id	sid	tid	token	lemma	pos
ejpam-3209	1	1	european	european	PROPN
ejpam-3209	1	2	journal	journal	PROPN
ejpam-3209	1	3	of	of	ADP
ejpam-3209	1	4	pure	pure	ADJ
ejpam-3209	1	5	and	and	CCONJ
ejpam-3209	1	6	applied	apply	VERB
ejpam-3209	1	7	mathematics	mathematic	NOUN
ejpam-3209	1	8	vol	vol	NOUN
ejpam-3209	1	9	.	.	PUNCT
ejpam-3209	2	1	11	11	NUM
ejpam-3209	2	2	,	,	PUNCT
ejpam-3209	2	3	no	no	INTJ
ejpam-3209	2	4	.	.	NOUN
ejpam-3209	2	5	2	2	NUM
ejpam-3209	2	6	,	,	PUNCT
ejpam-3209	2	7	2018	2018	NUM
ejpam-3209	2	8	,	,	PUNCT
ejpam-3209	2	9	375	375	NUM
ejpam-3209	2	10	-	-	SYM
ejpam-3209	2	11	389	389	NUM
ejpam-3209	2	12	issn	issn	PROPN
ejpam-3209	2	13	1307	1307	NUM
ejpam-3209	2	14	-	-	SYM
ejpam-3209	2	15	5543	5543	NUM
ejpam-3209	2	16	–	–	PUNCT
ejpam-3209	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3209	2	18	published	publish	VERB
ejpam-3209	2	19	by	by	ADP
ejpam-3209	2	20	new	new	PROPN
ejpam-3209	2	21	york	york	PROPN
ejpam-3209	2	22	business	business	PROPN
ejpam-3209	2	23	global	global	PROPN
ejpam-3209	2	24	the	the	DET
ejpam-3209	2	25	proofs	proof	NOUN
ejpam-3209	2	26	of	of	ADP
ejpam-3209	2	27	product	product	NOUN
ejpam-3209	2	28	inequalities	inequality	NOUN
ejpam-3209	2	29	in	in	ADP
ejpam-3209	2	30	a	a	DET
ejpam-3209	2	31	generalized	generalized	ADJ
ejpam-3209	2	32	vector	vector	NOUN
ejpam-3209	2	33	space	space	NOUN
ejpam-3209	2	34	benedict	benedict	PROPN
ejpam-3209	2	35	barnes1,∗	barnes1,∗	PROPN
ejpam-3209	2	36	,	,	PUNCT
ejpam-3209	2	37	e.	e.	PROPN
ejpam-3209	2	38	d.	d.	PROPN
ejpam-3209	2	39	j.	j.	PROPN
ejpam-3209	2	40	owusu	owusu	PROPN
ejpam-3209	2	41	-	-	PROPN
ejpam-3209	2	42	ansah1	ansah1	PROPN
ejpam-3209	2	43	,	,	PUNCT
ejpam-3209	2	44	s.	s.	PROPN
ejpam-3209	2	45	k.	k.	PROPN
ejpam-3209	2	46	amponsah1	amponsah1	PROPN
ejpam-3209	2	47	,	,	PUNCT
ejpam-3209	2	48	c.	c.	PROPN
ejpam-3209	2	49	sebil1	sebil1	PROPN
ejpam-3209	3	1	1	1	NUM
ejpam-3209	3	2	department	department	NOUN
ejpam-3209	3	3	of	of	ADP
ejpam-3209	3	4	mathematics	mathematics	PROPN
ejpam-3209	3	5	,	,	PUNCT
ejpam-3209	3	6	kwame	kwame	PROPN
ejpam-3209	3	7	nkrumah	nkrumah	PROPN
ejpam-3209	3	8	university	university	PROPN
ejpam-3209	3	9	of	of	ADP
ejpam-3209	3	10	science	science	NOUN
ejpam-3209	3	11	and	and	CCONJ
ejpam-3209	3	12	technology	technology	NOUN
ejpam-3209	3	13	,	,	PUNCT
ejpam-3209	3	14	kumasi	kumasi	PROPN
ejpam-3209	3	15	,	,	PUNCT
ejpam-3209	3	16	ghana	ghana	PROPN
ejpam-3209	3	17	abstract	abstract	NOUN
ejpam-3209	3	18	.	.	PUNCT
ejpam-3209	4	1	in	in	ADP
ejpam-3209	4	2	this	this	DET
ejpam-3209	4	3	paper	paper	NOUN
ejpam-3209	4	4	,	,	PUNCT
ejpam-3209	4	5	we	we	PRON
ejpam-3209	4	6	introduce	introduce	VERB
ejpam-3209	4	7	the	the	DET
ejpam-3209	4	8	proofs	proof	NOUN
ejpam-3209	4	9	of	of	ADP
ejpam-3209	4	10	product	product	NOUN
ejpam-3209	4	11	inequalities	inequality	NOUN
ejpam-3209	4	12	:	:	PUNCT
ejpam-3209	4	13	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3209	4	14	≤	≤	NOUN
ejpam-3209	4	15	‖u‖+	‖u‖+	PROPN
ejpam-3209	4	16	‖v‖	‖v‖	PROPN
ejpam-3209	4	17	,	,	PUNCT
ejpam-3209	4	18	for	for	ADP
ejpam-3209	4	19	all	all	DET
ejpam-3209	4	20	u	u	NOUN
ejpam-3209	4	21	,	,	PUNCT
ejpam-3209	4	22	v	v	NOUN
ejpam-3209	4	23	∈	∈	PROPN
ejpam-3209	5	1	[	[	X
ejpam-3209	5	2	0	0	NUM
ejpam-3209	5	3	,	,	PUNCT
ejpam-3209	5	4	2	2	NUM
ejpam-3209	5	5	]	]	PUNCT
ejpam-3209	5	6	,	,	PUNCT
ejpam-3209	5	7	and	and	CCONJ
ejpam-3209	5	8	‖u‖+	‖u‖+	PROPN
ejpam-3209	5	9	‖v‖	‖v‖	PROPN
ejpam-3209	5	10	≤	≤	NUM
ejpam-3209	5	11	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3209	5	12	,	,	PUNCT
ejpam-3209	5	13	for	for	ADP
ejpam-3209	5	14	all	all	DET
ejpam-3209	5	15	u	u	NOUN
ejpam-3209	5	16	,	,	PUNCT
ejpam-3209	5	17	v	v	NOUN
ejpam-3209	5	18	∈	∈	PROPN
ejpam-3209	5	19	[	[	X
ejpam-3209	5	20	2,∞	2,∞	NUM
ejpam-3209	5	21	)	)	PUNCT
ejpam-3209	5	22	.	.	PUNCT
ejpam-3209	6	1	by	by	ADP
ejpam-3209	6	2	applying	apply	VERB
ejpam-3209	6	3	the	the	DET
ejpam-3209	6	4	first	first	ADJ
ejpam-3209	6	5	product	product	NOUN
ejpam-3209	6	6	inequality	inequality	NOUN
ejpam-3209	6	7	to	to	ADP
ejpam-3209	6	8	the	the	DET
ejpam-3209	6	9	lp	lp	PROPN
ejpam-3209	6	10	spaces	space	NOUN
ejpam-3209	6	11	,	,	PUNCT
ejpam-3209	6	12	we	we	PRON
ejpam-3209	6	13	observed	observe	VERB
ejpam-3209	6	14	that	that	SCONJ
ejpam-3209	6	15	if	if	SCONJ
ejpam-3209	6	16	f	f	PROPN
ejpam-3209	6	17	:	:	PUNCT
ejpam-3209	6	18	ω	ω	X
ejpam-3209	6	19	→	→	PUNCT
ejpam-3209	7	1	[	[	X
ejpam-3209	7	2	0	0	NUM
ejpam-3209	7	3	,	,	PUNCT
ejpam-3209	7	4	1	1	NUM
ejpam-3209	7	5	]	]	PUNCT
ejpam-3209	7	6	,	,	PUNCT
ejpam-3209	7	7	and	and	CCONJ
ejpam-3209	7	8	g	g	NOUN
ejpam-3209	7	9	:	:	PUNCT
ejpam-3209	7	10	ω	ω	PROPN
ejpam-3209	7	11	→	→	SYM
ejpam-3209	7	12	r	r	NOUN
ejpam-3209	7	13	,	,	PUNCT
ejpam-3209	7	14	then	then	ADV
ejpam-3209	7	15	‖f‖p‖g‖p	‖f‖p‖g‖p	PRON
ejpam-3209	7	16	≤	≤	NUM
ejpam-3209	7	17	‖f‖p	‖f‖p	NOUN
ejpam-3209	7	18	+	+	CCONJ
ejpam-3209	7	19	‖g‖p	‖g‖p	NOUN
ejpam-3209	7	20	.	.	PUNCT
ejpam-3209	8	1	also	also	ADV
ejpam-3209	8	2	,	,	PUNCT
ejpam-3209	8	3	if	if	SCONJ
ejpam-3209	8	4	f	f	X
ejpam-3209	8	5	,	,	PUNCT
ejpam-3209	8	6	g	g	PROPN
ejpam-3209	8	7	:	:	PUNCT
ejpam-3209	8	8	ω→	ω→	PUNCT
ejpam-3209	8	9	r	r	NOUN
ejpam-3209	8	10	,	,	PUNCT
ejpam-3209	8	11	then	then	ADV
ejpam-3209	8	12	‖f‖p	‖f‖p	NOUN
ejpam-3209	8	13	+	+	CCONJ
ejpam-3209	8	14	‖g‖p	‖g‖p	VERB
ejpam-3209	8	15	≤	≤	NUM
ejpam-3209	8	16	‖f‖p‖g‖p	‖f‖p‖g‖p	PRON
ejpam-3209	8	17	.	.	PUNCT
ejpam-3209	9	1	2010	2010	NUM
ejpam-3209	9	2	mathematics	mathematic	NOUN
ejpam-3209	9	3	subject	subject	NOUN
ejpam-3209	9	4	classifications	classification	NOUN
ejpam-3209	9	5	:	:	PUNCT
ejpam-3209	9	6	44b45	44b45	NUM
ejpam-3209	9	7	,	,	PUNCT
ejpam-3209	9	8	44b46	44b46	NUM
ejpam-3209	9	9	key	key	ADJ
ejpam-3209	9	10	words	word	NOUN
ejpam-3209	9	11	and	and	CCONJ
ejpam-3209	9	12	phrases	phrase	NOUN
ejpam-3209	9	13	:	:	PUNCT
ejpam-3209	9	14	product	product	NOUN
ejpam-3209	9	15	inequality	inequality	NOUN
ejpam-3209	9	16	,	,	PUNCT
ejpam-3209	9	17	first	first	ADJ
ejpam-3209	9	18	product	product	NOUN
ejpam-3209	9	19	nequality	nequality	NOUN
ejpam-3209	9	20	,	,	PUNCT
ejpam-3209	9	21	second	second	ADJ
ejpam-3209	9	22	product	product	NOUN
ejpam-3209	9	23	inequality	inequality	NOUN
ejpam-3209	9	24	,	,	PUNCT
ejpam-3209	9	25	generalized	generalized	ADJ
ejpam-3209	9	26	vector	vector	NOUN
ejpam-3209	9	27	space	space	NOUN
ejpam-3209	9	28	,	,	PUNCT
ejpam-3209	9	29	cauchy	cauchy	NOUN
ejpam-3209	9	30	-	-	PUNCT
ejpam-3209	9	31	schwarz	schwarz	PROPN
ejpam-3209	9	32	inequality	inequality	NOUN
ejpam-3209	9	33	1	1	NUM
ejpam-3209	9	34	.	.	PUNCT
ejpam-3209	10	1	introduction	introduction	NOUN
ejpam-3209	10	2	inequalities	inequality	NOUN
ejpam-3209	10	3	are	be	AUX
ejpam-3209	10	4	inevitable	inevitable	ADJ
ejpam-3209	10	5	tools	tool	NOUN
ejpam-3209	10	6	in	in	ADP
ejpam-3209	10	7	the	the	DET
ejpam-3209	10	8	mathematical	mathematical	ADJ
ejpam-3209	10	9	analysis	analysis	NOUN
ejpam-3209	10	10	as	as	SCONJ
ejpam-3209	10	11	they	they	PRON
ejpam-3209	10	12	provide	provide	VERB
ejpam-3209	10	13	bases	basis	NOUN
ejpam-3209	10	14	for	for	ADP
ejpam-3209	10	15	sound	sound	ADJ
ejpam-3209	10	16	arguments	argument	NOUN
ejpam-3209	10	17	.	.	PUNCT
ejpam-3209	11	1	due	due	ADP
ejpam-3209	11	2	to	to	ADP
ejpam-3209	11	3	the	the	DET
ejpam-3209	11	4	enormous	enormous	ADJ
ejpam-3209	11	5	applications	application	NOUN
ejpam-3209	11	6	of	of	ADP
ejpam-3209	11	7	inequalities	inequality	NOUN
ejpam-3209	11	8	,	,	PUNCT
ejpam-3209	11	9	most	most	ADJ
ejpam-3209	11	10	researchers	researcher	NOUN
ejpam-3209	11	11	are	be	AUX
ejpam-3209	11	12	shifting	shift	VERB
ejpam-3209	11	13	to	to	ADP
ejpam-3209	11	14	this	this	DET
ejpam-3209	11	15	line	line	NOUN
ejpam-3209	11	16	of	of	ADP
ejpam-3209	11	17	research	research	NOUN
ejpam-3209	11	18	by	by	ADP
ejpam-3209	11	19	introducing	introduce	VERB
ejpam-3209	11	20	new	new	ADJ
ejpam-3209	11	21	inequalities	inequality	NOUN
ejpam-3209	11	22	.	.	PUNCT
ejpam-3209	12	1	from	from	ADP
ejpam-3209	12	2	the	the	DET
ejpam-3209	12	3	historical	historical	ADJ
ejpam-3209	12	4	standpoint	standpoint	NOUN
ejpam-3209	12	5	,	,	PUNCT
ejpam-3209	12	6	the	the	DET
ejpam-3209	12	7	triangle	triangle	NOUN
ejpam-3209	12	8	inequality	inequality	NOUN
ejpam-3209	12	9	was	be	AUX
ejpam-3209	12	10	first	first	ADV
ejpam-3209	12	11	discovered	discover	VERB
ejpam-3209	12	12	,	,	PUNCT
ejpam-3209	12	13	see	see	VERB
ejpam-3209	12	14	[	[	X
ejpam-3209	12	15	1	1	NUM
ejpam-3209	12	16	]	]	PUNCT
ejpam-3209	12	17	.	.	PUNCT
ejpam-3209	13	1	since	since	SCONJ
ejpam-3209	13	2	then	then	ADV
ejpam-3209	13	3	many	many	ADJ
ejpam-3209	13	4	researchers	researcher	NOUN
ejpam-3209	13	5	across	across	ADP
ejpam-3209	13	6	the	the	DET
ejpam-3209	13	7	globe	globe	NOUN
ejpam-3209	13	8	had	have	AUX
ejpam-3209	13	9	obtained	obtain	VERB
ejpam-3209	13	10	different	different	ADJ
ejpam-3209	13	11	ways	way	NOUN
ejpam-3209	13	12	of	of	ADP
ejpam-3209	13	13	proving	prove	VERB
ejpam-3209	13	14	triangle	triangle	NOUN
ejpam-3209	13	15	inequality	inequality	NOUN
ejpam-3209	13	16	.	.	PUNCT
ejpam-3209	14	1	for	for	ADP
ejpam-3209	14	2	example	example	NOUN
ejpam-3209	14	3	,	,	PUNCT
ejpam-3209	14	4	see	see	VERB
ejpam-3209	14	5	a	a	DET
ejpam-3209	14	6	research	research	NOUN
ejpam-3209	14	7	paper	paper	NOUN
ejpam-3209	14	8	by	by	ADP
ejpam-3209	14	9	authors	author	NOUN
ejpam-3209	14	10	in	in	ADP
ejpam-3209	14	11	[	[	X
ejpam-3209	14	12	2	2	NUM
ejpam-3209	14	13	]	]	PUNCT
ejpam-3209	14	14	.	.	PUNCT
ejpam-3209	15	1	after	after	ADP
ejpam-3209	15	2	the	the	DET
ejpam-3209	15	3	discovery	discovery	NOUN
ejpam-3209	15	4	of	of	ADP
ejpam-3209	15	5	triangle	triangle	NOUN
ejpam-3209	15	6	inequality	inequality	NOUN
ejpam-3209	15	7	the	the	DET
ejpam-3209	15	8	so	so	ADV
ejpam-3209	15	9	-	-	PUNCT
ejpam-3209	15	10	called	call	VERB
ejpam-3209	15	11	arithmetic	arithmetic	ADJ
ejpam-3209	15	12	-	-	PUNCT
ejpam-3209	15	13	geometric	geometric	ADJ
ejpam-3209	15	14	mean	mean	NOUN
ejpam-3209	15	15	agm	agm	PROPN
ejpam-3209	15	16	inequality	inequality	PROPN
ejpam-3209	15	17	:(	:(	PROPN
ejpam-3209	15	18	n∏	n∏	PROPN
ejpam-3209	15	19	i=1	i=1	PROPN
ejpam-3209	15	20	xi	xi	X
ejpam-3209	15	21	)	)	PUNCT
ejpam-3209	15	22	1	1	NUM
ejpam-3209	15	23	n	n	CCONJ
ejpam-3209	15	24	≤	≤	NUM
ejpam-3209	15	25	1	1	NUM
ejpam-3209	15	26	n	n	NUM
ejpam-3209	15	27	n∑	n∑	NOUN
ejpam-3209	15	28	i=1	i=1	PROPN
ejpam-3209	15	29	xi	xi	PROPN
ejpam-3209	15	30	,	,	PUNCT
ejpam-3209	15	31	∀	∀	VERB
ejpam-3209	15	32	n	n	PRON
ejpam-3209	15	33	∈	∈	PROPN
ejpam-3209	15	34	n	n	NOUN
ejpam-3209	15	35	and	and	CCONJ
ejpam-3209	15	36	xi	xi	X
ejpam-3209	15	37	∈	∈	PROPN
ejpam-3209	15	38	r+	r+	NOUN
ejpam-3209	15	39	,	,	PUNCT
ejpam-3209	15	40	with	with	ADP
ejpam-3209	15	41	equality	equality	NOUN
ejpam-3209	15	42	occurs	occur	VERB
ejpam-3209	15	43	when	when	SCONJ
ejpam-3209	15	44	x1	x1	PROPN
ejpam-3209	15	45	=	=	PUNCT
ejpam-3209	15	46	x2	x2	PROPN
ejpam-3209	16	1	=	=	X
ejpam-3209	16	2	.	.	PUNCT
ejpam-3209	16	3	.	.	PUNCT
ejpam-3209	16	4	.	.	PUNCT
ejpam-3209	17	1	=	=	PUNCT
ejpam-3209	17	2	xn	xn	PROPN
ejpam-3209	17	3	,	,	PUNCT
ejpam-3209	17	4	was	be	AUX
ejpam-3209	17	5	observed	observe	VERB
ejpam-3209	17	6	.	.	PUNCT
ejpam-3209	18	1	specifically	specifically	ADV
ejpam-3209	18	2	,	,	PUNCT
ejpam-3209	18	3	for	for	ADP
ejpam-3209	18	4	any	any	DET
ejpam-3209	18	5	two	two	NUM
ejpam-3209	18	6	positive	positive	ADJ
ejpam-3209	18	7	real	real	ADJ
ejpam-3209	18	8	numbers	number	NOUN
ejpam-3209	18	9	a	a	PRON
ejpam-3209	18	10	and	and	CCONJ
ejpam-3209	18	11	b	b	NOUN
ejpam-3209	18	12	the	the	DET
ejpam-3209	18	13	agm	agm	PROPN
ejpam-3209	18	14	inequality	inequality	NOUN
ejpam-3209	18	15	becomes	become	VERB
ejpam-3209	18	16	ab	ab	PROPN
ejpam-3209	18	17	≤	≤	PROPN
ejpam-3209	18	18	(	(	PUNCT
ejpam-3209	18	19	a+	a+	PUNCT
ejpam-3209	18	20	b	b	NOUN
ejpam-3209	18	21	2	2	NUM
ejpam-3209	18	22	)	)	SYM
ejpam-3209	18	23	2	2	NUM
ejpam-3209	18	24	,	,	PUNCT
ejpam-3209	18	25	∗corresponding	∗corresponde	VERB
ejpam-3209	18	26	author	author	NOUN
ejpam-3209	18	27	.	.	PUNCT
ejpam-3209	19	1	email	email	NOUN
ejpam-3209	19	2	addresses	address	NOUN
ejpam-3209	19	3	:	:	PUNCT
ejpam-3209	19	4	bbarnes.cos@knust.edu.gh	bbarnes.cos@knust.edu.gh	NOUN
ejpam-3209	19	5	(	(	PUNCT
ejpam-3209	19	6	b.	b.	PROPN
ejpam-3209	19	7	barnes	barnes	PROPN
ejpam-3209	19	8	)	)	PUNCT
ejpam-3209	19	9	,	,	PUNCT
ejpam-3209	19	10	degraftt@gmail.com	degraftt@gmail.com	X
ejpam-3209	19	11	(	(	PUNCT
ejpam-3209	19	12	e.	e.	PROPN
ejpam-3209	19	13	d.	d.	PROPN
ejpam-3209	19	14	j.	j.	PROPN
ejpam-3209	19	15	owusu	owusu	PROPN
ejpam-3209	19	16	-	-	PROPN
ejpam-3209	19	17	ansah	ansah	PROPN
ejpam-3209	19	18	)	)	PUNCT
ejpam-3209	19	19	,	,	PUNCT
ejpam-3209	19	20	skamponsah@knust.edu.gh	skamponsah@knust.edu.gh	NOUN
ejpam-3209	19	21	(	(	PUNCT
ejpam-3209	19	22	s.	s.	PROPN
ejpam-3209	19	23	k.	k.	PROPN
ejpam-3209	19	24	amponsah	amponsah	PROPN
ejpam-3209	19	25	)	)	PUNCT
ejpam-3209	19	26	,	,	PUNCT
ejpam-3209	19	27	sebicharles@yahoo.com	sebicharles@yahoo.com	X
ejpam-3209	20	1	(	(	PUNCT
ejpam-3209	20	2	c.	c.	PROPN
ejpam-3209	20	3	sebil	sebil	PROPN
ejpam-3209	20	4	)	)	PUNCT
ejpam-3209	20	5	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3209	21	1	375	375	NUM
ejpam-3209	21	2	c	c	NOUN
ejpam-3209	21	3	©	©	PROPN
ejpam-3209	21	4	2018	2018	NUM
ejpam-3209	21	5	ejpam	ejpam	VERB
ejpam-3209	21	6	all	all	DET
ejpam-3209	21	7	rights	right	NOUN
ejpam-3209	21	8	reserved	reserve	VERB
ejpam-3209	21	9	.	.	PUNCT
ejpam-3209	22	1	barnes	barnes	PROPN
ejpam-3209	22	2	et	et	PROPN
ejpam-3209	22	3	al	al	PROPN
ejpam-3209	22	4	.	.	PUNCT
ejpam-3209	22	5	/	/	SYM
ejpam-3209	22	6	eur	eur	PROPN
ejpam-3209	22	7	.	.	PUNCT
ejpam-3209	23	1	j.	j.	PROPN
ejpam-3209	23	2	pure	pure	PROPN
ejpam-3209	23	3	appl	appl	PROPN
ejpam-3209	23	4	.	.	PROPN
ejpam-3209	23	5	math	math	PROPN
ejpam-3209	23	6	,	,	PUNCT
ejpam-3209	23	7	11	11	NUM
ejpam-3209	23	8	(	(	PUNCT
ejpam-3209	23	9	2	2	NUM
ejpam-3209	23	10	)	)	PUNCT
ejpam-3209	23	11	(	(	PUNCT
ejpam-3209	23	12	2018	2018	NUM
ejpam-3209	23	13	)	)	PUNCT
ejpam-3209	23	14	,	,	PUNCT
ejpam-3209	23	15	375	375	NUM
ejpam-3209	23	16	-	-	SYM
ejpam-3209	23	17	389	389	NUM
ejpam-3209	23	18	376	376	NUM
ejpam-3209	23	19	see	see	VERB
ejpam-3209	23	20	[	[	X
ejpam-3209	23	21	3	3	NUM
ejpam-3209	23	22	]	]	PUNCT
ejpam-3209	23	23	.	.	PUNCT
ejpam-3209	24	1	several	several	ADJ
ejpam-3209	24	2	ways	way	NOUN
ejpam-3209	24	3	of	of	ADP
ejpam-3209	24	4	proving	prove	VERB
ejpam-3209	24	5	the	the	DET
ejpam-3209	24	6	agm	agm	PROPN
ejpam-3209	24	7	inequality	inequality	NOUN
ejpam-3209	24	8	have	have	AUX
ejpam-3209	24	9	been	be	AUX
ejpam-3209	24	10	observed	observe	VERB
ejpam-3209	24	11	,	,	PUNCT
ejpam-3209	24	12	see	see	VERB
ejpam-3209	24	13	research	research	NOUN
ejpam-3209	24	14	papers	paper	NOUN
ejpam-3209	24	15	by	by	ADP
ejpam-3209	24	16	authors	author	NOUN
ejpam-3209	24	17	in	in	ADP
ejpam-3209	24	18	[	[	X
ejpam-3209	24	19	4	4	NUM
ejpam-3209	24	20	]	]	PUNCT
ejpam-3209	24	21	and	and	CCONJ
ejpam-3209	24	22	[	[	X
ejpam-3209	24	23	5	5	NUM
ejpam-3209	24	24	]	]	PUNCT
ejpam-3209	24	25	.	.	PUNCT
ejpam-3209	25	1	however	however	ADV
ejpam-3209	25	2	,	,	PUNCT
ejpam-3209	25	3	the	the	DET
ejpam-3209	25	4	author	author	NOUN
ejpam-3209	25	5	in	in	ADP
ejpam-3209	25	6	[	[	X
ejpam-3209	25	7	6	6	NUM
ejpam-3209	25	8	]	]	PUNCT
ejpam-3209	25	9	proved	prove	VERB
ejpam-3209	25	10	that	that	SCONJ
ejpam-3209	25	11	the	the	DET
ejpam-3209	25	12	sum	sum	NOUN
ejpam-3209	25	13	of	of	ADP
ejpam-3209	25	14	vector	vector	NOUN
ejpam-3209	25	15	points	point	NOUN
ejpam-3209	25	16	and	and	CCONJ
ejpam-3209	25	17	one	one	NUM
ejpam-3209	25	18	is	be	AUX
ejpam-3209	25	19	less	less	ADJ
ejpam-3209	25	20	than	than	ADP
ejpam-3209	25	21	the	the	DET
ejpam-3209	25	22	product	product	NOUN
ejpam-3209	25	23	of	of	ADP
ejpam-3209	25	24	sum	sum	NOUN
ejpam-3209	25	25	of	of	ADP
ejpam-3209	25	26	one	one	NUM
ejpam-3209	25	27	and	and	CCONJ
ejpam-3209	25	28	the	the	DET
ejpam-3209	25	29	vector	vector	NOUN
ejpam-3209	25	30	points	point	NOUN
ejpam-3209	25	31	:	:	PUNCT
ejpam-3209	25	32	1	1	NUM
ejpam-3209	26	1	+	+	NUM
ejpam-3209	26	2	n∑	n∑	ADJ
ejpam-3209	26	3	i=1	i=1	X
ejpam-3209	26	4	xi	xi	X
ejpam-3209	26	5	<	<	X
ejpam-3209	26	6	πn	πn	X
ejpam-3209	26	7	i=1(1	i=1(1	PROPN
ejpam-3209	26	8	+	+	NUM
ejpam-3209	26	9	xi	xi	NUM
ejpam-3209	26	10	)	)	PUNCT
ejpam-3209	26	11	.	.	PUNCT
ejpam-3209	27	1	the	the	DET
ejpam-3209	27	2	agm	agm	PROPN
ejpam-3209	27	3	inequality	inequality	NOUN
ejpam-3209	27	4	has	have	AUX
ejpam-3209	27	5	been	be	AUX
ejpam-3209	27	6	applied	apply	VERB
ejpam-3209	27	7	to	to	PART
ejpam-3209	27	8	establish	establish	VERB
ejpam-3209	27	9	the	the	DET
ejpam-3209	27	10	relationships	relationship	NOUN
ejpam-3209	27	11	between	between	ADP
ejpam-3209	27	12	areas	area	NOUN
ejpam-3209	27	13	of	of	ADP
ejpam-3209	27	14	plane	plane	NOUN
ejpam-3209	27	15	figures	figure	NOUN
ejpam-3209	27	16	and	and	CCONJ
ejpam-3209	27	17	their	their	PRON
ejpam-3209	27	18	perimeters	perimeter	NOUN
ejpam-3209	27	19	of	of	ADP
ejpam-3209	27	20	what	what	PRON
ejpam-3209	27	21	is	be	AUX
ejpam-3209	27	22	called	call	VERB
ejpam-3209	27	23	isoperimetric	isoperimetric	ADJ
ejpam-3209	27	24	inequality	inequality	NOUN
ejpam-3209	27	25	.	.	PUNCT
ejpam-3209	28	1	for	for	ADP
ejpam-3209	28	2	example	example	NOUN
ejpam-3209	28	3	,	,	PUNCT
ejpam-3209	28	4	see	see	VERB
ejpam-3209	28	5	[	[	X
ejpam-3209	28	6	7	7	X
ejpam-3209	28	7	]	]	PUNCT
ejpam-3209	28	8	.	.	PUNCT
ejpam-3209	29	1	although	although	SCONJ
ejpam-3209	29	2	the	the	DET
ejpam-3209	29	3	inequality	inequality	NOUN
ejpam-3209	29	4	of	of	ADP
ejpam-3209	29	5	two	two	NUM
ejpam-3209	29	6	real	real	ADJ
ejpam-3209	29	7	numbers	number	NOUN
ejpam-3209	29	8	has	have	AUX
ejpam-3209	29	9	been	be	AUX
ejpam-3209	29	10	in	in	ADP
ejpam-3209	29	11	history	history	NOUN
ejpam-3209	29	12	for	for	ADP
ejpam-3209	29	13	long	long	ADJ
ejpam-3209	29	14	time	time	NOUN
ejpam-3209	29	15	with	with	ADP
ejpam-3209	29	16	many	many	ADJ
ejpam-3209	29	17	researchers	researcher	NOUN
ejpam-3209	29	18	looking	look	VERB
ejpam-3209	29	19	at	at	ADP
ejpam-3209	29	20	inequalities	inequality	NOUN
ejpam-3209	29	21	involving	involve	VERB
ejpam-3209	29	22	products	product	NOUN
ejpam-3209	29	23	of	of	ADP
ejpam-3209	29	24	real	real	ADJ
ejpam-3209	29	25	numbers	number	NOUN
ejpam-3209	29	26	.	.	PUNCT
ejpam-3209	30	1	newton	newton	PROPN
ejpam-3209	30	2	showed	show	VERB
ejpam-3209	30	3	that	that	SCONJ
ejpam-3209	30	4	the	the	DET
ejpam-3209	30	5	square	square	NOUN
ejpam-3209	30	6	of	of	ADP
ejpam-3209	30	7	a	a	DET
ejpam-3209	30	8	real	real	ADJ
ejpam-3209	30	9	number	number	NOUN
ejpam-3209	30	10	between	between	ADP
ejpam-3209	30	11	the	the	DET
ejpam-3209	30	12	first	first	ADJ
ejpam-3209	30	13	and	and	CCONJ
ejpam-3209	30	14	third	third	ADJ
ejpam-3209	30	15	consecutive	consecutive	ADJ
ejpam-3209	30	16	real	real	ADJ
ejpam-3209	30	17	numbers	number	NOUN
ejpam-3209	30	18	is	be	AUX
ejpam-3209	30	19	greater	great	ADJ
ejpam-3209	30	20	than	than	ADP
ejpam-3209	30	21	their	their	PRON
ejpam-3209	30	22	product	product	NOUN
ejpam-3209	31	1	pr−1pr+1	pr−1pr+1	X
ejpam-3209	31	2	<	<	X
ejpam-3209	31	3	p	p	X
ejpam-3209	31	4	2	2	NUM
ejpam-3209	31	5	r	r	NOUN
ejpam-3209	31	6	,	,	PUNCT
ejpam-3209	31	7	∀	∀	NOUN
ejpam-3209	31	8	1	1	NUM
ejpam-3209	31	9	≤	≤	NOUN
ejpam-3209	31	10	r	r	NOUN
ejpam-3209	31	11	<	<	X
ejpam-3209	31	12	n	n	CCONJ
ejpam-3209	31	13	,	,	PUNCT
ejpam-3209	31	14	see	see	VERB
ejpam-3209	31	15	[	[	X
ejpam-3209	31	16	8	8	NUM
ejpam-3209	31	17	]	]	PUNCT
ejpam-3209	31	18	.	.	PUNCT
ejpam-3209	32	1	in	in	ADP
ejpam-3209	32	2	[	[	X
ejpam-3209	32	3	9	9	NUM
ejpam-3209	32	4	]	]	PUNCT
ejpam-3209	32	5	,	,	PUNCT
ejpam-3209	32	6	the	the	DET
ejpam-3209	32	7	author	author	NOUN
ejpam-3209	32	8	obtained	obtain	VERB
ejpam-3209	32	9	another	another	DET
ejpam-3209	32	10	way	way	NOUN
ejpam-3209	32	11	of	of	ADP
ejpam-3209	32	12	proving	prove	VERB
ejpam-3209	32	13	the	the	DET
ejpam-3209	32	14	weierstrass	weierstrass	NOUN
ejpam-3209	32	15	inequality	inequality	NOUN
ejpam-3209	32	16	by	by	ADP
ejpam-3209	32	17	applying	apply	VERB
ejpam-3209	32	18	agm	agm	PROPN
ejpam-3209	32	19	inequality	inequality	NOUN
ejpam-3209	32	20	and	and	CCONJ
ejpam-3209	32	21	also	also	ADV
ejpam-3209	32	22	,	,	PUNCT
ejpam-3209	32	23	extended	extend	VERB
ejpam-3209	32	24	the	the	DET
ejpam-3209	32	25	weierstrass	weierstrass	NOUN
ejpam-3209	32	26	inequalities	inequality	NOUN
ejpam-3209	32	27	by	by	ADP
ejpam-3209	32	28	the	the	DET
ejpam-3209	32	29	use	use	NOUN
ejpam-3209	32	30	of	of	ADP
ejpam-3209	32	31	majorization	majorization	NOUN
ejpam-3209	32	32	.	.	PUNCT
ejpam-3209	33	1	the	the	DET
ejpam-3209	33	2	author	author	NOUN
ejpam-3209	33	3	in	in	ADP
ejpam-3209	33	4	[	[	X
ejpam-3209	33	5	10	10	NUM
ejpam-3209	33	6	]	]	PUNCT
ejpam-3209	33	7	,	,	PUNCT
ejpam-3209	33	8	generalized	generalize	VERB
ejpam-3209	33	9	the	the	DET
ejpam-3209	33	10	weierstrass	weierstrass	NOUN
ejpam-3209	33	11	inequality	inequality	NOUN
ejpam-3209	33	12	in	in	ADP
ejpam-3209	33	13	the	the	DET
ejpam-3209	33	14	euclidean	euclidean	ADJ
ejpam-3209	33	15	space	space	NOUN
ejpam-3209	33	16	.	.	PUNCT
ejpam-3209	34	1	inequalities	inequality	NOUN
ejpam-3209	34	2	involving	involve	VERB
ejpam-3209	34	3	functions	function	NOUN
ejpam-3209	34	4	have	have	AUX
ejpam-3209	34	5	received	receive	VERB
ejpam-3209	34	6	much	much	ADJ
ejpam-3209	34	7	attention	attention	NOUN
ejpam-3209	34	8	in	in	ADP
ejpam-3209	34	9	the	the	DET
ejpam-3209	34	10	21st	21st	ADJ
ejpam-3209	34	11	century	century	NOUN
ejpam-3209	34	12	.	.	PUNCT
ejpam-3209	35	1	jordan	jordan	PROPN
ejpam-3209	35	2	as	as	SCONJ
ejpam-3209	35	3	cited	cite	VERB
ejpam-3209	35	4	in	in	ADP
ejpam-3209	35	5	[	[	X
ejpam-3209	35	6	11	11	NUM
ejpam-3209	35	7	]	]	PUNCT
ejpam-3209	35	8	introduced	introduce	VERB
ejpam-3209	35	9	fractional	fractional	ADJ
ejpam-3209	35	10	inequality	inequality	NOUN
ejpam-3209	35	11	:	:	PUNCT
ejpam-3209	35	12	2	2	NUM
ejpam-3209	35	13	π	π	NOUN
ejpam-3209	35	14	≤	≤	NUM
ejpam-3209	35	15	sin(x	sin(x	PROPN
ejpam-3209	35	16	)	)	PUNCT
ejpam-3209	35	17	x	x	X
ejpam-3209	35	18	<	<	X
ejpam-3209	35	19	1	1	NUM
ejpam-3209	35	20	,	,	PUNCT
ejpam-3209	35	21	0	0	NUM
ejpam-3209	35	22	<	<	X
ejpam-3209	35	23	x	x	SYM
ejpam-3209	35	24	≤	≤	NUM
ejpam-3209	35	25	π	π	PROPN
ejpam-3209	35	26	2	2	NUM
ejpam-3209	35	27	.	.	PUNCT
ejpam-3209	36	1	hilbert	hilbert	PROPN
ejpam-3209	36	2	constructed	construct	VERB
ejpam-3209	36	3	double	double	ADJ
ejpam-3209	36	4	series	series	NOUN
ejpam-3209	36	5	inequality	inequality	NOUN
ejpam-3209	36	6	:	:	PUNCT
ejpam-3209	36	7	∞∑	∞∑	NUM
ejpam-3209	36	8	m	m	NOUN
ejpam-3209	36	9	,	,	PUNCT
ejpam-3209	36	10	n=1	n=1	PROPN
ejpam-3209	36	11	ambn	ambn	VERB
ejpam-3209	36	12	m+	m+	NUM
ejpam-3209	36	13	n	n	NOUN
ejpam-3209	36	14	≤	≤	NOUN
ejpam-3209	36	15	π	π	PROPN
ejpam-3209	36	16	(	(	PUNCT
ejpam-3209	36	17	∞∑	∞∑	PROPN
ejpam-3209	36	18	m=1	m=1	PROPN
ejpam-3209	36	19	a2	a2	PROPN
ejpam-3209	36	20	m	m	PROPN
ejpam-3209	36	21	)	)	PUNCT
ejpam-3209	36	22	1	1	NUM
ejpam-3209	36	23	2	2	NUM
ejpam-3209	36	24	(	(	PUNCT
ejpam-3209	36	25	∞∑	∞∑	PROPN
ejpam-3209	36	26	m=1	m=1	PROPN
ejpam-3209	36	27	b2	b2	NOUN
ejpam-3209	36	28	m	m	NOUN
ejpam-3209	36	29	)	)	PUNCT
ejpam-3209	36	30	1	1	NUM
ejpam-3209	36	31	2	2	NUM
ejpam-3209	36	32	,	,	PUNCT
ejpam-3209	36	33	see	see	VERB
ejpam-3209	36	34	[	[	X
ejpam-3209	36	35	12	12	NUM
ejpam-3209	36	36	]	]	PUNCT
