id	sid	tid	token	lemma	pos
ma-382	1	1	2025	2025	NUM
ma-382	1	2	ada	ada	PROPN
ma-382	1	3	academica	academica	PROPN
ma-382	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-382	1	5	.	.	PUNCT
ma-382	2	1	j.	j.	PROPN
ma-382	2	2	math	math	PROPN
ma-382	2	3	.	.	PUNCT
ma-382	3	1	anal	anal	ADJ
ma-382	3	2	.	.	PUNCT
ma-382	4	1	5	5	NUM
ma-382	4	2	(	(	PUNCT
ma-382	4	3	2025	2025	NUM
ma-382	4	4	)	)	PUNCT
ma-382	4	5	382doi	382doi	NUM
ma-382	4	6	:	:	PUNCT
ma-382	4	7	10.28924	10.28924	NUM
ma-382	4	8	/	/	SYM
ma-382	4	9	ada	ada	NOUN
ma-382	4	10	/	/	SYM
ma-382	4	11	ma.5.382	ma.5.382	PROPN
ma-382	4	12	computational	computational	ADJ
ma-382	4	13	theory	theory	NOUN
ma-382	4	14	of	of	ADP
ma-382	4	15	norm	norm	NOUN
ma-382	4	16	-	-	PUNCT
ma-382	4	17	attaining	attain	VERB
ma-382	4	18	functionals	functional	NOUN
ma-382	4	19	:	:	PUNCT
ma-382	4	20	algorithms	algorithm	NOUN
ma-382	4	21	,	,	PUNCT
ma-382	4	22	stability	stability	NOUN
ma-382	4	23	,	,	PUNCT
ma-382	4	24	and	and	CCONJ
ma-382	4	25	applications	application	NOUN
ma-382	4	26	in	in	ADP
ma-382	4	27	banach	banach	NOUN
ma-382	4	28	spaces	space	NOUN
ma-382	4	29	mogoi	mogoi	NOUN
ma-382	4	30	n.	n.	PROPN
ma-382	4	31	evans1	evans1	PROPN
ma-382	4	32	,	,	PUNCT
ma-382	4	33	robert	robert	PROPN
ma-382	4	34	obogi2,∗	obogi2,∗	PROPN
ma-382	4	35	1department	1department	NUM
ma-382	4	36	of	of	ADP
ma-382	4	37	pure	pure	ADJ
ma-382	4	38	and	and	CCONJ
ma-382	4	39	applied	applied	ADJ
ma-382	4	40	mathematics	mathematic	NOUN
ma-382	4	41	,	,	PUNCT
ma-382	4	42	jaramogi	jaramogi	PROPN
ma-382	4	43	oginga	oginga	PROPN
ma-382	4	44	odinga	odinga	PROPN
ma-382	4	45	university	university	PROPN
ma-382	4	46	of	of	ADP
ma-382	4	47	science	science	NOUN
ma-382	4	48	and	and	CCONJ
ma-382	4	49	technology	technology	NOUN
ma-382	4	50	,	,	PUNCT
ma-382	4	51	kenya	kenya	PROPN
ma-382	4	52	mogoievans4020@gmail.com	mogoievans4020@gmail.com	X
ma-382	5	1	2department	2department	NUM
ma-382	5	2	of	of	ADP
ma-382	5	3	mathematics	mathematic	NOUN
ma-382	5	4	and	and	CCONJ
ma-382	5	5	actuarial	actuarial	ADJ
ma-382	5	6	science	science	NOUN
ma-382	5	7	,	,	PUNCT
ma-382	5	8	kisii	kisii	PROPN
ma-382	5	9	university	university	PROPN
ma-382	5	10	,	,	PUNCT
ma-382	5	11	kenya	kenya	PROPN
ma-382	5	12	robogi@kisiiuniversity.ac.ke	robogi@kisiiuniversity.ac.ke	PROPN
ma-382	5	13	∗correspondence	∗correspondence	NOUN
ma-382	5	14	:	:	PUNCT
ma-382	5	15	robogi@kisiiuniversity.ac.ke	robogi@kisiiuniversity.ac.ke	NOUN
ma-382	5	16	abstract	abstract	NOUN
ma-382	5	17	.	.	PUNCT
ma-382	6	1	this	this	DET
ma-382	6	2	paper	paper	NOUN
ma-382	6	3	develops	develop	VERB
ma-382	6	4	novel	novel	ADJ
ma-382	6	5	computational	computational	ADJ
ma-382	6	6	methods	method	NOUN
ma-382	6	7	for	for	ADP
ma-382	6	8	studying	study	VERB
ma-382	6	9	norm	norm	NOUN
ma-382	6	10	-	-	PUNCT
ma-382	6	11	attaining	attain	VERB
ma-382	6	12	functionalsin	functionalsin	NOUN
ma-382	6	13	infinite	infinite	ADV
ma-382	6	14	-	-	PUNCT
ma-382	6	15	dimensional	dimensional	ADJ
ma-382	6	16	banach	banach	NOUN
ma-382	6	17	spaces	space	NOUN
ma-382	6	18	.	.	PUNCT
ma-382	7	1	we	we	PRON
ma-382	7	2	present	present	VERB
ma-382	7	3	constructive	constructive	ADJ
ma-382	7	4	approximation	approximation	NOUN
ma-382	7	5	algorithms	algorithm	NOUN
ma-382	7	6	with	with	ADP
ma-382	7	7	ex	ex	ADJ
ma-382	7	8	-	-	ADJ
ma-382	7	9	plicit	plicit	ADJ
ma-382	7	10	convergence	convergence	NOUN
ma-382	7	11	rates	rate	NOUN
ma-382	7	12	,	,	PUNCT
ma-382	7	13	stability	stability	NOUN
ma-382	7	14	analysis	analysis	NOUN
ma-382	7	15	under	under	ADP
ma-382	7	16	discretization	discretization	NOUN
ma-382	7	17	and	and	CCONJ
ma-382	7	18	perturbations	perturbation	NOUN
ma-382	7	19	,	,	PUNCT
ma-382	7	20	and	and	CCONJ
ma-382	7	21	new	new	ADJ
ma-382	7	22	geometriccharacterizations	geometriccharacterization	NOUN
ma-382	7	23	of	of	ADP
ma-382	7	24	norm	norm	NOUN
ma-382	7	25	attainment	attainment	NOUN
ma-382	7	26	.	.	PUNCT
ma-382	8	1	key	key	ADJ
ma-382	8	2	results	result	NOUN
ma-382	8	3	include	include	VERB
ma-382	8	4	:	:	PUNCT
ma-382	8	5	(	(	PUNCT
ma-382	8	6	1	1	X
ma-382	8	7	)	)	PUNCT
ma-382	8	8	efficient	efficient	ADJ
ma-382	8	9	procedures	procedure	NOUN
ma-382	8	10	to	to	PART
ma-382	8	11	compute	compute	VERB
ma-382	8	12	norm	norm	NOUN
ma-382	8	13	-	-	PUNCT
ma-382	8	14	attaining	attain	VERB
ma-382	8	15	approximations	approximation	NOUN
ma-382	8	16	of	of	ADP
ma-382	8	17	functionals	functional	NOUN
ma-382	8	18	in	in	ADP
ma-382	8	19	uniformly	uniformly	ADV
ma-382	8	20	convex	convex	ADJ
ma-382	8	21	spaces	space	NOUN
ma-382	8	22	,	,	PUNCT
ma-382	8	23	with	with	ADP
ma-382	8	24	quantitative	quantitative	ADJ
ma-382	8	25	error	error	NOUN
ma-382	8	26	bounds;(2	bounds;(2	PROPN
ma-382	8	27	)	)	PUNCT
ma-382	8	28	stability	stability	NOUN
ma-382	8	29	theorems	theorem	VERB
ma-382	8	30	for	for	ADP
ma-382	8	31	finite	finite	ADJ
ma-382	8	32	-	-	ADJ
ma-382	8	33	dimensional	dimensional	ADJ
ma-382	8	34	projections	projection	NOUN
ma-382	8	35	in	in	ADP
ma-382	8	36	reflexive	reflexive	ADJ
ma-382	8	37	spaces	space	NOUN
ma-382	8	38	;	;	PUNCT
ma-382	8	39	(	(	PUNCT
ma-382	8	40	3	3	X
ma-382	8	41	)	)	PUNCT
ma-382	8	42	perturbation	perturbation	NOUN
ma-382	8	43	resilienceestimates	resilienceestimate	NOUN
ma-382	8	44	relating	relate	VERB
ma-382	8	45	to	to	ADP
ma-382	8	46	the	the	DET
ma-382	8	47	modulus	modulus	NOUN
ma-382	8	48	of	of	ADP
ma-382	8	49	convexity	convexity	NOUN
ma-382	8	50	;	;	PUNCT
ma-382	8	51	and	and	CCONJ
ma-382	8	52	(	(	PUNCT
ma-382	8	53	4	4	X
ma-382	8	54	)	)	PUNCT
ma-382	8	55	applications	application	NOUN
ma-382	8	56	to	to	PART
ma-382	8	57	pde	pde	NOUN
ma-382	8	58	-	-	PUNCT
ma-382	8	59	constrained	constrain	VERB
ma-382	8	60	optimizationand	optimizationand	NOUN
ma-382	8	61	functional	functional	ADJ
ma-382	8	62	regression	regression	NOUN
ma-382	8	63	.	.	PUNCT
ma-382	9	1	our	our	PRON
ma-382	9	2	approach	approach	NOUN
ma-382	9	3	combines	combine	VERB
ma-382	9	4	techniques	technique	NOUN
ma-382	9	5	from	from	ADP
ma-382	9	6	functional	functional	ADJ
ma-382	9	7	analysis	analysis	NOUN
ma-382	9	8	,	,	PUNCT
ma-382	9	9	approximationtheory	approximationtheory	NOUN
ma-382	9	10	,	,	PUNCT
ma-382	9	11	and	and	CCONJ
ma-382	9	12	computational	computational	ADJ
ma-382	9	13	mathematics	mathematic	NOUN
ma-382	9	14	,	,	PUNCT
ma-382	9	15	yielding	yield	VERB
ma-382	9	16	both	both	DET
ma-382	9	17	theoretical	theoretical	ADJ
ma-382	9	18	insights	insight	NOUN
ma-382	9	19	and	and	CCONJ
ma-382	9	20	practical	practical	ADJ
ma-382	9	21	algorithms.the	algorithms.the	DET
ma-382	9	22	results	result	NOUN
ma-382	9	23	significantly	significantly	ADV
ma-382	9	24	extend	extend	VERB
ma-382	9	25	the	the	DET
ma-382	9	26	classical	classical	ADJ
ma-382	9	27	bishop	bishop	NOUN
ma-382	9	28	-	-	PUNCT
ma-382	9	29	phelps	phelps	PROPN
ma-382	9	30	theorem	theorem	VERB
ma-382	9	31	by	by	ADP
ma-382	9	32	providing	provide	VERB
ma-382	9	33	computable	computable	ADJ
ma-382	9	34	versionsand	versionsand	NOUN
ma-382	9	35	quantitative	quantitative	ADJ
ma-382	9	36	estimates	estimate	NOUN
ma-382	9	37	in	in	ADP
ma-382	9	38	various	various	ADJ
ma-382	9	39	banach	banach	NOUN
ma-382	9	40	space	space	NOUN
ma-382	9	41	geometries	geometry	NOUN
ma-382	9	42	.	.	PUNCT
ma-382	10	1	1	1	X
ma-382	10	2	.	.	X
ma-382	10	3	introduction	introduction	NOUN
ma-382	10	4	and	and	CCONJ
ma-382	10	5	relation	relation	NOUN
ma-382	10	6	to	to	PART
ma-382	10	7	prior	prior	ADV
ma-382	10	8	work	work	VERB
ma-382	10	9	the	the	DET
ma-382	10	10	study	study	NOUN
ma-382	10	11	of	of	ADP
ma-382	10	12	norm	norm	NOUN
ma-382	10	13	-	-	PUNCT
ma-382	10	14	attaining	attain	VERB
ma-382	10	15	functionals	functional	NOUN
ma-382	10	16	has	have	AUX
ma-382	10	17	been	be	AUX
ma-382	10	18	central	central	ADJ
ma-382	10	19	to	to	PART
ma-382	10	20	banach	banach	NOUN
ma-382	10	21	space	space	NOUN
ma-382	10	22	theory	theory	NOUN
ma-382	10	23	since	since	SCONJ
ma-382	10	24	bishopand	bishopand	PROPN
ma-382	10	25	phelps	phelps	PROPN
ma-382	10	26	’	'	PUNCT
ma-382	10	27	seminal	seminal	ADJ
ma-382	10	28	result	result	NOUN
ma-382	11	1	[	[	X
ma-382	11	2	2	2	X
ma-382	11	3	]	]	PUNCT
ma-382	11	4	established	establish	VERB
ma-382	11	5	their	their	PRON
ma-382	11	6	density	density	NOUN
ma-382	11	7	in	in	ADP
ma-382	11	8	arbitrary	arbitrary	ADJ
ma-382	11	9	banach	banach	NOUN
ma-382	11	10	spaces	space	NOUN
ma-382	11	11	.	.	PUNCT
ma-382	12	1	while	while	SCONJ
ma-382	12	2	lin	lin	PROPN
ma-382	12	3	-	-	PUNCT
ma-382	12	4	denstrauss	denstrauss	NOUN
ma-382	12	5	[	[	X
ma-382	12	6	12	12	NUM
ma-382	12	7	]	]	PUNCT
ma-382	12	8	later	later	ADV
ma-382	12	9	characterized	characterize	VERB
ma-382	12	10	geometric	geometric	ADJ
ma-382	12	11	obstructions	obstruction	NOUN
ma-382	12	12	to	to	ADP
ma-382	12	13	attainment	attainment	NOUN
ma-382	12	14	and	and	CCONJ
ma-382	12	15	bourgain	bourgain	VERB
ma-382	12	16	[	[	X
ma-382	12	17	3	3	NUM
ma-382	12	18	]	]	PUNCT
ma-382	12	19	analyzedperturbation	analyzedperturbation	NOUN
ma-382	12	20	stability	stability	NOUN
ma-382	12	21	,	,	PUNCT
ma-382	12	22	the	the	DET
ma-382	12	23	computational	computational	ADJ
ma-382	12	24	aspects	aspect	NOUN
ma-382	12	25	remained	remain	VERB
ma-382	12	26	largely	largely	ADV
ma-382	12	27	unexplored	unexplored	ADJ
ma-382	12	28	until	until	ADP
ma-382	12	29	recent	recent	ADJ
ma-382	12	30	advancesin	advancesin	NOUN
ma-382	12	31	computable	computable	ADJ
ma-382	12	32	analysis	analysis	NOUN
ma-382	12	33	[	[	X
ma-382	12	34	4	4	NUM
ma-382	12	35	,	,	PUNCT
ma-382	12	36	13	13	NUM
ma-382	12	37	]	]	PUNCT
ma-382	12	38	.	.	PUNCT
ma-382	13	1	our	our	PRON
ma-382	13	2	work	work	NOUN
ma-382	13	3	bridges	bridge	NOUN
ma-382	13	4	this	this	DET
ma-382	13	5	gap	gap	NOUN
ma-382	13	6	by	by	ADP
ma-382	13	7	developing	develop	VERB
ma-382	13	8	constructive	constructive	ADJ
ma-382	13	9	methods	method	NOUN
ma-382	13	10	thatextend	thatextend	VERB
ma-382	13	11	these	these	DET
ma-382	13	12	classical	classical	ADJ
ma-382	13	13	results	result	NOUN
ma-382	13	14	while	while	SCONJ
ma-382	13	15	addressing	address	VERB
ma-382	13	16	three	three	NUM
ma-382	13	17	key	key	ADJ
ma-382	13	18	limitations	limitation	NOUN
ma-382	13	19	in	in	ADP
ma-382	13	20	the	the	DET
ma-382	13	21	literature	literature	NOUN
ma-382	13	22	:	:	PUNCT
ma-382	13	23	(	(	PUNCT
ma-382	13	24	i	i	NOUN
ma-382	13	25	)	)	PUNCT
ma-382	13	26	the	the	DET
ma-382	13	27	lackof	lackof	ADJ
ma-382	13	28	quantitative	quantitative	ADJ
ma-382	13	29	rates	rate	NOUN
ma-382	13	30	in	in	ADP
ma-382	13	31	the	the	DET
ma-382	13	32	bishop	bishop	PROPN
ma-382	13	33	-	-	PUNCT
ma-382	13	34	phelps	phelps	PROPN
ma-382	13	35	theorem	theorem	NOUN
ma-382	13	36	(	(	PUNCT
ma-382	13	37	as	as	SCONJ
ma-382	13	38	noted	note	VERB
ma-382	13	39	in	in	ADP
ma-382	13	40	[	[	X
ma-382	13	41	5	5	NUM
ma-382	13	42	]	]	NUM
ma-382	13	43	)	)	PUNCT
ma-382	13	44	,	,	PUNCT
ma-382	13	45	(	(	PUNCT
ma-382	13	46	ii	ii	NOUN
ma-382	13	47	)	)	PUNCT
ma-382	13	48	the	the	DET
ma-382	13	49	absence	absence	NOUN
ma-382	13	50	of	of	ADP
ma-382	13	51	stabilityguarantees	stabilityguarantee	NOUN
ma-382	13	52	for	for	ADP
ma-382	13	53	finite	finite	ADJ
ma-382	13	54	-	-	ADJ
ma-382	13	55	dimensional	dimensional	ADJ
ma-382	13	56	approximations	approximation	NOUN
ma-382	13	57	(	(	PUNCT
ma-382	13	58	a	a	DET
ma-382	13	59	problem	problem	NOUN
ma-382	13	60	implicit	implicit	ADJ
ma-382	13	61	in	in	ADP
ma-382	13	62	[	[	X
ma-382	13	63	1	1	NUM
ma-382	13	64	]	]	NUM
ma-382	13	65	)	)	PUNCT
ma-382	13	66	,	,	PUNCT
ma-382	13	67	and	and	CCONJ
ma-382	13	68	(	(	PUNCT
ma-382	13	69	iii	iii	X
ma-382	13	70	)	)	PUNCT
ma-382	13	71	the	the	DET
ma-382	13	72	need	need	NOUN
ma-382	13	73	for	for	SCONJ
ma-382	13	74	received	receive	VERB
ma-382	13	75	:	:	PUNCT
ma-382	13	76	14	14	NUM
ma-382	13	77	may	may	PROPN
ma-382	13	78	2025	2025	NUM
ma-382	13	79	.	.	PUNCT
ma-382	14	1	key	key	ADJ
ma-382	14	2	words	word	NOUN
ma-382	14	3	and	and	CCONJ
ma-382	14	4	phrases	phrase	NOUN
ma-382	14	5	.	.	PUNCT
ma-382	15	1	norm	norm	NOUN
ma-382	15	2	-	-	PUNCT
ma-382	15	3	attaining	attain	VERB
ma-382	15	4	functionals	functional	NOUN
ma-382	15	5	;	;	PUNCT
ma-382	15	6	computable	computable	ADJ
ma-382	15	7	analysis	analysis	NOUN
ma-382	15	8	;	;	PUNCT
ma-382	15	9	reflexive	reflexive	ADJ
ma-382	15	10	banach	banach	NOUN
ma-382	15	11	spaces	space	NOUN
ma-382	15	12	;	;	PUNCT
ma-382	15	13	uniform	uniform	ADJ
ma-382	15	14	convex	convex	PROPN
ma-382	15	15	-	-	PUNCT
ma-382	15	16	ity	ity	NOUN
ma-382	15	17	;	;	PUNCT
ma-382	15	18	approximation	approximation	NOUN
ma-382	15	19	theory	theory	NOUN
ma-382	15	20	;	;	PUNCT
ma-382	15	21	stability	stability	NOUN
ma-382	15	22	analysis	analysis	NOUN
ma-382	15	23	;	;	PUNCT
ma-382	15	24	perturbation	perturbation	NOUN
ma-382	15	25	of	of	ADP
ma-382	15	26	operators	operator	NOUN
ma-382	15	27	;	;	PUNCT
ma-382	15	28	pde	pde	NOUN
ma-382	15	29	optimization	optimization	NOUN
ma-382	15	30	;	;	PUNCT
ma-382	15	31	functional	functional	ADJ
ma-382	15	32	regression.1	regression.1	PROPN
ma-382	15	33	https://adac.ee	https://adac.ee	PROPN
ma-382	15	34	https://doi.org/10.28924/ada/ma.5.382	https://doi.org/10.28924/ada/ma.5.382	PUNCT
ma-382	15	35	eur	eur	PROPN
ma-382	15	36	.	.	PUNCT
ma-382	16	1	j.	j.	PROPN
ma-382	16	2	math	math	PROPN
ma-382	16	3	.	.	PUNCT
ma-382	17	1	anal	anal	PROPN
ma-382	17	2	.	.	PUNCT
ma-382	18	1	10.28924	10.28924	NUM
ma-382	18	2	/	/	SYM
ma-382	18	3	ada	ada	NOUN
ma-382	18	4	/	/	SYM
ma-382	18	5	ma.5.382	ma.5.382	PROPN
ma-382	18	6	2computable	2computable	NUM
ma-382	18	7	versions	version	NOUN
ma-382	18	8	of	of	ADP
ma-382	18	9	geometric	geometric	ADJ
ma-382	18	10	characterizations	characterization	NOUN
ma-382	18	11	(	(	PUNCT
ma-382	18	12	extending	extend	VERB
ma-382	18	13	questions	question	NOUN
ma-382	18	14	raised	raise	VERB
ma-382	18	15	in	in	ADP
ma-382	18	16	[	[	X
ma-382	18	17	10	10	NUM
ma-382	18	18	]	]	NUM
ma-382	18	19	)	)	PUNCT
ma-382	18	20	.	.	PUNCT
ma-382	19	1	buildingon	buildingon	VERB
ma-382	19	2	our	our	PRON
ma-382	19	3	prior	prior	ADJ
ma-382	19	4	work	work	NOUN
ma-382	19	5	on	on	ADP
ma-382	19	6	operator	operator	NOUN
ma-382	19	7	norm	norm	NOUN
ma-382	19	8	-	-	PUNCT
ma-382	19	9	attainment	attainment	NOUN
ma-382	19	10	[	[	X
ma-382	19	11	6	6	NUM
ma-382	19	12	,	,	PUNCT
ma-382	19	13	8	8	NUM
ma-382	19	14	,	,	PUNCT
ma-382	19	15	9	9	NUM
ma-382	19	16	]	]	PUNCT
ma-382	19	17	,	,	PUNCT
ma-382	19	18	we	we	PRON
ma-382	19	19	introduce	introduce	VERB
ma-382	19	20	novel	novel	ADJ
ma-382	19	21	approximation	approximation	NOUN
ma-382	19	22	algo	algo	NOUN
ma-382	19	23	-	-	PUNCT
ma-382	19	24	rithms	rithm	NOUN
ma-382	19	25	with	with	ADP
ma-382	19	26	explicit	explicit	ADJ
ma-382	19	27	convergence	convergence	NOUN
ma-382	19	28	rates	rate	NOUN
ma-382	19	29	(	(	PUNCT
ma-382	19	30	theorems	theorem	NOUN
ma-382	19	31	1	1	NUM
ma-382	19	32	-	-	SYM
ma-382	19	33	2	2	NUM
ma-382	19	34	)	)	PUNCT
ma-382	19	35	,	,	PUNCT
ma-382	19	36	demonstrating	demonstrate	VERB
ma-382	19	37	that	that	SCONJ
ma-382	19	38	the	the	DET
ma-382	19	39	modulus	modulus	NOUN
ma-382	19	40	of	of	ADP
ma-382	19	41	convexitygoverns	convexitygovern	NOUN
ma-382	19	42	both	both	CCONJ
ma-382	19	43	theoretical	theoretical	ADJ
ma-382	19	44	and	and	CCONJ
ma-382	19	45	computational	computational	ADJ
ma-382	19	46	aspects	aspect	NOUN
ma-382	19	47	of	of	ADP
ma-382	19	48	attainment	attainment	NOUN
ma-382	19	49	.	.	PUNCT
ma-382	20	1	this	this	PRON
ma-382	20	2	provides	provide	VERB
ma-382	20	3	a	a	DET
ma-382	20	4	quantitativecounterpart	quantitativecounterpart	NOUN
ma-382	20	5	to	to	ADP
ma-382	20	6	hinze	hinze	PROPN
ma-382	20	7	’s	’s	PART
ma-382	20	8	pde	pde	PROPN
ma-382	20	9	optimization	optimization	NOUN
ma-382	20	10	framework	framework	NOUN
ma-382	20	11	[	[	X
ma-382	20	12	11	11	NUM
ma-382	20	13	]	]	PUNCT
ma-382	20	14	while	while	SCONJ
ma-382	20	15	resolving	resolve	VERB
ma-382	20	16	the	the	DET
ma-382	20	17	stability	stability	NOUN
ma-382	20	18	questions	question	NOUN
ma-382	20	19	leftopen	leftopen	VERB
ma-382	20	20	by	by	ADP
ma-382	20	21	[	[	X
ma-382	20	22	2	2	NUM
ma-382	20	23	]	]	PUNCT
ma-382	20	24	for	for	ADP
ma-382	20	25	reflexive	reflexive	ADJ
ma-382	20	26	spaces	space	NOUN
ma-382	20	27	.	.	PUNCT
ma-382	21	1	the	the	DET
ma-382	21	2	synthesis	synthesis	NOUN
ma-382	21	3	of	of	ADP
ma-382	21	4	computable	computable	ADJ
ma-382	21	5	analysis	analysis	NOUN
ma-382	21	6	techniques	technique	NOUN
ma-382	21	7	from	from	ADP
ma-382	21	8	[	[	X
ma-382	21	9	4	4	X
ma-382	21	10	]	]	PUNCT
ma-382	21	11	with	with	ADP
ma-382	21	12	geo	geo	PROPN
ma-382	21	13	-	-	PUNCT
ma-382	21	14	metric	metric	ADJ
ma-382	21	15	insights	insight	NOUN
ma-382	21	16	from	from	ADP
ma-382	21	17	[	[	X
ma-382	21	18	3	3	X
ma-382	21	19	]	]	PUNCT
ma-382	21	20	yields	yield	VERB
ma-382	21	21	new	new	ADJ
ma-382	21	22	applications	application	NOUN
ma-382	21	23	in	in	ADP
ma-382	21	24	functional	functional	ADJ
ma-382	21	25	regression	regression	NOUN
ma-382	21	26	and	and	CCONJ
ma-382	21	27	adaptive	adaptive	ADJ
ma-382	21	28	discretization(theorems	discretization(theorem	NOUN
ma-382	21	29	6	6	NUM
ma-382	21	30	-	-	SYM
ma-382	21	31	7	7	NUM
ma-382	21	32	)	)	PUNCT
ma-382	21	33	,	,	PUNCT
ma-382	21	34	advancing	advance	VERB
ma-382	21	35	beyond	beyond	ADP
ma-382	21	36	the	the	DET
ma-382	21	37	existential	existential	ADJ
ma-382	21	38	results	result	NOUN
ma-382	21	39	that	that	PRON
ma-382	21	40	dominated	dominate	VERB
ma-382	21	41	earlier	early	ADJ
ma-382	21	42	studies	study	NOUN
ma-382	21	43	[	[	X
ma-382	21	44	12	12	NUM
ma-382	21	45	]	]	PUNCT
ma-382	21	46	.	.	PUNCT
ma-382	22	1	ourunified	ourunifie	VERB
ma-382	22	2	approach	approach	NOUN
ma-382	22	3	not	not	PART
ma-382	22	4	only	only	ADV
ma-382	22	5	answers	answer	VERB
ma-382	22	6	longstanding	longstanding	ADJ
ma-382	22	7	questions	question	NOUN
ma-382	22	8	about	about	ADP
ma-382	22	9	effective	effective	ADJ
ma-382	22	10	norm	norm	NOUN
ma-382	22	11	-	-	PUNCT
ma-382	22	12	attainment	attainment	NOUN
ma-382	22	13	but	but	CCONJ
ma-382	22	14	alsoestablishes	alsoestablishe	VERB
ma-382	22	15	a	a	DET
ma-382	22	16	foundation	foundation	NOUN
ma-382	22	17	for	for	ADP
ma-382	22	18	computational	computational	ADJ
ma-382	22	19	applications	application	NOUN
ma-382	22	20	in	in	ADP
ma-382	22	21	data	datum	NOUN
ma-382	22	22	science	science	NOUN
ma-382	22	23	and	and	CCONJ
ma-382	22	24	infinite	infinite	NOUN
ma-382	22	25	-	-	PUNCT
ma-382	22	26	dimensionaloptimization	dimensionaloptimization	NOUN
ma-382	22	27	.	.	PUNCT
ma-382	23	1	2	2	X
ma-382	23	2	.	.	X
ma-382	23	3	preliminaries	preliminary	NOUN
ma-382	23	4	we	we	PRON
ma-382	23	5	recall	recall	VERB
ma-382	23	6	fundamental	fundamental	ADJ
ma-382	23	7	concepts	concept	NOUN
ma-382	23	8	from	from	ADP
ma-382	23	9	functional	functional	ADJ
ma-382	23	10	analysis	analysis	NOUN
ma-382	23	11	,	,	PUNCT
ma-382	23	12	approximation	approximation	NOUN
ma-382	23	13	theory	theory	NOUN
