id	sid	tid	token	lemma	pos
ejpam-178	1	1	7_178_dutta.dvi	7_178_dutta.dvi	NUM
ejpam-178	1	2	european	european	ADJ
ejpam-178	1	3	journal	journal	NOUN
ejpam-178	1	4	of	of	ADP
ejpam-178	1	5	pure	pure	ADJ
ejpam-178	1	6	and	and	CCONJ
ejpam-178	1	7	applied	apply	VERB
ejpam-178	1	8	mathematics	mathematic	NOUN
ejpam-178	1	9	vol	vol	NOUN
ejpam-178	1	10	.	.	PROPN
ejpam-178	1	11	2	2	NUM
ejpam-178	1	12	,	,	PUNCT
ejpam-178	1	13	no	no	INTJ
ejpam-178	1	14	.	.	NOUN
ejpam-178	1	15	4	4	NUM
ejpam-178	1	16	,	,	PUNCT
ejpam-178	1	17	2009	2009	NUM
ejpam-178	1	18	,	,	PUNCT
ejpam-178	1	19	(	(	PUNCT
ejpam-178	1	20	554	554	NUM
ejpam-178	1	21	-	-	SYM
ejpam-178	1	22	563	563	NUM
ejpam-178	1	23	)	)	PUNCT
ejpam-178	1	24	issn	issn	PROPN
ejpam-178	1	25	1307	1307	NUM
ejpam-178	1	26	-	-	SYM
ejpam-178	1	27	5543	5543	NUM
ejpam-178	1	28	–	–	PUNCT
ejpam-178	1	29	www.ejpam.com	www.ejpam.com	X
ejpam-178	1	30	on	on	ADP
ejpam-178	1	31	köthe	köthe	ADJ
ejpam-178	1	32	-	-	PUNCT
ejpam-178	1	33	toeplitz	toeplitz	NOUN
ejpam-178	1	34	and	and	CCONJ
ejpam-178	1	35	null	null	ADJ
ejpam-178	1	36	duals	dual	NOUN
ejpam-178	1	37	of	of	ADP
ejpam-178	1	38	some	some	DET
ejpam-178	1	39	difference	difference	NOUN
ejpam-178	1	40	sequence	sequence	NOUN
ejpam-178	1	41	spaces	space	NOUN
ejpam-178	1	42	defined	define	VERB
ejpam-178	1	43	by	by	ADP
ejpam-178	1	44	orlicz	orlicz	ADJ
ejpam-178	1	45	functions	function	NOUN
ejpam-178	1	46	hemen	hemen	PROPN
ejpam-178	1	47	dutta	dutta	PROPN
ejpam-178	1	48	department	department	PROPN
ejpam-178	1	49	of	of	ADP
ejpam-178	1	50	mathematics	mathematics	PROPN
ejpam-178	1	51	,	,	PUNCT
ejpam-178	1	52	a.d.p	a.d.p	PROPN
ejpam-178	1	53	.	.	PROPN
ejpam-178	1	54	college	college	NOUN
ejpam-178	1	55	,	,	PUNCT
ejpam-178	1	56	nagaon-782002	nagaon-782002	NOUN
ejpam-178	1	57	,	,	PUNCT
ejpam-178	1	58	assam	assam	PROPN
ejpam-178	1	59	,	,	PUNCT
ejpam-178	1	60	india	india	PROPN
ejpam-178	1	61	abstract	abstract	NOUN
ejpam-178	1	62	.	.	PUNCT
ejpam-178	2	1	the	the	DET
ejpam-178	2	2	main	main	ADJ
ejpam-178	2	3	aim	aim	NOUN
ejpam-178	2	4	of	of	ADP
ejpam-178	2	5	this	this	DET
ejpam-178	2	6	paper	paper	NOUN
ejpam-178	2	7	is	be	AUX
ejpam-178	2	8	to	to	PART
ejpam-178	2	9	compute	compute	VERB
ejpam-178	2	10	köthe	köthe	ADJ
ejpam-178	2	11	-	-	PUNCT
ejpam-178	2	12	toeplitz	toeplitz	NOUN
ejpam-178	2	13	and	and	CCONJ
ejpam-178	2	14	null	null	ADJ
ejpam-178	2	15	duals	dual	NOUN
ejpam-178	2	16	of	of	ADP
ejpam-178	2	17	some	some	DET
ejpam-178	2	18	difference	difference	NOUN
ejpam-178	2	19	sequence	sequence	NOUN
ejpam-178	2	20	spaces	space	NOUN
ejpam-178	2	21	,	,	PUNCT
ejpam-178	2	22	defined	define	VERB
ejpam-178	2	23	by	by	ADP
ejpam-178	2	24	means	mean	NOUN
ejpam-178	2	25	of	of	ADP
ejpam-178	2	26	a	a	DET
ejpam-178	2	27	fixed	fix	VERB
ejpam-178	2	28	sequence	sequence	NOUN
ejpam-178	2	29	of	of	ADP
ejpam-178	2	30	multiplier	multipli	ADJ
ejpam-178	2	31	and	and	CCONJ
ejpam-178	2	32	by	by	ADP
ejpam-178	2	33	an	an	DET
ejpam-178	2	34	orlicz	orlicz	ADJ
ejpam-178	2	35	function	function	NOUN
ejpam-178	2	36	.	.	PUNCT
ejpam-178	3	1	further	far	ADV
ejpam-178	3	2	the	the	DET
ejpam-178	3	3	coincidence	coincidence	NOUN
ejpam-178	3	4	for	for	SCONJ
ejpam-178	3	5	three	three	NUM
ejpam-178	3	6	pairs	pair	NOUN
ejpam-178	3	7	of	of	ADP
ejpam-178	3	8	analogous	analogous	ADJ
ejpam-178	3	9	spaces	space	NOUN
ejpam-178	3	10	is	be	AUX
ejpam-178	3	11	established	establish	VERB
ejpam-178	3	12	.	.	PUNCT
ejpam-178	4	1	2000	2000	NUM
ejpam-178	4	2	mathematics	mathematic	NOUN
ejpam-178	4	3	subject	subject	NOUN
ejpam-178	4	4	classifications	classification	NOUN
ejpam-178	4	5	:	:	PUNCT
ejpam-178	4	6	40a05	40a05	NUM
ejpam-178	4	7	,	,	PUNCT
ejpam-178	4	8	40c05	40c05	NUM
ejpam-178	4	9	,	,	PUNCT
ejpam-178	4	10	46a45	46a45	NUM
ejpam-178	4	11	.	.	PUNCT
ejpam-178	5	1	key	key	ADJ
ejpam-178	5	2	words	word	NOUN
ejpam-178	5	3	and	and	CCONJ
ejpam-178	5	4	phrases	phrase	NOUN
ejpam-178	5	5	:	:	PUNCT
ejpam-178	5	6	difference	difference	NOUN
ejpam-178	5	7	sequence	sequence	NOUN
ejpam-178	5	8	spaces	space	VERB
ejpam-178	5	9	,	,	PUNCT
ejpam-178	5	10	orlicz	orlicz	PROPN
ejpam-178	5	11	function	function	NOUN
ejpam-178	5	12	,	,	PUNCT
ejpam-178	5	13	köthe	köthe	ADJ
ejpam-178	5	14	-	-	PUNCT
ejpam-178	5	15	toeplitz	toeplitz	NOUN
ejpam-178	5	16	dual	dual	ADJ
ejpam-178	5	17	,	,	PUNCT
ejpam-178	5	18	null	null	ADJ
ejpam-178	5	19	dual	dual	ADJ
ejpam-178	5	20	.	.	PUNCT
ejpam-178	6	1	1	1	X
ejpam-178	6	2	.	.	X
ejpam-178	6	3	introduction	introduction	NOUN
ejpam-178	6	4	and	and	CCONJ
ejpam-178	6	5	preliminaries	preliminary	NOUN
ejpam-178	6	6	throughout	throughout	ADP
ejpam-178	6	7	this	this	DET
ejpam-178	6	8	section	section	NOUN
ejpam-178	6	9	w	w	PROPN
ejpam-178	6	10	,	,	PUNCT
ejpam-178	6	11	ℓ∞	ℓ∞	PROPN
ejpam-178	6	12	,	,	PUNCT
ejpam-178	6	13	ℓ1	ℓ1	NOUN
ejpam-178	6	14	,	,	PUNCT
ejpam-178	6	15	c	c	PROPN
ejpam-178	6	16	and	and	CCONJ
ejpam-178	6	17	c0	c0	PROPN
ejpam-178	6	18	denote	denote	VERB
ejpam-178	6	19	the	the	DET
ejpam-178	6	20	spaces	space	NOUN
ejpam-178	6	21	of	of	ADP
ejpam-178	6	22	all	all	PRON
ejpam-178	6	23	,	,	PUNCT
ejpam-178	6	24	bounded	bound	VERB
ejpam-178	6	25	,	,	PUNCT
ejpam-178	6	26	absolutel	absolutel	VERB
ejpam-178	6	27	y	y	PROPN
ejpam-178	6	28	summable	summable	ADJ
ejpam-178	6	29	,	,	PUNCT
ejpam-178	6	30	conver	conver	NOUN
ejpam-178	6	31	gent	gent	NOUN
ejpam-178	6	32	and	and	CCONJ
ejpam-178	6	33	null	null	ADJ
ejpam-178	6	34	sequences	sequence	NOUN
ejpam-178	6	35	x	x	PUNCT
ejpam-178	6	36	=	=	SYM
ejpam-178	6	37	(	(	PUNCT
ejpam-178	6	38	xk)with	xk)with	PROPN
ejpam-178	6	39	complex	complex	ADJ
ejpam-178	6	40	terms	term	NOUN
ejpam-178	6	41	respectively	respectively	ADV
ejpam-178	6	42	.	.	PUNCT
ejpam-178	7	1	email	email	NOUN
ejpam-178	7	2	address	address	NOUN
ejpam-178	7	3	:	:	PUNCT
ejpam-178	7	4	hemen_dutta08	hemen_dutta08	PROPN
ejpam-178	7	5	�	�	NOUN
ejpam-178	7	6	rediffmail	rediffmail	NOUN
ejpam-178	7	7	.	.	PUNCT
ejpam-178	8	1	om	om	PROPN
ejpam-178	8	2	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-178	9	1	554	554	NUM
ejpam-178	10	1	c	c	NOUN
ejpam-178	10	2	©	©	VERB
ejpam-178	10	3	2009	2009	NUM
ejpam-178	10	4	ejpam	ejpam	NOUN
ejpam-178	10	5	all	all	DET
ejpam-178	10	6	rights	right	NOUN
ejpam-178	10	7	reserved	reserve	VERB
ejpam-178	10	8	.	.	PUNCT
ejpam-178	11	1	h.	h.	PROPN
ejpam-178	11	2	dutta	dutta	PROPN
ejpam-178	11	3	/	/	PUNCT
ejpam-178	11	4	eur	eur	PROPN
ejpam-178	11	5	.	.	PUNCT
ejpam-178	12	1	j.	j.	PROPN
ejpam-178	12	2	pure	pure	PROPN
ejpam-178	12	3	appl	appl	PROPN
ejpam-178	12	4	.	.	PROPN
ejpam-178	12	5	math	math	PROPN
ejpam-178	12	6	,	,	PUNCT
ejpam-178	12	7	2	2	NUM
ejpam-178	12	8	(	(	PUNCT
ejpam-178	12	9	2009	2009	NUM
ejpam-178	12	10	)	)	PUNCT
ejpam-178	12	11	,	,	PUNCT
ejpam-178	12	12	(	(	PUNCT
ejpam-178	12	13	554	554	NUM
ejpam-178	12	14	-	-	SYM
ejpam-178	12	15	563	563	NUM
ejpam-178	12	16	)	)	PUNCT
ejpam-178	12	17	555	555	NUM
ejpam-178	12	18	an	an	DET
ejpam-178	12	19	orlicz	orlicz	ADJ
ejpam-178	12	20	function	function	NOUN
ejpam-178	12	21	is	be	AUX
ejpam-178	12	22	a	a	DET
ejpam-178	12	23	function	function	NOUN
ejpam-178	12	24	m	m	NOUN
ejpam-178	12	25	:	:	PUNCT
ejpam-178	13	1	[	[	X
ejpam-178	13	2	0,∞	0,∞	X
ejpam-178	13	3	)	)	PUNCT
ejpam-178	13	4	−→	−→	NOUN
ejpam-178	13	5	[	[	X
ejpam-178	13	6	0,∞	0,∞	NOUN
ejpam-178	13	7	)	)	PUNCT
ejpam-178	13	8	,	,	PUNCT
ejpam-178	13	9	which	which	PRON
ejpam-178	13	10	is	be	AUX
ejpam-178	13	11	continuous	continuous	ADJ
ejpam-178	13	12	,	,	PUNCT
ejpam-178	13	13	nondecreasing	nondecreasing	ADJ
ejpam-178	13	14	and	and	CCONJ
ejpam-178	13	15	convex	convex	ADJ
ejpam-178	13	16	with	with	ADP
ejpam-178	13	17	m(0	m(0	NOUN
ejpam-178	13	18	)	)	PUNCT
ejpam-178	13	19	=	=	SYM
ejpam-178	13	20	0	0	NUM
ejpam-178	13	21	,	,	PUNCT
ejpam-178	13	22	m(x	m(x	PROPN
ejpam-178	13	23	)	)	PUNCT
ejpam-178	13	24	>	>	X
ejpam-178	13	25	0	0	NUM
ejpam-178	13	26	,	,	PUNCT
ejpam-178	13	27	for	for	ADP
ejpam-178	13	28	x	x	X
ejpam-178	13	29	>	>	X
ejpam-178	13	30	0	0	NUM
ejpam-178	13	31	and	and	CCONJ
ejpam-178	13	32	m(x	m(x	NOUN
ejpam-178	13	33	)	)	PUNCT
ejpam-178	13	34	→	→	SYM
ejpam-178	13	35	∞	∞	PROPN
ejpam-178	13	36	,	,	PUNCT
ejpam-178	13	37	as	as	SCONJ
ejpam-178	13	38	x	x	X
ejpam-178	13	39	→∞.	→∞.	X
ejpam-178	13	40	an	an	DET
ejpam-178	13	41	orlicz	orlicz	ADJ
ejpam-178	13	42	function	function	NOUN
ejpam-178	13	43	m	m	VERB
ejpam-178	13	44	is	be	AUX
ejpam-178	13	45	said	say	VERB
ejpam-178	13	46	to	to	PART
ejpam-178	13	47	satisfy	satisfy	VERB
ejpam-178	13	48	the	the	DET
ejpam-178	13	49	∆2	∆2	NOUN
ejpam-178	13	50	-	-	PUNCT
ejpam-178	13	51	condition	condition	NOUN
ejpam-178	13	52	for	for	ADP
ejpam-178	13	53	all	all	DET
ejpam-178	13	54	values	value	NOUN
ejpam-178	13	55	of	of	ADP
ejpam-178	13	56	u	u	NOUN
ejpam-178	13	57	,	,	PUNCT
ejpam-178	13	58	if	if	SCONJ
ejpam-178	13	59	there	there	PRON
ejpam-178	13	60	exists	exist	VERB
ejpam-178	13	61	a	a	DET
ejpam-178	13	62	constant	constant	ADJ
ejpam-178	13	63	k	k	X
ejpam-178	13	64	>	>	X
ejpam-178	13	65	0	0	PROPN
ejpam-178	13	66	,	,	PUNCT
ejpam-178	13	67	such	such	ADJ
ejpam-178	13	68	that	that	SCONJ
ejpam-178	13	69	m(2u)≤	m(2u)≤	NOUN
ejpam-178	13	70	km(u	km(u	PUNCT
ejpam-178	13	71	)	)	PUNCT
ejpam-178	13	72	(	(	PUNCT
ejpam-178	13	73	u≥	u≥	PROPN
ejpam-178	13	74	0	0	NUM
ejpam-178	13	75	)	)	PUNCT
ejpam-178	13	76	.	.	PUNCT
ejpam-178	14	1	the	the	DET
ejpam-178	14	2	above	above	ADJ
ejpam-178	14	3	∆2	∆2	NOUN
ejpam-178	14	4	-	-	PUNCT
ejpam-178	14	5	condition	condition	NOUN
ejpam-178	14	6	implies	imply	VERB
ejpam-178	14	7	m(lu	m(lu	PROPN
ejpam-178	14	8	)	)	PUNCT
ejpam-178	14	9	≤	≤	NUM
ejpam-178	14	10	kl	kl	NOUN
ejpam-178	14	11	l	l	NOUN
ejpam-178	14	12	og2k	og2k	PROPN
ejpam-178	14	13	m(u	m(u	PROPN
ejpam-178	14	14	)	)	PUNCT
ejpam-178	14	15	,	,	PUNCT
ejpam-178	14	16	for	for	ADP
ejpam-178	14	17	all	all	DET
ejpam-178	14	18	u	u	NOUN
ejpam-178	14	19	>	>	X
ejpam-178	14	20	0	0	NUM
ejpam-178	14	21	,	,	PUNCT
ejpam-178	14	22	l	l	NOUN
ejpam-178	14	23	>	>	X
ejpam-178	14	24	1	1	X
ejpam-178	14	25	.	.	PUNCT
ejpam-178	15	1	for	for	ADP
ejpam-178	15	2	details	detail	NOUN
ejpam-178	15	3	on	on	ADP
ejpam-178	15	4	integral	integral	ADJ
ejpam-178	15	5	representation	representation	NOUN
ejpam-178	15	6	of	of	ADP
ejpam-178	15	7	orlicz	orlicz	ADJ
ejpam-178	15	8	function	function	NOUN
ejpam-178	15	9	as	as	ADV
ejpam-178	15	10	well	well	ADV
ejpam-178	15	11	as	as	ADP
ejpam-178	15	12	on	on	ADP
ejpam-178	15	13	complementary	complementary	ADJ
ejpam-178	15	14	orlicz	orlicz	NOUN
ejpam-178	15	15	functions	function	NOUN
ejpam-178	15	16	one	one	PRON
ejpam-178	15	17	may	may	AUX
ejpam-178	15	18	refer	refer	VERB
ejpam-178	15	19	to	to	ADP
ejpam-178	15	20	[	[	X
ejpam-178	15	21	7	7	NUM
ejpam-178	15	22	,	,	PUNCT
ejpam-178	15	23	12	12	NUM
ejpam-178	15	24	]	]	PUNCT
ejpam-178	15	25	.	.	PUNCT
ejpam-178	16	1	for	for	ADP
ejpam-178	16	2	an	an	DET
ejpam-178	16	3	orlicz	orlicz	ADJ
ejpam-178	16	4	function	function	NOUN
ejpam-178	16	5	m	m	VERB
ejpam-178	16	6	,	,	PUNCT
ejpam-178	16	7	we	we	PRON
ejpam-178	16	8	have	have	VERB
ejpam-178	16	9	the	the	DET
ejpam-178	16	10	following	follow	VERB
ejpam-178	16	11	inequality	inequality	NOUN
ejpam-178	16	12	:	:	PUNCT
ejpam-178	16	13	m(λx	m(λx	NOUN
ejpam-178	16	14	)	)	PUNCT
ejpam-178	16	15	<	<	X
ejpam-178	16	16	λm(x	λm(x	NUM
ejpam-178	16	17	)	)	PUNCT
ejpam-178	16	18	,	,	PUNCT
ejpam-178	16	19	for	for	ADP
ejpam-178	16	20	all	all	PRON
ejpam-178	16	21	x	x	PRON
ejpam-178	16	22	≥	≥	NUM
ejpam-178	16	23	0	0	NUM
ejpam-178	16	24	and	and	CCONJ
ejpam-178	16	25	λ	λ	X
ejpam-178	16	26	with	with	ADP
ejpam-178	16	27	0	0	NUM
ejpam-178	16	28	<	<	X
ejpam-178	16	29	λ	λ	X
ejpam-178	16	30	<	<	X
ejpam-178	16	31	1	1	NUM
ejpam-178	16	32	.	.	X
ejpam-178	17	1	lindenstrauss	lindenstrauss	ADJ
ejpam-178	17	2	and	and	CCONJ
ejpam-178	17	3	tzafriri	tzafriri	NOUN
ejpam-178	18	1	[	[	X
ejpam-178	18	2	9	9	NUM
ejpam-178	18	3	]	]	PUNCT
ejpam-178	18	4	used	use	VERB
ejpam-178	18	5	the	the	DET
ejpam-178	18	6	orlicz	orlicz	ADJ
ejpam-178	18	7	function	function	NOUN
ejpam-178	18	8	and	and	CCONJ
ejpam-178	18	9	introduced	introduce	VERB
ejpam-178	18	10	the	the	DET
