id	sid	tid	token	lemma	pos
ejpam-3284	1	1	european	european	PROPN
ejpam-3284	1	2	journal	journal	PROPN
ejpam-3284	1	3	of	of	ADP
ejpam-3284	1	4	pure	pure	ADJ
ejpam-3284	1	5	and	and	CCONJ
ejpam-3284	1	6	applied	apply	VERB
ejpam-3284	1	7	mathematics	mathematic	NOUN
ejpam-3284	1	8	vol	vol	NOUN
ejpam-3284	1	9	.	.	PUNCT
ejpam-3284	2	1	11	11	NUM
ejpam-3284	2	2	,	,	PUNCT
ejpam-3284	2	3	no	no	INTJ
ejpam-3284	2	4	.	.	NOUN
ejpam-3284	2	5	3	3	NUM
ejpam-3284	2	6	,	,	PUNCT
ejpam-3284	2	7	2018	2018	NUM
ejpam-3284	2	8	,	,	PUNCT
ejpam-3284	2	9	740	740	NUM
ejpam-3284	2	10	-	-	SYM
ejpam-3284	2	11	750	750	NUM
ejpam-3284	2	12	issn	issn	PROPN
ejpam-3284	2	13	1307	1307	NUM
ejpam-3284	2	14	-	-	SYM
ejpam-3284	2	15	5543	5543	NUM
ejpam-3284	2	16	–	–	PUNCT
ejpam-3284	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3284	2	18	published	publish	VERB
ejpam-3284	2	19	by	by	ADP
ejpam-3284	2	20	new	new	PROPN
ejpam-3284	2	21	york	york	PROPN
ejpam-3284	2	22	business	business	PROPN
ejpam-3284	2	23	global	global	PROPN
ejpam-3284	2	24	the	the	DET
ejpam-3284	2	25	product	product	NOUN
ejpam-3284	2	26	-	-	PUNCT
ejpam-3284	2	27	normed	norme	VERB
ejpam-3284	2	28	linear	linear	ADJ
ejpam-3284	2	29	space	space	NOUN
ejpam-3284	2	30	benedict	benedict	PROPN
ejpam-3284	2	31	barnes1∗	barnes1∗	PROPN
ejpam-3284	2	32	,	,	PUNCT
ejpam-3284	2	33	i.	i.	PROPN
ejpam-3284	2	34	a.	a.	NOUN
ejpam-3284	2	35	adjei2	adjei2	PROPN
ejpam-3284	2	36	,	,	PUNCT
ejpam-3284	2	37	c.	c.	PROPN
ejpam-3284	2	38	sebil3	sebil3	PROPN
ejpam-3284	2	39	,	,	PUNCT
ejpam-3284	3	1	e.	e.	PROPN
ejpam-3284	3	2	harris4	harris4	PROPN
ejpam-3284	3	3	1,2,3,4	1,2,3,4	NUM
ejpam-3284	3	4	department	department	NOUN
ejpam-3284	3	5	of	of	ADP
ejpam-3284	3	6	mathematics	mathematics	PROPN
ejpam-3284	3	7	,	,	PUNCT
ejpam-3284	3	8	kwame	kwame	PROPN
ejpam-3284	3	9	nkrumah	nkrumah	PROPN
ejpam-3284	3	10	university	university	PROPN
ejpam-3284	3	11	of	of	ADP
ejpam-3284	3	12	science	science	NOUN
ejpam-3284	3	13	and	and	CCONJ
ejpam-3284	3	14	technology	technology	NOUN
ejpam-3284	3	15	,	,	PUNCT
ejpam-3284	3	16	kumasi	kumasi	PROPN
ejpam-3284	3	17	,	,	PUNCT
ejpam-3284	3	18	ghana	ghana	PROPN
ejpam-3284	3	19	abstract	abstract	NOUN
ejpam-3284	3	20	.	.	PUNCT
ejpam-3284	4	1	in	in	ADP
ejpam-3284	4	2	this	this	DET
ejpam-3284	4	3	paper	paper	NOUN
ejpam-3284	4	4	,	,	PUNCT
ejpam-3284	4	5	both	both	CCONJ
ejpam-3284	4	6	the	the	DET
ejpam-3284	4	7	product	product	NOUN
ejpam-3284	4	8	-	-	PUNCT
ejpam-3284	4	9	normed	norme	VERB
ejpam-3284	4	10	linear	linear	ADJ
ejpam-3284	4	11	space	space	NOUN
ejpam-3284	4	12	p	p	NOUN
ejpam-3284	4	13	−nls	−nls	NOUN
ejpam-3284	4	14	(	(	PUNCT
ejpam-3284	4	15	product	product	NOUN
ejpam-3284	4	16	-	-	PUNCT
ejpam-3284	4	17	banach	banach	NOUN
ejpam-3284	4	18	space	space	NOUN
ejpam-3284	4	19	)	)	PUNCT
ejpam-3284	4	20	and	and	CCONJ
ejpam-3284	4	21	product	product	NOUN
ejpam-3284	4	22	-	-	PUNCT
ejpam-3284	4	23	semi	semi	ADV
ejpam-3284	4	24	-	-	ADJ
ejpam-3284	4	25	normed	normed	ADJ
ejpam-3284	4	26	linear	linear	ADJ
ejpam-3284	4	27	space	space	NOUN
ejpam-3284	4	28	(	(	PUNCT
ejpam-3284	4	29	product	product	NOUN
ejpam-3284	4	30	-	-	PUNCT
ejpam-3284	4	31	semi	semi	ADJ
ejpam-3284	4	32	-	-	ADJ
ejpam-3284	4	33	banch	banch	ADJ
ejpam-3284	4	34	space	space	NOUN
ejpam-3284	4	35	)	)	PUNCT
ejpam-3284	4	36	are	be	AUX
ejpam-3284	4	37	introduced	introduce	VERB
ejpam-3284	4	38	.	.	PUNCT
ejpam-3284	5	1	these	these	DET
ejpam-3284	5	2	normed	normed	PROPN
ejpam-3284	5	3	linear	linear	PROPN
ejpam-3284	5	4	spaces	space	NOUN
ejpam-3284	5	5	are	be	AUX
ejpam-3284	5	6	endowed	endow	VERB
ejpam-3284	5	7	with	with	ADP
ejpam-3284	5	8	the	the	DET
ejpam-3284	5	9	first	first	ADJ
ejpam-3284	5	10	and	and	CCONJ
ejpam-3284	5	11	second	second	ADJ
ejpam-3284	5	12	product	product	NOUN
ejpam-3284	5	13	inequalities	inequality	NOUN
ejpam-3284	5	14	,	,	PUNCT
ejpam-3284	5	15	which	which	PRON
ejpam-3284	5	16	have	have	VERB
ejpam-3284	5	17	a	a	DET
ejpam-3284	5	18	lot	lot	NOUN
ejpam-3284	5	19	of	of	ADP
ejpam-3284	5	20	applications	application	NOUN
ejpam-3284	5	21	in	in	ADP
ejpam-3284	5	22	linear	linear	ADJ
ejpam-3284	5	23	algebra	algebra	NOUN
ejpam-3284	5	24	and	and	CCONJ
ejpam-3284	5	25	differential	differential	ADJ
ejpam-3284	5	26	equations	equation	NOUN
ejpam-3284	5	27	.	.	PUNCT
ejpam-3284	6	1	in	in	ADP
ejpam-3284	6	2	addition	addition	NOUN
ejpam-3284	6	3	,	,	PUNCT
ejpam-3284	6	4	we	we	PRON
ejpam-3284	6	5	showed	show	VERB
ejpam-3284	6	6	that	that	SCONJ
ejpam-3284	6	7	p	p	NOUN
ejpam-3284	6	8	−	−	PROPN
ejpam-3284	6	9	nls	nls	NOUN
ejpam-3284	6	10	admits	admit	VERB
ejpam-3284	6	11	functional	functional	ADJ
ejpam-3284	6	12	properties	property	NOUN
ejpam-3284	6	13	such	such	ADJ
ejpam-3284	6	14	as	as	ADP
ejpam-3284	6	15	completeness	completeness	NOUN
ejpam-3284	6	16	,	,	PUNCT
ejpam-3284	6	17	continuity	continuity	NOUN
ejpam-3284	6	18	and	and	CCONJ
ejpam-3284	6	19	the	the	DET
ejpam-3284	6	20	fixed	fix	VERB
ejpam-3284	6	21	point	point	NOUN
ejpam-3284	6	22	.	.	PUNCT
ejpam-3284	7	1	2010	2010	NUM
ejpam-3284	7	2	mathematics	mathematic	NOUN
ejpam-3284	7	3	subject	subject	NOUN
ejpam-3284	7	4	classifications	classification	NOUN
ejpam-3284	7	5	:	:	PUNCT
ejpam-3284	7	6	4b47	4b47	NOUN
ejpam-3284	7	7	,	,	PUNCT
ejpam-3284	7	8	44b49	44b49	NUM
ejpam-3284	7	9	key	key	ADJ
ejpam-3284	7	10	words	word	NOUN
ejpam-3284	7	11	and	and	CCONJ
ejpam-3284	7	12	phrases	phrase	NOUN
ejpam-3284	7	13	:	:	PUNCT
ejpam-3284	7	14	product	product	NOUN
ejpam-3284	7	15	-	-	PUNCT
ejpam-3284	7	16	normed	norme	VERB
ejpam-3284	7	17	linear	linear	ADJ
ejpam-3284	7	18	space	space	NOUN
ejpam-3284	7	19	,	,	PUNCT
ejpam-3284	7	20	product	product	NOUN
ejpam-3284	7	21	-	-	PUNCT
ejpam-3284	7	22	semi	semi	ADV
ejpam-3284	7	23	-	-	ADJ
ejpam-3284	7	24	normed	normed	ADJ
ejpam-3284	7	25	linear	linear	ADJ
ejpam-3284	7	26	space	space	NOUN
ejpam-3284	7	27	,	,	PUNCT
ejpam-3284	7	28	completeness	completeness	NOUN
ejpam-3284	7	29	,	,	PUNCT
ejpam-3284	7	30	fixed	fix	VERB
ejpam-3284	7	31	point	point	NOUN
ejpam-3284	7	32	theorem	theorem	NOUN
ejpam-3284	7	33	1	1	NUM
ejpam-3284	7	34	.	.	PUNCT
ejpam-3284	7	35	introduction	introduction	NOUN
ejpam-3284	7	36	in	in	ADP
ejpam-3284	7	37	recent	recent	ADJ
ejpam-3284	7	38	times	time	NOUN
ejpam-3284	7	39	,	,	PUNCT
ejpam-3284	7	40	the	the	DET
ejpam-3284	7	41	solution	solution	NOUN
ejpam-3284	7	42	space	space	NOUN
ejpam-3284	7	43	of	of	ADP
ejpam-3284	7	44	a	a	DET
ejpam-3284	7	45	mathematical	mathematical	ADJ
ejpam-3284	7	46	problem	problem	NOUN
ejpam-3284	7	47	has	have	AUX
ejpam-3284	7	48	become	become	VERB
ejpam-3284	7	49	necessary	necessary	ADJ
ejpam-3284	7	50	condition	condition	NOUN
ejpam-3284	7	51	for	for	ADP
ejpam-3284	7	52	its	its	PRON
ejpam-3284	7	53	existence	existence	NOUN
ejpam-3284	7	54	.	.	PUNCT
ejpam-3284	8	1	without	without	ADP
ejpam-3284	8	2	the	the	DET
ejpam-3284	8	3	solution	solution	NOUN
ejpam-3284	8	4	space	space	NOUN
ejpam-3284	8	5	,	,	PUNCT
ejpam-3284	8	6	the	the	DET
ejpam-3284	8	7	qualitative	qualitative	NOUN
ejpam-3284	8	8	,	,	PUNCT
ejpam-3284	8	9	as	as	ADV
ejpam-3284	8	10	well	well	ADV
ejpam-3284	8	11	as	as	ADP
ejpam-3284	8	12	,	,	PUNCT
ejpam-3284	8	13	quantitative	quantitative	ADJ
ejpam-3284	8	14	property	property	NOUN
ejpam-3284	8	15	of	of	ADP
ejpam-3284	8	16	the	the	DET
ejpam-3284	8	17	mathematical	mathematical	ADJ
ejpam-3284	8	18	structure	structure	NOUN
ejpam-3284	8	19	can	can	AUX
ejpam-3284	8	20	not	not	PART
ejpam-3284	8	21	be	be	AUX
ejpam-3284	8	22	established	establish	VERB
ejpam-3284	8	23	.	.	PUNCT
ejpam-3284	9	1	in	in	ADP
ejpam-3284	9	2	[	[	X
ejpam-3284	9	3	1	1	NUM
ejpam-3284	9	4	]	]	PUNCT
ejpam-3284	9	5	,	,	PUNCT
ejpam-3284	9	6	the	the	DET
ejpam-3284	9	7	author	author	NOUN
ejpam-3284	9	8	introduced	introduce	VERB
ejpam-3284	9	9	the	the	DET
ejpam-3284	9	10	n−metrics	n−metric	NOUN
ejpam-3284	9	11	.	.	PUNCT
ejpam-3284	10	1	about	about	ADV
ejpam-3284	10	2	30	30	NUM
ejpam-3284	10	3	decades	decade	NOUN
ejpam-3284	10	4	afterwards	afterwards	ADV
ejpam-3284	10	5	,	,	PUNCT
ejpam-3284	10	6	the	the	DET
ejpam-3284	10	7	author	author	NOUN
ejpam-3284	10	8	in	in	ADP
ejpam-3284	10	9	[	[	X
ejpam-3284	10	10	2	2	NUM
ejpam-3284	10	11	]	]	PUNCT
ejpam-3284	10	12	narrowed	narrow	VERB
ejpam-3284	10	13	the	the	DET
ejpam-3284	10	14	n−metric	n−metric	ADJ
ejpam-3284	10	15	space	space	NOUN
ejpam-3284	10	16	to	to	ADP
ejpam-3284	10	17	two	two	NUM
ejpam-3284	10	18	-	-	PUNCT
ejpam-3284	10	19	dimensional	dimensional	ADJ
ejpam-3284	10	20	space	space	NOUN
ejpam-3284	10	21	and	and	CCONJ
ejpam-3284	10	22	also	also	ADV
ejpam-3284	10	23	,	,	PUNCT
ejpam-3284	10	24	obtained	obtain	VERB
ejpam-3284	10	25	some	some	DET
ejpam-3284	10	26	topological	topological	ADJ
ejpam-3284	10	27	properties	property	NOUN
ejpam-3284	10	28	associated	associate	VERB
ejpam-3284	10	29	with	with	ADP
ejpam-3284	10	30	this	this	DET
ejpam-3284	10	31	space	space	NOUN
ejpam-3284	10	32	.	.	PUNCT
ejpam-3284	11	1	definition	definition	NOUN
ejpam-3284	11	2	1	1	NUM
ejpam-3284	11	3	(	(	PUNCT
ejpam-3284	11	4	norm	norm	NOUN
ejpam-3284	11	5	)	)	PUNCT
ejpam-3284	11	6	.	.	PUNCT
ejpam-3284	12	1	let	let	VERB
ejpam-3284	12	2	v	v	PART
ejpam-3284	12	3	be	be	AUX
ejpam-3284	12	4	a	a	DET
ejpam-3284	12	5	linear	linear	ADJ
ejpam-3284	12	6	space	space	NOUN
ejpam-3284	12	7	over	over	ADP
ejpam-3284	12	8	the	the	DET
ejpam-3284	12	9	real	real	ADJ
ejpam-3284	12	10	number	number	NOUN
ejpam-3284	12	11	field	field	NOUN
ejpam-3284	12	12	r.	r.	NOUN
ejpam-3284	12	13	a	a	DET
ejpam-3284	12	14	norm	norm	NOUN
ejpam-3284	12	15	on	on	ADP
ejpam-3284	12	16	v	v	NUM
ejpam-3284	12	17	is	be	AUX
ejpam-3284	12	18	a	a	DET
ejpam-3284	12	19	real	real	ADV
ejpam-3284	12	20	-	-	PUNCT
ejpam-3284	12	21	valued	value	VERB
ejpam-3284	12	22	function	function	NOUN
ejpam-3284	12	23	‖	‖	PROPN
ejpam-3284	12	24	·	·	PUNCT
ejpam-3284	12	25	‖	‖	ADJ
ejpam-3284	12	26	:	:	PUNCT
ejpam-3284	12	27	v	v	X
ejpam-3284	12	28	→	→	SYM
ejpam-3284	12	29	[	[	X
ejpam-3284	12	30	0,∞	0,∞	NOUN
ejpam-3284	12	31	)	)	PUNCT
ejpam-3284	12	32	such	such	ADJ
ejpam-3284	12	33	that	that	PRON
ejpam-3284	12	34	for	for	ADP
ejpam-3284	12	35	any	any	DET
ejpam-3284	12	36	u	u	NOUN
ejpam-3284	12	37	,	,	PUNCT
ejpam-3284	12	38	v	v	NOUN
ejpam-3284	12	39	∈	∈	PROPN
ejpam-3284	12	40	v	v	NOUN
ejpam-3284	12	41	and	and	CCONJ
ejpam-3284	12	42	α	α	NOUN
ejpam-3284	12	43	∈	∈	NOUN
ejpam-3284	12	44	r	r	NOUN
ejpam-3284	12	45	the	the	DET
ejpam-3284	12	46	following	follow	VERB
ejpam-3284	12	47	conditions	condition	NOUN
ejpam-3284	12	48	are	be	AUX
ejpam-3284	12	49	met	meet	VERB
ejpam-3284	12	50	:	:	PUNCT
ejpam-3284	12	51	1.‖u‖	1.‖u‖	NUM
ejpam-3284	12	52	≥	≥	NOUN
ejpam-3284	12	53	0	0	NUM
ejpam-3284	12	54	,	,	PUNCT
ejpam-3284	12	55	and	and	CCONJ
ejpam-3284	12	56	‖u‖	‖u‖	PROPN
ejpam-3284	12	57	=	=	SYM
ejpam-3284	12	58	0	0	PROPN
ejpam-3284	12	59	,	,	PUNCT
ejpam-3284	12	60	iff	iff	PROPN
ejpam-3284	12	61	u	u	NOUN
ejpam-3284	12	62	=	=	NOUN
ejpam-3284	12	63	0	0	NUM
ejpam-3284	12	64	2.‖αu‖	2.‖αu‖	NUM
ejpam-3284	12	65	=	=	SYM
ejpam-3284	12	66	|α|‖u‖	|α|‖u‖	NOUN
ejpam-3284	12	67	,	,	PUNCT
ejpam-3284	12	68	∀	∀	X
ejpam-3284	12	69	u	u	NOUN
ejpam-3284	12	70	∈	∈	PROPN
ejpam-3284	12	71	v	v	NOUN
ejpam-3284	12	72	and	and	CCONJ
ejpam-3284	12	73	α	α	NOUN
ejpam-3284	12	74	∈	∈	NOUN
ejpam-3284	12	75	r	r	NOUN
ejpam-3284	12	76	3.‖u+	3.‖u+	NUM
ejpam-3284	12	77	v‖	v‖	NOUN
ejpam-3284	12	78	≤	≤	NOUN
ejpam-3284	12	79	‖u‖+	‖u‖+	PRON
ejpam-3284	12	80	‖v‖	‖v‖	PROPN
ejpam-3284	12	81	,	,	PUNCT
ejpam-3284	12	82	∀	∀	X
ejpam-3284	12	83	u	u	NOUN
ejpam-3284	12	84	,	,	PUNCT
ejpam-3284	12	85	v	v	PROPN
ejpam-3284	12	86	∈	∈	PROPN
ejpam-3284	12	87	v	v	PART
ejpam-3284	12	88	see	see	VERB
ejpam-3284	12	89	[	[	X
ejpam-3284	12	90	3	3	NUM
ejpam-3284	12	91	]	]	PUNCT
ejpam-3284	12	92	.	.	PUNCT
ejpam-3284	13	1	∗corresponding	∗corresponde	VERB
ejpam-3284	13	2	author	author	NOUN
ejpam-3284	13	3	.	.	PUNCT
ejpam-3284	14	1	doi	doi	NOUN
ejpam-3284	14	2	:	:	PUNCT
ejpam-3284	14	3	https://doi.org/10.29020/nybg.ejpam.v11i3.3284	https://doi.org/10.29020/nybg.ejpam.v11i3.3284	ADJ
ejpam-3284	14	4	email	email	NOUN
ejpam-3284	14	5	addresses	address	VERB
ejpam-3284	14	6	:	:	PUNCT
ejpam-3284	15	1	ewiekwamina@gmail.com	ewiekwamina@gmail.com	X
ejpam-3284	15	2	bbarnes.cos@knust.edu.gh	bbarnes.cos@knust.edu.gh	PROPN
ejpam-3284	15	3	(	(	PUNCT
ejpam-3284	15	4	b.	b.	PROPN
ejpam-3284	15	5	barnes	barnes	PROPN
ejpam-3284	15	6	)	)	PUNCT
ejpam-3284	15	7	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3284	16	1	740	740	NUM
ejpam-3284	16	2	c	c	NOUN
ejpam-3284	16	3	©	©	PROPN
ejpam-3284	16	4	2018	2018	NUM
ejpam-3284	16	5	ejpam	ejpam	VERB
ejpam-3284	16	6	all	all	DET
ejpam-3284	16	7	rights	right	NOUN
ejpam-3284	16	8	reserved	reserve	VERB
ejpam-3284	16	9	.	.	PUNCT
ejpam-3284	17	1	barnes	barnes	PROPN
ejpam-3284	17	2	et	et	PROPN
ejpam-3284	17	3	al	al	PROPN
ejpam-3284	17	4	.	.	PUNCT
ejpam-3284	17	5	/	/	SYM
ejpam-3284	17	6	eur	eur	PROPN
ejpam-3284	17	7	.	.	PUNCT
ejpam-3284	18	1	j.	j.	PROPN
ejpam-3284	18	2	pure	pure	PROPN
ejpam-3284	18	3	appl	appl	PROPN
ejpam-3284	18	4	.	.	PROPN
ejpam-3284	18	5	math	math	PROPN
ejpam-3284	18	6	,	,	PUNCT
ejpam-3284	18	7	11	11	NUM
ejpam-3284	18	8	(	(	PUNCT
ejpam-3284	18	9	3	3	NUM
ejpam-3284	18	10	)	)	PUNCT
ejpam-3284	18	11	(	(	PUNCT
ejpam-3284	18	12	2018	2018	NUM
ejpam-3284	18	13	)	)	PUNCT
ejpam-3284	18	14	,	,	PUNCT
ejpam-3284	18	15	740	740	NUM
ejpam-3284	18	16	-	-	SYM
ejpam-3284	18	17	750	750	NUM
ejpam-3284	18	18	741	741	NUM
ejpam-3284	18	19	in	in	ADP
ejpam-3284	18	20	[	[	X
ejpam-3284	18	21	4	4	NUM
ejpam-3284	18	22	]	]	PUNCT
ejpam-3284	18	23	,	,	PUNCT
ejpam-3284	18	24	the	the	DET
ejpam-3284	18	25	author	author	NOUN
ejpam-3284	18	26	generalized	generalize	VERB
ejpam-3284	18	27	the	the	DET
ejpam-3284	18	28	normed	normed	ADJ
ejpam-3284	18	29	linear	linear	ADJ
ejpam-3284	18	30	space	space	NOUN
ejpam-3284	18	31	by	by	ADP
ejpam-3284	18	32	replacing	replace	VERB
ejpam-3284	18	33	the	the	DET
ejpam-3284	18	34	third	third	ADJ
ejpam-3284	18	35	axiom	axiom	NOUN
ejpam-3284	18	36	of	of	ADP
ejpam-3284	18	37	the	the	DET
ejpam-3284	18	38	norm	norm	NOUN
ejpam-3284	18	39	,	,	PUNCT
ejpam-3284	18	40	triangle	triangle	NOUN
ejpam-3284	18	41	inequality	inequality	NOUN
ejpam-3284	18	42	,	,	PUNCT
ejpam-3284	18	43	with	with	ADP
ejpam-3284	18	44	tetrahedral	tetrahedral	ADJ
ejpam-3284	18	45	inequality	inequality	NOUN
ejpam-3284	18	46	.	.	PUNCT
ejpam-3284	19	1	the	the	DET
ejpam-3284	19	2	authors	author	NOUN
ejpam-3284	19	3	in	in	ADP
ejpam-3284	19	4	[	[	X
ejpam-3284	19	5	5	5	NUM
ejpam-3284	19	6	]	]	PUNCT
ejpam-3284	19	7	and	and	CCONJ
ejpam-3284	19	8	[	[	AUX
ejpam-3284	19	9	6	6	NUM
ejpam-3284	19	10	]	]	PUNCT
ejpam-3284	19	11	independently	independently	ADV
ejpam-3284	19	12	extended	extend	VERB
ejpam-3284	19	13	the	the	DET
ejpam-3284	19	14	linear	linear	ADJ
ejpam-3284	19	15	normed	normed	ADJ
ejpam-3284	19	16	space	space	NOUN
ejpam-3284	19	17	to	to	PART
ejpam-3284	19	18	quasinormed	quasinorme	VERB
ejpam-3284	19	19	linear	linear	PROPN
ejpam-3284	19	20	space	space	NOUN
ejpam-3284	19	21	.	.	PUNCT
ejpam-3284	20	1	they	they	PRON
ejpam-3284	20	2	replaced	replace	VERB
ejpam-3284	20	3	the	the	DET
ejpam-3284	20	4	third	third	ADJ
ejpam-3284	20	5	axiom	axiom	NOUN
ejpam-3284	20	6	of	of	ADP
ejpam-3284	20	7	the	the	DET
ejpam-3284	20	8	normed	normed	ADJ
ejpam-3284	20	9	linear	linear	PROPN
ejpam-3284	20	10	space	space	NOUN
ejpam-3284	20	11	with	with	ADP
ejpam-3284	20	12	quasi	quasi	ADJ
ejpam-3284	20	13	-	-	ADJ
ejpam-3284	20	14	norm	norm	ADJ
ejpam-3284	20	15	‖u+	‖u+	NOUN
ejpam-3284	20	16	v‖	v‖	NOUN
ejpam-3284	20	17	≤	≤	PUNCT
ejpam-3284	21	1	k	k	PROPN
ejpam-3284	21	2	(	(	PUNCT
ejpam-3284	21	3	‖u‖+	‖u‖+	PROPN
ejpam-3284	21	4	‖v‖	‖v‖	PROPN
ejpam-3284	21	5	)	)	PUNCT
ejpam-3284	21	6	,	,	PUNCT
ejpam-3284	21	7	∀	∀	X
ejpam-3284	21	8	u	u	NOUN
ejpam-3284	21	9	,	,	PUNCT
ejpam-3284	21	10	v	v	NOUN
ejpam-3284	21	11	∈	∈	PROPN
ejpam-3284	21	12	v	v	NOUN
ejpam-3284	21	13	,	,	PUNCT
ejpam-3284	21	14	where	where	SCONJ
ejpam-3284	21	15	k	k	PROPN
ejpam-3284	21	16	≥	≥	NUM
ejpam-3284	21	17	1	1	NUM
ejpam-3284	21	18	is	be	AUX
ejpam-3284	21	19	the	the	DET
ejpam-3284	21	20	quasi	quasi	NOUN
ejpam-3284	21	21	-	-	NOUN
ejpam-3284	21	22	number	number	NOUN
ejpam-3284	21	23	.	.	PUNCT
ejpam-3284	22	1	in	in	ADP
ejpam-3284	22	2	[	[	X
ejpam-3284	22	3	7	7	NUM
ejpam-3284	22	4	]	]	PUNCT
ejpam-3284	22	5	,	,	PUNCT
ejpam-3284	22	6	the	the	DET
ejpam-3284	22	7	authors	author	NOUN
ejpam-3284	22	8	extended	extend	VERB
ejpam-3284	22	9	the	the	DET
ejpam-3284	22	10	quasi	quasi	ADJ
ejpam-3284	22	11	-	-	ADJ
ejpam-3284	22	12	normed	norme	VERB
ejpam-3284	22	13	linear	linear	ADJ
ejpam-3284	22	14	space	space	NOUN
ejpam-3284	22	15	to	to	ADP
ejpam-3284	22	16	the	the	DET
ejpam-3284	22	17	complex	complex	ADJ
ejpam-3284	22	18	version	version	NOUN
ejpam-3284	22	19	and	and	CCONJ
ejpam-3284	22	20	also	also	ADV
ejpam-3284	22	21	,	,	PUNCT
ejpam-3284	22	22	included	include	VERB
ejpam-3284	22	23	a	a	DET
ejpam-3284	22	24	parameter	parameter	NOUN
ejpam-3284	22	25	p	p	X
ejpam-3284	22	26	∈	∈	PROPN
ejpam-3284	22	27	(	(	PUNCT
ejpam-3284	22	28	0,∞	0,∞	NOUN
ejpam-3284	22	29	)	)	PUNCT
ejpam-3284	22	30	in	in	ADP
ejpam-3284	22	31	this	this	DET
ejpam-3284	22	32	space	space	NOUN
ejpam-3284	22	33	,	,	PUNCT
ejpam-3284	22	34	which	which	PRON
ejpam-3284	22	35	is	be	AUX
ejpam-3284	22	36	used	use	VERB
ejpam-3284	22	37	to	to	PART
ejpam-3284	22	38	establish	establish	VERB
ejpam-3284	22	39	p−	p−	NOUN
ejpam-3284	22	40	uniformly	uniformly	ADV
ejpam-3284	22	41	pl−	pl−	PUNCT
ejpam-3284	22	42	convex	convex	PROPN
ejpam-3284	22	43	.	.	PUNCT
ejpam-3284	23	1	the	the	DET
ejpam-3284	23	2	author	author	NOUN
ejpam-3284	23	3	in	in	ADP
ejpam-3284	23	4	[	[	X
ejpam-3284	23	5	8	8	NUM
ejpam-3284	23	6	]	]	PUNCT
ejpam-3284	23	7	,	,	PUNCT
ejpam-3284	23	8	observed	observe	VERB
ejpam-3284	23	9	that	that	SCONJ
ejpam-3284	23	10	the	the	DET
ejpam-3284	23	11	quasi	quasi	ADJ
ejpam-3284	23	12	-	-	ADJ
ejpam-3284	23	13	normed	normed	ADJ
ejpam-3284	23	14	linear	linear	ADJ
ejpam-3284	23	15	space	space	NOUN
ejpam-3284	23	16	becomes	become	VERB
ejpam-3284	23	17	the	the	DET
ejpam-3284	23	18	p−normed	p−norme	VERB
