id	sid	tid	token	lemma	pos
ejpam-6846	1	1	european	european	PROPN
ejpam-6846	1	2	journal	journal	PROPN
ejpam-6846	1	3	of	of	ADP
ejpam-6846	1	4	pure	pure	ADJ
ejpam-6846	1	5	and	and	CCONJ
ejpam-6846	1	6	applied	applied	ADJ
ejpam-6846	1	7	mathematics	mathematic	NOUN
ejpam-6846	1	8	2025	2025	NUM
ejpam-6846	1	9	,	,	PUNCT
ejpam-6846	1	10	vol	vol	NOUN
ejpam-6846	1	11	.	.	PROPN
ejpam-6846	1	12	18	18	NUM
ejpam-6846	1	13	,	,	PUNCT
ejpam-6846	1	14	issue	issue	NOUN
ejpam-6846	1	15	4	4	NUM
ejpam-6846	1	16	,	,	PUNCT
ejpam-6846	1	17	article	article	NOUN
ejpam-6846	1	18	number	number	NOUN
ejpam-6846	1	19	6846	6846	NUM
ejpam-6846	1	20	issn	issn	VERB
ejpam-6846	1	21	1307	1307	NUM
ejpam-6846	1	22	-	-	SYM
ejpam-6846	1	23	5543	5543	NUM
ejpam-6846	1	24	–	–	PUNCT
ejpam-6846	1	25	ejpam.com	ejpam.com	X
ejpam-6846	1	26	published	publish	VERB
ejpam-6846	1	27	by	by	ADP
ejpam-6846	1	28	new	new	PROPN
ejpam-6846	1	29	york	york	PROPN
ejpam-6846	1	30	business	business	PROPN
ejpam-6846	1	31	global	global	ADJ
ejpam-6846	1	32	on	on	ADP
ejpam-6846	1	33	generalised	generalise	VERB
ejpam-6846	1	34	f	f	PROPN
ejpam-6846	1	35	-	-	PUNCT
ejpam-6846	1	36	projection	projection	NOUN
ejpam-6846	1	37	operator	operator	NOUN
ejpam-6846	1	38	over	over	ADP
ejpam-6846	1	39	nonconvex	nonconvex	NOUN
ejpam-6846	1	40	set	set	VERB
ejpam-6846	1	41	ali	ali	PROPN
ejpam-6846	1	42	al	al	PROPN
ejpam-6846	1	43	tane1,∗	tane1,∗	PROPN
ejpam-6846	1	44	,	,	PUNCT
ejpam-6846	1	45	lee	lee	PROPN
ejpam-6846	1	46	see	see	VERB
ejpam-6846	1	47	keong1	keong1	PROPN
ejpam-6846	1	48	1	1	NUM
ejpam-6846	1	49	school	school	NOUN
ejpam-6846	1	50	of	of	ADP
ejpam-6846	1	51	mathematical	mathematical	ADJ
ejpam-6846	1	52	sciences	science	NOUN
ejpam-6846	1	53	,	,	PUNCT
ejpam-6846	1	54	universiti	universiti	PROPN
ejpam-6846	1	55	sains	sain	VERB
ejpam-6846	1	56	malaysia	malaysia	PROPN
ejpam-6846	1	57	,	,	PUNCT
ejpam-6846	1	58	11800	11800	NUM
ejpam-6846	1	59	usm	usm	ADJ
ejpam-6846	1	60	,	,	PUNCT
ejpam-6846	1	61	penang	penang	PROPN
ejpam-6846	1	62	,	,	PUNCT
ejpam-6846	1	63	malaysia	malaysia	PROPN
ejpam-6846	1	64	.	.	PUNCT
ejpam-6846	2	1	abstract	abstract	PROPN
ejpam-6846	2	2	.	.	PUNCT
ejpam-6846	3	1	this	this	DET
ejpam-6846	3	2	work	work	NOUN
ejpam-6846	3	3	examines	examine	VERB
ejpam-6846	3	4	the	the	DET
ejpam-6846	3	5	generalized	generalize	VERB
ejpam-6846	3	6	f	f	PROPN
ejpam-6846	3	7	-projection	-projection	PROPN
ejpam-6846	3	8	operators	operator	NOUN
ejpam-6846	3	9	πf	πf	NOUN
ejpam-6846	3	10	s	s	VERB
ejpam-6846	3	11	.	.	PUNCT
ejpam-6846	3	12	by	by	ADP
ejpam-6846	3	13	proving	prove	VERB
ejpam-6846	3	14	the	the	DET
ejpam-6846	3	15	local	local	ADJ
ejpam-6846	3	16	lipschitz	lipschitz	NOUN
ejpam-6846	3	17	continuity	continuity	NOUN
ejpam-6846	3	18	of	of	ADP
ejpam-6846	3	19	the	the	DET
ejpam-6846	3	20	generalized	generalized	ADJ
ejpam-6846	3	21	projection	projection	NOUN
ejpam-6846	3	22	operator	operator	NOUN
ejpam-6846	4	1	πf	πf	NOUN
ejpam-6846	4	2	s	s	VERB
ejpam-6846	4	3	for	for	ADP
ejpam-6846	4	4	s	s	PRON
ejpam-6846	4	5	nonempty	nonempty	X
ejpam-6846	4	6	closed	close	VERB
ejpam-6846	4	7	sets	set	NOUN
ejpam-6846	4	8	that	that	PRON
ejpam-6846	4	9	are	be	AUX
ejpam-6846	4	10	not	not	PART
ejpam-6846	4	11	necessarily	necessarily	ADV
ejpam-6846	4	12	convex	convex	ADJ
ejpam-6846	4	13	,	,	PUNCT
ejpam-6846	4	14	and	and	CCONJ
ejpam-6846	4	15	using	use	VERB
ejpam-6846	4	16	convex	convex	PROPN
ejpam-6846	4	17	subdifferential	subdifferential	ADJ
ejpam-6846	4	18	∂conf	∂conf	PROPN
ejpam-6846	4	19	,	,	PUNCT
ejpam-6846	4	20	the	the	DET
ejpam-6846	4	21	fréchet	fréchet	ADJ
ejpam-6846	4	22	subdifferential	subdifferential	ADJ
ejpam-6846	4	23	∂f	∂f	PROPN
ejpam-6846	4	24	f(x	f(x	PROPN
ejpam-6846	4	25	)	)	PUNCT
ejpam-6846	4	26	,	,	PUNCT
ejpam-6846	4	27	and	and	CCONJ
ejpam-6846	4	28	the	the	DET
ejpam-6846	4	29	clarke	clarke	PROPN
ejpam-6846	4	30	subdifferential	subdifferential	PROPN
ejpam-6846	4	31	,	,	PUNCT
ejpam-6846	4	32	we	we	PRON
ejpam-6846	4	33	extend	extend	VERB
ejpam-6846	4	34	many	many	ADJ
ejpam-6846	4	35	properties	property	NOUN
ejpam-6846	4	36	of	of	ADP
ejpam-6846	4	37	πs	πs	ADP
ejpam-6846	4	38	to	to	PART
ejpam-6846	4	39	πf	πf	PROPN
ejpam-6846	4	40	s	s	PART
ejpam-6846	4	41	.	.	PUNCT
ejpam-6846	4	42	2020	2020	NUM
ejpam-6846	4	43	mathematics	mathematic	NOUN
ejpam-6846	4	44	subject	subject	NOUN
ejpam-6846	4	45	classifications	classification	NOUN
ejpam-6846	4	46	:	:	PUNCT
ejpam-6846	4	47	49j52	49j52	NUM
ejpam-6846	4	48	,	,	PUNCT
ejpam-6846	4	49	49j45	49j45	NUM
ejpam-6846	4	50	key	key	ADJ
ejpam-6846	4	51	words	word	NOUN
ejpam-6846	4	52	and	and	CCONJ
ejpam-6846	4	53	phrases	phrase	NOUN
ejpam-6846	4	54	:	:	PUNCT
ejpam-6846	4	55	generalized	generalized	ADJ
ejpam-6846	4	56	f	f	PROPN
ejpam-6846	4	57	-projection	-projection	PROPN
ejpam-6846	4	58	,	,	PUNCT
ejpam-6846	4	59	lipschitz	lipschitz	VERB
ejpam-6846	4	60	continuity	continuity	NOUN
ejpam-6846	4	61	1	1	NUM
ejpam-6846	4	62	.	.	PUNCT
ejpam-6846	5	1	introduction	introduction	NOUN
ejpam-6846	5	2	for	for	ADP
ejpam-6846	5	3	x	x	SYM
ejpam-6846	5	4	uniformly	uniformly	ADV
ejpam-6846	5	5	convex	convex	VERB
ejpam-6846	5	6	and	and	CCONJ
ejpam-6846	5	7	uniformly	uniformly	ADV
ejpam-6846	5	8	smooth	smooth	ADJ
ejpam-6846	5	9	banach	banach	NOUN
ejpam-6846	5	10	spaces	space	NOUN
ejpam-6846	6	1	,	,	PUNCT
ejpam-6846	6	2	alber	alber	PROPN
ejpam-6846	6	3	presented	present	VERB
ejpam-6846	6	4	the	the	DET
ejpam-6846	6	5	generalized	generalized	ADJ
ejpam-6846	6	6	projection	projection	NOUN
ejpam-6846	6	7	operators	operator	NOUN
ejpam-6846	6	8	πs	πs	ADV
ejpam-6846	6	9	.	.	PUNCT
ejpam-6846	7	1	additionally	additionally	ADV
ejpam-6846	7	2	,	,	PUNCT
ejpam-6846	7	3	he	he	PRON
ejpam-6846	7	4	thoroughly	thoroughly	ADV
ejpam-6846	7	5	examined	examine	VERB
ejpam-6846	7	6	their	their	PRON
ejpam-6846	7	7	characteristics	characteristic	NOUN
ejpam-6846	7	8	and	and	CCONJ
ejpam-6846	7	9	offered	offer	VERB
ejpam-6846	7	10	a	a	DET
ejpam-6846	7	11	number	number	NOUN
ejpam-6846	7	12	of	of	ADP
ejpam-6846	7	13	applications	application	NOUN
ejpam-6846	7	14	for	for	ADP
ejpam-6846	7	15	the	the	DET
ejpam-6846	7	16	generalized	generalized	ADJ
ejpam-6846	7	17	projections	projection	NOUN
ejpam-6846	7	18	,	,	PUNCT
ejpam-6846	7	19	including	include	VERB
ejpam-6846	7	20	approximating	approximate	VERB
ejpam-6846	7	21	the	the	DET
ejpam-6846	7	22	solution	solution	NOUN
ejpam-6846	7	23	to	to	ADP
ejpam-6846	7	24	variational	variational	ADJ
ejpam-6846	7	25	-	-	PUNCT
ejpam-6846	7	26	inequalities	inequality	NOUN
ejpam-6846	7	27	(	(	PUNCT
ejpam-6846	7	28	see	see	VERB
ejpam-6846	7	29	[	[	X
ejpam-6846	7	30	1	1	NUM
ejpam-6846	7	31	]	]	NUM
ejpam-6846	7	32	)	)	PUNCT
ejpam-6846	7	33	.	.	PUNCT
ejpam-6846	8	1	li	li	PROPN
ejpam-6846	8	2	explored	explore	VERB
ejpam-6846	8	3	the	the	DET
ejpam-6846	8	4	second	second	ADJ
ejpam-6846	8	5	direction	direction	NOUN
ejpam-6846	8	6	in	in	ADP
ejpam-6846	8	7	[	[	X
ejpam-6846	8	8	2	2	NUM
ejpam-6846	8	9	]	]	PUNCT
ejpam-6846	8	10	,	,	PUNCT
ejpam-6846	8	11	who	who	PRON
ejpam-6846	8	12	expanded	expand	VERB
ejpam-6846	8	13	the	the	DET
ejpam-6846	8	14	definition	definition	NOUN
ejpam-6846	8	15	of	of	ADP
ejpam-6846	8	16	πs	πs	INTJ
ejpam-6846	8	17	to	to	PART
ejpam-6846	8	18	where	where	SCONJ
ejpam-6846	8	19	x	x	PRON
ejpam-6846	8	20	is	be	AUX
ejpam-6846	8	21	a	a	DET
ejpam-6846	8	22	reflexive	reflexive	ADJ
ejpam-6846	8	23	banach	banach	NOUN
ejpam-6846	8	24	space	space	NOUN
ejpam-6846	8	25	.	.	PUNCT
ejpam-6846	9	1	also	also	ADV
ejpam-6846	9	2	,	,	PUNCT
ejpam-6846	9	3	examined	examine	VERB
ejpam-6846	9	4	some	some	PRON
ejpam-6846	9	5	of	of	ADP
ejpam-6846	9	6	its	its	PRON
ejpam-6846	9	7	properties	property	NOUN
ejpam-6846	9	8	and	and	CCONJ
ejpam-6846	9	9	applied	apply	VERB
ejpam-6846	9	10	them	they	PRON
ejpam-6846	9	11	to	to	ADP
ejpam-6846	9	12	the	the	DET
ejpam-6846	9	13	solution	solution	NOUN
ejpam-6846	9	14	of	of	ADP
ejpam-6846	9	15	variationalinequalities	variationalinequalitie	NOUN
ejpam-6846	9	16	.	.	PUNCT
ejpam-6846	10	1	since	since	SCONJ
ejpam-6846	10	2	lie	lie	NOUN
ejpam-6846	10	3	and	and	CCONJ
ejpam-6846	10	4	alber	alber	PROPN
ejpam-6846	10	5	’s	’s	PART
ejpam-6846	10	6	research	research	NOUN
ejpam-6846	10	7	was	be	AUX
ejpam-6846	10	8	based	base	VERB
ejpam-6846	10	9	on	on	ADP
ejpam-6846	10	10	the	the	DET
ejpam-6846	10	11	ideas	idea	NOUN
ejpam-6846	10	12	that	that	SCONJ
ejpam-6846	10	13	using	use	VERB
ejpam-6846	10	14	v	v	ADP
ejpam-6846	10	15	-functional	-functional	ADJ
ejpam-6846	10	16	,	,	PUNCT
ejpam-6846	10	17	and	and	CCONJ
ejpam-6846	10	18	s	s	VERB
ejpam-6846	10	19	is	be	AUX
ejpam-6846	10	20	a	a	DET
ejpam-6846	10	21	closed	closed	ADJ
ejpam-6846	10	22	,	,	PUNCT
ejpam-6846	10	23	and	and	CCONJ
ejpam-6846	10	24	convex	convex	PROPN
ejpam-6846	10	25	subset	subset	NOUN
ejpam-6846	10	26	of	of	ADP
ejpam-6846	10	27	reflexve	reflexve	NOUN
ejpam-6846	10	28	banach	banach	NOUN
ejpam-6846	10	29	space	space	NOUN
ejpam-6846	10	30	.	.	PUNCT
ejpam-6846	11	1	there	there	PRON
ejpam-6846	11	2	have	have	AUX
ejpam-6846	11	3	been	be	AUX
ejpam-6846	11	4	studies	study	NOUN
ejpam-6846	11	5	to	to	PART
ejpam-6846	11	6	generalize	generalize	VERB
ejpam-6846	11	7	projection	projection	NOUN
ejpam-6846	11	8	operators	operator	NOUN
ejpam-6846	11	9	by	by	ADP
ejpam-6846	11	10	either	either	CCONJ
ejpam-6846	11	11	accepting	accept	VERB
ejpam-6846	11	12	s	s	PRON
ejpam-6846	11	13	as	as	SCONJ
ejpam-6846	11	14	not	not	PART
ejpam-6846	11	15	necessarily	necessarily	ADV
ejpam-6846	11	16	convex	convex	VERB
ejpam-6846	11	17	or	or	CCONJ
ejpam-6846	11	18	by	by	ADP
ejpam-6846	11	19	extending	extend	VERB
ejpam-6846	11	20	v	v	ADP
ejpam-6846	11	21	-functional	-functional	ADJ
ejpam-6846	11	22	to	to	ADP
ejpam-6846	11	23	v	v	NOUN
ejpam-6846	11	24	f	f	PROPN
ejpam-6846	11	25	-functional	-functional	PROPN
ejpam-6846	11	26	.	.	PUNCT
ejpam-6846	12	1	in	in	ADP
ejpam-6846	12	2	[	[	X
ejpam-6846	12	3	3	3	NUM
ejpam-6846	12	4	]	]	PUNCT
ejpam-6846	12	5	,	,	PUNCT
ejpam-6846	12	6	k	k	PROPN
ejpam-6846	12	7	wu	wu	PROPN
ejpam-6846	12	8	and	and	CCONJ
ejpam-6846	12	9	n	n	PROPN
ejpam-6846	12	10	huang	huang	PROPN
ejpam-6846	12	11	presented	present	VERB
ejpam-6846	12	12	the	the	DET
ejpam-6846	12	13	generalized	generalize	VERB
ejpam-6846	12	14	f	f	PROPN
ejpam-6846	12	15	-projection	-projection	PROPN
ejpam-6846	12	16	operator	operator	NOUN
ejpam-6846	12	17	πf	πf	NOUN
ejpam-6846	12	18	s	s	PROPN
ejpam-6846	12	19	,	,	PUNCT
ejpam-6846	12	20	which	which	PRON
ejpam-6846	12	21	is	be	AUX
ejpam-6846	12	22	an	an	DET
ejpam-6846	12	23	extension	extension	NOUN
ejpam-6846	12	24	of	of	ADP
ejpam-6846	12	25	the	the	DET
ejpam-6846	12	26	generalized	generalized	ADJ
ejpam-6846	12	27	projection	projection	NOUN
ejpam-6846	12	28	operator	operator	NOUN
ejpam-6846	12	29	.	.	PUNCT
ejpam-6846	13	1	it	it	PRON
ejpam-6846	13	2	was	be	AUX
ejpam-6846	13	3	demonstrated	demonstrate	VERB
ejpam-6846	13	4	that	that	SCONJ
ejpam-6846	13	5	πf	πf	PRON
ejpam-6846	13	6	s	s	VERB
ejpam-6846	13	7	is	be	AUX
ejpam-6846	13	8	well	well	ADV
ejpam-6846	13	9	-	-	PUNCT
ejpam-6846	13	10	defined	define	VERB
ejpam-6846	13	11	for	for	ADP
ejpam-6846	13	12	reflexive	reflexive	ADJ
ejpam-6846	13	13	banach	banach	NOUN
ejpam-6846	13	14	spaces	space	NOUN
ejpam-6846	13	15	by	by	ADP
ejpam-6846	13	16	providing	provide	VERB
ejpam-6846	13	17	certain	certain	ADJ
ejpam-6846	13	18	properties	property	NOUN
ejpam-6846	13	19	,	,	PUNCT
ejpam-6846	13	20	where	where	SCONJ
ejpam-6846	13	21	they	they	PRON
ejpam-6846	13	22	used	use	VERB
ejpam-6846	13	23	the	the	DET
ejpam-6846	13	24	fankkm	fankkm	NOUN
ejpam-6846	13	25	theorem	theorem	NOUN
ejpam-6846	13	26	to	to	PART
ejpam-6846	13	27	look	look	VERB
ejpam-6846	13	28	into	into	ADP
ejpam-6846	13	29	the	the	DET
ejpam-6846	13	30	existence	existence	NOUN
ejpam-6846	13	31	of	of	ADP
ejpam-6846	13	32	solutions	solution	NOUN
ejpam-6846	13	33	to	to	ADP
ejpam-6846	13	34	a	a	DET
ejpam-6846	13	35	few	few	ADJ
ejpam-6846	13	36	variational	variational	ADJ
ejpam-6846	13	37	-	-	PUNCT
ejpam-6846	13	38	inequality	inequality	NOUN
ejpam-6846	13	39	issues	issue	NOUN
ejpam-6846	13	40	as	as	ADP
ejpam-6846	13	41	an	an	DET
ejpam-6846	13	42	application	application	NOUN
ejpam-6846	13	43	of	of	ADP
ejpam-6846	13	44	their	their	PRON
ejpam-6846	13	45	findings	finding	NOUN
ejpam-6846	13	46	.	.	PUNCT
ejpam-6846	14	1	in	in	ADP
ejpam-6846	14	2	their	their	PRON
ejpam-6846	14	3	study	study	NOUN
ejpam-6846	14	4	in	in	ADP
ejpam-6846	14	5	[	[	X
ejpam-6846	14	6	4	4	NUM
ejpam-6846	14	7	]	]	PUNCT
ejpam-6846	14	8	,	,	PUNCT
ejpam-6846	14	9	m.	m.	NOUN
ejpam-6846	14	10	bounkhel	bounkhel	PROPN
ejpam-6846	14	11	and	and	CCONJ
ejpam-6846	14	12	r.	r.	PROPN
ejpam-6846	14	13	al	al	PROPN
ejpam-6846	14	14	-	-	PUNCT
ejpam-6846	14	15	yusof	yusof	PROPN
ejpam-6846	14	16	used	use	VERB
ejpam-6846	14	17	the	the	DET
ejpam-6846	14	18	generalized	generalized	ADJ
ejpam-6846	14	19	projection	projection	NOUN
ejpam-6846	14	20	operator	operator	NOUN
ejpam-6846	14	21	πs	πs	ADP
ejpam-6846	14	22	to	to	PART
ejpam-6846	14	23	introduce	introduce	VERB
ejpam-6846	14	24	the	the	DET
ejpam-6846	14	25	new	new	ADJ
ejpam-6846	14	26	generalized	generalized	ADJ
ejpam-6846	14	27	proxmal	proxmal	NOUN
ejpam-6846	14	28	normal	normal	ADJ
ejpam-6846	14	29	cone	cone	NOUN
ejpam-6846	14	30	in	in	ADP
ejpam-6846	14	31	reflexive	reflexive	ADJ
ejpam-6846	14	32	smooth	smooth	ADJ
ejpam-6846	14	33	banach	banach	NOUN
ejpam-6846	14	34	spaces	space	VERB
ejpam-6846	14	35	with	with	ADP
ejpam-6846	14	36	s	s	VERB
ejpam-6846	14	37	not	not	PART
ejpam-6846	14	38	necessarily	necessarily	ADV
ejpam-6846	14	39	convex	convex	ADJ
ejpam-6846	14	40	.	.	PUNCT
ejpam-6846	15	1	however	however	ADV
ejpam-6846	15	2	,	,	PUNCT
ejpam-6846	15	3	begin	begin	VERB
ejpam-6846	15	4	with	with	ADP
ejpam-6846	15	5	m.	m.	NOUN
ejpam-6846	15	6	bounkhel	bounkhel	PROPN
ejpam-6846	15	7	(	(	PUNCT
ejpam-6846	15	8	see	see	VERB
ejpam-6846	15	9	[	[	X
ejpam-6846	15	10	5	5	NUM
ejpam-6846	15	11	]	]	PUNCT
ejpam-6846	15	12	)	)	PUNCT
ejpam-6846	15	13	∗corresponding	∗corresponde	VERB
ejpam-6846	15	14	author	author	NOUN
ejpam-6846	15	15	.	.	PUNCT
ejpam-6846	16	1	doi	doi	NOUN
ejpam-6846	16	2	:	:	PUNCT
ejpam-6846	16	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6846	https://doi.org/10.29020/nybg.ejpam.v18i4.6846	NOUN
ejpam-6846	16	4	email	email	NOUN
ejpam-6846	16	5	addresses	address	NOUN
ejpam-6846	16	6	:	:	PUNCT
ejpam-6846	16	7	alialtane@student.usm.my	alialtane@student.usm.my	NOUN
ejpam-6846	16	8	(	(	PUNCT
ejpam-6846	16	9	a.	a.	PROPN
ejpam-6846	16	10	al	al	PROPN
ejpam-6846	16	11	tane	tane	PROPN
ejpam-6846	16	12	)	)	PUNCT
ejpam-6846	16	13	,	,	PUNCT
ejpam-6846	16	14	sklee@usm.my	sklee@usm.my	PROPN
ejpam-6846	16	15	(	(	PUNCT
ejpam-6846	16	16	l.	l.	PROPN
ejpam-6846	16	17	s.	s.	PROPN
ejpam-6846	16	18	keong	keong	PROPN
ejpam-6846	16	19	)	)	PUNCT
ejpam-6846	16	20	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6846	17	1	1	1	NUM
ejpam-6846	17	2	copyright	copyright	NOUN
ejpam-6846	17	3	:	:	PUNCT
ejpam-6846	17	4	©	©	PROPN
ejpam-6846	17	5	2025	2025	NUM
ejpam-6846	17	6	the	the	DET
ejpam-6846	17	7	author(s	author(s	NOUN
ejpam-6846	17	8	)	)	PUNCT
ejpam-6846	17	9	.	.	PUNCT
ejpam-6846	18	1	(	(	PUNCT
ejpam-6846	18	2	cc	cc	NOUN
ejpam-6846	18	3	by	by	ADP
ejpam-6846	18	4	-	-	PUNCT
ejpam-6846	18	5	nc	nc	PROPN
ejpam-6846	18	6	4.0	4.0	NUM
ejpam-6846	18	7	)	)	PUNCT
ejpam-6846	18	8	a.	a.	NOUN
ejpam-6846	18	9	al	al	PROPN
ejpam-6846	18	10	tane	tane	PROPN
ejpam-6846	18	11	,	,	PUNCT
ejpam-6846	18	12	l.	l.	PROPN
ejpam-6846	18	13	s.	s.	PROPN
ejpam-6846	18	14	keong	keong	PROPN
ejpam-6846	18	15	/	/	PUNCT
ejpam-6846	18	16	eur	eur	PROPN
ejpam-6846	18	17	.	.	PUNCT
ejpam-6846	19	1	j.	j.	PROPN
ejpam-6846	19	2	pure	pure	PROPN
ejpam-6846	19	3	appl	appl	PROPN
ejpam-6846	19	4	.	.	PROPN
ejpam-6846	19	5	math	math	PROPN
ejpam-6846	19	6	,	,	PUNCT
ejpam-6846	19	7	18	18	NUM
ejpam-6846	19	8	(	(	PUNCT
ejpam-6846	19	9	4	4	NUM
ejpam-6846	19	10	)	)	PUNCT
ejpam-6846	19	11	(	(	PUNCT
ejpam-6846	19	12	2025	2025	NUM
ejpam-6846	19	13	)	)	PUNCT
ejpam-6846	19	14	,	,	PUNCT
ejpam-6846	19	15	6846	6846	NUM
ejpam-6846	19	16	2	2	NUM
ejpam-6846	19	17	of	of	ADP
ejpam-6846	19	18	13	13	NUM
ejpam-6846	19	19	he	he	PRON
ejpam-6846	19	20	 	 	SPACE
ejpam-6846	19	21	examined	examine	VERB
ejpam-6846	19	22	the	the	DET
ejpam-6846	19	23	existence	existence	NOUN
ejpam-6846	19	24	of	of	ADP
ejpam-6846	19	25	the	the	DET
ejpam-6846	19	26	generalized	generalized	ADJ
ejpam-6846	19	27	projection	projection	NOUN
ejpam-6846	19	28	operator	operator	NOUN
ejpam-6846	19	29	πs	πs	ADV
ejpam-6846	19	30	for	for	ADP
ejpam-6846	19	31	s	s	PRON
ejpam-6846	19	32	set	set	NOUN
ejpam-6846	19	33	that	that	PRON
ejpam-6846	19	34	are	be	AUX
ejpam-6846	19	35	not	not	PART
ejpam-6846	19	36	necessarily	necessarily	ADV
ejpam-6846	19	37	convex	convex	ADJ
ejpam-6846	19	38	.	.	PUNCT
ejpam-6846	20	1	in	in	ADP
ejpam-6846	20	2	uniformly	uniformly	ADV
ejpam-6846	20	3	convex	convex	VERB
ejpam-6846	20	4	and	and	CCONJ
ejpam-6846	20	5	smooth	smooth	ADJ
ejpam-6846	20	6	banach	banach	NOUN
ejpam-6846	20	7	spaces	space	NOUN
ejpam-6846	20	8	,	,	PUNCT
ejpam-6846	20	9	the	the	DET
ejpam-6846	20	10	author	author	NOUN
ejpam-6846	20	11	demonstrates	demonstrate	VERB
ejpam-6846	20	12	that	that	SCONJ
ejpam-6846	20	13	πs	πs	ADV
ejpam-6846	20	14	may	may	AUX
ejpam-6846	20	15	be	be	AUX
ejpam-6846	20	16	empty	empty	ADJ
ejpam-6846	20	17	for	for	ADP
ejpam-6846	20	18	a	a	DET
ejpam-6846	20	19	nonconvex	nonconvex	NOUN
ejpam-6846	20	20	closed	close	VERB
ejpam-6846	20	21	set	set	NOUN
ejpam-6846	20	22	.	.	PUNCT
ejpam-6846	21	1	nevertheless	nevertheless	ADV
ejpam-6846	21	2	,	,	PUNCT
ejpam-6846	21	3	he	he	PRON
ejpam-6846	21	4	demonstrated	demonstrate	VERB
ejpam-6846	21	5	that	that	SCONJ
ejpam-6846	21	6	the	the	DET
ejpam-6846	21	7	set	set	NOUN
ejpam-6846	21	8	of	of	ADP
ejpam-6846	21	9	points	point	NOUN
ejpam-6846	21	10	x∗	x∗	PROPN
ejpam-6846	21	11	∈	∈	PROPN
ejpam-6846	21	12	x∗	x∗	PROPN
ejpam-6846	21	13	s.t	s.t	PROPN
ejpam-6846	21	14	πs(x	πs(x	X
ejpam-6846	21	15	∗	∗	NOUN
ejpam-6846	21	16	)	)	PUNCT
ejpam-6846	21	17	6=	6=	ADP
ejpam-6846	22	1	ϕ	ϕ	NOUN
ejpam-6846	22	2	is	be	AUX
ejpam-6846	22	3	dense	dense	ADJ
ejpam-6846	22	4	in	in	ADP
ejpam-6846	22	5	x∗	x∗	PROPN
ejpam-6846	22	6	for	for	ADP
ejpam-6846	22	7	closed	closed	ADJ
ejpam-6846	22	8	nonconvex	nonconvex	NOUN
ejpam-6846	22	9	sets	set	NOUN
ejpam-6846	22	10	.	.	PUNCT
ejpam-6846	23	1	in	in	ADP
ejpam-6846	23	2	[	[	X
ejpam-6846	23	3	6	6	NUM
ejpam-6846	23	4	]	]	PUNCT
ejpam-6846	23	5	,	,	PUNCT
ejpam-6846	23	6	the	the	DET
ejpam-6846	23	7	properties	property	NOUN
ejpam-6846	23	8	of	of	ADP
ejpam-6846	23	9	the	the	DET
ejpam-6846	23	10	generalized	generalized	ADJ
ejpam-6846	23	11	projection	projection	NOUN
ejpam-6846	23	12	on	on	ADP
ejpam-6846	23	13	nonconvex	nonconvex	NOUN
ejpam-6846	23	14	sets	set	NOUN
ejpam-6846	23	15	in	in	ADP
ejpam-6846	23	16	reflexive	reflexive	ADJ
ejpam-6846	23	17	smooth	smooth	ADJ
ejpam-6846	23	18	banach	banach	NOUN
ejpam-6846	23	19	spaces	space	VERB
ejpam-6846	23	20	with	with	ADP
ejpam-6846	23	21	the	the	DET
ejpam-6846	23	22	smooth	smooth	ADJ
ejpam-6846	23	23	dual	dual	ADJ
ejpam-6846	23	24	norm	norm	NOUN
ejpam-6846	23	25	are	be	AUX
ejpam-6846	23	26	further	far	ADV
ejpam-6846	23	27	examined	examine	VERB
ejpam-6846	23	28	.	.	PUNCT
ejpam-6846	24	1	the	the	DET
ejpam-6846	24	2	local	local	ADJ
ejpam-6846	24	3	lipschtz	lipschtz	ADJ
ejpam-6846	24	4	continuity	continuity	NOUN
ejpam-6846	24	5	of	of	ADP
ejpam-6846	24	6	πs	πs	NOUN
ejpam-6846	24	7	for	for	ADP
ejpam-6846	24	8	s	s	PRON
ejpam-6846	24	9	not	not	PART
ejpam-6846	24	10	necessarily	necessarily	ADV
ejpam-6846	24	11	convex	convex	ADJ
ejpam-6846	24	12	was	be	AUX
ejpam-6846	24	13	established	establish	VERB
ejpam-6846	24	14	by	by	ADP
ejpam-6846	24	15	m.	m.	NOUN
ejpam-6846	24	16	bounkhel	bounkhel	PROPN
ejpam-6846	24	17	and	and	CCONJ
ejpam-6846	24	18	m.	m.	PROPN
ejpam-6846	24	19	bachar	bachar	PROPN
ejpam-6846	24	20	.	.	PUNCT
ejpam-6846	25	1	additionally	additionally	ADV
ejpam-6846	25	2	,	,	PUNCT
ejpam-6846	25	3	they	they	PRON
ejpam-6846	25	4	proved	prove	VERB
ejpam-6846	25	5	that	that	SCONJ
ejpam-6846	25	6	many	many	ADJ
ejpam-6846	25	7	of	of	ADP
ejpam-6846	25	8	the	the	DET
ejpam-6846	25	9	features	feature	NOUN
ejpam-6846	25	10	of	of	ADP
ejpam-6846	25	11	πs	πs	NOUN
ejpam-6846	25	12	on	on	ADP
ejpam-6846	25	13	open	open	ADJ
ejpam-6846	25	14	subsets	subset	NOUN
ejpam-6846	25	15	in	in	ADP
ejpam-6846	25	16	x∗	x∗	PROPN
ejpam-6846	25	17	are	be	AUX
ejpam-6846	25	18	equivalent	equivalent	ADJ
ejpam-6846	25	19	.	.	PUNCT
ejpam-6846	26	1	initiated	initiate	VERB
ejpam-6846	26	2	in	in	ADP
ejpam-6846	26	3	the	the	DET
ejpam-6846	26	4	nonconvex	nonconvex	NOUN
ejpam-6846	26	5	case	case	NOUN
ejpam-6846	26	6	by	by	ADP
ejpam-6846	26	7	bounkhel	bounkhel	PROPN
ejpam-6846	26	8	in	in	ADP
ejpam-6846	26	9	[	[	X
ejpam-6846	26	10	5–7	5–7	NOUN
ejpam-6846	26	11	]	]	PUNCT
ejpam-6846	26	12	,	,	PUNCT
ejpam-6846	26	13	the	the	DET
ejpam-6846	26	14	present	present	ADJ
ejpam-6846	26	15	work	work	NOUN
ejpam-6846	26	16	continues	continue	VERB
ejpam-6846	26	17	the	the	DET
ejpam-6846	26	18	study	study	NOUN
ejpam-6846	26	19	of	of	ADP
ejpam-6846	26	20	the	the	DET
ejpam-6846	26	21	πf	πf	NOUN
ejpam-6846	26	22	s	s	PROPN
ejpam-6846	26	23	.	.	PUNCT
ejpam-6846	27	1	by	by	ADP
ejpam-6846	27	2	proving	prove	VERB
ejpam-6846	27	3	the	the	DET
ejpam-6846	27	4	local	local	ADJ
ejpam-6846	27	5	lipschtz	lipschtz	ADJ
ejpam-6846	27	6	continuity	continuity	NOUN
ejpam-6846	27	7	of	of	ADP
ejpam-6846	27	8	the	the	DET
ejpam-6846	27	9	generalized	generalized	ADJ
ejpam-6846	27	10	projection	projection	NOUN
ejpam-6846	27	11	operator	operator	NOUN
ejpam-6846	27	12	πf	πf	NOUN
ejpam-6846	27	13	s	s	VERB
ejpam-6846	27	14	for	for	ADP
ejpam-6846	27	15	s	s	PRON
ejpam-6846	27	16	nonempty	nonempty	X
ejpam-6846	27	17	closed	close	VERB
ejpam-6846	27	18	sets	set	NOUN
ejpam-6846	27	19	not	not	PART
ejpam-6846	27	20	necessarily	necessarily	ADV
ejpam-6846	27	21	convex	convex	VERB
ejpam-6846	27	22	,	,	PUNCT
ejpam-6846	27	23	and	and	CCONJ
ejpam-6846	27	24	extending	extend	VERB
ejpam-6846	27	25	many	many	ADJ
ejpam-6846	27	26	properties	property	NOUN
ejpam-6846	27	27	of	of	ADP
ejpam-6846	27	28	πs	πs	INTJ
ejpam-6846	27	29	to	to	PART
ejpam-6846	27	30	πf	πf	VERB
ejpam-6846	27	31	s	s	VERB
ejpam-6846	27	32	on	on	ADP
ejpam-6846	27	33	open	open	ADJ
ejpam-6846	27	34	subsets	subset	NOUN
ejpam-6846	27	35	in	in	ADP
ejpam-6846	27	36	x∗.	x∗.	PROPN
ejpam-6846	27	37	2	2	X
ejpam-6846	27	38	.	.	X
ejpam-6846	27	39	mathematical	mathematical	ADJ
ejpam-6846	27	40	preliminaries	preliminary	NOUN
ejpam-6846	27	41	let	let	VERB
ejpam-6846	27	42	x∗	x∗	PROPN
ejpam-6846	27	43	be	be	AUX
ejpam-6846	27	44	a	a	DET
ejpam-6846	27	45	topological	topological	ADJ
ejpam-6846	27	46	dual	dual	ADJ
ejpam-6846	27	47	space	space	NOUN
ejpam-6846	27	48	of	of	ADP
ejpam-6846	27	49	a	a	DET
ejpam-6846	27	50	banach	banach	NOUN
ejpam-6846	27	51	space	space	NOUN
ejpam-6846	27	52	x.	x.	NOUN
ejpam-6846	27	53	in	in	ADP
ejpam-6846	27	54	x∗	x∗	PROPN
ejpam-6846	27	55	and	and	CCONJ
ejpam-6846	27	56	x	x	NOUN
ejpam-6846	27	57	,	,	PUNCT
ejpam-6846	27	58	the	the	DET
ejpam-6846	27	59	closed	closed	ADJ
ejpam-6846	27	60	unit	unit	NOUN
ejpam-6846	27	61	balls	ball	NOUN
ejpam-6846	27	62	are	be	AUX
ejpam-6846	27	63	b∗	b∗	ADJ
ejpam-6846	27	64	and	and	CCONJ
ejpam-6846	27	65	b	b	NOUN
ejpam-6846	27	66	,	,	PUNCT
ejpam-6846	27	67	respectively	respectively	ADV
ejpam-6846	27	68	.	.	PUNCT
ejpam-6846	28	1	for	for	ADP
ejpam-6846	28	2	definitions	definition	NOUN
ejpam-6846	28	3	and	and	CCONJ
ejpam-6846	28	4	some	some	DET
ejpam-6846	28	5	results	result	NOUN
ejpam-6846	28	6	concerning	concern	VERB
ejpam-6846	28	7	uniformly	uniformly	ADV
ejpam-6846	28	8	smooth	smooth	ADJ
ejpam-6846	28	9	,	,	PUNCT
ejpam-6846	28	10	uniformly	uniformly	ADV
ejpam-6846	28	11	convex	convex	VERB
ejpam-6846	28	12	banach	banach	NOUN
ejpam-6846	28	13	spaces	space	NOUN
ejpam-6846	28	14	,	,	PUNCT
ejpam-6846	28	15	and	and	CCONJ
ejpam-6846	28	16	strictly	strictly	ADV
ejpam-6846	28	17	convex	convex	ADJ
ejpam-6846	28	18	spaces	space	NOUN
ejpam-6846	28	19	,	,	PUNCT
ejpam-6846	28	20	we	we	PRON
ejpam-6846	28	21	refer	refer	VERB
ejpam-6846	28	22	to	to	ADP
ejpam-6846	28	23	(	(	PUNCT
ejpam-6846	28	24	[	[	X
ejpam-6846	28	25	8]),[9	8]),[9	NUM
ejpam-6846	28	26	]	]	SYM
ejpam-6846	28	27	)	)	PUNCT
ejpam-6846	28	28	.	.	PUNCT
ejpam-6846	29	1	for	for	ADP
ejpam-6846	29	2	j	j	PROPN
ejpam-6846	29	3	:	:	PUNCT
ejpam-6846	29	4	x−→	x−→	PROPN
ejpam-6846	29	5	−→x∗	−→x∗	NUM
ejpam-6846	29	6	,	,	PUNCT
ejpam-6846	29	7	called	call	VERB
ejpam-6846	29	8	the	the	DET
ejpam-6846	29	9	normalzed	normalze	VERB
ejpam-6846	29	10	duality	duality	NOUN
ejpam-6846	29	11	mapping	mapping	NOUN
ejpam-6846	29	12	defined	define	VERB
ejpam-6846	29	13	by	by	ADP
ejpam-6846	29	14	j(u	j(u	PROPN
ejpam-6846	29	15	)	)	PUNCT
ejpam-6846	30	1	=	=	PRON
ejpam-6846	30	2	{	{	PUNCT
ejpam-6846	30	3	u∗	u∗	NOUN
ejpam-6846	30	4	∈	∈	PROPN
ejpam-6846	30	5	x∗	x∗	PROPN
ejpam-6846	30	6	:	:	PUNCT
ejpam-6846	30	7	〈	〈	PROPN
ejpam-6846	30	8	u∗	u∗	ADV
ejpam-6846	30	9	,	,	PUNCT
ejpam-6846	30	10	u	u	NOUN
ejpam-6846	30	11	〉	〉	NOUN
ejpam-6846	30	12	=	=	NOUN
ejpam-6846	30	13	‖u∗‖.‖u‖	‖u∗‖.‖u‖	NOUN
ejpam-6846	30	14	=	=	SYM
ejpam-6846	30	15	‖u∗‖2	‖u∗‖2	PROPN
ejpam-6846	30	16	=	=	PUNCT
ejpam-6846	30	17	‖u‖2	‖u‖2	ADJ
ejpam-6846	30	18	}	}	PUNCT
ejpam-6846	30	19	.	.	PUNCT
ejpam-6846	31	1	it	it	PRON
ejpam-6846	31	2	is	be	AUX
ejpam-6846	31	3	clear	clear	ADJ
ejpam-6846	31	4	that	that	SCONJ
ejpam-6846	31	5	‖j(u)‖	‖j(u)‖	PROPN
ejpam-6846	31	6	is	be	AUX
ejpam-6846	31	7	the	the	DET
ejpam-6846	31	8	norm	norm	NOUN
ejpam-6846	31	9	defined	define	VERB
ejpam-6846	31	10	on	on	ADP
ejpam-6846	31	11	x∗	x∗	PROPN
ejpam-6846	31	12	,	,	PUNCT
ejpam-6846	31	13	and	and	CCONJ
ejpam-6846	31	14	‖u‖	‖u‖	PROPN
ejpam-6846	31	15	is	be	AUX
ejpam-6846	31	16	the	the	DET
ejpam-6846	31	17	norm	norm	NOUN
ejpam-6846	31	18	defined	define	VERB
ejpam-6846	31	19	on	on	ADP
ejpam-6846	31	20	x.	x.	NOUN
ejpam-6846	31	21	these	these	PRON
ejpam-6846	31	22	are	be	AUX
ejpam-6846	31	23	a	a	DET
ejpam-6846	31	24	few	few	ADJ
ejpam-6846	31	25	of	of	ADP
ejpam-6846	31	26	the	the	DET
ejpam-6846	31	27	j(x	j(x	PROPN
ejpam-6846	31	28	)	)	PUNCT
ejpam-6846	31	29	map	map	NOUN
ejpam-6846	31	30	’s	’s	PART
ejpam-6846	31	31	features	feature	NOUN
ejpam-6846	31	32	.	.	PUNCT
ejpam-6846	32	1	consider	consider	VERB
ejpam-6846	32	2	[	[	X
ejpam-6846	32	3	10	10	NUM
ejpam-6846	32	4	]	]	PUNCT
ejpam-6846	32	5	or	or	CCONJ
ejpam-6846	32	6	[	[	X
ejpam-6846	32	7	11	11	NUM
ejpam-6846	32	8	]	]	PUNCT
ejpam-6846	32	9	for	for	ADP
ejpam-6846	32	10	more	more	ADJ
ejpam-6846	32	11	details	detail	NOUN
ejpam-6846	32	12	.	.	PUNCT
ejpam-6846	33	1	(	(	PUNCT
ejpam-6846	33	2	i	i	NOUN
ejpam-6846	33	3	)	)	PUNCT
ejpam-6846	33	4	if	if	SCONJ
ejpam-6846	33	5	x	x	PRON
ejpam-6846	33	6	smooth	smooth	ADJ
ejpam-6846	33	7	banach	banach	NOUN
ejpam-6846	33	8	spaces	space	VERB
ejpam-6846	33	9	,	,	PUNCT
ejpam-6846	33	10	then	then	ADV
ejpam-6846	33	11	j	j	PROPN
ejpam-6846	33	12	continuous	continuous	ADJ
ejpam-6846	33	13	operator	operator	NOUN
ejpam-6846	33	14	.	.	PUNCT
ejpam-6846	34	1	(	(	PUNCT
ejpam-6846	34	2	ii	ii	X
ejpam-6846	34	3	)	)	PUNCT
ejpam-6846	34	4	j	j	PROPN
ejpam-6846	34	5	is	be	AUX
ejpam-6846	34	6	a	a	DET
ejpam-6846	34	7	single	single	ADJ
ejpam-6846	34	8	valued	value	VERB
ejpam-6846	34	9	mapping	mapping	NOUN
ejpam-6846	34	10	,	,	PUNCT
ejpam-6846	34	11	whenever	whenever	SCONJ
ejpam-6846	34	12	x∗	x∗	PROPN
ejpam-6846	34	13	is	be	AUX
ejpam-6846	34	14	strictly	strictly	ADV
ejpam-6846	34	15	convex	convex	ADJ
