id	sid	tid	token	lemma	pos
ejpam-4800	1	1	european	european	PROPN
ejpam-4800	1	2	journal	journal	PROPN
ejpam-4800	1	3	of	of	ADP
ejpam-4800	1	4	pure	pure	ADJ
ejpam-4800	1	5	and	and	CCONJ
ejpam-4800	1	6	applied	apply	VERB
ejpam-4800	1	7	mathematics	mathematic	NOUN
ejpam-4800	1	8	vol	vol	NOUN
ejpam-4800	1	9	.	.	PUNCT
ejpam-4800	2	1	16	16	NUM
ejpam-4800	2	2	,	,	PUNCT
ejpam-4800	2	3	no	no	INTJ
ejpam-4800	2	4	.	.	NOUN
ejpam-4800	2	5	3	3	NUM
ejpam-4800	2	6	,	,	PUNCT
ejpam-4800	2	7	2023	2023	NUM
ejpam-4800	2	8	,	,	PUNCT
ejpam-4800	2	9	1543	1543	NUM
ejpam-4800	2	10	-	-	SYM
ejpam-4800	2	11	1551	1551	NUM
ejpam-4800	2	12	issn	issn	PROPN
ejpam-4800	2	13	1307	1307	NUM
ejpam-4800	2	14	-	-	SYM
ejpam-4800	2	15	5543	5543	NUM
ejpam-4800	2	16	–	–	PUNCT
ejpam-4800	3	1	ejpam.com	ejpam.com	X
ejpam-4800	3	2	published	publish	VERB
ejpam-4800	3	3	by	by	ADP
ejpam-4800	3	4	new	new	PROPN
ejpam-4800	3	5	york	york	PROPN
ejpam-4800	3	6	business	business	PROPN
ejpam-4800	3	7	global	global	ADJ
ejpam-4800	3	8	strong	strong	ADJ
ejpam-4800	3	9	coproximinality	coproximinality	NOUN
ejpam-4800	3	10	in	in	ADP
ejpam-4800	3	11	bochner	bochner	NOUN
ejpam-4800	3	12	lp	lp	NOUN
ejpam-4800	3	13	-	-	PUNCT
ejpam-4800	3	14	spaces	space	NOUN
ejpam-4800	3	15	and	and	CCONJ
ejpam-4800	3	16	in	in	ADP
ejpam-4800	3	17	köthe	köthe	ADJ
ejpam-4800	3	18	spaces	space	NOUN
ejpam-4800	3	19	jamila	jamila	PROPN
ejpam-4800	3	20	jawdat	jawdat	PROPN
ejpam-4800	3	21	department	department	PROPN
ejpam-4800	3	22	of	of	ADP
ejpam-4800	3	23	mathematics	mathematic	NOUN
ejpam-4800	3	24	,	,	PUNCT
ejpam-4800	3	25	faculty	faculty	NOUN
ejpam-4800	3	26	of	of	ADP
ejpam-4800	3	27	science	science	NOUN
ejpam-4800	3	28	,	,	PUNCT
ejpam-4800	3	29	zarqa	zarqa	PROPN
ejpam-4800	3	30	university	university	PROPN
ejpam-4800	3	31	,	,	PUNCT
ejpam-4800	3	32	zarqa	zarqa	PROPN
ejpam-4800	3	33	,	,	PUNCT
ejpam-4800	3	34	jordan	jordan	PROPN
ejpam-4800	3	35	abstract	abstract	PROPN
ejpam-4800	3	36	.	.	PUNCT
ejpam-4800	4	1	in	in	ADP
ejpam-4800	4	2	this	this	DET
ejpam-4800	4	3	paper	paper	NOUN
ejpam-4800	4	4	,	,	PUNCT
ejpam-4800	4	5	we	we	PRON
ejpam-4800	4	6	study	study	VERB
ejpam-4800	4	7	strong	strong	ADJ
ejpam-4800	4	8	coproximinality	coproximinality	NOUN
ejpam-4800	4	9	in	in	ADP
ejpam-4800	4	10	bochner	bochner	NOUN
ejpam-4800	4	11	lp	lp	NOUN
ejpam-4800	4	12	-	-	PUNCT
ejpam-4800	4	13	spaces	space	NOUN
ejpam-4800	4	14	and	and	CCONJ
ejpam-4800	4	15	in	in	ADP
ejpam-4800	4	16	the	the	DET
ejpam-4800	4	17	köthe	köthe	PROPN
ejpam-4800	4	18	bochner	bochner	NOUN
ejpam-4800	4	19	function	function	NOUN
ejpam-4800	4	20	space	space	NOUN
ejpam-4800	4	21	e(x	e(x	NUM
ejpam-4800	4	22	)	)	PUNCT
ejpam-4800	4	23	.	.	PUNCT
ejpam-4800	5	1	we	we	PRON
ejpam-4800	5	2	investigate	investigate	VERB
ejpam-4800	5	3	some	some	DET
ejpam-4800	5	4	conditions	condition	NOUN
ejpam-4800	5	5	to	to	PART
ejpam-4800	5	6	be	be	AUX
ejpam-4800	5	7	imposed	impose	VERB
ejpam-4800	5	8	on	on	ADP
ejpam-4800	5	9	the	the	DET
ejpam-4800	5	10	subspace	subspace	NOUN
ejpam-4800	5	11	g	g	PROPN
ejpam-4800	5	12	of	of	ADP
ejpam-4800	5	13	the	the	DET
ejpam-4800	5	14	banach	banach	NOUN
ejpam-4800	5	15	space	space	NOUN
ejpam-4800	5	16	x	x	INTJ
ejpam-4800	5	17	such	such	ADJ
ejpam-4800	5	18	that	that	PRON
ejpam-4800	5	19	lp	lp	PROPN
ejpam-4800	5	20	(	(	PUNCT
ejpam-4800	5	21	µ,g	µ,g	PROPN
ejpam-4800	5	22	)	)	PUNCT
ejpam-4800	5	23	is	be	AUX
ejpam-4800	5	24	strongly	strongly	ADV
ejpam-4800	5	25	coproximinal	coproximinal	ADJ
ejpam-4800	5	26	in	in	ADP
ejpam-4800	5	27	lp	lp	PROPN
ejpam-4800	5	28	(	(	PUNCT
ejpam-4800	5	29	µ,x	µ,x	NOUN
ejpam-4800	5	30	)	)	PUNCT
ejpam-4800	5	31	,	,	PUNCT
ejpam-4800	5	32	1	1	NUM
ejpam-4800	5	33	≤	≤	NOUN
ejpam-4800	5	34	p	p	NOUN
ejpam-4800	5	35	<	<	X
ejpam-4800	5	36	∞.	∞.	PROPN
ejpam-4800	5	37	on	on	ADP
ejpam-4800	5	38	the	the	DET
ejpam-4800	5	39	other	other	ADJ
ejpam-4800	5	40	hand	hand	NOUN
ejpam-4800	5	41	,	,	PUNCT
ejpam-4800	5	42	we	we	PRON
ejpam-4800	5	43	prove	prove	VERB
ejpam-4800	5	44	that	that	SCONJ
ejpam-4800	5	45	if	if	SCONJ
ejpam-4800	5	46	g	g	PROPN
ejpam-4800	5	47	is	be	AUX
ejpam-4800	5	48	a	a	DET
ejpam-4800	5	49	separable	separable	ADJ
ejpam-4800	5	50	subspace	subspace	NOUN
ejpam-4800	5	51	of	of	ADP
ejpam-4800	5	52	x	x	PRON
ejpam-4800	5	53	then	then	ADV
ejpam-4800	5	54	g	g	PROPN
ejpam-4800	5	55	is	be	AUX
ejpam-4800	5	56	strongly	strongly	ADV
ejpam-4800	5	57	coproximinal	coproximinal	ADJ
ejpam-4800	5	58	in	in	ADP
ejpam-4800	5	59	x	x	SYM
ejpam-4800	5	60	if	if	SCONJ
ejpam-4800	5	61	and	and	CCONJ
ejpam-4800	5	62	only	only	ADV
ejpam-4800	5	63	if	if	SCONJ
ejpam-4800	5	64	e(g	e(g	PROPN
ejpam-4800	5	65	)	)	PUNCT
ejpam-4800	5	66	is	be	AUX
ejpam-4800	5	67	strongly	strongly	ADV
ejpam-4800	5	68	coproximinal	coproximinal	ADJ
ejpam-4800	5	69	in	in	ADP
ejpam-4800	5	70	e(x	e(x	NUM
ejpam-4800	5	71	)	)	PUNCT
ejpam-4800	5	72	,	,	PUNCT
ejpam-4800	5	73	provided	provide	VERB
ejpam-4800	5	74	that	that	SCONJ
ejpam-4800	5	75	e	e	NOUN
ejpam-4800	5	76	is	be	AUX
ejpam-4800	5	77	a	a	DET
ejpam-4800	5	78	strictly	strictly	ADV
ejpam-4800	5	79	monotone	monotone	ADJ
ejpam-4800	5	80	köthe	köthe	ADJ
ejpam-4800	5	81	space	space	NOUN
ejpam-4800	5	82	.	.	PUNCT
ejpam-4800	6	1	this	this	PRON
ejpam-4800	6	2	generalizes	generalize	VERB
ejpam-4800	6	3	some	some	DET
ejpam-4800	6	4	results	result	NOUN
ejpam-4800	6	5	in	in	ADP
ejpam-4800	6	6	the	the	DET
ejpam-4800	6	7	literature	literature	NOUN
ejpam-4800	6	8	.	.	PUNCT
ejpam-4800	7	1	some	some	DET
ejpam-4800	7	2	other	other	ADJ
ejpam-4800	7	3	results	result	NOUN
ejpam-4800	7	4	in	in	ADP
ejpam-4800	7	5	this	this	DET
ejpam-4800	7	6	direction	direction	NOUN
ejpam-4800	7	7	are	be	AUX
ejpam-4800	7	8	also	also	ADV
ejpam-4800	7	9	presented	present	VERB
ejpam-4800	7	10	.	.	PUNCT
ejpam-4800	8	1	2020	2020	NUM
ejpam-4800	8	2	mathematics	mathematics	PROPN
ejpam-4800	8	3	subject	subject	NOUN
ejpam-4800	8	4	classifications	classification	NOUN
ejpam-4800	8	5	:	:	PUNCT
ejpam-4800	8	6	41a50	41a50	NUM
ejpam-4800	8	7	,	,	PUNCT
ejpam-4800	8	8	46e30	46e30	NUM
ejpam-4800	8	9	,	,	PUNCT
ejpam-4800	8	10	46b20	46b20	NUM
ejpam-4800	8	11	key	key	ADJ
ejpam-4800	8	12	words	word	NOUN
ejpam-4800	8	13	and	and	CCONJ
ejpam-4800	8	14	phrases	phrase	NOUN
ejpam-4800	8	15	:	:	PUNCT
ejpam-4800	8	16	strong	strong	ADJ
ejpam-4800	8	17	coapproximation	coapproximation	NOUN
ejpam-4800	8	18	,	,	PUNCT
ejpam-4800	8	19	bochner	bochner	NOUN
ejpam-4800	8	20	spaces	space	NOUN
ejpam-4800	8	21	,	,	PUNCT
ejpam-4800	8	22	köthe	köthe	PRON
ejpam-4800	8	23	function	function	NOUN
ejpam-4800	8	24	space	space	NOUN
ejpam-4800	8	25	1	1	NUM
ejpam-4800	8	26	.	.	PUNCT
ejpam-4800	9	1	introduction	introduction	NOUN
ejpam-4800	9	2	and	and	CCONJ
ejpam-4800	9	3	some	some	DET
ejpam-4800	9	4	preliminaries	preliminary	NOUN
ejpam-4800	9	5	best	good	ADJ
ejpam-4800	9	6	approximation	approximation	NOUN
ejpam-4800	9	7	theory	theory	NOUN
ejpam-4800	9	8	in	in	ADP
ejpam-4800	9	9	normed	normed	ADJ
ejpam-4800	9	10	linear	linear	PROPN
ejpam-4800	9	11	spaces	space	NOUN
ejpam-4800	9	12	and	and	CCONJ
ejpam-4800	9	13	that	that	PRON
ejpam-4800	9	14	of	of	ADP
ejpam-4800	9	15	best	good	ADJ
ejpam-4800	9	16	coapproximation	coapproximation	NOUN
ejpam-4800	9	17	are	be	AUX
ejpam-4800	9	18	counterparts	counterpart	NOUN
ejpam-4800	9	19	.	.	PUNCT
ejpam-4800	10	1	since	since	SCONJ
ejpam-4800	10	2	1970	1970	NUM
ejpam-4800	10	3	,	,	PUNCT
ejpam-4800	10	4	[	[	X
ejpam-4800	10	5	14	14	NUM
ejpam-4800	10	6	]	]	PUNCT
ejpam-4800	10	7	,	,	PUNCT
ejpam-4800	10	8	this	this	DET
ejpam-4800	10	9	topic	topic	NOUN
ejpam-4800	10	10	had	have	AUX
ejpam-4800	10	11	been	be	AUX
ejpam-4800	10	12	intensively	intensively	ADV
ejpam-4800	10	13	studied	study	VERB
ejpam-4800	10	14	,	,	PUNCT
ejpam-4800	10	15	and	and	CCONJ
ejpam-4800	10	16	a	a	DET
ejpam-4800	10	17	huge	huge	ADJ
ejpam-4800	10	18	work	work	NOUN
ejpam-4800	10	19	have	have	AUX
ejpam-4800	10	20	been	be	AUX
ejpam-4800	10	21	published	publish	VERB
ejpam-4800	10	22	,	,	PUNCT
ejpam-4800	10	23	see	see	VERB
ejpam-4800	10	24	for	for	ADP
ejpam-4800	10	25	example	example	NOUN
ejpam-4800	11	1	[	[	X
ejpam-4800	11	2	1	1	NUM
ejpam-4800	11	3	,	,	PUNCT
ejpam-4800	11	4	2	2	NUM
ejpam-4800	11	5	,	,	PUNCT
ejpam-4800	11	6	4	4	NUM
ejpam-4800	11	7	,	,	PUNCT
ejpam-4800	11	8	6	6	NUM
ejpam-4800	11	9	,	,	PUNCT
ejpam-4800	11	10	8–10	8–10	NOUN
ejpam-4800	11	11	,	,	PUNCT
ejpam-4800	11	12	12	12	NUM
ejpam-4800	11	13	,	,	PUNCT
ejpam-4800	11	14	13	13	NUM
ejpam-4800	11	15	]	]	PUNCT
ejpam-4800	11	16	.	.	PUNCT
ejpam-4800	12	1	if	if	SCONJ
ejpam-4800	12	2	x	x	PRON
ejpam-4800	12	3	is	be	AUX
ejpam-4800	12	4	a	a	DET
ejpam-4800	12	5	banach	banach	NOUN
ejpam-4800	12	6	space	space	NOUN
ejpam-4800	12	7	with	with	ADP
ejpam-4800	12	8	g	g	PROPN
ejpam-4800	12	9	a	a	DET
ejpam-4800	12	10	closed	closed	ADJ
ejpam-4800	12	11	subspace	subspace	NOUN
ejpam-4800	12	12	,	,	PUNCT
ejpam-4800	12	13	then	then	ADV
ejpam-4800	12	14	g	g	PROPN
ejpam-4800	12	15	is	be	AUX
ejpam-4800	12	16	called	call	VERB
ejpam-4800	12	17	proximinal	proximinal	ADJ
ejpam-4800	12	18	in	in	ADP
ejpam-4800	12	19	x	x	SYM
ejpam-4800	12	20	,	,	PUNCT
ejpam-4800	12	21	if	if	SCONJ
ejpam-4800	12	22	for	for	ADP
ejpam-4800	12	23	each	each	DET
ejpam-4800	12	24	x	x	SYM
ejpam-4800	12	25	∈	∈	PROPN
ejpam-4800	12	26	x	x	NOUN
ejpam-4800	12	27	,	,	PUNCT
ejpam-4800	12	28	there	there	PRON
ejpam-4800	12	29	is	be	VERB
ejpam-4800	12	30	g0	g0	ADJ
ejpam-4800	12	31	in	in	ADP
ejpam-4800	12	32	g	g	NOUN
ejpam-4800	12	33	satisfying	satisfy	VERB
ejpam-4800	12	34	||g0	||g0	PROPN
ejpam-4800	12	35	−	−	PROPN
ejpam-4800	12	36	x||	x||	NOUN
ejpam-4800	12	37	≤	≤	PROPN
ejpam-4800	13	1	||x−	||x−	PROPN
ejpam-4800	13	2	g||	g||	NOUN
ejpam-4800	13	3	,	,	PUNCT
ejpam-4800	13	4	for	for	ADP
ejpam-4800	13	5	all	all	DET
ejpam-4800	13	6	g	g	PROPN
ejpam-4800	13	7	∈	∈	PROPN
ejpam-4800	13	8	g.	g.	NOUN
ejpam-4800	13	9	(	(	PUNCT
ejpam-4800	13	10	1	1	NUM
ejpam-4800	13	11	)	)	PUNCT
ejpam-4800	13	12	g0	g0	NOUN
ejpam-4800	13	13	is	be	AUX
ejpam-4800	13	14	called	call	VERB
ejpam-4800	13	15	an	an	DET
ejpam-4800	13	16	element	element	NOUN
ejpam-4800	13	17	of	of	ADP
ejpam-4800	13	18	best	good	ADJ
ejpam-4800	13	19	approximation	approximation	NOUN
ejpam-4800	13	20	to	to	ADP
ejpam-4800	13	21	x	x	PUNCT
ejpam-4800	13	22	from	from	ADP
ejpam-4800	13	23	g.	g.	PROPN
ejpam-4800	13	24	it	it	PRON
ejpam-4800	13	25	is	be	AUX
ejpam-4800	13	26	well	well	ADV
ejpam-4800	13	27	-	-	PUNCT
ejpam-4800	13	28	known	know	VERB
ejpam-4800	13	29	that	that	SCONJ
ejpam-4800	13	30	,	,	PUNCT
ejpam-4800	13	31	d(x	d(x	PROPN
ejpam-4800	13	32	,	,	PUNCT
ejpam-4800	13	33	g	g	NOUN
ejpam-4800	13	34	)	)	PUNCT
ejpam-4800	13	35	=	=	SYM
ejpam-4800	13	36	inf{||x	inf{||x	PROPN
ejpam-4800	13	37	−	−	NOUN
ejpam-4800	13	38	g||,∀g	g||,∀g	NOUN
ejpam-4800	13	39	∈	∈	PROPN
ejpam-4800	13	40	g	g	NOUN
ejpam-4800	13	41	}	}	PUNCT
ejpam-4800	13	42	.	.	PUNCT
ejpam-4800	14	1	hence	hence	ADV
ejpam-4800	14	2	,	,	PUNCT
ejpam-4800	14	3	g	g	PROPN
ejpam-4800	14	4	is	be	AUX
ejpam-4800	14	5	proximinal	proximinal	ADJ
ejpam-4800	14	6	in	in	ADP
ejpam-4800	14	7	x	x	PUNCT
ejpam-4800	14	8	if	if	SCONJ
ejpam-4800	14	9	for	for	ADP
ejpam-4800	14	10	each	each	PRON
ejpam-4800	14	11	x	x	PUNCT
ejpam-4800	14	12	in	in	ADP
ejpam-4800	14	13	x	x	SYM
ejpam-4800	14	14	,	,	PUNCT
ejpam-4800	14	15	there	there	PRON
ejpam-4800	14	16	exists	exist	VERB
ejpam-4800	14	17	g0	g0	ADJ
ejpam-4800	14	18	in	in	ADP
ejpam-4800	14	19	g	g	PROPN
ejpam-4800	14	20	that	that	SCONJ
ejpam-4800	14	21	satisfies	satisfie	NOUN
ejpam-4800	14	22	,	,	PUNCT
ejpam-4800	14	23	||g0	||g0	PROPN
ejpam-4800	14	24	−	−	NOUN
ejpam-4800	14	25	x||	x||	PUNCT
ejpam-4800	15	1	=	=	SYM
ejpam-4800	15	2	d(x	d(x	PROPN
ejpam-4800	15	3	,	,	PUNCT
ejpam-4800	15	4	g	g	NOUN
ejpam-4800	15	5	)	)	PUNCT
ejpam-4800	15	6	on	on	ADP
ejpam-4800	15	7	the	the	DET
ejpam-4800	15	8	other	other	ADJ
ejpam-4800	15	9	hand	hand	NOUN
ejpam-4800	15	10	,	,	PUNCT
ejpam-4800	15	11	g	g	PROPN
ejpam-4800	15	12	is	be	AUX
ejpam-4800	15	13	called	call	VERB
ejpam-4800	15	14	coproximinal	coproximinal	ADJ
ejpam-4800	15	15	in	in	ADP
ejpam-4800	15	16	x	x	SYM
ejpam-4800	15	17	,	,	PUNCT
ejpam-4800	15	18	if	if	SCONJ
ejpam-4800	15	19	for	for	ADP
ejpam-4800	15	20	each	each	DET
ejpam-4800	15	21	x	x	SYM
ejpam-4800	15	22	∈	∈	PROPN
ejpam-4800	15	23	x	x	NOUN
ejpam-4800	15	24	,	,	PUNCT
ejpam-4800	15	25	there	there	PRON
ejpam-4800	15	26	is	be	VERB
ejpam-4800	15	27	g0	g0	ADJ
ejpam-4800	15	28	in	in	ADP
ejpam-4800	15	29	g	g	NOUN
ejpam-4800	15	30	satisfying	satisfy	VERB
ejpam-4800	15	31	||g0	||g0	PROPN
ejpam-4800	16	1	−	−	PROPN
ejpam-4800	16	2	g||	g||	NOUN
ejpam-4800	16	3	≤	≤	NOUN
ejpam-4800	17	1	||x−	||x−	PROPN
ejpam-4800	17	2	g||	g||	NOUN
ejpam-4800	17	3	,	,	PUNCT
ejpam-4800	17	4	for	for	ADP
ejpam-4800	17	5	all	all	DET
ejpam-4800	17	6	g	g	PROPN
ejpam-4800	17	7	∈	∈	PROPN
ejpam-4800	17	8	g.	g.	NOUN
ejpam-4800	17	9	(	(	PUNCT
ejpam-4800	17	10	2	2	NUM
ejpam-4800	17	11	)	)	PUNCT
ejpam-4800	17	12	doi	doi	NOUN
ejpam-4800	17	13	:	:	PUNCT
ejpam-4800	17	14	https://doi.org/10.29020/nybg.ejpam.v16i3.4800	https://doi.org/10.29020/nybg.ejpam.v16i3.4800	PROPN
ejpam-4800	17	15	email	email	NOUN
ejpam-4800	17	16	address	address	NOUN
ejpam-4800	17	17	:	:	PUNCT
ejpam-4800	18	1	jjawdat@zu.edu.jo	jjawdat@zu.edu.jo	PROPN
ejpam-4800	18	2	(	(	PUNCT
ejpam-4800	18	3	j.	j.	PROPN
ejpam-4800	18	4	jawdat	jawdat	PROPN
ejpam-4800	18	5	)	)	PUNCT
ejpam-4800	18	6	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4800	18	7	1543	1543	NUM
ejpam-4800	18	8	©	©	PROPN
ejpam-4800	18	9	2023	2023	NUM
ejpam-4800	18	10	ejpam	ejpam	NOUN
ejpam-4800	18	11	all	all	DET
ejpam-4800	18	12	rights	right	NOUN
ejpam-4800	18	13	reserved	reserve	VERB
ejpam-4800	18	14	.	.	PUNCT
ejpam-4800	19	1	j.	j.	PROPN
ejpam-4800	19	2	jawdat	jawdat	PROPN
ejpam-4800	19	3	/	/	SYM
ejpam-4800	19	4	eur	eur	PROPN
ejpam-4800	19	5	.	.	PUNCT
ejpam-4800	20	1	j.	j.	PROPN
ejpam-4800	20	2	pure	pure	PROPN
ejpam-4800	20	3	appl	appl	PROPN
ejpam-4800	20	4	.	.	PROPN
ejpam-4800	20	5	math	math	PROPN
ejpam-4800	20	6	,	,	PUNCT
ejpam-4800	20	7	16	16	NUM
ejpam-4800	20	8	(	(	PUNCT
ejpam-4800	20	9	3	3	NUM
ejpam-4800	20	10	)	)	PUNCT
ejpam-4800	20	11	(	(	PUNCT
ejpam-4800	20	12	2023	2023	NUM
ejpam-4800	20	13	)	)	PUNCT
ejpam-4800	20	14	,	,	PUNCT
ejpam-4800	20	15	1543	1543	NUM
ejpam-4800	20	16	-	-	SYM
ejpam-4800	20	17	1551	1551	NUM
ejpam-4800	20	18	1544	1544	NUM
ejpam-4800	20	19	again	again	ADV
ejpam-4800	20	20	,	,	PUNCT
ejpam-4800	20	21	g0	g0	PROPN
ejpam-4800	20	22	is	be	AUX
ejpam-4800	20	23	called	call	VERB
ejpam-4800	20	24	an	an	DET
ejpam-4800	20	25	element	element	NOUN
ejpam-4800	20	26	of	of	ADP
ejpam-4800	20	27	best	good	ADJ
ejpam-4800	20	28	coapproximation	coapproximation	NOUN
ejpam-4800	20	29	to	to	ADP
ejpam-4800	20	30	x	x	PUNCT
ejpam-4800	20	31	from	from	ADP
ejpam-4800	20	32	g.	g.	PROPN
ejpam-4800	20	33	let	let	VERB
ejpam-4800	20	34	pg(x	pg(x	NUM
ejpam-4800	20	35	)	)	PUNCT
ejpam-4800	20	36	(	(	PUNCT
ejpam-4800	20	37	resp	resp	NOUN
ejpam-4800	20	38	.	.	PUNCT
ejpam-4800	21	1	rg(x	rg(x	PUNCT
ejpam-4800	21	2	)	)	PUNCT
ejpam-4800	21	3	)	)	PUNCT
ejpam-4800	21	4	be	be	AUX
ejpam-4800	21	5	the	the	DET
ejpam-4800	21	6	set	set	NOUN
ejpam-4800	21	7	of	of	ADP
ejpam-4800	21	8	all	all	DET
ejpam-4800	21	9	elements	element	NOUN
ejpam-4800	21	10	in	in	ADP
ejpam-4800	21	11	g	g	PROPN
ejpam-4800	21	12	that	that	DET
ejpam-4800	21	13	satisfy	satisfy	NOUN
ejpam-4800	21	14	(	(	PUNCT
ejpam-4800	21	15	1	1	NUM
ejpam-4800	21	16	)	)	PUNCT
ejpam-4800	21	17	(	(	PUNCT
ejpam-4800	21	18	resp	resp	NOUN
ejpam-4800	21	19	.	.	PUNCT
ejpam-4800	22	1	(	(	PUNCT
ejpam-4800	22	2	2	2	NUM
ejpam-4800	22	3	)	)	PUNCT
ejpam-4800	22	4	)	)	PUNCT
ejpam-4800	22	5	.	.	PUNCT
ejpam-4800	23	1	the	the	DET
ejpam-4800	23	2	notion	notion	NOUN
ejpam-4800	23	3	of	of	ADP
ejpam-4800	23	4	strong	strong	ADJ
ejpam-4800	23	5	proximinality	proximinality	NOUN
ejpam-4800	23	6	in	in	ADP
ejpam-4800	23	7	general	general	ADJ
ejpam-4800	23	8	banach	banach	NOUN
ejpam-4800	23	9	spaces	space	NOUN
ejpam-4800	23	10	,	,	PUNCT
ejpam-4800	23	11	was	be	AUX
ejpam-4800	23	12	first	first	ADV
ejpam-4800	23	13	studied	study	VERB
ejpam-4800	23	14	by	by	ADP
ejpam-4800	23	15	godefroy	godefroy	ADJ
