id	sid	tid	token	lemma	pos
ejpam-3384	1	1	european	european	PROPN
ejpam-3384	1	2	journal	journal	PROPN
ejpam-3384	1	3	of	of	ADP
ejpam-3384	1	4	pure	pure	ADJ
ejpam-3384	1	5	and	and	CCONJ
ejpam-3384	1	6	applied	apply	VERB
ejpam-3384	1	7	mathematics	mathematic	NOUN
ejpam-3384	1	8	vol	vol	NOUN
ejpam-3384	1	9	.	.	PROPN
ejpam-3384	2	1	12	12	NUM
ejpam-3384	2	2	,	,	PUNCT
ejpam-3384	2	3	no	no	INTJ
ejpam-3384	2	4	.	.	NOUN
ejpam-3384	2	5	2	2	NUM
ejpam-3384	2	6	,	,	PUNCT
ejpam-3384	2	7	2019	2019	NUM
ejpam-3384	2	8	,	,	PUNCT
ejpam-3384	2	9	469	469	NUM
ejpam-3384	2	10	-	-	SYM
ejpam-3384	2	11	485	485	NUM
ejpam-3384	2	12	issn	issn	PROPN
ejpam-3384	2	13	1307	1307	NUM
ejpam-3384	2	14	-	-	SYM
ejpam-3384	2	15	5543	5543	NUM
ejpam-3384	2	16	–	–	PUNCT
ejpam-3384	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3384	2	18	published	publish	VERB
ejpam-3384	2	19	by	by	ADP
ejpam-3384	2	20	new	new	PROPN
ejpam-3384	2	21	york	york	PROPN
ejpam-3384	2	22	business	business	PROPN
ejpam-3384	2	23	global	global	PROPN
ejpam-3384	2	24	the	the	DET
ejpam-3384	2	25	quotient	quotient	NOUN
ejpam-3384	2	26	inequalities	inequality	NOUN
ejpam-3384	2	27	benedict	benedict	PROPN
ejpam-3384	2	28	barnes1,∗	barnes1,∗	PROPN
ejpam-3384	2	29	,	,	PUNCT
ejpam-3384	2	30	c.	c.	PROPN
ejpam-3384	2	31	sebil1	sebil1	PROPN
ejpam-3384	2	32	,	,	PUNCT
ejpam-3384	3	1	i.	i.	PROPN
ejpam-3384	3	2	k.	k.	PROPN
ejpam-3384	3	3	dontwi1	dontwi1	PROPN
ejpam-3384	3	4	1	1	NUM
ejpam-3384	3	5	department	department	NOUN
ejpam-3384	3	6	of	of	ADP
ejpam-3384	3	7	mathematics	mathematics	PROPN
ejpam-3384	3	8	,	,	PUNCT
ejpam-3384	3	9	kwame	kwame	PROPN
ejpam-3384	3	10	nkrumah	nkrumah	PROPN
ejpam-3384	3	11	university	university	PROPN
ejpam-3384	3	12	of	of	ADP
ejpam-3384	3	13	science	science	NOUN
ejpam-3384	3	14	and	and	CCONJ
ejpam-3384	3	15	technology	technology	NOUN
ejpam-3384	3	16	,	,	PUNCT
ejpam-3384	3	17	kumasi	kumasi	PROPN
ejpam-3384	3	18	,	,	PUNCT
ejpam-3384	3	19	ghana	ghana	PROPN
ejpam-3384	3	20	abstract	abstract	PROPN
ejpam-3384	3	21	.	.	PUNCT
ejpam-3384	4	1	this	this	DET
ejpam-3384	4	2	paper	paper	NOUN
ejpam-3384	4	3	contributes	contribute	VERB
ejpam-3384	4	4	to	to	ADP
ejpam-3384	4	5	the	the	DET
ejpam-3384	4	6	field	field	NOUN
ejpam-3384	4	7	of	of	ADP
ejpam-3384	4	8	inequalities	inequality	NOUN
ejpam-3384	4	9	,	,	PUNCT
ejpam-3384	4	10	specifically	specifically	ADV
ejpam-3384	4	11	,	,	PUNCT
ejpam-3384	4	12	the	the	DET
ejpam-3384	4	13	relationships	relationship	NOUN
ejpam-3384	4	14	among	among	ADP
ejpam-3384	4	15	the	the	DET
ejpam-3384	4	16	norms	norm	NOUN
ejpam-3384	4	17	of	of	ADP
ejpam-3384	4	18	products	product	NOUN
ejpam-3384	4	19	of	of	ADP
ejpam-3384	4	20	elements	element	NOUN
ejpam-3384	4	21	or	or	CCONJ
ejpam-3384	4	22	vectors	vector	NOUN
ejpam-3384	4	23	or	or	CCONJ
ejpam-3384	4	24	functions	function	NOUN
ejpam-3384	4	25	and	and	CCONJ
ejpam-3384	4	26	their	their	PRON
ejpam-3384	4	27	quotients	quotient	NOUN
ejpam-3384	4	28	.	.	PUNCT
ejpam-3384	5	1	thus	thus	ADV
ejpam-3384	5	2	,	,	PUNCT
ejpam-3384	5	3	we	we	PRON
ejpam-3384	5	4	established	establish	VERB
ejpam-3384	5	5	that	that	SCONJ
ejpam-3384	5	6	the	the	DET
ejpam-3384	5	7	norm	norm	NOUN
ejpam-3384	5	8	of	of	ADP
ejpam-3384	5	9	product	product	NOUN
ejpam-3384	5	10	of	of	ADP
ejpam-3384	5	11	two	two	NUM
ejpam-3384	5	12	vectors	vector	NOUN
ejpam-3384	5	13	or	or	CCONJ
ejpam-3384	5	14	functions	function	NOUN
ejpam-3384	5	15	is	be	AUX
ejpam-3384	5	16	less	less	ADJ
ejpam-3384	5	17	than	than	ADP
ejpam-3384	5	18	or	or	CCONJ
ejpam-3384	5	19	equal	equal	ADJ
ejpam-3384	5	20	to	to	ADP
ejpam-3384	5	21	the	the	DET
ejpam-3384	5	22	norm	norm	NOUN
ejpam-3384	5	23	of	of	ADP
ejpam-3384	5	24	its	its	PRON
ejpam-3384	5	25	quotient	quotient	NOUN
ejpam-3384	5	26	if	if	SCONJ
ejpam-3384	5	27	the	the	DET
ejpam-3384	5	28	norm	norm	NOUN
ejpam-3384	5	29	of	of	ADP
ejpam-3384	5	30	the	the	DET
ejpam-3384	5	31	denominator	denominator	NOUN
ejpam-3384	5	32	is	be	AUX
ejpam-3384	5	33	less	less	ADJ
ejpam-3384	5	34	than	than	ADP
ejpam-3384	5	35	or	or	CCONJ
ejpam-3384	5	36	equal	equal	ADJ
ejpam-3384	5	37	to	to	ADP
ejpam-3384	5	38	one	one	NUM
ejpam-3384	5	39	.	.	PUNCT
ejpam-3384	6	1	on	on	ADP
ejpam-3384	6	2	the	the	DET
ejpam-3384	6	3	other	other	ADJ
ejpam-3384	6	4	hand	hand	NOUN
ejpam-3384	6	5	,	,	PUNCT
ejpam-3384	6	6	we	we	PRON
ejpam-3384	6	7	prove	prove	VERB
ejpam-3384	6	8	that	that	SCONJ
ejpam-3384	6	9	the	the	DET
ejpam-3384	6	10	norm	norm	NOUN
ejpam-3384	6	11	of	of	ADP
ejpam-3384	6	12	the	the	DET
ejpam-3384	6	13	quotient	quotient	NOUN
ejpam-3384	6	14	of	of	ADP
ejpam-3384	6	15	two	two	NUM
ejpam-3384	6	16	vectors	vector	NOUN
ejpam-3384	6	17	or	or	CCONJ
ejpam-3384	6	18	functions	function	NOUN
ejpam-3384	6	19	is	be	AUX
ejpam-3384	6	20	less	less	ADJ
ejpam-3384	6	21	than	than	ADP
ejpam-3384	6	22	or	or	CCONJ
ejpam-3384	6	23	equal	equal	ADJ
ejpam-3384	6	24	to	to	ADP
ejpam-3384	6	25	the	the	DET
ejpam-3384	6	26	norm	norm	NOUN
ejpam-3384	6	27	of	of	ADP
ejpam-3384	6	28	their	their	PRON
ejpam-3384	6	29	product	product	NOUN
ejpam-3384	6	30	.	.	PUNCT
ejpam-3384	7	1	in	in	ADP
ejpam-3384	7	2	addition	addition	NOUN
ejpam-3384	7	3	,	,	PUNCT
ejpam-3384	7	4	we	we	PRON
ejpam-3384	7	5	introduce	introduce	VERB
ejpam-3384	7	6	the	the	DET
ejpam-3384	7	7	proofs	proof	NOUN
ejpam-3384	7	8	of	of	ADP
ejpam-3384	7	9	inequalities	inequality	NOUN
ejpam-3384	7	10	including	include	VERB
ejpam-3384	7	11	the	the	DET
ejpam-3384	7	12	norm	norm	NOUN
ejpam-3384	7	13	of	of	ADP
ejpam-3384	7	14	index	index	NOUN
ejpam-3384	7	15	power	power	NOUN
ejpam-3384	7	16	of	of	ADP
ejpam-3384	7	17	products	product	NOUN
ejpam-3384	7	18	and	and	CCONJ
ejpam-3384	7	19	their	their	PRON
ejpam-3384	7	20	quotients	quotient	NOUN
ejpam-3384	7	21	,	,	PUNCT
ejpam-3384	7	22	and	and	CCONJ
ejpam-3384	7	23	then	then	ADV
ejpam-3384	7	24	applied	apply	VERB
ejpam-3384	7	25	these	these	DET
ejpam-3384	7	26	inequalities	inequality	NOUN
ejpam-3384	7	27	to	to	PART
ejpam-3384	7	28	estabish	estabish	VERB
ejpam-3384	7	29	properties	property	NOUN
ejpam-3384	7	30	of	of	ADP
ejpam-3384	7	31	some	some	DET
ejpam-3384	7	32	functional	functional	ADJ
ejpam-3384	7	33	spaces	space	NOUN
ejpam-3384	7	34	,	,	PUNCT
ejpam-3384	7	35	as	as	ADV
ejpam-3384	7	36	well	well	ADV
ejpam-3384	7	37	as	as	ADP
ejpam-3384	7	38	,	,	PUNCT
ejpam-3384	7	39	extention	extention	NOUN
ejpam-3384	7	40	of	of	ADP
ejpam-3384	7	41	some	some	PRON
ejpam-3384	7	42	of	of	ADP
ejpam-3384	7	43	the	the	DET
ejpam-3384	7	44	results	result	NOUN
ejpam-3384	7	45	in	in	ADP
ejpam-3384	7	46	these	these	DET
ejpam-3384	7	47	functional	functional	ADJ
ejpam-3384	7	48	spaces	space	NOUN
ejpam-3384	7	49	.	.	PUNCT
ejpam-3384	8	1	2010	2010	NUM
ejpam-3384	8	2	mathematics	mathematic	NOUN
ejpam-3384	8	3	subject	subject	NOUN
ejpam-3384	8	4	classifications	classification	NOUN
ejpam-3384	8	5	:	:	PUNCT
ejpam-3384	8	6	44b56	44b56	NUM
ejpam-3384	8	7	,	,	PUNCT
ejpam-3384	8	8	44b57	44b57	NOUN
ejpam-3384	8	9	key	key	ADJ
ejpam-3384	8	10	words	word	NOUN
ejpam-3384	8	11	and	and	CCONJ
ejpam-3384	8	12	phrases	phrase	NOUN
ejpam-3384	8	13	:	:	PUNCT
ejpam-3384	8	14	quotient	quotient	NOUN
ejpam-3384	8	15	inequalities	inequality	NOUN
ejpam-3384	8	16	,	,	PUNCT
ejpam-3384	8	17	first	first	ADJ
ejpam-3384	8	18	quotient	quotient	NOUN
ejpam-3384	8	19	inequality	inequality	NOUN
ejpam-3384	8	20	,	,	PUNCT
ejpam-3384	8	21	second	second	ADJ
ejpam-3384	8	22	quotient	quotient	NOUN
ejpam-3384	8	23	inequality	inequality	NOUN
ejpam-3384	8	24	,	,	PUNCT
ejpam-3384	8	25	index	index	NOUN
ejpam-3384	8	26	power	power	NOUN
ejpam-3384	8	27	quotient	quotient	NOUN
ejpam-3384	8	28	inequalities	inequality	NOUN
ejpam-3384	8	29	1	1	NUM
ejpam-3384	8	30	.	.	PUNCT
ejpam-3384	9	1	introduction	introduction	NOUN
ejpam-3384	9	2	inequalities	inequality	NOUN
ejpam-3384	9	3	play	play	VERB
ejpam-3384	9	4	a	a	DET
ejpam-3384	9	5	central	central	ADJ
ejpam-3384	9	6	role	role	NOUN
ejpam-3384	9	7	in	in	ADP
ejpam-3384	9	8	mathematical	mathematical	ADJ
ejpam-3384	9	9	analysis	analysis	NOUN
ejpam-3384	9	10	with	with	ADP
ejpam-3384	9	11	numerous	numerous	ADJ
ejpam-3384	9	12	applications	application	NOUN
ejpam-3384	9	13	in	in	ADP
ejpam-3384	9	14	solving	solve	VERB
ejpam-3384	9	15	ill	ill	ADV
ejpam-3384	9	16	-	-	PUNCT
ejpam-3384	9	17	posed	pose	VERB
ejpam-3384	9	18	differential	differential	ADJ
ejpam-3384	9	19	equations	equation	NOUN
ejpam-3384	9	20	,	,	PUNCT
ejpam-3384	9	21	approximation	approximation	NOUN
ejpam-3384	9	22	theory	theory	NOUN
ejpam-3384	9	23	,	,	PUNCT
ejpam-3384	9	24	optimization	optimization	NOUN
ejpam-3384	9	25	theory	theory	NOUN
ejpam-3384	9	26	,	,	PUNCT
ejpam-3384	9	27	numerical	numerical	ADJ
ejpam-3384	9	28	analysis	analysis	NOUN
ejpam-3384	9	29	,	,	PUNCT
ejpam-3384	9	30	probability	probability	NOUN
ejpam-3384	9	31	theory	theory	NOUN
ejpam-3384	9	32	and	and	CCONJ
ejpam-3384	9	33	statistics	statistic	NOUN
ejpam-3384	9	34	.	.	PUNCT
ejpam-3384	10	1	the	the	DET
ejpam-3384	10	2	authors	author	NOUN
ejpam-3384	10	3	in	in	ADP
ejpam-3384	10	4	[	[	X
ejpam-3384	10	5	1	1	NUM
ejpam-3384	10	6	]	]	PUNCT
ejpam-3384	10	7	obtained	obtain	VERB
ejpam-3384	10	8	estimates	estimate	NOUN
ejpam-3384	10	9	relating	relate	VERB
ejpam-3384	10	10	martingale	martingale	NOUN
ejpam-3384	10	11	difference	difference	NOUN
ejpam-3384	10	12	sequences	sequence	NOUN
ejpam-3384	10	13	in	in	ADP
ejpam-3384	10	14	the	the	DET
ejpam-3384	10	15	complex	complex	ADJ
ejpam-3384	10	16	uniformly	uniformly	ADJ
ejpam-3384	10	17	convex	convex	NOUN
ejpam-3384	10	18	spaces	space	NOUN
ejpam-3384	10	19	.	.	PUNCT
ejpam-3384	11	1	in	in	ADP
ejpam-3384	11	2	[	[	X
ejpam-3384	11	3	2	2	NUM
ejpam-3384	11	4	]	]	PUNCT
ejpam-3384	11	5	,	,	PUNCT
ejpam-3384	11	6	they	they	PRON
ejpam-3384	11	7	obtained	obtain	VERB
ejpam-3384	11	8	some	some	DET
ejpam-3384	11	9	estimates	estimate	NOUN
ejpam-3384	11	10	for	for	ADP
ejpam-3384	11	11	geometric	geometric	ADJ
ejpam-3384	11	12	inequalities	inequality	NOUN
ejpam-3384	11	13	and	and	CCONJ
ejpam-3384	11	14	compared	compare	VERB
ejpam-3384	11	15	these	these	DET
ejpam-3384	11	16	inequalities	inequality	NOUN
ejpam-3384	11	17	.	.	PUNCT
ejpam-3384	12	1	the	the	DET
ejpam-3384	12	2	authors	author	NOUN
ejpam-3384	12	3	in	in	ADP
ejpam-3384	12	4	[	[	X
ejpam-3384	12	5	3	3	NUM
ejpam-3384	12	6	]	]	PUNCT
ejpam-3384	12	7	provided	provide	VERB
ejpam-3384	12	8	an	an	DET
ejpam-3384	12	9	alternative	alternative	ADJ
ejpam-3384	12	10	way	way	NOUN
ejpam-3384	12	11	of	of	ADP
ejpam-3384	12	12	proving	prove	VERB
ejpam-3384	12	13	the	the	DET
ejpam-3384	12	14	quasi	quasi	ADJ
ejpam-3384	12	15	-	-	ADJ
ejpam-3384	12	16	normed	normed	ADJ
ejpam-3384	12	17	linear	linear	ADJ
ejpam-3384	12	18	space	space	NOUN
ejpam-3384	12	19	through	through	ADP
ejpam-3384	12	20	binomial	binomial	ADJ
ejpam-3384	12	21	inequalities	inequality	NOUN
ejpam-3384	12	22	.	.	PUNCT
ejpam-3384	13	1	in	in	ADP
ejpam-3384	13	2	this	this	DET
ejpam-3384	13	3	paper	paper	NOUN
ejpam-3384	13	4	,	,	PUNCT
ejpam-3384	13	5	the	the	DET
ejpam-3384	13	6	new	new	ADJ
ejpam-3384	13	7	inequalities	inequality	NOUN
ejpam-3384	13	8	;	;	PUNCT
ejpam-3384	13	9	first	first	ADJ
ejpam-3384	13	10	and	and	CCONJ
ejpam-3384	13	11	second	second	ADJ
ejpam-3384	13	12	quotient	quotient	NOUN
ejpam-3384	13	13	inequalities	inequality	NOUN
ejpam-3384	13	14	,	,	PUNCT
ejpam-3384	13	15	for	for	ADP
ejpam-3384	13	16	the	the	DET
ejpam-3384	13	17	continuous	continuous	ADJ
ejpam-3384	13	18	mappings	mapping	NOUN
ejpam-3384	13	19	or	or	CCONJ
ejpam-3384	13	20	operators	operator	NOUN
ejpam-3384	13	21	on	on	ADP
ejpam-3384	13	22	a	a	DET
ejpam-3384	13	23	real	real	ADJ
ejpam-3384	13	24	hilbert	hilbert	NOUN
ejpam-3384	13	25	space	space	NOUN
ejpam-3384	13	26	,	,	PUNCT
ejpam-3384	13	27	complex	complex	ADJ
ejpam-3384	13	28	hilbert	hilbert	NOUN
ejpam-3384	13	29	space	space	NOUN
ejpam-3384	13	30	,	,	PUNCT
ejpam-3384	13	31	banach	banach	NOUN
ejpam-3384	13	32	space	space	NOUN
ejpam-3384	13	33	and	and	CCONJ
ejpam-3384	13	34	hölder	hölder	NOUN
ejpam-3384	13	35	’s	’s	PART
ejpam-3384	13	36	spaces	space	NOUN
ejpam-3384	13	37	and	and	CCONJ
ejpam-3384	13	38	sobolev	sobolev	NOUN
ejpam-3384	13	39	spaces	space	NOUN
ejpam-3384	13	40	are	be	AUX
ejpam-3384	13	41	provided	provide	VERB
ejpam-3384	13	42	.	.	PUNCT
ejpam-3384	14	1	these	these	DET
ejpam-3384	14	2	quotient	quotient	NOUN
ejpam-3384	14	3	inequalities	inequality	NOUN
ejpam-3384	14	4	are	be	AUX
ejpam-3384	14	5	used	use	VERB
ejpam-3384	14	6	in	in	ADP
ejpam-3384	14	7	obtaining	obtain	VERB
ejpam-3384	14	8	various	various	ADJ
ejpam-3384	14	9	new	new	ADJ
ejpam-3384	14	10	inequalities	inequality	NOUN
ejpam-3384	14	11	for	for	ADP
ejpam-3384	14	12	continuous	continuous	ADJ
ejpam-3384	14	13	functions	function	NOUN
ejpam-3384	14	14	of	of	ADP
ejpam-3384	14	15	both	both	PRON
ejpam-3384	14	16	selfadjoint	selfadjoint	NOUN
ejpam-3384	14	17	and	and	CCONJ
ejpam-3384	14	18	adjoint	adjoint	NOUN
ejpam-3384	14	19	operators	operator	NOUN
ejpam-3384	14	20	.	.	PUNCT
ejpam-3384	15	1	the	the	DET
ejpam-3384	15	2	section	section	NOUN
ejpam-3384	15	3	one	one	NUM
ejpam-3384	15	4	of	of	ADP
ejpam-3384	15	5	this	this	DET
ejpam-3384	15	6	paper	paper	NOUN
ejpam-3384	15	7	contains	contain	VERB
ejpam-3384	15	8	the	the	DET
ejpam-3384	15	9	general	general	ADJ
ejpam-3384	15	10	overview	overview	NOUN
ejpam-3384	15	11	of	of	ADP
ejpam-3384	15	12	the	the	DET
ejpam-3384	15	13	inequalities	inequality	NOUN
ejpam-3384	15	14	with	with	ADP
ejpam-3384	15	15	emphasizes	emphasize	VERB
ejpam-3384	15	16	on	on	ADP
ejpam-3384	15	17	the	the	DET
ejpam-3384	15	18	quotient	quotient	NOUN
ejpam-3384	15	19	inequalities	inequality	NOUN
ejpam-3384	15	20	.	.	PUNCT
ejpam-3384	16	1	in	in	ADP
ejpam-3384	16	2	section	section	NOUN
ejpam-3384	16	3	2	2	NUM
ejpam-3384	16	4	,	,	PUNCT
ejpam-3384	16	5	the	the	DET
ejpam-3384	16	6	preliminary	preliminary	ADJ
ejpam-3384	16	7	results	result	NOUN
ejpam-3384	16	8	including	include	VERB
ejpam-3384	16	9	∗corresponding	∗corresponde	VERB
ejpam-3384	16	10	author	author	NOUN
ejpam-3384	16	11	.	.	PUNCT
ejpam-3384	17	1	doi	doi	NOUN
ejpam-3384	17	2	:	:	PUNCT
ejpam-3384	17	3	https://doi.org/10.29020/nybg.ejpam.v12i2.3384	https://doi.org/10.29020/nybg.ejpam.v12i2.3384	ADJ
ejpam-3384	17	4	email	email	NOUN
ejpam-3384	17	5	addresses	address	NOUN
ejpam-3384	17	6	:	:	PUNCT
ejpam-3384	17	7	ewiekwamina@gmail.com	ewiekwamina@gmail.com	X
ejpam-3384	17	8	(	(	PUNCT
ejpam-3384	17	9	b.	b.	PROPN
ejpam-3384	17	10	barnes	barnes	PROPN
ejpam-3384	17	11	)	)	PUNCT
ejpam-3384	17	12	,	,	PUNCT
ejpam-3384	17	13	bbarnes.cos@knust.edu.gh	bbarnes.cos@knust.edu.gh	NOUN
ejpam-3384	17	14	(	(	PUNCT
ejpam-3384	17	15	b.	b.	PROPN
ejpam-3384	17	16	barnes	barnes	PROPN
ejpam-3384	17	17	)	)	PUNCT
ejpam-3384	17	18	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3384	18	1	469	469	NUM
ejpam-3384	18	2	c	c	X
ejpam-3384	18	3	©	©	PROPN
ejpam-3384	18	4	2019	2019	NUM
ejpam-3384	18	5	ejpam	ejpam	NOUN
ejpam-3384	18	6	all	all	DET
ejpam-3384	18	7	rights	right	NOUN
ejpam-3384	18	8	reserved	reserve	VERB
ejpam-3384	18	9	.	.	PUNCT
ejpam-3384	19	1	b.	b.	PROPN
ejpam-3384	19	2	barnes	barnes	PROPN
ejpam-3384	19	3	,	,	PUNCT
ejpam-3384	19	4	c.	c.	PROPN
ejpam-3384	19	5	sebil	sebil	PROPN
ejpam-3384	19	6	,	,	PUNCT
ejpam-3384	19	7	i.	i.	PROPN
ejpam-3384	19	8	k.	k.	PROPN
ejpam-3384	19	9	dontwi	dontwi	PROPN
ejpam-3384	19	10	/	/	SYM
ejpam-3384	19	11	eur	eur	PROPN
ejpam-3384	19	12	.	.	PUNCT
ejpam-3384	20	1	j.	j.	PROPN
ejpam-3384	20	2	pure	pure	PROPN
ejpam-3384	20	3	appl	appl	PROPN
ejpam-3384	20	4	.	.	PROPN
ejpam-3384	20	5	math	math	PROPN
ejpam-3384	20	6	,	,	PUNCT
ejpam-3384	20	7	12	12	NUM
ejpam-3384	20	8	(	(	PUNCT
ejpam-3384	20	9	2	2	NUM
ejpam-3384	20	10	)	)	PUNCT
ejpam-3384	20	11	(	(	PUNCT
ejpam-3384	20	12	2019	2019	NUM
ejpam-3384	20	13	)	)	PUNCT
ejpam-3384	20	14	,	,	PUNCT
ejpam-3384	20	15	469	469	NUM
ejpam-3384	20	16	-	-	SYM
ejpam-3384	20	17	485	485	NUM
ejpam-3384	20	18	470	470	NUM
ejpam-3384	20	19	definitions	definition	NOUN
ejpam-3384	20	20	and	and	CCONJ
ejpam-3384	20	21	theorem	theorem	NOUN
ejpam-3384	20	22	which	which	PRON
ejpam-3384	20	23	are	be	AUX
ejpam-3384	20	24	necessary	necessary	ADJ
ejpam-3384	20	25	in	in	ADP
ejpam-3384	20	26	establishing	establish	VERB
ejpam-3384	20	27	the	the	DET
ejpam-3384	20	28	quotient	quotient	NOUN
ejpam-3384	20	29	inequalities	inequality	NOUN
ejpam-3384	20	30	are	be	AUX
ejpam-3384	20	31	provided	provide	VERB
ejpam-3384	20	32	.	.	PUNCT
ejpam-3384	21	1	the	the	DET
ejpam-3384	21	2	quotient	quotient	NOUN
ejpam-3384	21	3	inequalities	inequality	NOUN
ejpam-3384	21	4	;	;	PUNCT
ejpam-3384	21	5	the	the	DET
ejpam-3384	21	6	first	first	ADJ
ejpam-3384	21	7	and	and	CCONJ
ejpam-3384	21	8	second	second	ADJ
ejpam-3384	21	9	quotient	quotient	NOUN
ejpam-3384	21	10	inequalities	inequality	NOUN
ejpam-3384	21	11	,	,	PUNCT
ejpam-3384	21	12	are	be	AUX
ejpam-3384	21	13	introduced	introduce	VERB
ejpam-3384	21	14	in	in	ADP
ejpam-3384	21	15	section	section	NOUN
ejpam-3384	21	16	3	3	NUM
ejpam-3384	21	17	of	of	ADP
ejpam-3384	21	18	this	this	DET
ejpam-3384	21	19	paper	paper	NOUN
ejpam-3384	21	20	.	.	PUNCT
ejpam-3384	22	1	these	these	DET
ejpam-3384	22	2	inequalities	inequality	NOUN
ejpam-3384	22	3	are	be	AUX
ejpam-3384	22	4	used	use	VERB
ejpam-3384	22	5	to	to	PART
ejpam-3384	22	6	establish	establish	VERB
ejpam-3384	22	7	the	the	DET
ejpam-3384	22	8	relationships	relationship	NOUN
ejpam-3384	22	9	among	among	ADP
ejpam-3384	22	10	the	the	DET
ejpam-3384	22	11	norms	norm	NOUN
ejpam-3384	22	12	of	of	ADP
ejpam-3384	22	13	product	product	NOUN
ejpam-3384	22	14	of	of	ADP
ejpam-3384	22	15	functions	function	NOUN
ejpam-3384	22	16	and	and	CCONJ
ejpam-3384	22	17	their	their	PRON
ejpam-3384	22	18	quotient	quotient	NOUN
ejpam-3384	22	19	counterparts	counterpart	NOUN
ejpam-3384	22	20	in	in	ADP
ejpam-3384	22	21	lp	lp	ADJ
ejpam-3384	22	22	spaces	space	NOUN
ejpam-3384	22	23	,	,	PUNCT
ejpam-3384	22	24	hilbert	hilbert	NOUN
ejpam-3384	22	25	space	space	NOUN
ejpam-3384	22	26	,	,	PUNCT
ejpam-3384	22	27	sobolev	sobolev	NOUN
ejpam-3384	22	28	spaces	space	NOUN
ejpam-3384	22	29	,	,	PUNCT
ejpam-3384	22	30	holder	holder	NOUN
ejpam-3384	22	31	’s	’s	PART
ejpam-3384	22	32	space	space	NOUN
ejpam-3384	22	33	,	,	PUNCT
ejpam-3384	22	34	banach	banach	NOUN
ejpam-3384	22	35	space	space	NOUN
ejpam-3384	22	36	and	and	CCONJ
ejpam-3384	22	37	unitary	unitary	ADJ
ejpam-3384	22	38	space	space	NOUN
ejpam-3384	22	39	.	.	PUNCT
ejpam-3384	23	1	2	2	X
ejpam-3384	23	2	.	.	X
ejpam-3384	23	3	some	some	DET
ejpam-3384	23	4	preliminary	preliminary	ADJ
ejpam-3384	23	5	results	result	NOUN
ejpam-3384	23	6	in	in	ADP
ejpam-3384	23	7	this	this	DET
ejpam-3384	23	8	section	section	NOUN
ejpam-3384	23	9	,	,	PUNCT
ejpam-3384	23	10	the	the	DET
ejpam-3384	23	11	definitions	definition	NOUN
ejpam-3384	23	12	regarding	regard	VERB
ejpam-3384	23	13	the	the	DET
ejpam-3384	23	14	quotient	quotient	NOUN
ejpam-3384	23	15	inequalities	inequality	NOUN
ejpam-3384	23	16	and	and	CCONJ
ejpam-3384	23	17	index	index	NOUN
ejpam-3384	23	18	power	power	NOUN
ejpam-3384	23	19	inequalities	inequality	NOUN
ejpam-3384	23	20	are	be	AUX
ejpam-3384	23	21	provided	provide	VERB
ejpam-3384	23	22	.	.	PUNCT
ejpam-3384	24	1	definition	definition	NOUN
ejpam-3384	24	2	1	1	NUM
ejpam-3384	24	3	(	(	PUNCT
ejpam-3384	24	4	first	first	ADJ
ejpam-3384	24	5	and	and	CCONJ
ejpam-3384	24	6	second	second	ADJ
ejpam-3384	24	7	product	product	NOUN
ejpam-3384	24	8	inequalities	inequality	NOUN
ejpam-3384	24	9	)	)	PUNCT
ejpam-3384	24	10	.	.	PUNCT
ejpam-3384	25	1	let	let	VERB
ejpam-3384	25	2	a1	a1	NOUN
ejpam-3384	25	3	and	and	CCONJ
ejpam-3384	25	4	a2	a2	PROPN
ejpam-3384	25	5	be	be	VERB
ejpam-3384	25	6	any	any	DET
ejpam-3384	25	7	two	two	NUM
ejpam-3384	25	8	positive	positive	ADJ
ejpam-3384	25	9	real	real	ADJ
ejpam-3384	25	10	numbers	number	NOUN
ejpam-3384	25	11	,	,	PUNCT
ejpam-3384	25	12	then	then	ADV
ejpam-3384	25	13	(	(	PUNCT
ejpam-3384	25	14	i)‖a1‖‖a2‖	i)‖a1‖‖a2‖	NOUN
ejpam-3384	25	15	≤	≤	ADJ
ejpam-3384	25	16	‖a1‖+	‖a1‖+	NUM
ejpam-3384	25	17	‖a2‖	‖a2‖	PROPN
ejpam-3384	25	18	,	,	PUNCT
ejpam-3384	25	19	∀a1	∀a1	PROPN
ejpam-3384	25	20	,	,	PUNCT
ejpam-3384	25	21	a2	a2	PROPN
ejpam-3384	25	22	∈	∈	PROPN
ejpam-3384	26	1	[	[	X
ejpam-3384	26	2	0	0	NUM
ejpam-3384	26	3	,	,	PUNCT
ejpam-3384	26	4	2	2	NUM
ejpam-3384	26	5	]	]	PUNCT
ejpam-3384	26	6	.	.	PUNCT
ejpam-3384	27	1	(	(	PUNCT
ejpam-3384	27	2	ii)‖a1‖+	ii)‖a1‖+	NOUN
ejpam-3384	27	3	‖a2‖	‖a2‖	PROPN
ejpam-3384	27	4	≤	≤	PROPN
ejpam-3384	27	5	‖a1‖‖a2‖	‖a1‖‖a2‖	NUM
ejpam-3384	27	6	,	,	PUNCT
ejpam-3384	27	7	∀a1	∀a1	PROPN
ejpam-3384	27	8	,	,	PUNCT
ejpam-3384	27	9	a2	a2	PROPN
ejpam-3384	27	10	∈	∈	PROPN
ejpam-3384	28	1	[	[	X
ejpam-3384	28	2	2,∞	2,∞	NUM
ejpam-3384	28	3	)	)	PUNCT
ejpam-3384	28	4	.	.	PUNCT
ejpam-3384	29	1	see	see	VERB
ejpam-3384	29	2	[	[	X
ejpam-3384	29	3	4	4	NUM
ejpam-3384	29	4	]	]	PUNCT
ejpam-3384	29	5	.	.	PUNCT
ejpam-3384	30	1	definition	definition	NOUN
ejpam-3384	30	2	2	2	NUM
ejpam-3384	30	3	(	(	PUNCT
ejpam-3384	30	4	product	product	NOUN
ejpam-3384	30	5	-	-	PUNCT
ejpam-3384	30	6	normed	norme	VERB
ejpam-3384	30	7	linear	linear	ADJ
ejpam-3384	30	8	space	space	NOUN
ejpam-3384	30	9	)	)	PUNCT
ejpam-3384	30	10	.	.	PUNCT
ejpam-3384	31	1	let	let	VERB
ejpam-3384	31	2	u	u	PRON
ejpam-3384	31	3	be	be	AUX
ejpam-3384	31	4	a	a	DET
ejpam-3384	31	5	linear	linear	ADJ
ejpam-3384	31	6	space	space	NOUN
ejpam-3384	31	7	over	over	ADP
ejpam-3384	31	8	[	[	X
ejpam-3384	31	9	0	0	NUM
ejpam-3384	31	10	,	,	PUNCT
ejpam-3384	31	11	2	2	NUM
ejpam-3384	31	12	]	]	PUNCT
ejpam-3384	31	13	⊆	⊆	NUM
ejpam-3384	31	14	r.	r.	NOUN
ejpam-3384	31	15	a	a	DET
ejpam-3384	31	16	vector	vector	NOUN
ejpam-3384	31	17	space	space	NOUN
ejpam-3384	31	18	with	with	ADP
ejpam-3384	31	19	a	a	DET
ejpam-3384	31	20	product	product	NOUN
ejpam-3384	31	21	-	-	PUNCT
ejpam-3384	31	22	norm	norm	NOUN
ejpam-3384	31	23	u→	u→	PROPN
ejpam-3384	31	24	‖u‖	‖u‖	PROPN
ejpam-3384	31	25	satisfying	satisfy	VERB
ejpam-3384	31	26	real	real	ADV
ejpam-3384	31	27	-	-	PUNCT
ejpam-3384	31	28	valued	value	VERB
ejpam-3384	31	29	function	function	NOUN
ejpam-3384	31	30	‖	‖	PROPN
ejpam-3384	31	31	·	·	PUNCT
ejpam-3384	31	32	‖	‖	PROPN
ejpam-3384	31	33	,	,	PUNCT
ejpam-3384	31	34	‖	‖	ADJ
ejpam-3384	31	35	·	·	PUNCT
ejpam-3384	31	36	‖pn	‖pn	NUM
ejpam-3384	31	37	:	:	PUNCT
ejpam-3384	31	38	u	u	NOUN
ejpam-3384	31	39	→	→	PUNCT
ejpam-3384	31	40	[	[	X
ejpam-3384	31	41	0,∞	0,∞	NOUN
ejpam-3384	31	42	)	)	PUNCT
ejpam-3384	31	43	,	,	PUNCT
ejpam-3384	31	44	such	such	ADJ
ejpam-3384	31	45	that	that	SCONJ
ejpam-3384	31	46	for	for	ADP
ejpam-3384	31	47	arbitrary	arbitrary	ADJ
ejpam-3384	31	48	u	u	NOUN
ejpam-3384	31	49	,	,	PUNCT
ejpam-3384	31	50	v	v	PROPN
ejpam-3384	31	51	∈	∈	PROPN
ejpam-3384	31	52	u	u	NOUN
ejpam-3384	31	53	,	,	PUNCT
ejpam-3384	31	54	α	α	PROPN
ejpam-3384	31	55	∈	∈	PROPN
ejpam-3384	32	1	[	[	X
ejpam-3384	32	2	0	0	NUM
ejpam-3384	32	3	,	,	PUNCT
ejpam-3384	32	4	2	2	NUM
ejpam-3384	32	5	]	]	PUNCT
ejpam-3384	32	6	,	,	PUNCT
ejpam-3384	32	7	the	the	DET
ejpam-3384	32	8	following	follow	VERB
ejpam-3384	32	9	conditions	condition	NOUN
ejpam-3384	32	10	are	be	AUX
ejpam-3384	32	11	satisfied	satisfied	ADJ
ejpam-3384	32	12	:	:	PUNCT
ejpam-3384	32	13	p1	p1	NOUN
ejpam-3384	32	14	.	.	PUNCT
ejpam-3384	33	1	‖u‖	‖u‖	PROPN
ejpam-3384	33	2	≥	≥	NOUN
ejpam-3384	33	3	0	0	NUM
ejpam-3384	33	4	,	,	PUNCT
ejpam-3384	33	5	and	and	CCONJ
ejpam-3384	33	6	‖u‖	‖u‖	PROPN