ejpam-3209	36	37	.	.	PUNCT
ejpam-3209	37	1	unlike	unlike	ADP
ejpam-3209	37	2	agm	agm	PROPN
ejpam-3209	37	3	inequality	inequality	NOUN
ejpam-3209	37	4	,	,	PUNCT
ejpam-3209	37	5	the	the	DET
ejpam-3209	37	6	product	product	NOUN
ejpam-3209	37	7	of	of	ADP
ejpam-3209	37	8	two	two	NUM
ejpam-3209	37	9	positive	positive	ADJ
ejpam-3209	37	10	real	real	ADJ
ejpam-3209	37	11	numbers	number	NOUN
ejpam-3209	37	12	and	and	CCONJ
ejpam-3209	37	13	their	their	PRON
ejpam-3209	37	14	sum	sum	NOUN
ejpam-3209	37	15	without	without	ADP
ejpam-3209	37	16	any	any	DET
ejpam-3209	37	17	factor	factor	NOUN
ejpam-3209	37	18	multiplier	multiplier	ADV
ejpam-3209	37	19	,	,	PUNCT
ejpam-3209	37	20	in	in	ADP
ejpam-3209	37	21	euclidean	euclidean	ADJ
ejpam-3209	37	22	space	space	NOUN
ejpam-3209	37	23	,	,	PUNCT
ejpam-3209	37	24	delineate	delineate	VERB
ejpam-3209	37	25	unique	unique	ADJ
ejpam-3209	37	26	inequalities	inequality	NOUN
ejpam-3209	37	27	within	within	ADP
ejpam-3209	37	28	certain	certain	ADJ
ejpam-3209	37	29	intervals	interval	NOUN
ejpam-3209	37	30	.	.	PUNCT
ejpam-3209	38	1	in	in	ADP
ejpam-3209	38	2	this	this	DET
ejpam-3209	38	3	paper	paper	NOUN
ejpam-3209	38	4	,	,	PUNCT
ejpam-3209	38	5	we	we	PRON
ejpam-3209	38	6	provide	provide	VERB
ejpam-3209	38	7	two	two	NUM
ejpam-3209	38	8	inequalities	inequality	NOUN
ejpam-3209	38	9	in	in	ADP
ejpam-3209	38	10	the	the	DET
ejpam-3209	38	11	interval	interval	NOUN
ejpam-3209	38	12	[	[	X
ejpam-3209	38	13	0,∞	0,∞	NOUN
ejpam-3209	38	14	)	)	PUNCT
ejpam-3209	38	15	;	;	PUNCT
ejpam-3209	38	16	the	the	DET
ejpam-3209	38	17	first	first	ADJ
ejpam-3209	38	18	product	product	NOUN
ejpam-3209	38	19	inequality	inequality	NOUN
ejpam-3209	38	20	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3209	38	21	≤	≤	PUNCT
ejpam-3209	38	22	‖u‖	‖u‖	PROPN
ejpam-3209	38	23	+	+	CCONJ
ejpam-3209	38	24	‖v‖	‖v‖	PROPN
ejpam-3209	38	25	holds	hold	VERB
ejpam-3209	38	26	for	for	ADP
ejpam-3209	38	27	all	all	DET
ejpam-3209	38	28	values	value	NOUN
ejpam-3209	38	29	of	of	ADP
ejpam-3209	38	30	u	u	NOUN
ejpam-3209	38	31	and	and	CCONJ
ejpam-3209	38	32	v	v	NOUN
ejpam-3209	38	33	in	in	ADP
ejpam-3209	38	34	the	the	DET
ejpam-3209	38	35	interval	interval	NOUN
ejpam-3209	39	1	[	[	X
ejpam-3209	39	2	0	0	NUM
ejpam-3209	39	3	,	,	PUNCT
ejpam-3209	39	4	2	2	NUM
ejpam-3209	39	5	]	]	PUNCT
ejpam-3209	39	6	,	,	PUNCT
ejpam-3209	39	7	and	and	CCONJ
ejpam-3209	39	8	the	the	DET
ejpam-3209	39	9	second	second	ADJ
ejpam-3209	39	10	product	product	NOUN
ejpam-3209	39	11	inequality	inequality	NOUN
ejpam-3209	39	12	‖u‖+	‖u‖+	PROPN
ejpam-3209	39	13	‖v‖	‖v‖	PROPN
ejpam-3209	39	14	≤	≤	NUM
ejpam-3209	39	15	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3209	39	16	holds	hold	VERB
ejpam-3209	39	17	for	for	ADP
ejpam-3209	39	18	all	all	DET
ejpam-3209	39	19	u	u	NOUN
ejpam-3209	39	20	and	and	CCONJ
ejpam-3209	39	21	v	v	NOUN
ejpam-3209	39	22	in	in	ADP
ejpam-3209	39	23	interval	interval	NOUN
ejpam-3209	39	24	[	[	X
ejpam-3209	39	25	2,∞	2,∞	NUM
ejpam-3209	39	26	)	)	PUNCT
ejpam-3209	39	27	.	.	PUNCT
ejpam-3209	40	1	each	each	PRON
ejpam-3209	40	2	of	of	ADP
ejpam-3209	40	3	these	these	DET
ejpam-3209	40	4	inequalities	inequality	NOUN
ejpam-3209	40	5	is	be	AUX
ejpam-3209	40	6	proved	prove	VERB
ejpam-3209	40	7	by	by	ADP
ejpam-3209	40	8	induction	induction	NOUN
ejpam-3209	40	9	.	.	PUNCT
ejpam-3209	41	1	the	the	DET
ejpam-3209	41	2	section	section	NOUN
ejpam-3209	41	3	1	1	NUM
ejpam-3209	41	4	contains	contain	VERB
ejpam-3209	41	5	the	the	DET
ejpam-3209	41	6	introduction	introduction	NOUN
ejpam-3209	41	7	.	.	PUNCT
ejpam-3209	42	1	the	the	DET
ejpam-3209	42	2	section	section	NOUN
ejpam-3209	42	3	2	2	NUM
ejpam-3209	42	4	contains	contain	VERB
ejpam-3209	42	5	the	the	DET
ejpam-3209	42	6	proofs	proof	NOUN
ejpam-3209	42	7	of	of	ADP
ejpam-3209	42	8	the	the	DET
ejpam-3209	42	9	first	first	ADJ
ejpam-3209	42	10	product	product	NOUN
ejpam-3209	42	11	inequality	inequality	NOUN
ejpam-3209	42	12	in	in	ADP
ejpam-3209	42	13	generalized	generalized	ADJ
ejpam-3209	42	14	linear	linear	ADJ
ejpam-3209	42	15	space	space	NOUN
ejpam-3209	42	16	and	and	CCONJ
ejpam-3209	42	17	the	the	DET
ejpam-3209	42	18	second	second	ADJ
ejpam-3209	42	19	product	product	NOUN
ejpam-3209	42	20	inequality	inequality	NOUN
ejpam-3209	42	21	in	in	ADP
ejpam-3209	42	22	the	the	DET
ejpam-3209	42	23	linear	linear	ADJ
ejpam-3209	42	24	space	space	NOUN
ejpam-3209	42	25	.	.	PUNCT
ejpam-3209	43	1	in	in	ADP
ejpam-3209	43	2	section	section	NOUN
ejpam-3209	43	3	3	3	NUM
ejpam-3209	43	4	,	,	PUNCT
ejpam-3209	43	5	we	we	PRON
ejpam-3209	43	6	illustrated	illustrate	VERB
ejpam-3209	43	7	the	the	DET
ejpam-3209	43	8	first	first	ADJ
ejpam-3209	43	9	product	product	NOUN
ejpam-3209	43	10	inequality	inequality	NOUN
ejpam-3209	43	11	to	to	ADP
ejpam-3209	43	12	a	a	DET
ejpam-3209	43	13	real	real	ADJ
ejpam-3209	43	14	line	line	NOUN
ejpam-3209	43	15	,	,	PUNCT
ejpam-3209	43	16	norms	norm	NOUN
ejpam-3209	43	17	and	and	CCONJ
ejpam-3209	43	18	trigonometric	trigonometric	ADJ
ejpam-3209	43	19	functions	function	NOUN
ejpam-3209	43	20	.	.	PUNCT
ejpam-3209	44	1	we	we	PRON
ejpam-3209	44	2	extend	extend	VERB
ejpam-3209	44	3	and	and	CCONJ
ejpam-3209	44	4	also	also	ADV
ejpam-3209	44	5	,	,	PUNCT
ejpam-3209	44	6	apply	apply	VERB
ejpam-3209	44	7	the	the	DET
ejpam-3209	44	8	second	second	ADJ
ejpam-3209	44	9	product	product	NOUN
ejpam-3209	44	10	inequality	inequality	NOUN
ejpam-3209	44	11	to	to	ADP
ejpam-3209	44	12	the	the	DET
ejpam-3209	44	13	lp	lp	ADJ
ejpam-3209	44	14	spaces	space	NOUN
ejpam-3209	44	15	in	in	ADP
ejpam-3209	44	16	section	section	NOUN
ejpam-3209	44	17	4	4	NUM
ejpam-3209	44	18	of	of	ADP
ejpam-3209	44	19	this	this	DET
ejpam-3209	44	20	paper	paper	NOUN
ejpam-3209	44	21	.	.	PUNCT
ejpam-3209	45	1	in	in	ADP
ejpam-3209	45	2	section	section	NOUN
ejpam-3209	45	3	5	5	NUM
ejpam-3209	45	4	,	,	PUNCT
ejpam-3209	45	5	we	we	PRON
ejpam-3209	45	6	discuss	discuss	VERB
ejpam-3209	45	7	our	our	PRON
ejpam-3209	45	8	main	main	ADJ
ejpam-3209	45	9	findings	finding	NOUN
ejpam-3209	45	10	and	and	CCONJ
ejpam-3209	45	11	summarize	summarize	VERB
ejpam-3209	45	12	these	these	DET
ejpam-3209	45	13	results	result	NOUN
ejpam-3209	45	14	in	in	ADP
ejpam-3209	45	15	section	section	NOUN
ejpam-3209	45	16	6	6	NUM
ejpam-3209	45	17	.	.	PUNCT
ejpam-3209	46	1	barnes	barnes	PROPN
ejpam-3209	46	2	et	et	PROPN
ejpam-3209	46	3	al	al	PROPN
ejpam-3209	46	4	.	.	PUNCT
ejpam-3209	46	5	/	/	SYM
ejpam-3209	46	6	eur	eur	PROPN
ejpam-3209	46	7	.	.	PUNCT
ejpam-3209	47	1	j.	j.	PROPN
ejpam-3209	47	2	pure	pure	PROPN
ejpam-3209	47	3	appl	appl	PROPN
ejpam-3209	47	4	.	.	PROPN
ejpam-3209	47	5	math	math	PROPN
ejpam-3209	47	6	,	,	PUNCT
ejpam-3209	47	7	11	11	NUM
ejpam-3209	47	8	(	(	PUNCT
ejpam-3209	47	9	2	2	NUM
ejpam-3209	47	10	)	)	PUNCT
ejpam-3209	47	11	(	(	PUNCT
ejpam-3209	47	12	2018	2018	NUM
ejpam-3209	47	13	)	)	PUNCT
ejpam-3209	47	14	,	,	PUNCT
ejpam-3209	47	15	375	375	NUM
ejpam-3209	47	16	-	-	SYM
ejpam-3209	47	17	389	389	NUM
ejpam-3209	47	18	377	377	NUM
ejpam-3209	47	19	2	2	NUM
ejpam-3209	47	20	.	.	PUNCT
ejpam-3209	47	21	main	main	ADJ
ejpam-3209	47	22	result	result	NOUN
ejpam-3209	47	23	in	in	ADP
ejpam-3209	47	24	this	this	DET
ejpam-3209	47	25	section	section	NOUN
ejpam-3209	47	26	,	,	PUNCT
ejpam-3209	47	27	the	the	DET
ejpam-3209	47	28	two	two	NUM
ejpam-3209	47	29	product	product	NOUN
ejpam-3209	47	30	inequalities	inequality	NOUN
ejpam-3209	47	31	are	be	AUX
ejpam-3209	47	32	introduced	introduce	VERB
ejpam-3209	47	33	:	:	PUNCT
ejpam-3209	47	34	the	the	DET
ejpam-3209	47	35	first	first	ADJ
ejpam-3209	47	36	product	product	NOUN
ejpam-3209	47	37	inequality	inequality	NOUN
ejpam-3209	47	38	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3209	47	39	≤	≤	NUM
ejpam-3209	47	40	‖u‖+	‖u‖+	PROPN
ejpam-3209	47	41	‖v‖	‖v‖	PROPN
ejpam-3209	47	42	,	,	PUNCT
ejpam-3209	47	43	∀	∀	X
ejpam-3209	47	44	u	u	NOUN
ejpam-3209	47	45	,	,	PUNCT
ejpam-3209	47	46	v	v	NOUN
ejpam-3209	47	47	∈	∈	PROPN
ejpam-3209	48	1	[	[	X
ejpam-3209	48	2	0	0	NUM
ejpam-3209	48	3	,	,	PUNCT
ejpam-3209	48	4	2	2	NUM
ejpam-3209	48	5	]	]	PUNCT
ejpam-3209	48	6	,	,	PUNCT
ejpam-3209	48	7	(	(	PUNCT
ejpam-3209	48	8	1	1	X
ejpam-3209	48	9	)	)	PUNCT
ejpam-3209	48	10	and	and	CCONJ
ejpam-3209	48	11	the	the	DET
ejpam-3209	48	12	second	second	ADJ
ejpam-3209	48	13	product	product	NOUN
ejpam-3209	48	14	inequality	inequality	NOUN
ejpam-3209	48	15	,	,	PUNCT
ejpam-3209	48	16	‖u‖+	‖u‖+	PROPN
ejpam-3209	48	17	‖v‖	‖v‖	PROPN
ejpam-3209	48	18	≤	≤	NUM
ejpam-3209	48	19	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3209	48	20	,	,	PUNCT
ejpam-3209	48	21	∀	∀	X
ejpam-3209	48	22	u	u	NOUN
ejpam-3209	48	23	,	,	PUNCT
ejpam-3209	48	24	v	v	NOUN
ejpam-3209	48	25	∈	∈	PROPN
ejpam-3209	49	1	[	[	X
ejpam-3209	49	2	0	0	NUM
ejpam-3209	49	3	,	,	PUNCT
ejpam-3209	49	4	2	2	NUM
ejpam-3209	49	5	]	]	PUNCT
ejpam-3209	49	6	.	.	PUNCT
ejpam-3209	50	1	(	(	PUNCT
ejpam-3209	50	2	2	2	X
ejpam-3209	50	3	)	)	PUNCT
ejpam-3209	50	4	before	before	SCONJ
ejpam-3209	50	5	we	we	PRON
ejpam-3209	50	6	proceed	proceed	VERB
ejpam-3209	50	7	to	to	PART
ejpam-3209	50	8	provide	provide	VERB
ejpam-3209	50	9	the	the	DET
ejpam-3209	50	10	proofs	proof	NOUN
ejpam-3209	50	11	of	of	ADP
ejpam-3209	50	12	the	the	DET
ejpam-3209	50	13	product	product	NOUN
ejpam-3209	50	14	inequalities	inequality	NOUN
ejpam-3209	50	15	some	some	PRON
ejpam-3209	50	16	of	of	ADP
ejpam-3209	50	17	relevant	relevant	ADJ
ejpam-3209	50	18	derintions	derintion	NOUN
ejpam-3209	50	19	regarding	regard	VERB
ejpam-3209	50	20	the	the	DET
ejpam-3209	50	21	introduction	introduction	NOUN
ejpam-3209	50	22	of	of	ADP
ejpam-3209	50	23	product	product	NOUN
ejpam-3209	50	24	inequalities	inequality	NOUN
ejpam-3209	50	25	are	be	AUX
ejpam-3209	50	26	given	give	VERB
ejpam-3209	50	27	in	in	ADP
ejpam-3209	50	28	this	this	DET
ejpam-3209	50	29	paper	paper	NOUN
ejpam-3209	50	30	.	.	PUNCT
ejpam-3209	51	1	definition	definition	NOUN
ejpam-3209	51	2	1	1	NUM
ejpam-3209	51	3	(	(	PUNCT
ejpam-3209	51	4	inner	inner	ADJ
ejpam-3209	51	5	product	product	NOUN
ejpam-3209	51	6	)	)	PUNCT
ejpam-3209	51	7	.	.	PUNCT
ejpam-3209	52	1	let	let	VERB
ejpam-3209	52	2	v	v	PART
ejpam-3209	52	3	be	be	AUX
ejpam-3209	52	4	a	a	DET
ejpam-3209	52	5	linear	linear	ADJ
ejpam-3209	52	6	vector	vector	NOUN
ejpam-3209	52	7	space	space	NOUN
ejpam-3209	52	8	defined	define	VERB
ejpam-3209	52	9	over	over	ADP
ejpam-3209	52	10	the	the	DET
ejpam-3209	52	11	real	real	ADJ
ejpam-3209	52	12	number	number	NOUN
ejpam-3209	52	13	field	field	NOUN
ejpam-3209	52	14	r.	r.	VERB
ejpam-3209	52	15	a	a	DET
ejpam-3209	52	16	scalar	scalar	ADV
ejpam-3209	52	17	-	-	PUNCT
ejpam-3209	52	18	valued	value	VERB
ejpam-3209	52	19	function	function	NOUN
ejpam-3209	52	20	p	p	NOUN
ejpam-3209	52	21	:	:	PUNCT
ejpam-3209	52	22	v	v	NUM
ejpam-3209	52	23	×	×	NOUN
ejpam-3209	52	24	v	v	NOUN
ejpam-3209	52	25	→	→	SYM
ejpam-3209	52	26	r	r	NOUN
ejpam-3209	52	27	that	that	PRON
ejpam-3209	52	28	associates	associate	NOUN
ejpam-3209	52	29	with	with	ADP
ejpam-3209	52	30	each	each	DET
ejpam-3209	52	31	pair	pair	NOUN
ejpam-3209	52	32	u	u	NOUN
ejpam-3209	52	33	,	,	PUNCT
ejpam-3209	52	34	v	v	NOUN
ejpam-3209	52	35	of	of	ADP
ejpam-3209	52	36	vectors	vector	NOUN
ejpam-3209	52	37	in	in	ADP
ejpam-3209	52	38	v	v	ADP
ejpam-3209	52	39	a	a	DET
ejpam-3209	52	40	scalar	scalar	ADJ
ejpam-3209	52	41	,	,	PUNCT
ejpam-3209	52	42	denoted	denote	VERB
ejpam-3209	52	43	(	(	PUNCT
ejpam-3209	52	44	u	u	NOUN
ejpam-3209	52	45	,	,	PUNCT
ejpam-3209	52	46	v	v	NOUN
ejpam-3209	52	47	)	)	PUNCT
ejpam-3209	52	48	,	,	PUNCT
ejpam-3209	52	49	is	be	AUX
ejpam-3209	52	50	called	call	VERB
ejpam-3209	52	51	an	an	DET
ejpam-3209	52	52	inner	inner	ADJ
ejpam-3209	52	53	product	product	NOUN
ejpam-3209	52	54	on	on	ADP
ejpam-3209	52	55	v	v	PRON
ejpam-3209	52	56	if	if	SCONJ
ejpam-3209	53	1	and	and	CCONJ
ejpam-3209	53	2	only	only	ADV
ejpam-3209	53	3	if	if	SCONJ
ejpam-3209	53	4	(	(	PUNCT
ejpam-3209	53	5	i	i	NOUN
ejpam-3209	53	6	)	)	PUNCT
ejpam-3209	53	7	(	(	PUNCT
ejpam-3209	53	8	u	u	NOUN
ejpam-3209	53	9	,	,	PUNCT
ejpam-3209	53	10	u	u	NOUN
ejpam-3209	53	11	)	)	PUNCT
ejpam-3209	53	12	>	>	X
ejpam-3209	53	13	0	0	PUNCT
ejpam-3209	54	1	whenever	whenever	SCONJ
ejpam-3209	54	2	u	u	PROPN
ejpam-3209	54	3	6=	6=	PROPN
ejpam-3209	54	4	0	0	NUM
ejpam-3209	54	5	,	,	PUNCT
ejpam-3209	54	6	and	and	CCONJ
ejpam-3209	54	7	(	(	PUNCT
ejpam-3209	54	8	u	u	NOUN
ejpam-3209	54	9	,	,	PUNCT
ejpam-3209	54	10	u	u	NOUN
ejpam-3209	54	11	)	)	PUNCT
ejpam-3209	54	12	=	=	SYM
ejpam-3209	54	13	0	0	PUNCT
ejpam-3209	54	14	if	if	SCONJ
ejpam-3209	54	15	and	and	CCONJ
ejpam-3209	54	16	only	only	ADV
ejpam-3209	54	17	if	if	SCONJ
ejpam-3209	54	18	u	u	NOUN
ejpam-3209	54	19	=	=	NOUN
ejpam-3209	54	20	0	0	NUM
ejpam-3209	54	21	(	(	PUNCT
ejpam-3209	54	22	ii	ii	NOUN
ejpam-3209	54	23	)	)	PUNCT
ejpam-3209	54	24	(	(	PUNCT
ejpam-3209	54	25	u	u	NOUN
ejpam-3209	54	26	,	,	PUNCT
ejpam-3209	54	27	v	v	NOUN
ejpam-3209	54	28	)	)	PUNCT
ejpam-3209	54	29	=	=	SYM
ejpam-3209	54	30	(	(	PUNCT
ejpam-3209	54	31	v	v	NOUN
ejpam-3209	54	32	,	,	PUNCT
ejpam-3209	54	33	u	u	NOUN
ejpam-3209	54	34	)	)	PUNCT
ejpam-3209	54	35	,	,	PUNCT
ejpam-3209	54	36	∀	∀	X
ejpam-3209	54	37	u	u	NOUN
ejpam-3209	54	38	,	,	PUNCT
ejpam-3209	54	39	v	v	NOUN
ejpam-3209	54	40	∈	∈	PROPN
ejpam-3209	54	41	v	v	ADP
ejpam-3209	54	42	(	(	PUNCT
ejpam-3209	54	43	iii	iii	NOUN
ejpam-3209	54	44	)	)	PUNCT
ejpam-3209	54	45	(	(	PUNCT
ejpam-3209	54	46	αu1	αu1	NOUN
ejpam-3209	54	47	+	+	CCONJ
ejpam-3209	54	48	βu2	βu2	ADJ
ejpam-3209	54	49	,	,	PUNCT
ejpam-3209	54	50	v	v	NOUN
ejpam-3209	54	51	)	)	PUNCT
ejpam-3209	54	52	=	=	SYM
ejpam-3209	54	53	α(u1	α(u1	PROPN
ejpam-3209	54	54	,	,	PUNCT
ejpam-3209	54	55	v	v	NOUN
ejpam-3209	54	56	)	)	PUNCT
ejpam-3209	54	57	+	+	CCONJ
ejpam-3209	54	58	β(u2	β(u2	NOUN
ejpam-3209	54	59	,	,	PUNCT
ejpam-3209	54	60	v	v	NOUN
ejpam-3209	54	61	)	)	PUNCT
ejpam-3209	54	62	,	,	PUNCT
ejpam-3209	54	63	∀	∀	X
ejpam-3209	54	64	α	α	NOUN
ejpam-3209	54	65	,	,	PUNCT
ejpam-3209	54	66	β	β	X
ejpam-3209	54	67	∈	∈	NOUN
ejpam-3209	54	68	r	r	NOUN
ejpam-3209	54	69	,	,	PUNCT
ejpam-3209	54	70	and	and	CCONJ
ejpam-3209	54	71	u1	u1	NOUN
ejpam-3209	54	72	,	,	PUNCT
ejpam-3209	54	73	u2	u2	PROPN
ejpam-3209	54	74	,	,	PUNCT
ejpam-3209	54	75	v	v	NOUN
ejpam-3209	54	76	∈	∈	NOUN
ejpam-3209	54	77	v	v	NOUN
ejpam-3209	54	78	,	,	PUNCT
ejpam-3209	54	79	see	see	VERB
ejpam-3209	54	80	[	[	X
ejpam-3209	54	81	13	13	NUM
ejpam-3209	54	82	]	]	PUNCT
ejpam-3209	54	83	.	.	PUNCT
ejpam-3209	55	1	definition	definition	NOUN
ejpam-3209	55	2	2	2	NUM
ejpam-3209	55	3	(	(	PUNCT
ejpam-3209	55	4	norm	norm	NOUN
ejpam-3209	55	5	)	)	PUNCT
ejpam-3209	55	6	.	.	PUNCT
ejpam-3209	56	1	let	let	VERB
ejpam-3209	56	2	v	v	PART
ejpam-3209	56	3	be	be	AUX
ejpam-3209	56	4	a	a	DET
ejpam-3209	56	5	linear	linear	ADJ
ejpam-3209	56	6	space	space	NOUN
ejpam-3209	56	7	over	over	ADP
ejpam-3209	56	8	r.	r.	PROPN
ejpam-3209	56	9	a	a	DET
ejpam-3209	56	10	norm	norm	NOUN
ejpam-3209	56	11	on	on	ADP
ejpam-3209	56	12	v	v	NUM
ejpam-3209	56	13	is	be	AUX
ejpam-3209	56	14	a	a	DET
ejpam-3209	56	15	real	real	ADV
ejpam-3209	56	16	-	-	PUNCT
ejpam-3209	56	17	valued	value	VERB
ejpam-3209	56	18	function	function	NOUN
ejpam-3209	56	19	‖	‖	PROPN
ejpam-3209	56	20	·	·	PUNCT
ejpam-3209	56	21	‖	‖	ADJ
ejpam-3209	56	22	:	:	PUNCT
ejpam-3209	56	23	v	v	X
ejpam-3209	56	24	→	→	SYM
ejpam-3209	56	25	[	[	X
ejpam-3209	56	26	0,∞	0,∞	NOUN
ejpam-3209	56	27	)	)	PUNCT
ejpam-3209	56	28	such	such	ADJ
ejpam-3209	56	29	that	that	PRON
ejpam-3209	56	30	for	for	ADP
ejpam-3209	56	31	any	any	DET
ejpam-3209	56	32	u	u	NOUN
ejpam-3209	56	33	,	,	PUNCT
ejpam-3209	56	34	v	v	NOUN
ejpam-3209	56	35	∈	∈	PROPN
ejpam-3209	56	36	v	v	NOUN
ejpam-3209	56	37	and	and	CCONJ
ejpam-3209	56	38	α	α	NOUN
ejpam-3209	56	39	∈	∈	NOUN
ejpam-3209	56	40	r	r	NOUN
ejpam-3209	56	41	the	the	DET
ejpam-3209	56	42	following	follow	VERB
ejpam-3209	56	43	conditions	condition	NOUN
ejpam-3209	56	44	are	be	AUX
ejpam-3209	56	45	met	meet	VERB
ejpam-3209	56	46	:	:	PUNCT
ejpam-3209	56	47	‖u‖	‖u‖	PROPN
ejpam-3209	56	48	≥	≥	NOUN
ejpam-3209	56	49	0	0	NUM
ejpam-3209	56	50	,	,	PUNCT
ejpam-3209	56	51	and	and	CCONJ
ejpam-3209	56	52	‖u‖	‖u‖	PROPN
ejpam-3209	56	53	=	=	SYM
ejpam-3209	56	54	0	0	PROPN
ejpam-3209	56	55	,	,	PUNCT
ejpam-3209	56	56	iff	iff	PROPN
ejpam-3209	56	57	u	u	NOUN
ejpam-3209	56	58	=	=	NOUN
ejpam-3209	56	59	0	0	NUM
ejpam-3209	56	60	‖αu‖	‖αu‖	ADJ
ejpam-3209	56	61	=	=	SYM
ejpam-3209	56	62	|α|‖u‖	|α|‖u‖	NOUN
ejpam-3209	56	63	,	,	PUNCT
ejpam-3209	56	64	∀	∀	X
ejpam-3209	56	65	u	u	NOUN
ejpam-3209	56	66	∈	∈	PROPN
ejpam-3209	56	67	v	v	NOUN
ejpam-3209	56	68	and	and	CCONJ
ejpam-3209	56	69	α	α	NOUN
ejpam-3209	56	70	∈	∈	NOUN
ejpam-3209	56	71	r	r	NOUN
ejpam-3209	56	72	‖u±	‖u±	NOUN
ejpam-3209	56	73	v‖	v‖	NOUN
ejpam-3209	56	74	≤	≤	NOUN
ejpam-3209	56	75	‖u‖+	‖u‖+	PRON
ejpam-3209	56	76	‖v‖	‖v‖	PROPN
ejpam-3209	56	77	,	,	PUNCT
ejpam-3209	56	78	∀	∀	X
ejpam-3209	56	79	u	u	NOUN
ejpam-3209	56	80	,	,	PUNCT
ejpam-3209	56	81	v	v	PROPN
ejpam-3209	56	82	∈	∈	PROPN
ejpam-3209	56	83	v	v	ADP
ejpam-3209	56	84	the	the	DET
ejpam-3209	56	85	norm	norm	NOUN
ejpam-3209	56	86	of	of	ADP
ejpam-3209	56	87	a	a	DET
ejpam-3209	56	88	vector	vector	NOUN
ejpam-3209	56	89	u	u	NOUN
ejpam-3209	56	90	can	can	AUX
ejpam-3209	56	91	be	be	AUX
ejpam-3209	56	92	generated	generate	VERB
ejpam-3209	56	93	by	by	ADP
ejpam-3209	56	94	the	the	DET
ejpam-3209	56	95	inner	inner	ADJ
ejpam-3209	56	96	product	product	NOUN
ejpam-3209	56	97	(	(	PUNCT
ejpam-3209	56	98	,	,	PUNCT
ejpam-3209	56	99	)	)	PUNCT
ejpam-3209	57	1	‖u‖	‖u‖	PROPN
ejpam-3209	57	2	=	=	PUNCT
ejpam-3209	58	1	√	√	PROPN
ejpam-3209	58	2	(	(	PUNCT
ejpam-3209	58	3	u	u	NOUN
ejpam-3209	58	4	,	,	PUNCT
ejpam-3209	58	5	u	u	NOUN
ejpam-3209	58	6	)	)	PUNCT
ejpam-3209	58	7	.	.	PUNCT
ejpam-3209	59	1	see	see	VERB
ejpam-3209	59	2	[	[	X
ejpam-3209	59	3	14	14	NUM
ejpam-3209	59	4	]	]	PUNCT
ejpam-3209	59	5	.	.	PUNCT
ejpam-3209	60	1	definition	definition	NOUN
ejpam-3209	60	2	3	3	NUM
ejpam-3209	60	3	(	(	PUNCT
ejpam-3209	60	4	cauchy	cauchy	NOUN
ejpam-3209	60	5	-	-	PUNCT
ejpam-3209	60	6	schwarz	schwarz	PROPN
ejpam-3209	60	7	inequality	inequality	NOUN
ejpam-3209	60	8	)	)	PUNCT
ejpam-3209	60	9	.	.	PUNCT
ejpam-3209	61	1	let	let	VERB
ejpam-3209	61	2	e	e	PRON
ejpam-3209	61	3	be	be	AUX
ejpam-3209	61	4	an	an	DET
ejpam-3209	61	5	inner	inner	ADJ
ejpam-3209	61	6	product	product	NOUN
ejpam-3209	61	7	space	space	NOUN
ejpam-3209	61	8	.	.	PUNCT
ejpam-3209	62	1	then	then	ADV
ejpam-3209	62	2	|〈u	|〈u	VERB
ejpam-3209	62	3	,	,	PUNCT
ejpam-3209	62	4	v〉|2	v〉|2	VERB
ejpam-3209	62	5	≤	≤	NOUN
ejpam-3209	62	6	‖x‖.‖y‖	‖x‖.‖y‖	NOUN
ejpam-3209	62	7	,	,	PUNCT
ejpam-3209	62	8	∀	∀	X
ejpam-3209	62	9	x	x	NOUN
ejpam-3209	62	10	,	,	PUNCT
ejpam-3209	62	11	y	y	PROPN
ejpam-3209	62	12	∈	∈	PROPN
ejpam-3209	62	13	e.	e.	PROPN
ejpam-3209	63	1	the	the	DET
ejpam-3209	63	2	equality	equality	NOUN
ejpam-3209	63	3	holds	hold	VERB
ejpam-3209	63	4	if	if	SCONJ
ejpam-3209	63	5	and	and	CCONJ
ejpam-3209	63	6	only	only	ADV
ejpam-3209	63	7	if	if	SCONJ
ejpam-3209	63	8	x	x	PRON
ejpam-3209	63	9	and	and	CCONJ
ejpam-3209	63	10	y	y	PROPN
ejpam-3209	63	11	are	be	AUX
ejpam-3209	63	12	linearly	linearly	ADV
ejpam-3209	63	13	dependent	dependent	ADJ
ejpam-3209	63	14	,	,	PUNCT
ejpam-3209	63	15	see	see	VERB
ejpam-3209	63	16	[	[	X
ejpam-3209	63	17	15	15	NUM
ejpam-3209	63	18	]	]	PUNCT
ejpam-3209	63	19	.	.	PUNCT
ejpam-3209	64	1	barnes	barnes	PROPN
ejpam-3209	64	2	et	et	PROPN
ejpam-3209	64	3	al	al	PROPN
ejpam-3209	64	4	.	.	PUNCT
ejpam-3209	64	5	/	/	SYM
ejpam-3209	64	6	eur	eur	PROPN
ejpam-3209	64	7	.	.	PUNCT
ejpam-3209	65	1	j.	j.	PROPN
ejpam-3209	65	2	pure	pure	PROPN
ejpam-3209	65	3	appl	appl	PROPN
ejpam-3209	65	4	.	.	PROPN
ejpam-3209	65	5	math	math	PROPN
ejpam-3209	65	6	,	,	PUNCT
ejpam-3209	65	7	11	11	NUM
ejpam-3209	65	8	(	(	PUNCT
ejpam-3209	65	9	2	2	NUM
ejpam-3209	65	10	)	)	PUNCT
ejpam-3209	65	11	(	(	PUNCT
ejpam-3209	65	12	2018	2018	NUM
ejpam-3209	65	13	)	)	PUNCT
ejpam-3209	65	14	,	,	PUNCT
ejpam-3209	65	15	375	375	NUM
ejpam-3209	65	16	-	-	SYM
ejpam-3209	65	17	389	389	NUM
ejpam-3209	65	18	378	378	NUM
ejpam-3209	65	19	definition	definition	NOUN
ejpam-3209	65	20	4	4	NUM
ejpam-3209	65	21	(	(	PUNCT
ejpam-3209	65	22	boundedness	boundedness	NOUN
ejpam-3209	65	23	)	)	PUNCT
ejpam-3209	65	24	.	.	PUNCT
ejpam-3209	66	1	let	let	VERB
ejpam-3209	66	2	t	t	NOUN
ejpam-3209	66	3	:	:	PUNCT
ejpam-3209	66	4	x	x	X
ejpam-3209	66	5	→	→	SYM
ejpam-3209	66	6	y	y	X
ejpam-3209	66	7	be	be	AUX
ejpam-3209	66	8	a	a	DET
ejpam-3209	66	9	linear	linear	ADJ
ejpam-3209	66	10	map	map	NOUN
ejpam-3209	66	11	.	.	PUNCT
ejpam-3209	67	1	then	then	ADV
ejpam-3209	67	2	t	t	PROPN
ejpam-3209	67	3	is	be	AUX
ejpam-3209	67	4	said	say	VERB
ejpam-3209	67	5	to	to	PART
ejpam-3209	67	6	be	be	AUX
ejpam-3209	67	7	a	a	DET
ejpam-3209	67	8	bounded	bounded	ADJ
ejpam-3209	67	9	linear	linear	ADJ
ejpam-3209	67	10	map	map	NOUN
ejpam-3209	67	11	if	if	SCONJ
ejpam-3209	67	12	there	there	PRON
ejpam-3209	67	13	exists	exist	VERB
ejpam-3209	67	14	k	k	PROPN
ejpam-3209	67	15	≥	≥	X
ejpam-3209	67	16	0	0	NUM
ejpam-3209	67	17	such	such	ADJ
ejpam-3209	67	18	that	that	DET
ejpam-3209	67	19	‖tx‖	‖tx‖	ADJ
ejpam-3209	67	20	≤	≤	NUM
ejpam-3209	67	21	k‖x‖	k‖x‖	NOUN
ejpam-3209	67	22	,	,	PUNCT
ejpam-3209	67	23	∀	∀	X
ejpam-3209	67	24	x	x	SYM
ejpam-3209	67	25	∈	∈	NOUN
ejpam-3209	67	26	x	x	NOUN
ejpam-3209	67	27	,	,	PUNCT
ejpam-3209	67	28	and	and	CCONJ
ejpam-3209	67	29	k	k	PROPN
ejpam-3209	67	30	is	be	AUX
ejpam-3209	67	31	the	the	DET
ejpam-3209	67	32	boundedness	boundedness	NOUN
ejpam-3209	67	33	constant	constant	ADJ
ejpam-3209	67	34	for	for	ADP
ejpam-3209	67	35	t	t	PROPN
ejpam-3209	67	36	.	.	PUNCT
ejpam-3209	68	1	the	the	DET
ejpam-3209	68	2	boundedness	boundedness	NOUN
ejpam-3209	68	3	of	of	ADP
ejpam-3209	68	4	the	the	DET
ejpam-3209	68	5	linear	linear	ADJ
ejpam-3209	68	6	map	map	NOUN
ejpam-3209	68	7	implies	imply	VERB
ejpam-3209	68	8	continuity	continuity	NOUN
ejpam-3209	68	9	of	of	ADP
ejpam-3209	68	10	t	t	PROPN
ejpam-3209	68	11	.	.	PUNCT
ejpam-3209	69	1	see	see	VERB
ejpam-3209	69	2	[	[	X
ejpam-3209	69	3	16	16	NUM
ejpam-3209	69	4	]	]	PUNCT
ejpam-3209	69	5	definition	definition	NOUN
ejpam-3209	69	6	5	5	NUM
ejpam-3209	69	7	(	(	PUNCT
ejpam-3209	69	8	integral	integral	ADJ
ejpam-3209	69	9	operator	operator	NOUN
ejpam-3209	69	10	and	and	CCONJ
ejpam-3209	69	11	monotonicity	monotonicity	NOUN
ejpam-3209	69	12	of	of	ADP
ejpam-3209	69	13	the	the	DET
ejpam-3209	69	14	integral	integral	ADJ
ejpam-3209	69	15	operator	operator	NOUN
ejpam-3209	69	16	)	)	PUNCT
ejpam-3209	69	17	.	.	PUNCT
ejpam-3209	70	1	let	let	VERB
ejpam-3209	70	2	f(x	f(x	PROPN
ejpam-3209	70	3	)	)	PUNCT
ejpam-3209	70	4	and	and	CCONJ
ejpam-3209	70	5	g(x	g(x	NOUN
ejpam-3209	70	6	)	)	PUNCT
ejpam-3209	70	7	be	be	VERB
ejpam-3209	70	8	simple	simple	ADJ
ejpam-3209	70	9	functions	function	NOUN
ejpam-3209	70	10	defined	define	VERB
ejpam-3209	70	11	on	on	ADP
ejpam-3209	70	12	a	a	DET
ejpam-3209	70	13	set	set	NOUN
ejpam-3209	70	14	of	of	ADP
ejpam-3209	70	15	finite	finite	ADJ
ejpam-3209	70	16	measure	measure	NOUN
ejpam-3209	70	17	e.	e.	PROPN
ejpam-3209	70	18	then	then	ADV
ejpam-3209	70	19	for	for	ADP
ejpam-3209	70	20	any	any	DET
ejpam-3209	70	21	α	α	NOUN
ejpam-3209	70	22	and	and	CCONJ
ejpam-3209	70	23	β,∫	β,∫	NOUN
ejpam-3209	70	24	e	e	X
ejpam-3209	70	25	(	(	PUNCT
ejpam-3209	70	26	αf(x	αf(x	X
