j	X
ma-125	94	16	are	be	AUX
ma-125	94	17	two	two	NUM
ma-125	94	18	k	k	ADJ
ma-125	94	19	−	−	NOUN
ma-125	94	20	g−fusion	g−fusion	NOUN
ma-125	94	21	frames	frame	NOUN
ma-125	94	22	for	for	ADP
ma-125	94	23	h	h	NOUN
ma-125	94	24	where	where	SCONJ
ma-125	94	25	λj	λj	PROPN
ma-125	94	26	∈	∈	PROPN
ma-125	94	27	end∗a(h	end∗a(h	NOUN
ma-125	94	28	,	,	PUNCT
ma-125	94	29	hj	hj	PROPN
ma-125	94	30	)	)	PUNCT
ma-125	94	31	and	and	CCONJ
ma-125	94	32	θj	θj	NOUN
ma-125	94	33	∈	∈	PROPN
ma-125	94	34	end∗a(h	end∗a(h	NOUN
ma-125	94	35	,	,	PUNCT
ma-125	94	36	hj	hj	PROPN
ma-125	94	37	)	)	PUNCT
ma-125	94	38	for	for	ADP
ma-125	94	39	any	any	DET
ma-125	94	40	j	j	PROPN
ma-125	94	41	∈	∈	PROPN
ma-125	94	42	j	j	PROPN
ma-125	94	43	,	,	PUNCT
ma-125	94	44	the	the	DET
ma-125	94	45	following	follow	VERB
ma-125	94	46	assertions	assertion	NOUN
ma-125	94	47	are	be	AUX
ma-125	94	48	equivalent.(1	equivalent.(1	PROPN
ma-125	94	49	)	)	PUNCT
ma-125	94	50	{	{	PUNCT
ma-125	94	51	wj	wj	PROPN
ma-125	94	52	,	,	PUNCT
ma-125	94	53	λj	λj	INTJ
ma-125	94	54	,	,	PUNCT
ma-125	94	55	vj}j∈j	vj}j∈j	X
ma-125	94	56	and	and	CCONJ
ma-125	94	57	{	{	PUNCT
ma-125	94	58	vj	vj	INTJ
ma-125	94	59	,	,	PUNCT
ma-125	94	60	θj	θj	INTJ
ma-125	94	61	,	,	PUNCT
ma-125	94	62	µj}j∈j	µj}j∈j	X
ma-125	94	63	are	be	AUX
ma-125	94	64	k	k	PROPN
ma-125	94	65	−	−	PROPN
ma-125	94	66	g−fusion	g−fusion	NOUN
ma-125	94	67	woven.(2	woven.(2	NOUN
ma-125	94	68	)	)	PUNCT
ma-125	94	69	there	there	PRON
ma-125	94	70	exists	exist	VERB
ma-125	94	71	α	α	PROPN
ma-125	94	72	>	>	X
ma-125	94	73	0	0	NUM
ma-125	94	74	such	such	ADJ
ma-125	94	75	that	that	PRON
ma-125	94	76	for	for	ADP
ma-125	94	77	each	each	DET
ma-125	94	78	σ	σ	PROPN
ma-125	94	79	⊂	⊂	PROPN
ma-125	94	80	j	j	PROPN
ma-125	94	81	there	there	PRON
ma-125	94	82	exists	exist	VERB
ma-125	94	83	a	a	DET
ma-125	94	84	bounded	bounded	ADJ
ma-125	94	85	linear	linear	ADJ
ma-125	94	86	operator	operator	NOUN
ma-125	94	87	ψσ	ψσ	ADP
ma-125	94	88	:	:	PUNCT
ma-125	94	89	lσ2	lσ2	PROPN
ma-125	94	90	(	(	PUNCT
ma-125	94	91	{	{	PUNCT
ma-125	94	92	hj}j∈j)→	hj}j∈j)→	NOUN
ma-125	94	93	h	h	NOUN
ma-125	94	94	,	,	PUNCT
ma-125	94	95	ψσ{xj}j∈j	ψσ{xj}j∈j	ADP
ma-125	94	96	=	=	SYM
ma-125	94	97	∑	∑	PROPN
ma-125	94	98	j∈σ	j∈σ	PROPN
ma-125	94	99	vjpwj	vjpwj	PROPN
ma-125	94	100	λ∗j	λ∗j	PUNCT
ma-125	94	101	xj	xj	PROPN
ma-125	94	102	+	+	PROPN
ma-125	94	103	∑	∑	PUNCT
ma-125	94	104	j∈σc	j∈σc	NOUN
ma-125	94	105	µjpvj	µjpvj	NOUN
ma-125	94	106	θ	θ	PROPN
ma-125	94	107	∗	∗	PROPN
ma-125	94	108	j	j	PROPN
ma-125	94	109	xj	xj	PROPN
ma-125	94	110	,	,	PUNCT
ma-125	94	111	such	such	ADJ
ma-125	94	112	that	that	SCONJ
ma-125	94	113	αkk∗	αkk∗	ADP
ma-125	94	114	≤	≤	ADJ
ma-125	94	115	ψσψ∗σ	ψσψ∗σ	NUM
ma-125	94	116	,	,	PUNCT
ma-125	94	117	where	where	SCONJ
ma-125	94	118	lσ2	lσ2	PROPN
ma-125	94	119	(	(	PUNCT
ma-125	94	120	{	{	PUNCT
ma-125	94	121	hj}j∈j	hj}j∈j	NOUN
ma-125	94	122	)	)	PUNCT
ma-125	94	123	=	=	PRON
ma-125	94	124	{	{	PUNCT
ma-125	94	125	{	{	PUNCT
ma-125	94	126	xj}j∈j	xj}j∈j	NOUN
ma-125	94	127	=	=	SYM
ma-125	94	128	{	{	PUNCT
ma-125	94	129	fj}j∈σ	fj}j∈σ	X
ma-125	94	130	∪	∪	X
ma-125	94	131	{	{	PUNCT
ma-125	94	132	gj}j∈σc	gj}j∈σc	NOUN
ma-125	94	133	:	:	PUNCT
ma-125	94	134	fj	fj	PROPN
ma-125	94	135	∈	∈	PROPN
ma-125	94	136	hj	hj	PROPN
ma-125	94	137	,	,	PUNCT
ma-125	94	138	gj	gj	PROPN
ma-125	94	139	∈	∈	PROPN
ma-125	94	140	hj	hj	PROPN
ma-125	94	141	,	,	PUNCT
ma-125	94	142	‖	‖	PROPN
ma-125	94	143	∑	∑	PROPN
ma-125	94	144	j∈j	j∈j	PROPN
ma-125	94	145	〈	〈	PROPN
ma-125	94	146	xj	xj	PROPN
ma-125	94	147	,	,	PUNCT
ma-125	94	148	xj〉‖	xj〉‖	PROPN
ma-125	94	149	<	<	X
ma-125	94	150	∞	∞	NUM
ma-125	94	151	}	}	PUNCT
ma-125	94	152	.	.	PUNCT
ma-125	95	1	proof	proof	NOUN
ma-125	95	2	.	.	PUNCT
ma-125	96	1	(	(	PUNCT
ma-125	96	2	1	1	X
ma-125	96	3	)	)	PUNCT
ma-125	96	4	=	=	NOUN
ma-125	96	5	⇒	⇒	NOUN
ma-125	96	6	(	(	PUNCT
ma-125	96	7	2	2	NUM
ma-125	96	8	):	):	PUNCT
ma-125	96	9	suppose	suppose	VERB
ma-125	96	10	that	that	SCONJ
ma-125	96	11	a	a	PRON
ma-125	96	12	is	be	AUX
ma-125	96	13	an	an	DET
ma-125	96	14	universal	universal	ADJ
ma-125	96	15	lower	low	ADJ
ma-125	96	16	frame	frame	NOUN
ma-125	96	17	bound	bind	VERB
ma-125	96	18	for	for	ADP
ma-125	96	19	{	{	PUNCT
ma-125	96	20	wj	wj	PROPN
ma-125	96	21	,	,	PUNCT
ma-125	96	22	λj	λj	INTJ
ma-125	96	23	,	,	PUNCT
ma-125	96	24	vj}j∈j	vj}j∈j	X
ma-125	97	1	and	and	CCONJ
ma-125	97	2	{	{	PUNCT
ma-125	97	3	vj	vj	INTJ
ma-125	97	4	,	,	PUNCT
ma-125	97	5	θj	θj	INTJ
ma-125	97	6	,	,	PUNCT
ma-125	97	7	µj}j∈j	µj}j∈j	PROPN
ma-125	97	8	.	.	PUNCT
ma-125	97	9	choose	choose	VERB
ma-125	97	10	α	α	NOUN
ma-125	97	11	=	=	PUNCT
ma-125	97	12	a	a	PRON
ma-125	97	13	and	and	CCONJ
ma-125	97	14	ψσ	ψσ	ADJ
ma-125	97	15	=	=	PUNCT
ma-125	97	16	tσ	tσ	NOUN
ma-125	97	17	for	for	ADP
ma-125	97	18	every	every	DET
ma-125	97	19	σ	σ	PROPN
ma-125	97	20	⊂	⊂	PROPN
ma-125	97	21	j	j	PROPN
ma-125	97	22	,	,	PUNCT
ma-125	97	23	where	where	SCONJ
ma-125	97	24	tσ	tσ	PROPN
ma-125	97	25	is	be	AUX
ma-125	97	26	the	the	DET
ma-125	97	27	synthesis	synthesis	NOUN
ma-125	97	28	operator	operator	NOUN
ma-125	97	29	of	of	ADP
ma-125	97	30	{	{	PUNCT
ma-125	97	31	wj	wj	PROPN
ma-125	97	32	,	,	PUNCT
ma-125	97	33	λj	λj	PROPN
ma-125	97	34	,	,	PUNCT
ma-125	97	35	vj}j∈σ	vj}j∈σ	X
ma-125	97	36	∪	∪	X
ma-125	97	37	{	{	PUNCT
ma-125	97	38	vj	vj	INTJ
ma-125	97	39	,	,	PUNCT
ma-125	97	40	θj	θj	INTJ
ma-125	97	41	,	,	PUNCT
ma-125	97	42	µj}j∈σc	µj}j∈σc	PROPN
ma-125	97	43	.	.	PUNCT
ma-125	98	1	then	then	ADV
ma-125	98	2	,	,	PUNCT
ma-125	98	3	for	for	ADP
ma-125	98	4	any	any	DET
ma-125	98	5	{	{	PUNCT
ma-125	98	6	xj}j∈j	xj}j∈j	PROPN
ma-125	98	7	∈	∈	PROPN
ma-125	98	8	lσ2	lσ2	PROPN
ma-125	98	9	(	(	PUNCT
ma-125	98	10	{	{	PUNCT
ma-125	98	11	hj}j∈j	hj}j∈j	NOUN
ma-125	98	12	)	)	PUNCT
ma-125	98	13	we	we	PRON
ma-125	98	14	have	have	AUX
ma-125	98	15	ψσ{xj}j∈j	ψσ{xj}j∈j	ADP
ma-125	98	16	=	=	PRON
ma-125	98	17	tσ{xj}j∈j	tσ{xj}j∈j	NOUN
ma-125	98	18	=	=	PUNCT
ma-125	98	19	∑	∑	PUNCT
ma-125	98	20	j∈σ	j∈σ	PROPN
ma-125	98	21	vjpwj	vjpwj	PROPN
ma-125	98	22	λ∗j	λ∗j	PUNCT
ma-125	98	23	xj	xj	PROPN
ma-125	98	24	+	+	PROPN
ma-125	98	25	∑	∑	PUNCT
ma-125	98	26	j∈σc	j∈σc	NOUN
ma-125	98	27	µjpvj	µjpvj	NOUN
ma-125	98	28	θ	θ	PROPN
ma-125	98	29	∗	∗	PROPN
ma-125	98	30	j	j	PROPN
ma-125	98	31	xj	xj	PROPN
ma-125	98	32	,	,	PUNCT
ma-125	98	33	and	and	CCONJ
ma-125	98	34	also	also	ADV
ma-125	98	35	,	,	PUNCT
ma-125	98	36	for	for	ADP
ma-125	98	37	each	each	DET
ma-125	98	38	f	f	PROPN
ma-125	98	39	∈	∈	PROPN
ma-125	98	40	h	h	NOUN
ma-125	98	41	,	,	PUNCT
ma-125	98	42	a〈k∗f	a〈k∗f	PUNCT
ma-125	98	43	,	,	PUNCT
ma-125	98	44	k∗f	k∗f	PROPN
ma-125	98	45	〉	〉	PROPN
ma-125	98	46	≤	≤	PROPN
ma-125	99	1	〈	〈	PROPN
ma-125	99	2	t	t	PROPN
ma-125	99	3	∗σ	∗σ	PROPN
ma-125	99	4	f	f	PROPN
ma-125	99	5	,	,	PUNCT
ma-125	99	6	t	t	PROPN
ma-125	99	7	∗σ	∗σ	PROPN
ma-125	99	8	f	f	PROPN
ma-125	99	9	〉	〉	PROPN
ma-125	99	10	=	=	SYM
ma-125	99	11	〈	〈	PROPN
ma-125	99	12	ψ∗σf	ψ∗σf	PROPN
ma-125	99	13	,	,	PUNCT
ma-125	99	14	ψ∗σf	ψ∗σf	PROPN
ma-125	99	15	〉	〉	PROPN
ma-125	99	16	.thus	.thus	PROPN
ma-125	99	17	,	,	PUNCT
ma-125	99	18	αkk∗	αkk∗	ADP
ma-125	99	19	≤	≤	NUM
ma-125	99	20	ψσψ∗σ	ψσψ∗σ	NUM
ma-125	99	21	.	.	PUNCT
ma-125	100	1	(	(	PUNCT
ma-125	100	2	2	2	X
ma-125	100	3	)	)	PUNCT
ma-125	100	4	=	=	NOUN
ma-125	100	5	⇒	⇒	NOUN
ma-125	100	6	(	(	PUNCT
ma-125	100	7	1	1	NUM
ma-125	100	8	):	):	PUNCT
ma-125	100	9	let	let	VERB
ma-125	100	10	σ	σ	PROPN
ma-125	100	11	⊂	⊂	PROPN
ma-125	100	12	j	j	PROPN
ma-125	100	13	and	and	CCONJ
ma-125	100	14	f	f	PROPN
ma-125	100	15	∈	∈	PROPN
ma-125	100	16	h	h	NOUN
ma-125	100	17	,	,	PUNCT
ma-125	100	18	so	so	SCONJ
ma-125	100	19	it	it	PRON
ma-125	100	20	is	be	AUX
ma-125	100	21	easy	easy	ADJ
ma-125	100	22	to	to	PART
ma-125	100	23	check	check	VERB
ma-125	100	24	that	that	PRON
ma-125	100	25	ψ∗σf	ψ∗σf	PROPN
ma-125	100	26	=	=	PRON
ma-125	100	27	{	{	PUNCT
ma-125	100	28	vjλjpwj	vjλjpwj	NOUN
ma-125	100	29	f	f	PROPN
ma-125	100	30	}	}	PUNCT
ma-125	100	31	j∈σ	j∈σ	PROPN
ma-125	100	32	∪	∪	X
ma-125	100	33	{	{	PUNCT
ma-125	100	34	µjθjpvj	µjθjpvj	PROPN
ma-125	100	35	f	f	PROPN
ma-125	100	36	}	}	PUNCT
ma-125	100	37	j∈σc	j∈σc	NOUN
ma-125	100	38	.	.	PUNCT
ma-125	101	1	https://doi.org/10.28924/ada/ma.3.11	https://doi.org/10.28924/ada/ma.3.11	PROPN
ma-125	101	2	eur	eur	PROPN
ma-125	101	3	.	.	PUNCT
ma-125	102	1	j.	j.	PROPN
ma-125	102	2	math	math	PROPN
ma-125	102	3	.	.	PUNCT
ma-125	103	1	anal	anal	PROPN
ma-125	103	2	.	.	PUNCT
ma-125	104	1	10.28924	10.28924	NUM
ma-125	104	2	/	/	SYM
ma-125	104	3	ada	ada	PROPN
ma-125	104	4	/	/	SYM
ma-125	104	5	ma.3.11	ma.3.11	ADJ
ma-125	105	1	5therefore	5therefore	NUM
ma-125	105	2	,	,	PUNCT
ma-125	105	3	α〈k∗f	α〈k∗f	NOUN
ma-125	105	4	,	,	PUNCT
ma-125	105	5	k∗f	k∗f	PROPN
ma-125	105	6	〉	〉	PROPN
ma-125	105	7	=	=	SYM
ma-125	105	8	〈	〈	PROPN
ma-125	105	9	αkk∗f	αkk∗f	NOUN
ma-125	105	10	,	,	PUNCT
ma-125	105	11	f	f	PROPN
ma-125	105	12	〉	〉	PROPN
ma-125	105	13	≤	≤	PUNCT
ma-125	106	1	〈	〈	PROPN
ma-125	106	2	ψσψ∗σf	ψσψ∗σf	X
ma-125	106	3	,	,	PUNCT
ma-125	106	4	f	f	PROPN
ma-125	106	5	〉	〉	PROPN
ma-125	106	6	=	=	SYM
ma-125	106	7	〈	〈	PROPN
ma-125	106	8	ψ∗σf	ψ∗σf	PROPN
ma-125	106	9	,	,	PUNCT
ma-125	106	10	ψ∗σf	ψ∗σf	PROPN
ma-125	106	11	〉	〉	PROPN
ma-125	106	12	=	=	PUNCT
ma-125	106	13	∑	∑	PUNCT
ma-125	106	14	j∈σ	j∈σ	PROPN
ma-125	107	1	v2	v2	PROPN
ma-125	107	2	j	j	PROPN
ma-125	107	3	〈	〈	PROPN
ma-125	107	4	λjpwj	λjpwj	VERB
ma-125	107	5	f	f	PROPN
ma-125	107	6	,	,	PUNCT
ma-125	107	7	λjpwj	λjpwj	VERB
ma-125	107	8	f	f	PROPN
ma-125	107	9	〉	〉	PROPN
ma-125	107	10	+	+	PROPN
ma-125	107	11	∑	∑	PUNCT
ma-125	107	12	j∈σc	j∈σc	NOUN
ma-125	107	13	µ2	µ2	PROPN
ma-125	107	14	j	j	PROPN
ma-125	107	15	〈	〈	PROPN
ma-125	107	16	θjpvj	θjpvj	VERB
ma-125	107	17	f	f	PROPN
ma-125	107	18	,	,	PUNCT
ma-125	107	19	θjpvj	θjpvj	PROPN
ma-125	107	20	f	f	PROPN
ma-125	107	21	〉	〉	PROPN
ma-125	107	22	.	.	PUNCT
ma-125	108	1	this	this	PRON
ma-125	108	2	gives	give	VERB
ma-125	108	3	that	that	SCONJ
ma-125	108	4	α	α	PRON
ma-125	108	5	is	be	AUX
ma-125	108	6	an	an	DET
ma-125	108	7	universal	universal	ADJ
ma-125	108	8	lower	low	ADJ
ma-125	108	9	frame	frame	NOUN
ma-125	108	10	bound	bind	VERB
ma-125	108	11	of	of	ADP
ma-125	108	12	{	{	PUNCT
ma-125	108	13	wj	wj	PROPN
ma-125	108	14	,	,	PUNCT
ma-125	108	15	λj	λj	INTJ
ma-125	108	16	,	,	PUNCT
ma-125	108	17	vj}j∈j	vj}j∈j	X
ma-125	109	1	and	and	CCONJ
ma-125	109	2	{	{	PUNCT
ma-125	109	3	vj	vj	INTJ
ma-125	109	4	,	,	PUNCT
ma-125	109	5	θj	θj	INTJ
ma-125	109	6	,	,	PUNCT
ma-125	109	7	µj}j∈j	µj}j∈j	PROPN
ma-125	109	8	.	.	PUNCT
ma-125	109	9	�	�	PROPN
ma-125	109	10	in	in	ADP
ma-125	109	11	next	next	ADJ
ma-125	109	12	results	result	NOUN
ma-125	109	13	,	,	PUNCT
ma-125	109	14	we	we	PRON
ma-125	109	15	construct	construct	VERB
ma-125	109	16	a	a	DET
ma-125	109	17	k	k	PROPN
ma-125	109	18	−	−	PROPN
ma-125	109	19	g−fusion	g−fusion	NOUN
ma-125	109	20	woven	weave	VERB
ma-125	109	21	by	by	ADP
ma-125	109	22	using	use	VERB
ma-125	109	23	a	a	DET
ma-125	109	24	bounded	bounded	ADJ
ma-125	109	25	linear	linear	ADJ
ma-125	109	26	operator	operator	NOUN
ma-125	109	27	.	.	PUNCT
ma-125	110	1	theorem	theorem	VERB
ma-125	110	2	2.3	2.3	NUM
ma-125	110	3	.	.	PUNCT
ma-125	111	1	let	let	VERB
ma-125	111	2	{	{	PUNCT
ma-125	111	3	wi	wi	PROPN
ma-125	111	4	j	j	PROPN
ma-125	111	5	,	,	PUNCT
ma-125	111	6	λi	λi	PROPN
ma-125	111	7	j	j	PROPN
ma-125	111	8	,	,	PUNCT
ma-125	111	9	vi	vi	PROPN
ma-125	111	10	j}j∈j	j}j∈j	PROPN
ma-125	111	11	,	,	PUNCT
ma-125	111	12	i∈[m	i∈[m	PROPN
ma-125	111	13	]	]	PUNCT
ma-125	111	14	be	be	VERB
ma-125	111	15	a	a	DET
ma-125	111	16	k−g−fusion	k−g−fusion	NOUN
ma-125	111	17	woven	weave	VERB
ma-125	111	18	for	for	ADP
ma-125	111	19	h	h	NOUN
ma-125	111	20	with	with	ADP
ma-125	111	21	common	common	ADJ
ma-125	111	22	frame	frame	NOUN
ma-125	111	23	bounds	bound	VERB
ma-125	111	24	a	a	DET
ma-125	111	25	,	,	PUNCT
ma-125	111	26	b	b	NOUN
ma-125	111	27	and	and	CCONJ
ma-125	111	28	assume	assume	VERB
ma-125	111	29	that	that	SCONJ
ma-125	111	30	u	u	PROPN
ma-125	111	31	∈	∈	PROPN
ma-125	111	32	end∗a(h	end∗a(h	NOUN
ma-125	111	33	)	)	PUNCT
ma-125	111	34	has	have	AUX
ma-125	111	35	closed	close	VERB
ma-125	111	36	range	range	NOUN
ma-125	112	1	so	so	SCONJ
ma-125	112	2	that	that	SCONJ
ma-125	112	3	r(k∗	r(k∗	X
ma-125	112	4	)	)	PUNCT
ma-125	112	5	⊂	⊂	PROPN
ma-125	112	6	r(u	r(u	PROPN
ma-125	112	7	)	)	PUNCT
ma-125	112	8	and	and	CCONJ
ma-125	112	9	ku	ku	PROPN
ma-125	112	10	=	=	PROPN
ma-125	112	11	uk	uk	PROPN
ma-125	112	12	.	.	PROPN
ma-125	112	13	then	then	ADV
ma-125	112	14	{	{	PUNCT
ma-125	112	15	uwi	uwi	PROPN
ma-125	112	16	j	j	PROPN
ma-125	112	17	,	,	PUNCT
ma-125	112	18	λi	λi	ADP
ma-125	112	19	jpwi	jpwi	PROPN
ma-125	112	20	j	j	PROPN
ma-125	112	21	u∗	u∗	PROPN
ma-125	112	22	,	,	PUNCT
ma-125	112	23	vi	vi	PROPN
ma-125	112	24	j}j∈j	j}j∈j	PROPN
ma-125	112	25	,	,	PUNCT
ma-125	112	26	i∈[m	i∈[m	PROPN
ma-125	112	27	]	]	PUNCT
ma-125	112	28	is	be	AUX
ma-125	112	29	also	also	ADV
ma-125	112	30	k	k	PROPN
ma-125	112	31	−	−	PROPN
ma-125	112	32	g−fusion	g−fusion	NOUN
ma-125	112	33	woven	weave	VERB
