id	sid	tid	token	lemma	pos
ejpam-4831	1	1	european	european	PROPN
ejpam-4831	1	2	journal	journal	PROPN
ejpam-4831	1	3	of	of	ADP
ejpam-4831	1	4	pure	pure	ADJ
ejpam-4831	1	5	and	and	CCONJ
ejpam-4831	1	6	applied	apply	VERB
ejpam-4831	1	7	mathematics	mathematic	NOUN
ejpam-4831	1	8	vol	vol	NOUN
ejpam-4831	1	9	.	.	PUNCT
ejpam-4831	2	1	16	16	NUM
ejpam-4831	2	2	,	,	PUNCT
ejpam-4831	2	3	no	no	INTJ
ejpam-4831	2	4	.	.	NOUN
ejpam-4831	2	5	3	3	NUM
ejpam-4831	2	6	,	,	PUNCT
ejpam-4831	2	7	2023	2023	NUM
ejpam-4831	2	8	,	,	PUNCT
ejpam-4831	2	9	1762	1762	NUM
ejpam-4831	2	10	-	-	SYM
ejpam-4831	2	11	1771	1771	NUM
ejpam-4831	2	12	issn	issn	PROPN
ejpam-4831	2	13	1307	1307	NUM
ejpam-4831	2	14	-	-	SYM
ejpam-4831	2	15	5543	5543	NUM
ejpam-4831	2	16	–	–	PUNCT
ejpam-4831	2	17	ejpam.com	ejpam.com	X
ejpam-4831	2	18	published	publish	VERB
ejpam-4831	2	19	by	by	ADP
ejpam-4831	2	20	new	new	PROPN
ejpam-4831	2	21	york	york	PROPN
ejpam-4831	2	22	business	business	PROPN
ejpam-4831	2	23	global	global	ADJ
ejpam-4831	2	24	nuclearity	nuclearity	NOUN
ejpam-4831	2	25	of	of	ADP
ejpam-4831	2	26	a	a	DET
ejpam-4831	2	27	class	class	NOUN
ejpam-4831	2	28	of	of	ADP
ejpam-4831	2	29	vector	vector	NOUN
ejpam-4831	2	30	-	-	PUNCT
ejpam-4831	2	31	valued	value	VERB
ejpam-4831	2	32	sequence	sequence	NOUN
ejpam-4831	2	33	spaces	space	NOUN
ejpam-4831	2	34	mohamed	mohamed	PROPN
ejpam-4831	2	35	ahmed	ahmed	PROPN
ejpam-4831	3	1	ould	ould	AUX
ejpam-4831	3	2	sidaty1,2	sidaty1,2	PROPN
ejpam-4831	3	3	1	1	NUM
ejpam-4831	3	4	department	department	NOUN
ejpam-4831	3	5	of	of	ADP
ejpam-4831	3	6	mathematics	mathematic	NOUN
ejpam-4831	3	7	and	and	CCONJ
ejpam-4831	3	8	statistics	statistic	NOUN
ejpam-4831	3	9	,	,	PUNCT
ejpam-4831	3	10	college	college	NOUN
ejpam-4831	3	11	of	of	ADP
ejpam-4831	3	12	science	science	NOUN
ejpam-4831	3	13	,	,	PUNCT
ejpam-4831	3	14	imam	imam	PROPN
ejpam-4831	3	15	mohammad	mohammad	PROPN
ejpam-4831	3	16	ibn	ibn	PROPN
ejpam-4831	3	17	saud	saud	PROPN
ejpam-4831	3	18	islamic	islamic	PROPN
ejpam-4831	3	19	university	university	PROPN
ejpam-4831	3	20	,	,	PUNCT
ejpam-4831	3	21	riyadh	riyadh	NOUN
ejpam-4831	3	22	,	,	PUNCT
ejpam-4831	3	23	kingdom	kingdom	NOUN
ejpam-4831	3	24	of	of	ADP
ejpam-4831	3	25	saudi	saudi	PROPN
ejpam-4831	3	26	arabia	arabia	PROPN
ejpam-4831	3	27	2	2	NUM
ejpam-4831	3	28	école	école	ADJ
ejpam-4831	3	29	normale	normale	PROPN
ejpam-4831	3	30	supérieure	supérieure	PROPN
ejpam-4831	3	31	de	de	PROPN
ejpam-4831	3	32	nouakchott	nouakchott	PROPN
ejpam-4831	3	33	,	,	PUNCT
ejpam-4831	3	34	mauritanie	mauritanie	NOUN
ejpam-4831	3	35	abstract	abstract	ADJ
ejpam-4831	3	36	.	.	PUNCT
ejpam-4831	4	1	in	in	ADP
ejpam-4831	4	2	this	this	DET
ejpam-4831	4	3	note	note	NOUN
ejpam-4831	4	4	,	,	PUNCT
ejpam-4831	4	5	we	we	PRON
ejpam-4831	4	6	deal	deal	VERB
ejpam-4831	4	7	with	with	ADP
ejpam-4831	4	8	a	a	DET
ejpam-4831	4	9	perfect	perfect	ADJ
ejpam-4831	4	10	sequence	sequence	NOUN
ejpam-4831	4	11	space	space	NOUN
ejpam-4831	4	12	λ	λ	NOUN
ejpam-4831	4	13	and	and	CCONJ
ejpam-4831	4	14	a	a	DET
ejpam-4831	4	15	convex	convex	ADJ
ejpam-4831	4	16	bornological	bornological	ADJ
ejpam-4831	4	17	space	space	NOUN
ejpam-4831	4	18	e	e	NOUN
ejpam-4831	4	19	to	to	PART
ejpam-4831	4	20	introduce	introduce	VERB
ejpam-4831	4	21	and	and	CCONJ
ejpam-4831	4	22	study	study	VERB
ejpam-4831	4	23	the	the	DET
ejpam-4831	4	24	space	space	NOUN
ejpam-4831	4	25	λ(e	λ(e	VERB
ejpam-4831	4	26	)	)	PUNCT
ejpam-4831	4	27	of	of	ADP
ejpam-4831	4	28	all	all	DET
ejpam-4831	4	29	totally	totally	ADV
ejpam-4831	4	30	λ	λ	ADJ
ejpam-4831	4	31	-	-	ADJ
ejpam-4831	4	32	summable	summable	ADJ
ejpam-4831	4	33	sequences	sequence	NOUN
ejpam-4831	4	34	from	from	ADP
ejpam-4831	4	35	e.	e.	PROPN
ejpam-4831	4	36	we	we	PRON
ejpam-4831	4	37	prove	prove	VERB
ejpam-4831	4	38	that	that	SCONJ
ejpam-4831	4	39	λ(e	λ(e	VERB
ejpam-4831	4	40	)	)	PUNCT
ejpam-4831	4	41	is	be	AUX
ejpam-4831	4	42	complete	complete	ADJ
ejpam-4831	4	43	if	if	SCONJ
ejpam-4831	4	44	and	and	CCONJ
ejpam-4831	4	45	only	only	ADV
ejpam-4831	4	46	if	if	SCONJ
ejpam-4831	4	47	λ	λ	PROPN
ejpam-4831	4	48	and	and	CCONJ
ejpam-4831	4	49	e	e	NOUN
ejpam-4831	4	50	are	be	AUX
ejpam-4831	4	51	complete	complete	ADJ
ejpam-4831	4	52	,	,	PUNCT
ejpam-4831	4	53	nuclear	nuclear	ADJ
ejpam-4831	4	54	if	if	SCONJ
ejpam-4831	4	55	and	and	CCONJ
ejpam-4831	4	56	only	only	ADV
ejpam-4831	4	57	if	if	SCONJ
ejpam-4831	4	58	λ	λ	PROPN
ejpam-4831	4	59	and	and	CCONJ
ejpam-4831	4	60	e	e	NOUN
ejpam-4831	4	61	are	be	AUX
ejpam-4831	4	62	nuclear	nuclear	ADJ
ejpam-4831	4	63	,	,	PUNCT
ejpam-4831	4	64	and	and	CCONJ
ejpam-4831	4	65	we	we	PRON
ejpam-4831	4	66	make	make	VERB
ejpam-4831	4	67	use	use	NOUN
ejpam-4831	4	68	of	of	ADP
ejpam-4831	4	69	a	a	DET
ejpam-4831	4	70	result	result	NOUN
ejpam-4831	4	71	of	of	ADP
ejpam-4831	4	72	ronald	ronald	PROPN
ejpam-4831	4	73	c.	c.	PROPN
ejpam-4831	4	74	rosier	rosier	PUNCT
ejpam-4831	5	1	[	[	X
ejpam-4831	5	2	10	10	NUM
ejpam-4831	5	3	]	]	PUNCT
ejpam-4831	5	4	to	to	PART
ejpam-4831	5	5	give	give	VERB
ejpam-4831	5	6	a	a	DET
ejpam-4831	5	7	similar	similar	ADJ
ejpam-4831	5	8	characterization	characterization	NOUN
ejpam-4831	5	9	of	of	ADP
ejpam-4831	5	10	the	the	DET
ejpam-4831	5	11	nuclearity	nuclearity	NOUN
ejpam-4831	5	12	of	of	ADP
ejpam-4831	5	13	the	the	DET
ejpam-4831	5	14	space	space	NOUN
ejpam-4831	5	15	λ{e	λ{e	PROPN
ejpam-4831	5	16	}	}	PUNCT
ejpam-4831	5	17	of	of	ADP
ejpam-4831	5	18	all	all	DET
ejpam-4831	5	19	absolutely	absolutely	ADV
ejpam-4831	5	20	λ−summable	λ−summable	VERB
ejpam-4831	5	21	sequences	sequence	NOUN
ejpam-4831	5	22	in	in	ADP
ejpam-4831	5	23	a	a	DET
ejpam-4831	5	24	locally	locally	ADV
ejpam-4831	5	25	convex	convex	PROPN
ejpam-4831	5	26	e.	e.	PROPN
ejpam-4831	5	27	2020	2020	NUM
ejpam-4831	5	28	mathematics	mathematics	PROPN
ejpam-4831	5	29	subject	subject	NOUN
ejpam-4831	5	30	classifications	classification	NOUN
ejpam-4831	5	31	:	:	PUNCT
ejpam-4831	5	32	46a17	46a17	NUM
ejpam-4831	5	33	,	,	PUNCT
ejpam-4831	5	34	46a45	46a45	NUM
ejpam-4831	5	35	,	,	PUNCT
ejpam-4831	5	36	47b37	47b37	NUM
ejpam-4831	5	37	,	,	PUNCT
ejpam-4831	5	38	46b45	46b45	PRON
ejpam-4831	5	39	key	key	ADJ
ejpam-4831	5	40	words	word	NOUN
ejpam-4831	5	41	and	and	CCONJ
ejpam-4831	5	42	phrases	phrase	NOUN
ejpam-4831	5	43	:	:	PUNCT
ejpam-4831	5	44	sequence	sequence	NOUN
ejpam-4831	5	45	spaces	space	NOUN
ejpam-4831	5	46	,	,	PUNCT
ejpam-4831	5	47	convex	convex	ADJ
ejpam-4831	5	48	bornological	bornological	ADJ
ejpam-4831	5	49	spaces	space	NOUN
ejpam-4831	5	50	,	,	PUNCT
ejpam-4831	5	51	locally	locally	ADV
ejpam-4831	5	52	convex	convex	ADJ
ejpam-4831	5	53	sequence	sequence	NOUN
ejpam-4831	5	54	spaces	space	NOUN
ejpam-4831	5	55	,	,	PUNCT
ejpam-4831	5	56	nuclearity	nuclearity	NOUN
ejpam-4831	5	57	,	,	PUNCT
ejpam-4831	5	58	summability	summability	NOUN
ejpam-4831	5	59	introduction	introduction	NOUN
ejpam-4831	5	60	in	in	ADP
ejpam-4831	5	61	connection	connection	NOUN
ejpam-4831	5	62	with	with	ADP
ejpam-4831	5	63	the	the	DET
ejpam-4831	5	64	nuclearity	nuclearity	NOUN
ejpam-4831	5	65	of	of	ADP
ejpam-4831	5	66	a	a	DET
ejpam-4831	5	67	locally	locally	ADV
ejpam-4831	5	68	convex	convex	ADJ
ejpam-4831	5	69	space	space	NOUN
ejpam-4831	5	70	e	e	NOUN
ejpam-4831	5	71	,	,	PUNCT
ejpam-4831	5	72	a.	a.	NOUN
ejpam-4831	5	73	pietsch	pietsch	VERB
ejpam-4831	5	74	in	in	ADP
ejpam-4831	5	75	[	[	X
ejpam-4831	5	76	9	9	NUM
ejpam-4831	5	77	]	]	PUNCT
ejpam-4831	5	78	introduced	introduce	VERB
ejpam-4831	5	79	the	the	DET
ejpam-4831	5	80	spaces	space	NOUN
ejpam-4831	5	81	ℓp(e	ℓp(e	NUM
ejpam-4831	5	82	)	)	PUNCT
ejpam-4831	5	83	and	and	CCONJ
ejpam-4831	5	84	ℓp{e	ℓp{e	NOUN
ejpam-4831	5	85	}	}	PUNCT
ejpam-4831	5	86	respectively	respectively	ADV
ejpam-4831	5	87	of	of	ADP
ejpam-4831	5	88	weakly	weakly	ADJ
ejpam-4831	5	89	ℓp	ℓp	ADJ
ejpam-4831	5	90	-	-	PUNCT
ejpam-4831	5	91	summable	summable	ADJ
ejpam-4831	5	92	and	and	CCONJ
ejpam-4831	5	93	absolutely	absolutely	ADV
ejpam-4831	5	94	ℓp	ℓp	ADJ
ejpam-4831	5	95	-	-	PUNCT
ejpam-4831	5	96	summable	summable	ADJ
ejpam-4831	5	97	sequences	sequence	NOUN
ejpam-4831	5	98	in	in	ADP
ejpam-4831	5	99	e.	e.	PROPN
ejpam-4831	5	100	in	in	ADP
ejpam-4831	5	101	[	[	X
ejpam-4831	5	102	8	8	NUM
ejpam-4831	5	103	]	]	PUNCT
ejpam-4831	5	104	,	,	PUNCT
ejpam-4831	5	105	he	he	PRON
ejpam-4831	5	106	used	use	VERB
ejpam-4831	5	107	these	these	DET
ejpam-4831	5	108	spaces	space	NOUN
ejpam-4831	5	109	to	to	PART
ejpam-4831	5	110	study	study	VERB
ejpam-4831	5	111	the	the	DET
ejpam-4831	5	112	absolutely	absolutely	ADV
ejpam-4831	5	113	p	p	NOUN
ejpam-4831	5	114	-	-	PUNCT
ejpam-4831	5	115	summing	sum	VERB
ejpam-4831	5	116	operators	operator	NOUN
ejpam-4831	5	117	.	.	PUNCT
ejpam-4831	6	1	later	later	ADV
ejpam-4831	6	2	,	,	PUNCT
ejpam-4831	6	3	he	he	PRON
ejpam-4831	6	4	introduced	introduce	VERB
ejpam-4831	6	5	and	and	CCONJ
ejpam-4831	6	6	studied	study	VERB
ejpam-4831	6	7	also	also	ADV
ejpam-4831	6	8	the	the	DET
ejpam-4831	6	9	space	space	NOUN
ejpam-4831	6	10	λ{e	λ{e	PROPN
ejpam-4831	6	11	}	}	PUNCT
ejpam-4831	6	12	of	of	ADP
ejpam-4831	6	13	λ	λ	NOUN
ejpam-4831	6	14	-	-	ADJ
ejpam-4831	6	15	summable	summable	ADJ
ejpam-4831	6	16	sequences	sequence	NOUN
ejpam-4831	6	17	in	in	ADP
ejpam-4831	6	18	e	e	NOUN
ejpam-4831	6	19	,	,	PUNCT
ejpam-4831	6	20	for	for	ADP
ejpam-4831	6	21	a	a	DET
ejpam-4831	6	22	perfect	perfect	ADJ
ejpam-4831	6	23	sequence	sequence	NOUN
ejpam-4831	6	24	space	space	NOUN
ejpam-4831	6	25	λ	λ	NOUN
ejpam-4831	6	26	in	in	ADP
ejpam-4831	6	27	the	the	DET
ejpam-4831	6	28	sense	sense	NOUN
ejpam-4831	6	29	of	of	ADP
ejpam-4831	6	30	köthe	köthe	NOUN
ejpam-4831	6	31	endowed	endow	VERB
ejpam-4831	6	32	with	with	ADP
ejpam-4831	6	33	its	its	PRON
ejpam-4831	6	34	normal	normal	ADJ
ejpam-4831	6	35	topology	topology	NOUN
ejpam-4831	6	36	.	.	PUNCT
ejpam-4831	7	1	many	many	ADJ
ejpam-4831	7	2	other	other	ADJ
ejpam-4831	7	3	authors	author	NOUN
ejpam-4831	7	4	were	be	AUX
ejpam-4831	7	5	interested	interested	ADJ
ejpam-4831	7	6	in	in	ADP
ejpam-4831	7	7	the	the	DET
ejpam-4831	7	8	study	study	NOUN
ejpam-4831	7	9	of	of	ADP
ejpam-4831	7	10	these	these	DET
ejpam-4831	7	11	spaces	space	NOUN
ejpam-4831	7	12	.	.	PUNCT
ejpam-4831	8	1	ronald	ronald	PROPN
ejpam-4831	8	2	c.	c.	PROPN
ejpam-4831	8	3	rosier	rosier	ADV
ejpam-4831	8	4	in	in	ADP
ejpam-4831	8	5	[	[	X
ejpam-4831	8	6	10	10	NUM
ejpam-4831	8	7	]	]	PUNCT
ejpam-4831	8	8	considered	consider	VERB
ejpam-4831	8	9	a	a	DET
ejpam-4831	8	10	general	general	ADJ
ejpam-4831	8	11	polar	polar	ADJ
ejpam-4831	8	12	topology	topology	NOUN
ejpam-4831	8	13	on	on	ADP
ejpam-4831	8	14	λ{e	λ{e	NOUN
ejpam-4831	8	15	}	}	PUNCT
ejpam-4831	8	16	and	and	CCONJ
ejpam-4831	8	17	got	get	VERB
ejpam-4831	8	18	a	a	DET
ejpam-4831	8	19	precise	precise	ADJ
ejpam-4831	8	20	description	description	NOUN
ejpam-4831	8	21	of	of	ADP
ejpam-4831	8	22	the	the	DET
ejpam-4831	8	23	topological	topological	ADJ
ejpam-4831	8	24	dual	dual	ADJ
ejpam-4831	8	25	and	and	CCONJ
ejpam-4831	8	26	its	its	PRON
ejpam-4831	8	27	equicontinuous	equicontinuous	ADJ
ejpam-4831	8	28	subsets	subset	NOUN
ejpam-4831	8	29	.	.	PUNCT
ejpam-4831	9	1	m.	m.	NOUN
ejpam-4831	9	2	florencio	florencio	PROPN
ejpam-4831	9	3	and	and	CCONJ
ejpam-4831	9	4	p.	p.	PROPN
ejpam-4831	9	5	j.	j.	PROPN
ejpam-4831	10	1	paúl	paúl	PROPN
ejpam-4831	11	1	[	[	X
ejpam-4831	11	2	3	3	NUM
ejpam-4831	11	3	]	]	PUNCT
ejpam-4831	11	4	,	,	PUNCT
ejpam-4831	11	5	considering	consider	VERB
ejpam-4831	11	6	general	general	ADJ
ejpam-4831	11	7	polar	polar	ADJ
ejpam-4831	11	8	topologies	topology	NOUN
ejpam-4831	11	9	,	,	PUNCT
ejpam-4831	11	10	obtained	obtain	VERB
ejpam-4831	11	11	many	many	ADJ
ejpam-4831	11	12	interesting	interesting	ADJ
ejpam-4831	11	13	results	result	NOUN
ejpam-4831	11	14	such	such	ADJ
ejpam-4831	11	15	as	as	ADP
ejpam-4831	11	16	barreledness	barreledness	ADJ
ejpam-4831	11	17	conditions	condition	NOUN
ejpam-4831	11	18	.	.	PUNCT
ejpam-4831	12	1	in	in	ADP
ejpam-4831	12	2	[	[	X
ejpam-4831	12	3	1	1	NUM
ejpam-4831	12	4	]	]	PUNCT
ejpam-4831	12	5	and	and	CCONJ
ejpam-4831	12	6	[	[	X
ejpam-4831	12	7	2	2	NUM
ejpam-4831	12	8	]	]	PUNCT
ejpam-4831	12	9	,	,	PUNCT
ejpam-4831	12	10	they	they	PRON
ejpam-4831	12	11	studied	study	VERB
ejpam-4831	12	12	the	the	DET
ejpam-4831	12	13	space	space	NOUN
ejpam-4831	12	14	λ(e	λ(e	VERB
ejpam-4831	12	15	)	)	PUNCT
ejpam-4831	12	16	of	of	ADP
ejpam-4831	12	17	weakly	weakly	ADJ
ejpam-4831	12	18	λ−summables	λ−summable	NOUN
ejpam-4831	12	19	sequences	sequence	NOUN
ejpam-4831	12	20	in	in	ADP
ejpam-4831	12	21	e	e	NOUN
ejpam-4831	12	22	and	and	CCONJ
ejpam-4831	12	23	represented	represent	VERB
ejpam-4831	12	24	this	this	DET
ejpam-4831	12	25	space	space	NOUN
ejpam-4831	12	26	as	as	ADP
ejpam-4831	12	27	the	the	DET
ejpam-4831	12	28	completion	completion	NOUN
ejpam-4831	12	29	of	of	ADP
ejpam-4831	12	30	the	the	DET
ejpam-4831	12	31	injective	injective	ADJ
ejpam-4831	12	32	tensor	tensor	NOUN
ejpam-4831	12	33	product	product	NOUN
ejpam-4831	12	34	λ⊗̃ϵe	λ⊗̃ϵe	NOUN
ejpam-4831	12	35	.	.	PUNCT
ejpam-4831	13	1	in	in	ADP
ejpam-4831	13	2	[	[	X
ejpam-4831	13	3	6	6	NUM
ejpam-4831	13	4	]	]	PUNCT
ejpam-4831	13	5	and	and	CCONJ
ejpam-4831	13	6	[	[	X
ejpam-4831	13	7	7	7	NUM
ejpam-4831	13	8	]	]	PUNCT
ejpam-4831	13	9	,	,	PUNCT
ejpam-4831	13	10	l.	l.	PROPN
ejpam-4831	13	11	oubbi	oubbi	PROPN
ejpam-4831	13	12	and	and	CCONJ
ejpam-4831	13	13	m.	m.	PROPN
ejpam-4831	13	14	a.	a.	NOUN
ejpam-4831	13	15	ould	ould	AUX
ejpam-4831	13	16	sidaty	sidaty	VERB
ejpam-4831	13	17	reconsidered	reconsider	VERB
ejpam-4831	13	18	the	the	DET
ejpam-4831	13	19	space	space	NOUN
ejpam-4831	13	20	λ(e	λ(e	VERB
ejpam-4831	13	21	)	)	PUNCT
ejpam-4831	13	22	and	and	CCONJ
ejpam-4831	13	23	obtained	obtain	VERB
ejpam-4831	13	24	some	some	PRON
ejpam-4831	13	25	of	of	ADP
ejpam-4831	13	26	its	its	PRON
ejpam-4831	13	27	properties	property	NOUN
ejpam-4831	13	28	.	.	PUNCT
ejpam-4831	14	1	they	they	PRON
ejpam-4831	14	2	mainly	mainly	ADV
ejpam-4831	14	3	described	describe	VERB
ejpam-4831	14	4	the	the	DET
ejpam-4831	14	5	continuous	continuous	ADJ
ejpam-4831	14	6	dual	dual	ADJ
ejpam-4831	14	7	space	space	NOUN
ejpam-4831	14	8	of	of	ADP
ejpam-4831	14	9	λ(e	λ(e	NOUN
ejpam-4831	14	10	)	)	PUNCT
ejpam-4831	14	11	.	.	PUNCT
ejpam-4831	15	1	while	while	SCONJ
ejpam-4831	15	2	in	in	ADP
ejpam-4831	15	3	[	[	NOUN
ejpam-4831	15	4	11	11	NUM
ejpam-4831	15	5	]	]	PUNCT
ejpam-4831	15	6	and	and	CCONJ
ejpam-4831	15	7	[	[	X
ejpam-4831	15	8	13	13	NUM
ejpam-4831	15	9	]	]	PUNCT
ejpam-4831	15	10	,	,	PUNCT
ejpam-4831	15	11	characterizations	characterization	NOUN
ejpam-4831	15	12	of	of	ADP
ejpam-4831	15	13	the	the	DET
ejpam-4831	15	14	reflexivity	reflexivity	NOUN
ejpam-4831	15	15	of	of	ADP
ejpam-4831	15	16	λ(e	λ(e	NOUN
ejpam-4831	15	17	)	)	PUNCT
ejpam-4831	15	18	in	in	ADP
ejpam-4831	15	19	terms	term	NOUN
ejpam-4831	15	20	of	of	ADP
ejpam-4831	15	21	that	that	PRON
ejpam-4831	15	22	of	of	ADP
ejpam-4831	15	23	λ	λ	PROPN
ejpam-4831	15	24	and	and	CCONJ
ejpam-4831	15	25	e	e	PROPN
ejpam-4831	15	26	and	and	CCONJ
ejpam-4831	15	27	the	the	DET
ejpam-4831	15	28	ak	ak	PROPN
ejpam-4831	15	29	-	-	PUNCT
ejpam-4831	15	30	property	property	NOUN
ejpam-4831	15	31	are	be	AUX
ejpam-4831	15	32	given	give	VERB
ejpam-4831	15	33	.	.	PUNCT
ejpam-4831	16	1	a	a	DET
ejpam-4831	16	2	characterization	characterization	NOUN
ejpam-4831	16	3	of	of	ADP
ejpam-4831	16	4	the	the	DET
ejpam-4831	16	5	nuclearity	nuclearity	NOUN
ejpam-4831	16	6	of	of	ADP
ejpam-4831	16	7	of	of	ADP
ejpam-4831	16	8	the	the	DET
ejpam-4831	16	9	space	space	NOUN
ejpam-4831	16	10	of	of	ADP
ejpam-4831	16	11	weakly	weakly	ADJ
ejpam-4831	16	12	λ−summable	λ−summable	ADJ
ejpam-4831	16	13	sequences	sequence	NOUN
ejpam-4831	16	14	is	be	AUX
ejpam-4831	16	15	given	give	VERB
ejpam-4831	16	16	in	in	ADP
ejpam-4831	16	17	[	[	X
ejpam-4831	16	18	12	12	NUM
ejpam-4831	16	19	]	]	PUNCT
ejpam-4831	16	20	.	.	PUNCT
ejpam-4831	17	1	doi	doi	NOUN
ejpam-4831	17	2	:	:	PUNCT
ejpam-4831	17	3	https://doi.org/10.29020/nybg.ejpam.v16i3.4831	https://doi.org/10.29020/nybg.ejpam.v16i3.4831	PROPN
ejpam-4831	17	4	email	email	NOUN
ejpam-4831	17	5	address	address	NOUN
ejpam-4831	17	6	:	:	PUNCT
ejpam-4831	17	7	sidaty1@hotmail.com	sidaty1@hotmail.com	X
ejpam-4831	17	8	(	(	PUNCT
ejpam-4831	17	9	m.	m.	NOUN
ejpam-4831	17	10	a.	a.	NOUN
ejpam-4831	17	11	sidaty	sidaty	PROPN
ejpam-4831	17	12	)	)	PUNCT
ejpam-4831	17	13	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4831	17	14	1762	1762	NUM
ejpam-4831	18	1	©	©	ADP
ejpam-4831	18	2	2023	2023	NUM
ejpam-4831	18	3	ejpam	ejpam	NOUN
ejpam-4831	18	4	all	all	DET
ejpam-4831	18	5	rights	right	NOUN
ejpam-4831	18	6	reserved	reserve	VERB
ejpam-4831	18	7	.	.	PUNCT
ejpam-4831	19	1	m.	m.	NOUN
ejpam-4831	19	2	a.	a.	PROPN
ejpam-4831	19	3	sidaty	sidaty	PROPN
ejpam-4831	19	4	/	/	SYM
ejpam-4831	19	5	eur	eur	PROPN
ejpam-4831	19	6	.	.	PUNCT
ejpam-4831	20	1	j.	j.	PROPN
ejpam-4831	20	2	pure	pure	PROPN
ejpam-4831	20	3	appl	appl	PROPN
ejpam-4831	20	4	.	.	PROPN
ejpam-4831	20	5	math	math	PROPN
ejpam-4831	20	6	,	,	PUNCT
ejpam-4831	20	7	16	16	NUM
ejpam-4831	20	8	(	(	PUNCT
ejpam-4831	20	9	3	3	NUM
ejpam-4831	20	10	)	)	PUNCT
ejpam-4831	20	11	(	(	PUNCT
ejpam-4831	20	12	2023	2023	NUM
ejpam-4831	20	13	)	)	PUNCT
ejpam-4831	20	14	,	,	PUNCT
ejpam-4831	20	15	1762	1762	NUM
ejpam-4831	20	16	-	-	SYM
ejpam-4831	20	17	1771	1771	NUM
ejpam-4831	20	18	1763	1763	NUM
ejpam-4831	20	19	in	in	ADP
ejpam-4831	20	20	this	this	DET
ejpam-4831	20	21	note	note	NOUN
ejpam-4831	20	22	,	,	PUNCT
ejpam-4831	20	23	we	we	PRON
ejpam-4831	20	24	are	be	AUX
ejpam-4831	20	25	concerned	concerned	ADJ
ejpam-4831	20	26	with	with	ADP
ejpam-4831	20	27	the	the	DET
ejpam-4831	20	28	nuclearity	nuclearity	NOUN
ejpam-4831	20	29	of	of	ADP
ejpam-4831	20	30	the	the	DET
ejpam-4831	20	31	convex	convex	ADJ
ejpam-4831	20	32	bornological	bornological	ADJ
ejpam-4831	20	33	space	space	NOUN
ejpam-4831	20	34	λ(e	λ(e	VERB
ejpam-4831	20	35	)	)	PUNCT
ejpam-4831	20	36	of	of	ADP
ejpam-4831	20	37	all	all	DET
ejpam-4831	20	38	totally	totally	ADV
ejpam-4831	20	39	λ−summable	λ−summable	VERB
ejpam-4831	20	40	sequences	sequence	NOUN
ejpam-4831	20	41	in	in	ADP
ejpam-4831	20	42	e	e	NOUN
ejpam-4831	20	43	,	,	PUNCT
ejpam-4831	20	44	in	in	ADP
ejpam-4831	20	45	the	the	DET
ejpam-4831	20	46	sense	sense	NOUN
ejpam-4831	20	47	of	of	ADP
ejpam-4831	20	48	[	[	X
ejpam-4831	20	49	3	3	NUM
ejpam-4831	20	50	]	]	PUNCT
ejpam-4831	20	51	,	,	PUNCT
ejpam-4831	20	52	where	where	SCONJ
ejpam-4831	20	53	e	e	NOUN
ejpam-4831	20	54	is	be	AUX
ejpam-4831	20	55	a	a	DET
ejpam-4831	20	56	convex	convex	ADJ
ejpam-4831	20	57	bornological	bornological	ADJ
ejpam-4831	20	58	space	space	NOUN
ejpam-4831	20	59	.	.	PUNCT
ejpam-4831	21	1	in	in	ADP
ejpam-4831	21	2	sections	section	NOUN
ejpam-4831	21	3	1	1	NUM
ejpam-4831	21	4	and	and	CCONJ
ejpam-4831	21	5	2	2	NUM
ejpam-4831	21	6	,	,	PUNCT
ejpam-4831	21	7	we	we	PRON
ejpam-4831	21	8	endow	endow	VERB
ejpam-4831	21	9	this	this	DET
ejpam-4831	21	10	space	space	NOUN
ejpam-4831	21	11	with	with	ADP
ejpam-4831	21	12	a	a	DET
ejpam-4831	21	13	structure	structure	NOUN
ejpam-4831	21	14	of	of	ADP
ejpam-4831	21	15	b	b	NOUN
ejpam-4831	21	16	-	-	PUNCT
ejpam-4831	21	17	space	space	NOUN
ejpam-4831	21	18	,	,	PUNCT
ejpam-4831	21	19	and	and	CCONJ
ejpam-4831	21	20	study	study	VERB
ejpam-4831	21	21	some	some	PRON
ejpam-4831	21	22	of	of	ADP
ejpam-4831	21	23	its	its	PRON
ejpam-4831	21	24	properties	property	NOUN
ejpam-4831	21	25	.	.	PUNCT
ejpam-4831	22	1	the	the	DET
ejpam-4831	22	2	section	section	NOUN
ejpam-4831	22	3	3	3	NUM
ejpam-4831	22	4	is	be	AUX
ejpam-4831	22	5	devoted	devote	VERB
ejpam-4831	22	6	to	to	ADP
ejpam-4831	22	7	the	the	DET
ejpam-4831	22	8	nuclearity	nuclearity	NOUN
ejpam-4831	22	9	of	of	ADP
ejpam-4831	22	10	λ(e	λ(e	NOUN
ejpam-4831	22	11	)	)	PUNCT
ejpam-4831	22	12	.	.	PUNCT
ejpam-4831	23	1	we	we	PRON
ejpam-4831	23	2	prove	prove	VERB
ejpam-4831	23	3	mainly	mainly	ADV
ejpam-4831	23	4	that	that	SCONJ
ejpam-4831	23	5	λ(e	λ(e	PUNCT
ejpam-4831	23	6	)	)	PUNCT
ejpam-4831	23	7	possesses	possess	VERB
ejpam-4831	23	8	this	this	DET
ejpam-4831	23	9	property	property	NOUN
ejpam-4831	23	10	if	if	SCONJ
ejpam-4831	23	11	and	and	CCONJ
ejpam-4831	23	12	only	only	ADV
ejpam-4831	23	13	if	if	SCONJ
ejpam-4831	23	14	both	both	PRON
ejpam-4831	23	15	of	of	ADP
ejpam-4831	23	16	λ	λ	PROPN
ejpam-4831	23	17	and	and	CCONJ
ejpam-4831	23	18	e	e	PRON
ejpam-4831	23	19	have	have	VERB
ejpam-4831	23	20	.	.	PUNCT
ejpam-4831	24	1	in	in	ADP
ejpam-4831	24	2	section	section	NOUN
ejpam-4831	24	3	4	4	NUM
ejpam-4831	24	4	,	,	PUNCT
ejpam-4831	24	5	we	we	PRON
ejpam-4831	24	6	provide	provide	VERB
ejpam-4831	24	7	an	an	DET
ejpam-4831	24	8	application	application	NOUN
ejpam-4831	24	9	of	of	ADP
ejpam-4831	24	10	the	the	DET
ejpam-4831	24	11	results	result	NOUN
ejpam-4831	24	12	of	of	ADP
ejpam-4831	24	13	section	section	NOUN
ejpam-4831	24	14	3	3	NUM
ejpam-4831	24	15	on	on	ADP
ejpam-4831	24	16	the	the	DET
ejpam-4831	24	17	nuclearity	nuclearity	NOUN
ejpam-4831	24	18	of	of	ADP