ejpam-4800	23	16	and	and	CCONJ
ejpam-4800	23	17	indumathi	indumathi	NOUN
ejpam-4800	23	18	,	,	PUNCT
ejpam-4800	23	19	[	[	X
ejpam-4800	23	20	3	3	NUM
ejpam-4800	23	21	]	]	PUNCT
ejpam-4800	23	22	,	,	PUNCT
ejpam-4800	23	23	and	and	CCONJ
ejpam-4800	23	24	is	be	AUX
ejpam-4800	23	25	defined	define	VERB
ejpam-4800	23	26	as	as	ADP
ejpam-4800	23	27	follows	follow	VERB
ejpam-4800	23	28	.	.	PUNCT
ejpam-4800	24	1	definition	definition	NOUN
ejpam-4800	24	2	1	1	NUM
ejpam-4800	24	3	.	.	PUNCT
ejpam-4800	25	1	a	a	DET
ejpam-4800	25	2	proximinal	proximinal	ADJ
ejpam-4800	25	3	subspace	subspace	NOUN
ejpam-4800	25	4	g	g	PROPN
ejpam-4800	25	5	of	of	ADP
ejpam-4800	25	6	x	x	PROPN
ejpam-4800	25	7	is	be	AUX
ejpam-4800	25	8	called	call	VERB
ejpam-4800	25	9	strongly	strongly	ADV
ejpam-4800	25	10	proximinal	proximinal	ADJ
ejpam-4800	25	11	at	at	ADP
ejpam-4800	25	12	x	x	X
ejpam-4800	25	13	∈	∈	PROPN
ejpam-4800	25	14	x	x	INTJ
ejpam-4800	25	15	if	if	SCONJ
ejpam-4800	25	16	for	for	ADP
ejpam-4800	25	17	any	any	DET
ejpam-4800	25	18	ε	ε	PROPN
ejpam-4800	25	19	>	>	X
ejpam-4800	25	20	0	0	PROPN
ejpam-4800	25	21	,	,	PUNCT
ejpam-4800	25	22	there	there	PRON
ejpam-4800	25	23	exists	exist	VERB
ejpam-4800	25	24	δ	δ	PROPN
ejpam-4800	25	25	>	>	X
ejpam-4800	25	26	0	0	NUM
ejpam-4800	26	1	such	such	ADJ
ejpam-4800	26	2	that	that	SCONJ
ejpam-4800	26	3	pg(x	pg(x	NOUN
ejpam-4800	26	4	,	,	PUNCT
ejpam-4800	26	5	δ	δ	PROPN
ejpam-4800	26	6	)	)	PUNCT
ejpam-4800	26	7	⊆	⊆	NUM
ejpam-4800	26	8	pg(x)+	pg(x)+	VERB
ejpam-4800	26	9	εbx	εbx	NOUN
ejpam-4800	26	10	,	,	PUNCT
ejpam-4800	26	11	where	where	SCONJ
ejpam-4800	26	12	bx	bx	PROPN
ejpam-4800	26	13	is	be	AUX
ejpam-4800	26	14	the	the	DET
ejpam-4800	26	15	unit	unit	NOUN
ejpam-4800	26	16	ball	ball	NOUN
ejpam-4800	26	17	of	of	ADP
ejpam-4800	26	18	x	x	PROPN
ejpam-4800	26	19	and	and	CCONJ
ejpam-4800	26	20	pg(x	pg(x	NUM
ejpam-4800	26	21	,	,	PUNCT
ejpam-4800	26	22	δ	δ	PROPN
ejpam-4800	26	23	)	)	PUNCT
ejpam-4800	26	24	=	=	PRON
ejpam-4800	26	25	{	{	PUNCT
ejpam-4800	26	26	z	z	NOUN
ejpam-4800	26	27	∈	∈	PROPN
ejpam-4800	26	28	g	g	NOUN
ejpam-4800	26	29	:	:	PUNCT
ejpam-4800	26	30	||x−	||x−	PROPN
ejpam-4800	26	31	z||	z||	PROPN
ejpam-4800	26	32	<	<	X
ejpam-4800	26	33	d(x	d(x	PROPN
ejpam-4800	26	34	,	,	PUNCT
ejpam-4800	26	35	g	g	NOUN
ejpam-4800	26	36	)	)	PUNCT
ejpam-4800	26	37	+	+	NUM
ejpam-4800	26	38	δ	δ	X
ejpam-4800	26	39	}	}	PUNCT
ejpam-4800	26	40	.	.	PUNCT
ejpam-4800	27	1	usually	usually	ADV
ejpam-4800	27	2	,	,	PUNCT
ejpam-4800	27	3	pg(x	pg(x	X
ejpam-4800	27	4	,	,	PUNCT
ejpam-4800	27	5	δ	δ	PROPN
ejpam-4800	27	6	)	)	PUNCT
ejpam-4800	27	7	is	be	AUX
ejpam-4800	27	8	referred	refer	VERB
ejpam-4800	27	9	to	to	ADP
ejpam-4800	27	10	as	as	ADP
ejpam-4800	27	11	the	the	DET
ejpam-4800	27	12	set	set	NOUN
ejpam-4800	27	13	of	of	ADP
ejpam-4800	27	14	near	near	ADV
ejpam-4800	27	15	best	good	ADJ
ejpam-4800	27	16	approximation	approximation	NOUN
ejpam-4800	27	17	points	point	NOUN
ejpam-4800	27	18	to	to	ADP
ejpam-4800	27	19	x	x	PUNCT
ejpam-4800	27	20	from	from	ADP
ejpam-4800	27	21	g.	g.	PROPN
ejpam-4800	27	22	in	in	ADP
ejpam-4800	27	23	addition	addition	NOUN
ejpam-4800	27	24	,	,	PUNCT
ejpam-4800	27	25	if	if	SCONJ
ejpam-4800	27	26	g	g	PROPN
ejpam-4800	27	27	is	be	AUX
ejpam-4800	27	28	strongly	strongly	ADV
ejpam-4800	27	29	proximinal	proximinal	ADJ
ejpam-4800	27	30	at	at	ADP
ejpam-4800	27	31	each	each	DET
ejpam-4800	27	32	x	x	SYM
ejpam-4800	27	33	∈	∈	PROPN
ejpam-4800	27	34	x	x	X
ejpam-4800	27	35	then	then	ADV
ejpam-4800	27	36	it	it	PRON
ejpam-4800	27	37	is	be	AUX
ejpam-4800	27	38	called	call	VERB
ejpam-4800	27	39	strongly	strongly	ADV
ejpam-4800	27	40	proximinal	proximinal	ADJ
ejpam-4800	27	41	in	in	ADP
ejpam-4800	27	42	x.	x.	PROPN
ejpam-4800	27	43	an	an	DET
ejpam-4800	27	44	equivalent	equivalent	ADJ
ejpam-4800	27	45	definition	definition	NOUN
ejpam-4800	27	46	for	for	ADP
ejpam-4800	27	47	strong	strong	ADJ
ejpam-4800	27	48	proximinality	proximinality	NOUN
ejpam-4800	27	49	in	in	ADP
ejpam-4800	27	50	banach	banach	NOUN
ejpam-4800	27	51	spaces	space	NOUN
ejpam-4800	27	52	is	be	AUX
ejpam-4800	27	53	given	give	VERB
ejpam-4800	27	54	using	use	VERB
ejpam-4800	27	55	the	the	DET
ejpam-4800	27	56	notion	notion	NOUN
ejpam-4800	27	57	of	of	ADP
ejpam-4800	27	58	minimizing	minimize	VERB
ejpam-4800	27	59	sequences	sequence	NOUN
ejpam-4800	27	60	,	,	PUNCT
ejpam-4800	27	61	defined	define	VERB
ejpam-4800	27	62	as	as	ADP
ejpam-4800	27	63	below	below	ADV
ejpam-4800	27	64	.	.	PUNCT
ejpam-4800	28	1	definition	definition	NOUN
ejpam-4800	28	2	2	2	NUM
ejpam-4800	28	3	.	.	PUNCT
ejpam-4800	29	1	let	let	VERB
ejpam-4800	29	2	g	g	PRON
ejpam-4800	29	3	be	be	AUX
ejpam-4800	29	4	a	a	DET
ejpam-4800	29	5	subspace	subspace	NOUN
ejpam-4800	29	6	of	of	ADP
ejpam-4800	29	7	x	x	PRON
ejpam-4800	29	8	,	,	PUNCT
ejpam-4800	29	9	that	that	PRON
ejpam-4800	29	10	is	be	AUX
ejpam-4800	29	11	proximinal	proximinal	ADJ
ejpam-4800	29	12	in	in	ADP
ejpam-4800	29	13	x.	x.	PROPN
ejpam-4800	29	14	a	a	DET
ejpam-4800	29	15	sequence	sequence	NOUN
ejpam-4800	29	16	{	{	PUNCT
ejpam-4800	29	17	yn	yn	NOUN
ejpam-4800	29	18	}	}	PUNCT
ejpam-4800	29	19	in	in	ADP
ejpam-4800	29	20	g	g	PROPN
ejpam-4800	29	21	is	be	AUX
ejpam-4800	29	22	called	call	VERB
ejpam-4800	29	23	a	a	DET
ejpam-4800	29	24	minimizing	minimize	VERB
ejpam-4800	29	25	sequence	sequence	NOUN
ejpam-4800	29	26	for	for	ADP
ejpam-4800	29	27	an	an	DET
ejpam-4800	29	28	element	element	NOUN
ejpam-4800	29	29	x	x	PUNCT
ejpam-4800	29	30	in	in	ADP
ejpam-4800	29	31	x	x	SYM
ejpam-4800	29	32	if	if	SCONJ
ejpam-4800	29	33	limn→∞||	limn→∞||	NUM
ejpam-4800	29	34	x−	x−	NOUN
ejpam-4800	29	35	yn||	yn||	PUNCT
ejpam-4800	30	1	=	=	PUNCT
ejpam-4800	30	2	d(x	d(x	PROPN
ejpam-4800	30	3	,	,	PUNCT
ejpam-4800	30	4	g	g	NOUN
ejpam-4800	30	5	)	)	PUNCT
ejpam-4800	30	6	definition	definition	NOUN
ejpam-4800	30	7	3	3	NUM
ejpam-4800	30	8	.	.	PUNCT
ejpam-4800	31	1	a	a	DET
ejpam-4800	31	2	subset	subset	NOUN
ejpam-4800	31	3	g	g	NOUN
ejpam-4800	31	4	,	,	PUNCT
ejpam-4800	31	5	that	that	PRON
ejpam-4800	31	6	is	be	AUX
ejpam-4800	31	7	proximinal	proximinal	ADJ
ejpam-4800	31	8	in	in	ADP
ejpam-4800	31	9	x	x	X
ejpam-4800	31	10	,	,	PUNCT
ejpam-4800	31	11	is	be	AUX
ejpam-4800	31	12	called	call	VERB
ejpam-4800	31	13	strongly	strongly	ADV
ejpam-4800	31	14	proximinal	proximinal	ADJ
ejpam-4800	31	15	in	in	ADP
ejpam-4800	31	16	x	x	SYM
ejpam-4800	31	17	,	,	PUNCT
ejpam-4800	31	18	if	if	SCONJ
ejpam-4800	31	19	∀x	∀x	VERB
ejpam-4800	31	20	∈	∈	PROPN
ejpam-4800	31	21	x	x	X
ejpam-4800	31	22	and	and	CCONJ
ejpam-4800	31	23	any	any	DET
ejpam-4800	31	24	minimizing	minimize	VERB
ejpam-4800	31	25	sequence	sequence	NOUN
ejpam-4800	31	26	{	{	PUNCT
ejpam-4800	31	27	yn	yn	NOUN
ejpam-4800	31	28	}	}	PUNCT
ejpam-4800	31	29	in	in	ADP
ejpam-4800	31	30	g	g	NOUN
ejpam-4800	31	31	for	for	ADP
ejpam-4800	31	32	x	x	SYM
ejpam-4800	31	33	,	,	PUNCT
ejpam-4800	31	34	∃	∃	PROPN
ejpam-4800	31	35	a	a	DET
ejpam-4800	31	36	subsequence	subsequence	NOUN
ejpam-4800	31	37	{	{	PUNCT
ejpam-4800	31	38	ynk	ynk	NOUN
ejpam-4800	31	39	}	}	PUNCT
ejpam-4800	31	40	of	of	ADP
ejpam-4800	31	41	{	{	PUNCT
ejpam-4800	31	42	yn	yn	NOUN
ejpam-4800	31	43	}	}	PUNCT
ejpam-4800	31	44	and	and	CCONJ
ejpam-4800	31	45	a	a	DET
ejpam-4800	31	46	sequence	sequence	NOUN
ejpam-4800	31	47	{	{	PUNCT
ejpam-4800	31	48	zn	zn	NOUN
ejpam-4800	31	49	}	}	PUNCT
ejpam-4800	31	50	in	in	ADP
ejpam-4800	31	51	pg(x	pg(x	NOUN
ejpam-4800	31	52	)	)	PUNCT
ejpam-4800	31	53	satisfying	satisfy	VERB
ejpam-4800	31	54	||ynk	||ynk	ADP
ejpam-4800	31	55	−	−	PROPN
ejpam-4800	31	56	zn||	zn||	PROPN
ejpam-4800	31	57	→	→	X
ejpam-4800	31	58	0	0	X
ejpam-4800	31	59	.	.	PUNCT
ejpam-4800	32	1	in	in	ADP
ejpam-4800	32	2	other	other	ADJ
ejpam-4800	32	3	words	word	NOUN
ejpam-4800	32	4	,	,	PUNCT
ejpam-4800	32	5	the	the	DET
ejpam-4800	32	6	sequence	sequence	NOUN
ejpam-4800	32	7	{	{	PUNCT
ejpam-4800	32	8	ynk	ynk	NOUN
ejpam-4800	32	9	}	}	PUNCT
ejpam-4800	32	10	satisfies	satisfie	NOUN
ejpam-4800	32	11	d(ynk	d(ynk	NOUN
ejpam-4800	32	12	,	,	PUNCT
ejpam-4800	32	13	pg(x	pg(x	NUM
ejpam-4800	32	14	)	)	PUNCT
ejpam-4800	32	15	)	)	PUNCT
ejpam-4800	33	1	→	→	SYM
ejpam-4800	33	2	0	0	NUM
ejpam-4800	33	3	whenever	whenever	SCONJ
ejpam-4800	33	4	||x	||x	PROPN
ejpam-4800	33	5	−	−	PROPN
ejpam-4800	33	6	yn||	yn||	PROPN
ejpam-4800	33	7	→	→	SYM
ejpam-4800	33	8	d(x	d(x	PROPN
ejpam-4800	33	9	,	,	PUNCT
ejpam-4800	33	10	g	g	NOUN
ejpam-4800	33	11	)	)	PUNCT
ejpam-4800	33	12	.	.	PUNCT
ejpam-4800	34	1	in	in	ADP
ejpam-4800	34	2	recent	recent	ADJ
ejpam-4800	34	3	years	year	NOUN
ejpam-4800	34	4	,	,	PUNCT
ejpam-4800	34	5	strong	strong	ADJ
ejpam-4800	34	6	proximinality	proximinality	NOUN
ejpam-4800	34	7	has	have	AUX
ejpam-4800	34	8	become	become	VERB
ejpam-4800	34	9	a	a	DET
ejpam-4800	34	10	topic	topic	NOUN
ejpam-4800	34	11	of	of	ADP
ejpam-4800	34	12	much	much	ADJ
ejpam-4800	34	13	interest	interest	NOUN
ejpam-4800	34	14	,	,	PUNCT
ejpam-4800	34	15	see	see	VERB
ejpam-4800	34	16	[	[	X
ejpam-4800	34	17	3	3	NUM
ejpam-4800	34	18	,	,	PUNCT
ejpam-4800	34	19	5	5	NUM
ejpam-4800	34	20	,	,	PUNCT
ejpam-4800	34	21	7	7	NUM
ejpam-4800	34	22	,	,	PUNCT
ejpam-4800	34	23	15	15	NUM
ejpam-4800	34	24	]	]	PUNCT
ejpam-4800	34	25	and	and	CCONJ
ejpam-4800	34	26	the	the	DET
ejpam-4800	34	27	references	reference	NOUN
ejpam-4800	34	28	therein	therein	ADV
ejpam-4800	34	29	.	.	PUNCT
ejpam-4800	35	1	it	it	PRON
ejpam-4800	35	2	is	be	AUX
ejpam-4800	35	3	well	well	ADV
ejpam-4800	35	4	known	know	VERB
ejpam-4800	35	5	that	that	SCONJ
ejpam-4800	35	6	if	if	SCONJ
ejpam-4800	35	7	g	g	PROPN
ejpam-4800	35	8	is	be	AUX
ejpam-4800	35	9	finite	finite	ADJ
ejpam-4800	35	10	dimensional	dimensional	ADJ
ejpam-4800	35	11	in	in	ADP
ejpam-4800	35	12	x	x	SYM
ejpam-4800	35	13	then	then	ADV
ejpam-4800	35	14	g	g	PROPN
ejpam-4800	35	15	is	be	AUX
ejpam-4800	35	16	strongly	strongly	ADV
ejpam-4800	35	17	proximinal	proximinal	ADJ
ejpam-4800	35	18	in	in	ADP
ejpam-4800	35	19	x.	x.	NOUN
ejpam-4800	35	20	moreover	moreover	ADV
ejpam-4800	35	21	,	,	PUNCT
ejpam-4800	35	22	if	if	SCONJ
ejpam-4800	35	23	g	g	PROPN
ejpam-4800	35	24	is	be	AUX
ejpam-4800	35	25	an	an	DET
ejpam-4800	35	26	m	m	NOUN
ejpam-4800	35	27	-ideal	-ideal	ADJ
ejpam-4800	35	28	in	in	ADP
ejpam-4800	35	29	x	x	SYM
ejpam-4800	35	30	then	then	ADV
ejpam-4800	35	31	g	g	PROPN
ejpam-4800	35	32	is	be	AUX
ejpam-4800	35	33	also	also	ADV
ejpam-4800	35	34	strongly	strongly	ADV
ejpam-4800	35	35	proximinal	proximinal	ADJ
ejpam-4800	35	36	in	in	ADP
ejpam-4800	35	37	x.	x.	NOUN
ejpam-4800	35	38	the	the	DET
ejpam-4800	35	39	question	question	NOUN
ejpam-4800	35	40	to	to	PART
ejpam-4800	35	41	be	be	AUX
ejpam-4800	35	42	proposed	propose	VERB
ejpam-4800	35	43	here	here	ADV
ejpam-4800	35	44	is	be	AUX
ejpam-4800	35	45	that	that	SCONJ
ejpam-4800	35	46	whether	whether	SCONJ
ejpam-4800	35	47	strong	strong	ADJ
ejpam-4800	35	48	proximinality	proximinality	NOUN
ejpam-4800	35	49	of	of	ADP
ejpam-4800	35	50	g	g	PROPN
ejpam-4800	35	51	in	in	ADP
ejpam-4800	35	52	x	x	PRON
ejpam-4800	35	53	can	can	AUX
ejpam-4800	35	54	be	be	AUX
ejpam-4800	35	55	lifted	lift	VERB
ejpam-4800	35	56	to	to	ADP
ejpam-4800	35	57	the	the	DET
ejpam-4800	35	58	lp	lp	NOUN
ejpam-4800	35	59	-	-	PUNCT
ejpam-4800	35	60	space	space	NOUN
ejpam-4800	35	61	or	or	CCONJ
ejpam-4800	35	62	to	to	ADP
ejpam-4800	35	63	the	the	DET
ejpam-4800	35	64	köthe	köthe	ADJ
ejpam-4800	35	65	space	space	NOUN
ejpam-4800	35	66	under	under	ADP
ejpam-4800	35	67	certain	certain	ADJ
ejpam-4800	35	68	conditions	condition	NOUN
ejpam-4800	35	69	on	on	ADP
ejpam-4800	35	70	g	g	PRON
ejpam-4800	35	71	?	?	PUNCT
ejpam-4800	36	1	in	in	ADP
ejpam-4800	36	2	[	[	X
ejpam-4800	36	3	15	15	NUM
ejpam-4800	36	4	]	]	PUNCT
ejpam-4800	36	5	,	,	PUNCT
ejpam-4800	36	6	the	the	DET
ejpam-4800	36	7	author	author	NOUN
ejpam-4800	36	8	proved	prove	VERB
ejpam-4800	36	9	that	that	SCONJ
ejpam-4800	36	10	“	"	PUNCT
ejpam-4800	36	11	if	if	SCONJ
ejpam-4800	36	12	g	g	PROPN
ejpam-4800	36	13	is	be	AUX
ejpam-4800	36	14	separable	separable	ADJ
ejpam-4800	36	15	then	then	ADV
ejpam-4800	36	16	g	g	PROPN
ejpam-4800	36	17	is	be	AUX
ejpam-4800	36	18	strongly	strongly	ADV
ejpam-4800	36	19	proximinal	proximinal	ADJ
ejpam-4800	36	20	subspace	subspace	NOUN
ejpam-4800	36	21	in	in	ADP
ejpam-4800	36	22	x	x	SYM
ejpam-4800	36	23	if	if	SCONJ
ejpam-4800	36	24	and	and	CCONJ
ejpam-4800	36	25	only	only	ADV
ejpam-4800	36	26	if	if	SCONJ
ejpam-4800	36	27	lp(µ,g	lp(µ,g	NUM
ejpam-4800	36	28	)	)	PUNCT
ejpam-4800	36	29	is	be	AUX
ejpam-4800	36	30	strongly	strongly	ADV
ejpam-4800	36	31	proximinal	proximinal	ADJ
ejpam-4800	36	32	in	in	ADP
ejpam-4800	36	33	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	36	34	)	)	PUNCT
ejpam-4800	36	35	,	,	PUNCT
ejpam-4800	36	36	1	1	NUM
ejpam-4800	36	37	≤	≤	NOUN
ejpam-4800	36	38	p	p	X
ejpam-4800	36	39	<	<	X
ejpam-4800	36	40	∞	∞	PROPN
ejpam-4800	36	41	”	"	PUNCT
ejpam-4800	36	42	.	.	PUNCT
ejpam-4800	37	1	we	we	PRON
ejpam-4800	37	2	proved	prove	VERB
ejpam-4800	37	3	a	a	DET
ejpam-4800	37	4	similar	similar	ADJ
ejpam-4800	37	5	result	result	NOUN
ejpam-4800	37	6	for	for	ADP
ejpam-4800	37	7	the	the	DET
ejpam-4800	37	8	case	case	NOUN
ejpam-4800	37	9	where	where	SCONJ
ejpam-4800	37	10	0	0	X
ejpam-4800	37	11	<	<	X
ejpam-4800	37	12	p	p	X
ejpam-4800	37	13	<	<	X
ejpam-4800	37	14	1	1	NUM
ejpam-4800	37	15	,	,	PUNCT
ejpam-4800	37	16	see	see	VERB
ejpam-4800	37	17	theorem	theorem	VERB
ejpam-4800	37	18	3.3	3.3	NUM
ejpam-4800	37	19	in	in	ADP
ejpam-4800	37	20	[	[	X
ejpam-4800	37	21	7	7	NUM
ejpam-4800	37	22	]	]	PUNCT
ejpam-4800	37	23	.	.	PUNCT
ejpam-4800	38	1	on	on	ADP
ejpam-4800	38	2	the	the	DET
ejpam-4800	38	3	other	other	ADJ
ejpam-4800	38	4	hand	hand	NOUN
ejpam-4800	38	5	,	,	PUNCT
ejpam-4800	38	6	strong	strong	ADJ
ejpam-4800	38	7	coproximinality	coproximinality	NOUN
ejpam-4800	38	8	in	in	ADP
ejpam-4800	38	9	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	38	10	)	)	PUNCT
ejpam-4800	38	11	,	,	PUNCT
ejpam-4800	38	12	was	be	AUX
ejpam-4800	38	13	first	first	ADV
ejpam-4800	38	14	studied	study	VERB
ejpam-4800	38	15	in	in	ADP
ejpam-4800	38	16	[	[	X
ejpam-4800	38	17	5	5	NUM
ejpam-4800	38	18	]	]	PUNCT
ejpam-4800	38	19	.	.	PUNCT
ejpam-4800	39	1	in	in	ADP
ejpam-4800	39	2	this	this	DET
ejpam-4800	39	3	paper	paper	NOUN
ejpam-4800	39	4	,	,	PUNCT
ejpam-4800	39	5	we	we	PRON
ejpam-4800	39	6	will	will	AUX
ejpam-4800	39	7	study	study	VERB
ejpam-4800	39	8	more	more	ADJ
ejpam-4800	39	9	properties	property	NOUN
ejpam-4800	39	10	in	in	ADP
ejpam-4800	39	11	this	this	DET
ejpam-4800	39	12	direction	direction	NOUN
ejpam-4800	39	13	and	and	CCONJ
ejpam-4800	39	14	prove	prove	VERB
ejpam-4800	39	15	some	some	DET
ejpam-4800	39	16	new	new	ADJ
ejpam-4800	39	17	results	result	NOUN
ejpam-4800	39	18	.	.	PUNCT
ejpam-4800	40	1	this	this	PRON
ejpam-4800	40	2	will	will	AUX
ejpam-4800	40	3	be	be	AUX
ejpam-4800	40	4	done	do	VERB
ejpam-4800	40	5	in	in	ADP
ejpam-4800	40	6	section	section	NOUN
ejpam-4800	40	7	two	two	NUM
ejpam-4800	40	8	,	,	PUNCT
ejpam-4800	40	9	in	in	ADP
ejpam-4800	40	10	which	which	PRON
ejpam-4800	40	11	we	we	PRON
ejpam-4800	40	12	are	be	AUX
ejpam-4800	40	13	interested	interested	ADJ
ejpam-4800	40	14	with	with	ADP
ejpam-4800	40	15	the	the	DET
ejpam-4800	40	16	spaces	space	NOUN
ejpam-4800	40	17	of	of	ADP
ejpam-4800	40	18	p	p	NOUN
ejpam-4800	40	19	-	-	PUNCT
ejpam-4800	40	20	bochner	bochner	NOUN
ejpam-4800	40	21	integrable	integrable	ADJ
ejpam-4800	40	22	functions	function	NOUN
ejpam-4800	40	23	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	40	24	)	)	PUNCT
ejpam-4800	40	25	,	,	PUNCT
ejpam-4800	40	26	1	1	NUM
ejpam-4800	40	27	≤	≤	NOUN
ejpam-4800	40	28	p	p	X
ejpam-4800	40	29	<	<	X
ejpam-4800	40	30	∞	∞	PROPN
ejpam-4800	40	31	,	,	PUNCT
ejpam-4800	40	32	where	where	SCONJ
ejpam-4800	40	33	(	(	PUNCT
ejpam-4800	40	34	t	t	PROPN
ejpam-4800	40	35	,	,	PUNCT
ejpam-4800	40	36	σ	σ	PROPN
ejpam-4800	40	37	,	,	PUNCT
ejpam-4800	40	38	µ	µ	NOUN
ejpam-4800	40	39	)	)	PUNCT
ejpam-4800	40	40	is	be	AUX
ejpam-4800	40	41	a	a	DET
ejpam-4800	40	42	finite	finite	ADJ
ejpam-4800	40	43	measure	measure	NOUN
ejpam-4800	40	44	space	space	NOUN
ejpam-4800	40	45	.	.	PUNCT
ejpam-4800	41	1	the	the	DET
ejpam-4800	41	2	p	p	NOUN