ejpam-6846	34	16	.	.	PUNCT
ejpam-6846	35	1	(	(	PUNCT
ejpam-6846	35	2	iii	iii	X
ejpam-6846	35	3	)	)	PUNCT
ejpam-6846	35	4	j	j	PROPN
ejpam-6846	35	5	is	be	AUX
ejpam-6846	35	6	a	a	DET
ejpam-6846	35	7	single	single	ADJ
ejpam-6846	35	8	valued	value	VERB
ejpam-6846	35	9	mapping	mapping	NOUN
ejpam-6846	35	10	,	,	PUNCT
ejpam-6846	35	11	whenever	whenever	SCONJ
ejpam-6846	35	12	x	x	PRON
ejpam-6846	35	13	is	be	AUX
ejpam-6846	35	14	reflexive	reflexive	ADJ
ejpam-6846	35	15	smooth	smooth	ADJ
ejpam-6846	35	16	banach	banach	NOUN
ejpam-6846	35	17	space	space	NOUN
ejpam-6846	35	18	.	.	PUNCT
ejpam-6846	36	1	we	we	PRON
ejpam-6846	36	2	now	now	ADV
ejpam-6846	36	3	review	review	VERB
ejpam-6846	36	4	a	a	DET
ejpam-6846	36	5	number	number	NOUN
ejpam-6846	36	6	of	of	ADP
ejpam-6846	36	7	crucial	crucial	ADJ
ejpam-6846	36	8	terms	term	NOUN
ejpam-6846	36	9	and	and	CCONJ
ejpam-6846	36	10	symbols	symbol	NOUN
ejpam-6846	36	11	that	that	PRON
ejpam-6846	36	12	are	be	AUX
ejpam-6846	36	13	necessary	necessary	ADJ
ejpam-6846	36	14	for	for	ADP
ejpam-6846	36	15	our	our	PRON
ejpam-6846	36	16	work	work	NOUN
ejpam-6846	36	17	.	.	PUNCT
ejpam-6846	37	1	we	we	PRON
ejpam-6846	37	2	begin	begin	VERB
ejpam-6846	37	3	with	with	ADP
ejpam-6846	37	4	the	the	DET
ejpam-6846	37	5	widely	widely	ADV
ejpam-6846	37	6	recognized	recognize	VERB
ejpam-6846	37	7	concepts	concept	NOUN
ejpam-6846	37	8	of	of	ADP
ejpam-6846	37	9	the	the	DET
ejpam-6846	37	10	convex	convex	PROPN
ejpam-6846	37	11	subdifferential	subdifferential	PROPN
ejpam-6846	37	12	∂conf	∂conf	PROPN
ejpam-6846	37	13	,	,	PUNCT
ejpam-6846	37	14	the	the	DET
ejpam-6846	37	15	fréchet	fréchet	ADJ
ejpam-6846	37	16	subdifferential	subdifferential	ADJ
ejpam-6846	37	17	∂f	∂f	PROPN
ejpam-6846	37	18	f(x	f(x	PROPN
ejpam-6846	37	19	)	)	PUNCT
ejpam-6846	37	20	,	,	PUNCT
ejpam-6846	37	21	and	and	CCONJ
ejpam-6846	37	22	the	the	DET
ejpam-6846	37	23	clarke	clarke	PROPN
ejpam-6846	37	24	subdifferential	subdifferential	PROPN
ejpam-6846	37	25	(	(	PUNCT
ejpam-6846	37	26	see	see	VERB
ejpam-6846	37	27	[	[	X
ejpam-6846	37	28	12	12	NUM
ejpam-6846	37	29	]	]	PUNCT
ejpam-6846	37	30	)	)	PUNCT
ejpam-6846	37	31	.	.	PUNCT
ejpam-6846	38	1	(	(	PUNCT
ejpam-6846	38	2	i	i	NOUN
ejpam-6846	38	3	)	)	PUNCT
ejpam-6846	38	4	let	let	VERB
ejpam-6846	38	5	u	u	PRON
ejpam-6846	38	6	∈	∈	PROPN
ejpam-6846	38	7	x	x	PUNCT
ejpam-6846	38	8	and	and	CCONJ
ejpam-6846	38	9	consider	consider	VERB
ejpam-6846	38	10	f	f	PRON
ejpam-6846	38	11	to	to	PART
ejpam-6846	38	12	be	be	AUX
ejpam-6846	38	13	a	a	DET
ejpam-6846	38	14	convex	convex	NOUN
ejpam-6846	38	15	and	and	CCONJ
ejpam-6846	38	16	continuous	continuous	ADJ
ejpam-6846	38	17	function	function	NOUN
ejpam-6846	38	18	defined	define	VERB
ejpam-6846	38	19	on	on	ADP
ejpam-6846	38	20	x.	x.	NOUN
ejpam-6846	38	21	the	the	DET
ejpam-6846	38	22	convex	convex	PROPN
ejpam-6846	38	23	subdifferential	subdifferential	NOUN
ejpam-6846	38	24	of	of	ADP
ejpam-6846	38	25	f	f	PROPN
ejpam-6846	38	26	at	at	ADP
ejpam-6846	38	27	u	u	NOUN
ejpam-6846	38	28	is	be	AUX
ejpam-6846	38	29	given	give	VERB
ejpam-6846	38	30	by	by	ADP
ejpam-6846	38	31	the	the	DET
ejpam-6846	38	32	set	set	NOUN
ejpam-6846	38	33	:	:	PUNCT
ejpam-6846	38	34	∂conf(u	∂conf(u	X
ejpam-6846	38	35	)	)	PUNCT
ejpam-6846	38	36	=	=	PRON
ejpam-6846	39	1	{	{	PUNCT
ejpam-6846	39	2	u∗	u∗	NOUN
ejpam-6846	39	3	∈	∈	PROPN
ejpam-6846	39	4	x∗	x∗	PROPN
ejpam-6846	39	5	|	|	ADV
ejpam-6846	39	6	〈	〈	PROPN
ejpam-6846	39	7	u∗	u∗	PROPN
ejpam-6846	39	8	,	,	PUNCT
ejpam-6846	39	9	u−	u−	PROPN
ejpam-6846	39	10	v	v	PROPN
ejpam-6846	39	11	〉	〉	PROPN
ejpam-6846	39	12	≥	≥	NOUN
ejpam-6846	39	13	f(u)−	f(u)−	PROPN
ejpam-6846	39	14	f(v	f(v	NOUN
ejpam-6846	39	15	)	)	PUNCT
ejpam-6846	39	16	,	,	PUNCT
ejpam-6846	39	17	∀v	∀v	PROPN
ejpam-6846	39	18	∈	∈	PROPN
ejpam-6846	39	19	x	x	X
ejpam-6846	39	20	}	}	PUNCT
ejpam-6846	39	21	.	.	PUNCT
ejpam-6846	40	1	(	(	PUNCT
ejpam-6846	40	2	ii	ii	X
ejpam-6846	40	3	)	)	PUNCT
ejpam-6846	40	4	we	we	PRON
ejpam-6846	40	5	say	say	VERB
ejpam-6846	41	1	that	that	DET
ejpam-6846	41	2	u∗	u∗	PROPN
ejpam-6846	41	3	∈	∈	PROPN
ejpam-6846	41	4	∂f	∂f	PROPN
ejpam-6846	41	5	f(u	f(u	PROPN
ejpam-6846	41	6	)	)	PUNCT
ejpam-6846	41	7	iff	iff	NOUN
ejpam-6846	41	8	,	,	PUNCT
ejpam-6846	41	9	for	for	ADP
ejpam-6846	41	10	every	every	DET
ejpam-6846	41	11	ϵ	ϵ	PROPN
ejpam-6846	41	12	>	>	X
ejpam-6846	41	13	0	0	PROPN
ejpam-6846	41	14	,	,	PUNCT
ejpam-6846	41	15	∃	∃	PROPN
ejpam-6846	41	16	δ	δ	PROPN
ejpam-6846	41	17	>	>	X
ejpam-6846	41	18	0	0	PUNCT
ejpam-6846	42	1	s.t	s.t	PROPN
ejpam-6846	42	2	〈	〈	PROPN
ejpam-6846	42	3	u∗	u∗	PROPN
ejpam-6846	42	4	,	,	PUNCT
ejpam-6846	42	5	v	v	ADP
ejpam-6846	42	6	−	−	PROPN
ejpam-6846	42	7	u	u	NOUN
ejpam-6846	42	8	〉	〉	PROPN
ejpam-6846	42	9	≤	≤	PROPN
ejpam-6846	42	10	f(v)−	f(v)−	PROPN
ejpam-6846	42	11	f(u	f(u	PROPN
ejpam-6846	42	12	)	)	PUNCT
ejpam-6846	43	1	+	+	NUM
ejpam-6846	43	2	ϵ‖u−	ϵ‖u−	NUM
ejpam-6846	43	3	v‖	v‖	NOUN
ejpam-6846	43	4	,	,	PUNCT
ejpam-6846	43	5	∀v	∀v	PROPN
ejpam-6846	43	6	∈	∈	PROPN
ejpam-6846	43	7	u+	u+	PRON
ejpam-6846	43	8	δb	δb	NOUN
ejpam-6846	43	9	.	.	PUNCT
ejpam-6846	44	1	moreover	moreover	ADV
ejpam-6846	44	2	,	,	PUNCT
ejpam-6846	44	3	if	if	SCONJ
ejpam-6846	44	4	f	f	PROPN
ejpam-6846	44	5	is	be	AUX
ejpam-6846	44	6	a	a	DET
ejpam-6846	44	7	convex	convex	NOUN
ejpam-6846	44	8	extended	extend	VERB
ejpam-6846	44	9	real	real	ADV
ejpam-6846	44	10	-	-	PUNCT
ejpam-6846	44	11	valued	value	VERB
ejpam-6846	44	12	functional	functional	NOUN
ejpam-6846	44	13	that	that	PRON
ejpam-6846	44	14	is	be	AUX
ejpam-6846	44	15	lower	low	ADJ
ejpam-6846	44	16	semi	semi	ADJ
ejpam-6846	44	17	-	-	ADJ
ejpam-6846	44	18	continuous	continuous	ADJ
ejpam-6846	44	19	and	and	CCONJ
ejpam-6846	44	20	defined	define	VERB
ejpam-6846	44	21	at	at	ADP
ejpam-6846	44	22	u	u	PROPN
ejpam-6846	44	23	∈	∈	PROPN
ejpam-6846	44	24	s	s	NOUN
ejpam-6846	44	25	,	,	PUNCT
ejpam-6846	44	26	then	then	ADV
ejpam-6846	44	27	the	the	DET
ejpam-6846	44	28	fréchet	fréchet	ADJ
ejpam-6846	44	29	normal	normal	ADJ
ejpam-6846	44	30	cone	cone	NOUN
ejpam-6846	44	31	of	of	ADP
ejpam-6846	44	32	a	a	DET
ejpam-6846	44	33	nonempty	nonempty	ADV
ejpam-6846	44	34	closed	close	VERB
ejpam-6846	44	35	set	set	NOUN
ejpam-6846	44	36	s	s	PART
ejpam-6846	44	37	is	be	AUX
ejpam-6846	44	38	given	give	VERB
ejpam-6846	44	39	by	by	ADP
ejpam-6846	44	40	a.	a.	PROPN
ejpam-6846	44	41	al	al	PROPN
ejpam-6846	44	42	tane	tane	PROPN
ejpam-6846	44	43	,	,	PUNCT
ejpam-6846	44	44	l.	l.	PROPN
ejpam-6846	44	45	s.	s.	PROPN
ejpam-6846	44	46	keong	keong	PROPN
ejpam-6846	44	47	/	/	PUNCT
ejpam-6846	44	48	eur	eur	PROPN
ejpam-6846	44	49	.	.	PUNCT
ejpam-6846	45	1	j.	j.	PROPN
ejpam-6846	45	2	pure	pure	PROPN
ejpam-6846	45	3	appl	appl	PROPN
ejpam-6846	45	4	.	.	PROPN
ejpam-6846	45	5	math	math	PROPN
ejpam-6846	45	6	,	,	PUNCT
ejpam-6846	45	7	18	18	NUM
ejpam-6846	45	8	(	(	PUNCT
ejpam-6846	45	9	4	4	NUM
ejpam-6846	45	10	)	)	PUNCT
ejpam-6846	45	11	(	(	PUNCT
ejpam-6846	45	12	2025	2025	NUM
ejpam-6846	45	13	)	)	PUNCT
ejpam-6846	45	14	,	,	PUNCT
ejpam-6846	45	15	6846	6846	NUM
ejpam-6846	45	16	3	3	NUM
ejpam-6846	45	17	of	of	ADP
ejpam-6846	45	18	13	13	NUM
ejpam-6846	45	19	nf	nf	NOUN
ejpam-6846	45	20	(	(	PUNCT
ejpam-6846	45	21	s	s	PROPN
ejpam-6846	45	22	,	,	PUNCT
ejpam-6846	45	23	u	u	NOUN
ejpam-6846	45	24	)	)	PUNCT
ejpam-6846	45	25	=	=	PRON
ejpam-6846	45	26	{	{	PUNCT
ejpam-6846	45	27	u∗	u∗	PROPN
ejpam-6846	45	28	∈	∈	PROPN
ejpam-6846	45	29	x∗	x∗	NOUN
ejpam-6846	46	1	|	|	ADV
ejpam-6846	46	2	∀ϵ	∀ϵ	VERB
ejpam-6846	46	3	>	>	X
ejpam-6846	46	4	0	0	PROPN
ejpam-6846	46	5	,	,	PUNCT
ejpam-6846	46	6	∃δ	∃δ	PROPN
ejpam-6846	46	7	>	>	SYM
ejpam-6846	46	8	0	0	NUM
ejpam-6846	47	1	s.t	s.t	PROPN
ejpam-6846	47	2	〈	〈	PROPN
ejpam-6846	47	3	u∗	u∗	PROPN
ejpam-6846	47	4	,	,	PUNCT
ejpam-6846	47	5	v	v	ADP
ejpam-6846	47	6	−	−	PROPN
ejpam-6846	47	7	u	u	NOUN
ejpam-6846	47	8	〉	〉	NOUN
ejpam-6846	47	9	≤	≤	PROPN
ejpam-6846	47	10	ϵ‖u−	ϵ‖u−	PRON
ejpam-6846	47	11	v‖	v‖	NOUN
ejpam-6846	47	12	,	,	PUNCT
ejpam-6846	47	13	∀v	∀v	PROPN
ejpam-6846	47	14	∈	∈	PROPN
ejpam-6846	47	15	u+	u+	NOUN
ejpam-6846	47	16	δb	δb	NOUN
ejpam-6846	47	17	}	}	PUNCT
ejpam-6846	47	18	.	.	PUNCT
ejpam-6846	48	1	(	(	PUNCT
ejpam-6846	48	2	iii	iii	X
ejpam-6846	48	3	)	)	PUNCT
ejpam-6846	48	4	we	we	PRON
ejpam-6846	48	5	say	say	VERB
ejpam-6846	48	6	that	that	DET
ejpam-6846	48	7	u∗	u∗	PROPN
ejpam-6846	48	8	∈	∈	PROPN
ejpam-6846	48	9	∂cf(u	∂cf(u	NOUN
ejpam-6846	48	10	)	)	PUNCT
ejpam-6846	48	11	iff	iff	PROPN
ejpam-6846	48	12	〈	〈	PROPN
ejpam-6846	48	13	u∗	u∗	PROPN
ejpam-6846	48	14	,	,	PUNCT
ejpam-6846	48	15	u	u	NOUN
ejpam-6846	48	16	〉	〉	NOUN
ejpam-6846	48	17	≤	≤	X
ejpam-6846	48	18	lim	lim	PROPN
ejpam-6846	48	19	t→0,v→u	t→0,v→u	NUM
ejpam-6846	48	20	sup	sup	NUM
ejpam-6846	48	21	t−1[f(v	t−1[f(v	NOUN
ejpam-6846	48	22	+	+	CCONJ
ejpam-6846	48	23	tu)−	tu)−	NOUN
ejpam-6846	48	24	f(v	f(v	NOUN
ejpam-6846	48	25	)	)	PUNCT
ejpam-6846	48	26	]	]	PUNCT
ejpam-6846	48	27	,	,	PUNCT
ejpam-6846	48	28	u	u	PROPN
ejpam-6846	48	29	∈	∈	PROPN
ejpam-6846	48	30	x.	x.	NOUN
ejpam-6846	48	31	definition	definition	NOUN
ejpam-6846	48	32	1	1	NUM
ejpam-6846	48	33	.	.	PUNCT
ejpam-6846	49	1	let	let	VERB
ejpam-6846	49	2	x	x	PRON
ejpam-6846	49	3	be	be	AUX
ejpam-6846	49	4	a	a	DET
ejpam-6846	49	5	banach	banach	NOUN
ejpam-6846	49	6	space	space	NOUN
ejpam-6846	49	7	with	with	ADP
ejpam-6846	49	8	dual	dual	ADJ
ejpam-6846	49	9	space	space	NOUN
ejpam-6846	49	10	x∗	x∗	PROPN
ejpam-6846	49	11	and	and	CCONJ
ejpam-6846	49	12	let	let	VERB
ejpam-6846	49	13	f	f	NOUN
ejpam-6846	49	14	:	:	PUNCT
ejpam-6846	49	15	x	x	X
ejpam-6846	49	16	→	→	SYM
ejpam-6846	49	17	r	r	NOUN
ejpam-6846	49	18	∪	∪	X
ejpam-6846	49	19	{	{	PUNCT
ejpam-6846	49	20	∞	∞	NOUN
ejpam-6846	49	21	}	}	PUNCT
ejpam-6846	49	22	is	be	AUX
ejpam-6846	49	23	proper	proper	ADJ
ejpam-6846	49	24	function	function	NOUN
ejpam-6846	49	25	.	.	PUNCT
ejpam-6846	50	1	then	then	ADV
ejpam-6846	50	2	we	we	PRON
ejpam-6846	50	3	define	define	VERB
ejpam-6846	50	4	v	v	ADP
ejpam-6846	50	5	f	f	NOUN
ejpam-6846	50	6	:	:	PUNCT
ejpam-6846	50	7	x∗	x∗	PROPN
ejpam-6846	50	8	×x	×x	VERB
ejpam-6846	50	9	→	→	SYM
ejpam-6846	50	10	r	r	NOUN
ejpam-6846	50	11	∪	∪	X
ejpam-6846	50	12	{	{	PUNCT
ejpam-6846	50	13	∞	∞	NOUN
ejpam-6846	50	14	}	}	PUNCT
ejpam-6846	50	15	as	as	ADP
ejpam-6846	50	16	v	v	NOUN
ejpam-6846	50	17	f	f	X
ejpam-6846	50	18	(	(	PUNCT
ejpam-6846	50	19	u∗	u∗	PROPN
ejpam-6846	50	20	,	,	PUNCT
ejpam-6846	50	21	u	u	NOUN
ejpam-6846	50	22	)	)	PUNCT
ejpam-6846	50	23	=	=	SYM
ejpam-6846	50	24	‖u∗‖2	‖u∗‖2	PROPN
ejpam-6846	51	1	+	+	CCONJ
ejpam-6846	51	2	‖u‖2	‖u‖2	ADJ
ejpam-6846	51	3	−	−	PROPN
ejpam-6846	51	4	2〈u∗	2〈u∗	NUM
ejpam-6846	51	5	,	,	PUNCT
ejpam-6846	51	6	u〉+	u〉+	PROPN
ejpam-6846	51	7	f(u	f(u	PROPN
ejpam-6846	51	8	)	)	PUNCT
ejpam-6846	51	9	,	,	PUNCT
ejpam-6846	51	10	u∗	u∗	PROPN
ejpam-6846	51	11	∈	∈	PROPN
ejpam-6846	51	12	x∗	x∗	NOUN
ejpam-6846	51	13	,	,	PUNCT
ejpam-6846	51	14	u	u	PROPN
ejpam-6846	51	15	∈	∈	NOUN
ejpam-6846	51	16	x.	x.	NOUN
ejpam-6846	52	1	it	it	PRON
ejpam-6846	52	2	’s	’	VERB
ejpam-6846	52	3	straight	straight	ADV
ejpam-6846	52	4	to	to	PART
ejpam-6846	52	5	demonstrate	demonstrate	VERB
ejpam-6846	52	6	that	that	SCONJ
ejpam-6846	52	7	f(u	f(u	PROPN
ejpam-6846	52	8	)	)	PUNCT
ejpam-6846	53	1	+	+	CCONJ
ejpam-6846	53	2	(	(	PUNCT
ejpam-6846	53	3	‖u∗	‖u∗	NOUN
ejpam-6846	53	4	−	−	VERB
ejpam-6846	53	5	u‖)2	u‖)2	INTJ
ejpam-6846	53	6	≤	≤	NOUN
ejpam-6846	53	7	v	v	NUM
ejpam-6846	53	8	f	f	X
ejpam-6846	53	9	(	(	PUNCT
ejpam-6846	53	10	u∗	u∗	PROPN
ejpam-6846	53	11	,	,	PUNCT
ejpam-6846	53	12	u	u	NOUN
ejpam-6846	53	13	)	)	PUNCT
ejpam-6846	53	14	≤	≤	PROPN
ejpam-6846	53	15	f(u	f(u	PROPN
ejpam-6846	53	16	)	)	PUNCT
ejpam-6846	53	17	+	+	CCONJ
ejpam-6846	53	18	(	(	PUNCT
ejpam-6846	53	19	‖u∗	‖u∗	NOUN
ejpam-6846	53	20	+	+	CCONJ
ejpam-6846	53	21	u‖)2	u‖)2	ADJ
ejpam-6846	53	22	.	.	PUNCT
ejpam-6846	54	1	(	(	PUNCT
ejpam-6846	54	2	1	1	X
ejpam-6846	54	3	)	)	PUNCT
ejpam-6846	54	4	definition	definition	NOUN
ejpam-6846	54	5	2	2	NUM
ejpam-6846	54	6	.	.	PUNCT
ejpam-6846	54	7	given	give	VERB
ejpam-6846	54	8	a	a	DET
ejpam-6846	54	9	reflexive	reflexive	ADJ
ejpam-6846	54	10	banach	banach	NOUN
ejpam-6846	54	11	space	space	NOUN
ejpam-6846	54	12	x.	x.	NOUN
ejpam-6846	54	13	let	let	VERB
ejpam-6846	54	14	s	s	PRON
ejpam-6846	54	15	be	be	AUX
ejpam-6846	54	16	a	a	DET
ejpam-6846	54	17	nonempty	nonempty	ADJ
ejpam-6846	54	18	closed	close	VERB
ejpam-6846	54	19	subset	subset	NOUN
ejpam-6846	54	20	of	of	ADP
ejpam-6846	54	21	x.	x.	NOUN
ejpam-6846	54	22	we	we	PRON
ejpam-6846	54	23	denote	denote	VERB
ejpam-6846	54	24	by	by	ADP
ejpam-6846	54	25	mf	mf	NOUN
ejpam-6846	54	26	,	,	PUNCT
ejpam-6846	54	27	s(x	s(x	NOUN
ejpam-6846	54	28	∗	∗	NOUN
ejpam-6846	54	29	)	)	PUNCT
ejpam-6846	54	30	:	:	PUNCT
ejpam-6846	54	31	=	=	SYM
ejpam-6846	54	32	inf	inf	PROPN
ejpam-6846	54	33	v∈s	v∈s	NOUN
ejpam-6846	54	34	v	v	ADP
ejpam-6846	54	35	f	f	X
ejpam-6846	54	36	(	(	PUNCT
ejpam-6846	54	37	x∗	x∗	PROPN
ejpam-6846	54	38	,	,	PUNCT
ejpam-6846	54	39	v	v	NOUN
ejpam-6846	54	40	)	)	PUNCT
ejpam-6846	54	41	,	,	PUNCT
ejpam-6846	54	42	and	and	CCONJ
ejpam-6846	54	43	we	we	PRON
ejpam-6846	54	44	define	define	VERB
ejpam-6846	54	45	the	the	DET
ejpam-6846	54	46	operator	operator	NOUN
ejpam-6846	54	47	πf	πf	NOUN
ejpam-6846	54	48	s	s	PART
ejpam-6846	54	49	:	:	PUNCT
ejpam-6846	54	50	x∗−→	x∗−→	PROPN
ejpam-6846	54	51	−→x	−→x	NUM
ejpam-6846	54	52	as	as	ADP
ejpam-6846	54	53	πf	πf	INTJ
ejpam-6846	54	54	s(u	s(u	PROPN
ejpam-6846	54	55	∗	∗	NOUN
ejpam-6846	54	56	)	)	PUNCT
ejpam-6846	54	57	=	=	PRON
ejpam-6846	54	58	{	{	PUNCT
ejpam-6846	54	59	u	u	NOUN
ejpam-6846	54	60	∈	∈	PROPN
ejpam-6846	54	61	s	s	PART
ejpam-6846	54	62	:	:	PUNCT
ejpam-6846	54	63	v	v	NUM
ejpam-6846	54	64	f	f	X
ejpam-6846	54	65	(	(	PUNCT
ejpam-6846	54	66	u∗	u∗	PROPN
ejpam-6846	54	67	,	,	PUNCT
ejpam-6846	54	68	u	u	NOUN
ejpam-6846	54	69	)	)	PUNCT
ejpam-6846	54	70	=	=	PUNCT
ejpam-6846	54	71	mf	mf	X
ejpam-6846	54	72	,	,	PUNCT
ejpam-6846	54	73	s(u	s(u	PROPN
ejpam-6846	54	74	∗	∗	NOUN
ejpam-6846	54	75	)	)	PUNCT
ejpam-6846	54	76	}	}	PUNCT
ejpam-6846	54	77	∀u∗	∀u∗	PROPN
ejpam-6846	54	78	∈	∈	PROPN
ejpam-6846	54	79	x∗.	x∗.	PROPN
ejpam-6846	55	1	this	this	DET
ejpam-6846	55	2	operator	operator	NOUN
ejpam-6846	55	3	is	be	AUX
ejpam-6846	55	4	called	call	VERB
ejpam-6846	55	5	a	a	DET
ejpam-6846	55	6	generalized	generalized	ADJ
ejpam-6846	55	7	f	f	PROPN
ejpam-6846	55	8	-projection	-projection	PROPN
ejpam-6846	55	9	.	.	PUNCT
ejpam-6846	56	1	if	if	SCONJ
ejpam-6846	56	2	f(x	f(x	PROPN
ejpam-6846	56	3	)	)	PUNCT
ejpam-6846	57	1	=	=	SYM
ejpam-6846	57	2	0	0	NUM
ejpam-6846	58	1	for	for	ADP
ejpam-6846	58	2	every	every	DET
ejpam-6846	58	3	x	x	SYM
ejpam-6846	58	4	∈	∈	PROPN
ejpam-6846	58	5	x	x	NOUN
ejpam-6846	58	6	,	,	PUNCT
ejpam-6846	58	7	then	then	ADV
ejpam-6846	58	8	πf	πf	INTJ
ejpam-6846	58	9	s(x	s(x	PROPN
ejpam-6846	58	10	∗	∗	NOUN
ejpam-6846	58	11	)	)	PUNCT
ejpam-6846	58	12	coincides	coincide	VERB
ejpam-6846	58	13	with	with	ADP
ejpam-6846	58	14	the	the	DET
ejpam-6846	58	15	generalized	generalized	ADJ
ejpam-6846	58	16	projection	projection	NOUN
ejpam-6846	58	17	πs(x	πs(x	PUNCT
ejpam-6846	58	18	∗	∗	NOUN
ejpam-6846	58	19	)	)	PUNCT
ejpam-6846	58	20	,	,	PUNCT
ejpam-6846	58	21	which	which	PRON
ejpam-6846	58	22	was	be	AUX
ejpam-6846	58	23	introduced	introduce	VERB
ejpam-6846	58	24	and	and	CCONJ
ejpam-6846	58	25	analyzed	analyze	VERB
ejpam-6846	58	26	in	in	ADP
ejpam-6846	58	27	alber	alber	PROPN
ejpam-6846	59	1	[	[	X
ejpam-6846	59	2	1	1	NUM
ejpam-6846	59	3	]	]	PUNCT
ejpam-6846	59	4	and	and	CCONJ
ejpam-6846	59	5	li	li	X
ejpam-6846	60	1	[	[	X
ejpam-6846	60	2	2	2	NUM
ejpam-6846	60	3	]	]	PUNCT
ejpam-6846	60	4	for	for	ADP
ejpam-6846	60	5	closed	closed	ADJ
ejpam-6846	60	6	convex	convex	NOUN
ejpam-6846	60	7	sets	set	NOUN
ejpam-6846	60	8	and	and	CCONJ
ejpam-6846	60	9	by	by	ADP
ejpam-6846	60	10	[	[	X
ejpam-6846	60	11	5	5	NUM
ejpam-6846	60	12	,	,	PUNCT
ejpam-6846	60	13	6	6	NUM
ejpam-6846	60	14	]	]	PUNCT
ejpam-6846	60	15	for	for	ADP
ejpam-6846	60	16	nonempty	nonempty	ADV
ejpam-6846	60	17	closed	close	VERB
ejpam-6846	60	18	(	(	PUNCT
ejpam-6846	60	19	and	and	CCONJ
ejpam-6846	60	20	not	not	PART
ejpam-6846	60	21	necessarily	necessarily	ADV
ejpam-6846	60	22	convex	convex	VERB
ejpam-6846	60	23	)	)	PUNCT
ejpam-6846	60	24	sets	set	NOUN
ejpam-6846	60	25	.	.	PUNCT
ejpam-6846	61	1	we	we	PRON
ejpam-6846	61	2	would	would	AUX
ejpam-6846	61	3	want	want	VERB
ejpam-6846	61	4	to	to	PART
ejpam-6846	61	5	draw	draw	VERB
ejpam-6846	61	6	attention	attention	NOUN
ejpam-6846	61	7	to	to	ADP
ejpam-6846	61	8	the	the	DET
ejpam-6846	61	9	fact	fact	NOUN
ejpam-6846	61	10	that	that	SCONJ
ejpam-6846	61	11	,	,	PUNCT
ejpam-6846	61	12	in	in	ADP
ejpam-6846	61	13	some	some	DET
ejpam-6846	61	14	situations	situation	NOUN
ejpam-6846	61	15	,	,	PUNCT
ejpam-6846	61	16	it	it	PRON
ejpam-6846	61	17	is	be	AUX
ejpam-6846	61	18	possible	possible	ADJ
ejpam-6846	61	19	to	to	PART
ejpam-6846	61	20	find	find	VERB
ejpam-6846	61	21	a	a	DET
ejpam-6846	61	22	function	function	NOUN
ejpam-6846	61	23	f(x	f(x	PROPN
ejpam-6846	61	24	)	)	PUNCT
ejpam-6846	61	25	6=	6=	ADP
ejpam-6846	61	26	0	0	NUM
ejpam-6846	61	27	for	for	ADP
ejpam-6846	61	28	some	some	DET
ejpam-6846	61	29	x	x	SYM
ejpam-6846	61	30	∈	∈	PROPN
ejpam-6846	61	31	x	x	SYM
ejpam-6846	61	32	s.t	s.t	PROPN
ejpam-6846	61	33	πs(x	πs(x	NUM
ejpam-6846	61	34	∗	∗	NOUN
ejpam-6846	61	35	)	)	PUNCT
ejpam-6846	62	1	=	=	SYM
ejpam-6846	62	2	πf	πf	NUM
ejpam-6846	62	3	s(x	s(x	PROPN
ejpam-6846	62	4	∗	∗	NOUN
ejpam-6846	62	5	)	)	PUNCT
ejpam-6846	62	6	,	,	PUNCT
ejpam-6846	62	7	as	as	SCONJ
ejpam-6846	62	8	demonstrated	demonstrate	VERB
ejpam-6846	62	9	in	in	ADP
ejpam-6846	62	10	the	the	DET
ejpam-6846	62	11	example	example	NOUN
ejpam-6846	62	12	that	that	PRON
ejpam-6846	62	13	follows	follow	VERB
ejpam-6846	62	14	.	.	PUNCT
ejpam-6846	63	1	example	example	NOUN
ejpam-6846	64	1	1	1	NUM
ejpam-6846	64	2	.	.	PUNCT
ejpam-6846	65	1	jinlu	jinlu	PROPN
ejpam-6846	65	2	li	li	PROPN
ejpam-6846	66	1	[	[	X
ejpam-6846	66	2	2	2	NUM
ejpam-6846	66	3	]	]	PUNCT
ejpam-6846	66	4	proved	prove	VERB
ejpam-6846	66	5	in	in	ADP
ejpam-6846	66	6	the	the	DET
ejpam-6846	66	7	example	example	NOUN
ejpam-6846	66	8	(	(	PUNCT
ejpam-6846	66	9	1.2	1.2	NUM
ejpam-6846	66	10	)	)	PUNCT
ejpam-6846	67	1	that	that	SCONJ
ejpam-6846	67	2	for	for	ADP
ejpam-6846	67	3	x	x	PROPN
ejpam-6846	67	4	=	=	PROPN
ejpam-6846	67	5	l1	l1	PROPN
ejpam-6846	67	6	,	,	PUNCT
ejpam-6846	67	7	and	and	CCONJ
ejpam-6846	67	8	x∗	x∗	PROPN
ejpam-6846	67	9	=	=	SYM
ejpam-6846	67	10	l∞	l∞	PROPN
ejpam-6846	67	11	the	the	DET
ejpam-6846	67	12	πs(0	πs(0	NOUN
ejpam-6846	67	13	)	)	PUNCT
ejpam-6846	67	14	=	=	SYM
ejpam-6846	67	15	co{u1	co{u1	NOUN
ejpam-6846	67	16	,	,	PUNCT
ejpam-6846	67	17	u2	u2	NOUN
ejpam-6846	67	18	}	}	PUNCT
ejpam-6846	67	19	,	,	PUNCT
ejpam-6846	67	20	where	where	SCONJ
ejpam-6846	67	21	0	0	X
ejpam-6846	67	22	=	=	SYM
ejpam-6846	67	23	(	(	PUNCT
ejpam-6846	67	24	0	0	NUM
ejpam-6846	67	25	,	,	PUNCT
ejpam-6846	67	26	0	0	NUM
ejpam-6846	67	27	,	,	PUNCT
ejpam-6846	67	28	0	0	NUM
ejpam-6846	67	29	,	,	PUNCT
ejpam-6846	67	30	·	·	PUNCT
ejpam-6846	67	31	·	·	PUNCT
ejpam-6846	67	32	·	·	PUNCT
ejpam-6846	67	33	)	)	PUNCT
ejpam-6846	67	34	,	,	PUNCT
ejpam-6846	67	35	u1	u1	NOUN
ejpam-6846	67	36	=	=	SYM
ejpam-6846	67	37	(	(	PUNCT
ejpam-6846	67	38	1	1	NUM
ejpam-6846	67	39	,	,	PUNCT
ejpam-6846	67	40	1	1	NUM
ejpam-6846	67	41	,	,	PUNCT
ejpam-6846	67	42	0	0	NUM
ejpam-6846	67	43	,	,	PUNCT
ejpam-6846	67	44	0	0	NUM
ejpam-6846	67	45	,	,	PUNCT
ejpam-6846	67	46	·	·	PUNCT
ejpam-6846	67	47	·	·	PUNCT
ejpam-6846	67	48	·	·	PUNCT
ejpam-6846	67	49	)	)	PUNCT
ejpam-6846	67	50	,	,	PUNCT
ejpam-6846	67	51	u2	u2	NOUN
ejpam-6846	67	52	=	=	PUNCT
ejpam-6846	67	53	(	(	PUNCT
ejpam-6846	67	54	1	1	NUM
ejpam-6846	67	55	,	,	PUNCT
ejpam-6846	67	56	0	0	NUM
ejpam-6846	67	57	,	,	PUNCT
ejpam-6846	67	58	1	1	NUM
ejpam-6846	67	59	,	,	PUNCT
ejpam-6846	67	60	0	0	NUM
ejpam-6846	67	61	,	,	PUNCT
ejpam-6846	67	62	·	·	PUNCT
ejpam-6846	67	63	·	·	PUNCT
ejpam-6846	67	64	·	·	PUNCT
ejpam-6846	67	65	)	)	PUNCT
ejpam-6846	67	66	,	,	PUNCT
ejpam-6846	67	67	and	and	CCONJ
ejpam-6846	67	68	u3	u3	NOUN
ejpam-6846	67	69	=	=	SYM
ejpam-6846	67	70	(	(	PUNCT
ejpam-6846	67	71	2	2	NUM
ejpam-6846	67	72	,	,	PUNCT
ejpam-6846	67	73	0	0	NUM
ejpam-6846	67	74	,	,	PUNCT
ejpam-6846	67	75	0	0	NUM
ejpam-6846	67	76	,	,	PUNCT
ejpam-6846	67	77	1	1	NUM
ejpam-6846	67	78	,	,	PUNCT
ejpam-6846	67	79	0	0	NUM
ejpam-6846	67	80	,	,	PUNCT
ejpam-6846	67	81	·	·	PUNCT
ejpam-6846	67	82	·	·	PUNCT
ejpam-6846	67	83	·	·	PUNCT
ejpam-6846	67	84	)	)	PUNCT
ejpam-6846	68	1	∈	∈	PROPN
ejpam-6846	68	2	l1	l1	PROPN
ejpam-6846	68	3	with	with	ADP
ejpam-6846	68	4	s	s	NOUN
ejpam-6846	68	5	=	=	NOUN
ejpam-6846	68	6	co{u1	co{u1	NOUN
ejpam-6846	68	7	,	,	PUNCT
ejpam-6846	68	8	u2	u2	NOUN
ejpam-6846	68	9	,	,	PUNCT
ejpam-6846	68	10	u3	u3	NOUN
ejpam-6846	68	11	}	}	PUNCT
ejpam-6846	68	12	.	.	PUNCT
ejpam-6846	69	1	for	for	ADP
ejpam-6846	69	2	us	we	PRON
ejpam-6846	69	3	we	we	PRON
ejpam-6846	69	4	define	define	VERB
ejpam-6846	69	5	f(u	f(u	PROPN
ejpam-6846	69	6	)	)	PUNCT
ejpam-6846	69	7	=	=	PUNCT
ejpam-6846	70	1	‖u‖2	‖u‖2	ADJ
ejpam-6846	70	2	for	for	ADP
ejpam-6846	70	3	all	all	DET
ejpam-6846	70	4	u	u	PROPN
ejpam-6846	70	5	∈	∈	PROPN
ejpam-6846	70	6	l1	l1	PROPN
ejpam-6846	70	7	.	.	PUNCT
ejpam-6846	71	1	then	then	ADV
ejpam-6846	71	2	v	v	X
ejpam-6846	71	3	f	f	X
ejpam-6846	71	4	(	(	PUNCT
ejpam-6846	71	5	0	0	NUM
ejpam-6846	71	6	,	,	PUNCT
ejpam-6846	71	7	u1	u1	NOUN
ejpam-6846	71	8	)	)	PUNCT
ejpam-6846	71	9	=	=	PUNCT
ejpam-6846	72	1	v	v	NUM
ejpam-6846	72	2	f	f	X
ejpam-6846	72	3	(	(	PUNCT
ejpam-6846	72	4	0	0	NUM
ejpam-6846	72	5	,	,	PUNCT
ejpam-6846	72	6	u2	u2	NOUN
ejpam-6846	72	7	)	)	PUNCT
ejpam-6846	72	8	=	=	SYM
ejpam-6846	72	9	8	8	NUM
ejpam-6846	72	10	,	,	PUNCT
ejpam-6846	72	11	v	v	NOUN
ejpam-6846	72	12	f	f	X
ejpam-6846	72	13	(	(	PUNCT
ejpam-6846	72	14	0	0	NUM
ejpam-6846	72	15	,	,	PUNCT
ejpam-6846	72	16	u3	u3	NOUN
ejpam-6846	72	17	)	)	PUNCT
ejpam-6846	72	18	=	=	SYM
ejpam-6846	72	19	18	18	NUM
ejpam-6846	72	20	.	.	PUNCT
ejpam-6846	72	21	now	now	ADV
ejpam-6846	72	22	let	let	VERB
ejpam-6846	72	23	µ	µ	X
ejpam-6846	72	24	∈	∈	NOUN
ejpam-6846	72	25	[	[	X
ejpam-6846	72	26	0	0	NUM
ejpam-6846	72	27	,	,	PUNCT
ejpam-6846	72	28	1	1	NUM
ejpam-6846	72	29	]	]	PUNCT
ejpam-6846	72	30	and	and	CCONJ
ejpam-6846	72	31	v	v	ADP
ejpam-6846	72	32	∈	∈	NOUN
ejpam-6846	72	33	co{u1	co{u1	NOUN
ejpam-6846	72	34	,	,	PUNCT
ejpam-6846	72	35	u2	u2	NOUN
ejpam-6846	72	36	}	}	PUNCT
ejpam-6846	72	37	.	.	PUNCT
ejpam-6846	73	1	then	then	ADV
ejpam-6846	73	2	v	v	X
ejpam-6846	73	3	=	=	PUNCT
ejpam-6846	73	4	µu1	µu1	NOUN
ejpam-6846	73	5	+	+	CCONJ
ejpam-6846	73	6	(	(	PUNCT
ejpam-6846	73	7	1−	1−	NUM
ejpam-6846	73	8	µ)u2	µ)u2	PROPN
ejpam-6846	73	9	with	with	ADP
ejpam-6846	73	10	v	v	PROPN
ejpam-6846	73	11	f	f	X
ejpam-6846	73	12	(	(	PUNCT
ejpam-6846	73	13	0	0	NUM
ejpam-6846	73	14	,	,	PUNCT
ejpam-6846	73	15	v	v	NOUN
ejpam-6846	73	16	)	)	PUNCT
ejpam-6846	73	17	=	=	SYM
ejpam-6846	73	18	2‖(1	2‖(1	NUM
ejpam-6846	73	19	,	,	PUNCT
ejpam-6846	73	20	µ	µ	NOUN
ejpam-6846	73	21	,	,	PUNCT
ejpam-6846	73	22	1−	1−	NUM
ejpam-6846	73	23	µ	µ	NUM
ejpam-6846	73	24	,	,	PUNCT
ejpam-6846	73	25	0	0	NUM
ejpam-6846	73	26	,	,	PUNCT
ejpam-6846	73	27	·	·	PUNCT
ejpam-6846	73	28	·	·	PUNCT
ejpam-6846	73	29	·	·	PUNCT
ejpam-6846	73	30	)	)	PUNCT
ejpam-6846	73	31	‖2	‖2	NOUN
ejpam-6846	74	1	=	=	PUNCT
ejpam-6846	74	2	8	8	X
ejpam-6846	74	3	.	.	PUNCT
ejpam-6846	75	1	now	now	ADV
ejpam-6846	75	2	for	for	ADP
ejpam-6846	75	3	any	any	DET
ejpam-6846	75	4	y	y	PROPN
ejpam-6846	75	5	∈	∈	PROPN
ejpam-6846	75	6	s	s	PART
ejpam-6846	75	7	and	and	CCONJ
ejpam-6846	75	8	µj	µj	PROPN
ejpam-6846	75	9	∈	∈	PROPN
ejpam-6846	76	1	[	[	X
ejpam-6846	76	2	0	0	NUM
ejpam-6846	76	3	,	,	PUNCT
ejpam-6846	76	4	1	1	NUM
ejpam-6846	76	5	]	]	PUNCT
ejpam-6846	76	6	;	;	PUNCT
ejpam-6846	76	7	j	j	PROPN
ejpam-6846	76	8	=	=	SYM
ejpam-6846	76	9	1	1	NUM
ejpam-6846	76	10	,	,	PUNCT
ejpam-6846	76	11	2	2	NUM
ejpam-6846	76	12	,	,	PUNCT
ejpam-6846	76	13	3	3	NUM
ejpam-6846	76	14	with	with	ADP
ejpam-6846	76	15	3∑	3∑	NUM
ejpam-6846	76	16	j=1	j=1	NOUN
ejpam-6846	76	17	µj	µj	PROPN
ejpam-6846	76	18	=	=	SYM
ejpam-6846	76	19	1	1	NUM
ejpam-6846	76	20	,	,	PUNCT
ejpam-6846	76	21	then	then	ADV
ejpam-6846	76	22	y	y	PROPN
ejpam-6846	76	23	=	=	SYM
ejpam-6846	76	24	3∑	3∑	NUM
ejpam-6846	76	25	j=1	j=1	NOUN
ejpam-6846	76	26	µjuj	µjuj	ADV
ejpam-6846	76	27	.	.	PUNCT
ejpam-6846	77	1	hence	hence	ADV
ejpam-6846	77	2	v	v	X
ejpam-6846	77	3	f	f	X
ejpam-6846	77	4	(	(	PUNCT
ejpam-6846	77	5	0	0	NUM
ejpam-6846	77	6	,	,	PUNCT
ejpam-6846	77	7	y	y	NOUN
ejpam-6846	77	8	)	)	PUNCT
ejpam-6846	77	9	=	=	PUNCT
ejpam-6846	78	1	2‖(1	2‖(1	NUM
ejpam-6846	78	2	+	+	NUM
ejpam-6846	78	3	µ3	µ3	NOUN
ejpam-6846	78	4	,	,	PUNCT
ejpam-6846	78	5	µ1	µ1	PROPN
ejpam-6846	78	6	,	,	PUNCT
ejpam-6846	78	7	µ2	µ2	NOUN
ejpam-6846	78	8	,	,	PUNCT
ejpam-6846	78	9	µ3	µ3	NOUN
ejpam-6846	78	10	,	,	PUNCT
ejpam-6846	78	11	0	0	NUM
ejpam-6846	78	12	,	,	PUNCT
ejpam-6846	78	13	·	·	PUNCT
ejpam-6846	78	14	·	·	PUNCT
ejpam-6846	78	15	·	·	PUNCT
ejpam-6846	78	16	)	)	PUNCT
ejpam-6846	78	17	‖2	‖2	NOUN
ejpam-6846	79	1	=	=	PUNCT
ejpam-6846	80	1	2(2	2(2	NUM
ejpam-6846	80	2	+	+	NUM
ejpam-6846	80	3	µ3	µ3	NUM
ejpam-6846	80	4	)	)	PUNCT
ejpam-6846	80	5	2	2	NUM
ejpam-6846	80	6	≥	≥	NOUN
ejpam-6846	80	7	8	8	NUM
ejpam-6846	80	8	.	.	PUNCT
ejpam-6846	81	1	we	we	PRON
ejpam-6846	81	2	can	can	AUX
ejpam-6846	81	3	observe	observe	VERB
ejpam-6846	81	4	from	from	ADP
ejpam-6846	81	5	the	the	DET
ejpam-6846	81	6	aforementioned	aforementioned	ADJ
ejpam-6846	81	7	inequality	inequality	NOUN
ejpam-6846	81	8	that	that	SCONJ
ejpam-6846	81	9	v	v	ADP
ejpam-6846	81	10	f	f	X
ejpam-6846	81	11	(	(	PUNCT
ejpam-6846	81	12	0	0	NUM
ejpam-6846	81	13	,	,	PUNCT
ejpam-6846	81	14	y	y	NOUN
ejpam-6846	81	15	)	)	PUNCT
ejpam-6846	81	16	=	=	SYM
ejpam-6846	81	17	8	8	NUM
ejpam-6846	81	18	iff	iff	PROPN
ejpam-6846	81	19	µ3	µ3	NOUN
ejpam-6846	81	20	=	=	SYM
ejpam-6846	81	21	0	0	NUM
ejpam-6846	81	22	;	;	PUNCT
ejpam-6846	81	23	this	this	PRON
ejpam-6846	81	24	implies	imply	VERB
ejpam-6846	81	25	y	y	PROPN
ejpam-6846	81	26	∈	∈	PROPN
ejpam-6846	81	27	co{u1	co{u1	NOUN
ejpam-6846	81	28	,	,	PUNCT
ejpam-6846	81	29	u2	u2	NOUN
ejpam-6846	81	30	}	}	PUNCT
ejpam-6846	81	31	.	.	PUNCT
ejpam-6846	82	1	therefore	therefore	ADV
ejpam-6846	82	2	we	we	PRON
ejpam-6846	82	3	get	get	VERB
ejpam-6846	82	4	πf	πf	INTJ
ejpam-6846	82	5	s(0	s(0	PROPN
ejpam-6846	82	6	)	)	PUNCT
ejpam-6846	82	7	=	=	PRON
ejpam-6846	83	1	{	{	PUNCT
ejpam-6846	83	2	v	v	NUM
ejpam-6846	83	3	∈	∈	NOUN
ejpam-6846	83	4	s	s	PART
ejpam-6846	83	5	:	:	PUNCT
ejpam-6846	83	6	v	v	NUM
ejpam-6846	83	7	f	f	X
ejpam-6846	83	8	(	(	PUNCT
ejpam-6846	83	9	0	0	NUM
ejpam-6846	83	10	,	,	PUNCT
ejpam-6846	83	11	v	v	NOUN
ejpam-6846	83	12	)	)	PUNCT
ejpam-6846	83	13	=	=	SYM
ejpam-6846	83	14	inf	inf	PROPN
ejpam-6846	83	15	u∈s	u∈s	NOUN
ejpam-6846	83	16	v	v	ADP
ejpam-6846	83	17	f	f	PROPN
ejpam-6846	83	18	(	(	PUNCT
ejpam-6846	83	19	0	0	NUM
ejpam-6846	83	20	,	,	PUNCT
ejpam-6846	83	21	u	u	NOUN
ejpam-6846	83	22	)	)	PUNCT
ejpam-6846	83	23	=	=	SYM
ejpam-6846	83	24	8	8	NUM
ejpam-6846	83	25	}	}	PUNCT
ejpam-6846	83	26	=	=	SYM
ejpam-6846	83	27	co{u1	co{u1	NOUN
ejpam-6846	83	28	,	,	PUNCT
ejpam-6846	83	29	u2	u2	NOUN
ejpam-6846	83	30	}	}	PUNCT
ejpam-6846	83	31	=	=	SYM
ejpam-6846	83	32	πs(0	πs(0	NOUN
ejpam-6846	83	33	)	)	PUNCT
ejpam-6846	83	34	a.	a.	NOUN
ejpam-6846	83	35	al	al	PROPN
ejpam-6846	83	36	tane	tane	PROPN
ejpam-6846	83	37	,	,	PUNCT
ejpam-6846	83	38	l.	l.	PROPN
ejpam-6846	83	39	s.	s.	PROPN
ejpam-6846	83	40	keong	keong	PROPN
ejpam-6846	83	41	/	/	PUNCT
ejpam-6846	83	42	eur	eur	PROPN
ejpam-6846	83	43	.	.	PUNCT