ejpam-4831	24	19	the	the	DET
ejpam-4831	24	20	space	space	NOUN
ejpam-4831	24	21	λ{e	λ{e	PROPN
ejpam-4831	24	22	}	}	PUNCT
ejpam-4831	24	23	of	of	ADP
ejpam-4831	24	24	absolutely	absolutely	ADV
ejpam-4831	24	25	λ−summable	λ−summable	ADJ
ejpam-4831	24	26	sequences	sequence	NOUN
ejpam-4831	24	27	in	in	ADP
ejpam-4831	24	28	a	a	DET
ejpam-4831	24	29	locally	locally	ADV
ejpam-4831	24	30	convex	convex	ADJ
ejpam-4831	24	31	space	space	NOUN
ejpam-4831	24	32	e.	e.	PROPN
ejpam-4831	24	33	1	1	NUM
ejpam-4831	24	34	.	.	PUNCT
ejpam-4831	24	35	preliminaries	preliminary	NOUN
ejpam-4831	24	36	for	for	ADP
ejpam-4831	24	37	a	a	DET
ejpam-4831	24	38	linear	linear	ADJ
ejpam-4831	24	39	space	space	NOUN
ejpam-4831	24	40	e	e	NOUN
ejpam-4831	24	41	,	,	PUNCT
ejpam-4831	24	42	we	we	PRON
ejpam-4831	24	43	mean	mean	VERB
ejpam-4831	24	44	by	by	ADP
ejpam-4831	24	45	a	a	DET
ejpam-4831	24	46	convex	convex	ADJ
ejpam-4831	24	47	bornology	bornology	NOUN
ejpam-4831	24	48	on	on	ADP
ejpam-4831	24	49	e	e	NOUN
ejpam-4831	24	50	,	,	PUNCT
ejpam-4831	24	51	a	a	DET
ejpam-4831	24	52	collection	collection	NOUN
ejpam-4831	24	53	of	of	ADP
ejpam-4831	24	54	subsets	subset	NOUN
ejpam-4831	24	55	of	of	ADP
ejpam-4831	24	56	e	e	NOUN
ejpam-4831	24	57	covering	cover	VERB
ejpam-4831	24	58	e	e	NOUN
ejpam-4831	24	59	,	,	PUNCT
ejpam-4831	24	60	hereditary	hereditary	ADJ
ejpam-4831	24	61	for	for	ADP
ejpam-4831	24	62	the	the	DET
ejpam-4831	24	63	inclusion	inclusion	NOUN
ejpam-4831	24	64	,	,	PUNCT
ejpam-4831	24	65	and	and	CCONJ
ejpam-4831	24	66	closed	close	VERB
ejpam-4831	24	67	for	for	ADP
ejpam-4831	24	68	the	the	DET
ejpam-4831	24	69	finite	finite	ADJ
ejpam-4831	24	70	unions	union	NOUN
ejpam-4831	24	71	,	,	PUNCT
ejpam-4831	24	72	the	the	DET
ejpam-4831	24	73	addition	addition	NOUN
ejpam-4831	24	74	,	,	PUNCT
ejpam-4831	24	75	the	the	DET
ejpam-4831	24	76	scalar	scalar	ADJ
ejpam-4831	24	77	multiplication	multiplication	NOUN
ejpam-4831	24	78	and	and	CCONJ
ejpam-4831	24	79	the	the	DET
ejpam-4831	24	80	formation	formation	NOUN
ejpam-4831	24	81	of	of	ADP
ejpam-4831	24	82	absolutely	absolutely	ADV
ejpam-4831	24	83	convex	convex	ADJ
ejpam-4831	24	84	hulls	hull	NOUN
ejpam-4831	24	85	.	.	PUNCT
ejpam-4831	25	1	we	we	PRON
ejpam-4831	25	2	say	say	VERB
ejpam-4831	25	3	then	then	ADV
ejpam-4831	25	4	that	that	SCONJ
ejpam-4831	25	5	e	e	NOUN
ejpam-4831	25	6	is	be	AUX
ejpam-4831	25	7	a	a	DET
ejpam-4831	25	8	convex	convex	ADJ
ejpam-4831	25	9	bornological	bornological	ADJ
ejpam-4831	25	10	space	space	NOUN
ejpam-4831	25	11	or	or	CCONJ
ejpam-4831	25	12	simply	simply	ADV
ejpam-4831	25	13	a	a	DET
ejpam-4831	25	14	b	b	NOUN
ejpam-4831	25	15	-	-	NOUN
ejpam-4831	25	16	space	space	NOUN
ejpam-4831	25	17	.	.	PUNCT
ejpam-4831	26	1	the	the	DET
ejpam-4831	26	2	elements	element	NOUN
ejpam-4831	26	3	of	of	ADP
ejpam-4831	26	4	the	the	DET
ejpam-4831	26	5	bornology	bornology	NOUN
ejpam-4831	26	6	of	of	ADP
ejpam-4831	26	7	e	e	PROPN
ejpam-4831	26	8	are	be	AUX
ejpam-4831	26	9	called	call	VERB
ejpam-4831	26	10	bounded	bounded	ADJ
ejpam-4831	26	11	sets	set	NOUN
ejpam-4831	26	12	of	of	ADP
ejpam-4831	26	13	e.	e.	PROPN
ejpam-4831	26	14	a	a	DET
ejpam-4831	26	15	collection	collection	NOUN
ejpam-4831	26	16	b	b	PROPN
ejpam-4831	26	17	of	of	ADP
ejpam-4831	26	18	bounded	bounded	ADJ
ejpam-4831	26	19	sets	set	NOUN
ejpam-4831	26	20	of	of	ADP
ejpam-4831	26	21	e	e	NOUN
ejpam-4831	26	22	is	be	AUX
ejpam-4831	26	23	a	a	DET
ejpam-4831	26	24	basis	basis	NOUN
ejpam-4831	26	25	for	for	ADP
ejpam-4831	26	26	its	its	PRON
ejpam-4831	26	27	bornology	bornology	NOUN
ejpam-4831	26	28	if	if	SCONJ
ejpam-4831	26	29	every	every	DET
ejpam-4831	26	30	bounded	bound	VERB
ejpam-4831	26	31	set	set	VERB
ejpam-4831	26	32	in	in	ADP
ejpam-4831	26	33	e	e	PROPN
ejpam-4831	26	34	is	be	AUX
ejpam-4831	26	35	contained	contain	VERB
ejpam-4831	26	36	in	in	ADP
ejpam-4831	26	37	an	an	DET
ejpam-4831	26	38	element	element	NOUN
ejpam-4831	26	39	of	of	ADP
ejpam-4831	26	40	b.	b.	PROPN
ejpam-4831	26	41	in	in	ADP
ejpam-4831	26	42	the	the	DET
ejpam-4831	26	43	sequel	sequel	NOUN
ejpam-4831	26	44	,	,	PUNCT
ejpam-4831	26	45	we	we	PRON
ejpam-4831	26	46	assume	assume	VERB
ejpam-4831	26	47	that	that	SCONJ
ejpam-4831	26	48	the	the	DET
ejpam-4831	26	49	members	member	NOUN
ejpam-4831	26	50	of	of	ADP
ejpam-4831	26	51	b	b	NOUN
ejpam-4831	26	52	are	be	AUX
ejpam-4831	26	53	absolutely	absolutely	ADV
ejpam-4831	26	54	convex	convex	ADJ
ejpam-4831	26	55	.	.	PUNCT
ejpam-4831	27	1	a	a	DET
ejpam-4831	27	2	b	b	NOUN
ejpam-4831	27	3	-	-	PUNCT
ejpam-4831	27	4	space	space	NOUN
ejpam-4831	27	5	e	e	NOUN
ejpam-4831	27	6	is	be	AUX
ejpam-4831	27	7	said	say	VERB
ejpam-4831	27	8	to	to	PART
ejpam-4831	27	9	be	be	AUX
ejpam-4831	27	10	hausdorff	hausdorff	NOUN
ejpam-4831	27	11	if	if	SCONJ
ejpam-4831	27	12	the	the	DET
ejpam-4831	27	13	only	only	ADJ
ejpam-4831	27	14	bounded	bound	VERB
ejpam-4831	27	15	linear	linear	ADJ
ejpam-4831	27	16	subspace	subspace	NOUN
ejpam-4831	27	17	of	of	ADP
ejpam-4831	27	18	e	e	PROPN
ejpam-4831	27	19	is	be	AUX
ejpam-4831	27	20	{	{	PUNCT
ejpam-4831	27	21	0	0	NUM
ejpam-4831	27	22	}	}	PUNCT
ejpam-4831	27	23	.	.	PUNCT
ejpam-4831	28	1	we	we	PRON
ejpam-4831	28	2	say	say	VERB
ejpam-4831	28	3	that	that	SCONJ
ejpam-4831	28	4	a	a	DET
ejpam-4831	28	5	sequence	sequence	NOUN
ejpam-4831	28	6	{	{	PUNCT
ejpam-4831	28	7	xn}∞n=1	xn}∞n=1	PUNCT
ejpam-4831	28	8	⊂	⊂	PROPN
ejpam-4831	28	9	e	e	PROPN
ejpam-4831	28	10	converges	converge	VERB
ejpam-4831	28	11	to	to	ADP
ejpam-4831	28	12	x	x	SYM
ejpam-4831	28	13	∈	∈	PROPN
ejpam-4831	28	14	e	e	NOUN
ejpam-4831	28	15	,	,	PUNCT
ejpam-4831	28	16	or	or	CCONJ
ejpam-4831	28	17	that	that	SCONJ
ejpam-4831	28	18	x	x	PRON
ejpam-4831	28	19	is	be	AUX
ejpam-4831	28	20	a	a	DET
ejpam-4831	28	21	limit	limit	NOUN
ejpam-4831	28	22	of	of	ADP
ejpam-4831	28	23	{	{	PUNCT
ejpam-4831	28	24	xn}∞n=1	xn}∞n=1	PROPN
ejpam-4831	28	25	in	in	ADP
ejpam-4831	28	26	e	e	PROPN
ejpam-4831	28	27	if	if	SCONJ
ejpam-4831	28	28	there	there	PRON
ejpam-4831	28	29	exists	exist	VERB
ejpam-4831	28	30	an	an	DET
ejpam-4831	28	31	element	element	NOUN
ejpam-4831	28	32	b	b	PROPN
ejpam-4831	28	33	∈	∈	PROPN
ejpam-4831	28	34	b	b	NOUN
ejpam-4831	28	35	such	such	ADJ
ejpam-4831	28	36	that	that	SCONJ
ejpam-4831	28	37	{	{	PUNCT
ejpam-4831	28	38	xn	xn	NUM
ejpam-4831	28	39	−	−	PROPN
ejpam-4831	28	40	x}∞n=1	x}∞n=1	PROPN
ejpam-4831	28	41	is	be	AUX
ejpam-4831	28	42	contained	contain	VERB
ejpam-4831	28	43	and	and	CCONJ
ejpam-4831	28	44	convergent	convergent	NOUN
ejpam-4831	28	45	to	to	ADP
ejpam-4831	28	46	0	0	NUM
ejpam-4831	28	47	in	in	ADP
ejpam-4831	28	48	the	the	DET
ejpam-4831	28	49	normed	normed	ADJ
ejpam-4831	28	50	space	space	NOUN
ejpam-4831	28	51	(	(	PUNCT
ejpam-4831	28	52	eb	eb	PROPN
ejpam-4831	28	53	,	,	PUNCT
ejpam-4831	28	54	∥	∥	X
ejpam-4831	28	55	·	·	PUNCT
ejpam-4831	29	1	∥b	∥b	X
ejpam-4831	29	2	)	)	PUNCT
ejpam-4831	29	3	,	,	PUNCT
ejpam-4831	29	4	where	where	SCONJ
ejpam-4831	29	5	eb	eb	PROPN
ejpam-4831	29	6	is	be	AUX
ejpam-4831	29	7	the	the	DET
ejpam-4831	29	8	subspace	subspace	NOUN
ejpam-4831	29	9	of	of	ADP
ejpam-4831	29	10	e	e	PROPN
ejpam-4831	29	11	generated	generate	VERB
ejpam-4831	29	12	by	by	ADP
ejpam-4831	29	13	b	b	PROPN
ejpam-4831	29	14	and	and	CCONJ
ejpam-4831	29	15	∥	∥	NUM
ejpam-4831	29	16	·	·	PUNCT
ejpam-4831	30	1	∥b	∥b	NOUN
ejpam-4831	30	2	is	be	AUX
ejpam-4831	30	3	the	the	DET
ejpam-4831	30	4	gauge	gauge	NOUN
ejpam-4831	30	5	of	of	ADP
ejpam-4831	30	6	b.	b.	PROPN
ejpam-4831	30	7	a	a	DET
ejpam-4831	30	8	subset	subset	NOUN
ejpam-4831	30	9	of	of	ADP
ejpam-4831	30	10	a	a	DET
ejpam-4831	30	11	b	b	NOUN
ejpam-4831	30	12	-	-	PUNCT
ejpam-4831	30	13	space	space	NOUN
ejpam-4831	30	14	e	e	NOUN
ejpam-4831	30	15	will	will	AUX
ejpam-4831	30	16	be	be	AUX
ejpam-4831	30	17	said	say	VERB
ejpam-4831	30	18	to	to	PART
ejpam-4831	30	19	be	be	AUX
ejpam-4831	30	20	closed	close	VERB
ejpam-4831	30	21	if	if	SCONJ
ejpam-4831	30	22	it	it	PRON
ejpam-4831	30	23	contains	contain	VERB
ejpam-4831	30	24	the	the	DET
ejpam-4831	30	25	limits	limit	NOUN
ejpam-4831	30	26	of	of	ADP
ejpam-4831	30	27	all	all	DET
ejpam-4831	30	28	its	its	PRON
ejpam-4831	30	29	sequences	sequence	NOUN
ejpam-4831	30	30	.	.	PUNCT
ejpam-4831	31	1	a	a	DET
ejpam-4831	31	2	banach	banach	NOUN
ejpam-4831	31	3	disk	disk	NOUN
ejpam-4831	31	4	in	in	ADP
ejpam-4831	31	5	a	a	DET
ejpam-4831	31	6	b	b	NOUN
ejpam-4831	31	7	-	-	PUNCT
ejpam-4831	31	8	space	space	NOUN
ejpam-4831	31	9	e	e	NOUN
ejpam-4831	31	10	is	be	AUX
ejpam-4831	31	11	an	an	DET
ejpam-4831	31	12	element	element	NOUN
ejpam-4831	31	13	b	b	PROPN
ejpam-4831	31	14	∈	∈	PROPN
ejpam-4831	31	15	b	b	PROPN
ejpam-4831	31	16	for	for	ADP
ejpam-4831	31	17	which	which	PRON
ejpam-4831	31	18	the	the	DET
ejpam-4831	31	19	normed	normed	PROPN
ejpam-4831	31	20	space	space	PROPN
ejpam-4831	31	21	eb	eb	PROPN
ejpam-4831	31	22	is	be	AUX
ejpam-4831	31	23	complete	complete	ADJ
ejpam-4831	31	24	.	.	PUNCT
ejpam-4831	32	1	e	e	NOUN
ejpam-4831	32	2	is	be	AUX
ejpam-4831	32	3	said	say	VERB
ejpam-4831	32	4	to	to	PART
ejpam-4831	32	5	be	be	AUX
ejpam-4831	32	6	b	b	NOUN
ejpam-4831	32	7	-	-	PUNCT
ejpam-4831	32	8	complete	complete	ADJ
ejpam-4831	32	9	or	or	CCONJ
ejpam-4831	32	10	simply	simply	ADV
ejpam-4831	32	11	complete	complete	ADJ
ejpam-4831	32	12	if	if	SCONJ
ejpam-4831	32	13	every	every	DET
ejpam-4831	32	14	bounded	bound	VERB
ejpam-4831	32	15	set	set	VERB
ejpam-4831	32	16	in	in	ADP
ejpam-4831	32	17	e	e	PROPN
ejpam-4831	32	18	is	be	AUX
ejpam-4831	32	19	contained	contain	VERB
ejpam-4831	32	20	in	in	ADP
ejpam-4831	32	21	a	a	DET
ejpam-4831	32	22	banach	banach	NOUN
ejpam-4831	32	23	disk	disk	NOUN
ejpam-4831	32	24	in	in	ADP
ejpam-4831	32	25	e.	e.	PROPN
ejpam-4831	32	26	a	a	DET
ejpam-4831	32	27	linear	linear	ADJ
ejpam-4831	32	28	mapping	mapping	NOUN
ejpam-4831	32	29	between	between	ADP
ejpam-4831	32	30	two	two	NUM
ejpam-4831	32	31	b	b	NUM
ejpam-4831	32	32	-	-	PUNCT
ejpam-4831	32	33	spaces	space	NOUN
ejpam-4831	32	34	e	e	NOUN
ejpam-4831	32	35	and	and	CCONJ
ejpam-4831	32	36	f	f	PROPN
ejpam-4831	32	37	is	be	AUX
ejpam-4831	32	38	said	say	VERB
ejpam-4831	32	39	to	to	PART
ejpam-4831	32	40	be	be	AUX
ejpam-4831	32	41	bounded	bound	VERB
ejpam-4831	32	42	if	if	SCONJ
ejpam-4831	32	43	it	it	PRON
ejpam-4831	32	44	transforms	transform	VERB
ejpam-4831	32	45	bounded	bounded	ADJ
ejpam-4831	32	46	sets	set	NOUN
ejpam-4831	32	47	of	of	ADP
ejpam-4831	32	48	e	e	NOUN
ejpam-4831	32	49	to	to	PART
ejpam-4831	32	50	bounded	bound	VERB
ejpam-4831	32	51	sets	set	NOUN
ejpam-4831	32	52	of	of	ADP
ejpam-4831	32	53	f	f	PROPN
ejpam-4831	32	54	.	.	PUNCT
ejpam-4831	33	1	a	a	DET
ejpam-4831	33	2	bounded	bounded	ADJ
ejpam-4831	33	3	linear	linear	PROPN
ejpam-4831	33	4	mapping	mapping	NOUN
ejpam-4831	33	5	transforms	transform	VERB
ejpam-4831	33	6	convergent	convergent	NOUN
ejpam-4831	33	7	sequences	sequence	NOUN
ejpam-4831	33	8	to	to	ADP
ejpam-4831	33	9	convergent	convergent	NOUN
ejpam-4831	33	10	ones	one	NOUN
ejpam-4831	33	11	.	.	PUNCT
ejpam-4831	34	1	a	a	DET
ejpam-4831	34	2	bornological	bornological	ADJ
ejpam-4831	34	3	isomorphism	isomorphism	NOUN
ejpam-4831	34	4	is	be	AUX
ejpam-4831	34	5	a	a	DET
ejpam-4831	34	6	bounded	bounded	ADJ
ejpam-4831	34	7	linear	linear	ADJ
ejpam-4831	34	8	bijection	bijection	NOUN
ejpam-4831	34	9	whose	whose	DET
ejpam-4831	34	10	inverse	inverse	NOUN
ejpam-4831	34	11	is	be	AUX
ejpam-4831	34	12	also	also	ADV
ejpam-4831	34	13	bounded	bound	VERB
ejpam-4831	34	14	.	.	PUNCT
ejpam-4831	35	1	the	the	DET
ejpam-4831	35	2	köthe	köthe	PROPN
ejpam-4831	35	3	dual	dual	ADJ
ejpam-4831	35	4	of	of	ADP
ejpam-4831	35	5	a	a	DET
ejpam-4831	35	6	sequence	sequence	NOUN
ejpam-4831	35	7	space	space	NOUN
ejpam-4831	35	8	λ	λ	PROPN
ejpam-4831	35	9	is	be	AUX
ejpam-4831	35	10	defined	define	VERB
ejpam-4831	35	11	as	as	ADP
ejpam-4831	35	12	λ×	λ×	PROPN
ejpam-4831	35	13	=	=	PRON
ejpam-4831	35	14	{	{	PUNCT
ejpam-4831	35	15	(	(	PUNCT
ejpam-4831	35	16	βn	βn	NOUN
ejpam-4831	35	17	)	)	PUNCT
ejpam-4831	36	1	⊂	⊂	PROPN
ejpam-4831	36	2	c	c	NOUN
ejpam-4831	36	3	:	:	PUNCT
ejpam-4831	37	1	∞∑	∞∑	NUM
ejpam-4831	37	2	n=1	n=1	NUM
ejpam-4831	37	3	|αnβn|	|αnβn|	PRON
ejpam-4831	37	4	converges	converge	NOUN
ejpam-4831	37	5	for	for	ADP
ejpam-4831	37	6	all	all	DET
ejpam-4831	37	7	(	(	PUNCT
ejpam-4831	37	8	αn	αn	NOUN
ejpam-4831	37	9	)	)	PUNCT
ejpam-4831	37	10	∈	∈	PROPN
ejpam-4831	37	11	λ	λ	PROPN
ejpam-4831	37	12	}	}	PUNCT
ejpam-4831	37	13	.	.	PUNCT
ejpam-4831	38	1	we	we	PRON
ejpam-4831	38	2	see	see	VERB
ejpam-4831	38	3	that	that	SCONJ
ejpam-4831	38	4	λ	λ	PROPN
ejpam-4831	38	5	⊂	⊂	X
ejpam-4831	38	6	λ××	λ××	X
ejpam-4831	39	1	=	=	PRON
ejpam-4831	39	2	:	:	PUNCT
ejpam-4831	39	3	(	(	PUNCT
ejpam-4831	39	4	λ×)×	λ×)×	NOUN
ejpam-4831	39	5	;	;	PUNCT
ejpam-4831	39	6	we	we	PRON
ejpam-4831	39	7	say	say	VERB
ejpam-4831	39	8	that	that	SCONJ
ejpam-4831	39	9	λ	λ	PROPN
ejpam-4831	39	10	is	be	AUX
ejpam-4831	39	11	perfect	perfect	ADJ
ejpam-4831	39	12	if	if	SCONJ
ejpam-4831	39	13	the	the	DET
ejpam-4831	39	14	equality	equality	NOUN
ejpam-4831	39	15	holds	hold	VERB
ejpam-4831	39	16	.	.	PUNCT
ejpam-4831	40	1	the	the	DET
ejpam-4831	40	2	normal	normal	ADJ
ejpam-4831	40	3	cover	cover	NOUN
ejpam-4831	40	4	of	of	ADP
ejpam-4831	40	5	a	a	DET
ejpam-4831	40	6	subset	subset	NOUN
ejpam-4831	40	7	s	s	NOUN
ejpam-4831	40	8	of	of	ADP
ejpam-4831	40	9	λ	λ	PROPN
ejpam-4831	40	10	is	be	AUX
ejpam-4831	40	11	the	the	DET
ejpam-4831	40	12	subset	subset	NOUN
ejpam-4831	40	13	of	of	ADP
ejpam-4831	40	14	λ	λ	PROPN
ejpam-4831	40	15	formed	form	VERB
ejpam-4831	40	16	by	by	ADP
ejpam-4831	40	17	the	the	DET
ejpam-4831	40	18	sequences	sequence	NOUN
ejpam-4831	40	19	of	of	ADP
ejpam-4831	40	20	the	the	DET
ejpam-4831	40	21	m.	m.	NOUN
ejpam-4831	40	22	a.	a.	PROPN
ejpam-4831	40	23	sidaty	sidaty	PROPN
ejpam-4831	40	24	/	/	SYM
ejpam-4831	40	25	eur	eur	PROPN
ejpam-4831	40	26	.	.	PUNCT
ejpam-4831	41	1	j.	j.	PROPN
ejpam-4831	41	2	pure	pure	PROPN
ejpam-4831	41	3	appl	appl	PROPN
ejpam-4831	41	4	.	.	PROPN
ejpam-4831	41	5	math	math	PROPN
ejpam-4831	41	6	,	,	PUNCT
ejpam-4831	41	7	16	16	NUM
ejpam-4831	41	8	(	(	PUNCT
ejpam-4831	41	9	3	3	NUM
ejpam-4831	41	10	)	)	PUNCT
ejpam-4831	41	11	(	(	PUNCT
ejpam-4831	41	12	2023	2023	NUM
ejpam-4831	41	13	)	)	PUNCT
ejpam-4831	41	14	,	,	PUNCT
ejpam-4831	41	15	1762	1762	NUM
ejpam-4831	41	16	-	-	SYM
ejpam-4831	41	17	1771	1771	NUM
ejpam-4831	41	18	1764	1764	NUM
ejpam-4831	41	19	form	form	NOUN
ejpam-4831	41	20	(	(	PUNCT
ejpam-4831	41	21	εnαn)n	εnαn)n	PROPN
ejpam-4831	41	22	where	where	SCONJ
ejpam-4831	41	23	(	(	PUNCT
ejpam-4831	41	24	αn)n	αn)n	NOUN
ejpam-4831	41	25	∈	∈	NOUN
ejpam-4831	41	26	s	s	PART
ejpam-4831	41	27	and	and	CCONJ
ejpam-4831	41	28	(	(	PUNCT
ejpam-4831	41	29	εn)n	εn)n	PROPN
ejpam-4831	41	30	⊂	⊂	PROPN
ejpam-4831	41	31	c	c	PROPN
ejpam-4831	41	32	with	with	ADP
ejpam-4831	41	33	|εn|	|εn|	PRON
ejpam-4831	41	34	≤	≤	ADV
ejpam-4831	41	35	1	1	NUM
ejpam-4831	41	36	,	,	PUNCT
ejpam-4831	41	37	for	for	ADP
ejpam-4831	41	38	all	all	DET
ejpam-4831	41	39	n.	n.	NOUN
ejpam-4831	41	40	we	we	PRON
ejpam-4831	41	41	see	see	VERB
ejpam-4831	41	42	that	that	PRON
ejpam-4831	41	43	s	s	VERB
ejpam-4831	41	44	is	be	AUX
ejpam-4831	41	45	contained	contain	VERB
ejpam-4831	41	46	in	in	ADP
ejpam-4831	41	47	its	its	PRON
ejpam-4831	41	48	normal	normal	ADJ
ejpam-4831	41	49	cover	cover	NOUN
ejpam-4831	41	50	.	.	PUNCT
ejpam-4831	42	1	s	s	PART
ejpam-4831	42	2	is	be	AUX
ejpam-4831	42	3	said	say	VERB
ejpam-4831	42	4	to	to	PART
ejpam-4831	42	5	be	be	AUX
ejpam-4831	42	6	normal	normal	ADJ
ejpam-4831	42	7	or	or	CCONJ
ejpam-4831	42	8	solid	solid	ADJ
ejpam-4831	42	9	if	if	SCONJ
ejpam-4831	42	10	it	it	PRON
ejpam-4831	42	11	coincides	coincide	VERB
ejpam-4831	42	12	with	with	ADP
ejpam-4831	42	13	its	its	PRON
ejpam-4831	42	14	normal	normal	ADJ
ejpam-4831	42	15	cover	cover	NOUN
ejpam-4831	42	16	.	.	PUNCT
ejpam-4831	43	1	for	for	ADP
ejpam-4831	43	2	the	the	DET
ejpam-4831	43	3	general	general	ADJ
ejpam-4831	43	4	theory	theory	NOUN
ejpam-4831	43	5	of	of	ADP
ejpam-4831	43	6	locally	locally	ADV
ejpam-4831	43	7	convex	convex	ADJ
ejpam-4831	43	8	spaces	space	NOUN
ejpam-4831	43	9	and	and	CCONJ
ejpam-4831	43	10	köthe	köthe	DET
ejpam-4831	43	11	sequence	sequence	NOUN
ejpam-4831	43	12	spaces	space	VERB
ejpam-4831	43	13	,	,	PUNCT
ejpam-4831	43	14	we	we	PRON
ejpam-4831	43	15	refer	refer	VERB
ejpam-4831	43	16	the	the	DET
ejpam-4831	43	17	reader	reader	NOUN
ejpam-4831	43	18	to	to	ADP
ejpam-4831	43	19	[	[	X
ejpam-4831	43	20	5	5	NUM
ejpam-4831	43	21	]	]	PUNCT
ejpam-4831	43	22	.	.	PUNCT
ejpam-4831	44	1	throughout	throughout	ADP
ejpam-4831	44	2	this	this	DET
ejpam-4831	44	3	paper	paper	NOUN
ejpam-4831	44	4	,	,	PUNCT
ejpam-4831	44	5	λ	λ	PROPN
ejpam-4831	44	6	will	will	AUX
ejpam-4831	44	7	be	be	AUX
ejpam-4831	44	8	a	a	DET
ejpam-4831	44	9	perfect	perfect	ADJ
ejpam-4831	44	10	(	(	PUNCT
ejpam-4831	44	11	and	and	CCONJ
ejpam-4831	44	12	then	then	ADV
ejpam-4831	44	13	a	a	DET
ejpam-4831	44	14	normal	normal	ADJ
ejpam-4831	44	15	)	)	PUNCT
ejpam-4831	44	16	sequence	sequence	NOUN
ejpam-4831	44	17	space	space	NOUN
ejpam-4831	44	18	endowed	endow	VERB
ejpam-4831	44	19	with	with	ADP
ejpam-4831	44	20	a	a	DET
ejpam-4831	44	21	normal	normal	ADJ
ejpam-4831	44	22	bornology	bornology	NOUN
ejpam-4831	44	23	,	,	PUNCT
ejpam-4831	44	24	that	that	PRON
ejpam-4831	44	25	is	be	AUX
ejpam-4831	44	26	a	a	DET
ejpam-4831	44	27	convex	convex	ADJ
ejpam-4831	44	28	bornology	bornology	NOUN
ejpam-4831	44	29	having	have	VERB
ejpam-4831	44	30	a	a	DET
ejpam-4831	44	31	basis	basis	NOUN
ejpam-4831	44	32	s	s	NOUN
ejpam-4831	44	33	of	of	ADP
ejpam-4831	44	34	solid	solid	ADJ
ejpam-4831	44	35	sets	set	NOUN
ejpam-4831	44	36	,	,	PUNCT
ejpam-4831	44	37	and	and	CCONJ
ejpam-4831	44	38	for	for	ADP
ejpam-4831	44	39	which	which	PRON
ejpam-4831	44	40	the	the	DET
ejpam-4831	44	41	standard	standard	ADJ
ejpam-4831	44	42	coordinate	coordinate	NOUN
ejpam-4831	44	43	projections	projection	NOUN
ejpam-4831	44	44	from	from	ADP
ejpam-4831	44	45	λ	λ	PROPN
ejpam-4831	44	46	to	to	ADP
ejpam-4831	44	47	c	c	PROPN
ejpam-4831	44	48	are	be	AUX
ejpam-4831	44	49	bounded	bound	VERB
ejpam-4831	44	50	.	.	PUNCT
ejpam-4831	45	1	following	follow	VERB
ejpam-4831	45	2	the	the	DET
ejpam-4831	45	3	terminology	terminology	NOUN
ejpam-4831	45	4	of	of	ADP
ejpam-4831	45	5	[	[	X
ejpam-4831	45	6	3	3	NUM
ejpam-4831	45	7	]	]	PUNCT
ejpam-4831	45	8	,	,	PUNCT
ejpam-4831	45	9	a	a	DET
ejpam-4831	45	10	sequence	sequence	NOUN
ejpam-4831	45	11	(	(	PUNCT
ejpam-4831	45	12	xn)n	xn)n	PROPN
ejpam-4831	45	13	⊂	⊂	PROPN
ejpam-4831	45	14	e	e	PROPN
ejpam-4831	45	15	is	be	AUX
ejpam-4831	45	16	said	say	VERB
ejpam-4831	45	17	to	to	PART
ejpam-4831	45	18	be	be	AUX
ejpam-4831	45	19	totally	totally	ADV
ejpam-4831	45	20	λ−summable	λ−summable	ADJ
ejpam-4831	45	21	in	in	ADP
ejpam-4831	45	22	e	e	NOUN
ejpam-4831	45	23	if	if	SCONJ
ejpam-4831	45	24	there	there	PRON
ejpam-4831	45	25	exists	exist	VERB
ejpam-4831	45	26	an	an	DET
ejpam-4831	45	27	absolutely	absolutely	ADV
ejpam-4831	45	28	convex	convex	ADJ
ejpam-4831	45	29	element	element	NOUN
ejpam-4831	45	30	b	b	PROPN
ejpam-4831	45	31	∈	∈	PROPN
ejpam-4831	45	32	b	b	NOUN
ejpam-4831	45	33	such	such	ADJ
ejpam-4831	45	34	that	that	PRON
ejpam-4831	45	35	(	(	PUNCT
ejpam-4831	45	36	xn)n	xn)n	PROPN
ejpam-4831	45	37	⊂	⊂	PROPN
ejpam-4831	45	38	eb	eb	PROPN
ejpam-4831	45	39	and	and	CCONJ
ejpam-4831	45	40	(	(	PUNCT
ejpam-4831	45	41	∥xn∥b)n	∥xn∥b)n	PROPN
ejpam-4831	45	42	∈	∈	PROPN
ejpam-4831	45	43	λ	λ	PROPN
ejpam-4831	45	44	.	.	PUNCT
ejpam-4831	46	1	in	in	ADP
ejpam-4831	46	2	other	other	ADJ
ejpam-4831	46	3	words	word	NOUN
ejpam-4831	46	4	,	,	PUNCT
ejpam-4831	46	5	(	(	PUNCT
ejpam-4831	46	6	xn)n	xn)n	PROPN
ejpam-4831	46	7	=	=	SYM
ejpam-4831	46	8	(	(	PUNCT
ejpam-4831	46	9	αnbn)n	αnbn)n	NOUN
ejpam-4831	46	10	,	,	PUNCT
ejpam-4831	46	11	with	with	ADP
ejpam-4831	46	12	(	(	PUNCT
ejpam-4831	46	13	αn)n	αn)n	NOUN
ejpam-4831	46	14	∈	∈	PROPN
ejpam-4831	46	15	λ	λ	PROPN
ejpam-4831	46	16	and	and	CCONJ
ejpam-4831	46	17	{	{	PUNCT
ejpam-4831	46	18	bn}∞n=1	bn}∞n=1	X
ejpam-4831	46	19	⊂	⊂	PROPN
ejpam-4831	46	20	b.	b.	PROPN
ejpam-4831	46	21	starting	start	VERB
ejpam-4831	46	22	from	from	ADP
ejpam-4831	46	23	this	this	DET
ejpam-4831	46	24	definition	definition	NOUN
ejpam-4831	46	25	,	,	PUNCT
ejpam-4831	46	26	we	we	PRON
ejpam-4831	46	27	introduce	introduce	VERB
ejpam-4831	46	28	the	the	DET
ejpam-4831	46	29	vector	vector	NOUN
ejpam-4831	46	30	valued	value	VERB
ejpam-4831	46	31	sequence	sequence	NOUN
ejpam-4831	46	32	space	space	NOUN
ejpam-4831	46	33	λ(e	λ(e	VERB
ejpam-4831	46	34	)	)	PUNCT
ejpam-4831	46	35	=	=	PRON
ejpam-4831	46	36	{	{	PUNCT
ejpam-4831	46	37	(	(	PUNCT
ejpam-4831	46	38	xn)n	xn)n	PROPN
ejpam-4831	46	39	⊂	⊂	PROPN
ejpam-4831	46	40	e	e	X
ejpam-4831	46	41	:	:	PUNCT
ejpam-4831	46	42	∃b	∃b	PROPN
ejpam-4831	46	43	∈	∈	PROPN
ejpam-4831	46	44	b	b	PROPN
ejpam-4831	46	45	,	,	PUNCT
ejpam-4831	46	46	(	(	PUNCT
ejpam-4831	46	47	xn)n	xn)n	PROPN
ejpam-4831	46	48	⊂	⊂	PROPN
ejpam-4831	46	49	eb	eb	PROPN