ejpam-3209	70	27	)	)	PUNCT
ejpam-3209	71	1	+	+	CCONJ
ejpam-3209	71	2	βg(x))dx	βg(x))dx	NOUN
ejpam-3209	71	3	=	=	PUNCT
ejpam-3209	71	4	α	α	NUM
ejpam-3209	71	5	∫	∫	X
ejpam-3209	71	6	e	e	X
ejpam-3209	71	7	f(x)dx+	f(x)dx+	NOUN
ejpam-3209	71	8	β	β	X
ejpam-3209	71	9	∫	∫	PROPN
ejpam-3209	71	10	e	e	PROPN
ejpam-3209	71	11	g(x)dx	g(x)dx	NOUN
ejpam-3209	71	12	,	,	PUNCT
ejpam-3209	71	13	∀	∀	X
ejpam-3209	71	14	f	f	NOUN
ejpam-3209	71	15	,	,	PUNCT
ejpam-3209	71	16	g	g	PROPN
ejpam-3209	71	17	∈	∈	PROPN
ejpam-3209	71	18	e.	e.	PROPN
ejpam-3209	71	19	and	and	CCONJ
ejpam-3209	71	20	α	α	PROPN
ejpam-3209	71	21	,	,	PUNCT
ejpam-3209	71	22	β	β	X
ejpam-3209	71	23	are	be	AUX
ejpam-3209	71	24	scalars	scalar	NOUN
ejpam-3209	71	25	.	.	PUNCT
ejpam-3209	72	1	moreover	moreover	ADV
ejpam-3209	72	2	,	,	PUNCT
ejpam-3209	72	3	if	if	SCONJ
ejpam-3209	72	4	f(x	f(x	PROPN
ejpam-3209	72	5	)	)	PUNCT
ejpam-3209	72	6	≤	≤	NOUN
ejpam-3209	72	7	g(x	g(x	NOUN
ejpam-3209	72	8	)	)	PUNCT
ejpam-3209	72	9	,	,	PUNCT
ejpam-3209	72	10	then∫	then∫	NOUN
ejpam-3209	72	11	e	e	PROPN
ejpam-3209	72	12	f(x)dx	f(x)dx	VERB
ejpam-3209	72	13	≤	≤	NUM
ejpam-3209	72	14	∫	∫	PROPN
ejpam-3209	72	15	e	e	NOUN
ejpam-3209	72	16	g(x)dx	g(x)dx	NOUN
ejpam-3209	72	17	,	,	PUNCT
ejpam-3209	72	18	see	see	VERB
ejpam-3209	72	19	[	[	X
ejpam-3209	72	20	17	17	NUM
ejpam-3209	72	21	]	]	SYM
ejpam-3209	72	22	.	.	PUNCT
ejpam-3209	73	1	2.1	2.1	NUM
ejpam-3209	73	2	.	.	PUNCT
ejpam-3209	74	1	the	the	DET
ejpam-3209	74	2	proof	proof	NOUN
ejpam-3209	74	3	of	of	ADP
ejpam-3209	74	4	the	the	DET
ejpam-3209	74	5	first	first	ADJ
ejpam-3209	74	6	product	product	NOUN
ejpam-3209	74	7	inequality	inequality	NOUN
ejpam-3209	74	8	we	we	PRON
ejpam-3209	74	9	provide	provide	VERB
ejpam-3209	74	10	the	the	DET
ejpam-3209	74	11	proofs	proof	NOUN
ejpam-3209	74	12	of	of	ADP
ejpam-3209	74	13	product	product	NOUN
ejpam-3209	74	14	inequalities	inequality	NOUN
ejpam-3209	74	15	,	,	PUNCT
ejpam-3209	74	16	through	through	ADP
ejpam-3209	74	17	binomial	binomial	ADJ
ejpam-3209	74	18	inequalities	inequality	NOUN
ejpam-3209	74	19	,	,	PUNCT
ejpam-3209	74	20	by	by	ADP
ejpam-3209	74	21	induction	induction	NOUN
ejpam-3209	74	22	.	.	PUNCT
ejpam-3209	75	1	firstly	firstly	ADV
ejpam-3209	75	2	,	,	PUNCT
ejpam-3209	75	3	we	we	PRON
ejpam-3209	75	4	consider	consider	VERB
ejpam-3209	75	5	a	a	DET
ejpam-3209	75	6	positive	positive	ADJ
ejpam-3209	75	7	integer	integer	NOUN
ejpam-3209	75	8	n	n	NOUN
ejpam-3209	75	9	=	=	SYM
ejpam-3209	75	10	2	2	NUM
ejpam-3209	75	11	as	as	SCONJ
ejpam-3209	75	12	follows	follow	VERB
ejpam-3209	75	13	.	.	PUNCT
ejpam-3209	76	1	(	(	PUNCT
ejpam-3209	76	2	u+	u+	NUM
ejpam-3209	76	3	v)2	v)2	PROPN
ejpam-3209	76	4	≥	≥	NOUN
ejpam-3209	76	5	0	0	NUM
ejpam-3209	76	6	(	(	PUNCT
ejpam-3209	76	7	u	u	NOUN
ejpam-3209	76	8	,	,	PUNCT
ejpam-3209	76	9	u	u	NOUN
ejpam-3209	76	10	)	)	PUNCT
ejpam-3209	76	11	+	+	CCONJ
ejpam-3209	76	12	2(u	2(u	NUM
ejpam-3209	76	13	,	,	PUNCT
ejpam-3209	76	14	v	v	NOUN
ejpam-3209	76	15	)	)	PUNCT
ejpam-3209	76	16	+	+	CCONJ
ejpam-3209	76	17	(	(	PUNCT
ejpam-3209	76	18	v	v	NOUN
ejpam-3209	76	19	,	,	PUNCT
ejpam-3209	76	20	v	v	NOUN
ejpam-3209	76	21	)	)	PUNCT
ejpam-3209	76	22	≥	≥	NOUN
ejpam-3209	76	23	0	0	NUM
ejpam-3209	76	24	−2(u	−2(u	NUM
ejpam-3209	76	25	,	,	PUNCT
ejpam-3209	76	26	v	v	NOUN
ejpam-3209	76	27	)	)	PUNCT
ejpam-3209	76	28	≤	≤	NOUN
ejpam-3209	76	29	{	{	PUNCT
ejpam-3209	76	30	(	(	PUNCT
ejpam-3209	76	31	u	u	NOUN
ejpam-3209	76	32	,	,	PUNCT
ejpam-3209	76	33	u	u	NOUN
ejpam-3209	76	34	)	)	PUNCT
ejpam-3209	76	35	+	+	CCONJ
ejpam-3209	76	36	(	(	PUNCT
ejpam-3209	76	37	v	v	NOUN
ejpam-3209	76	38	,	,	PUNCT
ejpam-3209	76	39	v	v	NOUN
ejpam-3209	76	40	)	)	PUNCT
ejpam-3209	76	41	}	}	PUNCT
ejpam-3209	76	42	−2(u	−2(u	PROPN
ejpam-3209	76	43	,	,	PUNCT
ejpam-3209	76	44	v	v	NOUN
ejpam-3209	76	45	)	)	PUNCT
ejpam-3209	76	46	≤	≤	NOUN
ejpam-3209	76	47	(	(	PUNCT
ejpam-3209	76	48	u	u	NOUN
ejpam-3209	76	49	,	,	PUNCT
ejpam-3209	76	50	u	u	NOUN
ejpam-3209	76	51	)	)	PUNCT
ejpam-3209	77	1	+	+	CCONJ
ejpam-3209	77	2	(	(	PUNCT
ejpam-3209	77	3	v	v	NOUN
ejpam-3209	77	4	,	,	PUNCT
ejpam-3209	77	5	v	v	NOUN
ejpam-3209	77	6	)	)	PUNCT
ejpam-3209	78	1	+	+	CCONJ
ejpam-3209	78	2	2(u	2(u	NUM
ejpam-3209	78	3	,	,	PUNCT
ejpam-3209	78	4	v	v	NOUN
ejpam-3209	78	5	)	)	PUNCT
ejpam-3209	78	6	−2(u	−2(u	PROPN
ejpam-3209	78	7	,	,	PUNCT
ejpam-3209	78	8	v	v	NOUN
ejpam-3209	78	9	)	)	PUNCT
ejpam-3209	78	10	1	1	NUM
ejpam-3209	78	11	2	2	NUM
ejpam-3209	78	12	(	(	PUNCT
ejpam-3209	78	13	u	u	NOUN
ejpam-3209	78	14	,	,	PUNCT
ejpam-3209	78	15	v	v	NOUN
ejpam-3209	78	16	)	)	PUNCT
ejpam-3209	78	17	≤	≤	NOUN
ejpam-3209	78	18	(	(	PUNCT
ejpam-3209	78	19	u+	u+	NUM
ejpam-3209	78	20	v)2	v)2	PROPN
ejpam-3209	78	21	−(u	−(u	NOUN
ejpam-3209	78	22	,	,	PUNCT
ejpam-3209	78	23	v)2	v)2	ADJ
ejpam-3209	78	24	=	=	SYM
ejpam-3209	78	25	(	(	PUNCT
ejpam-3209	78	26	u+	u+	NUM
ejpam-3209	78	27	v)2	v)2	PROPN
ejpam-3209	78	28	‖	‖	PROPN
ejpam-3209	78	29	−	−	PROPN
ejpam-3209	78	30	(	(	PUNCT
ejpam-3209	78	31	u	u	NOUN
ejpam-3209	78	32	,	,	PUNCT
ejpam-3209	78	33	v)2‖	v)2‖	NOUN
ejpam-3209	78	34	=	=	PUNCT
ejpam-3209	78	35	‖(u+	‖(u+	NUM
ejpam-3209	78	36	v)2‖	v)2‖	NOUN
ejpam-3209	78	37	‖(u	‖(u	NOUN
ejpam-3209	78	38	,	,	PUNCT
ejpam-3209	78	39	v)‖2	v)‖2	NOUN
ejpam-3209	78	40	≤	≤	NOUN
ejpam-3209	78	41	‖(u+	‖(u+	PUNCT
ejpam-3209	78	42	v)‖2	v)‖2	NOUN
ejpam-3209	78	43	⇒	⇒	VERB
ejpam-3209	78	44	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3209	78	45	≤	≤	ADJ
ejpam-3209	78	46	‖u‖+	‖u‖+	PROPN
ejpam-3209	78	47	‖v‖.	‖v‖.	NOUN
ejpam-3209	78	48	for	for	ADP
ejpam-3209	78	49	n	n	NOUN
ejpam-3209	78	50	=	=	SYM
ejpam-3209	78	51	4	4	NUM
ejpam-3209	78	52	,	,	PUNCT
ejpam-3209	78	53	we	we	PRON
ejpam-3209	78	54	observe	observe	VERB
ejpam-3209	78	55	the	the	DET
ejpam-3209	78	56	following	follow	VERB
ejpam-3209	78	57	inequalities	inequality	NOUN
ejpam-3209	78	58	:	:	PUNCT
ejpam-3209	78	59	(	(	PUNCT
ejpam-3209	78	60	u+	u+	NUM
ejpam-3209	78	61	v)4	v)4	PROPN
ejpam-3209	78	62	≥	≥	X
ejpam-3209	78	63	0	0	NUM
ejpam-3209	78	64	−6(u	−6(u	NOUN
ejpam-3209	78	65	,	,	PUNCT
ejpam-3209	78	66	u)(v	u)(v	PROPN
ejpam-3209	78	67	,	,	PUNCT
ejpam-3209	78	68	v	v	NOUN
ejpam-3209	78	69	)	)	PUNCT
ejpam-3209	78	70	≤	≤	NOUN
ejpam-3209	78	71	{	{	PUNCT
ejpam-3209	78	72	(	(	PUNCT
ejpam-3209	78	73	u	u	NOUN
ejpam-3209	78	74	,	,	PUNCT
ejpam-3209	78	75	u)2	u)2	PROPN
ejpam-3209	78	76	+	+	CCONJ
ejpam-3209	78	77	4(u	4(u	NOUN
ejpam-3209	78	78	,	,	PUNCT
ejpam-3209	78	79	u)(u	u)(u	ADJ
ejpam-3209	78	80	,	,	PUNCT
ejpam-3209	78	81	v	v	NOUN
ejpam-3209	78	82	)	)	PUNCT
ejpam-3209	78	83	+	+	CCONJ
ejpam-3209	78	84	4(u	4(u	NOUN
ejpam-3209	78	85	,	,	PUNCT
ejpam-3209	78	86	v)(v	v)(v	NOUN
ejpam-3209	78	87	,	,	PUNCT
ejpam-3209	78	88	v	v	NOUN
ejpam-3209	78	89	)	)	PUNCT
ejpam-3209	79	1	+	+	CCONJ
ejpam-3209	79	2	(	(	PUNCT
ejpam-3209	79	3	v	v	NOUN
ejpam-3209	79	4	,	,	PUNCT
ejpam-3209	79	5	v)2	v)2	PROPN
ejpam-3209	79	6	}	}	PUNCT
ejpam-3209	79	7	−6(u	−6(u	NOUN
ejpam-3209	79	8	,	,	PUNCT
ejpam-3209	79	9	u)(v	u)(v	PROPN
ejpam-3209	79	10	,	,	PUNCT
ejpam-3209	79	11	v	v	NOUN
ejpam-3209	79	12	)	)	PUNCT
ejpam-3209	79	13	≤	≤	NOUN
ejpam-3209	79	14	{	{	PUNCT
ejpam-3209	79	15	(	(	PUNCT
ejpam-3209	79	16	u	u	NOUN
ejpam-3209	79	17	,	,	PUNCT
ejpam-3209	79	18	u)2	u)2	PROPN
ejpam-3209	79	19	+	+	CCONJ
ejpam-3209	79	20	4(u	4(u	NOUN
ejpam-3209	79	21	,	,	PUNCT
ejpam-3209	79	22	u)(u	u)(u	ADJ
ejpam-3209	79	23	,	,	PUNCT
ejpam-3209	79	24	v	v	NOUN
ejpam-3209	79	25	)	)	PUNCT
ejpam-3209	79	26	+	+	CCONJ
ejpam-3209	79	27	6(u	6(u	NUM
ejpam-3209	79	28	,	,	PUNCT
ejpam-3209	79	29	u)(v	u)(v	PROPN
ejpam-3209	79	30	,	,	PUNCT
ejpam-3209	79	31	v	v	NOUN
ejpam-3209	79	32	)	)	PUNCT
ejpam-3209	80	1	+	+	CCONJ
ejpam-3209	80	2	4(u	4(u	NOUN
ejpam-3209	80	3	,	,	PUNCT
ejpam-3209	80	4	v)(v	v)(v	NOUN
ejpam-3209	80	5	,	,	PUNCT
ejpam-3209	80	6	v	v	NOUN
ejpam-3209	80	7	)	)	PUNCT
ejpam-3209	80	8	+	+	CCONJ
ejpam-3209	80	9	(	(	PUNCT
ejpam-3209	80	10	v	v	NOUN
ejpam-3209	80	11	,	,	PUNCT
ejpam-3209	80	12	v)2	v)2	PROPN
ejpam-3209	80	13	}	}	PUNCT
ejpam-3209	80	14	−6(u	−6(u	NOUN
ejpam-3209	80	15	,	,	PUNCT
ejpam-3209	80	16	u)(v	u)(v	PROPN
ejpam-3209	80	17	,	,	PUNCT
ejpam-3209	80	18	v	v	NOUN
ejpam-3209	80	19	)	)	PUNCT
ejpam-3209	80	20	.	.	PUNCT
ejpam-3209	81	1	1	1	NUM
ejpam-3209	81	2	6	6	NUM
ejpam-3209	81	3	(	(	PUNCT
ejpam-3209	81	4	u	u	NOUN
ejpam-3209	81	5	,	,	PUNCT
ejpam-3209	81	6	u)(v	u)(v	PROPN
ejpam-3209	81	7	,	,	PUNCT
ejpam-3209	81	8	v	v	NOUN
ejpam-3209	81	9	)	)	PUNCT
ejpam-3209	81	10	≤	≤	NOUN
ejpam-3209	81	11	(	(	PUNCT
ejpam-3209	81	12	u+	u+	NUM
ejpam-3209	81	13	v)4	v)4	PROPN
ejpam-3209	81	14	−{(u	−{(u	NOUN
ejpam-3209	81	15	,	,	PUNCT
ejpam-3209	81	16	u)(v	u)(v	PROPN
ejpam-3209	81	17	,	,	PUNCT
ejpam-3209	81	18	v)}2	v)}2	PROPN
ejpam-3209	81	19	=	=	PUNCT
ejpam-3209	81	20	(	(	PUNCT
ejpam-3209	81	21	u+	u+	NUM
ejpam-3209	81	22	v)4	v)4	PROPN
ejpam-3209	81	23	barnes	barne	NOUN
ejpam-3209	81	24	et	et	PROPN
ejpam-3209	81	25	al	al	PROPN
ejpam-3209	81	26	.	.	PUNCT
ejpam-3209	81	27	/	/	SYM
ejpam-3209	81	28	eur	eur	PROPN
ejpam-3209	81	29	.	.	PUNCT
ejpam-3209	82	1	j.	j.	PROPN
ejpam-3209	82	2	pure	pure	PROPN
ejpam-3209	82	3	appl	appl	PROPN
ejpam-3209	82	4	.	.	PROPN
ejpam-3209	82	5	math	math	PROPN
ejpam-3209	82	6	,	,	PUNCT
ejpam-3209	82	7	11	11	NUM
ejpam-3209	82	8	(	(	PUNCT
ejpam-3209	82	9	2	2	NUM
ejpam-3209	82	10	)	)	PUNCT
ejpam-3209	82	11	(	(	PUNCT
ejpam-3209	82	12	2018	2018	NUM
ejpam-3209	82	13	)	)	PUNCT
ejpam-3209	82	14	,	,	PUNCT
ejpam-3209	82	15	375	375	NUM
ejpam-3209	82	16	-	-	SYM
ejpam-3209	82	17	389	389	NUM
ejpam-3209	82	18	379	379	NUM
ejpam-3209	82	19	‖	‖	PROPN
ejpam-3209	82	20	−	−	PROPN
ejpam-3209	82	21	{	{	PUNCT
ejpam-3209	82	22	(	(	PUNCT
ejpam-3209	82	23	u	u	NOUN
ejpam-3209	82	24	,	,	PUNCT
ejpam-3209	82	25	u)(v	u)(v	NOUN
ejpam-3209	82	26	,	,	PUNCT
ejpam-3209	82	27	v)}2‖	v)}2‖	NOUN
ejpam-3209	82	28	=	=	SYM
ejpam-3209	82	29	‖(u+	‖(u+	PUNCT
ejpam-3209	82	30	v)4‖	v)4‖	NOUN
ejpam-3209	82	31	{	{	PUNCT
ejpam-3209	82	32	‖u‖‖v‖}4	‖u‖‖v‖}4	NOUN
ejpam-3209	82	33	=	=	SYM
ejpam-3209	82	34	‖(u+	‖(u+	NUM
ejpam-3209	82	35	v)‖4	v)‖4	NOUN
ejpam-3209	82	36	⇒	⇒	VERB
ejpam-3209	82	37	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3209	82	38	≤	≤	ADJ
ejpam-3209	82	39	‖u‖+	‖u‖+	PROPN
ejpam-3209	82	40	‖v‖.	‖v‖.	NOUN
ejpam-3209	82	41	in	in	ADP
ejpam-3209	82	42	order	order	NOUN
ejpam-3209	82	43	to	to	PART
ejpam-3209	82	44	generalize	generalize	VERB
ejpam-3209	82	45	the	the	DET
ejpam-3209	82	46	first	first	ADJ
ejpam-3209	82	47	product	product	NOUN
ejpam-3209	82	48	inequality	inequality	NOUN
ejpam-3209	82	49	,	,	PUNCT
ejpam-3209	82	50	we	we	PRON
ejpam-3209	82	51	observed	observe	VERB
ejpam-3209	82	52	that	that	SCONJ
ejpam-3209	82	53	for	for	ADP
ejpam-3209	82	54	any	any	DET
ejpam-3209	82	55	positive	positive	ADJ
ejpam-3209	82	56	integer	integer	NOUN
ejpam-3209	82	57	n	n	CCONJ
ejpam-3209	82	58	,	,	PUNCT
ejpam-3209	82	59	we	we	PRON
ejpam-3209	82	60	obtain	obtain	VERB
ejpam-3209	82	61	:	:	PUNCT
ejpam-3209	82	62	(	(	PUNCT
ejpam-3209	83	1	u+	u+	NUM
ejpam-3209	83	2	v)n	v)n	ADJ
ejpam-3209	83	3	≥	≥	NOUN
ejpam-3209	83	4	0	0	NUM
ejpam-3209	83	5	⇒	⇒	NOUN
ejpam-3209	83	6	(	(	PUNCT
ejpam-3209	83	7	u	u	NOUN
ejpam-3209	83	8	,	,	PUNCT
ejpam-3209	83	9	u	u	NOUN
ejpam-3209	83	10	)	)	PUNCT
ejpam-3209	83	11	n	n	ADV
ejpam-3209	83	12	2	2	NUM
ejpam-3209	83	13	+	+	NOUN
ejpam-3209	83	14	n	n	PRON
ejpam-3209	83	15	c1(u	c1(u	NOUN
ejpam-3209	83	16	,	,	PUNCT
ejpam-3209	83	17	v)(u	v)(u	NUM
ejpam-3209	83	18	,	,	PUNCT
ejpam-3209	83	19	u	u	NOUN
ejpam-3209	83	20	)	)	PUNCT
ejpam-3209	83	21	n−2	n−2	PROPN
ejpam-3209	83	22	2	2	NUM
ejpam-3209	83	23	+	+	NOUN
ejpam-3209	83	24	n	n	PROPN
ejpam-3209	83	25	c2(u	c2(u	PROPN
ejpam-3209	83	26	,	,	PUNCT
ejpam-3209	83	27	u	u	NOUN
ejpam-3209	83	28	)	)	PUNCT
ejpam-3209	83	29	n−2	n−2	PROPN
ejpam-3209	83	30	2	2	NUM
ejpam-3209	83	31	(	(	PUNCT
ejpam-3209	83	32	v	v	NOUN
ejpam-3209	83	33	,	,	PUNCT
ejpam-3209	83	34	v	v	NOUN
ejpam-3209	83	35	)	)	PUNCT
ejpam-3209	83	36	+	+	NOUN
ejpam-3209	83	37	n	n	SYM
ejpam-3209	83	38	c3(u	c3(u	PROPN
ejpam-3209	83	39	,	,	PUNCT
ejpam-3209	83	40	u	u	NOUN
ejpam-3209	83	41	)	)	PUNCT
ejpam-3209	83	42	n−4	n−4	PROPN
ejpam-3209	83	43	2	2	NUM
ejpam-3209	83	44	(	(	PUNCT
ejpam-3209	83	45	v	v	NOUN
ejpam-3209	83	46	,	,	PUNCT
ejpam-3209	83	47	v)(u	v)(u	ADJ
ejpam-3209	83	48	,	,	PUNCT
ejpam-3209	83	49	v	v	NOUN
ejpam-3209	83	50	)	)	PUNCT
ejpam-3209	83	51	+	+	CCONJ
ejpam-3209	83	52	nc4(u	nc4(u	PROPN
ejpam-3209	83	53	,	,	PUNCT
ejpam-3209	83	54	u	u	NOUN
ejpam-3209	83	55	)	)	PUNCT
ejpam-3209	83	56	n−4	n−4	PROPN
ejpam-3209	83	57	2	2	NUM
ejpam-3209	83	58	(	(	PUNCT
ejpam-3209	83	59	v	v	NOUN
ejpam-3209	83	60	,	,	PUNCT
ejpam-3209	83	61	v)2	v)2	ADJ
ejpam-3209	83	62	+	+	NOUN
ejpam-3209	83	63	n	n	PROPN
ejpam-3209	83	64	c5(u	c5(u	PROPN
ejpam-3209	83	65	,	,	PUNCT
ejpam-3209	83	66	u	u	NOUN
ejpam-3209	83	67	)	)	PUNCT
ejpam-3209	83	68	n−6	n−6	PROPN
ejpam-3209	83	69	2	2	NUM
ejpam-3209	83	70	(	(	PUNCT
ejpam-3209	83	71	v	v	NOUN
ejpam-3209	83	72	,	,	PUNCT
ejpam-3209	83	73	v)2(u	v)2(u	NUM
ejpam-3209	83	74	,	,	PUNCT
ejpam-3209	83	75	v	v	NOUN
ejpam-3209	83	76	)	)	PUNCT
ejpam-3209	83	77	+	+	NOUN
ejpam-3209	83	78	n	n	PRON
ejpam-3209	83	79	c6(u	c6(u	PROPN
ejpam-3209	83	80	,	,	PUNCT
ejpam-3209	83	81	u	u	NOUN
ejpam-3209	83	82	)	)	PUNCT
ejpam-3209	83	83	n−6	n−6	PROPN
ejpam-3209	83	84	2	2	NUM
ejpam-3209	83	85	(	(	PUNCT
ejpam-3209	83	86	v	v	NOUN
ejpam-3209	83	87	,	,	PUNCT
ejpam-3209	83	88	v)3	v)3	PROPN
ejpam-3209	83	89	+	+	CCONJ
ejpam-3209	83	90	nc7(u	nc7(u	PROPN
ejpam-3209	83	91	,	,	PUNCT
ejpam-3209	83	92	u	u	NOUN
ejpam-3209	83	93	)	)	PUNCT
ejpam-3209	83	94	n−8	n−8	PROPN
ejpam-3209	83	95	2	2	NUM
ejpam-3209	83	96	(	(	PUNCT
ejpam-3209	83	97	v	v	NOUN
ejpam-3209	83	98	,	,	PUNCT
ejpam-3209	83	99	v)3(u	v)3(u	NOUN
ejpam-3209	83	100	,	,	PUNCT
ejpam-3209	83	101	v	v	NOUN
ejpam-3209	83	102	)	)	PUNCT
ejpam-3209	83	103	+	+	CCONJ
ejpam-3209	83	104	.	.	PUNCT
ejpam-3209	83	105	.	.	PUNCT
ejpam-3209	84	1	.+n	.+n	PROPN
ejpam-3209	84	2	cn	cn	NOUN
ejpam-3209	84	3	2	2	NUM
ejpam-3209	84	4	(	(	PUNCT
ejpam-3209	84	5	u	u	NOUN
ejpam-3209	84	6	,	,	PUNCT
ejpam-3209	84	7	u	u	NOUN
ejpam-3209	84	8	)	)	PUNCT
ejpam-3209	84	9	n	n	ADV
ejpam-3209	84	10	4	4	NUM
ejpam-3209	84	11	(	(	PUNCT
ejpam-3209	84	12	v	v	NOUN
ejpam-3209	84	13	,	,	PUNCT
ejpam-3209	84	14	v	v	NOUN
ejpam-3209	84	15	)	)	PUNCT
ejpam-3209	84	16	n	n	ADV
ejpam-3209	84	17	4	4	NUM
ejpam-3209	84	18	+	+	CCONJ
ejpam-3209	84	19	.	.	PUNCT
ejpam-3209	84	20	.	.	PUNCT
ejpam-3209	85	1	.+	.+	NOUN
ejpam-3209	85	2	(	(	PUNCT
ejpam-3209	85	3	u	u	NOUN
ejpam-3209	85	4	,	,	PUNCT
ejpam-3209	85	5	u	u	NOUN
ejpam-3209	85	6	)	)	PUNCT
ejpam-3209	85	7	n	n	DET
ejpam-3209	85	8	2	2	NUM
ejpam-3209	85	9	≥	≥	NOUN
ejpam-3209	85	10	0	0	NUM
ejpam-3209	85	11	⇒	⇒	NOUN
ejpam-3209	85	12	−ncn	−ncn	NOUN
ejpam-3209	85	13	2	2	NUM
ejpam-3209	85	14	(	(	PUNCT
ejpam-3209	85	15	u	u	NOUN
ejpam-3209	85	16	,	,	PUNCT
ejpam-3209	85	17	u	u	NOUN
ejpam-3209	85	18	)	)	PUNCT
ejpam-3209	85	19	n	n	ADV
ejpam-3209	85	20	4	4	NUM
ejpam-3209	85	21	(	(	PUNCT
ejpam-3209	85	22	v	v	NOUN
ejpam-3209	85	23	,	,	PUNCT
ejpam-3209	85	24	v	v	NOUN
ejpam-3209	85	25	)	)	PUNCT
ejpam-3209	85	26	n	n	PRON
ejpam-3209	85	27	4	4	NUM
ejpam-3209	85	28	≤	≤	NUM
ejpam-3209	85	29	{	{	PUNCT
ejpam-3209	85	30	(	(	PUNCT
ejpam-3209	85	31	u	u	NOUN
ejpam-3209	85	32	,	,	PUNCT
ejpam-3209	85	33	u	u	NOUN
ejpam-3209	85	34	)	)	PUNCT
ejpam-3209	85	35	n	n	ADV
ejpam-3209	85	36	2	2	NUM
ejpam-3209	85	37	+	+	NOUN
ejpam-3209	85	38	n	n	PRON
ejpam-3209	85	39	c1(u	c1(u	NOUN
ejpam-3209	85	40	,	,	PUNCT
ejpam-3209	85	41	v)(u	v)(u	NUM
ejpam-3209	85	42	,	,	PUNCT
ejpam-3209	85	43	u	u	NOUN
ejpam-3209	85	44	)	)	PUNCT
ejpam-3209	85	45	n−2	n−2	PROPN
ejpam-3209	85	46	2	2	NUM
ejpam-3209	85	47	+	+	NOUN
ejpam-3209	85	48	n	n	PROPN
ejpam-3209	85	49	c2(u	c2(u	PROPN
ejpam-3209	85	50	,	,	PUNCT
ejpam-3209	85	51	u	u	NOUN
ejpam-3209	85	52	)	)	PUNCT
ejpam-3209	85	53	n−2	n−2	PROPN
ejpam-3209	85	54	2	2	NUM
ejpam-3209	85	55	(	(	PUNCT
ejpam-3209	85	56	v	v	NOUN
ejpam-3209	85	57	,	,	PUNCT
ejpam-3209	85	58	v	v	NOUN
ejpam-3209	85	59	)	)	PUNCT
ejpam-3209	85	60	+	+	CCONJ
ejpam-3209	85	61	nc3(u	nc3(u	PROPN
ejpam-3209	85	62	,	,	PUNCT
ejpam-3209	85	63	u	u	NOUN
ejpam-3209	85	64	)	)	PUNCT
ejpam-3209	85	65	n−4	n−4	PROPN
ejpam-3209	85	66	2	2	NUM
ejpam-3209	85	67	(	(	PUNCT
ejpam-3209	85	68	v	v	NOUN
ejpam-3209	85	69	,	,	PUNCT
ejpam-3209	85	70	v)(u	v)(u	ADJ
ejpam-3209	85	71	,	,	PUNCT
ejpam-3209	85	72	v	v	NOUN
ejpam-3209	85	73	)	)	PUNCT
ejpam-3209	85	74	+	+	PROPN
ejpam-3209	85	75	n	n	PRON
ejpam-3209	85	76	c4(u	c4(u	PROPN
ejpam-3209	85	77	,	,	PUNCT
ejpam-3209	85	78	u	u	NOUN
ejpam-3209	85	79	)	)	PUNCT
ejpam-3209	85	80	n−4	n−4	PROPN
ejpam-3209	85	81	2	2	NUM
ejpam-3209	85	82	(	(	PUNCT
ejpam-3209	85	83	v	v	NOUN
ejpam-3209	85	84	,	,	PUNCT
ejpam-3209	85	85	v)2	v)2	ADJ
ejpam-3209	85	86	+	+	NOUN
ejpam-3209	85	87	n	n	PROPN
ejpam-3209	85	88	c5(u	c5(u	PROPN
ejpam-3209	85	89	,	,	PUNCT
ejpam-3209	85	90	u	u	NOUN
ejpam-3209	85	91	)	)	PUNCT
ejpam-3209	85	92	n−6	n−6	PROPN
ejpam-3209	85	93	2	2	NUM
ejpam-3209	85	94	(	(	PUNCT
ejpam-3209	85	95	v	v	NOUN
ejpam-3209	85	96	,	,	PUNCT
ejpam-3209	85	97	v)2(u	v)2(u	NUM
ejpam-3209	85	98	,	,	PUNCT
ejpam-3209	85	99	v	v	NOUN
ejpam-3209	85	100	)	)	PUNCT
ejpam-3209	85	101	+	+	CCONJ
ejpam-3209	85	102	nc6(u	nc6(u	PROPN
ejpam-3209	85	103	,	,	PUNCT
ejpam-3209	85	104	u	u	NOUN
ejpam-3209	85	105	)	)	PUNCT
ejpam-3209	85	106	n−6	n−6	PROPN
ejpam-3209	85	107	2	2	NUM
ejpam-3209	85	108	(	(	PUNCT
ejpam-3209	85	109	v	v	NOUN
ejpam-3209	85	110	,	,	PUNCT
ejpam-3209	85	111	v)3	v)3	PROPN
ejpam-3209	85	112	+	+	PROPN
ejpam-3209	85	113	n	n	PROPN
ejpam-3209	85	114	c7(u	c7(u	NOUN
ejpam-3209	85	115	,	,	PUNCT
ejpam-3209	85	116	u	u	NOUN
ejpam-3209	85	117	)	)	PUNCT
ejpam-3209	85	118	n−8	n−8	PROPN
ejpam-3209	85	119	2	2	NUM
ejpam-3209	85	120	(	(	PUNCT
ejpam-3209	85	121	v	v	NOUN
ejpam-3209	85	122	,	,	PUNCT
ejpam-3209	85	123	v)3(u	v)3(u	NOUN
ejpam-3209	85	124	,	,	PUNCT
ejpam-3209	85	125	v	v	NOUN
ejpam-3209	85	126	)	)	PUNCT
ejpam-3209	85	127	+	+	CCONJ
ejpam-3209	85	128	.	.	PUNCT
ejpam-3209	85	129	.	.	PUNCT
ejpam-3209	86	1	.+	.+	NOUN
ejpam-3209	86	2	(	(	PUNCT
ejpam-3209	86	3	u	u	NOUN
ejpam-3209	86	4	,	,	PUNCT
ejpam-3209	86	5	u	u	NOUN
ejpam-3209	86	6	)	)	PUNCT
ejpam-3209	86	7	n	n	DET
ejpam-3209	86	8	2	2	NUM
ejpam-3209	86	9	}	}	PUNCT
ejpam-3209	86	10	⇒	⇒	VERB
ejpam-3209	86	11	−ncn	−ncn	NOUN
ejpam-3209	86	12	2	2	NUM
ejpam-3209	86	13	(	(	PUNCT
ejpam-3209	86	14	u	u	NOUN
ejpam-3209	86	15	,	,	PUNCT
ejpam-3209	86	16	u	u	NOUN
ejpam-3209	86	17	)	)	PUNCT
ejpam-3209	86	18	n	n	ADV
ejpam-3209	86	19	4	4	NUM
ejpam-3209	86	20	(	(	PUNCT
ejpam-3209	86	21	v	v	NOUN
ejpam-3209	86	22	,	,	PUNCT
ejpam-3209	86	23	v	v	NOUN
ejpam-3209	86	24	)	)	PUNCT
ejpam-3209	86	25	n	n	PRON
ejpam-3209	86	26	4	4	NUM
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ejpam-3209	86	28	{	{	PUNCT
ejpam-3209	86	29	(	(	PUNCT
ejpam-3209	86	30	u	u	NOUN
ejpam-3209	86	31	,	,	PUNCT
ejpam-3209	86	32	u	u	NOUN
ejpam-3209	86	33	)	)	PUNCT
ejpam-3209	86	34	n	n	ADV
ejpam-3209	86	35	2	2	NUM
ejpam-3209	86	36	+	+	NOUN
ejpam-3209	86	37	n	n	PRON
ejpam-3209	86	38	c1(u	c1(u	NOUN
ejpam-3209	86	39	,	,	PUNCT
ejpam-3209	86	40	v)(u	v)(u	NUM
ejpam-3209	86	41	,	,	PUNCT
ejpam-3209	86	42	u	u	NOUN
ejpam-3209	86	43	)	)	PUNCT
ejpam-3209	86	44	n−2	n−2	PROPN
ejpam-3209	86	45	2	2	NUM
ejpam-3209	86	46	+	+	NOUN
ejpam-3209	86	47	n	n	PROPN
ejpam-3209	86	48	c2(u	c2(u	PROPN
ejpam-3209	86	49	,	,	PUNCT
ejpam-3209	86	50	u	u	NOUN
ejpam-3209	86	51	)	)	PUNCT
ejpam-3209	86	52	n−2	n−2	PROPN
ejpam-3209	86	53	2	2	NUM
ejpam-3209	86	54	(	(	PUNCT
ejpam-3209	86	55	v	v	NOUN
ejpam-3209	86	56	,	,	PUNCT
ejpam-3209	86	57	v	v	NOUN
ejpam-3209	86	58	)	)	PUNCT
ejpam-3209	86	59	+	+	CCONJ
ejpam-3209	87	1	nc3(u	nc3(u	PROPN
ejpam-3209	87	2	,	,	PUNCT
ejpam-3209	87	3	u	u	NOUN
ejpam-3209	87	4	)	)	PUNCT
ejpam-3209	87	5	n−4	n−4	PROPN
ejpam-3209	87	6	2	2	NUM
ejpam-3209	87	7	(	(	PUNCT
ejpam-3209	87	8	v	v	NOUN
ejpam-3209	87	9	,	,	PUNCT
ejpam-3209	87	10	v)(u	v)(u	ADJ
ejpam-3209	87	11	,	,	PUNCT
ejpam-3209	87	12	v	v	NOUN
ejpam-3209	87	13	)	)	PUNCT
ejpam-3209	87	14	+	+	PROPN
ejpam-3209	87	15	n	n	PRON
ejpam-3209	87	16	c4(u	c4(u	PROPN
ejpam-3209	87	17	,	,	PUNCT
ejpam-3209	87	18	u	u	NOUN
ejpam-3209	87	19	)	)	PUNCT
ejpam-3209	87	20	n−4	n−4	PROPN
ejpam-3209	87	21	2	2	NUM
ejpam-3209	87	22	(	(	PUNCT
ejpam-3209	87	23	v	v	NOUN
ejpam-3209	87	24	,	,	PUNCT
ejpam-3209	87	25	v)2	v)2	ADJ
ejpam-3209	87	26	+	+	NOUN
ejpam-3209	87	27	n	n	PROPN
ejpam-3209	87	28	c5(u	c5(u	PROPN
ejpam-3209	87	29	,	,	PUNCT
ejpam-3209	87	30	u	u	NOUN
ejpam-3209	87	31	)	)	PUNCT
ejpam-3209	87	32	n−6	n−6	PROPN
ejpam-3209	87	33	2	2	NUM
ejpam-3209	87	34	(	(	PUNCT
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ejpam-3209	87	36	,	,	PUNCT
ejpam-3209	87	37	v)2(u	v)2(u	NUM
ejpam-3209	87	38	,	,	PUNCT
ejpam-3209	87	39	v	v	NOUN
ejpam-3209	87	40	)	)	PUNCT
ejpam-3209	87	41	+	+	CCONJ
ejpam-3209	87	42	nc6(u	nc6(u	PROPN
ejpam-3209	87	43	,	,	PUNCT
ejpam-3209	87	44	u	u	NOUN
ejpam-3209	87	45	)	)	PUNCT
ejpam-3209	87	46	n−6	n−6	PROPN
ejpam-3209	87	47	2	2	NUM
ejpam-3209	87	48	(	(	PUNCT
ejpam-3209	87	49	v	v	NOUN
ejpam-3209	87	50	,	,	PUNCT
ejpam-3209	87	51	v)3	v)3	PROPN
ejpam-3209	87	52	+	+	PROPN
ejpam-3209	87	53	n	n	PROPN
ejpam-3209	87	54	c7(u	c7(u	NOUN
ejpam-3209	87	55	,	,	PUNCT
ejpam-3209	87	56	u	u	NOUN
ejpam-3209	87	57	)	)	PUNCT
ejpam-3209	87	58	n−8	n−8	PROPN
ejpam-3209	87	59	2	2	NUM
ejpam-3209	87	60	(	(	PUNCT
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ejpam-3209	87	64	,	,	PUNCT
ejpam-3209	87	65	v	v	NOUN
ejpam-3209	87	66	)	)	PUNCT
ejpam-3209	87	67	+	+	CCONJ
ejpam-3209	87	68	.	.	PUNCT
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ejpam-3209	88	1	.+n	.+n	PROPN
ejpam-3209	88	2	cn	cn	NOUN
ejpam-3209	88	3	2	2	NUM
ejpam-3209	88	4	(	(	PUNCT
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ejpam-3209	88	7	u	u	NOUN
ejpam-3209	88	8	)	)	PUNCT
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ejpam-3209	88	11	(	(	PUNCT
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ejpam-3209	88	13	,	,	PUNCT
ejpam-3209	88	14	v	v	NOUN
ejpam-3209	88	15	)	)	PUNCT
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ejpam-3209	88	17	4	4	NUM
ejpam-3209	88	18	+	+	CCONJ
ejpam-3209	88	19	.	.	PUNCT
ejpam-3209	88	20	.	.	PUNCT
ejpam-3209	89	1	.+	.+	NOUN
ejpam-3209	89	2	(	(	PUNCT
ejpam-3209	89	3	u	u	NOUN
ejpam-3209	89	4	,	,	PUNCT
ejpam-3209	89	5	u	u	NOUN
ejpam-3209	89	6	)	)	PUNCT
ejpam-3209	89	7	n	n	DET
ejpam-3209	89	8	2	2	NUM
ejpam-3209	89	9	}	}	PUNCT
ejpam-3209	89	10	⇒	⇒	VERB
ejpam-3209	89	11	−ncn	−ncn	NOUN
ejpam-3209	89	12	2	2	NUM
ejpam-3209	89	13	(	(	PUNCT
ejpam-3209	89	14	u	u	NOUN
ejpam-3209	89	15	,	,	PUNCT
ejpam-3209	89	16	u	u	NOUN
ejpam-3209	89	17	)	)	PUNCT
ejpam-3209	89	18	n	n	ADV
ejpam-3209	89	19	4	4	NUM
ejpam-3209	89	20	(	(	PUNCT
ejpam-3209	89	21	v	v	NOUN
ejpam-3209	89	22	,	,	PUNCT
ejpam-3209	89	23	v	v	NOUN
ejpam-3209	89	24	)	)	PUNCT
ejpam-3209	89	25	n	n	ADV
ejpam-3209	89	26	4	4	NUM
ejpam-3209	89	27	1	1	NUM
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ejpam-3209	89	30	(	(	PUNCT
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ejpam-3209	89	32	,	,	PUNCT
ejpam-3209	89	33	u	u	NOUN
ejpam-3209	89	34	)	)	PUNCT
ejpam-3209	89	35	n	n	ADV
ejpam-3209	89	36	4	4	NUM
ejpam-3209	89	37	(	(	PUNCT
ejpam-3209	89	38	v	v	NOUN
ejpam-3209	89	39	,	,	PUNCT