ejpam-4800	41	3	-	-	PUNCT
ejpam-4800	41	4	norm	norm	NOUN
ejpam-4800	41	5	,	,	PUNCT
ejpam-4800	41	6	defined	define	VERB
ejpam-4800	41	7	on	on	ADP
ejpam-4800	41	8	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	41	9	)	)	PUNCT
ejpam-4800	41	10	for	for	ADP
ejpam-4800	41	11	1	1	NUM
ejpam-4800	41	12	≤	≤	NOUN
ejpam-4800	41	13	p	p	NOUN
ejpam-4800	41	14	<	<	X
ejpam-4800	41	15	∞	∞	PROPN
ejpam-4800	41	16	,	,	PUNCT
ejpam-4800	41	17	is	be	AUX
ejpam-4800	41	18	given	give	VERB
ejpam-4800	41	19	by	by	ADP
ejpam-4800	41	20	:	:	PUNCT
ejpam-4800	41	21	||f	||f	NOUN
ejpam-4800	41	22	||p	||p	NOUN
ejpam-4800	41	23	=	=	SYM
ejpam-4800	41	24	(	(	PUNCT
ejpam-4800	41	25	∫	∫	PROPN
ejpam-4800	41	26	t	t	PROPN
ejpam-4800	41	27	||f	||f	PROPN
ejpam-4800	41	28	||pdt)1	||pdt)1	PROPN
ejpam-4800	41	29	/	/	SYM
ejpam-4800	41	30	p.	p.	NOUN
ejpam-4800	41	31	in	in	ADP
ejpam-4800	41	32	the	the	DET
ejpam-4800	41	33	third	third	ADJ
ejpam-4800	41	34	section	section	NOUN
ejpam-4800	41	35	,	,	PUNCT
ejpam-4800	41	36	we	we	PRON
ejpam-4800	41	37	study	study	VERB
ejpam-4800	41	38	strong	strong	ADJ
ejpam-4800	41	39	coproximinality	coproximinality	NOUN
ejpam-4800	41	40	in	in	ADP
ejpam-4800	41	41	köthe	köthe	DET
ejpam-4800	41	42	bochner	bochner	NOUN
ejpam-4800	41	43	function	function	NOUN
ejpam-4800	41	44	spaces	space	NOUN
ejpam-4800	41	45	.	.	PUNCT
ejpam-4800	42	1	first	first	ADV
ejpam-4800	42	2	consider	consider	VERB
ejpam-4800	42	3	e	e	NOUN
ejpam-4800	42	4	to	to	PART
ejpam-4800	42	5	be	be	AUX
ejpam-4800	42	6	the	the	DET
ejpam-4800	42	7	space	space	NOUN
ejpam-4800	42	8	of	of	ADP
ejpam-4800	42	9	all	all	DET
ejpam-4800	42	10	“	"	PUNCT
ejpam-4800	42	11	equivalence	equivalence	NOUN
ejpam-4800	42	12	classes	class	NOUN
ejpam-4800	42	13	”	"	PUNCT
ejpam-4800	42	14	of	of	ADP
ejpam-4800	42	15	µ-measurable	µ-measurable	ADJ
ejpam-4800	42	16	real	real	ADV
ejpam-4800	42	17	-	-	PUNCT
ejpam-4800	42	18	valued	value	VERB
ejpam-4800	42	19	j.	j.	PROPN
ejpam-4800	42	20	jawdat	jawdat	PROPN
ejpam-4800	42	21	/	/	SYM
ejpam-4800	42	22	eur	eur	PROPN
ejpam-4800	42	23	.	.	PUNCT
ejpam-4800	43	1	j.	j.	PROPN
ejpam-4800	43	2	pure	pure	PROPN
ejpam-4800	43	3	appl	appl	PROPN
ejpam-4800	43	4	.	.	PROPN
ejpam-4800	43	5	math	math	PROPN
ejpam-4800	43	6	,	,	PUNCT
ejpam-4800	43	7	16	16	NUM
ejpam-4800	43	8	(	(	PUNCT
ejpam-4800	43	9	3	3	NUM
ejpam-4800	43	10	)	)	PUNCT
ejpam-4800	43	11	(	(	PUNCT
ejpam-4800	43	12	2023	2023	NUM
ejpam-4800	43	13	)	)	PUNCT
ejpam-4800	43	14	,	,	PUNCT
ejpam-4800	43	15	1543	1543	NUM
ejpam-4800	43	16	-	-	SYM
ejpam-4800	43	17	1551	1551	NUM
ejpam-4800	43	18	1545	1545	NUM
ejpam-4800	43	19	functions	function	NOUN
ejpam-4800	43	20	on	on	ADP
ejpam-4800	43	21	t	t	PROPN
ejpam-4800	43	22	.	.	PUNCT
ejpam-4800	44	1	this	this	PRON
ejpam-4800	44	2	means	mean	VERB
ejpam-4800	44	3	for	for	ADP
ejpam-4800	44	4	h	h	NOUN
ejpam-4800	44	5	and	and	CCONJ
ejpam-4800	44	6	g	g	NOUN
ejpam-4800	44	7	in	in	ADP
ejpam-4800	44	8	e	e	NOUN
ejpam-4800	44	9	then	then	ADV
ejpam-4800	44	10	h	h	PROPN
ejpam-4800	45	1	=	=	SYM
ejpam-4800	45	2	g	g	PROPN
ejpam-4800	45	3	if	if	SCONJ
ejpam-4800	45	4	and	and	CCONJ
ejpam-4800	45	5	only	only	ADV
ejpam-4800	45	6	if	if	SCONJ
ejpam-4800	45	7	h(t	h(t	NUM
ejpam-4800	45	8	)	)	PUNCT
ejpam-4800	45	9	=	=	SYM
ejpam-4800	45	10	g(t	g(t	PROPN
ejpam-4800	45	11	)	)	PUNCT
ejpam-4800	45	12	,	,	PUNCT
ejpam-4800	45	13	µ-almost	µ-almost	ADV
ejpam-4800	45	14	everywhere	everywhere	ADV
ejpam-4800	45	15	t	t	PROPN
ejpam-4800	45	16	in	in	ADP
ejpam-4800	45	17	t	t	PROPN
ejpam-4800	45	18	(	(	PUNCT
ejpam-4800	45	19	for	for	ADP
ejpam-4800	45	20	simplicity	simplicity	NOUN
ejpam-4800	45	21	we	we	PRON
ejpam-4800	45	22	write	write	VERB
ejpam-4800	45	23	a.e	a.e	PROPN
ejpam-4800	45	24	.	.	PROPN
ejpam-4800	45	25	t	t	PROPN
ejpam-4800	45	26	∈	∈	PROPN
ejpam-4800	45	27	t	t	PROPN
ejpam-4800	45	28	)	)	PUNCT
ejpam-4800	45	29	.	.	PUNCT
ejpam-4800	46	1	when	when	SCONJ
ejpam-4800	46	2	e	e	NOUN
ejpam-4800	46	3	is	be	AUX
ejpam-4800	46	4	equipped	equip	VERB
ejpam-4800	46	5	with	with	ADP
ejpam-4800	46	6	a	a	DET
ejpam-4800	46	7	norm	norm	NOUN
ejpam-4800	46	8	||.||e	||.||e	PROPN
ejpam-4800	46	9	under	under	ADP
ejpam-4800	46	10	which	which	PRON
ejpam-4800	46	11	it	it	PRON
ejpam-4800	46	12	is	be	AUX
ejpam-4800	46	13	complete	complete	ADJ
ejpam-4800	46	14	then	then	ADV
ejpam-4800	46	15	e	e	PROPN
ejpam-4800	46	16	is	be	AUX
ejpam-4800	46	17	known	know	VERB
ejpam-4800	46	18	as	as	ADP
ejpam-4800	46	19	a	a	DET
ejpam-4800	46	20	real	real	ADJ
ejpam-4800	46	21	köthe	köthe	PRON
ejpam-4800	46	22	function	function	NOUN
ejpam-4800	46	23	space	space	NOUN
ejpam-4800	46	24	.	.	PUNCT
ejpam-4800	47	1	finally	finally	ADV
ejpam-4800	47	2	,	,	PUNCT
ejpam-4800	47	3	e	e	PROPN
ejpam-4800	47	4	becomes	become	VERB
ejpam-4800	47	5	a	a	DET
ejpam-4800	47	6	banach	banach	NOUN
ejpam-4800	47	7	lattice	lattice	NOUN
ejpam-4800	48	1	[	[	X
ejpam-4800	48	2	11	11	NUM
ejpam-4800	48	3	]	]	PUNCT
ejpam-4800	48	4	,	,	PUNCT
ejpam-4800	48	5	if	if	SCONJ
ejpam-4800	48	6	it	it	PRON
ejpam-4800	48	7	satisfies	satisfy	VERB
ejpam-4800	48	8	the	the	DET
ejpam-4800	48	9	two	two	NUM
ejpam-4800	48	10	conditions	condition	NOUN
ejpam-4800	48	11	below	below	ADV
ejpam-4800	48	12	.	.	PUNCT
ejpam-4800	49	1	(	(	PUNCT
ejpam-4800	49	2	i	i	NOUN
ejpam-4800	49	3	)	)	PUNCT
ejpam-4800	49	4	for	for	ADP
ejpam-4800	49	5	each	each	DET
ejpam-4800	49	6	measurable	measurable	NOUN
ejpam-4800	49	7	subset	subset	VERB
ejpam-4800	49	8	a	a	PRON
ejpam-4800	49	9	of	of	ADP
ejpam-4800	49	10	t	t	NOUN
ejpam-4800	49	11	,	,	PUNCT
ejpam-4800	49	12	with	with	ADP
ejpam-4800	49	13	µ(a	µ(a	PROPN
ejpam-4800	49	14	)	)	PUNCT
ejpam-4800	49	15	<	<	X
ejpam-4800	49	16	∞	∞	PROPN
ejpam-4800	49	17	,	,	PUNCT
ejpam-4800	49	18	the	the	DET
ejpam-4800	49	19	characteristic	characteristic	ADJ
ejpam-4800	49	20	function	function	NOUN
ejpam-4800	49	21	χa	χa	NOUN
ejpam-4800	49	22	is	be	AUX
ejpam-4800	49	23	again	again	ADV
ejpam-4800	49	24	in	in	ADP
ejpam-4800	49	25	e.	e.	PROPN
ejpam-4800	49	26	(	(	PUNCT
ejpam-4800	49	27	ii	ii	PROPN
ejpam-4800	49	28	)	)	PUNCT
ejpam-4800	49	29	for	for	ADP
ejpam-4800	49	30	any	any	DET
ejpam-4800	49	31	two	two	NUM
ejpam-4800	49	32	functions	function	NOUN
ejpam-4800	49	33	h	h	NOUN
ejpam-4800	49	34	and	and	CCONJ
ejpam-4800	49	35	g	g	NOUN
ejpam-4800	49	36	such	such	ADJ
ejpam-4800	49	37	that	that	DET
ejpam-4800	49	38	|h|	|h|	PROPN
ejpam-4800	49	39	≤	≤	X
ejpam-4800	49	40	|g|	|g|	PROPN
ejpam-4800	49	41	and	and	CCONJ
ejpam-4800	49	42	g	g	PROPN
ejpam-4800	49	43	∈	∈	PROPN
ejpam-4800	50	1	e	e	NOUN
ejpam-4800	50	2	then	then	ADV
ejpam-4800	50	3	h	h	PROPN
ejpam-4800	50	4	∈	∈	PROPN
ejpam-4800	50	5	e	e	PROPN
ejpam-4800	50	6	and	and	CCONJ
ejpam-4800	50	7	||h||e	||h||e	PROPN
ejpam-4800	50	8	≤	≤	PROPN
ejpam-4800	50	9	||g||e	||g||e	PROPN
ejpam-4800	50	10	.	.	PUNCT
ejpam-4800	51	1	a	a	DET
ejpam-4800	51	2	köthe	köthe	NOUN
ejpam-4800	51	3	space	space	NOUN
ejpam-4800	51	4	e	e	NOUN
ejpam-4800	51	5	is	be	AUX
ejpam-4800	51	6	said	say	VERB
ejpam-4800	51	7	to	to	PART
ejpam-4800	51	8	be	be	AUX
ejpam-4800	51	9	strictly	strictly	ADV
ejpam-4800	51	10	monotone	monotone	ADJ
ejpam-4800	51	11	if	if	SCONJ
ejpam-4800	51	12	the	the	DET
ejpam-4800	51	13	inequality	inequality	NOUN
ejpam-4800	51	14	in	in	ADP
ejpam-4800	51	15	(	(	PUNCT
ejpam-4800	51	16	ii	ii	NOUN
ejpam-4800	51	17	)	)	PUNCT
ejpam-4800	51	18	above	above	ADV
ejpam-4800	51	19	is	be	AUX
ejpam-4800	51	20	strict	strict	ADJ
ejpam-4800	51	21	.	.	PUNCT
ejpam-4800	52	1	in	in	ADP
ejpam-4800	52	2	other	other	ADJ
ejpam-4800	52	3	words	word	NOUN
ejpam-4800	52	4	,	,	PUNCT
ejpam-4800	52	5	if	if	SCONJ
ejpam-4800	52	6	h	h	PROPN
ejpam-4800	52	7	≥	≥	AUX
ejpam-4800	52	8	g	g	PROPN
ejpam-4800	52	9	≥	≥	NOUN
ejpam-4800	52	10	0	0	NUM
ejpam-4800	52	11	in	in	ADP
ejpam-4800	52	12	e	e	PROPN
ejpam-4800	52	13	and	and	CCONJ
ejpam-4800	52	14	||h||e	||h||e	PROPN
ejpam-4800	52	15	=	=	SYM
ejpam-4800	52	16	||g||e	||g||e	PROPN
ejpam-4800	52	17	imply	imply	VERB
ejpam-4800	52	18	h	h	NOUN
ejpam-4800	52	19	=	=	PUNCT
ejpam-4800	52	20	g.	g.	PROPN
ejpam-4800	52	21	for	for	ADP
ejpam-4800	52	22	a	a	DET
ejpam-4800	52	23	real	real	ADJ
ejpam-4800	52	24	banach	banach	NOUN
ejpam-4800	52	25	space	space	NOUN
ejpam-4800	52	26	(	(	PUNCT
ejpam-4800	52	27	x	x	X
ejpam-4800	52	28	,	,	PUNCT
ejpam-4800	52	29	||.||x	||.||x	PROPN
ejpam-4800	52	30	)	)	PUNCT
ejpam-4800	52	31	and	and	CCONJ
ejpam-4800	52	32	a	a	DET
ejpam-4800	52	33	real	real	ADJ
ejpam-4800	52	34	köthe	köthe	PRON
ejpam-4800	52	35	space	space	NOUN
ejpam-4800	52	36	e	e	NOUN
ejpam-4800	52	37	,	,	PUNCT
ejpam-4800	52	38	consider	consider	VERB
ejpam-4800	52	39	e(x	e(x	NUM
ejpam-4800	52	40	)	)	PUNCT
ejpam-4800	52	41	to	to	PART
ejpam-4800	52	42	be	be	AUX
ejpam-4800	52	43	the	the	DET
ejpam-4800	52	44	space	space	NOUN
ejpam-4800	52	45	of	of	ADP
ejpam-4800	52	46	(	(	PUNCT
ejpam-4800	52	47	equivalence	equivalence	NOUN
ejpam-4800	52	48	classes	class	NOUN
ejpam-4800	52	49	of	of	ADP
ejpam-4800	52	50	)	)	PUNCT
ejpam-4800	52	51	strongly	strongly	ADV
ejpam-4800	52	52	-	-	PUNCT
ejpam-4800	52	53	measurable	measurable	ADJ
ejpam-4800	52	54	functions	function	NOUN
ejpam-4800	52	55	f	f	X
ejpam-4800	52	56	:	:	PUNCT
ejpam-4800	52	57	t	t	PROPN
ejpam-4800	52	58	→	→	PUNCT
ejpam-4800	52	59	x	x	SYM
ejpam-4800	52	60	where	where	SCONJ
ejpam-4800	52	61	||f(.)||x	||f(.)||x	NOUN
ejpam-4800	52	62	∈	∈	PROPN
ejpam-4800	52	63	e.	e.	PROPN
ejpam-4800	52	64	define	define	VERB
ejpam-4800	52	65	a	a	DET
ejpam-4800	52	66	norm	norm	NOUN
ejpam-4800	52	67	on	on	ADP
ejpam-4800	52	68	e(x	e(x	NUM
ejpam-4800	52	69	)	)	PUNCT
ejpam-4800	52	70	as	as	SCONJ
ejpam-4800	52	71	follows	follow	VERB
ejpam-4800	52	72	.	.	PUNCT
ejpam-4800	53	1	|||f	|||f	PROPN
ejpam-4800	53	2	|||	|||	NOUN
ejpam-4800	54	1	=	=	PUNCT
ejpam-4800	55	1	||	||	NOUN
ejpam-4800	55	2	||f(.)||x	||f(.)||x	NOUN
ejpam-4800	55	3	||e	||e	NOUN
ejpam-4800	55	4	.	.	PUNCT
ejpam-4800	56	1	then	then	ADV
ejpam-4800	56	2	(	(	PUNCT
ejpam-4800	56	3	e(x	e(x	NUM
ejpam-4800	56	4	)	)	PUNCT
ejpam-4800	56	5	,	,	PUNCT
ejpam-4800	56	6	|||.|||	|||.|||	NUM
ejpam-4800	56	7	)	)	PUNCT
ejpam-4800	56	8	is	be	AUX
ejpam-4800	56	9	called	call	VERB
ejpam-4800	56	10	the	the	DET
ejpam-4800	56	11	köthe	köthe	PROPN
ejpam-4800	56	12	bochner	bochner	NOUN
ejpam-4800	56	13	function	function	NOUN
ejpam-4800	56	14	space	space	NOUN
ejpam-4800	56	15	,	,	PUNCT
ejpam-4800	56	16	see	see	VERB
ejpam-4800	56	17	[	[	X
ejpam-4800	56	18	11	11	NUM
ejpam-4800	56	19	]	]	PUNCT
ejpam-4800	56	20	,	,	PUNCT
ejpam-4800	56	21	which	which	PRON
ejpam-4800	56	22	is	be	AUX
ejpam-4800	56	23	a	a	DET
ejpam-4800	56	24	banach	banach	NOUN
ejpam-4800	56	25	space	space	NOUN
ejpam-4800	56	26	under	under	ADP
ejpam-4800	56	27	the	the	DET
ejpam-4800	56	28	above	above	ADJ
ejpam-4800	56	29	norm	norm	NOUN
ejpam-4800	56	30	.	.	PUNCT
ejpam-4800	57	1	the	the	DET
ejpam-4800	57	2	köthe	köthe	PROPN
ejpam-4800	57	3	bochner	bochner	NOUN
ejpam-4800	57	4	function	function	NOUN
ejpam-4800	57	5	spaces	space	NOUN
ejpam-4800	57	6	that	that	PRON
ejpam-4800	57	7	are	be	AUX
ejpam-4800	57	8	most	most	ADV
ejpam-4800	57	9	well	well	ADV
ejpam-4800	57	10	-	-	PUNCT
ejpam-4800	57	11	known	know	VERB
ejpam-4800	57	12	classes	class	NOUN
ejpam-4800	57	13	are	be	AUX
ejpam-4800	57	14	the	the	DET
ejpam-4800	57	15	lebesgue	lebesgue	NOUN
ejpam-4800	57	16	-	-	PUNCT
ejpam-4800	57	17	bochner	bochner	NOUN
ejpam-4800	57	18	spaces	space	NOUN
ejpam-4800	57	19	lp(µ,x	lp(µ,x	ADJ
ejpam-4800	57	20	)	)	PUNCT
ejpam-4800	57	21	,	,	PUNCT
ejpam-4800	57	22	1	1	NUM
ejpam-4800	57	23	≤	≤	NOUN
ejpam-4800	57	24	p	p	X
ejpam-4800	57	25	<	<	X
ejpam-4800	57	26	∞	∞	PROPN
ejpam-4800	57	27	and	and	CCONJ
ejpam-4800	57	28	the	the	DET
ejpam-4800	57	29	orlicz	orlicz	ADJ
ejpam-4800	57	30	-	-	PUNCT
ejpam-4800	57	31	bochner	bochner	NOUN
ejpam-4800	57	32	spaces	space	NOUN
ejpam-4800	57	33	lϕ(µ,x	lϕ(µ,x	NOUN
ejpam-4800	57	34	)	)	PUNCT
ejpam-4800	57	35	.	.	PUNCT
ejpam-4800	58	1	let	let	VERB
ejpam-4800	58	2	e(x	e(x	NUM
ejpam-4800	58	3	)	)	PUNCT
ejpam-4800	58	4	be	be	AUX
ejpam-4800	58	5	the	the	DET
ejpam-4800	58	6	köthe	köthe	PROPN
ejpam-4800	58	7	bochner	bochner	NOUN
ejpam-4800	58	8	function	function	NOUN
ejpam-4800	58	9	space	space	NOUN
ejpam-4800	58	10	on	on	ADP
ejpam-4800	58	11	x.	x.	NOUN
ejpam-4800	58	12	several	several	ADJ
ejpam-4800	58	13	authors	author	NOUN
ejpam-4800	58	14	studied	study	VERB
ejpam-4800	58	15	the	the	DET
ejpam-4800	58	16	problem	problem	NOUN
ejpam-4800	58	17	under	under	ADP
ejpam-4800	58	18	what	what	DET
ejpam-4800	58	19	conditions	condition	NOUN
ejpam-4800	58	20	the	the	DET
ejpam-4800	58	21	subspace	subspace	PROPN
ejpam-4800	58	22	e(g	e(g	PROPN
ejpam-4800	58	23	)	)	PUNCT
ejpam-4800	58	24	is	be	AUX
ejpam-4800	58	25	proximinal	proximinal	ADJ
ejpam-4800	58	26	(	(	PUNCT
ejpam-4800	58	27	resp	resp	NOUN
ejpam-4800	58	28	.	.	PUNCT
ejpam-4800	59	1	coproximinal	coproximinal	PROPN
ejpam-4800	59	2	)	)	PUNCT
ejpam-4800	59	3	in	in	ADP
ejpam-4800	59	4	e(x	e(x	NUM
ejpam-4800	59	5	)	)	PUNCT
ejpam-4800	59	6	,	,	PUNCT
ejpam-4800	59	7	see	see	VERB
ejpam-4800	59	8	for	for	ADP
ejpam-4800	59	9	example	example	NOUN
ejpam-4800	59	10	,	,	PUNCT
ejpam-4800	59	11	[	[	X
ejpam-4800	59	12	9	9	NUM
ejpam-4800	59	13	]	]	PUNCT
ejpam-4800	59	14	and	and	CCONJ
ejpam-4800	59	15	[	[	X
ejpam-4800	59	16	6	6	NUM
ejpam-4800	59	17	]	]	PUNCT
ejpam-4800	59	18	,	,	PUNCT
ejpam-4800	59	19	but	but	CCONJ
ejpam-4800	59	20	no	no	DET
ejpam-4800	59	21	work	work	NOUN
ejpam-4800	59	22	has	have	AUX
ejpam-4800	59	23	been	be	AUX
ejpam-4800	59	24	conducted	conduct	VERB
ejpam-4800	59	25	in	in	ADP
ejpam-4800	59	26	the	the	DET
ejpam-4800	59	27	direction	direction	NOUN
ejpam-4800	59	28	of	of	ADP
ejpam-4800	59	29	strong	strong	ADJ
ejpam-4800	59	30	proximinality	proximinality	NOUN
ejpam-4800	59	31	(	(	PUNCT
ejpam-4800	59	32	resp	resp	NOUN
ejpam-4800	59	33	.	.	PUNCT
ejpam-4800	60	1	coproximinality	coproximinality	PROPN
ejpam-4800	60	2	)	)	PUNCT
ejpam-4800	60	3	in	in	ADP
ejpam-4800	60	4	these	these	DET
ejpam-4800	60	5	spaces	space	NOUN
ejpam-4800	60	6	.	.	PUNCT
ejpam-4800	61	1	one	one	NUM
ejpam-4800	61	2	of	of	ADP
ejpam-4800	61	3	the	the	DET
ejpam-4800	61	4	main	main	ADJ
ejpam-4800	61	5	results	result	NOUN
ejpam-4800	61	6	of	of	ADP
ejpam-4800	61	7	this	this	DET
ejpam-4800	61	8	paper	paper	NOUN
ejpam-4800	61	9	is	be	AUX
ejpam-4800	61	10	to	to	PART
ejpam-4800	61	11	prove	prove	VERB
ejpam-4800	61	12	that	that	SCONJ
ejpam-4800	61	13	if	if	SCONJ
ejpam-4800	61	14	g	g	PROPN
ejpam-4800	61	15	is	be	AUX
ejpam-4800	61	16	a	a	DET
ejpam-4800	61	17	separable	separable	ADJ
ejpam-4800	61	18	subspace	subspace	NOUN
ejpam-4800	61	19	ofx	ofx	PROPN
ejpam-4800	61	20	then	then	ADV
ejpam-4800	61	21	g	g	PROPN
ejpam-4800	61	22	is	be	AUX
ejpam-4800	61	23	strongly	strongly	ADV
ejpam-4800	61	24	coproximinal	coproximinal	ADJ
ejpam-4800	61	25	in	in	ADP
ejpam-4800	61	26	x	x	SYM
ejpam-4800	61	27	if	if	SCONJ
ejpam-4800	61	28	and	and	CCONJ
ejpam-4800	61	29	only	only	ADV
ejpam-4800	61	30	if	if	SCONJ
ejpam-4800	61	31	e(g	e(g	PROPN
ejpam-4800	61	32	)	)	PUNCT
ejpam-4800	61	33	is	be	AUX
ejpam-4800	61	34	strongly	strongly	ADV
ejpam-4800	61	35	coproximinal	coproximinal	ADJ
ejpam-4800	61	36	in	in	ADP
ejpam-4800	61	37	e(x	e(x	NUM
ejpam-4800	61	38	)	)	PUNCT
ejpam-4800	61	39	,	,	PUNCT
ejpam-4800	61	40	provided	provide	VERB
ejpam-4800	61	41	that	that	SCONJ
ejpam-4800	61	42	e	e	NOUN
ejpam-4800	61	43	is	be	AUX
ejpam-4800	61	44	a	a	DET
ejpam-4800	61	45	strictly	strictly	ADV
ejpam-4800	61	46	monotone	monotone	ADJ
ejpam-4800	61	47	köthe	köthe	ADJ
ejpam-4800	61	48	space	space	NOUN
ejpam-4800	61	49	.	.	PUNCT
ejpam-4800	62	1	this	this	PRON
ejpam-4800	62	2	generalizes	generalize	VERB
ejpam-4800	62	3	the	the	DET
ejpam-4800	62	4	results	result	NOUN
ejpam-4800	62	5	for	for	ADP
ejpam-4800	62	6	bochner	bochn	ADJ
ejpam-4800	62	7	lp	lp	NOUN
ejpam-4800	62	8	-	-	NOUN
ejpam-4800	62	9	spaces	space	NOUN
ejpam-4800	62	10	.	.	PUNCT
ejpam-4800	63	1	2	2	X
ejpam-4800	63	2	.	.	X
ejpam-4800	63	3	strong	strong	ADJ
ejpam-4800	63	4	coproximinality	coproximinality	NOUN
ejpam-4800	63	5	of	of	ADP
ejpam-4800	63	6	lp(µ,g	lp(µ,g	NUM
ejpam-4800	63	7	)	)	PUNCT
ejpam-4800	63	8	in	in	ADP