ejpam-3284	23	19	linear	linear	ADJ
ejpam-3284	23	20	space	space	NOUN
ejpam-3284	23	21	if	if	SCONJ
ejpam-3284	23	22	the	the	DET
ejpam-3284	23	23	quasi	quasi	NOUN
ejpam-3284	23	24	number	number	NOUN
ejpam-3284	23	25	k	k	PROPN
ejpam-3284	23	26	=	=	SYM
ejpam-3284	23	27	2	2	NUM
ejpam-3284	23	28	1	1	NUM
ejpam-3284	23	29	p	p	NOUN
ejpam-3284	23	30	−1	−1	NOUN
ejpam-3284	23	31	.	.	PUNCT
ejpam-3284	24	1	in	in	ADP
ejpam-3284	24	2	[	[	X
ejpam-3284	24	3	9	9	NUM
ejpam-3284	24	4	]	]	PUNCT
ejpam-3284	24	5	,	,	PUNCT
ejpam-3284	24	6	the	the	DET
ejpam-3284	24	7	author	author	NOUN
ejpam-3284	24	8	established	establish	VERB
ejpam-3284	24	9	that	that	SCONJ
ejpam-3284	24	10	the	the	DET
ejpam-3284	24	11	quasi	quasi	ADJ
ejpam-3284	24	12	-	-	ADJ
ejpam-3284	24	13	normed	normed	ADJ
ejpam-3284	24	14	linear	linear	ADJ
ejpam-3284	24	15	space	space	NOUN
ejpam-3284	24	16	is	be	AUX
ejpam-3284	24	17	endowed	endow	VERB
ejpam-3284	24	18	with	with	ADP
ejpam-3284	24	19	a	a	DET
ejpam-3284	24	20	unique	unique	ADJ
ejpam-3284	24	21	fixed	fix	VERB
ejpam-3284	24	22	point	point	NOUN
ejpam-3284	24	23	.	.	PUNCT
ejpam-3284	25	1	in	in	ADP
ejpam-3284	25	2	[	[	X
ejpam-3284	25	3	10	10	NUM
ejpam-3284	25	4	]	]	PUNCT
ejpam-3284	25	5	,	,	PUNCT
ejpam-3284	25	6	he	he	PRON
ejpam-3284	25	7	narrowed	narrow	VERB
ejpam-3284	25	8	the	the	DET
ejpam-3284	25	9	quasi	quasi	ADJ
ejpam-3284	25	10	-	-	ADJ
ejpam-3284	25	11	normed	norme	VERB
ejpam-3284	25	12	linear	linear	ADJ
ejpam-3284	25	13	space	space	NOUN
ejpam-3284	25	14	to	to	ADP
ejpam-3284	25	15	quasi-2normed	quasi-2norme	VERB
ejpam-3284	25	16	linear	linear	ADJ
ejpam-3284	25	17	space	space	NOUN
ejpam-3284	25	18	q-2	q-2	NUM
ejpam-3284	25	19	-	-	PUNCT
ejpam-3284	25	20	nls	nls	NOUN
ejpam-3284	25	21	.	.	PUNCT
ejpam-3284	25	22	addition	addition	NOUN
ejpam-3284	25	23	properties	property	NOUN
ejpam-3284	25	24	of	of	ADP
ejpam-3284	25	25	the	the	DET
ejpam-3284	25	26	q-2	q-2	NUM
ejpam-3284	25	27	-	-	PUNCT
ejpam-3284	25	28	nls	nls	NOUN
ejpam-3284	25	29	such	such	ADJ
ejpam-3284	25	30	as	as	ADP
ejpam-3284	25	31	the	the	DET
ejpam-3284	25	32	convergence	convergence	NOUN
ejpam-3284	25	33	of	of	ADP
ejpam-3284	25	34	the	the	DET
ejpam-3284	25	35	two	two	NUM
ejpam-3284	25	36	cauchy	cauchy	ADJ
ejpam-3284	25	37	sequences	sequence	NOUN
ejpam-3284	25	38	and	and	CCONJ
ejpam-3284	25	39	a	a	DET
ejpam-3284	25	40	pseudo	pseudo	NOUN
ejpam-3284	25	41	-	-	ADJ
ejpam-3284	25	42	quasi-2	quasi-2	NOUN
ejpam-3284	25	43	-	-	PUNCT
ejpam-3284	25	44	norm	norm	NOUN
ejpam-3284	25	45	of	of	ADP
ejpam-3284	25	46	the	the	DET
ejpam-3284	25	47	q-2	q-2	NUM
ejpam-3284	25	48	-	-	PUNCT
ejpam-3284	25	49	nls	nls	NOUN
ejpam-3284	25	50	were	be	AUX
ejpam-3284	25	51	observed	observe	VERB
ejpam-3284	25	52	by	by	ADP
ejpam-3284	25	53	the	the	DET
ejpam-3284	25	54	authors	author	NOUN
ejpam-3284	25	55	in	in	ADP
ejpam-3284	25	56	[	[	X
ejpam-3284	25	57	11	11	NUM
ejpam-3284	25	58	]	]	PUNCT
ejpam-3284	25	59	.	.	PUNCT
ejpam-3284	26	1	undoubtedly	undoubtedly	ADV
ejpam-3284	26	2	,	,	PUNCT
ejpam-3284	26	3	the	the	DET
ejpam-3284	26	4	normed	normed	ADJ
ejpam-3284	26	5	linear	linear	ADJ
ejpam-3284	26	6	space	space	NOUN
ejpam-3284	26	7	is	be	AUX
ejpam-3284	26	8	called	call	VERB
ejpam-3284	26	9	a	a	DET
ejpam-3284	26	10	p−normed	p−norme	VERB
ejpam-3284	26	11	linear	linear	NOUN
ejpam-3284	26	12	if	if	SCONJ
ejpam-3284	26	13	‖u+	‖u+	NOUN
ejpam-3284	26	14	v‖p	v‖p	VERB
ejpam-3284	26	15	≤	≤	NUM
ejpam-3284	26	16	‖u‖p	‖u‖p	NOUN
ejpam-3284	26	17	+	+	CCONJ
ejpam-3284	26	18	‖v‖p	‖v‖p	NOUN
ejpam-3284	26	19	,	,	PUNCT
ejpam-3284	26	20	∀	∀	X
ejpam-3284	26	21	u	u	NOUN
ejpam-3284	26	22	,	,	PUNCT
ejpam-3284	26	23	v	v	NOUN
ejpam-3284	26	24	∈	∈	PROPN
ejpam-3284	26	25	v	v	NOUN
ejpam-3284	26	26	,	,	PUNCT
ejpam-3284	26	27	and	and	CCONJ
ejpam-3284	26	28	0	0	NUM
ejpam-3284	26	29	<	<	X
ejpam-3284	26	30	p	p	X
ejpam-3284	26	31	≤	≤	NOUN
ejpam-3284	26	32	1	1	NUM
ejpam-3284	26	33	,	,	PUNCT
ejpam-3284	26	34	is	be	AUX
ejpam-3284	26	35	included	include	VERB
ejpam-3284	26	36	to	to	ADP
ejpam-3284	26	37	the	the	DET
ejpam-3284	26	38	three	three	NUM
ejpam-3284	26	39	axioms	axiom	NOUN
ejpam-3284	26	40	of	of	ADP
ejpam-3284	26	41	the	the	DET
ejpam-3284	26	42	linear	linear	ADJ
ejpam-3284	26	43	normed	normed	ADJ
ejpam-3284	26	44	space	space	NOUN
ejpam-3284	26	45	.	.	PUNCT
ejpam-3284	27	1	for	for	ADP
ejpam-3284	27	2	example	example	NOUN
ejpam-3284	27	3	,	,	PUNCT
ejpam-3284	27	4	see	see	VERB
ejpam-3284	27	5	the	the	DET
ejpam-3284	27	6	author	author	NOUN
ejpam-3284	27	7	in	in	ADP
ejpam-3284	27	8	[	[	X
ejpam-3284	27	9	12	12	NUM
ejpam-3284	27	10	]	]	PUNCT
ejpam-3284	27	11	.	.	PUNCT
ejpam-3284	28	1	there	there	PRON
ejpam-3284	28	2	are	be	VERB
ejpam-3284	28	3	,	,	PUNCT
ejpam-3284	28	4	however	however	ADV
ejpam-3284	28	5	,	,	PUNCT
ejpam-3284	28	6	situations	situation	NOUN
ejpam-3284	28	7	where	where	SCONJ
ejpam-3284	28	8	it	it	PRON
ejpam-3284	28	9	is	be	AUX
ejpam-3284	28	10	more	more	ADV
ejpam-3284	28	11	convenient	convenient	ADJ
ejpam-3284	28	12	to	to	PART
ejpam-3284	28	13	work	work	VERB
ejpam-3284	28	14	with	with	ADP
ejpam-3284	28	15	a	a	DET
ejpam-3284	28	16	subspace	subspace	NOUN
ejpam-3284	28	17	than	than	ADP
ejpam-3284	28	18	the	the	DET
ejpam-3284	28	19	entire	entire	ADJ
ejpam-3284	28	20	functional	functional	ADJ
ejpam-3284	28	21	space	space	NOUN
ejpam-3284	28	22	.	.	PUNCT
ejpam-3284	29	1	for	for	ADP
ejpam-3284	29	2	example	example	NOUN
ejpam-3284	29	3	,	,	PUNCT
ejpam-3284	29	4	the	the	DET
ejpam-3284	29	5	imposing	imposing	NOUN
ejpam-3284	29	6	of	of	ADP
ejpam-3284	29	7	absolute	absolute	ADJ
ejpam-3284	29	8	value	value	NOUN
ejpam-3284	29	9	of	of	ADP
ejpam-3284	29	10	a	a	DET
ejpam-3284	29	11	variable	variable	NOUN
ejpam-3284	29	12	x	x	NOUN
ejpam-3284	29	13	as	as	ADP
ejpam-3284	29	14	an	an	DET
ejpam-3284	29	15	inhomogeneous	inhomogeneous	ADJ
ejpam-3284	29	16	boundary	boundary	ADJ
ejpam-3284	29	17	condition	condition	NOUN
ejpam-3284	29	18	of	of	ADP
ejpam-3284	29	19	the	the	DET
ejpam-3284	29	20	laplace	laplace	NOUN
ejpam-3284	29	21	equation	equation	NOUN
ejpam-3284	29	22	does	do	AUX
ejpam-3284	29	23	not	not	PART
ejpam-3284	29	24	yield	yield	VERB
ejpam-3284	29	25	a	a	DET
ejpam-3284	29	26	solution	solution	NOUN
ejpam-3284	29	27	in	in	ADP
ejpam-3284	29	28	the	the	DET
ejpam-3284	29	29	hilbert	hilbert	NOUN
ejpam-3284	29	30	space	space	NOUN
ejpam-3284	29	31	,	,	PUNCT
ejpam-3284	29	32	but	but	CCONJ
ejpam-3284	29	33	in	in	ADP
ejpam-3284	29	34	sobolev	sobolev	NOUN
ejpam-3284	29	35	spaces	space	NOUN
ejpam-3284	29	36	.	.	PUNCT
ejpam-3284	30	1	thus	thus	ADV
ejpam-3284	30	2	,	,	PUNCT
ejpam-3284	30	3	the	the	DET
ejpam-3284	30	4	continuity	continuity	NOUN
ejpam-3284	30	5	of	of	ADP
ejpam-3284	30	6	the	the	DET
ejpam-3284	30	7	partial	partial	ADJ
ejpam-3284	30	8	differential	differential	NOUN
ejpam-3284	30	9	operator	operator	NOUN
ejpam-3284	30	10	is	be	AUX
ejpam-3284	30	11	not	not	PART
ejpam-3284	30	12	guaranteed	guarantee	VERB
ejpam-3284	30	13	in	in	ADP
ejpam-3284	30	14	the	the	DET
ejpam-3284	30	15	hilbert	hilbert	NOUN
ejpam-3284	30	16	space	space	NOUN
ejpam-3284	30	17	.	.	PUNCT
ejpam-3284	31	1	in	in	ADP
ejpam-3284	31	2	order	order	NOUN
ejpam-3284	31	3	for	for	ADP
ejpam-3284	31	4	a	a	DET
ejpam-3284	31	5	functional	functional	ADJ
ejpam-3284	31	6	space	space	NOUN
ejpam-3284	31	7	to	to	PART
ejpam-3284	31	8	admit	admit	VERB
ejpam-3284	31	9	such	such	DET
ejpam-3284	31	10	a	a	DET
ejpam-3284	31	11	solution	solution	NOUN
ejpam-3284	31	12	,	,	PUNCT
ejpam-3284	31	13	we	we	PRON
ejpam-3284	31	14	impose	impose	VERB
ejpam-3284	31	15	some	some	DET
ejpam-3284	31	16	continuity	continuity	NOUN
ejpam-3284	31	17	conditions	condition	NOUN
ejpam-3284	31	18	on	on	ADP
ejpam-3284	31	19	the	the	DET
ejpam-3284	31	20	partial	partial	ADJ
ejpam-3284	31	21	differential	differential	NOUN
ejpam-3284	31	22	operator	operator	NOUN
ejpam-3284	31	23	which	which	PRON
ejpam-3284	31	24	lead	lead	VERB
ejpam-3284	31	25	to	to	ADP
ejpam-3284	31	26	analysis	analysis	NOUN
ejpam-3284	31	27	in	in	ADP
ejpam-3284	31	28	sobolev	sobolev	NOUN
ejpam-3284	31	29	spaces	space	NOUN
ejpam-3284	31	30	.	.	PUNCT
ejpam-3284	32	1	again	again	ADV
ejpam-3284	32	2	,	,	PUNCT
ejpam-3284	32	3	much	much	ADJ
ejpam-3284	32	4	attention	attention	NOUN
ejpam-3284	32	5	has	have	AUX
ejpam-3284	32	6	been	be	AUX
ejpam-3284	32	7	paid	pay	VERB
ejpam-3284	32	8	to	to	ADP
ejpam-3284	32	9	the	the	DET
ejpam-3284	32	10	extension	extension	NOUN
ejpam-3284	32	11	of	of	ADP
ejpam-3284	32	12	the	the	DET
ejpam-3284	32	13	triangle	triangle	NOUN
ejpam-3284	32	14	inequality	inequality	NOUN
ejpam-3284	32	15	,	,	PUNCT
ejpam-3284	32	16	the	the	DET
ejpam-3284	32	17	third	third	ADJ
ejpam-3284	32	18	axiom	axiom	NOUN
ejpam-3284	32	19	of	of	ADP
ejpam-3284	32	20	the	the	DET
ejpam-3284	32	21	normed	normed	ADJ
ejpam-3284	32	22	linear	linear	PROPN
ejpam-3284	32	23	space	space	NOUN
ejpam-3284	32	24	,	,	PUNCT
ejpam-3284	32	25	to	to	PART
ejpam-3284	32	26	obtain	obtain	VERB
ejpam-3284	32	27	the	the	DET
ejpam-3284	32	28	different	different	ADJ
ejpam-3284	32	29	normed	normed	ADJ
ejpam-3284	32	30	linear	linear	PROPN
ejpam-3284	32	31	spaces	space	NOUN
ejpam-3284	32	32	with	with	ADP
ejpam-3284	32	33	their	their	PRON
ejpam-3284	32	34	functional	functional	ADJ
ejpam-3284	32	35	properties	property	NOUN
ejpam-3284	32	36	.	.	PUNCT
ejpam-3284	33	1	in	in	ADP
ejpam-3284	33	2	this	this	DET
ejpam-3284	33	3	paper	paper	NOUN
ejpam-3284	33	4	,	,	PUNCT
ejpam-3284	33	5	we	we	PRON
ejpam-3284	33	6	include	include	VERB
ejpam-3284	33	7	another	another	DET
ejpam-3284	33	8	axiom	axiom	NOUN
ejpam-3284	33	9	to	to	ADP
ejpam-3284	33	10	the	the	DET
ejpam-3284	33	11	normed	normed	ADJ
ejpam-3284	33	12	linear	linear	PROPN
ejpam-3284	33	13	space	space	NOUN
ejpam-3284	33	14	as	as	ADP
ejpam-3284	33	15	a	a	DET
ejpam-3284	33	16	fourth	fourth	ADJ
ejpam-3284	33	17	axiom	axiom	NOUN
ejpam-3284	33	18	,	,	PUNCT
ejpam-3284	33	19	first	first	ADJ
ejpam-3284	33	20	product	product	NOUN
ejpam-3284	33	21	inequality	inequality	NOUN
ejpam-3284	33	22	,	,	PUNCT
ejpam-3284	33	23	from	from	ADP
ejpam-3284	33	24	which	which	PRON
ejpam-3284	33	25	we	we	PRON
ejpam-3284	33	26	coined	coin	VERB
ejpam-3284	33	27	the	the	DET
ejpam-3284	33	28	name	name	NOUN
ejpam-3284	33	29	product	product	NOUN
ejpam-3284	33	30	-	-	PUNCT
ejpam-3284	33	31	normed	norme	VERB
ejpam-3284	33	32	linear	linear	ADJ
ejpam-3284	33	33	space	space	NOUN
ejpam-3284	33	34	p	p	NOUN
ejpam-3284	33	35	−	−	PROPN
ejpam-3284	33	36	nls	nls	NOUN
ejpam-3284	33	37	,	,	PUNCT
ejpam-3284	33	38	‖.‖pn	‖.‖pn	NOUN
ejpam-3284	33	39	.	.	PUNCT
ejpam-3284	34	1	this	this	DET
ejpam-3284	34	2	functional	functional	ADJ
ejpam-3284	34	3	space	space	NOUN
ejpam-3284	34	4	narrows	narrow	VERB
ejpam-3284	34	5	the	the	DET
ejpam-3284	34	6	normed	normed	ADJ
ejpam-3284	34	7	linear	linear	PROPN
ejpam-3284	34	8	space	space	NOUN
ejpam-3284	34	9	which	which	PRON
ejpam-3284	34	10	allows	allow	VERB
ejpam-3284	34	11	the	the	DET
ejpam-3284	34	12	vectors	vector	NOUN
ejpam-3284	34	13	to	to	PART
ejpam-3284	34	14	be	be	AUX
ejpam-3284	34	15	defined	define	VERB
ejpam-3284	34	16	on	on	ADP
ejpam-3284	34	17	[	[	X
ejpam-3284	34	18	0	0	NUM
ejpam-3284	34	19	,	,	PUNCT
ejpam-3284	34	20	2	2	NUM
ejpam-3284	34	21	]	]	PUNCT
ejpam-3284	34	22	.	.	PUNCT
ejpam-3284	35	1	the	the	DET
ejpam-3284	35	2	p	p	PROPN
ejpam-3284	35	3	−nls	−nls	NOUN
ejpam-3284	35	4	admits	admit	VERB
ejpam-3284	35	5	the	the	DET
ejpam-3284	35	6	first	first	ADJ
ejpam-3284	35	7	product	product	NOUN
ejpam-3284	35	8	inequality	inequality	NOUN
ejpam-3284	35	9	,	,	PUNCT
ejpam-3284	35	10	see	see	VERB
ejpam-3284	35	11	[	[	X
ejpam-3284	35	12	13	13	NUM
ejpam-3284	35	13	]	]	PUNCT
ejpam-3284	35	14	,	,	PUNCT
ejpam-3284	35	15	and	and	CCONJ
ejpam-3284	35	16	useful	useful	ADJ
ejpam-3284	35	17	properties	property	NOUN
ejpam-3284	35	18	which	which	PRON
ejpam-3284	35	19	we	we	PRON
ejpam-3284	35	20	shall	shall	AUX
ejpam-3284	35	21	establish	establish	VERB
ejpam-3284	35	22	them	they	PRON
ejpam-3284	35	23	later	later	ADV
ejpam-3284	35	24	in	in	ADP
ejpam-3284	35	25	this	this	DET
ejpam-3284	35	26	paper	paper	NOUN
ejpam-3284	35	27	.	.	PUNCT
ejpam-3284	36	1	also	also	ADV
ejpam-3284	36	2	,	,	PUNCT
ejpam-3284	36	3	the	the	DET
ejpam-3284	36	4	product	product	NOUN
ejpam-3284	36	5	-	-	PUNCT
ejpam-3284	36	6	semi	semi	ADV
ejpam-3284	36	7	-	-	ADJ
ejpam-3284	36	8	normed	normed	ADJ
ejpam-3284	36	9	linear	linear	ADJ
ejpam-3284	36	10	space	space	NOUN
ejpam-3284	36	11	‖.‖pn	‖.‖pn	NOUN
ejpam-3284	36	12	is	be	AUX
ejpam-3284	36	13	introduced	introduce	VERB
ejpam-3284	36	14	in	in	ADP
ejpam-3284	36	15	this	this	DET
ejpam-3284	36	16	paper	paper	NOUN
ejpam-3284	36	17	.	.	PUNCT
ejpam-3284	37	1	2	2	X
ejpam-3284	37	2	.	.	X
ejpam-3284	37	3	some	some	DET
ejpam-3284	37	4	preliminary	preliminary	ADJ
ejpam-3284	37	5	results	result	NOUN
ejpam-3284	37	6	in	in	ADP
ejpam-3284	37	7	this	this	DET
ejpam-3284	37	8	section	section	NOUN
ejpam-3284	37	9	,	,	PUNCT
ejpam-3284	37	10	we	we	PRON
ejpam-3284	37	11	give	give	VERB
ejpam-3284	37	12	the	the	DET
ejpam-3284	37	13	theorems	theorem	NOUN
ejpam-3284	37	14	and	and	CCONJ
ejpam-3284	37	15	definitions	definition	NOUN
ejpam-3284	37	16	regarding	regard	VERB
ejpam-3284	37	17	to	to	ADP
ejpam-3284	37	18	the	the	DET
ejpam-3284	37	19	product	product	NOUN
ejpam-3284	37	20	-	-	PUNCT
ejpam-3284	37	21	normed	norme	VERB
ejpam-3284	37	22	linear	linear	ADJ
ejpam-3284	37	23	space	space	NOUN
ejpam-3284	37	24	and	and	CCONJ
ejpam-3284	37	25	product	product	NOUN
ejpam-3284	37	26	-	-	PUNCT
ejpam-3284	37	27	semi	semi	ADV
ejpam-3284	37	28	-	-	ADJ
ejpam-3284	37	29	normed	normed	ADJ
ejpam-3284	37	30	linear	linear	ADJ
ejpam-3284	37	31	space	space	NOUN
ejpam-3284	37	32	.	.	PUNCT
ejpam-3284	38	1	definition	definition	NOUN
ejpam-3284	38	2	2	2	NUM
ejpam-3284	38	3	(	(	PUNCT
ejpam-3284	38	4	inner	inner	ADJ
ejpam-3284	38	5	product	product	NOUN
ejpam-3284	38	6	space	space	NOUN
ejpam-3284	38	7	)	)	PUNCT
ejpam-3284	38	8	.	.	PUNCT
ejpam-3284	39	1	let	let	VERB
ejpam-3284	39	2	v	v	PART
ejpam-3284	39	3	be	be	AUX
ejpam-3284	39	4	a	a	DET
ejpam-3284	39	5	linear	linear	ADJ
ejpam-3284	39	6	vector	vector	NOUN
ejpam-3284	39	7	space	space	NOUN
ejpam-3284	39	8	defined	define	VERB
ejpam-3284	39	9	over	over	ADP
ejpam-3284	39	10	the	the	DET
ejpam-3284	39	11	real	real	ADJ
ejpam-3284	39	12	number	number	NOUN
ejpam-3284	39	13	field	field	NOUN
ejpam-3284	39	14	r.	r.	VERB
ejpam-3284	39	15	a	a	DET
ejpam-3284	39	16	scalar	scalar	ADV
ejpam-3284	39	17	-	-	PUNCT
ejpam-3284	39	18	valued	value	VERB
ejpam-3284	39	19	function	function	NOUN
ejpam-3284	39	20	p	p	NOUN
ejpam-3284	39	21	:	:	PUNCT
ejpam-3284	39	22	v	v	NUM
ejpam-3284	39	23	×	×	NOUN
ejpam-3284	39	24	v	v	NOUN
ejpam-3284	39	25	→	→	SYM
ejpam-3284	39	26	r	r	NOUN
ejpam-3284	39	27	that	that	PRON
ejpam-3284	39	28	associates	associate	NOUN
ejpam-3284	39	29	with	with	ADP
ejpam-3284	39	30	each	each	DET
ejpam-3284	39	31	pair	pair	NOUN
ejpam-3284	39	32	barnes	barne	VERB
ejpam-3284	39	33	et	et	PROPN
ejpam-3284	39	34	al	al	PROPN
ejpam-3284	39	35	.	.	PUNCT
ejpam-3284	39	36	/	/	SYM
ejpam-3284	39	37	eur	eur	PROPN
ejpam-3284	39	38	.	.	PUNCT
ejpam-3284	40	1	j.	j.	PROPN
ejpam-3284	40	2	pure	pure	PROPN
ejpam-3284	40	3	appl	appl	PROPN
ejpam-3284	40	4	.	.	PROPN
ejpam-3284	40	5	math	math	PROPN
ejpam-3284	40	6	,	,	PUNCT
ejpam-3284	40	7	11	11	NUM
ejpam-3284	40	8	(	(	PUNCT
ejpam-3284	40	9	3	3	NUM
ejpam-3284	40	10	)	)	PUNCT
ejpam-3284	40	11	(	(	PUNCT
ejpam-3284	40	12	2018	2018	NUM
ejpam-3284	40	13	)	)	PUNCT
ejpam-3284	40	14	,	,	PUNCT
ejpam-3284	40	15	740	740	NUM
ejpam-3284	40	16	-	-	SYM
ejpam-3284	40	17	750	750	NUM
ejpam-3284	40	18	742	742	NUM
ejpam-3284	40	19	u	u	NOUN
ejpam-3284	40	20	,	,	PUNCT
ejpam-3284	40	21	v	v	NOUN
ejpam-3284	40	22	of	of	ADP
ejpam-3284	40	23	vectors	vector	NOUN
ejpam-3284	40	24	in	in	ADP
ejpam-3284	40	25	v	v	ADP
ejpam-3284	40	26	a	a	DET
ejpam-3284	40	27	scalar	scalar	ADJ
ejpam-3284	40	28	,	,	PUNCT
ejpam-3284	40	29	denoted	denote	VERB
ejpam-3284	40	30	(	(	PUNCT
ejpam-3284	40	31	u	u	NOUN
ejpam-3284	40	32	,	,	PUNCT
ejpam-3284	40	33	v	v	NOUN
ejpam-3284	40	34	)	)	PUNCT
ejpam-3284	40	35	,	,	PUNCT
ejpam-3284	40	36	is	be	AUX
ejpam-3284	40	37	called	call	VERB
ejpam-3284	40	38	an	an	DET
ejpam-3284	40	39	inner	inner	ADJ
ejpam-3284	40	40	product	product	NOUN
ejpam-3284	40	41	on	on	ADP
ejpam-3284	40	42	v	v	PRON
ejpam-3284	40	43	if	if	SCONJ
ejpam-3284	41	1	and	and	CCONJ
ejpam-3284	41	2	only	only	ADV
ejpam-3284	41	3	if	if	SCONJ
ejpam-3284	41	4	(	(	PUNCT
ejpam-3284	41	5	i	i	NOUN
ejpam-3284	41	6	)	)	PUNCT
ejpam-3284	41	7	(	(	PUNCT
ejpam-3284	41	8	u	u	NOUN
ejpam-3284	41	9	,	,	PUNCT
ejpam-3284	41	10	u	u	NOUN
ejpam-3284	41	11	)	)	PUNCT
ejpam-3284	41	12	>	>	X
ejpam-3284	41	13	0	0	PUNCT
ejpam-3284	42	1	whenever	whenever	SCONJ
ejpam-3284	42	2	u	u	PROPN
ejpam-3284	42	3	6=	6=	PROPN
ejpam-3284	42	4	0	0	NUM
ejpam-3284	42	5	,	,	PUNCT
ejpam-3284	42	6	and	and	CCONJ
ejpam-3284	42	7	(	(	PUNCT
ejpam-3284	42	8	u	u	NOUN
ejpam-3284	42	9	,	,	PUNCT
ejpam-3284	42	10	u	u	NOUN
ejpam-3284	42	11	)	)	PUNCT
ejpam-3284	42	12	=	=	SYM
ejpam-3284	42	13	0	0	PUNCT
ejpam-3284	42	14	if	if	SCONJ
ejpam-3284	42	15	and	and	CCONJ
ejpam-3284	42	16	only	only	ADV
ejpam-3284	42	17	if	if	SCONJ
ejpam-3284	42	18	u	u	NOUN
ejpam-3284	42	19	=	=	NOUN
ejpam-3284	42	20	0	0	NUM
ejpam-3284	42	21	(	(	PUNCT
ejpam-3284	42	22	ii	ii	NOUN
ejpam-3284	42	23	)	)	PUNCT
ejpam-3284	42	24	(	(	PUNCT
ejpam-3284	42	25	u	u	NOUN
ejpam-3284	42	26	,	,	PUNCT
ejpam-3284	42	27	v	v	NOUN
ejpam-3284	42	28	)	)	PUNCT
ejpam-3284	42	29	=	=	SYM
ejpam-3284	42	30	(	(	PUNCT
ejpam-3284	42	31	v	v	NOUN
ejpam-3284	42	32	,	,	PUNCT
ejpam-3284	42	33	u	u	NOUN
ejpam-3284	42	34	)	)	PUNCT
ejpam-3284	42	35	,	,	PUNCT
ejpam-3284	42	36	∀	∀	X
ejpam-3284	42	37	u	u	NOUN
ejpam-3284	42	38	,	,	PUNCT
ejpam-3284	42	39	v	v	NOUN
ejpam-3284	42	40	∈	∈	PROPN
ejpam-3284	42	41	v	v	ADP
ejpam-3284	42	42	(	(	PUNCT
ejpam-3284	42	43	iii	iii	NOUN
ejpam-3284	42	44	)	)	PUNCT
ejpam-3284	42	45	(	(	PUNCT
ejpam-3284	42	46	αu1	αu1	NOUN
ejpam-3284	42	47	+	+	CCONJ
ejpam-3284	42	48	βu2	βu2	ADJ
ejpam-3284	42	49	,	,	PUNCT
ejpam-3284	42	50	v	v	NOUN
ejpam-3284	42	51	)	)	PUNCT
