id	sid	tid	token	lemma	pos
ma-30	1	1	2021	2021	NUM
ma-30	1	2	ada	ada	PROPN
ma-30	1	3	academica	academica	PROPN
ma-30	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-30	1	5	.	.	PUNCT
ma-30	2	1	j.	j.	PROPN
ma-30	2	2	math	math	PROPN
ma-30	2	3	.	.	PUNCT
ma-30	3	1	anal	anal	ADJ
ma-30	3	2	.	.	PUNCT
ma-30	4	1	1	1	NUM
ma-30	4	2	(	(	PUNCT
ma-30	4	3	2021	2021	NUM
ma-30	4	4	)	)	PUNCT
ma-30	4	5	151	151	NUM
ma-30	4	6	-	-	SYM
ma-30	4	7	163doi	163doi	NUM
ma-30	4	8	:	:	PUNCT
ma-30	4	9	10.28924	10.28924	NUM
ma-30	4	10	/	/	SYM
ma-30	4	11	ada	ada	PROPN
ma-30	4	12	/	/	SYM
ma-30	4	13	ma.1.151	ma.1.151	PROPN
ma-30	4	14	some	some	DET
ma-30	4	15	aspects	aspect	NOUN
ma-30	4	16	of	of	ADP
ma-30	4	17	geometric	geometric	ADJ
ma-30	4	18	constants	constant	NOUN
ma-30	4	19	in	in	ADP
ma-30	4	20	modular	modular	ADJ
ma-30	4	21	spaces	space	NOUN
ma-30	4	22	zhijian	zhijian	PROPN
ma-30	4	23	yang	yang	PROPN
ma-30	4	24	,	,	PUNCT
ma-30	4	25	qi	qi	PROPN
ma-30	4	26	liu	liu	PROPN
ma-30	4	27	,	,	PUNCT
ma-30	4	28	muhammad	muhammad	PROPN
ma-30	4	29	sarfraz	sarfraz	PROPN
ma-30	4	30	,	,	PUNCT
ma-30	4	31	yongjin	yongjin	PROPN
ma-30	4	32	li∗	li∗	PROPN
ma-30	4	33	department	department	PROPN
ma-30	4	34	of	of	ADP
ma-30	4	35	mathematics	mathematics	PROPN
ma-30	4	36	,	,	PUNCT
ma-30	4	37	sun	sun	PROPN
ma-30	4	38	yat	yat	PROPN
ma-30	4	39	-	-	PUNCT
ma-30	4	40	sen	sen	PROPN
ma-30	4	41	university	university	PROPN
ma-30	4	42	,	,	PUNCT
ma-30	4	43	guangzhou	guangzhou	PROPN
ma-30	4	44	,	,	PUNCT
ma-30	4	45	510275	510275	NUM
ma-30	4	46	,	,	PUNCT
ma-30	4	47	p.	p.	PROPN
ma-30	4	48	r.	r.	PROPN
ma-30	4	49	china	china	PROPN
ma-30	4	50	yangzhj55@mail2.sysu.edu.cn	yangzhj55@mail2.sysu.edu.cn	PROPN
ma-30	4	51	,	,	PUNCT
ma-30	4	52	liuq325@mail2.sysu.edu.cn	liuq325@mail2.sysu.edu.cn	PROPN
ma-30	4	53	,	,	PUNCT
ma-30	4	54	sarfraz@mail2.sysu.edu.cn	sarfraz@mail2.sysu.edu.cn	NOUN
ma-30	4	55	,	,	PUNCT
ma-30	4	56	stslyj@mail.sysu.edu.cn	stslyj@mail.sysu.edu.cn	NOUN
ma-30	4	57	∗correspondence	∗correspondence	NOUN
ma-30	4	58	:	:	PUNCT
ma-30	4	59	stslyj@mail.sysu.edu.cn	stslyj@mail.sysu.edu.cn	NOUN
ma-30	4	60	abstract	abstract	NOUN
ma-30	4	61	.	.	PUNCT
ma-30	5	1	in	in	ADP
ma-30	5	2	this	this	DET
ma-30	5	3	paper	paper	NOUN
ma-30	5	4	,	,	PUNCT
ma-30	5	5	we	we	PRON
ma-30	5	6	generalize	generalize	VERB
ma-30	5	7	the	the	DET
ma-30	5	8	typical	typical	ADJ
ma-30	5	9	geometric	geometric	ADJ
ma-30	5	10	constants	constant	NOUN
ma-30	5	11	of	of	ADP
ma-30	5	12	banach	banach	NOUN
ma-30	5	13	spaces	space	NOUN
ma-30	5	14	to	to	ADP
ma-30	5	15	modularspaces	modularspace	NOUN
ma-30	5	16	.	.	PUNCT
ma-30	6	1	we	we	PRON
ma-30	6	2	study	study	VERB
ma-30	6	3	the	the	DET
ma-30	6	4	equivalence	equivalence	NOUN
ma-30	6	5	between	between	ADP
ma-30	6	6	the	the	DET
ma-30	6	7	convexity	convexity	NOUN
ma-30	6	8	of	of	ADP
ma-30	6	9	modular	modular	ADJ
ma-30	6	10	and	and	CCONJ
ma-30	6	11	normed	normed	ADJ
ma-30	6	12	spaces	space	NOUN
ma-30	6	13	,	,	PUNCT
ma-30	6	14	and	and	CCONJ
ma-30	6	15	obtainthe	obtainthe	NOUN
ma-30	6	16	relationship	relationship	NOUN
ma-30	6	17	between	between	ADP
ma-30	6	18	ρ	ρ	PROPN
ma-30	6	19	-	-	PUNCT
ma-30	6	20	neumann	neumann	PROPN
ma-30	6	21	-	-	PUNCT
ma-30	6	22	jordan	jordan	PROPN
ma-30	6	23	constant	constant	PROPN
ma-30	6	24	and	and	CCONJ
ma-30	6	25	ρ	ρ	PROPN
ma-30	6	26	-	-	PUNCT
ma-30	6	27	james	james	PROPN
ma-30	6	28	constant	constant	PROPN
ma-30	6	29	.	.	PUNCT
ma-30	7	1	in	in	ADP
ma-30	7	2	particular	particular	ADJ
ma-30	7	3	,	,	PUNCT
ma-30	7	4	we	we	PRON
ma-30	7	5	extendthe	extendthe	VERB
ma-30	7	6	convexity	convexity	NOUN
ma-30	7	7	and	and	CCONJ
ma-30	7	8	smoothness	smoothness	ADJ
ma-30	7	9	modular	modular	NOUN
ma-30	7	10	,	,	PUNCT
ma-30	7	11	and	and	CCONJ
ma-30	7	12	obtain	obtain	VERB
ma-30	7	13	the	the	DET
ma-30	7	14	criterion	criterion	NOUN
ma-30	7	15	theorems	theorem	NOUN
ma-30	7	16	of	of	ADP
ma-30	7	17	the	the	DET
ma-30	7	18	uniform	uniform	ADJ
ma-30	7	19	convexity	convexity	PROPN
ma-30	7	20	andstrict	andstrict	NOUN
ma-30	7	21	convexity	convexity	NOUN
ma-30	7	22	.	.	PUNCT
ma-30	8	1	1	1	X
ma-30	8	2	.	.	X
ma-30	8	3	introduction	introduction	NOUN
ma-30	8	4	in	in	ADP
ma-30	8	5	the	the	DET
ma-30	8	6	recent	recent	ADJ
ma-30	8	7	years	year	NOUN
ma-30	8	8	,	,	PUNCT
ma-30	8	9	the	the	DET
ma-30	8	10	geometric	geometric	ADJ
ma-30	8	11	theory	theory	NOUN
ma-30	8	12	of	of	ADP
ma-30	8	13	banach	banach	NOUN
ma-30	8	14	spaces	space	NOUN
ma-30	8	15	has	have	AUX
ma-30	8	16	been	be	AUX
ma-30	8	17	fully	fully	ADV
ma-30	8	18	developed	develop	VERB
ma-30	8	19	,	,	PUNCT
ma-30	8	20	especiallythe	especiallythe	DET
ma-30	8	21	geometric	geometric	ADJ
ma-30	8	22	constant	constant	NOUN
ma-30	8	23	,	,	PUNCT
ma-30	8	24	which	which	PRON
ma-30	8	25	is	be	AUX
ma-30	8	26	a	a	DET
ma-30	8	27	powerful	powerful	ADJ
ma-30	8	28	tool	tool	NOUN
ma-30	8	29	to	to	PART
ma-30	8	30	characterize	characterize	VERB
ma-30	8	31	the	the	DET
ma-30	8	32	geometric	geometric	ADJ
ma-30	8	33	properties	property	NOUN
ma-30	8	34	of	of	ADP
ma-30	8	35	thespace	thespace	NOUN
ma-30	8	36	sphere	sphere	ADV
ma-30	8	37	.	.	PUNCT
ma-30	9	1	as	as	ADV
ma-30	9	2	early	early	ADV
ma-30	9	3	as	as	ADP
ma-30	9	4	1936	1936	NUM
ma-30	9	5	,	,	PUNCT
ma-30	9	6	clarkson	clarkson	PROPN
ma-30	9	7	introduced	introduce	VERB
ma-30	9	8	the	the	DET
ma-30	9	9	convexity	convexity	NOUN
ma-30	9	10	modular	modular	NOUN
ma-30	9	11	of	of	ADP
ma-30	9	12	space	space	NOUN
ma-30	9	13	[	[	X
ma-30	9	14	1	1	NUM
ma-30	9	15	]	]	PUNCT
ma-30	9	16	.	.	PUNCT
ma-30	10	1	in	in	ADP
ma-30	10	2	1963,lindenstrauss	1963,lindenstrauss	PROPN
ma-30	10	3	introduced	introduce	VERB
ma-30	10	4	the	the	DET
ma-30	10	5	smoothness	smoothness	NOUN
ma-30	10	6	modular	modular	NOUN
ma-30	10	7	,	,	PUNCT
ma-30	10	8	and	and	CCONJ
ma-30	10	9	obtained	obtain	VERB
ma-30	10	10	the	the	DET
ma-30	10	11	close	close	ADJ
ma-30	10	12	relationship	relationship	NOUN
ma-30	10	13	betweenthe	betweenthe	PRON
ma-30	10	14	two	two	NUM
ma-30	10	15	constants	constant	NOUN
ma-30	10	16	[	[	X
ma-30	10	17	2	2	NUM
ma-30	10	18	]	]	PUNCT
ma-30	10	19	.	.	PUNCT
ma-30	11	1	in	in	ADP
ma-30	11	2	1937	1937	NUM
ma-30	11	3	,	,	PUNCT
ma-30	11	4	in	in	ADP
ma-30	11	5	order	order	NOUN
ma-30	11	6	to	to	PART
ma-30	11	7	better	well	ADV
ma-30	11	8	characterize	characterize	VERB
ma-30	11	9	jordan	jordan	PROPN
ma-30	11	10	and	and	CCONJ
ma-30	11	11	von	von	PROPN
ma-30	11	12	-	-	PUNCT
ma-30	11	13	nuemann	nuemann	PROPN
ma-30	11	14	’s	’s	PART
ma-30	11	15	famouswork	famouswork	NOUN
ma-30	11	16	in	in	ADP
ma-30	11	17	inner	inner	ADJ
ma-30	11	18	product	product	NOUN
ma-30	11	19	spaces	space	NOUN
ma-30	11	20	,	,	PUNCT
ma-30	11	21	clarkson	clarkson	PROPN
ma-30	11	22	defines	define	VERB
ma-30	11	23	the	the	DET
ma-30	11	24	von	von	PROPN
ma-30	11	25	-	-	PUNCT
ma-30	11	26	nuemann	nuemann	PROPN
ma-30	11	27	constant	constant	ADJ
ma-30	12	1	[	[	X
ma-30	12	2	3	3	NUM
ma-30	12	3	]	]	PUNCT
ma-30	12	4	which	which	PRON
ma-30	12	5	is	be	AUX
ma-30	12	6	the	the	DET
ma-30	12	7	minimumconstant	minimumconstant	PROPN
ma-30	12	8	c	c	PROPN
ma-30	12	9	for	for	ADP
ma-30	12	10	all	all	DET
ma-30	12	11	x	x	NOUN
ma-30	12	12	,	,	PUNCT
ma-30	12	13	y	y	PROPN
ma-30	12	14	∈	∈	PROPN
ma-30	12	15	x	x	X
ma-30	12	16	and	and	CCONJ
ma-30	12	17	(	(	PUNCT
ma-30	12	18	x	x	NOUN
ma-30	12	19	,	,	PUNCT
ma-30	12	20	y	y	PROPN
ma-30	12	21	)	)	PUNCT
ma-30	12	22	6=	6=	ADP
ma-30	12	23	(	(	PUNCT
ma-30	12	24	0	0	NUM
ma-30	12	25	,	,	PUNCT
ma-30	12	26	0	0	NUM
ma-30	12	27	)	)	PUNCT
ma-30	12	28	of	of	ADP
ma-30	12	29	the	the	DET
ma-30	12	30	following	follow	VERB
ma-30	12	31	equations	equation	NOUN
ma-30	12	32	:	:	PUNCT
ma-30	12	33	1	1	NUM
ma-30	12	34	c	c	NOUN
ma-30	12	35	≤	≤	NUM
ma-30	12	36	‖x	‖x	PUNCT
ma-30	13	1	+	+	PUNCT
ma-30	13	2	y‖2	y‖2	X
ma-30	13	3	+	+	CCONJ
ma-30	13	4	‖x	‖x	NOUN
ma-30	13	5	−	−	PROPN
ma-30	14	1	y‖2	y‖2	PROPN
ma-30	14	2	2(‖x‖2	2(‖x‖2	NUM
ma-30	14	3	+	+	CCONJ
ma-30	14	4	‖y‖2	‖y‖2	X
ma-30	14	5	)	)	PUNCT
ma-30	14	6	≤	≤	NOUN
ma-30	14	7	c.	c.	NOUN
ma-30	14	8	in	in	ADP
ma-30	14	9	1964	1964	NUM
ma-30	14	10	,	,	PUNCT
ma-30	14	11	james	james	PROPN
ma-30	14	12	introduced	introduce	VERB
ma-30	14	13	james	james	PROPN
ma-30	14	14	constant	constant	ADJ
ma-30	15	1	[	[	X
ma-30	15	2	4	4	NUM
ma-30	15	3	]	]	PUNCT
ma-30	15	4	in	in	ADP
ma-30	15	5	order	order	NOUN
ma-30	15	6	to	to	PART
ma-30	15	7	study	study	VERB
ma-30	15	8	the	the	DET
ma-30	15	9	normal	normal	ADJ
ma-30	15	10	structure	structure	NOUN
ma-30	15	11	of	of	ADP
ma-30	15	12	space.after	space.aft	ADJ
ma-30	15	13	the	the	DET
ma-30	15	14	appearance	appearance	NOUN
ma-30	15	15	of	of	ADP
ma-30	15	16	these	these	DET
ma-30	15	17	constants	constant	NOUN
ma-30	15	18	,	,	PUNCT
ma-30	15	19	many	many	ADJ
ma-30	15	20	scholars	scholar	NOUN
ma-30	15	21	paid	pay	VERB
ma-30	15	22	attention	attention	NOUN
ma-30	15	23	to	to	ADP
ma-30	15	24	them	they	PRON
ma-30	15	25	and	and	CCONJ
ma-30	15	26	obtained	obtain	VERB
ma-30	15	27	manywonderful	manywonderful	ADJ
ma-30	15	28	properties	property	NOUN
ma-30	15	29	[	[	X
ma-30	15	30	5].modular	5].modular	NUM
ma-30	15	31	space	space	NOUN
ma-30	15	32	problems	problem	NOUN
ma-30	15	33	have	have	AUX
ma-30	15	34	been	be	AUX
ma-30	15	35	considered	consider	VERB
ma-30	15	36	by	by	ADP
ma-30	15	37	h.	h.	PROPN
ma-30	15	38	nakano	nakano	PROPN
ma-30	15	39	,	,	PUNCT
ma-30	15	40	musielak	musielak	NOUN
ma-30	15	41	and	and	CCONJ
ma-30	15	42	orlicz	orlicz	NOUN
ma-30	16	1	[	[	X
ma-30	16	2	6	6	X
ma-30	16	3	]	]	PUNCT
ma-30	16	4	underthe	underthe	PRON
ma-30	16	5	additional	additional	ADJ
ma-30	16	6	hypothesis	hypothesis	NOUN
ma-30	16	7	of	of	ADP
ma-30	16	8	convexity	convexity	NOUN
ma-30	16	9	or	or	CCONJ
ma-30	16	10	subadditivity	subadditivity	NOUN
ma-30	16	11	of	of	ADP
ma-30	16	12	the	the	DET
ma-30	16	13	modular	modular	ADJ
ma-30	16	14	ρ	ρ	NOUN
ma-30	16	15	:	:	PUNCT
ma-30	16	16	x	x	X
ma-30	17	1	→	→	PUNCT
ma-30	17	2	[	[	X
ma-30	17	3	0,+∞	0,+∞	NUM
ma-30	17	4	)	)	PUNCT
ma-30	17	5	.	.	PUNCT
ma-30	18	1	moreoverthe	moreoverthe	PRON
ma-30	18	2	case	case	NOUN
ma-30	18	3	of	of	ADP
ma-30	18	4	semi	semi	ADJ
ma-30	18	5	-	-	ADJ
ma-30	18	6	ordered	ordered	ADJ
ma-30	18	7	linear	linear	NOUN
ma-30	18	8	spaces	space	NOUN
ma-30	18	9	and	and	CCONJ
ma-30	18	10	that	that	SCONJ
ma-30	18	11	of	of	ADP
ma-30	18	12	b	b	NOUN
ma-30	18	13	-	-	PUNCT
ma-30	18	14	norms	norm	NOUN
ma-30	18	15	have	have	AUX
ma-30	18	16	been	be	AUX
ma-30	18	17	chiefly	chiefly	ADV
ma-30	18	18	investigated	investigate	VERB
ma-30	18	19	.	.	PUNCT
ma-30	19	1	under	under	ADP
ma-30	19	2	received	receive	VERB
ma-30	19	3	:	:	PUNCT
ma-30	19	4	13	13	NUM
ma-30	19	5	sep	sep	NOUN
ma-30	19	6	2021	2021	NUM
ma-30	19	7	.	.	PUNCT
ma-30	20	1	key	key	ADJ
ma-30	20	2	words	word	NOUN
ma-30	20	3	and	and	CCONJ
ma-30	20	4	phrases	phrase	NOUN
ma-30	20	5	.	.	PUNCT
ma-30	21	1	banach	banach	NOUN
ma-30	21	2	spaces	space	NOUN
ma-30	21	3	;	;	PUNCT
ma-30	21	4	geometric	geometric	ADJ
ma-30	21	5	constants	constant	NOUN
ma-30	21	6	;	;	PUNCT
ma-30	21	7	modular	modular	ADJ
ma-30	21	8	spaces.151	spaces.151	NOUN
ma-30	21	9	https://adac.ee	https://adac.ee	PROPN
ma-30	21	10	https://doi.org/10.28924/ada/ma.1.151	https://doi.org/10.28924/ada/ma.1.151	PROPN
ma-30	21	11	eur	eur	PROPN
ma-30	21	12	.	.	PUNCT
ma-30	22	1	j.	j.	PROPN
ma-30	22	2	math	math	PROPN
ma-30	22	3	.	.	PUNCT
ma-30	23	1	anal	anal	ADJ
ma-30	23	2	.	.	PUNCT
ma-30	24	1	1	1	NUM
ma-30	24	2	(	(	PUNCT
ma-30	24	3	2021	2021	NUM
ma-30	24	4	)	)	PUNCT
ma-30	25	1	152weaker	152weaker	PROPN
ma-30	25	2	assumptions	assumption	NOUN
ma-30	25	3	,	,	PUNCT
ma-30	25	4	they	they	PRON
ma-30	25	5	investigated	investigate	VERB
ma-30	25	6	the	the	DET
ma-30	25	7	structure	structure	NOUN
ma-30	25	8	of	of	ADP
ma-30	25	9	the	the	DET
ma-30	25	10	spaces	space	NOUN
ma-30	25	11	under	under	ADP
ma-30	25	12	consideration	consideration	NOUN
ma-30	25	13	.	.	PUNCT
ma-30	26	1	neitherconvexity	neitherconvexity	NOUN
ma-30	26	2	nor	nor	CCONJ
ma-30	26	3	subadditivity	subadditivity	NOUN
ma-30	26	4	of	of	ADP
ma-30	26	5	the	the	DET
ma-30	26	6	modular	modular	NOUN
ma-30	26	7	be	be	AUX
ma-30	26	8	assumed	assume	VERB
ma-30	26	9	.	.	PUNCT
ma-30	27	1	in	in	ADP
ma-30	27	2	introducing	introduce	VERB
ma-30	27	3	the	the	DET
ma-30	27	4	norm	norm	NOUN
ma-30	27	5	,	,	PUNCT
ma-30	27	6	a	a	DET
ma-30	27	7	certain	certain	ADJ
ma-30	27	8	naturalconnection	naturalconnection	NOUN
ma-30	27	9	between	between	ADP
ma-30	27	10	the	the	DET
ma-30	27	11	modular	modular	NOUN
ma-30	27	12	and	and	CCONJ
ma-30	27	13	the	the	DET
ma-30	27	14	norm	norm	NOUN
ma-30	27	15	convergence	convergence	NOUN
ma-30	27	16	will	will	AUX
ma-30	27	17	be	be	AUX
ma-30	27	18	required	require	VERB
ma-30	27	19	:	:	PUNCT
ma-30	27	20	norm	norm	NOUN
ma-30	27	21	convergenceshould	convergenceshould	AUX
ma-30	27	22	imply	imply	VERB
ma-30	27	23	modular	modular	ADJ
ma-30	27	24	convergence.through	convergence.through	ADV
ma-30	27	25	their	their	PRON
ma-30	27	26	researches	research	NOUN
ma-30	27	27	,	,	PUNCT
ma-30	27	28	they	they	PRON
ma-30	27	29	found	find	VERB
ma-30	27	30	that	that	SCONJ
ma-30	27	31	although	although	SCONJ
ma-30	27	32	modular	modular	ADJ
ma-30	27	33	spaces	space	NOUN
ma-30	27	34	are	be	AUX
ma-30	27	35	not	not	PART
ma-30	27	36	generally	generally	ADV
ma-30	27	37	normedspaces	normedspace	NOUN
ma-30	27	38	,	,	PUNCT
ma-30	27	39	they	they	PRON
ma-30	27	40	still	still	ADV
ma-30	27	41	have	have	VERB
ma-30	27	42	many	many	ADJ
ma-30	27	43	wonderful	wonderful	ADJ
ma-30	27	44	properties	property	NOUN
ma-30	27	45	,	,	PUNCT
ma-30	27	46	such	such	ADJ
ma-30	27	47	as	as	ADP
ma-30	27	48	convergence	convergence	NOUN
ma-30	27	49	,	,	PUNCT
ma-30	27	50	completeness	completeness	NOUN
ma-30	27	51	,	,	PUNCT
ma-30	27	52	convexity	convexity	NOUN
ma-30	27	53	andadditivity	andadditivity	NOUN
ma-30	27	54	.	.	PUNCT
ma-30	28	1	in	in	ADP
ma-30	28	2	view	view	NOUN
ma-30	28	3	of	of	ADP
ma-30	28	4	these	these	DET
ma-30	28	5	properties	property	NOUN
ma-30	28	6	,	,	PUNCT
ma-30	28	7	poom	poom	NOUN
ma-30	28	8	kumam	kumam	NOUN
ma-30	28	9	extended	extend	VERB
ma-30	28	10	jordan	jordan	PROPN
ma-30	28	11	von	von	PROPN
ma-30	28	12	-	-	PUNCT
ma-30	28	13	neumann	neumann	PROPN
ma-30	28	14	constant	constant	ADJ
ma-30	28	15	andjames	andjame	NOUN
ma-30	28	16	constant	constant	ADJ
ma-30	28	17	in	in	ADP
ma-30	28	18	banach	banach	NOUN
ma-30	28	19	spaces	space	NOUN
ma-30	28	20	to	to	ADP
ma-30	28	21	modular	modular	ADJ
ma-30	28	22	spaces	space	NOUN
ma-30	28	23	,	,	PUNCT
ma-30	28	24	and	and	CCONJ
ma-30	28	25	obtained	obtain	VERB
ma-30	28	26	uniform	uniform	ADJ
ma-30	28	27	convexity	convexity	NOUN
ma-30	28	28	and	and	CCONJ
ma-30	28	29	uniformnon	uniformnon	NOUN
ma-30	28	30	-	-	PUNCT
ma-30	28	31	squareness	squareness	NOUN
ma-30	28	32	of	of	ADP
ma-30	28	33	modular	modular	ADJ
ma-30	28	34	spaces	space	NOUN
ma-30	29	1	[	[	X
ma-30	29	2	10].in	10].in	NUM
ma-30	29	3	this	this	DET
ma-30	29	4	paper	paper	NOUN
ma-30	29	5	,	,	PUNCT
ma-30	29	6	based	base	VERB
ma-30	29	7	on	on	ADP
ma-30	29	8	the	the	DET
ma-30	29	9	idea	idea	NOUN
ma-30	29	10	of	of	ADP
ma-30	29	11	generalizing	generalize	VERB
ma-30	29	12	geometric	geometric	ADJ
ma-30	29	13	constants	constant	NOUN
ma-30	29	14	in	in	ADP
ma-30	29	15	banach	banach	NOUN
ma-30	29	16	spaces	space	NOUN
ma-30	29	17	to	to	ADP
ma-30	29	18	modularspaces	modularspace	NOUN
ma-30	29	19	,	,	PUNCT
ma-30	29	20	we	we	PRON
ma-30	29	21	generalize	generalize	VERB
ma-30	29	22	the	the	DET
ma-30	29	23	properties	property	NOUN
ma-30	29	24	of	of	ADP
ma-30	29	25	von	von	PROPN
ma-30	29	26	-	-	PUNCT
ma-30	29	27	neumann	neumann	PROPN
ma-30	29	28	constant	constant	PROPN
ma-30	29	29	and	and	CCONJ
ma-30	29	30	james	james	PROPN
ma-30	29	31	constant	constant	ADJ
ma-30	29	32	in	in	ADP
ma-30	29	33	[	[	X
ma-30	29	34	10	10	NUM
ma-30	29	35	]	]	PUNCT
ma-30	29	36	.	.	PUNCT
ma-30	30	1	bydefining	bydefine	VERB
ma-30	30	2	convexity	convexity	NOUN
ma-30	30	3	modules	module	NOUN
ma-30	30	4	and	and	CCONJ
ma-30	30	5	smoothness	smoothness	ADJ
ma-30	30	6	modular	modular	NOUN
ma-30	30	7	,	,	PUNCT
ma-30	30	8	we	we	PRON
ma-30	30	9	derive	derive	VERB
ma-30	30	10	the	the	DET
ma-30	30	11	relationships	relationship	NOUN
ma-30	30	12	between	between	ADP
ma-30	30	13	jamesconstant	jamesconstant	ADJ
ma-30	30	14	,	,	PUNCT
ma-30	30	15	convexity	convexity	NOUN
ma-30	30	16	modular	modular	NOUN
ma-30	30	17	and	and	CCONJ
ma-30	30	18	the	the	DET
ma-30	30	19	strict	strict	ADJ
ma-30	30	20	convexity	convexity	NOUN
ma-30	30	21	of	of	ADP
ma-30	30	22	modular	modular	ADJ
ma-30	30	23	spaces	space	NOUN
ma-30	30	24	.	.	PUNCT
ma-30	31	1	2	2	X
ma-30	31	2	.	.	X
ma-30	31	3	preliminaries	preliminary	NOUN
ma-30	31	4	we	we	PRON
ma-30	31	5	first	first	ADV
ma-30	31	6	give	give	VERB
ma-30	31	7	some	some	DET
ma-30	31	8	basic	basic	ADJ
ma-30	31	9	facts	fact	NOUN
ma-30	31	10	about	about	ADP
ma-30	31	11	modular	modular	ADJ
ma-30	31	12	spaces	space	NOUN
ma-30	31	13	formulated	formulate	VERB
ma-30	31	14	by	by	ADP
ma-30	31	15	musielak	musielak	NOUN
ma-30	31	16	and	and	CCONJ
ma-30	31	17	orlicz	orlicz	NOUN
ma-30	32	1	[	[	X
ma-30	32	2	6	6	NUM
ma-30	32	3	]	]	PUNCT
ma-30	32	4	.	.	PUNCT
ma-30	33	1	definition	definition	NOUN
ma-30	33	2	1.[8	1.[8	NUM
ma-30	33	3	]	]	PUNCT
ma-30	33	4	let	let	VERB
ma-30	33	5	x	x	PRON
ma-30	33	6	be	be	AUX
ma-30	33	7	a	a	DET
ma-30	33	8	vector	vector	NOUN
ma-30	33	9	space	space	NOUN
ma-30	33	10	over	over	ADP
ma-30	33	11	f	f	PROPN
ma-30	33	12	(	(	PUNCT
ma-30	33	13	r	r	NOUN
ma-30	33	14	or	or	CCONJ
ma-30	33	15	c	c	NOUN
ma-30	33	16	)	)	PUNCT
ma-30	33	17	.	.	PUNCT
ma-30	34	1	then	then	ADV
ma-30	34	2	a	a	DET
ma-30	34	3	function	function	NOUN
ma-30	34	4	ρ	ρ	NOUN
ma-30	34	5	:	:	PUNCT
ma-30	34	6	x	x	X
ma-30	34	7	→	→	PUNCT
ma-30	34	8	[	[	X
ma-30	34	9	0,∞	0,∞	X
ma-30	34	10	]	]	PUNCT
ma-30	34	11	is	be	AUX
ma-30	34	12	calleda	calleda	NOUN
ma-30	34	13	modular	modular	ADJ
ma-30	34	14	on	on	ADP
ma-30	34	15	x	x	SYM
ma-30	34	16	if	if	SCONJ
ma-30	34	17	for	for	ADP
ma-30	34	18	arbitrary	arbitrary	ADJ
ma-30	34	19	x	x	NOUN
ma-30	34	20	,	,	PUNCT
ma-30	34	21	y	y	PROPN
ma-30	34	22	in	in	ADP
ma-30	34	23	x	x	X
ma-30	34	24	,	,	PUNCT
ma-30	34	25	(	(	PUNCT
ma-30	34	26	i	i	NOUN
ma-30	34	27	)	)	PUNCT
ma-30	34	28	ρ(x	ρ(x	PROPN
ma-30	34	29	)	)	PUNCT
ma-30	35	1	=	=	SYM
ma-30	35	2	0	0	PUNCT
ma-30	36	1	if	if	SCONJ
ma-30	36	2	and	and	CCONJ
ma-30	36	3	only	only	ADV
ma-30	36	4	if	if	SCONJ
ma-30	36	5	x	x	X
ma-30	36	6	=	=	SYM
ma-30	36	7	0,(ii	0,(ii	ADJ
ma-30	36	8	)	)	PUNCT
ma-30	36	9	ρ(αx	ρ(αx	NUM
ma-30	36	10	)	)	PUNCT
ma-30	36	11	=	=	SYM
ma-30	36	12	ρ(x	ρ(x	PROPN
ma-30	36	13	)	)	PUNCT
ma-30	36	14	for	for	ADP
ma-30	36	15	every	every	DET
ma-30	36	16	scalar	scalar	ADJ
ma-30	36	17	α	α	NOUN
ma-30	36	18	with	with	ADP
ma-30	36	19	|α|	|α|	PROPN
ma-30	36	20	=	=	SYM
ma-30	36	21	1,(iii	1,(iii	NUM
ma-30	36	22	)	)	PUNCT
ma-30	36	23	ρ(αx	ρ(αx	NOUN
ma-30	36	24	+	+	NUM
ma-30	36	25	βy	βy	ADJ
ma-30	36	26	)	)	PUNCT
ma-30	36	27	≤	≤	NOUN
ma-30	36	28	ρ(x	ρ(x	NOUN
ma-30	36	29	)	)	PUNCT
ma-30	37	1	+	+	CCONJ
ma-30	37	2	ρ(y	ρ(y	NOUN
ma-30	37	3	)	)	PUNCT
ma-30	37	4	if	if	SCONJ
ma-30	37	5	α+	α+	DET
ma-30	37	6	β	β	X
ma-30	37	7	=	=	SYM
ma-30	37	8	1	1	NUM
ma-30	37	9	and	and	CCONJ
ma-30	37	10	α	α	NOUN
ma-30	37	11	,	,	PUNCT
ma-30	37	12	β	β	X
ma-30	37	13	≥	≥	NUM
ma-30	37	14	0.if	0.if	NUM
ma-30	37	15	(	(	PUNCT
ma-30	37	16	iii	iii	NOUN
ma-30	37	17	)	)	PUNCT
ma-30	37	18	is	be	AUX
ma-30	37	19	replaced	replace	VERB
ma-30	37	20	by	by	ADP
ma-30	37	21	(	(	PUNCT
ma-30	37	22	iv	iv	NUM
ma-30	37	23	):	):	PUNCT
ma-30	37	24	ρ(αx	ρ(αx	NUM
ma-30	37	25	+	+	NOUN
ma-30	37	26	βy	βy	ADJ
ma-30	37	27	)	)	PUNCT
ma-30	37	28	≤	≤	NOUN
ma-30	37	29	αρ(x	αρ(x	PUNCT
ma-30	37	30	)	)	PUNCT
ma-30	38	1	+	+	CCONJ
ma-30	38	2	βρ(y	βρ(y	X
ma-30	38	3	)	)	PUNCT
ma-30	38	4	if	if	SCONJ
ma-30	38	5	α	α	X
ma-30	38	6	,	,	PUNCT
ma-30	38	7	β	β	X
ma-30	38	8	≥	≥	NOUN
ma-30	38	9	0	0	NUM
ma-30	38	10	and	and	CCONJ
ma-30	38	11	α+β	α+β	NUM
ma-30	38	12	=	=	SYM
ma-30	38	13	1	1	X
ma-30	38	14	.	.	X
ma-30	39	1	we	we	PRON
ma-30	39	2	now	now	ADV
ma-30	39	3	callthat	callthat	PRON
ma-30	39	4	ρ	ρ	NOUN
ma-30	39	5	is	be	AUX
ma-30	39	6	a	a	DET
ma-30	39	7	convex	convex	NOUN
ma-30	39	8	modular.a	modular.a	NOUN
ma-30	39	9	modular	modular	ADJ
ma-30	39	10	ρ	ρ	PROPN
ma-30	39	11	can	can	AUX
ma-30	39	12	be	be	AUX
ma-30	39	13	used	use	VERB
ma-30	39	14	to	to	PART
ma-30	39	15	define	define	VERB
ma-30	39	16	a	a	DET
ma-30	39	17	corresponding	corresponding	ADJ
ma-30	39	18	modular	modular	ADJ
ma-30	39	19	space	space	NOUN
ma-30	39	20	,	,	PUNCT
ma-30	39	21	i.e	i.e	PRON