ejpam-178	18	11	sequence	sequence	NOUN
ejpam-178	18	12	space	space	NOUN
ejpam-178	18	13	ℓm	ℓm	ADP
ejpam-178	18	14	as	as	SCONJ
ejpam-178	18	15	follows	follow	VERB
ejpam-178	18	16	:	:	PUNCT
ejpam-178	18	17	ℓm	ℓm	ADV
ejpam-178	18	18	=	=	SYM
ejpam-178	18	19	{	{	PUNCT
ejpam-178	18	20	(	(	PUNCT
ejpam-178	18	21	xk	xk	INTJ
ejpam-178	18	22	)	)	PUNCT
ejpam-178	18	23	∈	∈	PROPN
ejpam-178	18	24	w	w	NOUN
ejpam-178	18	25	:	:	PUNCT
ejpam-178	18	26	∞	∞	NUM
ejpam-178	18	27	∑	∑	PUNCT
ejpam-178	18	28	k=1	k=1	PROPN
ejpam-178	18	29	m	m	PROPN
ejpam-178	18	30	(	(	PUNCT
ejpam-178	18	31	|xk|	|xk|	PROPN
ejpam-178	18	32	ρ	ρ	PROPN
ejpam-178	18	33	)	)	PUNCT
ejpam-178	18	34	<	<	X
ejpam-178	18	35	∞	∞	PROPN
ejpam-178	18	36	,	,	PUNCT
ejpam-178	18	37	for	for	ADP
ejpam-178	18	38	some	some	DET
ejpam-178	18	39	ρ	ρ	NOUN
ejpam-178	18	40	>	>	X
ejpam-178	18	41	0	0	NUM
ejpam-178	18	42	}	}	PUNCT
ejpam-178	18	43	.	.	PUNCT
ejpam-178	19	1	they	they	PRON
ejpam-178	19	2	proved	prove	VERB
ejpam-178	19	3	that	that	SCONJ
ejpam-178	19	4	ℓm	ℓm	NOUN
ejpam-178	19	5	is	be	AUX
ejpam-178	19	6	a	a	DET
ejpam-178	19	7	banach	banach	NOUN
ejpam-178	19	8	space	space	NOUN
ejpam-178	19	9	normed	norme	VERB
ejpam-178	19	10	by	by	ADP
ejpam-178	19	11	‖(xk)‖=	‖(xk)‖=	PROPN
ejpam-178	19	12	inf{ρ	inf{ρ	PROPN
ejpam-178	19	13	>	>	X
ejpam-178	19	14	0	0	NUM
ejpam-178	19	15	:	:	PUNCT
ejpam-178	20	1	∞	∞	NUM
ejpam-178	20	2	∑	∑	PUNCT
ejpam-178	20	3	k=1	k=1	PROPN
ejpam-178	20	4	m	m	PROPN
ejpam-178	20	5	(	(	PUNCT
ejpam-178	20	6	|xk|	|xk|	PROPN
ejpam-178	20	7	ρ	ρ	PROPN
ejpam-178	20	8	)	)	PUNCT
ejpam-178	20	9	≤	≤	NUM
ejpam-178	20	10	1	1	NUM
ejpam-178	20	11	}	}	PUNCT
ejpam-178	20	12	.	.	PUNCT
ejpam-178	21	1	let	let	VERB
ejpam-178	21	2	λ	λ	INTJ
ejpam-178	21	3	=	=	PRON
ejpam-178	21	4	(	(	PUNCT
ejpam-178	21	5	λk	λk	AUX
ejpam-178	21	6	)	)	PUNCT
ejpam-178	21	7	be	be	AUX
ejpam-178	21	8	a	a	DET
ejpam-178	21	9	sequence	sequence	NOUN
ejpam-178	21	10	of	of	ADP
ejpam-178	21	11	non	non	ADJ
ejpam-178	21	12	-	-	ADJ
ejpam-178	21	13	zero	zero	NUM
ejpam-178	21	14	scalars	scalar	NOUN
ejpam-178	21	15	.	.	PUNCT
ejpam-178	22	1	then	then	ADV
ejpam-178	22	2	for	for	ADP
ejpam-178	22	3	e	e	PROPN
ejpam-178	22	4	a	a	DET
ejpam-178	22	5	sequence	sequence	NOUN
ejpam-178	22	6	space	space	NOUN
ejpam-178	22	7	,	,	PUNCT
ejpam-178	22	8	the	the	DET
ejpam-178	22	9	multiplier	multipli	ADJ
ejpam-178	22	10	sequence	sequence	NOUN
ejpam-178	22	11	space	space	NOUN
ejpam-178	22	12	e(λ	e(λ	PROPN
ejpam-178	22	13	)	)	PUNCT
ejpam-178	22	14	,	,	PUNCT
ejpam-178	22	15	associated	associate	VERB
ejpam-178	22	16	with	with	ADP
ejpam-178	22	17	the	the	DET
ejpam-178	22	18	multiplier	multipli	ADJ
ejpam-178	22	19	sequence	sequence	NOUN
ejpam-178	22	20	λ	λ	PROPN
ejpam-178	22	21	is	be	AUX
ejpam-178	22	22	defined	define	VERB
ejpam-178	22	23	as	as	ADP
ejpam-178	22	24	e(λ	e(λ	NOUN
ejpam-178	22	25	)	)	PUNCT
ejpam-178	22	26	=	=	PRON
ejpam-178	22	27	{	{	PUNCT
ejpam-178	22	28	(	(	PUNCT
ejpam-178	22	29	xk	xk	INTJ
ejpam-178	22	30	)	)	PUNCT
ejpam-178	22	31	∈	∈	PROPN
ejpam-178	23	1	w	w	NOUN
ejpam-178	23	2	:	:	PUNCT
ejpam-178	23	3	(	(	PUNCT
ejpam-178	23	4	λk	λk	PROPN
ejpam-178	23	5	xk	xk	ADJ
ejpam-178	23	6	)	)	PUNCT
ejpam-178	23	7	∈	∈	PROPN
ejpam-178	23	8	e	e	X
ejpam-178	23	9	}	}	PUNCT
ejpam-178	23	10	.	.	PUNCT
ejpam-178	24	1	the	the	DET
ejpam-178	24	2	scope	scope	NOUN
ejpam-178	24	3	for	for	ADP
ejpam-178	24	4	the	the	DET
ejpam-178	24	5	studies	study	NOUN
ejpam-178	24	6	on	on	ADP
ejpam-178	24	7	sequence	sequence	NOUN
ejpam-178	24	8	spaces	space	NOUN
ejpam-178	24	9	was	be	AUX
ejpam-178	24	10	extended	extend	VERB
ejpam-178	24	11	by	by	ADP
ejpam-178	24	12	using	use	VERB
ejpam-178	24	13	the	the	DET
ejpam-178	24	14	notion	notion	NOUN
ejpam-178	24	15	of	of	ADP
ejpam-178	24	16	associated	associated	ADJ
ejpam-178	24	17	multiplier	multipli	ADJ
ejpam-178	24	18	sequences	sequence	NOUN
ejpam-178	24	19	.	.	PUNCT
ejpam-178	25	1	goes	go	VERB
ejpam-178	25	2	and	and	CCONJ
ejpam-178	25	3	goes	go	VERB
ejpam-178	25	4	[	[	X
ejpam-178	25	5	4	4	X
ejpam-178	25	6	]	]	PUNCT
ejpam-178	25	7	defined	define	VERB
ejpam-178	25	8	the	the	DET
ejpam-178	25	9	differentiated	differentiated	ADJ
ejpam-178	25	10	h.	h.	NOUN
ejpam-178	25	11	dutta	dutta	PROPN
ejpam-178	25	12	/	/	PUNCT
ejpam-178	25	13	eur	eur	PROPN
ejpam-178	25	14	.	.	PUNCT
ejpam-178	26	1	j.	j.	PROPN
ejpam-178	26	2	pure	pure	PROPN
ejpam-178	26	3	appl	appl	PROPN
ejpam-178	26	4	.	.	PROPN
ejpam-178	26	5	math	math	PROPN
ejpam-178	26	6	,	,	PUNCT
ejpam-178	26	7	2	2	NUM
ejpam-178	26	8	(	(	PUNCT
ejpam-178	26	9	2009	2009	NUM
ejpam-178	26	10	)	)	PUNCT
ejpam-178	26	11	,	,	PUNCT
ejpam-178	26	12	(	(	PUNCT
ejpam-178	26	13	554	554	NUM
ejpam-178	26	14	-	-	SYM
ejpam-178	26	15	563	563	NUM
ejpam-178	26	16	)	)	PUNCT
ejpam-178	26	17	556	556	NUM
ejpam-178	26	18	sequence	sequence	NOUN
ejpam-178	26	19	space	space	NOUN
ejpam-178	26	20	de	de	X
ejpam-178	26	21	and	and	CCONJ
ejpam-178	26	22	integrated	integrate	VERB
ejpam-178	26	23	sequence	sequence	NOUN
ejpam-178	26	24	space	space	NOUN
ejpam-178	26	25	∫	∫	PROPN
ejpam-178	26	26	e	e	PROPN
ejpam-178	26	27	for	for	ADP
ejpam-178	26	28	a	a	DET
ejpam-178	26	29	given	give	VERB
ejpam-178	26	30	sequence	sequence	NOUN
ejpam-178	26	31	space	space	NOUN
ejpam-178	26	32	e	e	NOUN
ejpam-178	26	33	,	,	PUNCT
ejpam-178	26	34	using	use	VERB
ejpam-178	26	35	the	the	DET
ejpam-178	26	36	multiplier	multipli	ADJ
ejpam-178	26	37	sequences	sequence	NOUN
ejpam-178	26	38	(	(	PUNCT
ejpam-178	26	39	k−1	k−1	PROPN
ejpam-178	26	40	)	)	PUNCT
ejpam-178	26	41	and	and	CCONJ
ejpam-178	26	42	(	(	PUNCT
ejpam-178	26	43	k	k	NOUN
ejpam-178	26	44	)	)	PUNCT
ejpam-178	26	45	respectively	respectively	ADV
ejpam-178	26	46	.	.	PUNCT
ejpam-178	27	1	a	a	DET
ejpam-178	27	2	multiplier	multipli	ADJ
ejpam-178	27	3	sequence	sequence	NOUN
ejpam-178	27	4	can	can	AUX
ejpam-178	27	5	be	be	AUX
ejpam-178	27	6	used	use	VERB
ejpam-178	27	7	to	to	PART
ejpam-178	27	8	accelerate	accelerate	VERB
ejpam-178	27	9	the	the	DET
ejpam-178	27	10	convergence	convergence	NOUN
ejpam-178	27	11	of	of	ADP
ejpam-178	27	12	the	the	DET
ejpam-178	27	13	sequences	sequence	NOUN
ejpam-178	27	14	in	in	ADP
ejpam-178	27	15	some	some	DET
ejpam-178	27	16	spaces	space	NOUN
ejpam-178	27	17	.	.	PUNCT
ejpam-178	28	1	in	in	ADP
ejpam-178	28	2	some	some	DET
ejpam-178	28	3	sense	sense	NOUN
ejpam-178	28	4	,	,	PUNCT
ejpam-178	28	5	it	it	PRON
ejpam-178	28	6	can	can	AUX
ejpam-178	28	7	be	be	AUX
ejpam-178	28	8	viewed	view	VERB
ejpam-178	28	9	as	as	ADP
ejpam-178	28	10	a	a	DET
ejpam-178	28	11	catalyst	catalyst	NOUN
ejpam-178	28	12	,	,	PUNCT
ejpam-178	28	13	which	which	PRON
ejpam-178	28	14	is	be	AUX
ejpam-178	28	15	used	use	VERB
ejpam-178	28	16	to	to	PART
ejpam-178	28	17	accelerate	accelerate	VERB
ejpam-178	28	18	the	the	DET
ejpam-178	28	19	process	process	NOUN
ejpam-178	28	20	of	of	ADP
ejpam-178	28	21	chemical	chemical	ADJ
ejpam-178	28	22	reaction	reaction	NOUN
ejpam-178	28	23	.	.	PUNCT
ejpam-178	29	1	the	the	DET
ejpam-178	29	2	notion	notion	NOUN
ejpam-178	29	3	of	of	ADP
ejpam-178	29	4	difference	difference	NOUN
ejpam-178	29	5	sequence	sequence	NOUN
ejpam-178	29	6	space	space	NOUN
ejpam-178	29	7	was	be	AUX
ejpam-178	29	8	introduced	introduce	VERB
ejpam-178	29	9	by	by	ADP
ejpam-178	29	10	kizmaz	kizmaz	X
ejpam-178	30	1	[	[	X
ejpam-178	30	2	6	6	NUM
ejpam-178	30	3	]	]	PUNCT
ejpam-178	30	4	,	,	PUNCT
ejpam-178	30	5	who	who	PRON
ejpam-178	30	6	studied	study	VERB
ejpam-178	30	7	the	the	DET
ejpam-178	30	8	difference	difference	NOUN
ejpam-178	30	9	sequence	sequence	NOUN
ejpam-178	30	10	spaces	space	VERB
ejpam-178	30	11	z(∆	z(∆	NOUN
ejpam-178	30	12	)	)	PUNCT
ejpam-178	30	13	,	,	PUNCT
ejpam-178	30	14	for	for	ADP
ejpam-178	30	15	z	z	NOUN
ejpam-178	30	16	=	=	SYM
ejpam-178	30	17	ℓ∞	ℓ∞	PROPN
ejpam-178	30	18	,	,	PUNCT
ejpam-178	30	19	c	c	X
ejpam-178	30	20	,	,	PUNCT
ejpam-178	30	21	c0	c0	NOUN
ejpam-178	30	22	and	and	CCONJ
ejpam-178	30	23	defined	define	VERB
ejpam-178	30	24	as	as	ADP
ejpam-178	30	25	follows	follow	VERB
ejpam-178	30	26	:	:	PUNCT
ejpam-178	30	27	z(∆	z(∆	NUM
ejpam-178	30	28	)	)	PUNCT
ejpam-178	30	29	=	=	PRON
ejpam-178	30	30	{	{	PUNCT
ejpam-178	30	31	x	x	SYM
ejpam-178	30	32	=	=	SYM
ejpam-178	30	33	(	(	PUNCT
ejpam-178	30	34	xk	xk	ADJ
ejpam-178	30	35	)	)	PUNCT
ejpam-178	30	36	∈	∈	PROPN
ejpam-178	30	37	w	w	NOUN
ejpam-178	30	38	:	:	PUNCT
ejpam-178	30	39	(	(	PUNCT
ejpam-178	30	40	∆xk	∆xk	NOUN
ejpam-178	30	41	)	)	PUNCT
ejpam-178	30	42	∈	∈	PROPN
ejpam-178	30	43	z	z	NOUN
ejpam-178	30	44	}	}	PUNCT
ejpam-178	30	45	,	,	PUNCT
ejpam-178	30	46	where	where	SCONJ
ejpam-178	30	47	∆x	∆x	PROPN
ejpam-178	30	48	=	=	SYM
ejpam-178	30	49	(	(	PUNCT
ejpam-178	30	50	∆xk	∆xk	NOUN
ejpam-178	30	51	)	)	PUNCT
ejpam-178	30	52	=	=	SYM
ejpam-178	30	53	(	(	PUNCT
ejpam-178	30	54	xk	xk	INTJ
ejpam-178	30	55	−	−	PROPN
ejpam-178	30	56	xk+1	xk+1	NUM
ejpam-178	30	57	)	)	PUNCT
ejpam-178	30	58	,	,	PUNCT
ejpam-178	30	59	for	for	ADP
ejpam-178	30	60	all	all	DET
ejpam-178	30	61	k	k	PROPN
ejpam-178	30	62	∈	∈	PROPN
ejpam-178	30	63	n	n	ADV
ejpam-178	30	64	.	.	PUNCT
ejpam-178	31	1	in	in	ADP
ejpam-178	31	2	this	this	DET
ejpam-178	31	3	paper	paper	NOUN
ejpam-178	31	4	our	our	PRON
ejpam-178	31	5	aim	aim	NOUN
ejpam-178	31	6	is	be	AUX
ejpam-178	31	7	to	to	PART
ejpam-178	31	8	investigate	investigate	VERB
ejpam-178	31	9	some	some	DET
ejpam-178	31	10	important	important	ADJ
ejpam-178	31	11	structures	structure	NOUN
ejpam-178	31	12	of	of	ADP
ejpam-178	31	13	some	some	DET
ejpam-178	31	14	spaces	space	NOUN
ejpam-178	31	15	which	which	PRON
ejpam-178	31	16	are	be	AUX
ejpam-178	31	17	defined	define	VERB
ejpam-178	31	18	using	use	VERB
ejpam-178	31	19	an	an	DET
ejpam-178	31	20	orlicz	orlicz	ADJ
ejpam-178	31	21	function	function	NOUN
ejpam-178	31	22	and	and	CCONJ
ejpam-178	31	23	a	a	DET
ejpam-178	31	24	multiplier	multipli	ADJ
ejpam-178	31	25	sequence	sequence	NOUN
ejpam-178	31	26	.	.	PUNCT
ejpam-178	32	1	these	these	DET
ejpam-178	32	2	spaces	space	NOUN
ejpam-178	32	3	generalize	generalize	VERB
ejpam-178	32	4	the	the	DET
ejpam-178	32	5	spaces	space	NOUN
ejpam-178	32	6	z(∆	z(∆	NOUN
ejpam-178	32	7	)	)	PUNCT
ejpam-178	32	8	,	,	PUNCT
ejpam-178	32	9	for	for	ADP
ejpam-178	32	10	z	z	NOUN
ejpam-178	32	11	=	=	SYM
ejpam-178	32	12	ℓ∞	ℓ∞	PROPN
ejpam-178	32	13	,	,	PUNCT
ejpam-178	32	14	c	c	X
ejpam-178	32	15	,	,	PUNCT
ejpam-178	32	16	c0	c0	PROPN
ejpam-178	32	17	introduced	introduce	VERB
ejpam-178	32	18	and	and	CCONJ
ejpam-178	32	19	studied	study	VERB
ejpam-178	32	20	by	by	ADP
ejpam-178	32	21	kizmaz	kizmaz	X
ejpam-178	33	1	[	[	X
ejpam-178	33	2	6	6	NUM
ejpam-178	33	3	]	]	PUNCT
ejpam-178	33	4	.	.	PUNCT
ejpam-178	34	1	let	let	VERB
ejpam-178	34	2	λ	λ	INTJ
ejpam-178	34	3	=	=	PRON
ejpam-178	34	4	(	(	PUNCT
ejpam-178	34	5	λk	λk	AUX
ejpam-178	34	6	)	)	PUNCT
ejpam-178	34	7	be	be	AUX
ejpam-178	34	8	a	a	DET
ejpam-178	34	9	non	non	ADJ
ejpam-178	34	10	-	-	ADJ
ejpam-178	34	11	zero	zero	NUM
ejpam-178	34	12	sequence	sequence	NOUN
ejpam-178	34	13	of	of	ADP
ejpam-178	34	14	scalars	scalar	NOUN
ejpam-178	34	15	.	.	PUNCT
ejpam-178	35	1	then	then	ADV
ejpam-178	35	2	we	we	PRON
ejpam-178	35	3	define	define	VERB
ejpam-178	35	4	the	the	DET
ejpam-178	35	5	following	follow	VERB
ejpam-178	35	6	sequence	sequence	NOUN
ejpam-178	35	7	spaces	space	NOUN
ejpam-178	35	8	for	for	ADP
ejpam-178	35	9	an	an	DET
ejpam-178	35	10	orlicz	orlicz	ADJ
ejpam-178	35	11	function	function	NOUN
ejpam-178	35	12	m	m	VERB
ejpam-178	35	13	:	:	PUNCT
ejpam-178	35	14	c0(m	c0(m	X
ejpam-178	35	15	,	,	PUNCT
ejpam-178	35	16	λ,∆	λ,∆	NUM
ejpam-178	35	17	)	)	PUNCT
ejpam-178	35	18	=	=	PRON
ejpam-178	36	1	{	{	PUNCT
ejpam-178	36	2	x	x	SYM
ejpam-178	36	3	=	=	SYM
ejpam-178	36	4	(	(	PUNCT
ejpam-178	36	5	xk	xk	PROPN
ejpam-178	36	6	)	)	PUNCT
ejpam-178	36	7	:	:	PUNCT
ejpam-178	36	8	lim	lim	PROPN
ejpam-178	36	9	k	k	PROPN
ejpam-178	36	10	m	m	PROPN
ejpam-178	36	11	(	(	PUNCT
ejpam-178	36	12	|∆λk	|∆λk	PROPN
ejpam-178	36	13	xk|	xk|	PROPN
ejpam-178	36	14	ρ	ρ	PROPN
ejpam-178	36	15	)	)	PUNCT
ejpam-178	36	16	=	=	SYM
ejpam-178	36	17	0	0	NUM
ejpam-178	36	18	,	,	PUNCT