ejpam-4831	46	50	and	and	CCONJ
ejpam-4831	46	51	(	(	PUNCT
ejpam-4831	46	52	∥xn∥)n	∥xn∥)n	NUM
ejpam-4831	46	53	∈	∈	PROPN
ejpam-4831	46	54	λ	λ	PROPN
ejpam-4831	46	55	}	}	PUNCT
ejpam-4831	46	56	.	.	PUNCT
ejpam-4831	47	1	due	due	ADP
ejpam-4831	47	2	to	to	ADP
ejpam-4831	47	3	the	the	DET
ejpam-4831	47	4	properties	property	NOUN
ejpam-4831	47	5	of	of	ADP
ejpam-4831	47	6	b	b	NOUN
ejpam-4831	47	7	,	,	PUNCT
ejpam-4831	47	8	the	the	DET
ejpam-4831	47	9	triangle	triangle	NOUN
ejpam-4831	47	10	inequality	inequality	NOUN
ejpam-4831	47	11	of	of	ADP
ejpam-4831	47	12	the	the	DET
ejpam-4831	47	13	norms	norm	NOUN
ejpam-4831	47	14	∥	∥	X
ejpam-4831	47	15	·	·	PUNCT
ejpam-4831	47	16	∥b	∥b	NOUN
ejpam-4831	47	17	and	and	CCONJ
ejpam-4831	47	18	the	the	DET
ejpam-4831	47	19	fact	fact	NOUN
ejpam-4831	47	20	that	that	SCONJ
ejpam-4831	47	21	λ	λ	NOUN
ejpam-4831	47	22	is	be	AUX
ejpam-4831	47	23	normal	normal	ADJ
ejpam-4831	47	24	,	,	PUNCT
ejpam-4831	47	25	we	we	PRON
ejpam-4831	47	26	see	see	VERB
ejpam-4831	47	27	that	that	SCONJ
ejpam-4831	47	28	λ(e	λ(e	PROPN
ejpam-4831	47	29	)	)	PUNCT
ejpam-4831	47	30	is	be	AUX
ejpam-4831	47	31	a	a	DET
ejpam-4831	47	32	linear	linear	ADJ
ejpam-4831	47	33	space	space	NOUN
ejpam-4831	47	34	.	.	PUNCT
ejpam-4831	48	1	for	for	ADP
ejpam-4831	48	2	s	s	PROPN
ejpam-4831	48	3	∈	∈	PROPN
ejpam-4831	48	4	s	s	X
ejpam-4831	48	5	and	and	CCONJ
ejpam-4831	48	6	b	b	PROPN
ejpam-4831	48	7	∈	∈	PROPN
ejpam-4831	48	8	b	b	NOUN
ejpam-4831	48	9	,	,	PUNCT
ejpam-4831	48	10	we	we	PRON
ejpam-4831	48	11	define	define	VERB
ejpam-4831	48	12	s(b	s(b	NOUN
ejpam-4831	48	13	)	)	PUNCT
ejpam-4831	49	1	=	=	PRON
ejpam-4831	49	2	{	{	PUNCT
ejpam-4831	49	3	(	(	PUNCT
ejpam-4831	49	4	xn)n	xn)n	PROPN
ejpam-4831	49	5	⊂	⊂	PROPN
ejpam-4831	49	6	eb	eb	PROPN
ejpam-4831	49	7	,	,	PUNCT
ejpam-4831	49	8	(	(	PUNCT
ejpam-4831	49	9	∥xn∥b)n	∥xn∥b)n	PROPN
ejpam-4831	49	10	∈	∈	PROPN
ejpam-4831	49	11	s	s	PART
ejpam-4831	49	12	}	}	PUNCT
ejpam-4831	49	13	.	.	PUNCT
ejpam-4831	50	1	2	2	X
ejpam-4831	50	2	.	.	X
ejpam-4831	50	3	properties	property	NOUN
ejpam-4831	50	4	of	of	ADP
ejpam-4831	50	5	λ(e	λ(e	NOUN
ejpam-4831	50	6	)	)	PUNCT
ejpam-4831	50	7	in	in	ADP
ejpam-4831	50	8	the	the	DET
ejpam-4831	50	9	sequel	sequel	NOUN
ejpam-4831	50	10	,	,	PUNCT
ejpam-4831	50	11	the	the	PRON
ejpam-4831	50	12	b	b	NOUN
ejpam-4831	50	13	-	-	PUNCT
ejpam-4831	50	14	spaces	space	NOUN
ejpam-4831	50	15	e	e	NOUN
ejpam-4831	50	16	equipped	equip	VERB
ejpam-4831	50	17	with	with	ADP
ejpam-4831	50	18	the	the	DET
ejpam-4831	50	19	convex	convex	ADJ
ejpam-4831	50	20	bornology	bornology	NOUN
ejpam-4831	50	21	with	with	ADP
ejpam-4831	50	22	basis	basis	NOUN
ejpam-4831	50	23	b	b	NOUN
ejpam-4831	50	24	and	and	CCONJ
ejpam-4831	50	25	λ	λ	X
ejpam-4831	50	26	with	with	ADP
ejpam-4831	50	27	the	the	DET
ejpam-4831	50	28	normal	normal	ADJ
ejpam-4831	50	29	bornology	bornology	NOUN
ejpam-4831	50	30	with	with	ADP
ejpam-4831	50	31	basis	basis	NOUN
ejpam-4831	50	32	s	s	PART
ejpam-4831	50	33	,	,	PUNCT
ejpam-4831	50	34	will	will	AUX
ejpam-4831	50	35	be	be	AUX
ejpam-4831	50	36	supposed	suppose	VERB
ejpam-4831	50	37	to	to	PART
ejpam-4831	50	38	be	be	AUX
ejpam-4831	50	39	hausdorff	hausdorff	NOUN
ejpam-4831	50	40	spaces	space	NOUN
ejpam-4831	50	41	.	.	PUNCT
ejpam-4831	51	1	starting	start	VERB
ejpam-4831	51	2	from	from	ADP
ejpam-4831	51	3	this	this	DET
ejpam-4831	51	4	setting	setting	NOUN
ejpam-4831	51	5	,	,	PUNCT
ejpam-4831	51	6	one	one	PRON
ejpam-4831	51	7	can	can	AUX
ejpam-4831	51	8	define	define	VERB
ejpam-4831	51	9	,	,	PUNCT
ejpam-4831	51	10	in	in	ADP
ejpam-4831	51	11	a	a	DET
ejpam-4831	51	12	natural	natural	ADJ
ejpam-4831	51	13	way	way	NOUN
ejpam-4831	51	14	,	,	PUNCT
ejpam-4831	51	15	a	a	DET
ejpam-4831	51	16	convex	convex	ADJ
ejpam-4831	51	17	bornology	bornology	NOUN
ejpam-4831	51	18	on	on	ADP
ejpam-4831	51	19	λ(e	λ(e	NOUN
ejpam-4831	51	20	)	)	PUNCT
ejpam-4831	51	21	with	with	ADP
ejpam-4831	51	22	basis	basis	NOUN
ejpam-4831	51	23	s(b	s(b	NOUN
ejpam-4831	51	24	)	)	PUNCT
ejpam-4831	51	25	by	by	ADP
ejpam-4831	51	26	setting	set	VERB
ejpam-4831	51	27	s(b	s(b	NOUN
ejpam-4831	51	28	)	)	PUNCT
ejpam-4831	52	1	=	=	PRON
ejpam-4831	52	2	{	{	PUNCT
ejpam-4831	52	3	h	h	NOUN
ejpam-4831	52	4	⊂	⊂	X
ejpam-4831	52	5	λ(e	λ(e	PROPN
ejpam-4831	52	6	)	)	PUNCT
ejpam-4831	52	7	:	:	PUNCT
ejpam-4831	52	8	∃s	∃s	PROPN
ejpam-4831	52	9	∈	∈	PROPN
ejpam-4831	52	10	s	s	PROPN
ejpam-4831	52	11	,	,	PUNCT
ejpam-4831	52	12	b	b	PROPN
ejpam-4831	52	13	∈	∈	PROPN
ejpam-4831	52	14	b	b	NOUN
ejpam-4831	52	15	such	such	ADJ
ejpam-4831	52	16	that	that	DET
ejpam-4831	52	17	h	h	NOUN
ejpam-4831	52	18	=	=	SYM
ejpam-4831	52	19	s(b	s(b	NOUN
ejpam-4831	52	20	)	)	PUNCT
ejpam-4831	52	21	}	}	PUNCT
ejpam-4831	52	22	.	.	PUNCT
ejpam-4831	53	1	in	in	ADP
ejpam-4831	53	2	view	view	NOUN
ejpam-4831	53	3	of	of	ADP
ejpam-4831	53	4	the	the	DET
ejpam-4831	53	5	hypothesis	hypothesis	NOUN
ejpam-4831	53	6	made	make	VERB
ejpam-4831	53	7	on	on	ADP
ejpam-4831	53	8	s	s	PRON
ejpam-4831	53	9	and	and	CCONJ
ejpam-4831	53	10	b	b	NOUN
ejpam-4831	53	11	,	,	PUNCT
ejpam-4831	53	12	s(b	s(b	NOUN
ejpam-4831	53	13	)	)	PUNCT
ejpam-4831	53	14	is	be	AUX
ejpam-4831	53	15	indeed	indeed	ADV
ejpam-4831	53	16	a	a	DET
ejpam-4831	53	17	basis	basis	NOUN
ejpam-4831	53	18	for	for	ADP
ejpam-4831	53	19	a	a	DET
ejpam-4831	53	20	convex	convex	ADJ
ejpam-4831	53	21	bornology	bornology	NOUN
ejpam-4831	53	22	on	on	ADP
ejpam-4831	53	23	λ(e	λ(e	NOUN
ejpam-4831	53	24	)	)	PUNCT
ejpam-4831	53	25	for	for	ADP
ejpam-4831	53	26	which	which	PRON
ejpam-4831	53	27	λ(e	λ(e	NUM
ejpam-4831	53	28	)	)	PUNCT
ejpam-4831	53	29	is	be	AUX
ejpam-4831	53	30	a	a	DET
ejpam-4831	53	31	hausdorff	hausdorff	NOUN
ejpam-4831	53	32	space	space	NOUN
ejpam-4831	53	33	.	.	PUNCT
ejpam-4831	54	1	lemma	lemma	PROPN
ejpam-4831	54	2	1	1	NUM
ejpam-4831	54	3	.	.	PUNCT
ejpam-4831	55	1	for	for	ADP
ejpam-4831	55	2	a	a	DET
ejpam-4831	55	3	fixed	fix	VERB
ejpam-4831	55	4	k	k	PROPN
ejpam-4831	55	5	∈	∈	PROPN
ejpam-4831	55	6	n	n	CCONJ
ejpam-4831	55	7	,	,	PUNCT
ejpam-4831	55	8	denote	denote	VERB
ejpam-4831	55	9	by	by	ADP
ejpam-4831	55	10	πk	πk	ADP
ejpam-4831	55	11	the	the	DET
ejpam-4831	55	12	projection	projection	NOUN
ejpam-4831	55	13	from	from	ADP
ejpam-4831	55	14	λ(e	λ(e	PROPN
ejpam-4831	55	15	)	)	PUNCT
ejpam-4831	55	16	on	on	ADP
ejpam-4831	55	17	e	e	NOUN
ejpam-4831	55	18	defined	define	VERB
ejpam-4831	55	19	by	by	ADP
ejpam-4831	55	20	πk(x	πk(x	NOUN
ejpam-4831	55	21	)	)	PUNCT
ejpam-4831	56	1	=	=	SYM
ejpam-4831	56	2	xk	xk	PROPN
ejpam-4831	56	3	,	,	PUNCT
ejpam-4831	56	4	for	for	ADP
ejpam-4831	56	5	all	all	PRON
ejpam-4831	56	6	x	x	X
ejpam-4831	56	7	=	=	SYM
ejpam-4831	56	8	(	(	PUNCT
ejpam-4831	56	9	xn	xn	X
ejpam-4831	56	10	)	)	PUNCT
ejpam-4831	56	11	∈	∈	PROPN
ejpam-4831	56	12	λ(e	λ(e	PROPN
ejpam-4831	56	13	)	)	PUNCT
ejpam-4831	56	14	.	.	PUNCT
ejpam-4831	57	1	then	then	ADV
ejpam-4831	57	2	,	,	PUNCT
ejpam-4831	57	3	πk	πk	PROPN
ejpam-4831	57	4	is	be	AUX
ejpam-4831	57	5	a	a	DET
ejpam-4831	57	6	bounded	bounded	ADJ
ejpam-4831	57	7	linear	linear	PROPN
ejpam-4831	57	8	map	map	NOUN
ejpam-4831	57	9	.	.	PUNCT
ejpam-4831	58	1	proof	proof	NOUN
ejpam-4831	58	2	.	.	PUNCT
ejpam-4831	59	1	let	let	VERB
ejpam-4831	59	2	b	b	X
ejpam-4831	59	3	∈	∈	PROPN
ejpam-4831	59	4	b	b	PROPN
ejpam-4831	59	5	and	and	CCONJ
ejpam-4831	59	6	s	s	PROPN
ejpam-4831	59	7	∈	∈	NOUN
ejpam-4831	59	8	s	s	PART
ejpam-4831	59	9	and	and	CCONJ
ejpam-4831	59	10	fix	fix	VERB
ejpam-4831	59	11	k	k	PROPN
ejpam-4831	59	12	∈	∈	PROPN
ejpam-4831	59	13	n.	n.	NOUN
ejpam-4831	59	14	since	since	SCONJ
ejpam-4831	59	15	the	the	DET
ejpam-4831	59	16	bornology	bornology	NOUN
ejpam-4831	59	17	of	of	ADP
ejpam-4831	59	18	λ	λ	PROPN
ejpam-4831	59	19	is	be	AUX
ejpam-4831	59	20	normal	normal	ADJ
ejpam-4831	59	21	,	,	PUNCT
ejpam-4831	59	22	the	the	DET
ejpam-4831	59	23	set	set	NOUN
ejpam-4831	59	24	{	{	PUNCT
ejpam-4831	59	25	αk	αk	NOUN
ejpam-4831	59	26	:	:	PUNCT
ejpam-4831	59	27	(	(	PUNCT
ejpam-4831	59	28	αn)n	αn)n	NOUN
ejpam-4831	59	29	∈	∈	NOUN
ejpam-4831	59	30	s	s	AUX
ejpam-4831	59	31	}	}	PUNCT
ejpam-4831	59	32	is	be	AUX
ejpam-4831	59	33	bounded	bound	VERB
ejpam-4831	59	34	in	in	ADP
ejpam-4831	59	35	c	c	PROPN
ejpam-4831	59	36	,	,	PUNCT
ejpam-4831	59	37	and	and	CCONJ
ejpam-4831	59	38	then	then	ADV
ejpam-4831	59	39	so	so	ADV
ejpam-4831	59	40	is	be	AUX
ejpam-4831	59	41	{	{	PUNCT
ejpam-4831	59	42	∥xk∥	∥xk∥	ADJ
ejpam-4831	59	43	:	:	PUNCT
ejpam-4831	59	44	(	(	PUNCT
ejpam-4831	59	45	xn)n	xn)n	PROPN
ejpam-4831	59	46	∈	∈	PROPN
ejpam-4831	59	47	s(b	s(b	NOUN
ejpam-4831	59	48	)	)	PUNCT
ejpam-4831	59	49	}	}	PUNCT
ejpam-4831	59	50	.	.	PUNCT
ejpam-4831	60	1	this	this	PRON
ejpam-4831	60	2	means	mean	VERB
ejpam-4831	60	3	that	that	SCONJ
ejpam-4831	60	4	{	{	PUNCT
ejpam-4831	60	5	xk	xk	INTJ
ejpam-4831	60	6	:	:	PUNCT
ejpam-4831	60	7	(	(	PUNCT
ejpam-4831	60	8	xn)n	xn)n	PROPN
ejpam-4831	60	9	∈	∈	PROPN
ejpam-4831	60	10	s(b	s(b	NOUN
ejpam-4831	60	11	)	)	PUNCT
ejpam-4831	60	12	}	}	PUNCT
ejpam-4831	60	13	is	be	AUX
ejpam-4831	60	14	bounded	bound	VERB
ejpam-4831	60	15	in	in	ADP
ejpam-4831	60	16	eb	eb	PROPN
ejpam-4831	60	17	.	.	PUNCT
ejpam-4831	61	1	thus	thus	ADV
ejpam-4831	61	2	,	,	PUNCT
ejpam-4831	61	3	πk	πk	PROPN
ejpam-4831	61	4	is	be	AUX
ejpam-4831	61	5	bounded	bound	VERB
ejpam-4831	61	6	.	.	PUNCT
ejpam-4831	62	1	■	■	PUNCT
ejpam-4831	62	2	m.	m.	NOUN
ejpam-4831	62	3	a.	a.	NOUN
ejpam-4831	62	4	sidaty	sidaty	PROPN
ejpam-4831	62	5	/	/	SYM
ejpam-4831	62	6	eur	eur	PROPN
ejpam-4831	62	7	.	.	PUNCT
ejpam-4831	63	1	j.	j.	PROPN
ejpam-4831	63	2	pure	pure	PROPN
ejpam-4831	63	3	appl	appl	PROPN
ejpam-4831	63	4	.	.	PROPN
ejpam-4831	63	5	math	math	PROPN
ejpam-4831	63	6	,	,	PUNCT
ejpam-4831	63	7	16	16	NUM
ejpam-4831	63	8	(	(	PUNCT
ejpam-4831	63	9	3	3	NUM
ejpam-4831	63	10	)	)	PUNCT
ejpam-4831	63	11	(	(	PUNCT
ejpam-4831	63	12	2023	2023	NUM
ejpam-4831	63	13	)	)	PUNCT
ejpam-4831	63	14	,	,	PUNCT
ejpam-4831	63	15	1762	1762	NUM
ejpam-4831	63	16	-	-	SYM
ejpam-4831	63	17	1771	1771	NUM
ejpam-4831	63	18	1765	1765	NUM
ejpam-4831	63	19	proposition	proposition	NOUN
ejpam-4831	63	20	1	1	NUM
ejpam-4831	63	21	.	.	PUNCT
ejpam-4831	64	1	the	the	DET
ejpam-4831	64	2	spaces	space	NOUN
ejpam-4831	64	3	λ	λ	PROPN
ejpam-4831	64	4	and	and	CCONJ
ejpam-4831	64	5	e	e	NOUN
ejpam-4831	64	6	can	can	AUX
ejpam-4831	64	7	be	be	AUX
ejpam-4831	64	8	identified	identify	VERB
ejpam-4831	64	9	with	with	ADP
ejpam-4831	64	10	closed	closed	ADJ
ejpam-4831	64	11	subspaces	subspace	NOUN
ejpam-4831	64	12	of	of	ADP
ejpam-4831	64	13	λ(e	λ(e	NOUN
ejpam-4831	64	14	)	)	PUNCT
ejpam-4831	64	15	.	.	PUNCT
ejpam-4831	65	1	proof	proof	NOUN
ejpam-4831	65	2	.	.	PUNCT
ejpam-4831	66	1	let	let	VERB
ejpam-4831	66	2	i	i	PRON
ejpam-4831	66	3	:	:	PUNCT
ejpam-4831	66	4	e	e	AUX
ejpam-4831	66	5	−→	−→	NOUN
ejpam-4831	66	6	λ(e	λ(e	VERB
ejpam-4831	66	7	)	)	PUNCT
ejpam-4831	66	8	,	,	PUNCT
ejpam-4831	66	9	t	t	PROPN
ejpam-4831	66	10	−→	−→	PROPN
ejpam-4831	66	11	te1	te1	PROPN
ejpam-4831	66	12	,	,	PUNCT
ejpam-4831	66	13	where	where	SCONJ
ejpam-4831	66	14	t	t	PROPN
ejpam-4831	66	15	is	be	AUX
ejpam-4831	66	16	at	at	ADP
ejpam-4831	66	17	the	the	DET
ejpam-4831	66	18	first	first	ADJ
ejpam-4831	66	19	component	component	NOUN
ejpam-4831	66	20	.	.	PUNCT
ejpam-4831	67	1	it	it	PRON
ejpam-4831	67	2	is	be	AUX
ejpam-4831	67	3	clear	clear	ADJ
ejpam-4831	67	4	that	that	SCONJ
ejpam-4831	67	5	i	i	PRON
ejpam-4831	67	6	is	be	AUX
ejpam-4831	67	7	linear	linear	ADJ
ejpam-4831	67	8	and	and	CCONJ
ejpam-4831	67	9	one	one	NUM
ejpam-4831	67	10	to	to	ADP
ejpam-4831	67	11	one	one	NUM
ejpam-4831	67	12	.	.	PUNCT
ejpam-4831	68	1	let	let	VERB
ejpam-4831	68	2	b	b	X
ejpam-4831	68	3	∈	∈	PROPN
ejpam-4831	68	4	b	b	PROPN
ejpam-4831	68	5	,	,	PUNCT
ejpam-4831	68	6	and	and	CCONJ
ejpam-4831	68	7	s	s	PROPN
ejpam-4831	68	8	∈	∈	NOUN
ejpam-4831	68	9	s	s	VERB
ejpam-4831	68	10	such	such	ADJ
ejpam-4831	68	11	that	that	PRON
ejpam-4831	68	12	e1	e1	NOUN
ejpam-4831	68	13	∈	∈	PROPN
ejpam-4831	68	14	s	s	NOUN
ejpam-4831	68	15	,	,	PUNCT
ejpam-4831	68	16	then	then	ADV
ejpam-4831	68	17	i(b	i(b	PROPN
ejpam-4831	68	18	)	)	PUNCT
ejpam-4831	69	1	⊂	⊂	PART
ejpam-4831	69	2	s(b	s(b	NOUN
ejpam-4831	69	3	)	)	PUNCT
ejpam-4831	69	4	and	and	CCONJ
ejpam-4831	69	5	i	i	PRON
ejpam-4831	69	6	is	be	AUX
ejpam-4831	69	7	bounded	bound	VERB
ejpam-4831	69	8	.	.	PUNCT
ejpam-4831	70	1	inversely	inversely	ADV
ejpam-4831	70	2	,	,	PUNCT
ejpam-4831	70	3	i−1	i−1	PROPN
ejpam-4831	70	4	:	:	PUNCT
ejpam-4831	70	5	i(e	i(e	PROPN
ejpam-4831	70	6	)	)	PUNCT
ejpam-4831	71	1	=	=	SYM
ejpam-4831	71	2	ee1	ee1	PROPN
ejpam-4831	72	1	→	→	PUNCT
ejpam-4831	72	2	e	e	NOUN
ejpam-4831	72	3	is	be	AUX
ejpam-4831	72	4	the	the	DET
ejpam-4831	72	5	restriction	restriction	NOUN
ejpam-4831	72	6	of	of	ADP
ejpam-4831	72	7	π1	π1	NOUN
ejpam-4831	72	8	to	to	ADP
ejpam-4831	72	9	the	the	DET
ejpam-4831	72	10	subspace	subspace	PROPN
ejpam-4831	72	11	i(e	i(e	PROPN
ejpam-4831	72	12	)	)	PUNCT
ejpam-4831	72	13	,	,	PUNCT
ejpam-4831	72	14	and	and	CCONJ
ejpam-4831	72	15	then	then	ADV
ejpam-4831	72	16	it	it	PRON
ejpam-4831	72	17	is	be	AUX
ejpam-4831	72	18	bounded	bound	VERB
ejpam-4831	72	19	by	by	ADP
ejpam-4831	72	20	lemma	lemma	PROPN
ejpam-4831	72	21	1	1	NUM
ejpam-4831	72	22	.	.	PUNCT
ejpam-4831	73	1	it	it	PRON
ejpam-4831	73	2	remains	remain	VERB
ejpam-4831	73	3	to	to	PART
ejpam-4831	73	4	show	show	VERB
ejpam-4831	73	5	that	that	SCONJ
ejpam-4831	73	6	i(e	i(e	NOUN
ejpam-4831	73	7	)	)	PUNCT
ejpam-4831	73	8	is	be	AUX
ejpam-4831	73	9	closed	close	VERB
ejpam-4831	73	10	in	in	ADP
ejpam-4831	73	11	λ(e	λ(e	NOUN
ejpam-4831	73	12	)	)	PUNCT
ejpam-4831	73	13	.	.	PUNCT
ejpam-4831	74	1	we	we	PRON
ejpam-4831	74	2	have	have	VERB
ejpam-4831	74	3	i(e	i(e	NOUN
ejpam-4831	74	4	)	)	PUNCT
ejpam-4831	75	1	=	=	SYM
ejpam-4831	76	1	⋂	⋂	PROPN
ejpam-4831	76	2	k	k	PROPN
ejpam-4831	76	3	̸=1	̸=1	NOUN
ejpam-4831	77	1	π	π	PROPN
ejpam-4831	77	2	−1	−1	VERB
ejpam-4831	77	3	k	k	X
ejpam-4831	77	4	(	(	PUNCT
ejpam-4831	77	5	{	{	PUNCT
ejpam-4831	77	6	0	0	NUM
ejpam-4831	77	7	}	}	PUNCT
ejpam-4831	77	8	)	)	PUNCT
ejpam-4831	77	9	.	.	PUNCT
ejpam-4831	78	1	since	since	SCONJ
ejpam-4831	78	2	e	e	PROPN
ejpam-4831	78	3	is	be	AUX
ejpam-4831	78	4	supposed	suppose	VERB
ejpam-4831	78	5	to	to	PART
ejpam-4831	78	6	be	be	AUX
ejpam-4831	78	7	a	a	DET
ejpam-4831	78	8	hausdorff	hausdorff	NOUN
ejpam-4831	78	9	space	space	NOUN
ejpam-4831	78	10	,	,	PUNCT
ejpam-4831	78	11	then	then	ADV
ejpam-4831	78	12	{	{	PUNCT
ejpam-4831	78	13	0	0	X
ejpam-4831	78	14	}	}	PUNCT
ejpam-4831	78	15	is	be	AUX
ejpam-4831	78	16	closed	closed	ADJ
ejpam-4831	78	17	and	and	CCONJ
ejpam-4831	78	18	so	so	ADV
ejpam-4831	78	19	is	be	AUX
ejpam-4831	78	20	i(e	i(e	NOUN
ejpam-4831	78	21	)	)	PUNCT
ejpam-4831	78	22	.	.	PUNCT
ejpam-4831	79	1	now	now	ADV
ejpam-4831	79	2	,	,	PUNCT
ejpam-4831	79	3	fix	fix	VERB
ejpam-4831	79	4	0	0	NUM
ejpam-4831	79	5	̸=	̸=	NOUN
ejpam-4831	79	6	x0	x0	PROPN
ejpam-4831	79	7	∈	∈	PROPN
ejpam-4831	79	8	e	e	X
ejpam-4831	79	9	and	and	CCONJ
ejpam-4831	79	10	let	let	VERB
ejpam-4831	79	11	g	g	NOUN
ejpam-4831	79	12	:	:	PUNCT
ejpam-4831	79	13	λ	λ	VERB
ejpam-4831	79	14	−→	−→	NOUN
ejpam-4831	79	15	λ(e	λ(e	ADJ
ejpam-4831	79	16	)	)	PUNCT
ejpam-4831	79	17	,	,	PUNCT
ejpam-4831	79	18	α	α	NOUN
ejpam-4831	79	19	=	=	X
ejpam-4831	79	20	(	(	PUNCT
ejpam-4831	79	21	αn)n	αn)n	NOUN
ejpam-4831	79	22	−→	−→	NOUN
ejpam-4831	79	23	(	(	PUNCT
ejpam-4831	79	24	αnx0)n	αnx0)n	NOUN
ejpam-4831	79	25	=	=	SYM
ejpam-4831	79	26	αx0	αx0	NOUN
ejpam-4831	79	27	.	.	PUNCT
ejpam-4831	80	1	it	it	PRON
ejpam-4831	80	2	is	be	AUX
ejpam-4831	80	3	clear	clear	ADJ
ejpam-4831	80	4	that	that	SCONJ
ejpam-4831	80	5	g	g	PROPN
ejpam-4831	80	6	is	be	AUX
ejpam-4831	80	7	linear	linear	ADJ
ejpam-4831	80	8	and	and	CCONJ
ejpam-4831	80	9	one	one	NUM
ejpam-4831	80	10	to	to	ADP
ejpam-4831	80	11	one	one	NUM
ejpam-4831	80	12	.	.	PUNCT
ejpam-4831	81	1	let	let	VERB
ejpam-4831	81	2	s	s	PRON
ejpam-4831	81	3	∈	∈	VERB
ejpam-4831	81	4	s	s	NOUN
ejpam-4831	81	5	,	,	PUNCT
ejpam-4831	81	6	and	and	CCONJ
ejpam-4831	81	7	b	b	X
ejpam-4831	81	8	∈	∈	PROPN
ejpam-4831	81	9	b	b	PROPN
ejpam-4831	81	10	with	with	ADP
ejpam-4831	81	11	x0	x0	PROPN
ejpam-4831	81	12	∈	∈	PROPN
ejpam-4831	81	13	b.	b.	PROPN
ejpam-4831	81	14	then	then	ADV
ejpam-4831	81	15	,	,	PUNCT
ejpam-4831	81	16	g(s	g(s	PROPN
ejpam-4831	81	17	)	)	PUNCT
ejpam-4831	81	18	⊂	⊂	PART
ejpam-4831	81	19	s(b	s(b	NOUN
ejpam-4831	81	20	)	)	PUNCT
ejpam-4831	82	1	;	;	PUNCT
ejpam-4831	82	2	so	so	CCONJ
ejpam-4831	82	3	g	g	PROPN
ejpam-4831	82	4	is	be	AUX
ejpam-4831	82	5	bounded	bound	VERB
ejpam-4831	82	6	.	.	PUNCT
ejpam-4831	83	1	inversely	inversely	ADV
ejpam-4831	83	2	,	,	PUNCT
ejpam-4831	83	3	if	if	SCONJ
ejpam-4831	83	4	s	s	VERB
ejpam-4831	83	5	∈	∈	PROPN
ejpam-4831	83	6	s	s	X
ejpam-4831	83	7	and	and	CCONJ
ejpam-4831	83	8	b	b	PROPN
ejpam-4831	83	9	∈	∈	PROPN
ejpam-4831	83	10	b	b	PROPN
ejpam-4831	83	11	,	,	PUNCT
ejpam-4831	83	12	then	then	ADV
ejpam-4831	83	13	g−1(s(b	g−1(s(b	NOUN
ejpam-4831	83	14	)	)	PUNCT
ejpam-4831	83	15	∩	∩	ADJ
ejpam-4831	83	16	λx0	λx0	NOUN
ejpam-4831	83	17	)	)	PUNCT
ejpam-4831	83	18	=	=	SYM
ejpam-4831	84	1	1	1	NUM
ejpam-4831	84	2	∥x0∥b	∥x0∥b	PROPN
ejpam-4831	84	3	s	s	PROPN
ejpam-4831	84	4	,	,	PUNCT
ejpam-4831	84	5	and	and	CCONJ
ejpam-4831	84	6	then	then	ADV
ejpam-4831	84	7	g−1	g−1	PROPN
ejpam-4831	84	8	:	:	PUNCT
ejpam-4831	84	9	g(e	g(e	PROPN
ejpam-4831	84	10	)	)	PUNCT
ejpam-4831	85	1	=	=	SYM
ejpam-4831	85	2	λx0	λx0	NOUN
ejpam-4831	85	3	→	→	SYM
ejpam-4831	85	4	λ	λ	PROPN
ejpam-4831	85	5	is	be	AUX
ejpam-4831	85	6	bounded	bound	VERB
ejpam-4831	85	7	.	.	PUNCT
ejpam-4831	86	1	it	it	PRON
ejpam-4831	86	2	remains	remain	VERB
ejpam-4831	86	3	to	to	PART
ejpam-4831	86	4	show	show	VERB
ejpam-4831	86	5	that	that	SCONJ
ejpam-4831	86	6	g(λ	g(λ	PROPN
ejpam-4831	86	7	)	)	PUNCT
ejpam-4831	86	8	is	be	AUX
ejpam-4831	86	9	closed	close	VERB
ejpam-4831	86	10	in	in	ADP
ejpam-4831	86	11	λ(e	λ(e	NOUN
ejpam-4831	86	12	)	)	PUNCT
ejpam-4831	86	13	.	.	PUNCT
ejpam-4831	87	1	let	let	VERB
ejpam-4831	87	2	{	{	PUNCT
ejpam-4831	87	3	α(k)x0	α(k)x0	NOUN
ejpam-4831	87	4	=	=	SYM
ejpam-4831	87	5	(	(	PUNCT
ejpam-4831	87	6	α	α	X
ejpam-4831	87	7	(	(	PUNCT
ejpam-4831	87	8	k	k	NOUN
ejpam-4831	87	9	)	)	PUNCT
ejpam-4831	87	10	n	n	CCONJ
ejpam-4831	87	11	x0)n}∞k=1	x0)n}∞k=1	PROPN
ejpam-4831	87	12	be	be	AUX
ejpam-4831	87	13	a	a	DET
ejpam-4831	87	14	sequence	sequence	NOUN
ejpam-4831	87	15	in	in	ADP
ejpam-4831	87	16	λx0	λx0	NOUN
ejpam-4831	87	17	which	which	PRON
ejpam-4831	87	18	converges	converge	VERB
ejpam-4831	87	19	to	to	ADP
ejpam-4831	87	20	x	x	SYM
ejpam-4831	87	21	=	=	SYM
ejpam-4831	87	22	(	(	PUNCT
ejpam-4831	87	23	xn)n	xn)n	PROPN
ejpam-4831	87	24	∈	∈	PROPN
ejpam-4831	87	25	λ(e	λ(e	PROPN
ejpam-4831	87	26	)	)	PUNCT
ejpam-4831	87	27	.	.	PUNCT
ejpam-4831	88	1	by	by	ADP
ejpam-4831	88	2	lemma	lemma	PROPN
ejpam-4831	88	3	1	1	NUM
ejpam-4831	88	4	,	,	PUNCT
ejpam-4831	88	5	{	{	PUNCT
ejpam-4831	88	6	α(k	α(k	NOUN
ejpam-4831	88	7	)	)	PUNCT
ejpam-4831	88	8	n	n	NOUN
ejpam-4831	88	9	x0}∞k=1	x0}∞k=1	NOUN
ejpam-4831	88	10	converges	converge	VERB
ejpam-4831	88	11	to	to	ADP
ejpam-4831	88	12	xn	xn	PROPN
ejpam-4831	88	13	in	in	ADP
ejpam-4831	88	14	e	e	NOUN
ejpam-4831	88	15	,	,	PUNCT
ejpam-4831	88	16	for	for	ADP
ejpam-4831	88	17	every	every	DET
ejpam-4831	88	18	n.	n.	NOUN
ejpam-4831	88	19	as	as	ADP
ejpam-4831	88	20	,	,	PUNCT
ejpam-4831	88	21	the	the	DET
ejpam-4831	88	22	subspace	subspace	NOUN
ejpam-4831	88	23	cx0	cx0	PROPN
ejpam-4831	88	24	of	of	ADP
ejpam-4831	88	25	e	e	PROPN
ejpam-4831	88	26	is	be	AUX
ejpam-4831	88	27	closed	close	VERB
ejpam-4831	88	28	in	in	ADP
ejpam-4831	88	29	e	e	NOUN
ejpam-4831	88	30	,	,	PUNCT
ejpam-4831	88	31	xn	xn	PROPN
ejpam-4831	88	32	must	must	AUX
ejpam-4831	88	33	belong	belong	VERB
ejpam-4831	88	34	to	to	ADP
ejpam-4831	88	35	cx0	cx0	PROPN
ejpam-4831	88	36	.	.	PUNCT
ejpam-4831	89	1	then	then	ADV
ejpam-4831	89	2	,	,	PUNCT
ejpam-4831	89	3	there	there	PRON
ejpam-4831	89	4	is	be	VERB
ejpam-4831	89	5	α	α	NOUN
ejpam-4831	89	6	=	=	PUNCT
ejpam-4831	89	7	(	(	PUNCT
ejpam-4831	89	8	αn	αn	NOUN
ejpam-4831	89	9	)	)	PUNCT
ejpam-4831	89	10	such	such	ADJ
ejpam-4831	89	11	that	that	SCONJ
ejpam-4831	89	12	x	x	X
ejpam-4831	89	13	=	=	SYM
ejpam-4831	89	14	(	(	PUNCT
ejpam-4831	89	15	xn)n	xn)n	PROPN
ejpam-4831	89	16	=	=	SYM
ejpam-4831	89	17	αx0	αx0	NOUN
ejpam-4831	89	18	.	.	PUNCT
ejpam-4831	90	1	it	it	PRON
ejpam-4831	90	2	is	be	AUX
ejpam-4831	90	3	easy	easy	ADJ
ejpam-4831	90	4	to	to	PART
ejpam-4831	90	5	see	see	VERB
ejpam-4831	90	6	that	that	SCONJ
ejpam-4831	90	7	α	α	PROPN
ejpam-4831	90	8	∈	∈	PROPN
ejpam-4831	90	9	λ	λ	PROPN
ejpam-4831	90	10	.	.	PUNCT
ejpam-4831	91	1	we	we	PRON
ejpam-4831	91	2	conclude	conclude	VERB
ejpam-4831	91	3	that	that	SCONJ
ejpam-4831	91	4	λx0	λx0	NOUN
ejpam-4831	91	5	is	be	AUX
ejpam-4831	91	6	closed	close	VERB
ejpam-4831	91	7	in	in	ADP
ejpam-4831	91	8	λ(e	λ(e	NOUN
ejpam-4831	91	9	)	)	PUNCT
ejpam-4831	91	10	.	.	PUNCT
ejpam-4831	92	1	■	■	PUNCT
ejpam-4831	92	2	proposition	proposition	NOUN
ejpam-4831	92	3	2	2	NUM