ejpam-4800	63	9	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	63	10	)	)	PUNCT
ejpam-4800	63	11	in	in	ADP
ejpam-4800	63	12	this	this	DET
ejpam-4800	63	13	section	section	NOUN
ejpam-4800	63	14	,	,	PUNCT
ejpam-4800	63	15	we	we	PRON
ejpam-4800	63	16	first	first	ADV
ejpam-4800	63	17	recall	recall	VERB
ejpam-4800	63	18	the	the	DET
ejpam-4800	63	19	definition	definition	NOUN
ejpam-4800	63	20	of	of	ADP
ejpam-4800	63	21	strong	strong	ADJ
ejpam-4800	63	22	coproximinality	coproximinality	NOUN
ejpam-4800	63	23	in	in	ADP
ejpam-4800	63	24	general	general	ADJ
ejpam-4800	63	25	banach	banach	NOUN
ejpam-4800	63	26	spaces	space	VERB
ejpam-4800	63	27	,	,	PUNCT
ejpam-4800	63	28	[	[	X
ejpam-4800	63	29	4	4	NUM
ejpam-4800	63	30	]	]	PUNCT
ejpam-4800	63	31	.	.	PUNCT
ejpam-4800	64	1	some	some	DET
ejpam-4800	64	2	new	new	ADJ
ejpam-4800	64	3	results	result	NOUN
ejpam-4800	64	4	are	be	AUX
ejpam-4800	64	5	also	also	ADV
ejpam-4800	64	6	given	give	VERB
ejpam-4800	64	7	.	.	PUNCT
ejpam-4800	65	1	let	let	VERB
ejpam-4800	65	2	g	g	NOUN
ejpam-4800	65	3	be	be	AUX
ejpam-4800	65	4	coproximinal	coproximinal	ADJ
ejpam-4800	65	5	in	in	ADP
ejpam-4800	65	6	x	x	NOUN
ejpam-4800	65	7	,	,	PUNCT
ejpam-4800	65	8	hence	hence	ADV
ejpam-4800	65	9	the	the	DET
ejpam-4800	65	10	set	set	NOUN
ejpam-4800	65	11	of	of	ADP
ejpam-4800	65	12	best	good	ADJ
ejpam-4800	65	13	coapproximation	coapproximation	NOUN
ejpam-4800	65	14	points	point	NOUN
ejpam-4800	65	15	to	to	ADP
ejpam-4800	65	16	x	x	PRON
ejpam-4800	65	17	,	,	PUNCT
ejpam-4800	65	18	which	which	PRON
ejpam-4800	65	19	is	be	AUX
ejpam-4800	65	20	denoted	denote	VERB
ejpam-4800	65	21	by	by	ADP
ejpam-4800	65	22	rg(x	rg(x	NOUN
ejpam-4800	65	23	)	)	PUNCT
ejpam-4800	65	24	,	,	PUNCT
ejpam-4800	65	25	is	be	AUX
ejpam-4800	65	26	nonempty	nonempty	ADJ
ejpam-4800	65	27	for	for	ADP
ejpam-4800	65	28	each	each	PRON
ejpam-4800	65	29	x	x	PUNCT
ejpam-4800	65	30	in	in	ADP
ejpam-4800	65	31	x.	x.	NOUN
ejpam-4800	65	32	for	for	ADP
ejpam-4800	65	33	some	some	DET
ejpam-4800	65	34	δ	δ	PROPN
ejpam-4800	65	35	>	>	X
ejpam-4800	65	36	0	0	PROPN
ejpam-4800	65	37	,	,	PUNCT
ejpam-4800	65	38	define	define	VERB
ejpam-4800	65	39	rg(x	rg(x	ADP
ejpam-4800	65	40	,	,	PUNCT
ejpam-4800	65	41	δ	δ	PROPN
ejpam-4800	65	42	)	)	PUNCT
ejpam-4800	65	43	to	to	PART
ejpam-4800	65	44	be	be	AUX
ejpam-4800	65	45	the	the	DET
ejpam-4800	65	46	set	set	NOUN
ejpam-4800	65	47	of	of	ADP
ejpam-4800	65	48	“	"	PUNCT
ejpam-4800	65	49	near	near	ADP
ejpam-4800	65	50	best	good	ADJ
ejpam-4800	65	51	coapproximation	coapproximation	NOUN
ejpam-4800	65	52	points	point	NOUN
ejpam-4800	65	53	”	"	PUNCT
ejpam-4800	65	54	to	to	ADP
ejpam-4800	65	55	x	x	PROPN
ejpam-4800	65	56	from	from	ADP
ejpam-4800	65	57	g	g	PROPN
ejpam-4800	65	58	,	,	PUNCT
ejpam-4800	65	59	as	as	SCONJ
ejpam-4800	65	60	follows	follow	VERB
ejpam-4800	65	61	.	.	PUNCT
ejpam-4800	66	1	rg(x	rg(x	NUM
ejpam-4800	66	2	,	,	PUNCT
ejpam-4800	66	3	δ	δ	X
ejpam-4800	66	4	)	)	PUNCT
ejpam-4800	66	5	=	=	PRON
ejpam-4800	67	1	{	{	PUNCT
ejpam-4800	67	2	z	z	PROPN
ejpam-4800	67	3	∈	∈	PROPN
ejpam-4800	67	4	g	g	NOUN
ejpam-4800	67	5	:	:	PUNCT
ejpam-4800	67	6	||z	||z	NOUN
ejpam-4800	67	7	−	−	PROPN
ejpam-4800	67	8	g||	g||	NOUN
ejpam-4800	67	9	<	<	X
ejpam-4800	67	10	||x−	||x−	PROPN
ejpam-4800	67	11	g||+	g||+	PROPN
ejpam-4800	67	12	δ	δ	PROPN
ejpam-4800	67	13	,	,	PUNCT
ejpam-4800	67	14	∀g	∀g	X
ejpam-4800	67	15	∈	∈	PROPN
ejpam-4800	67	16	g	g	NOUN
ejpam-4800	67	17	}	}	PUNCT
ejpam-4800	67	18	.	.	PUNCT
ejpam-4800	68	1	(	(	PUNCT
ejpam-4800	68	2	3	3	X
ejpam-4800	68	3	)	)	PUNCT
ejpam-4800	68	4	definition	definition	NOUN
ejpam-4800	68	5	4	4	NUM
ejpam-4800	68	6	.	.	PUNCT
ejpam-4800	69	1	a	a	DET
ejpam-4800	69	2	coproximinal	coproximinal	ADJ
ejpam-4800	69	3	subset	subset	VERB
ejpam-4800	69	4	g	g	NOUN
ejpam-4800	69	5	in	in	ADP
ejpam-4800	69	6	x	x	X
ejpam-4800	69	7	,	,	PUNCT
ejpam-4800	69	8	is	be	AUX
ejpam-4800	69	9	called	call	VERB
ejpam-4800	69	10	strongly	strongly	ADV
ejpam-4800	69	11	coproximinal	coproximinal	ADJ
ejpam-4800	69	12	in	in	ADP
ejpam-4800	69	13	x	x	SYM
ejpam-4800	69	14	,	,	PUNCT
ejpam-4800	69	15	if	if	SCONJ
ejpam-4800	69	16	for	for	ADP
ejpam-4800	69	17	each	each	DET
ejpam-4800	69	18	x	x	SYM
ejpam-4800	69	19	∈	∈	PROPN
ejpam-4800	69	20	x	x	NOUN
ejpam-4800	69	21	,	,	PUNCT
ejpam-4800	69	22	the	the	DET
ejpam-4800	69	23	following	follow	VERB
ejpam-4800	69	24	is	be	AUX
ejpam-4800	69	25	satisfied	satisfied	ADJ
ejpam-4800	69	26	:	:	PUNCT
ejpam-4800	69	27	for	for	ADP
ejpam-4800	69	28	any	any	DET
ejpam-4800	69	29	ε	ε	PROPN
ejpam-4800	69	30	>	>	X
ejpam-4800	69	31	0	0	PROPN
ejpam-4800	69	32	,	,	PUNCT
ejpam-4800	69	33	there	there	PRON
ejpam-4800	69	34	exists	exist	VERB
ejpam-4800	69	35	δ	δ	PROPN
ejpam-4800	69	36	>	>	X
ejpam-4800	69	37	0	0	NUM
ejpam-4800	70	1	such	such	ADJ
ejpam-4800	70	2	that	that	PRON
ejpam-4800	70	3	rg(x	rg(x	NOUN
ejpam-4800	70	4	,	,	PUNCT
ejpam-4800	70	5	δ	δ	PROPN
ejpam-4800	70	6	)	)	PUNCT
ejpam-4800	70	7	⊂	⊂	PROPN
ejpam-4800	70	8	rg(x	rg(x	PUNCT
ejpam-4800	70	9	)	)	PUNCT
ejpam-4800	71	1	+	+	CCONJ
ejpam-4800	71	2	εbx	εbx	NOUN
ejpam-4800	71	3	,	,	PUNCT
ejpam-4800	71	4	where	where	SCONJ
ejpam-4800	71	5	bx	bx	PROPN
ejpam-4800	71	6	again	again	ADV
ejpam-4800	71	7	is	be	AUX
ejpam-4800	71	8	the	the	DET
ejpam-4800	71	9	unit	unit	NOUN
ejpam-4800	71	10	ball	ball	NOUN
ejpam-4800	71	11	of	of	ADP
ejpam-4800	71	12	x	x	PROPN
ejpam-4800	71	13	and	and	CCONJ
ejpam-4800	71	14	rg(x	rg(x	NUM
ejpam-4800	71	15	,	,	PUNCT
ejpam-4800	71	16	δ	δ	PROPN
ejpam-4800	71	17	)	)	PUNCT
ejpam-4800	71	18	as	as	SCONJ
ejpam-4800	71	19	defined	define	VERB
ejpam-4800	71	20	above	above	ADV
ejpam-4800	71	21	.	.	PUNCT
ejpam-4800	72	1	an	an	DET
ejpam-4800	72	2	alternative	alternative	ADJ
ejpam-4800	72	3	definition	definition	NOUN
ejpam-4800	72	4	is	be	AUX
ejpam-4800	72	5	the	the	DET
ejpam-4800	72	6	following	following	NOUN
ejpam-4800	72	7	.	.	PUNCT
ejpam-4800	73	1	j.	j.	PROPN
ejpam-4800	73	2	jawdat	jawdat	PROPN
ejpam-4800	73	3	/	/	SYM
ejpam-4800	73	4	eur	eur	PROPN
ejpam-4800	73	5	.	.	PUNCT
ejpam-4800	74	1	j.	j.	PROPN
ejpam-4800	74	2	pure	pure	PROPN
ejpam-4800	74	3	appl	appl	PROPN
ejpam-4800	74	4	.	.	PROPN
ejpam-4800	74	5	math	math	PROPN
ejpam-4800	74	6	,	,	PUNCT
ejpam-4800	74	7	16	16	NUM
ejpam-4800	74	8	(	(	PUNCT
ejpam-4800	74	9	3	3	NUM
ejpam-4800	74	10	)	)	PUNCT
ejpam-4800	74	11	(	(	PUNCT
ejpam-4800	74	12	2023	2023	NUM
ejpam-4800	74	13	)	)	PUNCT
ejpam-4800	74	14	,	,	PUNCT
ejpam-4800	74	15	1543	1543	NUM
ejpam-4800	74	16	-	-	SYM
ejpam-4800	74	17	1551	1551	NUM
ejpam-4800	74	18	1546	1546	NUM
ejpam-4800	74	19	definition	definition	NOUN
ejpam-4800	74	20	5	5	NUM
ejpam-4800	74	21	.	.	PUNCT
ejpam-4800	75	1	a	a	DET
ejpam-4800	75	2	coproximinal	coproximinal	ADJ
ejpam-4800	75	3	subset	subset	VERB
ejpam-4800	75	4	g	g	PROPN
ejpam-4800	75	5	of	of	ADP
ejpam-4800	75	6	x	x	PROPN
ejpam-4800	75	7	is	be	AUX
ejpam-4800	75	8	called	call	VERB
ejpam-4800	75	9	strongly	strongly	ADV
ejpam-4800	75	10	coproximinal	coproximinal	ADJ
ejpam-4800	75	11	at	at	ADP
ejpam-4800	75	12	x	x	X
ejpam-4800	75	13	∈	∈	PROPN
ejpam-4800	75	14	x	x	NOUN
ejpam-4800	75	15	,	,	PUNCT
ejpam-4800	75	16	if	if	SCONJ
ejpam-4800	75	17	given	give	VERB
ejpam-4800	75	18	ε	ε	PROPN
ejpam-4800	75	19	>	>	X
ejpam-4800	75	20	0	0	PUNCT
ejpam-4800	76	1	there	there	PRON
ejpam-4800	76	2	exists	exist	VERB
ejpam-4800	76	3	some	some	DET
ejpam-4800	76	4	δ	δ	PROPN
ejpam-4800	76	5	>	>	X
ejpam-4800	76	6	0	0	PROPN
ejpam-4800	76	7	,	,	PUNCT
ejpam-4800	76	8	such	such	ADJ
ejpam-4800	76	9	that	that	PRON
ejpam-4800	76	10	for	for	ADP
ejpam-4800	76	11	each	each	DET
ejpam-4800	76	12	z	z	NOUN
ejpam-4800	76	13	∈	∈	PROPN
ejpam-4800	76	14	rg(x	rg(x	X
ejpam-4800	76	15	,	,	PUNCT
ejpam-4800	76	16	δ	δ	PROPN
ejpam-4800	76	17	)	)	PUNCT
ejpam-4800	76	18	there	there	PRON
ejpam-4800	76	19	exists	exist	VERB
ejpam-4800	76	20	y0	y0	PROPN
ejpam-4800	76	21	∈	∈	NOUN
ejpam-4800	76	22	rg(x	rg(x	NUM
ejpam-4800	76	23	)	)	PUNCT
ejpam-4800	76	24	satisfying	satisfy	VERB
ejpam-4800	76	25	||z	||z	NOUN
ejpam-4800	76	26	−	−	PROPN
ejpam-4800	76	27	y0||	y0||	PROPN
ejpam-4800	76	28	<	<	X
ejpam-4800	76	29	ε	ε	PROPN
ejpam-4800	76	30	.	.	PUNCT
ejpam-4800	77	1	in	in	ADP
ejpam-4800	77	2	addition	addition	NOUN
ejpam-4800	77	3	,	,	PUNCT
ejpam-4800	77	4	g	g	PROPN
ejpam-4800	77	5	is	be	AUX
ejpam-4800	77	6	said	say	VERB
ejpam-4800	77	7	to	to	PART
ejpam-4800	77	8	be	be	AUX
ejpam-4800	77	9	strongly	strongly	ADV
ejpam-4800	77	10	coproximinal	coproximinal	ADJ
ejpam-4800	77	11	in	in	ADP
ejpam-4800	77	12	x	x	PRON
ejpam-4800	77	13	,	,	PUNCT
ejpam-4800	77	14	if	if	SCONJ
ejpam-4800	77	15	it	it	PRON
ejpam-4800	77	16	is	be	AUX
ejpam-4800	77	17	so	so	ADV
ejpam-4800	77	18	for	for	ADP
ejpam-4800	77	19	all	all	DET
ejpam-4800	77	20	x	x	NOUN
ejpam-4800	77	21	in	in	ADP
ejpam-4800	77	22	x.	x.	NOUN
ejpam-4800	77	23	the	the	DET
ejpam-4800	77	24	main	main	ADJ
ejpam-4800	77	25	result	result	NOUN
ejpam-4800	77	26	in	in	ADP
ejpam-4800	77	27	this	this	DET
ejpam-4800	77	28	section	section	NOUN
ejpam-4800	77	29	is	be	AUX
ejpam-4800	77	30	that	that	SCONJ
ejpam-4800	77	31	,	,	PUNCT
ejpam-4800	77	32	given	give	VERB
ejpam-4800	77	33	g	g	PRON
ejpam-4800	77	34	separable	separable	NOUN
ejpam-4800	77	35	in	in	ADP
ejpam-4800	77	36	x	x	SYM
ejpam-4800	77	37	and	and	CCONJ
ejpam-4800	77	38	1	1	NUM
ejpam-4800	77	39	≤	≤	NOUN
ejpam-4800	77	40	p	p	X
ejpam-4800	77	41	<	<	X
ejpam-4800	77	42	∞	∞	PROPN
ejpam-4800	77	43	,	,	PUNCT
ejpam-4800	77	44	then	then	ADV
ejpam-4800	77	45	lp(µ,g	lp(µ,g	NUM
ejpam-4800	77	46	)	)	PUNCT
ejpam-4800	77	47	is	be	AUX
ejpam-4800	77	48	strongly	strongly	ADV
ejpam-4800	77	49	coproximinal	coproximinal	ADJ
ejpam-4800	77	50	in	in	ADP
ejpam-4800	77	51	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	77	52	)	)	PUNCT
ejpam-4800	78	1	if	if	SCONJ
ejpam-4800	78	2	and	and	CCONJ
ejpam-4800	78	3	only	only	ADV
ejpam-4800	78	4	if	if	SCONJ
ejpam-4800	78	5	g	g	PROPN
ejpam-4800	78	6	is	be	AUX
ejpam-4800	78	7	strongly	strongly	ADV
ejpam-4800	78	8	coproximinal	coproximinal	ADJ
ejpam-4800	78	9	subspace	subspace	NOUN
ejpam-4800	78	10	of	of	ADP
ejpam-4800	78	11	x.	x.	NOUN
ejpam-4800	78	12	we	we	PRON
ejpam-4800	78	13	first	first	ADV
ejpam-4800	78	14	prove	prove	VERB
ejpam-4800	78	15	some	some	DET
ejpam-4800	78	16	results	result	NOUN
ejpam-4800	78	17	for	for	ADP
ejpam-4800	78	18	the	the	DET
ejpam-4800	78	19	case	case	NOUN
ejpam-4800	78	20	p	p	X
ejpam-4800	78	21	=	=	NOUN
ejpam-4800	78	22	1	1	NUM
ejpam-4800	78	23	,	,	PUNCT
ejpam-4800	78	24	then	then	ADV
ejpam-4800	78	25	these	these	DET
ejpam-4800	78	26	results	result	NOUN
ejpam-4800	78	27	can	can	AUX
ejpam-4800	78	28	be	be	AUX
ejpam-4800	78	29	extended	extend	VERB
ejpam-4800	78	30	easily	easily	ADV
ejpam-4800	78	31	to	to	ADP
ejpam-4800	78	32	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	78	33	)	)	PUNCT
ejpam-4800	78	34	,	,	PUNCT
ejpam-4800	78	35	for	for	ADP
ejpam-4800	78	36	1	1	NUM
ejpam-4800	78	37	<	<	X
ejpam-4800	78	38	p	p	X
ejpam-4800	78	39	<	<	X
ejpam-4800	78	40	∞.	∞.	PROPN
ejpam-4800	78	41	the	the	DET
ejpam-4800	78	42	absence	absence	NOUN
ejpam-4800	78	43	of	of	ADP
ejpam-4800	78	44	the	the	DET
ejpam-4800	78	45	distance	distance	NOUN
ejpam-4800	78	46	formula	formula	NOUN
ejpam-4800	78	47	in	in	ADP
ejpam-4800	78	48	the	the	DET
ejpam-4800	78	49	theory	theory	NOUN
ejpam-4800	78	50	of	of	ADP
ejpam-4800	78	51	best	good	ADJ
ejpam-4800	78	52	coapproximation	coapproximation	NOUN
ejpam-4800	78	53	leads	lead	VERB
ejpam-4800	78	54	to	to	ADP
ejpam-4800	78	55	a	a	DET
ejpam-4800	78	56	variant	variant	ADJ
ejpam-4800	78	57	way	way	NOUN
ejpam-4800	78	58	of	of	ADP
ejpam-4800	78	59	dealing	deal	VERB
ejpam-4800	78	60	with	with	ADP
ejpam-4800	78	61	the	the	DET
ejpam-4800	78	62	proofs	proof	NOUN
ejpam-4800	78	63	.	.	PUNCT
ejpam-4800	79	1	since	since	SCONJ
ejpam-4800	79	2	we	we	PRON
ejpam-4800	79	3	can	can	AUX
ejpam-4800	79	4	not	not	PART
ejpam-4800	79	5	use	use	VERB
ejpam-4800	79	6	the	the	DET
ejpam-4800	79	7	minimizing	minimize	VERB
ejpam-4800	79	8	sequence	sequence	NOUN
ejpam-4800	79	9	definition	definition	NOUN
ejpam-4800	79	10	(	(	PUNCT
ejpam-4800	79	11	as	as	ADP
ejpam-4800	79	12	for	for	ADP
ejpam-4800	79	13	the	the	DET
ejpam-4800	79	14	case	case	NOUN
ejpam-4800	79	15	of	of	ADP
ejpam-4800	79	16	strong	strong	ADJ
ejpam-4800	79	17	proximinality	proximinality	NOUN
ejpam-4800	79	18	)	)	PUNCT
ejpam-4800	80	1	but	but	CCONJ
ejpam-4800	80	2	,	,	PUNCT
ejpam-4800	80	3	however	however	ADV
ejpam-4800	80	4	,	,	PUNCT
ejpam-4800	80	5	we	we	PRON
ejpam-4800	80	6	will	will	AUX
ejpam-4800	80	7	make	make	VERB
ejpam-4800	80	8	use	use	NOUN
ejpam-4800	80	9	of	of	ADP
ejpam-4800	80	10	the	the	DET
ejpam-4800	80	11	following	follow	VERB
ejpam-4800	80	12	lemma	lemma	PROPN
ejpam-4800	80	13	.	.	PUNCT
ejpam-4800	81	1	lemma	lemma	PROPN
ejpam-4800	81	2	1	1	X
ejpam-4800	81	3	.	.	PUNCT
ejpam-4800	82	1	let	let	VERB
ejpam-4800	82	2	f	f	PROPN
ejpam-4800	82	3	∈	∈	PROPN
ejpam-4800	82	4	l1(µ,x	l1(µ,x	NOUN
ejpam-4800	82	5	)	)	PUNCT
ejpam-4800	82	6	,	,	PUNCT
ejpam-4800	82	7	and	and	CCONJ
ejpam-4800	82	8	g	g	PROPN
ejpam-4800	82	9	∈	∈	PROPN
ejpam-4800	82	10	l1(µ,g	l1(µ,g	NUM
ejpam-4800	82	11	)	)	PUNCT
ejpam-4800	82	12	.	.	PUNCT
ejpam-4800	83	1	let	let	VERB
ejpam-4800	83	2	g	g	PRON
ejpam-4800	83	3	be	be	AUX
ejpam-4800	83	4	separable	separable	ADJ
ejpam-4800	83	5	and	and	CCONJ
ejpam-4800	83	6	coproximinal	coproximinal	ADJ
ejpam-4800	83	7	in	in	ADP
ejpam-4800	83	8	x.	x.	NOUN
ejpam-4800	83	9	then	then	ADV
ejpam-4800	83	10	g	g	PROPN
ejpam-4800	83	11	∈	∈	PROPN
ejpam-4800	83	12	rl1(µ,g)(f	rl1(µ,g)(f	VERB
ejpam-4800	83	13	,	,	PUNCT
ejpam-4800	83	14	δ	δ	PROPN
ejpam-4800	83	15	′	′	NOUN
ejpam-4800	83	16	)	)	PUNCT
ejpam-4800	84	1	if	if	SCONJ
ejpam-4800	84	2	and	and	CCONJ
ejpam-4800	84	3	only	only	ADV
ejpam-4800	84	4	if	if	SCONJ
ejpam-4800	84	5	g(t	g(t	PROPN
ejpam-4800	84	6	)	)	PUNCT
ejpam-4800	84	7	∈	∈	PROPN
ejpam-4800	84	8	rg(f(t	rg(f(t	NOUN
ejpam-4800	84	9	)	)	PUNCT
ejpam-4800	84	10	,	,	PUNCT
ejpam-4800	84	11	δ	δ	PROPN
ejpam-4800	84	12	)	)	PUNCT
ejpam-4800	84	13	,	,	PUNCT
ejpam-4800	84	14	a.e	a.e	PROPN
ejpam-4800	84	15	.	.	PROPN
ejpam-4800	84	16	t	t	PROPN
ejpam-4800	84	17	∈	∈	PROPN
ejpam-4800	84	18	t	t	PROPN
ejpam-4800	84	19	,	,	PUNCT
ejpam-4800	84	20	and	and	CCONJ
ejpam-4800	84	21	for	for	ADP
ejpam-4800	84	22	some	some	DET
ejpam-4800	84	23	δ	δ	PROPN
ejpam-4800	84	24	,	,	PUNCT
ejpam-4800	84	25	δ′	δ′	PROPN
ejpam-4800	84	26	>	>	X
ejpam-4800	84	27	0	0	X
ejpam-4800	84	28	.	.	PUNCT
ejpam-4800	85	1	proof	proof	NOUN
ejpam-4800	85	2	.	.	PUNCT
ejpam-4800	86	1	given	give	VERB
ejpam-4800	86	2	g	g	PROPN
ejpam-4800	86	3	∈	∈	PROPN
ejpam-4800	86	4	l1(µ,g	l1(µ,g	CCONJ
ejpam-4800	86	5	)	)	PUNCT
ejpam-4800	86	6	such	such	ADJ
ejpam-4800	86	7	that	that	SCONJ
ejpam-4800	86	8	g(t	g(t	PROPN
ejpam-4800	86	9	)	)	PUNCT
ejpam-4800	86	10	∈	∈	PROPN
ejpam-4800	86	11	rg(f(t	rg(f(t	NOUN
ejpam-4800	86	12	)	)	PUNCT
ejpam-4800	86	13	,	,	PUNCT
ejpam-4800	86	14	δ	δ	PROPN
ejpam-4800	86	15	)	)	PUNCT
ejpam-4800	86	16	,	,	PUNCT
ejpam-4800	86	17	a.e	a.e	PROPN
ejpam-4800	86	18	.	.	PROPN
ejpam-4800	86	19	t	t	PROPN
ejpam-4800	86	20	∈	∈	PROPN
ejpam-4800	86	21	t	t	PROPN
ejpam-4800	86	22	.	.	PUNCT
ejpam-4800	87	1	then	then	ADV
ejpam-4800	87	2	from	from	ADP
ejpam-4800	87	3	the	the	DET
ejpam-4800	87	4	definition	definition	NOUN
ejpam-4800	87	5	of	of	ADP
ejpam-4800	87	6	rg(f(t	rg(f(t	NOUN
ejpam-4800	87	7	)	)	PUNCT
ejpam-4800	87	8	,	,	PUNCT
ejpam-4800	87	9	δ	δ	PROPN
ejpam-4800	87	10	)	)	PUNCT
ejpam-4800	87	11	,	,	PUNCT
ejpam-4800	87	12	we	we	PRON
ejpam-4800	87	13	have	have	VERB
ejpam-4800	87	14	||g(t)−	||g(t)−	PROPN
ejpam-4800	87	15	y||	y||	PROPN
ejpam-4800	87	16	<	<	X
ejpam-4800	87	17	||f(t)−	||f(t)−	PROPN
ejpam-4800	87	18	y||+	y||+	PROPN
ejpam-4800	87	19	δ	δ	PROPN
ejpam-4800	87	20	,	,	PUNCT
ejpam-4800	87	21	for	for	ADP
ejpam-4800	87	22	all	all	DET
ejpam-4800	87	23	y	y	PROPN
ejpam-4800	87	24	∈	∈	PROPN
ejpam-4800	87	25	g	g	PROPN
ejpam-4800	87	26	,	,	PUNCT
ejpam-4800	87	27	a.e	a.e	PROPN
ejpam-4800	87	28	.	.	PROPN
ejpam-4800	87	29	t	t	PROPN
ejpam-4800	87	30	∈	∈	PROPN
ejpam-4800	87	31	t.	t.	NOUN
ejpam-4800	87	32	this	this	PRON
ejpam-4800	87	33	implies	imply	VERB
ejpam-4800	87	34	as	as	ADP