ejpam-6846	84	1	j.	j.	PROPN
ejpam-6846	84	2	pure	pure	PROPN
ejpam-6846	84	3	appl	appl	PROPN
ejpam-6846	84	4	.	.	PROPN
ejpam-6846	84	5	math	math	PROPN
ejpam-6846	84	6	,	,	PUNCT
ejpam-6846	84	7	18	18	NUM
ejpam-6846	84	8	(	(	PUNCT
ejpam-6846	84	9	4	4	NUM
ejpam-6846	84	10	)	)	PUNCT
ejpam-6846	84	11	(	(	PUNCT
ejpam-6846	84	12	2025	2025	NUM
ejpam-6846	84	13	)	)	PUNCT
ejpam-6846	84	14	,	,	PUNCT
ejpam-6846	84	15	6846	6846	NUM
ejpam-6846	84	16	4	4	NUM
ejpam-6846	84	17	of	of	ADP
ejpam-6846	84	18	13	13	NUM
ejpam-6846	84	19	furthermore	furthermore	ADV
ejpam-6846	84	20	,	,	PUNCT
ejpam-6846	84	21	bounkhel	bounkhel	PROPN
ejpam-6846	84	22	(	(	PUNCT
ejpam-6846	84	23	see	see	VERB
ejpam-6846	84	24	[	[	X
ejpam-6846	84	25	5	5	NUM
ejpam-6846	84	26	]	]	PUNCT
ejpam-6846	84	27	)	)	PUNCT
ejpam-6846	84	28	shows	show	VERB
ejpam-6846	84	29	that	that	SCONJ
ejpam-6846	84	30	πs(x	πs(x	NUM
ejpam-6846	84	31	∗	∗	NOUN
ejpam-6846	84	32	)	)	PUNCT
ejpam-6846	84	33	=	=	SYM
ejpam-6846	84	34	ϕ	ϕ	NOUN
ejpam-6846	84	35	by	by	ADP
ejpam-6846	84	36	using	use	VERB
ejpam-6846	84	37	an	an	DET
ejpam-6846	84	38	example	example	NOUN
ejpam-6846	84	39	considering	consider	VERB
ejpam-6846	84	40	nonconvex	nonconvex	NOUN
ejpam-6846	84	41	closed	close	VERB
ejpam-6846	84	42	sets	set	NOUN
ejpam-6846	84	43	s	s	PRON
ejpam-6846	84	44	in	in	ADP
ejpam-6846	84	45	banach	banach	NOUN
ejpam-6846	84	46	spaces	space	NOUN
ejpam-6846	84	47	that	that	PRON
ejpam-6846	84	48	are	be	AUX
ejpam-6846	84	49	uniformly	uniformly	ADV
ejpam-6846	84	50	smooth	smooth	ADJ
ejpam-6846	84	51	and	and	CCONJ
ejpam-6846	84	52	uniformly	uniformly	ADV
ejpam-6846	84	53	convex	convex	NOUN
ejpam-6846	84	54	.	.	PUNCT
ejpam-6846	85	1	using	use	VERB
ejpam-6846	85	2	a	a	DET
ejpam-6846	85	3	similar	similar	ADJ
ejpam-6846	85	4	method	method	NOUN
ejpam-6846	85	5	and	and	CCONJ
ejpam-6846	85	6	by	by	ADP
ejpam-6846	85	7	taking	take	VERB
ejpam-6846	85	8	x	x	PUNCT
ejpam-6846	85	9	=	=	SYM
ejpam-6846	85	10	lp	lp	NOUN
ejpam-6846	85	11	with	with	ADP
ejpam-6846	85	12	(	(	PUNCT
ejpam-6846	85	13	p	p	NOUN
ejpam-6846	85	14	≥	≥	NOUN
ejpam-6846	85	15	1	1	NUM
ejpam-6846	85	16	)	)	PUNCT
ejpam-6846	85	17	,	,	PUNCT
ejpam-6846	85	18	0	0	X
ejpam-6846	85	19	=	=	SYM
ejpam-6846	85	20	(	(	PUNCT
ejpam-6846	85	21	0	0	NUM
ejpam-6846	85	22	,	,	PUNCT
ejpam-6846	85	23	0	0	NUM
ejpam-6846	85	24	,	,	PUNCT
ejpam-6846	85	25	0	0	NUM
ejpam-6846	85	26	,	,	PUNCT
ejpam-6846	85	27	.	.	PUNCT
ejpam-6846	85	28	.	.	PUNCT
ejpam-6846	85	29	.	.	PUNCT
ejpam-6846	85	30	)	)	PUNCT
ejpam-6846	86	1	∈	∈	NOUN
ejpam-6846	86	2	lp	lp	NOUN
ejpam-6846	86	3	and	and	CCONJ
ejpam-6846	86	4	let	let	VERB
ejpam-6846	86	5	s	s	PRON
ejpam-6846	86	6	=	=	NOUN
ejpam-6846	86	7	{	{	PUNCT
ejpam-6846	86	8	s1	s1	NOUN
ejpam-6846	86	9	,	,	PUNCT
ejpam-6846	86	10	s2	s2	NOUN
ejpam-6846	86	11	,	,	PUNCT
ejpam-6846	86	12	.	.	PUNCT
ejpam-6846	86	13	.	.	PUNCT
ejpam-6846	86	14	.	.	PUNCT
ejpam-6846	87	1	,	,	PUNCT
ejpam-6846	87	2	sm	sm	INTJ
ejpam-6846	87	3	,	,	PUNCT
ejpam-6846	87	4	.	.	PUNCT
ejpam-6846	87	5	.	.	PUNCT
ejpam-6846	88	1	.	.	PUNCT
ejpam-6846	88	2	}	}	PUNCT
ejpam-6846	89	1	;	;	PUNCT
ejpam-6846	89	2	sm	sm	X
ejpam-6846	89	3	=	=	SYM
ejpam-6846	89	4	(	(	PUNCT
ejpam-6846	89	5	0	0	NUM
ejpam-6846	89	6	,	,	PUNCT
ejpam-6846	89	7	0	0	NUM
ejpam-6846	89	8	,	,	PUNCT
ejpam-6846	89	9	.	.	PUNCT
ejpam-6846	89	10	.	.	PUNCT
ejpam-6846	89	11	.	.	PUNCT
ejpam-6846	90	1	,	,	PUNCT
ejpam-6846	90	2	1	1	NUM
ejpam-6846	90	3	+	+	SYM
ejpam-6846	90	4	1	1	NUM
ejpam-6846	90	5	m	m	NOUN
ejpam-6846	90	6	,	,	PUNCT
ejpam-6846	90	7	.	.	PUNCT
ejpam-6846	90	8	.	.	PUNCT
ejpam-6846	90	9	.	.	PUNCT
ejpam-6846	90	10	)	)	PUNCT
ejpam-6846	91	1	define	define	VERB
ejpam-6846	91	2	f	f	PROPN
ejpam-6846	91	3	as	as	ADP
ejpam-6846	91	4	f(x	f(x	PROPN
ejpam-6846	91	5	)	)	PUNCT
ejpam-6846	91	6	=	=	PUNCT
ejpam-6846	92	1	‖x‖	‖x‖	VERB
ejpam-6846	92	2	the	the	DET
ejpam-6846	92	3	projection	projection	NOUN
ejpam-6846	92	4	πf	πf	INTJ
ejpam-6846	92	5	s(u	s(u	PROPN
ejpam-6846	92	6	∗	∗	PROPN
ejpam-6846	92	7	)	)	PUNCT
ejpam-6846	92	8	may	may	AUX
ejpam-6846	92	9	in	in	ADP
ejpam-6846	92	10	fact	fact	NOUN
ejpam-6846	92	11	be	be	AUX
ejpam-6846	92	12	empty	empty	ADJ
ejpam-6846	92	13	for	for	SCONJ
ejpam-6846	92	14	nonconvex	nonconvex	NOUN
ejpam-6846	92	15	closed	close	VERB
ejpam-6846	92	16	sets	set	NOUN
ejpam-6846	92	17	s	s	VERB
ejpam-6846	92	18	in	in	ADP
ejpam-6846	92	19	a	a	DET
ejpam-6846	92	20	uniformly	uniformly	ADV
ejpam-6846	92	21	smoth	smoth	NOUN
ejpam-6846	92	22	and	and	CCONJ
ejpam-6846	92	23	uniformly	uniformly	ADV
ejpam-6846	92	24	convex	convex	VERB
ejpam-6846	92	25	banach	banach	NOUN
ejpam-6846	92	26	space	space	NOUN
ejpam-6846	92	27	.	.	PUNCT
ejpam-6846	93	1	this	this	PRON
ejpam-6846	93	2	shows	show	VERB
ejpam-6846	93	3	that	that	SCONJ
ejpam-6846	93	4	the	the	DET
ejpam-6846	93	5	generalized	generalize	VERB
ejpam-6846	93	6	f	f	PROPN
ejpam-6846	93	7	-projection	-projection	PROPN
ejpam-6846	93	8	over	over	ADP
ejpam-6846	93	9	nonconvex	nonconvex	NOUN
ejpam-6846	93	10	sets	set	NOUN
ejpam-6846	93	11	may	may	AUX
ejpam-6846	93	12	be	be	AUX
ejpam-6846	93	13	empty	empty	ADJ
ejpam-6846	93	14	even	even	ADV
ejpam-6846	93	15	if	if	SCONJ
ejpam-6846	93	16	the	the	DET
ejpam-6846	93	17	function	function	NOUN
ejpam-6846	93	18	f	f	PROPN
ejpam-6846	93	19	is	be	AUX
ejpam-6846	93	20	convex	convex	ADJ
ejpam-6846	93	21	continuous	continuous	ADJ
ejpam-6846	93	22	and	and	CCONJ
ejpam-6846	93	23	the	the	DET
ejpam-6846	93	24	space	space	NOUN
ejpam-6846	93	25	x	x	PUNCT
ejpam-6846	93	26	is	be	AUX
ejpam-6846	93	27	smooth	smooth	ADJ
ejpam-6846	93	28	reflexive	reflexive	ADJ
ejpam-6846	93	29	.	.	PUNCT
ejpam-6846	94	1	additionally	additionally	ADV
ejpam-6846	94	2	,	,	PUNCT
ejpam-6846	94	3	it	it	PRON
ejpam-6846	94	4	recently	recently	ADV
ejpam-6846	94	5	demonstrated	demonstrate	VERB
ejpam-6846	94	6	that	that	SCONJ
ejpam-6846	94	7	,	,	PUNCT
ejpam-6846	94	8	whenever	whenever	SCONJ
ejpam-6846	94	9	the	the	DET
ejpam-6846	94	10	space	space	NOUN
ejpam-6846	94	11	x	x	PUNCT
ejpam-6846	94	12	is	be	AUX
ejpam-6846	94	13	taken	take	VERB
ejpam-6846	94	14	to	to	PART
ejpam-6846	94	15	be	be	AUX
ejpam-6846	94	16	a	a	DET
ejpam-6846	94	17	reflexive	reflexive	ADJ
ejpam-6846	94	18	banach	banach	NOUN
ejpam-6846	94	19	space	space	NOUN
ejpam-6846	94	20	with	with	ADP
ejpam-6846	94	21	the	the	DET
ejpam-6846	94	22	smooth	smooth	ADJ
ejpam-6846	94	23	dual	dual	ADJ
ejpam-6846	94	24	norm	norm	NOUN
ejpam-6846	94	25	,	,	PUNCT
ejpam-6846	94	26	the	the	DET
ejpam-6846	94	27	set	set	NOUN
ejpam-6846	94	28	of	of	ADP
ejpam-6846	94	29	points	point	NOUN
ejpam-6846	94	30	u∗	u∗	ADJ
ejpam-6846	94	31	in	in	ADP
ejpam-6846	94	32	x∗	x∗	PROPN
ejpam-6846	94	33	with	with	ADP
ejpam-6846	94	34	generalized	generalize	VERB
ejpam-6846	94	35	f	f	PROPN
ejpam-6846	94	36	-projection	-projection	PROPN
ejpam-6846	94	37	is	be	AUX
ejpam-6846	94	38	dense	dense	ADJ
ejpam-6846	94	39	in	in	ADP
ejpam-6846	94	40	x∗	x∗	PROPN
ejpam-6846	94	41	(	(	PUNCT
ejpam-6846	94	42	see	see	VERB
ejpam-6846	94	43	theorem	theorem	VERB
ejpam-6846	94	44	2.1	2.1	NUM
ejpam-6846	94	45	in	in	ADP
ejpam-6846	94	46	[	[	X
ejpam-6846	94	47	7	7	NUM
ejpam-6846	94	48	]	]	NUM
ejpam-6846	94	49	)	)	PUNCT
ejpam-6846	94	50	.	.	PUNCT
ejpam-6846	95	1	now	now	ADV
ejpam-6846	95	2	,	,	PUNCT
ejpam-6846	95	3	we	we	PRON
ejpam-6846	95	4	consider	consider	VERB
ejpam-6846	95	5	the	the	DET
ejpam-6846	95	6	vector	vector	NOUN
ejpam-6846	95	7	space	space	NOUN
ejpam-6846	95	8	of	of	ADP
ejpam-6846	95	9	all	all	DET
ejpam-6846	95	10	convergent	convergent	ADJ
ejpam-6846	95	11	sequences	sequence	NOUN
ejpam-6846	95	12	of	of	ADP
ejpam-6846	95	13	real	real	ADJ
ejpam-6846	95	14	numbers	number	NOUN
ejpam-6846	95	15	denoted	denote	VERB
ejpam-6846	95	16	by	by	ADP
ejpam-6846	95	17	c.	c.	NOUN
ejpam-6846	95	18	that	that	PRON
ejpam-6846	95	19	equipped	equip	VERB
ejpam-6846	95	20	with	with	ADP
ejpam-6846	95	21	the	the	DET
ejpam-6846	95	22	norm	norm	NOUN
ejpam-6846	95	23	‖u‖∞	‖u‖∞	NOUN
ejpam-6846	95	24	=	=	SYM
ejpam-6846	95	25	sup	sup	NOUN
ejpam-6846	95	26	n	n	PRON
ejpam-6846	95	27	|un|	|un|	NOUN
ejpam-6846	95	28	,	,	PUNCT
ejpam-6846	95	29	and	and	CCONJ
ejpam-6846	95	30	the	the	DET
ejpam-6846	95	31	dual	dual	ADJ
ejpam-6846	95	32	space	space	NOUN
ejpam-6846	95	33	c∗	c∗	PROPN
ejpam-6846	95	34	=	=	SYM
ejpam-6846	95	35	l1	l1	PROPN
ejpam-6846	95	36	(	(	PUNCT
ejpam-6846	95	37	note	note	VERB
ejpam-6846	95	38	that	that	SCONJ
ejpam-6846	95	39	l1	l1	PROPN
ejpam-6846	95	40	is	be	AUX
ejpam-6846	95	41	neither	neither	CCONJ
ejpam-6846	95	42	reflexive	reflexive	ADJ
ejpam-6846	95	43	nor	nor	CCONJ
ejpam-6846	95	44	strictly	strictly	ADV
ejpam-6846	95	45	convex	convex	ADJ
ejpam-6846	95	46	)	)	PUNCT
ejpam-6846	95	47	.	.	PUNCT
ejpam-6846	96	1	if	if	SCONJ
ejpam-6846	96	2	u	u	PRON
ejpam-6846	96	3	=	=	X
ejpam-6846	96	4	(	(	PUNCT
ejpam-6846	96	5	u0	u0	ADJ
ejpam-6846	96	6	,	,	PUNCT
ejpam-6846	96	7	u1	u1	NOUN
ejpam-6846	96	8	,	,	PUNCT
ejpam-6846	96	9	.	.	PUNCT
ejpam-6846	96	10	.	.	PUNCT
ejpam-6846	96	11	.	.	PUNCT
ejpam-6846	96	12	)	)	PUNCT
ejpam-6846	97	1	∈	∈	PROPN
ejpam-6846	97	2	ℓ1	ℓ1	NOUN
ejpam-6846	97	3	,	,	PUNCT
ejpam-6846	97	4	,	,	PUNCT
ejpam-6846	97	5	and	and	CCONJ
ejpam-6846	97	6	λ	λ	X
ejpam-6846	97	7	=	=	SYM
ejpam-6846	97	8	(	(	PUNCT
ejpam-6846	97	9	λ0	λ0	NOUN
ejpam-6846	97	10	,	,	PUNCT
ejpam-6846	97	11	λ1	λ1	ADJ
ejpam-6846	97	12	,	,	PUNCT
ejpam-6846	97	13	.	.	PUNCT
ejpam-6846	97	14	.	.	PUNCT
ejpam-6846	97	15	.	.	PUNCT
ejpam-6846	97	16	)	)	PUNCT
ejpam-6846	98	1	∈	∈	PROPN
ejpam-6846	98	2	c.	c.	NOUN
ejpam-6846	98	3	then	then	ADV
ejpam-6846	98	4	the	the	DET
ejpam-6846	98	5	duality	duality	NOUN
ejpam-6846	98	6	pairing	pairing	NOUN
ejpam-6846	98	7	of	of	ADP
ejpam-6846	98	8	c∗	c∗	PROPN
ejpam-6846	98	9	,	,	PUNCT
ejpam-6846	98	10	c	c	PROPN
ejpam-6846	98	11	is	be	AUX
ejpam-6846	98	12	given	give	VERB
ejpam-6846	98	13	by	by	ADP
ejpam-6846	98	14	〈	〈	PROPN
ejpam-6846	98	15	u	u	PROPN
ejpam-6846	98	16	,	,	PUNCT
ejpam-6846	98	17	λ	λ	NOUN
ejpam-6846	98	18	〉	〉	NOUN
ejpam-6846	98	19	=	=	SYM
ejpam-6846	98	20	u0	u0	PROPN
ejpam-6846	98	21	lim	lim	PROPN
ejpam-6846	98	22	n→∞	n→∞	PRON
ejpam-6846	98	23	λn	λn	X
ejpam-6846	98	24	+	+	CCONJ
ejpam-6846	98	25	∞∑	∞∑	PROPN
ejpam-6846	98	26	i=1	i=1	PROPN
ejpam-6846	98	27	uiλi	uiλi	PROPN
ejpam-6846	98	28	.	.	PUNCT
ejpam-6846	99	1	note	note	VERB
ejpam-6846	99	2	that	that	SCONJ
ejpam-6846	99	3	c0	c0	PROPN
ejpam-6846	99	4	denotes	denote	VERB
ejpam-6846	99	5	the	the	DET
ejpam-6846	99	6	closed	closed	ADJ
ejpam-6846	99	7	subspace	subspace	NOUN
ejpam-6846	99	8	of	of	ADP
ejpam-6846	99	9	c	c	PROPN
ejpam-6846	99	10	that	that	PRON
ejpam-6846	99	11	contains	contain	VERB
ejpam-6846	99	12	all	all	DET
ejpam-6846	99	13	convergent	convergent	ADJ
ejpam-6846	99	14	real	real	ADJ
ejpam-6846	99	15	sequences	sequence	NOUN
ejpam-6846	99	16	with	with	ADP
ejpam-6846	99	17	limit	limit	NOUN
ejpam-6846	99	18	zero	zero	NUM
ejpam-6846	99	19	.	.	PUNCT
ejpam-6846	100	1	(	(	PUNCT
ejpam-6846	100	2	for	for	ADP
ejpam-6846	100	3	additional	additional	ADJ
ejpam-6846	100	4	details	detail	NOUN
ejpam-6846	100	5	,	,	PUNCT
ejpam-6846	100	6	see[13	see[13	PROPN
ejpam-6846	100	7	]	]	PUNCT
ejpam-6846	100	8	)	)	PUNCT
ejpam-6846	100	9	.	.	PUNCT
ejpam-6846	101	1	using	use	VERB
ejpam-6846	101	2	c	c	PROPN
ejpam-6846	101	3	,	,	PUNCT
ejpam-6846	101	4	c0	c0	NOUN
ejpam-6846	101	5	,	,	PUNCT
ejpam-6846	101	6	ℓ1	ℓ1	VERB
ejpam-6846	101	7	one	one	NUM
ejpam-6846	101	8	can	can	AUX
ejpam-6846	101	9	show	show	VERB
ejpam-6846	101	10	that	that	SCONJ
ejpam-6846	101	11	if	if	SCONJ
ejpam-6846	101	12	the	the	DET
ejpam-6846	101	13	banach	banach	NOUN
ejpam-6846	101	14	space	space	NOUN
ejpam-6846	101	15	is	be	AUX
ejpam-6846	101	16	not	not	PART
ejpam-6846	101	17	reflexive	reflexive	ADJ
ejpam-6846	101	18	,	,	PUNCT
ejpam-6846	101	19	πf	πf	PROPN
ejpam-6846	101	20	s	s	VERB
ejpam-6846	101	21	may	may	AUX
ejpam-6846	101	22	be	be	AUX
ejpam-6846	101	23	empty	empty	ADJ
ejpam-6846	101	24	for	for	ADP
ejpam-6846	101	25	some	some	DET
ejpam-6846	101	26	elements	element	NOUN
ejpam-6846	101	27	x∗	x∗	PROPN
ejpam-6846	101	28	∈	∈	PROPN
ejpam-6846	101	29	x∗	x∗	PROPN
ejpam-6846	101	30	even	even	ADV
ejpam-6846	101	31	for	for	ADP
ejpam-6846	101	32	f(u	f(u	PROPN
ejpam-6846	101	33	)	)	PUNCT
ejpam-6846	101	34	6=	6=	ADP
ejpam-6846	101	35	0	0	NUM
ejpam-6846	101	36	.	.	PUNCT
ejpam-6846	102	1	(	(	PUNCT
ejpam-6846	102	2	take	take	VERB
ejpam-6846	102	3	note	note	NOUN
ejpam-6846	102	4	of	of	ADP
ejpam-6846	102	5	my	my	PRON
ejpam-6846	102	6	example	example	NOUN
ejpam-6846	102	7	,	,	PUNCT
ejpam-6846	102	8	which	which	PRON
ejpam-6846	102	9	uses	use	VERB
ejpam-6846	102	10	the	the	DET
ejpam-6846	102	11	same	same	ADJ
ejpam-6846	102	12	reasoning	reasoning	NOUN
ejpam-6846	102	13	as	as	ADP
ejpam-6846	102	14	example	example	NOUN
ejpam-6846	102	15	2.6	2.6	NUM
ejpam-6846	103	1	[	[	X
ejpam-6846	103	2	14	14	NUM
ejpam-6846	103	3	]	]	PUNCT
ejpam-6846	103	4	for	for	ADP
ejpam-6846	103	5	πs	πs	PROPN
ejpam-6846	103	6	)	)	PUNCT
ejpam-6846	103	7	.	.	PUNCT
ejpam-6846	104	1	example	example	NOUN
ejpam-6846	105	1	2	2	NUM
ejpam-6846	105	2	.	.	X
ejpam-6846	105	3	let	let	VERB
ejpam-6846	105	4	f(u	f(u	PROPN
ejpam-6846	105	5	)	)	PUNCT
ejpam-6846	105	6	=	=	SYM
ejpam-6846	105	7	4	4	NUM
ejpam-6846	105	8	and	and	CCONJ
ejpam-6846	105	9	u∗	u∗	NOUN
ejpam-6846	105	10	=	=	SYM
ejpam-6846	105	11	(	(	PUNCT
ejpam-6846	105	12	0	0	NUM
ejpam-6846	105	13	,	,	PUNCT
ejpam-6846	105	14	1	1	NUM
ejpam-6846	105	15	,	,	PUNCT
ejpam-6846	105	16	1	1	NUM
ejpam-6846	105	17	2	2	NUM
ejpam-6846	105	18	,	,	PUNCT
ejpam-6846	105	19	1	1	NUM
ejpam-6846	105	20	22	22	NUM
ejpam-6846	105	21	,	,	PUNCT
ejpam-6846	105	22	1	1	NUM
ejpam-6846	105	23	23	23	NUM
ejpam-6846	105	24	,	,	PUNCT
ejpam-6846	105	25	·	·	PUNCT
ejpam-6846	105	26	·	·	PUNCT
ejpam-6846	105	27	·	·	PUNCT
ejpam-6846	105	28	)	)	PUNCT
ejpam-6846	106	1	∈	∈	PROPN
ejpam-6846	106	2	ℓ1	ℓ1	NOUN
ejpam-6846	106	3	.	.	PUNCT
ejpam-6846	107	1	then	then	ADV
ejpam-6846	107	2	πf	πf	INTJ
ejpam-6846	107	3	c0	c0	PROPN
ejpam-6846	107	4	(	(	PUNCT
ejpam-6846	107	5	u∗	u∗	PROPN
ejpam-6846	107	6	)	)	PUNCT
ejpam-6846	107	7	=	=	SYM
ejpam-6846	108	1	ϕ.	ϕ.	NOUN
ejpam-6846	108	2	proof	proof	NOUN
ejpam-6846	108	3	:	:	PUNCT
ejpam-6846	108	4	we	we	PRON
ejpam-6846	108	5	define	define	VERB
ejpam-6846	108	6	λn	λn	PROPN
ejpam-6846	108	7	∈	∈	PROPN
ejpam-6846	108	8	c0	c0	NOUN
ejpam-6846	108	9	for	for	ADP
ejpam-6846	108	10	every	every	DET
ejpam-6846	108	11	positive	positive	ADJ
ejpam-6846	108	12	integer	integer	NOUN
ejpam-6846	108	13	n	n	CCONJ
ejpam-6846	108	14	such	such	ADJ
ejpam-6846	108	15	that	that	SCONJ
ejpam-6846	108	16	its	its	PRON
ejpam-6846	108	17	first	first	ADJ
ejpam-6846	108	18	n	n	PRON
ejpam-6846	108	19	components	component	NOUN
ejpam-6846	108	20	are	be	AUX
ejpam-6846	108	21	two	two	NUM
ejpam-6846	108	22	and	and	CCONJ
ejpam-6846	108	23	all	all	DET
ejpam-6846	108	24	others	other	NOUN
ejpam-6846	108	25	are	be	AUX
ejpam-6846	108	26	0	0	NUM
ejpam-6846	108	27	.	.	PUNCT
ejpam-6846	109	1	then	then	ADV
ejpam-6846	109	2	‖u∗‖	‖u∗‖	X
ejpam-6846	109	3	=	=	SYM
ejpam-6846	109	4	(	(	PUNCT
ejpam-6846	109	5	0	0	NUM
ejpam-6846	109	6	,	,	PUNCT
ejpam-6846	109	7	1	1	NUM
ejpam-6846	109	8	,	,	PUNCT
ejpam-6846	109	9	1	1	NUM
ejpam-6846	109	10	2	2	NUM
ejpam-6846	109	11	,	,	PUNCT
ejpam-6846	109	12	1	1	NUM
ejpam-6846	109	13	22	22	NUM
ejpam-6846	109	14	,	,	PUNCT
ejpam-6846	109	15	1	1	NUM
ejpam-6846	109	16	23	23	NUM
ejpam-6846	109	17	,	,	PUNCT
ejpam-6846	109	18	·	·	PUNCT
ejpam-6846	109	19	·	·	PUNCT
ejpam-6846	109	20	·	·	PUNCT
ejpam-6846	109	21	)	)	PUNCT
ejpam-6846	110	1	=	=	PUNCT
ejpam-6846	111	1	∞∑	∞∑	NUM
ejpam-6846	111	2	i=0	i=0	ADJ
ejpam-6846	111	3	1	1	NUM
ejpam-6846	111	4	2i	2i	NOUN
ejpam-6846	111	5	=	=	SYM
ejpam-6846	111	6	2	2	NUM
ejpam-6846	111	7	,	,	PUNCT
ejpam-6846	111	8	‖λn‖	‖λn‖	PROPN
ejpam-6846	111	9	=	=	SYM
ejpam-6846	111	10	‖(2	‖(2	PROPN
ejpam-6846	111	11	,	,	PUNCT
ejpam-6846	111	12	2	2	NUM
ejpam-6846	111	13	,	,	PUNCT
ejpam-6846	111	14	.	.	PUNCT
ejpam-6846	111	15	.	.	PUNCT
ejpam-6846	111	16	.	.	PUNCT
ejpam-6846	112	1	,	,	PUNCT
ejpam-6846	112	2	2	2	NUM
ejpam-6846	112	3	,	,	PUNCT
ejpam-6846	112	4	0	0	NUM
ejpam-6846	112	5	,	,	PUNCT
ejpam-6846	112	6	0	0	NUM
ejpam-6846	112	7	,	,	PUNCT
ejpam-6846	112	8	.	.	PUNCT
ejpam-6846	112	9	.	.	PUNCT
ejpam-6846	112	10	.	.	PUNCT
ejpam-6846	112	11	)	)	PUNCT
ejpam-6846	113	1	‖	‖	PROPN
ejpam-6846	113	2	=	=	SYM
ejpam-6846	113	3	2	2	NUM
ejpam-6846	113	4	,	,	PUNCT
ejpam-6846	113	5	and	and	CCONJ
ejpam-6846	113	6	so	so	ADV
ejpam-6846	113	7	〈	〈	PROPN
ejpam-6846	113	8	u	u	NOUN
ejpam-6846	113	9	,	,	PUNCT
ejpam-6846	113	10	λn	λn	PROPN
ejpam-6846	113	11	〉	〉	NUM
ejpam-6846	113	12	=	=	SYM
ejpam-6846	114	1	x0	x0	PROPN
ejpam-6846	114	2	lim	lim	PROPN
ejpam-6846	114	3	n→∞	n→∞	PRON
ejpam-6846	114	4	λn	λn	X
ejpam-6846	114	5	+	+	CCONJ
ejpam-6846	114	6	∞∑	∞∑	NUM
ejpam-6846	114	7	i=1	i=1	ADV
ejpam-6846	114	8	uiλi	uiλi	PROPN
ejpam-6846	114	9	=	=	SYM
ejpam-6846	114	10	0×	0×	PROPN
ejpam-6846	114	11	2	2	NUM
ejpam-6846	115	1	+	+	CCONJ
ejpam-6846	115	2	n∑	n∑	ADJ
ejpam-6846	115	3	i=1	i=1	ADP
ejpam-6846	115	4	1	1	NUM
ejpam-6846	115	5	2i−1	2i−1	NUM
ejpam-6846	115	6	=	=	SYM
ejpam-6846	115	7	4(1−	4(1−	NUM
ejpam-6846	115	8	2−n+1	2−n+1	NUM
ejpam-6846	115	9	)	)	PUNCT
ejpam-6846	115	10	.	.	PUNCT
ejpam-6846	116	1	which	which	PRON
ejpam-6846	116	2	implies	imply	VERB
ejpam-6846	116	3	v	v	PROPN
ejpam-6846	116	4	f	f	X
ejpam-6846	116	5	(	(	PUNCT
ejpam-6846	116	6	u∗	u∗	PROPN
ejpam-6846	116	7	,	,	PUNCT
ejpam-6846	116	8	λn	λn	NOUN
ejpam-6846	116	9	)	)	PUNCT
ejpam-6846	116	10	=	=	PUNCT
ejpam-6846	117	1	4−	4−	NOUN
ejpam-6846	117	2	8(1−	8(1−	NUM
ejpam-6846	117	3	2−n+1	2−n+1	NUM
ejpam-6846	117	4	)	)	PUNCT
ejpam-6846	118	1	+	+	CCONJ
ejpam-6846	118	2	8	8	NUM
ejpam-6846	118	3	→	→	SYM
ejpam-6846	118	4	4	4	NUM
ejpam-6846	118	5	,	,	PUNCT
ejpam-6846	118	6	as	as	ADP
ejpam-6846	118	7	,	,	PUNCT
ejpam-6846	118	8	n	n	CCONJ
ejpam-6846	118	9	→	→	SYM
ejpam-6846	118	10	∞	∞	PROPN
ejpam-6846	118	11	(	(	PUNCT
ejpam-6846	118	12	2	2	NUM
ejpam-6846	118	13	)	)	PUNCT
ejpam-6846	118	14	a.	a.	NOUN
ejpam-6846	118	15	al	al	PROPN
ejpam-6846	118	16	tane	tane	PROPN
ejpam-6846	118	17	,	,	PUNCT
ejpam-6846	118	18	l.	l.	PROPN
ejpam-6846	118	19	s.	s.	PROPN
ejpam-6846	118	20	keong	keong	PROPN
ejpam-6846	118	21	/	/	PUNCT
ejpam-6846	118	22	eur	eur	PROPN
ejpam-6846	118	23	.	.	PUNCT
ejpam-6846	119	1	j.	j.	PROPN
ejpam-6846	119	2	pure	pure	PROPN
ejpam-6846	119	3	appl	appl	PROPN
ejpam-6846	119	4	.	.	PROPN
ejpam-6846	119	5	math	math	PROPN
ejpam-6846	119	6	,	,	PUNCT
ejpam-6846	119	7	18	18	NUM
ejpam-6846	119	8	(	(	PUNCT
ejpam-6846	119	9	4	4	NUM
ejpam-6846	119	10	)	)	PUNCT
ejpam-6846	119	11	(	(	PUNCT
ejpam-6846	119	12	2025	2025	NUM
ejpam-6846	119	13	)	)	PUNCT
ejpam-6846	119	14	,	,	PUNCT
ejpam-6846	119	15	6846	6846	NUM
ejpam-6846	119	16	5	5	NUM
ejpam-6846	119	17	of	of	ADP
ejpam-6846	119	18	13	13	NUM
ejpam-6846	119	19	now	now	ADV
ejpam-6846	119	20	we	we	PRON
ejpam-6846	119	21	will	will	AUX
ejpam-6846	119	22	prove	prove	VERB
ejpam-6846	119	23	v	v	ADP
ejpam-6846	119	24	f	f	X
ejpam-6846	119	25	(	(	PUNCT
ejpam-6846	119	26	u∗	u∗	PROPN
ejpam-6846	119	27	,	,	PUNCT
ejpam-6846	119	28	λ	λ	NOUN
ejpam-6846	119	29	)	)	PUNCT
ejpam-6846	119	30	>	>	X
ejpam-6846	120	1	4	4	NUM
ejpam-6846	120	2	,	,	PUNCT
ejpam-6846	120	3	for	for	ADP
ejpam-6846	120	4	all	all	DET
ejpam-6846	120	5	λ	λ	PROPN
ejpam-6846	120	6	∈	∈	PROPN
ejpam-6846	120	7	c0	c0	NOUN
ejpam-6846	120	8	so	so	SCONJ
ejpam-6846	120	9	that	that	SCONJ
ejpam-6846	120	10	the	the	DET
ejpam-6846	120	11	infµ∈c0v	infµ∈c0v	PROPN
ejpam-6846	120	12	f	f	PROPN
ejpam-6846	120	13	(	(	PUNCT
ejpam-6846	120	14	u∗	u∗	PROPN
ejpam-6846	120	15	,	,	PUNCT
ejpam-6846	120	16	µ	µ	NOUN
ejpam-6846	120	17	)	)	PUNCT
ejpam-6846	120	18	by	by	ADP
ejpam-6846	120	19	2	2	NUM
ejpam-6846	120	20	not	not	PART
ejpam-6846	120	21	exists	exist	VERB
ejpam-6846	120	22	.	.	PUNCT
ejpam-6846	121	1	suppose	suppose	VERB
ejpam-6846	122	1	that	that	SCONJ
ejpam-6846	122	2	v	v	PROPN
ejpam-6846	122	3	f	f	X
ejpam-6846	122	4	(	(	PUNCT
ejpam-6846	122	5	u∗	u∗	PROPN
ejpam-6846	122	6	,	,	PUNCT
ejpam-6846	122	7	λ	λ	NOUN
ejpam-6846	122	8	)	)	PUNCT
ejpam-6846	122	9	=	=	SYM
ejpam-6846	122	10	4	4	X
ejpam-6846	122	11	.	.	PUNCT
ejpam-6846	122	12	then	then	ADV
ejpam-6846	122	13	by	by	ADP
ejpam-6846	122	14	(	(	PUNCT
ejpam-6846	122	15	1	1	NUM
ejpam-6846	122	16	)	)	PUNCT
ejpam-6846	122	17	(	(	PUNCT
ejpam-6846	122	18	‖u∗‖	‖u∗‖	INTJ
ejpam-6846	122	19	−	−	X
ejpam-6846	122	20	‖λ‖)2	‖λ‖)2	PROPN
ejpam-6846	122	21	+	+	NUM
ejpam-6846	122	22	f(λ	f(λ	NOUN
ejpam-6846	122	23	)	)	PUNCT
ejpam-6846	122	24	≤	≤	NOUN
ejpam-6846	122	25	v	v	ADP
ejpam-6846	122	26	f	f	X
ejpam-6846	122	27	(	(	PUNCT
ejpam-6846	122	28	u∗	u∗	PROPN
ejpam-6846	122	29	,	,	PUNCT
ejpam-6846	122	30	λ	λ	NOUN
ejpam-6846	122	31	)	)	PUNCT
ejpam-6846	122	32	≤	≤	NOUN
ejpam-6846	122	33	(	(	PUNCT
ejpam-6846	122	34	‖u∗‖+	‖u∗‖+	NOUN
ejpam-6846	122	35	‖λ‖)2	‖λ‖)2	X
ejpam-6846	122	36	+	+	CCONJ
ejpam-6846	122	37	f(λ	f(λ	ADJ
ejpam-6846	122	38	)	)	PUNCT
ejpam-6846	122	39	⇔	⇔	NOUN
ejpam-6846	122	40	(	(	PUNCT
ejpam-6846	122	41	‖u∗‖	‖u∗‖	INTJ
ejpam-6846	122	42	−	−	X
ejpam-6846	122	43	‖λ‖)2	‖λ‖)2	NOUN
ejpam-6846	122	44	+	+	CCONJ
ejpam-6846	122	45	4	4	NUM
ejpam-6846	122	46	≤	≤	NUM
ejpam-6846	122	47	4	4	NUM
ejpam-6846	122	48	≤	≤	NOUN
ejpam-6846	122	49	(	(	PUNCT
ejpam-6846	122	50	‖u∗‖+	‖u∗‖+	NOUN
ejpam-6846	122	51	‖λ‖)2	‖λ‖)2	PROPN
ejpam-6846	122	52	+	+	NUM
ejpam-6846	122	53	4	4	NUM
ejpam-6846	122	54	⇔	⇔	X
ejpam-6846	122	55	2	2	NUM
ejpam-6846	122	56	=	=	SYM
ejpam-6846	122	57	‖u∗‖	‖u∗‖	NOUN
ejpam-6846	123	1	=	=	SYM
ejpam-6846	123	2	‖λ‖	‖λ‖	PROPN
ejpam-6846	123	3	hence	hence	ADV
ejpam-6846	123	4	−2	−2	NOUN
ejpam-6846	123	5	≤	≤	NUM
ejpam-6846	123	6	λn	λn	NOUN
ejpam-6846	123	7	≤	≤	NUM
ejpam-6846	123	8	2	2	NUM
ejpam-6846	123	9	for	for	ADP
ejpam-6846	123	10	n	n	NOUN
ejpam-6846	123	11	=	=	SYM
ejpam-6846	123	12	1	1	NUM
ejpam-6846	123	13	,	,	PUNCT
ejpam-6846	123	14	2	2	NUM
ejpam-6846	123	15	,	,	PUNCT
ejpam-6846	123	16	.	.	PUNCT
ejpam-6846	123	17	.	.	PUNCT
ejpam-6846	123	18	.	.	PUNCT
ejpam-6846	124	1	.	.	PUNCT
ejpam-6846	125	1	but	but	CCONJ
ejpam-6846	125	2	limn→∞λn	limn→∞λn	PROPN
ejpam-6846	125	3	=	=	SYM
ejpam-6846	125	4	0	0	NUM
ejpam-6846	125	5	,	,	PUNCT
ejpam-6846	125	6	so	so	SCONJ
ejpam-6846	125	7	there	there	PRON
ejpam-6846	125	8	are	be	VERB
ejpam-6846	125	9	infinitely	infinitely	ADV
ejpam-6846	125	10	many	many	ADJ
ejpam-6846	125	11	n	n	ADP
ejpam-6846	125	12	such	such	ADJ
ejpam-6846	125	13	that	that	SCONJ
ejpam-6846	125	14	λn	λn	NOUN
ejpam-6846	125	15	<	<	X
ejpam-6846	125	16	2	2	NUM
ejpam-6846	125	17	.	.	PUNCT
ejpam-6846	126	1	that	that	PRON
ejpam-6846	126	2	implies	imply	VERB
ejpam-6846	126	3	v	v	PROPN
ejpam-6846	126	4	f	f	X
ejpam-6846	126	5	(	(	PUNCT
ejpam-6846	126	6	u∗	u∗	PROPN
ejpam-6846	126	7	,	,	PUNCT
ejpam-6846	126	8	λ	λ	NOUN
ejpam-6846	126	9	)	)	PUNCT
ejpam-6846	127	1	=	=	PUNCT
ejpam-6846	127	2	‖u∗‖	‖u∗‖	INTJ
ejpam-6846	128	1	−	−	NUM
ejpam-6846	128	2	2〈u∗	2〈u∗	NUM
ejpam-6846	128	3	,	,	PUNCT
ejpam-6846	128	4	λ〉+	λ〉+	NOUN
ejpam-6846	128	5	‖λ‖2	‖λ‖2	NOUN
ejpam-6846	128	6	+	+	CCONJ
ejpam-6846	128	7	f(λ	f(λ	NOUN
ejpam-6846	128	8	)	)	PUNCT
ejpam-6846	128	9	=	=	PUNCT
ejpam-6846	129	1	4−	4−	NOUN
ejpam-6846	129	2	2	2	NUM
ejpam-6846	130	1	∞∑	∞∑	NUM
ejpam-6846	130	2	i=1	i=1	PROPN
ejpam-6846	130	3	1	1	NUM
ejpam-6846	130	4	2i−1	2i−1	NUM
ejpam-6846	130	5	λi	λi	ADP
ejpam-6846	130	6	+	+	CCONJ
ejpam-6846	130	7	8	8	NUM
ejpam-6846	130	8	>	>	SYM
ejpam-6846	130	9	4−	4−	NUM
ejpam-6846	130	10	2	2	NUM
ejpam-6846	131	1	∞∑	∞∑	NUM
ejpam-6846	131	2	i=1	i=1	PROPN
ejpam-6846	131	3	1	1	NUM
ejpam-6846	131	4	2i−1	2i−1	NUM
ejpam-6846	131	5	2	2	NUM
ejpam-6846	131	6	+	+	CCONJ
ejpam-6846	131	7	8	8	NUM
ejpam-6846	131	8	=	=	SYM
ejpam-6846	131	9	4	4	NUM
ejpam-6846	131	10	which	which	PRON
ejpam-6846	131	11	is	be	AUX
ejpam-6846	131	12	a	a	DET
ejpam-6846	131	13	contraction	contraction	NOUN
ejpam-6846	131	14	to	to	ADP
ejpam-6846	131	15	our	our	PRON
ejpam-6846	131	16	assumption	assumption	NOUN
ejpam-6846	131	17	v	v	ADP
ejpam-6846	131	18	f	f	X
ejpam-6846	131	19	(	(	PUNCT
ejpam-6846	131	20	u∗	u∗	PROPN
ejpam-6846	131	21	,	,	PUNCT
ejpam-6846	131	22	λ	λ	NOUN
ejpam-6846	131	23	)	)	PUNCT
ejpam-6846	131	24	=	=	SYM
ejpam-6846	131	25	4	4	X
ejpam-6846	131	26	.	.	X
ejpam-6846	131	27	therefor	therefor	ADP
ejpam-6846	131	28	πf	πf	X
ejpam-6846	131	29	c0	c0	PROPN
ejpam-6846	131	30	(	(	PUNCT
ejpam-6846	131	31	u∗	u∗	PROPN
ejpam-6846	131	32	)	)	PUNCT
ejpam-6846	132	1	=	=	PUNCT
ejpam-6846	132	2	ϕ.	ϕ.	VERB
ejpam-6846	132	3	these	these	DET
ejpam-6846	132	4	observations	observation	NOUN
ejpam-6846	132	5	force	force	VERB
ejpam-6846	132	6	us	we	PRON
ejpam-6846	132	7	to	to	PART
ejpam-6846	132	8	assume	assume	VERB
ejpam-6846	132	9	that	that	SCONJ
ejpam-6846	132	10	the	the	DET
ejpam-6846	132	11	space	space	NOUN
ejpam-6846	132	12	x	x	PUNCT
ejpam-6846	132	13	complies	comply	VERB
ejpam-6846	132	14	with	with	ADP
ejpam-6846	132	15	this	this	DET
ejpam-6846	132	16	assumption	assumption	NOUN
ejpam-6846	132	17	going	go	VERB
ejpam-6846	132	18	onward	onward	ADV
ejpam-6846	132	19	;	;	PUNCT
ejpam-6846	132	20	in	in	ADP
ejpam-6846	132	21	other	other	ADJ
ejpam-6846	132	22	words	word	NOUN
ejpam-6846	132	23	,	,	PUNCT
ejpam-6846	132	24	we	we	PRON
ejpam-6846	132	25	will	will	AUX
ejpam-6846	132	26	presume	presume	VERB
ejpam-6846	132	27	that	that	SCONJ
ejpam-6846	132	28	x	x	PRON
ejpam-6846	132	29	is	be	AUX
ejpam-6846	132	30	a	a	DET
ejpam-6846	132	31	reflexive	reflexive	ADJ
ejpam-6846	132	32	banach	banach	NOUN
ejpam-6846	132	33	space	space	NOUN
ejpam-6846	132	34	with	with	ADP
ejpam-6846	132	35	a	a	DET
ejpam-6846	132	36	smooth	smooth	ADJ
ejpam-6846	132	37	dual	dual	ADJ
ejpam-6846	132	38	norm	norm	NOUN
ejpam-6846	132	39	for	for	ADP
ejpam-6846	132	40	the	the	DET
ejpam-6846	132	41	remainder	remainder	NOUN
ejpam-6846	132	42	of	of	ADP
ejpam-6846	132	43	the	the	DET
ejpam-6846	132	44	work	work	NOUN
ejpam-6846	132	45	.	.	PUNCT
ejpam-6846	133	1	3	3	X
ejpam-6846	133	2	.	.	X
ejpam-6846	133	3	on	on	ADP
ejpam-6846	133	4	πf	πf	PROPN
ejpam-6846	133	5	s(x	s(x	PROPN
ejpam-6846	133	6	∗	∗	NOUN
ejpam-6846	133	7	)	)	PUNCT
ejpam-6846	133	8	for	for	ADP
ejpam-6846	133	9	s	s	PRON
ejpam-6846	133	10	nonconvex	nonconvex	NOUN
ejpam-6846	133	11	set	set	VERB
ejpam-6846	133	12	in	in	ADP
ejpam-6846	133	13	this	this	DET
ejpam-6846	133	14	section	section	NOUN
ejpam-6846	133	15	,	,	PUNCT
ejpam-6846	133	16	we	we	PRON
ejpam-6846	133	17	generalized	generalize	VERB
ejpam-6846	133	18	results	result	NOUN
ejpam-6846	133	19	related	relate	VERB
ejpam-6846	133	20	to	to	ADP
ejpam-6846	133	21	generalized	generalized	ADJ
ejpam-6846	133	22	projection	projection	NOUN
ejpam-6846	133	23	on	on	ADP
ejpam-6846	133	24	closed	close	VERB
ejpam-6846	133	25	nonconvex	nonconvex	NOUN
ejpam-6846	133	26	sets	set	NOUN
ejpam-6846	133	27	πs	πs	ADP
ejpam-6846	133	28	to	to	PART
ejpam-6846	133	29	πf	πf	VERB
ejpam-6846	133	30	s	s	VERB
ejpam-6846	133	31	on	on	ADP
ejpam-6846	133	32	closed	close	VERB
ejpam-6846	133	33	nonconvex	nonconvex	NOUN
ejpam-6846	133	34	sets	set	NOUN
ejpam-6846	133	35	with	with	ADP
ejpam-6846	133	36	f	f	PROPN
ejpam-6846	133	37	is	be	AUX
ejpam-6846	133	38	a	a	DET
ejpam-6846	133	39	proper	proper	ADJ
ejpam-6846	133	40	,	,	PUNCT
ejpam-6846	133	41	lower	low	ADJ
ejpam-6846	133	42	semi	semi	ADJ
ejpam-6846	133	43	-	-	ADJ
ejpam-6846	133	44	continuous	continuous	ADJ
ejpam-6846	133	45	function	function	NOUN
ejpam-6846	133	46	(	(	PUNCT
ejpam-6846	133	47	unless	unless	SCONJ
ejpam-6846	133	48	we	we	PRON
ejpam-6846	133	49	specified	specify	VERB
ejpam-6846	133	50	otherwise	otherwise	ADV
ejpam-6846	133	51	,	,	PUNCT