ma-382	23	14	,	,	PUNCT
ma-382	23	15	and	and	CCONJ
ma-382	23	16	computableanalysis	computableanalysis	NOUN
ma-382	23	17	that	that	PRON
ma-382	23	18	will	will	AUX
ma-382	23	19	be	be	AUX
ma-382	23	20	used	use	VERB
ma-382	23	21	throughout	throughout	ADP
ma-382	23	22	this	this	DET
ma-382	23	23	work	work	NOUN
ma-382	23	24	.	.	PUNCT
ma-382	24	1	banach	banach	NOUN
ma-382	24	2	space	space	NOUN
ma-382	24	3	geometry	geometry	NOUN
ma-382	24	4	.	.	PUNCT
ma-382	25	1	let	let	VERB
ma-382	25	2	x	x	PRON
ma-382	25	3	be	be	AUX
ma-382	25	4	a	a	DET
ma-382	25	5	real	real	ADJ
ma-382	25	6	banach	banach	NOUN
ma-382	25	7	space	space	NOUN
ma-382	25	8	with	with	ADP
ma-382	25	9	dual	dual	ADJ
ma-382	25	10	space	space	NOUN
ma-382	25	11	x∗.	x∗.	PUNCT
ma-382	26	1	the	the	DET
ma-382	26	2	duality	duality	NOUN
ma-382	26	3	pairingis	pairingis	NOUN
ma-382	26	4	denoted	denote	VERB
ma-382	26	5	〈	〈	PROPN
ma-382	26	6	f	f	PRON
ma-382	26	7	,	,	PUNCT
ma-382	26	8	x	x	NOUN
ma-382	26	9	〉	〉	NUM
ma-382	26	10	=	=	SYM
ma-382	26	11	f	f	PROPN
ma-382	26	12	(	(	PUNCT
ma-382	26	13	x	x	NOUN
ma-382	26	14	)	)	PUNCT
ma-382	26	15	for	for	ADP
ma-382	26	16	f	f	PROPN
ma-382	26	17	∈	∈	PROPN
ma-382	26	18	x∗	x∗	PROPN
ma-382	26	19	,	,	PUNCT
ma-382	26	20	x	x	X
ma-382	26	21	∈	∈	PROPN
ma-382	26	22	x	x	X
ma-382	26	23	.	.	PUNCT
ma-382	27	1	key	key	ADJ
ma-382	27	2	geometric	geometric	ADJ
ma-382	27	3	properties	property	NOUN
ma-382	27	4	include	include	VERB
ma-382	27	5	:	:	PUNCT
ma-382	27	6	definition	definition	NOUN
ma-382	27	7	1	1	NUM
ma-382	27	8	(	(	PUNCT
ma-382	27	9	uniform	uniform	ADJ
ma-382	27	10	convexity	convexity	NOUN
ma-382	27	11	)	)	PUNCT
ma-382	27	12	.	.	PUNCT
ma-382	28	1	x	x	PRON
ma-382	28	2	is	be	AUX
ma-382	28	3	uniformly	uniformly	ADV
ma-382	28	4	convex	convex	ADJ
ma-382	28	5	if	if	SCONJ
ma-382	28	6	for	for	ADP
ma-382	28	7	every	every	DET
ma-382	28	8	ε	ε	PROPN
ma-382	28	9	>	>	X
ma-382	28	10	0	0	PROPN
ma-382	28	11	,	,	PUNCT
ma-382	28	12	there	there	PRON
ma-382	28	13	exists	exist	VERB
ma-382	28	14	δ(ε	δ(ε	NOUN
ma-382	28	15	)	)	PUNCT
ma-382	28	16	>	>	X
ma-382	28	17	0	0	PUNCT
ma-382	29	1	such	such	ADJ
ma-382	29	2	that	that	PRON
ma-382	29	3	for	for	ADP
ma-382	29	4	all	all	DET
ma-382	29	5	x	x	NOUN
ma-382	29	6	,	,	PUNCT
ma-382	29	7	y	y	PROPN
ma-382	29	8	∈	∈	PROPN
ma-382	29	9	sx	sx	PROPN
ma-382	29	10	,	,	PUNCT
ma-382	29	11	‖x	‖x	PROPN
ma-382	29	12	−	−	PROPN
ma-382	30	1	y‖	y‖	PROPN
ma-382	30	2	≥	≥	X
ma-382	30	3	ε	ε	PROPN
ma-382	30	4	=	=	PRON
ma-382	30	5	⇒	⇒	NOUN
ma-382	30	6	∥∥∥∥x	∥∥∥∥x	VERB
ma-382	31	1	+	+	X
ma-382	31	2	y	y	PROPN
ma-382	31	3	2	2	NUM
ma-382	31	4	∥∥∥∥	∥∥∥∥	NUM
ma-382	31	5	≤	≤	NUM
ma-382	31	6	1−	1−	NUM
ma-382	31	7	δ(ε	δ(ε	NOUN
ma-382	31	8	)	)	PUNCT
ma-382	31	9	.	.	PUNCT
ma-382	32	1	the	the	DET
ma-382	32	2	function	function	PROPN
ma-382	32	3	δ	δ	PROPN
ma-382	32	4	(	(	PUNCT
ma-382	32	5	·	·	PUNCT
ma-382	32	6	)	)	PUNCT
ma-382	32	7	is	be	AUX
ma-382	32	8	called	call	VERB
ma-382	32	9	the	the	DET
ma-382	32	10	modulus	modulus	NOUN
ma-382	32	11	of	of	ADP
ma-382	32	12	convexity	convexity	NOUN
ma-382	32	13	.	.	PUNCT
ma-382	33	1	definition	definition	NOUN
ma-382	33	2	2	2	NUM
ma-382	33	3	(	(	PUNCT
ma-382	33	4	reflexivity	reflexivity	NOUN
ma-382	33	5	and	and	CCONJ
ma-382	33	6	rnp	rnp	PROPN
ma-382	33	7	)	)	PUNCT
ma-382	33	8	.	.	PUNCT
ma-382	34	1	x	x	PUNCT
ma-382	34	2	is	be	AUX
ma-382	34	3	reflexive	reflexive	ADJ
ma-382	34	4	if	if	SCONJ
ma-382	34	5	the	the	DET
ma-382	34	6	natural	natural	ADJ
ma-382	34	7	embedding	embed	VERB
ma-382	34	8	x	x	X
ma-382	34	9	↪	↪	PROPN
ma-382	34	10	→	→	SYM
ma-382	34	11	x∗∗	x∗∗	PROPN
ma-382	34	12	is	be	AUX
ma-382	34	13	surjective	surjective	ADJ
ma-382	34	14	.	.	PUNCT
ma-382	35	1	it	it	PRON
ma-382	35	2	has	have	VERB
ma-382	35	3	the	the	DET
ma-382	35	4	radon	radon	PROPN
ma-382	35	5	-	-	PUNCT
ma-382	35	6	nikodym	nikodym	NOUN
ma-382	35	7	property	property	NOUN
ma-382	35	8	(	(	PUNCT
ma-382	35	9	rnp	rnp	PROPN
ma-382	35	10	)	)	PUNCT
ma-382	35	11	if	if	SCONJ
ma-382	35	12	every	every	DET
ma-382	35	13	bounded	bound	VERB
ma-382	35	14	subset	subset	NOUN
ma-382	35	15	is	be	AUX
ma-382	35	16	dentable	dentable	ADJ
ma-382	35	17	.	.	PUNCT
ma-382	36	1	norm	norm	NOUN
ma-382	36	2	-	-	PUNCT
ma-382	36	3	attaining	attain	VERB
ma-382	36	4	functionals	functional	NOUN
ma-382	36	5	.	.	PUNCT
ma-382	37	1	the	the	DET
ma-382	37	2	core	core	ADJ
ma-382	37	3	object	object	NOUN
ma-382	37	4	of	of	ADP
ma-382	37	5	our	our	PRON
ma-382	37	6	study	study	NOUN
ma-382	37	7	is	be	AUX
ma-382	37	8	:	:	PUNCT
ma-382	37	9	definition	definition	NOUN
ma-382	37	10	3	3	NUM
ma-382	37	11	(	(	PUNCT
ma-382	37	12	norm	norm	NOUN
ma-382	37	13	-	-	PUNCT
ma-382	37	14	attaining	attain	VERB
ma-382	37	15	functional	functional	ADJ
ma-382	37	16	)	)	PUNCT
ma-382	37	17	.	.	PUNCT
ma-382	38	1	a	a	DET
ma-382	38	2	functional	functional	ADJ
ma-382	38	3	f	f	PROPN
ma-382	38	4	∈	∈	PROPN
ma-382	38	5	x∗	x∗	PROPN
ma-382	38	6	norm	norm	NOUN
ma-382	38	7	-	-	PUNCT
ma-382	38	8	attains	attain	NOUN
ma-382	38	9	if	if	SCONJ
ma-382	38	10	there	there	PRON
ma-382	38	11	exists	exist	VERB
ma-382	38	12	x0	x0	PROPN
ma-382	38	13	∈	∈	PROPN
ma-382	38	14	sx	sx	PROPN
ma-382	38	15	(	(	PUNCT
ma-382	38	16	called	call	VERB
ma-382	38	17	an	an	DET
ma-382	38	18	attaining	attain	VERB
ma-382	38	19	point	point	NOUN
ma-382	38	20	)	)	PUNCT
ma-382	38	21	such	such	ADJ
ma-382	38	22	that	that	PRON
ma-382	38	23	|f	|f	PROPN
ma-382	39	1	(	(	PUNCT
ma-382	39	2	x0)|	x0)|	NUM
ma-382	39	3	=	=	SYM
ma-382	39	4	‖f‖.	‖f‖.	ADP
ma-382	39	5	the	the	DET
ma-382	39	6	foundational	foundational	ADJ
ma-382	39	7	result	result	NOUN
ma-382	39	8	is	be	AUX
ma-382	39	9	:	:	PUNCT
ma-382	39	10	theorem	theorem	ADJ
ma-382	39	11	1	1	NUM
ma-382	39	12	(	(	PUNCT
ma-382	39	13	bishop	bishop	NOUN
ma-382	39	14	-	-	PUNCT
ma-382	39	15	phelps	phelps	PROPN
ma-382	39	16	)	)	PUNCT
ma-382	39	17	.	.	PUNCT
ma-382	40	1	for	for	ADP
ma-382	40	2	any	any	DET
ma-382	40	3	banach	banach	NOUN
ma-382	40	4	space	space	NOUN
ma-382	40	5	x	x	X
ma-382	40	6	,	,	PUNCT
ma-382	40	7	the	the	DET
ma-382	40	8	norm	norm	NOUN
ma-382	40	9	-	-	PUNCT
ma-382	40	10	attaining	attain	VERB
ma-382	40	11	functionals	functional	NOUN
ma-382	40	12	are	be	AUX
ma-382	40	13	dense	dense	ADJ
ma-382	40	14	in	in	ADP
ma-382	40	15	x∗.	x∗.	PROPN
ma-382	40	16	https://doi.org/10.28924/ada/ma.5.382	https://doi.org/10.28924/ada/ma.5.382	PROPN
ma-382	40	17	eur	eur	PROPN
ma-382	40	18	.	.	PUNCT
ma-382	41	1	j.	j.	PROPN
ma-382	41	2	math	math	PROPN
ma-382	41	3	.	.	PUNCT
ma-382	42	1	anal	anal	PROPN
ma-382	42	2	.	.	PUNCT
ma-382	43	1	10.28924	10.28924	NUM
ma-382	43	2	/	/	SYM
ma-382	43	3	ada	ada	PROPN
ma-382	43	4	/	/	SYM
ma-382	43	5	ma.5.382	ma.5.382	PROPN
ma-382	43	6	3	3	NUM
ma-382	43	7	computational	computational	ADJ
ma-382	43	8	framework	framework	NOUN
ma-382	43	9	.	.	PUNCT
ma-382	44	1	for	for	ADP
ma-382	44	2	algorithmic	algorithmic	ADJ
ma-382	44	3	results	result	NOUN
ma-382	44	4	,	,	PUNCT
ma-382	44	5	we	we	PRON
ma-382	44	6	work	work	VERB
ma-382	44	7	in	in	ADP
ma-382	44	8	the	the	DET
ma-382	44	9	type-2	type-2	NUM
ma-382	44	10	effectivity	effectivity	NOUN
ma-382	44	11	(	(	PUNCT
ma-382	44	12	tte	tte	PROPN
ma-382	44	13	)	)	PUNCT
ma-382	44	14	model	model	NOUN
ma-382	44	15	:	:	PUNCT
ma-382	44	16	definition	definition	NOUN
ma-382	44	17	4	4	NUM
ma-382	44	18	(	(	PUNCT
ma-382	44	19	computable	computable	ADJ
ma-382	44	20	banach	banach	NOUN
ma-382	44	21	space	space	NOUN
ma-382	44	22	)	)	PUNCT
ma-382	44	23	.	.	PUNCT
ma-382	45	1	a	a	DET
ma-382	45	2	banach	banach	NOUN
ma-382	45	3	space	space	NOUN
ma-382	45	4	(	(	PUNCT
ma-382	45	5	x	x	X
ma-382	45	6	,	,	PUNCT
ma-382	45	7	‖	‖	PROPN
ma-382	45	8	·	·	PUNCT
ma-382	45	9	‖	‖	NUM
ma-382	45	10	)	)	PUNCT
ma-382	45	11	is	be	AUX
ma-382	45	12	computable	computable	ADJ
ma-382	45	13	if	if	SCONJ
ma-382	45	14	there	there	PRON
ma-382	45	15	exists	exist	VERB
ma-382	45	16	a	a	DET
ma-382	45	17	dense	dense	ADJ
ma-382	45	18	sequence	sequence	NOUN
ma-382	45	19	{	{	PUNCT
ma-382	45	20	en	en	X
ma-382	45	21	}	}	PUNCT
ma-382	45	22	(	(	PUNCT
ma-382	45	23	the	the	DET
ma-382	45	24	computable	computable	ADJ
ma-382	45	25	points	point	NOUN
ma-382	45	26	)	)	PUNCT
ma-382	45	27	and	and	CCONJ
ma-382	45	28	an	an	DET
ma-382	45	29	algorithm	algorithm	NOUN
ma-382	45	30	that	that	PRON
ma-382	45	31	computes	compute	VERB
ma-382	45	32	the	the	DET
ma-382	45	33	norm	norm	NOUN
ma-382	45	34	‖	‖	ADJ
ma-382	45	35	∑n	∑n	NOUN
ma-382	45	36	k=1	k=1	X
ma-382	45	37	akek‖	akek‖	PROPN
ma-382	45	38	to	to	ADP
ma-382	45	39	arbitrary	arbitrary	ADJ
ma-382	45	40	precision	precision	NOUN
ma-382	45	41	.	.	PUNCT
ma-382	46	1	definition	definition	NOUN
ma-382	46	2	5	5	NUM
ma-382	46	3	(	(	PUNCT
ma-382	46	4	computable	computable	ADJ
ma-382	46	5	functional	functional	ADJ
ma-382	46	6	)	)	PUNCT
ma-382	46	7	.	.	PUNCT
ma-382	47	1	f	f	PROPN
ma-382	47	2	∈	∈	PROPN
ma-382	47	3	x∗	x∗	PROPN
ma-382	47	4	is	be	AUX
ma-382	47	5	computable	computable	ADJ
ma-382	47	6	if	if	SCONJ
ma-382	47	7	there	there	PRON
ma-382	47	8	exists	exist	VERB
ma-382	47	9	an	an	DET
ma-382	47	10	algorithm	algorithm	NOUN
ma-382	47	11	that	that	PRON
ma-382	47	12	,	,	PUNCT
ma-382	47	13	given	give	VERB
ma-382	47	14	a	a	DET
ma-382	47	15	computable	computable	ADJ
ma-382	47	16	x	x	SYM
ma-382	47	17	∈	∈	PROPN
ma-382	47	18	x	x	X
ma-382	47	19	and	and	CCONJ
ma-382	47	20	n	n	CCONJ
ma-382	47	21	∈	∈	PROPN
ma-382	47	22	n	n	CCONJ
ma-382	47	23	,	,	PUNCT
ma-382	47	24	outputs	outputs	PROPN
ma-382	47	25	f	f	PROPN
ma-382	47	26	(	(	PUNCT
ma-382	47	27	x	x	X
ma-382	47	28	)	)	PUNCT
ma-382	47	29	with	with	ADP
ma-382	47	30	error	error	NOUN
ma-382	47	31	<	<	X
ma-382	47	32	2−n	2−n	NUM
ma-382	47	33	.	.	PUNCT
ma-382	47	34	approximation	approximation	NOUN
ma-382	47	35	theory	theory	NOUN
ma-382	47	36	.	.	PUNCT
ma-382	48	1	key	key	ADJ
ma-382	48	2	tools	tool	NOUN
ma-382	48	3	for	for	ADP
ma-382	48	4	our	our	PRON
ma-382	48	5	discretization	discretization	NOUN
ma-382	48	6	results	result	NOUN
ma-382	48	7	include	include	VERB
ma-382	48	8	:	:	PUNCT
ma-382	48	9	definition	definition	NOUN
ma-382	48	10	6	6	NUM
ma-382	48	11	(	(	PUNCT
ma-382	48	12	projection	projection	NOUN
ma-382	48	13	operators	operator	NOUN
ma-382	48	14	)	)	PUNCT
ma-382	48	15	.	.	PUNCT
ma-382	49	1	a	a	DET
ma-382	49	2	sequence	sequence	NOUN
ma-382	49	3	{	{	PUNCT
ma-382	49	4	pn	pn	NOUN
ma-382	49	5	}	}	PUNCT
ma-382	49	6	on	on	ADP
ma-382	49	7	x	x	PUNCT
ma-382	49	8	with	with	ADP
ma-382	49	9	dimpn(x	dimpn(x	NOUN
ma-382	49	10	)	)	PUNCT
ma-382	49	11	=	=	SYM
ma-382	49	12	n	n	PROPN
ma-382	49	13	is	be	AUX
ma-382	49	14	called	call	VERB
ma-382	49	15	:	:	PUNCT
ma-382	49	16	•	•	NUM
ma-382	49	17	finite	finite	NOUN
ma-382	49	18	-	-	NOUN
ma-382	49	19	rank	rank	NOUN
ma-382	49	20	if	if	SCONJ
ma-382	49	21	each	each	DET
ma-382	49	22	pn	pn	PROPN
ma-382	49	23	has	have	VERB
ma-382	49	24	finite	finite	ADJ
ma-382	49	25	-	-	ADJ
ma-382	49	26	dimensional	dimensional	ADJ
ma-382	49	27	range	range	NOUN
ma-382	49	28	•	•	NOUN
ma-382	49	29	admissible	admissible	ADJ
ma-382	49	30	if	if	SCONJ
ma-382	49	31	‖pn‖	‖pn‖	PROPN
ma-382	49	32	≤	≤	NOUN
ma-382	49	33	c	c	NOUN
ma-382	49	34	uniformly	uniformly	ADV
ma-382	49	35	and	and	CCONJ
ma-382	49	36	pn	pn	PROPN
ma-382	49	37	→	→	PUNCT
ma-382	49	38	i	i	PRON
ma-382	49	39	strongly	strongly	ADV
ma-382	49	40	definition	definition	VERB
ma-382	49	41	7	7	NUM
ma-382	49	42	(	(	PUNCT
ma-382	49	43	modulus	modulus	NOUN
ma-382	49	44	of	of	ADP
ma-382	49	45	smoothness	smoothness	NOUN
ma-382	49	46	)	)	PUNCT
ma-382	49	47	.	.	PUNCT
ma-382	50	1	for	for	ADP
ma-382	50	2	f	f	PROPN
ma-382	50	3	∈	∈	PROPN
ma-382	50	4	x∗	x∗	PROPN
ma-382	50	5	,	,	PUNCT
ma-382	50	6	its	its	PRON
ma-382	50	7	modulus	modulus	NOUN
ma-382	50	8	of	of	ADP
ma-382	50	9	smoothness	smoothness	NOUN
ma-382	50	10	on	on	ADP
ma-382	50	11	a	a	DET
ma-382	50	12	subset	subset	NOUN
ma-382	50	13	a	a	DET
ma-382	50	14	⊂	⊂	X
ma-382	50	15	x	x	X
ma-382	50	16	is	be	AUX
ma-382	50	17	:	:	PUNCT
ma-382	50	18	ω(f	ω(f	ADJ
ma-382	50	19	,	,	PUNCT
ma-382	50	20	δ;a	δ;a	NUM
ma-382	50	21	)	)	PUNCT
ma-382	50	22	:	:	PUNCT
ma-382	51	1	=	=	SYM
ma-382	51	2	sup{|f	sup{|f	PROPN
ma-382	51	3	(	(	PUNCT
ma-382	51	4	x)−	x)−	PROPN
ma-382	51	5	f	f	PROPN
ma-382	51	6	(	(	PUNCT
ma-382	51	7	y)|	y)|	NOUN
ma-382	51	8	:	:	PUNCT
ma-382	51	9	x	x	X
ma-382	51	10	,	,	PUNCT
ma-382	51	11	y	y	PROPN
ma-382	51	12	∈	∈	PROPN
ma-382	51	13	a	a	PRON
ma-382	51	14	,	,	PUNCT
ma-382	51	15	‖x	‖x	NOUN
ma-382	51	16	−	−	PROPN
ma-382	52	1	y‖	y‖	PROPN
ma-382	52	2	≤	≤	PROPN
ma-382	52	3	δ	δ	PROPN
ma-382	52	4	}	}	PUNCT
ma-382	52	5	.	.	PUNCT
ma-382	53	1	this	this	DET
ma-382	53	2	framework	framework	NOUN
ma-382	53	3	combines	combine	VERB
ma-382	53	4	classical	classical	ADJ
ma-382	53	5	banach	banach	NOUN
ma-382	53	6	space	space	NOUN
ma-382	53	7	theory	theory	NOUN
ma-382	53	8	with	with	ADP
ma-382	53	9	modern	modern	ADJ
ma-382	53	10	computational	computational	ADJ
ma-382	53	11	perspectives	perspective	NOUN
ma-382	53	12	,	,	PUNCT
ma-382	53	13	enabling	enable	VERB
ma-382	53	14	our	our	PRON
ma-382	53	15	subsequent	subsequent	ADJ
ma-382	53	16	analysis	analysis	NOUN
ma-382	53	17	of	of	ADP
ma-382	53	18	constructive	constructive	ADJ
ma-382	53	19	norm	norm	NOUN
ma-382	53	20	attainment	attainment	NOUN
ma-382	53	21	.	.	PUNCT
ma-382	54	1	3	3	X
ma-382	54	2	.	.	X
ma-382	54	3	main	main	ADJ
ma-382	54	4	results	result	NOUN
ma-382	54	5	and	and	CCONJ
ma-382	54	6	discussions	discussion	NOUN
ma-382	54	7	theorem	theorem	VERB
ma-382	54	8	2	2	X
ma-382	54	9	.	.	PUNCT
ma-382	55	1	let	let	VERB
ma-382	55	2	x	x	PRON
ma-382	55	3	be	be	AUX
ma-382	55	4	a	a	DET
ma-382	55	5	uniformly	uniformly	ADV
ma-382	55	6	convex	convex	NOUN
ma-382	55	7	banach	banach	NOUN
ma-382	55	8	space	space	NOUN
ma-382	55	9	with	with	ADP
ma-382	55	10	modulus	modulus	NOUN
ma-382	55	11	of	of	ADP
ma-382	55	12	convexity	convexity	NOUN
ma-382	55	13	δ(ε	δ(ε	NOUN
ma-382	55	14	)	)	PUNCT
ma-382	55	15	,	,	PUNCT
ma-382	55	16	and	and	CCONJ
ma-382	55	17	let	let	VERB
ma-382	55	18	{	{	PUNCT
ma-382	55	19	fn	fn	AUX
ma-382	55	20	}	}	PUNCT
ma-382	55	21	be	be	AUX
ma-382	55	22	a	a	DET
ma-382	55	23	sequence	sequence	NOUN
ma-382	55	24	of	of	ADP
ma-382	55	25	computable	computable	ADJ
ma-382	55	26	functionals	functional	NOUN
ma-382	55	27	converging	converge	VERB
ma-382	55	28	weakly	weakly	ADV
ma-382	55	29	to	to	ADP
ma-382	55	30	f	f	PROPN
ma-382	55	31	∈	∈	PROPN
ma-382	55	32	x∗.	x∗.	PUNCT
ma-382	56	1	then	then	ADV
ma-382	56	2	:	:	PUNCT
ma-382	56	3	(	(	PUNCT
ma-382	56	4	1	1	X
ma-382	56	5	)	)	PUNCT
ma-382	56	6	there	there	PRON
ma-382	56	7	exists	exist	VERB
ma-382	56	8	an	an	DET
ma-382	56	9	algorithm	algorithm	NOUN
ma-382	56	10	constructing	construct	VERB
ma-382	56	11	{	{	PUNCT
ma-382	56	12	f̃n	f̃n	X
ma-382	56	13	}	}	PUNCT
ma-382	56	14	with	with	ADP
ma-382	56	15	‖f̃n	‖f̃n	X
ma-382	56	16	−	−	NUM
ma-382	56	17	fn‖x∗	fn‖x∗	NUM
ma-382	56	18	<	<	X
ma-382	56	19	2−n	2−n	NUM
ma-382	56	20	that	that	SCONJ
ma-382	56	21	norm	norm	NOUN
ma-382	56	22	-	-	PUNCT
ma-382	56	23	attains	attain	NOUN
ma-382	56	24	at	at	ADP
ma-382	56	25	computable	computable	ADJ
ma-382	56	26	points	point	NOUN
ma-382	56	27	{	{	PUNCT
ma-382	56	28	xn	xn	NOUN
ma-382	56	29	}	}	PUNCT
ma-382	56	30	⊂	⊂	NOUN
ma-382	56	31	x	x	X
ma-382	56	32	(	(	PUNCT
ma-382	56	33	2	2	X
ma-382	56	34	)	)	PUNCT
ma-382	56	35	the	the	DET
ma-382	56	36	convergence	convergence	NOUN
ma-382	56	37	rate	rate	NOUN
ma-382	56	38	satisfies	satisfie	NOUN
ma-382	56	39	‖f	‖f	ADP
ma-382	56	40	−	−	PROPN
ma-382	56	41	f̃n‖x∗	f̃n‖x∗	PROPN
ma-382	56	42	≤	≤	PROPN
ma-382	56	43	cx	cx	PROPN
ma-382	56	44	·	·	PUNCT
ma-382	56	45	δ−1(2−n	δ−1(2−n	NUM
ma-382	56	46	)	)	PUNCT
ma-382	57	1	+	+	CCONJ
ma-382	57	2	‖f	‖f	ADP
ma-382	57	3	−	−	NUM
ma-382	57	4	fn‖x∗	fn‖x∗	SYM
ma-382	57	5	(	(	PUNCT
ma-382	57	6	3	3	NUM
ma-382	57	7	)	)	PUNCT
ma-382	57	8	for	for	ADP
ma-382	57	9	hilbert	hilbert	NOUN
ma-382	57	10	spaces	space	NOUN
ma-382	57	11	,	,	PUNCT
ma-382	57	12	the	the	DET
ma-382	57	13	convergence	convergence	NOUN
ma-382	57	14	becomes	become	VERB
ma-382	57	15	‖f	‖f	ADP
ma-382	57	16	−	−	PROPN
ma-382	57	17	f̃n‖x∗	f̃n‖x∗	PROPN
ma-382	57	18	≤	≤	PROPN
ma-382	57	19	√	√	ADP
ma-382	57	20	2−n	2−n	NUM
ma-382	58	1	+	+	CCONJ
ma-382	58	2	‖f	‖f	ADP
ma-382	58	3	−	−	NOUN
ma-382	58	4	fn‖x∗	fn‖x∗	ADP
ma-382	58	5	proof	proof	NOUN
ma-382	58	6	.	.	PUNCT
ma-382	59	1	since	since	SCONJ
ma-382	59	2	x	x	PRON
ma-382	59	3	is	be	AUX
ma-382	59	4	uniformly	uniformly	ADV
ma-382	59	5	convex	convex	NOUN
ma-382	59	6	,	,	PUNCT
ma-382	59	7	every	every	DET
ma-382	59	8	bounded	bounded	ADJ
ma-382	59	9	sequence	sequence	NOUN
ma-382	59	10	has	have	VERB
ma-382	59	11	unique	unique	ADJ
ma-382	59	12	asymptotic	asymptotic	ADJ
ma-382	59	13	limits	limit	NOUN
ma-382	59	14	andthe	andthe	ADJ
ma-382	59	15	dual	dual	ADJ
ma-382	59	16	space	space	NOUN
ma-382	59	17	x∗	x∗	PROPN
ma-382	59	18	is	be	AUX
ma-382	59	19	strictly	strictly	ADV
ma-382	59	20	convex	convex	ADJ
ma-382	59	21	.	.	PUNCT
ma-382	60	1	given	give	VERB
ma-382	60	2	a	a	DET
ma-382	60	3	sequence	sequence	NOUN
ma-382	60	4	{	{	PUNCT
ma-382	60	5	fn	fn	NOUN
ma-382	60	6	}	}	PUNCT
ma-382	60	7	converging	converge	VERB
ma-382	60	8	weakly	weakly	ADV
ma-382	60	9	to	to	ADP
ma-382	60	10	f	f	PROPN
ma-382	60	11	,	,	PUNCT
ma-382	60	12	by	by	ADP
ma-382	60	13	mazur’slemma	mazur’slemma	ADV
ma-382	60	14	,	,	PUNCT
ma-382	60	15	convex	convex	ADJ
ma-382	60	16	combinations	combination	NOUN
ma-382	60	17	of	of	ADP
ma-382	60	18	{	{	PUNCT
ma-382	60	19	fn	fn	NOUN
ma-382	60	20	}	}	PUNCT
ma-382	60	21	converge	converge	VERB
ma-382	60	22	strongly	strongly	ADV
ma-382	60	23	to	to	ADP
ma-382	60	24	f	f	PROPN
ma-382	60	25	in	in	ADP
ma-382	60	26	x∗.	x∗.	PROPN
ma-382	60	27	for	for	ADP
ma-382	60	28	each	each	DET
ma-382	60	29	n	n	CCONJ
ma-382	60	30	,	,	PUNCT
ma-382	60	31	choose	choose	VERB
ma-382	60	32	a	a	DET
ma-382	60	33	computableconvex	computableconvex	NOUN
ma-382	60	34	combination	combination	NOUN
ma-382	60	35	f̃n	f̃n	ADJ
ma-382	60	36	of	of	ADP
ma-382	60	37	the	the	DET
ma-382	60	38	form	form	NOUN
ma-382	60	39	:	:	PUNCT
ma-382	60	40	f̃n	f̃n	X
ma-382	60	41	=	=	PUNCT
ma-382	60	42	n(n)∑	n(n)∑	PUNCT
ma-382	60	43	k=1	k=1	PUNCT
ma-382	61	1	α	α	X
ma-382	61	2	(	(	PUNCT
ma-382	61	3	n	n	CCONJ
ma-382	61	4	)	)	PUNCT
ma-382	62	1	k	k	NOUN
ma-382	62	2	fk	fk	INTJ
ma-382	62	3	,	,	PUNCT
ma-382	62	4	∑	∑	PROPN
ma-382	62	5	α	α	PROPN
ma-382	62	6	(	(	PUNCT
ma-382	62	7	n	n	CCONJ
ma-382	62	8	)	)	PUNCT
ma-382	62	9	k	k	NOUN
ma-382	63	1	=	=	SYM
ma-382	63	2	1	1	NUM
ma-382	63	3	,	,	PUNCT
ma-382	63	4	α	α	PROPN
ma-382	63	5	(	(	PUNCT
ma-382	63	6	n	n	CCONJ
ma-382	63	7	)	)	PUNCT
ma-382	63	8	k	k	NOUN
ma-382	63	9	≥	≥	NOUN
ma-382	63	10	0	0	NUM
ma-382	63	11	,	,	PUNCT
ma-382	63	12	such	such	ADJ
ma-382	63	13	that	that	PRON
ma-382	63	14	‖f̃n−fn‖x∗	‖f̃n−fn‖x∗	NOUN
ma-382	63	15	<	<	X
ma-382	63	16	2−n	2−n	NUM
ma-382	63	17	.	.	PUNCT
ma-382	64	1	we	we	PRON
ma-382	64	2	now	now	ADV
ma-382	64	3	argue	argue	VERB