ejpam-4831	92	4	.	.	PUNCT
ejpam-4831	93	1	λ(e	λ(e	VERB
ejpam-4831	93	2	)	)	PUNCT
ejpam-4831	94	1	is	be	AUX
ejpam-4831	94	2	complete	complete	ADJ
ejpam-4831	94	3	if	if	SCONJ
ejpam-4831	94	4	and	and	CCONJ
ejpam-4831	94	5	only	only	ADV
ejpam-4831	94	6	if	if	SCONJ
ejpam-4831	94	7	λ	λ	PROPN
ejpam-4831	94	8	and	and	CCONJ
ejpam-4831	94	9	e	e	NOUN
ejpam-4831	94	10	are	be	AUX
ejpam-4831	94	11	complete	complete	ADJ
ejpam-4831	94	12	.	.	PUNCT
ejpam-4831	95	1	proof	proof	NOUN
ejpam-4831	95	2	.	.	PUNCT
ejpam-4831	96	1	if	if	SCONJ
ejpam-4831	96	2	λ(e	λ(e	VERB
ejpam-4831	96	3	)	)	PUNCT
ejpam-4831	96	4	is	be	AUX
ejpam-4831	96	5	complete	complete	ADJ
ejpam-4831	96	6	,	,	PUNCT
ejpam-4831	96	7	then	then	ADV
ejpam-4831	96	8	so	so	ADV
ejpam-4831	96	9	are	be	AUX
ejpam-4831	96	10	λ	λ	PROPN
ejpam-4831	96	11	and	and	CCONJ
ejpam-4831	96	12	e	e	NOUN
ejpam-4831	96	13	by	by	ADP
ejpam-4831	96	14	proposition	proposition	NOUN
ejpam-4831	96	15	1	1	NUM
ejpam-4831	96	16	.	.	PUNCT
ejpam-4831	97	1	inversely	inversely	ADV
ejpam-4831	97	2	,	,	PUNCT
ejpam-4831	97	3	suppose	suppose	VERB
ejpam-4831	97	4	that	that	SCONJ
ejpam-4831	97	5	λ	λ	PROPN
ejpam-4831	97	6	and	and	CCONJ
ejpam-4831	97	7	e	e	NOUN
ejpam-4831	97	8	are	be	AUX
ejpam-4831	97	9	complete	complete	ADJ
ejpam-4831	97	10	.	.	PUNCT
ejpam-4831	98	1	we	we	PRON
ejpam-4831	98	2	only	only	ADV
ejpam-4831	98	3	show	show	VERB
ejpam-4831	98	4	that	that	SCONJ
ejpam-4831	98	5	if	if	SCONJ
ejpam-4831	98	6	b	b	PROPN
ejpam-4831	98	7	and	and	CCONJ
ejpam-4831	98	8	s	s	NOUN
ejpam-4831	98	9	are	be	AUX
ejpam-4831	98	10	banach	banach	NOUN
ejpam-4831	98	11	disks	disk	NOUN
ejpam-4831	98	12	in	in	ADP
ejpam-4831	98	13	e	e	PROPN
ejpam-4831	98	14	and	and	CCONJ
ejpam-4831	98	15	λ	λ	PROPN
ejpam-4831	98	16	respectively	respectively	ADV
ejpam-4831	98	17	,	,	PUNCT
ejpam-4831	98	18	then	then	ADV
ejpam-4831	98	19	s(b	s(b	NOUN
ejpam-4831	98	20	)	)	PUNCT
ejpam-4831	98	21	is	be	AUX
ejpam-4831	98	22	a	a	DET
ejpam-4831	98	23	banach	banach	NOUN
ejpam-4831	98	24	disk	disk	NOUN
ejpam-4831	98	25	in	in	ADP
ejpam-4831	98	26	λ(e	λ(e	NOUN
ejpam-4831	98	27	)	)	PUNCT
ejpam-4831	98	28	.	.	PUNCT
ejpam-4831	99	1	to	to	PART
ejpam-4831	99	2	simplify	simplify	VERB
ejpam-4831	99	3	the	the	DET
ejpam-4831	99	4	notations	notation	NOUN
ejpam-4831	99	5	,	,	PUNCT
ejpam-4831	99	6	we	we	PRON
ejpam-4831	99	7	set	set	VERB
ejpam-4831	99	8	f	f	PROPN
ejpam-4831	99	9	=	=	PUNCT
ejpam-4831	99	10	λ(e	λ(e	PROPN
ejpam-4831	99	11	)	)	PUNCT
ejpam-4831	99	12	,	,	PUNCT
ejpam-4831	99	13	h	h	NOUN
ejpam-4831	99	14	=	=	SYM
ejpam-4831	99	15	s(b	s(b	NOUN
ejpam-4831	99	16	)	)	PUNCT
ejpam-4831	99	17	and	and	CCONJ
ejpam-4831	99	18	π	π	X
ejpam-4831	99	19	the	the	DET
ejpam-4831	99	20	gauge	gauge	NOUN
ejpam-4831	99	21	of	of	ADP
ejpam-4831	99	22	h.	h.	PROPN
ejpam-4831	99	23	let	let	VERB
ejpam-4831	99	24	{	{	PUNCT
ejpam-4831	99	25	(	(	PUNCT
ejpam-4831	99	26	xi)i}∞i=1	xi)i}∞i=1	ADV
ejpam-4831	99	27	be	be	AUX
ejpam-4831	99	28	a	a	DET
ejpam-4831	99	29	cauchy	cauchy	ADJ
ejpam-4831	99	30	sequence	sequence	NOUN
ejpam-4831	99	31	in	in	ADP
ejpam-4831	99	32	(	(	PUNCT
ejpam-4831	99	33	fh	fh	PROPN
ejpam-4831	99	34	,	,	PUNCT
ejpam-4831	99	35	π	π	PROPN
ejpam-4831	99	36	)	)	PUNCT
ejpam-4831	99	37	.	.	PUNCT
ejpam-4831	100	1	we	we	PRON
ejpam-4831	100	2	have∣∣∣∣∥∥(∥xin∥b)n∥∥s	have∣∣∣∣∥∥(∥xin∥b)n∥∥s	ADV
ejpam-4831	100	3	−	−	PROPN
ejpam-4831	100	4	∥∥(∥xjn∥b)n∥∥s∣∣∣∣	∥∥(∥xjn∥b)n∥∥s∣∣∣∣	ADJ
ejpam-4831	100	5	≤	≤	NOUN
ejpam-4831	100	6	∣∣∣∣∥∥(∥xin∥b)n	∣∣∣∣∥∥(∥xin∥b)n	X
ejpam-4831	100	7	−	−	PROPN
ejpam-4831	100	8	(	(	PUNCT
ejpam-4831	100	9	∥xjn∥b)n	∥xjn∥b)n	X
ejpam-4831	100	10	∥∥	∥∥	X
ejpam-4831	100	11	s	s	PART
ejpam-4831	100	12	∣∣∣∣	∣∣∣∣	PROPN
ejpam-4831	100	13	≤	≤	PUNCT
ejpam-4831	100	14	∥∥(∥xin∥b	∥∥(∥xin∥b	PUNCT
ejpam-4831	101	1	−	−	PROPN
ejpam-4831	101	2	∥xjn∥b)n	∥xjn∥b)n	X
ejpam-4831	101	3	∥∥	∥∥	X
ejpam-4831	101	4	s	s	PART
ejpam-4831	101	5	≤	≤	NUM
ejpam-4831	101	6	∥∥(∥xin	∥∥(∥xin	X
ejpam-4831	101	7	−	−	PROPN
ejpam-4831	101	8	xjn∥b)n	xjn∥b)n	PROPN
ejpam-4831	102	1	∥∥	∥∥	X
ejpam-4831	102	2	s	s	PART
ejpam-4831	102	3	=	=	PUNCT
ejpam-4831	102	4	π((xi	π((xi	NOUN
ejpam-4831	102	5	−	−	PROPN
ejpam-4831	102	6	xj)n	xj)n	NUM
ejpam-4831	102	7	)	)	PUNCT
ejpam-4831	102	8	.	.	PUNCT
ejpam-4831	103	1	this	this	PRON
ejpam-4831	103	2	means	mean	VERB
ejpam-4831	103	3	that	that	SCONJ
ejpam-4831	103	4	{	{	PUNCT
ejpam-4831	103	5	(	(	PUNCT
ejpam-4831	103	6	∥xi∥b)i}∞i=1	∥xi∥b)i}∞i=1	NOUN
ejpam-4831	103	7	is	be	AUX
ejpam-4831	103	8	a	a	DET
ejpam-4831	103	9	cauchy	cauchy	ADJ
ejpam-4831	103	10	sequence	sequence	NOUN
ejpam-4831	103	11	in	in	ADP
ejpam-4831	103	12	the	the	DET
ejpam-4831	103	13	complete	complete	ADJ
ejpam-4831	103	14	space	space	NOUN
ejpam-4831	103	15	(	(	PUNCT
ejpam-4831	103	16	λs	λs	NOUN
ejpam-4831	103	17	,	,	PUNCT
ejpam-4831	103	18	∥	∥	X
ejpam-4831	103	19	·	·	PUNCT
ejpam-4831	103	20	∥s	∥s	NOUN
ejpam-4831	103	21	)	)	PUNCT
ejpam-4831	103	22	;	;	PUNCT
ejpam-4831	103	23	let	let	VERB
ejpam-4831	103	24	α	α	NOUN
ejpam-4831	103	25	=	=	PUNCT
ejpam-4831	103	26	(	(	PUNCT
ejpam-4831	103	27	αn)n	αn)n	NOUN
ejpam-4831	103	28	be	be	AUX
ejpam-4831	103	29	its	its	PRON
ejpam-4831	103	30	limit	limit	NOUN
ejpam-4831	103	31	in	in	ADP
ejpam-4831	103	32	λs	λs	PROPN
ejpam-4831	103	33	.	.	PUNCT
ejpam-4831	104	1	fix	fix	VERB
ejpam-4831	104	2	n	n	CCONJ
ejpam-4831	104	3	∈	∈	PROPN
ejpam-4831	104	4	n.	n.	NOUN
ejpam-4831	104	5	due	due	ADP
ejpam-4831	104	6	to	to	ADP
ejpam-4831	104	7	the	the	DET
ejpam-4831	104	8	boundedness	boundedness	NOUN
ejpam-4831	104	9	of	of	ADP
ejpam-4831	104	10	the	the	DET
ejpam-4831	104	11	projections	projection	NOUN
ejpam-4831	104	12	,	,	PUNCT
ejpam-4831	104	13	{	{	PUNCT
ejpam-4831	104	14	∥xin∥b}∞i=1	∥xin∥b}∞i=1	NOUN
ejpam-4831	104	15	converges	converge	VERB
ejpam-4831	104	16	to	to	ADP
ejpam-4831	104	17	αn	αn	NOUN
ejpam-4831	104	18	and	and	CCONJ
ejpam-4831	104	19	{	{	PUNCT
ejpam-4831	104	20	xin}∞i=1	xin}∞i=1	PROPN
ejpam-4831	104	21	is	be	AUX
ejpam-4831	104	22	a	a	DET
ejpam-4831	104	23	cauchy	cauchy	ADJ
ejpam-4831	104	24	sequence	sequence	NOUN
ejpam-4831	104	25	in	in	ADP
ejpam-4831	104	26	the	the	DET
ejpam-4831	104	27	complete	complete	ADJ
ejpam-4831	104	28	space	space	NOUN
ejpam-4831	104	29	eb	eb	PROPN
ejpam-4831	104	30	;	;	PUNCT
ejpam-4831	104	31	denote	denote	VERB
ejpam-4831	104	32	by	by	ADP
ejpam-4831	104	33	xn	xn	PROPN
ejpam-4831	104	34	its	its	PRON
ejpam-4831	104	35	limit	limit	NOUN
ejpam-4831	104	36	.	.	PUNCT
ejpam-4831	105	1	thus	thus	ADV
ejpam-4831	105	2	,	,	PUNCT
ejpam-4831	105	3	∥xn∥b	∥xn∥b	PROPN
ejpam-4831	105	4	=	=	SYM
ejpam-4831	105	5	αn	αn	PROPN
ejpam-4831	105	6	,	,	PUNCT
ejpam-4831	105	7	and	and	CCONJ
ejpam-4831	105	8	x	x	X
ejpam-4831	105	9	=	=	SYM
ejpam-4831	105	10	(	(	PUNCT
ejpam-4831	105	11	xn)n	xn)n	PROPN
ejpam-4831	105	12	∈	∈	PROPN
ejpam-4831	105	13	λ(e	λ(e	PROPN
ejpam-4831	105	14	)	)	PUNCT
ejpam-4831	105	15	.	.	PUNCT
ejpam-4831	106	1	it	it	PRON
ejpam-4831	106	2	remains	remain	VERB
ejpam-4831	106	3	to	to	PART
ejpam-4831	106	4	prove	prove	VERB
ejpam-4831	106	5	the	the	DET
ejpam-4831	106	6	convergence	convergence	NOUN
ejpam-4831	106	7	of	of	ADP
ejpam-4831	106	8	{	{	PUNCT
ejpam-4831	106	9	(	(	PUNCT
ejpam-4831	106	10	xi)i}∞i=1	xi)i}∞i=1	INTJ
ejpam-4831	106	11	to	to	PART
ejpam-4831	106	12	x.	x.	VERB
ejpam-4831	106	13	this	this	PRON
ejpam-4831	106	14	derives	derive	VERB
ejpam-4831	106	15	from	from	ADP
ejpam-4831	106	16	the	the	DET
ejpam-4831	106	17	fact	fact	NOUN
ejpam-4831	106	18	that	that	SCONJ
ejpam-4831	106	19	{	{	PUNCT
ejpam-4831	106	20	(	(	PUNCT
ejpam-4831	106	21	∥xi	∥xi	NOUN
ejpam-4831	106	22	−	−	NOUN
ejpam-4831	106	23	x∥b)i}∞i=1	x∥b)i}∞i=1	PROPN
ejpam-4831	106	24	is	be	AUX
ejpam-4831	106	25	a	a	DET
ejpam-4831	106	26	cauchy	cauchy	ADJ
ejpam-4831	106	27	sequence	sequence	NOUN
ejpam-4831	106	28	in	in	ADP
ejpam-4831	106	29	(	(	PUNCT
ejpam-4831	106	30	λs	λs	ADV
ejpam-4831	106	31	,	,	PUNCT
ejpam-4831	106	32	∥	∥	X
ejpam-4831	106	33	·	·	PUNCT
ejpam-4831	106	34	∥s	∥s	NOUN
ejpam-4831	106	35	)	)	PUNCT
ejpam-4831	106	36	and	and	CCONJ
ejpam-4831	106	37	its	its	PRON
ejpam-4831	106	38	limit	limit	NOUN
ejpam-4831	106	39	is	be	AUX
ejpam-4831	106	40	nothing	nothing	PRON
ejpam-4831	106	41	but	but	SCONJ
ejpam-4831	106	42	the	the	DET
ejpam-4831	106	43	zero	zero	NUM
ejpam-4831	106	44	sequence	sequence	NOUN
ejpam-4831	106	45	in	in	ADP
ejpam-4831	106	46	λ	λ	PROPN
ejpam-4831	106	47	.	.	PUNCT
ejpam-4831	107	1	■	■	PUNCT
ejpam-4831	107	2	3	3	X
ejpam-4831	107	3	.	.	X
ejpam-4831	107	4	nuclearity	nuclearity	NOUN
ejpam-4831	107	5	of	of	ADP
ejpam-4831	107	6	λ(e	λ(e	PROPN
ejpam-4831	107	7	)	)	PUNCT
ejpam-4831	107	8	a	a	DET
ejpam-4831	107	9	linear	linear	ADJ
ejpam-4831	107	10	mapping	mapping	NOUN
ejpam-4831	107	11	f	f	NOUN
ejpam-4831	107	12	:	:	PUNCT
ejpam-4831	107	13	e	e	X
ejpam-4831	107	14	→	→	SYM
ejpam-4831	107	15	f	f	PROPN
ejpam-4831	107	16	between	between	ADP
ejpam-4831	107	17	complete	complete	ADJ
ejpam-4831	107	18	normed	norme	VERB
ejpam-4831	107	19	spaces	space	NOUN
ejpam-4831	107	20	is	be	AUX
ejpam-4831	107	21	said	say	VERB
ejpam-4831	107	22	to	to	PART
ejpam-4831	107	23	be	be	AUX
ejpam-4831	107	24	nuclear	nuclear	ADJ
ejpam-4831	107	25	if	if	SCONJ
ejpam-4831	107	26	there	there	PRON
ejpam-4831	107	27	exist	exist	VERB
ejpam-4831	107	28	(	(	PUNCT
ejpam-4831	107	29	εn)n	εn)n	PROPN
ejpam-4831	107	30	∈	∈	PROPN
ejpam-4831	107	31	ℓ1	ℓ1	NOUN
ejpam-4831	107	32	,	,	PUNCT
ejpam-4831	107	33	a	a	DET
ejpam-4831	107	34	bounded	bounded	ADJ
ejpam-4831	107	35	sequence	sequence	NOUN
ejpam-4831	107	36	(	(	PUNCT
ejpam-4831	107	37	an)n	an)n	PROPN
ejpam-4831	107	38	in	in	ADP
ejpam-4831	107	39	the	the	DET
ejpam-4831	107	40	continuous	continuous	ADJ
ejpam-4831	107	41	dual	dual	ADJ
ejpam-4831	107	42	e′	e′	PROPN
ejpam-4831	107	43	of	of	ADP
ejpam-4831	107	44	e	e	PROPN
ejpam-4831	107	45	and	and	CCONJ
ejpam-4831	107	46	a	a	DET
ejpam-4831	107	47	m.	m.	NOUN
ejpam-4831	107	48	a.	a.	NOUN
ejpam-4831	107	49	sidaty	sidaty	PROPN
ejpam-4831	107	50	/	/	SYM
ejpam-4831	107	51	eur	eur	PROPN
ejpam-4831	107	52	.	.	PUNCT
ejpam-4831	108	1	j.	j.	PROPN
ejpam-4831	108	2	pure	pure	PROPN
ejpam-4831	108	3	appl	appl	PROPN
ejpam-4831	108	4	.	.	PROPN
ejpam-4831	108	5	math	math	PROPN
ejpam-4831	108	6	,	,	PUNCT
ejpam-4831	108	7	16	16	NUM
ejpam-4831	108	8	(	(	PUNCT
ejpam-4831	108	9	3	3	NUM
ejpam-4831	108	10	)	)	PUNCT
ejpam-4831	108	11	(	(	PUNCT
ejpam-4831	108	12	2023	2023	NUM
ejpam-4831	108	13	)	)	PUNCT
ejpam-4831	108	14	,	,	PUNCT
ejpam-4831	108	15	1762	1762	NUM
ejpam-4831	108	16	-	-	SYM
ejpam-4831	108	17	1771	1771	NUM
ejpam-4831	108	18	1766	1766	NUM
ejpam-4831	108	19	bounded	bound	VERB
ejpam-4831	108	20	sequence	sequence	NOUN
ejpam-4831	108	21	(	(	PUNCT
ejpam-4831	108	22	yn)n	yn)n	PROPN
ejpam-4831	108	23	⊂	⊂	PROPN
ejpam-4831	108	24	f	f	PROPN
ejpam-4831	109	1	such	such	ADJ
ejpam-4831	109	2	that	that	SCONJ
ejpam-4831	109	3	f(x	f(x	NOUN
ejpam-4831	109	4	)	)	PUNCT
ejpam-4831	109	5	=	=	PUNCT
ejpam-4831	110	1	∞∑	∞∑	NUM
ejpam-4831	110	2	n=1	n=1	PROPN
ejpam-4831	110	3	εnan(x)yn	εnan(x)yn	PROPN
ejpam-4831	110	4	,	,	PUNCT
ejpam-4831	110	5	for	for	ADP
ejpam-4831	110	6	all	all	DET
ejpam-4831	110	7	x	x	SYM
ejpam-4831	110	8	∈	∈	PROPN
ejpam-4831	110	9	e.	e.	PROPN
ejpam-4831	110	10	a	a	DET
ejpam-4831	110	11	b	b	NOUN
ejpam-4831	110	12	-	-	PUNCT
ejpam-4831	110	13	space	space	NOUN
ejpam-4831	110	14	e	e	NOUN
ejpam-4831	110	15	is	be	AUX
ejpam-4831	110	16	said	say	VERB
ejpam-4831	110	17	to	to	PART
ejpam-4831	110	18	be	be	AUX
ejpam-4831	110	19	nuclear	nuclear	ADJ
ejpam-4831	110	20	(	(	PUNCT
ejpam-4831	110	21	a	a	DET
ejpam-4831	110	22	schwartz	schwartz	PROPN
ejpam-4831	110	23	space	space	NOUN
ejpam-4831	110	24	)	)	PUNCT
ejpam-4831	110	25	if	if	SCONJ
ejpam-4831	110	26	for	for	ADP
ejpam-4831	110	27	every	every	DET
ejpam-4831	110	28	banach	banach	NOUN
ejpam-4831	110	29	disk	disk	NOUN
ejpam-4831	110	30	a	a	PRON
ejpam-4831	110	31	in	in	ADP
ejpam-4831	110	32	e	e	NOUN
ejpam-4831	110	33	there	there	PRON
ejpam-4831	110	34	is	be	VERB
ejpam-4831	110	35	a	a	DET
ejpam-4831	110	36	banach	banach	NOUN
ejpam-4831	110	37	disk	disk	NOUN
ejpam-4831	110	38	b	b	PROPN
ejpam-4831	110	39	⊃	⊃	NOUN
ejpam-4831	110	40	a	a	PRON
ejpam-4831	110	41	in	in	ADP
ejpam-4831	110	42	e	e	NOUN
ejpam-4831	110	43	such	such	ADJ
ejpam-4831	110	44	that	that	SCONJ
ejpam-4831	110	45	the	the	DET
ejpam-4831	110	46	inclusion	inclusion	NOUN
ejpam-4831	110	47	mapping	mapping	NOUN
ejpam-4831	110	48	ea	ea	PROPN
ejpam-4831	110	49	→	→	SYM
ejpam-4831	110	50	eb	eb	PROPN
ejpam-4831	110	51	is	be	AUX
ejpam-4831	110	52	nuclear	nuclear	ADJ
ejpam-4831	110	53	(	(	PUNCT
ejpam-4831	110	54	compact	compact	ADJ
ejpam-4831	110	55	)	)	PUNCT
ejpam-4831	110	56	.	.	PUNCT
ejpam-4831	111	1	proposition	proposition	NOUN
ejpam-4831	111	2	3	3	NUM
ejpam-4831	111	3	.	.	PUNCT
ejpam-4831	112	1	the	the	DET
ejpam-4831	112	2	tensor	tensor	NOUN
ejpam-4831	112	3	product	product	NOUN
ejpam-4831	112	4	λ⊗	λ⊗	NOUN
ejpam-4831	112	5	e	e	NOUN
ejpam-4831	112	6	is	be	AUX
ejpam-4831	112	7	identifiable	identifiable	ADJ
ejpam-4831	112	8	with	with	ADP
ejpam-4831	112	9	a	a	DET
ejpam-4831	112	10	subspace	subspace	NOUN
ejpam-4831	112	11	of	of	ADP
ejpam-4831	112	12	λ(e	λ(e	NOUN
ejpam-4831	112	13	)	)	PUNCT
ejpam-4831	112	14	.	.	PUNCT
ejpam-4831	113	1	proof	proof	NOUN
ejpam-4831	113	2	.	.	PUNCT
ejpam-4831	114	1	we	we	PRON
ejpam-4831	114	2	see	see	VERB
ejpam-4831	114	3	that	that	PRON
ejpam-4831	114	4	for	for	ADP
ejpam-4831	114	5	all	all	DET
ejpam-4831	114	6	α	α	NOUN
ejpam-4831	114	7	=	=	PUNCT
ejpam-4831	114	8	(	(	PUNCT
ejpam-4831	114	9	αn)n	αn)n	NOUN
ejpam-4831	114	10	∈	∈	PROPN
ejpam-4831	114	11	λ	λ	PROPN
ejpam-4831	114	12	and	and	CCONJ
ejpam-4831	114	13	x	x	SYM
ejpam-4831	114	14	∈	∈	PROPN
ejpam-4831	114	15	e	e	NOUN
ejpam-4831	114	16	,	,	PUNCT
ejpam-4831	114	17	(	(	PUNCT
ejpam-4831	114	18	αnx)n	αnx)n	PROPN
ejpam-4831	114	19	∈	∈	PROPN
ejpam-4831	114	20	λ(e	λ(e	VERB
ejpam-4831	114	21	)	)	PUNCT
ejpam-4831	114	22	.	.	PUNCT
ejpam-4831	115	1	define	define	VERB
ejpam-4831	115	2	the	the	DET
ejpam-4831	115	3	bilinear	bilinear	NOUN
ejpam-4831	115	4	mapping	mapping	NOUN
ejpam-4831	115	5	φ	φ	NOUN
ejpam-4831	115	6	:	:	PUNCT
ejpam-4831	116	1	λ	λ	X
ejpam-4831	116	2	×	×	NOUN
ejpam-4831	116	3	e	e	X
ejpam-4831	116	4	→	→	SYM
ejpam-4831	116	5	λ(e	λ(e	PROPN
ejpam-4831	116	6	)	)	PUNCT
ejpam-4831	116	7	,	,	PUNCT
ejpam-4831	116	8	such	such	ADJ
ejpam-4831	116	9	that	that	SCONJ
ejpam-4831	116	10	φ(α	φ(α	PROPN
ejpam-4831	116	11	,	,	PUNCT
ejpam-4831	116	12	x	x	X
ejpam-4831	116	13	)	)	PUNCT
ejpam-4831	116	14	=	=	SYM
ejpam-4831	117	1	(	(	PUNCT
ejpam-4831	117	2	αnx)n	αnx)n	PROPN
ejpam-4831	117	3	.	.	PUNCT
ejpam-4831	118	1	there	there	PRON
ejpam-4831	118	2	exists	exist	VERB
ejpam-4831	118	3	a	a	DET
ejpam-4831	118	4	linear	linear	ADJ
ejpam-4831	118	5	mapping	mapping	NOUN
ejpam-4831	118	6	ℓ	ℓ	NOUN
ejpam-4831	118	7	:	:	PUNCT
ejpam-4831	119	1	λ	λ	X
ejpam-4831	119	2	⊗	⊗	PROPN
ejpam-4831	119	3	e	e	X
ejpam-4831	119	4	→	→	SYM
ejpam-4831	119	5	λ(e	λ(e	PROPN
ejpam-4831	119	6	)	)	PUNCT
ejpam-4831	119	7	,	,	PUNCT
ejpam-4831	119	8	with	with	ADP
ejpam-4831	119	9	ℓ(α	ℓ(α	PROPN
ejpam-4831	119	10	⊗	⊗	PROPN
ejpam-4831	119	11	x	x	NOUN
ejpam-4831	119	12	)	)	PUNCT
ejpam-4831	120	1	=	=	SYM
ejpam-4831	120	2	(	(	PUNCT
ejpam-4831	120	3	αnx)n	αnx)n	PROPN
ejpam-4831	120	4	.	.	PUNCT
ejpam-4831	120	5	let	let	VERB
ejpam-4831	120	6	us	we	PRON
ejpam-4831	120	7	show	show	VERB
ejpam-4831	120	8	that	that	SCONJ
ejpam-4831	120	9	ℓ	ℓ	PROPN
ejpam-4831	120	10	is	be	AUX
ejpam-4831	120	11	one	one	NUM
ejpam-4831	120	12	to	to	ADP
ejpam-4831	120	13	one	one	NUM
ejpam-4831	120	14	.	.	PUNCT
ejpam-4831	121	1	suppose	suppose	VERB
ejpam-4831	121	2	that	that	SCONJ
ejpam-4831	121	3	z	z	PROPN
ejpam-4831	121	4	∈	∈	PROPN
ejpam-4831	121	5	λ⊗	λ⊗	NOUN
ejpam-4831	121	6	e	e	NOUN
ejpam-4831	121	7	such	such	ADJ
ejpam-4831	121	8	that	that	SCONJ
ejpam-4831	121	9	ℓ(z	ℓ(z	NOUN
ejpam-4831	121	10	)	)	PUNCT
ejpam-4831	121	11	=	=	SYM
ejpam-4831	122	1	0	0	X
ejpam-4831	122	2	.	.	PUNCT
ejpam-4831	123	1	we	we	PRON
ejpam-4831	123	2	can	can	AUX
ejpam-4831	123	3	write	write	VERB
ejpam-4831	123	4	z	z	NOUN
ejpam-4831	123	5	=	=	PUNCT
ejpam-4831	124	1	∑k	∑k	PROPN
ejpam-4831	124	2	i=1(α	i=1(α	INTJ
ejpam-4831	125	1	i	i	PRON
ejpam-4831	125	2	n)n	n)n	PUNCT
ejpam-4831	126	1	⊗	⊗	PROPN
ejpam-4831	126	2	xi	xi	X
ejpam-4831	126	3	,	,	PUNCT
ejpam-4831	126	4	for	for	ADP
ejpam-4831	126	5	which	which	PRON
ejpam-4831	126	6	{	{	PUNCT
ejpam-4831	126	7	(	(	PUNCT
ejpam-4831	126	8	αi	αi	NOUN
ejpam-4831	126	9	n)n}ki=1	n)n}ki=1	PUNCT
ejpam-4831	127	1	and	and	CCONJ
ejpam-4831	127	2	{	{	PUNCT
ejpam-4831	127	3	xi}ki=1	xi}ki=1	X
ejpam-4831	127	4	are	be	AUX
ejpam-4831	127	5	linearly	linearly	ADV
ejpam-4831	127	6	independent	independent	ADJ
ejpam-4831	127	7	.	.	PUNCT
ejpam-4831	128	1	but	but	CCONJ
ejpam-4831	128	2	,	,	PUNCT
ejpam-4831	128	3	ℓ(z	ℓ(z	PROPN
ejpam-4831	128	4	)	)	PUNCT
ejpam-4831	129	1	=	=	PRON
ejpam-4831	129	2	k∑	k∑	VERB
ejpam-4831	129	3	i=1	i=1	PROPN
ejpam-4831	129	4	ℓ(αi	ℓ(αi	PROPN
ejpam-4831	129	5	⊗	⊗	PROPN
ejpam-4831	129	6	xi	xi	PROPN
ejpam-4831	129	7	)	)	PUNCT
ejpam-4831	130	1	=	=	VERB
ejpam-4831	130	2	k∑	k∑	PROPN
ejpam-4831	130	3	i=1	i=1	X
ejpam-4831	131	1	(	(	PUNCT
ejpam-4831	131	2	αi	αi	PROPN
ejpam-4831	131	3	nxi)n	nxi)n	PROPN
ejpam-4831	131	4	=	=	PRON
ejpam-4831	131	5	(	(	PUNCT
ejpam-4831	131	6	k∑	k∑	VERB
ejpam-4831	131	7	i=1	i=1	PROPN
ejpam-4831	131	8	αi	αi	PROPN
ejpam-4831	131	9	nxi	nxi	NOUN
ejpam-4831	131	10	)	)	PUNCT
ejpam-4831	131	11	n	n	CCONJ
ejpam-4831	131	12	.	.	PUNCT
ejpam-4831	132	1	since	since	SCONJ
ejpam-4831	132	2	ℓ(z	ℓ(z	PROPN
ejpam-4831	132	3	)	)	PUNCT
ejpam-4831	132	4	=	=	SYM
ejpam-4831	132	5	0	0	PUNCT
ejpam-4831	133	1	then	then	ADV
ejpam-4831	133	2	(	(	PUNCT
ejpam-4831	133	3	∑k	∑k	PROPN
ejpam-4831	133	4	i=1	i=1	PROPN
ejpam-4831	134	1	α	α	INTJ
ejpam-4831	135	1	i	i	PRON
ejpam-4831	135	2	nxi	nxi	ADJ
ejpam-4831	135	3	)	)	PUNCT
ejpam-4831	135	4	n	n	NOUN
ejpam-4831	135	5	=	=	SYM
ejpam-4831	135	6	0	0	NUM
ejpam-4831	135	7	and	and	CCONJ
ejpam-4831	135	8	∑k	∑k	PROPN
ejpam-4831	135	9	i=1	i=1	PROPN
ejpam-4831	136	1	α	α	INTJ
ejpam-4831	137	1	i	i	PRON
ejpam-4831	137	2	nxi	nxi	VERB
ejpam-4831	138	1	=	=	NOUN
ejpam-4831	138	2	0	0	NUM
ejpam-4831	138	3	,	,	PUNCT
ejpam-4831	138	4	for	for	ADP
ejpam-4831	138	5	every	every	DET
ejpam-4831	138	6	n.	n.	NOUN
ejpam-4831	138	7	but	but	CCONJ
ejpam-4831	138	8	,	,	PUNCT
ejpam-4831	138	9	as	as	SCONJ
ejpam-4831	138	10	{	{	PUNCT
ejpam-4831	138	11	xi}ki=1	xi}ki=1	PROPN
ejpam-4831	138	12	is	be	AUX
ejpam-4831	138	13	linearly	linearly	ADV
ejpam-4831	138	14	independent	independent	ADJ
ejpam-4831	138	15	,	,	PUNCT
ejpam-4831	138	16	αi	αi	NOUN
ejpam-4831	138	17	n	n	NOUN
ejpam-4831	138	18	=	=	SYM
ejpam-4831	138	19	0	0	NUM
ejpam-4831	138	20	,	,	PUNCT
ejpam-4831	138	21	for	for	ADP
ejpam-4831	138	22	all	all	DET
ejpam-4831	138	23	1	1	NUM
ejpam-4831	138	24	≤	≤	NUM
ejpam-4831	139	1	i	i	NOUN
ejpam-4831	139	2	≤	≤	NUM
ejpam-4831	140	1	k	k	PROPN
ejpam-4831	140	2	and	and	CCONJ
ejpam-4831	140	3	n	n	DET
ejpam-4831	140	4	∈	∈	PROPN
ejpam-4831	140	5	n.	n.	NOUN
ejpam-4831	140	6	thus	thus	ADV
ejpam-4831	140	7	,	,	PUNCT
ejpam-4831	140	8	z	z	PROPN
ejpam-4831	140	9	=	=	PUNCT
ejpam-4831	140	10	∑k	∑k	PROPN
ejpam-4831	140	11	i=1(α	i=1(α	INTJ
ejpam-4831	140	12	i	i	PRON
ejpam-4831	140	13	n)n	n)n	PUNCT
ejpam-4831	141	1	⊗	⊗	X
ejpam-4831	141	2	xi	xi	X
ejpam-4831	142	1	=	=	SYM
ejpam-4831	142	2	0	0	NUM
ejpam-4831	142	3	,	,	PUNCT
ejpam-4831	142	4	and	and	CCONJ
ejpam-4831	142	5	ℓ	ℓ	PROPN
ejpam-4831	142	6	is	be	AUX
ejpam-4831	142	7	one	one	NUM
ejpam-4831	142	8	to	to	ADP
ejpam-4831	142	9	one	one	NUM
ejpam-4831	142	10	.	.	PUNCT
ejpam-4831	143	1	■	■	PUNCT
ejpam-4831	143	2	lemma	lemma	PROPN
ejpam-4831	143	3	2	2	X
ejpam-4831	143	4	.	.	PUNCT
ejpam-4831	144	1	let	let	VERB
ejpam-4831	144	2	s	s	PRON
ejpam-4831	144	3	and	and	CCONJ
ejpam-4831	144	4	b	b	NOUN
ejpam-4831	144	5	be	be	AUX
ejpam-4831	144	6	banach	banach	NOUN
ejpam-4831	144	7	disks	disk	NOUN
ejpam-4831	144	8	in	in	ADP
ejpam-4831	144	9	λ	λ	PROPN
ejpam-4831	144	10	and	and	CCONJ
ejpam-4831	144	11	e	e	NOUN
ejpam-4831	144	12	respectively	respectively	ADV
ejpam-4831	144	13	,	,	PUNCT
ejpam-4831	144	14	n(x	n(x	PROPN
ejpam-4831	144	15	)	)	PUNCT
ejpam-4831	144	16	=	=	SYM
ejpam-4831	144	17	∥∥(∥xn∥b)n∥∥s	∥∥(∥xn∥b)n∥∥s	NOUN
ejpam-4831	144	18	for	for	ADP
ejpam-4831	144	19	all	all	PRON
ejpam-4831	145	1	x	x	PUNCT
ejpam-4831	145	2	=	=	SYM
ejpam-4831	145	3	(	(	PUNCT
ejpam-4831	145	4	xn)n	xn)n	PROPN
ejpam-4831	145	5	∈	∈	PROPN
ejpam-4831	145	6	λs(eb	λs(eb	PROPN
ejpam-4831	145	7	)	)	PUNCT
ejpam-4831	145	8	and	and	CCONJ
ejpam-4831	145	9	n1(z	n1(z	X
ejpam-4831	145	10	)	)	PUNCT
ejpam-4831	145	11	=	=	SYM
ejpam-4831	145	12	n(ℓ(z	n(ℓ(z	PROPN
ejpam-4831	145	13	)	)	PUNCT
ejpam-4831	145	14	)	)	PUNCT
ejpam-4831	146	1	for	for	ADP
ejpam-4831	146	2	all	all	DET
ejpam-4831	146	3	z	z	NOUN
ejpam-4831	146	4	∈	∈	PROPN
ejpam-4831	146	5	λs	λs	ADP