ejpam-6846	133	52	)	)	PUNCT
ejpam-6846	133	53	.	.	PUNCT
ejpam-6846	134	1	so	so	ADV
ejpam-6846	134	2	we	we	PRON
ejpam-6846	134	3	start	start	VERB
ejpam-6846	134	4	with	with	ADP
ejpam-6846	134	5	the	the	DET
ejpam-6846	134	6	following	follow	VERB
ejpam-6846	134	7	lemma	lemma	PROPN
ejpam-6846	134	8	that	that	PRON
ejpam-6846	134	9	is	be	AUX
ejpam-6846	134	10	analog	analog	NOUN
ejpam-6846	134	11	with	with	ADP
ejpam-6846	134	12	lemma	lemma	PROPN
ejpam-6846	134	13	2.1	2.1	NUM
ejpam-6846	134	14	[	[	SYM
ejpam-6846	134	15	15	15	NUM
ejpam-6846	134	16	]	]	PUNCT
ejpam-6846	134	17	for	for	ADP
ejpam-6846	134	18	metric	metric	ADJ
ejpam-6846	134	19	projection	projection	NOUN
ejpam-6846	134	20	,	,	PUNCT
ejpam-6846	134	21	proposition	proposition	NOUN
ejpam-6846	134	22	1.3	1.3	NUM
ejpam-6846	134	23	in	in	ADP
ejpam-6846	134	24	[	[	X
ejpam-6846	134	25	16	16	NUM
ejpam-6846	134	26	]	]	PUNCT
ejpam-6846	134	27	for	for	ADP
ejpam-6846	134	28	hilbert	hilbert	PROPN
ejpam-6846	134	29	spaces	space	NOUN
ejpam-6846	134	30	,	,	PUNCT
ejpam-6846	134	31	lemma	lemma	PROPN
ejpam-6846	134	32	2.2	2.2	NUM
ejpam-6846	135	1	[	[	X
ejpam-6846	135	2	6	6	NUM
ejpam-6846	135	3	]	]	PUNCT
ejpam-6846	135	4	for	for	ADP
ejpam-6846	135	5	generalized	generalized	ADJ
ejpam-6846	135	6	projection	projection	NOUN
ejpam-6846	135	7	on	on	ADP
ejpam-6846	135	8	closed	close	VERB
ejpam-6846	135	9	nonconvex	nonconvex	NOUN
ejpam-6846	135	10	sets	set	NOUN
ejpam-6846	135	11	and	and	CCONJ
ejpam-6846	135	12	recently	recently	ADV
ejpam-6846	135	13	this	this	DET
ejpam-6846	135	14	lemma	lemma	PROPN
ejpam-6846	135	15	proved	prove	VERB
ejpam-6846	135	16	for	for	ADP
ejpam-6846	135	17	generalized	generalized	ADJ
ejpam-6846	135	18	(	(	PUNCT
ejpam-6846	135	19	f	f	X
ejpam-6846	135	20	,	,	PUNCT
ejpam-6846	135	21	λ)-projection	λ)-projection	PUNCT
ejpam-6846	135	22	operator	operator	NOUN
ejpam-6846	135	23	see	see	VERB
ejpam-6846	135	24	[	[	X
ejpam-6846	135	25	7	7	NUM
ejpam-6846	135	26	]	]	PUNCT
ejpam-6846	135	27	.	.	PUNCT
ejpam-6846	136	1	lemma	lemma	PROPN
ejpam-6846	136	2	1	1	X
ejpam-6846	136	3	.	.	PUNCT
ejpam-6846	137	1	let	let	VERB
ejpam-6846	137	2	x∗	x∗	PROPN
ejpam-6846	137	3	be	be	AUX
ejpam-6846	137	4	a	a	DET
ejpam-6846	137	5	smooth	smooth	ADJ
ejpam-6846	137	6	dual	dual	ADJ
ejpam-6846	137	7	norm	norm	NOUN
ejpam-6846	137	8	space	space	NOUN
ejpam-6846	137	9	of	of	ADP
ejpam-6846	137	10	a	a	DET
ejpam-6846	137	11	reflexive	reflexive	ADJ
ejpam-6846	137	12	banach	banach	NOUN
ejpam-6846	137	13	space	space	NOUN
ejpam-6846	137	14	x.	x.	NOUN
ejpam-6846	137	15	then	then	ADV
ejpam-6846	137	16	for	for	ADP
ejpam-6846	137	17	every	every	DET
ejpam-6846	137	18	β	β	X
ejpam-6846	137	19	∈	∈	PROPN
ejpam-6846	137	20	(	(	PUNCT
ejpam-6846	137	21	0	0	NUM
ejpam-6846	137	22	,	,	PUNCT
ejpam-6846	137	23	1	1	NUM
ejpam-6846	137	24	)	)	PUNCT
ejpam-6846	137	25	and	and	CCONJ
ejpam-6846	137	26	for	for	ADP
ejpam-6846	137	27	all	all	DET
ejpam-6846	137	28	u	u	PROPN
ejpam-6846	137	29	∈	∈	PROPN
ejpam-6846	137	30	πf	πf	NOUN
ejpam-6846	137	31	s(j(v	s(j(v	NOUN
ejpam-6846	137	32	)	)	PUNCT
ejpam-6846	137	33	)	)	PUNCT
ejpam-6846	137	34	,	,	PUNCT
ejpam-6846	137	35	we	we	PRON
ejpam-6846	137	36	get	get	VERB
ejpam-6846	137	37	πf	πf	INTJ
ejpam-6846	137	38	s	s	PART
ejpam-6846	137	39	(	(	PUNCT
ejpam-6846	137	40	(	(	PUNCT
ejpam-6846	137	41	1−	1−	NUM
ejpam-6846	137	42	β)j(u	β)j(u	NUM
ejpam-6846	137	43	)	)	PUNCT
ejpam-6846	137	44	+	+	CCONJ
ejpam-6846	137	45	βj(v	βj(v	NOUN
ejpam-6846	137	46	)	)	PUNCT
ejpam-6846	137	47	)	)	PUNCT
ejpam-6846	138	1	=	=	PRON
ejpam-6846	138	2	{	{	PUNCT
ejpam-6846	138	3	u	u	NOUN
ejpam-6846	138	4	}	}	PUNCT
ejpam-6846	138	5	.	.	PUNCT
ejpam-6846	139	1	proof	proof	NOUN
ejpam-6846	139	2	:	:	PUNCT
ejpam-6846	139	3	for	for	ADP
ejpam-6846	139	4	proving	prove	VERB
ejpam-6846	139	5	u	u	PRON
ejpam-6846	139	6	∈	∈	PROPN
ejpam-6846	139	7	πf	πf	NOUN
ejpam-6846	139	8	s	s	X
ejpam-6846	139	9	(	(	PUNCT
ejpam-6846	139	10	(	(	PUNCT
ejpam-6846	139	11	1−	1−	NUM
ejpam-6846	139	12	β)j(u	β)j(u	NUM
ejpam-6846	139	13	)	)	PUNCT
ejpam-6846	139	14	+	+	CCONJ
ejpam-6846	139	15	βj(v	βj(v	NUM
ejpam-6846	139	16	)	)	PUNCT
ejpam-6846	139	17	)	)	PUNCT
ejpam-6846	139	18	proceeds	proceed	VERB
ejpam-6846	139	19	like	like	ADP
ejpam-6846	139	20	theorem	theorem	VERB
ejpam-6846	139	21	3.1	3.1	NUM
ejpam-6846	139	22	part	part	NOUN
ejpam-6846	139	23	3	3	NUM
ejpam-6846	139	24	in	in	ADP
ejpam-6846	139	25	[	[	X
ejpam-6846	139	26	7	7	NUM
ejpam-6846	139	27	]	]	PUNCT
ejpam-6846	139	28	by	by	ADP
ejpam-6846	139	29	taking	take	VERB
ejpam-6846	139	30	the	the	DET
ejpam-6846	139	31	constant	constant	ADJ
ejpam-6846	139	32	λ	λ	NOUN
ejpam-6846	139	33	defined	define	VERB
ejpam-6846	139	34	there	there	ADV
ejpam-6846	139	35	,	,	PUNCT
ejpam-6846	139	36	equal	equal	ADJ
ejpam-6846	139	37	half	half	NOUN
ejpam-6846	139	38	.	.	PUNCT
ejpam-6846	140	1	unfortunately	unfortunately	ADV
ejpam-6846	140	2	,	,	PUNCT
ejpam-6846	140	3	we	we	PRON
ejpam-6846	140	4	ca	can	AUX
ejpam-6846	140	5	n’t	not	PART
ejpam-6846	140	6	proceed	proceed	VERB
ejpam-6846	140	7	only	only	ADV
ejpam-6846	140	8	with	with	ADP
ejpam-6846	140	9	λ	λ	PROPN
ejpam-6846	140	10	=	=	NOUN
ejpam-6846	140	11	1	1	NUM
ejpam-6846	140	12	2	2	NUM
ejpam-6846	140	13	for	for	ADP
ejpam-6846	140	14	the	the	DET
ejpam-6846	140	15	uniqueness	uniqueness	ADJ
ejpam-6846	140	16	part	part	NOUN
ejpam-6846	140	17	,	,	PUNCT
ejpam-6846	140	18	so	so	ADV
ejpam-6846	140	19	we	we	PRON
ejpam-6846	140	20	assume	assume	VERB
ejpam-6846	140	21	that	that	SCONJ
ejpam-6846	140	22	u0	u0	PROPN
ejpam-6846	140	23	∈	∈	PROPN
ejpam-6846	141	1	πf	πf	NOUN
ejpam-6846	141	2	s	s	PROPN
ejpam-6846	141	3	(	(	PUNCT
ejpam-6846	141	4	(	(	PUNCT
ejpam-6846	141	5	1−	1−	NUM
ejpam-6846	141	6	β)j(u	β)j(u	NUM
ejpam-6846	141	7	)	)	PUNCT
ejpam-6846	141	8	+	+	CCONJ
ejpam-6846	141	9	βj(v	βj(v	NOUN
ejpam-6846	141	10	)	)	PUNCT
ejpam-6846	141	11	)	)	PUNCT
ejpam-6846	141	12	,	,	PUNCT
ejpam-6846	141	13	where	where	SCONJ
ejpam-6846	141	14	u	u	PROPN
ejpam-6846	141	15	6=	6=	SYM
ejpam-6846	141	16	u0	u0	PROPN
ejpam-6846	141	17	.	.	PUNCT
ejpam-6846	142	1	thus	thus	ADV
ejpam-6846	142	2	,	,	PUNCT
ejpam-6846	142	3	we	we	PRON
ejpam-6846	142	4	have	have	VERB
ejpam-6846	142	5	two	two	NUM
ejpam-6846	142	6	cases	case	NOUN
ejpam-6846	142	7	:	:	PUNCT
ejpam-6846	142	8	case1	case1	NOUN
ejpam-6846	142	9	:	:	PUNCT
ejpam-6846	142	10	f(u0)−	f(u0)−	PROPN
ejpam-6846	142	11	f(u	f(u	PROPN
ejpam-6846	142	12	)	)	PUNCT
ejpam-6846	142	13	≥	≥	NOUN
ejpam-6846	142	14	0	0	NUM
ejpam-6846	142	15	,	,	PUNCT
ejpam-6846	142	16	we	we	PRON
ejpam-6846	142	17	get	get	VERB
ejpam-6846	142	18	v	v	ADP
ejpam-6846	142	19	f	f	X
ejpam-6846	142	20	(	(	PUNCT
ejpam-6846	142	21	(	(	PUNCT
ejpam-6846	142	22	1−	1−	NUM
ejpam-6846	142	23	β)j(u	β)j(u	NUM
ejpam-6846	142	24	)	)	PUNCT
ejpam-6846	142	25	+	+	X
ejpam-6846	142	26	βj(v	βj(v	NOUN
ejpam-6846	142	27	)	)	PUNCT
ejpam-6846	142	28	,	,	PUNCT
ejpam-6846	142	29	u	u	NOUN
ejpam-6846	142	30	)	)	PUNCT
ejpam-6846	142	31	=	=	SYM
ejpam-6846	143	1	v	v	NUM
ejpam-6846	143	2	f	f	X
ejpam-6846	143	3	(	(	PUNCT
ejpam-6846	143	4	(	(	PUNCT
ejpam-6846	143	5	1−	1−	NUM
ejpam-6846	143	6	β)j(u	β)j(u	NUM
ejpam-6846	143	7	)	)	PUNCT
ejpam-6846	143	8	+	+	X
ejpam-6846	143	9	βj(v	βj(v	NOUN
ejpam-6846	143	10	)	)	PUNCT
ejpam-6846	143	11	,	,	PUNCT
ejpam-6846	143	12	u0	u0	ADJ
ejpam-6846	143	13	)	)	PUNCT
ejpam-6846	143	14	.	.	PUNCT
ejpam-6846	144	1	a.	a.	PROPN
ejpam-6846	144	2	al	al	PROPN
ejpam-6846	144	3	tane	tane	PROPN
ejpam-6846	144	4	,	,	PUNCT
ejpam-6846	144	5	l.	l.	PROPN
ejpam-6846	144	6	s.	s.	PROPN
ejpam-6846	144	7	keong	keong	PROPN
ejpam-6846	144	8	/	/	PUNCT
ejpam-6846	144	9	eur	eur	PROPN
ejpam-6846	144	10	.	.	PUNCT
ejpam-6846	145	1	j.	j.	PROPN
ejpam-6846	145	2	pure	pure	PROPN
ejpam-6846	145	3	appl	appl	PROPN
ejpam-6846	145	4	.	.	PROPN
ejpam-6846	145	5	math	math	PROPN
ejpam-6846	145	6	,	,	PUNCT
ejpam-6846	145	7	18	18	NUM
ejpam-6846	145	8	(	(	PUNCT
ejpam-6846	145	9	4	4	NUM
ejpam-6846	145	10	)	)	PUNCT
ejpam-6846	145	11	(	(	PUNCT
ejpam-6846	145	12	2025	2025	NUM
ejpam-6846	145	13	)	)	PUNCT
ejpam-6846	145	14	,	,	PUNCT
ejpam-6846	145	15	6846	6846	NUM
ejpam-6846	145	16	6	6	NUM
ejpam-6846	145	17	of	of	ADP
ejpam-6846	145	18	13	13	NUM
ejpam-6846	145	19	so	so	ADV
ejpam-6846	145	20	v	v	NOUN
ejpam-6846	145	21	(	(	PUNCT
ejpam-6846	145	22	(	(	PUNCT
ejpam-6846	145	23	1−	1−	NUM
ejpam-6846	145	24	β)j(u	β)j(u	NUM
ejpam-6846	145	25	)	)	PUNCT
ejpam-6846	145	26	+	+	X
ejpam-6846	145	27	βj(v	βj(v	NOUN
ejpam-6846	145	28	)	)	PUNCT
ejpam-6846	145	29	,	,	PUNCT
ejpam-6846	145	30	u)−v	u)−v	PROPN
ejpam-6846	145	31	(	(	PUNCT
ejpam-6846	145	32	(	(	PUNCT
ejpam-6846	145	33	1−	1−	NUM
ejpam-6846	145	34	β)j(u	β)j(u	NUM
ejpam-6846	145	35	)	)	PUNCT
ejpam-6846	145	36	+	+	X
ejpam-6846	145	37	βj(v	βj(v	NOUN
ejpam-6846	145	38	)	)	PUNCT
ejpam-6846	145	39	,	,	PUNCT
ejpam-6846	145	40	u0	u0	ADJ
ejpam-6846	145	41	)	)	PUNCT
ejpam-6846	145	42	=	=	SYM
ejpam-6846	145	43	f(u0)−f(u	f(u0)−f(u	ADJ
ejpam-6846	145	44	)	)	PUNCT
ejpam-6846	145	45	≥	≥	NOUN
ejpam-6846	145	46	β[f(u0)−f(u	β[f(u0)−f(u	NUM
ejpam-6846	145	47	)	)	PUNCT
ejpam-6846	145	48	]	]	PUNCT
ejpam-6846	145	49	.	.	PUNCT
ejpam-6846	146	1	on	on	ADP
ejpam-6846	146	2	the	the	DET
ejpam-6846	146	3	other	other	ADJ
ejpam-6846	146	4	hand	hand	NOUN
ejpam-6846	146	5	v	v	NOUN
ejpam-6846	146	6	(	(	PUNCT
ejpam-6846	146	7	(	(	PUNCT
ejpam-6846	146	8	1−	1−	NUM
ejpam-6846	146	9	β)j(u	β)j(u	NUM
ejpam-6846	146	10	)	)	PUNCT
ejpam-6846	146	11	+	+	X
ejpam-6846	146	12	βj(v	βj(v	NOUN
ejpam-6846	146	13	)	)	PUNCT
ejpam-6846	146	14	,	,	PUNCT
ejpam-6846	146	15	u)−	u)−	PROPN
ejpam-6846	146	16	v	v	PROPN
ejpam-6846	146	17	(	(	PUNCT
ejpam-6846	146	18	(	(	PUNCT
ejpam-6846	146	19	1−	1−	NUM
ejpam-6846	146	20	β)j(u	β)j(u	NUM
ejpam-6846	146	21	)	)	PUNCT
ejpam-6846	146	22	+	+	X
ejpam-6846	146	23	βj(v	βj(v	NOUN
ejpam-6846	146	24	)	)	PUNCT
ejpam-6846	146	25	,	,	PUNCT
ejpam-6846	146	26	u0	u0	ADJ
ejpam-6846	146	27	)	)	PUNCT
ejpam-6846	146	28	=	=	SYM
ejpam-6846	146	29	β‖u‖2	β‖u‖2	PROPN
ejpam-6846	146	30	+	+	CCONJ
ejpam-6846	146	31	(	(	PUNCT
ejpam-6846	146	32	1−	1−	NUM
ejpam-6846	146	33	β)‖u‖2	β)‖u‖2	PUNCT
ejpam-6846	146	34	−	−	PROPN
ejpam-6846	146	35	β‖u0‖2	β‖u0‖2	PRON
ejpam-6846	146	36	−	−	PROPN
ejpam-6846	147	1	(	(	PUNCT
ejpam-6846	147	2	1−	1−	NUM
ejpam-6846	147	3	β)‖u0‖2	β)‖u0‖2	PUNCT
ejpam-6846	147	4	−	−	PROPN
ejpam-6846	147	5	2(1−	2(1−	NUM
ejpam-6846	147	6	β)〈j(u	β)〈j(u	NUM
ejpam-6846	147	7	)	)	PUNCT
ejpam-6846	147	8	,	,	PUNCT
ejpam-6846	147	9	u〉+	u〉+	PROPN
ejpam-6846	147	10	2(1−	2(1−	NUM
ejpam-6846	147	11	β)〈j(u	β)〈j(u	NUM
ejpam-6846	147	12	)	)	PUNCT
ejpam-6846	147	13	,	,	PUNCT
ejpam-6846	147	14	u0	u0	PROPN
ejpam-6846	147	15	〉	〉	PROPN
ejpam-6846	147	16	−	−	PROPN
ejpam-6846	147	17	2β〈j(v	2β〈j(v	NUM
ejpam-6846	147	18	)	)	PUNCT
ejpam-6846	147	19	,	,	PUNCT
ejpam-6846	147	20	u〉+	u〉+	PROPN
ejpam-6846	147	21	2β〈j(v	2β〈j(v	NUM
ejpam-6846	147	22	)	)	PUNCT
ejpam-6846	147	23	,	,	PUNCT
ejpam-6846	147	24	u0	u0	PROPN
ejpam-6846	147	25	〉	〉	PROPN
ejpam-6846	147	26	=	=	SYM
ejpam-6846	147	27	β	β	X
ejpam-6846	147	28	(	(	PUNCT
ejpam-6846	147	29	v	v	NOUN
ejpam-6846	147	30	(	(	PUNCT
ejpam-6846	147	31	j(v	j(v	NOUN
ejpam-6846	147	32	)	)	PUNCT
ejpam-6846	147	33	,	,	PUNCT
ejpam-6846	147	34	u)−	u)−	PROPN
ejpam-6846	147	35	v	v	PROPN
ejpam-6846	147	36	(	(	PUNCT
ejpam-6846	147	37	j(v	j(v	PROPN
ejpam-6846	147	38	)	)	PUNCT
ejpam-6846	147	39	,	,	PUNCT
ejpam-6846	147	40	u0))−	u0))−	PROPN
ejpam-6846	147	41	(	(	PUNCT
ejpam-6846	147	42	1−	1−	NUM
ejpam-6846	147	43	β)v	β)v	X
ejpam-6846	147	44	(	(	PUNCT
ejpam-6846	147	45	j(u	j(u	PROPN
ejpam-6846	147	46	)	)	PUNCT
ejpam-6846	147	47	,	,	PUNCT
ejpam-6846	147	48	u0	u0	ADJ
ejpam-6846	147	49	)	)	PUNCT
ejpam-6846	147	50	.	.	PUNCT
ejpam-6846	148	1	and	and	CCONJ
ejpam-6846	148	2	so	so	ADV
ejpam-6846	148	3	β	β	X
ejpam-6846	148	4	(	(	PUNCT
ejpam-6846	148	5	v	v	NOUN
ejpam-6846	148	6	(	(	PUNCT
ejpam-6846	148	7	j(v	j(v	NOUN
ejpam-6846	148	8	)	)	PUNCT
ejpam-6846	148	9	,	,	PUNCT
ejpam-6846	148	10	u)−	u)−	PROPN
ejpam-6846	148	11	v	v	PROPN
ejpam-6846	148	12	(	(	PUNCT
ejpam-6846	148	13	j(v	j(v	PROPN
ejpam-6846	148	14	)	)	PUNCT
ejpam-6846	148	15	,	,	PUNCT
ejpam-6846	148	16	u0))−	u0))−	PROPN
ejpam-6846	148	17	(	(	PUNCT
ejpam-6846	148	18	1−	1−	NUM
ejpam-6846	148	19	β)v	β)v	X
ejpam-6846	148	20	(	(	PUNCT
ejpam-6846	148	21	j(u	j(u	PROPN
ejpam-6846	148	22	)	)	PUNCT
ejpam-6846	148	23	,	,	PUNCT
ejpam-6846	148	24	u0	u0	PROPN
ejpam-6846	148	25	)	)	PUNCT
ejpam-6846	148	26	≥	≥	PROPN
ejpam-6846	148	27	β[f(u0)−	β[f(u0)−	NUM
ejpam-6846	148	28	f(u	f(u	PROPN
ejpam-6846	148	29	)	)	PUNCT
ejpam-6846	148	30	]	]	PUNCT
ejpam-6846	148	31	.	.	PUNCT
ejpam-6846	149	1	hence	hence	ADV
ejpam-6846	149	2	,	,	PUNCT
ejpam-6846	149	3	(	(	PUNCT
ejpam-6846	149	4	1−	1−	NUM
ejpam-6846	149	5	β)v	β)v	X
ejpam-6846	149	6	(	(	PUNCT
ejpam-6846	149	7	j(u	j(u	PROPN
ejpam-6846	149	8	)	)	PUNCT
ejpam-6846	149	9	,	,	PUNCT
ejpam-6846	149	10	u0	u0	ADJ
ejpam-6846	149	11	)	)	PUNCT
ejpam-6846	149	12	≤	≤	NOUN
ejpam-6846	150	1	β	β	X
ejpam-6846	150	2	(	(	PUNCT
ejpam-6846	150	3	v	v	NOUN
ejpam-6846	150	4	(	(	PUNCT
ejpam-6846	150	5	j(v	j(v	NOUN
ejpam-6846	150	6	)	)	PUNCT
ejpam-6846	150	7	,	,	PUNCT
ejpam-6846	150	8	u	u	NOUN
ejpam-6846	150	9	)	)	PUNCT
ejpam-6846	150	10	+	+	CCONJ
ejpam-6846	150	11	f(u)−	f(u)−	PROPN
ejpam-6846	150	12	v	v	NOUN
ejpam-6846	150	13	(	(	PUNCT
ejpam-6846	150	14	j(v	j(v	PROPN
ejpam-6846	150	15	)	)	PUNCT
ejpam-6846	150	16	,	,	PUNCT
ejpam-6846	150	17	u0)−	u0)−	PROPN
ejpam-6846	150	18	f(u0	f(u0	ADJ
ejpam-6846	150	19	)	)	PUNCT
ejpam-6846	150	20	)	)	PUNCT
ejpam-6846	150	21	=	=	SYM
ejpam-6846	151	1	β	β	X
ejpam-6846	151	2	(	(	PUNCT
ejpam-6846	151	3	v	v	NOUN
ejpam-6846	151	4	f	f	X
ejpam-6846	151	5	(	(	PUNCT
ejpam-6846	151	6	j(v	j(v	PROPN
ejpam-6846	151	7	)	)	PUNCT
ejpam-6846	151	8	,	,	PUNCT
ejpam-6846	151	9	u)−	u)−	PROPN
ejpam-6846	151	10	v	v	PROPN
ejpam-6846	151	11	f	f	X
ejpam-6846	151	12	(	(	PUNCT
ejpam-6846	151	13	j(v	j(v	PROPN
ejpam-6846	151	14	)	)	PUNCT
ejpam-6846	151	15	,	,	PUNCT
ejpam-6846	151	16	u0	u0	ADJ
ejpam-6846	151	17	)	)	PUNCT
ejpam-6846	151	18	)	)	PUNCT
ejpam-6846	151	19	≤	≤	ADV
ejpam-6846	151	20	0	0	X
ejpam-6846	151	21	.	.	PUNCT
ejpam-6846	152	1	since	since	SCONJ
ejpam-6846	152	2	u	u	PROPN
ejpam-6846	152	3	∈	∈	PROPN
ejpam-6846	152	4	πf	πf	INTJ
ejpam-6846	152	5	s(j(v	s(j(v	NOUN
ejpam-6846	152	6	)	)	PUNCT
ejpam-6846	152	7	)	)	PUNCT
ejpam-6846	152	8	,	,	PUNCT
ejpam-6846	152	9	and	and	CCONJ
ejpam-6846	152	10	so	so	ADV
ejpam-6846	152	11	v	v	ADP
ejpam-6846	152	12	f	f	X
ejpam-6846	152	13	(	(	PUNCT
ejpam-6846	152	14	j(v	j(v	PROPN
ejpam-6846	152	15	)	)	PUNCT
ejpam-6846	152	16	,	,	PUNCT
ejpam-6846	152	17	u	u	NOUN
ejpam-6846	152	18	)	)	PUNCT
ejpam-6846	152	19	≤	≤	NOUN
ejpam-6846	152	20	v	v	ADP
ejpam-6846	152	21	f	f	PROPN
ejpam-6846	152	22	(	(	PUNCT
ejpam-6846	152	23	j(v	j(v	PROPN
ejpam-6846	152	24	)	)	PUNCT
ejpam-6846	152	25	,	,	PUNCT
ejpam-6846	152	26	u0	u0	PROPN
ejpam-6846	152	27	)	)	PUNCT
ejpam-6846	152	28	.	.	PUNCT
ejpam-6846	153	1	which	which	PRON
ejpam-6846	153	2	implies	imply	VERB
ejpam-6846	153	3	that	that	SCONJ
ejpam-6846	153	4	v	v	NOUN
ejpam-6846	153	5	(	(	PUNCT
ejpam-6846	153	6	j(u	j(u	PROPN
ejpam-6846	153	7	)	)	PUNCT
ejpam-6846	153	8	,	,	PUNCT
ejpam-6846	153	9	u0	u0	ADJ
ejpam-6846	153	10	)	)	PUNCT
ejpam-6846	153	11	≤	≤	NOUN
ejpam-6846	153	12	0	0	NUM
ejpam-6846	153	13	and	and	CCONJ
ejpam-6846	153	14	hence	hence	ADV
ejpam-6846	153	15	v	v	NOUN
ejpam-6846	153	16	(	(	PUNCT
ejpam-6846	153	17	j(u	j(u	PROPN
ejpam-6846	153	18	)	)	PUNCT
ejpam-6846	153	19	,	,	PUNCT
ejpam-6846	153	20	u0	u0	ADJ
ejpam-6846	153	21	)	)	PUNCT
ejpam-6846	153	22	=	=	SYM
ejpam-6846	153	23	0	0	NUM
ejpam-6846	153	24	,	,	PUNCT
ejpam-6846	153	25	and	and	CCONJ
ejpam-6846	153	26	so	so	ADV
ejpam-6846	153	27	u	u	X
ejpam-6846	153	28	=	=	PUNCT
ejpam-6846	153	29	u0	u0	PROPN
ejpam-6846	153	30	.	.	PUNCT
ejpam-6846	153	31	case2	case2	PROPN
ejpam-6846	153	32	:	:	PUNCT
ejpam-6846	153	33	f(u)−	f(u)−	NOUN
ejpam-6846	153	34	f(u0	f(u0	ADJ
ejpam-6846	153	35	)	)	PUNCT
ejpam-6846	153	36	≥	≥	NOUN
ejpam-6846	153	37	0	0	NUM
ejpam-6846	153	38	,	,	PUNCT
ejpam-6846	153	39	we	we	PRON
ejpam-6846	153	40	get	get	VERB
ejpam-6846	153	41	v	v	ADP
ejpam-6846	153	42	f	f	X
ejpam-6846	153	43	(	(	PUNCT
ejpam-6846	153	44	(	(	PUNCT
ejpam-6846	153	45	1−	1−	NUM
ejpam-6846	153	46	β)j(u	β)j(u	NUM
ejpam-6846	153	47	)	)	PUNCT
ejpam-6846	153	48	+	+	X
ejpam-6846	153	49	βj(v	βj(v	NOUN
ejpam-6846	153	50	)	)	PUNCT
ejpam-6846	153	51	,	,	PUNCT
ejpam-6846	153	52	u	u	NOUN
ejpam-6846	153	53	)	)	PUNCT
ejpam-6846	153	54	=	=	SYM
ejpam-6846	153	55	v	v	NUM
ejpam-6846	153	56	f	f	X
ejpam-6846	153	57	(	(	PUNCT
ejpam-6846	153	58	(	(	PUNCT
ejpam-6846	153	59	1−	1−	NUM
ejpam-6846	153	60	β)j(u	β)j(u	NUM
ejpam-6846	153	61	)	)	PUNCT
ejpam-6846	153	62	+	+	X
ejpam-6846	153	63	βj(v	βj(v	NOUN
ejpam-6846	153	64	)	)	PUNCT
ejpam-6846	153	65	,	,	PUNCT
ejpam-6846	153	66	u0	u0	ADJ
ejpam-6846	153	67	)	)	PUNCT
ejpam-6846	153	68	.	.	PUNCT
ejpam-6846	154	1	so	so	ADV
ejpam-6846	154	2	v	v	INTJ
ejpam-6846	154	3	(	(	PUNCT
ejpam-6846	154	4	(	(	PUNCT
ejpam-6846	154	5	1−β)j(u)+βj(v	1−β)j(u)+βj(v	NUM
ejpam-6846	154	6	)	)	PUNCT
ejpam-6846	154	7	,	,	PUNCT
ejpam-6846	154	8	u0)−v	u0)−v	PROPN
ejpam-6846	154	9	(	(	PUNCT
ejpam-6846	154	10	(	(	PUNCT
ejpam-6846	154	11	1−β)j(u)+βj(v	1−β)j(u)+βj(v	NUM
ejpam-6846	154	12	)	)	PUNCT
ejpam-6846	154	13	,	,	PUNCT
ejpam-6846	154	14	u	u	NOUN
ejpam-6846	154	15	)	)	PUNCT
ejpam-6846	154	16	=	=	PUNCT
ejpam-6846	154	17	f(u)−	f(u)−	NOUN
ejpam-6846	154	18	f(u0	f(u0	ADJ
ejpam-6846	154	19	)	)	PUNCT
ejpam-6846	154	20	≥	≥	NOUN
ejpam-6846	154	21	β[f(u)−	β[f(u)−	ADP
ejpam-6846	154	22	f(u0	f(u0	ADJ
ejpam-6846	154	23	)	)	PUNCT
ejpam-6846	154	24	]	]	PUNCT
ejpam-6846	154	25	.	.	PUNCT
ejpam-6846	155	1	proceed	proceed	VERB
ejpam-6846	155	2	like	like	ADP
ejpam-6846	155	3	case1	case1	PROPN
ejpam-6846	155	4	,	,	PUNCT
ejpam-6846	155	5	we	we	PRON
ejpam-6846	155	6	get	get	VERB
ejpam-6846	155	7	the	the	DET
ejpam-6846	155	8	same	same	ADJ
ejpam-6846	155	9	result	result	NOUN
ejpam-6846	155	10	,	,	PUNCT
ejpam-6846	155	11	and	and	CCONJ
ejpam-6846	155	12	this	this	PRON
ejpam-6846	155	13	ends	end	VERB
ejpam-6846	155	14	our	our	PRON
ejpam-6846	155	15	prove	prove	NOUN
ejpam-6846	155	16	□	□	PUNCT
ejpam-6846	155	17	now	now	ADV
ejpam-6846	155	18	,	,	PUNCT
ejpam-6846	155	19	using	use	VERB
ejpam-6846	155	20	concepts	concept	NOUN
ejpam-6846	155	21	from	from	ADP
ejpam-6846	155	22	[	[	X
ejpam-6846	155	23	17	17	NUM
ejpam-6846	155	24	]	]	PUNCT
ejpam-6846	155	25	and	and	CCONJ
ejpam-6846	155	26	[	[	X
ejpam-6846	155	27	6	6	NUM
ejpam-6846	155	28	]	]	PUNCT
ejpam-6846	155	29	.	.	PUNCT
ejpam-6846	156	1	we	we	PRON
ejpam-6846	156	2	provide	provide	VERB
ejpam-6846	156	3	a	a	DET
ejpam-6846	156	4	necessary	necessary	ADJ
ejpam-6846	156	5	and	and	CCONJ
ejpam-6846	156	6	sufficient	sufficient	ADJ
ejpam-6846	156	7	condition	condition	NOUN
ejpam-6846	156	8	for	for	ADP
ejpam-6846	156	9	the	the	DET
ejpam-6846	156	10	existence	existence	NOUN
ejpam-6846	156	11	and	and	CCONJ
ejpam-6846	156	12	uniqueness	uniqueness	NOUN
ejpam-6846	156	13	of	of	ADP
ejpam-6846	156	14	the	the	DET
ejpam-6846	156	15	mf	mf	PROPN
ejpam-6846	156	16	,	,	PUNCT
ejpam-6846	156	17	s(x	s(x	NOUN
ejpam-6846	156	18	∗	∗	NOUN
ejpam-6846	156	19	)	)	PUNCT
ejpam-6846	156	20	in	in	ADP
ejpam-6846	156	21	the	the	DET
ejpam-6846	156	22	following	follow	VERB
ejpam-6846	156	23	lemma	lemma	PROPN
ejpam-6846	156	24	,	,	PUNCT
ejpam-6846	156	25	in	in	ADP
ejpam-6846	156	26	terms	term	NOUN
ejpam-6846	156	27	of	of	ADP
ejpam-6846	156	28	∂fmf	∂fmf	NUM
ejpam-6846	156	29	,	,	PUNCT
ejpam-6846	156	30	s(x	s(x	NOUN
ejpam-6846	156	31	∗	∗	NOUN
ejpam-6846	156	32	)	)	PUNCT
ejpam-6846	156	33	.	.	PUNCT
ejpam-6846	157	1	lemma	lemma	PROPN
ejpam-6846	157	2	2	2	X
ejpam-6846	157	3	.	.	PUNCT
ejpam-6846	158	1	let	let	VERB
ejpam-6846	158	2	x∗	x∗	PROPN
ejpam-6846	158	3	be	be	AUX
ejpam-6846	158	4	an	an	DET
ejpam-6846	158	5	element	element	NOUN
ejpam-6846	158	6	of	of	ADP
ejpam-6846	158	7	smooth	smooth	ADJ
ejpam-6846	158	8	dual	dual	ADJ
ejpam-6846	158	9	norm	norm	NOUN
ejpam-6846	158	10	space	space	NOUN
ejpam-6846	158	11	x∗	x∗	PROPN
ejpam-6846	158	12	of	of	ADP
ejpam-6846	158	13	reflexive	reflexive	ADJ
ejpam-6846	158	14	banach	banach	NOUN
ejpam-6846	158	15	space	space	NOUN
ejpam-6846	159	1	x.	x.	NOUN
ejpam-6846	160	1	then	then	ADV
ejpam-6846	160	2	πf	πf	AUX
ejpam-6846	160	3	s(x	s(x	PROPN
ejpam-6846	160	4	∗	∗	NOUN
ejpam-6846	160	5	)	)	PUNCT
ejpam-6846	160	6	is	be	AUX
ejpam-6846	160	7	exists	exist	NOUN
ejpam-6846	160	8	and	and	CCONJ
ejpam-6846	160	9	unique	unique	ADJ
ejpam-6846	160	10	,	,	PUNCT
ejpam-6846	160	11	whenever	whenever	SCONJ
ejpam-6846	160	12	∂fmf	∂fmf	NUM
ejpam-6846	160	13	,	,	PUNCT
ejpam-6846	160	14	s(x	s(x	NOUN
ejpam-6846	160	15	∗	∗	NOUN
ejpam-6846	160	16	)	)	PUNCT
ejpam-6846	160	17	6=	6=	PUNCT
ejpam-6846	161	1	ϕ.	ϕ.	PROPN
ejpam-6846	161	2	also	also	ADV
ejpam-6846	161	3	,	,	PUNCT
ejpam-6846	161	4	we	we	PRON
ejpam-6846	161	5	get	get	VERB
ejpam-6846	161	6	that	that	PRON
ejpam-6846	161	7	∂fmf	∂fmf	NUM
ejpam-6846	161	8	,	,	PUNCT
ejpam-6846	161	9	s(x	s(x	NOUN
ejpam-6846	161	10	∗	∗	NOUN
ejpam-6846	161	11	)	)	PUNCT
ejpam-6846	161	12	=	=	PRON
ejpam-6846	162	1	{	{	PUNCT
ejpam-6846	162	2	2(j∗x∗	2(j∗x∗	NUM
ejpam-6846	162	3	−	−	NOUN
ejpam-6846	162	4	πf	πf	NOUN
ejpam-6846	162	5	s(x	s(x	VERB
ejpam-6846	162	6	∗	∗	NOUN
ejpam-6846	162	7	)	)	PUNCT
ejpam-6846	162	8	)	)	PUNCT
ejpam-6846	162	9	}	}	PUNCT
ejpam-6846	162	10	.	.	PUNCT
ejpam-6846	163	1	proof	proof	NOUN
ejpam-6846	163	2	:	:	PUNCT
ejpam-6846	163	3	let	let	VERB
ejpam-6846	163	4	∂fmf	∂fmf	NUM
ejpam-6846	163	5	,	,	PUNCT
ejpam-6846	163	6	s(x	s(x	NOUN
ejpam-6846	163	7	∗	∗	NOUN
ejpam-6846	163	8	)	)	PUNCT
ejpam-6846	163	9	6=	6=	ADP
ejpam-6846	163	10	ϕ	ϕ	NOUN
ejpam-6846	163	11	and	and	CCONJ
ejpam-6846	163	12	let	let	VERB
ejpam-6846	163	13	x1	x1	PROPN
ejpam-6846	163	14	∈	∈	PROPN
ejpam-6846	163	15	∂fmf	∂fmf	NUM
ejpam-6846	163	16	,	,	PUNCT
ejpam-6846	163	17	s(x	s(x	NOUN
ejpam-6846	163	18	∗	∗	NOUN
ejpam-6846	163	19	)	)	PUNCT
ejpam-6846	163	20	.	.	PUNCT
ejpam-6846	164	1	then	then	ADV
ejpam-6846	164	2	for	for	ADP
ejpam-6846	164	3	every	every	DET
ejpam-6846	164	4	ϵ	ϵ	X
ejpam-6846	164	5	>	>	X
ejpam-6846	164	6	0	0	PUNCT
ejpam-6846	164	7	and	and	CCONJ
ejpam-6846	164	8	applying	apply	VERB
ejpam-6846	164	9	the	the	DET
ejpam-6846	164	10	concept	concept	NOUN
ejpam-6846	164	11	of	of	ADP
ejpam-6846	164	12	∂f	∂f	PROPN
ejpam-6846	164	13	to	to	PART
ejpam-6846	164	14	obtain	obtain	VERB
ejpam-6846	164	15	,	,	PUNCT
ejpam-6846	164	16	∃δ	∃δ	PROPN
ejpam-6846	164	17	>	>	X
ejpam-6846	164	18	0	0	NUM
ejpam-6846	165	1	such	such	ADJ
ejpam-6846	165	2	that	that	SCONJ
ejpam-6846	165	3	given	give	VERB
ejpam-6846	165	4	any	any	DET
ejpam-6846	165	5	u∗	u∗	ADJ
ejpam-6846	165	6	∈	∈	NOUN
ejpam-6846	165	7	b∗	b∗	ADJ
ejpam-6846	165	8	and	and	CCONJ
ejpam-6846	165	9	any	any	DET
ejpam-6846	165	10	µ	µ	X
ejpam-6846	165	11	∈	∈	NOUN
ejpam-6846	165	12	(	(	PUNCT
ejpam-6846	165	13	0	0	NUM
ejpam-6846	165	14	,	,	PUNCT
ejpam-6846	165	15	δ	δ	PROPN
ejpam-6846	165	16	)	)	PUNCT
ejpam-6846	165	17	,	,	PUNCT
ejpam-6846	165	18	we	we	PRON
ejpam-6846	165	19	get	get	VERB
ejpam-6846	165	20	:	:	PUNCT
ejpam-6846	166	1	〈	〈	PROPN
ejpam-6846	166	2	x1;x∗	x1;x∗	PROPN
ejpam-6846	166	3	+	+	CCONJ
ejpam-6846	166	4	µu∗	µu∗	ADJ
ejpam-6846	166	5	−	−	PROPN
ejpam-6846	166	6	x∗	x∗	PROPN
ejpam-6846	166	7	〉	〉	NOUN
ejpam-6846	166	8	=	=	SYM
ejpam-6846	166	9	〈	〈	PROPN
ejpam-6846	166	10	x1;µu∗	x1;µu∗	NOUN
ejpam-6846	166	11	〉	〉	PROPN
ejpam-6846	166	12	≤	≤	NUM
ejpam-6846	166	13	mf	mf	X
ejpam-6846	166	14	,	,	PUNCT
ejpam-6846	166	15	s(x	s(x	PROPN
ejpam-6846	166	16	∗	∗	NOUN
ejpam-6846	166	17	+	+	NUM
ejpam-6846	166	18	µu∗)−mf	µu∗)−mf	NOUN
ejpam-6846	166	19	,	,	PUNCT
ejpam-6846	166	20	s(x	s(x	VERB
ejpam-6846	166	21	∗	∗	NOUN
ejpam-6846	166	22	)	)	PUNCT
ejpam-6846	167	1	+	+	CCONJ
ejpam-6846	167	2	ϵµ.	ϵµ.	PROPN
ejpam-6846	167	3	a.	a.	NOUN
ejpam-6846	167	4	al	al	PROPN
ejpam-6846	167	5	tane	tane	PROPN
ejpam-6846	167	6	,	,	PUNCT
ejpam-6846	167	7	l.	l.	PROPN
ejpam-6846	167	8	s.	s.	PROPN
ejpam-6846	167	9	keong	keong	PROPN
ejpam-6846	167	10	/	/	PUNCT
ejpam-6846	167	11	eur	eur	PROPN
ejpam-6846	167	12	.	.	PUNCT
ejpam-6846	168	1	j.	j.	PROPN
ejpam-6846	168	2	pure	pure	PROPN
ejpam-6846	168	3	appl	appl	PROPN
ejpam-6846	168	4	.	.	PROPN
ejpam-6846	168	5	math	math	PROPN
ejpam-6846	168	6	,	,	PUNCT
ejpam-6846	168	7	18	18	NUM
ejpam-6846	168	8	(	(	PUNCT
ejpam-6846	168	9	4	4	NUM
ejpam-6846	168	10	)	)	PUNCT
ejpam-6846	168	11	(	(	PUNCT
ejpam-6846	168	12	2025	2025	NUM
ejpam-6846	168	13	)	)	PUNCT
ejpam-6846	168	14	,	,	PUNCT
ejpam-6846	168	15	6846	6846	NUM
ejpam-6846	168	16	7	7	NUM
ejpam-6846	168	17	of	of	ADP
ejpam-6846	168	18	13	13	NUM
ejpam-6846	168	19	therefore	therefore	ADV
ejpam-6846	169	1	,	,	PUNCT
ejpam-6846	169	2	we	we	PRON
ejpam-6846	169	3	can	can	AUX
ejpam-6846	169	4	find	find	VERB
ejpam-6846	169	5	a	a	DET
ejpam-6846	169	6	sufficiently	sufficiently	ADV
ejpam-6846	169	7	large	large	ADJ
ejpam-6846	169	8	β	β	X
ejpam-6846	169	9	∈	∈	PROPN
ejpam-6846	169	10	n	n	PRON
ejpam-6846	169	11	s.t	s.t	PROPN
ejpam-6846	169	12	:	:	PUNCT
ejpam-6846	169	13	〈	〈	PROPN
ejpam-6846	169	14	x1;β−1u∗	x1;β−1u∗	PROPN
ejpam-6846	169	15	〉	〉	PROPN
ejpam-6846	169	16	≤	≤	NUM
ejpam-6846	169	17	mf	mf	X
ejpam-6846	169	18	,	,	PUNCT
ejpam-6846	169	19	s(x	s(x	PROPN
ejpam-6846	169	20	∗	∗	NOUN
ejpam-6846	169	21	+	+	SYM
ejpam-6846	169	22	β−1u∗)−mf	β−1u∗)−mf	NUM
ejpam-6846	169	23	,	,	PUNCT
ejpam-6846	169	24	s(x	s(x	NOUN
ejpam-6846	169	25	∗	∗	NOUN
ejpam-6846	169	26	)	)	PUNCT
ejpam-6846	170	1	+	+	CCONJ
ejpam-6846	170	2	ϵβ−1	ϵβ−1	NOUN
ejpam-6846	170	3	,	,	PUNCT
ejpam-6846	170	4	∀u∗	∀u∗	PROPN
ejpam-6846	170	5	∈	∈	PROPN
ejpam-6846	170	6	b∗.	b∗.	NOUN
ejpam-6846	170	7	(	(	PUNCT
ejpam-6846	170	8	3	3	X
ejpam-6846	170	9	)	)	PUNCT
ejpam-6846	170	10	using	use	VERB
ejpam-6846	170	11	definition	definition	NOUN
ejpam-6846	170	12	of	of	ADP
ejpam-6846	170	13	mf	mf	PROPN
ejpam-6846	170	14	,	,	PUNCT
ejpam-6846	170	15	s	s	PART
ejpam-6846	170	16	,	,	PUNCT
ejpam-6846	170	17	∃vn	∃vn	NOUN
ejpam-6846	170	18	∈	∈	PROPN
ejpam-6846	170	19	s	s	X
ejpam-6846	170	20	for	for	ADP
ejpam-6846	170	21	all	all	DET
ejpam-6846	170	22	n	n	PRON
ejpam-6846	170	23	≥	≥	NUM
ejpam-6846	170	24	1	1	NUM
ejpam-6846	170	25	.	.	PUNCT
ejpam-6846	171	1	mf	mf	PROPN
ejpam-6846	171	2	,	,	PUNCT
ejpam-6846	171	3	s(x	s(x	NOUN
ejpam-6846	171	4	∗	∗	NOUN
ejpam-6846	171	5	)	)	PUNCT
ejpam-6846	171	6	≤	≤	NOUN
ejpam-6846	171	7	v	v	ADP
ejpam-6846	171	8	f	f	X
ejpam-6846	171	9	(	(	PUNCT
ejpam-6846	171	10	x∗	x∗	PROPN
ejpam-6846	171	11	,	,	PUNCT
ejpam-6846	171	12	vn	vn	NOUN
ejpam-6846	171	13	)	)	PUNCT
ejpam-6846	171	14	≤	≤	NOUN
ejpam-6846	171	15	mf	mf	X
ejpam-6846	171	16	,	,	PUNCT
ejpam-6846	171	17	s(x	s(x	NOUN
ejpam-6846	171	18	∗	∗	NOUN
ejpam-6846	171	19	)	)	PUNCT
ejpam-6846	172	1	+	+	CCONJ
ejpam-6846	172	2	1	1	NUM
ejpam-6846	172	3	n2	n2	NOUN
ejpam-6846	172	4	.	.	PUNCT
ejpam-6846	173	1	(	(	PUNCT
ejpam-6846	173	2	4	4	X
ejpam-6846	173	3	)	)	PUNCT
ejpam-6846	173	4	hence	hence	ADV
ejpam-6846	173	5	,	,	PUNCT
ejpam-6846	173	6	for	for	ADP
ejpam-6846	173	7	sufficiently	sufficiently	ADV
ejpam-6846	173	8	large	large	ADJ
ejpam-6846	173	9	β	β	X
ejpam-6846	173	10	∈	∈	PROPN
ejpam-6846	173	11	n	n	CCONJ
ejpam-6846	173	12	,	,	PUNCT
ejpam-6846	173	13	we	we	PRON
ejpam-6846	173	14	can	can	AUX
ejpam-6846	173	15	combine	combine	VERB
ejpam-6846	173	16	inequalities	inequality	NOUN
ejpam-6846	173	17	(	(	PUNCT
ejpam-6846	173	18	3	3	NUM
ejpam-6846	173	19	)	)	PUNCT
ejpam-6846	173	20	and	and	CCONJ
ejpam-6846	173	21	(	(	PUNCT
ejpam-6846	173	22	4	4	X