ejpam-4800	87	35	a	a	DET
ejpam-4800	87	36	special	special	ADJ
ejpam-4800	87	37	case	case	NOUN
ejpam-4800	87	38	,	,	PUNCT
ejpam-4800	87	39	||g(t)−	||g(t)−	PROPN
ejpam-4800	87	40	h(t)||	h(t)||	VERB
ejpam-4800	87	41	<	<	X
ejpam-4800	87	42	||f(t)−	||f(t)−	PROPN
ejpam-4800	87	43	h(t)||+	h(t)||+	PROPN
ejpam-4800	87	44	δ	δ	PROPN
ejpam-4800	87	45	,	,	PUNCT
ejpam-4800	87	46	∀h	∀h	PROPN
ejpam-4800	87	47	∈	∈	PROPN
ejpam-4800	87	48	l1(µ,g	l1(µ,g	NUM
ejpam-4800	87	49	)	)	PUNCT
ejpam-4800	87	50	,	,	PUNCT
ejpam-4800	87	51	a.e	a.e	PROPN
ejpam-4800	87	52	.	.	PROPN
ejpam-4800	87	53	t	t	PROPN
ejpam-4800	87	54	∈	∈	PROPN
ejpam-4800	87	55	t.	t.	PROPN
ejpam-4800	87	56	hence	hence	ADV
ejpam-4800	87	57	,	,	PUNCT
ejpam-4800	87	58	we	we	PRON
ejpam-4800	87	59	get∫	get∫	VERB
ejpam-4800	87	60	t	t	X
ejpam-4800	88	1	||g(t)−	||g(t)−	PROPN
ejpam-4800	88	2	h(t)||dt	h(t)||dt	PROPN
ejpam-4800	88	3	<	<	X
ejpam-4800	88	4	∫	∫	PROPN
ejpam-4800	88	5	t	t	PROPN
ejpam-4800	88	6	||f(t)−	||f(t)−	PROPN
ejpam-4800	88	7	h(t)||dt+	h(t)||dt+	NOUN
ejpam-4800	88	8	δ	δ	PROPN
ejpam-4800	88	9	·	·	PUNCT
ejpam-4800	88	10	µ(t	µ(t	ADJ
ejpam-4800	88	11	)	)	PUNCT
ejpam-4800	88	12	,	,	PUNCT
ejpam-4800	88	13	∀h	∀h	PROPN
ejpam-4800	88	14	∈	∈	PROPN
ejpam-4800	88	15	l1(µ,g	l1(µ,g	NUM
ejpam-4800	88	16	)	)	PUNCT
ejpam-4800	88	17	.	.	PUNCT
ejpam-4800	89	1	and	and	CCONJ
ejpam-4800	89	2	since	since	SCONJ
ejpam-4800	89	3	µ(t	µ(t	ADJ
ejpam-4800	89	4	)	)	PUNCT
ejpam-4800	89	5	<	<	X
ejpam-4800	89	6	∞	∞	PROPN
ejpam-4800	89	7	,	,	PUNCT
ejpam-4800	89	8	we	we	PRON
ejpam-4800	89	9	can	can	AUX
ejpam-4800	89	10	take	take	VERB
ejpam-4800	89	11	δ′	δ′	NOUN
ejpam-4800	89	12	=	=	SYM
ejpam-4800	89	13	δ	δ	PROPN
ejpam-4800	89	14	·	·	PUNCT
ejpam-4800	89	15	µ(t	µ(t	ADJ
ejpam-4800	89	16	)	)	PUNCT
ejpam-4800	89	17	,	,	PUNCT
ejpam-4800	89	18	hence	hence	ADV
ejpam-4800	89	19	,	,	PUNCT
ejpam-4800	89	20	we	we	PRON
ejpam-4800	89	21	obtain	obtain	VERB
ejpam-4800	89	22	||g	||g	NOUN
ejpam-4800	89	23	−	−	PROPN
ejpam-4800	89	24	h||	h||	NOUN
ejpam-4800	89	25	<	<	X
ejpam-4800	89	26	||f	||f	PROPN
ejpam-4800	89	27	−	−	PROPN
ejpam-4800	89	28	h||+	h||+	ADJ
ejpam-4800	89	29	δ′	δ′	PROPN
ejpam-4800	89	30	,	,	PUNCT
ejpam-4800	89	31	∀h	∀h	PROPN
ejpam-4800	89	32	∈	∈	PROPN
ejpam-4800	89	33	l1(µ,g	l1(µ,g	NUM
ejpam-4800	89	34	)	)	PUNCT
ejpam-4800	89	35	.	.	PUNCT
ejpam-4800	90	1	so	so	ADV
ejpam-4800	90	2	,	,	PUNCT
ejpam-4800	90	3	we	we	PRON
ejpam-4800	90	4	get	get	VERB
ejpam-4800	90	5	g	g	PROPN
ejpam-4800	90	6	∈	∈	PROPN
ejpam-4800	90	7	rl1(µ,g)(f	rl1(µ,g)(f	VERB
ejpam-4800	90	8	,	,	PUNCT
ejpam-4800	90	9	δ	δ	PROPN
ejpam-4800	90	10	′	′	NOUN
ejpam-4800	90	11	)	)	PUNCT
ejpam-4800	90	12	.	.	PUNCT
ejpam-4800	91	1	for	for	ADP
ejpam-4800	91	2	the	the	DET
ejpam-4800	91	3	other	other	ADJ
ejpam-4800	91	4	direction	direction	NOUN
ejpam-4800	91	5	,	,	PUNCT
ejpam-4800	91	6	we	we	PRON
ejpam-4800	91	7	proceed	proceed	VERB
ejpam-4800	91	8	as	as	SCONJ
ejpam-4800	91	9	follows	follow	VERB
ejpam-4800	91	10	.	.	PUNCT
ejpam-4800	92	1	given	give	VERB
ejpam-4800	92	2	g	g	PROPN
ejpam-4800	92	3	∈	∈	PROPN
ejpam-4800	92	4	rl1(µ,g)(f	rl1(µ,g)(f	VERB
ejpam-4800	92	5	,	,	PUNCT
ejpam-4800	92	6	δ	δ	PROPN
ejpam-4800	92	7	′	′	NOUN
ejpam-4800	92	8	)	)	PUNCT
ejpam-4800	92	9	,	,	PUNCT
ejpam-4800	92	10	for	for	ADP
ejpam-4800	92	11	some	some	DET
ejpam-4800	92	12	δ′	δ′	NOUN
ejpam-4800	92	13	>	>	X
ejpam-4800	92	14	0	0	X
ejpam-4800	92	15	.	.	PUNCT
ejpam-4800	93	1	then	then	ADV
ejpam-4800	93	2	,	,	PUNCT
ejpam-4800	93	3	from	from	ADP
ejpam-4800	93	4	(	(	PUNCT
ejpam-4800	93	5	3	3	NUM
ejpam-4800	93	6	)	)	PUNCT
ejpam-4800	93	7	,	,	PUNCT
ejpam-4800	93	8	we	we	PRON
ejpam-4800	93	9	have	have	AUX
ejpam-4800	93	10	,	,	PUNCT
ejpam-4800	93	11	||g	||g	VERB
ejpam-4800	93	12	−	−	PROPN
ejpam-4800	93	13	k||	k||	X
ejpam-4800	93	14	<	<	X
ejpam-4800	93	15	||f	||f	PROPN
ejpam-4800	93	16	−	−	PROPN
ejpam-4800	93	17	k||+	k||+	PROPN
ejpam-4800	93	18	δ′	δ′	PROPN
ejpam-4800	93	19	,	,	PUNCT
ejpam-4800	93	20	∀k	∀k	NOUN
ejpam-4800	93	21	∈	∈	PROPN
ejpam-4800	93	22	l1(µ,g	l1(µ,g	VERB
ejpam-4800	93	23	)	)	PUNCT
ejpam-4800	93	24	.	.	PUNCT
ejpam-4800	94	1	but	but	CCONJ
ejpam-4800	94	2	since	since	SCONJ
ejpam-4800	94	3	g	g	PROPN
ejpam-4800	94	4	is	be	AUX
ejpam-4800	94	5	separable	separable	ADJ
ejpam-4800	94	6	and	and	CCONJ
ejpam-4800	94	7	coproximinal	coproximinal	ADJ
ejpam-4800	94	8	in	in	ADP
ejpam-4800	94	9	x	x	NOUN
ejpam-4800	94	10	,	,	PUNCT
ejpam-4800	94	11	then	then	ADV
ejpam-4800	94	12	l1(µ,g	l1(µ,g	CCONJ
ejpam-4800	94	13	)	)	PUNCT
ejpam-4800	94	14	is	be	AUX
ejpam-4800	94	15	coproximinal	coproximinal	ADJ
ejpam-4800	94	16	in	in	ADP
ejpam-4800	94	17	l1(µ,x	l1(µ,x	NOUN
ejpam-4800	94	18	)	)	PUNCT
ejpam-4800	94	19	,	,	PUNCT
ejpam-4800	94	20	see	see	VERB
ejpam-4800	94	21	[	[	X
ejpam-4800	94	22	3	3	NUM
ejpam-4800	94	23	]	]	PUNCT
ejpam-4800	94	24	.	.	PUNCT
ejpam-4800	95	1	so	so	ADV
ejpam-4800	95	2	,	,	PUNCT
ejpam-4800	95	3	let	let	VERB
ejpam-4800	95	4	h	h	PRON
ejpam-4800	95	5	∈	∈	PROPN
ejpam-4800	95	6	rl1(µ,g)(f	rl1(µ,g)(f	VERB
ejpam-4800	95	7	)	)	PUNCT
ejpam-4800	95	8	satisfying	satisfy	VERB
ejpam-4800	95	9	||g	||g	NOUN
ejpam-4800	95	10	−	−	PROPN
ejpam-4800	95	11	h||	h||	NOUN
ejpam-4800	95	12	<	<	X
ejpam-4800	95	13	δ′.	δ′.	X
ejpam-4800	95	14	again	again	ADV
ejpam-4800	95	15	,	,	PUNCT
ejpam-4800	95	16	from	from	ADP
ejpam-4800	95	17	[	[	X
ejpam-4800	95	18	4	4	NUM
ejpam-4800	95	19	]	]	PUNCT
ejpam-4800	95	20	,	,	PUNCT
ejpam-4800	95	21	h(t	h(t	PROPN
ejpam-4800	95	22	)	)	PUNCT
ejpam-4800	95	23	is	be	AUX
ejpam-4800	95	24	a	a	DET
ejpam-4800	95	25	best	good	ADJ
ejpam-4800	95	26	coapproximation	coapproximation	NOUN
ejpam-4800	95	27	for	for	ADP
ejpam-4800	95	28	f(t	f(t	NOUN
ejpam-4800	95	29	)	)	PUNCT
ejpam-4800	95	30	,	,	PUNCT
ejpam-4800	96	1	a.e	a.e	PROPN
ejpam-4800	96	2	.	.	PROPN
ejpam-4800	96	3	t	t	PROPN
ejpam-4800	96	4	∈	∈	PROPN
ejpam-4800	96	5	t	t	PROPN
ejpam-4800	96	6	.	.	PUNCT
ejpam-4800	97	1	moreover	moreover	ADV
ejpam-4800	97	2	,	,	PUNCT
ejpam-4800	97	3	||g(t)−	||g(t)−	PROPN
ejpam-4800	97	4	h(t)||	h(t)||	VERB
ejpam-4800	97	5	<	<	X
ejpam-4800	97	6	δ′/µ(t	δ′/µ(t	NOUN
ejpam-4800	97	7	)	)	PUNCT
ejpam-4800	97	8	,	,	PUNCT
ejpam-4800	97	9	a.e	a.e	PROPN
ejpam-4800	97	10	.	.	PROPN
ejpam-4800	97	11	t	t	PROPN
ejpam-4800	97	12	∈	∈	PROPN
ejpam-4800	97	13	t.	t.	PROPN
ejpam-4800	97	14	j.	j.	PROPN
ejpam-4800	97	15	jawdat	jawdat	PROPN
ejpam-4800	97	16	/	/	SYM
ejpam-4800	97	17	eur	eur	PROPN
ejpam-4800	97	18	.	.	PUNCT
ejpam-4800	98	1	j.	j.	PROPN
ejpam-4800	98	2	pure	pure	PROPN
ejpam-4800	98	3	appl	appl	PROPN
ejpam-4800	98	4	.	.	PROPN
ejpam-4800	98	5	math	math	PROPN
ejpam-4800	98	6	,	,	PUNCT
ejpam-4800	98	7	16	16	NUM
ejpam-4800	98	8	(	(	PUNCT
ejpam-4800	98	9	3	3	NUM
ejpam-4800	98	10	)	)	PUNCT
ejpam-4800	98	11	(	(	PUNCT
ejpam-4800	98	12	2023	2023	NUM
ejpam-4800	98	13	)	)	PUNCT
ejpam-4800	98	14	,	,	PUNCT
ejpam-4800	98	15	1543	1543	NUM
ejpam-4800	98	16	-	-	SYM
ejpam-4800	98	17	1551	1551	NUM
ejpam-4800	98	18	1547	1547	NUM
ejpam-4800	98	19	hence	hence	ADV
ejpam-4800	98	20	,	,	PUNCT
ejpam-4800	98	21	for	for	ADP
ejpam-4800	98	22	any	any	DET
ejpam-4800	98	23	y	y	PROPN
ejpam-4800	98	24	∈	∈	PROPN
ejpam-4800	98	25	g	g	PROPN
ejpam-4800	98	26	,	,	PUNCT
ejpam-4800	98	27	we	we	PRON
ejpam-4800	98	28	have	have	VERB
ejpam-4800	98	29	||g(t)−	||g(t)−	PROPN
ejpam-4800	98	30	y||	y||	PROPN
ejpam-4800	99	1	=	=	SYM
ejpam-4800	99	2	||g(t)−	||g(t)−	PROPN
ejpam-4800	99	3	h(t	h(t	PROPN
ejpam-4800	99	4	)	)	PUNCT
ejpam-4800	100	1	+	+	CCONJ
ejpam-4800	100	2	h(t)−	h(t)−	PROPN
ejpam-4800	100	3	y||	y||	PROPN
ejpam-4800	100	4	≤	≤	PROPN
ejpam-4800	100	5	||g(t)−	||g(t)−	PROPN
ejpam-4800	100	6	h(t)||+	h(t)||+	PROPN
ejpam-4800	100	7	||h(t)−	||h(t)−	PROPN
ejpam-4800	101	1	y||	y||	NOUN
ejpam-4800	101	2	<	<	X
ejpam-4800	101	3	||f(t)−	||f(t)−	PROPN
ejpam-4800	101	4	y||+	y||+	NOUN
ejpam-4800	101	5	δ′/µ(t	δ′/µ(t	PROPN
ejpam-4800	101	6	)	)	PUNCT
ejpam-4800	101	7	,	,	PUNCT
ejpam-4800	101	8	∀y	∀y	PROPN
ejpam-4800	101	9	∈	∈	PROPN
ejpam-4800	101	10	g	g	PROPN
ejpam-4800	101	11	,	,	PUNCT
ejpam-4800	101	12	a.e	a.e	PROPN
ejpam-4800	101	13	.	.	PROPN
ejpam-4800	101	14	t	t	PROPN
ejpam-4800	101	15	∈	∈	PROPN
ejpam-4800	101	16	t.	t.	PROPN
ejpam-4800	101	17	finally	finally	ADV
ejpam-4800	101	18	,	,	PUNCT
ejpam-4800	101	19	taking	take	VERB
ejpam-4800	101	20	δ	δ	NOUN
ejpam-4800	101	21	=	=	SYM
ejpam-4800	101	22	δ′/µ(t	δ′/µ(t	PROPN
ejpam-4800	101	23	)	)	PUNCT
ejpam-4800	101	24	,	,	PUNCT
ejpam-4800	101	25	we	we	PRON
ejpam-4800	101	26	get	get	VERB
ejpam-4800	101	27	,	,	PUNCT
ejpam-4800	101	28	g(t	g(t	PROPN
ejpam-4800	101	29	)	)	PUNCT
ejpam-4800	101	30	∈	∈	PROPN
ejpam-4800	101	31	rg(f(t	rg(f(t	NOUN
ejpam-4800	101	32	)	)	PUNCT
ejpam-4800	101	33	,	,	PUNCT
ejpam-4800	101	34	δ	δ	PROPN
ejpam-4800	101	35	)	)	PUNCT
ejpam-4800	101	36	,	,	PUNCT
ejpam-4800	101	37	for	for	ADP
ejpam-4800	101	38	a.e	a.e	PROPN
ejpam-4800	101	39	.	.	PROPN
ejpam-4800	101	40	t	t	PROPN
ejpam-4800	101	41	∈	∈	PROPN
ejpam-4800	101	42	t.	t.	NOUN
ejpam-4800	101	43	remark	remark	NOUN
ejpam-4800	101	44	1	1	NUM
ejpam-4800	101	45	.	.	PUNCT
ejpam-4800	102	1	the	the	DET
ejpam-4800	102	2	result	result	NOUN
ejpam-4800	102	3	in	in	ADP
ejpam-4800	102	4	lemma	lemma	PROPN
ejpam-4800	102	5	1	1	NUM
ejpam-4800	102	6	can	can	AUX
ejpam-4800	102	7	be	be	AUX
ejpam-4800	102	8	easily	easily	ADV
ejpam-4800	102	9	extended	extend	VERB
ejpam-4800	102	10	for	for	ADP
ejpam-4800	102	11	the	the	DET
ejpam-4800	102	12	case	case	NOUN
ejpam-4800	102	13	of	of	ADP
ejpam-4800	102	14	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	102	15	)	)	PUNCT
ejpam-4800	102	16	,	,	PUNCT
ejpam-4800	102	17	1	1	NUM
ejpam-4800	102	18	<	<	X
ejpam-4800	102	19	p	p	X
ejpam-4800	102	20	<	<	X
ejpam-4800	102	21	∞.	∞.	PROPN
ejpam-4800	102	22	one	one	NUM
ejpam-4800	102	23	main	main	ADJ
ejpam-4800	102	24	result	result	NOUN
ejpam-4800	102	25	in	in	ADP
ejpam-4800	102	26	this	this	DET
ejpam-4800	102	27	paper	paper	NOUN
ejpam-4800	102	28	,	,	PUNCT
ejpam-4800	102	29	is	be	AUX
ejpam-4800	102	30	the	the	DET
ejpam-4800	102	31	following	following	NOUN
ejpam-4800	102	32	.	.	PUNCT
ejpam-4800	103	1	theorem	theorem	NOUN
ejpam-4800	103	2	1	1	NUM
ejpam-4800	103	3	.	.	PUNCT
ejpam-4800	104	1	if	if	SCONJ
ejpam-4800	104	2	g	g	PROPN
ejpam-4800	104	3	is	be	AUX
ejpam-4800	104	4	separable	separable	ADJ
ejpam-4800	104	5	and	and	CCONJ
ejpam-4800	104	6	strongly	strongly	ADV
ejpam-4800	104	7	coproximinal	coproximinal	ADJ
ejpam-4800	104	8	in	in	ADP
ejpam-4800	104	9	x	x	PRON
ejpam-4800	104	10	,	,	PUNCT
ejpam-4800	104	11	then	then	ADV
ejpam-4800	104	12	l1(µ,g	l1(µ,g	CCONJ
ejpam-4800	104	13	)	)	PUNCT
ejpam-4800	104	14	is	be	AUX
ejpam-4800	104	15	strongly	strongly	ADV
ejpam-4800	104	16	coproximinal	coproximinal	ADJ
ejpam-4800	104	17	in	in	ADP
ejpam-4800	104	18	l1(µ,x	l1(µ,x	NOUN
ejpam-4800	104	19	)	)	PUNCT
ejpam-4800	104	20	.	.	PUNCT
ejpam-4800	105	1	proof	proof	NOUN
ejpam-4800	105	2	.	.	PUNCT
ejpam-4800	106	1	given	give	VERB
ejpam-4800	106	2	g	g	PROPN
ejpam-4800	106	3	in	in	ADP
ejpam-4800	106	4	x	x	DET
ejpam-4800	106	5	a	a	DET
ejpam-4800	106	6	strongly	strongly	ADV
ejpam-4800	106	7	coproximinal	coproximinal	ADJ
ejpam-4800	106	8	subspace	subspace	NOUN
ejpam-4800	106	9	then	then	ADV
ejpam-4800	106	10	g	g	PROPN
ejpam-4800	106	11	is	be	AUX
ejpam-4800	106	12	coproximinal	coproximinal	ADJ
ejpam-4800	106	13	in	in	ADP
ejpam-4800	106	14	x	x	PROPN
ejpam-4800	106	15	(	(	PUNCT
ejpam-4800	106	16	by	by	ADP
ejpam-4800	106	17	definition	definition	NOUN
ejpam-4800	106	18	)	)	PUNCT
ejpam-4800	106	19	.	.	PUNCT
ejpam-4800	107	1	also	also	ADV
ejpam-4800	107	2	g	g	ADP
ejpam-4800	107	3	being	be	AUX
ejpam-4800	107	4	separable	separable	ADJ
ejpam-4800	107	5	then	then	ADV
ejpam-4800	107	6	l1(µ,g	l1(µ,g	CCONJ
ejpam-4800	107	7	)	)	PUNCT
ejpam-4800	108	1	is	be	AUX
ejpam-4800	108	2	coproximinal	coproximinal	ADJ
ejpam-4800	108	3	in	in	ADP
ejpam-4800	108	4	l1(µ,x	l1(µ,x	NOUN
ejpam-4800	108	5	)	)	PUNCT
ejpam-4800	108	6	,	,	PUNCT
ejpam-4800	108	7	see	see	VERB
ejpam-4800	108	8	[	[	X
ejpam-4800	108	9	4	4	NUM
ejpam-4800	108	10	]	]	PUNCT
ejpam-4800	108	11	.	.	PUNCT
ejpam-4800	109	1	now	now	ADV
ejpam-4800	109	2	,	,	PUNCT
ejpam-4800	109	3	let	let	VERB
ejpam-4800	109	4	f	f	PROPN
ejpam-4800	109	5	∈	∈	PROPN
ejpam-4800	109	6	l1(µ,x	l1(µ,x	NOUN
ejpam-4800	109	7	)	)	PUNCT
ejpam-4800	109	8	and	and	CCONJ
ejpam-4800	109	9	ε	ε	PROPN
ejpam-4800	109	10	>	>	X
ejpam-4800	109	11	0	0	NUM
ejpam-4800	109	12	be	be	AUX
ejpam-4800	109	13	arbitrary	arbitrary	ADJ
ejpam-4800	109	14	.	.	PUNCT
ejpam-4800	110	1	let	let	VERB
ejpam-4800	110	2	g	g	PROPN
ejpam-4800	110	3	∈	∈	PROPN
ejpam-4800	110	4	rl1(µ,g)(f	rl1(µ,g)(f	VERB
ejpam-4800	110	5	,	,	PUNCT
ejpam-4800	110	6	δ	δ	PROPN
ejpam-4800	110	7	)	)	PUNCT
ejpam-4800	110	8	,	,	PUNCT
ejpam-4800	110	9	for	for	ADP
ejpam-4800	110	10	some	some	DET
ejpam-4800	110	11	δ	δ	PROPN
ejpam-4800	110	12	>	>	X
ejpam-4800	110	13	0	0	PROPN
ejpam-4800	110	14	,	,	PUNCT
ejpam-4800	110	15	then	then	ADV
ejpam-4800	110	16	by	by	ADP
ejpam-4800	110	17	lemma	lemma	PROPN
ejpam-4800	110	18	1	1	NUM
ejpam-4800	110	19	,	,	PUNCT
ejpam-4800	110	20	g(t	g(t	PROPN
ejpam-4800	110	21	)	)	PUNCT
ejpam-4800	110	22	∈	∈	PROPN
ejpam-4800	110	23	rg(f(t	rg(f(t	NOUN
ejpam-4800	110	24	)	)	PUNCT
ejpam-4800	110	25	,	,	PUNCT
ejpam-4800	110	26	δt	δt	PROPN
ejpam-4800	110	27	)	)	PUNCT
ejpam-4800	110	28	,	,	PUNCT
ejpam-4800	110	29	for	for	ADP
ejpam-4800	110	30	some	some	PRON
ejpam-4800	110	31	δt	δt	NOUN
ejpam-4800	110	32	>	>	X
ejpam-4800	110	33	0	0	PROPN
ejpam-4800	110	34	,	,	PUNCT
ejpam-4800	110	35	a.e	a.e	PROPN
ejpam-4800	110	36	.	.	PROPN
ejpam-4800	110	37	t	t	PROPN
ejpam-4800	110	38	in	in	ADP
ejpam-4800	110	39	t	t	PROPN
ejpam-4800	110	40	.	.	PUNCT
ejpam-4800	111	1	again	again	ADV
ejpam-4800	111	2	,	,	PUNCT
ejpam-4800	111	3	since	since	SCONJ
ejpam-4800	111	4	g	g	PROPN
ejpam-4800	111	5	is	be	AUX
ejpam-4800	111	6	strongly	strongly	ADV
ejpam-4800	111	7	coproximinal	coproximinal	ADJ
ejpam-4800	111	8	in	in	ADP
ejpam-4800	111	9	x	x	X
ejpam-4800	111	10	then	then	ADV
ejpam-4800	111	11	,	,	PUNCT
ejpam-4800	111	12	from	from	ADP
ejpam-4800	111	13	definition	definition	NOUN
ejpam-4800	111	14	5	5	NUM
ejpam-4800	111	15	,	,	PUNCT
ejpam-4800	111	16	there	there	PRON
ejpam-4800	111	17	exist	exist	VERB
ejpam-4800	111	18	yt	yt	PRON
ejpam-4800	111	19	∈	∈	PROPN
ejpam-4800	111	20	rg(f(t	rg(f(t	NOUN
ejpam-4800	111	21	)	)	PUNCT
ejpam-4800	111	22	)	)	PUNCT
ejpam-4800	112	1	satisfying	satisfy	VERB
ejpam-4800	112	2	||g(t	||g(t	NOUN
ejpam-4800	112	3	)	)	PUNCT
ejpam-4800	112	4	−	−	PROPN
ejpam-4800	113	1	yt||	yt||	PROPN
ejpam-4800	113	2	<	<	X
ejpam-4800	113	3	ε/µ(t	ε/µ(t	PROPN
ejpam-4800	113	4	)	)	PUNCT
ejpam-4800	113	5	,	,	PUNCT
ejpam-4800	113	6	a.e	a.e	PROPN
ejpam-4800	113	7	.	.	PROPN
ejpam-4800	113	8	t	t	PROPN
ejpam-4800	113	9	∈	∈	PROPN
ejpam-4800	113	10	t	t	PROPN
ejpam-4800	113	11	.	.	PUNCT
ejpam-4800	114	1	since	since	SCONJ
ejpam-4800	114	2	g	g	PROPN
ejpam-4800	114	3	separable	separable	PROPN
ejpam-4800	114	4	,	,	PUNCT
ejpam-4800	114	5	we	we	PRON
ejpam-4800	114	6	may	may	AUX
ejpam-4800	114	7	define	define	VERB
ejpam-4800	114	8	a	a	DET
ejpam-4800	114	9	function	function	NOUN
ejpam-4800	114	10	h	h	NOUN
ejpam-4800	114	11	,	,	PUNCT
ejpam-4800	114	12	such	such	ADJ
ejpam-4800	114	13	that	that	SCONJ
ejpam-4800	114	14	h(t	h(t	PROPN
ejpam-4800	114	15	)	)	PUNCT
ejpam-4800	114	16	=	=	SYM
ejpam-4800	114	17	yt	yt	NOUN
ejpam-4800	114	18	,	,	PUNCT
ejpam-4800	114	19	for	for	ADP
ejpam-4800	114	20	all	all	DET
ejpam-4800	114	21	t	t	NOUN
ejpam-4800	114	22	∈	∈	PROPN
ejpam-4800	114	23	t	t	PROPN
ejpam-4800	114	24	.	.	PUNCT
ejpam-4800	115	1	so	so	ADV
ejpam-4800	115	2	,	,	PUNCT
ejpam-4800	115	3	||g(t)−	||g(t)−	PROPN
ejpam-4800	115	4	h(t)||	h(t)||	VERB
ejpam-4800	115	5	<	<	X
ejpam-4800	115	6	ε/µ(t	ε/µ(t	PROPN
ejpam-4800	115	7	)	)	PUNCT
ejpam-4800	115	8	,	,	PUNCT
ejpam-4800	115	9	a.e	a.e	PROPN
ejpam-4800	115	10	.	.	PROPN
ejpam-4800	115	11	t	t	PROPN
ejpam-4800	115	12	∈	∈	PROPN
ejpam-4800	115	13	t.	t.	PROPN
ejpam-4800	115	14	(	(	PUNCT
ejpam-4800	115	15	4	4	NUM
ejpam-4800	115	16	)	)	PUNCT
ejpam-4800	115	17	then	then	ADV
ejpam-4800	115	18	h	h	NOUN
ejpam-4800	115	19	can	can	AUX
