id	sid	tid	token	lemma	pos
ma-361	1	1	2025	2025	NUM
ma-361	1	2	ada	ada	PROPN
ma-361	1	3	academica	academica	PROPN
ma-361	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-361	1	5	.	.	PUNCT
ma-361	2	1	j.	j.	PROPN
ma-361	2	2	math	math	PROPN
ma-361	2	3	.	.	PUNCT
ma-361	3	1	anal	anal	ADJ
ma-361	3	2	.	.	PUNCT
ma-361	4	1	5	5	NUM
ma-361	4	2	(	(	PUNCT
ma-361	4	3	2025	2025	NUM
ma-361	4	4	)	)	PUNCT
ma-361	4	5	17doi	17doi	NUM
ma-361	4	6	:	:	PUNCT
ma-361	4	7	10.28924	10.28924	NUM
ma-361	4	8	/	/	SYM
ma-361	4	9	ada	ada	PROPN
ma-361	4	10	/	/	SYM
ma-361	4	11	ma.5.17	ma.5.17	ADJ
ma-361	4	12	nonlinear	nonlinear	ADJ
ma-361	4	13	geometry	geometry	NOUN
ma-361	4	14	of	of	ADP
ma-361	4	15	norm	norm	NOUN
ma-361	4	16	-	-	PUNCT
ma-361	4	17	attaining	attain	VERB
ma-361	4	18	functionals	functional	NOUN
ma-361	4	19	:	:	PUNCT
ma-361	4	20	variational	variational	ADJ
ma-361	4	21	principles	principle	NOUN
ma-361	4	22	,	,	PUNCT
ma-361	4	23	subdifferential	subdifferential	ADJ
ma-361	4	24	calculus	calculus	NOUN
ma-361	4	25	,	,	PUNCT
ma-361	4	26	and	and	CCONJ
ma-361	4	27	polynomial	polynomial	ADJ
ma-361	4	28	optimization	optimization	NOUN
ma-361	4	29	in	in	ADP
ma-361	4	30	locally	locally	ADV
ma-361	4	31	convex	convex	NOUN
ma-361	4	32	spaces	space	NOUN
ma-361	4	33	mogoi	mogoi	NOUN
ma-361	4	34	n.	n.	PROPN
ma-361	4	35	evans1	evans1	PROPN
ma-361	4	36	,	,	PUNCT
ma-361	4	37	priscah	priscah	VERB
ma-361	4	38	moraa2,∗	moraa2,∗	PROPN
ma-361	4	39	1department	1department	NUM
ma-361	4	40	of	of	ADP
ma-361	4	41	pure	pure	ADJ
ma-361	4	42	and	and	CCONJ
ma-361	4	43	applied	applied	ADJ
ma-361	4	44	mathematics	mathematic	NOUN
ma-361	4	45	,	,	PUNCT
ma-361	4	46	jaramogi	jaramogi	PROPN
ma-361	4	47	oginga	oginga	PROPN
ma-361	4	48	odinga	odinga	PROPN
ma-361	4	49	university	university	PROPN
ma-361	4	50	of	of	ADP
ma-361	4	51	science	science	NOUN
ma-361	4	52	and	and	CCONJ
ma-361	4	53	technology	technology	NOUN
ma-361	4	54	,	,	PUNCT
ma-361	4	55	kenya	kenya	PROPN
ma-361	4	56	mogoievans4020@gmail.com	mogoievans4020@gmail.com	X
ma-361	5	1	2department	2department	NUM
ma-361	5	2	of	of	ADP
ma-361	5	3	mathematics	mathematic	NOUN
ma-361	5	4	and	and	CCONJ
ma-361	5	5	actuarial	actuarial	ADJ
ma-361	5	6	science	science	NOUN
ma-361	5	7	,	,	PUNCT
ma-361	5	8	kisii	kisii	PROPN
ma-361	5	9	university	university	PROPN
ma-361	5	10	,	,	PUNCT
ma-361	5	11	kenya	kenya	PROPN
ma-361	5	12	priscahmoraa@kisiiuniversity.ac.ke	priscahmoraa@kisiiuniversity.ac.ke	PROPN
ma-361	5	13	∗correspondence	∗correspondence	NOUN
ma-361	5	14	:	:	PUNCT
ma-361	5	15	priscahmoraa@kisiiuniversity.ac.ke	priscahmoraa@kisiiuniversity.ac.ke	NOUN
ma-361	5	16	abstract	abstract	ADJ
ma-361	5	17	.	.	PUNCT
ma-361	6	1	we	we	PRON
ma-361	6	2	develop	develop	VERB
ma-361	6	3	a	a	DET
ma-361	6	4	unified	unified	ADJ
ma-361	6	5	theory	theory	NOUN
ma-361	6	6	of	of	ADP
ma-361	6	7	norm	norm	NOUN
ma-361	6	8	-	-	PUNCT
ma-361	6	9	attainment	attainment	NOUN
ma-361	6	10	for	for	ADP
ma-361	6	11	nonlinear	nonlinear	ADJ
ma-361	6	12	functionals	functional	NOUN
ma-361	6	13	in	in	ADP
ma-361	6	14	locally	locally	ADV
ma-361	6	15	con	con	ADJ
ma-361	6	16	-	-	PUNCT
ma-361	6	17	vex	vex	NOUN
ma-361	6	18	spaces	space	NOUN
ma-361	6	19	,	,	PUNCT
ma-361	6	20	extending	extend	VERB
ma-361	6	21	classical	classical	ADJ
ma-361	6	22	results	result	NOUN
ma-361	6	23	to	to	PART
ma-361	6	24	sublinear	sublinear	VERB
ma-361	6	25	,	,	PUNCT
ma-361	6	26	quasiconvex	quasiconvex	ADJ
ma-361	6	27	,	,	PUNCT
ma-361	6	28	and	and	CCONJ
ma-361	6	29	polynomial	polynomial	ADJ
ma-361	6	30	settings	setting	NOUN
ma-361	6	31	.	.	PUNCT
ma-361	7	1	our	our	PRON
ma-361	7	2	maincontributions	maincontribution	NOUN
ma-361	7	3	include	include	VERB
ma-361	7	4	:	:	PUNCT
ma-361	7	5	(	(	PUNCT
ma-361	7	6	1	1	X
ma-361	7	7	)	)	PUNCT
ma-361	7	8	nonlinear	nonlinear	ADJ
ma-361	7	9	bishop	bishop	PROPN
ma-361	7	10	-	-	PUNCT
ma-361	7	11	phelps	phelps	PROPN
ma-361	7	12	theorems	theorem	NOUN
ma-361	7	13	establishing	establish	VERB
ma-361	7	14	density	density	NOUN
ma-361	7	15	of	of	ADP
ma-361	7	16	norm	norm	NOUN
ma-361	7	17	-	-	PUNCT
ma-361	7	18	attainingfunctionals	attainingfunctional	NOUN
ma-361	7	19	,	,	PUNCT
ma-361	7	20	(	(	PUNCT
ma-361	7	21	2	2	X
ma-361	7	22	)	)	PUNCT
ma-361	7	23	a	a	DET
ma-361	7	24	subdifferential	subdifferential	ADJ
ma-361	7	25	characterization	characterization	NOUN
ma-361	7	26	of	of	ADP
ma-361	7	27	attainment	attainment	NOUN
ma-361	7	28	via	via	ADP
ma-361	7	29	interiority	interiority	NOUN
ma-361	7	30	conditions	condition	NOUN
ma-361	7	31	,	,	PUNCT
ma-361	7	32	(	(	PUNCT
ma-361	7	33	3	3	X
ma-361	7	34	)	)	PUNCT
ma-361	7	35	a	a	DET
ma-361	7	36	krein	krein	NOUN
ma-361	7	37	-	-	PUNCT
ma-361	7	38	milman	milman	NOUN
ma-361	7	39	principle	principle	NOUN
ma-361	7	40	for	for	ADP
ma-361	7	41	convex	convex	ADJ
ma-361	7	42	functionals	functional	NOUN
ma-361	7	43	on	on	ADP
ma-361	7	44	compact	compact	ADJ
ma-361	7	45	sets	set	NOUN
ma-361	7	46	,	,	PUNCT
ma-361	7	47	and	and	CCONJ
ma-361	7	48	(	(	PUNCT
ma-361	7	49	4	4	X
ma-361	7	50	)	)	PUNCT
ma-361	7	51	a	a	DET
ma-361	7	52	complete	complete	ADJ
ma-361	7	53	solution	solution	NOUN
ma-361	7	54	to	to	ADP
ma-361	7	55	the	the	DET
ma-361	7	56	poly	poly	ADJ
ma-361	7	57	-	-	PUNCT
ma-361	7	58	nomial	nomial	ADJ
ma-361	7	59	norm	norm	NOUN
ma-361	7	60	-	-	PUNCT
ma-361	7	61	attainment	attainment	ADJ
ma-361	7	62	problem	problem	NOUN
ma-361	7	63	through	through	ADP
ma-361	7	64	tensor	tensor	NOUN
ma-361	7	65	product	product	NOUN
ma-361	7	66	geometry	geometry	NOUN
ma-361	7	67	.	.	PUNCT
ma-361	8	1	the	the	DET
ma-361	8	2	work	work	NOUN
ma-361	8	3	combines	combine	VERB
ma-361	8	4	innovativeapplications	innovativeapplication	NOUN
ma-361	8	5	of	of	ADP
ma-361	8	6	choquet	choquet	NOUN
ma-361	8	7	theory	theory	NOUN
ma-361	8	8	,	,	PUNCT
ma-361	8	9	variational	variational	ADJ
ma-361	8	10	analysis	analysis	NOUN
ma-361	8	11	,	,	PUNCT
ma-361	8	12	and	and	CCONJ
ma-361	8	13	complex	complex	ADJ
ma-361	8	14	-	-	PUNCT
ma-361	8	15	geometric	geometric	ADJ
ma-361	8	16	methods	method	NOUN
ma-361	8	17	to	to	PART
ma-361	8	18	reveal	reveal	VERB
ma-361	8	19	newconnections	newconnection	NOUN
ma-361	8	20	between	between	ADP
ma-361	8	21	functional	functional	ADJ
ma-361	8	22	analysis	analysis	NOUN
ma-361	8	23	and	and	CCONJ
ma-361	8	24	optimization	optimization	NOUN
ma-361	8	25	.	.	PUNCT
ma-361	9	1	key	key	ADJ
ma-361	9	2	applications	application	NOUN
ma-361	9	3	address	address	VERB
ma-361	9	4	stochastic	stochastic	ADJ
ma-361	9	5	vari	vari	ADJ
ma-361	9	6	-	-	ADJ
ma-361	9	7	ational	ational	ADJ
ma-361	9	8	principles	principle	NOUN
ma-361	9	9	and	and	CCONJ
ma-361	9	10	reproducing	reproduce	VERB
ma-361	9	11	kernel	kernel	PROPN
ma-361	9	12	hilbert	hilbert	PROPN
ma-361	9	13	space	space	NOUN
ma-361	9	14	optimization	optimization	NOUN
ma-361	9	15	,	,	PUNCT
ma-361	9	16	with	with	ADP
ma-361	9	17	tools	tool	NOUN
ma-361	9	18	applicable	applicable	ADJ
ma-361	9	19	to	to	ADP
ma-361	9	20	pdeconstraints	pdeconstraint	NOUN
ma-361	9	21	and	and	CCONJ
ma-361	9	22	high	high	ADV
ma-361	9	23	-	-	PUNCT
ma-361	9	24	dimensional	dimensional	ADJ
ma-361	9	25	data	datum	NOUN
ma-361	9	26	science	science	NOUN
ma-361	9	27	.	.	PUNCT
ma-361	10	1	these	these	DET
ma-361	10	2	results	result	NOUN
ma-361	10	3	collectively	collectively	ADV
ma-361	10	4	bridge	bridge	VERB
ma-361	10	5	fundamental	fundamental	ADJ
ma-361	10	6	gapsbetween	gapsbetween	ADJ
ma-361	10	7	linear	linear	NOUN
ma-361	10	8	and	and	CCONJ
ma-361	10	9	nonlinear	nonlinear	ADJ
ma-361	10	10	functional	functional	ADJ
ma-361	10	11	analysis	analysis	NOUN
ma-361	10	12	while	while	SCONJ
ma-361	10	13	providing	provide	VERB
ma-361	10	14	fresh	fresh	ADJ
ma-361	10	15	geometric	geometric	ADJ
ma-361	10	16	insight	insight	NOUN
ma-361	10	17	into	into	ADP
ma-361	10	18	infinite	infinite	ADJ
ma-361	10	19	-	-	PUNCT
ma-361	10	20	dimensional	dimensional	ADJ
ma-361	10	21	phenomena	phenomenon	NOUN
ma-361	10	22	.	.	PUNCT
ma-361	11	1	introduction	introduction	NOUN
ma-361	11	2	the	the	DET
ma-361	11	3	study	study	NOUN
ma-361	11	4	of	of	ADP
ma-361	11	5	norm	norm	NOUN
ma-361	11	6	-	-	PUNCT
ma-361	11	7	attaining	attain	VERB
ma-361	11	8	functionals	functional	NOUN
ma-361	11	9	originated	originate	VERB
ma-361	11	10	with	with	ADP
ma-361	11	11	the	the	DET
ma-361	11	12	seminal	seminal	ADJ
ma-361	11	13	bishop	bishop	PROPN
ma-361	11	14	-	-	PUNCT
ma-361	11	15	phelps	phelps	PROPN
ma-361	11	16	theorem	theorem	VERB
ma-361	11	17	[	[	X
ma-361	11	18	13],which	13],which	PROPN
ma-361	11	19	established	establish	VERB
ma-361	11	20	the	the	DET
ma-361	11	21	density	density	NOUN
ma-361	11	22	of	of	ADP
ma-361	11	23	norm	norm	NOUN
ma-361	11	24	-	-	PUNCT
ma-361	11	25	attaining	attain	VERB
ma-361	11	26	linear	linear	ADJ
ma-361	11	27	functionals	functional	NOUN
ma-361	11	28	in	in	ADP
ma-361	11	29	banach	banach	NOUN
ma-361	11	30	spaces	space	NOUN
ma-361	11	31	.	.	PUNCT
ma-361	12	1	while	while	SCONJ
ma-361	12	2	thisresult	thisresult	NOUN
ma-361	12	3	has	have	AUX
ma-361	12	4	been	be	AUX
ma-361	12	5	extended	extend	VERB
ma-361	12	6	in	in	ADP
ma-361	12	7	various	various	ADJ
ma-361	12	8	directions	direction	NOUN
ma-361	12	9	[	[	X
ma-361	12	10	10	10	NUM
ma-361	12	11	,	,	PUNCT
ma-361	12	12	11	11	NUM
ma-361	12	13	]	]	PUNCT
ma-361	12	14	,	,	PUNCT
ma-361	12	15	existing	exist	VERB
ma-361	12	16	theories	theory	NOUN
ma-361	12	17	remain	remain	VERB
ma-361	12	18	constrained	constrained	ADJ
ma-361	12	19	bybanach	bybanach	NOUN
ma-361	12	20	space	space	NOUN
ma-361	12	21	limitations	limitation	NOUN
ma-361	12	22	and	and	CCONJ
ma-361	12	23	lack	lack	VERB
ma-361	12	24	comprehensive	comprehensive	ADJ
ma-361	12	25	frameworks	framework	NOUN
ma-361	12	26	for	for	ADP
ma-361	12	27	nonlinear	nonlinear	ADJ
ma-361	12	28	functionals	functional	NOUN
ma-361	12	29	.	.	PUNCT
ma-361	13	1	our	our	PRON
ma-361	13	2	workovercomes	workovercome	VERB
ma-361	13	3	these	these	DET
ma-361	13	4	limitations	limitation	NOUN
ma-361	13	5	by	by	ADP
ma-361	13	6	developing	develop	VERB
ma-361	13	7	a	a	DET
ma-361	13	8	unified	unified	ADJ
ma-361	13	9	theory	theory	NOUN
ma-361	13	10	in	in	ADP
ma-361	13	11	locally	locally	ADV
ma-361	13	12	convex	convex	ADJ
ma-361	13	13	spaces	space	NOUN
ma-361	13	14	,	,	PUNCT
ma-361	13	15	combininginnovative	combininginnovative	ADJ
ma-361	13	16	tools	tool	NOUN
ma-361	13	17	from	from	ADP
ma-361	13	18	variational	variational	ADJ
ma-361	13	19	analysis	analysis	NOUN
ma-361	13	20	[	[	X
ma-361	13	21	5	5	NUM
ma-361	13	22	]	]	PUNCT
ma-361	13	23	,	,	PUNCT
ma-361	13	24	convex	convex	ADJ
ma-361	13	25	geometry	geometry	NOUN
ma-361	13	26	[	[	X
ma-361	13	27	13	13	NUM
ma-361	13	28	]	]	PUNCT
ma-361	13	29	,	,	PUNCT
ma-361	13	30	and	and	CCONJ
ma-361	13	31	polynomial	polynomial	ADJ
ma-361	13	32	functional	functional	ADJ
ma-361	13	33	anal	anal	NOUN
ma-361	13	34	-	-	PUNCT
ma-361	13	35	ysis	ysis	NOUN
ma-361	13	36	[	[	X
ma-361	13	37	8	8	NUM
ma-361	13	38	]	]	PUNCT
ma-361	13	39	.	.	PUNCT
ma-361	14	1	the	the	DET
ma-361	14	2	classical	classical	ADJ
ma-361	14	3	bishop	bishop	PROPN
ma-361	14	4	-	-	PUNCT
ma-361	14	5	phelps	phelps	PROPN
ma-361	14	6	theorem	theorem	NOUN
ma-361	14	7	has	have	AUX
ma-361	14	8	seen	see	VERB
ma-361	14	9	partial	partial	ADJ
ma-361	14	10	extensions	extension	NOUN
ma-361	14	11	to	to	ADP
ma-361	14	12	nonlinear	nonlinear	ADJ
ma-361	14	13	settings	setting	NOUN
ma-361	14	14	,	,	PUNCT
ma-361	14	15	received	receive	VERB
ma-361	14	16	:	:	PUNCT
ma-361	14	17	16	16	NUM
ma-361	14	18	apr	apr	NOUN
ma-361	14	19	2025	2025	NUM
ma-361	14	20	.	.	PUNCT
ma-361	15	1	key	key	ADJ
ma-361	15	2	words	word	NOUN
ma-361	15	3	and	and	CCONJ
ma-361	15	4	phrases	phrase	NOUN
ma-361	15	5	.	.	PUNCT
ma-361	16	1	norm	norm	NOUN
ma-361	16	2	-	-	PUNCT
ma-361	16	3	attaining	attain	VERB
ma-361	16	4	functionals	functional	NOUN
ma-361	16	5	;	;	PUNCT
ma-361	16	6	nonlinear	nonlinear	ADJ
ma-361	16	7	functional	functional	ADJ
ma-361	16	8	analysis	analysis	NOUN
ma-361	16	9	;	;	PUNCT
ma-361	16	10	locally	locally	ADV
ma-361	16	11	convex	convex	NOUN
ma-361	16	12	spaces	space	NOUN
ma-361	16	13	;	;	PUNCT
ma-361	16	14	variationalprinciples	variationalprinciple	NOUN
ma-361	16	15	;	;	PUNCT
ma-361	16	16	subdifferential	subdifferential	ADJ
ma-361	16	17	geometry	geometry	NOUN
ma-361	16	18	;	;	PUNCT
ma-361	16	19	polynomial	polynomial	ADJ
ma-361	16	20	optimization	optimization	NOUN
ma-361	16	21	;	;	PUNCT
ma-361	16	22	bishop	bishop	PROPN
ma-361	16	23	-	-	PUNCT
ma-361	16	24	phelps	phelps	PROPN
ma-361	16	25	theorem	theorem	VERB
ma-361	16	26	;	;	PUNCT
ma-361	16	27	infinite	infinite	ADJ
ma-361	16	28	-	-	PUNCT
ma-361	16	29	dimensional	dimensional	ADJ
ma-361	16	30	convexity.1	convexity.1	PROPN
ma-361	16	31	https://adac.ee	https://adac.ee	PROPN
ma-361	16	32	https://doi.org/10.28924/ada/ma.5.17	https://doi.org/10.28924/ada/ma.5.17	PROPN
ma-361	16	33	eur	eur	PROPN
ma-361	16	34	.	.	PUNCT
ma-361	17	1	j.	j.	PROPN
ma-361	17	2	math	math	PROPN
ma-361	17	3	.	.	PUNCT
ma-361	18	1	anal	anal	PROPN
ma-361	18	2	.	.	PUNCT
ma-361	19	1	10.28924	10.28924	NUM
ma-361	19	2	/	/	SYM
ma-361	19	3	ada	ada	PROPN
ma-361	19	4	/	/	SYM
ma-361	19	5	ma.5.17	ma.5.17	X
ma-361	19	6	2including	2including	NUM
ma-361	19	7	sublinear	sublinear	NOUN
ma-361	19	8	functionals	functional	NOUN
ma-361	19	9	[	[	X
ma-361	19	10	3	3	NUM
ma-361	19	11	]	]	PUNCT
ma-361	19	12	and	and	CCONJ
ma-361	19	13	quasiconvex	quasiconvex	NOUN
ma-361	19	14	cases	case	NOUN
ma-361	19	15	[	[	X
ma-361	19	16	6	6	NUM
ma-361	19	17	]	]	PUNCT
ma-361	19	18	,	,	PUNCT
ma-361	19	19	but	but	CCONJ
ma-361	19	20	these	these	DET
ma-361	19	21	advances	advance	NOUN
ma-361	19	22	have	have	AUX
ma-361	19	23	been	be	AUX
ma-361	19	24	re	re	VERB
ma-361	19	25	-	-	VERB
ma-361	19	26	stricted	stricted	ADJ
ma-361	19	27	to	to	ADP
ma-361	19	28	either	either	CCONJ
ma-361	19	29	banach	banach	NOUN
ma-361	19	30	spaces	space	NOUN
ma-361	19	31	or	or	CCONJ
ma-361	19	32	specific	specific	ADJ
ma-361	19	33	functional	functional	ADJ
ma-361	19	34	classes	class	NOUN
ma-361	19	35	.	.	PUNCT
ma-361	20	1	our	our	PRON
ma-361	20	2	theorem	theorem	ADJ
ma-361	20	3	1	1	NUM
ma-361	20	4	breaks	break	NOUN
ma-361	20	5	new	new	ADJ
ma-361	20	6	groundby	groundby	NOUN
ma-361	20	7	characterizing	characterize	VERB
ma-361	20	8	the	the	DET
ma-361	20	9	nonlinear	nonlinear	ADJ
ma-361	20	10	bishop	bishop	NOUN
ma-361	20	11	-	-	PUNCT
ma-361	20	12	phelps	phelps	PROPN
ma-361	20	13	property	property	NOUN
ma-361	20	14	in	in	ADP
ma-361	20	15	general	general	ADJ
ma-361	20	16	locally	locally	ADV
ma-361	20	17	convex	convex	ADJ
ma-361	20	18	spaces	space	NOUN
ma-361	20	19	,	,	PUNCT
ma-361	20	20	revealingan	revealingan	VERB
ma-361	20	21	essential	essential	ADJ
ma-361	20	22	connection	connection	NOUN
ma-361	20	23	with	with	ADP
ma-361	20	24	lattice	lattice	NOUN
ma-361	20	25	norms	norm	NOUN
ma-361	20	26	and	and	CCONJ
ma-361	20	27	τ	τ	NOUN
ma-361	20	28	-	-	PUNCT
ma-361	20	29	lower	low	ADJ
ma-361	20	30	semicontinuity	semicontinuity	NOUN
ma-361	20	31	that	that	PRON
ma-361	20	32	extends	extend	VERB
ma-361	20	33	both	both	CCONJ
ma-361	20	34	the	the	DET
ma-361	20	35	orig	orig	NOUN
ma-361	20	36	-	-	PUNCT
ma-361	20	37	inal	inal	ADJ
ma-361	20	38	results	result	NOUN
ma-361	20	39	and	and	CCONJ
ma-361	20	40	their	their	PRON
ma-361	20	41	convex	convex	ADJ
ma-361	20	42	generalizations	generalization	NOUN
ma-361	20	43	[	[	X
ma-361	20	44	7	7	NUM
ma-361	20	45	]	]	PUNCT
ma-361	20	46	.	.	PUNCT
ma-361	21	1	the	the	DET
ma-361	21	2	proof	proof	NOUN
ma-361	21	3	introduces	introduce	VERB
ma-361	21	4	novel	novel	ADJ
ma-361	21	5	variational	variational	NOUN
ma-361	21	6	techniquesinspired	techniquesinspire	VERB
ma-361	21	7	by	by	ADP
ma-361	21	8	the	the	DET
ma-361	21	9	borwein	borwein	ADJ
ma-361	21	10	-	-	PUNCT
ma-361	21	11	preiss	preiss	NOUN
ma-361	21	12	principle	principle	NOUN
ma-361	21	13	[	[	X
ma-361	21	14	5	5	NUM
ma-361	21	15	]	]	PUNCT
ma-361	21	16	.	.	PUNCT
ma-361	22	1	in	in	ADP
ma-361	22	2	subdifferential	subdifferential	ADJ
ma-361	22	3	geometry	geometry	NOUN
ma-361	22	4	,	,	PUNCT
ma-361	22	5	theorem	theorem	VERB
ma-361	22	6	2	2	NUM
ma-361	22	7	establishes	establishe	NOUN
ma-361	22	8	thefirst	thefirst	VERB
ma-361	22	9	complete	complete	ADJ
ma-361	22	10	characterization	characterization	NOUN
ma-361	22	11	of	of	ADP
ma-361	22	12	norm	norm	NOUN
ma-361	22	13	-	-	PUNCT
ma-361	22	14	attainment	attainment	NOUN
ma-361	22	15	for	for	ADP
ma-361	22	16	sublinear	sublinear	NOUN
ma-361	22	17	functionals	functional	NOUN
ma-361	22	18	through	through	ADP
ma-361	22	19	weak∗-exposedpoints	weak∗-exposedpoint	NOUN
ma-361	22	20	and	and	CCONJ
ma-361	22	21	subdifferential	subdifferential	ADJ
ma-361	22	22	monotonicity	monotonicity	NOUN
ma-361	22	23	.	.	PUNCT
ma-361	23	1	this	this	DET
ma-361	23	2	bridges	bridge	NOUN
ma-361	23	3	classical	classical	ADJ
ma-361	23	4	subdifferential	subdifferential	ADJ
ma-361	23	5	calculus	calculus	NOUN
ma-361	23	6	[	[	X
ma-361	23	7	14	14	NUM
ma-361	23	8	]	]	PUNCT
ma-361	23	9	withmodern	withmodern	NOUN
ma-361	23	10	theories	theory	NOUN
ma-361	23	11	of	of	ADP
ma-361	23	12	barrelled	barrelled	ADJ
ma-361	23	13	spaces	space	NOUN
ma-361	23	14	[	[	X
ma-361	23	15	9	9	NUM
ma-361	23	16	]	]	PUNCT
ma-361	23	17	,	,	PUNCT
ma-361	23	18	while	while	SCONJ
ma-361	23	19	theorem	theorem	VERB
ma-361	23	20	8	8	NUM
ma-361	23	21	extends	extend	VERB
ma-361	23	22	the	the	DET
ma-361	23	23	brondsted	brondste	VERB
ma-361	23	24	-	-	PUNCT
ma-361	23	25	rockafellar	rockafellar	ADJ
ma-361	23	26	theo	theo	PROPN
ma-361	23	27	-	-	PUNCT
ma-361	23	28	rem	rem	VERB
ma-361	23	29	with	with	ADP
ma-361	23	30	new	new	ADJ
ma-361	23	31	interiority	interiority	NOUN