ejpam-3384	33	7	=	=	PUNCT
ejpam-3384	34	1	0	0	NUM
ejpam-3384	34	2	,	,	PUNCT
ejpam-3384	34	3	if	if	SCONJ
ejpam-3384	34	4	and	and	CCONJ
ejpam-3384	34	5	only	only	ADV
ejpam-3384	34	6	if	if	SCONJ
ejpam-3384	34	7	u	u	NOUN
ejpam-3384	34	8	=	=	NOUN
ejpam-3384	34	9	0	0	NUM
ejpam-3384	34	10	p2	p2	NOUN
ejpam-3384	34	11	.	.	PUNCT
ejpam-3384	35	1	‖αu‖	‖αu‖	ADJ
ejpam-3384	35	2	=	=	SYM
ejpam-3384	35	3	|α|‖u‖	|α|‖u‖	NOUN
ejpam-3384	35	4	,	,	PUNCT
ejpam-3384	35	5	α	α	NOUN
ejpam-3384	35	6	∈	∈	PROPN
ejpam-3384	36	1	[	[	X
ejpam-3384	36	2	0	0	NUM
ejpam-3384	36	3	,	,	PUNCT
ejpam-3384	36	4	2	2	NUM
ejpam-3384	36	5	]	]	PUNCT
ejpam-3384	36	6	,	,	PUNCT
ejpam-3384	36	7	and	and	CCONJ
ejpam-3384	36	8	u	u	PROPN
ejpam-3384	36	9	∈	∈	PROPN
ejpam-3384	36	10	u	u	PROPN
ejpam-3384	36	11	p3	p3	PROPN
ejpam-3384	36	12	.	.	PUNCT
ejpam-3384	37	1	‖u+	‖u+	NOUN
ejpam-3384	37	2	v‖pn	v‖pn	VERB
ejpam-3384	37	3	≤	≤	NUM
ejpam-3384	37	4	‖u‖pn	‖u‖pn	PUNCT
ejpam-3384	38	1	+	+	CCONJ
ejpam-3384	38	2	‖v‖pn	‖v‖pn	PROPN
ejpam-3384	38	3	,	,	PUNCT
ejpam-3384	38	4	∀u	∀u	NOUN
ejpam-3384	38	5	,	,	PUNCT
ejpam-3384	38	6	v	v	X
ejpam-3384	38	7	∈	∈	NOUN
ejpam-3384	38	8	u	u	NOUN
ejpam-3384	38	9	we	we	PRON
ejpam-3384	38	10	call	call	VERB
ejpam-3384	38	11	‖	‖	PROPN
ejpam-3384	38	12	·	·	PUNCT
ejpam-3384	38	13	‖pn	‖pn	NUM
ejpam-3384	38	14	a	a	DET
ejpam-3384	38	15	product	product	NOUN
ejpam-3384	38	16	norm	norm	NOUN
ejpam-3384	38	17	,	,	PUNCT
ejpam-3384	38	18	if	if	SCONJ
ejpam-3384	38	19	in	in	ADP
ejpam-3384	38	20	addition	addition	NOUN
ejpam-3384	38	21	to	to	ADP
ejpam-3384	38	22	p1−	p1−	PROPN
ejpam-3384	38	23	p3	p3	PROPN
ejpam-3384	38	24	,	,	PUNCT
ejpam-3384	38	25	the	the	DET
ejpam-3384	38	26	u	u	NOUN
ejpam-3384	38	27	and	and	CCONJ
ejpam-3384	38	28	v	v	ADP
ejpam-3384	38	29	satisfy	satisfy	NOUN
ejpam-3384	38	30	p4(a	p4(a	NOUN
ejpam-3384	38	31	)	)	PUNCT
ejpam-3384	38	32	.	.	PUNCT
ejpam-3384	39	1	‖u‖pn‖y‖pn	‖u‖pn‖y‖pn	NOUN
ejpam-3384	39	2	≤	≤	NUM
ejpam-3384	39	3	‖u‖pn	‖u‖pn	VERB
ejpam-3384	40	1	+	+	NUM
ejpam-3384	40	2	‖v‖pn	‖v‖pn	NUM
ejpam-3384	40	3	∀u	∀u	NOUN
ejpam-3384	40	4	,	,	PUNCT
ejpam-3384	40	5	v	v	NOUN
ejpam-3384	40	6	∈	∈	PROPN
ejpam-3384	41	1	[	[	X
ejpam-3384	41	2	0	0	NUM
ejpam-3384	41	3	,	,	PUNCT
ejpam-3384	41	4	2	2	NUM
ejpam-3384	41	5	]	]	PUNCT
ejpam-3384	41	6	(	(	PUNCT
ejpam-3384	41	7	first	first	ADJ
ejpam-3384	41	8	product	product	NOUN
ejpam-3384	41	9	inequality	inequality	NOUN
ejpam-3384	41	10	)	)	PUNCT
ejpam-3384	41	11	p4(b).‖u‖+	p4(b).‖u‖+	PUNCT
ejpam-3384	42	1	‖v‖	‖v‖	PROPN
ejpam-3384	42	2	≤	≤	NUM
ejpam-3384	42	3	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3384	42	4	,	,	PUNCT
ejpam-3384	42	5	∀u	∀u	NOUN
ejpam-3384	42	6	,	,	PUNCT
ejpam-3384	42	7	v	v	NOUN
ejpam-3384	42	8	∈	∈	PROPN
ejpam-3384	43	1	[	[	X
ejpam-3384	43	2	2,∞	2,∞	NUM
ejpam-3384	43	3	)	)	PUNCT
ejpam-3384	43	4	(	(	PUNCT
ejpam-3384	43	5	second	second	ADJ
ejpam-3384	43	6	product	product	NOUN
ejpam-3384	43	7	inequality	inequality	NOUN
ejpam-3384	43	8	)	)	PUNCT
ejpam-3384	43	9	.	.	PUNCT
ejpam-3384	44	1	see	see	VERB
ejpam-3384	44	2	[	[	X
ejpam-3384	44	3	5	5	NUM
ejpam-3384	44	4	]	]	PUNCT
ejpam-3384	44	5	.	.	PUNCT
ejpam-3384	45	1	definition	definition	NOUN
ejpam-3384	45	2	3	3	NUM
ejpam-3384	45	3	(	(	PUNCT
ejpam-3384	45	4	young	young	PROPN
ejpam-3384	45	5	’s	’s	PART
ejpam-3384	45	6	inequality	inequality	NOUN
ejpam-3384	45	7	)	)	PUNCT
ejpam-3384	45	8	.	.	PUNCT
ejpam-3384	46	1	for	for	ADP
ejpam-3384	46	2	1	1	NUM
ejpam-3384	46	3	<	<	X
ejpam-3384	46	4	p	p	X
ejpam-3384	46	5	<	<	X
ejpam-3384	46	6	∞	∞	PROPN
ejpam-3384	46	7	,	,	PUNCT
ejpam-3384	46	8	q	q	X
ejpam-3384	46	9	the	the	DET
ejpam-3384	46	10	conjugate	conjugate	NOUN
ejpam-3384	46	11	of	of	ADP
ejpam-3384	46	12	p	p	NOUN
ejpam-3384	46	13	,	,	PUNCT
ejpam-3384	46	14	and	and	CCONJ
ejpam-3384	46	15	any	any	DET
ejpam-3384	46	16	two	two	NUM
ejpam-3384	46	17	positive	positive	ADJ
ejpam-3384	46	18	numbers	number	NOUN
ejpam-3384	46	19	a	a	PRON
ejpam-3384	46	20	and	and	CCONJ
ejpam-3384	46	21	b	b	NOUN
ejpam-3384	46	22	,	,	PUNCT
ejpam-3384	46	23	then	then	ADV
ejpam-3384	46	24	ab	ab	PROPN
ejpam-3384	46	25	≤	≤	NUM
ejpam-3384	46	26	1	1	NUM
ejpam-3384	46	27	p	p	NOUN
ejpam-3384	46	28	ap	ap	PROPN
ejpam-3384	47	1	+	+	CCONJ
ejpam-3384	47	2	1	1	NUM
ejpam-3384	47	3	q	q	NOUN
ejpam-3384	47	4	aq	aq	NOUN
ejpam-3384	47	5	,	,	PUNCT
ejpam-3384	47	6	p	p	X
ejpam-3384	47	7	,	,	PUNCT
ejpam-3384	47	8	q	q	PROPN
ejpam-3384	47	9	≤	≤	NUM
ejpam-3384	47	10	1	1	NUM
ejpam-3384	47	11	.	.	PUNCT
ejpam-3384	48	1	see	see	VERB
ejpam-3384	48	2	[	[	X
ejpam-3384	48	3	6	6	NUM
ejpam-3384	48	4	]	]	PUNCT
ejpam-3384	48	5	.	.	PUNCT
ejpam-3384	49	1	b.	b.	PROPN
ejpam-3384	49	2	barnes	barnes	PROPN
ejpam-3384	49	3	,	,	PUNCT
ejpam-3384	49	4	c.	c.	PROPN
ejpam-3384	49	5	sebil	sebil	PROPN
ejpam-3384	49	6	,	,	PUNCT
ejpam-3384	49	7	i.	i.	PROPN
ejpam-3384	49	8	k.	k.	PROPN
ejpam-3384	49	9	dontwi	dontwi	PROPN
ejpam-3384	49	10	/	/	SYM
ejpam-3384	49	11	eur	eur	PROPN
ejpam-3384	49	12	.	.	PUNCT
ejpam-3384	50	1	j.	j.	PROPN
ejpam-3384	50	2	pure	pure	PROPN
ejpam-3384	50	3	appl	appl	PROPN
ejpam-3384	50	4	.	.	PROPN
ejpam-3384	50	5	math	math	PROPN
ejpam-3384	50	6	,	,	PUNCT
ejpam-3384	50	7	12	12	NUM
ejpam-3384	50	8	(	(	PUNCT
ejpam-3384	50	9	2	2	NUM
ejpam-3384	50	10	)	)	PUNCT
ejpam-3384	50	11	(	(	PUNCT
ejpam-3384	50	12	2019	2019	NUM
ejpam-3384	50	13	)	)	PUNCT
ejpam-3384	50	14	,	,	PUNCT
ejpam-3384	50	15	469	469	NUM
ejpam-3384	50	16	-	-	SYM
ejpam-3384	50	17	485	485	NUM
ejpam-3384	50	18	471	471	NUM
ejpam-3384	50	19	definition	definition	NOUN
ejpam-3384	50	20	4	4	NUM
ejpam-3384	50	21	(	(	PUNCT
ejpam-3384	50	22	contractive	contractive	ADJ
ejpam-3384	50	23	mapping	mapping	NOUN
ejpam-3384	50	24	)	)	PUNCT
ejpam-3384	50	25	.	.	PUNCT
ejpam-3384	51	1	let	let	VERB
ejpam-3384	51	2	t	t	NOUN
ejpam-3384	51	3	:	:	PUNCT
ejpam-3384	51	4	x	x	X
ejpam-3384	51	5	→	→	PUNCT
ejpam-3384	51	6	x	x	PUNCT
ejpam-3384	51	7	be	be	AUX
ejpam-3384	51	8	a	a	DET
ejpam-3384	51	9	mapping	mapping	NOUN
ejpam-3384	51	10	from	from	ADP
ejpam-3384	51	11	a	a	DET
ejpam-3384	51	12	complete	complete	ADJ
ejpam-3384	51	13	normed	norme	VERB
ejpam-3384	51	14	linear	linear	ADJ
ejpam-3384	51	15	space	space	NOUN
ejpam-3384	51	16	x	x	PUNCT
ejpam-3384	51	17	into	into	ADP
ejpam-3384	51	18	itself	itself	PRON
ejpam-3384	51	19	.	.	PUNCT
ejpam-3384	52	1	the	the	DET
ejpam-3384	52	2	lipschitz	lipschitz	ADJ
ejpam-3384	52	3	continuity	continuity	NOUN
ejpam-3384	52	4	on	on	ADP
ejpam-3384	52	5	t	t	PROPN
ejpam-3384	52	6	is	be	AUX
ejpam-3384	52	7	said	say	VERB
ejpam-3384	52	8	to	to	PART
ejpam-3384	52	9	be	be	AUX
ejpam-3384	52	10	a	a	DET
ejpam-3384	52	11	contraction	contraction	NOUN
ejpam-3384	52	12	if	if	SCONJ
ejpam-3384	52	13	‖t	‖t	PROPN
ejpam-3384	52	14	(	(	PUNCT
ejpam-3384	52	15	u)−	u)−	PROPN
ejpam-3384	52	16	t	t	PROPN
ejpam-3384	52	17	(	(	PUNCT
ejpam-3384	52	18	v)‖	v)‖	NOUN
ejpam-3384	52	19	≤	≤	NOUN
ejpam-3384	52	20	λ‖u−	λ‖u−	ADJ
ejpam-3384	52	21	v‖	v‖	NOUN
ejpam-3384	52	22	,	,	PUNCT
ejpam-3384	52	23	∀	∀	X
ejpam-3384	52	24	u	u	NOUN
ejpam-3384	52	25	,	,	PUNCT
ejpam-3384	52	26	v	v	NOUN
ejpam-3384	52	27	∈	∈	NOUN
ejpam-3384	52	28	x	x	NOUN
ejpam-3384	52	29	,	,	PUNCT
ejpam-3384	52	30	and	and	CCONJ
ejpam-3384	52	31	0	0	NUM
ejpam-3384	52	32	<	<	X
ejpam-3384	52	33	λ	λ	X
ejpam-3384	52	34	<	<	X
ejpam-3384	52	35	1	1	NUM
ejpam-3384	52	36	.	.	PUNCT
ejpam-3384	52	37	see	see	VERB
ejpam-3384	53	1	[	[	X
ejpam-3384	53	2	7	7	NUM
ejpam-3384	53	3	]	]	PUNCT
ejpam-3384	53	4	.	.	PUNCT
ejpam-3384	54	1	definition	definition	NOUN
ejpam-3384	54	2	5	5	NUM
ejpam-3384	54	3	.	.	PUNCT
ejpam-3384	55	1	let	let	VERB
ejpam-3384	55	2	γ	γ	X
ejpam-3384	55	3	∈	∈	PROPN
ejpam-3384	55	4	(	(	PUNCT
ejpam-3384	55	5	0	0	NUM
ejpam-3384	55	6	,	,	PUNCT
ejpam-3384	55	7	1	1	NUM
ejpam-3384	55	8	]	]	PUNCT
ejpam-3384	55	9	.	.	PUNCT
ejpam-3384	56	1	we	we	PRON
ejpam-3384	56	2	say	say	VERB
ejpam-3384	56	3	a	a	DET
ejpam-3384	56	4	function	function	NOUN
ejpam-3384	56	5	t	t	NOUN
ejpam-3384	56	6	:	:	PUNCT
ejpam-3384	56	7	x	x	X
ejpam-3384	56	8	→	→	SYM
ejpam-3384	56	9	y	y	PROPN
ejpam-3384	56	10	is	be	AUX
ejpam-3384	56	11	hölder	hölder	NOUN
ejpam-3384	56	12	continuous	continuous	ADJ
ejpam-3384	56	13	of	of	ADP
ejpam-3384	56	14	exponent	exponent	PROPN
ejpam-3384	56	15	γ	γ	PROPN
ejpam-3384	56	16	at	at	ADP
ejpam-3384	56	17	xo	xo	PROPN
ejpam-3384	56	18	∈	∈	PROPN
ejpam-3384	56	19	x	x	PRON
ejpam-3384	56	20	,	,	PUNCT
ejpam-3384	56	21	if	if	SCONJ
ejpam-3384	56	22	‖u(x)−	‖u(x)−	PROPN
ejpam-3384	56	23	u(xo)‖	u(xo)‖	VERB
ejpam-3384	56	24	≤	≤	ADJ
ejpam-3384	56	25	l‖x−	l‖x−	NOUN
ejpam-3384	56	26	xo‖γ	xo‖γ	PROPN
ejpam-3384	56	27	,	,	PUNCT
ejpam-3384	56	28	where	where	SCONJ
ejpam-3384	56	29	l	l	NOUN
ejpam-3384	56	30	is	be	AUX
ejpam-3384	56	31	boundedness	boundedness	ADJ
ejpam-3384	56	32	constant	constant	ADJ
ejpam-3384	56	33	may	may	AUX
ejpam-3384	56	34	depend	depend	VERB
ejpam-3384	56	35	on	on	ADP
ejpam-3384	56	36	x	x	PROPN
ejpam-3384	56	37	,	,	PUNCT
ejpam-3384	56	38	xo	xo	PROPN
ejpam-3384	56	39	,	,	PUNCT
ejpam-3384	56	40	γ	γ	PROPN
ejpam-3384	56	41	and	and	CCONJ
ejpam-3384	56	42	t	t	PROPN
ejpam-3384	56	43	.	.	PUNCT
ejpam-3384	57	1	see	see	VERB
ejpam-3384	57	2	[	[	X
ejpam-3384	57	3	8	8	NUM
ejpam-3384	57	4	]	]	PUNCT
ejpam-3384	57	5	.	.	PUNCT
ejpam-3384	58	1	theorem	theorem	ADJ
ejpam-3384	58	2	1	1	NUM
ejpam-3384	58	3	(	(	PUNCT
ejpam-3384	58	4	gagliardo	gagliardo	NOUN
ejpam-3384	58	5	-	-	PUNCT
ejpam-3384	58	6	nirenberg	nirenberg	PROPN
ejpam-3384	58	7	-	-	PUNCT
ejpam-3384	58	8	sobolev	sobolev	NOUN
ejpam-3384	58	9	inequality	inequality	NOUN
ejpam-3384	58	10	)	)	PUNCT
ejpam-3384	58	11	.	.	PUNCT
ejpam-3384	59	1	let	let	VERB
ejpam-3384	59	2	1	1	NUM
ejpam-3384	59	3	≤	≤	NOUN
ejpam-3384	59	4	p	p	PRON
ejpam-3384	59	5	<	<	X
ejpam-3384	59	6	n.	n.	NOUN
ejpam-3384	59	7	then	then	ADV
ejpam-3384	59	8	there	there	PRON
ejpam-3384	59	9	exists	exist	VERB
ejpam-3384	59	10	a	a	DET
ejpam-3384	59	11	constant	constant	ADJ
ejpam-3384	59	12	c	c	NOUN
ejpam-3384	59	13	>	>	X
ejpam-3384	59	14	0	0	PUNCT
ejpam-3384	60	1	(	(	PUNCT
ejpam-3384	60	2	depending	depend	VERB
ejpam-3384	60	3	on	on	ADP
ejpam-3384	60	4	p	p	NOUN
ejpam-3384	60	5	and	and	CCONJ
ejpam-3384	60	6	n	n	CCONJ
ejpam-3384	60	7	)	)	PUNCT
ejpam-3384	60	8	such	such	ADJ
ejpam-3384	60	9	that	that	SCONJ
ejpam-3384	60	10	‖u‖p∗,rn	‖u‖p∗,rn	DET
ejpam-3384	60	11	≤	≤	PROPN
ejpam-3384	60	12	c‖∇u‖p	c‖∇u‖p	NOUN
ejpam-3384	60	13	,	,	PUNCT
ejpam-3384	60	14	rn	rn	PROPN
ejpam-3384	60	15	,	,	PUNCT
ejpam-3384	60	16	u	u	PROPN
ejpam-3384	60	17	∈w	∈w	NOUN
ejpam-3384	60	18	1,p(rn	1,p(rn	NUM
ejpam-3384	60	19	)	)	PUNCT
ejpam-3384	60	20	.	.	PUNCT
ejpam-3384	61	1	in	in	ADP
ejpam-3384	61	2	particular	particular	ADJ
ejpam-3384	61	3	,	,	PUNCT
ejpam-3384	61	4	we	we	PRON
ejpam-3384	61	5	have	have	VERB
ejpam-3384	61	6	the	the	DET
ejpam-3384	61	7	continuous	continuous	ADJ
ejpam-3384	61	8	imbedding	imbedding	NOUN
ejpam-3384	61	9	w	w	PROPN
ejpam-3384	61	10	1,p(rn	1,p(rn	NUM
ejpam-3384	61	11	)	)	PUNCT
ejpam-3384	62	1	↪	↪	PROPN
ejpam-3384	62	2	→	→	SYM
ejpam-3384	62	3	lp∗(rn	lp∗(rn	NOUN
ejpam-3384	62	4	)	)	PUNCT
ejpam-3384	62	5	.	.	PUNCT
ejpam-3384	63	1	for	for	ADP
ejpam-3384	63	2	example	example	NOUN
ejpam-3384	63	3	,	,	PUNCT
ejpam-3384	63	4	see	see	VERB
ejpam-3384	63	5	authors	author	NOUN
ejpam-3384	63	6	in	in	ADP
ejpam-3384	63	7	[	[	X
ejpam-3384	63	8	9	9	NUM
ejpam-3384	63	9	]	]	PUNCT
ejpam-3384	63	10	.	.	PUNCT
ejpam-3384	64	1	definition	definition	NOUN
ejpam-3384	64	2	6	6	NUM
ejpam-3384	64	3	.	.	PUNCT
ejpam-3384	65	1	let	let	VERB
ejpam-3384	65	2	a	a	PRON
ejpam-3384	65	3	and	and	CCONJ
ejpam-3384	65	4	b	b	NOUN
ejpam-3384	65	5	be	be	AUX
ejpam-3384	65	6	selfadjoint	selfadjoint	NOUN
ejpam-3384	65	7	operators	operator	NOUN
ejpam-3384	65	8	with	with	ADP
ejpam-3384	65	9	sp(a	sp(a	PROPN
ejpam-3384	65	10	)	)	PUNCT
ejpam-3384	65	11	,	,	PUNCT
ejpam-3384	65	12	sp(b	sp(b	PROPN
ejpam-3384	65	13	)	)	PUNCT
ejpam-3384	66	1	⊆	⊆	NUM
ejpam-3384	67	1	[	[	X
ejpam-3384	67	2	m	m	NOUN
ejpam-3384	67	3	,	,	PUNCT
ejpam-3384	67	4	m	m	VERB
ejpam-3384	67	5	]	]	PUNCT
ejpam-3384	67	6	for	for	ADP
ejpam-3384	67	7	some	some	DET
ejpam-3384	67	8	real	real	ADJ
ejpam-3384	67	9	numbers	number	NOUN
ejpam-3384	67	10	m	m	VERB
ejpam-3384	67	11	<	<	X
ejpam-3384	67	12	m	m	VERB
ejpam-3384	67	13	.	.	PUNCT
ejpam-3384	68	1	if	if	SCONJ
ejpam-3384	68	2	f	f	X
ejpam-3384	68	3	:	:	PUNCT
ejpam-3384	69	1	[	[	X
ejpam-3384	69	2	m	m	X
ejpam-3384	69	3	,	,	PUNCT
ejpam-3384	69	4	m	m	VERB
ejpam-3384	69	5	]	]	PUNCT
ejpam-3384	69	6	→	→	PUNCT
ejpam-3384	69	7	r	r	NOUN
ejpam-3384	69	8	is	be	AUX
ejpam-3384	69	9	of	of	ADP
ejpam-3384	69	10	r	r	NOUN
ejpam-3384	69	11	−	−	NOUN
ejpam-3384	69	12	l−hölder	l−hölder	NOUN
ejpam-3384	69	13	type	type	NOUN
ejpam-3384	69	14	.	.	PUNCT
ejpam-3384	70	1	thus	thus	ADV
ejpam-3384	70	2	,	,	PUNCT
ejpam-3384	70	3	for	for	ADP
ejpam-3384	70	4	a	a	DET
ejpam-3384	70	5	given	give	VERB
ejpam-3384	70	6	r	r	NOUN
ejpam-3384	70	7	∈	∈	PROPN
ejpam-3384	70	8	(	(	PUNCT
ejpam-3384	70	9	0	0	NUM
ejpam-3384	70	10	,	,	PUNCT
ejpam-3384	70	11	1	1	NUM
ejpam-3384	70	12	]	]	PUNCT
ejpam-3384	70	13	and	and	CCONJ
ejpam-3384	70	14	l	l	NOUN
ejpam-3384	70	15	>	>	X
ejpam-3384	70	16	0	0	NUM
ejpam-3384	70	17	,	,	PUNCT
ejpam-3384	70	18	we	we	PRON
ejpam-3384	70	19	have∣∣∣f(s)−	have∣∣∣f(s)−	PROPN
ejpam-3384	70	20	f(t	f(t	PROPN
ejpam-3384	70	21	)	)	PUNCT
ejpam-3384	70	22	∣∣∣	∣∣∣	NOUN
ejpam-3384	70	23	≤	≤	PROPN
ejpam-3384	71	1	l∣∣∣s−	l∣∣∣s−	PROPN
ejpam-3384	71	2	t∣∣∣r	t∣∣∣r	NOUN
ejpam-3384	71	3	,	,	PUNCT
ejpam-3384	71	4	∀	∀	X
ejpam-3384	71	5	s	s	X
ejpam-3384	71	6	,	,	PUNCT
ejpam-3384	71	7	t	t	PROPN
ejpam-3384	71	8	∈	∈	PROPN
ejpam-3384	72	1	[	[	X
ejpam-3384	72	2	m	m	X
ejpam-3384	72	3	,	,	PUNCT
ejpam-3384	72	4	m	m	VERB
ejpam-3384	72	5	]	]	PUNCT
ejpam-3384	72	6	,	,	PUNCT
ejpam-3384	72	7	then	then	ADV
ejpam-3384	72	8	we	we	PRON
ejpam-3384	72	9	have	have	VERB
ejpam-3384	72	10	the	the	DET
ejpam-3384	72	11	ostrowski	ostrowski	ADJ
ejpam-3384	72	12	type	type	NOUN
ejpam-3384	72	13	inequality	inequality	NOUN
ejpam-3384	72	14	for	for	ADP
ejpam-3384	72	15	selfadjoint	selfadjoint	NOUN
ejpam-3384	72	16	operators:∣∣∣f(s)−	operators:∣∣∣f(s)−	ADP
ejpam-3384	72	17	〈	〈	PROPN
ejpam-3384	72	18	f(a)x	f(a)x	PROPN
ejpam-3384	72	19	,	,	PUNCT
ejpam-3384	72	20	x	x	NOUN
ejpam-3384	72	21	〉	〉	NOUN
ejpam-3384	72	22	∣∣∣	∣∣∣	NOUN
ejpam-3384	72	23	≤	≤	PUNCT
ejpam-3384	73	1	[	[	X
ejpam-3384	73	2	1	1	NUM
ejpam-3384	73	3	2	2	NUM
ejpam-3384	73	4	(	(	PUNCT
ejpam-3384	73	5	m	m	NOUN
ejpam-3384	73	6	−m	−m	NOUN
ejpam-3384	73	7	)	)	PUNCT
ejpam-3384	74	1	+	+	CCONJ
ejpam-3384	74	2	|s−	|s−	ADJ
ejpam-3384	74	3	m+m	m+m	NOUN
ejpam-3384	74	4	2	2	NUM
ejpam-3384	75	1	|	|	NOUN
ejpam-3384	75	2	]	]	X
ejpam-3384	75	3	r	r	NOUN
ejpam-3384	75	4	,	,	PUNCT
ejpam-3384	75	5	∀	∀	NOUN
ejpam-3384	75	6	s	s	PART
ejpam-3384	75	7	∈	∈	PROPN
ejpam-3384	75	8	[	[	X
ejpam-3384	75	9	m	m	NOUN
ejpam-3384	75	10	,	,	PUNCT
ejpam-3384	75	11	m	m	VERB
ejpam-3384	75	12	]	]	PUNCT
ejpam-3384	75	13	and	and	CCONJ
ejpam-3384	75	14	x	x	PUNCT
ejpam-3384	75	15	∈	∈	NOUN
ejpam-3384	75	16	h	h	NOUN
ejpam-3384	75	17	with	with	ADP
ejpam-3384	75	18	‖x‖	‖x‖	PROPN
ejpam-3384	75	19	=	=	SYM
ejpam-3384	75	20	1	1	X
ejpam-3384	75	21	.	.	PUNCT
ejpam-3384	76	1	moreover	moreover	ADV
ejpam-3384	76	2	,	,	PUNCT
ejpam-3384	76	3	we	we	PRON
ejpam-3384	76	4	have∣∣∣〈f(b)y	have∣∣∣〈f(b)y	VERB
ejpam-3384	76	5	,	,	PUNCT
ejpam-3384	76	6	y	y	PROPN
ejpam-3384	76	7	〉	〉	PROPN
ejpam-3384	76	8	−	−	PROPN
ejpam-3384	77	1	〈	〈	PROPN
ejpam-3384	77	2	f(a)x	f(a)x	PROPN
ejpam-3384	77	3	,	,	PUNCT
ejpam-3384	77	4	x	x	NOUN
ejpam-3384	77	5	〉	〉	NOUN
ejpam-3384	77	6	∣∣∣	∣∣∣	NOUN
ejpam-3384	77	7	≤	≤	NUM
ejpam-3384	77	8	〈	〈	PROPN
ejpam-3384	77	9	∣∣∣f(b)−	∣∣∣f(b)−	NOUN
ejpam-3384	77	10	〈	〈	PROPN
ejpam-3384	77	11	f(a)x	f(a)x	PROPN
ejpam-3384	77	12	,	,	PUNCT
ejpam-3384	77	13	x	x	NOUN
ejpam-3384	77	14	〉	〉	NUM
ejpam-3384	77	15	·	·	PUNCT
ejpam-3384	77	16	1h	1h	NUM
ejpam-3384	77	17	∣∣∣	∣∣∣	PROPN
ejpam-3384	77	18	y	y	PROPN
ejpam-3384	77	19	,	,	PUNCT
ejpam-3384	77	20	y	y	PROPN
ejpam-3384	77	21	〉	〉	NOUN
ejpam-3384	77	22	.	.	PUNCT
ejpam-3384	78	1	≤	≤	NUM
ejpam-3384	78	2	l	l	NOUN
ejpam-3384	79	1	[	[	X
ejpam-3384	79	2	1	1	NUM
ejpam-3384	79	3	2	2	NUM
ejpam-3384	79	4	(	(	PUNCT
ejpam-3384	79	5	m	m	NOUN
ejpam-3384	79	6	−m	−m	NOUN
ejpam-3384	79	7	)	)	PUNCT
ejpam-3384	79	8	+	+	CCONJ
ejpam-3384	79	9	〈	〈	PROPN
ejpam-3384	79	10	|b	|b	ADJ
ejpam-3384	79	11	−	−	PROPN
ejpam-3384	79	12	m+m	m+m	PROPN
ejpam-3384	79	13	2	2	NUM
ejpam-3384	79	14	·	·	SYM
ejpam-3384	79	15	1h	1h	NUM
ejpam-3384	79	16	|y	|y	NOUN
ejpam-3384	79	17	,	,	PUNCT
ejpam-3384	79	18	y	y	PROPN
ejpam-3384	79	19	〉	〉	NOUN
ejpam-3384	79	20	]	]	X
ejpam-3384	79	21	r	r	NOUN
ejpam-3384	79	22	,	,	PUNCT
ejpam-3384	79	23	∀	∀	X
ejpam-3384	79	24	x	x	NOUN
ejpam-3384	79	25	,	,	PUNCT
ejpam-3384	79	26	y	y	PROPN
ejpam-3384	79	27	∈	∈	PROPN
ejpam-3384	79	28	h	h	NOUN
ejpam-3384	79	29	with	with	ADP
ejpam-3384	79	30	‖x‖	‖x‖	PROPN
ejpam-3384	79	31	=	=	SYM
ejpam-3384	79	32	‖y‖	‖y‖	PROPN
ejpam-3384	79	33	=	=	NOUN
ejpam-3384	79	34	1	1	X
ejpam-3384	79	35	.	.	X
ejpam-3384	79	36	see	see	VERB
ejpam-3384	79	37	[	[	X
ejpam-3384	79	38	10	10	NUM
ejpam-3384	79	39	]	]	PUNCT
ejpam-3384	79	40	.	.	PUNCT
ejpam-3384	80	1	b.	b.	PROPN
ejpam-3384	80	2	barnes	barnes	PROPN
ejpam-3384	80	3	,	,	PUNCT
ejpam-3384	80	4	c.	c.	PROPN
ejpam-3384	80	5	sebil	sebil	PROPN
ejpam-3384	80	6	,	,	PUNCT
ejpam-3384	80	7	i.	i.	PROPN
ejpam-3384	80	8	k.	k.	PROPN
ejpam-3384	80	9	dontwi	dontwi	PROPN
ejpam-3384	80	10	/	/	SYM
ejpam-3384	80	11	eur	eur	PROPN
ejpam-3384	80	12	.	.	PUNCT
ejpam-3384	81	1	j.	j.	PROPN
ejpam-3384	81	2	pure	pure	PROPN
ejpam-3384	81	3	appl	appl	PROPN
ejpam-3384	81	4	.	.	PROPN
ejpam-3384	81	5	math	math	PROPN
ejpam-3384	81	6	,	,	PUNCT
ejpam-3384	81	7	12	12	NUM
ejpam-3384	81	8	(	(	PUNCT
ejpam-3384	81	9	2	2	NUM
ejpam-3384	81	10	)	)	PUNCT
ejpam-3384	81	11	(	(	PUNCT
ejpam-3384	81	12	2019	2019	NUM
ejpam-3384	81	13	)	)	PUNCT
ejpam-3384	81	14	,	,	PUNCT
ejpam-3384	81	15	469	469	NUM
ejpam-3384	81	16	-	-	SYM
ejpam-3384	81	17	485	485	NUM
ejpam-3384	81	18	472	472	NUM
ejpam-3384	81	19	3	3	NUM
ejpam-3384	81	20	.	.	PUNCT
ejpam-3384	81	21	main	main	ADJ
ejpam-3384	81	22	result	result	NOUN
ejpam-3384	81	23	in	in	ADP
ejpam-3384	81	24	this	this	DET
ejpam-3384	81	25	section	section	NOUN
ejpam-3384	81	26	,	,	PUNCT
ejpam-3384	81	27	we	we	PRON
ejpam-3384	81	28	derive	derive	VERB
ejpam-3384	81	29	both	both	CCONJ
ejpam-3384	81	30	the	the	DET
ejpam-3384	81	31	first	first	ADJ
ejpam-3384	81	32	and	and	CCONJ
ejpam-3384	81	33	second	second	ADJ
ejpam-3384	81	34	quotient	quotient	NOUN
ejpam-3384	81	35	inequalities	inequality	NOUN
ejpam-3384	81	36	in	in	ADP
ejpam-3384	81	37	a	a	DET
ejpam-3384	81	38	suitable	suitable	ADJ
ejpam-3384	81	39	functional	functional	ADJ
ejpam-3384	81	40	space	space	NOUN
ejpam-3384	81	41	.	.	PUNCT
ejpam-3384	82	1	theorem	theorem	ADJ
ejpam-3384	82	2	2	2	NUM
ejpam-3384	82	3	(	(	PUNCT
ejpam-3384	82	4	first	first	ADJ
ejpam-3384	82	5	quotient	quotient	NOUN
ejpam-3384	82	6	inequality	inequality	NOUN
ejpam-3384	82	7	)	)	PUNCT
ejpam-3384	82	8	.	.	PUNCT
ejpam-3384	83	1	suppose	suppose	VERB
ejpam-3384	83	2	that	that	SCONJ
ejpam-3384	83	3	x	x	PROPN
ejpam-3384	83	4	and	and	CCONJ
ejpam-3384	83	5	y	y	PROPN
ejpam-3384	83	6	are	be	AUX
ejpam-3384	83	7	two	two	NUM
ejpam-3384	83	8	real	real	ADJ
ejpam-3384	83	9	numbers	number	NOUN
ejpam-3384	83	10	or	or	CCONJ
ejpam-3384	83	11	two	two	NUM
ejpam-3384	83	12	real	real	ADV
ejpam-3384	83	13	-	-	PUNCT
ejpam-3384	83	14	valued	value	VERB
ejpam-3384	83	15	vectors	vector	NOUN
ejpam-3384	83	16	,	,	PUNCT
ejpam-3384	83	17	then	then	ADV
ejpam-3384	83	18	‖x‖‖y‖	‖x‖‖y‖	VERB
ejpam-3384	83	19	≤	≤	X
ejpam-3384	84	1	‖x‖	‖x‖	PROPN
ejpam-3384	84	2	‖y‖	‖y‖	PROPN
ejpam-3384	84	3	,	,	PUNCT
ejpam-3384	84	4	∀	∀	X
ejpam-3384	84	5	0	0	NUM
ejpam-3384	85	1	≤	≤	NUM
ejpam-3384	85	2	‖x‖	‖x‖	PROPN
ejpam-3384	85	3	<	<	X
ejpam-3384	85	4	∞	∞	NUM
ejpam-3384	85	5	and	and	CCONJ
ejpam-3384	85	6	0	0	NUM
ejpam-3384	85	7	<	<	X
ejpam-3384	86	1	‖y‖	‖y‖	PROPN
ejpam-3384	86	2	≤	≤	ADV
ejpam-3384	86	3	1	1	NUM
ejpam-3384	86	4	,	,	PUNCT
ejpam-3384	86	5	where	where	SCONJ
ejpam-3384	86	6	the	the	DET
ejpam-3384	86	7	equality	equality	NOUN
ejpam-3384	86	8	occurs	occur	VERB
ejpam-3384	86	9	at	at	ADP
ejpam-3384	86	10	either	either	CCONJ
ejpam-3384	86	11	‖x‖	‖x‖	PROPN
ejpam-3384	86	12	=	=	SYM
ejpam-3384	86	13	0	0	NUM
ejpam-3384	86	14	or	or	CCONJ
ejpam-3384	86	15	‖y‖	‖y‖	PROPN
ejpam-3384	86	16	=	=	SYM
ejpam-3384	86	17	1	1	X
ejpam-3384	86	18	.	.	PUNCT
ejpam-3384	87	1	proof	proof	NOUN
ejpam-3384	87	2	:	:	PUNCT
ejpam-3384	87	3	setting	set	VERB
ejpam-3384	87	4	f(x	f(x	PROPN
ejpam-3384	87	5	,	,	PUNCT
ejpam-3384	87	6	y	y	NOUN
ejpam-3384	87	7	)	)	PUNCT
ejpam-3384	87	8	=	=	PRON
ejpam-3384	88	1	−νx2	−νx2	PROPN
ejpam-3384	88	2	−	−	PROPN
ejpam-3384	88	3	2νx2y2	2νx2y2	NUM
ejpam-3384	88	4	,	,	PUNCT
ejpam-3384	88	5	the	the	DET
ejpam-3384	88	6	function	function	NOUN
ejpam-3384	88	7	attains	attain	VERB
ejpam-3384	88	8	its	its	PRON
ejpam-3384	88	9	maximum	maximum	ADJ
ejpam-3384	88	10	value	value	NOUN
ejpam-3384	88	11	at	at	ADP
ejpam-3384	88	12	zero	zero	NUM
ejpam-3384	88	13	,	,	PUNCT
ejpam-3384	88	14	for	for	ADP
ejpam-3384	88	15	all	all	DET
ejpam-3384	88	16	(	(	PUNCT
ejpam-3384	88	17	x	x	NOUN
ejpam-3384	88	18	,	,	PUNCT
ejpam-3384	88	19	y	y	NOUN
ejpam-3384	88	20	)	)	PUNCT
ejpam-3384	88	21	∈	∈	PROPN
ejpam-3384	88	22	r2	r2	NOUN
ejpam-3384	88	23	and	and	CCONJ
ejpam-3384	88	24	ν	ν	X
ejpam-3384	88	25	∈	∈	PROPN
ejpam-3384	89	1	[	[	X
ejpam-3384	89	2	0	0	NUM
ejpam-3384	89	3	,	,	PUNCT
ejpam-3384	89	4	1	1	NUM
ejpam-3384	89	5	]	]	PUNCT
ejpam-3384	89	6	.	.	PUNCT
ejpam-3384	90	1	we	we	PRON
ejpam-3384	90	2	can	can	AUX
ejpam-3384	90	3	see	see	VERB
ejpam-3384	90	4	that	that	PRON
ejpam-3384	90	5	:	:	PUNCT
ejpam-3384	90	6	f(x	f(x	PROPN
ejpam-3384	90	7	,	,	PUNCT
ejpam-3384	90	8	y	y	PROPN
ejpam-3384	90	9	)	)	PUNCT
ejpam-3384	90	10	=	=	SYM
ejpam-3384	91	1	−	−	PROPN
ejpam-3384	91	2	(	(	PUNCT
ejpam-3384	91	3	νx2	νx2	NOUN
ejpam-3384	91	4	+	+	CCONJ
ejpam-3384	91	5	2νx2y2	2νx2y2	NUM
ejpam-3384	91	6	)	)	PUNCT
ejpam-3384	91	7	≤	≤	NOUN
ejpam-3384	91	8	0	0	NUM
ejpam-3384	92	1	f(x	f(x	PROPN
ejpam-3384	92	2	,	,	PUNCT
ejpam-3384	92	3	y	y	NOUN
ejpam-3384	92	4	)	)	PUNCT
ejpam-3384	92	5	=	=	SYM
ejpam-3384	93	1	−	−	PROPN
ejpam-3384	93	2	(	(	PUNCT
ejpam-3384	93	3	νx2	νx2	NOUN
ejpam-3384	93	4	+	+	CCONJ
ejpam-3384	93	5	x2(2νy2	x2(2νy2	PROPN
ejpam-3384	93	6	)	)	PUNCT
ejpam-3384	93	7	)	)	PUNCT
ejpam-3384	94	1	≤	≤	NOUN
ejpam-3384	94	2	0	0	NUM
ejpam-3384	95	1	f(x	f(x	PROPN
ejpam-3384	95	2	,	,	PUNCT
ejpam-3384	95	3	y	y	NOUN
ejpam-3384	95	4	)	)	PUNCT
ejpam-3384	95	5	=	=	SYM
ejpam-3384	96	1	−	−	PROPN
ejpam-3384	96	2	(	(	PUNCT
ejpam-3384	96	3	νx2	νx2	NOUN
ejpam-3384	96	4	+	+	CCONJ
ejpam-3384	96	5	x2(2νy2	x2(2νy2	PROPN
ejpam-3384	96	6	)	)	PUNCT
ejpam-3384	96	7	)	)	PUNCT
ejpam-3384	97	1	≤	≤	ADV
ejpam-3384	97	2	−	−	PROPN
ejpam-3384	97	3	(	(	PUNCT
ejpam-3384	97	4	x2ν	x2ν	X
ejpam-3384	97	5	+	+	CCONJ
ejpam-3384	97	6	x2y4ν	x2y4ν	PUNCT
ejpam-3384	97	7	)	)	PUNCT
ejpam-3384	97	8	≤	≤	NOUN
ejpam-3384	97	9	0	0	NUM
ejpam-3384	97	10	by	by	ADP
ejpam-3384	97	11	transitivity	transitivity	NOUN
ejpam-3384	97	12	law	law	NOUN
ejpam-3384	97	13	,	,	PUNCT
ejpam-3384	97	14	we	we	PRON
ejpam-3384	97	15	obtain	obtain	VERB
ejpam-3384	97	16	−	−	PROPN
ejpam-3384	97	17	(	(	PUNCT
ejpam-3384	97	18	x2ν	x2ν	X