ma-125	112	34	for	for	ADP
ma-125	112	35	r(u	r(u	NOUN
ma-125	112	36	)	)	PUNCT
ma-125	112	37	.	.	PUNCT
ma-125	113	1	proof	proof	NOUN
ma-125	113	2	.	.	PUNCT
ma-125	114	1	by	by	ADP
ma-125	114	2	the	the	DET
ma-125	114	3	open	open	ADJ
ma-125	114	4	mapping	mapping	NOUN
ma-125	114	5	theorem	theorem	NOUN
ma-125	114	6	,	,	PUNCT
ma-125	114	7	uwi	uwi	PROPN
ma-125	114	8	j	j	PROPN
ma-125	114	9	is	be	AUX
ma-125	114	10	closed	close	VERB
ma-125	114	11	for	for	ADP
ma-125	114	12	any	any	DET
ma-125	114	13	j	j	PROPN
ma-125	114	14	∈	∈	PROPN
ma-125	114	15	j	j	PROPN
ma-125	114	16	and	and	CCONJ
ma-125	114	17	i	i	PRON
ma-125	114	18	∈	∈	PROPN
ma-125	115	1	[	[	X
ma-125	116	1	m	m	X
ma-125	116	2	]	]	X
ma-125	116	3	.	.	PUNCT
ma-125	117	1	using	use	VERB
ma-125	117	2	lemme(refk	lemme(refk	PROPN
ma-125	117	3	-	-	PUNCT
ma-125	117	4	g	g	NOUN
ma-125	117	5	-	-	PUNCT
ma-125	117	6	fusion	fusion	NOUN
ma-125	117	7	)	)	PUNCT
ma-125	117	8	,	,	PUNCT
ma-125	117	9	we	we	PRON
ma-125	117	10	can	can	AUX
ma-125	117	11	write	write	VERB
ma-125	117	12	for	for	ADP
ma-125	117	13	each	each	DET
ma-125	117	14	f	f	PROPN
ma-125	117	15	∈	∈	PROPN
ma-125	117	16	r(u	r(u	PROPN
ma-125	117	17	)	)	PUNCT
ma-125	117	18	,	,	PUNCT
ma-125	117	19	a〈k∗f	a〈k∗f	PUNCT
ma-125	117	20	,	,	PUNCT
ma-125	117	21	k∗f	k∗f	PROPN
ma-125	117	22	〉	〉	PROPN
ma-125	117	23	=	=	PUNCT
ma-125	117	24	a〈(u+)∗u∗k∗f	a〈(u+)∗u∗k∗f	NOUN
ma-125	117	25	,	,	PUNCT
ma-125	117	26	(	(	PUNCT
ma-125	117	27	u+)∗u∗k∗f	u+)∗u∗k∗f	NOUN
ma-125	117	28	〉	〉	PROPN
ma-125	117	29	≤	≤	PROPN
ma-125	117	30	a‖u+‖2〈k∗u∗f	a‖u+‖2〈k∗u∗f	NOUN
ma-125	117	31	,	,	PUNCT
ma-125	117	32	k∗u∗f	k∗u∗f	VERB
ma-125	117	33	〉	〉	NOUN
ma-125	117	34	≤	≤	PROPN
ma-125	117	35	‖u+‖2	‖u+‖2	PUNCT
ma-125	117	36	∑	∑	PUNCT
ma-125	117	37	i∈[m	i∈[m	PROPN
ma-125	117	38	]	]	PUNCT
ma-125	117	39	∑	∑	PUNCT
ma-125	117	40	j∈j	j∈j	NOUN
ma-125	117	41	v2	v2	PROPN
ma-125	118	1	i	i	PRON
ma-125	118	2	j	j	PROPN
ma-125	119	1	〈	〈	PROPN
ma-125	119	2	λi	λi	PROPN
ma-125	119	3	jpwi	jpwi	PROPN
ma-125	119	4	j	j	PROPN
ma-125	119	5	u∗f	u∗f	PROPN
ma-125	119	6	,	,	PUNCT
ma-125	119	7	λi	λi	ADP
ma-125	119	8	jpwi	jpwi	NOUN
ma-125	119	9	j	j	PROPN
ma-125	119	10	u∗f	u∗f	NOUN
ma-125	119	11	〉	〉	NUM
ma-125	119	12	=	=	SYM
ma-125	119	13	‖u+‖2	‖u+‖2	PROPN
ma-125	119	14	∑	∑	PUNCT
ma-125	119	15	i∈[m	i∈[m	NOUN
ma-125	119	16	]	]	PUNCT
ma-125	119	17	∑	∑	PUNCT
ma-125	119	18	j∈j	j∈j	NOUN
ma-125	119	19	v2	v2	PROPN
ma-125	120	1	i	i	PRON
ma-125	120	2	j	j	PROPN
ma-125	121	1	〈	〈	PROPN
ma-125	121	2	λi	λi	PROPN
ma-125	121	3	jpwi	jpwi	PROPN
ma-125	121	4	j	j	PROPN
ma-125	121	5	u∗puwi	u∗puwi	PROPN
ma-125	121	6	j	j	PROPN
ma-125	121	7	f	f	PROPN
ma-125	121	8	,	,	PUNCT
ma-125	121	9	λi	λi	ADP
ma-125	121	10	jpwi	jpwi	PROPN
ma-125	121	11	j	j	PROPN
ma-125	121	12	u∗puwi	u∗puwi	PROPN
ma-125	121	13	j	j	PROPN
ma-125	121	14	f	f	PROPN
ma-125	121	15	〉	〉	PROPN
ma-125	121	16	.	.	PUNCT
ma-125	122	1	the	the	DET
ma-125	122	2	upper	upper	ADJ
ma-125	122	3	bound	bind	VERB
ma-125	122	4	is	be	AUX
ma-125	122	5	obvious	obvious	ADJ
ma-125	122	6	.	.	PUNCT
ma-125	123	1	�	�	PROPN
ma-125	123	2	theorem	theorem	VERB
ma-125	123	3	2.4	2.4	NUM
ma-125	123	4	.	.	PUNCT
ma-125	124	1	let	let	VERB
ma-125	124	2	k	k	PROPN
ma-125	124	3	have	have	AUX
ma-125	124	4	closed	close	VERB
ma-125	124	5	range	range	NOUN
ma-125	124	6	,	,	PUNCT
ma-125	124	7	{	{	PUNCT
ma-125	124	8	wi	wi	PROPN
ma-125	124	9	j	j	PROPN
ma-125	124	10	,	,	PUNCT
ma-125	124	11	λi	λi	PROPN
ma-125	124	12	j	j	PROPN
ma-125	124	13	,	,	PUNCT
ma-125	124	14	vi	vi	PROPN
ma-125	124	15	j}j∈j	j}j∈j	PROPN
ma-125	124	16	,	,	PUNCT
ma-125	124	17	i∈[m	i∈[m	PROPN
ma-125	124	18	]	]	PUNCT
ma-125	124	19	be	be	VERB
ma-125	124	20	a	a	DET
ma-125	124	21	k−	k−	PROPN
ma-125	124	22	g−fusion	g−fusion	NOUN
ma-125	124	23	woven	weave	VERB
ma-125	124	24	for	for	ADP
ma-125	124	25	h	h	NOUN
ma-125	124	26	with	with	ADP
ma-125	124	27	the	the	DET
ma-125	124	28	universal	universal	ADJ
ma-125	124	29	bounds	bound	NOUN
ma-125	124	30	a	a	PRON
ma-125	124	31	,	,	PUNCT
ma-125	124	32	b	b	NOUN
ma-125	124	33	and	and	CCONJ
ma-125	124	34	u	u	PROPN
ma-125	124	35	∈	∈	PROPN
ma-125	124	36	end∗a(h	end∗a(h	NOUN
ma-125	124	37	)	)	PUNCT
ma-125	124	38	has	have	AUX
ma-125	124	39	closed	close	VERB
ma-125	124	40	range	range	NOUN
ma-125	124	41	so	so	SCONJ
ma-125	124	42	that	that	SCONJ
ma-125	124	43	r(u∗	r(u∗	X
ma-125	124	44	)	)	PUNCT
ma-125	124	45	⊂	⊂	PUNCT
ma-125	124	46	r(k	r(k	PROPN
ma-125	124	47	)	)	PUNCT
ma-125	124	48	.	.	PUNCT
ma-125	125	1	then	then	ADV
ma-125	125	2	{	{	PUNCT
ma-125	125	3	uwi	uwi	PROPN
ma-125	125	4	j	j	PROPN
ma-125	125	5	,	,	PUNCT
ma-125	125	6	λi	λi	ADP
ma-125	125	7	jpwi	jpwi	PROPN
ma-125	125	8	j	j	PROPN
ma-125	125	9	u∗	u∗	PROPN
ma-125	125	10	,	,	PUNCT
ma-125	125	11	vi	vi	PROPN
ma-125	125	12	j}j∈j	j}j∈j	PROPN
ma-125	125	13	,	,	PUNCT
ma-125	125	14	i∈[m	i∈[m	PROPN
ma-125	125	15	]	]	PUNCT
ma-125	125	16	is	be	AUX
ma-125	125	17	a	a	DET
ma-125	125	18	k	k	PROPN
ma-125	125	19	−	−	NOUN
ma-125	125	20	g−fusion	g−fusion	NOUN
ma-125	125	21	woven	weave	VERB
ma-125	125	22	for	for	ADP
ma-125	126	1	h	h	NOUN
ma-125	126	2	if	if	SCONJ
ma-125	127	1	and	and	CCONJ
ma-125	127	2	only	only	ADV
ma-125	127	3	if	if	SCONJ
ma-125	127	4	there	there	PRON
ma-125	127	5	exists	exist	VERB
ma-125	127	6	a	a	DET
ma-125	127	7	δ	δ	PROPN
ma-125	127	8	>	>	X
ma-125	127	9	0	0	NUM
ma-125	128	1	such	such	ADJ
ma-125	128	2	that	that	PRON
ma-125	128	3	for	for	ADP
ma-125	128	4	every	every	DET
ma-125	128	5	f	f	PROPN
ma-125	128	6	∈	∈	PROPN
ma-125	128	7	h	h	NOUN
ma-125	128	8	,	,	PUNCT
ma-125	128	9	〈	〈	PROPN
ma-125	128	10	u∗f	u∗f	NUM
ma-125	128	11	,	,	PUNCT
ma-125	128	12	u∗f	u∗f	PROPN
ma-125	128	13	〉	〉	NUM
ma-125	128	14	≥	≥	NUM
ma-125	128	15	δ〈k∗f	δ〈k∗f	NOUN
ma-125	128	16	,	,	PUNCT
ma-125	128	17	k∗f	k∗f	PROPN
ma-125	128	18	〉	〉	PROPN
ma-125	128	19	.	.	PUNCT
ma-125	128	20	proof	proof	NOUN
ma-125	128	21	.	.	PUNCT
ma-125	129	1	let	let	VERB
ma-125	129	2	f	f	PRON
ma-125	129	3	∈	∈	PROPN
ma-125	129	4	h	h	NOUN
ma-125	129	5	and	and	CCONJ
ma-125	129	6	{	{	PUNCT
ma-125	129	7	uwi	uwi	PROPN
ma-125	129	8	j	j	PROPN
ma-125	129	9	,	,	PUNCT
ma-125	129	10	λi	λi	ADP
ma-125	129	11	jpwi	jpwi	PROPN
ma-125	129	12	j	j	PROPN
ma-125	129	13	u∗	u∗	PROPN
ma-125	129	14	,	,	PUNCT
ma-125	129	15	vi	vi	PROPN
ma-125	129	16	j}j∈j	j}j∈j	PROPN
ma-125	129	17	,	,	PUNCT
ma-125	129	18	i∈[m	i∈[m	PROPN
ma-125	129	19	]	]	PUNCT
ma-125	129	20	is	be	AUX
ma-125	129	21	a	a	DET
ma-125	129	22	k	k	PROPN
ma-125	129	23	−	−	NOUN
ma-125	129	24	g−fusion	g−fusion	NOUN
ma-125	129	25	woven	weave	VERB
ma-125	129	26	for	for	ADP
ma-125	129	27	h	h	NOUN
ma-125	129	28	with	with	ADP
ma-125	129	29	lowerbound	lowerbound	NOUN
ma-125	129	30	c	c	NOUN
ma-125	129	31	,	,	PUNCT
ma-125	129	32	we	we	PRON
ma-125	129	33	get	get	VERB
ma-125	129	34	c〈k∗f	c〈k∗f	PROPN
ma-125	129	35	,	,	PUNCT
ma-125	129	36	k∗f	k∗f	PROPN
ma-125	129	37	〉	〉	PROPN
ma-125	129	38	≤	≤	PROPN
ma-125	129	39	∑	∑	PUNCT
ma-125	129	40	i∈[m	i∈[m	PROPN
ma-125	129	41	]	]	PUNCT
ma-125	129	42	∑	∑	PUNCT
ma-125	129	43	j∈j	j∈j	NOUN
ma-125	129	44	v2	v2	PROPN
ma-125	130	1	i	i	PRON
ma-125	130	2	j	j	PROPN
ma-125	131	1	〈	〈	PROPN
ma-125	131	2	λi	λi	PROPN
ma-125	131	3	jpwi	jpwi	PROPN
ma-125	131	4	j	j	PROPN
ma-125	131	5	u∗puwi	u∗puwi	PROPN
ma-125	131	6	j	j	PROPN
ma-125	131	7	f	f	PROPN
ma-125	131	8	,	,	PUNCT
ma-125	131	9	λi	λi	ADP
ma-125	131	10	jpwi	jpwi	PROPN
ma-125	131	11	j	j	PROPN
ma-125	131	12	u∗puwi	u∗puwi	PROPN
ma-125	131	13	j	j	PROPN
ma-125	131	14	f	f	PROPN
ma-125	131	15	〉	〉	PROPN
ma-125	131	16	=	=	SYM
ma-125	131	17	∑	∑	PUNCT
ma-125	131	18	i∈[m	i∈[m	PROPN
ma-125	131	19	]	]	PUNCT
ma-125	131	20	∑	∑	PUNCT
ma-125	131	21	j∈j	j∈j	NOUN
ma-125	131	22	v2	v2	PROPN
ma-125	132	1	i	i	PRON
ma-125	132	2	j	j	PROPN
ma-125	133	1	〈	〈	PROPN
ma-125	133	2	λi	λi	PROPN
ma-125	133	3	jpwi	jpwi	PROPN
ma-125	133	4	j	j	PROPN
ma-125	133	5	u∗f	u∗f	PROPN
ma-125	133	6	,	,	PUNCT
ma-125	133	7	λi	λi	ADP
ma-125	133	8	jpwi	jpwi	NOUN
ma-125	133	9	j	j	PROPN
ma-125	133	10	u∗f	u∗f	NOUN
ma-125	133	11	〉	〉	NOUN
ma-125	133	12	≤	≤	NUM
ma-125	133	13	b〈u∗f	b〈u∗f	NOUN
ma-125	133	14	,	,	PUNCT
ma-125	133	15	u∗f	u∗f	PROPN
ma-125	133	16	〉	〉	NUM
ma-125	133	17	.	.	PUNCT
ma-125	134	1	https://doi.org/10.28924/ada/ma.3.11	https://doi.org/10.28924/ada/ma.3.11	PROPN
ma-125	134	2	eur	eur	PROPN
ma-125	134	3	.	.	PUNCT
ma-125	135	1	j.	j.	PROPN
ma-125	135	2	math	math	PROPN
ma-125	135	3	.	.	PUNCT
ma-125	136	1	anal	anal	PROPN
ma-125	136	2	.	.	PUNCT
ma-125	137	1	10.28924	10.28924	NUM
ma-125	137	2	/	/	SYM
ma-125	137	3	ada	ada	PROPN
ma-125	137	4	/	/	SYM
ma-125	137	5	ma.3.11	ma.3.11	ADJ
ma-125	137	6	6	6	NUM
ma-125	137	7	therefore	therefore	ADV
ma-125	137	8	,	,	PUNCT
ma-125	137	9	〈	〈	PROPN
ma-125	137	10	u∗f	u∗f	NUM
ma-125	137	11	,	,	PUNCT
ma-125	138	1	u∗f	u∗f	PROPN
ma-125	138	2	〉	〉	NUM
ma-125	138	3	≥√c	≥√c	SYM
ma-125	138	4	b	b	NOUN
ma-125	138	5	〈	〈	PROPN
ma-125	138	6	k	k	PROPN
ma-125	138	7	∗f	∗f	PROPN
ma-125	138	8	,	,	PUNCT
ma-125	138	9	k∗f	k∗f	PROPN
ma-125	138	10	〉	〉	PROPN
ma-125	138	11	.	.	PUNCT
ma-125	139	1	for	for	ADP
ma-125	139	2	the	the	DET
ma-125	139	3	opposite	opposite	ADJ
ma-125	139	4	implication	implication	NOUN
ma-125	139	5	,	,	PUNCT
ma-125	139	6	we	we	PRON
ma-125	139	7	can	can	AUX
ma-125	139	8	write	write	VERB
ma-125	139	9	for	for	ADP
ma-125	139	10	all	all	DET
ma-125	139	11	f	f	PROPN
ma-125	139	12	∈	∈	PROPN
ma-125	139	13	h	h	NOUN
ma-125	139	14	,	,	PUNCT
ma-125	139	15	〈	〈	PROPN
ma-125	139	16	u∗f	u∗f	NUM
ma-125	139	17	,	,	PUNCT
ma-125	139	18	u∗f	u∗f	NOUN
ma-125	139	19	〉	〉	NOUN
ma-125	139	20	=	=	SYM
ma-125	139	21	〈	〈	PROPN
ma-125	139	22	(	(	PUNCT
ma-125	139	23	k+)∗k∗u∗f	k+)∗k∗u∗f	PROPN
ma-125	139	24	,	,	PUNCT
ma-125	139	25	(	(	PUNCT
ma-125	139	26	k+)∗k∗u∗f	k+)∗k∗u∗f	PROPN
ma-125	139	27	〉	〉	NOUN
ma-125	139	28	≤	≤	NOUN
ma-125	139	29	‖k+‖2〈k∗u∗f	‖k+‖2〈k∗u∗f	PUNCT
ma-125	139	30	,	,	PUNCT
ma-125	139	31	k∗u∗f	k∗u∗f	PROPN
ma-125	139	32	〉	〉	PROPN
ma-125	139	33	.	.	PUNCT
ma-125	140	1	hence	hence	ADV
ma-125	140	2	,	,	PUNCT
ma-125	140	3	we	we	PRON
ma-125	140	4	have	have	VERB
ma-125	140	5	aδ‖k+‖−2〈k∗f	aδ‖k+‖−2〈k∗f	NOUN
ma-125	140	6	,	,	PUNCT
ma-125	140	7	k∗f	k∗f	PROPN
ma-125	140	8	〉	〉	PROPN
ma-125	140	9	≤	≤	PROPN
ma-125	140	10	a‖k+‖−2〈u∗f	a‖k+‖−2〈u∗f	PROPN
ma-125	140	11	,	,	PUNCT
ma-125	141	1	u∗f	u∗f	NOUN
ma-125	141	2	〉	〉	NOUN
ma-125	141	3	≤	≤	NOUN
ma-125	141	4	a〈k∗u∗f	a〈k∗u∗f	PROPN
ma-125	141	5	,	,	PUNCT
ma-125	141	6	k∗u∗f	k∗u∗f	VERB
ma-125	141	7	〉	〉	NOUN
ma-125	141	8	≤	≤	PROPN
ma-125	141	9	∑	∑	PUNCT
ma-125	141	10	i∈[m	i∈[m	PROPN
ma-125	141	11	]	]	PUNCT
ma-125	141	12	∑	∑	PUNCT
ma-125	141	13	j∈j	j∈j	NOUN
ma-125	141	14	v2	v2	PROPN
ma-125	142	1	i	i	PRON
ma-125	142	2	j	j	PROPN
ma-125	143	1	〈	〈	PROPN
ma-125	143	2	λi	λi	PROPN
ma-125	143	3	jpwi	jpwi	PROPN
ma-125	143	4	j	j	PROPN
ma-125	143	5	u∗f	u∗f	PROPN
ma-125	143	6	,	,	PUNCT
ma-125	143	7	λi	λi	ADP
ma-125	143	8	jpwi	jpwi	NOUN
ma-125	143	9	j	j	PROPN
ma-125	143	10	u∗f	u∗f	NOUN
ma-125	143	11	〉	〉	NUM
ma-125	143	12	=	=	SYM
ma-125	143	13	∑	∑	PUNCT
ma-125	143	14	i∈[m	i∈[m	PROPN
ma-125	143	15	]	]	PUNCT
ma-125	143	16	∑	∑	PUNCT
ma-125	143	17	j∈j	j∈j	NOUN
ma-125	143	18	v2	v2	PROPN
ma-125	144	1	i	i	PRON
ma-125	144	2	j	j	PROPN
ma-125	145	1	〈	〈	PROPN
ma-125	145	2	λi	λi	PROPN
ma-125	145	3	jpwi	jpwi	PROPN
ma-125	145	4	j	j	PROPN
ma-125	145	5	u∗puwi	u∗puwi	PROPN
ma-125	145	6	j	j	PROPN
ma-125	145	7	f	f	PROPN
ma-125	145	8	,	,	PUNCT
ma-125	145	9	λi	λi	ADP
ma-125	145	10	jpwi	jpwi	PROPN
ma-125	145	11	j	j	PROPN
ma-125	145	12	u∗puwi	u∗puwi	PROPN
ma-125	145	13	j	j	PROPN
ma-125	145	14	f	f	PROPN
ma-125	145	15	〉	〉	PROPN
ma-125	145	16	≤	≤	NOUN
ma-125	145	17	b‖u‖2〈f	b‖u‖2〈f	NUM
ma-125	145	18	,	,	PUNCT
ma-125	145	19	f	f	PROPN
ma-125	145	20	〉	〉	PROPN
ma-125	145	21	.	.	PUNCT
ma-125	146	1	so	so	ADV
ma-125	146	2	,	,	PUNCT
ma-125	146	3	{	{	PUNCT
ma-125	146	4	uwi	uwi	PROPN
ma-125	146	5	j	j	PROPN
ma-125	146	6	,	,	PUNCT
ma-125	146	7	λi	λi	ADP
ma-125	146	8	jpwi	jpwi	PROPN
ma-125	146	9	j	j	PROPN
ma-125	146	10	u∗	u∗	PROPN
ma-125	146	11	,	,	PUNCT
ma-125	146	12	vi	vi	PROPN
ma-125	146	13	j}j∈j	j}j∈j	PROPN
ma-125	146	14	,	,	PUNCT
ma-125	146	15	i∈[m	i∈[m	PROPN
ma-125	146	16	]	]	PUNCT
ma-125	146	17	is	be	AUX
ma-125	146	18	a	a	DET
ma-125	146	19	k	k	PROPN
ma-125	146	20	−	−	NOUN
ma-125	146	21	g−fusion	g−fusion	NOUN
ma-125	146	22	woven	weave	VERB
ma-125	146	23	for	for	ADP
ma-125	146	24	h	h	NOUN
ma-125	146	25	with	with	ADP
ma-125	146	26	frame	frame	NOUN
ma-125	146	27	bounds	bound	NOUN
ma-125	146	28	aδ‖k+‖−2and	aδ‖k+‖−2and	ADP
ma-125	146	29	b‖u‖2	b‖u‖2	PROPN
ma-125	146	30	.	.	PUNCT
ma-125	147	1	�	�	PROPN
ma-125	147	2	theorem	theorem	VERB
ma-125	147	3	2.5	2.5	NUM
ma-125	147	4	.	.	PUNCT
ma-125	148	1	let	let	AUX
ma-125	148	2	{	{	PUNCT
ma-125	148	3	wi	wi	PROPN
ma-125	148	4	j	j	PROPN
ma-125	148	5	,	,	PUNCT
ma-125	148	6	λi	λi	PROPN
ma-125	148	7	j	j	PROPN
ma-125	148	8	,	,	PUNCT
ma-125	148	9	vi	vi	PROPN
ma-125	148	10	j}j∈j	j}j∈j	PROPN
ma-125	148	11	,	,	PUNCT
ma-125	148	12	i∈[m	i∈[m	PROPN
ma-125	148	13	]	]	PUNCT
ma-125	148	14	be	be	VERB