ejpam-4800	115	20	be	be	AUX
ejpam-4800	115	21	proved	prove	VERB
ejpam-4800	115	22	to	to	PART
ejpam-4800	115	23	be	be	AUX
ejpam-4800	115	24	a	a	DET
ejpam-4800	115	25	measurable	measurable	ADJ
ejpam-4800	115	26	function	function	NOUN
ejpam-4800	115	27	using	use	VERB
ejpam-4800	115	28	a	a	DET
ejpam-4800	115	29	technique	technique	NOUN
ejpam-4800	115	30	similar	similar	ADJ
ejpam-4800	115	31	to	to	ADP
ejpam-4800	115	32	that	that	PRON
ejpam-4800	115	33	of	of	ADP
ejpam-4800	115	34	theorem	theorem	NOUN
ejpam-4800	115	35	7	7	NUM
ejpam-4800	115	36	in	in	ADP
ejpam-4800	115	37	[	[	X
ejpam-4800	115	38	6	6	NUM
ejpam-4800	115	39	]	]	PUNCT
ejpam-4800	115	40	.	.	PUNCT
ejpam-4800	116	1	also	also	ADV
ejpam-4800	116	2	,	,	PUNCT
ejpam-4800	116	3	h	h	NOUN
ejpam-4800	116	4	∈	∈	PROPN
ejpam-4800	116	5	l1(µ,g	l1(µ,g	CCONJ
ejpam-4800	116	6	)	)	PUNCT
ejpam-4800	116	7	since	since	SCONJ
ejpam-4800	116	8	||h(t)||	||h(t)||	ADJ
ejpam-4800	116	9	≤	≤	NUM
ejpam-4800	116	10	||h(t	||h(t	NOUN
ejpam-4800	116	11	)	)	PUNCT
ejpam-4800	116	12	−	−	NOUN
ejpam-4800	117	1	g(t)||	g(t)||	NOUN
ejpam-4800	118	1	+	+	NUM
ejpam-4800	118	2	||g(t)||	||g(t)||	NOUN
ejpam-4800	118	3	<	<	X
ejpam-4800	118	4	ε/µ(t	ε/µ(t	PROPN
ejpam-4800	118	5	)	)	PUNCT
ejpam-4800	119	1	+	+	CCONJ
ejpam-4800	119	2	||g(t)||	||g(t)||	NOUN
ejpam-4800	119	3	,	,	PUNCT
ejpam-4800	119	4	a.e	a.e	PROPN
ejpam-4800	119	5	.	.	PROPN
ejpam-4800	119	6	t	t	PROPN
ejpam-4800	119	7	in	in	ADP
ejpam-4800	119	8	t	t	PROPN
ejpam-4800	119	9	.	.	PUNCT
ejpam-4800	120	1	finally	finally	ADV
ejpam-4800	120	2	,	,	PUNCT
ejpam-4800	120	3	by	by	ADP
ejpam-4800	120	4	the	the	DET
ejpam-4800	120	5	way	way	NOUN
ejpam-4800	120	6	h	h	NOUN
ejpam-4800	120	7	was	be	AUX
ejpam-4800	120	8	defined	define	VERB
ejpam-4800	120	9	,	,	PUNCT
ejpam-4800	120	10	it	it	PRON
ejpam-4800	120	11	follows	follow	VERB
ejpam-4800	120	12	that	that	SCONJ
ejpam-4800	120	13	h	h	PROPN
ejpam-4800	120	14	∈	∈	PROPN
ejpam-4800	120	15	rl1(µ,g)(f	rl1(µ,g)(f	VERB
ejpam-4800	120	16	)	)	PUNCT
ejpam-4800	120	17	and	and	CCONJ
ejpam-4800	120	18	,	,	PUNCT
ejpam-4800	120	19	from	from	ADP
ejpam-4800	120	20	(	(	PUNCT
ejpam-4800	120	21	4	4	NUM
ejpam-4800	120	22	)	)	PUNCT
ejpam-4800	120	23	,	,	PUNCT
ejpam-4800	120	24	h	h	NOUN
ejpam-4800	120	25	satisfies	satisfy	VERB
ejpam-4800	120	26	||g−	||g−	ADP
ejpam-4800	120	27	h||	h||	X
ejpam-4800	120	28	<	<	X
ejpam-4800	120	29	ε	ε	PROPN
ejpam-4800	120	30	.	.	PUNCT
ejpam-4800	121	1	hence	hence	ADV
ejpam-4800	121	2	,	,	PUNCT
ejpam-4800	121	3	definition	definition	NOUN
ejpam-4800	121	4	5	5	NUM
ejpam-4800	121	5	is	be	AUX
ejpam-4800	121	6	satisfied	satisfied	ADJ
ejpam-4800	121	7	and	and	CCONJ
ejpam-4800	121	8	we	we	PRON
ejpam-4800	121	9	get	get	VERB
ejpam-4800	121	10	that	that	PRON
ejpam-4800	121	11	l1(µ,g	l1(µ,g	CCONJ
ejpam-4800	121	12	)	)	PUNCT
ejpam-4800	121	13	is	be	AUX
ejpam-4800	121	14	strongly	strongly	ADV
ejpam-4800	121	15	coproximinal	coproximinal	ADJ
ejpam-4800	121	16	in	in	ADP
ejpam-4800	121	17	l1(µ,x	l1(µ,x	NOUN
ejpam-4800	121	18	)	)	PUNCT
ejpam-4800	121	19	.	.	PUNCT
ejpam-4800	122	1	theorem	theorem	NOUN
ejpam-4800	122	2	2	2	NUM
ejpam-4800	122	3	.	.	PUNCT
ejpam-4800	123	1	let	let	AUX
ejpam-4800	123	2	lp(µ,g	lp(µ,g	NUM
ejpam-4800	123	3	)	)	PUNCT
ejpam-4800	123	4	be	be	AUX
ejpam-4800	123	5	strongly	strongly	ADV
ejpam-4800	123	6	coproximinal	coproximinal	ADJ
ejpam-4800	123	7	in	in	ADP
ejpam-4800	123	8	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	123	9	)	)	PUNCT
ejpam-4800	123	10	,	,	PUNCT
ejpam-4800	123	11	1	1	NUM
ejpam-4800	123	12	≤	≤	NOUN
ejpam-4800	123	13	p	p	X
ejpam-4800	123	14	<	<	X
ejpam-4800	123	15	∞	∞	PROPN
ejpam-4800	123	16	,	,	PUNCT
ejpam-4800	123	17	then	then	ADV
ejpam-4800	123	18	g	g	PROPN
ejpam-4800	123	19	is	be	AUX
ejpam-4800	123	20	strongly	strongly	ADV
ejpam-4800	123	21	coproximinal	coproximinal	ADJ
ejpam-4800	123	22	in	in	ADP
ejpam-4800	123	23	x.	x.	NOUN
ejpam-4800	123	24	proof	proof	NOUN
ejpam-4800	123	25	.	.	PUNCT
ejpam-4800	124	1	by	by	ADP
ejpam-4800	124	2	relating	relate	VERB
ejpam-4800	124	3	each	each	DET
ejpam-4800	124	4	x	x	PUNCT
ejpam-4800	124	5	in	in	ADP
ejpam-4800	124	6	x	x	PUNCT
ejpam-4800	124	7	with	with	ADP
ejpam-4800	124	8	a	a	DET
ejpam-4800	124	9	function	function	NOUN
ejpam-4800	124	10	fx	fx	NOUN
ejpam-4800	124	11	=	=	PUNCT
ejpam-4800	124	12	x	x	SYM
ejpam-4800	124	13	·	·	PUNCT
ejpam-4800	124	14	χt	χt	SCONJ
ejpam-4800	124	15	,	,	PUNCT
ejpam-4800	124	16	in	in	ADP
ejpam-4800	124	17	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	124	18	)	)	PUNCT
ejpam-4800	124	19	,	,	PUNCT
ejpam-4800	124	20	where	where	SCONJ
ejpam-4800	124	21	χt	χt	ADV
ejpam-4800	124	22	is	be	AUX
ejpam-4800	124	23	the	the	DET
ejpam-4800	124	24	characteristic	characteristic	ADJ
ejpam-4800	124	25	function	function	NOUN
ejpam-4800	124	26	on	on	ADP
ejpam-4800	124	27	t	t	PROPN
ejpam-4800	124	28	and	and	CCONJ
ejpam-4800	124	29	since	since	SCONJ
ejpam-4800	124	30	lp(µ,g	lp(µ,g	NUM
ejpam-4800	124	31	)	)	PUNCT
ejpam-4800	124	32	is	be	AUX
ejpam-4800	124	33	strongly	strongly	ADV
ejpam-4800	124	34	coproximinal	coproximinal	ADJ
ejpam-4800	124	35	in	in	ADP
ejpam-4800	124	36	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	124	37	)	)	PUNCT
ejpam-4800	124	38	then	then	ADV
ejpam-4800	124	39	by	by	ADP
ejpam-4800	124	40	the	the	DET
ejpam-4800	124	41	definition	definition	NOUN
ejpam-4800	124	42	of	of	ADP
ejpam-4800	124	43	strong	strong	ADJ
ejpam-4800	124	44	coproximinality	coproximinality	NOUN
ejpam-4800	124	45	,	,	PUNCT
ejpam-4800	124	46	lemma	lemma	PROPN
ejpam-4800	124	47	1	1	NUM
ejpam-4800	124	48	and	and	CCONJ
ejpam-4800	124	49	remark	remark	NOUN
ejpam-4800	124	50	1	1	NUM
ejpam-4800	124	51	,	,	PUNCT
ejpam-4800	124	52	thereafter	thereafter	ADV
ejpam-4800	124	53	,	,	PUNCT
ejpam-4800	124	54	the	the	DET
ejpam-4800	124	55	result	result	NOUN
ejpam-4800	124	56	follows	follow	VERB
ejpam-4800	124	57	.	.	PUNCT
ejpam-4800	125	1	another	another	DET
ejpam-4800	125	2	main	main	ADJ
ejpam-4800	125	3	result	result	NOUN
ejpam-4800	125	4	is	be	AUX
ejpam-4800	125	5	the	the	DET
ejpam-4800	125	6	following	following	NOUN
ejpam-4800	125	7	.	.	PUNCT
ejpam-4800	126	1	theorem	theorem	NOUN
ejpam-4800	126	2	3	3	X
ejpam-4800	126	3	.	.	PUNCT
ejpam-4800	127	1	let	let	AUX
ejpam-4800	127	2	l1(µ,g	l1(µ,g	X
ejpam-4800	127	3	)	)	PUNCT
ejpam-4800	127	4	be	be	AUX
ejpam-4800	127	5	strongly	strongly	ADV
ejpam-4800	127	6	coproximinal	coproximinal	ADJ
ejpam-4800	127	7	in	in	ADP
ejpam-4800	127	8	l1(µ,x	l1(µ,x	NOUN
ejpam-4800	127	9	)	)	PUNCT
ejpam-4800	127	10	then	then	ADV
ejpam-4800	127	11	lp(µ,g	lp(µ,g	NUM
ejpam-4800	127	12	)	)	PUNCT
ejpam-4800	127	13	is	be	AUX
ejpam-4800	127	14	strongly	strongly	ADV
ejpam-4800	127	15	coproximinal	coproximinal	ADJ
ejpam-4800	127	16	in	in	ADP
ejpam-4800	127	17	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	127	18	)	)	PUNCT
ejpam-4800	127	19	,	,	PUNCT
ejpam-4800	127	20	for	for	ADP
ejpam-4800	127	21	1	1	NUM
ejpam-4800	127	22	<	<	X
ejpam-4800	127	23	p	p	X
ejpam-4800	127	24	<	<	X
ejpam-4800	127	25	∞.	∞.	PROPN
ejpam-4800	127	26	j.	j.	PROPN
ejpam-4800	127	27	jawdat	jawdat	PROPN
ejpam-4800	127	28	/	/	SYM
ejpam-4800	127	29	eur	eur	PROPN
ejpam-4800	127	30	.	.	PUNCT
ejpam-4800	128	1	j.	j.	PROPN
ejpam-4800	128	2	pure	pure	PROPN
ejpam-4800	128	3	appl	appl	PROPN
ejpam-4800	128	4	.	.	PROPN
ejpam-4800	128	5	math	math	PROPN
ejpam-4800	128	6	,	,	PUNCT
ejpam-4800	128	7	16	16	NUM
ejpam-4800	128	8	(	(	PUNCT
ejpam-4800	128	9	3	3	NUM
ejpam-4800	128	10	)	)	PUNCT
ejpam-4800	128	11	(	(	PUNCT
ejpam-4800	128	12	2023	2023	NUM
ejpam-4800	128	13	)	)	PUNCT
ejpam-4800	128	14	,	,	PUNCT
ejpam-4800	128	15	1543	1543	NUM
ejpam-4800	128	16	-	-	SYM
ejpam-4800	128	17	1551	1551	NUM
ejpam-4800	128	18	1548	1548	NUM
ejpam-4800	128	19	proof	proof	NOUN
ejpam-4800	128	20	.	.	PUNCT
ejpam-4800	129	1	it	it	PRON
ejpam-4800	129	2	has	have	AUX
ejpam-4800	129	3	been	be	AUX
ejpam-4800	129	4	proved	prove	VERB
ejpam-4800	129	5	,	,	PUNCT
ejpam-4800	129	6	in	in	ADP
ejpam-4800	129	7	[	[	PUNCT
ejpam-4800	129	8	3	3	NUM
ejpam-4800	129	9	]	]	PUNCT
ejpam-4800	129	10	,	,	PUNCT
ejpam-4800	129	11	that	that	SCONJ
ejpam-4800	129	12	l1(µ,g	l1(µ,g	CCONJ
ejpam-4800	129	13	)	)	PUNCT
ejpam-4800	129	14	is	be	AUX
ejpam-4800	129	15	coproximinal	coproximinal	ADJ
ejpam-4800	129	16	in	in	ADP
ejpam-4800	129	17	l1(µ,x	l1(µ,x	NOUN
ejpam-4800	129	18	)	)	PUNCT
ejpam-4800	130	1	if	if	SCONJ
ejpam-4800	130	2	and	and	CCONJ
ejpam-4800	130	3	only	only	ADV
ejpam-4800	130	4	if	if	SCONJ
ejpam-4800	130	5	lp(µ,g	lp(µ,g	NUM
ejpam-4800	130	6	)	)	PUNCT
ejpam-4800	130	7	is	be	AUX
ejpam-4800	130	8	coproximinal	coproximinal	ADJ
ejpam-4800	130	9	in	in	ADP
ejpam-4800	130	10	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	130	11	)	)	PUNCT
ejpam-4800	130	12	,	,	PUNCT
ejpam-4800	130	13	1	1	NUM
ejpam-4800	130	14	<	<	X
ejpam-4800	130	15	p	p	X
ejpam-4800	130	16	<	<	X
ejpam-4800	130	17	∞.	∞.	PROPN
ejpam-4800	130	18	now	now	ADV
ejpam-4800	130	19	,	,	PUNCT
ejpam-4800	130	20	let	let	VERB
ejpam-4800	130	21	l1(µ,g	l1(µ,g	X
ejpam-4800	130	22	)	)	PUNCT
ejpam-4800	130	23	be	be	AUX
ejpam-4800	130	24	strongly	strongly	ADV
ejpam-4800	130	25	coproximinal	coproximinal	ADJ
ejpam-4800	130	26	in	in	ADP
ejpam-4800	130	27	l1(µ,x	l1(µ,x	NOUN
ejpam-4800	130	28	)	)	PUNCT
ejpam-4800	130	29	and	and	CCONJ
ejpam-4800	130	30	f	f	PROPN
ejpam-4800	130	31	∈	∈	PROPN
ejpam-4800	130	32	lp(µ,x	lp(µ,x	ADJ
ejpam-4800	130	33	)	)	PUNCT
ejpam-4800	130	34	.	.	PUNCT
ejpam-4800	131	1	take	take	VERB
ejpam-4800	131	2	h	h	NOUN
ejpam-4800	131	3	∈	∈	PROPN
ejpam-4800	131	4	rlp(µ,g)(f	rlp(µ,g)(f	PROPN
ejpam-4800	131	5	,	,	PUNCT
ejpam-4800	131	6	δ	δ	PROPN
ejpam-4800	131	7	)	)	PUNCT
ejpam-4800	131	8	,	,	PUNCT
ejpam-4800	131	9	for	for	ADP
ejpam-4800	131	10	some	some	DET
ejpam-4800	131	11	δ	δ	PROPN
ejpam-4800	131	12	>	>	X
ejpam-4800	131	13	0	0	PROPN
ejpam-4800	131	14	.	.	PUNCT
ejpam-4800	132	1	since	since	SCONJ
ejpam-4800	132	2	µ(t	µ(t	ADJ
ejpam-4800	132	3	)	)	PUNCT
ejpam-4800	132	4	<	<	X
ejpam-4800	132	5	∞	∞	PROPN
ejpam-4800	132	6	,	,	PUNCT
ejpam-4800	132	7	then	then	ADV
ejpam-4800	132	8	lp(µ,x	lp(µ,x	ADJ
ejpam-4800	132	9	)	)	PUNCT
ejpam-4800	132	10	⊂	⊂	PROPN
ejpam-4800	132	11	l1(µ,x	l1(µ,x	NOUN
ejpam-4800	132	12	)	)	PUNCT
ejpam-4800	132	13	,	,	PUNCT
ejpam-4800	132	14	1	1	NUM
ejpam-4800	132	15	<	<	X
ejpam-4800	132	16	p	p	X
ejpam-4800	132	17	<	<	X
ejpam-4800	132	18	∞	∞	PROPN
ejpam-4800	132	19	and	and	CCONJ
ejpam-4800	132	20	so	so	ADV
ejpam-4800	132	21	f	f	PROPN
ejpam-4800	132	22	∈	∈	PROPN
ejpam-4800	132	23	l1(µ,x	l1(µ,x	NOUN
ejpam-4800	132	24	)	)	PUNCT
ejpam-4800	132	25	and	and	CCONJ
ejpam-4800	132	26	h	h	PROPN
ejpam-4800	132	27	∈	∈	PROPN
ejpam-4800	132	28	rl1(µ,g)(f	rl1(µ,g)(f	VERB
ejpam-4800	132	29	,	,	PUNCT
ejpam-4800	132	30	δ	δ	PROPN
ejpam-4800	132	31	′	′	NUM
ejpam-4800	132	32	)	)	PUNCT
ejpam-4800	132	33	,	,	PUNCT
ejpam-4800	132	34	for	for	ADP
ejpam-4800	132	35	some	some	DET
ejpam-4800	132	36	δ	δ	NOUN
ejpam-4800	133	1	′	′	NUM
ejpam-4800	133	2	>	>	X
ejpam-4800	133	3	0	0	X
ejpam-4800	133	4	.	.	PUNCT
ejpam-4800	134	1	but	but	CCONJ
ejpam-4800	134	2	l1(µ,g	l1(µ,g	CCONJ
ejpam-4800	134	3	)	)	PUNCT
ejpam-4800	134	4	is	be	AUX
ejpam-4800	134	5	strongly	strongly	ADV
ejpam-4800	134	6	coproxminal	coproxminal	ADJ
ejpam-4800	134	7	in	in	ADP
ejpam-4800	134	8	l1(µ,x	l1(µ,x	NOUN
ejpam-4800	134	9	)	)	PUNCT
ejpam-4800	134	10	,	,	PUNCT
ejpam-4800	134	11	which	which	PRON
ejpam-4800	134	12	implies	imply	VERB
ejpam-4800	134	13	that	that	SCONJ
ejpam-4800	134	14	for	for	ADP
ejpam-4800	134	15	any	any	DET
ejpam-4800	134	16	ε	ε	PROPN
ejpam-4800	134	17	>	>	X
ejpam-4800	134	18	0	0	PROPN
ejpam-4800	134	19	,	,	PUNCT
ejpam-4800	134	20	there	there	PRON
ejpam-4800	134	21	exists	exist	VERB
ejpam-4800	134	22	g0	g0	PROPN
ejpam-4800	134	23	∈	∈	PROPN
ejpam-4800	134	24	rl1(µ,g)(f	rl1(µ,g)(f	VERB
ejpam-4800	134	25	)	)	PUNCT
ejpam-4800	135	1	such	such	ADJ
ejpam-4800	135	2	that	that	SCONJ
ejpam-4800	135	3	||h−	||h−	PROPN
ejpam-4800	135	4	g0||	g0||	PROPN
ejpam-4800	135	5	<	<	X
ejpam-4800	135	6	ε	ε	PROPN
ejpam-4800	135	7	.	.	PUNCT
ejpam-4800	135	8	but	but	CCONJ
ejpam-4800	135	9	,	,	PUNCT
ejpam-4800	135	10	by	by	ADP
ejpam-4800	135	11	theorem	theorem	NOUN
ejpam-4800	135	12	2	2	NUM
ejpam-4800	135	13	and	and	CCONJ
ejpam-4800	135	14	lemma	lemma	PROPN
ejpam-4800	135	15	1	1	NUM
ejpam-4800	135	16	,	,	PUNCT
ejpam-4800	135	17	we	we	PRON
ejpam-4800	135	18	have	have	AUX
ejpam-4800	135	19	g	g	PROPN
ejpam-4800	135	20	is	be	AUX
ejpam-4800	135	21	strongly	strongly	ADV
ejpam-4800	135	22	coproximinal	coproximinal	ADJ
ejpam-4800	135	23	in	in	ADP
ejpam-4800	135	24	x	x	NOUN
ejpam-4800	135	25	,	,	PUNCT
ejpam-4800	135	26	and	and	CCONJ
ejpam-4800	135	27	||h(t)−	||h(t)−	PROPN
ejpam-4800	135	28	g0(t)||	g0(t)||	NOUN
ejpam-4800	135	29	<	<	X
ejpam-4800	135	30	ε/µ(t	ε/µ(t	PROPN
ejpam-4800	135	31	)	)	PUNCT
ejpam-4800	135	32	,	,	PUNCT
ejpam-4800	135	33	a.e	a.e	PROPN
ejpam-4800	135	34	.	.	PROPN
ejpam-4800	135	35	t	t	PROPN
ejpam-4800	135	36	∈	∈	PROPN
ejpam-4800	135	37	t.	t.	PROPN
ejpam-4800	135	38	(	(	PUNCT
ejpam-4800	135	39	5	5	NUM
ejpam-4800	135	40	)	)	PUNCT
ejpam-4800	135	41	on	on	ADP
ejpam-4800	135	42	the	the	DET
ejpam-4800	135	43	other	other	ADJ
ejpam-4800	135	44	hand	hand	NOUN
ejpam-4800	135	45	,	,	PUNCT
ejpam-4800	135	46	we	we	PRON
ejpam-4800	135	47	have	have	AUX
ejpam-4800	135	48	g0(t	g0(t	VERB
ejpam-4800	135	49	)	)	PUNCT
ejpam-4800	135	50	is	be	AUX
ejpam-4800	135	51	a	a	DET
ejpam-4800	135	52	best	good	ADJ
ejpam-4800	135	53	coapproximation	coapproximation	NOUN
ejpam-4800	135	54	for	for	ADP
ejpam-4800	135	55	f(t	f(t	NOUN
ejpam-4800	135	56	)	)	PUNCT
ejpam-4800	135	57	,	,	PUNCT
ejpam-4800	135	58	a.e	a.e	PROPN
ejpam-4800	135	59	.	.	PROPN
ejpam-4800	135	60	t	t	PROPN
ejpam-4800	135	61	∈	∈	PROPN
ejpam-4800	135	62	t	t	PROPN
ejpam-4800	135	63	.	.	PUNCT
ejpam-4800	136	1	so	so	ADV
ejpam-4800	136	2	,	,	PUNCT
ejpam-4800	136	3	for	for	ADP
ejpam-4800	136	4	w	w	ADP
ejpam-4800	136	5	an	an	DET
ejpam-4800	136	6	arbitrary	arbitrary	ADJ
ejpam-4800	136	7	element	element	NOUN
ejpam-4800	136	8	of	of	ADP
ejpam-4800	136	9	lp(µ,g	lp(µ,g	NUM
ejpam-4800	136	10	)	)	PUNCT
ejpam-4800	136	11	,	,	PUNCT
ejpam-4800	136	12	we	we	PRON
ejpam-4800	136	13	can	can	AUX
ejpam-4800	136	14	write	write	VERB
ejpam-4800	136	15	||w(t)−	||w(t)−	PROPN
ejpam-4800	136	16	g0(t)||	g0(t)||	NOUN
ejpam-4800	136	17	≤	≤	PROPN
ejpam-4800	136	18	||w(t)−	||w(t)−	PROPN
ejpam-4800	136	19	f(t)||	f(t)||	NOUN
ejpam-4800	136	20	,	,	PUNCT
ejpam-4800	136	21	a.e	a.e	PROPN
ejpam-4800	136	22	.	.	PROPN
ejpam-4800	136	23	t	t	PROPN
ejpam-4800	136	24	∈	∈	PROPN
ejpam-4800	136	25	t.	t.	PROPN
ejpam-4800	136	26	(	(	PUNCT
ejpam-4800	136	27	6	6	NUM
ejpam-4800	136	28	)	)	PUNCT
ejpam-4800	136	29	this	this	PRON
ejpam-4800	136	30	gives	give	VERB
ejpam-4800	136	31	,	,	PUNCT
ejpam-4800	136	32	||g0(t)||	||g0(t)||	NOUN
ejpam-4800	136	33	≤	≤	NOUN
ejpam-4800	136	34	||f(t)||	||f(t)||	NOUN
ejpam-4800	136	35	,	,	PUNCT
ejpam-4800	136	36	a.e	a.e	PROPN
ejpam-4800	136	37	.	.	PROPN
ejpam-4800	136	38	t	t	PROPN
ejpam-4800	136	39	∈	∈	PROPN
ejpam-4800	136	40	t.	t.	PROPN
ejpam-4800	136	41	therefore	therefore	ADV
ejpam-4800	136	42	,	,	PUNCT
ejpam-4800	136	43	g0	g0	PROPN
ejpam-4800	136	44	∈	∈	PROPN
ejpam-4800	136	45	lp(µ,g	lp(µ,g	PROPN
ejpam-4800	136	46	)	)	PUNCT
ejpam-4800	136	47	and	and	CCONJ
ejpam-4800	136	48	consequently	consequently	ADV
ejpam-4800	136	49	,	,	PUNCT
ejpam-4800	136	50	from	from	ADP
ejpam-4800	136	51	(	(	PUNCT
ejpam-4800	136	52	6	6	NUM
ejpam-4800	136	53	)	)	PUNCT
ejpam-4800	136	54	,	,	PUNCT
ejpam-4800	136	55	we	we	PRON
ejpam-4800	136	56	have	have	VERB
ejpam-4800	136	57	,	,	PUNCT
ejpam-4800	136	58	||w	||w	ADJ
ejpam-4800	136	59	−	−	PROPN
ejpam-4800	136	60	g0||p	g0||p	NOUN
ejpam-4800	136	61	≤	≤	NOUN
ejpam-4800	136	62	||w	||w	NOUN
ejpam-4800	136	63	−	−	PROPN
ejpam-4800	136	64	f	f	PROPN
ejpam-4800	136	65	||p	||p	PROPN
ejpam-4800	136	66	,	,	PUNCT
ejpam-4800	136	67	for	for	ADP
ejpam-4800	136	68	all	all	DET
ejpam-4800	136	69	w	w	NOUN
ejpam-4800	136	70	in	in	ADP
ejpam-4800	136	71	lp(µ,g	lp(µ,g	NUM
ejpam-4800	136	72	)	)	PUNCT
ejpam-4800	136	73	.	.	PUNCT
ejpam-4800	137	1	this	this	PRON