ejpam-4831	146	6	⊗	⊗	PROPN
ejpam-4831	146	7	eb	eb	PROPN
ejpam-4831	146	8	.	.	PUNCT
ejpam-4831	147	1	then	then	ADV
ejpam-4831	147	2	,	,	PUNCT
ejpam-4831	147	3	1	1	X
ejpam-4831	147	4	.	.	X
ejpam-4831	147	5	n1	n1	PROPN
ejpam-4831	147	6	is	be	AUX
ejpam-4831	147	7	a	a	DET
ejpam-4831	147	8	cross	cross	NOUN
ejpam-4831	147	9	-	-	NOUN
ejpam-4831	147	10	norm	norm	NOUN
ejpam-4831	147	11	on	on	ADP
ejpam-4831	147	12	λs	λs	PROPN
ejpam-4831	147	13	⊗	⊗	PROPN
ejpam-4831	147	14	eb	eb	PROPN
ejpam-4831	147	15	,	,	PUNCT
ejpam-4831	147	16	that	that	PRON
ejpam-4831	147	17	is	be	AUX
ejpam-4831	147	18	n(α⊗	n(α⊗	ADJ
ejpam-4831	147	19	x	x	SYM
ejpam-4831	147	20	)	)	PUNCT
ejpam-4831	147	21	=	=	PUNCT
ejpam-4831	147	22	∥α∥s∥x∥b	∥α∥s∥x∥b	NOUN
ejpam-4831	147	23	,	,	PUNCT
ejpam-4831	147	24	for	for	ADP
ejpam-4831	147	25	every	every	DET
ejpam-4831	147	26	α	α	NOUN
ejpam-4831	147	27	∈	∈	NOUN
ejpam-4831	147	28	λs	λs	PROPN
ejpam-4831	148	1	and	and	CCONJ
ejpam-4831	148	2	x	x	PROPN
ejpam-4831	148	3	∈	∈	PROPN
ejpam-4831	148	4	eb	eb	PROPN
ejpam-4831	148	5	.	.	PROPN
ejpam-4831	149	1	2	2	NUM
ejpam-4831	149	2	.	.	X
ejpam-4831	150	1	the	the	DET
ejpam-4831	150	2	mapping	mapping	NOUN
ejpam-4831	150	3	ℓ	ℓ	PROPN
ejpam-4831	150	4	:	:	PUNCT
ejpam-4831	150	5	λs	λs	PROPN
ejpam-4831	150	6	⊗	⊗	PROPN
ejpam-4831	150	7	eb	eb	PROPN
ejpam-4831	150	8	→	→	SYM
ejpam-4831	150	9	λs(eb	λs(eb	PROPN
ejpam-4831	150	10	)	)	PUNCT
ejpam-4831	150	11	is	be	AUX
ejpam-4831	150	12	isometric	isometric	ADJ
ejpam-4831	150	13	and	and	CCONJ
ejpam-4831	150	14	can	can	AUX
ejpam-4831	150	15	be	be	AUX
ejpam-4831	150	16	extended	extend	VERB
ejpam-4831	150	17	to	to	ADP
ejpam-4831	150	18	a	a	DET
ejpam-4831	150	19	unique	unique	ADJ
ejpam-4831	150	20	linear	linear	NOUN
ejpam-4831	150	21	mapping	mapping	NOUN
ejpam-4831	150	22	ℓ̂	ℓ̂	NOUN
ejpam-4831	150	23	:	:	PUNCT
ejpam-4831	150	24	λs⊗̂n1eb	λs⊗̂n1eb	X
ejpam-4831	150	25	→	→	SYM
ejpam-4831	150	26	λs(eb	λs(eb	NOUN
ejpam-4831	150	27	)	)	PUNCT
ejpam-4831	150	28	,	,	PUNCT
ejpam-4831	150	29	where	where	SCONJ
ejpam-4831	150	30	λs⊗̂n1eb	λs⊗̂n1eb	ADP
ejpam-4831	150	31	the	the	DET
ejpam-4831	150	32	completion	completion	NOUN
ejpam-4831	150	33	of	of	ADP
ejpam-4831	150	34	the	the	DET
ejpam-4831	150	35	normed	normed	ADJ
ejpam-4831	150	36	space	space	NOUN
ejpam-4831	150	37	(	(	PUNCT
ejpam-4831	150	38	λs	λs	PROPN
ejpam-4831	150	39	⊗n1	⊗n1	PROPN
ejpam-4831	150	40	eb	eb	PROPN
ejpam-4831	150	41	,	,	PUNCT
ejpam-4831	150	42	n1	n1	PROPN
ejpam-4831	150	43	)	)	PUNCT
ejpam-4831	150	44	.	.	PUNCT
ejpam-4831	151	1	proof	proof	NOUN
ejpam-4831	151	2	.	.	PUNCT
ejpam-4831	152	1	since	since	SCONJ
ejpam-4831	152	2	n	n	NUM
ejpam-4831	152	3	is	be	AUX
ejpam-4831	152	4	a	a	DET
ejpam-4831	152	5	solid	solid	ADJ
ejpam-4831	152	6	norm	norm	NOUN
ejpam-4831	152	7	and	and	CCONJ
ejpam-4831	152	8	ℓ	ℓ	PROPN
ejpam-4831	152	9	is	be	AUX
ejpam-4831	152	10	a	a	DET
ejpam-4831	152	11	one	one	NUM
ejpam-4831	152	12	to	to	ADP
ejpam-4831	152	13	one	one	NUM
ejpam-4831	152	14	linear	linear	ADJ
ejpam-4831	152	15	mapping	mapping	NOUN
ejpam-4831	152	16	,	,	PUNCT
ejpam-4831	152	17	n1	n1	PROPN
ejpam-4831	152	18	is	be	AUX
ejpam-4831	152	19	a	a	DET
ejpam-4831	152	20	norm	norm	NOUN
ejpam-4831	152	21	.	.	PUNCT
ejpam-4831	153	1	it	it	PRON
ejpam-4831	153	2	is	be	AUX
ejpam-4831	153	3	clear	clear	ADJ
ejpam-4831	153	4	that	that	SCONJ
ejpam-4831	153	5	n1(α⊗	n1(α⊗	PROPN
ejpam-4831	153	6	x	x	X
ejpam-4831	153	7	)	)	PUNCT
ejpam-4831	153	8	=	=	PUNCT
ejpam-4831	153	9	∥α∥s∥x∥b	∥α∥s∥x∥b	NOUN
ejpam-4831	153	10	,	,	PUNCT
ejpam-4831	153	11	and	and	CCONJ
ejpam-4831	153	12	1	1	X
ejpam-4831	153	13	.	.	PUNCT
ejpam-4831	153	14	holds	hold	NOUN
ejpam-4831	153	15	.	.	PUNCT
ejpam-4831	154	1	by	by	ADP
ejpam-4831	154	2	the	the	DET
ejpam-4831	154	3	definition	definition	NOUN
ejpam-4831	154	4	of	of	ADP
ejpam-4831	154	5	n1	n1	NOUN
ejpam-4831	154	6	,	,	PUNCT
ejpam-4831	154	7	we	we	PRON
ejpam-4831	154	8	see	see	VERB
ejpam-4831	154	9	that	that	SCONJ
ejpam-4831	154	10	ℓ	ℓ	PROPN
ejpam-4831	154	11	is	be	AUX
ejpam-4831	154	12	isometric	isometric	ADJ
ejpam-4831	154	13	from	from	ADP
ejpam-4831	154	14	λs	λs	NOUN
ejpam-4831	154	15	⊗eb	⊗eb	ADP
ejpam-4831	154	16	to	to	ADP
ejpam-4831	154	17	the	the	DET
ejpam-4831	154	18	complete	complete	ADJ
ejpam-4831	154	19	space	space	NOUN
ejpam-4831	154	20	λs(eb	λs(eb	PROPN
ejpam-4831	154	21	)	)	PUNCT
ejpam-4831	154	22	,	,	PUNCT
ejpam-4831	154	23	and	and	CCONJ
ejpam-4831	154	24	then	then	ADV
ejpam-4831	154	25	it	it	PRON
ejpam-4831	154	26	has	have	VERB
ejpam-4831	154	27	an	an	DET
ejpam-4831	154	28	extension	extension	NOUN
ejpam-4831	154	29	to	to	ADP
ejpam-4831	154	30	the	the	DET
ejpam-4831	154	31	completion	completion	NOUN
ejpam-4831	154	32	λs⊗̂n1eb	λs⊗̂n1eb	PROPN
ejpam-4831	154	33	of	of	ADP
ejpam-4831	154	34	λs	λs	PROPN
ejpam-4831	154	35	⊗n1	⊗n1	PROPN
ejpam-4831	154	36	eb	eb	PROPN
ejpam-4831	154	37	.	.	PUNCT
ejpam-4831	155	1	this	this	PRON
ejpam-4831	155	2	gives	give	VERB
ejpam-4831	155	3	the	the	DET
ejpam-4831	155	4	second	second	ADJ
ejpam-4831	155	5	item	item	NOUN
ejpam-4831	155	6	.	.	PUNCT
ejpam-4831	156	1	■	■	PUNCT
ejpam-4831	156	2	we	we	PRON
ejpam-4831	156	3	will	will	AUX
ejpam-4831	156	4	make	make	VERB
ejpam-4831	156	5	use	use	NOUN
ejpam-4831	156	6	of	of	ADP
ejpam-4831	156	7	the	the	DET
ejpam-4831	156	8	following	following	ADJ
ejpam-4831	156	9	result	result	NOUN
ejpam-4831	156	10	to	to	PART
ejpam-4831	156	11	represent	represent	VERB
ejpam-4831	156	12	λ(e	λ(e	PROPN
ejpam-4831	156	13	)	)	PUNCT
ejpam-4831	156	14	as	as	ADP
ejpam-4831	156	15	a	a	DET
ejpam-4831	156	16	bornological	bornological	ADJ
ejpam-4831	156	17	tensor	tensor	NOUN
ejpam-4831	156	18	product	product	NOUN
ejpam-4831	156	19	.	.	PUNCT
ejpam-4831	157	1	proposition	proposition	NOUN
ejpam-4831	157	2	4	4	NUM
ejpam-4831	157	3	.	.	PUNCT
ejpam-4831	158	1	[	[	X
ejpam-4831	158	2	4	4	NUM
ejpam-4831	158	3	,	,	PUNCT
ejpam-4831	158	4	ch	ch	NOUN
ejpam-4831	158	5	viii	viii	NOUN
ejpam-4831	158	6	,	,	PUNCT
ejpam-4831	158	7	prop	prop	NOUN
ejpam-4831	158	8	.	.	PUNCT
ejpam-4831	159	1	4	4	NUM
ejpam-4831	159	2	]	]	SYM
ejpam-4831	159	3	1	1	X
ejpam-4831	159	4	.	.	X
ejpam-4831	160	1	there	there	PRON
ejpam-4831	160	2	is	be	VERB
ejpam-4831	160	3	a	a	DET
ejpam-4831	160	4	convex	convex	ADJ
ejpam-4831	160	5	bornology	bornology	NOUN
ejpam-4831	160	6	b	b	NOUN
ejpam-4831	160	7	on	on	ADP
ejpam-4831	160	8	λ⊗	λ⊗	NOUN
ejpam-4831	160	9	e	e	X
ejpam-4831	160	10	(	(	PUNCT
ejpam-4831	160	11	the	the	DET
ejpam-4831	160	12	finest	fine	ADJ
ejpam-4831	160	13	one	one	NUM
ejpam-4831	160	14	)	)	PUNCT
ejpam-4831	160	15	making	make	VERB
ejpam-4831	160	16	bounded	bound	VERB
ejpam-4831	160	17	the	the	DET
ejpam-4831	160	18	inclusion	inclusion	NOUN
ejpam-4831	160	19	mappings	mapping	NOUN
ejpam-4831	160	20	λs	λs	ADP
ejpam-4831	160	21	⊗n1	⊗n1	PROPN
ejpam-4831	160	22	eb	eb	PROPN
ejpam-4831	160	23	→	→	SYM
ejpam-4831	160	24	λ(e	λ(e	PROPN
ejpam-4831	160	25	)	)	PUNCT
ejpam-4831	160	26	.	.	PUNCT
ejpam-4831	161	1	moreover	moreover	ADV
ejpam-4831	161	2	,	,	PUNCT
ejpam-4831	161	3	λ⊗b	λ⊗b	NOUN
ejpam-4831	161	4	e	e	X
ejpam-4831	161	5	=	=	SYM
ejpam-4831	161	6	lim−→λs	lim−→λs	PROPN
ejpam-4831	161	7	⊗n1	⊗n1	PROPN
ejpam-4831	161	8	eb	eb	PROPN
ejpam-4831	161	9	.	.	PROPN
ejpam-4831	162	1	2	2	NUM
ejpam-4831	162	2	.	.	X
ejpam-4831	162	3	b	b	NOUN
ejpam-4831	162	4	is	be	AUX
ejpam-4831	162	5	located	locate	VERB
ejpam-4831	162	6	between	between	ADP
ejpam-4831	162	7	the	the	DET
ejpam-4831	162	8	projective	projective	ADJ
ejpam-4831	162	9	bornology	bornology	NOUN
ejpam-4831	162	10	π	π	PROPN
ejpam-4831	162	11	and	and	CCONJ
ejpam-4831	162	12	the	the	DET
ejpam-4831	162	13	injective	injective	ADJ
ejpam-4831	162	14	bornology	bornology	NOUN
ejpam-4831	162	15	ε	ε	PROPN
ejpam-4831	162	16	.	.	PROPN
ejpam-4831	162	17	3	3	NUM
ejpam-4831	162	18	.	.	PUNCT
ejpam-4831	163	1	if	if	SCONJ
ejpam-4831	163	2	λ	λ	PROPN
ejpam-4831	163	3	or	or	CCONJ
ejpam-4831	163	4	e	e	NOUN
ejpam-4831	163	5	is	be	AUX
ejpam-4831	163	6	nuclear	nuclear	ADJ
ejpam-4831	163	7	,	,	PUNCT
ejpam-4831	163	8	then	then	ADV
ejpam-4831	163	9	π	π	PROPN
ejpam-4831	163	10	=	=	SYM
ejpam-4831	163	11	b	b	PROPN
ejpam-4831	163	12	=	=	SYM
ejpam-4831	163	13	ε	ε	PROPN
ejpam-4831	163	14	.	.	PROPN
ejpam-4831	163	15	4	4	NUM
ejpam-4831	163	16	.	.	PUNCT
ejpam-4831	164	1	if	if	SCONJ
ejpam-4831	164	2	λ	λ	PROPN
ejpam-4831	164	3	and	and	CCONJ
ejpam-4831	164	4	e	e	PROPN
ejpam-4831	164	5	are	be	AUX
ejpam-4831	164	6	nuclear	nuclear	ADJ
ejpam-4831	164	7	,	,	PUNCT
ejpam-4831	164	8	the	the	DET
ejpam-4831	164	9	bornological	bornological	ADJ
ejpam-4831	164	10	completion	completion	NOUN
ejpam-4831	164	11	λ⊗̃be	λ⊗̃be	NOUN
ejpam-4831	164	12	of	of	ADP
ejpam-4831	164	13	λ	λ	PROPN
ejpam-4831	164	14	⊗b	⊗b	PROPN
ejpam-4831	164	15	e	e	NOUN
ejpam-4831	164	16	is	be	AUX
ejpam-4831	164	17	the	the	DET
ejpam-4831	164	18	inductive	inductive	ADJ
ejpam-4831	164	19	limit	limit	NOUN
ejpam-4831	164	20	of	of	ADP
ejpam-4831	164	21	the	the	DET
ejpam-4831	164	22	banach	banach	NOUN
ejpam-4831	164	23	spaces	space	VERB
ejpam-4831	164	24	λs⊗̂n1eb	λs⊗̂n1eb	NOUN
ejpam-4831	164	25	.	.	PUNCT
ejpam-4831	164	26	m.	m.	NOUN
ejpam-4831	164	27	a.	a.	PROPN
ejpam-4831	164	28	sidaty	sidaty	PROPN
ejpam-4831	164	29	/	/	SYM
ejpam-4831	164	30	eur	eur	PROPN
ejpam-4831	164	31	.	.	PUNCT
ejpam-4831	165	1	j.	j.	PROPN
ejpam-4831	165	2	pure	pure	PROPN
ejpam-4831	165	3	appl	appl	PROPN
ejpam-4831	165	4	.	.	PROPN
ejpam-4831	165	5	math	math	PROPN
ejpam-4831	165	6	,	,	PUNCT
ejpam-4831	165	7	16	16	NUM
ejpam-4831	165	8	(	(	PUNCT
ejpam-4831	165	9	3	3	NUM
ejpam-4831	165	10	)	)	PUNCT
ejpam-4831	165	11	(	(	PUNCT
ejpam-4831	165	12	2023	2023	NUM
ejpam-4831	165	13	)	)	PUNCT
ejpam-4831	165	14	,	,	PUNCT
ejpam-4831	165	15	1762	1762	NUM
ejpam-4831	165	16	-	-	SYM
ejpam-4831	165	17	1771	1771	NUM
ejpam-4831	165	18	1767	1767	NUM
ejpam-4831	165	19	now	now	ADV
ejpam-4831	165	20	,	,	PUNCT
ejpam-4831	165	21	we	we	PRON
ejpam-4831	165	22	prove	prove	VERB
ejpam-4831	165	23	theorem	theorem	ADJ
ejpam-4831	165	24	1	1	X
ejpam-4831	165	25	.	.	PUNCT
ejpam-4831	166	1	if	if	SCONJ
ejpam-4831	166	2	λ	λ	PROPN
ejpam-4831	166	3	and	and	CCONJ
ejpam-4831	166	4	e	e	PROPN
ejpam-4831	166	5	are	be	AUX
ejpam-4831	166	6	nuclear	nuclear	ADJ
ejpam-4831	166	7	,	,	PUNCT
ejpam-4831	166	8	the	the	DET
ejpam-4831	166	9	equality	equality	NOUN
ejpam-4831	166	10	λ(e	λ(e	PART
ejpam-4831	166	11	)	)	PUNCT
ejpam-4831	166	12	=	=	NOUN
ejpam-4831	166	13	λ⊗̃be	λ⊗̃be	NOUN
ejpam-4831	166	14	holds	hold	VERB
ejpam-4831	166	15	algebraically	algebraically	ADV
ejpam-4831	166	16	and	and	CCONJ
ejpam-4831	166	17	bornologically	bornologically	ADV
ejpam-4831	166	18	.	.	PUNCT
ejpam-4831	167	1	proof	proof	NOUN
ejpam-4831	167	2	.	.	PUNCT
ejpam-4831	168	1	consider	consider	VERB
ejpam-4831	168	2	the	the	DET
ejpam-4831	168	3	linear	linear	ADJ
ejpam-4831	168	4	mapping	mapping	NOUN
ejpam-4831	168	5	ℓ	ℓ	NOUN
ejpam-4831	168	6	:	:	PUNCT
ejpam-4831	169	1	λ⊗b	λ⊗b	PROPN
ejpam-4831	169	2	e	e	X
ejpam-4831	169	3	→	→	PUNCT
ejpam-4831	169	4	λ(e	λ(e	ADJ
ejpam-4831	169	5	)	)	PUNCT
ejpam-4831	169	6	defined	define	VERB
ejpam-4831	169	7	in	in	ADP
ejpam-4831	169	8	the	the	DET
ejpam-4831	169	9	proof	proof	NOUN
ejpam-4831	169	10	of	of	ADP
ejpam-4831	169	11	proposition	proposition	NOUN
ejpam-4831	169	12	3	3	NUM
ejpam-4831	169	13	.	.	PUNCT
ejpam-4831	169	14	according	accord	VERB
ejpam-4831	169	15	to	to	ADP
ejpam-4831	169	16	the	the	DET
ejpam-4831	169	17	definition	definition	NOUN
ejpam-4831	169	18	of	of	ADP
ejpam-4831	169	19	the	the	DET
ejpam-4831	169	20	norms	norm	NOUN
ejpam-4831	169	21	n	n	NOUN
ejpam-4831	169	22	and	and	CCONJ
ejpam-4831	169	23	n1	n1	NOUN
ejpam-4831	169	24	,	,	PUNCT
ejpam-4831	169	25	we	we	PRON
ejpam-4831	169	26	see	see	VERB
ejpam-4831	169	27	that	that	SCONJ
ejpam-4831	169	28	ℓ	ℓ	PROPN
ejpam-4831	169	29	is	be	AUX
ejpam-4831	169	30	bounded	bound	VERB
ejpam-4831	169	31	,	,	PUNCT
ejpam-4831	169	32	and	and	CCONJ
ejpam-4831	169	33	since	since	SCONJ
ejpam-4831	169	34	λ(e	λ(e	VERB
ejpam-4831	169	35	)	)	PUNCT
ejpam-4831	169	36	is	be	AUX
ejpam-4831	169	37	complete	complete	ADJ
ejpam-4831	169	38	,	,	PUNCT
ejpam-4831	169	39	ℓ	ℓ	PROPN
ejpam-4831	169	40	can	can	AUX
ejpam-4831	169	41	be	be	AUX
ejpam-4831	169	42	extended	extend	VERB
ejpam-4831	169	43	to	to	ADP
ejpam-4831	169	44	a	a	DET
ejpam-4831	169	45	bounded	bounded	ADJ
ejpam-4831	169	46	linear	linear	PROPN
ejpam-4831	169	47	mapping	mapping	NOUN
ejpam-4831	169	48	ℓ̃	ℓ̃	PROPN
ejpam-4831	169	49	from	from	ADP
ejpam-4831	169	50	the	the	DET
ejpam-4831	169	51	bornological	bornological	ADJ
ejpam-4831	169	52	completion	completion	NOUN
ejpam-4831	169	53	λ⊗̃be	λ⊗̃be	NOUN
ejpam-4831	169	54	of	of	ADP
ejpam-4831	169	55	λ⊗b	λ⊗b	NOUN
ejpam-4831	169	56	e	e	NOUN
ejpam-4831	169	57	to	to	ADP
ejpam-4831	169	58	λ(e	λ(e	NOUN
ejpam-4831	169	59	)	)	PUNCT
ejpam-4831	169	60	.	.	PUNCT
ejpam-4831	170	1	we	we	PRON
ejpam-4831	170	2	will	will	AUX
ejpam-4831	170	3	prove	prove	VERB
ejpam-4831	170	4	that	that	SCONJ
ejpam-4831	170	5	ℓ̃	ℓ̃	PROPN
ejpam-4831	170	6	makes	make	VERB
ejpam-4831	170	7	λ⊗̃be	λ⊗̃be	NOUN
ejpam-4831	170	8	and	and	CCONJ
ejpam-4831	170	9	λ(e	λ(e	PROPN
ejpam-4831	170	10	)	)	PUNCT
ejpam-4831	170	11	bornologically	bornologically	ADV
ejpam-4831	170	12	isomorphic	isomorphic	ADJ
ejpam-4831	170	13	.	.	PUNCT
ejpam-4831	171	1	let	let	VERB
ejpam-4831	171	2	z	z	NOUN
ejpam-4831	171	3	∈	∈	PROPN
ejpam-4831	171	4	λ⊗̃be	λ⊗̃be	NOUN
ejpam-4831	171	5	be	be	VERB
ejpam-4831	171	6	such	such	ADJ
ejpam-4831	171	7	that	that	SCONJ
ejpam-4831	171	8	ℓ̃(z	ℓ̃(z	PROPN
ejpam-4831	171	9	)	)	PUNCT
ejpam-4831	172	1	=	=	PUNCT
ejpam-4831	172	2	0	0	X
ejpam-4831	172	3	.	.	PUNCT
ejpam-4831	173	1	by	by	ADP
ejpam-4831	173	2	[	[	X
ejpam-4831	173	3	4	4	NUM
ejpam-4831	173	4	,	,	PUNCT
ejpam-4831	173	5	ch	ch	NOUN
ejpam-4831	173	6	viii	viii	NOUN
ejpam-4831	173	7	,	,	PUNCT
ejpam-4831	173	8	prop	prop	NOUN
ejpam-4831	173	9	.	.	PUNCT
ejpam-4831	174	1	2	2	NUM
ejpam-4831	174	2	]	]	PUNCT
ejpam-4831	174	3	,	,	PUNCT
ejpam-4831	174	4	a	a	DET
ejpam-4831	174	5	sequence	sequence	NOUN
ejpam-4831	174	6	{	{	PUNCT
ejpam-4831	174	7	zk}∞k=1	zk}∞k=1	NUM
ejpam-4831	174	8	of	of	ADP
ejpam-4831	174	9	elements	element	NOUN
ejpam-4831	174	10	of	of	ADP
ejpam-4831	174	11	λ⊗b	λ⊗b	NOUN
ejpam-4831	174	12	e	e	NOUN
ejpam-4831	174	13	converges	converge	VERB
ejpam-4831	174	14	to	to	ADP
ejpam-4831	174	15	z.	z.	PROPN
ejpam-4831	174	16	then	then	ADV
ejpam-4831	174	17	{	{	PUNCT
ejpam-4831	174	18	zk	zk	PROPN
ejpam-4831	174	19	−	−	PROPN
ejpam-4831	174	20	z}∞k=1	z}∞k=1	PROPN
ejpam-4831	174	21	is	be	AUX
ejpam-4831	174	22	a	a	DET
ejpam-4831	174	23	null	null	ADJ
ejpam-4831	174	24	sequence	sequence	NOUN
ejpam-4831	174	25	in	in	ADP
ejpam-4831	174	26	some	some	DET
ejpam-4831	174	27	subspace	subspace	NOUN
ejpam-4831	174	28	λs⊗̃beb	λs⊗̃beb	PROPN
ejpam-4831	174	29	.	.	PUNCT
ejpam-4831	175	1	thus	thus	ADV
ejpam-4831	175	2	,	,	PUNCT
ejpam-4831	175	3	ℓ̂(z	ℓ̂(z	NOUN
ejpam-4831	175	4	)	)	PUNCT
ejpam-4831	175	5	=	=	SYM
ejpam-4831	175	6	ℓ̂(lim	ℓ̂(lim	PROPN
ejpam-4831	175	7	k	k	PROPN
ejpam-4831	175	8	ι(zk	ι(zk	PROPN
ejpam-4831	175	9	)	)	PUNCT
ejpam-4831	175	10	)	)	PUNCT
ejpam-4831	176	1	=	=	SYM
ejpam-4831	176	2	lim	lim	PROPN
ejpam-4831	176	3	k	k	PROPN
ejpam-4831	176	4	(	(	PUNCT
ejpam-4831	176	5	ℓ̂	ℓ̂	X
ejpam-4831	176	6	◦	◦	NOUN
ejpam-4831	176	7	ι)(zk	ι)(zk	NOUN
ejpam-4831	176	8	)	)	PUNCT
ejpam-4831	177	1	=	=	PROPN
ejpam-4831	177	2	lim	lim	PROPN
ejpam-4831	177	3	k	k	PROPN
ejpam-4831	177	4	ℓ(zk	ℓ(zk	PROPN
ejpam-4831	177	5	)	)	PUNCT
ejpam-4831	178	1	=	=	PROPN
ejpam-4831	179	1	lim	lim	PROPN
ejpam-4831	179	2	k	k	PROPN
ejpam-4831	179	3	(	(	PUNCT
ejpam-4831	179	4	ℓ̃	ℓ̃	PROPN
ejpam-4831	179	5	◦	◦	NOUN
ejpam-4831	179	6	ι)(zk	ι)(zk	NOUN
ejpam-4831	179	7	)	)	PUNCT
ejpam-4831	179	8	=	=	PUNCT
ejpam-4831	180	1	ℓ̃(lim	ℓ̃(lim	PROPN
ejpam-4831	180	2	k	k	PROPN
ejpam-4831	180	3	zk	zk	PROPN
ejpam-4831	180	4	)	)	PUNCT
ejpam-4831	180	5	=	=	SYM
ejpam-4831	180	6	ℓ̃(z	ℓ̃(z	PROPN
ejpam-4831	180	7	)	)	PUNCT
ejpam-4831	180	8	=	=	PUNCT
ejpam-4831	181	1	0	0	X
ejpam-4831	181	2	.	.	PUNCT
ejpam-4831	182	1	here	here	ADV
ejpam-4831	182	2	ι	ι	X
ejpam-4831	182	3	is	be	AUX
ejpam-4831	182	4	the	the	DET
ejpam-4831	182	5	canonical	canonical	ADJ
ejpam-4831	182	6	injection	injection	NOUN
ejpam-4831	182	7	from	from	ADP
ejpam-4831	182	8	λ⊗b	λ⊗b	NOUN
ejpam-4831	182	9	e	e	NOUN
ejpam-4831	182	10	to	to	ADP
ejpam-4831	182	11	its	its	PRON
ejpam-4831	182	12	completion	completion	NOUN
ejpam-4831	182	13	λ⊗̃be	λ⊗̃be	NOUN
ejpam-4831	182	14	.	.	PUNCT
ejpam-4831	183	1	by	by	ADP
ejpam-4831	183	2	lemma	lemma	PROPN
ejpam-4831	183	3	2	2	NUM
ejpam-4831	183	4	,	,	PUNCT
ejpam-4831	183	5	ℓ̂	ℓ̂	X
ejpam-4831	183	6	is	be	AUX
ejpam-4831	183	7	isometric	isometric	ADJ
ejpam-4831	183	8	and	and	CCONJ
ejpam-4831	183	9	then	then	ADV
ejpam-4831	183	10	it	it	PRON
ejpam-4831	183	11	is	be	AUX
ejpam-4831	183	12	one	one	NUM
ejpam-4831	183	13	to	to	ADP
ejpam-4831	183	14	one	one	NUM
ejpam-4831	183	15	,	,	PUNCT
ejpam-4831	183	16	then	then	ADV
ejpam-4831	183	17	z	z	NOUN
ejpam-4831	183	18	=	=	SYM
ejpam-4831	183	19	0	0	NUM
ejpam-4831	183	20	,	,	PUNCT
ejpam-4831	183	21	and	and	CCONJ
ejpam-4831	183	22	ℓ̃	ℓ̃	PROPN
ejpam-4831	183	23	is	be	AUX
ejpam-4831	183	24	one	one	NUM
ejpam-4831	183	25	to	to	ADP
ejpam-4831	183	26	one	one	NUM
ejpam-4831	183	27	.	.	PUNCT
ejpam-4831	184	1	we	we	PRON
ejpam-4831	184	2	will	will	AUX
ejpam-4831	184	3	prove	prove	VERB
ejpam-4831	184	4	that	that	SCONJ
ejpam-4831	184	5	ℓ̃	ℓ̃	PROPN
ejpam-4831	184	6	is	be	AUX
ejpam-4831	184	7	onto	onto	ADP
ejpam-4831	184	8	as	as	SCONJ
ejpam-4831	184	9	follows	follow	VERB
ejpam-4831	184	10	.	.	PUNCT
ejpam-4831	185	1	let	let	VERB
ejpam-4831	185	2	a	a	DET
ejpam-4831	185	3	∈	∈	PROPN
ejpam-4831	185	4	b	b	AUX
ejpam-4831	185	5	be	be	AUX
ejpam-4831	185	6	a	a	DET
ejpam-4831	185	7	banach	banach	NOUN
ejpam-4831	185	8	disk	disk	NOUN
ejpam-4831	185	9	;	;	PUNCT
ejpam-4831	185	10	since	since	SCONJ
ejpam-4831	185	11	e	e	NOUN
ejpam-4831	185	12	is	be	AUX
ejpam-4831	185	13	nuclear	nuclear	ADJ
ejpam-4831	185	14	we	we	PRON
ejpam-4831	185	15	can	can	AUX
ejpam-4831	185	16	select	select	VERB
ejpam-4831	185	17	a	a	DET
ejpam-4831	185	18	banach	banach	NOUN
ejpam-4831	185	19	disk	disk	NOUN
ejpam-4831	185	20	b	b	PROPN
ejpam-4831	185	21	∈	∈	PROPN
ejpam-4831	185	22	b	b	NOUN
ejpam-4831	185	23	containing	contain	VERB
ejpam-4831	185	24	a	a	DET
ejpam-4831	185	25	such	such	ADJ
ejpam-4831	185	26	that	that	SCONJ
ejpam-4831	185	27	the	the	DET
ejpam-4831	185	28	inclusion	inclusion	NOUN
ejpam-4831	185	29	ea	ea	PROPN
ejpam-4831	185	30	→	→	SYM
ejpam-4831	185	31	eb	eb	PROPN
ejpam-4831	185	32	is	be	AUX
ejpam-4831	185	33	nuclear	nuclear	ADJ
ejpam-4831	185	34	.	.	PUNCT
ejpam-4831	186	1	there	there	PRON
ejpam-4831	186	2	are	be	VERB
ejpam-4831	186	3	(	(	PUNCT
ejpam-4831	186	4	εk)k	εk)k	PROPN
ejpam-4831	186	5	∈	∈	PROPN
ejpam-4831	186	6	ℓ1	ℓ1	NOUN
ejpam-4831	186	7	,	,	PUNCT
ejpam-4831	186	8	a	a	DET
ejpam-4831	186	9	bounded	bounded	ADJ
ejpam-4831	186	10	sequence	sequence	NOUN
ejpam-4831	186	11	(	(	PUNCT
ejpam-4831	186	12	ak)k	ak)k	PROPN
ejpam-4831	186	13	in	in	ADP
ejpam-4831	186	14	the	the	DET
ejpam-4831	186	15	continuous	continuous	ADJ
ejpam-4831	186	16	dual	dual	ADJ
ejpam-4831	186	17	(	(	PUNCT
ejpam-4831	186	18	ea	ea	NOUN
ejpam-4831	186	19	)	)	PUNCT
ejpam-4831	186	20	′	′	NUM
ejpam-4831	186	21	of	of	ADP
ejpam-4831	186	22	ea	ea	NOUN
ejpam-4831	186	23	and	and	CCONJ
ejpam-4831	186	24	a	a	DET
ejpam-4831	186	25	bounded	bounded	ADJ
ejpam-4831	186	26	sequence	sequence	NOUN
ejpam-4831	186	27	(	(	PUNCT
ejpam-4831	186	28	yk)k	yk)k	PROPN
ejpam-4831	186	29	⊂	⊂	PROPN
ejpam-4831	186	30	eb	eb	PROPN
ejpam-4831	186	31	such	such	ADJ
ejpam-4831	186	32	that	that	SCONJ
ejpam-4831	186	33	x	x	X
ejpam-4831	187	1	=	=	SYM
ejpam-4831	187	2	∞∑	∞∑	NUM
ejpam-4831	187	3	k=1	k=1	ADJ
ejpam-4831	187	4	εkak(x)yk	εkak(x)yk	NOUN
ejpam-4831	187	5	,	,	PUNCT
ejpam-4831	187	6	for	for	ADP
ejpam-4831	187	7	all	all	DET
ejpam-4831	187	8	x	x	SYM
ejpam-4831	187	9	∈	∈	PROPN
ejpam-4831	187	10	ea	ea	PROPN
ejpam-4831	187	11	.	.	PUNCT
ejpam-4831	188	1	(	(	PUNCT
ejpam-4831	188	2	1	1	X
ejpam-4831	188	3	)	)	PUNCT
ejpam-4831	188	4	let	let	VERB
ejpam-4831	188	5	x	x	PUNCT
ejpam-4831	188	6	=	=	SYM
ejpam-4831	188	7	(	(	PUNCT
ejpam-4831	188	8	xn)n	xn)n	PROPN
ejpam-4831	188	9	∈	∈	PROPN
ejpam-4831	188	10	λs(ea	λs(ea	PROPN
ejpam-4831	188	11	)	)	PUNCT
ejpam-4831	188	12	,	,	PUNCT
ejpam-4831	188	13	and	and	CCONJ
ejpam-4831	188	14	αk	αk	INTJ
ejpam-4831	188	15	=	=	NOUN
ejpam-4831	188	16	(	(	PUNCT
ejpam-4831	188	17	αk	αk	NOUN
ejpam-4831	188	18	n)n	n)n	NOUN
ejpam-4831	189	1	=	=	NOUN
ejpam-4831	189	2	:	:	PUNCT
ejpam-4831	189	3	(	(	PUNCT
ejpam-4831	189	4	ak(xn))n	ak(xn))n	NOUN
ejpam-4831	189	5	.	.	PUNCT
ejpam-4831	190	1	we	we	PRON
ejpam-4831	190	2	have	have	VERB
ejpam-4831	190	3	|αk	|αk	ADP
ejpam-4831	190	4	n|	n|	NOUN
ejpam-4831	190	5	=	=	SYM
ejpam-4831	190	6	|ak(xn)|	|ak(xn)|	NOUN
ejpam-4831	190	7	≤	≤	PUNCT
ejpam-4831	190	8	∥ak∥∥xn∥a	∥ak∥∥xn∥a	ADV