ma-382	64	4	that	that	SCONJ
ma-382	64	5	each	each	PRON
ma-382	64	6	f̃n	f̃n	AUX
ma-382	64	7	norm	norm	NOUN
ma-382	64	8	-	-	PUNCT
ma-382	64	9	attains	attain	NOUN
ma-382	64	10	.	.	PUNCT
ma-382	65	1	since	since	SCONJ
ma-382	65	2	x	x	PRON
ma-382	65	3	is	be	AUX
ma-382	65	4	uniformly	uniformly	ADV
ma-382	65	5	convex	convex	NOUN
ma-382	65	6	,	,	PUNCT
ma-382	65	7	its	its	PRON
ma-382	65	8	dual	dual	ADJ
ma-382	65	9	x∗	x∗	NOUN
ma-382	65	10	is	be	AUX
ma-382	65	11	reflexive	reflexive	ADJ
ma-382	65	12	.	.	PUNCT
ma-382	66	1	then	then	ADV
ma-382	66	2	,	,	PUNCT
ma-382	66	3	the	the	DET
ma-382	66	4	bishop	bishop	PROPN
ma-382	66	5	-	-	PUNCT
ma-382	66	6	phelps	phelps	PROPN
ma-382	66	7	theorem	theorem	NOUN
ma-382	66	8	(	(	PUNCT
ma-382	66	9	or	or	CCONJ
ma-382	66	10	in	in	ADP
ma-382	66	11	effective	effective	ADJ
ma-382	66	12	terms	term	NOUN
ma-382	66	13	,	,	PUNCT
ma-382	66	14	its	its	PRON
ma-382	66	15	computableversion	computableversion	NOUN
ma-382	66	16	)	)	PUNCT
ma-382	66	17	ensures	ensure	VERB
ma-382	66	18	that	that	SCONJ
ma-382	66	19	norm	norm	NOUN
ma-382	66	20	-	-	PUNCT
ma-382	66	21	attaining	attain	VERB
ma-382	66	22	functionals	functional	NOUN
ma-382	66	23	are	be	AUX
ma-382	66	24	dense	dense	ADJ
ma-382	66	25	in	in	ADP
ma-382	66	26	x∗.	x∗.	PROPN
ma-382	66	27	thus	thus	ADV
ma-382	66	28	,	,	PUNCT
ma-382	66	29	for	for	ADP
ma-382	66	30	each	each	DET
ma-382	66	31	f̃n	f̃n	NOUN
ma-382	66	32	,	,	PUNCT
ma-382	66	33	we	we	PRON
ma-382	66	34	can	can	AUX
ma-382	66	35	finda	finda	ADV
ma-382	66	36	computable	computable	ADJ
ma-382	66	37	xn	xn	PROPN
ma-382	66	38	∈	∈	PROPN
ma-382	66	39	sx	sx	PROPN
ma-382	66	40	such	such	ADJ
ma-382	66	41	that	that	PRON
ma-382	66	42	f̃n(xn	f̃n(xn	X
ma-382	66	43	)	)	PUNCT
ma-382	66	44	=	=	SYM
ma-382	67	1	‖f̃n‖	‖f̃n‖	NOUN
ma-382	67	2	and	and	CCONJ
ma-382	67	3	xn	xn	PROPN
ma-382	67	4	is	be	AUX
ma-382	67	5	effectively	effectively	ADV
ma-382	67	6	computable	computable	ADJ
ma-382	67	7	by	by	ADP
ma-382	67	8	exhaustive	exhaustive	ADJ
ma-382	67	9	https://doi.org/10.28924/ada/ma.5.382	https://doi.org/10.28924/ada/ma.5.382	NOUN
ma-382	67	10	eur	eur	PROPN
ma-382	67	11	.	.	PUNCT
ma-382	68	1	j.	j.	PROPN
ma-382	68	2	math	math	PROPN
ma-382	68	3	.	.	PUNCT
ma-382	69	1	anal	anal	PROPN
ma-382	69	2	.	.	PUNCT
ma-382	70	1	10.28924	10.28924	NUM
ma-382	70	2	/	/	SYM
ma-382	70	3	ada	ada	PROPN
ma-382	70	4	/	/	SYM
ma-382	70	5	ma.5.382	ma.5.382	PROPN
ma-382	70	6	4search	4search	NUM
ma-382	70	7	on	on	ADP
ma-382	70	8	a	a	DET
ma-382	70	9	computable	computable	ADJ
ma-382	70	10	dense	dense	ADJ
ma-382	70	11	set	set	NOUN
ma-382	70	12	.	.	PUNCT
ma-382	71	1	for	for	ADP
ma-382	71	2	the	the	DET
ma-382	71	3	convergence	convergence	NOUN
ma-382	71	4	estimate	estimate	NOUN
ma-382	71	5	,	,	PUNCT
ma-382	71	6	recall	recall	VERB
ma-382	71	7	the	the	DET
ma-382	71	8	definition	definition	NOUN
ma-382	71	9	of	of	ADP
ma-382	71	10	modulusof	modulusof	NOUN
ma-382	71	11	convexity	convexity	NOUN
ma-382	71	12	:	:	PUNCT
ma-382	71	13	for	for	ADP
ma-382	71	14	any	any	DET
ma-382	71	15	ε	ε	PROPN
ma-382	71	16	>	>	X
ma-382	71	17	0	0	PROPN
ma-382	71	18	,	,	PUNCT
ma-382	71	19	if	if	SCONJ
ma-382	71	20	‖x‖	‖x‖	PROPN
ma-382	71	21	=	=	SYM
ma-382	71	22	‖y‖	‖y‖	PROPN
ma-382	71	23	=	=	SYM
ma-382	71	24	1	1	NUM
ma-382	71	25	and	and	CCONJ
ma-382	71	26	‖x	‖x	NOUN
ma-382	72	1	−	−	PROPN
ma-382	72	2	y‖	y‖	PROPN
ma-382	72	3	≥	≥	NUM
ma-382	72	4	ε	ε	PROPN
ma-382	72	5	,	,	PUNCT
ma-382	72	6	then	then	ADV
ma-382	72	7	∥∥	∥∥	X
ma-382	72	8	x+y	x+y	PROPN
ma-382	72	9	2	2	NUM
ma-382	72	10	∥∥	∥∥	X
ma-382	72	11	≤	≤	NUM
ma-382	72	12	1	1	NUM
ma-382	72	13	−	−	PROPN
ma-382	72	14	δ(ε	δ(ε	NOUN
ma-382	72	15	)	)	PUNCT
ma-382	72	16	.	.	PUNCT
ma-382	73	1	thisquantifies	thisquantifie	VERB
ma-382	73	2	the	the	DET
ma-382	73	3	deviation	deviation	NOUN
ma-382	73	4	from	from	ADP
ma-382	73	5	norm	norm	NOUN
ma-382	73	6	in	in	ADP
ma-382	73	7	non	non	ADJ
ma-382	73	8	-	-	ADJ
ma-382	73	9	convex	convex	ADJ
ma-382	73	10	directions	direction	NOUN
ma-382	73	11	.	.	PUNCT
ma-382	74	1	it	it	PRON
ma-382	74	2	follows	follow	VERB
ma-382	74	3	from	from	ADP
ma-382	74	4	this	this	PRON
ma-382	74	5	that	that	SCONJ
ma-382	74	6	:	:	PUNCT
ma-382	75	1	‖f	‖f	ADP
ma-382	75	2	−	−	PROPN
ma-382	75	3	f̃n‖x∗	f̃n‖x∗	PROPN
ma-382	75	4	≤	≤	NOUN
ma-382	75	5	‖f	‖f	ADP
ma-382	75	6	−	−	ADP
ma-382	75	7	fn‖x∗	fn‖x∗	SYM
ma-382	75	8	+	+	NUM
ma-382	75	9	‖fn	‖fn	NUM
ma-382	75	10	−	−	PROPN
ma-382	75	11	f̃n‖x∗	f̃n‖x∗	PROPN
ma-382	75	12	≤	≤	NOUN
ma-382	75	13	‖f	‖f	ADP
ma-382	75	14	−	−	ADP
ma-382	75	15	fn‖x∗	fn‖x∗	NUM
ma-382	75	16	+	+	NOUN
ma-382	75	17	2−n	2−n	NUM
ma-382	75	18	.	.	PUNCT
ma-382	76	1	now	now	ADV
ma-382	76	2	,	,	PUNCT
ma-382	76	3	using	use	VERB
ma-382	76	4	convexity	convexity	NOUN
ma-382	76	5	and	and	CCONJ
ma-382	76	6	duality	duality	NOUN
ma-382	76	7	bounds	bound	NOUN
ma-382	76	8	,	,	PUNCT
ma-382	76	9	and	and	CCONJ
ma-382	76	10	inverting	invert	VERB
ma-382	76	11	the	the	DET
ma-382	76	12	modulus	modulus	NOUN
ma-382	76	13	,	,	PUNCT
ma-382	76	14	we	we	PRON
ma-382	76	15	obtain	obtain	VERB
ma-382	76	16	‖f	‖f	PUNCT
ma-382	76	17	−	−	PUNCT
ma-382	76	18	f̃n‖x∗	f̃n‖x∗	PROPN
ma-382	76	19	≤	≤	PROPN
ma-382	76	20	cx	cx	PROPN
ma-382	76	21	·	·	PUNCT
ma-382	76	22	δ−1(2−n	δ−1(2−n	NUM
ma-382	76	23	)	)	PUNCT
ma-382	77	1	+	+	CCONJ
ma-382	77	2	‖f	‖f	ADP
ma-382	77	3	−	−	NOUN
ma-382	77	4	fn‖x∗for	fn‖x∗for	ADP
ma-382	77	5	some	some	DET
ma-382	77	6	constant	constant	ADJ
ma-382	77	7	cx	cx	NOUN
ma-382	77	8	depending	depend	VERB
ma-382	77	9	on	on	ADP
ma-382	77	10	x	x	X
ma-382	77	11	.	.	PUNCT
ma-382	78	1	in	in	ADP
ma-382	78	2	the	the	DET
ma-382	78	3	special	special	ADJ
ma-382	78	4	case	case	NOUN
ma-382	78	5	where	where	SCONJ
ma-382	78	6	x	x	PRON
ma-382	78	7	is	be	AUX
ma-382	78	8	a	a	DET
ma-382	78	9	hilbert	hilbert	NOUN
ma-382	78	10	space	space	NOUN
ma-382	78	11	,	,	PUNCT
ma-382	78	12	the	the	DET
ma-382	78	13	modulusof	modulusof	NOUN
ma-382	78	14	convexity	convexity	NOUN
ma-382	78	15	satisfies	satisfy	VERB
ma-382	78	16	δ(ε	δ(ε	NOUN
ma-382	78	17	)	)	PUNCT
ma-382	78	18	≥	≥	NOUN
ma-382	78	19	ε2	ε2	ADJ
ma-382	78	20	8	8	NUM
ma-382	78	21	.	.	PUNCT
ma-382	79	1	inverting	invert	VERB
ma-382	79	2	this	this	PRON
ma-382	79	3	gives	give	VERB
ma-382	79	4	δ−1(t	δ−1(t	PROPN
ma-382	79	5	)	)	PUNCT
ma-382	79	6	≤	≤	NOUN
ma-382	79	7	√	√	NUM
ma-382	79	8	8	8	NUM
ma-382	79	9	t	t	NOUN
ma-382	79	10	.	.	PUNCT
ma-382	80	1	thus	thus	ADV
ma-382	80	2	,	,	PUNCT
ma-382	80	3	‖f	‖f	ADP
ma-382	80	4	−	−	NOUN
ma-382	80	5	f̃n‖x∗	f̃n‖x∗	PROPN
ma-382	80	6	≤	≤	PROPN
ma-382	80	7	√	√	ADP
ma-382	80	8	2−n	2−n	NUM
ma-382	80	9	+	+	CCONJ
ma-382	80	10	‖f	‖f	ADP
ma-382	80	11	−	−	NOUN
ma-382	80	12	fn‖x∗	fn‖x∗	NUM
ma-382	80	13	,	,	PUNCT
ma-382	80	14	as	as	SCONJ
ma-382	80	15	required	require	VERB
ma-382	80	16	.	.	PUNCT
ma-382	81	1	�	�	PROPN
ma-382	81	2	theorem	theorem	VERB
ma-382	81	3	3	3	NUM
ma-382	81	4	.	.	X
ma-382	82	1	for	for	ADP
ma-382	82	2	any	any	DET
ma-382	82	3	reflexive	reflexive	ADJ
ma-382	82	4	banach	banach	NOUN
ma-382	82	5	space	space	NOUN
ma-382	82	6	x	x	NOUN
ma-382	82	7	and	and	CCONJ
ma-382	82	8	norm	norm	NOUN
ma-382	82	9	-	-	PUNCT
ma-382	82	10	attaining	attain	VERB
ma-382	82	11	f	f	PROPN
ma-382	82	12	∈	∈	PROPN
ma-382	82	13	x∗	x∗	PROPN
ma-382	82	14	,	,	PUNCT
ma-382	82	15	consider	consider	VERB
ma-382	82	16	finitedimensional	finitedimensional	ADJ
ma-382	82	17	subspaces	subspace	NOUN
ma-382	82	18	{	{	PUNCT
ma-382	82	19	xn	xn	NUM
ma-382	82	20	}	}	PUNCT
ma-382	82	21	with	with	ADP
ma-382	82	22	dimxn	dimxn	NOUN
ma-382	82	23	=	=	SYM
ma-382	82	24	n	n	NOUN
ma-382	82	25	and	and	CCONJ
ma-382	82	26	xn	xn	PROPN
ma-382	82	27	↪	↪	PROPN
ma-382	82	28	→	→	SYM
ma-382	82	29	xn+1	xn+1	NUM
ma-382	82	30	.	.	PUNCT
ma-382	83	1	then	then	ADV
ma-382	83	2	:	:	PUNCT
ma-382	83	3	(	(	PUNCT
ma-382	83	4	1	1	X
ma-382	83	5	)	)	PUNCT
ma-382	83	6	the	the	DET
ma-382	83	7	projected	project	VERB
ma-382	83	8	functionals	functional	NOUN
ma-382	83	9	fn	fn	NOUN
ma-382	83	10	=	=	SYM
ma-382	83	11	f	f	X
ma-382	83	12	|xn	|xn	X
ma-382	83	13	norm	norm	NOUN
ma-382	83	14	-	-	PUNCT
ma-382	83	15	attain	attain	VERB
ma-382	83	16	with	with	ADP
ma-382	83	17	probability	probability	NOUN
ma-382	83	18	1	1	NUM
ma-382	83	19	under	under	ADP
ma-382	83	20	any	any	DET
ma-382	83	21	reasonable	reasonable	ADJ
ma-382	83	22	sampling	sampling	NOUN
ma-382	83	23	measure	measure	NOUN
ma-382	83	24	(	(	PUNCT
ma-382	83	25	2	2	X
ma-382	83	26	)	)	PUNCT
ma-382	83	27	the	the	DET
ma-382	83	28	stability	stability	NOUN
ma-382	83	29	estimate	estimate	NOUN
ma-382	83	30	holds	hold	VERB
ma-382	83	31	:	:	PUNCT
ma-382	83	32	sup‖x‖=1	sup‖x‖=1	PROPN
ma-382	83	33	|f	|f	PROPN
ma-382	84	1	(	(	PUNCT
ma-382	84	2	x)−	x)−	PROPN
ma-382	84	3	fn(pnx)|	fn(pnx)|	VERB
ma-382	84	4	≤	≤	NUM
ma-382	84	5	ω(f	ω(f	NUM
ma-382	84	6	,	,	PUNCT
ma-382	84	7	dist(x	dist(x	PROPN
ma-382	84	8	,	,	PUNCT
ma-382	84	9	xn	xn	PROPN
ma-382	84	10	)	)	PUNCT
ma-382	84	11	)	)	PUNCT
ma-382	85	1	(	(	PUNCT
ma-382	85	2	3	3	X
ma-382	85	3	)	)	PUNCT
ma-382	85	4	for	for	ADP
ma-382	85	5	lp	lp	ADJ
ma-382	85	6	spaces	space	NOUN
ma-382	85	7	,	,	PUNCT
ma-382	85	8	explicit	explicit	ADJ
ma-382	85	9	convergence	convergence	NOUN
ma-382	85	10	rates	rate	NOUN
ma-382	85	11	are	be	AUX
ma-382	85	12	o(n−α(p	o(n−α(p	VERB
ma-382	85	13	)	)	PUNCT
ma-382	85	14	)	)	PUNCT
ma-382	85	15	where	where	SCONJ
ma-382	85	16	α(p	α(p	ADV
ma-382	85	17	)	)	PUNCT
ma-382	85	18	>	>	SYM
ma-382	86	1	0	0	NUM
ma-382	86	2	proof	proof	NOUN
ma-382	86	3	.	.	PUNCT
ma-382	87	1	let	let	VERB
ma-382	87	2	x	x	PRON
ma-382	87	3	be	be	AUX
ma-382	87	4	a	a	DET
ma-382	87	5	reflexive	reflexive	ADJ
ma-382	87	6	banach	banach	NOUN
ma-382	87	7	space	space	NOUN
ma-382	87	8	.	.	PUNCT
ma-382	88	1	then	then	ADV
ma-382	88	2	the	the	DET
ma-382	88	3	dual	dual	ADJ
ma-382	88	4	x∗	x∗	PROPN
ma-382	88	5	is	be	AUX
ma-382	88	6	also	also	ADV
ma-382	88	7	reflexive	reflexive	ADJ
ma-382	88	8	.	.	PUNCT
ma-382	89	1	let	let	VERB
ma-382	89	2	f	f	PROPN
ma-382	89	3	∈	∈	PROPN
ma-382	89	4	x∗	x∗	PROPN
ma-382	89	5	bea	bea	PROPN
ma-382	89	6	norm	norm	NOUN
ma-382	89	7	-	-	PUNCT
ma-382	89	8	attaining	attain	VERB
ma-382	89	9	functional	functional	ADJ
ma-382	89	10	.	.	PUNCT
ma-382	90	1	define	define	VERB
ma-382	90	2	fn	fn	NOUN
ma-382	90	3	:	:	PUNCT
ma-382	90	4	=	=	SYM
ma-382	90	5	f	f	X
ma-382	90	6	|xn	|xn	X
ma-382	90	7	,	,	PUNCT
ma-382	90	8	the	the	DET
ma-382	90	9	restriction	restriction	NOUN
ma-382	90	10	of	of	ADP
ma-382	90	11	f	f	PROPN
ma-382	90	12	to	to	ADP
ma-382	90	13	the	the	DET
ma-382	90	14	finite	finite	NOUN
ma-382	90	15	-	-	NOUN
ma-382	90	16	dimensionalsubspace	dimensionalsubspace	NOUN
ma-382	90	17	xn	xn	NOUN
ma-382	90	18	.	.	PUNCT
ma-382	91	1	(	(	PUNCT
ma-382	91	2	1	1	X
ma-382	91	3	)	)	PUNCT
ma-382	91	4	since	since	SCONJ
ma-382	91	5	xn	xn	PROPN
ma-382	91	6	is	be	AUX
ma-382	91	7	finite	finite	ADJ
ma-382	91	8	-	-	ADJ
ma-382	91	9	dimensional	dimensional	ADJ
ma-382	91	10	,	,	PUNCT
ma-382	91	11	the	the	DET
ma-382	91	12	norm	norm	NOUN
ma-382	91	13	on	on	ADP
ma-382	91	14	x∗n	x∗n	NUM
ma-382	91	15	is	be	AUX
ma-382	91	16	attained	attain	VERB
ma-382	91	17	at	at	ADP
ma-382	91	18	some	some	DET
ma-382	91	19	xn	xn	NOUN
ma-382	91	20	∈	∈	PROPN
ma-382	91	21	sxn	sxn	NOUN
ma-382	91	22	.	.	PUNCT
ma-382	92	1	by	by	ADP
ma-382	92	2	the	the	DET
ma-382	92	3	rieszrepresentation	rieszrepresentation	NOUN
ma-382	92	4	theorem	theorem	NOUN
ma-382	92	5	(	(	PUNCT
ma-382	92	6	or	or	CCONJ
ma-382	92	7	simply	simply	ADV
ma-382	92	8	compactness	compactness	NOUN
ma-382	92	9	of	of	ADP
ma-382	92	10	the	the	DET
ma-382	92	11	unit	unit	NOUN
ma-382	92	12	sphere	sphere	ADV
ma-382	92	13	in	in	ADP
ma-382	92	14	finite	finite	ADJ
ma-382	92	15	dimensions	dimension	NOUN
ma-382	92	16	)	)	PUNCT
ma-382	92	17	,	,	PUNCT
ma-382	92	18	there	there	PRON
ma-382	92	19	exists	exist	VERB
ma-382	92	20	xn	xn	PROPN
ma-382	92	21	∈	∈	PROPN
ma-382	92	22	xn	xn	PROPN
ma-382	92	23	such	such	ADJ
ma-382	92	24	that	that	SCONJ
ma-382	92	25	fn(xn	fn(xn	NOUN
ma-382	92	26	)	)	PUNCT
ma-382	92	27	=	=	PUNCT
ma-382	92	28	‖fn‖.	‖fn‖.	VERB
ma-382	92	29	moreover	moreover	ADV
ma-382	92	30	,	,	PUNCT
ma-382	92	31	if	if	SCONJ
ma-382	92	32	f	f	PROPN
ma-382	92	33	is	be	AUX
ma-382	92	34	randomly	randomly	ADV
ma-382	92	35	selected	select	VERB
ma-382	92	36	(	(	PUNCT
ma-382	92	37	e.g.	e.g.	ADV
ma-382	92	38	,	,	PUNCT
ma-382	92	39	under	under	ADP
ma-382	92	40	a	a	DET
ma-382	92	41	gaussian	gaussian	ADJ
ma-382	92	42	orhaar	orhaar	NOUN
ma-382	92	43	measure	measure	NOUN
ma-382	92	44	)	)	PUNCT
ma-382	92	45	,	,	PUNCT
ma-382	92	46	the	the	DET
ma-382	92	47	probability	probability	NOUN
ma-382	92	48	that	that	SCONJ
ma-382	92	49	f	f	PROPN
ma-382	92	50	lies	lie	VERB
ma-382	92	51	in	in	ADP
ma-382	92	52	the	the	DET
ma-382	92	53	set	set	NOUN
ma-382	92	54	of	of	ADP
ma-382	92	55	functionals	functional	NOUN
ma-382	92	56	whose	whose	DET
ma-382	92	57	restriction	restriction	NOUN
ma-382	92	58	fails	fail	VERB
ma-382	92	59	to	to	PART
ma-382	92	60	attainthe	attainthe	DET
ma-382	92	61	norm	norm	NOUN
ma-382	92	62	is	be	AUX
ma-382	92	63	zero	zero	NUM
ma-382	92	64	,	,	PUNCT
ma-382	92	65	due	due	ADJ
ma-382	92	66	to	to	ADP
ma-382	92	67	density	density	NOUN
ma-382	92	68	and	and	CCONJ
ma-382	92	69	baire	baire	NOUN
ma-382	92	70	category	category	NOUN
ma-382	92	71	arguments	argument	NOUN
ma-382	92	72	.	.	PUNCT
ma-382	93	1	(	(	PUNCT
ma-382	93	2	2	2	X
ma-382	93	3	)	)	PUNCT
ma-382	93	4	for	for	ADP
ma-382	93	5	stability	stability	NOUN
ma-382	93	6	,	,	PUNCT
ma-382	93	7	let	let	VERB
ma-382	93	8	pn	pn	PART
ma-382	93	9	be	be	AUX
ma-382	93	10	the	the	DET
ma-382	93	11	nearest	near	ADJ
ma-382	93	12	-	-	PUNCT
ma-382	93	13	point	point	NOUN
ma-382	93	14	projection	projection	NOUN
ma-382	93	15	onto	onto	ADP
ma-382	93	16	xn	xn	PROPN
ma-382	93	17	.	.	PUNCT
ma-382	94	1	then	then	ADV
ma-382	94	2	for	for	ADP
ma-382	94	3	x	x	SYM
ma-382	94	4	∈	∈	PROPN
ma-382	94	5	x	x	PUNCT
ma-382	94	6	with	with	ADP
ma-382	94	7	‖x‖	‖x‖	PROPN
ma-382	94	8	=	=	SYM
ma-382	94	9	1	1	NUM
ma-382	94	10	,	,	PUNCT
ma-382	94	11	|f	|f	PROPN
ma-382	94	12	(	(	PUNCT
ma-382	94	13	x)−	x)−	PROPN
ma-382	94	14	fn(pnx)|	fn(pnx)|	PROPN
ma-382	94	15	=	=	SYM
ma-382	94	16	|f	|f	PROPN
ma-382	94	17	(	(	PUNCT
ma-382	94	18	x	x	SYM
ma-382	94	19	−	−	NOUN
ma-382	94	20	pnx)|	pnx)|	VERB
ma-382	94	21	≤	≤	PUNCT
ma-382	94	22	‖f‖	‖f‖	PROPN
ma-382	94	23	·	·	PUNCT
ma-382	94	24	‖x	‖x	NUM
ma-382	95	1	−	−	NOUN
ma-382	95	2	pnx‖	pnx‖	PROPN
ma-382	96	1	≤	≤	NOUN
ma-382	96	2	ω(f	ω(f	ADJ
ma-382	96	3	,	,	PUNCT
ma-382	96	4	dist(x	dist(x	PROPN
ma-382	96	5	,	,	PUNCT
ma-382	96	6	xn	xn	PROPN
ma-382	96	7	)	)	PUNCT
ma-382	96	8	)	)	PUNCT
ma-382	96	9	,	,	PUNCT
ma-382	96	10	where	where	SCONJ
ma-382	96	11	ω(f	ω(f	ADJ
ma-382	96	12	,	,	PUNCT
ma-382	96	13	·	·	PUNCT
ma-382	96	14	)	)	PUNCT
ma-382	96	15	denotes	denote	VERB
ma-382	96	16	the	the	DET
ma-382	96	17	modulus	modulus	NOUN
ma-382	96	18	of	of	ADP
ma-382	96	19	continuity	continuity	NOUN
ma-382	96	20	of	of	ADP
ma-382	96	21	f	f	PROPN
ma-382	96	22	on	on	ADP
ma-382	96	23	bounded	bounded	ADJ
ma-382	96	24	sets	set	NOUN
ma-382	96	25	,	,	PUNCT
ma-382	96	26	which	which	PRON
ma-382	96	27	exists	exist	VERB
ma-382	96	28	since	since	SCONJ
ma-382	96	29	f	f	PROPN
ma-382	96	30	iscontinuous	iscontinuous	ADJ
ma-382	96	31	.	.	PUNCT
ma-382	97	1	(	(	PUNCT
ma-382	97	2	3	3	X
ma-382	97	3	)	)	PUNCT
ma-382	97	4	for	for	ADP
ma-382	97	5	x	x	SYM
ma-382	97	6	=	=	SYM
ma-382	97	7	lp([0	lp([0	NOUN
ma-382	97	8	,	,	PUNCT
ma-382	97	9	1	1	NUM
ma-382	97	10	]	]	NUM
ma-382	97	11	)	)	PUNCT
ma-382	97	12	,	,	PUNCT
ma-382	97	13	the	the	DET
ma-382	97	14	rate	rate	NOUN
ma-382	97	15	of	of	ADP
ma-382	97	16	best	good	ADJ
ma-382	97	17	approximation	approximation	NOUN
ma-382	97	18	by	by	ADP
ma-382	97	19	finite	finite	ADJ
ma-382	97	20	-	-	ADJ
ma-382	97	21	dimensional	dimensional	ADJ
ma-382	97	22	subspaces	subspace	NOUN
ma-382	97	23	is	be	AUX
ma-382	97	24	well	well	ADV
ma-382	97	25	-	-	PUNCT
ma-382	97	26	known	know	VERB
ma-382	97	27	:	:	PUNCT
ma-382	97	28	for	for	ADP
ma-382	97	29	spline	spline	NOUN
ma-382	97	30	or	or	CCONJ
ma-382	97	31	fourier	fourier	ADJ
ma-382	97	32	-	-	PUNCT
ma-382	97	33	type	type	NOUN
ma-382	97	34	subspaces	subspace	NOUN
ma-382	97	35	xn	xn	X
ma-382	97	36	,	,	PUNCT
ma-382	97	37	the	the	DET
ma-382	97	38	projection	projection	NOUN
ma-382	97	39	error	error	NOUN
ma-382	97	40	‖x−pnx‖lp	‖x−pnx‖lp	ADV
ma-382	97	41	is	be	AUX
ma-382	97	42	o(n−α(p	o(n−α(p	VERB
ma-382	97	43	)	)	PUNCT
ma-382	97	44	)	)	PUNCT
ma-382	97	45	,	,	PUNCT
ma-382	97	46	with	with	ADP
ma-382	97	47	α(p	α(p	NUM
ma-382	97	48	)	)	PUNCT
ma-382	97	49	depending	depend	VERB
ma-382	97	50	on	on	ADP
ma-382	97	51	smoothness	smoothness	ADJ
ma-382	97	52	assumptions	assumption	NOUN
ma-382	97	53	and	and	CCONJ
ma-382	97	54	the	the	DET
ma-382	97	55	space	space	NOUN
ma-382	97	56	structure	structure	NOUN
ma-382	97	57	(	(	PUNCT
ma-382	97	58	e.g.	e.g.	ADV
ma-382	97	59	,	,	PUNCT
ma-382	97	60	α(p	α(p	NUM
ma-382	97	61	)	)	PUNCT
ma-382	98	1	=	=	SYM
ma-382	98	2	1	1	NUM
ma-382	98	3	/	/	SYM
ma-382	98	4	p	p	NOUN
ma-382	98	5	for	for	ADP
ma-382	98	6	piecewisepolynomial	piecewisepolynomial	ADJ
ma-382	98	7	approximations	approximation	NOUN
ma-382	98	8	under	under	ADP
ma-382	98	9	certain	certain	ADJ
ma-382	98	10	regularity	regularity	NOUN
ma-382	98	11	)	)	PUNCT
ma-382	98	12	.	.	PUNCT
ma-382	99	1	this	this	DET
ma-382	99	2	rate	rate	NOUN
ma-382	99	3	carries	carry	VERB
ma-382	99	4	over	over	ADP
ma-382	99	5	to	to	ADP
ma-382	99	6	the	the	DET
ma-382	99	7	convergence	convergence	NOUN
ma-382	99	8	of	of	ADP
ma-382	99	9	fn(pnx	fn(pnx	NOUN
ma-382	99	10	)	)	PUNCT
ma-382	99	11	to	to	ADP
ma-382	99	12	f	f	PROPN
ma-382	99	13	(	(	PUNCT
ma-382	99	14	x	x	X
ma-382	99	15	)	)	PUNCT
ma-382	99	16	by	by	ADP
ma-382	99	17	the	the	DET
ma-382	99	18	continuity	continuity	NOUN
ma-382	99	19	of	of	ADP
ma-382	99	20	f	f	PROPN
ma-382	99	21	.	.	PUNCT
ma-382	100	1	�	�	PROPN
ma-382	100	2	https://doi.org/10.28924/ada/ma.5.382	https://doi.org/10.28924/ada/ma.5.382	NUM
ma-382	100	3	eur	eur	PROPN
ma-382	100	4	.	.	PUNCT
ma-382	101	1	j.	j.	PROPN
ma-382	101	2	math	math	PROPN
ma-382	101	3	.	.	PUNCT
ma-382	102	1	anal	anal	PROPN
ma-382	102	2	.	.	PUNCT
ma-382	103	1	10.28924	10.28924	NUM
ma-382	103	2	/	/	SYM
ma-382	103	3	ada	ada	PROPN
ma-382	103	4	/	/	SYM
ma-382	103	5	ma.5.382	ma.5.382	PROPN
ma-382	103	6	5	5	NUM
ma-382	103	7	theorem	theorem	NOUN
ma-382	103	8	4	4	NUM
ma-382	103	9	.	.	PUNCT
ma-382	104	1	let	let	VERB
ma-382	104	2	f	f	PROPN
ma-382	104	3	∈	∈	PROPN
ma-382	104	4	x∗	x∗	PROPN
ma-382	104	5	norm	norm	NOUN
ma-382	104	6	-	-	PUNCT
ma-382	104	7	attain	attain	NOUN
ma-382	104	8	at	at	ADP
ma-382	104	9	x0	x0	PROPN
ma-382	104	10	∈	∈	PROPN
ma-382	104	11	sx	sx	PROPN
ma-382	104	12	with	with	ADP
ma-382	104	13	f	f	PROPN
ma-382	104	14	(	(	PUNCT
ma-382	104	15	x0	x0	PROPN
ma-382	104	16	)	)	PUNCT
ma-382	105	1	=	=	SYM
ma-382	105	2	‖f‖.	‖f‖.	PROPN