ma-125	148	15	a	a	DET
ma-125	148	16	k	k	NOUN
ma-125	148	17	−	−	NOUN
ma-125	148	18	g−fusion	g−fusion	NOUN
ma-125	148	19	woven	weave	VERB
ma-125	148	20	for	for	ADP
ma-125	148	21	h	h	NOUN
ma-125	148	22	with	with	ADP
ma-125	148	23	common	common	ADJ
ma-125	148	24	frame	frame	NOUN
ma-125	148	25	bounds	bound	VERB
ma-125	148	26	a	a	PRON
ma-125	148	27	and	and	CCONJ
ma-125	148	28	b.	b.	PROPN
ma-125	148	29	suppose	suppose	VERB
ma-125	148	30	that	that	SCONJ
ma-125	148	31	0	0	NUM
ma-125	148	32	≤	≤	NUM
ma-125	148	33	c	c	NOUN
ma-125	148	34	≤	≤	X
ma-125	148	35	|w	|w	NOUN
ma-125	148	36	(	(	PUNCT
ma-125	148	37	i	i	NOUN
ma-125	148	38	)	)	PUNCT
ma-125	148	39	j	j	PROPN
ma-125	149	1	|	|	ADV
ma-125	149	2	2	2	NUM
ma-125	149	3	≤	≤	NUM
ma-125	150	1	d	d	ADP
ma-125	150	2	<	<	X
ma-125	150	3	∞	∞	PROPN
ma-125	150	4	for	for	ADP
ma-125	150	5	any	any	DET
ma-125	150	6	i	i	PRON
ma-125	150	7	∈	∈	PROPN
ma-125	151	1	[	[	X
ma-125	151	2	m	m	X
ma-125	151	3	]	]	X
ma-125	151	4	and	and	CCONJ
ma-125	151	5	j	j	PROPN
ma-125	151	6	∈	∈	PROPN
ma-125	151	7	j	j	PROPN
ma-125	151	8	,	,	PUNCT
ma-125	151	9	then	then	ADV
ma-125	151	10	{	{	PUNCT
ma-125	151	11	wi	wi	PROPN
ma-125	151	12	j	j	PROPN
ma-125	151	13	,	,	PUNCT
ma-125	151	14	w	w	PROPN
ma-125	151	15	(	(	PUNCT
ma-125	151	16	i	i	NOUN
ma-125	151	17	)	)	PUNCT
ma-125	151	18	j	j	PROPN
ma-125	151	19	λi	λi	ADP
ma-125	151	20	j	j	PROPN
ma-125	151	21	,	,	PUNCT
ma-125	151	22	vi	vi	PROPN
ma-125	151	23	j}j∈j	j}j∈j	PROPN
ma-125	151	24	,	,	PUNCT
ma-125	151	25	i∈[m	i∈[m	PROPN
ma-125	151	26	]	]	PUNCT
ma-125	151	27	is	be	AUX
ma-125	151	28	a	a	DET
ma-125	151	29	k	k	PROPN
ma-125	151	30	−	−	NOUN
ma-125	151	31	g−fusion	g−fusion	NOUN
ma-125	151	32	woven	weave	VERB
ma-125	151	33	for	for	ADP
ma-125	151	34	h	h	NOUN
ma-125	151	35	with	with	ADP
ma-125	151	36	frame	frame	NOUN
ma-125	151	37	bounds	bound	NOUN
ma-125	152	1	ac	ac	PROPN
ma-125	152	2	and	and	CCONJ
ma-125	152	3	bd	bd	PROPN
ma-125	152	4	.	.	PROPN
ma-125	152	5	proof	proof	NOUN
ma-125	152	6	.	.	PUNCT
ma-125	153	1	for	for	ADP
ma-125	153	2	any	any	DET
ma-125	153	3	partition	partition	NOUN
ma-125	153	4	{	{	PUNCT
ma-125	153	5	σi}i∈[m	σi}i∈[m	NOUN
ma-125	153	6	]	]	PUNCT
ma-125	153	7	of	of	ADP
ma-125	153	8	j	j	PROPN
ma-125	153	9	and	and	CCONJ
ma-125	153	10	f	f	PROPN
ma-125	153	11	∈	∈	PROPN
ma-125	153	12	h	h	NOUN
ma-125	153	13	,	,	PUNCT
ma-125	153	14	we	we	PRON
ma-125	153	15	get	get	VERB
ma-125	153	16	ac〈k∗f	ac〈k∗f	NOUN
ma-125	153	17	,	,	PUNCT
ma-125	153	18	k∗f	k∗f	PROPN
ma-125	153	19	〉	〉	PROPN
ma-125	153	20	=	=	SYM
ma-125	153	21	min	min	PROPN
ma-125	153	22	i∈[m	i∈[m	PROPN
ma-125	153	23	]	]	PUNCT
ma-125	153	24	|w	|w	NOUN
ma-125	153	25	(	(	PUNCT
ma-125	153	26	i	i	NOUN
ma-125	153	27	)	)	PUNCT
ma-125	153	28	j	j	PROPN
ma-125	154	1	|	|	ADV
ma-125	154	2	2a〈k∗f	2a〈k∗f	NUM
ma-125	154	3	,	,	PUNCT
ma-125	154	4	k∗f	k∗f	PROPN
ma-125	154	5	〉	〉	PROPN
ma-125	154	6	≤	≤	PROPN
ma-125	154	7	∑	∑	PUNCT
ma-125	154	8	i∈[m	i∈[m	PROPN
ma-125	154	9	]	]	PUNCT
ma-125	154	10	∑	∑	PUNCT
ma-125	154	11	j∈σi	j∈σi	PROPN
ma-125	154	12	v2	v2	PROPN
ma-125	155	1	i	i	PRON
ma-125	155	2	j	j	PROPN
ma-125	156	1	〈	〈	PROPN
ma-125	156	2	w	w	PROPN
ma-125	156	3	(	(	PUNCT
ma-125	156	4	i	i	NOUN
ma-125	156	5	)	)	PUNCT
ma-125	156	6	j	j	PROPN
ma-125	156	7	λi	λi	NUM
ma-125	156	8	jpwi	jpwi	PROPN
ma-125	156	9	j	j	PROPN
ma-125	156	10	f	f	PROPN
ma-125	156	11	,	,	PUNCT
ma-125	156	12	w	w	PROPN
ma-125	156	13	(	(	PUNCT
ma-125	156	14	i	i	NOUN
ma-125	156	15	)	)	PUNCT
ma-125	156	16	j	j	PROPN
ma-125	156	17	λi	λi	NUM
ma-125	156	18	jpwi	jpwi	PROPN
ma-125	156	19	j	j	PROPN
ma-125	157	1	f	f	PROPN
ma-125	157	2	〉	〉	PROPN
ma-125	157	3	≤	≤	PROPN
ma-125	157	4	max	max	PROPN
ma-125	157	5	i∈[m	i∈[m	PROPN
ma-125	157	6	]	]	PUNCT
ma-125	157	7	|w	|w	NOUN
ma-125	157	8	(	(	PUNCT
ma-125	157	9	i	i	NOUN
ma-125	157	10	)	)	PUNCT
ma-125	157	11	j	j	PROPN
ma-125	158	1	|	|	ADV
ma-125	158	2	2b〈f	2b〈f	NUM
ma-125	158	3	,	,	PUNCT
ma-125	158	4	f	f	PROPN
ma-125	158	5	〉	〉	PROPN
ma-125	158	6	=	=	SYM
ma-125	158	7	bd〈f	bd〈f	NOUN
ma-125	158	8	,	,	PUNCT
ma-125	158	9	f	f	PROPN
ma-125	158	10	〉	〉	PROPN
ma-125	158	11	.	.	PUNCT
ma-125	159	1	�	�	PROPN
ma-125	159	2	theorem	theorem	VERB
ma-125	159	3	2.6	2.6	NUM
ma-125	159	4	.	.	PUNCT
ma-125	160	1	let	let	VERB
ma-125	161	1	i	i	PRON
ma-125	161	2	⊂	⊂	PROPN
ma-125	161	3	j	j	PROPN
ma-125	161	4	be	be	AUX
ma-125	161	5	arbitrary	arbitrary	ADJ
ma-125	161	6	and	and	CCONJ
ma-125	161	7	{	{	PUNCT
ma-125	161	8	wi	wi	PROPN
ma-125	161	9	j	j	PROPN
ma-125	161	10	,	,	PUNCT
ma-125	161	11	λi	λi	PROPN
ma-125	161	12	j	j	PROPN
ma-125	161	13	,	,	PUNCT
ma-125	161	14	vi	vi	PROPN
ma-125	161	15	j}j∈i	j}j∈i	PROPN
ma-125	161	16	,	,	PUNCT
ma-125	161	17	i∈[m	i∈[m	PROPN
ma-125	161	18	]	]	PUNCT
ma-125	161	19	be	be	VERB
ma-125	161	20	a	a	DET
ma-125	161	21	k	k	NOUN
ma-125	161	22	−	−	NOUN
ma-125	161	23	g−fusion	g−fusion	NOUN
ma-125	161	24	woven	weave	VERB
ma-125	161	25	for	for	ADP
ma-125	161	26	h.	h.	PROPN
ma-125	161	27	then	then	ADV
ma-125	161	28	{	{	PUNCT
ma-125	161	29	wi	wi	PROPN
ma-125	161	30	j	j	PROPN
ma-125	161	31	,	,	PUNCT
ma-125	161	32	λi	λi	PROPN
ma-125	161	33	j	j	PROPN
ma-125	161	34	,	,	PUNCT
ma-125	161	35	vi	vi	PROPN
ma-125	161	36	j}j∈j	j}j∈j	PROPN
ma-125	161	37	,	,	PUNCT
ma-125	161	38	i∈[m	i∈[m	PROPN
ma-125	161	39	]	]	PUNCT
ma-125	161	40	is	be	AUX
ma-125	161	41	a	a	DET
ma-125	161	42	k	k	PROPN
ma-125	161	43	−	−	PROPN
ma-125	161	44	g−fusion	g−fusion	NOUN
ma-125	161	45	woven	weave	VERB
ma-125	161	46	.	.	PUNCT
ma-125	162	1	proof	proof	NOUN
ma-125	162	2	.	.	PUNCT
ma-125	163	1	assume	assume	VERB
ma-125	163	2	that	that	SCONJ
ma-125	163	3	σi	σi	PROPN
ma-125	163	4	⊂	⊂	PROPN
ma-125	163	5	j	j	PROPN
ma-125	163	6	,	,	PUNCT
ma-125	163	7	so	so	SCONJ
ma-125	163	8	σi	σi	ADP
ma-125	163	9	∩	∩	PROPN
ma-125	163	10	i	i	PRON
ma-125	163	11	⊂	⊂	PROPN
ma-125	163	12	i	i	PRON
ma-125	163	13	and	and	CCONJ
ma-125	163	14	a	a	PRON
ma-125	163	15	is	be	AUX
ma-125	163	16	the	the	DET
ma-125	163	17	lower	low	ADJ
ma-125	163	18	bound	bind	VERB
ma-125	163	19	of	of	ADP
ma-125	163	20	{	{	PUNCT
ma-125	163	21	wi	wi	PROPN
ma-125	163	22	j	j	PROPN
ma-125	163	23	,	,	PUNCT
ma-125	163	24	λi	λi	PROPN
ma-125	163	25	j	j	PROPN
ma-125	163	26	,	,	PUNCT
ma-125	163	27	vi	vi	X
ma-125	163	28	j}j∈σi∩i	j}j∈σi∩i	NOUN
ma-125	163	29	,	,	PUNCT
ma-125	163	30	i∈[m	i∈[m	PROPN
ma-125	163	31	]	]	PUNCT
ma-125	163	32	,	,	PUNCT
ma-125	163	33	thenfor	thenfor	VERB
ma-125	163	34	every	every	DET
ma-125	163	35	f	f	PROPN
ma-125	163	36	∈	∈	PROPN
ma-125	163	37	h	h	NOUN
ma-125	163	38	we	we	PRON
ma-125	163	39	have	have	VERB
ma-125	163	40	a〈k∗f	a〈k∗f	PUNCT
ma-125	163	41	,	,	PUNCT
ma-125	163	42	k∗f	k∗f	PROPN
ma-125	163	43	〉	〉	PROPN
ma-125	163	44	≤	≤	PROPN
ma-125	163	45	∑	∑	PUNCT
ma-125	163	46	i∈[m	i∈[m	VERB
ma-125	163	47	]	]	PUNCT
ma-125	163	48	∑	∑	PART
ma-125	164	1	j∈σi∩i	j∈σi∩i	PROPN
ma-125	164	2	v2	v2	PROPN
ma-125	165	1	i	i	PRON
ma-125	165	2	j	j	PROPN
ma-125	166	1	〈	〈	PROPN
ma-125	166	2	λi	λi	ADP
ma-125	166	3	jpwi	jpwi	PROPN
ma-125	166	4	j	j	PROPN
ma-125	166	5	f	f	PROPN
ma-125	166	6	,	,	PUNCT
ma-125	166	7	λi	λi	ADP
ma-125	166	8	jpwi	jpwi	PROPN
ma-125	166	9	j	j	PROPN
ma-125	166	10	f	f	PROPN
ma-125	166	11	〉	〉	PROPN
ma-125	166	12	≤	≤	PROPN
ma-125	166	13	∑	∑	PUNCT
ma-125	166	14	i∈[m	i∈[m	PROPN
ma-125	166	15	]	]	PUNCT
ma-125	166	16	∑	∑	PUNCT
ma-125	166	17	j∈σi	j∈σi	PROPN
ma-125	166	18	v2	v2	PROPN
ma-125	167	1	i	i	PRON
ma-125	167	2	j	j	PROPN
ma-125	168	1	〈	〈	PROPN
ma-125	168	2	λi	λi	ADP
ma-125	168	3	jpwi	jpwi	PROPN
ma-125	168	4	j	j	PROPN
ma-125	168	5	f	f	PROPN
ma-125	168	6	,	,	PUNCT
ma-125	168	7	λi	λi	ADP
ma-125	168	8	jpwi	jpwi	PROPN
ma-125	168	9	j	j	PROPN
ma-125	168	10	f	f	PROPN
ma-125	168	11	〉	〉	PROPN
ma-125	168	12	.	.	PUNCT
ma-125	169	1	this	this	PRON
ma-125	169	2	implies	imply	VERB
ma-125	169	3	the	the	DET
ma-125	169	4	statement	statement	NOUN
ma-125	169	5	.	.	PUNCT
ma-125	170	1	�	�	PROPN
ma-125	170	2	next	next	ADJ
ma-125	170	3	theorem	theorem	NOUN
ma-125	170	4	is	be	AUX
ma-125	170	5	shows	show	NOUN
ma-125	170	6	that	that	SCONJ
ma-125	170	7	even	even	ADV
ma-125	170	8	if	if	SCONJ
ma-125	170	9	one	one	NUM
ma-125	170	10	subspace	subspace	NOUN
ma-125	170	11	is	be	AUX
ma-125	170	12	deleted	delete	VERB
ma-125	170	13	,	,	PUNCT
ma-125	170	14	it	it	PRON
ma-125	170	15	dose	dose	VERB
ma-125	170	16	not	not	PART
ma-125	170	17	still	still	ADV
ma-125	170	18	remain	remain	VERB
ma-125	170	19	a	a	DET
ma-125	170	20	k−g−fusionwoven	k−g−fusionwoven	NOUN
ma-125	170	21	.	.	PUNCT
ma-125	171	1	https://doi.org/10.28924/ada/ma.3.11	https://doi.org/10.28924/ada/ma.3.11	PROPN
ma-125	171	2	eur	eur	PROPN
ma-125	171	3	.	.	PUNCT
ma-125	172	1	j.	j.	PROPN
ma-125	172	2	math	math	PROPN
ma-125	172	3	.	.	PUNCT
ma-125	173	1	anal	anal	PROPN
ma-125	173	2	.	.	PUNCT
ma-125	174	1	10.28924	10.28924	NUM
ma-125	174	2	/	/	SYM
ma-125	174	3	ada	ada	PROPN
ma-125	174	4	/	/	SYM
ma-125	174	5	ma.3.11	ma.3.11	ADJ
ma-125	174	6	7	7	NUM
ma-125	174	7	theorem	theorem	ADJ
ma-125	174	8	2.7	2.7	NUM
ma-125	174	9	.	.	PUNCT
ma-125	175	1	let	let	VERB
ma-125	175	2	k	k	PROPN
ma-125	175	3	has	have	AUX
ma-125	175	4	closed	close	VERB
ma-125	175	5	range	range	NOUN
ma-125	175	6	,	,	PUNCT
ma-125	175	7	i	i	PRON
ma-125	175	8	⊂	⊂	PROPN
ma-125	175	9	j	j	PROPN
ma-125	175	10	and	and	CCONJ
ma-125	175	11	{	{	PUNCT
ma-125	175	12	wi	wi	PROPN
ma-125	175	13	j	j	PROPN
ma-125	175	14	,	,	PUNCT
ma-125	175	15	λi	λi	PROPN
ma-125	175	16	j	j	PROPN
ma-125	175	17	,	,	PUNCT
ma-125	175	18	vi	vi	PROPN
ma-125	175	19	j}j∈j	j}j∈j	PROPN
ma-125	175	20	,	,	PUNCT
ma-125	175	21	i∈[m	i∈[m	PROPN
ma-125	175	22	]	]	PUNCT
ma-125	175	23	be	be	VERB
ma-125	175	24	a	a	DET
ma-125	175	25	k	k	NOUN
ma-125	175	26	−	−	NOUN
ma-125	175	27	g−fusion	g−fusion	NOUN
ma-125	175	28	woven	weave	VERB
ma-125	175	29	for	for	ADP
ma-125	175	30	h	h	NOUN
ma-125	175	31	with	with	ADP
ma-125	175	32	the	the	DET
ma-125	175	33	bounds	bound	NOUN
ma-125	175	34	a	a	PRON
ma-125	175	35	,	,	PUNCT
ma-125	175	36	b.	b.	NOUN
ma-125	176	1	if	if	SCONJ
ma-125	176	2	c	c	PROPN
ma-125	176	3	=	=	SYM
ma-125	176	4	∑	∑	PUNCT
ma-125	176	5	i∈[m	i∈[m	PROPN
ma-125	176	6	]	]	PUNCT
ma-125	176	7	∑	∑	PART
ma-125	176	8	j∈i	j∈i	PROPN
ma-125	176	9	v2	v2	PROPN
ma-125	176	10	i	i	PRON
ma-125	177	1	j‖λi	j‖λi	PROPN
ma-125	177	2	jpwi	jpwi	NOUN
ma-125	177	3	j	j	PROPN
ma-125	177	4	‖2	‖2	NOUN
ma-125	177	5	<	<	X
ma-125	177	6	a‖k+‖2	a‖k+‖2	NOUN
ma-125	177	7	,	,	PUNCT
ma-125	177	8	then	then	ADV
ma-125	177	9	{	{	PUNCT
ma-125	177	10	wi	wi	PROPN
ma-125	177	11	j	j	PROPN
ma-125	177	12	,	,	PUNCT
ma-125	177	13	λi	λi	PROPN
ma-125	177	14	j	j	PROPN
ma-125	177	15	,	,	PUNCT
ma-125	177	16	vi	vi	PROPN
ma-125	177	17	j}j∈j−i	j}j∈j−i	PROPN
ma-125	177	18	,	,	PUNCT
ma-125	177	19	i∈[m	i∈[m	PROPN
ma-125	177	20	]	]	PUNCT
ma-125	177	21	is	be	AUX
ma-125	177	22	a	a	DET
ma-125	177	23	k	k	PROPN
ma-125	177	24	−	−	NOUN
ma-125	177	25	g−fusion	g−fusion	NOUN
ma-125	177	26	woven	weave	VERB
ma-125	177	27	for	for	ADP
ma-125	177	28	r(k	r(k	PROPN
ma-125	177	29	)	)	PUNCT
ma-125	177	30	.	.	PUNCT
ma-125	178	1	proof	proof	NOUN
ma-125	178	2	.	.	PUNCT
ma-125	179	1	the	the	DET
ma-125	179	2	upper	upper	ADJ
ma-125	179	3	bound	bind	VERB
ma-125	179	4	is	be	AUX
ma-125	179	5	obvious	obvious	ADJ
ma-125	179	6	.	.	PUNCT
ma-125	180	1	suppose	suppose	VERB
ma-125	180	2	that	that	SCONJ
ma-125	180	3	σi	σi	PROPN
ma-125	180	4	i∈[m	i∈[m	VERB
ma-125	180	5	]	]	X
ma-125	181	1	⊂	⊂	PROPN
ma-125	181	2	j−	j−	VERB
ma-125	181	3	i	i	PRON
ma-125	181	4	and	and	CCONJ
ma-125	181	5	f	f	PROPN
ma-125	181	6	∈	∈	PROPN
ma-125	181	7	r(k	r(k	PROPN
ma-125	181	8	)	)	PUNCT
ma-125	181	9	,	,	PUNCT
ma-125	181	10	so	so	SCONJ
ma-125	181	11	we	we	PRON
ma-125	181	12	get∑	get∑	NUM
ma-125	181	13	i∈[m	i∈[m	VERB
ma-125	181	14	]	]	PUNCT
ma-125	181	15	∑	∑	PUNCT
ma-125	181	16	j∈σi	j∈σi	PROPN
ma-125	181	17	v2	v2	PROPN
ma-125	182	1	i	i	PRON
ma-125	182	2	j	j	PROPN
ma-125	183	1	〈	〈	PROPN
ma-125	183	2	λi	λi	ADP
ma-125	183	3	jpwi	jpwi	PROPN
ma-125	183	4	j	j	PROPN
ma-125	183	5	f	f	PROPN
ma-125	183	6	,	,	PUNCT
ma-125	183	7	λi	λi	ADP
ma-125	183	8	jpwi	jpwi	PROPN
ma-125	183	9	j	j	PROPN
ma-125	184	1	f	f	PROPN
ma-125	184	2	〉	〉	PROPN
ma-125	184	3	=	=	SYM
ma-125	184	4	∑	∑	PUNCT
ma-125	184	5	i∈[m	i∈[m	PROPN
ma-125	184	6	]	]	PUNCT
ma-125	184	7	∑	∑	PUNCT
ma-125	184	8	j∈σi∪i	j∈σi∪i	VERB
ma-125	184	9	v2	v2	VERB
ma-125	185	1	i	i	PRON
ma-125	185	2	j	j	PROPN
ma-125	186	1	〈	〈	PROPN
ma-125	186	2	λi	λi	ADP
ma-125	186	3	jpwi	jpwi	PROPN
ma-125	186	4	j	j	PROPN
ma-125	186	5	f	f	PROPN
ma-125	186	6	,	,	PUNCT
ma-125	186	7	λi	λi	ADP
ma-125	186	8	jpwi	jpwi	PROPN
ma-125	186	9	j	j	PROPN
ma-125	186	10	f	f	PROPN
ma-125	186	11	〉	〉	PROPN
ma-125	186	12	−	−	PROPN
ma-125	186	13	∑	∑	PUNCT
ma-125	186	14	i∈[m	i∈[m	PROPN
ma-125	186	15	]	]	PUNCT
ma-125	186	16	∑	∑	PART
ma-125	186	17	j∈i	j∈i	PROPN
ma-125	186	18	v2	v2	PROPN
ma-125	187	1	i	i	PRON
ma-125	187	2	j	j	PROPN
ma-125	188	1	〈	〈	PROPN
ma-125	188	2	λi	λi	ADP
ma-125	188	3	jpwi	jpwi	PROPN
ma-125	188	4	j	j	PROPN
ma-125	188	5	f	f	PROPN
ma-125	188	6	,	,	PUNCT