ma-361	23	32	conditions	condition	NOUN
ma-361	23	33	for	for	ADP
ma-361	23	34	norm	norm	NOUN
ma-361	23	35	-	-	PUNCT
ma-361	23	36	attainment	attainment	NOUN
ma-361	23	37	.	.	PUNCT
ma-361	24	1	the	the	DET
ma-361	24	2	quasilinear	quasilinear	PROPN
ma-361	24	3	separation	separation	NOUN
ma-361	24	4	in	in	ADP
ma-361	24	5	theorem	theorem	NOUN
ma-361	24	6	4generalizes	4generalizes	NUM
ma-361	24	7	the	the	DET
ma-361	24	8	hahn	hahn	NOUN
ma-361	24	9	-	-	PUNCT
ma-361	24	10	banach	banach	NOUN
ma-361	24	11	theorem	theorem	ADJ
ma-361	24	12	while	while	SCONJ
ma-361	24	13	preserving	preserve	VERB
ma-361	24	14	norm	norm	NOUN
ma-361	24	15	-	-	PUNCT
ma-361	24	16	attainment	attainment	NOUN
ma-361	24	17	,	,	PUNCT
ma-361	24	18	with	with	ADP
ma-361	24	19	immediate	immediate	ADJ
ma-361	24	20	applica	applica	PROPN
ma-361	24	21	-	-	PUNCT
ma-361	24	22	tions	tion	NOUN
ma-361	24	23	to	to	ADP
ma-361	24	24	game	game	NOUN
ma-361	24	25	theory	theory	NOUN
ma-361	24	26	and	and	CCONJ
ma-361	24	27	economic	economic	ADJ
ma-361	24	28	equilibrium	equilibrium	NOUN
ma-361	24	29	[	[	X
ma-361	24	30	4	4	NUM
ma-361	24	31	]	]	PUNCT
ma-361	24	32	.	.	PUNCT
ma-361	25	1	for	for	ADP
ma-361	25	2	polynomial	polynomial	ADJ
ma-361	25	3	functionals	functional	NOUN
ma-361	25	4	,	,	PUNCT
ma-361	25	5	theorem	theorem	VERB
ma-361	25	6	10	10	NUM
ma-361	25	7	solvesthe	solvesthe	DET
ma-361	25	8	long	long	ADV
ma-361	25	9	-	-	PUNCT
ma-361	25	10	standing	stand	VERB
ma-361	25	11	attainment	attainment	ADJ
ma-361	25	12	problem	problem	NOUN
ma-361	25	13	through	through	ADP
ma-361	25	14	projective	projective	ADJ
ma-361	25	15	tensor	tensor	NOUN
ma-361	25	16	products	product	NOUN
ma-361	25	17	and	and	CCONJ
ma-361	25	18	radon	radon	PROPN
ma-361	25	19	-	-	PUNCT
ma-361	25	20	nikodym	nikodym	NOUN
ma-361	25	21	prop	prop	NOUN
ma-361	25	22	-	-	PUNCT
ma-361	25	23	erties	ertie	NOUN
ma-361	25	24	[	[	X
ma-361	25	25	1	1	NUM
ma-361	25	26	,	,	PUNCT
ma-361	25	27	8	8	NUM
ma-361	25	28	]	]	PUNCT
ma-361	25	29	.	.	PUNCT
ma-361	26	1	this	this	DET
ma-361	26	2	complements	complement	NOUN
ma-361	26	3	theorem	theorem	VERB
ma-361	26	4	6	6	NUM
ma-361	26	5	’s	’s	PART
ma-361	26	6	surprising	surprising	ADJ
ma-361	26	7	link	link	NOUN
ma-361	26	8	between	between	ADP
ma-361	26	9	operator	operator	NOUN
ma-361	26	10	norm	norm	NOUN
ma-361	26	11	-	-	PUNCT
ma-361	26	12	attainment	attainment	ADJ
ma-361	26	13	andplurisubharmonic	andplurisubharmonic	NOUN
ma-361	26	14	norms	norm	NOUN
ma-361	26	15	,	,	PUNCT
ma-361	26	16	combining	combine	VERB
ma-361	26	17	operator	operator	NOUN
ma-361	26	18	theory	theory	NOUN
ma-361	26	19	with	with	ADP
ma-361	26	20	complex	complex	ADJ
ma-361	26	21	analysis	analysis	NOUN
ma-361	26	22	[	[	X
ma-361	26	23	12	12	NUM
ma-361	26	24	]	]	PUNCT
ma-361	26	25	.	.	PUNCT
ma-361	27	1	the	the	DET
ma-361	27	2	nonlinearkrein	nonlinearkrein	ADJ
ma-361	27	3	-	-	PUNCT
ma-361	27	4	milman	milman	NOUN
ma-361	27	5	theorem	theorem	NOUN
ma-361	27	6	(	(	PUNCT
ma-361	27	7	theorem	theorem	NOUN
ma-361	27	8	9	9	NUM
ma-361	27	9	)	)	PUNCT
ma-361	27	10	and	and	CCONJ
ma-361	27	11	james	james	PROPN
ma-361	27	12	-	-	PUNCT
ma-361	27	13	type	type	NOUN
ma-361	27	14	characterization	characterization	NOUN
ma-361	27	15	(	(	PUNCT
ma-361	27	16	theorem	theorem	VERB
ma-361	27	17	7	7	NUM
ma-361	27	18	)	)	PUNCT
ma-361	27	19	complete	complete	ADJ
ma-361	27	20	thepicture	thepicture	NOUN
ma-361	27	21	,	,	PUNCT
ma-361	27	22	employing	employ	VERB
ma-361	27	23	choquet	choquet	NOUN
ma-361	27	24	theory	theory	NOUN
ma-361	27	25	[	[	X
ma-361	27	26	13	13	NUM
ma-361	27	27	]	]	PUNCT
ma-361	27	28	and	and	CCONJ
ma-361	27	29	geometric	geometric	ADJ
ma-361	27	30	methods	method	NOUN
ma-361	27	31	[	[	X
ma-361	27	32	2	2	X
ma-361	27	33	]	]	PUNCT
ma-361	27	34	to	to	PART
ma-361	27	35	extend	extend	VERB
ma-361	27	36	fundamental	fundamental	ADJ
ma-361	27	37	results	result	NOUN
ma-361	27	38	togeneral	togeneral	ADJ
ma-361	27	39	locally	locally	ADV
ma-361	27	40	convex	convex	NOUN
ma-361	27	41	spaces	space	NOUN
ma-361	27	42	.	.	PUNCT
ma-361	28	1	collectively	collectively	ADV
ma-361	28	2	,	,	PUNCT
ma-361	28	3	these	these	DET
ma-361	28	4	advances	advance	NOUN
ma-361	28	5	bridge	bridge	NOUN
ma-361	28	6	critical	critical	ADJ
ma-361	28	7	gaps	gap	NOUN
ma-361	28	8	between	between	ADP
ma-361	28	9	linear	linear	PROPN
ma-361	28	10	andnonlinear	andnonlinear	ADJ
ma-361	28	11	functional	functional	ADJ
ma-361	28	12	analysis	analysis	NOUN
ma-361	28	13	while	while	SCONJ
ma-361	28	14	providing	provide	VERB
ma-361	28	15	powerful	powerful	ADJ
ma-361	28	16	new	new	ADJ
ma-361	28	17	tools	tool	NOUN
ma-361	28	18	for	for	ADP
ma-361	28	19	optimization	optimization	NOUN
ma-361	28	20	,	,	PUNCT
ma-361	28	21	stochastic	stochastic	ADJ
ma-361	28	22	pdes	pde	NOUN
ma-361	28	23	,	,	PUNCT
ma-361	28	24	and	and	CCONJ
ma-361	28	25	high	high	ADJ
ma-361	28	26	-	-	PUNCT
ma-361	28	27	dimensional	dimensional	ADJ
ma-361	28	28	statistics	statistic	NOUN
ma-361	28	29	.	.	PUNCT
ma-361	29	1	the	the	DET
ma-361	29	2	synthesis	synthesis	NOUN
ma-361	29	3	of	of	ADP
ma-361	29	4	variational	variational	ADJ
ma-361	29	5	principles	principle	NOUN
ma-361	29	6	,	,	PUNCT
ma-361	29	7	geometric	geometric	ADJ
ma-361	29	8	methods	method	NOUN
ma-361	29	9	,	,	PUNCT
ma-361	29	10	andcomplex	andcomplex	ADJ
ma-361	29	11	-	-	PUNCT
ma-361	29	12	analytic	analytic	ADJ
ma-361	29	13	techniques	technique	NOUN
ma-361	29	14	reveals	reveal	VERB
ma-361	29	15	previously	previously	ADV
ma-361	29	16	unrecognized	unrecognize	VERB
ma-361	29	17	connections	connection	NOUN
ma-361	29	18	across	across	ADP
ma-361	29	19	these	these	DET
ma-361	29	20	domains	domain	NOUN
ma-361	29	21	.	.	PUNCT
ma-361	30	1	notation	notation	NOUN
ma-361	30	2	x	x	PUNCT
ma-361	30	3	locally	locally	ADV
ma-361	30	4	convex	convex	ADJ
ma-361	30	5	space	space	NOUN
ma-361	30	6	⊗̂nπ	⊗̂nπ	PROPN
ma-361	30	7	projective	projective	ADJ
ma-361	30	8	tensor	tensor	NOUN
ma-361	30	9	product	product	NOUN
ma-361	30	10	∂p(x	∂p(x	PROPN
ma-361	30	11	)	)	PUNCT
ma-361	30	12	subdifferential	subdifferential	NOUN
ma-361	30	13	at	at	ADP
ma-361	30	14	x	x	NOUN
ma-361	30	15	preliminaries	preliminary	NOUN
ma-361	30	16	throughout	throughout	ADP
ma-361	30	17	this	this	DET
ma-361	30	18	work	work	NOUN
ma-361	30	19	,	,	PUNCT
ma-361	30	20	we	we	PRON
ma-361	30	21	consider	consider	VERB
ma-361	30	22	x	x	PART
ma-361	30	23	to	to	PART
ma-361	30	24	be	be	AUX
ma-361	30	25	a	a	DET
ma-361	30	26	hausdorff	hausdorff	NOUN
ma-361	30	27	locally	locally	ADV
ma-361	30	28	convex	convex	ADJ
ma-361	30	29	space	space	NOUN
ma-361	30	30	(	(	PUNCT
ma-361	30	31	lcs	lcs	NOUN
ma-361	30	32	)	)	PUNCT
ma-361	30	33	over	over	ADP
ma-361	30	34	r	r	NOUN
ma-361	30	35	or	or	CCONJ
ma-361	30	36	c	c	NOUN
ma-361	30	37	,	,	PUNCT
ma-361	30	38	with	with	ADP
ma-361	30	39	topology	topology	NOUN
ma-361	30	40	τ	τ	PROPN
ma-361	30	41	generated	generate	VERB
ma-361	30	42	by	by	ADP
ma-361	30	43	a	a	DET
ma-361	30	44	separating	separate	VERB
ma-361	30	45	family	family	NOUN
ma-361	30	46	of	of	ADP
ma-361	30	47	seminorms	seminorm	NOUN
ma-361	30	48	{	{	PUNCT
ma-361	30	49	ρα}α∈i	ρα}α∈i	X
ma-361	30	50	.	.	PUNCT
ma-361	31	1	we	we	PRON
ma-361	31	2	denote	denote	VERB
ma-361	31	3	by	by	ADP
ma-361	31	4	x∗	x∗	PROPN
ma-361	31	5	thetopological	thetopological	ADJ
ma-361	31	6	dual	dual	ADJ
ma-361	31	7	space	space	NOUN
ma-361	31	8	,	,	PUNCT
ma-361	31	9	equipped	equip	VERB
ma-361	31	10	with	with	ADP
ma-361	31	11	the	the	DET
ma-361	31	12	weak-∗	weak-∗	PROPN
ma-361	31	13	topology	topology	NOUN
ma-361	31	14	σ(x∗	σ(x∗	NOUN
ma-361	31	15	,	,	PUNCT
ma-361	31	16	x	x	NOUN
ma-361	31	17	)	)	PUNCT
ma-361	31	18	.	.	PUNCT
ma-361	32	1	fundamental	fundamental	ADJ
ma-361	32	2	references	reference	NOUN
ma-361	32	3	forthese	forthese	VERB
ma-361	32	4	concepts	concept	NOUN
ma-361	32	5	include	include	VERB
ma-361	32	6	[	[	X
ma-361	32	7	9	9	NUM
ma-361	32	8	]	]	PUNCT
ma-361	32	9	and	and	CCONJ
ma-361	32	10	[	[	X
ma-361	32	11	4	4	NUM
ma-361	32	12	]	]	PUNCT
ma-361	32	13	.	.	PUNCT
ma-361	33	1	functional	functional	ADJ
ma-361	33	2	analytic	analytic	ADJ
ma-361	33	3	foundations	foundation	NOUN
ma-361	33	4	.	.	PUNCT
ma-361	34	1	definition	definition	NOUN
ma-361	34	2	1	1	NUM
ma-361	34	3	(	(	PUNCT
ma-361	34	4	continuous	continuous	ADJ
ma-361	34	5	sublinear	sublinear	NOUN
ma-361	34	6	functionals	functional	NOUN
ma-361	34	7	)	)	PUNCT
ma-361	34	8	.	.	PUNCT
ma-361	35	1	a	a	DET
ma-361	35	2	functional	functional	ADJ
ma-361	35	3	p	p	X
ma-361	35	4	:	:	PUNCT
ma-361	35	5	x	x	X
ma-361	35	6	→	→	SYM
ma-361	35	7	r	r	NOUN
ma-361	35	8	is	be	AUX
ma-361	35	9	called	call	VERB
ma-361	35	10	sublinear	sublinear	NOUN
ma-361	35	11	if	if	SCONJ
ma-361	35	12	:	:	PUNCT
ma-361	35	13	•	•	NUM
ma-361	35	14	p(λx	p(λx	NOUN
ma-361	35	15	)	)	PUNCT
ma-361	35	16	=	=	PUNCT
ma-361	35	17	λp(x	λp(x	X
ma-361	35	18	)	)	PUNCT
ma-361	35	19	for	for	ADP
ma-361	35	20	all	all	DET
ma-361	35	21	λ	λ	PROPN
ma-361	35	22	≥	≥	X
ma-361	35	23	0	0	NUM
ma-361	35	24	(	(	PUNCT
ma-361	35	25	positive	positive	ADJ
ma-361	35	26	homogeneity	homogeneity	NOUN
ma-361	35	27	)	)	PUNCT
ma-361	35	28	•	•	NUM
ma-361	35	29	p(x	p(x	PROPN
ma-361	35	30	+	+	CCONJ
ma-361	35	31	y	y	NOUN
ma-361	35	32	)	)	PUNCT
ma-361	35	33	≤	≤	NOUN
ma-361	35	34	p(x	p(x	PROPN
ma-361	35	35	)	)	PUNCT
ma-361	35	36	+	+	NUM
ma-361	36	1	p(y	p(y	NOUN
ma-361	36	2	)	)	PUNCT
ma-361	36	3	(	(	PUNCT
ma-361	36	4	subadditivity	subadditivity	NOUN
ma-361	36	5	)	)	PUNCT
ma-361	36	6	https://doi.org/10.28924/ada/ma.5.17	https://doi.org/10.28924/ada/ma.5.17	PROPN
ma-361	36	7	eur	eur	PROPN
ma-361	36	8	.	.	PUNCT
ma-361	37	1	j.	j.	PROPN
ma-361	37	2	math	math	PROPN
ma-361	37	3	.	.	PUNCT
ma-361	38	1	anal	anal	PROPN
ma-361	38	2	.	.	PUNCT
ma-361	39	1	10.28924	10.28924	NUM
ma-361	39	2	/	/	SYM
ma-361	39	3	ada	ada	PROPN
ma-361	39	4	/	/	SYM
ma-361	39	5	ma.5.17	ma.5.17	ADJ
ma-361	39	6	3	3	NUM
ma-361	39	7	such	such	ADJ
ma-361	39	8	p	p	NOUN
ma-361	39	9	is	be	AUX
ma-361	39	10	τ	τ	NOUN
ma-361	39	11	-	-	ADJ
ma-361	39	12	continuous	continuous	ADJ
ma-361	39	13	if	if	SCONJ
ma-361	40	1	and	and	CCONJ
ma-361	40	2	only	only	ADV
ma-361	40	3	if	if	SCONJ
ma-361	40	4	it	it	PRON
ma-361	40	5	is	be	AUX
ma-361	40	6	dominated	dominate	VERB
ma-361	40	7	by	by	ADP
ma-361	40	8	some	some	DET
ma-361	40	9	continuous	continuous	ADJ
ma-361	40	10	seminorm	seminorm	NOUN
ma-361	40	11	ρα	ρα	PROPN
ma-361	41	1	[	[	X
ma-361	41	2	14	14	NUM
ma-361	41	3	]	]	PUNCT
ma-361	41	4	.	.	PUNCT
ma-361	42	1	definition	definition	NOUN
ma-361	42	2	2	2	NUM
ma-361	42	3	(	(	PUNCT
ma-361	42	4	norm	norm	NOUN
ma-361	42	5	-	-	PUNCT
ma-361	42	6	attainment	attainment	NOUN
ma-361	42	7	)	)	PUNCT
ma-361	42	8	.	.	PUNCT
ma-361	43	1	for	for	ADP
ma-361	43	2	a	a	DET
ma-361	43	3	functional	functional	ADJ
ma-361	43	4	f	f	NOUN
ma-361	43	5	:	:	PUNCT
ma-361	43	6	x	x	X
ma-361	43	7	→	→	SYM
ma-361	43	8	r	r	NOUN
ma-361	43	9	,	,	PUNCT
ma-361	43	10	we	we	PRON
ma-361	43	11	say	say	VERB
ma-361	43	12	f	f	PROPN
ma-361	43	13	attains	attain	VERB
ma-361	43	14	its	its	PRON
ma-361	43	15	p	p	NOUN
ma-361	43	16	-	-	PUNCT
ma-361	43	17	norm	norm	NOUN
ma-361	43	18	at	at	ADP
ma-361	43	19	x0	x0	PROPN
ma-361	43	20	∈	∈	PROPN
ma-361	43	21	x	x	INTJ
ma-361	44	1	if	if	SCONJ
ma-361	44	2	:	:	PUNCT
ma-361	44	3	f	f	PROPN
ma-361	44	4	(	(	PUNCT
ma-361	44	5	x0	x0	PROPN
ma-361	44	6	)	)	PUNCT
ma-361	45	1	=	=	SYM
ma-361	45	2	‖f	‖f	ADP
ma-361	45	3	‖p	‖p	NOUN
ma-361	45	4	:	:	PUNCT
ma-361	46	1	=	=	PUNCT
ma-361	46	2	sup	sup	NOUN
ma-361	46	3	x∈x	x∈x	NOUN
ma-361	46	4	p(x)≤1	p(x)≤1	NOUN
ma-361	46	5	|f	|f	PROPN
ma-361	46	6	(	(	PUNCT
ma-361	46	7	x)|	x)|	INTJ
ma-361	46	8	where	where	SCONJ
ma-361	46	9	p	p	NOUN
ma-361	46	10	is	be	AUX
ma-361	46	11	a	a	DET
ma-361	46	12	given	give	VERB
ma-361	46	13	continuous	continuous	ADJ
ma-361	46	14	sublinear	sublinear	NOUN
ma-361	46	15	functional	functional	ADJ
ma-361	46	16	.	.	PUNCT
ma-361	47	1	when	when	SCONJ
ma-361	47	2	p	p	NOUN
ma-361	47	3	is	be	AUX
ma-361	47	4	the	the	DET
ma-361	47	5	minkowski	minkowski	ADJ
ma-361	47	6	functional	functional	NOUN
ma-361	47	7	of	of	ADP
ma-361	47	8	a	a	DET
ma-361	47	9	bounded	bound	VERB
ma-361	47	10	set	set	NOUN
ma-361	47	11	b	b	NOUN
ma-361	47	12	,	,	PUNCT
ma-361	47	13	we	we	PRON
ma-361	47	14	write	write	VERB
ma-361	47	15	‖f	‖f	PRON
ma-361	47	16	‖b	‖b	NOUN
ma-361	47	17	.	.	PUNCT
ma-361	48	1	convex	convex	VERB
ma-361	48	2	analysis	analysis	NOUN
ma-361	48	3	tools	tool	NOUN
ma-361	48	4	.	.	PUNCT
ma-361	49	1	definition	definition	NOUN
ma-361	49	2	3	3	NUM
ma-361	49	3	(	(	PUNCT
ma-361	49	4	subdifferentials	subdifferential	NOUN
ma-361	49	5	)	)	PUNCT
ma-361	49	6	.	.	PUNCT
ma-361	50	1	for	for	ADP
ma-361	50	2	f	f	PROPN
ma-361	50	3	:	:	PUNCT
ma-361	50	4	x	x	X
ma-361	50	5	→	→	SYM
ma-361	50	6	r	r	NOUN
ma-361	50	7	∪	∪	X
ma-361	50	8	{	{	PUNCT
ma-361	50	9	+	+	NOUN
ma-361	50	10	∞	∞	NUM
ma-361	50	11	}	}	PUNCT
ma-361	50	12	,	,	PUNCT
ma-361	50	13	the	the	DET
ma-361	50	14	subdifferential	subdifferential	NOUN
ma-361	50	15	at	at	ADP
ma-361	50	16	x	x	PROPN
ma-361	50	17	∈	∈	PROPN
ma-361	50	18	dom(f	dom(f	PROPN
ma-361	50	19	)	)	PUNCT
ma-361	50	20	is	be	AUX
ma-361	50	21	:	:	PUNCT
ma-361	50	22	∂f	∂f	PROPN
ma-361	50	23	(	(	PUNCT
ma-361	50	24	x	x	NOUN
ma-361	50	25	)	)	PUNCT
ma-361	50	26	:	:	PUNCT
ma-361	51	1	=	=	SYM
ma-361	51	2	{	{	PUNCT
ma-361	51	3	φ	φ	PROPN
ma-361	51	4	∈	∈	PROPN
ma-361	51	5	x∗	x∗	PROPN
ma-361	51	6	:	:	PUNCT
ma-361	51	7	f	f	X
ma-361	51	8	(	(	PUNCT
ma-361	51	9	y	y	PROPN
ma-361	51	10	)	)	PUNCT
ma-361	51	11	≥	≥	NOUN
ma-361	51	12	f	f	X
ma-361	51	13	(	(	PUNCT
ma-361	51	14	x	x	X
ma-361	51	15	)	)	PUNCT
ma-361	51	16	+	+	CCONJ
ma-361	51	17	φ(y	φ(y	ADJ
ma-361	51	18	−	−	NUM
ma-361	51	19	x	x	SYM
ma-361	51	20	)	)	PUNCT
ma-361	51	21	∀y	∀y	PROPN
ma-361	51	22	∈	∈	PROPN
ma-361	51	23	x	x	X
ma-361	51	24	}	}	PUNCT
ma-361	51	25	when	when	SCONJ
ma-361	51	26	f	f	PROPN
ma-361	51	27	is	be	AUX
ma-361	51	28	continuous	continuous	ADJ
ma-361	51	29	and	and	CCONJ
ma-361	51	30	convex	convex	ADJ
ma-361	51	31	,	,	PUNCT
ma-361	51	32	∂f	∂f	PROPN
ma-361	51	33	(	(	PUNCT
ma-361	51	34	x	x	X
ma-361	51	35	)	)	PUNCT
ma-361	51	36	is	be	AUX
ma-361	51	37	nonempty	nonempty	ADJ
ma-361	51	38	and	and	CCONJ
ma-361	51	39	σ(x∗	σ(x∗	NOUN
ma-361	51	40	,	,	PUNCT
ma-361	51	41	x)-compact	x)-compact	PUNCT
ma-361	52	1	[	[	X
ma-361	52	2	14	14	NUM
ma-361	52	3	]	]	PUNCT
ma-361	52	4	.	.	PUNCT
ma-361	53	1	proposition	proposition	NOUN
ma-361	53	2	1	1	NUM
ma-361	53	3	(	(	PUNCT
ma-361	53	4	borwein	borwein	NOUN
ma-361	53	5	-	-	PUNCT
ma-361	53	6	preiss	preiss	NOUN
ma-361	53	7	variational	variational	ADJ
ma-361	53	8	principle	principle	NOUN
ma-361	53	9	[	[	X
ma-361	53	10	5	5	NUM
ma-361	53	11	]	]	PUNCT
ma-361	53	12	)	)	PUNCT
ma-361	53	13	.	.	PUNCT
ma-361	54	1	let	let	VERB
ma-361	54	2	(	(	PUNCT
ma-361	54	3	x	x	NOUN
ma-361	54	4	,	,	PUNCT
ma-361	54	5	τ	τ	X
ma-361	54	6	)	)	PUNCT
ma-361	54	7	be	be	VERB
ma-361	54	8	a	a	DET
ma-361	54	9	complete	complete	ADJ
ma-361	54	10	lcs	lcs	NOUN
ma-361	54	11	and	and	CCONJ
ma-361	54	12	f	f	NOUN
ma-361	54	13	:	:	PUNCT
ma-361	54	14	x	x	X
ma-361	54	15	→	→	SYM
ma-361	54	16	r	r	NOUN
ma-361	54	17	∪	∪	X
ma-361	54	18	{	{	PUNCT
ma-361	54	19	+	+	NOUN
ma-361	54	20	∞	∞	NOUN
ma-361	54	21	}	}	PUNCT
ma-361	54	22	proper	proper	ADJ
ma-361	54	23	,	,	PUNCT
ma-361	54	24	lower	low	ADJ
ma-361	54	25	semicontinuous	semicontinuous	ADJ
ma-361	54	26	,	,	PUNCT
ma-361	54	27	and	and	CCONJ
ma-361	54	28	bounded	bound	VERB
ma-361	54	29	below	below	ADV
ma-361	54	30	.	.	PUNCT
ma-361	55	1	then	then	ADV
ma-361	55	2	there	there	PRON
ma-361	55	3	exists	exist	VERB
ma-361	55	4	a	a	DET
ma-361	55	5	τ	τ	NOUN
ma-361	55	6	-	-	PUNCT
ma-361	55	7	dense	dense	ADJ
ma-361	55	8	gδ	gδ	NOUN
ma-361	55	9	set	set	VERB
ma-361	55	10	g	g	PROPN
ma-361	55	11	⊂	⊂	PROPN
ma-361	55	12	x	x	PUNCT
ma-361	55	13	such	such	ADJ
ma-361	55	14	that	that	PRON
ma-361	55	15	for	for	ADP
ma-361	55	16	all	all	PRON
ma-361	55	17	ξ	ξ	PROPN
ma-361	55	18	∈	∈	PROPN
ma-361	55	19	g	g	NOUN
ma-361	55	20	,	,	PUNCT
ma-361	55	21	the	the	DET
ma-361	55	22	perturbed	perturb	VERB
ma-361	55	23	functional	functional	ADJ
ma-361	55	24	f	f	PROPN
ma-361	55	25	+	+	CCONJ
ma-361	55	26	ξ	ξ	PROPN
ma-361	55	27	attains	attain	NOUN
ma-361	55	28	its	its	PRON
ma-361	55	29	strong	strong	ADJ
ma-361	55	30	minimum	minimum	NOUN
ma-361	55	31	on	on	ADP
ma-361	55	32	x	x	X
ma-361	55	33	.	.	PUNCT
ma-361	56	1	geometric	geometric	ADJ
ma-361	56	2	properties	property	NOUN
ma-361	56	3	.	.	PUNCT
ma-361	57	1	definition	definition	NOUN
ma-361	57	2	4	4	NUM
ma-361	57	3	(	(	PUNCT
ma-361	57	4	plurisubharmonic	plurisubharmonic	ADJ
ma-361	57	5	norms	norm	NOUN
ma-361	57	6	)	)	PUNCT
ma-361	57	7	.	.	PUNCT
ma-361	58	1	a	a	DET
ma-361	58	2	norm	norm	NOUN
ma-361	58	3	‖	‖	PROPN
ma-361	58	4	·	·	PUNCT
ma-361	58	5	‖	‖	ADJ
ma-361	58	6	on	on	ADP
ma-361	58	7	complex	complex	ADJ
ma-361	58	8	x	x	PUNCT
ma-361	58	9	is	be	AUX
ma-361	58	10	plurisubharmonic	plurisubharmonic	ADJ
ma-361	58	11	if	if	SCONJ
ma-361	58	12	for	for	ADP
ma-361	58	13	all	all	DET
ma-361	58	14	x	x	NOUN
ma-361	58	15	,	,	PUNCT
ma-361	58	16	y	y	PROPN
ma-361	58	17	∈	∈	PROPN
ma-361	58	18	x	x	X
ma-361	58	19	,	,	PUNCT
ma-361	58	20	the	the	DET
ma-361	58	21	function	function	NOUN
ma-361	58	22	:	:	PUNCT
ma-361	59	1	λ	λ	X
ma-361	59	2	7→	7→	NUM
ma-361	59	3	‖x	‖x	PUNCT
ma-361	60	1	+	+	CCONJ
ma-361	60	2	λy‖	λy‖	PROPN
ma-361	60	3	is	be	AUX
ma-361	60	4	subharmonic	subharmonic	ADJ
ma-361	60	5	on	on	ADP
ma-361	60	6	c.	c.	PROPN
ma-361	60	7	this	this	PRON
ma-361	60	8	generalizes	generalize	VERB
ma-361	60	9	the	the	DET
ma-361	60	10	notion	notion	NOUN
ma-361	60	11	of	of	ADP
ma-361	60	12	complex	complex	ADJ
ma-361	60	13	convexity	convexity	NOUN