ma-382	105	3	for	for	ADP
ma-382	105	4	any	any	DET
ma-382	105	5	ε	ε	NOUN
ma-382	105	6	-	-	PUNCT
ma-382	105	7	perturbation	perturbation	NOUN
ma-382	105	8	g	g	NOUN
ma-382	105	9	∈	∈	PROPN
ma-382	105	10	x∗	x∗	PROPN
ma-382	105	11	with	with	ADP
ma-382	105	12	‖f	‖f	PRON
ma-382	105	13	−	−	NOUN
ma-382	105	14	g‖	g‖	NOUN
ma-382	105	15	<	<	X
ma-382	105	16	ε	ε	PROPN
ma-382	105	17	:	:	PUNCT
ma-382	105	18	(	(	PUNCT
ma-382	105	19	1	1	X
ma-382	105	20	)	)	PUNCT
ma-382	105	21	there	there	PRON
ma-382	105	22	exists	exist	VERB
ma-382	105	23	a	a	DET
ma-382	105	24	nearby	nearby	ADJ
ma-382	105	25	point	point	NOUN
ma-382	105	26	xε	xε	PUNCT
ma-382	106	1	where	where	SCONJ
ma-382	106	2	g	g	PROPN
ma-382	106	3	norm	norm	NOUN
ma-382	106	4	-	-	PUNCT
ma-382	106	5	attains	attain	NOUN
ma-382	106	6	with	with	ADP
ma-382	106	7	‖x0	‖x0	PROPN
ma-382	106	8	−	−	PROPN
ma-382	106	9	xε‖	xε‖	PROPN
ma-382	106	10	≤	≤	NUM
ma-382	106	11	√	√	NUM
ma-382	106	12	2ε	2ε	NUM
ma-382	106	13	/	/	SYM
ma-382	106	14	δx(ε	δx(ε	NOUN
ma-382	106	15	)	)	PUNCT
ma-382	106	16	(	(	PUNCT
ma-382	106	17	2	2	X
ma-382	106	18	)	)	PUNCT
ma-382	106	19	the	the	DET
ma-382	106	20	norm	norm	NOUN
ma-382	106	21	ratio	ratio	NOUN
ma-382	106	22	satisfies	satisfie	NOUN
ma-382	106	23	1−	1−	NUM
ma-382	106	24	ε	ε	PROPN
ma-382	106	25	‖f‖	‖f‖	PROPN
ma-382	106	26	≤	≤	NOUN
ma-382	106	27	‖g‖	‖g‖	NUM
ma-382	106	28	‖f‖	‖f‖	PROPN
ma-382	106	29	≤	≤	NUM
ma-382	106	30	1	1	NUM
ma-382	106	31	+	+	CCONJ
ma-382	106	32	ε	ε	PROPN
ma-382	106	33	‖f‖	‖f‖	PROPN
ma-382	106	34	(	(	PUNCT
ma-382	106	35	3	3	NUM
ma-382	106	36	)	)	PUNCT
ma-382	106	37	for	for	ADP
ma-382	106	38	uniformly	uniformly	ADV
ma-382	106	39	smooth	smooth	ADJ
ma-382	106	40	spaces	space	NOUN
ma-382	106	41	,	,	PUNCT
ma-382	106	42	the	the	DET
ma-382	106	43	attaining	attain	VERB
ma-382	106	44	point	point	NOUN
ma-382	106	45	moves	move	VERB
ma-382	106	46	continuously	continuously	ADV
ma-382	106	47	:	:	PUNCT
ma-382	107	1	limε→0	limε→0	NOUN
ma-382	107	2	xε	xε	PUNCT
ma-382	108	1	=	=	SYM
ma-382	108	2	x0	x0	PROPN
ma-382	108	3	proof	proof	NOUN
ma-382	108	4	.	.	PUNCT
ma-382	109	1	let	let	VERB
ma-382	109	2	f	f	PRON
ma-382	109	3	∈	∈	PROPN
ma-382	109	4	x∗	x∗	VERB
ma-382	109	5	such	such	ADJ
ma-382	109	6	that	that	SCONJ
ma-382	109	7	f	f	PROPN
ma-382	109	8	norm	norm	NOUN
ma-382	109	9	-	-	PUNCT
ma-382	109	10	attains	attain	NOUN
ma-382	109	11	at	at	ADP
ma-382	109	12	x0	x0	PROPN
ma-382	109	13	∈	∈	PROPN
ma-382	109	14	sx	sx	PROPN
ma-382	109	15	,	,	PUNCT
ma-382	109	16	i.e.	i.e.	X
ma-382	109	17	,	,	PUNCT
ma-382	109	18	f	f	PROPN
ma-382	109	19	(	(	PUNCT
ma-382	109	20	x0	x0	PROPN
ma-382	109	21	)	)	PUNCT
ma-382	109	22	=	=	SYM
ma-382	109	23	‖f‖.	‖f‖.	PART
ma-382	109	24	now	now	ADV
ma-382	109	25	consider	consider	VERB
ma-382	109	26	g	g	PROPN
ma-382	109	27	∈	∈	PROPN
ma-382	110	1	x∗such	x∗such	ADP
ma-382	110	2	that	that	SCONJ
ma-382	110	3	‖f	‖f	ADP
ma-382	110	4	−	−	PROPN
ma-382	110	5	g‖	g‖	PROPN
ma-382	110	6	<	<	X
ma-382	110	7	ε	ε	PROPN
ma-382	110	8	.	.	PUNCT
ma-382	110	9	(	(	PUNCT
ma-382	110	10	1	1	X
ma-382	110	11	)	)	PUNCT
ma-382	110	12	existence	existence	NOUN
ma-382	110	13	of	of	ADP
ma-382	110	14	nearby	nearby	ADJ
ma-382	110	15	norm	norm	NOUN
ma-382	110	16	-	-	PUNCT
ma-382	110	17	attaining	attain	VERB
ma-382	110	18	point	point	NOUN
ma-382	110	19	.	.	PUNCT
ma-382	111	1	define	define	VERB
ma-382	111	2	the	the	DET
ma-382	111	3	duality	duality	NOUN
ma-382	111	4	map	map	NOUN
ma-382	111	5	j	j	NOUN
ma-382	111	6	:	:	PUNCT
ma-382	111	7	x	x	X
ma-382	111	8	→	→	SYM
ma-382	111	9	2x	2x	NUM
ma-382	111	10	∗	∗	NOUN
ma-382	111	11	by	by	ADP
ma-382	111	12	j(x	j(x	NOUN
ma-382	111	13	)	)	PUNCT
ma-382	111	14	=	=	PRON
ma-382	112	1	{	{	PUNCT
ma-382	112	2	x∗	x∗	PROPN
ma-382	112	3	∈	∈	PROPN
ma-382	112	4	x∗	x∗	PROPN
ma-382	112	5	:	:	PUNCT
ma-382	112	6	‖x∗‖	‖x∗‖	X
ma-382	112	7	=	=	SYM
ma-382	112	8	‖x‖	‖x‖	PROPN
ma-382	112	9	,	,	PUNCT
ma-382	112	10	x∗(x	x∗(x	PROPN
ma-382	112	11	)	)	PUNCT
ma-382	112	12	=	=	PRON
ma-382	112	13	‖x‖2	‖x‖2	ADJ
ma-382	112	14	}	}	PUNCT
ma-382	112	15	.	.	PUNCT
ma-382	113	1	if	if	SCONJ
ma-382	113	2	f	f	PROPN
ma-382	113	3	∈	∈	PROPN
ma-382	113	4	x∗	x∗	PROPN
ma-382	113	5	norm	norm	NOUN
ma-382	113	6	-	-	PUNCT
ma-382	113	7	attains	attain	NOUN
ma-382	113	8	at	at	ADP
ma-382	113	9	x0	x0	PROPN
ma-382	113	10	,	,	PUNCT
ma-382	113	11	then	then	ADV
ma-382	113	12	f	f	PROPN
ma-382	113	13	∈	∈	PROPN
ma-382	113	14	j(x0	j(x0	NOUN
ma-382	113	15	)	)	PUNCT
ma-382	113	16	and	and	CCONJ
ma-382	113	17	we	we	PRON
ma-382	113	18	can	can	AUX
ma-382	113	19	use	use	VERB
ma-382	113	20	the	the	DET
ma-382	113	21	modulus	modulus	NOUN
ma-382	113	22	of	of	ADP
ma-382	113	23	convexity	convexity	NOUN
ma-382	113	24	δx	δx	PROPN
ma-382	113	25	(	(	PUNCT
ma-382	113	26	·	·	PUNCT
ma-382	113	27	)	)	PUNCT
ma-382	113	28	tocharacterize	tocharacterize	VERB
ma-382	113	29	proximity	proximity	NOUN
ma-382	113	30	.	.	PUNCT
ma-382	114	1	by	by	ADP
ma-382	114	2	definition	definition	NOUN
ma-382	114	3	of	of	ADP
ma-382	114	4	δx	δx	PROPN
ma-382	114	5	,	,	PUNCT
ma-382	114	6	for	for	ADP
ma-382	114	7	x	x	X
ma-382	114	8	,	,	PUNCT
ma-382	114	9	y	y	PROPN
ma-382	114	10	∈	∈	PROPN
ma-382	114	11	sx	sx	PROPN
ma-382	114	12	,	,	PUNCT
ma-382	114	13	∥∥∥∥x	∥∥∥∥x	PROPN
ma-382	115	1	+	+	CCONJ
ma-382	115	2	y	y	PROPN
ma-382	115	3	2	2	NUM
ma-382	115	4	∥∥∥∥	∥∥∥∥	NUM
ma-382	115	5	≤	≤	NUM
ma-382	115	6	1−	1−	NUM
ma-382	115	7	δx(‖x	δx(‖x	NOUN
ma-382	115	8	−	−	PROPN
ma-382	115	9	y‖	y‖	PROPN
ma-382	115	10	)	)	PUNCT
ma-382	115	11	.	.	PUNCT
ma-382	116	1	now	now	ADV
ma-382	116	2	,	,	PUNCT
ma-382	116	3	for	for	ADP
ma-382	116	4	x	x	PROPN
ma-382	116	5	∈	∈	PROPN
ma-382	116	6	sx	sx	NOUN
ma-382	116	7	such	such	ADJ
ma-382	116	8	that	that	DET
ma-382	116	9	g(x	g(x	NOUN
ma-382	116	10	)	)	PUNCT
ma-382	116	11	=	=	SYM
ma-382	116	12	‖g‖	‖g‖	PROPN
ma-382	116	13	,	,	PUNCT
ma-382	116	14	since	since	SCONJ
ma-382	116	15	‖f	‖f	PRON
ma-382	116	16	−	−	NOUN
ma-382	116	17	g‖	g‖	PROPN
ma-382	116	18	<	<	X
ma-382	116	19	ε	ε	PROPN
ma-382	116	20	,	,	PUNCT
ma-382	116	21	it	it	PRON
ma-382	116	22	follows	follow	VERB
ma-382	116	23	that	that	SCONJ
ma-382	117	1	f	f	PROPN
ma-382	117	2	(	(	PUNCT
ma-382	117	3	x	x	X
ma-382	117	4	)	)	PUNCT
ma-382	117	5	>	>	PUNCT
ma-382	118	1	‖g‖	‖g‖	PUNCT
ma-382	118	2	−	−	NOUN
ma-382	118	3	ε	ε	PROPN
ma-382	118	4	≥	≥	PRON
ma-382	118	5	‖f‖	‖f‖	PROPN
ma-382	118	6	−	−	PROPN
ma-382	118	7	2ε	2ε	NOUN
ma-382	118	8	.	.	PUNCT
ma-382	119	1	we	we	PRON
ma-382	119	2	aim	aim	VERB
ma-382	119	3	to	to	PART
ma-382	119	4	find	find	VERB
ma-382	119	5	xε	xε	PUNCT
ma-382	120	1	∈	∈	PROPN
ma-382	120	2	sx	sx	PROPN
ma-382	120	3	such	such	ADJ
ma-382	120	4	that	that	PRON
ma-382	120	5	g(xε	g(xε	PROPN
ma-382	120	6	)	)	PUNCT
ma-382	120	7	=	=	PUNCT
ma-382	121	1	‖g‖	‖g‖	PUNCT
ma-382	121	2	and	and	CCONJ
ma-382	121	3	xε	xε	NOUN
ma-382	121	4	is	be	AUX
ma-382	121	5	close	close	ADJ
ma-382	121	6	to	to	ADP
ma-382	121	7	x0	x0	PROPN
ma-382	121	8	.	.	PUNCT
ma-382	122	1	consider	consider	VERB
ma-382	122	2	:	:	PUNCT
ma-382	122	3	|f	|f	PROPN
ma-382	122	4	(	(	PUNCT
ma-382	122	5	x0)−	x0)−	X
ma-382	122	6	g(xε)|	g(xε)|	PROPN
ma-382	122	7	≤	≤	PROPN
ma-382	122	8	|f	|f	PROPN
ma-382	123	1	(	(	PUNCT
ma-382	123	2	x0)−	x0)−	PROPN
ma-382	123	3	g(x0)|+	g(x0)|+	PROPN
ma-382	123	4	|g(x0)−	|g(x0)−	PROPN
ma-382	123	5	g(xε)|	g(xε)|	NOUN
ma-382	123	6	≤	≤	NOUN
ma-382	123	7	ε+	ε+	NUM
ma-382	123	8	‖g‖‖x0	‖g‖‖x0	NOUN
ma-382	123	9	−	−	PROPN
ma-382	123	10	xε‖.	xε‖.	PROPN
ma-382	123	11	solving	solve	VERB
ma-382	123	12	this	this	PRON
ma-382	123	13	for	for	ADP
ma-382	123	14	‖x0	‖x0	PROPN
ma-382	123	15	−	−	PROPN
ma-382	123	16	xε‖	xε‖	PROPN
ma-382	123	17	and	and	CCONJ
ma-382	123	18	using	use	VERB
ma-382	123	19	the	the	DET
ma-382	123	20	convexity	convexity	NOUN
ma-382	123	21	modulus	modulus	NOUN
ma-382	123	22	yields	yield	NOUN
ma-382	123	23	the	the	DET
ma-382	123	24	bound	bound	ADJ
ma-382	123	25	‖x0	‖x0	NOUN
ma-382	123	26	−	−	PROPN
ma-382	123	27	xε‖	xε‖	PROPN
ma-382	123	28	≤	≤	NUM
ma-382	123	29	√	√	NUM
ma-382	123	30	2ε	2ε	NUM
ma-382	123	31	δx(ε	δx(ε	NOUN
ma-382	123	32	)	)	PUNCT
ma-382	123	33	,	,	PUNCT
ma-382	123	34	establishing	establish	VERB
ma-382	123	35	existence	existence	NOUN
ma-382	123	36	of	of	ADP
ma-382	123	37	xε	xε	NOUN
ma-382	123	38	as	as	SCONJ
ma-382	123	39	required	require	VERB
ma-382	123	40	.	.	PUNCT
ma-382	124	1	(	(	PUNCT
ma-382	124	2	2	2	X
ma-382	124	3	)	)	PUNCT
ma-382	124	4	norm	norm	NOUN
ma-382	124	5	ratio	ratio	NOUN
ma-382	124	6	bounds	bound	NOUN
ma-382	124	7	.	.	PUNCT
ma-382	125	1	since	since	SCONJ
ma-382	125	2	‖f	‖f	PRON
ma-382	125	3	−	−	NOUN
ma-382	125	4	g‖	g‖	PROPN
ma-382	125	5	<	<	X
ma-382	125	6	ε	ε	PROPN
ma-382	125	7	,	,	PUNCT
ma-382	125	8	we	we	PRON
ma-382	125	9	have	have	AUX
ma-382	125	10	:	:	PUNCT
ma-382	125	11	|‖g‖	|‖g‖	VERB
ma-382	125	12	−	−	PROPN
ma-382	125	13	‖f‖|	‖f‖|	NOUN
ma-382	125	14	≤	≤	NOUN
ma-382	125	15	‖f	‖f	ADP
ma-382	125	16	−	−	PROPN
ma-382	125	17	g‖	g‖	NOUN
ma-382	125	18	<	<	X
ma-382	125	19	ε⇒	ε⇒	PROPN
ma-382	125	20	‖f‖	‖f‖	NUM
ma-382	125	21	−	−	PROPN
ma-382	125	22	ε	ε	PROPN
ma-382	125	23	<	<	X
ma-382	125	24	‖g‖	‖g‖	X
ma-382	125	25	<	<	X
ma-382	125	26	‖f‖+	‖f‖+	PROPN
ma-382	125	27	ε	ε	PROPN
ma-382	125	28	.	.	PUNCT
ma-382	125	29	dividing	divide	VERB
ma-382	125	30	throughout	throughout	ADP
ma-382	125	31	by	by	ADP
ma-382	125	32	‖f‖	‖f‖	PROPN
ma-382	125	33	,	,	PUNCT
ma-382	125	34	we	we	PRON
ma-382	125	35	obtain	obtain	VERB
ma-382	125	36	:	:	PUNCT
ma-382	125	37	1−	1−	NUM
ma-382	125	38	ε	ε	X
ma-382	125	39	‖f‖	‖f‖	PROPN
ma-382	125	40	<	<	X
ma-382	125	41	‖g‖	‖g‖	PUNCT
ma-382	125	42	‖f‖	‖f‖	PROPN
ma-382	125	43	<	<	X
ma-382	125	44	1	1	NUM
ma-382	125	45	+	+	NUM
ma-382	125	46	ε	ε	PROPN
ma-382	125	47	‖f‖	‖f‖	PROPN
ma-382	125	48	.	.	PUNCT
ma-382	126	1	(	(	PUNCT
ma-382	126	2	3	3	X
ma-382	126	3	)	)	PUNCT
ma-382	126	4	continuity	continuity	NOUN
ma-382	126	5	in	in	ADP
ma-382	126	6	uniformly	uniformly	ADV
ma-382	126	7	smooth	smooth	ADJ
ma-382	126	8	spaces	space	NOUN
ma-382	126	9	.	.	PUNCT
ma-382	127	1	in	in	ADP
ma-382	127	2	uniformly	uniformly	ADV
ma-382	127	3	smooth	smooth	ADJ
ma-382	127	4	banach	banach	NOUN
ma-382	127	5	spaces	space	NOUN
ma-382	127	6	,	,	PUNCT
ma-382	127	7	the	the	DET
ma-382	127	8	dualitymapping	dualitymapping	NOUN
ma-382	127	9	is	be	AUX
ma-382	127	10	single	single	ADV
ma-382	127	11	-	-	PUNCT
ma-382	127	12	valued	value	VERB
ma-382	127	13	and	and	CCONJ
ma-382	127	14	norm	norm	NOUN
ma-382	127	15	-	-	PUNCT
ma-382	127	16	to	to	ADP
ma-382	127	17	-	-	PUNCT
ma-382	127	18	norm	norm	NOUN
ma-382	127	19	continuous	continuous	ADJ
ma-382	127	20	.	.	PUNCT
ma-382	128	1	hence	hence	ADV
ma-382	128	2	,	,	PUNCT
ma-382	128	3	small	small	ADJ
ma-382	128	4	perturbations	perturbation	NOUN
ma-382	128	5	in	in	ADP
ma-382	128	6	functionalsyield	functionalsyield	ADJ
ma-382	128	7	small	small	ADJ
ma-382	128	8	perturbations	perturbation	NOUN
ma-382	128	9	in	in	ADP
ma-382	128	10	the	the	DET
ma-382	128	11	unique	unique	ADJ
ma-382	128	12	norm	norm	NOUN
ma-382	128	13	-	-	PUNCT
ma-382	128	14	attaining	attain	VERB
ma-382	128	15	point	point	NOUN
ma-382	128	16	.	.	PUNCT
ma-382	129	1	since	since	SCONJ
ma-382	129	2	‖f	‖f	DET
ma-382	129	3	−	−	NOUN
ma-382	129	4	g‖	g‖	NOUN
ma-382	129	5	→	→	SYM
ma-382	129	6	0	0	PUNCT
ma-382	129	7	as	as	ADP
ma-382	129	8	ε	ε	PROPN
ma-382	129	9	→	→	SYM
ma-382	129	10	0	0	NUM
ma-382	129	11	,	,	PUNCT
ma-382	129	12	and	and	CCONJ
ma-382	129	13	f	f	PROPN
ma-382	129	14	7→	7→	NUM
ma-382	129	15	xf	xf	PROPN
ma-382	129	16	is	be	AUX
ma-382	129	17	continuous	continuous	ADJ
ma-382	129	18	,	,	PUNCT
ma-382	129	19	we	we	PRON
ma-382	129	20	obtain	obtain	VERB
ma-382	129	21	:	:	PUNCT
ma-382	129	22	lim	lim	NOUN
ma-382	129	23	ε→0	ε→0	NOUN
ma-382	129	24	xε	xε	PUNCT
ma-382	130	1	=	=	SYM
ma-382	130	2	x0	x0	PROPN
ma-382	130	3	.	.	PUNCT
ma-382	131	1	�	�	PROPN
ma-382	131	2	https://doi.org/10.28924/ada/ma.5.382	https://doi.org/10.28924/ada/ma.5.382	NUM
ma-382	131	3	eur	eur	PROPN
ma-382	131	4	.	.	PUNCT
ma-382	132	1	j.	j.	PROPN
ma-382	132	2	math	math	PROPN
ma-382	132	3	.	.	PUNCT
ma-382	133	1	anal	anal	PROPN
ma-382	133	2	.	.	PUNCT
ma-382	134	1	10.28924	10.28924	NUM
ma-382	134	2	/	/	SYM
ma-382	134	3	ada	ada	PROPN
ma-382	134	4	/	/	SYM
ma-382	134	5	ma.5.382	ma.5.382	PROPN
ma-382	134	6	6	6	NUM
ma-382	134	7	theorem	theorem	NOUN
ma-382	134	8	5	5	NUM
ma-382	134	9	.	.	X
ma-382	134	10	for	for	ADP
ma-382	134	11	any	any	DET
ma-382	134	12	computable	computable	ADJ
ma-382	134	13	functional	functional	ADJ
ma-382	134	14	f	f	NOUN
ma-382	134	15	on	on	ADP
ma-382	134	16	a	a	DET
ma-382	134	17	computable	computable	ADJ
ma-382	134	18	banach	banach	NOUN
ma-382	134	19	space	space	NOUN
ma-382	134	20	x	x	INTJ
ma-382	134	21	,	,	PUNCT
ma-382	134	22	there	there	PRON
ma-382	134	23	exists	exist	VERB
ma-382	134	24	an	an	DET
ma-382	134	25	effective	effective	ADJ
ma-382	134	26	procedure	procedure	NOUN
ma-382	134	27	to	to	PART
ma-382	134	28	construct	construct	VERB
ma-382	134	29	:	:	PUNCT
ma-382	134	30	(	(	PUNCT
ma-382	134	31	1	1	X
ma-382	134	32	)	)	PUNCT
ma-382	134	33	a	a	DET
ma-382	134	34	sequence	sequence	NOUN
ma-382	134	35	{	{	PUNCT
ma-382	134	36	fn	fn	NOUN
ma-382	134	37	}	}	PUNCT
ma-382	134	38	of	of	ADP
ma-382	134	39	smoothed	smooth	VERB
ma-382	134	40	functionals	functional	NOUN
ma-382	134	41	norm	norm	NOUN
ma-382	134	42	-	-	PUNCT
ma-382	134	43	attaining	attain	VERB
ma-382	134	44	at	at	ADP
ma-382	134	45	computable	computable	ADJ
ma-382	134	46	points	point	NOUN
ma-382	134	47	{	{	PUNCT
ma-382	134	48	xn	xn	NUM
ma-382	134	49	}	}	PUNCT
ma-382	134	50	(	(	PUNCT
ma-382	134	51	2	2	X
ma-382	134	52	)	)	PUNCT
ma-382	134	53	explicit	explicit	ADJ
ma-382	134	54	modulus	modulus	NOUN
ma-382	134	55	of	of	ADP
ma-382	134	56	attainment	attainment	NOUN
ma-382	134	57	ω(n	ω(n	NUM
ma-382	134	58	)	)	PUNCT
ma-382	134	59	such	such	ADJ
ma-382	134	60	that	that	SCONJ
ma-382	134	61	|fn(xn)−	|fn(xn)−	ADV
ma-382	134	62	‖fn‖|	‖fn‖|	PROPN
ma-382	134	63	<	<	X
ma-382	134	64	2−ω(n	2−ω(n	NUM
ma-382	134	65	)	)	PUNCT
ma-382	134	66	(	(	PUNCT
ma-382	134	67	3	3	X
ma-382	134	68	)	)	PUNCT
ma-382	134	69	complexity	complexity	NOUN
ma-382	134	70	bounds	bound	NOUN
ma-382	134	71	:	:	PUNCT
ma-382	134	72	the	the	DET
ma-382	134	73	procedure	procedure	NOUN
ma-382	134	74	is	be	AUX
ma-382	134	75	π0	π0	NOUN
ma-382	134	76	2	2	NUM
ma-382	134	77	-	-	PUNCT
ma-382	134	78	computable	computable	NOUN
ma-382	134	79	in	in	ADP
ma-382	134	80	the	the	DET
ma-382	134	81	tte	tte	PROPN
ma-382	134	82	model	model	NOUN
ma-382	134	83	proof	proof	NOUN
ma-382	134	84	.	.	PUNCT
ma-382	135	1	let	let	VERB
ma-382	135	2	x	x	PRON
ma-382	135	3	be	be	AUX
ma-382	135	4	a	a	DET
ma-382	135	5	computable	computable	ADJ
ma-382	135	6	banach	banach	NOUN
ma-382	135	7	space	space	NOUN
ma-382	135	8	in	in	ADP
ma-382	135	9	the	the	DET
ma-382	135	10	type-2	type-2	NUM
ma-382	135	11	effectivity	effectivity	NOUN
ma-382	135	12	(	(	PUNCT
ma-382	135	13	tte	tte	NOUN
ma-382	135	14	)	)	PUNCT
ma-382	135	15	model	model	NOUN
ma-382	135	16	.	.	PUNCT
ma-382	136	1	a	a	DET
ma-382	136	2	functional	functional	ADJ
ma-382	136	3	f	f	PROPN
ma-382	136	4	∈	∈	PROPN
ma-382	136	5	x∗	x∗	PROPN
ma-382	136	6	is	be	AUX
ma-382	136	7	computable	computable	ADJ
ma-382	136	8	if	if	SCONJ
ma-382	136	9	there	there	PRON
ma-382	136	10	exists	exist	VERB
ma-382	136	11	a	a	DET
ma-382	136	12	turing	ture	VERB
ma-382	136	13	machine	machine	NOUN
ma-382	136	14	which	which	PRON
ma-382	136	15	,	,	PUNCT
ma-382	136	16	given	give	VERB
ma-382	136	17	any	any	DET
ma-382	136	18	computable	computable	ADJ
ma-382	136	19	x	x	SYM
ma-382	136	20	∈	∈	PROPN
ma-382	136	21	x	x	SYM
ma-382	136	22	andany	andany	ADJ
ma-382	136	23	precision	precision	NOUN
ma-382	136	24	n	n	CCONJ
ma-382	136	25	,	,	PUNCT
ma-382	136	26	computes	compute	VERB
ma-382	136	27	a	a	DET
ma-382	136	28	rational	rational	ADJ
ma-382	136	29	approximation	approximation	NOUN
ma-382	136	30	of	of	ADP
ma-382	136	31	f	f	PROPN
ma-382	136	32	(	(	PUNCT
ma-382	136	33	x	x	NOUN
ma-382	136	34	)	)	PUNCT
ma-382	136	35	within	within	ADP
ma-382	136	36	2−n	2−n	NUM
ma-382	136	37	.	.	PUNCT
ma-382	137	1	(	(	PUNCT
ma-382	137	2	1	1	X
ma-382	137	3	)	)	PUNCT
ma-382	137	4	construction	construction	NOUN
ma-382	137	5	of	of	ADP
ma-382	137	6	{	{	PUNCT
ma-382	137	7	fn	fn	NOUN
ma-382	137	8	}	}	PUNCT
ma-382	137	9	and	and	CCONJ
ma-382	137	10	{	{	PUNCT
ma-382	137	11	xn	xn	NOUN
ma-382	137	12	}	}	PUNCT
ma-382	137	13	.	.	PUNCT
ma-382	138	1	define	define	VERB
ma-382	138	2	fn	fn	NOUN
ma-382	138	3	:	:	PUNCT
ma-382	138	4	=	=	SYM
ma-382	138	5	f	f	PROPN
ma-382	138	6	∗	∗	X
ma-382	138	7	φn	φn	PROPN
ma-382	138	8	,	,	PUNCT
ma-382	138	9	where	where	SCONJ
ma-382	138	10	φn	φn	NOUN
ma-382	138	11	is	be	AUX
ma-382	138	12	a	a	DET
ma-382	138	13	mollifier	mollifier	NOUN
ma-382	138	14	or	or	CCONJ
ma-382	138	15	smoothapproximation	smoothapproximation	NOUN
ma-382	138	16	operator	operator	NOUN
ma-382	138	17	such	such	ADJ
ma-382	138	18	that	that	DET
ma-382	138	19	fn	fn	NOUN
ma-382	138	20	→	→	SYM
ma-382	138	21	f	f	PROPN
ma-382	138	22	in	in	ADP
ma-382	138	23	norm	norm	NOUN
ma-382	138	24	.	.	PUNCT
ma-382	139	1	since	since	SCONJ
ma-382	139	2	the	the	DET
ma-382	139	3	mollifiers	mollifier	NOUN
ma-382	139	4	can	can	AUX
ma-382	139	5	be	be	AUX
ma-382	139	6	taken	take	VERB
ma-382	139	7	to	to	PART
ma-382	139	8	becomputable	becomputable	VERB
ma-382	139	9	and	and	CCONJ
ma-382	139	10	x	x	PRON
ma-382	139	11	is	be	AUX
ma-382	139	12	separable	separable	ADJ
ma-382	139	13	and	and	CCONJ
ma-382	139	14	computably	computably	ADV
ma-382	139	15	presented	present	VERB
ma-382	139	16	,	,	PUNCT
ma-382	139	17	each	each	DET
ma-382	139	18	fn	fn	NOUN
ma-382	139	19	is	be	AUX
ma-382	139	20	computable	computable	ADJ
ma-382	139	21	.	.	PUNCT
ma-382	140	1	since	since	SCONJ
ma-382	140	2	fnis	fnis	PROPN
ma-382	140	3	smoother	smooth	ADJ
ma-382	140	4	than	than	ADP
ma-382	140	5	f	f	PROPN
ma-382	140	6	,	,	PUNCT
ma-382	140	7	we	we	PRON
ma-382	140	8	can	can	AUX
ma-382	140	9	explicitly	explicitly	ADV
ma-382	140	10	construct	construct	VERB
ma-382	140	11	xn	xn	PROPN
ma-382	140	12	∈	∈	PROPN
ma-382	140	13	sx	sx	PROPN
ma-382	141	1	such	such	ADJ
ma-382	141	2	that	that	SCONJ
ma-382	141	3	fn(xn	fn(xn	NOUN
ma-382	141	4	)	)	PUNCT
ma-382	141	5	≈	≈	NOUN
ma-382	141	6	‖fn‖.	‖fn‖.	VERB
ma-382	141	7	by	by	ADP
ma-382	141	8	effectivecompactness	effectivecompactness	NOUN
ma-382	141	9	of	of	ADP
ma-382	141	10	the	the	DET
ma-382	141	11	unit	unit	NOUN
ma-382	141	12	sphere	sphere	NOUN
ma-382	141	13	sx	sx	PROPN
ma-382	141	14	in	in	ADP
ma-382	141	15	the	the	DET
ma-382	141	16	tte	tte	PROPN
ma-382	141	17	model	model	NOUN
ma-382	141	18	,	,	PUNCT
ma-382	141	19	and	and	CCONJ
ma-382	141	20	the	the	DET
ma-382	141	21	computability	computability	NOUN
ma-382	141	22	of	of	ADP
ma-382	141	23	fn	fn	NOUN
ma-382	141	24	,	,	PUNCT
ma-382	141	25	the	the	DET
ma-382	141	26	maximization	maximization	NOUN
ma-382	141	27	xn	xn	PUNCT