ma-125	188	7	λi	λi	ADP
ma-125	188	8	jpwi	jpwi	PROPN
ma-125	188	9	j	j	PROPN
ma-125	188	10	f	f	PROPN
ma-125	188	11	〉	〉	PROPN
ma-125	188	12	≥	≥	PROPN
ma-125	188	13	a〈k∗f	a〈k∗f	PUNCT
ma-125	188	14	,	,	PUNCT
ma-125	188	15	k∗f	k∗f	PROPN
ma-125	188	16	〉	〉	PROPN
ma-125	188	17	−	−	PROPN
ma-125	188	18	∑	∑	PUNCT
ma-125	188	19	i∈[m	i∈[m	PROPN
ma-125	188	20	]	]	PUNCT
ma-125	188	21	∑	∑	PART
ma-125	188	22	j∈i	j∈i	PROPN
ma-125	188	23	v2	v2	PROPN
ma-125	188	24	i	i	PRON
ma-125	189	1	j‖λi	j‖λi	PROPN
ma-125	189	2	jpwi	jpwi	NOUN
ma-125	189	3	j	j	PROPN
ma-125	189	4	‖2〈f	‖2〈f	PROPN
ma-125	189	5	,	,	PUNCT
ma-125	189	6	f	f	PROPN
ma-125	189	7	〉	〉	PROPN
ma-125	189	8	≥	≥	PROPN
ma-125	189	9	(	(	PUNCT
ma-125	189	10	a−	a−	PROPN
ma-125	189	11	c‖k+‖2)〈k∗f	c‖k+‖2)〈k∗f	X
ma-125	189	12	,	,	PUNCT
ma-125	189	13	k∗f	k∗f	PROPN
ma-125	189	14	〉	〉	PROPN
ma-125	189	15	.	.	PUNCT
ma-125	190	1	�	�	PROPN
ma-125	190	2	theorem	theorem	VERB
ma-125	190	3	2.8	2.8	NUM
ma-125	190	4	.	.	PUNCT
ma-125	191	1	let	let	VERB
ma-125	191	2	{	{	PUNCT
ma-125	191	3	wi	wi	PROPN
ma-125	191	4	j	j	PROPN
ma-125	191	5	,	,	PUNCT
ma-125	191	6	λi	λi	PROPN
ma-125	191	7	j	j	PROPN
ma-125	191	8	,	,	PUNCT
ma-125	191	9	vi	vi	PROPN
ma-125	191	10	j}j∈j	j}j∈j	PROPN
ma-125	191	11	,	,	PUNCT
ma-125	191	12	i∈[m	i∈[m	PROPN
ma-125	191	13	]	]	PUNCT
ma-125	191	14	be	be	VERB
ma-125	191	15	a	a	DET
ma-125	191	16	k	k	NOUN
ma-125	191	17	−	−	NOUN
ma-125	191	18	g−fusion	g−fusion	NOUN
ma-125	191	19	woven	weave	VERB
ma-125	191	20	for	for	ADP
ma-125	191	21	h	h	NOUN
ma-125	191	22	with	with	ADP
ma-125	191	23	bounds	bound	NOUN
ma-125	191	24	a	a	DET
ma-125	191	25	,	,	PUNCT
ma-125	191	26	b.	b.	PROPN
ma-125	191	27	for	for	ADP
ma-125	191	28	each	each	DET
ma-125	191	29	i	i	PRON
ma-125	191	30	∈	∈	PROPN
ma-125	192	1	[	[	AUX
ma-125	192	2	m],j	m],j	PROPN
ma-125	192	3	∈	∈	PROPN
ma-125	192	4	j	j	PROPN
ma-125	192	5	and	and	CCONJ
ma-125	192	6	a	a	DET
ma-125	192	7	index	index	NOUN
ma-125	192	8	set	set	VERB
ma-125	192	9	ii	ii	PROPN
ma-125	192	10	j	j	PROPN
ma-125	192	11	,	,	PUNCT
ma-125	192	12	suppose	suppose	VERB
ma-125	192	13	that	that	SCONJ
ma-125	192	14	{	{	PUNCT
ma-125	192	15	f	f	X
ma-125	192	16	(	(	PUNCT
ma-125	192	17	k	k	NOUN
ma-125	192	18	)	)	PUNCT
ma-125	192	19	i	i	PRON
ma-125	192	20	j	j	PROPN
ma-125	192	21	}	}	PUNCT
ma-125	192	22	k∈ii	k∈ii	VERB
ma-125	192	23	j	j	PROPN
ma-125	192	24	∈	∈	PROPN
ma-125	192	25	λi	λi	ADP
ma-125	192	26	j(wi	j(wi	PROPN
ma-125	192	27	j	j	PROPN
ma-125	192	28	)	)	PUNCT
ma-125	192	29	is	be	AUX
ma-125	192	30	a	a	DET
ma-125	192	31	parseval	parseval	NOUN
ma-125	192	32	frame	frame	NOUN
ma-125	192	33	for	for	ADP
ma-125	192	34	hi	hi	INTJ
ma-125	192	35	j	j	PROPN
ma-125	192	36	such	such	ADJ
ma-125	192	37	that	that	PRON
ma-125	192	38	for	for	ADP
ma-125	192	39	every	every	DET
ma-125	192	40	finite	finite	NOUN
ma-125	192	41	subset	subset	VERB
ma-125	193	1	ki	ki	PROPN
ma-125	193	2	j	j	PROPN
ma-125	193	3	⊂	⊂	PROPN
ma-125	193	4	ii	ii	PROPN
ma-125	193	5	j	j	PROPN
ma-125	193	6	,	,	PUNCT
ma-125	193	7	the	the	DET
ma-125	193	8	set	set	NOUN
ma-125	193	9	{	{	PUNCT
ma-125	193	10	f	f	PROPN
ma-125	193	11	kij	kij	PROPN
ma-125	193	12	}	}	PUNCT
ma-125	193	13	k∈ii	k∈ii	PROPN
ma-125	193	14	j−ki	j−ki	PROPN
ma-125	193	15	j	j	PROPN
ma-125	193	16	is	be	AUX
ma-125	193	17	a	a	DET
ma-125	193	18	frame	frame	NOUN
ma-125	193	19	with	with	ADP
ma-125	193	20	the	the	DET
ma-125	193	21	lower	lower	ADV
ma-125	193	22	bound	bind	VERB
ma-125	193	23	ci	ci	PROPN
ma-125	193	24	j	j	PROPN
ma-125	193	25	.	.	PUNCT
ma-125	194	1	let	let	VERB
ma-125	194	2	w̃i	w̃i	PROPN
ma-125	194	3	j	j	PROPN
ma-125	194	4	=	=	SYM
ma-125	194	5	span{λ∗i	span{λ∗i	PROPN
ma-125	195	1	j	j	PROPN
ma-125	195	2	f	f	X
ma-125	195	3	(	(	PUNCT
ma-125	195	4	k	k	X
ma-125	195	5	)	)	PUNCT
ma-125	195	6	i	i	PRON
ma-125	195	7	j	j	PROPN
ma-125	195	8	}	}	PUNCT
ma-125	195	9	k∈ii	k∈ii	PROPN
ma-125	195	10	j−ki	j−ki	PROPN
ma-125	195	11	j	j	PROPN
ma-125	195	12	for	for	ADP
ma-125	195	13	any	any	DET
ma-125	195	14	i	i	PRON
ma-125	195	15	∈	∈	PROPN
ma-125	196	1	[	[	X
ma-125	196	2	m	m	X
ma-125	196	3	]	]	X
ma-125	196	4	and	and	CCONJ
ma-125	196	5	j	j	PROPN
ma-125	196	6	∈	∈	PROPN
ma-125	196	7	j	j	PROPN
ma-125	196	8	,	,	PUNCT
ma-125	196	9	then	then	ADV
ma-125	196	10	{	{	PUNCT
ma-125	196	11	w̃i	w̃i	PROPN
ma-125	196	12	j	j	PROPN
ma-125	196	13	,	,	PUNCT
ma-125	196	14	λi	λi	PROPN
ma-125	196	15	j	j	PROPN
ma-125	196	16	,	,	PUNCT
ma-125	196	17	vi	vi	PROPN
ma-125	196	18	j}j∈j	j}j∈j	PROPN
ma-125	196	19	,	,	PUNCT
ma-125	196	20	i∈[m	i∈[m	PROPN
ma-125	196	21	]	]	PUNCT
ma-125	196	22	is	be	AUX
ma-125	196	23	a	a	DET
ma-125	196	24	k	k	PROPN
ma-125	196	25	−	−	NOUN
ma-125	196	26	g−fusion	g−fusion	NOUN
ma-125	196	27	woven	weave	VERB
ma-125	196	28	for	for	ADP
ma-125	196	29	h	h	NOUN
ma-125	196	30	with	with	ADP
ma-125	196	31	the	the	DET
ma-125	196	32	bounds	bound	NOUN
ma-125	196	33	(	(	PUNCT
ma-125	196	34	mini∈[m],j∈j	mini∈[m],j∈j	NOUN
ma-125	196	35	ci	ci	PROPN
ma-125	196	36	j)a	j)a	PROPN
ma-125	196	37	and	and	CCONJ
ma-125	196	38	b.	b.	PROPN
ma-125	196	39	proof	proof	NOUN
ma-125	196	40	.	.	PUNCT
ma-125	197	1	obviously	obviously	ADV
ma-125	197	2	,	,	PUNCT
ma-125	197	3	b	b	PROPN
ma-125	197	4	is	be	AUX
ma-125	197	5	the	the	DET
ma-125	197	6	upper	upper	ADJ
ma-125	197	7	bound	bind	VERB
ma-125	197	8	of	of	ADP
ma-125	197	9	{	{	PUNCT
ma-125	197	10	w̃i	w̃i	PROPN
ma-125	197	11	j	j	PROPN
ma-125	197	12	,	,	PUNCT
ma-125	197	13	λi	λi	PROPN
ma-125	197	14	j	j	PROPN
ma-125	197	15	,	,	PUNCT
ma-125	197	16	vi	vi	PROPN
ma-125	197	17	j}j∈j	j}j∈j	PROPN
ma-125	197	18	,	,	PUNCT
ma-125	197	19	i∈[m	i∈[m	PROPN
ma-125	197	20	]	]	PUNCT
ma-125	197	21	.	.	PUNCT
ma-125	198	1	assume	assume	VERB
ma-125	198	2	that	that	SCONJ
ma-125	198	3	f	f	PROPN
ma-125	198	4	∈	∈	PROPN
ma-125	198	5	h	h	NOUN
ma-125	198	6	and	and	CCONJ
ma-125	198	7	{	{	PUNCT
ma-125	198	8	σi}i∈[m	σi}i∈[m	PROPN
ma-125	198	9	]	]	X
ma-125	198	10	∈	∈	PROPN
ma-125	198	11	j	j	PROPN
ma-125	198	12	,	,	PUNCT
ma-125	198	13	so	so	ADV
ma-125	198	14	∑	∑	PUNCT
ma-125	198	15	i∈[m	i∈[m	VERB
ma-125	198	16	]	]	PUNCT
ma-125	198	17	∑	∑	PUNCT
ma-125	198	18	j∈σi	j∈σi	PROPN
ma-125	198	19	v2	v2	PROPN
ma-125	199	1	i	i	PRON
ma-125	199	2	j	j	PROPN
ma-125	200	1	〈	〈	PROPN
ma-125	200	2	λi	λi	PROPN
ma-125	200	3	jpw̃i	jpw̃i	PROPN
ma-125	200	4	j	j	PROPN
ma-125	200	5	f	f	PROPN
ma-125	200	6	,	,	PUNCT
ma-125	200	7	λi	λi	ADP
ma-125	200	8	jpw̃i	jpw̃i	PROPN
ma-125	200	9	j	j	PROPN
ma-125	200	10	f	f	PROPN
ma-125	200	11	〉	〉	PROPN
ma-125	200	12	=	=	SYM
ma-125	200	13	∑	∑	PUNCT
ma-125	200	14	i∈[m	i∈[m	PROPN
ma-125	200	15	]	]	PUNCT
ma-125	200	16	∑	∑	PUNCT
ma-125	200	17	j∈σi	j∈σi	PROPN
ma-125	200	18	v2	v2	PROPN
ma-125	201	1	i	i	PRON
ma-125	201	2	j	j	PROPN
ma-125	201	3	∑	∑	PUNCT
ma-125	201	4	k∈ii	k∈ii	VERB
ma-125	201	5	j	j	PROPN
ma-125	201	6	〈	〈	PROPN
ma-125	201	7	λi	λi	PROPN
ma-125	201	8	jpw̃i	jpw̃i	PROPN
ma-125	201	9	j	j	PROPN
ma-125	201	10	f	f	PROPN
ma-125	201	11	,	,	PUNCT
ma-125	201	12	f	f	PROPN
ma-125	201	13	(	(	PUNCT
ma-125	201	14	k	k	X
ma-125	201	15	)	)	PUNCT
ma-125	202	1	i	i	PRON
ma-125	203	1	j	j	PROPN
ma-125	203	2	〉	〉	PROPN
ma-125	203	3	〈	〈	PROPN
ma-125	203	4	f	f	PROPN
ma-125	203	5	(	(	PUNCT
ma-125	203	6	k	k	X
ma-125	203	7	)	)	PUNCT
ma-125	204	1	i	i	PRON
ma-125	204	2	j	j	PROPN
ma-125	204	3	,	,	PUNCT
ma-125	204	4	λi	λi	ADP
ma-125	204	5	jpw̃i	jpw̃i	PROPN
ma-125	205	1	j	j	PROPN
ma-125	205	2	f	f	PROPN
ma-125	205	3	〉	〉	PROPN
ma-125	205	4	≥	≥	PROPN
ma-125	205	5	∑	∑	ADV
ma-125	205	6	i∈[m	i∈[m	VERB
ma-125	205	7	]	]	PUNCT
ma-125	205	8	∑	∑	PUNCT
ma-125	205	9	j∈σi	j∈σi	PROPN
ma-125	205	10	v2	v2	PROPN
ma-125	206	1	i	i	PRON
ma-125	206	2	j	j	PROPN
ma-125	206	3	∑	∑	PUNCT
ma-125	206	4	k∈ii	k∈ii	PROPN
ma-125	206	5	j−ki	j−ki	PROPN
ma-125	206	6	j	j	PROPN
ma-125	206	7	〈	〈	PROPN
ma-125	206	8	λi	λi	PROPN
ma-125	206	9	jpw̃i	jpw̃i	PROPN
ma-125	206	10	j	j	PROPN
ma-125	206	11	f	f	PROPN
ma-125	206	12	,	,	PUNCT
ma-125	206	13	f	f	PROPN
ma-125	206	14	(	(	PUNCT
ma-125	206	15	k	k	X
ma-125	206	16	)	)	PUNCT
ma-125	206	17	i	i	PRON
ma-125	207	1	j	j	PROPN
ma-125	207	2	〉	〉	PROPN
ma-125	207	3	〈	〈	PROPN
ma-125	207	4	f	f	PROPN
ma-125	207	5	(	(	PUNCT
ma-125	207	6	k	k	X
ma-125	207	7	)	)	PUNCT
ma-125	208	1	i	i	PRON
ma-125	208	2	j	j	PROPN
ma-125	208	3	,	,	PUNCT
ma-125	208	4	λi	λi	ADP
ma-125	208	5	jpw̃i	jpw̃i	PROPN
ma-125	209	1	j	j	PROPN
ma-125	209	2	f	f	PROPN
ma-125	209	3	〉	〉	PROPN
ma-125	209	4	≥	≥	PROPN
ma-125	209	5	∑	∑	ADV
ma-125	209	6	i∈[m	i∈[m	VERB
ma-125	209	7	]	]	PUNCT
ma-125	209	8	∑	∑	PUNCT
ma-125	209	9	j∈σi	j∈σi	PROPN
ma-125	209	10	v2	v2	PROPN
ma-125	210	1	i	i	PRON
ma-125	210	2	jci	jci	PROPN
ma-125	210	3	j〈λi	j〈λi	PROPN
ma-125	210	4	jpwi	jpwi	PROPN
ma-125	210	5	j	j	PROPN
ma-125	210	6	f	f	PROPN
ma-125	210	7	,	,	PUNCT
ma-125	210	8	λi	λi	ADP
ma-125	210	9	jpwi	jpwi	PROPN
ma-125	211	1	j	j	PROPN
ma-125	212	1	f	f	PROPN
ma-125	212	2	〉	〉	PROPN
ma-125	212	3	≥	≥	NUM
ma-125	212	4	(	(	PUNCT
ma-125	212	5	min	min	PROPN
ma-125	212	6	i∈[m],j∈j	i∈[m],j∈j	ADP
ma-125	212	7	ci	ci	PROPN
ma-125	212	8	j	j	PROPN
ma-125	212	9	)	)	PUNCT
ma-125	212	10	∑	∑	PUNCT
ma-125	212	11	i∈[m	i∈[m	PROPN
ma-125	212	12	]	]	PUNCT
ma-125	212	13	∑	∑	PUNCT
ma-125	212	14	j∈σi	j∈σi	PROPN
ma-125	212	15	v2	v2	PROPN
ma-125	213	1	i	i	PRON
ma-125	213	2	j	j	PROPN
ma-125	214	1	〈	〈	PROPN
ma-125	214	2	λi	λi	ADP
ma-125	214	3	jpwi	jpwi	PROPN
ma-125	214	4	j	j	PROPN
ma-125	214	5	f	f	PROPN
ma-125	214	6	,	,	PUNCT
ma-125	214	7	λi	λi	ADP
ma-125	214	8	jpwi	jpwi	PROPN
ma-125	214	9	j	j	PROPN
ma-125	215	1	f	f	PROPN
ma-125	215	2	〉	〉	PROPN
ma-125	215	3	≥	≥	NUM
ma-125	215	4	(	(	PUNCT
ma-125	215	5	min	min	PROPN
ma-125	215	6	i∈[m],j∈j	i∈[m],j∈j	ADP
ma-125	215	7	ci	ci	PROPN
ma-125	215	8	j)a〈k∗f	j)a〈k∗f	PROPN
ma-125	215	9	,	,	PUNCT
ma-125	215	10	k∗f	k∗f	PROPN
ma-125	215	11	〉	〉	PROPN
ma-125	215	12	.	.	PUNCT
ma-125	215	13	�	�	PROPN
ma-125	215	14	theorem	theorem	VERB
ma-125	215	15	2.9	2.9	NUM
ma-125	215	16	.	.	PUNCT
ma-125	216	1	let	let	VERB
ma-125	216	2	{	{	PUNCT
ma-125	216	3	wi	wi	PROPN
ma-125	216	4	j	j	PROPN
ma-125	216	5	,	,	PUNCT
ma-125	216	6	λi	λi	PROPN
ma-125	216	7	j	j	PROPN
ma-125	216	8	,	,	PUNCT
ma-125	216	9	vi	vi	PROPN
ma-125	217	1	j}j∈j	j}j∈j	PROPN
ma-125	217	2	is	be	AUX
ma-125	217	3	a	a	DET
ma-125	217	4	k−g−fusion	k−g−fusion	NOUN
ma-125	217	5	frame	frame	NOUN
ma-125	217	6	for	for	ADP
ma-125	217	7	h	h	NOUN
ma-125	217	8	for	for	ADP
ma-125	217	9	each	each	DET
ma-125	217	10	i	i	PRON
ma-125	217	11	∈	∈	PROPN
ma-125	218	1	[	[	X
ma-125	219	1	m	m	X
ma-125	219	2	]	]	X
ma-125	219	3	.	.	PUNCT
ma-125	220	1	suppose	suppose	VERB
ma-125	220	2	that	that	SCONJ
ma-125	220	3	for	for	ADP
ma-125	220	4	a	a	DET
ma-125	220	5	partition	partition	NOUN
ma-125	220	6	collection	collection	NOUN
ma-125	220	7	of	of	ADP
ma-125	220	8	disjoint	disjoint	PROPN
ma-125	220	9	finite	finite	PROPN
ma-125	220	10	sets	set	NOUN
ma-125	220	11	{	{	PUNCT
ma-125	220	12	δi}i∈[m	δi}i∈[m	NOUN
ma-125	220	13	]	]	X
ma-125	220	14	of	of	ADP
ma-125	220	15	j	j	PROPN
ma-125	220	16	and	and	CCONJ
ma-125	220	17	for	for	ADP
ma-125	220	18	any	any	DET
ma-125	220	19	ε	ε	PROPN
ma-125	220	20	>	>	X
ma-125	220	21	0	0	PUNCT
ma-125	220	22	there	there	PRON
ma-125	220	23	exists	exist	VERB
ma-125	220	24	a	a	DET
ma-125	220	25	partition	partition	NOUN
ma-125	220	26	{	{	PUNCT
ma-125	220	27	σi}i∈[m	σi}i∈[m	NOUN
ma-125	220	28	]	]	PUNCT
ma-125	220	29	of	of	ADP
ma-125	220	30	the	the	DET
ma-125	220	31	set	set	NOUN
ma-125	220	32	j	j	PROPN
ma-125	220	33	−	−	PROPN
ma-125	220	34	∪i∈[m]δi	∪i∈[m]δi	PROPN
ma-125	220	35	such	such	ADJ
ma-125	220	36	that	that	SCONJ
ma-125	220	37	{	{	PUNCT
ma-125	220	38	wi	wi	PROPN
ma-125	220	39	j	j	PROPN
ma-125	220	40	,	,	PUNCT
ma-125	220	41	λi	λi	PROPN
ma-125	220	42	j	j	PROPN
ma-125	220	43	,	,	PUNCT
ma-125	220	44	vi	vi	PROPN
ma-125	220	45	j}j∈(σi∪δi	j}j∈(σi∪δi	PUNCT
ma-125	220	46	)	)	PUNCT
ma-125	220	47	,	,	PUNCT
ma-125	220	48	i∈[m	i∈[m	PROPN
ma-125	220	49	]	]	PUNCT
ma-125	220	50	has	have	VERB
ma-125	220	51	a	a	DET
ma-125	220	52	lower	low	ADJ
ma-125	220	53	k	k	NOUN
ma-125	220	54	−	−	NOUN
ma-125	220	55	g−fusion	g−fusion	NOUN
ma-125	220	56	frame	frame	NOUN
ma-125	220	57	bound	bind	VERB
ma-125	220	58	less	less	ADJ
ma-125	220	59	than	than	ADP
ma-125	220	60	ε	ε	PROPN
ma-125	220	61	.	.	PUNCT
ma-125	221	1	then	then	ADV
ma-125	221	2	{	{	PUNCT
ma-125	221	3	wi	wi	PROPN
ma-125	221	4	j	j	PROPN
ma-125	221	5	,	,	PUNCT
ma-125	221	6	λi	λi	PROPN
ma-125	221	7	j	j	PROPN
ma-125	221	8	,	,	PUNCT
ma-125	221	9	vi	vi	PROPN
ma-125	221	10	j}j∈j	j}j∈j	PROPN
ma-125	221	11	,	,	PUNCT
ma-125	221	12	i∈[m	i∈[m	PROPN
ma-125	221	13	]	]	PUNCT
ma-125	221	14	is	be	AUX
ma-125	221	15	not	not	PART
ma-125	221	16	a	a	DET
ma-125	221	17	woven	weave	VERB
ma-125	221	18	.	.	PUNCT
ma-125	222	1	https://doi.org/10.28924/ada/ma.3.11	https://doi.org/10.28924/ada/ma.3.11	PROPN
ma-125	222	2	eur	eur	PROPN
ma-125	222	3	.	.	PUNCT
ma-125	223	1	j.	j.	PROPN
ma-125	223	2	math	math	PROPN
ma-125	223	3	.	.	PUNCT
ma-125	224	1	anal	anal	PROPN
ma-125	224	2	.	.	PUNCT
ma-125	225	1	10.28924	10.28924	NUM