ma-361	61	1	[	[	X
ma-361	61	2	8	8	NUM
ma-361	61	3	]	]	PUNCT
ma-361	61	4	.	.	PUNCT
ma-361	62	1	definition	definition	NOUN
ma-361	62	2	5	5	NUM
ma-361	62	3	(	(	PUNCT
ma-361	62	4	radon	radon	NOUN
ma-361	62	5	-	-	PUNCT
ma-361	62	6	nikodym	nikodym	NOUN
ma-361	62	7	property	property	NOUN
ma-361	62	8	)	)	PUNCT
ma-361	62	9	.	.	PUNCT
ma-361	63	1	a	a	DET
ma-361	63	2	locally	locally	ADV
ma-361	63	3	convex	convex	ADJ
ma-361	63	4	space	space	NOUN
ma-361	63	5	x	x	PUNCT
ma-361	63	6	has	have	VERB
ma-361	63	7	the	the	DET
ma-361	63	8	radon	radon	PROPN
ma-361	63	9	-	-	PUNCT
ma-361	63	10	nikodym	nikodym	ADJ
ma-361	63	11	prop	prop	PROPN
ma-361	63	12	-	-	PUNCT
ma-361	63	13	erty	erty	NOUN
ma-361	63	14	(	(	PUNCT
ma-361	63	15	rnp	rnp	PROPN
ma-361	63	16	)	)	PUNCT
ma-361	63	17	if	if	SCONJ
ma-361	63	18	every	every	DET
ma-361	63	19	continuous	continuous	ADJ
ma-361	63	20	linear	linear	NOUN
ma-361	63	21	operator	operator	NOUN
ma-361	63	22	t	t	NOUN
ma-361	63	23	:	:	PUNCT
ma-361	63	24	l1[0	l1[0	ADV
ma-361	63	25	,	,	PUNCT
ma-361	63	26	1]→	1]→	NOUN
ma-361	63	27	x	x	PUNCT
ma-361	63	28	is	be	AUX
ma-361	63	29	representable	representable	ADJ
ma-361	63	30	by	by	ADP
ma-361	63	31	an	an	DET
ma-361	63	32	x	x	ADV
ma-361	63	33	-	-	PUNCT
ma-361	63	34	valued	value	VERB
ma-361	63	35	bochner	bochner	NOUN
ma-361	63	36	integrable	integrable	ADJ
ma-361	63	37	function	function	NOUN
ma-361	64	1	[	[	X
ma-361	64	2	9	9	NUM
ma-361	64	3	]	]	PUNCT
ma-361	64	4	.	.	PUNCT
ma-361	65	1	polynomial	polynomial	ADJ
ma-361	65	2	mappings	mapping	NOUN
ma-361	65	3	.	.	PUNCT
ma-361	66	1	definition	definition	NOUN
ma-361	66	2	6	6	NUM
ma-361	66	3	(	(	PUNCT
ma-361	66	4	n	n	CCONJ
ma-361	66	5	-	-	PUNCT
ma-361	66	6	homogeneous	homogeneous	ADJ
ma-361	66	7	polynomials	polynomial	NOUN
ma-361	66	8	)	)	PUNCT
ma-361	66	9	.	.	PUNCT
ma-361	67	1	a	a	DET
ma-361	67	2	mapping	mapping	NOUN
ma-361	67	3	p	p	X
ma-361	67	4	:	:	PUNCT
ma-361	67	5	x	x	X
ma-361	67	6	→	→	SYM
ma-361	67	7	c	c	NOUN
ma-361	67	8	is	be	AUX
ma-361	67	9	a	a	DET
ma-361	67	10	continuous	continuous	ADJ
ma-361	67	11	n	n	CCONJ
ma-361	67	12	-	-	PUNCT
ma-361	67	13	homogeneouspolynomial	homogeneouspolynomial	ADJ
ma-361	67	14	if	if	SCONJ
ma-361	67	15	there	there	PRON
ma-361	67	16	exists	exist	VERB
ma-361	67	17	a	a	DET
ma-361	67	18	continuous	continuous	ADJ
ma-361	67	19	n	n	CCONJ
ma-361	67	20	-	-	PUNCT
ma-361	67	21	linear	linear	NOUN
ma-361	67	22	form	form	NOUN
ma-361	67	23	l	l	NOUN
ma-361	67	24	:	:	PUNCT
ma-361	67	25	x	x	SYM
ma-361	67	26	×	×	PROPN
ma-361	67	27	·	·	PUNCT
ma-361	67	28	·	·	PUNCT
ma-361	67	29	·	·	PUNCT
ma-361	67	30	×x︸	×x︸	VERB
ma-361	67	31	︷︷	︷︷	NOUN
ma-361	67	32	︸	︸	PUNCT
ma-361	67	33	n	n	PROPN
ma-361	67	34	→	→	SYM
ma-361	67	35	c	c	NOUN
ma-361	67	36	such	such	ADJ
ma-361	67	37	that	that	PRON
ma-361	67	38	:	:	PUNCT
ma-361	67	39	p	p	X
ma-361	67	40	(	(	PUNCT
ma-361	67	41	x	x	NOUN
ma-361	67	42	)	)	PUNCT
ma-361	67	43	=	=	SYM
ma-361	67	44	l(x	l(x	PROPN
ma-361	67	45	,	,	PUNCT
ma-361	67	46	.	.	PUNCT
ma-361	67	47	.	.	PUNCT
ma-361	67	48	.	.	PUNCT
ma-361	68	1	,	,	PUNCT
ma-361	68	2	x	x	X
ma-361	68	3	)	)	PUNCT
ma-361	68	4	the	the	DET
ma-361	68	5	space	space	NOUN
ma-361	68	6	p(nx	p(nx	NOUN
ma-361	68	7	)	)	PUNCT
ma-361	68	8	carries	carry	VERB
ma-361	68	9	the	the	DET
ma-361	68	10	topology	topology	NOUN
ma-361	68	11	of	of	ADP
ma-361	68	12	uniform	uniform	ADJ
ma-361	68	13	convergence	convergence	NOUN
ma-361	68	14	on	on	ADP
ma-361	68	15	bounded	bounded	ADJ
ma-361	68	16	sets	set	NOUN
ma-361	68	17	[	[	X
ma-361	68	18	1	1	NUM
ma-361	68	19	]	]	PUNCT
ma-361	68	20	.	.	PUNCT
ma-361	69	1	https://doi.org/10.28924/ada/ma.5.17	https://doi.org/10.28924/ada/ma.5.17	PROPN
ma-361	69	2	eur	eur	PROPN
ma-361	69	3	.	.	PUNCT
ma-361	70	1	j.	j.	PROPN
ma-361	70	2	math	math	PROPN
ma-361	70	3	.	.	PUNCT
ma-361	71	1	anal	anal	PROPN
ma-361	71	2	.	.	PUNCT
ma-361	72	1	10.28924	10.28924	NUM
ma-361	72	2	/	/	SYM
ma-361	72	3	ada	ada	PROPN
ma-361	72	4	/	/	SYM
ma-361	72	5	ma.5.17	ma.5.17	ADJ
ma-361	72	6	4	4	NUM
ma-361	72	7	proposition	proposition	NOUN
ma-361	72	8	2	2	NUM
ma-361	72	9	(	(	PUNCT
ma-361	72	10	projective	projective	ADJ
ma-361	72	11	tensor	tensor	NOUN
ma-361	72	12	representation	representation	NOUN
ma-361	72	13	)	)	PUNCT
ma-361	72	14	.	.	PUNCT
ma-361	73	1	for	for	ADP
ma-361	73	2	any	any	DET
ma-361	73	3	n	n	CCONJ
ma-361	73	4	-	-	PUNCT
ma-361	73	5	homogeneous	homogeneous	ADJ
ma-361	73	6	polynomial	polynomial	ADJ
ma-361	73	7	p	p	NOUN
ma-361	73	8	,	,	PUNCT
ma-361	73	9	there	there	PRON
ma-361	73	10	exists	exist	VERB
ma-361	73	11	a	a	DET
ma-361	73	12	unique	unique	ADJ
ma-361	73	13	linear	linear	ADJ
ma-361	73	14	functional	functional	ADJ
ma-361	73	15	p̃	p̃	PROPN
ma-361	73	16	on	on	ADP
ma-361	73	17	the	the	DET
ma-361	73	18	projective	projective	ADJ
ma-361	73	19	tensor	tensor	NOUN
ma-361	73	20	product	product	NOUN
ma-361	73	21	⊗̂nπx	⊗̂nπx	VERB
ma-361	73	22	such	such	ADJ
ma-361	73	23	that	that	PRON
ma-361	73	24	:	:	PUNCT
ma-361	74	1	p	p	X
ma-361	74	2	(	(	PUNCT
ma-361	74	3	x	x	NOUN
ma-361	74	4	)	)	PUNCT
ma-361	74	5	=	=	SYM
ma-361	75	1	p̃	p̃	PROPN
ma-361	75	2	(	(	PUNCT
ma-361	75	3	x	x	PROPN
ma-361	75	4	⊗	⊗	PROPN
ma-361	75	5	·	·	PUNCT
ma-361	75	6	·	·	PUNCT
ma-361	75	7	·	·	PUNCT
ma-361	76	1	⊗	⊗	NUM
ma-361	76	2	x︸	x︸	VERB
ma-361	76	3	︷︷	︷︷	PROPN
ma-361	76	4	︸	︸	ADP
ma-361	76	5	n	n	PROPN
ma-361	76	6	)	)	PUNCT
ma-361	76	7	norm	norm	NOUN
ma-361	76	8	-	-	PUNCT
ma-361	76	9	attainment	attainment	NOUN
ma-361	76	10	of	of	ADP
ma-361	76	11	p	p	NOUN
ma-361	76	12	corresponds	correspond	NOUN
ma-361	76	13	to	to	ADP
ma-361	76	14	norm	norm	NOUN
ma-361	76	15	-	-	PUNCT
ma-361	76	16	attainment	attainment	NOUN
ma-361	76	17	of	of	ADP
ma-361	76	18	p̃	p̃	PROPN
ma-361	76	19	[	[	X
ma-361	76	20	8	8	NUM
ma-361	76	21	]	]	PUNCT
ma-361	76	22	.	.	PUNCT
ma-361	77	1	key	key	ADJ
ma-361	77	2	topological	topological	ADJ
ma-361	77	3	concepts	concept	NOUN
ma-361	77	4	.	.	PUNCT
ma-361	78	1	definition	definition	NOUN
ma-361	78	2	7	7	NUM
ma-361	78	3	(	(	PUNCT
ma-361	78	4	mackey	mackey	NOUN
ma-361	78	5	-	-	PUNCT
ma-361	78	6	arens	arens	PROPN
ma-361	78	7	property	property	NOUN
ma-361	78	8	)	)	PUNCT
ma-361	78	9	.	.	PUNCT
ma-361	79	1	a	a	DET
ma-361	79	2	lcs	lcs	PROPN
ma-361	79	3	(	(	PUNCT
ma-361	79	4	x	x	PROPN
ma-361	79	5	,	,	PUNCT
ma-361	79	6	τ	τ	X
ma-361	79	7	)	)	PUNCT
ma-361	79	8	satisfies	satisfy	VERB
ma-361	79	9	the	the	DET
ma-361	79	10	mackey	mackey	PROPN
ma-361	79	11	-	-	PUNCT
ma-361	79	12	arens	aren	NOUN
ma-361	79	13	property	property	NOUN
ma-361	79	14	if	if	SCONJ
ma-361	79	15	τ	τ	PROPN
ma-361	79	16	coincides	coincide	VERB
ma-361	79	17	with	with	ADP
ma-361	79	18	the	the	DET
ma-361	79	19	mackey	mackey	PROPN
ma-361	79	20	topology	topology	PROPN
ma-361	79	21	τ(x	τ(x	PROPN
ma-361	79	22	,	,	PUNCT
ma-361	79	23	x∗	x∗	NOUN
ma-361	79	24	)	)	PUNCT
ma-361	80	1	[	[	X
ma-361	80	2	9	9	NUM
ma-361	80	3	]	]	PUNCT
ma-361	80	4	.	.	PUNCT
ma-361	81	1	definition	definition	NOUN
ma-361	81	2	8	8	NUM
ma-361	81	3	(	(	PUNCT
ma-361	81	4	quasi	quasi	NOUN
ma-361	81	5	-	-	NOUN
ma-361	81	6	completeness	completeness	NOUN
ma-361	81	7	)	)	PUNCT
ma-361	81	8	.	.	PUNCT
ma-361	82	1	x	x	PUNCT
ma-361	82	2	is	be	AUX
ma-361	82	3	quasi	quasi	ADJ
ma-361	82	4	-	-	NOUN
ma-361	82	5	complete	complete	ADJ
ma-361	82	6	if	if	SCONJ
ma-361	82	7	every	every	DET
ma-361	82	8	bounded	bounded	ADJ
ma-361	82	9	cauchy	cauchy	ADJ
ma-361	82	10	net	net	NOUN
ma-361	82	11	converges	converge	NOUN
ma-361	82	12	.	.	PUNCT
ma-361	83	1	this	this	PRON
ma-361	83	2	generalizes	generalize	VERB
ma-361	83	3	completeness	completeness	NOUN
ma-361	83	4	for	for	ADP
ma-361	83	5	non	non	ADJ
ma-361	83	6	-	-	ADJ
ma-361	83	7	metrizable	metrizable	ADJ
ma-361	83	8	spaces	space	NOUN
ma-361	83	9	[	[	X
ma-361	83	10	3	3	NUM
ma-361	83	11	]	]	PUNCT
ma-361	83	12	.	.	PUNCT
ma-361	84	1	definition	definition	NOUN
ma-361	84	2	9	9	NUM
ma-361	84	3	(	(	PUNCT
ma-361	84	4	barrelled	barrelled	ADJ
ma-361	84	5	spaces	space	NOUN
ma-361	84	6	)	)	PUNCT
ma-361	84	7	.	.	PUNCT
ma-361	85	1	x	x	PUNCT
ma-361	85	2	is	be	AUX
ma-361	85	3	barrelled	barrel	VERB
ma-361	85	4	if	if	SCONJ
ma-361	85	5	every	every	DET
ma-361	85	6	closed	closed	ADJ
ma-361	85	7	,	,	PUNCT
ma-361	85	8	absolutely	absolutely	ADV
ma-361	85	9	convex	convex	ADJ
ma-361	85	10	,	,	PUNCT
ma-361	85	11	absorbing	absorb	VERB
ma-361	85	12	set	set	NOUN
ma-361	85	13	is	be	AUX
ma-361	85	14	a	a	DET
ma-361	85	15	τ	τ	NOUN
ma-361	85	16	-	-	PUNCT
ma-361	85	17	neighborhood	neighborhood	NOUN
ma-361	85	18	of	of	ADP
ma-361	85	19	0	0	NUM
ma-361	85	20	.	.	PUNCT
ma-361	86	1	this	this	PRON
ma-361	86	2	ensures	ensure	VERB
ma-361	86	3	the	the	DET
ma-361	86	4	uniform	uniform	ADJ
ma-361	86	5	boundedness	boundedness	PROPN
ma-361	86	6	principle	principle	NOUN
ma-361	86	7	holds	hold	VERB
ma-361	86	8	[	[	X
ma-361	86	9	9	9	NUM
ma-361	86	10	]	]	PUNCT
ma-361	86	11	.	.	PUNCT
ma-361	87	1	these	these	DET
ma-361	87	2	foundational	foundational	ADJ
ma-361	87	3	concepts	concept	NOUN
ma-361	87	4	will	will	AUX
ma-361	87	5	be	be	AUX
ma-361	87	6	essential	essential	ADJ
ma-361	87	7	throughout	throughout	ADP
ma-361	87	8	our	our	PRON
ma-361	87	9	analysis	analysis	NOUN
ma-361	87	10	of	of	ADP
ma-361	87	11	nonlinear	nonlinear	ADJ
ma-361	87	12	norm	norm	NOUN
ma-361	87	13	-	-	PUNCT
ma-361	87	14	attainment	attainment	ADJ
ma-361	87	15	phenomena	phenomenon	NOUN
ma-361	87	16	in	in	ADP
ma-361	87	17	the	the	DET
ma-361	87	18	subsequent	subsequent	ADJ
ma-361	87	19	sections	section	NOUN
ma-361	87	20	.	.	PUNCT
ma-361	88	1	main	main	ADJ
ma-361	88	2	results	result	NOUN
ma-361	88	3	and	and	CCONJ
ma-361	88	4	discussions	discussion	NOUN
ma-361	88	5	theorem	theorem	VERB
ma-361	88	6	1	1	NUM
ma-361	88	7	(	(	PUNCT
ma-361	88	8	nonlinear	nonlinear	ADJ
ma-361	88	9	bishop	bishop	NOUN
ma-361	88	10	-	-	PUNCT
ma-361	88	11	phelps	phelps	PROPN
ma-361	88	12	property	property	NOUN
ma-361	88	13	)	)	PUNCT
ma-361	88	14	.	.	PUNCT
ma-361	89	1	let	let	VERB
ma-361	89	2	x	x	PRON
ma-361	89	3	be	be	AUX
ma-361	89	4	a	a	DET
ma-361	89	5	locally	locally	ADV
ma-361	89	6	convex	convex	ADJ
ma-361	89	7	space	space	NOUN
ma-361	89	8	and	and	CCONJ
ma-361	89	9	p	p	NOUN
ma-361	89	10	:	:	PUNCT
ma-361	89	11	x	x	X
ma-361	89	12	→	→	PUNCT
ma-361	89	13	r	r	NOUN
ma-361	89	14	a	a	DET
ma-361	89	15	continuous	continuous	ADJ
ma-361	89	16	sublinear	sublinear	NOUN
ma-361	89	17	functional	functional	ADJ
ma-361	89	18	.	.	PUNCT
ma-361	90	1	the	the	DET
ma-361	90	2	following	follow	VERB
ma-361	90	3	are	be	AUX
ma-361	90	4	equivalent	equivalent	ADJ
ma-361	90	5	:	:	PUNCT
ma-361	90	6	(	(	PUNCT
ma-361	90	7	1	1	X
ma-361	90	8	)	)	PUNCT
ma-361	90	9	every	every	PRON
ma-361	90	10	bounded	bound	VERB
ma-361	90	11	below	below	ADP
ma-361	90	12	p	p	NOUN
ma-361	90	13	-	-	PUNCT
ma-361	90	14	dominated	dominate	VERB
ma-361	90	15	convex	convex	NOUN
ma-361	90	16	functional	functional	ADJ
ma-361	90	17	attains	attain	NOUN
ma-361	90	18	its	its	PRON
ma-361	90	19	strong	strong	ADJ
ma-361	90	20	minimum	minimum	NOUN
ma-361	90	21	(	(	PUNCT
ma-361	90	22	2	2	NUM
ma-361	90	23	)	)	PUNCT
ma-361	90	24	the	the	DET
ma-361	90	25	set	set	NOUN
ma-361	90	26	{	{	PUNCT
ma-361	90	27	f	f	PROPN
ma-361	90	28	∈	∈	PROPN
ma-361	90	29	x∗	x∗	PROPN
ma-361	90	30	:	:	PUNCT
ma-361	90	31	f	f	PROPN
ma-361	90	32	attains	attain	VERB
ma-361	90	33	its	its	PRON
ma-361	90	34	p	p	NOUN
ma-361	90	35	-	-	PUNCT
ma-361	90	36	norm	norm	NOUN
ma-361	90	37	}	}	PUNCT
ma-361	90	38	is	be	AUX
ma-361	90	39	dense	dense	ADJ
ma-361	90	40	in	in	ADP
ma-361	90	41	(	(	PUNCT
ma-361	90	42	x∗	x∗	PROPN
ma-361	90	43	,	,	PUNCT
ma-361	90	44	β(x∗	β(x∗	ADV
ma-361	90	45	,	,	PUNCT
ma-361	90	46	x	x	NOUN
ma-361	90	47	)	)	PUNCT
ma-361	90	48	)	)	PUNCT
ma-361	91	1	(	(	PUNCT
ma-361	91	2	3	3	X
ma-361	91	3	)	)	PUNCT
ma-361	91	4	x	x	PRON
ma-361	91	5	admits	admit	VERB
ma-361	91	6	an	an	DET
ma-361	91	7	equivalent	equivalent	ADJ
ma-361	91	8	τ	τ	NOUN
ma-361	91	9	-	-	PUNCT
ma-361	91	10	lower	low	ADJ
ma-361	91	11	semicontinuous	semicontinuous	ADJ
ma-361	91	12	lattice	lattice	NOUN
ma-361	91	13	norm	norm	NOUN
ma-361	91	14	proof	proof	NOUN
ma-361	91	15	of	of	ADP
ma-361	91	16	theorem	theorem	ADJ
ma-361	91	17	1	1	NUM
ma-361	91	18	(	(	PUNCT
ma-361	91	19	nonlinear	nonlinear	ADJ
ma-361	91	20	bishop	bishop	NOUN
ma-361	91	21	-	-	PUNCT
ma-361	91	22	phelps	phelps	PROPN
ma-361	91	23	property	property	NOUN
ma-361	91	24	)	)	PUNCT
ma-361	91	25	.	.	PUNCT
ma-361	92	1	we	we	PRON
ma-361	92	2	employ	employ	VERB
ma-361	92	3	a	a	DET
ma-361	92	4	nonlinear	nonlinear	ADJ
ma-361	92	5	geometric	geometric	ADJ
ma-361	92	6	ap	ap	VERB
ma-361	92	7	-	-	ADJ
ma-361	92	8	proach	proach	ADV
ma-361	92	9	inspired	inspire	VERB
ma-361	92	10	by	by	ADP
ma-361	92	11	borwein	borwein	PROPN
ma-361	92	12	’s	’s	PART
ma-361	92	13	variational	variational	ADJ
ma-361	92	14	techniques	technique	NOUN
ma-361	92	15	.	.	PUNCT
ma-361	93	1	step	step	NOUN
ma-361	93	2	1	1	NUM
ma-361	93	3	:	:	PUNCT
ma-361	93	4	(	(	PUNCT
ma-361	93	5	1	1	X
ma-361	93	6	)	)	PUNCT
ma-361	93	7	⇒	⇒	NOUN
ma-361	93	8	(	(	PUNCT
ma-361	93	9	2	2	NUM
ma-361	93	10	)	)	PUNCT
ma-361	93	11	.	.	PUNCT
ma-361	94	1	for	for	ADP
ma-361	94	2	any	any	DET
ma-361	94	3	f	f	PROPN
ma-361	94	4	∈	∈	PROPN
ma-361	94	5	x∗	x∗	PROPN
ma-361	94	6	and	and	CCONJ
ma-361	94	7	ε	ε	PROPN
ma-361	94	8	>	>	X
ma-361	94	9	0	0	PROPN
ma-361	94	10	,	,	PUNCT
ma-361	94	11	define	define	VERB
ma-361	94	12	g(x	g(x	NOUN
ma-361	94	13	)	)	PUNCT
ma-361	95	1	:	:	PUNCT
ma-361	95	2	=	=	SYM
ma-361	95	3	p(x	p(x	PROPN
ma-361	95	4	)	)	PUNCT
ma-361	95	5	−	−	PROPN
ma-361	95	6	f	f	PROPN
ma-361	95	7	(	(	PUNCT
ma-361	95	8	x	x	NOUN
ma-361	95	9	)	)	PUNCT
ma-361	95	10	.	.	PUNCT
ma-361	96	1	by	by	ADP
ma-361	96	2	(	(	PUNCT
ma-361	96	3	1	1	NUM
ma-361	96	4	)	)	PUNCT
ma-361	96	5	,	,	PUNCT
ma-361	96	6	there	there	PRON
ma-361	96	7	exists	exist	VERB
ma-361	96	8	xε	xε	X
ma-361	96	9	∈	∈	PROPN
ma-361	96	10	x	x	SYM
ma-361	96	11	attaining	attain	VERB
ma-361	96	12	infx∈x	infx∈x	PROPN
ma-361	96	13	g(x	g(x	PROPN
ma-361	96	14	)	)	PUNCT
ma-361	96	15	.	.	PUNCT
ma-361	97	1	the	the	DET
ma-361	97	2	perturbed	perturb	VERB
ma-361	97	3	functional	functional	ADJ
ma-361	97	4	:	:	PUNCT
ma-361	97	5	fε	fε	NOUN
ma-361	97	6	:	:	PUNCT
ma-361	97	7	=	=	SYM
ma-361	97	8	f	f	PROPN
ma-361	97	9	+	+	NUM
ma-361	97	10	ε∂p(xε	ε∂p(xε	NOUN
ma-361	97	11	)	)	PUNCT
ma-361	97	12	attains	attain	VERB
ma-361	97	13	its	its	PRON
ma-361	97	14	p	p	NOUN
ma-361	97	15	-	-	PUNCT
ma-361	97	16	norm	norm	NOUN
ma-361	97	17	at	at	ADP
ma-361	97	18	xε	xε	NOUN
ma-361	97	19	since	since	SCONJ
ma-361	97	20	:	:	PUNCT
ma-361	98	1	‖fε‖p	‖fε‖p	PROPN
ma-361	98	2	=	=	PUNCT
ma-361	98	3	sup	sup	NOUN
ma-361	98	4	x	x	PUNCT
ma-361	99	1	[	[	X
ma-361	99	2	f	f	X
ma-361	99	3	(	(	PUNCT
ma-361	99	4	x	x	X
ma-361	99	5	)	)	PUNCT
ma-361	99	6	+	+	CCONJ
ma-361	99	7	εp(xε)−	εp(xε)−	X
ma-361	99	8	εp(x	εp(x	NUM
ma-361	99	9	−	−	PROPN
ma-361	99	10	xε	xε	NOUN
ma-361	99	11	)	)	PUNCT
ma-361	99	12	]	]	PUNCT
ma-361	100	1	=	=	SYM
ma-361	100	2	f	f	X
ma-361	100	3	(	(	PUNCT
ma-361	100	4	xε	xε	NOUN
ma-361	100	5	)	)	PUNCT
ma-361	100	6	+	+	CCONJ
ma-361	100	7	εp(xε	εp(xε	NOUN
ma-361	100	8	)	)	PUNCT
ma-361	100	9	moreover	moreover	ADV
ma-361	100	10	,	,	PUNCT
ma-361	100	11	‖fε	‖fε	NUM
ma-361	100	12	−	−	PROPN
ma-361	100	13	f	f	PROPN
ma-361	100	14	‖	‖	PROPN
ma-361	100	15	≤	≤	PROPN
ma-361	100	16	2ε	2ε	NOUN
ma-361	100	17	by	by	ADP
ma-361	100	18	construction	construction	NOUN
ma-361	100	19	.	.	PUNCT
ma-361	101	1	step	step	NOUN
ma-361	101	2	2	2	NUM
ma-361	101	3	:	:	PUNCT
ma-361	101	4	(	(	PUNCT
ma-361	101	5	2)⇒	2)⇒	NUM
ma-361	101	6	(	(	PUNCT
ma-361	101	7	3	3	NUM
ma-361	101	8	)	)	PUNCT
ma-361	101	9	.	.	PUNCT
ma-361	102	1	using	use	VERB
ma-361	102	2	the	the	DET
ma-361	102	3	density	density	NOUN
ma-361	102	4	,	,	PUNCT
ma-361	102	5	construct	construct	VERB
ma-361	102	6	a	a	DET
ma-361	102	7	sequence	sequence	NOUN
ma-361	102	8	(	(	PUNCT
ma-361	102	9	fn	fn	NOUN
ma-361	102	10	)	)	PUNCT
ma-361	102	11	⊂	⊂	PROPN
ma-361	102	12	x∗	x∗	PROPN
ma-361	102	13	of	of	ADP
ma-361	102	14	norm	norm	NOUN
ma-361	102	15	-	-	PUNCT
ma-361	102	16	attaining	attain	VERB
ma-361	102	17	functionalsseparating	functionalsseparate	VERB
ma-361	102	18	points	point	NOUN
ma-361	102	19	.	.	PUNCT
ma-361	103	1	the	the	DET
ma-361	103	2	lattice	lattice	PROPN
ma-361	103	3	norm	norm	NOUN
ma-361	103	4	:	:	PUNCT
ma-361	103	5	‖x‖	‖x‖	VERB
ma-361	103	6	:	:	PUNCT
ma-361	103	7	=	=	SYM
ma-361	103	8	sup	sup	NOUN
ma-361	103	9	n	n	NOUN
ma-361	103	10	|fn(x)|	|fn(x)|	VERB
ma-361	103	11	‖fn‖p	‖fn‖p	X
ma-361	103	12	+	+	NUM
ma-361	103	13	ρ(x	ρ(x	PROPN
ma-361	103	14	)	)	PUNCT
ma-361	103	15	https://doi.org/10.28924/ada/ma.5.17	https://doi.org/10.28924/ada/ma.5.17	PROPN
ma-361	103	16	eur	eur	PROPN
ma-361	103	17	.	.	PUNCT
ma-361	104	1	j.	j.	PROPN
ma-361	104	2	math	math	PROPN
ma-361	104	3	.	.	PUNCT
ma-361	105	1	anal	anal	PROPN
ma-361	105	2	.	.	PUNCT
ma-361	106	1	10.28924	10.28924	NUM
ma-361	106	2	/	/	SYM
ma-361	106	3	ada	ada	PROPN
ma-361	106	4	/	/	SYM
ma-361	106	5	ma.5.17	ma.5.17	PROPN
ma-361	106	6	5where	5where	NUM
ma-361	106	7	ρ	ρ	NOUN
ma-361	106	8	is	be	AUX
ma-361	106	9	the	the	DET
ma-361	106	10	original	original	ADJ
ma-361	106	11	τ	τ	PROPN
ma-361	106	12	-	-	PUNCT
ma-361	106	13	lsc	lsc	PROPN
ma-361	106	14	seminorm	seminorm	NOUN
ma-361	106	15	,	,	PUNCT
ma-361	106	16	has	have	VERB
ma-361	106	17	the	the	DET
ma-361	106	18	required	require	VERB
ma-361	106	19	properties	property	NOUN
ma-361	106	20	by	by	ADP
ma-361	106	21	the	the	DET
ma-361	106	22	nachbin	nachbin	NOUN
ma-361	106	23	-	-	PUNCT
ma-361	106	24	shirota	shirota	NOUN
ma-361	106	25	theorem	theorem	NOUN
ma-361	106	26	.	.	PUNCT
ma-361	107	1	step	step	NOUN
ma-361	107	2	3	3	NUM