ejpam-3384	97	19	+	+	CCONJ
ejpam-3384	97	20	x2y4ν	x2y4ν	PUNCT
ejpam-3384	97	21	)	)	PUNCT
ejpam-3384	97	22	≤	≤	NOUN
ejpam-3384	97	23	0	0	NUM
ejpam-3384	97	24	⇒	⇒	NOUN
ejpam-3384	97	25	−x2y4ν	−x2y4ν	PROPN
ejpam-3384	97	26	≤	≤	NUM
ejpam-3384	97	27	x2ν	x2ν	PROPN
ejpam-3384	98	1	⇒	⇒	VERB
ejpam-3384	98	2	‖−	‖−	NOUN
ejpam-3384	98	3	x2νy2ν‖	x2νy2ν‖	PROPN
ejpam-3384	98	4	=	=	PUNCT
ejpam-3384	98	5	‖	‖	PROPN
ejpam-3384	98	6	x2ν	x2ν	PROPN
ejpam-3384	99	1	x2(1−ν)y2ν	x2(1−ν)y2ν	PROPN
ejpam-3384	99	2	‖	‖	ADJ
ejpam-3384	99	3	⇒	⇒	NOUN
ejpam-3384	99	4	‖x‖2ν‖y‖2ν	‖x‖2ν‖y‖2ν	NOUN
ejpam-3384	99	5	≤	≤	NUM
ejpam-3384	99	6	‖x‖2ν	‖x‖2ν	PROPN
ejpam-3384	99	7	‖x‖2(1−ν)‖y‖2ν	‖x‖2(1−ν)‖y‖2ν	PROPN
ejpam-3384	99	8	.	.	PUNCT
ejpam-3384	100	1	(	(	PUNCT
ejpam-3384	100	2	1	1	X
ejpam-3384	100	3	)	)	PUNCT
ejpam-3384	100	4	setting	set	VERB
ejpam-3384	100	5	ν	ν	NOUN
ejpam-3384	100	6	=	=	SYM
ejpam-3384	100	7	1	1	NUM
ejpam-3384	100	8	in	in	ADP
ejpam-3384	100	9	inequality	inequality	NOUN
ejpam-3384	100	10	(	(	PUNCT
ejpam-3384	100	11	1	1	X
ejpam-3384	100	12	)	)	PUNCT
ejpam-3384	100	13	yields	yield	NOUN
ejpam-3384	100	14	‖x‖‖y‖	‖x‖‖y‖	PROPN
ejpam-3384	100	15	≤	≤	PUNCT
ejpam-3384	101	1	‖x‖	‖x‖	VERB
ejpam-3384	101	2	‖y‖	‖y‖	PROPN
ejpam-3384	101	3	.	.	PUNCT
ejpam-3384	102	1	(	(	PUNCT
ejpam-3384	102	2	2	2	X
ejpam-3384	102	3	)	)	PUNCT
ejpam-3384	102	4	we	we	PRON
ejpam-3384	102	5	search	search	VERB
ejpam-3384	102	6	for	for	ADP
ejpam-3384	102	7	the	the	DET
ejpam-3384	102	8	regions	region	NOUN
ejpam-3384	102	9	for	for	ADP
ejpam-3384	102	10	which	which	PRON
ejpam-3384	102	11	‖x‖	‖x‖	PROPN
ejpam-3384	102	12	and	and	CCONJ
ejpam-3384	102	13	‖y‖	‖y‖	PROPN
ejpam-3384	102	14	hold	hold	VERB
ejpam-3384	102	15	.	.	PUNCT
ejpam-3384	103	1	setting	set	VERB
ejpam-3384	103	2	ν	ν	NOUN
ejpam-3384	103	3	=	=	SYM
ejpam-3384	103	4	1	1	NUM
ejpam-3384	103	5	4	4	NUM
ejpam-3384	103	6	in	in	ADP
ejpam-3384	103	7	inequality	inequality	NOUN
ejpam-3384	103	8	(	(	PUNCT
ejpam-3384	103	9	1	1	NUM
ejpam-3384	103	10	)	)	PUNCT
ejpam-3384	103	11	,	,	PUNCT
ejpam-3384	103	12	we	we	PRON
ejpam-3384	103	13	obtain	obtain	VERB
ejpam-3384	103	14	‖x‖3	‖x‖3	ADP
ejpam-3384	103	15	≤	≤	ADJ
ejpam-3384	103	16	1	1	NUM
ejpam-3384	103	17	‖y‖2	‖y‖2	ADV
ejpam-3384	103	18	.	.	PUNCT
ejpam-3384	104	1	(	(	PUNCT
ejpam-3384	104	2	3	3	X
ejpam-3384	104	3	)	)	PUNCT
ejpam-3384	104	4	again	again	ADV
ejpam-3384	104	5	,	,	PUNCT
ejpam-3384	104	6	plugging	plug	VERB
ejpam-3384	104	7	ν	ν	NOUN
ejpam-3384	104	8	=	=	SYM
ejpam-3384	104	9	1	1	NUM
ejpam-3384	104	10	2	2	NUM
ejpam-3384	104	11	into	into	ADP
ejpam-3384	104	12	inequality	inequality	NOUN
ejpam-3384	104	13	(	(	PUNCT
ejpam-3384	104	14	1	1	X
ejpam-3384	104	15	)	)	PUNCT
ejpam-3384	104	16	yields	yield	NOUN
ejpam-3384	104	17	‖x‖	‖x‖	VERB
ejpam-3384	104	18	≤	≤	NUM
ejpam-3384	104	19	1	1	NUM
ejpam-3384	104	20	‖y‖2	‖y‖2	ADV
ejpam-3384	104	21	.	.	PUNCT
ejpam-3384	105	1	(	(	PUNCT
ejpam-3384	105	2	4	4	X
ejpam-3384	105	3	)	)	PUNCT
ejpam-3384	105	4	b.	b.	PROPN
ejpam-3384	105	5	barnes	barnes	PROPN
ejpam-3384	105	6	,	,	PUNCT
ejpam-3384	105	7	c.	c.	PROPN
ejpam-3384	105	8	sebil	sebil	PROPN
ejpam-3384	105	9	,	,	PUNCT
ejpam-3384	105	10	i.	i.	PROPN
ejpam-3384	105	11	k.	k.	PROPN
ejpam-3384	105	12	dontwi	dontwi	PROPN
ejpam-3384	105	13	/	/	SYM
ejpam-3384	105	14	eur	eur	PROPN
ejpam-3384	105	15	.	.	PUNCT
ejpam-3384	106	1	j.	j.	PROPN
ejpam-3384	106	2	pure	pure	PROPN
ejpam-3384	106	3	appl	appl	PROPN
ejpam-3384	106	4	.	.	PROPN
ejpam-3384	106	5	math	math	PROPN
ejpam-3384	106	6	,	,	PUNCT
ejpam-3384	106	7	12	12	NUM
ejpam-3384	106	8	(	(	PUNCT
ejpam-3384	106	9	2	2	NUM
ejpam-3384	106	10	)	)	PUNCT
ejpam-3384	106	11	(	(	PUNCT
ejpam-3384	106	12	2019	2019	NUM
ejpam-3384	106	13	)	)	PUNCT
ejpam-3384	106	14	,	,	PUNCT
ejpam-3384	106	15	469	469	NUM
ejpam-3384	106	16	-	-	SYM
ejpam-3384	106	17	485	485	NUM
ejpam-3384	106	18	473	473	NUM
ejpam-3384	106	19	from	from	ADP
ejpam-3384	106	20	the	the	DET
ejpam-3384	106	21	inequalities	inequality	NOUN
ejpam-3384	106	22	(	(	PUNCT
ejpam-3384	106	23	3	3	NUM
ejpam-3384	106	24	)	)	PUNCT
ejpam-3384	106	25	and	and	CCONJ
ejpam-3384	106	26	(	(	PUNCT
ejpam-3384	106	27	4	4	NUM
ejpam-3384	106	28	)	)	PUNCT
ejpam-3384	107	1	,	,	PUNCT
ejpam-3384	107	2	we	we	PRON
ejpam-3384	107	3	consider	consider	VERB
ejpam-3384	107	4	three	three	NUM
ejpam-3384	107	5	situations	situation	NOUN
ejpam-3384	107	6	for	for	ADP
ejpam-3384	107	7	which	which	DET
ejpam-3384	107	8	inequality	inequality	NOUN
ejpam-3384	107	9	(	(	PUNCT
ejpam-3384	107	10	2	2	NUM
ejpam-3384	107	11	)	)	PUNCT
ejpam-3384	107	12	holds	hold	VERB
ejpam-3384	107	13	.	.	PUNCT
ejpam-3384	108	1	we	we	PRON
ejpam-3384	108	2	observe	observe	VERB
ejpam-3384	108	3	that	that	SCONJ
ejpam-3384	108	4	‖x‖3	‖x‖3	ADP
ejpam-3384	108	5	<	<	X
ejpam-3384	108	6	‖x‖	‖x‖	PROPN
ejpam-3384	108	7	⇒	⇒	NOUN
ejpam-3384	108	8	x(x2	x(x2	VERB
ejpam-3384	109	1	−	−	NOUN
ejpam-3384	109	2	1	1	NUM
ejpam-3384	109	3	)	)	PUNCT
ejpam-3384	109	4	<	<	X
ejpam-3384	109	5	0	0	NUM
ejpam-3384	109	6	⇒	⇒	NOUN
ejpam-3384	109	7	0	0	PUNCT
ejpam-3384	109	8	<	<	X
ejpam-3384	109	9	‖x‖	‖x‖	X
ejpam-3384	109	10	<	<	X
ejpam-3384	109	11	1	1	NUM
ejpam-3384	109	12	.	.	PUNCT
ejpam-3384	109	13	(	(	PUNCT
ejpam-3384	109	14	5	5	NUM
ejpam-3384	109	15	)	)	PUNCT
ejpam-3384	109	16	also	also	ADV
ejpam-3384	109	17	,	,	PUNCT
ejpam-3384	109	18	we	we	PRON
ejpam-3384	109	19	can	can	AUX
ejpam-3384	109	20	see	see	VERB
ejpam-3384	109	21	that	that	SCONJ
ejpam-3384	109	22	the	the	DET
ejpam-3384	109	23	two	two	NUM
ejpam-3384	109	24	inequalities	inequality	NOUN
ejpam-3384	109	25	in	in	ADP
ejpam-3384	109	26	(	(	PUNCT
ejpam-3384	109	27	3	3	NUM
ejpam-3384	109	28	)	)	PUNCT
ejpam-3384	109	29	and	and	CCONJ
ejpam-3384	109	30	(	(	PUNCT
ejpam-3384	109	31	4	4	X
ejpam-3384	109	32	)	)	PUNCT
ejpam-3384	109	33	are	be	AUX
ejpam-3384	109	34	equal	equal	ADJ
ejpam-3384	109	35	if	if	SCONJ
ejpam-3384	109	36	,	,	PUNCT
ejpam-3384	109	37	‖x‖3	‖x‖3	ADP
ejpam-3384	109	38	=	=	PUNCT
ejpam-3384	109	39	‖x‖	‖x‖	PROPN
ejpam-3384	109	40	⇒	⇒	NOUN
ejpam-3384	109	41	x(x2	x(x2	VERB
ejpam-3384	110	1	−	−	NOUN
ejpam-3384	110	2	1	1	NUM
ejpam-3384	110	3	)	)	PUNCT
ejpam-3384	110	4	=	=	SYM
ejpam-3384	110	5	0	0	NUM
ejpam-3384	110	6	⇒	⇒	NOUN
ejpam-3384	110	7	‖x‖	‖x‖	PROPN
ejpam-3384	110	8	=	=	SYM
ejpam-3384	110	9	0	0	NUM
ejpam-3384	110	10	,	,	PUNCT
ejpam-3384	110	11	or	or	CCONJ
ejpam-3384	110	12	‖x‖	‖x‖	PROPN
ejpam-3384	110	13	=	=	SYM
ejpam-3384	110	14	1	1	X
ejpam-3384	110	15	.	.	PUNCT
ejpam-3384	110	16	(	(	PUNCT
ejpam-3384	110	17	6	6	NUM
ejpam-3384	110	18	)	)	PUNCT
ejpam-3384	110	19	lastly	lastly	ADV
ejpam-3384	110	20	,	,	PUNCT
ejpam-3384	110	21	the	the	DET
ejpam-3384	110	22	norms	norm	NOUN
ejpam-3384	110	23	on	on	ADP
ejpam-3384	110	24	the	the	DET
ejpam-3384	110	25	left	left	ADJ
ejpam-3384	110	26	hand	hand	NOUN
ejpam-3384	110	27	sides	side	NOUN
ejpam-3384	110	28	of	of	ADP
ejpam-3384	110	29	inequalities	inequality	NOUN
ejpam-3384	110	30	(	(	PUNCT
ejpam-3384	110	31	3	3	NUM
ejpam-3384	110	32	)	)	PUNCT
ejpam-3384	110	33	and	and	CCONJ
ejpam-3384	110	34	(	(	PUNCT
ejpam-3384	110	35	4	4	X
ejpam-3384	110	36	)	)	PUNCT
ejpam-3384	110	37	can	can	AUX
ejpam-3384	110	38	be	be	AUX
ejpam-3384	110	39	written	write	VERB
ejpam-3384	110	40	as	as	ADP
ejpam-3384	110	41	:	:	PUNCT
ejpam-3384	110	42	‖x‖3	‖x‖3	ADV
ejpam-3384	110	43	>	>	PUNCT
ejpam-3384	110	44	‖x‖.	‖x‖.	PROPN
ejpam-3384	110	45	⇒	⇒	PROPN
ejpam-3384	110	46	x(x2	x(x2	PROPN
ejpam-3384	111	1	−	−	NOUN
ejpam-3384	111	2	1	1	NUM
ejpam-3384	111	3	)	)	PUNCT
ejpam-3384	111	4	>	>	SYM
ejpam-3384	111	5	0	0	PUNCT
ejpam-3384	111	6	⇒	⇒	PROPN
ejpam-3384	111	7	‖x‖	‖x‖	PROPN
ejpam-3384	111	8	>	>	X
ejpam-3384	111	9	0	0	NUM
ejpam-3384	111	10	,	,	PUNCT
ejpam-3384	111	11	or	or	CCONJ
ejpam-3384	111	12	‖x‖	‖x‖	VERB
ejpam-3384	111	13	>	>	X
ejpam-3384	112	1	1	1	X
ejpam-3384	112	2	.	.	PUNCT
ejpam-3384	112	3	(	(	PUNCT
ejpam-3384	112	4	7	7	X
ejpam-3384	112	5	)	)	PUNCT
ejpam-3384	112	6	the	the	DET
ejpam-3384	112	7	region	region	NOUN
ejpam-3384	112	8	for	for	ADP
ejpam-3384	112	9	which	which	PRON
ejpam-3384	112	10	‖y‖	‖y‖	PROPN
ejpam-3384	112	11	holds	hold	VERB
ejpam-3384	112	12	is	be	AUX
ejpam-3384	112	13	as	as	SCONJ
ejpam-3384	112	14	follows	follow	VERB
ejpam-3384	112	15	.	.	PUNCT
ejpam-3384	113	1	‖y‖2	‖y‖2	PROPN
ejpam-3384	113	2	≤	≤	ADJ
ejpam-3384	113	3	1	1	NUM
ejpam-3384	113	4	‖x‖	‖x‖	PROPN
ejpam-3384	113	5	⇒	⇒	NOUN
ejpam-3384	113	6	‖y‖2	‖y‖2	PROPN
ejpam-3384	113	7	≤	≤	NUM
ejpam-3384	113	8	1	1	NUM
ejpam-3384	113	9	,	,	PUNCT
ejpam-3384	113	10	∀	∀	PUNCT
ejpam-3384	113	11	‖x‖	‖x‖	ADJ
ejpam-3384	113	12	=	=	SYM
ejpam-3384	113	13	1	1	NUM
ejpam-3384	113	14	⇒	⇒	NOUN
ejpam-3384	113	15	0	0	PUNCT
ejpam-3384	114	1	<	<	X
ejpam-3384	114	2	‖y‖	‖y‖	PROPN
ejpam-3384	114	3	≤	≤	ADV
ejpam-3384	114	4	1	1	NUM
ejpam-3384	114	5	.	.	PUNCT
ejpam-3384	115	1	(	(	PUNCT
ejpam-3384	115	2	8)	8)	NUM
ejpam-3384	115	3	combining	combine	VERB
ejpam-3384	115	4	the	the	DET
ejpam-3384	115	5	inequalities	inequality	NOUN
ejpam-3384	115	6	(	(	PUNCT
ejpam-3384	115	7	5	5	NUM
ejpam-3384	115	8	)	)	PUNCT
ejpam-3384	115	9	,	,	PUNCT
ejpam-3384	115	10	(	(	PUNCT
ejpam-3384	115	11	7	7	X
ejpam-3384	115	12	)	)	PUNCT
ejpam-3384	115	13	and	and	CCONJ
ejpam-3384	115	14	(	(	PUNCT
ejpam-3384	115	15	8)	8)	NUM
ejpam-3384	115	16	,	,	PUNCT
ejpam-3384	115	17	and	and	CCONJ
ejpam-3384	115	18	equation	equation	NOUN
ejpam-3384	115	19	(	(	PUNCT
ejpam-3384	115	20	6	6	NUM
ejpam-3384	115	21	)	)	PUNCT
ejpam-3384	115	22	together	together	ADV
ejpam-3384	115	23	with	with	ADP
ejpam-3384	115	24	inequality	inequality	NOUN
ejpam-3384	115	25	(	(	PUNCT
ejpam-3384	115	26	2	2	NUM
ejpam-3384	115	27	)	)	PUNCT
ejpam-3384	115	28	,	,	PUNCT
ejpam-3384	115	29	we	we	PRON
ejpam-3384	115	30	obtain	obtain	VERB
ejpam-3384	115	31	‖x‖‖y‖	‖x‖‖y‖	NOUN
ejpam-3384	115	32	≤	≤	PUNCT
ejpam-3384	116	1	‖x‖	‖x‖	PROPN
ejpam-3384	116	2	‖y‖	‖y‖	PROPN
ejpam-3384	116	3	,	,	PUNCT
ejpam-3384	116	4	∀	∀	X
ejpam-3384	116	5	0	0	NUM
ejpam-3384	117	1	≤	≤	NUM
ejpam-3384	117	2	‖x‖	‖x‖	PROPN
ejpam-3384	117	3	<	<	X
ejpam-3384	117	4	∞	∞	NUM
ejpam-3384	117	5	and	and	CCONJ
ejpam-3384	117	6	0	0	NUM
ejpam-3384	117	7	<	<	X
ejpam-3384	117	8	‖y‖	‖y‖	PROPN
ejpam-3384	117	9	≤	≤	ADV
ejpam-3384	117	10	1	1	NUM
ejpam-3384	117	11	.	.	PUNCT
ejpam-3384	118	1	theorem	theorem	VERB
ejpam-3384	118	2	3	3	NUM
ejpam-3384	118	3	(	(	PUNCT
ejpam-3384	118	4	second	second	ADJ
ejpam-3384	118	5	quotient	quotient	NOUN
ejpam-3384	118	6	inequality	inequality	NOUN
ejpam-3384	118	7	)	)	PUNCT
ejpam-3384	118	8	.	.	PUNCT
ejpam-3384	119	1	suppose	suppose	VERB
ejpam-3384	119	2	that	that	SCONJ
ejpam-3384	119	3	x	x	PROPN
ejpam-3384	119	4	and	and	CCONJ
ejpam-3384	119	5	y	y	PROPN
ejpam-3384	119	6	are	be	AUX
ejpam-3384	119	7	any	any	DET
ejpam-3384	119	8	two	two	NUM
ejpam-3384	119	9	real	real	ADJ
ejpam-3384	119	10	numbers	number	NOUN
ejpam-3384	119	11	or	or	CCONJ
ejpam-3384	119	12	any	any	DET
ejpam-3384	119	13	two	two	NUM
ejpam-3384	119	14	real	real	ADV
ejpam-3384	119	15	-	-	PUNCT
ejpam-3384	119	16	valued	value	VERB
ejpam-3384	119	17	vectors	vector	NOUN
ejpam-3384	119	18	,	,	PUNCT
ejpam-3384	119	19	then	then	ADV
ejpam-3384	119	20	‖x‖	‖x‖	VERB
ejpam-3384	119	21	‖y‖	‖y‖	PROPN
ejpam-3384	119	22	≤	≤	PROPN
ejpam-3384	119	23	‖x‖‖y‖	‖x‖‖y‖	NOUN
ejpam-3384	119	24	,	,	PUNCT
ejpam-3384	119	25	∀	∀	PUNCT
ejpam-3384	119	26	‖x‖	‖x‖	VERB
ejpam-3384	119	27	≥	≥	NOUN
ejpam-3384	119	28	1	1	NUM
ejpam-3384	119	29	and	and	CCONJ
ejpam-3384	119	30	‖y‖	‖y‖	PROPN
ejpam-3384	119	31	≥	≥	NUM
ejpam-3384	119	32	1	1	NUM
ejpam-3384	119	33	,	,	PUNCT
ejpam-3384	119	34	where	where	SCONJ
ejpam-3384	119	35	the	the	DET
ejpam-3384	119	36	equality	equality	NOUN
ejpam-3384	119	37	occurs	occur	VERB
ejpam-3384	119	38	at	at	ADP
ejpam-3384	119	39	either	either	CCONJ
ejpam-3384	119	40	‖x‖	‖x‖	PROPN
ejpam-3384	119	41	=	=	SYM
ejpam-3384	119	42	‖y‖	‖y‖	PROPN
ejpam-3384	119	43	=	=	SYM
ejpam-3384	119	44	1	1	NUM
ejpam-3384	119	45	or	or	CCONJ
ejpam-3384	119	46	‖y‖	‖y‖	PROPN
ejpam-3384	119	47	=	=	SYM
ejpam-3384	119	48	1	1	X
ejpam-3384	120	1	.	.	X
ejpam-3384	120	2	proof	proof	NOUN
ejpam-3384	120	3	:	:	PUNCT
ejpam-3384	120	4	we	we	PRON
ejpam-3384	120	5	can	can	AUX
ejpam-3384	120	6	see	see	VERB
ejpam-3384	120	7	that	that	SCONJ
ejpam-3384	120	8	f(x	f(x	PROPN
ejpam-3384	120	9	,	,	PUNCT
ejpam-3384	120	10	y	y	NOUN
ejpam-3384	120	11	)	)	PUNCT
ejpam-3384	120	12	=	=	NOUN
ejpam-3384	121	1	νx2	νx2	NOUN
ejpam-3384	121	2	+	+	CCONJ
ejpam-3384	121	3	2νx2y2	2νx2y2	NUM
ejpam-3384	121	4	attains	attain	VERB
ejpam-3384	121	5	its	its	PRON
ejpam-3384	121	6	minimum	minimum	ADJ
ejpam-3384	121	7	value	value	NOUN
ejpam-3384	121	8	at	at	ADP
ejpam-3384	121	9	zero	zero	NUM
ejpam-3384	121	10	,	,	PUNCT
ejpam-3384	121	11	for	for	ADP
ejpam-3384	121	12	all	all	DET
ejpam-3384	121	13	(	(	PUNCT
ejpam-3384	121	14	x	x	NOUN
ejpam-3384	121	15	,	,	PUNCT
ejpam-3384	121	16	y	y	NOUN
ejpam-3384	121	17	)	)	PUNCT
ejpam-3384	121	18	∈	∈	PROPN
ejpam-3384	121	19	r2	r2	NOUN
ejpam-3384	121	20	and	and	CCONJ
ejpam-3384	121	21	ν	ν	X
ejpam-3384	121	22	∈	∈	PROPN
ejpam-3384	122	1	[	[	X
ejpam-3384	122	2	0	0	NUM
ejpam-3384	122	3	,	,	PUNCT
ejpam-3384	122	4	1	1	NUM
ejpam-3384	122	5	]	]	PUNCT
ejpam-3384	122	6	.	.	PUNCT
ejpam-3384	123	1	f(x	f(x	PROPN
ejpam-3384	123	2	,	,	PUNCT
ejpam-3384	123	3	y	y	NOUN
ejpam-3384	123	4	)	)	PUNCT
ejpam-3384	123	5	=	=	NOUN
ejpam-3384	124	1	νx2	νx2	NOUN
ejpam-3384	124	2	+	+	CCONJ
ejpam-3384	124	3	2νx2y2	2νx2y2	NUM
ejpam-3384	124	4	≥	≥	NOUN
ejpam-3384	124	5	0	0	NUM
ejpam-3384	124	6	b.	b.	PROPN
ejpam-3384	124	7	barnes	barnes	PROPN
ejpam-3384	124	8	,	,	PUNCT
ejpam-3384	124	9	c.	c.	PROPN
ejpam-3384	124	10	sebil	sebil	PROPN
ejpam-3384	124	11	,	,	PUNCT
ejpam-3384	125	1	i.	i.	PROPN
ejpam-3384	125	2	k.	k.	PROPN
ejpam-3384	125	3	dontwi	dontwi	PROPN
ejpam-3384	125	4	/	/	SYM
ejpam-3384	125	5	eur	eur	PROPN
ejpam-3384	125	6	.	.	PUNCT
ejpam-3384	126	1	j.	j.	PROPN
ejpam-3384	126	2	pure	pure	PROPN
ejpam-3384	126	3	appl	appl	PROPN
ejpam-3384	126	4	.	.	PROPN
ejpam-3384	126	5	math	math	PROPN
ejpam-3384	126	6	,	,	PUNCT
ejpam-3384	126	7	12	12	NUM
ejpam-3384	126	8	(	(	PUNCT
ejpam-3384	126	9	2	2	NUM
ejpam-3384	126	10	)	)	PUNCT
ejpam-3384	126	11	(	(	PUNCT
ejpam-3384	126	12	2019	2019	NUM
ejpam-3384	126	13	)	)	PUNCT
ejpam-3384	126	14	,	,	PUNCT
ejpam-3384	126	15	469	469	NUM
ejpam-3384	126	16	-	-	SYM
ejpam-3384	126	17	485	485	NUM
ejpam-3384	126	18	474	474	NUM
ejpam-3384	126	19	f(x	f(x	PROPN
ejpam-3384	126	20	,	,	PUNCT
ejpam-3384	126	21	y	y	NOUN
ejpam-3384	126	22	)	)	PUNCT
ejpam-3384	126	23	=	=	NOUN
ejpam-3384	127	1	νx2	νx2	NOUN
ejpam-3384	127	2	+	+	CCONJ
ejpam-3384	127	3	x2(2νy2	x2(2νy2	PROPN
ejpam-3384	127	4	)	)	PUNCT
ejpam-3384	127	5	≥	≥	NOUN
ejpam-3384	127	6	0	0	NUM
ejpam-3384	127	7	f(x	f(x	PROPN
ejpam-3384	127	8	,	,	PUNCT
ejpam-3384	127	9	y	y	NOUN
ejpam-3384	127	10	)	)	PUNCT
ejpam-3384	127	11	=	=	NOUN
ejpam-3384	127	12	νx2	νx2	NOUN
ejpam-3384	127	13	+	+	CCONJ
ejpam-3384	127	14	x2(2νy2	x2(2νy2	PROPN
ejpam-3384	127	15	)	)	PUNCT
ejpam-3384	127	16	≥	≥	NUM
ejpam-3384	127	17	x2ν	x2ν	PUNCT
ejpam-3384	128	1	+	+	CCONJ
ejpam-3384	128	2	x2y4ν	x2y4ν	PUNCT
ejpam-3384	128	3	≥	≥	X
ejpam-3384	128	4	0	0	NUM
ejpam-3384	128	5	by	by	ADP
ejpam-3384	128	6	transitivity	transitivity	NOUN
ejpam-3384	128	7	law	law	NOUN
ejpam-3384	128	8	,	,	PUNCT
ejpam-3384	128	9	we	we	PRON
ejpam-3384	128	10	obtain	obtain	VERB
ejpam-3384	128	11	x2ν	x2ν	X
ejpam-3384	129	1	+	+	CCONJ
ejpam-3384	129	2	x2y4ν	x2y4ν	PUNCT
ejpam-3384	129	3	≥	≥	NOUN
ejpam-3384	129	4	0	0	NUM
ejpam-3384	129	5	⇒	⇒	PROPN
ejpam-3384	129	6	−x2ν	−x2ν	NUM
ejpam-3384	129	7	≤	≤	NUM
ejpam-3384	129	8	x2y4ν	x2y4ν	PUNCT
ejpam-3384	129	9	⇒	⇒	PROPN
ejpam-3384	129	10	‖−	‖−	PROPN
ejpam-3384	129	11	x2ν	x2ν	X
ejpam-3384	130	1	x2(1−ν)y2ν	x2(1−ν)y2ν	PROPN
ejpam-3384	130	2	‖	‖	PROPN
ejpam-3384	131	1	=	=	PUNCT
ejpam-3384	131	2	‖x2νy2ν‖	‖x2νy2ν‖	NOUN
ejpam-3384	131	3	⇒	⇒	NOUN
ejpam-3384	131	4	‖x‖2ν	‖x‖2ν	PROPN
ejpam-3384	131	5	‖x‖2(1−ν)‖y‖2ν	‖x‖2(1−ν)‖y‖2ν	ADJ
ejpam-3384	131	6	≤	≤	NUM
ejpam-3384	131	7	‖x‖2ν‖y‖2ν	‖x‖2ν‖y‖2ν	NOUN
ejpam-3384	131	8	.	.	PUNCT
ejpam-3384	132	1	(	(	PUNCT
ejpam-3384	132	2	9	9	X
ejpam-3384	132	3	)	)	PUNCT
ejpam-3384	132	4	setting	set	VERB
ejpam-3384	132	5	ν	ν	NOUN
ejpam-3384	132	6	=	=	SYM
ejpam-3384	132	7	1	1	NUM
ejpam-3384	132	8	in	in	ADP
ejpam-3384	132	9	inequality	inequality	NOUN
ejpam-3384	132	10	(	(	PUNCT
ejpam-3384	132	11	9	9	NUM
ejpam-3384	132	12	)	)	PUNCT
ejpam-3384	132	13	yields	yield	NOUN
ejpam-3384	132	14	‖x‖	‖x‖	VERB
ejpam-3384	132	15	‖y‖	‖y‖	PROPN
ejpam-3384	132	16	≤	≤	NOUN
ejpam-3384	132	17	‖x‖‖y‖.	‖x‖‖y‖.	NUM
ejpam-3384	132	18	the	the	DET
ejpam-3384	132	19	regions	region	NOUN
ejpam-3384	132	20	in	in	ADP
ejpam-3384	132	21	which	which	PRON
ejpam-3384	132	22	the	the	DET
ejpam-3384	132	23	above	above	ADJ
ejpam-3384	132	24	inequality	inequality	NOUN
ejpam-3384	132	25	holds	hold	NOUN
ejpam-3384	132	26	are	be	AUX
ejpam-3384	132	27	as	as	SCONJ
ejpam-3384	132	28	follows	follow	VERB
ejpam-3384	132	29	.	.	PUNCT
ejpam-3384	133	1	setting	set	VERB
ejpam-3384	133	2	ν	ν	X
ejpam-3384	133	3	=	=	SYM
ejpam-3384	133	4	0	0	NUM
ejpam-3384	133	5	in	in	ADP
ejpam-3384	133	6	inequality	inequality	NOUN
ejpam-3384	133	7	(	(	PUNCT
ejpam-3384	133	8	9	9	NUM
ejpam-3384	133	9	)	)	PUNCT
ejpam-3384	133	10	yields	yield	NOUN
ejpam-3384	133	11	‖x‖2	‖x‖2	VERB
ejpam-3384	133	12	≥	≥	NUM
ejpam-3384	133	13	1	1	NUM
ejpam-3384	133	14	⇒	⇒	NOUN
ejpam-3384	133	15	‖x‖	‖x‖	PROPN
ejpam-3384	133	16	≥	≥	NOUN
ejpam-3384	133	17	1	1	NUM
ejpam-3384	133	18	.	.	PUNCT
ejpam-3384	134	1	in	in	ADP
ejpam-3384	134	2	order	order	NOUN
ejpam-3384	134	3	to	to	PART
ejpam-3384	134	4	obtain	obtain	VERB
ejpam-3384	134	5	the	the	DET
ejpam-3384	134	6	region	region	NOUN
ejpam-3384	134	7	for	for	ADP
ejpam-3384	134	8	‖y‖	‖y‖	PROPN
ejpam-3384	134	9	,	,	PUNCT
ejpam-3384	134	10	we	we	PRON
ejpam-3384	134	11	set	set	VERB
ejpam-3384	134	12	f(x	f(x	PROPN
ejpam-3384	134	13	,	,	PUNCT
ejpam-3384	134	14	y	y	NOUN
ejpam-3384	134	15	)	)	PUNCT
ejpam-3384	134	16	=	=	NOUN
ejpam-3384	135	1	νx2	νx2	NOUN
ejpam-3384	135	2	+	+	CCONJ
ejpam-3384	135	3	(	(	PUNCT
ejpam-3384	135	4	2νx2)y2	2νx2)y2	NUM
ejpam-3384	135	5	≥	≥	NOUN
ejpam-3384	135	6	x2ν	x2ν	X
ejpam-3384	136	1	+	+	CCONJ
ejpam-3384	136	2	x2y4ν	x2y4ν	PUNCT
ejpam-3384	136	3	≥	≥	NOUN
ejpam-3384	136	4	0	0	NUM
ejpam-3384	136	5	⇒	⇒	NOUN
ejpam-3384	136	6	−x2ν	−x2ν	NUM
ejpam-3384	136	7	≤	≤	NUM
ejpam-3384	136	8	x4νy2	x4νy2	ADJ
ejpam-3384	136	9	⇒	⇒	NOUN
ejpam-3384	136	10	∥∥∥	∥∥∥	PROPN
ejpam-3384	136	11	−1	−1	NOUN
ejpam-3384	136	12	y2(1−ν	y2(1−ν	NOUN
ejpam-3384	136	13	)	)	PUNCT
ejpam-3384	136	14	∥∥∥	∥∥∥	PROPN
ejpam-3384	136	15	=	=	PUNCT
ejpam-3384	136	16	∥∥∥x2νy2ν	∥∥∥x2νy2ν	ADJ
ejpam-3384	136	17	∥∥∥	∥∥∥	PROPN
ejpam-3384	136	18	⇒	⇒	NOUN
ejpam-3384	136	19	1	1	NUM
ejpam-3384	136	20	‖y‖2(1−ν	‖y‖2(1−ν	NOUN
ejpam-3384	136	21	)	)	PUNCT
ejpam-3384	136	22	≤	≤	NUM
ejpam-3384	136	23	‖x‖2ν‖y‖2ν	‖x‖2ν‖y‖2ν	NOUN
ejpam-3384	136	24	.	.	PUNCT
ejpam-3384	137	1	(	(	PUNCT
ejpam-3384	137	2	10	10	X
ejpam-3384	137	3	)	)	PUNCT
ejpam-3384	137	4	setting	set	VERB
ejpam-3384	137	5	ν	ν	X
ejpam-3384	137	6	=	=	SYM
ejpam-3384	137	7	0	0	NUM
ejpam-3384	137	8	in	in	ADP
ejpam-3384	137	9	inequality	inequality	NOUN
ejpam-3384	137	10	(	(	PUNCT
ejpam-3384	137	11	10	10	NUM
ejpam-3384	137	12	)	)	PUNCT
ejpam-3384	137	13	yields	yield	NOUN
ejpam-3384	137	14	‖y‖2	‖y‖2	VERB
ejpam-3384	137	15	≥	≥	NUM
ejpam-3384	137	16	1	1	NUM
ejpam-3384	137	17	⇒	⇒	NOUN
ejpam-3384	137	18	‖y‖	‖y‖	PROPN
ejpam-3384	137	19	≥	≥	NUM
ejpam-3384	137	20	1	1	NUM
ejpam-3384	137	21	.	.	PUNCT
ejpam-3384	138	1	we	we	PRON
ejpam-3384	138	2	observed	observe	VERB
ejpam-3384	138	3	that	that	SCONJ
ejpam-3384	138	4	the	the	DET
ejpam-3384	138	5	equality	equality	NOUN
ejpam-3384	138	6	occurs	occur	VERB
ejpam-3384	138	7	at	at	ADP
ejpam-3384	138	8	either	either	CCONJ
ejpam-3384	138	9	‖x‖	‖x‖	PROPN
ejpam-3384	138	10	=	=	SYM
ejpam-3384	138	11	‖y‖	‖y‖	PROPN
ejpam-3384	138	12	=	=	SYM
ejpam-3384	138	13	1	1	NUM
ejpam-3384	138	14	or	or	CCONJ
ejpam-3384	138	15	‖y‖	‖y‖	PROPN
ejpam-3384	138	16	=	=	SYM
ejpam-3384	138	17	1	1	NUM
ejpam-3384	138	18	.	.	NOUN
ejpam-3384	138	19	3.1	3.1	NUM
ejpam-3384	138	20	.	.	PUNCT
ejpam-3384	139	1	illustration	illustration	NOUN
ejpam-3384	139	2	of	of	ADP
ejpam-3384	139	3	the	the	DET
ejpam-3384	139	4	first	first	ADJ
ejpam-3384	139	5	quotient	quotient	NOUN
ejpam-3384	139	6	inequality	inequality	NOUN
ejpam-3384	139	7	to	to	ADP
ejpam-3384	139	8	the	the	DET
ejpam-3384	139	9	real	real	ADJ
ejpam-3384	139	10	line	line	NOUN
ejpam-3384	139	11	in	in	ADP
ejpam-3384	139	12	this	this	DET
ejpam-3384	139	13	subsection	subsection	NOUN
ejpam-3384	139	14	,	,	PUNCT
ejpam-3384	139	15	the	the	DET
ejpam-3384	139	16	illustration	illustration	NOUN
ejpam-3384	139	17	of	of	ADP
ejpam-3384	139	18	the	the	DET
ejpam-3384	139	19	first	first	ADJ
ejpam-3384	139	20	quotient	quotient	NOUN
ejpam-3384	139	21	inequality	inequality	NOUN
ejpam-3384	139	22	is	be	AUX
ejpam-3384	139	23	provided	provide	VERB
ejpam-3384	139	24	.	.	PUNCT
ejpam-3384	140	1	example	example	NOUN
ejpam-3384	141	1	1	1	NUM
ejpam-3384	141	2	.	.	PUNCT
ejpam-3384	141	3	.	.	PUNCT
ejpam-3384	142	1	setting	set	VERB
ejpam-3384	142	2	a	a	DET
ejpam-3384	142	3	=	=	SYM
ejpam-3384	142	4	2	2	NUM
ejpam-3384	142	5	3	3	NUM
ejpam-3384	142	6	and	and	CCONJ
ejpam-3384	142	7	b	b	NOUN
ejpam-3384	142	8	=	=	SYM
ejpam-3384	142	9	4	4	NUM
ejpam-3384	142	10	5	5	NUM
ejpam-3384	142	11	,	,	PUNCT
ejpam-3384	142	12	then∥∥∥2	then∥∥∥2	PROPN
ejpam-3384	142	13	3	3	NUM
ejpam-3384	142	14	∥∥∥∥∥∥4	∥∥∥∥∥∥4	NOUN
ejpam-3384	142	15	5	5	NUM
ejpam-3384	142	16	∥∥∥	∥∥∥	NOUN
ejpam-3384	142	17	<	<	X
ejpam-3384	142	18	‖2	‖2	NOUN
ejpam-3384	142	19	3‖	3‖	NUM
ejpam-3384	142	20	‖4	‖4	ADJ
ejpam-3384	142	21	5‖	5‖	PROPN
ejpam-3384	142	22	⇒	⇒	NOUN
ejpam-3384	142	23	8	8	NUM
ejpam-3384	142	24	15	15	NUM
ejpam-3384	142	25	<	<	SYM
ejpam-3384	142	26	5	5	NUM
ejpam-3384	142	27	6	6	NUM
ejpam-3384	142	28	.	.	PUNCT
ejpam-3384	143	1	b.	b.	PROPN
ejpam-3384	143	2	barnes	barnes	PROPN
ejpam-3384	143	3	,	,	PUNCT
ejpam-3384	143	4	c.	c.	PROPN
ejpam-3384	143	5	sebil	sebil	PROPN
ejpam-3384	143	6	,	,	PUNCT
ejpam-3384	143	7	i.	i.	PROPN
ejpam-3384	143	8	k.	k.	PROPN
ejpam-3384	143	9	dontwi	dontwi	PROPN
ejpam-3384	143	10	/	/	SYM
ejpam-3384	143	11	eur	eur	PROPN
ejpam-3384	143	12	.	.	PUNCT
ejpam-3384	144	1	j.	j.	PROPN
ejpam-3384	144	2	pure	pure	PROPN
ejpam-3384	144	3	appl	appl	PROPN
ejpam-3384	144	4	.	.	PROPN
ejpam-3384	144	5	math	math	PROPN
ejpam-3384	144	6	,	,	PUNCT
ejpam-3384	144	7	12	12	NUM
ejpam-3384	144	8	(	(	PUNCT
ejpam-3384	144	9	2	2	NUM
ejpam-3384	144	10	)	)	PUNCT
ejpam-3384	144	11	(	(	PUNCT
ejpam-3384	144	12	2019	2019	NUM
ejpam-3384	144	13	)	)	PUNCT
ejpam-3384	144	14	,	,	PUNCT
ejpam-3384	144	15	469	469	NUM
ejpam-3384	144	16	-	-	SYM
ejpam-3384	144	17	485	485	NUM
ejpam-3384	144	18	475	475	NUM
ejpam-3384	144	19	3.2	3.2	NUM
ejpam-3384	144	20	.	.	PUNCT
ejpam-3384	145	1	the	the	DET
ejpam-3384	145	2	applications	application	NOUN
ejpam-3384	145	3	of	of	ADP
ejpam-3384	145	4	the	the	DET
ejpam-3384	145	5	first	first	ADJ
ejpam-3384	145	6	quotient	quotient	NOUN
ejpam-3384	145	7	inequality	inequality	NOUN
ejpam-3384	145	8	to	to	ADP
ejpam-3384	145	9	lp	lp	PROPN
ejpam-3384	145	10	spaces	space	NOUN
ejpam-3384	145	11	theorem	theorem	VERB
ejpam-3384	145	12	4	4	NUM
ejpam-3384	145	13	.	.	PUNCT
ejpam-3384	145	14	suppose	suppose	VERB
ejpam-3384	145	15	that	that	SCONJ