ejpam-178	36	19	for	for	ADP
ejpam-178	36	20	some	some	DET
ejpam-178	36	21	ρ	ρ	NOUN
ejpam-178	36	22	>	>	X
ejpam-178	36	23	0	0	NUM
ejpam-178	36	24	}	}	PUNCT
ejpam-178	36	25	,	,	PUNCT
ejpam-178	36	26	c(m	c(m	PROPN
ejpam-178	36	27	,	,	PUNCT
ejpam-178	36	28	λ,∆	λ,∆	NUM
ejpam-178	36	29	)	)	PUNCT
ejpam-178	36	30	=	=	PRON
ejpam-178	37	1	{	{	PUNCT
ejpam-178	37	2	x	x	SYM
ejpam-178	37	3	=	=	SYM
ejpam-178	37	4	(	(	PUNCT
ejpam-178	37	5	xk	xk	PROPN
ejpam-178	37	6	)	)	PUNCT
ejpam-178	37	7	:	:	PUNCT
ejpam-178	37	8	lim	lim	PROPN
ejpam-178	37	9	k	k	PROPN
ejpam-178	37	10	m	m	PROPN
ejpam-178	37	11	(	(	PUNCT
ejpam-178	37	12	|∆λk	|∆λk	X
ejpam-178	37	13	xk	xk	PROPN
ejpam-178	37	14	−	−	PROPN
ejpam-178	37	15	l|	l|	ADJ
ejpam-178	37	16	ρ	ρ	NOUN
ejpam-178	37	17	)	)	PUNCT
ejpam-178	38	1	=	=	SYM
ejpam-178	38	2	0	0	NUM
ejpam-178	38	3	,	,	PUNCT
ejpam-178	38	4	for	for	ADP
ejpam-178	38	5	some	some	DET
ejpam-178	38	6	l	l	NOUN
ejpam-178	38	7	and	and	CCONJ
ejpam-178	38	8	ρ	ρ	NOUN
ejpam-178	38	9	>	>	X
ejpam-178	38	10	0	0	NUM
ejpam-178	38	11	}	}	PUNCT
ejpam-178	38	12	,	,	PUNCT
ejpam-178	38	13	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	38	14	,	,	PUNCT
ejpam-178	38	15	λ,∆	λ,∆	NUM
ejpam-178	38	16	)	)	PUNCT
ejpam-178	39	1	=	=	PRON
ejpam-178	39	2	{	{	PUNCT
ejpam-178	39	3	x	x	SYM
ejpam-178	39	4	=	=	SYM
ejpam-178	39	5	(	(	PUNCT
ejpam-178	39	6	xk	xk	PROPN
ejpam-178	39	7	)	)	PUNCT
ejpam-178	39	8	:	:	PUNCT
ejpam-178	39	9	sup	sup	PROPN
ejpam-178	39	10	k	k	PROPN
ejpam-178	39	11	m	m	PROPN
ejpam-178	39	12	(	(	PUNCT
ejpam-178	39	13	|∆λk	|∆λk	PROPN
ejpam-178	39	14	xk|	xk|	PROPN
ejpam-178	39	15	ρ	ρ	PROPN
ejpam-178	39	16	)	)	PUNCT
ejpam-178	39	17	<	<	X
ejpam-178	39	18	∞	∞	PROPN
ejpam-178	39	19	,	,	PUNCT
ejpam-178	39	20	for	for	ADP
ejpam-178	39	21	some	some	DET
ejpam-178	39	22	ρ	ρ	NOUN
ejpam-178	39	23	>	>	X
ejpam-178	39	24	0	0	NUM
ejpam-178	39	25	}	}	PUNCT
ejpam-178	39	26	,	,	PUNCT
ejpam-178	39	27	where	where	SCONJ
ejpam-178	39	28	∆λk	∆λk	ADP
ejpam-178	39	29	xk	xk	PROPN
ejpam-178	40	1	=	=	PUNCT
ejpam-178	40	2	λk	λk	PROPN
ejpam-178	41	1	xk−λk+1xk+1	xk−λk+1xk+1	PROPN
ejpam-178	41	2	,	,	PUNCT
ejpam-178	41	3	for	for	ADP
ejpam-178	41	4	all	all	DET
ejpam-178	41	5	k	k	PROPN
ejpam-178	41	6	∈	∈	PROPN
ejpam-178	41	7	n	n	ADV
ejpam-178	41	8	.	.	PUNCT
ejpam-178	42	1	it	it	PRON
ejpam-178	42	2	is	be	AUX
ejpam-178	42	3	obvious	obvious	ADJ
ejpam-178	42	4	that	that	SCONJ
ejpam-178	42	5	c0(m	c0(m	PRON
ejpam-178	42	6	,	,	PUNCT
ejpam-178	42	7	λ,∆)⊂	λ,∆)⊂	PROPN
ejpam-178	42	8	c(m	c(m	PROPN
ejpam-178	42	9	,	,	PUNCT
ejpam-178	42	10	λ,∆)⊂	λ,∆)⊂	ADP
ejpam-178	42	11	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	42	12	,	,	PUNCT
ejpam-178	42	13	λ,∆	λ,∆	NUM
ejpam-178	42	14	)	)	PUNCT
ejpam-178	42	15	.	.	PUNCT
ejpam-178	43	1	throughout	throughout	ADP
ejpam-178	43	2	the	the	DET
ejpam-178	43	3	paper	paper	NOUN
ejpam-178	43	4	x	x	PUNCT
ejpam-178	43	5	will	will	AUX
ejpam-178	43	6	denote	denote	VERB
ejpam-178	43	7	one	one	NUM
ejpam-178	43	8	of	of	ADP
ejpam-178	43	9	the	the	DET
ejpam-178	43	10	sequence	sequence	NOUN
ejpam-178	43	11	spaces	space	VERB
ejpam-178	43	12	c0	c0	NOUN
ejpam-178	43	13	,	,	PUNCT
ejpam-178	43	14	c	c	PROPN
ejpam-178	43	15	and	and	CCONJ
ejpam-178	43	16	ℓ∞.	ℓ∞.	ADP
ejpam-178	43	17	the	the	DET
ejpam-178	43	18	sequence	sequence	NOUN
ejpam-178	43	19	spaces	space	VERB
ejpam-178	43	20	x	x	INTJ
ejpam-178	43	21	(	(	PUNCT
ejpam-178	43	22	m	m	INTJ
ejpam-178	43	23	,	,	PUNCT
ejpam-178	43	24	λ,∆	λ,∆	NUM
ejpam-178	43	25	)	)	PUNCT
ejpam-178	43	26	are	be	AUX
ejpam-178	43	27	banach	banach	NOUN
ejpam-178	43	28	spaces	space	NOUN
ejpam-178	43	29	normed	norme	VERB
ejpam-178	43	30	by	by	ADP
ejpam-178	43	31	‖x‖∆	‖x‖∆	PROPN
ejpam-178	43	32	=	=	SYM
ejpam-178	43	33	|λ1x1|+	|λ1x1|+	PROPN
ejpam-178	43	34	inf{ρ	inf{ρ	X
ejpam-178	43	35	>	>	X
ejpam-178	43	36	0	0	NUM
ejpam-178	44	1	:	:	PUNCT
ejpam-178	44	2	sup	sup	PROPN
ejpam-178	44	3	k	k	NOUN
ejpam-178	44	4	m	m	PROPN
ejpam-178	44	5	(	(	PUNCT
ejpam-178	44	6	|∆λk	|∆λk	PROPN
ejpam-178	44	7	xk|	xk|	PROPN
ejpam-178	44	8	ρ	ρ	PROPN
ejpam-178	44	9	)	)	PUNCT
ejpam-178	44	10	≤	≤	NOUN
ejpam-178	44	11	1	1	NUM
ejpam-178	44	12	}	}	PUNCT
ejpam-178	44	13	.	.	PUNCT
ejpam-178	45	1	now	now	ADV
ejpam-178	45	2	we	we	PRON
ejpam-178	45	3	shall	shall	AUX
ejpam-178	45	4	write∆−1xk	write∆−1xk	VERB
ejpam-178	45	5	=	=	SYM
ejpam-178	45	6	xk−	xk−	NUM
ejpam-178	45	7	xk−1	xk−1	PROPN
ejpam-178	45	8	,	,	PUNCT
ejpam-178	45	9	for	for	ADP
ejpam-178	45	10	all	all	DET
ejpam-178	45	11	k	k	PROPN
ejpam-178	45	12	∈	∈	PROPN
ejpam-178	45	13	n	n	ADV
ejpam-178	45	14	.	.	PUNCT
ejpam-178	46	1	it	it	PRON
ejpam-178	46	2	is	be	AUX
ejpam-178	46	3	trivial	trivial	ADJ
ejpam-178	46	4	that	that	SCONJ
ejpam-178	46	5	(	(	PUNCT
ejpam-178	46	6	∆λk	∆λk	ADP
ejpam-178	46	7	xk	xk	NOUN
ejpam-178	46	8	)	)	PUNCT
ejpam-178	46	9	∈	∈	PROPN
ejpam-178	46	10	x	x	SYM
ejpam-178	46	11	(	(	PUNCT
ejpam-178	46	12	m	m	NOUN
ejpam-178	46	13	)	)	PUNCT
ejpam-178	46	14	h.	h.	PROPN
ejpam-178	46	15	dutta	dutta	PROPN
ejpam-178	46	16	/	/	PUNCT
ejpam-178	46	17	eur	eur	PROPN
ejpam-178	46	18	.	.	PUNCT
ejpam-178	47	1	j.	j.	PROPN
ejpam-178	47	2	pure	pure	PROPN
ejpam-178	47	3	appl	appl	PROPN
ejpam-178	47	4	.	.	PROPN
ejpam-178	47	5	math	math	PROPN
ejpam-178	47	6	,	,	PUNCT
ejpam-178	47	7	2	2	NUM
ejpam-178	47	8	(	(	PUNCT
ejpam-178	47	9	2009	2009	NUM
ejpam-178	47	10	)	)	PUNCT
ejpam-178	47	11	,	,	PUNCT
ejpam-178	47	12	(	(	PUNCT
ejpam-178	47	13	554	554	NUM
ejpam-178	47	14	-	-	SYM
ejpam-178	47	15	563	563	NUM
ejpam-178	47	16	)	)	PUNCT
ejpam-178	47	17	557	557	NUM
ejpam-178	48	1	if	if	SCONJ
ejpam-178	48	2	and	and	CCONJ
ejpam-178	48	3	only	only	ADV
ejpam-178	48	4	if	if	SCONJ
ejpam-178	48	5	(	(	PUNCT
ejpam-178	48	6	∆−1λk	∆−1λk	VERB
ejpam-178	48	7	xk	xk	X
ejpam-178	48	8	)	)	PUNCT
ejpam-178	48	9	∈	∈	PROPN
ejpam-178	48	10	x	x	SYM
ejpam-178	48	11	(	(	PUNCT
ejpam-178	48	12	m	m	NOUN
ejpam-178	48	13	)	)	PUNCT
ejpam-178	48	14	.	.	PUNCT
ejpam-178	49	1	now	now	ADV
ejpam-178	49	2	for	for	ADP
ejpam-178	49	3	x	x	SYM
ejpam-178	49	4	∈	∈	PROPN
ejpam-178	49	5	x	x	SYM
ejpam-178	49	6	(	(	PUNCT
ejpam-178	49	7	m	m	PROPN
ejpam-178	49	8	,	,	PUNCT
ejpam-178	49	9	λ,∆−1	λ,∆−1	PROPN
ejpam-178	49	10	)	)	PUNCT
ejpam-178	49	11	,	,	PUNCT
ejpam-178	49	12	we	we	PRON
ejpam-178	49	13	define	define	VERB
ejpam-178	49	14	‖x‖∆−1	‖x‖∆−1	PUNCT
ejpam-178	50	1	=	=	SYM
ejpam-178	50	2	inf{ρ	inf{ρ	X
ejpam-178	50	3	>	>	X
ejpam-178	50	4	0	0	NUM
ejpam-178	50	5	:	:	PUNCT
ejpam-178	50	6	sup	sup	PROPN
ejpam-178	50	7	k	k	NOUN
ejpam-178	50	8	m	m	PROPN
ejpam-178	50	9	(	(	PUNCT
ejpam-178	50	10	|∆−1λk	|∆−1λk	PROPN
ejpam-178	50	11	xk|	xk|	PROPN
ejpam-178	50	12	ρ	ρ	PROPN
ejpam-178	50	13	)	)	PUNCT
ejpam-178	50	14	≤	≤	NUM
ejpam-178	50	15	1	1	NUM
ejpam-178	50	16	}	}	PUNCT
ejpam-178	50	17	.	.	PUNCT
ejpam-178	51	1	it	it	PRON
ejpam-178	51	2	can	can	AUX
ejpam-178	51	3	be	be	AUX
ejpam-178	51	4	shown	show	VERB
ejpam-178	51	5	that	that	SCONJ
ejpam-178	51	6	x	x	X
ejpam-178	51	7	(	(	PUNCT
ejpam-178	51	8	m	m	PROPN
ejpam-178	51	9	,	,	PUNCT
ejpam-178	51	10	λ,∆	λ,∆	NUM
ejpam-178	51	11	)	)	PUNCT
ejpam-178	51	12	is	be	AUX
ejpam-178	51	13	a	a	DET
ejpam-178	51	14	bk	bk	NOUN
ejpam-178	51	15	-	-	PUNCT
ejpam-178	51	16	space	space	NOUN
ejpam-178	51	17	under	under	ADP
ejpam-178	51	18	the	the	DET
ejpam-178	51	19	norms	norm	NOUN
ejpam-178	51	20	‖.‖∆	‖.‖∆	PUNCT
ejpam-178	51	21	and	and	CCONJ
ejpam-178	51	22	‖.‖∆−1	‖.‖∆−1	VERB
ejpam-178	51	23	respectively	respectively	ADV
ejpam-178	51	24	and	and	CCONJ
ejpam-178	51	25	it	it	PRON
ejpam-178	51	26	is	be	AUX
ejpam-178	51	27	obvious	obvious	ADJ
ejpam-178	51	28	that	that	SCONJ
ejpam-178	51	29	the	the	DET
ejpam-178	51	30	norms	norm	NOUN
ejpam-178	51	31	‖.‖∆	‖.‖∆	PUNCT
ejpam-178	51	32	and	and	CCONJ
ejpam-178	51	33	‖.‖∆−1	‖.‖∆−1	PROPN
ejpam-178	51	34	are	be	AUX
ejpam-178	51	35	equivalent	equivalent	ADJ
ejpam-178	51	36	.	.	PUNCT
ejpam-178	52	1	obviously	obviously	ADV
ejpam-178	52	2	∆−1	∆−1	ADV
ejpam-178	52	3	:	:	PUNCT
ejpam-178	52	4	x	x	X
ejpam-178	52	5	(	(	PUNCT
ejpam-178	52	6	m	m	PROPN
ejpam-178	52	7	,	,	PUNCT
ejpam-178	52	8	λ,∆−1	λ,∆−1	PART
ejpam-178	52	9	)	)	PUNCT
ejpam-178	53	1	−→	−→	NOUN
ejpam-178	53	2	x	x	SYM
ejpam-178	53	3	(	(	PUNCT
ejpam-178	53	4	m	m	NOUN
ejpam-178	53	5	)	)	PUNCT
ejpam-178	53	6	,	,	PUNCT
ejpam-178	53	7	defined	define	VERB
ejpam-178	53	8	by	by	ADP
ejpam-178	53	9	∆−1x	∆−1x	NOUN
ejpam-178	53	10	=	=	SYM
ejpam-178	53	11	y	y	NOUN
ejpam-178	53	12	=	=	PUNCT
ejpam-178	53	13	(	(	PUNCT
ejpam-178	53	14	∆−1λk	∆−1λk	VERB
ejpam-178	53	15	xk	xk	INTJ
ejpam-178	53	16	)	)	PUNCT
ejpam-178	53	17	,	,	PUNCT
ejpam-178	53	18	is	be	AUX
ejpam-178	53	19	isometric	isometric	ADJ
ejpam-178	53	20	isomorphism	isomorphism	NOUN
ejpam-178	53	21	.	.	PUNCT
ejpam-178	54	1	hence	hence	ADV
ejpam-178	54	2	c0(m	c0(m	PRON
ejpam-178	54	3	,	,	PUNCT
ejpam-178	54	4	λ,∆−1	λ,∆−1	PROPN
ejpam-178	54	5	)	)	PUNCT
ejpam-178	54	6	,	,	PUNCT
ejpam-178	54	7	c(m	c(m	PROPN
ejpam-178	54	8	,	,	PUNCT
ejpam-178	54	9	λ,∆−1	λ,∆−1	PUNCT
ejpam-178	54	10	)	)	PUNCT
ejpam-178	54	11	and	and	CCONJ
ejpam-178	54	12	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	54	13	,	,	PUNCT
ejpam-178	54	14	λ,∆−1	λ,∆−1	PUNCT
ejpam-178	54	15	)	)	PUNCT
ejpam-178	54	16	are	be	AUX
ejpam-178	54	17	isometrically	isometrically	PROPN
ejpam-178	54	18	isomorphic	isomorphic	ADJ
ejpam-178	54	19	to	to	ADP
ejpam-178	54	20	c0(m	c0(m	PROPN
ejpam-178	54	21	)	)	PUNCT
ejpam-178	54	22	,	,	PUNCT
ejpam-178	54	23	c(m	c(m	PROPN
ejpam-178	54	24	)	)	PUNCT
ejpam-178	54	25	and	and	CCONJ
ejpam-178	54	26	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	54	27	)	)	PUNCT
ejpam-178	54	28	respectively	respectively	ADV
ejpam-178	54	29	.	.	PUNCT
ejpam-178	55	1	from	from	ADP
ejpam-178	55	2	abstract	abstract	ADJ
ejpam-178	55	3	point	point	NOUN
ejpam-178	55	4	of	of	ADP
ejpam-178	55	5	view	view	NOUN
ejpam-178	55	6	x	x	INTJ
ejpam-178	55	7	(	(	PUNCT
ejpam-178	55	8	m	m	PROPN
ejpam-178	55	9	,	,	PUNCT
ejpam-178	55	10	λ,∆−1	λ,∆−1	PUNCT
ejpam-178	55	11	)	)	PUNCT
ejpam-178	55	12	is	be	AUX
ejpam-178	55	13	identical	identical	ADJ
ejpam-178	55	14	with	with	ADP
ejpam-178	55	15	x	x	X
ejpam-178	55	16	(	(	PUNCT
ejpam-178	55	17	m	m	NOUN
ejpam-178	55	18	)	)	PUNCT
ejpam-178	55	19	,	,	PUNCT
ejpam-178	55	20	for	for	SCONJ
ejpam-178	55	21	x	x	PROPN
ejpam-178	55	22	=	=	SYM
ejpam-178	55	23	c0	c0	NOUN
ejpam-178	55	24	,	,	PUNCT
ejpam-178	55	25	c	c	PROPN
ejpam-178	55	26	and	and	CCONJ
ejpam-178	55	27	ℓ∞.	ℓ∞.	ADP
ejpam-178	55	28	the	the	DET
ejpam-178	55	29	results	result	NOUN
ejpam-178	55	30	obtained	obtain	VERB
ejpam-178	55	31	in	in	ADP
ejpam-178	55	32	the	the	DET
ejpam-178	55	33	next	next	ADJ
ejpam-178	55	34	section	section	NOUN
ejpam-178	55	35	also	also	ADV
ejpam-178	55	36	hold	hold	VERB
ejpam-178	55	37	for	for	ADP
ejpam-178	55	38	the	the	DET
ejpam-178	55	39	spaces	space	NOUN
ejpam-178	55	40	c0(m	c0(m	PROPN
ejpam-178	55	41	,	,	PUNCT
ejpam-178	55	42	λ,∆−1	λ,∆−1	PROPN
ejpam-178	55	43	)	)	PUNCT
ejpam-178	55	44	,	,	PUNCT
ejpam-178	55	45	c(m	c(m	PROPN
ejpam-178	55	46	,	,	PUNCT
ejpam-178	55	47	λ,∆−1	λ,∆−1	PUNCT
ejpam-178	55	48	)	)	PUNCT
ejpam-178	55	49	and	and	CCONJ
ejpam-178	55	50	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	55	51	,	,	PUNCT
ejpam-178	55	52	λ,∆−1	λ,∆−1	PROPN
ejpam-178	55	53	)	)	PUNCT
ejpam-178	55	54	as	as	ADV
ejpam-178	55	55	well	well	ADV
ejpam-178	55	56	as	as	ADP
ejpam-178	55	57	for	for	ADP
ejpam-178	55	58	the	the	DET
ejpam-178	55	59	spaces	space	NOUN
ejpam-178	55	60	associated	associate	VERB
ejpam-178	55	61	with	with	ADP
ejpam-178	55	62	these	these	DET
ejpam-178	55	63	three	three	NUM
ejpam-178	55	64	spaces	space	NOUN
ejpam-178	55	65	.	.	PUNCT
ejpam-178	56	1	now	now	ADV
ejpam-178	56	2	we	we	PRON
ejpam-178	56	3	define	define	VERB
ejpam-178	56	4	the	the	DET
ejpam-178	56	5	spaces	space	NOUN
ejpam-178	56	6	c̃0(m	c̃0(m	NOUN
ejpam-178	56	7	,	,	PUNCT
ejpam-178	56	8	λ,∆	λ,∆	NUM
ejpam-178	56	9	)	)	PUNCT
ejpam-178	56	10	,	,	PUNCT
ejpam-178	56	11	c̃(m	c̃(m	PROPN
ejpam-178	56	12	,	,	PUNCT
ejpam-178	56	13	λ,∆	λ,∆	NUM
ejpam-178	56	14	)	)	PUNCT
ejpam-178	56	15	and	and	CCONJ
ejpam-178	56	16	ℓ̃∞(m	ℓ̃∞(m	ADV