ejpam-3284	42	52	=	=	SYM
ejpam-3284	42	53	α(u1	α(u1	PROPN
ejpam-3284	42	54	,	,	PUNCT
ejpam-3284	42	55	v	v	NOUN
ejpam-3284	42	56	)	)	PUNCT
ejpam-3284	42	57	+	+	CCONJ
ejpam-3284	42	58	β(u2	β(u2	NOUN
ejpam-3284	42	59	,	,	PUNCT
ejpam-3284	42	60	v	v	NOUN
ejpam-3284	42	61	)	)	PUNCT
ejpam-3284	42	62	,	,	PUNCT
ejpam-3284	42	63	∀	∀	X
ejpam-3284	42	64	α	α	NOUN
ejpam-3284	42	65	,	,	PUNCT
ejpam-3284	42	66	β	β	X
ejpam-3284	42	67	∈	∈	NOUN
ejpam-3284	42	68	r	r	NOUN
ejpam-3284	42	69	,	,	PUNCT
ejpam-3284	42	70	and	and	CCONJ
ejpam-3284	42	71	u1	u1	NOUN
ejpam-3284	42	72	,	,	PUNCT
ejpam-3284	42	73	u2	u2	PROPN
ejpam-3284	42	74	,	,	PUNCT
ejpam-3284	42	75	v	v	NOUN
ejpam-3284	42	76	∈	∈	PROPN
ejpam-3284	42	77	v	v	NOUN
ejpam-3284	42	78	.	.	PUNCT
ejpam-3284	43	1	see	see	VERB
ejpam-3284	43	2	[	[	X
ejpam-3284	43	3	14	14	NUM
ejpam-3284	43	4	]	]	PUNCT
ejpam-3284	43	5	definition	definition	NOUN
ejpam-3284	43	6	3	3	NUM
ejpam-3284	43	7	(	(	PUNCT
ejpam-3284	43	8	subspace	subspace	NOUN
ejpam-3284	43	9	)	)	PUNCT
ejpam-3284	43	10	.	.	PUNCT
ejpam-3284	44	1	let	let	VERB
ejpam-3284	44	2	v	v	PART
ejpam-3284	44	3	be	be	AUX
ejpam-3284	44	4	a	a	DET
ejpam-3284	44	5	vector	vector	NOUN
ejpam-3284	44	6	space	space	NOUN
ejpam-3284	44	7	and	and	CCONJ
ejpam-3284	44	8	ω	ω	NUM
ejpam-3284	44	9	a	a	DET
ejpam-3284	44	10	nonempty	nonempty	NOUN
ejpam-3284	44	11	subset	subset	NOUN
ejpam-3284	44	12	of	of	ADP
ejpam-3284	44	13	v	v	NOUN
ejpam-3284	44	14	.	.	PUNCT
ejpam-3284	45	1	if	if	SCONJ
ejpam-3284	45	2	ω	ω	PROPN
ejpam-3284	45	3	is	be	AUX
ejpam-3284	45	4	a	a	DET
ejpam-3284	45	5	vector	vector	NOUN
ejpam-3284	45	6	space	space	NOUN
ejpam-3284	45	7	with	with	ADP
ejpam-3284	45	8	respect	respect	NOUN
ejpam-3284	45	9	to	to	ADP
ejpam-3284	45	10	the	the	DET
ejpam-3284	45	11	operations	operation	NOUN
ejpam-3284	45	12	in	in	ADP
ejpam-3284	45	13	v	v	NUM
ejpam-3284	45	14	,	,	PUNCT
ejpam-3284	45	15	then	then	ADV
ejpam-3284	45	16	ω	ω	PROPN
ejpam-3284	45	17	is	be	AUX
ejpam-3284	45	18	called	call	VERB
ejpam-3284	45	19	a	a	DET
ejpam-3284	45	20	subspace	subspace	NOUN
ejpam-3284	45	21	of	of	ADP
ejpam-3284	45	22	v	v	NOUN
ejpam-3284	45	23	.	.	PUNCT
ejpam-3284	46	1	see	see	VERB
ejpam-3284	46	2	[	[	X
ejpam-3284	46	3	15	15	NUM
ejpam-3284	46	4	]	]	PUNCT
ejpam-3284	46	5	.	.	PUNCT
ejpam-3284	47	1	definition	definition	NOUN
ejpam-3284	47	2	4	4	NUM
ejpam-3284	47	3	(	(	PUNCT
ejpam-3284	47	4	continuity	continuity	NOUN
ejpam-3284	47	5	at	at	ADP
ejpam-3284	47	6	a	a	DET
ejpam-3284	47	7	point	point	NOUN
ejpam-3284	47	8	)	)	PUNCT
ejpam-3284	47	9	.	.	PUNCT
ejpam-3284	48	1	let	let	VERB
ejpam-3284	48	2	u	u	PRON
ejpam-3284	48	3	and	and	CCONJ
ejpam-3284	48	4	v	v	NOUN
ejpam-3284	48	5	be	be	AUX
ejpam-3284	48	6	two	two	NUM
ejpam-3284	48	7	normed	normed	ADJ
ejpam-3284	48	8	linear	linear	ADJ
ejpam-3284	48	9	spaces	space	NOUN
ejpam-3284	48	10	.	.	PUNCT
ejpam-3284	49	1	an	an	DET
ejpam-3284	49	2	operator	operator	NOUN
ejpam-3284	49	3	a	a	DET
ejpam-3284	49	4	:	:	PUNCT
ejpam-3284	49	5	u	u	NOUN
ejpam-3284	49	6	→	→	SYM
ejpam-3284	49	7	v	v	NUM
ejpam-3284	49	8	is	be	AUX
ejpam-3284	49	9	continuous	continuous	ADJ
ejpam-3284	49	10	at	at	ADP
ejpam-3284	49	11	uo	uo	PROPN
ejpam-3284	49	12	∈	∈	PROPN
ejpam-3284	49	13	u	u	NOUN
ejpam-3284	49	14	,	,	PUNCT
ejpam-3284	49	15	if	if	SCONJ
ejpam-3284	49	16	for	for	ADP
ejpam-3284	49	17	every	every	DET
ejpam-3284	49	18	ε	ε	PROPN
ejpam-3284	49	19	>	>	X
ejpam-3284	49	20	0	0	PROPN
ejpam-3284	49	21	,	,	PUNCT
ejpam-3284	49	22	there	there	PRON
ejpam-3284	49	23	exists	exist	VERB
ejpam-3284	49	24	a	a	DET
ejpam-3284	49	25	δ	δ	PROPN
ejpam-3284	49	26	=	=	PUNCT
ejpam-3284	49	27	δ(ε	δ(ε	PROPN
ejpam-3284	49	28	,	,	PUNCT
ejpam-3284	49	29	uo	uo	NOUN
ejpam-3284	49	30	)	)	PUNCT
ejpam-3284	49	31	such	such	ADJ
ejpam-3284	50	1	that	that	SCONJ
ejpam-3284	50	2	‖au−auo‖	‖au−auo‖	PRON
ejpam-3284	50	3	<	<	X
ejpam-3284	50	4	ε	ε	PROPN
ejpam-3284	50	5	,	,	PUNCT
ejpam-3284	50	6	whenever	whenever	SCONJ
ejpam-3284	50	7	‖u−	‖u−	PROPN
ejpam-3284	50	8	uo‖	uo‖	NOUN
ejpam-3284	50	9	<	<	X
ejpam-3284	50	10	δ	δ	PROPN
ejpam-3284	50	11	.	.	PUNCT
ejpam-3284	50	12	see	see	VERB
ejpam-3284	50	13	[	[	X
ejpam-3284	50	14	16	16	NUM
ejpam-3284	50	15	]	]	PUNCT
ejpam-3284	50	16	.	.	PUNCT
ejpam-3284	51	1	definition	definition	NOUN
ejpam-3284	51	2	5	5	NUM
ejpam-3284	51	3	(	(	PUNCT
ejpam-3284	51	4	convergent	convergent	NOUN
ejpam-3284	51	5	sequence	sequence	NOUN
ejpam-3284	51	6	)	)	PUNCT
ejpam-3284	51	7	.	.	PUNCT
ejpam-3284	52	1	a	a	DET
ejpam-3284	52	2	sequence	sequence	NOUN
ejpam-3284	52	3	{	{	PUNCT
ejpam-3284	52	4	un}∞n=1	un}∞n=1	NUM
ejpam-3284	52	5	in	in	ADP
ejpam-3284	52	6	a	a	DET
ejpam-3284	52	7	normed	normed	ADJ
ejpam-3284	52	8	linear	linear	ADJ
ejpam-3284	52	9	space	space	NOUN
ejpam-3284	52	10	(	(	PUNCT
ejpam-3284	52	11	rn	rn	PROPN
ejpam-3284	52	12	,	,	PUNCT
ejpam-3284	52	13	‖	‖	PROPN
ejpam-3284	52	14	·	·	PUNCT
ejpam-3284	52	15	‖	‖	ADJ
ejpam-3284	52	16	)	)	PUNCT
ejpam-3284	52	17	converges	converge	VERB
ejpam-3284	52	18	to	to	ADP
ejpam-3284	52	19	a	a	DET
ejpam-3284	52	20	point	point	NOUN
ejpam-3284	52	21	u	u	PROPN
ejpam-3284	52	22	∈	∈	PROPN
ejpam-3284	52	23	rn	rn	PROPN
ejpam-3284	52	24	,	,	PUNCT
ejpam-3284	52	25	if	if	SCONJ
ejpam-3284	52	26	and	and	CCONJ
ejpam-3284	52	27	only	only	ADV
ejpam-3284	52	28	if	if	SCONJ
ejpam-3284	52	29	,	,	PUNCT
ejpam-3284	52	30	for	for	ADP
ejpam-3284	52	31	every	every	DET
ejpam-3284	52	32	real	real	ADJ
ejpam-3284	52	33	number	number	NOUN
ejpam-3284	52	34	ε	ε	PROPN
ejpam-3284	52	35	>	>	X
ejpam-3284	52	36	0	0	PROPN
ejpam-3284	52	37	,	,	PUNCT
ejpam-3284	52	38	there	there	PRON
ejpam-3284	52	39	exists	exist	VERB
ejpam-3284	52	40	an	an	DET
ejpam-3284	52	41	integer	integer	NOUN
ejpam-3284	52	42	n	n	CCONJ
ejpam-3284	52	43	such	such	ADJ
ejpam-3284	52	44	that	that	SCONJ
ejpam-3284	52	45	‖un	‖un	PROPN
ejpam-3284	52	46	−	−	PROPN
ejpam-3284	52	47	u‖	u‖	NOUN
ejpam-3284	52	48	<	<	X
ejpam-3284	52	49	ε	ε	PROPN
ejpam-3284	52	50	,	,	PUNCT
ejpam-3284	52	51	n	n	PROPN
ejpam-3284	52	52	>	>	X
ejpam-3284	52	53	n.	n.	PROPN
ejpam-3284	52	54	see	see	VERB
ejpam-3284	53	1	[	[	X
ejpam-3284	53	2	17	17	NUM
ejpam-3284	53	3	]	]	PUNCT
ejpam-3284	53	4	.	.	PUNCT
ejpam-3284	54	1	definition	definition	NOUN
ejpam-3284	54	2	6	6	NUM
ejpam-3284	54	3	(	(	PUNCT
ejpam-3284	54	4	closed	closed	ADJ
ejpam-3284	54	5	set	set	NOUN
ejpam-3284	54	6	)	)	PUNCT
ejpam-3284	54	7	.	.	PUNCT
ejpam-3284	55	1	a	a	DET
ejpam-3284	55	2	set	set	NOUN
ejpam-3284	55	3	u	u	NOUN
ejpam-3284	55	4	=	=	PUNCT
ejpam-3284	55	5	[	[	X
ejpam-3284	55	6	0	0	NUM
ejpam-3284	55	7	,	,	PUNCT
ejpam-3284	55	8	2	2	NUM
ejpam-3284	55	9	]	]	PUNCT
ejpam-3284	55	10	in	in	ADP
ejpam-3284	55	11	a	a	DET
ejpam-3284	55	12	normed	normed	ADJ
ejpam-3284	55	13	linear	linear	ADJ
ejpam-3284	55	14	space	space	NOUN
ejpam-3284	55	15	(	(	PUNCT
ejpam-3284	55	16	rn	rn	PROPN
ejpam-3284	55	17	,	,	PUNCT
ejpam-3284	55	18	‖	‖	PROPN
ejpam-3284	55	19	·	·	PUNCT
ejpam-3284	55	20	‖	‖	PROPN
ejpam-3284	55	21	)	)	PUNCT
ejpam-3284	55	22	is	be	AUX
ejpam-3284	55	23	closed	close	VERB
ejpam-3284	55	24	if	if	SCONJ
ejpam-3284	55	25	and	and	CCONJ
ejpam-3284	55	26	only	only	ADV
ejpam-3284	55	27	if	if	SCONJ
ejpam-3284	55	28	every	every	DET
ejpam-3284	55	29	sequence	sequence	NOUN
ejpam-3284	55	30	of	of	ADP
ejpam-3284	55	31	points	point	NOUN
ejpam-3284	55	32	{	{	PUNCT
ejpam-3284	55	33	un}∞n=1	un}∞n=1	X
ejpam-3284	55	34	⊂	⊂	X
ejpam-3284	55	35	u	u	NOUN
ejpam-3284	55	36	convergent	convergent	NOUN
ejpam-3284	55	37	in	in	ADP
ejpam-3284	55	38	rn	rn	PROPN
ejpam-3284	55	39	has	have	VERB
ejpam-3284	55	40	a	a	DET
ejpam-3284	55	41	limit	limit	NOUN
ejpam-3284	55	42	in	in	ADP
ejpam-3284	55	43	u	u	PROPN
ejpam-3284	55	44	.	.	PUNCT
ejpam-3284	56	1	for	for	ADP
ejpam-3284	56	2	example	example	NOUN
ejpam-3284	56	3	,	,	PUNCT
ejpam-3284	56	4	see	see	VERB
ejpam-3284	56	5	[	[	X
ejpam-3284	56	6	18	18	NUM
ejpam-3284	56	7	]	]	PUNCT
ejpam-3284	56	8	definition	definition	NOUN
ejpam-3284	56	9	7	7	NUM
ejpam-3284	56	10	(	(	PUNCT
ejpam-3284	56	11	contractive	contractive	ADJ
ejpam-3284	56	12	mapping	mapping	NOUN
ejpam-3284	56	13	)	)	PUNCT
ejpam-3284	56	14	.	.	PUNCT
ejpam-3284	57	1	let	let	VERB
ejpam-3284	57	2	t	t	NOUN
ejpam-3284	57	3	:	:	PUNCT
ejpam-3284	57	4	u	u	PROPN
ejpam-3284	57	5	→	→	SYM
ejpam-3284	57	6	u	u	X
ejpam-3284	57	7	be	be	VERB
ejpam-3284	57	8	a	a	DET
ejpam-3284	57	9	mapping	mapping	NOUN
ejpam-3284	57	10	from	from	ADP
ejpam-3284	57	11	a	a	DET
ejpam-3284	57	12	complete	complete	ADJ
ejpam-3284	57	13	normed	norme	VERB
ejpam-3284	57	14	linear	linear	ADJ
ejpam-3284	57	15	space	space	NOUN
ejpam-3284	57	16	u	u	NOUN
ejpam-3284	57	17	into	into	ADP
ejpam-3284	57	18	itself	itself	PRON
ejpam-3284	57	19	.	.	PUNCT
ejpam-3284	58	1	the	the	DET
ejpam-3284	58	2	lipschitz	lipschitz	ADJ
ejpam-3284	58	3	continuity	continuity	NOUN
ejpam-3284	58	4	on	on	ADP
ejpam-3284	58	5	t	t	PROPN
ejpam-3284	58	6	is	be	AUX
ejpam-3284	58	7	said	say	VERB
ejpam-3284	58	8	to	to	PART
ejpam-3284	58	9	be	be	AUX
ejpam-3284	58	10	contraction	contraction	NOUN
ejpam-3284	58	11	if	if	SCONJ
ejpam-3284	58	12	‖t	‖t	PROPN
ejpam-3284	58	13	(	(	PUNCT
ejpam-3284	58	14	u)−	u)−	PROPN
ejpam-3284	58	15	t	t	PROPN
ejpam-3284	58	16	(	(	PUNCT
ejpam-3284	58	17	v)‖	v)‖	NOUN
ejpam-3284	58	18	≤	≤	NOUN
ejpam-3284	58	19	λ‖u−	λ‖u−	ADJ
ejpam-3284	58	20	v‖	v‖	NOUN
ejpam-3284	58	21	,	,	PUNCT
ejpam-3284	58	22	∀	∀	X
ejpam-3284	58	23	u	u	NOUN
ejpam-3284	58	24	,	,	PUNCT
ejpam-3284	58	25	v	v	NOUN
ejpam-3284	58	26	∈	∈	PROPN
ejpam-3284	58	27	u	u	NOUN
ejpam-3284	58	28	,	,	PUNCT
ejpam-3284	58	29	and	and	CCONJ
ejpam-3284	58	30	λ	λ	X
ejpam-3284	58	31	<	<	X
ejpam-3284	58	32	1	1	X
ejpam-3284	58	33	.	.	PUNCT
ejpam-3284	58	34	see	see	VERB
ejpam-3284	58	35	[	[	X
ejpam-3284	58	36	19	19	NUM
ejpam-3284	58	37	]	]	PUNCT
ejpam-3284	58	38	.	.	PUNCT
ejpam-3284	59	1	3	3	X
ejpam-3284	59	2	.	.	X
ejpam-3284	59	3	main	main	ADJ
ejpam-3284	59	4	result	result	NOUN
ejpam-3284	59	5	in	in	ADP
ejpam-3284	59	6	this	this	DET
ejpam-3284	59	7	section	section	NOUN
ejpam-3284	59	8	,	,	PUNCT
ejpam-3284	59	9	we	we	PRON
ejpam-3284	59	10	introduce	introduce	VERB
ejpam-3284	59	11	product	product	NOUN
ejpam-3284	59	12	-	-	PUNCT
ejpam-3284	59	13	normed	norme	VERB
ejpam-3284	59	14	linear	linear	ADJ
ejpam-3284	59	15	space	space	NOUN
ejpam-3284	59	16	and	and	CCONJ
ejpam-3284	59	17	product	product	NOUN
ejpam-3284	59	18	-	-	PUNCT
ejpam-3284	59	19	semi	semi	ADV
ejpam-3284	59	20	-	-	ADJ
ejpam-3284	59	21	normed	normed	ADJ
ejpam-3284	59	22	linear	linear	ADJ
ejpam-3284	59	23	space	space	NOUN
ejpam-3284	59	24	.	.	PUNCT
ejpam-3284	60	1	the	the	DET
ejpam-3284	60	2	product	product	NOUN
ejpam-3284	60	3	-	-	PUNCT
ejpam-3284	60	4	normed	norme	VERB
ejpam-3284	60	5	linear	linear	ADJ
ejpam-3284	60	6	space	space	NOUN
ejpam-3284	60	7	is	be	AUX
ejpam-3284	60	8	presented	present	VERB
ejpam-3284	60	9	as	as	SCONJ
ejpam-3284	60	10	follows	follow	VERB
ejpam-3284	60	11	.	.	PUNCT
ejpam-3284	61	1	barnes	barnes	PROPN
ejpam-3284	61	2	et	et	PROPN
ejpam-3284	61	3	al	al	PROPN
ejpam-3284	61	4	.	.	PUNCT
ejpam-3284	61	5	/	/	SYM
ejpam-3284	61	6	eur	eur	PROPN
ejpam-3284	61	7	.	.	PUNCT
ejpam-3284	62	1	j.	j.	PROPN
ejpam-3284	62	2	pure	pure	PROPN
ejpam-3284	62	3	appl	appl	PROPN
ejpam-3284	62	4	.	.	PROPN
ejpam-3284	62	5	math	math	PROPN
ejpam-3284	62	6	,	,	PUNCT
ejpam-3284	62	7	11	11	NUM
ejpam-3284	62	8	(	(	PUNCT
ejpam-3284	62	9	3	3	NUM
ejpam-3284	62	10	)	)	PUNCT
ejpam-3284	62	11	(	(	PUNCT
ejpam-3284	62	12	2018	2018	NUM
ejpam-3284	62	13	)	)	PUNCT
ejpam-3284	62	14	,	,	PUNCT
ejpam-3284	62	15	740	740	NUM
ejpam-3284	62	16	-	-	SYM
ejpam-3284	62	17	750	750	NUM
ejpam-3284	62	18	743	743	NUM
ejpam-3284	62	19	theorem	theorem	NOUN
ejpam-3284	62	20	1	1	NUM
ejpam-3284	62	21	(	(	PUNCT
ejpam-3284	62	22	product	product	NOUN
ejpam-3284	62	23	-	-	PUNCT
ejpam-3284	62	24	normed	norme	VERB
ejpam-3284	62	25	linear	linear	ADJ
ejpam-3284	62	26	space	space	NOUN
ejpam-3284	62	27	)	)	PUNCT
ejpam-3284	62	28	.	.	PUNCT
ejpam-3284	63	1	let	let	VERB
ejpam-3284	63	2	u	u	PRON
ejpam-3284	63	3	be	be	AUX
ejpam-3284	63	4	a	a	DET
ejpam-3284	63	5	linear	linear	ADJ
ejpam-3284	63	6	space	space	NOUN
ejpam-3284	63	7	over	over	ADP
ejpam-3284	63	8	[	[	X
ejpam-3284	63	9	0	0	NUM
ejpam-3284	63	10	,	,	PUNCT
ejpam-3284	63	11	2	2	NUM
ejpam-3284	63	12	]	]	PUNCT
ejpam-3284	63	13	⊆	⊆	NUM
ejpam-3284	63	14	r.	r.	NOUN
ejpam-3284	63	15	a	a	DET
ejpam-3284	63	16	vector	vector	NOUN
ejpam-3284	63	17	space	space	NOUN
ejpam-3284	63	18	with	with	ADP
ejpam-3284	63	19	a	a	DET
ejpam-3284	63	20	product	product	NOUN
ejpam-3284	63	21	-	-	PUNCT
ejpam-3284	63	22	norm	norm	NOUN
ejpam-3284	63	23	u→	u→	PROPN
ejpam-3284	63	24	‖u‖	‖u‖	PROPN
ejpam-3284	63	25	satisfying	satisfy	VERB
ejpam-3284	63	26	real	real	ADV
ejpam-3284	63	27	-	-	PUNCT
ejpam-3284	63	28	valued	value	VERB
ejpam-3284	63	29	function	function	NOUN
ejpam-3284	63	30	‖	‖	PROPN
ejpam-3284	63	31	·	·	PUNCT
ejpam-3284	63	32	‖	‖	PROPN
ejpam-3284	63	33	,	,	PUNCT
ejpam-3284	63	34	‖	‖	ADJ
ejpam-3284	63	35	·	·	PUNCT
ejpam-3284	63	36	‖pn	‖pn	NUM
ejpam-3284	63	37	:	:	PUNCT
ejpam-3284	63	38	u	u	NOUN
ejpam-3284	63	39	→	→	PUNCT
ejpam-3284	63	40	[	[	X
ejpam-3284	63	41	0,∞	0,∞	NOUN
ejpam-3284	63	42	)	)	PUNCT
ejpam-3284	63	43	,	,	PUNCT
ejpam-3284	63	44	such	such	ADJ
ejpam-3284	63	45	that	that	SCONJ
ejpam-3284	63	46	for	for	ADP
ejpam-3284	63	47	arbitrary	arbitrary	ADJ
ejpam-3284	63	48	u	u	NOUN
ejpam-3284	63	49	,	,	PUNCT
ejpam-3284	63	50	v	v	PROPN
ejpam-3284	63	51	∈	∈	PROPN
ejpam-3284	63	52	u	u	NOUN
ejpam-3284	63	53	,	,	PUNCT
ejpam-3284	63	54	α	α	PROPN
ejpam-3284	63	55	∈	∈	PROPN
ejpam-3284	64	1	[	[	X
ejpam-3284	64	2	0	0	NUM
ejpam-3284	64	3	,	,	PUNCT
ejpam-3284	64	4	2	2	NUM
ejpam-3284	64	5	]	]	PUNCT
ejpam-3284	64	6	,	,	PUNCT
ejpam-3284	64	7	the	the	DET
ejpam-3284	64	8	following	follow	VERB
ejpam-3284	64	9	conditions	condition	NOUN
ejpam-3284	64	10	are	be	AUX
ejpam-3284	64	11	satisfied	satisfied	ADJ
ejpam-3284	64	12	:	:	PUNCT
ejpam-3284	64	13	p1	p1	NOUN
ejpam-3284	64	14	.	.	PUNCT
ejpam-3284	65	1	‖u‖	‖u‖	PROPN
ejpam-3284	65	2	≥	≥	NOUN
ejpam-3284	65	3	0	0	NUM
ejpam-3284	65	4	,	,	PUNCT
ejpam-3284	65	5	and	and	CCONJ
ejpam-3284	65	6	‖u‖	‖u‖	PROPN
ejpam-3284	65	7	=	=	PUNCT
ejpam-3284	66	1	0	0	NUM
ejpam-3284	66	2	,	,	PUNCT
ejpam-3284	66	3	if	if	SCONJ
ejpam-3284	66	4	and	and	CCONJ
ejpam-3284	66	5	only	only	ADV
ejpam-3284	66	6	if	if	SCONJ
ejpam-3284	66	7	u	u	NOUN
ejpam-3284	66	8	=	=	NOUN
ejpam-3284	66	9	0	0	NUM
ejpam-3284	66	10	p2	p2	NOUN
ejpam-3284	66	11	.	.	PUNCT
ejpam-3284	67	1	‖αu‖	‖αu‖	ADJ
ejpam-3284	67	2	=	=	SYM
ejpam-3284	67	3	|α|‖u‖	|α|‖u‖	NOUN
ejpam-3284	67	4	,	,	PUNCT
ejpam-3284	67	5	α	α	NOUN
ejpam-3284	67	6	∈	∈	PROPN
ejpam-3284	68	1	[	[	X
ejpam-3284	68	2	0	0	NUM
ejpam-3284	68	3	,	,	PUNCT
ejpam-3284	68	4	2	2	NUM
ejpam-3284	68	5	]	]	PUNCT
ejpam-3284	68	6	,	,	PUNCT
ejpam-3284	68	7	and	and	CCONJ
ejpam-3284	68	8	u	u	PROPN
ejpam-3284	68	9	∈	∈	PROPN
ejpam-3284	68	10	u	u	PROPN
ejpam-3284	68	11	p3	p3	PROPN
ejpam-3284	68	12	.	.	PUNCT
ejpam-3284	69	1	‖u+	‖u+	NOUN
ejpam-3284	69	2	v‖pn	v‖pn	VERB
ejpam-3284	69	3	≤	≤	NUM
ejpam-3284	69	4	‖u‖pn	‖u‖pn	PUNCT
ejpam-3284	70	1	+	+	CCONJ
ejpam-3284	70	2	‖v‖pn	‖v‖pn	PROPN
ejpam-3284	70	3	,	,	PUNCT
ejpam-3284	70	4	∀u	∀u	NOUN
ejpam-3284	70	5	,	,	PUNCT
ejpam-3284	70	6	v	v	X
ejpam-3284	70	7	∈	∈	NOUN
ejpam-3284	70	8	u	u	NOUN
ejpam-3284	70	9	we	we	PRON
ejpam-3284	70	10	call	call	VERB
ejpam-3284	70	11	‖	‖	PROPN
ejpam-3284	70	12	·	·	PUNCT
ejpam-3284	70	13	‖pn	‖pn	NUM
ejpam-3284	70	14	a	a	DET
ejpam-3284	70	15	product	product	NOUN
ejpam-3284	70	16	norm	norm	NOUN
ejpam-3284	70	17	if	if	SCONJ
ejpam-3284	70	18	in	in	ADP
ejpam-3284	70	19	addition	addition	NOUN
ejpam-3284	70	20	to	to	ADP
ejpam-3284	70	21	p1−	p1−	PROPN
ejpam-3284	70	22	p3	p3	PROPN
ejpam-3284	70	23	,	,	PUNCT
ejpam-3284	70	24	the	the	DET
ejpam-3284	70	25	u	u	NOUN
ejpam-3284	70	26	and	and	CCONJ
ejpam-3284	70	27	v	v	ADP
ejpam-3284	70	28	satisfy	satisfy	NOUN
ejpam-3284	70	29	p4(a	p4(a	NOUN
ejpam-3284	70	30	)	)	PUNCT
ejpam-3284	70	31	.	.	PUNCT
ejpam-3284	71	1	‖u‖pn‖y‖pn	‖u‖pn‖y‖pn	NOUN
ejpam-3284	71	2	≤	≤	NUM
ejpam-3284	71	3	‖u‖pn	‖u‖pn	VERB
ejpam-3284	72	1	+	+	NUM
ejpam-3284	72	2	‖v‖pn	‖v‖pn	NUM
ejpam-3284	72	3	∀u	∀u	NOUN
ejpam-3284	72	4	,	,	PUNCT
ejpam-3284	72	5	v	v	NOUN
ejpam-3284	72	6	∈	∈	PROPN
ejpam-3284	73	1	[	[	X
ejpam-3284	73	2	0	0	NUM
ejpam-3284	73	3	,	,	PUNCT
ejpam-3284	73	4	2	2	NUM
ejpam-3284	73	5	]	]	PUNCT
ejpam-3284	73	6	(	(	PUNCT
ejpam-3284	73	7	first	first	ADJ
ejpam-3284	73	8	product	product	NOUN
ejpam-3284	73	9	inequality	inequality	NOUN
ejpam-3284	73	10	)	)	PUNCT
ejpam-3284	73	11	p4(b).‖u‖+	p4(b).‖u‖+	PUNCT
ejpam-3284	74	1	‖v‖	‖v‖	PROPN
ejpam-3284	74	2	≤	≤	NUM
ejpam-3284	74	3	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3284	74	4	,	,	PUNCT
ejpam-3284	74	5	∀u	∀u	NOUN
ejpam-3284	74	6	,	,	PUNCT
ejpam-3284	74	7	v	v	NOUN
ejpam-3284	74	8	∈	∈	PROPN
ejpam-3284	75	1	[	[	X
ejpam-3284	75	2	2,∞	2,∞	NUM
ejpam-3284	75	3	)	)	PUNCT
ejpam-3284	75	4	(	(	PUNCT
ejpam-3284	75	5	second	second	ADJ
ejpam-3284	75	6	product	product	NOUN
ejpam-3284	75	7	inequality	inequality	NOUN
ejpam-3284	75	8	)	)	PUNCT
ejpam-3284	75	9	.	.	PUNCT
ejpam-3284	76	1	p	p	NOUN
ejpam-3284	76	2	roof	roof	NOUN
ejpam-3284	76	3	:	:	PUNCT
ejpam-3284	76	4	we	we	PRON
ejpam-3284	76	5	start	start	VERB
ejpam-3284	76	6	at	at	ADP
ejpam-3284	76	7	p1	p1	PROPN
ejpam-3284	76	8	.	.	PUNCT
ejpam-3284	76	9	suppose	suppose	VERB
ejpam-3284	76	10	that	that	SCONJ
ejpam-3284	76	11	u	u	NOUN
ejpam-3284	76	12	=	=	NOUN
ejpam-3284	76	13	0	0	NUM
ejpam-3284	76	14	,	,	PUNCT
ejpam-3284	76	15	then	then	ADV
ejpam-3284	76	16	‖u‖2pn	‖u‖2pn	NOUN
ejpam-3284	76	17	=	=	SYM
ejpam-3284	76	18	〈	〈	PROPN
ejpam-3284	76	19	0	0	NUM
ejpam-3284	76	20	,	,	PUNCT
ejpam-3284	77	1	0	0	NUM
ejpam-3284	77	2	〉	〉	PROPN
ejpam-3284	77	3	⇒	⇒	NOUN
ejpam-3284	77	4	‖u‖2pn	‖u‖2pn	NOUN
ejpam-3284	78	1	=	=	SYM
ejpam-3284	78	2	0	0	NUM
ejpam-3284	78	3	⇒	⇒	NOUN
ejpam-3284	78	4	‖u‖pn	‖u‖pn	VERB
ejpam-3284	79	1	=	=	NOUN
ejpam-3284	79	2	0	0	PROPN
ejpam-3284	79	3	.	.	PUNCT
ejpam-3284	80	1	on	on	ADP
ejpam-3284	80	2	the	the	DET
ejpam-3284	80	3	other	other	ADJ
ejpam-3284	80	4	hand	hand	NOUN
ejpam-3284	80	5	,	,	PUNCT