ejpam-3209	89	40	v	v	NOUN
ejpam-3209	89	41	)	)	PUNCT
ejpam-3209	89	42	n	n	PRON
ejpam-3209	89	43	4	4	NUM
ejpam-3209	89	44	≤	≤	NUM
ejpam-3209	89	45	(	(	PUNCT
ejpam-3209	89	46	u+	u+	NUM
ejpam-3209	89	47	v)n	v)n	NOUN
ejpam-3209	89	48	⇒	⇒	NOUN
ejpam-3209	89	49	‖−	‖−	PROPN
ejpam-3209	89	50	(	(	PUNCT
ejpam-3209	89	51	u	u	NOUN
ejpam-3209	89	52	,	,	PUNCT
ejpam-3209	89	53	u	u	NOUN
ejpam-3209	89	54	)	)	PUNCT
ejpam-3209	89	55	n	n	ADV
ejpam-3209	89	56	2	2	NUM
ejpam-3209	89	57	(	(	PUNCT
ejpam-3209	89	58	v	v	NOUN
ejpam-3209	89	59	,	,	PUNCT
ejpam-3209	89	60	v	v	NOUN
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ejpam-3209	89	62	n	n	PRON
ejpam-3209	89	63	2	2	NUM
ejpam-3209	89	64	‖	‖	PROPN
ejpam-3209	89	65	=	=	SYM
ejpam-3209	89	66	‖(u+	‖(u+	PUNCT
ejpam-3209	89	67	v)n‖	v)n‖	NUM
ejpam-3209	89	68	⇒	⇒	NOUN
ejpam-3209	89	69	{	{	PUNCT
ejpam-3209	89	70	‖u‖‖v‖}n	‖u‖‖v‖}n	PUNCT
ejpam-3209	89	71	=	=	SYM
ejpam-3209	89	72	‖(u+	‖(u+	NUM
ejpam-3209	89	73	v)‖n	v)‖n	NOUN
ejpam-3209	89	74	⇒	⇒	VERB
ejpam-3209	89	75	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3209	89	76	≤	≤	PROPN
ejpam-3209	89	77	‖u‖+	‖u‖+	PROPN
ejpam-3209	89	78	‖v‖	‖v‖	PROPN
ejpam-3209	89	79	,	,	PUNCT
ejpam-3209	89	80	∀	∀	X
ejpam-3209	89	81	u	u	NOUN
ejpam-3209	89	82	,	,	PUNCT
ejpam-3209	89	83	v	v	NOUN
ejpam-3209	89	84	∈	∈	PROPN
ejpam-3209	90	1	[	[	X
ejpam-3209	90	2	0	0	NUM
ejpam-3209	90	3	,	,	PUNCT
ejpam-3209	90	4	2	2	NUM
ejpam-3209	90	5	]	]	PUNCT
ejpam-3209	90	6	.	.	PUNCT
ejpam-3209	91	1	(	(	PUNCT
ejpam-3209	91	2	3	3	X
ejpam-3209	91	3	)	)	PUNCT
ejpam-3209	91	4	2.2	2.2	NUM
ejpam-3209	91	5	.	.	PUNCT
ejpam-3209	92	1	the	the	DET
ejpam-3209	92	2	proof	proof	NOUN
ejpam-3209	92	3	of	of	ADP
ejpam-3209	92	4	the	the	DET
ejpam-3209	92	5	second	second	ADJ
ejpam-3209	92	6	product	product	NOUN
ejpam-3209	92	7	inequality	inequality	NOUN
ejpam-3209	92	8	in	in	ADP
ejpam-3209	92	9	this	this	DET
ejpam-3209	92	10	subsection	subsection	NOUN
ejpam-3209	92	11	,	,	PUNCT
ejpam-3209	92	12	we	we	PRON
ejpam-3209	92	13	show	show	VERB
ejpam-3209	92	14	that	that	SCONJ
ejpam-3209	92	15	the	the	DET
ejpam-3209	92	16	sum	sum	NOUN
ejpam-3209	92	17	of	of	ADP
ejpam-3209	92	18	norms	norm	NOUN
ejpam-3209	92	19	of	of	ADP
ejpam-3209	92	20	two	two	NUM
ejpam-3209	92	21	vectors	vector	NOUN
ejpam-3209	92	22	less	less	ADJ
ejpam-3209	92	23	than	than	ADP
ejpam-3209	92	24	or	or	CCONJ
ejpam-3209	92	25	equal	equal	ADJ
ejpam-3209	92	26	to	to	ADP
ejpam-3209	92	27	the	the	DET
ejpam-3209	92	28	norms	norm	NOUN
ejpam-3209	92	29	of	of	ADP
ejpam-3209	92	30	product	product	NOUN
ejpam-3209	92	31	of	of	ADP
ejpam-3209	92	32	their	their	PRON
ejpam-3209	92	33	vectors	vector	NOUN
ejpam-3209	92	34	.	.	PUNCT
ejpam-3209	93	1	setting	set	VERB
ejpam-3209	93	2	n	n	NOUN
ejpam-3209	93	3	=	=	SYM
ejpam-3209	93	4	2	2	NUM
ejpam-3209	93	5	,	,	PUNCT
ejpam-3209	93	6	we	we	PRON
ejpam-3209	93	7	obtain	obtain	VERB
ejpam-3209	93	8	:	:	PUNCT
ejpam-3209	93	9	(	(	PUNCT
ejpam-3209	93	10	u+	u+	NUM
ejpam-3209	93	11	v)2	v)2	PROPN
ejpam-3209	93	12	≥	≥	NOUN
ejpam-3209	93	13	0	0	NUM
ejpam-3209	93	14	(	(	PUNCT
ejpam-3209	93	15	u	u	NOUN
ejpam-3209	93	16	,	,	PUNCT
ejpam-3209	93	17	u	u	NOUN
ejpam-3209	93	18	)	)	PUNCT
ejpam-3209	93	19	+	+	CCONJ
ejpam-3209	93	20	2(u	2(u	NUM
ejpam-3209	93	21	,	,	PUNCT
ejpam-3209	93	22	v	v	NOUN
ejpam-3209	93	23	)	)	PUNCT
ejpam-3209	94	1	+	+	CCONJ
ejpam-3209	94	2	(	(	PUNCT
ejpam-3209	94	3	v	v	NOUN
ejpam-3209	94	4	,	,	PUNCT
ejpam-3209	94	5	v	v	NOUN
ejpam-3209	94	6	)	)	PUNCT
ejpam-3209	94	7	≥	≥	NOUN
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ejpam-3209	94	9	−{(u	−{(u	NOUN
ejpam-3209	94	10	,	,	PUNCT
ejpam-3209	94	11	u	u	NOUN
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ejpam-3209	94	13	+	+	CCONJ
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ejpam-3209	94	16	,	,	PUNCT
ejpam-3209	94	17	v	v	NOUN
ejpam-3209	94	18	)	)	PUNCT
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ejpam-3209	94	20	≤	≤	NUM
ejpam-3209	94	21	2(u	2(u	NUM
ejpam-3209	94	22	,	,	PUNCT
ejpam-3209	94	23	v	v	NOUN
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ejpam-3209	94	25	−{(u	−{(u	NOUN
ejpam-3209	94	26	,	,	PUNCT
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ejpam-3209	94	37	,	,	PUNCT
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ejpam-3209	94	41	≤	≤	NUM
ejpam-3209	94	42	2(u	2(u	NUM
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ejpam-3209	94	46	−{(u	−{(u	NOUN
ejpam-3209	94	47	,	,	PUNCT
ejpam-3209	94	48	u	u	NOUN
ejpam-3209	94	49	)	)	PUNCT
ejpam-3209	94	50	+	+	CCONJ
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ejpam-3209	94	53	,	,	PUNCT
ejpam-3209	94	54	v	v	NOUN
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ejpam-3209	94	58	,	,	PUNCT
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ejpam-3209	94	60	)	)	PUNCT
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ejpam-3209	94	62	≤	≤	NUM
ejpam-3209	94	63	2(u	2(u	NUM
ejpam-3209	94	64	,	,	PUNCT
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ejpam-3209	94	66	)	)	PUNCT
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ejpam-3209	95	1	1	1	NUM
ejpam-3209	95	2	2	2	NUM
ejpam-3209	95	3	(	(	PUNCT
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ejpam-3209	95	5	,	,	PUNCT
ejpam-3209	95	6	v	v	NOUN
ejpam-3209	95	7	)	)	PUNCT
ejpam-3209	95	8	−(u+	−(u+	NUM
ejpam-3209	95	9	v)2	v)2	NOUN
ejpam-3209	95	10	=	=	SYM
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ejpam-3209	95	14	v)2	v)2	PROPN
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ejpam-3209	95	17	(	(	PUNCT
ejpam-3209	95	18	u+	u+	NUM
ejpam-3209	95	19	v)2‖	v)2‖	PROPN
ejpam-3209	95	20	=	=	SYM
ejpam-3209	95	21	‖(u	‖(u	NOUN
ejpam-3209	95	22	,	,	PUNCT
ejpam-3209	95	23	v)2‖	v)2‖	NOUN
ejpam-3209	95	24	‖(u+	‖(u+	PUNCT
ejpam-3209	95	25	v)‖2	v)‖2	NOUN
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ejpam-3209	95	27	‖(u	‖(u	NOUN
ejpam-3209	95	28	,	,	PUNCT
ejpam-3209	95	29	v)‖2	v)‖2	NOUN
ejpam-3209	95	30	barnes	barne	NOUN
ejpam-3209	95	31	et	et	PROPN
ejpam-3209	95	32	al	al	PROPN
ejpam-3209	95	33	.	.	PUNCT
ejpam-3209	95	34	/	/	SYM
ejpam-3209	95	35	eur	eur	PROPN
ejpam-3209	95	36	.	.	PUNCT
ejpam-3209	96	1	j.	j.	PROPN
ejpam-3209	96	2	pure	pure	PROPN
ejpam-3209	96	3	appl	appl	PROPN
ejpam-3209	96	4	.	.	PROPN
ejpam-3209	96	5	math	math	PROPN
ejpam-3209	96	6	,	,	PUNCT
ejpam-3209	96	7	11	11	NUM
ejpam-3209	96	8	(	(	PUNCT
ejpam-3209	96	9	2	2	NUM
ejpam-3209	96	10	)	)	PUNCT
ejpam-3209	96	11	(	(	PUNCT
ejpam-3209	96	12	2018	2018	NUM
ejpam-3209	96	13	)	)	PUNCT
ejpam-3209	96	14	,	,	PUNCT
ejpam-3209	96	15	375	375	NUM
ejpam-3209	96	16	-	-	SYM
ejpam-3209	96	17	389	389	NUM
ejpam-3209	96	18	380	380	NUM
ejpam-3209	96	19	⇒	⇒	NOUN
ejpam-3209	96	20	‖u‖+	‖u‖+	PROPN
ejpam-3209	96	21	‖v‖	‖v‖	PROPN
ejpam-3209	96	22	≤	≤	NOUN
ejpam-3209	96	23	‖u‖‖v‖.	‖u‖‖v‖.	ADJ
ejpam-3209	96	24	also	also	ADV
ejpam-3209	96	25	,	,	PUNCT
ejpam-3209	96	26	we	we	PRON
ejpam-3209	96	27	set	set	VERB
ejpam-3209	96	28	n	n	NOUN
ejpam-3209	96	29	=	=	SYM
ejpam-3209	96	30	4	4	NUM
ejpam-3209	96	31	,	,	PUNCT
ejpam-3209	96	32	which	which	PRON
ejpam-3209	96	33	gives	give	VERB
ejpam-3209	96	34	as	as	ADP
ejpam-3209	96	35	the	the	DET
ejpam-3209	96	36	following	following	ADJ
ejpam-3209	96	37	result	result	NOUN
ejpam-3209	96	38	:	:	PUNCT
ejpam-3209	96	39	(	(	PUNCT
ejpam-3209	96	40	u+	u+	NUM
ejpam-3209	96	41	v)4	v)4	PROPN
ejpam-3209	96	42	≥	≥	NOUN
ejpam-3209	96	43	0	0	NUM
ejpam-3209	96	44	−{(u	−{(u	NOUN
ejpam-3209	96	45	,	,	PUNCT
ejpam-3209	96	46	u)2	u)2	PROPN
ejpam-3209	96	47	+	+	CCONJ
ejpam-3209	96	48	4(u	4(u	NOUN
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ejpam-3209	96	50	u)(u	u)(u	ADJ
ejpam-3209	96	51	,	,	PUNCT
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ejpam-3209	96	53	)	)	PUNCT
ejpam-3209	97	1	+	+	CCONJ
ejpam-3209	97	2	4(u	4(u	NOUN
ejpam-3209	97	3	,	,	PUNCT
ejpam-3209	97	4	v)(v	v)(v	NOUN
ejpam-3209	97	5	,	,	PUNCT
ejpam-3209	97	6	v	v	NOUN
ejpam-3209	97	7	)	)	PUNCT
ejpam-3209	97	8	+	+	CCONJ
ejpam-3209	97	9	(	(	PUNCT
ejpam-3209	97	10	v	v	NOUN
ejpam-3209	97	11	,	,	PUNCT
ejpam-3209	97	12	v)2	v)2	ADJ
ejpam-3209	97	13	}	}	PUNCT
ejpam-3209	97	14	≤	≤	NUM
ejpam-3209	97	15	6(u	6(u	NUM
ejpam-3209	97	16	,	,	PUNCT
ejpam-3209	97	17	u)(v	u)(v	PROPN
ejpam-3209	97	18	,	,	PUNCT
ejpam-3209	97	19	v	v	NOUN
ejpam-3209	97	20	)	)	PUNCT
ejpam-3209	97	21	−{(u	−{(u	NOUN
ejpam-3209	97	22	,	,	PUNCT
ejpam-3209	97	23	u)2	u)2	ADV
ejpam-3209	97	24	+	+	CCONJ
ejpam-3209	97	25	4(u	4(u	NOUN
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ejpam-3209	97	27	u)(u	u)(u	ADJ
ejpam-3209	97	28	,	,	PUNCT
ejpam-3209	97	29	v	v	NOUN
ejpam-3209	97	30	)	)	PUNCT
ejpam-3209	97	31	+	+	CCONJ
ejpam-3209	97	32	6(u	6(u	NUM
ejpam-3209	97	33	,	,	PUNCT
ejpam-3209	97	34	u)(v	u)(v	PROPN
ejpam-3209	97	35	,	,	PUNCT
ejpam-3209	97	36	v	v	NOUN
ejpam-3209	97	37	)	)	PUNCT
ejpam-3209	97	38	+	+	CCONJ
ejpam-3209	97	39	4(u	4(u	NOUN
ejpam-3209	97	40	,	,	PUNCT
ejpam-3209	97	41	v)(v	v)(v	NOUN
ejpam-3209	97	42	,	,	PUNCT
ejpam-3209	97	43	v	v	NOUN
ejpam-3209	97	44	)	)	PUNCT
ejpam-3209	97	45	+	+	CCONJ
ejpam-3209	97	46	(	(	PUNCT
ejpam-3209	97	47	v	v	NOUN
ejpam-3209	97	48	,	,	PUNCT
ejpam-3209	97	49	v)2	v)2	ADJ
ejpam-3209	97	50	}	}	PUNCT
ejpam-3209	97	51	≤	≤	NUM
ejpam-3209	97	52	6(u	6(u	NUM
ejpam-3209	97	53	,	,	PUNCT
ejpam-3209	97	54	u)(v	u)(v	PROPN
ejpam-3209	97	55	,	,	PUNCT
ejpam-3209	97	56	v	v	NOUN
ejpam-3209	97	57	)	)	PUNCT
ejpam-3209	97	58	−(u+	−(u+	PROPN
ejpam-3209	97	59	v)4	v)4	PROPN
ejpam-3209	97	60	≤	≤	NUM
ejpam-3209	97	61	6(u	6(u	NUM
ejpam-3209	97	62	,	,	PUNCT
ejpam-3209	97	63	u)(v	u)(v	PROPN
ejpam-3209	97	64	,	,	PUNCT
ejpam-3209	97	65	v	v	NOUN
ejpam-3209	97	66	)	)	PUNCT
ejpam-3209	97	67	.	.	PUNCT
ejpam-3209	98	1	1	1	NUM
ejpam-3209	98	2	6	6	NUM
ejpam-3209	98	3	(	(	PUNCT
ejpam-3209	98	4	u	u	NOUN
ejpam-3209	98	5	,	,	PUNCT
ejpam-3209	98	6	u)(v	u)(v	PROPN
ejpam-3209	98	7	,	,	PUNCT
ejpam-3209	98	8	v	v	NOUN
ejpam-3209	98	9	)	)	PUNCT
ejpam-3209	98	10	−(u+	−(u+	NUM
ejpam-3209	98	11	v)4	v)4	NOUN
ejpam-3209	98	12	=	=	SYM
ejpam-3209	98	13	{	{	PUNCT
ejpam-3209	98	14	(	(	PUNCT
ejpam-3209	98	15	u	u	NOUN
ejpam-3209	98	16	,	,	PUNCT
ejpam-3209	98	17	u)(v	u)(v	PROPN
ejpam-3209	98	18	,	,	PUNCT
ejpam-3209	98	19	v)}2	v)}2	PROPN
ejpam-3209	98	20	‖	‖	PROPN
ejpam-3209	98	21	−	−	PROPN
ejpam-3209	98	22	(	(	PUNCT
ejpam-3209	98	23	u+	u+	NUM
ejpam-3209	98	24	v)4‖	v)4‖	PROPN
ejpam-3209	98	25	=	=	SYM
ejpam-3209	98	26	‖{(u	‖{(u	PROPN
ejpam-3209	98	27	,	,	PUNCT
ejpam-3209	98	28	u)(v	u)(v	PROPN
ejpam-3209	98	29	,	,	PUNCT
ejpam-3209	98	30	v)}2‖	v)}2‖	VERB
ejpam-3209	98	31	‖(u+	‖(u+	NUM
ejpam-3209	98	32	v)‖4	v)‖4	NOUN
ejpam-3209	98	33	=	=	PUNCT
ejpam-3209	98	34	{	{	PUNCT
ejpam-3209	98	35	‖u‖‖v‖}4	‖u‖‖v‖}4	PUNCT
ejpam-3209	98	36	⇒	⇒	VERB
ejpam-3209	98	37	‖u‖+	‖u‖+	X
ejpam-3209	98	38	‖v‖	‖v‖	PROPN
ejpam-3209	98	39	≤	≤	PROPN
ejpam-3209	98	40	‖u‖‖v‖.	‖u‖‖v‖.	ADV
ejpam-3209	98	41	similarly	similarly	ADV
ejpam-3209	98	42	,	,	PUNCT
ejpam-3209	98	43	we	we	PRON
ejpam-3209	98	44	generalize	generalize	VERB
ejpam-3209	98	45	the	the	DET
ejpam-3209	98	46	second	second	ADJ
ejpam-3209	98	47	product	product	NOUN
ejpam-3209	98	48	inequality	inequality	NOUN
ejpam-3209	98	49	by	by	ADP
ejpam-3209	98	50	observing	observe	VERB
ejpam-3209	98	51	the	the	DET
ejpam-3209	98	52	following	following	NOUN
ejpam-3209	98	53	:	:	PUNCT
ejpam-3209	98	54	(	(	PUNCT
ejpam-3209	98	55	u+	u+	NUM
ejpam-3209	98	56	v)n	v)n	ADJ
ejpam-3209	98	57	≥	≥	NOUN
ejpam-3209	98	58	0	0	NUM
ejpam-3209	98	59	⇒	⇒	NOUN
ejpam-3209	98	60	(	(	PUNCT
ejpam-3209	98	61	u	u	NOUN
ejpam-3209	98	62	,	,	PUNCT
ejpam-3209	98	63	u	u	NOUN
ejpam-3209	98	64	)	)	PUNCT
ejpam-3209	98	65	n	n	ADV
ejpam-3209	98	66	2	2	NUM
ejpam-3209	98	67	+	+	NOUN
ejpam-3209	98	68	n	n	PRON
ejpam-3209	98	69	c1(u	c1(u	NOUN
ejpam-3209	98	70	,	,	PUNCT
ejpam-3209	98	71	v)(u.u	v)(u.u	ADJ
ejpam-3209	98	72	)	)	PUNCT
ejpam-3209	98	73	n−2	n−2	PROPN
ejpam-3209	98	74	2	2	NUM
ejpam-3209	98	75	+	+	NOUN
ejpam-3209	98	76	n	n	PROPN
ejpam-3209	98	77	c2(u	c2(u	PROPN
ejpam-3209	98	78	,	,	PUNCT
ejpam-3209	98	79	u	u	NOUN
ejpam-3209	98	80	)	)	PUNCT
ejpam-3209	98	81	n−2	n−2	PROPN
ejpam-3209	98	82	2	2	NUM
ejpam-3209	98	83	(	(	PUNCT
ejpam-3209	98	84	v	v	NOUN
ejpam-3209	98	85	,	,	PUNCT
ejpam-3209	98	86	v	v	NOUN
ejpam-3209	98	87	)	)	PUNCT
ejpam-3209	98	88	+	+	NOUN
ejpam-3209	98	89	n	n	SYM
ejpam-3209	98	90	c3(u	c3(u	PROPN
ejpam-3209	98	91	,	,	PUNCT
ejpam-3209	98	92	u	u	NOUN
ejpam-3209	98	93	)	)	PUNCT
ejpam-3209	98	94	n−4	n−4	PROPN
ejpam-3209	98	95	2	2	NUM
ejpam-3209	98	96	(	(	PUNCT
ejpam-3209	98	97	v	v	NOUN
ejpam-3209	98	98	,	,	PUNCT
ejpam-3209	98	99	v)(u	v)(u	ADJ
ejpam-3209	98	100	,	,	PUNCT
ejpam-3209	98	101	v	v	NOUN
ejpam-3209	98	102	)	)	PUNCT
ejpam-3209	98	103	+	+	CCONJ
ejpam-3209	98	104	nc4(u	nc4(u	PROPN
ejpam-3209	98	105	,	,	PUNCT
ejpam-3209	98	106	u	u	NOUN
ejpam-3209	98	107	)	)	PUNCT
ejpam-3209	98	108	n−4	n−4	PROPN
ejpam-3209	98	109	2	2	NUM
ejpam-3209	98	110	(	(	PUNCT
ejpam-3209	98	111	v	v	NOUN
ejpam-3209	98	112	,	,	PUNCT
ejpam-3209	98	113	v)2	v)2	ADJ
ejpam-3209	98	114	+	+	NOUN
ejpam-3209	98	115	n	n	PROPN
ejpam-3209	98	116	c5(u	c5(u	PROPN
ejpam-3209	98	117	,	,	PUNCT
ejpam-3209	98	118	u	u	NOUN
ejpam-3209	98	119	)	)	PUNCT
ejpam-3209	98	120	n−6	n−6	PROPN
ejpam-3209	98	121	2	2	NUM
ejpam-3209	98	122	(	(	PUNCT
ejpam-3209	98	123	v	v	NOUN
ejpam-3209	98	124	,	,	PUNCT
ejpam-3209	98	125	v)2(u	v)2(u	NUM
ejpam-3209	98	126	,	,	PUNCT
ejpam-3209	98	127	v	v	NOUN
ejpam-3209	98	128	)	)	PUNCT
ejpam-3209	98	129	+	+	NOUN
ejpam-3209	98	130	n	n	PRON
ejpam-3209	98	131	c6(u	c6(u	PROPN
ejpam-3209	98	132	,	,	PUNCT
ejpam-3209	98	133	u	u	NOUN
ejpam-3209	98	134	)	)	PUNCT
ejpam-3209	98	135	n−6	n−6	PROPN
ejpam-3209	98	136	2	2	NUM
ejpam-3209	98	137	(	(	PUNCT
ejpam-3209	98	138	v	v	NOUN
ejpam-3209	98	139	,	,	PUNCT
ejpam-3209	98	140	v)3	v)3	PROPN
ejpam-3209	98	141	+	+	CCONJ
ejpam-3209	98	142	nc7(u	nc7(u	PROPN
ejpam-3209	98	143	,	,	PUNCT
ejpam-3209	98	144	u	u	NOUN
ejpam-3209	98	145	)	)	PUNCT
ejpam-3209	98	146	n−8	n−8	PROPN
ejpam-3209	98	147	2	2	NUM
ejpam-3209	98	148	(	(	PUNCT
ejpam-3209	98	149	v	v	NOUN
ejpam-3209	98	150	,	,	PUNCT
ejpam-3209	98	151	v)3(u	v)3(u	NOUN
ejpam-3209	98	152	,	,	PUNCT
ejpam-3209	98	153	v	v	NOUN
ejpam-3209	98	154	)	)	PUNCT
ejpam-3209	98	155	+	+	CCONJ
ejpam-3209	98	156	.	.	PUNCT
ejpam-3209	98	157	.	.	PUNCT
ejpam-3209	99	1	.+n	.+n	PROPN
ejpam-3209	99	2	cn	cn	NOUN
ejpam-3209	99	3	2	2	NUM
ejpam-3209	99	4	(	(	PUNCT
ejpam-3209	99	5	u	u	NOUN
ejpam-3209	99	6	,	,	PUNCT
ejpam-3209	99	7	u	u	NOUN
ejpam-3209	99	8	)	)	PUNCT
ejpam-3209	99	9	n	n	ADV
ejpam-3209	99	10	4	4	NUM
ejpam-3209	99	11	(	(	PUNCT
ejpam-3209	99	12	v	v	NOUN
ejpam-3209	99	13	,	,	PUNCT
ejpam-3209	99	14	v	v	NOUN
ejpam-3209	99	15	)	)	PUNCT
ejpam-3209	99	16	n	n	ADV
ejpam-3209	99	17	4	4	NUM
ejpam-3209	99	18	+	+	CCONJ
ejpam-3209	99	19	.	.	PUNCT
ejpam-3209	99	20	.	.	PUNCT
ejpam-3209	100	1	.+	.+	NOUN
ejpam-3209	100	2	(	(	PUNCT
ejpam-3209	100	3	u	u	NOUN
ejpam-3209	100	4	,	,	PUNCT
ejpam-3209	100	5	u	u	NOUN
ejpam-3209	100	6	)	)	PUNCT
ejpam-3209	100	7	n	n	PRON
ejpam-3209	100	8	2	2	NUM
ejpam-3209	100	9	≥	≥	NOUN
ejpam-3209	100	10	0	0	NUM
ejpam-3209	100	11	⇒	⇒	PROPN
ejpam-3209	100	12	−{(u	−{(u	NOUN
ejpam-3209	100	13	,	,	PUNCT
ejpam-3209	100	14	u	u	NOUN
ejpam-3209	100	15	)	)	PUNCT
ejpam-3209	100	16	n	n	ADV
ejpam-3209	100	17	2	2	NUM
ejpam-3209	100	18	+	+	NOUN
ejpam-3209	100	19	n	n	PRON
ejpam-3209	100	20	c1(u	c1(u	NOUN
ejpam-3209	100	21	,	,	PUNCT
ejpam-3209	100	22	v)(u	v)(u	NUM
ejpam-3209	100	23	,	,	PUNCT
ejpam-3209	100	24	u	u	NOUN
ejpam-3209	100	25	)	)	PUNCT
ejpam-3209	100	26	n−2	n−2	PROPN
ejpam-3209	100	27	2	2	NUM
ejpam-3209	100	28	+	+	NOUN
ejpam-3209	100	29	n	n	PROPN
ejpam-3209	100	30	c2(u	c2(u	PROPN
ejpam-3209	100	31	,	,	PUNCT
ejpam-3209	100	32	u	u	NOUN
ejpam-3209	100	33	)	)	PUNCT
ejpam-3209	100	34	n−2	n−2	PROPN
ejpam-3209	100	35	2	2	NUM
ejpam-3209	100	36	(	(	PUNCT
ejpam-3209	100	37	v	v	NOUN
ejpam-3209	100	38	,	,	PUNCT
ejpam-3209	100	39	v	v	NOUN
ejpam-3209	100	40	)	)	PUNCT
ejpam-3209	100	41	+	+	CCONJ
ejpam-3209	100	42	nc3(u	nc3(u	PROPN
ejpam-3209	100	43	,	,	PUNCT
ejpam-3209	100	44	u	u	NOUN
ejpam-3209	100	45	)	)	PUNCT
ejpam-3209	100	46	n−4	n−4	PROPN
ejpam-3209	100	47	2	2	NUM
ejpam-3209	100	48	(	(	PUNCT
ejpam-3209	100	49	v	v	NOUN
ejpam-3209	100	50	,	,	PUNCT
ejpam-3209	100	51	v)(u	v)(u	ADJ
ejpam-3209	100	52	,	,	PUNCT
ejpam-3209	100	53	v	v	NOUN
ejpam-3209	100	54	)	)	PUNCT
ejpam-3209	100	55	+	+	PROPN
ejpam-3209	100	56	n	n	PRON
ejpam-3209	100	57	c4(u	c4(u	PROPN
ejpam-3209	100	58	,	,	PUNCT
ejpam-3209	100	59	u	u	NOUN
ejpam-3209	100	60	)	)	PUNCT
ejpam-3209	100	61	n−4	n−4	PROPN
ejpam-3209	100	62	2	2	NUM
ejpam-3209	100	63	(	(	PUNCT
ejpam-3209	100	64	v	v	NOUN
ejpam-3209	100	65	,	,	PUNCT
ejpam-3209	100	66	v)2	v)2	ADJ
ejpam-3209	100	67	+	+	NOUN
ejpam-3209	100	68	n	n	PROPN
ejpam-3209	100	69	c5(u	c5(u	PROPN
ejpam-3209	100	70	,	,	PUNCT
ejpam-3209	100	71	u	u	NOUN
ejpam-3209	100	72	)	)	PUNCT
ejpam-3209	100	73	n−6	n−6	PROPN
ejpam-3209	100	74	2	2	NUM
ejpam-3209	100	75	(	(	PUNCT
ejpam-3209	100	76	v	v	NOUN
ejpam-3209	100	77	,	,	PUNCT
ejpam-3209	100	78	v)2(u	v)2(u	NUM
ejpam-3209	100	79	,	,	PUNCT
ejpam-3209	100	80	v	v	NOUN
ejpam-3209	100	81	)	)	PUNCT
ejpam-3209	100	82	+	+	CCONJ
ejpam-3209	100	83	nc6(u	nc6(u	PROPN
ejpam-3209	100	84	,	,	PUNCT
ejpam-3209	100	85	u	u	NOUN
ejpam-3209	100	86	)	)	PUNCT
ejpam-3209	100	87	n−6	n−6	PROPN
ejpam-3209	100	88	2	2	NUM
ejpam-3209	100	89	(	(	PUNCT
ejpam-3209	100	90	v	v	NOUN
ejpam-3209	100	91	,	,	PUNCT
ejpam-3209	100	92	v)3	v)3	PROPN
ejpam-3209	100	93	+	+	PROPN
ejpam-3209	100	94	n	n	PROPN
ejpam-3209	100	95	c7(u	c7(u	NOUN
ejpam-3209	100	96	,	,	PUNCT
ejpam-3209	100	97	u	u	NOUN
ejpam-3209	100	98	)	)	PUNCT
ejpam-3209	100	99	n−8	n−8	PROPN
ejpam-3209	100	100	2	2	NUM
ejpam-3209	100	101	(	(	PUNCT
ejpam-3209	100	102	v	v	NOUN
ejpam-3209	100	103	,	,	PUNCT
ejpam-3209	100	104	v)3(u	v)3(u	NOUN
ejpam-3209	100	105	,	,	PUNCT
ejpam-3209	100	106	v	v	NOUN
ejpam-3209	100	107	)	)	PUNCT
ejpam-3209	100	108	+	+	CCONJ
ejpam-3209	100	109	.	.	PUNCT
ejpam-3209	100	110	.	.	PUNCT
ejpam-3209	101	1	.+	.+	NOUN
ejpam-3209	101	2	(	(	PUNCT
ejpam-3209	101	3	u	u	NOUN
ejpam-3209	101	4	,	,	PUNCT
ejpam-3209	101	5	u	u	NOUN
ejpam-3209	101	6	)	)	PUNCT
ejpam-3209	101	7	n	n	ADV
ejpam-3209	101	8	2	2	NUM
ejpam-3209	101	9	}	}	PUNCT
ejpam-3209	101	10	≤	≤	NOUN
ejpam-3209	101	11	ncn	ncn	NOUN
ejpam-3209	101	12	2	2	NUM
ejpam-3209	101	13	(	(	PUNCT
ejpam-3209	101	14	u	u	NOUN
ejpam-3209	101	15	,	,	PUNCT
ejpam-3209	101	16	u	u	NOUN
ejpam-3209	101	17	)	)	PUNCT
ejpam-3209	101	18	n	n	ADV
ejpam-3209	101	19	4	4	NUM
ejpam-3209	101	20	(	(	PUNCT
ejpam-3209	101	21	v	v	NOUN
ejpam-3209	101	22	,	,	PUNCT
ejpam-3209	101	23	v	v	NOUN
ejpam-3209	101	24	)	)	PUNCT
ejpam-3209	101	25	n	n	PRON
ejpam-3209	101	26	4	4	NUM
ejpam-3209	101	27	⇒	⇒	NOUN
ejpam-3209	101	28	−	−	PROPN
ejpam-3209	101	29	{	{	PUNCT
ejpam-3209	101	30	(	(	PUNCT
ejpam-3209	101	31	u	u	NOUN
ejpam-3209	101	32	,	,	PUNCT
ejpam-3209	101	33	u	u	NOUN
ejpam-3209	101	34	)	)	PUNCT
ejpam-3209	101	35	n	n	ADV
ejpam-3209	101	36	2	2	NUM
ejpam-3209	101	37	+	+	NOUN
ejpam-3209	101	38	n	n	PRON
ejpam-3209	101	39	c1(u	c1(u	NOUN
ejpam-3209	101	40	,	,	PUNCT
ejpam-3209	101	41	v)(u	v)(u	NUM
ejpam-3209	101	42	,	,	PUNCT
ejpam-3209	101	43	u	u	NOUN
ejpam-3209	101	44	)	)	PUNCT
ejpam-3209	101	45	n−2	n−2	PROPN
ejpam-3209	101	46	2	2	NUM
ejpam-3209	101	47	+	+	NOUN
ejpam-3209	101	48	n	n	PROPN
ejpam-3209	101	49	c2(u	c2(u	PROPN
ejpam-3209	101	50	,	,	PUNCT
ejpam-3209	101	51	u	u	NOUN
ejpam-3209	101	52	)	)	PUNCT
ejpam-3209	101	53	n−2	n−2	PROPN
ejpam-3209	101	54	2	2	NUM
ejpam-3209	101	55	(	(	PUNCT
ejpam-3209	101	56	v	v	NOUN
ejpam-3209	101	57	,	,	PUNCT
ejpam-3209	101	58	v	v	NOUN
ejpam-3209	101	59	)	)	PUNCT
ejpam-3209	101	60	+	+	CCONJ
ejpam-3209	101	61	nc3(u	nc3(u	PROPN
ejpam-3209	101	62	,	,	PUNCT
ejpam-3209	101	63	u	u	NOUN
ejpam-3209	101	64	)	)	PUNCT
ejpam-3209	101	65	n−4	n−4	PROPN
ejpam-3209	101	66	2	2	NUM
ejpam-3209	101	67	(	(	PUNCT
ejpam-3209	101	68	v	v	NOUN
ejpam-3209	101	69	,	,	PUNCT
ejpam-3209	101	70	v)(u	v)(u	ADJ
ejpam-3209	101	71	,	,	PUNCT
ejpam-3209	101	72	v	v	NOUN
ejpam-3209	101	73	)	)	PUNCT
ejpam-3209	101	74	+	+	PROPN
ejpam-3209	101	75	n	n	PRON
ejpam-3209	101	76	c4(u	c4(u	PROPN
ejpam-3209	101	77	,	,	PUNCT
ejpam-3209	101	78	u	u	NOUN
ejpam-3209	101	79	)	)	PUNCT
ejpam-3209	101	80	n−4	n−4	PROPN
ejpam-3209	101	81	2	2	NUM
ejpam-3209	101	82	(	(	PUNCT
ejpam-3209	101	83	v	v	NOUN
ejpam-3209	101	84	,	,	PUNCT
ejpam-3209	101	85	v)2	v)2	ADJ
ejpam-3209	101	86	+	+	NOUN
ejpam-3209	101	87	n	n	PROPN
ejpam-3209	101	88	c5(u	c5(u	PROPN
ejpam-3209	101	89	,	,	PUNCT
ejpam-3209	101	90	u	u	NOUN
ejpam-3209	101	91	)	)	PUNCT
ejpam-3209	101	92	n−6	n−6	PROPN
ejpam-3209	101	93	2	2	NUM
ejpam-3209	101	94	(	(	PUNCT
ejpam-3209	101	95	v	v	NOUN
ejpam-3209	101	96	,	,	PUNCT
ejpam-3209	101	97	v)2(u	v)2(u	NUM
ejpam-3209	101	98	,	,	PUNCT
ejpam-3209	101	99	v	v	NOUN
ejpam-3209	101	100	)	)	PUNCT
ejpam-3209	101	101	+	+	CCONJ
ejpam-3209	101	102	nc6(u	nc6(u	PROPN
ejpam-3209	101	103	,	,	PUNCT
ejpam-3209	101	104	u	u	NOUN
ejpam-3209	101	105	)	)	PUNCT
ejpam-3209	101	106	n−6	n−6	PROPN
ejpam-3209	101	107	2	2	NUM
ejpam-3209	101	108	(	(	PUNCT
ejpam-3209	101	109	v	v	NOUN
ejpam-3209	101	110	,	,	PUNCT
ejpam-3209	101	111	v)3	v)3	PROPN
ejpam-3209	101	112	+	+	PROPN
ejpam-3209	101	113	n	n	PROPN
ejpam-3209	101	114	c7(u	c7(u	NOUN
ejpam-3209	101	115	,	,	PUNCT
ejpam-3209	101	116	u	u	NOUN
ejpam-3209	101	117	)	)	PUNCT
ejpam-3209	101	118	n−8	n−8	PROPN
ejpam-3209	101	119	2	2	NUM
ejpam-3209	101	120	(	(	PUNCT
ejpam-3209	101	121	v	v	NOUN
ejpam-3209	101	122	,	,	PUNCT
ejpam-3209	101	123	v)3(u	v)3(u	NOUN
ejpam-3209	101	124	,	,	PUNCT
ejpam-3209	101	125	v	v	NOUN
ejpam-3209	101	126	)	)	PUNCT
ejpam-3209	101	127	+	+	CCONJ
ejpam-3209	101	128	.	.	PUNCT
ejpam-3209	101	129	.	.	PUNCT
ejpam-3209	102	1	.+n	.+n	PROPN
ejpam-3209	102	2	cn	cn	NOUN
ejpam-3209	102	3	2	2	NUM
ejpam-3209	102	4	(	(	PUNCT
ejpam-3209	102	5	u	u	NOUN
ejpam-3209	102	6	,	,	PUNCT
ejpam-3209	102	7	u	u	NOUN
ejpam-3209	102	8	)	)	PUNCT
ejpam-3209	102	9	n	n	ADV
ejpam-3209	102	10	4	4	NUM
ejpam-3209	102	11	(	(	PUNCT
ejpam-3209	102	12	v	v	NOUN
ejpam-3209	102	13	,	,	PUNCT
ejpam-3209	102	14	v	v	NOUN
ejpam-3209	102	15	)	)	PUNCT
ejpam-3209	102	16	n	n	ADV
ejpam-3209	102	17	4	4	NUM
ejpam-3209	102	18	+	+	CCONJ
ejpam-3209	102	19	.	.	PUNCT
ejpam-3209	102	20	.	.	PUNCT
ejpam-3209	103	1	.+	.+	NOUN
ejpam-3209	103	2	(	(	PUNCT
ejpam-3209	103	3	u	u	NOUN
ejpam-3209	103	4	,	,	PUNCT
ejpam-3209	103	5	u	u	NOUN
ejpam-3209	103	6	)	)	PUNCT
ejpam-3209	103	7	n	n	ADV
ejpam-3209	103	8	2	2	NUM
ejpam-3209	103	9	}	}	PUNCT
ejpam-3209	103	10	≤	≤	NOUN
ejpam-3209	103	11	ncn	ncn	NOUN
ejpam-3209	103	12	2	2	NUM
ejpam-3209	103	13	(	(	PUNCT
ejpam-3209	103	14	u	u	NOUN
ejpam-3209	103	15	,	,	PUNCT
ejpam-3209	103	16	u	u	NOUN
ejpam-3209	103	17	)	)	PUNCT
ejpam-3209	103	18	n	n	ADV
ejpam-3209	103	19	4	4	NUM
ejpam-3209	103	20	(	(	PUNCT
ejpam-3209	103	21	v	v	NOUN
ejpam-3209	103	22	,	,	PUNCT
ejpam-3209	103	23	v	v	NOUN
ejpam-3209	103	24	)	)	PUNCT
ejpam-3209	103	25	n	n	PRON
ejpam-3209	103	26	4	4	NUM
ejpam-3209	103	27	⇒	⇒	NOUN
ejpam-3209	103	28	−(u+	−(u+	NUM
ejpam-3209	103	29	v)n	v)n	NOUN
ejpam-3209	103	30	≤n	≤n	NOUN
ejpam-3209	103	31	cn	cn	PROPN
ejpam-3209	103	32	2	2	NUM
ejpam-3209	103	33	(	(	PUNCT
ejpam-3209	103	34	u	u	NOUN
ejpam-3209	103	35	,	,	PUNCT
ejpam-3209	103	36	u	u	NOUN