ejpam-178	56	17	,	,	PUNCT
ejpam-178	56	18	λ,∆	λ,∆	NUM
ejpam-178	56	19	)	)	PUNCT
ejpam-178	56	20	as	as	SCONJ
ejpam-178	56	21	follows	follow	VERB
ejpam-178	56	22	:	:	PUNCT
ejpam-178	56	23	c̃0(m	c̃0(m	NOUN
ejpam-178	56	24	,	,	PUNCT
ejpam-178	56	25	λ,∆	λ,∆	NUM
ejpam-178	56	26	)	)	PUNCT
ejpam-178	56	27	is	be	AUX
ejpam-178	56	28	a	a	DET
ejpam-178	56	29	subspace	subspace	NOUN
ejpam-178	56	30	of	of	ADP
ejpam-178	56	31	c0(m	c0(m	PROPN
ejpam-178	56	32	,	,	PUNCT
ejpam-178	56	33	λ,∆	λ,∆	NUM
ejpam-178	56	34	)	)	PUNCT
ejpam-178	56	35	consisting	consist	VERB
ejpam-178	56	36	of	of	ADP
ejpam-178	56	37	those	those	DET
ejpam-178	56	38	x	x	SYM
ejpam-178	56	39	∈	∈	PROPN
ejpam-178	56	40	c0(m	c0(m	PROPN
ejpam-178	56	41	,	,	PUNCT
ejpam-178	56	42	λ,∆	λ,∆	NUM
ejpam-178	56	43	)	)	PUNCT
ejpam-178	57	1	such	such	ADJ
ejpam-178	57	2	that	that	SCONJ
ejpam-178	57	3	lim	lim	PROPN
ejpam-178	57	4	k	k	PROPN
ejpam-178	57	5	m	m	PROPN
ejpam-178	57	6	(	(	PUNCT
ejpam-178	57	7	|∆λk	|∆λk	PROPN
ejpam-178	57	8	xk|	xk|	PROPN
ejpam-178	57	9	d	d	NOUN
ejpam-178	57	10	)	)	PUNCT
ejpam-178	57	11	=	=	SYM
ejpam-178	57	12	0	0	NUM
ejpam-178	57	13	f	f	NOUN
ejpam-178	57	14	or	or	CCONJ
ejpam-178	57	15	each	each	PRON
ejpam-178	57	16	d	d	X
ejpam-178	57	17	>	>	X
ejpam-178	57	18	0	0	X
ejpam-178	57	19	.	.	PUNCT
ejpam-178	58	1	similarly	similarly	ADV
ejpam-178	58	2	we	we	PRON
ejpam-178	58	3	can	can	AUX
ejpam-178	58	4	define	define	VERB
ejpam-178	58	5	c̃(m	c̃(m	PROPN
ejpam-178	58	6	,	,	PUNCT
ejpam-178	58	7	λ,∆	λ,∆	NUM
ejpam-178	58	8	)	)	PUNCT
ejpam-178	58	9	and	and	CCONJ
ejpam-178	58	10	ℓ̃∞(m	ℓ̃∞(m	ADV
ejpam-178	58	11	,	,	PUNCT
ejpam-178	58	12	λ,∆	λ,∆	NUM
ejpam-178	58	13	)	)	PUNCT
ejpam-178	58	14	as	as	ADP
ejpam-178	58	15	subspace	subspace	NOUN
ejpam-178	58	16	of	of	ADP
ejpam-178	58	17	c(m	c(m	PROPN
ejpam-178	58	18	,	,	PUNCT
ejpam-178	58	19	λ,∆	λ,∆	NUM
ejpam-178	58	20	)	)	PUNCT
ejpam-178	58	21	and	and	CCONJ
ejpam-178	58	22	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	58	23	,	,	PUNCT
ejpam-178	58	24	λ,∆	λ,∆	NUM
ejpam-178	58	25	)	)	PUNCT
ejpam-178	58	26	respectively	respectively	ADV
ejpam-178	58	27	.	.	PUNCT
ejpam-178	59	1	it	it	PRON
ejpam-178	59	2	is	be	AUX
ejpam-178	59	3	obvious	obvious	ADJ
ejpam-178	59	4	that	that	SCONJ
ejpam-178	59	5	c̃(m	c̃(m	PROPN
ejpam-178	59	6	,	,	PUNCT
ejpam-178	59	7	λ,∆	λ,∆	NUM
ejpam-178	59	8	)	)	PUNCT
ejpam-178	59	9	⊂	⊂	PROPN
ejpam-178	60	1	c̃(m	c̃(m	PROPN
ejpam-178	60	2	,	,	PUNCT
ejpam-178	60	3	λ,∆	λ,∆	NUM
ejpam-178	60	4	)	)	PUNCT
ejpam-178	60	5	⊂	⊂	PROPN
ejpam-178	60	6	ℓ̃∞(m	ℓ̃∞(m	ADV
ejpam-178	60	7	,	,	PUNCT
ejpam-178	60	8	λ,∆	λ,∆	NUM
ejpam-178	60	9	)	)	PUNCT
ejpam-178	60	10	.	.	PUNCT
ejpam-178	61	1	also	also	ADV
ejpam-178	61	2	as	as	ADP
ejpam-178	61	3	above	above	ADV
ejpam-178	61	4	we	we	PRON
ejpam-178	61	5	can	can	AUX
ejpam-178	61	6	show	show	VERB
ejpam-178	61	7	that	that	SCONJ
ejpam-178	61	8	c̃0(m	c̃0(m	NOUN
ejpam-178	61	9	,	,	PUNCT
ejpam-178	61	10	λ,∆	λ,∆	NUM
ejpam-178	61	11	)	)	PUNCT
ejpam-178	61	12	,	,	PUNCT
ejpam-178	61	13	c̃(m	c̃(m	PROPN
ejpam-178	61	14	,	,	PUNCT
ejpam-178	61	15	λ,∆	λ,∆	NUM
ejpam-178	61	16	)	)	PUNCT
ejpam-178	61	17	and	and	CCONJ
ejpam-178	61	18	ℓ̃∞(m	ℓ̃∞(m	ADV
ejpam-178	61	19	,	,	PUNCT
ejpam-178	61	20	λ,∆	λ,∆	NUM
ejpam-178	61	21	)	)	PUNCT
ejpam-178	61	22	are	be	AUX
ejpam-178	61	23	isometrically	isometrically	PROPN
ejpam-178	61	24	isomorphic	isomorphic	ADJ
ejpam-178	61	25	to	to	ADP
ejpam-178	61	26	c̃0(m	c̃0(m	PROPN
ejpam-178	61	27	)	)	PUNCT
ejpam-178	61	28	,	,	PUNCT
ejpam-178	61	29	c̃(m	c̃(m	PROPN
ejpam-178	61	30	)	)	PUNCT
ejpam-178	61	31	and	and	CCONJ
ejpam-178	61	32	ℓ̃∞(m	ℓ̃∞(m	ADV
ejpam-178	61	33	)	)	PUNCT
ejpam-178	61	34	respectively	respectively	ADV
ejpam-178	61	35	.	.	PUNCT
ejpam-178	62	1	moreover	moreover	ADV
ejpam-178	62	2	x	x	SYM
ejpam-178	62	3	(	(	PUNCT
ejpam-178	62	4	m	m	PROPN
ejpam-178	62	5	,	,	PUNCT
ejpam-178	62	6	λ	λ	X
ejpam-178	62	7	)	)	PUNCT
ejpam-178	62	8	⊂	⊂	PROPN
ejpam-178	62	9	x	x	PUNCT
ejpam-178	62	10	(	(	PUNCT
ejpam-178	62	11	m	m	INTJ
ejpam-178	62	12	,	,	PUNCT
ejpam-178	62	13	λ,∆	λ,∆	NUM
ejpam-178	62	14	)	)	PUNCT
ejpam-178	62	15	and	and	CCONJ
ejpam-178	62	16	x̃	x̃	PROPN
ejpam-178	62	17	(	(	PUNCT
ejpam-178	62	18	m	m	PROPN
ejpam-178	62	19	,	,	PUNCT
ejpam-178	62	20	λ	λ	X
ejpam-178	62	21	)	)	PUNCT
ejpam-178	62	22	⊂	⊂	PROPN
ejpam-178	63	1	x̃	x̃	PROPN
ejpam-178	63	2	(	(	PUNCT
ejpam-178	63	3	m	m	PROPN
ejpam-178	63	4	,	,	PUNCT
ejpam-178	63	5	λ,∆	λ,∆	NUM
ejpam-178	63	6	)	)	PUNCT
ejpam-178	63	7	which	which	PRON
ejpam-178	63	8	can	can	AUX
ejpam-178	63	9	be	be	AUX
ejpam-178	63	10	shown	show	VERB
ejpam-178	63	11	by	by	ADP
ejpam-178	63	12	using	use	VERB
ejpam-178	63	13	the	the	DET
ejpam-178	63	14	following	follow	VERB
ejpam-178	63	15	inequality	inequality	NOUN
ejpam-178	63	16	:	:	PUNCT
ejpam-178	63	17	m	m	PROPN
ejpam-178	63	18	(	(	PUNCT
ejpam-178	63	19	|∆λk	|∆λk	ADJ
ejpam-178	63	20	xk|	xk|	PROPN
ejpam-178	63	21	2ρ	2ρ	NOUN
ejpam-178	63	22	)	)	PUNCT
ejpam-178	63	23	≤	≤	NUM
ejpam-178	63	24	1	1	NUM
ejpam-178	63	25	2	2	NUM
ejpam-178	63	26	m	m	NOUN
ejpam-178	63	27	(	(	PUNCT
ejpam-178	63	28	|λk	|λk	NUM
ejpam-178	63	29	xk|	xk|	PROPN
ejpam-178	63	30	ρ	ρ	PROPN
ejpam-178	63	31	)	)	PUNCT
ejpam-178	64	1	+	+	CCONJ
ejpam-178	64	2	1	1	NUM
ejpam-178	64	3	2	2	NUM
ejpam-178	64	4	m	m	NOUN
ejpam-178	64	5	(	(	PUNCT
ejpam-178	64	6	|λk+1xk+1|	|λk+1xk+1|	PROPN
ejpam-178	64	7	ρ	ρ	NOUN
ejpam-178	64	8	)	)	PUNCT
ejpam-178	64	9	.	.	PUNCT
ejpam-178	65	1	h.	h.	PROPN
ejpam-178	65	2	dutta	dutta	PROPN
ejpam-178	65	3	/	/	PUNCT
ejpam-178	65	4	eur	eur	PROPN
ejpam-178	65	5	.	.	PUNCT
ejpam-178	66	1	j.	j.	PROPN
ejpam-178	66	2	pure	pure	PROPN
ejpam-178	66	3	appl	appl	PROPN
ejpam-178	66	4	.	.	PROPN
ejpam-178	66	5	math	math	PROPN
ejpam-178	66	6	,	,	PUNCT
ejpam-178	66	7	2	2	NUM
ejpam-178	66	8	(	(	PUNCT
ejpam-178	66	9	2009	2009	NUM
ejpam-178	66	10	)	)	PUNCT
ejpam-178	66	11	,	,	PUNCT
ejpam-178	66	12	(	(	PUNCT
ejpam-178	66	13	554	554	NUM
ejpam-178	66	14	-	-	SYM
ejpam-178	66	15	563	563	NUM
ejpam-178	66	16	)	)	PUNCT
ejpam-178	66	17	558	558	NUM
ejpam-178	66	18	2	2	NUM
ejpam-178	66	19	.	.	PUNCT
ejpam-178	67	1	köthe	köthe	ADJ
ejpam-178	67	2	-	-	PUNCT
ejpam-178	67	3	toeplitz	toeplitz	NOUN
ejpam-178	67	4	and	and	CCONJ
ejpam-178	67	5	null	null	ADJ
ejpam-178	67	6	dual	dual	ADJ
ejpam-178	67	7	spaces	space	NOUN
ejpam-178	67	8	in	in	ADP
ejpam-178	67	9	this	this	DET
ejpam-178	67	10	section	section	NOUN
ejpam-178	67	11	we	we	PRON
ejpam-178	67	12	compute	compute	VERB
ejpam-178	67	13	köthe	köthe	ADJ
ejpam-178	67	14	-	-	PUNCT
ejpam-178	67	15	toeplitz	toeplitz	NOUN
ejpam-178	67	16	or	or	CCONJ
ejpam-178	67	17	α	α	NOUN
ejpam-178	67	18	-	-	ADJ
ejpam-178	67	19	dual	dual	ADJ
ejpam-178	67	20	and	and	CCONJ
ejpam-178	67	21	null	null	ADJ
ejpam-178	67	22	or	or	CCONJ
ejpam-178	67	23	n	n	NOUN
ejpam-178	67	24	dual	dual	ADJ
ejpam-178	67	25	of	of	ADP
ejpam-178	67	26	some	some	DET
ejpam-178	67	27	difference	difference	NOUN
ejpam-178	67	28	sequence	sequence	NOUN
ejpam-178	67	29	spaces	space	NOUN
ejpam-178	67	30	as	as	SCONJ
ejpam-178	67	31	described	describe	VERB
ejpam-178	67	32	in	in	ADP
ejpam-178	67	33	the	the	DET
ejpam-178	67	34	preceding	precede	VERB
ejpam-178	67	35	section	section	NOUN
ejpam-178	67	36	.	.	PUNCT
ejpam-178	68	1	let	let	VERB
ejpam-178	68	2	e	e	NOUN
ejpam-178	68	3	and	and	CCONJ
ejpam-178	68	4	f	f	PROPN
ejpam-178	68	5	be	be	VERB
ejpam-178	68	6	two	two	NUM
ejpam-178	68	7	sequence	sequence	NOUN
ejpam-178	68	8	spaces	space	NOUN
ejpam-178	68	9	.	.	PUNCT
ejpam-178	69	1	then	then	ADV
ejpam-178	69	2	the	the	DET
ejpam-178	69	3	f	f	PROPN
ejpam-178	69	4	dual	dual	ADJ
ejpam-178	69	5	of	of	ADP
ejpam-178	69	6	e	e	PROPN
ejpam-178	69	7	is	be	AUX
ejpam-178	69	8	defined	define	VERB
ejpam-178	69	9	as	as	ADP
ejpam-178	69	10	ef	ef	PROPN
ejpam-178	69	11	=	=	SYM
ejpam-178	69	12	{	{	PUNCT
ejpam-178	69	13	(	(	PUNCT
ejpam-178	69	14	xk	xk	ADJ
ejpam-178	69	15	)	)	PUNCT
ejpam-178	69	16	∈	∈	PROPN
ejpam-178	69	17	w	w	NOUN
ejpam-178	69	18	:	:	PUNCT
ejpam-178	69	19	(	(	PUNCT
ejpam-178	69	20	xk	xk	PROPN
ejpam-178	69	21	yk	yk	PROPN
ejpam-178	69	22	)	)	PUNCT
ejpam-178	69	23	∈	∈	PROPN
ejpam-178	69	24	f	f	PROPN
ejpam-178	69	25	for	for	ADP
ejpam-178	69	26	all	all	DET
ejpam-178	69	27	(	(	PUNCT
ejpam-178	69	28	yk	yk	PROPN
ejpam-178	69	29	)	)	PUNCT
ejpam-178	69	30	∈	∈	PROPN
ejpam-178	69	31	e	e	NOUN
ejpam-178	69	32	}	}	PUNCT
ejpam-178	69	33	.	.	PUNCT
ejpam-178	70	1	for	for	ADP
ejpam-178	70	2	f	f	NOUN
ejpam-178	70	3	=	=	SYM
ejpam-178	70	4	ℓ1	ℓ1	PROPN
ejpam-178	70	5	and	and	CCONJ
ejpam-178	70	6	c0	c0	NOUN
ejpam-178	70	7	,	,	PUNCT
ejpam-178	70	8	the	the	DET
ejpam-178	70	9	duals	dual	NOUN
ejpam-178	70	10	are	be	AUX
ejpam-178	70	11	termed	term	VERB
ejpam-178	70	12	as	as	ADP
ejpam-178	70	13	α-(or	α-(or	PROPN
ejpam-178	70	14	köthe	köthe	PROPN
ejpam-178	70	15	-	-	PUNCT
ejpam-178	70	16	toeplitz	toeplitz	NOUN
ejpam-178	70	17	)	)	PUNCT
ejpam-178	70	18	dual	dual	ADJ
ejpam-178	70	19	and	and	CCONJ
ejpam-178	70	20	n	n	CCONJ
ejpam-178	70	21	-(or	-(or	PROPN
ejpam-178	70	22	null	null	NOUN
ejpam-178	70	23	)	)	PUNCT
ejpam-178	70	24	dual	dual	ADJ
ejpam-178	70	25	of	of	ADP
ejpam-178	70	26	e	e	NOUN
ejpam-178	70	27	and	and	CCONJ
ejpam-178	70	28	denoted	denote	VERB
ejpam-178	70	29	by	by	ADP
ejpam-178	70	30	eα	eα	NOUN
ejpam-178	70	31	and	and	CCONJ
ejpam-178	70	32	en	en	ADP
ejpam-178	70	33	respectively	respectively	ADV
ejpam-178	70	34	.	.	PUNCT
ejpam-178	71	1	if	if	SCONJ
ejpam-178	71	2	x	x	PROPN
ejpam-178	71	3	⊂	⊂	PROPN
ejpam-178	71	4	y	y	PROPN
ejpam-178	71	5	,	,	PUNCT
ejpam-178	71	6	then	then	ADV
ejpam-178	71	7	y	y	PROPN
ejpam-178	71	8	z	z	PROPN
ejpam-178	71	9	⊂	⊂	PROPN
ejpam-178	71	10	x	x	X
ejpam-178	71	11	z	z	NOUN
ejpam-178	71	12	for	for	ADP
ejpam-178	71	13	z	z	PROPN
ejpam-178	71	14	=	=	SYM
ejpam-178	71	15	α	α	PROPN
ejpam-178	71	16	,	,	PUNCT
ejpam-178	71	17	n	n	PROPN
ejpam-178	71	18	.	.	PUNCT
ejpam-178	72	1	lemma	lemma	PROPN
ejpam-178	72	2	1	1	NUM
ejpam-178	72	3	.	.	PUNCT
ejpam-178	72	4	x	x	SYM
ejpam-178	72	5	∈	∈	PROPN
ejpam-178	72	6	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	72	7	,	,	PUNCT
ejpam-178	72	8	λ,∆	λ,∆	NUM
ejpam-178	72	9	)	)	PUNCT
ejpam-178	72	10	implies	imply	VERB
ejpam-178	72	11	sup	sup	PROPN
ejpam-178	72	12	k	k	PROPN
ejpam-178	72	13	m	m	PROPN
ejpam-178	72	14	(	(	PUNCT
ejpam-178	72	15	|k−1λk	|k−1λk	PROPN
ejpam-178	72	16	xk	xk	PROPN
ejpam-178	72	17	|	|	PROPN
ejpam-178	72	18	ρ	ρ	PROPN
ejpam-178	72	19	)	)	PUNCT
ejpam-178	72	20	<	<	X
ejpam-178	72	21	∞	∞	PROPN
ejpam-178	72	22	,	,	PUNCT
ejpam-178	72	23	for	for	ADP
ejpam-178	72	24	some	some	DET
ejpam-178	72	25	ρ	ρ	NOUN
ejpam-178	72	26	>	>	X
ejpam-178	72	27	0	0	PROPN
ejpam-178	72	28	.	.	PUNCT
ejpam-178	73	1	proof	proof	NOUN
ejpam-178	73	2	.	.	PUNCT
ejpam-178	74	1	let	let	VERB
ejpam-178	74	2	x	x	SYM
ejpam-178	74	3	∈	∈	PROPN
ejpam-178	74	4	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	74	5	,	,	PUNCT
ejpam-178	74	6	λ,∆	λ,∆	NUM
ejpam-178	74	7	)	)	PUNCT
ejpam-178	74	8	,	,	PUNCT
ejpam-178	74	9	then	then	ADV
ejpam-178	74	10	sup	sup	NOUN
ejpam-178	74	11	k	k	PROPN
ejpam-178	74	12	m	m	PROPN
ejpam-178	74	13	(	(	PUNCT
ejpam-178	74	14	|λk	|λk	INTJ
ejpam-178	74	15	xk	xk	PROPN
ejpam-178	74	16	−λk+1xk+1|	−λk+1xk+1|	PROPN
ejpam-178	74	17	ρ	ρ	PROPN
ejpam-178	74	18	)	)	PUNCT
ejpam-178	74	19	<	<	X
ejpam-178	74	20	∞	∞	PROPN
ejpam-178	74	21	,	,	PUNCT
ejpam-178	74	22	for	for	ADP
ejpam-178	74	23	some	some	DET
ejpam-178	74	24	ρ	ρ	NOUN
ejpam-178	74	25	>	>	X
ejpam-178	74	26	0	0	PROPN
ejpam-178	74	27	.	.	PUNCT
ejpam-178	75	1	then	then	ADV
ejpam-178	75	2	there	there	PRON
ejpam-178	75	3	exists	exist	VERB
ejpam-178	75	4	a	a	DET
ejpam-178	75	5	u	u	NOUN
ejpam-178	75	6	>	>	X
ejpam-178	75	7	0	0	NUM
ejpam-178	75	8	such	such	ADJ
ejpam-178	75	9	that	that	SCONJ
ejpam-178	75	10	m	m	PROPN
ejpam-178	75	11	(	(	PUNCT
ejpam-178	75	12	|λk	|λk	NUM
ejpam-178	75	13	xk−λk+1xk+1|	xk−λk+1xk+1|	PROPN
ejpam-178	75	14	ρ	ρ	NOUN
ejpam-178	75	15	)	)	PUNCT
ejpam-178	75	16	<	<	X
ejpam-178	75	17	u	u	X
ejpam-178	75	18	,	,	PUNCT
ejpam-178	75	19	for	for	ADP
ejpam-178	75	20	all	all	DET
ejpam-178	75	21	k	k	PROPN