ejpam-3284	80	6	we	we	PRON
ejpam-3284	80	7	suppose	suppose	VERB
ejpam-3284	80	8	that	that	SCONJ
ejpam-3284	80	9	‖u‖pn	‖u‖pn	PROPN
ejpam-3284	80	10	=	=	NOUN
ejpam-3284	80	11	0	0	PROPN
ejpam-3284	80	12	.	.	PUNCT
ejpam-3284	81	1	but	but	CCONJ
ejpam-3284	81	2	,	,	PUNCT
ejpam-3284	81	3	‖u‖2pn	‖u‖2pn	NOUN
ejpam-3284	81	4	=	=	SYM
ejpam-3284	82	1	〈	〈	PROPN
ejpam-3284	82	2	u	u	NOUN
ejpam-3284	82	3	,	,	PUNCT
ejpam-3284	82	4	u	u	NOUN
ejpam-3284	82	5	〉	〉	NOUN
ejpam-3284	82	6	⇒	⇒	NOUN
ejpam-3284	82	7	0	0	NUM
ejpam-3284	82	8	=	=	SYM
ejpam-3284	82	9	〈	〈	PROPN
ejpam-3284	82	10	uu	uu	NOUN
ejpam-3284	82	11	〉	〉	PROPN
ejpam-3284	82	12	.	.	PUNCT
ejpam-3284	83	1	the	the	DET
ejpam-3284	83	2	above	above	ADJ
ejpam-3284	83	3	equality	equality	NOUN
ejpam-3284	83	4	holds	hold	VERB
ejpam-3284	83	5	if	if	SCONJ
ejpam-3284	83	6	u	u	NOUN
ejpam-3284	83	7	=	=	NOUN
ejpam-3284	83	8	0	0	NUM
ejpam-3284	83	9	.	.	PUNCT
ejpam-3284	84	1	hence	hence	ADV
ejpam-3284	84	2	,	,	PUNCT
ejpam-3284	84	3	‖u‖	‖u‖	PROPN
ejpam-3284	84	4	=	=	PUNCT
ejpam-3284	84	5	0	0	PUNCT
ejpam-3284	85	1	if	if	SCONJ
ejpam-3284	85	2	and	and	CCONJ
ejpam-3284	85	3	only	only	ADV
ejpam-3284	85	4	if	if	SCONJ
ejpam-3284	85	5	u	u	NOUN
ejpam-3284	85	6	=	=	NOUN
ejpam-3284	85	7	0	0	NUM
ejpam-3284	85	8	.	.	PUNCT
ejpam-3284	86	1	without	without	ADP
ejpam-3284	86	2	loss	loss	NOUN
ejpam-3284	86	3	of	of	ADP
ejpam-3284	86	4	generality	generality	NOUN
ejpam-3284	86	5	,	,	PUNCT
ejpam-3284	86	6	let	let	VERB
ejpam-3284	86	7	the	the	DET
ejpam-3284	86	8	‖u‖pn	‖u‖pn	PROPN
ejpam-3284	86	9	=	=	SYM
ejpam-3284	86	10	|u|pn	|u|pn	PROPN
ejpam-3284	86	11	,	,	PUNCT
ejpam-3284	86	12	then	then	ADV
ejpam-3284	86	13	|u|pn	|u|pn	ADJ
ejpam-3284	86	14	=	=	PUNCT
ejpam-3284	86	15	0	0	NUM
ejpam-3284	86	16	,	,	PUNCT
ejpam-3284	86	17	which	which	PRON
ejpam-3284	86	18	implies	imply	VERB
ejpam-3284	86	19	that	that	SCONJ
ejpam-3284	86	20	‖u‖pn	‖u‖pn	PROPN
ejpam-3284	86	21	=	=	SYM
ejpam-3284	86	22	0	0	PROPN
ejpam-3284	86	23	.	.	PUNCT
ejpam-3284	87	1	but	but	CCONJ
ejpam-3284	87	2	‖u‖2pn	‖u‖2pn	NOUN
ejpam-3284	87	3	=	=	SYM
ejpam-3284	88	1	〈	〈	PROPN
ejpam-3284	88	2	u	u	NOUN
ejpam-3284	88	3	,	,	PUNCT
ejpam-3284	88	4	u	u	NOUN
ejpam-3284	88	5	〉	〉	NOUN
ejpam-3284	88	6	.	.	PUNCT
ejpam-3284	89	1	since	since	SCONJ
ejpam-3284	89	2	the	the	DET
ejpam-3284	89	3	〈	〈	PROPN
ejpam-3284	89	4	u	u	NOUN
ejpam-3284	89	5	,	,	PUNCT
ejpam-3284	89	6	u	u	NOUN
ejpam-3284	89	7	〉	〉	NOUN
ejpam-3284	89	8	can	can	AUX
ejpam-3284	89	9	not	not	PART
ejpam-3284	89	10	be	be	AUX
ejpam-3284	89	11	negative	negative	ADJ
ejpam-3284	89	12	.	.	PUNCT
ejpam-3284	90	1	this	this	PRON
ejpam-3284	90	2	implies	imply	VERB
ejpam-3284	90	3	that	that	SCONJ
ejpam-3284	90	4	‖u‖pn	‖u‖pn	INTJ
ejpam-3284	90	5	>	>	X
ejpam-3284	90	6	0	0	NUM
ejpam-3284	90	7	.	.	PUNCT
ejpam-3284	91	1	thus	thus	ADV
ejpam-3284	91	2	,	,	PUNCT
ejpam-3284	91	3	p1	p1	PROPN
ejpam-3284	91	4	is	be	AUX
ejpam-3284	91	5	satisfied	satisfied	ADJ
ejpam-3284	91	6	.	.	PUNCT
ejpam-3284	92	1	in	in	ADP
ejpam-3284	92	2	order	order	NOUN
ejpam-3284	92	3	to	to	PART
ejpam-3284	92	4	prove	prove	VERB
ejpam-3284	92	5	axiom	axiom	NOUN
ejpam-3284	92	6	p2	p2	NOUN
ejpam-3284	92	7	,	,	PUNCT
ejpam-3284	92	8	we	we	PRON
ejpam-3284	92	9	take	take	VERB
ejpam-3284	92	10	the	the	DET
ejpam-3284	92	11	expression	expression	NOUN
ejpam-3284	92	12	on	on	ADP
ejpam-3284	92	13	the	the	DET
ejpam-3284	92	14	left	left	ADJ
ejpam-3284	92	15	hand	hand	NOUN
ejpam-3284	92	16	side	side	NOUN
ejpam-3284	92	17	yields	yield	NOUN
ejpam-3284	92	18	‖αu‖2	‖αu‖2	PROPN
ejpam-3284	92	19	=	=	SYM
ejpam-3284	92	20	〈	〈	PROPN
ejpam-3284	92	21	αu	αu	NOUN
ejpam-3284	92	22	,	,	PUNCT
ejpam-3284	92	23	αu	αu	PROPN
ejpam-3284	92	24	〉	〉	NOUN
ejpam-3284	92	25	⇒	⇒	VERB
ejpam-3284	92	26	‖αu‖2	‖αu‖2	PROPN
ejpam-3284	92	27	=	=	SYM
ejpam-3284	92	28	|α|2〈u	|α|2〈u	PROPN
ejpam-3284	92	29	,	,	PUNCT
ejpam-3284	92	30	u	u	NOUN
ejpam-3284	93	1	〉	〉	NOUN
ejpam-3284	93	2	⇒	⇒	VERB
ejpam-3284	93	3	‖αu‖2	‖αu‖2	PROPN
ejpam-3284	93	4	=	=	SYM
ejpam-3284	93	5	|α|2‖u‖2	|α|2‖u‖2	PROPN
ejpam-3284	93	6	barnes	barnes	PROPN
ejpam-3284	93	7	et	et	PROPN
ejpam-3284	93	8	al	al	PROPN
ejpam-3284	93	9	.	.	PUNCT
ejpam-3284	93	10	/	/	SYM
ejpam-3284	93	11	eur	eur	PROPN
ejpam-3284	93	12	.	.	PUNCT
ejpam-3284	94	1	j.	j.	PROPN
ejpam-3284	94	2	pure	pure	PROPN
ejpam-3284	94	3	appl	appl	PROPN
ejpam-3284	94	4	.	.	PROPN
ejpam-3284	94	5	math	math	PROPN
ejpam-3284	94	6	,	,	PUNCT
ejpam-3284	94	7	11	11	NUM
ejpam-3284	94	8	(	(	PUNCT
ejpam-3284	94	9	3	3	NUM
ejpam-3284	94	10	)	)	PUNCT
ejpam-3284	94	11	(	(	PUNCT
ejpam-3284	94	12	2018	2018	NUM
ejpam-3284	94	13	)	)	PUNCT
ejpam-3284	94	14	,	,	PUNCT
ejpam-3284	94	15	740	740	NUM
ejpam-3284	94	16	-	-	SYM
ejpam-3284	94	17	750	750	NUM
ejpam-3284	94	18	744	744	NUM
ejpam-3284	94	19	⇒	⇒	NOUN
ejpam-3284	94	20	‖αu‖2	‖αu‖2	PROPN
ejpam-3284	94	21	=	=	PUNCT
ejpam-3284	94	22	(	(	PUNCT
ejpam-3284	94	23	|α|‖u‖	|α|‖u‖	X
ejpam-3284	94	24	)	)	PUNCT
ejpam-3284	94	25	2	2	NUM
ejpam-3284	94	26	⇒	⇒	NOUN
ejpam-3284	94	27	‖αu‖	‖αu‖	ADJ
ejpam-3284	94	28	=	=	X
ejpam-3284	94	29	|α|‖u‖.	|α|‖u‖.	PUNCT
ejpam-3284	94	30	also	also	ADV
ejpam-3284	94	31	,	,	PUNCT
ejpam-3284	94	32	the	the	DET
ejpam-3284	94	33	axiom	axiom	NOUN
ejpam-3284	94	34	p3	p3	PROPN
ejpam-3284	94	35	is	be	AUX
ejpam-3284	94	36	proved	prove	VERB
ejpam-3284	94	37	as	as	SCONJ
ejpam-3284	94	38	follows	follow	VERB
ejpam-3284	94	39	.	.	PUNCT
ejpam-3284	95	1	‖u+	‖u+	NOUN
ejpam-3284	95	2	v‖2	v‖2	NOUN
ejpam-3284	95	3	=	=	SYM
ejpam-3284	95	4	〈	〈	PROPN
ejpam-3284	95	5	u+	u+	NOUN
ejpam-3284	95	6	v	v	NOUN
ejpam-3284	95	7	,	,	PUNCT
ejpam-3284	95	8	u+	u+	NUM
ejpam-3284	95	9	v	v	ADP
ejpam-3284	95	10	〉	〉	PROPN
ejpam-3284	95	11	⇒	⇒	NOUN
ejpam-3284	95	12	‖u+	‖u+	NOUN
ejpam-3284	95	13	v‖2	v‖2	PROPN
ejpam-3284	95	14	=	=	SYM
ejpam-3284	95	15	〈	〈	PROPN
ejpam-3284	95	16	u	u	NOUN
ejpam-3284	95	17	,	,	PUNCT
ejpam-3284	95	18	u〉+	u〉+	PROPN
ejpam-3284	95	19	〈	〈	PROPN
ejpam-3284	95	20	u	u	PROPN
ejpam-3284	95	21	,	,	PUNCT
ejpam-3284	95	22	v〉+	v〉+	PROPN
ejpam-3284	95	23	〈	〈	PROPN
ejpam-3284	95	24	v	v	NOUN
ejpam-3284	95	25	,	,	PUNCT
ejpam-3284	95	26	u〉+	u〉+	PROPN
ejpam-3284	95	27	〈	〈	PROPN
ejpam-3284	95	28	v	v	NOUN
ejpam-3284	95	29	,	,	PUNCT
ejpam-3284	95	30	v	v	NOUN
ejpam-3284	95	31	〉	〉	PROPN
ejpam-3284	95	32	⇒	⇒	NOUN
ejpam-3284	95	33	‖u+	‖u+	NOUN
ejpam-3284	95	34	v‖2	v‖2	PROPN
ejpam-3284	95	35	=	=	SYM
ejpam-3284	95	36	〈	〈	PROPN
ejpam-3284	95	37	u	u	NOUN
ejpam-3284	95	38	,	,	PUNCT
ejpam-3284	95	39	u〉+	u〉+	PROPN
ejpam-3284	95	40	〈	〈	PROPN
ejpam-3284	95	41	u	u	PROPN
ejpam-3284	95	42	,	,	PUNCT
ejpam-3284	95	43	v〉+	v〉+	PROPN
ejpam-3284	95	44	〈	〈	PROPN
ejpam-3284	95	45	u	u	NOUN
ejpam-3284	95	46	,	,	PUNCT
ejpam-3284	95	47	v〉+	v〉+	PROPN
ejpam-3284	95	48	〈	〈	PROPN
ejpam-3284	95	49	v	v	NOUN
ejpam-3284	95	50	,	,	PUNCT
ejpam-3284	95	51	v	v	NOUN
ejpam-3284	95	52	〉	〉	PROPN
ejpam-3284	95	53	⇒	⇒	NOUN
ejpam-3284	95	54	‖u+	‖u+	NOUN
ejpam-3284	95	55	v‖2	v‖2	PROPN
ejpam-3284	95	56	=	=	SYM
ejpam-3284	95	57	〈	〈	PROPN
ejpam-3284	95	58	u	u	NOUN
ejpam-3284	95	59	,	,	PUNCT
ejpam-3284	95	60	u〉+	u〉+	PROPN
ejpam-3284	95	61	2|〈u	2|〈u	PROPN
ejpam-3284	95	62	,	,	PUNCT
ejpam-3284	95	63	v〉|+	v〉|+	PRON
ejpam-3284	95	64	〈	〈	PROPN
ejpam-3284	95	65	v	v	NOUN
ejpam-3284	95	66	,	,	PUNCT
ejpam-3284	95	67	v	v	NOUN
ejpam-3284	95	68	〉	〉	PROPN
ejpam-3284	95	69	⇒	⇒	NOUN
ejpam-3284	95	70	‖u+	‖u+	NOUN
ejpam-3284	95	71	v‖2	v‖2	PROPN
ejpam-3284	95	72	=	=	PUNCT
ejpam-3284	95	73	‖u‖2	‖u‖2	PROPN
ejpam-3284	95	74	+	+	PROPN
ejpam-3284	95	75	2|〈u	2|〈u	NUM
ejpam-3284	95	76	,	,	PUNCT
ejpam-3284	95	77	v〉|+	v〉|+	PRON
ejpam-3284	95	78	‖v‖2	‖v‖2	NOUN
ejpam-3284	95	79	.	.	PUNCT
ejpam-3284	96	1	applying	apply	VERB
ejpam-3284	96	2	the	the	DET
ejpam-3284	96	3	cauchy	cauchy	NOUN
ejpam-3284	96	4	-	-	PUNCT
ejpam-3284	96	5	schwarz	schwarz	PROPN
ejpam-3284	96	6	inequality	inequality	NOUN
ejpam-3284	96	7	to	to	ADP
ejpam-3284	96	8	the	the	DET
ejpam-3284	96	9	middle	middle	ADJ
ejpam-3284	96	10	term	term	NOUN
ejpam-3284	96	11	on	on	ADP
ejpam-3284	96	12	the	the	DET
ejpam-3284	96	13	right	right	ADJ
ejpam-3284	96	14	hand	hand	NOUN
ejpam-3284	96	15	side	side	NOUN
ejpam-3284	96	16	,	,	PUNCT
ejpam-3284	96	17	we	we	PRON
ejpam-3284	96	18	obtain	obtain	VERB
ejpam-3284	96	19	‖u+	‖u+	NOUN
ejpam-3284	96	20	v‖2	v‖2	NOUN
ejpam-3284	96	21	≤	≤	PUNCT
ejpam-3284	96	22	‖u‖2	‖u‖2	PROPN
ejpam-3284	97	1	+	+	CCONJ
ejpam-3284	97	2	2‖u‖‖v‖+	2‖u‖‖v‖+	PROPN
ejpam-3284	97	3	‖v‖2	‖v‖2	NOUN
ejpam-3284	97	4	⇒	⇒	VERB
ejpam-3284	97	5	‖u+	‖u+	NOUN
ejpam-3284	97	6	v‖2	v‖2	NOUN
ejpam-3284	97	7	=	=	SYM
ejpam-3284	97	8	(	(	PUNCT
ejpam-3284	97	9	‖u‖+	‖u‖+	PROPN
ejpam-3284	97	10	‖v‖	‖v‖	PROPN
ejpam-3284	97	11	)	)	PUNCT
ejpam-3284	97	12	2	2	NUM
ejpam-3284	97	13	⇒	⇒	NOUN
ejpam-3284	97	14	‖u+	‖u+	PROPN
ejpam-3284	97	15	v‖	v‖	NOUN
ejpam-3284	97	16	≤	≤	ADV
ejpam-3284	97	17	‖u‖+	‖u‖+	PROPN
ejpam-3284	97	18	‖v‖.	‖v‖.	NOUN
ejpam-3284	97	19	the	the	DET
ejpam-3284	97	20	axiom	axiom	NOUN
ejpam-3284	97	21	p4	p4	NOUN
ejpam-3284	97	22	has	have	AUX
ejpam-3284	97	23	been	be	AUX
ejpam-3284	97	24	discussed	discuss	VERB
ejpam-3284	97	25	by	by	ADP
ejpam-3284	97	26	the	the	DET
ejpam-3284	97	27	authors	author	NOUN
ejpam-3284	97	28	in	in	ADP
ejpam-3284	97	29	[	[	X
ejpam-3284	97	30	13	13	NUM
ejpam-3284	97	31	]	]	PUNCT
ejpam-3284	97	32	.	.	PUNCT
ejpam-3284	98	1	we	we	PRON
ejpam-3284	98	2	present	present	VERB
ejpam-3284	98	3	the	the	DET
ejpam-3284	98	4	proof	proof	NOUN
ejpam-3284	98	5	as	as	SCONJ
ejpam-3284	98	6	follows	follow	VERB
ejpam-3284	98	7	.	.	PUNCT
ejpam-3284	99	1	(	(	PUNCT
ejpam-3284	99	2	u+	u+	NUM
ejpam-3284	99	3	v)n	v)n	ADJ
ejpam-3284	99	4	≥	≥	NOUN
ejpam-3284	99	5	0	0	NUM
ejpam-3284	99	6	⇒	⇒	NOUN
ejpam-3284	99	7	〈	〈	PROPN
ejpam-3284	99	8	u	u	PROPN
ejpam-3284	99	9	,	,	PUNCT
ejpam-3284	99	10	u	u	NOUN
ejpam-3284	99	11	〉	〉	NOUN
ejpam-3284	99	12	n	n	PRON
ejpam-3284	99	13	2	2	NUM
ejpam-3284	99	14	+	+	NOUN
ejpam-3284	99	15	n	n	NOUN
ejpam-3284	99	16	c1〈u	c1〈u	ADJ
ejpam-3284	99	17	,	,	PUNCT
ejpam-3284	99	18	v〉〈u	v〉〈u	NOUN
ejpam-3284	99	19	,	,	PUNCT
ejpam-3284	99	20	u	u	NOUN
ejpam-3284	99	21	〉	〉	NOUN
ejpam-3284	99	22	n−2	n−2	PROPN
ejpam-3284	99	23	2	2	NUM
ejpam-3284	99	24	+	+	NOUN
ejpam-3284	99	25	n	n	PRON
ejpam-3284	99	26	c2〈u	c2〈u	NOUN
ejpam-3284	99	27	,	,	PUNCT
ejpam-3284	99	28	u	u	NOUN
ejpam-3284	99	29	〉	〉	NOUN
ejpam-3284	99	30	n−2	n−2	PROPN
ejpam-3284	99	31	2	2	NUM
ejpam-3284	99	32	〈	〈	PROPN
ejpam-3284	99	33	v	v	NOUN
ejpam-3284	99	34	,	,	PUNCT
ejpam-3284	99	35	v〉+n	v〉+n	NOUN
ejpam-3284	99	36	c3〈u	c3〈u	NOUN
ejpam-3284	99	37	,	,	PUNCT
ejpam-3284	99	38	u	u	NOUN
ejpam-3284	99	39	〉	〉	PROPN
ejpam-3284	99	40	n−4	n−4	PROPN
ejpam-3284	99	41	2	2	NUM
ejpam-3284	99	42	〈	〈	PROPN
ejpam-3284	99	43	v	v	NOUN
ejpam-3284	99	44	,	,	PUNCT
ejpam-3284	99	45	v〉〈u	v〉〈u	NOUN
ejpam-3284	99	46	,	,	PUNCT
ejpam-3284	99	47	v	v	NOUN
ejpam-3284	99	48	〉	〉	NOUN
ejpam-3284	99	49	+	+	CCONJ
ejpam-3284	99	50	nc4〈u	nc4〈u	PROPN
ejpam-3284	99	51	,	,	PUNCT
ejpam-3284	99	52	u	u	NOUN
ejpam-3284	99	53	〉	〉	NOUN
ejpam-3284	99	54	n−4	n−4	PROPN
ejpam-3284	99	55	2	2	NUM
ejpam-3284	99	56	〈	〈	PROPN
ejpam-3284	99	57	v	v	NOUN
ejpam-3284	99	58	,	,	PUNCT
ejpam-3284	99	59	v〉2	v〉2	VERB
ejpam-3284	99	60	+	+	NOUN
ejpam-3284	99	61	n	n	NOUN
ejpam-3284	99	62	c5〈u	c5〈u	NOUN
ejpam-3284	99	63	,	,	PUNCT
ejpam-3284	99	64	u	u	NOUN
ejpam-3284	99	65	〉	〉	NUM
ejpam-3284	99	66	n−6	n−6	PROPN
ejpam-3284	99	67	2	2	NUM
ejpam-3284	99	68	〈	〈	PROPN
ejpam-3284	99	69	v	v	NOUN
ejpam-3284	99	70	,	,	PUNCT
ejpam-3284	99	71	v〉2〈u	v〉2〈u	PROPN
ejpam-3284	99	72	,	,	PUNCT
ejpam-3284	99	73	v〉+n	v〉+n	NOUN
ejpam-3284	99	74	c6〈u	c6〈u	NOUN
ejpam-3284	99	75	,	,	PUNCT
ejpam-3284	99	76	u	u	NOUN
ejpam-3284	99	77	〉	〉	PROPN
ejpam-3284	99	78	n−6	n−6	PROPN
ejpam-3284	99	79	2	2	NUM
ejpam-3284	99	80	〈	〈	PROPN
ejpam-3284	99	81	v	v	NOUN
ejpam-3284	99	82	,	,	PUNCT
ejpam-3284	99	83	v〉3	v〉3	PROPN
ejpam-3284	99	84	+	+	CCONJ
ejpam-3284	99	85	nc7〈u	nc7〈u	PROPN
ejpam-3284	99	86	,	,	PUNCT
ejpam-3284	99	87	u	u	NOUN
ejpam-3284	99	88	〉	〉	PROPN
ejpam-3284	99	89	n−8	n−8	PROPN
ejpam-3284	99	90	2	2	NUM
ejpam-3284	99	91	〈	〈	PROPN
ejpam-3284	99	92	v	v	NOUN
ejpam-3284	99	93	,	,	PUNCT
ejpam-3284	99	94	v〉3〈u	v〉3〈u	PROPN
ejpam-3284	99	95	,	,	PUNCT
ejpam-3284	99	96	v〉+	v〉+	PROPN
ejpam-3284	99	97	.	.	PUNCT
ejpam-3284	99	98	.	.	PUNCT
ejpam-3284	100	1	.+n	.+n	PROPN
ejpam-3284	100	2	cn	cn	PROPN
ejpam-3284	100	3	2	2	NUM
ejpam-3284	100	4	〈	〈	PROPN
ejpam-3284	100	5	u	u	NOUN
ejpam-3284	100	6	,	,	PUNCT
ejpam-3284	100	7	u	u	NOUN
ejpam-3284	100	8	〉	〉	NOUN
ejpam-3284	100	9	n	n	PRON
ejpam-3284	100	10	4	4	NUM
ejpam-3284	100	11	〈	〈	PROPN
ejpam-3284	100	12	v	v	NOUN
ejpam-3284	100	13	,	,	PUNCT
ejpam-3284	100	14	v	v	NOUN
ejpam-3284	100	15	〉	〉	NOUN
ejpam-3284	100	16	n	n	PRON
ejpam-3284	100	17	4	4	NUM
ejpam-3284	100	18	+	+	CCONJ
ejpam-3284	100	19	.	.	PUNCT
ejpam-3284	100	20	.	.	PUNCT
ejpam-3284	101	1	.+	.+	NOUN
ejpam-3284	102	1	〈	〈	PROPN
ejpam-3284	102	2	u	u	NOUN
ejpam-3284	102	3	,	,	PUNCT
ejpam-3284	102	4	u	u	NOUN
ejpam-3284	102	5	〉	〉	NOUN
ejpam-3284	102	6	n	n	NUM
ejpam-3284	102	7	2	2	NUM
ejpam-3284	102	8	≥	≥	NOUN
ejpam-3284	102	9	0	0	NUM
ejpam-3284	102	10	⇒	⇒	NOUN
ejpam-3284	102	11	−ncn	−ncn	NOUN
ejpam-3284	102	12	2	2	NUM
ejpam-3284	102	13	〈	〈	PROPN
ejpam-3284	102	14	u	u	NOUN
ejpam-3284	102	15	,	,	PUNCT
ejpam-3284	102	16	u	u	NOUN
ejpam-3284	102	17	〉	〉	NOUN
ejpam-3284	102	18	n	n	PRON
ejpam-3284	102	19	4	4	NUM
ejpam-3284	102	20	〈	〈	PROPN
ejpam-3284	102	21	v	v	NOUN
ejpam-3284	102	22	,	,	PUNCT
ejpam-3284	102	23	v	v	NOUN
ejpam-3284	102	24	〉	〉	NOUN
ejpam-3284	102	25	n	n	PRON
ejpam-3284	102	26	4	4	NUM
ejpam-3284	102	27	≤	≤	NOUN
ejpam-3284	102	28	{	{	PUNCT
ejpam-3284	102	29	〈	〈	NOUN
ejpam-3284	102	30	u	u	NOUN
ejpam-3284	102	31	,	,	PUNCT
ejpam-3284	102	32	u	u	NOUN
ejpam-3284	102	33	〉	〉	NOUN
ejpam-3284	102	34	n	n	PRON
ejpam-3284	102	35	2	2	NUM
ejpam-3284	102	36	+	+	NOUN
ejpam-3284	102	37	n	n	NOUN
ejpam-3284	102	38	c1〈u	c1〈u	ADJ
ejpam-3284	102	39	,	,	PUNCT
ejpam-3284	102	40	v〉〈u	v〉〈u	NOUN
ejpam-3284	102	41	,	,	PUNCT
ejpam-3284	102	42	u	u	NOUN
ejpam-3284	102	43	〉	〉	NOUN
ejpam-3284	102	44	n−2	n−2	PROPN
ejpam-3284	102	45	2	2	NUM
ejpam-3284	102	46	+	+	NOUN
ejpam-3284	102	47	n	n	PRON
ejpam-3284	102	48	c2〈u	c2〈u	NOUN
ejpam-3284	102	49	,	,	PUNCT
ejpam-3284	102	50	u	u	NOUN
ejpam-3284	102	51	〉	〉	NOUN
ejpam-3284	102	52	n−2	n−2	PROPN
ejpam-3284	102	53	2	2	NUM
ejpam-3284	102	54	〈	〈	PROPN
ejpam-3284	102	55	v	v	NOUN
ejpam-3284	102	56	,	,	PUNCT
ejpam-3284	102	57	v	v	NOUN
ejpam-3284	102	58	〉	〉	NOUN
ejpam-3284	102	59	+	+	CCONJ
ejpam-3284	102	60	nc3〈u	nc3〈u	NOUN
ejpam-3284	102	61	,	,	PUNCT
ejpam-3284	102	62	u	u	NOUN
ejpam-3284	102	63	〉	〉	NOUN
ejpam-3284	102	64	n−4	n−4	PROPN
ejpam-3284	102	65	2	2	NUM
ejpam-3284	102	66	〈	〈	PROPN
ejpam-3284	102	67	v	v	NOUN
ejpam-3284	102	68	,	,	PUNCT
ejpam-3284	102	69	v〉〈u	v〉〈u	NOUN
ejpam-3284	102	70	,	,	PUNCT
ejpam-3284	102	71	v〉+n	v〉+n	NOUN
ejpam-3284	102	72	c4〈u	c4〈u	VERB
ejpam-3284	102	73	,	,	PUNCT
ejpam-3284	102	74	u	u	NOUN
ejpam-3284	102	75	〉	〉	PROPN
ejpam-3284	102	76	n−4	n−4	PROPN
ejpam-3284	102	77	2	2	NUM
ejpam-3284	102	78	〈	〈	PROPN
ejpam-3284	102	79	v	v	NOUN
ejpam-3284	102	80	,	,	PUNCT
ejpam-3284	102	81	v〉2	v〉2	VERB
ejpam-3284	102	82	+	+	NOUN
ejpam-3284	102	83	n	n	NOUN
ejpam-3284	102	84	c5〈u	c5〈u	NOUN
ejpam-3284	102	85	,	,	PUNCT
ejpam-3284	102	86	u	u	NOUN
ejpam-3284	102	87	〉	〉	NUM
ejpam-3284	102	88	n−6	n−6	PROPN
ejpam-3284	102	89	2	2	NUM
ejpam-3284	102	90	〈	〈	PROPN
ejpam-3284	102	91	v	v	NOUN
ejpam-3284	102	92	,	,	PUNCT
ejpam-3284	102	93	v〉2〈u	v〉2〈u	NOUN
ejpam-3284	102	94	,	,	PUNCT
ejpam-3284	102	95	v	v	NOUN
ejpam-3284	102	96	〉	〉	NOUN
ejpam-3284	102	97	+	+	CCONJ
ejpam-3284	102	98	nc6〈u	nc6〈u	PROPN
ejpam-3284	102	99	,	,	PUNCT
ejpam-3284	102	100	u	u	NOUN
ejpam-3284	102	101	〉	〉	NOUN
ejpam-3284	102	102	n−6	n−6	PROPN
ejpam-3284	102	103	2	2	NUM
ejpam-3284	102	104	〈	〈	PROPN
ejpam-3284	102	105	v	v	NOUN
ejpam-3284	102	106	,	,	PUNCT
ejpam-3284	102	107	v〉3	v〉3	PROPN
ejpam-3284	102	108	+	+	PROPN
ejpam-3284	102	109	n	n	PRON
ejpam-3284	102	110	c7〈u	c7〈u	NOUN
ejpam-3284	102	111	,	,	PUNCT
ejpam-3284	102	112	u	u	NOUN
ejpam-3284	102	113	〉	〉	PROPN
ejpam-3284	102	114	n−8	n−8	PROPN
ejpam-3284	102	115	2	2	NUM
ejpam-3284	102	116	〈	〈	PROPN
ejpam-3284	102	117	v	v	NOUN
ejpam-3284	102	118	,	,	PUNCT
ejpam-3284	102	119	v〉3〈u	v〉3〈u	PROPN
ejpam-3284	102	120	,	,	PUNCT
ejpam-3284	102	121	v〉+	v〉+	PROPN
ejpam-3284	102	122	.	.	PUNCT
ejpam-3284	102	123	.	.	PUNCT
ejpam-3284	103	1	.+	.+	NOUN
ejpam-3284	104	1	〈	〈	PROPN
ejpam-3284	104	2	u	u	NOUN
ejpam-3284	104	3	,	,	PUNCT
ejpam-3284	104	4	u	u	NOUN
ejpam-3284	104	5	〉	〉	NOUN
ejpam-3284	104	6	n	n	PRON
ejpam-3284	104	7	2	2	NUM
ejpam-3284	104	8	}	}	PUNCT
ejpam-3284	104	9	⇒	⇒	NOUN
ejpam-3284	104	10	−ncn	−ncn	NOUN
ejpam-3284	104	11	2	2	NUM
ejpam-3284	104	12	(	(	PUNCT
ejpam-3284	104	13	u	u	NOUN
ejpam-3284	104	14	,	,	PUNCT
ejpam-3284	104	15	u	u	NOUN
ejpam-3284	104	16	)	)	PUNCT
ejpam-3284	104	17	n	n	ADV
ejpam-3284	104	18	4	4	NUM
ejpam-3284	104	19	(	(	PUNCT
ejpam-3284	104	20	v	v	NOUN
ejpam-3284	104	21	,	,	PUNCT
ejpam-3284	104	22	v	v	NOUN
ejpam-3284	104	23	)	)	PUNCT
ejpam-3284	104	24	n	n	PRON
ejpam-3284	104	25	4	4	NUM
ejpam-3284	104	26	≤	≤	NOUN
ejpam-3284	104	27	{	{	PUNCT
ejpam-3284	104	28	(	(	PUNCT
ejpam-3284	104	29	u	u	NOUN
ejpam-3284	104	30	,	,	PUNCT
ejpam-3284	104	31	u	u	NOUN
ejpam-3284	104	32	)	)	PUNCT
ejpam-3284	104	33	n	n	ADV
ejpam-3284	104	34	2	2	NUM
ejpam-3284	104	35	+	+	NOUN
ejpam-3284	104	36	n	n	PRON
ejpam-3284	104	37	c1(u	c1(u	NOUN
ejpam-3284	104	38	,	,	PUNCT
ejpam-3284	104	39	v)(u	v)(u	NUM
ejpam-3284	104	40	,	,	PUNCT
ejpam-3284	104	41	u	u	NOUN
ejpam-3284	104	42	)	)	PUNCT
ejpam-3284	104	43	n−2	n−2	PROPN
ejpam-3284	104	44	2	2	NUM
ejpam-3284	104	45	+	+	NOUN
ejpam-3284	104	46	n	n	PROPN
ejpam-3284	104	47	c2(u	c2(u	PROPN
ejpam-3284	104	48	,	,	PUNCT
ejpam-3284	104	49	u	u	NOUN
ejpam-3284	104	50	)	)	PUNCT
ejpam-3284	104	51	n−2	n−2	PROPN
ejpam-3284	104	52	2	2	NUM
ejpam-3284	104	53	(	(	PUNCT
ejpam-3284	104	54	v	v	NOUN
ejpam-3284	104	55	,	,	PUNCT
ejpam-3284	104	56	v	v	NOUN
ejpam-3284	104	57	)	)	PUNCT
ejpam-3284	104	58	+	+	CCONJ
ejpam-3284	105	1	nc3(u	nc3(u	PROPN
ejpam-3284	105	2	,	,	PUNCT