ma-30	39	22	,	,	PUNCT
ma-30	39	23	the	the	DET
ma-30	39	24	vector	vector	NOUN
ma-30	39	25	space	space	NOUN
ma-30	39	26	xρ	xρ	PROPN
ma-30	39	27	asgiven	asgiven	VERB
ma-30	39	28	by	by	ADP
ma-30	39	29	xρ	xρ	PROPN
ma-30	39	30	=	=	SYM
ma-30	39	31	{	{	PUNCT
ma-30	39	32	x	x	PUNCT
ma-30	39	33	∈	∈	PROPN
ma-30	39	34	x	x	X
ma-30	39	35	:	:	PUNCT
ma-30	39	36	ρ(λx)→	ρ(λx)→	PROPN
ma-30	39	37	0	0	PUNCT
ma-30	39	38	as	as	ADP
ma-30	39	39	λ→	λ→	PROPN
ma-30	39	40	0	0	NUM
ma-30	39	41	}	}	PUNCT
ma-30	39	42	,	,	PUNCT
ma-30	39	43	where	where	SCONJ
ma-30	39	44	xρ	xρ	PROPN
ma-30	39	45	is	be	AUX
ma-30	39	46	a	a	DET
ma-30	39	47	linear	linear	ADJ
ma-30	39	48	subspace	subspace	NOUN
ma-30	39	49	of	of	ADP
ma-30	39	50	x	x	PROPN
ma-30	39	51	.in	.in	X
ma-30	39	52	general	general	ADJ
ma-30	39	53	,	,	PUNCT
ma-30	39	54	the	the	DET
ma-30	39	55	modular	modular	ADJ
ma-30	39	56	ρ	ρ	NOUN
ma-30	39	57	is	be	AUX
ma-30	39	58	not	not	PART
ma-30	39	59	necessarily	necessarily	ADV
ma-30	39	60	subadditive	subadditive	ADJ
ma-30	39	61	and	and	CCONJ
ma-30	39	62	therefore	therefore	ADV
ma-30	39	63	it	it	PRON
ma-30	39	64	does	do	AUX
ma-30	39	65	not	not	PART
ma-30	39	66	behave	behave	VERB
ma-30	39	67	as	as	ADP
ma-30	39	68	anorm	anorm	NOUN
ma-30	39	69	or	or	CCONJ
ma-30	39	70	a	a	DET
ma-30	39	71	distance	distance	NOUN
ma-30	39	72	.	.	PUNCT
ma-30	40	1	but	but	CCONJ
ma-30	40	2	we	we	PRON
ma-30	40	3	can	can	AUX
ma-30	40	4	associate	associate	VERB
ma-30	40	5	it	it	PRON
ma-30	40	6	to	to	ADP
ma-30	40	7	a	a	DET
ma-30	40	8	modular	modular	ADJ
ma-30	40	9	f	f	NOUN
ma-30	40	10	-norm.the	-norm.the	DET
ma-30	40	11	modular	modular	ADJ
ma-30	40	12	space	space	NOUN
ma-30	40	13	xρ	xρ	PROPN
ma-30	40	14	can	can	AUX
ma-30	40	15	be	be	AUX
ma-30	40	16	equipped	equip	VERB
ma-30	40	17	with	with	ADP
ma-30	40	18	a	a	DET
ma-30	40	19	f	f	PROPN
ma-30	40	20	-norm	-norm	NOUN
ma-30	40	21	defined	define	VERB
ma-30	40	22	by	by	ADP
ma-30	40	23	‖x‖ρ	‖x‖ρ	NOUN
ma-30	40	24	=	=	SYM
ma-30	40	25	inf	inf	NOUN
ma-30	40	26	{	{	PUNCT
ma-30	40	27	α	α	NOUN
ma-30	40	28	>	>	X
ma-30	40	29	0	0	NUM
ma-30	40	30	;	;	PUNCT
ma-30	40	31	ρ	ρ	PROPN
ma-30	40	32	(	(	PUNCT
ma-30	40	33	x	x	PART
ma-30	40	34	λ	λ	PROPN
ma-30	40	35	)	)	PUNCT
ma-30	40	36	≤	≤	NOUN
ma-30	40	37	α	α	PROPN
ma-30	40	38	}	}	PUNCT
ma-30	40	39	,	,	PUNCT
ma-30	40	40	when	when	SCONJ
ma-30	40	41	ρ	ρ	PROPN
ma-30	40	42	is	be	AUX
ma-30	40	43	convex	convex	PROPN
ma-30	40	44	.	.	PUNCT
ma-30	41	1	then	then	ADV
ma-30	41	2	norm	norm	VERB
ma-30	41	3	‖	‖	PROPN
ma-30	41	4	·	·	PUNCT
ma-30	41	5	‖ρ	‖ρ	NOUN
ma-30	41	6	is	be	AUX
ma-30	41	7	frequently	frequently	ADV
ma-30	41	8	called	call	VERB
ma-30	41	9	the	the	DET
ma-30	41	10	luxemburg	luxemburg	PROPN
ma-30	41	11	norm	norm	NOUN
ma-30	41	12	.	.	PUNCT
ma-30	42	1	if	if	SCONJ
ma-30	42	2	ρ	ρ	PROPN
ma-30	42	3	is	be	AUX
ma-30	42	4	convex	convex	NOUN
ma-30	42	5	,	,	PUNCT
ma-30	42	6	thenthe	thenthe	ADJ
ma-30	42	7	functional	functional	ADJ
ma-30	42	8	‖x‖ρ	‖x‖ρ	NOUN
ma-30	42	9	=	=	SYM
ma-30	42	10	inf	inf	NOUN
ma-30	42	11	{	{	PUNCT
ma-30	42	12	α	α	X
ma-30	42	13	>	>	X
ma-30	42	14	0	0	NUM
ma-30	42	15	;	;	PUNCT
ma-30	42	16	ρ	ρ	PROPN
ma-30	42	17	(	(	PUNCT
ma-30	42	18	xλ	xλ	NOUN
ma-30	42	19	)	)	PUNCT
ma-30	42	20	≤	≤	NUM
ma-30	42	21	1	1	NUM
ma-30	42	22	}	}	PUNCT
ma-30	42	23	is	be	AUX
ma-30	42	24	a	a	DET
ma-30	42	25	norm	norm	NOUN
ma-30	42	26	in	in	ADP
ma-30	42	27	xρ	xρ	PROPN
ma-30	42	28	which	which	PRON
ma-30	42	29	is	be	AUX
ma-30	42	30	equivalent	equivalent	ADJ
ma-30	42	31	to	to	ADP
ma-30	42	32	the	the	DET
ma-30	42	33	f	f	PROPN
ma-30	42	34	-norm	-norm	PROPN
ma-30	42	35	‖	‖	PROPN
ma-30	42	36	·	·	PUNCT
ma-30	42	37	‖ρ	‖ρ	PROPN
ma-30	42	38	.	.	PUNCT
ma-30	43	1	eur	eur	PROPN
ma-30	43	2	.	.	PUNCT
ma-30	44	1	j.	j.	PROPN
ma-30	44	2	math	math	PROPN
ma-30	44	3	.	.	PUNCT
ma-30	45	1	anal	anal	ADJ
ma-30	45	2	.	.	PUNCT
ma-30	46	1	1	1	NUM
ma-30	46	2	(	(	PUNCT
ma-30	46	3	2021	2021	NUM
ma-30	46	4	)	)	PUNCT
ma-30	46	5	153	153	NUM
ma-30	46	6	proposition	proposition	NOUN
ma-30	46	7	1	1	NUM
ma-30	46	8	.	.	PUNCT
ma-30	47	1	let	let	VERB
ma-30	47	2	xρ	xρ	PROPN
ma-30	47	3	be	be	AUX
ma-30	47	4	a	a	DET
ma-30	47	5	modular	modular	ADJ
ma-30	47	6	space	space	NOUN
ma-30	47	7	.	.	PUNCT
ma-30	48	1	then	then	ADV
ma-30	48	2	ρ	ρ	PROPN
ma-30	48	3	is	be	AUX
ma-30	48	4	convex	convex	ADJ
ma-30	48	5	if	if	SCONJ
ma-30	48	6	and	and	CCONJ
ma-30	48	7	only	only	ADV
ma-30	48	8	if	if	SCONJ
ma-30	48	9	xρ	xρ	PROPN
ma-30	48	10	is	be	AUX
ma-30	48	11	a	a	DET
ma-30	48	12	normed	norme	VERB
ma-30	48	13	spacewith	spacewith	PROPN
ma-30	48	14	ρ	ρ	PROPN
ma-30	48	15	as	as	ADP
ma-30	48	16	norm	norm	NOUN
ma-30	48	17	.	.	PUNCT
ma-30	49	1	proof	proof	NOUN
ma-30	49	2	.	.	PUNCT
ma-30	50	1	the	the	DET
ma-30	50	2	proof	proof	NOUN
ma-30	50	3	of	of	ADP
ma-30	50	4	sufficiency	sufficiency	NOUN
ma-30	50	5	is	be	AUX
ma-30	50	6	obvious.conversely	obvious.conversely	ADV
ma-30	50	7	,	,	PUNCT
ma-30	50	8	assume	assume	VERB
ma-30	50	9	ρ	ρ	PROPN
ma-30	50	10	is	be	AUX
ma-30	50	11	convex	convex	PROPN
ma-30	50	12	,	,	PUNCT
ma-30	50	13	then	then	ADV
ma-30	50	14	we	we	PRON
ma-30	50	15	can	can	AUX
ma-30	50	16	obtain	obtain	VERB
ma-30	50	17	ρ(x	ρ(x	NOUN
ma-30	50	18	)	)	PUNCT
ma-30	51	1	=	=	SYM
ma-30	51	2	0	0	PUNCT
ma-30	52	1	if	if	SCONJ
ma-30	52	2	and	and	CCONJ
ma-30	52	3	only	only	ADV
ma-30	52	4	if	if	SCONJ
ma-30	52	5	x	x	X
ma-30	52	6	=	=	SYM
ma-30	52	7	0.(i	0.(i	NUM
ma-30	52	8	)	)	PUNCT
ma-30	52	9	according	accord	VERB
ma-30	52	10	to	to	ADP
ma-30	52	11	the	the	DET
ma-30	52	12	definition	definition	NOUN
ma-30	52	13	1	1	NUM
ma-30	52	14	,	,	PUNCT
ma-30	52	15	if	if	SCONJ
ma-30	52	16	α	α	PROPN
ma-30	52	17	>	>	X
ma-30	52	18	0	0	NUM
ma-30	52	19	,	,	PUNCT
ma-30	52	20	then	then	ADV
ma-30	52	21	ρ	ρ	PROPN
ma-30	52	22	(	(	PUNCT
ma-30	52	23	1	1	NUM
ma-30	52	24	α	α	NOUN
ma-30	52	25	x	x	NOUN
ma-30	52	26	)	)	PUNCT
ma-30	52	27	=	=	SYM
ma-30	52	28	ρ	ρ	PROPN
ma-30	52	29	(	(	PUNCT
ma-30	52	30	1	1	NUM
ma-30	52	31	α	α	NOUN
ma-30	52	32	x	x	PUNCT
ma-30	53	1	+	+	NUM
ma-30	53	2	1−	1−	NUM
ma-30	53	3	α	α	NOUN
ma-30	53	4	α	α	NOUN
ma-30	53	5	·	·	PUNCT
ma-30	53	6	0	0	NUM
ma-30	53	7	)	)	PUNCT
ma-30	53	8	≤	≤	NOUN
ma-30	53	9	1	1	NUM
ma-30	53	10	α	α	NOUN
ma-30	53	11	ρ(x	ρ(x	NOUN
ma-30	53	12	)	)	PUNCT
ma-30	53	13	+	+	CCONJ
ma-30	53	14	1−	1−	NUM
ma-30	53	15	α	α	NUM
ma-30	53	16	α	α	NOUN
ma-30	53	17	ρ(0	ρ(0	PROPN
ma-30	53	18	)	)	PUNCT
ma-30	53	19	=	=	SYM
ma-30	53	20	1	1	NUM
ma-30	53	21	α	α	NOUN
ma-30	53	22	ρ(x	ρ(x	NOUN
ma-30	53	23	)	)	PUNCT
ma-30	53	24	and	and	CCONJ
ma-30	53	25	αρ	αρ	PROPN
ma-30	53	26	(	(	PUNCT
ma-30	53	27	1	1	NUM
ma-30	53	28	α	α	NOUN
ma-30	53	29	x	x	NOUN
ma-30	53	30	)	)	PUNCT
ma-30	53	31	=	=	SYM
ma-30	53	32	αρ	αρ	PROPN
ma-30	53	33	(	(	PUNCT
ma-30	53	34	1	1	NUM
ma-30	53	35	α	α	NOUN
ma-30	53	36	x	x	NOUN
ma-30	53	37	)	)	PUNCT
ma-30	54	1	+	+	CCONJ
ma-30	54	2	(	(	PUNCT
ma-30	54	3	1−	1−	NUM
ma-30	54	4	α)ρ(0	α)ρ(0	NOUN
ma-30	54	5	)	)	PUNCT
ma-30	54	6	≥	≥	NOUN
ma-30	54	7	ρ(α	ρ(α	NOUN
ma-30	54	8	·	·	PUNCT
ma-30	54	9	1	1	NUM
ma-30	54	10	α	α	NOUN
ma-30	54	11	+	+	X
ma-30	54	12	(	(	PUNCT
ma-30	54	13	1−	1−	NUM
ma-30	54	14	α	α	NOUN
ma-30	54	15	)	)	PUNCT
ma-30	54	16	·	·	PUNCT
ma-30	54	17	0	0	X
ma-30	54	18	)	)	PUNCT
ma-30	54	19	=	=	SYM
ma-30	54	20	ρ(x	ρ(x	NOUN
ma-30	54	21	)	)	PUNCT
ma-30	54	22	.	.	PUNCT
ma-30	55	1	this	this	PRON
ma-30	55	2	show	show	VERB
ma-30	55	3	that	that	SCONJ
ma-30	55	4	ρ	ρ	PROPN
ma-30	55	5	(	(	PUNCT
ma-30	55	6	1αx	1αx	ADJ
ma-30	55	7	)	)	PUNCT
ma-30	55	8	≥	≥	NOUN
ma-30	55	9	1	1	NUM
ma-30	55	10	αρ(x	αρ(x	NUM
ma-30	55	11	)	)	PUNCT
ma-30	55	12	and	and	CCONJ
ma-30	55	13	hence	hence	ADV
ma-30	55	14	ρ	ρ	PROPN
ma-30	55	15	(	(	PUNCT
ma-30	55	16	1	1	NUM
ma-30	55	17	α	α	NOUN
ma-30	55	18	x	x	NOUN
ma-30	55	19	)	)	PUNCT
ma-30	55	20	=	=	SYM
ma-30	55	21	1	1	NUM
ma-30	55	22	α	α	NOUN
ma-30	55	23	ρ(x	ρ(x	NOUN
ma-30	55	24	)	)	PUNCT
ma-30	55	25	for	for	ADP
ma-30	55	26	α	α	PROPN
ma-30	55	27	>	>	X
ma-30	55	28	0.suppose	0.suppose	NUM
ma-30	55	29	α	α	PRON
ma-30	55	30	6=	6=	ADP
ma-30	55	31	0	0	NUM
ma-30	55	32	,	,	PUNCT
ma-30	55	33	then	then	ADV
ma-30	55	34	|α|	|α|	PROPN
ma-30	55	35	>	>	X
ma-30	55	36	0	0	NUM
ma-30	55	37	.	.	PUNCT
ma-30	56	1	according	accord	VERB
ma-30	56	2	to	to	ADP
ma-30	56	3	the	the	DET
ma-30	56	4	definition	definition	NOUN
ma-30	56	5	1	1	NUM
ma-30	56	6	,	,	PUNCT
ma-30	56	7	we	we	PRON
ma-30	56	8	have	have	VERB
ma-30	56	9	ρ	ρ	NOUN
ma-30	56	10	(	(	PUNCT
ma-30	56	11	|α|	|α|	PROPN
ma-30	56	12	·	·	SYM
ma-30	56	13	1	1	NUM
ma-30	56	14	|α|αx	|α|αx	PROPN
ma-30	56	15	)	)	PUNCT
ma-30	57	1	=	=	PUNCT
ma-30	57	2	|α|ρ	|α|ρ	NOUN
ma-30	57	3	(	(	PUNCT
ma-30	57	4	1	1	NUM
ma-30	57	5	|α|αx	|α|αx	PROPN
ma-30	57	6	)	)	PUNCT
ma-30	57	7	=	=	SYM
ma-30	57	8	|α|ρ(x	|α|ρ(x	PROPN
ma-30	57	9	)	)	PUNCT
ma-30	57	10	which	which	PRON
ma-30	57	11	shows	show	VERB
ma-30	57	12	that	that	SCONJ
ma-30	57	13	ρ(αx	ρ(αx	NUM
ma-30	57	14	)	)	PUNCT
ma-30	57	15	=	=	SYM
ma-30	58	1	|α|ρ	|α|ρ	NOUN
ma-30	58	2	(	(	PUNCT
ma-30	58	3	1|α|αx	1|α|αx	NUM
ma-30	58	4	)	)	PUNCT
ma-30	58	5	=	=	SYM
ma-30	58	6	|α|ρ(x).(ii	|α|ρ(x).(ii	NUM
ma-30	58	7	)	)	PUNCT
ma-30	58	8	since	since	SCONJ
ma-30	58	9	ρ(x	ρ(x	PROPN
ma-30	58	10	+	+	PROPN
ma-30	58	11	y	y	NOUN
ma-30	58	12	)	)	PUNCT
ma-30	58	13	=	=	SYM
ma-30	58	14	ρ	ρ	PROPN
ma-30	58	15	(	(	PUNCT
ma-30	58	16	2	2	NUM
ma-30	58	17	(	(	PUNCT
ma-30	58	18	x	x	SYM
ma-30	58	19	2	2	NUM
ma-30	58	20	+	+	NUM
ma-30	58	21	y	y	PROPN
ma-30	58	22	2	2	NUM
ma-30	58	23	)	)	PUNCT
ma-30	58	24	)	)	PUNCT
ma-30	59	1	=	=	SYM
ma-30	59	2	2ρ	2ρ	NOUN
ma-30	59	3	(	(	PUNCT
ma-30	59	4	x	x	SYM
ma-30	59	5	2	2	NUM
ma-30	59	6	+	+	NUM
ma-30	59	7	y	y	PROPN
ma-30	59	8	2	2	NUM
ma-30	59	9	)	)	PUNCT
ma-30	59	10	≤	≤	NOUN
ma-30	59	11	ρ(x	ρ(x	NOUN
ma-30	59	12	)	)	PUNCT
ma-30	60	1	+	+	CCONJ
ma-30	60	2	ρ(y),then	ρ(y),then	NOUN
ma-30	60	3	xρ	xρ	PROPN
ma-30	60	4	is	be	AUX
ma-30	60	5	a	a	DET
ma-30	60	6	normed	normed	ADJ
ma-30	60	7	space	space	NOUN
ma-30	60	8	with	with	ADP
ma-30	60	9	ρ	ρ	PROPN
ma-30	60	10	as	as	ADP
ma-30	60	11	norm	norm	NOUN
ma-30	60	12	.	.	PUNCT
ma-30	61	1	3	3	X
ma-30	61	2	.	.	X
ma-30	61	3	the	the	DET
ma-30	61	4	ρ	ρ	PROPN
ma-30	61	5	-	-	PUNCT
ma-30	61	6	neumann	neumann	PROPN
ma-30	61	7	–	–	PUNCT
ma-30	61	8	jordan	jordan	PROPN
ma-30	61	9	constant	constant	PROPN
ma-30	61	10	and	and	CCONJ
ma-30	61	11	the	the	DET
ma-30	61	12	ρ	ρ	PROPN
ma-30	61	13	-	-	PUNCT
ma-30	61	14	james	james	PROPN
ma-30	61	15	constant	constant	ADJ
ma-30	61	16	in	in	ADP
ma-30	61	17	2006	2006	NUM
ma-30	61	18	,	,	PUNCT
ma-30	61	19	poom	poom	NOUN
ma-30	61	20	kumam	kumam	NOUN
ma-30	62	1	[	[	X
ma-30	62	2	10	10	NUM
ma-30	62	3	]	]	X
ma-30	62	4	generalized	generalize	VERB
ma-30	62	5	two	two	NUM
ma-30	62	6	typical	typical	ADJ
ma-30	62	7	constants	constant	NOUN
ma-30	62	8	cnj(x	cnj(x	PROPN
ma-30	62	9	)	)	PUNCT
ma-30	62	10	=	=	PUNCT
ma-30	63	1	sup{‖x	sup{‖x	PROPN
ma-30	64	1	+	+	NUM
ma-30	64	2	y‖2	y‖2	X
ma-30	64	3	+	+	CCONJ
ma-30	64	4	‖x	‖x	NOUN
ma-30	64	5	−	−	PROPN
ma-30	65	1	y‖2	y‖2	X
ma-30	65	2	2‖x‖2	2‖x‖2	NOUN
ma-30	66	1	+	+	CCONJ
ma-30	66	2	2‖y‖2	2‖y‖2	NOUN
ma-30	66	3	:	:	PUNCT
ma-30	66	4	x	x	X
ma-30	66	5	,	,	PUNCT
ma-30	66	6	y	y	PROPN
ma-30	66	7	∈	∈	PROPN
ma-30	66	8	x	x	X
ma-30	66	9	,	,	PUNCT
ma-30	66	10	(	(	PUNCT
ma-30	66	11	x	x	NOUN
ma-30	66	12	,	,	PUNCT
ma-30	66	13	y	y	PROPN
ma-30	66	14	)	)	PUNCT
ma-30	66	15	6=	6=	ADP
ma-30	66	16	(	(	PUNCT
ma-30	66	17	0	0	NUM
ma-30	66	18	,	,	PUNCT
ma-30	66	19	0	0	NUM
ma-30	66	20	)	)	PUNCT
ma-30	66	21	}	}	PUNCT
ma-30	66	22	and	and	CCONJ
ma-30	66	23	j(x	j(x	PROPN
ma-30	66	24	)	)	PUNCT
ma-30	66	25	=	=	PUNCT
ma-30	67	1	sup{min{‖x	sup{min{‖x	PROPN
ma-30	67	2	+	+	NUM
ma-30	67	3	y‖	y‖	PROPN
ma-30	67	4	,	,	PUNCT
ma-30	67	5	‖x	‖x	NOUN
ma-30	68	1	−	−	PROPN
ma-30	68	2	y‖	y‖	PROPN
ma-30	68	3	}	}	PUNCT
ma-30	68	4	:	:	PUNCT
ma-30	69	1	x	x	X
ma-30	69	2	,	,	PUNCT
ma-30	69	3	y	y	PROPN
ma-30	69	4	∈	∈	PROPN
ma-30	69	5	x	x	X
ma-30	69	6	,	,	PUNCT
ma-30	69	7	‖x‖	‖x‖	PROPN
ma-30	69	8	=	=	SYM
ma-30	69	9	‖y‖	‖y‖	PROPN
ma-30	69	10	=	=	SYM
ma-30	69	11	1}and	1}and	PROPN
ma-30	69	12	introduced	introduce	VERB
ma-30	69	13	two	two	NUM
ma-30	69	14	new	new	ADJ
ma-30	69	15	geometric	geometric	ADJ
ma-30	69	16	constants	constant	NOUN
ma-30	69	17	cnj(xρ	cnj(xρ	NOUN
ma-30	69	18	)	)	PUNCT
ma-30	69	19	and	and	CCONJ
ma-30	69	20	j(xρ	j(xρ	PROPN
ma-30	69	21	)	)	PUNCT
ma-30	69	22	defined	define	VERB
ma-30	69	23	on	on	ADP
ma-30	69	24	modular	modular	ADJ
ma-30	69	25	spaces	space	NOUN
ma-30	69	26	.	.	PUNCT
ma-30	70	1	definition	definition	NOUN
ma-30	70	2	2.[10	2.[10	NUM
ma-30	70	3	]	]	PUNCT
ma-30	70	4	the	the	DET
ma-30	70	5	ρ	ρ	PROPN
ma-30	70	6	-	-	PUNCT
ma-30	70	7	neumann	neumann	PROPN
ma-30	70	8	-	-	PUNCT
ma-30	70	9	jordan	jordan	PROPN
ma-30	70	10	constant	constant	PROPN
ma-30	70	11	cnj(xρ	cnj(xρ	NOUN
ma-30	70	12	)	)	PUNCT
ma-30	70	13	of	of	ADP
ma-30	70	14	a	a	DET
ma-30	70	15	modular	modular	ADJ
ma-30	70	16	space	space	NOUN
ma-30	70	17	xρ	xρ	PROPN
ma-30	70	18	is	be	AUX
ma-30	70	19	defined	define	VERB
ma-30	70	20	by	by	ADP
ma-30	70	21	cnj(xρ	cnj(xρ	NOUN
ma-30	70	22	)	)	PUNCT
ma-30	70	23	=	=	SYM
ma-30	70	24	2	2	NUM
ma-30	70	25	sup	sup	NOUN
ma-30	70	26	{	{	PUNCT
ma-30	70	27	ρ2	ρ2	NOUN
ma-30	70	28	(	(	PUNCT
ma-30	70	29	x+y2	x+y2	PROPN
ma-30	70	30	)	)	PUNCT
ma-30	71	1	+	+	CCONJ
ma-30	72	1	ρ	ρ	PROPN
ma-30	72	2	2	2	NUM
ma-30	72	3	(	(	PUNCT
ma-30	72	4	x−y2	x−y2	PROPN
ma-30	72	5	)	)	PUNCT
ma-30	72	6	ρ2(x	ρ2(x	NUM
ma-30	72	7	)	)	PUNCT
ma-30	72	8	+	+	NUM
ma-30	72	9	ρ2(y	ρ2(y	NUM
ma-30	72	10	)	)	PUNCT
ma-30	72	11	:	:	PUNCT
ma-30	73	1	x	x	X
ma-30	73	2	,	,	PUNCT
ma-30	73	3	y	y	PROPN
ma-30	73	4	∈	∈	PROPN
ma-30	73	5	xρ	xρ	PROPN
ma-30	73	6	,	,	PUNCT
ma-30	73	7	ρ(x	ρ(x	PROPN
ma-30	73	8	)	)	PUNCT
ma-30	73	9	=	=	SYM
ma-30	73	10	1	1	NUM
ma-30	73	11	,	,	PUNCT
ma-30	73	12	ρ(y	ρ(y	NOUN
ma-30	73	13	)	)	PUNCT
ma-30	73	14	≤	≤	NOUN
ma-30	73	15	1	1	NUM
ma-30	73	16	}	}	PUNCT
ma-30	73	17	.	.	PUNCT
ma-30	74	1	definition	definition	NOUN
ma-30	74	2	3.[10	3.[10	NUM
ma-30	74	3	]	]	PUNCT
ma-30	74	4	the	the	DET
ma-30	74	5	ρ	ρ	PROPN
ma-30	74	6	-	-	PUNCT
ma-30	74	7	james	james	PROPN
ma-30	74	8	constant	constant	PROPN
ma-30	74	9	j(xρ	j(xρ	PROPN
ma-30	74	10	)	)	PUNCT
ma-30	74	11	of	of	ADP
ma-30	74	12	a	a	DET
ma-30	74	13	modular	modular	ADJ
ma-30	74	14	space	space	NOUN
ma-30	74	15	xρ	xρ	PROPN
ma-30	74	16	is	be	AUX
ma-30	74	17	defined	define	VERB
ma-30	74	18	by	by	ADP
ma-30	74	19	j(xρ	j(xρ	PROPN
ma-30	74	20	)	)	PUNCT
ma-30	74	21	=	=	SYM
ma-30	74	22	2	2	NUM
ma-30	74	23	sup	sup	NOUN
ma-30	74	24	{	{	PUNCT
ma-30	74	25	min{ρ	min{ρ	PROPN
ma-30	74	26	(	(	PUNCT
ma-30	74	27	x	x	PROPN
ma-30	74	28	+	+	NUM
ma-30	74	29	y	y	PROPN
ma-30	74	30	2	2	NUM
ma-30	74	31	)	)	PUNCT
ma-30	74	32	,	,	PUNCT
ma-30	74	33	ρ	ρ	PROPN
ma-30	74	34	(	(	PUNCT
ma-30	74	35	x	x	SYM
ma-30	74	36	−	−	PROPN
ma-30	74	37	y	y	PROPN
ma-30	74	38	2	2	NUM
ma-30	74	39	)	)	PUNCT
ma-30	74	40	}	}	PUNCT
ma-30	74	41	:	:	PUNCT
ma-30	75	1	x	x	X
ma-30	75	2	,	,	PUNCT
ma-30	75	3	y	y	PROPN
ma-30	75	4	∈	∈	PROPN
ma-30	75	5	xρ	xρ	PROPN
ma-30	75	6	,	,	PUNCT
ma-30	75	7	ρ(x	ρ(x	PROPN
ma-30	75	8	)	)	PUNCT
ma-30	75	9	=	=	SYM
ma-30	75	10	1	1	NUM
ma-30	75	11	,	,	PUNCT
ma-30	75	12	ρ(y	ρ(y	NOUN
ma-30	75	13	)	)	PUNCT
ma-30	75	14	≤	≤	NOUN
ma-30	75	15	1	1	NUM
ma-30	75	16	}	}	PUNCT
ma-30	75	17	.	.	PUNCT
ma-30	76	1	in	in	ADP
ma-30	76	2	the	the	DET
ma-30	76	3	following	follow	VERB
ma-30	76	4	section	section	NOUN
ma-30	76	5	,	,	PUNCT
ma-30	76	6	we	we	PRON
ma-30	76	7	extend	extend	VERB
ma-30	76	8	the	the	DET
ma-30	76	9	proposition	proposition	NOUN
ma-30	76	10	3.5	3.5	NUM
ma-30	76	11	in	in	ADP
ma-30	76	12	[	[	X
ma-30	76	13	10	10	NUM
ma-30	76	14	]	]	PUNCT
ma-30	76	15	and	and	CCONJ
ma-30	76	16	obtain	obtain	VERB
ma-30	76	17	inequalities	inequality	NOUN
ma-30	76	18	of	of	ADP
ma-30	76	19	cnj(xρ)and	cnj(xρ)and	PROPN
ma-30	76	20	j(xρ	j(xρ	PROPN
ma-30	76	21	)	)	PUNCT
ma-30	76	22	.	.	PUNCT
ma-30	77	1	theorem	theorem	NOUN
ma-30	77	2	1	1	NUM
ma-30	77	3	.	.	PUNCT
ma-30	78	1	let	let	VERB
ma-30	78	2	xρ	xρ	PROPN
ma-30	78	3	be	be	AUX
ma-30	78	4	a	a	DET
ma-30	78	5	modular	modular	ADJ
ma-30	78	6	space	space	NOUN
ma-30	78	7	,	,	PUNCT
ma-30	78	8	then	then	ADV
ma-30	78	9	eur	eur	PROPN
ma-30	78	10	.	.	PUNCT
ma-30	79	1	j.	j.	PROPN
ma-30	79	2	math	math	PROPN
ma-30	79	3	.	.	PUNCT
ma-30	80	1	anal	anal	ADJ
ma-30	80	2	.	.	PUNCT
ma-30	81	1	1	1	NUM
ma-30	81	2	(	(	PUNCT
ma-30	81	3	2021	2021	NUM
ma-30	81	4	)	)	PUNCT
ma-30	81	5	154(i	154(i	NUM
ma-30	81	6	)	)	PUNCT
ma-30	81	7	0	0	PUNCT
ma-30	82	1	<	<	X
ma-30	82	2	j	j	PROPN
ma-30	82	3	(	(	PUNCT
ma-30	82	4	xρ	xρ	PROPN
ma-30	82	5	)	)	PUNCT
ma-30	82	6	≤	≤	NOUN
ma-30	82	7	4	4	NUM
ma-30	82	8	and	and	CCONJ
ma-30	82	9	1	1	NUM
ma-30	82	10	≤	≤	NOUN
ma-30	82	11	cnj	cnj	NOUN
ma-30	82	12	(	(	PUNCT
ma-30	82	13	xρ	xρ	PROPN
ma-30	82	14	)	)	PUNCT
ma-30	82	15	≤	≤	NOUN
ma-30	82	16	8	8	NUM
ma-30	82	17	,	,	PUNCT
ma-30	82	18	in	in	ADP
ma-30	82	19	particular	particular	ADJ
ma-30	82	20	,	,	PUNCT
ma-30	82	21	if	if	SCONJ
ma-30	82	22	ρ	ρ	PROPN
ma-30	82	23	is	be	AUX
ma-30	82	24	convex	convex	NOUN
ma-30	82	25	,	,	PUNCT
ma-30	82	26	then	then	ADV
ma-30	82	27	1	1	NUM
ma-30	82	28	≤	≤	NUM
ma-30	82	29	j	j	PROPN
ma-30	82	30	(	(	PUNCT
ma-30	82	31	xρ	xρ	PROPN
ma-30	82	32	)	)	PUNCT
ma-30	82	33	≤	≤	NOUN
ma-30	82	34	2	2	NUM
ma-30	82	35	and	and	CCONJ
ma-30	82	36	1	1	NUM
ma-30	82	37	≤	≤	NOUN
ma-30	82	38	cnj	cnj	NOUN
ma-30	82	39	(	(	PUNCT
ma-30	82	40	xρ	xρ	PROPN
ma-30	82	41	)	)	PUNCT
ma-30	82	42	≤	≤	NOUN
ma-30	82	43	2;(ii)12j2(xρ	2;(ii)12j2(xρ	NOUN
ma-30	82	44	)	)	PUNCT
ma-30	82	45	≤	≤	NUM
ma-30	82	46	cnj(xρ	cnj(xρ	NOUN
ma-30	82	47	)	)	PUNCT
ma-30	82	48	≤	≤	NOUN
ma-30	82	49	64	64	NUM
ma-30	82	50	j2(xρ	j2(xρ	NOUN
ma-30	82	51	)	)	PUNCT
ma-30	83	1	+	+	NUM
ma-30	83	2	4	4	NUM
ma-30	83	3	,	,	PUNCT
ma-30	83	4	in	in	ADP
ma-30	83	5	particular	particular	ADJ
ma-30	83	6	,	,	PUNCT
ma-30	83	7	if	if	SCONJ
ma-30	83	8	ρ	ρ	PROPN
ma-30	83	9	is	be	AUX
ma-30	83	10	convex	convex	NOUN
ma-30	83	11	,	,	PUNCT
ma-30	83	12	then	then	ADV
ma-30	83	13	1	1	NUM
ma-30	83	14	2j	2j	NUM
ma-30	83	15	2(xρ	2(xρ	NUM
ma-30	83	16	)	)	PUNCT
ma-30	83	17	≤	≤	NUM
ma-30	83	18	cnj(xρ	cnj(xρ	NOUN
ma-30	83	19	)	)	PUNCT
ma-30	83	20	≤	≤	NUM
ma-30	83	21	4	4	NUM
ma-30	83	22	j2(xρ	j2(xρ	NOUN
ma-30	83	23	)	)	PUNCT
ma-30	84	1	+	+	NOUN
ma-30	84	2	1	1	X
ma-30	84	3	.	.	X
ma-30	84	4	proof	proof	NOUN
ma-30	84	5	.	.	PUNCT
ma-30	85	1	(	(	PUNCT
ma-30	85	2	i	i	NOUN
ma-30	85	3	)	)	PUNCT
ma-30	85	4	let	let	VERB
ma-30	85	5	y	y	PROPN
ma-30	85	6	=	=	SYM
ma-30	85	7	0	0	PROPN
ma-30	85	8	,	,	PUNCT
ma-30	85	9	then	then	ADV
ma-30	85	10	j(xρ	j(xρ	PROPN
ma-30	85	11	)	)	PUNCT
ma-30	85	12	≥	≥	NOUN
ma-30	85	13	2	2	NUM
ma-30	85	14	sup{ρ	sup{ρ	PROPN
ma-30	85	15	(	(	PUNCT
ma-30	85	16	x	x	NOUN
ma-30	85	17	2	2	NUM
ma-30	85	18	)	)	PUNCT
ma-30	85	19	:	:	PUNCT
ma-30	86	1	x	x	X
ma-30	86	2	∈	∈	NOUN
ma-30	86	3	xρ	xρ	PROPN
ma-30	86	4	,	,	PUNCT
ma-30	86	5	ρ(x	ρ(x	PROPN
ma-30	86	6	)	)	PUNCT
ma-30	86	7	=	=	SYM
ma-30	86	8	1	1	NUM
ma-30	86	9	}	}	PUNCT
ma-30	86	10	.	.	PUNCT
ma-30	87	1	since	since	SCONJ
ma-30	87	2	ρ(x	ρ(x	NUM
ma-30	87	3	)	)	PUNCT
ma-30	87	4	=	=	SYM
ma-30	87	5	1	1	NUM
ma-30	87	6	,	,	PUNCT
ma-30	87	7	then	then	ADV
ma-30	87	8	ρ(x	ρ(x	PROPN
ma-30	87	9	2	2	NUM
ma-30	87	10	)	)	PUNCT
ma-30	87	11	>	>	SYM
ma-30	87	12	0	0	NUM
ma-30	87	13	implies	imply	VERB
ma-30	87	14	j(xρ	j(xρ	PROPN
ma-30	87	15	)	)	PUNCT
ma-30	87	16	>	>	X
ma-30	88	1	0	0	X
ma-30	88	2	.	.	PUNCT
ma-30	89	1	since	since	SCONJ
ma-30	89	2	ρ	ρ	PROPN
ma-30	89	3	(	(	PUNCT
ma-30	89	4	x±y2	x±y2	PROPN
ma-30	89	5	)	)	PUNCT
ma-30	89	6	≤	≤	NUM
ma-30	89	7	ρ(x	ρ(x	NOUN
ma-30	89	8	)	)	PUNCT
ma-30	89	9	+	+	CCONJ
ma-30	89	10	ρ(y	ρ(y	NOUN
ma-30	89	11	)	)	PUNCT
ma-30	89	12	≤	≤	NOUN
ma-30	89	13	2	2	NUM
ma-30	89	14	,	,	PUNCT
ma-30	89	15	then	then	ADV
ma-30	89	16	0	0	NUM
ma-30	89	17	<	<	X
ma-30	89	18	j(xρ	j(xρ	PROPN
ma-30	89	19	)	)	PUNCT
ma-30	89	20	≤	≤	NOUN
ma-30	89	21	4.let	4.let	NOUN
ma-30	89	22	x	x	X
ma-30	89	23	=	=	SYM
ma-30	89	24	y	y	PROPN
ma-30	89	25	,	,	PUNCT
ma-30	89	26	then	then	ADV
ma-30	89	27	cnj(xρ	cnj(xρ	VERB
ma-30	89	28	)	)	PUNCT
ma-30	89	29	≥	≥	NOUN
ma-30	89	30	2	2	NUM
ma-30	89	31	sup	sup	NOUN
ma-30	89	32	{	{	PUNCT
ma-30	89	33	ρ2	ρ2	PROPN
ma-30	89	34	(	(	PUNCT
ma-30	89	35	x+x2	x+x2	PROPN
ma-30	89	36	)	)	PUNCT
ma-30	90	1	+	+	CCONJ
ma-30	91	1	ρ	ρ	PROPN
ma-30	91	2	2	2	NUM
ma-30	91	3	(	(	PUNCT
ma-30	91	4	x−x2	x−x2	X
ma-30	91	5	)	)	PUNCT
ma-30	91	6	ρ2(x	ρ2(x	NUM
ma-30	91	7	)	)	PUNCT
ma-30	91	8	+	+	NUM
ma-30	91	9	ρ2(x	ρ2(x	NOUN
ma-30	91	10	)	)	PUNCT
ma-30	91	11	:	:	PUNCT
ma-30	91	12	x	x	SYM
ma-30	91	13	∈	∈	NOUN
ma-30	91	14	xρ	xρ	PROPN
ma-30	91	15	,	,	PUNCT
ma-30	91	16	ρ(x	ρ(x	PROPN