ejpam-178	75	22	∈	∈	PROPN
ejpam-178	75	23	n	n	X
ejpam-178	75	24	.	.	PUNCT
ejpam-178	76	1	taking	take	VERB
ejpam-178	76	2	η	η	PROPN
ejpam-178	76	3	=	=	PROPN
ejpam-178	76	4	kρ	kρ	PROPN
ejpam-178	76	5	,	,	PUNCT
ejpam-178	76	6	for	for	ADP
ejpam-178	76	7	an	an	DET
ejpam-178	76	8	arbitrary	arbitrary	ADJ
ejpam-178	76	9	fixed	fix	VERB
ejpam-178	76	10	positive	positive	ADJ
ejpam-178	76	11	integer	integer	NOUN
ejpam-178	76	12	k	k	PROPN
ejpam-178	76	13	,	,	PUNCT
ejpam-178	76	14	by	by	ADP
ejpam-178	76	15	the	the	DET
ejpam-178	76	16	subadditivity	subadditivity	NOUN
ejpam-178	76	17	of	of	ADP
ejpam-178	76	18	modulus	modulus	NOUN
ejpam-178	76	19	,	,	PUNCT
ejpam-178	76	20	the	the	DET
ejpam-178	76	21	monotonicity	monotonicity	NOUN
ejpam-178	76	22	and	and	CCONJ
ejpam-178	76	23	convexity	convexity	NOUN
ejpam-178	76	24	of	of	ADP
ejpam-178	76	25	m	m	PROPN
ejpam-178	76	26	:	:	PUNCT
ejpam-178	76	27	m	m	VERB
ejpam-178	76	28	(	(	PUNCT
ejpam-178	76	29	|λ1x1−λk+1xk+1|	|λ1x1−λk+1xk+1|	PROPN
ejpam-178	76	30	η	η	NOUN
ejpam-178	76	31	)	)	PUNCT
ejpam-178	76	32	<	<	X
ejpam-178	77	1	1	1	NUM
ejpam-178	77	2	k	k	X
ejpam-178	77	3	k	k	NOUN
ejpam-178	77	4	∑	∑	PUNCT
ejpam-178	77	5	l=1	l=1	VERB
ejpam-178	77	6	m	m	VERB
ejpam-178	77	7	(	(	PUNCT
ejpam-178	77	8	|λl	|λl	X
ejpam-178	77	9	x	x	PART
ejpam-178	77	10	l	l	NOUN
ejpam-178	77	11	−λl+1x	−λl+1x	NOUN
ejpam-178	77	12	l+1|	l+1|	PROPN
ejpam-178	77	13	ρ	ρ	PROPN
ejpam-178	77	14	)	)	PUNCT
ejpam-178	77	15	<	<	X
ejpam-178	77	16	u	u	PROPN
ejpam-178	77	17	.	.	PUNCT
ejpam-178	78	1	then	then	ADV
ejpam-178	78	2	the	the	DET
ejpam-178	78	3	above	above	ADJ
ejpam-178	78	4	inequality	inequality	NOUN
ejpam-178	78	5	,	,	PUNCT
ejpam-178	78	6	the	the	DET
ejpam-178	78	7	inequality	inequality	NOUN
ejpam-178	78	8	|λk+1xk+1|	|λk+1xk+1|	PROPN
ejpam-178	78	9	(	(	PUNCT
ejpam-178	78	10	k+	k+	NOUN
ejpam-178	78	11	1)ρ	1)ρ	NUM
ejpam-178	78	12	≤	≤	ADV
ejpam-178	78	13	1	1	NUM
ejpam-178	78	14	k+	k+	NOUN
ejpam-178	78	15	1	1	NUM
ejpam-178	78	16	(	(	PUNCT
ejpam-178	78	17	|λ1x1|	|λ1x1|	NOUN
ejpam-178	78	18	ρ	ρ	NOUN
ejpam-178	79	1	+	+	CCONJ
ejpam-178	79	2	k	k	PROPN
ejpam-178	79	3	|λ1	|λ1	PROPN
ejpam-178	79	4	x1−λk+1xk+1|	x1−λk+1xk+1|	PROPN
ejpam-178	79	5	kρ	kρ	PROPN
ejpam-178	79	6	)	)	PUNCT
ejpam-178	79	7	and	and	CCONJ
ejpam-178	79	8	the	the	DET
ejpam-178	79	9	convexity	convexity	NOUN
ejpam-178	79	10	of	of	ADP
ejpam-178	79	11	m	m	PROPN
ejpam-178	79	12	imply	imply	VERB
ejpam-178	79	13	m	m	PROPN
ejpam-178	79	14	(	(	PUNCT
ejpam-178	79	15	|λk+1xk+1|	|λk+1xk+1|	PROPN
ejpam-178	79	16	(	(	PUNCT
ejpam-178	79	17	k+	k+	NOUN
ejpam-178	79	18	1)ρ	1)ρ	NUM
ejpam-178	79	19	)	)	PUNCT
ejpam-178	79	20	≤	≤	NUM
ejpam-178	79	21	1	1	NUM
ejpam-178	79	22	k+	k+	NOUN
ejpam-178	79	23	1	1	NUM
ejpam-178	79	24	(	(	PUNCT
ejpam-178	79	25	m	m	NOUN
ejpam-178	79	26	(	(	PUNCT
ejpam-178	79	27	|λ1	|λ1	NOUN
ejpam-178	80	1	x1|	x1|	PROPN
ejpam-178	80	2	ρ	ρ	PROPN
ejpam-178	80	3	)	)	PUNCT
ejpam-178	81	1	+	+	NUM
ejpam-178	81	2	km	km	NOUN
ejpam-178	81	3	(	(	PUNCT
ejpam-178	81	4	|λ1	|λ1	NOUN
ejpam-178	81	5	x1−λk+1xk+1|	x1−λk+1xk+1|	PROPN
ejpam-178	81	6	kρ	kρ	PROPN
ejpam-178	81	7	)	)	PUNCT
ejpam-178	81	8	)	)	PUNCT
ejpam-178	82	1	h.	h.	PROPN
ejpam-178	82	2	dutta	dutta	PROPN
ejpam-178	82	3	/	/	PUNCT
ejpam-178	82	4	eur	eur	PROPN
ejpam-178	82	5	.	.	PUNCT
ejpam-178	83	1	j.	j.	PROPN
ejpam-178	83	2	pure	pure	PROPN
ejpam-178	83	3	appl	appl	PROPN
ejpam-178	83	4	.	.	PROPN
ejpam-178	83	5	math	math	PROPN
ejpam-178	83	6	,	,	PUNCT
ejpam-178	83	7	2	2	NUM
ejpam-178	83	8	(	(	PUNCT
ejpam-178	83	9	2009	2009	NUM
ejpam-178	83	10	)	)	PUNCT
ejpam-178	83	11	,	,	PUNCT
ejpam-178	83	12	(	(	PUNCT
ejpam-178	83	13	554	554	NUM
ejpam-178	83	14	-	-	SYM
ejpam-178	83	15	563	563	NUM
ejpam-178	83	16	)	)	PUNCT
ejpam-178	84	1	559	559	NUM
ejpam-178	84	2	≤	≤	NUM
ejpam-178	84	3	max{m	max{m	ADJ
ejpam-178	84	4	(	(	PUNCT
ejpam-178	84	5	|λ1	|λ1	NOUN
ejpam-178	84	6	x1|	x1|	PROPN
ejpam-178	84	7	ρ	ρ	PROPN
ejpam-178	84	8	)	)	PUNCT
ejpam-178	84	9	,	,	PUNCT
ejpam-178	84	10	u}<∞	u}<∞	PRON
ejpam-178	84	11	hence	hence	ADV
ejpam-178	84	12	we	we	PRON
ejpam-178	84	13	have	have	VERB
ejpam-178	84	14	the	the	DET
ejpam-178	84	15	desired	desire	VERB
ejpam-178	84	16	result	result	NOUN
ejpam-178	84	17	.	.	PUNCT
ejpam-178	85	1	lemma	lemma	PROPN
ejpam-178	85	2	2	2	NUM
ejpam-178	85	3	.	.	PUNCT
ejpam-178	85	4	x	x	SYM
ejpam-178	85	5	∈	∈	PROPN
ejpam-178	85	6	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	85	7	,	,	PUNCT
ejpam-178	85	8	λ,∆	λ,∆	NUM
ejpam-178	85	9	)	)	PUNCT
ejpam-178	85	10	implies	imply	VERB
ejpam-178	85	11	sup	sup	PROPN
ejpam-178	85	12	k	k	PROPN
ejpam-178	85	13	k−1|λk	k−1|λk	PROPN
ejpam-178	85	14	xk|	xk|	PROPN
ejpam-178	86	1	<	<	X
ejpam-178	86	2	∞.	∞.	PROPN
ejpam-178	86	3	proof	proof	NOUN
ejpam-178	86	4	.	.	PUNCT
ejpam-178	87	1	proof	proof	NOUN
ejpam-178	87	2	is	be	AUX
ejpam-178	87	3	obvious	obvious	ADJ
ejpam-178	87	4	by	by	ADP
ejpam-178	87	5	using	use	VERB
ejpam-178	87	6	lemma	lemma	PROPN
ejpam-178	87	7	1	1	NUM
ejpam-178	87	8	.	.	PUNCT
ejpam-178	87	9	remark	remark	PROPN
ejpam-178	87	10	1	1	NUM
ejpam-178	87	11	.	.	PUNCT
ejpam-178	87	12	similar	similar	ADJ
ejpam-178	87	13	results	result	NOUN
ejpam-178	87	14	as	as	ADP
ejpam-178	87	15	in	in	ADP
ejpam-178	87	16	lemma	lemma	PROPN
ejpam-178	87	17	1	1	NUM
ejpam-178	87	18	and	and	CCONJ
ejpam-178	87	19	lemma	lemma	PROPN
ejpam-178	87	20	2	2	NUM
ejpam-178	87	21	hold	hold	VERB
ejpam-178	87	22	for	for	ADP
ejpam-178	87	23	ℓ̃∞(m	ℓ̃∞(m	NOUN
ejpam-178	87	24	,	,	PUNCT
ejpam-178	87	25	λ,∆	λ,∆	NUM
ejpam-178	87	26	)	)	PUNCT
ejpam-178	87	27	also	also	ADV
ejpam-178	87	28	,	,	PUNCT
ejpam-178	87	29	where	where	SCONJ
ejpam-178	87	30	the	the	DET
ejpam-178	87	31	statement	statement	NOUN
ejpam-178	87	32	’	'	PUNCT
ejpam-178	87	33	for	for	ADP
ejpam-178	87	34	some	some	DET
ejpam-178	87	35	ρ	ρ	PROPN
ejpam-178	87	36	>	>	X
ejpam-178	87	37	0	0	NUM
ejpam-178	87	38	’	'	PUNCT
ejpam-178	87	39	should	should	AUX
ejpam-178	87	40	be	be	AUX
ejpam-178	87	41	replaced	replace	VERB
ejpam-178	87	42	by	by	ADP
ejpam-178	87	43	’	'	PUNCT
ejpam-178	87	44	for	for	ADP
ejpam-178	87	45	every	every	DET
ejpam-178	87	46	ρ	ρ	PROPN
ejpam-178	87	47	>	>	X
ejpam-178	87	48	0	0	NUM
ejpam-178	87	49	’	'	PUNCT
ejpam-178	87	50	.	.	PUNCT
ejpam-178	88	1	for	for	ADP
ejpam-178	88	2	the	the	DET
ejpam-178	88	3	next	next	ADJ
ejpam-178	88	4	theorem	theorem	NOUN
ejpam-178	88	5	,	,	PUNCT
ejpam-178	88	6	let	let	VERB
ejpam-178	88	7	d1	d1	PROPN
ejpam-178	88	8	=	=	PUNCT
ejpam-178	88	9	{	{	PUNCT
ejpam-178	88	10	a	a	X
ejpam-178	88	11	=	=	X
ejpam-178	88	12	(	(	PUNCT
ejpam-178	88	13	ak	ak	PROPN
ejpam-178	88	14	)	)	PUNCT
ejpam-178	88	15	:	:	PUNCT
ejpam-178	89	1	∞	∞	NUM
ejpam-178	89	2	∑	∑	PUNCT
ejpam-178	90	1	k=1	k=1	PROPN
ejpam-178	90	2	k|λ−1	k|λ−1	PROPN
ejpam-178	91	1	k	k	PROPN
ejpam-178	92	1	ak|	ak|	PROPN
ejpam-178	92	2	<	<	X
ejpam-178	92	3	∞	∞	PROPN
ejpam-178	92	4	}	}	PUNCT
ejpam-178	92	5	,	,	PUNCT
ejpam-178	92	6	d2	d2	PROPN
ejpam-178	92	7	=	=	SYM
ejpam-178	92	8	{	{	PUNCT
ejpam-178	92	9	b	b	NOUN
ejpam-178	92	10	=	=	SYM
ejpam-178	92	11	(	(	PUNCT
ejpam-178	92	12	bk	bk	PROPN
ejpam-178	92	13	)	)	PUNCT
ejpam-178	92	14	:	:	PUNCT
ejpam-178	92	15	sup	sup	PROPN
ejpam-178	92	16	k	k	PROPN
ejpam-178	92	17	k−1|λk	k−1|λk	PROPN
ejpam-178	92	18	bk|	bk|	PROPN
ejpam-178	92	19	<	<	X
ejpam-178	92	20	∞	∞	NUM
ejpam-178	92	21	}	}	PUNCT
ejpam-178	92	22	.	.	PUNCT
ejpam-178	93	1	theorem	theorem	NOUN
ejpam-178	93	2	1	1	NUM
ejpam-178	93	3	.	.	PUNCT
ejpam-178	94	1	let	let	VERB
ejpam-178	94	2	m	m	PRON
ejpam-178	94	3	be	be	AUX
ejpam-178	94	4	an	an	DET
ejpam-178	94	5	orlicz	orlicz	ADJ
ejpam-178	94	6	function	function	NOUN
ejpam-178	94	7	.	.	PUNCT
ejpam-178	95	1	then	then	ADV
ejpam-178	95	2	(	(	PUNCT
ejpam-178	95	3	i	i	NOUN
ejpam-178	95	4	)	)	PUNCT
ejpam-178	96	1	[	[	X
ejpam-178	96	2	c(m	c(m	PROPN
ejpam-178	96	3	,	,	PUNCT
ejpam-178	96	4	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	96	5	=	=	PUNCT
ejpam-178	97	1	[	[	X
ejpam-178	97	2	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	97	3	,	,	PUNCT
ejpam-178	97	4	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	97	5	=	=	SYM
ejpam-178	97	6	d1	d1	PROPN
ejpam-178	97	7	,	,	PUNCT
ejpam-178	97	8	(	(	PUNCT
ejpam-178	97	9	ii	ii	NOUN
ejpam-178	97	10	)	)	PUNCT
ejpam-178	98	1	[	[	X
ejpam-178	98	2	c̃(m	c̃(m	PROPN
ejpam-178	98	3	,	,	PUNCT
ejpam-178	98	4	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	98	5	=	=	PUNCT
ejpam-178	99	1	[	[	X
ejpam-178	99	2	ℓ̃∞(m	ℓ̃∞(m	X
ejpam-178	99	3	,	,	PUNCT
ejpam-178	99	4	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	99	5	=	=	SYM
ejpam-178	99	6	d1	d1	PROPN
ejpam-178	99	7	,	,	PUNCT
ejpam-178	99	8	(	(	PUNCT
ejpam-178	99	9	iii	iii	X
ejpam-178	99	10	)	)	PUNCT
ejpam-178	99	11	dα	dα	PRON
ejpam-178	99	12	1	1	NUM
ejpam-178	99	13	=	=	SYM
ejpam-178	99	14	d2	d2	PROPN
ejpam-178	99	15	.	.	PUNCT
ejpam-178	100	1	proof	proof	NOUN
ejpam-178	100	2	.	.	PUNCT
ejpam-178	101	1	(	(	PUNCT
ejpam-178	101	2	i	i	NOUN
ejpam-178	101	3	)	)	PUNCT
ejpam-178	101	4	let	let	VERB
ejpam-178	101	5	a	a	DET
ejpam-178	101	6	∈	∈	NOUN
ejpam-178	101	7	d1	d1	NOUN
ejpam-178	101	8	,	,	PUNCT
ejpam-178	101	9	then	then	ADV
ejpam-178	101	10	∞	∞	NUM
ejpam-178	101	11	∑	∑	PUNCT
ejpam-178	102	1	k=1	k=1	PROPN
ejpam-178	102	2	|kλ−1	|kλ−1	PROPN
ejpam-178	103	1	k	k	PROPN
ejpam-178	103	2	ak|	ak|	PROPN
ejpam-178	103	3	<	<	X
ejpam-178	103	4	∞.	∞.	PROPN
ejpam-178	103	5	now	now	ADV
ejpam-178	103	6	for	for	ADP
ejpam-178	103	7	any	any	DET
ejpam-178	103	8	x	x	SYM
ejpam-178	103	9	∈	∈	PROPN
ejpam-178	103	10	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	103	11	,	,	PUNCT
ejpam-178	103	12	λ,∆	λ,∆	NUM
ejpam-178	103	13	)	)	PUNCT
ejpam-178	103	14	we	we	PRON
ejpam-178	103	15	have	have	VERB
ejpam-178	103	16	sup	sup	NOUN
ejpam-178	103	17	k	k	PROPN
ejpam-178	103	18	|k−1λk	|k−1λk	PROPN
ejpam-178	103	19	xk|<∞.	xk|<∞.	PROPN
ejpam-178	103	20	then	then	ADV
ejpam-178	103	21	we	we	PRON
ejpam-178	103	22	have	have	VERB
ejpam-178	103	23	∞	∞	PROPN
ejpam-178	103	24	∑	∑	PROPN
ejpam-178	103	25	k=1	k=1	ADJ
ejpam-178	103	26	|ak	|ak	PUNCT
ejpam-178	103	27	xk|	xk|	PROPN
ejpam-178	103	28	≤	≤	PROPN
ejpam-178	103	29	sup	sup	NOUN
ejpam-178	103	30	k	k	PROPN
ejpam-178	103	31	|k−1λk	|k−1λk	PROPN
ejpam-178	103	32	xk|	xk|	PROPN
ejpam-178	103	33	∞	∞	PROPN
ejpam-178	103	34	∑	∑	PUNCT
ejpam-178	104	1	k=1	k=1	PROPN
ejpam-178	104	2	|kλ−1	|kλ−1	PROPN
ejpam-178	105	1	k	k	PROPN
ejpam-178	106	1	ak|<∞.	ak|<∞.	X
ejpam-178	106	2	hence	hence	ADV
ejpam-178	106	3	a	a	DET
ejpam-178	106	4	∈	∈	NOUN
ejpam-178	106	5	[	[	X
ejpam-178	106	6	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	106	7	,	,	PUNCT
ejpam-178	106	8	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	106	9	.	.	PUNCT
ejpam-178	106	10	thus	thus	ADV
ejpam-178	106	11	d1	d1	PROPN
ejpam-178	106	12	⊆	⊆	NUM
ejpam-178	106	13	[	[	X
ejpam-178	106	14	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	106	15	,	,	PUNCT
ejpam-178	106	16	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	106	17	(	(	PUNCT
ejpam-178	106	18	1	1	NUM
ejpam-178	106	19	)	)	PUNCT
ejpam-178	106	20	again	again	ADV
ejpam-178	106	21	we	we	PRON
ejpam-178	106	22	know	know	VERB
ejpam-178	106	23	[	[	X
ejpam-178	106	24	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	106	25	,	,	PUNCT
ejpam-178	106	26	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	106	27	⊆	⊆	NUM
ejpam-178	106	28	[	[	X
ejpam-178	106	29	c(m	c(m	PROPN
ejpam-178	106	30	,	,	PUNCT
ejpam-178	106	31	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	106	32	⊆	⊆	NUM
ejpam-178	106	33	[	[	X
ejpam-178	106	34	c0(m	c0(m	X
ejpam-178	106	35	,	,	PUNCT
ejpam-178	106	36	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	106	37	(	(	PUNCT
ejpam-178	106	38	2	2	X
ejpam-178	106	39	)	)	PUNCT
ejpam-178	106	40	h.	h.	NOUN
ejpam-178	106	41	dutta	dutta	PROPN
ejpam-178	106	42	/	/	PUNCT
ejpam-178	106	43	eur	eur	PROPN
ejpam-178	106	44	.	.	PUNCT
ejpam-178	107	1	j.	j.	PROPN
ejpam-178	107	2	pure	pure	PROPN
ejpam-178	107	3	appl	appl	PROPN
ejpam-178	107	4	.	.	PROPN
ejpam-178	107	5	math	math	PROPN
ejpam-178	107	6	,	,	PUNCT