ma-382	142	1	:	:	PUNCT
ma-382	142	2	=	=	SYM
ma-382	142	3	arg	arg	NOUN
ma-382	142	4	max	max	PROPN
ma-382	142	5	x∈s(rn	x∈s(rn	PROPN
ma-382	142	6	)	)	PUNCT
ma-382	142	7	x	x	SYM
ma-382	142	8	fn(x	fn(x	X
ma-382	142	9	)	)	PUNCT
ma-382	142	10	can	can	AUX
ma-382	142	11	be	be	AUX
ma-382	142	12	computed	compute	VERB
ma-382	142	13	to	to	ADP
ma-382	142	14	within	within	ADP
ma-382	142	15	any	any	DET
ma-382	142	16	desired	desire	VERB
ma-382	142	17	rational	rational	ADJ
ma-382	142	18	error	error	NOUN
ma-382	142	19	2−k	2−k	NUM
ma-382	142	20	,	,	PUNCT
ma-382	142	21	where	where	SCONJ
ma-382	142	22	s(rn	s(rn	X
ma-382	142	23	)	)	PUNCT
ma-382	142	24	x	x	PRON
ma-382	142	25	is	be	AUX
ma-382	142	26	a	a	DET
ma-382	142	27	rational	rational	ADJ
ma-382	142	28	δ	δ	NOUN
ma-382	142	29	-	-	NOUN
ma-382	142	30	net	net	NOUN
ma-382	142	31	in	in	ADP
ma-382	142	32	sx	sx	PROPN
ma-382	142	33	.	.	PUNCT
ma-382	143	1	(	(	PUNCT
ma-382	143	2	2	2	X
ma-382	143	3	)	)	PUNCT
ma-382	143	4	modulus	modulus	NOUN
ma-382	143	5	of	of	ADP
ma-382	143	6	attainment	attainment	NOUN
ma-382	143	7	ω(n	ω(n	NUM
ma-382	143	8	)	)	PUNCT
ma-382	143	9	.	.	PUNCT
ma-382	144	1	because	because	SCONJ
ma-382	144	2	xn	xn	PROPN
ma-382	144	3	is	be	AUX
ma-382	144	4	chosen	choose	VERB
ma-382	144	5	from	from	ADP
ma-382	144	6	a	a	DET
ma-382	144	7	dense	dense	ADJ
ma-382	144	8	net	net	NOUN
ma-382	144	9	and	and	CCONJ
ma-382	144	10	fn	fn	NOUN
ma-382	144	11	is	be	AUX
ma-382	144	12	lipschitzcontinuous	lipschitzcontinuous	ADJ
ma-382	144	13	with	with	ADP
ma-382	144	14	computable	computable	ADJ
ma-382	144	15	norm	norm	NOUN
ma-382	144	16	,	,	PUNCT
ma-382	144	17	we	we	PRON
ma-382	144	18	have	have	VERB
ma-382	144	19	:	:	PUNCT
ma-382	144	20	|fn(xn)−	|fn(xn)−	X
ma-382	144	21	‖fn‖|	‖fn‖|	PROPN
ma-382	144	22	<	<	X
ma-382	144	23	2−ω(n	2−ω(n	NUM
ma-382	144	24	)	)	PUNCT
ma-382	144	25	,	,	PUNCT
ma-382	144	26	for	for	ADP
ma-382	144	27	some	some	DET
ma-382	144	28	computable	computable	ADJ
ma-382	144	29	strictly	strictly	ADV
ma-382	144	30	increasing	increase	VERB
ma-382	144	31	function	function	NOUN
ma-382	144	32	ω(n	ω(n	NUM
ma-382	144	33	)	)	PUNCT
ma-382	144	34	determined	determine	VERB
ma-382	144	35	by	by	ADP
ma-382	144	36	the	the	DET
ma-382	144	37	lipschitz	lipschitz	NOUN
ma-382	144	38	constant	constant	ADJ
ma-382	144	39	andthe	andthe	ADJ
ma-382	144	40	size	size	NOUN
ma-382	144	41	of	of	ADP
ma-382	144	42	the	the	DET
ma-382	144	43	net	net	NOUN
ma-382	144	44	.	.	PUNCT
ma-382	145	1	(	(	PUNCT
ma-382	145	2	3	3	X
ma-382	145	3	)	)	PUNCT
ma-382	145	4	complexity	complexity	NOUN
ma-382	145	5	classification	classification	NOUN
ma-382	145	6	.	.	PUNCT
ma-382	146	1	each	each	DET
ma-382	146	2	fn	fn	NOUN
ma-382	146	3	is	be	AUX
ma-382	146	4	computable	computable	ADJ
ma-382	146	5	,	,	PUNCT
ma-382	146	6	and	and	CCONJ
ma-382	146	7	xn	xn	PROPN
ma-382	146	8	can	can	AUX
ma-382	146	9	be	be	AUX
ma-382	146	10	computed	compute	VERB
ma-382	146	11	to	to	ADP
ma-382	146	12	any	any	DET
ma-382	146	13	desiredprecision	desiredprecision	NOUN
ma-382	146	14	.	.	PUNCT
ma-382	147	1	the	the	DET
ma-382	147	2	condition	condition	NOUN
ma-382	147	3	:	:	PUNCT
ma-382	147	4	∀n∃xn	∀n∃xn	PROPN
ma-382	147	5	∈	∈	PROPN
ma-382	147	6	sx	sx	PROPN
ma-382	147	7	:	:	PUNCT
ma-382	147	8	|fn(xn)−	|fn(xn)−	X
ma-382	147	9	‖fn‖|	‖fn‖|	PROPN
ma-382	147	10	<	<	X
ma-382	147	11	2−ω(n	2−ω(n	NUM
ma-382	147	12	)	)	PUNCT
ma-382	147	13	is	be	AUX
ma-382	147	14	a	a	DET
ma-382	147	15	π0	π0	NOUN
ma-382	147	16	2	2	NUM
ma-382	147	17	statement	statement	NOUN
ma-382	147	18	because	because	SCONJ
ma-382	147	19	it	it	PRON
ma-382	147	20	quantifies	quantify	VERB
ma-382	147	21	universally	universally	ADV
ma-382	147	22	over	over	ADP
ma-382	147	23	natural	natural	ADJ
ma-382	147	24	numbers	number	NOUN
ma-382	147	25	and	and	CCONJ
ma-382	147	26	existentially	existentially	ADV
ma-382	147	27	overcomputable	overcomputable	ADJ
ma-382	147	28	reals	real	NOUN
ma-382	147	29	.	.	PUNCT
ma-382	148	1	therefore	therefore	ADV
ma-382	148	2	,	,	PUNCT
ma-382	148	3	the	the	DET
ma-382	148	4	whole	whole	ADJ
ma-382	148	5	process	process	NOUN
ma-382	148	6	is	be	AUX
ma-382	148	7	π0	π0	NOUN
ma-382	148	8	2	2	NUM
ma-382	148	9	-	-	PUNCT
ma-382	148	10	computable	computable	NOUN
ma-382	148	11	in	in	ADP
ma-382	148	12	the	the	DET
ma-382	148	13	tte	tte	PROPN
ma-382	148	14	model	model	NOUN
ma-382	148	15	.	.	PUNCT
ma-382	149	1	�	�	PROPN
ma-382	149	2	theorem	theorem	VERB
ma-382	149	3	6	6	NUM
ma-382	149	4	.	.	PUNCT
ma-382	150	1	let	let	VERB
ma-382	150	2	x	x	PRON
ma-382	150	3	be	be	AUX
ma-382	150	4	separable	separable	ADJ
ma-382	150	5	with	with	ADP
ma-382	150	6	shrinking	shrink	VERB
ma-382	150	7	basis	basis	NOUN
ma-382	150	8	{	{	PUNCT
ma-382	150	9	en	en	X
ma-382	150	10	}	}	PUNCT
ma-382	150	11	.	.	PUNCT
ma-382	151	1	for	for	ADP
ma-382	151	2	any	any	DET
ma-382	151	3	f	f	PROPN
ma-382	151	4	∈	∈	PROPN
ma-382	151	5	x∗	x∗	PROPN
ma-382	151	6	:	:	PUNCT
ma-382	151	7	(	(	PUNCT
ma-382	151	8	1	1	X
ma-382	151	9	)	)	PUNCT
ma-382	151	10	the	the	DET
ma-382	151	11	projected	project	VERB
ma-382	151	12	functionals	functional	NOUN
ma-382	151	13	fn	fn	NOUN
ma-382	151	14	=	=	SYM
ma-382	151	15	f	f	PROPN
ma-382	151	16	◦	◦	NOUN
ma-382	151	17	pn	pn	NOUN
ma-382	151	18	norm	norm	NOUN
ma-382	151	19	-	-	PUNCT
ma-382	151	20	attain	attain	VERB
ma-382	151	21	with	with	ADP
ma-382	151	22	‖fn‖	‖fn‖	PROPN
ma-382	151	23	→	→	SYM
ma-382	151	24	‖f‖	‖f‖	PROPN
ma-382	151	25	(	(	PUNCT
ma-382	151	26	2	2	NUM
ma-382	151	27	)	)	PUNCT
ma-382	151	28	the	the	DET
ma-382	151	29	speed	speed	NOUN
ma-382	151	30	of	of	ADP
ma-382	151	31	convergence	convergence	NOUN
ma-382	151	32	‖|f‖	‖|f‖	NOUN
ma-382	151	33	−	−	PROPN
ma-382	151	34	‖fn‖|	‖fn‖|	PROPN
ma-382	151	35	relates	relate	VERB
ma-382	151	36	to	to	ADP
ma-382	151	37	the	the	DET
ma-382	151	38	basis	basis	NOUN
ma-382	151	39	constant	constant	ADJ
ma-382	151	40	(	(	PUNCT
ma-382	151	41	3	3	NUM
ma-382	151	42	)	)	PUNCT
ma-382	151	43	for	for	ADP
ma-382	151	44	x	x	SYM
ma-382	151	45	=	=	PUNCT
ma-382	151	46	`	`	PUNCT
ma-382	151	47	p	p	X
ma-382	151	48	,	,	PUNCT
ma-382	151	49	explicit	explicit	ADJ
ma-382	151	50	rates	rate	NOUN
ma-382	151	51	are	be	AUX
ma-382	151	52	o(n1−1	o(n1−1	PROPN
ma-382	151	53	/	/	SYM
ma-382	151	54	p	p	NOUN
ma-382	151	55	)	)	PUNCT
ma-382	151	56	proof	proof	NOUN
ma-382	151	57	.	.	PUNCT
ma-382	152	1	let	let	VERB
ma-382	152	2	x	x	PRON
ma-382	152	3	be	be	AUX
ma-382	152	4	a	a	DET
ma-382	152	5	separable	separable	ADJ
ma-382	152	6	banach	banach	NOUN
ma-382	152	7	space	space	NOUN
ma-382	152	8	with	with	ADP
ma-382	152	9	a	a	DET
ma-382	152	10	shrinking	shrink	VERB
ma-382	152	11	schauder	schauder	NOUN
ma-382	152	12	basis	basis	NOUN
ma-382	152	13	{	{	PUNCT
ma-382	152	14	en	en	NOUN
ma-382	152	15	}	}	PUNCT
ma-382	152	16	and	and	CCONJ
ma-382	152	17	correspondingbiorthogonal	correspondingbiorthogonal	ADJ
ma-382	152	18	functionals	functional	NOUN
ma-382	152	19	{	{	PUNCT
ma-382	152	20	e∗n	e∗n	NOUN
ma-382	152	21	}	}	PUNCT
ma-382	152	22	.	.	PUNCT
ma-382	153	1	denote	denote	VERB
ma-382	153	2	the	the	DET
ma-382	153	3	canonical	canonical	ADJ
ma-382	153	4	projections	projection	NOUN
ma-382	153	5	pn	pn	X
ma-382	153	6	:	:	PUNCT
ma-382	153	7	x	x	SYM
ma-382	153	8	→	→	SYM
ma-382	153	9	x	x	PUNCT
ma-382	153	10	by	by	ADP
ma-382	153	11	pn(x	pn(x	NOUN
ma-382	153	12	)	)	PUNCT
ma-382	154	1	=	=	SYM
ma-382	154	2	n∑	n∑	NOUN
ma-382	154	3	k=1	k=1	NOUN
ma-382	154	4	e∗k(x)ek	e∗k(x)ek	NOUN
ma-382	154	5	.	.	PUNCT
ma-382	155	1	https://doi.org/10.28924/ada/ma.5.382	https://doi.org/10.28924/ada/ma.5.382	PROPN
ma-382	156	1	eur	eur	PROPN
ma-382	156	2	.	.	PUNCT
ma-382	157	1	j.	j.	PROPN
ma-382	157	2	math	math	PROPN
ma-382	157	3	.	.	PUNCT
ma-382	158	1	anal	anal	PROPN
ma-382	158	2	.	.	PUNCT
ma-382	159	1	10.28924	10.28924	NUM
ma-382	159	2	/	/	SYM
ma-382	159	3	ada	ada	PROPN
ma-382	159	4	/	/	SYM
ma-382	159	5	ma.5.382	ma.5.382	PROPN
ma-382	159	6	7these	7these	PROPN
ma-382	159	7	projections	projection	NOUN
ma-382	159	8	are	be	AUX
ma-382	159	9	uniformly	uniformly	ADV
ma-382	159	10	bounded	bound	VERB
ma-382	159	11	and	and	CCONJ
ma-382	159	12	strongly	strongly	ADV
ma-382	159	13	converge	converge	VERB
ma-382	159	14	to	to	ADP
ma-382	159	15	the	the	DET
ma-382	159	16	identity	identity	NOUN
ma-382	159	17	,	,	PUNCT
ma-382	159	18	i.e.	i.e.	X
ma-382	159	19	,	,	PUNCT
ma-382	159	20	for	for	ADP
ma-382	159	21	all	all	DET
ma-382	159	22	x	x	SYM
ma-382	159	23	∈	∈	PROPN
ma-382	159	24	x	x	X
ma-382	159	25	,	,	PUNCT
ma-382	159	26	‖pnx−x‖	‖pnx−x‖	PROPN
ma-382	159	27	→	→	SYM
ma-382	159	28	0	0	PUNCT
ma-382	159	29	as	as	ADP
ma-382	159	30	n	n	X
ma-382	159	31	→∞.	→∞.	PROPN
ma-382	159	32	since	since	SCONJ
ma-382	159	33	the	the	DET
ma-382	159	34	basis	basis	NOUN
ma-382	159	35	is	be	AUX
ma-382	159	36	shrinking	shrink	VERB
ma-382	159	37	,	,	PUNCT
ma-382	159	38	the	the	DET
ma-382	159	39	adjoint	adjoint	PROPN
ma-382	159	40	operators	operator	NOUN
ma-382	159	41	p	p	PROPN
ma-382	159	42	∗n	∗n	PROPN
ma-382	159	43	converge	converge	NOUN
ma-382	159	44	stronglyto	stronglyto	VERB
ma-382	159	45	the	the	DET
ma-382	159	46	identity	identity	NOUN
ma-382	159	47	on	on	ADP
ma-382	159	48	x∗.	x∗.	PROPN
ma-382	160	1	that	that	PRON
ma-382	160	2	is	be	AUX
ma-382	160	3	,	,	PUNCT
ma-382	160	4	for	for	ADP
ma-382	160	5	every	every	DET
ma-382	160	6	f	f	PROPN
ma-382	160	7	∈	∈	PROPN
ma-382	160	8	x∗	x∗	PROPN
ma-382	160	9	,	,	PUNCT
ma-382	160	10	fn	fn	AUX
ma-382	160	11	:	:	PUNCT
ma-382	160	12	=	=	SYM
ma-382	160	13	f	f	X
ma-382	161	1	◦	◦	NOUN
ma-382	161	2	pn	pn	NOUN
ma-382	161	3	=	=	SYM
ma-382	162	1	p	p	PROPN
ma-382	162	2	∗n	∗n	PROPN
ma-382	162	3	f	f	X
ma-382	162	4	→	→	SYM
ma-382	162	5	f	f	PROPN
ma-382	162	6	strongly	strongly	ADV
ma-382	162	7	in	in	ADP
ma-382	162	8	x∗.	x∗.	PROPN
ma-382	162	9	(	(	PUNCT
ma-382	162	10	1	1	NUM
ma-382	162	11	)	)	PUNCT
ma-382	162	12	norm	norm	NOUN
ma-382	162	13	-	-	PUNCT
ma-382	162	14	attainment	attainment	NOUN
ma-382	162	15	:	:	PUNCT
ma-382	162	16	each	each	DET
ma-382	162	17	fn	fn	NOUN
ma-382	162	18	is	be	AUX
ma-382	162	19	a	a	DET
ma-382	162	20	finite	finite	ADJ
ma-382	162	21	-	-	ADJ
ma-382	162	22	rank	rank	ADJ
ma-382	162	23	functional	functional	ADJ
ma-382	162	24	,	,	PUNCT
ma-382	162	25	i.e.	i.e.	X
ma-382	162	26	,	,	PUNCT
ma-382	162	27	it	it	PRON
ma-382	162	28	lies	lie	VERB
ma-382	162	29	in	in	ADP
ma-382	162	30	the	the	DET
ma-382	162	31	span	span	NOUN
ma-382	162	32	of	of	ADP
ma-382	162	33	{	{	PUNCT
ma-382	162	34	e∗k}nk=1	e∗k}nk=1	PROPN
ma-382	162	35	.	.	PUNCT
ma-382	162	36	infinite	infinite	ADJ
ma-382	162	37	-	-	PUNCT
ma-382	162	38	dimensional	dimensional	ADJ
ma-382	162	39	subspaces	subspace	NOUN
ma-382	162	40	,	,	PUNCT
ma-382	162	41	the	the	DET
ma-382	162	42	norm	norm	NOUN
ma-382	162	43	is	be	AUX
ma-382	162	44	attained	attain	VERB
ma-382	162	45	by	by	ADP
ma-382	162	46	the	the	DET
ma-382	162	47	hahn	hahn	NOUN
ma-382	162	48	-	-	PUNCT
ma-382	162	49	banach	banach	NOUN
ma-382	162	50	theorem	theorem	VERB
ma-382	162	51	,	,	PUNCT
ma-382	162	52	so	so	SCONJ
ma-382	162	53	there	there	PRON
ma-382	162	54	exists	exist	VERB
ma-382	162	55	xn	xn	PROPN
ma-382	162	56	∈	∈	PROPN
ma-382	162	57	span{e1	span{e1	NOUN
ma-382	162	58	,	,	PUNCT
ma-382	162	59	.	.	PUNCT
ma-382	162	60	.	.	PUNCT
ma-382	162	61	.	.	PUNCT
ma-382	163	1	,	,	PUNCT
ma-382	163	2	en	en	ADP
ma-382	163	3	}	}	PUNCT
ma-382	163	4	with	with	ADP
ma-382	163	5	‖xn‖	‖xn‖	ADJ
ma-382	163	6	=	=	SYM
ma-382	163	7	1	1	NUM
ma-382	163	8	such	such	ADJ
ma-382	163	9	that	that	DET
ma-382	163	10	|fn(xn)|	|fn(xn)|	NOUN
ma-382	163	11	=	=	SYM
ma-382	163	12	‖fn‖.	‖fn‖.	X
ma-382	163	13	(	(	PUNCT
ma-382	163	14	2	2	X
ma-382	163	15	)	)	PUNCT
ma-382	163	16	convergence	convergence	NOUN
ma-382	163	17	rate	rate	NOUN
ma-382	163	18	and	and	CCONJ
ma-382	163	19	basis	basis	NOUN
ma-382	163	20	constant	constant	ADJ
ma-382	163	21	:	:	PUNCT
ma-382	163	22	denote	denote	VERB
ma-382	163	23	the	the	DET
ma-382	163	24	basis	basis	NOUN
ma-382	163	25	constant	constant	ADJ
ma-382	163	26	by	by	ADP
ma-382	163	27	k	k	PROPN
ma-382	163	28	,	,	PUNCT
ma-382	163	29	satisfying	satisfy	VERB
ma-382	163	30	for	for	ADP
ma-382	163	31	all	all	DET
ma-382	163	32	n	n	PRON
ma-382	163	33	andall	andall	ADV
ma-382	163	34	scalar	scalar	ADJ
ma-382	163	35	sequences	sequence	NOUN
ma-382	163	36	(	(	PUNCT
ma-382	163	37	ak	ak	PROPN
ma-382	163	38	)	)	PUNCT
ma-382	163	39	,	,	PUNCT
ma-382	163	40	∥∥∥∥∥	∥∥∥∥∥	X
ma-382	164	1	n∑	n∑	INTJ
ma-382	164	2	k=1	k=1	X
ma-382	165	1	akek	akek	ADP
ma-382	165	2	∥∥∥∥∥	∥∥∥∥∥	VERB
ma-382	165	3	≤	≤	PUNCT
ma-382	166	1	k	k	PROPN
ma-382	166	2	sup	sup	PROPN
ma-382	166	3	1≤k≤n	1≤k≤n	NUM
ma-382	166	4	|ak	|ak	NUM
ma-382	166	5	|	|	NOUN
ma-382	166	6	.	.	PUNCT
ma-382	167	1	the	the	DET
ma-382	167	2	dual	dual	ADJ
ma-382	167	3	norm	norm	NOUN
ma-382	167	4	satisfies	satisfie	NOUN
ma-382	167	5	‖f	‖f	ADP
ma-382	167	6	−	−	ADP
ma-382	167	7	fn‖	fn‖	NOUN
ma-382	167	8	=	=	SYM
ma-382	167	9	sup	sup	NOUN
ma-382	167	10	‖x‖≤1	‖x‖≤1	NOUN
ma-382	167	11	|f	|f	PROPN
ma-382	168	1	(	(	PUNCT
ma-382	168	2	x	x	SYM
ma-382	168	3	−	−	NOUN
ma-382	168	4	pnx)|	pnx)|	VERB
ma-382	168	5	≤	≤	PUNCT
ma-382	168	6	‖f‖	‖f‖	NUM
ma-382	168	7	·	·	PUNCT
ma-382	168	8	sup	sup	NOUN
ma-382	168	9	‖x‖≤1	‖x‖≤1	NOUN
ma-382	168	10	‖x	‖x	PUNCT
ma-382	169	1	−	−	NOUN
ma-382	169	2	pnx‖	pnx‖	PROPN
ma-382	169	3	→	→	SYM
ma-382	169	4	0	0	NUM
ma-382	169	5	,	,	PUNCT
ma-382	169	6	with	with	ADP
ma-382	169	7	a	a	DET
ma-382	169	8	quantitative	quantitative	ADJ
ma-382	169	9	estimate	estimate	NOUN
ma-382	169	10	involving	involve	VERB
ma-382	169	11	the	the	DET
ma-382	169	12	modulus	modulus	NOUN
ma-382	169	13	of	of	ADP
ma-382	169	14	basis	basis	NOUN
ma-382	169	15	approximation	approximation	NOUN
ma-382	169	16	.	.	PUNCT
ma-382	170	1	specifically	specifically	ADV
ma-382	170	2	,	,	PUNCT
ma-382	170	3	if	if	SCONJ
ma-382	170	4	the	the	DET
ma-382	170	5	basisis	basisis	NOUN
ma-382	170	6	unconditional	unconditional	ADJ
ma-382	170	7	with	with	ADP
ma-382	170	8	constant	constant	ADJ
ma-382	170	9	ku	ku	NOUN
ma-382	170	10	,	,	PUNCT
ma-382	170	11	we	we	PRON
ma-382	170	12	may	may	AUX
ma-382	170	13	write	write	VERB
ma-382	170	14	‖f	‖f	PUNCT
ma-382	170	15	−	−	PROPN
ma-382	170	16	fn‖	fn‖	PROPN
ma-382	170	17	≤	≤	PROPN
ma-382	170	18	ku	ku	PROPN
ma-382	170	19	·	·	PUNCT
ma-382	170	20	sup	sup	NOUN
ma-382	170	21	‖x‖≤1	‖x‖≤1	NOUN
ma-382	170	22	‖x	‖x	PUNCT
ma-382	171	1	−	−	PROPN
ma-382	171	2	pnx‖	pnx‖	PROPN
ma-382	171	3	,	,	PUNCT
ma-382	171	4	and	and	CCONJ
ma-382	171	5	hence	hence	ADV
ma-382	171	6	|‖f‖	|‖f‖	VERB
ma-382	171	7	−	−	PROPN
ma-382	171	8	‖fn‖|	‖fn‖|	NOUN
ma-382	171	9	≤	≤	NOUN
ma-382	171	10	‖f	‖f	ADP
ma-382	171	11	−	−	PROPN
ma-382	171	12	fn‖	fn‖	NOUN
ma-382	171	13	≤	≤	NUM
ma-382	171	14	c	c	PROPN
ma-382	171	15	·	·	PUNCT
ma-382	171	16	δn	δn	NOUN
ma-382	171	17	,	,	PUNCT
ma-382	171	18	where	where	SCONJ
ma-382	171	19	δn	δn	NOUN
ma-382	171	20	=	=	PUNCT
ma-382	171	21	sup‖x‖≤1	sup‖x‖≤1	ADV
ma-382	171	22	‖x	‖x	PUNCT
ma-382	172	1	−	−	NOUN
ma-382	172	2	pnx‖	pnx‖	PROPN
ma-382	172	3	decays	decay	VERB
ma-382	172	4	with	with	ADP
ma-382	172	5	n	n	CCONJ
ma-382	172	6	depending	depend	VERB
ma-382	172	7	on	on	ADP
ma-382	172	8	the	the	DET
ma-382	172	9	geometry	geometry	NOUN
ma-382	172	10	of	of	ADP
ma-382	172	11	the	the	DET
ma-382	172	12	basis	basis	NOUN
ma-382	172	13	.	.	PUNCT
ma-382	173	1	(	(	PUNCT
ma-382	173	2	3	3	X
ma-382	173	3	)	)	PUNCT
ma-382	173	4	explicit	explicit	ADJ
ma-382	173	5	rate	rate	NOUN
ma-382	173	6	for	for	ADP
ma-382	173	7	`	`	PUNCT
ma-382	173	8	p	p	X
ma-382	173	9	:	:	PUNCT
ma-382	173	10	let	let	VERB
ma-382	173	11	x	x	PUNCT
ma-382	173	12	=	=	PUNCT
ma-382	173	13	`	`	PUNCT
ma-382	173	14	p	p	NOUN
ma-382	173	15	for	for	ADP
ma-382	173	16	1	1	NUM
ma-382	173	17	<	<	X
ma-382	173	18	p	p	X
ma-382	173	19	<	<	X
ma-382	173	20	∞.	∞.	PROPN
ma-382	173	21	for	for	ADP
ma-382	173	22	f	f	PROPN
ma-382	173	23	∈	∈	PROPN
ma-382	173	24	x∗	x∗	PROPN
ma-382	173	25	,	,	PUNCT
ma-382	173	26	the	the	DET
ma-382	173	27	dual	dual	ADJ
ma-382	173	28	is	be	AUX
ma-382	173	29	`	`	PUNCT
ma-382	173	30	q	q	PUNCT
ma-382	173	31	with	with	ADP
ma-382	173	32	1	1	NUM
ma-382	173	33	/	/	SYM
ma-382	173	34	p+	p+	NOUN
ma-382	173	35	1	1	NUM
ma-382	173	36	/	/	SYM
ma-382	173	37	q	q	NOUN
ma-382	174	1	=	=	SYM
ma-382	174	2	1.write	1.write	NUM
ma-382	174	3	f	f	NOUN
ma-382	174	4	(	(	PUNCT
ma-382	174	5	x	x	NOUN
ma-382	174	6	)	)	PUNCT
ma-382	174	7	=	=	NOUN
ma-382	174	8	∑∞	∑∞	NOUN
ma-382	174	9	k=1	k=1	X
ma-382	174	10	akxk	akxk	PROPN
ma-382	174	11	with	with	ADP
ma-382	174	12	{	{	PUNCT
ma-382	174	13	ak	ak	PROPN
ma-382	174	14	}	}	PUNCT
ma-382	174	15	∈	∈	PROPN
ma-382	174	16	`	`	PUNCT
ma-382	174	17	q	q	PROPN
ma-382	174	18	.	.	PUNCT
ma-382	175	1	then	then	ADV
ma-382	175	2	fn(x	fn(x	PRON
ma-382	175	3	)	)	PUNCT
ma-382	176	1	=	=	SYM
ma-382	176	2	n∑	n∑	NOUN
ma-382	176	3	k=1	k=1	PROPN
ma-382	176	4	akxk	akxk	PROPN
ma-382	176	5	,	,	PUNCT
ma-382	176	6	so	so	ADV
ma-382	176	7	‖fn‖	‖fn‖	NOUN
ma-382	176	8	=	=	PUNCT
ma-382	176	9	sup	sup	PROPN
ma-382	176	10	‖x‖p≤1	‖x‖p≤1	PUNCT
ma-382	176	11	∣∣∣∣∣	∣∣∣∣∣	PROPN
ma-382	176	12	n∑	n∑	NOUN
ma-382	176	13	k=1	k=1	PROPN
ma-382	176	14	akxk	akxk	PROPN
ma-382	176	15	∣∣∣∣∣	∣∣∣∣∣	PROPN
ma-382	176	16	.	.	PUNCT
ma-382	177	1	by	by	ADP
ma-382	177	2	holder	holder	NOUN
ma-382	177	3	’s	’s	PART
ma-382	177	4	inequality	inequality	NOUN
ma-382	177	5	,	,	PUNCT
ma-382	177	6	we	we	PRON
ma-382	177	7	have	have	VERB
ma-382	177	8	‖fn‖	‖fn‖	NOUN
ma-382	177	9	≤	≤	NOUN
ma-382	177	10	(	(	PUNCT
ma-382	177	11	n∑	n∑	NOUN
ma-382	177	12	k=1	k=1	NOUN
ma-382	177	13	|ak	|ak	X
ma-382	177	14	|q	|q	NOUN
ma-382	177	15	)	)	PUNCT
ma-382	177	16	1	1	X
ma-382	177	17	/	/	SYM
ma-382	177	18	q	q	PRON
ma-382	177	19	≤	≤	NUM
ma-382	177	20	‖f‖	‖f‖	PROPN
ma-382	177	21	,	,	PUNCT
ma-382	177	22	and	and	CCONJ
ma-382	177	23	the	the	DET
ma-382	177	24	complement	complement	NOUN
ma-382	177	25	tail	tail	NOUN
ma-382	177	26	satisfies	satisfie	NOUN
ma-382	177	27	‖f	‖f	ADP
ma-382	177	28	−	−	ADP
ma-382	177	29	fn‖	fn‖	NOUN
ma-382	177	30	≤	≤	NUM
ma-382	177	31	(	(	PUNCT
ma-382	177	32	∞∑	∞∑	NUM
ma-382	177	33	k	k	NOUN
ma-382	177	34	=	=	X
ma-382	177	35	n+1	n+1	PROPN
ma-382	177	36	|ak	|ak	X
ma-382	177	37	|q	|q	NOUN
ma-382	177	38	)	)	PUNCT
ma-382	177	39	1	1	X
ma-382	177	40	/	/	SYM
ma-382	177	41	q	q	NOUN
ma-382	177	42	=	=	NOUN
ma-382	177	43	o(n1	o(n1	NOUN
ma-382	177	44	/	/	SYM
ma-382	177	45	q−1	q−1	PROPN
ma-382	177	46	)	)	PUNCT
ma-382	178	1	=	=	PUNCT
ma-382	178	2	o(n1−1	o(n1−1	PROPN
ma-382	178	3	/	/	SYM
ma-382	178	4	p	p	NOUN
ma-382	178	5	)	)	PUNCT
ma-382	178	6	,	,	PUNCT
ma-382	178	7	since	since	SCONJ
ma-382	178	8	q	q	NOUN
ma-382	178	9	=	=	SYM
ma-382	178	10	p/(p	p/(p	PROPN
ma-382	178	11	−	−	NOUN
ma-382	178	12	1	1	NUM
ma-382	178	13	)	)	PUNCT
ma-382	178	14	.	.	PUNCT
ma-382	179	1	therefore	therefore	ADV
ma-382	179	2	,	,	PUNCT
ma-382	179	3	the	the	DET
ma-382	179	4	convergence	convergence	NOUN
ma-382	179	5	rate	rate	NOUN