ma-30	91	17	)	)	PUNCT
ma-30	91	18	=	=	SYM
ma-30	91	19	1	1	X
ma-30	91	20	}	}	PUNCT
ma-30	91	21	≥	≥	NOUN
ma-30	91	22	1	1	NUM
ma-30	91	23	.	.	PUNCT
ma-30	92	1	since	since	SCONJ
ma-30	92	2	ρ2(x	ρ2(x	PROPN
ma-30	92	3	+	+	CCONJ
ma-30	92	4	y	y	PROPN
ma-30	92	5	2	2	NUM
ma-30	92	6	)	)	PUNCT
ma-30	92	7	+	+	CCONJ
ma-30	92	8	ρ2	ρ2	NOUN
ma-30	92	9	(	(	PUNCT
ma-30	92	10	x	x	SYM
ma-30	92	11	−	−	PROPN
ma-30	92	12	y	y	PROPN
ma-30	92	13	2	2	NUM
ma-30	92	14	)	)	PUNCT
ma-30	92	15	≤	≤	NOUN
ma-30	92	16	2(1	2(1	NUM
ma-30	93	1	+	+	CCONJ
ma-30	93	2	ρ(y))2	ρ(y))2	NOUN
ma-30	93	3	,	,	PUNCT
ma-30	93	4	we	we	PRON
ma-30	93	5	have	have	VERB
ma-30	93	6	ρ2	ρ2	NOUN
ma-30	93	7	(	(	PUNCT
ma-30	93	8	x+y	x+y	NUM
ma-30	93	9	2	2	NUM
ma-30	93	10	)	)	PUNCT
ma-30	94	1	+	+	CCONJ
ma-30	94	2	ρ2	ρ2	NOUN
ma-30	94	3	(	(	PUNCT
ma-30	94	4	x−y	x−y	PROPN
ma-30	94	5	2	2	NUM
ma-30	94	6	)	)	PUNCT
ma-30	94	7	ρ2(x	ρ2(x	NOUN
ma-30	94	8	)	)	PUNCT
ma-30	94	9	+	+	NUM
ma-30	94	10	ρ2(y	ρ2(y	X
ma-30	94	11	)	)	PUNCT
ma-30	94	12	≤	≤	NOUN
ma-30	94	13	2	2	NUM
ma-30	94	14	(	(	PUNCT
ma-30	94	15	1	1	NUM
ma-30	94	16	+	+	CCONJ
ma-30	94	17	2ρ(y	2ρ(y	NUM
ma-30	94	18	)	)	PUNCT
ma-30	94	19	1	1	NUM
ma-30	94	20	+	+	NUM
ma-30	94	21	ρ2(y	ρ2(y	NUM
ma-30	94	22	)	)	PUNCT
ma-30	94	23	)	)	PUNCT
ma-30	94	24	≤	≤	NOUN
ma-30	94	25	4	4	NUM
ma-30	94	26	,	,	PUNCT
ma-30	94	27	thus	thus	ADV
ma-30	94	28	1	1	NUM
ma-30	94	29	≤	≤	NUM
ma-30	94	30	cnj(xρ	cnj(xρ	NOUN
ma-30	94	31	)	)	PUNCT
ma-30	94	32	≤	≤	NOUN
ma-30	94	33	8.in	8.in	NUM
ma-30	94	34	particular	particular	ADJ
ma-30	94	35	,	,	PUNCT
ma-30	94	36	if	if	SCONJ
ma-30	94	37	ρ	ρ	PROPN
ma-30	94	38	is	be	AUX
ma-30	94	39	convex	convex	ADJ
ma-30	94	40	and	and	CCONJ
ma-30	94	41	let	let	VERB
ma-30	94	42	x	x	SYM
ma-30	94	43	=	=	SYM
ma-30	94	44	y	y	PROPN
ma-30	94	45	,	,	PUNCT
ma-30	94	46	then	then	ADV
ma-30	94	47	j(xρ	j(xρ	PROPN
ma-30	94	48	)	)	PUNCT
ma-30	94	49	≥	≥	NOUN
ma-30	94	50	2sup{ρ(x	2sup{ρ(x	NUM
ma-30	94	51	2	2	NUM
ma-30	94	52	)	)	PUNCT
ma-30	94	53	:	:	PUNCT
ma-30	95	1	x	x	X
ma-30	95	2	∈	∈	NOUN
ma-30	95	3	xρ	xρ	PROPN
ma-30	95	4	,	,	PUNCT
ma-30	95	5	ρ(x	ρ(x	PROPN
ma-30	95	6	)	)	PUNCT
ma-30	95	7	=	=	SYM
ma-30	95	8	1	1	X
ma-30	95	9	}	}	PUNCT
ma-30	95	10	=	=	SYM
ma-30	95	11	1	1	X
ma-30	95	12	.	.	PUNCT
ma-30	95	13	since	since	SCONJ
ma-30	95	14	ρ	ρ	PROPN
ma-30	95	15	(	(	PUNCT
ma-30	95	16	x±y2	x±y2	PROPN
ma-30	95	17	)	)	PUNCT
ma-30	95	18	≤	≤	NUM
ma-30	95	19	1	1	NUM
ma-30	95	20	2ρ(x	2ρ(x	NUM
ma-30	95	21	)	)	PUNCT
ma-30	96	1	+	+	CCONJ
ma-30	96	2	1	1	NUM
ma-30	96	3	2ρ(y	2ρ(y	NUM
ma-30	96	4	)	)	PUNCT
ma-30	96	5	≤	≤	NUM
ma-30	96	6	1	1	NUM
ma-30	96	7	,	,	PUNCT
ma-30	96	8	then	then	ADV
ma-30	96	9	1	1	NUM
ma-30	96	10	≤	≤	PROPN
ma-30	96	11	j(xρ	j(xρ	PROPN
ma-30	96	12	)	)	PUNCT
ma-30	96	13	≤	≤	NOUN
ma-30	96	14	2	2	NUM
ma-30	96	15	.	.	PUNCT
ma-30	97	1	we	we	PRON
ma-30	97	2	also	also	ADV
ma-30	97	3	can	can	AUX
ma-30	97	4	prove	prove	VERB
ma-30	97	5	1	1	NUM
ma-30	97	6	≤	≤	NUM
ma-30	97	7	cnj(xρ	cnj(xρ	NOUN
ma-30	97	8	)	)	PUNCT
ma-30	97	9	≤	≤	PUNCT
ma-30	98	1	2by	2by	ADJ
ma-30	98	2	the	the	DET
ma-30	98	3	same	same	ADJ
ma-30	98	4	way.(ii	way.(ii	PROPN
ma-30	98	5	)	)	PUNCT
ma-30	98	6	since	since	SCONJ
ma-30	98	7	ρ2	ρ2	NOUN
ma-30	98	8	(	(	PUNCT
ma-30	98	9	x	x	SYM
ma-30	98	10	+	+	NUM
ma-30	98	11	y	y	PROPN
ma-30	98	12	2	2	NUM
ma-30	98	13	)	)	PUNCT
ma-30	99	1	+	+	CCONJ
ma-30	99	2	ρ2	ρ2	NOUN
ma-30	99	3	(	(	PUNCT
ma-30	99	4	x	x	SYM
ma-30	99	5	−	−	PROPN
ma-30	99	6	y	y	PROPN
ma-30	99	7	2	2	NUM
ma-30	99	8	)	)	PUNCT
ma-30	99	9	≤	≤	NOUN
ma-30	99	10	2[1	2[1	NUM
ma-30	100	1	+	+	CCONJ
ma-30	100	2	ρ(y)]2	ρ(y)]2	ADJ
ma-30	100	3	≤	≤	ADV
ma-30	100	4	4	4	NUM
ma-30	100	5	(	(	PUNCT
ma-30	100	6	1	1	NUM
ma-30	100	7	+	+	NUM
ma-30	100	8	ρ2(y	ρ2(y	NUM
ma-30	100	9	)	)	PUNCT
ma-30	100	10	)	)	PUNCT
ma-30	100	11	,	,	PUNCT
ma-30	100	12	then	then	ADV
ma-30	100	13	ρ2	ρ2	PROPN
ma-30	100	14	(	(	PUNCT
ma-30	100	15	x+y	x+y	NUM
ma-30	100	16	2	2	NUM
ma-30	100	17	)	)	PUNCT
ma-30	101	1	+	+	CCONJ
ma-30	101	2	ρ2	ρ2	NOUN
ma-30	101	3	(	(	PUNCT
ma-30	101	4	x−y	x−y	PROPN
ma-30	101	5	2	2	NUM
ma-30	101	6	)	)	PUNCT
ma-30	101	7	ρ2(x	ρ2(x	NOUN
ma-30	101	8	)	)	PUNCT
ma-30	101	9	+	+	NUM
ma-30	101	10	ρ2(y	ρ2(y	X
ma-30	101	11	)	)	PUNCT
ma-30	101	12	−	−	PROPN
ma-30	101	13	2	2	NUM
ma-30	101	14	≤	≤	NOUN
ma-30	101	15	2(1	2(1	NUM
ma-30	102	1	+	+	CCONJ
ma-30	102	2	ρ(y))2	ρ(y))2	NOUN
ma-30	102	3	1	1	NUM
ma-30	102	4	+	+	NUM
ma-30	102	5	ρ2(y	ρ2(y	NUM
ma-30	102	6	)	)	PUNCT
ma-30	102	7	−	−	PROPN
ma-30	102	8	2	2	NUM
ma-30	102	9	=	=	SYM
ma-30	102	10	4ρ(y	4ρ(y	NUM
ma-30	102	11	)	)	PUNCT
ma-30	102	12	1	1	NUM
ma-30	102	13	+	+	NUM
ma-30	102	14	ρ2(y	ρ2(y	NUM
ma-30	102	15	)	)	PUNCT
ma-30	102	16	.	.	PUNCT
ma-30	103	1	since	since	SCONJ
ma-30	103	2	14(ρ2	14(ρ2	NUM
ma-30	103	3	(	(	PUNCT
ma-30	103	4	x+y2	x+y2	PROPN
ma-30	103	5	)	)	PUNCT
ma-30	103	6	+	+	CCONJ
ma-30	103	7	ρ2	ρ2	PROPN
ma-30	103	8	(	(	PUNCT
ma-30	103	9	x−y2	x−y2	PROPN
ma-30	103	10	)	)	PUNCT
ma-30	103	11	)	)	PUNCT
ma-30	103	12	≤	≤	ADV
ma-30	103	13	1	1	NUM
ma-30	103	14	+	+	NUM
ma-30	103	15	ρ2(y	ρ2(y	NUM
ma-30	103	16	)	)	PUNCT
ma-30	103	17	,	,	PUNCT
ma-30	103	18	then	then	ADV
ma-30	103	19	4ρ(y	4ρ(y	NUM
ma-30	103	20	)	)	PUNCT
ma-30	103	21	1	1	NUM
ma-30	103	22	+	+	NUM
ma-30	103	23	ρ2(y	ρ2(y	NUM
ma-30	103	24	)	)	PUNCT
ma-30	103	25	≤	≤	NOUN
ma-30	103	26	16ρ(y	16ρ(y	NUM
ma-30	103	27	)	)	PUNCT
ma-30	103	28	ρ2	ρ2	NOUN
ma-30	103	29	(	(	PUNCT
ma-30	103	30	x+y2	x+y2	PROPN
ma-30	103	31	)	)	PUNCT
ma-30	103	32	+	+	CCONJ
ma-30	103	33	ρ	ρ	PROPN
ma-30	103	34	2	2	NUM
ma-30	103	35	(	(	PUNCT
ma-30	103	36	x−y2	x−y2	PROPN
ma-30	103	37	)	)	PUNCT
ma-30	103	38	,	,	PUNCT
ma-30	103	39	that	that	PRON
ma-30	103	40	is	be	AUX
ma-30	103	41	ρ2	ρ2	PROPN
ma-30	103	42	(	(	PUNCT
ma-30	103	43	x+y	x+y	NUM
ma-30	103	44	2	2	NUM
ma-30	103	45	)	)	PUNCT
ma-30	104	1	+	+	CCONJ
ma-30	104	2	ρ2	ρ2	NOUN
ma-30	104	3	(	(	PUNCT
ma-30	104	4	x−y	x−y	PROPN
ma-30	104	5	2	2	NUM
ma-30	104	6	)	)	PUNCT
ma-30	104	7	ρ2(x	ρ2(x	NOUN
ma-30	104	8	)	)	PUNCT
ma-30	104	9	+	+	NUM
ma-30	104	10	ρ2(y	ρ2(y	X
ma-30	104	11	)	)	PUNCT
ma-30	104	12	−	−	PROPN
ma-30	104	13	2	2	NUM
ma-30	104	14	≤	≤	NUM
ma-30	104	15	16ρ(y	16ρ(y	NUM
ma-30	104	16	)	)	PUNCT
ma-30	104	17	ρ2	ρ2	NOUN
ma-30	104	18	(	(	PUNCT
ma-30	104	19	x+y	x+y	NUM
ma-30	104	20	2	2	NUM
ma-30	104	21	)	)	PUNCT
ma-30	105	1	+	+	CCONJ
ma-30	105	2	ρ2	ρ2	NOUN
ma-30	105	3	(	(	PUNCT
ma-30	105	4	x−y	x−y	PROPN
ma-30	105	5	2	2	NUM
ma-30	105	6	)	)	PUNCT
ma-30	105	7	≤	≤	NOUN
ma-30	105	8	16	16	NUM
ma-30	105	9	ρ2	ρ2	NOUN
ma-30	105	10	(	(	PUNCT
ma-30	105	11	x+y	x+y	NUM
ma-30	105	12	2	2	NUM
ma-30	105	13	)	)	PUNCT
ma-30	105	14	+	+	CCONJ
ma-30	106	1	ρ2	ρ2	NOUN
ma-30	106	2	(	(	PUNCT
ma-30	106	3	x−y	x−y	NOUN
ma-30	106	4	2	2	NUM
ma-30	106	5	)	)	PUNCT
ma-30	106	6	.	.	PUNCT
ma-30	107	1	finally	finally	ADV
ma-30	107	2	12cnj(xρ)−	12cnj(xρ)−	NUM
ma-30	107	3	2	2	NUM
ma-30	107	4	≤	≤	NUM
ma-30	107	5	16	16	NUM
ma-30	107	6	1	1	NUM
ma-30	107	7	2	2	NUM
ma-30	107	8	j2(xρ	j2(xρ	PROPN
ma-30	107	9	)	)	PUNCT
ma-30	107	10	implies	imply	VERB
ma-30	107	11	that	that	SCONJ
ma-30	107	12	cnj(xρ	cnj(xρ	VERB
ma-30	107	13	)	)	PUNCT
ma-30	107	14	≤	≤	NOUN
ma-30	107	15	64	64	NUM
ma-30	107	16	j2(xρ	j2(xρ	NOUN
ma-30	107	17	)	)	PUNCT
ma-30	108	1	+	+	CCONJ
ma-30	108	2	4	4	X
ma-30	108	3	.	.	X
ma-30	108	4	eur	eur	PROPN
ma-30	108	5	.	.	PUNCT
ma-30	109	1	j.	j.	PROPN
ma-30	109	2	math	math	PROPN
ma-30	109	3	.	.	PUNCT
ma-30	110	1	anal	anal	ADJ
ma-30	110	2	.	.	PUNCT
ma-30	111	1	1	1	NUM
ma-30	111	2	(	(	PUNCT
ma-30	111	3	2021	2021	NUM
ma-30	111	4	)	)	PUNCT
ma-30	112	1	155according	155according	NOUN
ma-30	112	2	to	to	ADP
ma-30	112	3	the	the	DET
ma-30	112	4	proof	proof	NOUN
ma-30	112	5	of	of	ADP
ma-30	112	6	proposition	proposition	NOUN
ma-30	112	7	3.5	3.5	NUM
ma-30	112	8	in	in	ADP
ma-30	112	9	[	[	X
ma-30	112	10	12	12	NUM
ma-30	112	11	]	]	PUNCT
ma-30	112	12	,	,	PUNCT
ma-30	112	13	we	we	PRON
ma-30	112	14	can	can	AUX
ma-30	112	15	prove	prove	VERB
ma-30	112	16	12j2(xρ	12j2(xρ	NUM
ma-30	112	17	)	)	PUNCT
ma-30	112	18	≤	≤	NUM
ma-30	112	19	cnj(xρ	cnj(xρ	NOUN
ma-30	112	20	)	)	PUNCT
ma-30	112	21	,	,	PUNCT
ma-30	112	22	thus	thus	ADV
ma-30	112	23	1	1	NUM
ma-30	112	24	2	2	NUM
ma-30	112	25	j2(xρ	j2(xρ	PROPN
ma-30	112	26	)	)	PUNCT
ma-30	112	27	≤	≤	NUM
ma-30	112	28	cnj(xρ	cnj(xρ	NOUN
ma-30	112	29	)	)	PUNCT
ma-30	112	30	≤	≤	NOUN
ma-30	112	31	64	64	NUM
ma-30	112	32	j2(xρ	j2(xρ	NOUN
ma-30	112	33	)	)	PUNCT
ma-30	113	1	+	+	NUM
ma-30	113	2	4	4	X
ma-30	113	3	.	.	X
ma-30	113	4	in	in	ADP
ma-30	113	5	particular	particular	ADJ
ma-30	113	6	,	,	PUNCT
ma-30	113	7	if	if	SCONJ
ma-30	113	8	ρ	ρ	PROPN
ma-30	113	9	is	be	AUX
ma-30	113	10	convex	convex	NOUN
ma-30	113	11	,	,	PUNCT
ma-30	113	12	then	then	ADV
ma-30	113	13	ρ2	ρ2	VERB
ma-30	113	14	(	(	PUNCT
ma-30	113	15	x	x	PROPN
ma-30	113	16	+	+	NUM
ma-30	113	17	y	y	PROPN
ma-30	113	18	2	2	NUM
ma-30	113	19	)	)	PUNCT
ma-30	114	1	+	+	CCONJ
ma-30	114	2	ρ2	ρ2	NOUN
ma-30	114	3	(	(	PUNCT
ma-30	114	4	x	x	SYM
ma-30	114	5	−	−	PROPN
ma-30	114	6	y	y	PROPN
ma-30	114	7	2	2	NUM
ma-30	114	8	)	)	PUNCT
ma-30	114	9	≤	≤	NOUN
ma-30	114	10	1	1	NUM
ma-30	114	11	2	2	NUM
ma-30	114	12	(	(	PUNCT
ma-30	114	13	1	1	NUM
ma-30	114	14	+	+	NUM
ma-30	114	15	ρ(y))2	ρ(y))2	NOUN
ma-30	114	16	≤	≤	X
ma-30	114	17	1	1	NUM
ma-30	114	18	+	+	NUM
ma-30	114	19	ρ2(y	ρ2(y	NUM
ma-30	114	20	)	)	PUNCT
ma-30	114	21	,	,	PUNCT
ma-30	114	22	thus	thus	ADV
ma-30	114	23	ρ2	ρ2	NOUN
ma-30	114	24	(	(	PUNCT
ma-30	114	25	x+y	x+y	NUM
ma-30	114	26	2	2	NUM
ma-30	114	27	)	)	PUNCT
ma-30	115	1	+	+	CCONJ
ma-30	115	2	ρ2	ρ2	NOUN
ma-30	115	3	(	(	PUNCT
ma-30	115	4	x−y	x−y	PROPN
ma-30	115	5	2	2	NUM
ma-30	115	6	)	)	PUNCT
ma-30	115	7	ρ2(x	ρ2(x	NOUN
ma-30	115	8	)	)	PUNCT
ma-30	115	9	+	+	NUM
ma-30	115	10	ρ2(y	ρ2(y	X
ma-30	115	11	)	)	PUNCT
ma-30	115	12	−	−	NOUN
ma-30	115	13	1	1	NUM
ma-30	115	14	2	2	NUM
ma-30	115	15	≤	≤	NOUN
ma-30	115	16	ρ(y	ρ(y	NOUN
ma-30	115	17	)	)	PUNCT
ma-30	115	18	1	1	NUM
ma-30	115	19	+	+	NUM
ma-30	115	20	ρ2(y	ρ2(y	NUM
ma-30	115	21	)	)	PUNCT
ma-30	115	22	≤	≤	NOUN
ma-30	115	23	1	1	NUM
ma-30	115	24	ρ2	ρ2	NOUN
ma-30	115	25	(	(	PUNCT
ma-30	115	26	x+y	x+y	NUM
ma-30	115	27	2	2	NUM
ma-30	115	28	)	)	PUNCT
ma-30	116	1	+	+	CCONJ
ma-30	116	2	ρ2	ρ2	NOUN
ma-30	116	3	(	(	PUNCT
ma-30	116	4	x−y	x−y	NOUN
ma-30	116	5	2	2	NUM
ma-30	116	6	)	)	PUNCT
ma-30	116	7	.	.	PUNCT
ma-30	117	1	therefore	therefore	ADV
ma-30	117	2	cnj	cnj	PROPN
ma-30	117	3	(	(	PUNCT
ma-30	117	4	xρ	xρ	PROPN
ma-30	117	5	)	)	PUNCT
ma-30	117	6	≤	≤	NOUN
ma-30	117	7	4	4	NUM
ma-30	117	8	j2	j2	NOUN
ma-30	117	9	(	(	PUNCT
ma-30	117	10	xρ	xρ	PROPN
ma-30	117	11	)	)	PUNCT
ma-30	117	12	+	+	CCONJ
ma-30	117	13	1	1	X
ma-30	117	14	.	.	NOUN
ma-30	117	15	example	example	NOUN
ma-30	117	16	1	1	NUM
ma-30	117	17	.	.	PUNCT
ma-30	118	1	(	(	PUNCT
ma-30	118	2	i	i	NOUN
ma-30	118	3	)	)	PUNCT
ma-30	118	4	consider	consider	VERB
ma-30	118	5	x	x	NOUN
ma-30	118	6	=	=	SYM
ma-30	118	7	r2	r2	NOUN
ma-30	118	8	,	,	PUNCT
ma-30	118	9	ρ(x	ρ(x	NOUN
ma-30	118	10	)	)	PUNCT
ma-30	118	11	=	=	PRON
ma-30	118	12	{	{	PUNCT
ma-30	118	13	0	0	NUM
ma-30	118	14	,	,	PUNCT
ma-30	118	15	x	x	SYM
ma-30	118	16	=	=	SYM
ma-30	118	17	0	0	NUM
ma-30	118	18	1	1	NUM
ma-30	118	19	‖x‖1	‖x‖1	NOUN
ma-30	118	20	,	,	PUNCT
ma-30	118	21	x	x	PROPN
ma-30	118	22	6=	6=	ADP
ma-30	118	23	0	0	NUM
ma-30	118	24	,	,	PUNCT
ma-30	118	25	where	where	SCONJ
ma-30	118	26	‖x‖1	‖x‖1	PROPN
ma-30	118	27	=	=	SYM
ma-30	118	28	‖(x1	‖(x1	PROPN
ma-30	118	29	,	,	PUNCT
ma-30	118	30	x2)‖1	x2)‖1	PROPN
ma-30	118	31	=	=	SYM
ma-30	118	32	|x1|	|x1|	NOUN
ma-30	118	33	+	+	CCONJ
ma-30	118	34	|x2|.obviously	|x2|.obviously	ADV
ma-30	118	35	,	,	PUNCT
ma-30	118	36	xρ	xρ	PROPN
ma-30	118	37	is	be	AUX
ma-30	118	38	a	a	DET
ma-30	118	39	modular	modular	ADJ
ma-30	118	40	space.we	space.we	PRON
ma-30	118	41	choose	choose	VERB
ma-30	118	42	x0	x0	PROPN
ma-30	118	43	=	=	PUNCT
ma-30	118	44	(	(	PUNCT
ma-30	118	45	1	1	NUM
ma-30	118	46	2	2	NUM
ma-30	118	47	,	,	PUNCT
ma-30	118	48	1	1	NUM
ma-30	118	49	2	2	NUM
ma-30	118	50	)	)	PUNCT
ma-30	118	51	,	,	PUNCT
ma-30	118	52	y0	y0	NOUN
ma-30	118	53	=	=	SYM
ma-30	118	54	(	(	PUNCT
ma-30	118	55	1	1	NUM
ma-30	118	56	2	2	NUM
ma-30	118	57	,	,	PUNCT
ma-30	118	58	−	−	PROPN
ma-30	118	59	1	1	NUM
ma-30	118	60	2	2	NUM
ma-30	118	61	)	)	PUNCT
ma-30	118	62	,	,	PUNCT
ma-30	118	63	then	then	ADV
ma-30	118	64	ρ	ρ	PROPN
ma-30	118	65	(	(	PUNCT
ma-30	118	66	x0	x0	PROPN
ma-30	118	67	)	)	PUNCT
ma-30	118	68	=	=	SYM
ma-30	118	69	ρ	ρ	PROPN
ma-30	118	70	(	(	PUNCT
ma-30	118	71	y0	y0	NOUN
ma-30	118	72	)	)	PUNCT
ma-30	118	73	=	=	SYM
ma-30	118	74	1	1	NUM
ma-30	118	75	,	,	PUNCT
ma-30	118	76	ρ	ρ	PROPN
ma-30	118	77	(	(	PUNCT
ma-30	118	78	x0	x0	PROPN
ma-30	119	1	+	+	CCONJ
ma-30	119	2	y0	y0	NOUN
ma-30	119	3	2	2	NUM
ma-30	119	4	)	)	PUNCT
ma-30	120	1	=	=	SYM
ma-30	120	2	ρ	ρ	PROPN
ma-30	120	3	(	(	PUNCT
ma-30	120	4	x0	x0	PROPN
ma-30	120	5	−	−	PUNCT
ma-30	121	1	y0	y0	NOUN
ma-30	121	2	2	2	NUM
ma-30	121	3	)	)	PUNCT
ma-30	121	4	=	=	SYM
ma-30	121	5	2	2	NUM
ma-30	121	6	,	,	PUNCT
ma-30	121	7	thus	thus	ADV
ma-30	121	8	j	j	PROPN
ma-30	121	9	(	(	PUNCT
ma-30	121	10	xρ	xρ	PROPN
ma-30	121	11	)	)	PUNCT
ma-30	121	12	≥	≥	NOUN
ma-30	121	13	4	4	NUM
ma-30	121	14	.	.	PUNCT
ma-30	122	1	since	since	SCONJ
ma-30	122	2	j	j	PROPN
ma-30	122	3	(	(	PUNCT
ma-30	122	4	xρ	xρ	PROPN
ma-30	122	5	)	)	PUNCT
ma-30	122	6	≤	≤	NOUN
ma-30	122	7	4	4	NUM
ma-30	122	8	,	,	PUNCT
ma-30	122	9	then	then	ADV
ma-30	122	10	j	j	PROPN
ma-30	122	11	(	(	PUNCT
ma-30	122	12	xρ	xρ	PROPN
ma-30	122	13	)	)	PUNCT
ma-30	122	14	=	=	PUNCT
ma-30	122	15	4	4	X
ma-30	122	16	.	.	PUNCT
ma-30	122	17	according	accord	VERB
ma-30	122	18	to	to	ADP
ma-30	122	19	(	(	PUNCT
ma-30	122	20	ii	ii	NOUN
ma-30	122	21	)	)	PUNCT
ma-30	122	22	of	of	ADP
ma-30	122	23	theorem	theorem	NOUN
ma-30	122	24	1	1	NUM
ma-30	122	25	,	,	PUNCT
ma-30	122	26	we	we	PRON
ma-30	122	27	know	know	VERB
ma-30	122	28	that	that	DET
ma-30	122	29	cnj(xρ	cnj(xρ	VERB
ma-30	122	30	)	)	PUNCT
ma-30	122	31	=	=	SYM
ma-30	122	32	8	8	NUM
ma-30	122	33	in	in	ADP
ma-30	122	34	this	this	DET
ma-30	122	35	example.(ii	example.(ii	NOUN
ma-30	122	36	)	)	PUNCT
ma-30	122	37	consider	consider	VERB
ma-30	122	38	x	x	NOUN
ma-30	122	39	=	=	SYM
ma-30	122	40	r2	r2	NOUN
ma-30	122	41	,	,	PUNCT
ma-30	122	42	ρ(x	ρ(x	NOUN
ma-30	122	43	)	)	PUNCT
ma-30	122	44	=	=	SYM
ma-30	123	1	‖x‖1	‖x‖1	ADJ
ma-30	123	2	.	.	PUNCT
ma-30	124	1	obviously	obviously	ADV
ma-30	124	2	,	,	PUNCT
ma-30	124	3	xρ	xρ	PROPN
ma-30	124	4	is	be	AUX
ma-30	124	5	a	a	DET
ma-30	124	6	modular	modular	ADJ
ma-30	124	7	space	space	NOUN
ma-30	124	8	and	and	CCONJ
ma-30	124	9	ρ	ρ	NOUN
ma-30	124	10	is	be	AUX
ma-30	124	11	convex	convex	ADJ
ma-30	124	12	.	.	PUNCT
ma-30	125	1	we	we	PRON
ma-30	125	2	have	have	VERB
ma-30	125	3	j	j	PROPN
ma-30	125	4	(	(	PUNCT
ma-30	125	5	xρ	xρ	PROPN
ma-30	125	6	)	)	PUNCT
ma-30	125	7	=	=	SYM
ma-30	125	8	sup	sup	NOUN
ma-30	125	9	{	{	PUNCT
ma-30	125	10	min{‖x	min{‖x	PROPN
ma-30	125	11	+	+	CCONJ
ma-30	125	12	y‖1	y‖1	PROPN
ma-30	125	13	,	,	PUNCT
ma-30	125	14	‖x	‖x	NOUN
ma-30	125	15	−	−	PROPN
ma-30	125	16	y‖1	y‖1	PROPN
ma-30	125	17	}	}	PUNCT
ma-30	125	18	:	:	PUNCT
ma-30	125	19	x	x	X
ma-30	125	20	,	,	PUNCT
ma-30	125	21	y	y	PROPN
ma-30	125	22	∈	∈	PROPN
ma-30	125	23	xρ	xρ	PROPN
ma-30	125	24	,	,	PUNCT
ma-30	125	25	‖x‖	‖x‖	PROPN
ma-30	125	26	=	=	SYM
ma-30	125	27	1	1	NUM
ma-30	125	28	,	,	PUNCT
ma-30	125	29	‖y‖	‖y‖	PROPN
ma-30	125	30	≤	≤	ADV
ma-30	125	31	1}.we	1}.we	NUM
ma-30	125	32	choose	choose	VERB
ma-30	125	33	x0	x0	PROPN
ma-30	125	34	=	=	PUNCT
ma-30	125	35	(	(	PUNCT
ma-30	125	36	1	1	NUM
ma-30	125	37	,	,	PUNCT
ma-30	125	38	0	0	NUM
ma-30	125	39	)	)	PUNCT
ma-30	125	40	,	,	PUNCT
ma-30	125	41	y0	y0	NOUN
ma-30	125	42	=	=	SYM
ma-30	125	43	(	(	PUNCT
ma-30	125	44	0	0	NUM
ma-30	125	45	,	,	PUNCT
ma-30	125	46	1	1	NUM
ma-30	125	47	)	)	PUNCT
ma-30	125	48	,	,	PUNCT
ma-30	125	49	then	then	ADV
ma-30	125	50	‖x0‖1	‖x0‖1	PROPN
ma-30	126	1	=	=	PUNCT
ma-30	126	2	‖y0‖1	‖y0‖1	NOUN
ma-30	126	3	=	=	SYM
ma-30	126	4	1	1	NUM
ma-30	126	5	and	and	CCONJ
ma-30	126	6	‖x0	‖x0	PROPN
ma-30	127	1	+	+	CCONJ
ma-30	127	2	y0‖1	y0‖1	NUM
ma-30	128	1	=	=	SYM
ma-30	128	2	‖x0	‖x0	NOUN
ma-30	129	1	−	−	NOUN
ma-30	129	2	y0‖1	y0‖1	NUM
ma-30	130	1	=	=	SYM
ma-30	130	2	2,thus	2,thus	NUM
ma-30	130	3	j	j	PROPN
ma-30	130	4	(	(	PUNCT
ma-30	130	5	xρ	xρ	PROPN
ma-30	130	6	)	)	PUNCT
ma-30	130	7	=	=	SYM
ma-30	130	8	2	2	X
ma-30	130	9	.	.	PUNCT
ma-30	130	10	according	accord	VERB
ma-30	130	11	to	to	ADP
ma-30	130	12	(	(	PUNCT
ma-30	130	13	ii	ii	NOUN
ma-30	130	14	)	)	PUNCT
ma-30	130	15	of	of	ADP
ma-30	130	16	theorem	theorem	NOUN
ma-30	130	17	1	1	NUM
ma-30	130	18	,	,	PUNCT
ma-30	130	19	we	we	PRON
ma-30	130	20	can	can	AUX
ma-30	130	21	get	get	VERB
ma-30	130	22	that	that	DET
ma-30	130	23	cnj(xρ	cnj(xρ	NOUN
ma-30	130	24	)	)	PUNCT
ma-30	130	25	=	=	SYM
ma-30	130	26	2	2	NUM
ma-30	130	27	in	in	ADP
ma-30	130	28	this	this	DET
ma-30	130	29	example	example	NOUN
ma-30	130	30	.	.	PUNCT
ma-30	131	1	4	4	X
ma-30	131	2	.	.	X
ma-30	131	3	the	the	DET
ma-30	131	4	ρ	ρ	ADJ
ma-30	131	5	-	-	PUNCT
ma-30	131	6	convex	convex	NOUN
ma-30	131	7	modular	modular	NOUN
ma-30	131	8	and	and	CCONJ
ma-30	131	9	the	the	DET
ma-30	131	10	ρ	ρ	ADJ
ma-30	131	11	-	-	ADJ
ma-30	131	12	smooth	smooth	ADJ
ma-30	131	13	modular	modular	NOUN
ma-30	131	14	in	in	ADP
ma-30	131	15	order	order	NOUN
ma-30	131	16	to	to	PART
ma-30	131	17	study	study	VERB
ma-30	131	18	the	the	DET
ma-30	131	19	uniform	uniform	ADJ
ma-30	131	20	convexity	convexity	NOUN
ma-30	131	21	of	of	ADP
ma-30	131	22	banach	banach	NOUN
ma-30	131	23	spaces	space	NOUN
ma-30	131	24	,	,	PUNCT
ma-30	131	25	clarkson	clarkson	PROPN
ma-30	131	26	introduced	introduce	VERB
ma-30	131	27	the	the	DET
ma-30	131	28	modular	modular	ADJ
ma-30	131	29	ofconvexity	ofconvexity	NOUN
ma-30	131	30	δx(ε	δx(ε	NOUN
ma-30	131	31	)	)	PUNCT
ma-30	132	1	=	=	SYM
ma-30	132	2	inf{1−	inf{1−	ADJ
ma-30	132	3	1	1	NUM
ma-30	132	4	2	2	NUM
ma-30	132	5	‖x	‖x	NOUN
ma-30	132	6	+	+	CCONJ
ma-30	132	7	y‖	y‖	NOUN
ma-30	132	8	:	:	PUNCT
ma-30	133	1	‖x‖	‖x‖	PROPN
ma-30	133	2	=	=	PUNCT
ma-30	133	3	‖y‖	‖y‖	PROPN
ma-30	133	4	=	=	SYM
ma-30	133	5	1	1	NUM
ma-30	133	6	,	,	PUNCT
ma-30	133	7	‖x	‖x	NOUN
ma-30	134	1	−	−	PROPN
ma-30	134	2	y‖	y‖	PROPN
ma-30	134	3	≥	≥	NOUN
ma-30	134	4	ε	ε	PROPN
ma-30	134	5	}	}	PUNCT
ma-30	134	6	.	.	PUNCT
ma-30	135	1	goebel	goebel	NOUN
ma-30	135	2	called	call	VERB
ma-30	135	3	ε0	ε0	NOUN
ma-30	135	4	=	=	PUNCT
ma-30	135	5	sup{ε	sup{ε	NOUN
ma-30	135	6	∈	∈	PROPN
ma-30	135	7	[	[	X
ma-30	135	8	0	0	NUM
ma-30	135	9	,	,	PUNCT
ma-30	135	10	2	2	NUM
ma-30	135	11	]	]	PUNCT
ma-30	135	12	:	:	PUNCT
ma-30	135	13	δx(ε	δx(ε	X
ma-30	135	14	)	)	PUNCT
ma-30	136	1	=	=	SYM
ma-30	136	2	0	0	X
ma-30	136	3	}	}	PUNCT
ma-30	136	4	as	as	ADP
ma-30	136	5	the	the	DET
ma-30	136	6	characteristic	characteristic	NOUN
ma-30	136	7	of	of	ADP
ma-30	136	8	convexity	convexity	NOUN
ma-30	136	9	.	.	PUNCT
ma-30	137	1	based	base	VERB
ma-30	137	2	on	on	ADP
ma-30	137	3	thegeometric	thegeometric	ADJ
ma-30	137	4	intuitionistic	intuitionistic	ADJ
ma-30	137	5	meaning	meaning	NOUN
ma-30	137	6	of	of	ADP
ma-30	137	7	convexity	convexity	NOUN
ma-30	137	8	of	of	ADP
ma-30	137	9	banach	banach	NOUN
ma-30	137	10	spaces	space	NOUN
ma-30	137	11	and	and	CCONJ
ma-30	137	12	its	its	PRON
ma-30	137	13	application	application	NOUN
ma-30	137	14	in	in	ADP
ma-30	137	15	fixed	fix	VERB
ma-30	137	16	pointtheory	pointtheory	NOUN
ma-30	137	17	,	,	PUNCT
ma-30	137	18	this	this	DET
ma-30	137	19	paper	paper	NOUN
ma-30	137	20	gives	give	VERB
ma-30	137	21	the	the	DET
ma-30	137	22	ρ	ρ	ADJ
ma-30	137	23	-	-	PUNCT
ma-30	137	24	convex	convex	ADJ
ma-30	137	25	modular	modular	NOUN
ma-30	137	26	of	of	ADP
ma-30	137	27	modular	modular	ADJ
ma-30	137	28	spaces	space	NOUN
ma-30	137	29	with	with	ADP
ma-30	137	30	reference	reference	NOUN
ma-30	137	31	to	to	ADP
ma-30	137	32	the	the	DET
ma-30	137	33	definition	definition	NOUN
ma-30	137	34	of	of	ADP
ma-30	137	35	δx(ε	δx(ε	NOUN
ma-30	137	36	)	)	PUNCT
ma-30	137	37	.	.	PUNCT
ma-30	138	1	definition	definition	NOUN
ma-30	138	2	4	4	NUM
ma-30	138	3	.	.	PUNCT
ma-30	139	1	the	the	DET
ma-30	139	2	ρ	ρ	ADJ
ma-30	139	3	-	-	PUNCT
ma-30	139	4	convex	convex	ADJ
ma-30	139	5	modular	modular	NOUN
ma-30	139	6	δxρ(ε	δxρ(ε	NOUN
ma-30	139	7	)	)	PUNCT
ma-30	139	8	of	of	ADP
ma-30	139	9	a	a	DET
ma-30	139	10	modular	modular	ADJ
ma-30	139	11	space	space	NOUN
ma-30	139	12	xρ	xρ	PROPN
ma-30	139	13	is	be	AUX
ma-30	139	14	defined	define	VERB
ma-30	139	15	by	by	ADP
ma-30	139	16	δxρ(ε	δxρ(ε	X
ma-30	139	17	)	)	PUNCT
ma-30	139	18	=	=	SYM
ma-30	139	19	inf	inf	NOUN
ma-30	139	20	{	{	PUNCT