ejpam-3284	105	3	u	u	NOUN
ejpam-3284	105	4	)	)	PUNCT
ejpam-3284	105	5	n−4	n−4	PROPN
ejpam-3284	105	6	2	2	NUM
ejpam-3284	105	7	(	(	PUNCT
ejpam-3284	105	8	v	v	NOUN
ejpam-3284	105	9	,	,	PUNCT
ejpam-3284	105	10	v)(u	v)(u	ADJ
ejpam-3284	105	11	,	,	PUNCT
ejpam-3284	105	12	v	v	NOUN
ejpam-3284	105	13	)	)	PUNCT
ejpam-3284	105	14	+	+	PROPN
ejpam-3284	105	15	n	n	PRON
ejpam-3284	105	16	c4(u	c4(u	PROPN
ejpam-3284	105	17	,	,	PUNCT
ejpam-3284	105	18	u	u	NOUN
ejpam-3284	105	19	)	)	PUNCT
ejpam-3284	105	20	n−4	n−4	PROPN
ejpam-3284	105	21	2	2	NUM
ejpam-3284	105	22	(	(	PUNCT
ejpam-3284	105	23	v	v	NOUN
ejpam-3284	105	24	,	,	PUNCT
ejpam-3284	105	25	v)2	v)2	ADJ
ejpam-3284	105	26	+	+	NOUN
ejpam-3284	105	27	n	n	PROPN
ejpam-3284	105	28	c5(u	c5(u	PROPN
ejpam-3284	105	29	,	,	PUNCT
ejpam-3284	105	30	u	u	NOUN
ejpam-3284	105	31	)	)	PUNCT
ejpam-3284	105	32	n−6	n−6	PROPN
ejpam-3284	105	33	2	2	NUM
ejpam-3284	105	34	(	(	PUNCT
ejpam-3284	105	35	v	v	NOUN
ejpam-3284	105	36	,	,	PUNCT
ejpam-3284	105	37	v)2(u	v)2(u	NUM
ejpam-3284	105	38	,	,	PUNCT
ejpam-3284	105	39	v	v	NOUN
ejpam-3284	105	40	)	)	PUNCT
ejpam-3284	105	41	+	+	CCONJ
ejpam-3284	105	42	nc6(u	nc6(u	PROPN
ejpam-3284	105	43	,	,	PUNCT
ejpam-3284	105	44	u	u	NOUN
ejpam-3284	105	45	)	)	PUNCT
ejpam-3284	105	46	n−6	n−6	PROPN
ejpam-3284	105	47	2	2	NUM
ejpam-3284	105	48	(	(	PUNCT
ejpam-3284	105	49	v	v	NOUN
ejpam-3284	105	50	,	,	PUNCT
ejpam-3284	105	51	v)3	v)3	PROPN
ejpam-3284	105	52	+	+	PROPN
ejpam-3284	105	53	n	n	PROPN
ejpam-3284	105	54	c7(u	c7(u	NOUN
ejpam-3284	105	55	,	,	PUNCT
ejpam-3284	105	56	u	u	NOUN
ejpam-3284	105	57	)	)	PUNCT
ejpam-3284	105	58	n−8	n−8	PROPN
ejpam-3284	105	59	2	2	NUM
ejpam-3284	105	60	(	(	PUNCT
ejpam-3284	105	61	v	v	NOUN
ejpam-3284	105	62	,	,	PUNCT
ejpam-3284	105	63	v)3(u	v)3(u	NOUN
ejpam-3284	105	64	,	,	PUNCT
ejpam-3284	105	65	v	v	NOUN
ejpam-3284	105	66	)	)	PUNCT
ejpam-3284	105	67	+	+	CCONJ
ejpam-3284	105	68	.	.	PUNCT
ejpam-3284	105	69	.	.	PUNCT
ejpam-3284	106	1	.+n	.+n	PROPN
ejpam-3284	106	2	cn	cn	NOUN
ejpam-3284	106	3	2	2	NUM
ejpam-3284	106	4	(	(	PUNCT
ejpam-3284	106	5	u	u	NOUN
ejpam-3284	106	6	,	,	PUNCT
ejpam-3284	106	7	u	u	NOUN
ejpam-3284	106	8	)	)	PUNCT
ejpam-3284	106	9	n	n	ADV
ejpam-3284	106	10	4	4	NUM
ejpam-3284	106	11	(	(	PUNCT
ejpam-3284	106	12	v	v	NOUN
ejpam-3284	106	13	,	,	PUNCT
ejpam-3284	106	14	v	v	NOUN
ejpam-3284	106	15	)	)	PUNCT
ejpam-3284	106	16	n	n	ADV
ejpam-3284	106	17	4	4	NUM
ejpam-3284	106	18	+	+	CCONJ
ejpam-3284	106	19	.	.	PUNCT
ejpam-3284	106	20	.	.	PUNCT
ejpam-3284	107	1	.+	.+	NOUN
ejpam-3284	107	2	(	(	PUNCT
ejpam-3284	107	3	u	u	NOUN
ejpam-3284	107	4	,	,	PUNCT
ejpam-3284	107	5	u	u	NOUN
ejpam-3284	107	6	)	)	PUNCT
ejpam-3284	107	7	n	n	DET
ejpam-3284	107	8	2	2	NUM
ejpam-3284	107	9	}	}	PUNCT
ejpam-3284	107	10	⇒	⇒	VERB
ejpam-3284	107	11	−ncn	−ncn	NOUN
ejpam-3284	107	12	2	2	NUM
ejpam-3284	107	13	(	(	PUNCT
ejpam-3284	107	14	u	u	NOUN
ejpam-3284	107	15	,	,	PUNCT
ejpam-3284	107	16	u	u	NOUN
ejpam-3284	107	17	)	)	PUNCT
ejpam-3284	107	18	n	n	ADV
ejpam-3284	107	19	4	4	NUM
ejpam-3284	107	20	(	(	PUNCT
ejpam-3284	107	21	v	v	NOUN
ejpam-3284	107	22	,	,	PUNCT
ejpam-3284	107	23	v	v	NOUN
ejpam-3284	107	24	)	)	PUNCT
ejpam-3284	107	25	n	n	ADV
ejpam-3284	107	26	4	4	NUM
ejpam-3284	107	27	1	1	NUM
ejpam-3284	107	28	ncn	ncn	NOUN
ejpam-3284	107	29	2	2	NUM
ejpam-3284	107	30	(	(	PUNCT
ejpam-3284	107	31	u	u	NOUN
ejpam-3284	107	32	,	,	PUNCT
ejpam-3284	107	33	u	u	NOUN
ejpam-3284	107	34	)	)	PUNCT
ejpam-3284	107	35	n	n	ADV
ejpam-3284	107	36	4	4	NUM
ejpam-3284	107	37	(	(	PUNCT
ejpam-3284	107	38	v	v	NOUN
ejpam-3284	107	39	,	,	PUNCT
ejpam-3284	107	40	v	v	NOUN
ejpam-3284	107	41	)	)	PUNCT
ejpam-3284	107	42	n	n	PRON
ejpam-3284	107	43	4	4	NUM
ejpam-3284	107	44	≤	≤	NUM
ejpam-3284	107	45	(	(	PUNCT
ejpam-3284	107	46	u+	u+	NUM
ejpam-3284	107	47	v)n	v)n	NOUN
ejpam-3284	107	48	⇒	⇒	NOUN
ejpam-3284	107	49	‖−	‖−	PROPN
ejpam-3284	107	50	(	(	PUNCT
ejpam-3284	107	51	u	u	NOUN
ejpam-3284	107	52	,	,	PUNCT
ejpam-3284	107	53	u	u	NOUN
ejpam-3284	107	54	)	)	PUNCT
ejpam-3284	107	55	n	n	ADV
ejpam-3284	107	56	2	2	NUM
ejpam-3284	107	57	(	(	PUNCT
ejpam-3284	107	58	v	v	NOUN
ejpam-3284	107	59	,	,	PUNCT
ejpam-3284	107	60	v	v	NOUN
ejpam-3284	107	61	)	)	PUNCT
ejpam-3284	107	62	n	n	PRON
ejpam-3284	107	63	2	2	NUM
ejpam-3284	107	64	‖	‖	PROPN
ejpam-3284	107	65	=	=	SYM
ejpam-3284	107	66	‖(u+	‖(u+	PUNCT
ejpam-3284	107	67	v)n‖	v)n‖	NUM
ejpam-3284	107	68	⇒	⇒	NOUN
ejpam-3284	107	69	{	{	PUNCT
ejpam-3284	107	70	‖u‖‖v‖}n	‖u‖‖v‖}n	PUNCT
ejpam-3284	107	71	=	=	SYM
ejpam-3284	107	72	‖(u+	‖(u+	NUM
ejpam-3284	107	73	v)‖n	v)‖n	NOUN
ejpam-3284	107	74	⇒	⇒	VERB
ejpam-3284	107	75	‖u‖‖v‖	‖u‖‖v‖	PROPN
ejpam-3284	107	76	≤	≤	PROPN
ejpam-3284	107	77	‖u‖+	‖u‖+	PROPN
ejpam-3284	107	78	‖v‖	‖v‖	PROPN
ejpam-3284	107	79	,	,	PUNCT
ejpam-3284	107	80	∀	∀	X
ejpam-3284	107	81	u	u	NOUN
ejpam-3284	107	82	,	,	PUNCT
ejpam-3284	107	83	v	v	NOUN
ejpam-3284	107	84	∈	∈	PROPN
ejpam-3284	108	1	[	[	X
ejpam-3284	108	2	0	0	NUM
ejpam-3284	108	3	,	,	PUNCT
ejpam-3284	108	4	2	2	NUM
ejpam-3284	108	5	]	]	PUNCT
ejpam-3284	108	6	.	.	PUNCT
ejpam-3284	109	1	barnes	barnes	PROPN
ejpam-3284	109	2	et	et	PROPN
ejpam-3284	109	3	al	al	PROPN
ejpam-3284	109	4	.	.	PUNCT
ejpam-3284	109	5	/	/	SYM
ejpam-3284	109	6	eur	eur	PROPN
ejpam-3284	109	7	.	.	PUNCT
ejpam-3284	110	1	j.	j.	PROPN
ejpam-3284	110	2	pure	pure	PROPN
ejpam-3284	110	3	appl	appl	PROPN
ejpam-3284	110	4	.	.	PROPN
ejpam-3284	110	5	math	math	PROPN
ejpam-3284	110	6	,	,	PUNCT
ejpam-3284	110	7	11	11	NUM
ejpam-3284	110	8	(	(	PUNCT
ejpam-3284	110	9	3	3	NUM
ejpam-3284	110	10	)	)	PUNCT
ejpam-3284	110	11	(	(	PUNCT
ejpam-3284	110	12	2018	2018	NUM
ejpam-3284	110	13	)	)	PUNCT
ejpam-3284	110	14	,	,	PUNCT
ejpam-3284	110	15	740	740	NUM
ejpam-3284	110	16	-	-	SYM
ejpam-3284	110	17	750	750	NUM
ejpam-3284	110	18	745	745	NUM
ejpam-3284	110	19	the	the	DET
ejpam-3284	110	20	proof	proof	NOUN
ejpam-3284	110	21	of	of	ADP
ejpam-3284	110	22	p4(b	p4(b	NOUN
ejpam-3284	110	23	)	)	PUNCT
ejpam-3284	110	24	,	,	PUNCT
ejpam-3284	110	25	see	see	VERB
ejpam-3284	110	26	[	[	X
ejpam-3284	110	27	13	13	NUM
ejpam-3284	110	28	]	]	PUNCT
ejpam-3284	110	29	.	.	PUNCT
ejpam-3284	111	1	we	we	PRON
ejpam-3284	111	2	show	show	VERB
ejpam-3284	111	3	that	that	SCONJ
ejpam-3284	111	4	the	the	DET
ejpam-3284	111	5	product	product	NOUN
ejpam-3284	111	6	-	-	PUNCT
ejpam-3284	111	7	normed	norme	VERB
ejpam-3284	111	8	space	space	NOUN
ejpam-3284	111	9	is	be	AUX
ejpam-3284	111	10	valid	valid	ADJ
ejpam-3284	111	11	for	for	ADP
ejpam-3284	111	12	1−	1−	NUM
ejpam-3284	111	13	norm	norm	NOUN
ejpam-3284	111	14	,	,	PUNCT
ejpam-3284	111	15	2−norm	2−norm	NUM
ejpam-3284	111	16	,	,	PUNCT
ejpam-3284	111	17	p−	p−	NOUN
ejpam-3284	111	18	norms	norm	NOUN
ejpam-3284	111	19	and	and	CCONJ
ejpam-3284	111	20	∞−norm	∞−norm	NOUN
ejpam-3284	111	21	.	.	PUNCT
ejpam-3284	112	1	we	we	PRON
ejpam-3284	112	2	give	give	VERB
ejpam-3284	112	3	the	the	DET
ejpam-3284	112	4	proof	proof	NOUN
ejpam-3284	112	5	for	for	ADP
ejpam-3284	112	6	p−norm	p−norm	NOUN
ejpam-3284	112	7	which	which	PRON
ejpam-3284	112	8	can	can	AUX
ejpam-3284	112	9	be	be	AUX
ejpam-3284	112	10	used	use	VERB
ejpam-3284	112	11	to	to	PART
ejpam-3284	112	12	generate	generate	VERB
ejpam-3284	112	13	the	the	DET
ejpam-3284	112	14	other	other	ADJ
ejpam-3284	112	15	norms	norm	NOUN
ejpam-3284	112	16	.	.	PUNCT
ejpam-3284	113	1	theorem	theorem	NOUN
ejpam-3284	113	2	2	2	NUM
ejpam-3284	113	3	.	.	PUNCT
ejpam-3284	113	4	suppose	suppose	VERB
ejpam-3284	113	5	that	that	SCONJ
ejpam-3284	113	6	f(x	f(x	PROPN
ejpam-3284	113	7	)	)	PUNCT
ejpam-3284	113	8	and	and	CCONJ
ejpam-3284	113	9	g(x	g(x	NOUN
ejpam-3284	113	10	)	)	PUNCT
ejpam-3284	113	11	are	be	AUX
ejpam-3284	113	12	measurable	measurable	ADJ
ejpam-3284	113	13	functions	function	NOUN
ejpam-3284	113	14	over	over	ADP
ejpam-3284	113	15	the	the	DET
ejpam-3284	113	16	domain	domain	NOUN
ejpam-3284	114	1	[	[	X
ejpam-3284	114	2	0	0	NUM
ejpam-3284	114	3	,	,	PUNCT
ejpam-3284	114	4	2	2	NUM
ejpam-3284	114	5	]	]	PUNCT
ejpam-3284	114	6	,	,	PUNCT
ejpam-3284	114	7	then	then	ADV
ejpam-3284	114	8	‖f‖p‖g‖p	‖f‖p‖g‖p	PRON
ejpam-3284	114	9	≤	≤	NUM
ejpam-3284	114	10	‖f‖p	‖f‖p	NOUN
ejpam-3284	114	11	+	+	CCONJ
ejpam-3284	114	12	‖g‖p	‖g‖p	NOUN
ejpam-3284	114	13	,	,	PUNCT
ejpam-3284	114	14	where	where	SCONJ
ejpam-3284	114	15	1	1	NUM
ejpam-3284	114	16	p	p	NOUN
ejpam-3284	114	17	+	+	NOUN
ejpam-3284	114	18	1	1	NUM
ejpam-3284	114	19	q	q	NOUN
ejpam-3284	114	20	=	=	NOUN
ejpam-3284	114	21	1	1	X
ejpam-3284	114	22	.	.	X
ejpam-3284	115	1	proof	proof	NOUN
ejpam-3284	115	2	:	:	PUNCT
ejpam-3284	115	3	we	we	PRON
ejpam-3284	115	4	see	see	VERB
ejpam-3284	115	5	that	that	SCONJ
ejpam-3284	115	6	(	(	PUNCT
ejpam-3284	115	7	‖f‖‖g‖)p	‖f‖‖g‖)p	NOUN
ejpam-3284	115	8	=	=	SYM
ejpam-3284	115	9	(	(	PUNCT
ejpam-3284	115	10	‖f‖‖g‖)p−1(‖f‖‖g‖	‖f‖‖g‖)p−1(‖f‖‖g‖	NOUN
ejpam-3284	115	11	)	)	PUNCT
ejpam-3284	115	12	applying	apply	VERB
ejpam-3284	115	13	the	the	DET
ejpam-3284	115	14	first	first	ADJ
ejpam-3284	115	15	product	product	NOUN
ejpam-3284	115	16	inequality	inequality	NOUN
ejpam-3284	115	17	to	to	ADP
ejpam-3284	115	18	the	the	DET
ejpam-3284	115	19	right	right	ADJ
ejpam-3284	115	20	hand	hand	NOUN
ejpam-3284	115	21	side	side	NOUN
ejpam-3284	115	22	of	of	ADP
ejpam-3284	115	23	the	the	DET
ejpam-3284	115	24	above	above	ADJ
ejpam-3284	115	25	equation	equation	NOUN
ejpam-3284	115	26	yields	yield	NOUN
ejpam-3284	115	27	(	(	PUNCT
ejpam-3284	115	28	‖f‖‖g‖)p	‖f‖‖g‖)p	PROPN
ejpam-3284	115	29	≤	≤	NOUN
ejpam-3284	115	30	(	(	PUNCT
ejpam-3284	115	31	‖f‖+	‖f‖+	X
ejpam-3284	115	32	‖g‖)p−1(‖f‖+	‖g‖)p−1(‖f‖+	NOUN
ejpam-3284	115	33	‖g‖	‖g‖	NUM
ejpam-3284	115	34	)	)	PUNCT
ejpam-3284	115	35	(	(	PUNCT
ejpam-3284	115	36	‖f‖‖g‖)p	‖f‖‖g‖)p	NOUN
ejpam-3284	115	37	=	=	SYM
ejpam-3284	115	38	(	(	PUNCT
ejpam-3284	115	39	‖f‖+	‖f‖+	PRON
ejpam-3284	115	40	‖g‖)p−1‖f‖+	‖g‖)p−1‖f‖+	NOUN
ejpam-3284	115	41	(	(	PUNCT
ejpam-3284	115	42	‖f‖+	‖f‖+	NOUN
ejpam-3284	115	43	‖g‖)p−1‖g‖	‖g‖)p−1‖g‖	NOUN
ejpam-3284	115	44	taking	take	VERB
ejpam-3284	115	45	the	the	DET
ejpam-3284	115	46	lebesgue	lebesgue	NOUN
ejpam-3284	115	47	integral	integral	ADJ
ejpam-3284	115	48	of	of	ADP
ejpam-3284	115	49	both	both	DET
ejpam-3284	115	50	sides	side	NOUN
ejpam-3284	115	51	of	of	ADP
ejpam-3284	115	52	the	the	DET
ejpam-3284	115	53	above	above	ADJ
ejpam-3284	115	54	equation	equation	NOUN
ejpam-3284	115	55	with	with	ADP
ejpam-3284	115	56	respect	respect	NOUN
ejpam-3284	115	57	µ	µ	NUM
ejpam-3284	115	58	,	,	PUNCT
ejpam-3284	115	59	we	we	PRON
ejpam-3284	115	60	obtain∫	obtain∫	VERB
ejpam-3284	115	61	(	(	PUNCT
ejpam-3284	115	62	‖f‖‖g‖)pdµ	‖f‖‖g‖)pdµ	NOUN
ejpam-3284	115	63	=	=	SYM
ejpam-3284	115	64	∫	∫	NOUN
ejpam-3284	115	65	{	{	PUNCT
ejpam-3284	115	66	(	(	PUNCT
ejpam-3284	115	67	‖f‖+	‖f‖+	PRON
ejpam-3284	115	68	‖g‖)p−1‖f‖+	‖g‖)p−1‖f‖+	NOUN
ejpam-3284	115	69	(	(	PUNCT
ejpam-3284	115	70	‖f‖+	‖f‖+	NOUN
ejpam-3284	115	71	‖g‖)p−1‖g‖	‖g‖)p−1‖g‖	NOUN
ejpam-3284	115	72	}	}	PUNCT
ejpam-3284	115	73	dµ	dµ	ADP
ejpam-3284	115	74	⇒	⇒	PROPN
ejpam-3284	115	75	∫	∫	PROPN
ejpam-3284	115	76	(	(	PUNCT
ejpam-3284	115	77	‖f‖‖g‖)pdµ	‖f‖‖g‖)pdµ	PROPN
ejpam-3284	115	78	=	=	SYM
ejpam-3284	115	79	∫	∫	PROPN
ejpam-3284	115	80	(	(	PUNCT
ejpam-3284	115	81	‖f‖+	‖f‖+	PRON
ejpam-3284	115	82	‖g‖)p−1‖f‖dµ+	‖g‖)p−1‖f‖dµ+	PROPN
ejpam-3284	115	83	∫	∫	PROPN
ejpam-3284	115	84	(	(	PUNCT
ejpam-3284	115	85	‖f‖+	‖f‖+	PROPN
ejpam-3284	115	86	‖g‖)p−1‖g‖dµ	‖g‖)p−1‖g‖dµ	PROPN
ejpam-3284	115	87	⇒	⇒	PROPN
ejpam-3284	115	88	(	(	PUNCT
ejpam-3284	115	89	∫	∫	PROPN
ejpam-3284	115	90	(	(	PUNCT
ejpam-3284	115	91	‖f‖‖g‖)p	‖f‖‖g‖)p	PROPN
ejpam-3284	115	92	)	)	PUNCT
ejpam-3284	116	1	1	1	NUM
ejpam-3284	116	2	p	p	NOUN
ejpam-3284	116	3	dµ	dµ	X
ejpam-3284	116	4	=	=	PUNCT
ejpam-3284	116	5	[	[	PUNCT
ejpam-3284	116	6	∫	∫	PROPN
ejpam-3284	116	7	(	(	PUNCT
ejpam-3284	116	8	‖f‖+	‖f‖+	PRON
ejpam-3284	116	9	‖g‖)p−1‖f‖dµ+	‖g‖)p−1‖f‖dµ+	PROPN
ejpam-3284	116	10	∫	∫	PROPN
ejpam-3284	116	11	(	(	PUNCT
ejpam-3284	116	12	‖f‖+	‖f‖+	PROPN
ejpam-3284	116	13	‖g‖)p−1‖g‖dµ	‖g‖)p−1‖g‖dµ	X
ejpam-3284	116	14	]	]	PUNCT
ejpam-3284	116	15	1	1	NUM
ejpam-3284	116	16	p	p	NOUN
ejpam-3284	116	17	⇒	⇒	NOUN
ejpam-3284	116	18	‖f‖p‖g‖p	‖f‖p‖g‖p	PUNCT
ejpam-3284	117	1	=	=	PUNCT
ejpam-3284	118	1	[	[	X
ejpam-3284	118	2	(	(	PUNCT
ejpam-3284	118	3	∫	∫	PROPN
ejpam-3284	118	4	(	(	PUNCT
ejpam-3284	118	5	(	(	PUNCT
ejpam-3284	118	6	‖f‖+	‖f‖+	PROPN
ejpam-3284	118	7	‖g‖)p−1‖f‖)pdµ	‖g‖)p−1‖f‖)pdµ	NOUN
ejpam-3284	118	8	)	)	PUNCT
ejpam-3284	118	9	1	1	NUM
ejpam-3284	118	10	p	p	NOUN
ejpam-3284	118	11	+	+	X
ejpam-3284	118	12	(	(	PUNCT
ejpam-3284	118	13	∫	∫	PROPN
ejpam-3284	118	14	(	(	PUNCT
ejpam-3284	118	15	(	(	PUNCT
ejpam-3284	118	16	‖f‖+	‖f‖+	X
ejpam-3284	118	17	‖g‖)p−1‖g‖)pdµ	‖g‖)p−1‖g‖)pdµ	NUM
ejpam-3284	118	18	)	)	PUNCT
ejpam-3284	118	19	1	1	NUM
ejpam-3284	118	20	p	p	NOUN
ejpam-3284	118	21	]	]	PUNCT
ejpam-3284	118	22	1	1	NUM
ejpam-3284	118	23	p	p	NOUN
ejpam-3284	118	24	⇒	⇒	NOUN
ejpam-3284	118	25	‖f‖p‖g‖p	‖f‖p‖g‖p	PRON
ejpam-3284	118	26	≤	≤	PUNCT
ejpam-3284	119	1	[	[	X
ejpam-3284	119	2	{	{	PUNCT
ejpam-3284	119	3	(	(	PUNCT
ejpam-3284	119	4	∫	∫	PROPN
ejpam-3284	119	5	‖f‖pdµ	‖f‖pdµ	PROPN
ejpam-3284	119	6	)	)	PUNCT
ejpam-3284	119	7	1	1	NUM
ejpam-3284	119	8	p	p	NOUN
ejpam-3284	119	9	+	+	X
ejpam-3284	119	10	(	(	PUNCT
ejpam-3284	119	11	∫	∫	PROPN
ejpam-3284	119	12	‖g‖pdµ	‖g‖pdµ	NUM
ejpam-3284	119	13	)	)	PUNCT
ejpam-3284	119	14	1	1	NUM
ejpam-3284	119	15	p	p	NOUN
ejpam-3284	119	16	}	}	PUNCT
ejpam-3284	119	17	{	{	PUNCT
ejpam-3284	119	18	(	(	PUNCT
ejpam-3284	119	19	∫	∫	PROPN
ejpam-3284	119	20	(	(	PUNCT
ejpam-3284	119	21	‖f‖+	‖f‖+	PROPN
ejpam-3284	119	22	‖g‖)pdµ	‖g‖)pdµ	NUM
ejpam-3284	119	23	)	)	PUNCT
ejpam-3284	119	24	1	1	NUM
ejpam-3284	119	25	p	p	NOUN
ejpam-3284	119	26	}	}	PUNCT
ejpam-3284	119	27	p−1	p−1	PROPN
ejpam-3284	119	28	]	]	PUNCT
ejpam-3284	119	29	1	1	NUM
ejpam-3284	119	30	p	p	NOUN
ejpam-3284	119	31	⇒	⇒	NOUN
ejpam-3284	119	32	‖f‖p‖g‖p	‖f‖p‖g‖p	PRON
ejpam-3284	119	33	≤	≤	PUNCT
ejpam-3284	120	1	[	[	X
ejpam-3284	120	2	(	(	PUNCT
ejpam-3284	120	3	‖f‖p	‖f‖p	NOUN
ejpam-3284	120	4	+	+	CCONJ
ejpam-3284	120	5	‖g‖p	‖g‖p	NOUN
ejpam-3284	120	6	)	)	PUNCT
ejpam-3284	120	7	(	(	PUNCT
ejpam-3284	120	8	‖f‖p	‖f‖p	NOUN
ejpam-3284	120	9	+	+	CCONJ
ejpam-3284	120	10	‖g‖p	‖g‖p	X
ejpam-3284	120	11	)	)	PUNCT
ejpam-3284	120	12	p−1	p−1	NOUN
ejpam-3284	120	13	]	]	X
ejpam-3284	120	14	1	1	NUM
ejpam-3284	120	15	p	p	NOUN
ejpam-3284	120	16	⇒	⇒	NOUN
ejpam-3284	120	17	‖f‖p‖g‖p	‖f‖p‖g‖p	PROPN
ejpam-3284	120	18	=	=	SYM
ejpam-3284	120	19	(	(	PUNCT
ejpam-3284	120	20	‖f‖p	‖f‖p	NOUN
ejpam-3284	120	21	+	+	CCONJ
ejpam-3284	120	22	‖g‖p	‖g‖p	X
ejpam-3284	120	23	)	)	PUNCT
ejpam-3284	120	24	1	1	NUM
ejpam-3284	120	25	p	p	NOUN
ejpam-3284	120	26	(	(	PUNCT
ejpam-3284	120	27	‖f‖p	‖f‖p	NOUN
ejpam-3284	120	28	+	+	CCONJ
ejpam-3284	120	29	‖g‖p	‖g‖p	X
ejpam-3284	120	30	)	)	PUNCT
ejpam-3284	121	1	p−1	p−1	PROPN
ejpam-3284	121	2	p	p	PROPN
ejpam-3284	121	3	⇒	⇒	VERB
ejpam-3284	121	4	‖f‖p‖g‖p	‖f‖p‖g‖p	NUM
ejpam-3284	121	5	≤	≤	ADJ
ejpam-3284	121	6	‖f‖p	‖f‖p	NOUN
ejpam-3284	121	7	+	+	CCONJ
ejpam-3284	121	8	‖g‖p	‖g‖p	NOUN
ejpam-3284	121	9	.	.	PUNCT
ejpam-3284	122	1	(	(	PUNCT
ejpam-3284	122	2	1	1	X
ejpam-3284	122	3	)	)	PUNCT
ejpam-3284	122	4	choosing	choose	VERB
ejpam-3284	122	5	p	p	NOUN
ejpam-3284	122	6	=	=	NOUN
ejpam-3284	122	7	1	1	NUM
ejpam-3284	122	8	and	and	CCONJ
ejpam-3284	122	9	p	p	NOUN
ejpam-3284	122	10	=	=	SYM
ejpam-3284	122	11	2	2	NUM
ejpam-3284	122	12	in	in	ADP
ejpam-3284	122	13	inequality	inequality	NOUN
ejpam-3284	122	14	(	(	PUNCT
ejpam-3284	122	15	1	1	NUM
ejpam-3284	122	16	)	)	PUNCT
ejpam-3284	122	17	,	,	PUNCT
ejpam-3284	122	18	we	we	PRON
ejpam-3284	122	19	obtain	obtain	VERB
ejpam-3284	122	20	the	the	DET
ejpam-3284	122	21	following	follow	VERB
ejpam-3284	122	22	‖f‖1‖g‖1	‖f‖1‖g‖1	PUNCT
ejpam-3284	122	23	≤	≤	X
ejpam-3284	122	24	‖f‖1	‖f‖1	NOUN
ejpam-3284	122	25	+	+	CCONJ
ejpam-3284	122	26	‖g‖1	‖g‖1	NOUN
ejpam-3284	122	27	and	and	CCONJ
ejpam-3284	122	28	‖f‖2‖g‖2	‖f‖2‖g‖2	SYM
ejpam-3284	122	29	≤	≤	NUM
ejpam-3284	122	30	‖f‖2	‖f‖2	PUNCT
ejpam-3284	123	1	+	+	PUNCT
ejpam-3284	123	2	‖g‖2	‖g‖2	NOUN
ejpam-3284	123	3	,	,	PUNCT
ejpam-3284	123	4	barnes	barne	VERB
ejpam-3284	123	5	et	et	PROPN
ejpam-3284	123	6	al	al	PROPN
ejpam-3284	123	7	.	.	PUNCT
ejpam-3284	123	8	/	/	SYM
ejpam-3284	123	9	eur	eur	PROPN
ejpam-3284	123	10	.	.	PUNCT
ejpam-3284	124	1	j.	j.	PROPN
ejpam-3284	124	2	pure	pure	PROPN
ejpam-3284	124	3	appl	appl	PROPN
ejpam-3284	124	4	.	.	PROPN
ejpam-3284	124	5	math	math	PROPN
ejpam-3284	124	6	,	,	PUNCT
ejpam-3284	124	7	11	11	NUM
ejpam-3284	124	8	(	(	PUNCT
ejpam-3284	124	9	3	3	NUM
ejpam-3284	124	10	)	)	PUNCT
ejpam-3284	124	11	(	(	PUNCT
ejpam-3284	124	12	2018	2018	NUM
ejpam-3284	124	13	)	)	PUNCT
ejpam-3284	124	14	,	,	PUNCT
ejpam-3284	124	15	740	740	NUM
ejpam-3284	124	16	-	-	SYM
ejpam-3284	124	17	750	750	NUM
ejpam-3284	124	18	746	746	NUM
ejpam-3284	124	19	respectively	respectively	ADV
ejpam-3284	124	20	.	.	PUNCT
ejpam-3284	125	1	but	but	CCONJ
ejpam-3284	125	2	if	if	SCONJ
ejpam-3284	125	3	,	,	PUNCT
ejpam-3284	125	4	p→∞	p→∞	ADV
ejpam-3284	125	5	then	then	ADV
ejpam-3284	125	6	the	the	DET
ejpam-3284	125	7	inequality	inequality	NOUN
ejpam-3284	125	8	(	(	PUNCT
ejpam-3284	125	9	1	1	X
ejpam-3284	125	10	)	)	PUNCT
ejpam-3284	125	11	becomes	become	VERB
ejpam-3284	125	12	‖f‖∞‖g‖∞	‖f‖∞‖g‖∞	NOUN
ejpam-3284	125	13	≤	≤	PUNCT
ejpam-3284	125	14	‖f‖∞	‖f‖∞	PROPN
ejpam-3284	126	1	+	+	CCONJ
ejpam-3284	126	2	‖g‖∞.	‖g‖∞.	PROPN
ejpam-3284	126	3	theorem	theorem	VERB
ejpam-3284	126	4	3	3	NUM
ejpam-3284	126	5	(	(	PUNCT
ejpam-3284	126	6	completeness	completeness	NOUN
ejpam-3284	126	7	of	of	ADP
ejpam-3284	126	8	the	the	DET
ejpam-3284	126	9	product	product	NOUN
ejpam-3284	126	10	-	-	PUNCT
ejpam-3284	126	11	normed	norme	VERB
ejpam-3284	126	12	linear	linear	ADJ
ejpam-3284	126	13	space	space	NOUN
ejpam-3284	126	14	)	)	PUNCT
ejpam-3284	126	15	.	.	PUNCT
ejpam-3284	127	1	the	the	DET
ejpam-3284	127	2	product	product	NOUN
ejpam-3284	127	3	-	-	PUNCT
ejpam-3284	127	4	normed	norme	VERB
ejpam-3284	127	5	linear	linear	ADJ
ejpam-3284	127	6	space	space	NOUN
ejpam-3284	127	7	(	(	PUNCT
ejpam-3284	127	8	u	u	NOUN