ejpam-3209	103	37	)	)	PUNCT
ejpam-3209	103	38	n	n	ADV
ejpam-3209	103	39	4	4	NUM
ejpam-3209	103	40	(	(	PUNCT
ejpam-3209	103	41	v	v	NOUN
ejpam-3209	103	42	,	,	PUNCT
ejpam-3209	103	43	v	v	NOUN
ejpam-3209	103	44	)	)	PUNCT
ejpam-3209	103	45	n	n	ADV
ejpam-3209	103	46	4	4	NUM
ejpam-3209	103	47	1	1	NUM
ejpam-3209	103	48	ncn	ncn	NOUN
ejpam-3209	103	49	2	2	NUM
ejpam-3209	103	50	(	(	PUNCT
ejpam-3209	103	51	u	u	NOUN
ejpam-3209	103	52	,	,	PUNCT
ejpam-3209	103	53	u	u	NOUN
ejpam-3209	103	54	)	)	PUNCT
ejpam-3209	103	55	n	n	ADV
ejpam-3209	103	56	4	4	NUM
ejpam-3209	103	57	(	(	PUNCT
ejpam-3209	103	58	v	v	NOUN
ejpam-3209	103	59	,	,	PUNCT
ejpam-3209	103	60	v	v	NOUN
ejpam-3209	103	61	)	)	PUNCT
ejpam-3209	103	62	n	n	DET
ejpam-3209	103	63	4	4	NUM
ejpam-3209	103	64	⇒	⇒	NOUN
ejpam-3209	103	65	‖−	‖−	NOUN
ejpam-3209	103	66	(	(	PUNCT
ejpam-3209	103	67	u+	u+	NUM
ejpam-3209	103	68	v)n‖	v)n‖	NUM
ejpam-3209	103	69	=	=	SYM
ejpam-3209	103	70	‖(u	‖(u	PROPN
ejpam-3209	103	71	,	,	PUNCT
ejpam-3209	103	72	u	u	NOUN
ejpam-3209	103	73	)	)	PUNCT
ejpam-3209	103	74	n	n	ADV
ejpam-3209	103	75	2	2	NUM
ejpam-3209	103	76	(	(	PUNCT
ejpam-3209	103	77	v	v	NOUN
ejpam-3209	103	78	,	,	PUNCT
ejpam-3209	103	79	v	v	NOUN
ejpam-3209	103	80	)	)	PUNCT
ejpam-3209	103	81	n	n	PRON
ejpam-3209	103	82	2	2	NUM
ejpam-3209	103	83	‖	‖	ADJ
ejpam-3209	103	84	⇒	⇒	NOUN
ejpam-3209	103	85	‖(u+	‖(u+	NUM
ejpam-3209	103	86	v)‖n	v)‖n	NOUN
ejpam-3209	103	87	=	=	SYM
ejpam-3209	103	88	{	{	PUNCT
ejpam-3209	103	89	‖u‖‖v‖}n	‖u‖‖v‖}n	VERB
ejpam-3209	103	90	⇒	⇒	NOUN
ejpam-3209	103	91	‖u‖+	‖u‖+	PROPN
ejpam-3209	103	92	‖v‖	‖v‖	PROPN
ejpam-3209	103	93	≤	≤	PROPN
ejpam-3209	103	94	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3209	103	95	,	,	PUNCT
ejpam-3209	103	96	∀	∀	X
ejpam-3209	103	97	u	u	NOUN
ejpam-3209	103	98	,	,	PUNCT
ejpam-3209	103	99	v	v	NOUN
ejpam-3209	103	100	∈	∈	PROPN
ejpam-3209	103	101	[	[	X
ejpam-3209	103	102	2,∞	2,∞	NUM
ejpam-3209	103	103	)	)	PUNCT
ejpam-3209	103	104	.	.	PUNCT
ejpam-3209	104	1	(	(	PUNCT
ejpam-3209	104	2	4	4	X
ejpam-3209	104	3	)	)	PUNCT
ejpam-3209	104	4	3	3	NUM
ejpam-3209	104	5	.	.	X
ejpam-3209	104	6	illustration	illustration	NOUN
ejpam-3209	104	7	of	of	ADP
ejpam-3209	104	8	the	the	DET
ejpam-3209	104	9	first	first	ADJ
ejpam-3209	104	10	product	product	NOUN
ejpam-3209	104	11	inequality	inequality	NOUN
ejpam-3209	104	12	in	in	ADP
ejpam-3209	104	13	this	this	DET
ejpam-3209	104	14	section	section	NOUN
ejpam-3209	104	15	of	of	ADP
ejpam-3209	104	16	the	the	DET
ejpam-3209	104	17	paper	paper	NOUN
ejpam-3209	104	18	,	,	PUNCT
ejpam-3209	104	19	we	we	PRON
ejpam-3209	104	20	illustrate	illustrate	VERB
ejpam-3209	104	21	some	some	PRON
ejpam-3209	104	22	of	of	ADP
ejpam-3209	104	23	the	the	DET
ejpam-3209	104	24	aspects	aspect	NOUN
ejpam-3209	104	25	of	of	ADP
ejpam-3209	104	26	mathematics	mathematic	NOUN
ejpam-3209	104	27	where	where	SCONJ
ejpam-3209	104	28	the	the	DET
ejpam-3209	104	29	first	first	ADJ
ejpam-3209	104	30	product	product	NOUN
ejpam-3209	104	31	inequality	inequality	NOUN
ejpam-3209	104	32	is	be	AUX
ejpam-3209	104	33	feasible	feasible	ADJ
ejpam-3209	104	34	.	.	PUNCT
ejpam-3209	105	1	notwithstanding	notwithstanding	ADV
ejpam-3209	105	2	,	,	PUNCT
ejpam-3209	105	3	the	the	DET
ejpam-3209	105	4	areas	area	NOUN
ejpam-3209	105	5	of	of	ADP
ejpam-3209	105	6	applications	application	NOUN
ejpam-3209	105	7	of	of	ADP
ejpam-3209	105	8	the	the	DET
ejpam-3209	105	9	barnes	barnes	PROPN
ejpam-3209	105	10	et	et	PROPN
ejpam-3209	105	11	al	al	PROPN
ejpam-3209	105	12	.	.	PUNCT
ejpam-3209	105	13	/	/	SYM
ejpam-3209	105	14	eur	eur	PROPN
ejpam-3209	105	15	.	.	PUNCT
ejpam-3209	106	1	j.	j.	PROPN
ejpam-3209	106	2	pure	pure	PROPN
ejpam-3209	106	3	appl	appl	PROPN
ejpam-3209	106	4	.	.	PROPN
ejpam-3209	106	5	math	math	PROPN
ejpam-3209	106	6	,	,	PUNCT
ejpam-3209	106	7	11	11	NUM
ejpam-3209	106	8	(	(	PUNCT
ejpam-3209	106	9	2	2	NUM
ejpam-3209	106	10	)	)	PUNCT
ejpam-3209	106	11	(	(	PUNCT
ejpam-3209	106	12	2018	2018	NUM
ejpam-3209	106	13	)	)	PUNCT
ejpam-3209	106	14	,	,	PUNCT
ejpam-3209	106	15	375	375	NUM
ejpam-3209	106	16	-	-	SYM
ejpam-3209	106	17	389	389	NUM
ejpam-3209	106	18	381	381	NUM
ejpam-3209	106	19	first	first	ADJ
ejpam-3209	106	20	product	product	NOUN
ejpam-3209	106	21	inequality	inequality	NOUN
ejpam-3209	106	22	are	be	AUX
ejpam-3209	106	23	highlighted	highlight	VERB
ejpam-3209	106	24	.	.	PUNCT
ejpam-3209	107	1	3.1	3.1	NUM
ejpam-3209	107	2	.	.	PUNCT
ejpam-3209	107	3	illustration	illustration	NOUN
ejpam-3209	107	4	of	of	ADP
ejpam-3209	107	5	the	the	DET
ejpam-3209	107	6	first	first	ADJ
ejpam-3209	107	7	product	product	NOUN
ejpam-3209	107	8	inequality	inequality	NOUN
ejpam-3209	107	9	to	to	ADP
ejpam-3209	107	10	the	the	DET
ejpam-3209	107	11	real	real	ADJ
ejpam-3209	107	12	line	line	NOUN
ejpam-3209	107	13	firstly	firstly	ADV
ejpam-3209	107	14	,	,	PUNCT
ejpam-3209	107	15	we	we	PRON
ejpam-3209	107	16	illustrate	illustrate	VERB
ejpam-3209	107	17	the	the	DET
ejpam-3209	107	18	first	first	ADJ
ejpam-3209	107	19	product	product	NOUN
ejpam-3209	107	20	inequality	inequality	NOUN
ejpam-3209	107	21	in	in	ADP
ejpam-3209	107	22	(	(	PUNCT
ejpam-3209	107	23	1	1	NUM
ejpam-3209	107	24	)	)	PUNCT
ejpam-3209	107	25	to	to	ADP
ejpam-3209	107	26	any	any	DET
ejpam-3209	107	27	two	two	NUM
ejpam-3209	107	28	real	real	ADJ
ejpam-3209	107	29	numbers	number	NOUN
ejpam-3209	107	30	a	a	PRON
ejpam-3209	107	31	,	,	PUNCT
ejpam-3209	107	32	b	b	X
ejpam-3209	107	33	∈	∈	PROPN
ejpam-3209	108	1	[	[	X
ejpam-3209	108	2	0	0	NUM
ejpam-3209	108	3	,	,	PUNCT
ejpam-3209	108	4	2	2	NUM
ejpam-3209	108	5	]	]	PUNCT
ejpam-3209	108	6	.	.	PUNCT
ejpam-3209	109	1	example	example	NOUN
ejpam-3209	110	1	1	1	NUM
ejpam-3209	110	2	.	.	PUNCT
ejpam-3209	110	3	let	let	VERB
ejpam-3209	110	4	a	a	DET
ejpam-3209	110	5	=	=	SYM
ejpam-3209	110	6	5	5	NUM
ejpam-3209	110	7	6	6	NUM
ejpam-3209	110	8	and	and	CCONJ
ejpam-3209	110	9	b	b	X
ejpam-3209	110	10	=	=	SYM
ejpam-3209	110	11	13	13	NUM
ejpam-3209	110	12	18	18	NUM
ejpam-3209	110	13	,	,	PUNCT
ejpam-3209	110	14	then∥∥∥5	then∥∥∥5	PROPN
ejpam-3209	110	15	6	6	NUM
ejpam-3209	110	16	∥∥∥∥∥∥13	∥∥∥∥∥∥13	PROPN
ejpam-3209	110	17	18	18	NUM
ejpam-3209	110	18	∥∥∥	∥∥∥	PROPN
ejpam-3209	110	19	≤	≤	NUM
ejpam-3209	110	20	∥∥∥5	∥∥∥5	X
ejpam-3209	110	21	6	6	NUM
ejpam-3209	110	22	∥∥∥+	∥∥∥+	PROPN
ejpam-3209	110	23	∥∥∥13	∥∥∥13	PROPN
ejpam-3209	110	24	18	18	NUM
ejpam-3209	110	25	∥∥∥	∥∥∥	PROPN
ejpam-3209	110	26	65	65	NUM
ejpam-3209	110	27	108	108	NUM
ejpam-3209	110	28	<	<	X
ejpam-3209	110	29	28	28	NUM
ejpam-3209	110	30	18	18	NUM
ejpam-3209	110	31	.	.	PUNCT
ejpam-3209	110	32	example	example	NOUN
ejpam-3209	111	1	2	2	NUM
ejpam-3209	111	2	.	.	PUNCT
ejpam-3209	111	3	let	let	VERB
ejpam-3209	111	4	a	a	DET
ejpam-3209	111	5	=	=	NOUN
ejpam-3209	111	6	−1	−1	NOUN
ejpam-3209	111	7	2	2	NUM
ejpam-3209	111	8	and	and	CCONJ
ejpam-3209	111	9	b	b	X
ejpam-3209	111	10	=	=	X
ejpam-3209	111	11	−3	−3	PROPN
ejpam-3209	111	12	5	5	NUM
ejpam-3209	111	13	,	,	PUNCT
ejpam-3209	111	14	then∥∥∥−1	then∥∥∥−1	NOUN
ejpam-3209	111	15	2	2	NUM
ejpam-3209	111	16	∥∥∥∥∥∥−3	∥∥∥∥∥∥−3	PROPN
ejpam-3209	111	17	5	5	NUM
ejpam-3209	111	18	∥∥∥	∥∥∥	PROPN
ejpam-3209	111	19	≤	≤	NUM
ejpam-3209	111	20	∥∥∥−1	∥∥∥−1	PROPN
ejpam-3209	111	21	2	2	NUM
ejpam-3209	111	22	∥∥∥+	∥∥∥+	NOUN
ejpam-3209	111	23	∥∥∥−3	∥∥∥−3	PROPN
ejpam-3209	111	24	5	5	NUM
ejpam-3209	111	25	∥∥∥	∥∥∥	PROPN
ejpam-3209	111	26	3	3	NUM
ejpam-3209	111	27	10	10	NUM
ejpam-3209	111	28	<	<	SYM
ejpam-3209	111	29	11	11	NUM
ejpam-3209	111	30	10	10	NUM
ejpam-3209	111	31	.	.	PUNCT
ejpam-3209	111	32	example	example	NOUN
ejpam-3209	112	1	3	3	X
ejpam-3209	112	2	.	.	PUNCT
ejpam-3209	112	3	let	let	VERB
ejpam-3209	112	4	a	a	DET
ejpam-3209	112	5	=	=	SYM
ejpam-3209	112	6	19	19	NUM
ejpam-3209	112	7	10	10	NUM
ejpam-3209	112	8	and	and	CCONJ
ejpam-3209	112	9	b	b	NOUN
ejpam-3209	112	10	=	=	SYM
ejpam-3209	112	11	9	9	NUM
ejpam-3209	112	12	5	5	NUM
ejpam-3209	112	13	,	,	PUNCT
ejpam-3209	112	14	then∥∥∥19	then∥∥∥19	PROPN
ejpam-3209	112	15	10	10	NUM
ejpam-3209	112	16	∥∥∥∥∥∥9	∥∥∥∥∥∥9	PROPN
ejpam-3209	112	17	5	5	NUM
ejpam-3209	112	18	∥∥∥	∥∥∥	PROPN
ejpam-3209	112	19	≤	≤	NOUN
ejpam-3209	112	20	∥∥∥19	∥∥∥19	NOUN
ejpam-3209	112	21	10	10	NUM
ejpam-3209	112	22	∥∥∥+	∥∥∥+	PROPN
ejpam-3209	112	23	∥∥∥	∥∥∥	PROPN
ejpam-3209	112	24	9	9	NUM
ejpam-3209	112	25	10	10	NUM
ejpam-3209	112	26	∥∥∥	∥∥∥	PROPN
ejpam-3209	112	27	171	171	NUM
ejpam-3209	112	28	50	50	NUM
ejpam-3209	112	29	<	<	X
ejpam-3209	112	30	37	37	NUM
ejpam-3209	112	31	10	10	NUM
ejpam-3209	112	32	.	.	PUNCT
ejpam-3209	113	1	3.2	3.2	NUM
ejpam-3209	113	2	.	.	PUNCT
ejpam-3209	113	3	illustration	illustration	NOUN
ejpam-3209	113	4	of	of	ADP
ejpam-3209	113	5	the	the	DET
ejpam-3209	113	6	first	first	ADJ
ejpam-3209	113	7	product	product	NOUN
ejpam-3209	113	8	inequality	inequality	NOUN
ejpam-3209	113	9	to	to	ADP
ejpam-3209	113	10	the	the	DET
ejpam-3209	113	11	euclidean	euclidean	ADJ
ejpam-3209	113	12	space	space	NOUN
ejpam-3209	113	13	again	again	ADV
ejpam-3209	113	14	,	,	PUNCT
ejpam-3209	113	15	the	the	DET
ejpam-3209	113	16	first	first	ADJ
ejpam-3209	113	17	product	product	NOUN
ejpam-3209	113	18	inequality	inequality	NOUN
ejpam-3209	113	19	is	be	AUX
ejpam-3209	113	20	illustrated	illustrate	VERB
ejpam-3209	113	21	to	to	ADP
ejpam-3209	113	22	the	the	DET
ejpam-3209	113	23	vector	vector	NOUN
ejpam-3209	113	24	points	point	NOUN
ejpam-3209	113	25	of	of	ADP
ejpam-3209	113	26	real	real	ADJ
ejpam-3209	113	27	numbers	number	NOUN
ejpam-3209	113	28	as	as	SCONJ
ejpam-3209	113	29	follows	follow	VERB
ejpam-3209	113	30	.	.	PUNCT
ejpam-3209	114	1	in	in	ADP
ejpam-3209	114	2	this	this	DET
ejpam-3209	114	3	paper	paper	NOUN
ejpam-3209	114	4	,	,	PUNCT
ejpam-3209	114	5	three	three	NUM
ejpam-3209	114	6	basic	basic	ADJ
ejpam-3209	114	7	norms	norm	NOUN
ejpam-3209	114	8	:	:	PUNCT
ejpam-3209	114	9	1−	1−	NUM
ejpam-3209	114	10	norm	norm	NOUN
ejpam-3209	114	11	,	,	PUNCT
ejpam-3209	114	12	2−	2−	NUM
ejpam-3209	114	13	norm	norm	NOUN
ejpam-3209	114	14	and∞-norm	and∞-norm	NOUN
ejpam-3209	114	15	are	be	AUX
ejpam-3209	114	16	considered	consider	VERB
ejpam-3209	114	17	.	.	PUNCT
ejpam-3209	115	1	example	example	NOUN
ejpam-3209	115	2	4	4	NUM
ejpam-3209	115	3	.	.	PUNCT
ejpam-3209	116	1	let	let	VERB
ejpam-3209	116	2	a	a	DET
ejpam-3209	116	3	=	=	X
ejpam-3209	116	4	(	(	PUNCT
ejpam-3209	116	5	1	1	NUM
ejpam-3209	116	6	2	2	NUM
ejpam-3209	116	7	2	2	NUM
ejpam-3209	116	8	3	3	NUM
ejpam-3209	116	9	)	)	PUNCT
ejpam-3209	116	10	and	and	CCONJ
ejpam-3209	116	11	b	b	X
ejpam-3209	116	12	=	=	SYM
ejpam-3209	116	13	(	(	PUNCT
ejpam-3209	116	14	4	4	NUM
ejpam-3209	116	15	5	5	NUM
ejpam-3209	116	16	−1	−1	NOUN
ejpam-3209	116	17	3	3	NUM
ejpam-3209	116	18	)	)	PUNCT
ejpam-3209	116	19	,	,	PUNCT
ejpam-3209	116	20	then	then	ADV
ejpam-3209	116	21	∥∥∥	∥∥∥	PROPN
ejpam-3209	116	22	(	(	PUNCT
ejpam-3209	116	23	1	1	NUM
ejpam-3209	116	24	2	2	NUM
ejpam-3209	116	25	2	2	NUM
ejpam-3209	116	26	3	3	NUM
ejpam-3209	116	27	)	)	PUNCT
ejpam-3209	116	28	∥∥∥	∥∥∥	PROPN
ejpam-3209	116	29	1	1	NUM
ejpam-3209	116	30	∥∥∥	∥∥∥	NUM
ejpam-3209	116	31	(	(	PUNCT
ejpam-3209	116	32	4	4	NUM
ejpam-3209	116	33	5	5	NUM
ejpam-3209	116	34	−1	−1	NOUN
ejpam-3209	116	35	3	3	NUM
ejpam-3209	116	36	)	)	PUNCT
ejpam-3209	116	37	∥∥∥	∥∥∥	PROPN
ejpam-3209	116	38	1	1	NUM
ejpam-3209	116	39	≤	≤	NUM
ejpam-3209	116	40	∥∥∥	∥∥∥	PROPN
ejpam-3209	116	41	(	(	PUNCT
ejpam-3209	116	42	1	1	NUM
ejpam-3209	116	43	2	2	NUM
ejpam-3209	116	44	2	2	NUM
ejpam-3209	116	45	3	3	NUM
ejpam-3209	116	46	)	)	PUNCT
ejpam-3209	116	47	∥∥∥	∥∥∥	PROPN
ejpam-3209	116	48	1	1	NUM
ejpam-3209	116	49	+	+	NUM
ejpam-3209	116	50	∥∥∥	∥∥∥	PROPN
ejpam-3209	116	51	(	(	PUNCT
ejpam-3209	116	52	4	4	NUM
ejpam-3209	116	53	5	5	NUM
ejpam-3209	116	54	−1	−1	NOUN
ejpam-3209	116	55	3	3	NUM
ejpam-3209	116	56	)	)	PUNCT
ejpam-3209	116	57	∥∥∥	∥∥∥	PROPN
ejpam-3209	116	58	1	1	NUM
ejpam-3209	116	59	⇒	⇒	NOUN
ejpam-3209	116	60	119	119	NUM
ejpam-3209	116	61	90	90	NUM
ejpam-3209	116	62	<	<	SYM
ejpam-3209	116	63	69	69	NUM
ejpam-3209	116	64	30	30	NUM
ejpam-3209	116	65	.	.	PUNCT
ejpam-3209	117	1	example	example	NOUN
ejpam-3209	117	2	5	5	NUM
ejpam-3209	117	3	.	.	PUNCT
ejpam-3209	118	1	let	let	VERB
ejpam-3209	118	2	a	a	DET
ejpam-3209	118	3	=	=	X
ejpam-3209	118	4	(	(	PUNCT
ejpam-3209	118	5	1	1	NUM
ejpam-3209	118	6	2	2	NUM
ejpam-3209	118	7	2	2	NUM
ejpam-3209	118	8	3	3	NUM
ejpam-3209	118	9	)	)	PUNCT
ejpam-3209	118	10	and	and	CCONJ
ejpam-3209	118	11	b	b	X
ejpam-3209	118	12	=	=	SYM
ejpam-3209	118	13	(	(	PUNCT
ejpam-3209	118	14	4	4	NUM
ejpam-3209	118	15	5	5	NUM
ejpam-3209	118	16	−1	−1	NOUN
ejpam-3209	118	17	3	3	NUM
ejpam-3209	118	18	)	)	PUNCT
ejpam-3209	118	19	,	,	PUNCT
ejpam-3209	118	20	then	then	ADV
ejpam-3209	118	21	∥∥∥	∥∥∥	PROPN
ejpam-3209	118	22	(	(	PUNCT
ejpam-3209	118	23	1	1	NUM
ejpam-3209	118	24	2	2	NUM
ejpam-3209	118	25	2	2	NUM
ejpam-3209	118	26	3	3	NUM
ejpam-3209	118	27	)	)	PUNCT
ejpam-3209	118	28	∥∥∥	∥∥∥	PROPN
ejpam-3209	118	29	2	2	NUM
ejpam-3209	118	30	∥∥∥	∥∥∥	NUM
ejpam-3209	118	31	(	(	PUNCT
ejpam-3209	118	32	4	4	NUM
ejpam-3209	118	33	5	5	NUM
ejpam-3209	118	34	−1	−1	NOUN
ejpam-3209	118	35	3	3	NUM
ejpam-3209	118	36	)	)	PUNCT
ejpam-3209	118	37	∥∥∥	∥∥∥	PROPN
ejpam-3209	118	38	2	2	NUM
ejpam-3209	118	39	≤	≤	NUM
ejpam-3209	118	40	∥∥∥	∥∥∥	PROPN
ejpam-3209	118	41	(	(	PUNCT
ejpam-3209	118	42	1	1	NUM
ejpam-3209	118	43	2	2	NUM
ejpam-3209	118	44	2	2	NUM
ejpam-3209	118	45	3	3	NUM
ejpam-3209	118	46	)	)	PUNCT
ejpam-3209	118	47	∥∥∥	∥∥∥	PROPN
ejpam-3209	118	48	2	2	NUM
ejpam-3209	118	49	+	+	NUM
ejpam-3209	118	50	∥∥∥	∥∥∥	PROPN
ejpam-3209	118	51	(	(	PUNCT
ejpam-3209	118	52	4	4	NUM
ejpam-3209	118	53	5	5	NUM
ejpam-3209	118	54	−1	−1	NOUN
ejpam-3209	118	55	3	3	NUM
ejpam-3209	118	56	)	)	PUNCT
ejpam-3209	118	57	∥∥∥	∥∥∥	PROPN
ejpam-3209	118	58	2	2	NUM
ejpam-3209	118	59	⇒	⇒	NOUN
ejpam-3209	118	60	13	13	NUM
ejpam-3209	118	61	18	18	NUM
ejpam-3209	118	62	<	<	X
ejpam-3209	118	63	28	28	NUM
ejpam-3209	118	64	18	18	NUM
ejpam-3209	118	65	.	.	PUNCT
ejpam-3209	119	1	barnes	barnes	PROPN
ejpam-3209	119	2	et	et	PROPN
ejpam-3209	119	3	al	al	PROPN
ejpam-3209	119	4	.	.	PUNCT
ejpam-3209	119	5	/	/	SYM
ejpam-3209	119	6	eur	eur	PROPN
ejpam-3209	119	7	.	.	PUNCT
ejpam-3209	120	1	j.	j.	PROPN
ejpam-3209	120	2	pure	pure	PROPN
ejpam-3209	120	3	appl	appl	PROPN
ejpam-3209	120	4	.	.	PROPN
ejpam-3209	120	5	math	math	PROPN
ejpam-3209	120	6	,	,	PUNCT
ejpam-3209	120	7	11	11	NUM
ejpam-3209	120	8	(	(	PUNCT
ejpam-3209	120	9	2	2	NUM
ejpam-3209	120	10	)	)	PUNCT
ejpam-3209	120	11	(	(	PUNCT
ejpam-3209	120	12	2018	2018	NUM
ejpam-3209	120	13	)	)	PUNCT
ejpam-3209	120	14	,	,	PUNCT
ejpam-3209	121	1	375	375	NUM
ejpam-3209	121	2	-	-	SYM
ejpam-3209	121	3	389	389	NUM
ejpam-3209	121	4	382	382	NUM
ejpam-3209	121	5	example	example	NOUN
ejpam-3209	121	6	6	6	NUM
ejpam-3209	121	7	.	.	PUNCT
ejpam-3209	122	1	let	let	VERB
ejpam-3209	122	2	a	a	DET
ejpam-3209	122	3	=	=	X
ejpam-3209	122	4	(	(	PUNCT
ejpam-3209	122	5	1	1	NUM
ejpam-3209	122	6	2	2	NUM
ejpam-3209	122	7	2	2	NUM
ejpam-3209	122	8	3	3	NUM
ejpam-3209	122	9	)	)	PUNCT
ejpam-3209	122	10	and	and	CCONJ
ejpam-3209	122	11	b	b	X
ejpam-3209	122	12	=	=	SYM
ejpam-3209	122	13	(	(	PUNCT
ejpam-3209	122	14	4	4	NUM
ejpam-3209	122	15	5	5	NUM
ejpam-3209	122	16	−1	−1	NOUN
ejpam-3209	122	17	3	3	NUM
ejpam-3209	122	18	)	)	PUNCT
ejpam-3209	122	19	,	,	PUNCT
ejpam-3209	122	20	then	then	ADV
ejpam-3209	122	21	∥∥∥	∥∥∥	PROPN
ejpam-3209	122	22	(	(	PUNCT
ejpam-3209	122	23	1	1	NUM
ejpam-3209	122	24	2	2	NUM
ejpam-3209	122	25	2	2	NUM
ejpam-3209	122	26	3	3	NUM
ejpam-3209	122	27	)	)	PUNCT
ejpam-3209	122	28	∥∥∥	∥∥∥	PROPN
ejpam-3209	122	29	∞	∞	NUM
ejpam-3209	122	30	∥∥∥	∥∥∥	PROPN
ejpam-3209	122	31	(	(	PUNCT
ejpam-3209	122	32	4	4	NUM
ejpam-3209	122	33	5	5	NUM
ejpam-3209	122	34	−1	−1	NOUN
ejpam-3209	122	35	3	3	NUM
ejpam-3209	122	36	)	)	PUNCT
ejpam-3209	122	37	∥∥∥	∥∥∥	PROPN
ejpam-3209	122	38	∞	∞	PROPN
ejpam-3209	122	39	≤	≤	PROPN
ejpam-3209	122	40	∥∥∥	∥∥∥	PROPN
ejpam-3209	122	41	(	(	PUNCT
ejpam-3209	122	42	1	1	NUM
ejpam-3209	122	43	2	2	NUM
ejpam-3209	122	44	2	2	NUM
ejpam-3209	122	45	3	3	NUM
ejpam-3209	122	46	)	)	PUNCT
ejpam-3209	122	47	∥∥∥	∥∥∥	PROPN
ejpam-3209	122	48	∞	∞	PROPN
ejpam-3209	122	49	+	+	CCONJ
ejpam-3209	122	50	∥∥∥	∥∥∥	PROPN
ejpam-3209	122	51	(	(	PUNCT
ejpam-3209	122	52	4	4	NUM
ejpam-3209	122	53	5	5	NUM
ejpam-3209	122	54	−1	−1	NOUN
ejpam-3209	122	55	3	3	NUM
ejpam-3209	122	56	)	)	PUNCT
ejpam-3209	122	57	∥∥∥	∥∥∥	NOUN
ejpam-3209	122	58	∞	∞	PROPN
ejpam-3209	122	59	⇒	⇒	NOUN
ejpam-3209	122	60	8	8	NUM
ejpam-3209	122	61	15	15	NUM
ejpam-3209	122	62	<	<	X
ejpam-3209	122	63	22	22	NUM
ejpam-3209	122	64	15	15	NUM
ejpam-3209	122	65	.	.	PUNCT
ejpam-3209	123	1	3.3	3.3	NUM
ejpam-3209	123	2	.	.	PUNCT
ejpam-3209	124	1	illustration	illustration	NOUN
ejpam-3209	124	2	of	of	ADP
ejpam-3209	124	3	the	the	DET
ejpam-3209	124	4	first	first	ADJ
ejpam-3209	124	5	product	product	NOUN
ejpam-3209	124	6	inequality	inequality	NOUN
ejpam-3209	124	7	to	to	ADP
ejpam-3209	124	8	trigonometric	trigonometric	ADJ
ejpam-3209	124	9	functions	function	NOUN
ejpam-3209	124	10	undoubedly	undoubedly	ADV
ejpam-3209	124	11	,	,	PUNCT
ejpam-3209	124	12	we	we	PRON
ejpam-3209	124	13	construct	construct	VERB
ejpam-3209	124	14	the	the	DET
ejpam-3209	124	15	inequalities	inequality	NOUN
ejpam-3209	124	16	involving	involve	VERB
ejpam-3209	124	17	the	the	DET
ejpam-3209	124	18	product	product	NOUN
ejpam-3209	124	19	of	of	ADP
ejpam-3209	124	20	sine	sine	NOUN
ejpam-3209	124	21	of	of	ADP
ejpam-3209	124	22	an	an	DET
ejpam-3209	124	23	angle	angle	NOUN
ejpam-3209	124	24	and	and	CCONJ
ejpam-3209	124	25	cosine	cosine	NOUN
ejpam-3209	124	26	of	of	ADP
ejpam-3209	124	27	an	an	DET
ejpam-3209	124	28	angle	angle	NOUN
ejpam-3209	124	29	.	.	PUNCT
ejpam-3209	125	1	we	we	PRON
ejpam-3209	125	2	can	can	AUX
ejpam-3209	125	3	see	see	VERB
ejpam-3209	125	4	that	that	PRON
ejpam-3209	125	5	:	:	PUNCT
ejpam-3209	126	1	‖	‖	ADJ
ejpam-3209	126	2	sin(θ)‖	sin(θ)‖	NOUN
ejpam-3209	126	3	≤	≤	ADJ
ejpam-3209	126	4	1	1	NUM
ejpam-3209	126	5	,	,	PUNCT
ejpam-3209	126	6	and	and	CCONJ
ejpam-3209	126	7	‖	‖	PROPN
ejpam-3209	126	8	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	126	9	≤	≤	NOUN
ejpam-3209	126	10	1	1	NUM
ejpam-3209	126	11	.	.	PUNCT
ejpam-3209	127	1	by	by	ADP
ejpam-3209	127	2	induction	induction	NOUN
ejpam-3209	127	3	,	,	PUNCT
ejpam-3209	127	4	we	we	PRON
ejpam-3209	127	5	start	start	VERB
ejpam-3209	127	6	at	at	ADP
ejpam-3209	127	7	n	n	PROPN
ejpam-3209	127	8	=	=	SYM
ejpam-3209	127	9	0	0	NUM
ejpam-3209	127	10	,	,	PUNCT
ejpam-3209	127	11	by	by	ADP
ejpam-3209	127	12	setting	set	VERB
ejpam-3209	127	13	(	(	PUNCT
ejpam-3209	127	14	sin(θ	sin(θ	PROPN
ejpam-3209	127	15	)	)	PUNCT
ejpam-3209	127	16	cos(θ	cos(θ	PROPN
ejpam-3209	127	17	)	)	PUNCT
ejpam-3209	127	18	)	)	PUNCT
ejpam-3209	127	19	0	0	PUNCT
ejpam-3209	128	1	taking	take	VERB
ejpam-3209	128	2	the	the	DET
ejpam-3209	128	3	magnitude	magnitude	NOUN
ejpam-3209	128	4	of	of	ADP
ejpam-3209	128	5	(	(	PUNCT
ejpam-3209	128	6	sin(θ	sin(θ	PROPN
ejpam-3209	128	7	)	)	PUNCT
ejpam-3209	128	8	cos(θ	cos(θ	PROPN
ejpam-3209	128	9	)	)	PUNCT
ejpam-3209	128	10	)	)	PUNCT
ejpam-3209	128	11	0	0	PUNCT
ejpam-3209	129	1	and	and	CCONJ
ejpam-3209	129	2	applying	apply	VERB
ejpam-3209	129	3	the	the	DET
ejpam-3209	129	4	cauchy	cauchy	NOUN
ejpam-3209	129	5	-	-	PUNCT
ejpam-3209	129	6	schwarz	schwarz	PROPN
ejpam-3209	129	7	inequality	inequality	NOUN
ejpam-3209	129	8	to	to	ADP
ejpam-3209	129	9	it	it	PRON
ejpam-3209	129	10	,	,	PUNCT
ejpam-3209	129	11	we	we	PRON
ejpam-3209	129	12	obtain	obtain	VERB
ejpam-3209	129	13	:(	:(	X
ejpam-3209	129	14	|〈sin(θ	|〈sin(θ	NOUN
ejpam-3209	129	15	)	)	PUNCT
ejpam-3209	129	16	,	,	PUNCT
ejpam-3209	129	17	cos(θ)〉|	cos(θ)〉|	PROPN
ejpam-3209	129	18	)	)	PUNCT
ejpam-3209	129	19	0	0	NUM
ejpam-3209	130	1	≤	≤	NUM
ejpam-3209	130	2	(	(	PUNCT
ejpam-3209	130	3	‖	‖	PROPN
ejpam-3209	130	4	sin(θ)‖‖	sin(θ)‖‖	PROPN
ejpam-3209	130	5	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	130	6	)	)	PUNCT
ejpam-3209	130	7	0	0	PUNCT
ejpam-3209	130	8	.	.	PUNCT
ejpam-3209	131	1	(	(	PUNCT
ejpam-3209	131	2	5	5	X
ejpam-3209	131	3	)	)	PUNCT
ejpam-3209	131	4	applying	apply	VERB
ejpam-3209	131	5	the	the	DET
ejpam-3209	131	6	first	first	ADJ
ejpam-3209	131	7	product	product	NOUN
ejpam-3209	131	8	inequality	inequality	NOUN
ejpam-3209	131	9	to	to	ADP
ejpam-3209	131	10	the	the	DET
ejpam-3209	131	11	expression	expression	NOUN
ejpam-3209	131	12	on	on	ADP
ejpam-3209	131	13	the	the	DET
ejpam-3209	131	14	right	right	ADJ
ejpam-3209	131	15	hand	hand	NOUN
ejpam-3209	131	16	side	side	NOUN
ejpam-3209	131	17	of	of	ADP
ejpam-3209	131	18	equation	equation	NOUN
ejpam-3209	131	19	(	(	PUNCT
ejpam-3209	131	20	5	5	NUM
ejpam-3209	131	21	)	)	PUNCT
ejpam-3209	131	22	yields	yield	NOUN
ejpam-3209	131	23	(	(	PUNCT
ejpam-3209	131	24	‖	‖	PROPN
ejpam-3209	131	25	sin(θ)‖‖	sin(θ)‖‖	PROPN
ejpam-3209	131	26	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	131	27	)	)	PUNCT
ejpam-3209	131	28	0	0	X
ejpam-3209	131	29	≤	≤	NUM
ejpam-3209	131	30	(	(	PUNCT
ejpam-3209	131	31	‖	‖	PROPN
ejpam-3209	131	32	sin(θ)‖+	sin(θ)‖+	X
ejpam-3209	131	33	‖	‖	PROPN
ejpam-3209	131	34	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	131	35	)	)	PUNCT
ejpam-3209	131	36	0	0	PUNCT
ejpam-3209	132	1	(	(	PUNCT
ejpam-3209	132	2	‖	‖	PROPN
ejpam-3209	132	3	sin(θ)‖‖	sin(θ)‖‖	PROPN
ejpam-3209	132	4	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	132	5	)	)	PUNCT
ejpam-3209	132	6	0	0	PUNCT
ejpam-3209	133	1	=	=	SYM
ejpam-3209	133	2	(	(	PUNCT
ejpam-3209	133	3	1	1	NUM
ejpam-3209	133	4	+	+	CCONJ
ejpam-3209	133	5	1)0	1)0	NUM
ejpam-3209	133	6	(	(	PUNCT
ejpam-3209	133	7	‖	‖	PROPN
ejpam-3209	133	8	sin(θ)‖‖	sin(θ)‖‖	PROPN
ejpam-3209	133	9	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	133	10	)	)	PUNCT
ejpam-3209	133	11	0	0	PUNCT
ejpam-3209	134	1	=	=	SYM
ejpam-3209	134	2	20	20	NUM
ejpam-3209	134	3	for	for	ADP
ejpam-3209	134	4	n	n	NOUN
ejpam-3209	134	5	=	=	SYM
ejpam-3209	134	6	1	1	NUM
ejpam-3209	134	7	,	,	PUNCT
ejpam-3209	134	8	the	the	DET
ejpam-3209	134	9	following	following	ADJ
ejpam-3209	134	10	result	result	NOUN
ejpam-3209	134	11	is	be	AUX
ejpam-3209	134	12	obtained	obtain	VERB
ejpam-3209	134	13	:(	:(	PUNCT
ejpam-3209	134	14	sin(θ	sin(θ	PROPN
ejpam-3209	134	15	)	)	PUNCT
ejpam-3209	134	16	cos(θ	cos(θ	PROPN
ejpam-3209	134	17	)	)	PUNCT
ejpam-3209	134	18	)	)	PUNCT
ejpam-3209	134	19	1	1	NUM
ejpam-3209	134	20	⇒	⇒	NOUN
ejpam-3209	134	21	(	(	PUNCT
ejpam-3209	134	22	|〈sin(θ	|〈sin(θ	NOUN
ejpam-3209	134	23	)	)	PUNCT
ejpam-3209	134	24	,	,	PUNCT
ejpam-3209	134	25	cos(θ)〉|	cos(θ)〉|	PROPN
ejpam-3209	134	26	)	)	PUNCT
ejpam-3209	134	27	1	1	NUM
ejpam-3209	134	28	≤	≤	NUM
ejpam-3209	134	29	(	(	PUNCT
ejpam-3209	134	30	‖	‖	PROPN
ejpam-3209	134	31	sin(θ)‖‖	sin(θ)‖‖	PROPN
ejpam-3209	134	32	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	134	33	)	)	PUNCT
ejpam-3209	134	34	1	1	X
ejpam-3209	134	35	.	.	PUNCT
ejpam-3209	135	1	(	(	PUNCT
ejpam-3209	135	2	6	6	X
ejpam-3209	135	3	)	)	PUNCT
ejpam-3209	135	4	barnes	barne	NOUN
ejpam-3209	135	5	et	et	PROPN
ejpam-3209	135	6	al	al	PROPN
ejpam-3209	135	7	.	.	PUNCT
ejpam-3209	135	8	/	/	SYM
ejpam-3209	135	9	eur	eur	PROPN
ejpam-3209	135	10	.	.	PUNCT
ejpam-3209	136	1	j.	j.	PROPN
ejpam-3209	136	2	pure	pure	PROPN
ejpam-3209	136	3	appl	appl	PROPN
ejpam-3209	136	4	.	.	PROPN
ejpam-3209	136	5	math	math	PROPN
ejpam-3209	136	6	,	,	PUNCT