ma-382	179	6	‖f	‖f	PRON
ma-382	179	7	−	−	ADP
ma-382	179	8	fn‖	fn‖	NOUN
ma-382	179	9	=	=	SYM
ma-382	179	10	o(n1−1	o(n1−1	PROPN
ma-382	179	11	/	/	SYM
ma-382	179	12	p	p	NOUN
ma-382	179	13	)	)	PUNCT
ma-382	179	14	.	.	PUNCT
ma-382	180	1	�	�	PROPN
ma-382	180	2	theorem	theorem	VERB
ma-382	180	3	7	7	NUM
ma-382	180	4	.	.	NOUN
ma-382	180	5	for	for	ADP
ma-382	180	6	pde	pde	NOUN
ma-382	180	7	-	-	PUNCT
ma-382	180	8	constrained	constrain	VERB
ma-382	180	9	optimization	optimization	NOUN
ma-382	180	10	problems	problem	NOUN
ma-382	180	11	minu∈u	minu∈u	PROPN
ma-382	180	12	j(u	j(u	PROPN
ma-382	180	13	)	)	PUNCT
ma-382	180	14	with	with	SCONJ
ma-382	180	15	u	u	PROPN
ma-382	180	16	⊂	⊂	PROPN
ma-382	180	17	x	x	X
ma-382	180	18	:	:	PUNCT
ma-382	180	19	(	(	PUNCT
ma-382	180	20	1	1	X
ma-382	180	21	)	)	PUNCT
ma-382	180	22	norm	norm	NOUN
ma-382	180	23	-	-	PUNCT
ma-382	180	24	attaining	attain	VERB
ma-382	180	25	functionals	functional	NOUN
ma-382	180	26	in	in	ADP
ma-382	180	27	x∗	x∗	PROPN
ma-382	180	28	yield	yield	VERB
ma-382	180	29	minimizers	minimizer	NOUN
ma-382	180	30	with	with	ADP
ma-382	180	31	extremal	extremal	ADJ
ma-382	180	32	properties	property	NOUN
ma-382	180	33	https://doi.org/10.28924/ada/ma.5.382	https://doi.org/10.28924/ada/ma.5.382	VERB
ma-382	180	34	eur	eur	NOUN
ma-382	180	35	.	.	PUNCT
ma-382	181	1	j.	j.	PROPN
ma-382	181	2	math	math	PROPN
ma-382	181	3	.	.	PUNCT
ma-382	182	1	anal	anal	PROPN
ma-382	182	2	.	.	PUNCT
ma-382	183	1	10.28924	10.28924	NUM
ma-382	183	2	/	/	SYM
ma-382	183	3	ada	ada	PROPN
ma-382	183	4	/	/	SYM
ma-382	183	5	ma.5.382	ma.5.382	NOUN
ma-382	183	6	8	8	NUM
ma-382	183	7	(	(	PUNCT
ma-382	183	8	2	2	NUM
ma-382	183	9	)	)	PUNCT
ma-382	183	10	the	the	DET
ma-382	183	11	euler	euler	ADJ
ma-382	183	12	-	-	PUNCT
ma-382	183	13	lagrange	lagrange	PROPN
ma-382	183	14	equations	equation	NOUN
ma-382	183	15	admit	admit	VERB
ma-382	183	16	stabilized	stabilize	VERB
ma-382	183	17	discrete	discrete	ADJ
ma-382	183	18	approximations	approximation	NOUN
ma-382	183	19	(	(	PUNCT
ma-382	183	20	3	3	X
ma-382	183	21	)	)	PUNCT
ma-382	183	22	adaptive	adaptive	ADJ
ma-382	183	23	algorithms	algorithm	NOUN
ma-382	183	24	can	can	AUX
ma-382	183	25	achieve	achieve	VERB
ma-382	183	26	ε	ε	NOUN
ma-382	183	27	-	-	NOUN
ma-382	183	28	attainment	attainment	NOUN
ma-382	183	29	in	in	ADP
ma-382	183	30	o(ε−α	o(ε−α	NOUN
ma-382	183	31	)	)	PUNCT
ma-382	183	32	steps	step	NOUN
ma-382	183	33	proof	proof	NOUN
ma-382	183	34	.	.	PUNCT
ma-382	184	1	let	let	VERB
ma-382	184	2	j	j	NOUN
ma-382	184	3	:	:	PUNCT
ma-382	184	4	u	u	X
ma-382	184	5	⊂	⊂	PROPN
ma-382	184	6	x	x	PUNCT
ma-382	185	1	→	→	PUNCT
ma-382	185	2	r	r	NOUN
ma-382	185	3	be	be	VERB
ma-382	185	4	a	a	DET
ma-382	185	5	frechet	frechet	NOUN
ma-382	185	6	differentiable	differentiable	ADJ
ma-382	185	7	cost	cost	NOUN
ma-382	185	8	functional	functional	ADJ
ma-382	185	9	with	with	ADP
ma-382	185	10	u	u	NOUN
ma-382	185	11	convex	convex	NOUN
ma-382	185	12	and	and	CCONJ
ma-382	185	13	closed.suppose	closed.suppose	NOUN
ma-382	185	14	x	x	NOUN
ma-382	185	15	is	be	AUX
ma-382	185	16	a	a	DET
ma-382	185	17	reflexive	reflexive	ADJ
ma-382	185	18	banach	banach	NOUN
ma-382	185	19	space	space	NOUN
ma-382	185	20	and	and	CCONJ
ma-382	185	21	j	j	PROPN
ma-382	185	22	is	be	AUX
ma-382	185	23	coercive	coercive	ADJ
ma-382	185	24	and	and	CCONJ
ma-382	185	25	weakly	weakly	ADV
ma-382	185	26	lower	low	ADJ
ma-382	185	27	semi	semi	ADJ
ma-382	185	28	-	-	ADJ
ma-382	185	29	continuous	continuous	ADJ
ma-382	185	30	.	.	PUNCT
ma-382	186	1	thenstandard	thenstandard	NOUN
ma-382	186	2	variational	variational	ADJ
ma-382	186	3	arguments	argument	NOUN
ma-382	186	4	guarantee	guarantee	VERB
ma-382	186	5	the	the	DET
ma-382	186	6	existence	existence	NOUN
ma-382	186	7	of	of	ADP
ma-382	186	8	minimizers	minimizer	NOUN
ma-382	186	9	.	.	PUNCT
ma-382	187	1	(	(	PUNCT
ma-382	187	2	1	1	X
ma-382	187	3	)	)	PUNCT
ma-382	187	4	norm	norm	NOUN
ma-382	187	5	-	-	PUNCT
ma-382	187	6	attaining	attain	VERB
ma-382	187	7	functionals	functional	NOUN
ma-382	187	8	yield	yield	VERB
ma-382	187	9	extremal	extremal	ADJ
ma-382	187	10	minimizers	minimizer	NOUN
ma-382	187	11	:	:	PUNCT
ma-382	187	12	let	let	VERB
ma-382	187	13	f	f	PROPN
ma-382	187	14	∈	∈	PROPN
ma-382	187	15	x∗	x∗	PROPN
ma-382	187	16	norm	norm	NOUN
ma-382	187	17	-	-	PUNCT
ma-382	187	18	attain	attain	NOUN
ma-382	187	19	at	at	ADP
ma-382	187	20	u∗	u∗	PROPN
ma-382	187	21	∈	∈	PROPN
ma-382	187	22	u	u	NOUN
ma-382	187	23	,	,	PUNCT
ma-382	187	24	i.e.	i.e.	X
ma-382	187	25	,	,	PUNCT
ma-382	187	26	‖f‖	‖f‖	PROPN
ma-382	187	27	=	=	SYM
ma-382	187	28	|f	|f	PROPN
ma-382	187	29	(	(	PUNCT
ma-382	187	30	u∗)|	u∗)|	PROPN
ma-382	187	31	=	=	SYM
ma-382	187	32	sup‖u‖≤1	sup‖u‖≤1	PROPN
ma-382	188	1	|f	|f	PROPN
ma-382	188	2	(	(	PUNCT
ma-382	188	3	u)|	u)|	NOUN
ma-382	188	4	.	.	PUNCT
ma-382	188	5	define	define	VERB
ma-382	188	6	j(u	j(u	PROPN
ma-382	188	7	)	)	PUNCT
ma-382	189	1	=	=	SYM
ma-382	189	2	−f	−f	NOUN
ma-382	189	3	(	(	PUNCT
ma-382	189	4	u	u	NOUN
ma-382	189	5	)	)	PUNCT
ma-382	189	6	+	+	CCONJ
ma-382	189	7	r(u	r(u	NOUN
ma-382	189	8	)	)	PUNCT
ma-382	189	9	,	,	PUNCT
ma-382	189	10	where	where	SCONJ
ma-382	189	11	r	r	NOUN
ma-382	189	12	is	be	AUX
ma-382	189	13	convex	convex	ADJ
ma-382	189	14	and	and	CCONJ
ma-382	189	15	coercive.then	coercive.then	PROPN
ma-382	189	16	j	j	PROPN
ma-382	189	17	admits	admit	VERB
ma-382	189	18	a	a	DET
ma-382	189	19	minimizer	minimizer	NOUN
ma-382	189	20	at	at	ADP
ma-382	189	21	u∗	u∗	ADJ
ma-382	189	22	due	due	ADP
ma-382	189	23	to	to	ADP
ma-382	189	24	the	the	DET
ma-382	189	25	extremality	extremality	NOUN
ma-382	189	26	of	of	ADP
ma-382	189	27	f	f	PROPN
ma-382	189	28	and	and	CCONJ
ma-382	189	29	convexity	convexity	NOUN
ma-382	189	30	of	of	ADP
ma-382	189	31	r.	r.	PROPN
ma-382	189	32	the	the	DET
ma-382	189	33	minimizerinherits	minimizerinherit	NOUN
ma-382	189	34	the	the	DET
ma-382	189	35	extremal	extremal	ADJ
ma-382	189	36	nature	nature	NOUN
ma-382	189	37	of	of	ADP
ma-382	189	38	f	f	PROPN
ma-382	189	39	through	through	ADP
ma-382	189	40	the	the	DET
ma-382	189	41	dual	dual	ADJ
ma-382	189	42	representation	representation	NOUN
ma-382	189	43	of	of	ADP
ma-382	189	44	j	j	PROPN
ma-382	189	45	.	.	PUNCT
ma-382	190	1	(	(	PUNCT
ma-382	190	2	2	2	X
ma-382	190	3	)	)	PUNCT
ma-382	190	4	discrete	discrete	ADJ
ma-382	190	5	euler	euler	NOUN
ma-382	190	6	-	-	PUNCT
ma-382	190	7	lagrange	lagrange	NOUN
ma-382	190	8	approximation	approximation	NOUN
ma-382	190	9	:	:	PUNCT
ma-382	190	10	let	let	VERB
ma-382	190	11	xh	xh	PROPN
ma-382	190	12	⊂	⊂	PROPN
ma-382	190	13	x	x	PUNCT
ma-382	190	14	be	be	AUX
ma-382	190	15	a	a	DET
ma-382	190	16	finite	finite	ADJ
ma-382	190	17	-	-	ADJ
ma-382	190	18	dimensional	dimensional	ADJ
ma-382	190	19	subspace	subspace	NOUN
ma-382	190	20	(	(	PUNCT
ma-382	190	21	e.g.	e.g.	ADV
ma-382	190	22	,galerkin	,galerkin	PUNCT
ma-382	190	23	approximation	approximation	NOUN
ma-382	190	24	)	)	PUNCT
ma-382	190	25	,	,	PUNCT
ma-382	190	26	and	and	CCONJ
ma-382	190	27	let	let	VERB
ma-382	190	28	jh	jh	PROPN
ma-382	190	29	=	=	PROPN
ma-382	190	30	j|xh	j|xh	PROPN
ma-382	190	31	.	.	PUNCT
ma-382	191	1	then	then	ADV
ma-382	191	2	minimizers	minimizer	NOUN
ma-382	191	3	uh	uh	INTJ
ma-382	191	4	∈	∈	PROPN
ma-382	191	5	xh	xh	PROPN
ma-382	191	6	satisfy	satisfy	VERB
ma-382	191	7	the	the	DET
ma-382	191	8	discrete	discrete	ADJ
ma-382	191	9	euler	euler	NOUN
ma-382	191	10	-	-	PUNCT
ma-382	191	11	lagrange	lagrange	NOUN
ma-382	191	12	equation	equation	NOUN
ma-382	191	13	:	:	PUNCT
ma-382	191	14	j	j	PROPN
ma-382	191	15	′h(uh)(v	′h(uh)(v	PROPN
ma-382	191	16	)	)	PUNCT
ma-382	191	17	=	=	SYM
ma-382	191	18	0	0	NUM
ma-382	191	19	∀v	∀v	PROPN
ma-382	191	20	∈	∈	PROPN
ma-382	191	21	xh.by	xh.by	PROPN
ma-382	191	22	cea	cea	PROPN
ma-382	191	23	’s	’s	PART
ma-382	191	24	lemma	lemma	PROPN
ma-382	191	25	and	and	CCONJ
ma-382	191	26	coercivity	coercivity	NOUN
ma-382	191	27	of	of	ADP
ma-382	191	28	j	j	PROPN
ma-382	191	29	′′	′′	PROPN
ma-382	191	30	,	,	PUNCT
ma-382	191	31	we	we	PRON
ma-382	191	32	have	have	VERB
ma-382	191	33	‖uh	‖uh	NUM
ma-382	191	34	−	−	NOUN
ma-382	191	35	u‖	u‖	ADJ
ma-382	191	36	≤	≤	NUM
ma-382	191	37	c	c	PROPN
ma-382	191	38	inf	inf	PROPN
ma-382	191	39	v∈xh	v∈xh	PROPN
ma-382	191	40	‖u	‖u	PROPN
ma-382	191	41	−	−	PROPN
ma-382	191	42	v‖	v‖	NOUN
ma-382	191	43	,	,	PUNCT
ma-382	191	44	and	and	CCONJ
ma-382	191	45	the	the	DET
ma-382	191	46	convergence	convergence	NOUN
ma-382	191	47	rate	rate	NOUN
ma-382	191	48	improves	improve	VERB
ma-382	191	49	as	as	ADP
ma-382	191	50	h	h	NOUN
ma-382	191	51	→	→	SYM
ma-382	191	52	0	0	NUM
ma-382	191	53	depending	depend	VERB
ma-382	191	54	on	on	ADP
ma-382	191	55	the	the	DET
ma-382	191	56	regularity	regularity	NOUN
ma-382	191	57	of	of	ADP
ma-382	191	58	u.	u.	NOUN
ma-382	191	59	(	(	PUNCT
ma-382	191	60	3	3	NUM
ma-382	191	61	)	)	PUNCT
ma-382	191	62	adaptive	adaptive	ADJ
ma-382	191	63	algorithms	algorithm	NOUN
ma-382	191	64	and	and	CCONJ
ma-382	191	65	ε	ε	NOUN
ma-382	191	66	-	-	PUNCT
ma-382	191	67	attainment	attainment	NOUN
ma-382	191	68	:	:	PUNCT
ma-382	191	69	let	let	VERB
ma-382	191	70	a	a	PRON
ma-382	191	71	be	be	AUX
ma-382	191	72	an	an	DET
ma-382	191	73	adaptive	adaptive	ADJ
ma-382	191	74	refinement	refinement	NOUN
ma-382	191	75	procedure	procedure	NOUN
ma-382	191	76	,	,	PUNCT
ma-382	191	77	selectingsubspaces	selectingsubspace	NOUN
ma-382	191	78	xhk	xhk	VERB
ma-382	191	79	based	base	VERB
ma-382	191	80	on	on	ADP
ma-382	191	81	a	a	DET
ma-382	191	82	posteriori	posteriori	NOUN
ma-382	191	83	error	error	NOUN
ma-382	191	84	indicators	indicator	NOUN
ma-382	191	85	.	.	PUNCT
ma-382	192	1	at	at	ADP
ma-382	192	2	each	each	DET
ma-382	192	3	step	step	NOUN
ma-382	192	4	k	k	PROPN
ma-382	192	5	,	,	PUNCT
ma-382	192	6	we	we	PRON
ma-382	192	7	compute	compute	VERB
ma-382	192	8	uk	uk	PROPN
ma-382	192	9	∈	∈	PROPN
ma-382	192	10	xhkminimizing	xhkminimizing	NOUN
ma-382	192	11	jhk	jhk	VERB
ma-382	192	12	such	such	DET
ma-382	192	13	that	that	DET
ma-382	192	14	|j(uk)−	|j(uk)−	ADJ
ma-382	192	15	inf	inf	ADJ
ma-382	192	16	j|	j|	PROPN
ma-382	192	17	≤	≤	ADJ
ma-382	192	18	εk	εk	PROPN
ma-382	193	1	.under	.under	ADP
ma-382	193	2	assumptions	assumption	NOUN
ma-382	193	3	of	of	ADP
ma-382	193	4	ellipticity	ellipticity	NOUN
ma-382	193	5	,	,	PUNCT
ma-382	193	6	local	local	ADJ
ma-382	193	7	approximability	approximability	NOUN
ma-382	193	8	,	,	PUNCT
ma-382	193	9	and	and	CCONJ
ma-382	193	10	stability	stability	NOUN
ma-382	193	11	,	,	PUNCT
ma-382	193	12	we	we	PRON
ma-382	193	13	have	have	VERB
ma-382	193	14	a	a	DET
ma-382	193	15	convergence	convergence	NOUN
ma-382	193	16	com	com	NOUN
ma-382	193	17	-	-	PUNCT
ma-382	193	18	plexity	plexity	NOUN
ma-382	193	19	εk	εk	NOUN
ma-382	193	20	≤	≤	X
ma-382	193	21	ck−β	ck−β	NOUN
ma-382	193	22	⇒	⇒	NOUN
ma-382	193	23	k	k	PROPN
ma-382	194	1	=	=	PUNCT
ma-382	194	2	o(ε−1	o(ε−1	PROPN
ma-382	194	3	/	/	SYM
ma-382	194	4	β	β	NOUN
ma-382	194	5	)	)	PUNCT
ma-382	194	6	=	=	SYM
ma-382	194	7	o(ε−α	o(ε−α	NOUN
ma-382	194	8	)	)	PUNCT
ma-382	194	9	.	.	PUNCT
ma-382	195	1	here	here	ADV
ma-382	195	2	α	α	X
ma-382	195	3	=	=	SYM
ma-382	195	4	1	1	NUM
ma-382	195	5	/	/	SYM
ma-382	195	6	β	β	NOUN
ma-382	195	7	depends	depend	VERB
ma-382	195	8	on	on	ADP
ma-382	195	9	the	the	DET
ma-382	195	10	spatial	spatial	ADJ
ma-382	195	11	adaptivity	adaptivity	NOUN
ma-382	195	12	and	and	CCONJ
ma-382	195	13	smoothness	smoothness	NOUN
ma-382	195	14	of	of	ADP
ma-382	195	15	the	the	DET
ma-382	195	16	minimizer	minimizer	NOUN
ma-382	195	17	.	.	PUNCT
ma-382	196	1	for	for	ADP
ma-382	196	2	example	example	NOUN
ma-382	196	3	,	,	PUNCT
ma-382	196	4	in	in	ADP
ma-382	196	5	second	second	ADJ
ma-382	196	6	-	-	PUNCT
ma-382	196	7	order	order	NOUN
ma-382	196	8	elliptic	elliptic	ADJ
ma-382	196	9	pdes	pde	NOUN
ma-382	196	10	with	with	ADP
ma-382	196	11	h1	h1	PROPN
ma-382	196	12	regularity	regularity	NOUN
ma-382	196	13	,	,	PUNCT
ma-382	196	14	α	α	PROPN
ma-382	196	15	∈	∈	PROPN
ma-382	197	1	[	[	X
ma-382	197	2	1	1	NUM
ma-382	197	3	,	,	PUNCT
ma-382	197	4	2	2	NUM
ma-382	197	5	]	]	PUNCT
ma-382	197	6	depending	depend	VERB
ma-382	197	7	on	on	ADP
ma-382	197	8	the	the	DET
ma-382	197	9	mesh	mesh	NOUN
ma-382	197	10	refinementstrategy	refinementstrategy	NOUN
ma-382	197	11	.	.	PUNCT
ma-382	198	1	thus	thus	ADV
ma-382	198	2	,	,	PUNCT
ma-382	198	3	adaptive	adaptive	ADJ
ma-382	198	4	optimization	optimization	NOUN
ma-382	198	5	transfers	transfer	VERB
ma-382	198	6	the	the	DET
ma-382	198	7	functional	functional	ADJ
ma-382	198	8	attainment	attainment	ADJ
ma-382	198	9	structure	structure	NOUN
ma-382	198	10	into	into	ADP
ma-382	198	11	an	an	DET
ma-382	198	12	efficientcomputational	efficientcomputational	ADJ
ma-382	198	13	framework	framework	NOUN
ma-382	198	14	.	.	PUNCT
ma-382	199	1	�	�	PROPN
ma-382	199	2	theorem	theorem	VERB
ma-382	199	3	8	8	NUM
ma-382	199	4	.	.	PUNCT
ma-382	200	1	for	for	ADP
ma-382	200	2	regression	regression	NOUN
ma-382	200	3	models	model	NOUN
ma-382	200	4	y	y	NOUN
ma-382	200	5	=	=	SYM
ma-382	200	6	f	f	PROPN
ma-382	200	7	(	(	PUNCT
ma-382	200	8	x	x	X
ma-382	200	9	)	)	PUNCT
ma-382	200	10	+	+	CCONJ
ma-382	200	11	ε	ε	PROPN
ma-382	200	12	with	with	ADP
ma-382	200	13	f	f	PROPN
ma-382	200	14	∈	∈	PROPN
ma-382	200	15	x∗	x∗	PROPN
ma-382	200	16	:	:	PUNCT
ma-382	200	17	(	(	PUNCT
ma-382	200	18	1	1	X
ma-382	200	19	)	)	PUNCT
ma-382	200	20	the	the	DET
ma-382	200	21	empirical	empirical	ADJ
ma-382	200	22	risk	risk	NOUN
ma-382	200	23	minimizer	minimizer	NOUN
ma-382	200	24	f̂n	f̂n	NOUN
ma-382	200	25	norm	norm	NOUN
ma-382	200	26	-	-	PUNCT
ma-382	200	27	attains	attain	NOUN
ma-382	200	28	with	with	ADP
ma-382	200	29	high	high	ADJ
ma-382	200	30	probability	probability	NOUN
ma-382	200	31	(	(	PUNCT
ma-382	200	32	2	2	NUM
ma-382	200	33	)	)	PUNCT
ma-382	200	34	the	the	DET
ma-382	200	35	attainment	attainment	ADJ
ma-382	200	36	gap	gap	NOUN
ma-382	200	37	decays	decay	VERB
ma-382	200	38	as	as	ADP
ma-382	200	39	e[‖f̂n‖	e[‖f̂n‖	NOUN
ma-382	200	40	−	−	NOUN
ma-382	200	41	sup‖x‖≤1	sup‖x‖≤1	ADV
ma-382	200	42	f̂n(x)|	f̂n(x)|	ADP
ma-382	200	43	]	]	PUNCT
ma-382	200	44	≤	≤	NUM
ma-382	200	45	c/	c/	NOUN
ma-382	200	46	√	√	PROPN
ma-382	200	47	n	n	CCONJ
ma-382	200	48	(	(	PUNCT
ma-382	200	49	3	3	X
ma-382	200	50	)	)	PUNCT
ma-382	200	51	adaptive	adaptive	ADJ
ma-382	200	52	sampling	sampling	NOUN
ma-382	200	53	improves	improve	VERB
ma-382	200	54	convergence	convergence	NOUN
ma-382	200	55	to	to	ADP
ma-382	200	56	o(1	o(1	VERB
ma-382	200	57	/	/	SYM
ma-382	200	58	n	n	CCONJ
ma-382	200	59	)	)	PUNCT
ma-382	200	60	in	in	ADP
ma-382	200	61	smooth	smooth	ADJ
ma-382	200	62	cases	case	NOUN
ma-382	200	63	proof	proof	NOUN
ma-382	200	64	.	.	PUNCT
ma-382	201	1	consider	consider	VERB
ma-382	201	2	the	the	DET
ma-382	201	3	standard	standard	ADJ
ma-382	201	4	empirical	empirical	ADJ
ma-382	201	5	risk	risk	NOUN
ma-382	201	6	minimization	minimization	NOUN
ma-382	201	7	framework	framework	NOUN
ma-382	201	8	.	.	PUNCT
ma-382	202	1	we	we	PRON
ma-382	202	2	observe	observe	VERB
ma-382	202	3	data	datum	NOUN
ma-382	202	4	{	{	PUNCT
ma-382	202	5	(	(	PUNCT
ma-382	202	6	xi	xi	INTJ
ma-382	202	7	,	,	PUNCT
ma-382	202	8	yi)}ni=1where	yi)}ni=1where	INTJ
ma-382	202	9	xi	xi	PROPN
ma-382	202	10	∈	∈	PROPN
ma-382	202	11	x	x	X
ma-382	202	12	and	and	CCONJ
ma-382	202	13	yi	yi	NOUN
ma-382	202	14	=	=	SYM
ma-382	202	15	f	f	PROPN
ma-382	202	16	(	(	PUNCT
ma-382	202	17	xi	xi	PROPN
ma-382	202	18	)	)	PUNCT
ma-382	203	1	+	+	NOUN
ma-382	203	2	εi	εi	VERB
ma-382	203	3	with	with	ADP
ma-382	203	4	i.i.d	i.i.d	PROPN
ma-382	203	5	.	.	PUNCT
ma-382	203	6	noise	noise	PROPN
ma-382	203	7	εi	εi	NOUN
ma-382	203	8	of	of	ADP
ma-382	203	9	mean	mean	NOUN
ma-382	203	10	zero	zero	NUM
ma-382	203	11	and	and	CCONJ
ma-382	203	12	finite	finite	ADJ
ma-382	203	13	variance	variance	NOUN
ma-382	203	14	.	.	PUNCT
ma-382	204	1	the	the	DET
ma-382	204	2	empirical	empirical	ADJ
ma-382	204	3	https://doi.org/10.28924/ada/ma.5.382	https://doi.org/10.28924/ada/ma.5.382	INTJ
ma-382	204	4	eur	eur	PROPN
ma-382	204	5	.	.	PUNCT
ma-382	205	1	j.	j.	PROPN
ma-382	205	2	math	math	PROPN
ma-382	205	3	.	.	PUNCT
ma-382	206	1	anal	anal	PROPN
ma-382	206	2	.	.	PUNCT
ma-382	207	1	10.28924	10.28924	NUM
ma-382	207	2	/	/	SYM
ma-382	207	3	ada	ada	PROPN
ma-382	207	4	/	/	SYM
ma-382	207	5	ma.5.382	ma.5.382	PROPN
ma-382	207	6	9risk	9risk	NUM
ma-382	207	7	minimizer	minimizer	NOUN
ma-382	207	8	is	be	AUX
ma-382	207	9	defined	define	VERB
ma-382	207	10	as	as	ADP
ma-382	207	11	:	:	PUNCT
ma-382	207	12	f̂n	f̂n	NOUN
ma-382	207	13	=	=	PUNCT
ma-382	207	14	arg	arg	NOUN
ma-382	207	15	min	min	PROPN
ma-382	207	16	g∈x∗	g∈x∗	NOUN
ma-382	207	17	1	1	NUM
ma-382	207	18	n	n	NUM
ma-382	207	19	n∑	n∑	NOUN
ma-382	207	20	i=1	i=1	PROPN
ma-382	207	21	(	(	PUNCT
ma-382	207	22	yi	yi	NOUN
ma-382	207	23	−	−	PROPN
ma-382	207	24	g(xi))2	g(xi))2	PROPN
ma-382	207	25	.	.	PUNCT
ma-382	208	1	by	by	ADP
ma-382	208	2	the	the	DET
ma-382	208	3	representer	representer	PROPN
ma-382	208	4	theorem	theorem	VERB
ma-382	208	5	in	in	ADP
ma-382	208	6	a	a	DET
ma-382	208	7	dual	dual	ADJ
ma-382	208	8	banach	banach	NOUN
ma-382	208	9	setting	setting	NOUN
ma-382	208	10	,	,	PUNCT
ma-382	208	11	under	under	ADP
ma-382	208	12	mild	mild	ADJ
ma-382	208	13	assumptions	assumption	NOUN
ma-382	208	14	on	on	ADP
ma-382	208	15	x	x	SYM
ma-382	208	16	(	(	PUNCT
ma-382	208	17	e.g.	e.g.	ADV
ma-382	208	18	,	,	PUNCT
ma-382	208	19	separa	separa	NOUN
ma-382	208	20	-	-	PUNCT
ma-382	208	21	bility	bility	NOUN
ma-382	208	22	,	,	PUNCT
ma-382	208	23	reflexivity	reflexivity	NOUN
ma-382	208	24	)	)	PUNCT
ma-382	208	25	,	,	PUNCT
ma-382	208	26	the	the	DET
ma-382	208	27	minimizer	minimizer	NOUN
ma-382	208	28	f̂n	f̂n	NOUN
ma-382	208	29	lies	lie	VERB
ma-382	208	30	in	in	ADP
ma-382	208	31	a	a	DET
ma-382	208	32	finite	finite	ADJ
ma-382	208	33	-	-	ADJ
ma-382	208	34	dimensional	dimensional	ADJ
ma-382	208	35	subspace	subspace	NOUN
ma-382	208	36	of	of	ADP
ma-382	208	37	x∗	x∗	PROPN
ma-382	208	38	spanned	span	VERB
ma-382	208	39	by	by	ADP
ma-382	208	40	{	{	PUNCT
ma-382	208	41	xi}ni=1.in	xi}ni=1.in	INTJ
ma-382	208	42	such	such	DET
ma-382	208	43	a	a	DET
ma-382	208	44	subspace	subspace	NOUN
ma-382	208	45	,	,	PUNCT
ma-382	208	46	the	the	DET
ma-382	208	47	supremum	supremum	ADJ
ma-382	208	48	sup‖x‖≤1	sup‖x‖≤1	ADV
ma-382	208	49	|f̂n(x)|	|f̂n(x)|	NOUN
ma-382	208	50	is	be	AUX
ma-382	208	51	attained	attain	VERB
ma-382	208	52	due	due	ADJ
ma-382	208	53	to	to	ADP
ma-382	208	54	compactness	compactness	NOUN
ma-382	208	55	of	of	ADP
ma-382	208	56	the	the	DET
ma-382	208	57	unit	unit	NOUN
ma-382	208	58	balland	balland	VERB
ma-382	208	59	continuity	continuity	NOUN
ma-382	208	60	of	of	ADP
ma-382	208	61	f̂n	f̂n	PROPN
ma-382	208	62	.	.	PUNCT
ma-382	209	1	since	since	SCONJ
ma-382	209	2	the	the	DET
ma-382	209	3	optimization	optimization	NOUN
ma-382	209	4	occurs	occur	VERB
ma-382	209	5	in	in	ADP
ma-382	209	6	finite	finite	ADJ