ma-30	139	21	1−	1−	NUM
ma-30	139	22	ρ	ρ	PROPN
ma-30	139	23	(	(	PUNCT
ma-30	139	24	x+y	x+y	PROPN
ma-30	139	25	2	2	NUM
ma-30	139	26	)	)	PUNCT
ma-30	139	27	:	:	PUNCT
ma-30	139	28	x	x	X
ma-30	139	29	,	,	PUNCT
ma-30	139	30	y	y	PROPN
ma-30	139	31	∈	∈	PROPN
ma-30	139	32	xρ	xρ	PROPN
ma-30	139	33	,	,	PUNCT
ma-30	139	34	ρ(x	ρ(x	PROPN
ma-30	139	35	)	)	PUNCT
ma-30	139	36	,	,	PUNCT
ma-30	139	37	ρ(y	ρ(y	NOUN
ma-30	139	38	)	)	PUNCT
ma-30	139	39	≤	≤	NUM
ma-30	139	40	1	1	NUM
ma-30	139	41	,	,	PUNCT
ma-30	139	42	ρ(x	ρ(x	PROPN
ma-30	139	43	−	−	PROPN
ma-30	139	44	y	y	PROPN
ma-30	139	45	)	)	PUNCT
ma-30	139	46	≥	≥	NOUN
ma-30	139	47	ε	ε	PROPN
ma-30	139	48	}	}	PUNCT
ma-30	139	49	,	,	PUNCT
ma-30	139	50	0	0	NUM
ma-30	139	51	≤	≤	NUM
ma-30	139	52	ε	ε	PROPN
ma-30	139	53	≤	≤	NOUN
ma-30	139	54	2.in	2.in	NUM
ma-30	139	55	particular	particular	ADJ
ma-30	139	56	,	,	PUNCT
ma-30	139	57	if	if	SCONJ
ma-30	139	58	ρ	ρ	PROPN
ma-30	139	59	is	be	AUX
ma-30	139	60	convex	convex	PROPN
ma-30	139	61	,	,	PUNCT
ma-30	139	62	the	the	DET
ma-30	139	63	ρ	ρ	PROPN
ma-30	139	64	-	-	PUNCT
ma-30	139	65	uniform	uniform	NOUN
ma-30	139	66	convexity	convexity	NOUN
ma-30	139	67	of	of	ADP
ma-30	139	68	xρ	xρ	PROPN
ma-30	139	69	is	be	AUX
ma-30	139	70	defined	define	VERB
ma-30	139	71	as	as	ADP
ma-30	139	72	ε0	ε0	PROPN
ma-30	139	73	(	(	PUNCT
ma-30	139	74	xρ	xρ	PROPN
ma-30	139	75	)	)	PUNCT
ma-30	140	1	=	=	SYM
ma-30	140	2	sup	sup	NOUN
ma-30	140	3	{	{	PUNCT
ma-30	140	4	ε	ε	PROPN
ma-30	140	5	∈	∈	PROPN
ma-30	141	1	[	[	X
ma-30	141	2	0	0	NUM
ma-30	141	3	,	,	PUNCT
ma-30	141	4	2	2	NUM
ma-30	141	5	]	]	PUNCT
ma-30	141	6	:	:	PUNCT
ma-30	141	7	δxρ(ε	δxρ(ε	X
ma-30	141	8	)	)	PUNCT
ma-30	141	9	=	=	SYM
ma-30	141	10	0	0	NUM
ma-30	141	11	}	}	PUNCT
ma-30	141	12	.	.	PUNCT
ma-30	142	1	eur	eur	PROPN
ma-30	142	2	.	.	PUNCT
ma-30	143	1	j.	j.	PROPN
ma-30	143	2	math	math	PROPN
ma-30	143	3	.	.	PUNCT
ma-30	144	1	anal	anal	ADJ
ma-30	144	2	.	.	PUNCT
ma-30	145	1	1	1	NUM
ma-30	145	2	(	(	PUNCT
ma-30	145	3	2021	2021	NUM
ma-30	145	4	)	)	PUNCT
ma-30	145	5	156	156	NUM
ma-30	145	6	remark	remark	NOUN
ma-30	145	7	1	1	NUM
ma-30	145	8	.	.	PUNCT
ma-30	146	1	we	we	PRON
ma-30	146	2	can	can	AUX
ma-30	146	3	easily	easily	ADV
ma-30	146	4	prove	prove	VERB
ma-30	146	5	that	that	DET
ma-30	146	6	−1	−1	NOUN
ma-30	146	7	≤	≤	PUNCT
ma-30	146	8	δxρ(ε	δxρ(ε	X
ma-30	146	9	)	)	PUNCT
ma-30	146	10	≤	≤	NUM
ma-30	146	11	1	1	NUM
ma-30	146	12	and	and	CCONJ
ma-30	146	13	δxρ(0	δxρ(0	NOUN
ma-30	146	14	)	)	PUNCT
ma-30	146	15	≤	≤	NOUN
ma-30	146	16	0.in	0.in	NUM
ma-30	146	17	banach	banach	NOUN
ma-30	146	18	spaces	space	NOUN
ma-30	146	19	,	,	PUNCT
ma-30	146	20	the	the	DET
ma-30	146	21	convexity	convexity	NOUN
ma-30	146	22	modular	modular	ADJ
ma-30	146	23	δx(ε	δx(ε	NOUN
ma-30	146	24	)	)	PUNCT
ma-30	146	25	and	and	CCONJ
ma-30	146	26	the	the	DET
ma-30	146	27	smoothness	smoothness	ADJ
ma-30	146	28	modular	modular	NOUN
ma-30	146	29	ρx(t	ρx(t	NOUN
ma-30	146	30	)	)	PUNCT
ma-30	146	31	=	=	PUNCT
ma-30	146	32	sup{‖x	sup{‖x	PROPN
ma-30	147	1	+	+	CCONJ
ma-30	147	2	y‖+	y‖+	ADJ
ma-30	147	3	‖x	‖x	PUNCT
ma-30	147	4	−	−	PROPN
ma-30	147	5	y‖	y‖	NOUN
ma-30	147	6	2	2	NUM
ma-30	147	7	−	−	NOUN
ma-30	147	8	1	1	NUM
ma-30	147	9	:	:	PUNCT
ma-30	147	10	‖x‖	‖x‖	VERB
ma-30	147	11	=	=	SYM
ma-30	147	12	1	1	NUM
ma-30	147	13	,	,	PUNCT
ma-30	147	14	‖y‖	‖y‖	PROPN
ma-30	147	15	=	=	SYM
ma-30	147	16	1	1	NUM
ma-30	147	17	,	,	PUNCT
ma-30	147	18	t	t	PROPN
ma-30	147	19	≥	≥	NOUN
ma-30	147	20	0	0	NUM
ma-30	147	21	}	}	PUNCT
ma-30	147	22	are	be	AUX
ma-30	147	23	conjugate	conjugate	ADJ
ma-30	147	24	concepts	concept	NOUN
ma-30	147	25	.	.	PUNCT
ma-30	148	1	therefore	therefore	ADV
ma-30	148	2	,	,	PUNCT
ma-30	148	3	this	this	DET
ma-30	148	4	paper	paper	NOUN
ma-30	148	5	gives	give	VERB
ma-30	148	6	the	the	DET
ma-30	148	7	definition	definition	NOUN
ma-30	148	8	of	of	ADP
ma-30	148	9	ρ	ρ	PROPN
ma-30	148	10	-	-	ADJ
ma-30	148	11	smooth	smooth	ADJ
ma-30	148	12	modular	modular	NOUN
ma-30	148	13	of	of	ADP
ma-30	148	14	modularspaces	modularspace	NOUN
ma-30	148	15	by	by	ADP
ma-30	148	16	referring	refer	VERB
ma-30	148	17	to	to	ADP
ma-30	148	18	the	the	DET
ma-30	148	19	definition	definition	NOUN
ma-30	148	20	of	of	ADP
ma-30	148	21	smoothness	smoothness	ADJ
ma-30	148	22	modular	modular	NOUN
ma-30	148	23	ρx(t	ρx(t	NOUN
ma-30	148	24	)	)	PUNCT
ma-30	148	25	.	.	PUNCT
ma-30	149	1	definition	definition	NOUN
ma-30	149	2	5	5	NUM
ma-30	149	3	.	.	PUNCT
ma-30	150	1	the	the	DET
ma-30	150	2	ρ	ρ	ADJ
ma-30	150	3	-	-	ADJ
ma-30	150	4	smooth	smooth	ADJ
ma-30	150	5	modular	modular	NOUN
ma-30	150	6	ρxρ(t	ρxρ(t	NOUN
ma-30	150	7	)	)	PUNCT
ma-30	150	8	of	of	ADP
ma-30	150	9	a	a	DET
ma-30	150	10	modular	modular	ADJ
ma-30	150	11	space	space	NOUN
ma-30	150	12	xρ	xρ	PROPN
ma-30	150	13	is	be	AUX
ma-30	150	14	defined	define	VERB
ma-30	150	15	by	by	ADP
ma-30	150	16	ρxρ(t	ρxρ(t	NUM
ma-30	150	17	)	)	PUNCT
ma-30	151	1	=	=	SYM
ma-30	151	2	sup	sup	NOUN
ma-30	151	3	{	{	PUNCT
ma-30	151	4	ρ	ρ	PROPN
ma-30	151	5	(	(	PUNCT
ma-30	151	6	x	x	PROPN
ma-30	151	7	+	+	NUM
ma-30	151	8	y	y	PROPN
ma-30	151	9	2	2	NUM
ma-30	151	10	)	)	PUNCT
ma-30	151	11	+	+	CCONJ
ma-30	151	12	ρ	ρ	PROPN
ma-30	151	13	(	(	PUNCT
ma-30	151	14	x	x	SYM
ma-30	151	15	−	−	PROPN
ma-30	151	16	y	y	PROPN
ma-30	151	17	2	2	NUM
ma-30	151	18	)	)	PUNCT
ma-30	151	19	−	−	PROPN
ma-30	151	20	1	1	NUM
ma-30	151	21	:	:	SYM
ma-30	151	22	x	x	X
ma-30	151	23	,	,	PUNCT
ma-30	151	24	y	y	PROPN
ma-30	151	25	∈	∈	PROPN
ma-30	151	26	xρ	xρ	PROPN
ma-30	151	27	,	,	PUNCT
ma-30	151	28	ρ(x	ρ(x	NOUN
ma-30	151	29	)	)	PUNCT
ma-30	151	30	≤	≤	NOUN
ma-30	151	31	1	1	NUM
ma-30	151	32	,	,	PUNCT
ma-30	151	33	ρ(y	ρ(y	NOUN
ma-30	151	34	)	)	PUNCT
ma-30	151	35	≤	≤	NOUN
ma-30	151	36	t	t	PROPN
ma-30	151	37	}	}	PUNCT
ma-30	151	38	,	,	PUNCT
ma-30	151	39	t	t	PROPN
ma-30	151	40	≥	≥	NUM
ma-30	151	41	0	0	NUM
ma-30	151	42	.	.	PUNCT
ma-30	151	43	remark	remark	PROPN
ma-30	151	44	2	2	NUM
ma-30	151	45	.	.	PUNCT
ma-30	152	1	it	it	PRON
ma-30	152	2	is	be	AUX
ma-30	152	3	true	true	ADJ
ma-30	152	4	that	that	SCONJ
ma-30	152	5	min{0	min{0	PROPN
ma-30	152	6	,	,	PUNCT
ma-30	152	7	t	t	NOUN
ma-30	152	8	−	−	PROPN
ma-30	152	9	1	1	NUM
ma-30	152	10	}	}	PUNCT
ma-30	152	11	≤	≤	NUM
ma-30	152	12	ρxρ(t	ρxρ(t	NUM
ma-30	152	13	)	)	PUNCT
ma-30	152	14	≤	≤	NUM
ma-30	152	15	1	1	NUM
ma-30	153	1	+	+	SYM
ma-30	153	2	2	2	NUM
ma-30	153	3	t	t	NOUN
ma-30	153	4	and	and	CCONJ
ma-30	153	5	ρxρ(t	ρxρ(t	NUM
ma-30	153	6	)	)	PUNCT
ma-30	153	7	is	be	AUX
ma-30	153	8	increasing	increase	VERB
ma-30	153	9	of	of	ADP
ma-30	153	10	t	t	PROPN
ma-30	153	11	.	.	PUNCT
ma-30	154	1	theorem	theorem	NOUN
ma-30	154	2	2	2	NUM
ma-30	154	3	.	.	PUNCT
ma-30	155	1	let	let	VERB
ma-30	155	2	x	x	PRON
ma-30	155	3	be	be	AUX
ma-30	155	4	a	a	DET
ma-30	155	5	modular	modular	ADJ
ma-30	155	6	space	space	NOUN
ma-30	155	7	,	,	PUNCT
ma-30	155	8	then(i	then(i	NOUN
ma-30	155	9	)	)	PUNCT
ma-30	155	10	j(xρ	j(xρ	PROPN
ma-30	155	11	)	)	PUNCT
ma-30	155	12	<	<	X
ma-30	155	13	2ε	2ε	PROPN
ma-30	156	1	if	if	SCONJ
ma-30	156	2	and	and	CCONJ
ma-30	156	3	only	only	ADV
ma-30	156	4	if	if	SCONJ
ma-30	156	5	δxρ(ε	δxρ(ε	X
ma-30	156	6	)	)	PUNCT
ma-30	156	7	>	>	X
ma-30	156	8	1	1	NUM
ma-30	156	9	−	−	ADP
ma-30	156	10	ε	ε	PROPN
ma-30	156	11	,	,	PUNCT
ma-30	156	12	in	in	ADP
ma-30	156	13	particular	particular	ADJ
ma-30	156	14	,	,	PUNCT
ma-30	156	15	if	if	SCONJ
ma-30	156	16	ρ	ρ	PROPN
ma-30	156	17	is	be	AUX
ma-30	156	18	convex	convex	NOUN
ma-30	156	19	,	,	PUNCT
ma-30	156	20	then	then	ADV
ma-30	156	21	j(xρ	j(xρ	PROPN
ma-30	156	22	)	)	PUNCT
ma-30	156	23	<	<	X
ma-30	156	24	ε	ε	PROPN
ma-30	157	1	if	if	SCONJ
ma-30	157	2	andonly	andonly	ADV
ma-30	157	3	if	if	SCONJ
ma-30	157	4	δxρ(ε	δxρ(ε	X
ma-30	157	5	)	)	PUNCT
ma-30	157	6	>	>	X
ma-30	157	7	1−	1−	NUM
ma-30	157	8	ε	ε	PROPN
ma-30	157	9	2	2	NUM
ma-30	157	10	;	;	PUNCT
ma-30	157	11	(	(	PUNCT
ma-30	157	12	ii	ii	NOUN
ma-30	157	13	)	)	PUNCT
ma-30	157	14	j(xρ	j(xρ	PROPN
ma-30	157	15	)	)	PUNCT
ma-30	157	16	=	=	SYM
ma-30	157	17	2	2	NUM
ma-30	157	18	sup{ε	sup{ε	NOUN
ma-30	157	19	∈	∈	PROPN
ma-30	157	20	(	(	PUNCT
ma-30	157	21	0	0	NUM
ma-30	157	22	,	,	PUNCT
ma-30	157	23	2	2	NUM
ma-30	157	24	)	)	PUNCT
ma-30	157	25	:	:	PUNCT
ma-30	157	26	δxρ(ε	δxρ(ε	X
ma-30	157	27	)	)	PUNCT
ma-30	157	28	≤	≤	NOUN
ma-30	157	29	1−ε	1−ε	NUM
ma-30	157	30	}	}	PUNCT
ma-30	157	31	,	,	PUNCT
ma-30	157	32	in	in	ADP
ma-30	157	33	particular	particular	ADJ
ma-30	157	34	,	,	PUNCT
ma-30	157	35	if	if	SCONJ
ma-30	157	36	ρ	ρ	PROPN
ma-30	157	37	is	be	AUX
ma-30	157	38	convex	convex	NOUN
ma-30	157	39	,	,	PUNCT
ma-30	157	40	then	then	ADV
ma-30	157	41	j(xρ	j(xρ	PROPN
ma-30	157	42	)	)	PUNCT
ma-30	158	1	=	=	PUNCT
ma-30	158	2	sup{ε	sup{ε	NOUN
ma-30	158	3	∈	∈	PROPN
ma-30	158	4	(	(	PUNCT
ma-30	158	5	0	0	NUM
ma-30	158	6	,	,	PUNCT
ma-30	158	7	2	2	NUM
ma-30	158	8	)	)	PUNCT
ma-30	158	9	:	:	PUNCT
ma-30	159	1	δxρ(ε	δxρ(ε	X
ma-30	159	2	)	)	PUNCT
ma-30	159	3	<	<	X
ma-30	159	4	1−	1−	NUM
ma-30	159	5	ε	ε	PROPN
ma-30	159	6	2	2	NUM
ma-30	159	7	}	}	PUNCT
ma-30	159	8	.	.	PUNCT
ma-30	160	1	proof	proof	NOUN
ma-30	160	2	.	.	PUNCT
ma-30	161	1	(	(	PUNCT
ma-30	161	2	i	i	NOUN
ma-30	161	3	)	)	PUNCT
ma-30	161	4	note	note	VERB
ma-30	161	5	α	α	NOUN
ma-30	161	6	=	=	SYM
ma-30	161	7	j(xρ	j(xρ	PROPN
ma-30	161	8	)	)	PUNCT
ma-30	161	9	<	<	X
ma-30	161	10	2ε	2ε	NUM
ma-30	161	11	,	,	PUNCT
ma-30	161	12	thus	thus	ADV
ma-30	161	13	min	min	X
ma-30	161	14	{	{	PUNCT
ma-30	161	15	ρ	ρ	PROPN
ma-30	161	16	(	(	PUNCT
ma-30	161	17	x	x	PROPN
ma-30	161	18	+	+	NUM
ma-30	161	19	y	y	PROPN
ma-30	161	20	2	2	NUM
ma-30	161	21	)	)	PUNCT
ma-30	161	22	,	,	PUNCT
ma-30	161	23	ρ	ρ	PROPN
ma-30	161	24	(	(	PUNCT
ma-30	161	25	x	x	SYM
ma-30	161	26	−	−	PROPN
ma-30	161	27	y	y	PROPN
ma-30	161	28	2	2	NUM
ma-30	161	29	)	)	PUNCT
ma-30	161	30	}	}	PUNCT
ma-30	161	31	≤	≤	NUM
ma-30	161	32	α	α	PRON
ma-30	161	33	2	2	NUM
ma-30	161	34	,	,	PUNCT
ma-30	161	35	shows	show	VERB
ma-30	161	36	that	that	SCONJ
ma-30	161	37	1−	1−	NUM
ma-30	161	38	ρ	ρ	PROPN
ma-30	161	39	(	(	PUNCT
ma-30	161	40	x+y2	x+y2	PROPN
ma-30	161	41	)	)	PUNCT
ma-30	161	42	≥	≥	NOUN
ma-30	161	43	1−	1−	NUM
ma-30	161	44	α	α	NOUN
ma-30	161	45	2	2	NUM
ma-30	161	46	>	>	SYM
ma-30	161	47	1−	1−	NUM
ma-30	161	48	ε	ε	PROPN
ma-30	161	49	.	.	PUNCT
ma-30	161	50	therefore	therefore	ADV
ma-30	161	51	δxρ(ε	δxρ(ε	PROPN
ma-30	161	52	)	)	PUNCT
ma-30	161	53	>	>	X
ma-30	162	1	1−	1−	NUM
ma-30	162	2	ε.note	ε.note	NOUN
ma-30	162	3	β	β	X
ma-30	162	4	=	=	PUNCT
ma-30	162	5	δxρ(ε	δxρ(ε	PROPN
ma-30	162	6	)	)	PUNCT
ma-30	162	7	>	>	X
ma-30	163	1	1−	1−	NUM
ma-30	163	2	ε	ε	PROPN
ma-30	163	3	,	,	PUNCT
ma-30	163	4	then	then	ADV
ma-30	163	5	1−	1−	NUM
ma-30	163	6	ρ	ρ	PROPN
ma-30	163	7	(	(	PUNCT
ma-30	163	8	x+y2	x+y2	PROPN
ma-30	163	9	)	)	PUNCT
ma-30	163	10	≥	≥	PROPN
ma-30	163	11	β	β	PROPN
ma-30	163	12	implies	imply	VERB
ma-30	163	13	ρ	ρ	PROPN
ma-30	163	14	(	(	PUNCT
ma-30	163	15	x+y2	x+y2	PROPN
ma-30	163	16	)	)	PUNCT
ma-30	163	17	≤	≤	NOUN
ma-30	163	18	1−	1−	NUM
ma-30	163	19	β	β	X
ma-30	163	20	<	<	X
ma-30	163	21	ε	ε	PROPN
ma-30	163	22	.	.	PUNCT
ma-30	164	1	thus	thus	ADV
ma-30	164	2	min	min	X
ma-30	164	3	{	{	PUNCT
ma-30	164	4	ρ	ρ	PROPN
ma-30	164	5	(	(	PUNCT
ma-30	164	6	x	x	PROPN
ma-30	164	7	+	+	NUM
ma-30	164	8	y	y	PROPN
ma-30	164	9	2	2	NUM
ma-30	164	10	)	)	PUNCT
ma-30	164	11	,	,	PUNCT
ma-30	164	12	ρ	ρ	PROPN
ma-30	164	13	(	(	PUNCT
ma-30	164	14	x	x	SYM
ma-30	164	15	−	−	PROPN
ma-30	164	16	y	y	PROPN
ma-30	164	17	2	2	NUM
ma-30	164	18	)	)	PUNCT
ma-30	164	19	}	}	PUNCT
ma-30	164	20	=	=	SYM
ma-30	164	21	ρ	ρ	PROPN
ma-30	164	22	(	(	PUNCT
ma-30	164	23	x	x	PROPN
ma-30	164	24	+	+	NUM
ma-30	164	25	y	y	PROPN
ma-30	164	26	2	2	NUM
ma-30	164	27	)	)	PUNCT
ma-30	164	28	.	.	PUNCT
ma-30	165	1	then	then	ADV
ma-30	165	2	j	j	PROPN
ma-30	165	3	(	(	PUNCT
ma-30	165	4	xρ	xρ	PROPN
ma-30	165	5	)	)	PUNCT
ma-30	165	6	=	=	SYM
ma-30	165	7	2	2	NUM
ma-30	165	8	sup	sup	NOUN
ma-30	165	9	{	{	PUNCT
ma-30	165	10	ρ	ρ	PROPN
ma-30	165	11	(	(	PUNCT
ma-30	165	12	x	x	PROPN
ma-30	165	13	+	+	NUM
ma-30	165	14	y	y	PROPN
ma-30	165	15	2	2	NUM
ma-30	165	16	)	)	PUNCT
ma-30	165	17	:	:	PUNCT
ma-30	165	18	x	x	X
ma-30	165	19	,	,	PUNCT
ma-30	165	20	y	y	PROPN
ma-30	165	21	∈	∈	PROPN
ma-30	165	22	xρ	xρ	PROPN
ma-30	165	23	,	,	PUNCT
ma-30	165	24	ρ(x	ρ(x	PROPN
ma-30	165	25	)	)	PUNCT
ma-30	165	26	=	=	SYM
ma-30	165	27	1	1	NUM
ma-30	165	28	,	,	PUNCT
ma-30	165	29	ρ(y	ρ(y	NOUN
ma-30	165	30	)	)	PUNCT
ma-30	165	31	≤	≤	NUM
ma-30	165	32	1	1	NUM
ma-30	165	33	}	}	PUNCT
ma-30	165	34	≤	≤	NOUN
ma-30	165	35	2−	2−	NUM
ma-30	165	36	2β	2β	NOUN
ma-30	165	37	<	<	X
ma-30	165	38	2ε.in	2ε.in	PROPN
ma-30	165	39	particular	particular	ADJ
ma-30	165	40	,	,	PUNCT
ma-30	165	41	if	if	SCONJ
ma-30	165	42	ρ	ρ	PROPN
ma-30	165	43	is	be	AUX
ma-30	165	44	convex	convex	ADJ
ma-30	165	45	and	and	CCONJ
ma-30	165	46	let	let	VERB
ma-30	165	47	λ	λ	PROPN
ma-30	165	48	=	=	SYM
ma-30	165	49	j(xρ	j(xρ	PROPN
ma-30	165	50	)	)	PUNCT
ma-30	165	51	<	<	X
ma-30	165	52	ε	ε	PROPN
ma-30	165	53	,	,	PUNCT
ma-30	165	54	then	then	ADV
ma-30	165	55	j(xρ	j(xρ	PROPN
ma-30	165	56	)	)	PUNCT
ma-30	165	57	<	<	X
ma-30	165	58	ε	ε	PROPN
ma-30	165	59	if	if	SCONJ
ma-30	165	60	and	and	CCONJ
ma-30	165	61	only	only	ADV
ma-30	165	62	if	if	SCONJ
ma-30	165	63	∀x	∀x	NUM
ma-30	165	64	,	,	PUNCT
ma-30	165	65	y	y	PROPN
ma-30	165	66	∈	∈	PROPN
ma-30	165	67	xρ	xρ	PROPN
ma-30	165	68	,	,	PUNCT
ma-30	165	69	ρ(x	ρ(x	PROPN
ma-30	165	70	)	)	PUNCT
ma-30	165	71	,	,	PUNCT
ma-30	165	72	ρ(y	ρ(y	NOUN
ma-30	165	73	)	)	PUNCT
ma-30	165	74	≤	≤	NUM
ma-30	165	75	1	1	NUM
ma-30	165	76	,	,	PUNCT
ma-30	165	77	we	we	PRON
ma-30	165	78	have	have	VERB
ma-30	165	79	ρ(x	ρ(x	NOUN
ma-30	165	80	+	+	CCONJ
ma-30	165	81	y	y	NOUN
ma-30	165	82	)	)	PUNCT
ma-30	165	83	≤	≤	NUM
ma-30	165	84	λ	λ	NOUN
ma-30	165	85	or	or	CCONJ
ma-30	165	86	ρ(x	ρ(x	PROPN
ma-30	165	87	−	−	PROPN
ma-30	165	88	y	y	PROPN
ma-30	165	89	)	)	PUNCT
ma-30	165	90	≤	≤	NOUN
ma-30	165	91	λ.according	λ.accorde	VERB
ma-30	165	92	to	to	ADP
ma-30	165	93	the	the	DET
ma-30	165	94	definition	definition	NOUN
ma-30	165	95	of	of	ADP
ma-30	165	96	δxρ(ε	δxρ(ε	PROPN
ma-30	165	97	)	)	PUNCT
ma-30	165	98	,	,	PUNCT
ma-30	165	99	we	we	PRON
ma-30	165	100	obtain	obtain	VERB
ma-30	165	101	ρ(x	ρ(x	PROPN
ma-30	165	102	+	+	CCONJ
ma-30	165	103	y	y	NOUN
ma-30	165	104	)	)	PUNCT
ma-30	165	105	≥	≥	X
ma-30	165	106	ε	ε	PROPN
ma-30	165	107	>	>	X
ma-30	165	108	λ	λ	PROPN
ma-30	165	109	,	,	PUNCT
ma-30	165	110	thus	thus	ADV
ma-30	165	111	ρ(x	ρ(x	PROPN
ma-30	165	112	−	−	PROPN
ma-30	165	113	y	y	SYM
ma-30	165	114	)	)	PUNCT
ma-30	165	115	≤	≤	NUM
ma-30	166	1	λ	λ	PROPN
ma-30	166	2	shows	show	VERB
ma-30	166	3	that	that	SCONJ
ma-30	166	4	δxρ(ε	δxρ(ε	PROPN
ma-30	166	5	)	)	PUNCT
ma-30	166	6	≥	≥	NOUN
ma-30	166	7	1−	1−	NUM
ma-30	166	8	α	α	NOUN
ma-30	166	9	2	2	NUM
ma-30	166	10	>	>	SYM
ma-30	166	11	1−	1−	NUM
ma-30	166	12	ε	ε	PROPN
ma-30	166	13	2	2	NUM
ma-30	166	14	.	.	PUNCT
ma-30	167	1	(	(	PUNCT
ma-30	167	2	ii	ii	NOUN
ma-30	167	3	)	)	PUNCT
ma-30	167	4	note	note	VERB
ma-30	167	5	ε0	ε0	NOUN
ma-30	167	6	=	=	PUNCT
ma-30	167	7	sup{ε	sup{ε	NOUN
ma-30	167	8	∈	∈	PROPN
ma-30	167	9	(	(	PUNCT
ma-30	167	10	0	0	NUM
ma-30	167	11	,	,	PUNCT
ma-30	167	12	2	2	NUM
ma-30	167	13	)	)	PUNCT
ma-30	167	14	:	:	PUNCT
ma-30	168	1	δxρ(ε	δxρ(ε	X
ma-30	168	2	)	)	PUNCT
ma-30	168	3	≤	≤	NOUN
ma-30	168	4	1−	1−	NUM
ma-30	168	5	ε}.suppose	ε}.suppose	NUM
ma-30	168	6	ε0	ε0	PROPN
ma-30	168	7	<	<	X
ma-30	168	8	2	2	NUM
ma-30	168	9	.	.	PUNCT
ma-30	169	1	∀ε	∀ε	X
ma-30	169	2	∈	∈	PROPN
ma-30	169	3	(	(	PUNCT
ma-30	169	4	ε0	ε0	PROPN
ma-30	169	5	,	,	PUNCT
ma-30	169	6	2	2	NUM
ma-30	169	7	)	)	PUNCT
ma-30	169	8	,	,	PUNCT
ma-30	169	9	for	for	ADP
ma-30	169	10	any	any	DET
ma-30	169	11	x	x	NOUN
ma-30	169	12	,	,	PUNCT
ma-30	169	13	y	y	PROPN
ma-30	169	14	∈	∈	PROPN
ma-30	169	15	xρ	xρ	PROPN
ma-30	169	16	and	and	CCONJ
ma-30	169	17	ρ(x	ρ(x	NUM
ma-30	169	18	)	)	PUNCT
ma-30	169	19	,	,	PUNCT
ma-30	169	20	ρ(y	ρ(y	NOUN
ma-30	169	21	)	)	PUNCT
ma-30	169	22	≤	≤	NUM
ma-30	169	23	1	1	NUM
ma-30	169	24	,	,	PUNCT
ma-30	169	25	we	we	PRON
ma-30	169	26	have	have	VERB
ma-30	169	27	ρ(x	ρ(x	PROPN
ma-30	169	28	−	−	PROPN
ma-30	169	29	y	y	PROPN
ma-30	169	30	)	)	PUNCT
ma-30	169	31	>	>	X
ma-30	169	32	ε	ε	PROPN
ma-30	169	33	or	or	CCONJ
ma-30	169	34	ρ(x	ρ(x	PROPN
ma-30	169	35	−	−	PROPN
ma-30	169	36	y	y	PROPN
ma-30	169	37	)	)	PUNCT
ma-30	169	38	≤	≤	NOUN
ma-30	170	1	ε	ε	PROPN
ma-30	170	2	.	.	PUNCT
ma-30	171	1	if	if	SCONJ
ma-30	171	2	ρ(x	ρ(x	PROPN
ma-30	171	3	−	−	PROPN
ma-30	171	4	y	y	PROPN
ma-30	171	5	)	)	PUNCT
ma-30	171	6	>	>	X
ma-30	172	1	ε	ε	PROPN
ma-30	172	2	,	,	PUNCT
ma-30	172	3	then	then	ADV
ma-30	172	4	δxρ(ε	δxρ(ε	X
ma-30	172	5	)	)	PUNCT
ma-30	172	6	≥	≥	NOUN
ma-30	172	7	1−	1−	NUM
ma-30	172	8	ε	ε	PROPN
ma-30	172	9	implies	imply	VERB
ma-30	172	10	ρ	ρ	PROPN
ma-30	172	11	(	(	PUNCT
ma-30	172	12	x+y2	x+y2	PROPN
ma-30	172	13	)	)	PUNCT
ma-30	172	14	≤	≤	NUM
ma-30	172	15	ε	ε	PROPN
ma-30	172	16	.	.	PUNCT
ma-30	173	1	thus	thus	ADV
ma-30	173	2	j(xρ	j(xρ	PROPN
ma-30	173	3	)	)	PUNCT
ma-30	173	4	≤	≤	NUM
ma-30	173	5	2ε	2ε	NOUN
ma-30	173	6	.	.	PUNCT
ma-30	174	1	eur	eur	PROPN
ma-30	174	2	.	.	PUNCT
ma-30	175	1	j.	j.	PROPN
ma-30	175	2	math	math	PROPN
ma-30	175	3	.	.	PUNCT
ma-30	176	1	anal	anal	ADJ
ma-30	176	2	.	.	PUNCT
ma-30	177	1	1	1	NUM
ma-30	177	2	(	(	PUNCT
ma-30	177	3	2021	2021	NUM
ma-30	177	4	)	)	PUNCT
ma-30	178	1	157since	157since	PROPN
ma-30	178	2	δxρ(ε	δxρ(ε	PROPN
ma-30	178	3	)	)	PUNCT
ma-30	178	4	≤	≤	NOUN
ma-30	178	5	1−	1−	NUM
ma-30	178	6	ε	ε	PROPN
ma-30	178	7	,	,	PUNCT
ma-30	178	8	then	then	ADV
ma-30	178	9	j(xρ	j(xρ	PROPN
ma-30	178	10	)	)	PUNCT
ma-30	178	11	≤	≤	NOUN
ma-30	178	12	2ε0	2ε0	NUM
ma-30	178	13	shows	show	VERB
ma-30	178	14	that	that	SCONJ
ma-30	178	15	j(xρ	j(xρ	PROPN
ma-30	178	16	)	)	PUNCT
ma-30	178	17	=	=	SYM
ma-30	178	18	2	2	NUM
ma-30	178	19	sup{ε	sup{ε	NOUN
ma-30	178	20	∈	∈	PROPN
ma-30	178	21	(	(	PUNCT
ma-30	178	22	0	0	NUM
ma-30	178	23	,	,	PUNCT
ma-30	178	24	2	2	NUM
ma-30	178	25	)	)	PUNCT
ma-30	178	26	:	:	PUNCT
ma-30	178	27	δxρ(ε	δxρ(ε	X
ma-30	178	28	)	)	PUNCT
ma-30	178	29	≤	≤	NOUN
ma-30	178	30	1−	1−	NUM
ma-30	178	31	ε	ε	PROPN
ma-30	178	32	}	}	PUNCT
ma-30	178	33	.	.	PUNCT
ma-30	179	1	in	in	ADP
ma-30	179	2	particular	particular	ADJ
ma-30	179	3	,	,	PUNCT
ma-30	179	4	if	if	SCONJ
ma-30	179	5	ρ	ρ	PROPN
ma-30	179	6	is	be	AUX
ma-30	179	7	convex	convex	ADJ
ma-30	179	8	and	and	CCONJ
ma-30	179	9	let	let	VERB
ma-30	179	10	α	α	PROPN
ma-30	179	11	=	=	PUNCT
ma-30	179	12	j(xρ	j(xρ	PROPN
ma-30	179	13	)	)	PUNCT
ma-30	179	14	∈	∈	PROPN
ma-30	180	1	[	[	X
ma-30	180	2	1	1	NUM
ma-30	180	3	,	,	PUNCT
ma-30	180	4	2	2	NUM
ma-30	180	5	]	]	PUNCT
ma-30	180	6	,	,	PUNCT
ma-30	180	7	then	then	ADV
ma-30	180	8	∀x	∀x	X
ma-30	180	9	,	,	PUNCT
ma-30	180	10	y	y	PROPN
ma-30	180	11	∈	∈	PROPN
ma-30	180	12	xρ	xρ	PROPN
ma-30	180	13	,	,	PUNCT
ma-30	180	14	ρ(x	ρ(x	PROPN
ma-30	180	15	)	)	PUNCT
ma-30	180	16	,	,	PUNCT
ma-30	180	17	ρ(y	ρ(y	NOUN
ma-30	180	18	)	)	PUNCT
ma-30	180	19	≤	≤	NUM
ma-30	180	20	1	1	NUM
ma-30	180	21	,	,	PUNCT
ma-30	180	22	we	we	PRON
ma-30	180	23	have	have	VERB
ma-30	180	24	ρ(x	ρ(x	NOUN
ma-30	180	25	+	+	CCONJ
ma-30	180	26	y	y	NOUN
ma-30	180	27	)	)	PUNCT
ma-30	180	28	≤	≤	NOUN
ma-30	180	29	α	α	NOUN
ma-30	180	30	or	or	CCONJ
ma-30	180	31	ρ(x	ρ(x	PROPN
ma-30	180	32	−	−	PROPN
ma-30	180	33	y	y	PROPN
ma-30	180	34	)	)	PUNCT
ma-30	180	35	≤	≤	NOUN
ma-30	181	1	α	α	X
ma-30	181	2	.	.	PUNCT
ma-30	182	1	what	what	PRON
ma-30	182	2	’s	’	VERB
ma-30	182	3	more	more	ADJ
ma-30	182	4	,	,	PUNCT
ma-30	182	5	∀η	∀η	X
ma-30	182	6	>	>	X
ma-30	182	7	0	0	PROPN
ma-30	182	8	,	,	PUNCT
ma-30	182	9	there	there	PRON
ma-30	182	10	exist	exist	VERB
ma-30	182	11	x	x	NOUN
ma-30	182	12	′	′	NUM
ma-30	182	13	,	,	PUNCT
ma-30	182	14	y	y	PROPN
ma-30	182	15	′	′	NUM
ma-30	182	16	∈	∈	PROPN
ma-30	182	17	xρ	xρ	PROPN
ma-30	183	1	and	and	CCONJ
ma-30	183	2	ρ(x	ρ(x	PROPN
ma-30	183	3	′	′	NOUN
ma-30	183	4	)	)	PUNCT
ma-30	183	5	,	,	PUNCT
ma-30	184	1	ρ(y	ρ(y	PROPN
ma-30	184	2	′	′	NUM
ma-30	184	3	)	)	PUNCT
ma-30	184	4	≤	≤	NUM
ma-30	184	5	1	1	NUM
ma-30	184	6	such	such	ADJ
ma-30	184	7	that	that	SCONJ
ma-30	184	8	ρ	ρ	PROPN
ma-30	184	9	(	(	PUNCT
ma-30	184	10	x	x	SYM
ma-30	184	11	′	′	NUM
ma-30	185	1	+	+	CCONJ
ma-30	185	2	y	y	PROPN
ma-30	185	3	′	′	NUM
ma-30	185	4	)	)	PUNCT
ma-30	185	5	>	>	X
ma-30	186	1	α−	α−	ADP
ma-30	186	2	η	η	PROPN
ma-30	186	3	and	and	CCONJ
ma-30	186	4	ρ	ρ	PROPN
ma-30	186	5	(	(	PUNCT
ma-30	186	6	x	x	NOUN
ma-30	186	7	′	′	NUM
ma-30	186	8	−	−	NOUN
ma-30	186	9	y	y	PROPN
ma-30	186	10	′	′	PROPN
ma-30	186	11	)	)	PUNCT
ma-30	186	12	>	>	X
ma-30	186	13	α−	α−	ADP
ma-30	186	14	η	η	PROPN
ma-30	186	15	.	.	PROPN