ejpam-3384	145	16	f(x	f(x	PROPN
ejpam-3384	145	17	)	)	PUNCT
ejpam-3384	145	18	and	and	CCONJ
ejpam-3384	145	19	g(x	g(x	NOUN
ejpam-3384	145	20	)	)	PUNCT
ejpam-3384	145	21	are	be	AUX
ejpam-3384	145	22	measurable	measurable	ADJ
ejpam-3384	145	23	functions	function	NOUN
ejpam-3384	145	24	over	over	ADP
ejpam-3384	145	25	the	the	DET
ejpam-3384	145	26	domain	domain	NOUN
ejpam-3384	145	27	ω	ω	NOUN
ejpam-3384	145	28	such	such	ADJ
ejpam-3384	145	29	that	that	DET
ejpam-3384	145	30	∫	∫	PROPN
ejpam-3384	145	31	ω	ω	PROPN
ejpam-3384	145	32	|f(x)|dx	|f(x)|dx	PRON
ejpam-3384	145	33	≤	≤	X
ejpam-3384	146	1	+	+	SYM
ejpam-3384	146	2	∞.	∞.	PROPN
ejpam-3384	146	3	and	and	CCONJ
ejpam-3384	146	4	∫	∫	PROPN
ejpam-3384	146	5	ω	ω	PROPN
ejpam-3384	146	6	|g(x)|dx	|g(x)|dx	PROPN
ejpam-3384	146	7	≤	≤	NUM
ejpam-3384	146	8	1	1	NUM
ejpam-3384	146	9	,	,	PUNCT
ejpam-3384	146	10	then	then	ADV
ejpam-3384	146	11	‖f‖p‖g‖p	‖f‖p‖g‖p	PRON
ejpam-3384	146	12	≤	≤	NUM
ejpam-3384	146	13	‖f‖p	‖f‖p	NOUN
ejpam-3384	146	14	‖g‖p	‖g‖p	NOUN
ejpam-3384	146	15	.	.	PUNCT
ejpam-3384	147	1	proof	proof	NOUN
ejpam-3384	147	2	:	:	PUNCT
ejpam-3384	147	3	we	we	PRON
ejpam-3384	147	4	observe	observe	VERB
ejpam-3384	147	5	that:∣∣∣	that:∣∣∣	PROPN
ejpam-3384	147	6	∫	∫	PROPN
ejpam-3384	147	7	ω	ω	PROPN
ejpam-3384	147	8	f(x)g(x)dx	f(x)g(x)dx	VERB
ejpam-3384	147	9	∣∣∣	∣∣∣	ADJ
ejpam-3384	147	10	≤	≤	NUM
ejpam-3384	147	11	∫	∫	PROPN
ejpam-3384	147	12	ω	ω	NUM
ejpam-3384	147	13	|f(x)||g(x)|dx	|f(x)||g(x)|dx	PROPN
ejpam-3384	147	14	.	.	PUNCT
ejpam-3384	148	1	applying	apply	VERB
ejpam-3384	148	2	the	the	DET
ejpam-3384	148	3	first	first	ADJ
ejpam-3384	148	4	quotient	quotient	NOUN
ejpam-3384	148	5	inequality	inequality	NOUN
ejpam-3384	148	6	to	to	ADP
ejpam-3384	148	7	the	the	DET
ejpam-3384	148	8	term	term	NOUN
ejpam-3384	148	9	on	on	ADP
ejpam-3384	148	10	the	the	DET
ejpam-3384	148	11	right	right	ADJ
ejpam-3384	148	12	hand	hand	NOUN
ejpam-3384	148	13	side	side	NOUN
ejpam-3384	148	14	of	of	ADP
ejpam-3384	148	15	the	the	DET
ejpam-3384	148	16	above	above	ADJ
ejpam-3384	148	17	inequality	inequality	NOUN
ejpam-3384	148	18	,	,	PUNCT
ejpam-3384	148	19	we	we	PRON
ejpam-3384	148	20	obtain	obtain	VERB
ejpam-3384	148	21	∣∣∣	∣∣∣	ADJ
ejpam-3384	148	22	∫	∫	PROPN
ejpam-3384	148	23	ω	ω	PROPN
ejpam-3384	148	24	f(x)g(x)dx	f(x)g(x)dx	NOUN
ejpam-3384	148	25	∣∣∣	∣∣∣	ADJ
ejpam-3384	148	26	≤	≤	NUM
ejpam-3384	148	27	∫	∫	PROPN
ejpam-3384	148	28	ω	ω	PROPN
ejpam-3384	148	29	|f(x)||g(x)|dx	|f(x)||g(x)|dx	PROPN
ejpam-3384	148	30	≤	≤	NUM
ejpam-3384	148	31	∫	∫	PROPN
ejpam-3384	148	32	ω	ω	PROPN
ejpam-3384	148	33	|f(x)|	|f(x)|	PROPN
ejpam-3384	148	34	|g(x)|	|g(x)|	PROPN
ejpam-3384	148	35	dx	dx	PROPN
ejpam-3384	148	36	⇒	⇒	PROPN
ejpam-3384	148	37	∫	∫	PROPN
ejpam-3384	148	38	ω	ω	PROPN
ejpam-3384	148	39	|f(x)||g(x)|dx	|f(x)||g(x)|dx	PROPN
ejpam-3384	148	40	≤	≤	NUM
ejpam-3384	148	41	∫	∫	PROPN
ejpam-3384	148	42	ω	ω	PROPN
ejpam-3384	148	43	|f(x)|	|f(x)|	PROPN
ejpam-3384	148	44	|g(x)|	|g(x)|	PROPN
ejpam-3384	148	45	dx	dx	PROPN
ejpam-3384	148	46	⇒	⇒	PROPN
ejpam-3384	148	47	∫	∫	PROPN
ejpam-3384	148	48	ω	ω	PROPN
ejpam-3384	148	49	|f(x)||g(x)|dx	|f(x)||g(x)|dx	PROPN
ejpam-3384	148	50	≤	≤	NUM
ejpam-3384	148	51	∫	∫	PROPN
ejpam-3384	149	1	ω	ω	PROPN
ejpam-3384	149	2	|f(x)|dx∫	|f(x)|dx∫	PROPN
ejpam-3384	149	3	ω	ω	PROPN
ejpam-3384	149	4	|g(x)|dx	|g(x)|dx	X
ejpam-3384	149	5	⇒	⇒	PROPN
ejpam-3384	149	6	(	(	PUNCT
ejpam-3384	149	7	∫	∫	PROPN
ejpam-3384	149	8	ω	ω	PROPN
ejpam-3384	149	9	|f(x)|pdx	|f(x)|pdx	NOUN
ejpam-3384	149	10	)	)	PUNCT
ejpam-3384	149	11	1	1	NUM
ejpam-3384	149	12	p	p	NOUN
ejpam-3384	149	13	(	(	PUNCT
ejpam-3384	149	14	∫	∫	PROPN
ejpam-3384	149	15	ω	ω	PROPN
ejpam-3384	149	16	|g(x)|pdx	|g(x)|pdx	PROPN
ejpam-3384	149	17	)	)	PUNCT
ejpam-3384	149	18	1	1	NUM
ejpam-3384	149	19	p	p	NOUN
ejpam-3384	149	20	=	=	PUNCT
ejpam-3384	149	21	(	(	PUNCT
ejpam-3384	149	22	∫	∫	PROPN
ejpam-3384	149	23	ω	ω	PROPN
ejpam-3384	149	24	|f(x)|pdx	|f(x)|pdx	NOUN
ejpam-3384	149	25	)	)	PUNCT
ejpam-3384	149	26	1	1	NUM
ejpam-3384	149	27	p	p	NOUN
ejpam-3384	149	28	(	(	PUNCT
ejpam-3384	149	29	∫	∫	PROPN
ejpam-3384	149	30	ω	ω	PROPN
ejpam-3384	149	31	|g(x)|pdx	|g(x)|pdx	PROPN
ejpam-3384	149	32	)	)	PUNCT
ejpam-3384	149	33	1	1	NUM
ejpam-3384	149	34	p	p	NOUN
ejpam-3384	149	35	⇒	⇒	NOUN
ejpam-3384	149	36	‖f‖p‖g‖p	‖f‖p‖g‖p	NUM
ejpam-3384	149	37	≤	≤	NUM
ejpam-3384	149	38	‖f‖p	‖f‖p	NOUN
ejpam-3384	149	39	‖g‖p	‖g‖p	NOUN
ejpam-3384	149	40	.	.	PUNCT
ejpam-3384	150	1	3.3	3.3	NUM
ejpam-3384	150	2	.	.	PUNCT
ejpam-3384	151	1	the	the	DET
ejpam-3384	151	2	applications	application	NOUN
ejpam-3384	151	3	of	of	ADP
ejpam-3384	151	4	the	the	DET
ejpam-3384	151	5	second	second	ADJ
ejpam-3384	151	6	quotient	quotient	NOUN
ejpam-3384	151	7	inequality	inequality	NOUN
ejpam-3384	151	8	to	to	ADP
ejpam-3384	151	9	lp	lp	PROPN
ejpam-3384	151	10	spaces	space	NOUN
ejpam-3384	151	11	in	in	ADP
ejpam-3384	151	12	this	this	DET
ejpam-3384	151	13	subsection	subsection	NOUN
ejpam-3384	151	14	,	,	PUNCT
ejpam-3384	151	15	the	the	DET
ejpam-3384	151	16	second	second	ADJ
ejpam-3384	151	17	quotient	quotient	NOUN
ejpam-3384	151	18	inequality	inequality	NOUN
ejpam-3384	151	19	is	be	AUX
ejpam-3384	151	20	used	use	VERB
ejpam-3384	151	21	to	to	PART
ejpam-3384	151	22	estimate	estimate	VERB
ejpam-3384	151	23	the	the	DET
ejpam-3384	151	24	integrals	integral	NOUN
ejpam-3384	151	25	in	in	ADP
ejpam-3384	151	26	lp	lp	ADJ
ejpam-3384	151	27	spaces	space	NOUN
ejpam-3384	151	28	.	.	PUNCT
ejpam-3384	152	1	theorem	theorem	ADJ
ejpam-3384	152	2	5	5	NUM
ejpam-3384	152	3	.	.	PUNCT
ejpam-3384	152	4	suppose	suppose	VERB
ejpam-3384	152	5	that	that	SCONJ
ejpam-3384	152	6	f(x	f(x	PROPN
ejpam-3384	152	7	)	)	PUNCT
ejpam-3384	152	8	and	and	CCONJ
ejpam-3384	152	9	g(x	g(x	NOUN
ejpam-3384	152	10	)	)	PUNCT
ejpam-3384	152	11	are	be	AUX
ejpam-3384	152	12	measurable	measurable	ADJ
ejpam-3384	152	13	functions	function	NOUN
ejpam-3384	152	14	over	over	ADP
ejpam-3384	152	15	the	the	DET
ejpam-3384	152	16	domain	domain	NOUN
ejpam-3384	152	17	ω	ω	NOUN
ejpam-3384	152	18	such	such	ADJ
ejpam-3384	152	19	that	that	DET
ejpam-3384	152	20	∫	∫	PROPN
ejpam-3384	152	21	ω	ω	PROPN
ejpam-3384	152	22	|f(x)|dx	|f(x)|dx	PROPN
ejpam-3384	152	23	≥	≥	PROPN
ejpam-3384	152	24	1	1	NUM
ejpam-3384	152	25	b.	b.	PROPN
ejpam-3384	152	26	barnes	barnes	PROPN
ejpam-3384	152	27	,	,	PUNCT
ejpam-3384	152	28	c.	c.	PROPN
ejpam-3384	152	29	sebil	sebil	PROPN
ejpam-3384	152	30	,	,	PUNCT
ejpam-3384	152	31	i.	i.	PROPN
ejpam-3384	152	32	k.	k.	PROPN
ejpam-3384	152	33	dontwi	dontwi	PROPN
ejpam-3384	152	34	/	/	SYM
ejpam-3384	152	35	eur	eur	PROPN
ejpam-3384	152	36	.	.	PUNCT
ejpam-3384	153	1	j.	j.	PROPN
ejpam-3384	153	2	pure	pure	PROPN
ejpam-3384	153	3	appl	appl	PROPN
ejpam-3384	153	4	.	.	PROPN
ejpam-3384	153	5	math	math	PROPN
ejpam-3384	153	6	,	,	PUNCT
ejpam-3384	153	7	12	12	NUM
ejpam-3384	153	8	(	(	PUNCT
ejpam-3384	153	9	2	2	NUM
ejpam-3384	153	10	)	)	PUNCT
ejpam-3384	153	11	(	(	PUNCT
ejpam-3384	153	12	2019	2019	NUM
ejpam-3384	153	13	)	)	PUNCT
ejpam-3384	153	14	,	,	PUNCT
ejpam-3384	153	15	469	469	NUM
ejpam-3384	153	16	-	-	SYM
ejpam-3384	153	17	485	485	NUM
ejpam-3384	153	18	476	476	NUM
ejpam-3384	153	19	and	and	CCONJ
ejpam-3384	153	20	∫	∫	PROPN
ejpam-3384	153	21	ω	ω	PROPN
ejpam-3384	153	22	|g(x)|dx	|g(x)|dx	X
ejpam-3384	153	23	≥	≥	PROPN
ejpam-3384	153	24	1	1	NUM
ejpam-3384	153	25	,	,	PUNCT
ejpam-3384	153	26	then	then	ADV
ejpam-3384	153	27	‖f‖p	‖f‖p	NOUN
ejpam-3384	153	28	‖g‖p	‖g‖p	VERB
ejpam-3384	153	29	≤	≤	NUM
ejpam-3384	153	30	‖f‖p‖g‖p	‖f‖p‖g‖p	PRON
ejpam-3384	153	31	.	.	PUNCT
ejpam-3384	154	1	proof	proof	NOUN
ejpam-3384	154	2	:	:	PUNCT
ejpam-3384	154	3	we	we	PRON
ejpam-3384	154	4	can	can	AUX
ejpam-3384	154	5	see	see	VERB
ejpam-3384	154	6	that	that	PRON
ejpam-3384	154	7	:	:	PUNCT
ejpam-3384	154	8	∣∣∣	∣∣∣	PROPN
ejpam-3384	154	9	∫	∫	PROPN
ejpam-3384	154	10	ω	ω	NUM
ejpam-3384	154	11	f(x	f(x	PROPN
ejpam-3384	154	12	)	)	PUNCT
ejpam-3384	154	13	g(x	g(x	NOUN
ejpam-3384	154	14	)	)	PUNCT
ejpam-3384	154	15	dx	dx	PROPN
ejpam-3384	155	1	∣∣∣	∣∣∣	PROPN
ejpam-3384	155	2	≤	≤	NUM
ejpam-3384	155	3	∫	∫	PROPN
ejpam-3384	155	4	ω	ω	PROPN
ejpam-3384	155	5	|f(x)|	|f(x)|	PROPN
ejpam-3384	155	6	|g(x)|	|g(x)|	PROPN
ejpam-3384	155	7	dx∣∣∣	dx∣∣∣	PROPN
ejpam-3384	155	8	∫	∫	PROPN
ejpam-3384	155	9	ω	ω	PROPN
ejpam-3384	155	10	f(x	f(x	PROPN
ejpam-3384	155	11	)	)	PUNCT
ejpam-3384	155	12	g(x	g(x	NOUN
ejpam-3384	155	13	)	)	PUNCT
ejpam-3384	155	14	dx	dx	PROPN
ejpam-3384	155	15	∣∣∣	∣∣∣	PROPN
ejpam-3384	155	16	≤	≤	NUM
ejpam-3384	155	17	∫	∫	PROPN
ejpam-3384	155	18	ω	ω	PROPN
ejpam-3384	155	19	|f(x)|dx∫	|f(x)|dx∫	PROPN
ejpam-3384	155	20	ω	ω	X
ejpam-3384	155	21	|g(x)|dx	|g(x)|dx	VERB
ejpam-3384	155	22	applying	apply	VERB
ejpam-3384	155	23	the	the	DET
ejpam-3384	155	24	second	second	ADJ
ejpam-3384	155	25	quotient	quotient	NOUN
ejpam-3384	155	26	inequality	inequality	NOUN
ejpam-3384	155	27	to	to	ADP
ejpam-3384	155	28	the	the	DET
ejpam-3384	155	29	term	term	NOUN
ejpam-3384	155	30	on	on	ADP
ejpam-3384	155	31	the	the	DET
ejpam-3384	155	32	right	right	ADJ
ejpam-3384	155	33	hand	hand	NOUN
ejpam-3384	155	34	side	side	NOUN
ejpam-3384	155	35	of	of	ADP
ejpam-3384	155	36	the	the	DET
ejpam-3384	155	37	above	above	ADJ
ejpam-3384	155	38	inequality	inequality	NOUN
ejpam-3384	155	39	yields	yield	NOUN
ejpam-3384	155	40	∣∣∣	∣∣∣	ADJ
ejpam-3384	155	41	∫	∫	PROPN
ejpam-3384	155	42	ω	ω	PROPN
ejpam-3384	155	43	f(x	f(x	PROPN
ejpam-3384	155	44	)	)	PUNCT
ejpam-3384	155	45	g(x	g(x	NOUN
ejpam-3384	155	46	)	)	PUNCT
ejpam-3384	155	47	dx	dx	PROPN
ejpam-3384	155	48	∣∣∣	∣∣∣	PROPN
ejpam-3384	155	49	≤	≤	NUM
ejpam-3384	155	50	∫	∫	PROPN
ejpam-3384	155	51	ω	ω	PROPN
ejpam-3384	155	52	|f(x)|dx∫	|f(x)|dx∫	PROPN
ejpam-3384	155	53	ω	ω	PROPN
ejpam-3384	155	54	|g(x)|dx	|g(x)|dx	PROPN
ejpam-3384	155	55	≤	≤	NUM
ejpam-3384	155	56	∫	∫	PROPN
ejpam-3384	155	57	ω	ω	PROPN
ejpam-3384	155	58	|f(x)||g(x)|dx	|f(x)||g(x)|dx	PROPN
ejpam-3384	155	59	⇒	⇒	PROPN
ejpam-3384	155	60	∫	∫	PROPN
ejpam-3384	155	61	ω	ω	PROPN
ejpam-3384	155	62	|f(x)|dx∫	|f(x)|dx∫	PROPN
ejpam-3384	155	63	ω	ω	PROPN
ejpam-3384	155	64	|g(x)|dx	|g(x)|dx	PROPN
ejpam-3384	155	65	≤	≤	NUM
ejpam-3384	155	66	∫	∫	PROPN
ejpam-3384	155	67	ω	ω	PROPN
ejpam-3384	155	68	|f(x)||g(x)|dx	|f(x)||g(x)|dx	PROPN
ejpam-3384	155	69	⇒	⇒	PROPN
ejpam-3384	155	70	(	(	PUNCT
ejpam-3384	155	71	∫	∫	PROPN
ejpam-3384	155	72	ω	ω	PROPN
ejpam-3384	155	73	|f(x)|pdx	|f(x)|pdx	NOUN
ejpam-3384	155	74	)	)	PUNCT
ejpam-3384	155	75	1	1	NUM
ejpam-3384	155	76	p	p	NOUN
ejpam-3384	155	77	(	(	PUNCT
ejpam-3384	155	78	∫	∫	PROPN
ejpam-3384	155	79	ω	ω	PROPN
ejpam-3384	155	80	|g(x)|pdx	|g(x)|pdx	PROPN
ejpam-3384	155	81	)	)	PUNCT
ejpam-3384	155	82	1	1	NUM
ejpam-3384	155	83	p	p	NOUN
ejpam-3384	155	84	=	=	PUNCT
ejpam-3384	155	85	(	(	PUNCT
ejpam-3384	155	86	∫	∫	PROPN
ejpam-3384	155	87	ω	ω	PROPN
ejpam-3384	155	88	|f(x)|pdx	|f(x)|pdx	NOUN
ejpam-3384	155	89	)	)	PUNCT
ejpam-3384	155	90	1	1	NUM
ejpam-3384	155	91	p	p	NOUN
ejpam-3384	155	92	(	(	PUNCT
ejpam-3384	155	93	∫	∫	PROPN
ejpam-3384	155	94	ω	ω	PROPN
ejpam-3384	155	95	|g(x)|pdx	|g(x)|pdx	PROPN
ejpam-3384	155	96	)	)	PUNCT
ejpam-3384	155	97	1	1	NUM
ejpam-3384	155	98	p	p	NOUN
ejpam-3384	155	99	⇒	⇒	NOUN
ejpam-3384	155	100	‖f‖p	‖f‖p	NOUN
ejpam-3384	155	101	‖g‖p	‖g‖p	VERB
ejpam-3384	155	102	≤	≤	PUNCT
ejpam-3384	155	103	‖f‖p‖g‖p	‖f‖p‖g‖p	PRON
ejpam-3384	155	104	.	.	PUNCT
ejpam-3384	156	1	corollary	corollary	ADJ
ejpam-3384	156	2	1	1	PROPN
ejpam-3384	156	3	.	.	PUNCT
ejpam-3384	156	4	suppose	suppose	VERB
ejpam-3384	156	5	that	that	SCONJ
ejpam-3384	156	6	f(x	f(x	PROPN
ejpam-3384	156	7	)	)	PUNCT
ejpam-3384	156	8	and	and	CCONJ
ejpam-3384	156	9	g(x	g(x	NOUN
ejpam-3384	156	10	)	)	PUNCT
ejpam-3384	156	11	are	be	AUX
ejpam-3384	156	12	any	any	DET
ejpam-3384	156	13	two	two	NUM
ejpam-3384	156	14	measurable	measurable	ADJ
ejpam-3384	156	15	functions	function	NOUN
ejpam-3384	156	16	defined	define	VERB
ejpam-3384	156	17	on	on	ADP
ejpam-3384	156	18	r	r	NOUN
ejpam-3384	156	19	with	with	ADP
ejpam-3384	156	20	‖f(x)‖	‖f(x)‖	ADJ
ejpam-3384	156	21	≤	≤	NUM
ejpam-3384	156	22	1	1	NUM
ejpam-3384	156	23	and	and	CCONJ
ejpam-3384	156	24	‖g(x)‖	‖g(x)‖	PUNCT
ejpam-3384	156	25	≤	≤	NUM
ejpam-3384	156	26	1	1	NUM
ejpam-3384	156	27	,	,	PUNCT
ejpam-3384	157	1	then∫	then∫	NOUN
ejpam-3384	157	2	b	b	PROPN
ejpam-3384	157	3	a	a	DET
ejpam-3384	157	4	‖f(x)‖	‖f(x)‖	NOUN
ejpam-3384	157	5	‖g(x)‖	‖g(x)‖	NOUN
ejpam-3384	157	6	dx	dx	PROPN
ejpam-3384	157	7	≤	≤	PROPN
ejpam-3384	158	1	∫	∫	PROPN
ejpam-3384	159	1	b	b	PROPN
ejpam-3384	159	2	a	a	DET
ejpam-3384	159	3	‖f(x)‖‖g(x)‖dx	‖f(x)‖‖g(x)‖dx	PROPN
ejpam-3384	159	4	.	.	PUNCT
ejpam-3384	160	1	proof	proof	NOUN
ejpam-3384	160	2	:	:	PUNCT
ejpam-3384	160	3	we	we	PRON
ejpam-3384	160	4	observe	observe	VERB
ejpam-3384	160	5	that:∣∣∣	that:∣∣∣	PROPN
ejpam-3384	160	6	∫	∫	PROPN
ejpam-3384	160	7	b	b	PROPN
ejpam-3384	160	8	a	a	DET
ejpam-3384	160	9	f(x	f(x	PROPN
ejpam-3384	160	10	)	)	PUNCT
ejpam-3384	160	11	g(x	g(x	NOUN
ejpam-3384	160	12	)	)	PUNCT
ejpam-3384	160	13	dx	dx	PROPN
ejpam-3384	160	14	∣∣∣	∣∣∣	PROPN
ejpam-3384	160	15	≤	≤	NUM
ejpam-3384	161	1	∫	∫	PROPN
ejpam-3384	161	2	b	b	PROPN
ejpam-3384	161	3	a	a	DET
ejpam-3384	161	4	∣∣∣f(x	∣∣∣f(x	NOUN
ejpam-3384	161	5	)	)	PUNCT
ejpam-3384	161	6	g(x	g(x	NOUN
ejpam-3384	161	7	)	)	PUNCT
ejpam-3384	162	1	∣∣∣dx	∣∣∣dx	PRON
ejpam-3384	162	2	⇒	⇒	VERB
ejpam-3384	162	3	∣∣∣	∣∣∣	PROPN
ejpam-3384	162	4	∫	∫	PROPN
ejpam-3384	162	5	b	b	PROPN
ejpam-3384	162	6	a	a	DET
ejpam-3384	162	7	f(x	f(x	PROPN
ejpam-3384	162	8	)	)	PUNCT
ejpam-3384	162	9	g(x	g(x	NOUN
ejpam-3384	162	10	)	)	PUNCT
ejpam-3384	162	11	dx	dx	PROPN
ejpam-3384	162	12	∣∣∣	∣∣∣	NOUN
ejpam-3384	162	13	=	=	SYM
ejpam-3384	162	14	∫	∫	PROPN
ejpam-3384	162	15	b	b	PROPN
ejpam-3384	162	16	a	a	DET
ejpam-3384	162	17	‖f(x)‖	‖f(x)‖	NOUN
ejpam-3384	162	18	‖g(x)‖	‖g(x)‖	NOUN
ejpam-3384	162	19	dx	dx	PROPN
ejpam-3384	162	20	≤	≤	PROPN
ejpam-3384	162	21	∫	∫	PROPN
ejpam-3384	163	1	b	b	PROPN
ejpam-3384	163	2	a	a	PRON
ejpam-3384	163	3	‖f(x)‖‖g(x)‖dx	‖f(x)‖‖g(x)‖dx	X
ejpam-3384	163	4	⇒	⇒	PROPN
ejpam-3384	163	5	∫	∫	PROPN
ejpam-3384	163	6	b	b	PROPN
ejpam-3384	163	7	a	a	DET
ejpam-3384	163	8	‖f(x)‖	‖f(x)‖	NOUN
ejpam-3384	163	9	‖g(x)‖	‖g(x)‖	NOUN
ejpam-3384	163	10	dx	dx	PROPN
ejpam-3384	163	11	≤	≤	PROPN
ejpam-3384	163	12	∫	∫	PROPN
ejpam-3384	163	13	b	b	PROPN
ejpam-3384	163	14	a	a	DET
ejpam-3384	163	15	‖f(x)‖‖g(x)‖dx	‖f(x)‖‖g(x)‖dx	PROPN
ejpam-3384	163	16	.	.	PUNCT
ejpam-3384	163	17	b.	b.	PROPN
ejpam-3384	163	18	barnes	barnes	PROPN
ejpam-3384	163	19	,	,	PUNCT
ejpam-3384	163	20	c.	c.	PROPN
ejpam-3384	163	21	sebil	sebil	PROPN
ejpam-3384	163	22	,	,	PUNCT
ejpam-3384	163	23	i.	i.	PROPN
ejpam-3384	163	24	k.	k.	PROPN
ejpam-3384	163	25	dontwi	dontwi	PROPN
ejpam-3384	163	26	/	/	SYM
ejpam-3384	163	27	eur	eur	PROPN
ejpam-3384	163	28	.	.	PUNCT
ejpam-3384	164	1	j.	j.	PROPN
ejpam-3384	164	2	pure	pure	PROPN
ejpam-3384	164	3	appl	appl	PROPN
ejpam-3384	164	4	.	.	PROPN
ejpam-3384	164	5	math	math	PROPN
ejpam-3384	164	6	,	,	PUNCT
ejpam-3384	164	7	12	12	NUM
ejpam-3384	164	8	(	(	PUNCT
ejpam-3384	164	9	2	2	NUM
ejpam-3384	164	10	)	)	PUNCT
ejpam-3384	164	11	(	(	PUNCT
ejpam-3384	164	12	2019	2019	NUM
ejpam-3384	164	13	)	)	PUNCT
ejpam-3384	164	14	,	,	PUNCT
ejpam-3384	164	15	469	469	NUM
ejpam-3384	164	16	-	-	SYM
ejpam-3384	164	17	485	485	NUM
ejpam-3384	164	18	477	477	NUM
ejpam-3384	164	19	corollary	corollary	ADJ
ejpam-3384	164	20	2	2	NUM
ejpam-3384	164	21	(	(	PUNCT
ejpam-3384	164	22	isotonic	isotonic	ADJ
ejpam-3384	164	23	linear	linear	PROPN
ejpam-3384	164	24	functional	functional	NOUN
ejpam-3384	164	25	)	)	PUNCT
ejpam-3384	164	26	.	.	PUNCT
ejpam-3384	165	1	let	let	VERB
ejpam-3384	165	2	f	f	PROPN
ejpam-3384	165	3	(	(	PUNCT
ejpam-3384	165	4	t	t	PROPN
ejpam-3384	165	5	)	)	PUNCT
ejpam-3384	165	6	be	be	AUX
ejpam-3384	165	7	an	an	DET
ejpam-3384	165	8	algebra	algebra	NOUN
ejpam-3384	165	9	of	of	ADP
ejpam-3384	165	10	real	real	ADJ
ejpam-3384	165	11	functions	function	NOUN
ejpam-3384	165	12	defined	define	VERB
ejpam-3384	165	13	on	on	ADP
ejpam-3384	165	14	t	t	PROPN
ejpam-3384	165	15	and	and	CCONJ
ejpam-3384	165	16	l	l	NOUN
ejpam-3384	165	17	a	a	DET
ejpam-3384	165	18	subclass	subclass	NOUN
ejpam-3384	165	19	of	of	ADP
ejpam-3384	165	20	f	f	PROPN
ejpam-3384	165	21	(	(	PUNCT
ejpam-3384	165	22	t	t	PROPN
ejpam-3384	165	23	)	)	PUNCT
ejpam-3384	165	24	satisfying	satisfy	VERB
ejpam-3384	165	25	the	the	DET
ejpam-3384	165	26	axioms	axiom	NOUN
ejpam-3384	165	27	:	:	PUNCT
ejpam-3384	165	28	(	(	PUNCT
ejpam-3384	165	29	i	i	NOUN
ejpam-3384	165	30	)	)	PUNCT
ejpam-3384	166	1	f	f	X
ejpam-3384	166	2	,	,	PUNCT
ejpam-3384	166	3	g	g	PROPN
ejpam-3384	166	4	∈	∈	PROPN
ejpam-3384	166	5	l⇒	l⇒	PROPN
ejpam-3384	166	6	f	f	PROPN
ejpam-3384	167	1	+	+	CCONJ
ejpam-3384	167	2	g	g	PROPN
ejpam-3384	167	3	∈	∈	PROPN
ejpam-3384	167	4	l	l	NOUN
ejpam-3384	167	5	;	;	PUNCT
ejpam-3384	167	6	(	(	PUNCT
ejpam-3384	167	7	ii	ii	NOUN
ejpam-3384	167	8	)	)	PUNCT
ejpam-3384	167	9	f	f	PROPN
ejpam-3384	167	10	∈	∈	PROPN
ejpam-3384	167	11	l	l	NOUN
ejpam-3384	167	12	,	,	PUNCT
ejpam-3384	167	13	α	α	PROPN
ejpam-3384	167	14	∈	∈	NOUN
ejpam-3384	167	15	r⇒	r⇒	X
ejpam-3384	167	16	αf	αf	NOUN
ejpam-3384	167	17	∈	∈	PROPN
ejpam-3384	167	18	l.	l.	NOUN
ejpam-3384	167	19	a	a	DET
ejpam-3384	167	20	functional	functional	ADJ
ejpam-3384	167	21	a	a	PRON
ejpam-3384	167	22	defined	define	VERB
ejpam-3384	167	23	on	on	ADP
ejpam-3384	167	24	l	l	NOUN
ejpam-3384	167	25	is	be	AUX
ejpam-3384	167	26	an	an	DET
ejpam-3384	167	27	isotonic	isotonic	ADJ
ejpam-3384	167	28	linear	linear	ADJ
ejpam-3384	167	29	functional	functional	ADJ
ejpam-3384	167	30	on	on	ADP
ejpam-3384	167	31	l	l	NOUN
ejpam-3384	167	32	provided	provide	VERB
ejpam-3384	167	33	that	that	SCONJ
ejpam-3384	167	34	:	:	PUNCT
ejpam-3384	167	35	(	(	PUNCT
ejpam-3384	167	36	a)a(αf	a)a(αf	X
ejpam-3384	167	37	+	+	CCONJ
ejpam-3384	167	38	βg	βg	X
ejpam-3384	167	39	)	)	PUNCT
ejpam-3384	167	40	=	=	PUNCT
ejpam-3384	167	41	αa(f	αa(f	NUM
ejpam-3384	167	42	)	)	PUNCT
ejpam-3384	167	43	+	+	CCONJ
ejpam-3384	167	44	βa(g	βa(g	NUM
ejpam-3384	167	45	)	)	PUNCT
ejpam-3384	167	46	,	,	PUNCT
ejpam-3384	167	47	∀	∀	X
ejpam-3384	167	48	α	α	NOUN
ejpam-3384	167	49	,	,	PUNCT
ejpam-3384	167	50	β	β	X
ejpam-3384	167	51	∈	∈	NOUN
ejpam-3384	167	52	r	r	NOUN
ejpam-3384	167	53	and	and	CCONJ
ejpam-3384	167	54	f	f	NOUN
ejpam-3384	167	55	,	,	PUNCT
ejpam-3384	167	56	g	g	PROPN
ejpam-3384	167	57	∈	∈	PROPN
ejpam-3384	167	58	l	l	X
ejpam-3384	167	59	(	(	PUNCT
ejpam-3384	167	60	b	b	NOUN
ejpam-3384	167	61	)	)	PUNCT
ejpam-3384	167	62	f(t	f(t	PROPN
ejpam-3384	167	63	)	)	PUNCT
ejpam-3384	167	64	≥	≥	NOUN
ejpam-3384	167	65	g(t)⇒	g(t)⇒	CCONJ
ejpam-3384	167	66	a(f	a(f	PROPN
ejpam-3384	167	67	)	)	PUNCT
ejpam-3384	167	68	≥	≥	NOUN
ejpam-3384	167	69	a(g	a(g	PROPN
ejpam-3384	167	70	)	)	PUNCT
ejpam-3384	167	71	,	,	PUNCT
ejpam-3384	167	72	t	t	PROPN
ejpam-3384	167	73	∈	∈	PROPN
ejpam-3384	167	74	t.	t.	NOUN
ejpam-3384	167	75	proof	proof	NOUN
ejpam-3384	167	76	:	:	PUNCT
ejpam-3384	167	77	to	to	PART
ejpam-3384	167	78	prove	prove	VERB
ejpam-3384	167	79	the	the	DET
ejpam-3384	167	80	result	result	NOUN
ejpam-3384	167	81	in	in	ADP
ejpam-3384	167	82	corollary	corollary	ADJ
ejpam-3384	167	83	2(a	2(a	NUM
ejpam-3384	167	84	)	)	PUNCT
ejpam-3384	167	85	,	,	PUNCT
ejpam-3384	167	86	see	see	VERB
ejpam-3384	167	87	for	for	ADP
ejpam-3384	167	88	example	example	NOUN
ejpam-3384	167	89	,	,	PUNCT
ejpam-3384	167	90	author	author	NOUN
ejpam-3384	167	91	in	in	ADP
ejpam-3384	167	92	[	[	X
ejpam-3384	167	93	11	11	NUM
ejpam-3384	167	94	]	]	PUNCT
ejpam-3384	167	95	.	.	PUNCT
ejpam-3384	168	1	we	we	PRON
ejpam-3384	168	2	prove	prove	VERB
ejpam-3384	168	3	the	the	DET
ejpam-3384	168	4	finding	finding	NOUN
ejpam-3384	168	5	in	in	ADP
ejpam-3384	168	6	corollary	corollary	ADJ
ejpam-3384	168	7	2(b	2(b	NUM
ejpam-3384	168	8	)	)	PUNCT
ejpam-3384	168	9	by	by	ADP
ejpam-3384	168	10	setting	set	VERB
ejpam-3384	168	11	‖f(t)g(t)‖	‖f(t)g(t)‖	NOUN
ejpam-3384	168	12	≥	≥	NOUN
ejpam-3384	168	13	1	1	NUM
ejpam-3384	168	14	,	,	PUNCT
ejpam-3384	168	15	then	then	ADV
ejpam-3384	168	16	‖f(t)‖	‖f(t)‖	VERB
ejpam-3384	168	17	≥	≥	NOUN
ejpam-3384	168	18	1	1	NUM
ejpam-3384	168	19	‖g(t)‖	‖g(t)‖	NOUN
ejpam-3384	168	20	.	.	PUNCT
ejpam-3384	169	1	applying	apply	VERB
ejpam-3384	169	2	the	the	DET
ejpam-3384	169	3	second	second	ADJ
ejpam-3384	169	4	quotient	quotient	NOUN
ejpam-3384	169	5	inequality	inequality	NOUN
ejpam-3384	169	6	to	to	ADP
ejpam-3384	169	7	the	the	DET
ejpam-3384	169	8	right	right	ADJ
ejpam-3384	169	9	hand	hand	NOUN
ejpam-3384	169	10	side	side	NOUN
ejpam-3384	169	11	of	of	ADP
ejpam-3384	169	12	the	the	DET
ejpam-3384	169	13	above	above	ADJ
ejpam-3384	169	14	inequality	inequality	NOUN
ejpam-3384	169	15	yields	yield	NOUN
ejpam-3384	169	16	‖f(t)‖	‖f(t)‖	VERB
ejpam-3384	169	17	≥	≥	NOUN
ejpam-3384	169	18	1	1	NUM
ejpam-3384	169	19	‖g(t)‖	‖g(t)‖	NOUN
ejpam-3384	169	20	≤	≤	NUM
ejpam-3384	169	21	‖g(t)‖	‖g(t)‖	NOUN
ejpam-3384	169	22	⇒	⇒	PROPN
ejpam-3384	169	23	‖f(t)‖	‖f(t)‖	VERB
ejpam-3384	169	24	≥	≥	NOUN
ejpam-3384	169	25	‖g(t)‖.	‖g(t)‖.	NUM
ejpam-3384	169	26	this	this	PRON
ejpam-3384	169	27	completes	complete	VERB
ejpam-3384	169	28	the	the	DET
ejpam-3384	169	29	proof	proof	NOUN
ejpam-3384	169	30	.	.	PUNCT
ejpam-3384	170	1	3.4	3.4	NUM
ejpam-3384	170	2	.	.	PUNCT
ejpam-3384	170	3	using	use	VERB
ejpam-3384	170	4	the	the	DET
ejpam-3384	170	5	quotient	quotient	NOUN
ejpam-3384	170	6	inequalities	inequality	NOUN
ejpam-3384	170	7	to	to	PART
ejpam-3384	170	8	obtain	obtain	VERB
ejpam-3384	170	9	the	the	DET
ejpam-3384	170	10	estimate	estimate	NOUN
ejpam-3384	170	11	of	of	ADP
ejpam-3384	170	12	hahnbanach	hahnbanach	ADJ
ejpam-3384	170	13	contraction	contraction	NOUN
ejpam-3384	170	14	mapping	mapping	NOUN
ejpam-3384	170	15	theorem	theorem	VERB
ejpam-3384	170	16	in	in	ADP
ejpam-3384	170	17	this	this	DET
ejpam-3384	170	18	section	section	NOUN
ejpam-3384	170	19	,	,	PUNCT
ejpam-3384	170	20	the	the	DET
ejpam-3384	170	21	first	first	ADJ
ejpam-3384	170	22	product	product	NOUN
ejpam-3384	170	23	inequality	inequality	NOUN
ejpam-3384	170	24	is	be	AUX
ejpam-3384	170	25	used	use	VERB
ejpam-3384	170	26	to	to	PART
ejpam-3384	170	27	obtain	obtain	VERB
ejpam-3384	170	28	an	an	DET
ejpam-3384	170	29	alternative	alternative	ADJ
ejpam-3384	170	30	way	way	NOUN
ejpam-3384	170	31	for	for	ADP
ejpam-3384	170	32	proving	prove	VERB
ejpam-3384	170	33	hahn	hahn	PROPN
ejpam-3384	170	34	banach	banach	NOUN
ejpam-3384	170	35	contraction	contraction	NOUN
ejpam-3384	170	36	mapping	mapping	NOUN
ejpam-3384	170	37	theorem	theorem	NOUN
ejpam-3384	170	38	.	.	PUNCT
ejpam-3384	171	1	proposition	proposition	NOUN
ejpam-3384	171	2	1	1	NUM
ejpam-3384	171	3	.	.	PUNCT
ejpam-3384	172	1	a	a	DET
ejpam-3384	172	2	contraction	contraction	NOUN
ejpam-3384	172	3	mapping	mapping	NOUN
ejpam-3384	172	4	t	t	NOUN
ejpam-3384	172	5	,	,	PUNCT
ejpam-3384	172	6	defined	define	VERB
ejpam-3384	172	7	on	on	ADP
ejpam-3384	172	8	a	a	DET
ejpam-3384	172	9	complete	complete	ADJ
ejpam-3384	172	10	normed	norme	VERB
ejpam-3384	172	11	linear	linear	ADJ
ejpam-3384	172	12	space	space	NOUN
ejpam-3384	172	13	,	,	PUNCT
ejpam-3384	172	14	has	have	VERB
ejpam-3384	172	15	unique	unique	ADJ
ejpam-3384	172	16	fixed	fix	VERB
ejpam-3384	172	17	point	point	NOUN
ejpam-3384	172	18	.	.	PUNCT
ejpam-3384	173	1	proof	proof	NOUN
ejpam-3384	173	2	:	:	PUNCT