ma-361	107	3	:	:	PUNCT
ma-361	107	4	(	(	PUNCT
ma-361	107	5	3)⇒	3)⇒	NUM
ma-361	107	6	(	(	PUNCT
ma-361	107	7	1	1	NUM
ma-361	107	8	)	)	PUNCT
ma-361	107	9	.	.	PUNCT
ma-361	108	1	for	for	ADP
ma-361	108	2	any	any	DET
ma-361	108	3	p	p	NOUN
ma-361	108	4	-	-	PUNCT
ma-361	108	5	dominated	dominate	VERB
ma-361	108	6	convex	convex	NOUN
ma-361	108	7	h	h	NOUN
ma-361	108	8	,	,	PUNCT
ma-361	108	9	the	the	DET
ma-361	108	10	sublevel	sublevel	NOUN
ma-361	108	11	sets	set	NOUN
ma-361	108	12	{	{	PUNCT
ma-361	108	13	x	x	NOUN
ma-361	108	14	:	:	PUNCT
ma-361	108	15	h(x	h(x	PROPN
ma-361	108	16	)	)	PUNCT
ma-361	108	17	≤	≤	NOUN
ma-361	108	18	α	α	X
ma-361	108	19	}	}	PUNCT
ma-361	108	20	are	be	AUX
ma-361	108	21	τ	τ	X
ma-361	108	22	-	-	PUNCT
ma-361	108	23	closed	close	VERB
ma-361	108	24	andbounded	andbounde	VERB
ma-361	108	25	in	in	ADP
ma-361	108	26	the	the	DET
ma-361	108	27	lattice	lattice	NOUN
ma-361	108	28	norm	norm	NOUN
ma-361	108	29	,	,	PUNCT
ma-361	108	30	hence	hence	ADV
ma-361	108	31	τ	τ	PROPN
ma-361	108	32	-	-	ADJ
ma-361	108	33	compact	compact	ADJ
ma-361	108	34	by	by	ADP
ma-361	108	35	the	the	DET
ma-361	108	36	generalized	generalized	ADJ
ma-361	108	37	alaoglu	alaoglu	NOUN
ma-361	108	38	theorem	theorem	NOUN
ma-361	108	39	.	.	PUNCT
ma-361	109	1	the	the	DET
ma-361	109	2	attainmentfollows	attainmentfollow	NOUN
ma-361	109	3	from	from	ADP
ma-361	109	4	lower	low	ADJ
ma-361	109	5	semicontinuity	semicontinuity	NOUN
ma-361	109	6	.	.	PUNCT
ma-361	110	1	�	�	PROPN
ma-361	110	2	example	example	NOUN
ma-361	110	3	1	1	NUM
ma-361	110	4	(	(	PUNCT
ma-361	110	5	non	non	ADJ
ma-361	110	6	-	-	ADJ
ma-361	110	7	attaining	attaining	ADJ
ma-361	110	8	sublinear	sublinear	NOUN
ma-361	110	9	functional	functional	ADJ
ma-361	110	10	)	)	PUNCT
ma-361	110	11	.	.	PUNCT
ma-361	111	1	let	let	VERB
ma-361	111	2	x	x	SYM
ma-361	111	3	=	=	SYM
ma-361	111	4	c[0	c[0	PROPN
ma-361	111	5	,	,	PUNCT
ma-361	111	6	1	1	NUM
ma-361	111	7	]	]	PUNCT
ma-361	111	8	with	with	ADP
ma-361	111	9	p(f	p(f	PROPN
ma-361	111	10	)	)	PUNCT
ma-361	112	1	=	=	PUNCT
ma-361	112	2	supx∈[0,1/2	supx∈[0,1/2	PROPN
ma-361	112	3	]	]	X
ma-361	112	4	|f	|f	PROPN
ma-361	112	5	(	(	PUNCT
ma-361	112	6	x)|	x)|	PROPN
ma-361	112	7	.	.	PUNCT
ma-361	113	1	the	the	DET
ma-361	113	2	functional	functional	PROPN
ma-361	113	3	φ(f	φ(f	PROPN
ma-361	113	4	)	)	PUNCT
ma-361	114	1	=	=	PUNCT
ma-361	114	2	∫	∫	PROPN
ma-361	115	1	1	1	NUM
ma-361	115	2	0	0	NUM
ma-361	115	3	f	f	PROPN
ma-361	115	4	fails	fail	VERB
ma-361	115	5	to	to	PART
ma-361	115	6	attain	attain	VERB
ma-361	115	7	its	its	PRON
ma-361	115	8	p	p	NOUN
ma-361	115	9	-	-	PUNCT
ma-361	115	10	norm	norm	NOUN
ma-361	115	11	,	,	PUNCT
ma-361	115	12	illustrating	illustrate	VERB
ma-361	115	13	theorem	theorem	ADJ
ma-361	115	14	1	1	NUM
ma-361	115	15	’s	’s	PART
ma-361	115	16	lattice	lattice	NOUN
ma-361	115	17	condition	condition	NOUN
ma-361	115	18	necessity	necessity	NOUN
ma-361	115	19	.	.	PUNCT
ma-361	116	1	theorem	theorem	ADJ
ma-361	116	2	2	2	NUM
ma-361	116	3	(	(	PUNCT
ma-361	116	4	characterization	characterization	NOUN
ma-361	116	5	of	of	ADP
ma-361	116	6	sublinear	sublinear	NOUN
ma-361	116	7	norm	norm	NOUN
ma-361	116	8	-	-	PUNCT
ma-361	116	9	attainment	attainment	NOUN
ma-361	116	10	)	)	PUNCT
ma-361	116	11	.	.	PUNCT
ma-361	117	1	for	for	ADP
ma-361	117	2	a	a	DET
ma-361	117	3	sublinear	sublinear	NOUN
ma-361	117	4	p	p	X
ma-361	117	5	:	:	PUNCT
ma-361	117	6	x	x	SYM
ma-361	117	7	→	→	SYM
ma-361	117	8	r	r	NOUN
ma-361	117	9	on	on	ADP
ma-361	117	10	a	a	DET
ma-361	117	11	barrelled	barrelled	ADJ
ma-361	117	12	space	space	NOUN
ma-361	117	13	,	,	PUNCT
ma-361	117	14	the	the	DET
ma-361	117	15	following	follow	VERB
ma-361	117	16	are	be	AUX
ma-361	117	17	equivalent	equivalent	ADJ
ma-361	117	18	:	:	PUNCT
ma-361	117	19	•	•	ADP
ma-361	117	20	p	p	NOUN
ma-361	117	21	attains	attain	VERB
ma-361	117	22	its	its	PRON
ma-361	117	23	norm	norm	NOUN
ma-361	117	24	at	at	ADP
ma-361	117	25	some	some	DET
ma-361	117	26	x0	x0	PROPN
ma-361	117	27	∈	∈	PROPN
ma-361	117	28	x	x	SYM
ma-361	117	29	•	•	NUM
ma-361	117	30	∂p(0	∂p(0	NOUN
ma-361	117	31	)	)	PUNCT
ma-361	117	32	∩x∗	∩x∗	PUNCT
ma-361	117	33	contains	contain	VERB
ma-361	117	34	a	a	DET
ma-361	117	35	weak∗-exposed	weak∗-expose	VERB
ma-361	117	36	point	point	NOUN
ma-361	117	37	•	•	ADP
ma-361	117	38	the	the	DET
ma-361	117	39	subdifferential	subdifferential	ADJ
ma-361	117	40	map	map	NOUN
ma-361	117	41	x	x	PROPN
ma-361	117	42	7→	7→	NUM
ma-361	117	43	∂p(x	∂p(x	NOUN
ma-361	117	44	)	)	PUNCT
ma-361	117	45	is	be	AUX
ma-361	117	46	not	not	PART
ma-361	117	47	norm	norm	NOUN
ma-361	117	48	-	-	PUNCT
ma-361	117	49	decreasing	decrease	VERB
ma-361	117	50	proof	proof	NOUN
ma-361	117	51	of	of	ADP
ma-361	117	52	theorem	theorem	ADJ
ma-361	117	53	2	2	NUM
ma-361	117	54	(	(	PUNCT
ma-361	117	55	characterization	characterization	NOUN
ma-361	117	56	of	of	ADP
ma-361	117	57	sublinear	sublinear	NOUN
ma-361	117	58	norm	norm	NOUN
ma-361	117	59	-	-	PUNCT
ma-361	117	60	attainment	attainment	NOUN
ma-361	117	61	)	)	PUNCT
ma-361	117	62	.	.	PUNCT
ma-361	118	1	we	we	PRON
ma-361	118	2	develop	develop	VERB
ma-361	118	3	a	a	DET
ma-361	118	4	new	new	ADJ
ma-361	118	5	subdif	subdif	NOUN
ma-361	118	6	-	-	PUNCT
ma-361	118	7	ferential	ferential	ADJ
ma-361	118	8	calculus	calculus	NOUN
ma-361	118	9	approach	approach	NOUN
ma-361	118	10	:	:	PUNCT
ma-361	118	11	(	(	PUNCT
ma-361	118	12	i	i	NOUN
ma-361	118	13	)	)	PUNCT
ma-361	118	14	⇒	⇒	PROPN
ma-361	118	15	(	(	PUNCT
ma-361	118	16	ii	ii	PROPN
ma-361	118	17	):	):	PUNCT
ma-361	118	18	if	if	SCONJ
ma-361	118	19	p	p	NOUN
ma-361	118	20	attains	attain	VERB
ma-361	118	21	its	its	PRON
ma-361	118	22	norm	norm	NOUN
ma-361	118	23	at	at	ADP
ma-361	118	24	x0	x0	PROPN
ma-361	118	25	,	,	PUNCT
ma-361	118	26	then	then	ADV
ma-361	118	27	for	for	ADP
ma-361	118	28	any	any	DET
ma-361	118	29	f	f	PROPN
ma-361	118	30	∈	∈	PROPN
ma-361	118	31	∂p(x0	∂p(x0	NOUN
ma-361	118	32	)	)	PUNCT
ma-361	118	33	we	we	PRON
ma-361	118	34	have	have	VERB
ma-361	118	35	:	:	PUNCT
ma-361	119	1	f	f	PROPN
ma-361	119	2	(	(	PUNCT
ma-361	119	3	x0	x0	PROPN
ma-361	119	4	)	)	PUNCT
ma-361	119	5	=	=	SYM
ma-361	119	6	p(x0	p(x0	NOUN
ma-361	119	7	)	)	PUNCT
ma-361	119	8	=	=	SYM
ma-361	119	9	‖p‖	‖p‖	PROPN
ma-361	119	10	thus	thus	ADV
ma-361	119	11	f	f	PROPN
ma-361	119	12	exposes	expose	VERB
ma-361	119	13	∂p(0	∂p(0	NOUN
ma-361	119	14	)	)	PUNCT
ma-361	119	15	at	at	ADP
ma-361	119	16	x0	x0	PROPN
ma-361	119	17	in	in	ADP
ma-361	119	18	the	the	DET
ma-361	119	19	weak∗	weak∗	NOUN
ma-361	119	20	topology	topology	NOUN
ma-361	119	21	.	.	PUNCT
ma-361	120	1	(	(	PUNCT
ma-361	120	2	ii	ii	NOUN
ma-361	120	3	)	)	PUNCT
ma-361	120	4	⇒	⇒	NOUN
ma-361	120	5	(	(	PUNCT
ma-361	120	6	iii	iii	NOUN
ma-361	120	7	):	):	PUNCT
ma-361	120	8	let	let	VERB
ma-361	120	9	f	f	PRON
ma-361	120	10	be	be	AUX
ma-361	120	11	a	a	DET
ma-361	120	12	weak∗-exposed	weak∗-exposed	ADJ
ma-361	120	13	point	point	NOUN
ma-361	120	14	of	of	ADP
ma-361	120	15	∂p(0	∂p(0	NOUN
ma-361	120	16	)	)	PUNCT
ma-361	120	17	.	.	PUNCT
ma-361	121	1	there	there	PRON
ma-361	121	2	exists	exist	VERB
ma-361	121	3	x0	x0	PROPN
ma-361	121	4	∈	∈	PROPN
ma-361	121	5	x	x	PUNCT
ma-361	121	6	such	such	ADJ
ma-361	121	7	that	that	SCONJ
ma-361	121	8	:	:	PUNCT
ma-361	122	1	f	f	X
ma-361	122	2	(	(	PUNCT
ma-361	122	3	x0	x0	PROPN
ma-361	122	4	)	)	PUNCT
ma-361	122	5	>	>	X
ma-361	122	6	g(x0	g(x0	NOUN
ma-361	122	7	)	)	PUNCT
ma-361	122	8	∀g	∀g	NOUN
ma-361	122	9	∈	∈	PROPN
ma-361	122	10	∂p(0	∂p(0	NOUN
ma-361	122	11	)	)	PUNCT
ma-361	122	12	\	\	NOUN
ma-361	123	1	{	{	PUNCT
ma-361	123	2	f	f	X
ma-361	123	3	}	}	PUNCT
ma-361	123	4	this	this	PRON
ma-361	123	5	implies	imply	VERB
ma-361	123	6	‖∂p(x0)‖	‖∂p(x0)‖	NOUN
ma-361	123	7	=	=	SYM
ma-361	123	8	‖f	‖f	PRON
ma-361	123	9	‖	‖	ADJ
ma-361	123	10	since	since	SCONJ
ma-361	123	11	any	any	DET
ma-361	123	12	other	other	ADJ
ma-361	123	13	subgradient	subgradient	NOUN
ma-361	123	14	would	would	AUX
ma-361	123	15	violate	violate	VERB
ma-361	123	16	the	the	DET
ma-361	123	17	exposing	expose	VERB
ma-361	123	18	property	property	NOUN
ma-361	123	19	.	.	PUNCT
ma-361	124	1	(	(	PUNCT
ma-361	124	2	iii)⇒	iii)⇒	PROPN
ma-361	124	3	(	(	PUNCT
ma-361	124	4	i	i	NOUN
ma-361	124	5	):	):	PUNCT
ma-361	124	6	by	by	ADP
ma-361	124	7	the	the	DET
ma-361	124	8	barrelledness	barrelledness	NOUN
ma-361	124	9	assumption	assumption	NOUN
ma-361	124	10	,	,	PUNCT
ma-361	124	11	the	the	DET
ma-361	124	12	subdifferential	subdifferential	ADJ
ma-361	124	13	map	map	NOUN
ma-361	124	14	is	be	AUX
ma-361	124	15	locally	locally	ADV
ma-361	124	16	bounded	bound	VERB
ma-361	124	17	.	.	PUNCT
ma-361	125	1	if	if	SCONJ
ma-361	125	2	‖∂p(x)‖is	‖∂p(x)‖is	ADJ
ma-361	125	3	non	non	ADJ
ma-361	125	4	-	-	ADJ
ma-361	125	5	decreasing	decrease	VERB
ma-361	125	6	,	,	PUNCT
ma-361	125	7	then	then	ADV
ma-361	125	8	for	for	ADP
ma-361	125	9	some	some	DET
ma-361	125	10	x0	x0	PROPN
ma-361	125	11	we	we	PRON
ma-361	125	12	must	must	AUX
ma-361	125	13	have	have	VERB
ma-361	125	14	:	:	PUNCT
ma-361	125	15	‖∂p(x0)‖	‖∂p(x0)‖	NOUN
ma-361	126	1	=	=	PUNCT
ma-361	126	2	sup	sup	NOUN
ma-361	126	3	x∈x	x∈x	NOUN
ma-361	126	4	‖∂p(x)‖	‖∂p(x)‖	PROPN
ma-361	127	1	=	=	SYM
ma-361	127	2	‖p‖	‖p‖	PROPN
ma-361	127	3	the	the	DET
ma-361	127	4	attainment	attainment	NOUN
ma-361	127	5	follows	follow	VERB
ma-361	127	6	from	from	ADP
ma-361	127	7	the	the	DET
ma-361	127	8	hahn	hahn	NOUN
ma-361	127	9	-	-	PUNCT
ma-361	127	10	banach	banach	ADV
ma-361	127	11	theorem	theorem	NOUN
ma-361	127	12	applied	apply	VERB
ma-361	127	13	to	to	ADP
ma-361	127	14	∂p(x0	∂p(x0	NOUN
ma-361	127	15	)	)	PUNCT
ma-361	127	16	.	.	PUNCT
ma-361	128	1	�	�	PROPN
ma-361	128	2	theorem	theorem	VERB
ma-361	128	3	3	3	NUM
ma-361	128	4	(	(	PUNCT
ma-361	128	5	non	non	ADJ
ma-361	128	6	-	-	ADJ
ma-361	128	7	convex	convex	ADJ
ma-361	128	8	variational	variational	ADJ
ma-361	128	9	principle	principle	NOUN
ma-361	128	10	)	)	PUNCT
ma-361	128	11	.	.	PUNCT
ma-361	129	1	let	let	VERB
ma-361	129	2	x	x	PRON
ma-361	129	3	be	be	AUX
ma-361	129	4	a	a	DET
ma-361	129	5	quasi	quasi	NOUN
ma-361	129	6	-	-	ADJ
ma-361	129	7	complete	complete	ADJ
ma-361	129	8	locally	locally	ADV
ma-361	129	9	convex	convex	ADJ
ma-361	129	10	space	space	NOUN
ma-361	129	11	and	and	CCONJ
ma-361	129	12	f	f	NOUN
ma-361	129	13	:	:	PUNCT
ma-361	130	1	x	x	X
ma-361	131	1	→	→	SYM
ma-361	131	2	r	r	NOUN
ma-361	131	3	gateaux	gateaux	ADV
ma-361	131	4	differentiable	differentiable	ADJ
ma-361	131	5	.	.	PUNCT
ma-361	132	1	if	if	SCONJ
ma-361	132	2	f	f	PROPN
ma-361	132	3	is	be	AUX
ma-361	132	4	τ	τ	NOUN
ma-361	132	5	-	-	PUNCT
ma-361	132	6	lower	low	ADJ
ma-361	132	7	semicontinuous	semicontinuous	ADJ
ma-361	132	8	and	and	CCONJ
ma-361	132	9	coercive	coercive	ADJ
ma-361	132	10	,	,	PUNCT
ma-361	132	11	then	then	ADV
ma-361	132	12	there	there	PRON
ma-361	132	13	exists	exist	VERB
ma-361	132	14	a	a	DET
ma-361	132	15	dense	dense	ADJ
ma-361	132	16	gδ	gδ	NOUN
ma-361	132	17	set	set	VERB
ma-361	132	18	g	g	PROPN
ma-361	132	19	⊂	⊂	PROPN
ma-361	132	20	x∗	x∗	PROPN
ma-361	132	21	such	such	ADJ
ma-361	132	22	that	that	PRON
ma-361	132	23	for	for	ADP
ma-361	132	24	all	all	PRON
ma-361	132	25	ξ	ξ	PROPN
ma-361	132	26	∈	∈	PROPN
ma-361	132	27	g	g	NOUN
ma-361	132	28	,	,	PUNCT
ma-361	132	29	the	the	DET
ma-361	132	30	perturbed	perturb	VERB
ma-361	132	31	functional	functional	ADJ
ma-361	132	32	f	f	PROPN
ma-361	132	33	+	+	CCONJ
ma-361	132	34	ξ	ξ	PROPN
ma-361	132	35	attains	attain	VERB
ma-361	132	36	its	its	PRON
ma-361	132	37	exact	exact	ADJ
ma-361	132	38	norm	norm	NOUN
ma-361	132	39	on	on	ADP
ma-361	132	40	x	x	X
ma-361	132	41	.	.	PUNCT
ma-361	133	1	proof	proof	NOUN
ma-361	133	2	of	of	ADP
ma-361	133	3	theorem	theorem	ADJ
ma-361	133	4	3	3	NUM
ma-361	133	5	(	(	PUNCT
ma-361	133	6	non	non	ADJ
ma-361	133	7	-	-	ADJ
ma-361	133	8	convex	convex	ADJ
ma-361	133	9	variational	variational	ADJ
ma-361	133	10	principle	principle	NOUN
ma-361	133	11	)	)	PUNCT
ma-361	133	12	.	.	PUNCT
ma-361	134	1	we	we	PRON
ma-361	134	2	combine	combine	VERB
ma-361	134	3	phelps	phelps	PROPN
ma-361	134	4	’	'	PUNCT
ma-361	134	5	perturbed	perturb	VERB
ma-361	134	6	minimiza	minimiza	ADJ
ma-361	134	7	-	-	PUNCT
ma-361	134	8	tion	tion	NOUN
ma-361	134	9	with	with	ADP
ma-361	134	10	christensen	christensen	PROPN
ma-361	134	11	’s	’s	PART
ma-361	134	12	category	category	NOUN
ma-361	134	13	methods	method	NOUN
ma-361	134	14	:	:	PUNCT
ma-361	134	15	step	step	NOUN
ma-361	134	16	1	1	NUM
ma-361	134	17	:	:	PUNCT
ma-361	134	18	define	define	VERB
ma-361	134	19	the	the	DET
ma-361	134	20	family	family	NOUN
ma-361	134	21	:	:	PUNCT
ma-361	134	22	f	f	X
ma-361	134	23	:	:	PUNCT
ma-361	134	24	=	=	SYM
ma-361	134	25	{	{	PUNCT
ma-361	134	26	ξ	ξ	X
ma-361	134	27	∈	∈	PROPN
ma-361	134	28	x∗	x∗	NOUN
ma-361	134	29	:	:	PUNCT
ma-361	134	30	f	f	X
ma-361	134	31	+	+	SYM
ma-361	134	32	ξ	ξ	PROPN
ma-361	134	33	attains	attain	NOUN
ma-361	134	34	its	its	PRON
ma-361	134	35	norm	norm	NOUN
ma-361	134	36	}	}	PUNCT
ma-361	134	37	https://doi.org/10.28924/ada/ma.5.17	https://doi.org/10.28924/ada/ma.5.17	PROPN
ma-361	134	38	eur	eur	PROPN
ma-361	134	39	.	.	PUNCT
ma-361	135	1	j.	j.	PROPN
ma-361	135	2	math	math	PROPN
ma-361	135	3	.	.	PUNCT
ma-361	136	1	anal	anal	PROPN
ma-361	136	2	.	.	PUNCT
ma-361	137	1	10.28924	10.28924	NUM
ma-361	137	2	/	/	SYM
ma-361	137	3	ada	ada	PROPN
ma-361	137	4	/	/	SYM
ma-361	137	5	ma.5.17	ma.5.17	ADJ
ma-361	137	6	6	6	NUM
ma-361	137	7	step	step	NOUN
ma-361	137	8	2	2	NUM
ma-361	137	9	:	:	PUNCT
ma-361	137	10	for	for	ADP
ma-361	137	11	each	each	DET
ma-361	137	12	n	n	PRON
ma-361	137	13	∈	∈	PROPN
ma-361	137	14	n	n	CCONJ
ma-361	137	15	,	,	PUNCT
ma-361	137	16	consider	consider	VERB
ma-361	137	17	the	the	DET
ma-361	137	18	open	open	ADJ
ma-361	137	19	sets	set	NOUN
ma-361	137	20	:	:	PUNCT
ma-361	137	21	un	un	PROPN
ma-361	137	22	:	:	PUNCT
ma-361	137	23	=	=	SYM
ma-361	138	1	⋃	⋃	NOUN
ma-361	138	2	x∈x	x∈x	NOUN
ma-361	138	3	‖x‖>n	‖x‖>n	PUNCT
ma-361	138	4	{	{	PUNCT
ma-361	138	5	ξ	ξ	X
ma-361	138	6	∈	∈	NOUN
ma-361	138	7	x∗	x∗	X
ma-361	138	8	:	:	PUNCT
ma-361	138	9	(	(	PUNCT
ma-361	138	10	f	f	X
ma-361	138	11	+	+	CCONJ
ma-361	138	12	ξ)(x	ξ)(x	PROPN
ma-361	138	13	)	)	PUNCT
ma-361	138	14	>	>	PUNCT
ma-361	139	1	‖f	‖f	ADP
ma-361	139	2	+	+	CCONJ
ma-361	139	3	ξ‖	ξ‖	ADJ
ma-361	139	4	−	−	NUM
ma-361	139	5	1	1	NUM
ma-361	139	6	n	n	PROPN
ma-361	139	7	}	}	PUNCT
ma-361	139	8	these	these	PRON
ma-361	139	9	are	be	AUX
ma-361	139	10	dense	dense	ADJ
ma-361	139	11	by	by	ADP
ma-361	139	12	the	the	DET
ma-361	139	13	quasi	quasi	NOUN
ma-361	139	14	-	-	NOUN
ma-361	139	15	completeness	completeness	NOUN
ma-361	139	16	and	and	CCONJ
ma-361	139	17	the	the	DET
ma-361	139	18	ekeland	ekeland	NOUN
ma-361	139	19	variational	variational	ADJ
ma-361	139	20	principle	principle	NOUN
ma-361	139	21	.	.	PUNCT
ma-361	140	1	step	step	NOUN
ma-361	140	2	3	3	NUM
ma-361	140	3	:	:	PUNCT
ma-361	140	4	the	the	DET
ma-361	140	5	set	set	NOUN
ma-361	140	6	g	g	NOUN
ma-361	140	7	:	:	PUNCT
ma-361	140	8	=	=	SYM
ma-361	140	9	⋂	⋂	NUM
ma-361	140	10	n∈n	n∈n	NOUN
ma-361	140	11	un	un	PROPN
ma-361	140	12	is	be	AUX
ma-361	140	13	a	a	DET
ma-361	140	14	dense	dense	ADJ
ma-361	140	15	gδ	gδ	NOUN
ma-361	140	16	by	by	ADP
ma-361	140	17	baire	baire	NOUN
ma-361	140	18	’s	’s	PART
ma-361	140	19	theorem	theorem	PROPN
ma-361	140	20	.	.	PROPN
ma-361	141	1	for	for	ADP
ma-361	141	2	any	any	DET
ma-361	141	3	ξ	ξ	PROPN
ma-361	141	4	∈	∈	PROPN
ma-361	141	5	g	g	NOUN
ma-361	141	6	,	,	PUNCT
ma-361	141	7	take	take	VERB
ma-361	141	8	a	a	DET
ma-361	141	9	sequence	sequence	NOUN
ma-361	141	10	(	(	PUNCT
ma-361	141	11	xn	xn	PROPN
ma-361	141	12	)	)	PUNCT
ma-361	141	13	with	with	ADP
ma-361	141	14	‖xn‖	‖xn‖	ADJ
ma-361	141	15	→	→	SYM
ma-361	141	16	∞	∞	PROPN
ma-361	141	17	and	and	CCONJ
ma-361	141	18	:	:	PUNCT
ma-361	141	19	(	(	PUNCT
ma-361	141	20	f	f	X
ma-361	141	21	+	+	CCONJ
ma-361	142	1	ξ)(xn)→	ξ)(xn)→	PROPN
ma-361	142	2	‖f	‖f	PRON
ma-361	142	3	+	+	CCONJ
ma-361	142	4	ξ‖the	ξ‖the	DET
ma-361	142	5	coercivity	coercivity	NOUN
ma-361	142	6	and	and	CCONJ
ma-361	142	7	lower	low	ADJ
ma-361	142	8	semicontinuity	semicontinuity	NOUN
ma-361	142	9	ensure	ensure	VERB
ma-361	142	10	the	the	DET
ma-361	142	11	existence	existence	NOUN
ma-361	142	12	of	of	ADP
ma-361	142	13	a	a	DET
ma-361	142	14	norm	norm	NOUN
ma-361	142	15	-	-	PUNCT
ma-361	142	16	attaining	attain	VERB
ma-361	142	17	point	point	NOUN
ma-361	142	18	.	.	PUNCT
ma-361	143	1	�	�	PROPN
ma-361	143	2	theorem	theorem	VERB
ma-361	143	3	4	4	NUM
ma-361	143	4	(	(	PUNCT
ma-361	143	5	quasilinear	quasilinear	NOUN
ma-361	143	6	separation	separation	NOUN
ma-361	143	7	)	)	PUNCT
ma-361	143	8	.	.	PUNCT
ma-361	144	1	let	let	VERB
ma-361	144	2	a	a	PRON
ma-361	144	3	,	,	PUNCT
ma-361	144	4	b	b	X
ma-361	144	5	⊂	⊂	PROPN
ma-361	144	6	x	x	VERB
ma-361	144	7	be	be	AUX
ma-361	144	8	disjoint	disjoint	ADJ
ma-361	144	9	convex	convex	NOUN
ma-361	144	10	sets	set	NOUN
ma-361	144	11	in	in	ADP
ma-361	144	12	a	a	DET
ma-361	144	13	locally	locally	ADV
ma-361	144	14	convex	convex	ADJ
ma-361	144	15	space	space	NOUN
ma-361	144	16	,	,	PUNCT
ma-361	144	17	with	with	ADP
ma-361	144	18	a	a	DET
ma-361	144	19	open	open	NOUN
ma-361	144	20	.	.	PUNCT
ma-361	145	1	for	for	ADP
ma-361	145	2	any	any	DET
ma-361	145	3	continuous	continuous	ADJ
ma-361	145	4	quasilinear	quasilinear	NOUN
ma-361	145	5	p	p	NOUN
ma-361	145	6	:	:	PUNCT
ma-361	145	7	x	x	SYM
ma-361	145	8	→	→	SYM
ma-361	145	9	r	r	NOUN
ma-361	145	10	,	,	PUNCT
ma-361	145	11	there	there	PRON
ma-361	145	12	exists	exist	VERB
ma-361	145	13	f	f	PROPN
ma-361	145	14	∈	∈	PROPN
ma-361	145	15	x∗	x∗	NOUN
ma-361	145	16	attaining	attain	VERB
ma-361	145	17	its	its	PRON
ma-361	145	18	p	p	NOUN
ma-361	145	19	-	-	PUNCT
ma-361	145	20	norm	norm	NOUN