ma-30	186	16	fix	fix	PROPN
ma-30	186	17	η	η	PROPN
ma-30	186	18	>	>	X
ma-30	186	19	0	0	PROPN
ma-30	186	20	,	,	PUNCT
ma-30	186	21	then	then	ADV
ma-30	186	22	1−	1−	NUM
ma-30	186	23	ρ(x	ρ(x	NOUN
ma-30	186	24	′	′	NUM
ma-30	187	1	+	+	CCONJ
ma-30	187	2	y	y	PROPN
ma-30	187	3	′	′	NUM
ma-30	187	4	2	2	NUM
ma-30	187	5	)	)	PUNCT
ma-30	187	6	<	<	X
ma-30	187	7	1−	1−	NUM
ma-30	187	8	α−	α−	ADP
ma-30	187	9	η	η	PROPN
ma-30	187	10	2	2	PROPN
ma-30	187	11	implies	imply	VERB
ma-30	187	12	δxρ(ε	δxρ(ε	X
ma-30	187	13	)	)	PUNCT
ma-30	187	14	<	<	X
ma-30	187	15	1−	1−	NUM
ma-30	187	16	α−	α−	ADP
ma-30	187	17	η2	η2	NOUN
ma-30	187	18	,	,	PUNCT
ma-30	187	19	therefore	therefore	ADV
ma-30	187	20	sup	sup	PROPN
ma-30	187	21	{	{	PUNCT
ma-30	187	22	ε	ε	PROPN
ma-30	187	23	∈	∈	PROPN
ma-30	187	24	(	(	PUNCT
ma-30	187	25	0	0	NUM
ma-30	187	26	,	,	PUNCT
ma-30	187	27	2	2	NUM
ma-30	187	28	)	)	PUNCT
ma-30	187	29	:	:	PUNCT
ma-30	188	1	δxρ(ε	δxρ(ε	X
ma-30	188	2	)	)	PUNCT
ma-30	188	3	<	<	X
ma-30	188	4	1−	1−	NUM
ma-30	188	5	ε	ε	PROPN
ma-30	188	6	2	2	NUM
ma-30	188	7	}	}	PUNCT
ma-30	188	8	≥	≥	NOUN
ma-30	188	9	α−	α−	ADP
ma-30	188	10	η	η	PROPN
ma-30	188	11	.	.	PROPN
ma-30	188	12	∀ε	∀ε	X
ma-30	188	13	∈	∈	PROPN
ma-30	188	14	(	(	PUNCT
ma-30	188	15	0	0	NUM
ma-30	188	16	,	,	PUNCT
ma-30	188	17	2	2	NUM
ma-30	188	18	)	)	PUNCT
ma-30	188	19	,	,	PUNCT
ma-30	188	20	if	if	SCONJ
ma-30	188	21	ε	ε	PROPN
ma-30	188	22	≤	≤	NOUN
ma-30	188	23	α	α	NUM
ma-30	188	24	,	,	PUNCT
ma-30	188	25	thus	thus	ADV
ma-30	188	26	sup	sup	NOUN
ma-30	188	27	{	{	PUNCT
ma-30	188	28	ε	ε	PROPN
ma-30	188	29	∈	∈	PROPN
ma-30	188	30	(	(	PUNCT
ma-30	188	31	0	0	NUM
ma-30	188	32	,	,	PUNCT
ma-30	188	33	2	2	NUM
ma-30	188	34	)	)	PUNCT
ma-30	188	35	:	:	PUNCT
ma-30	189	1	δxρ(ε	δxρ(ε	X
ma-30	189	2	)	)	PUNCT
ma-30	189	3	<	<	X
ma-30	189	4	1−	1−	NUM
ma-30	189	5	ε	ε	PROPN
ma-30	189	6	2	2	NUM
ma-30	189	7	}	}	PUNCT
ma-30	189	8	≤	≤	NUM
ma-30	189	9	α	α	X
ma-30	189	10	.	.	PUNCT
ma-30	190	1	if	if	SCONJ
ma-30	190	2	ε	ε	PROPN
ma-30	190	3	>	>	X
ma-30	190	4	α	α	PROPN
ma-30	190	5	,	,	PUNCT
ma-30	190	6	then	then	ADV
ma-30	190	7	ρ(x	ρ(x	PROPN
ma-30	190	8	+	+	PROPN
ma-30	190	9	y	y	NOUN
ma-30	190	10	)	)	PUNCT
ma-30	190	11	≤	≤	NOUN
ma-30	191	1	α	α	PROPN
ma-30	191	2	shows	show	VERB
ma-30	191	3	that	that	SCONJ
ma-30	191	4	δxρ(ε	δxρ(ε	PROPN
ma-30	191	5	)	)	PUNCT
ma-30	191	6	≥	≥	NOUN
ma-30	191	7	1−	1−	NUM
ma-30	191	8	α	α	NOUN
ma-30	191	9	2	2	NUM
ma-30	191	10	.	.	PUNCT
ma-30	192	1	in	in	ADP
ma-30	192	2	(	(	PUNCT
ma-30	192	3	0	0	NUM
ma-30	192	4	,	,	PUNCT
ma-30	192	5	2	2	NUM
ma-30	192	6	)	)	PUNCT
ma-30	192	7	,	,	PUNCT
ma-30	192	8	we	we	PRON
ma-30	192	9	know	know	VERB
ma-30	192	10	sup	sup	NOUN
ma-30	192	11	{	{	PUNCT
ma-30	192	12	ε	ε	PROPN
ma-30	192	13	∈	∈	PROPN
ma-30	192	14	(	(	PUNCT
ma-30	192	15	0	0	NUM
ma-30	192	16	,	,	PUNCT
ma-30	192	17	2	2	NUM
ma-30	192	18	)	)	PUNCT
ma-30	192	19	:	:	PUNCT
ma-30	193	1	δxρ(ε	δxρ(ε	X
ma-30	193	2	)	)	PUNCT
ma-30	193	3	<	<	X
ma-30	193	4	1−	1−	NUM
ma-30	193	5	ε	ε	PROPN
ma-30	193	6	2	2	NUM
ma-30	193	7	}	}	PUNCT
ma-30	193	8	≤	≤	NUM
ma-30	193	9	α	α	NOUN
ma-30	193	10	,	,	PUNCT
ma-30	193	11	thus	thus	ADV
ma-30	193	12	α−	α−	ADP
ma-30	193	13	η	η	NOUN
ma-30	193	14	≤	≤	NOUN
ma-30	193	15	sup{ε	sup{ε	NOUN
ma-30	193	16	∈	∈	PROPN
ma-30	193	17	(	(	PUNCT
ma-30	193	18	0	0	NUM
ma-30	193	19	,	,	PUNCT
ma-30	193	20	2	2	NUM
ma-30	193	21	)	)	PUNCT
ma-30	193	22	:	:	PUNCT
ma-30	194	1	δxρ(ε	δxρ(ε	X
ma-30	194	2	)	)	PUNCT
ma-30	194	3	<	<	X
ma-30	194	4	1−	1−	NUM
ma-30	194	5	ε	ε	PROPN
ma-30	194	6	2	2	NUM
ma-30	194	7	}	}	PUNCT
ma-30	194	8	≤	≤	NUM
ma-30	194	9	α.let	α.let	PROPN
ma-30	194	10	η	η	PROPN
ma-30	194	11	→	→	X
ma-30	194	12	0	0	PROPN
ma-30	194	13	,	,	PUNCT
ma-30	194	14	then	then	ADV
ma-30	194	15	sup	sup	AUX
ma-30	194	16	{	{	PUNCT
ma-30	194	17	ε	ε	PROPN
ma-30	194	18	∈	∈	PROPN
ma-30	194	19	(	(	PUNCT
ma-30	194	20	0	0	NUM
ma-30	194	21	,	,	PUNCT
ma-30	194	22	2	2	NUM
ma-30	194	23	)	)	PUNCT
ma-30	194	24	:	:	PUNCT
ma-30	194	25	δxρ(ε	δxρ(ε	X
ma-30	194	26	)	)	PUNCT
ma-30	194	27	<	<	X
ma-30	194	28	1−	1−	NUM
ma-30	194	29	ε	ε	PROPN
ma-30	194	30	2	2	NUM
ma-30	194	31	}	}	PUNCT
ma-30	194	32	=	=	SYM
ma-30	194	33	α	α	X
ma-30	194	34	.	.	PUNCT
ma-30	194	35	theorem	theorem	NOUN
ma-30	194	36	3	3	X
ma-30	194	37	.	.	PUNCT
ma-30	195	1	let	let	VERB
ma-30	195	2	xρ	xρ	PROPN
ma-30	195	3	be	be	AUX
ma-30	195	4	a	a	DET
ma-30	195	5	modular	modular	ADJ
ma-30	195	6	space	space	NOUN
ma-30	195	7	,	,	PUNCT
ma-30	195	8	then(i	then(i	NOUN
ma-30	195	9	)	)	PUNCT
ma-30	195	10	j(xρ	j(xρ	PROPN
ma-30	195	11	)	)	PUNCT
ma-30	195	12	≤	≤	NUM
ma-30	195	13	ρxρ(1	ρxρ(1	NUM
ma-30	195	14	)	)	PUNCT
ma-30	196	1	+	+	CCONJ
ma-30	196	2	1;(ii	1;(ii	NUM
ma-30	196	3	)	)	PUNCT
ma-30	196	4	cnj(xρ	cnj(xρ	NOUN
ma-30	196	5	)	)	PUNCT
ma-30	196	6	≤	≤	NUM
ma-30	196	7	2(√12	2(√12	NUM
ma-30	197	1	+	+	CCONJ
ma-30	197	2	(	(	PUNCT
ma-30	197	3	1	1	NUM
ma-30	197	4	+	+	NUM
ma-30	197	5	ρxρ(1))2	ρxρ(1))2	NOUN
ma-30	197	6	−	−	PROPN
ma-30	197	7	2)2	2)2	NUM
ma-30	197	8	.	.	PUNCT
ma-30	198	1	proof	proof	NOUN
ma-30	198	2	.	.	PUNCT
ma-30	199	1	(	(	PUNCT
ma-30	199	2	i	i	NOUN
ma-30	199	3	)	)	PUNCT
ma-30	199	4	we	we	PRON
ma-30	199	5	can	can	AUX
ma-30	199	6	deduce	deduce	VERB
ma-30	199	7	that	that	DET
ma-30	199	8	j	j	PROPN
ma-30	199	9	(	(	PUNCT
ma-30	199	10	xρ	xρ	PROPN
ma-30	199	11	)	)	PUNCT
ma-30	199	12	≤	≤	NUM
ma-30	199	13	sup	sup	NOUN
ma-30	199	14	{	{	PUNCT
ma-30	199	15	ρ	ρ	PROPN
ma-30	199	16	(	(	PUNCT
ma-30	199	17	x	x	PROPN
ma-30	199	18	+	+	NUM
ma-30	199	19	y	y	PROPN
ma-30	199	20	2	2	NUM
ma-30	199	21	)	)	PUNCT
ma-30	200	1	+	+	CCONJ
ma-30	200	2	ρ	ρ	PROPN
ma-30	200	3	(	(	PUNCT
ma-30	200	4	x	x	SYM
ma-30	200	5	−	−	PROPN
ma-30	200	6	y	y	PROPN
ma-30	200	7	2	2	NUM
ma-30	200	8	)	)	PUNCT
ma-30	200	9	:	:	PUNCT
ma-30	201	1	x	x	X
ma-30	201	2	,	,	PUNCT
ma-30	201	3	y	y	PROPN
ma-30	201	4	∈	∈	PROPN
ma-30	201	5	xρ	xρ	PROPN
ma-30	201	6	,	,	PUNCT
ma-30	201	7	ρ(x	ρ(x	PROPN
ma-30	201	8	)	)	PUNCT
ma-30	201	9	=	=	SYM
ma-30	201	10	1	1	NUM
ma-30	201	11	,	,	PUNCT
ma-30	201	12	ρ(y	ρ(y	NOUN
ma-30	201	13	)	)	PUNCT
ma-30	201	14	≤	≤	NOUN
ma-30	201	15	1	1	NUM
ma-30	201	16	}	}	PUNCT
ma-30	201	17	=	=	SYM
ma-30	201	18	ρxρ(1	ρxρ(1	NUM
ma-30	201	19	)	)	PUNCT
ma-30	202	1	+	+	CCONJ
ma-30	202	2	1	1	X
ma-30	202	3	.	.	X
ma-30	202	4	(	(	PUNCT
ma-30	202	5	ii	ii	NOUN
ma-30	202	6	)	)	PUNCT
ma-30	202	7	we	we	PRON
ma-30	202	8	know	know	VERB
ma-30	202	9	that	that	SCONJ
ma-30	202	10	a2	a2	PROPN
ma-30	202	11	+	+	CCONJ
ma-30	202	12	b2	b2	NOUN
ma-30	202	13	≤	≤	NOUN
ma-30	202	14	(	(	PUNCT
ma-30	202	15	a	a	DET
ma-30	202	16	+	+	NUM
ma-30	202	17	b)2	b)2	ADJ
ma-30	202	18	−	−	PROPN
ma-30	202	19	4(a	4(a	NUM
ma-30	202	20	+	+	CCONJ
ma-30	202	21	b	b	X
ma-30	202	22	)	)	PUNCT
ma-30	203	1	+	+	CCONJ
ma-30	203	2	8	8	NUM
ma-30	203	3	for	for	ADP
ma-30	203	4	0	0	NUM
ma-30	203	5	<	<	X
ma-30	203	6	a	a	PROPN
ma-30	203	7	,	,	PUNCT
ma-30	203	8	b	b	PROPN
ma-30	203	9	≤	≤	ADV
ma-30	203	10	2	2	NUM
ma-30	203	11	.	.	PUNCT
ma-30	204	1	thus	thus	ADV
ma-30	204	2	ρ2	ρ2	VERB
ma-30	204	3	(	(	PUNCT
ma-30	204	4	x	x	SYM
ma-30	204	5	+	+	NUM
ma-30	204	6	y	y	PROPN
ma-30	204	7	2	2	NUM
ma-30	204	8	)	)	PUNCT
ma-30	205	1	+	+	CCONJ
ma-30	205	2	ρ2	ρ2	NOUN
ma-30	205	3	(	(	PUNCT
ma-30	205	4	x	x	SYM
ma-30	205	5	−	−	PROPN
ma-30	205	6	y	y	PROPN
ma-30	205	7	2	2	NUM
ma-30	205	8	)	)	PUNCT
ma-30	205	9	≤	≤	NOUN
ma-30	205	10	(	(	PUNCT
ma-30	205	11	ρ	ρ	NOUN
ma-30	205	12	(	(	PUNCT
ma-30	205	13	x	x	PROPN
ma-30	205	14	+	+	NUM
ma-30	205	15	y	y	PROPN
ma-30	205	16	2	2	NUM
ma-30	205	17	)	)	PUNCT
ma-30	205	18	+	+	CCONJ
ma-30	205	19	ρ	ρ	PROPN
ma-30	205	20	(	(	PUNCT
ma-30	205	21	x	x	SYM
ma-30	205	22	−	−	PROPN
ma-30	205	23	y	y	PROPN
ma-30	205	24	2	2	NUM
ma-30	205	25	)	)	PUNCT
ma-30	205	26	)	)	PUNCT
ma-30	205	27	2	2	NUM
ma-30	205	28	−	−	NOUN
ma-30	205	29	4	4	NUM
ma-30	205	30	(	(	PUNCT
ma-30	205	31	ρ	ρ	PROPN
ma-30	205	32	(	(	PUNCT
ma-30	205	33	x	x	PROPN
ma-30	205	34	+	+	NUM
ma-30	205	35	y	y	PROPN
ma-30	205	36	2	2	NUM
ma-30	205	37	)	)	PUNCT
ma-30	205	38	+	+	CCONJ
ma-30	205	39	ρ	ρ	PROPN
ma-30	205	40	(	(	PUNCT
ma-30	205	41	x	x	SYM
ma-30	205	42	−	−	PROPN
ma-30	205	43	y	y	PROPN
ma-30	205	44	2	2	NUM
ma-30	205	45	)	)	PUNCT
ma-30	205	46	)	)	PUNCT
ma-30	206	1	+	+	CCONJ
ma-30	206	2	8	8	X
ma-30	206	3	.	.	PUNCT
ma-30	206	4	since	since	SCONJ
ma-30	206	5	ρ	ρ	PROPN
ma-30	206	6	(	(	PUNCT
ma-30	206	7	x	x	PROPN
ma-30	206	8	+	+	NUM
ma-30	206	9	y	y	PROPN
ma-30	206	10	2	2	NUM
ma-30	206	11	)	)	PUNCT
ma-30	206	12	+	+	CCONJ
ma-30	206	13	ρ	ρ	PROPN
ma-30	206	14	(	(	PUNCT
ma-30	206	15	x	x	SYM
ma-30	206	16	−	−	PROPN
ma-30	206	17	y	y	PROPN
ma-30	206	18	2	2	NUM
ma-30	206	19	)	)	PUNCT
ma-30	206	20	≥	≥	NOUN
ma-30	206	21	√	√	NUM
ma-30	206	22	ρ2	ρ2	PROPN
ma-30	206	23	(	(	PUNCT
ma-30	206	24	x	x	SYM
ma-30	206	25	+	+	NUM
ma-30	206	26	y	y	PROPN
ma-30	206	27	2	2	NUM
ma-30	206	28	)	)	PUNCT
ma-30	207	1	+	+	CCONJ
ma-30	207	2	ρ2	ρ2	NOUN
ma-30	207	3	(	(	PUNCT
ma-30	207	4	x	x	SYM
ma-30	207	5	−	−	PROPN
ma-30	207	6	y	y	PROPN
ma-30	207	7	2	2	NUM
ma-30	207	8	)	)	PUNCT
ma-30	207	9	,	,	PUNCT
ma-30	207	10	eur	eur	PROPN
ma-30	207	11	.	.	PUNCT
ma-30	208	1	j.	j.	PROPN
ma-30	208	2	math	math	PROPN
ma-30	208	3	.	.	PUNCT
ma-30	209	1	anal	anal	ADJ
ma-30	209	2	.	.	PUNCT
ma-30	210	1	1	1	NUM
ma-30	210	2	(	(	PUNCT
ma-30	210	3	2021	2021	NUM
ma-30	210	4	)	)	PUNCT
ma-30	211	1	158then	158then	PROPN
ma-30	211	2	ρ2	ρ2	NOUN
ma-30	211	3	(	(	PUNCT
ma-30	211	4	x	x	SYM
ma-30	211	5	+	+	NUM
ma-30	211	6	y	y	PROPN
ma-30	211	7	2	2	NUM
ma-30	211	8	)	)	PUNCT
ma-30	211	9	+	+	CCONJ
ma-30	211	10	ρ2	ρ2	NOUN
ma-30	211	11	(	(	PUNCT
ma-30	211	12	x	x	SYM
ma-30	211	13	−	−	PROPN
ma-30	211	14	y	y	PROPN
ma-30	211	15	2	2	NUM
ma-30	211	16	)	)	PUNCT
ma-30	211	17	+	+	CCONJ
ma-30	211	18	4	4	NUM
ma-30	211	19	√	√	NOUN
ma-30	211	20	ρ2	ρ2	NOUN
ma-30	211	21	(	(	PUNCT
ma-30	211	22	x	x	SYM
ma-30	211	23	+	+	NUM
ma-30	211	24	y	y	PROPN
ma-30	211	25	2	2	NUM
ma-30	211	26	)	)	PUNCT
ma-30	211	27	+	+	CCONJ
ma-30	211	28	ρ2	ρ2	NOUN
ma-30	211	29	(	(	PUNCT
ma-30	211	30	x	x	SYM
ma-30	211	31	−	−	PROPN
ma-30	211	32	y	y	PROPN
ma-30	211	33	2	2	NUM
ma-30	211	34	)	)	PUNCT
ma-30	211	35	−	−	NOUN
ma-30	211	36	8	8	NUM
ma-30	211	37	≤	≤	NOUN
ma-30	211	38	(	(	PUNCT
ma-30	211	39	ρ	ρ	NOUN
ma-30	211	40	(	(	PUNCT
ma-30	211	41	x	x	PROPN
ma-30	211	42	+	+	NUM
ma-30	211	43	y	y	PROPN
ma-30	211	44	2	2	NUM
ma-30	211	45	)	)	PUNCT
ma-30	211	46	+	+	CCONJ
ma-30	211	47	ρ	ρ	PROPN
ma-30	211	48	(	(	PUNCT
ma-30	211	49	x	x	SYM
ma-30	211	50	−	−	PROPN
ma-30	211	51	y	y	PROPN
ma-30	211	52	2	2	NUM
ma-30	211	53	)	)	PUNCT
ma-30	211	54	)	)	PUNCT
ma-30	211	55	2	2	NUM
ma-30	211	56	≤	≤	NOUN
ma-30	211	57	(	(	PUNCT
ma-30	211	58	1	1	NUM
ma-30	211	59	+	+	NUM
ma-30	211	60	ρxρ(1	ρxρ(1	NUM
ma-30	211	61	)	)	PUNCT
ma-30	211	62	)	)	PUNCT
ma-30	211	63	2	2	NUM
ma-30	211	64	.thus	.thus	PRON
ma-30	211	65	ρ2	ρ2	NOUN
ma-30	211	66	(	(	PUNCT
ma-30	211	67	x	x	SYM
ma-30	211	68	+	+	NUM
ma-30	211	69	y	y	PROPN
ma-30	211	70	2	2	NUM
ma-30	211	71	)	)	PUNCT
ma-30	211	72	+	+	CCONJ
ma-30	211	73	ρ2	ρ2	NOUN
ma-30	211	74	(	(	PUNCT
ma-30	211	75	x	x	SYM
ma-30	211	76	−	−	PROPN
ma-30	211	77	y	y	PROPN
ma-30	211	78	2	2	NUM
ma-30	211	79	)	)	PUNCT
ma-30	211	80	≤	≤	NOUN
ma-30	211	81	(	(	PUNCT
ma-30	211	82	√	√	PROPN
ma-30	211	83	12	12	NUM
ma-30	211	84	+	+	CCONJ
ma-30	211	85	(	(	PUNCT
ma-30	211	86	1	1	NUM
ma-30	211	87	+	+	NUM
ma-30	211	88	ρxρ(1	ρxρ(1	NUM
ma-30	211	89	)	)	PUNCT
ma-30	211	90	)	)	PUNCT
ma-30	211	91	2	2	NUM
ma-30	211	92	−	−	PROPN
ma-30	211	93	2)2	2)2	NUM
ma-30	211	94	which	which	PRON
ma-30	211	95	shows	show	VERB
ma-30	211	96	that	that	SCONJ
ma-30	211	97	12cnj(xρ	12cnj(xρ	NUM
ma-30	211	98	)	)	PUNCT
ma-30	211	99	≤	≤	NOUN
ma-30	211	100	(	(	PUNCT
ma-30	211	101	√12	√12	PROPN
ma-30	211	102	+	+	CCONJ
ma-30	211	103	(	(	PUNCT
ma-30	211	104	1	1	NUM
ma-30	211	105	+	+	NUM
ma-30	211	106	ρxρ(1))2	ρxρ(1))2	NOUN
ma-30	211	107	−	−	PROPN
ma-30	211	108	2)2	2)2	NUM
ma-30	211	109	.	.	PUNCT
ma-30	212	1	5	5	X
ma-30	212	2	.	.	X
ma-30	212	3	convexity	convexity	NOUN
ma-30	212	4	and	and	CCONJ
ma-30	212	5	non	non	ADJ
ma-30	212	6	-	-	ADJ
ma-30	212	7	squareness	squareness	ADJ
ma-30	212	8	clarkson	clarkson	PROPN
ma-30	212	9	introduced	introduce	VERB
ma-30	212	10	uniform	uniform	ADJ
ma-30	212	11	convexity	convexity	NOUN
ma-30	212	12	in	in	ADP
ma-30	212	13	1936	1936	NUM
ma-30	212	14	,	,	PUNCT
ma-30	212	15	proved	prove	VERB
ma-30	212	16	that	that	SCONJ
ma-30	212	17	lp(1	lp(1	PROPN
ma-30	212	18	≤	≤	PROPN
ma-30	213	1	p	p	PROPN
ma-30	213	2	<	<	X
ma-30	213	3	∞	∞	NOUN
ma-30	213	4	)	)	PUNCT
ma-30	213	5	spaces	space	NOUN
ma-30	213	6	are	be	AUX
ma-30	213	7	uniformlyconvex	uniformlyconvex	ADJ
ma-30	213	8	banach	banach	NOUN
ma-30	213	9	spaces	space	NOUN
ma-30	213	10	and	and	CCONJ
ma-30	213	11	uniformly	uniformly	ADV
ma-30	213	12	convex	convex	NOUN
ma-30	213	13	banach	banach	NOUN
ma-30	213	14	spaces	space	NOUN
ma-30	213	15	have	have	VERB
ma-30	213	16	radon	radon	VERB
ma-30	213	17	-	-	PUNCT
ma-30	213	18	nikodym	nikodym	ADJ
ma-30	213	19	properties	property	NOUN
ma-30	213	20	.	.	PUNCT
ma-30	214	1	dueto	dueto	VERB
ma-30	214	2	the	the	DET
ma-30	214	3	geometrical	geometrical	ADJ
ma-30	214	4	intuitiveness	intuitiveness	NOUN
ma-30	214	5	of	of	ADP
ma-30	214	6	convexity	convexity	NOUN
ma-30	214	7	,	,	PUNCT
ma-30	214	8	poom	poom	NOUN
ma-30	214	9	kumam	kumam	NOUN
ma-30	215	1	[	[	X
ma-30	215	2	10	10	NUM
ma-30	215	3	]	]	PUNCT
ma-30	215	4	gave	give	VERB
ma-30	215	5	the	the	DET
ma-30	215	6	definitions	definition	NOUN
ma-30	215	7	of	of	ADP
ma-30	215	8	ρr	ρr	PROPN
ma-30	215	9	-uniformlyconvex	-uniformlyconvex	PROPN
ma-30	215	10	,	,	PUNCT
ma-30	215	11	ρ	ρ	NOUN
ma-30	215	12	-	-	PUNCT
ma-30	215	13	uniformly	uniformly	ADJ
ma-30	215	14	non	non	ADJ
ma-30	215	15	-	-	ADJ
ma-30	215	16	square	square	ADJ
ma-30	215	17	and	and	CCONJ
ma-30	215	18	ρ	ρ	VERB
ma-30	215	19	-	-	PUNCT
ma-30	215	20	strictly	strictly	ADV
ma-30	215	21	convex	convex	NOUN
ma-30	215	22	of	of	ADP
ma-30	215	23	modular	modular	ADJ
ma-30	215	24	spaces	space	NOUN
ma-30	215	25	in	in	ADP
ma-30	215	26	2006.on	2006.on	NUM
ma-30	215	27	the	the	DET
ma-30	215	28	basis	basis	NOUN
ma-30	215	29	of	of	ADP
ma-30	215	30	literature	literature	NOUN
ma-30	215	31	[	[	X
ma-30	215	32	10	10	NUM
ma-30	215	33	]	]	PUNCT
ma-30	215	34	,	,	PUNCT
ma-30	215	35	this	this	DET
ma-30	215	36	paper	paper	NOUN
ma-30	215	37	studies	study	VERB
ma-30	215	38	the	the	DET
ma-30	215	39	relationships	relationship	NOUN
ma-30	215	40	between	between	ADP
ma-30	215	41	convexity	convexity	NOUN
ma-30	215	42	,	,	PUNCT
ma-30	215	43	non	non	ADJ
ma-30	215	44	-	-	ADJ
ma-30	215	45	squareness	squareness	ADJ
ma-30	215	46	and	and	CCONJ
ma-30	215	47	geometric	geometric	ADJ
ma-30	215	48	constants	constant	NOUN
ma-30	215	49	of	of	ADP
ma-30	215	50	modular	modular	ADJ
ma-30	215	51	spaces	space	NOUN
ma-30	215	52	.	.	PUNCT
ma-30	216	1	definition	definition	NOUN
ma-30	216	2	6.[10	6.[10	NUM
ma-30	216	3	]	]	PUNCT
ma-30	216	4	for	for	ADP
ma-30	216	5	r	r	NOUN
ma-30	216	6	>	>	X
ma-30	216	7	0	0	NUM
ma-30	216	8	,	,	PUNCT
ma-30	216	9	a	a	DET
ma-30	216	10	modular	modular	ADJ
ma-30	216	11	space	space	NOUN
ma-30	216	12	xρ	xρ	PROPN
ma-30	216	13	is	be	AUX
ma-30	216	14	said	say	VERB
ma-30	216	15	to	to	PART
ma-30	216	16	be	be	AUX
ma-30	216	17	ρr	ρr	PRON
ma-30	216	18	-uniformly	-uniformly	ADJ
ma-30	216	19	convex	convex	NOUN
ma-30	216	20	if	if	SCONJ
ma-30	216	21	for	for	ADP
ma-30	216	22	each	each	DET
ma-30	216	23	ε	ε	PROPN
ma-30	216	24	>	>	X
ma-30	216	25	0,there	0,there	PROPN
ma-30	216	26	exists	exist	VERB
ma-30	216	27	δ	δ	PROPN
ma-30	216	28	>	>	X
ma-30	216	29	0	0	NUM
ma-30	216	30	such	such	ADJ
ma-30	216	31	that	that	PRON
ma-30	216	32	for	for	ADP
ma-30	216	33	any	any	DET
ma-30	216	34	x	x	NOUN
ma-30	216	35	,	,	PUNCT
ma-30	216	36	y	y	PROPN
ma-30	216	37	∈	∈	PROPN
ma-30	216	38	xρ	xρ	PROPN
ma-30	216	39	,	,	PUNCT
ma-30	216	40	the	the	DET
ma-30	216	41	conditions	condition	NOUN
ma-30	216	42	ρ(x	ρ(x	NOUN
ma-30	216	43	)	)	PUNCT
ma-30	216	44	≤	≤	NOUN
ma-30	216	45	r	r	NOUN
ma-30	216	46	,	,	PUNCT
ma-30	216	47	ρ(y	ρ(y	NOUN
ma-30	216	48	)	)	PUNCT
ma-30	216	49	≤	≤	NOUN
ma-30	216	50	r	r	NOUN
ma-30	216	51	and	and	CCONJ
ma-30	216	52	ρ(x	ρ(x	PROPN
ma-30	216	53	−	−	PROPN
ma-30	216	54	y	y	PROPN
ma-30	216	55	)	)	PUNCT
ma-30	216	56	≥	≥	AUX
ma-30	216	57	rεimply	rεimply	VERB
ma-30	216	58	that	that	SCONJ
ma-30	216	59	ρ	ρ	PROPN
ma-30	216	60	(	(	PUNCT
ma-30	216	61	x+y2	x+y2	PROPN
ma-30	216	62	)	)	PUNCT
ma-30	216	63	≤	≤	NOUN
ma-30	216	64	(	(	PUNCT
ma-30	216	65	1−	1−	NUM
ma-30	216	66	δ)r	δ)r	NOUN
ma-30	216	67	.	.	PUNCT
ma-30	217	1	definition	definition	NOUN
ma-30	217	2	7.[10	7.[10	NUM
ma-30	217	3	]	]	PUNCT
ma-30	217	4	the	the	DET
ma-30	217	5	modular	modular	ADJ
ma-30	217	6	space	space	NOUN
ma-30	217	7	xρ	xρ	PROPN
ma-30	217	8	is	be	AUX
ma-30	217	9	said	say	VERB
ma-30	217	10	to	to	PART
ma-30	217	11	be	be	AUX
ma-30	217	12	ρ	ρ	VERB
ma-30	217	13	-	-	PUNCT
ma-30	217	14	uniformly	uniformly	ADJ
ma-30	217	15	non	non	ADJ
ma-30	217	16	-	-	ADJ
ma-30	217	17	square	square	ADJ
ma-30	217	18	if	if	SCONJ
ma-30	217	19	there	there	PRON
ma-30	217	20	exists	exist	VERB
ma-30	217	21	δ	δ	PROPN
ma-30	217	22	∈	∈	PROPN
ma-30	217	23	(	(	PUNCT
ma-30	217	24	0	0	NUM
ma-30	217	25	,	,	PUNCT
ma-30	217	26	1)such	1)such	NUM
ma-30	217	27	that	that	SCONJ
ma-30	217	28	for	for	ADP
ma-30	217	29	any	any	DET
ma-30	217	30	x	x	NOUN
ma-30	217	31	,	,	PUNCT
ma-30	217	32	y	y	PROPN
ma-30	217	33	∈	∈	PROPN
ma-30	217	34	xρ	xρ	PROPN
ma-30	217	35	with	with	ADP
ma-30	217	36	ρ(x	ρ(x	NOUN
ma-30	217	37	)	)	PUNCT
ma-30	217	38	=	=	SYM
ma-30	218	1	1	1	NUM
ma-30	218	2	and	and	CCONJ
ma-30	218	3	ρ(y	ρ(y	NOUN
ma-30	218	4	)	)	PUNCT
ma-30	218	5	≤	≤	NUM
ma-30	218	6	1	1	NUM
ma-30	218	7	,	,	PUNCT
ma-30	218	8	ρ	ρ	PROPN
ma-30	218	9	(	(	PUNCT
ma-30	218	10	x+y2	x+y2	PROPN
ma-30	218	11	)	)	PUNCT
ma-30	218	12	≤	≤	NOUN
ma-30	218	13	1−	1−	NUM
ma-30	218	14	δ	δ	PROPN
ma-30	218	15	or	or	CCONJ
ma-30	218	16	ρ	ρ	PROPN
ma-30	218	17	(	(	PUNCT
ma-30	218	18	x−y2	x−y2	PROPN
ma-30	218	19	)	)	PUNCT
ma-30	219	1	≤	≤	NOUN
ma-30	219	2	1−	1−	NUM
ma-30	219	3	δ	δ	PROPN
ma-30	219	4	.	.	PUNCT
ma-30	220	1	definition	definition	NOUN
ma-30	220	2	8.[10	8.[10	NUM
ma-30	220	3	]	]	PUNCT
ma-30	220	4	the	the	DET
ma-30	220	5	modular	modular	ADJ
ma-30	220	6	space	space	NOUN
ma-30	220	7	xρ	xρ	PROPN
ma-30	220	8	is	be	AUX
ma-30	220	9	said	say	VERB
ma-30	220	10	to	to	PART
ma-30	220	11	be	be	AUX
ma-30	220	12	ρ	ρ	NOUN
ma-30	220	13	-	-	PUNCT
ma-30	220	14	strictly	strictly	ADV
ma-30	220	15	convex	convex	ADJ
ma-30	220	16	if	if	SCONJ
ma-30	220	17	for	for	ADP
ma-30	220	18	any	any	DET
ma-30	220	19	x	x	NOUN
ma-30	220	20	,	,	PUNCT
ma-30	220	21	y	y	PROPN
ma-30	220	22	∈	∈	PROPN
ma-30	220	23	xρ	xρ	PROPN
ma-30	220	24	,	,	PUNCT
ma-30	220	25	theconditions	thecondition	NOUN
ma-30	220	26	ρ(x	ρ(x	NOUN
ma-30	220	27	)	)	PUNCT
ma-30	220	28	≤	≤	NUM
ma-30	220	29	1	1	NUM
ma-30	220	30	,	,	PUNCT
ma-30	220	31	ρ(y	ρ(y	NOUN
ma-30	220	32	)	)	PUNCT
ma-30	220	33	≤	≤	NOUN
ma-30	220	34	1	1	NUM
ma-30	220	35	and	and	CCONJ
ma-30	220	36	x	x	SYM
ma-30	220	37	6=	6=	NOUN
ma-30	220	38	y	y	PRON
ma-30	220	39	imply	imply	VERB
ma-30	220	40	that	that	SCONJ
ma-30	220	41	ρ	ρ	PROPN
ma-30	220	42	(	(	PUNCT
ma-30	220	43	x+y2	x+y2	PROPN
ma-30	220	44	)	)	PUNCT
ma-30	220	45	<	<	X
ma-30	220	46	1	1	X
ma-30	220	47	.	.	PUNCT
ma-30	220	48	theorem	theorem	NOUN
ma-30	220	49	4	4	NUM
ma-30	220	50	.	.	PUNCT
ma-30	221	1	let	let	VERB
ma-30	221	2	xρ	xρ	PROPN
ma-30	221	3	be	be	AUX
ma-30	221	4	a	a	DET
ma-30	221	5	modular	modular	ADJ
ma-30	221	6	space	space	NOUN
ma-30	221	7	,	,	PUNCT
ma-30	221	8	then	then	ADV
ma-30	221	9	the	the	DET
ma-30	221	10	following	follow	VERB
ma-30	221	11	conditions	condition	NOUN
ma-30	221	12	are	be	AUX
ma-30	221	13	equivalent.(i	equivalent.(i	NOUN
ma-30	221	14	)	)	PUNCT
ma-30	221	15	j(xρ	j(xρ	PROPN
ma-30	221	16	)	)	PUNCT
ma-30	221	17	<	<	X
ma-30	221	18	2;(ii	2;(ii	NUM
ma-30	221	19	)	)	PUNCT
ma-30	221	20	ε0(xρ	ε0(xρ	NUM
ma-30	221	21	)	)	PUNCT
ma-30	221	22	<	<	X
ma-30	221	23	2	2	NUM
ma-30	221	24	for	for	ADP
ma-30	221	25	all	all	PRON
ma-30	221	26	0	0	NUM
ma-30	221	27	<	<	X
ma-30	221	28	ε	ε	PROPN
ma-30	221	29	≤	≤	NUM
ma-30	221	30	2;(ii	2;(ii	NUM
ma-30	221	31	)	)	PUNCT
ma-30	221	32	xρ	xρ	PROPN
ma-30	221	33	is	be	AUX
ma-30	221	34	ρ	ρ	NOUN
ma-30	221	35	-	-	PUNCT
ma-30	221	36	uniformly	uniformly	ADJ
ma-30	221	37	non	non	ADJ
ma-30	221	38	-	-	ADJ
ma-30	221	39	square	square	ADJ
ma-30	221	40	.	.	PUNCT
ma-30	222	1	proof	proof	NOUN
ma-30	222	2	.	.	PUNCT
ma-30	223	1	suppose	suppose	VERB
ma-30	223	2	j(xρ	j(xρ	PROPN
ma-30	223	3	)	)	PUNCT
ma-30	223	4	<	<	X
ma-30	223	5	2	2	X