ejpam-3384	173	3	setting	set	VERB
ejpam-3384	173	4	a	a	DET
ejpam-3384	173	5	mapping	mapping	NOUN
ejpam-3384	173	6	t	t	NOUN
ejpam-3384	173	7	:	:	PUNCT
ejpam-3384	173	8	x	x	X
ejpam-3384	173	9	→	→	SYM
ejpam-3384	173	10	x	x	PART
ejpam-3384	173	11	,	,	PUNCT
ejpam-3384	173	12	and	and	CCONJ
ejpam-3384	173	13	let	let	VERB
ejpam-3384	173	14	xo	xo	PROPN
ejpam-3384	173	15	∈	∈	PROPN
ejpam-3384	173	16	x	x	PUNCT
ejpam-3384	173	17	such	such	ADJ
ejpam-3384	173	18	that	that	DET
ejpam-3384	173	19	txn−1	txn−1	PROPN
ejpam-3384	173	20	=	=	SYM
ejpam-3384	173	21	xn	xn	PROPN
ejpam-3384	173	22	,	,	PUNCT
ejpam-3384	173	23	∀	∀	X
ejpam-3384	173	24	n	n	NOUN
ejpam-3384	173	25	=	=	SYM
ejpam-3384	173	26	1	1	NUM
ejpam-3384	173	27	,	,	PUNCT
ejpam-3384	173	28	2	2	NUM
ejpam-3384	173	29	,	,	PUNCT
ejpam-3384	173	30	.	.	PUNCT
ejpam-3384	173	31	.	.	PUNCT
ejpam-3384	173	32	.	.	PUNCT
ejpam-3384	173	33	.	.	PUNCT
ejpam-3384	174	1	for	for	ADP
ejpam-3384	174	2	any	any	DET
ejpam-3384	174	3	positive	positive	ADJ
ejpam-3384	174	4	integer	integer	NOUN
ejpam-3384	174	5	n	n	CCONJ
ejpam-3384	174	6	,	,	PUNCT
ejpam-3384	174	7	then	then	ADV
ejpam-3384	174	8	‖xm+1	‖xm+1	ADP
ejpam-3384	174	9	−	−	PROPN
ejpam-3384	174	10	xm‖	xm‖	PROPN
ejpam-3384	175	1	=	=	PUNCT
ejpam-3384	175	2	‖txm	‖txm	PROPN
ejpam-3384	175	3	−	−	X
ejpam-3384	176	1	txm−1‖	txm−1‖	NOUN
ejpam-3384	176	2	≤	≤	PUNCT
ejpam-3384	176	3	α‖xm	α‖xm	PROPN
ejpam-3384	176	4	−	−	PROPN
ejpam-3384	176	5	xm−1‖	xm−1‖	PROPN
ejpam-3384	176	6	...	...	PUNCT
ejpam-3384	177	1	‖xm+1	‖xm+1	PUNCT
ejpam-3384	177	2	−	−	PROPN
ejpam-3384	177	3	xm‖	xm‖	PROPN
ejpam-3384	177	4	≤	≤	PROPN
ejpam-3384	177	5	αm‖x1	αm‖x1	NOUN
ejpam-3384	177	6	−	−	PROPN
ejpam-3384	177	7	xo‖	xo‖	PROPN
ejpam-3384	178	1	‖xm+1	‖xm+1	NUM
ejpam-3384	179	1	−	−	PROPN
ejpam-3384	179	2	xm‖	xm‖	PROPN
ejpam-3384	179	3	≤	≤	PROPN
ejpam-3384	179	4	(	(	PUNCT
ejpam-3384	179	5	αn+k−1	αn+k−1	NOUN
ejpam-3384	179	6	+	+	CCONJ
ejpam-3384	179	7	αn+k−2	αn+k−2	PROPN
ejpam-3384	179	8	+	+	X
ejpam-3384	179	9	.	.	PUNCT
ejpam-3384	179	10	.	.	PUNCT
ejpam-3384	180	1	.+	.+	NOUN
ejpam-3384	180	2	αn	αn	NOUN
ejpam-3384	180	3	)	)	PUNCT
ejpam-3384	180	4	‖x1	‖x1	NOUN
ejpam-3384	180	5	−	−	PROPN
ejpam-3384	180	6	xo‖.	xo‖.	PROPN
ejpam-3384	180	7	b.	b.	PROPN
ejpam-3384	180	8	barnes	barnes	PROPN
ejpam-3384	180	9	,	,	PUNCT
ejpam-3384	180	10	c.	c.	PROPN
ejpam-3384	180	11	sebil	sebil	PROPN
ejpam-3384	180	12	,	,	PUNCT
ejpam-3384	180	13	i.	i.	PROPN
ejpam-3384	180	14	k.	k.	PROPN
ejpam-3384	180	15	dontwi	dontwi	PROPN
ejpam-3384	181	1	/	/	SYM
ejpam-3384	181	2	eur	eur	PROPN
ejpam-3384	181	3	.	.	PUNCT
ejpam-3384	182	1	j.	j.	PROPN
ejpam-3384	182	2	pure	pure	PROPN
ejpam-3384	182	3	appl	appl	PROPN
ejpam-3384	182	4	.	.	PROPN
ejpam-3384	182	5	math	math	PROPN
ejpam-3384	182	6	,	,	PUNCT
ejpam-3384	182	7	12	12	NUM
ejpam-3384	182	8	(	(	PUNCT
ejpam-3384	182	9	2	2	NUM
ejpam-3384	182	10	)	)	PUNCT
ejpam-3384	182	11	(	(	PUNCT
ejpam-3384	182	12	2019	2019	NUM
ejpam-3384	182	13	)	)	PUNCT
ejpam-3384	182	14	,	,	PUNCT
ejpam-3384	182	15	469	469	NUM
ejpam-3384	182	16	-	-	SYM
ejpam-3384	182	17	485	485	NUM
ejpam-3384	182	18	478	478	NUM
ejpam-3384	182	19	where	where	SCONJ
ejpam-3384	182	20	,	,	PUNCT
ejpam-3384	182	21	α	α	X
ejpam-3384	182	22	,	,	PUNCT
ejpam-3384	182	23	is	be	AUX
ejpam-3384	182	24	the	the	DET
ejpam-3384	182	25	boundedness	boundedness	NOUN
ejpam-3384	182	26	constant	constant	ADJ
ejpam-3384	182	27	.	.	PUNCT
ejpam-3384	183	1	using	use	VERB
ejpam-3384	183	2	the	the	DET
ejpam-3384	183	3	first	first	ADJ
ejpam-3384	183	4	quotient	quotient	NOUN
ejpam-3384	183	5	inequality	inequality	NOUN
ejpam-3384	183	6	on	on	ADP
ejpam-3384	183	7	the	the	DET
ejpam-3384	183	8	right	right	ADJ
ejpam-3384	183	9	hand	hand	NOUN
ejpam-3384	183	10	side	side	NOUN
ejpam-3384	183	11	of	of	ADP
ejpam-3384	183	12	the	the	DET
ejpam-3384	183	13	above	above	ADJ
ejpam-3384	183	14	inequality	inequality	NOUN
ejpam-3384	183	15	yields	yield	NOUN
ejpam-3384	183	16	‖xm+1	‖xm+1	PUNCT
ejpam-3384	183	17	−	−	PROPN
ejpam-3384	183	18	xm‖	xm‖	PROPN
ejpam-3384	183	19	≤	≤	NUM
ejpam-3384	183	20	1	1	NUM
ejpam-3384	183	21	(	(	PUNCT
ejpam-3384	183	22	αn+k−1	αn+k−1	NOUN
ejpam-3384	183	23	+	+	CCONJ
ejpam-3384	183	24	αn+k−2	αn+k−2	NOUN
ejpam-3384	183	25	+	+	X
ejpam-3384	183	26	.	.	PUNCT
ejpam-3384	183	27	.	.	PUNCT
ejpam-3384	184	1	.+	.+	NOUN
ejpam-3384	184	2	αn	αn	NOUN
ejpam-3384	184	3	)	)	PUNCT
ejpam-3384	184	4	‖x1	‖x1	NOUN
ejpam-3384	184	5	−	−	NOUN
ejpam-3384	184	6	xo‖	xo‖	PROPN
ejpam-3384	185	1	‖xm+1	‖xm+1	NUM
ejpam-3384	186	1	−	−	PROPN
ejpam-3384	186	2	xm‖	xm‖	PROPN
ejpam-3384	187	1	=	=	SYM
ejpam-3384	188	1	(	(	PUNCT
ejpam-3384	188	2	1−	1−	NUM
ejpam-3384	188	3	α	α	NOUN
ejpam-3384	188	4	)	)	PUNCT
ejpam-3384	188	5	αn	αn	NOUN
ejpam-3384	188	6	‖x1	‖x1	NOUN
ejpam-3384	188	7	−	−	PROPN
ejpam-3384	188	8	xo‖.	xo‖.	NOUN
ejpam-3384	189	1	we	we	PRON
ejpam-3384	189	2	can	can	AUX
ejpam-3384	189	3	see	see	VERB
ejpam-3384	189	4	that	that	PRON
ejpam-3384	189	5	(	(	PUNCT
ejpam-3384	189	6	1−α	1−α	NUM
ejpam-3384	189	7	)	)	PUNCT
ejpam-3384	189	8	αn	αn	NOUN
ejpam-3384	189	9	→	→	SYM
ejpam-3384	189	10	0	0	NUM
ejpam-3384	189	11	as	as	ADP
ejpam-3384	189	12	n→∞	n→∞	NUM
ejpam-3384	189	13	for	for	ADP
ejpam-3384	189	14	all	all	DET
ejpam-3384	189	15	α	α	PRON
ejpam-3384	189	16	≥	≥	NUM
ejpam-3384	189	17	1	1	NUM
ejpam-3384	189	18	.	.	PUNCT
ejpam-3384	190	1	the	the	DET
ejpam-3384	190	2	sequence	sequence	NOUN
ejpam-3384	190	3	{	{	PUNCT
ejpam-3384	190	4	xn}∞n=1	xn}∞n=1	PROPN
ejpam-3384	190	5	is	be	AUX
ejpam-3384	190	6	convergent	convergent	NOUN
ejpam-3384	190	7	.	.	PUNCT
ejpam-3384	191	1	the	the	DET
ejpam-3384	191	2	normed	normed	ADJ
ejpam-3384	191	3	space	space	NOUN
ejpam-3384	191	4	x	x	PUNCT
ejpam-3384	191	5	is	be	AUX
ejpam-3384	191	6	complete	complete	ADJ
ejpam-3384	191	7	since	since	SCONJ
ejpam-3384	191	8	{	{	PUNCT
ejpam-3384	191	9	xn}∞n=1	xn}∞n=1	PROPN
ejpam-3384	191	10	has	have	VERB
ejpam-3384	191	11	a	a	DET
ejpam-3384	191	12	limit	limit	NOUN
ejpam-3384	191	13	point	point	NOUN
ejpam-3384	191	14	in	in	ADP
ejpam-3384	191	15	x.	x.	NOUN
ejpam-3384	191	16	let	let	VERB
ejpam-3384	191	17	x	x	PRON
ejpam-3384	191	18	be	be	AUX
ejpam-3384	191	19	the	the	DET
ejpam-3384	191	20	element	element	NOUN
ejpam-3384	191	21	of	of	ADP
ejpam-3384	191	22	x	x	SYM
ejpam-3384	191	23	such	such	ADJ
ejpam-3384	191	24	that	that	SCONJ
ejpam-3384	191	25	lim	lim	PROPN
ejpam-3384	191	26	n→∞	n→∞	X
ejpam-3384	191	27	xn	xn	PROPN
ejpam-3384	192	1	=	=	PUNCT
ejpam-3384	192	2	x.	x.	PUNCT
ejpam-3384	192	3	thus	thus	ADV
ejpam-3384	192	4	,	,	PUNCT
ejpam-3384	192	5	tx	tx	PROPN
ejpam-3384	192	6	=	=	SYM
ejpam-3384	192	7	t	t	PROPN
ejpam-3384	192	8	(	(	PUNCT
ejpam-3384	192	9	lim	lim	PROPN
ejpam-3384	192	10	n→∞	n→∞	NUM
ejpam-3384	192	11	xn	xn	X
ejpam-3384	192	12	)	)	PUNCT
ejpam-3384	192	13	tx	tx	PROPN
ejpam-3384	193	1	=	=	SYM
ejpam-3384	193	2	lim	lim	PROPN
ejpam-3384	193	3	n→∞	n→∞	NUM
ejpam-3384	193	4	txn	txn	NOUN
ejpam-3384	193	5	.	.	PUNCT
ejpam-3384	194	1	by	by	ADP
ejpam-3384	194	2	the	the	DET
ejpam-3384	194	3	continuity	continuity	NOUN
ejpam-3384	194	4	of	of	ADP
ejpam-3384	194	5	t	t	PROPN
ejpam-3384	194	6	.	.	PUNCT
ejpam-3384	195	1	we	we	PRON
ejpam-3384	195	2	can	can	AUX
ejpam-3384	195	3	see	see	VERB
ejpam-3384	195	4	that	that	PRON
ejpam-3384	195	5	:	:	PUNCT
ejpam-3384	195	6	lim	lim	PROPN
ejpam-3384	195	7	n→∞	n→∞	NUM
ejpam-3384	196	1	tyn	tyn	PROPN
ejpam-3384	196	2	=	=	SYM
ejpam-3384	196	3	lim	lim	PROPN
ejpam-3384	196	4	n→∞	n→∞	NUM
ejpam-3384	196	5	tyn+1	tyn+1	NOUN
ejpam-3384	196	6	=	=	SYM
ejpam-3384	196	7	y.	y.	NOUN
ejpam-3384	196	8	suppose	suppose	VERB
ejpam-3384	196	9	further	far	ADV
ejpam-3384	196	10	that	that	SCONJ
ejpam-3384	196	11	ty1	ty1	NOUN
ejpam-3384	196	12	=	=	PUNCT
ejpam-3384	196	13	y1	y1	NOUN
ejpam-3384	196	14	and	and	CCONJ
ejpam-3384	196	15	ty2	ty2	NOUN
ejpam-3384	196	16	=	=	ADJ
ejpam-3384	196	17	y2	y2	PROPN
ejpam-3384	196	18	.	.	PUNCT
ejpam-3384	197	1	then	then	ADV
ejpam-3384	197	2	‖y1	‖y1	PRON
ejpam-3384	197	3	−	−	PROPN
ejpam-3384	197	4	y2‖	y2‖	PROPN
ejpam-3384	197	5	=	=	SYM
ejpam-3384	197	6	‖ty1	‖ty1	PROPN
ejpam-3384	197	7	−	−	PROPN
ejpam-3384	198	1	ty2‖	ty2‖	PROPN
ejpam-3384	199	1	‖y1	‖y1	PRON
ejpam-3384	199	2	−	−	PROPN
ejpam-3384	199	3	y2‖	y2‖	PROPN
ejpam-3384	199	4	≤	≤	PUNCT
ejpam-3384	200	1	‖y1	‖y1	PRON
ejpam-3384	200	2	−	−	PROPN
ejpam-3384	201	1	y2‖	y2‖	PROPN
ejpam-3384	202	1	‖y1	‖y1	PRON
ejpam-3384	202	2	−	−	PROPN
ejpam-3384	203	1	y2‖	y2‖	X
ejpam-3384	203	2	<	<	X
ejpam-3384	204	1	‖y1	‖y1	PRON
ejpam-3384	204	2	−	−	PROPN
ejpam-3384	204	3	y2‖	y2‖	PROPN
ejpam-3384	204	4	,	,	PUNCT
ejpam-3384	204	5	which	which	PRON
ejpam-3384	204	6	is	be	AUX
ejpam-3384	204	7	a	a	DET
ejpam-3384	204	8	contradiction	contradiction	NOUN
ejpam-3384	204	9	.	.	PUNCT
ejpam-3384	205	1	thus	thus	ADV
ejpam-3384	205	2	,	,	PUNCT
ejpam-3384	205	3	the	the	DET
ejpam-3384	205	4	fixed	fix	VERB
ejpam-3384	205	5	point	point	NOUN
ejpam-3384	205	6	theorem	theorem	NOUN
ejpam-3384	205	7	is	be	AUX
ejpam-3384	205	8	unique	unique	ADJ
ejpam-3384	205	9	.	.	PUNCT
ejpam-3384	206	1	3.5	3.5	NUM
ejpam-3384	206	2	.	.	PUNCT
ejpam-3384	207	1	using	use	VERB
ejpam-3384	207	2	the	the	DET
ejpam-3384	207	3	product	product	NOUN
ejpam-3384	207	4	and	and	CCONJ
ejpam-3384	207	5	quotient	quotient	NOUN
ejpam-3384	207	6	inequalities	inequality	NOUN
ejpam-3384	207	7	to	to	PART
ejpam-3384	207	8	obtain	obtain	VERB
ejpam-3384	207	9	sharp	sharp	ADJ
ejpam-3384	207	10	inequalities	inequality	NOUN
ejpam-3384	207	11	in	in	ADP
ejpam-3384	207	12	sobolev	sobolev	NOUN
ejpam-3384	207	13	spaces	space	NOUN
ejpam-3384	207	14	lemma	lemma	PROPN
ejpam-3384	207	15	1	1	NUM
ejpam-3384	207	16	.	.	PUNCT
ejpam-3384	208	1	for	for	ADP
ejpam-3384	208	2	any	any	DET
ejpam-3384	208	3	1	1	NUM
ejpam-3384	208	4	≤	≤	NOUN
ejpam-3384	208	5	p	p	NOUN
ejpam-3384	208	6	<	<	X
ejpam-3384	208	7	n	n	CCONJ
ejpam-3384	208	8	,	,	PUNCT
ejpam-3384	208	9	w	w	PROPN
ejpam-3384	208	10	1,p(rn	1,p(rn	NUM
ejpam-3384	208	11	)	)	PUNCT
ejpam-3384	208	12	↪	↪	PROPN
ejpam-3384	208	13	→	→	SYM
ejpam-3384	208	14	lr(rn	lr(rn	PROPN
ejpam-3384	208	15	)	)	PUNCT
ejpam-3384	208	16	is	be	AUX
ejpam-3384	208	17	continuously	continuously	ADV
ejpam-3384	208	18	imbedded	imbed	VERB
ejpam-3384	208	19	,	,	PUNCT
ejpam-3384	208	20	∀	∀	X
ejpam-3384	208	21	α	α	NOUN
ejpam-3384	208	22	∈	∈	PROPN
ejpam-3384	208	23	(	(	PUNCT
ejpam-3384	208	24	0	0	NUM
ejpam-3384	208	25	,	,	PUNCT
ejpam-3384	208	26	1	1	NUM
ejpam-3384	208	27	]	]	PUNCT
ejpam-3384	208	28	and	and	CCONJ
ejpam-3384	208	29	‖u‖	‖u‖	PROPN
ejpam-3384	208	30	≥	≥	NOUN
ejpam-3384	208	31	2	2	NUM
ejpam-3384	208	32	.	.	PUNCT
ejpam-3384	209	1	proof	proof	NOUN
ejpam-3384	209	2	:	:	PUNCT
ejpam-3384	209	3	setting	set	VERB
ejpam-3384	209	4	α	α	PROPN
ejpam-3384	209	5	∈	∈	PROPN
ejpam-3384	209	6	(	(	PUNCT
ejpam-3384	209	7	0	0	NUM
ejpam-3384	209	8	,	,	PUNCT
ejpam-3384	209	9	1	1	NUM
ejpam-3384	209	10	]	]	PUNCT
ejpam-3384	209	11	and	and	CCONJ
ejpam-3384	209	12	u	u	PROPN
ejpam-3384	209	13	∈w	∈w	NOUN
ejpam-3384	209	14	1,p(rn	1,p(rn	PROPN
ejpam-3384	209	15	)	)	PUNCT
ejpam-3384	209	16	,	,	PUNCT
ejpam-3384	209	17	we	we	PRON
ejpam-3384	209	18	see	see	VERB
ejpam-3384	209	19	that	that	SCONJ
ejpam-3384	209	20	u	u	PROPN
ejpam-3384	209	21	∈	∈	NOUN
ejpam-3384	209	22	lp∗(rn	lp∗(rn	PROPN
ejpam-3384	209	23	)	)	PUNCT
ejpam-3384	209	24	.	.	PUNCT
ejpam-3384	210	1	then	then	ADV
ejpam-3384	210	2	‖u‖rr	‖u‖rr	VERB
ejpam-3384	210	3	≤	≤	NOUN
ejpam-3384	210	4	α	α	PRON
ejpam-3384	210	5	∫	∫	PROPN
ejpam-3384	210	6	rn	rn	PROPN
ejpam-3384	210	7	(	(	PUNCT
ejpam-3384	210	8	|u|r	|u|r	PROPN
ejpam-3384	210	9	+	+	CCONJ
ejpam-3384	210	10	|u|r)dx	|u|r)dx	VERB
ejpam-3384	210	11	‖u‖rr	‖u‖rr	ADP
ejpam-3384	210	12	=	=	SYM
ejpam-3384	210	13	α	α	PROPN
ejpam-3384	210	14	(	(	PUNCT
ejpam-3384	210	15	∫	∫	PROPN
ejpam-3384	210	16	rn	rn	PROPN
ejpam-3384	210	17	|u|rdx+	|u|rdx+	PROPN
ejpam-3384	210	18	∫	∫	PROPN
ejpam-3384	210	19	rn	rn	PROPN
ejpam-3384	210	20	|u|rdx	|u|rdx	PUNCT
ejpam-3384	210	21	)	)	PUNCT
ejpam-3384	210	22	‖u‖rr	‖u‖rr	VERB
ejpam-3384	210	23	≤	≤	NOUN
ejpam-3384	211	1	α	α	PRON
ejpam-3384	211	2	(	(	PUNCT
ejpam-3384	211	3	(	(	PUNCT
ejpam-3384	211	4	∫	∫	PROPN
ejpam-3384	211	5	rn	rn	PROPN
ejpam-3384	211	6	|u|pdx	|u|pdx	NUM
ejpam-3384	211	7	)	)	PUNCT
ejpam-3384	211	8	r	r	NOUN
ejpam-3384	211	9	p	p	NOUN
ejpam-3384	212	1	+	+	CCONJ
ejpam-3384	212	2	(	(	PUNCT
ejpam-3384	212	3	∫	∫	PROPN
ejpam-3384	212	4	rn	rn	PROPN
ejpam-3384	212	5	|u|p∗dx	|u|p∗dx	PROPN
ejpam-3384	212	6	)	)	PUNCT
ejpam-3384	212	7	r	r	NOUN
ejpam-3384	212	8	p∗	p∗	PROPN
ejpam-3384	212	9	)	)	PUNCT
ejpam-3384	212	10	b.	b.	PROPN
ejpam-3384	212	11	barnes	barnes	PROPN
ejpam-3384	212	12	,	,	PUNCT
ejpam-3384	212	13	c.	c.	PROPN
ejpam-3384	212	14	sebil	sebil	PROPN
ejpam-3384	212	15	,	,	PUNCT
ejpam-3384	212	16	i.	i.	PROPN
ejpam-3384	212	17	k.	k.	PROPN
ejpam-3384	212	18	dontwi	dontwi	PROPN
ejpam-3384	212	19	/	/	SYM
ejpam-3384	212	20	eur	eur	PROPN
ejpam-3384	212	21	.	.	PUNCT
ejpam-3384	213	1	j.	j.	PROPN
ejpam-3384	213	2	pure	pure	PROPN
ejpam-3384	213	3	appl	appl	PROPN
ejpam-3384	213	4	.	.	PROPN
ejpam-3384	213	5	math	math	PROPN
ejpam-3384	213	6	,	,	PUNCT
ejpam-3384	213	7	12	12	NUM
ejpam-3384	213	8	(	(	PUNCT
ejpam-3384	213	9	2	2	NUM
ejpam-3384	213	10	)	)	PUNCT
ejpam-3384	213	11	(	(	PUNCT
ejpam-3384	213	12	2019	2019	NUM
ejpam-3384	213	13	)	)	PUNCT
ejpam-3384	213	14	,	,	PUNCT
ejpam-3384	213	15	469	469	NUM
ejpam-3384	213	16	-	-	SYM
ejpam-3384	213	17	485	485	NUM
ejpam-3384	213	18	479	479	NUM
ejpam-3384	213	19	‖u‖r	‖u‖r	NOUN
ejpam-3384	213	20	≤	≤	ADJ
ejpam-3384	213	21	α	α	NOUN
ejpam-3384	213	22	(	(	PUNCT
ejpam-3384	213	23	‖u‖p	‖u‖p	NOUN
ejpam-3384	213	24	+	+	CCONJ
ejpam-3384	213	25	‖u‖p∗	‖u‖p∗	NUM
ejpam-3384	213	26	)	)	PUNCT
ejpam-3384	213	27	.	.	PUNCT
ejpam-3384	214	1	using	use	VERB
ejpam-3384	214	2	the	the	DET
ejpam-3384	214	3	second	second	ADJ
ejpam-3384	214	4	product	product	NOUN
ejpam-3384	214	5	inequality	inequality	NOUN
ejpam-3384	214	6	,	,	PUNCT
ejpam-3384	214	7	we	we	PRON
ejpam-3384	214	8	obtain	obtain	VERB
ejpam-3384	214	9	‖u‖r	‖u‖r	NOUN
ejpam-3384	214	10	≤	≤	ADJ
ejpam-3384	214	11	α	α	PROPN
ejpam-3384	214	12	(	(	PUNCT
ejpam-3384	214	13	‖u‖p‖u‖p∗	‖u‖p‖u‖p∗	PROPN
ejpam-3384	214	14	)	)	PUNCT
ejpam-3384	214	15	⇒	⇒	NOUN
ejpam-3384	214	16	‖u‖r	‖u‖r	VERB
ejpam-3384	214	17	≤	≤	ADJ
ejpam-3384	214	18	α	α	PROPN
ejpam-3384	214	19	(	(	PUNCT
ejpam-3384	214	20	c‖u‖p‖∇u‖p∗	c‖u‖p‖∇u‖p∗	ADV
ejpam-3384	214	21	)	)	PUNCT
ejpam-3384	214	22	⇒	⇒	NOUN
ejpam-3384	214	23	‖u‖r	‖u‖r	VERB
ejpam-3384	214	24	≤	≤	ADJ
ejpam-3384	214	25	αc‖u‖21,p	αc‖u‖21,p	NOUN
ejpam-3384	214	26	.	.	PUNCT
ejpam-3384	215	1	this	this	PRON
ejpam-3384	215	2	completes	complete	VERB
ejpam-3384	215	3	the	the	DET
ejpam-3384	215	4	proof	proof	NOUN
ejpam-3384	215	5	.	.	PUNCT
ejpam-3384	216	1	4	4	X
ejpam-3384	216	2	.	.	X
ejpam-3384	216	3	the	the	DET
ejpam-3384	216	4	applications	application	NOUN
ejpam-3384	216	5	of	of	ADP
ejpam-3384	216	6	the	the	DET
ejpam-3384	216	7	first	first	ADJ
ejpam-3384	216	8	quotient	quotient	NOUN
ejpam-3384	216	9	inequality	inequality	NOUN
ejpam-3384	216	10	to	to	ADP
ejpam-3384	216	11	unitary	unitary	ADJ
ejpam-3384	216	12	space	space	NOUN
ejpam-3384	216	13	in	in	ADP
ejpam-3384	216	14	this	this	DET
ejpam-3384	216	15	section	section	NOUN
ejpam-3384	216	16	,	,	PUNCT
ejpam-3384	216	17	the	the	DET
ejpam-3384	216	18	estimates	estimate	NOUN
ejpam-3384	216	19	involving	involve	VERB
ejpam-3384	216	20	the	the	DET
ejpam-3384	216	21	quotients	quotient	NOUN
ejpam-3384	216	22	of	of	ADP
ejpam-3384	216	23	norms	norm	NOUN
ejpam-3384	216	24	in	in	ADP
ejpam-3384	216	25	the	the	DET
ejpam-3384	216	26	unitary	unitary	ADJ
ejpam-3384	216	27	space	space	NOUN
ejpam-3384	216	28	are	be	AUX
ejpam-3384	216	29	obtained	obtain	VERB
ejpam-3384	216	30	by	by	ADP
ejpam-3384	216	31	using	use	VERB
ejpam-3384	216	32	both	both	CCONJ
ejpam-3384	216	33	the	the	DET
ejpam-3384	216	34	first	first	ADJ
ejpam-3384	216	35	and	and	CCONJ
ejpam-3384	216	36	second	second	ADJ
ejpam-3384	216	37	product	product	NOUN
ejpam-3384	216	38	inequalities	inequality	NOUN
ejpam-3384	216	39	.	.	PUNCT
ejpam-3384	217	1	definition	definition	NOUN
ejpam-3384	217	2	7	7	NUM
ejpam-3384	217	3	.	.	PUNCT
ejpam-3384	217	4	setting	set	VERB
ejpam-3384	217	5	he	he	PRON
ejpam-3384	217	6	p	p	X
ejpam-3384	217	7	(	(	PUNCT
ejpam-3384	217	8	ε	ε	PROPN
ejpam-3384	217	9	)	)	PUNCT
ejpam-3384	217	10	=	=	SYM
ejpam-3384	217	11	inf	inf	NOUN
ejpam-3384	217	12	{	{	PUNCT
ejpam-3384	217	13	sup	sup	NOUN
ejpam-3384	217	14	(	(	PUNCT
ejpam-3384	217	15	1	1	NUM
ejpam-3384	217	16	2π	2π	NUM
ejpam-3384	217	17	∫	∫	PROPN
ejpam-3384	217	18	2π	2π	PROPN
ejpam-3384	217	19	0	0	PUNCT
ejpam-3384	218	1	‖x+	‖x+	NOUN
ejpam-3384	218	2	eiθy‖pdθ	eiθy‖pdθ	PUNCT
ejpam-3384	218	3	)	)	PUNCT
ejpam-3384	218	4	1	1	NUM
ejpam-3384	218	5	p	p	NOUN
ejpam-3384	218	6	−	−	NOUN
ejpam-3384	218	7	1	1	NUM
ejpam-3384	218	8	:	:	PUNCT
ejpam-3384	218	9	‖y‖	‖y‖	PROPN
ejpam-3384	218	10	=	=	SYM
ejpam-3384	218	11	1	1	NUM
ejpam-3384	218	12	,	,	PUNCT
ejpam-3384	218	13	‖x‖	‖x‖	PROPN
ejpam-3384	218	14	=	=	SYM
ejpam-3384	218	15	ε	ε	PROPN
ejpam-3384	218	16	}	}	PUNCT
ejpam-3384	218	17	,	,	PUNCT
ejpam-3384	218	18	0	0	PUNCT
ejpam-3384	218	19	<	<	X
ejpam-3384	218	20	p	p	X
ejpam-3384	218	21	<	<	X
ejpam-3384	218	22	∞	∞	PROPN
ejpam-3384	218	23	,	,	PUNCT
ejpam-3384	218	24	and	and	CCONJ
ejpam-3384	218	25	ε	ε	PROPN
ejpam-3384	218	26	≥	≥	NUM
ejpam-3384	218	27	1	1	NUM
ejpam-3384	218	28	.	.	PUNCT
ejpam-3384	218	29	theorem	theorem	NOUN
ejpam-3384	218	30	6	6	NUM
ejpam-3384	218	31	.	.	PUNCT
ejpam-3384	218	32	suppose	suppose	VERB
ejpam-3384	218	33	that	that	SCONJ
ejpam-3384	218	34	(	(	PUNCT
ejpam-3384	218	35	x	x	X
ejpam-3384	218	36	,	,	PUNCT
ejpam-3384	218	37	‖	‖	PROPN
ejpam-3384	218	38	·	·	SYM
ejpam-3384	218	39	‖p	‖p	PROPN
ejpam-3384	218	40	)	)	PUNCT
ejpam-3384	218	41	is	be	AUX
ejpam-3384	218	42	a	a	DET
ejpam-3384	218	43	continuously	continuously	ADV
ejpam-3384	218	44	quasi	quasi	ADJ
ejpam-3384	218	45	-	-	ADJ
ejpam-3384	218	46	normed	normed	ADJ
ejpam-3384	218	47	space	space	NOUN
ejpam-3384	218	48	.	.	PUNCT
ejpam-3384	219	1	then	then	ADV
ejpam-3384	219	2	for	for	ADP
ejpam-3384	219	3	any	any	DET
ejpam-3384	219	4	x	x	SYM
ejpam-3384	219	5	∈	∈	PROPN
ejpam-3384	219	6	x	x	NOUN
ejpam-3384	219	7	,	,	PUNCT
ejpam-3384	219	8	there	there	PRON
ejpam-3384	219	9	exists	exist	VERB
ejpam-3384	219	10	δ	δ	PROPN
ejpam-3384	219	11	>	>	X
ejpam-3384	219	12	0	0	NUM
ejpam-3384	220	1	such	such	ADJ
ejpam-3384	220	2	that	that	SCONJ
ejpam-3384	220	3	1	1	NUM
ejpam-3384	220	4	2π	2π	NUM
ejpam-3384	220	5	∫	∫	PROPN
ejpam-3384	221	1	π	π	NOUN
ejpam-3384	221	2	0	0	NUM
ejpam-3384	221	3	‖x+	‖x+	PROPN
ejpam-3384	221	4	reiθy‖dθ	reiθy‖dθ	PROPN
ejpam-3384	221	5	≤	≤	ADV
ejpam-3384	221	6	1	1	NUM
ejpam-3384	221	7	2π	2π	NUM
ejpam-3384	221	8	∫	∫	PROPN
ejpam-3384	222	1	π	π	X
ejpam-3384	222	2	0	0	PUNCT
ejpam-3384	222	3	‖x‖	‖x‖	PROPN
ejpam-3384	222	4	‖reiθy‖	‖reiθy‖	PROPN
ejpam-3384	222	5	dθ	dθ	NOUN
ejpam-3384	222	6	.	.	PUNCT
ejpam-3384	223	1	proof	proof	NOUN
ejpam-3384	223	2	:	:	PUNCT
ejpam-3384	223	3	setting	set	VERB
ejpam-3384	223	4	0	0	PUNCT
ejpam-3384	223	5	<	<	X
ejpam-3384	223	6	r	r	NOUN
ejpam-3384	223	7	≤	≤	NUM
ejpam-3384	223	8	1	1	NUM
ejpam-3384	223	9	,	,	PUNCT
ejpam-3384	223	10	we	we	PRON
ejpam-3384	223	11	have	have	AUX
ejpam-3384	223	12	(	(	PUNCT
ejpam-3384	223	13	1	1	NUM
ejpam-3384	223	14	2π	2π	NUM
ejpam-3384	224	1	∫	∫	PROPN
ejpam-3384	225	1	π	π	NOUN
ejpam-3384	225	2	0	0	NUM
ejpam-3384	225	3	‖x+	‖x+	PROPN
ejpam-3384	225	4	reiθy‖pdθ	reiθy‖pdθ	PROPN
ejpam-3384	225	5	)	)	PUNCT
ejpam-3384	225	6	1	1	NUM
ejpam-3384	225	7	p	p	NOUN
ejpam-3384	225	8	≤	≤	NOUN
ejpam-3384	225	9	(	(	PUNCT
ejpam-3384	225	10	1	1	NUM
ejpam-3384	225	11	2π	2π	NUM
ejpam-3384	225	12	∫	∫	PROPN
ejpam-3384	226	1	π	π	X
ejpam-3384	226	2	0	0	PUNCT
ejpam-3384	227	1	(	(	PUNCT
ejpam-3384	227	2	‖x‖+	‖x‖+	PROPN
ejpam-3384	227	3	‖reiθy‖)pdθ	‖reiθy‖)pdθ	PUNCT
ejpam-3384	227	4	)	)	PUNCT
ejpam-3384	227	5	1	1	NUM
ejpam-3384	227	6	p	p	NOUN
ejpam-3384	227	7	⇒	⇒	NOUN
ejpam-3384	227	8	(	(	PUNCT
ejpam-3384	227	9	1	1	NUM
ejpam-3384	227	10	2π	2π	NUM
ejpam-3384	227	11	∫	∫	PROPN
ejpam-3384	227	12	π	π	X
ejpam-3384	227	13	0	0	PUNCT
ejpam-3384	228	1	(	(	PUNCT
ejpam-3384	228	2	‖x+	‖x+	PROPN
ejpam-3384	228	3	reiθy‖)pdθ	reiθy‖)pdθ	PROPN
ejpam-3384	228	4	)	)	PUNCT
ejpam-3384	228	5	1	1	NUM
ejpam-3384	228	6	p	p	NOUN
ejpam-3384	228	7	≤	≤	NOUN
ejpam-3384	228	8	(	(	PUNCT
ejpam-3384	228	9	1	1	NUM
ejpam-3384	228	10	2π	2π	NUM
ejpam-3384	228	11	∫	∫	PROPN
ejpam-3384	228	12	π	π	X
ejpam-3384	228	13	0	0	PUNCT
ejpam-3384	228	14	(	(	PUNCT
ejpam-3384	228	15	‖x‖p	‖x‖p	NOUN
ejpam-3384	228	16	+	+	CCONJ
ejpam-3384	228	17	‖r‖p)dθ	‖r‖p)dθ	NUM
ejpam-3384	228	18	)	)	PUNCT
ejpam-3384	228	19	1	1	NUM
ejpam-3384	228	20	p	p	NOUN
ejpam-3384	228	21	≤	≤	NOUN
ejpam-3384	228	22	(	(	PUNCT
ejpam-3384	228	23	1	1	NUM
ejpam-3384	228	24	2π	2π	NUM
ejpam-3384	229	1	∫	∫	PROPN
ejpam-3384	230	1	π	π	NOUN
ejpam-3384	230	2	0	0	NUM
ejpam-3384	230	3	‖x‖p	‖x‖p	PROPN
ejpam-3384	230	4	‖r‖p	‖r‖p	PROPN
ejpam-3384	230	5	dθ	dθ	PROPN
ejpam-3384	230	6	)	)	PUNCT
ejpam-3384	230	7	1	1	NUM
ejpam-3384	230	8	p	p	NOUN
ejpam-3384	230	9	⇒	⇒	NOUN
ejpam-3384	230	10	(	(	PUNCT
ejpam-3384	230	11	1	1	NUM
ejpam-3384	230	12	2π	2π	NUM
ejpam-3384	230	13	∫	∫	PROPN
ejpam-3384	231	1	π	π	NOUN
ejpam-3384	231	2	0	0	PUNCT
ejpam-3384	231	3	‖x‖pdθ	‖x‖pdθ	X
ejpam-3384	231	4	)	)	PUNCT
ejpam-3384	231	5	1	1	NUM
ejpam-3384	231	6	p	p	NOUN
ejpam-3384	231	7	≤	≤	NOUN
ejpam-3384	231	8	(	(	PUNCT
ejpam-3384	231	9	1	1	NUM
ejpam-3384	231	10	2π	2π	NUM
ejpam-3384	231	11	∫	∫	PROPN
ejpam-3384	232	1	π	π	NOUN
ejpam-3384	232	2	0	0	NUM
ejpam-3384	232	3	‖x‖p	‖x‖p	PROPN
ejpam-3384	232	4	‖r‖p	‖r‖p	PROPN
ejpam-3384	232	5	dθ	dθ	PROPN
ejpam-3384	232	6	)	)	PUNCT
ejpam-3384	232	7	1	1	NUM
ejpam-3384	232	8	p	p	NOUN
ejpam-3384	232	9	⇒	⇒	NOUN
ejpam-3384	232	10	‖x‖p	‖x‖p	VERB
ejpam-3384	232	11	≤	≤	NUM
ejpam-3384	232	12	‖x‖p	‖x‖p	PROPN
ejpam-3384	232	13	‖r‖p	‖r‖p	PROPN
ejpam-3384	232	14	.	.	PUNCT
ejpam-3384	233	1	hence	hence	ADV
ejpam-3384	233	2	,	,	PUNCT
ejpam-3384	233	3	1	1	NUM
ejpam-3384	233	4	2π	2π	NUM
ejpam-3384	233	5	∫	∫	PROPN
ejpam-3384	234	1	π	π	NOUN
ejpam-3384	234	2	0	0	NUM
ejpam-3384	234	3	‖x+	‖x+	PROPN
ejpam-3384	234	4	reiθy‖dθ	reiθy‖dθ	PROPN
ejpam-3384	234	5	≤	≤	ADV
ejpam-3384	234	6	1	1	NUM
ejpam-3384	234	7	2π	2π	NUM
ejpam-3384	234	8	∫	∫	PROPN
ejpam-3384	235	1	π	π	X
ejpam-3384	235	2	0	0	PUNCT
ejpam-3384	235	3	‖x‖	‖x‖	PROPN
ejpam-3384	235	4	‖reiθy‖	‖reiθy‖	PROPN
ejpam-3384	235	5	dθ	dθ	NOUN
ejpam-3384	235	6	.	.	PUNCT
ejpam-3384	236	1	the	the	DET
ejpam-3384	236	2	converse	converse	NOUN
ejpam-3384	236	3	inequality	inequality	NOUN
ejpam-3384	236	4	of	of	ADP
ejpam-3384	236	5	theorem	theorem	NOUN
ejpam-3384	236	6	(	(	PUNCT
ejpam-3384	236	7	6	6	NUM
ejpam-3384	236	8	)	)	PUNCT
ejpam-3384	236	9	is	be	AUX
ejpam-3384	236	10	obvious	obvious	ADJ
ejpam-3384	236	11	.	.	PUNCT
ejpam-3384	237	1	thus	thus	ADV
ejpam-3384	237	2	,	,	PUNCT
ejpam-3384	237	3	define	define	VERB
ejpam-3384	237	4	b.	b.	PROPN
ejpam-3384	237	5	barnes	barnes	PROPN
ejpam-3384	237	6	,	,	PUNCT
ejpam-3384	237	7	c.	c.	PROPN
ejpam-3384	237	8	sebil	sebil	PROPN
ejpam-3384	237	9	,	,	PUNCT
ejpam-3384	237	10	i.	i.	PROPN
ejpam-3384	237	11	k.	k.	PROPN
ejpam-3384	237	12	dontwi	dontwi	PROPN
ejpam-3384	237	13	/	/	SYM
ejpam-3384	237	14	eur	eur	PROPN
ejpam-3384	237	15	.	.	PUNCT
ejpam-3384	238	1	j.	j.	PROPN