ma-125	225	2	/	/	SYM
ma-125	225	3	ada	ada	PROPN
ma-125	225	4	/	/	SYM
ma-125	225	5	ma.3.11	ma.3.11	ADJ
ma-125	225	6	8	8	NUM
ma-125	225	7	proof	proof	NOUN
ma-125	225	8	.	.	PUNCT
ma-125	226	1	we	we	PRON
ma-125	226	2	can	can	AUX
ma-125	226	3	write	write	VERB
ma-125	226	4	j	j	PROPN
ma-125	226	5	=	=	PUNCT
ma-125	226	6	∪j∈njj	∪j∈njj	PROPN
ma-125	226	7	,	,	PUNCT
ma-125	226	8	where	where	SCONJ
ma-125	226	9	jj	jj	PROPN
ma-125	226	10	are	be	AUX
ma-125	226	11	disjoint	disjoint	NOUN
ma-125	226	12	index	index	NOUN
ma-125	226	13	sets	set	NOUN
ma-125	226	14	.	.	PUNCT
ma-125	227	1	assume	assume	VERB
ma-125	227	2	that	that	SCONJ
ma-125	227	3	δ1j	δ1j	PROPN
ma-125	227	4	=	=	X
ma-125	227	5	∅	∅	NOUN
ma-125	227	6	for	for	ADP
ma-125	227	7	all	all	PRON
ma-125	227	8	i	i	PRON
ma-125	227	9	∈	∈	PROPN
ma-125	228	1	[	[	X
ma-125	228	2	m	m	X
ma-125	228	3	]	]	X
ma-125	228	4	and	and	CCONJ
ma-125	228	5	ε	ε	PROPN
ma-125	228	6	=	=	SYM
ma-125	228	7	1	1	X
ma-125	228	8	.	.	PUNCT
ma-125	229	1	then	then	ADV
ma-125	229	2	,	,	PUNCT
ma-125	229	3	there	there	PRON
ma-125	229	4	exists	exist	VERB
ma-125	229	5	a	a	DET
ma-125	229	6	partition	partition	NOUN
ma-125	229	7	σi1i∈[m	σi1i∈[m	NOUN
ma-125	229	8	]	]	PUNCT
ma-125	229	9	of	of	ADP
ma-125	229	10	j	j	PROPN
ma-125	229	11	such	such	ADJ
ma-125	229	12	that	that	SCONJ
ma-125	229	13	{	{	PUNCT
ma-125	229	14	wi	wi	PROPN
ma-125	229	15	j	j	PROPN
ma-125	229	16	,	,	PUNCT
ma-125	229	17	λi	λi	PROPN
ma-125	229	18	j	j	PROPN
ma-125	229	19	,	,	PUNCT
ma-125	229	20	vi	vi	PROPN
ma-125	229	21	j}j∈(σi1∪δi1),i∈[m]has	j}j∈(σi1∪δi1),i∈[m]ha	VERB
ma-125	229	22	a	a	DET
ma-125	229	23	lower	lower	ADV
ma-125	229	24	bound	bind	VERB
ma-125	229	25	(	(	PUNCT
ma-125	229	26	also	also	ADV
ma-125	229	27	,	,	PUNCT
ma-125	229	28	optimal	optimal	ADJ
ma-125	229	29	lower	lower	ADV
ma-125	229	30	bound	bind	VERB
ma-125	229	31	)	)	PUNCT
ma-125	229	32	less	less	ADJ
ma-125	229	33	than	than	ADP
ma-125	229	34	1	1	NUM
ma-125	229	35	.	.	PUNCT
ma-125	230	1	thus	thus	ADV
ma-125	230	2	,	,	PUNCT
ma-125	230	3	there	there	PRON
ma-125	230	4	is	be	VERB
ma-125	230	5	a	a	DET
ma-125	230	6	f1	f1	ADJ
ma-125	230	7	∈	∈	NOUN
ma-125	230	8	h	h	NOUN
ma-125	230	9	such	such	ADJ
ma-125	230	10	that∑	that∑	PROPN
ma-125	230	11	i∈[m	i∈[m	NOUN
ma-125	230	12	]	]	PUNCT
ma-125	230	13	∑	∑	PART
ma-125	230	14	j∈(σi1∪δi1	j∈(σi1∪δi1	NOUN
ma-125	230	15	)	)	PUNCT
ma-125	230	16	v2	v2	NOUN
ma-125	231	1	i	i	PRON
ma-125	231	2	j	j	PROPN
ma-125	232	1	〈	〈	PROPN
ma-125	232	2	λi	λi	ADP
ma-125	232	3	jpwi	jpwi	PROPN
ma-125	232	4	j	j	PROPN
ma-125	232	5	f1,λi	f1,λi	PROPN
ma-125	232	6	jpwi	jpwi	PROPN
ma-125	232	7	j	j	PROPN
ma-125	232	8	f1	f1	PROPN
ma-125	232	9	〉	〉	PROPN
ma-125	232	10	<	<	X
ma-125	232	11	〈	〈	PROPN
ma-125	232	12	k∗f1	k∗f1	NOUN
ma-125	232	13	,	,	PUNCT
ma-125	232	14	k∗f1	k∗f1	PROPN
ma-125	232	15	〉	〉	NUM
ma-125	232	16	.	.	PUNCT
ma-125	233	1	since	since	SCONJ
ma-125	233	2	∑	∑	PART
ma-125	233	3	i∈[m	i∈[m	VERB
ma-125	233	4	]	]	PUNCT
ma-125	233	5	∑	∑	PUNCT
ma-125	233	6	j∈j	j∈j	NOUN
ma-125	233	7	v2	v2	PROPN
ma-125	234	1	i	i	PRON
ma-125	234	2	j	j	PROPN
ma-125	235	1	〈	〈	PROPN
ma-125	235	2	λi	λi	ADP
ma-125	235	3	jpwi	jpwi	PROPN
ma-125	235	4	j	j	PROPN
ma-125	235	5	f1,λi	f1,λi	PROPN
ma-125	235	6	jpwi	jpwi	PROPN
ma-125	235	7	j	j	PROPN
ma-125	235	8	f1	f1	PROPN
ma-125	235	9	〉	〉	PROPN
ma-125	235	10	<	<	X
ma-125	235	11	∞	∞	PROPN
ma-125	235	12	,	,	PUNCT
ma-125	235	13	so	so	ADV
ma-125	235	14	,	,	PUNCT
ma-125	235	15	there	there	PRON
ma-125	235	16	is	be	VERB
ma-125	235	17	a	a	DET
ma-125	235	18	k1	k1	NOUN
ma-125	235	19	∈	∈	NOUN
ma-125	235	20	n	n	DET
ma-125	235	21	such	such	ADJ
ma-125	235	22	that∑	that∑	NOUN
ma-125	235	23	i∈[m	i∈[m	NOUN
ma-125	235	24	]	]	PUNCT
ma-125	235	25	∑	∑	PUNCT
ma-125	235	26	j∈k1	j∈k1	VERB
ma-125	235	27	v2	v2	NOUN
ma-125	236	1	i	i	PRON
ma-125	236	2	j	j	PROPN
ma-125	237	1	〈	〈	PROPN
ma-125	237	2	λi	λi	ADP
ma-125	237	3	jpwi	jpwi	PROPN
ma-125	237	4	j	j	PROPN
ma-125	237	5	f1,λi	f1,λi	PROPN
ma-125	237	6	jpwi	jpwi	PROPN
ma-125	237	7	j	j	PROPN
ma-125	237	8	f1	f1	PROPN
ma-125	237	9	〉	〉	PROPN
ma-125	237	10	<	<	X
ma-125	237	11	〈	〈	PROPN
ma-125	237	12	k∗f1	k∗f1	NOUN
ma-125	237	13	,	,	PUNCT
ma-125	237	14	k∗f1	k∗f1	PROPN
ma-125	237	15	〉	〉	PROPN
ma-125	237	16	,	,	PUNCT
ma-125	237	17	where	where	SCONJ
ma-125	237	18	,	,	PUNCT
ma-125	237	19	k1	k1	NOUN
ma-125	237	20	=	=	SYM
ma-125	237	21	∪i≥k1	∪i≥k1	NOUN
ma-125	237	22	+	+	NOUN
ma-125	237	23	1jj	1jj	ADJ
ma-125	237	24	.	.	PUNCT
ma-125	238	1	continuing	continue	VERB
ma-125	238	2	this	this	DET
ma-125	238	3	way	way	NOUN
ma-125	238	4	,	,	PUNCT
ma-125	238	5	for	for	ADP
ma-125	238	6	ε	ε	PROPN
ma-125	238	7	=	=	SYM
ma-125	238	8	1	1	NUM
ma-125	238	9	n	n	NUM
ma-125	238	10	and	and	CCONJ
ma-125	238	11	a	a	DET
ma-125	238	12	partition	partition	NOUN
ma-125	238	13	{	{	PUNCT
ma-125	238	14	δni}i∈[m	δni}i∈[m	PROPN
ma-125	238	15	]	]	PUNCT
ma-125	238	16	of	of	ADP
ma-125	238	17	j1	j1	PROPN
ma-125	238	18	∪	∪	ADV
ma-125	238	19	...	...	PUNCT
ma-125	238	20	∪	∪	ADJ
ma-125	238	21	jkn−1such	jkn−1such	ADV
ma-125	238	22	that	that	SCONJ
ma-125	238	23	δni	δni	PROPN
ma-125	238	24	=	=	PRON
ma-125	238	25	δ(n−1)i	δ(n−1)i	PROPN
ma-125	238	26	∪	∪	VERB
ma-125	238	27	(	(	PUNCT
ma-125	238	28	σ(n−1)i	σ(n−1)i	NOUN
ma-125	238	29	∩	∩	NOUN
ma-125	238	30	(	(	PUNCT
ma-125	238	31	j1	j1	PROPN
ma-125	238	32	∪	∪	NOUN
ma-125	238	33	...	...	PUNCT
ma-125	238	34	∪	∪	ADP
ma-125	238	35	jkn−1	jkn−1	PROPN
ma-125	238	36	)	)	PUNCT
ma-125	238	37	)	)	PUNCT
ma-125	238	38	for	for	ADP
ma-125	238	39	all	all	PRON
ma-125	238	40	i	i	PRON
ma-125	238	41	∈	∈	PROPN
ma-125	239	1	[	[	X
ma-125	239	2	m	m	X
ma-125	239	3	]	]	X
ma-125	239	4	,	,	PUNCT
ma-125	239	5	there	there	PRON
ma-125	239	6	exists	exist	VERB
ma-125	239	7	a	a	DET
ma-125	239	8	partition	partition	NOUN
ma-125	239	9	{	{	PUNCT
ma-125	239	10	σni}i∈[m	σni}i∈[m	PROPN
ma-125	239	11	]	]	PUNCT
ma-125	239	12	of	of	ADP
ma-125	239	13	j	j	PROPN
ma-125	239	14	−	−	PROPN
ma-125	239	15	(	(	PUNCT
ma-125	239	16	j1	j1	PROPN
ma-125	239	17	∪	∪	ADV
ma-125	239	18	...	...	PUNCT
ma-125	239	19	∪	∪	ADP
ma-125	239	20	jkn−1	jkn−1	PROPN
ma-125	239	21	)	)	PUNCT
ma-125	239	22	such	such	ADJ
ma-125	239	23	that	that	SCONJ
ma-125	239	24	{	{	PUNCT
ma-125	239	25	wi	wi	PROPN
ma-125	239	26	j	j	PROPN
ma-125	239	27	,	,	PUNCT
ma-125	239	28	λi	λi	PROPN
ma-125	239	29	j	j	PROPN
ma-125	239	30	,	,	PUNCT
ma-125	239	31	vi	vi	PROPN
ma-125	239	32	j}j∈(σni∪δni	j}j∈(σni∪δni	PROPN
ma-125	239	33	)	)	PUNCT
ma-125	239	34	,	,	PUNCT
ma-125	239	35	i∈[m	i∈[m	PROPN
ma-125	239	36	]	]	PUNCT
ma-125	239	37	has	have	VERB
ma-125	239	38	a	a	DET
ma-125	239	39	lower	lower	ADV
ma-125	239	40	bound	bind	VERB
ma-125	239	41	less	less	ADJ
ma-125	239	42	than	than	ADP
ma-125	239	43	1	1	NUM
ma-125	239	44	n	n	NOUN
ma-125	239	45	.	.	PUNCT
ma-125	240	1	therefore	therefore	ADV
ma-125	240	2	,	,	PUNCT
ma-125	240	3	there	there	PRON
ma-125	240	4	is	be	VERB
ma-125	240	5	a	a	DET
ma-125	240	6	fn	fn	NOUN
ma-125	240	7	∈	∈	PROPN
ma-125	240	8	h	h	NOUN
ma-125	240	9	and	and	CCONJ
ma-125	240	10	kn	kn	PROPN
ma-125	240	11	∈	∈	PROPN
ma-125	240	12	n	n	PRON
ma-125	240	13	such	such	ADJ
ma-125	240	14	that	that	SCONJ
ma-125	240	15	kn	kn	PROPN
ma-125	240	16	>	>	X
ma-125	240	17	kn−1	kn−1	PROPN
ma-125	240	18	and∑	and∑	PROPN
ma-125	240	19	i∈[m	i∈[m	PROPN
ma-125	240	20	]	]	PUNCT
ma-125	240	21	∑	∑	PUNCT
ma-125	240	22	j∈kn	j∈kn	PROPN
ma-125	240	23	v2	v2	PROPN
ma-125	241	1	i	i	PRON
ma-125	241	2	j	j	PROPN
ma-125	242	1	〈	〈	PROPN
ma-125	242	2	λi	λi	ADP
ma-125	242	3	jpwi	jpwi	PROPN
ma-125	242	4	j	j	PROPN
ma-125	242	5	fn	fn	PROPN
ma-125	242	6	,	,	PUNCT
ma-125	242	7	λi	λi	ADP
ma-125	242	8	jpwi	jpwi	PROPN
ma-125	242	9	j	j	PROPN
ma-125	243	1	fn	fn	PROPN
ma-125	243	2	〉	〉	PROPN
ma-125	243	3	<	<	X
ma-125	243	4	1	1	NUM
ma-125	243	5	n	n	NUM
ma-125	243	6	〈	〈	PROPN
ma-125	243	7	k∗fn	k∗fn	PROPN
ma-125	243	8	,	,	PUNCT
ma-125	243	9	k∗f1	k∗f1	PROPN
ma-125	243	10	〉	〉	PROPN
ma-125	243	11	,	,	PUNCT
ma-125	243	12	where	where	SCONJ
ma-125	243	13	,	,	PUNCT
ma-125	243	14	kn	kn	PROPN
ma-125	243	15	=	=	NOUN
ma-125	243	16	∪i≥kn+1jj	∪i≥kn+1jj	PROPN
ma-125	243	17	.	.	PUNCT
ma-125	244	1	choose	choose	VERB
ma-125	244	2	a	a	DET
ma-125	244	3	partition	partition	NOUN
ma-125	244	4	{	{	PUNCT
ma-125	244	5	αi}i∈[m	αi}i∈[m	NOUN
ma-125	244	6	]	]	X
ma-125	244	7	of	of	ADP
ma-125	244	8	j	j	PROPN
ma-125	244	9	,	,	PUNCT
ma-125	244	10	where	where	SCONJ
ma-125	244	11	αi	αi	PRON
ma-125	244	12	=	=	SYM
ma-125	244	13	∪j∈n{δj	∪j∈n{δj	PROPN
ma-125	244	14	i	i	PRON
ma-125	244	15	}	}	PUNCT
ma-125	244	16	=	=	PUNCT
ma-125	244	17	δ(n+1)i	δ(n+1)i	NOUN
ma-125	244	18	∪	∪	ADV
ma-125	244	19	(	(	PUNCT
ma-125	244	20	αi	αi	NOUN
ma-125	244	21	∩	∩	PROPN
ma-125	244	22	j	j	PROPN
ma-125	244	23	−	−	PROPN
ma-125	244	24	(	(	PUNCT
ma-125	244	25	j1	j1	PROPN
ma-125	244	26	∪	∪	ADV
ma-125	244	27	...	...	PUNCT
ma-125	244	28	∪	∪	PROPN
ma-125	244	29	jn	jn	PROPN
ma-125	244	30	)	)	PUNCT
ma-125	244	31	)	)	PUNCT
ma-125	244	32	.	.	PUNCT
ma-125	245	1	assume	assume	VERB
ma-125	245	2	that	that	SCONJ
ma-125	245	3	{	{	PUNCT
ma-125	245	4	wi	wi	PROPN
ma-125	245	5	j	j	PROPN
ma-125	245	6	,	,	PUNCT
ma-125	245	7	λi	λi	PROPN
ma-125	245	8	j	j	PROPN
ma-125	245	9	,	,	PUNCT
ma-125	245	10	vi	vi	PROPN
ma-125	245	11	j}j∈αi	j}j∈αi	PROPN
ma-125	245	12	,	,	PUNCT
ma-125	245	13	i∈[m	i∈[m	PROPN
ma-125	245	14	]	]	PUNCT
ma-125	245	15	is	be	AUX
ma-125	245	16	a	a	DET
ma-125	245	17	k	k	NOUN
ma-125	245	18	−	−	PROPN
ma-125	245	19	g−fusion	g−fusion	NOUN
ma-125	245	20	frame	frame	NOUN
ma-125	245	21	for	for	ADP
ma-125	245	22	h	h	NOUN
ma-125	245	23	with	with	ADP
ma-125	245	24	theoptimal	theoptimal	ADJ
ma-125	245	25	lower	lower	ADV
ma-125	245	26	bound	bind	VERB
ma-125	245	27	a.	a.	NOUN
ma-125	245	28	then	then	ADV
ma-125	245	29	,	,	PUNCT
ma-125	245	30	by	by	ADP
ma-125	245	31	the	the	DET
ma-125	245	32	archimedean	archimedean	PROPN
ma-125	245	33	property	property	NOUN
ma-125	245	34	,	,	PUNCT
ma-125	245	35	there	there	PRON
ma-125	245	36	exits	exit	VERB
ma-125	245	37	a	a	DET
ma-125	245	38	r	r	NOUN
ma-125	245	39	∈	∈	NOUN
ma-125	245	40	n	n	PRON
ma-125	246	1	such	such	ADJ
ma-125	246	2	that	that	SCONJ
ma-125	246	3	r	r	NOUN
ma-125	246	4	>	>	X
ma-125	246	5	2	2	NUM
ma-125	246	6	a	a	DET
ma-125	246	7	.now	.now	NOUN
ma-125	246	8	,	,	PUNCT
ma-125	246	9	there	there	PRON
ma-125	246	10	exists	exist	VERB
ma-125	246	11	a	a	DET
ma-125	246	12	fr	fr	ADJ
ma-125	246	13	∈	∈	PROPN
ma-125	246	14	h	h	NOUN
ma-125	246	15	such	such	ADJ
ma-125	246	16	that∑	that∑	PROPN
ma-125	246	17	i∈[m	i∈[m	NOUN
ma-125	246	18	]	]	PUNCT
ma-125	246	19	∑	∑	PUNCT
ma-125	246	20	j∈αi	j∈αi	PROPN
ma-125	246	21	v2	v2	VERB
ma-125	246	22	i	i	PRON
ma-125	247	1	j	j	PROPN
ma-125	247	2	〈	〈	PROPN
ma-125	248	1	λi	λi	ADP
ma-125	248	2	jpwi	jpwi	PROPN
ma-125	248	3	j	j	PROPN
ma-125	248	4	fr	fr	INTJ
ma-125	248	5	,	,	PUNCT
ma-125	248	6	λi	λi	ADP
ma-125	248	7	jpwi	jpwi	PROPN
ma-125	249	1	j	j	PROPN
ma-125	250	1	fr	fr	PROPN
ma-125	250	2	〉	〉	PROPN
ma-125	250	3	=	=	PUNCT
ma-125	250	4	∑	∑	PUNCT
ma-125	250	5	i∈[m	i∈[m	PROPN
ma-125	250	6	]	]	PUNCT
ma-125	250	7	∑	∑	PUNCT
ma-125	250	8	j∈δ(r+1)i	j∈δ(r+1)i	PROPN
ma-125	251	1	v2	v2	INTJ
ma-125	252	1	i	i	PRON
ma-125	252	2	j	j	PROPN
ma-125	253	1	〈	〈	PROPN
ma-125	253	2	λi	λi	ADP
ma-125	253	3	jpwi	jpwi	PROPN
ma-125	253	4	j	j	PROPN
ma-125	253	5	fr	fr	INTJ
ma-125	253	6	,	,	PUNCT
ma-125	253	7	λi	λi	ADP
ma-125	253	8	jpwi	jpwi	PROPN
ma-125	253	9	j	j	PROPN
ma-125	253	10	fr	fr	PROPN
ma-125	253	11	〉	〉	PROPN
ma-125	253	12	+	+	CCONJ
ma-125	253	13	∑	∑	PART
ma-125	253	14	i∈[m	i∈[m	PROPN
ma-125	253	15	]	]	PUNCT
ma-125	253	16	∑	∑	PUNCT
ma-125	253	17	j∈αi∩j−(j1∪	j∈αi∩j−(j1∪	PROPN
ma-125	253	18	...	...	PUNCT
ma-125	253	19	∪jr	∪jr	PRON
ma-125	253	20	)	)	PUNCT
ma-125	254	1	v2	v2	VERB
ma-125	255	1	i	i	PRON
ma-125	255	2	j	j	PROPN
ma-125	256	1	〈	〈	PROPN
ma-125	256	2	λi	λi	ADP
ma-125	256	3	jpwi	jpwi	PROPN
ma-125	256	4	j	j	PROPN
ma-125	256	5	fr	fr	INTJ
ma-125	256	6	,	,	PUNCT
ma-125	256	7	λi	λi	ADP
ma-125	256	8	jpwi	jpwi	PROPN
ma-125	256	9	j	j	PROPN
ma-125	256	10	fr	fr	PROPN
ma-125	256	11	〉	〉	PROPN
ma-125	256	12	≤	≤	PROPN
ma-125	256	13	∑	∑	PUNCT
ma-125	256	14	i∈[m	i∈[m	PROPN
ma-125	256	15	]	]	PUNCT
ma-125	256	16	∑	∑	PUNCT
ma-125	256	17	j∈(σr	j∈(σr	NOUN
ma-125	256	18	i∪δr	i∪δr	PUNCT
ma-125	257	1	i	i	PRON
ma-125	257	2	)	)	PUNCT
ma-125	258	1	v2	v2	VERB
ma-125	259	1	i	i	PRON
ma-125	259	2	j	j	PROPN
ma-125	260	1	〈	〈	PROPN
ma-125	260	2	λi	λi	ADP
ma-125	260	3	jpwi	jpwi	PROPN
ma-125	260	4	j	j	PROPN
ma-125	260	5	fr	fr	INTJ
ma-125	260	6	,	,	PUNCT
ma-125	260	7	λi	λi	ADP
ma-125	260	8	jpwi	jpwi	PROPN
ma-125	260	9	j	j	PROPN
ma-125	260	10	fr	fr	PROPN
ma-125	260	11	〉	〉	PROPN
ma-125	260	12	+	+	CCONJ
ma-125	260	13	∑	∑	PART
ma-125	260	14	i∈[m	i∈[m	VERB
ma-125	260	15	]	]	PUNCT
ma-125	260	16	∑	∑	PUNCT
ma-125	260	17	j∈∪k≥r+1jk	j∈∪k≥r+1jk	PROPN
ma-125	260	18	v2	v2	PROPN
ma-125	261	1	i	i	PRON
ma-125	261	2	j	j	PROPN
ma-125	262	1	〈	〈	PROPN
ma-125	262	2	λi	λi	ADP
ma-125	262	3	jpwi	jpwi	PROPN
ma-125	262	4	j	j	PROPN