ma-30	223	6	.	.	X
ma-30	223	7	there	there	PRON
ma-30	223	8	exists	exist	VERB
ma-30	223	9	ε	ε	PROPN
ma-30	223	10	>	>	X
ma-30	223	11	0	0	PROPN
ma-30	223	12	,	,	PUNCT
ma-30	223	13	for	for	ADP
ma-30	223	14	any	any	DET
ma-30	223	15	x	x	NOUN
ma-30	223	16	,	,	PUNCT
ma-30	223	17	y	y	PROPN
ma-30	223	18	∈	∈	PROPN
ma-30	223	19	xρ	xρ	PROPN
ma-30	223	20	with	with	ADP
ma-30	223	21	ρ(x	ρ(x	NOUN
ma-30	223	22	)	)	PUNCT
ma-30	223	23	=	=	SYM
ma-30	223	24	1	1	NUM
ma-30	223	25	and	and	CCONJ
ma-30	223	26	ρ(y	ρ(y	NOUN
ma-30	223	27	)	)	PUNCT
ma-30	224	1	≤	≤	NOUN
ma-30	224	2	1,such	1,such	NUM
ma-30	224	3	that	that	SCONJ
ma-30	224	4	ρ	ρ	PROPN
ma-30	224	5	(	(	PUNCT
ma-30	224	6	x	x	PROPN
ma-30	224	7	+	+	NUM
ma-30	224	8	y	y	PROPN
ma-30	224	9	2	2	NUM
ma-30	224	10	)	)	PUNCT
ma-30	224	11	≤	≤	NOUN
ma-30	224	12	j(xρ	j(xρ	PROPN
ma-30	224	13	)	)	PUNCT
ma-30	224	14	2	2	NUM
ma-30	224	15	−	−	NOUN
ma-30	224	16	ε	ε	X
ma-30	224	17	<	<	X
ma-30	224	18	1−	1−	NUM
ma-30	224	19	ε	ε	PROPN
ma-30	224	20	or	or	CCONJ
ma-30	224	21	ρ(x	ρ(x	PROPN
ma-30	224	22	−	−	PROPN
ma-30	224	23	y	y	PROPN
ma-30	224	24	2	2	NUM
ma-30	224	25	)	)	PUNCT
ma-30	224	26	≤	≤	NOUN
ma-30	224	27	j(xρ	j(xρ	PROPN
ma-30	224	28	)	)	PUNCT
ma-30	224	29	2	2	NUM
ma-30	224	30	−	−	NOUN
ma-30	224	31	ε	ε	PROPN
ma-30	224	32	<	<	X
ma-30	224	33	1−	1−	PROPN
ma-30	224	34	ε	ε	PROPN
ma-30	224	35	,	,	PUNCT
ma-30	224	36	implies	imply	VERB
ma-30	224	37	xρ	xρ	PROPN
ma-30	224	38	is	be	AUX
ma-30	224	39	ρ	ρ	NOUN
ma-30	224	40	-	-	PUNCT
ma-30	224	41	uniformly	uniformly	ADV
ma-30	224	42	non-square.suppose	non-square.suppose	NOUN
ma-30	224	43	xρ	xρ	PROPN
ma-30	224	44	is	be	AUX
ma-30	224	45	ρ	ρ	NOUN
ma-30	224	46	-	-	PUNCT
ma-30	224	47	uniformly	uniformly	ADJ
ma-30	224	48	non	non	ADJ
ma-30	224	49	-	-	ADJ
ma-30	224	50	square	square	ADJ
ma-30	224	51	,	,	PUNCT
ma-30	224	52	then	then	ADV
ma-30	224	53	we	we	PRON
ma-30	224	54	can	can	AUX
ma-30	224	55	prove	prove	VERB
ma-30	224	56	j(xρ	j(xρ	PROPN
ma-30	224	57	)	)	PUNCT
ma-30	224	58	<	<	X
ma-30	224	59	2	2	NUM
ma-30	224	60	by	by	ADP
ma-30	224	61	the	the	DET
ma-30	224	62	same	same	ADJ
ma-30	224	63	way	way	NOUN
ma-30	224	64	.	.	PUNCT
ma-30	225	1	thus(i	thus(i	NOUN
ma-30	225	2	)	)	PUNCT
ma-30	225	3	and	and	CCONJ
ma-30	225	4	(	(	PUNCT
ma-30	225	5	iii	iii	X
ma-30	225	6	)	)	PUNCT
ma-30	225	7	are	be	AUX
ma-30	225	8	equivalent.next	equivalent.next	NOUN
ma-30	225	9	,	,	PUNCT
ma-30	225	10	we	we	PRON
ma-30	225	11	know	know	VERB
ma-30	225	12	that	that	PRON
ma-30	225	13	ε0(xρ	ε0(xρ	NUM
ma-30	225	14	)	)	PUNCT
ma-30	225	15	<	<	X
ma-30	225	16	2	2	NUM
ma-30	225	17	if	if	SCONJ
ma-30	225	18	and	and	CCONJ
ma-30	225	19	only	only	ADV
ma-30	225	20	if	if	SCONJ
ma-30	225	21	δxρ(2	δxρ(2	NOUN
ma-30	225	22	)	)	PUNCT
ma-30	225	23	>	>	X
ma-30	226	1	0	0	X
ma-30	226	2	.	.	PUNCT
ma-30	227	1	let	let	VERB
ma-30	227	2	α	α	NOUN
ma-30	227	3	=	=	SYM
ma-30	227	4	δxρ(2	δxρ(2	NOUN
ma-30	227	5	)	)	PUNCT
ma-30	227	6	,	,	PUNCT
ma-30	227	7	then	then	ADV
ma-30	227	8	∀x	∀x	X
ma-30	227	9	,	,	PUNCT
ma-30	227	10	y	y	PROPN
ma-30	227	11	∈	∈	PROPN
ma-30	227	12	xρ	xρ	PROPN
ma-30	227	13	and	and	CCONJ
ma-30	227	14	ρ(x	ρ(x	NUM
ma-30	227	15	)	)	PUNCT
ma-30	227	16	=	=	SYM
ma-30	228	1	1	1	NUM
ma-30	228	2	,	,	PUNCT
ma-30	228	3	ρ(y	ρ(y	NOUN
ma-30	228	4	)	)	PUNCT
ma-30	228	5	≤	≤	NUM
ma-30	228	6	1	1	NUM
ma-30	228	7	,	,	PUNCT
ma-30	228	8	we	we	PRON
ma-30	228	9	can	can	AUX
ma-30	228	10	get	get	VERB
ma-30	228	11	ρ	ρ	PROPN
ma-30	228	12	(	(	PUNCT
ma-30	228	13	x±y2	x±y2	PROPN
ma-30	228	14	)	)	PUNCT
ma-30	228	15	≤	≤	NOUN
ma-30	228	16	1−	1−	NUM
ma-30	228	17	α	α	NOUN
ma-30	228	18	,	,	PUNCT
ma-30	228	19	thus	thus	ADV
ma-30	228	20	xρ	xρ	PROPN
ma-30	228	21	is	be	AUX
ma-30	228	22	ρ	ρ	VERB
ma-30	228	23	-	-	PUNCT
ma-30	228	24	uniformly	uniformly	ADJ
ma-30	228	25	non	non	ADJ
ma-30	228	26	-	-	ADJ
ma-30	228	27	square	square	ADJ
ma-30	228	28	.	.	PUNCT
ma-30	229	1	thus	thus	ADV
ma-30	229	2	eur	eur	PROPN
ma-30	229	3	.	.	PUNCT
ma-30	230	1	j.	j.	PROPN
ma-30	230	2	math	math	PROPN
ma-30	230	3	.	.	PUNCT
ma-30	231	1	anal	anal	ADJ
ma-30	231	2	.	.	PUNCT
ma-30	232	1	1	1	NUM
ma-30	232	2	(	(	PUNCT
ma-30	232	3	2021	2021	NUM
ma-30	232	4	)	)	PUNCT
ma-30	232	5	159(ii	159(ii	NUM
ma-30	232	6	)	)	PUNCT
ma-30	232	7	and	and	CCONJ
ma-30	232	8	(	(	PUNCT
ma-30	232	9	iii	iii	X
ma-30	232	10	)	)	PUNCT
ma-30	232	11	are	be	AUX
ma-30	232	12	equivalent	equivalent	ADJ
ma-30	232	13	.	.	PUNCT
ma-30	233	1	remark	remark	NOUN
ma-30	233	2	3	3	NUM
ma-30	233	3	.	.	PUNCT
ma-30	234	1	in	in	ADP
ma-30	234	2	fact	fact	NOUN
ma-30	234	3	,	,	PUNCT
ma-30	234	4	this	this	DET
ma-30	234	5	theorem	theorem	NOUN
ma-30	234	6	is	be	AUX
ma-30	234	7	a	a	DET
ma-30	234	8	generalization	generalization	NOUN
ma-30	234	9	of	of	ADP
ma-30	234	10	theorem	theorem	NOUN
ma-30	234	11	3.8	3.8	NUM
ma-30	234	12	in	in	ADP
ma-30	234	13	[	[	PUNCT
ma-30	234	14	11	11	NUM
ma-30	234	15	]	]	PUNCT
ma-30	234	16	.	.	PUNCT
ma-30	235	1	theorem	theorem	NOUN
ma-30	235	2	5	5	NUM
ma-30	235	3	.	.	PUNCT
ma-30	236	1	let	let	VERB
ma-30	236	2	x	x	PRON
ma-30	236	3	be	be	AUX
ma-30	236	4	a	a	DET
ma-30	236	5	modular	modular	ADJ
ma-30	236	6	space	space	NOUN
ma-30	236	7	,	,	PUNCT
ma-30	236	8	then(i	then(i	NOUN
ma-30	236	9	)	)	PUNCT
ma-30	236	10	x	x	PUNCT
ma-30	236	11	is	be	AUX
ma-30	236	12	ρ1	ρ1	NOUN
ma-30	236	13	-	-	PUNCT
ma-30	236	14	uniformly	uniformly	ADV
ma-30	236	15	convex	convex	NOUN
ma-30	236	16	if	if	SCONJ
ma-30	236	17	and	and	CCONJ
ma-30	236	18	only	only	ADV
ma-30	236	19	if	if	SCONJ
ma-30	236	20	δxρ(ε	δxρ(ε	X
ma-30	236	21	)	)	PUNCT
ma-30	236	22	>	>	X
ma-30	236	23	0	0	PUNCT
ma-30	237	1	for	for	ADP
ma-30	237	2	0	0	NUM
ma-30	237	3	<	<	X
ma-30	237	4	ε	ε	PROPN
ma-30	237	5	≤	≤	NUM
ma-30	237	6	2;(ii	2;(ii	NUM
ma-30	237	7	)	)	PUNCT
ma-30	237	8	if	if	SCONJ
ma-30	237	9	δxρ(2	δxρ(2	NOUN
ma-30	237	10	)	)	PUNCT
ma-30	237	11	=	=	SYM
ma-30	237	12	1	1	NUM
ma-30	237	13	,	,	PUNCT
ma-30	237	14	then	then	ADV
ma-30	237	15	xρ	xρ	PROPN
ma-30	237	16	is	be	AUX
ma-30	237	17	ρ	ρ	PROPN
ma-30	237	18	-	-	PUNCT
ma-30	237	19	strictly	strictly	ADV
ma-30	237	20	convex	convex	ADJ
ma-30	237	21	.	.	PUNCT
ma-30	238	1	proof	proof	NOUN
ma-30	238	2	.	.	PUNCT
ma-30	239	1	(	(	PUNCT
ma-30	239	2	i	i	NOUN
ma-30	239	3	)	)	PUNCT
ma-30	239	4	denote	denote	VERB
ma-30	239	5	δε	δε	X
ma-30	239	6	=	=	X
ma-30	239	7	δxρ(ε	δxρ(ε	PROPN
ma-30	239	8	)	)	PUNCT
ma-30	239	9	.	.	PUNCT
ma-30	240	1	then	then	ADV
ma-30	240	2	δxρ(ε	δxρ(ε	X
ma-30	240	3	)	)	PUNCT
ma-30	240	4	>	>	X
ma-30	240	5	0	0	PUNCT
ma-30	241	1	if	if	SCONJ
ma-30	241	2	and	and	CCONJ
ma-30	241	3	only	only	ADV
ma-30	241	4	if	if	SCONJ
ma-30	241	5	∀x	∀x	NUM
ma-30	241	6	,	,	PUNCT
ma-30	241	7	y	y	PROPN
ma-30	241	8	∈	∈	PROPN
ma-30	241	9	xρ	xρ	PROPN
ma-30	241	10	,	,	PUNCT
ma-30	241	11	ρ(x	ρ(x	PROPN
ma-30	241	12	)	)	PUNCT
ma-30	241	13	,	,	PUNCT
ma-30	241	14	ρ(y	ρ(y	NOUN
ma-30	241	15	)	)	PUNCT
ma-30	241	16	≤	≤	NUM
ma-30	241	17	1	1	NUM
ma-30	241	18	and	and	CCONJ
ma-30	241	19	ρ(x	ρ(x	PROPN
ma-30	241	20	−	−	PROPN
ma-30	241	21	y	y	PROPN
ma-30	241	22	)	)	PUNCT
ma-30	241	23	≥	≥	NOUN
ma-30	241	24	ε	ε	PROPN
ma-30	241	25	,	,	PUNCT
ma-30	241	26	we	we	PRON
ma-30	241	27	have	have	VERB
ma-30	241	28	ρ	ρ	NOUN
ma-30	241	29	(	(	PUNCT
ma-30	241	30	x+y2	x+y2	PROPN
ma-30	241	31	)	)	PUNCT
ma-30	241	32	≤	≤	NOUN
ma-30	241	33	1−	1−	NUM
ma-30	241	34	δε	δε	NOUN
ma-30	241	35	.	.	PUNCT
ma-30	242	1	thus	thus	ADV
ma-30	242	2	xρ	xρ	PROPN
ma-30	242	3	is	be	AUX
ma-30	242	4	ρ1	ρ1	NOUN
ma-30	242	5	-	-	PUNCT
ma-30	242	6	uniformly	uniformly	NOUN
ma-30	242	7	convex.(ii	convex.(ii	NOUN
ma-30	242	8	)	)	PUNCT
ma-30	242	9	since	since	SCONJ
ma-30	242	10	δxρ(2	δxρ(2	NOUN
ma-30	242	11	)	)	PUNCT
ma-30	243	1	=	=	SYM
ma-30	243	2	1	1	NUM
ma-30	243	3	,	,	PUNCT
ma-30	243	4	then	then	ADV
ma-30	243	5	∀x	∀x	NUM
ma-30	243	6	,	,	PUNCT
ma-30	243	7	y	y	PROPN
ma-30	243	8	∈	∈	PROPN
ma-30	243	9	xρ	xρ	PROPN
ma-30	243	10	,	,	PUNCT
ma-30	243	11	ρ(x	ρ(x	PROPN
ma-30	243	12	)	)	PUNCT
ma-30	243	13	,	,	PUNCT
ma-30	243	14	ρ(y	ρ(y	NOUN
ma-30	243	15	)	)	PUNCT
ma-30	243	16	≤	≤	NUM
ma-30	243	17	1	1	NUM
ma-30	243	18	and	and	CCONJ
ma-30	243	19	ρ(x	ρ(x	PROPN
ma-30	243	20	−	−	PROPN
ma-30	243	21	y	y	PROPN
ma-30	243	22	)	)	PUNCT
ma-30	243	23	≥	≥	NOUN
ma-30	243	24	2	2	NUM
ma-30	243	25	,	,	PUNCT
ma-30	243	26	we	we	PRON
ma-30	243	27	have	have	VERB
ma-30	243	28	ρ	ρ	NUM
ma-30	243	29	(	(	PUNCT
ma-30	243	30	x	x	PROPN
ma-30	244	1	+	+	NUM
ma-30	244	2	y	y	PROPN
ma-30	244	3	2	2	NUM
ma-30	244	4	)	)	PUNCT
ma-30	244	5	=	=	PUNCT
ma-30	244	6	0	0	PUNCT
ma-30	244	7	<	<	X
ma-30	244	8	1	1	NUM
ma-30	244	9	,	,	PUNCT
ma-30	244	10	implies	imply	VERB
ma-30	244	11	xρ	xρ	PROPN
ma-30	244	12	is	be	AUX
ma-30	244	13	ρ	ρ	PROPN
ma-30	244	14	-	-	PUNCT
ma-30	244	15	strictly	strictly	ADV
ma-30	244	16	convex	convex	NOUN
ma-30	244	17	.	.	PUNCT
ma-30	245	1	6	6	NUM
ma-30	245	2	.	.	X
ma-30	245	3	midpoint	midpoint	NOUN
ma-30	245	4	convexity	convexity	NOUN
ma-30	245	5	in	in	ADP
ma-30	245	6	the	the	DET
ma-30	245	7	following	follow	VERB
ma-30	245	8	section	section	NOUN
ma-30	245	9	,	,	PUNCT
ma-30	245	10	we	we	PRON
ma-30	245	11	discuss	discuss	VERB
ma-30	245	12	a	a	DET
ma-30	245	13	special	special	ADJ
ma-30	245	14	type	type	NOUN
ma-30	245	15	of	of	ADP
ma-30	245	16	modular	modular	NOUN
ma-30	245	17	and	and	CCONJ
ma-30	245	18	study	study	VERB
ma-30	245	19	its	its	PRON
ma-30	245	20	properties	property	NOUN
ma-30	245	21	in	in	ADP
ma-30	245	22	termsof	termsof	NOUN
ma-30	245	23	geometric	geometric	ADJ
ma-30	245	24	constants	constant	NOUN
ma-30	245	25	.	.	PUNCT
ma-30	246	1	definition	definition	NOUN
ma-30	246	2	9.[6	9.[6	X
ma-30	246	3	]	]	X
ma-30	246	4	let	let	VERB
ma-30	246	5	(	(	PUNCT
ma-30	246	6	x	x	NOUN
ma-30	246	7	,	,	PUNCT
ma-30	246	8	‖	‖	PROPN
ma-30	246	9	·	·	PUNCT
ma-30	246	10	‖	‖	NUM
ma-30	246	11	)	)	PUNCT
ma-30	246	12	be	be	AUX
ma-30	246	13	a	a	DET
ma-30	246	14	normed	normed	ADJ
ma-30	246	15	space	space	NOUN
ma-30	246	16	and	and	CCONJ
ma-30	246	17	xρ	xρ	PROPN
ma-30	246	18	be	be	AUX
ma-30	246	19	a	a	DET
ma-30	246	20	modular	modular	ADJ
ma-30	246	21	space	space	NOUN
ma-30	246	22	.	.	PUNCT
ma-30	247	1	then	then	ADV
ma-30	247	2	ρ	ρ	PROPN
ma-30	247	3	is	be	AUX
ma-30	247	4	said	say	VERB
ma-30	247	5	to	to	PART
ma-30	247	6	bestrongly	bestrongly	ADV
ma-30	247	7	midpoint	midpoint	VERB
ma-30	247	8	convex	convex	NOUN
ma-30	247	9	with	with	ADP
ma-30	247	10	non	non	ADJ
ma-30	247	11	-	-	ADJ
ma-30	247	12	negtive	negtive	ADJ
ma-30	247	13	constant	constant	ADJ
ma-30	247	14	c	c	NOUN
ma-30	247	15	if	if	SCONJ
ma-30	247	16	ρ	ρ	PROPN
ma-30	247	17	(	(	PUNCT
ma-30	247	18	x+y2	x+y2	PROPN
ma-30	247	19	)	)	PUNCT
ma-30	247	20	≤	≤	NUM
ma-30	247	21	ρ(x)+ρ(y	ρ(x)+ρ(y	NOUN
ma-30	247	22	)	)	PUNCT
ma-30	247	23	2	2	NUM
ma-30	247	24	−	−	NOUN
ma-30	247	25	c	c	NOUN
ma-30	247	26	4	4	NUM
ma-30	247	27	‖x	‖x	NOUN
ma-30	247	28	−	−	PROPN
ma-30	247	29	y‖	y‖	NOUN
ma-30	247	30	2	2	NUM
ma-30	247	31	.	.	PUNCT
ma-30	247	32	theorem	theorem	NOUN
ma-30	247	33	6	6	NUM
ma-30	247	34	.	.	PUNCT
ma-30	248	1	let	let	AUX
ma-30	248	2	(	(	PUNCT
ma-30	248	3	x	x	NOUN
ma-30	248	4	,	,	PUNCT
ma-30	248	5	‖	‖	PROPN
ma-30	248	6	·	·	PUNCT
ma-30	248	7	‖	‖	NUM
ma-30	248	8	)	)	PUNCT
ma-30	248	9	be	be	AUX
ma-30	248	10	a	a	DET
ma-30	248	11	normed	normed	ADJ
ma-30	248	12	space	space	NOUN
ma-30	248	13	and	and	CCONJ
ma-30	248	14	xρ	xρ	PROPN
ma-30	248	15	be	be	AUX
ma-30	248	16	a	a	DET
ma-30	248	17	modular	modular	ADJ
ma-30	248	18	space	space	NOUN
ma-30	248	19	.	.	PUNCT
ma-30	249	1	if	if	SCONJ
ma-30	249	2	there	there	PRON
ma-30	249	3	exists	exist	VERB
ma-30	249	4	c	c	PROPN
ma-30	249	5	≥	≥	NOUN
ma-30	249	6	0such	0such	NUM
ma-30	249	7	that	that	PRON
ma-30	249	8	c‖x‖2	c‖x‖2	VERB
ma-30	249	9	≤	≤	NUM
ma-30	249	10	1	1	NUM
ma-30	249	11	2	2	NUM
ma-30	249	12	ρ(x	ρ(x	NUM
ma-30	249	13	)	)	PUNCT
ma-30	249	14	for	for	ADP
ma-30	249	15	all	all	PRON
ma-30	249	16	x	x	SYM
ma-30	249	17	∈	∈	PROPN
ma-30	249	18	bxρand	bxρand	NOUN
ma-30	249	19	ρ	ρ	PROPN
ma-30	249	20	is	be	AUX
ma-30	249	21	strongly	strongly	ADV
ma-30	249	22	midpoint	midpoint	ADJ
ma-30	249	23	convex	convex	NOUN
ma-30	249	24	with	with	ADP
ma-30	249	25	constant	constant	ADJ
ma-30	249	26	c	c	NOUN
ma-30	249	27	,	,	PUNCT
ma-30	249	28	then	then	ADV
ma-30	249	29	cnj	cnj	PROPN
ma-30	249	30	(	(	PUNCT
ma-30	249	31	xρ	xρ	PROPN
ma-30	249	32	)	)	PUNCT
ma-30	249	33	≤	≤	NOUN
ma-30	249	34	3	3	NUM
ma-30	249	35	.	.	PUNCT
ma-30	249	36	proof	proof	NOUN
ma-30	249	37	.	.	PUNCT
ma-30	250	1	since	since	SCONJ
ma-30	250	2	ρ	ρ	PROPN
ma-30	250	3	(	(	PUNCT
ma-30	250	4	x+y2	x+y2	PROPN
ma-30	250	5	)	)	PUNCT
ma-30	250	6	≤	≤	NUM
ma-30	250	7	ρ(x)+ρ(y	ρ(x)+ρ(y	NOUN
ma-30	250	8	)	)	PUNCT
ma-30	251	1	2	2	NUM
ma-30	251	2	−	−	NOUN
ma-30	251	3	c	c	NOUN
ma-30	251	4	4	4	NUM
ma-30	251	5	‖x	‖x	NOUN
ma-30	251	6	−	−	PROPN
ma-30	251	7	y‖	y‖	NOUN
ma-30	251	8	2	2	NUM
ma-30	251	9	and	and	CCONJ
ma-30	251	10	ρ	ρ	PROPN
ma-30	251	11	(	(	PUNCT
ma-30	251	12	x−y2	x−y2	PROPN
ma-30	251	13	)	)	PUNCT
ma-30	251	14	≤	≤	NUM
ma-30	251	15	ρ(x)+ρ(−y	ρ(x)+ρ(−y	NOUN
ma-30	251	16	)	)	PUNCT
ma-30	251	17	2	2	NUM
ma-30	251	18	−	−	NOUN
ma-30	251	19	c	c	NOUN
ma-30	251	20	4	4	NUM
ma-30	251	21	‖x	‖x	NOUN
ma-30	251	22	+	+	CCONJ
ma-30	251	23	y‖	y‖	PROPN
ma-30	251	24	2	2	NUM
ma-30	251	25	,	,	PUNCT
ma-30	251	26	then	then	ADV
ma-30	251	27	ρ2	ρ2	VERB
ma-30	251	28	(	(	PUNCT
ma-30	251	29	x	x	SYM
ma-30	251	30	+	+	NUM
ma-30	251	31	y	y	PROPN
ma-30	251	32	2	2	NUM
ma-30	251	33	)	)	PUNCT
ma-30	251	34	≤	≤	NOUN
ma-30	251	35	1	1	NUM
ma-30	251	36	4	4	NUM
ma-30	251	37	(	(	PUNCT
ma-30	251	38	ρ(x	ρ(x	NOUN
ma-30	251	39	)	)	PUNCT
ma-30	251	40	+	+	CCONJ
ma-30	252	1	ρ(y))2	ρ(y))2	NOUN
ma-30	252	2	−	−	NOUN
ma-30	252	3	c	c	NOUN
ma-30	252	4	4	4	NUM
ma-30	252	5	‖x	‖x	NOUN
ma-30	252	6	−	−	PROPN
ma-30	252	7	y‖2(ρ(x	y‖2(ρ(x	PROPN
ma-30	252	8	)	)	PUNCT
ma-30	253	1	+	+	CCONJ
ma-30	253	2	ρ(y	ρ(y	NOUN
ma-30	253	3	)	)	PUNCT
ma-30	253	4	)	)	PUNCT
ma-30	254	1	+	+	CCONJ
ma-30	254	2	c2	c2	PROPN
ma-30	254	3	16	16	NUM
ma-30	254	4	‖x	‖x	NOUN
ma-30	255	1	−	−	PROPN
ma-30	255	2	y‖4	y‖4	NOUN
ma-30	255	3	and	and	CCONJ
ma-30	255	4	ρ2	ρ2	PROPN
ma-30	255	5	(	(	PUNCT
ma-30	255	6	x	x	SYM
ma-30	255	7	−	−	PROPN
ma-30	255	8	y	y	PROPN
ma-30	255	9	2	2	NUM
ma-30	255	10	)	)	PUNCT
ma-30	255	11	≤	≤	NOUN
ma-30	255	12	1	1	NUM
ma-30	255	13	4	4	NUM
ma-30	255	14	(	(	PUNCT
ma-30	255	15	ρ(x	ρ(x	NOUN
ma-30	255	16	)	)	PUNCT
ma-30	255	17	+	+	CCONJ
ma-30	255	18	ρ(y))2	ρ(y))2	NOUN
ma-30	256	1	−	−	NOUN
ma-30	256	2	c	c	NOUN
ma-30	256	3	4	4	NUM
ma-30	256	4	‖x	‖x	NOUN
ma-30	256	5	+	+	CCONJ
ma-30	256	6	y‖2(ρ(x	y‖2(ρ(x	NOUN
ma-30	256	7	)	)	PUNCT
ma-30	257	1	+	+	CCONJ
ma-30	257	2	ρ(y	ρ(y	NOUN
ma-30	257	3	)	)	PUNCT
ma-30	257	4	)	)	PUNCT
ma-30	258	1	+	+	CCONJ
ma-30	258	2	c2	c2	PROPN
ma-30	258	3	16	16	NUM
ma-30	258	4	‖x	‖x	PUNCT
ma-30	259	1	+	+	CCONJ
ma-30	260	1	y‖4	y‖4	INTJ
ma-30	260	2	.	.	PUNCT
ma-30	261	1	therefore	therefore	ADV
ma-30	261	2	,	,	PUNCT
ma-30	261	3	for	for	ADP
ma-30	261	4	x	x	PROPN
ma-30	261	5	∈	∈	PROPN
ma-30	261	6	sxρ	sxρ	NOUN
ma-30	261	7	and	and	CCONJ
ma-30	261	8	y	y	PROPN
ma-30	261	9	∈	∈	PROPN
ma-30	261	10	bxρ	bxρ	NOUN
ma-30	261	11	,	,	PUNCT
ma-30	261	12	we	we	PRON
ma-30	261	13	have	have	VERB
ma-30	261	14	ρ2	ρ2	NOUN
ma-30	261	15	(	(	PUNCT
ma-30	261	16	x	x	SYM
ma-30	262	1	+	+	NUM
ma-30	262	2	y	y	PROPN
ma-30	262	3	2	2	NUM
ma-30	262	4	)	)	PUNCT
ma-30	262	5	+	+	CCONJ
ma-30	262	6	ρ2	ρ2	NOUN
ma-30	262	7	(	(	PUNCT
ma-30	262	8	x	x	SYM
ma-30	262	9	−	−	PROPN
ma-30	262	10	y	y	PROPN
ma-30	262	11	2	2	NUM
ma-30	262	12	)	)	PUNCT
ma-30	262	13	≤	≤	NOUN
ma-30	262	14	1	1	NUM
ma-30	262	15	2	2	NUM
ma-30	262	16	(	(	PUNCT
ma-30	262	17	1	1	NUM
ma-30	262	18	+	+	NUM
ma-30	262	19	ρ(y))2	ρ(y))2	NOUN
ma-30	262	20	−	−	NOUN
ma-30	262	21	c	c	NOUN
ma-30	262	22	4	4	NUM
ma-30	262	23	(	(	PUNCT
ma-30	262	24	1	1	NUM
ma-30	262	25	+	+	CCONJ
ma-30	262	26	ρ(y	ρ(y	NOUN
ma-30	262	27	)	)	PUNCT
ma-30	262	28	)	)	PUNCT
ma-30	263	1	(	(	PUNCT
ma-30	263	2	‖x	‖x	NOUN
ma-30	263	3	+	+	NUM
ma-30	263	4	y‖2	y‖2	X
ma-30	263	5	+	+	CCONJ
ma-30	263	6	‖x	‖x	NOUN
ma-30	263	7	−	−	PROPN
ma-30	263	8	y‖2	y‖2	PROPN
ma-30	263	9	)	)	PUNCT
ma-30	264	1	+	+	CCONJ
ma-30	264	2	c2	c2	PROPN
ma-30	264	3	16	16	NUM
ma-30	264	4	(	(	PUNCT
ma-30	264	5	‖x	‖x	NOUN
ma-30	264	6	+	+	NUM
ma-30	264	7	y‖4	y‖4	NOUN
ma-30	265	1	+	+	NUM
ma-30	265	2	‖x	‖x	NOUN
ma-30	265	3	−	−	PROPN
ma-30	265	4	y‖4	y‖4	PROPN
ma-30	265	5	)	)	PUNCT
ma-30	265	6	.	.	PUNCT
ma-30	266	1	next	next	ADV
ma-30	266	2	,	,	PUNCT
ma-30	266	3	we	we	PRON
ma-30	266	4	only	only	ADV
ma-30	266	5	need	need	VERB
ma-30	266	6	to	to	PART
ma-30	266	7	prove	prove	VERB
ma-30	266	8	ρ(y	ρ(y	NOUN
ma-30	266	9	)	)	PUNCT
ma-30	267	1	+	+	CCONJ
ma-30	267	2	c2	c2	PROPN
ma-30	267	3	16	16	NUM
ma-30	267	4	(	(	PUNCT
ma-30	267	5	‖x	‖x	NOUN
ma-30	267	6	+	+	NUM
ma-30	267	7	y‖4	y‖4	NOUN
ma-30	268	1	+	+	NUM
ma-30	268	2	‖x	‖x	NOUN
ma-30	268	3	−	−	NOUN
ma-30	268	4	y‖4	y‖4	NOUN
ma-30	268	5	)	)	PUNCT
ma-30	269	1	−	−	PROPN
ma-30	269	2	c	c	NOUN
ma-30	269	3	4	4	NUM
ma-30	269	4	(	(	PUNCT
ma-30	269	5	‖x	‖x	NOUN
ma-30	269	6	+	+	NUM
ma-30	269	7	y‖2	y‖2	X
ma-30	269	8	+	+	CCONJ
ma-30	269	9	‖x	‖x	NOUN
ma-30	269	10	−	−	PROPN
ma-30	269	11	y‖2	y‖2	PROPN
ma-30	269	12	)	)	PUNCT
ma-30	269	13	(	(	PUNCT
ma-30	269	14	1	1	NUM
ma-30	269	15	+	+	CCONJ
ma-30	269	16	ρ(y))−	ρ(y))−	VERB
ma-30	269	17	1−	1−	NUM
ma-30	269	18	ρ2(y	ρ2(y	NUM
ma-30	269	19	)	)	PUNCT
ma-30	269	20	≤	≤	NOUN
ma-30	269	21	0	0	NUM
ma-30	269	22	.	.	PUNCT
ma-30	270	1	eur	eur	PROPN
ma-30	270	2	.	.	PUNCT
ma-30	271	1	j.	j.	PROPN
ma-30	271	2	math	math	PROPN
ma-30	271	3	.	.	PUNCT
ma-30	272	1	anal	anal	ADJ
ma-30	272	2	.	.	PUNCT
ma-30	273	1	1	1	NUM
ma-30	273	2	(	(	PUNCT
ma-30	273	3	2021	2021	NUM
ma-30	273	4	)	)	PUNCT
ma-30	273	5	160	160	NUM
ma-30	273	6	let	let	VERB
ma-30	273	7	t	t	NOUN
ma-30	273	8	=	=	PUNCT
ma-30	273	9	‖x	‖x	X
ma-30	274	1	+	+	NUM
ma-30	274	2	y‖2	y‖2	X
ma-30	274	3	+	+	CCONJ
ma-30	274	4	‖x	‖x	NOUN
ma-30	274	5	−	−	PROPN
ma-30	274	6	y‖2	y‖2	PROPN
ma-30	274	7	,	,	PUNCT
ma-30	274	8	s	s	PART
ma-30	274	9	=	=	NOUN
ma-30	274	10	‖x	‖x	NOUN
ma-30	275	1	+	+	PUNCT
ma-30	275	2	y‖‖x	y‖‖x	INTJ
ma-30	275	3	−	−	PROPN
ma-30	275	4	y‖	y‖	PROPN
ma-30	275	5	and	and	CCONJ
ma-30	275	6	i1	i1	PROPN
ma-30	275	7	=	=	PUNCT
ma-30	275	8	ρ(y	ρ(y	PROPN
ma-30	275	9	)	)	PUNCT
ma-30	275	10	+	+	CCONJ
ma-30	275	11	c2	c2	PROPN
ma-30	275	12	16	16	NUM
ma-30	275	13	(	(	PUNCT
ma-30	275	14	t2	t2	PROPN
ma-30	275	15	−	−	PROPN
ma-30	275	16	2s2	2s2	NUM
ma-30	275	17	)	)	PUNCT
ma-30	276	1	−	−	PROPN
ma-30	276	2	c	c	NOUN
ma-30	276	3	4	4	NUM
ma-30	276	4	t(1	t(1	NOUN
ma-30	276	5	+	+	CCONJ
ma-30	276	6	ρ(y))−	ρ(y))−	VERB
ma-30	276	7	1−	1−	NUM
ma-30	276	8	ρ2(y	ρ2(y	NUM
ma-30	276	9	)	)	PUNCT
ma-30	276	10	,	,	PUNCT
ma-30	276	11	then	then	ADV
ma-30	276	12	i1	i1	PROPN
ma-30	276	13	≤	≤	PROPN
ma-30	276	14	t2	t2	PROPN
ma-30	276	15	−	−	PROPN
ma-30	276	16	2s2	2s2	NUM
ma-30	276	17	16	16	NUM
ma-30	276	18	c2	c2	PROPN
ma-30	276	19	−	−	PROPN
ma-30	276	20	t	t	PROPN
ma-30	276	21	4	4	NUM
ma-30	276	22	c	c	NOUN
ma-30	276	23	=	=	SYM
ma-30	276	24	c	c	PROPN
ma-30	276	25	(	(	PUNCT
ma-30	276	26	t2	t2	PROPN
ma-30	276	27	−	−	PROPN
ma-30	276	28	2s2	2s2	NUM
ma-30	276	29	)	)	PUNCT
ma-30	276	30	16	16	NUM
ma-30	276	31	(	(	PUNCT
ma-30	276	32	c	c	NOUN
ma-30	276	33	−	−	PROPN
ma-30	276	34	4	4	NUM
ma-30	276	35	t	t	NOUN
ma-30	276	36	t2	t2	NOUN
ma-30	276	37	−	−	PROPN
ma-30	276	38	2s2	2s2	NUM
ma-30	276	39	)	)	PUNCT
ma-30	276	40	.	.	PUNCT
ma-30	277	1	since	since	SCONJ
ma-30	277	2	c‖x‖2	c‖x‖2	VERB
ma-30	277	3	≤	≤	NUM
ma-30	277	4	1	1	NUM
ma-30	277	5	2ρ(x	2ρ(x	NUM
ma-30	277	6	)	)	PUNCT
ma-30	277	7	,	,	PUNCT
ma-30	277	8	then	then	ADV
ma-30	277	9	c‖	c‖	PROPN
ma-30	277	10	x+y2	x+y2	PROPN
ma-30	277	11	‖2	‖2	NOUN
ma-30	277	12	≤	≤	NUM
ma-30	277	13	1	1	NUM
ma-30	277	14	2ρ	2ρ	NOUN
ma-30	277	15	(	(	PUNCT
ma-30	277	16	x+y	x+y	NUM
ma-30	277	17	2	2	NUM
ma-30	277	18	)	)	PUNCT
ma-30	277	19	≤	≤	NOUN
ma-30	277	20	1	1	NUM
ma-30	277	21	and	and	CCONJ
ma-30	277	22	c‖	c‖	PROPN
ma-30	277	23	x−y2	x−y2	PROPN
ma-30	278	1	‖2	‖2	NOUN
ma-30	278	2	≤	≤	NUM
ma-30	279	1	1	1	NUM
ma-30	279	2	2ρ	2ρ	NOUN
ma-30	279	3	(	(	PUNCT
ma-30	279	4	x−y	x−y	PROPN
ma-30	279	5	2	2	NUM
ma-30	279	6	)	)	PUNCT
ma-30	279	7	≤	≤	NOUN
ma-30	279	8	1.therefore	1.therefore	NUM
ma-30	279	9	4	4	NUM
ma-30	279	10	t	t	NOUN
ma-30	279	11	t2	t2	NOUN
ma-30	279	12	−	−	PROPN
ma-30	279	13	2s2	2s2	NUM
ma-30	279	14	=	=	SYM
ma-30	279	15	∥∥	∥∥	X
ma-30	279	16	x+y	x+y	NUM
ma-30	279	17	2	2	NUM
ma-30	279	18	∥∥2	∥∥2	NOUN
ma-30	279	19	+	+	CCONJ
ma-30	279	20	‖	‖	PROPN
ma-30	279	21	x−y2	x−y2	PROPN
ma-30	279	22	‖2∥∥	‖2∥∥	NUM
ma-30	279	23	x+y	x+y	NUM
ma-30	279	24	2	2	NUM