ejpam-3209	136	7	11	11	NUM
ejpam-3209	136	8	(	(	PUNCT
ejpam-3209	136	9	2	2	NUM
ejpam-3209	136	10	)	)	PUNCT
ejpam-3209	136	11	(	(	PUNCT
ejpam-3209	136	12	2018	2018	NUM
ejpam-3209	136	13	)	)	PUNCT
ejpam-3209	136	14	,	,	PUNCT
ejpam-3209	136	15	375	375	NUM
ejpam-3209	136	16	-	-	SYM
ejpam-3209	136	17	389	389	NUM
ejpam-3209	136	18	383	383	NUM
ejpam-3209	136	19	applying	apply	VERB
ejpam-3209	136	20	the	the	DET
ejpam-3209	136	21	first	first	ADJ
ejpam-3209	136	22	product	product	NOUN
ejpam-3209	136	23	inequality	inequality	NOUN
ejpam-3209	136	24	to	to	ADP
ejpam-3209	136	25	the	the	DET
ejpam-3209	136	26	expression	expression	NOUN
ejpam-3209	136	27	on	on	ADP
ejpam-3209	136	28	the	the	DET
ejpam-3209	136	29	right	right	ADJ
ejpam-3209	136	30	hand	hand	NOUN
ejpam-3209	136	31	side	side	NOUN
ejpam-3209	136	32	of	of	ADP
ejpam-3209	136	33	equation	equation	NOUN
ejpam-3209	136	34	(	(	PUNCT
ejpam-3209	136	35	6	6	NUM
ejpam-3209	136	36	)	)	PUNCT
ejpam-3209	136	37	yields	yield	NOUN
ejpam-3209	136	38	(	(	PUNCT
ejpam-3209	136	39	‖	‖	PROPN
ejpam-3209	136	40	sin(θ)‖‖	sin(θ)‖‖	PROPN
ejpam-3209	136	41	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	136	42	)	)	PUNCT
ejpam-3209	136	43	1	1	NUM
ejpam-3209	136	44	≤	≤	NUM
ejpam-3209	136	45	(	(	PUNCT
ejpam-3209	136	46	‖	‖	PROPN
ejpam-3209	136	47	sin(θ)‖+	sin(θ)‖+	X
ejpam-3209	136	48	‖	‖	PROPN
ejpam-3209	136	49	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	136	50	)	)	PUNCT
ejpam-3209	136	51	1	1	NUM
ejpam-3209	136	52	(	(	PUNCT
ejpam-3209	136	53	‖	‖	PROPN
ejpam-3209	136	54	sin(θ)‖‖	sin(θ)‖‖	PROPN
ejpam-3209	136	55	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	136	56	)	)	PUNCT
ejpam-3209	137	1	1	1	NUM
ejpam-3209	137	2	=	=	SYM
ejpam-3209	137	3	(	(	PUNCT
ejpam-3209	137	4	1	1	NUM
ejpam-3209	137	5	+	+	NUM
ejpam-3209	137	6	1)1	1)1	NUM
ejpam-3209	137	7	(	(	PUNCT
ejpam-3209	137	8	‖	‖	PROPN
ejpam-3209	137	9	sin(θ)‖‖	sin(θ)‖‖	ADJ
ejpam-3209	137	10	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	137	11	)	)	PUNCT
ejpam-3209	137	12	1	1	NUM
ejpam-3209	137	13	<	<	X
ejpam-3209	137	14	21	21	NUM
ejpam-3209	137	15	similarly	similarly	ADV
ejpam-3209	137	16	,	,	PUNCT
ejpam-3209	137	17	we	we	PRON
ejpam-3209	137	18	observed	observe	VERB
ejpam-3209	137	19	that	that	SCONJ
ejpam-3209	137	20	when	when	SCONJ
ejpam-3209	137	21	n	n	X
ejpam-3209	137	22	=	=	SYM
ejpam-3209	137	23	2	2	NUM
ejpam-3209	137	24	,	,	PUNCT
ejpam-3209	137	25	the	the	DET
ejpam-3209	137	26	following	follow	VERB
ejpam-3209	137	27	expression	expression	NOUN
ejpam-3209	137	28	is	be	AUX
ejpam-3209	137	29	obtained	obtain	VERB
ejpam-3209	137	30	:(	:(	PUNCT
ejpam-3209	137	31	sin(θ	sin(θ	PROPN
ejpam-3209	137	32	)	)	PUNCT
ejpam-3209	138	1	cos(θ	cos(θ	PROPN
ejpam-3209	138	2	)	)	PUNCT
ejpam-3209	138	3	)	)	PUNCT
ejpam-3209	138	4	2	2	NUM
ejpam-3209	138	5	⇒	⇒	NOUN
ejpam-3209	138	6	(	(	PUNCT
ejpam-3209	138	7	|〈sin(θ	|〈sin(θ	NOUN
ejpam-3209	138	8	)	)	PUNCT
ejpam-3209	138	9	,	,	PUNCT
ejpam-3209	138	10	cos(θ)〉|	cos(θ)〉|	NOUN
ejpam-3209	138	11	)	)	PUNCT
ejpam-3209	138	12	2	2	NUM
ejpam-3209	138	13	≤	≤	NOUN
ejpam-3209	138	14	(	(	PUNCT
ejpam-3209	138	15	‖	‖	PROPN
ejpam-3209	138	16	sin(θ)‖‖	sin(θ)‖‖	ADJ
ejpam-3209	138	17	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	138	18	)	)	PUNCT
ejpam-3209	138	19	2	2	NUM
ejpam-3209	138	20	(	(	PUNCT
ejpam-3209	138	21	7	7	X
ejpam-3209	138	22	)	)	PUNCT
ejpam-3209	138	23	applying	apply	VERB
ejpam-3209	138	24	the	the	DET
ejpam-3209	138	25	first	first	ADJ
ejpam-3209	138	26	product	product	NOUN
ejpam-3209	138	27	inequality	inequality	NOUN
ejpam-3209	138	28	to	to	ADP
ejpam-3209	138	29	the	the	DET
ejpam-3209	138	30	expression	expression	NOUN
ejpam-3209	138	31	on	on	ADP
ejpam-3209	138	32	the	the	DET
ejpam-3209	138	33	right	right	ADJ
ejpam-3209	138	34	hand	hand	NOUN
ejpam-3209	138	35	side	side	NOUN
ejpam-3209	138	36	of	of	ADP
ejpam-3209	138	37	equation	equation	NOUN
ejpam-3209	138	38	(	(	PUNCT
ejpam-3209	138	39	7	7	X
ejpam-3209	138	40	)	)	PUNCT
ejpam-3209	138	41	yields	yield	NOUN
ejpam-3209	138	42	(	(	PUNCT
ejpam-3209	138	43	‖	‖	PROPN
ejpam-3209	138	44	sin(θ)‖‖	sin(θ)‖‖	PROPN
ejpam-3209	138	45	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	138	46	)	)	PUNCT
ejpam-3209	138	47	2	2	NUM
ejpam-3209	138	48	≤	≤	NOUN
ejpam-3209	138	49	(	(	PUNCT
ejpam-3209	138	50	‖	‖	PROPN
ejpam-3209	138	51	sin(θ)‖+	sin(θ)‖+	X
ejpam-3209	138	52	‖	‖	PROPN
ejpam-3209	138	53	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	138	54	)	)	PUNCT
ejpam-3209	138	55	2	2	NUM
ejpam-3209	138	56	(	(	PUNCT
ejpam-3209	138	57	‖	‖	PROPN
ejpam-3209	138	58	sin(θ)‖‖	sin(θ)‖‖	PROPN
ejpam-3209	138	59	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	138	60	)	)	PUNCT
ejpam-3209	138	61	2	2	NUM
ejpam-3209	138	62	=	=	SYM
ejpam-3209	138	63	(	(	PUNCT
ejpam-3209	138	64	1	1	NUM
ejpam-3209	138	65	+	+	SYM
ejpam-3209	138	66	1)2	1)2	NUM
ejpam-3209	138	67	(	(	PUNCT
ejpam-3209	138	68	‖	‖	PROPN
ejpam-3209	138	69	sin(θ)‖‖	sin(θ)‖‖	PROPN
ejpam-3209	138	70	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	138	71	)	)	PUNCT
ejpam-3209	138	72	2	2	NUM
ejpam-3209	138	73	<	<	SYM
ejpam-3209	138	74	22	22	NUM
ejpam-3209	138	75	generalizing	generalize	VERB
ejpam-3209	138	76	the	the	DET
ejpam-3209	138	77	product	product	NOUN
ejpam-3209	138	78	inequality	inequality	NOUN
ejpam-3209	138	79	of	of	ADP
ejpam-3209	138	80	sin(θ	sin(θ	PROPN
ejpam-3209	138	81	)	)	PUNCT
ejpam-3209	138	82	and	and	CCONJ
ejpam-3209	138	83	cos(θ	cos(θ	PROPN
ejpam-3209	138	84	)	)	PUNCT
ejpam-3209	139	1	,	,	PUNCT
ejpam-3209	139	2	we	we	PRON
ejpam-3209	139	3	observed	observe	VERB
ejpam-3209	139	4	that	that	SCONJ
ejpam-3209	139	5	for	for	ADP
ejpam-3209	139	6	any	any	DET
ejpam-3209	139	7	integer	integer	NOUN
ejpam-3209	139	8	n	n	PRON
ejpam-3209	139	9	∈	∈	PROPN
ejpam-3209	140	1	[	[	X
ejpam-3209	140	2	0,∞	0,∞	NOUN
ejpam-3209	140	3	)	)	PUNCT
ejpam-3209	140	4	,	,	PUNCT
ejpam-3209	140	5	we	we	PRON
ejpam-3209	140	6	obtain	obtain	VERB
ejpam-3209	140	7	:(	:(	PUNCT
ejpam-3209	140	8	sin(θ	sin(θ	PROPN
ejpam-3209	140	9	)	)	PUNCT
ejpam-3209	140	10	cos(θ	cos(θ	PROPN
ejpam-3209	140	11	)	)	PUNCT
ejpam-3209	140	12	)	)	PUNCT
ejpam-3209	141	1	n	n	CCONJ
ejpam-3209	141	2	⇒	⇒	NOUN
ejpam-3209	141	3	(	(	PUNCT
ejpam-3209	141	4	|〈sin(θ	|〈sin(θ	NOUN
ejpam-3209	141	5	)	)	PUNCT
ejpam-3209	141	6	,	,	PUNCT
ejpam-3209	141	7	cos(θ)〉|	cos(θ)〉|	NOUN
ejpam-3209	141	8	)	)	PUNCT
ejpam-3209	142	1	n	n	NOUN
ejpam-3209	142	2	≤	≤	NOUN
ejpam-3209	142	3	(	(	PUNCT
ejpam-3209	142	4	‖	‖	PROPN
ejpam-3209	142	5	sin(θ)‖‖	sin(θ)‖‖	ADJ
ejpam-3209	142	6	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	142	7	)	)	PUNCT
ejpam-3209	143	1	n	n	X
ejpam-3209	143	2	.	.	PUNCT
ejpam-3209	144	1	(	(	PUNCT
ejpam-3209	144	2	8)	8)	NUM
ejpam-3209	144	3	applying	apply	VERB
ejpam-3209	144	4	the	the	DET
ejpam-3209	144	5	first	first	ADJ
ejpam-3209	144	6	product	product	NOUN
ejpam-3209	144	7	inequality	inequality	NOUN
ejpam-3209	144	8	to	to	ADP
ejpam-3209	144	9	the	the	DET
ejpam-3209	144	10	expression	expression	NOUN
ejpam-3209	144	11	on	on	ADP
ejpam-3209	144	12	the	the	DET
ejpam-3209	144	13	right	right	ADJ
ejpam-3209	144	14	hand	hand	NOUN
ejpam-3209	144	15	side	side	NOUN
ejpam-3209	144	16	of	of	ADP
ejpam-3209	144	17	equation	equation	NOUN
ejpam-3209	144	18	(	(	PUNCT
ejpam-3209	144	19	6	6	NUM
ejpam-3209	144	20	)	)	PUNCT
ejpam-3209	144	21	,	,	PUNCT
ejpam-3209	144	22	we	we	PRON
ejpam-3209	144	23	obtain	obtain	VERB
ejpam-3209	144	24	:(	:(	PUNCT
ejpam-3209	144	25	‖	‖	PROPN
ejpam-3209	144	26	sin(θ)‖‖	sin(θ)‖‖	ADJ
ejpam-3209	144	27	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	144	28	)	)	PUNCT
ejpam-3209	144	29	n	n	X
ejpam-3209	144	30	≤	≤	NOUN
ejpam-3209	144	31	(	(	PUNCT
ejpam-3209	144	32	‖	‖	PROPN
ejpam-3209	144	33	sin(θ)‖+	sin(θ)‖+	X
ejpam-3209	144	34	‖	‖	PROPN
ejpam-3209	144	35	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	144	36	)	)	PUNCT
ejpam-3209	145	1	n	n	CCONJ
ejpam-3209	145	2	(	(	PUNCT
ejpam-3209	145	3	‖	‖	PROPN
ejpam-3209	145	4	sin(θ)‖‖	sin(θ)‖‖	PROPN
ejpam-3209	145	5	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	145	6	)	)	PUNCT
ejpam-3209	146	1	n	n	X
ejpam-3209	146	2	=	=	SYM
ejpam-3209	146	3	(	(	PUNCT
ejpam-3209	146	4	1	1	NUM
ejpam-3209	146	5	+	+	SYM
ejpam-3209	146	6	1)n	1)n	NUM
ejpam-3209	146	7	(	(	PUNCT
ejpam-3209	146	8	‖	‖	PROPN
ejpam-3209	146	9	sin(θ)‖‖	sin(θ)‖‖	ADJ
ejpam-3209	146	10	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	146	11	)	)	PUNCT
ejpam-3209	147	1	n	n	X
ejpam-3209	147	2	≤	≤	NUM
ejpam-3209	147	3	2n	2n	NUM
ejpam-3209	147	4	.	.	PUNCT
ejpam-3209	148	1	(	(	PUNCT
ejpam-3209	148	2	9	9	NUM
ejpam-3209	148	3	)	)	PUNCT
ejpam-3209	148	4	again	again	ADV
ejpam-3209	148	5	,	,	PUNCT
ejpam-3209	148	6	the	the	DET
ejpam-3209	148	7	first	first	ADJ
ejpam-3209	148	8	product	product	NOUN
ejpam-3209	148	9	inequality	inequality	NOUN
ejpam-3209	148	10	is	be	AUX
ejpam-3209	148	11	applied	apply	VERB
ejpam-3209	148	12	to	to	ADP
ejpam-3209	148	13	functions	function	NOUN
ejpam-3209	148	14	which	which	PRON
ejpam-3209	148	15	give	give	VERB
ejpam-3209	148	16	the	the	DET
ejpam-3209	148	17	following	follow	VERB
ejpam-3209	148	18	results	result	NOUN
ejpam-3209	148	19	.	.	PUNCT
ejpam-3209	149	1	theorem	theorem	NOUN
ejpam-3209	149	2	1	1	NUM
ejpam-3209	149	3	.	.	PUNCT
ejpam-3209	149	4	suppose	suppose	VERB
ejpam-3209	149	5	that	that	SCONJ
ejpam-3209	149	6	f	f	PROPN
ejpam-3209	149	7	be	be	AUX
ejpam-3209	149	8	the	the	DET
ejpam-3209	149	9	space	space	NOUN
ejpam-3209	149	10	of	of	ADP
ejpam-3209	149	11	all	all	DET
ejpam-3209	149	12	measurable	measurable	ADJ
ejpam-3209	149	13	functions	function	NOUN
ejpam-3209	149	14	f	f	NOUN
ejpam-3209	149	15	:	:	PUNCT
ejpam-3209	150	1	[	[	X
ejpam-3209	150	2	0	0	NUM
ejpam-3209	150	3	,	,	PUNCT
ejpam-3209	150	4	2	2	NUM
ejpam-3209	150	5	]	]	PUNCT
ejpam-3209	150	6	→	→	SYM
ejpam-3209	150	7	r	r	NOUN
ejpam-3209	150	8	,	,	PUNCT
ejpam-3209	150	9	and	and	CCONJ
ejpam-3209	150	10	g	g	NOUN
ejpam-3209	150	11	:	:	PUNCT
ejpam-3209	150	12	ω→	ω→	PUNCT
ejpam-3209	151	1	[	[	X
ejpam-3209	151	2	0	0	NUM
ejpam-3209	151	3	,	,	PUNCT
ejpam-3209	151	4	1	1	NUM
ejpam-3209	151	5	]	]	PUNCT
ejpam-3209	151	6	,	,	PUNCT
ejpam-3209	151	7	then	then	ADV
ejpam-3209	151	8	following	follow	VERB
ejpam-3209	151	9	inequalities	inequality	NOUN
ejpam-3209	151	10	hold	hold	VERB
ejpam-3209	151	11	:	:	PUNCT
ejpam-3209	151	12	‖ex‖‖	‖ex‖‖	NOUN
ejpam-3209	151	13	sin(θ)‖	sin(θ)‖	NOUN
ejpam-3209	151	14	≤	≤	ADJ
ejpam-3209	151	15	‖ex‖+	‖ex‖+	PROPN
ejpam-3209	151	16	‖	‖	PROPN
ejpam-3209	151	17	sin(θ)‖	sin(θ)‖	NOUN
ejpam-3209	151	18	barnes	barne	VERB
ejpam-3209	151	19	et	et	PROPN
ejpam-3209	151	20	al	al	PROPN
ejpam-3209	151	21	.	.	PUNCT
ejpam-3209	151	22	/	/	SYM
ejpam-3209	151	23	eur	eur	PROPN
ejpam-3209	151	24	.	.	PUNCT
ejpam-3209	152	1	j.	j.	PROPN
ejpam-3209	152	2	pure	pure	PROPN
ejpam-3209	152	3	appl	appl	PROPN
ejpam-3209	152	4	.	.	PROPN
ejpam-3209	152	5	math	math	PROPN
ejpam-3209	152	6	,	,	PUNCT
ejpam-3209	152	7	11	11	NUM
ejpam-3209	152	8	(	(	PUNCT
ejpam-3209	152	9	2	2	NUM
ejpam-3209	152	10	)	)	PUNCT
ejpam-3209	152	11	(	(	PUNCT
ejpam-3209	152	12	2018	2018	NUM
ejpam-3209	152	13	)	)	PUNCT
ejpam-3209	152	14	,	,	PUNCT
ejpam-3209	152	15	375	375	NUM
ejpam-3209	152	16	-	-	SYM
ejpam-3209	152	17	389	389	NUM
ejpam-3209	152	18	384	384	NUM
ejpam-3209	152	19	‖ex‖‖	‖ex‖‖	NOUN
ejpam-3209	152	20	cos(θ)‖	cos(θ)‖	PART
ejpam-3209	152	21	≤	≤	NUM
ejpam-3209	153	1	‖ex‖+	‖ex‖+	PROPN
ejpam-3209	153	2	‖	‖	PROPN
ejpam-3209	153	3	cos(θ)‖	cos(θ)‖	X
ejpam-3209	154	1	‖	‖	PROPN
ejpam-3209	154	2	tan(θ)‖‖	tan(θ)‖‖	PROPN
ejpam-3209	154	3	sin(θ)‖	sin(θ)‖	NOUN
ejpam-3209	155	1	≤	≤	PROPN
ejpam-3209	155	2	‖	‖	NUM
ejpam-3209	155	3	tan(θ)‖+	tan(θ)‖+	X
ejpam-3209	155	4	‖	‖	PROPN
ejpam-3209	155	5	sin(θ)‖	sin(θ)‖	NOUN
ejpam-3209	155	6	‖	‖	ADJ
ejpam-3209	155	7	tan(θ)‖‖	tan(θ)‖‖	PROPN
ejpam-3209	155	8	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	155	9	≤	≤	NOUN
ejpam-3209	155	10	‖	‖	PROPN
ejpam-3209	155	11	tan(θ)‖+	tan(θ)‖+	X
ejpam-3209	155	12	‖	‖	ADJ
ejpam-3209	155	13	cos(θ)‖	cos(θ)‖	X
ejpam-3209	155	14	‖sec(θ)‖‖	‖sec(θ)‖‖	ADJ
ejpam-3209	155	15	sin(θ)‖	sin(θ)‖	NOUN
ejpam-3209	155	16	≤	≤	NOUN
ejpam-3209	155	17	‖sec(θ)‖+	‖sec(θ)‖+	PUNCT
ejpam-3209	155	18	‖	‖	PROPN
ejpam-3209	155	19	sin(θ)‖	sin(θ)‖	NOUN
ejpam-3209	155	20	‖cosec(θ)‖‖	‖cosec(θ)‖‖	NOUN
ejpam-3209	155	21	cos(θ)‖	cos(θ)‖	X
ejpam-3209	155	22	≤	≤	NUM
ejpam-3209	155	23	‖cosec(θ)‖+	‖cosec(θ)‖+	NUM
ejpam-3209	155	24	‖	‖	ADJ
ejpam-3209	155	25	cos(θ)‖	cos(θ)‖	X
ejpam-3209	155	26	‖cot(θ)‖‖	‖cot(θ)‖‖	ADJ
ejpam-3209	155	27	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	155	28	≤	≤	NOUN
ejpam-3209	155	29	‖cot(θ)‖+	‖cot(θ)‖+	X
ejpam-3209	155	30	‖	‖	X
ejpam-3209	155	31	cos(θ)‖	cos(θ)‖	NOUN
ejpam-3209	155	32	we	we	PRON
ejpam-3209	155	33	observed	observe	VERB
ejpam-3209	155	34	that	that	SCONJ
ejpam-3209	155	35	the	the	DET
ejpam-3209	155	36	norms	norm	NOUN
ejpam-3209	155	37	of	of	ADP
ejpam-3209	155	38	the	the	DET
ejpam-3209	155	39	trigonometric	trigonometric	ADJ
ejpam-3209	155	40	functions	function	NOUN
ejpam-3209	155	41	:	:	PUNCT
ejpam-3209	155	42	sin(θ	sin(θ	PROPN
ejpam-3209	155	43	)	)	PUNCT
ejpam-3209	155	44	and	and	CCONJ
ejpam-3209	155	45	cos(θ	cos(θ	PROPN
ejpam-3209	155	46	)	)	PUNCT
ejpam-3209	155	47	are	be	AUX
ejpam-3209	155	48	seen	see	VERB
ejpam-3209	155	49	as	as	ADP
ejpam-3209	155	50	delayed	delay	VERB
ejpam-3209	155	51	growth	growth	NOUN
ejpam-3209	155	52	norms	norm	NOUN
ejpam-3209	155	53	.	.	PUNCT
ejpam-3209	156	1	definition	definition	NOUN
ejpam-3209	156	2	6	6	NUM
ejpam-3209	156	3	.	.	PUNCT
ejpam-3209	157	1	if	if	SCONJ
ejpam-3209	157	2	u	u	PROPN
ejpam-3209	157	3	and	and	CCONJ
ejpam-3209	157	4	v	v	NOUN
ejpam-3209	157	5	are	be	AUX
ejpam-3209	157	6	any	any	DET
ejpam-3209	157	7	two	two	NUM
ejpam-3209	157	8	vectors	vector	NOUN
ejpam-3209	157	9	in	in	ADP
ejpam-3209	157	10	the	the	DET
ejpam-3209	157	11	euclidean	euclidean	ADJ
ejpam-3209	157	12	space	space	NOUN
ejpam-3209	157	13	,	,	PUNCT
ejpam-3209	157	14	then	then	ADV
ejpam-3209	157	15	|u+	|u+	NOUN
ejpam-3209	157	16	v|	v|	ADV
ejpam-3209	157	17	≤	≤	NUM
ejpam-3209	157	18	‖u‖+	‖u‖+	PRON
ejpam-3209	157	19	‖v‖	‖v‖	PROPN
ejpam-3209	157	20	≤	≤	NUM
ejpam-3209	157	21	2	2	NUM
ejpam-3209	157	22	max{|u|	max{|u|	NOUN
ejpam-3209	157	23	,	,	PUNCT
ejpam-3209	157	24	|v|	|v|	VERB
ejpam-3209	157	25	}	}	PUNCT
ejpam-3209	157	26	⇒	⇒	NOUN
ejpam-3209	157	27	|u+	|u+	PROPN
ejpam-3209	157	28	v|p	v|p	PROPN
ejpam-3209	157	29	≤	≤	NUM
ejpam-3209	157	30	2p{|u|p	2p{|u|p	NUM
ejpam-3209	157	31	+	+	CCONJ
ejpam-3209	157	32	|v|p	|v|p	PROPN
ejpam-3209	157	33	}	}	PUNCT
ejpam-3209	157	34	,	,	PUNCT
ejpam-3209	157	35	see	see	VERB
ejpam-3209	157	36	[	[	X
ejpam-3209	157	37	17	17	NUM
ejpam-3209	157	38	]	]	PUNCT
ejpam-3209	157	39	.	.	PUNCT
ejpam-3209	158	1	with	with	ADP
ejpam-3209	158	2	the	the	DET
ejpam-3209	158	3	above	above	ADJ
ejpam-3209	158	4	inequalities	inequality	NOUN
ejpam-3209	158	5	,	,	PUNCT
ejpam-3209	158	6	the	the	DET
ejpam-3209	158	7	first	first	ADJ
ejpam-3209	158	8	product	product	NOUN
ejpam-3209	158	9	inequality	inequality	NOUN
ejpam-3209	158	10	is	be	AUX
ejpam-3209	158	11	extended	extend	VERB
ejpam-3209	158	12	to	to	ADP
ejpam-3209	158	13	p−normed	p−norme	VERB
ejpam-3209	158	14	spaces	space	NOUN
ejpam-3209	158	15	as	as	SCONJ
ejpam-3209	158	16	follows	follow	VERB
ejpam-3209	158	17	:	:	PUNCT
ejpam-3209	158	18	theorem	theorem	NOUN
ejpam-3209	158	19	2	2	NUM
ejpam-3209	158	20	.	.	PUNCT
ejpam-3209	159	1	if	if	SCONJ
ejpam-3209	159	2	p	p	NOUN
ejpam-3209	159	3	is	be	AUX
ejpam-3209	159	4	any	any	DET
ejpam-3209	159	5	positive	positive	ADJ
ejpam-3209	159	6	integer	integer	NOUN
ejpam-3209	159	7	,	,	PUNCT
ejpam-3209	159	8	then	then	ADV
ejpam-3209	159	9	‖u‖p‖v‖p	‖u‖p‖v‖p	VERB
ejpam-3209	159	10	≤	≤	NUM
ejpam-3209	159	11	2p{‖u‖p	2p{‖u‖p	NUM
ejpam-3209	159	12	+	+	CCONJ
ejpam-3209	159	13	‖v‖p	‖v‖p	NOUN
ejpam-3209	159	14	}	}	PUNCT
ejpam-3209	159	15	.	.	PUNCT
ejpam-3209	160	1	(	(	PUNCT
ejpam-3209	160	2	10	10	NUM
ejpam-3209	160	3	)	)	PUNCT
ejpam-3209	160	4	proof	proof	NOUN
ejpam-3209	160	5	:	:	PUNCT
ejpam-3209	160	6	we	we	PRON
ejpam-3209	160	7	see	see	VERB
ejpam-3209	160	8	from	from	ADP
ejpam-3209	160	9	the	the	DET
ejpam-3209	160	10	above	above	ADJ
ejpam-3209	160	11	inequality	inequality	NOUN
ejpam-3209	160	12	that	that	SCONJ
ejpam-3209	160	13	:	:	PUNCT
ejpam-3209	160	14	|u+	|u+	NOUN
ejpam-3209	160	15	v|	v|	ADV
ejpam-3209	160	16	≤	≤	NOUN
ejpam-3209	160	17	‖u‖+	‖u‖+	PRON
ejpam-3209	160	18	‖v‖	‖v‖	PROPN
ejpam-3209	160	19	raising	raise	VERB
ejpam-3209	160	20	the	the	DET
ejpam-3209	160	21	expression	expression	NOUN
ejpam-3209	160	22	on	on	ADP
ejpam-3209	160	23	the	the	DET
ejpam-3209	160	24	both	both	DET
ejpam-3209	160	25	sides	side	NOUN
ejpam-3209	160	26	of	of	ADP
ejpam-3209	160	27	inequality	inequality	NOUN
ejpam-3209	160	28	to	to	ADP
ejpam-3209	160	29	the	the	DET
ejpam-3209	160	30	power	power	NOUN
ejpam-3209	160	31	p	p	NOUN
ejpam-3209	160	32	,	,	PUNCT
ejpam-3209	160	33	we	we	PRON
ejpam-3209	160	34	get	get	VERB
ejpam-3209	160	35	:	:	PUNCT
ejpam-3209	160	36	|u+	|u+	X
ejpam-3209	160	37	v|p	v|p	X
ejpam-3209	160	38	≤	≤	NOUN
ejpam-3209	160	39	∣∣∣‖u‖+	∣∣∣‖u‖+	PUNCT
ejpam-3209	160	40	‖v‖	‖v‖	PROPN
ejpam-3209	160	41	∣∣∣p	∣∣∣p	NOUN
ejpam-3209	160	42	⇒	⇒	NOUN
ejpam-3209	160	43	|u+	|u+	PROPN
ejpam-3209	160	44	v|p	v|p	ADJ
ejpam-3209	160	45	≤	≤	NUM
ejpam-3209	160	46	‖u‖p	‖u‖p	NOUN
ejpam-3209	160	47	+	+	CCONJ
ejpam-3209	160	48	‖v‖p	‖v‖p	ADJ
ejpam-3209	160	49	⇒	⇒	NOUN
ejpam-3209	160	50	|u+	|u+	PROPN
ejpam-3209	160	51	v|p	v|p	PROPN
ejpam-3209	161	1	≤	≤	NUM
ejpam-3209	162	1	2p{‖u‖p	2p{‖u‖p	NUM
ejpam-3209	162	2	+	+	CCONJ
ejpam-3209	162	3	‖v‖p	‖v‖p	PROPN
ejpam-3209	162	4	}	}	PUNCT
ejpam-3209	162	5	⇒	⇒	NOUN
ejpam-3209	162	6	‖u‖p	‖u‖p	NOUN
ejpam-3209	162	7	+	+	CCONJ
ejpam-3209	162	8	‖v‖p	‖v‖p	VERB
ejpam-3209	162	9	≤	≤	NOUN
ejpam-3209	162	10	2p{‖u‖p	2p{‖u‖p	NUM
ejpam-3209	162	11	+	+	CCONJ
ejpam-3209	162	12	‖v‖p	‖v‖p	NOUN
ejpam-3209	162	13	}	}	PUNCT
ejpam-3209	162	14	.	.	PUNCT
ejpam-3209	163	1	substituting	substitute	VERB
ejpam-3209	163	2	inequality	inequality	NOUN
ejpam-3209	163	3	in	in	ADP
ejpam-3209	163	4	(	(	PUNCT
ejpam-3209	163	5	3	3	NUM
ejpam-3209	163	6	)	)	PUNCT
ejpam-3209	163	7	into	into	ADP
ejpam-3209	163	8	the	the	DET
ejpam-3209	163	9	inequality	inequality	NOUN
ejpam-3209	163	10	in	in	ADP
ejpam-3209	163	11	(	(	PUNCT
ejpam-3209	163	12	10	10	NUM
ejpam-3209	163	13	)	)	PUNCT
ejpam-3209	163	14	yields	yield	NOUN
ejpam-3209	163	15	|‖u‖‖v‖|p	|‖u‖‖v‖|p	PROPN
ejpam-3209	163	16	≤	≤	NUM
ejpam-3209	164	1	2p{|u|p	2p{|u|p	NUM
ejpam-3209	164	2	+	+	CCONJ
ejpam-3209	164	3	|v|p	|v|p	PROPN
ejpam-3209	164	4	}	}	PUNCT
ejpam-3209	164	5	‖u‖p‖v‖p	‖u‖p‖v‖p	VERB
ejpam-3209	164	6	≤	≤	NUM
ejpam-3209	164	7	2p{‖u‖p	2p{‖u‖p	NUM
ejpam-3209	164	8	+	+	CCONJ
ejpam-3209	164	9	‖v‖p	‖v‖p	PROPN
ejpam-3209	164	10	}	}	PUNCT
ejpam-3209	164	11	,	,	PUNCT
ejpam-3209	164	12	∀	∀	X
ejpam-3209	164	13	u	u	NOUN
ejpam-3209	164	14	,	,	PUNCT
ejpam-3209	164	15	v	v	NOUN
ejpam-3209	164	16	∈	∈	PROPN
ejpam-3209	165	1	[	[	X
ejpam-3209	165	2	0	0	NUM
ejpam-3209	165	3	,	,	PUNCT
ejpam-3209	165	4	2	2	NUM
ejpam-3209	165	5	]	]	PUNCT
ejpam-3209	165	6	theorem	theorem	NOUN
ejpam-3209	165	7	3	3	NUM
ejpam-3209	165	8	.	.	PUNCT
ejpam-3209	166	1	if	if	SCONJ
ejpam-3209	166	2	u	u	PROPN
ejpam-3209	166	3	and	and	CCONJ
ejpam-3209	166	4	v	v	NOUN
ejpam-3209	166	5	are	be	AUX
ejpam-3209	166	6	any	any	DET
ejpam-3209	166	7	two	two	NUM
ejpam-3209	166	8	vectors	vector	NOUN
ejpam-3209	166	9	in	in	ADP
ejpam-3209	166	10	the	the	DET
ejpam-3209	166	11	euclidean	euclidean	ADJ
ejpam-3209	166	12	space	space	NOUN
ejpam-3209	166	13	,	,	PUNCT
ejpam-3209	166	14	then	then	ADV
ejpam-3209	166	15	‖u‖p‖v‖p	‖u‖p‖v‖p	VERB
ejpam-3209	166	16	≤	≤	ADJ
ejpam-3209	166	17	‖u‖p	‖u‖p	NOUN
ejpam-3209	166	18	+	+	CCONJ
ejpam-3209	166	19	‖v‖p	‖v‖p	NOUN
ejpam-3209	166	20	,	,	PUNCT
ejpam-3209	166	21	∀	∀	PUNCT
ejpam-3209	166	22	‖u‖	‖u‖	PROPN
ejpam-3209	166	23	,	,	PUNCT
ejpam-3209	166	24	‖v‖	‖v‖	PROPN
ejpam-3209	166	25	∈	∈	PROPN
ejpam-3209	167	1	[	[	X
ejpam-3209	167	2	0	0	NUM
ejpam-3209	167	3	,	,	PUNCT
ejpam-3209	167	4	2	2	NUM
ejpam-3209	167	5	]	]	PUNCT
ejpam-3209	167	6	.	.	PUNCT
ejpam-3209	168	1	p	p	NOUN
ejpam-3209	168	2	roof	roof	NOUN
ejpam-3209	168	3	:	:	PUNCT
ejpam-3209	168	4	the	the	DET
ejpam-3209	168	5	result	result	NOUN
ejpam-3209	168	6	in	in	ADP
ejpam-3209	168	7	theorem	theorem	NOUN
ejpam-3209	168	8	(	(	PUNCT
ejpam-3209	168	9	3	3	NUM
ejpam-3209	168	10	)	)	PUNCT
ejpam-3209	168	11	follows	follow	VERB
ejpam-3209	168	12	from	from	ADP
ejpam-3209	168	13	theorem	theorem	ADJ
ejpam-3209	168	14	(	(	PUNCT
ejpam-3209	168	15	2	2	NUM
ejpam-3209	168	16	)	)	PUNCT
ejpam-3209	168	17	.	.	PUNCT
ejpam-3209	169	1	barnes	barnes	PROPN
ejpam-3209	169	2	et	et	PROPN
ejpam-3209	169	3	al	al	PROPN
ejpam-3209	169	4	.	.	PUNCT
ejpam-3209	169	5	/	/	SYM
ejpam-3209	169	6	eur	eur	PROPN
ejpam-3209	169	7	.	.	PUNCT
ejpam-3209	170	1	j.	j.	PROPN
ejpam-3209	170	2	pure	pure	PROPN
ejpam-3209	170	3	appl	appl	PROPN
ejpam-3209	170	4	.	.	PROPN
ejpam-3209	170	5	math	math	PROPN
ejpam-3209	170	6	,	,	PUNCT
ejpam-3209	170	7	11	11	NUM
ejpam-3209	170	8	(	(	PUNCT
ejpam-3209	170	9	2	2	NUM
ejpam-3209	170	10	)	)	PUNCT
ejpam-3209	170	11	(	(	PUNCT
ejpam-3209	170	12	2018	2018	NUM
ejpam-3209	170	13	)	)	PUNCT
ejpam-3209	170	14	,	,	PUNCT
ejpam-3209	170	15	375	375	NUM
ejpam-3209	170	16	-	-	SYM
ejpam-3209	170	17	389	389	NUM
ejpam-3209	170	18	385	385	NUM
ejpam-3209	170	19	theorem	theorem	NOUN
ejpam-3209	170	20	4	4	NUM
ejpam-3209	170	21	.	.	PUNCT
ejpam-3209	170	22	suppose	suppose	VERB
ejpam-3209	170	23	that	that	SCONJ
ejpam-3209	170	24	at	at	ADV
ejpam-3209	170	25	least	least	ADJ
ejpam-3209	170	26	one	one	NUM
ejpam-3209	170	27	of	of	ADP
ejpam-3209	170	28	the	the	DET
ejpam-3209	170	29	measurable	measurable	ADJ
ejpam-3209	170	30	functions	function	NOUN
ejpam-3209	170	31	over	over	ADP
ejpam-3209	170	32	the	the	DET
ejpam-3209	170	33	domain	domain	NOUN
ejpam-3209	170	34	ω	ω	NOUN
ejpam-3209	170	35	is	be	AUX
ejpam-3209	170	36	such	such	ADJ
ejpam-3209	170	37	that	that	SCONJ
ejpam-3209	170	38	f	f	X
ejpam-3209	170	39	:	:	PUNCT
ejpam-3209	170	40	ω→	ω→	PUNCT
ejpam-3209	171	1	[	[	X
ejpam-3209	171	2	0	0	NUM
ejpam-3209	171	3	,	,	PUNCT
ejpam-3209	171	4	1	1	NUM
ejpam-3209	171	5	]	]	PUNCT
ejpam-3209	171	6	and	and	CCONJ
ejpam-3209	171	7	the	the	DET
ejpam-3209	171	8	other	other	ADJ
ejpam-3209	171	9	measurable	measurable	ADJ
ejpam-3209	171	10	function	function	NOUN
ejpam-3209	171	11	g	g	NOUN
ejpam-3209	171	12	:	:	PUNCT
ejpam-3209	171	13	ω→	ω→	PUNCT
ejpam-3209	171	14	r	r	NOUN
ejpam-3209	171	15	,	,	PUNCT
ejpam-3209	171	16	then	then	ADV
ejpam-3209	171	17	‖fg‖p	‖fg‖p	PROPN
ejpam-3209	171	18	≤	≤	ADJ
ejpam-3209	171	19	‖f‖p	‖f‖p	NOUN
ejpam-3209	171	20	+	+	CCONJ
ejpam-3209	171	21	‖g‖p	‖g‖p	NOUN
ejpam-3209	171	22	.	.	PUNCT
ejpam-3209	172	1	p	p	NOUN
ejpam-3209	172	2	roof	roof	NOUN
ejpam-3209	172	3	:	:	PUNCT
ejpam-3209	172	4	∣∣∣	∣∣∣	PROPN
ejpam-3209	172	5	∫	∫	PROPN
ejpam-3209	172	6	ω	ω	PROPN
ejpam-3209	172	7	f(x)g(x)dx	f(x)g(x)dx	PROPN
ejpam-3209	172	8	∣∣∣p	∣∣∣p	NOUN
ejpam-3209	172	9	≤	≤	NUM
ejpam-3209	172	10	∫	∫	PROPN
ejpam-3209	172	11	ω	ω	NUM
ejpam-3209	172	12	∣∣∣f(x)g(x	∣∣∣f(x)g(x	NOUN
ejpam-3209	172	13	)	)	PUNCT
ejpam-3209	172	14	∣∣∣pdx	∣∣∣pdx	PROPN
ejpam-3209	172	15	applying	apply	VERB