ma-125	262	5	fr	fr	INTJ
ma-125	262	6	,	,	PUNCT
ma-125	262	7	λi	λi	ADP
ma-125	262	8	jpwi	jpwi	PROPN
ma-125	262	9	j	j	PROPN
ma-125	262	10	fr	fr	PROPN
ma-125	262	11	〉	〉	PROPN
ma-125	262	12	<	<	X
ma-125	262	13	1	1	NUM
ma-125	262	14	r	r	NOUN
ma-125	262	15	〈	〈	NOUN
ma-125	262	16	k∗fr	k∗fr	NOUN
ma-125	262	17	,	,	PUNCT
ma-125	262	18	k∗fr	k∗fr	PROPN
ma-125	262	19	〉	〉	PROPN
ma-125	262	20	+	+	NOUN
ma-125	262	21	1	1	NUM
ma-125	262	22	r	r	NOUN
ma-125	262	23	〈	〈	NOUN
ma-125	262	24	k∗fr	k∗fr	NOUN
ma-125	262	25	,	,	PUNCT
ma-125	262	26	k∗fr	k∗fr	PROPN
ma-125	262	27	〉	〉	PROPN
ma-125	262	28	<	<	X
ma-125	262	29	a〈k∗fr	a〈k∗fr	PROPN
ma-125	262	30	,	,	PUNCT
ma-125	262	31	k∗fr	k∗fr	PROPN
ma-125	262	32	〉	〉	PROPN
ma-125	262	33	and	and	CCONJ
ma-125	262	34	this	this	PRON
ma-125	262	35	is	be	AUX
ma-125	262	36	a	a	DET
ma-125	262	37	contradiction	contradiction	NOUN
ma-125	262	38	with	with	ADP
ma-125	262	39	the	the	DET
ma-125	262	40	lower	low	ADJ
ma-125	262	41	bound	bind	VERB
ma-125	262	42	of	of	ADP
ma-125	262	43	a.	a.	PROPN
ma-125	262	44	�	�	PROPN
ma-125	262	45	https://doi.org/10.28924/ada/ma.3.11	https://doi.org/10.28924/ada/ma.3.11	PROPN
ma-125	262	46	eur	eur	PROPN
ma-125	262	47	.	.	PUNCT
ma-125	263	1	j.	j.	PROPN
ma-125	263	2	math	math	PROPN
ma-125	263	3	.	.	PUNCT
ma-125	264	1	anal	anal	PROPN
ma-125	264	2	.	.	PUNCT
ma-125	265	1	10.28924	10.28924	NUM
ma-125	265	2	/	/	SYM
ma-125	265	3	ada	ada	PROPN
ma-125	265	4	/	/	SYM
ma-125	265	5	ma.3.11	ma.3.11	ADJ
ma-125	265	6	9	9	NUM
ma-125	265	7	corollary	corollary	ADJ
ma-125	265	8	2.10	2.10	NUM
ma-125	265	9	.	.	PUNCT
ma-125	266	1	let	let	VERB
ma-125	266	2	{	{	PUNCT
ma-125	266	3	wi	wi	PROPN
ma-125	266	4	j	j	PROPN
ma-125	266	5	,	,	PUNCT
ma-125	266	6	λi	λi	PROPN
ma-125	266	7	j	j	PROPN
ma-125	266	8	,	,	PUNCT
ma-125	266	9	vi	vi	PROPN
ma-125	266	10	j}j∈j	j}j∈j	PROPN
ma-125	266	11	,	,	PUNCT
ma-125	266	12	i∈[m	i∈[m	PROPN
ma-125	266	13	]	]	PUNCT
ma-125	266	14	be	be	VERB
ma-125	266	15	a	a	DET
ma-125	266	16	k	k	NOUN
ma-125	266	17	−	−	NOUN
ma-125	266	18	g−fusion	g−fusion	NOUN
ma-125	266	19	woven	weave	VERB
ma-125	266	20	for	for	ADP
ma-125	266	21	h.	h.	PROPN
ma-125	266	22	then	then	ADV
ma-125	266	23	there	there	PRON
ma-125	266	24	exists	exist	VERB
ma-125	266	25	a	a	DET
ma-125	266	26	collection	collection	NOUN
ma-125	266	27	of	of	ADP
ma-125	266	28	disjoint	disjoint	PROPN
ma-125	266	29	finite	finite	PROPN
ma-125	266	30	subsets	subset	NOUN
ma-125	266	31	{	{	PUNCT
ma-125	266	32	δi}i∈[m	δi}i∈[m	NOUN
ma-125	266	33	]	]	X
ma-125	266	34	of	of	ADP
ma-125	266	35	j	j	PROPN
ma-125	266	36	and	and	CCONJ
ma-125	266	37	a	a	DET
ma-125	266	38	>	>	X
ma-125	266	39	0	0	NUM
ma-125	266	40	such	such	ADJ
ma-125	266	41	that	that	PRON
ma-125	266	42	for	for	ADP
ma-125	266	43	each	each	DET
ma-125	266	44	partition	partition	NOUN
ma-125	266	45	{	{	PUNCT
ma-125	266	46	σi}i∈[m	σi}i∈[m	NOUN
ma-125	266	47	]	]	PUNCT
ma-125	266	48	of	of	ADP
ma-125	266	49	the	the	DET
ma-125	266	50	set	set	NOUN
ma-125	266	51	j	j	PROPN
ma-125	266	52	−	−	PROPN
ma-125	266	53	∪i∈[m]δi	∪i∈[m]δi	PROPN
ma-125	266	54	,	,	PUNCT
ma-125	266	55	some	some	DET
ma-125	266	56	the	the	DET
ma-125	266	57	family	family	NOUN
ma-125	266	58	{	{	PUNCT
ma-125	266	59	wi	wi	PROPN
ma-125	266	60	j	j	PROPN
ma-125	266	61	,	,	PUNCT
ma-125	266	62	λi	λi	PROPN
ma-125	266	63	j	j	PROPN
ma-125	266	64	,	,	PUNCT
ma-125	266	65	vi	vi	PROPN
ma-125	266	66	j}j∈(σi∪δi	j}j∈(σi∪δi	PUNCT
ma-125	266	67	)	)	PUNCT
ma-125	266	68	,	,	PUNCT
ma-125	266	69	i∈[m	i∈[m	PROPN
ma-125	266	70	]	]	PUNCT
ma-125	266	71	is	be	AUX
ma-125	266	72	a	a	DET
ma-125	266	73	k	k	NOUN
ma-125	266	74	−	−	PROPN
ma-125	266	75	g−fusion	g−fusion	NOUN
ma-125	266	76	frame	frame	NOUN
ma-125	266	77	for	for	ADP
ma-125	266	78	h	h	NOUN
ma-125	266	79	with	with	ADP
ma-125	266	80	the	the	DET
ma-125	266	81	lower	low	ADJ
ma-125	266	82	frame	frame	NOUN
ma-125	266	83	bound	bind	VERB
ma-125	266	84	a.	a.	NOUN
ma-125	266	85	theorem	theorem	VERB
ma-125	266	86	2.11	2.11	NUM
ma-125	266	87	.	.	PUNCT
ma-125	267	1	let	let	VERB
ma-125	267	2	{	{	PUNCT
ma-125	267	3	wi	wi	PROPN
ma-125	267	4	j	j	PROPN
ma-125	267	5	,	,	PUNCT
ma-125	267	6	λi	λi	PROPN
ma-125	267	7	j	j	PROPN
ma-125	267	8	,	,	PUNCT
ma-125	267	9	vi	vi	PROPN
ma-125	268	1	j}j∈j	j}j∈j	X
ma-125	268	2	be	be	VERB
ma-125	268	3	a	a	DET
ma-125	268	4	k−g−fusion	k−g−fusion	NOUN
ma-125	268	5	frame	frame	NOUN
ma-125	268	6	for	for	SCONJ
ma-125	268	7	h	h	NOUN
ma-125	268	8	with	with	ADP
ma-125	268	9	bounds	bound	NOUN
ma-125	268	10	ai	ai	VERB
ma-125	268	11	and	and	CCONJ
ma-125	268	12	bi	bi	NOUN
ma-125	268	13	for	for	ADP
ma-125	268	14	each	each	DET
ma-125	268	15	i	i	PRON
ma-125	268	16	∈	∈	PROPN
ma-125	269	1	[	[	X
ma-125	270	1	m	m	X
ma-125	270	2	]	]	X
ma-125	270	3	.	.	PUNCT
ma-125	270	4	suppose	suppose	VERB
ma-125	270	5	that	that	SCONJ
ma-125	270	6	there	there	PRON
ma-125	270	7	exists	exist	VERB
ma-125	270	8	n	n	PROPN
ma-125	270	9	>	>	X
ma-125	270	10	0	0	NUM
ma-125	270	11	such	such	ADJ
ma-125	270	12	that	that	PRON
ma-125	270	13	for	for	ADP
ma-125	270	14	all	all	DET
ma-125	270	15	i	i	PRON
ma-125	270	16	,	,	PUNCT
ma-125	270	17	k	k	PROPN
ma-125	270	18	∈	∈	PROPN
ma-125	271	1	[	[	X
ma-125	271	2	m	m	X
ma-125	271	3	]	]	X
ma-125	271	4	with	with	ADP
ma-125	271	5	i	i	PRON
ma-125	271	6	6=	6=	PUNCT
ma-125	272	1	k	k	PROPN
ma-125	272	2	,	,	PUNCT
ma-125	272	3	i	i	PRON
ma-125	272	4	⊂	⊂	PROPN
ma-125	272	5	j	j	PROPN
ma-125	272	6	and	and	CCONJ
ma-125	272	7	f	f	PROPN
ma-125	272	8	∈	∈	PROPN
ma-125	273	1	h,∑	h,∑	PROPN
ma-125	273	2	j∈i	j∈i	PROPN
ma-125	273	3	〈	〈	PROPN
ma-125	273	4	(	(	PUNCT
ma-125	273	5	vi	vi	PROPN
ma-125	273	6	jλi	jλi	NOUN
ma-125	273	7	jpwi	jpwi	NOUN
ma-125	273	8	j	j	PROPN
ma-125	274	1	−	−	PROPN
ma-125	274	2	vkjλkjpwkj	vkjλkjpwkj	PROPN
ma-125	274	3	)	)	PUNCT
ma-125	274	4	f	f	PROPN
ma-125	274	5	,	,	PUNCT
ma-125	274	6	(	(	PUNCT
ma-125	274	7	vi	vi	NOUN
ma-125	274	8	jλi	jλi	NOUN
ma-125	274	9	jpwi	jpwi	NOUN
ma-125	274	10	j	j	PROPN
ma-125	275	1	−	−	PROPN
ma-125	275	2	vkjλkjpwkj	vkjλkjpwkj	PROPN
ma-125	275	3	)	)	PUNCT
ma-125	276	1	f	f	PROPN
ma-125	276	2	〉	〉	PROPN
ma-125	276	3	≤	≤	PROPN
ma-125	276	4	n	n	PRON
ma-125	276	5	min	min	NOUN
ma-125	276	6	{	{	PUNCT
ma-125	276	7	∑	∑	PROPN
ma-125	276	8	j∈i	j∈i	PROPN
ma-125	276	9	v2	v2	PROPN
ma-125	277	1	i	i	PRON
ma-125	277	2	j	j	PROPN
ma-125	278	1	〈	〈	PROPN
ma-125	278	2	λi	λi	ADP
ma-125	278	3	jpwi	jpwi	PROPN
ma-125	278	4	j	j	PROPN
ma-125	278	5	f	f	PROPN
ma-125	278	6	,	,	PUNCT
ma-125	278	7	λi	λi	ADP
ma-125	278	8	jpwi	jpwi	PROPN
ma-125	278	9	j	j	PROPN
ma-125	278	10	f	f	PROPN
ma-125	278	11	〉	〉	PROPN
ma-125	278	12	,	,	PUNCT
ma-125	278	13	∑	∑	PROPN
ma-125	278	14	j∈i	j∈i	PROPN
ma-125	278	15	v2	v2	PROPN
ma-125	278	16	kj〈λkjpwkj	kj〈λkjpwkj	PROPN
ma-125	278	17	f	f	PROPN
ma-125	278	18	,	,	PUNCT
ma-125	278	19	λkjpwkj	λkjpwkj	PROPN
ma-125	278	20	f	f	PROPN
ma-125	278	21	〉	〉	PROPN
ma-125	278	22	}	}	PUNCT
ma-125	278	23	.	.	PUNCT
ma-125	279	1	then	then	ADV
ma-125	279	2	the	the	DET
ma-125	279	3	family	family	NOUN
ma-125	279	4	{	{	PUNCT
ma-125	279	5	wi	wi	PROPN
ma-125	279	6	j	j	PROPN
ma-125	279	7	,	,	PUNCT
ma-125	279	8	λi	λi	PROPN
ma-125	279	9	j	j	PROPN
ma-125	279	10	,	,	PUNCT
ma-125	279	11	vi	vi	PROPN
ma-125	279	12	j}j∈j	j}j∈j	PROPN
ma-125	279	13	,	,	PUNCT
ma-125	279	14	i∈[m	i∈[m	PROPN
ma-125	279	15	]	]	PUNCT
ma-125	279	16	is	be	AUX
ma-125	279	17	woven	weave	VERB
ma-125	279	18	with	with	ADP
ma-125	279	19	universal	universal	ADJ
ma-125	279	20	bounds	bound	NOUN
ma-125	279	21	a	a	PRON
ma-125	279	22	(	(	PUNCT
ma-125	279	23	m	m	NOUN
ma-125	279	24	−	−	NOUN
ma-125	279	25	1)(n	1)(n	NUM
ma-125	279	26	+	+	CCONJ
ma-125	279	27	1	1	NUM
ma-125	279	28	)	)	PUNCT
ma-125	279	29	+	+	CCONJ
ma-125	279	30	1	1	NUM
ma-125	279	31	and	and	CCONJ
ma-125	279	32	b	b	NOUN
ma-125	279	33	,	,	PUNCT
ma-125	279	34	where	where	SCONJ
ma-125	279	35	a	a	DET
ma-125	279	36	=	=	X
ma-125	279	37	∑	∑	NOUN
ma-125	279	38	i∈[m	i∈[m	PROPN
ma-125	279	39	]	]	PUNCT
ma-125	279	40	ai	ai	VERB
ma-125	279	41	and	and	CCONJ
ma-125	279	42	b	b	X
ma-125	280	1	=	=	SYM
ma-125	280	2	∑	∑	PUNCT
ma-125	280	3	i∈[m]bi	i∈[m]bi	PROPN
ma-125	280	4	.	.	PUNCT
ma-125	281	1	proof	proof	NOUN
ma-125	281	2	.	.	PUNCT
ma-125	282	1	let	let	VERB
ma-125	282	2	{	{	PUNCT
ma-125	282	3	σi}i∈[m	σi}i∈[m	PROPN
ma-125	282	4	]	]	PUNCT
ma-125	282	5	be	be	VERB
ma-125	282	6	a	a	DET
ma-125	282	7	partition	partition	NOUN
ma-125	282	8	of	of	ADP
ma-125	282	9	j	j	PROPN
ma-125	282	10	and	and	CCONJ
ma-125	282	11	f	f	PROPN
ma-125	282	12	∈	∈	PROPN
ma-125	282	13	h.	h.	PROPN
ma-125	283	1	therefore,∑	therefore,∑	PROPN
ma-125	283	2	i∈[m	i∈[m	PROPN
ma-125	283	3	]	]	PUNCT
ma-125	283	4	ai〈k∗f	ai〈k∗f	ADV
ma-125	283	5	,	,	PUNCT
ma-125	283	6	k∗f	k∗f	PROPN
ma-125	283	7	〉	〉	PROPN
ma-125	283	8	∑	∑	PUNCT
ma-125	283	9	i∈[m	i∈[m	VERB
ma-125	283	10	]	]	PUNCT
ma-125	283	11	∑	∑	PUNCT
ma-125	283	12	j∈j	j∈j	NOUN
ma-125	283	13	v2	v2	PROPN
ma-125	284	1	i	i	PRON
ma-125	284	2	j	j	PROPN
ma-125	285	1	〈	〈	PROPN
ma-125	285	2	λi	λi	ADP
ma-125	285	3	jpwi	jpwi	PROPN
ma-125	285	4	j	j	PROPN
ma-125	285	5	f	f	PROPN
ma-125	285	6	,	,	PUNCT
ma-125	285	7	λi	λi	ADP
ma-125	285	8	jpwi	jpwi	PROPN
ma-125	285	9	j	j	PROPN
ma-125	286	1	f	f	PROPN
ma-125	286	2	〉	〉	PROPN
ma-125	286	3	=	=	SYM
ma-125	286	4	∑	∑	PUNCT
ma-125	286	5	i∈[m	i∈[m	PROPN
ma-125	286	6	]	]	PUNCT
ma-125	286	7	∑	∑	PART
ma-125	286	8	k∈[m	k∈[m	NOUN
ma-125	286	9	]	]	PUNCT
ma-125	286	10	∑	∑	PUNCT
ma-125	286	11	j∈σk	j∈σk	PROPN
ma-125	286	12	v2	v2	PROPN
ma-125	287	1	i	i	PRON
ma-125	287	2	j	j	PROPN
ma-125	288	1	〈	〈	PROPN
ma-125	288	2	λi	λi	ADP
ma-125	288	3	jpwi	jpwi	PROPN
ma-125	288	4	j	j	PROPN
ma-125	288	5	f	f	PROPN
ma-125	288	6	,	,	PUNCT
ma-125	288	7	λi	λi	ADP
ma-125	288	8	jpwi	jpwi	PROPN
ma-125	288	9	j	j	PROPN
ma-125	288	10	f	f	PROPN
ma-125	288	11	〉	〉	PROPN
ma-125	288	12	≤	≤	PROPN
ma-125	288	13	∑	∑	PUNCT
ma-125	288	14	i∈[m	i∈[m	PROPN
ma-125	288	15	]	]	X
ma-125	288	16	(	(	PUNCT
ma-125	288	17	∑	∑	PUNCT
ma-125	288	18	j∈σi	j∈σi	PROPN
ma-125	288	19	v2	v2	PROPN
ma-125	289	1	i	i	PRON
ma-125	289	2	j	j	PROPN
ma-125	290	1	〈	〈	PROPN
ma-125	290	2	λi	λi	ADP
ma-125	290	3	jpwi	jpwi	PROPN
ma-125	290	4	j	j	PROPN
ma-125	290	5	f	f	PROPN
ma-125	290	6	,	,	PUNCT
ma-125	290	7	λi	λi	ADP
ma-125	290	8	jpwi	jpwi	PROPN
ma-125	290	9	j	j	PROPN
ma-125	290	10	f	f	PROPN
ma-125	290	11	〉	〉	PROPN
ma-125	290	12	+	+	PROPN
ma-125	290	13	∑	∑	PROPN
ma-125	290	14	k∈[m],k	k∈[m],k	PROPN
ma-125	290	15	6	6	NUM
ma-125	290	16	=	=	PROPN
ma-125	290	17	i	i	PROPN
ma-125	290	18	∑	∑	PROPN
ma-125	290	19	j∈σk	j∈σk	PROPN
ma-125	290	20	{	{	PUNCT
ma-125	290	21	v2	v2	PROPN
ma-125	290	22	kj〈λkjpwkj	kj〈λkjpwkj	PROPN
ma-125	290	23	f	f	PROPN
ma-125	290	24	,	,	PUNCT
ma-125	290	25	λkjpwkj	λkjpwkj	PROPN
ma-125	290	26	f	f	PROPN
ma-125	290	27	〉	〉	PROPN
ma-125	291	1	+	+	CCONJ
ma-125	291	2	〈	〈	PROPN
ma-125	291	3	(	(	PUNCT
ma-125	291	4	vi	vi	ADJ
ma-125	291	5	jλi	jλi	NOUN
ma-125	291	6	jpwi	jpwi	NOUN
ma-125	291	7	j	j	PROPN
ma-125	292	1	−	−	PROPN
ma-125	292	2	vkjλkjpwkj	vkjλkjpwkj	PROPN
ma-125	292	3	)	)	PUNCT
ma-125	292	4	f	f	PROPN
ma-125	292	5	,	,	PUNCT
ma-125	292	6	(	(	PUNCT
ma-125	292	7	vi	vi	NOUN
ma-125	292	8	jλi	jλi	NOUN
ma-125	292	9	jpwi	jpwi	NOUN
ma-125	292	10	j	j	PROPN
ma-125	293	1	−	−	PROPN
ma-125	293	2	vkjλkjpwkj	vkjλkjpwkj	PROPN
ma-125	293	3	)	)	PUNCT
ma-125	293	4	f	f	PROPN
ma-125	293	5	〉	〉	PROPN
ma-125	293	6	}	}	PUNCT
ma-125	293	7	)	)	PUNCT
ma-125	293	8	≤	≤	NOUN
ma-125	293	9	∑	∑	PUNCT
ma-125	293	10	i∈[m	i∈[m	PROPN
ma-125	293	11	]	]	X
ma-125	293	12	(	(	PUNCT
ma-125	293	13	∑	∑	PUNCT
ma-125	293	14	j∈σi	j∈σi	PROPN
ma-125	293	15	v2	v2	PROPN
ma-125	294	1	i	i	PRON
ma-125	294	2	j	j	PROPN
ma-125	295	1	〈	〈	PROPN
ma-125	295	2	λi	λi	ADP
ma-125	295	3	jpwi	jpwi	PROPN
ma-125	295	4	j	j	PROPN
ma-125	295	5	f	f	PROPN
ma-125	295	6	,	,	PUNCT
ma-125	295	7	λi	λi	ADP
ma-125	295	8	jpwi	jpwi	PROPN
ma-125	295	9	j	j	PROPN
ma-125	295	10	f	f	PROPN
ma-125	295	11	〉	〉	PROPN
ma-125	296	1	+	+	CCONJ
ma-125	296	2	∑	∑	PROPN
ma-125	296	3	k∈[m],k	k∈[m],k	PROPN
ma-125	296	4	6	6	NUM
ma-125	296	5	=	=	PROPN
ma-125	296	6	i	i	PROPN
ma-125	296	7	∑	∑	PROPN
ma-125	296	8	j∈σk	j∈σk	PROPN
ma-125	296	9	(	(	PUNCT
ma-125	296	10	n	n	PROPN
ma-125	296	11	+	+	CCONJ
ma-125	296	12	1)v2	1)v2	NUM
ma-125	296	13	kj〈λkjpwkj	kj〈λkjpwkj	PROPN
ma-125	296	14	f	f	PROPN
ma-125	296	15	,	,	PUNCT
ma-125	296	16	λkjpwkj	λkjpwkj	PROPN
ma-125	296	17	f	f	PROPN
ma-125	296	18	〉	〉	PROPN
ma-125	296	19	)	)	PUNCT
ma-125	297	1	=	=	PRON
ma-125	297	2	{	{	PUNCT
ma-125	297	3	(	(	PUNCT
ma-125	297	4	m	m	NOUN
ma-125	297	5	−	−	NOUN
ma-125	297	6	1)(n	1)(n	NUM
ma-125	297	7	+	+	CCONJ
ma-125	297	8	1	1	NUM
ma-125	297	9	)	)	PUNCT
ma-125	297	10	+	+	CCONJ
ma-125	297	11	1	1	X
ma-125	297	12	}	}	PUNCT
ma-125	297	13	∑	∑	PUNCT
ma-125	297	14	i∈[m	i∈[m	NOUN
ma-125	297	15	]	]	X
ma-125	297	16	(	(	PUNCT
ma-125	297	17	∑	∑	PUNCT
ma-125	297	18	j∈σi	j∈σi	PROPN
ma-125	297	19	v2	v2	PROPN
ma-125	298	1	i	i	PRON
ma-125	298	2	j	j	PROPN
ma-125	299	1	〈	〈	PROPN
ma-125	299	2	λi	λi	ADP
ma-125	299	3	jpwi	jpwi	PROPN
ma-125	299	4	j	j	PROPN