ejpam-3284	127	9	,	,	PUNCT
ejpam-3284	127	10	‖	‖	PROPN
ejpam-3284	127	11	·	·	PUNCT
ejpam-3284	127	12	‖pn	‖pn	NUM
ejpam-3284	127	13	)	)	PUNCT
ejpam-3284	127	14	is	be	AUX
ejpam-3284	127	15	complete	complete	ADJ
ejpam-3284	127	16	.	.	PUNCT
ejpam-3284	128	1	proof	proof	NOUN
ejpam-3284	128	2	:	:	PUNCT
ejpam-3284	128	3	let	let	VERB
ejpam-3284	128	4	{	{	PUNCT
ejpam-3284	128	5	un}∞n=1	un}∞n=1	X
ejpam-3284	128	6	be	be	AUX
ejpam-3284	128	7	a	a	DET
ejpam-3284	128	8	sequence	sequence	NOUN
ejpam-3284	128	9	in	in	ADP
ejpam-3284	128	10	(	(	PUNCT
ejpam-3284	128	11	u	u	NOUN
ejpam-3284	128	12	,	,	PUNCT
ejpam-3284	128	13	‖	‖	PROPN
ejpam-3284	128	14	·	·	PUNCT
ejpam-3284	128	15	‖pn	‖pn	NUM
ejpam-3284	128	16	)	)	PUNCT
ejpam-3284	128	17	and	and	CCONJ
ejpam-3284	128	18	u	u	PROPN
ejpam-3284	128	19	∈	∈	PROPN
ejpam-3284	128	20	u	u	NOUN
ejpam-3284	128	21	.	.	PUNCT
ejpam-3284	129	1	for	for	ADP
ejpam-3284	129	2	every	every	DET
ejpam-3284	129	3	ε	ε	PROPN
ejpam-3284	129	4	>	>	X
ejpam-3284	129	5	0	0	PROPN
ejpam-3284	129	6	,	,	PUNCT
ejpam-3284	129	7	there	there	PRON
ejpam-3284	129	8	exists	exist	VERB
ejpam-3284	129	9	an	an	DET
ejpam-3284	129	10	integer	integer	NOUN
ejpam-3284	129	11	no	no	INTJ
ejpam-3284	129	12	>	>	X
ejpam-3284	129	13	n	n	CCONJ
ejpam-3284	129	14	such	such	ADJ
ejpam-3284	129	15	that	that	SCONJ
ejpam-3284	129	16	‖un	‖un	PROPN
ejpam-3284	129	17	−	−	PROPN
ejpam-3284	129	18	u‖	u‖	NOUN
ejpam-3284	129	19	<	<	X
ejpam-3284	129	20	ε	ε	PROPN
ejpam-3284	129	21	,	,	PUNCT
ejpam-3284	129	22	for	for	ADP
ejpam-3284	129	23	all	all	PRON
ejpam-3284	129	24	n	n	CCONJ
ejpam-3284	129	25	>	>	X
ejpam-3284	129	26	no	no	INTJ
ejpam-3284	129	27	,	,	PUNCT
ejpam-3284	129	28	then	then	ADV
ejpam-3284	129	29	{	{	PUNCT
ejpam-3284	129	30	un}∞n=1	un}∞n=1	X
ejpam-3284	129	31	converges	converge	VERB
ejpam-3284	129	32	u.	u.	VERB
ejpam-3284	129	33	thus	thus	ADV
ejpam-3284	129	34	,	,	PUNCT
ejpam-3284	129	35	limn→∞	limn→∞	PROPN
ejpam-3284	129	36	‖un	‖un	PROPN
ejpam-3284	129	37	−	−	PROPN
ejpam-3284	129	38	u‖pn	u‖pn	PROPN
ejpam-3284	129	39	=	=	SYM
ejpam-3284	129	40	0	0	X
ejpam-3284	129	41	.	.	PUNCT
ejpam-3284	130	1	also	also	ADV
ejpam-3284	130	2	,	,	PUNCT
ejpam-3284	130	3	we	we	PRON
ejpam-3284	130	4	see	see	VERB
ejpam-3284	130	5	that	that	PRON
ejpam-3284	130	6	for	for	ADP
ejpam-3284	130	7	every	every	DET
ejpam-3284	130	8	ε	ε	PROPN
ejpam-3284	130	9	>	>	X
ejpam-3284	130	10	0	0	PROPN
ejpam-3284	130	11	,	,	PUNCT
ejpam-3284	130	12	there	there	PRON
ejpam-3284	130	13	exists	exist	VERB
ejpam-3284	130	14	integers	integer	NOUN
ejpam-3284	130	15	m	m	PRON
ejpam-3284	130	16	,	,	PUNCT
ejpam-3284	130	17	n	n	CCONJ
ejpam-3284	130	18	>	>	X
ejpam-3284	131	1	n	n	PRON
ejpam-3284	131	2	such	such	ADJ
ejpam-3284	131	3	that	that	DET
ejpam-3284	131	4	limm	limm	NOUN
ejpam-3284	131	5	,	,	PUNCT
ejpam-3284	131	6	n→∞	n→∞	X
ejpam-3284	131	7	‖um	‖um	NUM
ejpam-3284	131	8	−	−	NOUN
ejpam-3284	131	9	un‖	un‖	NOUN
ejpam-3284	131	10	=	=	SYM
ejpam-3284	131	11	0	0	X
ejpam-3284	131	12	.	.	PUNCT
ejpam-3284	132	1	in	in	ADP
ejpam-3284	132	2	this	this	DET
ejpam-3284	132	3	case	case	NOUN
ejpam-3284	132	4	,	,	PUNCT
ejpam-3284	132	5	we	we	PRON
ejpam-3284	132	6	see	see	VERB
ejpam-3284	132	7	that	that	SCONJ
ejpam-3284	132	8	every	every	DET
ejpam-3284	132	9	cauchy	cauchy	ADJ
ejpam-3284	132	10	sequence	sequence	NOUN
ejpam-3284	132	11	converges	converge	VERB
ejpam-3284	132	12	to	to	ADP
ejpam-3284	132	13	a	a	DET
ejpam-3284	132	14	point	point	NOUN
ejpam-3284	132	15	(	(	PUNCT
ejpam-3284	132	16	u	u	NOUN
ejpam-3284	132	17	,	,	PUNCT
ejpam-3284	132	18	‖	‖	PROPN
ejpam-3284	132	19	·	·	PUNCT
ejpam-3284	132	20	‖pn	‖pn	NUM
ejpam-3284	132	21	)	)	PUNCT
ejpam-3284	132	22	,	,	PUNCT
ejpam-3284	132	23	then	then	ADV
ejpam-3284	132	24	(	(	PUNCT
ejpam-3284	132	25	u	u	NOUN
ejpam-3284	132	26	,	,	PUNCT
ejpam-3284	132	27	‖	‖	PROPN
ejpam-3284	132	28	·	·	PUNCT
ejpam-3284	132	29	‖pn	‖pn	NUM
ejpam-3284	132	30	)	)	PUNCT
ejpam-3284	132	31	is	be	AUX
ejpam-3284	132	32	a	a	DET
ejpam-3284	132	33	complete	complete	ADJ
ejpam-3284	132	34	product	product	NOUN
ejpam-3284	132	35	-	-	PUNCT
ejpam-3284	132	36	normed	norme	VERB
ejpam-3284	132	37	linear	linear	ADJ
ejpam-3284	132	38	space	space	NOUN
ejpam-3284	132	39	.	.	PUNCT
ejpam-3284	133	1	alternatively	alternatively	ADV
ejpam-3284	133	2	,	,	PUNCT
ejpam-3284	133	3	we	we	PRON
ejpam-3284	133	4	the	the	DET
ejpam-3284	133	5	following	following	ADJ
ejpam-3284	133	6	result	result	NOUN
ejpam-3284	133	7	shows	show	VERB
ejpam-3284	133	8	the	the	DET
ejpam-3284	133	9	product	product	NOUN
ejpam-3284	133	10	-	-	PUNCT
ejpam-3284	133	11	normed	norme	VERB
ejpam-3284	133	12	linear	linear	ADJ
ejpam-3284	133	13	space	space	NOUN
ejpam-3284	133	14	is	be	AUX
ejpam-3284	133	15	complete	complete	ADJ
ejpam-3284	133	16	.	.	PUNCT
ejpam-3284	134	1	theorem	theorem	ADJ
ejpam-3284	134	2	4	4	NUM
ejpam-3284	134	3	(	(	PUNCT
ejpam-3284	134	4	completeness	completeness	NOUN
ejpam-3284	134	5	of	of	ADP
ejpam-3284	134	6	the	the	DET
ejpam-3284	134	7	product	product	NOUN
ejpam-3284	134	8	-	-	PUNCT
ejpam-3284	134	9	normed	norme	VERB
ejpam-3284	134	10	linear	linear	ADJ
ejpam-3284	134	11	space	space	NOUN
ejpam-3284	134	12	using	use	VERB
ejpam-3284	134	13	closed	close	VERB
ejpam-3284	134	14	set	set	NOUN
ejpam-3284	134	15	)	)	PUNCT
ejpam-3284	134	16	.	.	PUNCT
ejpam-3284	135	1	let	let	VERB
ejpam-3284	135	2	u	u	PRON
ejpam-3284	135	3	be	be	AUX
ejpam-3284	135	4	a	a	DET
ejpam-3284	135	5	linear	linear	ADJ
ejpam-3284	135	6	space	space	NOUN
ejpam-3284	135	7	over	over	ADP
ejpam-3284	135	8	[	[	X
ejpam-3284	135	9	0	0	NUM
ejpam-3284	135	10	,	,	PUNCT
ejpam-3284	135	11	2	2	NUM
ejpam-3284	135	12	]	]	PUNCT
ejpam-3284	135	13	⊆	⊆	NUM
ejpam-3284	135	14	r.	r.	NOUN
ejpam-3284	135	15	a	a	DET
ejpam-3284	135	16	subspace	subspace	NOUN
ejpam-3284	135	17	(	(	PUNCT
ejpam-3284	135	18	u	u	NOUN
ejpam-3284	135	19	,	,	PUNCT
ejpam-3284	135	20	‖	‖	PROPN
ejpam-3284	135	21	·	·	PUNCT
ejpam-3284	135	22	‖pn	‖pn	NUM
ejpam-3284	135	23	)	)	PUNCT
ejpam-3284	135	24	of	of	ADP
ejpam-3284	135	25	a	a	DET
ejpam-3284	135	26	complete	complete	ADJ
ejpam-3284	135	27	normed	normed	ADJ
ejpam-3284	135	28	linear	linear	ADJ
ejpam-3284	135	29	space	space	NOUN
ejpam-3284	135	30	(	(	PUNCT
ejpam-3284	135	31	rn	rn	PROPN
ejpam-3284	135	32	,	,	PUNCT
ejpam-3284	135	33	‖	‖	PROPN
ejpam-3284	135	34	·	·	PUNCT
ejpam-3284	135	35	‖pn	‖pn	NUM
ejpam-3284	135	36	)	)	PUNCT
ejpam-3284	135	37	is	be	AUX
ejpam-3284	135	38	complete	complete	ADJ
ejpam-3284	136	1	if	if	SCONJ
ejpam-3284	137	1	and	and	CCONJ
ejpam-3284	137	2	only	only	ADV
ejpam-3284	137	3	if	if	SCONJ
ejpam-3284	137	4	u	u	NOUN
ejpam-3284	137	5	is	be	AUX
ejpam-3284	137	6	a	a	DET
ejpam-3284	137	7	closed	closed	ADJ
ejpam-3284	137	8	set	set	NOUN
ejpam-3284	137	9	.	.	PUNCT
ejpam-3284	138	1	proof	proof	NOUN
ejpam-3284	138	2	:	:	PUNCT
ejpam-3284	138	3	let	let	VERB
ejpam-3284	138	4	{	{	PUNCT
ejpam-3284	138	5	un}∞n=1	un}∞n=1	X
ejpam-3284	138	6	be	be	AUX
ejpam-3284	138	7	a	a	DET
ejpam-3284	138	8	sequence	sequence	NOUN
ejpam-3284	138	9	of	of	ADP
ejpam-3284	138	10	points	point	NOUN
ejpam-3284	138	11	of	of	ADP
ejpam-3284	138	12	u	u	NOUN
ejpam-3284	138	13	.	.	PUNCT
ejpam-3284	139	1	we	we	PRON
ejpam-3284	139	2	see	see	VERB
ejpam-3284	139	3	that	that	SCONJ
ejpam-3284	139	4	{	{	PUNCT
ejpam-3284	139	5	un}∞n=1	un}∞n=1	NUM
ejpam-3284	139	6	has	have	AUX
ejpam-3284	139	7	limit	limit	VERB
ejpam-3284	139	8	u	u	NOUN
ejpam-3284	139	9	in	in	ADP
ejpam-3284	139	10	u	u	NOUN
ejpam-3284	139	11	,	,	PUNCT
ejpam-3284	139	12	since	since	SCONJ
ejpam-3284	139	13	u	u	NOUN
ejpam-3284	139	14	is	be	AUX
ejpam-3284	139	15	closed	close	VERB
ejpam-3284	139	16	.	.	PUNCT
ejpam-3284	140	1	thus	thus	ADV
ejpam-3284	140	2	,	,	PUNCT
ejpam-3284	140	3	lim	lim	PROPN
ejpam-3284	140	4	n→∞	n→∞	X
ejpam-3284	141	1	‖un	‖un	PROPN
ejpam-3284	141	2	−	−	PROPN
ejpam-3284	141	3	u‖pn	u‖pn	PROPN
ejpam-3284	141	4	=	=	NOUN
ejpam-3284	141	5	0	0	X
ejpam-3284	141	6	.	.	PUNCT
ejpam-3284	142	1	since	since	SCONJ
ejpam-3284	142	2	u	u	PROPN
ejpam-3284	142	3	∈	∈	PROPN
ejpam-3284	142	4	u	u	NOUN
ejpam-3284	142	5	then	then	ADV
ejpam-3284	142	6	(	(	PUNCT
ejpam-3284	142	7	u	u	NOUN
ejpam-3284	142	8	,	,	PUNCT
ejpam-3284	142	9	‖	‖	PROPN
ejpam-3284	142	10	·	·	PUNCT
ejpam-3284	142	11	‖pn	‖pn	NUM
ejpam-3284	142	12	)	)	PUNCT
ejpam-3284	142	13	is	be	AUX
ejpam-3284	142	14	complete	complete	ADJ
ejpam-3284	142	15	.	.	PUNCT
ejpam-3284	143	1	conversely	conversely	ADV
ejpam-3284	143	2	,	,	PUNCT
ejpam-3284	143	3	assume	assume	VERB
ejpam-3284	143	4	that	that	SCONJ
ejpam-3284	143	5	(	(	PUNCT
ejpam-3284	143	6	u	u	NOUN
ejpam-3284	143	7	,	,	PUNCT
ejpam-3284	143	8	‖	‖	PROPN
ejpam-3284	143	9	·	·	PUNCT
ejpam-3284	143	10	‖pn	‖pn	NUM
ejpam-3284	143	11	)	)	PUNCT
ejpam-3284	143	12	is	be	AUX
ejpam-3284	143	13	complete	complete	ADJ
ejpam-3284	143	14	.	.	PUNCT
ejpam-3284	144	1	let	let	VERB
ejpam-3284	144	2	u	u	PRON
ejpam-3284	144	3	be	be	AUX
ejpam-3284	144	4	an	an	DET
ejpam-3284	144	5	accumulation	accumulation	NOUN
ejpam-3284	144	6	point	point	NOUN
ejpam-3284	144	7	of	of	ADP
ejpam-3284	144	8	u	u	PROPN
ejpam-3284	144	9	.	.	PUNCT
ejpam-3284	145	1	then	then	ADV
ejpam-3284	145	2	each	each	DET
ejpam-3284	145	3	open	open	ADJ
ejpam-3284	145	4	ball	ball	NOUN
ejpam-3284	145	5	centred	centre	VERB
ejpam-3284	145	6	at	at	ADP
ejpam-3284	145	7	u	u	NOUN
ejpam-3284	145	8	,	,	PUNCT
ejpam-3284	145	9	b1	b1	PROPN
ejpam-3284	145	10	/	/	SYM
ejpam-3284	145	11	n(u	n(u	PROPN
ejpam-3284	145	12	)	)	PUNCT
ejpam-3284	145	13	contains	contain	VERB
ejpam-3284	145	14	a	a	DET
ejpam-3284	145	15	point	point	NOUN
ejpam-3284	145	16	un	un	PROPN
ejpam-3284	145	17	∈	∈	PROPN
ejpam-3284	145	18	u	u	PROPN
ejpam-3284	145	19	.	.	PUNCT
ejpam-3284	146	1	that	that	PRON
ejpam-3284	146	2	is	be	AUX
ejpam-3284	146	3	,	,	PUNCT
ejpam-3284	146	4	‖un	‖un	PROPN
ejpam-3284	146	5	−	−	PROPN
ejpam-3284	146	6	u‖pn	u‖pn	VERB
ejpam-3284	146	7	<	<	NOUN
ejpam-3284	146	8	1	1	NUM
ejpam-3284	146	9	n	n	NOUN
ejpam-3284	146	10	.	.	PUNCT
ejpam-3284	147	1	this	this	PRON
ejpam-3284	147	2	implies	imply	VERB
ejpam-3284	147	3	that	that	SCONJ
ejpam-3284	147	4	{	{	PUNCT
ejpam-3284	147	5	un}∞n=1	un}∞n=1	NUM
ejpam-3284	147	6	converges	converge	VERB
ejpam-3284	147	7	to	to	PART
ejpam-3284	147	8	u.	u.	VERB
ejpam-3284	147	9	however	however	ADV
ejpam-3284	147	10	,	,	PUNCT
ejpam-3284	147	11	(	(	PUNCT
ejpam-3284	147	12	u	u	NOUN
ejpam-3284	147	13	,	,	PUNCT
ejpam-3284	147	14	‖·‖pn	‖·‖pn	NOUN
ejpam-3284	147	15	)	)	PUNCT
ejpam-3284	147	16	is	be	AUX
ejpam-3284	147	17	complete	complete	ADJ
ejpam-3284	147	18	on	on	ADP
ejpam-3284	147	19	the	the	DET
ejpam-3284	147	20	grounds	ground	NOUN
ejpam-3284	147	21	that	that	SCONJ
ejpam-3284	147	22	u	u	NOUN
ejpam-3284	147	23	is	be	AUX
ejpam-3284	147	24	in	in	ADP
ejpam-3284	147	25	u	u	PROPN
ejpam-3284	147	26	.	.	PUNCT
ejpam-3284	148	1	hence	hence	ADV
ejpam-3284	148	2	,	,	PUNCT
ejpam-3284	148	3	u	u	PROPN
ejpam-3284	148	4	is	be	AUX
ejpam-3284	148	5	closed	close	VERB
ejpam-3284	148	6	.	.	PUNCT
ejpam-3284	149	1	theorem	theorem	ADJ
ejpam-3284	149	2	5	5	NUM
ejpam-3284	149	3	(	(	PUNCT
ejpam-3284	149	4	continuity	continuity	NOUN
ejpam-3284	149	5	)	)	PUNCT
ejpam-3284	149	6	.	.	PUNCT
ejpam-3284	150	1	let	let	VERB
ejpam-3284	150	2	(	(	PUNCT
ejpam-3284	150	3	u	u	NOUN
ejpam-3284	150	4	,	,	PUNCT
ejpam-3284	150	5	‖	‖	PROPN
ejpam-3284	150	6	·	·	PUNCT
ejpam-3284	150	7	‖pn1	‖pn1	NOUN
ejpam-3284	150	8	)	)	PUNCT
ejpam-3284	150	9	and	and	CCONJ
ejpam-3284	150	10	(	(	PUNCT
ejpam-3284	150	11	u	u	NOUN
ejpam-3284	150	12	,	,	PUNCT
ejpam-3284	150	13	‖	‖	PROPN
ejpam-3284	150	14	·	·	PUNCT
ejpam-3284	150	15	‖pn2	‖pn2	NOUN
ejpam-3284	150	16	)	)	PUNCT
ejpam-3284	150	17	be	be	VERB
ejpam-3284	150	18	two	two	NUM
ejpam-3284	150	19	product	product	NOUN
ejpam-3284	150	20	normed	norme	VERB
ejpam-3284	150	21	linear	linear	PROPN
ejpam-3284	150	22	spaces	space	NOUN
ejpam-3284	150	23	,	,	PUNCT
ejpam-3284	150	24	and	and	CCONJ
ejpam-3284	150	25	t	t	PROPN
ejpam-3284	150	26	:	:	PUNCT
ejpam-3284	150	27	u1	u1	PROPN
ejpam-3284	150	28	→	→	SYM
ejpam-3284	150	29	u2	u2	PROPN
ejpam-3284	150	30	be	be	AUX
ejpam-3284	150	31	a	a	DET
ejpam-3284	150	32	linear	linear	ADJ
ejpam-3284	150	33	operator	operator	NOUN
ejpam-3284	150	34	.	.	PUNCT
ejpam-3284	151	1	then	then	ADV
ejpam-3284	151	2	t	t	PROPN
ejpam-3284	151	3	is	be	AUX
ejpam-3284	151	4	bounded	bound	VERB
ejpam-3284	151	5	if	if	SCONJ
ejpam-3284	151	6	and	and	CCONJ
ejpam-3284	151	7	only	only	ADV
ejpam-3284	151	8	if	if	SCONJ
ejpam-3284	151	9	t	t	PROPN
ejpam-3284	151	10	is	be	AUX
ejpam-3284	151	11	continuous	continuous	ADJ
ejpam-3284	151	12	.	.	PUNCT
ejpam-3284	152	1	proof	proof	NOUN
ejpam-3284	152	2	:	:	PUNCT
ejpam-3284	152	3	let	let	VERB
ejpam-3284	152	4	a	a	PRON
ejpam-3284	152	5	be	be	AUX
ejpam-3284	152	6	a	a	DET
ejpam-3284	152	7	bounded	bounded	ADJ
ejpam-3284	152	8	linear	linear	ADJ
ejpam-3284	152	9	operator	operator	NOUN
ejpam-3284	152	10	.	.	PUNCT
ejpam-3284	153	1	then	then	ADV
ejpam-3284	153	2	there	there	PRON
ejpam-3284	153	3	exists	exist	VERB
ejpam-3284	153	4	a	a	DET
ejpam-3284	153	5	constant	constant	ADJ
ejpam-3284	153	6	m	m	NOUN
ejpam-3284	153	7	such	such	ADJ
ejpam-3284	153	8	that	that	SCONJ
ejpam-3284	153	9	‖a(u)‖u2	‖a(u)‖u2	NUM
ejpam-3284	153	10	≤m‖x‖u1	≤m‖x‖u1	NUM
ejpam-3284	153	11	,	,	PUNCT
ejpam-3284	153	12	∀	∀	PUNCT
ejpam-3284	153	13	u	u	NOUN
ejpam-3284	153	14	∈	∈	PROPN
ejpam-3284	153	15	u1	u1	NOUN
ejpam-3284	153	16	.	.	PUNCT
ejpam-3284	154	1	let	let	VERB
ejpam-3284	154	2	{	{	PUNCT
ejpam-3284	154	3	un}∞n=1	un}∞n=1	PUNCT
ejpam-3284	154	4	be	be	AUX
ejpam-3284	154	5	a	a	DET
ejpam-3284	154	6	sequence	sequence	NOUN
ejpam-3284	154	7	in	in	ADP
ejpam-3284	154	8	u1	u1	NOUN
ejpam-3284	154	9	such	such	ADJ
ejpam-3284	154	10	that	that	SCONJ
ejpam-3284	154	11	lim	lim	PROPN
ejpam-3284	154	12	n→∞	n→∞	PRON
ejpam-3284	155	1	‖un	‖un	PROPN
ejpam-3284	155	2	−	−	PROPN
ejpam-3284	155	3	u‖u1	u‖u1	PROPN
ejpam-3284	155	4	=	=	SYM
ejpam-3284	155	5	0	0	X
ejpam-3284	155	6	.	.	PUNCT
ejpam-3284	156	1	barnes	barnes	PROPN
ejpam-3284	156	2	et	et	PROPN
ejpam-3284	156	3	al	al	PROPN
ejpam-3284	156	4	.	.	PUNCT
ejpam-3284	156	5	/	/	SYM
ejpam-3284	156	6	eur	eur	PROPN
ejpam-3284	156	7	.	.	PUNCT
ejpam-3284	157	1	j.	j.	PROPN
ejpam-3284	157	2	pure	pure	PROPN
ejpam-3284	157	3	appl	appl	PROPN
ejpam-3284	157	4	.	.	PROPN
ejpam-3284	157	5	math	math	PROPN
ejpam-3284	157	6	,	,	PUNCT
ejpam-3284	157	7	11	11	NUM
ejpam-3284	157	8	(	(	PUNCT
ejpam-3284	157	9	3	3	NUM
ejpam-3284	157	10	)	)	PUNCT
ejpam-3284	157	11	(	(	PUNCT
ejpam-3284	157	12	2018	2018	NUM
ejpam-3284	157	13	)	)	PUNCT
ejpam-3284	157	14	,	,	PUNCT
ejpam-3284	157	15	740	740	NUM
ejpam-3284	157	16	-	-	SYM
ejpam-3284	157	17	750	750	NUM
ejpam-3284	157	18	747	747	NUM
ejpam-3284	157	19	now	now	ADV
ejpam-3284	157	20	,	,	PUNCT
ejpam-3284	157	21	‖a(un)−a(u)‖u2	‖a(un)−a(u)‖u2	PUNCT
ejpam-3284	157	22	=	=	SYM
ejpam-3284	157	23	∥∥∥a(un	∥∥∥a(un	PROPN
ejpam-3284	157	24	−	−	PROPN
ejpam-3284	157	25	u)∥∥∥	u)∥∥∥	PROPN
ejpam-3284	157	26	u2	u2	NOUN
ejpam-3284	157	27	‖a(un)−a(u)‖u2	‖a(un)−a(u)‖u2	VERB
ejpam-3284	157	28	≤	≤	NUM
ejpam-3284	157	29	m	m	VERB
ejpam-3284	157	30	∥∥∥un	∥∥∥un	ADJ
ejpam-3284	157	31	−	−	ADP
ejpam-3284	157	32	u∥∥∥	u∥∥∥	NUM
ejpam-3284	157	33	u1	u1	NOUN
ejpam-3284	157	34	lim	lim	PROPN
ejpam-3284	157	35	n→∞	n→∞	NUM
ejpam-3284	157	36	‖a(un)−a(u)‖u2	‖a(un)−a(u)‖u2	VERB
ejpam-3284	157	37	≤	≤	X
ejpam-3284	157	38	0	0	NUM
ejpam-3284	158	1	lim	lim	PROPN
ejpam-3284	158	2	n→∞	n→∞	NUM
ejpam-3284	158	3	a(un)−	a(un)−	PUNCT
ejpam-3284	159	1	lim	lim	PROPN
ejpam-3284	159	2	n→∞	n→∞	X
ejpam-3284	159	3	a(u	a(u	X
ejpam-3284	159	4	)	)	PUNCT
ejpam-3284	160	1	=	=	SYM
ejpam-3284	160	2	0	0	NUM
ejpam-3284	161	1	lim	lim	PROPN
ejpam-3284	161	2	n→∞	n→∞	NUM
ejpam-3284	161	3	a(un)−a(u	a(un)−a(u	PROPN
ejpam-3284	161	4	)	)	PUNCT
ejpam-3284	162	1	=	=	SYM
ejpam-3284	162	2	0	0	PUNCT
ejpam-3284	163	1	a	a	PRON
ejpam-3284	163	2	(	(	PUNCT
ejpam-3284	163	3	lim	lim	PROPN
ejpam-3284	163	4	n→∞	n→∞	NUM
ejpam-3284	163	5	un	un	PROPN
ejpam-3284	163	6	)	)	PUNCT
ejpam-3284	163	7	→	→	SYM
ejpam-3284	163	8	a(u	a(u	PROPN
ejpam-3284	163	9	)	)	PUNCT
ejpam-3284	163	10	.	.	PUNCT
ejpam-3284	164	1	this	this	PRON
ejpam-3284	164	2	implies	imply	VERB
ejpam-3284	164	3	that	that	SCONJ
ejpam-3284	164	4	a	a	PRON
ejpam-3284	164	5	is	be	AUX
ejpam-3284	164	6	continuous	continuous	ADJ
ejpam-3284	164	7	on	on	ADP
ejpam-3284	164	8	u1	u1	NOUN
ejpam-3284	164	9	.	.	PUNCT
ejpam-3284	165	1	conversely	conversely	ADV
ejpam-3284	165	2	,	,	PUNCT
ejpam-3284	165	3	suppose	suppose	VERB
ejpam-3284	165	4	that	that	SCONJ
ejpam-3284	165	5	a	a	PRON
ejpam-3284	165	6	is	be	AUX
ejpam-3284	165	7	a	a	DET
ejpam-3284	165	8	continuous	continuous	ADJ
ejpam-3284	165	9	linear	linear	NOUN
ejpam-3284	165	10	operator	operator	NOUN
ejpam-3284	165	11	.	.	PUNCT
ejpam-3284	166	1	then	then	ADV
ejpam-3284	166	2	a	a	PRON
ejpam-3284	166	3	is	be	AUX
ejpam-3284	166	4	continuous	continuous	ADJ
ejpam-3284	166	5	at	at	ADP
ejpam-3284	166	6	0	0	NUM
ejpam-3284	166	7	.	.	PUNCT
ejpam-3284	167	1	thus	thus	ADV
ejpam-3284	167	2	,	,	PUNCT
ejpam-3284	167	3	for	for	ADP
ejpam-3284	167	4	every	every	DET
ejpam-3284	167	5	δ	δ	PROPN
ejpam-3284	167	6	>	>	X
ejpam-3284	167	7	0	0	PROPN
ejpam-3284	167	8	,	,	PUNCT
ejpam-3284	167	9	‖u‖u1	‖u‖u1	NUM
ejpam-3284	167	10	,	,	PUNCT
ejpam-3284	167	11	there	there	PRON
ejpam-3284	167	12	exists	exist	VERB
ejpam-3284	167	13	an	an	DET
ejpam-3284	167	14	ε	ε	PROPN
ejpam-3284	167	15	>	>	X
ejpam-3284	167	16	0	0	NUM
ejpam-3284	168	1	such	such	ADJ
ejpam-3284	168	2	that	that	SCONJ
ejpam-3284	168	3	‖au‖	‖au‖	PROPN
ejpam-3284	168	4	≤	≤	PUNCT
ejpam-3284	168	5	ε	ε	PROPN
ejpam-3284	168	6	.	.	PUNCT
ejpam-3284	169	1	(	(	PUNCT
ejpam-3284	169	2	2	2	X
ejpam-3284	169	3	)	)	PUNCT
ejpam-3284	169	4	setting	set	VERB
ejpam-3284	169	5	uo	uo	NOUN
ejpam-3284	169	6	6=	6=	PRON
ejpam-3284	169	7	0	0	NUM
ejpam-3284	169	8	,	,	PUNCT
ejpam-3284	169	9	be	be	AUX
ejpam-3284	169	10	an	an	DET
ejpam-3284	169	11	arbitrary	arbitrary	ADJ
ejpam-3284	169	12	element	element	NOUN
ejpam-3284	169	13	in	in	ADP
ejpam-3284	169	14	u1	u1	NOUN
ejpam-3284	169	15	and	and	CCONJ
ejpam-3284	169	16	u	u	NOUN
ejpam-3284	169	17	=	=	PROPN