ejpam-6846	173	23	)	)	PUNCT
ejpam-6846	173	24	for	for	ADP
ejpam-6846	173	25	all	all	DET
ejpam-6846	173	26	u∗	u∗	PROPN
ejpam-6846	173	27	∈	∈	PROPN
ejpam-6846	173	28	b∗.	b∗.	NOUN
ejpam-6846	173	29	〈	〈	PROPN
ejpam-6846	173	30	x1;β−1u∗	x1;β−1u∗	PROPN
ejpam-6846	173	31	〉	〉	PROPN
ejpam-6846	173	32	≤	≤	NOUN
ejpam-6846	173	33	v	v	ADP
ejpam-6846	173	34	f	f	X
ejpam-6846	173	35	(	(	PUNCT
ejpam-6846	173	36	x∗	x∗	X
ejpam-6846	173	37	+	+	CCONJ
ejpam-6846	173	38	β−1u∗	β−1u∗	ADJ
ejpam-6846	173	39	;	;	PUNCT
ejpam-6846	173	40	vβ)−	vβ)−	NUM
ejpam-6846	173	41	v	v	PROPN
ejpam-6846	173	42	f	f	X
ejpam-6846	173	43	(	(	PUNCT
ejpam-6846	173	44	x∗	x∗	PROPN
ejpam-6846	173	45	,	,	PUNCT
ejpam-6846	173	46	vβ	vβ	NOUN
ejpam-6846	173	47	)	)	PUNCT
ejpam-6846	173	48	+	+	CCONJ
ejpam-6846	173	49	β−2	β−2	ADJ
ejpam-6846	173	50	+	+	CCONJ
ejpam-6846	173	51	β−1ϵ	β−1ϵ	ADJ
ejpam-6846	173	52	≤	≤	NOUN
ejpam-6846	173	53	‖x∗	‖x∗	PUNCT
ejpam-6846	174	1	+	+	CCONJ
ejpam-6846	174	2	β−1u∗‖2	β−1u∗‖2	PROPN
ejpam-6846	174	3	−	−	PROPN
ejpam-6846	174	4	‖x∗‖2	‖x∗‖2	NOUN
ejpam-6846	174	5	−	−	PROPN
ejpam-6846	174	6	2〈β−1u∗	2〈β−1u∗	NUM
ejpam-6846	174	7	;	;	PUNCT
ejpam-6846	174	8	vβ〉+	vβ〉+	NOUN
ejpam-6846	174	9	β−2	β−2	PROPN
ejpam-6846	174	10	+	+	CCONJ
ejpam-6846	174	11	β−1ϵ.	β−1ϵ.	NUM
ejpam-6846	174	12	thus	thus	ADV
ejpam-6846	174	13	〈	〈	ADP
ejpam-6846	174	14	x1	x1	PROPN
ejpam-6846	174	15	+	+	X
ejpam-6846	174	16	2vβ	2vβ	ADJ
ejpam-6846	174	17	;	;	PUNCT
ejpam-6846	174	18	u	u	NOUN
ejpam-6846	174	19	∗	∗	NOUN
ejpam-6846	174	20	〉	〉	NOUN
ejpam-6846	174	21	≤	≤	NOUN
ejpam-6846	174	22	β‖x∗	β‖x∗	PUNCT
ejpam-6846	175	1	+	+	CCONJ
ejpam-6846	175	2	β−1u∗‖2	β−1u∗‖2	PROPN
ejpam-6846	175	3	−	−	NOUN
ejpam-6846	175	4	β‖x∗‖2	β‖x∗‖2	PUNCT
ejpam-6846	176	1	+	+	CCONJ
ejpam-6846	176	2	β−1	β−1	PUNCT
ejpam-6846	176	3	+	+	NUM
ejpam-6846	176	4	ϵ	ϵ	PRON
ejpam-6846	176	5	∀u∗	∀u∗	ADJ
ejpam-6846	176	6	∈	∈	PROPN
ejpam-6846	176	7	b∗.	b∗.	NOUN
ejpam-6846	176	8	for	for	ADP
ejpam-6846	176	9	β	β	PRON
ejpam-6846	176	10	large	large	ADJ
ejpam-6846	176	11	enough	enough	ADV
ejpam-6846	176	12	,	,	PUNCT
ejpam-6846	176	13	and	and	CCONJ
ejpam-6846	176	14	∇f	∇f	PROPN
ejpam-6846	176	15	‖.‖2(x∗	‖.‖2(x∗	NOUN
ejpam-6846	176	16	)	)	PUNCT
ejpam-6846	176	17	=	=	SYM
ejpam-6846	177	1	2j∗x∗	2j∗x∗	NUM
ejpam-6846	177	2	,	,	PUNCT
ejpam-6846	177	3	we	we	PRON
ejpam-6846	177	4	can	can	AUX
ejpam-6846	177	5	write	write	VERB
ejpam-6846	177	6	sup	sup	NOUN
ejpam-6846	177	7	u∗∈b∗	u∗∈b∗	PROPN
ejpam-6846	177	8	β	β	X
ejpam-6846	177	9	[	[	PUNCT
ejpam-6846	177	10	‖x∗	‖x∗	X
ejpam-6846	177	11	+	+	CCONJ
ejpam-6846	177	12	β−1u∗‖2	β−1u∗‖2	PROPN
ejpam-6846	177	13	−	−	PROPN
ejpam-6846	177	14	‖x∗‖2	‖x∗‖2	NOUN
ejpam-6846	177	15	−	−	PROPN
ejpam-6846	178	1	〈	〈	PROPN
ejpam-6846	178	2	2j∗x∗;β−1u∗	2j∗x∗;β−1u∗	NUM
ejpam-6846	178	3	〉	〉	NOUN
ejpam-6846	178	4	]	]	PUNCT
ejpam-6846	178	5	≤	≤	NUM
ejpam-6846	179	1	ϵ.	ϵ.	NOUN
ejpam-6846	179	2	therefore	therefore	ADV
ejpam-6846	179	3	,	,	PUNCT
ejpam-6846	179	4	for	for	ADP
ejpam-6846	179	5	β	β	PRON
ejpam-6846	179	6	large	large	ADJ
ejpam-6846	179	7	enough	enough	ADJ
ejpam-6846	179	8	‖x1	‖x1	NOUN
ejpam-6846	179	9	−	−	PROPN
ejpam-6846	179	10	2(j∗x∗	2(j∗x∗	PROPN
ejpam-6846	179	11	−	−	NOUN
ejpam-6846	179	12	vβ)‖|	vβ)‖|	NOUN
ejpam-6846	179	13	=	=	NOUN
ejpam-6846	179	14	sup	sup	NOUN
ejpam-6846	179	15	u∗∈b∗	u∗∈b∗	PROPN
ejpam-6846	179	16	〈	〈	PROPN
ejpam-6846	179	17	x1	x1	PROPN
ejpam-6846	180	1	−	−	PROPN
ejpam-6846	180	2	2(j∗x∗	2(j∗x∗	PROPN
ejpam-6846	180	3	−	−	NOUN
ejpam-6846	181	1	vβ);u	vβ);u	PROPN
ejpam-6846	181	2	∗	∗	NOUN
ejpam-6846	181	3	〉	〉	PROPN
ejpam-6846	181	4	≤	≤	NUM
ejpam-6846	181	5	3ϵ.	3ϵ.	NUM
ejpam-6846	182	1	(	(	PUNCT
ejpam-6846	182	2	5	5	NUM
ejpam-6846	182	3	)	)	PUNCT
ejpam-6846	182	4	we	we	PRON
ejpam-6846	182	5	conclude	conclude	VERB
ejpam-6846	182	6	that	that	SCONJ
ejpam-6846	182	7	this	this	DET
ejpam-6846	182	8	sequence	sequence	NOUN
ejpam-6846	182	9	(	(	PUNCT
ejpam-6846	182	10	vβ)β	vβ)β	NOUN
ejpam-6846	182	11	converges	converge	VERB
ejpam-6846	182	12	to	to	ADP
ejpam-6846	182	13	v	v	NOUN
ejpam-6846	182	14	:	:	PUNCT
ejpam-6846	182	15	=	=	SYM
ejpam-6846	182	16	j∗x∗	j∗x∗	NOUN
ejpam-6846	183	1	−	−	NUM
ejpam-6846	183	2	1	1	NUM
ejpam-6846	183	3	2x1	2x1	NUM
ejpam-6846	183	4	by	by	ADP
ejpam-6846	183	5	keeping	keep	VERB
ejpam-6846	183	6	in	in	ADP
ejpam-6846	183	7	mind	mind	NOUN
ejpam-6846	183	8	that	that	SCONJ
ejpam-6846	183	9	the	the	DET
ejpam-6846	183	10	sequence(vβ	sequence(vβ	NOUN
ejpam-6846	183	11	)	)	PUNCT
ejpam-6846	183	12	can	can	AUX
ejpam-6846	183	13	be	be	AUX
ejpam-6846	183	14	selected	select	VERB
ejpam-6846	183	15	arbitrarily	arbitrarily	ADV
ejpam-6846	183	16	and	and	CCONJ
ejpam-6846	183	17	that	that	SCONJ
ejpam-6846	183	18	for	for	ADP
ejpam-6846	183	19	arbitrary	arbitrary	ADJ
ejpam-6846	183	20	positive	positive	ADJ
ejpam-6846	183	21	ϵ	ϵ	NOUN
ejpam-6846	183	22	,	,	PUNCT
ejpam-6846	183	23	the	the	DET
ejpam-6846	183	24	aforementioned	aforementioned	ADJ
ejpam-6846	183	25	inequality	inequality	NOUN
ejpam-6846	183	26	(	(	PUNCT
ejpam-6846	183	27	5	5	NUM
ejpam-6846	183	28	)	)	PUNCT
ejpam-6846	183	29	is	be	AUX
ejpam-6846	183	30	holds	hold	NOUN
ejpam-6846	183	31	.	.	PUNCT
ejpam-6846	184	1	(	(	PUNCT
ejpam-6846	184	2	when	when	SCONJ
ejpam-6846	184	3	β	β	X
ejpam-6846	184	4	is	be	AUX
ejpam-6846	184	5	sufficiently	sufficiently	ADV
ejpam-6846	184	6	large	large	ADJ
ejpam-6846	184	7	)	)	PUNCT
ejpam-6846	184	8	.	.	PUNCT
ejpam-6846	185	1	additionally	additionally	ADV
ejpam-6846	185	2	,	,	PUNCT
ejpam-6846	185	3	since	since	SCONJ
ejpam-6846	185	4	v	v	NUM
ejpam-6846	185	5	f	f	X
ejpam-6846	185	6	(	(	PUNCT
ejpam-6846	185	7	x∗	x∗	PROPN
ejpam-6846	185	8	,	,	PUNCT
ejpam-6846	185	9	vn	vn	NOUN
ejpam-6846	185	10	)	)	PUNCT
ejpam-6846	185	11	→	→	SYM
ejpam-6846	185	12	mf	mf	PROPN
ejpam-6846	185	13	,	,	PUNCT
ejpam-6846	185	14	s(x	s(x	ADJ
ejpam-6846	185	15	∗	∗	NOUN
ejpam-6846	185	16	)	)	PUNCT
ejpam-6846	185	17	,	,	PUNCT
ejpam-6846	185	18	we	we	PRON
ejpam-6846	185	19	conclude	conclude	VERB
ejpam-6846	185	20	that	that	SCONJ
ejpam-6846	185	21	v	v	NUM
ejpam-6846	185	22	∈	∈	PROPN
ejpam-6846	185	23	πf	πf	NOUN
ejpam-6846	185	24	s(x	s(x	PROPN
ejpam-6846	185	25	∗	∗	NOUN
ejpam-6846	185	26	)	)	PUNCT
ejpam-6846	185	27	.	.	PUNCT
ejpam-6846	186	1	now	now	ADV
ejpam-6846	186	2	let	let	VERB
ejpam-6846	186	3	x̄	x̄	PRON
ejpam-6846	186	4	∈	∈	PROPN
ejpam-6846	186	5	πf	πf	X
ejpam-6846	186	6	s(x	s(x	PROPN
ejpam-6846	186	7	∗	∗	NOUN
ejpam-6846	186	8	)	)	PUNCT
ejpam-6846	186	9	.	.	PUNCT
ejpam-6846	187	1	then	then	ADV
ejpam-6846	187	2	carrying	carry	VERB
ejpam-6846	187	3	on	on	ADP
ejpam-6846	187	4	as	as	SCONJ
ejpam-6846	187	5	previously	previously	ADV
ejpam-6846	187	6	we	we	PRON
ejpam-6846	187	7	get	get	VERB
ejpam-6846	187	8	x̄	x̄	NOUN
ejpam-6846	187	9	:	:	PUNCT
ejpam-6846	187	10	=	=	SYM
ejpam-6846	187	11	j∗x∗	j∗x∗	NOUN
ejpam-6846	188	1	−	−	NUM
ejpam-6846	188	2	1	1	NUM
ejpam-6846	189	1	2x1	2x1	NUM
ejpam-6846	189	2	=	=	SYM
ejpam-6846	189	3	v.	v.	PROPN
ejpam-6846	189	4	□	□	PUNCT
ejpam-6846	189	5	from	from	ADP
ejpam-6846	189	6	lemma	lemma	PROPN
ejpam-6846	189	7	(	(	PUNCT
ejpam-6846	189	8	2	2	NUM
ejpam-6846	189	9	)	)	PUNCT
ejpam-6846	189	10	,	,	PUNCT
ejpam-6846	189	11	it	it	PRON
ejpam-6846	189	12	is	be	AUX
ejpam-6846	189	13	clearly	clearly	ADV
ejpam-6846	189	14	that	that	SCONJ
ejpam-6846	189	15	if	if	SCONJ
ejpam-6846	189	16	πf	πf	INTJ
ejpam-6846	189	17	s(u	s(u	PROPN
ejpam-6846	189	18	∗	∗	PROPN
ejpam-6846	189	19	)	)	PUNCT
ejpam-6846	189	20	6=	6=	ADP
ejpam-6846	190	1	ϕ	ϕ	PROPN
ejpam-6846	190	2	,	,	PUNCT
ejpam-6846	190	3	then	then	ADV
ejpam-6846	190	4	mf	mf	NOUN
ejpam-6846	190	5	,	,	PUNCT
ejpam-6846	190	6	s(u	s(u	PROPN
ejpam-6846	190	7	∗	∗	NOUN
ejpam-6846	190	8	)	)	PUNCT
ejpam-6846	191	1	⊂	⊂	PROPN
ejpam-6846	191	2	{	{	PUNCT
ejpam-6846	191	3	2(j∗u∗	2(j∗u∗	NUM
ejpam-6846	191	4	−	−	PROPN
ejpam-6846	191	5	u	u	NOUN
ejpam-6846	191	6	)	)	PUNCT
ejpam-6846	191	7	}	}	PUNCT
ejpam-6846	191	8	for	for	ADP
ejpam-6846	191	9	all	all	DET
ejpam-6846	191	10	u	u	PROPN
ejpam-6846	191	11	∈	∈	PROPN
ejpam-6846	191	12	πf	πf	NOUN
ejpam-6846	191	13	s(u	s(u	PROPN
ejpam-6846	191	14	∗	∗	PROPN
ejpam-6846	191	15	)	)	PUNCT
ejpam-6846	191	16	.	.	PUNCT
ejpam-6846	192	1	in	in	ADP
ejpam-6846	192	2	the	the	DET
ejpam-6846	192	3	upcoming	upcoming	ADJ
ejpam-6846	192	4	lemma	lemma	PROPN
ejpam-6846	192	5	,	,	PUNCT
ejpam-6846	192	6	we	we	PRON
ejpam-6846	192	7	will	will	AUX
ejpam-6846	192	8	show	show	VERB
ejpam-6846	192	9	that	that	SCONJ
ejpam-6846	192	10	mf	mf	NOUN
ejpam-6846	192	11	,	,	PUNCT
ejpam-6846	192	12	s	s	PART
ejpam-6846	192	13	is	be	AUX
ejpam-6846	192	14	locally	locally	ADV
ejpam-6846	192	15	lipschtz	lipschtz	ADV
ejpam-6846	192	16	continuous	continuous	ADJ
ejpam-6846	192	17	.	.	PUNCT
ejpam-6846	193	1	lemma	lemma	PROPN
ejpam-6846	193	2	3	3	X
ejpam-6846	193	3	.	.	PUNCT
ejpam-6846	194	1	let	let	VERB
ejpam-6846	194	2	x∗	x∗	PROPN
ejpam-6846	194	3	be	be	AUX
ejpam-6846	194	4	the	the	DET
ejpam-6846	194	5	dual	dual	ADJ
ejpam-6846	194	6	space	space	NOUN
ejpam-6846	194	7	of	of	ADP
ejpam-6846	194	8	the	the	DET
ejpam-6846	194	9	reflexive	reflexive	ADJ
ejpam-6846	194	10	banach	banach	NOUN
ejpam-6846	194	11	space	space	NOUN
ejpam-6846	194	12	x	x	PUNCT
ejpam-6846	194	13	and	and	CCONJ
ejpam-6846	194	14	f	f	PROPN
ejpam-6846	194	15	is	be	AUX
ejpam-6846	194	16	a	a	DET
ejpam-6846	194	17	proper	proper	ADJ
ejpam-6846	194	18	function	function	NOUN
ejpam-6846	194	19	,	,	PUNCT
ejpam-6846	194	20	and	and	CCONJ
ejpam-6846	194	21	bounded	bound	VERB
ejpam-6846	194	22	from	from	ADP
ejpam-6846	194	23	bellow	bellow	ADV
ejpam-6846	194	24	on	on	ADP
ejpam-6846	194	25	x	x	X
ejpam-6846	194	26	,	,	PUNCT
ejpam-6846	194	27	then	then	ADV
ejpam-6846	194	28	u∗	u∗	VERB
ejpam-6846	194	29	7→	7→	NUM
ejpam-6846	194	30	v	v	NOUN
ejpam-6846	194	31	f	f	X
ejpam-6846	194	32	(	(	PUNCT
ejpam-6846	194	33	u∗	u∗	PROPN
ejpam-6846	194	34	,	,	PUNCT
ejpam-6846	194	35	u	u	NOUN
ejpam-6846	194	36	)	)	PUNCT
ejpam-6846	194	37	is	be	AUX
ejpam-6846	194	38	locally	locally	ADV
ejpam-6846	194	39	lipschtz	lipschtz	ADV
ejpam-6846	194	40	on	on	ADP
ejpam-6846	194	41	x∗.	x∗.	ADJ
ejpam-6846	194	42	proof	proof	NOUN
ejpam-6846	194	43	:	:	PUNCT
ejpam-6846	194	44	fix	fix	VERB
ejpam-6846	194	45	δ	δ	PROPN
ejpam-6846	194	46	>	>	X
ejpam-6846	194	47	0	0	PUNCT
ejpam-6846	194	48	and	and	CCONJ
ejpam-6846	194	49	let	let	VERB
ejpam-6846	194	50	u∗	u∗	ADV
ejpam-6846	194	51	be	be	AUX
ejpam-6846	194	52	an	an	DET
ejpam-6846	194	53	element	element	NOUN
ejpam-6846	194	54	of	of	ADP
ejpam-6846	194	55	x∗.	x∗.	PROPN
ejpam-6846	194	56	choose	choose	VERB
ejpam-6846	194	57	a	a	DET
ejpam-6846	194	58	value	value	NOUN
ejpam-6846	194	59	ϵ	ϵ	ADP
ejpam-6846	194	60	such	such	ADJ
ejpam-6846	194	61	that	that	SCONJ
ejpam-6846	194	62	0	0	NUM
ejpam-6846	194	63	<	<	X
ejpam-6846	194	64	ϵ	ϵ	X
ejpam-6846	194	65	<	<	X
ejpam-6846	194	66	δ	δ	NOUN
ejpam-6846	194	67	and	and	CCONJ
ejpam-6846	194	68	fix	fix	VERB
ejpam-6846	194	69	two	two	NUM
ejpam-6846	194	70	elements	element	NOUN
ejpam-6846	194	71	v∗	v∗	ADJ
ejpam-6846	194	72	and	and	CCONJ
ejpam-6846	194	73	z∗	z∗	PROPN
ejpam-6846	194	74	in	in	ADP
ejpam-6846	194	75	u∗+	u∗+	ADJ
ejpam-6846	194	76	δb∗.	δb∗.	PROPN
ejpam-6846	194	77	by	by	ADP
ejpam-6846	194	78	definition	definition	NOUN
ejpam-6846	194	79	of	of	ADP
ejpam-6846	194	80	mf	mf	PROPN
ejpam-6846	194	81	,	,	PUNCT
ejpam-6846	194	82	s	s	PART
ejpam-6846	194	83	there	there	PRON
ejpam-6846	194	84	exists	exist	VERB
ejpam-6846	194	85	rϵ	rϵ	ADP
ejpam-6846	194	86	∈	∈	PROPN
ejpam-6846	194	87	s	s	VERB
ejpam-6846	194	88	such	such	ADJ
ejpam-6846	194	89	that	that	SCONJ
ejpam-6846	194	90	mf	mf	NOUN
ejpam-6846	194	91	,	,	PUNCT
ejpam-6846	194	92	s(v	s(v	PROPN
ejpam-6846	194	93	∗	∗	NOUN
ejpam-6846	194	94	)	)	PUNCT
ejpam-6846	194	95	≤	≤	NOUN
ejpam-6846	194	96	v	v	ADP
ejpam-6846	194	97	f	f	PROPN
ejpam-6846	194	98	(	(	PUNCT
ejpam-6846	194	99	v∗	v∗	PROPN
ejpam-6846	194	100	,	,	PUNCT
ejpam-6846	194	101	rϵ	rϵ	NOUN
ejpam-6846	194	102	)	)	PUNCT
ejpam-6846	194	103	<	<	X
ejpam-6846	194	104	mf	mf	PROPN
ejpam-6846	194	105	,	,	PUNCT
ejpam-6846	194	106	s(v	s(v	PROPN
ejpam-6846	194	107	∗	∗	NOUN
ejpam-6846	194	108	)	)	PUNCT
ejpam-6846	195	1	+	+	CCONJ
ejpam-6846	195	2	ϵ.	ϵ.	NOUN
ejpam-6846	195	3	a.	a.	NOUN
ejpam-6846	195	4	al	al	PROPN
ejpam-6846	195	5	tane	tane	PROPN
ejpam-6846	195	6	,	,	PUNCT
ejpam-6846	195	7	l.	l.	PROPN
ejpam-6846	195	8	s.	s.	PROPN
ejpam-6846	195	9	keong	keong	PROPN
ejpam-6846	195	10	/	/	PUNCT
ejpam-6846	195	11	eur	eur	PROPN
ejpam-6846	195	12	.	.	PUNCT
ejpam-6846	196	1	j.	j.	PROPN
ejpam-6846	196	2	pure	pure	PROPN
ejpam-6846	196	3	appl	appl	PROPN
ejpam-6846	196	4	.	.	PROPN
ejpam-6846	196	5	math	math	PROPN
ejpam-6846	196	6	,	,	PUNCT
ejpam-6846	196	7	18	18	NUM
ejpam-6846	196	8	(	(	PUNCT
ejpam-6846	196	9	4	4	NUM
ejpam-6846	196	10	)	)	PUNCT
ejpam-6846	196	11	(	(	PUNCT
ejpam-6846	196	12	2025	2025	NUM
ejpam-6846	196	13	)	)	PUNCT
ejpam-6846	196	14	,	,	PUNCT
ejpam-6846	196	15	6846	6846	NUM
ejpam-6846	196	16	8	8	NUM
ejpam-6846	196	17	of	of	ADP
ejpam-6846	196	18	13	13	NUM
ejpam-6846	196	19	so	so	ADV
ejpam-6846	196	20	mf	mf	NOUN
ejpam-6846	196	21	,	,	PUNCT
ejpam-6846	196	22	s(z	s(z	PROPN
ejpam-6846	196	23	∗)−mf	∗)−mf	PROPN
ejpam-6846	196	24	,	,	PUNCT
ejpam-6846	196	25	s(v	s(v	PROPN
ejpam-6846	196	26	∗	∗	NOUN
ejpam-6846	196	27	)	)	PUNCT
ejpam-6846	196	28	≤	≤	NOUN
ejpam-6846	196	29	v	v	ADP
ejpam-6846	196	30	f	f	X
ejpam-6846	196	31	(	(	PUNCT
ejpam-6846	196	32	z∗	z∗	PROPN
ejpam-6846	196	33	,	,	PUNCT
ejpam-6846	196	34	rϵ)−	rϵ)−	NOUN
ejpam-6846	196	35	v	v	ADP
ejpam-6846	196	36	f	f	X
ejpam-6846	196	37	(	(	PUNCT
ejpam-6846	196	38	v∗	v∗	PROPN
ejpam-6846	196	39	,	,	PUNCT
ejpam-6846	196	40	rϵ	rϵ	NOUN
ejpam-6846	196	41	)	)	PUNCT
ejpam-6846	197	1	+	+	CCONJ
ejpam-6846	197	2	ϵ	ϵ	X
ejpam-6846	197	3	=	=	SYM
ejpam-6846	197	4	v	v	PROPN
ejpam-6846	197	5	(	(	PUNCT
ejpam-6846	197	6	z∗	z∗	NOUN
ejpam-6846	197	7	,	,	PUNCT
ejpam-6846	197	8	rϵ)−	rϵ)−	ADJ
ejpam-6846	197	9	v	v	NOUN
ejpam-6846	197	10	(	(	PUNCT
ejpam-6846	197	11	v∗	v∗	ADJ
ejpam-6846	197	12	,	,	PUNCT
ejpam-6846	197	13	rϵ	rϵ	NOUN
ejpam-6846	197	14	)	)	PUNCT
ejpam-6846	197	15	+	+	CCONJ
ejpam-6846	197	16	ϵ	ϵ	X
ejpam-6846	197	17	≤	≤	NOUN
ejpam-6846	197	18	(	(	PUNCT
ejpam-6846	197	19	‖z∗‖+	‖z∗‖+	NUM
ejpam-6846	197	20	‖v∗‖+	‖v∗‖+	PUNCT
ejpam-6846	198	1	2‖rϵ‖)‖z∗	2‖rϵ‖)‖z∗	NUM
ejpam-6846	198	2	−	−	NOUN
ejpam-6846	198	3	v∗‖+	v∗‖+	NOUN
ejpam-6846	198	4	ϵ.	ϵ.	NOUN
ejpam-6846	198	5	let	let	VERB
ejpam-6846	198	6	r	r	NOUN
ejpam-6846	198	7	=	=	SYM
ejpam-6846	198	8	mf	mf	NOUN
ejpam-6846	198	9	,	,	PUNCT
ejpam-6846	198	10	s(u	s(u	PROPN
ejpam-6846	198	11	∗	∗	NOUN
ejpam-6846	198	12	)	)	PUNCT
ejpam-6846	198	13	.	.	PUNCT
ejpam-6846	199	1	then	then	ADV
ejpam-6846	199	2	∃uϵ	∃uϵ	VERB
ejpam-6846	199	3	∈	∈	PROPN
ejpam-6846	199	4	s	s	PART
ejpam-6846	199	5	s.t	s.t	PROPN
ejpam-6846	199	6	.	.	PROPN
ejpam-6846	199	7	mf	mf	PROPN
ejpam-6846	199	8	,	,	PUNCT
ejpam-6846	199	9	s(u	s(u	PROPN
ejpam-6846	199	10	∗	∗	NOUN
ejpam-6846	199	11	)	)	PUNCT
ejpam-6846	199	12	≤	≤	NOUN
ejpam-6846	199	13	v	v	ADP
ejpam-6846	199	14	f	f	X
ejpam-6846	199	15	(	(	PUNCT
ejpam-6846	199	16	u∗	u∗	PROPN
ejpam-6846	199	17	,	,	PUNCT
ejpam-6846	199	18	uϵ	uϵ	NOUN
ejpam-6846	199	19	)	)	PUNCT
ejpam-6846	199	20	<	<	X
ejpam-6846	199	21	mf	mf	X
ejpam-6846	199	22	,	,	PUNCT
ejpam-6846	199	23	s(u	s(u	PROPN
ejpam-6846	199	24	∗	∗	NOUN
ejpam-6846	199	25	)	)	PUNCT
ejpam-6846	200	1	+	+	CCONJ
ejpam-6846	200	2	ϵ.	ϵ.	NOUN
ejpam-6846	200	3	now	now	ADV
ejpam-6846	200	4	we	we	PRON
ejpam-6846	200	5	can	can	AUX
ejpam-6846	200	6	write	write	VERB
ejpam-6846	200	7	‖rϵ‖	‖rϵ‖	PROPN
ejpam-6846	200	8	≤	≤	PROPN
ejpam-6846	200	9	(	(	PUNCT
ejpam-6846	200	10	v	v	NOUN
ejpam-6846	200	11	(	(	PUNCT
ejpam-6846	200	12	v∗	v∗	ADJ
ejpam-6846	200	13	,	,	PUNCT
ejpam-6846	200	14	rϵ	rϵ	NOUN
ejpam-6846	200	15	)	)	PUNCT
ejpam-6846	200	16	)	)	PUNCT
ejpam-6846	200	17	1	1	NUM
ejpam-6846	200	18	2	2	NUM
ejpam-6846	200	19	+	+	CCONJ
ejpam-6846	200	20	‖v∗‖	‖v∗‖	NOUN
ejpam-6846	200	21	=	=	SYM
ejpam-6846	200	22	(	(	PUNCT
ejpam-6846	200	23	v	v	NUM
ejpam-6846	200	24	f	f	X
ejpam-6846	200	25	(	(	PUNCT
ejpam-6846	200	26	v∗	v∗	ADJ
ejpam-6846	200	27	,	,	PUNCT
ejpam-6846	200	28	rϵ)−	rϵ)−	NOUN
ejpam-6846	200	29	f(rϵ	f(rϵ	PROPN
ejpam-6846	200	30	)	)	PUNCT
ejpam-6846	200	31	)	)	PUNCT
ejpam-6846	200	32	1	1	NUM
ejpam-6846	200	33	2	2	NUM
ejpam-6846	200	34	+	+	CCONJ
ejpam-6846	200	35	‖v∗‖	‖v∗‖	NOUN
ejpam-6846	200	36	<	<	X
ejpam-6846	200	37	(	(	PUNCT
ejpam-6846	200	38	mf	mf	X
ejpam-6846	200	39	,	,	PUNCT
ejpam-6846	200	40	s(v	s(v	PROPN
ejpam-6846	200	41	∗)−	∗)−	ADV
ejpam-6846	200	42	f(rϵ	f(rϵ	PROPN
ejpam-6846	200	43	)	)	PUNCT
ejpam-6846	200	44	+	+	NUM
ejpam-6846	201	1	ϵ	ϵ	X
ejpam-6846	201	2	)	)	PUNCT
ejpam-6846	201	3	1	1	NUM
ejpam-6846	201	4	2	2	NUM
ejpam-6846	201	5	+	+	CCONJ
ejpam-6846	201	6	‖v∗‖	‖v∗‖	NOUN
ejpam-6846	201	7	≤	≤	NOUN
ejpam-6846	201	8	(	(	PUNCT
ejpam-6846	201	9	mf	mf	X
ejpam-6846	201	10	,	,	PUNCT
ejpam-6846	201	11	s(v	s(v	PROPN
ejpam-6846	201	12	∗)−	∗)−	ADV
ejpam-6846	201	13	f(rϵ	f(rϵ	PROPN
ejpam-6846	201	14	)	)	PUNCT
ejpam-6846	201	15	+	+	NUM
ejpam-6846	201	16	ϵ	ϵ	X
ejpam-6846	201	17	)	)	PUNCT
ejpam-6846	201	18	1	1	NUM
ejpam-6846	201	19	2	2	NUM
ejpam-6846	201	20	+	+	NUM
ejpam-6846	201	21	‖u∗‖+	‖u∗‖+	NOUN
ejpam-6846	201	22	δ	δ	PROPN
ejpam-6846	201	23	≤	≤	NUM
ejpam-6846	201	24	(	(	PUNCT
ejpam-6846	201	25	v	v	NUM
ejpam-6846	201	26	f	f	X
ejpam-6846	201	27	(	(	PUNCT
ejpam-6846	201	28	v∗	v∗	ADJ
ejpam-6846	201	29	,	,	PUNCT
ejpam-6846	201	30	uϵ)−	uϵ)−	ADJ
ejpam-6846	201	31	f(rϵ	f(rϵ	PROPN
ejpam-6846	201	32	)	)	PUNCT
ejpam-6846	201	33	+	+	NUM
ejpam-6846	201	34	ϵ	ϵ	X
ejpam-6846	201	35	)	)	PUNCT
ejpam-6846	201	36	1	1	NUM
ejpam-6846	201	37	2	2	NUM
ejpam-6846	201	38	+	+	NUM
ejpam-6846	201	39	‖u∗‖+	‖u∗‖+	NOUN
ejpam-6846	201	40	δ	δ	PROPN
ejpam-6846	201	41	≤	≤	NUM
ejpam-6846	201	42	(	(	PUNCT
ejpam-6846	201	43	v	v	NOUN
ejpam-6846	201	44	(	(	PUNCT
ejpam-6846	201	45	v∗	v∗	NOUN
ejpam-6846	201	46	,	,	PUNCT
ejpam-6846	201	47	uϵ	uϵ	NOUN
ejpam-6846	201	48	)	)	PUNCT
ejpam-6846	201	49	+	+	NUM
ejpam-6846	201	50	f(uϵ)−	f(uϵ)−	NOUN
ejpam-6846	201	51	f(rϵ	f(rϵ	PROPN
ejpam-6846	201	52	)	)	PUNCT
ejpam-6846	202	1	+	+	SYM
ejpam-6846	202	2	ϵ	ϵ	X
ejpam-6846	202	3	)	)	PUNCT
ejpam-6846	202	4	1	1	NUM
ejpam-6846	202	5	2	2	NUM
ejpam-6846	202	6	+	+	NUM
ejpam-6846	202	7	‖u∗‖+	‖u∗‖+	NOUN
ejpam-6846	202	8	δ	δ	PROPN
ejpam-6846	202	9	≤	≤	NUM
ejpam-6846	202	10	(	(	PUNCT
ejpam-6846	202	11	(	(	PUNCT
ejpam-6846	202	12	‖v∗‖+	‖v∗‖+	PUNCT
ejpam-6846	202	13	‖uϵ‖)2	‖uϵ‖)2	X
ejpam-6846	202	14	+	+	CCONJ
ejpam-6846	202	15	f(uϵ)−	f(uϵ)−	NOUN
ejpam-6846	202	16	f(rϵ	f(rϵ	PROPN
ejpam-6846	202	17	)	)	PUNCT
ejpam-6846	203	1	+	+	NUM
ejpam-6846	203	2	ϵ	ϵ	X
ejpam-6846	203	3	)	)	PUNCT
ejpam-6846	203	4	1	1	NUM
ejpam-6846	203	5	2	2	NUM
ejpam-6846	203	6	+	+	NUM
ejpam-6846	203	7	‖u∗‖+	‖u∗‖+	NOUN
ejpam-6846	203	8	δ	δ	PROPN
ejpam-6846	203	9	≤	≤	NUM
ejpam-6846	203	10	(	(	PUNCT
ejpam-6846	203	11	(	(	PUNCT
ejpam-6846	203	12	‖u∗‖+	‖u∗‖+	NOUN
ejpam-6846	203	13	δ	δ	PROPN
ejpam-6846	203	14	+	+	CCONJ
ejpam-6846	203	15	(	(	PUNCT
ejpam-6846	203	16	v	v	NOUN
ejpam-6846	203	17	(	(	PUNCT
ejpam-6846	203	18	u∗	u∗	PROPN
ejpam-6846	203	19	,	,	PUNCT
ejpam-6846	203	20	uϵ	uϵ	NOUN
ejpam-6846	203	21	)	)	PUNCT
ejpam-6846	203	22	)	)	PUNCT
ejpam-6846	203	23	1	1	NUM
ejpam-6846	203	24	2	2	NUM
ejpam-6846	203	25	+	+	CCONJ
ejpam-6846	203	26	‖u∗‖	‖u∗‖	X
ejpam-6846	203	27	)	)	PUNCT
ejpam-6846	203	28	2	2	NUM
ejpam-6846	203	29	+	+	NUM
ejpam-6846	203	30	f(uϵ)−	f(uϵ)−	NOUN
ejpam-6846	203	31	f(rϵ	f(rϵ	PROPN
ejpam-6846	203	32	)	)	PUNCT
ejpam-6846	203	33	+	+	NUM
ejpam-6846	203	34	ϵ	ϵ	X
ejpam-6846	203	35	)	)	PUNCT
ejpam-6846	203	36	1	1	NUM
ejpam-6846	203	37	2	2	NUM
ejpam-6846	203	38	+	+	NUM
ejpam-6846	203	39	‖u∗‖+	‖u∗‖+	NOUN
ejpam-6846	203	40	δ	δ	PROPN
ejpam-6846	203	41	≤	≤	NUM
ejpam-6846	203	42	(	(	PUNCT
ejpam-6846	203	43	(	(	PUNCT
ejpam-6846	203	44	2‖u∗‖+	2‖u∗‖+	NUM
ejpam-6846	203	45	δ	δ	NOUN
ejpam-6846	203	46	+	+	CCONJ
ejpam-6846	203	47	(	(	PUNCT
ejpam-6846	203	48	v	v	NUM
ejpam-6846	203	49	f	f	X
ejpam-6846	203	50	(	(	PUNCT
ejpam-6846	203	51	u∗	u∗	ADJ
ejpam-6846	203	52	,	,	PUNCT
ejpam-6846	203	53	uϵ)−	uϵ)−	ADJ
ejpam-6846	203	54	f(uϵ	f(uϵ	NUM
ejpam-6846	203	55	)	)	PUNCT
ejpam-6846	203	56	)	)	PUNCT
ejpam-6846	204	1	1	1	NUM
ejpam-6846	204	2	2	2	NUM
ejpam-6846	204	3	)	)	SYM
ejpam-6846	204	4	2	2	NUM
ejpam-6846	204	5	+	+	NUM
ejpam-6846	204	6	f(uϵ)−	f(uϵ)−	NOUN
ejpam-6846	204	7	f(rϵ	f(rϵ	PROPN
ejpam-6846	204	8	)	)	PUNCT
ejpam-6846	205	1	+	+	NUM
ejpam-6846	205	2	ϵ	ϵ	X
ejpam-6846	205	3	)	)	PUNCT
ejpam-6846	205	4	1	1	NUM
ejpam-6846	205	5	2	2	NUM
ejpam-6846	205	6	+	+	NUM
ejpam-6846	205	7	‖u∗‖+	‖u∗‖+	NOUN
ejpam-6846	205	8	δ	δ	PROPN
ejpam-6846	205	9	≤	≤	NUM
ejpam-6846	205	10	(	(	PUNCT
ejpam-6846	205	11	(	(	PUNCT
ejpam-6846	205	12	2‖u∗‖+	2‖u∗‖+	NUM
ejpam-6846	205	13	δ	δ	NOUN
ejpam-6846	205	14	+	+	X
ejpam-6846	205	15	(	(	PUNCT
ejpam-6846	205	16	r	r	NOUN
ejpam-6846	205	17	+	+	NUM
ejpam-6846	205	18	ϵ−	ϵ−	NUM
ejpam-6846	205	19	f(uϵ	f(uϵ	NOUN
ejpam-6846	205	20	)	)	PUNCT
ejpam-6846	205	21	)	)	PUNCT
ejpam-6846	205	22	1	1	NUM
ejpam-6846	205	23	2	2	NUM
ejpam-6846	205	24	)	)	SYM
ejpam-6846	205	25	2	2	NUM
ejpam-6846	205	26	+	+	NUM
ejpam-6846	205	27	f(uϵ)−	f(uϵ)−	NOUN
ejpam-6846	205	28	f(rϵ	f(rϵ	PROPN
ejpam-6846	205	29	)	)	PUNCT
ejpam-6846	206	1	+	+	NUM
ejpam-6846	206	2	ϵ	ϵ	X
ejpam-6846	206	3	)	)	PUNCT
ejpam-6846	206	4	1	1	NUM
ejpam-6846	206	5	2	2	NUM
ejpam-6846	206	6	+	+	NUM
ejpam-6846	206	7	‖u∗‖+	‖u∗‖+	NOUN
ejpam-6846	206	8	δ	δ	PROPN
ejpam-6846	206	9	≤	≤	NUM
ejpam-6846	206	10	(	(	PUNCT
ejpam-6846	206	11	(	(	PUNCT
ejpam-6846	206	12	2‖u∗‖+	2‖u∗‖+	NUM
ejpam-6846	206	13	δ)2	δ)2	NOUN
ejpam-6846	206	14	+	+	NOUN
ejpam-6846	206	15	r	r	NOUN
ejpam-6846	206	16	+	+	CCONJ
ejpam-6846	206	17	2ϵ+	2ϵ+	NUM
ejpam-6846	206	18	2(2‖u∗‖+	2(2‖u∗‖+	NUM
ejpam-6846	206	19	δ)(r	δ)(r	NOUN
ejpam-6846	206	20	+	+	CCONJ
ejpam-6846	206	21	ϵ−	ϵ−	NUM
ejpam-6846	206	22	f(uϵ	f(uϵ	NOUN
ejpam-6846	206	23	)	)	PUNCT
ejpam-6846	206	24	)	)	PUNCT
ejpam-6846	206	25	1	1	NUM
ejpam-6846	206	26	2	2	NUM
ejpam-6846	206	27	−	−	NOUN
ejpam-6846	206	28	f(rϵ	f(rϵ	PROPN
ejpam-6846	206	29	)	)	PUNCT
ejpam-6846	206	30	)	)	PUNCT
ejpam-6846	206	31	1	1	NUM
ejpam-6846	206	32	2	2	NUM
ejpam-6846	206	33	+	+	CCONJ
ejpam-6846	206	34	‖u∗‖+	‖u∗‖+	NOUN
ejpam-6846	206	35	δ	δ	PROPN
ejpam-6846	206	36	.	.	PUNCT
ejpam-6846	207	1	since	since	SCONJ
ejpam-6846	207	2	f(u	f(u	PROPN
ejpam-6846	207	3	)	)	PUNCT
ejpam-6846	207	4	≥	≥	NOUN
ejpam-6846	207	5	l	l	NOUN
ejpam-6846	207	6	for	for	ADP
ejpam-6846	207	7	some	some	DET
ejpam-6846	207	8	l	l	NOUN
ejpam-6846	207	9	∈	∈	PROPN
ejpam-6846	207	10	r	r	NOUN
ejpam-6846	207	11	,	,	PUNCT
ejpam-6846	207	12	so	so	SCONJ
ejpam-6846	207	13	we	we	PRON
ejpam-6846	207	14	can	can	AUX
ejpam-6846	207	15	write	write	VERB
ejpam-6846	207	16	‖rϵ‖	‖rϵ‖	PROPN
ejpam-6846	207	17	≤	≤	PROPN
ejpam-6846	207	18	(	(	PUNCT
ejpam-6846	207	19	(	(	PUNCT
ejpam-6846	207	20	2‖u∗‖+	2‖u∗‖+	NUM
ejpam-6846	207	21	δ)2	δ)2	NOUN
ejpam-6846	207	22	+	+	NOUN
ejpam-6846	207	23	r	r	NOUN
ejpam-6846	207	24	+	+	CCONJ
ejpam-6846	207	25	2ϵ+	2ϵ+	NUM
ejpam-6846	207	26	2(2‖u∗‖+	2(2‖u∗‖+	NUM
ejpam-6846	207	27	δ)(r	δ)(r	NOUN
ejpam-6846	207	28	+	+	CCONJ
ejpam-6846	207	29	ϵ−	ϵ−	NUM
ejpam-6846	207	30	l	l	NOUN
ejpam-6846	207	31	)	)	PUNCT
ejpam-6846	207	32	1	1	NUM
ejpam-6846	207	33	2	2	NUM
ejpam-6846	207	34	−	−	PROPN
ejpam-6846	207	35	l	l	NOUN
ejpam-6846	207	36	)	)	PUNCT
ejpam-6846	207	37	1	1	NUM
ejpam-6846	207	38	2	2	NUM
ejpam-6846	207	39	+	+	NUM
ejpam-6846	207	40	‖u∗‖+	‖u∗‖+	NOUN
ejpam-6846	207	41	δ	δ	NOUN
ejpam-6846	207	42	:	:	PUNCT
ejpam-6846	207	43	=	=	PUNCT
ejpam-6846	207	44	cδ,ϵ,u∗	cδ,ϵ,u∗	ADJ
ejpam-6846	207	45	.	.	PUNCT
ejpam-6846	208	1	now	now	ADV
ejpam-6846	208	2	letting	let	VERB
ejpam-6846	208	3	ϵ	ϵ	X
ejpam-6846	208	4	→	→	SYM
ejpam-6846	208	5	0	0	NUM
ejpam-6846	208	6	and	and	CCONJ
ejpam-6846	208	7	with	with	ADP
ejpam-6846	208	8	kδ	kδ	NOUN
ejpam-6846	208	9	,	,	PUNCT
ejpam-6846	208	10	u∗	u∗	PROPN
ejpam-6846	208	11	=	=	NOUN
ejpam-6846	208	12	:	:	PUNCT
ejpam-6846	208	13	2(‖u∗‖+	2(‖u∗‖+	NUM
ejpam-6846	208	14	cδ	cδ	NOUN
ejpam-6846	208	15	,	,	PUNCT
ejpam-6846	208	16	u∗	u∗	ADJ
ejpam-6846	208	17	+	+	CCONJ
ejpam-6846	208	18	δ	δ	NOUN
ejpam-6846	208	19	)	)	PUNCT
ejpam-6846	208	20	we	we	PRON
ejpam-6846	208	21	get	get	VERB
ejpam-6846	208	22	|mf	|mf	NOUN
ejpam-6846	208	23	,	,	PUNCT
ejpam-6846	208	24	s(z	s(z	PROPN
ejpam-6846	208	25	∗)−mf	∗)−mf	PROPN
ejpam-6846	208	26	,	,	PUNCT
ejpam-6846	208	27	s(v	s(v	PROPN
ejpam-6846	208	28	∗)|	∗)|	PROPN
ejpam-6846	208	29	≤	≤	NOUN
ejpam-6846	208	30	2(‖u∗‖+	2(‖u∗‖+	NUM
ejpam-6846	208	31	cδ	cδ	NOUN
ejpam-6846	208	32	,	,	PUNCT
ejpam-6846	208	33	u∗	u∗	ADJ
ejpam-6846	209	1	+	+	CCONJ
ejpam-6846	209	2	δ)‖z∗	δ)‖z∗	PROPN
ejpam-6846	209	3	−	−	PROPN
ejpam-6846	209	4	v∗‖	v∗‖	PROPN
ejpam-6846	209	5	≤	≤	ADJ
ejpam-6846	209	6	kδ	kδ	NOUN
ejpam-6846	209	7	,	,	PUNCT
ejpam-6846	209	8	u∗‖z∗	u∗‖z∗	VERB
ejpam-6846	209	9	−	−	PROPN
ejpam-6846	209	10	v∗‖.	v∗‖.	PROPN
ejpam-6846	209	11	a.	a.	NOUN
ejpam-6846	209	12	al	al	PROPN
ejpam-6846	209	13	tane	tane	PROPN
ejpam-6846	209	14	,	,	PUNCT
ejpam-6846	209	15	l.	l.	PROPN
ejpam-6846	209	16	s.	s.	PROPN
ejpam-6846	209	17	keong	keong	PROPN
ejpam-6846	209	18	/	/	PUNCT
ejpam-6846	209	19	eur	eur	PROPN
ejpam-6846	209	20	.	.	PUNCT
ejpam-6846	210	1	j.	j.	PROPN
ejpam-6846	210	2	pure	pure	PROPN
ejpam-6846	210	3	appl	appl	PROPN
ejpam-6846	210	4	.	.	PROPN
ejpam-6846	210	5	math	math	PROPN
ejpam-6846	210	6	,	,	PUNCT
ejpam-6846	210	7	18	18	NUM
ejpam-6846	210	8	(	(	PUNCT
ejpam-6846	210	9	4	4	NUM
ejpam-6846	210	10	)	)	PUNCT
ejpam-6846	210	11	(	(	PUNCT
ejpam-6846	210	12	2025	2025	NUM
ejpam-6846	210	13	)	)	PUNCT
ejpam-6846	210	14	,	,	PUNCT
ejpam-6846	210	15	6846	6846	NUM
ejpam-6846	210	16	9	9	NUM
ejpam-6846	210	17	of	of	ADP
ejpam-6846	210	18	13	13	NUM
ejpam-6846	210	19	so	so	SCONJ
ejpam-6846	210	20	the	the	DET
ejpam-6846	210	21	proof	proof	NOUN
ejpam-6846	210	22	is	be	AUX
ejpam-6846	210	23	complete	complete	ADJ
ejpam-6846	210	24	□	□	PUNCT
ejpam-6846	210	25	the	the	DET
ejpam-6846	210	26	fréchet	fréchet	ADJ
ejpam-6846	210	27	sub	sub	NOUN
ejpam-6846	210	28	-	-	NOUN
ejpam-6846	210	29	differentiability	differentiability	NOUN
ejpam-6846	210	30	of	of	ADP
ejpam-6846	210	31	mf	mf	NOUN
ejpam-6846	210	32	,	,	PUNCT
ejpam-6846	210	33	s	s	PART
ejpam-6846	210	34	is	be	AUX
ejpam-6846	210	35	equivalent	equivalent	ADJ
ejpam-6846	210	36	to	to	ADP
ejpam-6846	210	37	its	its	PRON
ejpam-6846	210	38	fréchet	fréchet	ADJ
ejpam-6846	210	39	differentiability	differentiability	NOUN
ejpam-6846	210	40	,	,	PUNCT
ejpam-6846	210	41	as	as	SCONJ
ejpam-6846	210	42	demonstrated	demonstrate	VERB
ejpam-6846	210	43	by	by	ADP
ejpam-6846	210	44	the	the	DET
ejpam-6846	210	45	following	follow	VERB
ejpam-6846	210	46	lemma	lemma	PROPN
ejpam-6846	210	47	.	.	PUNCT
ejpam-6846	211	1	lemma	lemma	PROPN
ejpam-6846	211	2	4	4	NUM
ejpam-6846	211	3	.	.	PUNCT
ejpam-6846	212	1	given	give	VERB
ejpam-6846	212	2	that	that	DET
ejpam-6846	212	3	x∗	x∗	PROPN
ejpam-6846	212	4	an	an	DET
ejpam-6846	212	5	element	element	NOUN
ejpam-6846	212	6	of	of	ADP
ejpam-6846	212	7	smooth	smooth	ADJ
ejpam-6846	212	8	dual	dual	ADJ
ejpam-6846	212	9	norm	norm	NOUN
ejpam-6846	212	10	space	space	NOUN
ejpam-6846	212	11	x∗	x∗	PROPN
ejpam-6846	212	12	of	of	ADP
ejpam-6846	212	13	reflexive	reflexive	ADJ
ejpam-6846	212	14	banach	banach	NOUN
ejpam-6846	212	15	space	space	NOUN
ejpam-6846	212	16	x.	x.	NOUN
ejpam-6846	212	17	then	then	ADV
ejpam-6846	212	18	,	,	PUNCT
ejpam-6846	212	19	the	the	DET
ejpam-6846	212	20	two	two	NUM
ejpam-6846	212	21	statements	statement	NOUN
ejpam-6846	212	22	that	that	PRON
ejpam-6846	212	23	follow	follow	VERB