ma-361	145	21	and	and	CCONJ
ma-361	145	22	separating	separate	VERB
ma-361	145	23	a	a	PRON
ma-361	145	24	from	from	ADP
ma-361	145	25	b	b	NOUN
ma-361	145	26	:	:	PUNCT
ma-361	145	27	sup	sup	PROPN
ma-361	145	28	a∈a	a∈a	PROPN
ma-361	145	29	f	f	PROPN
ma-361	145	30	(	(	PUNCT
ma-361	145	31	a	a	X
ma-361	145	32	)	)	PUNCT
ma-361	145	33	≤	≤	NUM
ma-361	145	34	inf	inf	PROPN
ma-361	145	35	b∈b	b∈b	NOUN
ma-361	145	36	f	f	PROPN
ma-361	145	37	(	(	PUNCT
ma-361	145	38	b	b	NOUN
ma-361	145	39	)	)	PUNCT
ma-361	145	40	proof	proof	NOUN
ma-361	145	41	.	.	PUNCT
ma-361	146	1	we	we	PRON
ma-361	146	2	proceed	proceed	VERB
ma-361	146	3	via	via	ADP
ma-361	146	4	a	a	DET
ma-361	146	5	nonlinear	nonlinear	ADJ
ma-361	146	6	geometric	geometric	ADJ
ma-361	146	7	approach	approach	NOUN
ma-361	146	8	:	:	PUNCT
ma-361	146	9	step	step	NOUN
ma-361	146	10	1	1	NUM
ma-361	146	11	:	:	PUNCT
ma-361	146	12	constructing	construct	VERB
ma-361	146	13	the	the	DET
ma-361	146	14	quasilinear	quasilinear	NOUN
ma-361	146	15	sandwichdefine	sandwichdefine	VERB
ma-361	146	16	the	the	DET
ma-361	146	17	functional	functional	ADJ
ma-361	146	18	φ(x	φ(x	NOUN
ma-361	146	19	)	)	PUNCT
ma-361	146	20	:	:	PUNCT
ma-361	146	21	=	=	PUNCT
ma-361	146	22	infa∈a	infa∈a	NOUN
ma-361	146	23	p(x	p(x	PROPN
ma-361	146	24	−	−	PROPN
ma-361	146	25	a	a	PRON
ma-361	146	26	)	)	PUNCT
ma-361	146	27	.	.	PUNCT
ma-361	147	1	by	by	ADP
ma-361	147	2	quasilinearity	quasilinearity	NOUN
ma-361	147	3	and	and	CCONJ
ma-361	147	4	continuity	continuity	NOUN
ma-361	147	5	of	of	ADP
ma-361	147	6	p	p	PRON
ma-361	147	7	,	,	PUNCT
ma-361	147	8	φ	φ	PROPN
ma-361	147	9	is	be	AUX
ma-361	147	10	:	:	PUNCT
ma-361	147	11	•	•	NOUN
ma-361	147	12	subadditive	subadditive	ADJ
ma-361	147	13	:	:	PUNCT
ma-361	147	14	φ(x	φ(x	PROPN
ma-361	147	15	+	+	CCONJ
ma-361	147	16	y	y	PROPN
ma-361	147	17	)	)	PUNCT
ma-361	147	18	≤	≤	ADJ
ma-361	147	19	φ(x	φ(x	NOUN
ma-361	147	20	)	)	PUNCT
ma-361	147	21	+	+	SYM
ma-361	147	22	φ(y	φ(y	NOUN
ma-361	147	23	)	)	PUNCT
ma-361	147	24	•	•	ADV
ma-361	147	25	positively	positively	ADV
ma-361	147	26	homogeneous	homogeneous	ADJ
ma-361	147	27	:	:	PUNCT
ma-361	147	28	φ(λx	φ(λx	NOUN
ma-361	147	29	)	)	PUNCT
ma-361	147	30	=	=	PUNCT
ma-361	147	31	λφ(x	λφ(x	X
ma-361	147	32	)	)	PUNCT
ma-361	147	33	for	for	ADP
ma-361	147	34	λ	λ	PROPN
ma-361	147	35	>	>	X
ma-361	147	36	0	0	NUM
ma-361	147	37	•	•	NUM
ma-361	147	38	τ	τ	NOUN
ma-361	147	39	-	-	NOUN
ma-361	147	40	continuous	continuous	ADJ
ma-361	147	41	on	on	ADP
ma-361	147	42	x	x	PART
ma-361	147	43	step	step	NOUN
ma-361	147	44	2	2	NUM
ma-361	147	45	:	:	PUNCT
ma-361	147	46	geometric	geometric	ADJ
ma-361	147	47	separation	separation	NOUN
ma-361	147	48	via	via	ADP
ma-361	147	49	nonlinear	nonlinear	PROPN
ma-361	147	50	hahn	hahn	PROPN
ma-361	147	51	-	-	PUNCT
ma-361	147	52	banachconsider	banachconsider	VERB
ma-361	147	53	the	the	DET
ma-361	147	54	sublevel	sublevel	NOUN
ma-361	147	55	set	set	NOUN
ma-361	148	1	k	k	PROPN
ma-361	148	2	:	:	PUNCT
ma-361	148	3	=	=	X
ma-361	148	4	{	{	PUNCT
ma-361	148	5	x	x	X
ma-361	148	6	:	:	PUNCT
ma-361	148	7	φ(x	φ(x	NOUN
ma-361	148	8	)	)	PUNCT
ma-361	148	9	<	<	X
ma-361	148	10	1	1	NUM
ma-361	148	11	}	}	PUNCT
ma-361	148	12	.	.	PUNCT
ma-361	149	1	since	since	SCONJ
ma-361	149	2	a	a	PRON
ma-361	149	3	is	be	AUX
ma-361	149	4	open	open	ADJ
ma-361	149	5	and	and	CCONJ
ma-361	149	6	a∩b	a∩b	ADJ
ma-361	149	7	=	=	SYM
ma-361	149	8	∅	∅	NOUN
ma-361	149	9	,	,	PUNCT
ma-361	149	10	we	we	PRON
ma-361	149	11	have	have	VERB
ma-361	149	12	0	0	NUM
ma-361	149	13	/∈	/∈	NOUN
ma-361	149	14	b−k.by	b−k.by	PROPN
ma-361	149	15	the	the	DET
ma-361	149	16	nonlinear	nonlinear	ADJ
ma-361	149	17	separation	separation	NOUN
ma-361	149	18	theorem	theorem	NOUN
ma-361	149	19	(	(	PUNCT
ma-361	149	20	see	see	VERB
ma-361	149	21	[	[	X
ma-361	149	22	1	1	NUM
ma-361	149	23	]	]	NUM
ma-361	149	24	)	)	PUNCT
ma-361	149	25	,	,	PUNCT
ma-361	149	26	there	there	PRON
ma-361	149	27	exists	exist	VERB
ma-361	149	28	f	f	PROPN
ma-361	149	29	∈	∈	PROPN
ma-361	149	30	x∗	x∗	PROPN
ma-361	149	31	with	with	ADP
ma-361	149	32	:	:	PUNCT
ma-361	149	33	sup	sup	NUM
ma-361	149	34	k∈k	k∈k	NOUN
ma-361	149	35	f	f	X
ma-361	149	36	(	(	PUNCT
ma-361	149	37	k	k	NOUN
ma-361	149	38	)	)	PUNCT
ma-361	149	39	≤	≤	NUM
ma-361	149	40	inf	inf	PROPN
ma-361	149	41	b∈b	b∈b	NOUN
ma-361	149	42	f	f	PROPN
ma-361	149	43	(	(	PUNCT
ma-361	149	44	b	b	NOUN
ma-361	149	45	)	)	PUNCT
ma-361	149	46	step	step	NOUN
ma-361	149	47	3	3	NUM
ma-361	149	48	:	:	PUNCT
ma-361	149	49	norm	norm	NOUN
ma-361	149	50	-	-	PUNCT
ma-361	149	51	attainment	attainment	NOUN
ma-361	149	52	verificationthe	verificationthe	DET
ma-361	149	53	critical	critical	ADJ
ma-361	149	54	observation	observation	NOUN
ma-361	149	55	is	be	AUX
ma-361	149	56	that	that	SCONJ
ma-361	149	57	f	f	PROPN
ma-361	149	58	attains	attain	VERB
ma-361	149	59	its	its	PRON
ma-361	149	60	p	p	NOUN
ma-361	149	61	-	-	PUNCT
ma-361	149	62	norm	norm	NOUN
ma-361	149	63	on	on	ADP
ma-361	149	64	∂k	∂k	PROPN
ma-361	149	65	:	:	PUNCT
ma-361	149	66	∃x0	∃x0	PROPN
ma-361	149	67	∈	∈	PROPN
ma-361	150	1	∂k	∂k	NOUN
ma-361	150	2	with	with	ADP
ma-361	150	3	f	f	PROPN
ma-361	150	4	(	(	PUNCT
ma-361	150	5	x0	x0	PROPN
ma-361	150	6	)	)	PUNCT
ma-361	151	1	=	=	SYM
ma-361	151	2	sup	sup	NOUN
ma-361	151	3	p(x)≤1	p(x)≤1	NOUN
ma-361	151	4	f	f	X
ma-361	151	5	(	(	PUNCT
ma-361	151	6	x	x	X
ma-361	151	7	)	)	PUNCT
ma-361	151	8	this	this	PRON
ma-361	151	9	follows	follow	VERB
ma-361	151	10	from	from	ADP
ma-361	151	11	the	the	DET
ma-361	151	12	τ	τ	NOUN
ma-361	151	13	-	-	NOUN
ma-361	151	14	compactness	compactness	NOUN
ma-361	151	15	of	of	ADP
ma-361	151	16	∂k	∂k	PROPN
ma-361	151	17	∩	∩	ADJ
ma-361	151	18	ker(f	ker(f	PROPN
ma-361	151	19	)	)	PUNCT
ma-361	151	20	⊥	⊥	PROPN
ma-361	151	21	and	and	CCONJ
ma-361	151	22	the	the	DET
ma-361	151	23	continuity	continuity	NOUN
ma-361	151	24	of	of	ADP
ma-361	151	25	p.	p.	NOUN
ma-361	151	26	the	the	DET
ma-361	151	27	separationinequality	separationinequality	NOUN
ma-361	151	28	follows	follow	VERB
ma-361	151	29	by	by	ADP
ma-361	151	30	scaling	scale	VERB
ma-361	151	31	arguments	argument	NOUN
ma-361	151	32	,	,	PUNCT
ma-361	151	33	completing	complete	VERB
ma-361	151	34	the	the	DET
ma-361	151	35	proof	proof	NOUN
ma-361	151	36	.	.	PUNCT
ma-361	152	1	�	�	PROPN
ma-361	152	2	theorem	theorem	VERB
ma-361	152	3	5	5	NUM
ma-361	152	4	(	(	PUNCT
ma-361	152	5	stability	stability	NOUN
ma-361	152	6	of	of	ADP
ma-361	152	7	nonlinear	nonlinear	ADJ
ma-361	152	8	norm	norm	NOUN
ma-361	152	9	-	-	PUNCT
ma-361	152	10	attainment	attainment	NOUN
ma-361	152	11	)	)	PUNCT
ma-361	152	12	.	.	PUNCT
ma-361	153	1	let	let	VERB
ma-361	153	2	(	(	PUNCT
ma-361	153	3	x	x	NOUN
ma-361	153	4	,	,	PUNCT
ma-361	153	5	τ	τ	X
ma-361	153	6	)	)	PUNCT
ma-361	153	7	be	be	AUX
ma-361	153	8	a	a	DET
ma-361	153	9	locally	locally	ADV
ma-361	153	10	convex	convex	ADJ
ma-361	153	11	space	space	NOUN
ma-361	153	12	with	with	ADP
ma-361	153	13	the	the	DET
ma-361	153	14	mackey	mackey	PROPN
ma-361	153	15	-	-	PUNCT
ma-361	153	16	arens	aren	NOUN
ma-361	153	17	property	property	NOUN
ma-361	153	18	.	.	PUNCT
ma-361	154	1	the	the	DET
ma-361	154	2	set	set	NOUN
ma-361	154	3	of	of	ADP
ma-361	154	4	τ	τ	PROPN
ma-361	154	5	-	-	ADJ
ma-361	154	6	continuous	continuous	ADJ
ma-361	154	7	convex	convex	NOUN
ma-361	154	8	functions	function	NOUN
ma-361	154	9	attaining	attain	VERB
ma-361	154	10	their	their	PRON
ma-361	154	11	norms	norm	NOUN
ma-361	154	12	is	be	AUX
ma-361	154	13	:	:	PUNCT
ma-361	154	14	•	•	ADP
ma-361	154	15	a	a	DET
ma-361	154	16	gδ	gδ	NOUN
ma-361	154	17	subset	subset	NOUN
ma-361	154	18	in	in	ADP
ma-361	154	19	the	the	DET
ma-361	154	20	topology	topology	NOUN
ma-361	154	21	of	of	ADP
ma-361	154	22	uniform	uniform	ADJ
ma-361	154	23	convergence	convergence	NOUN
ma-361	154	24	on	on	ADP
ma-361	154	25	bounded	bounded	ADJ
ma-361	154	26	sets	set	NOUN
ma-361	154	27	•	•	NOUN
ma-361	154	28	stable	stable	ADJ
ma-361	154	29	under	under	ADP
ma-361	154	30	finite	finite	PROPN
ma-361	154	31	inf	inf	NOUN
ma-361	154	32	-	-	PUNCT
ma-361	154	33	convolutions	convolution	NOUN
ma-361	154	34	•	•	NOUN
ma-361	154	35	not	not	PART
ma-361	154	36	preserved	preserve	VERB
ma-361	154	37	by	by	ADP
ma-361	154	38	epi	epi	NOUN
ma-361	154	39	-	-	NOUN
ma-361	154	40	sums	sum	NOUN
ma-361	154	41	in	in	ADP
ma-361	154	42	general	general	ADJ
ma-361	154	43	https://doi.org/10.28924/ada/ma.5.17	https://doi.org/10.28924/ada/ma.5.17	PROPN
ma-361	154	44	eur	eur	PROPN
ma-361	154	45	.	.	PUNCT
ma-361	155	1	j.	j.	PROPN
ma-361	155	2	math	math	PROPN
ma-361	155	3	.	.	PUNCT
ma-361	156	1	anal	anal	PROPN
ma-361	156	2	.	.	PUNCT
ma-361	157	1	10.28924	10.28924	NUM
ma-361	157	2	/	/	SYM
ma-361	157	3	ada	ada	PROPN
ma-361	157	4	/	/	SYM
ma-361	157	5	ma.5.17	ma.5.17	ADJ
ma-361	157	6	7	7	NUM
ma-361	157	7	proof	proof	NOUN
ma-361	157	8	.	.	PUNCT
ma-361	158	1	we	we	PRON
ma-361	158	2	employ	employ	VERB
ma-361	158	3	a	a	DET
ma-361	158	4	categorical	categorical	ADJ
ma-361	158	5	approach	approach	NOUN
ma-361	158	6	combined	combine	VERB
ma-361	158	7	with	with	ADP
ma-361	158	8	baire	baire	NOUN
ma-361	158	9	category	category	NOUN
ma-361	158	10	techniques	technique	NOUN
ma-361	158	11	:	:	PUNCT
ma-361	158	12	part	part	NOUN
ma-361	158	13	(	(	PUNCT
ma-361	158	14	i	i	NOUN
ma-361	158	15	):	):	PUNCT
ma-361	158	16	gδ	gδ	PROPN
ma-361	158	17	propertylet	propertylet	VERB
ma-361	158	18	an	an	PRON
ma-361	158	19	:	:	PUNCT
ma-361	158	20	=	=	SYM
ma-361	158	21	{	{	PUNCT
ma-361	158	22	f	f	NOUN
ma-361	158	23	:	:	PUNCT
ma-361	158	24	∃x	∃x	PROPN
ma-361	158	25	∈	∈	PROPN
ma-361	158	26	x	x	PUNCT
ma-361	158	27	with	with	ADP
ma-361	158	28	f	f	PROPN
ma-361	158	29	(	(	PUNCT
ma-361	158	30	x	x	X
ma-361	158	31	)	)	PUNCT
ma-361	158	32	>	>	PUNCT
ma-361	158	33	‖f	‖f	PUNCT
ma-361	158	34	‖	‖	ADJ
ma-361	158	35	−	−	NUM
ma-361	158	36	1	1	NUM
ma-361	158	37	/	/	SYM
ma-361	158	38	n	n	CCONJ
ma-361	158	39	}	}	PUNCT
ma-361	158	40	.	.	PUNCT
ma-361	159	1	each	each	PRON
ma-361	159	2	an	an	PRON
ma-361	159	3	is	be	AUX
ma-361	159	4	open	open	ADJ
ma-361	159	5	in	in	ADP
ma-361	159	6	the	the	DET
ma-361	159	7	topology	topology	NOUN
ma-361	159	8	of	of	ADP
ma-361	159	9	uniformconvergence	uniformconvergence	NOUN
ma-361	159	10	on	on	ADP
ma-361	159	11	bounded	bounded	ADJ
ma-361	159	12	sets	set	NOUN
ma-361	159	13	by	by	ADP
ma-361	159	14	the	the	DET
ma-361	159	15	mackey	mackey	PROPN
ma-361	159	16	-	-	PUNCT
ma-361	159	17	arens	aren	NOUN
ma-361	159	18	property	property	NOUN
ma-361	159	19	.	.	PUNCT
ma-361	160	1	the	the	DET
ma-361	160	2	attainment	attainment	ADJ
ma-361	160	3	set	set	NOUN
ma-361	160	4	is	be	AUX
ma-361	160	5	⋂nan	⋂nan	NOUN
ma-361	160	6	.	.	PUNCT
ma-361	161	1	part	part	NOUN
ma-361	161	2	(	(	PUNCT
ma-361	161	3	ii	ii	NOUN
ma-361	161	4	):	):	PUNCT
ma-361	161	5	inf	inf	ADJ
ma-361	161	6	-	-	PUNCT
ma-361	161	7	convolution	convolution	NOUN
ma-361	161	8	stabilitygiven	stabilitygiven	VERB
ma-361	161	9	norm	norm	NOUN
ma-361	161	10	-	-	PUNCT
ma-361	161	11	attaining	attain	VERB
ma-361	161	12	f	f	PROPN
ma-361	161	13	,	,	PUNCT
ma-361	161	14	g	g	PROPN
ma-361	161	15	,	,	PUNCT
ma-361	161	16	consider	consider	VERB
ma-361	161	17	(	(	PUNCT
ma-361	161	18	f	f	X
ma-361	161	19	�	�	PROPN
ma-361	161	20	g)(x	g)(x	PROPN
ma-361	161	21	)	)	PUNCT
ma-361	161	22	:	:	PUNCT
ma-361	162	1	=	=	SYM
ma-361	162	2	infy{f	infy{f	NOUN
ma-361	162	3	(	(	PUNCT
ma-361	162	4	y	y	NOUN
ma-361	162	5	)	)	PUNCT
ma-361	162	6	+	+	CCONJ
ma-361	163	1	g(x	g(x	NUM
ma-361	163	2	−	−	PROPN
ma-361	163	3	y	y	NOUN
ma-361	163	4	)	)	PUNCT
ma-361	163	5	}	}	PUNCT
ma-361	163	6	.	.	PUNCT
ma-361	164	1	let	let	VERB
ma-361	164	2	xf	xf	PROPN
ma-361	164	3	,	,	PUNCT
ma-361	164	4	xg	xg	PROPN
ma-361	164	5	be	be	AUX
ma-361	164	6	attainmentpoints	attainmentpoint	NOUN
ma-361	164	7	.	.	PUNCT
ma-361	165	1	then	then	ADV
ma-361	165	2	:	:	PUNCT
ma-361	165	3	(	(	PUNCT
ma-361	165	4	f	f	X
ma-361	165	5	�	�	PROPN
ma-361	165	6	g)(xf	g)(xf	ADJ
ma-361	165	7	+	+	NOUN
ma-361	165	8	xg	xg	NOUN
ma-361	165	9	)	)	PUNCT
ma-361	166	1	=	=	SYM
ma-361	166	2	f	f	PROPN
ma-361	166	3	(	(	PUNCT
ma-361	166	4	xf	xf	PROPN
ma-361	166	5	)	)	PUNCT
ma-361	167	1	+	+	CCONJ
ma-361	167	2	g(xg	g(xg	ADJ
ma-361	167	3	)	)	PUNCT
ma-361	167	4	=	=	SYM
ma-361	167	5	‖f	‖f	ADP
ma-361	167	6	‖+	‖+	NUM
ma-361	167	7	‖g‖	‖g‖	NUM
ma-361	167	8	=	=	SYM
ma-361	167	9	‖f	‖f	NOUN
ma-361	167	10	�	�	NOUN
ma-361	167	11	g‖	g‖	NOUN
ma-361	167	12	where	where	SCONJ
ma-361	167	13	the	the	DET
ma-361	167	14	last	last	ADJ
ma-361	167	15	equality	equality	NOUN
ma-361	167	16	uses	use	VERB
ma-361	167	17	the	the	DET
ma-361	167	18	hahn	hahn	NOUN
ma-361	167	19	-	-	PUNCT
ma-361	167	20	banach	banach	NOUN
ma-361	167	21	extension	extension	NOUN
ma-361	167	22	property	property	NOUN
ma-361	167	23	.	.	PUNCT
ma-361	168	1	part	part	NOUN
ma-361	168	2	(	(	PUNCT
ma-361	168	3	iii	iii	NOUN
ma-361	168	4	):	):	PUNCT
ma-361	168	5	epi	epi	ADJ
ma-361	168	6	-	-	ADJ
ma-361	168	7	sum	sum	NOUN
ma-361	168	8	counterexampleon	counterexampleon	NOUN
ma-361	168	9	`	`	PUNCT
ma-361	168	10	2	2	NUM
ma-361	168	11	,	,	PUNCT
ma-361	168	12	take	take	VERB
ma-361	168	13	f	f	PROPN
ma-361	168	14	(	(	PUNCT
ma-361	168	15	x	x	NOUN
ma-361	168	16	)	)	PUNCT
ma-361	168	17	=	=	PUNCT
ma-361	169	1	‖x‖	‖x‖	PROPN
ma-361	169	2	and	and	CCONJ
ma-361	169	3	g(x	g(x	NOUN
ma-361	169	4	)	)	PUNCT
ma-361	169	5	=	=	SYM
ma-361	169	6	δ{e1}⊥(x	δ{e1}⊥(x	ADJ
ma-361	169	7	)	)	PUNCT
ma-361	169	8	.	.	PUNCT
ma-361	170	1	both	both	PRON
ma-361	170	2	attain	attain	VERB
ma-361	170	3	their	their	PRON
ma-361	170	4	norms	norm	NOUN
ma-361	170	5	,	,	PUNCT
ma-361	170	6	but	but	CCONJ
ma-361	170	7	:	:	PUNCT
ma-361	170	8	(	(	PUNCT
ma-361	170	9	f	f	X
ma-361	170	10	+	+	CCONJ
ma-361	170	11	g)(x	g)(x	PROPN
ma-361	170	12	)	)	PUNCT
ma-361	170	13	=	=	PUNCT
ma-361	171	1	‖x‖	‖x‖	VERB
ma-361	171	2	if	if	SCONJ
ma-361	171	3	x1	x1	PROPN
ma-361	171	4	=	=	NOUN
ma-361	171	5	0	0	PUNCT
ma-361	172	1	+	+	NUM
ma-361	172	2	∞	∞	PROPN
ma-361	172	3	otherwise	otherwise	ADV
ma-361	172	4	does	do	AUX
ma-361	172	5	not	not	PART
ma-361	172	6	attain	attain	VERB
ma-361	172	7	its	its	PRON
ma-361	172	8	norm	norm	NOUN
ma-361	172	9	in	in	ADP
ma-361	172	10	`	`	PUNCT
ma-361	172	11	2	2	X
ma-361	172	12	.	.	X
ma-361	172	13	�	�	PROPN
ma-361	172	14	theorem	theorem	VERB
ma-361	172	15	6	6	NUM
ma-361	172	16	(	(	PUNCT
ma-361	172	17	density	density	NOUN
ma-361	172	18	of	of	ADP
ma-361	172	19	nonlinear	nonlinear	ADJ
ma-361	172	20	norm	norm	NOUN
ma-361	172	21	-	-	PUNCT
ma-361	172	22	attainers	attainer	NOUN
ma-361	172	23	)	)	PUNCT
ma-361	172	24	.	.	PUNCT
ma-361	173	1	for	for	ADP
ma-361	173	2	any	any	DET
ma-361	173	3	frechet	frechet	NOUN
ma-361	173	4	space	space	NOUN
ma-361	173	5	x	x	PUNCT
ma-361	173	6	and	and	CCONJ
ma-361	173	7	1	1	NUM
ma-361	173	8	<	<	X
ma-361	173	9	p	p	X
ma-361	173	10	<	<	X
ma-361	173	11	∞	∞	PROPN
ma-361	173	12	,	,	PUNCT
ma-361	173	13	the	the	DET
ma-361	173	14	set	set	NOUN
ma-361	173	15	:	:	PUNCT
ma-361	173	16	{	{	PUNCT
ma-361	173	17	f	f	PROPN
ma-361	173	18	∈	∈	PROPN
ma-361	173	19	lp(x	lp(x	PROPN
ma-361	173	20	)	)	PUNCT
ma-361	173	21	:	:	PUNCT
ma-361	173	22	f	f	PROPN
ma-361	173	23	attains	attain	VERB
ma-361	173	24	its	its	PRON
ma-361	173	25	operator	operator	NOUN
ma-361	173	26	p	p	NOUN
ma-361	173	27	-	-	PUNCT
ma-361	173	28	norm	norm	NOUN
ma-361	173	29	}	}	PUNCT
ma-361	173	30	is	be	AUX
ma-361	173	31	dense	dense	ADJ
ma-361	173	32	in	in	ADP
ma-361	173	33	the	the	DET
ma-361	173	34	strong	strong	ADJ
ma-361	173	35	operator	operator	NOUN
ma-361	173	36	topology	topology	NOUN
ma-361	173	37	if	if	SCONJ
ma-361	173	38	and	and	CCONJ
ma-361	173	39	only	only	ADV
ma-361	173	40	if	if	SCONJ
ma-361	173	41	x	x	PRON
ma-361	173	42	admits	admit	VERB
ma-361	173	43	an	an	DET
ma-361	173	44	equivalent	equivalent	ADJ
ma-361	173	45	plurisubharmonic	plurisubharmonic	ADJ
ma-361	173	46	norm	norm	NOUN
ma-361	173	47	.	.	PUNCT
ma-361	174	1	proof	proof	NOUN
ma-361	174	2	.	.	PUNCT
ma-361	175	1	the	the	DET
ma-361	175	2	proof	proof	NOUN
ma-361	175	3	combines	combine	VERB
ma-361	175	4	pluripotential	pluripotential	ADJ
ma-361	175	5	theory	theory	NOUN
ma-361	175	6	with	with	ADP
ma-361	175	7	operator	operator	NOUN
ma-361	175	8	algebra	algebra	NOUN
ma-361	175	9	techniques	technique	NOUN
ma-361	175	10	:	:	PUNCT
ma-361	175	11	necessity	necessity	NOUN
ma-361	175	12	(	(	PUNCT
ma-361	175	13	⇒)assume	⇒)assume	ADJ
ma-361	175	14	density	density	NOUN
ma-361	175	15	of	of	ADP
ma-361	175	16	norm	norm	NOUN
ma-361	175	17	-	-	PUNCT
ma-361	175	18	attainers	attainer	NOUN
ma-361	175	19	.	.	PUNCT
ma-361	176	1	for	for	ADP
ma-361	176	2	any	any	DET
ma-361	176	3	x∗∗	x∗∗	PROPN
ma-361	176	4	∈	∈	PROPN
ma-361	176	5	x∗∗	x∗∗	PROPN
ma-361	176	6	,	,	PUNCT
ma-361	176	7	the	the	DET
ma-361	176	8	evaluation	evaluation	NOUN
ma-361	176	9	functional	functional	ADJ
ma-361	176	10	δx∗∗(f	δx∗∗(f	NOUN
ma-361	176	11	)	)	PUNCT
ma-361	177	1	=	=	SYM
ma-361	177	2	f	f	PROPN
ma-361	177	3	(	(	PUNCT
ma-361	177	4	x∗∗)must	x∗∗)must	AUX
ma-361	177	5	be	be	AUX
ma-361	177	6	norm	norm	NOUN
ma-361	177	7	-	-	PUNCT
ma-361	177	8	attaining	attain	VERB
ma-361	177	9	on	on	ADP
ma-361	177	10	lp(x	lp(x	NOUN
ma-361	177	11	)	)	PUNCT
ma-361	177	12	.	.	PUNCT
ma-361	178	1	this	this	DET
ma-361	178	2	forces	force	NOUN
ma-361	178	3	x∗∗	x∗∗	PROPN
ma-361	178	4	∈	∈	PROPN
ma-361	178	5	x	x	PUNCT
ma-361	178	6	via	via	ADP
ma-361	178	7	the	the	DET
ma-361	178	8	plurisubharmonic	plurisubharmonic	ADJ
ma-361	178	9	maximum	maximum	ADJ
ma-361	178	10	principle	principle	NOUN
ma-361	178	11	.	.	PUNCT
ma-361	179	1	sufficiency	sufficiency	NOUN
ma-361	179	2	(	(	PUNCT
ma-361	179	3	⇐	⇐	ADJ
ma-361	179	4	)	)	PUNCT
ma-361	179	5	let	let	VERB
ma-361	179	6	x	x	PRON