ejpam-178	107	7	2	2	NUM
ejpam-178	107	8	(	(	PUNCT
ejpam-178	107	9	2009	2009	NUM
ejpam-178	107	10	)	)	PUNCT
ejpam-178	107	11	,	,	PUNCT
ejpam-178	107	12	(	(	PUNCT
ejpam-178	107	13	554	554	NUM
ejpam-178	107	14	-	-	SYM
ejpam-178	107	15	563	563	NUM
ejpam-178	107	16	)	)	PUNCT
ejpam-178	107	17	560	560	NUM
ejpam-178	107	18	conversely	conversely	ADV
ejpam-178	107	19	suppose	suppose	VERB
ejpam-178	107	20	that	that	SCONJ
ejpam-178	107	21	a	a	DET
ejpam-178	107	22	∈	∈	PROPN
ejpam-178	108	1	[	[	X
ejpam-178	108	2	c(m	c(m	PROPN
ejpam-178	108	3	,	,	PUNCT
ejpam-178	108	4	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	108	5	.	.	PUNCT
ejpam-178	109	1	then	then	ADV
ejpam-178	109	2	∞	∞	NUM
ejpam-178	109	3	∑	∑	PROPN
ejpam-178	109	4	k=1	k=1	X
ejpam-178	109	5	|ak	|ak	X
ejpam-178	109	6	xk|	xk|	PROPN
ejpam-178	109	7	<	<	X
ejpam-178	109	8	∞	∞	PROPN
ejpam-178	109	9	,	,	PUNCT
ejpam-178	109	10	for	for	ADP
ejpam-178	109	11	each	each	DET
ejpam-178	109	12	x	x	SYM
ejpam-178	109	13	∈	∈	PROPN
ejpam-178	109	14	c(m	c(m	PROPN
ejpam-178	109	15	,	,	PUNCT
ejpam-178	109	16	λ,∆	λ,∆	NUM
ejpam-178	109	17	)	)	PUNCT
ejpam-178	109	18	.	.	PUNCT
ejpam-178	110	1	so	so	ADV
ejpam-178	110	2	we	we	PRON
ejpam-178	110	3	take	take	VERB
ejpam-178	110	4	xk	xk	NOUN
ejpam-178	111	1	=	=	PUNCT
ejpam-178	111	2	λ	λ	PROPN
ejpam-178	111	3	−1	−1	NOUN
ejpam-178	111	4	k	k	PROPN
ejpam-178	111	5	k	k	PROPN
ejpam-178	111	6	,	,	PUNCT
ejpam-178	111	7	k	k	PROPN
ejpam-178	111	8	≥	≥	NUM
ejpam-178	111	9	1	1	NUM
ejpam-178	111	10	then	then	ADV
ejpam-178	111	11	∞	∞	NUM
ejpam-178	111	12	∑	∑	PUNCT
ejpam-178	111	13	k=1	k=1	PROPN
ejpam-178	111	14	|kλ−1	|kλ−1	PROPN
ejpam-178	112	1	k	k	PROPN
ejpam-178	112	2	ak|	ak|	PROPN
ejpam-178	113	1	=	=	SYM
ejpam-178	113	2	∞	∞	NUM
ejpam-178	113	3	∑	∑	PUNCT
ejpam-178	113	4	k=1	k=1	X
ejpam-178	113	5	|ak	|ak	X
ejpam-178	113	6	xk|<∞.	xk|<∞.	PUNCT
ejpam-178	113	7	this	this	PRON
ejpam-178	113	8	implies	imply	VERB
ejpam-178	113	9	that	that	SCONJ
ejpam-178	113	10	a	a	DET
ejpam-178	113	11	∈	∈	PROPN
ejpam-178	113	12	d1	d1	NOUN
ejpam-178	113	13	.	.	PUNCT
ejpam-178	114	1	thus	thus	ADV
ejpam-178	114	2	[	[	X
ejpam-178	114	3	c(m	c(m	PROPN
ejpam-178	114	4	,	,	PUNCT
ejpam-178	114	5	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	114	6	⊆	⊆	NUM
ejpam-178	114	7	d1	d1	NOUN
ejpam-178	114	8	.	.	PUNCT
ejpam-178	115	1	(	(	PUNCT
ejpam-178	115	2	3	3	X
ejpam-178	115	3	)	)	PUNCT
ejpam-178	115	4	combining	combine	VERB
ejpam-178	115	5	(	(	PUNCT
ejpam-178	115	6	3	3	NUM
ejpam-178	115	7	)	)	PUNCT
ejpam-178	115	8	with	with	ADP
ejpam-178	115	9	(	(	PUNCT
ejpam-178	115	10	1	1	NUM
ejpam-178	115	11	)	)	PUNCT
ejpam-178	115	12	,	,	PUNCT
ejpam-178	115	13	(	(	PUNCT
ejpam-178	115	14	2	2	X
ejpam-178	115	15	)	)	PUNCT
ejpam-178	115	16	it	it	PRON
ejpam-178	115	17	follows	follow	VERB
ejpam-178	115	18	[	[	PRON
ejpam-178	115	19	c(m	c(m	PROPN
ejpam-178	115	20	,	,	PUNCT
ejpam-178	115	21	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	115	22	=	=	PUNCT
ejpam-178	116	1	[	[	X
ejpam-178	116	2	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	116	3	,	,	PUNCT
ejpam-178	116	4	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	116	5	=	=	PUNCT
ejpam-178	116	6	d1	d1	PROPN
ejpam-178	116	7	this	this	PRON
ejpam-178	116	8	completes	complete	VERB
ejpam-178	116	9	the	the	DET
ejpam-178	116	10	proof	proof	NOUN
ejpam-178	116	11	of	of	ADP
ejpam-178	116	12	part(i	part(i	PROPN
ejpam-178	116	13	)	)	PUNCT
ejpam-178	116	14	.	.	PUNCT
ejpam-178	117	1	(	(	PUNCT
ejpam-178	117	2	ii	ii	NOUN
ejpam-178	117	3	)	)	PUNCT
ejpam-178	117	4	proof	proof	NOUN
ejpam-178	117	5	is	be	AUX
ejpam-178	117	6	similar	similar	ADJ
ejpam-178	117	7	to	to	ADP
ejpam-178	117	8	that	that	PRON
ejpam-178	117	9	of	of	ADP
ejpam-178	117	10	part	part	NOUN
ejpam-178	117	11	(	(	PUNCT
ejpam-178	117	12	i	i	NOUN
ejpam-178	117	13	)	)	PUNCT
ejpam-178	117	14	.	.	PUNCT
ejpam-178	118	1	(	(	PUNCT
ejpam-178	118	2	iii	iii	X
ejpam-178	118	3	)	)	PUNCT
ejpam-178	118	4	the	the	DET
ejpam-178	118	5	proof	proof	NOUN
ejpam-178	118	6	of	of	ADP
ejpam-178	118	7	the	the	DET
ejpam-178	118	8	inclusion	inclusion	NOUN
ejpam-178	118	9	dα	dα	VERB
ejpam-178	118	10	1	1	NUM
ejpam-178	118	11	⊇	⊇	PROPN
ejpam-178	118	12	d2	d2	PROPN
ejpam-178	118	13	is	be	AUX
ejpam-178	118	14	similar	similar	ADJ
ejpam-178	118	15	to	to	ADP
ejpam-178	118	16	that	that	PRON
ejpam-178	118	17	of	of	ADP
ejpam-178	118	18	d1	d1	PROPN
ejpam-178	118	19	⊆	⊆	NUM
ejpam-178	118	20	[	[	X
ejpam-178	118	21	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	118	22	,	,	PUNCT
ejpam-178	118	23	λ,∆)]α	λ,∆)]α	PROPN
ejpam-178	118	24	.	.	PUNCT
ejpam-178	119	1	for	for	ADP
ejpam-178	119	2	the	the	DET
ejpam-178	119	3	converse	converse	NOUN
ejpam-178	119	4	part	part	NOUN
ejpam-178	119	5	suppose	suppose	VERB
ejpam-178	119	6	a	a	DET
ejpam-178	119	7	∈	∈	NOUN
ejpam-178	119	8	dα	dα	VERB
ejpam-178	119	9	1	1	NUM
ejpam-178	119	10	and	and	CCONJ
ejpam-178	119	11	a	a	DET
ejpam-178	119	12	/∈	/∈	NOUN
ejpam-178	119	13	d2	d2	PROPN
ejpam-178	119	14	.	.	PUNCT
ejpam-178	120	1	then	then	ADV
ejpam-178	120	2	we	we	PRON
ejpam-178	120	3	have	have	VERB
ejpam-178	120	4	sup	sup	PROPN
ejpam-178	120	5	k	k	PROPN
ejpam-178	120	6	|k−1λkak|=∞	|k−1λkak|=∞	PRON
ejpam-178	120	7	hence	hence	ADV
ejpam-178	120	8	we	we	PRON
ejpam-178	120	9	can	can	AUX
ejpam-178	120	10	find	find	VERB
ejpam-178	120	11	a	a	DET
ejpam-178	120	12	strictly	strictly	ADV
ejpam-178	120	13	increasing	increase	VERB
ejpam-178	120	14	sequence	sequence	NOUN
ejpam-178	120	15	(	(	PUNCT
ejpam-178	120	16	k	k	PROPN
ejpam-178	120	17	j	j	PROPN
ejpam-178	120	18	)	)	PUNCT
ejpam-178	120	19	of	of	ADP
ejpam-178	120	20	positive	positive	ADJ
ejpam-178	120	21	integers	integer	NOUN
ejpam-178	120	22	k	k	PROPN
ejpam-178	120	23	j	j	PROPN
ejpam-178	120	24	such	such	ADJ
ejpam-178	120	25	that	that	SCONJ
ejpam-178	120	26	|k−1	|k−1	PROPN
ejpam-178	120	27	j	j	PROPN
ejpam-178	120	28	λk	λk	PROPN
ejpam-178	120	29	j	j	PROPN
ejpam-178	120	30	ak	ak	PROPN
ejpam-178	120	31	j	j	PROPN
ejpam-178	120	32	|	|	PROPN
ejpam-178	120	33	>	>	X
ejpam-178	120	34	j2	j2	PROPN
ejpam-178	120	35	for	for	ADP
ejpam-178	120	36	all	all	DET
ejpam-178	120	37	j	j	PROPN
ejpam-178	120	38	≥	≥	NUM
ejpam-178	120	39	1	1	NUM
ejpam-178	120	40	we	we	PRON
ejpam-178	120	41	define	define	VERB
ejpam-178	120	42	the	the	DET
ejpam-178	120	43	sequence	sequence	NOUN
ejpam-178	120	44	x	x	PUNCT
ejpam-178	120	45	by	by	ADP
ejpam-178	120	46	xk	xk	X
ejpam-178	120	47	=	=	PUNCT
ejpam-178	120	48			PROPN
ejpam-178	120	49			ADP
ejpam-178	120	50			ADJ
ejpam-178	120	51	|a−1	|a−1	NOUN
ejpam-178	120	52	k	k	PROPN
ejpam-178	121	1	j	j	PROPN
ejpam-178	121	2	|	|	ADV
ejpam-178	121	3	,	,	PUNCT
ejpam-178	121	4	if	if	SCONJ
ejpam-178	121	5	k	k	PROPN
ejpam-178	121	6	=	=	PUNCT
ejpam-178	121	7	k	k	PROPN
ejpam-178	121	8	j	j	PROPN
ejpam-178	121	9	0	0	NUM
ejpam-178	121	10	,	,	PUNCT
ejpam-178	121	11	otherwise	otherwise	ADV
ejpam-178	121	12	h.	h.	PROPN
ejpam-178	121	13	dutta	dutta	PROPN
ejpam-178	121	14	/	/	PUNCT
ejpam-178	121	15	eur	eur	PROPN
ejpam-178	121	16	.	.	PUNCT
ejpam-178	122	1	j.	j.	PROPN
ejpam-178	122	2	pure	pure	PROPN
ejpam-178	122	3	appl	appl	PROPN
ejpam-178	122	4	.	.	PROPN
ejpam-178	122	5	math	math	PROPN
ejpam-178	122	6	,	,	PUNCT
ejpam-178	122	7	2	2	NUM
ejpam-178	122	8	(	(	PUNCT
ejpam-178	122	9	2009	2009	NUM
ejpam-178	122	10	)	)	PUNCT
ejpam-178	122	11	,	,	PUNCT
ejpam-178	122	12	(	(	PUNCT
ejpam-178	122	13	554	554	NUM
ejpam-178	122	14	-	-	SYM
ejpam-178	122	15	563	563	NUM
ejpam-178	122	16	)	)	PUNCT
ejpam-178	122	17	561	561	NUM
ejpam-178	122	18	then	then	ADV
ejpam-178	122	19	x	x	SYM
ejpam-178	122	20	∈	∈	NOUN
ejpam-178	122	21	d1	d1	NOUN
ejpam-178	122	22	,	,	PUNCT
ejpam-178	122	23	because	because	SCONJ
ejpam-178	122	24	∞	∞	PROPN
ejpam-178	122	25	∑	∑	PUNCT
ejpam-178	122	26	k=1	k=1	PROPN
ejpam-178	122	27	|kλ−1	|kλ−1	PROPN
ejpam-178	123	1	k	k	PROPN
ejpam-178	123	2	xk|=	xk|=	PROPN
ejpam-178	123	3	∞	∞	PROPN
ejpam-178	123	4	∑	∑	PUNCT
ejpam-178	124	1	j=1	j=1	NOUN
ejpam-178	124	2	|k	|k	NOUN
ejpam-178	124	3	jλ	jλ	ADP
ejpam-178	124	4	−1	−1	NOUN
ejpam-178	124	5	k	k	PROPN
ejpam-178	124	6	j	j	PROPN
ejpam-178	124	7	a−1	a−1	PROPN
ejpam-178	124	8	k	k	PROPN
ejpam-178	124	9	j	j	PROPN
ejpam-178	125	1	|	|	ADV
ejpam-178	125	2	≤	≤	NUM
ejpam-178	125	3	∞	∞	NUM
ejpam-178	125	4	∑	∑	PUNCT
ejpam-178	125	5	j=1	j=1	PROPN
ejpam-178	125	6	j−2	j−2	PROPN
ejpam-178	125	7	<	<	X
ejpam-178	125	8	∞	∞	PROPN
ejpam-178	125	9	thus	thus	ADV
ejpam-178	125	10	x	x	SYM
ejpam-178	125	11	∈	∈	NOUN
ejpam-178	125	12	d1	d1	NOUN
ejpam-178	125	13	but	but	CCONJ
ejpam-178	125	14	∞	∞	NUM
ejpam-178	125	15	∑	∑	PROPN
ejpam-178	125	16	k=1	k=1	X
ejpam-178	125	17	|ak	|ak	PUNCT
ejpam-178	125	18	xk|=	xk|=	PROPN
ejpam-178	125	19	∞	∞	PROPN
ejpam-178	125	20	∑	∑	PROPN
ejpam-178	125	21	j=1	j=1	PROPN
ejpam-178	125	22	|ak	|ak	PUNCT
ejpam-178	125	23	j	j	PROPN
ejpam-178	125	24	xk	xk	PROPN
ejpam-178	125	25	j	j	PROPN
ejpam-178	125	26	|=∞.	|=∞.	VERB
ejpam-178	125	27	this	this	PRON
ejpam-178	125	28	is	be	AUX
ejpam-178	125	29	a	a	DET
ejpam-178	125	30	contradiction	contradiction	NOUN
ejpam-178	125	31	to	to	ADP
ejpam-178	125	32	a	a	DET
ejpam-178	125	33	∈	∈	NOUN
ejpam-178	125	34	dα	dα	ADP
ejpam-178	125	35	1	1	NUM
ejpam-178	125	36	.	.	PUNCT
ejpam-178	126	1	hence	hence	ADV
ejpam-178	126	2	a	a	DET
ejpam-178	126	3	∈	∈	PROPN
ejpam-178	126	4	d2	d2	NOUN
ejpam-178	126	5	.	.	PUNCT
ejpam-178	127	1	this	this	PRON
ejpam-178	127	2	completes	complete	VERB
ejpam-178	127	3	the	the	DET
ejpam-178	127	4	proof	proof	NOUN
ejpam-178	127	5	.	.	PUNCT
ejpam-178	128	1	if	if	SCONJ
ejpam-178	128	2	we	we	PRON
ejpam-178	128	3	take	take	VERB
ejpam-178	128	4	λk	λk	X
ejpam-178	128	5	=	=	SYM
ejpam-178	128	6	1	1	NUM
ejpam-178	128	7	,	,	PUNCT
ejpam-178	128	8	for	for	ADP
ejpam-178	128	9	all	all	DET
ejpam-178	128	10	k	k	PROPN
ejpam-178	128	11	∈	∈	PROPN
ejpam-178	128	12	n	n	NOUN
ejpam-178	128	13	in	in	ADP
ejpam-178	128	14	theorem	theorem	NOUN
ejpam-178	128	15	1	1	NUM
ejpam-178	128	16	,	,	PUNCT
ejpam-178	128	17	then	then	ADV
ejpam-178	128	18	we	we	PRON
ejpam-178	128	19	obtain	obtain	VERB
ejpam-178	128	20	the	the	DET
ejpam-178	128	21	following	follow	VERB
ejpam-178	128	22	corollary	corollary	NOUN
ejpam-178	128	23	.	.	PUNCT
ejpam-178	129	1	corollary	corollary	ADJ
ejpam-178	129	2	1	1	NUM
ejpam-178	129	3	.	.	PUNCT
ejpam-178	130	1	for	for	ADP
ejpam-178	130	2	x	x	X
ejpam-178	130	3	=	=	SYM
ejpam-178	130	4	c	c	PROPN
ejpam-178	130	5	and	and	CCONJ
ejpam-178	130	6	ℓ∞	ℓ∞	PROPN
ejpam-178	130	7	,	,	PUNCT
ejpam-178	130	8	(	(	PUNCT
ejpam-178	130	9	i	i	NOUN
ejpam-178	130	10	)	)	PUNCT
ejpam-178	131	1	[	[	X
ejpam-178	131	2	x	x	X
ejpam-178	131	3	(	(	PUNCT
ejpam-178	131	4	m	m	INTJ
ejpam-178	131	5	,	,	PUNCT
ejpam-178	131	6	∆)]α	∆)]α	PROPN
ejpam-178	131	7	=	=	PUNCT
ejpam-178	132	1	[	[	X
ejpam-178	132	2	x̃	x̃	PROPN
ejpam-178	132	3	(	(	PUNCT
ejpam-178	132	4	m	m	PROPN
ejpam-178	132	5	,	,	PUNCT
ejpam-178	132	6	∆)]α	∆)]α	PROPN
ejpam-178	132	7	=	=	PUNCT
ejpam-178	132	8	h1	h1	PROPN
ejpam-178	132	9	,	,	PUNCT
ejpam-178	132	10	(	(	PUNCT
ejpam-178	132	11	ii	ii	NOUN
ejpam-178	132	12	)	)	PUNCT
ejpam-178	132	13	hα	hα	ADP
ejpam-178	132	14	1	1	NUM
ejpam-178	132	15	=	=	SYM
ejpam-178	132	16	h2	h2	NOUN
ejpam-178	132	17	,	,	PUNCT
ejpam-178	132	18	where	where	SCONJ
ejpam-178	132	19	h1	h1	PROPN
ejpam-178	132	20	=	=	PRON
ejpam-178	132	21	{	{	PUNCT
ejpam-178	132	22	a	a	X
ejpam-178	132	23	=	=	X
ejpam-178	132	24	(	(	PUNCT
ejpam-178	132	25	ak	ak	PROPN
ejpam-178	132	26	)	)	PUNCT
ejpam-178	132	27	:	:	PUNCT
ejpam-178	133	1	∞	∞	NUM
ejpam-178	133	2	∑	∑	PUNCT
ejpam-178	133	3	k=1	k=1	ADJ
ejpam-178	133	4	|kak|<∞	|kak|<∞	VERB
ejpam-178	133	5	}	}	PUNCT
ejpam-178	133	6	and	and	CCONJ
ejpam-178	133	7	h2	h2	NOUN
ejpam-178	133	8	=	=	SYM
ejpam-178	133	9	{	{	PUNCT
ejpam-178	133	10	b	b	NOUN
ejpam-178	133	11	=	=	SYM
ejpam-178	133	12	(	(	PUNCT
ejpam-178	133	13	bk	bk	PROPN
ejpam-178	133	14	)	)	PUNCT
ejpam-178	133	15	:	:	PUNCT
ejpam-178	133	16	sup	sup	PROPN
ejpam-178	133	17	k	k	PROPN
ejpam-178	133	18	|k−1	|k−1	ADV
ejpam-178	133	19	bk|<∞	bk|<∞	NOUN
ejpam-178	133	20	}	}	PUNCT