ejpam-6846	212	24	are	be	AUX
ejpam-6846	212	25	equivalent	equivalent	ADJ
ejpam-6846	212	26	:	:	PUNCT
ejpam-6846	212	27	(	(	PUNCT
ejpam-6846	212	28	i	i	NOUN
ejpam-6846	212	29	)	)	PUNCT
ejpam-6846	212	30	∂fmf	∂fmf	NUM
ejpam-6846	212	31	,	,	PUNCT
ejpam-6846	212	32	s(x	s(x	NOUN
ejpam-6846	212	33	∗	∗	NOUN
ejpam-6846	212	34	)	)	PUNCT
ejpam-6846	212	35	6=	6=	ADP
ejpam-6846	212	36	ϕ.	ϕ.	PROPN
ejpam-6846	212	37	(	(	PUNCT
ejpam-6846	212	38	ii	ii	PROPN
ejpam-6846	212	39	)	)	PUNCT
ejpam-6846	212	40	mf	mf	PROPN
ejpam-6846	212	41	,	,	PUNCT
ejpam-6846	212	42	s	s	PART
ejpam-6846	212	43	is	be	AUX
ejpam-6846	212	44	fréchet	fréchet	VERB
ejpam-6846	212	45	differentiable	differentiable	ADJ
ejpam-6846	212	46	at	at	ADP
ejpam-6846	212	47	x∗.	x∗.	ADJ
ejpam-6846	212	48	proof	proof	NOUN
ejpam-6846	212	49	:	:	PUNCT
ejpam-6846	212	50	from	from	ADP
ejpam-6846	212	51	(	(	PUNCT
ejpam-6846	212	52	2	2	NUM
ejpam-6846	212	53	)	)	PUNCT
ejpam-6846	212	54	→	→	X
ejpam-6846	212	55	(	(	PUNCT
ejpam-6846	212	56	1	1	X
ejpam-6846	212	57	)	)	PUNCT
ejpam-6846	212	58	clear	clear	ADJ
ejpam-6846	212	59	,	,	PUNCT
ejpam-6846	212	60	so	so	SCONJ
ejpam-6846	212	61	we	we	PRON
ejpam-6846	212	62	merely	merely	ADV
ejpam-6846	212	63	need	need	VERB
ejpam-6846	212	64	to	to	PART
ejpam-6846	212	65	demonstrate	demonstrate	VERB
ejpam-6846	212	66	the	the	DET
ejpam-6846	212	67	opposite	opposite	ADJ
ejpam-6846	212	68	implication	implication	NOUN
ejpam-6846	212	69	.	.	PUNCT
ejpam-6846	213	1	suppose	suppose	VERB
ejpam-6846	213	2	∂fmf	∂fmf	NUM
ejpam-6846	213	3	,	,	PUNCT
ejpam-6846	213	4	s(x	s(x	NOUN
ejpam-6846	213	5	∗	∗	NOUN
ejpam-6846	213	6	)	)	PUNCT
ejpam-6846	213	7	6=	6=	ADP
ejpam-6846	213	8	ϕ	ϕ	NOUN
ejpam-6846	213	9	and	and	CCONJ
ejpam-6846	213	10	let	let	VERB
ejpam-6846	213	11	z	z	PROPN
ejpam-6846	213	12	∈	∈	PROPN
ejpam-6846	213	13	∂fmf	∂fmf	NUM
ejpam-6846	213	14	,	,	PUNCT
ejpam-6846	213	15	s(x	s(x	NOUN
ejpam-6846	213	16	∗	∗	NOUN
ejpam-6846	213	17	)	)	PUNCT
ejpam-6846	213	18	.	.	PUNCT
ejpam-6846	214	1	by	by	ADP
ejpam-6846	214	2	lemma	lemma	PROPN
ejpam-6846	214	3	(	(	PUNCT
ejpam-6846	214	4	2	2	X
ejpam-6846	214	5	)	)	PUNCT
ejpam-6846	214	6	we	we	PRON
ejpam-6846	214	7	get	get	VERB
ejpam-6846	214	8	z	z	NOUN
ejpam-6846	214	9	=	=	SYM
ejpam-6846	214	10	2(j∗x∗	2(j∗x∗	NUM
ejpam-6846	214	11	−	−	PROPN
ejpam-6846	214	12	v	v	NOUN
ejpam-6846	214	13	)	)	PUNCT
ejpam-6846	214	14	with	with	ADP
ejpam-6846	214	15	v	v	NUM
ejpam-6846	214	16	∈	∈	PROPN
ejpam-6846	214	17	πf	πf	NOUN
ejpam-6846	214	18	s(x	s(x	NOUN
ejpam-6846	214	19	∗	∗	NOUN
ejpam-6846	214	20	)	)	PUNCT
ejpam-6846	214	21	.	.	PUNCT
ejpam-6846	215	1	for	for	ADP
ejpam-6846	215	2	some	some	DET
ejpam-6846	215	3	ϵ	ϵ	PROPN
ejpam-6846	215	4	>	>	X
ejpam-6846	215	5	0	0	NUM
ejpam-6846	215	6	.	.	PUNCT
ejpam-6846	215	7	using	use	VERB
ejpam-6846	215	8	concept	concept	NOUN
ejpam-6846	215	9	of	of	ADP
ejpam-6846	215	10	∂f	∂f	PROPN
ejpam-6846	215	11	,	,	PUNCT
ejpam-6846	215	12	∃β1	∃β1	ADJ
ejpam-6846	215	13	>	>	X
ejpam-6846	215	14	0	0	NUM
ejpam-6846	215	15	,	,	PUNCT
ejpam-6846	215	16	and	and	CCONJ
ejpam-6846	215	17	so	so	ADV
ejpam-6846	215	18	µ	µ	X
ejpam-6846	215	19	∈	∈	NOUN
ejpam-6846	215	20	(	(	PUNCT
ejpam-6846	215	21	0	0	NUM
ejpam-6846	215	22	,	,	PUNCT
ejpam-6846	215	23	β1	β1	PROPN
ejpam-6846	215	24	)	)	PUNCT
ejpam-6846	215	25	and	and	CCONJ
ejpam-6846	215	26	all	all	DET
ejpam-6846	215	27	u∗	u∗	NOUN
ejpam-6846	215	28	∈	∈	PROPN
ejpam-6846	215	29	b∗	b∗	ADJ
ejpam-6846	215	30	,	,	PUNCT
ejpam-6846	215	31	we	we	PRON
ejpam-6846	215	32	get	get	VERB
ejpam-6846	215	33	〈	〈	PROPN
ejpam-6846	215	34	2(j∗x∗	2(j∗x∗	PROPN
ejpam-6846	215	35	−	−	PRON
ejpam-6846	215	36	v);µu∗	v);µu∗	NOUN
ejpam-6846	215	37	〉	〉	PROPN
ejpam-6846	215	38	≤	≤	NOUN
ejpam-6846	215	39	mf	mf	X
ejpam-6846	215	40	,	,	PUNCT
ejpam-6846	215	41	s(x	s(x	PROPN
ejpam-6846	215	42	∗	∗	NOUN
ejpam-6846	215	43	+	+	NUM
ejpam-6846	215	44	µu∗)−mf	µu∗)−mf	NOUN
ejpam-6846	215	45	,	,	PUNCT
ejpam-6846	215	46	s(x	s(x	VERB
ejpam-6846	215	47	∗	∗	NOUN
ejpam-6846	215	48	)	)	PUNCT
ejpam-6846	216	1	+	+	NUM
ejpam-6846	217	1	ϵµ.	ϵµ.	NOUN
ejpam-6846	217	2	which	which	PRON
ejpam-6846	217	3	implies	imply	VERB
ejpam-6846	217	4	µ−1	µ−1	PROPN
ejpam-6846	217	5	[	[	X
ejpam-6846	217	6	mf	mf	INTJ
ejpam-6846	217	7	,	,	PUNCT
ejpam-6846	217	8	s(x	s(x	PROPN
ejpam-6846	217	9	∗	∗	NOUN
ejpam-6846	217	10	+	+	NUM
ejpam-6846	217	11	µu∗)−mf	µu∗)−mf	NOUN
ejpam-6846	217	12	,	,	PUNCT
ejpam-6846	217	13	s(x	s(x	PROPN
ejpam-6846	217	14	∗)]−	∗)]−	PROPN
ejpam-6846	217	15	〈	〈	PROPN
ejpam-6846	217	16	2(j∗x∗	2(j∗x∗	PROPN
ejpam-6846	217	17	−	−	PRON
ejpam-6846	217	18	v);u∗	v);u∗	PROPN
ejpam-6846	217	19	〉	〉	PROPN
ejpam-6846	217	20	≥	≥	NUM
ejpam-6846	217	21	−ϵ	−ϵ	VERB
ejpam-6846	217	22	;	;	PUNCT
ejpam-6846	217	23	∀µ	∀µ	PROPN
ejpam-6846	217	24	∈	∈	PROPN
ejpam-6846	217	25	(	(	PUNCT
ejpam-6846	217	26	0	0	NUM
ejpam-6846	217	27	,	,	PUNCT
ejpam-6846	217	28	β1	β1	PROPN
ejpam-6846	217	29	)	)	PUNCT
ejpam-6846	217	30	,	,	PUNCT
ejpam-6846	217	31	∀u∗	∀u∗	PROPN
ejpam-6846	217	32	∈	∈	PROPN
ejpam-6846	217	33	b∗.	b∗.	NOUN
ejpam-6846	217	34	also	also	ADV
ejpam-6846	217	35	,	,	PUNCT
ejpam-6846	217	36	by	by	ADP
ejpam-6846	217	37	definition	definition	NOUN
ejpam-6846	217	38	of	of	ADP
ejpam-6846	217	39	mf	mf	PROPN
ejpam-6846	217	40	,	,	PUNCT
ejpam-6846	217	41	s	s	PART
ejpam-6846	217	42	we	we	PRON
ejpam-6846	217	43	have	have	AUX
ejpam-6846	217	44	µ−1	µ−1	VERB
ejpam-6846	217	45	[	[	X
ejpam-6846	217	46	mf	mf	INTJ
ejpam-6846	217	47	,	,	PUNCT
ejpam-6846	217	48	s(x	s(x	PROPN
ejpam-6846	217	49	∗	∗	NOUN
ejpam-6846	217	50	+	+	NUM
ejpam-6846	217	51	µu∗)−mf	µu∗)−mf	NOUN
ejpam-6846	217	52	,	,	PUNCT
ejpam-6846	217	53	s(x	s(x	VERB
ejpam-6846	217	54	∗	∗	NOUN
ejpam-6846	217	55	)	)	PUNCT
ejpam-6846	217	56	]	]	PUNCT
ejpam-6846	217	57	≤	≤	NUM
ejpam-6846	218	1	µ−1	µ−1	PROPN
ejpam-6846	218	2	[	[	PUNCT
ejpam-6846	218	3	v	v	X
ejpam-6846	218	4	f	f	X
ejpam-6846	218	5	(	(	PUNCT
ejpam-6846	218	6	x∗	x∗	PROPN
ejpam-6846	218	7	+	+	CCONJ
ejpam-6846	218	8	µu∗	µu∗	ADV
ejpam-6846	218	9	,	,	PUNCT
ejpam-6846	218	10	y)−	y)−	PROPN
ejpam-6846	218	11	v	v	ADP
ejpam-6846	218	12	f	f	X
ejpam-6846	218	13	(	(	PUNCT
ejpam-6846	218	14	x∗	x∗	PROPN
ejpam-6846	218	15	,	,	PUNCT
ejpam-6846	218	16	y	y	NOUN
ejpam-6846	218	17	)	)	PUNCT
ejpam-6846	218	18	]	]	PUNCT
ejpam-6846	218	19	.	.	PUNCT
ejpam-6846	219	1	since	since	SCONJ
ejpam-6846	219	2	∇fv	∇fv	PROPN
ejpam-6846	219	3	f	f	PROPN
ejpam-6846	219	4	(	(	PUNCT
ejpam-6846	219	5	·	·	PUNCT
ejpam-6846	219	6	;	;	PUNCT
ejpam-6846	219	7	v	v	X
ejpam-6846	219	8	)	)	PUNCT
ejpam-6846	219	9	=	=	SYM
ejpam-6846	219	10	2(j∗x∗	2(j∗x∗	NUM
ejpam-6846	219	11	−	−	NOUN
ejpam-6846	219	12	v	v	NOUN
ejpam-6846	219	13	)	)	PUNCT
ejpam-6846	219	14	,	,	PUNCT
ejpam-6846	219	15	there	there	PRON
ejpam-6846	219	16	exists	exist	VERB
ejpam-6846	219	17	β2	β2	NOUN
ejpam-6846	219	18	>	>	X
ejpam-6846	219	19	0	0	NUM
ejpam-6846	220	1	such	such	ADJ
ejpam-6846	220	2	that	that	PRON
ejpam-6846	220	3	for	for	ADP
ejpam-6846	220	4	any	any	DET
ejpam-6846	220	5	µ	µ	X
ejpam-6846	220	6	∈	∈	NOUN
ejpam-6846	220	7	(	(	PUNCT
ejpam-6846	220	8	0	0	NUM
ejpam-6846	220	9	,	,	PUNCT
ejpam-6846	220	10	β2	β2	NOUN
ejpam-6846	220	11	)	)	PUNCT
ejpam-6846	220	12	and	and	CCONJ
ejpam-6846	220	13	for	for	ADP
ejpam-6846	220	14	all	all	DET
ejpam-6846	220	15	u∗	u∗	NOUN
ejpam-6846	220	16	∈	∈	PROPN
ejpam-6846	220	17	b∗	b∗	ADV
ejpam-6846	221	1	we	we	PRON
ejpam-6846	221	2	have.∣∣∣µ−1	have.∣∣∣µ−1	PROPN
ejpam-6846	221	3	[	[	PUNCT
ejpam-6846	221	4	v	v	NOUN
ejpam-6846	221	5	f	f	X
ejpam-6846	221	6	(	(	PUNCT
ejpam-6846	221	7	x∗	x∗	PROPN
ejpam-6846	221	8	+	+	CCONJ
ejpam-6846	221	9	µu∗	µu∗	ADV
ejpam-6846	221	10	,	,	PUNCT
ejpam-6846	221	11	y)−	y)−	PROPN
ejpam-6846	221	12	v	v	ADP
ejpam-6846	221	13	f	f	X
ejpam-6846	221	14	(	(	PUNCT
ejpam-6846	221	15	x∗	x∗	PROPN
ejpam-6846	221	16	,	,	PUNCT
ejpam-6846	221	17	y	y	NOUN
ejpam-6846	221	18	)	)	PUNCT
ejpam-6846	221	19	]	]	PUNCT
ejpam-6846	222	1	−	−	PROPN
ejpam-6846	223	1	〈	〈	PROPN
ejpam-6846	223	2	2(j∗x∗	2(j∗x∗	PROPN
ejpam-6846	223	3	−	−	PROPN
ejpam-6846	223	4	v	v	NOUN
ejpam-6846	223	5	)	)	PUNCT
ejpam-6846	224	1	,	,	PUNCT
ejpam-6846	224	2	u∗	u∗	PROPN
ejpam-6846	224	3	〉	〉	PROPN
ejpam-6846	224	4	∣∣∣	∣∣∣	NOUN
ejpam-6846	224	5	≤	≤	NUM
ejpam-6846	224	6	ϵ.	ϵ.	NOUN
ejpam-6846	225	1	and	and	CCONJ
ejpam-6846	225	2	so	so	ADV
ejpam-6846	225	3	µ−1	µ−1	PROPN
ejpam-6846	226	1	[	[	X
ejpam-6846	226	2	mf	mf	INTJ
ejpam-6846	226	3	,	,	PUNCT
ejpam-6846	226	4	s(x	s(x	PROPN
ejpam-6846	226	5	∗	∗	NOUN
ejpam-6846	226	6	+	+	NUM
ejpam-6846	226	7	µu∗)−mf	µu∗)−mf	NOUN
ejpam-6846	226	8	,	,	PUNCT
ejpam-6846	226	9	s(x	s(x	PROPN
ejpam-6846	226	10	∗)]−	∗)]−	PROPN
ejpam-6846	226	11	〈	〈	PROPN
ejpam-6846	226	12	2(j∗x∗	2(j∗x∗	NUM
ejpam-6846	226	13	−	−	PRON
ejpam-6846	226	14	v);u∗	v);u∗	PROPN
ejpam-6846	226	15	〉	〉	PROPN
ejpam-6846	226	16	≤	≤	NUM
ejpam-6846	226	17	ϵ.	ϵ.	NOUN
ejpam-6846	227	1	hence∣∣µ−1	hence∣∣µ−1	PROPN
ejpam-6846	228	1	[	[	X
ejpam-6846	228	2	mf	mf	INTJ
ejpam-6846	228	3	,	,	PUNCT
ejpam-6846	228	4	s(x	s(x	PROPN
ejpam-6846	228	5	∗	∗	NOUN
ejpam-6846	228	6	+	+	NUM
ejpam-6846	228	7	µu∗)−mf	µu∗)−mf	NOUN
ejpam-6846	228	8	,	,	PUNCT
ejpam-6846	228	9	s(x	s(x	PROPN
ejpam-6846	228	10	∗)]−	∗)]−	PROPN
ejpam-6846	228	11	〈	〈	PROPN
ejpam-6846	228	12	2(j∗x∗	2(j∗x∗	NUM
ejpam-6846	228	13	−	−	PRON
ejpam-6846	229	1	v);u∗	v);u∗	ADJ
ejpam-6846	229	2	〉	〉	PROPN
ejpam-6846	229	3	∣∣	∣∣	NUM
ejpam-6846	229	4	≤	≤	PROPN
ejpam-6846	229	5	ϵ	ϵ	ADP
ejpam-6846	229	6	,	,	PUNCT
ejpam-6846	229	7	µ	µ	X
ejpam-6846	229	8	∈	∈	NOUN
ejpam-6846	229	9	(	(	PUNCT
ejpam-6846	229	10	0	0	NUM
ejpam-6846	229	11	,	,	PUNCT
ejpam-6846	229	12	β	β	NOUN
ejpam-6846	229	13	)	)	PUNCT
ejpam-6846	229	14	,	,	PUNCT
ejpam-6846	229	15	∀u∗	∀u∗	PROPN
ejpam-6846	229	16	∈	∈	PROPN
ejpam-6846	229	17	b∗.	b∗.	NOUN
ejpam-6846	229	18	with	with	ADP
ejpam-6846	229	19	β	β	X
ejpam-6846	229	20	=	=	SYM
ejpam-6846	229	21	min{β1	min{β1	ADJ
ejpam-6846	229	22	,	,	PUNCT
ejpam-6846	229	23	β2	β2	NOUN
ejpam-6846	229	24	}	}	PUNCT
ejpam-6846	229	25	,	,	PUNCT
ejpam-6846	229	26	since	since	SCONJ
ejpam-6846	229	27	ϵ	ϵ	X
ejpam-6846	229	28	>	>	X
ejpam-6846	229	29	0	0	NUM
ejpam-6846	229	30	is	be	AUX
ejpam-6846	229	31	arbitrary	arbitrary	ADJ
ejpam-6846	229	32	,	,	PUNCT
ejpam-6846	229	33	so	so	ADV
ejpam-6846	229	34	∇fmf	∇fmf	PROPN
ejpam-6846	229	35	,	,	PUNCT
ejpam-6846	229	36	s(x	s(x	ADJ
ejpam-6846	229	37	∗	∗	NOUN
ejpam-6846	229	38	)	)	PUNCT
ejpam-6846	230	1	=	=	SYM
ejpam-6846	230	2	2(j∗x∗	2(j∗x∗	NUM
ejpam-6846	230	3	−	−	NOUN
ejpam-6846	230	4	v	v	NOUN
ejpam-6846	230	5	)	)	PUNCT
ejpam-6846	230	6	.	.	PUNCT
ejpam-6846	231	1	thus	thus	ADV
ejpam-6846	231	2	,	,	PUNCT
ejpam-6846	231	3	our	our	PRON
ejpam-6846	231	4	proof	proof	NOUN
ejpam-6846	231	5	is	be	AUX
ejpam-6846	231	6	finished	finish	VERB
ejpam-6846	231	7	.	.	PUNCT
ejpam-6846	232	1	□	□	PUNCT
ejpam-6846	232	2	the	the	DET
ejpam-6846	232	3	‖.‖	‖.‖	NOUN
ejpam-6846	232	4	−	−	PROPN
ejpam-6846	232	5	‖.‖	‖.‖	NOUN
ejpam-6846	232	6	continuity	continuity	NOUN
ejpam-6846	232	7	of	of	ADP
ejpam-6846	232	8	πf	πf	NOUN
ejpam-6846	232	9	s	s	VERB
ejpam-6846	232	10	are	be	AUX
ejpam-6846	232	11	related	relate	VERB
ejpam-6846	232	12	to	to	ADP
ejpam-6846	232	13	the	the	DET
ejpam-6846	232	14	continuou	continuou	ADJ
ejpam-6846	232	15	∇fmf	∇fmf	PROPN
ejpam-6846	232	16	,	,	PUNCT
ejpam-6846	232	17	s	s	VERB
ejpam-6846	232	18	by	by	ADP
ejpam-6846	232	19	the	the	DET
ejpam-6846	232	20	following	follow	VERB
ejpam-6846	232	21	lemma	lemma	PROPN
ejpam-6846	232	22	.	.	PUNCT
ejpam-6846	232	23	a.	a.	PROPN
ejpam-6846	232	24	al	al	PROPN
ejpam-6846	232	25	tane	tane	PROPN
ejpam-6846	232	26	,	,	PUNCT
ejpam-6846	232	27	l.	l.	PROPN
ejpam-6846	232	28	s.	s.	PROPN
ejpam-6846	232	29	keong	keong	PROPN
ejpam-6846	232	30	/	/	PUNCT
ejpam-6846	232	31	eur	eur	PROPN
ejpam-6846	232	32	.	.	PUNCT
ejpam-6846	233	1	j.	j.	PROPN
ejpam-6846	233	2	pure	pure	PROPN
ejpam-6846	233	3	appl	appl	PROPN
ejpam-6846	233	4	.	.	PROPN
ejpam-6846	233	5	math	math	PROPN
ejpam-6846	233	6	,	,	PUNCT
ejpam-6846	233	7	18	18	NUM
ejpam-6846	233	8	(	(	PUNCT
ejpam-6846	233	9	4	4	NUM
ejpam-6846	233	10	)	)	PUNCT
ejpam-6846	233	11	(	(	PUNCT
ejpam-6846	233	12	2025	2025	NUM
ejpam-6846	233	13	)	)	PUNCT
ejpam-6846	233	14	,	,	PUNCT
ejpam-6846	233	15	6846	6846	NUM
ejpam-6846	233	16	10	10	NUM
ejpam-6846	233	17	of	of	ADP
ejpam-6846	233	18	13	13	NUM
ejpam-6846	233	19	lemma	lemma	PROPN
ejpam-6846	233	20	5	5	NUM
ejpam-6846	233	21	.	.	PUNCT
ejpam-6846	234	1	given	give	VERB
ejpam-6846	234	2	that	that	DET
ejpam-6846	234	3	u∗	u∗	NOUN
ejpam-6846	234	4	is	be	AUX
ejpam-6846	234	5	an	an	DET
ejpam-6846	234	6	open	open	ADJ
ejpam-6846	234	7	set	set	NOUN
ejpam-6846	234	8	of	of	ADP
ejpam-6846	234	9	smooth	smooth	ADJ
ejpam-6846	234	10	dual	dual	ADJ
ejpam-6846	234	11	norm	norm	NOUN
ejpam-6846	234	12	space	space	NOUN
ejpam-6846	234	13	x∗	x∗	PROPN
ejpam-6846	234	14	of	of	ADP
ejpam-6846	234	15	reflexive	reflexive	ADJ
ejpam-6846	234	16	banach	banach	NOUN
ejpam-6846	234	17	space	space	NOUN
ejpam-6846	234	18	x.	x.	NOUN
ejpam-6846	234	19	then	then	ADV
ejpam-6846	234	20	,	,	PUNCT
ejpam-6846	234	21	the	the	DET
ejpam-6846	234	22	two	two	NUM
ejpam-6846	234	23	claims	claim	NOUN
ejpam-6846	234	24	that	that	PRON
ejpam-6846	234	25	follow	follow	VERB
ejpam-6846	234	26	are	be	AUX
ejpam-6846	234	27	equivalent	equivalent	ADJ
ejpam-6846	234	28	:	:	PUNCT
ejpam-6846	234	29	(	(	PUNCT
ejpam-6846	234	30	i	i	NOUN
ejpam-6846	234	31	)	)	PUNCT
ejpam-6846	234	32	the	the	DET
ejpam-6846	234	33	function	function	NOUN
ejpam-6846	234	34	mf	mf	VERB
ejpam-6846	234	35	,	,	PUNCT
ejpam-6846	234	36	s	s	PART
ejpam-6846	234	37	is	be	AUX
ejpam-6846	234	38	c1	c1	NOUN
ejpam-6846	234	39	on	on	ADP
ejpam-6846	234	40	u∗.	u∗.	PROPN
ejpam-6846	234	41	(	(	PUNCT
ejpam-6846	234	42	ii	ii	PROPN
ejpam-6846	234	43	)	)	PUNCT
ejpam-6846	234	44	the	the	DET
ejpam-6846	234	45	operator	operator	NOUN
ejpam-6846	235	1	πf	πf	INTJ
ejpam-6846	235	2	s	s	VERB
ejpam-6846	235	3	is	be	AUX
ejpam-6846	235	4	single	single	ADV
ejpam-6846	235	5	-	-	PUNCT
ejpam-6846	235	6	valued	value	VERB
ejpam-6846	235	7	and	and	CCONJ
ejpam-6846	235	8	‖.‖	‖.‖	NOUN
ejpam-6846	235	9	−	−	PROPN
ejpam-6846	235	10	‖.‖	‖.‖	NOUN
ejpam-6846	235	11	continuous	continuous	ADJ
ejpam-6846	235	12	on	on	ADP
ejpam-6846	235	13	u∗.	u∗.	PROPN
ejpam-6846	235	14	proof	proof	NOUN
ejpam-6846	235	15	:	:	PUNCT
ejpam-6846	235	16	(	(	PUNCT
ejpam-6846	235	17	1	1	X
ejpam-6846	235	18	)	)	PUNCT
ejpam-6846	235	19	⇒	⇒	NOUN
ejpam-6846	235	20	(	(	PUNCT
ejpam-6846	235	21	2	2	X
ejpam-6846	235	22	)	)	PUNCT
ejpam-6846	235	23	assuming	assume	VERB
ejpam-6846	235	24	that	that	SCONJ
ejpam-6846	235	25	mf	mf	NOUN
ejpam-6846	235	26	,	,	PUNCT
ejpam-6846	235	27	s	s	PART
ejpam-6846	235	28	is	be	AUX
ejpam-6846	235	29	c1	c1	NOUN
ejpam-6846	235	30	on	on	ADP
ejpam-6846	235	31	u∗	u∗	PROPN
ejpam-6846	235	32	,	,	PUNCT
ejpam-6846	235	33	πf	πf	PROPN
ejpam-6846	235	34	s	s	VERB
ejpam-6846	235	35	is	be	AUX
ejpam-6846	235	36	single	single	ADJ
ejpam-6846	235	37	-	-	PUNCT
ejpam-6846	235	38	value	value	NOUN
ejpam-6846	235	39	functional	functional	ADJ
ejpam-6846	235	40	on	on	ADP
ejpam-6846	235	41	u∗	u∗	NOUN
ejpam-6846	235	42	and	and	CCONJ
ejpam-6846	235	43	for	for	ADP
ejpam-6846	235	44	every	every	DET
ejpam-6846	235	45	u∗	u∗	PROPN
ejpam-6846	235	46	∈	∈	PROPN
ejpam-6846	235	47	u∗	u∗	NOUN
ejpam-6846	235	48	,	,	PUNCT
ejpam-6846	235	49	πf	πf	INTJ
ejpam-6846	235	50	s(u	s(u	PROPN
ejpam-6846	235	51	∗	∗	NOUN
ejpam-6846	235	52	)	)	PUNCT
ejpam-6846	236	1	=	=	SYM
ejpam-6846	236	2	j∗u∗	j∗u∗	NOUN
ejpam-6846	237	1	−	−	NOUN
ejpam-6846	237	2	1	1	NUM
ejpam-6846	237	3	2∇	2∇	NUM
ejpam-6846	237	4	fmf	fmf	NOUN
ejpam-6846	237	5	,	,	PUNCT
ejpam-6846	237	6	s(u	s(u	PROPN
ejpam-6846	237	7	∗	∗	NOUN
ejpam-6846	237	8	)	)	PUNCT
ejpam-6846	237	9	by	by	ADP
ejpam-6846	237	10	lemma	lemma	PROPN
ejpam-6846	237	11	(	(	PUNCT
ejpam-6846	237	12	4	4	NUM
ejpam-6846	237	13	)	)	PUNCT
ejpam-6846	237	14	.	.	PUNCT
ejpam-6846	238	1	since	since	SCONJ
ejpam-6846	238	2	j∗	j∗	PROPN
ejpam-6846	238	3	and	and	CCONJ
ejpam-6846	238	4	∇fmf	∇fmf	PROPN
ejpam-6846	238	5	,	,	PUNCT
ejpam-6846	238	6	s	s	VERB
ejpam-6846	238	7	are	be	AUX
ejpam-6846	238	8	‖.‖	‖.‖	NOUN
ejpam-6846	238	9	−	−	PROPN
ejpam-6846	238	10	‖.‖	‖.‖	NOUN
ejpam-6846	238	11	continuous	continuous	ADJ
ejpam-6846	238	12	,	,	PUNCT
ejpam-6846	238	13	so	so	ADV
ejpam-6846	238	14	πf	πf	INTJ
ejpam-6846	238	15	s	s	VERB
ejpam-6846	238	16	is	be	AUX
ejpam-6846	238	17	‖.‖	‖.‖	NOUN
ejpam-6846	238	18	−	−	PROPN
ejpam-6846	238	19	‖.‖	‖.‖	NOUN
ejpam-6846	238	20	continuous	continuous	ADJ
ejpam-6846	238	21	on	on	ADP
ejpam-6846	238	22	u∗.	u∗.	PROPN
ejpam-6846	238	23	(	(	PUNCT
ejpam-6846	238	24	2	2	NUM
ejpam-6846	238	25	)	)	PUNCT
ejpam-6846	238	26	⇒	⇒	NOUN
ejpam-6846	238	27	(	(	PUNCT
ejpam-6846	238	28	1	1	X
ejpam-6846	238	29	)	)	PUNCT
ejpam-6846	238	30	first	first	ADV
ejpam-6846	238	31	,	,	PUNCT
ejpam-6846	238	32	we	we	PRON
ejpam-6846	238	33	observe	observe	VERB
ejpam-6846	238	34	that	that	SCONJ
ejpam-6846	238	35	mf	mf	NOUN
ejpam-6846	238	36	,	,	PUNCT
ejpam-6846	238	37	s	s	PART
ejpam-6846	238	38	is	be	AUX
ejpam-6846	238	39	locally	locally	ADV
ejpam-6846	238	40	lipschtz	lipschtz	ADV
ejpam-6846	238	41	on	on	ADP
ejpam-6846	238	42	u∗	u∗	ADV
ejpam-6846	238	43	(	(	PUNCT
ejpam-6846	238	44	by	by	ADP
ejpam-6846	238	45	lemma	lemma	PROPN
ejpam-6846	238	46	3	3	NUM
ejpam-6846	238	47	)	)	PUNCT
ejpam-6846	238	48	hence	hence	ADV
ejpam-6846	238	49	according	accord	VERB
ejpam-6846	238	50	to	to	ADP
ejpam-6846	238	51	mordukhovich	mordukhovich	NOUN
ejpam-6846	238	52	-	-	PUNCT
ejpam-6846	238	53	shao	shao	PROPN
ejpam-6846	238	54	(	(	PUNCT
ejpam-6846	238	55	see	see	VERB
ejpam-6846	238	56	[	[	X
ejpam-6846	238	57	18	18	NUM
ejpam-6846	238	58	]	]	NUM
ejpam-6846	238	59	)	)	PUNCT
ejpam-6846	238	60	,	,	PUNCT
ejpam-6846	238	61	for	for	ADP
ejpam-6846	238	62	all	all	DET
ejpam-6846	238	63	u∗	u∗	NOUN
ejpam-6846	238	64	∈	∈	PROPN
ejpam-6846	238	65	u∗	u∗	INTJ
ejpam-6846	238	66	we	we	PRON
ejpam-6846	238	67	get	get	VERB
ejpam-6846	238	68	∂cmf	∂cmf	NOUN
ejpam-6846	238	69	,	,	PUNCT
ejpam-6846	238	70	s(u	s(u	PROPN
ejpam-6846	238	71	∗	∗	NOUN
ejpam-6846	238	72	)	)	PUNCT
ejpam-6846	239	1	=	=	SYM
ejpam-6846	239	2	c̄o{weak−	c̄o{weak−	NOUN
ejpam-6846	239	3	lim	lim	PROPN
ejpam-6846	239	4	v∗→u∗	v∗→u∗	PROPN
ejpam-6846	239	5	sup∂fmf	sup∂fmf	PROPN
ejpam-6846	239	6	,	,	PUNCT
ejpam-6846	239	7	s(v	s(v	PROPN
ejpam-6846	239	8	∗	∗	NOUN
ejpam-6846	239	9	)	)	PUNCT
ejpam-6846	239	10	}	}	PUNCT
ejpam-6846	239	11	=	=	SYM
ejpam-6846	239	12	c̄o{weak−limun	c̄o{weak−limun	PROPN
ejpam-6846	239	13	:	:	PUNCT
ejpam-6846	239	14	un	un	PROPN
ejpam-6846	239	15	∈	∈	PROPN
ejpam-6846	239	16	∂fmf	∂fmf	NUM
ejpam-6846	239	17	,	,	PUNCT
ejpam-6846	239	18	s(u	s(u	PROPN
ejpam-6846	239	19	∗	∗	NOUN
ejpam-6846	239	20	n);u	n);u	NOUN
ejpam-6846	239	21	∗	∗	NOUN
ejpam-6846	239	22	n	n	X
ejpam-6846	239	23	→	→	SYM
ejpam-6846	239	24	u∗	u∗	ADJ
ejpam-6846	239	25	}	}	PUNCT
ejpam-6846	239	26	.	.	PUNCT
ejpam-6846	240	1	now	now	ADV
ejpam-6846	240	2	by	by	ADP
ejpam-6846	240	3	lemma	lemma	PROPN
ejpam-6846	240	4	(	(	PUNCT
ejpam-6846	240	5	2	2	NUM
ejpam-6846	240	6	)	)	PUNCT
ejpam-6846	240	7	for	for	ADP
ejpam-6846	240	8	any	any	DET
ejpam-6846	240	9	u∗n	u∗n	NUM
ejpam-6846	240	10	→	→	SYM
ejpam-6846	240	11	u∗	u∗	ADV
ejpam-6846	240	12	with	with	ADP
ejpam-6846	240	13	un	un	PROPN
ejpam-6846	240	14	∈	∈	PROPN
ejpam-6846	240	15	∂fmf	∂fmf	NUM
ejpam-6846	240	16	,	,	PUNCT
ejpam-6846	240	17	s(u	s(u	PROPN
ejpam-6846	240	18	∗	∗	NOUN
ejpam-6846	240	19	n	n	CCONJ
ejpam-6846	240	20	)	)	PUNCT
ejpam-6846	240	21	we	we	PRON
ejpam-6846	240	22	have	have	VERB
ejpam-6846	240	23	xn	xn	NOUN
ejpam-6846	241	1	=	=	SYM
ejpam-6846	241	2	2(j∗u∗−πf	2(j∗u∗−πf	PROPN
ejpam-6846	241	3	s(u	s(u	PROPN
ejpam-6846	241	4	∗	∗	NOUN
ejpam-6846	241	5	n	n	CCONJ
ejpam-6846	241	6	)	)	PUNCT
ejpam-6846	241	7	)	)	PUNCT
ejpam-6846	241	8	.	.	PUNCT
ejpam-6846	242	1	therefore	therefore	ADV
ejpam-6846	242	2	∂cmf	∂cmf	NOUN
ejpam-6846	242	3	,	,	PUNCT
ejpam-6846	242	4	s(u	s(u	PROPN
ejpam-6846	242	5	∗	∗	NOUN
ejpam-6846	242	6	)	)	PUNCT
ejpam-6846	242	7	=	=	SYM
ejpam-6846	242	8	c̄o{weak	c̄o{weak	PROPN
ejpam-6846	242	9	−	−	PROPN
ejpam-6846	242	10	limun	limun	NOUN
ejpam-6846	242	11	:	:	PUNCT
ejpam-6846	243	1	un	un	PROPN
ejpam-6846	243	2	=	=	PROPN
ejpam-6846	243	3	2(j∗u∗n	2(j∗u∗n	PROPN
ejpam-6846	243	4	−	−	PROPN
ejpam-6846	243	5	πf	πf	INTJ
ejpam-6846	243	6	s(u	s(u	PROPN
ejpam-6846	243	7	∗	∗	NOUN
ejpam-6846	243	8	n	n	CCONJ
ejpam-6846	243	9	)	)	PUNCT
ejpam-6846	243	10	)	)	PUNCT
ejpam-6846	243	11	,	,	PUNCT
ejpam-6846	243	12	u	u	NOUN
ejpam-6846	243	13	∗	∗	NOUN
ejpam-6846	243	14	n	n	X
ejpam-6846	243	15	→	→	SYM
ejpam-6846	243	16	u∗	u∗	ADJ
ejpam-6846	243	17	}	}	PUNCT
ejpam-6846	243	18	.	.	PUNCT
ejpam-6846	244	1	we	we	PRON
ejpam-6846	244	2	now	now	ADV
ejpam-6846	244	3	obtain	obtain	VERB
ejpam-6846	244	4	∂cmf	∂cmf	NOUN
ejpam-6846	244	5	,	,	PUNCT
ejpam-6846	244	6	s(u	s(u	NOUN
ejpam-6846	244	7	∗	∗	NOUN
ejpam-6846	244	8	)	)	PUNCT
ejpam-6846	244	9	=	=	SYM
ejpam-6846	244	10	{	{	PUNCT
ejpam-6846	244	11	2(j∗u∗	2(j∗u∗	NUM
ejpam-6846	244	12	−	−	NOUN
ejpam-6846	244	13	πf	πf	INTJ
ejpam-6846	244	14	s(u	s(u	PROPN
ejpam-6846	244	15	∗	∗	PROPN
ejpam-6846	244	16	)	)	PUNCT
ejpam-6846	244	17	}	}	PUNCT
ejpam-6846	244	18	using	use	VERB
ejpam-6846	244	19	the	the	DET
ejpam-6846	244	20	‖.‖−	‖.‖−	PROPN
ejpam-6846	244	21	‖.‖	‖.‖	NOUN
ejpam-6846	244	22	continuity	continuity	NOUN
ejpam-6846	244	23	of	of	ADP
ejpam-6846	244	24	πf	πf	NOUN
ejpam-6846	244	25	s	s	PART
ejpam-6846	244	26	and	and	CCONJ
ejpam-6846	244	27	j∗	j∗	ADJ
ejpam-6846	244	28	on	on	ADP
ejpam-6846	244	29	x∗.	x∗.	PROPN
ejpam-6846	245	1	as	as	ADP
ejpam-6846	245	2	a	a	DET
ejpam-6846	245	3	result	result	NOUN
ejpam-6846	245	4	,	,	PUNCT
ejpam-6846	245	5	mf	mf	X
ejpam-6846	245	6	,	,	PUNCT
ejpam-6846	245	7	s	s	PART
ejpam-6846	245	8	will	will	AUX
ejpam-6846	245	9	definitely	definitely	ADV
ejpam-6846	245	10	be	be	AUX
ejpam-6846	245	11	continuously	continuously	ADV
ejpam-6846	245	12	gâteaux	gâteaux	VERB
ejpam-6846	245	13	differentiable	differentiable	ADJ
ejpam-6846	245	14	on	on	ADP
ejpam-6846	245	15	u∗.	u∗.	PROPN
ejpam-6846	245	16	we	we	PRON
ejpam-6846	245	17	may	may	AUX
ejpam-6846	245	18	finally	finally	ADV
ejpam-6846	245	19	detect	detect	VERB
ejpam-6846	245	20	that	that	SCONJ
ejpam-6846	245	21	it	it	PRON
ejpam-6846	245	22	is	be	AUX
ejpam-6846	245	23	c1	c1	NOUN
ejpam-6846	245	24	on	on	ADP
ejpam-6846	245	25	u∗	u∗	PROPN
ejpam-6846	245	26	since	since	SCONJ
ejpam-6846	245	27	mf	mf	NOUN
ejpam-6846	245	28	,	,	PUNCT
ejpam-6846	245	29	s	s	PART
ejpam-6846	245	30	is	be	AUX
ejpam-6846	245	31	locally	locally	ADV
ejpam-6846	245	32	lipschtz	lipschtz	ADV
ejpam-6846	245	33	on	on	ADP
ejpam-6846	245	34	u∗.	u∗.	PROPN
ejpam-6846	245	35	□	□	PUNCT
ejpam-6846	245	36	the	the	DET
ejpam-6846	245	37	following	follow	VERB
ejpam-6846	245	38	lemma	lemma	PROPN
ejpam-6846	245	39	proves	prove	VERB
ejpam-6846	245	40	that	that	SCONJ
ejpam-6846	245	41	,	,	PUNCT
ejpam-6846	245	42	whenever	whenever	SCONJ
ejpam-6846	245	43	theπf	theπf	NOUN
ejpam-6846	245	44	s	s	X
ejpam-6846	245	45	is	be	AUX
ejpam-6846	245	46	single	single	ADJ
ejpam-6846	245	47	valued	value	VERB
ejpam-6846	245	48	on	on	ADP
ejpam-6846	245	49	reflexive	reflexive	ADJ
ejpam-6846	245	50	banach	banach	NOUN
ejpam-6846	245	51	spaces	space	NOUN
ejpam-6846	245	52	with	with	ADP
ejpam-6846	245	53	the	the	DET
ejpam-6846	245	54	kadec	kadec	PROPN
ejpam-6846	245	55	condition(i.e	condition(i.e	PROPN
ejpam-6846	245	56	.	.	PUNCT
ejpam-6846	246	1	for	for	ADP
ejpam-6846	246	2	any	any	DET
ejpam-6846	246	3	sequence	sequence	NOUN
ejpam-6846	246	4	(	(	PUNCT
ejpam-6846	246	5	un	un	PROPN
ejpam-6846	246	6	)	)	PUNCT
ejpam-6846	246	7	⇀	⇀	NUM
ejpam-6846	247	1	u	u	NOUN
ejpam-6846	247	2	weakly	weakly	ADV
ejpam-6846	247	3	in	in	ADP
ejpam-6846	247	4	x	x	PUNCT
ejpam-6846	247	5	with	with	ADP
ejpam-6846	247	6	‖un‖	‖un‖	NOUN
ejpam-6846	247	7	→	→	SYM
ejpam-6846	247	8	‖u‖	‖u‖	PROPN
ejpam-6846	247	9	,	,	PUNCT
ejpam-6846	247	10	then	then	ADV
ejpam-6846	247	11	‖un	‖un	PROPN
ejpam-6846	247	12	−	−	PROPN
ejpam-6846	247	13	u‖	u‖	NOUN
ejpam-6846	247	14	→	→	SYM
ejpam-6846	247	15	0	0	NUM
ejpam-6846	247	16	,	,	PUNCT
ejpam-6846	247	17	see	see	VERB
ejpam-6846	247	18	[	[	X
ejpam-6846	247	19	8	8	NUM
ejpam-6846	247	20	]	]	PUNCT
ejpam-6846	247	21	)	)	PUNCT
ejpam-6846	247	22	,	,	PUNCT
ejpam-6846	247	23	its	its	PRON
ejpam-6846	247	24	norm	norm	NOUN
ejpam-6846	247	25	-	-	PUNCT
ejpam-6846	247	26	to	to	ADP
ejpam-6846	247	27	-	-	PUNCT
ejpam-6846	247	28	weak	weak	ADJ
ejpam-6846	247	29	continuity	continuity	NOUN
ejpam-6846	247	30	and	and	CCONJ
ejpam-6846	247	31	‖.‖	‖.‖	NOUN
ejpam-6846	247	32	−	−	PROPN
ejpam-6846	247	33	‖.‖	‖.‖	NOUN
ejpam-6846	247	34	continuity	continuity	NOUN
ejpam-6846	247	35	are	be	AUX
ejpam-6846	247	36	equivalent	equivalent	ADJ
ejpam-6846	247	37	.	.	PUNCT
ejpam-6846	248	1	its	its	PRON
ejpam-6846	248	2	proof	proof	NOUN
ejpam-6846	248	3	is	be	AUX
ejpam-6846	248	4	based	base	VERB
ejpam-6846	248	5	on	on	ADP
ejpam-6846	248	6	a	a	DET
ejpam-6846	248	7	concept	concept	NOUN
ejpam-6846	248	8	from	from	ADP
ejpam-6846	248	9	lemma	lemma	PROPN
ejpam-6846	248	10	5.1	5.1	NUM
ejpam-6846	248	11	proof	proof	NOUN
ejpam-6846	248	12	in	in	ADP
ejpam-6846	248	13	[	[	X
ejpam-6846	248	14	15	15	NUM
ejpam-6846	248	15	]	]	PUNCT
ejpam-6846	248	16	for	for	ADP
ejpam-6846	248	17	metric	metric	ADJ
ejpam-6846	248	18	projection	projection	NOUN
ejpam-6846	248	19	,	,	PUNCT
ejpam-6846	248	20	and	and	CCONJ
ejpam-6846	248	21	lemma	lemma	PROPN
ejpam-6846	248	22	2.8	2.8	NUM
ejpam-6846	248	23	[	[	NOUN
ejpam-6846	248	24	6	6	NUM
ejpam-6846	248	25	]	]	PUNCT
ejpam-6846	248	26	for	for	ADP
ejpam-6846	248	27	generalized	generalized	ADJ
ejpam-6846	248	28	projection	projection	NOUN
ejpam-6846	248	29	operator	operator	NOUN
ejpam-6846	248	30	.	.	PUNCT
ejpam-6846	249	1	lemma	lemma	PROPN
ejpam-6846	249	2	6	6	NUM
ejpam-6846	249	3	.	.	PUNCT
ejpam-6846	250	1	given	give	VERB
ejpam-6846	250	2	that	that	DET
ejpam-6846	250	3	u∗	u∗	NOUN
ejpam-6846	250	4	is	be	AUX
ejpam-6846	250	5	an	an	DET
ejpam-6846	250	6	open	open	ADJ
ejpam-6846	250	7	set	set	NOUN
ejpam-6846	250	8	in	in	ADP
ejpam-6846	250	9	x∗	x∗	PROPN
ejpam-6846	250	10	,	,	PUNCT
ejpam-6846	250	11	where	where	SCONJ
ejpam-6846	250	12	x	x	X
ejpam-6846	250	13	banach	banach	NOUN
ejpam-6846	250	14	space	space	NOUN
ejpam-6846	250	15	is	be	AUX
ejpam-6846	250	16	reflexive	reflexive	ADJ
ejpam-6846	250	17	with	with	ADP
ejpam-6846	250	18	the	the	DET
ejpam-6846	250	19	kadec	kadec	PROPN
ejpam-6846	250	20	condition	condition	NOUN
ejpam-6846	250	21	.	.	PUNCT
ejpam-6846	251	1	suppose	suppose	VERB
ejpam-6846	251	2	that	that	SCONJ
ejpam-6846	251	3	πf	πf	PRON
ejpam-6846	251	4	s	s	VERB
ejpam-6846	251	5	is	be	AUX
ejpam-6846	251	6	a	a	DET
ejpam-6846	251	7	single	single	ADV
ejpam-6846	251	8	-	-	PUNCT
ejpam-6846	251	9	valued	value	VERB
ejpam-6846	251	10	function	function	NOUN
ejpam-6846	251	11	on	on	ADP
ejpam-6846	251	12	u∗	u∗	PROPN
ejpam-6846	251	13	,	,	PUNCT
ejpam-6846	251	14	and	and	CCONJ
ejpam-6846	251	15	that	that	SCONJ
ejpam-6846	251	16	f	f	X
ejpam-6846	251	17	:	:	PUNCT
ejpam-6846	251	18	x	x	X
ejpam-6846	251	19	→	→	SYM
ejpam-6846	251	20	r	r	NOUN
ejpam-6846	251	21	∪	∪	X
ejpam-6846	251	22	{	{	PUNCT
ejpam-6846	251	23	∞	∞	NOUN
ejpam-6846	251	24	}	}	PUNCT
ejpam-6846	251	25	is	be	AUX
ejpam-6846	251	26	proper	proper	ADJ
ejpam-6846	251	27	continuous	continuous	ADJ
ejpam-6846	251	28	function	function	NOUN
ejpam-6846	251	29	.	.	PUNCT
ejpam-6846	252	1	then	then	ADV
ejpam-6846	252	2	,	,	PUNCT
ejpam-6846	252	3	πf	πf	PROPN
ejpam-6846	252	4	s	s	VERB
ejpam-6846	252	5	is	be	AUX
ejpam-6846	252	6	‖.‖	‖.‖	NOUN