ma-361	179	7	have	have	VERB
ma-361	179	8	a	a	DET
ma-361	179	9	plurisubharmonic	plurisubharmonic	ADJ
ma-361	179	10	norm	norm	NOUN
ma-361	179	11	‖	‖	PROPN
ma-361	179	12	·	·	PUNCT
ma-361	179	13	‖psh	‖psh	PROPN
ma-361	179	14	.	.	PUNCT
ma-361	180	1	for	for	ADP
ma-361	180	2	any	any	DET
ma-361	180	3	t	t	PROPN
ma-361	180	4	∈	∈	PROPN
ma-361	180	5	lp(x	lp(x	PUNCT
ma-361	180	6	)	)	PUNCT
ma-361	180	7	and	and	CCONJ
ma-361	180	8	ε	ε	PROPN
ma-361	180	9	>	>	SYM
ma-361	180	10	0:(1	0:(1	PROPN
ma-361	180	11	)	)	PUNCT
ma-361	180	12	approximate	approximate	PROPN
ma-361	180	13	t	t	PROPN
ma-361	180	14	by	by	ADP
ma-361	180	15	finite	finite	ADJ
ma-361	180	16	-	-	ADJ
ma-361	180	17	rank	rank	ADJ
ma-361	180	18	operators	operator	NOUN
ma-361	180	19	tn	tn	PROPN
ma-361	180	20	→	→	SYM
ma-361	180	21	t	t	PROPN
ma-361	180	22	in	in	ADP
ma-361	180	23	sot(2	sot(2	NOUN
ma-361	180	24	)	)	PUNCT
ma-361	180	25	solve	solve	VERB
ma-361	180	26	the	the	DET
ma-361	180	27	∂-equation	∂-equation	NOUN
ma-361	180	28	on	on	ADP
ma-361	180	29	ran(tn	ran(tn	NOUN
ma-361	180	30	)	)	PUNCT
ma-361	180	31	to	to	PART
ma-361	180	32	get	get	VERB
ma-361	180	33	attainment	attainment	ADJ
ma-361	180	34	points	point	NOUN
ma-361	180	35	xn(3	xn(3	X
ma-361	180	36	)	)	PUNCT
ma-361	180	37	use	use	VERB
ma-361	180	38	the	the	DET
ma-361	180	39	hormander	hormander	ADJ
ma-361	180	40	estimate	estimate	NOUN
ma-361	180	41	to	to	PART
ma-361	180	42	show	show	VERB
ma-361	180	43	lim	lim	PROPN
ma-361	180	44	sup	sup	PROPN
ma-361	180	45	‖tnxn‖	‖tnxn‖	VERB
ma-361	181	1	≥	≥	NOUN
ma-361	181	2	‖t‖	‖t‖	ADP
ma-361	181	3	−	−	PROPN
ma-361	181	4	εthe	εthe	ADJ
ma-361	181	5	key	key	NOUN
ma-361	181	6	is	be	AUX
ma-361	181	7	the	the	DET
ma-361	181	8	inequality	inequality	NOUN
ma-361	181	9	:	:	PUNCT
ma-361	181	10	log	log	VERB
ma-361	181	11	‖t‖op	‖t‖op	PROPN
ma-361	181	12	≤	≤	PROPN
ma-361	181	13	sup	sup	NOUN
ma-361	181	14	‖x‖psh=1	‖x‖psh=1	PROPN
ma-361	181	15	log	log	NOUN
ma-361	181	16	‖tx‖+	‖tx‖+	NOUN
ma-361	181	17	cpcapp(sp(t	cpcapp(sp(t	NOUN
ma-361	181	18	)	)	PUNCT
ma-361	181	19	)	)	PUNCT
ma-361	181	20	where	where	SCONJ
ma-361	181	21	capp	capp	PROPN
ma-361	181	22	is	be	AUX
ma-361	181	23	the	the	DET
ma-361	181	24	p	p	NOUN
ma-361	181	25	-	-	PUNCT
ma-361	181	26	capacity	capacity	NOUN
ma-361	181	27	.	.	PUNCT
ma-361	182	1	the	the	DET
ma-361	182	2	plurisubharmonicity	plurisubharmonicity	NOUN
ma-361	182	3	condition	condition	NOUN
ma-361	182	4	makes	make	VERB
ma-361	182	5	the	the	DET
ma-361	182	6	capacity	capacity	NOUN
ma-361	182	7	term	term	NOUN
ma-361	182	8	vanish	vanish	VERB
ma-361	182	9	.	.	PUNCT
ma-361	183	1	�	�	PROPN
ma-361	183	2	https://doi.org/10.28924/ada/ma.5.17	https://doi.org/10.28924/ada/ma.5.17	PROPN
ma-361	183	3	eur	eur	PROPN
ma-361	183	4	.	.	PUNCT
ma-361	184	1	j.	j.	PROPN
ma-361	184	2	math	math	PROPN
ma-361	184	3	.	.	PUNCT
ma-361	185	1	anal	anal	PROPN
ma-361	185	2	.	.	PUNCT
ma-361	186	1	10.28924	10.28924	NUM
ma-361	186	2	/	/	SYM
ma-361	186	3	ada	ada	NOUN
ma-361	186	4	/	/	SYM
ma-361	186	5	ma.5.17	ma.5.17	ADJ
ma-361	186	6	8	8	NUM
ma-361	186	7	theorem	theorem	VERB
ma-361	186	8	7	7	NUM
ma-361	186	9	(	(	PUNCT
ma-361	186	10	nonlinear	nonlinear	PROPN
ma-361	186	11	james	james	PROPN
ma-361	186	12	’	'	PUNCT
ma-361	186	13	theorem	theorem	NOUN
ma-361	186	14	)	)	PUNCT
ma-361	186	15	.	.	PUNCT
ma-361	187	1	a	a	DET
ma-361	187	2	bounded	bound	VERB
ma-361	187	3	complete	complete	NOUN
ma-361	187	4	locally	locally	ADV
ma-361	187	5	convex	convex	ADJ
ma-361	187	6	space	space	NOUN
ma-361	187	7	x	x	PUNCT
ma-361	187	8	is	be	AUX
ma-361	187	9	semireflexive	semireflexive	ADJ
ma-361	187	10	if	if	SCONJ
ma-361	187	11	and	and	CCONJ
ma-361	187	12	only	only	ADV
ma-361	187	13	if	if	SCONJ
ma-361	187	14	every	every	DET
ma-361	187	15	continuous	continuous	ADJ
ma-361	187	16	quasiconvex	quasiconvex	ADJ
ma-361	187	17	coercive	coercive	ADJ
ma-361	187	18	functional	functional	ADJ
ma-361	187	19	f	f	NOUN
ma-361	187	20	:	:	PUNCT
ma-361	187	21	x	x	X
ma-361	187	22	→	→	SYM
ma-361	187	23	r	r	NOUN
ma-361	187	24	attains	attain	VERB
ma-361	187	25	its	its	PRON
ma-361	187	26	supremum	supremum	NOUN
ma-361	187	27	on	on	ADP
ma-361	187	28	closed	closed	ADJ
ma-361	187	29	bounded	bound	VERB
ma-361	187	30	sets	set	NOUN
ma-361	187	31	.	.	PUNCT
ma-361	188	1	proof	proof	NOUN
ma-361	188	2	.	.	PUNCT
ma-361	189	1	(	(	PUNCT
ma-361	189	2	⇒	⇒	PROPN
ma-361	189	3	)	)	PUNCT
ma-361	189	4	suppose	suppose	VERB
ma-361	189	5	x	x	PRON
ma-361	189	6	is	be	AUX
ma-361	189	7	semi	semi	ADJ
ma-361	189	8	-	-	ADJ
ma-361	189	9	reflexive	reflexive	ADJ
ma-361	189	10	.	.	PUNCT
ma-361	190	1	let	let	VERB
ma-361	190	2	f	f	NOUN
ma-361	190	3	:	:	PUNCT
ma-361	190	4	x	x	X
ma-361	190	5	→	→	SYM
ma-361	190	6	r	r	NOUN
ma-361	190	7	be	be	VERB
ma-361	190	8	quasiconvex	quasiconvex	NOUN
ma-361	190	9	coercive	coercive	ADJ
ma-361	190	10	and	and	CCONJ
ma-361	190	11	continuous.for	continuous.for	ADP
ma-361	190	12	any	any	DET
ma-361	190	13	closed	closed	ADJ
ma-361	190	14	bounded	bounded	PROPN
ma-361	190	15	b	b	PROPN
ma-361	190	16	⊂	⊂	PROPN
ma-361	190	17	x	x	X
ma-361	190	18	,	,	PUNCT
ma-361	190	19	the	the	DET
ma-361	190	20	set	set	PROPN
ma-361	190	21	b	b	PROPN
ma-361	190	22	is	be	AUX
ma-361	190	23	weakly	weakly	ADV
ma-361	190	24	compact	compact	ADJ
ma-361	190	25	by	by	ADP
ma-361	190	26	semi	semi	ADJ
ma-361	190	27	-	-	ADJ
ma-361	190	28	reflexivity	reflexivity	ADJ
ma-361	190	29	.	.	PUNCT
ma-361	191	1	define	define	NOUN
ma-361	191	2	:	:	PUNCT
ma-361	192	1	f	f	X
ma-361	192	2	=	=	PRON
ma-361	192	3	{	{	PUNCT
ma-361	192	4	{	{	PUNCT
ma-361	192	5	x	x	PROPN
ma-361	192	6	∈	∈	PROPN
ma-361	192	7	b	b	PROPN
ma-361	192	8	:	:	PUNCT
ma-361	192	9	f	f	PROPN
ma-361	192	10	(	(	PUNCT
ma-361	192	11	x	x	X
ma-361	192	12	)	)	PUNCT
ma-361	192	13	≥	≥	NOUN
ma-361	192	14	α	α	NOUN
ma-361	192	15	}	}	PUNCT
ma-361	192	16	:	:	PUNCT
ma-361	192	17	α	α	X
ma-361	192	18	<	<	X
ma-361	192	19	sup	sup	PROPN
ma-361	192	20	b	b	PROPN
ma-361	192	21	f	f	PROPN
ma-361	192	22	}	}	PUNCT
ma-361	192	23	this	this	PRON
ma-361	192	24	is	be	AUX
ma-361	192	25	a	a	DET
ma-361	192	26	family	family	NOUN
ma-361	192	27	of	of	ADP
ma-361	192	28	weakly	weakly	ADJ
ma-361	192	29	closed	closed	ADJ
ma-361	192	30	sets	set	NOUN
ma-361	192	31	with	with	ADP
ma-361	192	32	finite	finite	ADJ
ma-361	192	33	intersection	intersection	NOUN
ma-361	192	34	property	property	NOUN
ma-361	192	35	by	by	ADP
ma-361	192	36	quasiconvexity	quasiconvexity	PROPN
ma-361	192	37	.	.	PUNCT
ma-361	193	1	by	by	ADP
ma-361	193	2	weakcompactness	weakcompactness	NOUN
ma-361	193	3	,	,	PUNCT
ma-361	193	4	⋂f	⋂f	PROPN
ma-361	193	5	6=	6=	PROPN
ma-361	193	6	∅	∅	NOUN
ma-361	193	7	,	,	PUNCT
ma-361	193	8	yielding	yield	VERB
ma-361	193	9	a	a	DET
ma-361	193	10	maximizer	maximizer	NOUN
ma-361	193	11	.	.	PUNCT
ma-361	194	1	(	(	PUNCT
ma-361	194	2	⇐	⇐	ADJ
ma-361	194	3	)	)	PUNCT
ma-361	194	4	assume	assume	VERB
ma-361	194	5	norm	norm	NOUN
ma-361	194	6	-	-	PUNCT
ma-361	194	7	attainment	attainment	NOUN
ma-361	194	8	holds	hold	NOUN
ma-361	194	9	.	.	PUNCT
ma-361	195	1	suppose	suppose	VERB
ma-361	195	2	x	x	PRON
ma-361	195	3	is	be	AUX
ma-361	195	4	not	not	PART
ma-361	195	5	semi	semi	ADJ
ma-361	195	6	-	-	ADJ
ma-361	195	7	reflexive	reflexive	ADJ
ma-361	195	8	.	.	PUNCT
ma-361	196	1	then	then	ADV
ma-361	196	2	there	there	PRON
ma-361	196	3	exists	exist	VERB
ma-361	196	4	a	a	DET
ma-361	196	5	σ(x	σ(x	PROPN
ma-361	196	6	,	,	PUNCT
ma-361	196	7	x∗)-closed	x∗)-closed	PROPN
ma-361	196	8	bounded	bound	VERB
ma-361	196	9	set	set	VERB
ma-361	196	10	b	b	NOUN
ma-361	196	11	not	not	PART
ma-361	196	12	weakly	weakly	ADV
ma-361	196	13	compact	compact	ADJ
ma-361	196	14	.	.	PUNCT
ma-361	197	1	using	use	VERB
ma-361	197	2	a	a	DET
ma-361	197	3	construction	construction	NOUN
ma-361	197	4	from	from	ADP
ma-361	197	5	[	[	X
ma-361	197	6	2	2	NUM
ma-361	197	7	]	]	PUNCT
ma-361	197	8	,	,	PUNCT
ma-361	197	9	build	build	VERB
ma-361	197	10	a	a	DET
ma-361	197	11	coercive	coercive	ADJ
ma-361	197	12	continuousquasiconvex	continuousquasiconvex	NOUN
ma-361	197	13	function	function	NOUN
ma-361	197	14	:	:	PUNCT
ma-361	198	1	f	f	PROPN
ma-361	198	2	(	(	PUNCT
ma-361	198	3	x	x	X
ma-361	198	4	)	)	PUNCT
ma-361	198	5	=	=	SYM
ma-361	198	6	inf{λ	inf{λ	X
ma-361	198	7	>	>	X
ma-361	198	8	0	0	NUM
ma-361	199	1	:	:	PUNCT
ma-361	199	2	x	x	X
ma-361	199	3	∈	∈	NOUN
ma-361	200	1	λc}where	λc}where	INTJ
ma-361	200	2	c	c	NOUN
ma-361	200	3	is	be	AUX
ma-361	200	4	a	a	DET
ma-361	200	5	carefully	carefully	ADV
ma-361	200	6	chosen	choose	VERB
ma-361	200	7	barrel	barrel	NOUN
ma-361	200	8	containing	contain	VERB
ma-361	200	9	b.	b.	PROPN
ma-361	200	10	by	by	ADP
ma-361	200	11	hypothesis	hypothesis	NOUN
ma-361	200	12	,	,	PUNCT
ma-361	200	13	f	f	PROPN
ma-361	200	14	attains	attain	VERB
ma-361	200	15	its	its	PRON
ma-361	200	16	supremum	supremum	NOUN
ma-361	200	17	on	on	ADP
ma-361	200	18	b	b	NOUN
ma-361	200	19	,	,	PUNCT
ma-361	200	20	contradicting	contradict	VERB
ma-361	200	21	james	jame	NOUN
ma-361	200	22	’	'	PUNCT
ma-361	200	23	weak	weak	ADJ
ma-361	200	24	compactness	compactness	NOUN
ma-361	200	25	theorem	theorem	NOUN
ma-361	200	26	in	in	ADP
ma-361	200	27	its	its	PRON
ma-361	200	28	generalized	generalized	ADJ
ma-361	200	29	form	form	NOUN
ma-361	200	30	[	[	X
ma-361	200	31	9	9	NUM
ma-361	200	32	]	]	PUNCT
ma-361	200	33	.	.	PUNCT
ma-361	201	1	�	�	PROPN
ma-361	201	2	theorem	theorem	VERB
ma-361	201	3	8	8	NUM
ma-361	201	4	(	(	PUNCT
ma-361	201	5	subdifferential	subdifferential	ADJ
ma-361	201	6	characterization	characterization	NOUN
ma-361	201	7	)	)	PUNCT
ma-361	201	8	.	.	PUNCT
ma-361	202	1	for	for	ADP
ma-361	202	2	a	a	DET
ma-361	202	3	proper	proper	ADJ
ma-361	202	4	convex	convex	NOUN
ma-361	202	5	τ	τ	PROPN
ma-361	202	6	-	-	PUNCT
ma-361	202	7	lsc	lsc	PROPN
ma-361	202	8	function	function	PROPN
ma-361	202	9	f	f	NOUN
ma-361	202	10	:	:	PUNCT
ma-361	202	11	x	x	X
ma-361	202	12	→	→	SYM
ma-361	202	13	r	r	NOUN
ma-361	202	14	∪	∪	X
ma-361	202	15	{	{	PUNCT
ma-361	202	16	+	+	NOUN
ma-361	202	17	∞	∞	NOUN
ma-361	202	18	}	}	PUNCT
ma-361	202	19	on	on	ADP
ma-361	202	20	a	a	DET
ma-361	202	21	locally	locally	ADV
ma-361	202	22	convex	convex	ADJ
ma-361	202	23	space	space	NOUN
ma-361	202	24	:	:	PUNCT
ma-361	202	25	f	f	PROPN
ma-361	202	26	attains	attain	VERB
ma-361	202	27	its	its	PRON
ma-361	202	28	norm	norm	NOUN
ma-361	202	29	at	at	ADP
ma-361	202	30	x0	x0	PROPN
ma-361	202	31	⇐	⇐	ADJ
ma-361	202	32	⇒	⇒	NOUN
ma-361	202	33	0	0	NUM
ma-361	202	34	∈	∈	PROPN
ma-361	202	35	int(∂f	int(∂f	NOUN
ma-361	203	1	(	(	PUNCT
ma-361	203	2	x0)−	x0)−	PROPN
ma-361	203	3	∂f	∂f	PROPN
ma-361	203	4	(	(	PUNCT
ma-361	203	5	0	0	NUM
ma-361	203	6	)	)	PUNCT
ma-361	203	7	)	)	PUNCT
ma-361	203	8	moreover	moreover	ADV
ma-361	203	9	,	,	PUNCT
ma-361	203	10	the	the	DET
ma-361	203	11	attainment	attainment	ADJ
ma-361	203	12	set	set	NOUN
ma-361	203	13	is	be	AUX
ma-361	203	14	always	always	ADV
ma-361	203	15	a	a	DET
ma-361	203	16	τ	τ	PROPN
ma-361	203	17	-	-	PUNCT
ma-361	203	18	borel	borel	NOUN
ma-361	203	19	subset	subset	NOUN
ma-361	203	20	of	of	ADP
ma-361	203	21	x	x	X
ma-361	203	22	.	.	PUNCT
ma-361	204	1	proof	proof	NOUN
ma-361	204	2	.	.	PUNCT
ma-361	205	1	(	(	PUNCT
ma-361	205	2	⇒	⇒	PROPN
ma-361	205	3	)	)	PUNCT
ma-361	205	4	if	if	SCONJ
ma-361	205	5	f	f	PROPN
ma-361	205	6	attains	attain	VERB
ma-361	205	7	its	its	PRON
ma-361	205	8	norm	norm	NOUN
ma-361	205	9	at	at	ADP
ma-361	205	10	x0	x0	PROPN
ma-361	205	11	,	,	PUNCT
ma-361	205	12	then	then	ADV
ma-361	205	13	0	0	NUM
ma-361	205	14	∈	∈	PROPN
ma-361	205	15	∂(f	∂(f	NOUN
ma-361	205	16	−	−	PROPN
ma-361	205	17	‖f	‖f	PRON
ma-361	205	18	‖)(x0	‖)(x0	NOUN
ma-361	205	19	)	)	PUNCT
ma-361	205	20	.	.	PUNCT
ma-361	206	1	by	by	ADP
ma-361	206	2	the	the	DET
ma-361	206	3	brondsted	brondste	VERB
ma-361	206	4	-	-	PUNCT
ma-361	206	5	rockafellartheorem	rockafellartheorem	VERB
ma-361	206	6	[	[	X
ma-361	206	7	14	14	NUM
ma-361	206	8	]	]	X
ma-361	206	9	,	,	PUNCT
ma-361	206	10	there	there	PRON
ma-361	206	11	exist	exist	VERB
ma-361	206	12	sequences	sequence	NOUN
ma-361	206	13	xn	xn	PUNCT
ma-361	207	1	→	→	SYM
ma-361	207	2	x0	x0	PROPN
ma-361	207	3	and	and	CCONJ
ma-361	207	4	x∗n	x∗n	PUNCT
ma-361	207	5	∈	∈	PROPN
ma-361	207	6	∂f	∂f	PROPN
ma-361	207	7	(	(	PUNCT
ma-361	207	8	xn	xn	PROPN
ma-361	207	9	)	)	PUNCT
ma-361	207	10	with	with	ADP
ma-361	207	11	x∗n	x∗n	NUM
ma-361	207	12	→	→	SYM
ma-361	207	13	0	0	NUM
ma-361	207	14	.	.	PUNCT
ma-361	208	1	the	the	DET
ma-361	208	2	interioritycondition	interioritycondition	NOUN
ma-361	208	3	follows	follow	VERB
ma-361	208	4	from	from	ADP
ma-361	208	5	the	the	DET
ma-361	208	6	multidirectional	multidirectional	ADJ
ma-361	208	7	mean	mean	NOUN
ma-361	208	8	value	value	NOUN
ma-361	208	9	inequality	inequality	NOUN
ma-361	208	10	.	.	PUNCT
ma-361	209	1	(	(	PUNCT
ma-361	209	2	⇐	⇐	ADJ
ma-361	209	3	)	)	PUNCT
ma-361	209	4	assume	assume	VERB
ma-361	209	5	0	0	NUM
ma-361	209	6	∈	∈	PROPN
ma-361	209	7	int(∂f	int(∂f	NOUN
ma-361	209	8	(	(	PUNCT
ma-361	209	9	x0)−	x0)−	PROPN
ma-361	209	10	∂f	∂f	PROPN
ma-361	209	11	(	(	PUNCT
ma-361	209	12	0	0	NUM
ma-361	209	13	)	)	PUNCT
ma-361	209	14	)	)	PUNCT
ma-361	209	15	.	.	PUNCT
ma-361	210	1	by	by	ADP
ma-361	210	2	the	the	DET
ma-361	210	3	borwein	borwein	ADJ
ma-361	210	4	-	-	PUNCT
ma-361	210	5	preiss	preiss	NOUN
ma-361	210	6	variational	variational	ADJ
ma-361	210	7	principle	principle	NOUN
ma-361	210	8	[	[	X
ma-361	210	9	5	5	NUM
ma-361	210	10	]	]	PUNCT
ma-361	210	11	,	,	PUNCT
ma-361	210	12	there	there	PRON
ma-361	210	13	exists	exist	VERB
ma-361	210	14	v	v	ADP
ma-361	210	15	∈	∈	PROPN
ma-361	210	16	x	x	PUNCT
ma-361	210	17	such	such	ADJ
ma-361	210	18	that	that	SCONJ
ma-361	210	19	:	:	PUNCT
ma-361	210	20	f	f	X
ma-361	210	21	(	(	PUNCT
ma-361	210	22	x0	x0	PROPN
ma-361	210	23	+	+	PROPN
ma-361	210	24	v)−	v)−	PROPN
ma-361	210	25	f	f	X
ma-361	210	26	(	(	PUNCT
ma-361	210	27	x0	x0	PROPN
ma-361	210	28	)	)	PUNCT
ma-361	210	29	≥	≥	NOUN
ma-361	210	30	δ‖v‖for	δ‖v‖for	ADP
ma-361	210	31	some	some	DET
ma-361	210	32	δ	δ	PROPN
ma-361	210	33	>	>	X
ma-361	210	34	0	0	PROPN
ma-361	210	35	.	.	PUNCT
ma-361	211	1	this	this	DET
ma-361	211	2	gradient	gradient	ADJ
ma-361	211	3	inequality	inequality	NOUN
ma-361	211	4	forces	force	NOUN
ma-361	211	5	norm	norm	NOUN
ma-361	211	6	-	-	PUNCT
ma-361	211	7	attainment	attainment	NOUN
ma-361	211	8	.	.	PUNCT
ma-361	212	1	for	for	ADP
ma-361	212	2	the	the	DET
ma-361	212	3	borel	borel	PROPN
ma-361	212	4	claim	claim	NOUN
ma-361	212	5	:	:	PUNCT
ma-361	212	6	theattainment	theattainment	NOUN
ma-361	212	7	set	set	NOUN
ma-361	212	8	equals	equal	VERB
ma-361	212	9	:	:	PUNCT
ma-361	212	10	⋃	⋃	VERB
ma-361	212	11	n∈n	n∈n	NOUN
ma-361	212	12	(	(	PUNCT
ma-361	212	13	n−1∂f	n−1∂f	NOUN
ma-361	212	14	∗(bx∗(0	∗(bx∗(0	NOUN
ma-361	212	15	,	,	PUNCT
ma-361	212	16	n	n	CCONJ
ma-361	212	17	)	)	PUNCT
ma-361	212	18	)	)	PUNCT
ma-361	212	19	)	)	PUNCT
ma-361	212	20	where	where	SCONJ
ma-361	212	21	f	f	PROPN
ma-361	212	22	∗	∗	NOUN
ma-361	212	23	is	be	AUX
ma-361	212	24	the	the	DET
ma-361	212	25	fenchel	fenchel	PROPN
ma-361	212	26	conjugate	conjugate	NOUN
ma-361	212	27	.	.	PUNCT
ma-361	213	1	this	this	PRON
ma-361	213	2	is	be	AUX
ma-361	213	3	a	a	DET
ma-361	213	4	countable	countable	ADJ
ma-361	213	5	union	union	NOUN
ma-361	213	6	of	of	ADP
ma-361	213	7	τ	τ	PROPN
ma-361	213	8	-	-	ADJ
ma-361	213	9	continuous	continuous	ADJ
ma-361	213	10	images	image	NOUN
ma-361	213	11	of	of	ADP
ma-361	213	12	weak∗-compact	weak∗-compact	PROPN
ma-361	213	13	sets	set	NOUN
ma-361	213	14	,	,	PUNCT
ma-361	213	15	hence	hence	ADV
ma-361	213	16	τ	τ	PROPN
ma-361	213	17	-	-	PUNCT
ma-361	213	18	borel	borel	NOUN
ma-361	213	19	.	.	PUNCT
ma-361	214	1	�	�	PROPN
ma-361	214	2	theorem	theorem	VERB
ma-361	214	3	9	9	NUM
ma-361	214	4	(	(	PUNCT
ma-361	214	5	nonlinear	nonlinear	ADJ
ma-361	214	6	krein	krein	NOUN
ma-361	214	7	-	-	PUNCT
ma-361	214	8	milman	milman	NOUN
ma-361	214	9	property	property	NOUN
ma-361	214	10	)	)	PUNCT
ma-361	214	11	.	.	PUNCT
ma-361	215	1	let	let	VERB
ma-361	215	2	k	k	PRON
ma-361	215	3	be	be	AUX
ma-361	215	4	a	a	DET
ma-361	215	5	τ	τ	NOUN
ma-361	215	6	-	-	ADJ
ma-361	215	7	compact	compact	ADJ
ma-361	215	8	convex	convex	NOUN
ma-361	215	9	set	set	VERB
ma-361	215	10	in	in	ADP
ma-361	215	11	a	a	DET
ma-361	215	12	locally	locally	ADV
ma-361	215	13	convex	convex	ADJ
ma-361	215	14	space	space	NOUN
ma-361	215	15	.	.	PUNCT
ma-361	216	1	every	every	DET
ma-361	216	2	τ	τ	PROPN
ma-361	216	3	-	-	ADJ
ma-361	216	4	continuous	continuous	ADJ
ma-361	216	5	convex	convex	NOUN
ma-361	216	6	function	function	NOUN
ma-361	216	7	on	on	ADP
ma-361	216	8	k	k	PROPN
ma-361	216	9	attains	attain	NOUN
ma-361	216	10	its	its	PRON
ma-361	216	11	maximum	maximum	NOUN
ma-361	216	12	at	at	ADP
ma-361	216	13	some	some	DET
ma-361	216	14	extreme	extreme	ADJ
ma-361	216	15	point	point	NOUN
ma-361	216	16	if	if	SCONJ
ma-361	216	17	and	and	CCONJ
ma-361	216	18	only	only	ADV
ma-361	216	19	if	if	SCONJ
ma-361	216	20	k	k	PROPN
ma-361	216	21	is	be	AUX
ma-361	216	22	the	the	DET
ma-361	216	23	closed	closed	ADJ
ma-361	216	24	convex	convex	NOUN
ma-361	216	25	hull	hull	NOUN
ma-361	216	26	of	of	ADP
ma-361	216	27	its	its	PRON
ma-361	216	28	exposed	expose	VERB
ma-361	216	29	points	point	NOUN
ma-361	216	30	.	.	PUNCT
ma-361	217	1	https://doi.org/10.28924/ada/ma.5.17	https://doi.org/10.28924/ada/ma.5.17	PROPN
ma-361	217	2	eur	eur	PROPN
ma-361	217	3	.	.	PUNCT
ma-361	218	1	j.	j.	PROPN
ma-361	218	2	math	math	PROPN
ma-361	218	3	.	.	PUNCT
ma-361	219	1	anal	anal	PROPN
ma-361	219	2	.	.	PUNCT
ma-361	220	1	10.28924	10.28924	NUM
ma-361	220	2	/	/	SYM
ma-361	220	3	ada	ada	PROPN
ma-361	220	4	/	/	SYM
ma-361	220	5	ma.5.17	ma.5.17	ADJ
ma-361	220	6	9	9	NUM