ma-125	299	5	f	f	PROPN
ma-125	299	6	,	,	PUNCT
ma-125	299	7	λi	λi	ADP
ma-125	299	8	jpwi	jpwi	PROPN
ma-125	299	9	j	j	PROPN
ma-125	299	10	f	f	PROPN
ma-125	299	11	〉	〉	PROPN
ma-125	299	12	)	)	PUNCT
ma-125	299	13	.	.	PUNCT
ma-125	300	1	thus	thus	ADV
ma-125	300	2	,	,	PUNCT
ma-125	300	3	we	we	PRON
ma-125	300	4	get	get	VERB
ma-125	300	5	a	a	DET
ma-125	300	6	(	(	PUNCT
ma-125	300	7	m	m	NOUN
ma-125	300	8	−	−	NOUN
ma-125	300	9	1)(n	1)(n	NUM
ma-125	300	10	+	+	CCONJ
ma-125	300	11	1	1	NUM
ma-125	300	12	)	)	PUNCT
ma-125	300	13	+	+	CCONJ
ma-125	301	1	1	1	NUM
ma-125	301	2	〈	〈	PROPN
ma-125	301	3	k∗f	k∗f	X
ma-125	301	4	,	,	PUNCT
ma-125	301	5	k∗f	k∗f	PROPN
ma-125	301	6	〉	〉	PROPN
ma-125	301	7	≤	≤	PROPN
ma-125	301	8	∑	∑	PUNCT
ma-125	301	9	i∈[m	i∈[m	PROPN
ma-125	301	10	]	]	X
ma-125	301	11	(	(	PUNCT
ma-125	301	12	∑	∑	PUNCT
ma-125	301	13	j∈σi	j∈σi	PROPN
ma-125	301	14	v2	v2	PROPN
ma-125	302	1	i	i	PRON
ma-125	302	2	j	j	PROPN
ma-125	303	1	〈	〈	PROPN
ma-125	303	2	λi	λi	ADP
ma-125	303	3	jpwi	jpwi	PROPN
ma-125	303	4	j	j	PROPN
ma-125	303	5	f	f	PROPN
ma-125	303	6	,	,	PUNCT
ma-125	303	7	λi	λi	ADP
ma-125	303	8	jpwi	jpwi	PROPN
ma-125	303	9	j	j	PROPN
ma-125	303	10	f	f	PROPN
ma-125	303	11	〉	〉	PROPN
ma-125	303	12	)	)	PUNCT
ma-125	303	13	≤	≤	NUM
ma-125	303	14	b〈f	b〈f	PUNCT
ma-125	303	15	,	,	PUNCT
ma-125	303	16	f	f	PROPN
ma-125	303	17	〉	〉	PROPN
ma-125	303	18	.	.	PUNCT
ma-125	304	1	�	�	PROPN
ma-125	304	2	in	in	ADP
ma-125	304	3	next	next	ADJ
ma-125	304	4	theorem	theorem	NOUN
ma-125	304	5	we	we	PRON
ma-125	304	6	study	study	VERB
ma-125	304	7	a	a	DET
ma-125	304	8	paley	paley	ADJ
ma-125	304	9	-	-	PUNCT
ma-125	304	10	wiener	wiener	NOUN
ma-125	304	11	type	type	NOUN
ma-125	304	12	perturbation	perturbation	NOUN
ma-125	304	13	for	for	ADP
ma-125	304	14	weaving	weave	VERB
ma-125	304	15	k	k	PROPN
ma-125	304	16	−	−	PROPN
ma-125	304	17	g−fusion	g−fusion	NOUN
ma-125	304	18	frames	frame	NOUN
ma-125	304	19	.	.	PUNCT
ma-125	305	1	https://doi.org/10.28924/ada/ma.3.11	https://doi.org/10.28924/ada/ma.3.11	PROPN
ma-125	305	2	eur	eur	PROPN
ma-125	305	3	.	.	PUNCT
ma-125	306	1	j.	j.	PROPN
ma-125	306	2	math	math	PROPN
ma-125	306	3	.	.	PUNCT
ma-125	307	1	anal	anal	PROPN
ma-125	307	2	.	.	PUNCT
ma-125	308	1	10.28924	10.28924	NUM
ma-125	308	2	/	/	SYM
ma-125	308	3	ada	ada	PROPN
ma-125	308	4	/	/	SYM
ma-125	308	5	ma.3.11	ma.3.11	ADJ
ma-125	308	6	10	10	NUM
ma-125	308	7	theorem	theorem	VERB
ma-125	308	8	2.12	2.12	NUM
ma-125	308	9	.	.	PUNCT
ma-125	309	1	let	let	VERB
ma-125	309	2	{	{	PUNCT
ma-125	309	3	wj	wj	PROPN
ma-125	309	4	,	,	PUNCT
ma-125	309	5	λj	λj	INTJ
ma-125	309	6	,	,	PUNCT
ma-125	309	7	wj}j∈j	wj}j∈j	ADP
ma-125	309	8	and	and	CCONJ
ma-125	309	9	{	{	PUNCT
ma-125	309	10	vj	vj	INTJ
ma-125	309	11	,	,	PUNCT
ma-125	309	12	θj	θj	INTJ
ma-125	309	13	,	,	PUNCT
ma-125	309	14	vj}j∈j	vj}j∈j	PART
ma-125	309	15	be	be	AUX
ma-125	309	16	two	two	NUM
ma-125	309	17	k−g−fusion	k−g−fusion	NOUN
ma-125	309	18	frames	frame	NOUN
ma-125	309	19	for	for	ADP
ma-125	309	20	h	h	NOUN
ma-125	309	21	with	with	ADP
ma-125	309	22	frame	frame	NOUN
ma-125	309	23	bounds	bound	NOUN
ma-125	309	24	a1	a1	PROPN
ma-125	309	25	,	,	PUNCT
ma-125	309	26	b1	b1	NOUN
ma-125	309	27	and	and	CCONJ
ma-125	309	28	a2	a2	PROPN
ma-125	309	29	,	,	PUNCT
ma-125	309	30	b2	b2	NOUN
ma-125	309	31	,	,	PUNCT
ma-125	309	32	respectively	respectively	ADV
ma-125	309	33	.	.	PUNCT
ma-125	310	1	suppose	suppose	VERB
ma-125	310	2	that	that	SCONJ
ma-125	310	3	there	there	PRON
ma-125	310	4	exist	exist	VERB
ma-125	310	5	non	non	ADJ
ma-125	310	6	-	-	ADJ
ma-125	310	7	negative	negative	ADJ
ma-125	310	8	scalers	scaler	NOUN
ma-125	310	9	µ	µ	NOUN
ma-125	310	10	and	and	CCONJ
ma-125	310	11	0	0	NUM
ma-125	310	12	≤	≤	NUM
ma-125	311	1	λ	λ	X
ma-125	311	2	<	<	X
ma-125	311	3	1	1	NUM
ma-125	311	4	2	2	NUM
ma-125	311	5	such	such	ADJ
ma-125	311	6	that	that	SCONJ
ma-125	311	7	(	(	PUNCT
ma-125	311	8	1	1	NUM
ma-125	311	9	2	2	NUM
ma-125	311	10	−	−	NOUN
ma-125	311	11	λ)a1	λ)a1	PROPN
ma-125	311	12	>	>	X
ma-125	311	13	µ	µ	NOUN
ma-125	311	14	and	and	CCONJ
ma-125	311	15	for	for	ADP
ma-125	311	16	each	each	DET
ma-125	311	17	f	f	PROPN
ma-125	311	18	∈	∈	PROPN
ma-125	311	19	h,∑	h,∑	X
ma-125	311	20	j∈j	j∈j	NOUN
ma-125	311	21	〈	〈	PROPN
ma-125	311	22	(	(	PUNCT
ma-125	311	23	wjλjpwj	wjλjpwj	VERB
ma-125	311	24	−	−	PROPN
ma-125	311	25	vjθjpvj	vjθjpvj	NOUN
ma-125	311	26	)	)	PUNCT
ma-125	311	27	f	f	NOUN
ma-125	311	28	,	,	PUNCT
ma-125	311	29	(	(	PUNCT
ma-125	311	30	wjλjpwj	wjλjpwj	VERB
ma-125	311	31	−	−	PROPN
ma-125	311	32	vjθjpvj	vjθjpvj	NOUN
ma-125	311	33	)	)	PUNCT
ma-125	312	1	f	f	PROPN
ma-125	312	2	〉	〉	PROPN
ma-125	312	3	≤	≤	PUNCT
ma-125	312	4	λ	λ	PROPN
ma-125	312	5	∑	∑	PROPN
ma-125	312	6	j∈j	j∈j	PROPN
ma-125	312	7	〈	〈	PROPN
ma-125	312	8	wjλjpwj	wjλjpwj	VERB
ma-125	312	9	f	f	PROPN
ma-125	312	10	,	,	PUNCT
ma-125	312	11	wjλjpwj	wjλjpwj	VERB
ma-125	312	12	f	f	PROPN
ma-125	312	13	〉	〉	PROPN
ma-125	312	14	+	+	NUM
ma-125	312	15	µ〈k∗f	µ〈k∗f	X
ma-125	312	16	,	,	PUNCT
ma-125	312	17	k∗f	k∗f	PROPN
ma-125	312	18	〉	〉	PROPN
ma-125	312	19	.	.	PUNCT
ma-125	313	1	then	then	ADV
ma-125	313	2	,	,	PUNCT
ma-125	313	3	{	{	PUNCT
ma-125	313	4	wj	wj	X
ma-125	313	5	,	,	PUNCT
ma-125	313	6	λj	λj	PROPN
ma-125	313	7	,	,	PUNCT
ma-125	313	8	wj}j∈j	wj}j∈j	ADP
ma-125	313	9	and	and	CCONJ
ma-125	313	10	{	{	PUNCT
ma-125	313	11	vj	vj	INTJ
ma-125	313	12	,	,	PUNCT
ma-125	313	13	θj	θj	INTJ
ma-125	313	14	,	,	PUNCT
ma-125	313	15	vj}j∈j	vj}j∈j	X
ma-125	313	16	are	be	AUX
ma-125	313	17	k−g−fusion	k−g−fusion	NOUN
ma-125	313	18	woven	weave	VERB
ma-125	313	19	for	for	ADP
ma-125	313	20	h	h	NOUN
ma-125	313	21	with	with	ADP
ma-125	313	22	universal	universal	ADJ
ma-125	313	23	frame	frame	NOUN
ma-125	313	24	bounds	bound	NOUN
ma-125	313	25	(	(	PUNCT
ma-125	313	26	1	1	NUM
ma-125	313	27	2	2	NUM
ma-125	313	28	−	−	NOUN
ma-125	313	29	λ)a1	λ)a1	PROPN
ma-125	313	30	−	−	PROPN
ma-125	313	31	µ	µ	NOUN
ma-125	313	32	and	and	CCONJ
ma-125	313	33	b1	b1	NOUN
ma-125	313	34	+	+	CCONJ
ma-125	313	35	b2	b2	NOUN
ma-125	313	36	.	.	PUNCT
ma-125	314	1	proof	proof	NOUN
ma-125	314	2	.	.	PUNCT
ma-125	315	1	the	the	DET
ma-125	315	2	upper	upper	ADJ
ma-125	315	3	frame	frame	NOUN
ma-125	315	4	bound	bind	VERB
ma-125	315	5	is	be	AUX
ma-125	315	6	clear	clear	ADJ
ma-125	315	7	.	.	PUNCT
ma-125	316	1	for	for	ADP
ma-125	316	2	the	the	DET
ma-125	316	3	lower	low	ADJ
ma-125	316	4	frame	frame	NOUN
ma-125	316	5	bound	bind	VERB
ma-125	316	6	,	,	PUNCT
ma-125	316	7	assume	assume	VERB
ma-125	316	8	that	that	SCONJ
ma-125	316	9	σ	σ	PROPN
ma-125	316	10	⊂	⊂	PROPN
ma-125	316	11	j	j	PROPN
ma-125	316	12	and	and	CCONJ
ma-125	316	13	we	we	PRON
ma-125	316	14	get	get	VERB
ma-125	316	15	,	,	PUNCT
ma-125	316	16	by	by	ADP
ma-125	316	17	the	the	DET
ma-125	316	18	arithmetic	arithmetic	ADJ
ma-125	316	19	-	-	PUNCT
ma-125	316	20	quadratic	quadratic	ADJ
ma-125	316	21	mean	mean	NOUN
ma-125	316	22	,	,	PUNCT
ma-125	316	23	for	for	ADP
ma-125	316	24	any	any	DET
ma-125	316	25	f	f	PROPN
ma-125	316	26	∈	∈	PROPN
ma-125	316	27	h∑	h∑	ADP
ma-125	317	1	j∈σ	j∈σ	PROPN
ma-125	317	2	w2	w2	PROPN
ma-125	317	3	j	j	PROPN
ma-125	317	4	〈	〈	PROPN
ma-125	317	5	λjpwj	λjpwj	VERB
ma-125	317	6	f	f	PROPN
ma-125	317	7	,	,	PUNCT
ma-125	317	8	λjpwj	λjpwj	VERB
ma-125	317	9	f	f	PROPN
ma-125	317	10	〉	〉	PROPN
ma-125	317	11	+	+	PROPN
ma-125	317	12	∑	∑	PUNCT
ma-125	317	13	j∈σc	j∈σc	PROPN
ma-125	317	14	v2	v2	PROPN
ma-125	317	15	j	j	PROPN
ma-125	317	16	〈	〈	PROPN
ma-125	317	17	θjpvj	θjpvj	VERB
ma-125	317	18	f	f	PROPN
ma-125	317	19	,	,	PUNCT
ma-125	317	20	θjpvj	θjpvj	VERB
ma-125	317	21	f	f	PROPN
ma-125	317	22	〉	〉	PROPN
ma-125	318	1	=	=	SYM
ma-125	318	2	∑	∑	PROPN
ma-125	318	3	j∈σ	j∈σ	PROPN
ma-125	318	4	w2	w2	PROPN
ma-125	318	5	j	j	PROPN
ma-125	318	6	〈	〈	PROPN
ma-125	318	7	λjpwj	λjpwj	VERB
ma-125	318	8	f	f	PROPN
ma-125	318	9	,	,	PUNCT
ma-125	318	10	λjpwj	λjpwj	VERB
ma-125	318	11	f	f	PROPN
ma-125	318	12	〉	〉	PROPN
ma-125	319	1	+	+	PROPN
ma-125	319	2	∑	∑	PROPN
ma-125	319	3	j∈σc	j∈σc	NOUN
ma-125	319	4	〈	〈	PROPN
ma-125	319	5	wjλjpwj	wjλjpwj	VERB
ma-125	319	6	f	f	PROPN
ma-125	319	7	−	−	PROPN
ma-125	319	8	(	(	PUNCT
ma-125	319	9	wjλjpwj	wjλjpwj	VERB
ma-125	319	10	−	−	PROPN
ma-125	319	11	vjθjpvj	vjθjpvj	NOUN
ma-125	319	12	)	)	PUNCT
ma-125	319	13	f	f	PROPN
ma-125	319	14	,	,	PUNCT
ma-125	319	15	wjλjpwj	wjλjpwj	VERB
ma-125	319	16	f	f	PROPN
ma-125	319	17	−	−	PROPN
ma-125	319	18	(	(	PUNCT
ma-125	319	19	wjλjpwj	wjλjpwj	VERB
ma-125	319	20	−	−	PROPN
ma-125	319	21	vjθjpvj	vjθjpvj	NOUN
ma-125	319	22	)	)	PUNCT
ma-125	320	1	f	f	PROPN
ma-125	320	2	〉	〉	PROPN
ma-125	320	3	≥	≥	PROPN
ma-125	321	1	∑	∑	PROPN
ma-125	321	2	j∈σ	j∈σ	PROPN
ma-125	321	3	w2	w2	PROPN
ma-125	321	4	j	j	PROPN
ma-125	321	5	〈	〈	PROPN
ma-125	321	6	λjpwj	λjpwj	VERB
ma-125	321	7	f	f	PROPN
ma-125	321	8	,	,	PUNCT
ma-125	321	9	λjpwj	λjpwj	VERB
ma-125	321	10	f	f	PROPN
ma-125	321	11	〉	〉	PROPN
ma-125	321	12	+	+	PROPN
ma-125	321	13	1	1	NUM
ma-125	321	14	2	2	NUM
ma-125	321	15	∑	∑	NOUN
ma-125	321	16	j∈σc	j∈σc	NOUN
ma-125	321	17	w2	w2	PROPN
ma-125	321	18	j	j	PROPN
ma-125	321	19	〈	〈	PROPN
ma-125	321	20	λjpwj	λjpwj	VERB
ma-125	321	21	f	f	PROPN
ma-125	321	22	,	,	PUNCT
ma-125	321	23	λjpwj	λjpwj	VERB
ma-125	321	24	f	f	PROPN
ma-125	321	25	〉	〉	PROPN
ma-125	321	26	−	−	PROPN
ma-125	321	27	∑	∑	PUNCT
ma-125	321	28	j∈σc	j∈σc	PROPN
ma-125	321	29	〈	〈	PROPN
ma-125	321	30	(	(	PUNCT
ma-125	321	31	wjλjpwj	wjλjpwj	VERB
ma-125	321	32	−	−	PROPN
ma-125	321	33	vjθjpvj	vjθjpvj	NOUN
ma-125	321	34	)	)	PUNCT
ma-125	321	35	f	f	NOUN
ma-125	321	36	,	,	PUNCT
ma-125	321	37	(	(	PUNCT
ma-125	321	38	wjλjpwj	wjλjpwj	VERB
ma-125	321	39	−	−	PROPN
ma-125	321	40	vjθjpvj	vjθjpvj	NOUN
ma-125	321	41	)	)	PUNCT
ma-125	322	1	f	f	PROPN
ma-125	322	2	〉	〉	NUM
ma-125	322	3	=	=	SYM
ma-125	322	4	1	1	NUM
ma-125	322	5	2	2	NUM
ma-125	322	6	∑	∑	NOUN
ma-125	322	7	j∈j	j∈j	NOUN
ma-125	322	8	w2	w2	NOUN
ma-125	322	9	j	j	PROPN
ma-125	322	10	〈	〈	PROPN
ma-125	322	11	λjpwj	λjpwj	VERB
ma-125	322	12	f	f	PROPN
ma-125	322	13	,	,	PUNCT
ma-125	322	14	λjpwj	λjpwj	VERB
ma-125	322	15	f	f	PROPN
ma-125	322	16	〉	〉	PROPN
ma-125	322	17	+	+	PROPN
ma-125	322	18	1	1	NUM
ma-125	322	19	2	2	NUM
ma-125	322	20	∑	∑	PROPN
ma-125	322	21	j∈σ	j∈σ	PROPN
ma-125	322	22	w2	w2	PROPN
ma-125	322	23	j	j	PROPN
ma-125	322	24	〈	〈	PROPN
ma-125	322	25	λjpwj	λjpwj	VERB
ma-125	322	26	f	f	PROPN
ma-125	322	27	,	,	PUNCT
ma-125	322	28	λjpwj	λjpwj	VERB
ma-125	322	29	f	f	PROPN
ma-125	322	30	〉	〉	PROPN
ma-125	322	31	−	−	PROPN
ma-125	322	32	∑	∑	PUNCT
ma-125	322	33	j∈σc	j∈σc	PROPN
ma-125	322	34	〈	〈	PROPN
ma-125	322	35	(	(	PUNCT
ma-125	322	36	wjλjpwj	wjλjpwj	VERB
ma-125	322	37	−	−	PROPN
ma-125	322	38	vjθjpvj	vjθjpvj	NOUN
ma-125	322	39	)	)	PUNCT
ma-125	322	40	f	f	NOUN
ma-125	322	41	,	,	PUNCT
ma-125	322	42	(	(	PUNCT
ma-125	322	43	wjλjpwj	wjλjpwj	VERB
ma-125	322	44	−	−	PROPN
ma-125	322	45	vjθjpvj	vjθjpvj	NOUN
ma-125	322	46	)	)	PUNCT
ma-125	323	1	f	f	PROPN
ma-125	323	2	〉	〉	PROPN
ma-125	323	3	≥	≥	NUM
ma-125	323	4	1	1	NUM
ma-125	323	5	2	2	NUM
ma-125	323	6	∑	∑	NOUN
ma-125	323	7	j∈j	j∈j	NOUN
ma-125	323	8	w2	w2	NOUN
ma-125	323	9	j	j	PROPN
ma-125	323	10	〈	〈	PROPN
ma-125	323	11	λjpwj	λjpwj	VERB
ma-125	323	12	f	f	PROPN
ma-125	323	13	,	,	PUNCT
ma-125	323	14	λjpwj	λjpwj	VERB
ma-125	323	15	f	f	PROPN
ma-125	323	16	〉	〉	PROPN
ma-125	323	17	−	−	PROPN
ma-125	323	18	∑	∑	PUNCT
ma-125	323	19	j∈σc	j∈σc	PROPN
ma-125	323	20	〈	〈	PROPN
ma-125	323	21	(	(	PUNCT
ma-125	323	22	wjλjpwj	wjλjpwj	VERB
ma-125	323	23	−	−	PROPN
ma-125	323	24	vjθjpvj	vjθjpvj	NOUN
ma-125	323	25	)	)	PUNCT
ma-125	323	26	f	f	NOUN
ma-125	323	27	,	,	PUNCT
ma-125	323	28	(	(	PUNCT
ma-125	323	29	wjλjpwj	wjλjpwj	VERB
ma-125	323	30	−	−	PROPN
ma-125	323	31	vjθjpvj	vjθjpvj	NOUN
ma-125	323	32	)	)	PUNCT
ma-125	324	1	f	f	PROPN
ma-125	324	2	〉	〉	PROPN
ma-125	324	3	≥	≥	NUM
ma-125	324	4	1	1	NUM
ma-125	324	5	2	2	NUM
ma-125	324	6	∑	∑	NOUN
ma-125	324	7	j∈j	j∈j	NOUN
ma-125	324	8	w2	w2	NOUN
ma-125	324	9	j	j	PROPN
ma-125	324	10	〈	〈	PROPN
ma-125	324	11	λjpwj	λjpwj	VERB
ma-125	324	12	f	f	PROPN
ma-125	324	13	,	,	PUNCT
ma-125	324	14	λjpwj	λjpwj	VERB
ma-125	324	15	f	f	PROPN
ma-125	324	16	〉	〉	NOUN
ma-125	325	1	−	−	PROPN
ma-125	325	2	λ	λ	PROPN
ma-125	325	3	∑	∑	PROPN
ma-125	325	4	j∈j	j∈j	PROPN
ma-125	325	5	w2	w2	PROPN
ma-125	325	6	j	j	PROPN
ma-125	325	7	〈	〈	PROPN
ma-125	325	8	λjpwj	λjpwj	VERB
ma-125	325	9	f	f	PROPN
ma-125	325	10	,	,	PUNCT
ma-125	325	11	λjpwj	λjpwj	VERB
ma-125	325	12	f	f	PROPN
ma-125	325	13	〉	〉	PROPN
ma-125	325	14	−	−	PROPN
ma-125	325	15	µ〈k∗f	µ〈k∗f	NOUN
ma-125	325	16	,	,	PUNCT
ma-125	325	17	k∗f	k∗f	PROPN
ma-125	325	18	〉	〉	PROPN
ma-125	325	19	≥	≥	NUM
ma-125	325	20	(	(	PUNCT
ma-125	325	21	(	(	PUNCT
ma-125	325	22	1	1	NUM
ma-125	325	23	2−	2−	NUM
ma-125	325	24	λ)a1	λ)a1	PROPN
ma-125	325	25	−	−	PROPN
ma-125	325	26	µ	µ	NOUN
ma-125	325	27	)	)	PUNCT
ma-125	325	28	〈	〈	PROPN
ma-125	325	29	k∗f	k∗f	X
ma-125	325	30	,	,	PUNCT
ma-125	325	31	k∗f	k∗f	PROPN
ma-125	325	32	〉	〉	PROPN
ma-125	325	33	.	.	PUNCT
ma-125	326	1	this	this	PRON
ma-125	326	2	completes	complete	VERB