ejpam-3284	169	18	αuo	αuo	PROPN
ejpam-3284	169	19	,	,	PUNCT
ejpam-3284	169	20	where	where	SCONJ
ejpam-3284	169	21	α	α	PROPN
ejpam-3284	169	22	=	=	SYM
ejpam-3284	169	23	δ	δ	PROPN
ejpam-3284	169	24	‖uo‖u1	‖uo‖u1	PROPN
ejpam-3284	169	25	.	.	PUNCT
ejpam-3284	170	1	we	we	PRON
ejpam-3284	170	2	can	can	AUX
ejpam-3284	170	3	see	see	VERB
ejpam-3284	170	4	that	that	PRON
ejpam-3284	170	5	‖au‖u2	‖au‖u2	PROPN
ejpam-3284	171	1	=	=	PUNCT
ejpam-3284	171	2	‖a(αuo)‖u2	‖a(αuo)‖u2	NOUN
ejpam-3284	171	3	‖au‖u2	‖au‖u2	PUNCT
ejpam-3284	171	4	=	=	PUNCT
ejpam-3284	171	5	|α|‖auo‖u2	|α|‖auo‖u2	NOUN
ejpam-3284	171	6	,	,	PUNCT
ejpam-3284	171	7	∀	∀	NOUN
ejpam-3284	171	8	uo	uo	NOUN
ejpam-3284	171	9	∈	∈	PROPN
ejpam-3284	171	10	u1	u1	NOUN
ejpam-3284	171	11	.	.	PUNCT
ejpam-3284	172	1	(	(	PUNCT
ejpam-3284	172	2	3	3	X
ejpam-3284	172	3	)	)	PUNCT
ejpam-3284	172	4	comparing	compare	VERB
ejpam-3284	172	5	(	(	PUNCT
ejpam-3284	172	6	2	2	NUM
ejpam-3284	172	7	)	)	PUNCT
ejpam-3284	172	8	and	and	CCONJ
ejpam-3284	172	9	(	(	PUNCT
ejpam-3284	172	10	3	3	NUM
ejpam-3284	172	11	)	)	PUNCT
ejpam-3284	172	12	,	,	PUNCT
ejpam-3284	172	13	we	we	PRON
ejpam-3284	172	14	obtain	obtain	AUX
ejpam-3284	172	15	|α|‖auo‖u2	|α|‖auo‖u2	NOUN
ejpam-3284	172	16	≤	≤	NUM
ejpam-3284	172	17	ε	ε	PROPN
ejpam-3284	172	18	⇒	⇒	PROPN
ejpam-3284	172	19	‖auo‖u2	‖auo‖u2	PROPN
ejpam-3284	172	20	≤	≤	NUM
ejpam-3284	172	21	ε	ε	PROPN
ejpam-3284	172	22	|α|	|α|	PROPN
ejpam-3284	172	23	⇒	⇒	PROPN
ejpam-3284	172	24	‖auo‖u2	‖auo‖u2	VERB
ejpam-3284	172	25	≤	≤	NUM
ejpam-3284	172	26	ε	ε	PROPN
ejpam-3284	172	27	|δ|	|δ|	VERB
ejpam-3284	172	28	‖uo‖u1	‖uo‖u1	PROPN
ejpam-3284	172	29	.	.	PUNCT
ejpam-3284	173	1	hence	hence	ADV
ejpam-3284	173	2	a	a	PRON
ejpam-3284	173	3	is	be	AUX
ejpam-3284	173	4	bounded	bound	VERB
ejpam-3284	173	5	.	.	PUNCT
ejpam-3284	174	1	this	this	PRON
ejpam-3284	174	2	completes	complete	VERB
ejpam-3284	174	3	the	the	DET
ejpam-3284	174	4	proof	proof	NOUN
ejpam-3284	174	5	.	.	PUNCT
ejpam-3284	175	1	theorem	theorem	ADJ
ejpam-3284	175	2	6	6	NUM
ejpam-3284	175	3	(	(	PUNCT
ejpam-3284	175	4	unique	unique	ADJ
ejpam-3284	175	5	fixed	fix	VERB
ejpam-3284	175	6	point	point	NOUN
ejpam-3284	175	7	)	)	PUNCT
ejpam-3284	175	8	.	.	PUNCT
ejpam-3284	176	1	let	let	VERB
ejpam-3284	176	2	(	(	PUNCT
ejpam-3284	176	3	u	u	NOUN
ejpam-3284	176	4	,	,	PUNCT
ejpam-3284	176	5	‖	‖	PROPN
ejpam-3284	176	6	·	·	PUNCT
ejpam-3284	176	7	‖pn	‖pn	NUM
ejpam-3284	176	8	)	)	PUNCT
ejpam-3284	176	9	be	be	AUX
ejpam-3284	176	10	a	a	DET
ejpam-3284	176	11	product	product	NOUN
ejpam-3284	176	12	-	-	PUNCT
ejpam-3284	176	13	normed	norme	VERB
ejpam-3284	176	14	linear	linear	ADJ
ejpam-3284	176	15	space	space	NOUN
ejpam-3284	176	16	and	and	CCONJ
ejpam-3284	176	17	setting	set	VERB
ejpam-3284	176	18	t	t	NOUN
ejpam-3284	176	19	:	:	PUNCT
ejpam-3284	176	20	u	u	X
ejpam-3284	176	21	→	→	SYM
ejpam-3284	176	22	u	u	X
ejpam-3284	176	23	be	be	VERB
ejpam-3284	176	24	a	a	DET
ejpam-3284	176	25	contractive	contractive	ADJ
ejpam-3284	176	26	mapping	mapping	NOUN
ejpam-3284	176	27	.	.	PUNCT
ejpam-3284	177	1	then	then	ADV
ejpam-3284	177	2	t	t	PROPN
ejpam-3284	177	3	has	have	VERB
ejpam-3284	177	4	a	a	DET
ejpam-3284	177	5	unique	unique	ADJ
ejpam-3284	177	6	fixed	fix	VERB
ejpam-3284	177	7	point	point	NOUN
ejpam-3284	177	8	ū	ū	NOUN
ejpam-3284	177	9	∈	∈	PROPN
ejpam-3284	177	10	u	u	NOUN
ejpam-3284	177	11	.	.	PUNCT
ejpam-3284	178	1	proof	proof	NOUN
ejpam-3284	178	2	:	:	PUNCT
ejpam-3284	178	3	letting	let	VERB
ejpam-3284	178	4	uo	uo	X
ejpam-3284	178	5	∈	∈	PROPN
ejpam-3284	178	6	u	u	NOUN
ejpam-3284	178	7	,	,	PUNCT
ejpam-3284	178	8	then	then	ADV
ejpam-3284	178	9	un	un	PROPN
ejpam-3284	178	10	=	=	PROPN
ejpam-3284	178	11	t	t	PROPN
ejpam-3284	178	12	(	(	PUNCT
ejpam-3284	178	13	un	un	PROPN
ejpam-3284	178	14	)	)	PUNCT
ejpam-3284	178	15	⇒	⇒	PROPN
ejpam-3284	178	16	un	un	PROPN
ejpam-3284	178	17	=	=	PROPN
ejpam-3284	178	18	tn(uo	tn(uo	PROPN
ejpam-3284	178	19	)	)	PUNCT
ejpam-3284	178	20	,	,	PUNCT
ejpam-3284	178	21	∀	∀	X
ejpam-3284	178	22	n	n	PRON
ejpam-3284	178	23	∈	∈	PROPN
ejpam-3284	178	24	n.	n.	NOUN
ejpam-3284	178	25	setting	set	VERB
ejpam-3284	178	26	{	{	PUNCT
ejpam-3284	178	27	u}∞n	u}∞n	NUM
ejpam-3284	178	28	a	a	DET
ejpam-3284	178	29	cauchy	cauchy	ADJ
ejpam-3284	178	30	sequence	sequence	NOUN
ejpam-3284	178	31	in	in	ADP
ejpam-3284	178	32	(	(	PUNCT
ejpam-3284	178	33	u	u	NOUN
ejpam-3284	178	34	,	,	PUNCT
ejpam-3284	178	35	‖	‖	PROPN
ejpam-3284	178	36	·	·	PUNCT
ejpam-3284	178	37	‖pn	‖pn	NUM
ejpam-3284	178	38	)	)	PUNCT
ejpam-3284	178	39	.	.	PUNCT
ejpam-3284	179	1	since	since	SCONJ
ejpam-3284	179	2	(	(	PUNCT
ejpam-3284	179	3	u	u	NOUN
ejpam-3284	179	4	,	,	PUNCT
ejpam-3284	179	5	‖	‖	PROPN
ejpam-3284	179	6	·	·	PUNCT
ejpam-3284	179	7	‖pn	‖pn	NUM
ejpam-3284	179	8	)	)	PUNCT
ejpam-3284	179	9	is	be	AUX
ejpam-3284	179	10	a	a	DET
ejpam-3284	179	11	complete	complete	ADJ
ejpam-3284	179	12	,	,	PUNCT
ejpam-3284	179	13	which	which	PRON
ejpam-3284	179	14	implies	imply	VERB
ejpam-3284	179	15	that	that	SCONJ
ejpam-3284	179	16	u	u	PROPN
ejpam-3284	179	17	∈	∈	PROPN
ejpam-3284	179	18	u	u	NOUN
ejpam-3284	179	19	such	such	ADJ
ejpam-3284	179	20	that	that	SCONJ
ejpam-3284	179	21	lim	lim	PROPN
ejpam-3284	179	22	n→∞	n→∞	PRON
ejpam-3284	180	1	un	un	PROPN
ejpam-3284	180	2	=	=	PROPN
ejpam-3284	180	3	u.	u.	PROPN
ejpam-3284	180	4	barnes	barnes	PROPN
ejpam-3284	180	5	et	et	PROPN
ejpam-3284	180	6	al	al	PROPN
ejpam-3284	180	7	.	.	PUNCT
ejpam-3284	180	8	/	/	SYM
ejpam-3284	180	9	eur	eur	PROPN
ejpam-3284	180	10	.	.	PUNCT
ejpam-3284	181	1	j.	j.	PROPN
ejpam-3284	181	2	pure	pure	PROPN
ejpam-3284	181	3	appl	appl	PROPN
ejpam-3284	181	4	.	.	PROPN
ejpam-3284	181	5	math	math	PROPN
ejpam-3284	181	6	,	,	PUNCT
ejpam-3284	181	7	11	11	NUM
ejpam-3284	181	8	(	(	PUNCT
ejpam-3284	181	9	3	3	NUM
ejpam-3284	181	10	)	)	PUNCT
ejpam-3284	181	11	(	(	PUNCT
ejpam-3284	181	12	2018	2018	NUM
ejpam-3284	181	13	)	)	PUNCT
ejpam-3284	181	14	,	,	PUNCT
ejpam-3284	181	15	740	740	NUM
ejpam-3284	181	16	-	-	SYM
ejpam-3284	181	17	750	750	NUM
ejpam-3284	181	18	748	748	NUM
ejpam-3284	181	19	again	again	ADV
ejpam-3284	181	20	,	,	PUNCT
ejpam-3284	181	21	we	we	PRON
ejpam-3284	181	22	see	see	VERB
ejpam-3284	181	23	that	that	SCONJ
ejpam-3284	181	24	t	t	PROPN
ejpam-3284	181	25	is	be	AUX
ejpam-3284	181	26	a	a	DET
ejpam-3284	181	27	contractive	contractive	ADJ
ejpam-3284	181	28	mapping	mapping	NOUN
ejpam-3284	181	29	,	,	PUNCT
ejpam-3284	181	30	then	then	ADV
ejpam-3284	181	31	it	it	PRON
ejpam-3284	181	32	is	be	AUX
ejpam-3284	181	33	continuous	continuous	ADJ
ejpam-3284	181	34	.	.	PUNCT
ejpam-3284	182	1	thus	thus	ADV
ejpam-3284	182	2	,	,	PUNCT
ejpam-3284	182	3	lim	lim	PROPN
ejpam-3284	182	4	n→∞	n→∞	PROPN
ejpam-3284	182	5	t	t	PROPN
ejpam-3284	182	6	(	(	PUNCT
ejpam-3284	182	7	un	un	PROPN
ejpam-3284	182	8	)	)	PUNCT
ejpam-3284	182	9	=	=	SYM
ejpam-3284	182	10	tu	tu	PROPN
ejpam-3284	182	11	.	.	PUNCT
ejpam-3284	183	1	we	we	PRON
ejpam-3284	183	2	can	can	AUX
ejpam-3284	183	3	see	see	VERB
ejpam-3284	183	4	that	that	SCONJ
ejpam-3284	183	5	tu	tu	PROPN
ejpam-3284	183	6	=	=	PROPN
ejpam-3284	183	7	lim	lim	PROPN
ejpam-3284	183	8	n→∞	n→∞	PROPN
ejpam-3284	183	9	t	t	PROPN
ejpam-3284	183	10	(	(	PUNCT
ejpam-3284	183	11	un	un	PROPN
ejpam-3284	183	12	)	)	PUNCT
ejpam-3284	183	13	⇒	⇒	NOUN
ejpam-3284	183	14	tu	tu	PROPN
ejpam-3284	183	15	=	=	PROPN
ejpam-3284	183	16	lim	lim	PROPN
ejpam-3284	183	17	n→∞	n→∞	X
ejpam-3284	184	1	un+1	un+1	PROPN
ejpam-3284	184	2	⇒	⇒	X
ejpam-3284	184	3	tu	tu	PROPN
ejpam-3284	184	4	=	=	PUNCT
ejpam-3284	184	5	u.	u.	PROPN
ejpam-3284	184	6	hence	hence	ADV
ejpam-3284	184	7	,	,	PUNCT
ejpam-3284	184	8	u	u	PROPN
ejpam-3284	184	9	is	be	AUX
ejpam-3284	184	10	a	a	DET
ejpam-3284	184	11	fixed	fix	VERB
ejpam-3284	184	12	point	point	NOUN
ejpam-3284	184	13	of	of	ADP
ejpam-3284	184	14	the	the	DET
ejpam-3284	184	15	mapping	mapping	NOUN
ejpam-3284	184	16	t	t	NOUN
ejpam-3284	184	17	.	.	PUNCT
ejpam-3284	185	1	in	in	ADP
ejpam-3284	185	2	addition	addition	NOUN
ejpam-3284	185	3	,	,	PUNCT
ejpam-3284	185	4	we	we	PRON
ejpam-3284	185	5	show	show	VERB
ejpam-3284	185	6	that	that	SCONJ
ejpam-3284	185	7	the	the	DET
ejpam-3284	185	8	fixed	fix	VERB
ejpam-3284	185	9	point	point	NOUN
ejpam-3284	185	10	is	be	AUX
ejpam-3284	185	11	unique	unique	ADJ
ejpam-3284	185	12	in	in	ADP
ejpam-3284	185	13	(	(	PUNCT
ejpam-3284	185	14	u	u	NOUN
ejpam-3284	185	15	,	,	PUNCT
ejpam-3284	185	16	‖	‖	PROPN
ejpam-3284	185	17	·	·	PUNCT
ejpam-3284	185	18	‖pn	‖pn	NUM
ejpam-3284	185	19	)	)	PUNCT
ejpam-3284	185	20	.	.	PUNCT
ejpam-3284	186	1	setting	set	VERB
ejpam-3284	186	2	u1	u1	NOUN
ejpam-3284	186	3	,	,	PUNCT
ejpam-3284	186	4	u2	u2	PROPN
ejpam-3284	186	5	∈	∈	PROPN
ejpam-3284	186	6	u	u	NOUN
ejpam-3284	186	7	be	be	VERB
ejpam-3284	186	8	the	the	DET
ejpam-3284	186	9	fixed	fix	VERB
ejpam-3284	186	10	points	point	NOUN
ejpam-3284	186	11	of	of	ADP
ejpam-3284	186	12	t	t	PROPN
ejpam-3284	186	13	,	,	PUNCT
ejpam-3284	186	14	then	then	ADV
ejpam-3284	186	15	‖t	‖t	NOUN
ejpam-3284	186	16	(	(	PUNCT
ejpam-3284	186	17	u1)−	u1)−	PROPN
ejpam-3284	186	18	t	t	PROPN
ejpam-3284	186	19	(	(	PUNCT
ejpam-3284	186	20	u2)‖pn	u2)‖pn	PROPN
ejpam-3284	186	21	=	=	SYM
ejpam-3284	186	22	‖u1	‖u1	NOUN
ejpam-3284	186	23	−	−	PROPN
ejpam-3284	186	24	u2‖pn	u2‖pn	PROPN
ejpam-3284	186	25	.	.	PUNCT
ejpam-3284	187	1	(	(	PUNCT
ejpam-3284	187	2	2	2	X
ejpam-3284	187	3	)	)	PUNCT
ejpam-3284	187	4	also	also	ADV
ejpam-3284	187	5	,	,	PUNCT
ejpam-3284	187	6	we	we	PRON
ejpam-3284	187	7	see	see	VERB
ejpam-3284	187	8	that	that	PRON
ejpam-3284	187	9	:	:	PUNCT
ejpam-3284	187	10	‖t	‖t	NOUN
ejpam-3284	187	11	(	(	PUNCT
ejpam-3284	187	12	u1)−	u1)−	PROPN
ejpam-3284	187	13	t	t	PROPN
ejpam-3284	187	14	(	(	PUNCT
ejpam-3284	187	15	u2)‖pn	u2)‖pn	PROPN
ejpam-3284	187	16	≤	≤	PROPN
ejpam-3284	187	17	λ‖u1	λ‖u1	PROPN
ejpam-3284	187	18	−	−	PROPN
ejpam-3284	187	19	u2‖pn	u2‖pn	PROPN
ejpam-3284	187	20	,	,	PUNCT
ejpam-3284	187	21	∀	∀	PUNCT
ejpam-3284	187	22	λ	λ	X
ejpam-3284	187	23	<	<	X
ejpam-3284	187	24	1	1	NUM
ejpam-3284	187	25	(	(	PUNCT
ejpam-3284	187	26	3	3	NUM
ejpam-3284	187	27	)	)	PUNCT
ejpam-3284	187	28	substituting	substitute	VERB
ejpam-3284	187	29	equation	equation	NOUN
ejpam-3284	187	30	(	(	PUNCT
ejpam-3284	187	31	2	2	NUM
ejpam-3284	187	32	)	)	PUNCT
ejpam-3284	187	33	into	into	ADP
ejpam-3284	187	34	inequality	inequality	NOUN
ejpam-3284	187	35	(	(	PUNCT
ejpam-3284	187	36	3	3	NUM
ejpam-3284	187	37	)	)	PUNCT
ejpam-3284	187	38	yields	yield	NOUN
ejpam-3284	187	39	(	(	PUNCT
ejpam-3284	187	40	1−	1−	NUM
ejpam-3284	187	41	λ	λ	PROPN
ejpam-3284	187	42	)	)	PUNCT
ejpam-3284	187	43	‖u1	‖u1	ADV
ejpam-3284	187	44	−	−	PROPN
ejpam-3284	187	45	u2‖pn	u2‖pn	VERB
ejpam-3284	187	46	≤	≤	NOUN
ejpam-3284	187	47	0	0	NUM
ejpam-3284	187	48	⇒	⇒	NOUN
ejpam-3284	187	49	‖u1	‖u1	ADV
ejpam-3284	187	50	−	−	PROPN
ejpam-3284	187	51	u2‖pn	u2‖pn	PROPN
ejpam-3284	188	1	=	=	NOUN
ejpam-3284	188	2	0	0	PROPN
ejpam-3284	188	3	.	.	PUNCT
ejpam-3284	189	1	the	the	DET
ejpam-3284	189	2	above	above	ADJ
ejpam-3284	189	3	equation	equation	NOUN
ejpam-3284	189	4	holds	hold	VERB
ejpam-3284	189	5	if	if	SCONJ
ejpam-3284	189	6	u1	u1	NOUN
ejpam-3284	189	7	=	=	SYM
ejpam-3284	189	8	u2	u2	PROPN
ejpam-3284	189	9	.	.	PUNCT
ejpam-3284	190	1	hence	hence	ADV
ejpam-3284	190	2	,	,	PUNCT
ejpam-3284	190	3	the	the	DET
ejpam-3284	190	4	fixed	fixed	ADJ
ejpam-3284	190	5	point	point	NOUN
ejpam-3284	190	6	x	x	VERB
ejpam-3284	190	7	is	be	AUX
ejpam-3284	190	8	unique	unique	ADJ
ejpam-3284	190	9	in	in	ADP
ejpam-3284	190	10	the	the	DET
ejpam-3284	190	11	productnormed	productnorme	VERB
ejpam-3284	190	12	linear	linear	PROPN
ejpam-3284	190	13	space	space	NOUN
ejpam-3284	190	14	.	.	PUNCT
ejpam-3284	191	1	another	another	DET
ejpam-3284	191	2	normed	normed	PROPN
ejpam-3284	191	3	linear	linear	ADJ
ejpam-3284	191	4	space	space	NOUN
ejpam-3284	191	5	is	be	AUX
ejpam-3284	191	6	introduced	introduce	VERB
ejpam-3284	191	7	as	as	ADP
ejpam-3284	191	8	:	:	PUNCT
ejpam-3284	191	9	theorem	theorem	ADJ
ejpam-3284	191	10	7	7	NUM
ejpam-3284	191	11	(	(	PUNCT
ejpam-3284	191	12	product	product	NOUN
ejpam-3284	191	13	semi	semi	ADJ
ejpam-3284	191	14	-	-	ADJ
ejpam-3284	191	15	normed	normed	ADJ
ejpam-3284	191	16	linear	linear	ADJ
ejpam-3284	191	17	space	space	NOUN
ejpam-3284	191	18	)	)	PUNCT
ejpam-3284	191	19	.	.	PUNCT
ejpam-3284	192	1	let	let	VERB
ejpam-3284	192	2	u	u	PRON
ejpam-3284	192	3	be	be	AUX
ejpam-3284	192	4	a	a	DET
ejpam-3284	192	5	linear	linear	ADJ
ejpam-3284	192	6	space	space	NOUN
ejpam-3284	192	7	over	over	ADP
ejpam-3284	192	8	[	[	X
ejpam-3284	192	9	2,∞	2,∞	NUM
ejpam-3284	192	10	]	]	PUNCT
ejpam-3284	192	11	⊆	⊆	NUM
ejpam-3284	192	12	r.	r.	NOUN
ejpam-3284	192	13	a	a	DET
ejpam-3284	192	14	product	product	NOUN
ejpam-3284	192	15	norm	norm	NOUN
ejpam-3284	192	16	on	on	ADP
ejpam-3284	192	17	u	u	PROPN
ejpam-3284	192	18	is	be	AUX
ejpam-3284	192	19	a	a	DET
ejpam-3284	192	20	real	real	ADV
ejpam-3284	192	21	-	-	PUNCT
ejpam-3284	192	22	valued	value	VERB
ejpam-3284	192	23	function	function	NOUN
ejpam-3284	192	24	‖	‖	PROPN
ejpam-3284	192	25	·	·	PUNCT
ejpam-3284	192	26	‖	‖	PROPN
ejpam-3284	192	27	,	,	PUNCT
ejpam-3284	192	28	‖	‖	PROPN
ejpam-3284	192	29	·	·	PUNCT
ejpam-3284	192	30	‖	‖	ADJ
ejpam-3284	192	31	:	:	PUNCT
ejpam-3284	192	32	u	u	X
ejpam-3284	192	33	→	→	SYM
ejpam-3284	192	34	[	[	X
ejpam-3284	192	35	0,∞	0,∞	NOUN
ejpam-3284	192	36	)	)	PUNCT
ejpam-3284	192	37	,	,	PUNCT
ejpam-3284	192	38	such	such	ADJ
ejpam-3284	192	39	that	that	SCONJ
ejpam-3284	192	40	for	for	ADP
ejpam-3284	192	41	arbitrary	arbitrary	ADJ
ejpam-3284	192	42	u	u	NOUN
ejpam-3284	192	43	,	,	PUNCT
ejpam-3284	192	44	v	v	PROPN
ejpam-3284	192	45	∈	∈	PROPN
ejpam-3284	192	46	u	u	NOUN
ejpam-3284	192	47	,	,	PUNCT
ejpam-3284	192	48	α	α	PROPN
ejpam-3284	192	49	∈	∈	PROPN
ejpam-3284	193	1	[	[	X
ejpam-3284	193	2	0	0	NUM
ejpam-3284	193	3	,	,	PUNCT
ejpam-3284	193	4	2	2	NUM
ejpam-3284	193	5	]	]	PUNCT
ejpam-3284	193	6	,	,	PUNCT
ejpam-3284	193	7	the	the	DET
ejpam-3284	193	8	following	follow	VERB
ejpam-3284	193	9	conditions	condition	NOUN
ejpam-3284	193	10	are	be	AUX
ejpam-3284	193	11	satisfied	satisfied	ADJ
ejpam-3284	193	12	:	:	PUNCT
ejpam-3284	193	13	1	1	X
ejpam-3284	193	14	.	.	X
ejpam-3284	194	1	‖u‖	‖u‖	PROPN
ejpam-3284	194	2	>	>	X
ejpam-3284	194	3	0	0	NUM
ejpam-3284	194	4	,	,	PUNCT
ejpam-3284	194	5	and	and	CCONJ
ejpam-3284	194	6	‖u‖	‖u‖	PROPN
ejpam-3284	194	7	=	=	PUNCT
ejpam-3284	194	8	0	0	NUM
ejpam-3284	194	9	,	,	PUNCT
ejpam-3284	194	10	if	if	SCONJ
ejpam-3284	194	11	and	and	CCONJ
ejpam-3284	194	12	only	only	ADV
ejpam-3284	194	13	if	if	SCONJ
ejpam-3284	194	14	u	u	NOUN
ejpam-3284	194	15	=	=	NOUN
ejpam-3284	194	16	0	0	NUM
ejpam-3284	194	17	2	2	NUM
ejpam-3284	194	18	.	.	PUNCT
ejpam-3284	195	1	‖αu‖	‖αu‖	ADJ
ejpam-3284	195	2	=	=	SYM
ejpam-3284	195	3	|α|‖u‖	|α|‖u‖	NOUN
ejpam-3284	195	4	,	,	PUNCT
ejpam-3284	195	5	α	α	NOUN
ejpam-3284	195	6	∈	∈	PROPN
ejpam-3284	196	1	[	[	X
ejpam-3284	196	2	0	0	NUM
ejpam-3284	196	3	,	,	PUNCT
ejpam-3284	196	4	2	2	NUM
ejpam-3284	196	5	]	]	PUNCT
ejpam-3284	196	6	,	,	PUNCT
ejpam-3284	196	7	and	and	CCONJ
ejpam-3284	196	8	u	u	PROPN
ejpam-3284	196	9	∈	∈	PROPN
ejpam-3284	196	10	u	u	NOUN
ejpam-3284	196	11	3	3	NUM
ejpam-3284	196	12	.	.	PUNCT
ejpam-3284	196	13	‖u+	‖u+	NOUN
ejpam-3284	196	14	v‖	v‖	NOUN
ejpam-3284	196	15	≤	≤	NOUN
ejpam-3284	196	16	‖u‖+	‖u‖+	PRON
ejpam-3284	196	17	‖v‖	‖v‖	PROPN
ejpam-3284	196	18	,	,	PUNCT
ejpam-3284	196	19	∀u	∀u	NOUN
ejpam-3284	196	20	,	,	PUNCT
ejpam-3284	196	21	v	v	X
ejpam-3284	196	22	∈	∈	PROPN
ejpam-3284	196	23	u	u	NOUN
ejpam-3284	196	24	4	4	NUM
ejpam-3284	196	25	.	.	PUNCT
ejpam-3284	197	1	‖u‖+	‖u‖+	PRON
ejpam-3284	197	2	‖v‖	‖v‖	PROPN
ejpam-3284	197	3	≤	≤	NUM
ejpam-3284	197	4	‖u‖‖v‖	‖u‖‖v‖	ADJ
ejpam-3284	197	5	∀u	∀u	NOUN
ejpam-3284	197	6	,	,	PUNCT
ejpam-3284	197	7	v	v	ADP
ejpam-3284	197	8	∈	∈	PROPN
ejpam-3284	197	9	v	v	NOUN
ejpam-3284	197	10	(	(	PUNCT
ejpam-3284	197	11	product	product	NOUN
ejpam-3284	197	12	inequality	inequality	NOUN
ejpam-3284	197	13	)	)	PUNCT
ejpam-3284	197	14	.	.	PUNCT
ejpam-3284	198	1	4	4	X
ejpam-3284	198	2	.	.	X
ejpam-3284	198	3	conclusion	conclusion	NOUN
ejpam-3284	198	4	in	in	ADP
ejpam-3284	198	5	a	a	DET
ejpam-3284	198	6	nutshell	nutshell	NOUN
ejpam-3284	198	7	,	,	PUNCT
ejpam-3284	198	8	we	we	PRON
ejpam-3284	198	9	have	have	AUX
ejpam-3284	198	10	introduced	introduce	VERB
ejpam-3284	198	11	the	the	DET
ejpam-3284	198	12	product	product	NOUN
ejpam-3284	198	13	-	-	PUNCT
ejpam-3284	198	14	normed	norme	VERB
ejpam-3284	198	15	linear	linear	ADJ
ejpam-3284	198	16	space	space	NOUN
ejpam-3284	198	17	and	and	CCONJ
ejpam-3284	198	18	product	product	NOUN
ejpam-3284	198	19	-	-	PUNCT
ejpam-3284	198	20	seminormed	seminorme	VERB
ejpam-3284	198	21	linear	linear	ADJ
ejpam-3284	198	22	space	space	NOUN
ejpam-3284	198	23	(	(	PUNCT
ejpam-3284	198	24	product	product	NOUN
ejpam-3284	198	25	-	-	PUNCT
ejpam-3284	198	26	semi	semi	ADJ
ejpam-3284	198	27	-	-	ADJ
ejpam-3284	198	28	banch	banch	ADJ
ejpam-3284	198	29	space	space	NOUN
ejpam-3284	198	30	)	)	PUNCT
ejpam-3284	198	31	which	which	PRON
ejpam-3284	198	32	are	be	AUX
ejpam-3284	198	33	endowed	endow	VERB
ejpam-3284	198	34	with	with	ADP
ejpam-3284	198	35	some	some	DET
ejpam-3284	198	36	functional	functional	ADJ
ejpam-3284	198	37	space	space	NOUN
ejpam-3284	198	38	.	.	PUNCT
ejpam-3284	199	1	in	in	ADP
ejpam-3284	199	2	addition	addition	NOUN
ejpam-3284	199	3	,	,	PUNCT
ejpam-3284	199	4	p	p	NOUN
ejpam-3284	199	5	−nls	−nls	NOUN
ejpam-3284	199	6	is	be	AUX
ejpam-3284	199	7	endowed	endow	VERB
ejpam-3284	199	8	with	with	ADP
ejpam-3284	199	9	fixed	fix	VERB
ejpam-3284	199	10	point	point	NOUN
ejpam-3284	199	11	.	.	PUNCT
ejpam-3284	200	1	references	reference	NOUN
ejpam-3284	200	2	749	749	NUM
ejpam-3284	200	3	references	reference	NOUN
ejpam-3284	200	4	[	[	X
ejpam-3284	200	5	1	1	NUM
ejpam-3284	200	6	]	]	PUNCT
ejpam-3284	200	7	i.	i.	PROPN
ejpam-3284	200	8	k.	k.	PROPN
ejpam-3284	200	9	menger	menger	PROPN
ejpam-3284	200	10	,	,	PUNCT
ejpam-3284	200	11	untersuchungen	untersuchungen	PROPN
ejpam-3284	200	12	veber	veber	PROPN
ejpam-3284	200	13	allgeine	allgeine	PROPN
ejpam-3284	200	14	matrik	matrik	PROPN
ejpam-3284	200	15	,	,	PUNCT