ejpam-6846	252	7	−	−	PROPN
ejpam-6846	252	8	‖.‖	‖.‖	NOUN
ejpam-6846	252	9	continuous	continuous	ADJ
ejpam-6846	252	10	on	on	ADP
ejpam-6846	252	11	u∗	u∗	PROPN
ejpam-6846	252	12	iff	iff	PROPN
ejpam-6846	252	13	it	it	PRON
ejpam-6846	252	14	is	be	AUX
ejpam-6846	252	15	‖.‖	‖.‖	NOUN
ejpam-6846	252	16	−	−	PROPN
ejpam-6846	252	17	to−	to−	PUNCT
ejpam-6846	252	18	weak	weak	ADJ
ejpam-6846	252	19	continuous	continuous	ADJ
ejpam-6846	252	20	on	on	ADP
ejpam-6846	252	21	u∗.	u∗.	PROPN
ejpam-6846	252	22	proof	proof	NOUN
ejpam-6846	252	23	:	:	PUNCT
ejpam-6846	252	24	πf	πf	INTJ
ejpam-6846	252	25	s	s	VERB
ejpam-6846	252	26	is	be	AUX
ejpam-6846	252	27	‖.‖	‖.‖	NOUN
ejpam-6846	252	28	−	−	PROPN
ejpam-6846	252	29	‖.‖	‖.‖	NOUN
ejpam-6846	252	30	continuous	continuous	ADJ
ejpam-6846	252	31	on	on	ADP
ejpam-6846	252	32	u∗	u∗	NOUN
ejpam-6846	252	33	follows	follow	VERB
ejpam-6846	252	34	naturally	naturally	ADV
ejpam-6846	252	35	from	from	ADP
ejpam-6846	252	36	the	the	DET
ejpam-6846	252	37	‖.‖	‖.‖	NOUN
ejpam-6846	252	38	−	−	NOUN
ejpam-6846	252	39	to	to	PART
ejpam-6846	252	40	−	−	VERB
ejpam-6846	252	41	weak	weak	ADJ
ejpam-6846	252	42	continuity	continuity	NOUN
ejpam-6846	252	43	on	on	ADP
ejpam-6846	252	44	u∗.	u∗.	PROPN
ejpam-6846	252	45	all	all	PRON
ejpam-6846	252	46	we	we	PRON
ejpam-6846	252	47	have	have	VERB
ejpam-6846	252	48	to	to	PART
ejpam-6846	252	49	do	do	VERB
ejpam-6846	252	50	is	be	AUX
ejpam-6846	252	51	demonstrate	demonstrate	VERB
ejpam-6846	252	52	the	the	DET
ejpam-6846	252	53	opposite	opposite	NOUN
ejpam-6846	252	54	.	.	PUNCT
ejpam-6846	253	1	let	let	VERB
ejpam-6846	253	2	πf	πf	INTJ
ejpam-6846	253	3	s(u	s(u	PROPN
ejpam-6846	253	4	∗	∗	NOUN
ejpam-6846	253	5	n	n	CCONJ
ejpam-6846	253	6	)	)	PUNCT
ejpam-6846	253	7	→	→	SYM
ejpam-6846	253	8	πf	πf	INTJ
ejpam-6846	253	9	s(u	s(u	PROPN
ejpam-6846	253	10	∗	∗	NOUN
ejpam-6846	253	11	)	)	PUNCT
ejpam-6846	253	12	weakly	weakly	ADV
ejpam-6846	253	13	in	in	ADP
ejpam-6846	253	14	x	x	NOUN
ejpam-6846	253	15	,	,	PUNCT
ejpam-6846	253	16	and	and	CCONJ
ejpam-6846	253	17	lim‖u∗n	lim‖u∗n	PRON
ejpam-6846	254	1	−	−	NOUN
ejpam-6846	254	2	x∗‖	x∗‖	X
ejpam-6846	254	3	=	=	SYM
ejpam-6846	255	1	0	0	X
ejpam-6846	255	2	.	.	PUNCT
ejpam-6846	256	1	then	then	ADV
ejpam-6846	256	2	,	,	PUNCT
ejpam-6846	256	3	by	by	ADP
ejpam-6846	256	4	lipschtz	lipschtz	ADJ
ejpam-6846	256	5	continuity	continuity	NOUN
ejpam-6846	256	6	of	of	ADP
ejpam-6846	256	7	mf	mf	NOUN
ejpam-6846	256	8	,	,	PUNCT
ejpam-6846	256	9	s	s	PART
ejpam-6846	256	10	we	we	PRON
ejpam-6846	256	11	have	have	VERB
ejpam-6846	256	12	v	v	NUM
ejpam-6846	256	13	f	f	X
ejpam-6846	256	14	(	(	PUNCT
ejpam-6846	256	15	u∗n	u∗n	PROPN
ejpam-6846	256	16	,	,	PUNCT
ejpam-6846	256	17	π	π	PROPN
ejpam-6846	256	18	f	f	PROPN
ejpam-6846	256	19	s(u	s(u	PROPN
ejpam-6846	256	20	∗	∗	NOUN
ejpam-6846	256	21	n	n	CCONJ
ejpam-6846	256	22	)	)	PUNCT
ejpam-6846	256	23	)	)	PUNCT
ejpam-6846	257	1	=	=	PUNCT
ejpam-6846	257	2	mf	mf	X
ejpam-6846	257	3	,	,	PUNCT
ejpam-6846	257	4	s(u	s(u	PROPN
ejpam-6846	257	5	∗	∗	NOUN
ejpam-6846	257	6	n	n	CCONJ
ejpam-6846	257	7	)	)	PUNCT
ejpam-6846	257	8	→	→	SYM
ejpam-6846	257	9	mf	mf	X
ejpam-6846	257	10	,	,	PUNCT
ejpam-6846	257	11	s(u	s(u	NOUN
ejpam-6846	257	12	∗	∗	NOUN
ejpam-6846	257	13	)	)	PUNCT
ejpam-6846	257	14	=	=	SYM
ejpam-6846	258	1	v	v	NUM
ejpam-6846	258	2	f	f	X
ejpam-6846	258	3	(	(	PUNCT
ejpam-6846	258	4	x∗	x∗	PROPN
ejpam-6846	258	5	,	,	PUNCT
ejpam-6846	258	6	πf	πf	INTJ
ejpam-6846	258	7	s(u	s(u	PROPN
ejpam-6846	258	8	∗	∗	PROPN
ejpam-6846	258	9	)	)	PUNCT
ejpam-6846	258	10	)	)	PUNCT
ejpam-6846	258	11	.	.	PUNCT
ejpam-6846	259	1	now	now	ADV
ejpam-6846	259	2	notice	notice	VERB
ejpam-6846	259	3	that	that	SCONJ
ejpam-6846	259	4	:	:	PUNCT
ejpam-6846	259	5	‖πf	‖πf	NUM
ejpam-6846	259	6	s(u	s(u	PROPN
ejpam-6846	259	7	∗	∗	NOUN
ejpam-6846	259	8	n)‖2	n)‖2	ADJ
ejpam-6846	259	9	−	−	PROPN
ejpam-6846	259	10	‖πf	‖πf	NUM
ejpam-6846	259	11	s(u	s(u	PROPN
ejpam-6846	259	12	∗)‖2	∗)‖2	NOUN
ejpam-6846	259	13	=	=	PUNCT
ejpam-6846	259	14	[	[	PUNCT
ejpam-6846	259	15	v	v	X
ejpam-6846	259	16	f	f	X
ejpam-6846	259	17	(	(	PUNCT
ejpam-6846	259	18	u∗n	u∗n	PROPN
ejpam-6846	259	19	,	,	PUNCT
ejpam-6846	259	20	π	π	PROPN
ejpam-6846	259	21	f	f	PROPN
ejpam-6846	259	22	s(u	s(u	PROPN
ejpam-6846	259	23	∗	∗	X
ejpam-6846	259	24	n))−	n))−	VERB
ejpam-6846	259	25	‖u∗n‖2	‖u∗n‖2	NOUN
ejpam-6846	260	1	+	+	NOUN
ejpam-6846	260	2	2〈u∗n;π	2〈u∗n;π	PROPN
ejpam-6846	260	3	f	f	PROPN
ejpam-6846	260	4	s(u	s(u	PROPN
ejpam-6846	260	5	∗	∗	NOUN
ejpam-6846	260	6	n	n	CCONJ
ejpam-6846	260	7	)	)	PUNCT
ejpam-6846	260	8	〉	〉	NOUN
ejpam-6846	260	9	−	−	NOUN
ejpam-6846	260	10	f(πf	f(πf	NOUN
ejpam-6846	260	11	s(u	s(u	PROPN
ejpam-6846	260	12	∗	∗	NOUN
ejpam-6846	260	13	n	n	CCONJ
ejpam-6846	260	14	)	)	PUNCT
ejpam-6846	260	15	)	)	PUNCT
ejpam-6846	260	16	]	]	PUNCT
ejpam-6846	261	1	−	−	PROPN
ejpam-6846	261	2	[	[	PUNCT
ejpam-6846	261	3	v	v	X
ejpam-6846	261	4	f	f	X
ejpam-6846	261	5	(	(	PUNCT
ejpam-6846	261	6	u∗	u∗	PROPN
ejpam-6846	261	7	,	,	PUNCT
ejpam-6846	261	8	πf	πf	INTJ
ejpam-6846	261	9	s(u	s(u	PROPN
ejpam-6846	261	10	∗))−	∗))−	VERB
ejpam-6846	261	11	‖u∗‖2	‖u∗‖2	NOUN
ejpam-6846	261	12	+	+	NOUN
ejpam-6846	261	13	2〈u∗;πf	2〈u∗;πf	ADJ
ejpam-6846	261	14	s(u	s(u	PROPN
ejpam-6846	261	15	∗	∗	NOUN
ejpam-6846	261	16	)	)	PUNCT
ejpam-6846	262	1	〉	〉	NOUN
ejpam-6846	262	2	−	−	NOUN
ejpam-6846	262	3	f(πf	f(πf	NOUN
ejpam-6846	262	4	s(u	s(u	PROPN
ejpam-6846	262	5	∗	∗	NOUN
ejpam-6846	262	6	)	)	PUNCT
ejpam-6846	262	7	]	]	PUNCT
ejpam-6846	263	1	a.	a.	PROPN
ejpam-6846	263	2	al	al	PROPN
ejpam-6846	263	3	tane	tane	PROPN
ejpam-6846	263	4	,	,	PUNCT
ejpam-6846	263	5	l.	l.	PROPN
ejpam-6846	263	6	s.	s.	PROPN
ejpam-6846	263	7	keong	keong	PROPN
ejpam-6846	263	8	/	/	PUNCT
ejpam-6846	263	9	eur	eur	PROPN
ejpam-6846	263	10	.	.	PUNCT
ejpam-6846	264	1	j.	j.	PROPN
ejpam-6846	264	2	pure	pure	PROPN
ejpam-6846	264	3	appl	appl	PROPN
ejpam-6846	264	4	.	.	PROPN
ejpam-6846	264	5	math	math	PROPN
ejpam-6846	264	6	,	,	PUNCT
ejpam-6846	264	7	18	18	NUM
ejpam-6846	264	8	(	(	PUNCT
ejpam-6846	264	9	4	4	NUM
ejpam-6846	264	10	)	)	PUNCT
ejpam-6846	264	11	(	(	PUNCT
ejpam-6846	264	12	2025	2025	NUM
ejpam-6846	264	13	)	)	PUNCT
ejpam-6846	264	14	,	,	PUNCT
ejpam-6846	264	15	6846	6846	NUM
ejpam-6846	264	16	11	11	NUM
ejpam-6846	264	17	of	of	ADP
ejpam-6846	264	18	13	13	NUM
ejpam-6846	264	19	=	=	SYM
ejpam-6846	264	20	[	[	PUNCT
ejpam-6846	264	21	v	v	X
ejpam-6846	264	22	f	f	X
ejpam-6846	264	23	(	(	PUNCT
ejpam-6846	264	24	u∗n	u∗n	PROPN
ejpam-6846	264	25	,	,	PUNCT
ejpam-6846	264	26	π	π	PROPN
ejpam-6846	264	27	f	f	PROPN
ejpam-6846	264	28	s(u	s(u	PROPN
ejpam-6846	264	29	∗	∗	X
ejpam-6846	264	30	n))−	n))−	VERB
ejpam-6846	264	31	v	v	ADP
ejpam-6846	264	32	f	f	X
ejpam-6846	264	33	(	(	PUNCT
ejpam-6846	264	34	u∗	u∗	PROPN
ejpam-6846	264	35	,	,	PUNCT
ejpam-6846	264	36	πf	πf	INTJ
ejpam-6846	264	37	s(u	s(u	PROPN
ejpam-6846	264	38	∗	∗	NOUN
ejpam-6846	264	39	)	)	PUNCT
ejpam-6846	264	40	)	)	PUNCT
ejpam-6846	264	41	]	]	PUNCT
ejpam-6846	265	1	+	+	CCONJ
ejpam-6846	265	2	[	[	PUNCT
ejpam-6846	265	3	‖u∗‖2	‖u∗‖2	NOUN
ejpam-6846	265	4	−	−	NOUN
ejpam-6846	265	5	‖u∗n‖2	‖u∗n‖2	NOUN
ejpam-6846	265	6	]	]	PUNCT
ejpam-6846	266	1	+	+	CCONJ
ejpam-6846	266	2	2	2	NUM
ejpam-6846	266	3	[	[	PUNCT
ejpam-6846	266	4	〈	〈	PROPN
ejpam-6846	266	5	u∗n;π	u∗n;π	ADJ
ejpam-6846	266	6	f	f	PROPN
ejpam-6846	266	7	s(u	s(u	PROPN
ejpam-6846	266	8	∗	∗	NOUN
ejpam-6846	266	9	n	n	CCONJ
ejpam-6846	266	10	)	)	PUNCT
ejpam-6846	266	11	〉	〉	NOUN
ejpam-6846	266	12	−	−	NOUN
ejpam-6846	266	13	〈	〈	PROPN
ejpam-6846	266	14	u∗;πf	u∗;πf	VERB
ejpam-6846	266	15	s(u	s(u	PROPN
ejpam-6846	266	16	∗	∗	NOUN
ejpam-6846	266	17	)	)	PUNCT
ejpam-6846	266	18	〉	〉	NOUN
ejpam-6846	266	19	]	]	PUNCT
ejpam-6846	267	1	+	+	CCONJ
ejpam-6846	267	2	[	[	PUNCT
ejpam-6846	267	3	f(πf	f(πf	NOUN
ejpam-6846	267	4	s(u	s(u	ADJ
ejpam-6846	267	5	∗)−	∗)−	ADJ
ejpam-6846	267	6	f(πf	f(πf	NOUN
ejpam-6846	267	7	s(u	s(u	PROPN
ejpam-6846	267	8	∗	∗	NOUN
ejpam-6846	267	9	n	n	CCONJ
ejpam-6846	267	10	)	)	PUNCT
ejpam-6846	267	11	]	]	PUNCT
ejpam-6846	267	12	.	.	PUNCT
ejpam-6846	268	1	by	by	ADP
ejpam-6846	268	2	continuity	continuity	NOUN
ejpam-6846	268	3	of	of	ADP
ejpam-6846	268	4	f	f	PROPN
ejpam-6846	268	5	,	,	PUNCT
ejpam-6846	268	6	we	we	PRON
ejpam-6846	268	7	get	get	VERB
ejpam-6846	268	8	that	that	SCONJ
ejpam-6846	268	9	|πf	|πf	PROPN
ejpam-6846	268	10	s(u	s(u	PROPN
ejpam-6846	268	11	∗	∗	X
ejpam-6846	268	12	n)‖	n)‖	NOUN
ejpam-6846	268	13	→	→	SYM
ejpam-6846	268	14	‖πf	‖πf	NUM
ejpam-6846	268	15	s(u	s(u	PROPN
ejpam-6846	268	16	∗)‖	∗)‖	PROPN
ejpam-6846	268	17	and	and	CCONJ
ejpam-6846	268	18	hence	hence	ADV
ejpam-6846	268	19	‖πf	‖πf	NUM
ejpam-6846	268	20	s(u	s(u	PROPN
ejpam-6846	268	21	∗	∗	NOUN
ejpam-6846	268	22	n	n	CCONJ
ejpam-6846	268	23	)	)	PUNCT
ejpam-6846	268	24	−	−	PROPN
ejpam-6846	269	1	πf	πf	INTJ
ejpam-6846	269	2	s(u	s(u	PROPN
ejpam-6846	269	3	∗)‖	∗)‖	PROPN
ejpam-6846	269	4	→	→	SYM
ejpam-6846	269	5	0	0	NUM
ejpam-6846	269	6	is	be	AUX
ejpam-6846	269	7	guaranteed	guarantee	VERB
ejpam-6846	269	8	since	since	SCONJ
ejpam-6846	269	9	the	the	DET
ejpam-6846	269	10	space	space	NOUN
ejpam-6846	269	11	x	x	PUNCT
ejpam-6846	269	12	has	have	VERB
ejpam-6846	269	13	kadec	kadec	NOUN
ejpam-6846	269	14	property	property	NOUN
ejpam-6846	269	15	,	,	PUNCT
ejpam-6846	269	16	this	this	PRON
ejpam-6846	269	17	concluding	conclude	VERB
ejpam-6846	269	18	the	the	DET
ejpam-6846	269	19	proof	proof	NOUN
ejpam-6846	269	20	□	□	PUNCT
ejpam-6846	269	21	.	.	PUNCT
ejpam-6846	270	1	the	the	DET
ejpam-6846	270	2	∇fmf	∇fmf	PROPN
ejpam-6846	270	3	,	,	PUNCT
ejpam-6846	270	4	s	s	PART
ejpam-6846	270	5	and	and	CCONJ
ejpam-6846	270	6	continuous	continuous	ADJ
ejpam-6846	270	7	fréchet	fréchet	NOUN
ejpam-6846	270	8	differentiability	differentiability	NOUN
ejpam-6846	270	9	of	of	ADP
ejpam-6846	270	10	mf	mf	PROPN
ejpam-6846	270	11	,	,	PUNCT
ejpam-6846	270	12	s	s	PART
ejpam-6846	270	13	are	be	AUX
ejpam-6846	270	14	equivalent	equivalent	ADJ
ejpam-6846	270	15	 	 	SPACE
ejpam-6846	270	16	on	on	ADP
ejpam-6846	270	17	reflexive	reflexive	ADJ
ejpam-6846	270	18	banach	banach	NOUN
ejpam-6846	270	19	spaces	space	NOUN
ejpam-6846	270	20	with	with	ADP
ejpam-6846	270	21	the	the	DET
ejpam-6846	270	22	kadec	kadec	NOUN
ejpam-6846	270	23	condition	condition	NOUN
ejpam-6846	270	24	.	.	PUNCT
ejpam-6846	271	1	this	this	PRON
ejpam-6846	271	2	is	be	AUX
ejpam-6846	271	3	shown	show	VERB
ejpam-6846	271	4	by	by	ADP
ejpam-6846	271	5	the	the	DET
ejpam-6846	271	6	following	follow	VERB
ejpam-6846	271	7	lemma	lemma	PROPN
ejpam-6846	271	8	,	,	PUNCT
ejpam-6846	271	9	based	base	VERB
ejpam-6846	271	10	on	on	ADP
ejpam-6846	271	11	a	a	DET
ejpam-6846	271	12	concept	concept	NOUN
ejpam-6846	271	13	from	from	ADP
ejpam-6846	271	14	lemma	lemma	PROPN
ejpam-6846	271	15	4.2	4.2	NUM
ejpam-6846	271	16	in	in	ADP
ejpam-6846	271	17	[	[	X
ejpam-6846	271	18	15	15	NUM
ejpam-6846	271	19	]	]	PUNCT
ejpam-6846	271	20	and	and	CCONJ
ejpam-6846	271	21	lemma	lemma	PROPN
ejpam-6846	271	22	2.9	2.9	NUM
ejpam-6846	271	23	in	in	ADP
ejpam-6846	271	24	[	[	X
ejpam-6846	271	25	6	6	NUM
ejpam-6846	271	26	]	]	PUNCT
ejpam-6846	271	27	.	.	PUNCT
ejpam-6846	272	1	lemma	lemma	PROPN
ejpam-6846	272	2	7	7	X
ejpam-6846	272	3	.	.	PUNCT
ejpam-6846	272	4	given	give	VERB
ejpam-6846	272	5	u∗	u∗	PROPN
ejpam-6846	272	6	as	as	ADP
ejpam-6846	272	7	an	an	DET
ejpam-6846	272	8	open	open	ADJ
ejpam-6846	272	9	set	set	NOUN
ejpam-6846	272	10	in	in	ADP
ejpam-6846	272	11	x∗	x∗	PROPN
ejpam-6846	272	12	,	,	PUNCT
ejpam-6846	272	13	with	with	ADP
ejpam-6846	272	14	x	x	PUNCT
ejpam-6846	272	15	a	a	DET
ejpam-6846	272	16	reflexive	reflexive	ADJ
ejpam-6846	272	17	banach	banach	NOUN
ejpam-6846	272	18	space	space	NOUN
ejpam-6846	272	19	with	with	ADP
ejpam-6846	272	20	a	a	DET
ejpam-6846	272	21	smooth	smooth	ADJ
ejpam-6846	272	22	dual	dual	ADJ
ejpam-6846	272	23	norm	norm	NOUN
ejpam-6846	272	24	and	and	CCONJ
ejpam-6846	272	25	the	the	DET
ejpam-6846	272	26	kadec	kadec	PROPN
ejpam-6846	272	27	condition	condition	NOUN
ejpam-6846	272	28	,	,	PUNCT
ejpam-6846	272	29	and	and	CCONJ
ejpam-6846	272	30	f	f	PROPN
ejpam-6846	272	31	is	be	AUX
ejpam-6846	272	32	proper	proper	ADJ
ejpam-6846	272	33	,	,	PUNCT
ejpam-6846	272	34	convex	convex	NOUN
ejpam-6846	272	35	,	,	PUNCT
ejpam-6846	272	36	and	and	CCONJ
ejpam-6846	272	37	fréchet	fréchet	VERB
ejpam-6846	272	38	differentiable	differentiable	ADJ
ejpam-6846	272	39	on	on	ADP
ejpam-6846	272	40	u∗	u∗	PROPN
ejpam-6846	272	41	,	,	PUNCT
ejpam-6846	272	42	the	the	DET
ejpam-6846	272	43	two	two	NUM
ejpam-6846	272	44	assertions	assertion	NOUN
ejpam-6846	272	45	that	that	PRON
ejpam-6846	272	46	follow	follow	NOUN
ejpam-6846	272	47	are	be	AUX
ejpam-6846	272	48	equivalent	equivalent	ADJ
ejpam-6846	272	49	:	:	PUNCT
ejpam-6846	272	50	(	(	PUNCT
ejpam-6846	272	51	i	i	NOUN
ejpam-6846	272	52	)	)	PUNCT
ejpam-6846	272	53	mf	mf	PROPN
ejpam-6846	272	54	,	,	PUNCT
ejpam-6846	272	55	s	s	PART
ejpam-6846	272	56	is	be	AUX
ejpam-6846	272	57	c1	c1	PROPN
ejpam-6846	272	58	in	in	ADP
ejpam-6846	272	59	u∗.	u∗.	PROPN
ejpam-6846	272	60	(	(	PUNCT
ejpam-6846	272	61	ii	ii	NOUN
ejpam-6846	272	62	)	)	PUNCT
ejpam-6846	272	63	mf	mf	PROPN
ejpam-6846	272	64	,	,	PUNCT
ejpam-6846	272	65	s	s	PART
ejpam-6846	272	66	is	be	AUX
ejpam-6846	272	67	fréchet	fréchet	VERB
ejpam-6846	272	68	differentiable	differentiable	ADJ
ejpam-6846	272	69	on	on	ADP
ejpam-6846	272	70	u∗.	u∗.	PROPN
ejpam-6846	272	71	proof	proof	NOUN
ejpam-6846	272	72	:	:	PUNCT
ejpam-6846	272	73	from	from	ADP
ejpam-6846	272	74	(	(	PUNCT
ejpam-6846	272	75	1	1	NUM
ejpam-6846	272	76	)	)	PUNCT
ejpam-6846	272	77	→	→	X
ejpam-6846	272	78	(	(	PUNCT
ejpam-6846	272	79	2	2	X
ejpam-6846	272	80	)	)	PUNCT
ejpam-6846	272	81	clear	clear	ADJ
ejpam-6846	272	82	,	,	PUNCT
ejpam-6846	272	83	we	we	PRON
ejpam-6846	272	84	need	need	VERB
ejpam-6846	272	85	only	only	ADV
ejpam-6846	272	86	to	to	PART
ejpam-6846	272	87	prove	prove	VERB
ejpam-6846	272	88	(	(	PUNCT
ejpam-6846	272	89	2	2	NUM
ejpam-6846	272	90	)	)	PUNCT
ejpam-6846	272	91	→	→	X
ejpam-6846	272	92	(	(	PUNCT
ejpam-6846	272	93	1	1	NUM
ejpam-6846	272	94	)	)	PUNCT
ejpam-6846	272	95	.	.	PUNCT
ejpam-6846	273	1	let	let	VERB
ejpam-6846	273	2	u∗n	u∗n	PRON
ejpam-6846	273	3	be	be	AUX
ejpam-6846	273	4	a	a	DET
ejpam-6846	273	5	sequence	sequence	NOUN
ejpam-6846	273	6	that	that	PRON
ejpam-6846	273	7	converges	converge	VERB
ejpam-6846	273	8	to	to	PART
ejpam-6846	273	9	u∗	u∗	VERB
ejpam-6846	273	10	in	in	ADP
ejpam-6846	273	11	x.	x.	NOUN
ejpam-6846	274	1	we	we	PRON
ejpam-6846	274	2	want	want	VERB
ejpam-6846	274	3	to	to	PART
ejpam-6846	274	4	demonstrate	demonstrate	VERB
ejpam-6846	274	5	that	that	SCONJ
ejpam-6846	274	6	∇fmf	∇fmf	PROPN
ejpam-6846	274	7	,	,	PUNCT
ejpam-6846	274	8	s(u	s(u	PROPN
ejpam-6846	274	9	∗	∗	NOUN
ejpam-6846	274	10	n	n	CCONJ
ejpam-6846	274	11	)	)	PUNCT
ejpam-6846	274	12	→	→	SYM
ejpam-6846	274	13	∇fmf	∇fmf	PROPN
ejpam-6846	274	14	,	,	PUNCT
ejpam-6846	274	15	s(u	s(u	PROPN
ejpam-6846	274	16	∗	∗	NOUN
ejpam-6846	274	17	)	)	PUNCT
ejpam-6846	274	18	.	.	PUNCT
ejpam-6846	275	1	notice	notice	VERB
ejpam-6846	275	2	that	that	SCONJ
ejpam-6846	275	3	mf	mf	VERB
ejpam-6846	275	4	,	,	PUNCT
ejpam-6846	275	5	s(v	s(v	PROPN
ejpam-6846	275	6	∗	∗	NOUN
ejpam-6846	275	7	)	)	PUNCT
ejpam-6846	276	1	=	=	SYM
ejpam-6846	276	2	inf	inf	NOUN
ejpam-6846	276	3	s∈s	s∈s	NOUN
ejpam-6846	276	4	v	v	PROPN
ejpam-6846	276	5	f	f	PROPN
ejpam-6846	276	6	(	(	PUNCT
ejpam-6846	276	7	v∗	v∗	PROPN
ejpam-6846	276	8	,	,	PUNCT
ejpam-6846	276	9	s	s	NOUN
ejpam-6846	276	10	)	)	PUNCT
ejpam-6846	276	11	=	=	SYM
ejpam-6846	276	12	inf	inf	PROPN
ejpam-6846	276	13	s∈s	s∈s	NOUN
ejpam-6846	276	14	{	{	PUNCT
ejpam-6846	276	15	‖v∗‖2	‖v∗‖2	NOUN
ejpam-6846	276	16	−	−	PROPN
ejpam-6846	276	17	2〈v∗	2〈v∗	PROPN
ejpam-6846	276	18	,	,	PUNCT
ejpam-6846	276	19	s〉+	s〉+	NOUN
ejpam-6846	276	20	‖s‖2	‖s‖2	PROPN
ejpam-6846	276	21	+	+	X
ejpam-6846	276	22	f(s	f(	NOUN
ejpam-6846	276	23	)	)	PUNCT
ejpam-6846	276	24	}	}	PUNCT
ejpam-6846	276	25	=	=	SYM
ejpam-6846	276	26	‖v∗‖2	‖v∗‖2	NOUN
ejpam-6846	276	27	−	−	NOUN
ejpam-6846	277	1	sup	sup	NOUN
ejpam-6846	277	2	s∈s	s∈s	NOUN
ejpam-6846	277	3	{	{	PUNCT
ejpam-6846	277	4	−‖s‖2	−‖s‖2	PROPN
ejpam-6846	277	5	+	+	CCONJ
ejpam-6846	277	6	2〈v∗	2〈v∗	PROPN
ejpam-6846	277	7	,	,	PUNCT
ejpam-6846	277	8	s	s	PROPN
ejpam-6846	277	9	〉	〉	NOUN
ejpam-6846	277	10	−	−	NOUN
ejpam-6846	277	11	f(s	f(	NOUN
ejpam-6846	277	12	)	)	PUNCT
ejpam-6846	277	13	}	}	PUNCT
ejpam-6846	277	14	.	.	PUNCT
ejpam-6846	278	1	the	the	DET
ejpam-6846	278	2	function	function	NOUN
ejpam-6846	278	3	f	f	PROPN
ejpam-6846	278	4	is	be	AUX
ejpam-6846	278	5	convex	convex	ADJ
ejpam-6846	278	6	,	,	PUNCT
ejpam-6846	278	7	and	and	CCONJ
ejpam-6846	278	8	it	it	PRON
ejpam-6846	278	9	is	be	AUX
ejpam-6846	278	10	clear	clear	ADJ
ejpam-6846	278	11	that	that	SCONJ
ejpam-6846	278	12	both	both	DET
ejpam-6846	278	13	functions	function	NOUN
ejpam-6846	278	14	v∗	v∗	PROPN
ejpam-6846	278	15	7→	7→	NUM
ejpam-6846	278	16	vs(v	vs(v	PUNCT
ejpam-6846	278	17	∗	∗	NOUN
ejpam-6846	278	18	)	)	PUNCT
ejpam-6846	279	1	=	=	SYM
ejpam-6846	279	2	sup	sup	NOUN
ejpam-6846	279	3	s∈s	s∈s	NOUN
ejpam-6846	279	4	{	{	PUNCT
ejpam-6846	279	5	−‖s‖2	−‖s‖2	PROPN
ejpam-6846	279	6	+	+	CCONJ
ejpam-6846	279	7	2〈v∗	2〈v∗	PROPN
ejpam-6846	279	8	,	,	PUNCT
ejpam-6846	279	9	s	s	PROPN
ejpam-6846	279	10	〉	〉	NOUN
ejpam-6846	279	11	−	−	NOUN
ejpam-6846	279	12	f(s	f(	NOUN
ejpam-6846	279	13	)	)	PUNCT
ejpam-6846	279	14	}	}	PUNCT
ejpam-6846	279	15	.	.	PUNCT
ejpam-6846	280	1	mf	mf	NOUN
ejpam-6846	280	2	,	,	PUNCT
ejpam-6846	280	3	s(v	s(v	PROPN
ejpam-6846	280	4	∗	∗	NOUN
ejpam-6846	280	5	)	)	PUNCT
ejpam-6846	281	1	=	=	SYM
ejpam-6846	281	2	‖v∗‖2	‖v∗‖2	NOUN
ejpam-6846	281	3	+	+	CCONJ
ejpam-6846	281	4	vs(v	vs(v	NOUN
ejpam-6846	281	5	∗	∗	NOUN
ejpam-6846	281	6	)	)	PUNCT
ejpam-6846	281	7	are	be	AUX
ejpam-6846	281	8	convex	convex	ADJ
ejpam-6846	281	9	.	.	PUNCT
ejpam-6846	282	1	f	f	PROPN
ejpam-6846	282	2	is	be	AUX
ejpam-6846	282	3	fréchet	fréchet	VERB
ejpam-6846	282	4	differentiable	differentiable	ADJ
ejpam-6846	282	5	.	.	PUNCT
ejpam-6846	283	1	then	then	ADV
ejpam-6846	283	2	,	,	PUNCT
ejpam-6846	283	3	vs	vs	ADP
ejpam-6846	283	4	is	be	AUX
ejpam-6846	283	5	functional	functional	ADJ
ejpam-6846	283	6	and	and	CCONJ
ejpam-6846	283	7	is	be	AUX
ejpam-6846	283	8	both	both	PRON
ejpam-6846	283	9	convex	convex	ADJ
ejpam-6846	283	10	and	and	CCONJ
ejpam-6846	283	11	fréchet	fréchet	VERB
ejpam-6846	283	12	differentiable	differentiable	ADJ
ejpam-6846	283	13	.	.	PUNCT
ejpam-6846	284	1	so	so	ADV
ejpam-6846	284	2	the	the	DET
ejpam-6846	284	3	derivative	derivative	NOUN
ejpam-6846	284	4	of	of	ADP
ejpam-6846	284	5	vs	vs	ADP
ejpam-6846	284	6	is	be	AUX
ejpam-6846	284	7	norm	norm	NOUN
ejpam-6846	284	8	-	-	PUNCT
ejpam-6846	284	9	toweak	toweak	NOUN
ejpam-6846	284	10	continuous	continuous	ADJ
ejpam-6846	284	11	(	(	PUNCT
ejpam-6846	284	12	see[19	see[19	NOUN
ejpam-6846	284	13	]	]	X
ejpam-6846	284	14	)	)	PUNCT
ejpam-6846	284	15	,	,	PUNCT
ejpam-6846	284	16	which	which	PRON
ejpam-6846	284	17	implies	imply	VERB
ejpam-6846	284	18	∇fvs(u	∇fvs(u	PROPN
ejpam-6846	284	19	∗	∗	NOUN
ejpam-6846	284	20	n	n	CCONJ
ejpam-6846	284	21	)	)	PUNCT
ejpam-6846	284	22	converges	converge	VERB
ejpam-6846	284	23	weakly	weakly	ADJ
ejpam-6846	284	24	to	to	ADP
ejpam-6846	284	25	∇fvs(u	∇fvs(u	PROPN
ejpam-6846	284	26	∗	∗	NOUN
ejpam-6846	284	27	)	)	PUNCT
ejpam-6846	284	28	,	,	PUNCT
ejpam-6846	284	29	which	which	PRON
ejpam-6846	284	30	means	mean	VERB
ejpam-6846	284	31	that	that	SCONJ
ejpam-6846	284	32	∇fmf	∇fmf	PROPN
ejpam-6846	284	33	,	,	PUNCT
ejpam-6846	284	34	s(u	s(u	PROPN
ejpam-6846	284	35	∗	∗	NOUN
ejpam-6846	284	36	n	n	CCONJ
ejpam-6846	284	37	)	)	PUNCT
ejpam-6846	284	38	approaching	approach	VERB
ejpam-6846	284	39	∇fmf	∇fmf	PROPN
ejpam-6846	284	40	,	,	PUNCT
ejpam-6846	284	41	s(u	s(u	PROPN
ejpam-6846	284	42	∗	∗	NOUN
ejpam-6846	284	43	)	)	PUNCT
ejpam-6846	284	44	weakly	weakly	ADV
ejpam-6846	284	45	.	.	PUNCT
ejpam-6846	285	1	lemma	lemma	PROPN
ejpam-6846	285	2	4	4	NUM
ejpam-6846	285	3	allows	allow	VERB
ejpam-6846	285	4	us	we	PRON
ejpam-6846	285	5	to	to	PART
ejpam-6846	285	6	write	write	VERB
ejpam-6846	285	7	∇fmf	∇fmf	PROPN
ejpam-6846	285	8	,	,	PUNCT
ejpam-6846	285	9	s(u	s(u	PROPN
ejpam-6846	285	10	∗	∗	NOUN
ejpam-6846	285	11	n	n	CCONJ
ejpam-6846	285	12	)	)	PUNCT
ejpam-6846	285	13	=	=	SYM
ejpam-6846	285	14	2(j∗u∗n	2(j∗u∗n	NUM
ejpam-6846	285	15	−	−	PROPN
ejpam-6846	285	16	πf	πf	INTJ
ejpam-6846	285	17	s(u	s(u	PROPN
ejpam-6846	285	18	∗	∗	NOUN
ejpam-6846	285	19	n	n	CCONJ
ejpam-6846	285	20	)	)	PUNCT
ejpam-6846	285	21	)	)	PUNCT
ejpam-6846	285	22	and	and	CCONJ
ejpam-6846	285	23	∇fmf	∇fmf	PROPN
ejpam-6846	285	24	,	,	PUNCT
ejpam-6846	285	25	s(u	s(u	PROPN
ejpam-6846	285	26	∗	∗	NOUN
ejpam-6846	285	27	)	)	PUNCT
ejpam-6846	285	28	=	=	SYM
ejpam-6846	286	1	2(j∗u∗	2(j∗u∗	NUM
ejpam-6846	286	2	−	−	NOUN
ejpam-6846	286	3	πf	πf	INTJ
ejpam-6846	286	4	s(u	s(u	PROPN
ejpam-6846	286	5	∗	∗	PROPN
ejpam-6846	286	6	)	)	PUNCT
ejpam-6846	286	7	)	)	PUNCT
ejpam-6846	286	8	.	.	PUNCT
ejpam-6846	287	1	thus	thus	ADV
ejpam-6846	287	2	,	,	PUNCT
ejpam-6846	287	3	πf	πf	INTJ
ejpam-6846	287	4	s(u	s(u	PROPN
ejpam-6846	287	5	∗	∗	NOUN
ejpam-6846	287	6	n	n	CCONJ
ejpam-6846	287	7	)	)	PUNCT
ejpam-6846	287	8	converges	converge	VERB
ejpam-6846	287	9	weakly	weakly	ADJ
ejpam-6846	287	10	to	to	ADP
ejpam-6846	287	11	πf	πf	VERB
ejpam-6846	287	12	s(u	s(u	PROPN
ejpam-6846	287	13	∗	∗	PROPN
ejpam-6846	287	14	)	)	PUNCT
ejpam-6846	287	15	.	.	PUNCT
ejpam-6846	288	1	by	by	ADP
ejpam-6846	288	2	6	6	NUM
ejpam-6846	288	3	we	we	PRON
ejpam-6846	288	4	get	get	VERB
ejpam-6846	288	5	the	the	DET
ejpam-6846	288	6	strong	strong	ADJ
ejpam-6846	288	7	convergence	convergence	NOUN
ejpam-6846	288	8	of	of	ADP
ejpam-6846	288	9	πf	πf	INTJ
ejpam-6846	288	10	s(u	s(u	PROPN
ejpam-6846	288	11	∗	∗	NOUN
ejpam-6846	288	12	n	n	CCONJ
ejpam-6846	288	13	)	)	PUNCT
ejpam-6846	288	14	to	to	PART
ejpam-6846	288	15	πf	πf	VERB
ejpam-6846	288	16	s(u	s(u	PROPN
ejpam-6846	288	17	∗).the	∗).the	DET
ejpam-6846	288	18	continuity	continuity	NOUN
ejpam-6846	288	19	of	of	ADP
ejpam-6846	288	20	∇fmf	∇fmf	PROPN
ejpam-6846	288	21	,	,	PUNCT
ejpam-6846	288	22	s	s	PART
ejpam-6846	288	23	has	have	AUX
ejpam-6846	288	24	been	be	AUX
ejpam-6846	288	25	established	establish	VERB
ejpam-6846	288	26	,	,	PUNCT
ejpam-6846	288	27	which	which	PRON
ejpam-6846	288	28	brings	bring	VERB
ejpam-6846	288	29	the	the	DET
ejpam-6846	288	30	proof	proof	NOUN
ejpam-6846	288	31	to	to	ADP
ejpam-6846	288	32	an	an	DET
ejpam-6846	288	33	end	end	NOUN
ejpam-6846	288	34	.	.	PUNCT
ejpam-6846	289	1	□	□	PUNCT
ejpam-6846	289	2	the	the	DET
ejpam-6846	289	3	theorem	theorem	NOUN
ejpam-6846	289	4	below	below	ADV
ejpam-6846	289	5	establishes	establish	VERB
ejpam-6846	289	6	an	an	DET
ejpam-6846	289	7	equivalence	equivalence	NOUN
ejpam-6846	289	8	between	between	ADP
ejpam-6846	289	9	the	the	DET
ejpam-6846	289	10	properties	property	NOUN
ejpam-6846	289	11	of	of	ADP
ejpam-6846	289	12	πf	πf	NOUN
ejpam-6846	289	13	s	s	PROPN
ejpam-6846	289	14	and	and	CCONJ
ejpam-6846	289	15	mf	mf	NOUN
ejpam-6846	289	16	,	,	PUNCT
ejpam-6846	289	17	s	s	PART
ejpam-6846	289	18	on	on	ADP
ejpam-6846	289	19	an	an	DET
ejpam-6846	289	20	open	open	ADJ
ejpam-6846	289	21	subset	subset	NOUN
ejpam-6846	289	22	of	of	ADP
ejpam-6846	289	23	x∗.	x∗.	PROPN
ejpam-6846	289	24	theorem	theorem	NOUN
ejpam-6846	289	25	1	1	X
ejpam-6846	289	26	.	.	PUNCT
ejpam-6846	290	1	let	let	VERB
ejpam-6846	290	2	x	x	PRON
ejpam-6846	290	3	be	be	AUX
ejpam-6846	290	4	a	a	DET
ejpam-6846	290	5	reflexive	reflexive	ADJ
ejpam-6846	290	6	banach	banach	NOUN
ejpam-6846	290	7	space	space	NOUN
ejpam-6846	290	8	with	with	ADP
ejpam-6846	290	9	smooth	smooth	ADJ
ejpam-6846	290	10	dual	dual	ADJ
ejpam-6846	290	11	norm	norm	NOUN
ejpam-6846	290	12	and	and	CCONJ
ejpam-6846	290	13	kadec	kadec	NOUN
ejpam-6846	290	14	property	property	NOUN
ejpam-6846	290	15	.	.	PUNCT
ejpam-6846	291	1	let	let	VERB
ejpam-6846	291	2	u∗	u∗	ADV
ejpam-6846	291	3	be	be	AUX
ejpam-6846	291	4	a	a	DET
ejpam-6846	291	5	subset	subset	NOUN
ejpam-6846	291	6	of	of	ADP
ejpam-6846	291	7	x∗	x∗	PROPN
ejpam-6846	291	8	that	that	PRON
ejpam-6846	291	9	is	be	AUX
ejpam-6846	291	10	open	open	ADJ
ejpam-6846	291	11	.	.	PUNCT
ejpam-6846	292	1	let	let	VERB
ejpam-6846	292	2	f	f	PROPN
ejpam-6846	292	3	is	be	AUX
ejpam-6846	292	4	proper	proper	ADJ
ejpam-6846	292	5	,	,	PUNCT
ejpam-6846	292	6	convex	convex	ADJ
ejpam-6846	292	7	,	,	PUNCT
ejpam-6846	292	8	continuous	continuous	ADJ
ejpam-6846	292	9	,	,	PUNCT
ejpam-6846	292	10	bounded	bound	VERB
ejpam-6846	292	11	from	from	ADP
ejpam-6846	292	12	bellow	bellow	ADJ
ejpam-6846	292	13	,	,	PUNCT
ejpam-6846	292	14	and	and	CCONJ
ejpam-6846	292	15	fréchet	fréchet	VERB
ejpam-6846	292	16	differentiable	differentiable	ADJ
ejpam-6846	292	17	function	function	NOUN
ejpam-6846	292	18	.	.	PUNCT
ejpam-6846	293	1	the	the	DET
ejpam-6846	293	2	statements	statement	NOUN
ejpam-6846	293	3	that	that	PRON
ejpam-6846	293	4	follow	follow	VERB
ejpam-6846	293	5	are	be	AUX
ejpam-6846	293	6	equivalent	equivalent	ADJ
ejpam-6846	293	7	.	.	PUNCT
ejpam-6846	294	1	a.	a.	PROPN
ejpam-6846	294	2	al	al	PROPN
ejpam-6846	294	3	tane	tane	PROPN
ejpam-6846	294	4	,	,	PUNCT
ejpam-6846	294	5	l.	l.	PROPN
ejpam-6846	294	6	s.	s.	PROPN
ejpam-6846	294	7	keong	keong	PROPN
ejpam-6846	294	8	/	/	PUNCT
ejpam-6846	294	9	eur	eur	PROPN
ejpam-6846	294	10	.	.	PUNCT
ejpam-6846	295	1	j.	j.	PROPN
ejpam-6846	295	2	pure	pure	PROPN
ejpam-6846	295	3	appl	appl	PROPN
ejpam-6846	295	4	.	.	PROPN
ejpam-6846	295	5	math	math	PROPN
ejpam-6846	295	6	,	,	PUNCT
ejpam-6846	295	7	18	18	NUM
ejpam-6846	295	8	(	(	PUNCT
ejpam-6846	295	9	4	4	NUM
ejpam-6846	295	10	)	)	PUNCT
ejpam-6846	295	11	(	(	PUNCT
ejpam-6846	295	12	2025	2025	NUM
ejpam-6846	295	13	)	)	PUNCT
ejpam-6846	295	14	,	,	PUNCT
ejpam-6846	295	15	6846	6846	NUM
ejpam-6846	295	16	12	12	NUM
ejpam-6846	295	17	of	of	ADP
ejpam-6846	295	18	13	13	NUM
ejpam-6846	295	19	(	(	PUNCT
ejpam-6846	295	20	i	i	NOUN
ejpam-6846	295	21	)	)	PUNCT
ejpam-6846	295	22	mf	mf	PROPN
ejpam-6846	295	23	,	,	PUNCT
ejpam-6846	295	24	s	s	PART
ejpam-6846	295	25	is	be	AUX
ejpam-6846	295	26	c1	c1	PROPN
ejpam-6846	295	27	in	in	ADP
ejpam-6846	295	28	u∗.	u∗.	PROPN
ejpam-6846	295	29	(	(	PUNCT
ejpam-6846	295	30	ii	ii	NOUN
ejpam-6846	295	31	)	)	PUNCT
ejpam-6846	295	32	mf	mf	PROPN
ejpam-6846	295	33	,	,	PUNCT
ejpam-6846	295	34	s	s	PART
ejpam-6846	295	35	is	be	AUX
ejpam-6846	295	36	fréchet	fréchet	VERB
ejpam-6846	295	37	differentiable	differentiable	ADJ
ejpam-6846	295	38	on	on	ADP
ejpam-6846	295	39	u∗.	u∗.	PROPN
ejpam-6846	295	40	(	(	PUNCT
ejpam-6846	295	41	iii	iii	NOUN
ejpam-6846	295	42	)	)	PUNCT
ejpam-6846	295	43	∂fmf	∂fmf	NUM
ejpam-6846	295	44	,	,	PUNCT
ejpam-6846	295	45	s	s	PROPN
ejpam-6846	295	46	6=	6=	PROPN
ejpam-6846	295	47	ϕ	ϕ	NOUN
ejpam-6846	295	48	on	on	ADP
ejpam-6846	295	49	u∗.	u∗.	PROPN
ejpam-6846	295	50	(	(	PUNCT
ejpam-6846	295	51	iv	iv	X
ejpam-6846	295	52	)	)	PUNCT
ejpam-6846	296	1	πf	πf	NOUN
ejpam-6846	296	2	s	s	NOUN
ejpam-6846	296	3	is	be	AUX
ejpam-6846	296	4	single	single	ADV
ejpam-6846	296	5	-	-	PUNCT
ejpam-6846	296	6	valued	value	VERB
ejpam-6846	296	7	and	and	CCONJ
ejpam-6846	296	8	‖.‖	‖.‖	NOUN
ejpam-6846	296	9	−	−	PROPN
ejpam-6846	296	10	to−	to−	PUNCT
ejpam-6846	296	11	weak	weak	ADJ
ejpam-6846	296	12	continuous	continuous	ADJ
ejpam-6846	296	13	on	on	ADP
ejpam-6846	296	14	u∗.	u∗.	PROPN
ejpam-6846	296	15	(	(	PUNCT
ejpam-6846	296	16	v	v	NOUN
ejpam-6846	296	17	)	)	PUNCT
ejpam-6846	296	18	πf	πf	NOUN
ejpam-6846	296	19	s	s	VERB
ejpam-6846	296	20	is	be	AUX
ejpam-6846	296	21	single	single	ADV
ejpam-6846	296	22	-	-	PUNCT
ejpam-6846	296	23	valued	value	VERB
ejpam-6846	296	24	and	and	CCONJ