ma-382	209	7	dimensions	dimension	NOUN
ma-382	209	8	,	,	PUNCT
ma-382	209	9	f̂n	f̂n	NOUN
ma-382	209	10	norm	norm	NOUN
ma-382	209	11	-	-	PUNCT
ma-382	209	12	attains	attain	NOUN
ma-382	209	13	withhigh	withhigh	ADJ
ma-382	209	14	probability	probability	NOUN
ma-382	209	15	as	as	ADP
ma-382	209	16	n	n	PROPN
ma-382	209	17	→	→	SYM
ma-382	209	18	∞	∞	PROPN
ma-382	209	19	,	,	PUNCT
ma-382	209	20	because	because	SCONJ
ma-382	209	21	the	the	DET
ma-382	209	22	data	datum	NOUN
ma-382	209	23	becomes	become	VERB
ma-382	209	24	dense	dense	ADJ
ma-382	209	25	in	in	ADP
ma-382	209	26	x	x	PUNCT
ma-382	209	27	and	and	CCONJ
ma-382	209	28	the	the	DET
ma-382	209	29	empirical	empirical	ADJ
ma-382	209	30	geometryapproximates	geometryapproximate	VERB
ma-382	209	31	the	the	DET
ma-382	209	32	full	full	ADJ
ma-382	209	33	geometry	geometry	NOUN
ma-382	209	34	of	of	ADP
ma-382	209	35	x	x	X
ma-382	209	36	.	.	PUNCT
ma-382	210	1	to	to	PART
ma-382	210	2	establish	establish	VERB
ma-382	210	3	the	the	DET
ma-382	210	4	attainment	attainment	ADJ
ma-382	210	5	gap	gap	NOUN
ma-382	210	6	bound	bind	VERB
ma-382	210	7	,	,	PUNCT
ma-382	210	8	define	define	VERB
ma-382	210	9	the	the	DET
ma-382	210	10	norm	norm	NOUN
ma-382	210	11	gapas	gapas	NOUN
ma-382	210	12	:	:	PUNCT
ma-382	210	13	gn	gn	X
ma-382	210	14	:	:	PUNCT
ma-382	210	15	=	=	PUNCT
ma-382	210	16	‖f̂n‖	‖f̂n‖	PROPN
ma-382	210	17	−	−	NOUN
ma-382	210	18	sup	sup	NOUN
ma-382	210	19	‖x‖≤1	‖x‖≤1	NOUN
ma-382	210	20	|f̂n(x)|	|f̂n(x)|	NOUN
ma-382	210	21	.	.	PUNCT
ma-382	211	1	we	we	PRON
ma-382	211	2	interpret	interpret	VERB
ma-382	211	3	this	this	PRON
ma-382	211	4	as	as	ADP
ma-382	211	5	a	a	DET
ma-382	211	6	deviation	deviation	NOUN
ma-382	211	7	measure	measure	NOUN
ma-382	211	8	of	of	ADP
ma-382	211	9	how	how	SCONJ
ma-382	211	10	close	close	ADJ
ma-382	211	11	f̂n	f̂n	NOUN
ma-382	211	12	comes	come	VERB
ma-382	211	13	to	to	ADP
ma-382	211	14	attaining	attain	VERB
ma-382	211	15	its	its	PRON
ma-382	211	16	norm	norm	NOUN
ma-382	211	17	.	.	PUNCT
ma-382	212	1	since	since	SCONJ
ma-382	212	2	f̂napproximates	f̂napproximate	NOUN
ma-382	212	3	f	f	PROPN
ma-382	212	4	and	and	CCONJ
ma-382	212	5	lives	live	VERB
ma-382	212	6	in	in	ADP
ma-382	212	7	the	the	DET
ma-382	212	8	empirical	empirical	ADJ
ma-382	212	9	subspace	subspace	NOUN
ma-382	212	10	,	,	PUNCT
ma-382	212	11	and	and	CCONJ
ma-382	212	12	since	since	SCONJ
ma-382	212	13	the	the	DET
ma-382	212	14	unit	unit	NOUN
ma-382	212	15	ball	ball	NOUN
ma-382	212	16	in	in	ADP
ma-382	212	17	x	x	PROPN
ma-382	212	18	is	be	AUX
ma-382	212	19	compact	compact	ADJ
ma-382	212	20	underweak	underweak	NOUN
ma-382	212	21	topology	topology	NOUN
ma-382	212	22	,	,	PUNCT
ma-382	212	23	standard	standard	ADJ
ma-382	212	24	empirical	empirical	ADJ
ma-382	212	25	process	process	NOUN
ma-382	212	26	theory	theory	NOUN
ma-382	212	27	(	(	PUNCT
ma-382	212	28	e.g.	e.g.	ADV
ma-382	212	29	,	,	PUNCT
ma-382	212	30	symmetrization	symmetrization	NOUN
ma-382	212	31	,	,	PUNCT
ma-382	212	32	rademacher	rademacher	ADJ
ma-382	212	33	complexity	complexity	NOUN
ma-382	212	34	,	,	PUNCT
ma-382	212	35	concentration	concentration	NOUN
ma-382	212	36	inequalities	inequality	NOUN
ma-382	212	37	)	)	PUNCT
ma-382	212	38	yields	yield	NOUN
ma-382	212	39	:	:	PUNCT
ma-382	212	40	e[gn	e[gn	PROPN
ma-382	212	41	]	]	PUNCT
ma-382	212	42	≤	≤	NUM
ma-382	212	43	c√	c√	NUM
ma-382	212	44	n	n	CCONJ
ma-382	212	45	,	,	PUNCT
ma-382	212	46	for	for	ADP
ma-382	212	47	some	some	DET
ma-382	212	48	constant	constant	ADJ
ma-382	212	49	c	c	NOUN
ma-382	212	50	depending	depend	VERB
ma-382	212	51	on	on	ADP
ma-382	212	52	the	the	DET
ma-382	212	53	complexity	complexity	NOUN
ma-382	212	54	of	of	ADP
ma-382	212	55	the	the	DET
ma-382	212	56	function	function	NOUN
ma-382	212	57	class	class	NOUN
ma-382	212	58	{	{	PUNCT
ma-382	212	59	x	x	PROPN
ma-382	212	60	7→	7→	NUM
ma-382	212	61	g(x	g(x	NOUN
ma-382	212	62	)	)	PUNCT
ma-382	212	63	:	:	PUNCT
ma-382	213	1	g	g	PROPN
ma-382	213	2	∈	∈	PROPN
ma-382	213	3	x∗}and	x∗}and	CCONJ
ma-382	213	4	the	the	DET
ma-382	213	5	distribution	distribution	NOUN
ma-382	213	6	of	of	ADP
ma-382	213	7	x	x	X
ma-382	213	8	.	.	PUNCT
ma-382	214	1	for	for	ADP
ma-382	214	2	smooth	smooth	ADJ
ma-382	214	3	cases	case	NOUN
ma-382	214	4	,	,	PUNCT
ma-382	214	5	where	where	SCONJ
ma-382	214	6	f	f	PROPN
ma-382	214	7	belongs	belong	VERB
ma-382	214	8	to	to	ADP
ma-382	214	9	a	a	DET
ma-382	214	10	sobolev	sobolev	NOUN
ma-382	214	11	-	-	PUNCT
ma-382	214	12	type	type	NOUN
ma-382	214	13	or	or	CCONJ
ma-382	214	14	kernel	kernel	NOUN
ma-382	214	15	-	-	PUNCT
ma-382	214	16	smoothsubspace	smoothsubspace	NOUN
ma-382	214	17	of	of	ADP
ma-382	214	18	x∗	x∗	PROPN
ma-382	214	19	,	,	PUNCT
ma-382	214	20	adaptive	adaptive	ADJ
ma-382	214	21	sampling	sample	VERB
ma-382	214	22	schemes	scheme	NOUN
ma-382	214	23	(	(	PUNCT
ma-382	214	24	e.g.	e.g.	ADV
ma-382	214	25	,	,	PUNCT
ma-382	214	26	greedily	greedily	ADV
ma-382	214	27	selecting	select	VERB
ma-382	214	28	xi	xi	NOUN
ma-382	214	29	to	to	PART
ma-382	214	30	maximize	maximize	VERB
ma-382	214	31	information	information	NOUN
ma-382	214	32	gainor	gainor	NOUN
ma-382	214	33	leverage	leverage	NOUN
ma-382	214	34	scores	score	NOUN
ma-382	214	35	)	)	PUNCT
ma-382	214	36	reduce	reduce	VERB
ma-382	214	37	the	the	DET
ma-382	214	38	effective	effective	ADJ
ma-382	214	39	dimension	dimension	NOUN
ma-382	214	40	faster	fast	ADV
ma-382	214	41	.	.	PUNCT
ma-382	215	1	this	this	PRON
ma-382	215	2	improves	improve	VERB
ma-382	215	3	the	the	DET
ma-382	215	4	convergence	convergence	NOUN
ma-382	215	5	rate	rate	NOUN
ma-382	215	6	of	of	ADP
ma-382	215	7	f̂n	f̂n	NOUN
ma-382	215	8	in	in	ADP
ma-382	215	9	operator	operator	NOUN
ma-382	215	10	norm	norm	NOUN
ma-382	215	11	and	and	CCONJ
ma-382	215	12	sharpens	sharpen	VERB
ma-382	215	13	the	the	DET
ma-382	215	14	norm	norm	NOUN
ma-382	215	15	-	-	PUNCT
ma-382	215	16	attainment	attainment	NOUN
ma-382	215	17	,	,	PUNCT
ma-382	215	18	yielding	yield	VERB
ma-382	215	19	an	an	DET
ma-382	215	20	improved	improved	ADJ
ma-382	215	21	convergence	convergence	NOUN
ma-382	215	22	of	of	ADP
ma-382	215	23	theattainment	theattainment	NOUN
ma-382	215	24	gap	gap	NOUN
ma-382	215	25	:	:	PUNCT
ma-382	215	26	e[gn	e[gn	PROPN
ma-382	215	27	]	]	PUNCT
ma-382	215	28	≤	≤	NUM
ma-382	215	29	c′	c′	NUM
ma-382	215	30	n	n	NOUN
ma-382	215	31	,	,	PUNCT
ma-382	215	32	with	with	ADP
ma-382	215	33	c′	c′	VERB
ma-382	215	34	depending	depend	VERB
ma-382	215	35	on	on	ADP
ma-382	215	36	smoothness	smoothness	ADJ
ma-382	215	37	parameters	parameter	NOUN
ma-382	215	38	and	and	CCONJ
ma-382	215	39	sampling	sample	VERB
ma-382	215	40	design	design	NOUN
ma-382	215	41	.	.	PUNCT
ma-382	216	1	this	this	PRON
ma-382	216	2	concludes	conclude	VERB
ma-382	216	3	the	the	DET
ma-382	216	4	proof	proof	NOUN
ma-382	216	5	.	.	PUNCT
ma-382	217	1	�	�	PROPN
ma-382	217	2	theorem	theorem	VERB
ma-382	217	3	9	9	NUM
ma-382	217	4	.	.	PUNCT
ma-382	218	1	for	for	ADP
ma-382	218	2	x	x	PUNCT
ma-382	218	3	with	with	ADP
ma-382	218	4	rnp	rnp	PROPN
ma-382	218	5	and	and	CCONJ
ma-382	218	6	f	f	PROPN
ma-382	218	7	∈	∈	PROPN
ma-382	218	8	x∗	x∗	PROPN
ma-382	218	9	,	,	PUNCT
ma-382	218	10	the	the	DET
ma-382	218	11	following	follow	VERB
ma-382	218	12	are	be	AUX
ma-382	218	13	equivalent	equivalent	ADJ
ma-382	218	14	:	:	PUNCT
ma-382	218	15	(	(	PUNCT
ma-382	218	16	1	1	X
ma-382	218	17	)	)	PUNCT
ma-382	218	18	f	f	PROPN
ma-382	218	19	norm	norm	NOUN
ma-382	218	20	-	-	PUNCT
ma-382	218	21	attains	attain	NOUN
ma-382	218	22	(	(	PUNCT
ma-382	218	23	2	2	NUM
ma-382	218	24	)	)	PUNCT
ma-382	218	25	the	the	DET
ma-382	218	26	subdifferential	subdifferential	ADJ
ma-382	218	27	∂‖f‖	∂‖f‖	PROPN
ma-382	218	28	contains	contain	VERB
ma-382	218	29	a	a	DET
ma-382	218	30	weak∗	weak∗	NOUN
ma-382	218	31	exposed	expose	VERB
ma-382	218	32	point	point	NOUN
ma-382	218	33	(	(	PUNCT
ma-382	218	34	3	3	X
ma-382	218	35	)	)	PUNCT
ma-382	218	36	there	there	PRON
ma-382	218	37	exists	exist	VERB
ma-382	218	38	a	a	DET
ma-382	218	39	computable	computable	ADJ
ma-382	218	40	minimizing	minimize	VERB
ma-382	218	41	sequence	sequence	NOUN
ma-382	218	42	with	with	ADP
ma-382	218	43	effective	effective	ADJ
ma-382	218	44	modulus	modulus	NOUN
ma-382	218	45	(	(	PUNCT
ma-382	218	46	4	4	NUM
ma-382	218	47	)	)	PUNCT
ma-382	218	48	all	all	DET
ma-382	218	49	ultrapowers	ultrapower	NOUN
ma-382	218	50	fu	fu	NOUN
ma-382	218	51	in	in	ADP
ma-382	218	52	x∗u	x∗u	NUM
ma-382	218	53	simultaneously	simultaneously	ADV
ma-382	218	54	attain	attain	VERB
ma-382	218	55	proof	proof	NOUN
ma-382	218	56	.	.	PUNCT
ma-382	219	1	(	(	PUNCT
ma-382	219	2	1	1	X
ma-382	219	3	)	)	PUNCT
ma-382	219	4	⇒	⇒	NOUN
ma-382	219	5	(	(	PUNCT
ma-382	219	6	2	2	NUM
ma-382	219	7	):	):	PUNCT
ma-382	219	8	suppose	suppose	VERB
ma-382	219	9	f	f	PROPN
ma-382	219	10	norm	norm	NOUN
ma-382	219	11	-	-	PUNCT
ma-382	219	12	attains	attain	NOUN
ma-382	219	13	,	,	PUNCT
ma-382	219	14	i.e.	i.e.	X
ma-382	219	15	,	,	PUNCT
ma-382	219	16	there	there	PRON
ma-382	219	17	exists	exist	VERB
ma-382	219	18	x0	x0	PROPN
ma-382	219	19	∈	∈	PROPN
ma-382	219	20	x	x	PUNCT
ma-382	219	21	with	with	ADP
ma-382	219	22	‖x0‖	‖x0‖	NOUN
ma-382	219	23	=	=	SYM
ma-382	219	24	1	1	NUM
ma-382	219	25	such	such	ADJ
ma-382	219	26	that	that	SCONJ
ma-382	219	27	f	f	PROPN
ma-382	219	28	(	(	PUNCT
ma-382	219	29	x0	x0	PROPN
ma-382	219	30	)	)	PUNCT
ma-382	219	31	=	=	SYM
ma-382	220	1	‖f‖.	‖f‖.	PROPN
ma-382	220	2	by	by	ADP
ma-382	220	3	duality	duality	NOUN
ma-382	220	4	,	,	PUNCT
ma-382	220	5	x0	x0	PROPN
ma-382	220	6	lies	lie	VERB
ma-382	220	7	in	in	ADP
ma-382	220	8	the	the	DET
ma-382	220	9	subdifferential	subdifferential	ADJ
ma-382	220	10	∂‖f‖	∂‖f‖	PROPN
ma-382	220	11	of	of	ADP
ma-382	220	12	the	the	DET
ma-382	220	13	dual	dual	ADJ
ma-382	220	14	norm	norm	NOUN
ma-382	220	15	.	.	PUNCT
ma-382	221	1	if	if	SCONJ
ma-382	221	2	x0	x0	PROPN
ma-382	221	3	is	be	AUX
ma-382	221	4	an	an	DET
ma-382	221	5	extremepoint	extremepoint	NOUN
ma-382	221	6	,	,	PUNCT
ma-382	221	7	it	it	PRON
ma-382	221	8	is	be	AUX
ma-382	221	9	also	also	ADV
ma-382	221	10	weak∗	weak∗	NOUN
ma-382	221	11	exposed	expose	VERB
ma-382	221	12	by	by	ADP
ma-382	221	13	the	the	DET
ma-382	221	14	functional	functional	ADJ
ma-382	221	15	f	f	PROPN
ma-382	221	16	.	.	PUNCT
ma-382	222	1	hence	hence	ADV
ma-382	222	2	,	,	PUNCT
ma-382	222	3	the	the	DET
ma-382	222	4	subdifferential	subdifferential	NOUN
ma-382	222	5	contains	contain	VERB
ma-382	222	6	a	a	DET
ma-382	222	7	weak∗exposed	weak∗expose	VERB
ma-382	222	8	point	point	NOUN
ma-382	222	9	.	.	PUNCT
ma-382	223	1	(	(	PUNCT
ma-382	223	2	2)⇒	2)⇒	NUM
ma-382	223	3	(	(	PUNCT
ma-382	223	4	3	3	NUM
ma-382	223	5	):	):	PUNCT
ma-382	223	6	suppose	suppose	VERB
ma-382	223	7	∂‖f‖	∂‖f‖	PROPN
ma-382	223	8	contains	contain	VERB
ma-382	223	9	a	a	DET
ma-382	223	10	weak∗	weak∗	NOUN
ma-382	223	11	exposed	expose	VERB
ma-382	223	12	point	point	NOUN
ma-382	223	13	x0	x0	PROPN
ma-382	223	14	.	.	PUNCT
ma-382	224	1	then	then	ADV
ma-382	224	2	there	there	PRON
ma-382	224	3	exists	exist	VERB
ma-382	224	4	g	g	PROPN
ma-382	224	5	∈	∈	PROPN
ma-382	224	6	x∗	x∗	NOUN
ma-382	224	7	such	such	ADJ
ma-382	224	8	that	that	SCONJ
ma-382	224	9	https://doi.org/10.28924/ada/ma.5.382	https://doi.org/10.28924/ada/ma.5.382	PUNCT
ma-382	224	10	eur	eur	PROPN
ma-382	224	11	.	.	PUNCT
ma-382	225	1	j.	j.	PROPN
ma-382	225	2	math	math	PROPN
ma-382	225	3	.	.	PUNCT
ma-382	226	1	anal	anal	PROPN
ma-382	226	2	.	.	PUNCT
ma-382	227	1	10.28924	10.28924	NUM
ma-382	227	2	/	/	SYM
ma-382	227	3	ada	ada	PROPN
ma-382	227	4	/	/	SYM
ma-382	227	5	ma.5.382	ma.5.382	PROPN
ma-382	227	6	10	10	NUM
ma-382	227	7	x0	x0	PROPN
ma-382	227	8	maximizes	maximize	VERB
ma-382	227	9	g(x	g(x	NOUN
ma-382	227	10	)	)	PUNCT
ma-382	227	11	over	over	ADP
ma-382	227	12	the	the	DET
ma-382	227	13	unit	unit	NOUN
ma-382	227	14	ball	ball	NOUN
ma-382	227	15	,	,	PUNCT
ma-382	227	16	and	and	CCONJ
ma-382	227	17	the	the	DET
ma-382	227	18	maximum	maximum	NOUN
ma-382	227	19	is	be	AUX
ma-382	227	20	attained	attain	VERB
ma-382	227	21	only	only	ADV
ma-382	227	22	at	at	ADP
ma-382	227	23	x0	x0	PROPN
ma-382	227	24	.	.	PUNCT
ma-382	228	1	by	by	ADP
ma-382	228	2	continuity	continuity	NOUN
ma-382	228	3	andconvexity	andconvexity	NOUN
ma-382	228	4	,	,	PUNCT
ma-382	228	5	one	one	PRON
ma-382	228	6	can	can	AUX
ma-382	228	7	define	define	VERB
ma-382	228	8	a	a	DET
ma-382	228	9	sequence	sequence	NOUN
ma-382	228	10	(	(	PUNCT
ma-382	228	11	xn	xn	X
ma-382	228	12	)	)	PUNCT
ma-382	228	13	approaching	approach	VERB
ma-382	228	14	x0	x0	PROPN
ma-382	228	15	with	with	ADP
ma-382	228	16	f	f	PROPN
ma-382	228	17	(	(	PUNCT
ma-382	228	18	xn)→	xn)→	PROPN
ma-382	228	19	‖f‖	‖f‖	PROPN
ma-382	228	20	,	,	PUNCT
ma-382	228	21	and	and	CCONJ
ma-382	228	22	the	the	DET
ma-382	228	23	modulus	modulus	ADJ
ma-382	228	24	ofconvergence	ofconvergence	NOUN
ma-382	228	25	is	be	AUX
ma-382	228	26	governed	govern	VERB
ma-382	228	27	by	by	ADP
ma-382	228	28	the	the	DET
ma-382	228	29	modulus	modulus	NOUN
ma-382	228	30	of	of	ADP
ma-382	228	31	convexity	convexity	NOUN
ma-382	228	32	and	and	CCONJ
ma-382	228	33	smoothness	smoothness	NOUN
ma-382	228	34	of	of	ADP
ma-382	228	35	the	the	DET
ma-382	228	36	norm	norm	NOUN
ma-382	228	37	.	.	PUNCT
ma-382	229	1	this	this	DET
ma-382	229	2	yields	yield	NOUN
ma-382	229	3	acomputable	acomputable	ADJ
ma-382	229	4	minimizing	minimize	VERB
ma-382	229	5	sequence	sequence	NOUN
ma-382	229	6	.	.	PUNCT
ma-382	230	1	(	(	PUNCT
ma-382	230	2	3	3	X
ma-382	230	3	)	)	PUNCT
ma-382	230	4	⇒	⇒	NOUN
ma-382	230	5	(	(	PUNCT
ma-382	230	6	4	4	NUM
ma-382	230	7	):	):	PUNCT
ma-382	230	8	a	a	DET
ma-382	230	9	computable	computable	ADJ
ma-382	230	10	minimizing	minimize	VERB
ma-382	230	11	sequence	sequence	NOUN
ma-382	230	12	(	(	PUNCT
ma-382	230	13	xn	xn	PROPN
ma-382	230	14	)	)	PUNCT
ma-382	230	15	with	with	ADP
ma-382	230	16	effective	effective	ADJ
ma-382	230	17	modulus	modulus	NOUN
ma-382	230	18	ensures	ensure	VERB
ma-382	230	19	that	that	SCONJ
ma-382	230	20	for	for	ADP
ma-382	230	21	anynonprincipal	anynonprincipal	ADJ
ma-382	230	22	ultrafilter	ultrafilter	NOUN
ma-382	230	23	u	u	NOUN
ma-382	230	24	,	,	PUNCT
ma-382	230	25	the	the	DET
ma-382	230	26	image	image	NOUN
ma-382	230	27	of	of	ADP
ma-382	230	28	(	(	PUNCT
ma-382	230	29	xn	xn	X
ma-382	230	30	)	)	PUNCT
ma-382	230	31	in	in	ADP
ma-382	230	32	the	the	DET
ma-382	230	33	ultrapower	ultrapower	ADJ
ma-382	230	34	space	space	NOUN
ma-382	230	35	xu	xu	PROPN
ma-382	230	36	gives	give	VERB
ma-382	230	37	rise	rise	NOUN
ma-382	230	38	to	to	ADP
ma-382	230	39	a	a	DET
ma-382	230	40	point	point	NOUN
ma-382	230	41	xuwith	xuwith	NOUN
ma-382	230	42	‖xu‖	‖xu‖	NOUN
ma-382	230	43	=	=	PUNCT
ma-382	230	44	1	1	NUM
ma-382	230	45	and	and	CCONJ
ma-382	230	46	fu(xu	fu(xu	NOUN
ma-382	230	47	)	)	PUNCT
ma-382	230	48	=	=	PUNCT
ma-382	231	1	‖f‖.	‖f‖.	ADP
ma-382	231	2	thus	thus	ADV
ma-382	231	3	,	,	PUNCT
ma-382	231	4	fu	fu	PROPN
ma-382	231	5	attains	attain	VERB
ma-382	231	6	its	its	PRON
ma-382	231	7	norm	norm	NOUN
ma-382	231	8	.	.	PUNCT
ma-382	232	1	(	(	PUNCT
ma-382	232	2	4	4	X
ma-382	232	3	)	)	PUNCT
ma-382	232	4	⇒	⇒	NOUN
ma-382	232	5	(	(	PUNCT
ma-382	232	6	1	1	NUM
ma-382	232	7	):	):	PUNCT
ma-382	232	8	suppose	suppose	VERB
ma-382	232	9	all	all	DET
ma-382	232	10	ultrapowers	ultrapower	NOUN
ma-382	232	11	fu	fu	PROPN
ma-382	232	12	attain	attain	VERB
ma-382	232	13	their	their	PRON
ma-382	232	14	norm	norm	NOUN
ma-382	232	15	.	.	PUNCT
ma-382	233	1	then	then	ADV
ma-382	233	2	,	,	PUNCT
ma-382	233	3	in	in	ADP
ma-382	233	4	particular	particular	ADJ
ma-382	233	5	,	,	PUNCT
ma-382	233	6	the	the	DET
ma-382	233	7	canonicalembedding	canonicalembedding	NOUN
ma-382	233	8	of	of	ADP
ma-382	233	9	f	f	PROPN
ma-382	233	10	into	into	ADP
ma-382	233	11	x∗u	x∗u	NUM
ma-382	233	12	satisfies	satisfie	NOUN
ma-382	233	13	‖fu‖	‖fu‖	X
ma-382	233	14	=	=	SYM
ma-382	233	15	fu(xu	fu(xu	NOUN
ma-382	233	16	)	)	PUNCT
ma-382	233	17	for	for	ADP
ma-382	233	18	some	some	DET
ma-382	233	19	xu	xu	PROPN
ma-382	233	20	in	in	ADP
ma-382	233	21	xu	xu	PROPN
ma-382	233	22	with	with	ADP
ma-382	233	23	‖xu‖	‖xu‖	NOUN
ma-382	233	24	=	=	PUNCT
ma-382	233	25	1	1	X
ma-382	233	26	.	.	PUNCT
ma-382	234	1	since	since	SCONJ
ma-382	234	2	x	x	PROPN
ma-382	234	3	hasrnp	hasrnp	PROPN
ma-382	234	4	,	,	PUNCT
ma-382	234	5	it	it	PRON
ma-382	234	6	satisfies	satisfy	VERB
ma-382	234	7	the	the	DET
ma-382	234	8	local	local	ADJ
ma-382	234	9	reflexivity	reflexivity	NOUN
ma-382	234	10	property	property	NOUN
ma-382	234	11	.	.	PUNCT
ma-382	235	1	hence	hence	ADV
ma-382	235	2	,	,	PUNCT
ma-382	235	3	every	every	DET
ma-382	235	4	such	such	ADJ
ma-382	235	5	attainment	attainment	NOUN
ma-382	235	6	in	in	ADP
ma-382	235	7	ultrapowers	ultrapower	NOUN
ma-382	235	8	reflectsa	reflectsa	NOUN
ma-382	235	9	norm	norm	ADV
ma-382	235	10	-	-	PUNCT
ma-382	235	11	attaining	attain	VERB
ma-382	235	12	sequence	sequence	NOUN
ma-382	235	13	in	in	ADP
ma-382	235	14	x	x	X
ma-382	235	15	,	,	PUNCT
ma-382	235	16	and	and	CCONJ
ma-382	235	17	ultimately	ultimately	ADV
ma-382	235	18	shows	show	VERB
ma-382	235	19	that	that	SCONJ
ma-382	235	20	f	f	PROPN
ma-382	235	21	itself	itself	PRON
ma-382	235	22	norm	norm	NOUN
ma-382	235	23	-	-	PUNCT
ma-382	235	24	attains	attain	NOUN
ma-382	235	25	.	.	PUNCT
ma-382	236	1	therefore	therefore	ADV
ma-382	236	2	,	,	PUNCT
ma-382	236	3	allfour	allfour	NOUN
ma-382	236	4	conditions	condition	NOUN
ma-382	236	5	are	be	AUX
ma-382	236	6	equivalent	equivalent	ADJ
ma-382	236	7	under	under	ADP
ma-382	236	8	the	the	DET
ma-382	236	9	radon	radon	PROPN
ma-382	236	10	-	-	PUNCT
ma-382	236	11	nikodym	nikodym	NOUN
ma-382	236	12	property	property	NOUN
ma-382	236	13	,	,	PUNCT
ma-382	236	14	completing	complete	VERB
ma-382	236	15	the	the	DET
ma-382	236	16	proof	proof	NOUN
ma-382	236	17	.	.	PUNCT
ma-382	237	1	�	�	PROPN
ma-382	237	2	4	4	NUM
ma-382	237	3	.	.	PUNCT
ma-382	237	4	conclusion	conclusion	NOUN
ma-382	237	5	this	this	DET
ma-382	237	6	work	work	NOUN
ma-382	237	7	has	have	AUX
ma-382	237	8	established	establish	VERB
ma-382	237	9	a	a	DET
ma-382	237	10	comprehensive	comprehensive	ADJ
ma-382	237	11	framework	framework	NOUN
ma-382	237	12	for	for	ADP
ma-382	237	13	studying	study	VERB
ma-382	237	14	norm	norm	NOUN
ma-382	237	15	-	-	PUNCT
ma-382	237	16	attaining	attain	VERB
ma-382	237	17	functionalsthrough	functionalsthrough	ADJ
ma-382	237	18	computational	computational	ADJ
ma-382	237	19	,	,	PUNCT
ma-382	237	20	geometric	geometric	ADJ
ma-382	237	21	,	,	PUNCT
ma-382	237	22	and	and	CCONJ
ma-382	237	23	analytic	analytic	ADJ
ma-382	237	24	perspectives	perspective	NOUN
ma-382	237	25	.	.	PUNCT
ma-382	238	1	our	our	PRON
ma-382	238	2	main	main	ADJ
ma-382	238	3	contributions	contribution	NOUN
ma-382	238	4	include	include	VERB
ma-382	238	5	:	:	PUNCT
ma-382	238	6	(	(	PUNCT
ma-382	238	7	1)constructive	1)constructive	NUM
ma-382	238	8	approximation	approximation	NOUN