ejpam-3209	172	16	the	the	DET
ejpam-3209	172	17	first	first	ADJ
ejpam-3209	172	18	product	product	NOUN
ejpam-3209	172	19	inequality	inequality	NOUN
ejpam-3209	172	20	,	,	PUNCT
ejpam-3209	172	21	we	we	PRON
ejpam-3209	172	22	obtain∣∣∣	obtain∣∣∣	VERB
ejpam-3209	172	23	∫	∫	PROPN
ejpam-3209	172	24	ω	ω	PROPN
ejpam-3209	172	25	f(x)g(x)dx	f(x)g(x)dx	PROPN
ejpam-3209	172	26	∣∣∣p	∣∣∣p	NOUN
ejpam-3209	172	27	≤	≤	NUM
ejpam-3209	172	28	∫	∫	PROPN
ejpam-3209	172	29	ω	ω	PROPN
ejpam-3209	172	30	{	{	PUNCT
ejpam-3209	172	31	∣∣∣f(x)|p	∣∣∣f(x)|p	PROPN
ejpam-3209	172	32	+	+	CCONJ
ejpam-3209	172	33	|g(x	|g(x	NOUN
ejpam-3209	172	34	)	)	PUNCT
ejpam-3209	172	35	∣∣∣p}dx	∣∣∣p}dx	NOUN
ejpam-3209	172	36	⇒	⇒	NOUN
ejpam-3209	172	37	(	(	PUNCT
ejpam-3209	172	38	∣∣∣	∣∣∣	PROPN
ejpam-3209	172	39	∫	∫	PROPN
ejpam-3209	172	40	ω	ω	PROPN
ejpam-3209	172	41	f(x)g(x)dx	f(x)g(x)dx	PROPN
ejpam-3209	172	42	∣∣∣p	∣∣∣p	NOUN
ejpam-3209	172	43	)	)	PUNCT
ejpam-3209	172	44	1	1	NUM
ejpam-3209	172	45	p	p	NOUN
ejpam-3209	172	46	=	=	PUNCT
ejpam-3209	172	47	(	(	PUNCT
ejpam-3209	172	48	∫	∫	PROPN
ejpam-3209	172	49	ω	ω	PROPN
ejpam-3209	172	50	∣∣∣f(x)|pdx+	∣∣∣f(x)|pdx+	NUM
ejpam-3209	172	51	∫	∫	PROPN
ejpam-3209	172	52	ω	ω	PROPN
ejpam-3209	172	53	|g(x	|g(x	PROPN
ejpam-3209	172	54	)	)	PUNCT
ejpam-3209	172	55	∣∣∣pdx	∣∣∣pdx	PROPN
ejpam-3209	172	56	)	)	PUNCT
ejpam-3209	172	57	1	1	NUM
ejpam-3209	172	58	p	p	NOUN
ejpam-3209	172	59	⇒	⇒	NOUN
ejpam-3209	172	60	(	(	PUNCT
ejpam-3209	172	61	∣∣∣	∣∣∣	PROPN
ejpam-3209	172	62	∫	∫	PROPN
ejpam-3209	172	63	ω	ω	PROPN
ejpam-3209	172	64	f(x)g(x)dx	f(x)g(x)dx	PROPN
ejpam-3209	172	65	∣∣∣p	∣∣∣p	NOUN
ejpam-3209	172	66	)	)	PUNCT
ejpam-3209	172	67	1	1	NUM
ejpam-3209	172	68	p	p	NOUN
ejpam-3209	172	69	≤	≤	NUM
ejpam-3209	172	70	(	(	PUNCT
ejpam-3209	172	71	∫	∫	PROPN
ejpam-3209	172	72	ω	ω	PROPN
ejpam-3209	172	73	∣∣∣f(x)|pdx	∣∣∣f(x)|pdx	PROPN
ejpam-3209	172	74	)	)	PUNCT
ejpam-3209	173	1	1	1	NUM
ejpam-3209	173	2	p	p	NOUN
ejpam-3209	173	3	+	+	X
ejpam-3209	173	4	(	(	PUNCT
ejpam-3209	173	5	∫	∫	PROPN
ejpam-3209	173	6	ω	ω	PROPN
ejpam-3209	173	7	|g(x	|g(x	PROPN
ejpam-3209	173	8	)	)	PUNCT
ejpam-3209	173	9	∣∣∣pdx	∣∣∣pdx	PROPN
ejpam-3209	173	10	)	)	PUNCT
ejpam-3209	173	11	1	1	NUM
ejpam-3209	173	12	p	p	NOUN
ejpam-3209	173	13	⇒	⇒	NOUN
ejpam-3209	173	14	‖fg‖p	‖fg‖p	PROPN
ejpam-3209	173	15	≤	≤	ADJ
ejpam-3209	173	16	‖f‖p	‖f‖p	NOUN
ejpam-3209	173	17	+	+	CCONJ
ejpam-3209	173	18	‖g‖p	‖g‖p	NOUN
ejpam-3209	173	19	.	.	PUNCT
ejpam-3209	174	1	(	(	PUNCT
ejpam-3209	174	2	11	11	NUM
ejpam-3209	174	3	)	)	PUNCT
ejpam-3209	174	4	using	use	VERB
ejpam-3209	174	5	the	the	DET
ejpam-3209	174	6	cauchy	cauchy	NOUN
ejpam-3209	174	7	-	-	PUNCT
ejpam-3209	174	8	schwarz	schwarz	PROPN
ejpam-3209	174	9	inequality	inequality	NOUN
ejpam-3209	174	10	,	,	PUNCT
ejpam-3209	174	11	we	we	PRON
ejpam-3209	174	12	observe	observe	VERB
ejpam-3209	174	13	that	that	SCONJ
ejpam-3209	174	14	:	:	PUNCT
ejpam-3209	174	15	‖fg‖p	‖fg‖p	PROPN
ejpam-3209	174	16	≤	≤	ADJ
ejpam-3209	174	17	‖f‖p‖g‖p	‖f‖p‖g‖p	NOUN
ejpam-3209	174	18	.	.	PUNCT
ejpam-3209	175	1	(	(	PUNCT
ejpam-3209	175	2	12	12	NUM
ejpam-3209	175	3	)	)	PUNCT
ejpam-3209	175	4	substituting	substitute	VERB
ejpam-3209	175	5	inequality	inequality	NOUN
ejpam-3209	175	6	(	(	PUNCT
ejpam-3209	175	7	11	11	NUM
ejpam-3209	175	8	)	)	PUNCT
ejpam-3209	175	9	into	into	ADP
ejpam-3209	175	10	inequality	inequality	NOUN
ejpam-3209	175	11	(	(	PUNCT
ejpam-3209	175	12	12	12	NUM
ejpam-3209	175	13	)	)	PUNCT
ejpam-3209	175	14	yields	yield	VERB
ejpam-3209	175	15	‖f‖p‖g‖p	‖f‖p‖g‖p	PRON
ejpam-3209	175	16	≤	≤	NUM
ejpam-3209	175	17	‖f‖p	‖f‖p	NOUN
ejpam-3209	175	18	+	+	CCONJ
ejpam-3209	175	19	‖g‖p	‖g‖p	NOUN
ejpam-3209	175	20	.	.	PUNCT
ejpam-3209	176	1	we	we	PRON
ejpam-3209	176	2	show	show	VERB
ejpam-3209	176	3	how	how	SCONJ
ejpam-3209	176	4	theorem	theorem	ADJ
ejpam-3209	176	5	(	(	PUNCT
ejpam-3209	176	6	4	4	NUM
ejpam-3209	176	7	)	)	PUNCT
ejpam-3209	176	8	is	be	AUX
ejpam-3209	176	9	applied	apply	VERB
ejpam-3209	176	10	to	to	PART
ejpam-3209	176	11	estimate	estimate	VERB
ejpam-3209	176	12	two	two	NUM
ejpam-3209	176	13	smooth	smooth	ADJ
ejpam-3209	176	14	functions	function	NOUN
ejpam-3209	176	15	defined	define	VERB
ejpam-3209	176	16	on	on	ADP
ejpam-3209	176	17	domains	domain	NOUN
ejpam-3209	176	18	.	.	PUNCT
ejpam-3209	177	1	theorem	theorem	NOUN
ejpam-3209	177	2	5	5	NUM
ejpam-3209	177	3	.	.	PUNCT
ejpam-3209	177	4	suppose	suppose	VERB
ejpam-3209	177	5	that	that	SCONJ
ejpam-3209	177	6	f(x	f(x	PROPN
ejpam-3209	177	7	)	)	PUNCT
ejpam-3209	177	8	,	,	PUNCT
ejpam-3209	177	9	g(x	g(x	NOUN
ejpam-3209	177	10	)	)	PUNCT
ejpam-3209	177	11	are	be	AUX
ejpam-3209	177	12	c∞(ω	c∞(ω	NOUN
ejpam-3209	177	13	)	)	PUNCT
ejpam-3209	177	14	such	such	ADJ
ejpam-3209	177	15	that	that	SCONJ
ejpam-3209	177	16	f	f	X
ejpam-3209	177	17	:	:	PUNCT
ejpam-3209	178	1	ω	ω	X
ejpam-3209	178	2	→	→	PUNCT
ejpam-3209	179	1	[	[	X
ejpam-3209	179	2	0	0	NUM
ejpam-3209	179	3	,	,	PUNCT
ejpam-3209	179	4	1	1	NUM
ejpam-3209	179	5	]	]	PUNCT
ejpam-3209	179	6	and	and	CCONJ
ejpam-3209	179	7	g	g	PROPN
ejpam-3209	179	8	:	:	PUNCT
ejpam-3209	179	9	ω	ω	PROPN
ejpam-3209	179	10	→	→	PUNCT
ejpam-3209	179	11	[	[	X
ejpam-3209	179	12	0	0	NUM
ejpam-3209	179	13	,	,	PUNCT
ejpam-3209	179	14	1	1	NUM
ejpam-3209	179	15	]	]	PUNCT
ejpam-3209	179	16	,	,	PUNCT
ejpam-3209	179	17	then	then	ADV
ejpam-3209	179	18	the	the	DET
ejpam-3209	179	19	following	follow	VERB
ejpam-3209	179	20	inequality	inequality	NOUN
ejpam-3209	179	21	holds:∣∣∣	holds:∣∣∣	PROPN
ejpam-3209	180	1	∫	∫	PROPN
ejpam-3209	180	2	b	b	PROPN
ejpam-3209	180	3	a	a	DET
ejpam-3209	180	4	cos(nx	cos(nx	NOUN
ejpam-3209	180	5	)	)	PUNCT
ejpam-3209	180	6	sin(nx)dx	sin(nx)dx	VERB
ejpam-3209	180	7	∣∣∣	∣∣∣	ADJ
ejpam-3209	180	8	≤	≤	NUM
ejpam-3209	180	9	2(b−	2(b−	PROPN
ejpam-3209	180	10	a	a	PRON
ejpam-3209	180	11	)	)	PUNCT
ejpam-3209	180	12	.	.	PUNCT
ejpam-3209	181	1	barnes	barnes	PROPN
ejpam-3209	181	2	et	et	PROPN
ejpam-3209	181	3	al	al	PROPN
ejpam-3209	181	4	.	.	PUNCT
ejpam-3209	181	5	/	/	SYM
ejpam-3209	181	6	eur	eur	PROPN
ejpam-3209	181	7	.	.	PUNCT
ejpam-3209	182	1	j.	j.	PROPN
ejpam-3209	182	2	pure	pure	PROPN
ejpam-3209	182	3	appl	appl	PROPN
ejpam-3209	182	4	.	.	PROPN
ejpam-3209	182	5	math	math	PROPN
ejpam-3209	182	6	,	,	PUNCT
ejpam-3209	182	7	11	11	NUM
ejpam-3209	182	8	(	(	PUNCT
ejpam-3209	182	9	2	2	NUM
ejpam-3209	182	10	)	)	PUNCT
ejpam-3209	182	11	(	(	PUNCT
ejpam-3209	182	12	2018	2018	NUM
ejpam-3209	182	13	)	)	PUNCT
ejpam-3209	182	14	,	,	PUNCT
ejpam-3209	182	15	375	375	NUM
ejpam-3209	182	16	-	-	SYM
ejpam-3209	182	17	389	389	NUM
ejpam-3209	182	18	386	386	NUM
ejpam-3209	182	19	proof	proof	NOUN
ejpam-3209	182	20	:	:	PUNCT
ejpam-3209	182	21	∣∣∣	∣∣∣	PROPN
ejpam-3209	182	22	∫	∫	PROPN
ejpam-3209	182	23	b	b	PROPN
ejpam-3209	182	24	a	a	DET
ejpam-3209	182	25	cos(nx	cos(nx	NOUN
ejpam-3209	182	26	)	)	PUNCT
ejpam-3209	182	27	sin(nx)dx	sin(nx)dx	VERB
ejpam-3209	182	28	∣∣∣	∣∣∣	ADJ
ejpam-3209	182	29	≤	≤	NUM
ejpam-3209	182	30	∫	∫	PROPN
ejpam-3209	183	1	b	b	PROPN
ejpam-3209	183	2	a	a	DET
ejpam-3209	183	3	∣∣∣	∣∣∣	ADJ
ejpam-3209	183	4	cos(nx	cos(nx	NOUN
ejpam-3209	183	5	)	)	PUNCT
ejpam-3209	183	6	sin(nx	sin(nx	NOUN
ejpam-3209	183	7	)	)	PUNCT
ejpam-3209	183	8	∣∣∣dx∣∣∣	∣∣∣dx∣∣∣	PROPN
ejpam-3209	183	9	∫	∫	PROPN
ejpam-3209	184	1	b	b	PROPN
ejpam-3209	184	2	a	a	DET
ejpam-3209	184	3	cos(nx	cos(nx	NOUN
ejpam-3209	184	4	)	)	PUNCT
ejpam-3209	184	5	sin(nx)dx	sin(nx)dx	VERB
ejpam-3209	184	6	∣∣∣	∣∣∣	ADJ
ejpam-3209	184	7	≤	≤	NUM
ejpam-3209	184	8	∫	∫	PROPN
ejpam-3209	185	1	b	b	PROPN
ejpam-3209	185	2	a	a	DET
ejpam-3209	185	3	∣∣∣	∣∣∣	ADJ
ejpam-3209	185	4	cos(nx	cos(nx	NOUN
ejpam-3209	185	5	)	)	PUNCT
ejpam-3209	185	6	∣∣∣∣∣∣	∣∣∣∣∣∣	ADP
ejpam-3209	185	7	sin(nx	sin(nx	NOUN
ejpam-3209	185	8	)	)	PUNCT
ejpam-3209	186	1	∣∣∣dx	∣∣∣dx	NOUN
ejpam-3209	186	2	applying	apply	VERB
ejpam-3209	186	3	the	the	DET
ejpam-3209	186	4	first	first	ADJ
ejpam-3209	186	5	product	product	NOUN
ejpam-3209	186	6	inequality	inequality	NOUN
ejpam-3209	187	1	,	,	PUNCT
ejpam-3209	187	2	we	we	PRON
ejpam-3209	187	3	obtain:∣∣∣	obtain:∣∣∣	VERB
ejpam-3209	187	4	∫	∫	PROPN
ejpam-3209	187	5	b	b	PROPN
ejpam-3209	187	6	a	a	DET
ejpam-3209	187	7	cos(nx	cos(nx	NOUN
ejpam-3209	187	8	)	)	PUNCT
ejpam-3209	187	9	sin(nx)dx	sin(nx)dx	VERB
ejpam-3209	187	10	∣∣∣	∣∣∣	ADJ
ejpam-3209	187	11	≤	≤	NUM
ejpam-3209	187	12	∫	∫	PROPN
ejpam-3209	187	13	b	b	PROPN
ejpam-3209	187	14	a	a	DET
ejpam-3209	187	15	{	{	PUNCT
ejpam-3209	187	16	∣∣∣	∣∣∣	ADJ
ejpam-3209	187	17	cos(nx	cos(nx	NOUN
ejpam-3209	187	18	)	)	PUNCT
ejpam-3209	187	19	∣∣∣+	∣∣∣+	PROPN
ejpam-3209	187	20	∣∣∣	∣∣∣	ADJ
ejpam-3209	187	21	sin(nx	sin(nx	NOUN
ejpam-3209	187	22	)	)	PUNCT
ejpam-3209	187	23	∣∣∣}dx	∣∣∣}dx	PROPN
ejpam-3209	187	24	⇒	⇒	VERB
ejpam-3209	187	25	∣∣∣	∣∣∣	PROPN
ejpam-3209	187	26	∫	∫	PROPN
ejpam-3209	187	27	b	b	PROPN
ejpam-3209	187	28	a	a	DET
ejpam-3209	187	29	cos(nx	cos(nx	NOUN
ejpam-3209	187	30	)	)	PUNCT
ejpam-3209	187	31	sin(nx)dx	sin(nx)dx	VERB
ejpam-3209	187	32	∣∣∣	∣∣∣	NOUN
ejpam-3209	187	33	=	=	SYM
ejpam-3209	187	34	∫	∫	PROPN
ejpam-3209	187	35	b	b	PROPN
ejpam-3209	187	36	a	a	DET
ejpam-3209	187	37	∣∣∣	∣∣∣	ADJ
ejpam-3209	187	38	cos(nx	cos(nx	NOUN
ejpam-3209	187	39	)	)	PUNCT
ejpam-3209	187	40	∣∣∣dx+	∣∣∣dx+	ADV
ejpam-3209	187	41	∫	∫	PROPN
ejpam-3209	187	42	b	b	NOUN
ejpam-3209	187	43	a	a	DET
ejpam-3209	187	44	∣∣∣	∣∣∣	ADJ
ejpam-3209	187	45	sin(nx	sin(nx	NOUN
ejpam-3209	187	46	)	)	PUNCT
ejpam-3209	187	47	∣∣∣dx	∣∣∣dx	NOUN
ejpam-3209	187	48	⇒	⇒	VERB
ejpam-3209	187	49	∣∣∣	∣∣∣	PROPN
ejpam-3209	187	50	∫	∫	PROPN
ejpam-3209	188	1	b	b	PROPN
ejpam-3209	189	1	a	a	DET
ejpam-3209	189	2	cos(nx	cos(nx	NOUN
ejpam-3209	189	3	)	)	PUNCT
ejpam-3209	190	1	sin(nx)dx	sin(nx)dx	VERB
ejpam-3209	190	2	∣∣∣	∣∣∣	ADJ
ejpam-3209	190	3	≤	≤	NUM
ejpam-3209	190	4	∫	∫	PROPN
ejpam-3209	191	1	b	b	PROPN
ejpam-3209	191	2	a	a	DET
ejpam-3209	191	3	1dx+	1dx+	NUM
ejpam-3209	191	4	∫	∫	PROPN
ejpam-3209	191	5	b	b	PROPN
ejpam-3209	191	6	a	a	DET
ejpam-3209	191	7	1dx	1dx	ADJ
ejpam-3209	191	8	⇒	⇒	NOUN
ejpam-3209	191	9	∣∣∣	∣∣∣	PROPN
ejpam-3209	192	1	∫	∫	PROPN
ejpam-3209	192	2	b	b	PROPN
ejpam-3209	192	3	a	a	DET
ejpam-3209	192	4	cos(nx	cos(nx	NOUN
ejpam-3209	192	5	)	)	PUNCT
ejpam-3209	192	6	sin(nx)dx	sin(nx)dx	NOUN
ejpam-3209	192	7	∣∣∣	∣∣∣	NOUN
ejpam-3209	193	1	=	=	PUNCT
ejpam-3209	194	1	[	[	PUNCT
ejpam-3209	194	2	x	x	X
ejpam-3209	194	3	]	]	X
ejpam-3209	194	4	b	b	X
ejpam-3209	194	5	a	a	PRON
ejpam-3209	194	6	+	+	X
ejpam-3209	194	7	[	[	PUNCT
ejpam-3209	194	8	x	x	X
ejpam-3209	194	9	]	]	X
ejpam-3209	194	10	b	b	X
ejpam-3209	194	11	a	a	DET
ejpam-3209	194	12	⇒	⇒	NOUN
ejpam-3209	194	13	∣∣∣	∣∣∣	X
ejpam-3209	194	14	∫	∫	PROPN
ejpam-3209	195	1	b	b	PROPN
ejpam-3209	195	2	a	a	DET
ejpam-3209	195	3	cos(nx	cos(nx	NOUN
ejpam-3209	195	4	)	)	PUNCT
ejpam-3209	195	5	sin(nx)dx	sin(nx)dx	VERB
ejpam-3209	195	6	∣∣∣	∣∣∣	ADJ
ejpam-3209	195	7	≤	≤	NUM
ejpam-3209	195	8	2(b−	2(b−	PROPN
ejpam-3209	195	9	a	a	PRON
ejpam-3209	195	10	)	)	PUNCT
ejpam-3209	195	11	.	.	PUNCT
ejpam-3209	196	1	4	4	X
ejpam-3209	196	2	.	.	X
ejpam-3209	196	3	extension	extension	NOUN
ejpam-3209	196	4	and	and	CCONJ
ejpam-3209	196	5	applications	application	NOUN
ejpam-3209	196	6	of	of	ADP
ejpam-3209	196	7	the	the	DET
ejpam-3209	196	8	second	second	ADJ
ejpam-3209	196	9	product	product	NOUN
ejpam-3209	196	10	inequality	inequality	NOUN
ejpam-3209	196	11	to	to	ADP
ejpam-3209	196	12	the	the	DET
ejpam-3209	196	13	lp	lp	ADJ
ejpam-3209	196	14	spaces	space	NOUN
ejpam-3209	196	15	in	in	ADP
ejpam-3209	196	16	this	this	DET
ejpam-3209	196	17	section	section	NOUN
ejpam-3209	196	18	,	,	PUNCT
ejpam-3209	196	19	we	we	PRON
ejpam-3209	196	20	apply	apply	VERB
ejpam-3209	196	21	the	the	DET
ejpam-3209	196	22	second	second	ADJ
ejpam-3209	196	23	product	product	NOUN
ejpam-3209	196	24	inequality	inequality	NOUN
ejpam-3209	196	25	to	to	ADP
ejpam-3209	196	26	lp	lp	PROPN
ejpam-3209	196	27	spaces	space	NOUN
ejpam-3209	196	28	.	.	PUNCT
ejpam-3209	197	1	theorem	theorem	ADJ
ejpam-3209	197	2	6	6	NUM
ejpam-3209	197	3	.	.	PUNCT
ejpam-3209	198	1	if	if	SCONJ
ejpam-3209	198	2	u	u	PROPN
ejpam-3209	198	3	and	and	CCONJ
ejpam-3209	198	4	v	v	NOUN
ejpam-3209	198	5	are	be	AUX
ejpam-3209	198	6	any	any	DET
ejpam-3209	198	7	two	two	NUM
ejpam-3209	198	8	vectors	vector	NOUN
ejpam-3209	198	9	in	in	ADP
ejpam-3209	198	10	the	the	DET
ejpam-3209	198	11	euclidean	euclidean	ADJ
ejpam-3209	198	12	space	space	NOUN
ejpam-3209	198	13	,	,	PUNCT
ejpam-3209	198	14	then	then	ADV
ejpam-3209	198	15	‖u‖p	‖u‖p	NOUN
ejpam-3209	198	16	+	+	CCONJ
ejpam-3209	198	17	‖v‖p	‖v‖p	X
ejpam-3209	198	18	≤	≤	NOUN
ejpam-3209	198	19	‖u‖p‖v‖p	‖u‖p‖v‖p	VERB
ejpam-3209	198	20	,	,	PUNCT
ejpam-3209	198	21	∀	∀	X
ejpam-3209	198	22	u	u	NOUN
ejpam-3209	198	23	,	,	PUNCT
ejpam-3209	198	24	v	v	NOUN
ejpam-3209	198	25	∈	∈	PROPN
ejpam-3209	199	1	[	[	X
ejpam-3209	199	2	2,∞	2,∞	NUM
ejpam-3209	199	3	)	)	PUNCT
ejpam-3209	199	4	proof	proof	NOUN
ejpam-3209	199	5	:	:	PUNCT
ejpam-3209	199	6	we	we	PRON
ejpam-3209	199	7	can	can	AUX
ejpam-3209	199	8	see	see	VERB
ejpam-3209	199	9	that	that	PRON
ejpam-3209	199	10	:	:	PUNCT
ejpam-3209	199	11	‖u‖+	‖u‖+	PROPN
ejpam-3209	199	12	‖v‖	‖v‖	PROPN
ejpam-3209	199	13	≤	≤	NUM
ejpam-3209	199	14	‖u‖‖v‖	‖u‖‖v‖	ADJ
ejpam-3209	199	15	⇒	⇒	NOUN
ejpam-3209	199	16	(	(	PUNCT
ejpam-3209	199	17	‖u‖+	‖u‖+	PROPN
ejpam-3209	199	18	‖v‖	‖v‖	PROPN
ejpam-3209	199	19	)	)	PUNCT
ejpam-3209	199	20	p	p	NOUN
ejpam-3209	199	21	≤	≤	NOUN
ejpam-3209	199	22	(	(	PUNCT
ejpam-3209	199	23	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3209	199	24	)	)	PUNCT
ejpam-3209	199	25	p	p	NOUN
ejpam-3209	199	26	.	.	PUNCT
ejpam-3209	200	1	(	(	PUNCT
ejpam-3209	200	2	13	13	NUM
ejpam-3209	200	3	)	)	PUNCT
ejpam-3209	200	4	also	also	ADV
ejpam-3209	200	5	,	,	PUNCT
ejpam-3209	200	6	we	we	PRON
ejpam-3209	200	7	observed	observe	VERB
ejpam-3209	200	8	that	that	SCONJ
ejpam-3209	200	9	:	:	PUNCT
ejpam-3209	200	10	‖u‖p	‖u‖p	NOUN
ejpam-3209	200	11	+	+	CCONJ
ejpam-3209	200	12	‖v‖p	‖v‖p	X
ejpam-3209	200	13	≤	≤	NOUN
ejpam-3209	200	14	(	(	PUNCT
ejpam-3209	200	15	‖u‖+	‖u‖+	PROPN
ejpam-3209	200	16	‖v‖	‖v‖	PROPN
ejpam-3209	200	17	)	)	PUNCT
ejpam-3209	200	18	p	p	NOUN
ejpam-3209	200	19	.	.	PUNCT
ejpam-3209	201	1	(	(	PUNCT
ejpam-3209	201	2	14	14	NUM
ejpam-3209	201	3	)	)	PUNCT
ejpam-3209	201	4	by	by	ADP
ejpam-3209	201	5	the	the	DET
ejpam-3209	201	6	transitivity	transitivity	NOUN
ejpam-3209	201	7	of	of	ADP
ejpam-3209	201	8	the	the	DET
ejpam-3209	201	9	inequalities	inequality	NOUN
ejpam-3209	201	10	(	(	PUNCT
ejpam-3209	201	11	13	13	NUM
ejpam-3209	201	12	)	)	PUNCT
ejpam-3209	201	13	and	and	CCONJ
ejpam-3209	201	14	(	(	PUNCT
ejpam-3209	201	15	14	14	NUM
ejpam-3209	201	16	)	)	PUNCT
ejpam-3209	201	17	,	,	PUNCT
ejpam-3209	201	18	we	we	PRON
ejpam-3209	201	19	obtain	obtain	VERB
ejpam-3209	201	20	‖u‖p	‖u‖p	NOUN
ejpam-3209	201	21	+	+	CCONJ
ejpam-3209	201	22	‖v‖p	‖v‖p	X
ejpam-3209	201	23	≤	≤	NOUN
ejpam-3209	201	24	(	(	PUNCT
ejpam-3209	201	25	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3209	201	26	)	)	PUNCT
ejpam-3209	201	27	p	p	NOUN
ejpam-3209	201	28	‖u‖p	‖u‖p	NOUN
ejpam-3209	201	29	+	+	CCONJ
ejpam-3209	201	30	‖v‖p	‖v‖p	NOUN
ejpam-3209	201	31	≤	≤	PUNCT
ejpam-3209	201	32	‖u‖p‖v‖p	‖u‖p‖v‖p	VERB
ejpam-3209	201	33	.	.	PUNCT
ejpam-3209	202	1	barnes	barnes	PROPN
ejpam-3209	202	2	et	et	PROPN
ejpam-3209	202	3	al	al	PROPN
ejpam-3209	202	4	.	.	PUNCT
ejpam-3209	202	5	/	/	SYM
ejpam-3209	202	6	eur	eur	PROPN
ejpam-3209	202	7	.	.	PUNCT
ejpam-3209	203	1	j.	j.	PROPN
ejpam-3209	203	2	pure	pure	PROPN
ejpam-3209	203	3	appl	appl	PROPN
ejpam-3209	203	4	.	.	PROPN
ejpam-3209	203	5	math	math	PROPN
ejpam-3209	203	6	,	,	PUNCT
ejpam-3209	203	7	11	11	NUM
ejpam-3209	203	8	(	(	PUNCT
ejpam-3209	203	9	2	2	NUM
ejpam-3209	203	10	)	)	PUNCT
ejpam-3209	203	11	(	(	PUNCT
ejpam-3209	203	12	2018	2018	NUM
ejpam-3209	203	13	)	)	PUNCT
ejpam-3209	203	14	,	,	PUNCT
ejpam-3209	203	15	375	375	NUM
ejpam-3209	203	16	-	-	SYM
ejpam-3209	203	17	389	389	NUM
ejpam-3209	203	18	387	387	NUM
ejpam-3209	203	19	theorem	theorem	NOUN
ejpam-3209	203	20	7	7	NUM
ejpam-3209	203	21	.	.	PUNCT
ejpam-3209	203	22	suppose	suppose	VERB
ejpam-3209	203	23	that	that	SCONJ
ejpam-3209	203	24	the	the	DET
ejpam-3209	203	25	measurable	measurable	ADJ
ejpam-3209	203	26	functions	function	NOUN
ejpam-3209	203	27	over	over	ADP
ejpam-3209	203	28	the	the	DET
ejpam-3209	203	29	domain	domain	NOUN
ejpam-3209	203	30	ω	ω	NOUN
ejpam-3209	203	31	are	be	AUX
ejpam-3209	203	32	such	such	ADJ
ejpam-3209	203	33	that	that	SCONJ
ejpam-3209	203	34	f	f	X
ejpam-3209	203	35	:	:	PUNCT
ejpam-3209	203	36	ω→	ω→	PUNCT
ejpam-3209	203	37	r	r	NOUN
ejpam-3209	203	38	and	and	CCONJ
ejpam-3209	203	39	the	the	DET
ejpam-3209	203	40	other	other	ADJ
ejpam-3209	203	41	measurable	measurable	ADJ
ejpam-3209	203	42	function	function	NOUN
ejpam-3209	203	43	g	g	NOUN
ejpam-3209	203	44	:	:	PUNCT
ejpam-3209	203	45	ω→	ω→	PUNCT
ejpam-3209	203	46	r	r	NOUN
ejpam-3209	203	47	,	,	PUNCT
ejpam-3209	203	48	then	then	ADV
ejpam-3209	203	49	‖f‖p	‖f‖p	NOUN
ejpam-3209	203	50	+	+	CCONJ
ejpam-3209	203	51	‖g‖p	‖g‖p	VERB
ejpam-3209	203	52	≤	≤	NUM
ejpam-3209	203	53	‖f‖p‖g‖p	‖f‖p‖g‖p	NOUN
ejpam-3209	203	54	.	.	PUNCT
ejpam-3209	204	1	p	p	NOUN
ejpam-3209	204	2	roof	roof	NOUN
ejpam-3209	204	3	:	:	PUNCT
ejpam-3209	204	4	∣∣∣	∣∣∣	PROPN
ejpam-3209	204	5	∫	∫	PROPN
ejpam-3209	204	6	ω	ω	PROPN
ejpam-3209	204	7	f(x)g(x)dx	f(x)g(x)dx	PROPN
ejpam-3209	204	8	∣∣∣p	∣∣∣p	NOUN
ejpam-3209	204	9	≤	≤	NUM
ejpam-3209	204	10	∫	∫	PROPN
ejpam-3209	204	11	ω	ω	NUM
ejpam-3209	204	12	∣∣∣f(x)g(x	∣∣∣f(x)g(x	NOUN
ejpam-3209	204	13	)	)	PUNCT
ejpam-3209	204	14	∣∣∣pdx	∣∣∣pdx	PROPN
ejpam-3209	204	15	applying	apply	VERB
ejpam-3209	204	16	the	the	DET
ejpam-3209	204	17	second	second	ADJ
ejpam-3209	204	18	product	product	NOUN
ejpam-3209	204	19	inequality	inequality	NOUN
ejpam-3209	204	20	,	,	PUNCT
ejpam-3209	204	21	we	we	PRON
ejpam-3209	204	22	obtain∫	obtain∫	VERB
ejpam-3209	204	23	ω	ω	PROPN
ejpam-3209	204	24	(	(	PUNCT
ejpam-3209	204	25	|f(x)|+	|f(x)|+	ADJ
ejpam-3209	204	26	|g(x)|	|g(x)|	NOUN
ejpam-3209	204	27	)	)	PUNCT
ejpam-3209	204	28	dx	dx	PROPN
ejpam-3209	205	1	≤	≤	NUM
ejpam-3209	205	2	∫	∫	PROPN
ejpam-3209	205	3	ω	ω	PROPN
ejpam-3209	205	4	|f(x)g(x)|dx	|f(x)g(x)|dx	PROPN
ejpam-3209	205	5	⇒	⇒	PROPN
ejpam-3209	205	6	∫	∫	PROPN
ejpam-3209	205	7	ω	ω	PROPN
ejpam-3209	205	8	(	(	PUNCT
ejpam-3209	205	9	|f(x)|+	|f(x)|+	ADV
ejpam-3209	205	10	|g(x)|	|g(x)|	NOUN
ejpam-3209	205	11	)	)	PUNCT
ejpam-3209	205	12	p	p	NOUN
ejpam-3209	205	13	dx	dx	PROPN
ejpam-3209	205	14	=	=	SYM
ejpam-3209	205	15	∫	∫	PROPN
ejpam-3209	205	16	ω	ω	PROPN
ejpam-3209	205	17	(	(	PUNCT
ejpam-3209	205	18	|f(x)g(x)|	|f(x)g(x)|	NOUN
ejpam-3209	205	19	)	)	PUNCT
ejpam-3209	205	20	p	p	PRON
ejpam-3209	205	21	dx	dx	PROPN
ejpam-3209	205	22	.	.	PUNCT
ejpam-3209	206	1	(	(	PUNCT
ejpam-3209	206	2	15	15	X
ejpam-3209	206	3	)	)	PUNCT
ejpam-3209	206	4	we	we	PRON
ejpam-3209	206	5	see	see	VERB
ejpam-3209	206	6	that:∫	that:∫	NOUN
ejpam-3209	206	7	ω	ω	NUM
ejpam-3209	206	8	|f(x)|p	|f(x)|p	PROPN
ejpam-3209	206	9	+	+	NUM
ejpam-3209	207	1	|g(x)|pdx	|g(x)|pdx	X
ejpam-3209	207	2	≤	≤	NUM
ejpam-3209	207	3	∫	∫	PROPN
ejpam-3209	207	4	ω	ω	PROPN
ejpam-3209	207	5	(	(	PUNCT
ejpam-3209	207	6	|f(x)|+	|f(x)|+	ADV
ejpam-3209	207	7	|g(x)|	|g(x)|	NOUN
ejpam-3209	207	8	)	)	PUNCT
ejpam-3209	207	9	p	p	PROPN
ejpam-3209	207	10	dx	dx	PROPN
ejpam-3209	207	11	.	.	PUNCT
ejpam-3209	208	1	(	(	PUNCT
ejpam-3209	208	2	16	16	NUM
ejpam-3209	208	3	)	)	PUNCT
ejpam-3209	208	4	applying	apply	VERB
ejpam-3209	208	5	the	the	DET
ejpam-3209	208	6	transitive	transitive	ADJ
ejpam-3209	208	7	law	law	NOUN
ejpam-3209	208	8	to	to	ADP
ejpam-3209	208	9	inequalities	inequality	NOUN
ejpam-3209	208	10	(	(	PUNCT
ejpam-3209	208	11	15	15	NUM
ejpam-3209	208	12	)	)	PUNCT
ejpam-3209	208	13	and	and	CCONJ
ejpam-3209	208	14	(	(	PUNCT
ejpam-3209	208	15	16	16	NUM
ejpam-3209	208	16	)	)	PUNCT
ejpam-3209	208	17	,	,	PUNCT
ejpam-3209	208	18	we	we	PRON
ejpam-3209	208	19	obtain:∫	obtain:∫	VERB
ejpam-3209	208	20	ω	ω	NUM
ejpam-3209	208	21	|f(x)|p	|f(x)|p	PROPN
ejpam-3209	209	1	+	+	SYM
ejpam-3209	209	2	|g(x)|pdx	|g(x)|pdx	X
ejpam-3209	209	3	≤	≤	NUM
ejpam-3209	209	4	∫	∫	PROPN
ejpam-3209	209	5	ω	ω	PROPN
ejpam-3209	209	6	(	(	PUNCT
ejpam-3209	209	7	|f(x)g(x)|	|f(x)g(x)|	NOUN
ejpam-3209	209	8	)	)	PUNCT
ejpam-3209	209	9	p	p	PRON
ejpam-3209	209	10	dx	dx	PROPN
ejpam-3209	209	11	⇒	⇒	PROPN
ejpam-3209	209	12	∫	∫	PROPN
ejpam-3209	209	13	ω	ω	PROPN
ejpam-3209	209	14	|f(x)|p	|f(x)|p	PROPN
ejpam-3209	209	15	+	+	CCONJ
ejpam-3209	209	16	|g(x)|pdx	|g(x)|pdx	X
ejpam-3209	209	17	=	=	SYM
ejpam-3209	209	18	∫	∫	PROPN
ejpam-3209	209	19	ω	ω	NUM
ejpam-3209	209	20	|f(x)|p|g(x)|pdx	|f(x)|p|g(x)|pdx	PROPN
ejpam-3209	209	21	⇒	⇒	VERB
ejpam-3209	209	22	(	(	PUNCT
ejpam-3209	209	23	∫	∫	PROPN
ejpam-3209	209	24	ω	ω	PROPN
ejpam-3209	209	25	|f(x)|p	|f(x)|p	PROPN
ejpam-3209	209	26	+	+	NUM
ejpam-3209	209	27	|g(x)|pdx	|g(x)|pdx	NUM
ejpam-3209	209	28	)	)	PUNCT
ejpam-3209	209	29	1	1	NUM
ejpam-3209	209	30	p	p	NOUN
ejpam-3209	209	31	=	=	PUNCT
ejpam-3209	209	32	(	(	PUNCT
ejpam-3209	209	33	∫	∫	PROPN
ejpam-3209	209	34	ω	ω	PROPN
ejpam-3209	209	35	|f(x)|p|g(x)|pdx	|f(x)|p|g(x)|pdx	PROPN
ejpam-3209	209	36	)	)	PUNCT
ejpam-3209	209	37	1	1	NUM
ejpam-3209	209	38	p	p	NOUN
ejpam-3209	209	39	⇒	⇒	NOUN
ejpam-3209	209	40	(	(	PUNCT
ejpam-3209	209	41	∫	∫	PROPN
ejpam-3209	209	42	ω	ω	PROPN
ejpam-3209	209	43	|f(x)|pdx	|f(x)|pdx	NOUN
ejpam-3209	209	44	)	)	PUNCT
ejpam-3209	209	45	1	1	NUM
ejpam-3209	210	1	p	p	NOUN
ejpam-3209	210	2	+	+	X
ejpam-3209	210	3	(	(	PUNCT
ejpam-3209	210	4	∫	∫	PROPN
ejpam-3209	210	5	|g(x)|pdx	|g(x)|pdx	PROPN
ejpam-3209	210	6	)	)	PUNCT
ejpam-3209	210	7	1	1	NUM
ejpam-3209	210	8	p	p	NOUN
ejpam-3209	210	9	≤	≤	NUM
ejpam-3209	210	10	(	(	PUNCT
ejpam-3209	210	11	∫	∫	PROPN
ejpam-3209	210	12	ω	ω	PROPN
ejpam-3209	210	13	|f(x)|pdx	|f(x)|pdx	NOUN
ejpam-3209	210	14	)	)	PUNCT
ejpam-3209	210	15	1	1	NUM
ejpam-3209	210	16	p	p	NOUN
ejpam-3209	210	17	(	(	PUNCT
ejpam-3209	210	18	∫	∫	PROPN
ejpam-3209	210	19	|g(x)|pdx	|g(x)|pdx	PROPN
ejpam-3209	210	20	)	)	PUNCT
ejpam-3209	210	21	1	1	NUM
ejpam-3209	210	22	p	p	NOUN
ejpam-3209	210	23	⇒	⇒	NOUN
ejpam-3209	210	24	‖f‖p	‖f‖p	NOUN
ejpam-3209	211	1	+	+	CCONJ
ejpam-3209	211	2	‖g‖p	‖g‖p	VERB
ejpam-3209	211	3	≤	≤	NUM
ejpam-3209	211	4	‖f‖p‖g‖p	‖f‖p‖g‖p	NUM
ejpam-3209	211	5	.	.	PUNCT
ejpam-3209	212	1	5	5	X
ejpam-3209	212	2	.	.	X
ejpam-3209	212	3	discussion	discussion	NOUN
ejpam-3209	212	4	we	we	PRON
ejpam-3209	212	5	deduce	deduce	VERB
ejpam-3209	212	6	from	from	ADP
ejpam-3209	212	7	(	(	PUNCT
ejpam-3209	212	8	9	9	NUM
ejpam-3209	212	9	)	)	PUNCT
ejpam-3209	212	10	that	that	PRON
ejpam-3209	212	11	,	,	PUNCT
ejpam-3209	212	12	applying	apply	VERB
ejpam-3209	212	13	the	the	DET
ejpam-3209	212	14	first	first	ADJ
ejpam-3209	212	15	product	product	NOUN
ejpam-3209	212	16	inequality	inequality	NOUN
ejpam-3209	212	17	to	to	ADP
ejpam-3209	212	18	the	the	DET
ejpam-3209	212	19	trigonometric	trigonometric	ADJ
ejpam-3209	212	20	functions	function	NOUN
ejpam-3209	212	21	sin(θ	sin(θ	VERB
ejpam-3209	212	22	)	)	PUNCT
ejpam-3209	212	23	and	and	CCONJ
ejpam-3209	212	24	cos(θ	cos(θ	PROPN
ejpam-3209	212	25	)	)	PUNCT
ejpam-3209	212	26	,	,	PUNCT
ejpam-3209	212	27	another	another	DET
ejpam-3209	212	28	new	new	ADJ
ejpam-3209	212	29	inequality	inequality	NOUN
ejpam-3209	212	30	1	1	NUM
ejpam-3209	212	31	≤	≤	NUM