ejpam-3284	200	16	mathematische	mathematische	NOUN
ejpam-3284	200	17	annalen	annalen	PROPN
ejpam-3284	200	18	,	,	PUNCT
ejpam-3284	200	19	vol	vol	NOUN
ejpam-3284	200	20	.	.	PROPN
ejpam-3284	200	21	100	100	NUM
ejpam-3284	200	22	,	,	PUNCT
ejpam-3284	200	23	(	(	PUNCT
ejpam-3284	200	24	1928	1928	NUM
ejpam-3284	200	25	)	)	PUNCT
ejpam-3284	200	26	.	.	PUNCT
ejpam-3284	201	1	[	[	X
ejpam-3284	201	2	2	2	X
ejpam-3284	201	3	]	]	PUNCT
ejpam-3284	201	4	v.	v.	CCONJ
ejpam-3284	201	5	s.	s.	PROPN
ejpam-3284	201	6	gähler	gähler	PROPN
ejpam-3284	201	7	,	,	PUNCT
ejpam-3284	201	8	lineare	lineare	ADJ
ejpam-3284	201	9	2−normierte	2−normierte	NUM
ejpam-3284	201	10	räume	räume	PROPN
ejpam-3284	201	11	diese	diese	PROPN
ejpam-3284	201	12	nachr	nachr	PROPN
ejpam-3284	201	13	,	,	PUNCT
ejpam-3284	201	14	28	28	NUM
ejpam-3284	201	15	,	,	PUNCT
ejpam-3284	201	16	1	1	NUM
ejpam-3284	201	17	-	-	SYM
ejpam-3284	201	18	2	2	NUM
ejpam-3284	201	19	,	,	PUNCT
ejpam-3284	201	20	(	(	PUNCT
ejpam-3284	201	21	1963).pp.1	1963).pp.1	NUM
ejpam-3284	201	22	43	43	NUM
ejpam-3284	202	1	[	[	X
ejpam-3284	202	2	3	3	X
ejpam-3284	202	3	]	]	PUNCT
ejpam-3284	202	4	j.	j.	PROPN
ejpam-3284	202	5	t.	t.	PROPN
ejpam-3284	202	6	scheick	scheick	PROPN
ejpam-3284	202	7	,	,	PUNCT
ejpam-3284	202	8	linear	linear	ADJ
ejpam-3284	202	9	algebra	algebra	NOUN
ejpam-3284	202	10	with	with	ADP
ejpam-3284	202	11	applications	application	NOUN
ejpam-3284	202	12	,	,	PUNCT
ejpam-3284	202	13	prentice	prentice	NOUN
ejpam-3284	202	14	-	-	PUNCT
ejpam-3284	202	15	hall	hall	NOUN
ejpam-3284	202	16	,	,	PUNCT
ejpam-3284	202	17	inc	inc	PROPN
ejpam-3284	202	18	,	,	PUNCT
ejpam-3284	202	19	new	new	PROPN
ejpam-3284	202	20	york	york	PROPN
ejpam-3284	202	21	;	;	PUNCT
ejpam-3284	202	22	1997	1997	NUM
ejpam-3284	202	23	.	.	PUNCT
ejpam-3284	203	1	[	[	X
ejpam-3284	203	2	4	4	NUM
ejpam-3284	203	3	]	]	PUNCT
ejpam-3284	203	4	a.	a.	NOUN
ejpam-3284	203	5	branciari	branciari	PROPN
ejpam-3284	203	6	,	,	PUNCT
ejpam-3284	203	7	a	a	DET
ejpam-3284	203	8	fixed	fix	VERB
ejpam-3284	203	9	point	point	NOUN
ejpam-3284	203	10	theorem	theorem	NOUN
ejpam-3284	203	11	of	of	ADP
ejpam-3284	203	12	banach	banach	NOUN
ejpam-3284	203	13	-	-	PUNCT
ejpam-3284	203	14	caccippoli	caccippoli	NOUN
ejpam-3284	203	15	type	type	NOUN
ejpam-3284	203	16	on	on	ADP
ejpam-3284	203	17	a	a	DET
ejpam-3284	203	18	class	class	NOUN
ejpam-3284	203	19	of	of	ADP
ejpam-3284	203	20	generalized	generalized	ADJ
ejpam-3284	203	21	metric	metric	ADJ
ejpam-3284	203	22	spaces	space	NOUN
ejpam-3284	203	23	,	,	PUNCT
ejpam-3284	203	24	publ	publ	PROPN
ejpam-3284	203	25	.	.	PUNCT
ejpam-3284	203	26	math	math	NOUN
ejpam-3284	203	27	.	.	PUNCT
ejpam-3284	204	1	debrecen	debrecen	PROPN
ejpam-3284	204	2	57	57	NUM
ejpam-3284	204	3	(	(	PUNCT
ejpam-3284	204	4	2000	2000	NUM
ejpam-3284	204	5	)	)	PUNCT
ejpam-3284	204	6	.	.	PUNCT
ejpam-3284	205	1	pp	pp	ADV
ejpam-3284	205	2	31	31	NUM
ejpam-3284	205	3	-	-	SYM
ejpam-3284	205	4	37	37	NUM
ejpam-3284	205	5	.	.	PUNCT
ejpam-3284	206	1	[	[	X
ejpam-3284	206	2	5	5	NUM
ejpam-3284	206	3	]	]	PUNCT
ejpam-3284	206	4	m.	m.	NOUN
ejpam-3284	206	5	pavel	pavel	PROPN
ejpam-3284	206	6	,	,	PUNCT
ejpam-3284	206	7	on	on	ADP
ejpam-3284	206	8	quasi	quasi	NOUN
ejpam-3284	206	9	normed	normed	PROPN
ejpam-3284	206	10	spaces	space	VERB
ejpam-3284	206	11	bull	bull	NOUN
ejpam-3284	206	12	.	.	PUNCT
ejpam-3284	207	1	acad	acad	PROPN
ejpam-3284	207	2	.	.	PUNCT
ejpam-3284	208	1	polon	polon	PROPN
ejpam-3284	208	2	sci	sci	PROPN
ejpam-3284	208	3	.	.	PROPN
ejpam-3284	208	4	,	,	PUNCT
ejpam-3284	208	5	ci	ci	PROPN
ejpam-3284	208	6	iii	iii	PROPN
ejpam-3284	208	7	,	,	PUNCT
ejpam-3284	208	8	5	5	NUM
ejpam-3284	208	9	,	,	PUNCT
ejpam-3284	208	10	no	no	INTJ
ejpam-3284	208	11	.	.	NOUN
ejpam-3284	208	12	5	5	NUM
ejpam-3284	208	13	,	,	PUNCT
ejpam-3284	208	14	(	(	PUNCT
ejpam-3284	208	15	1957).pp	1957).pp	X
ejpam-3284	208	16	.	.	PUNCT
ejpam-3284	208	17	479	479	NUM
ejpam-3284	208	18	-	-	NUM
ejpam-3284	208	19	487	487	NUM
ejpam-3284	208	20	[	[	X
ejpam-3284	208	21	6	6	NUM
ejpam-3284	208	22	]	]	PUNCT
ejpam-3284	208	23	s.	s.	PROPN
ejpam-3284	208	24	rolewiez	rolewiez	PROPN
ejpam-3284	208	25	,	,	PUNCT
ejpam-3284	208	26	on	on	ADP
ejpam-3284	208	27	a	a	DET
ejpam-3284	208	28	certain	certain	ADJ
ejpam-3284	208	29	class	class	NOUN
ejpam-3284	208	30	of	of	ADP
ejpam-3284	208	31	linear	linear	PROPN
ejpam-3284	208	32	metric	metric	ADJ
ejpam-3284	208	33	space	space	NOUN
ejpam-3284	208	34	bull	bull	NOUN
ejpam-3284	208	35	.	.	PUNCT
ejpam-3284	209	1	acad	acad	PROPN
ejpam-3284	209	2	.	.	PUNCT
ejpam-3284	210	1	polon	polon	PROPN
ejpam-3284	210	2	sci	sci	PROPN
ejpam-3284	210	3	.	.	PROPN
ejpam-3284	210	4	,	,	PUNCT
ejpam-3284	210	5	ci	ci	PROPN
ejpam-3284	210	6	iii	iii	PROPN
ejpam-3284	210	7	,	,	PUNCT
ejpam-3284	210	8	5	5	NUM
ejpam-3284	210	9	,	,	PUNCT
ejpam-3284	210	10	no	no	INTJ
ejpam-3284	210	11	.	.	NOUN
ejpam-3284	210	12	5	5	NUM
ejpam-3284	210	13	,	,	PUNCT
ejpam-3284	210	14	(	(	PUNCT
ejpam-3284	210	15	1957).pp	1957).pp	X
ejpam-3284	210	16	.	.	PUNCT
ejpam-3284	210	17	471	471	NUM
ejpam-3284	210	18	-	-	SYM
ejpam-3284	210	19	473	473	NUM
ejpam-3284	210	20	[	[	X
ejpam-3284	210	21	7	7	NUM
ejpam-3284	210	22	]	]	PUNCT
ejpam-3284	210	23	w.	w.	PROPN
ejpam-3284	210	24	j.	j.	PROPN
ejpam-3284	210	25	davis	davis	PROPN
ejpam-3284	210	26	,	,	PUNCT
ejpam-3284	210	27	d.	d.	PROPN
ejpam-3284	210	28	j.	j.	PROPN
ejpam-3284	210	29	h.	h.	PROPN
ejpam-3284	210	30	garling	garling	PROPN
ejpam-3284	210	31	and	and	CCONJ
ejpam-3284	210	32	n.	n.	PROPN
ejpam-3284	210	33	tomczak	tomczak	NOUN
ejpam-3284	210	34	-	-	PUNCT
ejpam-3284	210	35	jaegermann	jaegermann	PROPN
ejpam-3284	210	36	,	,	PUNCT
ejpam-3284	210	37	the	the	DET
ejpam-3284	210	38	complex	complex	NOUN
ejpam-3284	210	39	of	of	ADP
ejpam-3284	210	40	quasinormed	quasinorme	VERB
ejpam-3284	210	41	linear	linear	PROPN
ejpam-3284	210	42	spaces	space	NOUN
ejpam-3284	210	43	journal	journal	NOUN
ejpam-3284	210	44	of	of	ADP
ejpam-3284	210	45	functional	functional	ADJ
ejpam-3284	210	46	analysis	analysis	NOUN
ejpam-3284	210	47	,	,	PUNCT
ejpam-3284	210	48	55	55	NUM
ejpam-3284	210	49	,	,	PUNCT
ejpam-3284	210	50	(	(	PUNCT
ejpam-3284	210	51	1984	1984	NUM
ejpam-3284	210	52	)	)	PUNCT
ejpam-3284	210	53	.	.	PUNCT
ejpam-3284	211	1	pp	pp	ADJ
ejpam-3284	211	2	.	.	PUNCT
ejpam-3284	212	1	110	110	NUM
ejpam-3284	212	2	-	-	SYM
ejpam-3284	212	3	150	150	NUM
ejpam-3284	212	4	.	.	PUNCT
ejpam-3284	213	1	[	[	X
ejpam-3284	213	2	8	8	NUM
ejpam-3284	213	3	]	]	PUNCT
ejpam-3284	213	4	s.	s.	PROPN
ejpam-3284	213	5	rolewicz	rolewicz	PROPN
ejpam-3284	213	6	,	,	PUNCT
ejpam-3284	213	7	metric	metric	ADJ
ejpam-3284	213	8	linear	linear	ADJ
ejpam-3284	213	9	spaces	space	NOUN
ejpam-3284	213	10	,	,	PUNCT
ejpam-3284	213	11	pmn	pmn	INTJ
ejpam-3284	213	12	,	,	PUNCT
ejpam-3284	213	13	warsaw(1972	warsaw(1972	PROPN
ejpam-3284	213	14	)	)	PUNCT
ejpam-3284	213	15	.	.	PUNCT
ejpam-3284	214	1	[	[	X
ejpam-3284	214	2	9	9	NUM
ejpam-3284	214	3	]	]	X
ejpam-3284	214	4	g.	g.	PROPN
ejpam-3284	214	5	rano	rano	PROPN
ejpam-3284	214	6	and	and	CCONJ
ejpam-3284	214	7	t.	t.	PROPN
ejpam-3284	214	8	bag	bag	NOUN
ejpam-3284	214	9	,	,	PUNCT
ejpam-3284	214	10	bounded	bound	VERB
ejpam-3284	214	11	linear	linear	PROPN
ejpam-3284	214	12	operators	operator	NOUN
ejpam-3284	214	13	in	in	ADP
ejpam-3284	214	14	quasi	quasi	ADJ
ejpam-3284	214	15	-	-	ADJ
ejpam-3284	214	16	normed	normed	ADJ
ejpam-3284	214	17	linear	linear	ADJ
ejpam-3284	214	18	space	space	NOUN
ejpam-3284	214	19	,	,	PUNCT
ejpam-3284	214	20	journal	journal	NOUN
ejpam-3284	214	21	of	of	ADP
ejpam-3284	214	22	the	the	DET
ejpam-3284	214	23	egyptian	egyptian	PROPN
ejpam-3284	214	24	mathematical	mathematical	PROPN
ejpam-3284	214	25	society	society	NOUN
ejpam-3284	214	26	,	,	PUNCT
ejpam-3284	214	27	23	23	NUM
ejpam-3284	214	28	,	,	PUNCT
ejpam-3284	214	29	(	(	PUNCT
ejpam-3284	214	30	2015).pp	2015).pp	NUM
ejpam-3284	214	31	.	.	PUNCT
ejpam-3284	214	32	303	303	NUM
ejpam-3284	214	33	-	-	SYM
ejpam-3284	214	34	308	308	NUM
ejpam-3284	214	35	.	.	PUNCT
ejpam-3284	215	1	[	[	X
ejpam-3284	215	2	10	10	NUM
ejpam-3284	215	3	]	]	X
ejpam-3284	215	4	c.	c.	PROPN
ejpam-3284	215	5	park	park	PROPN
ejpam-3284	215	6	,	,	PUNCT
ejpam-3284	215	7	generalized	generalize	VERB
ejpam-3284	215	8	quasi	quasi	ADJ
ejpam-3284	215	9	-	-	ADJ
ejpam-3284	215	10	banach	banach	ADJ
ejpam-3284	215	11	spaces	space	NOUN
ejpam-3284	215	12	and	and	CCONJ
ejpam-3284	215	13	quasi-(2	quasi-(2	ADV
ejpam-3284	215	14	,	,	PUNCT
ejpam-3284	215	15	p	p	NOUN
ejpam-3284	215	16	)	)	PUNCT
ejpam-3284	215	17	normed	normed	ADJ
ejpam-3284	215	18	space	space	NOUN
ejpam-3284	215	19	,	,	PUNCT
ejpam-3284	215	20	journal	journal	NOUN
ejpam-3284	215	21	of	of	ADP
ejpam-3284	215	22	the	the	DET
ejpam-3284	215	23	chungcheong	chungcheong	PROPN
ejpam-3284	215	24	matematical	matematical	ADJ
ejpam-3284	215	25	society	society	NOUN
ejpam-3284	215	26	,	,	PUNCT
ejpam-3284	215	27	19	19	NUM
ejpam-3284	215	28	,	,	PUNCT
ejpam-3284	215	29	no	no	INTJ
ejpam-3284	215	30	.	.	NOUN
ejpam-3284	215	31	2	2	NUM
ejpam-3284	215	32	(	(	PUNCT
ejpam-3284	215	33	2006	2006	NUM
ejpam-3284	215	34	)	)	PUNCT
ejpam-3284	215	35	.	.	PUNCT
ejpam-3284	216	1	[	[	X
ejpam-3284	216	2	11	11	NUM
ejpam-3284	216	3	]	]	PUNCT
ejpam-3284	216	4	m.	m.	NOUN
ejpam-3284	216	5	kir	kir	PROPN
ejpam-3284	216	6	and	and	CCONJ
ejpam-3284	216	7	m.	m.	NOUN
ejpam-3284	216	8	acikgoz	acikgoz	PROPN
ejpam-3284	216	9	,	,	PUNCT
ejpam-3284	216	10	a	a	DET
ejpam-3284	216	11	study	study	NOUN
ejpam-3284	216	12	involving	involve	VERB
ejpam-3284	216	13	the	the	DET
ejpam-3284	216	14	completion	completion	NOUN
ejpam-3284	216	15	of	of	ADP
ejpam-3284	216	16	a	a	DET
ejpam-3284	216	17	quasi-2	quasi-2	NUM
ejpam-3284	216	18	-	-	PUNCT
ejpam-3284	216	19	normed	normed	ADJ
ejpam-3284	216	20	space	space	NOUN
ejpam-3284	216	21	,	,	PUNCT
ejpam-3284	216	22	international	international	ADJ
ejpam-3284	216	23	journal	journal	NOUN
ejpam-3284	216	24	of	of	ADP
ejpam-3284	216	25	analysis	analysis	NOUN
ejpam-3284	216	26	,	,	PUNCT
ejpam-3284	216	27	(	(	PUNCT
ejpam-3284	216	28	2013	2013	NUM
ejpam-3284	216	29	)	)	PUNCT
ejpam-3284	216	30	.	.	PUNCT
ejpam-3284	217	1	http	http	PROPN
ejpam-3284	217	2	:	:	PUNCT
ejpam-3284	217	3	//dx.doi.org/10.1155/2013/512372	//dx.doi.org/10.1155/2013/512372	PUNCT
ejpam-3284	217	4	.	.	PUNCT
ejpam-3284	218	1	[	[	X
ejpam-3284	218	2	12	12	NUM
ejpam-3284	218	3	]	]	PUNCT
ejpam-3284	218	4	m.	m.	NOUN
ejpam-3284	218	5	mastylo	mastylo	NOUN
ejpam-3284	218	6	,	,	PUNCT
ejpam-3284	218	7	on	on	ADP
ejpam-3284	218	8	interpolation	interpolation	NOUN
ejpam-3284	218	9	of	of	ADP
ejpam-3284	218	10	some	some	DET
ejpam-3284	218	11	quasi	quasi	ADJ
ejpam-3284	218	12	-	-	ADJ
ejpam-3284	218	13	banach	banach	ADJ
ejpam-3284	218	14	spaces	space	NOUN
ejpam-3284	218	15	,	,	PUNCT
ejpam-3284	218	16	journal	journal	NOUN
ejpam-3284	218	17	of	of	ADP
ejpam-3284	218	18	mathematical	mathematical	ADJ
ejpam-3284	218	19	analysis	analysis	NOUN
ejpam-3284	218	20	and	and	CCONJ
ejpam-3284	218	21	applications	application	NOUN
ejpam-3284	218	22	,	,	PUNCT
ejpam-3284	218	23	147	147	NUM
ejpam-3284	218	24	,	,	PUNCT
ejpam-3284	218	25	(	(	PUNCT
ejpam-3284	218	26	1990).pp	1990).pp	NOUN
ejpam-3284	218	27	.	.	PUNCT
ejpam-3284	219	1	403	403	NUM
ejpam-3284	219	2	-	-	SYM
ejpam-3284	219	3	419	419	NUM
ejpam-3284	219	4	.	.	PUNCT
ejpam-3284	220	1	[	[	X
ejpam-3284	220	2	13	13	NUM
ejpam-3284	220	3	]	]	X
ejpam-3284	220	4	b.	b.	PROPN
ejpam-3284	220	5	barnes	barnes	PROPN
ejpam-3284	220	6	,	,	PUNCT
ejpam-3284	220	7	e.	e.	PROPN
ejpam-3284	220	8	d.	d.	PROPN
ejpam-3284	220	9	j.	j.	PROPN
ejpam-3284	220	10	owusu	owusu	PROPN
ejpam-3284	220	11	-	-	PUNCT
ejpam-3284	220	12	ansah	ansah	PROPN
ejpam-3284	220	13	,	,	PUNCT
ejpam-3284	220	14	s.	s.	PROPN
ejpam-3284	220	15	k.	k.	PROPN
ejpam-3284	220	16	amponsah	amponsah	PROPN
ejpam-3284	220	17	and	and	CCONJ
ejpam-3284	220	18	c.	c.	PROPN
ejpam-3284	220	19	sebil	sebil	PROPN
ejpam-3284	220	20	,	,	PUNCT
ejpam-3284	220	21	the	the	DET
ejpam-3284	220	22	proofs	proof	NOUN
ejpam-3284	220	23	of	of	ADP
ejpam-3284	220	24	product	product	NOUN
ejpam-3284	220	25	inequalities	inequality	NOUN
ejpam-3284	220	26	in	in	ADP
ejpam-3284	220	27	a	a	DET
ejpam-3284	220	28	generalized	generalized	ADJ
ejpam-3284	220	29	vector	vector	NOUN
ejpam-3284	220	30	spaces	space	NOUN
ejpam-3284	220	31	,	,	PUNCT
ejpam-3284	220	32	european	european	PROPN
ejpam-3284	220	33	journal	journal	PROPN
ejpam-3284	220	34	of	of	ADP
ejpam-3284	220	35	pure	pure	ADJ
ejpam-3284	220	36	and	and	CCONJ
ejpam-3284	220	37	applied	applied	ADJ
ejpam-3284	220	38	mathematics,11	mathematics,11	NOUN
ejpam-3284	220	39	,	,	PUNCT
ejpam-3284	220	40	no	no	INTJ
ejpam-3284	220	41	.	.	PUNCT
ejpam-3284	221	1	2(2018	2(2018	NUM
ejpam-3284	221	2	)	)	PUNCT
ejpam-3284	221	3	.	.	PUNCT
ejpam-3284	222	1	pp	pp	ADV
ejpam-3284	222	2	.	.	PUNCT
ejpam-3284	223	1	375	375	NUM
ejpam-3284	223	2	-	-	SYM
ejpam-3284	223	3	389	389	NUM
ejpam-3284	223	4	.	.	PUNCT
ejpam-3284	224	1	[	[	X
ejpam-3284	224	2	14	14	NUM
ejpam-3284	224	3	]	]	X
ejpam-3284	224	4	b.	b.	PROPN
ejpam-3284	224	5	kolman	kolman	PROPN
ejpam-3284	224	6	and	and	CCONJ
ejpam-3284	224	7	d.	d.	PROPN
ejpam-3284	224	8	r.	r.	PROPN
ejpam-3284	224	9	hill	hill	PROPN
ejpam-3284	224	10	,	,	PUNCT
ejpam-3284	224	11	elementary	elementary	ADJ
ejpam-3284	224	12	linear	linear	PROPN
ejpam-3284	224	13	algebra	algebra	PROPN
ejpam-3284	224	14	,	,	PUNCT
ejpam-3284	224	15	prentice	prentice	NOUN
ejpam-3284	224	16	-	-	PUNCT
ejpam-3284	224	17	hall	hall	NOUN
ejpam-3284	224	18	,	,	PUNCT
ejpam-3284	224	19	inc	inc	PROPN
ejpam-3284	224	20	,	,	PUNCT
ejpam-3284	224	21	new	new	PROPN
ejpam-3284	224	22	jersey	jersey	PROPN
ejpam-3284	224	23	;	;	PUNCT
ejpam-3284	224	24	2000	2000	NUM
ejpam-3284	224	25	.	.	PUNCT
ejpam-3284	225	1	[	[	X
ejpam-3284	225	2	15	15	NUM
ejpam-3284	225	3	]	]	X
ejpam-3284	225	4	h.	h.	NOUN
ejpam-3284	225	5	royden	royden	PROPN
ejpam-3284	225	6	and	and	CCONJ
ejpam-3284	225	7	p.	p.	PROPN
ejpam-3284	225	8	fitxpatrick	fitxpatrick	PROPN
ejpam-3284	225	9	,	,	PUNCT
ejpam-3284	225	10	real	real	ADJ
ejpam-3284	225	11	analysis	analysis	NOUN
ejpam-3284	225	12	.	.	PUNCT
ejpam-3284	226	1	pearson	pearson	PROPN
ejpam-3284	226	2	education	education	PROPN
ejpam-3284	226	3	,	,	PUNCT
ejpam-3284	226	4	inc	inc	PROPN
ejpam-3284	226	5	,	,	PUNCT
ejpam-3284	226	6	4th	4th	ADJ
ejpam-3284	226	7	ed	ed	NOUN
ejpam-3284	226	8	.	.	PROPN
ejpam-3284	226	9	,	,	PUNCT
ejpam-3284	226	10	upper	upper	ADJ
ejpam-3284	226	11	saddle	saddle	NOUN
ejpam-3284	226	12	river	river	NOUN
ejpam-3284	226	13	;	;	PUNCT
ejpam-3284	226	14	2010	2010	NUM
ejpam-3284	226	15	.	.	PUNCT
ejpam-3284	227	1	[	[	X
ejpam-3284	227	2	16	16	NUM
ejpam-3284	227	3	]	]	X
ejpam-3284	227	4	n.	n.	PROPN
ejpam-3284	227	5	dunford	dunford	PROPN
ejpam-3284	227	6	and	and	CCONJ
ejpam-3284	227	7	j.	j.	PROPN
ejpam-3284	227	8	t.	t.	PROPN
ejpam-3284	227	9	schwartz	schwartz	PROPN
ejpam-3284	227	10	,	,	PUNCT
ejpam-3284	227	11	linear	linear	PROPN
ejpam-3284	227	12	operators	operator	NOUN
ejpam-3284	227	13	.	.	PUNCT
ejpam-3284	228	1	vol	vol	NOUN
ejpam-3284	228	2	1	1	NUM
ejpam-3284	228	3	,	,	PUNCT
ejpam-3284	228	4	interscience	interscience	NOUN
ejpam-3284	228	5	,	,	PUNCT
ejpam-3284	228	6	new	new	PROPN
ejpam-3284	228	7	york	york	PROPN
ejpam-3284	228	8	;	;	PUNCT
ejpam-3284	228	9	1958	1958	NUM
ejpam-3284	228	10	.	.	PUNCT
ejpam-3284	229	1	references	reference	NOUN
ejpam-3284	229	2	750	750	NUM
ejpam-3284	229	3	[	[	X
ejpam-3284	229	4	17	17	NUM
ejpam-3284	229	5	]	]	X
ejpam-3284	229	6	b.	b.	PROPN
ejpam-3284	229	7	d.	d.	PROPN
ejpam-3284	229	8	maccluer	maccluer	PROPN
ejpam-3284	229	9	,	,	PUNCT
ejpam-3284	229	10	graduate	graduate	NOUN
ejpam-3284	229	11	texts	text	NOUN
ejpam-3284	229	12	in	in	ADP
ejpam-3284	229	13	mathematics	mathematic	NOUN
ejpam-3284	229	14	:	:	PUNCT
ejpam-3284	229	15	elementary	elementary	ADJ
ejpam-3284	229	16	functional	functional	ADJ
ejpam-3284	229	17	analysis	analysis	NOUN
ejpam-3284	229	18	,	,	PUNCT
ejpam-3284	229	19	springer	springer	NOUN
ejpam-3284	229	20	science	science	NOUN
ejpam-3284	229	21	+	+	CCONJ
ejpam-3284	229	22	business	business	NOUN
ejpam-3284	229	23	media	medium	NOUN
ejpam-3284	229	24	,	,	PUNCT
ejpam-3284	229	25	new	new	PROPN
ejpam-3284	229	26	york	york	PROPN
ejpam-3284	229	27	;	;	PUNCT
ejpam-3284	229	28	2009	2009	NUM
ejpam-3284	229	29	.	.	PUNCT
ejpam-3284	230	1	[	[	X
ejpam-3284	230	2	18	18	NUM
ejpam-3284	230	3	]	]	X
ejpam-3284	230	4	g.	g.	PROPN
ejpam-3284	230	5	bachman	bachman	PROPN
ejpam-3284	230	6	and	and	CCONJ
ejpam-3284	230	7	l.	l.	PROPN
ejpam-3284	230	8	narici	narici	PROPN
ejpam-3284	230	9	,	,	PUNCT
ejpam-3284	230	10	functional	functional	ADJ
ejpam-3284	230	11	analysis	analysis	NOUN
ejpam-3284	230	12	,	,	PUNCT
ejpam-3284	230	13	academic	academic	ADJ
ejpam-3284	230	14	press	press	NOUN
ejpam-3284	230	15	,	,	PUNCT
ejpam-3284	230	16	new	new	PROPN
ejpam-3284	230	17	york	york	PROPN
ejpam-3284	230	18	and	and	CCONJ
ejpam-3284	230	19	london	london	PROPN
ejpam-3284	230	20	;	;	PUNCT
ejpam-3284	230	21	1966	1966	NUM
ejpam-3284	230	22	.	.	PUNCT
ejpam-3284	231	1	[	[	X
ejpam-3284	231	2	19	19	NUM
ejpam-3284	231	3	]	]	X
ejpam-3284	231	4	j.	j.	PROPN
ejpam-3284	231	5	t.	t.	PROPN
ejpam-3284	231	6	oden	oden	PROPN
ejpam-3284	231	7	,	,	PUNCT
ejpam-3284	231	8	applied	apply	VERB
ejpam-3284	231	9	functional	functional	ADJ
ejpam-3284	231	10	analysis	analysis	NOUN
ejpam-3284	231	11	:	:	PUNCT
ejpam-3284	231	12	a	a	DET
ejpam-3284	231	13	first	first	ADJ
ejpam-3284	231	14	course	course	NOUN
ejpam-3284	231	15	for	for	ADP
ejpam-3284	231	16	students	student	NOUN
ejpam-3284	231	17	of	of	ADP
ejpam-3284	231	18	mechanics	mechanic	NOUN
ejpam-3284	231	19	and	and	CCONJ
ejpam-3284	231	20	engineering	engineering	NOUN
ejpam-3284	231	21	science	science	NOUN
ejpam-3284	231	22	,	,	PUNCT
ejpam-3284	231	23	prentice	prentice	NOUN
ejpam-3284	231	24	-	-	PUNCT
ejpam-3284	231	25	hall	hall	NOUN
ejpam-3284	231	26	,	,	PUNCT
ejpam-3284	231	27	inc	inc	PROPN
ejpam-3284	231	28	,	,	PUNCT
ejpam-3284	231	29	new	new	PROPN
ejpam-3284	231	30	jersey	jersey	PROPN
ejpam-3284	231	31	;	;	PUNCT
ejpam-3284	231	32	1979	1979	NUM
ejpam-3284	231	33	.	.	PUNCT