ma-361	220	7	proof	proof	NOUN
ma-361	220	8	.	.	PUNCT
ma-361	221	1	(	(	PUNCT
ma-361	221	2	⇒	⇒	PROPN
ma-361	221	3	)	)	PUNCT
ma-361	221	4	suppose	suppose	VERB
ma-361	221	5	every	every	DET
ma-361	221	6	τ	τ	PROPN
ma-361	221	7	-	-	ADJ
ma-361	221	8	continuous	continuous	ADJ
ma-361	221	9	convex	convex	NOUN
ma-361	221	10	function	function	NOUN
ma-361	221	11	attains	attain	VERB
ma-361	221	12	its	its	PRON
ma-361	221	13	maximum	maximum	NOUN
ma-361	221	14	at	at	ADP
ma-361	221	15	extreme	extreme	ADJ
ma-361	221	16	points	point	NOUN
ma-361	221	17	.	.	PUNCT
ma-361	222	1	let	let	VERB
ma-361	222	2	x	x	SYM
ma-361	222	3	∈	∈	PROPN
ma-361	222	4	k	k	PROPN
ma-361	222	5	\	\	PROPN
ma-361	222	6	co(expk	co(expk	PROPN
ma-361	222	7	)	)	PUNCT
ma-361	222	8	.	.	PUNCT
ma-361	223	1	by	by	ADP
ma-361	223	2	the	the	DET
ma-361	223	3	strong	strong	ADJ
ma-361	223	4	separation	separation	NOUN
ma-361	223	5	theorem	theorem	VERB
ma-361	223	6	,	,	PUNCT
ma-361	223	7	there	there	PRON
ma-361	223	8	exists	exist	VERB
ma-361	223	9	f	f	PROPN
ma-361	223	10	∈	∈	PROPN
ma-361	223	11	x∗	x∗	PROPN
ma-361	223	12	with	with	ADP
ma-361	223	13	:	:	PUNCT
ma-361	224	1	f	f	PROPN
ma-361	224	2	(	(	PUNCT
ma-361	224	3	x	x	X
ma-361	224	4	)	)	PUNCT
ma-361	224	5	>	>	X
ma-361	224	6	sup	sup	PROPN
ma-361	224	7	y∈co(expk	y∈co(expk	PROPN
ma-361	224	8	)	)	PUNCT
ma-361	224	9	f	f	PROPN
ma-361	224	10	(	(	PUNCT
ma-361	224	11	y	y	NOUN
ma-361	224	12	)	)	PUNCT
ma-361	224	13	define	define	VERB
ma-361	224	14	g(y	g(y	NOUN
ma-361	224	15	)	)	PUNCT
ma-361	224	16	=	=	SYM
ma-361	224	17	max(f	max(f	NOUN
ma-361	224	18	(	(	PUNCT
ma-361	224	19	y)−	y)−	PROPN
ma-361	224	20	f	f	X
ma-361	224	21	(	(	PUNCT
ma-361	224	22	x	x	NOUN
ma-361	224	23	)	)	PUNCT
ma-361	224	24	,	,	PUNCT
ma-361	224	25	0	0	NUM
ma-361	224	26	)	)	PUNCT
ma-361	224	27	.	.	PUNCT
ma-361	225	1	then	then	ADV
ma-361	225	2	g	g	PROPN
ma-361	225	3	is	be	AUX
ma-361	225	4	convex	convex	ADJ
ma-361	225	5	continuous	continuous	ADJ
ma-361	225	6	but	but	CCONJ
ma-361	225	7	attains	attain	NOUN
ma-361	225	8	no	no	DET
ma-361	225	9	maximum	maximum	NOUN
ma-361	225	10	on	on	ADP
ma-361	225	11	expk	expk	NOUN
ma-361	225	12	,	,	PUNCT
ma-361	225	13	a	a	DET
ma-361	225	14	contradiction	contradiction	NOUN
ma-361	225	15	.	.	PUNCT
ma-361	226	1	(	(	PUNCT
ma-361	226	2	⇐	⇐	ADJ
ma-361	226	3	)	)	PUNCT
ma-361	226	4	assume	assume	VERB
ma-361	226	5	k	k	X
ma-361	226	6	=	=	SYM
ma-361	226	7	co(expk	co(expk	PROPN
ma-361	226	8	)	)	PUNCT
ma-361	226	9	.	.	PUNCT
ma-361	227	1	for	for	ADP
ma-361	227	2	any	any	DET
ma-361	227	3	continuous	continuous	ADJ
ma-361	227	4	convex	convex	NOUN
ma-361	227	5	f	f	PROPN
ma-361	227	6	,	,	PUNCT
ma-361	227	7	consider	consider	VERB
ma-361	227	8	:	:	PUNCT
ma-361	227	9	m	m	VERB
ma-361	227	10	=	=	PUNCT
ma-361	227	11	{	{	PUNCT
ma-361	227	12	µ	µ	X
ma-361	227	13	∈	∈	NOUN
ma-361	227	14	p(k	p(k	NOUN
ma-361	227	15	)	)	PUNCT
ma-361	227	16	:	:	PUNCT
ma-361	227	17	µ	µ	X
ma-361	227	18	represents	represent	VERB
ma-361	227	19	a	a	DET
ma-361	227	20	maximizer	maximizer	NOUN
ma-361	227	21	}	}	PUNCT
ma-361	227	22	where	where	SCONJ
ma-361	227	23	p(k	p(k	NOUN
ma-361	227	24	)	)	PUNCT
ma-361	227	25	are	be	AUX
ma-361	227	26	radon	radon	NOUN
ma-361	227	27	probability	probability	NOUN
ma-361	227	28	measures	measure	NOUN
ma-361	227	29	.	.	PUNCT
ma-361	228	1	by	by	ADP
ma-361	228	2	choquet	choquet	PROPN
ma-361	228	3	’s	’s	PART
ma-361	228	4	theorem	theorem	NOUN
ma-361	228	5	[	[	PUNCT
ma-361	228	6	13	13	NUM
ma-361	228	7	]	]	PUNCT
ma-361	228	8	,	,	PUNCT
ma-361	228	9	each	each	DET
ma-361	228	10	µ	µ	NOUN
ma-361	228	11	is	be	AUX
ma-361	228	12	supported	support	VERB
ma-361	228	13	on	on	ADP
ma-361	228	14	expk	expk	NOUN
ma-361	228	15	.	.	PUNCT
ma-361	229	1	hence	hence	ADV
ma-361	229	2	:	:	PUNCT
ma-361	229	3	max	max	PROPN
ma-361	229	4	k	k	PROPN
ma-361	229	5	f	f	PROPN
ma-361	229	6	=	=	NOUN
ma-361	229	7	sup	sup	PROPN
ma-361	229	8	x∈expk	x∈expk	PROPN
ma-361	229	9	f	f	PROPN
ma-361	229	10	(	(	PUNCT
ma-361	229	11	x	x	NOUN
ma-361	229	12	)	)	PUNCT
ma-361	229	13	and	and	CCONJ
ma-361	229	14	the	the	DET
ma-361	229	15	supremum	supremum	NOUN
ma-361	229	16	is	be	AUX
ma-361	229	17	attained	attain	VERB
ma-361	229	18	by	by	ADP
ma-361	229	19	τ	τ	PROPN
ma-361	229	20	-	-	PUNCT
ma-361	229	21	continuity	continuity	NOUN
ma-361	229	22	and	and	CCONJ
ma-361	229	23	compactness	compactness	NOUN
ma-361	229	24	.	.	PUNCT
ma-361	230	1	�	�	PROPN
ma-361	230	2	theorem	theorem	VERB
ma-361	230	3	10	10	NUM
ma-361	230	4	(	(	PUNCT
ma-361	230	5	polynomial	polynomial	ADJ
ma-361	230	6	norm	norm	NOUN
ma-361	230	7	-	-	PUNCT
ma-361	230	8	attainment	attainment	NOUN
ma-361	230	9	)	)	PUNCT
ma-361	230	10	.	.	PUNCT
ma-361	231	1	for	for	ADP
ma-361	231	2	x	x	DET
ma-361	231	3	a	a	DET
ma-361	231	4	complex	complex	ADJ
ma-361	231	5	locally	locally	ADV
ma-361	231	6	convex	convex	ADJ
ma-361	231	7	space	space	NOUN
ma-361	231	8	,	,	PUNCT
ma-361	231	9	the	the	DET
ma-361	231	10	following	follow	VERB
ma-361	231	11	are	be	AUX
ma-361	231	12	equivalent	equivalent	ADJ
ma-361	231	13	:	:	PUNCT
ma-361	231	14	(	(	PUNCT
ma-361	231	15	1	1	X
ma-361	231	16	)	)	PUNCT
ma-361	231	17	all	all	DET
ma-361	231	18	continuous	continuous	ADJ
ma-361	231	19	n	n	CCONJ
ma-361	231	20	-	-	PUNCT
ma-361	231	21	homogeneous	homogeneous	ADJ
ma-361	231	22	polynomials	polynomial	NOUN
ma-361	231	23	attain	attain	VERB
ma-361	231	24	their	their	PRON
ma-361	231	25	norms	norm	NOUN
ma-361	231	26	(	(	PUNCT
ma-361	231	27	2	2	X
ma-361	231	28	)	)	PUNCT
ma-361	231	29	the	the	DET
ma-361	231	30	n	n	ADV
ma-361	231	31	-	-	ADJ
ma-361	231	32	fold	fold	ADJ
ma-361	231	33	projective	projective	ADJ
ma-361	231	34	tensor	tensor	NOUN
ma-361	231	35	product	product	NOUN
ma-361	231	36	⊗̂nπx	⊗̂nπx	NOUN
ma-361	231	37	has	have	VERB
ma-361	231	38	the	the	DET
ma-361	231	39	radon	radon	PROPN
ma-361	231	40	-	-	PUNCT
ma-361	231	41	nikodym	nikodym	NOUN
ma-361	231	42	property	property	NOUN
ma-361	231	43	(	(	PUNCT
ma-361	231	44	3	3	X
ma-361	231	45	)	)	PUNCT
ma-361	231	46	every	every	DET
ma-361	231	47	τ	τ	PROPN
ma-361	231	48	-	-	ADJ
ma-361	231	49	continuous	continuous	ADJ
ma-361	231	50	polynomial	polynomial	NOUN
ma-361	231	51	is	be	AUX
ma-361	231	52	frechet	frechet	NOUN
ma-361	231	53	differentiable	differentiable	ADJ
ma-361	231	54	at	at	ADP
ma-361	231	55	some	some	DET
ma-361	231	56	point	point	NOUN
ma-361	231	57	proof	proof	NOUN
ma-361	231	58	.	.	PUNCT
ma-361	232	1	(	(	PUNCT
ma-361	232	2	i)⇒(ii	i)⇒(ii	ADV
ma-361	232	3	):	):	PUNCT
ma-361	232	4	let	let	VERB
ma-361	232	5	p	p	NOUN
ma-361	232	6	:	:	PUNCT
ma-361	232	7	⊗̂nπx	⊗̂nπx	VERB
ma-361	232	8	→	→	SYM
ma-361	232	9	c	c	X
ma-361	232	10	be	be	AUX
ma-361	232	11	the	the	DET
ma-361	232	12	canonical	canonical	ADJ
ma-361	232	13	n	n	CCONJ
ma-361	232	14	-	-	PUNCT
ma-361	232	15	linear	linear	NOUN
ma-361	232	16	form	form	NOUN
ma-361	232	17	.	.	PUNCT
ma-361	233	1	if	if	SCONJ
ma-361	233	2	all	all	DET
ma-361	233	3	polynomials	polynomial	NOUN
ma-361	233	4	attain	attain	VERB
ma-361	233	5	norms	norm	NOUN
ma-361	233	6	,	,	PUNCT
ma-361	233	7	then	then	ADV
ma-361	233	8	p	p	PROPN
ma-361	233	9	attains	attain	VERB
ma-361	233	10	its	its	PRON
ma-361	233	11	projective	projective	ADJ
ma-361	233	12	norm	norm	NOUN
ma-361	233	13	,	,	PUNCT
ma-361	233	14	making	make	VERB
ma-361	233	15	⊗̂nπx	⊗̂nπx	NOUN
ma-361	233	16	reflexive	reflexive	VERB
ma-361	233	17	by	by	ADP
ma-361	233	18	a	a	DET
ma-361	233	19	polynomial	polynomial	ADJ
ma-361	233	20	version	version	NOUN
ma-361	233	21	of	of	ADP
ma-361	233	22	james	james	PROPN
ma-361	233	23	’	'	PUNCT
ma-361	233	24	theorem.the	theorem.the	DET
ma-361	233	25	radon	radon	NOUN
ma-361	233	26	-	-	PUNCT
ma-361	233	27	nikodym	nikodym	PROPN
ma-361	233	28	property	property	NOUN
ma-361	233	29	follows	follow	VERB
ma-361	233	30	from	from	ADP
ma-361	233	31	[	[	NOUN
ma-361	233	32	8].(ii)⇒(iii	8].(ii)⇒(iii	NUM
ma-361	233	33	):	):	PUNCT
ma-361	233	34	when	when	SCONJ
ma-361	233	35	⊗̂nπx	⊗̂nπx	NOUN
ma-361	233	36	has	have	VERB
ma-361	233	37	rnp	rnp	PROPN
ma-361	233	38	,	,	PUNCT
ma-361	233	39	the	the	DET
ma-361	233	40	aron	aron	PROPN
ma-361	233	41	-	-	PUNCT
ma-361	233	42	berner	berner	PROPN
ma-361	233	43	extension	extension	NOUN
ma-361	233	44	[	[	X
ma-361	233	45	1	1	X
ma-361	233	46	]	]	PUNCT
ma-361	233	47	shows	show	VERB
ma-361	233	48	that	that	SCONJ
ma-361	233	49	every	every	DET
ma-361	233	50	polynomial	polynomial	ADJ
ma-361	233	51	isfrechet	isfrechet	NOUN
ma-361	233	52	differentiable	differentiable	ADJ
ma-361	233	53	on	on	ADP
ma-361	233	54	a	a	DET
ma-361	233	55	dense	dense	ADJ
ma-361	233	56	set	set	NOUN
ma-361	233	57	by	by	ADP
ma-361	233	58	the	the	DET
ma-361	233	59	asplund	asplund	NOUN
ma-361	233	60	averaging	average	VERB
ma-361	233	61	technique.(iii)⇒(i	technique.(iii)⇒(i	NOUN
ma-361	233	62	):	):	PUNCT
ma-361	233	63	suppose	suppose	VERB
ma-361	233	64	p	p	NOUN
ma-361	233	65	is	be	AUX
ma-361	233	66	differentiable	differentiable	ADJ
ma-361	233	67	at	at	ADP
ma-361	233	68	x0	x0	PROPN
ma-361	233	69	.	.	PUNCT
ma-361	234	1	the	the	DET
ma-361	234	2	taylor	taylor	PROPN
ma-361	234	3	expansion	expansion	NOUN
ma-361	234	4	:	:	PUNCT
ma-361	234	5	p(x0	p(x0	NOUN
ma-361	234	6	+	+	CCONJ
ma-361	234	7	h	h	NOUN
ma-361	234	8	)	)	PUNCT
ma-361	234	9	=	=	SYM
ma-361	234	10	p(x0	p(x0	NOUN
ma-361	234	11	)	)	PUNCT
ma-361	235	1	+	+	NOUN
ma-361	235	2	dp(x0)(h	dp(x0)(h	X
ma-361	235	3	)	)	PUNCT
ma-361	235	4	+	+	CCONJ
ma-361	235	5	·	·	PUNCT
ma-361	235	6	·	·	PUNCT
ma-361	235	7	·	·	PUNCT
ma-361	235	8	+	+	CCONJ
ma-361	235	9	1	1	NUM
ma-361	235	10	n	n	NOUN
ma-361	235	11	!	!	PUNCT
ma-361	235	12	dnp(x0)(hn	dnp(x0)(hn	NOUN
ma-361	235	13	)	)	PUNCT
ma-361	235	14	allows	allow	VERB
ma-361	235	15	construction	construction	NOUN
ma-361	235	16	of	of	ADP
ma-361	235	17	a	a	DET
ma-361	235	18	norm	norm	NOUN
ma-361	235	19	-	-	PUNCT
ma-361	235	20	attaining	attain	VERB
ma-361	235	21	direction	direction	NOUN
ma-361	235	22	using	use	VERB
ma-361	235	23	the	the	DET
ma-361	235	24	polarization	polarization	NOUN
ma-361	235	25	constants	constant	NOUN
ma-361	235	26	from	from	ADP
ma-361	235	27	[	[	X
ma-361	235	28	12	12	NUM
ma-361	235	29	]	]	PUNCT
ma-361	235	30	.	.	PUNCT
ma-361	236	1	thenorm	thenorm	NOUN
ma-361	236	2	is	be	AUX
ma-361	236	3	attained	attain	VERB
ma-361	236	4	along	along	ADP
ma-361	236	5	a	a	DET
ma-361	236	6	complex	complex	ADJ
ma-361	236	7	line	line	NOUN
ma-361	236	8	through	through	ADP
ma-361	236	9	x0	x0	PROPN
ma-361	236	10	.	.	PUNCT
ma-361	237	1	�	�	PROPN
ma-361	237	2	example	example	NOUN
ma-361	237	3	2	2	NUM
ma-361	237	4	(	(	PUNCT
ma-361	237	5	polynomial	polynomial	ADJ
ma-361	237	6	attainment	attainment	NOUN
ma-361	237	7	)	)	PUNCT
ma-361	237	8	.	.	PUNCT
ma-361	238	1	on	on	ADP
ma-361	238	2	x	x	X
ma-361	238	3	=	=	PUNCT
ma-361	238	4	`	`	PUNCT
ma-361	238	5	2	2	NUM
ma-361	238	6	,	,	PUNCT
ma-361	238	7	the	the	DET
ma-361	238	8	2	2	NUM
ma-361	238	9	-	-	PUNCT
ma-361	238	10	homogeneous	homogeneous	ADJ
ma-361	238	11	polynomial	polynomial	ADJ
ma-361	238	12	p	p	NOUN
ma-361	238	13	(	(	PUNCT
ma-361	238	14	x	x	NOUN
ma-361	238	15	)	)	PUNCT
ma-361	238	16	=	=	SYM
ma-361	238	17	∑	∑	PUNCT
ma-361	238	18	(	(	PUNCT
ma-361	238	19	1	1	NUM
ma-361	238	20	−	−	NUM
ma-361	238	21	1	1	NUM
ma-361	238	22	n	n	NOUN
ma-361	238	23	)	)	PUNCT
ma-361	238	24	x2n	x2n	PRON
ma-361	238	25	attains	attain	VERB
ma-361	238	26	its	its	PRON
ma-361	238	27	norm	norm	NOUN
ma-361	238	28	at	at	ADP
ma-361	238	29	0	0	NUM
ma-361	238	30	,	,	PUNCT
ma-361	238	31	demonstrating	demonstrating	NOUN
ma-361	238	32	theorem	theorem	VERB
ma-361	238	33	10	10	NUM
ma-361	238	34	’s	’s	PART
ma-361	238	35	rnp	rnp	PROPN
ma-361	238	36	condition	condition	NOUN
ma-361	238	37	.	.	PUNCT
ma-361	239	1	conclusion	conclusion	NOUN
ma-361	239	2	this	this	DET
ma-361	239	3	work	work	NOUN
ma-361	239	4	has	have	AUX
ma-361	239	5	established	establish	VERB
ma-361	239	6	a	a	DET
ma-361	239	7	unified	unified	ADJ
ma-361	239	8	framework	framework	NOUN
ma-361	239	9	for	for	ADP
ma-361	239	10	nonlinear	nonlinear	ADJ
ma-361	239	11	norm	norm	NOUN
ma-361	239	12	-	-	PUNCT
ma-361	239	13	attainment	attainment	NOUN
ma-361	239	14	in	in	ADP
ma-361	239	15	locally	locally	ADJ
ma-361	239	16	convexspaces	convexspace	NOUN
ma-361	239	17	,	,	PUNCT
ma-361	239	18	resolving	resolve	VERB
ma-361	239	19	several	several	ADJ
ma-361	239	20	open	open	ADJ
ma-361	239	21	problems	problem	NOUN
ma-361	239	22	and	and	CCONJ
ma-361	239	23	extending	extend	VERB
ma-361	239	24	classical	classical	ADJ
ma-361	239	25	results	result	NOUN
ma-361	239	26	to	to	PART
ma-361	239	27	sublinear	sublinear	VERB
ma-361	239	28	,	,	PUNCT
ma-361	239	29	quasiconvex	quasiconvex	NOUN
ma-361	239	30	,	,	PUNCT
ma-361	239	31	and	and	CCONJ
ma-361	239	32	polynomial	polynomial	ADJ
ma-361	239	33	settings	setting	NOUN
ma-361	239	34	.	.	PUNCT
ma-361	240	1	our	our	PRON
ma-361	240	2	main	main	ADJ
ma-361	240	3	theorems	theorem	NOUN
ma-361	240	4	reveal	reveal	VERB
ma-361	240	5	deep	deep	ADJ
ma-361	240	6	connections	connection	NOUN
ma-361	240	7	between	between	ADP
ma-361	240	8	functional	functional	ADJ
ma-361	240	9	analysis	analysis	NOUN
ma-361	240	10	,	,	PUNCT
ma-361	240	11	convex	convex	NOUN
ma-361	240	12	geometry	geometry	NOUN
ma-361	240	13	,	,	PUNCT
ma-361	240	14	and	and	CCONJ
ma-361	240	15	optimization	optimization	NOUN
ma-361	240	16	:	:	PUNCT
ma-361	240	17	https://doi.org/10.28924/ada/ma.5.17	https://doi.org/10.28924/ada/ma.5.17	PROPN
ma-361	240	18	eur	eur	PROPN
ma-361	240	19	.	.	PUNCT
ma-361	241	1	j.	j.	PROPN
ma-361	241	2	math	math	PROPN
ma-361	241	3	.	.	PUNCT
ma-361	242	1	anal	anal	PROPN
ma-361	242	2	.	.	PUNCT
ma-361	243	1	10.28924	10.28924	NUM
ma-361	243	2	/	/	SYM
ma-361	243	3	ada	ada	PROPN
ma-361	243	4	/	/	SYM
ma-361	243	5	ma.5.17	ma.5.17	ADJ
ma-361	243	6	10	10	NUM
ma-361	243	7	•	•	NOUN
ma-361	244	1	nonlinear	nonlinear	ADJ
ma-361	244	2	density	density	NOUN
ma-361	244	3	theorems	theorem	VERB
ma-361	244	4	:	:	PUNCT
ma-361	244	5	the	the	DET
ma-361	244	6	equivalence	equivalence	NOUN
ma-361	244	7	between	between	ADP
ma-361	244	8	norm	norm	NOUN
ma-361	244	9	-	-	PUNCT
ma-361	244	10	attainment	attainment	NOUN
ma-361	244	11	density	density	NOUN
ma-361	244	12	and	and	CCONJ
ma-361	244	13	latticenorms	latticenorm	NOUN
ma-361	244	14	(	(	PUNCT
ma-361	244	15	theorem	theorem	NOUN
ma-361	244	16	1	1	NUM
ma-361	244	17	)	)	PUNCT
ma-361	244	18	subsumes	subsume	VERB
ma-361	244	19	the	the	DET
ma-361	244	20	classical	classical	ADJ
ma-361	244	21	bishop	bishop	NOUN
ma-361	244	22	-	-	PUNCT
ma-361	244	23	phelps	phelps	PROPN
ma-361	244	24	theorem	theorem	VERB
ma-361	244	25	while	while	SCONJ
ma-361	244	26	providing	provide	VERB
ma-361	244	27	new	new	ADJ
ma-361	244	28	tools	tool	NOUN
ma-361	244	29	fornon	fornon	NOUN
ma-361	244	30	-	-	PUNCT
ma-361	244	31	reflexive	reflexive	ADJ
ma-361	244	32	spaces	space	NOUN
ma-361	244	33	.	.	PUNCT
ma-361	245	1	this	this	PRON
ma-361	245	2	complements	complement	VERB
ma-361	245	3	recent	recent	ADJ
ma-361	245	4	advances	advance	NOUN
ma-361	245	5	in	in	ADP
ma-361	245	6	[	[	X
ma-361	245	7	6	6	NUM
ma-361	245	8	]	]	PUNCT
ma-361	245	9	and	and	CCONJ
ma-361	245	10	[	[	X
ma-361	245	11	7	7	X
ma-361	245	12	]	]	PUNCT
ma-361	245	13	on	on	ADP
ma-361	245	14	perturbed	perturb	VERB
ma-361	245	15	minimization	minimization	NOUN
ma-361	245	16	.	.	PUNCT
ma-361	246	1	•	•	NUM
ma-361	246	2	geometric	geometric	ADJ
ma-361	246	3	characterization	characterization	NOUN
ma-361	246	4	:	:	PUNCT
ma-361	246	5	the	the	DET
ma-361	246	6	subdifferential	subdifferential	ADJ
ma-361	246	7	characterization	characterization	NOUN
ma-361	246	8	of	of	ADP
ma-361	246	9	norm	norm	NOUN
ma-361	246	10	-	-	PUNCT
ma-361	246	11	attainment	attainment	NOUN
ma-361	246	12	(	(	PUNCT
ma-361	246	13	theorem	theorem	VERB
ma-361	246	14	2,theorem	2,theorem	NUM
ma-361	246	15	8)	8)	NUM
ma-361	246	16	extends	extend	VERB
ma-361	246	17	rockafellar	rockafellar	ADJ
ma-361	246	18	’s	’s	PART
ma-361	246	19	foundational	foundational	ADJ
ma-361	246	20	work	work	NOUN
ma-361	246	21	[	[	X
ma-361	246	22	14	14	NUM
ma-361	246	23	]	]	PUNCT
ma-361	246	24	to	to	ADP
ma-361	246	25	non	non	ADJ
ma-361	246	26	-	-	ADJ
ma-361	246	27	smooth	smooth	ADJ
ma-361	246	28	settings	setting	NOUN
ma-361	246	29	,	,	PUNCT
ma-361	246	30	with	with	ADP
ma-361	246	31	applicationsto	applicationsto	NOUN
ma-361	246	32	stochastic	stochastic	ADJ
ma-361	246	33	variational	variational	ADJ
ma-361	246	34	inequalities	inequality	NOUN
ma-361	246	35	.	.	PUNCT
ma-361	247	1	•	•	NUM
ma-361	247	2	polynomial	polynomial	ADJ
ma-361	247	3	optimization	optimization	NOUN
ma-361	247	4	:	:	PUNCT
ma-361	247	5	our	our	PRON
ma-361	247	6	tensor	tensor	NOUN
ma-361	247	7	product	product	NOUN
ma-361	247	8	approach	approach	NOUN
ma-361	247	9	(	(	PUNCT
ma-361	247	10	theorem	theorem	NOUN
ma-361	247	11	10	10	NUM
ma-361	247	12	)	)	PUNCT
ma-361	247	13	solves	solve	VERB
ma-361	247	14	the	the	DET
ma-361	247	15	polynomialnorm	polynomialnorm	NOUN
ma-361	247	16	-	-	PUNCT
ma-361	247	17	attainment	attainment	ADJ
ma-361	247	18	problem	problem	NOUN
ma-361	247	19	via	via	ADP
ma-361	247	20	the	the	DET
ma-361	247	21	radon	radon	PROPN
ma-361	247	22	-	-	PUNCT
ma-361	247	23	nikodym	nikodym	NOUN
ma-361	247	24	property	property	NOUN
ma-361	247	25	,	,	PUNCT
ma-361	247	26	bridging	bridging	NOUN
ma-361	247	27	complex	complex	ADJ
ma-361	247	28	analysis	analysis	NOUN
ma-361	247	29	[	[	X
ma-361	247	30	8	8	NUM
ma-361	247	31	]	]	SYM
ma-361	247	32	andmultilinear	andmultilinear	NOUN
ma-361	247	33	algebra	algebra	NOUN
ma-361	247	34	[	[	X
ma-361	247	35	1	1	NUM
ma-361	247	36	]	]	PUNCT
ma-361	247	37	.	.	PUNCT
ma-361	248	1	•	•	NUM
ma-361	248	2	category	category	NOUN
ma-361	248	3	-	-	PUNCT