ma-30	279	25	∥∥4	∥∥4	NOUN
ma-30	279	26	+	+	NUM
ma-30	279	27	∥∥	∥∥	PUNCT
ma-30	279	28	x−y2	x−y2	NUM
ma-30	279	29	∥∥4	∥∥4	PROPN
ma-30	279	30	≥	≥	NUM
ma-30	279	31	1	1	NUM
ma-30	279	32	‖	‖	PROPN
ma-30	279	33	x+y2	x+y2	PROPN
ma-30	279	34	‖2+‖	‖2+‖	ADJ
ma-30	279	35	x−y	x−y	PROPN
ma-30	279	36	2	2	NUM
ma-30	279	37	‖2	‖2	NOUN
ma-30	279	38	≥	≥	NOUN
ma-30	279	39	c	c	NOUN
ma-30	279	40	,	,	PUNCT
ma-30	279	41	then	then	ADV
ma-30	279	42	i1	i1	PROPN
ma-30	279	43	≤	≤	PROPN
ma-30	279	44	0	0	NUM
ma-30	279	45	.	.	PUNCT
ma-30	279	46	example	example	NOUN
ma-30	280	1	3	3	X
ma-30	280	2	.	.	PUNCT
ma-30	281	1	if	if	SCONJ
ma-30	281	2	ρ	ρ	PROPN
ma-30	281	3	is	be	AUX
ma-30	281	4	convex	convex	NOUN
ma-30	281	5	,	,	PUNCT
ma-30	281	6	then	then	ADV
ma-30	281	7	c	c	NOUN
ma-30	281	8	=	=	SYM
ma-30	281	9	0	0	NUM
ma-30	281	10	which	which	PRON
ma-30	281	11	satisfies	satisfy	VERB
ma-30	281	12	the	the	DET
ma-30	281	13	condition	condition	NOUN
ma-30	281	14	of	of	ADP
ma-30	281	15	theorem	theorem	ADJ
ma-30	281	16	6	6	NUM
ma-30	281	17	,	,	PUNCT
ma-30	281	18	and	and	CCONJ
ma-30	281	19	cnj	cnj	PROPN
ma-30	281	20	(	(	PUNCT
ma-30	281	21	xρ	xρ	PROPN
ma-30	281	22	)	)	PUNCT
ma-30	281	23	≤	≤	NOUN
ma-30	281	24	2	2	NUM
ma-30	281	25	<	<	SYM
ma-30	281	26	3	3	NUM
ma-30	281	27	.	.	PUNCT
ma-30	281	28	theorem	theorem	NOUN
ma-30	281	29	7	7	NUM
ma-30	281	30	.	.	PUNCT
ma-30	282	1	let	let	AUX
ma-30	282	2	(	(	PUNCT
ma-30	282	3	x	x	NOUN
ma-30	282	4	,	,	PUNCT
ma-30	282	5	‖·‖	‖·‖	NUM
ma-30	282	6	)	)	PUNCT
ma-30	282	7	be	be	VERB
ma-30	282	8	a	a	DET
ma-30	282	9	normed	normed	ADJ
ma-30	282	10	space	space	NOUN
ma-30	282	11	and	and	CCONJ
ma-30	282	12	xρ	xρ	PROPN
ma-30	282	13	be	be	AUX
ma-30	282	14	a	a	DET
ma-30	282	15	modular	modular	ADJ
ma-30	282	16	space	space	NOUN
ma-30	282	17	.	.	PUNCT
ma-30	283	1	if	if	SCONJ
ma-30	283	2	there	there	PRON
ma-30	283	3	exist	exist	VERB
ma-30	283	4	c	c	PROPN
ma-30	283	5	,	,	PUNCT
ma-30	283	6	λ	λ	PROPN
ma-30	283	7	,	,	PUNCT
ma-30	283	8	µ	µ	NOUN
ma-30	283	9	,	,	PUNCT
ma-30	283	10	γ	γ	X
ma-30	283	11	>	>	X
ma-30	283	12	0such	0such	PROPN
ma-30	283	13	that	that	DET
ma-30	283	14	2µγ	2µγ	ADJ
ma-30	283	15	≤	≤	ADV
ma-30	283	16	1	1	NUM
ma-30	283	17	≤	≤	NUM
ma-30	283	18	1	1	NUM
ma-30	283	19	2λ	2λ	NOUN
ma-30	283	20	+	+	CCONJ
ma-30	283	21	√	√	NUM
ma-30	283	22	6	6	NUM
ma-30	283	23	8µ	8µ	NUM
ma-30	283	24	and	and	CCONJ
ma-30	283	25	µρ(x	µρ(x	PUNCT
ma-30	283	26	)	)	PUNCT
ma-30	283	27	≤	≤	NOUN
ma-30	283	28	c‖x‖2	c‖x‖2	VERB
ma-30	283	29	≤	≤	NOUN
ma-30	283	30	λρ(x	λρ(x	NUM
ma-30	283	31	)	)	PUNCT
ma-30	283	32	for	for	ADP
ma-30	283	33	all	all	DET
ma-30	283	34	x	x	PROPN
ma-30	283	35	∈	∈	PROPN
ma-30	283	36	xρ.what’more	xρ.what’more	PROPN
ma-30	283	37	,	,	PUNCT
ma-30	283	38	ρ	ρ	PROPN
ma-30	283	39	is	be	AUX
ma-30	283	40	strongly	strongly	ADV
ma-30	283	41	midpoint	midpoint	ADJ
ma-30	283	42	convex	convex	NOUN
ma-30	283	43	with	with	ADP
ma-30	283	44	constant	constant	ADJ
ma-30	283	45	c	c	NOUN
ma-30	283	46	,	,	PUNCT
ma-30	283	47	then	then	ADV
ma-30	283	48	cnj	cnj	PROPN
ma-30	283	49	(	(	PUNCT
ma-30	283	50	xρ	xρ	PROPN
ma-30	283	51	)	)	PUNCT
ma-30	283	52	≤	≤	NOUN
ma-30	283	53	2	2	NUM
ma-30	283	54	.	.	PUNCT
ma-30	283	55	proof	proof	NOUN
ma-30	283	56	.	.	PUNCT
ma-30	284	1	by	by	ADP
ma-30	284	2	following	follow	VERB
ma-30	284	3	the	the	DET
ma-30	284	4	ideas	idea	NOUN
ma-30	284	5	in	in	ADP
ma-30	284	6	theorem	theorem	NOUN
ma-30	284	7	6	6	NUM
ma-30	284	8	,	,	PUNCT
ma-30	284	9	we	we	PRON
ma-30	284	10	can	can	AUX
ma-30	284	11	get	get	VERB
ma-30	284	12	ρ2	ρ2	NOUN
ma-30	284	13	(	(	PUNCT
ma-30	284	14	x+y	x+y	NUM
ma-30	284	15	2	2	NUM
ma-30	284	16	)	)	PUNCT
ma-30	285	1	+	+	CCONJ
ma-30	285	2	ρ2	ρ2	NOUN
ma-30	285	3	(	(	PUNCT
ma-30	285	4	x−y	x−y	PROPN
ma-30	285	5	2	2	NUM
ma-30	285	6	)	)	PUNCT
ma-30	285	7	1	1	NUM
ma-30	285	8	+	+	NUM
ma-30	285	9	ρ2(y	ρ2(y	NUM
ma-30	285	10	)	)	PUNCT
ma-30	285	11	≤	≤	NUM
ma-30	285	12	1	1	NUM
ma-30	285	13	2	2	NUM
ma-30	285	14	+	+	CCONJ
ma-30	285	15	1	1	NUM
ma-30	285	16	1	1	NUM
ma-30	285	17	+	+	NUM
ma-30	285	18	ρ2(y	ρ2(y	NUM
ma-30	285	19	)	)	PUNCT
ma-30	285	20	{	{	PUNCT
ma-30	285	21	ρ(y)−	ρ(y)−	PROPN
ma-30	285	22	c	c	PROPN
ma-30	285	23	4	4	NUM
ma-30	285	24	(	(	PUNCT
ma-30	285	25	1	1	NUM
ma-30	285	26	+	+	CCONJ
ma-30	285	27	ρ(y	ρ(y	NOUN
ma-30	285	28	)	)	PUNCT
ma-30	285	29	)	)	PUNCT
ma-30	286	1	(	(	PUNCT
ma-30	286	2	|	|	ADV
ma-30	286	3	x	x	X
ma-30	286	4	+	+	NUM
ma-30	286	5	y	y	PROPN
ma-30	286	6	∥∥2+∥∥	∥∥2+∥∥	NOUN
ma-30	286	7	x	x	PUNCT
ma-30	286	8	−	−	NOUN
ma-30	286	9	y‖2	y‖2	NOUN
ma-30	286	10	)	)	PUNCT
ma-30	286	11	}	}	PUNCT
ma-30	287	1	+	+	CCONJ
ma-30	288	1	1	1	NUM
ma-30	288	2	1	1	NUM
ma-30	288	3	+	+	NUM
ma-30	288	4	ρ2(y	ρ2(y	NUM
ma-30	288	5	)	)	PUNCT
ma-30	288	6	{	{	PUNCT
ma-30	288	7	c2	c2	PROPN
ma-30	288	8	16	16	NUM
ma-30	288	9	(	(	PUNCT
ma-30	288	10	‖x	‖x	NOUN
ma-30	288	11	+	+	PUNCT
ma-30	288	12	y	y	PROPN
ma-30	288	13	∥∥4+∥∥	∥∥4+∥∥	NUM
ma-30	288	14	x	x	X
ma-30	288	15	−	−	PROPN
ma-30	288	16	y‖4	y‖4	PROPN
ma-30	288	17	)	)	PUNCT
ma-30	288	18	}	}	PUNCT
ma-30	288	19	.	.	PUNCT
ma-30	289	1	let	let	VERB
ma-30	289	2	t	t	NOUN
ma-30	289	3	=	=	PUNCT
ma-30	289	4	‖x	‖x	X
ma-30	290	1	+	+	NUM
ma-30	290	2	y‖2	y‖2	X
ma-30	290	3	+	+	CCONJ
ma-30	290	4	‖x	‖x	NOUN
ma-30	290	5	−	−	PROPN
ma-30	290	6	y‖2	y‖2	PROPN
ma-30	290	7	,	,	PUNCT
ma-30	290	8	s	s	PART
ma-30	290	9	=	=	NOUN
ma-30	290	10	‖x	‖x	NOUN
ma-30	291	1	+	+	PUNCT
ma-30	291	2	y‖‖x	y‖‖x	INTJ
ma-30	291	3	−	−	PROPN
ma-30	291	4	y‖	y‖	PROPN
ma-30	291	5	and	and	CCONJ
ma-30	291	6	i2	i2	PROPN
ma-30	291	7	=	=	PROPN
ma-30	291	8	t2	t2	PROPN
ma-30	291	9	−	−	PROPN
ma-30	291	10	2s2	2s2	NUM
ma-30	291	11	16	16	NUM
ma-30	291	12	c2	c2	PROPN
ma-30	291	13	−	−	PROPN
ma-30	291	14	(	(	PUNCT
ma-30	291	15	1	1	NUM
ma-30	291	16	+	+	NUM
ma-30	291	17	ρ(y))t	ρ(y))t	PROPN
ma-30	291	18	4	4	NUM
ma-30	291	19	c	c	NOUN
ma-30	291	20	−	−	NOUN
ma-30	291	21	1	1	NUM
ma-30	291	22	2	2	NUM
ma-30	291	23	(	(	PUNCT
ma-30	291	24	1−	1−	NUM
ma-30	291	25	ρ(y))2	ρ(y))2	NOUN
ma-30	291	26	,	,	PUNCT
ma-30	291	27	thus	thus	ADV
ma-30	291	28	we	we	PRON
ma-30	291	29	only	only	ADV
ma-30	291	30	need	need	VERB
ma-30	291	31	to	to	PART
ma-30	291	32	prove	prove	VERB
ma-30	291	33	i2	i2	PROPN
ma-30	291	34	≤	≤	NUM
ma-30	291	35	0	0	NUM
ma-30	291	36	.	.	PUNCT
ma-30	292	1	since	since	SCONJ
ma-30	292	2	ρ(y	ρ(y	NOUN
ma-30	292	3	)	)	PUNCT
ma-30	292	4	≤	≤	NUM
ma-30	292	5	1	1	NUM
ma-30	292	6	,	,	PUNCT
ma-30	292	7	then	then	ADV
ma-30	292	8	(	(	PUNCT
ma-30	292	9	1	1	NUM
ma-30	292	10	+	+	NUM
ma-30	292	11	ρ(y))t	ρ(y))t	PROPN
ma-30	292	12	−	−	NOUN
ma-30	292	13	√	√	PROPN
ma-30	292	14	(	(	PUNCT
ma-30	292	15	1	1	NUM
ma-30	292	16	+	+	X
ma-30	292	17	ρ(y))2t2	ρ(y))2t2	PROPN
ma-30	292	18	+	+	CCONJ
ma-30	292	19	2	2	NUM
ma-30	292	20	(	(	PUNCT
ma-30	292	21	t2	t2	NOUN
ma-30	292	22	−	−	PROPN
ma-30	292	23	2s2	2s2	NUM
ma-30	292	24	)	)	PUNCT
ma-30	292	25	(	(	PUNCT
ma-30	292	26	1−	1−	NUM
ma-30	292	27	ρ(y))2	ρ(y))2	PROPN
ma-30	292	28	≤	≤	PROPN
ma-30	292	29	t	t	PROPN
ma-30	292	30	and	and	CCONJ
ma-30	292	31	(	(	PUNCT
ma-30	292	32	1	1	NUM
ma-30	292	33	+	+	NUM
ma-30	293	1	ρ(y))t	ρ(y))t	PROPN
ma-30	293	2	+	+	ADJ
ma-30	293	3	√(1	√(1	NOUN
ma-30	293	4	+	+	CCONJ
ma-30	293	5	ρ(y))2t2	ρ(y))2t2	NOUN
ma-30	293	6	+	+	CCONJ
ma-30	293	7	2	2	NUM
ma-30	293	8	(	(	PUNCT
ma-30	293	9	t2	t2	NOUN
ma-30	293	10	−	−	PROPN
ma-30	293	11	2s2	2s2	NUM
ma-30	293	12	)	)	PUNCT
ma-30	293	13	(	(	PUNCT
ma-30	293	14	1−	1−	NUM
ma-30	293	15	ρ(y))2	ρ(y))2	PROPN
ma-30	293	16	≥	≥	X
ma-30	293	17	t	t	PROPN
ma-30	293	18	+	+	PROPN
ma-30	293	19	√3t2	√3t2	PROPN
ma-30	293	20	−	−	PROPN
ma-30	293	21	4s2.thus	4s2.thus	NUM
ma-30	293	22	t	t	NOUN
ma-30	293	23	t2	t2	NOUN
ma-30	293	24	−	−	PROPN
ma-30	293	25	2s2	2s2	NUM
ma-30	293	26	=	=	SYM
ma-30	293	27	1	1	NUM
ma-30	293	28	4	4	NUM
ma-30	293	29	·	·	PUNCT
ma-30	293	30	‖	‖	PROPN
ma-30	293	31	x+y2	x+y2	PROPN
ma-30	293	32	∥∥2	∥∥2	PROPN
ma-30	293	33	+	+	CCONJ
ma-30	293	34	‖	‖	PROPN
ma-30	293	35	x−y2	x−y2	PROPN
ma-30	293	36	‖2	‖2	NOUN
ma-30	293	37	‖	‖	ADJ
ma-30	293	38	x+y2	x+y2	PROPN
ma-30	293	39	‖4	‖4	NOUN
ma-30	293	40	+	+	CCONJ
ma-30	293	41	∥∥	∥∥	PUNCT
ma-30	293	42	x−y	x−y	NUM
ma-30	293	43	2	2	NUM
ma-30	293	44	∥∥4and	∥∥4and	NOUN
ma-30	293	45	t	t	NOUN
ma-30	293	46	+	+	CCONJ
ma-30	293	47	√	√	PROPN
ma-30	293	48	3t2	3t2	NUM
ma-30	293	49	−	−	PROPN
ma-30	293	50	4s2	4s2	NUM
ma-30	293	51	t2	t2	PROPN
ma-30	293	52	−	−	PROPN
ma-30	293	53	2s2	2s2	NUM
ma-30	293	54	=	=	SYM
ma-30	293	55	1	1	NUM
ma-30	293	56	4	4	NUM
ma-30	293	57	·	·	PUNCT
ma-30	293	58	‖x	‖x	NOUN
ma-30	294	1	+	+	PUNCT
ma-30	294	2	y‖2	y‖2	X
ma-30	294	3	+	+	CCONJ
ma-30	294	4	‖	‖	PROPN
ma-30	294	5	x−y2	x−y2	PROPN
ma-30	294	6	‖	‖	PROPN
ma-30	294	7	2	2	NUM
ma-30	294	8	+	+	CCONJ
ma-30	294	9	√	√	NUM
ma-30	294	10	3‖	3‖	NUM
ma-30	295	1	x+y2	x+y2	NOUN
ma-30	295	2	‖4	‖4	NOUN
ma-30	295	3	+	+	CCONJ
ma-30	296	1	3‖	3‖	NUM
ma-30	296	2	x−y	x−y	NUM
ma-30	296	3	2	2	NUM
ma-30	296	4	‖4	‖4	NOUN
ma-30	296	5	+	+	X
ma-30	296	6	2‖	2‖	PROPN
ma-30	296	7	x+y	x+y	NUM
ma-30	296	8	2	2	NUM
ma-30	296	9	‖2‖	‖2‖	PROPN
ma-30	296	10	x−y	x−y	NUM
ma-30	296	11	2	2	NUM
ma-30	296	12	‖2	‖2	NOUN
ma-30	296	13	‖	‖	ADJ
ma-30	296	14	x+y2	x+y2	NOUN
ma-30	296	15	‖4	‖4	NOUN
ma-30	296	16	+	+	CCONJ
ma-30	296	17	‖	‖	PROPN
ma-30	296	18	x−y	x−y	ADJ
ma-30	296	19	2	2	NUM
ma-30	296	20	‖4	‖4	NOUN
ma-30	296	21	,	,	PUNCT
ma-30	296	22	eur	eur	PROPN
ma-30	296	23	.	.	PUNCT
ma-30	297	1	j.	j.	PROPN
ma-30	297	2	math	math	PROPN
ma-30	297	3	.	.	PUNCT
ma-30	298	1	anal	anal	ADJ
ma-30	298	2	.	.	PUNCT
ma-30	299	1	1	1	NUM
ma-30	299	2	(	(	PUNCT
ma-30	299	3	2021	2021	NUM
ma-30	299	4	)	)	PUNCT
ma-30	300	1	161then	161then	NOUN
ma-30	300	2	t+	t+	VERB
ma-30	300	3	√	√	PROPN
ma-30	300	4	3t2−4s2	3t2−4s2	NUM
ma-30	300	5	t2−2s2	t2−2s2	PROPN
ma-30	300	6	≥	≥	NUM
ma-30	300	7	1	1	NUM
ma-30	300	8	4‖	4‖	NUM
ma-30	300	9	x+y	x+y	NUM
ma-30	300	10	2	2	NUM
ma-30	300	11	‖2	‖2	NOUN
ma-30	300	12	+	+	NOUN
ma-30	300	13	4‖	4‖	NOUN
ma-30	300	14	x−y	x−y	NUM
ma-30	300	15	2	2	NUM
ma-30	300	16	‖2	‖2	NOUN
ma-30	300	17	+	+	CCONJ
ma-30	300	18	√	√	NUM
ma-30	300	19	3	3	NUM
ma-30	300	20	4	4	NUM
ma-30	300	21	√	√	NUM
ma-30	300	22	‖	‖	PROPN
ma-30	300	23	x+y	x+y	NUM
ma-30	300	24	2	2	NUM
ma-30	300	25	‖4+‖	‖4+‖	NOUN
ma-30	300	26	x−y	x−y	PROPN
ma-30	300	27	2	2	NUM
ma-30	300	28	‖4	‖4	NOUN
ma-30	300	29	≥	≥	NOUN
ma-30	300	30	c	c	PROPN
ma-30	301	1	λ(ρ	λ(ρ	PROPN
ma-30	301	2	(	(	PUNCT
ma-30	301	3	x+y2	x+y2	PROPN
ma-30	301	4	)	)	PUNCT
ma-30	301	5	+	+	NOUN
ma-30	301	6	ρ	ρ	PROPN
ma-30	301	7	(	(	PUNCT
ma-30	301	8	x−y	x−y	PROPN
ma-30	301	9	2	2	NUM
ma-30	301	10	)	)	PUNCT
ma-30	301	11	)	)	PUNCT
ma-30	302	1	+	+	CCONJ
ma-30	303	1	√	√	NUM
ma-30	303	2	3c	3c	NUM
ma-30	303	3	4µ	4µ	NOUN
ma-30	303	4	√	√	NOUN
ma-30	303	5	ρ2	ρ2	PROPN
ma-30	303	6	(	(	PUNCT
ma-30	303	7	x+y2	x+y2	PROPN
ma-30	303	8	)	)	PUNCT
ma-30	303	9	+	+	NOUN
ma-30	303	10	ρ2	ρ2	PROPN
ma-30	303	11	(	(	PUNCT
ma-30	303	12	x−y	x−y	NOUN
ma-30	303	13	2	2	NUM
ma-30	303	14	)	)	PUNCT
ma-30	303	15	≥	≥	NOUN
ma-30	303	16	c	c	NOUN
ma-30	303	17	4λ	4λ	PROPN
ma-30	303	18	+	+	CCONJ
ma-30	303	19	√	√	NUM
ma-30	303	20	3c	3c	NUM
ma-30	303	21	8	8	NUM
ma-30	303	22	√	√	NUM
ma-30	303	23	2µ	2µ	NUM
ma-30	303	24	≥	≥	NOUN
ma-30	303	25	c	c	NOUN
ma-30	303	26	2	2	NUM
ma-30	303	27	,	,	PUNCT
ma-30	303	28	and	and	CCONJ
ma-30	303	29	t	t	PROPN
ma-30	303	30	t2	t2	PROPN
ma-30	303	31	−	−	PROPN
ma-30	303	32	2s2	2s2	NUM
ma-30	303	33	≤	≤	NUM
ma-30	303	34	1	1	NUM
ma-30	303	35	2	2	NUM
ma-30	303	36	·	·	SYM
ma-30	303	37	1∥∥	1∥∥	PROPN
ma-30	303	38	x+y	x+y	NUM
ma-30	303	39	2	2	NUM
ma-30	303	40	∥∥2	∥∥2	NOUN
ma-30	304	1	+	+	CCONJ
ma-30	304	2	∥∥	∥∥	PUNCT
ma-30	304	3	x−y2	x−y2	PROPN
ma-30	304	4	∥∥2	∥∥2	PROPN
ma-30	305	1	≤	≤	PROPN
ma-30	305	2	c	c	PROPN
ma-30	305	3	4µ	4µ	NUM
ma-30	305	4	(	(	PUNCT
ma-30	305	5	ρ	ρ	PROPN
ma-30	305	6	(	(	PUNCT
ma-30	305	7	x+y	x+y	PROPN
ma-30	305	8	2	2	NUM
ma-30	305	9	)	)	PUNCT
ma-30	305	10	+	+	CCONJ
ma-30	305	11	ρ	ρ	PROPN
ma-30	305	12	(	(	PUNCT
ma-30	305	13	x−y	x−y	PROPN
ma-30	305	14	2	2	NUM
ma-30	305	15	)	)	PUNCT
ma-30	305	16	)	)	PUNCT
ma-30	305	17	≤	≤	NOUN
ma-30	306	1	c	c	NOUN
ma-30	306	2	4µγ	4µγ	NOUN
ma-30	306	3	≤	≤	NUM
ma-30	306	4	c	c	NOUN
ma-30	306	5	2	2	NUM
ma-30	306	6	.	.	PUNCT
ma-30	307	1	therefore	therefore	ADV
ma-30	307	2	(	(	PUNCT
ma-30	307	3	1	1	NUM
ma-30	307	4	+	+	NUM
ma-30	307	5	ρ(y))t	ρ(y))t	PROPN
ma-30	307	6	−	−	NOUN
ma-30	307	7	√	√	PROPN
ma-30	307	8	(	(	PUNCT
ma-30	307	9	1	1	NUM
ma-30	307	10	+	+	X
ma-30	307	11	ρ(y))2t2	ρ(y))2t2	PROPN
ma-30	307	12	+	+	CCONJ
ma-30	307	13	2	2	NUM
ma-30	307	14	(	(	PUNCT
ma-30	307	15	t2	t2	NOUN
ma-30	307	16	−	−	PROPN
ma-30	307	17	2s2	2s2	NUM
ma-30	307	18	)	)	PUNCT
ma-30	307	19	(	(	PUNCT
ma-30	307	20	1−	1−	NUM
ma-30	307	21	ρ(y))2	ρ(y))2	NOUN
ma-30	307	22	t2	t2	NOUN
ma-30	307	23	−	−	PROPN
ma-30	307	24	2s2	2s2	NUM
ma-30	307	25	≤	≤	NUM
ma-30	308	1	c	c	NOUN
ma-30	308	2	2	2	NUM
ma-30	308	3	≤	≤	NUM
ma-30	308	4	(	(	PUNCT
ma-30	308	5	1	1	NUM
ma-30	308	6	+	+	NUM
ma-30	308	7	ρ(y))t	ρ(y))t	PROPN
ma-30	308	8	+	+	CCONJ
ma-30	308	9	√	√	PROPN
ma-30	308	10	(	(	PUNCT
ma-30	308	11	1	1	NUM
ma-30	308	12	+	+	X
ma-30	308	13	ρ(y))2t2	ρ(y))2t2	PROPN
ma-30	308	14	+	+	CCONJ
ma-30	308	15	2	2	NUM
ma-30	308	16	(	(	PUNCT
ma-30	308	17	t2	t2	NOUN
ma-30	308	18	−	−	PROPN
ma-30	308	19	2s2	2s2	NUM
ma-30	308	20	)	)	PUNCT
ma-30	308	21	(	(	PUNCT
ma-30	308	22	1−	1−	NUM
ma-30	308	23	ρ(y))2	ρ(y))2	NOUN
ma-30	308	24	t2	t2	NOUN
ma-30	308	25	−	−	PROPN
ma-30	308	26	2s2thus	2s2thus	NUM
ma-30	308	27	i2	i2	PROPN
ma-30	308	28	=	=	PROPN
ma-30	308	29	t2	t2	PROPN
ma-30	308	30	−	−	PROPN
ma-30	308	31	2s2	2s2	NUM
ma-30	308	32	4	4	NUM
ma-30	308	33	(	(	PUNCT
ma-30	308	34	c	c	NOUN
ma-30	308	35	2	2	NUM
ma-30	308	36	−	−	NOUN
ma-30	308	37	(	(	PUNCT
ma-30	308	38	1	1	NUM
ma-30	308	39	+	+	NUM
ma-30	308	40	ρ(y))t	ρ(y))t	PROPN
ma-30	308	41	+	+	CCONJ
ma-30	308	42	√	√	PROPN
ma-30	308	43	(	(	PUNCT
ma-30	308	44	1	1	NUM
ma-30	308	45	+	+	X
ma-30	308	46	ρ(y))2t2	ρ(y))2t2	PROPN
ma-30	308	47	+	+	CCONJ
ma-30	308	48	2	2	NUM
ma-30	308	49	(	(	PUNCT
ma-30	308	50	t2	t2	NOUN
ma-30	308	51	−	−	PROPN
ma-30	308	52	2s2	2s2	NUM
ma-30	308	53	)	)	PUNCT
ma-30	308	54	(	(	PUNCT
ma-30	308	55	1−	1−	NUM
ma-30	308	56	ρ(y))2	ρ(y))2	NOUN
ma-30	308	57	t2	t2	NOUN
ma-30	308	58	−	−	PROPN
ma-30	308	59	2s2	2s2	NUM
ma-30	308	60	)	)	PUNCT
ma-30	309	1	(	(	PUNCT
ma-30	309	2	c	c	NOUN
ma-30	309	3	2	2	NUM
ma-30	309	4	−	−	NOUN
ma-30	309	5	(	(	PUNCT
ma-30	309	6	1	1	NUM
ma-30	309	7	+	+	NUM
ma-30	309	8	ρ(y))t	ρ(y))t	PROPN
ma-30	309	9	−	−	NOUN
ma-30	309	10	√	√	PROPN
ma-30	309	11	(	(	PUNCT
ma-30	309	12	1	1	NUM
ma-30	309	13	+	+	X
ma-30	309	14	ρ(y))2t2	ρ(y))2t2	PROPN
ma-30	309	15	+	+	CCONJ
ma-30	309	16	2	2	NUM
ma-30	309	17	(	(	PUNCT
ma-30	309	18	t2	t2	NOUN
ma-30	309	19	−	−	PROPN
ma-30	309	20	2s2	2s2	NUM
ma-30	309	21	)	)	PUNCT
ma-30	309	22	(	(	PUNCT
ma-30	309	23	1−	1−	NUM
ma-30	309	24	ρ(y))2	ρ(y))2	NOUN
ma-30	309	25	t2	t2	NOUN
ma-30	309	26	−	−	PROPN
ma-30	309	27	2s2	2s2	NUM
ma-30	309	28	)	)	PUNCT
ma-30	310	1	≤	≤	ADV
ma-30	310	2	0	0	NUM
ma-30	310	3	.	.	PUNCT
ma-30	310	4	example	example	NOUN
ma-30	311	1	4	4	X
ma-30	311	2	.	.	X
ma-30	311	3	consider	consider	VERB
ma-30	311	4	ρ(x	ρ(x	NOUN
ma-30	311	5	)	)	PUNCT
ma-30	311	6	=	=	PUNCT
ma-30	312	1	4c‖x‖2	4c‖x‖2	NUM
ma-30	312	2	and	and	CCONJ
ma-30	312	3	let	let	VERB
ma-30	312	4	λ	λ	X
ma-30	312	5	=	=	SYM
ma-30	312	6	µ	µ	X
ma-30	312	7	=	=	SYM
ma-30	312	8	1	1	NUM
ma-30	312	9	4	4	NUM
ma-30	312	10	,	,	PUNCT
ma-30	312	11	then	then	ADV
ma-30	312	12	µ2ρ(x	µ2ρ(x	X
ma-30	312	13	)	)	PUNCT
ma-30	312	14	≤	≤	NOUN
ma-30	312	15	c‖x‖2	c‖x‖2	VERB
ma-30	312	16	≤	≤	NOUN
ma-30	312	17	λρ(x	λρ(x	PUNCT
ma-30	312	18	)	)	PUNCT
ma-30	312	19	and	and	CCONJ
ma-30	312	20	µ2	µ2	PROPN
ma-30	312	21	=	=	PUNCT
ma-30	312	22	λ	λ	PROPN
ma-30	312	23	4	4	NUM
ma-30	312	24	.	.	PUNCT
ma-30	313	1	what’more	what’more	ADV
ma-30	313	2	,	,	PUNCT
ma-30	313	3	cnj	cnj	PROPN
ma-30	313	4	(	(	PUNCT
ma-30	313	5	xρ	xρ	PROPN
ma-30	313	6	)	)	PUNCT
ma-30	313	7	=	=	SYM
ma-30	313	8	2c	2c	NUM
ma-30	313	9	sup	sup	NOUN
ma-30	313	10	{	{	PUNCT
ma-30	313	11	‖x	‖x	NOUN
ma-30	313	12	+	+	NUM
ma-30	313	13	y‖4	y‖4	NOUN
ma-30	314	1	+	+	NUM
ma-30	314	2	‖x	‖x	NOUN
ma-30	315	1	−	−	NOUN
ma-30	315	2	y‖4	y‖4	INTJ
ma-30	315	3	‖x	‖x	PUNCT
ma-30	316	1	+	+	CCONJ
ma-30	316	2	y‖2	y‖2	X
ma-30	316	3	+	+	CCONJ
ma-30	316	4	‖x	‖x	NOUN
ma-30	316	5	−	−	PROPN
ma-30	316	6	y‖2	y‖2	X
ma-30	316	7	:	:	PUNCT
ma-30	317	1	x	x	X
ma-30	317	2	,	,	PUNCT
ma-30	317	3	y	y	PROPN
ma-30	317	4	∈	∈	PROPN
ma-30	317	5	xρ	xρ	PROPN
ma-30	317	6	,	,	PUNCT
ma-30	317	7	ρ(x	ρ(x	PROPN
ma-30	317	8	)	)	PUNCT
ma-30	317	9	=	=	SYM
ma-30	317	10	1	1	NUM
ma-30	317	11	,	,	PUNCT
ma-30	317	12	ρ(y	ρ(y	NOUN
ma-30	317	13	)	)	PUNCT
ma-30	317	14	≤	≤	ADV
ma-30	317	15	1	1	NUM
ma-30	317	16	}	}	PUNCT
ma-30	317	17	≤	≤	NUM
ma-30	317	18	2c	2c	NUM
ma-30	317	19	sup	sup	NOUN
ma-30	317	20	{	{	PUNCT
ma-30	317	21	‖x	‖x	NOUN
ma-30	317	22	+	+	PUNCT
ma-30	317	23	y	y	PROPN
ma-30	317	24	∥∥2+∥∥	∥∥2+∥∥	NOUN
ma-30	317	25	x	x	PUNCT
ma-30	317	26	−	−	X
ma-30	317	27	y‖2	y‖2	X
ma-30	317	28	:	:	PUNCT
ma-30	317	29	x	x	X
ma-30	317	30	,	,	PUNCT
ma-30	317	31	y	y	PROPN
ma-30	317	32	∈	∈	PROPN
ma-30	317	33	xρ	xρ	PROPN
ma-30	317	34	,	,	PUNCT
ma-30	317	35	ρ(x	ρ(x	PROPN
ma-30	317	36	)	)	PUNCT
ma-30	317	37	=	=	SYM
ma-30	317	38	1	1	NUM
ma-30	317	39	,	,	PUNCT
ma-30	317	40	ρ(y	ρ(y	NOUN
ma-30	317	41	)	)	PUNCT
ma-30	317	42	≤	≤	NUM
ma-30	317	43	1	1	NUM
ma-30	317	44	}	}	PUNCT
ma-30	317	45	≤	≤	NUM
ma-30	317	46	4c	4c	NOUN
ma-30	317	47	sup{‖x‖2	sup{‖x‖2	VERB
ma-30	317	48	+	+	CCONJ
ma-30	317	49	‖y‖2	‖y‖2	ADV
ma-30	317	50	:	:	PUNCT
ma-30	317	51	x	x	X
ma-30	317	52	,	,	PUNCT
ma-30	317	53	y	y	PROPN
ma-30	317	54	∈	∈	PROPN
ma-30	317	55	xρ	xρ	PROPN
ma-30	317	56	,	,	PUNCT
ma-30	317	57	ρ(x	ρ(x	PROPN
ma-30	317	58	)	)	PUNCT
ma-30	317	59	=	=	SYM
ma-30	317	60	1	1	NUM
ma-30	317	61	,	,	PUNCT
ma-30	317	62	ρ(y	ρ(y	NOUN
ma-30	317	63	)	)	PUNCT
ma-30	317	64	≤	≤	NOUN
ma-30	317	65	1	1	NUM
ma-30	317	66	}	}	PUNCT
ma-30	317	67	=	=	SYM
ma-30	317	68	sup{ρ(x	sup{ρ(x	PROPN
ma-30	317	69	)	)	PUNCT
ma-30	318	1	+	+	CCONJ
ma-30	318	2	ρ(y	ρ(y	NOUN
ma-30	318	3	)	)	PUNCT
ma-30	318	4	:	:	PUNCT
ma-30	319	1	x	x	X
ma-30	319	2	,	,	PUNCT
ma-30	319	3	y	y	PROPN
ma-30	319	4	∈	∈	PROPN
ma-30	319	5	xρ	xρ	PROPN
ma-30	319	6	,	,	PUNCT
ma-30	319	7	ρ(x	ρ(x	PROPN
ma-30	319	8	)	)	PUNCT
ma-30	319	9	=	=	SYM
ma-30	319	10	1	1	NUM
ma-30	319	11	,	,	PUNCT
ma-30	319	12	ρ(y	ρ(y	NOUN
ma-30	319	13	)	)	PUNCT
ma-30	319	14	≤	≤	NOUN
ma-30	319	15	1	1	NUM
ma-30	319	16	}	}	PUNCT
ma-30	319	17	=	=	SYM
ma-30	319	18	2	2	X
ma-30	319	19	.	.	X
ma-30	319	20	theorem	theorem	NOUN
ma-30	319	21	8	8	NUM
ma-30	319	22	.	.	PUNCT
ma-30	320	1	let	let	AUX
ma-30	320	2	(	(	PUNCT
ma-30	320	3	x	x	NOUN
ma-30	320	4	,	,	PUNCT
ma-30	320	5	‖	‖	PROPN
ma-30	320	6	·	·	PUNCT
ma-30	320	7	‖	‖	NUM
ma-30	320	8	)	)	PUNCT
ma-30	320	9	be	be	AUX
ma-30	320	10	a	a	DET
ma-30	320	11	normed	normed	ADJ
ma-30	320	12	space	space	NOUN
ma-30	320	13	,	,	PUNCT
ma-30	320	14	xρ	xρ	PROPN
ma-30	320	15	be	be	AUX
ma-30	320	16	a	a	DET
ma-30	320	17	modular	modular	ADJ
ma-30	320	18	space	space	NOUN
ma-30	320	19	and	and	CCONJ
ma-30	320	20	α0	α0	ADJ
ma-30	320	21	∈	∈	PROPN
ma-30	320	22	(	(	PUNCT
ma-30	320	23	0	0	NUM
ma-30	320	24	,	,	PUNCT
ma-30	320	25	2√2	2√2	NUM
ma-30	320	26	]	]	PUNCT
ma-30	320	27	.	.	PUNCT
ma-30	321	1	if	if	SCONJ
ma-30	321	2	thereexists	thereexist	NOUN
ma-30	321	3	c	c	PROPN
ma-30	321	4	>	>	X
ma-30	321	5	0	0	NUM
ma-30	321	6	such	such	ADJ
ma-30	321	7	that	that	SCONJ
ma-30	321	8	c	c	PROPN
ma-30	321	9	≥	≥	PUNCT
ma-30	321	10	4α0√	4α0√	NUM
ma-30	322	1	‖x0	‖x0	PROPN
ma-30	323	1	+	+	NUM
ma-30	323	2	y0‖4	y0‖4	X
ma-30	323	3	+	+	CCONJ
ma-30	323	4	‖x0	‖x0	ADJ
ma-30	323	5	−	−	PROPN
ma-30	323	6	y0‖4	y0‖4	NOUN
ma-30	323	7	for	for	ADP
ma-30	323	8	some	some	DET
ma-30	323	9	x0	x0	PROPN
ma-30	323	10	∈	∈	PROPN
ma-30	323	11	sxρ	sxρ	NOUN
ma-30	323	12	,	,	PUNCT
ma-30	323	13	y0	y0	PROPN
ma-30	323	14	∈	∈	NOUN
ma-30	323	15	bxρ	bxρ	NOUN
ma-30	323	16	and	and	CCONJ
ma-30	323	17	ρ	ρ	NOUN
ma-30	323	18	is	be	AUX
ma-30	323	19	strongly	strongly	ADV
ma-30	323	20	midpoint	midpoint	ADJ
ma-30	323	21	convex	convex	NOUN
ma-30	323	22	with	with	ADP
ma-30	323	23	positive	positive	ADJ
ma-30	323	24	constant	constant	ADJ
ma-30	323	25	c	c	NOUN
ma-30	323	26	,	,	PUNCT
ma-30	323	27	then	then	ADV
ma-30	323	28	cnj	cnj	PROPN
ma-30	323	29	(	(	PUNCT
ma-30	323	30	xρ	xρ	PROPN
ma-30	323	31	)	)	PUNCT
ma-30	323	32	≥	≥	NOUN
ma-30	323	33	α20	α20	NOUN
ma-30	323	34	.	.	PUNCT
ma-30	324	1	in	in	ADP
ma-30	324	2	particular	particular	ADJ
ma-30	324	3	,	,	PUNCT
ma-30	324	4	if	if	SCONJ
ma-30	324	5	α0	α0	ADJ
ma-30	324	6	=	=	SYM