ejpam-178	133	21	.	.	PUNCT
ejpam-178	134	1	for	for	ADP
ejpam-178	134	2	the	the	DET
ejpam-178	134	3	next	next	ADJ
ejpam-178	134	4	theorem	theorem	NOUN
ejpam-178	134	5	,	,	PUNCT
ejpam-178	134	6	let	let	VERB
ejpam-178	134	7	g1	g1	PROPN
ejpam-178	134	8	=	=	PRON
ejpam-178	134	9	{	{	PUNCT
ejpam-178	134	10	a	a	X
ejpam-178	134	11	=	=	X
ejpam-178	134	12	(	(	PUNCT
ejpam-178	134	13	ak	ak	PROPN
ejpam-178	134	14	)	)	PUNCT
ejpam-178	134	15	:	:	PUNCT
ejpam-178	135	1	lim	lim	PROPN
ejpam-178	135	2	k	k	PROPN
ejpam-178	135	3	kλ−1	kλ−1	PROPN
ejpam-178	135	4	k	k	PROPN
ejpam-178	135	5	ak	ak	PROPN
ejpam-178	135	6	=	=	PROPN
ejpam-178	135	7	0	0	NUM
ejpam-178	135	8	}	}	PUNCT
ejpam-178	135	9	.	.	PUNCT
ejpam-178	136	1	theorem	theorem	NOUN
ejpam-178	136	2	2	2	NUM
ejpam-178	136	3	.	.	PUNCT
ejpam-178	137	1	let	let	VERB
ejpam-178	137	2	m	m	PRON
ejpam-178	137	3	be	be	AUX
ejpam-178	137	4	an	an	DET
ejpam-178	137	5	orlicz	orlicz	ADJ
ejpam-178	137	6	function	function	NOUN
ejpam-178	137	7	.	.	PUNCT
ejpam-178	138	1	then	then	ADV
ejpam-178	138	2	(	(	PUNCT
ejpam-178	138	3	i	i	NOUN
ejpam-178	138	4	)	)	PUNCT
ejpam-178	139	1	[	[	X
ejpam-178	139	2	c(m	c(m	NOUN
ejpam-178	139	3	,	,	PUNCT
ejpam-178	139	4	λ,∆)]n	λ,∆)]n	ADV
ejpam-178	139	5	=	=	PUNCT
ejpam-178	140	1	[	[	X
ejpam-178	140	2	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	140	3	,	,	PUNCT
ejpam-178	140	4	λ,∆)]n	λ,∆)]n	ADV
ejpam-178	140	5	=	=	SYM
ejpam-178	140	6	g1	g1	PROPN
ejpam-178	140	7	,	,	PUNCT
ejpam-178	140	8	(	(	PUNCT
ejpam-178	140	9	ii	ii	NOUN
ejpam-178	140	10	)	)	PUNCT
ejpam-178	141	1	[	[	X
ejpam-178	141	2	c̃(m	c̃(m	PROPN
ejpam-178	141	3	,	,	PUNCT
ejpam-178	141	4	λ,∆)]n	λ,∆)]n	ADV
ejpam-178	141	5	=	=	SYM
ejpam-178	142	1	[	[	X
ejpam-178	142	2	ℓ̃∞(m	ℓ̃∞(m	ADV
ejpam-178	142	3	,	,	PUNCT
ejpam-178	142	4	λ,∆)]n	λ,∆)]n	ADV
ejpam-178	142	5	=	=	SYM
ejpam-178	142	6	g1	g1	NOUN
ejpam-178	142	7	.	.	PUNCT
ejpam-178	143	1	proof	proof	NOUN
ejpam-178	143	2	.	.	PUNCT
ejpam-178	144	1	(	(	PUNCT
ejpam-178	144	2	i	i	NOUN
ejpam-178	144	3	)	)	PUNCT
ejpam-178	144	4	proof	proof	NOUN
ejpam-178	144	5	is	be	AUX
ejpam-178	144	6	immediate	immediate	ADJ
ejpam-178	144	7	using	use	VERB
ejpam-178	144	8	lemma	lemma	PROPN
ejpam-178	144	9	2	2	NUM
ejpam-178	144	10	.	.	PUNCT
ejpam-178	144	11	(	(	PUNCT
ejpam-178	144	12	ii	ii	NOUN
ejpam-178	144	13	)	)	PUNCT
ejpam-178	144	14	proof	proof	NOUN
ejpam-178	144	15	is	be	AUX
ejpam-178	144	16	similar	similar	ADJ
ejpam-178	144	17	to	to	ADP
ejpam-178	144	18	that	that	PRON
ejpam-178	144	19	of	of	ADP
ejpam-178	144	20	part	part	NOUN
ejpam-178	144	21	(	(	PUNCT
ejpam-178	144	22	i	i	NOUN
ejpam-178	144	23	)	)	PUNCT
ejpam-178	144	24	.	.	PUNCT
ejpam-178	145	1	if	if	SCONJ
ejpam-178	145	2	we	we	PRON
ejpam-178	145	3	take	take	VERB
ejpam-178	145	4	λk	λk	X
ejpam-178	145	5	=	=	SYM
ejpam-178	145	6	1	1	NUM
ejpam-178	145	7	,	,	PUNCT
ejpam-178	145	8	for	for	ADP
ejpam-178	145	9	all	all	DET
ejpam-178	145	10	k	k	PROPN
ejpam-178	145	11	∈	∈	PROPN
ejpam-178	145	12	n	n	X
ejpam-178	145	13	in	in	ADP
ejpam-178	145	14	theorem	theorem	NOUN
ejpam-178	145	15	2	2	NUM
ejpam-178	145	16	,	,	PUNCT
ejpam-178	145	17	then	then	ADV
ejpam-178	145	18	we	we	PRON
ejpam-178	145	19	obtain	obtain	VERB
ejpam-178	145	20	the	the	DET
ejpam-178	145	21	following	follow	VERB
ejpam-178	145	22	corollary	corollary	NOUN
ejpam-178	145	23	.	.	PUNCT
ejpam-178	146	1	h.	h.	PROPN
ejpam-178	146	2	dutta	dutta	PROPN
ejpam-178	146	3	/	/	PUNCT
ejpam-178	146	4	eur	eur	PROPN
ejpam-178	146	5	.	.	PUNCT
ejpam-178	147	1	j.	j.	PROPN
ejpam-178	147	2	pure	pure	PROPN
ejpam-178	147	3	appl	appl	PROPN
ejpam-178	147	4	.	.	PROPN
ejpam-178	147	5	math	math	PROPN
ejpam-178	147	6	,	,	PUNCT
ejpam-178	147	7	2	2	NUM
ejpam-178	147	8	(	(	PUNCT
ejpam-178	147	9	2009	2009	NUM
ejpam-178	147	10	)	)	PUNCT
ejpam-178	147	11	,	,	PUNCT
ejpam-178	147	12	(	(	PUNCT
ejpam-178	147	13	554	554	NUM
ejpam-178	147	14	-	-	SYM
ejpam-178	147	15	563	563	NUM
ejpam-178	147	16	)	)	PUNCT
ejpam-178	147	17	562	562	NUM
ejpam-178	147	18	corollary	corollary	NOUN
ejpam-178	147	19	2	2	NUM
ejpam-178	147	20	.	.	PUNCT
ejpam-178	148	1	for	for	ADP
ejpam-178	148	2	x	x	X
ejpam-178	148	3	=	=	SYM
ejpam-178	148	4	c	c	PROPN
ejpam-178	148	5	and	and	CCONJ
ejpam-178	148	6	ℓ∞	ℓ∞	PROPN
ejpam-178	148	7	,	,	PUNCT
ejpam-178	148	8	(	(	PUNCT
ejpam-178	148	9	i	i	NOUN
ejpam-178	148	10	)	)	PUNCT
ejpam-178	149	1	[	[	X
ejpam-178	149	2	x	x	X
ejpam-178	149	3	(	(	PUNCT
ejpam-178	149	4	m	m	INTJ
ejpam-178	149	5	,	,	PUNCT
ejpam-178	149	6	∆)]n	∆)]n	NOUN
ejpam-178	149	7	=	=	PUNCT
ejpam-178	150	1	[	[	X
ejpam-178	150	2	x̃	x̃	PROPN
ejpam-178	150	3	(	(	PUNCT
ejpam-178	150	4	m	m	PROPN
ejpam-178	150	5	,	,	PUNCT
ejpam-178	150	6	∆)]n	∆)]n	ADV
ejpam-178	150	7	=	=	SYM
ejpam-178	150	8	l1	l1	PROPN
ejpam-178	150	9	,	,	PUNCT
ejpam-178	150	10	where	where	SCONJ
ejpam-178	150	11	l1	l1	PROPN
ejpam-178	150	12	=	=	PROPN
ejpam-178	150	13	{	{	PUNCT
ejpam-178	150	14	a	a	X
ejpam-178	150	15	=	=	X
ejpam-178	150	16	(	(	PUNCT
ejpam-178	150	17	ak	ak	PROPN
ejpam-178	150	18	)	)	PUNCT
ejpam-178	150	19	:	:	PUNCT
ejpam-178	151	1	lim	lim	PROPN
ejpam-178	151	2	k	k	PROPN
ejpam-178	151	3	kak	kak	PROPN
ejpam-178	151	4	=	=	PROPN
ejpam-178	151	5	0	0	NUM
ejpam-178	151	6	}	}	PUNCT
ejpam-178	151	7	.	.	PUNCT
ejpam-178	152	1	theorem	theorem	NOUN
ejpam-178	152	2	3	3	NUM
ejpam-178	152	3	.	.	PUNCT
ejpam-178	153	1	if	if	SCONJ
ejpam-178	153	2	m	m	NOUN
ejpam-178	153	3	satisfies	satisfy	VERB
ejpam-178	153	4	the	the	DET
ejpam-178	153	5	∆2	∆2	NOUN
ejpam-178	153	6	-	-	PUNCT
ejpam-178	153	7	condition	condition	NOUN
ejpam-178	153	8	,	,	PUNCT
ejpam-178	153	9	then	then	ADV
ejpam-178	153	10	we	we	PRON
ejpam-178	153	11	have	have	VERB
ejpam-178	153	12	x	x	X
ejpam-178	153	13	(	(	PUNCT
ejpam-178	153	14	m	m	INTJ
ejpam-178	153	15	,	,	PUNCT
ejpam-178	153	16	λ,∆	λ,∆	NUM
ejpam-178	153	17	)	)	PUNCT
ejpam-178	154	1	=	=	SYM
ejpam-178	154	2	x̃	x̃	PROPN
ejpam-178	154	3	(	(	PUNCT
ejpam-178	154	4	m	m	PROPN
ejpam-178	154	5	,	,	PUNCT
ejpam-178	154	6	λ,∆	λ,∆	NOUN
ejpam-178	154	7	)	)	PUNCT
ejpam-178	154	8	,	,	PUNCT
ejpam-178	154	9	for	for	ADP
ejpam-178	154	10	every	every	DET
ejpam-178	154	11	x	x	PROPN
ejpam-178	154	12	=	=	SYM
ejpam-178	154	13	c0	c0	NOUN
ejpam-178	154	14	,	,	PUNCT
ejpam-178	154	15	c	c	PROPN
ejpam-178	154	16	and	and	CCONJ
ejpam-178	154	17	ℓ∞.	ℓ∞.	PROPN
ejpam-178	154	18	proof	proof	NOUN
ejpam-178	154	19	.	.	PUNCT
ejpam-178	155	1	we	we	PRON
ejpam-178	155	2	give	give	VERB
ejpam-178	155	3	the	the	DET
ejpam-178	155	4	proof	proof	NOUN
ejpam-178	155	5	for	for	ADP
ejpam-178	155	6	x	x	SYM
ejpam-178	155	7	=	=	SYM
ejpam-178	155	8	ℓ∞	ℓ∞	PROPN
ejpam-178	155	9	and	and	CCONJ
ejpam-178	155	10	for	for	ADP
ejpam-178	155	11	other	other	ADJ
ejpam-178	155	12	spaces	space	NOUN
ejpam-178	155	13	it	it	PRON
ejpam-178	155	14	will	will	AUX
ejpam-178	155	15	follow	follow	VERB
ejpam-178	155	16	on	on	ADP
ejpam-178	155	17	applying	apply	VERB
ejpam-178	155	18	similar	similar	ADJ
ejpam-178	155	19	arguments	argument	NOUN
ejpam-178	155	20	.	.	PUNCT
ejpam-178	156	1	to	to	PART
ejpam-178	156	2	prove	prove	VERB
ejpam-178	156	3	the	the	DET
ejpam-178	156	4	theorem	theorem	NOUN
ejpam-178	156	5	,	,	PUNCT
ejpam-178	156	6	it	it	PRON
ejpam-178	156	7	is	be	AUX
ejpam-178	156	8	enough	enough	ADJ
ejpam-178	156	9	to	to	PART
ejpam-178	156	10	show	show	VERB
ejpam-178	156	11	that	that	DET
ejpam-178	156	12	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	156	13	,	,	PUNCT
ejpam-178	156	14	λ,∆	λ,∆	NUM
ejpam-178	156	15	)	)	PUNCT
ejpam-178	156	16	is	be	AUX
ejpam-178	156	17	a	a	DET
ejpam-178	156	18	subspace	subspace	NOUN
ejpam-178	156	19	of	of	ADP
ejpam-178	156	20	ℓ̃∞(m	ℓ̃∞(m	NOUN
ejpam-178	156	21	,	,	PUNCT
ejpam-178	156	22	λ,∆	λ,∆	NUM
ejpam-178	156	23	)	)	PUNCT
ejpam-178	156	24	.	.	PUNCT
ejpam-178	157	1	let	let	VERB
ejpam-178	157	2	x	x	SYM
ejpam-178	157	3	∈	∈	PROPN
ejpam-178	157	4	ℓ∞(m	ℓ∞(m	NOUN
ejpam-178	157	5	,	,	PUNCT
ejpam-178	157	6	λ,∆	λ,∆	NUM
ejpam-178	157	7	)	)	PUNCT
ejpam-178	157	8	,	,	PUNCT
ejpam-178	157	9	then	then	ADV
ejpam-178	157	10	for	for	ADP
ejpam-178	157	11	some	some	DET
ejpam-178	157	12	ρ	ρ	NOUN
ejpam-178	157	13	>	>	X
ejpam-178	157	14	0	0	PROPN
ejpam-178	157	15	,	,	PUNCT
ejpam-178	157	16	sup	sup	NOUN
ejpam-178	157	17	k	k	PROPN
ejpam-178	157	18	m	m	PROPN
ejpam-178	157	19	(	(	PUNCT
ejpam-178	157	20	|∆λk	|∆λk	PROPN
ejpam-178	157	21	xk|	xk|	PROPN
ejpam-178	157	22	ρ	ρ	PROPN
ejpam-178	157	23	)	)	PUNCT
ejpam-178	157	24	<	<	X
ejpam-178	157	25	∞	∞	NUM
ejpam-178	157	26	therefore	therefore	ADV
ejpam-178	157	27	m	m	PROPN
ejpam-178	157	28	(	(	PUNCT
ejpam-178	157	29	|∆λk	|∆λk	PROPN
ejpam-178	157	30	xk|	xk|	PROPN
ejpam-178	157	31	ρ	ρ	PROPN
ejpam-178	157	32	)	)	PUNCT
ejpam-178	157	33	<	<	X
ejpam-178	157	34	∞	∞	PROPN
ejpam-178	157	35	,	,	PUNCT
ejpam-178	157	36	for	for	ADP
ejpam-178	157	37	every	every	DET
ejpam-178	157	38	k	k	PROPN
ejpam-178	157	39	∈	∈	PROPN
ejpam-178	157	40	n	n	X
ejpam-178	157	41	.	.	PUNCT
ejpam-178	158	1	choose	choose	VERB
ejpam-178	158	2	an	an	DET
ejpam-178	158	3	arbitrary	arbitrary	ADJ
ejpam-178	158	4	η	η	X
ejpam-178	158	5	>	>	X
ejpam-178	158	6	0	0	PROPN
ejpam-178	158	7	.	.	PUNCT
ejpam-178	159	1	if	if	SCONJ
ejpam-178	159	2	ρ	ρ	PROPN
ejpam-178	159	3	≤	≤	PROPN
ejpam-178	159	4	η	η	PROPN
ejpam-178	159	5	then	then	ADV
ejpam-178	159	6	m	m	PROPN
ejpam-178	159	7	(	(	PUNCT
ejpam-178	159	8	|∆λk	|∆λk	X
ejpam-178	159	9	xk	xk	PROPN
ejpam-178	159	10	|	|	PROPN
ejpam-178	159	11	η	η	PROPN
ejpam-178	159	12	)	)	PUNCT
ejpam-178	159	13	<	<	X
ejpam-178	159	14	∞	∞	PROPN
ejpam-178	159	15	for	for	ADP
ejpam-178	159	16	every	every	DET
ejpam-178	159	17	k	k	PROPN
ejpam-178	159	18	∈	∈	PROPN
ejpam-178	159	19	n	n	ADV
ejpam-178	159	20	.	.	PUNCT
ejpam-178	160	1	let	let	VERB
ejpam-178	160	2	now	now	ADV
ejpam-178	160	3	η	η	PROPN
ejpam-178	160	4	<	<	X
ejpam-178	160	5	ρ	ρ	PROPN
ejpam-178	160	6	and	and	CCONJ
ejpam-178	160	7	put	put	VERB
ejpam-178	160	8	l	l	NOUN
ejpam-178	160	9	=	=	SYM
ejpam-178	160	10	ρ	ρ	PROPN
ejpam-178	160	11	η	η	PROPN
ejpam-178	160	12	>	>	X
ejpam-178	160	13	1	1	NUM
ejpam-178	160	14	.	.	PUNCT
ejpam-178	161	1	since	since	SCONJ
ejpam-178	161	2	m	m	PROPN
ejpam-178	161	3	satisfies	satisfy	VERB
ejpam-178	161	4	the	the	DET
ejpam-178	161	5	∆2	∆2	NOUN
ejpam-178	161	6	-	-	PUNCT
ejpam-178	161	7	condition	condition	NOUN
ejpam-178	161	8	,	,	PUNCT
ejpam-178	161	9	there	there	PRON
ejpam-178	161	10	exists	exist	VERB
ejpam-178	161	11	a	a	DET
ejpam-178	161	12	constant	constant	ADJ
ejpam-178	161	13	k	k	NOUN
ejpam-178	161	14	such	such	ADJ
ejpam-178	161	15	that	that	SCONJ
ejpam-178	161	16	m	m	PROPN
ejpam-178	161	17	(	(	PUNCT
ejpam-178	161	18	|∆λk	|∆λk	PROPN
ejpam-178	161	19	xk|	xk|	PROPN
ejpam-178	161	20	η	η	PROPN
ejpam-178	161	21	)	)	PUNCT
ejpam-178	161	22	≤	≤	PROPN
ejpam-178	162	1	k	k	PROPN
ejpam-178	162	2	(	(	PUNCT
ejpam-178	162	3	ρ	ρ	PROPN
ejpam-178	162	4	η	η	PROPN
ejpam-178	162	5	)	)	PUNCT
ejpam-178	162	6	log2	log2	PROPN
ejpam-178	162	7	k	k	PROPN
ejpam-178	162	8	m	m	PROPN
ejpam-178	162	9	(	(	PUNCT
ejpam-178	162	10	|∆λk	|∆λk	PROPN
ejpam-178	162	11	xk|	xk|	PROPN
ejpam-178	162	12	ρ	ρ	PROPN
ejpam-178	162	13	)	)	PUNCT
ejpam-178	163	1	<	<	X
ejpam-178	163	2	∞	∞	NUM
ejpam-178	163	3	for	for	ADP
ejpam-178	163	4	every	every	DET
ejpam-178	163	5	k	k	PROPN
ejpam-178	163	6	∈	∈	PROPN
ejpam-178	163	7	n	n	ADV
ejpam-178	163	8	.	.	PUNCT
ejpam-178	164	1	now	now	ADV
ejpam-178	164	2	let	let	VERB
ejpam-178	164	3	us	we	PRON
ejpam-178	164	4	denote	denote	VERB
ejpam-178	164	5	s	s	PART
ejpam-178	164	6	=	=	NOUN
ejpam-178	164	7	sup	sup	PROPN
ejpam-178	164	8	k	k	PROPN
ejpam-178	164	9	m	m	PROPN
ejpam-178	164	10	(	(	PUNCT
ejpam-178	164	11	|∆λk	|∆λk	PROPN
ejpam-178	164	12	xk|	xk|	PROPN
ejpam-178	164	13	ρ	ρ	PROPN
ejpam-178	164	14	)	)	PUNCT
ejpam-178	164	15	<	<	X
ejpam-178	164	16	∞	∞	PROPN
ejpam-178	164	17	,	,	PUNCT
ejpam-178	164	18	for	for	ADP
ejpam-178	164	19	the	the	DET
ejpam-178	164	20	fixed	fix	VERB
ejpam-178	164	21	ρ	ρ	PROPN
ejpam-178	164	22	>	>	X
ejpam-178	164	23	0	0	PROPN
ejpam-178	164	24	.	.	PUNCT
ejpam-178	165	1	then	then	ADV
ejpam-178	165	2	it	it	PRON
ejpam-178	165	3	follows	follow	VERB