ejpam-6846	296	25	‖.‖	‖.‖	NOUN
ejpam-6846	296	26	−	−	PROPN
ejpam-6846	296	27	‖.‖	‖.‖	NOUN
ejpam-6846	296	28	continuous	continuous	ADJ
ejpam-6846	296	29	on	on	ADP
ejpam-6846	296	30	u∗.	u∗.	PROPN
ejpam-6846	296	31	proof	proof	NOUN
ejpam-6846	296	32	:	:	PUNCT
ejpam-6846	296	33	combining	combine	VERB
ejpam-6846	296	34	the	the	DET
ejpam-6846	296	35	previously	previously	ADV
ejpam-6846	296	36	proven	prove	VERB
ejpam-6846	296	37	lemmas	lemmas	ADJ
ejpam-6846	296	38	to	to	PART
ejpam-6846	296	39	prove	prove	VERB
ejpam-6846	296	40	this	this	DET
ejpam-6846	296	41	theorem	theorem	NOUN
ejpam-6846	296	42	is	be	AUX
ejpam-6846	296	43	enough	enough	ADJ
ejpam-6846	296	44	.	.	PUNCT
ejpam-6846	297	1	4	4	X
ejpam-6846	297	2	.	.	X
ejpam-6846	297	3	conclusion	conclusion	NOUN
ejpam-6846	297	4	to	to	PART
ejpam-6846	297	5	investigate	investigate	VERB
ejpam-6846	297	6	the	the	DET
ejpam-6846	297	7	πf	πf	NOUN
ejpam-6846	297	8	s	s	PART
ejpam-6846	297	9	,	,	PUNCT
ejpam-6846	297	10	this	this	DET
ejpam-6846	297	11	study	study	NOUN
ejpam-6846	297	12	shows	show	VERB
ejpam-6846	297	13	the	the	DET
ejpam-6846	297	14	efficiency	efficiency	NOUN
ejpam-6846	297	15	of	of	ADP
ejpam-6846	297	16	the	the	DET
ejpam-6846	297	17	subdifferential	subdifferential	ADJ
ejpam-6846	297	18	∂conf	∂conf	PROPN
ejpam-6846	297	19	,	,	PUNCT
ejpam-6846	297	20	the	the	DET
ejpam-6846	297	21	fréchet	fréchet	ADJ
ejpam-6846	297	22	subdifferential	subdifferential	ADJ
ejpam-6846	297	23	∂f	∂f	PROPN
ejpam-6846	297	24	f(x	f(x	PROPN
ejpam-6846	297	25	)	)	PUNCT
ejpam-6846	297	26	,	,	PUNCT
ejpam-6846	297	27	and	and	CCONJ
ejpam-6846	297	28	the	the	DET
ejpam-6846	297	29	clarke	clarke	PROPN
ejpam-6846	297	30	subdifferential	subdifferential	PROPN
ejpam-6846	297	31	to	to	PART
ejpam-6846	297	32	extend	extend	VERB
ejpam-6846	297	33	many	many	ADJ
ejpam-6846	297	34	properties	property	NOUN
ejpam-6846	297	35	of	of	ADP
ejpam-6846	297	36	πs	πs	ADP
ejpam-6846	297	37	to	to	PART
ejpam-6846	297	38	πf	πf	VERB
ejpam-6846	297	39	s	s	VERB
ejpam-6846	297	40	in	in	ADP
ejpam-6846	297	41	addition	addition	NOUN
ejpam-6846	297	42	to	to	PART
ejpam-6846	297	43	provide	provide	VERB
ejpam-6846	297	44	a	a	DET
ejpam-6846	297	45	necessary	necessary	ADJ
ejpam-6846	297	46	and	and	CCONJ
ejpam-6846	297	47	sufficient	sufficient	ADJ
ejpam-6846	297	48	condition	condition	NOUN
ejpam-6846	297	49	for	for	ADP
ejpam-6846	297	50	the	the	DET
ejpam-6846	297	51	existence	existence	NOUN
ejpam-6846	297	52	and	and	CCONJ
ejpam-6846	297	53	uniqueness	uniqueness	NOUN
ejpam-6846	297	54	and	and	CCONJ
ejpam-6846	297	55	prove	prove	VERB
ejpam-6846	297	56	locally	locally	ADV
ejpam-6846	297	57	lipschtz	lipschtz	ADV
ejpam-6846	297	58	continuous	continuous	ADJ
ejpam-6846	297	59	of	of	ADP
ejpam-6846	297	60	the	the	DET
ejpam-6846	297	61	πf	πf	NOUN
ejpam-6846	297	62	s	s	PROPN
ejpam-6846	297	63	.	.	PUNCT
ejpam-6846	298	1	in	in	ADP
ejpam-6846	298	2	the	the	DET
ejpam-6846	298	3	future	future	NOUN
ejpam-6846	298	4	,	,	PUNCT
ejpam-6846	298	5	we	we	PRON
ejpam-6846	298	6	intend	intend	VERB
ejpam-6846	298	7	to	to	PART
ejpam-6846	298	8	introduce	introduce	VERB
ejpam-6846	298	9	a	a	DET
ejpam-6846	298	10	new	new	ADJ
ejpam-6846	298	11	set	set	NOUN
ejpam-6846	298	12	defined	define	VERB
ejpam-6846	298	13	in	in	ADP
ejpam-6846	298	14	terms	term	NOUN
ejpam-6846	298	15	of	of	ADP
ejpam-6846	298	16	the	the	DET
ejpam-6846	298	17	generalized	generalize	VERB
ejpam-6846	298	18	f	f	PROPN
ejpam-6846	298	19	-projection	-projection	PROPN
ejpam-6846	298	20	,	,	PUNCT
ejpam-6846	298	21	to	to	PART
ejpam-6846	298	22	extend	extend	VERB
ejpam-6846	298	23	many	many	ADJ
ejpam-6846	298	24	well	well	ADV
ejpam-6846	298	25	-	-	PUNCT
ejpam-6846	298	26	known	know	VERB
ejpam-6846	298	27	results	result	NOUN
ejpam-6846	298	28	on	on	ADP
ejpam-6846	298	29	the	the	DET
ejpam-6846	298	30	usual	usual	ADJ
ejpam-6846	298	31	v	v	NOUN
ejpam-6846	298	32	-proximal	-proximal	NOUN
ejpam-6846	298	33	normal	normal	ADJ
ejpam-6846	298	34	cone	cone	NOUN
ejpam-6846	298	35	in	in	ADP
ejpam-6846	298	36	the	the	DET
ejpam-6846	298	37	setting	setting	NOUN
ejpam-6846	298	38	of	of	ADP
ejpam-6846	298	39	banach	banach	NOUN
ejpam-6846	298	40	spaces	space	NOUN
ejpam-6846	298	41	.	.	PUNCT
ejpam-6846	299	1	furthermore	furthermore	ADV
ejpam-6846	299	2	,	,	PUNCT
ejpam-6846	299	3	we	we	PRON
ejpam-6846	299	4	will	will	AUX
ejpam-6846	299	5	extend	extend	VERB
ejpam-6846	299	6	the	the	DET
ejpam-6846	299	7	widely	widely	ADV
ejpam-6846	299	8	studied	study	VERB
ejpam-6846	299	9	notion	notion	NOUN
ejpam-6846	299	10	of	of	ADP
ejpam-6846	299	11	prox	prox	NOUN
ejpam-6846	299	12	-	-	PUNCT
ejpam-6846	299	13	regularity	regularity	NOUN
ejpam-6846	299	14	by	by	ADP
ejpam-6846	299	15	incorporating	incorporate	VERB
ejpam-6846	299	16	the	the	DET
ejpam-6846	299	17	flexibility	flexibility	NOUN
ejpam-6846	299	18	of	of	ADP
ejpam-6846	299	19	f	f	PROPN
ejpam-6846	299	20	-projections	-projection	NOUN
ejpam-6846	299	21	.	.	PUNCT
ejpam-6846	300	1	such	such	ADJ
ejpam-6846	300	2	as	as	ADP
ejpam-6846	300	3	in	in	ADP
ejpam-6846	300	4	[	[	X
ejpam-6846	300	5	6	6	NUM
ejpam-6846	300	6	,	,	PUNCT
ejpam-6846	300	7	7	7	NUM
ejpam-6846	300	8	]	]	PUNCT
ejpam-6846	300	9	.	.	PUNCT
ejpam-6846	301	1	conflicts	conflict	NOUN
ejpam-6846	301	2	of	of	ADP
ejpam-6846	301	3	interest	interest	NOUN
ejpam-6846	301	4	:	:	PUNCT
ejpam-6846	301	5	the	the	DET
ejpam-6846	301	6	authors	author	NOUN
ejpam-6846	301	7	declare	declare	VERB
ejpam-6846	301	8	that	that	SCONJ
ejpam-6846	301	9	they	they	PRON
ejpam-6846	301	10	have	have	VERB
ejpam-6846	301	11	no	no	DET
ejpam-6846	301	12	conflict	conflict	NOUN
ejpam-6846	301	13	of	of	ADP
ejpam-6846	301	14	interest	interest	NOUN
ejpam-6846	301	15	.	.	PUNCT
ejpam-6846	302	1	references	reference	NOUN
ejpam-6846	302	2	[	[	X
ejpam-6846	302	3	1	1	X
ejpam-6846	302	4	]	]	PUNCT
ejpam-6846	302	5	ya	ya	PROPN
ejpam-6846	302	6	i	i	PROPN
ejpam-6846	302	7	alber	alber	PROPN
ejpam-6846	302	8	.	.	PUNCT
ejpam-6846	303	1	metric	metric	ADJ
ejpam-6846	303	2	and	and	CCONJ
ejpam-6846	303	3	generalized	generalized	ADJ
ejpam-6846	303	4	projection	projection	NOUN
ejpam-6846	303	5	operators	operator	NOUN
ejpam-6846	303	6	in	in	ADP
ejpam-6846	303	7	banach	banach	NOUN
ejpam-6846	303	8	spaces	space	NOUN
ejpam-6846	303	9	:	:	PUNCT
ejpam-6846	303	10	properties	property	NOUN
ejpam-6846	303	11	and	and	CCONJ
ejpam-6846	303	12	applications	application	NOUN
ejpam-6846	303	13	.	.	PUNCT
ejpam-6846	304	1	arxiv	arxiv	PROPN
ejpam-6846	304	2	preprint	preprint	VERB
ejpam-6846	304	3	funct	funct	NOUN
ejpam-6846	304	4	-	-	PUNCT
ejpam-6846	304	5	an/9311001	an/9311001	NOUN
ejpam-6846	304	6	,	,	PUNCT
ejpam-6846	304	7	1993	1993	NUM
ejpam-6846	304	8	.	.	PUNCT
ejpam-6846	305	1	[	[	X
ejpam-6846	305	2	2	2	X
ejpam-6846	305	3	]	]	X
ejpam-6846	305	4	jinlu	jinlu	PROPN
ejpam-6846	305	5	li	li	PROPN
ejpam-6846	305	6	.	.	PUNCT
ejpam-6846	306	1	the	the	DET
ejpam-6846	306	2	generalized	generalized	ADJ
ejpam-6846	306	3	projection	projection	NOUN
ejpam-6846	306	4	operator	operator	NOUN
ejpam-6846	306	5	on	on	ADP
ejpam-6846	306	6	reflexive	reflexive	ADJ
ejpam-6846	306	7	banach	banach	NOUN
ejpam-6846	306	8	spaces	space	NOUN
ejpam-6846	306	9	and	and	CCONJ
ejpam-6846	306	10	its	its	PRON
ejpam-6846	306	11	applications	application	NOUN
ejpam-6846	306	12	.	.	PUNCT
ejpam-6846	307	1	journal	journal	PROPN
ejpam-6846	307	2	of	of	ADP
ejpam-6846	307	3	mathematical	mathematical	ADJ
ejpam-6846	307	4	analysis	analysis	NOUN
ejpam-6846	307	5	and	and	CCONJ
ejpam-6846	307	6	applications	application	NOUN
ejpam-6846	307	7	,	,	PUNCT
ejpam-6846	307	8	306(1):55–71	306(1):55–71	NOUN
ejpam-6846	307	9	,	,	PUNCT
ejpam-6846	307	10	2005	2005	NUM
ejpam-6846	307	11	,	,	PUNCT
ejpam-6846	307	12	https://doi.org/10.1016/j.jmaa.2004.11.007	https://doi.org/10.1016/j.jmaa.2004.11.007	ADJ
ejpam-6846	307	13	.	.	PUNCT
ejpam-6846	308	1	[	[	X
ejpam-6846	308	2	3	3	X
ejpam-6846	308	3	]	]	X
ejpam-6846	308	4	ke	ke	NOUN
ejpam-6846	308	5	-	-	PUNCT
ejpam-6846	308	6	qing	qe	VERB
ejpam-6846	308	7	wu	wu	PROPN
ejpam-6846	308	8	and	and	CCONJ
ejpam-6846	308	9	nan	nan	PROPN
ejpam-6846	308	10	-	-	PROPN
ejpam-6846	308	11	jing	jing	PROPN
ejpam-6846	308	12	huang	huang	PROPN
ejpam-6846	308	13	.	.	PUNCT
ejpam-6846	309	1	the	the	DET
ejpam-6846	309	2	generalised	generalise	VERB
ejpam-6846	309	3	f	f	PROPN
ejpam-6846	309	4	-	-	PUNCT
ejpam-6846	309	5	projection	projection	NOUN
ejpam-6846	309	6	operator	operator	NOUN
ejpam-6846	309	7	with	with	ADP
ejpam-6846	309	8	an	an	DET
ejpam-6846	309	9	application	application	NOUN
ejpam-6846	309	10	.	.	PUNCT
ejpam-6846	310	1	bulletin	bulletin	NOUN
ejpam-6846	310	2	of	of	ADP
ejpam-6846	310	3	the	the	DET
ejpam-6846	310	4	australian	australian	ADJ
ejpam-6846	310	5	mathematical	mathematical	ADJ
ejpam-6846	310	6	society	society	NOUN
ejpam-6846	310	7	,	,	PUNCT
ejpam-6846	310	8	73(2):307–317	73(2):307–317	NUM
ejpam-6846	310	9	,	,	PUNCT
ejpam-6846	310	10	2006	2006	NUM
ejpam-6846	310	11	,	,	PUNCT
ejpam-6846	310	12	https://doi.org/10.1017/s0004972700038892	https://doi.org/10.1017/s0004972700038892	NUM
ejpam-6846	310	13	.	.	PUNCT
ejpam-6846	311	1	[	[	X
ejpam-6846	311	2	4	4	NUM
ejpam-6846	311	3	]	]	SYM
ejpam-6846	311	4	m	m	VERB
ejpam-6846	311	5	bounkhel	bounkhel	NOUN
ejpam-6846	311	6	and	and	CCONJ
ejpam-6846	311	7	r	r	NOUN
ejpam-6846	311	8	al	al	PROPN
ejpam-6846	311	9	-	-	PUNCT
ejpam-6846	311	10	yusof	yusof	NOUN
ejpam-6846	311	11	.	.	PUNCT
ejpam-6846	312	1	proximal	proximal	ADJ
ejpam-6846	312	2	analysis	analysis	NOUN
ejpam-6846	312	3	in	in	ADP
ejpam-6846	312	4	reflexive	reflexive	ADJ
ejpam-6846	312	5	smooth	smooth	ADJ
ejpam-6846	312	6	banach	banach	NOUN
ejpam-6846	312	7	spaces	space	VERB
ejpam-6846	312	8	.	.	PUNCT
ejpam-6846	313	1	nonlinear	nonlinear	ADJ
ejpam-6846	313	2	analysis	analysis	NOUN
ejpam-6846	313	3	:	:	PUNCT
ejpam-6846	313	4	theory	theory	NOUN
ejpam-6846	313	5	,	,	PUNCT
ejpam-6846	313	6	methods	method	NOUN
ejpam-6846	313	7	&	&	CCONJ
ejpam-6846	313	8	applications	application	NOUN
ejpam-6846	313	9	,	,	PUNCT
ejpam-6846	313	10	73(7):1921–1939	73(7):1921–1939	NUM
ejpam-6846	313	11	,	,	PUNCT
ejpam-6846	313	12	2010	2010	NUM
ejpam-6846	313	13	,	,	PUNCT
ejpam-6846	313	14	https://doi.org/10.1016/j.na.2010.04.077	https://doi.org/10.1016/j.na.2010.04.077	PROPN
ejpam-6846	313	15	.	.	PUNCT
ejpam-6846	314	1	[	[	X
ejpam-6846	314	2	5	5	NUM
ejpam-6846	314	3	]	]	PUNCT
ejpam-6846	314	4	messaoud	messaoud	PROPN
ejpam-6846	314	5	bounkhel	bounkhel	PROPN
ejpam-6846	314	6	et	et	PROPN
ejpam-6846	314	7	al	al	PROPN
ejpam-6846	314	8	.	.	PROPN
ejpam-6846	315	1	generalized	generalized	ADJ
ejpam-6846	315	2	projections	projection	NOUN
ejpam-6846	315	3	on	on	ADP
ejpam-6846	315	4	closed	close	VERB
ejpam-6846	315	5	nonconvex	nonconvex	NOUN
ejpam-6846	315	6	sets	set	NOUN
ejpam-6846	315	7	in	in	ADP
ejpam-6846	315	8	uniformly	uniformly	ADV
ejpam-6846	315	9	convex	convex	NOUN
ejpam-6846	315	10	and	and	CCONJ
ejpam-6846	315	11	uniformly	uniformly	ADV
ejpam-6846	315	12	smooth	smooth	ADJ
ejpam-6846	315	13	banach	banach	NOUN
ejpam-6846	315	14	spaces	space	NOUN
ejpam-6846	315	15	.	.	PUNCT
ejpam-6846	316	1	journal	journal	NOUN
ejpam-6846	316	2	of	of	ADP
ejpam-6846	316	3	function	function	NOUN
ejpam-6846	316	4	spaces	space	NOUN
ejpam-6846	316	5	,	,	PUNCT
ejpam-6846	316	6	2015	2015	NUM
ejpam-6846	316	7	,	,	PUNCT
ejpam-6846	316	8	2015	2015	NUM
ejpam-6846	316	9	,	,	PUNCT
ejpam-6846	316	10	https://doi.org/10.1155/2015/478437	https://doi.org/10.1155/2015/478437	PROPN
ejpam-6846	316	11	.	.	PUNCT
ejpam-6846	317	1	[	[	X
ejpam-6846	317	2	6	6	NUM
ejpam-6846	317	3	]	]	SYM
ejpam-6846	317	4	messaoud	messaoud	PROPN
ejpam-6846	317	5	bounkhel	bounkhel	PROPN
ejpam-6846	317	6	and	and	CCONJ
ejpam-6846	317	7	mostafa	mostafa	PROPN
ejpam-6846	317	8	bachar	bachar	PROPN
ejpam-6846	317	9	.	.	PUNCT
ejpam-6846	318	1	generalised	generalise	VERB
ejpam-6846	318	2	-	-	PUNCT
ejpam-6846	318	3	prox	prox	NOUN
ejpam-6846	318	4	-	-	PUNCT
ejpam-6846	318	5	regularity	regularity	NOUN
ejpam-6846	318	6	in	in	ADP
ejpam-6846	318	7	reflexive	reflexive	ADJ
ejpam-6846	318	8	smooth	smooth	ADJ
ejpam-6846	318	9	banach	banach	NOUN
ejpam-6846	318	10	spaces	space	VERB
ejpam-6846	318	11	with	with	ADP
ejpam-6846	318	12	smooth	smooth	ADJ
ejpam-6846	318	13	dual	dual	ADJ
ejpam-6846	318	14	norm	norm	NOUN
ejpam-6846	318	15	.	.	PUNCT
ejpam-6846	319	1	journal	journal	PROPN
ejpam-6846	319	2	of	of	ADP
ejpam-6846	319	3	mathematical	mathematical	ADJ
ejpam-6846	319	4	analysis	analysis	NOUN
ejpam-6846	319	5	a.	a.	NOUN
ejpam-6846	319	6	al	al	PROPN
ejpam-6846	319	7	tane	tane	PROPN
ejpam-6846	319	8	,	,	PUNCT
ejpam-6846	319	9	l.	l.	PROPN
ejpam-6846	319	10	s.	s.	PROPN
ejpam-6846	319	11	keong	keong	PROPN
ejpam-6846	319	12	/	/	PUNCT
ejpam-6846	319	13	eur	eur	PROPN
ejpam-6846	319	14	.	.	PUNCT
ejpam-6846	320	1	j.	j.	PROPN
ejpam-6846	320	2	pure	pure	PROPN
ejpam-6846	320	3	appl	appl	PROPN
ejpam-6846	320	4	.	.	PROPN
ejpam-6846	320	5	math	math	PROPN
ejpam-6846	320	6	,	,	PUNCT
ejpam-6846	320	7	18	18	NUM
ejpam-6846	320	8	(	(	PUNCT
ejpam-6846	320	9	4	4	NUM
ejpam-6846	320	10	)	)	PUNCT
ejpam-6846	320	11	(	(	PUNCT
ejpam-6846	320	12	2025	2025	NUM
ejpam-6846	320	13	)	)	PUNCT
ejpam-6846	320	14	,	,	PUNCT
ejpam-6846	320	15	6846	6846	NUM
ejpam-6846	320	16	13	13	NUM
ejpam-6846	320	17	of	of	ADP
ejpam-6846	320	18	13	13	NUM
ejpam-6846	320	19	and	and	CCONJ
ejpam-6846	320	20	applications	application	NOUN
ejpam-6846	320	21	,	,	PUNCT
ejpam-6846	320	22	475(1):699–729	475(1):699–729	NUM
ejpam-6846	320	23	,	,	PUNCT
ejpam-6846	320	24	2019	2019	NUM
ejpam-6846	320	25	,	,	PUNCT
ejpam-6846	320	26	https://doi.org/10.1016/j.jmaa.2019.02.064	https://doi.org/10.1016/j.jmaa.2019.02.064	NOUN
ejpam-6846	320	27	.	.	PUNCT
ejpam-6846	321	1	[	[	X
ejpam-6846	321	2	7	7	NUM
ejpam-6846	321	3	]	]	X
ejpam-6846	321	4	messaoud	messaoud	PROPN
ejpam-6846	321	5	bounkhel	bounkhel	PROPN
ejpam-6846	321	6	.	.	PUNCT
ejpam-6846	322	1	generalized	generalize	VERB
ejpam-6846	322	2	(	(	PUNCT
ejpam-6846	322	3	f	f	X
ejpam-6846	322	4	,	,	PUNCT
ejpam-6846	322	5	λ)-projection	λ)-projection	PUNCT
ejpam-6846	322	6	operator	operator	NOUN
ejpam-6846	322	7	on	on	ADP
ejpam-6846	322	8	closed	close	VERB
ejpam-6846	322	9	nonconvex	nonconvex	NOUN
ejpam-6846	322	10	sets	set	NOUN
ejpam-6846	322	11	and	and	CCONJ
ejpam-6846	322	12	its	its	PRON
ejpam-6846	322	13	applications	application	NOUN
ejpam-6846	322	14	in	in	ADP
ejpam-6846	322	15	reflexive	reflexive	ADJ
ejpam-6846	322	16	smooth	smooth	ADJ
ejpam-6846	322	17	banach	banach	NOUN
ejpam-6846	322	18	spaces	space	VERB
ejpam-6846	322	19	.	.	PUNCT
ejpam-6846	323	1	aims	aim	VERB
ejpam-6846	323	2	mathematics	mathematic	NOUN
ejpam-6846	323	3	,	,	PUNCT
ejpam-6846	323	4	8(12):29555–29568	8(12):29555–29568	NUM
ejpam-6846	323	5	,	,	PUNCT
ejpam-6846	323	6	2023	2023	NUM
ejpam-6846	323	7	,	,	PUNCT
ejpam-6846	323	8	https://doi.org/10.3934/math.20231513	https://doi.org/10.3934/math.20231513	NOUN
ejpam-6846	323	9	.	.	PUNCT
ejpam-6846	324	1	[	[	X
ejpam-6846	324	2	8	8	NUM
ejpam-6846	324	3	]	]	X
ejpam-6846	324	4	robert	robert	PROPN
ejpam-6846	324	5	deville	deville	PROPN
ejpam-6846	324	6	,	,	PUNCT
ejpam-6846	324	7	gilles	gille	NOUN
ejpam-6846	324	8	godefroy	godefroy	PROPN
ejpam-6846	324	9	,	,	PUNCT
ejpam-6846	324	10	and	and	CCONJ
ejpam-6846	324	11	václav	václav	PROPN
ejpam-6846	324	12	zizler	zizler	PROPN
ejpam-6846	324	13	.	.	PUNCT
ejpam-6846	325	1	smoothness	smoothness	NOUN
ejpam-6846	325	2	and	and	CCONJ
ejpam-6846	325	3	renormings	renorming	NOUN
ejpam-6846	325	4	in	in	ADP
ejpam-6846	325	5	banach	banach	NOUN
ejpam-6846	325	6	spaces	space	NOUN
ejpam-6846	325	7	.	.	PUNCT
ejpam-6846	326	1	longman	longman	ADJ
ejpam-6846	326	2	scientific	scientific	PROPN
ejpam-6846	326	3	technical	technical	ADJ
ejpam-6846	326	4	,	,	PUNCT
ejpam-6846	326	5	harlow	harlow	PROPN
ejpam-6846	326	6	;	;	PUNCT
ejpam-6846	326	7	copublished	copublishe	VERB
ejpam-6846	326	8	in	in	ADP
ejpam-6846	326	9	the	the	DET
ejpam-6846	326	10	united	united	PROPN
ejpam-6846	326	11	states	states	PROPN
ejpam-6846	326	12	with	with	ADP
ejpam-6846	326	13	…	…	PUNCT
ejpam-6846	326	14	,	,	PUNCT
ejpam-6846	326	15	1993	1993	NUM
ejpam-6846	326	16	.	.	PUNCT
ejpam-6846	327	1	[	[	X
ejpam-6846	327	2	9	9	NUM
ejpam-6846	327	3	]	]	X
ejpam-6846	327	4	j	j	PROPN
ejpam-6846	327	5	diestel	diestel	PROPN
ejpam-6846	327	6	.	.	PUNCT
ejpam-6846	327	7	geometry	geometry	NOUN
ejpam-6846	327	8	of	of	ADP
ejpam-6846	327	9	banach	banach	NOUN
ejpam-6846	327	10	space	space	NOUN
ejpam-6846	327	11	-	-	PUNCT
ejpam-6846	327	12	selected	select	VERB
ejpam-6846	327	13	topics	topic	NOUN
ejpam-6846	327	14	lect	lect	ADJ
ejpam-6846	327	15	.	.	PUNCT
ejpam-6846	328	1	notes	note	NOUN
ejpam-6846	328	2	in	in	ADP
ejpam-6846	328	3	math	math	NOUN
ejpam-6846	328	4	,	,	PUNCT
ejpam-6846	328	5	485	485	NUM
ejpam-6846	328	6	,	,	PUNCT
ejpam-6846	328	7	1975	1975	NUM
ejpam-6846	328	8	.	.	PUNCT
ejpam-6846	329	1	[	[	X
ejpam-6846	329	2	10	10	NUM
ejpam-6846	329	3	]	]	X
ejpam-6846	329	4	wataru	wataru	PROPN
ejpam-6846	329	5	takahashi	takahashi	PROPN
ejpam-6846	329	6	.	.	PUNCT
ejpam-6846	330	1	nonlinear	nonlinear	ADJ
ejpam-6846	330	2	functional	functional	ADJ
ejpam-6846	330	3	analysis	analysis	NOUN
ejpam-6846	330	4	.	.	PUNCT
ejpam-6846	331	1	fixed	fix	VERB
ejpam-6846	331	2	point	point	NOUN
ejpam-6846	331	3	theory	theory	NOUN
ejpam-6846	331	4	and	and	CCONJ
ejpam-6846	331	5	its	its	PRON
ejpam-6846	331	6	applications	application	NOUN
ejpam-6846	331	7	,	,	PUNCT
ejpam-6846	331	8	2000	2000	NUM
ejpam-6846	331	9	.	.	PUNCT
ejpam-6846	332	1	[	[	X
ejpam-6846	332	2	11	11	NUM
ejpam-6846	332	3	]	]	PUNCT
ejpam-6846	332	4	mordukhaui	mordukhaui	NOUN
ejpam-6846	332	5	moiseevich	moiseevich	PROPN
ejpam-6846	332	6	vauinberg	vauinberg	PROPN
ejpam-6846	332	7	,	,	PUNCT
ejpam-6846	332	8	alexander	alexander	PROPN
ejpam-6846	332	9	libin	libin	PROPN
ejpam-6846	332	10	,	,	PUNCT
ejpam-6846	332	11	and	and	CCONJ
ejpam-6846	332	12	david	david	PROPN
ejpam-6846	332	13	louvish	louvish	PROPN
ejpam-6846	332	14	.	.	PUNCT
ejpam-6846	333	1	variational	variational	ADJ
ejpam-6846	333	2	method	method	NOUN
ejpam-6846	333	3	and	and	CCONJ
ejpam-6846	333	4	method	method	NOUN
ejpam-6846	333	5	of	of	ADP
ejpam-6846	333	6	monotone	monotone	ADJ
ejpam-6846	333	7	operators	operator	NOUN
ejpam-6846	333	8	in	in	ADP
ejpam-6846	333	9	the	the	DET
ejpam-6846	333	10	theory	theory	NOUN
ejpam-6846	333	11	of	of	ADP
ejpam-6846	333	12	nonlinear	nonlinear	ADJ
ejpam-6846	333	13	equations	equation	NOUN
ejpam-6846	333	14	.	.	PUNCT
ejpam-6846	334	1	(	(	PUNCT
ejpam-6846	334	2	no	no	DET
ejpam-6846	334	3	title	title	NOUN
ejpam-6846	334	4	)	)	PUNCT
ejpam-6846	334	5	,	,	PUNCT
ejpam-6846	334	6	1973	1973	NUM
ejpam-6846	334	7	,	,	PUNCT
ejpam-6846	334	8	https://lccn.loc.gov/73016383	https://lccn.loc.gov/73016383	NOUN
ejpam-6846	334	9	.	.	PUNCT
ejpam-6846	335	1	[	[	X
ejpam-6846	335	2	12	12	NUM
ejpam-6846	335	3	]	]	PUNCT
ejpam-6846	335	4	messaoud	messaoud	PROPN
ejpam-6846	335	5	bounkhel	bounkhel	PROPN
ejpam-6846	335	6	.	.	PUNCT
ejpam-6846	336	1	regularity	regularity	NOUN
ejpam-6846	336	2	concepts	concept	NOUN
ejpam-6846	336	3	in	in	ADP
ejpam-6846	336	4	nonsmooth	nonsmooth	ADJ
ejpam-6846	336	5	analysis	analysis	NOUN
ejpam-6846	336	6	:	:	PUNCT
ejpam-6846	336	7	theory	theory	NOUN
ejpam-6846	336	8	and	and	CCONJ
ejpam-6846	336	9	applications	application	NOUN
ejpam-6846	336	10	,	,	PUNCT
ejpam-6846	336	11	volume	volume	NOUN
ejpam-6846	336	12	59	59	NUM
ejpam-6846	336	13	.	.	PUNCT
ejpam-6846	337	1	springer	springer	PROPN
ejpam-6846	337	2	science	science	PROPN
ejpam-6846	337	3	&	&	CCONJ
ejpam-6846	337	4	business	business	NOUN
ejpam-6846	337	5	media	medium	NOUN
ejpam-6846	337	6	,	,	PUNCT
ejpam-6846	337	7	2011	2011	NUM
ejpam-6846	337	8	.	.	PUNCT
ejpam-6846	338	1	[	[	X
ejpam-6846	338	2	13	13	NUM
ejpam-6846	338	3	]	]	X
ejpam-6846	338	4	nelson	nelson	PROPN
ejpam-6846	338	5	dunford	dunford	PROPN
ejpam-6846	338	6	and	and	CCONJ
ejpam-6846	338	7	jacob	jacob	PROPN
ejpam-6846	338	8	t	t	PROPN
ejpam-6846	338	9	schwartz	schwartz	PROPN
ejpam-6846	338	10	.	.	PUNCT
ejpam-6846	339	1	linear	linear	PROPN
ejpam-6846	339	2	operators	operator	NOUN
ejpam-6846	339	3	,	,	PUNCT
ejpam-6846	339	4	part	part	NOUN
ejpam-6846	339	5	1	1	NUM
ejpam-6846	339	6	:	:	PUNCT
ejpam-6846	339	7	general	general	ADJ
ejpam-6846	339	8	theory	theory	NOUN
ejpam-6846	339	9	,	,	PUNCT
ejpam-6846	339	10	volume	volume	NOUN
ejpam-6846	339	11	10	10	NUM
ejpam-6846	339	12	.	.	PUNCT
ejpam-6846	340	1	john	john	PROPN
ejpam-6846	340	2	wiley	wiley	PROPN
ejpam-6846	340	3	&	&	CCONJ
ejpam-6846	340	4	sons	son	NOUN
ejpam-6846	340	5	,	,	PUNCT
ejpam-6846	340	6	1988	1988	NUM
ejpam-6846	340	7	.	.	PUNCT
ejpam-6846	341	1	[	[	X
ejpam-6846	341	2	14	14	NUM
ejpam-6846	341	3	]	]	X
ejpam-6846	341	4	akhtar	akhtar	PROPN
ejpam-6846	341	5	a	a	DET
ejpam-6846	341	6	khan	khan	PROPN
ejpam-6846	341	7	,	,	PUNCT
ejpam-6846	341	8	jinlu	jinlu	PROPN
ejpam-6846	341	9	li	li	PROPN
ejpam-6846	341	10	,	,	PUNCT
ejpam-6846	341	11	and	and	CCONJ
ejpam-6846	341	12	simeon	simeon	PROPN
ejpam-6846	341	13	reich	reich	PROPN
ejpam-6846	341	14	.	.	PUNCT
ejpam-6846	342	1	generalized	generalize	VERB
ejpam-6846	342	2	projections	projection	NOUN
ejpam-6846	342	3	on	on	ADP
ejpam-6846	342	4	general	general	ADJ
ejpam-6846	342	5	banach	banach	NOUN
ejpam-6846	342	6	spaces	space	VERB
ejpam-6846	342	7	.	.	PUNCT
ejpam-6846	343	1	arxiv	arxiv	PROPN
ejpam-6846	343	2	preprint	preprint	VERB
ejpam-6846	343	3	arxiv:2207.09317	arxiv:2207.09317	NOUN
ejpam-6846	343	4	,	,	PUNCT
ejpam-6846	343	5	2022	2022	NUM
ejpam-6846	343	6	,	,	PUNCT
ejpam-6846	343	7	https://doi.org/10.48550/arxiv.2207.09317	https://doi.org/10.48550/arxiv.2207.09317	PROPN
ejpam-6846	343	8	.	.	PUNCT
ejpam-6846	344	1	[	[	X
ejpam-6846	344	2	15	15	NUM
ejpam-6846	344	3	]	]	X
ejpam-6846	344	4	frédéric	frédéric	ADJ
ejpam-6846	344	5	bernard	bernard	PROPN
ejpam-6846	344	6	,	,	PUNCT
ejpam-6846	344	7	lionel	lionel	PROPN
ejpam-6846	344	8	thibault	thibault	PROPN
ejpam-6846	344	9	,	,	PUNCT
ejpam-6846	344	10	and	and	CCONJ
ejpam-6846	344	11	nadia	nadia	PROPN
ejpam-6846	344	12	zlateva	zlateva	PROPN
ejpam-6846	344	13	.	.	PUNCT
ejpam-6846	344	14	characterizations	characterization	NOUN
ejpam-6846	344	15	of	of	ADP
ejpam-6846	344	16	proxregular	proxregular	ADJ
ejpam-6846	344	17	sets	set	NOUN
ejpam-6846	344	18	in	in	ADP
ejpam-6846	344	19	uniformly	uniformly	ADV
ejpam-6846	344	20	convex	convex	NOUN
ejpam-6846	344	21	banach	banach	NOUN
ejpam-6846	344	22	spaces	space	NOUN
ejpam-6846	344	23	.	.	PUNCT
ejpam-6846	345	1	journal	journal	NOUN
ejpam-6846	345	2	of	of	ADP
ejpam-6846	345	3	convex	convex	PROPN
ejpam-6846	345	4	analysis	analysis	NOUN
ejpam-6846	345	5	,	,	PUNCT
ejpam-6846	345	6	13(3/4):525	13(3/4):525	NUM
ejpam-6846	345	7	,	,	PUNCT
ejpam-6846	345	8	2006	2006	NUM
ejpam-6846	345	9	,	,	PUNCT
ejpam-6846	345	10	.	.	PUNCT
ejpam-6846	346	1	[	[	X
ejpam-6846	346	2	16	16	NUM
ejpam-6846	346	3	]	]	X
ejpam-6846	346	4	francis	francis	PROPN
ejpam-6846	346	5	h	h	PROPN
ejpam-6846	346	6	clarke	clarke	PROPN
ejpam-6846	346	7	,	,	PUNCT
ejpam-6846	346	8	yuri	yuri	PROPN
ejpam-6846	346	9	s	s	PROPN
ejpam-6846	346	10	ledyaev	ledyaev	PROPN
ejpam-6846	346	11	,	,	PUNCT
ejpam-6846	346	12	ronald	ronald	PROPN
ejpam-6846	346	13	j	j	PROPN
ejpam-6846	346	14	stern	stern	PROPN
ejpam-6846	346	15	,	,	PUNCT
ejpam-6846	346	16	and	and	CCONJ
ejpam-6846	346	17	peter	peter	PROPN
ejpam-6846	346	18	r	r	PROPN
ejpam-6846	346	19	wolenski	wolenski	PROPN
ejpam-6846	346	20	.	.	PUNCT
ejpam-6846	347	1	nonsmooth	nonsmooth	NOUN
ejpam-6846	347	2	analysis	analysis	NOUN
ejpam-6846	347	3	and	and	CCONJ
ejpam-6846	347	4	control	control	NOUN
ejpam-6846	347	5	theory	theory	NOUN
ejpam-6846	347	6	,	,	PUNCT
ejpam-6846	347	7	volume	volume	NOUN
ejpam-6846	347	8	178	178	NUM
ejpam-6846	347	9	.	.	PUNCT
ejpam-6846	348	1	springer	springer	NOUN
ejpam-6846	348	2	science	science	PROPN
ejpam-6846	348	3	&	&	CCONJ
ejpam-6846	348	4	business	business	NOUN
ejpam-6846	348	5	media	medium	NOUN
ejpam-6846	348	6	,	,	PUNCT
ejpam-6846	348	7	2008	2008	NUM
ejpam-6846	348	8	.	.	PUNCT
ejpam-6846	349	1	[	[	X
ejpam-6846	349	2	17	17	NUM
ejpam-6846	349	3	]	]	X
ejpam-6846	349	4	jonathan	jonathan	PROPN
ejpam-6846	349	5	m	m	PROPN
ejpam-6846	349	6	borwein	borwein	NOUN
ejpam-6846	349	7	and	and	CCONJ
ejpam-6846	349	8	jr	jr	PROPN
ejpam-6846	349	9	giles	gile	NOUN
ejpam-6846	349	10	.	.	PUNCT
ejpam-6846	350	1	the	the	DET
ejpam-6846	350	2	proximal	proximal	ADJ
ejpam-6846	350	3	normal	normal	ADJ
ejpam-6846	350	4	formula	formula	NOUN
ejpam-6846	350	5	in	in	ADP
ejpam-6846	350	6	banach	banach	NOUN
ejpam-6846	350	7	space	space	NOUN
ejpam-6846	350	8	.	.	PUNCT
ejpam-6846	351	1	transactions	transaction	NOUN
ejpam-6846	351	2	of	of	ADP
ejpam-6846	351	3	the	the	DET
ejpam-6846	351	4	american	american	PROPN
ejpam-6846	351	5	mathematical	mathematical	PROPN
ejpam-6846	351	6	society	society	NOUN
ejpam-6846	351	7	,	,	PUNCT
ejpam-6846	351	8	302(1):371–381	302(1):371–381	NUM
ejpam-6846	351	9	,	,	PUNCT
ejpam-6846	351	10	1987	1987	NUM
ejpam-6846	351	11	,	,	PUNCT
ejpam-6846	351	12	https://doi.org/10.1090/s0002-9947-1987-0887515-5	https://doi.org/10.1090/s0002-9947-1987-0887515-5	X
ejpam-6846	351	13	.	.	PUNCT
ejpam-6846	352	1	[	[	X
ejpam-6846	352	2	18	18	NUM
ejpam-6846	352	3	]	]	PUNCT
ejpam-6846	352	4	boris	boris	PROPN
ejpam-6846	352	5	mordukhovich	mordukhovich	PROPN
ejpam-6846	352	6	and	and	CCONJ
ejpam-6846	352	7	yongheng	yongheng	PROPN
ejpam-6846	352	8	shao	shao	PROPN
ejpam-6846	352	9	.	.	PUNCT
ejpam-6846	353	1	nonsmooth	nonsmooth	PROPN
ejpam-6846	353	2	sequential	sequential	ADJ
ejpam-6846	353	3	analysis	analysis	NOUN
ejpam-6846	353	4	in	in	ADP
ejpam-6846	353	5	asplund	asplund	ADJ
ejpam-6846	353	6	spaces	space	NOUN
ejpam-6846	353	7	.	.	PUNCT
ejpam-6846	354	1	transactions	transaction	NOUN
ejpam-6846	354	2	of	of	ADP
ejpam-6846	354	3	the	the	DET
ejpam-6846	354	4	american	american	PROPN
ejpam-6846	354	5	mathematical	mathematical	PROPN
ejpam-6846	354	6	society	society	NOUN
ejpam-6846	354	7	,	,	PUNCT
ejpam-6846	354	8	348(4):1235–1280	348(4):1235–1280	NUM
ejpam-6846	354	9	,	,	PUNCT
ejpam-6846	354	10	1996	1996	NUM
ejpam-6846	354	11	,	,	PUNCT
ejpam-6846	354	12	https://doi.org/10.1090/s0002-9947-96-01543-7	https://doi.org/10.1090/s0002-9947-96-01543-7	X
ejpam-6846	354	13	.	.	PUNCT
ejpam-6846	355	1	[	[	X
ejpam-6846	355	2	19	19	NUM
ejpam-6846	355	3	]	]	PUNCT
ejpam-6846	355	4	ivar	ivar	NOUN
ejpam-6846	355	5	ekeland	ekeland	PROPN
ejpam-6846	355	6	and	and	CCONJ
ejpam-6846	355	7	gérard	gérard	PROPN
ejpam-6846	355	8	lebourg	lebourg	PROPN
ejpam-6846	355	9	.	.	PUNCT
ejpam-6846	356	1	generic	generic	ADJ
ejpam-6846	356	2	fréchet	fréchet	NOUN
ejpam-6846	356	3	-	-	PUNCT
ejpam-6846	356	4	differentiability	differentiability	NOUN
ejpam-6846	356	5	and	and	CCONJ
ejpam-6846	356	6	perturbed	perturb	VERB
ejpam-6846	356	7	optimization	optimization	NOUN
ejpam-6846	356	8	problems	problem	NOUN
ejpam-6846	356	9	in	in	ADP
ejpam-6846	356	10	banach	banach	NOUN
ejpam-6846	356	11	spaces	space	NOUN
ejpam-6846	356	12	.	.	PUNCT
ejpam-6846	357	1	transactions	transaction	NOUN
ejpam-6846	357	2	of	of	ADP
ejpam-6846	357	3	the	the	DET
ejpam-6846	357	4	american	american	PROPN
ejpam-6846	357	5	mathematical	mathematical	PROPN
ejpam-6846	357	6	society	society	NOUN
ejpam-6846	357	7	,	,	PUNCT
ejpam-6846	357	8	224(2):193–216	224(2):193–216	NUM
ejpam-6846	357	9	,	,	PUNCT
ejpam-6846	357	10	1976	1976	NUM
ejpam-6846	357	11	,	,	PUNCT
ejpam-6846	357	12	https://doi.org/10.1090/s0002-9947-1976-0431253-2	https://doi.org/10.1090/s0002-9947-1976-0431253-2	PROPN
ejpam-6846	357	13	.	.	PUNCT