ejpam-4831	190	9	≤	≤	NUM
ejpam-4831	190	10	(	(	PUNCT
ejpam-4831	190	11	sup	sup	NOUN
ejpam-4831	190	12	p	p	NOUN
ejpam-4831	190	13	∥ap∥	∥ap∥	ADJ
ejpam-4831	190	14	)	)	PUNCT
ejpam-4831	191	1	∥xn∥a	∥xn∥a	ADJ
ejpam-4831	191	2	,	,	PUNCT
ejpam-4831	191	3	for	for	ADP
ejpam-4831	191	4	all	all	DET
ejpam-4831	191	5	k	k	PROPN
ejpam-4831	191	6	,	,	PUNCT
ejpam-4831	191	7	n.	n.	NOUN
ejpam-4831	191	8	(	(	PUNCT
ejpam-4831	191	9	2	2	NUM
ejpam-4831	191	10	)	)	PUNCT
ejpam-4831	191	11	the	the	DET
ejpam-4831	191	12	sequence	sequence	NOUN
ejpam-4831	191	13	(	(	PUNCT
ejpam-4831	191	14	ak)k	ak)k	PROPN
ejpam-4831	191	15	being	be	AUX
ejpam-4831	191	16	bounded	bound	VERB
ejpam-4831	191	17	in	in	ADP
ejpam-4831	191	18	(	(	PUNCT
ejpam-4831	191	19	ea	ea	NOUN
ejpam-4831	191	20	)	)	PUNCT
ejpam-4831	191	21	′	′	NOUN
ejpam-4831	191	22	,	,	PUNCT
ejpam-4831	191	23	supp	supp	PROPN
ejpam-4831	191	24	∥ap∥	∥ap∥	NOUN
ejpam-4831	191	25	is	be	AUX
ejpam-4831	191	26	finite	finite	ADJ
ejpam-4831	191	27	,	,	PUNCT
ejpam-4831	191	28	αk	αk	ADP
ejpam-4831	191	29	=	=	SYM
ejpam-4831	191	30	(	(	PUNCT
ejpam-4831	191	31	αk	αk	ADP
ejpam-4831	191	32	n)n	n)n	NOUN
ejpam-4831	191	33	∈	∈	PROPN
ejpam-4831	191	34	λs(ea	λs(ea	NOUN
ejpam-4831	191	35	)	)	PUNCT
ejpam-4831	191	36	,	,	PUNCT
ejpam-4831	191	37	for	for	ADP
ejpam-4831	191	38	all	all	DET
ejpam-4831	191	39	k	k	NOUN
ejpam-4831	191	40	,	,	PUNCT
ejpam-4831	191	41	and	and	CCONJ
ejpam-4831	191	42	,	,	PUNCT
ejpam-4831	191	43	by	by	ADP
ejpam-4831	191	44	(	(	PUNCT
ejpam-4831	191	45	2	2	NUM
ejpam-4831	191	46	)	)	PUNCT
ejpam-4831	191	47	,	,	PUNCT
ejpam-4831	191	48	∥αk∥s	∥αk∥s	X
ejpam-4831	191	49	≤	≤	NUM
ejpam-4831	191	50	(	(	PUNCT
ejpam-4831	191	51	supp	supp	PROPN
ejpam-4831	191	52	∥ap∥)∥(∥xn∥a)n∥s	∥ap∥)∥(∥xn∥a)n∥s	PROPN
ejpam-4831	191	53	and	and	CCONJ
ejpam-4831	191	54	then	then	ADV
ejpam-4831	191	55	supk	supk	PROPN
ejpam-4831	191	56	∥αk∥s	∥αk∥s	PROPN
ejpam-4831	191	57	is	be	AUX
ejpam-4831	191	58	finite	finite	ADJ
ejpam-4831	191	59	.	.	PUNCT
ejpam-4831	192	1	then	then	ADV
ejpam-4831	192	2	,	,	PUNCT
ejpam-4831	192	3	r∑	r∑	ADP
ejpam-4831	192	4	k=1	k=1	PUNCT
ejpam-4831	192	5	n1(εkα	n1(εkα	PROPN
ejpam-4831	192	6	k	k	PROPN
ejpam-4831	192	7	⊗	⊗	PROPN
ejpam-4831	192	8	yk	yk	PROPN
ejpam-4831	192	9	)	)	PUNCT
ejpam-4831	193	1	=	=	PUNCT
ejpam-4831	194	1	r∑	r∑	NOUN
ejpam-4831	194	2	k=1	k=1	VERB
ejpam-4831	194	3	|εk|∥αk∥s∥yk∥b	|εk|∥αk∥s∥yk∥b	ADJ
ejpam-4831	194	4	≤	≤	NOUN
ejpam-4831	194	5	(	(	PUNCT
ejpam-4831	194	6	sup	sup	NOUN
ejpam-4831	194	7	p	p	NOUN
ejpam-4831	194	8	∥ap∥)(sup	∥ap∥)(sup	NOUN
ejpam-4831	194	9	p	p	NOUN
ejpam-4831	194	10	∥yp∥)n(x	∥yp∥)n(x	ADJ
ejpam-4831	194	11	)	)	PUNCT
ejpam-4831	195	1	r∑	r∑	NOUN
ejpam-4831	196	1	k=1	k=1	PROPN
ejpam-4831	196	2	εk	εk	NOUN
ejpam-4831	196	3	.	.	PUNCT
ejpam-4831	197	1	(	(	PUNCT
ejpam-4831	197	2	3	3	NUM
ejpam-4831	197	3	)	)	PUNCT
ejpam-4831	197	4	as	as	ADP
ejpam-4831	197	5	,	,	PUNCT
ejpam-4831	197	6	λs(eb	λs(eb	PROPN
ejpam-4831	197	7	)	)	PUNCT
ejpam-4831	197	8	is	be	AUX
ejpam-4831	197	9	a	a	DET
ejpam-4831	197	10	complete	complete	ADJ
ejpam-4831	197	11	normed	normed	ADJ
ejpam-4831	197	12	spaces	space	NOUN
ejpam-4831	197	13	,	,	PUNCT
ejpam-4831	197	14	the	the	DET
ejpam-4831	197	15	series	series	NOUN
ejpam-4831	197	16	∑∞	∑∞	PROPN
ejpam-4831	197	17	k=1	k=1	X
ejpam-4831	197	18	εkα	εkα	PROPN
ejpam-4831	197	19	k	k	PROPN
ejpam-4831	197	20	⊗	⊗	PROPN
ejpam-4831	197	21	yk	yk	PROPN
ejpam-4831	197	22	converges	converge	VERB
ejpam-4831	197	23	in	in	ADP
ejpam-4831	197	24	λs(eb	λs(eb	NOUN
ejpam-4831	197	25	)	)	PUNCT
ejpam-4831	197	26	to	to	ADP
ejpam-4831	197	27	a	a	DET
ejpam-4831	197	28	limit	limit	NOUN
ejpam-4831	197	29	g(x	g(x	NOUN
ejpam-4831	197	30	)	)	PUNCT
ejpam-4831	197	31	.	.	PUNCT
ejpam-4831	198	1	moreover	moreover	ADV
ejpam-4831	198	2	,	,	PUNCT
ejpam-4831	198	3	ℓ̃(g(x	ℓ̃(g(x	NOUN
ejpam-4831	198	4	)	)	PUNCT
ejpam-4831	198	5	)	)	PUNCT
ejpam-4831	199	1	=	=	PUNCT
ejpam-4831	199	2	x.	x.	NOUN
ejpam-4831	199	3	(	(	PUNCT
ejpam-4831	199	4	4	4	NUM
ejpam-4831	199	5	)	)	PUNCT
ejpam-4831	199	6	indeed	indeed	ADV
ejpam-4831	199	7	,	,	PUNCT
ejpam-4831	199	8	if	if	SCONJ
ejpam-4831	199	9	z	z	NOUN
ejpam-4831	199	10	=	=	SYM
ejpam-4831	199	11	(	(	PUNCT
ejpam-4831	199	12	zn)n	zn)n	PROPN
ejpam-4831	199	13	∈	∈	PROPN
ejpam-4831	199	14	λs(eb	λs(eb	PROPN
ejpam-4831	199	15	)	)	PUNCT
ejpam-4831	199	16	is	be	AUX
ejpam-4831	199	17	such	such	ADJ
ejpam-4831	199	18	that	that	SCONJ
ejpam-4831	199	19	z	z	NOUN
ejpam-4831	199	20	=	=	SYM
ejpam-4831	199	21	ℓ̃(g(x	ℓ̃(g(x	NUM
ejpam-4831	199	22	)	)	PUNCT
ejpam-4831	199	23	)	)	PUNCT
ejpam-4831	199	24	,	,	PUNCT
ejpam-4831	199	25	then	then	ADV
ejpam-4831	199	26	z	z	NOUN
ejpam-4831	199	27	=	=	PUNCT
ejpam-4831	199	28	(	(	PUNCT
ejpam-4831	199	29	zn)n	zn)n	PROPN
ejpam-4831	199	30	=	=	SYM
ejpam-4831	199	31	ℓ̃	ℓ̃	PROPN
ejpam-4831	199	32	(	(	PUNCT
ejpam-4831	199	33	∞∑	∞∑	NUM
ejpam-4831	199	34	k=1	k=1	PUNCT
ejpam-4831	199	35	εk(ak(xn))n	εk(ak(xn))n	PROPN
ejpam-4831	199	36	⊗	⊗	PROPN
ejpam-4831	199	37	yk	yk	PROPN
ejpam-4831	199	38	)	)	PUNCT
ejpam-4831	200	1	=	=	PUNCT
ejpam-4831	201	1	∞∑	∞∑	NUM
ejpam-4831	201	2	k=1	k=1	PUNCT
ejpam-4831	201	3	εk	εk	PROPN
ejpam-4831	201	4	ℓ̃((ak(xn))n	ℓ̃((ak(xn))n	PROPN
ejpam-4831	201	5	⊗	⊗	PROPN
ejpam-4831	201	6	yk	yk	PROPN
ejpam-4831	201	7	)	)	PUNCT
ejpam-4831	201	8	m.	m.	NOUN
ejpam-4831	201	9	a.	a.	PROPN
ejpam-4831	201	10	sidaty	sidaty	PROPN
ejpam-4831	201	11	/	/	SYM
ejpam-4831	201	12	eur	eur	PROPN
ejpam-4831	201	13	.	.	PUNCT
ejpam-4831	202	1	j.	j.	PROPN
ejpam-4831	202	2	pure	pure	PROPN
ejpam-4831	202	3	appl	appl	PROPN
ejpam-4831	202	4	.	.	PROPN
ejpam-4831	202	5	math	math	PROPN
ejpam-4831	202	6	,	,	PUNCT
ejpam-4831	202	7	16	16	NUM
ejpam-4831	202	8	(	(	PUNCT
ejpam-4831	202	9	3	3	NUM
ejpam-4831	202	10	)	)	PUNCT
ejpam-4831	202	11	(	(	PUNCT
ejpam-4831	202	12	2023	2023	NUM
ejpam-4831	202	13	)	)	PUNCT
ejpam-4831	202	14	,	,	PUNCT
ejpam-4831	202	15	1762	1762	NUM
ejpam-4831	202	16	-	-	SYM
ejpam-4831	202	17	1771	1771	NUM
ejpam-4831	202	18	1768	1768	NUM
ejpam-4831	202	19	=	=	PUNCT
ejpam-4831	203	1	∞∑	∞∑	NUM
ejpam-4831	203	2	k=1	k=1	NOUN
ejpam-4831	203	3	εkℓ((ak(xn))n	εkℓ((ak(xn))n	X
ejpam-4831	203	4	⊗	⊗	PROPN
ejpam-4831	203	5	yk	yk	PROPN
ejpam-4831	203	6	)	)	PUNCT
ejpam-4831	204	1	=	=	PUNCT
ejpam-4831	205	1	∞∑	∞∑	NUM
ejpam-4831	205	2	k=1	k=1	PUNCT
ejpam-4831	205	3	εk(ak(xn)yk)n	εk(ak(xn)yk)n	PROPN
ejpam-4831	205	4	.	.	PUNCT
ejpam-4831	206	1	but	but	CCONJ
ejpam-4831	206	2	the	the	DET
ejpam-4831	206	3	projections	projection	NOUN
ejpam-4831	206	4	are	be	AUX
ejpam-4831	206	5	bounded	bound	VERB
ejpam-4831	206	6	by	by	ADP
ejpam-4831	206	7	lemma	lemma	PROPN
ejpam-4831	206	8	1	1	NUM
ejpam-4831	206	9	,	,	PUNCT
ejpam-4831	206	10	then	then	ADV
ejpam-4831	206	11	zn	zn	PROPN
ejpam-4831	206	12	=	=	PUNCT
ejpam-4831	207	1	∞∑	∞∑	NUM
ejpam-4831	207	2	k=1	k=1	ADP
ejpam-4831	207	3	εkak(xn)yk	εkak(xn)yk	PROPN
ejpam-4831	207	4	,	,	PUNCT
ejpam-4831	207	5	for	for	ADP
ejpam-4831	207	6	all	all	DET
ejpam-4831	207	7	n.	n.	NOUN
ejpam-4831	207	8	by	by	ADP
ejpam-4831	207	9	(	(	PUNCT
ejpam-4831	207	10	1	1	NUM
ejpam-4831	207	11	)	)	PUNCT
ejpam-4831	207	12	,	,	PUNCT
ejpam-4831	207	13	zn	zn	PROPN
ejpam-4831	207	14	=	=	SYM
ejpam-4831	207	15	xn	xn	PROPN
ejpam-4831	207	16	,	,	PUNCT
ejpam-4831	207	17	for	for	ADP
ejpam-4831	207	18	all	all	DET
ejpam-4831	207	19	n	n	CCONJ
ejpam-4831	207	20	,	,	PUNCT
ejpam-4831	207	21	and	and	CCONJ
ejpam-4831	207	22	ℓ̃(g(x	ℓ̃(g(x	NUM
ejpam-4831	207	23	)	)	PUNCT
ejpam-4831	207	24	)	)	PUNCT
ejpam-4831	208	1	=	=	PUNCT
ejpam-4831	209	1	x.	x.	NOUN
ejpam-4831	209	2	this	this	PRON
ejpam-4831	209	3	means	mean	VERB
ejpam-4831	209	4	that	that	SCONJ
ejpam-4831	209	5	ℓ̃	ℓ̃	PROPN
ejpam-4831	209	6	is	be	AUX
ejpam-4831	209	7	onto	onto	ADP
ejpam-4831	209	8	.	.	PUNCT
ejpam-4831	210	1	in	in	ADP
ejpam-4831	210	2	the	the	DET
ejpam-4831	210	3	other	other	ADJ
ejpam-4831	210	4	hand	hand	NOUN
ejpam-4831	210	5	,	,	PUNCT
ejpam-4831	210	6	if	if	SCONJ
ejpam-4831	210	7	k	k	PROPN
ejpam-4831	210	8	is	be	AUX
ejpam-4831	210	9	bounded	bound	VERB
ejpam-4831	210	10	in	in	ADP
ejpam-4831	210	11	λ(e	λ(e	PROPN
ejpam-4831	210	12	)	)	PUNCT
ejpam-4831	210	13	,	,	PUNCT
ejpam-4831	210	14	then	then	ADV
ejpam-4831	210	15	k	k	PROPN
ejpam-4831	210	16	is	be	AUX
ejpam-4831	210	17	contained	contain	VERB
ejpam-4831	210	18	and	and	CCONJ
ejpam-4831	210	19	bounded	bound	VERB
ejpam-4831	210	20	in	in	ADP
ejpam-4831	210	21	some	some	DET
ejpam-4831	210	22	λs(eb	λs(eb	NOUN
ejpam-4831	210	23	)	)	PUNCT
ejpam-4831	210	24	,	,	PUNCT
ejpam-4831	210	25	and	and	CCONJ
ejpam-4831	210	26	ℓ̃(g(k	ℓ̃(g(k	NOUN
ejpam-4831	210	27	)	)	PUNCT
ejpam-4831	210	28	)	)	PUNCT
ejpam-4831	211	1	=	=	SYM
ejpam-4831	212	1	k	k	NOUN
ejpam-4831	212	2	,	,	PUNCT
ejpam-4831	212	3	from	from	ADP
ejpam-4831	212	4	what	what	PRON
ejpam-4831	212	5	,	,	PUNCT
ejpam-4831	212	6	we	we	PRON
ejpam-4831	212	7	conclude	conclude	VERB
ejpam-4831	212	8	that	that	SCONJ
ejpam-4831	212	9	the	the	DET
ejpam-4831	212	10	inverse	inverse	NOUN
ejpam-4831	212	11	of	of	ADP
ejpam-4831	212	12	ℓ̃	ℓ̃	PROPN
ejpam-4831	212	13	is	be	AUX
ejpam-4831	212	14	bounded	bound	VERB
ejpam-4831	212	15	.	.	PUNCT
ejpam-4831	213	1	■	■	PUNCT
ejpam-4831	213	2	we	we	PRON
ejpam-4831	213	3	are	be	AUX
ejpam-4831	213	4	now	now	ADV
ejpam-4831	213	5	ready	ready	ADJ
ejpam-4831	213	6	to	to	PART
ejpam-4831	213	7	prove	prove	VERB
ejpam-4831	213	8	the	the	DET
ejpam-4831	213	9	main	main	ADJ
ejpam-4831	213	10	result	result	NOUN
ejpam-4831	213	11	of	of	ADP
ejpam-4831	213	12	this	this	DET
ejpam-4831	213	13	section	section	NOUN
ejpam-4831	213	14	.	.	PUNCT
ejpam-4831	214	1	theorem	theorem	NOUN
ejpam-4831	214	2	2	2	NUM
ejpam-4831	214	3	.	.	PUNCT
ejpam-4831	215	1	let	let	VERB
ejpam-4831	215	2	e	e	PRON
ejpam-4831	215	3	be	be	AUX
ejpam-4831	215	4	a	a	DET
ejpam-4831	215	5	complete	complete	ADJ
ejpam-4831	215	6	b	b	NOUN
ejpam-4831	215	7	-	-	PUNCT
ejpam-4831	215	8	space	space	NOUN
ejpam-4831	215	9	and	and	CCONJ
ejpam-4831	215	10	λ	λ	PROPN
ejpam-4831	215	11	be	be	AUX
ejpam-4831	215	12	a	a	DET
ejpam-4831	215	13	normal	normal	ADJ
ejpam-4831	215	14	sequence	sequence	NOUN
ejpam-4831	215	15	space	space	NOUN
ejpam-4831	215	16	.	.	PUNCT
ejpam-4831	216	1	then	then	ADV
ejpam-4831	216	2	λ(e	λ(e	PROPN
ejpam-4831	216	3	)	)	PUNCT
ejpam-4831	216	4	is	be	AUX
ejpam-4831	216	5	nuclear	nuclear	ADJ
ejpam-4831	216	6	if	if	SCONJ
ejpam-4831	216	7	and	and	CCONJ
ejpam-4831	216	8	only	only	ADV
ejpam-4831	216	9	if	if	SCONJ
ejpam-4831	216	10	λ	λ	PROPN
ejpam-4831	216	11	and	and	CCONJ
ejpam-4831	216	12	e	e	NOUN
ejpam-4831	216	13	are	be	AUX
ejpam-4831	216	14	nuclear	nuclear	ADJ
ejpam-4831	216	15	.	.	PUNCT
ejpam-4831	217	1	proof	proof	NOUN
ejpam-4831	217	2	.	.	PUNCT
ejpam-4831	218	1	if	if	SCONJ
ejpam-4831	218	2	λ(e	λ(e	VERB
ejpam-4831	218	3	)	)	PUNCT
ejpam-4831	218	4	is	be	AUX
ejpam-4831	218	5	nuclear	nuclear	ADJ
ejpam-4831	218	6	then	then	ADV
ejpam-4831	218	7	,	,	PUNCT
ejpam-4831	218	8	by	by	ADP
ejpam-4831	218	9	proposition	proposition	NOUN
ejpam-4831	218	10	1	1	NUM
ejpam-4831	218	11	,	,	PUNCT
ejpam-4831	218	12	e	e	NOUN
ejpam-4831	218	13	and	and	CCONJ
ejpam-4831	218	14	λ	λ	NOUN
ejpam-4831	218	15	are	be	AUX
ejpam-4831	218	16	closed	close	VERB
ejpam-4831	218	17	subspaces	subspace	NOUN
ejpam-4831	218	18	of	of	ADP
ejpam-4831	218	19	λ(e	λ(e	ADJ
ejpam-4831	218	20	)	)	PUNCT
ejpam-4831	218	21	and	and	CCONJ
ejpam-4831	218	22	then	then	ADV
ejpam-4831	218	23	they	they	PRON
ejpam-4831	218	24	are	be	AUX
ejpam-4831	218	25	nuclear	nuclear	ADJ
ejpam-4831	218	26	also	also	ADV
ejpam-4831	218	27	.	.	PUNCT
ejpam-4831	219	1	inversely	inversely	ADV
ejpam-4831	219	2	,	,	PUNCT
ejpam-4831	219	3	suppose	suppose	VERB
ejpam-4831	219	4	that	that	SCONJ
ejpam-4831	219	5	e	e	PROPN
ejpam-4831	219	6	and	and	CCONJ
ejpam-4831	219	7	λ	λ	PROPN
ejpam-4831	219	8	are	be	AUX
ejpam-4831	219	9	nuclear	nuclear	ADJ
ejpam-4831	219	10	.	.	PUNCT
ejpam-4831	220	1	by	by	ADP
ejpam-4831	220	2	proposition	proposition	NOUN
ejpam-4831	220	3	4	4	NUM
ejpam-4831	220	4	,	,	PUNCT
ejpam-4831	220	5	λ⊗̃be	λ⊗̃be	NOUN
ejpam-4831	220	6	is	be	AUX
ejpam-4831	220	7	nuclear	nuclear	ADJ
ejpam-4831	220	8	.	.	PUNCT
ejpam-4831	221	1	so	so	ADV
ejpam-4831	221	2	by	by	ADP
ejpam-4831	221	3	theorem	theorem	NOUN
ejpam-4831	221	4	1	1	NUM
ejpam-4831	221	5	,	,	PUNCT
ejpam-4831	221	6	λ(e	λ(e	PROPN
ejpam-4831	221	7	)	)	PUNCT
ejpam-4831	221	8	is	be	AUX
ejpam-4831	221	9	nuclear	nuclear	ADJ
ejpam-4831	221	10	.	.	PUNCT
ejpam-4831	222	1	■	■	PUNCT
ejpam-4831	222	2	theorem	theorem	ADJ
ejpam-4831	222	3	3	3	X
ejpam-4831	222	4	.	.	PUNCT
ejpam-4831	223	1	let	let	VERB
ejpam-4831	223	2	e	e	PRON
ejpam-4831	223	3	be	be	AUX
ejpam-4831	223	4	a	a	DET
ejpam-4831	223	5	complete	complete	ADJ
ejpam-4831	223	6	b	b	NOUN
ejpam-4831	223	7	-	-	PUNCT
ejpam-4831	223	8	space	space	NOUN
ejpam-4831	223	9	and	and	CCONJ
ejpam-4831	223	10	λ	λ	PROPN
ejpam-4831	223	11	be	be	AUX
ejpam-4831	223	12	a	a	DET
ejpam-4831	223	13	normal	normal	ADJ
ejpam-4831	223	14	sequence	sequence	NOUN
ejpam-4831	223	15	space	space	NOUN
ejpam-4831	223	16	.	.	PUNCT
ejpam-4831	224	1	(	(	PUNCT
ejpam-4831	224	2	i	i	NOUN
ejpam-4831	224	3	)	)	PUNCT
ejpam-4831	224	4	if	if	SCONJ
ejpam-4831	224	5	λ	λ	NOUN
ejpam-4831	224	6	is	be	AUX
ejpam-4831	224	7	nuclear	nuclear	ADJ
ejpam-4831	224	8	then	then	ADV
ejpam-4831	224	9	,	,	PUNCT
ejpam-4831	224	10	λ(e	λ(e	PROPN
ejpam-4831	224	11	)	)	PUNCT
ejpam-4831	224	12	is	be	AUX
ejpam-4831	224	13	a	a	DET
ejpam-4831	224	14	schwartz	schwartz	NOUN
ejpam-4831	224	15	space	space	NOUN
ejpam-4831	224	16	if	if	SCONJ
ejpam-4831	224	17	and	and	CCONJ
ejpam-4831	224	18	only	only	ADV
ejpam-4831	224	19	if	if	SCONJ
ejpam-4831	224	20	e	e	NOUN
ejpam-4831	224	21	is	be	AUX
ejpam-4831	224	22	a	a	DET
ejpam-4831	224	23	schwartz	schwartz	NOUN
ejpam-4831	224	24	space	space	NOUN
ejpam-4831	224	25	.	.	PUNCT
ejpam-4831	225	1	(	(	PUNCT
ejpam-4831	225	2	ii	ii	NOUN
ejpam-4831	225	3	)	)	PUNCT
ejpam-4831	225	4	if	if	SCONJ
ejpam-4831	225	5	e	e	PRON
ejpam-4831	225	6	is	be	AUX
ejpam-4831	225	7	nuclear	nuclear	ADJ
ejpam-4831	225	8	then	then	ADV
ejpam-4831	225	9	,	,	PUNCT
ejpam-4831	225	10	λ(e	λ(e	PROPN
ejpam-4831	225	11	)	)	PUNCT
ejpam-4831	225	12	is	be	AUX
ejpam-4831	225	13	a	a	DET
ejpam-4831	225	14	schwartz	schwartz	NOUN
ejpam-4831	225	15	space	space	NOUN
ejpam-4831	225	16	if	if	SCONJ
ejpam-4831	225	17	and	and	CCONJ
ejpam-4831	225	18	only	only	ADV
ejpam-4831	225	19	if	if	SCONJ
ejpam-4831	225	20	λ	λ	NOUN
ejpam-4831	225	21	is	be	AUX
ejpam-4831	225	22	a	a	DET
ejpam-4831	225	23	schwartz	schwartz	NOUN
ejpam-4831	225	24	space	space	NOUN
ejpam-4831	225	25	.	.	PUNCT
ejpam-4831	226	1	proof	proof	NOUN
ejpam-4831	226	2	.	.	PUNCT
ejpam-4831	227	1	suppose	suppose	VERB
ejpam-4831	227	2	that	that	SCONJ
ejpam-4831	227	3	e	e	PROPN
ejpam-4831	227	4	is	be	AUX
ejpam-4831	227	5	nuclear	nuclear	ADJ
ejpam-4831	227	6	.	.	PUNCT
ejpam-4831	228	1	if	if	SCONJ
ejpam-4831	228	2	λ(e	λ(e	VERB
ejpam-4831	228	3	)	)	PUNCT
ejpam-4831	228	4	is	be	AUX
ejpam-4831	228	5	a	a	DET
ejpam-4831	228	6	schwartz	schwartz	PROPN
ejpam-4831	228	7	space	space	NOUN
ejpam-4831	228	8	,	,	PUNCT
ejpam-4831	228	9	then	then	ADV
ejpam-4831	228	10	λ	λ	X
ejpam-4831	228	11	,	,	PUNCT
ejpam-4831	228	12	being	be	AUX
ejpam-4831	228	13	a	a	DET
ejpam-4831	228	14	closed	closed	ADJ
ejpam-4831	228	15	subspace	subspace	NOUN
ejpam-4831	228	16	of	of	ADP
ejpam-4831	228	17	λ(e	λ(e	PROPN
ejpam-4831	228	18	)	)	PUNCT
ejpam-4831	228	19	by	by	ADP
ejpam-4831	228	20	proposition	proposition	NOUN
ejpam-4831	228	21	1	1	NUM
ejpam-4831	228	22	,	,	PUNCT
ejpam-4831	228	23	is	be	AUX
ejpam-4831	228	24	a	a	DET
ejpam-4831	228	25	schwartz	schwartz	NOUN
ejpam-4831	228	26	space	space	NOUN
ejpam-4831	228	27	.	.	PUNCT
ejpam-4831	229	1	inversely	inversely	ADV
ejpam-4831	229	2	,	,	PUNCT
ejpam-4831	229	3	suppose	suppose	VERB
ejpam-4831	229	4	that	that	SCONJ
ejpam-4831	229	5	e	e	PROPN
ejpam-4831	229	6	is	be	AUX
ejpam-4831	229	7	nuclear	nuclear	ADJ
ejpam-4831	229	8	and	and	CCONJ
ejpam-4831	229	9	λ	λ	PROPN
ejpam-4831	229	10	is	be	AUX
ejpam-4831	229	11	a	a	DET
ejpam-4831	229	12	schwartz	schwartz	NOUN
ejpam-4831	229	13	space	space	NOUN
ejpam-4831	229	14	.	.	PUNCT
ejpam-4831	230	1	let	let	VERB
ejpam-4831	230	2	a	a	DET
ejpam-4831	230	3	∈	∈	PROPN
ejpam-4831	230	4	b	b	NOUN
ejpam-4831	230	5	and	and	CCONJ
ejpam-4831	230	6	s	s	PROPN
ejpam-4831	230	7	∈	∈	NOUN
ejpam-4831	230	8	s	s	AUX
ejpam-4831	230	9	be	be	AUX
ejpam-4831	230	10	a	a	DET
ejpam-4831	230	11	banach	banach	NOUN
ejpam-4831	230	12	disks	disk	NOUN
ejpam-4831	230	13	in	in	ADP
ejpam-4831	230	14	e	e	NOUN
ejpam-4831	230	15	and	and	CCONJ
ejpam-4831	230	16	λ	λ	X
ejpam-4831	230	17	respectively	respectively	ADV
ejpam-4831	230	18	.	.	PUNCT
ejpam-4831	231	1	since	since	SCONJ
ejpam-4831	231	2	e	e	PROPN
ejpam-4831	231	3	is	be	AUX
ejpam-4831	231	4	nuclear	nuclear	ADJ
ejpam-4831	231	5	we	we	PRON
ejpam-4831	231	6	can	can	AUX
ejpam-4831	231	7	select	select	VERB
ejpam-4831	231	8	a	a	DET
ejpam-4831	231	9	banach	banach	NOUN
ejpam-4831	231	10	disk	disk	NOUN
ejpam-4831	231	11	b	b	PROPN
ejpam-4831	231	12	∈	∈	PROPN
ejpam-4831	231	13	b	b	NOUN
ejpam-4831	231	14	containing	contain	VERB
ejpam-4831	231	15	a	a	DET
ejpam-4831	231	16	such	such	ADJ
ejpam-4831	231	17	that	that	SCONJ
ejpam-4831	231	18	the	the	DET
ejpam-4831	231	19	inclusion	inclusion	NOUN
ejpam-4831	231	20	ea	ea	PROPN
ejpam-4831	231	21	→	→	SYM
ejpam-4831	231	22	eb	eb	PROPN
ejpam-4831	231	23	is	be	AUX
ejpam-4831	231	24	nuclear	nuclear	ADJ
ejpam-4831	231	25	.	.	PUNCT
ejpam-4831	232	1	so	so	ADV
ejpam-4831	232	2	,	,	PUNCT
ejpam-4831	232	3	there	there	PRON
ejpam-4831	232	4	are	be	VERB
ejpam-4831	232	5	(	(	PUNCT
ejpam-4831	232	6	εk)k	εk)k	PROPN
ejpam-4831	232	7	∈	∈	PROPN
ejpam-4831	232	8	ℓ1	ℓ1	NOUN
ejpam-4831	232	9	,	,	PUNCT
ejpam-4831	232	10	a	a	DET
ejpam-4831	232	11	bounded	bounded	ADJ
ejpam-4831	232	12	sequence	sequence	NOUN
ejpam-4831	232	13	(	(	PUNCT
ejpam-4831	232	14	ak)k	ak)k	PROPN
ejpam-4831	232	15	in	in	ADP
ejpam-4831	232	16	the	the	DET
ejpam-4831	232	17	continuous	continuous	ADJ
ejpam-4831	232	18	dual	dual	ADJ
ejpam-4831	232	19	(	(	PUNCT
ejpam-4831	232	20	ea	ea	NOUN
ejpam-4831	232	21	)	)	PUNCT
ejpam-4831	232	22	′	′	NUM
ejpam-4831	232	23	of	of	ADP
ejpam-4831	232	24	ea	ea	NOUN
ejpam-4831	232	25	and	and	CCONJ
ejpam-4831	232	26	a	a	DET
ejpam-4831	232	27	bounded	bounded	ADJ
ejpam-4831	232	28	sequence	sequence	NOUN
ejpam-4831	232	29	(	(	PUNCT
ejpam-4831	232	30	yk)k	yk)k	PROPN
ejpam-4831	232	31	⊂	⊂	PROPN
ejpam-4831	232	32	eb	eb	PROPN
ejpam-4831	232	33	such	such	ADJ
ejpam-4831	232	34	that	that	SCONJ
ejpam-4831	232	35	x	x	X
ejpam-4831	233	1	=	=	SYM
ejpam-4831	233	2	∞∑	∞∑	NUM
ejpam-4831	233	3	k=1	k=1	ADJ
ejpam-4831	233	4	εkak(x)yk	εkak(x)yk	NOUN
ejpam-4831	233	5	,	,	PUNCT
ejpam-4831	233	6	for	for	ADP
ejpam-4831	233	7	all	all	DET
ejpam-4831	233	8	x	x	SYM
ejpam-4831	233	9	∈	∈	PROPN
ejpam-4831	233	10	ea	ea	PROPN
ejpam-4831	233	11	.	.	PUNCT
ejpam-4831	234	1	(	(	PUNCT
ejpam-4831	234	2	5	5	NUM
ejpam-4831	234	3	)	)	PUNCT
ejpam-4831	234	4	since	since	SCONJ
ejpam-4831	234	5	λ	λ	PROPN
ejpam-4831	234	6	is	be	AUX
ejpam-4831	234	7	a	a	DET
ejpam-4831	234	8	schwartz	schwartz	NOUN
ejpam-4831	234	9	space	space	NOUN
ejpam-4831	234	10	,	,	PUNCT
ejpam-4831	234	11	there	there	PRON
ejpam-4831	234	12	is	be	VERB
ejpam-4831	234	13	a	a	DET
ejpam-4831	234	14	banach	banach	NOUN
ejpam-4831	234	15	disk	disk	NOUN
ejpam-4831	234	16	t	t	NOUN
ejpam-4831	234	17	in	in	ADP
ejpam-4831	234	18	λ	λ	PROPN
ejpam-4831	234	19	such	such	ADJ
ejpam-4831	234	20	that	that	SCONJ
ejpam-4831	234	21	the	the	DET
ejpam-4831	234	22	injection	injection	NOUN
ejpam-4831	234	23	λs	λs	NOUN
ejpam-4831	234	24	→	→	SYM
ejpam-4831	234	25	λt	λt	X
ejpam-4831	234	26	is	be	AUX
ejpam-4831	234	27	compact	compact	ADJ
ejpam-4831	234	28	.	.	PUNCT
ejpam-4831	235	1	we	we	PRON
ejpam-4831	235	2	will	will	AUX
ejpam-4831	235	3	show	show	VERB
ejpam-4831	235	4	that	that	SCONJ
ejpam-4831	235	5	the	the	DET
ejpam-4831	235	6	injection	injection	NOUN
ejpam-4831	235	7	λs(ea	λs(ea	NOUN
ejpam-4831	235	8	)	)	PUNCT
ejpam-4831	235	9	→	→	SYM
ejpam-4831	235	10	λt	λt	X
ejpam-4831	235	11	(	(	PUNCT
ejpam-4831	235	12	eb	eb	PROPN
ejpam-4831	235	13	)	)	PUNCT
ejpam-4831	235	14	is	be	AUX
ejpam-4831	235	15	compact	compact	ADJ
ejpam-4831	235	16	.	.	PUNCT
ejpam-4831	236	1	let	let	VERB
ejpam-4831	236	2	{	{	PUNCT
ejpam-4831	236	3	xi	xi	PROPN
ejpam-4831	236	4	=	=	SYM
ejpam-4831	236	5	(	(	PUNCT
ejpam-4831	236	6	xin)n}∞i=1	xin)n}∞i=1	X
ejpam-4831	236	7	(	(	PUNCT
ejpam-4831	236	8	6	6	NUM
ejpam-4831	236	9	)	)	PUNCT
ejpam-4831	236	10	m.	m.	NOUN
ejpam-4831	236	11	a.	a.	NOUN
ejpam-4831	236	12	sidaty	sidaty	PROPN
ejpam-4831	236	13	/	/	SYM
ejpam-4831	236	14	eur	eur	PROPN
ejpam-4831	236	15	.	.	PUNCT
ejpam-4831	237	1	j.	j.	PROPN
ejpam-4831	237	2	pure	pure	PROPN
ejpam-4831	237	3	appl	appl	PROPN
ejpam-4831	237	4	.	.	PROPN
ejpam-4831	237	5	math	math	PROPN
ejpam-4831	237	6	,	,	PUNCT
ejpam-4831	237	7	16	16	NUM