ma-30	324	7	2	2	NUM
ma-30	324	8	√	√	NUM
ma-30	324	9	2	2	NUM
ma-30	324	10	,	,	PUNCT
ma-30	324	11	then	then	ADV
ma-30	324	12	cnj	cnj	PROPN
ma-30	324	13	(	(	PUNCT
ma-30	324	14	xρ	xρ	PROPN
ma-30	324	15	)	)	PUNCT
ma-30	324	16	=	=	SYM
ma-30	324	17	8	8	X
ma-30	324	18	.	.	PUNCT
ma-30	325	1	proof	proof	NOUN
ma-30	325	2	.	.	PUNCT
ma-30	326	1	since	since	SCONJ
ma-30	326	2	2ρ(x	2ρ(x	NUM
ma-30	326	3	)	)	PUNCT
ma-30	326	4	≥	≥	NOUN
ma-30	326	5	c‖x‖2	c‖x‖2	NOUN
ma-30	326	6	,	,	PUNCT
ma-30	326	7	then	then	ADV
ma-30	326	8	ρ2	ρ2	PROPN
ma-30	326	9	(	(	PUNCT
ma-30	326	10	x±y2	x±y2	PROPN
ma-30	326	11	)	)	PUNCT
ma-30	326	12	≥	≥	PROPN
ma-30	326	13	c	c	NOUN
ma-30	326	14	2	2	NUM
ma-30	326	15	∥∥	∥∥	X
ma-30	326	16	x±y	x±y	PROPN
ma-30	326	17	2	2	NUM
ma-30	326	18	∥∥2	∥∥2	PROPN
ma-30	326	19	.	.	PUNCT
ma-30	327	1	therefore	therefore	ADV
ma-30	327	2	ρ2	ρ2	PROPN
ma-30	327	3	(	(	PUNCT
ma-30	327	4	x+y	x+y	NUM
ma-30	327	5	2	2	NUM
ma-30	327	6	)	)	PUNCT
ma-30	328	1	+	+	CCONJ
ma-30	328	2	ρ2	ρ2	NOUN
ma-30	328	3	(	(	PUNCT
ma-30	328	4	x−y	x−y	PROPN
ma-30	328	5	2	2	NUM
ma-30	328	6	)	)	PUNCT
ma-30	328	7	1	1	NUM
ma-30	328	8	+	+	NUM
ma-30	328	9	ρ2(y	ρ2(y	NUM
ma-30	328	10	)	)	PUNCT
ma-30	328	11	≥	≥	NOUN
ma-30	328	12	c2	c2	PROPN
ma-30	328	13	16	16	NUM
ma-30	328	14	(	(	PUNCT
ma-30	328	15	‖x	‖x	NOUN
ma-30	328	16	+	+	NUM
ma-30	328	17	y‖4	y‖4	NOUN
ma-30	329	1	+	+	NUM
ma-30	329	2	‖x	‖x	NOUN
ma-30	329	3	−	−	NOUN
ma-30	329	4	y‖4	y‖4	NOUN
ma-30	329	5	)	)	PUNCT
ma-30	329	6	1	1	NUM
ma-30	330	1	+	+	NUM
ma-30	330	2	ρ2(y	ρ2(y	NUM
ma-30	330	3	)	)	PUNCT
ma-30	330	4	,	,	PUNCT
ma-30	330	5	eur	eur	PROPN
ma-30	330	6	.	.	PUNCT
ma-30	331	1	j.	j.	PROPN
ma-30	331	2	math	math	PROPN
ma-30	331	3	.	.	PUNCT
ma-30	332	1	anal	anal	ADJ
ma-30	332	2	.	.	PUNCT
ma-30	333	1	1	1	NUM
ma-30	333	2	(	(	PUNCT
ma-30	333	3	2021	2021	NUM
ma-30	333	4	)	)	PUNCT
ma-30	334	1	162	162	NUM
ma-30	334	2	shows	show	VERB
ma-30	334	3	that	that	SCONJ
ma-30	334	4	ρ2	ρ2	NOUN
ma-30	334	5	(	(	PUNCT
ma-30	334	6	x+y2	x+y2	PROPN
ma-30	334	7	)	)	PUNCT
ma-30	334	8	+	+	NOUN
ma-30	334	9	ρ	ρ	NOUN
ma-30	334	10	2	2	NUM
ma-30	334	11	(	(	PUNCT
ma-30	334	12	x−y2	x−y2	NUM
ma-30	334	13	)	)	PUNCT
ma-30	334	14	1+ρ2(y	1+ρ2(y	PROPN
ma-30	334	15	)	)	PUNCT
ma-30	334	16	≥	≥	NOUN
ma-30	334	17	α20	α20	NOUN
ma-30	334	18	1+ρ2(y	1+ρ2(y	NUM
ma-30	334	19	)	)	PUNCT
ma-30	334	20	,	,	PUNCT
ma-30	334	21	then	then	ADV
ma-30	334	22	cnj	cnj	PROPN
ma-30	334	23	(	(	PUNCT
ma-30	334	24	xρ	xρ	PROPN
ma-30	334	25	)	)	PUNCT
ma-30	334	26	≥	≥	NOUN
ma-30	334	27	α20	α20	NOUN
ma-30	334	28	.	.	PUNCT
ma-30	335	1	if	if	SCONJ
ma-30	335	2	α0	α0	PROPN
ma-30	335	3	=	=	SYM
ma-30	335	4	2√2	2√2	NUM
ma-30	335	5	,	,	PUNCT
ma-30	335	6	then	then	ADV
ma-30	335	7	cn	cn	PROPN
ma-30	335	8	(	(	PUNCT
ma-30	335	9	xρ	xρ	PROPN
ma-30	335	10	)	)	PUNCT
ma-30	335	11	≥	≥	NOUN
ma-30	335	12	8implies	8implies	NUM
ma-30	335	13	cnj	cnj	NOUN
ma-30	335	14	(	(	PUNCT
ma-30	335	15	xρ	xρ	PROPN
ma-30	335	16	)	)	PUNCT
ma-30	335	17	=	=	NOUN
ma-30	335	18	8	8	NUM
ma-30	335	19	.	.	X
ma-30	335	20	7	7	X
ma-30	335	21	.	.	X
ma-30	335	22	data	datum	NOUN
ma-30	335	23	availability	availability	NOUN
ma-30	335	24	no	no	DET
ma-30	335	25	data	datum	NOUN
ma-30	335	26	were	be	AUX
ma-30	335	27	used	use	VERB
ma-30	335	28	to	to	PART
ma-30	335	29	support	support	VERB
ma-30	335	30	this	this	DET
ma-30	335	31	study	study	NOUN
ma-30	335	32	.	.	PUNCT
ma-30	336	1	8	8	X
ma-30	336	2	.	.	PUNCT
ma-30	337	1	conflicts	conflict	NOUN
ma-30	337	2	of	of	ADP
ma-30	337	3	interest	interest	NOUN
ma-30	337	4	the	the	DET
ma-30	337	5	author(s	author(s	NOUN
ma-30	337	6	)	)	PUNCT
ma-30	337	7	declare(s	declare(s	NOUN
ma-30	337	8	)	)	PUNCT
ma-30	337	9	that	that	SCONJ
ma-30	337	10	there	there	PRON
ma-30	337	11	is	be	VERB
ma-30	337	12	no	no	DET
ma-30	337	13	conflict	conflict	NOUN
ma-30	337	14	of	of	ADP
ma-30	337	15	interest	interest	NOUN
ma-30	337	16	regarding	regard	VERB
ma-30	337	17	the	the	DET
ma-30	337	18	publication	publication	NOUN
ma-30	337	19	of	of	ADP
ma-30	337	20	this	this	DET
ma-30	337	21	paper	paper	NOUN
ma-30	337	22	.	.	PUNCT
ma-30	338	1	9	9	X
ma-30	338	2	.	.	X
ma-30	338	3	funding	funding	NOUN
ma-30	338	4	statement	statement	NOUN
ma-30	338	5	this	this	DET
ma-30	338	6	work	work	NOUN
ma-30	338	7	was	be	AUX
ma-30	338	8	supported	support	VERB
ma-30	338	9	by	by	ADP
ma-30	338	10	the	the	DET
ma-30	338	11	national	national	ADJ
ma-30	338	12	natural	natural	PROPN
ma-30	338	13	science	science	PROPN
ma-30	338	14	foundation	foundation	NOUN
ma-30	338	15	of	of	ADP
ma-30	338	16	p.	p.	PROPN
ma-30	338	17	r.	r.	PROPN
ma-30	338	18	china	china	PROPN
ma-30	338	19	(	(	PUNCT
ma-30	338	20	nos.11971493	nos.11971493	NUM
ma-30	338	21	and	and	CCONJ
ma-30	338	22	12071491	12071491	NUM
ma-30	338	23	)	)	PUNCT
ma-30	338	24	.	.	PUNCT
ma-30	339	1	references	reference	NOUN
ma-30	339	2	[	[	X
ma-30	339	3	1	1	X
ma-30	339	4	]	]	PUNCT
ma-30	339	5	j.	j.	PROPN
ma-30	339	6	lindenstrauss	lindenstrauss	PROPN
ma-30	339	7	,	,	PUNCT
ma-30	339	8	on	on	ADP
ma-30	339	9	the	the	DET
ma-30	339	10	modulus	modulus	NOUN
ma-30	339	11	of	of	ADP
ma-30	339	12	smoothness	smoothness	ADJ
ma-30	339	13	and	and	CCONJ
ma-30	339	14	divergent	divergent	ADJ
ma-30	339	15	series	series	NOUN
ma-30	339	16	in	in	ADP
ma-30	339	17	banach	banach	NOUN
ma-30	339	18	spaces	space	NOUN
ma-30	339	19	,	,	PUNCT
ma-30	339	20	michigan	michigan	PROPN
ma-30	339	21	math	math	PROPN
ma-30	339	22	.	.	PUNCT
ma-30	340	1	j.	j.	PROPN
ma-30	340	2	10	10	NUM
ma-30	340	3	(	(	PUNCT
ma-30	340	4	1963	1963	NUM
ma-30	340	5	)	)	PUNCT
ma-30	340	6	.	.	PUNCT
ma-30	341	1	https://doi.org/10.1307/mmj/1028998906.[2	https://doi.org/10.1307/mmj/1028998906.[2	X
ma-30	341	2	]	]	X
ma-30	341	3	j.a	j.a	PROPN
ma-30	341	4	.	.	PROPN
ma-30	341	5	clarkson	clarkson	PROPN
ma-30	341	6	,	,	PUNCT
ma-30	341	7	uniformly	uniformly	ADV
ma-30	341	8	convex	convex	NOUN
ma-30	341	9	spaces	space	NOUN
ma-30	341	10	,	,	PUNCT
ma-30	341	11	trans	trans	PROPN
ma-30	341	12	.	.	PROPN
ma-30	342	1	amer	amer	PROPN
ma-30	342	2	.	.	PUNCT
ma-30	342	3	math	math	PROPN
ma-30	342	4	.	.	PUNCT
ma-30	343	1	soc	soc	PROPN
ma-30	343	2	.	.	PUNCT
ma-30	344	1	40	40	NUM
ma-30	344	2	(	(	PUNCT
ma-30	344	3	1936	1936	NUM
ma-30	344	4	)	)	PUNCT
ma-30	345	1	396–396	396–396	NUM
ma-30	345	2	.	.	PUNCT
ma-30	346	1	https://doi.org/10.1090/	https://doi.org/10.1090/	DET
ma-30	346	2	s0002	s0002	NOUN
ma-30	346	3	-	-	PUNCT
ma-30	346	4	9947	9947	NUM
ma-30	346	5	-	-	PUNCT
ma-30	346	6	1936	1936	NUM
ma-30	346	7	-	-	PUNCT
ma-30	346	8	1501880	1501880	NUM
ma-30	346	9	-	-	PUNCT
ma-30	346	10	4.[3	4.[3	NOUN
ma-30	346	11	]	]	X
ma-30	346	12	j.a	j.a	PROPN
ma-30	346	13	.	.	PROPN
ma-30	346	14	clarkson	clarkson	PROPN
ma-30	346	15	,	,	PUNCT
ma-30	346	16	the	the	DET
ma-30	346	17	von	von	PROPN
ma-30	346	18	neumann	neumann	PROPN
ma-30	346	19	-	-	PUNCT
ma-30	346	20	jordan	jordan	PROPN
ma-30	346	21	constant	constant	PROPN
ma-30	346	22	for	for	ADP
ma-30	346	23	the	the	DET
ma-30	346	24	lebesgue	lebesgue	NOUN
ma-30	346	25	spaces	space	NOUN
ma-30	346	26	,	,	PUNCT
ma-30	346	27	the	the	DET
ma-30	346	28	annals	annal	NOUN
ma-30	346	29	of	of	ADP
ma-30	346	30	mathematics	mathematic	NOUN
ma-30	346	31	.	.	PUNCT
ma-30	347	1	38	38	NUM
ma-30	347	2	(	(	PUNCT
ma-30	347	3	1937)114	1937)114	NUM
ma-30	347	4	.	.	PUNCT
ma-30	348	1	https://doi.org/10.2307/1968512.[4	https://doi.org/10.2307/1968512.[4	PRON
ma-30	348	2	]	]	X
ma-30	348	3	r.c	r.c	PROPN
ma-30	348	4	.	.	PROPN
ma-30	348	5	james	james	PROPN
ma-30	348	6	,	,	PUNCT
ma-30	348	7	uniformly	uniformly	ADV
ma-30	348	8	non	non	ADJ
ma-30	348	9	-	-	ADJ
ma-30	348	10	square	square	ADJ
ma-30	348	11	banach	banach	NOUN
ma-30	348	12	spaces	space	NOUN
ma-30	348	13	,	,	PUNCT
ma-30	348	14	the	the	DET
ma-30	348	15	annals	annal	NOUN
ma-30	348	16	of	of	ADP
ma-30	348	17	mathematics	mathematic	NOUN
ma-30	348	18	.	.	PUNCT
ma-30	349	1	80	80	NUM
ma-30	349	2	(	(	PUNCT
ma-30	349	3	1964	1964	NUM
ma-30	349	4	)	)	PUNCT
ma-30	349	5	542	542	NUM
ma-30	349	6	.	.	PUNCT
ma-30	350	1	https://doi.org/	https://doi.org/	VERB
ma-30	350	2	10.2307/1970663.[5	10.2307/1970663.[5	NUM
ma-30	350	3	]	]	X
ma-30	350	4	e.	e.	PROPN
ma-30	350	5	l.	l.	PROPN
ma-30	350	6	fuster	fuster	PROPN
ma-30	350	7	,	,	PUNCT
ma-30	350	8	moduli	modulus	NOUN
ma-30	350	9	and	and	CCONJ
ma-30	350	10	constants	constant	NOUN
ma-30	350	11	-	-	PUNCT
ma-30	350	12	what	what	PRON
ma-30	350	13	a	a	DET
ma-30	350	14	show	show	NOUN
ma-30	350	15	!	!	PUNCT
ma-30	351	1	(	(	PUNCT
ma-30	351	2	2006	2006	NUM
ma-30	351	3	)	)	PUNCT
ma-30	351	4	.	.	PUNCT
ma-30	352	1	https://www.uv.es/llorens/documento.pdf.[6	https://www.uv.es/llorens/documento.pdf.[6	PROPN
ma-30	352	2	]	]	X
ma-30	353	1	j.	j.	PROPN
ma-30	353	2	musielak	musielak	PROPN
ma-30	353	3	,	,	PUNCT
ma-30	353	4	w.	w.	PROPN
ma-30	353	5	orlicz	orlicz	PROPN
ma-30	353	6	,	,	PUNCT
ma-30	353	7	on	on	ADP
ma-30	353	8	modular	modular	ADJ
ma-30	353	9	spaces	space	NOUN
ma-30	353	10	,	,	PUNCT
ma-30	353	11	studia	studia	PROPN
ma-30	353	12	math	math	NOUN
ma-30	353	13	.	.	PUNCT
ma-30	354	1	18	18	NUM
ma-30	354	2	(	(	PUNCT
ma-30	354	3	1959	1959	NUM
ma-30	354	4	)	)	PUNCT
ma-30	354	5	49–65	49–65	NUM
ma-30	354	6	.	.	PUNCT
ma-30	355	1	https://doi.org/10.4064/	https://doi.org/10.4064/	NOUN
ma-30	355	2	sm-18	sm-18	NOUN
ma-30	355	3	-	-	PUNCT
ma-30	355	4	1	1	NUM
ma-30	355	5	-	-	PUNCT
ma-30	355	6	49	49	NUM
ma-30	355	7	-	-	PUNCT
ma-30	355	8	65.[7	65.[7	NUM
ma-30	355	9	]	]	PUNCT
ma-30	355	10	h.	h.	PROPN
ma-30	355	11	nakano	nakano	PROPN
ma-30	355	12	,	,	PUNCT
ma-30	355	13	modulared	modulare	VERB
ma-30	355	14	semi	semi	ADJ
ma-30	355	15	-	-	ADJ
ma-30	355	16	ordered	ordered	ADJ
ma-30	355	17	linear	linear	ADJ
ma-30	355	18	spaces	space	NOUN
ma-30	355	19	,	,	PUNCT
ma-30	355	20	tokyo	tokyo	PROPN
ma-30	355	21	,	,	PUNCT
ma-30	355	22	maruzen	maruzen	PROPN
ma-30	355	23	co.ltd	co.ltd	PROPN
ma-30	355	24	.	.	PUNCT
ma-30	356	1	(	(	PUNCT
ma-30	356	2	1950).[8	1950).[8	X
ma-30	356	3	]	]	X
ma-30	356	4	m.a	m.a	PROPN
ma-30	356	5	.	.	PROPN
ma-30	356	6	khamsi	khamsi	PROPN
ma-30	356	7	,	,	PUNCT
ma-30	356	8	w.m	w.m	PROPN
ma-30	356	9	.	.	PROPN
ma-30	356	10	kozlowski	kozlowski	PROPN
ma-30	356	11	,	,	PUNCT
ma-30	356	12	fixed	fix	VERB
ma-30	356	13	point	point	NOUN
ma-30	356	14	theory	theory	NOUN
ma-30	356	15	in	in	ADP
ma-30	356	16	modular	modular	ADJ
ma-30	356	17	function	function	NOUN
ma-30	356	18	spaces	space	NOUN
ma-30	356	19	,	,	PUNCT
ma-30	356	20	springer	springer	NOUN
ma-30	356	21	international	international	ADJ
ma-30	356	22	publishing	publishing	NOUN
ma-30	356	23	,	,	PUNCT
ma-30	356	24	cham	cham	NOUN
ma-30	356	25	,	,	PUNCT
ma-30	356	26	2015	2015	NUM
ma-30	356	27	.	.	PUNCT
ma-30	357	1	https://doi.org/10.1007/978-3-319-14051-3.[9	https://doi.org/10.1007/978-3-319-14051-3.[9	PROPN
ma-30	357	2	]	]	PUNCT
ma-30	357	3	j.	j.	PROPN
ma-30	357	4	musielak	musielak	PROPN
ma-30	357	5	,	,	PUNCT
ma-30	357	6	orlicz	orlicz	NOUN
ma-30	357	7	spaces	space	NOUN
ma-30	357	8	and	and	CCONJ
ma-30	357	9	modular	modular	ADJ
ma-30	357	10	spaces.lecture	spaces.lecture	NOUN
ma-30	357	11	note	note	NOUN
ma-30	357	12	in	in	ADP
ma-30	357	13	math	math	NOUN
ma-30	357	14	,	,	PUNCT
ma-30	357	15	springer	springer	NOUN
ma-30	357	16	-	-	PUNCT
ma-30	357	17	verlag	verlag	PROPN
ma-30	357	18	,	,	PUNCT
ma-30	357	19	berlin	berlin	PROPN
ma-30	357	20	,	,	PUNCT
ma-30	357	21	heidelberg	heidelberg	PROPN
ma-30	357	22	,	,	PUNCT
ma-30	357	23	newyork	newyork	PROPN
ma-30	357	24	,	,	PUNCT
ma-30	357	25	1034	1034	NUM
ma-30	357	26	.	.	PUNCT
ma-30	358	1	(	(	PUNCT
ma-30	358	2	1983).[10	1983).[10	NUM
ma-30	358	3	]	]	PUNCT
ma-30	358	4	p.	p.	NOUN
ma-30	358	5	kumam	kumam	PROPN
ma-30	358	6	,	,	PUNCT
ma-30	358	7	fixed	fix	VERB
ma-30	358	8	point	point	NOUN
ma-30	358	9	property	property	NOUN
ma-30	358	10	in	in	ADP
ma-30	358	11	modular	modular	NOUN
ma-30	358	12	spaces.master	spaces.master	CCONJ
ma-30	358	13	thesis	thesis	NOUN
ma-30	358	14	,	,	PUNCT
ma-30	358	15	chiang	chiang	PROPN
ma-30	358	16	mai	mai	PROPN
ma-30	358	17	university	university	PROPN
ma-30	358	18	thailand	thailand	PROPN
ma-30	358	19	,	,	PUNCT
ma-30	358	20	(	(	PUNCT
ma-30	358	21	2002).[11	2002).[11	NUM
ma-30	358	22	]	]	PUNCT
ma-30	358	23	p.	p.	NOUN
ma-30	358	24	kumam	kumam	PROPN
ma-30	358	25	,	,	PUNCT
ma-30	358	26	on	on	ADP
ma-30	358	27	nonsquare	nonsquare	NOUN
ma-30	358	28	and	and	CCONJ
ma-30	358	29	von	von	PROPN
ma-30	358	30	neumann	neumann	PROPN
ma-30	358	31	constants	constant	NOUN
ma-30	358	32	of	of	ADP
ma-30	358	33	modular	modular	ADJ
ma-30	358	34	spaces	space	NOUN
ma-30	358	35	,	,	PUNCT
ma-30	358	36	southeast	southeast	ADJ
ma-30	358	37	asian	asian	ADJ
ma-30	358	38	bull	bull	NOUN
ma-30	358	39	.	.	PUNCT
ma-30	359	1	math	math	NOUN
ma-30	359	2	.	.	PUNCT
ma-30	360	1	30	30	NUM
ma-30	360	2	(	(	PUNCT
ma-30	360	3	2006),69	2006),69	NUM
ma-30	360	4	-	-	SYM
ma-30	360	5	77.[12	77.[12	PROPN
ma-30	360	6	]	]	PUNCT
ma-30	360	7	p.	p.	NOUN
ma-30	360	8	kumam	kumam	PROPN
ma-30	360	9	,	,	PUNCT
ma-30	360	10	some	some	DET
ma-30	360	11	geometric	geometric	ADJ
ma-30	360	12	properties	property	NOUN
ma-30	360	13	and	and	CCONJ
ma-30	360	14	fixed	fix	VERB
ma-30	360	15	point	point	NOUN
ma-30	360	16	theorem	theorem	VERB
ma-30	360	17	in	in	ADP
ma-30	360	18	modular	modular	ADJ
ma-30	360	19	spaces	space	NOUN
ma-30	360	20	,	,	PUNCT
ma-30	360	21	international	international	ADJ
ma-30	360	22	conference	conference	NOUN
ma-30	360	23	onfixed	onfixe	VERB
ma-30	360	24	point	point	NOUN
ma-30	360	25	theory	theory	NOUN
ma-30	360	26	and	and	CCONJ
ma-30	360	27	applications	application	NOUN
ma-30	360	28	,	,	PUNCT
ma-30	360	29	173	173	NUM
ma-30	360	30	-	-	SYM
ma-30	360	31	188	188	NUM
ma-30	360	32	.	.	PUNCT
ma-30	361	1	(	(	PUNCT
ma-30	361	2	2003).[13	2003).[13	NUM
ma-30	361	3	]	]	X
ma-30	361	4	c.	c.	PROPN
ma-30	361	5	yang	yang	PROPN
ma-30	361	6	,	,	PUNCT
ma-30	361	7	a	a	DET
ma-30	361	8	note	note	NOUN
ma-30	361	9	on	on	ADP
ma-30	361	10	jordan	jordan	PROPN
ma-30	361	11	-	-	PUNCT
ma-30	361	12	von	von	PROPN
ma-30	361	13	neumann	neumann	PROPN
ma-30	361	14	constant	constant	PROPN
ma-30	361	15	and	and	CCONJ
ma-30	361	16	james	james	PROPN
ma-30	361	17	constant	constant	PROPN
ma-30	361	18	,	,	PUNCT
ma-30	361	19	j.	j.	PROPN
ma-30	361	20	math	math	PROPN
ma-30	361	21	.	.	PUNCT
ma-30	362	1	anal	anal	PROPN
ma-30	362	2	.	.	PUNCT
ma-30	363	1	appl	appl	PROPN
ma-30	363	2	.	.	PUNCT
ma-30	364	1	357	357	NUM
ma-30	364	2	(	(	PUNCT
ma-30	364	3	2009	2009	NUM
ma-30	364	4	)	)	PUNCT
ma-30	365	1	98–102	98–102	NUM
ma-30	365	2	.	.	PUNCT
ma-30	366	1	https://doi.org/10.1016/j.jmaa.2009.04.002.[14	https://doi.org/10.1016/j.jmaa.2009.04.002.[14	PROPN
ma-30	366	2	]	]	X
ma-30	366	3	c.	c.	PROPN
ma-30	366	4	yang	yang	PROPN
ma-30	366	5	,	,	PUNCT
ma-30	366	6	an	an	DET
ma-30	366	7	inequality	inequality	NOUN
ma-30	366	8	between	between	ADP
ma-30	366	9	the	the	DET
ma-30	366	10	james	james	PROPN
ma-30	366	11	type	type	NOUN
ma-30	366	12	constant	constant	ADJ
ma-30	366	13	and	and	CCONJ
ma-30	366	14	the	the	DET
ma-30	366	15	modulus	modulus	NOUN
ma-30	366	16	of	of	ADP
ma-30	366	17	smoothness	smoothness	NOUN
ma-30	366	18	,	,	PUNCT
ma-30	366	19	journal	journal	NOUN
ma-30	366	20	of	of	ADP
ma-30	366	21	mathematicalanalysis	mathematicalanalysis	NOUN
ma-30	366	22	and	and	CCONJ
ma-30	366	23	applications	application	NOUN
ma-30	366	24	.	.	PUNCT
ma-30	367	1	398	398	NUM
ma-30	367	2	(	(	PUNCT
ma-30	367	3	2013	2013	NUM
ma-30	367	4	)	)	PUNCT
ma-30	368	1	622–629	622–629	NUM
ma-30	368	2	.	.	PUNCT
ma-30	369	1	https://doi.org/10.1016/j.jmaa.2012.07.063	https://doi.org/10.1016/j.jmaa.2012.07.063	NOUN
ma-30	369	2	.	.	PUNCT
ma-30	370	1	https://doi.org/10.1307/mmj/1028998906	https://doi.org/10.1307/mmj/1028998906	PROPN
ma-30	370	2	https://doi.org/10.1090/s0002-9947-1936-1501880-4	https://doi.org/10.1090/s0002-9947-1936-1501880-4	PROPN
ma-30	370	3	https://doi.org/10.1090/s0002-9947-1936-1501880-4	https://doi.org/10.1090/s0002-9947-1936-1501880-4	PROPN
ma-30	370	4	https://doi.org/10.2307/1968512	https://doi.org/10.2307/1968512	NOUN
ma-30	370	5	https://doi.org/10.2307/1970663	https://doi.org/10.2307/1970663	X
ma-30	370	6	https://doi.org/10.2307/1970663	https://doi.org/10.2307/1970663	X
ma-30	370	7	https://www.uv.es/llorens/documento.pdf	https://www.uv.es/llorens/documento.pdf	PROPN
ma-30	370	8	https://doi.org/10.4064/sm-18-1-49-65	https://doi.org/10.4064/sm-18-1-49-65	NOUN
ma-30	370	9	https://doi.org/10.4064/sm-18-1-49-65	https://doi.org/10.4064/sm-18-1-49-65	PUNCT
ma-30	371	1	https://doi.org/10.1007/978-3-319-14051-3	https://doi.org/10.1007/978-3-319-14051-3	PROPN
ma-30	371	2	https://doi.org/10.1016/j.jmaa.2009.04.002	https://doi.org/10.1016/j.jmaa.2009.04.002	NOUN
ma-30	371	3	https://doi.org/10.1016/j.jmaa.2012.07.063	https://doi.org/10.1016/j.jmaa.2012.07.063	PROPN
ma-30	371	4	eur	eur	NOUN
ma-30	371	5	.	.	PUNCT
ma-30	372	1	j.	j.	PROPN
ma-30	372	2	math	math	PROPN
ma-30	372	3	.	.	PUNCT
ma-30	373	1	anal	anal	ADJ
ma-30	373	2	.	.	PUNCT
ma-30	374	1	1	1	NUM
ma-30	374	2	(	(	PUNCT
ma-30	374	3	2021	2021	NUM
ma-30	374	4	)	)	PUNCT
ma-30	374	5	163	163	NUM
ma-30	375	1	[	[	X
ma-30	375	2	15	15	NUM
ma-30	375	3	]	]	X
ma-30	375	4	g.	g.	NOUN
ma-30	375	5	nordlander	nordlander	PROPN
ma-30	375	6	,	,	PUNCT
ma-30	375	7	the	the	DET
ma-30	375	8	modulus	modulus	NOUN
ma-30	375	9	of	of	ADP
ma-30	375	10	convexity	convexity	NOUN
ma-30	375	11	in	in	ADP
ma-30	375	12	normed	normed	ADJ
ma-30	375	13	linear	linear	PROPN
ma-30	375	14	spaces	space	NOUN
ma-30	375	15	,	,	PUNCT
ma-30	376	1	ark	ark	PROPN
ma-30	376	2	.	.	PROPN
ma-30	376	3	mat	mat	PROPN
ma-30	376	4	.	.	PROPN
ma-30	376	5	4	4	NUM
ma-30	376	6	(	(	PUNCT
ma-30	376	7	1960	1960	NUM
ma-30	376	8	)	)	PUNCT
ma-30	376	9	15–17	15–17	NUM
ma-30	376	10	.	.	PUNCT
ma-30	377	1	https://doi.org/	https://doi.org/	VERB
ma-30	377	2	10.1007	10.1007	NUM
ma-30	377	3	/	/	SYM
ma-30	377	4	bf02591317.[16	bf02591317.[16	NOUN
ma-30	377	5	]	]	PUNCT
ma-30	377	6	k.	k.	PROPN
ma-30	378	1	nikodem	nikodem	PROPN
ma-30	378	2	,	,	PUNCT
ma-30	378	3	z.	z.	PROPN
ma-30	378	4	pales	pale	NOUN
ma-30	378	5	,	,	PUNCT
ma-30	378	6	characterizations	characterization	NOUN
ma-30	378	7	of	of	ADP
ma-30	378	8	inner	inner	ADJ
ma-30	378	9	product	product	NOUN
ma-30	378	10	spaces	space	NOUN
ma-30	378	11	by	by	ADP
ma-30	378	12	strongly	strongly	ADV
ma-30	378	13	convex	convex	NOUN
ma-30	378	14	functions	function	NOUN
ma-30	378	15	,	,	PUNCT
ma-30	378	16	banach	banach	NOUN
ma-30	378	17	j.	j.	PROPN
ma-30	378	18	math	math	PROPN
ma-30	378	19	.	.	PUNCT
ma-30	379	1	anal.5	anal.5	ADV
ma-30	379	2	(	(	PUNCT
ma-30	379	3	2011	2011	NUM
ma-30	379	4	)	)	PUNCT
ma-30	379	5	83–87	83–87	NUM
ma-30	379	6	.	.	PUNCT
ma-30	380	1	https://doi.org/10.15352/bjma/1313362982	https://doi.org/10.15352/bjma/1313362982	PROPN
ma-30	380	2	.	.	PROPN
ma-30	380	3	https://doi.org/10.1007/bf02591317	https://doi.org/10.1007/bf02591317	PROPN
ma-30	381	1	https://doi.org/10.1007/bf02591317	https://doi.org/10.1007/bf02591317	PROPN
ma-30	381	2	https://doi.org/10.15352/bjma/1313362982	https://doi.org/10.15352/bjma/1313362982	PROPN
ma-30	381	3	1	1	PROPN
ma-30	381	4	.	.	PUNCT
ma-30	382	1	introduction	introduction	NOUN
ma-30	382	2	2	2	NUM
ma-30	382	3	.	.	PUNCT
ma-30	382	4	preliminaries	preliminary	NOUN
ma-30	382	5	3	3	NUM
ma-30	382	6	.	.	PUNCT
ma-30	383	1	the	the	DET
ma-30	383	2	-neumann	-neumann	PROPN
ma-30	383	3	–	–	PUNCT
ma-30	383	4	jordan	jordan	PROPN
ma-30	383	5	constant	constant	ADJ
ma-30	383	6	and	and	CCONJ
ma-30	383	7	the	the	DET
ma-30	383	8	-james	-jame	NOUN
ma-30	383	9	constant	constant	ADJ
ma-30	383	10	4	4	NUM
ma-30	383	11	.	.	PUNCT
ma-30	384	1	the	the	DET
ma-30	384	2	-convex	-convex	NOUN
ma-30	384	3	modular	modular	ADJ
ma-30	384	4	and	and	CCONJ
ma-30	384	5	the	the	DET
ma-30	384	6	-smooth	-smooth	NOUN
ma-30	384	7	modular	modular	ADJ
ma-30	384	8	5	5	NUM
ma-30	384	9	.	.	PUNCT
ma-30	384	10	convexity	convexity	NOUN
ma-30	384	11	and	and	CCONJ
ma-30	384	12	non	non	ADJ
ma-30	384	13	-	-	NOUN
ma-30	384	14	squareness	squareness	ADJ
ma-30	384	15	6	6	NUM
ma-30	384	16	.	.	PUNCT
ma-30	384	17	midpoint	midpoint	NOUN
ma-30	384	18	convexity	convexity	NOUN
ma-30	384	19	7	7	NUM
ma-30	384	20	.	.	PUNCT
ma-30	384	21	data	datum	NOUN
ma-30	384	22	availability	availability	NOUN
ma-30	384	23	8	8	NUM
ma-30	384	24	.	.	PUNCT
ma-30	385	1	conflicts	conflict	NOUN
ma-30	385	2	of	of	ADP
ma-30	385	3	interest	interest	NOUN
ma-30	385	4	9	9	NUM
ma-30	385	5	.	.	PUNCT
ma-30	386	1	funding	funding	NOUN
ma-30	386	2	statement	statement	NOUN
ma-30	386	3	references	reference	NOUN