ma-382	238	9	algorithms	algorithm	NOUN
ma-382	238	10	with	with	ADP
ma-382	238	11	explicit	explicit	ADJ
ma-382	238	12	convergence	convergence	NOUN
ma-382	238	13	rates	rate	NOUN
ma-382	238	14	in	in	ADP
ma-382	238	15	uniformly	uniformly	ADV
ma-382	238	16	convex	convex	ADJ
ma-382	238	17	banachspaces	banachspace	NOUN
ma-382	238	18	(	(	PUNCT
ma-382	238	19	theorems	theorem	NOUN
ma-382	238	20	1	1	NUM
ma-382	238	21	-	-	SYM
ma-382	238	22	2	2	NUM
ma-382	238	23	)	)	PUNCT
ma-382	238	24	,	,	PUNCT
ma-382	238	25	(	(	PUNCT
ma-382	238	26	2	2	X
ma-382	238	27	)	)	PUNCT
ma-382	238	28	stability	stability	NOUN
ma-382	238	29	analysis	analysis	NOUN
ma-382	238	30	under	under	ADP
ma-382	238	31	discretization	discretization	NOUN
ma-382	238	32	and	and	CCONJ
ma-382	238	33	perturbations	perturbation	NOUN
ma-382	238	34	(	(	PUNCT
ma-382	238	35	theorems	theorem	NOUN
ma-382	238	36	3	3	NUM
ma-382	238	37	-	-	SYM
ma-382	238	38	4),and	4),and	NUM
ma-382	238	39	(	(	PUNCT
ma-382	238	40	3	3	X
ma-382	238	41	)	)	PUNCT
ma-382	238	42	new	new	ADJ
ma-382	238	43	applications	application	NOUN
ma-382	238	44	to	to	ADP
ma-382	238	45	optimization	optimization	NOUN
ma-382	238	46	and	and	CCONJ
ma-382	238	47	regression	regression	NOUN
ma-382	238	48	problems	problem	NOUN
ma-382	238	49	(	(	PUNCT
ma-382	238	50	theorems	theorem	NOUN
ma-382	238	51	6	6	NUM
ma-382	238	52	-	-	SYM
ma-382	238	53	7	7	NUM
ma-382	238	54	)	)	PUNCT
ma-382	238	55	.	.	PUNCT
ma-382	239	1	the	the	DET
ma-382	239	2	geometriccharacterization	geometriccharacterization	NOUN
ma-382	239	3	in	in	ADP
ma-382	239	4	theorem	theorem	ADJ
ma-382	239	5	5	5	NUM
ma-382	239	6	unifies	unify	VERB
ma-382	239	7	these	these	DET
ma-382	239	8	results	result	NOUN
ma-382	239	9	by	by	ADP
ma-382	239	10	connecting	connect	VERB
ma-382	239	11	attainment	attainment	NOUN
ma-382	239	12	to	to	ADP
ma-382	239	13	subdifferentialproperties	subdifferentialpropertie	NOUN
ma-382	239	14	and	and	CCONJ
ma-382	239	15	ultrapower	ultrapow	ADJ
ma-382	239	16	constructions	construction	NOUN
ma-382	239	17	.	.	PUNCT
ma-382	240	1	key	key	ADJ
ma-382	240	2	advances	advance	NOUN
ma-382	240	3	beyond	beyond	ADP
ma-382	240	4	prior	prior	ADJ
ma-382	240	5	work	work	NOUN
ma-382	240	6	[	[	X
ma-382	240	7	2	2	NUM
ma-382	240	8	,	,	PUNCT
ma-382	240	9	12	12	NUM
ma-382	240	10	]	]	PUNCT
ma-382	240	11	include	include	VERB
ma-382	240	12	:	:	PUNCT
ma-382	240	13	•	•	NUM
ma-382	240	14	quantitative	quantitative	ADJ
ma-382	240	15	versions	version	NOUN
ma-382	240	16	of	of	ADP
ma-382	240	17	the	the	DET
ma-382	240	18	bishop	bishop	PROPN
ma-382	240	19	-	-	PUNCT
ma-382	240	20	phelps	phelps	PROPN
ma-382	240	21	theorem	theorem	VERB
ma-382	240	22	with	with	ADP
ma-382	240	23	computable	computable	ADJ
ma-382	240	24	rates	rate	NOUN
ma-382	240	25	•	•	NOUN
ma-382	240	26	perturbation	perturbation	NOUN
ma-382	240	27	bounds	bound	NOUN
ma-382	240	28	tied	tie	VERB
ma-382	240	29	to	to	ADP
ma-382	240	30	moduli	modulus	NOUN
ma-382	240	31	of	of	ADP
ma-382	240	32	convexity	convexity	NOUN
ma-382	240	33	(	(	PUNCT
ma-382	240	34	extending	extend	VERB
ma-382	240	35	[	[	X
ma-382	240	36	3	3	NUM
ma-382	240	37	]	]	SYM
ma-382	240	38	)	)	PUNCT
ma-382	240	39	•	•	ADP
ma-382	240	40	adaptive	adaptive	ADJ
ma-382	240	41	algorithms	algorithm	NOUN
ma-382	240	42	for	for	ADP
ma-382	240	43	pde	pde	NOUN
ma-382	240	44	-	-	PUNCT
ma-382	240	45	constrained	constrain	VERB
ma-382	240	46	optimization	optimization	NOUN
ma-382	240	47	(	(	PUNCT
ma-382	240	48	building	build	VERB
ma-382	240	49	on	on	ADP
ma-382	240	50	[	[	X
ma-382	240	51	11])future	11])future	NOUN
ma-382	240	52	directions	direction	NOUN
ma-382	240	53	include	include	VERB
ma-382	240	54	:	:	PUNCT
ma-382	240	55	•	•	ADP
ma-382	240	56	extending	extend	VERB
ma-382	240	57	the	the	DET
ma-382	240	58	computational	computational	ADJ
ma-382	240	59	framework	framework	NOUN
ma-382	240	60	to	to	ADP
ma-382	240	61	non	non	ADJ
ma-382	240	62	-	-	ADJ
ma-382	240	63	reflexive	reflexive	ADJ
ma-382	240	64	spaces	space	NOUN
ma-382	240	65	•	•	ADP
ma-382	240	66	applications	application	NOUN
ma-382	240	67	to	to	ADP
ma-382	240	68	neural	neural	ADJ
ma-382	240	69	network	network	NOUN
ma-382	240	70	analysis	analysis	NOUN
ma-382	240	71	via	via	ADP
ma-382	240	72	infinite	infinite	ADJ
ma-382	240	73	-	-	PUNCT
ma-382	240	74	dimensional	dimensional	ADJ
ma-382	240	75	regression	regression	NOUN
ma-382	240	76	•	•	NOUN
ma-382	240	77	connections	connection	NOUN
ma-382	240	78	to	to	ADP
ma-382	240	79	the	the	DET
ma-382	240	80	james	james	PROPN
ma-382	240	81	theorem	theorem	PROPN
ma-382	240	82	in	in	ADP
ma-382	240	83	computable	computable	ADJ
ma-382	240	84	settingsthese	settingsthese	PROPN
ma-382	240	85	results	result	NOUN
ma-382	240	86	open	open	VERB
ma-382	240	87	new	new	ADJ
ma-382	240	88	avenues	avenue	NOUN
ma-382	240	89	for	for	ADP
ma-382	240	90	combining	combine	VERB
ma-382	240	91	functional	functional	ADJ
ma-382	240	92	-	-	PUNCT
ma-382	240	93	analytic	analytic	ADJ
ma-382	240	94	theory	theory	NOUN
ma-382	240	95	with	with	ADP
ma-382	240	96	computational	computational	ADJ
ma-382	240	97	prac	prac	NOUN
ma-382	240	98	-	-	PUNCT
ma-382	240	99	tice	tice	NOUN
ma-382	240	100	,	,	PUNCT
ma-382	240	101	particularly	particularly	ADV
ma-382	240	102	in	in	ADP
ma-382	240	103	problems	problem	NOUN
ma-382	240	104	requiring	require	VERB
ma-382	240	105	certified	certified	ADJ
ma-382	240	106	norm	norm	NOUN
ma-382	240	107	-	-	PUNCT
ma-382	240	108	attainment	attainment	NOUN
ma-382	240	109	.	.	PUNCT
ma-382	241	1	the	the	DET
ma-382	241	2	methods	method	NOUN
ma-382	241	3	developed	develop	VERB
ma-382	241	4	heremay	heremay	ADV
ma-382	241	5	also	also	ADV
ma-382	241	6	find	find	VERB
ma-382	241	7	applications	application	NOUN
ma-382	241	8	in	in	ADP
ma-382	241	9	quantum	quantum	ADJ
ma-382	241	10	information	information	NOUN
ma-382	241	11	theory	theory	NOUN
ma-382	241	12	and	and	CCONJ
ma-382	241	13	high	high	ADJ
ma-382	241	14	-	-	PUNCT
ma-382	241	15	dimensional	dimensional	ADJ
ma-382	241	16	statistics	statistic	NOUN
ma-382	241	17	,	,	PUNCT
ma-382	241	18	wherebanach	wherebanach	NOUN
ma-382	241	19	space	space	NOUN
ma-382	241	20	geometry	geometry	NOUN
ma-382	241	21	plays	play	VERB
ma-382	241	22	a	a	DET
ma-382	241	23	fundamental	fundamental	ADJ
ma-382	241	24	role	role	NOUN
ma-382	241	25	.	.	PUNCT
ma-382	242	1	references	reference	NOUN
ma-382	242	2	[	[	X
ma-382	242	3	1	1	NUM
ma-382	242	4	]	]	X
ma-382	242	5	m.d	m.d	PROPN
ma-382	242	6	.	.	PROPN
ma-382	242	7	acosta	acosta	PROPN
ma-382	242	8	,	,	PUNCT
ma-382	242	9	f.j	f.j	PROPN
ma-382	242	10	.	.	PROPN
ma-382	242	11	aguirre	aguirre	PROPN
ma-382	242	12	,	,	PUNCT
ma-382	242	13	r.	r.	PROPN
ma-382	242	14	paya	paya	PROPN
ma-382	242	15	,	,	PUNCT
ma-382	242	16	there	there	PRON
ma-382	242	17	is	be	VERB
ma-382	242	18	no	no	DET
ma-382	242	19	bilinear	bilinear	NOUN
ma-382	242	20	bishop	bishop	PROPN
ma-382	242	21	-	-	PUNCT
ma-382	242	22	phelps	phelps	PROPN
ma-382	242	23	theorem	theorem	PROPN
ma-382	242	24	,	,	PUNCT
ma-382	242	25	israel	israel	PROPN
ma-382	242	26	j.	j.	PROPN
ma-382	242	27	math	math	PROPN
ma-382	242	28	.	.	PUNCT
ma-382	243	1	154	154	NUM
ma-382	243	2	(	(	PUNCT
ma-382	243	3	2006	2006	NUM
ma-382	243	4	)	)	PUNCT
ma-382	243	5	,	,	PUNCT
ma-382	243	6	115–131.[2	115–131.[2	NUM
ma-382	243	7	]	]	PUNCT
ma-382	243	8	e.	e.	PROPN
ma-382	243	9	bishop	bishop	PROPN
ma-382	243	10	,	,	PUNCT
ma-382	243	11	r.r	r.r	PROPN
ma-382	243	12	.	.	PROPN
ma-382	243	13	phelps	phelps	PROPN
ma-382	243	14	,	,	PUNCT
ma-382	243	15	a	a	DET
ma-382	243	16	proof	proof	NOUN
ma-382	243	17	that	that	SCONJ
ma-382	243	18	every	every	DET
ma-382	243	19	banach	banach	NOUN
ma-382	243	20	space	space	NOUN
ma-382	243	21	is	be	AUX
ma-382	243	22	subreflexive	subreflexive	ADJ
ma-382	243	23	,	,	PUNCT
ma-382	243	24	bull	bull	NOUN
ma-382	243	25	.	.	PUNCT
ma-382	244	1	amer	amer	PROPN
ma-382	244	2	.	.	PUNCT
ma-382	244	3	math	math	PROPN
ma-382	244	4	.	.	PUNCT
ma-382	245	1	soc	soc	PROPN
ma-382	245	2	.	.	PUNCT
ma-382	246	1	67	67	NUM
ma-382	246	2	(	(	PUNCT
ma-382	246	3	1961	1961	NUM
ma-382	246	4	)	)	PUNCT
ma-382	246	5	,	,	PUNCT
ma-382	246	6	97–98.[3	97–98.[3	NUM
ma-382	246	7	]	]	PUNCT
ma-382	246	8	j.	j.	PROPN
ma-382	246	9	bourgain	bourgain	PROPN
ma-382	246	10	,	,	PUNCT
ma-382	246	11	on	on	ADP
ma-382	246	12	dentability	dentability	NOUN
ma-382	246	13	and	and	CCONJ
ma-382	246	14	the	the	DET
ma-382	246	15	bishop	bishop	PROPN
ma-382	246	16	-	-	PUNCT
ma-382	246	17	phelps	phelps	PROPN
ma-382	246	18	property	property	NOUN
ma-382	246	19	,	,	PUNCT
ma-382	246	20	israel	israel	PROPN
ma-382	246	21	j.	j.	PROPN
ma-382	246	22	math	math	PROPN
ma-382	246	23	.	.	PUNCT
ma-382	247	1	28	28	NUM
ma-382	247	2	(	(	PUNCT
ma-382	247	3	1977	1977	NUM
ma-382	247	4	)	)	PUNCT
ma-382	247	5	,	,	PUNCT
ma-382	247	6	265–271	265–271	NUM
ma-382	247	7	.	.	PUNCT
ma-382	248	1	https://doi.org/10.28924/ada/ma.5.382	https://doi.org/10.28924/ada/ma.5.382	PROPN
ma-382	249	1	eur	eur	PROPN
ma-382	249	2	.	.	PUNCT
ma-382	250	1	j.	j.	PROPN
ma-382	250	2	math	math	PROPN
ma-382	250	3	.	.	PUNCT
ma-382	251	1	anal	anal	PROPN
ma-382	251	2	.	.	PUNCT
ma-382	252	1	10.28924	10.28924	NUM
ma-382	252	2	/	/	SYM
ma-382	252	3	ada	ada	PROPN
ma-382	252	4	/	/	SYM
ma-382	252	5	ma.5.382	ma.5.382	PROPN
ma-382	252	6	11	11	NUM
ma-382	253	1	[	[	X
ma-382	253	2	4	4	X
ma-382	253	3	]	]	X
ma-382	253	4	v.	v.	CCONJ
ma-382	253	5	brattka	brattka	PROPN
ma-382	253	6	,	,	PUNCT
ma-382	253	7	p.	p.	PROPN
ma-382	253	8	hertling	hertling	PROPN
ma-382	253	9	,	,	PUNCT
ma-382	253	10	k.	k.	PROPN
ma-382	254	1	weihrauch	weihrauch	PROPN
ma-382	254	2	,	,	PUNCT
ma-382	254	3	a	a	DET
ma-382	254	4	tutorial	tutorial	NOUN
ma-382	254	5	on	on	ADP
ma-382	254	6	computable	computable	ADJ
ma-382	254	7	analysis	analysis	NOUN
ma-382	254	8	,	,	PUNCT
ma-382	254	9	in	in	ADP
ma-382	254	10	:	:	PUNCT
ma-382	254	11	s.b	s.b	PROPN
ma-382	254	12	.	.	PROPN
ma-382	254	13	cooper	cooper	PROPN
ma-382	254	14	,	,	PUNCT
ma-382	254	15	a.	a.	NOUN
ma-382	254	16	sorbi	sorbi	NOUN
ma-382	254	17	,	,	PUNCT
ma-382	254	18	(	(	PUNCT
ma-382	254	19	eds	ed	NOUN
ma-382	254	20	)	)	PUNCT
ma-382	254	21	,	,	PUNCT
ma-382	254	22	newcomputational	newcomputational	PROPN
ma-382	254	23	paradigms	paradigm	NOUN
ma-382	254	24	,	,	PUNCT
ma-382	254	25	springer	springer	NOUN
ma-382	254	26	,	,	PUNCT
ma-382	254	27	(	(	PUNCT
ma-382	254	28	2015).[5	2015).[5	NUM
ma-382	254	29	]	]	X
ma-382	254	30	n.l	n.l	PROPN
ma-382	254	31	.	.	PROPN
ma-382	254	32	carothers	carother	NOUN
ma-382	254	33	,	,	PUNCT
ma-382	254	34	a	a	DET
ma-382	254	35	short	short	ADJ
ma-382	254	36	course	course	NOUN
ma-382	254	37	on	on	ADP
ma-382	254	38	banach	banach	NOUN
ma-382	254	39	space	space	NOUN
ma-382	254	40	theory	theory	NOUN
ma-382	254	41	,	,	PUNCT
ma-382	254	42	cambridge	cambridge	PROPN
ma-382	254	43	university	university	PROPN
ma-382	254	44	press	press	NOUN
ma-382	254	45	,	,	PUNCT
ma-382	254	46	(	(	PUNCT
ma-382	254	47	2000).[6	2000).[6	X
ma-382	254	48	]	]	X
ma-382	254	49	m.n	m.n	PROPN
ma-382	254	50	.	.	PROPN
ma-382	254	51	evans	evans	PROPN
ma-382	254	52	,	,	PUNCT
ma-382	254	53	a.	a.	NOUN
ma-382	254	54	samwel	samwel	PROPN
ma-382	254	55	o.	o.	PROPN
ma-382	254	56	,	,	PUNCT
ma-382	254	57	norm	norm	NOUN
ma-382	254	58	attainability	attainability	NOUN
ma-382	254	59	of	of	ADP
ma-382	254	60	compact	compact	ADJ
ma-382	254	61	operators	operator	NOUN
ma-382	254	62	:	:	PUNCT
ma-382	254	63	spectral	spectral	ADJ
ma-382	254	64	,	,	PUNCT
ma-382	254	65	geometric	geometric	ADJ
ma-382	254	66	,	,	PUNCT
ma-382	254	67	and	and	CCONJ
ma-382	254	68	perturbation	perturbation	NOUN
ma-382	254	69	insights	insight	NOUN
ma-382	254	70	,	,	PUNCT
ma-382	254	71	arch	arch	NOUN
ma-382	254	72	.	.	PUNCT
ma-382	255	1	curr	curr	PROPN
ma-382	255	2	.	.	PUNCT
ma-382	256	1	res	res	PROPN
ma-382	256	2	.	.	PUNCT
ma-382	257	1	int	int	NOUN
ma-382	257	2	.	.	PUNCT
ma-382	258	1	25	25	NUM
ma-382	258	2	(	(	PUNCT
ma-382	258	3	2025	2025	NUM
ma-382	258	4	)	)	PUNCT
ma-382	258	5	,	,	PUNCT
ma-382	258	6	287–294.[7	287–294.[7	NUM
ma-382	258	7	]	]	X
ma-382	258	8	m.n	m.n	PROPN
ma-382	258	9	.	.	PROPN
ma-382	258	10	evans	evans	PROPN
ma-382	258	11	,	,	PUNCT
ma-382	258	12	p.	p.	NOUN
ma-382	258	13	moraa	moraa	NOUN
ma-382	258	14	,	,	PUNCT
ma-382	258	15	a	a	DET
ma-382	258	16	note	note	NOUN
ma-382	258	17	on	on	ADP
ma-382	258	18	norm	norm	NOUN
ma-382	258	19	-	-	PUNCT
ma-382	258	20	attaining	attain	VERB
ma-382	258	21	properties	property	NOUN
ma-382	258	22	for	for	ADP
ma-382	258	23	frame	frame	NOUN
ma-382	258	24	operators	operator	NOUN
ma-382	258	25	,	,	PUNCT
ma-382	258	26	asian	asian	PROPN
ma-382	258	27	j.	j.	PROPN
ma-382	258	28	adv	adv	PROPN
ma-382	258	29	.	.	PUNCT
ma-382	259	1	res	res	PROPN
ma-382	259	2	.	.	PUNCT
ma-382	260	1	rep	rep	PROPN
ma-382	260	2	.	.	PROPN
ma-382	260	3	19	19	NUM
ma-382	260	4	(	(	PUNCT
ma-382	260	5	2025),337–343.[8	2025),337–343.[8	NUM
ma-382	260	6	]	]	X
ma-382	260	7	m.n	m.n	PROPN
ma-382	260	8	.	.	PROPN
ma-382	260	9	evans	evans	PROPN
ma-382	260	10	,	,	PUNCT
ma-382	260	11	r.	r.	PROPN
ma-382	260	12	obogi	obogi	PROPN
ma-382	260	13	,	,	PUNCT
ma-382	260	14	the	the	DET
ma-382	260	15	geometry	geometry	NOUN
ma-382	260	16	and	and	CCONJ
ma-382	260	17	norm	norm	NOUN
ma-382	260	18	-	-	PUNCT
ma-382	260	19	attainability	attainability	NOUN
ma-382	260	20	of	of	ADP
ma-382	260	21	operators	operator	NOUN
ma-382	260	22	in	in	ADP
ma-382	260	23	operator	operator	NOUN
ma-382	260	24	ideals	ideal	NOUN
ma-382	260	25	:	:	PUNCT
ma-382	260	26	the	the	DET
ma-382	260	27	role	role	NOUN
ma-382	260	28	of	of	ADP
ma-382	260	29	singularvalues	singularvalue	NOUN
ma-382	260	30	and	and	CCONJ
ma-382	260	31	compactness	compactness	NOUN
ma-382	260	32	,	,	PUNCT
ma-382	260	33	open	open	ADJ
ma-382	260	34	j.	j.	PROPN
ma-382	260	35	math	math	PROPN
ma-382	260	36	.	.	PUNCT
ma-382	261	1	anal	anal	PROPN
ma-382	261	2	.	.	PUNCT
ma-382	262	1	8	8	NUM
ma-382	262	2	(	(	PUNCT
ma-382	262	3	2024	2024	NUM
ma-382	262	4	)	)	PUNCT
ma-382	262	5	,	,	PUNCT
ma-382	263	1	79–88.[9	79–88.[9	NUM
ma-382	263	2	]	]	X
ma-382	263	3	m.n	m.n	PROPN
ma-382	263	4	.	.	PROPN
ma-382	263	5	evans	evans	PROPN
ma-382	263	6	,	,	PUNCT
ma-382	263	7	r.	r.	PROPN
ma-382	263	8	obogi	obogi	PROPN
ma-382	263	9	,	,	PUNCT
ma-382	263	10	characterizing	characterize	VERB
ma-382	263	11	norm	norm	NOUN
ma-382	263	12	-	-	PUNCT
ma-382	263	13	attainability	attainability	NOUN
ma-382	263	14	in	in	ADP
ma-382	263	15	operator	operator	NOUN
ma-382	263	16	ideals	ideal	NOUN
ma-382	263	17	:	:	PUNCT
ma-382	263	18	necessary	necessary	ADJ
ma-382	263	19	and	and	CCONJ
ma-382	263	20	sufficient	sufficient	ADJ
ma-382	263	21	conditions	condition	NOUN
ma-382	263	22	foroperators	foroperator	NOUN
ma-382	263	23	in	in	ADP
ma-382	263	24	compact	compact	ADJ
ma-382	263	25	,	,	PUNCT
ma-382	263	26	hilbert	hilbert	NOUN
ma-382	263	27	-	-	PUNCT
ma-382	263	28	schmidt	schmidt	VERB
ma-382	263	29	,	,	PUNCT
ma-382	263	30	and	and	CCONJ
ma-382	263	31	schatten	schatten	NUM
ma-382	263	32	classes	class	NOUN
ma-382	263	33	,	,	PUNCT
ma-382	263	34	ann	ann	PROPN
ma-382	263	35	.	.	PUNCT
ma-382	264	1	pure	pure	ADJ
ma-382	264	2	appl	appl	PROPN
ma-382	264	3	.	.	PUNCT
ma-382	264	4	math	math	NOUN
ma-382	264	5	.	.	PUNCT
ma-382	265	1	31	31	NUM
ma-382	265	2	(	(	PUNCT
ma-382	265	3	2025	2025	NUM
ma-382	265	4	)	)	PUNCT
ma-382	266	1	,	,	PUNCT
ma-382	266	2	1–8.[10	1–8.[10	NUM
ma-382	266	3	]	]	X
ma-382	266	4	g.	g.	PROPN
ma-382	266	5	godefroy	godefroy	PROPN
ma-382	266	6	,	,	PUNCT
ma-382	266	7	some	some	DET
ma-382	266	8	applications	application	NOUN
ma-382	266	9	of	of	ADP
ma-382	266	10	simons	simon	NOUN
ma-382	266	11	’	'	PUNCT
ma-382	266	12	inequality	inequality	NOUN
ma-382	266	13	,	,	PUNCT
ma-382	266	14	seminar	seminar	NOUN
ma-382	266	15	funct	funct	NOUN
ma-382	266	16	.	.	PUNCT
ma-382	267	1	anal	anal	PROPN
ma-382	267	2	.	.	PUNCT
ma-382	268	1	31	31	NUM
ma-382	268	2	(	(	PUNCT
ma-382	268	3	1987	1987	NUM
ma-382	268	4	)	)	PUNCT
ma-382	269	1	,	,	PUNCT
ma-382	269	2	36–51.[11	36–51.[11	NUM
ma-382	269	3	]	]	PUNCT
ma-382	269	4	m.	m.	NOUN
ma-382	269	5	hinze	hinze	PROPN
ma-382	269	6	,	,	PUNCT
ma-382	269	7	r.	r.	PROPN
ma-382	269	8	pinnau	pinnau	PROPN
ma-382	269	9	,	,	PUNCT
ma-382	269	10	m.	m.	NOUN
ma-382	269	11	ulbrich	ulbrich	PROPN
ma-382	269	12	,	,	PUNCT
ma-382	269	13	s.	s.	PROPN
ma-382	269	14	ulbrich	ulbrich	PROPN
ma-382	269	15	,	,	PUNCT
ma-382	269	16	optimization	optimization	NOUN
ma-382	269	17	with	with	ADP
ma-382	269	18	pde	pde	NOUN
ma-382	269	19	constraints	constraint	NOUN
ma-382	269	20	,	,	PUNCT
ma-382	269	21	springer	springer	NOUN
ma-382	269	22	,	,	PUNCT
ma-382	269	23	(	(	PUNCT
ma-382	269	24	2009).[12	2009).[12	NOUN
ma-382	269	25	]	]	X
ma-382	269	26	j.	j.	PROPN
ma-382	269	27	lindenstrauss	lindenstrauss	PROPN
ma-382	269	28	,	,	PUNCT
ma-382	269	29	on	on	ADP
ma-382	269	30	operators	operator	NOUN
ma-382	269	31	which	which	PRON
ma-382	269	32	attain	attain	VERB
ma-382	269	33	their	their	PRON
ma-382	269	34	norm	norm	NOUN
ma-382	269	35	,	,	PUNCT
ma-382	269	36	israel	israel	PROPN
ma-382	269	37	j.	j.	PROPN
ma-382	269	38	math	math	PROPN
ma-382	269	39	.	.	PUNCT
ma-382	270	1	1	1	NUM
ma-382	270	2	(	(	PUNCT
ma-382	270	3	1963	1963	NUM
ma-382	270	4	)	)	PUNCT
ma-382	270	5	,	,	PUNCT
ma-382	270	6	139–148.[13	139–148.[13	NUM
ma-382	270	7	]	]	PUNCT
ma-382	270	8	k.	k.	NOUN
ma-382	271	1	weihrauch	weihrauch	PROPN
ma-382	271	2	,	,	PUNCT
ma-382	271	3	computable	computable	ADJ
ma-382	271	4	analysis	analysis	NOUN
ma-382	271	5	:	:	PUNCT
ma-382	271	6	an	an	DET
ma-382	271	7	introduction	introduction	NOUN
ma-382	271	8	,	,	PUNCT
ma-382	271	9	springer	springer	NOUN
ma-382	271	10	,	,	PUNCT
ma-382	271	11	(	(	PUNCT
ma-382	271	12	2000	2000	NUM
ma-382	271	13	)	)	PUNCT
ma-382	271	14	.	.	PUNCT
ma-382	272	1	https://doi.org/10.28924/ada/ma.5.382	https://doi.org/10.28924/ada/ma.5.382	PUNCT
ma-382	273	1	1	1	NUM
ma-382	273	2	.	.	PUNCT
ma-382	273	3	introduction	introduction	NOUN
ma-382	273	4	and	and	CCONJ
ma-382	273	5	relation	relation	NOUN
ma-382	273	6	to	to	PART
ma-382	273	7	prior	prior	ADJ
ma-382	273	8	work	work	NOUN
ma-382	273	9	2	2	NUM
ma-382	273	10	.	.	PUNCT
ma-382	273	11	preliminaries	preliminary	NOUN
ma-382	273	12	banach	banach	NOUN
ma-382	273	13	space	space	NOUN
ma-382	273	14	geometry	geometry	NOUN
ma-382	273	15	norm	norm	NOUN
ma-382	273	16	-	-	PUNCT
ma-382	273	17	attaining	attain	VERB
ma-382	273	18	functionals	functional	NOUN
ma-382	273	19	computational	computational	ADJ
ma-382	273	20	framework	framework	NOUN
ma-382	273	21	approximation	approximation	NOUN
ma-382	273	22	theory	theory	NOUN
ma-382	273	23	3	3	NUM
ma-382	273	24	.	.	PUNCT
ma-382	273	25	main	main	ADJ
ma-382	273	26	results	result	NOUN
ma-382	273	27	and	and	CCONJ
ma-382	273	28	discussions	discussion	NOUN
ma-382	273	29	4	4	NUM
ma-382	273	30	.	.	PUNCT
ma-382	274	1	conclusion	conclusion	NOUN
ma-382	274	2	references	reference	NOUN