ejpam-4800	137	2	implies	imply	VERB
ejpam-4800	137	3	that	that	SCONJ
ejpam-4800	137	4	g0	g0	PROPN
ejpam-4800	137	5	∈	∈	PROPN
ejpam-4800	137	6	rlp(µ,g)(f	rlp(µ,g)(f	PROPN
ejpam-4800	137	7	)	)	PUNCT
ejpam-4800	137	8	.	.	PUNCT
ejpam-4800	138	1	equation	equation	NOUN
ejpam-4800	138	2	(	(	PUNCT
ejpam-4800	138	3	5	5	NUM
ejpam-4800	138	4	)	)	PUNCT
ejpam-4800	138	5	also	also	ADV
ejpam-4800	138	6	gives	give	VERB
ejpam-4800	138	7	that	that	PRON
ejpam-4800	138	8	||h−	||h−	PROPN
ejpam-4800	138	9	g0||p	g0||p	NOUN
ejpam-4800	138	10	<	<	X
ejpam-4800	138	11	ε	ε	PROPN
ejpam-4800	138	12	.	.	PUNCT
ejpam-4800	139	1	hence	hence	ADV
ejpam-4800	139	2	,	,	PUNCT
ejpam-4800	139	3	lp(µ,g	lp(µ,g	NUM
ejpam-4800	139	4	)	)	PUNCT
ejpam-4800	140	1	is	be	AUX
ejpam-4800	140	2	strongly	strongly	ADV
ejpam-4800	140	3	coproximinal	coproximinal	ADJ
ejpam-4800	140	4	in	in	ADP
ejpam-4800	140	5	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	140	6	)	)	PUNCT
ejpam-4800	140	7	,	,	PUNCT
ejpam-4800	140	8	1	1	NUM
ejpam-4800	140	9	<	<	X
ejpam-4800	140	10	p	p	X
ejpam-4800	140	11	<	<	X
ejpam-4800	140	12	∞.	∞.	PROPN
ejpam-4800	140	13	the	the	DET
ejpam-4800	140	14	following	follow	VERB
ejpam-4800	140	15	corollary	corollary	NOUN
ejpam-4800	140	16	follows	follow	VERB
ejpam-4800	140	17	directly	directly	ADV
ejpam-4800	140	18	from	from	ADP
ejpam-4800	140	19	theorems	theorem	NOUN
ejpam-4800	140	20	1	1	NUM
ejpam-4800	140	21	,	,	PUNCT
ejpam-4800	140	22	2	2	NUM
ejpam-4800	140	23	and	and	CCONJ
ejpam-4800	140	24	3	3	NUM
ejpam-4800	140	25	.	.	PUNCT
ejpam-4800	140	26	corollary	corollary	ADJ
ejpam-4800	140	27	1	1	NUM
ejpam-4800	140	28	.	.	PUNCT
ejpam-4800	141	1	for	for	ADP
ejpam-4800	141	2	g	g	PROPN
ejpam-4800	141	3	separable	separable	NOUN
ejpam-4800	141	4	in	in	ADP
ejpam-4800	141	5	x	x	NOUN
ejpam-4800	141	6	,	,	PUNCT
ejpam-4800	141	7	then	then	ADV
ejpam-4800	141	8	g	g	PROPN
ejpam-4800	141	9	is	be	AUX
ejpam-4800	141	10	strongly	strongly	ADV
ejpam-4800	141	11	coproximinal	coproximinal	ADJ
ejpam-4800	141	12	in	in	ADP
ejpam-4800	141	13	x	x	SYM
ejpam-4800	141	14	if	if	SCONJ
ejpam-4800	142	1	and	and	CCONJ
ejpam-4800	142	2	only	only	ADV
ejpam-4800	142	3	if	if	SCONJ
ejpam-4800	142	4	lp(µ,g	lp(µ,g	NUM
ejpam-4800	142	5	)	)	PUNCT
ejpam-4800	142	6	is	be	AUX
ejpam-4800	142	7	strongly	strongly	ADV
ejpam-4800	142	8	coproximinal	coproximinal	ADJ
ejpam-4800	142	9	in	in	ADP
ejpam-4800	142	10	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	142	11	)	)	PUNCT
ejpam-4800	142	12	,	,	PUNCT
ejpam-4800	143	1	1	1	NUM
ejpam-4800	143	2	≤	≤	NOUN
ejpam-4800	143	3	p	p	NOUN
ejpam-4800	143	4	<	<	X
ejpam-4800	143	5	∞.	∞.	PROPN
ejpam-4800	143	6	3	3	NUM
ejpam-4800	143	7	.	.	PUNCT
ejpam-4800	143	8	strong	strong	ADJ
ejpam-4800	143	9	coproximinality	coproximinality	NOUN
ejpam-4800	143	10	of	of	ADP
ejpam-4800	143	11	e(g	e(g	PROPN
ejpam-4800	143	12	)	)	PUNCT
ejpam-4800	143	13	in	in	ADP
ejpam-4800	143	14	e(x	e(x	NUM
ejpam-4800	143	15	)	)	PUNCT
ejpam-4800	143	16	in	in	ADP
ejpam-4800	143	17	this	this	DET
ejpam-4800	143	18	section	section	NOUN
ejpam-4800	143	19	,	,	PUNCT
ejpam-4800	143	20	let	let	VERB
ejpam-4800	143	21	(	(	PUNCT
ejpam-4800	143	22	x	x	NOUN
ejpam-4800	143	23	,	,	PUNCT
ejpam-4800	143	24	||.||x	||.||x	PROPN
ejpam-4800	143	25	)	)	PUNCT
ejpam-4800	143	26	be	be	AUX
ejpam-4800	143	27	a	a	DET
ejpam-4800	143	28	real	real	ADJ
ejpam-4800	143	29	banach	banach	NOUN
ejpam-4800	143	30	space	space	NOUN
ejpam-4800	143	31	and	and	CCONJ
ejpam-4800	143	32	e	e	NOUN
ejpam-4800	143	33	a	a	DET
ejpam-4800	143	34	real	real	ADJ
ejpam-4800	143	35	köthe	köthe	PRON
ejpam-4800	143	36	space	space	NOUN
ejpam-4800	143	37	.	.	PUNCT
ejpam-4800	144	1	consider	consider	VERB
ejpam-4800	144	2	e(x	e(x	NUM
ejpam-4800	144	3	)	)	PUNCT
ejpam-4800	144	4	as	as	SCONJ
ejpam-4800	144	5	defined	define	VERB
ejpam-4800	144	6	in	in	ADP
ejpam-4800	144	7	the	the	DET
ejpam-4800	144	8	introduction	introduction	NOUN
ejpam-4800	144	9	section	section	NOUN
ejpam-4800	144	10	with	with	ADP
ejpam-4800	144	11	the	the	DET
ejpam-4800	144	12	following	follow	VERB
ejpam-4800	144	13	norm	norm	NOUN
ejpam-4800	144	14	,	,	PUNCT
ejpam-4800	144	15	|||f	|||f	NOUN
ejpam-4800	144	16	|||	|||	NOUN
ejpam-4800	145	1	=	=	SYM
ejpam-4800	145	2	||	||	PROPN
ejpam-4800	145	3	||f(·)||x	||f(·)||x	VERB
ejpam-4800	145	4	||e	||e	PROPN
ejpam-4800	145	5	.	.	PUNCT
ejpam-4800	146	1	then	then	ADV
ejpam-4800	146	2	(	(	PUNCT
ejpam-4800	146	3	e(x	e(x	NUM
ejpam-4800	146	4	)	)	PUNCT
ejpam-4800	146	5	,	,	PUNCT
ejpam-4800	146	6	|||.|||	|||.|||	NUM
ejpam-4800	146	7	)	)	PUNCT
ejpam-4800	146	8	is	be	AUX
ejpam-4800	146	9	a	a	DET
ejpam-4800	146	10	banach	banach	NOUN
ejpam-4800	146	11	space	space	NOUN
ejpam-4800	146	12	called	call	VERB
ejpam-4800	146	13	the	the	DET
ejpam-4800	146	14	köthe	köthe	PROPN
ejpam-4800	146	15	bochner	bochner	NOUN
ejpam-4800	146	16	function	function	NOUN
ejpam-4800	146	17	space	space	NOUN
ejpam-4800	146	18	.	.	PUNCT
ejpam-4800	147	1	for	for	ADP
ejpam-4800	147	2	more	more	ADJ
ejpam-4800	147	3	on	on	ADP
ejpam-4800	147	4	köthe	köthe	PRON
ejpam-4800	147	5	bochner	bochner	NOUN
ejpam-4800	147	6	function	function	NOUN
ejpam-4800	147	7	spaces	space	NOUN
ejpam-4800	147	8	,	,	PUNCT
ejpam-4800	147	9	see	see	VERB
ejpam-4800	147	10	[	[	X
ejpam-4800	147	11	11	11	NUM
ejpam-4800	147	12	]	]	PUNCT
ejpam-4800	147	13	.	.	PUNCT
ejpam-4800	148	1	the	the	DET
ejpam-4800	148	2	second	second	ADJ
ejpam-4800	148	3	goal	goal	NOUN
ejpam-4800	148	4	of	of	ADP
ejpam-4800	148	5	this	this	DET
ejpam-4800	148	6	paper	paper	NOUN
ejpam-4800	148	7	is	be	AUX
ejpam-4800	148	8	to	to	PART
ejpam-4800	148	9	extend	extend	VERB
ejpam-4800	148	10	the	the	DET
ejpam-4800	148	11	main	main	ADJ
ejpam-4800	148	12	theorem	theorem	NOUN
ejpam-4800	148	13	in	in	ADP
ejpam-4800	148	14	the	the	DET
ejpam-4800	148	15	previous	previous	ADJ
ejpam-4800	148	16	section	section	NOUN
ejpam-4800	148	17	to	to	ADP
ejpam-4800	148	18	the	the	DET
ejpam-4800	148	19	köthe	köthe	PROPN
ejpam-4800	148	20	bochner	bochner	NOUN
ejpam-4800	148	21	function	function	NOUN
ejpam-4800	148	22	spaces	space	NOUN
ejpam-4800	148	23	,	,	PUNCT
ejpam-4800	148	24	as	as	ADP
ejpam-4800	148	25	in	in	ADP
ejpam-4800	148	26	the	the	DET
ejpam-4800	148	27	following	follow	VERB
ejpam-4800	148	28	theorem	theorem	ADJ
ejpam-4800	148	29	.	.	PUNCT
ejpam-4800	148	30	main	main	ADJ
ejpam-4800	148	31	theorem	theorem	NOUN
ejpam-4800	148	32	(	(	PUNCT
ejpam-4800	148	33	theorem	theorem	NOUN
ejpam-4800	148	34	5	5	NUM
ejpam-4800	148	35	)	)	PUNCT
ejpam-4800	148	36	.	.	PUNCT
ejpam-4800	149	1	let	let	VERB
ejpam-4800	149	2	g	g	PRON
ejpam-4800	149	3	be	be	AUX
ejpam-4800	149	4	a	a	DET
ejpam-4800	149	5	separable	separable	ADJ
ejpam-4800	149	6	subspace	subspace	NOUN
ejpam-4800	149	7	of	of	ADP
ejpam-4800	149	8	x	x	SYM
ejpam-4800	149	9	such	such	ADJ
ejpam-4800	149	10	that	that	SCONJ
ejpam-4800	149	11	e	e	NOUN
ejpam-4800	149	12	is	be	AUX
ejpam-4800	149	13	strictly	strictly	ADV
ejpam-4800	149	14	monotone	monotone	ADJ
ejpam-4800	149	15	.	.	PUNCT
ejpam-4800	150	1	then	then	ADV
ejpam-4800	150	2	e(g	e(g	PROPN
ejpam-4800	150	3	)	)	PUNCT
ejpam-4800	150	4	is	be	AUX
ejpam-4800	150	5	strongly	strongly	ADV
ejpam-4800	150	6	coproximinal	coproximinal	ADJ
ejpam-4800	150	7	in	in	ADP
ejpam-4800	150	8	e(x	e(x	NUM
ejpam-4800	150	9	)	)	PUNCT
ejpam-4800	150	10	if	if	SCONJ
ejpam-4800	150	11	and	and	CCONJ
ejpam-4800	150	12	only	only	ADV
ejpam-4800	150	13	if	if	SCONJ
ejpam-4800	150	14	g	g	PROPN
ejpam-4800	150	15	is	be	AUX
ejpam-4800	150	16	strongly	strongly	ADV
ejpam-4800	150	17	coproximinal	coproximinal	ADJ
ejpam-4800	150	18	in	in	ADP
ejpam-4800	150	19	x.	x.	NOUN
ejpam-4800	150	20	to	to	PART
ejpam-4800	150	21	prove	prove	VERB
ejpam-4800	150	22	our	our	PRON
ejpam-4800	150	23	main	main	ADJ
ejpam-4800	150	24	theorem	theorem	NOUN
ejpam-4800	150	25	,	,	PUNCT
ejpam-4800	150	26	we	we	PRON
ejpam-4800	150	27	need	need	VERB
ejpam-4800	150	28	the	the	DET
ejpam-4800	150	29	following	follow	VERB
ejpam-4800	150	30	two	two	NUM
ejpam-4800	150	31	results	result	NOUN
ejpam-4800	150	32	.	.	PUNCT
ejpam-4800	151	1	j.	j.	PROPN
ejpam-4800	151	2	jawdat	jawdat	PROPN
ejpam-4800	151	3	/	/	SYM
ejpam-4800	151	4	eur	eur	PROPN
ejpam-4800	151	5	.	.	PUNCT
ejpam-4800	152	1	j.	j.	PROPN
ejpam-4800	152	2	pure	pure	PROPN
ejpam-4800	152	3	appl	appl	PROPN
ejpam-4800	152	4	.	.	PROPN
ejpam-4800	152	5	math	math	PROPN
ejpam-4800	152	6	,	,	PUNCT
ejpam-4800	152	7	16	16	NUM
ejpam-4800	152	8	(	(	PUNCT
ejpam-4800	152	9	3	3	NUM
ejpam-4800	152	10	)	)	PUNCT
ejpam-4800	152	11	(	(	PUNCT
ejpam-4800	152	12	2023	2023	NUM
ejpam-4800	152	13	)	)	PUNCT
ejpam-4800	152	14	,	,	PUNCT
ejpam-4800	152	15	1543	1543	NUM
ejpam-4800	152	16	-	-	SYM
ejpam-4800	152	17	1551	1551	NUM
ejpam-4800	152	18	1549	1549	NUM
ejpam-4800	152	19	theorem	theorem	VERB
ejpam-4800	152	20	4	4	NUM
ejpam-4800	152	21	.	.	PUNCT
ejpam-4800	153	1	let	let	VERB
ejpam-4800	153	2	g	g	NOUN
ejpam-4800	153	3	be	be	AUX
ejpam-4800	153	4	coproximinal	coproximinal	ADJ
ejpam-4800	153	5	in	in	ADP
ejpam-4800	153	6	x	x	PUNCT
ejpam-4800	153	7	and	and	CCONJ
ejpam-4800	153	8	e	e	NOUN
ejpam-4800	153	9	is	be	AUX
ejpam-4800	153	10	strictly	strictly	ADV
ejpam-4800	153	11	monotone	monotone	ADJ
ejpam-4800	153	12	köthe	köthe	ADJ
ejpam-4800	153	13	space	space	NOUN
ejpam-4800	153	14	.	.	PUNCT
ejpam-4800	154	1	for	for	ADP
ejpam-4800	154	2	f	f	PROPN
ejpam-4800	154	3	in	in	ADP
ejpam-4800	154	4	e(x	e(x	NUM
ejpam-4800	154	5	)	)	PUNCT
ejpam-4800	154	6	and	and	CCONJ
ejpam-4800	154	7	g	g	NOUN
ejpam-4800	154	8	in	in	ADP
ejpam-4800	154	9	e(g	e(g	PROPN
ejpam-4800	154	10	)	)	PUNCT
ejpam-4800	154	11	such	such	ADJ
ejpam-4800	154	12	that	that	PRON
ejpam-4800	154	13	for	for	ADP
ejpam-4800	154	14	each	each	DET
ejpam-4800	154	15	t	t	PROPN
ejpam-4800	154	16	,	,	PUNCT
ejpam-4800	154	17	g(t	g(t	PROPN
ejpam-4800	154	18	)	)	PUNCT
ejpam-4800	154	19	is	be	AUX
ejpam-4800	154	20	a	a	DET
ejpam-4800	154	21	near	near	ADV
ejpam-4800	154	22	best	good	ADJ
ejpam-4800	154	23	coapproximation	coapproximation	NOUN
ejpam-4800	154	24	point	point	NOUN
ejpam-4800	154	25	in	in	ADP
ejpam-4800	154	26	g	g	NOUN
ejpam-4800	154	27	to	to	ADP
ejpam-4800	154	28	f(t	f(t	NOUN
ejpam-4800	154	29	)	)	PUNCT
ejpam-4800	154	30	in	in	ADP
ejpam-4800	154	31	x	x	PROPN
ejpam-4800	154	32	,	,	PUNCT
ejpam-4800	154	33	a.e	a.e	PROPN
ejpam-4800	154	34	.	.	PROPN
ejpam-4800	154	35	t	t	PROPN
ejpam-4800	154	36	∈	∈	PROPN
ejpam-4800	154	37	t	t	PROPN
ejpam-4800	154	38	,	,	PUNCT
ejpam-4800	154	39	then	then	ADV
ejpam-4800	154	40	g	g	PROPN
ejpam-4800	154	41	is	be	AUX
ejpam-4800	154	42	a	a	DET
ejpam-4800	154	43	near	near	ADV
ejpam-4800	154	44	best	good	ADJ
ejpam-4800	154	45	coapproximation	coapproximation	NOUN
ejpam-4800	154	46	to	to	ADP
ejpam-4800	154	47	f	f	PROPN
ejpam-4800	154	48	.	.	PUNCT
ejpam-4800	155	1	proof	proof	NOUN
ejpam-4800	155	2	.	.	PUNCT
ejpam-4800	156	1	given	give	VERB
ejpam-4800	156	2	f	f	PROPN
ejpam-4800	156	3	and	and	CCONJ
ejpam-4800	156	4	g	g	PROPN
ejpam-4800	156	5	as	as	ADP
ejpam-4800	156	6	above	above	ADV
ejpam-4800	156	7	.	.	PUNCT
ejpam-4800	157	1	let	let	VERB
ejpam-4800	157	2	g(t	g(t	PROPN
ejpam-4800	157	3	)	)	PUNCT
ejpam-4800	157	4	be	be	AUX
ejpam-4800	157	5	a	a	DET
ejpam-4800	157	6	near	near	ADV
ejpam-4800	157	7	best	good	ADJ
ejpam-4800	157	8	coapproximation	coapproximation	NOUN
ejpam-4800	157	9	point	point	NOUN
ejpam-4800	157	10	in	in	ADP
ejpam-4800	157	11	g	g	NOUN
ejpam-4800	157	12	to	to	ADP
ejpam-4800	157	13	f(t	f(t	NOUN
ejpam-4800	157	14	)	)	PUNCT
ejpam-4800	157	15	in	in	ADP
ejpam-4800	157	16	x.	x.	NOUN
ejpam-4800	157	17	then	then	ADV
ejpam-4800	157	18	from	from	ADP
ejpam-4800	157	19	(	(	PUNCT
ejpam-4800	157	20	3	3	NUM
ejpam-4800	157	21	)	)	PUNCT
ejpam-4800	157	22	,	,	PUNCT
ejpam-4800	157	23	||g(t)−	||g(t)−	PROPN
ejpam-4800	157	24	y||	y||	PROPN
ejpam-4800	157	25	<	<	X
ejpam-4800	157	26	||f(t)−	||f(t)−	PROPN
ejpam-4800	157	27	y||+	y||+	PROPN
ejpam-4800	157	28	δ	δ	PROPN
ejpam-4800	157	29	,	,	PUNCT
ejpam-4800	157	30	for	for	ADP
ejpam-4800	157	31	some	some	DET
ejpam-4800	157	32	δ	δ	PROPN
ejpam-4800	157	33	>	>	X
ejpam-4800	157	34	0	0	PUNCT
ejpam-4800	157	35	and	and	CCONJ
ejpam-4800	157	36	for	for	ADP
ejpam-4800	157	37	all	all	DET
ejpam-4800	157	38	y	y	PROPN
ejpam-4800	157	39	∈	∈	PROPN
ejpam-4800	157	40	g.	g.	NOUN
ejpam-4800	158	1	so	so	ADV
ejpam-4800	158	2	,	,	PUNCT
ejpam-4800	158	3	if	if	SCONJ
ejpam-4800	158	4	for	for	ADP
ejpam-4800	158	5	any	any	DET
ejpam-4800	158	6	function	function	NOUN
ejpam-4800	158	7	h	h	NOUN
ejpam-4800	158	8	in	in	ADP
ejpam-4800	158	9	e(g	e(g	PROPN
ejpam-4800	158	10	)	)	PUNCT
ejpam-4800	158	11	,	,	PUNCT
ejpam-4800	158	12	we	we	PRON
ejpam-4800	158	13	have	have	VERB
ejpam-4800	158	14	||g(t)−	||g(t)−	PROPN
ejpam-4800	158	15	h(t)||	h(t)||	NOUN
ejpam-4800	158	16	<	<	X
ejpam-4800	158	17	||f(t)−	||f(t)−	PROPN
ejpam-4800	158	18	h(t)||+	h(t)||+	PROPN
ejpam-4800	158	19	δ	δ	PROPN
ejpam-4800	158	20	,	,	PUNCT
ejpam-4800	158	21	for	for	ADP
ejpam-4800	158	22	some	some	DET
ejpam-4800	158	23	δ	δ	PROPN
ejpam-4800	158	24	>	>	X
ejpam-4800	158	25	0	0	PROPN
ejpam-4800	158	26	.	.	PUNCT
ejpam-4800	159	1	this	this	PRON
ejpam-4800	159	2	implies	imply	VERB
ejpam-4800	159	3	,	,	PUNCT
ejpam-4800	159	4	from	from	ADP
ejpam-4800	159	5	the	the	DET
ejpam-4800	159	6	strict	strict	ADJ
ejpam-4800	159	7	monotonicity	monotonicity	NOUN
ejpam-4800	159	8	of	of	ADP
ejpam-4800	159	9	e	e	NOUN
ejpam-4800	159	10	,	,	PUNCT
ejpam-4800	159	11	that	that	PRON
ejpam-4800	159	12	|||g	|||g	VERB
ejpam-4800	159	13	−	−	PROPN
ejpam-4800	159	14	h|||	h|||	NOUN
ejpam-4800	159	15	<	<	X
ejpam-4800	159	16	|||f	|||f	PROPN
ejpam-4800	159	17	−	−	PROPN
ejpam-4800	159	18	h|||+	h|||+	PROPN
ejpam-4800	159	19	δ	δ	PROPN
ejpam-4800	159	20	µ(t	µ(t	PROPN
ejpam-4800	159	21	)	)	PUNCT
ejpam-4800	159	22	,	,	PUNCT
ejpam-4800	159	23	∀h	∀h	PROPN
ejpam-4800	159	24	∈	∈	PROPN
ejpam-4800	159	25	e(g	e(g	PROPN
ejpam-4800	159	26	)	)	PUNCT
ejpam-4800	159	27	.	.	PUNCT
ejpam-4800	160	1	finally	finally	ADV
ejpam-4800	160	2	,	,	PUNCT
ejpam-4800	160	3	since	since	SCONJ
ejpam-4800	160	4	the	the	DET
ejpam-4800	160	5	measure	measure	NOUN
ejpam-4800	160	6	space	space	NOUN
ejpam-4800	160	7	is	be	AUX
ejpam-4800	160	8	finite	finite	ADJ
ejpam-4800	160	9	then	then	ADV
ejpam-4800	160	10	g	g	PROPN
ejpam-4800	160	11	is	be	AUX
ejpam-4800	160	12	a	a	DET
ejpam-4800	160	13	near	near	ADV
ejpam-4800	160	14	best	good	ADJ
ejpam-4800	160	15	coapproximation	coapproximation	NOUN
ejpam-4800	160	16	to	to	ADP
ejpam-4800	160	17	f	f	PROPN
ejpam-4800	160	18	.	.	PUNCT
ejpam-4800	161	1	a	a	DET
ejpam-4800	161	2	simple	simple	ADJ
ejpam-4800	161	3	function	function	NOUN
ejpam-4800	161	4	in	in	ADP
ejpam-4800	161	5	e(x	e(x	NUM
ejpam-4800	161	6	)	)	PUNCT
ejpam-4800	161	7	is	be	AUX
ejpam-4800	161	8	a	a	DET
ejpam-4800	161	9	function	function	NOUN
ejpam-4800	161	10	f	f	NOUN
ejpam-4800	161	11	:	:	PUNCT
ejpam-4800	161	12	t	t	PROPN
ejpam-4800	161	13	→	→	SYM
ejpam-4800	161	14	x	x	X
ejpam-4800	161	15	of	of	ADP
ejpam-4800	161	16	the	the	DET
ejpam-4800	161	17	form	form	NOUN
ejpam-4800	162	1	f	f	NOUN
ejpam-4800	162	2	=	=	SYM
ejpam-4800	162	3	∑n	∑n	NOUN
ejpam-4800	162	4	k=1	k=1	PROPN
ejpam-4800	162	5	akχak	akχak	VERB
ejpam-4800	162	6	,	,	PUNCT
ejpam-4800	162	7	where	where	SCONJ
ejpam-4800	162	8	ak	ak	PROPN
ejpam-4800	162	9	’s	’s	PART
ejpam-4800	162	10	are	be	AUX
ejpam-4800	162	11	in	in	ADP
ejpam-4800	162	12	x	x	X
ejpam-4800	162	13	(	(	PUNCT
ejpam-4800	162	14	may	may	AUX
ejpam-4800	162	15	or	or	CCONJ
ejpam-4800	162	16	may	may	AUX
ejpam-4800	162	17	not	not	PART
ejpam-4800	162	18	be	be	AUX
ejpam-4800	162	19	distinct	distinct	ADJ
ejpam-4800	162	20	)	)	PUNCT
ejpam-4800	162	21	and	and	CCONJ
ejpam-4800	162	22	{	{	PUNCT
ejpam-4800	162	23	a1	a1	NOUN
ejpam-4800	162	24	,	,	PUNCT
ejpam-4800	162	25	.	.	PUNCT
ejpam-4800	162	26	.	.	PUNCT
ejpam-4800	163	1	.	.	PUNCT
ejpam-4800	164	1	,	,	PUNCT
ejpam-4800	164	2	an	an	PRON
ejpam-4800	164	3	}	}	PUNCT
ejpam-4800	164	4	is	be	AUX
ejpam-4800	164	5	a	a	DET
ejpam-4800	164	6	finite	finite	ADJ
ejpam-4800	164	7	collection	collection	NOUN
ejpam-4800	164	8	of	of	ADP
ejpam-4800	164	9	mutually	mutually	ADV
ejpam-4800	164	10	disjoint	disjoint	VERB