ejpam-3384	238	2	pure	pure	PROPN
ejpam-3384	238	3	appl	appl	PROPN
ejpam-3384	238	4	.	.	PROPN
ejpam-3384	238	5	math	math	PROPN
ejpam-3384	238	6	,	,	PUNCT
ejpam-3384	238	7	12	12	NUM
ejpam-3384	238	8	(	(	PUNCT
ejpam-3384	238	9	2	2	NUM
ejpam-3384	238	10	)	)	PUNCT
ejpam-3384	238	11	(	(	PUNCT
ejpam-3384	238	12	2019	2019	NUM
ejpam-3384	238	13	)	)	PUNCT
ejpam-3384	238	14	,	,	PUNCT
ejpam-3384	238	15	469	469	NUM
ejpam-3384	238	16	-	-	SYM
ejpam-3384	238	17	485	485	NUM
ejpam-3384	238	18	480	480	NUM
ejpam-3384	238	19	definition	definition	NOUN
ejpam-3384	238	20	8	8	NUM
ejpam-3384	238	21	.	.	PUNCT
ejpam-3384	239	1	he	he	PRON
ejpam-3384	239	2	p	p	X
ejpam-3384	239	3	(	(	PUNCT
ejpam-3384	239	4	ε	ε	PROPN
ejpam-3384	239	5	)	)	PUNCT
ejpam-3384	239	6	=	=	SYM
ejpam-3384	239	7	inf	inf	NOUN
ejpam-3384	239	8	{	{	PUNCT
ejpam-3384	239	9	sup	sup	NOUN
ejpam-3384	239	10	(	(	PUNCT
ejpam-3384	239	11	1	1	NUM
ejpam-3384	239	12	2π	2π	NUM
ejpam-3384	239	13	∫	∫	PROPN
ejpam-3384	239	14	2π	2π	PROPN
ejpam-3384	239	15	0	0	PUNCT
ejpam-3384	240	1	‖x+	‖x+	NOUN
ejpam-3384	240	2	eiθy‖pdθ	eiθy‖pdθ	PUNCT
ejpam-3384	240	3	)	)	PUNCT
ejpam-3384	240	4	1	1	NUM
ejpam-3384	240	5	p	p	NOUN
ejpam-3384	240	6	−	−	NOUN
ejpam-3384	240	7	1	1	NUM
ejpam-3384	240	8	:	:	PUNCT
ejpam-3384	240	9	‖y‖	‖y‖	PROPN
ejpam-3384	240	10	=	=	SYM
ejpam-3384	240	11	1	1	NUM
ejpam-3384	240	12	,	,	PUNCT
ejpam-3384	240	13	‖x‖	‖x‖	PROPN
ejpam-3384	240	14	=	=	SYM
ejpam-3384	240	15	ε	ε	PROPN
ejpam-3384	240	16	}	}	PUNCT
ejpam-3384	240	17	,	,	PUNCT
ejpam-3384	240	18	0	0	PUNCT
ejpam-3384	240	19	<	<	X
ejpam-3384	240	20	p	p	X
ejpam-3384	240	21	<	<	X
ejpam-3384	240	22	∞	∞	PROPN
ejpam-3384	240	23	,	,	PUNCT
ejpam-3384	240	24	and	and	CCONJ
ejpam-3384	240	25	ε	ε	PROPN
ejpam-3384	240	26	≤	≤	NUM
ejpam-3384	240	27	1	1	NUM
ejpam-3384	240	28	,	,	PUNCT
ejpam-3384	240	29	and	and	CCONJ
ejpam-3384	240	30	provide	provide	VERB
ejpam-3384	240	31	the	the	DET
ejpam-3384	240	32	converse	converse	NOUN
ejpam-3384	240	33	result	result	NOUN
ejpam-3384	240	34	in	in	ADP
ejpam-3384	240	35	corollary	corollary	ADJ
ejpam-3384	240	36	3	3	NUM
ejpam-3384	240	37	below	below	ADV
ejpam-3384	240	38	.	.	PUNCT
ejpam-3384	241	1	corollary	corollary	ADJ
ejpam-3384	241	2	3	3	X
ejpam-3384	241	3	.	.	PUNCT
ejpam-3384	241	4	suppose	suppose	VERB
ejpam-3384	241	5	that	that	SCONJ
ejpam-3384	241	6	(	(	PUNCT
ejpam-3384	241	7	x	x	X
ejpam-3384	241	8	,	,	PUNCT
ejpam-3384	241	9	‖	‖	PROPN
ejpam-3384	241	10	·	·	PUNCT
ejpam-3384	241	11	‖q	‖q	PRON
ejpam-3384	241	12	)	)	PUNCT
ejpam-3384	241	13	is	be	AUX
ejpam-3384	241	14	a	a	DET
ejpam-3384	241	15	continuously	continuously	ADV
ejpam-3384	241	16	quasi	quasi	ADJ
ejpam-3384	241	17	-	-	ADJ
ejpam-3384	241	18	normed	normed	ADJ
ejpam-3384	241	19	space	space	NOUN
ejpam-3384	241	20	.	.	PUNCT
ejpam-3384	242	1	then	then	ADV
ejpam-3384	242	2	there	there	PRON
ejpam-3384	242	3	exists	exist	VERB
ejpam-3384	242	4	0	0	PUNCT
ejpam-3384	242	5	<	<	X
ejpam-3384	242	6	p	p	X
ejpam-3384	242	7	<	<	X
ejpam-3384	242	8	∞	∞	PROPN
ejpam-3384	242	9	,	,	PUNCT
ejpam-3384	242	10	such	such	ADJ
ejpam-3384	242	11	that	that	SCONJ
ejpam-3384	242	12	whenever	whenever	SCONJ
ejpam-3384	242	13	x	x	PRON
ejpam-3384	242	14	and	and	CCONJ
ejpam-3384	242	15	y	y	PROPN
ejpam-3384	242	16	are	be	AUX
ejpam-3384	242	17	in	in	ADP
ejpam-3384	242	18	x	x	PUNCT
ejpam-3384	242	19	with	with	ADP
ejpam-3384	242	20	δ	δ	PROPN
ejpam-3384	242	21	=	=	SYM
ejpam-3384	242	22	δ(x	δ(x	PROPN
ejpam-3384	242	23	,	,	PUNCT
ejpam-3384	242	24	y	y	PROPN
ejpam-3384	242	25	)	)	PUNCT
ejpam-3384	242	26	>	>	X
ejpam-3384	242	27	0	0	NUM
ejpam-3384	242	28	,	,	PUNCT
ejpam-3384	242	29	then	then	ADV
ejpam-3384	242	30	‖x‖	‖x‖	VERB
ejpam-3384	242	31	≤	≤	NOUN
ejpam-3384	242	32	(	(	PUNCT
ejpam-3384	242	33	1	1	NUM
ejpam-3384	242	34	2π	2π	NUM
ejpam-3384	242	35	∫	∫	PROPN
ejpam-3384	242	36	2π	2π	NOUN
ejpam-3384	242	37	0	0	NUM
ejpam-3384	242	38	∥∥∥	∥∥∥	NUM
ejpam-3384	242	39	x	x	SYM
ejpam-3384	242	40	reiθy	reiθy	NOUN
ejpam-3384	242	41	∥∥∥p	∥∥∥p	NOUN
ejpam-3384	242	42	)	)	PUNCT
ejpam-3384	242	43	1	1	NUM
ejpam-3384	242	44	p	p	NOUN
ejpam-3384	242	45	,	,	PUNCT
ejpam-3384	242	46	0	0	PUNCT
ejpam-3384	242	47	<	<	X
ejpam-3384	242	48	r	r	NOUN
ejpam-3384	242	49	≤	≤	PUNCT
ejpam-3384	242	50	δ	δ	NOUN
ejpam-3384	242	51	≤	≤	NOUN
ejpam-3384	242	52	1	1	NUM
ejpam-3384	242	53	.	.	PUNCT
ejpam-3384	243	1	proof	proof	NOUN
ejpam-3384	243	2	:	:	PUNCT
ejpam-3384	243	3	we	we	PRON
ejpam-3384	243	4	see	see	VERB
ejpam-3384	243	5	that	that	SCONJ
ejpam-3384	243	6	:(	:(	PROPN
ejpam-3384	243	7	1	1	NUM
ejpam-3384	243	8	2π	2π	NUM
ejpam-3384	243	9	∫	∫	PROPN
ejpam-3384	243	10	2π	2π	NOUN
ejpam-3384	243	11	0	0	NUM
ejpam-3384	243	12	∥∥∥	∥∥∥	NUM
ejpam-3384	243	13	x	x	SYM
ejpam-3384	243	14	reiθy	reiθy	NOUN
ejpam-3384	243	15	∥∥∥pdθ	∥∥∥pdθ	PROPN
ejpam-3384	243	16	)	)	PUNCT
ejpam-3384	243	17	1	1	NUM
ejpam-3384	243	18	p	p	NOUN
ejpam-3384	243	19	=	=	PUNCT
ejpam-3384	243	20	(	(	PUNCT
ejpam-3384	243	21	1	1	NUM
ejpam-3384	243	22	2π	2π	NUM
ejpam-3384	243	23	∫	∫	PROPN
ejpam-3384	243	24	2π	2π	NOUN
ejpam-3384	243	25	0	0	NUM
ejpam-3384	243	26	‖x‖p	‖x‖p	PROPN
ejpam-3384	243	27	‖reiθy‖p	‖reiθy‖p	NOUN
ejpam-3384	243	28	dθ	dθ	PROPN
ejpam-3384	243	29	)	)	PUNCT
ejpam-3384	243	30	1	1	NUM
ejpam-3384	243	31	p	p	NOUN
ejpam-3384	243	32	.	.	PUNCT
ejpam-3384	244	1	using	use	VERB
ejpam-3384	244	2	the	the	DET
ejpam-3384	244	3	first	first	ADJ
ejpam-3384	244	4	quotient	quotient	NOUN
ejpam-3384	244	5	inequality	inequality	NOUN
ejpam-3384	244	6	on	on	ADP
ejpam-3384	244	7	the	the	DET
ejpam-3384	244	8	right	right	ADJ
ejpam-3384	244	9	hand	hand	NOUN
ejpam-3384	244	10	side	side	NOUN
ejpam-3384	244	11	of	of	ADP
ejpam-3384	244	12	the	the	DET
ejpam-3384	244	13	above	above	ADJ
ejpam-3384	244	14	equation	equation	NOUN
ejpam-3384	244	15	,	,	PUNCT
ejpam-3384	244	16	we	we	PRON
ejpam-3384	244	17	obtain	obtain	VERB
ejpam-3384	244	18	⇒	⇒	NOUN
ejpam-3384	244	19	(	(	PUNCT
ejpam-3384	244	20	1	1	NUM
ejpam-3384	244	21	2π	2π	NUM
ejpam-3384	244	22	∫	∫	PROPN
ejpam-3384	244	23	2π	2π	NOUN
ejpam-3384	244	24	0	0	NUM
ejpam-3384	244	25	∥∥∥	∥∥∥	NUM
ejpam-3384	244	26	x	x	SYM
ejpam-3384	244	27	reiθy	reiθy	NOUN
ejpam-3384	244	28	∥∥∥pdθ	∥∥∥pdθ	PROPN
ejpam-3384	244	29	)	)	PUNCT
ejpam-3384	244	30	1	1	NUM
ejpam-3384	244	31	p	p	NOUN
ejpam-3384	244	32	≥	≥	X
ejpam-3384	244	33	(	(	PUNCT
ejpam-3384	244	34	∫	∫	PROPN
ejpam-3384	244	35	2π	2π	PROPN
ejpam-3384	244	36	0	0	NUM
ejpam-3384	244	37	‖x‖p	‖x‖p	NOUN
ejpam-3384	244	38	‖r‖p	‖r‖p	X
ejpam-3384	244	39	)	)	PUNCT
ejpam-3384	244	40	1	1	NUM
ejpam-3384	244	41	p	p	NOUN
ejpam-3384	244	42	≥	≥	X
ejpam-3384	244	43	(	(	PUNCT
ejpam-3384	244	44	∫	∫	PROPN
ejpam-3384	244	45	2π	2π	PROPN
ejpam-3384	244	46	0	0	NUM
ejpam-3384	245	1	‖x‖p‖r‖p	‖x‖p‖r‖p	NOUN
ejpam-3384	245	2	)	)	PUNCT
ejpam-3384	245	3	1	1	NUM
ejpam-3384	245	4	p	p	NOUN
ejpam-3384	245	5	⇒	⇒	NOUN
ejpam-3384	245	6	(	(	PUNCT
ejpam-3384	245	7	∫	∫	PROPN
ejpam-3384	245	8	2π	2π	PROPN
ejpam-3384	245	9	0	0	NUM
ejpam-3384	246	1	‖x‖p‖r‖p	‖x‖p‖r‖p	NOUN
ejpam-3384	246	2	)	)	PUNCT
ejpam-3384	246	3	1	1	NUM
ejpam-3384	246	4	p	p	NOUN
ejpam-3384	246	5	≤	≤	NUM
ejpam-3384	246	6	(	(	PUNCT
ejpam-3384	246	7	∫	∫	PROPN
ejpam-3384	246	8	2π	2π	PROPN
ejpam-3384	246	9	0	0	NUM
ejpam-3384	246	10	‖x‖p	‖x‖p	NOUN
ejpam-3384	246	11	‖r‖p	‖r‖p	X
ejpam-3384	246	12	)	)	PUNCT
ejpam-3384	246	13	1	1	NUM
ejpam-3384	246	14	p	p	NOUN
ejpam-3384	246	15	⇒	⇒	NOUN
ejpam-3384	246	16	(	(	PUNCT
ejpam-3384	246	17	∫	∫	PROPN
ejpam-3384	246	18	2π	2π	PROPN
ejpam-3384	246	19	0	0	NUM
ejpam-3384	246	20	‖x‖p	‖x‖p	NOUN
ejpam-3384	246	21	)	)	PUNCT
ejpam-3384	246	22	1	1	NUM
ejpam-3384	246	23	p	p	NOUN
ejpam-3384	246	24	≤	≤	NUM
ejpam-3384	246	25	(	(	PUNCT
ejpam-3384	246	26	∫	∫	PROPN
ejpam-3384	246	27	2π	2π	PROPN
ejpam-3384	246	28	0	0	NUM
ejpam-3384	246	29	‖x‖p	‖x‖p	NOUN
ejpam-3384	246	30	‖r‖p	‖r‖p	X
ejpam-3384	246	31	)	)	PUNCT
ejpam-3384	247	1	1	1	NUM
ejpam-3384	248	1	p	p	NOUN
ejpam-3384	248	2	⇒	⇒	NOUN
ejpam-3384	248	3	‖x‖p	‖x‖p	VERB
ejpam-3384	248	4	≤	≤	NUM
ejpam-3384	248	5	‖x‖p	‖x‖p	PROPN
ejpam-3384	248	6	‖r‖p	‖r‖p	PROPN
ejpam-3384	248	7	.	.	PUNCT
ejpam-3384	249	1	it	it	PRON
ejpam-3384	249	2	follows	follow	VERB
ejpam-3384	249	3	that	that	SCONJ
ejpam-3384	249	4	,	,	PUNCT
ejpam-3384	249	5	‖x‖p	‖x‖p	VERB
ejpam-3384	249	6	≤	≤	NUM
ejpam-3384	249	7	(	(	PUNCT
ejpam-3384	249	8	∫	∫	PROPN
ejpam-3384	249	9	2π	2π	PROPN
ejpam-3384	249	10	0	0	NUM
ejpam-3384	249	11	‖x‖p	‖x‖p	NOUN
ejpam-3384	249	12	‖r‖p	‖r‖p	X
ejpam-3384	249	13	)	)	PUNCT
ejpam-3384	249	14	1	1	NUM
ejpam-3384	249	15	p	p	NOUN
ejpam-3384	249	16	.	.	PUNCT
ejpam-3384	250	1	hence	hence	ADV
ejpam-3384	250	2	,	,	PUNCT
ejpam-3384	250	3	‖x‖	‖x‖	PROPN
ejpam-3384	250	4	≤	≤	NOUN
ejpam-3384	250	5	(	(	PUNCT
ejpam-3384	250	6	1	1	NUM
ejpam-3384	250	7	2π	2π	NUM
ejpam-3384	250	8	∫	∫	PROPN
ejpam-3384	250	9	2π	2π	NOUN
ejpam-3384	250	10	0	0	NUM
ejpam-3384	250	11	∥∥∥	∥∥∥	NUM
ejpam-3384	250	12	x	x	SYM
ejpam-3384	250	13	reiθy	reiθy	NOUN
ejpam-3384	250	14	∥∥∥p	∥∥∥p	NOUN
ejpam-3384	250	15	)	)	PUNCT
ejpam-3384	250	16	1	1	NUM
ejpam-3384	250	17	p	p	NOUN
ejpam-3384	250	18	,	,	PUNCT
ejpam-3384	250	19	0	0	PUNCT
ejpam-3384	250	20	<	<	X
ejpam-3384	251	1	r	r	NOUN
ejpam-3384	251	2	≤	≤	PUNCT
ejpam-3384	251	3	δ	δ	NOUN
ejpam-3384	251	4	≤	≤	ADV
ejpam-3384	251	5	1	1	NUM
ejpam-3384	251	6	.	.	PUNCT
ejpam-3384	251	7	proposition	proposition	NOUN
ejpam-3384	251	8	2	2	NUM
ejpam-3384	251	9	.	.	PUNCT
ejpam-3384	251	10	suppose	suppose	VERB
ejpam-3384	251	11	that	that	SCONJ
ejpam-3384	251	12	(	(	PUNCT
ejpam-3384	251	13	x	x	X
ejpam-3384	251	14	,	,	PUNCT
ejpam-3384	251	15	‖	‖	PROPN
ejpam-3384	251	16	·	·	PUNCT
ejpam-3384	251	17	‖q	‖q	PRON
ejpam-3384	251	18	)	)	PUNCT
ejpam-3384	251	19	is	be	AUX
ejpam-3384	251	20	a	a	DET
ejpam-3384	251	21	continuously	continuously	ADV
ejpam-3384	251	22	quasi	quasi	ADJ
ejpam-3384	251	23	-	-	ADJ
ejpam-3384	251	24	normed	normed	ADJ
ejpam-3384	251	25	space	space	NOUN
ejpam-3384	251	26	.	.	PUNCT
ejpam-3384	252	1	the	the	DET
ejpam-3384	252	2	following	follow	VERB
ejpam-3384	252	3	statements	statement	NOUN
ejpam-3384	252	4	are	be	AUX
ejpam-3384	252	5	equivalent	equivalent	ADJ
ejpam-3384	252	6	:	:	PUNCT
ejpam-3384	252	7	(	(	PUNCT
ejpam-3384	252	8	i	i	NOUN
ejpam-3384	252	9	)	)	PUNCT
ejpam-3384	252	10	(	(	PUNCT
ejpam-3384	252	11	x	x	X
ejpam-3384	252	12	,	,	PUNCT
ejpam-3384	252	13	‖	‖	PROPN
ejpam-3384	252	14	·	·	PUNCT
ejpam-3384	252	15	‖	‖	NUM
ejpam-3384	252	16	)	)	PUNCT
ejpam-3384	252	17	is	be	AUX
ejpam-3384	252	18	locally	locally	ADV
ejpam-3384	252	19	pl	pl	NOUN
ejpam-3384	252	20	-	-	NOUN
ejpam-3384	252	21	convex	convex	NOUN
ejpam-3384	252	22	;	;	PUNCT
ejpam-3384	252	23	(	(	PUNCT
ejpam-3384	252	24	ii	ii	NOUN
ejpam-3384	252	25	)	)	PUNCT
ejpam-3384	252	26	there	there	PRON
ejpam-3384	252	27	exists	exist	VERB
ejpam-3384	252	28	0	0	PUNCT
ejpam-3384	252	29	<	<	X
ejpam-3384	252	30	p	p	X
ejpam-3384	252	31	<	<	X
ejpam-3384	252	32	∞	∞	PROPN
ejpam-3384	252	33	,	,	PUNCT
ejpam-3384	252	34	such	such	ADJ
ejpam-3384	252	35	that	that	SCONJ
ejpam-3384	252	36	whenever	whenever	SCONJ
ejpam-3384	252	37	x	x	PRON
ejpam-3384	252	38	and	and	CCONJ
ejpam-3384	252	39	y	y	PROPN
ejpam-3384	252	40	are	be	AUX
ejpam-3384	252	41	in	in	ADP
ejpam-3384	252	42	x	x	PUNCT
ejpam-3384	252	43	with	with	ADP
ejpam-3384	252	44	δ	δ	PROPN
ejpam-3384	252	45	=	=	SYM
ejpam-3384	252	46	δ(x	δ(x	PROPN
ejpam-3384	252	47	,	,	PUNCT
ejpam-3384	252	48	y	y	PROPN
ejpam-3384	252	49	)	)	PUNCT
ejpam-3384	252	50	>	>	X
ejpam-3384	252	51	0	0	NUM
ejpam-3384	253	1	such	such	ADJ
ejpam-3384	253	2	that	that	SCONJ
ejpam-3384	253	3	‖x‖	‖x‖	PROPN
ejpam-3384	253	4	≤	≤	X
ejpam-3384	253	5	(	(	PUNCT
ejpam-3384	253	6	1	1	NUM
ejpam-3384	253	7	2π	2π	NUM
ejpam-3384	253	8	∫	∫	PROPN
ejpam-3384	253	9	2π	2π	NOUN
ejpam-3384	253	10	0	0	NUM
ejpam-3384	253	11	∥∥∥	∥∥∥	NUM
ejpam-3384	253	12	x	x	SYM
ejpam-3384	253	13	reiθy	reiθy	NOUN
ejpam-3384	253	14	∥∥∥p	∥∥∥p	NOUN
ejpam-3384	253	15	)	)	PUNCT
ejpam-3384	253	16	1	1	NUM
ejpam-3384	253	17	p	p	NOUN
ejpam-3384	253	18	,	,	PUNCT
ejpam-3384	253	19	0	0	PUNCT
ejpam-3384	253	20	<	<	X
ejpam-3384	253	21	r	r	NOUN
ejpam-3384	253	22	≤	≤	PUNCT
ejpam-3384	253	23	δ	δ	NOUN
ejpam-3384	253	24	≤	≤	NOUN
ejpam-3384	253	25	1	1	NUM
ejpam-3384	253	26	;	;	PUNCT
ejpam-3384	253	27	(	(	PUNCT
ejpam-3384	253	28	iii	iii	X
ejpam-3384	253	29	)	)	PUNCT
ejpam-3384	253	30	ln	ln	PROPN
ejpam-3384	253	31	‖x‖	‖x‖	PROPN
ejpam-3384	253	32	is	be	AUX
ejpam-3384	253	33	a	a	DET
ejpam-3384	253	34	pluri	pluri	ADJ
ejpam-3384	253	35	-	-	ADJ
ejpam-3384	253	36	subharmonic	subharmonic	ADJ
ejpam-3384	253	37	function	function	NOUN
ejpam-3384	253	38	on	on	ADP
ejpam-3384	253	39	x.	x.	PROPN
ejpam-3384	253	40	b.	b.	PROPN
ejpam-3384	253	41	barnes	barnes	PROPN
ejpam-3384	253	42	,	,	PUNCT
ejpam-3384	253	43	c.	c.	PROPN
ejpam-3384	253	44	sebil	sebil	PROPN
ejpam-3384	253	45	,	,	PUNCT
ejpam-3384	253	46	i.	i.	PROPN
ejpam-3384	253	47	k.	k.	PROPN
ejpam-3384	253	48	dontwi	dontwi	PROPN
ejpam-3384	253	49	/	/	SYM
ejpam-3384	253	50	eur	eur	PROPN
ejpam-3384	253	51	.	.	PUNCT
ejpam-3384	254	1	j.	j.	PROPN
ejpam-3384	254	2	pure	pure	PROPN
ejpam-3384	254	3	appl	appl	PROPN
ejpam-3384	254	4	.	.	PROPN
ejpam-3384	254	5	math	math	PROPN
ejpam-3384	254	6	,	,	PUNCT
ejpam-3384	254	7	12	12	NUM
ejpam-3384	254	8	(	(	PUNCT
ejpam-3384	254	9	2	2	NUM
ejpam-3384	254	10	)	)	PUNCT
ejpam-3384	254	11	(	(	PUNCT
ejpam-3384	254	12	2019	2019	NUM
ejpam-3384	254	13	)	)	PUNCT
ejpam-3384	254	14	,	,	PUNCT
ejpam-3384	254	15	469	469	NUM
ejpam-3384	254	16	-	-	SYM
ejpam-3384	254	17	485	485	NUM
ejpam-3384	254	18	481	481	NUM
ejpam-3384	254	19	5	5	NUM
ejpam-3384	254	20	.	.	PUNCT
ejpam-3384	255	1	quotient	quotient	NOUN
ejpam-3384	255	2	inequalities	inequality	NOUN
ejpam-3384	255	3	involving	involve	VERB
ejpam-3384	255	4	powers	power	NOUN
ejpam-3384	255	5	in	in	ADP
ejpam-3384	255	6	this	this	DET
ejpam-3384	255	7	section	section	NOUN
ejpam-3384	255	8	,	,	PUNCT
ejpam-3384	255	9	the	the	DET
ejpam-3384	255	10	inequalities	inequality	NOUN
ejpam-3384	255	11	involving	involve	VERB
ejpam-3384	255	12	the	the	DET
ejpam-3384	255	13	relationships	relationship	NOUN
ejpam-3384	255	14	among	among	ADP
ejpam-3384	255	15	index	index	NOUN
ejpam-3384	255	16	product	product	NOUN
ejpam-3384	255	17	of	of	ADP
ejpam-3384	255	18	numbers	number	NOUN
ejpam-3384	255	19	and	and	CCONJ
ejpam-3384	255	20	its	its	PRON
ejpam-3384	255	21	quotient	quotient	NOUN
ejpam-3384	255	22	of	of	ADP
ejpam-3384	255	23	numbers	number	NOUN
ejpam-3384	255	24	as	as	ADP
ejpam-3384	255	25	bases	basis	NOUN
ejpam-3384	255	26	are	be	AUX
ejpam-3384	255	27	established	establish	VERB
ejpam-3384	255	28	for	for	ADP
ejpam-3384	255	29	given	give	VERB
ejpam-3384	255	30	regions	region	NOUN
ejpam-3384	255	31	of	of	ADP
ejpam-3384	255	32	validity	validity	NOUN
ejpam-3384	255	33	.	.	PUNCT
ejpam-3384	256	1	thus	thus	ADV
ejpam-3384	256	2	,	,	PUNCT
ejpam-3384	256	3	we	we	PRON
ejpam-3384	256	4	introduce	introduce	VERB
ejpam-3384	256	5	in	in	ADP
ejpam-3384	256	6	this	this	DET
ejpam-3384	256	7	paper	paper	NOUN
ejpam-3384	256	8	the	the	DET
ejpam-3384	256	9	index	index	NOUN
ejpam-3384	256	10	power	power	NOUN
ejpam-3384	256	11	quotient	quotient	NOUN
ejpam-3384	256	12	inequalities	inequality	NOUN
ejpam-3384	256	13	;	;	PUNCT
ejpam-3384	256	14	the	the	DET
ejpam-3384	256	15	first	first	ADJ
ejpam-3384	256	16	and	and	CCONJ
ejpam-3384	256	17	second	second	ADJ
ejpam-3384	256	18	index	index	NOUN
ejpam-3384	256	19	power	power	NOUN
ejpam-3384	256	20	quotient	quotient	NOUN
ejpam-3384	256	21	inequalities	inequality	NOUN
ejpam-3384	256	22	.	.	PUNCT
ejpam-3384	257	1	the	the	DET
ejpam-3384	257	2	main	main	ADJ
ejpam-3384	257	3	result	result	NOUN
ejpam-3384	257	4	is	be	AUX
ejpam-3384	257	5	expressed	express	VERB
ejpam-3384	257	6	in	in	ADP
ejpam-3384	257	7	theorem	theorem	NOUN
ejpam-3384	257	8	(	(	PUNCT
ejpam-3384	257	9	7	7	NUM
ejpam-3384	257	10	)	)	PUNCT
ejpam-3384	257	11	,	,	PUNCT
ejpam-3384	257	12	which	which	PRON
ejpam-3384	257	13	is	be	AUX
ejpam-3384	257	14	the	the	DET
ejpam-3384	257	15	fundamental	fundamental	ADJ
ejpam-3384	257	16	tool	tool	NOUN
ejpam-3384	257	17	for	for	ADP
ejpam-3384	257	18	proving	prove	VERB
ejpam-3384	257	19	other	other	ADJ
ejpam-3384	257	20	results	result	NOUN
ejpam-3384	257	21	in	in	ADP
ejpam-3384	257	22	this	this	DET
ejpam-3384	257	23	paper	paper	NOUN
ejpam-3384	257	24	.	.	PUNCT
ejpam-3384	258	1	involving	involve	VERB
ejpam-3384	258	2	propositions	proposition	NOUN
ejpam-3384	258	3	and	and	CCONJ
ejpam-3384	258	4	corollaries	corollary	NOUN
ejpam-3384	258	5	.	.	PUNCT
ejpam-3384	259	1	theorem	theorem	VERB
ejpam-3384	259	2	7	7	NUM
ejpam-3384	259	3	(	(	PUNCT
ejpam-3384	259	4	first	first	ADJ
ejpam-3384	259	5	index	index	NOUN
ejpam-3384	259	6	power	power	NOUN
ejpam-3384	259	7	quotient	quotient	NOUN
ejpam-3384	259	8	inequality	inequality	PROPN
ejpam-3384	259	9	)	)	PUNCT
ejpam-3384	259	10	.	.	PUNCT
ejpam-3384	260	1	let	let	VERB
ejpam-3384	260	2	t	t	NOUN
ejpam-3384	260	3	:	:	PUNCT
ejpam-3384	260	4	x	x	X
ejpam-3384	260	5	→	→	SYM
ejpam-3384	260	6	y	y	X
ejpam-3384	260	7	be	be	AUX
ejpam-3384	260	8	a	a	DET
ejpam-3384	260	9	banach	banach	NOUN
ejpam-3384	260	10	space	space	NOUN
ejpam-3384	260	11	.	.	PUNCT
ejpam-3384	261	1	then	then	ADV
ejpam-3384	261	2	‖x‖pq	‖x‖pq	VERB
ejpam-3384	261	3	≤	≤	X
ejpam-3384	262	1	‖x‖	‖x‖	PROPN
ejpam-3384	262	2	p	p	X
ejpam-3384	262	3	q	q	NOUN
ejpam-3384	262	4	,	,	PUNCT
ejpam-3384	262	5	∀	∀	X
ejpam-3384	262	6	x	x	X
ejpam-3384	262	7	∈	∈	NOUN
ejpam-3384	262	8	x	x	X
ejpam-3384	262	9	,	,	PUNCT
ejpam-3384	262	10	0	0	NUM
ejpam-3384	262	11	≤	≤	NOUN
ejpam-3384	263	1	p	p	X
ejpam-3384	263	2	<	<	X
ejpam-3384	263	3	∞	∞	PROPN
ejpam-3384	263	4	,	,	PUNCT
ejpam-3384	263	5	and	and	CCONJ
ejpam-3384	263	6	0	0	NUM
ejpam-3384	263	7	<	<	X
ejpam-3384	263	8	q	q	X
ejpam-3384	263	9	≤	≤	NUM
ejpam-3384	263	10	1	1	NUM
ejpam-3384	263	11	,	,	PUNCT
ejpam-3384	263	12	where	where	SCONJ
ejpam-3384	263	13	equality	equality	NOUN
ejpam-3384	263	14	occurs	occur	VERB
ejpam-3384	263	15	at	at	ADP
ejpam-3384	263	16	either	either	CCONJ
ejpam-3384	263	17	p	p	PROPN
ejpam-3384	263	18	=	=	NOUN
ejpam-3384	263	19	0	0	NUM
ejpam-3384	263	20	or	or	CCONJ
ejpam-3384	263	21	q	q	ADJ
ejpam-3384	263	22	=	=	NOUN
ejpam-3384	263	23	1	1	X
ejpam-3384	263	24	.	.	PUNCT
ejpam-3384	264	1	proof	proof	NOUN
ejpam-3384	264	2	:	:	PUNCT
ejpam-3384	264	3	setting	set	VERB
ejpam-3384	264	4	0	0	NUM
ejpam-3384	264	5	≤	≤	NOUN
ejpam-3384	265	1	p	p	X
ejpam-3384	265	2	<	<	X
ejpam-3384	265	3	∞	∞	PROPN
ejpam-3384	265	4	,	,	PUNCT
ejpam-3384	265	5	and	and	CCONJ
ejpam-3384	265	6	0	0	NUM
ejpam-3384	265	7	<	<	X
ejpam-3384	265	8	q	q	X
ejpam-3384	265	9	≤	≤	NUM
ejpam-3384	265	10	1	1	NUM
ejpam-3384	265	11	.	.	PUNCT
ejpam-3384	266	1	we	we	PRON
ejpam-3384	266	2	can	can	AUX
ejpam-3384	266	3	see	see	VERB
ejpam-3384	266	4	that	that	PRON
ejpam-3384	266	5	:	:	PUNCT
ejpam-3384	266	6	f(x	f(x	PROPN
ejpam-3384	266	7	)	)	PUNCT
ejpam-3384	267	1	=	=	SYM
ejpam-3384	267	2	−(pqx+	−(pqx+	NOUN
ejpam-3384	267	3	p	p	NOUN
ejpam-3384	267	4	q	q	NOUN
ejpam-3384	267	5	x	x	NOUN
ejpam-3384	267	6	)	)	PUNCT
ejpam-3384	267	7	≤	≤	NOUN
ejpam-3384	267	8	0	0	NUM
ejpam-3384	267	9	⇒	⇒	PROPN
ejpam-3384	267	10	f(x	f(x	PROPN
ejpam-3384	267	11	)	)	PUNCT
ejpam-3384	268	1	=	=	SYM
ejpam-3384	268	2	−(pqx+	−(pqx+	NOUN
ejpam-3384	268	3	p	p	NOUN
ejpam-3384	268	4	q	q	NOUN
ejpam-3384	268	5	x	x	NOUN
ejpam-3384	268	6	)	)	PUNCT
ejpam-3384	268	7	=	=	SYM
ejpam-3384	268	8	−(xpq	−(xpq	PROPN
ejpam-3384	269	1	+	+	NUM
ejpam-3384	269	2	x	x	X
ejpam-3384	269	3	p	p	X
ejpam-3384	269	4	q	q	PROPN
ejpam-3384	269	5	)	)	PUNCT
ejpam-3384	269	6	≤	≤	NOUN
ejpam-3384	269	7	0	0	NUM
ejpam-3384	269	8	⇒	⇒	NOUN
ejpam-3384	269	9	−(xpq	−(xpq	NOUN
ejpam-3384	269	10	+	+	PUNCT
ejpam-3384	269	11	x	x	X
ejpam-3384	269	12	p	p	X
ejpam-3384	269	13	q	q	PROPN
ejpam-3384	269	14	)	)	PUNCT
ejpam-3384	269	15	≤	≤	NOUN
ejpam-3384	269	16	0	0	NUM
ejpam-3384	269	17	⇒	⇒	NOUN
ejpam-3384	269	18	−xpq	−xpq	VERB
ejpam-3384	269	19	≤	≤	NOUN
ejpam-3384	270	1	−x	−x	NOUN
ejpam-3384	270	2	p	p	X
ejpam-3384	270	3	q	q	X
ejpam-3384	270	4	⇒	⇒	NOUN
ejpam-3384	270	5	‖x‖pq	‖x‖pq	VERB
ejpam-3384	270	6	≤	≤	PUNCT
ejpam-3384	271	1	‖x‖	‖x‖	PROPN
ejpam-3384	271	2	p	p	X
ejpam-3384	271	3	q	q	NOUN
ejpam-3384	271	4	,	,	PUNCT
ejpam-3384	271	5	∀	∀	X
ejpam-3384	272	1	x	x	SYM
ejpam-3384	272	2	∈	∈	NOUN
ejpam-3384	272	3	x.	x.	NOUN
ejpam-3384	272	4	this	this	PRON
ejpam-3384	272	5	completes	complete	VERB
ejpam-3384	272	6	the	the	DET
ejpam-3384	272	7	proof	proof	NOUN
ejpam-3384	272	8	.	.	PUNCT
ejpam-3384	273	1	the	the	DET
ejpam-3384	273	2	converse	converse	NOUN
ejpam-3384	273	3	of	of	ADP
ejpam-3384	273	4	theorem	theorem	NOUN
ejpam-3384	273	5	7	7	NUM
ejpam-3384	273	6	is	be	AUX
ejpam-3384	273	7	stated	state	VERB
ejpam-3384	273	8	in	in	ADP
ejpam-3384	273	9	theorem	theorem	NOUN
ejpam-3384	273	10	8	8	NUM
ejpam-3384	273	11	below	below	ADV
ejpam-3384	273	12	.	.	PUNCT
ejpam-3384	274	1	theorem	theorem	ADJ
ejpam-3384	274	2	8	8	NUM
ejpam-3384	274	3	(	(	PUNCT
ejpam-3384	274	4	second	second	ADJ
ejpam-3384	274	5	index	index	NOUN
ejpam-3384	274	6	power	power	NOUN
ejpam-3384	274	7	quotient	quotient	NOUN
ejpam-3384	274	8	inequality	inequality	PROPN
ejpam-3384	274	9	)	)	PUNCT
ejpam-3384	274	10	.	.	PUNCT
ejpam-3384	275	1	let	let	VERB
ejpam-3384	275	2	t	t	NOUN
ejpam-3384	275	3	:	:	PUNCT
ejpam-3384	275	4	x	x	X
ejpam-3384	275	5	→	→	SYM
ejpam-3384	275	6	y	y	X
ejpam-3384	275	7	be	be	AUX
ejpam-3384	275	8	a	a	DET
ejpam-3384	275	9	banach	banach	NOUN
ejpam-3384	275	10	space	space	NOUN
ejpam-3384	275	11	.	.	PUNCT
ejpam-3384	276	1	then	then	ADV
ejpam-3384	276	2	‖x‖	‖x‖	VERB
ejpam-3384	276	3	p	p	PROPN
ejpam-3384	276	4	q	q	PUNCT
ejpam-3384	276	5	≤	≤	NUM
ejpam-3384	276	6	‖x‖pq	‖x‖pq	NOUN
ejpam-3384	276	7	,	,	PUNCT
ejpam-3384	276	8	∀	∀	X
ejpam-3384	276	9	x	x	SYM
ejpam-3384	276	10	∈	∈	NOUN
ejpam-3384	276	11	x	x	SYM
ejpam-3384	276	12	,	,	PUNCT
ejpam-3384	276	13	1	1	NUM
ejpam-3384	276	14	≤	≤	NOUN
ejpam-3384	276	15	p	p	X
ejpam-3384	276	16	<	<	X
ejpam-3384	276	17	∞	∞	PROPN
ejpam-3384	276	18	,	,	PUNCT
ejpam-3384	276	19	and	and	CCONJ
ejpam-3384	276	20	1	1	NUM
ejpam-3384	276	21	≤	≤	NUM
ejpam-3384	276	22	q	q	NOUN
ejpam-3384	277	1	<	<	X
ejpam-3384	277	2	∞.	∞.	PROPN
ejpam-3384	277	3	proof	proof	NOUN
ejpam-3384	277	4	:	:	PUNCT
ejpam-3384	277	5	the	the	DET
ejpam-3384	277	6	proof	proof	NOUN
ejpam-3384	277	7	of	of	ADP
ejpam-3384	277	8	theorem	theorem	ADJ
ejpam-3384	277	9	8	8	NUM
ejpam-3384	277	10	is	be	AUX
ejpam-3384	277	11	similar	similar	ADJ
ejpam-3384	277	12	to	to	AUX
ejpam-3384	277	13	theorem	theorem	VERB
ejpam-3384	277	14	7	7	NUM
ejpam-3384	277	15	.	.	PUNCT
ejpam-3384	277	16	theorem	theorem	NOUN
ejpam-3384	277	17	9	9	NUM
ejpam-3384	277	18	.	.	PUNCT
ejpam-3384	278	1	let	let	VERB
ejpam-3384	278	2	t	t	NOUN
ejpam-3384	278	3	:	:	PUNCT
ejpam-3384	278	4	x	x	X
ejpam-3384	278	5	→	→	SYM
ejpam-3384	278	6	y	y	X
ejpam-3384	278	7	be	be	AUX
ejpam-3384	278	8	a	a	DET
ejpam-3384	278	9	banach	banach	NOUN
ejpam-3384	278	10	space	space	NOUN
ejpam-3384	278	11	.	.	PUNCT
ejpam-3384	279	1	then	then	ADV
ejpam-3384	279	2	‖a‖‖b‖	‖a‖‖b‖	ADJ
ejpam-3384	279	3	≤	≤	PUNCT
ejpam-3384	279	4	pq2‖a‖p	pq2‖a‖p	PROPN
ejpam-3384	279	5	+	+	CCONJ
ejpam-3384	279	6	p2q‖b‖q	p2q‖b‖q	PROPN
ejpam-3384	279	7	,	,	PUNCT
ejpam-3384	279	8	∀	∀	X
ejpam-3384	279	9	a	a	PRON
ejpam-3384	279	10	,	,	PUNCT
ejpam-3384	279	11	b	b	X
ejpam-3384	279	12	∈	∈	PROPN
ejpam-3384	279	13	x	x	SYM