ma-125	326	3	the	the	DET
ma-125	326	4	proof	proof	NOUN
ma-125	326	5	.	.	PUNCT
ma-125	327	1	�	�	PROPN
ma-125	327	2	declarations	declaration	NOUN
ma-125	327	3	availablity	availablity	NOUN
ma-125	327	4	of	of	ADP
ma-125	327	5	data	datum	NOUN
ma-125	327	6	and	and	CCONJ
ma-125	327	7	materialsnot	materialsnot	ADV
ma-125	327	8	applicable	applicable	ADJ
ma-125	327	9	.	.	PUNCT
ma-125	328	1	https://doi.org/10.28924/ada/ma.3.11	https://doi.org/10.28924/ada/ma.3.11	PROPN
ma-125	328	2	eur	eur	PROPN
ma-125	328	3	.	.	PUNCT
ma-125	329	1	j.	j.	PROPN
ma-125	329	2	math	math	PROPN
ma-125	329	3	.	.	PUNCT
ma-125	330	1	anal	anal	PROPN
ma-125	330	2	.	.	PUNCT
ma-125	331	1	10.28924	10.28924	NUM
ma-125	331	2	/	/	SYM
ma-125	331	3	ada	ada	PROPN
ma-125	331	4	/	/	SYM
ma-125	331	5	ma.3.11	ma.3.11	ADJ
ma-125	331	6	11	11	NUM
ma-125	331	7	human	human	ADJ
ma-125	331	8	and	and	CCONJ
ma-125	331	9	animal	animal	NOUN
ma-125	331	10	rightswe	rightswe	NOUN
ma-125	331	11	would	would	AUX
ma-125	331	12	like	like	VERB
ma-125	331	13	to	to	PART
ma-125	331	14	mention	mention	VERB
ma-125	331	15	that	that	SCONJ
ma-125	331	16	this	this	DET
ma-125	331	17	article	article	NOUN
ma-125	331	18	does	do	AUX
ma-125	331	19	not	not	PART
ma-125	331	20	contain	contain	VERB
ma-125	331	21	any	any	DET
ma-125	331	22	studies	study	NOUN
ma-125	331	23	with	with	ADP
ma-125	331	24	animals	animal	NOUN
ma-125	331	25	and	and	CCONJ
ma-125	331	26	does	do	AUX
ma-125	331	27	notinvolve	notinvolve	VERB
ma-125	331	28	any	any	DET
ma-125	331	29	studies	study	NOUN
ma-125	331	30	over	over	ADP
ma-125	331	31	human	human	ADJ
ma-125	331	32	being	being	NOUN
ma-125	331	33	.	.	PUNCT
ma-125	332	1	competing	compete	VERB
ma-125	332	2	intereston	intereston	NOUN
ma-125	332	3	behalf	behalf	NOUN
ma-125	332	4	of	of	ADP
ma-125	332	5	all	all	DET
ma-125	332	6	authors	author	NOUN
ma-125	332	7	,	,	PUNCT
ma-125	332	8	the	the	DET
ma-125	332	9	corresponding	corresponding	ADJ
ma-125	332	10	author	author	NOUN
ma-125	332	11	states	state	VERB
ma-125	332	12	that	that	SCONJ
ma-125	332	13	there	there	PRON
ma-125	332	14	is	be	VERB
ma-125	332	15	no	no	DET
ma-125	332	16	conflict	conflict	NOUN
ma-125	332	17	of	of	ADP
ma-125	332	18	interest	interest	NOUN
ma-125	332	19	.	.	PUNCT
ma-125	333	1	fundingsauthors	fundingsauthor	NOUN
ma-125	333	2	declare	declare	VERB
ma-125	333	3	that	that	SCONJ
ma-125	333	4	there	there	PRON
ma-125	333	5	is	be	VERB
ma-125	333	6	no	no	DET
ma-125	333	7	funding	funding	NOUN
ma-125	333	8	available	available	ADJ
ma-125	333	9	for	for	ADP
ma-125	333	10	this	this	DET
ma-125	333	11	article	article	NOUN
ma-125	333	12	.	.	PUNCT
ma-125	334	1	authors	author	NOUN
ma-125	334	2	’	'	PUNCT
ma-125	334	3	contributionsthe	contributionsthe	DET
ma-125	334	4	authors	author	NOUN
ma-125	334	5	equally	equally	ADV
ma-125	334	6	conceived	conceive	VERB
ma-125	334	7	of	of	ADP
ma-125	334	8	the	the	DET
ma-125	334	9	study	study	NOUN
ma-125	334	10	,	,	PUNCT
ma-125	334	11	participated	participate	VERB
ma-125	334	12	in	in	ADP
ma-125	334	13	its	its	PRON
ma-125	334	14	design	design	NOUN
ma-125	334	15	and	and	CCONJ
ma-125	334	16	coordination	coordination	NOUN
ma-125	334	17	,	,	PUNCT
ma-125	334	18	drafted	draft	VERB
ma-125	334	19	themanuscript	themanuscript	NOUN
ma-125	334	20	,	,	PUNCT
ma-125	334	21	participated	participate	VERB
ma-125	334	22	in	in	ADP
ma-125	334	23	the	the	DET
ma-125	334	24	sequence	sequence	NOUN
ma-125	334	25	alignment	alignment	NOUN
ma-125	334	26	,	,	PUNCT
ma-125	334	27	and	and	CCONJ
ma-125	334	28	read	read	VERB
ma-125	334	29	and	and	CCONJ
ma-125	334	30	approved	approve	VERB
ma-125	334	31	the	the	DET
ma-125	334	32	final	final	ADJ
ma-125	334	33	manuscript	manuscript	NOUN
ma-125	334	34	.	.	PUNCT
ma-125	335	1	references	reference	NOUN
ma-125	335	2	[	[	X
ma-125	335	3	1	1	NUM
ma-125	335	4	]	]	PUNCT
ma-125	335	5	a.	a.	NOUN
ma-125	335	6	alijani	alijani	PROPN
ma-125	335	7	,	,	PUNCT
ma-125	335	8	m.	m.	NOUN
ma-125	335	9	dehghan	dehghan	PROPN
ma-125	335	10	,	,	PUNCT
ma-125	335	11	∗-frames	∗-frames	PROPN
ma-125	335	12	in	in	ADP
ma-125	335	13	hilbert	hilbert	PROPN
ma-125	335	14	c∗modules	c∗module	NOUN
ma-125	335	15	,	,	PUNCT
ma-125	335	16	u.p.b	u.p.b	NOUN
ma-125	335	17	.	.	PUNCT
ma-125	336	1	sci	sci	PROPN
ma-125	336	2	.	.	PUNCT
ma-125	336	3	bull	bull	PROPN
ma-125	336	4	.	.	PUNCT
ma-125	336	5	,	,	PUNCT
ma-125	336	6	ser	ser	PROPN
ma-125	336	7	.	.	PUNCT
ma-125	337	1	a	a	PRON
ma-125	337	2	,	,	PUNCT
ma-125	337	3	73	73	NUM
ma-125	337	4	(	(	PUNCT
ma-125	337	5	2011	2011	NUM
ma-125	337	6	)	)	PUNCT
ma-125	337	7	,	,	PUNCT
ma-125	337	8	89	89	NUM
ma-125	337	9	-	-	SYM
ma-125	337	10	106.[2	106.[2	NUM
ma-125	337	11	]	]	X
ma-125	337	12	lj	lj	PROPN
ma-125	337	13	.	.	PUNCT
ma-125	338	1	arambašić	arambašić	NUM
ma-125	338	2	,	,	PUNCT
ma-125	338	3	on	on	ADP
ma-125	338	4	frames	frame	NOUN
ma-125	338	5	for	for	ADP
ma-125	338	6	countably	countably	ADV
ma-125	338	7	generated	generate	VERB
ma-125	338	8	hilbert	hilbert	PROPN
ma-125	338	9	c∗-modules	c∗-modules	PROPN
ma-125	338	10	,	,	PUNCT
ma-125	338	11	proc	proc	NOUN
ma-125	338	12	.	.	PUNCT
ma-125	339	1	amer	amer	PROPN
ma-125	339	2	.	.	PUNCT
ma-125	339	3	math	math	PROPN
ma-125	339	4	.	.	PUNCT
ma-125	340	1	soc	soc	PROPN
ma-125	340	2	.	.	PUNCT
ma-125	341	1	135	135	NUM
ma-125	341	2	(	(	PUNCT
ma-125	341	3	2007	2007	NUM
ma-125	341	4	)	)	PUNCT
ma-125	341	5	469	469	NUM
ma-125	341	6	-	-	SYM
ma-125	341	7	478	478	NUM
ma-125	341	8	.	.	PUNCT
ma-125	342	1	https://doi.org/10.1090/s0002-9939-06-08498-x.[3	https://doi.org/10.1090/s0002-9939-06-08498-x.[3	PROPN
ma-125	342	2	]	]	PUNCT
ma-125	342	3	r.j	r.j	PROPN
ma-125	342	4	.	.	PROPN
ma-125	342	5	duffin	duffin	PROPN
ma-125	342	6	,	,	PUNCT
ma-125	342	7	a.c	a.c	PROPN
ma-125	342	8	.	.	PROPN
ma-125	342	9	schaeffer	schaeffer	PROPN
ma-125	342	10	,	,	PUNCT
ma-125	342	11	a	a	DET
ma-125	342	12	class	class	NOUN
ma-125	342	13	of	of	ADP
ma-125	342	14	nonharmonic	nonharmonic	ADJ
ma-125	342	15	fourier	fourier	NOUN
ma-125	342	16	series	series	NOUN
ma-125	342	17	,	,	PUNCT
ma-125	342	18	trans	trans	PROPN
ma-125	342	19	.	.	PROPN
ma-125	343	1	amer	amer	PROPN
ma-125	343	2	.	.	PUNCT
ma-125	343	3	math	math	PROPN
ma-125	343	4	.	.	PUNCT
ma-125	344	1	soc	soc	PROPN
ma-125	344	2	.	.	PUNCT
ma-125	345	1	72	72	NUM
ma-125	345	2	(	(	PUNCT
ma-125	345	3	1952	1952	NUM
ma-125	345	4	)	)	PUNCT
ma-125	345	5	,	,	PUNCT
ma-125	346	1	341–366	341–366	NUM
ma-125	346	2	.	.	PUNCT
ma-125	347	1	https://doi.org/10.1090/s0002-9947-1952-0047179-6.[4	https://doi.org/10.1090/s0002-9947-1952-0047179-6.[4	PROPN
ma-125	347	2	]	]	X
ma-125	347	3	m.	m.	PROPN
ma-125	347	4	frank	frank	PROPN
ma-125	347	5	,	,	PUNCT
ma-125	347	6	d.r	d.r	PROPN
ma-125	347	7	.	.	PROPN
ma-125	347	8	larson	larson	PROPN
ma-125	347	9	,	,	PUNCT
ma-125	347	10	a	a	DET
ma-125	347	11	-	-	PUNCT
ma-125	347	12	module	module	NOUN
ma-125	347	13	frame	frame	NOUN
ma-125	347	14	concept	concept	NOUN
ma-125	347	15	for	for	ADP
ma-125	347	16	hilbert	hilbert	PROPN
ma-125	347	17	c∗-modules	c∗-modules	PROPN
ma-125	347	18	,	,	PUNCT
ma-125	347	19	funct	funct	ADJ
ma-125	347	20	.	.	PUNCT
ma-125	347	21	harm	harm	NOUN
ma-125	347	22	.	.	PUNCT
ma-125	348	1	anal	anal	PROPN
ma-125	348	2	.	.	PUNCT
ma-125	348	3	wavel	wavel	PROPN
ma-125	348	4	.	.	PUNCT
ma-125	349	1	contempt	contempt	NOUN
ma-125	349	2	.	.	PUNCT
ma-125	350	1	math.247	math.247	NOUN
ma-125	350	2	(	(	PUNCT
ma-125	350	3	2000	2000	NUM
ma-125	350	4	)	)	PUNCT
ma-125	350	5	207	207	NUM
ma-125	350	6	-	-	SYM
ma-125	350	7	233.[5	233.[5	NUM
ma-125	350	8	]	]	X
ma-125	350	9	d.	d.	PROPN
ma-125	350	10	gabor	gabor	PROPN
ma-125	350	11	,	,	PUNCT
ma-125	350	12	theory	theory	NOUN
ma-125	350	13	of	of	ADP
ma-125	350	14	communication	communication	NOUN
ma-125	350	15	.	.	PUNCT
ma-125	351	1	part	part	NOUN
ma-125	351	2	1	1	NUM
ma-125	351	3	:	:	PUNCT
ma-125	351	4	the	the	DET
ma-125	351	5	analysis	analysis	NOUN
ma-125	351	6	of	of	ADP
ma-125	351	7	information	information	NOUN
ma-125	351	8	,	,	PUNCT
ma-125	351	9	j.	j.	PROPN
ma-125	351	10	inst	inst	PROPN
ma-125	351	11	.	.	PUNCT
ma-125	351	12	electric	electric	PROPN
ma-125	351	13	.	.	PUNCT
ma-125	352	1	eng	eng	PROPN
ma-125	352	2	.	.	PROPN
ma-125	353	1	93	93	NUM
ma-125	353	2	(	(	PUNCT
ma-125	353	3	1946	1946	NUM
ma-125	353	4	)	)	PUNCT
ma-125	354	1	429–441	429–441	NUM
ma-125	354	2	.	.	PUNCT
ma-125	354	3	https://doi.org/10.1049/ji-3-2.1946.0074.[6	https://doi.org/10.1049/ji-3-2.1946.0074.[6	PRON
ma-125	354	4	]	]	X
ma-125	354	5	e.c	e.c	PROPN
ma-125	354	6	.	.	PROPN
ma-125	354	7	lance	lance	PROPN
ma-125	354	8	,	,	PUNCT
ma-125	354	9	hilbert	hilbert	PROPN
ma-125	354	10	c∗−modules	c∗−modules	PROPN
ma-125	354	11	:	:	PUNCT
ma-125	354	12	a	a	DET
ma-125	354	13	toolkit	toolkit	NOUN
ma-125	354	14	for	for	ADP
ma-125	354	15	operator	operator	NOUN
ma-125	354	16	algebraist	algebraist	NOUN
ma-125	354	17	,	,	PUNCT
ma-125	354	18	london	london	PROPN
ma-125	354	19	math	math	PROPN
ma-125	354	20	.	.	PUNCT
ma-125	355	1	soc	soc	PROPN
ma-125	355	2	.	.	PUNCT
ma-125	356	1	lecture	lecture	NOUN
ma-125	356	2	note	note	NOUN
ma-125	356	3	ser	ser	PROPN
ma-125	356	4	.	.	PUNCT
ma-125	356	5	cambridgeuniv	cambridgeuniv	PROPN
ma-125	356	6	.	.	PUNCT
ma-125	357	1	press	press	NOUN
ma-125	357	2	,	,	PUNCT
ma-125	357	3	cambridge	cambridge	PROPN
ma-125	357	4	,	,	PUNCT
ma-125	357	5	1995.[7	1995.[7	NUM
ma-125	357	6	]	]	PUNCT
ma-125	357	7	s.	s.	PROPN
ma-125	357	8	kabbaj	kabbaj	PROPN
ma-125	357	9	,	,	PUNCT
ma-125	357	10	m.	m.	NOUN
ma-125	357	11	rossafi	rossafi	NOUN
ma-125	357	12	,	,	PUNCT
ma-125	357	13	∗-operator	∗-operator	NOUN
ma-125	357	14	frame	frame	NOUN
ma-125	357	15	for	for	ADP
ma-125	357	16	end∗a(h	end∗a(h	NOUN
ma-125	357	17	)	)	PUNCT
ma-125	357	18	,	,	PUNCT
ma-125	357	19	wavel	wavel	NOUN
ma-125	357	20	.	.	PUNCT
ma-125	358	1	linear	linear	PROPN
ma-125	358	2	algebra	algebra	PROPN
ma-125	358	3	,	,	PUNCT
ma-125	358	4	5	5	NUM
ma-125	358	5	(	(	PUNCT
ma-125	358	6	2018	2018	NUM
ma-125	358	7	)	)	PUNCT
ma-125	358	8	1	1	NUM
ma-125	358	9	-	-	SYM
ma-125	358	10	13.[8	13.[8	NUM
ma-125	358	11	]	]	PUNCT
ma-125	358	12	i.	i.	NOUN
ma-125	358	13	kaplansky	kaplansky	PROPN
ma-125	358	14	,	,	PUNCT
ma-125	358	15	modules	module	NOUN
ma-125	358	16	over	over	ADP
ma-125	358	17	operator	operator	NOUN
ma-125	358	18	algebras	algebra	NOUN
ma-125	358	19	,	,	PUNCT
ma-125	358	20	amer	amer	PROPN
ma-125	358	21	.	.	PUNCT
ma-125	359	1	j.	j.	PROPN
ma-125	359	2	math	math	PROPN
ma-125	359	3	.	.	PUNCT
ma-125	360	1	75	75	NUM
ma-125	360	2	(	(	PUNCT
ma-125	360	3	1953	1953	NUM
ma-125	360	4	)	)	PUNCT
ma-125	360	5	839	839	NUM
ma-125	360	6	-	-	NUM
ma-125	360	7	858	858	NUM
ma-125	360	8	.	.	PUNCT
ma-125	361	1	https://doi.org/10.2307/	https://doi.org/10.2307/	PROPN
ma-125	361	2	2372552.[9	2372552.[9	NUM
ma-125	361	3	]	]	X
ma-125	361	4	a.	a.	NOUN
ma-125	361	5	khorsavi	khorsavi	PROPN
ma-125	361	6	,	,	PUNCT
ma-125	361	7	b.	b.	PROPN
ma-125	361	8	khorsavi	khorsavi	PROPN
ma-125	361	9	,	,	PUNCT
ma-125	361	10	fusion	fusion	NOUN
ma-125	361	11	frames	frame	NOUN
ma-125	361	12	and	and	CCONJ
ma-125	361	13	g	g	NOUN
ma-125	361	14	-	-	PUNCT
ma-125	361	15	frames	frame	NOUN
ma-125	361	16	in	in	ADP
ma-125	361	17	hilbert	hilbert	PROPN
ma-125	361	18	c∗-modules	c∗-modules	PROPN
ma-125	361	19	,	,	PUNCT
ma-125	361	20	int	int	NOUN
ma-125	361	21	.	.	PUNCT
ma-125	362	1	j.	j.	PROPN
ma-125	362	2	wavel	wavel	PROPN
ma-125	362	3	.	.	PUNCT
ma-125	363	1	multiresolut	multiresolut	PROPN
ma-125	363	2	.	.	PUNCT
ma-125	364	1	inf	inf	PROPN
ma-125	364	2	.	.	PUNCT
ma-125	364	3	proc	proc	PROPN
ma-125	364	4	.	.	PUNCT
ma-125	365	1	6(2008	6(2008	NOUN
ma-125	365	2	)	)	PUNCT
ma-125	365	3	433	433	NUM
ma-125	365	4	-	-	SYM
ma-125	365	5	446	446	NUM
ma-125	365	6	.	.	PUNCT
ma-125	366	1	https://doi.org/10.1142/s0219691308002458.[10	https://doi.org/10.1142/s0219691308002458.[10	X
ma-125	366	2	]	]	X
ma-125	366	3	f.d	f.d	PROPN
ma-125	366	4	.	.	PROPN
ma-125	366	5	nhari	nhari	PROPN
ma-125	366	6	,	,	PUNCT
ma-125	366	7	r.	r.	PROPN
ma-125	366	8	echarghaoui	echarghaoui	PROPN
ma-125	366	9	,	,	PUNCT
ma-125	366	10	m.	m.	NOUN
ma-125	366	11	rossafi	rossafi	PROPN
ma-125	366	12	,	,	PUNCT
ma-125	366	13	k	k	PROPN
ma-125	366	14	−	−	PROPN
ma-125	366	15	g−fusion	g−fusion	NOUN
ma-125	366	16	frames	frame	NOUN
ma-125	366	17	in	in	ADP
ma-125	366	18	hilbert	hilbert	PROPN
ma-125	366	19	c∗−modules	c∗−modules	PROPN
ma-125	366	20	,	,	PUNCT
ma-125	366	21	int	int	NOUN
ma-125	366	22	.	.	PUNCT
ma-125	367	1	j.	j.	PROPN
ma-125	367	2	anal	anal	PROPN
ma-125	367	3	.	.	PUNCT
ma-125	368	1	appl	appl	PROPN
ma-125	368	2	.	.	PROPN
ma-125	369	1	19	19	NUM
ma-125	369	2	(	(	PUNCT
ma-125	369	3	2021)836	2021)836	NOUN
ma-125	369	4	-	-	NOUN
ma-125	369	5	857	857	NUM
ma-125	369	6	.	.	PUNCT
ma-125	370	1	https://doi.org/10.28924/2291-8639-19-2021-836.[11	https://doi.org/10.28924/2291-8639-19-2021-836.[11	ADV
ma-125	370	2	]	]	PUNCT
ma-125	370	3	m.	m.	NOUN
ma-125	370	4	rossafi	rossafi	PROPN
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