ma-361	248	4	theoretic	theoretic	NOUN
ma-361	248	5	insights	insight	NOUN
ma-361	248	6	:	:	PUNCT
ma-361	248	7	the	the	DET
ma-361	248	8	stability	stability	NOUN
ma-361	248	9	results	result	VERB
ma-361	248	10	(	(	PUNCT
ma-361	248	11	theorem	theorem	NOUN
ma-361	248	12	5	5	NUM
ma-361	248	13	)	)	PUNCT
ma-361	248	14	and	and	CCONJ
ma-361	248	15	generic	generic	ADJ
ma-361	248	16	attainment	attainment	NOUN
ma-361	248	17	(	(	PUNCT
ma-361	248	18	theorem3	theorem3	NOUN
ma-361	248	19	)	)	PUNCT
ma-361	248	20	demonstrate	demonstrate	VERB
ma-361	248	21	that	that	SCONJ
ma-361	248	22	baire	baire	NOUN
ma-361	248	23	category	category	NOUN
ma-361	248	24	methods	method	NOUN
ma-361	248	25	remain	remain	VERB
ma-361	248	26	powerful	powerful	ADJ
ma-361	248	27	in	in	ADP
ma-361	248	28	nonlinear	nonlinear	ADJ
ma-361	248	29	settings	setting	NOUN
ma-361	248	30	,	,	PUNCT
ma-361	248	31	as	as	ADP
ma-361	248	32	conjecturedin	conjecturedin	VERB
ma-361	248	33	[	[	X
ma-361	248	34	5	5	NUM
ma-361	248	35	]	]	PUNCT
ma-361	248	36	.	.	PUNCT
ma-361	249	1	future	future	ADJ
ma-361	249	2	directions:(i	directions:(i	NOUN
ma-361	249	3	)	)	PUNCT
ma-361	249	4	infinite	infinite	ADJ
ma-361	249	5	-	-	PUNCT
ma-361	249	6	dimensional	dimensional	ADJ
ma-361	249	7	polynomial	polynomial	ADJ
ma-361	249	8	optimization	optimization	NOUN
ma-361	249	9	:	:	PUNCT
ma-361	249	10	theorem	theorem	NOUN
ma-361	249	11	10	10	NUM
ma-361	249	12	suggests	suggest	VERB
ma-361	249	13	a	a	DET
ma-361	249	14	program	program	NOUN
ma-361	249	15	to	to	ADP
ma-361	249	16	extendlasserre	extendlasserre	VERB
ma-361	249	17	’s	’s	PART
ma-361	249	18	hierarchy	hierarchy	NOUN
ma-361	249	19	to	to	ADP
ma-361	249	20	projective	projective	VERB
ma-361	249	21	tensor	tensor	NOUN
ma-361	249	22	products	product	NOUN
ma-361	249	23	,	,	PUNCT
ma-361	249	24	with	with	ADP
ma-361	249	25	applications	application	NOUN
ma-361	249	26	to	to	ADP
ma-361	249	27	pde	pde	NOUN
ma-361	249	28	-	-	PUNCT
ma-361	249	29	constrainedoptimal	constrainedoptimal	ADJ
ma-361	249	30	control.(ii	control.(ii	NOUN
ma-361	249	31	)	)	PUNCT
ma-361	249	32	stochastic	stochastic	ADJ
ma-361	249	33	variational	variational	ADJ
ma-361	249	34	principles	principle	NOUN
ma-361	249	35	:	:	PUNCT
ma-361	249	36	the	the	DET
ma-361	249	37	interiority	interiority	NOUN
ma-361	249	38	condition	condition	NOUN
ma-361	249	39	in	in	ADP
ma-361	249	40	theorem	theorem	NOUN
ma-361	249	41	8	8	NUM
ma-361	249	42	could	could	AUX
ma-361	249	43	yield	yield	VERB
ma-361	249	44	newexistence	newexistence	NOUN
ma-361	249	45	theorems	theorem	NOUN
ma-361	249	46	for	for	ADP
ma-361	249	47	random	random	ADJ
ma-361	249	48	functionals	functional	NOUN
ma-361	249	49	on	on	ADP
ma-361	249	50	frechet	frechet	PROPN
ma-361	249	51	spaces	space	NOUN
ma-361	249	52	,	,	PUNCT
ma-361	249	53	building	build	VERB
ma-361	249	54	on	on	ADP
ma-361	249	55	[	[	NOUN
ma-361	249	56	9].(iii	9].(iii	NUM
ma-361	249	57	)	)	PUNCT
ma-361	249	58	non	non	ADJ
ma-361	249	59	-	-	ADJ
ma-361	249	60	convex	convex	ADJ
ma-361	249	61	separation	separation	NOUN
ma-361	249	62	theory	theory	NOUN
ma-361	249	63	:	:	PUNCT
ma-361	249	64	theorem	theorem	VERB
ma-361	249	65	4	4	NUM
ma-361	249	66	’s	’s	PART
ma-361	249	67	quasilinear	quasilinear	NOUN
ma-361	249	68	separation	separation	NOUN
ma-361	249	69	may	may	AUX
ma-361	249	70	enable	enable	VERB
ma-361	249	71	nashequilibrium	nashequilibrium	NOUN
ma-361	249	72	analysis	analysis	NOUN
ma-361	249	73	in	in	ADP
ma-361	249	74	general	general	ADJ
ma-361	249	75	topological	topological	ADJ
ma-361	249	76	vector	vector	NOUN
ma-361	249	77	spaces	space	NOUN
ma-361	249	78	,	,	PUNCT
ma-361	249	79	beyond	beyond	ADP
ma-361	249	80	current	current	ADJ
ma-361	249	81	banach	banach	NOUN
ma-361	249	82	spacetechniques	spacetechnique	NOUN
ma-361	249	83	[	[	X
ma-361	249	84	4].(iv	4].(iv	NUM
ma-361	249	85	)	)	PUNCT
ma-361	249	86	computational	computational	ADJ
ma-361	249	87	aspects	aspect	NOUN
ma-361	249	88	:	:	PUNCT
ma-361	249	89	implementing	implement	VERB
ma-361	249	90	theorem	theorem	VERB
ma-361	249	91	7	7	NUM
ma-361	249	92	’s	’s	PART
ma-361	249	93	krein	krein	NOUN
ma-361	249	94	-	-	PUNCT
ma-361	249	95	milman	milman	NOUN
ma-361	249	96	property	property	NOUN
ma-361	249	97	for	for	ADP
ma-361	249	98	convex	convex	ADJ
ma-361	249	99	pro	pro	ADJ
ma-361	249	100	-	-	NOUN
ma-361	249	101	grams	gram	NOUN
ma-361	249	102	in	in	ADP
ma-361	249	103	sequence	sequence	NOUN
ma-361	249	104	spaces	space	NOUN
ma-361	249	105	(	(	PUNCT
ma-361	249	106	e.g.	e.g.	ADV
ma-361	249	107	,	,	PUNCT
ma-361	249	108	`	`	PUNCT
ma-361	249	109	p	p	NOUN
ma-361	249	110	with	with	ADP
ma-361	249	111	0	0	NUM
ma-361	249	112	<	<	X
ma-361	249	113	p	p	X
ma-361	249	114	<	<	X
ma-361	249	115	1	1	NUM
ma-361	249	116	)	)	PUNCT
ma-361	249	117	requires	require	VERB
ma-361	249	118	new	new	ADJ
ma-361	249	119	discretization	discretization	NOUN
ma-361	249	120	schemes.the	schemes.the	DET
ma-361	249	121	methods	method	NOUN
ma-361	249	122	developed	develop	VERB
ma-361	249	123	here	here	ADV
ma-361	249	124	-	-	PUNCT
ma-361	249	125	particularly	particularly	ADV
ma-361	249	126	the	the	DET
ma-361	249	127	interplay	interplay	NOUN
ma-361	249	128	between	between	ADP
ma-361	249	129	choquet	choquet	NOUN
ma-361	249	130	theory	theory	NOUN
ma-361	249	131	(	(	PUNCT
ma-361	249	132	theorem	theorem	NOUN
ma-361	249	133	9	9	NUM
ma-361	249	134	)	)	PUNCT
ma-361	249	135	,	,	PUNCT
ma-361	249	136	vari	vari	ADJ
ma-361	249	137	-	-	ADJ
ma-361	249	138	ational	ational	ADJ
ma-361	249	139	analysis	analysis	NOUN
ma-361	249	140	,	,	PUNCT
ma-361	249	141	and	and	CCONJ
ma-361	249	142	complex	complex	ADJ
ma-361	249	143	geometry	geometry	NOUN
ma-361	249	144	(	(	PUNCT
ma-361	249	145	theorem	theorem	NOUN
ma-361	249	146	6)-open	6)-open	NOUN
ma-361	249	147	pathways	pathway	NOUN
ma-361	249	148	to	to	ADP
ma-361	249	149	unifying	unify	VERB
ma-361	249	150	fragmented	fragmented	ADJ
ma-361	249	151	resultsin	resultsin	NOUN
ma-361	249	152	nonlinear	nonlinear	ADJ
ma-361	249	153	functional	functional	ADJ
ma-361	249	154	analysis	analysis	NOUN
ma-361	249	155	.	.	PUNCT
ma-361	250	1	further	further	ADJ
ma-361	250	2	exploration	exploration	NOUN
ma-361	250	3	of	of	ADP
ma-361	250	4	these	these	DET
ma-361	250	5	connections	connection	NOUN
ma-361	250	6	promises	promise	VERB
ma-361	250	7	advances	advance	NOUN
ma-361	250	8	inhigh	inhigh	ADJ
ma-361	250	9	-	-	PUNCT
ma-361	250	10	dimensional	dimensional	ADJ
ma-361	250	11	statistics	statistic	NOUN
ma-361	250	12	,	,	PUNCT
ma-361	250	13	mean	mean	ADJ
ma-361	250	14	-	-	PUNCT
ma-361	250	15	field	field	NOUN
ma-361	250	16	game	game	NOUN
ma-361	250	17	theory	theory	NOUN
ma-361	250	18	,	,	PUNCT
ma-361	250	19	and	and	CCONJ
ma-361	250	20	non	non	ADJ
ma-361	250	21	-	-	ADJ
ma-361	250	22	archimedean	archimedean	ADJ
ma-361	250	23	optimization	optimization	NOUN
ma-361	250	24	.	.	PUNCT
ma-361	251	1	references	reference	NOUN
ma-361	251	2	[	[	X
ma-361	251	3	1	1	NUM
ma-361	251	4	]	]	X
ma-361	251	5	r.m	r.m	PROPN
ma-361	251	6	.	.	PROPN
ma-361	251	7	aron	aron	PROPN
ma-361	251	8	,	,	PUNCT
ma-361	251	9	p.d	p.d	PROPN
ma-361	251	10	.	.	PROPN
ma-361	251	11	berner	berner	PROPN
ma-361	251	12	,	,	PUNCT
ma-361	251	13	a	a	DET
ma-361	251	14	hahn	hahn	NOUN
ma-361	251	15	-	-	PUNCT
ma-361	251	16	banach	banach	NOUN
ma-361	251	17	extension	extension	NOUN
ma-361	251	18	theorem	theorem	NOUN
ma-361	251	19	for	for	ADP
ma-361	251	20	analytic	analytic	ADJ
ma-361	251	21	mappings	mapping	NOUN
ma-361	251	22	,	,	PUNCT
ma-361	251	23	bull	bull	NOUN
ma-361	251	24	.	.	PUNCT
ma-361	252	1	soc	soc	PROPN
ma-361	252	2	.	.	PUNCT
ma-361	253	1	math	math	NOUN
ma-361	253	2	.	.	PUNCT
ma-361	254	1	fr	fr	PROPN
ma-361	254	2	.	.	PROPN
ma-361	255	1	115	115	NUM
ma-361	255	2	(	(	PUNCT
ma-361	255	3	1987),3–24.[2	1987),3–24.[2	NUM
ma-361	255	4	]	]	X
ma-361	255	5	b.	b.	PROPN
ma-361	255	6	beauzamy	beauzamy	PROPN
ma-361	255	7	,	,	PUNCT
ma-361	255	8	introduction	introduction	NOUN
ma-361	255	9	to	to	PART
ma-361	255	10	banach	banach	NOUN
ma-361	255	11	spaces	space	NOUN
ma-361	255	12	and	and	CCONJ
ma-361	255	13	their	their	PRON
ma-361	255	14	geometry	geometry	NOUN
ma-361	255	15	,	,	PUNCT
ma-361	255	16	north	north	NOUN
ma-361	255	17	-	-	PUNCT
ma-361	255	18	holland	holland	NOUN
ma-361	255	19	,	,	PUNCT
ma-361	255	20	(	(	PUNCT
ma-361	255	21	1982).[3	1982).[3	NUM
ma-361	255	22	]	]	X
ma-361	255	23	g.	g.	PROPN
ma-361	255	24	beer	beer	PROPN
ma-361	255	25	,	,	PUNCT
ma-361	255	26	topologies	topology	NOUN
ma-361	255	27	on	on	ADP
ma-361	255	28	closed	closed	ADJ
ma-361	255	29	and	and	CCONJ
ma-361	255	30	closed	close	VERB
ma-361	255	31	convex	convex	NOUN
ma-361	255	32	sets	set	NOUN
ma-361	255	33	,	,	PUNCT
ma-361	255	34	kluwer	kluwer	NOUN
ma-361	255	35	academic	academic	ADJ
ma-361	255	36	publishers	publisher	NOUN
ma-361	255	37	,	,	PUNCT
ma-361	255	38	1993.[4	1993.[4	NUM
ma-361	255	39	]	]	X
ma-361	255	40	y.	y.	PROPN
ma-361	255	41	benyamini	benyamini	PROPN
ma-361	255	42	,	,	PUNCT
ma-361	255	43	j.	j.	PROPN
ma-361	255	44	lindenstrauss	lindenstrauss	PROPN
ma-361	255	45	,	,	PUNCT
ma-361	255	46	geometric	geometric	ADJ
ma-361	255	47	nonlinear	nonlinear	ADJ
ma-361	255	48	functional	functional	ADJ
ma-361	255	49	analysis	analysis	NOUN
ma-361	255	50	,	,	PUNCT
ma-361	255	51	american	american	PROPN
ma-361	255	52	mathematical	mathematical	ADJ
ma-361	255	53	society	society	NOUN
ma-361	255	54	,	,	PUNCT
ma-361	255	55	(	(	PUNCT
ma-361	255	56	2000).[5	2000).[5	X
ma-361	255	57	]	]	X
ma-361	255	58	j.m	j.m	PROPN
ma-361	255	59	.	.	PROPN
ma-361	255	60	borwein	borwein	PROPN
ma-361	255	61	,	,	PUNCT
ma-361	255	62	d.	d.	PROPN
ma-361	255	63	preiss	preiss	PROPN
ma-361	255	64	,	,	PUNCT
ma-361	255	65	a	a	DET
ma-361	255	66	smooth	smooth	ADJ
ma-361	255	67	variational	variational	ADJ
ma-361	255	68	principle	principle	NOUN
ma-361	255	69	with	with	ADP
ma-361	255	70	applications	application	NOUN
ma-361	255	71	to	to	ADP
ma-361	255	72	subdifferentiability	subdifferentiability	NOUN
ma-361	255	73	and	and	CCONJ
ma-361	255	74	to	to	PART
ma-361	255	75	differentiabilityof	differentiabilityof	VERB
ma-361	255	76	convex	convex	NOUN
ma-361	255	77	functions	function	NOUN
ma-361	255	78	,	,	PUNCT
ma-361	255	79	trans	trans	PROPN
ma-361	255	80	.	.	PROPN
ma-361	255	81	am	be	AUX
ma-361	255	82	.	.	PUNCT
ma-361	256	1	math	math	NOUN
ma-361	256	2	.	.	PUNCT
ma-361	257	1	soc	soc	PROPN
ma-361	257	2	.	.	PUNCT
ma-361	258	1	303	303	NUM
ma-361	258	2	(	(	PUNCT
ma-361	258	3	1987	1987	NUM
ma-361	258	4	)	)	PUNCT
ma-361	258	5	,	,	PUNCT
ma-361	258	6	517–527.[6	517–527.[6	NUM
ma-361	258	7	]	]	X
ma-361	258	8	j.m	j.m	PROPN
ma-361	258	9	.	.	PROPN
ma-361	258	10	borwein	borwein	PROPN
ma-361	258	11	,	,	PUNCT
ma-361	258	12	j.d	j.d	PROPN
ma-361	258	13	.	.	PROPN
ma-361	258	14	vanderwerff	vanderwerff	PROPN
ma-361	258	15	,	,	PUNCT
ma-361	258	16	differentiability	differentiability	NOUN
ma-361	258	17	of	of	ADP
ma-361	258	18	conjugate	conjugate	ADJ
ma-361	258	19	functions	function	NOUN
ma-361	258	20	and	and	CCONJ
ma-361	258	21	perturbed	perturb	VERB
ma-361	258	22	minimization	minimization	NOUN
ma-361	258	23	principles	principle	NOUN
ma-361	258	24	,	,	PUNCT
ma-361	258	25	j.convex	j.convex	PROPN
ma-361	258	26	anal	anal	NOUN
ma-361	258	27	.	.	PUNCT
ma-361	259	1	6	6	NUM
ma-361	259	2	(	(	PUNCT
ma-361	259	3	2009	2009	NUM
ma-361	259	4	)	)	PUNCT
ma-361	259	5	,	,	PUNCT
ma-361	259	6	1–11	1–11	PROPN
ma-361	259	7	.	.	PUNCT
ma-361	260	1	https://doi.org/10.28924/ada/ma.5.17	https://doi.org/10.28924/ada/ma.5.17	PROPN
ma-361	260	2	eur	eur	PROPN
ma-361	260	3	.	.	PUNCT
ma-361	261	1	j.	j.	PROPN
ma-361	261	2	math	math	PROPN
ma-361	261	3	.	.	PUNCT
ma-361	262	1	anal	anal	PROPN
ma-361	262	2	.	.	PUNCT
ma-361	263	1	10.28924	10.28924	NUM
ma-361	263	2	/	/	SYM
ma-361	263	3	ada	ada	PROPN
ma-361	263	4	/	/	SYM
ma-361	263	5	ma.5.17	ma.5.17	ADJ
ma-361	263	6	11	11	NUM
ma-361	263	7	[	[	X
ma-361	263	8	7	7	NUM
ma-361	263	9	]	]	X
ma-361	263	10	r.	r.	PROPN
ma-361	263	11	deville	deville	PROPN
ma-361	263	12	,	,	PUNCT
ma-361	263	13	g.	g.	PROPN
ma-361	263	14	godefroy	godefroy	PROPN
ma-361	263	15	,	,	PUNCT
ma-361	263	16	v.	v.	PROPN
ma-361	263	17	zizler	zizler	NOUN
ma-361	263	18	,	,	PUNCT
ma-361	263	19	smoothness	smoothness	ADJ
ma-361	263	20	and	and	CCONJ
ma-361	263	21	renormings	renorming	NOUN
ma-361	263	22	in	in	ADP
ma-361	263	23	banach	banach	NOUN
ma-361	263	24	spaces	space	NOUN
ma-361	263	25	,	,	PUNCT
ma-361	263	26	longman	longman	NOUN
ma-361	263	27	,	,	PUNCT
ma-361	263	28	1993.[8	1993.[8	NUM
ma-361	263	29	]	]	PUNCT
ma-361	263	30	s.	s.	PROPN
ma-361	263	31	dineen	dineen	PROPN
ma-361	263	32	,	,	PUNCT
ma-361	263	33	complex	complex	ADJ
ma-361	263	34	analysis	analysis	NOUN
ma-361	263	35	on	on	ADP
ma-361	263	36	infinite	infinite	ADJ
ma-361	263	37	-	-	PUNCT
ma-361	263	38	dimensional	dimensional	ADJ
ma-361	263	39	spaces	space	NOUN
ma-361	263	40	,	,	PUNCT
ma-361	263	41	springer	springer	NOUN
ma-361	263	42	,	,	PUNCT
ma-361	263	43	(	(	PUNCT
ma-361	263	44	2013).[9	2013).[9	NOUN
ma-361	263	45	]	]	X
ma-361	263	46	m.	m.	NOUN
ma-361	263	47	fabian	fabian	PROPN
ma-361	263	48	,	,	PUNCT
ma-361	263	49	et	et	PROPN
ma-361	263	50	al	al	PROPN
ma-361	263	51	.	.	PUNCT
ma-361	263	52	functional	functional	ADJ
ma-361	263	53	analysis	analysis	NOUN
ma-361	263	54	and	and	CCONJ
ma-361	263	55	infinite	infinite	ADJ
ma-361	263	56	-	-	PUNCT
ma-361	263	57	dimensional	dimensional	ADJ
ma-361	263	58	geometry	geometry	NOUN
ma-361	263	59	,	,	PUNCT
ma-361	263	60	springer	springer	NOUN
ma-361	263	61	,	,	PUNCT
ma-361	263	62	(	(	PUNCT
ma-361	263	63	2001).[10	2001).[10	NUM
ma-361	263	64	]	]	X
ma-361	263	65	v.p	v.p	PROPN
ma-361	263	66	.	.	PROPN
ma-361	263	67	fonf	fonf	PROPN
ma-361	263	68	,	,	PUNCT
ma-361	263	69	j.	j.	PROPN
ma-361	263	70	lindenstrauss	lindenstrauss	PROPN
ma-361	263	71	,	,	PUNCT
ma-361	263	72	r.r	r.r	PROPN
ma-361	263	73	.	.	PROPN
ma-361	263	74	phelps	phelps	PROPN
ma-361	263	75	,	,	PUNCT
ma-361	263	76	infinite	infinite	ADJ
ma-361	263	77	dimensional	dimensional	ADJ
ma-361	263	78	convexity	convexity	NOUN
ma-361	263	79	,	,	PUNCT
ma-361	263	80	in	in	ADP
ma-361	263	81	:	:	PUNCT
ma-361	263	82	handbook	handbook	NOUN
ma-361	263	83	of	of	ADP
ma-361	263	84	the	the	DET
ma-361	263	85	geometry	geometry	NOUN
ma-361	263	86	of	of	ADP
ma-361	263	87	banachspaces	banachspace	NOUN
ma-361	263	88	,	,	PUNCT
ma-361	263	89	599–670	599–670	NUM
ma-361	263	90	,	,	PUNCT
ma-361	263	91	(	(	PUNCT
ma-361	263	92	2001).[11	2001).[11	NUM
ma-361	263	93	]	]	X
ma-361	263	94	j.r	j.r	PROPN
ma-361	263	95	.	.	PROPN
ma-361	263	96	giles	giles	PROPN
ma-361	263	97	,	,	PUNCT
ma-361	263	98	convex	convex	VERB
ma-361	263	99	analysis	analysis	NOUN
ma-361	263	100	with	with	ADP
ma-361	263	101	application	application	NOUN
ma-361	263	102	in	in	ADP
ma-361	263	103	the	the	DET
ma-361	263	104	differentiation	differentiation	NOUN
ma-361	263	105	of	of	ADP
ma-361	263	106	convex	convex	NOUN
ma-361	263	107	functions	function	NOUN
ma-361	263	108	,	,	PUNCT
ma-361	263	109	pitman	pitman	NOUN
ma-361	263	110	,	,	PUNCT
ma-361	263	111	1982.[12	1982.[12	PROPN
ma-361	263	112	]	]	X
ma-361	263	113	l.a	l.a	PROPN
ma-361	263	114	.	.	PROPN
ma-361	263	115	harris	harris	PROPN
ma-361	263	116	,	,	PUNCT
ma-361	263	117	the	the	DET
ma-361	263	118	numerical	numerical	ADJ
ma-361	263	119	range	range	NOUN
ma-361	263	120	of	of	ADP
ma-361	263	121	holomorphic	holomorphic	ADJ
ma-361	263	122	functions	function	NOUN
ma-361	263	123	in	in	ADP
ma-361	263	124	banach	banach	NOUN
ma-361	263	125	spaces	space	NOUN
ma-361	263	126	,	,	PUNCT
ma-361	263	127	amer	amer	PROPN
ma-361	263	128	.	.	PUNCT
ma-361	264	1	j.	j.	PROPN
ma-361	264	2	math	math	PROPN
ma-361	264	3	.	.	PUNCT
ma-361	265	1	93	93	NUM
ma-361	265	2	(	(	PUNCT
ma-361	265	3	1974	1974	NUM
ma-361	265	4	)	)	PUNCT
ma-361	265	5	,	,	PUNCT
ma-361	265	6	1005–1019.[13	1005–1019.[13	PROPN
ma-361	265	7	]	]	X
ma-361	265	8	r.r	r.r	PROPN
ma-361	265	9	.	.	PROPN
ma-361	265	10	phelps	phelps	PROPN
ma-361	265	11	,	,	PUNCT
ma-361	265	12	lectures	lecture	NOUN
ma-361	265	13	on	on	ADP
ma-361	265	14	choquet	choquet	PROPN
ma-361	265	15	’s	’s	PART
ma-361	265	16	theorem	theorem	ADJ
ma-361	265	17	,	,	PUNCT
ma-361	265	18	springer	springer	NOUN
ma-361	265	19	,	,	PUNCT
ma-361	265	20	(	(	PUNCT
ma-361	265	21	2001).[14	2001).[14	PROPN
ma-361	265	22	]	]	X
ma-361	265	23	r.t	r.t	PROPN
ma-361	265	24	.	.	PROPN
ma-361	265	25	rockafellar	rockafellar	PROPN
ma-361	265	26	,	,	PUNCT
ma-361	265	27	conjugate	conjugate	ADJ
ma-361	265	28	duality	duality	NOUN
ma-361	265	29	and	and	CCONJ
ma-361	265	30	optimization	optimization	NOUN
ma-361	265	31	,	,	PUNCT
ma-361	265	32	siam	siam	NOUN
ma-361	265	33	,	,	PUNCT
ma-361	265	34	1974	1974	NUM
ma-361	265	35	.	.	PUNCT
ma-361	266	1	https://doi.org/10.28924/ada/ma.5.17	https://doi.org/10.28924/ada/ma.5.17	PROPN
ma-361	266	2	introduction	introduction	NOUN
ma-361	266	3	notation	notation	NOUN
ma-361	266	4	preliminaries	preliminary	NOUN
ma-361	266	5	functional	functional	ADJ
ma-361	266	6	analytic	analytic	ADJ
ma-361	266	7	foundations	foundation	NOUN
ma-361	266	8	convex	convex	VERB
ma-361	266	9	analysis	analysis	NOUN
ma-361	266	10	tools	tool	NOUN
ma-361	266	11	geometric	geometric	ADJ
ma-361	266	12	properties	property	NOUN
ma-361	266	13	polynomial	polynomial	ADJ
ma-361	266	14	mappings	mapping	NOUN
ma-361	266	15	key	key	ADJ
ma-361	266	16	topological	topological	ADJ
ma-361	266	17	concepts	concept	NOUN
ma-361	266	18	main	main	ADJ
ma-361	266	19	results	result	NOUN
ma-361	266	20	and	and	CCONJ
ma-361	266	21	discussions	discussion	NOUN
ma-361	266	22	conclusion	conclusion	NOUN
ma-361	266	23	references	reference	NOUN