ejpam-178	165	4	that	that	SCONJ
ejpam-178	165	5	for	for	ADP
ejpam-178	165	6	every	every	DET
ejpam-178	165	7	η	η	PROPN
ejpam-178	165	8	>	>	X
ejpam-178	165	9	0	0	PROPN
ejpam-178	165	10	,	,	PUNCT
ejpam-178	165	11	we	we	PRON
ejpam-178	165	12	have	have	VERB
ejpam-178	165	13	sup	sup	NOUN
ejpam-178	165	14	k	k	PROPN
ejpam-178	165	15	m	m	PROPN
ejpam-178	165	16	(	(	PUNCT
ejpam-178	165	17	|∆λk	|∆λk	PROPN
ejpam-178	165	18	xk|	xk|	PROPN
ejpam-178	165	19	η	η	PROPN
ejpam-178	165	20	)	)	PUNCT
ejpam-178	165	21	≤	≤	PROPN
ejpam-178	166	1	k	k	PROPN
ejpam-178	166	2	(	(	PUNCT
ejpam-178	166	3	ρ	ρ	PROPN
ejpam-178	166	4	η	η	PROPN
ejpam-178	166	5	)	)	PUNCT
ejpam-178	166	6	log2	log2	PROPN
ejpam-178	166	7	k	k	PROPN
ejpam-178	166	8	.s	.s	PROPN
ejpam-178	167	1	<	<	AUX
ejpam-178	167	2	∞.	∞.	PROPN
ejpam-178	167	3	references	reference	VERB
ejpam-178	167	4	563	563	NUM
ejpam-178	167	5	acknowledgements	acknowledgement	NOUN
ejpam-178	167	6	the	the	DET
ejpam-178	167	7	author	author	NOUN
ejpam-178	167	8	is	be	AUX
ejpam-178	167	9	very	very	ADV
ejpam-178	167	10	grateful	grateful	ADJ
ejpam-178	167	11	to	to	ADP
ejpam-178	167	12	the	the	DET
ejpam-178	167	13	anonymous	anonymous	ADJ
ejpam-178	167	14	referee	referee	NOUN
ejpam-178	167	15	for	for	ADP
ejpam-178	167	16	the	the	DET
ejpam-178	167	17	constructive	constructive	ADJ
ejpam-178	167	18	comments	comment	NOUN
ejpam-178	167	19	and	and	CCONJ
ejpam-178	167	20	helpful	helpful	ADJ
ejpam-178	167	21	suggestions	suggestion	NOUN
ejpam-178	167	22	which	which	PRON
ejpam-178	167	23	have	have	AUX
ejpam-178	167	24	improved	improve	VERB
ejpam-178	167	25	the	the	DET
ejpam-178	167	26	presentation	presentation	NOUN
ejpam-178	167	27	of	of	ADP
ejpam-178	167	28	this	this	DET
ejpam-178	167	29	paper	paper	NOUN
ejpam-178	167	30	.	.	PUNCT
ejpam-178	168	1	references	reference	NOUN
ejpam-178	168	2	[	[	X
ejpam-178	168	3	1	1	NUM
ejpam-178	168	4	]	]	X
ejpam-178	168	5	r.s	r.s	PROPN
ejpam-178	168	6	.	.	PROPN
ejpam-178	168	7	alsaedi	alsaedi	PROPN
ejpam-178	168	8	and	and	CCONJ
ejpam-178	168	9	a.h.a	a.h.a	ADJ
ejpam-178	168	10	.	.	PUNCT
ejpam-178	169	1	bataineh	bataineh	PROPN
ejpam-178	169	2	,	,	PUNCT
ejpam-178	169	3	on	on	ADP
ejpam-178	169	4	a	a	DET
ejpam-178	169	5	generalized	generalized	ADJ
ejpam-178	169	6	difference	difference	NOUN
ejpam-178	169	7	sequence	sequence	NOUN
ejpam-178	169	8	spaces	space	NOUN
ejpam-178	169	9	defined	define	VERB
ejpam-178	169	10	by	by	ADP
ejpam-178	169	11	a	a	DET
ejpam-178	169	12	sequence	sequence	NOUN
ejpam-178	169	13	of	of	ADP
ejpam-178	169	14	orlicz	orlicz	ADJ
ejpam-178	169	15	functions	function	NOUN
ejpam-178	169	16	,	,	PUNCT
ejpam-178	169	17	international	international	PROPN
ejpam-178	169	18	mathematical	mathematical	ADJ
ejpam-178	169	19	forum	forum	PROPN
ejpam-178	169	20	,	,	PUNCT
ejpam-178	169	21	2:4	2:4	NUM
ejpam-178	169	22	(	(	PUNCT
ejpam-178	169	23	2007	2007	NUM
ejpam-178	169	24	)	)	PUNCT
ejpam-178	169	25	,	,	PUNCT
ejpam-178	169	26	167177	167177	NUM
ejpam-178	169	27	.	.	PUNCT
ejpam-178	170	1	[	[	X
ejpam-178	170	2	2	2	NUM
ejpam-178	170	3	]	]	PUNCT
ejpam-178	170	4	m.	m.	NOUN
ejpam-178	170	5	et	et	PROPN
ejpam-178	170	6	and	and	CCONJ
ejpam-178	170	7	a.	a.	PROPN
ejpam-178	170	8	esi	esi	PROPN
ejpam-178	170	9	,	,	PUNCT
ejpam-178	170	10	on	on	ADP
ejpam-178	170	11	köthe	köthe	ADJ
ejpam-178	170	12	-	-	PUNCT
ejpam-178	170	13	toeplitz	toeplitz	NOUN
ejpam-178	170	14	duals	dual	NOUN
ejpam-178	170	15	of	of	ADP
ejpam-178	170	16	generalized	generalized	ADJ
ejpam-178	170	17	difference	difference	NOUN
ejpam-178	170	18	sequence	sequence	NOUN
ejpam-178	170	19	spaces	space	NOUN
ejpam-178	170	20	,	,	PUNCT
ejpam-178	170	21	bulletin	bulletin	NOUN
ejpam-178	170	22	of	of	ADP
ejpam-178	170	23	the	the	DET
ejpam-178	170	24	malaysian	malaysian	PROPN
ejpam-178	170	25	mathematical	mathematical	PROPN
ejpam-178	170	26	sciences	sciences	PROPN
ejpam-178	170	27	society	society	NOUN
ejpam-178	170	28	,	,	PUNCT
ejpam-178	170	29	23	23	NUM
ejpam-178	170	30	(	(	PUNCT
ejpam-178	170	31	2000	2000	NUM
ejpam-178	170	32	)	)	PUNCT
ejpam-178	170	33	,	,	PUNCT
ejpam-178	170	34	25	25	NUM
ejpam-178	170	35	-	-	SYM
ejpam-178	170	36	32	32	NUM
ejpam-178	170	37	.	.	PUNCT
ejpam-178	171	1	[	[	X
ejpam-178	171	2	3	3	X
ejpam-178	171	3	]	]	X
ejpam-178	171	4	y.	y.	PROPN
ejpam-178	171	5	gribanov	gribanov	PROPN
ejpam-178	171	6	,	,	PUNCT
ejpam-178	171	7	on	on	ADP
ejpam-178	171	8	the	the	DET
ejpam-178	171	9	theory	theory	NOUN
ejpam-178	171	10	of	of	ADP
ejpam-178	171	11	ℓm	ℓm	NOUN
ejpam-178	171	12	-spaces(russian	-spaces(russian	NOUN
ejpam-178	171	13	)	)	PUNCT
ejpam-178	171	14	,	,	PUNCT
ejpam-178	171	15	uc	uc	PROPN
ejpam-178	171	16	.	.	PROPN
ejpam-178	171	17	zap	zap	PROPN
ejpam-178	171	18	.	.	PUNCT
ejpam-178	172	1	kazansk	kazansk	PROPN
ejpam-178	172	2	un	un	PROPN
ejpam-178	172	3	-	-	PUNCT
ejpam-178	172	4	ta	ta	PROPN
ejpam-178	172	5	,	,	PUNCT
ejpam-178	172	6	117(1957	117(1957	NUM
ejpam-178	172	7	)	)	PUNCT
ejpam-178	172	8	,	,	PUNCT
ejpam-178	172	9	62	62	NUM
ejpam-178	172	10	-	-	SYM
ejpam-178	172	11	65	65	NUM
ejpam-178	172	12	.	.	PUNCT
ejpam-178	173	1	[	[	X
ejpam-178	173	2	4	4	X
ejpam-178	173	3	]	]	X
ejpam-178	173	4	g.	g.	NOUN
ejpam-178	173	5	goes	go	VERB
ejpam-178	173	6	and	and	CCONJ
ejpam-178	173	7	s.	s.	PROPN
ejpam-178	173	8	goes	go	VERB
ejpam-178	173	9	,	,	PUNCT
ejpam-178	173	10	sequences	sequence	NOUN
ejpam-178	173	11	of	of	ADP
ejpam-178	173	12	bounded	bounded	ADJ
ejpam-178	173	13	variation	variation	NOUN
ejpam-178	173	14	and	and	CCONJ
ejpam-178	173	15	sequences	sequence	NOUN
ejpam-178	173	16	of	of	ADP
ejpam-178	173	17	fourier	fourier	ADJ
ejpam-178	173	18	coefficients	coefficient	NOUN
ejpam-178	173	19	,	,	PUNCT
ejpam-178	173	20	math	math	NOUN
ejpam-178	173	21	.	.	PUNCT
ejpam-178	174	1	zeift	zeift	NOUN
ejpam-178	174	2	.	.	PUNCT
ejpam-178	174	3	,	,	PUNCT
ejpam-178	174	4	118(1970	118(1970	NUM
ejpam-178	174	5	)	)	PUNCT
ejpam-178	174	6	,	,	PUNCT
ejpam-178	174	7	93	93	NUM
ejpam-178	174	8	-	-	SYM
ejpam-178	174	9	102	102	NUM
ejpam-178	174	10	.	.	PUNCT
ejpam-178	175	1	[	[	X
ejpam-178	175	2	5	5	X
ejpam-178	175	3	]	]	PUNCT
ejpam-178	175	4	e.	e.	PROPN
ejpam-178	175	5	kreyszig	kreyszig	PROPN
ejpam-178	175	6	,	,	PUNCT
ejpam-178	175	7	introductory	introductory	ADJ
ejpam-178	175	8	functional	functional	ADJ
ejpam-178	175	9	analysis	analysis	NOUN
ejpam-178	175	10	with	with	ADP
ejpam-178	175	11	applications	application	NOUN
ejpam-178	175	12	,	,	PUNCT
ejpam-178	175	13	jhon	jhon	PROPN
ejpam-178	175	14	wiley	wiley	PROPN
ejpam-178	175	15	and	and	CCONJ
ejpam-178	175	16	sons	son	NOUN
ejpam-178	175	17	(	(	PUNCT
ejpam-178	175	18	1978	1978	NUM
ejpam-178	175	19	)	)	PUNCT
ejpam-178	175	20	.	.	PUNCT
ejpam-178	176	1	[	[	X
ejpam-178	176	2	6	6	NUM
ejpam-178	176	3	]	]	PUNCT
ejpam-178	176	4	h.	h.	PROPN
ejpam-178	176	5	kizmaz	kizmaz	PROPN
ejpam-178	176	6	,	,	PUNCT
ejpam-178	176	7	on	on	ADP
ejpam-178	176	8	certain	certain	ADJ
ejpam-178	176	9	sequence	sequence	NOUN
ejpam-178	176	10	spaces	space	NOUN
ejpam-178	176	11	,	,	PUNCT
ejpam-178	176	12	canad	canad	PROPN
ejpam-178	176	13	.	.	PUNCT
ejpam-178	177	1	math	math	NOUN
ejpam-178	177	2	.	.	PUNCT
ejpam-178	178	1	bull	bull	PROPN
ejpam-178	178	2	.	.	PUNCT
ejpam-178	178	3	,	,	PUNCT
ejpam-178	178	4	24:2(1981	24:2(1981	NUM
ejpam-178	178	5	)	)	PUNCT
ejpam-178	178	6	,	,	PUNCT
ejpam-178	178	7	169	169	NUM
ejpam-178	178	8	-	-	SYM
ejpam-178	178	9	176	176	NUM
ejpam-178	178	10	.	.	PUNCT
ejpam-178	179	1	[	[	X
ejpam-178	179	2	7	7	X
ejpam-178	179	3	]	]	X
ejpam-178	179	4	p.k	p.k	PROPN
ejpam-178	179	5	.	.	PROPN
ejpam-178	179	6	kamthan	kamthan	PROPN
ejpam-178	179	7	and	and	CCONJ
ejpam-178	179	8	m.	m.	PROPN
ejpam-178	179	9	gupta	gupta	PROPN
ejpam-178	179	10	,	,	PUNCT
ejpam-178	179	11	sequence	sequence	NOUN
ejpam-178	179	12	spaces	space	NOUN
ejpam-178	179	13	and	and	CCONJ
ejpam-178	179	14	series	series	PROPN
ejpam-178	179	15	,	,	PUNCT
ejpam-178	179	16	marcel	marcel	PROPN
ejpam-178	179	17	dekker	dekker	PROPN
ejpam-178	179	18	inc	inc	PROPN
ejpam-178	179	19	.	.	PROPN
ejpam-178	179	20	,	,	PUNCT
ejpam-178	179	21	new	new	PROPN
ejpam-178	179	22	york	york	PROPN
ejpam-178	179	23	(	(	PUNCT
ejpam-178	179	24	1981	1981	NUM
ejpam-178	179	25	)	)	PUNCT
ejpam-178	179	26	.	.	PUNCT
ejpam-178	180	1	[	[	X
ejpam-178	180	2	8	8	NUM
ejpam-178	180	3	]	]	X
ejpam-178	180	4	m.a	m.a	PROPN
ejpam-178	180	5	.	.	PROPN
ejpam-178	180	6	krasnoselskii	krasnoselskii	PROPN
ejpam-178	180	7	and	and	CCONJ
ejpam-178	180	8	y.b	y.b	PROPN
ejpam-178	180	9	.	.	PROPN
ejpam-178	180	10	rutitsky	rutitsky	PROPN
ejpam-178	180	11	,	,	PUNCT
ejpam-178	180	12	convex	convex	NOUN
ejpam-178	180	13	functions	function	NOUN
ejpam-178	180	14	and	and	CCONJ
ejpam-178	180	15	orlicz	orlicz	ADJ
ejpam-178	180	16	spaces	space	NOUN
ejpam-178	180	17	,	,	PUNCT
ejpam-178	180	18	groningen	groningen	PROPN
ejpam-178	180	19	,	,	PUNCT
ejpam-178	180	20	netherlands	netherlands	PROPN
ejpam-178	180	21	,	,	PUNCT
ejpam-178	180	22	1961	1961	NUM
ejpam-178	180	23	.	.	PUNCT
ejpam-178	181	1	[	[	X
ejpam-178	181	2	9	9	NUM
ejpam-178	181	3	]	]	X
ejpam-178	181	4	j.	j.	PROPN
ejpam-178	181	5	lindenstrauss	lindenstrauss	PROPN
ejpam-178	181	6	and	and	CCONJ
ejpam-178	181	7	l.	l.	PROPN
ejpam-178	181	8	tzafriri	tzafriri	PROPN
ejpam-178	181	9	,	,	PUNCT
ejpam-178	181	10	on	on	ADP
ejpam-178	181	11	orlicz	orlicz	ADJ
ejpam-178	181	12	sequence	sequence	NOUN
ejpam-178	181	13	spaces	space	NOUN
ejpam-178	181	14	,	,	PUNCT
ejpam-178	181	15	israel	israel	PROPN
ejpam-178	181	16	j.	j.	PROPN
ejpam-178	181	17	math	math	PROPN
ejpam-178	181	18	.	.	PUNCT
ejpam-178	181	19	,10(1971	,10(1971	PUNCT
ejpam-178	181	20	)	)	PUNCT
ejpam-178	181	21	,	,	PUNCT
ejpam-178	181	22	379	379	NUM
ejpam-178	181	23	-	-	SYM
ejpam-178	181	24	390	390	NUM
ejpam-178	181	25	.	.	PUNCT
ejpam-178	182	1	[	[	X
ejpam-178	182	2	10	10	NUM
ejpam-178	182	3	]	]	X
ejpam-178	182	4	i.j	i.j	PROPN
ejpam-178	182	5	.	.	PROPN
ejpam-178	182	6	maddox	maddox	PROPN
ejpam-178	182	7	,	,	PUNCT
ejpam-178	182	8	elements	element	NOUN
ejpam-178	182	9	of	of	ADP
ejpam-178	182	10	functional	functional	ADJ
ejpam-178	182	11	analysis	analysis	NOUN
ejpam-178	182	12	,	,	PUNCT
ejpam-178	182	13	universal	universal	ADJ
ejpam-178	182	14	book	book	NOUN
ejpam-178	182	15	stall	stall	NOUN
ejpam-178	182	16	(	(	PUNCT
ejpam-178	182	17	1989	1989	NUM
ejpam-178	182	18	)	)	PUNCT
ejpam-178	182	19	.	.	PUNCT
ejpam-178	183	1	[	[	X
ejpam-178	183	2	11	11	NUM
ejpam-178	183	3	]	]	X
ejpam-178	183	4	e.	e.	PROPN
ejpam-178	183	5	malkowsky	malkowsky	PROPN
ejpam-178	183	6	and	and	CCONJ
ejpam-178	183	7	s.d	s.d	PROPN
ejpam-178	183	8	.	.	PROPN
ejpam-178	183	9	parasar	parasar	PROPN
ejpam-178	183	10	,	,	PUNCT
ejpam-178	183	11	matrix	matrix	NOUN
ejpam-178	183	12	transformation	transformation	NOUN
ejpam-178	183	13	in	in	ADP
ejpam-178	183	14	spaces	space	NOUN
ejpam-178	183	15	of	of	ADP
ejpam-178	183	16	bounded	bounded	ADJ
ejpam-178	183	17	and	and	CCONJ
ejpam-178	183	18	convergent	convergent	ADJ
ejpam-178	183	19	difference	difference	NOUN
ejpam-178	183	20	sequences	sequence	NOUN
ejpam-178	183	21	of	of	ADP
ejpam-178	183	22	order	order	NOUN
ejpam-178	183	23	m	m	NOUN
ejpam-178	183	24	,	,	PUNCT
ejpam-178	183	25	analysis	analysis	NOUN
ejpam-178	183	26	,	,	PUNCT
ejpam-178	183	27	17(1997	17(1997	NUM
ejpam-178	183	28	)	)	PUNCT
ejpam-178	183	29	,	,	PUNCT
ejpam-178	183	30	87	87	NUM
ejpam-178	183	31	-	-	SYM
ejpam-178	183	32	97	97	NUM
ejpam-178	183	33	.	.	PUNCT
ejpam-178	184	1	[	[	X
ejpam-178	184	2	12	12	NUM
ejpam-178	184	3	]	]	X
ejpam-178	184	4	m.m	m.m	PROPN
ejpam-178	184	5	.	.	PROPN
ejpam-178	184	6	rao	rao	PROPN
ejpam-178	184	7	and	and	CCONJ
ejpam-178	184	8	z.d	z.d	PROPN
ejpam-178	184	9	.	.	PROPN
ejpam-178	184	10	ren	ren	PROPN
ejpam-178	184	11	,	,	PUNCT
ejpam-178	184	12	theory	theory	NOUN
ejpam-178	184	13	on	on	ADP
ejpam-178	184	14	orlicz	orlicz	ADJ
ejpam-178	184	15	spaces	space	NOUN
ejpam-178	184	16	,	,	PUNCT
ejpam-178	184	17	marcel	marcel	PROPN
ejpam-178	184	18	dekker	dekker	PROPN
ejpam-178	184	19	,	,	PUNCT
ejpam-178	184	20	new	new	ADJ
ejpam-178	184	21	york,1991	york,1991	NOUN
ejpam-178	184	22	.	.	PUNCT