ejpam-4831	237	8	(	(	PUNCT
ejpam-4831	237	9	3	3	NUM
ejpam-4831	237	10	)	)	PUNCT
ejpam-4831	237	11	(	(	PUNCT
ejpam-4831	237	12	2023	2023	NUM
ejpam-4831	237	13	)	)	PUNCT
ejpam-4831	237	14	,	,	PUNCT
ejpam-4831	237	15	1762	1762	NUM
ejpam-4831	237	16	-	-	SYM
ejpam-4831	237	17	1771	1771	NUM
ejpam-4831	237	18	1769	1769	NUM
ejpam-4831	237	19	be	be	AUX
ejpam-4831	237	20	a	a	DET
ejpam-4831	237	21	sequence	sequence	NOUN
ejpam-4831	237	22	in	in	ADP
ejpam-4831	237	23	s(a	s(a	PROPN
ejpam-4831	237	24	)	)	PUNCT
ejpam-4831	237	25	.	.	PUNCT
ejpam-4831	238	1	by	by	ADP
ejpam-4831	238	2	(	(	PUNCT
ejpam-4831	238	3	5	5	NUM
ejpam-4831	238	4	)	)	PUNCT
ejpam-4831	238	5	,	,	PUNCT
ejpam-4831	238	6	we	we	PRON
ejpam-4831	238	7	have	have	VERB
ejpam-4831	238	8	xin	xin	PROPN
ejpam-4831	238	9	=	=	X
ejpam-4831	239	1	∞∑	∞∑	NUM
ejpam-4831	239	2	k=1	k=1	PUNCT
ejpam-4831	239	3	εkak(x	εkak(x	INTJ
ejpam-4831	239	4	i	i	PRON
ejpam-4831	239	5	n)yk	n)yk	PROPN
ejpam-4831	239	6	,	,	PUNCT
ejpam-4831	239	7	for	for	ADP
ejpam-4831	239	8	all	all	DET
ejpam-4831	239	9	n	n	NOUN
ejpam-4831	239	10	,	,	PUNCT
ejpam-4831	239	11	i.	i.	PROPN
ejpam-4831	239	12	(	(	PUNCT
ejpam-4831	239	13	7	7	NUM
ejpam-4831	239	14	)	)	PUNCT
ejpam-4831	239	15	the	the	DET
ejpam-4831	239	16	sequence	sequence	NOUN
ejpam-4831	239	17	(	(	PUNCT
ejpam-4831	239	18	ak)k	ak)k	PROPN
ejpam-4831	239	19	being	be	AUX
ejpam-4831	239	20	bounded	bound	VERB
ejpam-4831	239	21	in	in	ADP
ejpam-4831	239	22	(	(	PUNCT
ejpam-4831	239	23	ea	ea	NOUN
ejpam-4831	239	24	)	)	PUNCT
ejpam-4831	239	25	′	′	NOUN
ejpam-4831	239	26	,	,	PUNCT
ejpam-4831	239	27	there	there	PRON
ejpam-4831	239	28	is	be	VERB
ejpam-4831	239	29	a	a	DET
ejpam-4831	239	30	constant	constant	ADJ
ejpam-4831	239	31	c	c	NOUN
ejpam-4831	239	32	>	>	X
ejpam-4831	239	33	0	0	NUM
ejpam-4831	240	1	such	such	ADJ
ejpam-4831	240	2	that	that	SCONJ
ejpam-4831	240	3	|ak(xin)|	|ak(xin)|	ADJ
ejpam-4831	240	4	≤	≤	NOUN
ejpam-4831	240	5	c∥xin∥a	c∥xin∥a	X
ejpam-4831	240	6	for	for	ADP
ejpam-4831	240	7	all	all	DET
ejpam-4831	240	8	i	i	PROPN
ejpam-4831	240	9	,	,	PUNCT
ejpam-4831	240	10	k	k	PROPN
ejpam-4831	240	11	,	,	PUNCT
ejpam-4831	240	12	n.	n.	NOUN
ejpam-4831	240	13	this	this	PRON
ejpam-4831	240	14	means	mean	VERB
ejpam-4831	240	15	that	that	SCONJ
ejpam-4831	240	16	{	{	PUNCT
ejpam-4831	240	17	(	(	PUNCT
ejpam-4831	240	18	ak(xin))n}∞i=1	ak(xin))n}∞i=1	X
ejpam-4831	240	19	⊂	⊂	NOUN
ejpam-4831	240	20	λs	λs	NOUN
ejpam-4831	240	21	and	and	CCONJ
ejpam-4831	240	22	that	that	SCONJ
ejpam-4831	240	23	{	{	PUNCT
ejpam-4831	240	24	(	(	PUNCT
ejpam-4831	240	25	ak(xin))n}∞i=1	ak(xin))n}∞i=1	NUM
ejpam-4831	240	26	⊂	⊂	PROPN
ejpam-4831	240	27	cs	cs	PROPN
ejpam-4831	240	28	.	.	PROPN
ejpam-4831	241	1	(	(	PUNCT
ejpam-4831	241	2	8)	8)	NUM
ejpam-4831	241	3	a	a	DET
ejpam-4831	241	4	subsequence	subsequence	NOUN
ejpam-4831	241	5	{	{	PUNCT
ejpam-4831	241	6	(	(	PUNCT
ejpam-4831	241	7	ak(xjn))n}∞j=1	ak(xjn))n}∞j=1	PROPN
ejpam-4831	241	8	of	of	ADP
ejpam-4831	241	9	{	{	PUNCT
ejpam-4831	241	10	(	(	PUNCT
ejpam-4831	241	11	ak(xin))n}∞i=1	ak(xin))n}∞i=1	PRON
ejpam-4831	241	12	should	should	AUX
ejpam-4831	241	13	converge	converge	VERB
ejpam-4831	241	14	in	in	ADP
ejpam-4831	241	15	λt	λt	ADP
ejpam-4831	241	16	to	to	ADP
ejpam-4831	241	17	αk	αk	NOUN
ejpam-4831	241	18	=	=	NOUN
ejpam-4831	241	19	(	(	PUNCT
ejpam-4831	241	20	αk	αk	NOUN
ejpam-4831	241	21	n)n	n)n	NOUN
ejpam-4831	241	22	.	.	PUNCT
ejpam-4831	242	1	in	in	ADP
ejpam-4831	242	2	the	the	DET
ejpam-4831	242	3	other	other	ADJ
ejpam-4831	242	4	hand	hand	NOUN
ejpam-4831	242	5	,	,	PUNCT
ejpam-4831	242	6	the	the	DET
ejpam-4831	242	7	equation	equation	NOUN
ejpam-4831	242	8	(	(	PUNCT
ejpam-4831	242	9	8)	8)	NUM
ejpam-4831	242	10	shows	show	VERB
ejpam-4831	242	11	that	that	SCONJ
ejpam-4831	242	12	the	the	DET
ejpam-4831	242	13	sequence	sequence	NOUN
ejpam-4831	242	14	{	{	PUNCT
ejpam-4831	242	15	(	(	PUNCT
ejpam-4831	242	16	ak(xjn))n}∞k	ak(xjn))n}∞k	PROPN
ejpam-4831	242	17	,	,	PUNCT
ejpam-4831	242	18	j=1	j=1	PROPN
ejpam-4831	242	19	is	be	AUX
ejpam-4831	242	20	bounded	bound	VERB
ejpam-4831	242	21	in	in	ADP
ejpam-4831	242	22	λs	λs	PROPN
ejpam-4831	242	23	.	.	PUNCT
ejpam-4831	243	1	for	for	ADP
ejpam-4831	243	2	every	every	DET
ejpam-4831	243	3	n	n	PRON
ejpam-4831	243	4	∈	∈	PROPN
ejpam-4831	243	5	n	n	CCONJ
ejpam-4831	243	6	,	,	PUNCT
ejpam-4831	243	7	there	there	ADV
ejpam-4831	243	8	cn	cn	INTJ
ejpam-4831	243	9	>	>	X
ejpam-4831	243	10	0	0	NUM
ejpam-4831	244	1	such	such	ADJ
ejpam-4831	244	2	that	that	PRON
ejpam-4831	244	3	for	for	ADP
ejpam-4831	244	4	all	all	DET
ejpam-4831	244	5	j	j	PROPN
ejpam-4831	244	6	,	,	PUNCT
ejpam-4831	244	7	k	k	PROPN
ejpam-4831	244	8	|ak(xjn)|	|ak(xjn)|	NOUN
ejpam-4831	244	9	≤	≤	PROPN
ejpam-4831	244	10	cn	cn	PROPN
ejpam-4831	245	1	and	and	CCONJ
ejpam-4831	245	2	then	then	ADV
ejpam-4831	245	3	|αk	|αk	ADP
ejpam-4831	245	4	n|	n|	NOUN
ejpam-4831	245	5	≤	≤	X
ejpam-4831	245	6	cn	cn	PROPN
ejpam-4831	245	7	.	.	PUNCT
ejpam-4831	246	1	(	(	PUNCT
ejpam-4831	246	2	9	9	NUM
ejpam-4831	246	3	)	)	PUNCT
ejpam-4831	246	4	for	for	ADP
ejpam-4831	246	5	every	every	DET
ejpam-4831	246	6	n	n	PRON
ejpam-4831	246	7	∈	∈	PROPN
ejpam-4831	246	8	n	n	CCONJ
ejpam-4831	246	9	,	,	PUNCT
ejpam-4831	246	10	since	since	SCONJ
ejpam-4831	246	11	{	{	PUNCT
ejpam-4831	246	12	αk	αk	INTJ
ejpam-4831	246	13	nyk}∞k=1	nyk}∞k=1	PROPN
ejpam-4831	246	14	is	be	AUX
ejpam-4831	246	15	bounded	bound	VERB
ejpam-4831	246	16	in	in	ADP
ejpam-4831	246	17	the	the	DET
ejpam-4831	246	18	complete	complete	ADJ
ejpam-4831	246	19	normed	normed	PROPN
ejpam-4831	246	20	space	space	PROPN
ejpam-4831	246	21	eb	eb	PROPN
ejpam-4831	246	22	,	,	PUNCT
ejpam-4831	246	23	the	the	DET
ejpam-4831	246	24	series	series	NOUN
ejpam-4831	246	25	∑	∑	PROPN
ejpam-4831	246	26	k	k	PROPN
ejpam-4831	246	27	εkα	εkα	PROPN
ejpam-4831	246	28	k	k	PROPN
ejpam-4831	246	29	nyk	nyk	PROPN
ejpam-4831	246	30	converges	converge	NOUN
ejpam-4831	246	31	to	to	ADP
ejpam-4831	246	32	a	a	DET
ejpam-4831	246	33	limit	limit	NOUN
ejpam-4831	246	34	xn	xn	PUNCT
ejpam-4831	247	1	∈	∈	PROPN
ejpam-4831	248	1	eb	eb	PROPN
ejpam-4831	248	2	.	.	PUNCT
ejpam-4831	249	1	let	let	VERB
ejpam-4831	249	2	x	x	PUNCT
ejpam-4831	249	3	=	=	PRON
ejpam-4831	249	4	(	(	PUNCT
ejpam-4831	249	5	xn)n	xn)n	PROPN
ejpam-4831	249	6	.	.	PUNCT
ejpam-4831	250	1	since	since	SCONJ
ejpam-4831	250	2	{	{	PUNCT
ejpam-4831	250	3	(	(	PUNCT
ejpam-4831	250	4	αk	αk	INTJ
ejpam-4831	250	5	n)n}∞k=1	n)n}∞k=1	PROPN
ejpam-4831	250	6	is	be	AUX
ejpam-4831	250	7	bounded	bound	VERB
ejpam-4831	250	8	in	in	ADP
ejpam-4831	250	9	λs	λs	PROPN
ejpam-4831	250	10	and	and	CCONJ
ejpam-4831	250	11	{	{	PUNCT
ejpam-4831	250	12	yk}∞k=1	yk}∞k=1	PRON
ejpam-4831	250	13	is	be	AUX
ejpam-4831	250	14	bounded	bound	VERB
ejpam-4831	250	15	in	in	ADP
ejpam-4831	250	16	eb	eb	PROPN
ejpam-4831	250	17	,	,	PUNCT
ejpam-4831	250	18	the	the	DET
ejpam-4831	250	19	sequence	sequence	NOUN
ejpam-4831	250	20	{	{	PUNCT
ejpam-4831	250	21	(	(	PUNCT
ejpam-4831	250	22	αk	αk	CCONJ
ejpam-4831	250	23	nyk)n}∞k=1	nyk)n}∞k=1	PROPN
ejpam-4831	250	24	is	be	AUX
ejpam-4831	250	25	bounded	bound	VERB
ejpam-4831	250	26	in	in	ADP
ejpam-4831	250	27	λs(eb	λs(eb	PROPN
ejpam-4831	250	28	)	)	PUNCT
ejpam-4831	250	29	and	and	CCONJ
ejpam-4831	250	30	then	then	ADV
ejpam-4831	250	31	in	in	ADP
ejpam-4831	250	32	λt	λt	ADP
ejpam-4831	250	33	(	(	PUNCT
ejpam-4831	250	34	eb	eb	PROPN
ejpam-4831	250	35	)	)	PUNCT
ejpam-4831	250	36	.	.	PUNCT
ejpam-4831	251	1	thus	thus	ADV
ejpam-4831	251	2	,	,	PUNCT
ejpam-4831	251	3	the	the	DET
ejpam-4831	251	4	series	series	NOUN
ejpam-4831	251	5	∑	∑	PROPN
ejpam-4831	251	6	k	k	PROPN
ejpam-4831	251	7	εk(α	εk(α	ADV
ejpam-4831	251	8	k	k	PROPN
ejpam-4831	251	9	nyk)n	nyk)n	PROPN
ejpam-4831	251	10	converges	converge	VERB
ejpam-4831	251	11	in	in	ADP
ejpam-4831	251	12	λt	λt	ADP
ejpam-4831	251	13	(	(	PUNCT
ejpam-4831	251	14	eb	eb	PROPN
ejpam-4831	251	15	)	)	PUNCT
ejpam-4831	251	16	to	to	ADP
ejpam-4831	251	17	z	z	NOUN
ejpam-4831	251	18	=	=	SYM
ejpam-4831	251	19	(	(	PUNCT
ejpam-4831	251	20	zn)n	zn)n	PROPN
ejpam-4831	251	21	.	.	PUNCT
ejpam-4831	252	1	since	since	SCONJ
ejpam-4831	252	2	the	the	DET
ejpam-4831	252	3	projections	projection	NOUN
ejpam-4831	252	4	are	be	AUX
ejpam-4831	252	5	bounded	bound	VERB
ejpam-4831	252	6	by	by	ADP
ejpam-4831	252	7	lemma	lemma	PROPN
ejpam-4831	252	8	1	1	NUM
ejpam-4831	252	9	,	,	PUNCT
ejpam-4831	252	10	one	one	NUM
ejpam-4831	252	11	has	have	VERB
ejpam-4831	252	12	zn	zn	PROPN
ejpam-4831	252	13	=	=	SYM
ejpam-4831	252	14	∑	∑	PUNCT
ejpam-4831	252	15	k	k	PROPN
ejpam-4831	252	16	εkα	εkα	PROPN
ejpam-4831	253	1	k	k	X
ejpam-4831	253	2	nyk	nyk	ADP
ejpam-4831	253	3	for	for	ADP
ejpam-4831	253	4	all	all	DET
ejpam-4831	253	5	n	n	CCONJ
ejpam-4831	253	6	,	,	PUNCT
ejpam-4831	253	7	and	and	CCONJ
ejpam-4831	253	8	then	then	ADV
ejpam-4831	253	9	x	x	X
ejpam-4831	253	10	=	=	PUNCT
ejpam-4831	253	11	z	z	NOUN
ejpam-4831	253	12	∈	∈	PROPN
ejpam-4831	253	13	λt	λt	X
ejpam-4831	253	14	(	(	PUNCT
ejpam-4831	253	15	eb	eb	PROPN
ejpam-4831	253	16	)	)	PUNCT
ejpam-4831	253	17	.	.	PUNCT
ejpam-4831	254	1	it	it	PRON
ejpam-4831	254	2	remains	remain	VERB
ejpam-4831	254	3	to	to	PART
ejpam-4831	254	4	prove	prove	VERB
ejpam-4831	254	5	that	that	SCONJ
ejpam-4831	254	6	{	{	PUNCT
ejpam-4831	254	7	xj}∞i=1	xj}∞i=1	PRON
ejpam-4831	254	8	converges	converge	VERB
ejpam-4831	254	9	in	in	ADP
ejpam-4831	254	10	(	(	PUNCT
ejpam-4831	254	11	λt	λt	INTJ
ejpam-4831	254	12	(	(	PUNCT
ejpam-4831	254	13	eb	eb	PROPN
ejpam-4831	254	14	)	)	PUNCT
ejpam-4831	254	15	,	,	PUNCT
ejpam-4831	254	16	n	n	CCONJ
ejpam-4831	254	17	)	)	PUNCT
ejpam-4831	254	18	to	to	PART
ejpam-4831	254	19	x.	x.	NOUN
ejpam-4831	254	20	we	we	PRON
ejpam-4831	254	21	have	have	VERB
ejpam-4831	254	22	,	,	PUNCT
ejpam-4831	254	23	xj	xj	PROPN
ejpam-4831	254	24	−	−	NOUN
ejpam-4831	254	25	x	x	PUNCT
ejpam-4831	254	26	=	=	SYM
ejpam-4831	254	27	∑	∑	PUNCT
ejpam-4831	254	28	k	k	X
ejpam-4831	254	29	εk(an(x	εk(an(x	X
ejpam-4831	255	1	j	j	PROPN
ejpam-4831	255	2	n)−	n)−	NOUN
ejpam-4831	255	3	αj	αj	PUNCT
ejpam-4831	255	4	n)nyk	n)nyk	ADV
ejpam-4831	255	5	and	and	CCONJ
ejpam-4831	255	6	n(xj	n(xj	PROPN
ejpam-4831	255	7	−	−	PROPN
ejpam-4831	255	8	x	x	SYM
ejpam-4831	255	9	)	)	PUNCT
ejpam-4831	255	10	≤	≤	NOUN
ejpam-4831	255	11	∑	∑	PUNCT
ejpam-4831	255	12	k	k	PROPN
ejpam-4831	255	13	|εk|∥(an(xjn)−	|εk|∥(an(xjn)−	NOUN
ejpam-4831	255	14	αj	αj	PROPN
ejpam-4831	255	15	n)n∥s∥yk∥b	n)n∥s∥yk∥b	X
ejpam-4831	255	16	(	(	PUNCT
ejpam-4831	255	17	10	10	NUM
ejpam-4831	255	18	)	)	PUNCT
ejpam-4831	255	19	for	for	ADP
ejpam-4831	255	20	j	j	PROPN
ejpam-4831	255	21	,	,	PUNCT
ejpam-4831	255	22	k	k	PROPN
ejpam-4831	255	23	,	,	PUNCT
ejpam-4831	255	24	let	let	VERB
ejpam-4831	255	25	βj	βj	PRON
ejpam-4831	255	26	k	k	NOUN
ejpam-4831	255	27	=	=	PUNCT
ejpam-4831	255	28	∥ak(xjn)−	∥ak(xjn)−	PRON
ejpam-4831	256	1	αk	αk	CCONJ
ejpam-4831	256	2	n∥t	n∥t	ADJ
ejpam-4831	256	3	and	and	CCONJ
ejpam-4831	256	4	γk	γk	PROPN
ejpam-4831	256	5	=	=	SYM
ejpam-4831	256	6	∥yk∥b	∥yk∥b	PROPN
ejpam-4831	256	7	.	.	PUNCT
ejpam-4831	257	1	(	(	PUNCT
ejpam-4831	257	2	11	11	NUM
ejpam-4831	257	3	)	)	PUNCT
ejpam-4831	257	4	then	then	ADV
ejpam-4831	257	5	,	,	PUNCT
ejpam-4831	257	6	(	(	PUNCT
ejpam-4831	257	7	γk)k	γk)k	PROPN
ejpam-4831	257	8	∈	∈	PROPN
ejpam-4831	257	9	c0	c0	PROPN
ejpam-4831	257	10	and	and	CCONJ
ejpam-4831	257	11	{	{	PUNCT
ejpam-4831	257	12	(	(	PUNCT
ejpam-4831	257	13	εkβj	εkβj	NOUN
ejpam-4831	257	14	k)k	k)k	ADJ
ejpam-4831	257	15	}	}	PUNCT
ejpam-4831	257	16	∞	∞	NUM
ejpam-4831	257	17	j=1	j=1	PROPN
ejpam-4831	257	18	is	be	AUX
ejpam-4831	257	19	a	a	DET
ejpam-4831	257	20	sequence	sequence	NOUN
ejpam-4831	257	21	in	in	ADP
ejpam-4831	257	22	ℓ1	ℓ1	NOUN
ejpam-4831	257	23	which	which	PRON
ejpam-4831	257	24	is	be	AUX
ejpam-4831	257	25	σ(ℓ1	σ(ℓ1	NOUN
ejpam-4831	257	26	,	,	PUNCT
ejpam-4831	257	27	c0)−bounded	c0)−bounde	VERB
ejpam-4831	257	28	,	,	PUNCT
ejpam-4831	257	29	then	then	ADV
ejpam-4831	257	30	it	it	PRON
ejpam-4831	257	31	has	have	VERB
ejpam-4831	257	32	a	a	DET
ejpam-4831	257	33	convergent	convergent	NOUN
ejpam-4831	257	34	subsequence	subsequence	NOUN
ejpam-4831	257	35	say	say	VERB
ejpam-4831	257	36	,	,	PUNCT
ejpam-4831	257	37	{	{	PUNCT
ejpam-4831	257	38	(	(	PUNCT
ejpam-4831	257	39	εkβr	εkβr	PROPN
ejpam-4831	257	40	k)k}∞r=1	k)k}∞r=1	NOUN
ejpam-4831	257	41	.	.	PUNCT
ejpam-4831	258	1	(	(	PUNCT
ejpam-4831	258	2	12	12	NUM
ejpam-4831	258	3	)	)	PUNCT
ejpam-4831	258	4	as	as	ADP
ejpam-4831	258	5	,	,	PUNCT
ejpam-4831	258	6	lim	lim	PROPN
ejpam-4831	258	7	r→∞	r→∞	NUM
ejpam-4831	258	8	εkβ	εkβ	PROPN
ejpam-4831	258	9	r	r	NOUN
ejpam-4831	258	10	k	k	NOUN
ejpam-4831	258	11	=	=	SYM
ejpam-4831	258	12	0	0	PROPN
ejpam-4831	258	13	,	,	PUNCT
ejpam-4831	258	14	for	for	ADP
ejpam-4831	258	15	all	all	DET
ejpam-4831	258	16	k	k	NOUN
ejpam-4831	258	17	,	,	PUNCT
ejpam-4831	258	18	then	then	ADV
ejpam-4831	258	19	the	the	DET
ejpam-4831	258	20	sequence	sequence	NOUN
ejpam-4831	258	21	in	in	ADP
ejpam-4831	258	22	(	(	PUNCT
ejpam-4831	258	23	12	12	NUM
ejpam-4831	258	24	)	)	PUNCT
ejpam-4831	258	25	converges	converge	VERB
ejpam-4831	258	26	to	to	ADP
ejpam-4831	258	27	0	0	NUM
ejpam-4831	258	28	in	in	ADP
ejpam-4831	258	29	(	(	PUNCT
ejpam-4831	258	30	ℓ1	ℓ1	NOUN
ejpam-4831	258	31	,	,	PUNCT
ejpam-4831	258	32	σ(ℓ1	σ(ℓ1	NOUN
ejpam-4831	258	33	,	,	PUNCT
ejpam-4831	258	34	c0	c0	NOUN
ejpam-4831	258	35	)	)	PUNCT
ejpam-4831	258	36	)	)	PUNCT
ejpam-4831	258	37	.	.	PUNCT
ejpam-4831	259	1	by	by	ADP
ejpam-4831	259	2	(	(	PUNCT
ejpam-4831	259	3	11	11	NUM
ejpam-4831	259	4	)	)	PUNCT
ejpam-4831	259	5	and	and	CCONJ
ejpam-4831	259	6	(	(	PUNCT
ejpam-4831	259	7	10	10	NUM
ejpam-4831	259	8	)	)	PUNCT
ejpam-4831	259	9	,	,	PUNCT
ejpam-4831	259	10	we	we	PRON
ejpam-4831	259	11	have	have	VERB
ejpam-4831	259	12	n(xr	n(xr	NOUN
ejpam-4831	259	13	−	−	PROPN
ejpam-4831	259	14	x	x	NOUN
ejpam-4831	259	15	)	)	PUNCT
ejpam-4831	259	16	≤	≤	NOUN
ejpam-4831	259	17	∑	∑	PUNCT
ejpam-4831	259	18	k	k	PROPN
ejpam-4831	259	19	|εkβr	|εkβr	VERB
ejpam-4831	259	20	k|γk	k|γk	NOUN
ejpam-4831	259	21	,	,	PUNCT
ejpam-4831	259	22	for	for	ADP
ejpam-4831	259	23	all	all	DET
ejpam-4831	259	24	r	r	NOUN
ejpam-4831	259	25	∈	∈	PROPN
ejpam-4831	259	26	n.	n.	NOUN
ejpam-4831	259	27	thus	thus	ADV
ejpam-4831	259	28	,	,	PUNCT
ejpam-4831	259	29	{	{	PUNCT
ejpam-4831	259	30	xr	xr	PROPN
ejpam-4831	259	31	−	−	PROPN
ejpam-4831	259	32	x}∞r=1	x}∞r=1	PROPN
ejpam-4831	259	33	converges	converge	VERB
ejpam-4831	259	34	to	to	ADP
ejpam-4831	259	35	0	0	NUM
ejpam-4831	259	36	in	in	ADP
ejpam-4831	259	37	λt	λt	ADP
ejpam-4831	259	38	(	(	PUNCT
ejpam-4831	259	39	eb	eb	PROPN
ejpam-4831	259	40	)	)	PUNCT
ejpam-4831	259	41	,	,	PUNCT
ejpam-4831	259	42	and	and	CCONJ
ejpam-4831	259	43	(	(	PUNCT
ejpam-4831	259	44	6	6	NUM
ejpam-4831	259	45	)	)	PUNCT
ejpam-4831	259	46	has	have	VERB
ejpam-4831	259	47	a	a	DET
ejpam-4831	259	48	convergent	convergent	NOUN
ejpam-4831	259	49	subsequence	subsequence	NOUN
ejpam-4831	259	50	.	.	PUNCT
ejpam-4831	260	1	this	this	PRON
ejpam-4831	260	2	finishes	finish	VERB
ejpam-4831	260	3	the	the	DET
ejpam-4831	260	4	proof	proof	NOUN
ejpam-4831	260	5	of	of	ADP
ejpam-4831	260	6	(	(	PUNCT
ejpam-4831	260	7	i	i	PROPN
ejpam-4831	260	8	)	)	PUNCT
ejpam-4831	260	9	.	.	PUNCT
ejpam-4831	261	1	the	the	DET
ejpam-4831	261	2	proof	proof	NOUN
ejpam-4831	261	3	of	of	ADP
ejpam-4831	261	4	(	(	PUNCT
ejpam-4831	261	5	ii	ii	NOUN
ejpam-4831	261	6	)	)	PUNCT
ejpam-4831	261	7	is	be	AUX
ejpam-4831	261	8	similar	similar	ADJ
ejpam-4831	261	9	by	by	ADP
ejpam-4831	261	10	interchanging	interchange	VERB
ejpam-4831	261	11	the	the	DET
ejpam-4831	261	12	roles	role	NOUN
ejpam-4831	261	13	of	of	ADP
ejpam-4831	261	14	e	e	NOUN
ejpam-4831	261	15	and	and	CCONJ
ejpam-4831	261	16	λ	λ	PROPN
ejpam-4831	261	17	in	in	ADP
ejpam-4831	261	18	the	the	DET
ejpam-4831	261	19	proof	proof	NOUN
ejpam-4831	261	20	.	.	PUNCT
ejpam-4831	262	1	■	■	PUNCT
ejpam-4831	262	2	m.	m.	NOUN
ejpam-4831	262	3	a.	a.	NOUN
ejpam-4831	262	4	sidaty	sidaty	PROPN
ejpam-4831	262	5	/	/	SYM
ejpam-4831	262	6	eur	eur	PROPN
ejpam-4831	262	7	.	.	PUNCT
ejpam-4831	263	1	j.	j.	PROPN
ejpam-4831	263	2	pure	pure	PROPN
ejpam-4831	263	3	appl	appl	PROPN
ejpam-4831	263	4	.	.	PROPN
ejpam-4831	263	5	math	math	PROPN
ejpam-4831	263	6	,	,	PUNCT
ejpam-4831	263	7	16	16	NUM
ejpam-4831	263	8	(	(	PUNCT
ejpam-4831	263	9	3	3	NUM
ejpam-4831	263	10	)	)	PUNCT
ejpam-4831	263	11	(	(	PUNCT
ejpam-4831	263	12	2023	2023	NUM
ejpam-4831	263	13	)	)	PUNCT
ejpam-4831	263	14	,	,	PUNCT
ejpam-4831	263	15	1762	1762	NUM
ejpam-4831	263	16	-	-	SYM
ejpam-4831	263	17	1771	1771	NUM
ejpam-4831	263	18	1770	1770	NUM
ejpam-4831	263	19	4	4	NUM
ejpam-4831	263	20	.	.	PUNCT
ejpam-4831	264	1	nuclearity	nuclearity	NOUN
ejpam-4831	264	2	of	of	ADP
ejpam-4831	264	3	λ{e	λ{e	PROPN
ejpam-4831	264	4	}	}	PUNCT
ejpam-4831	264	5	notice	notice	VERB
ejpam-4831	264	6	that	that	SCONJ
ejpam-4831	264	7	a	a	DET
ejpam-4831	264	8	locally	locally	ADV
ejpam-4831	264	9	convex	convex	ADJ
ejpam-4831	264	10	space	space	NOUN
ejpam-4831	264	11	is	be	AUX
ejpam-4831	264	12	said	say	VERB
ejpam-4831	264	13	to	to	PART
ejpam-4831	264	14	be	be	AUX
ejpam-4831	264	15	nuclear	nuclear	ADJ
ejpam-4831	264	16	(	(	PUNCT
ejpam-4831	264	17	resp	resp	NOUN
ejpam-4831	264	18	.	.	PUNCT
ejpam-4831	265	1	a	a	DET
ejpam-4831	265	2	schwartz	schwartz	PROPN
ejpam-4831	265	3	space	space	NOUN
ejpam-4831	265	4	)	)	PUNCT
ejpam-4831	265	5	if	if	SCONJ
ejpam-4831	265	6	the	the	DET
ejpam-4831	265	7	convex	convex	ADJ
ejpam-4831	265	8	bornology	bornology	NOUN
ejpam-4831	265	9	of	of	ADP
ejpam-4831	265	10	equicontinuous	equicontinuous	ADJ
ejpam-4831	265	11	subsets	subset	NOUN
ejpam-4831	265	12	of	of	ADP
ejpam-4831	265	13	its	its	PRON
ejpam-4831	265	14	topological	topological	ADJ
ejpam-4831	265	15	dual	dual	NOUN
ejpam-4831	265	16	is	be	AUX
ejpam-4831	265	17	nuclear	nuclear	ADJ
ejpam-4831	265	18	(	(	PUNCT
ejpam-4831	265	19	resp	resp	NOUN
ejpam-4831	265	20	.	.	PUNCT
ejpam-4831	265	21	of	of	ADP
ejpam-4831	265	22	schwartz	schwartz	PROPN
ejpam-4831	265	23	)	)	PUNCT
ejpam-4831	265	24	.	.	PUNCT
ejpam-4831	266	1	let	let	VERB
ejpam-4831	266	2	λ	λ	PRON
ejpam-4831	266	3	be	be	AUX
ejpam-4831	266	4	a	a	DET
ejpam-4831	266	5	perfect	perfect	ADJ
ejpam-4831	266	6	sequence	sequence	NOUN
ejpam-4831	266	7	space	space	NOUN
ejpam-4831	266	8	and	and	CCONJ
ejpam-4831	266	9	e	e	NOUN
ejpam-4831	266	10	a	a	DET
ejpam-4831	266	11	locally	locally	ADV
ejpam-4831	266	12	convex	convex	ADJ
ejpam-4831	266	13	space	space	NOUN
ejpam-4831	266	14	whose	whose	DET
ejpam-4831	266	15	topology	topology	NOUN
ejpam-4831	266	16	is	be	AUX
ejpam-4831	266	17	defined	define	VERB
ejpam-4831	266	18	by	by	ADP
ejpam-4831	266	19	a	a	DET
ejpam-4831	266	20	family	family	NOUN
ejpam-4831	266	21	m	m	NOUN
ejpam-4831	266	22	of	of	ADP
ejpam-4831	266	23	absolutely	absolutely	ADV
ejpam-4831	266	24	convex	convex	ADJ
ejpam-4831	266	25	equicontinuous	equicontinuous	ADJ
ejpam-4831	266	26	subsets	subset	NOUN
ejpam-4831	266	27	of	of	ADP
ejpam-4831	266	28	its	its	PRON
ejpam-4831	266	29	topological	topological	ADJ
ejpam-4831	266	30	dual	dual	ADJ
ejpam-4831	266	31	e′.	e′.	NOUN
ejpam-4831	266	32	define	define	VERB
ejpam-4831	266	33	λ{e	λ{e	NOUN
ejpam-4831	266	34	}	}	PUNCT
ejpam-4831	266	35	=	=	SYM
ejpam-4831	266	36	{	{	PUNCT
ejpam-4831	266	37	(	(	PUNCT
ejpam-4831	266	38	xn)n	xn)n	PROPN
ejpam-4831	266	39	⊂	⊂	PROPN
ejpam-4831	266	40	e	e	X
ejpam-4831	266	41	:	:	PUNCT
ejpam-4831	266	42	(	(	PUNCT
ejpam-4831	266	43	pm	pm	NOUN
ejpam-4831	266	44	(	(	PUNCT
ejpam-4831	266	45	xn))n	xn))n	PROPN
ejpam-4831	266	46	∈	∈	PROPN
ejpam-4831	266	47	λ	λ	NOUN
ejpam-4831	266	48	}	}	PUNCT
ejpam-4831	266	49	,	,	PUNCT
ejpam-4831	266	50	where	where	SCONJ
ejpam-4831	266	51	pm	pm	NOUN
ejpam-4831	266	52	(	(	PUNCT
ejpam-4831	266	53	xn	xn	PROPN
ejpam-4831	266	54	)	)	PUNCT
ejpam-4831	266	55	=	=	SYM
ejpam-4831	266	56	sup	sup	NOUN
ejpam-4831	266	57	a∈m	a∈m	NOUN
ejpam-4831	266	58	|a(xn)|	|a(xn)|	NOUN
ejpam-4831	266	59	.	.	PUNCT
ejpam-4831	267	1	if	if	SCONJ
ejpam-4831	267	2	a	a	DET
ejpam-4831	267	3	topology	topology	NOUN
ejpam-4831	267	4	on	on	ADP
ejpam-4831	267	5	λ	λ	PROPN
ejpam-4831	267	6	is	be	AUX
ejpam-4831	267	7	defined	define	VERB
ejpam-4831	267	8	by	by	ADP
ejpam-4831	267	9	family	family	NOUN
ejpam-4831	267	10	s	s	PROPN
ejpam-4831	267	11	of	of	ADP
ejpam-4831	267	12	normal	normal	ADJ
ejpam-4831	267	13	,	,	PUNCT
ejpam-4831	267	14	absolutely	absolutely	ADV
ejpam-4831	267	15	convex	convex	ADJ
ejpam-4831	267	16	and	and	CCONJ
ejpam-4831	267	17	σ(λ×	σ(λ×	ADJ
ejpam-4831	267	18	,	,	PUNCT
ejpam-4831	267	19	λ)−bounded	λ)−bounde	VERB
ejpam-4831	267	20	subsets	subset	NOUN
ejpam-4831	267	21	of	of	ADP
ejpam-4831	267	22	λ×	λ×	PROPN
ejpam-4831	267	23	,	,	PUNCT
ejpam-4831	267	24	then	then	ADV
ejpam-4831	267	25	a	a	DET
ejpam-4831	267	26	locally	locally	ADV
ejpam-4831	267	27	convex	convex	ADJ
ejpam-4831	267	28	topology	topology	NOUN
ejpam-4831	267	29	can	can	AUX
ejpam-4831	267	30	be	be	AUX
ejpam-4831	267	31	defined	define	VERB