ejpam-3384	279	14	p	p	NOUN
ejpam-3384	279	15	,	,	PUNCT
ejpam-3384	279	16	q	q	PROPN
ejpam-3384	279	17	≤	≤	NUM
ejpam-3384	279	18	1	1	NUM
ejpam-3384	279	19	.	.	PUNCT
ejpam-3384	279	20	proof	proof	NOUN
ejpam-3384	279	21	:	:	PUNCT
ejpam-3384	279	22	using	use	VERB
ejpam-3384	279	23	the	the	DET
ejpam-3384	279	24	young	young	PROPN
ejpam-3384	279	25	’s	’s	PART
ejpam-3384	279	26	inequality	inequality	NOUN
ejpam-3384	279	27	,	,	PUNCT
ejpam-3384	279	28	we	we	PRON
ejpam-3384	279	29	have	have	AUX
ejpam-3384	279	30	‖ab‖	‖ab‖	PRON
ejpam-3384	279	31	≤	≤	PROPN
ejpam-3384	279	32	‖qa	‖qa	PROPN
ejpam-3384	279	33	p	p	NOUN
ejpam-3384	279	34	+	+	CCONJ
ejpam-3384	279	35	pbq	pbq	ADJ
ejpam-3384	280	1	pq	pq	NOUN
ejpam-3384	280	2	‖	‖	VERB
ejpam-3384	280	3	‖a‖‖b‖	‖a‖‖b‖	ADJ
ejpam-3384	280	4	≤	≤	NUM
ejpam-3384	280	5	‖qap	‖qap	NOUN
ejpam-3384	280	6	+	+	NUM
ejpam-3384	280	7	pbq‖	pbq‖	PROPN
ejpam-3384	280	8	‖pq‖	‖pq‖	PROPN
ejpam-3384	280	9	.	.	PUNCT
ejpam-3384	281	1	b.	b.	PROPN
ejpam-3384	281	2	barnes	barnes	PROPN
ejpam-3384	281	3	,	,	PUNCT
ejpam-3384	281	4	c.	c.	PROPN
ejpam-3384	281	5	sebil	sebil	PROPN
ejpam-3384	281	6	,	,	PUNCT
ejpam-3384	281	7	i.	i.	PROPN
ejpam-3384	281	8	k.	k.	PROPN
ejpam-3384	281	9	dontwi	dontwi	PROPN
ejpam-3384	281	10	/	/	SYM
ejpam-3384	281	11	eur	eur	PROPN
ejpam-3384	281	12	.	.	PUNCT
ejpam-3384	282	1	j.	j.	PROPN
ejpam-3384	282	2	pure	pure	PROPN
ejpam-3384	282	3	appl	appl	PROPN
ejpam-3384	282	4	.	.	PROPN
ejpam-3384	282	5	math	math	PROPN
ejpam-3384	282	6	,	,	PUNCT
ejpam-3384	282	7	12	12	NUM
ejpam-3384	282	8	(	(	PUNCT
ejpam-3384	282	9	2	2	NUM
ejpam-3384	282	10	)	)	PUNCT
ejpam-3384	282	11	(	(	PUNCT
ejpam-3384	282	12	2019	2019	NUM
ejpam-3384	282	13	)	)	PUNCT
ejpam-3384	282	14	,	,	PUNCT
ejpam-3384	282	15	469	469	NUM
ejpam-3384	282	16	-	-	SYM
ejpam-3384	282	17	485	485	NUM
ejpam-3384	282	18	482	482	NUM
ejpam-3384	282	19	applying	apply	VERB
ejpam-3384	282	20	the	the	DET
ejpam-3384	282	21	second	second	ADJ
ejpam-3384	282	22	quotient	quotient	NOUN
ejpam-3384	282	23	inequality	inequality	NOUN
ejpam-3384	282	24	on	on	ADP
ejpam-3384	282	25	the	the	DET
ejpam-3384	282	26	right	right	ADJ
ejpam-3384	282	27	hand	hand	NOUN
ejpam-3384	282	28	side	side	NOUN
ejpam-3384	282	29	of	of	ADP
ejpam-3384	282	30	the	the	DET
ejpam-3384	282	31	above	above	ADJ
ejpam-3384	282	32	inequality	inequality	NOUN
ejpam-3384	282	33	yields	yield	VERB
ejpam-3384	282	34	‖a‖‖b‖	‖a‖‖b‖	ADJ
ejpam-3384	282	35	≤	≤	NUM
ejpam-3384	282	36	(	(	PUNCT
ejpam-3384	282	37	‖qap	‖qap	NOUN
ejpam-3384	282	38	+	+	NUM
ejpam-3384	282	39	pbq‖	pbq‖	NOUN
ejpam-3384	282	40	)	)	PUNCT
ejpam-3384	283	1	‖pq‖	‖pq‖	PROPN
ejpam-3384	283	2	‖a‖‖b‖	‖a‖‖b‖	VERB
ejpam-3384	283	3	≤	≤	PUNCT
ejpam-3384	283	4	pq2‖a‖p	pq2‖a‖p	PROPN
ejpam-3384	283	5	+	+	CCONJ
ejpam-3384	284	1	p2q‖b‖q	p2q‖b‖q	PROPN
ejpam-3384	284	2	,	,	PUNCT
ejpam-3384	284	3	p	p	X
ejpam-3384	284	4	,	,	PUNCT
ejpam-3384	284	5	q	q	X
ejpam-3384	284	6	≥	≥	NOUN
ejpam-3384	284	7	1	1	NUM
ejpam-3384	284	8	.	.	PUNCT
ejpam-3384	284	9	theorem	theorem	NOUN
ejpam-3384	284	10	10	10	NUM
ejpam-3384	284	11	.	.	PUNCT
ejpam-3384	285	1	let	let	VERB
ejpam-3384	285	2	t	t	NOUN
ejpam-3384	285	3	:	:	PUNCT
ejpam-3384	285	4	x	x	X
ejpam-3384	285	5	→	→	SYM
ejpam-3384	285	6	y	y	PROPN
ejpam-3384	285	7	is	be	AUX
ejpam-3384	285	8	a	a	DET
ejpam-3384	285	9	banach	banach	NOUN
ejpam-3384	285	10	space	space	NOUN
ejpam-3384	285	11	.	.	PUNCT
ejpam-3384	286	1	then	then	ADV
ejpam-3384	286	2	pq2‖a‖p	pq2‖a‖p	PUNCT
ejpam-3384	286	3	+	+	CCONJ
ejpam-3384	286	4	p2q‖b‖q	p2q‖b‖q	ADJ
ejpam-3384	286	5	≤	≤	PROPN
ejpam-3384	286	6	‖a‖‖b‖,∀	‖a‖‖b‖,∀	NOUN
ejpam-3384	286	7	a	a	PRON
ejpam-3384	286	8	,	,	PUNCT
ejpam-3384	286	9	b	b	X
ejpam-3384	286	10	∈	∈	PROPN
ejpam-3384	286	11	x	x	SYM
ejpam-3384	286	12	p	p	NOUN
ejpam-3384	286	13	,	,	PUNCT
ejpam-3384	286	14	q	q	NOUN
ejpam-3384	286	15	∈	∈	PROPN
ejpam-3384	287	1	[	[	X
ejpam-3384	287	2	0	0	NUM
ejpam-3384	287	3	,	,	PUNCT
ejpam-3384	287	4	1	1	NUM
ejpam-3384	287	5	)	)	PUNCT
ejpam-3384	287	6	.	.	PUNCT
ejpam-3384	288	1	proof	proof	NOUN
ejpam-3384	288	2	:	:	PUNCT
ejpam-3384	288	3	we	we	PRON
ejpam-3384	288	4	see	see	VERB
ejpam-3384	288	5	that	that	PRON
ejpam-3384	288	6	:	:	PUNCT
ejpam-3384	288	7	‖ab‖	‖ab‖	PROPN
ejpam-3384	288	8	≤	≤	PROPN
ejpam-3384	289	1	‖qa	‖qa	VERB
ejpam-3384	289	2	p	p	NOUN
ejpam-3384	289	3	+	+	CCONJ
ejpam-3384	289	4	pbq	pbq	ADJ
ejpam-3384	289	5	pq	pq	NOUN
ejpam-3384	289	6	‖	‖	PROPN
ejpam-3384	289	7	⇒	⇒	NOUN
ejpam-3384	289	8	‖a‖‖b‖	‖a‖‖b‖	ADJ
ejpam-3384	289	9	≤	≤	NUM
ejpam-3384	289	10	‖qap	‖qap	NOUN
ejpam-3384	289	11	+	+	NUM
ejpam-3384	289	12	pbq‖	pbq‖	PROPN
ejpam-3384	289	13	‖pq‖	‖pq‖	PROPN
ejpam-3384	289	14	.	.	PUNCT
ejpam-3384	290	1	applying	apply	VERB
ejpam-3384	290	2	the	the	DET
ejpam-3384	290	3	first	first	ADJ
ejpam-3384	290	4	quotient	quotient	NOUN
ejpam-3384	290	5	inequality	inequality	NOUN
ejpam-3384	290	6	on	on	ADP
ejpam-3384	290	7	the	the	DET
ejpam-3384	290	8	right	right	ADJ
ejpam-3384	290	9	hand	hand	NOUN
ejpam-3384	290	10	side	side	NOUN
ejpam-3384	290	11	of	of	ADP
ejpam-3384	290	12	the	the	DET
ejpam-3384	290	13	above	above	ADJ
ejpam-3384	290	14	inequality	inequality	NOUN
ejpam-3384	290	15	yields	yield	NOUN
ejpam-3384	290	16	(	(	PUNCT
ejpam-3384	290	17	‖qap	‖qap	NOUN
ejpam-3384	290	18	+	+	NUM
ejpam-3384	290	19	pbq‖	pbq‖	NOUN
ejpam-3384	290	20	)	)	PUNCT
ejpam-3384	291	1	‖pq‖	‖pq‖	PROPN
ejpam-3384	291	2	≤	≤	NUM
ejpam-3384	291	3	‖a‖‖b‖	‖a‖‖b‖	ADJ
ejpam-3384	291	4	⇒	⇒	NOUN
ejpam-3384	291	5	pq2‖a‖p	pq2‖a‖p	PROPN
ejpam-3384	292	1	+	+	PUNCT
ejpam-3384	292	2	p2q‖b‖q	p2q‖b‖q	ADJ
ejpam-3384	292	3	≤	≤	NOUN
ejpam-3384	292	4	‖a‖‖b‖	‖a‖‖b‖	ADJ
ejpam-3384	292	5	,	,	PUNCT
ejpam-3384	292	6	p	p	X
ejpam-3384	292	7	,	,	PUNCT
ejpam-3384	292	8	q	q	NOUN
ejpam-3384	292	9	∈	∈	PROPN
ejpam-3384	293	1	[	[	X
ejpam-3384	293	2	0	0	NUM
ejpam-3384	293	3	,	,	PUNCT
ejpam-3384	293	4	1	1	NUM
ejpam-3384	293	5	)	)	PUNCT
ejpam-3384	293	6	.	.	PUNCT
ejpam-3384	294	1	6	6	X
ejpam-3384	294	2	.	.	X
ejpam-3384	295	1	the	the	DET
ejpam-3384	295	2	applications	application	NOUN
ejpam-3384	295	3	of	of	ADP
ejpam-3384	295	4	the	the	DET
ejpam-3384	295	5	index	index	NOUN
ejpam-3384	295	6	power	power	NOUN
ejpam-3384	295	7	quotient	quotient	NOUN
ejpam-3384	295	8	inequalities	inequality	NOUN
ejpam-3384	295	9	to	to	PART
ejpam-3384	295	10	hölder	hölder	VERB
ejpam-3384	295	11	spaces	space	NOUN
ejpam-3384	295	12	in	in	ADP
ejpam-3384	295	13	this	this	DET
ejpam-3384	295	14	section	section	NOUN
ejpam-3384	295	15	,	,	PUNCT
ejpam-3384	295	16	the	the	DET
ejpam-3384	295	17	quotient	quotient	NOUN
ejpam-3384	295	18	inequalities	inequality	NOUN
ejpam-3384	295	19	involving	involve	VERB
ejpam-3384	295	20	index	index	NOUN
ejpam-3384	295	21	are	be	AUX
ejpam-3384	295	22	introduced	introduce	VERB
ejpam-3384	295	23	.	.	PUNCT
ejpam-3384	296	1	theorem	theorem	VERB
ejpam-3384	296	2	11	11	NUM
ejpam-3384	296	3	.	.	PUNCT
ejpam-3384	297	1	let	let	VERB
ejpam-3384	297	2	0	0	PUNCT
ejpam-3384	297	3	<	<	X
ejpam-3384	298	1	p	p	X
ejpam-3384	298	2	<	<	X
ejpam-3384	298	3	+	+	PROPN
ejpam-3384	298	4	∞	∞	NUM
ejpam-3384	298	5	and	and	CCONJ
ejpam-3384	298	6	0	0	NUM
ejpam-3384	299	1	<	<	X
ejpam-3384	299	2	q	q	X
ejpam-3384	299	3	<	<	X
ejpam-3384	299	4	1	1	NUM
ejpam-3384	299	5	such	such	ADJ
ejpam-3384	299	6	that	that	SCONJ
ejpam-3384	299	7	|pq	|pq	NUM
ejpam-3384	299	8	|	|	ADV
ejpam-3384	299	9	<	<	X
ejpam-3384	299	10	1	1	NUM
ejpam-3384	299	11	.	.	PUNCT
ejpam-3384	300	1	then	then	ADV
ejpam-3384	300	2	a	a	DET
ejpam-3384	300	3	mapping	mapping	NOUN
ejpam-3384	300	4	t	t	NOUN
ejpam-3384	300	5	:	:	PUNCT
ejpam-3384	300	6	x	x	X
ejpam-3384	300	7	→	→	SYM
ejpam-3384	300	8	y	y	PROPN
ejpam-3384	300	9	is	be	AUX
ejpam-3384	300	10	hölder	hölder	NOUN
ejpam-3384	300	11	-	-	PUNCT
ejpam-3384	300	12	type	type	NOUN
ejpam-3384	300	13	continuous	continuous	ADJ
ejpam-3384	300	14	of	of	ADP
ejpam-3384	300	15	exponent	exponent	NOUN
ejpam-3384	300	16	p	p	X
ejpam-3384	300	17	q	q	PROPN
ejpam-3384	300	18	at	at	ADP
ejpam-3384	300	19	xo	xo	PROPN
ejpam-3384	300	20	,	,	PUNCT
ejpam-3384	300	21	if	if	SCONJ
ejpam-3384	300	22	‖t	‖t	PROPN
ejpam-3384	300	23	(	(	PUNCT
ejpam-3384	300	24	x)−	x)−	PROPN
ejpam-3384	300	25	t	t	PROPN
ejpam-3384	300	26	(	(	PUNCT
ejpam-3384	300	27	xo)‖	xo)‖	PROPN
ejpam-3384	300	28	≤	≤	PUNCT
ejpam-3384	300	29	l‖x−	l‖x−	NOUN
ejpam-3384	300	30	xo‖	xo‖	PUNCT
ejpam-3384	301	1	p	p	NOUN
ejpam-3384	301	2	q	q	X
ejpam-3384	301	3	,	,	PUNCT
ejpam-3384	301	4	∀	∀	X
ejpam-3384	301	5	x	x	NOUN
ejpam-3384	301	6	,	,	PUNCT
ejpam-3384	301	7	xo	xo	PROPN
ejpam-3384	301	8	∈	∈	PROPN
ejpam-3384	302	1	x	x	X
ejpam-3384	302	2	,	,	PUNCT
ejpam-3384	302	3	where	where	SCONJ
ejpam-3384	302	4	l	l	NOUN
ejpam-3384	302	5	is	be	AUX
ejpam-3384	302	6	the	the	DET
ejpam-3384	302	7	p	p	ADJ
ejpam-3384	302	8	q−th	q−th	PROPN
ejpam-3384	302	9	hölder	hölder	NOUN
ejpam-3384	302	10	coefficient	coefficient	NOUN
ejpam-3384	302	11	of	of	ADP
ejpam-3384	302	12	t	t	PROPN
ejpam-3384	302	13	.	.	PUNCT
ejpam-3384	303	1	proof	proof	NOUN
ejpam-3384	303	2	:	:	PUNCT
ejpam-3384	303	3	by	by	ADP
ejpam-3384	303	4	the	the	DET
ejpam-3384	303	5	hölder	hölder	NOUN
ejpam-3384	303	6	continuity	continuity	NOUN
ejpam-3384	303	7	of	of	ADP
ejpam-3384	303	8	t	t	PROPN
ejpam-3384	303	9	of	of	ADP
ejpam-3384	303	10	exponent	exponent	PROPN
ejpam-3384	303	11	γ	γ	PROPN
ejpam-3384	303	12	,	,	PUNCT
ejpam-3384	303	13	we	we	PRON
ejpam-3384	303	14	have	have	VERB
ejpam-3384	303	15	‖t	‖t	NOUN
ejpam-3384	303	16	(	(	PUNCT
ejpam-3384	303	17	x)−	x)−	PROPN
ejpam-3384	303	18	t	t	PROPN
ejpam-3384	303	19	(	(	PUNCT
ejpam-3384	303	20	xo)‖	xo)‖	PROPN
ejpam-3384	303	21	≤	≤	NOUN
ejpam-3384	303	22	l‖x−	l‖x−	NOUN
ejpam-3384	303	23	xo‖γ	xo‖γ	PROPN
ejpam-3384	303	24	,	,	PUNCT
ejpam-3384	303	25	∀	∀	X
ejpam-3384	303	26	x	x	NOUN
ejpam-3384	303	27	,	,	PUNCT
ejpam-3384	303	28	xo	xo	PROPN
ejpam-3384	303	29	∈	∈	PROPN
ejpam-3384	303	30	x	x	X
ejpam-3384	303	31	and	and	CCONJ
ejpam-3384	303	32	γ	γ	X
ejpam-3384	303	33	∈	∈	PROPN
ejpam-3384	303	34	(	(	PUNCT
ejpam-3384	303	35	0	0	NUM
ejpam-3384	303	36	,	,	PUNCT
ejpam-3384	303	37	1	1	NUM
ejpam-3384	303	38	]	]	PUNCT
ejpam-3384	303	39	.	.	PUNCT
ejpam-3384	304	1	setting	set	VERB
ejpam-3384	304	2	γ	γ	X
ejpam-3384	304	3	=	=	SYM
ejpam-3384	304	4	|p||q|	|p||q|	PROPN
ejpam-3384	304	5	and	and	CCONJ
ejpam-3384	304	6	applying	apply	VERB
ejpam-3384	304	7	the	the	DET
ejpam-3384	304	8	power	power	NOUN
ejpam-3384	304	9	quotient	quotient	NOUN
ejpam-3384	304	10	inequality	inequality	NOUN
ejpam-3384	304	11	,	,	PUNCT
ejpam-3384	304	12	we	we	PRON
ejpam-3384	304	13	obtain	obtain	VERB
ejpam-3384	304	14	‖t	‖t	NOUN
ejpam-3384	304	15	(	(	PUNCT
ejpam-3384	304	16	x)−	x)−	PROPN
ejpam-3384	304	17	t	t	PROPN
ejpam-3384	304	18	(	(	PUNCT
ejpam-3384	304	19	xo)‖	xo)‖	PROPN
ejpam-3384	304	20	≤	≤	PUNCT
ejpam-3384	304	21	l‖x−	l‖x−	NOUN
ejpam-3384	304	22	xo‖pq	xo‖pq	VERB
ejpam-3384	304	23	≤	≤	ADJ
ejpam-3384	304	24	l‖x−	l‖x−	NOUN
ejpam-3384	304	25	xo‖	xo‖	PUNCT
ejpam-3384	305	1	p	p	X
ejpam-3384	305	2	q	q	PROPN
ejpam-3384	305	3	‖t	‖t	NOUN
ejpam-3384	305	4	(	(	PUNCT
ejpam-3384	305	5	x)−	x)−	PROPN
ejpam-3384	305	6	t	t	PROPN
ejpam-3384	305	7	(	(	PUNCT
ejpam-3384	305	8	xo)‖	xo)‖	PROPN
ejpam-3384	305	9	≤	≤	PUNCT
ejpam-3384	305	10	l‖x−	l‖x−	NOUN
ejpam-3384	305	11	xo‖	xo‖	PUNCT
ejpam-3384	306	1	p	p	X
ejpam-3384	306	2	q	q	X
ejpam-3384	306	3	∀	∀	PUNCT
ejpam-3384	306	4	x	x	NOUN
ejpam-3384	306	5	,	,	PUNCT
ejpam-3384	306	6	xo	xo	PROPN
ejpam-3384	306	7	∈	∈	PROPN
ejpam-3384	306	8	x	x	PRON
ejpam-3384	306	9	,	,	PUNCT
ejpam-3384	306	10	∣∣∣p	∣∣∣p	PROPN
ejpam-3384	306	11	q	q	NOUN
ejpam-3384	306	12	∣∣∣	∣∣∣	ADJ
ejpam-3384	306	13	≤	≤	NUM
ejpam-3384	306	14	1	1	NUM
ejpam-3384	306	15	.	.	PUNCT
ejpam-3384	306	16	b.	b.	PROPN
ejpam-3384	306	17	barnes	barnes	PROPN
ejpam-3384	306	18	,	,	PUNCT
ejpam-3384	306	19	c.	c.	PROPN
ejpam-3384	306	20	sebil	sebil	PROPN
ejpam-3384	306	21	,	,	PUNCT
ejpam-3384	306	22	i.	i.	PROPN
ejpam-3384	306	23	k.	k.	PROPN
ejpam-3384	306	24	dontwi	dontwi	PROPN
ejpam-3384	306	25	/	/	SYM
ejpam-3384	306	26	eur	eur	PROPN
ejpam-3384	306	27	.	.	PUNCT
ejpam-3384	307	1	j.	j.	PROPN
ejpam-3384	307	2	pure	pure	PROPN
ejpam-3384	307	3	appl	appl	PROPN
ejpam-3384	307	4	.	.	PROPN
ejpam-3384	307	5	math	math	PROPN
ejpam-3384	307	6	,	,	PUNCT
ejpam-3384	307	7	12	12	NUM
ejpam-3384	307	8	(	(	PUNCT
ejpam-3384	307	9	2	2	NUM
ejpam-3384	307	10	)	)	PUNCT
ejpam-3384	307	11	(	(	PUNCT
ejpam-3384	307	12	2019	2019	NUM
ejpam-3384	307	13	)	)	PUNCT
ejpam-3384	307	14	,	,	PUNCT
ejpam-3384	307	15	469	469	NUM
ejpam-3384	307	16	-	-	SYM
ejpam-3384	307	17	485	485	NUM
ejpam-3384	307	18	483	483	NUM
ejpam-3384	307	19	theorem	theorem	NOUN
ejpam-3384	307	20	12	12	NUM
ejpam-3384	307	21	.	.	PUNCT
ejpam-3384	308	1	let	let	VERB
ejpam-3384	308	2	a	a	PRON
ejpam-3384	308	3	and	and	CCONJ
ejpam-3384	308	4	b	b	NOUN
ejpam-3384	308	5	be	be	AUX
ejpam-3384	308	6	selfadjoint	selfadjoint	NOUN
ejpam-3384	308	7	operators	operator	NOUN
ejpam-3384	308	8	with	with	ADP
ejpam-3384	308	9	sp(a	sp(a	PROPN
ejpam-3384	308	10	)	)	PUNCT
ejpam-3384	308	11	,	,	PUNCT
ejpam-3384	308	12	sp(b	sp(b	PROPN
ejpam-3384	308	13	)	)	PUNCT
ejpam-3384	309	1	⊆	⊆	NUM
ejpam-3384	310	1	[	[	X
ejpam-3384	310	2	m	m	NOUN
ejpam-3384	310	3	,	,	PUNCT
ejpam-3384	310	4	m	m	VERB
ejpam-3384	310	5	]	]	PUNCT
ejpam-3384	310	6	for	for	ADP
ejpam-3384	310	7	some	some	DET
ejpam-3384	310	8	real	real	ADJ
ejpam-3384	310	9	numbers	number	NOUN
ejpam-3384	310	10	m	m	VERB
ejpam-3384	310	11	<	<	X
ejpam-3384	310	12	m	m	VERB
ejpam-3384	310	13	.	.	PUNCT
ejpam-3384	311	1	if	if	SCONJ
ejpam-3384	311	2	f	f	X
ejpam-3384	311	3	:	:	PUNCT
ejpam-3384	312	1	[	[	X
ejpam-3384	312	2	m	m	X
ejpam-3384	312	3	,	,	PUNCT
ejpam-3384	312	4	m	m	VERB
ejpam-3384	312	5	]	]	PUNCT
ejpam-3384	312	6	→	→	PUNCT
ejpam-3384	312	7	r	r	NOUN
ejpam-3384	312	8	is	be	AUX
ejpam-3384	312	9	of	of	ADP
ejpam-3384	312	10	p	p	NOUN
ejpam-3384	312	11	q	q	NOUN
ejpam-3384	312	12	−	−	NOUN
ejpam-3384	312	13	l−hölder	l−hölder	NOUN
ejpam-3384	312	14	type	type	NOUN
ejpam-3384	312	15	.	.	PUNCT
ejpam-3384	313	1	thus	thus	ADV
ejpam-3384	313	2	,	,	PUNCT
ejpam-3384	313	3	for	for	ADP
ejpam-3384	313	4	a	a	DET
ejpam-3384	313	5	given	give	VERB
ejpam-3384	313	6	|p||q|	|p||q|	ADJ
ejpam-3384	313	7	≤	≤	NUM
ejpam-3384	313	8	1	1	NUM
ejpam-3384	313	9	and	and	CCONJ
ejpam-3384	313	10	l	l	NOUN
ejpam-3384	313	11	>	>	X
ejpam-3384	313	12	0	0	NUM
ejpam-3384	313	13	,	,	PUNCT
ejpam-3384	313	14	we	we	PRON
ejpam-3384	313	15	have:∣∣∣f(s)−	have:∣∣∣f(s)−	VERB
ejpam-3384	313	16	f(t	f(t	NOUN
ejpam-3384	313	17	)	)	PUNCT
ejpam-3384	313	18	∣∣∣	∣∣∣	NOUN
ejpam-3384	313	19	≤	≤	NUM
ejpam-3384	314	1	l∣∣∣s−	l∣∣∣s−	PROPN
ejpam-3384	314	2	t∣∣∣	t∣∣∣	PROPN
ejpam-3384	314	3	pq	pq	PROPN
ejpam-3384	314	4	,	,	PUNCT
ejpam-3384	314	5	∀	∀	X
ejpam-3384	314	6	s	s	NOUN
ejpam-3384	314	7	,	,	PUNCT
ejpam-3384	314	8	t	t	PROPN
ejpam-3384	314	9	∈	∈	PROPN
ejpam-3384	315	1	[	[	X
ejpam-3384	315	2	m	m	X
ejpam-3384	315	3	,	,	PUNCT
ejpam-3384	315	4	m	m	VERB
ejpam-3384	315	5	]	]	PUNCT
ejpam-3384	315	6	.	.	PUNCT
ejpam-3384	316	1	then	then	ADV
ejpam-3384	316	2	the	the	DET
ejpam-3384	316	3	ostrowski	ostrowski	ADJ
ejpam-3384	316	4	type	type	NOUN
ejpam-3384	316	5	inequality	inequality	NOUN
ejpam-3384	316	6	for	for	ADP
ejpam-3384	316	7	selfadjoint	selfadjoint	NOUN
ejpam-3384	316	8	operators	operator	NOUN
ejpam-3384	316	9	becomes:∣∣∣f(s)−	becomes:∣∣∣f(s)−	VERB
ejpam-3384	316	10	〈	〈	PROPN
ejpam-3384	316	11	f(a)x	f(a)x	PROPN
ejpam-3384	316	12	,	,	PUNCT
ejpam-3384	316	13	x	x	NOUN
ejpam-3384	316	14	〉	〉	NOUN
ejpam-3384	316	15	∣∣∣	∣∣∣	NOUN
ejpam-3384	316	16	≤	≤	PUNCT
ejpam-3384	317	1	[	[	X
ejpam-3384	317	2	1	1	NUM
ejpam-3384	317	3	2	2	NUM
ejpam-3384	317	4	(	(	PUNCT
ejpam-3384	317	5	m	m	NOUN
ejpam-3384	317	6	−m	−m	NOUN
ejpam-3384	317	7	)	)	PUNCT
ejpam-3384	318	1	+	+	CCONJ
ejpam-3384	318	2	|s−	|s−	ADJ
ejpam-3384	318	3	m+m	m+m	NOUN
ejpam-3384	318	4	2	2	NUM
ejpam-3384	319	1	|	|	ADV
ejpam-3384	319	2	]	]	X
ejpam-3384	320	1	p	p	X
ejpam-3384	320	2	q	q	X
ejpam-3384	320	3	,	,	PUNCT
ejpam-3384	320	4	∀	∀	NOUN
ejpam-3384	320	5	s	s	NOUN
ejpam-3384	320	6	∈	∈	PROPN
ejpam-3384	321	1	[	[	X
ejpam-3384	321	2	m	m	NOUN
ejpam-3384	321	3	,	,	PUNCT
ejpam-3384	321	4	m	m	VERB
ejpam-3384	321	5	]	]	PUNCT
ejpam-3384	321	6	and	and	CCONJ
ejpam-3384	321	7	x	x	PUNCT
ejpam-3384	321	8	∈	∈	NOUN
ejpam-3384	321	9	h	h	NOUN
ejpam-3384	321	10	with	with	ADP
ejpam-3384	321	11	‖x‖	‖x‖	PROPN
ejpam-3384	321	12	=	=	SYM
ejpam-3384	321	13	1	1	X
ejpam-3384	321	14	.	.	PUNCT
ejpam-3384	322	1	moreover	moreover	ADV
ejpam-3384	322	2	,	,	PUNCT
ejpam-3384	322	3	we	we	PRON
ejpam-3384	322	4	have:∣∣∣〈f(b)y	have:∣∣∣〈f(b)y	VERB
ejpam-3384	322	5	,	,	PUNCT
ejpam-3384	322	6	y	y	PROPN
ejpam-3384	322	7	〉	〉	PROPN
ejpam-3384	322	8	−	−	PROPN
ejpam-3384	323	1	〈	〈	PROPN
ejpam-3384	323	2	f(a)x	f(a)x	PROPN
ejpam-3384	323	3	,	,	PUNCT
ejpam-3384	323	4	x	x	NOUN
ejpam-3384	323	5	〉	〉	NOUN
ejpam-3384	323	6	∣∣∣	∣∣∣	NOUN
ejpam-3384	323	7	≤	≤	NUM
ejpam-3384	323	8	〈	〈	PROPN
ejpam-3384	323	9	∣∣∣f(b)−	∣∣∣f(b)−	NOUN
ejpam-3384	323	10	〈	〈	PROPN
ejpam-3384	323	11	f(a)x	f(a)x	PROPN
ejpam-3384	323	12	,	,	PUNCT
ejpam-3384	323	13	x	x	NOUN
ejpam-3384	323	14	〉	〉	NUM
ejpam-3384	323	15	·	·	PUNCT
ejpam-3384	323	16	1h	1h	NUM
ejpam-3384	323	17	∣∣∣	∣∣∣	PROPN
ejpam-3384	323	18	y	y	PROPN
ejpam-3384	323	19	,	,	PUNCT
ejpam-3384	323	20	y	y	PROPN
ejpam-3384	323	21	〉	〉	NOUN
ejpam-3384	323	22	.	.	PUNCT
ejpam-3384	324	1	≤	≤	NUM
ejpam-3384	324	2	l	l	NOUN
ejpam-3384	325	1	[	[	X
ejpam-3384	325	2	1	1	NUM
ejpam-3384	325	3	2	2	NUM
ejpam-3384	325	4	(	(	PUNCT
ejpam-3384	325	5	m	m	NOUN
ejpam-3384	325	6	−m	−m	NOUN
ejpam-3384	325	7	)	)	PUNCT
ejpam-3384	325	8	+	+	CCONJ
ejpam-3384	325	9	〈	〈	PROPN
ejpam-3384	325	10	|b	|b	ADJ
ejpam-3384	325	11	−	−	PROPN
ejpam-3384	325	12	m+m	m+m	PROPN
ejpam-3384	325	13	2	2	NUM
ejpam-3384	325	14	·	·	SYM
ejpam-3384	325	15	1h	1h	NUM
ejpam-3384	325	16	|y	|y	NOUN
ejpam-3384	325	17	,	,	PUNCT
ejpam-3384	325	18	y	y	PROPN
ejpam-3384	325	19	〉	〉	NOUN
ejpam-3384	325	20	]	]	PUNCT
ejpam-3384	325	21	p	p	X
ejpam-3384	325	22	q	q	X
ejpam-3384	325	23	,	,	PUNCT
ejpam-3384	325	24	∀	∀	X
ejpam-3384	325	25	x	x	NOUN
ejpam-3384	325	26	,	,	PUNCT
ejpam-3384	325	27	y	y	PROPN
ejpam-3384	325	28	∈	∈	PROPN
ejpam-3384	325	29	h	h	NOUN
ejpam-3384	325	30	with	with	ADP
ejpam-3384	325	31	‖x‖	‖x‖	PROPN
ejpam-3384	325	32	=	=	SYM
ejpam-3384	325	33	‖y‖	‖y‖	PROPN
ejpam-3384	325	34	=	=	SYM
ejpam-3384	325	35	1	1	NUM
ejpam-3384	325	36	,	,	PUNCT
ejpam-3384	325	37	and	and	CCONJ
ejpam-3384	325	38	h	h	NOUN
ejpam-3384	325	39	denotes	denote	VERB
ejpam-3384	325	40	hilbert	hilbert	PROPN
ejpam-3384	325	41	space	space	NOUN
ejpam-3384	325	42	.	.	PUNCT
ejpam-3384	326	1	proof	proof	NOUN
ejpam-3384	326	2	:	:	PUNCT
ejpam-3384	326	3	we	we	PRON
ejpam-3384	326	4	can	can	AUX
ejpam-3384	326	5	see	see	VERB
ejpam-3384	326	6	that	that	PRON
ejpam-3384	326	7	,	,	PUNCT
ejpam-3384	326	8	using	use	VERB
ejpam-3384	326	9	the	the	DET
ejpam-3384	326	10	ostrowski	ostrowski	ADJ
ejpam-3384	326	11	-	-	PUNCT
ejpam-3384	326	12	type	type	NOUN
ejpam-3384	326	13	inequality	inequality	NOUN
ejpam-3384	326	14	for	for	ADP
ejpam-3384	326	15	the	the	DET
ejpam-3384	326	16	riemann	riemann	PROPN
ejpam-3384	326	17	-	-	PUNCT
ejpam-3384	326	18	stieltjes	stieltjes	PROPN
ejpam-3384	326	19	integral	integral	ADJ
ejpam-3384	326	20	,	,	PUNCT
ejpam-3384	326	21	we	we	PRON
ejpam-3384	326	22	have	have	AUX
ejpam-3384	326	23	:	:	PUNCT
ejpam-3384	326	24	∣∣∣f(s)[u(b)−	∣∣∣f(s)[u(b)−	VERB
ejpam-3384	326	25	u(a)]−	u(a)]−	ADV
ejpam-3384	326	26	∫	∫	PROPN
ejpam-3384	326	27	b	b	PROPN
ejpam-3384	326	28	a	a	DET
ejpam-3384	326	29	f(t)du(t	f(t)du(t	NOUN
ejpam-3384	326	30	)	)	PUNCT
ejpam-3384	326	31	∣∣∣	∣∣∣	ADJ
ejpam-3384	326	32	≤	≤	X
ejpam-3384	326	33	l[1	l[1	ADJ
ejpam-3384	326	34	2	2	NUM
ejpam-3384	326	35	(	(	PUNCT
ejpam-3384	326	36	b−	b−	NOUN
ejpam-3384	326	37	a	a	PRON
ejpam-3384	326	38	)	)	PUNCT
ejpam-3384	327	1	+	+	CCONJ
ejpam-3384	327	2	|s−	|s−	ADJ
ejpam-3384	327	3	a+	a+	PRON
ejpam-3384	327	4	b	b	X
ejpam-3384	327	5	2	2	NUM
ejpam-3384	327	6	|	|	NOUN
ejpam-3384	327	7	]	]	X
ejpam-3384	327	8	r	r	NOUN
ejpam-3384	327	9	b∨	b∨	PROPN
ejpam-3384	327	10	a	a	DET
ejpam-3384	327	11	(	(	PUNCT
ejpam-3384	327	12	u	u	NOUN
ejpam-3384	327	13	)	)	PUNCT
ejpam-3384	327	14	.	.	PUNCT
ejpam-3384	328	1	setting	set	VERB
ejpam-3384	328	2	r	r	NOUN
ejpam-3384	328	3	=	=	PUNCT
ejpam-3384	328	4	|p||q|	|p||q|	ADJ
ejpam-3384	328	5	≤	≤	NUM
ejpam-3384	328	6	1	1	NUM
ejpam-3384	328	7	and	and	CCONJ
ejpam-3384	328	8	applying	apply	VERB
ejpam-3384	328	9	the	the	DET
ejpam-3384	328	10	first	first	ADJ
ejpam-3384	328	11	index	index	NOUN
ejpam-3384	328	12	power	power	NOUN
ejpam-3384	328	13	quotient	quotient	NOUN
ejpam-3384	328	14	inequality	inequality	NOUN
ejpam-3384	328	15	,	,	PUNCT
ejpam-3384	328	16	we	we	PRON
ejpam-3384	328	17	obtain	obtain	VERB
ejpam-3384	328	18	∣∣∣f(s)[u(b)−	∣∣∣f(s)[u(b)−	PRON
ejpam-3384	328	19	u(a)]−	u(a)]−	ADV
ejpam-3384	328	20	∫	∫	PROPN
ejpam-3384	328	21	b	b	PROPN
ejpam-3384	328	22	a	a	DET
ejpam-3384	328	23	f(t)du(t	f(t)du(t	NOUN
ejpam-3384	328	24	)	)	PUNCT
ejpam-3384	328	25	∣∣∣	∣∣∣	NOUN
ejpam-3384	328	26	≤	≤	NUM
ejpam-3384	328	27	l	l	NOUN
ejpam-3384	329	1	[	[	X
ejpam-3384	329	2	1	1	NUM
ejpam-3384	329	3	2	2	NUM
ejpam-3384	329	4	(	(	PUNCT
ejpam-3384	329	5	b−	b−	NOUN
ejpam-3384	329	6	a	a	PRON
ejpam-3384	329	7	)	)	PUNCT
ejpam-3384	330	1	+	+	CCONJ
ejpam-3384	330	2	|s−	|s−	ADJ
ejpam-3384	330	3	a+	a+	PRON
ejpam-3384	330	4	b	b	X
ejpam-3384	330	5	2	2	NUM
ejpam-3384	330	6	|	|	NOUN
ejpam-3384	330	7	]	]	X
ejpam-3384	330	8	pq	pq	PROPN
ejpam-3384	330	9	b∨	b∨	PROPN
ejpam-3384	330	10	a	a	DET
ejpam-3384	330	11	(	(	PUNCT
ejpam-3384	330	12	u	u	NOUN
ejpam-3384	330	13	)	)	PUNCT
ejpam-3384	330	14	≤	≤	PUNCT
ejpam-3384	330	15	l	l	NOUN
ejpam-3384	331	1	[	[	X
ejpam-3384	331	2	1	1	NUM
ejpam-3384	331	3	2	2	NUM
ejpam-3384	331	4	(	(	PUNCT
ejpam-3384	331	5	b−	b−	NOUN
ejpam-3384	331	6	a	a	PRON
ejpam-3384	331	7	)	)	PUNCT
ejpam-3384	332	1	+	+	CCONJ
ejpam-3384	332	2	|s−	|s−	ADJ
ejpam-3384	332	3	a+	a+	PRON
ejpam-3384	332	4	b	b	X
ejpam-3384	332	5	2	2	NUM
ejpam-3384	332	6	]	]	PUNCT
ejpam-3384	332	7	p	p	X
ejpam-3384	332	8	q	q	PROPN
ejpam-3384	332	9	b∨	b∨	PROPN
ejpam-3384	332	10	a	a	DET
ejpam-3384	332	11	(	(	PUNCT
ejpam-3384	332	12	u	u	NOUN
ejpam-3384	332	13	)	)	PUNCT
ejpam-3384	332	14	∣∣∣f(s)[u(b)−	∣∣∣f(s)[u(b)−	VERB
ejpam-3384	332	15	u(a)]−	u(a)]−	ADV
ejpam-3384	332	16	∫	∫	PROPN
ejpam-3384	332	17	b	b	PROPN
ejpam-3384	332	18	a	a	DET
ejpam-3384	332	19	f(t)du(t	f(t)du(t	NOUN
ejpam-3384	332	20	)	)	PUNCT
ejpam-3384	332	21	∣∣∣	∣∣∣	NOUN
ejpam-3384	332	22	≤	≤	NUM
ejpam-3384	332	23	l	l	NOUN
ejpam-3384	333	1	[	[	X
ejpam-3384	333	2	1	1	NUM
ejpam-3384	333	3	2	2	NUM
ejpam-3384	333	4	(	(	PUNCT
ejpam-3384	333	5	b−	b−	NOUN
ejpam-3384	333	6	a	a	PRON
ejpam-3384	333	7	)	)	PUNCT
ejpam-3384	334	1	+	+	CCONJ
ejpam-3384	334	2	|s−	|s−	ADJ
ejpam-3384	334	3	a+	a+	DET
ejpam-3384	334	4	b	b	X
ejpam-3384	334	5	2	2	NUM
ejpam-3384	334	6	|	|	NOUN
ejpam-3384	334	7	]	]	X
ejpam-3384	334	8	p	p	X
ejpam-3384	334	9	q	q	PROPN
ejpam-3384	334	10	b∨	b∨	PROPN
ejpam-3384	334	11	a	a	DET
ejpam-3384	334	12	(	(	PUNCT
ejpam-3384	334	13	u	u	NOUN
ejpam-3384	334	14	)	)	PUNCT
ejpam-3384	334	15	,	,	PUNCT
ejpam-3384	334	16	s	s	VERB
ejpam-3384	334	17	∈	∈	PROPN
ejpam-3384	335	1	[	[	X
ejpam-3384	335	2	a	a	X
ejpam-3384	335	3	,	,	PUNCT
ejpam-3384	335	4	b	b	NOUN
ejpam-3384	335	5	]	]	X
ejpam-3384	335	6	,	,	PUNCT