ejpam-3209	212	32	2n	2n	NUM
ejpam-3209	212	33	,	,	PUNCT
ejpam-3209	212	34	∀n	∀n	NUM
ejpam-3209	212	35	=	=	SYM
ejpam-3209	212	36	0	0	NUM
ejpam-3209	212	37	,	,	PUNCT
ejpam-3209	212	38	1	1	NUM
ejpam-3209	212	39	,	,	PUNCT
ejpam-3209	212	40	2	2	NUM
ejpam-3209	212	41	,	,	PUNCT
ejpam-3209	212	42	.	.	PUNCT
ejpam-3209	212	43	.	.	PUNCT
ejpam-3209	213	1	.	.	PUNCT
ejpam-3209	214	1	is	be	AUX
ejpam-3209	214	2	uncovered	uncover	VERB
ejpam-3209	214	3	.	.	PUNCT
ejpam-3209	215	1	the	the	DET
ejpam-3209	215	2	first	first	ADJ
ejpam-3209	215	3	and	and	CCONJ
ejpam-3209	215	4	second	second	ADJ
ejpam-3209	215	5	product	product	NOUN
ejpam-3209	215	6	inequalities	inequality	NOUN
ejpam-3209	215	7	have	have	AUX
ejpam-3209	215	8	given	give	VERB
ejpam-3209	215	9	birth	birth	NOUN
ejpam-3209	215	10	to	to	ADP
ejpam-3209	215	11	new	new	ADJ
ejpam-3209	215	12	inequalities	inequality	NOUN
ejpam-3209	215	13	in	in	ADP
ejpam-3209	215	14	the	the	DET
ejpam-3209	215	15	generalized	generalized	ADJ
ejpam-3209	215	16	space	space	NOUN
ejpam-3209	215	17	.	.	PUNCT
ejpam-3209	216	1	these	these	DET
ejpam-3209	216	2	new	new	ADJ
ejpam-3209	216	3	inequalities	inequality	NOUN
ejpam-3209	216	4	give	give	VERB
ejpam-3209	216	5	additional	additional	ADJ
ejpam-3209	216	6	information	information	NOUN
ejpam-3209	216	7	about	about	ADP
ejpam-3209	216	8	embeddings	embedding	NOUN
ejpam-3209	216	9	of	of	ADP
ejpam-3209	216	10	one	one	NUM
ejpam-3209	216	11	space	space	NOUN
ejpam-3209	216	12	into	into	ADP
ejpam-3209	216	13	another	another	DET
ejpam-3209	216	14	space	space	NOUN
ejpam-3209	216	15	.	.	PUNCT
ejpam-3209	217	1	references	reference	NOUN
ejpam-3209	217	2	388	388	NUM
ejpam-3209	217	3	6	6	NUM
ejpam-3209	217	4	.	.	PUNCT
ejpam-3209	217	5	conclusion	conclusion	NOUN
ejpam-3209	217	6	in	in	ADP
ejpam-3209	217	7	summary	summary	NOUN
ejpam-3209	217	8	,	,	PUNCT
ejpam-3209	217	9	we	we	PRON
ejpam-3209	217	10	observed	observe	VERB
ejpam-3209	217	11	that	that	SCONJ
ejpam-3209	217	12	the	the	DET
ejpam-3209	217	13	first	first	ADJ
ejpam-3209	217	14	product	product	NOUN
ejpam-3209	217	15	inequality	inequality	NOUN
ejpam-3209	217	16	holds	hold	VERB
ejpam-3209	217	17	for	for	ADP
ejpam-3209	217	18	any	any	DET
ejpam-3209	217	19	two	two	NUM
ejpam-3209	217	20	vectors	vector	NOUN
ejpam-3209	217	21	in	in	ADP
ejpam-3209	217	22	a	a	DET
ejpam-3209	217	23	genearalized	genearalized	ADJ
ejpam-3209	217	24	linear	linear	ADJ
ejpam-3209	217	25	space	space	NOUN
ejpam-3209	217	26	.	.	PUNCT
ejpam-3209	218	1	on	on	ADP
ejpam-3209	218	2	the	the	DET
ejpam-3209	218	3	other	other	ADJ
ejpam-3209	218	4	hand	hand	NOUN
ejpam-3209	218	5	,	,	PUNCT
ejpam-3209	218	6	the	the	DET
ejpam-3209	218	7	second	second	ADJ
ejpam-3209	218	8	product	product	NOUN
ejpam-3209	218	9	inequality	inequality	NOUN
ejpam-3209	218	10	holds	hold	VERB
ejpam-3209	218	11	for	for	ADP
ejpam-3209	218	12	any	any	DET
ejpam-3209	218	13	two	two	NUM
ejpam-3209	218	14	vectors	vector	NOUN
ejpam-3209	218	15	only	only	ADV
ejpam-3209	218	16	in	in	ADP
ejpam-3209	218	17	the	the	DET
ejpam-3209	218	18	euclidean	euclidean	ADJ
ejpam-3209	218	19	space	space	NOUN
ejpam-3209	218	20	.	.	PUNCT
ejpam-3209	219	1	in	in	ADP
ejpam-3209	219	2	addition	addition	NOUN
ejpam-3209	219	3	,	,	PUNCT
ejpam-3209	219	4	by	by	ADP
ejpam-3209	219	5	applying	apply	VERB
ejpam-3209	219	6	the	the	DET
ejpam-3209	219	7	first	first	ADJ
ejpam-3209	219	8	product	product	NOUN
ejpam-3209	219	9	inequality	inequality	NOUN
ejpam-3209	219	10	to	to	ADP
ejpam-3209	219	11	the	the	DET
ejpam-3209	219	12	lp	lp	PROPN
ejpam-3209	219	13	spaces	space	NOUN
ejpam-3209	219	14	,	,	PUNCT
ejpam-3209	219	15	we	we	PRON
ejpam-3209	219	16	observed	observe	VERB
ejpam-3209	219	17	that	that	SCONJ
ejpam-3209	219	18	if	if	SCONJ
ejpam-3209	219	19	,	,	PUNCT
ejpam-3209	219	20	f	f	X
ejpam-3209	219	21	:	:	PUNCT
ejpam-3209	219	22	ω	ω	X
ejpam-3209	219	23	→	→	PUNCT
ejpam-3209	220	1	[	[	X
ejpam-3209	220	2	0	0	NUM
ejpam-3209	220	3	,	,	PUNCT
ejpam-3209	220	4	1	1	NUM
ejpam-3209	220	5	]	]	PUNCT
ejpam-3209	220	6	,	,	PUNCT
ejpam-3209	220	7	and	and	CCONJ
ejpam-3209	220	8	g	g	NOUN
ejpam-3209	220	9	:	:	PUNCT
ejpam-3209	220	10	ω	ω	PROPN
ejpam-3209	220	11	→	→	SYM
ejpam-3209	220	12	r	r	NOUN
ejpam-3209	220	13	,	,	PUNCT
ejpam-3209	220	14	then	then	ADV
ejpam-3209	220	15	‖f‖p‖g‖p	‖f‖p‖g‖p	PRON
ejpam-3209	220	16	≤	≤	NUM
ejpam-3209	220	17	‖f‖p	‖f‖p	NOUN
ejpam-3209	221	1	+	+	NOUN
ejpam-3209	221	2	‖g‖p	‖g‖p	NOUN
ejpam-3209	221	3	.	.	PUNCT
ejpam-3209	222	1	but	but	CCONJ
ejpam-3209	222	2	if	if	SCONJ
ejpam-3209	222	3	,	,	PUNCT
ejpam-3209	222	4	f	f	X
ejpam-3209	222	5	,	,	PUNCT
ejpam-3209	222	6	g	g	PROPN
ejpam-3209	222	7	:	:	PUNCT
ejpam-3209	222	8	ω→	ω→	PUNCT
ejpam-3209	222	9	r	r	NOUN
ejpam-3209	222	10	,	,	PUNCT
ejpam-3209	222	11	then	then	ADV
ejpam-3209	222	12	‖f‖p	‖f‖p	NOUN
ejpam-3209	223	1	+	+	ADV
ejpam-3209	223	2	‖g‖p	‖g‖p	VERB
ejpam-3209	223	3	≤	≤	NUM
ejpam-3209	223	4	‖f‖p‖g‖p	‖f‖p‖g‖p	PRON
ejpam-3209	223	5	.	.	PUNCT
ejpam-3209	224	1	last	last	ADJ
ejpam-3209	224	2	but	but	CCONJ
ejpam-3209	224	3	not	not	PART
ejpam-3209	224	4	the	the	DET
ejpam-3209	224	5	least	least	ADJ
ejpam-3209	224	6	,	,	PUNCT
ejpam-3209	224	7	we	we	PRON
ejpam-3209	224	8	have	have	AUX
ejpam-3209	224	9	shown	show	VERB
ejpam-3209	224	10	in	in	ADP
ejpam-3209	224	11	this	this	DET
ejpam-3209	224	12	paper	paper	NOUN
ejpam-3209	224	13	that	that	SCONJ
ejpam-3209	224	14	the	the	DET
ejpam-3209	224	15	first	first	ADJ
ejpam-3209	224	16	product	product	NOUN
ejpam-3209	224	17	inequality	inequality	NOUN
ejpam-3209	224	18	holds	hold	VERB
ejpam-3209	224	19	for	for	ADP
ejpam-3209	224	20	c∞[0	c∞[0	NOUN
ejpam-3209	224	21	,	,	PUNCT
ejpam-3209	224	22	1	1	NUM
ejpam-3209	224	23	]	]	PUNCT
ejpam-3209	224	24	.	.	PUNCT
ejpam-3209	225	1	references	reference	NOUN
ejpam-3209	225	2	[	[	X
ejpam-3209	225	3	1	1	NUM
ejpam-3209	225	4	]	]	PUNCT
ejpam-3209	225	5	a.	a.	NOUN
ejpam-3209	225	6	m.	m.	PROPN
ejpam-3209	225	7	fink	fink	PROPN
ejpam-3209	225	8	,	,	PUNCT
ejpam-3209	225	9	an	an	DET
ejpam-3209	225	10	essay	essay	NOUN
ejpam-3209	225	11	on	on	ADP
ejpam-3209	225	12	the	the	DET
ejpam-3209	225	13	history	history	NOUN
ejpam-3209	225	14	of	of	ADP
ejpam-3209	225	15	inequalities	inequality	NOUN
ejpam-3209	225	16	,	,	PUNCT
ejpam-3209	225	17	journal	journal	NOUN
ejpam-3209	225	18	of	of	ADP
ejpam-3209	225	19	mathematical	mathematical	ADJ
ejpam-3209	225	20	analysis	analysis	NOUN
ejpam-3209	225	21	and	and	CCONJ
ejpam-3209	225	22	applications	application	NOUN
ejpam-3209	225	23	,	,	PUNCT
ejpam-3209	225	24	249	249	NUM
ejpam-3209	225	25	,	,	PUNCT
ejpam-3209	225	26	(	(	PUNCT
ejpam-3209	225	27	2000).pp	2000).pp	NUM
ejpam-3209	225	28	.	.	NOUN
ejpam-3209	225	29	118	118	NUM
ejpam-3209	225	30	-	-	SYM
ejpam-3209	225	31	134	134	NUM
ejpam-3209	225	32	.	.	PUNCT
ejpam-3209	226	1	[	[	X
ejpam-3209	226	2	2	2	NUM
ejpam-3209	226	3	]	]	X
ejpam-3209	226	4	b.	b.	PROPN
ejpam-3209	226	5	barnes	barnes	PROPN
ejpam-3209	226	6	,	,	PUNCT
ejpam-3209	226	7	e.	e.	PROPN
ejpam-3209	226	8	d.	d.	PROPN
ejpam-3209	226	9	j.	j.	PROPN
ejpam-3209	226	10	owusu	owusu	PROPN
ejpam-3209	226	11	-	-	PUNCT
ejpam-3209	226	12	ansah	ansah	PROPN
ejpam-3209	226	13	,	,	PUNCT
ejpam-3209	226	14	s.	s.	PROPN
ejpam-3209	226	15	k.	k.	PROPN
ejpam-3209	226	16	amponsah	amponsah	PROPN
ejpam-3209	226	17	and	and	CCONJ
ejpam-3209	226	18	i.	i.	PROPN
ejpam-3209	226	19	a.	a.	PROPN
ejpam-3209	226	20	adjei	adjei	PROPN
ejpam-3209	226	21	,	,	PUNCT
ejpam-3209	226	22	the	the	DET
ejpam-3209	226	23	proofs	proof	NOUN
ejpam-3209	226	24	of	of	ADP
ejpam-3209	226	25	triangle	triangle	NOUN
ejpam-3209	226	26	inequality	inequality	NOUN
ejpam-3209	226	27	using	use	VERB
ejpam-3209	226	28	binomial	binomial	ADJ
ejpam-3209	226	29	inequalities	inequality	NOUN
ejpam-3209	226	30	,	,	PUNCT
ejpam-3209	226	31	european	european	PROPN
ejpam-3209	226	32	journal	journal	PROPN
ejpam-3209	226	33	of	of	ADP
ejpam-3209	226	34	pure	pure	ADJ
ejpam-3209	226	35	and	and	CCONJ
ejpam-3209	226	36	applied	applied	ADJ
ejpam-3209	226	37	mathematics	mathematic	NOUN
ejpam-3209	226	38	,	,	PUNCT
ejpam-3209	226	39	11,1	11,1	NUM
ejpam-3209	226	40	,	,	PUNCT
ejpam-3209	226	41	(	(	PUNCT
ejpam-3209	226	42	2018	2018	NUM
ejpam-3209	226	43	)	)	PUNCT
ejpam-3209	226	44	.	.	PUNCT
ejpam-3209	227	1	pp	pp	ADV
ejpam-3209	227	2	.	.	PUNCT
ejpam-3209	228	1	352	352	NUM
ejpam-3209	228	2	-	-	SYM
ejpam-3209	228	3	361	361	NUM
ejpam-3209	228	4	.	.	PUNCT
ejpam-3209	229	1	[	[	X
ejpam-3209	229	2	3	3	X
ejpam-3209	229	3	]	]	X
ejpam-3209	229	4	d.	d.	PROPN
ejpam-3209	229	5	s.	s.	PROPN
ejpam-3209	229	6	mitrinović	mitrinović	PROPN
ejpam-3209	229	7	,	,	PUNCT
ejpam-3209	229	8	analytic	analytic	ADJ
ejpam-3209	229	9	inequalities	inequality	NOUN
ejpam-3209	229	10	,	,	PUNCT
ejpam-3209	229	11	springer	springer	NOUN
ejpam-3209	229	12	-	-	PUNCT
ejpam-3209	229	13	verlag	verlag	PROPN
ejpam-3209	229	14	,	,	PUNCT
ejpam-3209	229	15	new	new	PROPN
ejpam-3209	229	16	york	york	PROPN
ejpam-3209	229	17	/	/	SYM
ejpam-3209	229	18	berlin	berlin	PROPN
ejpam-3209	229	19	;	;	PUNCT
ejpam-3209	229	20	1961	1961	NUM
ejpam-3209	229	21	.	.	PUNCT
ejpam-3209	230	1	[	[	X
ejpam-3209	230	2	4	4	NUM
ejpam-3209	230	3	]	]	X
ejpam-3209	230	4	d.	d.	PROPN
ejpam-3209	230	5	rüthing	rüthing	PROPN
ejpam-3209	230	6	,	,	PUNCT
ejpam-3209	230	7	proofs	proof	NOUN
ejpam-3209	230	8	of	of	ADP
ejpam-3209	230	9	the	the	DET
ejpam-3209	230	10	arithmetic	arithmetic	ADJ
ejpam-3209	230	11	mean	mean	ADJ
ejpam-3209	230	12	-	-	PUNCT
ejpam-3209	230	13	geometric	geometric	ADJ
ejpam-3209	230	14	mean	mean	NOUN
ejpam-3209	230	15	inequality	inequality	NOUN
ejpam-3209	230	16	.	.	PUNCT
ejpam-3209	231	1	international	international	ADJ
ejpam-3209	231	2	journal	journal	PROPN
ejpam-3209	231	3	of	of	ADP
ejpam-3209	231	4	mathematical	mathematical	ADJ
ejpam-3209	231	5	education	education	NOUN
ejpam-3209	231	6	in	in	ADP
ejpam-3209	231	7	science	science	NOUN
ejpam-3209	231	8	and	and	CCONJ
ejpam-3209	231	9	technology	technology	NOUN
ejpam-3209	231	10	,	,	PUNCT
ejpam-3209	231	11	13	13	NUM
ejpam-3209	231	12	,	,	PUNCT
ejpam-3209	231	13	no	no	INTJ
ejpam-3209	231	14	.	.	NOUN
ejpam-3209	231	15	1	1	NUM
ejpam-3209	231	16	(	(	PUNCT
ejpam-3209	231	17	1982	1982	NUM
ejpam-3209	231	18	)	)	PUNCT
ejpam-3209	231	19	.	.	PUNCT
ejpam-3209	232	1	pp	pp	ADV
ejpam-3209	232	2	.	.	PUNCT
ejpam-3209	233	1	49	49	NUM
ejpam-3209	233	2	-	-	SYM
ejpam-3209	233	3	54	54	NUM
ejpam-3209	233	4	.	.	PUNCT
ejpam-3209	234	1	[	[	X
ejpam-3209	234	2	5	5	X
ejpam-3209	234	3	]	]	PUNCT
ejpam-3209	234	4	g.	g.	PROPN
ejpam-3209	234	5	h.	h.	PROPN
ejpam-3209	234	6	hardy	hardy	PROPN
ejpam-3209	234	7	,	,	PUNCT
ejpam-3209	234	8	j.	j.	PROPN
ejpam-3209	234	9	e.	e.	PROPN
ejpam-3209	234	10	littlewood	littlewood	PROPN
ejpam-3209	234	11	and	and	CCONJ
ejpam-3209	234	12	g.	g.	PROPN
ejpam-3209	234	13	polya	polya	PROPN
ejpam-3209	234	14	,	,	PUNCT
ejpam-3209	234	15	inequalities	inequality	NOUN
ejpam-3209	234	16	.	.	PUNCT
ejpam-3209	235	1	london	london	PROPN
ejpam-3209	235	2	,	,	PUNCT
ejpam-3209	235	3	new	new	PROPN
ejpam-3209	235	4	york	york	PROPN
ejpam-3209	235	5	;	;	PUNCT
ejpam-3209	235	6	cambridge	cambridge	PROPN
ejpam-3209	235	7	university	university	PROPN
ejpam-3209	235	8	press	press	NOUN
ejpam-3209	235	9	,	,	PUNCT
ejpam-3209	235	10	1978	1978	NUM
ejpam-3209	235	11	.	.	PUNCT
ejpam-3209	236	1	[	[	X
ejpam-3209	236	2	6	6	NUM
ejpam-3209	236	3	]	]	PUNCT
ejpam-3209	236	4	d.	d.	PROPN
ejpam-3209	236	5	s.	s.	PROPN
ejpam-3209	236	6	mitrinović	mitrinović	PROPN
ejpam-3209	236	7	,	,	PUNCT
ejpam-3209	236	8	j.	j.	PROPN
ejpam-3209	236	9	e.	e.	PROPN
ejpam-3209	236	10	pec̆arić	pec̆arić	PROPN
ejpam-3209	236	11	,	,	PUNCT
ejpam-3209	236	12	and	and	CCONJ
ejpam-3209	236	13	a.	a.	NOUN
ejpam-3209	236	14	m.	m.	PROPN
ejpam-3209	236	15	fink	fink	PROPN
ejpam-3209	236	16	,	,	PUNCT
ejpam-3209	236	17	classical	classical	ADJ
ejpam-3209	236	18	and	and	CCONJ
ejpam-3209	236	19	new	new	ADJ
ejpam-3209	236	20	inequalities	inequality	NOUN
ejpam-3209	236	21	in	in	ADP
ejpam-3209	236	22	analysis	analysis	NOUN
ejpam-3209	236	23	,	,	PUNCT
ejpam-3209	236	24	kluwer	kluwer	NOUN
ejpam-3209	236	25	academic	academic	NOUN
ejpam-3209	236	26	,	,	PUNCT
ejpam-3209	236	27	dordrecht	dordrecht	X
ejpam-3209	236	28	(	(	PUNCT
ejpam-3209	236	29	1993).pp	1993).pp	NUM
ejpam-3209	236	30	.	.	PUNCT
ejpam-3209	236	31	69	69	NUM
ejpam-3209	236	32	-	-	SYM
ejpam-3209	236	33	72	72	NUM
ejpam-3209	236	34	.	.	PUNCT
ejpam-3209	237	1	[	[	X
ejpam-3209	237	2	7	7	X
ejpam-3209	237	3	]	]	X
ejpam-3209	237	4	s.	s.	PROPN
ejpam-3209	237	5	alan	alan	PROPN
ejpam-3209	237	6	,	,	PUNCT
ejpam-3209	237	7	a	a	DET
ejpam-3209	237	8	historical	historical	ADJ
ejpam-3209	237	9	review	review	NOUN
ejpam-3209	237	10	of	of	ADP
ejpam-3209	237	11	the	the	DET
ejpam-3209	237	12	isoperimetric	isoperimetric	NOUN
ejpam-3209	237	13	theorem	theorem	NOUN
ejpam-3209	237	14	in	in	ADP
ejpam-3209	237	15	2−	2−	NUM
ejpam-3209	237	16	d	d	NOUN
ejpam-3209	237	17	,	,	PUNCT
ejpam-3209	237	18	and	and	CCONJ
ejpam-3209	237	19	its	its	PRON
ejpam-3209	237	20	place	place	NOUN
ejpam-3209	237	21	in	in	ADP
ejpam-3209	237	22	elementary	elementary	ADJ
ejpam-3209	237	23	plane	plane	NOUN
ejpam-3209	237	24	geometry	geometry	NOUN
ejpam-3209	237	25	,	,	PUNCT
ejpam-3209	237	26	courant	courant	PROPN
ejpam-3209	237	27	institute	institute	PROPN
ejpam-3209	237	28	of	of	ADP
ejpam-3209	237	29	mathematical	mathematical	PROPN
ejpam-3209	237	30	sciences	sciences	PROPN
ejpam-3209	237	31	,	,	PUNCT
ejpam-3209	237	32	new	new	PROPN
ejpam-3209	237	33	york	york	PROPN
ejpam-3209	237	34	university	university	PROPN
ejpam-3209	237	35	;	;	PUNCT
ejpam-3209	237	36	2013	2013	NUM
ejpam-3209	237	37	.	.	PUNCT
ejpam-3209	238	1	http	http	NOUN
ejpam-3209	238	2	:	:	PUNCT
ejpam-3209	238	3	//www.cs.nyu.edu	//www.cs.nyu.edu	PUNCT
ejpam-3209	238	4	/	/	SYM
ejpam-3209	238	5	faculty	faculty	NOUN
ejpam-3209	238	6	/	/	SYM
ejpam-3209	238	7	siegel	siegel	NOUN
ejpam-3209	238	8	/	/	SYM
ejpam-3209	238	9	sciam.pdf	sciam.pdf	X
ejpam-3209	238	10	.	.	PUNCT
ejpam-3209	239	1	[	[	X
ejpam-3209	239	2	8	8	X
ejpam-3209	239	3	]	]	PUNCT
ejpam-3209	239	4	j.	j.	PROPN
ejpam-3209	239	5	v.	v.	PROPN
ejpam-3209	239	6	grabiner	grabiner	PROPN
ejpam-3209	239	7	,	,	PUNCT
ejpam-3209	239	8	was	be	AUX
ejpam-3209	239	9	newton	newton	PROPN
ejpam-3209	239	10	’s	’s	PART
ejpam-3209	239	11	calculus	calculus	NOUN
ejpam-3209	239	12	a	a	DET
ejpam-3209	239	13	dead	dead	ADJ
ejpam-3209	239	14	end	end	NOUN
ejpam-3209	239	15	?	?	PUNCT
ejpam-3209	240	1	the	the	DET
ejpam-3209	240	2	continental	continental	ADJ
ejpam-3209	240	3	influence	influence	NOUN
ejpam-3209	240	4	of	of	ADP
ejpam-3209	240	5	maclaurin	maclaurin	NOUN
ejpam-3209	240	6	’s	’s	PART
ejpam-3209	240	7	treatise	treatise	NOUN
ejpam-3209	240	8	of	of	ADP
ejpam-3209	240	9	fluxions	fluxion	NOUN
ejpam-3209	240	10	,	,	PUNCT
ejpam-3209	240	11	american	american	ADJ
ejpam-3209	240	12	mathematics	mathematics	PROPN
ejpam-3209	240	13	monthly	monthly	ADV
ejpam-3209	240	14	,	,	PUNCT
ejpam-3209	240	15	104	104	NUM
ejpam-3209	240	16	,	,	PUNCT
ejpam-3209	240	17	(	(	PUNCT
ejpam-3209	240	18	1997).pp	1997).pp	NOUN
ejpam-3209	240	19	.	.	PUNCT
ejpam-3209	240	20	393	393	NUM
ejpam-3209	240	21	-	-	SYM
ejpam-3209	240	22	410	410	NUM
ejpam-3209	240	23	.	.	PUNCT
ejpam-3209	241	1	[	[	X
ejpam-3209	241	2	9	9	NUM
ejpam-3209	241	3	]	]	PUNCT
ejpam-3209	241	4	s.	s.	PROPN
ejpam-3209	241	5	wu	wu	PROPN
ejpam-3209	241	6	,	,	PUNCT
ejpam-3209	241	7	some	some	PRON
ejpam-3209	241	8	results	result	VERB
ejpam-3209	241	9	on	on	ADP
ejpam-3209	241	10	extending	extend	VERB
ejpam-3209	241	11	and	and	CCONJ
ejpam-3209	241	12	sharpening	sharpen	VERB
ejpam-3209	241	13	the	the	DET
ejpam-3209	241	14	weierstrass	weierstrass	NOUN
ejpam-3209	241	15	product	product	NOUN
ejpam-3209	241	16	inequalities	inequality	NOUN
ejpam-3209	241	17	,	,	PUNCT
ejpam-3209	241	18	journal	journal	NOUN
ejpam-3209	241	19	of	of	ADP
ejpam-3209	241	20	mathematical	mathematical	ADJ
ejpam-3209	241	21	analysis	analysis	NOUN
ejpam-3209	241	22	and	and	CCONJ
ejpam-3209	241	23	applications	application	NOUN
ejpam-3209	241	24	,	,	PUNCT
ejpam-3209	241	25	308	308	NUM
ejpam-3209	241	26	,	,	PUNCT
ejpam-3209	241	27	(	(	PUNCT
ejpam-3209	241	28	2005).pp	2005).pp	NOUN
ejpam-3209	241	29	.	.	PUNCT
ejpam-3209	241	30	689	689	NUM
ejpam-3209	241	31	-	-	SYM
ejpam-3209	241	32	702	702	NUM
ejpam-3209	241	33	.	.	PUNCT
ejpam-3209	242	1	[	[	X
ejpam-3209	242	2	10	10	NUM
ejpam-3209	242	3	]	]	PUNCT
ejpam-3209	242	4	m.	m.	NOUN
ejpam-3209	242	5	s.	s.	PROPN
ejpam-3209	242	6	klamkin	klamkin	PROPN
ejpam-3209	242	7	,	,	PUNCT
ejpam-3209	242	8	extension	extension	NOUN
ejpam-3209	242	9	of	of	ADP
ejpam-3209	242	10	the	the	DET
ejpam-3209	242	11	weierstrass	weierstrass	NOUN
ejpam-3209	242	12	product	product	NOUN
ejpam-3209	242	13	inequality	inequality	PROPN
ejpam-3209	242	14	ii	ii	PROPN
ejpam-3209	242	15	,	,	PUNCT
ejpam-3209	242	16	amer	amer	PROPN
ejpam-3209	242	17	.	.	PROPN
ejpam-3209	242	18	math	math	PROPN
ejpam-3209	242	19	.	.	PUNCT
ejpam-3209	242	20	,	,	PUNCT
ejpam-3209	242	21	82	82	NUM
ejpam-3209	242	22	,	,	PUNCT
ejpam-3209	242	23	(	(	PUNCT
ejpam-3209	242	24	1975).pp	1975).pp	NUM
ejpam-3209	242	25	.	.	PUNCT
ejpam-3209	243	1	741	741	NUM
ejpam-3209	243	2	-	-	SYM
ejpam-3209	243	3	742	742	NUM
ejpam-3209	243	4	.	.	PUNCT
ejpam-3209	244	1	[	[	X
ejpam-3209	244	2	11	11	NUM
ejpam-3209	244	3	]	]	X
ejpam-3209	244	4	d.	d.	PROPN
ejpam-3209	244	5	s.	s.	PROPN
ejpam-3209	244	6	mitrinović	mitrinović	PROPN
ejpam-3209	244	7	and	and	CCONJ
ejpam-3209	244	8	p.	p.	PROPN
ejpam-3209	244	9	m.	m.	NOUN
ejpam-3209	244	10	vasić	vasić	NOUN
ejpam-3209	244	11	,	,	PUNCT
ejpam-3209	244	12	analytic	analytic	ADJ
ejpam-3209	244	13	inequalities	inequality	NOUN
ejpam-3209	244	14	,	,	PUNCT
ejpam-3209	244	15	springer	springer	NOUN
ejpam-3209	244	16	-	-	PUNCT
ejpam-3209	244	17	verlag	verlag	PROPN
ejpam-3209	244	18	,	,	PUNCT
ejpam-3209	244	19	new	new	PROPN
ejpam-3209	244	20	york	york	PROPN
ejpam-3209	244	21	,	,	PUNCT
ejpam-3209	244	22	1970	1970	NUM
ejpam-3209	244	23	references	reference	NOUN
ejpam-3209	244	24	389	389	NUM
ejpam-3209	244	25	[	[	X
ejpam-3209	244	26	12	12	NUM
ejpam-3209	244	27	]	]	PUNCT
ejpam-3209	244	28	i.	i.	PROPN
ejpam-3209	244	29	schur	schur	PROPN
ejpam-3209	244	30	,	,	PUNCT
ejpam-3209	244	31	bemerkungen	bemerkungen	PROPN
ejpam-3209	244	32	zur	zur	PROPN
ejpam-3209	244	33	theorie	theorie	PROPN
ejpam-3209	244	34	de	de	X
ejpam-3209	244	35	beschränkten	beschränkten	PROPN
ejpam-3209	244	36	bilinearformen	bilinearforman	NOUN
ejpam-3209	244	37	mit	mit	PROPN
ejpam-3209	244	38	unendlich	unendlich	PROPN
ejpam-3209	244	39	vielen	vielen	VERB
ejpam-3209	244	40	veränderlichen	veränderlichen	PROPN
ejpam-3209	244	41	,	,	PUNCT
ejpam-3209	244	42	journal	journal	NOUN
ejpam-3209	244	43	of	of	ADP
ejpam-3209	244	44	mathematics	mathematic	NOUN
ejpam-3209	244	45	,	,	PUNCT
ejpam-3209	244	46	140	140	NUM
ejpam-3209	244	47	,	,	PUNCT
ejpam-3209	244	48	(	(	PUNCT
ejpam-3209	244	49	1911).pp	1911).pp	NUM
ejpam-3209	244	50	.	.	NOUN
ejpam-3209	245	1	1	1	NUM
ejpam-3209	245	2	-	-	SYM
ejpam-3209	245	3	28	28	NUM
ejpam-3209	245	4	.	.	PUNCT
ejpam-3209	246	1	[	[	X
ejpam-3209	246	2	13	13	NUM
ejpam-3209	246	3	]	]	X
ejpam-3209	246	4	b.	b.	PROPN
ejpam-3209	246	5	kolman	kolman	PROPN
ejpam-3209	246	6	and	and	CCONJ
ejpam-3209	246	7	d.	d.	PROPN
ejpam-3209	246	8	r.	r.	PROPN
ejpam-3209	246	9	hill	hill	PROPN
ejpam-3209	246	10	,	,	PUNCT
ejpam-3209	246	11	elementary	elementary	ADJ
ejpam-3209	246	12	linear	linear	PROPN
ejpam-3209	246	13	algebra	algebra	PROPN
ejpam-3209	246	14	,	,	PUNCT
ejpam-3209	246	15	prentice	prentice	NOUN
ejpam-3209	246	16	-	-	PUNCT
ejpam-3209	246	17	hall	hall	NOUN
ejpam-3209	246	18	,	,	PUNCT
ejpam-3209	246	19	inc	inc	PROPN
ejpam-3209	246	20	,	,	PUNCT
ejpam-3209	246	21	new	new	PROPN
ejpam-3209	246	22	jersey	jersey	PROPN
ejpam-3209	246	23	;	;	PUNCT
ejpam-3209	246	24	2000	2000	NUM
ejpam-3209	246	25	.	.	PUNCT
ejpam-3209	247	1	[	[	X
ejpam-3209	247	2	14	14	NUM
ejpam-3209	247	3	]	]	X
ejpam-3209	247	4	j.	j.	PROPN
ejpam-3209	247	5	t.	t.	PROPN
ejpam-3209	247	6	scheick	scheick	PROPN
ejpam-3209	247	7	,	,	PUNCT
ejpam-3209	247	8	linear	linear	ADJ
ejpam-3209	247	9	algebra	algebra	NOUN
ejpam-3209	247	10	with	with	ADP
ejpam-3209	247	11	applications	application	NOUN
ejpam-3209	247	12	,	,	PUNCT
ejpam-3209	247	13	prentice	prentice	NOUN
ejpam-3209	247	14	-	-	PUNCT
ejpam-3209	247	15	hall	hall	NOUN
ejpam-3209	247	16	,	,	PUNCT
ejpam-3209	247	17	inc	inc	PROPN
ejpam-3209	247	18	,	,	PUNCT
ejpam-3209	247	19	new	new	PROPN
ejpam-3209	247	20	york	york	PROPN
ejpam-3209	247	21	;	;	PUNCT
ejpam-3209	247	22	1997	1997	NUM
ejpam-3209	247	23	.	.	PUNCT
ejpam-3209	248	1	[	[	X
ejpam-3209	248	2	15	15	NUM
ejpam-3209	248	3	]	]	X
ejpam-3209	248	4	j.	j.	PROPN
ejpam-3209	248	5	t.	t.	PROPN
ejpam-3209	248	6	oden	oden	PROPN
ejpam-3209	248	7	,	,	PUNCT
ejpam-3209	248	8	applied	apply	VERB
ejpam-3209	248	9	functional	functional	ADJ
ejpam-3209	248	10	analysis	analysis	NOUN
ejpam-3209	248	11	:	:	PUNCT
ejpam-3209	248	12	a	a	DET
ejpam-3209	248	13	first	first	ADJ
ejpam-3209	248	14	course	course	NOUN
ejpam-3209	248	15	for	for	ADP
ejpam-3209	248	16	students	student	NOUN
ejpam-3209	248	17	of	of	ADP
ejpam-3209	248	18	mechanics	mechanic	NOUN
ejpam-3209	248	19	and	and	CCONJ
ejpam-3209	248	20	engineering	engineering	NOUN
ejpam-3209	248	21	science	science	NOUN
ejpam-3209	248	22	,	,	PUNCT
ejpam-3209	248	23	prentice	prentice	NOUN
ejpam-3209	248	24	-	-	PUNCT
ejpam-3209	248	25	hall	hall	NOUN
ejpam-3209	248	26	,	,	PUNCT
ejpam-3209	248	27	inc	inc	PROPN
ejpam-3209	248	28	,	,	PUNCT
ejpam-3209	248	29	new	new	PROPN
ejpam-3209	248	30	jersey	jersey	PROPN
ejpam-3209	248	31	;	;	PUNCT
ejpam-3209	248	32	1979	1979	NUM
ejpam-3209	248	33	.	.	PUNCT
ejpam-3209	249	1	[	[	X
ejpam-3209	249	2	16	16	NUM
ejpam-3209	249	3	]	]	X
ejpam-3209	249	4	c.	c.	PROPN
ejpam-3209	249	5	e.	e.	PROPN
ejpam-3209	249	6	chidume	chidume	PROPN
ejpam-3209	249	7	,	,	PUNCT
ejpam-3209	249	8	functional	functional	ADJ
ejpam-3209	249	9	analysis	analysis	NOUN
ejpam-3209	249	10	:	:	PUNCT
ejpam-3209	249	11	an	an	DET
ejpam-3209	249	12	introduction	introduction	NOUN
ejpam-3209	249	13	to	to	ADP
ejpam-3209	249	14	metric	metric	ADJ
ejpam-3209	249	15	spaces	space	NOUN
ejpam-3209	249	16	,	,	PUNCT
ejpam-3209	249	17	longman	longman	NOUN
ejpam-3209	249	18	,	,	PUNCT
ejpam-3209	249	19	nigeria	nigeria	PROPN
ejpam-3209	249	20	;	;	PUNCT
ejpam-3209	249	21	1989	1989	NUM
ejpam-3209	249	22	.	.	PUNCT
ejpam-3209	250	1	[	[	X
ejpam-3209	250	2	17	17	NUM
ejpam-3209	250	3	]	]	X
ejpam-3209	250	4	h.	h.	PROPN
ejpam-3209	250	5	royden	royden	PROPN
ejpam-3209	250	6	and	and	CCONJ
ejpam-3209	250	7	p.	p.	PROPN
ejpam-3209	250	8	fitxpatrick	fitxpatrick	PROPN
ejpam-3209	250	9	,	,	PUNCT
ejpam-3209	250	10	real	real	ADJ
ejpam-3209	250	11	analysis	analysis	NOUN
ejpam-3209	250	12	.	.	PUNCT
ejpam-3209	251	1	pearson	pearson	PROPN
ejpam-3209	251	2	education	education	PROPN
ejpam-3209	251	3	,	,	PUNCT
ejpam-3209	251	4	inc	inc	PROPN
ejpam-3209	251	5	,	,	PUNCT
ejpam-3209	251	6	4th	4th	ADJ
ejpam-3209	251	7	ed	ed	NOUN
ejpam-3209	251	8	.	.	PROPN
ejpam-3209	251	9	,	,	PUNCT
ejpam-3209	251	10	upper	upper	ADJ
ejpam-3209	251	11	saddle	saddle	NOUN
ejpam-3209	251	12	river	river	NOUN
ejpam-3209	251	13	;	;	PUNCT
ejpam-3209	251	14	2010	2010	NUM
ejpam-3209	251	15	.	.	PUNCT