ejpam-4831	267	32	on	on	ADP
ejpam-4831	267	33	λ{e	λ{e	NOUN
ejpam-4831	267	34	}	}	PUNCT
ejpam-4831	267	35	by	by	ADP
ejpam-4831	267	36	the	the	DET
ejpam-4831	267	37	family	family	NOUN
ejpam-4831	267	38	of	of	ADP
ejpam-4831	267	39	semi	semi	NOUN
ejpam-4831	267	40	-	-	NOUN
ejpam-4831	267	41	norms	norm	NOUN
ejpam-4831	267	42	(	(	PUNCT
ejpam-4831	267	43	πs	πs	INTJ
ejpam-4831	267	44	,	,	PUNCT
ejpam-4831	267	45	m	m	NOUN
ejpam-4831	267	46	)	)	PUNCT
ejpam-4831	267	47	s∈s	s∈s	NOUN
ejpam-4831	267	48	,	,	PUNCT
ejpam-4831	267	49	m∈m	m∈m	NOUN
ejpam-4831	267	50	,	,	PUNCT
ejpam-4831	267	51	such	such	ADJ
ejpam-4831	267	52	that	that	SCONJ
ejpam-4831	267	53	,	,	PUNCT
ejpam-4831	267	54	if	if	SCONJ
ejpam-4831	267	55	x	x	X
ejpam-4831	267	56	=	=	SYM
ejpam-4831	267	57	(	(	PUNCT
ejpam-4831	267	58	xn)n	xn)n	PROPN
ejpam-4831	267	59	∈	∈	PROPN
ejpam-4831	267	60	λ{e	λ{e	PROPN
ejpam-4831	267	61	}	}	PUNCT
ejpam-4831	267	62	then	then	ADV
ejpam-4831	267	63	πs	πs	ADP
ejpam-4831	267	64	,	,	PUNCT
ejpam-4831	267	65	m	m	VERB
ejpam-4831	267	66	(	(	PUNCT
ejpam-4831	267	67	(	(	PUNCT
ejpam-4831	267	68	xn)n	xn)n	PROPN
ejpam-4831	267	69	)	)	PUNCT
ejpam-4831	268	1	=	=	SYM
ejpam-4831	268	2	ps((pm	ps((pm	INTJ
ejpam-4831	268	3	(	(	PUNCT
ejpam-4831	268	4	xn	xn	NOUN
ejpam-4831	268	5	)	)	PUNCT
ejpam-4831	268	6	)	)	PUNCT
ejpam-4831	268	7	)	)	PUNCT
ejpam-4831	269	1	=	=	SYM
ejpam-4831	269	2	sup	sup	NOUN
ejpam-4831	269	3	{	{	PUNCT
ejpam-4831	269	4	∞∑	∞∑	PROPN
ejpam-4831	269	5	n=1	n=1	PROPN
ejpam-4831	269	6	|αnpm	|αnpm	PROPN
ejpam-4831	269	7	(	(	PUNCT
ejpam-4831	269	8	xn)|	xn)|	PROPN
ejpam-4831	269	9	:	:	PUNCT
ejpam-4831	269	10	(	(	PUNCT
ejpam-4831	269	11	αn)n	αn)n	NOUN
ejpam-4831	269	12	∈	∈	NOUN
ejpam-4831	269	13	s	s	PART
ejpam-4831	269	14	}	}	PUNCT
ejpam-4831	269	15	.	.	PUNCT
ejpam-4831	270	1	for	for	ADP
ejpam-4831	270	2	the	the	DET
ejpam-4831	270	3	topology	topology	NOUN
ejpam-4831	270	4	so	so	ADV
ejpam-4831	270	5	defined	define	VERB
ejpam-4831	270	6	,	,	PUNCT
ejpam-4831	270	7	ronald	ronald	PROPN
ejpam-4831	270	8	c.	c.	PROPN
ejpam-4831	270	9	rosier	rosier	ADV
ejpam-4831	270	10	in	in	ADP
ejpam-4831	270	11	[	[	X
ejpam-4831	270	12	10	10	NUM
ejpam-4831	270	13	]	]	PUNCT
ejpam-4831	270	14	proved	prove	VERB
ejpam-4831	270	15	that	that	SCONJ
ejpam-4831	270	16	the	the	DET
ejpam-4831	270	17	dual	dual	ADJ
ejpam-4831	270	18	space	space	NOUN
ejpam-4831	270	19	(	(	PUNCT
ejpam-4831	270	20	λ{e})∗	λ{e})∗	PROPN
ejpam-4831	270	21	of	of	ADP
ejpam-4831	270	22	λ{e	λ{e	PROPN
ejpam-4831	270	23	}	}	PUNCT
ejpam-4831	270	24	is	be	AUX
ejpam-4831	270	25	λ×(e′	λ×(e′	ADJ
ejpam-4831	270	26	)	)	PUNCT
ejpam-4831	270	27	and	and	CCONJ
ejpam-4831	270	28	that	that	SCONJ
ejpam-4831	270	29	a	a	DET
ejpam-4831	270	30	subset	subset	NOUN
ejpam-4831	270	31	of	of	ADP
ejpam-4831	270	32	(	(	PUNCT
ejpam-4831	270	33	λ{e})∗	λ{e})∗	PROPN
ejpam-4831	270	34	is	be	AUX
ejpam-4831	270	35	equicontinuous	equicontinuous	ADJ
ejpam-4831	270	36	if	if	SCONJ
ejpam-4831	270	37	and	and	CCONJ
ejpam-4831	270	38	only	only	ADV
ejpam-4831	270	39	if	if	SCONJ
ejpam-4831	270	40	it	it	PRON
ejpam-4831	270	41	is	be	AUX
ejpam-4831	270	42	contained	contain	VERB
ejpam-4831	270	43	in	in	ADP
ejpam-4831	270	44	some	some	DET
ejpam-4831	270	45	s(m	s(m	NOUN
ejpam-4831	270	46	)	)	PUNCT
ejpam-4831	270	47	for	for	ADP
ejpam-4831	270	48	s	s	PROPN
ejpam-4831	270	49	∈	∈	PROPN
ejpam-4831	270	50	s	s	X
ejpam-4831	270	51	and	and	CCONJ
ejpam-4831	270	52	m	m	PROPN
ejpam-4831	270	53	∈	∈	NOUN
ejpam-4831	270	54	m.	m.	NOUN
ejpam-4831	270	55	starting	start	VERB
ejpam-4831	270	56	from	from	ADP
ejpam-4831	270	57	this	this	DET
ejpam-4831	270	58	setting	setting	NOUN
ejpam-4831	270	59	,	,	PUNCT
ejpam-4831	270	60	theorem	theorem	ADJ
ejpam-4831	270	61	2	2	NUM
ejpam-4831	270	62	gives	give	NOUN
ejpam-4831	270	63	theorem	theorem	VERB
ejpam-4831	270	64	4	4	NUM
ejpam-4831	270	65	.	.	PUNCT
ejpam-4831	270	66	λ{e	λ{e	PROPN
ejpam-4831	270	67	}	}	PUNCT
ejpam-4831	270	68	is	be	AUX
ejpam-4831	270	69	nuclear	nuclear	ADJ
ejpam-4831	270	70	if	if	SCONJ
ejpam-4831	270	71	and	and	CCONJ
ejpam-4831	270	72	only	only	ADV
ejpam-4831	270	73	if	if	SCONJ
ejpam-4831	270	74	λ	λ	PROPN
ejpam-4831	270	75	and	and	CCONJ
ejpam-4831	270	76	e	e	NOUN
ejpam-4831	270	77	are	be	AUX
ejpam-4831	270	78	nuclear	nuclear	ADJ
ejpam-4831	270	79	.	.	PUNCT
ejpam-4831	271	1	also	also	ADV
ejpam-4831	271	2	,	,	PUNCT
ejpam-4831	271	3	theorem	theorem	ADJ
ejpam-4831	271	4	3	3	NUM
ejpam-4831	271	5	gives	give	VERB
ejpam-4831	271	6	theorem	theorem	VERB
ejpam-4831	271	7	5	5	NUM
ejpam-4831	271	8	.	.	PUNCT
ejpam-4831	272	1	if	if	SCONJ
ejpam-4831	272	2	e	e	PROPN
ejpam-4831	272	3	(	(	PUNCT
ejpam-4831	272	4	resp	resp	NOUN
ejpam-4831	272	5	.	.	PUNCT
ejpam-4831	273	1	λ	λ	X
ejpam-4831	273	2	)	)	PUNCT
ejpam-4831	273	3	is	be	AUX
ejpam-4831	273	4	nuclear	nuclear	ADJ
ejpam-4831	273	5	,	,	PUNCT
ejpam-4831	273	6	then	then	ADV
ejpam-4831	273	7	λ{e	λ{e	PROPN
ejpam-4831	273	8	}	}	PUNCT
ejpam-4831	273	9	is	be	AUX
ejpam-4831	273	10	a	a	DET
ejpam-4831	273	11	schwartz	schwartz	NOUN
ejpam-4831	273	12	space	space	NOUN
ejpam-4831	274	1	if	if	SCONJ
ejpam-4831	274	2	and	and	CCONJ
ejpam-4831	274	3	only	only	ADV
ejpam-4831	274	4	if	if	SCONJ
ejpam-4831	274	5	λ	λ	PROPN
ejpam-4831	274	6	(	(	PUNCT
ejpam-4831	274	7	resp	resp	NOUN
ejpam-4831	274	8	.	.	PUNCT
ejpam-4831	275	1	e	e	X
ejpam-4831	275	2	)	)	PUNCT
ejpam-4831	275	3	is	be	AUX
ejpam-4831	275	4	a	a	DET
ejpam-4831	275	5	schwartz	schwartz	PROPN
ejpam-4831	275	6	space	space	NOUN
ejpam-4831	275	7	.	.	PUNCT
ejpam-4831	276	1	5	5	X
ejpam-4831	276	2	.	.	X
ejpam-4831	276	3	conclusion	conclusion	NOUN
ejpam-4831	276	4	in	in	ADP
ejpam-4831	276	5	this	this	DET
ejpam-4831	276	6	paper	paper	NOUN
ejpam-4831	276	7	we	we	PRON
ejpam-4831	276	8	have	have	AUX
ejpam-4831	276	9	characterized	characterize	VERB
ejpam-4831	276	10	the	the	DET
ejpam-4831	276	11	bornological	bornological	ADJ
ejpam-4831	276	12	structure	structure	NOUN
ejpam-4831	276	13	,	,	PUNCT
ejpam-4831	276	14	the	the	DET
ejpam-4831	276	15	completeness	completeness	NOUN
ejpam-4831	276	16	and	and	CCONJ
ejpam-4831	276	17	the	the	DET
ejpam-4831	276	18	nuclearity	nuclearity	NOUN
ejpam-4831	276	19	of	of	ADP
ejpam-4831	276	20	λ(e	λ(e	PROPN
ejpam-4831	276	21	)	)	PUNCT
ejpam-4831	276	22	in	in	ADP
ejpam-4831	276	23	terms	term	NOUN
ejpam-4831	276	24	of	of	ADP
ejpam-4831	276	25	that	that	PRON
ejpam-4831	276	26	of	of	ADP
ejpam-4831	276	27	λ	λ	PROPN
ejpam-4831	276	28	and	and	CCONJ
ejpam-4831	276	29	e.	e.	PROPN
ejpam-4831	276	30	an	an	DET
ejpam-4831	276	31	application	application	NOUN
ejpam-4831	276	32	to	to	ADP
ejpam-4831	276	33	the	the	DET
ejpam-4831	276	34	nuclearity	nuclearity	NOUN
ejpam-4831	276	35	of	of	ADP
ejpam-4831	276	36	the	the	DET
ejpam-4831	276	37	locally	locally	ADV
ejpam-4831	276	38	convex	convex	ADJ
ejpam-4831	276	39	space	space	NOUN
ejpam-4831	276	40	λ{e	λ{e	PROPN
ejpam-4831	276	41	}	}	PUNCT
ejpam-4831	276	42	is	be	AUX
ejpam-4831	276	43	given	give	VERB
ejpam-4831	276	44	.	.	PUNCT
ejpam-4831	277	1	acknowledgements	acknowledgement	NOUN
ejpam-4831	277	2	the	the	DET
ejpam-4831	277	3	author	author	NOUN
ejpam-4831	277	4	is	be	AUX
ejpam-4831	277	5	grateful	grateful	ADJ
ejpam-4831	277	6	to	to	ADP
ejpam-4831	277	7	reviewers	reviewer	NOUN
ejpam-4831	277	8	for	for	ADP
ejpam-4831	277	9	their	their	PRON
ejpam-4831	277	10	suggestions	suggestion	NOUN
ejpam-4831	277	11	and	and	CCONJ
ejpam-4831	277	12	comments	comment	NOUN
ejpam-4831	277	13	which	which	PRON
ejpam-4831	277	14	improved	improve	VERB
ejpam-4831	277	15	the	the	DET
ejpam-4831	277	16	quality	quality	NOUN
ejpam-4831	277	17	of	of	ADP
ejpam-4831	277	18	the	the	DET
ejpam-4831	277	19	paper	paper	NOUN
ejpam-4831	277	20	.	.	PUNCT
ejpam-4831	278	1	references	reference	NOUN
ejpam-4831	278	2	1771	1771	NUM
ejpam-4831	278	3	references	reference	NOUN
ejpam-4831	278	4	[	[	X
ejpam-4831	278	5	1	1	NUM
ejpam-4831	278	6	]	]	PUNCT
ejpam-4831	278	7	m.	m.	NOUN
ejpam-4831	278	8	florencio	florencio	PROPN
ejpam-4831	278	9	and	and	CCONJ
ejpam-4831	278	10	pedro	pedro	PROPN
ejpam-4831	278	11	j.	j.	PROPN
ejpam-4831	278	12	paúl	paúl	PROPN
ejpam-4831	278	13	.	.	PROPN
ejpam-4831	278	14	una	una	PROPN
ejpam-4831	278	15	representación	representación	PROPN
ejpam-4831	278	16	de	de	X
ejpam-4831	278	17	cietros	cietros	PROPN
ejpam-4831	278	18	ϵ-productos	ϵ-producto	NOUN
ejpam-4831	278	19	tensoriales	tensoriale	NOUN
ejpam-4831	278	20	.	.	PUNCT
ejpam-4831	279	1	in	in	ADP
ejpam-4831	279	2	actas	actas	PROPN
ejpam-4831	279	3	de	de	PROPN
ejpam-4831	279	4	las	las	PROPN
ejpam-4831	279	5	jornadas	jornadas	PROPN
ejpam-4831	279	6	matematicas	matematicas	PROPN
ejpam-4831	279	7	hispano	hispano	PROPN
ejpam-4831	279	8	lusas	lusas	PROPN
ejpam-4831	279	9	,	,	PUNCT
ejpam-4831	279	10	murcia	murcia	PROPN
ejpam-4831	279	11	.	.	PROPN
ejpam-4831	279	12	,	,	PUNCT
ejpam-4831	279	13	pages	page	NOUN
ejpam-4831	279	14	191–203	191–203	NUM
ejpam-4831	279	15	,	,	PUNCT
ejpam-4831	279	16	murcia	murcia	PROPN
ejpam-4831	279	17	,	,	PUNCT
ejpam-4831	279	18	spain	spain	PROPN
ejpam-4831	279	19	,	,	PUNCT
ejpam-4831	279	20	1985	1985	NUM
ejpam-4831	279	21	.	.	PUNCT
ejpam-4831	280	1	universidad	universidad	PROPN
ejpam-4831	280	2	de	de	PROPN
ejpam-4831	280	3	murcia	murcia	PROPN
ejpam-4831	280	4	.	.	PUNCT
ejpam-4831	281	1	[	[	X
ejpam-4831	281	2	2	2	NUM
ejpam-4831	281	3	]	]	PUNCT
ejpam-4831	281	4	m.	m.	NOUN
ejpam-4831	281	5	florencio	florencio	PROPN
ejpam-4831	281	6	and	and	CCONJ
ejpam-4831	281	7	pedro	pedro	PROPN
ejpam-4831	281	8	j.	j.	PROPN
ejpam-4831	281	9	paúl	paúl	PROPN
ejpam-4831	281	10	.	.	PUNCT
ejpam-4831	281	11	la	la	PROPN
ejpam-4831	281	12	propiedad	propiedad	PROPN
ejpam-4831	281	13	ak	ak	PROPN
ejpam-4831	281	14	en	en	PROPN
ejpam-4831	281	15	ciertos	ciertos	PROPN
ejpam-4831	281	16	espacios	espacios	PROPN
ejpam-4831	281	17	de	de	PROPN
ejpam-4831	281	18	suecsiones	suecsione	NOUN
ejpam-4831	281	19	vectoriales	vectoriale	NOUN
ejpam-4831	281	20	.	.	PUNCT
ejpam-4831	282	1	in	in	ADP
ejpam-4831	282	2	dep	dep	PROPN
ejpam-4831	282	3	.	.	PUNCT
ejpam-4831	282	4	mat	mat	PROPN
ejpam-4831	282	5	.	.	PROPN
ejpam-4831	282	6	univ	univ	PROPN
ejpam-4831	282	7	.	.	PUNCT
ejpam-4831	282	8	extremadura	extremadura	PROPN
ejpam-4831	282	9	,	,	PUNCT
ejpam-4831	282	10	editor	editor	NOUN
ejpam-4831	282	11	,	,	PUNCT
ejpam-4831	282	12	proc	proc	NOUN
ejpam-4831	282	13	.	.	PUNCT
ejpam-4831	283	1	eleventh	eleventh	ADJ
ejpam-4831	283	2	spanish	spanish	ADJ
ejpam-4831	283	3	-	-	PUNCT
ejpam-4831	283	4	portuguese	portuguese	ADJ
ejpam-4831	283	5	conference	conference	NOUN
ejpam-4831	283	6	on	on	ADP
ejpam-4831	283	7	mathematics	mathematic	NOUN
ejpam-4831	283	8	,	,	PUNCT
ejpam-4831	283	9	pages	page	NOUN
ejpam-4831	283	10	197–203	197–203	NUM
ejpam-4831	283	11	,	,	PUNCT
ejpam-4831	283	12	1986	1986	NUM
ejpam-4831	283	13	.	.	PUNCT
ejpam-4831	284	1	[	[	X
ejpam-4831	284	2	3	3	X
ejpam-4831	284	3	]	]	X
ejpam-4831	284	4	m.	m.	NOUN
ejpam-4831	284	5	florencio	florencio	NOUN
ejpam-4831	284	6	and	and	CCONJ
ejpam-4831	284	7	pedro	pedro	PROPN
ejpam-4831	284	8	j.	j.	PROPN
ejpam-4831	284	9	paúl	paúl	PROPN
ejpam-4831	284	10	.	.	PUNCT
ejpam-4831	284	11	barrelledness	barrelledness	NOUN
ejpam-4831	284	12	conditions	condition	NOUN
ejpam-4831	284	13	on	on	ADP
ejpam-4831	284	14	vector	vector	NOUN
ejpam-4831	284	15	valued	value	VERB
ejpam-4831	284	16	sequence	sequence	NOUN
ejpam-4831	284	17	spaces	space	VERB
ejpam-4831	284	18	.	.	PUNCT
ejpam-4831	285	1	arch	arch	NOUN
ejpam-4831	285	2	.	.	PUNCT
ejpam-4831	286	1	math	math	NOUN
ejpam-4831	286	2	.	.	PUNCT
ejpam-4831	286	3	,	,	PUNCT
ejpam-4831	286	4	48:153–164	48:153–164	NOUN
ejpam-4831	286	5	,	,	PUNCT
ejpam-4831	286	6	1987	1987	NUM
ejpam-4831	286	7	.	.	PUNCT
ejpam-4831	287	1	[	[	X
ejpam-4831	287	2	4	4	X
ejpam-4831	287	3	]	]	PUNCT
ejpam-4831	287	4	h.	h.	PROPN
ejpam-4831	287	5	henri	henri	PROPN
ejpam-4831	287	6	-	-	PUNCT
ejpam-4831	287	7	hogbé-n’lend	hogbé-n’lend	NOUN
ejpam-4831	287	8	.	.	PUNCT
ejpam-4831	288	1	théorie	théorie	PROPN
ejpam-4831	288	2	de	de	PROPN
ejpam-4831	288	3	bornologies	bornologie	NOUN
ejpam-4831	288	4	et	et	NOUN
ejpam-4831	288	5	applications	application	NOUN
ejpam-4831	288	6	.	.	PUNCT
ejpam-4831	289	1	springer	springer	NOUN
ejpam-4831	289	2	-	-	PUNCT
ejpam-4831	289	3	verlag	verlag	PROPN
ejpam-4831	289	4	,	,	PUNCT
ejpam-4831	289	5	lecture	lecture	NOUN
ejpam-4831	289	6	notes	note	NOUN
ejpam-4831	289	7	,	,	PUNCT
ejpam-4831	289	8	213	213	NUM
ejpam-4831	289	9	,	,	PUNCT
ejpam-4831	289	10	berlin	berlin	PROPN
ejpam-4831	289	11	and	and	CCONJ
ejpam-4831	289	12	heidelberg	heidelberg	PROPN
ejpam-4831	289	13	,	,	PUNCT
ejpam-4831	289	14	1971	1971	NUM
ejpam-4831	289	15	.	.	PUNCT
ejpam-4831	290	1	[	[	X
ejpam-4831	290	2	5	5	X
ejpam-4831	290	3	]	]	PUNCT
ejpam-4831	290	4	g.	g.	PROPN
ejpam-4831	290	5	köthe	köthe	PROPN
ejpam-4831	290	6	.	.	PUNCT
ejpam-4831	291	1	topological	topological	ADJ
ejpam-4831	291	2	vector	vector	NOUN
ejpam-4831	291	3	spaces	space	NOUN
ejpam-4831	291	4	i	i	PRON
ejpam-4831	291	5	and	and	CCONJ
ejpam-4831	291	6	ii	ii	PROPN
ejpam-4831	291	7	.	.	PROPN
ejpam-4831	292	1	springer	springer	NOUN
ejpam-4831	292	2	-	-	PUNCT
ejpam-4831	292	3	verlag	verlag	PROPN
ejpam-4831	292	4	,	,	PUNCT
ejpam-4831	292	5	berlin	berlin	PROPN
ejpam-4831	292	6	,	,	PUNCT
ejpam-4831	292	7	heidelberg	heidelberg	PROPN
ejpam-4831	292	8	,	,	PUNCT
ejpam-4831	292	9	new	new	PROPN
ejpam-4831	292	10	york	york	PROPN
ejpam-4831	292	11	,	,	PUNCT
ejpam-4831	292	12	1979	1979	NUM
ejpam-4831	292	13	.	.	PUNCT
ejpam-4831	293	1	[	[	X
ejpam-4831	293	2	6	6	NUM
ejpam-4831	293	3	]	]	PUNCT
ejpam-4831	293	4	l.	l.	PROPN
ejpam-4831	293	5	oubbi	oubbi	PROPN
ejpam-4831	293	6	and	and	CCONJ
ejpam-4831	293	7	m.	m.	PROPN
ejpam-4831	293	8	a.	a.	PROPN
ejpam-4831	293	9	ould	ould	AUX
ejpam-4831	293	10	sidaty	sidaty	VERB
ejpam-4831	293	11	.	.	PUNCT
ejpam-4831	294	1	dual	dual	ADJ
ejpam-4831	294	2	space	space	NOUN
ejpam-4831	294	3	of	of	ADP
ejpam-4831	294	4	certain	certain	ADJ
ejpam-4831	294	5	locally	locally	ADV
ejpam-4831	294	6	convex	convex	ADJ
ejpam-4831	294	7	sequence	sequence	NOUN
ejpam-4831	294	8	spaces	space	VERB
ejpam-4831	294	9	.	.	PUNCT
ejpam-4831	295	1	revista	revista	PROPN
ejpam-4831	295	2	de	de	X
ejpam-4831	295	3	la	la	PROPN
ejpam-4831	295	4	real	real	PROPN
ejpam-4831	295	5	academia	academia	PROPN
ejpam-4831	295	6	de	de	PROPN
ejpam-4831	295	7	ciencias	ciencias	PROPN
ejpam-4831	295	8	de	de	PROPN
ejpam-4831	295	9	zargoza	zargoza	PROPN
ejpam-4831	295	10	,	,	PUNCT
ejpam-4831	295	11	59:79–88	59:79–88	NUM
ejpam-4831	295	12	,	,	PUNCT
ejpam-4831	295	13	2004	2004	NUM
ejpam-4831	295	14	.	.	PUNCT
ejpam-4831	296	1	[	[	X
ejpam-4831	296	2	7	7	X
ejpam-4831	296	3	]	]	X
ejpam-4831	296	4	l.	l.	PROPN
ejpam-4831	296	5	oubbi	oubbi	PROPN
ejpam-4831	296	6	and	and	CCONJ
ejpam-4831	296	7	m.	m.	PROPN
ejpam-4831	296	8	a.	a.	PROPN
ejpam-4831	296	9	ould	ould	AUX
ejpam-4831	296	10	sidaty	sidaty	VERB
ejpam-4831	296	11	.	.	PUNCT
ejpam-4831	297	1	reflexivity	reflexivity	NOUN
ejpam-4831	297	2	of	of	ADP
ejpam-4831	297	3	spaces	space	NOUN
ejpam-4831	297	4	of	of	ADP
ejpam-4831	297	5	weakly	weakly	ADJ
ejpam-4831	297	6	summable	summable	ADJ
ejpam-4831	297	7	sequences	sequence	NOUN
ejpam-4831	297	8	.	.	PUNCT
ejpam-4831	298	1	rev	rev	PROPN
ejpam-4831	298	2	.	.	PROPN
ejpam-4831	298	3	r.	r.	PROPN
ejpam-4831	298	4	acad	acad	PROPN
ejpam-4831	298	5	.	.	PUNCT
ejpam-4831	299	1	cien	cien	NOUN
ejpam-4831	299	2	.	.	PUNCT
ejpam-4831	300	1	serie	serie	PROPN
ejpam-4831	300	2	a.	a.	PROPN
ejpam-4831	300	3	mat	mat	PROPN
ejpam-4831	300	4	.	.	PROPN
ejpam-4831	300	5	,	,	PUNCT
ejpam-4831	300	6	101(1):51–62	101(1):51–62	NUM
ejpam-4831	300	7	,	,	PUNCT
ejpam-4831	300	8	2007	2007	NUM
ejpam-4831	300	9	.	.	PUNCT
ejpam-4831	301	1	[	[	X
ejpam-4831	301	2	8	8	NUM
ejpam-4831	301	3	]	]	X
ejpam-4831	301	4	e.	e.	PROPN
ejpam-4831	301	5	pietsch	pietsch	PROPN
ejpam-4831	301	6	.	.	PUNCT
ejpam-4831	302	1	verallgemeinerte	verallgemeinerte	PROPN
ejpam-4831	302	2	vollkommene	vollkommene	PROPN
ejpam-4831	302	3	folgenräume	folgenräume	PROPN
ejpam-4831	302	4	.	.	PUNCT
ejpam-4831	303	1	akademie	akademie	PROPN
ejpam-4831	303	2	-	-	PUNCT
ejpam-4831	303	3	verlag	verlag	PROPN
ejpam-4831	303	4	,	,	PUNCT
ejpam-4831	303	5	berlin	berlin	PROPN
ejpam-4831	303	6	,	,	PUNCT
ejpam-4831	303	7	heidelberg	heidelberg	PROPN
ejpam-4831	303	8	,	,	PUNCT
ejpam-4831	303	9	new	new	PROPN
ejpam-4831	303	10	york	york	PROPN
ejpam-4831	303	11	,	,	PUNCT
ejpam-4831	303	12	1962	1962	NUM
ejpam-4831	303	13	.	.	PUNCT
ejpam-4831	304	1	[	[	X
ejpam-4831	304	2	9	9	NUM
ejpam-4831	304	3	]	]	X
ejpam-4831	304	4	e.	e.	PROPN
ejpam-4831	304	5	pietsch	pietsch	PROPN
ejpam-4831	304	6	.	.	PUNCT
ejpam-4831	305	1	nuclear	nuclear	ADJ
ejpam-4831	305	2	locally	locally	ADV
ejpam-4831	305	3	convex	convex	PROPN
ejpam-4831	305	4	spaces	space	NOUN
ejpam-4831	305	5	.	.	PUNCT
ejpam-4831	306	1	springer	springer	NOUN
ejpam-4831	306	2	-	-	PUNCT
ejpam-4831	306	3	verlag	verlag	PROPN
ejpam-4831	306	4	,	,	PUNCT
ejpam-4831	306	5	berlin	berlin	PROPN
ejpam-4831	306	6	,	,	PUNCT
ejpam-4831	306	7	heidelberg	heidelberg	PROPN
ejpam-4831	306	8	,	,	PUNCT
ejpam-4831	306	9	new	new	PROPN
ejpam-4831	306	10	york	york	PROPN
ejpam-4831	306	11	,	,	PUNCT
ejpam-4831	306	12	1972	1972	NUM
ejpam-4831	306	13	.	.	PUNCT
ejpam-4831	307	1	[	[	X
ejpam-4831	307	2	10	10	NUM
ejpam-4831	307	3	]	]	X
ejpam-4831	307	4	r.	r.	PROPN
ejpam-4831	307	5	c.	c.	PROPN
ejpam-4831	307	6	rosier	rosier	PROPN
ejpam-4831	307	7	.	.	PUNCT
ejpam-4831	308	1	dual	dual	ADJ
ejpam-4831	308	2	space	space	NOUN
ejpam-4831	308	3	of	of	ADP
ejpam-4831	308	4	certain	certain	ADJ
ejpam-4831	308	5	vector	vector	NOUN
ejpam-4831	308	6	sequence	sequence	NOUN
ejpam-4831	308	7	spaces	space	VERB
ejpam-4831	308	8	.	.	PUNCT
ejpam-4831	309	1	pacific	pacific	PROPN
ejpam-4831	309	2	j.	j.	PROPN
ejpam-4831	309	3	math	math	PROPN
ejpam-4831	309	4	.	.	PUNCT
ejpam-4831	309	5	,	,	PUNCT
ejpam-4831	309	6	46(2):487–501	46(2):487–501	NOUN
ejpam-4831	309	7	,	,	PUNCT
ejpam-4831	309	8	1973	1973	NUM
ejpam-4831	309	9	.	.	PUNCT
ejpam-4831	310	1	[	[	X
ejpam-4831	310	2	11	11	NUM
ejpam-4831	310	3	]	]	PUNCT
ejpam-4831	310	4	m.	m.	NOUN
ejpam-4831	310	5	a.	a.	NOUN
ejpam-4831	310	6	ould	ould	AUX
ejpam-4831	310	7	sidaty	sidaty	VERB
ejpam-4831	310	8	.	.	PUNCT
ejpam-4831	311	1	reflexivity	reflexivity	NOUN
ejpam-4831	311	2	and	and	CCONJ
ejpam-4831	311	3	ak	ak	NOUN
ejpam-4831	311	4	-	-	PUNCT
ejpam-4831	311	5	property	property	NOUN
ejpam-4831	311	6	of	of	ADP
ejpam-4831	311	7	certain	certain	ADJ
ejpam-4831	311	8	vector	vector	NOUN
ejpam-4831	311	9	sequence	sequence	NOUN
ejpam-4831	311	10	spaces	space	VERB
ejpam-4831	311	11	.	.	PUNCT
ejpam-4831	312	1	bull	bull	NOUN
ejpam-4831	312	2	.	.	PUNCT
ejpam-4831	313	1	belg	belg	PROPN
ejpam-4831	313	2	.	.	PUNCT
ejpam-4831	314	1	math	math	NOUN
ejpam-4831	314	2	.	.	PUNCT
ejpam-4831	315	1	soc	soc	PROPN
ejpam-4831	315	2	.	.	PUNCT
ejpam-4831	315	3	,	,	PUNCT
ejpam-4831	315	4	10(4):579–583	10(4):579–583	PROPN
ejpam-4831	315	5	,	,	PUNCT
ejpam-4831	315	6	2003	2003	NUM
ejpam-4831	315	7	.	.	PUNCT
ejpam-4831	316	1	[	[	X
ejpam-4831	316	2	12	12	NUM
ejpam-4831	316	3	]	]	PUNCT
ejpam-4831	316	4	m.	m.	NOUN
ejpam-4831	316	5	a.	a.	NOUN
ejpam-4831	316	6	ould	ould	AUX
ejpam-4831	316	7	sidaty	sidaty	VERB
ejpam-4831	316	8	.	.	PUNCT
ejpam-4831	317	1	nuclearity	nuclearity	NOUN
ejpam-4831	317	2	of	of	ADP
ejpam-4831	317	3	certain	certain	ADJ
ejpam-4831	317	4	vector	vector	NOUN
ejpam-4831	317	5	-	-	PUNCT
ejpam-4831	317	6	valued	value	VERB
ejpam-4831	317	7	sequence	sequence	NOUN
ejpam-4831	317	8	spaces	space	VERB
ejpam-4831	317	9	.	.	PUNCT
ejpam-4831	318	1	rev	rev	PROPN
ejpam-4831	318	2	.	.	PUNCT
ejpam-4831	319	1	real	real	PROPN
ejpam-4831	319	2	academia	academia	PROPN
ejpam-4831	319	3	de	de	PROPN
ejpam-4831	319	4	ciencias	ciencias	PROPN
ejpam-4831	319	5	.	.	PUNCT
ejpam-4831	320	1	zaragoza	zaragoza	PROPN
ejpam-4831	320	2	.	.	PROPN
ejpam-4831	320	3	,	,	PUNCT
ejpam-4831	320	4	62:81–89	62:81–89	PROPN
ejpam-4831	320	5	,	,	PUNCT
ejpam-4831	320	6	2007	2007	NUM
ejpam-4831	320	7	.	.	PUNCT
ejpam-4831	321	1	[	[	X
ejpam-4831	321	2	13	13	NUM
ejpam-4831	321	3	]	]	PUNCT
ejpam-4831	321	4	m.	m.	NOUN
ejpam-4831	321	5	a.	a.	NOUN
ejpam-4831	321	6	ould	ould	AUX
ejpam-4831	321	7	sidaty	sidaty	VERB
ejpam-4831	321	8	.	.	PUNCT
ejpam-4831	322	1	reflexivity	reflexivity	NOUN
ejpam-4831	322	2	of	of	ADP
ejpam-4831	322	3	vector	vector	NOUN
ejpam-4831	322	4	-	-	PUNCT
ejpam-4831	322	5	valued	value	VERB
ejpam-4831	322	6	köthe	köthe	PROPN
ejpam-4831	322	7	-	-	PUNCT
ejpam-4831	322	8	orlicz	orlicz	ADJ
ejpam-4831	322	9	sequence	sequence	NOUN
ejpam-4831	322	10	spaces	space	VERB
ejpam-4831	322	11	.	.	PUNCT
ejpam-4831	323	1	turk	turk	PROPN
ejpam-4831	323	2	j	j	PROPN
ejpam-4831	323	3	math	math	PROPN
ejpam-4831	323	4	.	.	PUNCT
ejpam-4831	323	5	,	,	PUNCT
ejpam-4831	323	6	42(3):911–923	42(3):911–923	PROPN
ejpam-4831	323	7	,	,	PUNCT
ejpam-4831	323	8	2018	2018	NUM
ejpam-4831	323	9	.	.	PUNCT