ejpam-3384	335	7	u	u	NOUN
ejpam-3384	335	8	is	be	AUX
ejpam-3384	335	9	a	a	DET
ejpam-3384	335	10	bounded	bounded	ADJ
ejpam-3384	335	11	variation	variation	NOUN
ejpam-3384	335	12	on	on	ADP
ejpam-3384	335	13	[	[	X
ejpam-3384	335	14	a	a	X
ejpam-3384	335	15	,	,	PUNCT
ejpam-3384	335	16	b	b	NOUN
ejpam-3384	335	17	]	]	PUNCT
ejpam-3384	335	18	and	and	CCONJ
ejpam-3384	335	19	∨b	∨b	NOUN
ejpam-3384	335	20	a	a	PRON
ejpam-3384	335	21	is	be	AUX
ejpam-3384	335	22	the	the	DET
ejpam-3384	335	23	total	total	ADJ
ejpam-3384	335	24	variation	variation	NOUN
ejpam-3384	335	25	of	of	ADP
ejpam-3384	335	26	u	u	NOUN
ejpam-3384	335	27	on	on	ADP
ejpam-3384	335	28	[	[	X
ejpam-3384	335	29	a	a	X
ejpam-3384	335	30	,	,	PUNCT
ejpam-3384	335	31	b	b	NOUN
ejpam-3384	335	32	]	]	X
ejpam-3384	335	33	.	.	PUNCT
ejpam-3384	336	1	then	then	ADV
ejpam-3384	336	2	the	the	DET
ejpam-3384	336	3	functional	functional	ADJ
ejpam-3384	336	4	f(t	f(t	NOUN
ejpam-3384	336	5	)	)	PUNCT
ejpam-3384	336	6	is	be	AUX
ejpam-3384	336	7	of	of	ADP
ejpam-3384	336	8	p	p	NOUN
ejpam-3384	336	9	q	q	NOUN
ejpam-3384	336	10	−	−	NOUN
ejpam-3384	336	11	l−hölder	l−hölder	NOUN
ejpam-3384	336	12	-	-	PUNCT
ejpam-3384	336	13	type	type	NOUN
ejpam-3384	336	14	on	on	ADP
ejpam-3384	336	15	[	[	X
ejpam-3384	336	16	a	a	X
ejpam-3384	336	17	,	,	PUNCT
ejpam-3384	336	18	b	b	NOUN
ejpam-3384	336	19	]	]	X
ejpam-3384	336	20	.	.	PUNCT
ejpam-3384	337	1	also	also	ADV
ejpam-3384	337	2	,	,	PUNCT
ejpam-3384	337	3	setting	set	VERB
ejpam-3384	337	4	u(λ	u(λ	PROPN
ejpam-3384	337	5	)	)	PUNCT
ejpam-3384	337	6	=	=	SYM
ejpam-3384	337	7	gx(λ	gx(λ	X
ejpam-3384	337	8	)	)	PUNCT
ejpam-3384	337	9	=	=	PUNCT
ejpam-3384	338	1	〈	〈	PROPN
ejpam-3384	338	2	eλx	eλx	NOUN
ejpam-3384	338	3	,	,	PUNCT
ejpam-3384	338	4	x	x	PROPN
ejpam-3384	338	5	〉	〉	NOUN
ejpam-3384	338	6	,	,	PUNCT
ejpam-3384	338	7	where	where	SCONJ
ejpam-3384	338	8	x	x	PUNCT
ejpam-3384	338	9	∈	∈	PROPN
ejpam-3384	338	10	h	h	NOUN
ejpam-3384	338	11	with	with	ADP
ejpam-3384	338	12	‖x‖	‖x‖	PROPN
ejpam-3384	338	13	=	=	SYM
ejpam-3384	338	14	1	1	NUM
ejpam-3384	338	15	,	,	PUNCT
ejpam-3384	338	16	then	then	ADV
ejpam-3384	338	17	∣∣∣f(s)−	∣∣∣f(s)−	PROPN
ejpam-3384	338	18	∫	∫	PROPN
ejpam-3384	338	19	m	m	PROPN
ejpam-3384	338	20	m	m	VERB
ejpam-3384	338	21	f(λ)d(〈eλx	f(λ)d(〈eλx	NOUN
ejpam-3384	338	22	,	,	PUNCT
ejpam-3384	338	23	x	x	NOUN
ejpam-3384	338	24	〉	〉	NOUN
ejpam-3384	338	25	)	)	PUNCT
ejpam-3384	338	26	∣∣∣	∣∣∣	NOUN
ejpam-3384	338	27	≤	≤	NUM
ejpam-3384	339	1	l	l	NOUN
ejpam-3384	340	1	[	[	X
ejpam-3384	340	2	1	1	NUM
ejpam-3384	340	3	2	2	NUM
ejpam-3384	340	4	(	(	PUNCT
ejpam-3384	340	5	m	m	NOUN
ejpam-3384	340	6	−m	−m	NOUN
ejpam-3384	340	7	)	)	PUNCT
ejpam-3384	341	1	+	+	CCONJ
ejpam-3384	341	2	|s−	|s−	ADJ
ejpam-3384	341	3	m+m	m+m	NOUN
ejpam-3384	341	4	2	2	NUM
ejpam-3384	341	5	|	|	NOUN
ejpam-3384	341	6	]	]	X
ejpam-3384	341	7	pq	pq	NOUN
ejpam-3384	341	8	m∨	m∨	PROPN
ejpam-3384	341	9	m	m	PROPN
ejpam-3384	341	10	(	(	PUNCT
ejpam-3384	341	11	g(x	g(x	NOUN
ejpam-3384	341	12	)	)	PUNCT
ejpam-3384	341	13	)	)	PUNCT
ejpam-3384	341	14	references	reference	VERB
ejpam-3384	341	15	484	484	NUM
ejpam-3384	341	16	≤	≤	NUM
ejpam-3384	341	17	l	l	NOUN
ejpam-3384	342	1	[	[	X
ejpam-3384	342	2	1	1	NUM
ejpam-3384	342	3	2	2	NUM
ejpam-3384	342	4	(	(	PUNCT
ejpam-3384	342	5	m	m	NOUN
ejpam-3384	342	6	−m	−m	NOUN
ejpam-3384	342	7	)	)	PUNCT
ejpam-3384	343	1	+	+	CCONJ
ejpam-3384	343	2	|s−	|s−	ADJ
ejpam-3384	343	3	m+m	m+m	NOUN
ejpam-3384	343	4	2	2	NUM
ejpam-3384	344	1	|	|	ADV
ejpam-3384	344	2	]	]	X
ejpam-3384	345	1	p	p	X
ejpam-3384	345	2	q	q	X
ejpam-3384	345	3	m∨	m∨	PROPN
ejpam-3384	345	4	m	m	PROPN
ejpam-3384	345	5	(	(	PUNCT
ejpam-3384	345	6	g(x	g(x	NOUN
ejpam-3384	345	7	)	)	PUNCT
ejpam-3384	345	8	)	)	PUNCT
ejpam-3384	346	1	∣∣∣f(s)−	∣∣∣f(s)−	PROPN
ejpam-3384	346	2	∫	∫	PROPN
ejpam-3384	346	3	m	m	PROPN
ejpam-3384	346	4	m	m	VERB
ejpam-3384	346	5	f(λ)d(〈eλx	f(λ)d(〈eλx	NOUN
ejpam-3384	346	6	,	,	PUNCT
ejpam-3384	346	7	x	x	NOUN
ejpam-3384	346	8	〉	〉	NOUN
ejpam-3384	346	9	)	)	PUNCT
ejpam-3384	346	10	∣∣∣	∣∣∣	NOUN
ejpam-3384	346	11	≤	≤	NUM
ejpam-3384	347	1	l	l	NOUN
ejpam-3384	348	1	[	[	X
ejpam-3384	348	2	1	1	NUM
ejpam-3384	348	3	2	2	NUM
ejpam-3384	348	4	(	(	PUNCT
ejpam-3384	348	5	m	m	NOUN
ejpam-3384	348	6	−m	−m	NOUN
ejpam-3384	348	7	)	)	PUNCT
ejpam-3384	349	1	+	+	CCONJ
ejpam-3384	349	2	|s−	|s−	ADJ
ejpam-3384	349	3	m+m	m+m	NOUN
ejpam-3384	349	4	2	2	NUM
ejpam-3384	350	1	|	|	ADV
ejpam-3384	350	2	]	]	X
ejpam-3384	351	1	p	p	X
ejpam-3384	351	2	q	q	X
ejpam-3384	351	3	m∨	m∨	PROPN
ejpam-3384	351	4	m	m	PROPN
ejpam-3384	351	5	(	(	PUNCT
ejpam-3384	351	6	g(x	g(x	NOUN
ejpam-3384	351	7	)	)	PUNCT
ejpam-3384	351	8	)	)	PUNCT
ejpam-3384	351	9	.	.	PUNCT
ejpam-3384	352	1	again	again	ADV
ejpam-3384	352	2	,	,	PUNCT
ejpam-3384	352	3	we	we	PRON
ejpam-3384	352	4	see	see	VERB
ejpam-3384	352	5	that	that	PRON
ejpam-3384	352	6	:	:	PUNCT
ejpam-3384	352	7	〈	〈	PROPN
ejpam-3384	352	8	|f(b)−	|f(b)−	NOUN
ejpam-3384	352	9	〈	〈	PROPN
ejpam-3384	352	10	f(a)x	f(a)x	PROPN
ejpam-3384	352	11	,	,	PUNCT
ejpam-3384	352	12	x	x	NOUN
ejpam-3384	352	13	〉	〉	NUM
ejpam-3384	352	14	·	·	PUNCT
ejpam-3384	352	15	1h	1h	NUM
ejpam-3384	352	16	|y	|y	NOUN
ejpam-3384	352	17	,	,	PUNCT
ejpam-3384	352	18	y	y	PROPN
ejpam-3384	352	19	〉	〉	NOUN
ejpam-3384	352	20	≤	≤	PUNCT
ejpam-3384	352	21	〈	〈	PROPN
ejpam-3384	352	22	[	[	X
ejpam-3384	352	23	1	1	NUM
ejpam-3384	352	24	2	2	NUM
ejpam-3384	352	25	(	(	PUNCT
ejpam-3384	352	26	m	m	NOUN
ejpam-3384	352	27	−m	−m	NOUN
ejpam-3384	352	28	)	)	PUNCT
ejpam-3384	353	1	+	+	CCONJ
ejpam-3384	353	2	|b	|b	ADJ
ejpam-3384	353	3	−	−	PROPN
ejpam-3384	353	4	m+m	m+m	PROPN
ejpam-3384	353	5	2	2	NUM
ejpam-3384	353	6	·	·	SYM
ejpam-3384	353	7	1h	1h	NUM
ejpam-3384	354	1	|	|	ADV
ejpam-3384	354	2	]	]	X
ejpam-3384	354	3	pq	pq	NOUN
ejpam-3384	354	4	y	y	PROPN
ejpam-3384	354	5	,	,	PUNCT
ejpam-3384	354	6	y	y	PROPN
ejpam-3384	354	7	〉	〉	NOUN
ejpam-3384	354	8	≤	≤	PUNCT
ejpam-3384	354	9	〈	〈	PROPN
ejpam-3384	354	10	[	[	X
ejpam-3384	354	11	1	1	NUM
ejpam-3384	354	12	2	2	NUM
ejpam-3384	354	13	(	(	PUNCT
ejpam-3384	354	14	m	m	NOUN
ejpam-3384	354	15	−m	−m	NOUN
ejpam-3384	354	16	)	)	PUNCT
ejpam-3384	355	1	+	+	CCONJ
ejpam-3384	355	2	|b	|b	ADJ
ejpam-3384	355	3	−	−	PROPN
ejpam-3384	355	4	m+m	m+m	PROPN
ejpam-3384	355	5	2	2	NUM
ejpam-3384	355	6	·	·	SYM
ejpam-3384	355	7	1h	1h	NUM
ejpam-3384	355	8	|	|	ADV
ejpam-3384	355	9	]	]	X
ejpam-3384	355	10	p	p	X
ejpam-3384	355	11	q	q	PROPN
ejpam-3384	355	12	y	y	PROPN
ejpam-3384	355	13	,	,	PUNCT
ejpam-3384	355	14	y	y	PROPN
ejpam-3384	355	15	〉	〉	NOUN
ejpam-3384	355	16	,	,	PUNCT
ejpam-3384	355	17	∀	∀	X
ejpam-3384	355	18	x	x	NOUN
ejpam-3384	355	19	,	,	PUNCT
ejpam-3384	355	20	y	y	PROPN
ejpam-3384	355	21	∈	∈	PROPN
ejpam-3384	355	22	h	h	NOUN
ejpam-3384	355	23	,	,	PUNCT
ejpam-3384	355	24	‖x‖	‖x‖	PROPN
ejpam-3384	355	25	=	=	SYM
ejpam-3384	356	1	‖y‖	‖y‖	PROPN
ejpam-3384	356	2	=	=	NOUN
ejpam-3384	356	3	1	1	X
ejpam-3384	356	4	.	.	PUNCT
ejpam-3384	357	1	this	this	PRON
ejpam-3384	357	2	completes	complete	VERB
ejpam-3384	357	3	the	the	DET
ejpam-3384	357	4	proof	proof	NOUN
ejpam-3384	357	5	.	.	PUNCT
ejpam-3384	358	1	7	7	X
ejpam-3384	358	2	.	.	X
ejpam-3384	358	3	conclusion	conclusion	NOUN
ejpam-3384	358	4	in	in	ADP
ejpam-3384	358	5	a	a	DET
ejpam-3384	358	6	nutshell	nutshell	NOUN
ejpam-3384	358	7	,	,	PUNCT
ejpam-3384	358	8	the	the	DET
ejpam-3384	358	9	first	first	ADJ
ejpam-3384	358	10	and	and	CCONJ
ejpam-3384	358	11	second	second	ADJ
ejpam-3384	358	12	quotient	quotient	NOUN
ejpam-3384	358	13	inequalities	inequality	NOUN
ejpam-3384	358	14	are	be	AUX
ejpam-3384	358	15	introduced	introduce	VERB
ejpam-3384	358	16	which	which	PRON
ejpam-3384	358	17	extend	extend	VERB
ejpam-3384	358	18	some	some	DET
ejpam-3384	358	19	findings	finding	NOUN
ejpam-3384	358	20	in	in	ADP
ejpam-3384	358	21	functional	functional	ADJ
ejpam-3384	358	22	spaces	space	NOUN
ejpam-3384	358	23	such	such	ADJ
ejpam-3384	358	24	as	as	ADP
ejpam-3384	358	25	lp	lp	PROPN
ejpam-3384	358	26	space	space	NOUN
ejpam-3384	358	27	,	,	PUNCT
ejpam-3384	358	28	hilbert	hilbert	NOUN
ejpam-3384	358	29	space	space	NOUN
ejpam-3384	358	30	,	,	PUNCT
ejpam-3384	358	31	unitary	unitary	ADJ
ejpam-3384	358	32	space	space	NOUN
ejpam-3384	358	33	,	,	PUNCT
ejpam-3384	358	34	hölder	hölder	NOUN
ejpam-3384	358	35	spaces	space	NOUN
ejpam-3384	358	36	and	and	CCONJ
ejpam-3384	358	37	sobolev	sobolev	NOUN
ejpam-3384	358	38	spaces	space	NOUN
ejpam-3384	358	39	.	.	PUNCT
ejpam-3384	359	1	the	the	DET
ejpam-3384	359	2	relationship	relationship	NOUN
ejpam-3384	359	3	between	between	ADP
ejpam-3384	359	4	the	the	DET
ejpam-3384	359	5	inequality	inequality	NOUN
ejpam-3384	359	6	involving	involve	VERB
ejpam-3384	359	7	norm	norm	NOUN
ejpam-3384	359	8	of	of	ADP
ejpam-3384	359	9	quotient	quotient	NOUN
ejpam-3384	359	10	of	of	ADP
ejpam-3384	359	11	the	the	DET
ejpam-3384	359	12	vectors	vector	NOUN
ejpam-3384	359	13	or	or	CCONJ
ejpam-3384	359	14	functions	function	NOUN
ejpam-3384	359	15	and	and	CCONJ
ejpam-3384	359	16	its	its	PRON
ejpam-3384	359	17	product	product	NOUN
ejpam-3384	359	18	counterpart	counterpart	NOUN
ejpam-3384	359	19	has	have	AUX
ejpam-3384	359	20	not	not	PART
ejpam-3384	359	21	been	be	AUX
ejpam-3384	359	22	observed	observe	VERB
ejpam-3384	359	23	over	over	ADP
ejpam-3384	359	24	the	the	DET
ejpam-3384	359	25	years	year	NOUN
ejpam-3384	359	26	and	and	CCONJ
ejpam-3384	359	27	vice	vice	NOUN
ejpam-3384	359	28	versa	versa	ADV
ejpam-3384	359	29	.	.	PUNCT
ejpam-3384	360	1	these	these	DET
ejpam-3384	360	2	inequalities	inequality	NOUN
ejpam-3384	360	3	do	do	AUX
ejpam-3384	360	4	not	not	PART
ejpam-3384	360	5	only	only	ADV
ejpam-3384	360	6	establish	establish	VERB
ejpam-3384	360	7	the	the	DET
ejpam-3384	360	8	relationship	relationship	NOUN
ejpam-3384	360	9	between	between	ADP
ejpam-3384	360	10	two	two	NUM
ejpam-3384	360	11	mathematical	mathematical	ADJ
ejpam-3384	360	12	structures	structure	NOUN
ejpam-3384	360	13	but	but	CCONJ
ejpam-3384	360	14	also	also	ADV
ejpam-3384	360	15	,	,	PUNCT
ejpam-3384	360	16	useful	useful	ADJ
ejpam-3384	360	17	for	for	ADP
ejpam-3384	360	18	establishing	establish	VERB
ejpam-3384	360	19	many	many	ADJ
ejpam-3384	360	20	functional	functional	ADJ
ejpam-3384	360	21	properties	property	NOUN
ejpam-3384	360	22	such	such	ADJ
ejpam-3384	360	23	as	as	ADP
ejpam-3384	360	24	continuity	continuity	NOUN
ejpam-3384	360	25	,	,	PUNCT
ejpam-3384	360	26	boundedness	boundedness	NOUN
ejpam-3384	360	27	,	,	PUNCT
ejpam-3384	360	28	expansive	expansive	ADJ
ejpam-3384	360	29	of	of	ADP
ejpam-3384	360	30	the	the	DET
ejpam-3384	360	31	operators	operator	NOUN
ejpam-3384	360	32	in	in	ADP
ejpam-3384	360	33	functional	functional	ADJ
ejpam-3384	360	34	spaces	space	NOUN
ejpam-3384	360	35	.	.	PUNCT
ejpam-3384	361	1	references	reference	NOUN
ejpam-3384	361	2	[	[	X
ejpam-3384	361	3	1	1	NUM
ejpam-3384	361	4	]	]	PUNCT
ejpam-3384	361	5	w.	w.	PROPN
ejpam-3384	361	6	j.	j.	PROPN
ejpam-3384	361	7	davis	davis	PROPN
ejpam-3384	361	8	,	,	PUNCT
ejpam-3384	361	9	d.	d.	PROPN
ejpam-3384	361	10	j.	j.	PROPN
ejpam-3384	361	11	h.	h.	PROPN
ejpam-3384	361	12	garling	garling	PROPN
ejpam-3384	361	13	and	and	CCONJ
ejpam-3384	361	14	n.	n.	PROPN
ejpam-3384	361	15	tomczak	tomczak	NOUN
ejpam-3384	361	16	-	-	PUNCT
ejpam-3384	361	17	jaegermann	jaegermann	PROPN
ejpam-3384	361	18	,	,	PUNCT
ejpam-3384	361	19	the	the	DET
ejpam-3384	361	20	complex	complex	NOUN
ejpam-3384	361	21	of	of	ADP
ejpam-3384	361	22	quasinormed	quasinorme	VERB
ejpam-3384	361	23	linear	linear	PROPN
ejpam-3384	361	24	spaces	space	NOUN
ejpam-3384	361	25	journal	journal	NOUN
ejpam-3384	361	26	of	of	ADP
ejpam-3384	361	27	functional	functional	ADJ
ejpam-3384	361	28	analysis	analysis	NOUN
ejpam-3384	361	29	,	,	PUNCT
ejpam-3384	361	30	55	55	NUM
ejpam-3384	361	31	,	,	PUNCT
ejpam-3384	361	32	(	(	PUNCT
ejpam-3384	361	33	1984	1984	NUM
ejpam-3384	361	34	)	)	PUNCT
ejpam-3384	361	35	.	.	PUNCT
ejpam-3384	362	1	pp	pp	ADJ
ejpam-3384	362	2	.	.	PUNCT
ejpam-3384	363	1	110	110	NUM
ejpam-3384	363	2	-	-	SYM
ejpam-3384	363	3	150	150	NUM
ejpam-3384	363	4	.	.	PUNCT
ejpam-3384	364	1	[	[	X
ejpam-3384	364	2	2	2	NUM
ejpam-3384	364	3	]	]	PUNCT
ejpam-3384	364	4	m.	m.	NOUN
ejpam-3384	364	5	agueh	agueh	PROPN
ejpam-3384	364	6	,	,	PUNCT
ejpam-3384	364	7	n.	n.	PROPN
ejpam-3384	364	8	ghoussoub	ghoussoub	PROPN
ejpam-3384	364	9	and	and	CCONJ
ejpam-3384	364	10	x.	x.	PROPN
ejpam-3384	364	11	kang	kang	PROPN
ejpam-3384	364	12	,	,	PUNCT
ejpam-3384	364	13	geometric	geometric	ADJ
ejpam-3384	364	14	inequalities	inequality	NOUN
ejpam-3384	364	15	via	via	ADP
ejpam-3384	364	16	a	a	DET
ejpam-3384	364	17	general	general	ADJ
ejpam-3384	364	18	comparison	comparison	NOUN
ejpam-3384	364	19	principle	principle	NOUN
ejpam-3384	364	20	for	for	ADP
ejpam-3384	364	21	interacting	interact	VERB
ejpam-3384	364	22	gases	gas	NOUN
ejpam-3384	364	23	,	,	PUNCT
ejpam-3384	364	24	geom	geom	PROPN
ejpam-3384	364	25	.	.	PUNCT
ejpam-3384	365	1	funct	funct	PROPN
ejpam-3384	365	2	.	.	PUNCT
ejpam-3384	366	1	anal	anal	PROPN
ejpam-3384	366	2	.	.	PROPN
ejpam-3384	366	3	,	,	PUNCT
ejpam-3384	366	4	14,(2004	14,(2004	NUM
ejpam-3384	366	5	)	)	PUNCT
ejpam-3384	366	6	.	.	PUNCT
ejpam-3384	367	1	pp	pp	ADV
ejpam-3384	367	2	.	.	PUNCT
ejpam-3384	368	1	215	215	NUM
ejpam-3384	368	2	-	-	SYM
ejpam-3384	368	3	244	244	NUM
ejpam-3384	368	4	.	.	PUNCT
ejpam-3384	369	1	[	[	X
ejpam-3384	369	2	3	3	X
ejpam-3384	369	3	]	]	X
ejpam-3384	369	4	b.	b.	PROPN
ejpam-3384	369	5	barnes	barnes	PROPN
ejpam-3384	369	6	,	,	PUNCT
ejpam-3384	369	7	c.	c.	PROPN
ejpam-3384	369	8	sebil	sebil	PROPN
ejpam-3384	369	9	and	and	CCONJ
ejpam-3384	369	10	e.	e.	PROPN
ejpam-3384	369	11	harris	harris	PROPN
ejpam-3384	369	12	,	,	PUNCT
ejpam-3384	369	13	a	a	DET
ejpam-3384	369	14	note	note	NOUN
ejpam-3384	369	15	on	on	ADP
ejpam-3384	369	16	the	the	DET
ejpam-3384	369	17	quasi	quasi	ADJ
ejpam-3384	369	18	-	-	ADJ
ejpam-3384	369	19	normed	normed	ADJ
ejpam-3384	369	20	linear	linear	ADJ
ejpam-3384	369	21	space	space	NOUN
ejpam-3384	369	22	,	,	PUNCT
ejpam-3384	369	23	advances	advance	NOUN
ejpam-3384	369	24	in	in	ADP
ejpam-3384	369	25	theoretical	theoretical	ADJ
ejpam-3384	369	26	and	and	CCONJ
ejpam-3384	369	27	applied	applied	ADJ
ejpam-3384	369	28	mathematics,13	mathematics,13	NOUN
ejpam-3384	369	29	,	,	PUNCT
ejpam-3384	369	30	no	no	INTJ
ejpam-3384	369	31	.	.	PUNCT
ejpam-3384	370	1	2(2018	2(2018	NUM
ejpam-3384	370	2	)	)	PUNCT
ejpam-3384	370	3	.	.	PUNCT
ejpam-3384	371	1	pp	pp	X
ejpam-3384	371	2	.	.	PUNCT
ejpam-3384	372	1	107	107	NUM
ejpam-3384	372	2	-	-	SYM
ejpam-3384	372	3	114	114	NUM
ejpam-3384	372	4	.	.	PUNCT
ejpam-3384	373	1	[	[	X
ejpam-3384	373	2	4	4	X
ejpam-3384	373	3	]	]	X
ejpam-3384	373	4	b.	b.	PROPN
ejpam-3384	373	5	barnes	barnes	PROPN
ejpam-3384	373	6	,	,	PUNCT
ejpam-3384	373	7	e.	e.	PROPN
ejpam-3384	373	8	d.	d.	PROPN
ejpam-3384	373	9	j.	j.	PROPN
ejpam-3384	373	10	owusu	owusu	PROPN
ejpam-3384	373	11	-	-	PUNCT
ejpam-3384	373	12	ansah	ansah	PROPN
ejpam-3384	373	13	,	,	PUNCT
ejpam-3384	373	14	s.	s.	PROPN
ejpam-3384	373	15	k.	k.	PROPN
ejpam-3384	373	16	amponsah	amponsah	PROPN
ejpam-3384	373	17	and	and	CCONJ
ejpam-3384	373	18	c.	c.	PROPN
ejpam-3384	373	19	sebil	sebil	PROPN
ejpam-3384	373	20	,	,	PUNCT
ejpam-3384	373	21	the	the	DET
ejpam-3384	373	22	proofs	proof	NOUN
ejpam-3384	373	23	of	of	ADP
ejpam-3384	373	24	product	product	NOUN
ejpam-3384	373	25	inequalities	inequality	NOUN
ejpam-3384	373	26	in	in	ADP
ejpam-3384	373	27	a	a	DET
ejpam-3384	373	28	generalized	generalized	ADJ
ejpam-3384	373	29	vector	vector	NOUN
ejpam-3384	373	30	spaces	space	NOUN
ejpam-3384	373	31	,	,	PUNCT
ejpam-3384	373	32	european	european	PROPN
ejpam-3384	373	33	journal	journal	PROPN
ejpam-3384	373	34	of	of	ADP
ejpam-3384	373	35	pure	pure	ADJ
ejpam-3384	373	36	and	and	CCONJ
ejpam-3384	373	37	applied	applied	ADJ
ejpam-3384	373	38	mathematics,11	mathematics,11	NOUN
ejpam-3384	373	39	,	,	PUNCT
ejpam-3384	373	40	no	no	INTJ
ejpam-3384	373	41	.	.	PUNCT
ejpam-3384	374	1	2(2018	2(2018	NUM
ejpam-3384	374	2	)	)	PUNCT
ejpam-3384	374	3	.	.	PUNCT
ejpam-3384	375	1	pp	pp	ADV
ejpam-3384	375	2	.	.	PUNCT
ejpam-3384	376	1	375	375	NUM
ejpam-3384	376	2	-	-	SYM
ejpam-3384	376	3	389	389	NUM
ejpam-3384	376	4	.	.	PUNCT
ejpam-3384	377	1	[	[	X
ejpam-3384	377	2	5	5	X
ejpam-3384	377	3	]	]	PUNCT
ejpam-3384	377	4	b.	b.	PROPN
ejpam-3384	377	5	barnes	barnes	PROPN
ejpam-3384	377	6	,	,	PUNCT
ejpam-3384	377	7	i.	i.	PROPN
ejpam-3384	377	8	a.	a.	PROPN
ejpam-3384	377	9	adjei	adjei	PROPN
ejpam-3384	377	10	and	and	CCONJ
ejpam-3384	377	11	c.	c.	PROPN
ejpam-3384	377	12	sebil	sebil	PROPN
ejpam-3384	377	13	and	and	CCONJ
ejpam-3384	377	14	e.	e.	PROPN
ejpam-3384	377	15	harris	harris	PROPN
ejpam-3384	377	16	,	,	PUNCT
ejpam-3384	377	17	product	product	NOUN
ejpam-3384	377	18	-	-	PUNCT
ejpam-3384	377	19	normed	norme	VERB
ejpam-3384	377	20	linear	linear	ADJ
ejpam-3384	377	21	spaces	space	NOUN
ejpam-3384	377	22	,	,	PUNCT
ejpam-3384	377	23	european	european	PROPN
ejpam-3384	377	24	journal	journal	PROPN
ejpam-3384	377	25	of	of	ADP
ejpam-3384	377	26	pure	pure	ADJ
ejpam-3384	377	27	and	and	CCONJ
ejpam-3384	377	28	applied	applied	ADJ
ejpam-3384	377	29	mathematics,11	mathematics,11	NOUN
ejpam-3384	377	30	,	,	PUNCT
ejpam-3384	377	31	no	no	INTJ
ejpam-3384	377	32	.	.	PUNCT
ejpam-3384	378	1	3(2018	3(2018	NUM
ejpam-3384	378	2	)	)	PUNCT
ejpam-3384	378	3	.	.	PUNCT
ejpam-3384	379	1	pp	pp	ADV
ejpam-3384	379	2	.	.	PUNCT
ejpam-3384	380	1	740	740	NUM
ejpam-3384	380	2	-	-	SYM
ejpam-3384	380	3	750	750	NUM
ejpam-3384	380	4	.	.	PUNCT
ejpam-3384	381	1	[	[	X
ejpam-3384	381	2	6	6	NUM
ejpam-3384	381	3	]	]	PUNCT
ejpam-3384	381	4	h.	h.	NOUN
ejpam-3384	381	5	royden	royden	PROPN
ejpam-3384	381	6	and	and	CCONJ
ejpam-3384	381	7	p.	p.	PROPN
ejpam-3384	381	8	fitxpatrick	fitxpatrick	PROPN
ejpam-3384	381	9	,	,	PUNCT
ejpam-3384	381	10	real	real	ADJ
ejpam-3384	381	11	analysis	analysis	NOUN
ejpam-3384	381	12	.	.	PUNCT
ejpam-3384	382	1	pearson	pearson	PROPN
ejpam-3384	382	2	education	education	PROPN
ejpam-3384	382	3	,	,	PUNCT
ejpam-3384	382	4	inc	inc	PROPN
ejpam-3384	382	5	,	,	PUNCT
ejpam-3384	382	6	4th	4th	ADJ
ejpam-3384	382	7	ed	ed	NOUN
ejpam-3384	382	8	.	.	PROPN
ejpam-3384	382	9	,	,	PUNCT
ejpam-3384	382	10	upper	upper	ADJ
ejpam-3384	382	11	saddle	saddle	NOUN
ejpam-3384	382	12	river	river	NOUN
ejpam-3384	382	13	;	;	PUNCT
ejpam-3384	382	14	2010	2010	NUM
ejpam-3384	382	15	.	.	PUNCT
ejpam-3384	382	16	references	reference	NOUN
ejpam-3384	382	17	485	485	NUM
ejpam-3384	383	1	[	[	X
ejpam-3384	383	2	7	7	NUM
ejpam-3384	383	3	]	]	PUNCT
ejpam-3384	383	4	j.	j.	PROPN
ejpam-3384	383	5	t.	t.	PROPN
ejpam-3384	383	6	oden	oden	PROPN
ejpam-3384	383	7	,	,	PUNCT
ejpam-3384	383	8	applied	apply	VERB
ejpam-3384	383	9	functional	functional	ADJ
ejpam-3384	383	10	analysis	analysis	NOUN
ejpam-3384	383	11	:	:	PUNCT
ejpam-3384	383	12	a	a	DET
ejpam-3384	383	13	first	first	ADJ
ejpam-3384	383	14	course	course	NOUN
ejpam-3384	383	15	for	for	ADP
ejpam-3384	383	16	students	student	NOUN
ejpam-3384	383	17	of	of	ADP
ejpam-3384	383	18	mechanics	mechanic	NOUN
ejpam-3384	383	19	and	and	CCONJ
ejpam-3384	383	20	engineering	engineering	NOUN
ejpam-3384	383	21	science	science	NOUN
ejpam-3384	383	22	,	,	PUNCT
ejpam-3384	383	23	prentice	prentice	NOUN
ejpam-3384	383	24	-	-	PUNCT
ejpam-3384	383	25	hall	hall	NOUN
ejpam-3384	383	26	,	,	PUNCT
ejpam-3384	383	27	inc	inc	PROPN
ejpam-3384	383	28	,	,	PUNCT
ejpam-3384	383	29	new	new	PROPN
ejpam-3384	383	30	jersey	jersey	PROPN
ejpam-3384	383	31	;	;	PUNCT
ejpam-3384	383	32	1979	1979	NUM
ejpam-3384	383	33	.	.	PUNCT
ejpam-3384	384	1	[	[	X
ejpam-3384	384	2	8	8	X
ejpam-3384	384	3	]	]	PUNCT
ejpam-3384	384	4	j.	j.	PROPN
ejpam-3384	384	5	horvath	horvath	PROPN
ejpam-3384	384	6	,	,	PUNCT
ejpam-3384	384	7	an	an	DET
ejpam-3384	384	8	introduction	introduction	NOUN
ejpam-3384	384	9	to	to	ADP
ejpam-3384	384	10	distributions	distribution	NOUN
ejpam-3384	384	11	,	,	PUNCT
ejpam-3384	384	12	american	american	PROPN
ejpam-3384	384	13	mathematical	mathematical	PROPN
ejpam-3384	384	14	monthly	monthly	ADV
ejpam-3384	384	15	,	,	PUNCT
ejpam-3384	384	16	77(1970).pp	77(1970).pp	NUM
ejpam-3384	384	17	.	.	PUNCT
ejpam-3384	385	1	227	227	NUM
ejpam-3384	385	2	-	-	SYM
ejpam-3384	385	3	240	240	NUM
ejpam-3384	385	4	.	.	PUNCT
ejpam-3384	386	1	[	[	X
ejpam-3384	386	2	9	9	NUM
ejpam-3384	386	3	]	]	X
ejpam-3384	386	4	h.	h.	PROPN
ejpam-3384	386	5	kozono	kozono	PROPN
ejpam-3384	386	6	and	and	CCONJ
ejpam-3384	386	7	h.	h.	PROPN
ejpam-3384	386	8	wadade	wadade	PROPN
ejpam-3384	386	9	,	,	PUNCT
ejpam-3384	386	10	remarks	remark	NOUN
ejpam-3384	386	11	on	on	ADP
ejpam-3384	386	12	gagliardo	gagliardo	NOUN
ejpam-3384	386	13	-	-	PUNCT
ejpam-3384	386	14	nirenberg	nirenberg	PROPN
ejpam-3384	386	15	type	type	NOUN
ejpam-3384	386	16	inequality	inequality	NOUN
ejpam-3384	386	17	with	with	ADP
ejpam-3384	386	18	critical	critical	ADJ
ejpam-3384	386	19	sobolev	sobolev	NOUN
ejpam-3384	386	20	space	space	NOUN
ejpam-3384	386	21	and	and	CCONJ
ejpam-3384	386	22	bmo	bmo	NOUN
ejpam-3384	386	23	,	,	PUNCT
ejpam-3384	386	24	mathematische	mathematische	NOUN
ejpam-3384	386	25	zeitschrift,259(2008).pp	zeitschrift,259(2008).pp	NOUN
ejpam-3384	386	26	.	.	PUNCT
ejpam-3384	387	1	935	935	NUM
ejpam-3384	387	2	-	-	SYM
ejpam-3384	387	3	950	950	NUM
ejpam-3384	387	4	.	.	PUNCT
ejpam-3384	388	1	[	[	X
ejpam-3384	388	2	10	10	NUM
ejpam-3384	388	3	]	]	PUNCT
ejpam-3384	388	4	s.	s.	PROPN
ejpam-3384	388	5	s.	s.	PROPN
ejpam-3384	388	6	dragomir	dragomir	PROPN
ejpam-3384	388	7	,	,	PUNCT
ejpam-3384	388	8	operators	operator	NOUN
ejpam-3384	388	9	inequalities	inequality	NOUN
ejpam-3384	388	10	of	of	ADP
ejpam-3384	388	11	the	the	DET
ejpam-3384	388	12	jensen	jensen	PROPN
ejpam-3384	388	13	,	,	PUNCT
ejpam-3384	388	14	čhebyšev	čhebyšev	PROPN
ejpam-3384	388	15	and	and	CCONJ
ejpam-3384	388	16	grüss	grüss	PROPN
ejpam-3384	388	17	type	type	NOUN
ejpam-3384	388	18	,	,	PUNCT
ejpam-3384	388	19	springer	springer	NOUN
ejpam-3384	388	20	new	new	PROPN
ejpam-3384	388	21	york	york	PROPN
ejpam-3384	388	22	dordrecht	dordrecht	PROPN
ejpam-3384	388	23	heidelberg	heidelberg	PROPN
ejpam-3384	388	24	,	,	PUNCT
ejpam-3384	388	25	london	london	PROPN
ejpam-3384	388	26	;	;	PUNCT
ejpam-3384	388	27	2012	2012	NUM
ejpam-3384	388	28	.	.	PUNCT
ejpam-3384	389	1	[	[	X
ejpam-3384	389	2	11	11	NUM
ejpam-3384	389	3	]	]	PUNCT
ejpam-3384	389	4	s.	s.	PROPN
ejpam-3384	389	5	s.	s.	PROPN
ejpam-3384	389	6	dragomir	dragomir	PROPN
ejpam-3384	389	7	,	,	PUNCT
ejpam-3384	389	8	a	a	DET
ejpam-3384	389	9	generalization	generalization	NOUN
ejpam-3384	389	10	of	of	ADP
ejpam-3384	389	11	grüss	grüss	PROPN
ejpam-3384	389	12	’s	’s	PART
ejpam-3384	389	13	inequality	inequality	NOUN
ejpam-3384	389	14	in	in	ADP
ejpam-3384	389	15	inner	inner	ADJ
ejpam-3384	389	16	product	product	NOUN
ejpam-3384	389	17	spaces	space	NOUN
ejpam-3384	389	18	and	and	CCONJ
ejpam-3384	389	19	applications	application	NOUN
ejpam-3384	389	20	journal	journal	NOUN
ejpam-3384	389	21	of	of	ADP
ejpam-3384	389	22	mathematical	mathematical	ADJ
ejpam-3384	389	23	analysis	analysis	NOUN
ejpam-3384	389	24	and	and	CCONJ
ejpam-3384	389	25	applications	application	NOUN
ejpam-3384	389	26	237	237	NUM
ejpam-3384	389	27	,	,	PUNCT
ejpam-3384	389	28	(	(	PUNCT
ejpam-3384	389	29	2009).pp	2009).pp	NOUN
ejpam-3384	389	30	.	.	PUNCT
ejpam-3384	389	31	7482	7482	NUM
ejpam-3384	389	32	.	.	PUNCT