ejpam-4800	164	11	measurable	measurable	ADJ
ejpam-4800	164	12	subsets	subset	NOUN
ejpam-4800	164	13	of	of	ADP
ejpam-4800	164	14	t	t	NOUN
ejpam-4800	164	15	such	such	ADJ
ejpam-4800	164	16	that	that	DET
ejpam-4800	164	17	∪ak	∪ak	PROPN
ejpam-4800	164	18	=	=	SYM
ejpam-4800	164	19	t	t	PROPN
ejpam-4800	164	20	.	.	PUNCT
ejpam-4800	165	1	the	the	DET
ejpam-4800	165	2	following	follow	VERB
ejpam-4800	165	3	lemma	lemma	PROPN
ejpam-4800	165	4	follows	follow	VERB
ejpam-4800	165	5	directly	directly	ADV
ejpam-4800	165	6	from	from	ADP
ejpam-4800	165	7	theorem	theorem	ADJ
ejpam-4800	165	8	4	4	NUM
ejpam-4800	165	9	above	above	ADV
ejpam-4800	165	10	and	and	CCONJ
ejpam-4800	165	11	lemma	lemma	PROPN
ejpam-4800	165	12	3	3	NUM
ejpam-4800	165	13	in	in	ADP
ejpam-4800	165	14	[	[	X
ejpam-4800	165	15	6	6	NUM
ejpam-4800	165	16	]	]	PUNCT
ejpam-4800	165	17	.	.	PUNCT
ejpam-4800	166	1	lemma	lemma	PROPN
ejpam-4800	166	2	2	2	X
ejpam-4800	166	3	.	.	PUNCT
ejpam-4800	167	1	let	let	VERB
ejpam-4800	167	2	g	g	NOUN
ejpam-4800	167	3	be	be	AUX
ejpam-4800	167	4	strongly	strongly	ADV
ejpam-4800	167	5	coproximinal	coproximinal	ADJ
ejpam-4800	167	6	in	in	ADP
ejpam-4800	167	7	x.	x.	PROPN
ejpam-4800	167	8	then	then	ADV
ejpam-4800	167	9	e(g	e(g	PROPN
ejpam-4800	167	10	)	)	PUNCT
ejpam-4800	167	11	is	be	AUX
ejpam-4800	167	12	strongly	strongly	ADV
ejpam-4800	167	13	coproximinal	coproximinal	ADJ
ejpam-4800	167	14	at	at	ADP
ejpam-4800	167	15	any	any	DET
ejpam-4800	167	16	simple	simple	ADJ
ejpam-4800	167	17	function	function	NOUN
ejpam-4800	167	18	in	in	ADP
ejpam-4800	167	19	e(x	e(x	NUM
ejpam-4800	167	20	)	)	PUNCT
ejpam-4800	167	21	.	.	PUNCT
ejpam-4800	168	1	the	the	DET
ejpam-4800	168	2	following	follow	VERB
ejpam-4800	168	3	theorem	theorem	NOUN
ejpam-4800	168	4	is	be	AUX
ejpam-4800	168	5	another	another	DET
ejpam-4800	168	6	main	main	ADJ
ejpam-4800	168	7	result	result	NOUN
ejpam-4800	168	8	in	in	ADP
ejpam-4800	168	9	this	this	DET
ejpam-4800	168	10	paper	paper	NOUN
ejpam-4800	168	11	.	.	PUNCT
ejpam-4800	169	1	theorem	theorem	NOUN
ejpam-4800	169	2	5	5	NUM
ejpam-4800	169	3	.	.	PUNCT
ejpam-4800	170	1	let	let	VERB
ejpam-4800	170	2	g	g	PRON
ejpam-4800	170	3	be	be	AUX
ejpam-4800	170	4	a	a	DET
ejpam-4800	170	5	separable	separable	ADJ
ejpam-4800	170	6	subspace	subspace	NOUN
ejpam-4800	170	7	of	of	ADP
ejpam-4800	170	8	x	x	PUNCT
ejpam-4800	170	9	and	and	CCONJ
ejpam-4800	170	10	let	let	VERB
ejpam-4800	170	11	e	e	PRON
ejpam-4800	170	12	be	be	AUX
ejpam-4800	170	13	a	a	DET
ejpam-4800	170	14	strictly	strictly	ADV
ejpam-4800	170	15	monotone	monotone	ADJ
ejpam-4800	170	16	köthe	köthe	ADJ
ejpam-4800	170	17	space	space	NOUN
ejpam-4800	170	18	.	.	PUNCT
ejpam-4800	171	1	then	then	ADV
ejpam-4800	171	2	e(g	e(g	PROPN
ejpam-4800	171	3	)	)	PUNCT
ejpam-4800	171	4	is	be	AUX
ejpam-4800	171	5	strongly	strongly	ADV
ejpam-4800	171	6	coproximinal	coproximinal	ADJ
ejpam-4800	171	7	in	in	ADP
ejpam-4800	171	8	e(x	e(x	NUM
ejpam-4800	171	9	)	)	PUNCT
ejpam-4800	171	10	if	if	SCONJ
ejpam-4800	171	11	and	and	CCONJ
ejpam-4800	171	12	only	only	ADV
ejpam-4800	171	13	if	if	SCONJ
ejpam-4800	171	14	g	g	PROPN
ejpam-4800	171	15	is	be	AUX
ejpam-4800	171	16	strongly	strongly	ADV
ejpam-4800	171	17	coproximinal	coproximinal	ADJ
ejpam-4800	171	18	in	in	ADP
ejpam-4800	171	19	x.	x.	NOUN
ejpam-4800	171	20	proof	proof	NOUN
ejpam-4800	171	21	.	.	PUNCT
ejpam-4800	172	1	⇒	⇒	NOUN
ejpam-4800	172	2	)	)	PUNCT
ejpam-4800	172	3	let	let	VERB
ejpam-4800	172	4	x0	x0	PROPN
ejpam-4800	172	5	in	in	ADP
ejpam-4800	172	6	x.	x.	NOUN
ejpam-4800	172	7	by	by	ADP
ejpam-4800	172	8	taking	take	VERB
ejpam-4800	172	9	f	f	PROPN
ejpam-4800	172	10	=	=	SYM
ejpam-4800	172	11	x0χt	x0χt	PUNCT
ejpam-4800	172	12	,	,	PUNCT
ejpam-4800	172	13	then	then	ADV
ejpam-4800	172	14	clearly	clearly	ADV
ejpam-4800	172	15	f	f	PROPN
ejpam-4800	172	16	is	be	AUX
ejpam-4800	172	17	a	a	DET
ejpam-4800	172	18	simple	simple	ADJ
ejpam-4800	172	19	function	function	NOUN
ejpam-4800	172	20	in	in	ADP
ejpam-4800	172	21	e(x	e(x	NUM
ejpam-4800	172	22	)	)	PUNCT
ejpam-4800	172	23	,	,	PUNCT
ejpam-4800	172	24	since	since	SCONJ
ejpam-4800	172	25	it	it	PRON
ejpam-4800	172	26	can	can	AUX
ejpam-4800	172	27	be	be	AUX
ejpam-4800	172	28	represented	represent	VERB
ejpam-4800	172	29	as	as	ADP
ejpam-4800	172	30	f	f	PROPN
ejpam-4800	172	31	=	=	SYM
ejpam-4800	172	32	∑n	∑n	PROPN
ejpam-4800	172	33	k=1	k=1	PROPN
ejpam-4800	172	34	akχak	akχak	VERB
ejpam-4800	172	35	,	,	PUNCT
ejpam-4800	172	36	where	where	SCONJ
ejpam-4800	172	37	ak	ak	PROPN
ejpam-4800	172	38	=	=	PROPN
ejpam-4800	172	39	x0	x0	PROPN
ejpam-4800	172	40	,	,	PUNCT
ejpam-4800	172	41	for	for	ADP
ejpam-4800	172	42	each	each	DET
ejpam-4800	172	43	k.	k.	NOUN
ejpam-4800	172	44	the	the	DET
ejpam-4800	172	45	sequence	sequence	NOUN
ejpam-4800	172	46	{	{	PUNCT
ejpam-4800	172	47	a1	a1	NOUN
ejpam-4800	172	48	,	,	PUNCT
ejpam-4800	172	49	.	.	PUNCT
ejpam-4800	172	50	.	.	PUNCT
ejpam-4800	173	1	.	.	PUNCT
ejpam-4800	174	1	,	,	PUNCT
ejpam-4800	174	2	an	an	PRON
ejpam-4800	174	3	}	}	PUNCT
ejpam-4800	174	4	consists	consist	VERB
ejpam-4800	174	5	of	of	ADP
ejpam-4800	174	6	mutually	mutually	ADV
ejpam-4800	174	7	disjoint	disjoint	ADJ
ejpam-4800	174	8	measurable	measurable	ADJ
ejpam-4800	174	9	subsets	subset	NOUN
ejpam-4800	174	10	of	of	ADP
ejpam-4800	174	11	t	t	NOUN
ejpam-4800	174	12	such	such	ADJ
ejpam-4800	174	13	that	that	DET
ejpam-4800	174	14	∪ak	∪ak	PROPN
ejpam-4800	174	15	=	=	SYM
ejpam-4800	174	16	t	t	PROPN
ejpam-4800	174	17	.	.	PUNCT
ejpam-4800	175	1	now	now	ADV
ejpam-4800	175	2	,	,	PUNCT
ejpam-4800	175	3	since	since	SCONJ
ejpam-4800	175	4	e(g	e(g	PROPN
ejpam-4800	175	5	)	)	PUNCT
ejpam-4800	175	6	is	be	AUX
ejpam-4800	175	7	strongly	strongly	ADV
ejpam-4800	175	8	coproximinal	coproximinal	ADJ
ejpam-4800	175	9	in	in	ADP
ejpam-4800	175	10	e(x	e(x	NUM
ejpam-4800	175	11	)	)	PUNCT
ejpam-4800	175	12	,	,	PUNCT
ejpam-4800	175	13	then	then	ADV
ejpam-4800	175	14	it	it	PRON
ejpam-4800	175	15	is	be	AUX
ejpam-4800	175	16	coproximinal	coproximinal	ADJ
ejpam-4800	175	17	in	in	ADP
ejpam-4800	175	18	e(x	e(x	NUM
ejpam-4800	175	19	)	)	PUNCT
ejpam-4800	175	20	and	and	CCONJ
ejpam-4800	175	21	hence	hence	ADV
ejpam-4800	175	22	g	g	PROPN
ejpam-4800	175	23	is	be	AUX
ejpam-4800	175	24	coproximinal	coproximinal	ADJ
ejpam-4800	175	25	in	in	ADP
ejpam-4800	175	26	e	e	NOUN
ejpam-4800	175	27	,	,	PUNCT
ejpam-4800	175	28	see	see	VERB
ejpam-4800	175	29	[	[	X
ejpam-4800	175	30	6	6	NUM
ejpam-4800	175	31	]	]	PUNCT
ejpam-4800	175	32	.	.	PUNCT
ejpam-4800	176	1	also	also	ADV
ejpam-4800	176	2	,	,	PUNCT
ejpam-4800	176	3	e(g	e(g	PROPN
ejpam-4800	176	4	)	)	PUNCT
ejpam-4800	176	5	is	be	AUX
ejpam-4800	176	6	strongly	strongly	ADV
ejpam-4800	176	7	coproximinal	coproximinal	ADJ
ejpam-4800	176	8	at	at	ADP
ejpam-4800	176	9	f	f	PROPN
ejpam-4800	176	10	above	above	ADV
ejpam-4800	176	11	.	.	PUNCT
ejpam-4800	177	1	hence	hence	ADV
ejpam-4800	177	2	,	,	PUNCT
ejpam-4800	177	3	there	there	PRON
ejpam-4800	177	4	exist	exist	VERB
ejpam-4800	177	5	g0	g0	ADJ
ejpam-4800	177	6	∈	∈	PROPN
ejpam-4800	177	7	re(g)(f	re(g)(f	NOUN
ejpam-4800	177	8	)	)	PUNCT
ejpam-4800	177	9	and	and	CCONJ
ejpam-4800	177	10	h	h	NOUN
ejpam-4800	177	11	∈	∈	PROPN
ejpam-4800	177	12	re(g)(f	re(g)(f	VERB
ejpam-4800	177	13	,	,	PUNCT
ejpam-4800	177	14	δ	δ	PROPN
ejpam-4800	177	15	)	)	PUNCT
ejpam-4800	178	1	such	such	ADJ
ejpam-4800	178	2	that	that	SCONJ
ejpam-4800	178	3	|||g0	|||g0	PROPN
ejpam-4800	178	4	−	−	PROPN
ejpam-4800	178	5	h|||	h|||	VERB
ejpam-4800	178	6	<	<	X
ejpam-4800	178	7	ε	ε	PROPN
ejpam-4800	178	8	.	.	PROPN
ejpam-4800	179	1	since	since	SCONJ
ejpam-4800	179	2	g0	g0	ADJ
ejpam-4800	179	3	and	and	CCONJ
ejpam-4800	179	4	h	h	NOUN
ejpam-4800	179	5	can	can	AUX
ejpam-4800	179	6	be	be	AUX
ejpam-4800	179	7	taken	take	VERB
ejpam-4800	179	8	to	to	PART
ejpam-4800	179	9	be	be	AUX
ejpam-4800	179	10	simple	simple	ADJ
ejpam-4800	179	11	functions	function	NOUN
ejpam-4800	179	12	,	,	PUNCT
ejpam-4800	179	13	so	so	SCONJ
ejpam-4800	179	14	for	for	ADP
ejpam-4800	179	15	some	some	DET
ejpam-4800	179	16	y	y	PROPN
ejpam-4800	179	17	∈	∈	PROPN
ejpam-4800	179	18	rg(x0	rg(x0	NOUN
ejpam-4800	179	19	)	)	PUNCT
ejpam-4800	179	20	,	,	PUNCT
ejpam-4800	179	21	and	and	CCONJ
ejpam-4800	179	22	zk	zk	PROPN
ejpam-4800	179	23	in	in	ADP
ejpam-4800	179	24	g	g	PROPN
ejpam-4800	179	25	,	,	PUNCT
ejpam-4800	179	26	we	we	PRON
ejpam-4800	179	27	set	set	VERB
ejpam-4800	179	28	g0	g0	NOUN
ejpam-4800	180	1	=	=	SYM
ejpam-4800	180	2	n∑	n∑	NOUN
ejpam-4800	180	3	k=1	k=1	PUNCT
ejpam-4800	181	1	zk	zk	PROPN
ejpam-4800	181	2	.	.	PUNCT
ejpam-4800	182	1	χak	χak	PROPN
ejpam-4800	182	2	and	and	CCONJ
ejpam-4800	182	3	h	h	NOUN
ejpam-4800	183	1	=	=	SYM
ejpam-4800	183	2	n∑	n∑	NOUN
ejpam-4800	183	3	k=1	k=1	PUNCT
ejpam-4800	184	1	y	y	INTJ
ejpam-4800	184	2	.	.	PUNCT
ejpam-4800	185	1	χak	χak	PROPN
ejpam-4800	185	2	.	.	PUNCT
ejpam-4800	186	1	now	now	ADV
ejpam-4800	186	2	,	,	PUNCT
ejpam-4800	186	3	both	both	PRON
ejpam-4800	186	4	|||g0−h|||	|||g0−h|||	VERB
ejpam-4800	186	5	<	<	X
ejpam-4800	186	6	ε	ε	PROPN
ejpam-4800	186	7	and	and	CCONJ
ejpam-4800	187	1	the	the	DET
ejpam-4800	187	2	measure	measure	NOUN
ejpam-4800	187	3	space	space	NOUN
ejpam-4800	187	4	being	be	AUX
ejpam-4800	187	5	finite	finite	ADJ
ejpam-4800	187	6	,	,	PUNCT
ejpam-4800	187	7	imply	imply	VERB
ejpam-4800	187	8	that	that	SCONJ
ejpam-4800	187	9	||zk−y||	||zk−y||	NOUN
ejpam-4800	187	10	<	<	X
ejpam-4800	187	11	ε/µ(t	ε/µ(t	ADJ
ejpam-4800	187	12	)	)	PUNCT
ejpam-4800	187	13	.	.	PUNCT
ejpam-4800	188	1	hence	hence	ADV
ejpam-4800	188	2	,	,	PUNCT
ejpam-4800	188	3	the	the	DET
ejpam-4800	188	4	result	result	NOUN
ejpam-4800	188	5	follows	follow	VERB
ejpam-4800	188	6	.	.	PUNCT
ejpam-4800	189	1	⇐	⇐	ADJ
ejpam-4800	189	2	)	)	PUNCT
ejpam-4800	189	3	let	let	VERB
ejpam-4800	189	4	g	g	NOUN
ejpam-4800	189	5	be	be	AUX
ejpam-4800	189	6	strongly	strongly	ADV
ejpam-4800	189	7	coproximinal	coproximinal	ADJ
ejpam-4800	189	8	in	in	ADP
ejpam-4800	189	9	x.	x.	NOUN
ejpam-4800	189	10	by	by	ADP
ejpam-4800	189	11	lemma	lemma	PROPN
ejpam-4800	189	12	2	2	NUM
ejpam-4800	189	13	,	,	PUNCT
ejpam-4800	189	14	above	above	ADP
ejpam-4800	189	15	e(g	e(g	PROPN
ejpam-4800	189	16	)	)	PUNCT
ejpam-4800	189	17	is	be	AUX
ejpam-4800	189	18	strongly	strongly	ADV
ejpam-4800	189	19	coproximinal	coproximinal	ADJ
ejpam-4800	189	20	at	at	ADP
ejpam-4800	189	21	any	any	DET
ejpam-4800	189	22	simple	simple	ADJ
ejpam-4800	189	23	function	function	NOUN
ejpam-4800	189	24	in	in	ADP
ejpam-4800	189	25	e(x	e(x	NUM
ejpam-4800	189	26	)	)	PUNCT
ejpam-4800	189	27	.	.	PUNCT
ejpam-4800	190	1	but	but	CCONJ
ejpam-4800	190	2	since	since	SCONJ
ejpam-4800	190	3	simple	simple	ADJ
ejpam-4800	190	4	functions	function	NOUN
ejpam-4800	190	5	are	be	AUX
ejpam-4800	190	6	dense	dense	ADJ
ejpam-4800	190	7	in	in	ADP
ejpam-4800	190	8	the	the	DET
ejpam-4800	190	9	whole	whole	ADJ
ejpam-4800	190	10	space	space	NOUN
ejpam-4800	190	11	then	then	ADV
ejpam-4800	190	12	one	one	PRON
ejpam-4800	190	13	can	can	AUX
ejpam-4800	190	14	deduce	deduce	VERB
ejpam-4800	190	15	that	that	DET
ejpam-4800	190	16	e(g	e(g	NOUN
ejpam-4800	190	17	)	)	PUNCT
ejpam-4800	190	18	is	be	AUX
ejpam-4800	190	19	strongly	strongly	ADV
ejpam-4800	190	20	coproximinal	coproximinal	ADJ
ejpam-4800	190	21	at	at	ADP
ejpam-4800	190	22	any	any	DET
ejpam-4800	190	23	function	function	NOUN
ejpam-4800	190	24	in	in	ADP
ejpam-4800	190	25	e(x	e(x	NUM
ejpam-4800	190	26	)	)	PUNCT
ejpam-4800	190	27	.	.	PUNCT
ejpam-4800	191	1	references	reference	NOUN
ejpam-4800	191	2	1550	1550	NUM
ejpam-4800	191	3	corollary	corollary	ADJ
ejpam-4800	191	4	2	2	NUM
ejpam-4800	191	5	.	.	PUNCT
ejpam-4800	192	1	let	let	VERB
ejpam-4800	192	2	g	g	PRON
ejpam-4800	192	3	be	be	AUX
ejpam-4800	192	4	separable	separable	ADJ
ejpam-4800	192	5	in	in	ADP
ejpam-4800	192	6	x.	x.	PROPN
ejpam-4800	192	7	g	g	PROPN
ejpam-4800	192	8	is	be	AUX
ejpam-4800	192	9	strongly	strongly	ADV
ejpam-4800	192	10	coproximinal	coproximinal	ADJ
ejpam-4800	192	11	in	in	ADP
ejpam-4800	192	12	x	x	SYM
ejpam-4800	192	13	if	if	SCONJ
ejpam-4800	193	1	and	and	CCONJ
ejpam-4800	193	2	only	only	ADV
ejpam-4800	193	3	if	if	SCONJ
ejpam-4800	193	4	lp(µ,g	lp(µ,g	NUM
ejpam-4800	193	5	)	)	PUNCT
ejpam-4800	193	6	is	be	AUX
ejpam-4800	193	7	strongly	strongly	ADV
ejpam-4800	193	8	coproximinal	coproximinal	ADJ
ejpam-4800	193	9	in	in	ADP
ejpam-4800	193	10	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	193	11	)	)	PUNCT
ejpam-4800	193	12	,	,	PUNCT
ejpam-4800	193	13	for	for	ADP
ejpam-4800	193	14	1	1	NUM
ejpam-4800	193	15	≤	≤	NOUN
ejpam-4800	193	16	p	p	NOUN
ejpam-4800	193	17	<	<	X
ejpam-4800	193	18	∞.	∞.	PROPN
ejpam-4800	193	19	4	4	NUM
ejpam-4800	193	20	.	.	PUNCT
ejpam-4800	194	1	conclusion	conclusion	NOUN
ejpam-4800	194	2	in	in	ADP
ejpam-4800	194	3	this	this	DET
ejpam-4800	194	4	paper	paper	NOUN
ejpam-4800	194	5	,	,	PUNCT
ejpam-4800	194	6	strong	strong	ADJ
ejpam-4800	194	7	coproximinality	coproximinality	NOUN
ejpam-4800	194	8	was	be	AUX
ejpam-4800	194	9	studied	study	VERB
ejpam-4800	194	10	for	for	ADP
ejpam-4800	194	11	bochner	bochn	ADJ
ejpam-4800	194	12	function	function	NOUN
ejpam-4800	194	13	spaces	space	NOUN
ejpam-4800	194	14	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	194	15	)	)	PUNCT
ejpam-4800	194	16	,	,	PUNCT
ejpam-4800	194	17	for	for	ADP
ejpam-4800	194	18	1	1	NUM
ejpam-4800	194	19	≤	≤	NOUN
ejpam-4800	194	20	p	p	NOUN
ejpam-4800	194	21	<	<	X
ejpam-4800	194	22	∞	∞	PROPN
ejpam-4800	194	23	,	,	PUNCT
ejpam-4800	194	24	and	and	CCONJ
ejpam-4800	194	25	for	for	ADP
ejpam-4800	194	26	the	the	DET
ejpam-4800	194	27	köthe	köthe	PROPN
ejpam-4800	194	28	bochner	bochner	NOUN
ejpam-4800	194	29	function	function	NOUN
ejpam-4800	194	30	space	space	NOUN
ejpam-4800	194	31	e(x	e(x	NUM
ejpam-4800	194	32	)	)	PUNCT
ejpam-4800	194	33	.	.	PUNCT
ejpam-4800	195	1	the	the	DET
ejpam-4800	195	2	main	main	ADJ
ejpam-4800	195	3	result	result	NOUN
ejpam-4800	195	4	was	be	AUX
ejpam-4800	195	5	:	:	PUNCT
ejpam-4800	195	6	if	if	SCONJ
ejpam-4800	195	7	g	g	PROPN
ejpam-4800	195	8	is	be	AUX
ejpam-4800	195	9	separable	separable	ADJ
ejpam-4800	195	10	in	in	ADP
ejpam-4800	195	11	x	x	NOUN
ejpam-4800	195	12	,	,	PUNCT
ejpam-4800	195	13	then	then	ADV
ejpam-4800	195	14	lp(µ,g	lp(µ,g	NUM
ejpam-4800	195	15	)	)	PUNCT
ejpam-4800	195	16	(	(	PUNCT
ejpam-4800	195	17	resp	resp	NOUN
ejpam-4800	195	18	.	.	PUNCT
ejpam-4800	196	1	e(g	e(g	NOUN
ejpam-4800	196	2	)	)	PUNCT
ejpam-4800	196	3	)	)	PUNCT
ejpam-4800	196	4	is	be	AUX
ejpam-4800	196	5	strongly	strongly	ADV
ejpam-4800	196	6	coproximinal	coproximinal	ADJ
ejpam-4800	196	7	in	in	ADP
ejpam-4800	196	8	lp(µ,x	lp(µ,x	NOUN
ejpam-4800	196	9	)	)	PUNCT
ejpam-4800	196	10	(	(	PUNCT
ejpam-4800	196	11	resp	resp	NOUN
ejpam-4800	196	12	.	.	PUNCT
ejpam-4800	197	1	e(x	e(x	NUM
ejpam-4800	197	2	)	)	PUNCT
ejpam-4800	197	3	)	)	PUNCT
ejpam-4800	198	1	,	,	PUNCT
ejpam-4800	198	2	if	if	SCONJ
ejpam-4800	198	3	and	and	CCONJ
ejpam-4800	198	4	only	only	ADV
ejpam-4800	198	5	if	if	SCONJ
ejpam-4800	198	6	g	g	PROPN
ejpam-4800	198	7	is	be	AUX
ejpam-4800	198	8	strongly	strongly	ADV
ejpam-4800	198	9	coproximinal	coproximinal	ADJ
ejpam-4800	198	10	subspace	subspace	NOUN
ejpam-4800	198	11	of	of	ADP
ejpam-4800	198	12	x.	x.	NOUN
ejpam-4800	198	13	some	some	DET
ejpam-4800	198	14	other	other	ADJ
ejpam-4800	198	15	results	result	NOUN
ejpam-4800	198	16	were	be	AUX
ejpam-4800	198	17	also	also	ADV
ejpam-4800	198	18	given	give	VERB
ejpam-4800	198	19	and	and	CCONJ
ejpam-4800	198	20	proved	prove	VERB
ejpam-4800	198	21	for	for	ADP
ejpam-4800	198	22	strong	strong	ADJ
ejpam-4800	198	23	coproximinality	coproximinality	NOUN
ejpam-4800	198	24	in	in	ADP
ejpam-4800	198	25	these	these	DET
ejpam-4800	198	26	spaces	space	NOUN
ejpam-4800	198	27	.	.	PUNCT
ejpam-4800	199	1	acknowledgements	acknowledgement	NOUN
ejpam-4800	199	2	this	this	DET
ejpam-4800	199	3	research	research	NOUN
ejpam-4800	199	4	was	be	AUX
ejpam-4800	199	5	funded	fund	VERB
ejpam-4800	199	6	by	by	ADP
ejpam-4800	199	7	zarqa	zarqa	PROPN
ejpam-4800	199	8	university	university	PROPN
ejpam-4800	199	9	.	.	PUNCT
ejpam-4800	200	1	funding	funding	NOUN
ejpam-4800	200	2	agency	agency	NOUN
ejpam-4800	200	3	names	name	NOUN
ejpam-4800	200	4	at	at	ADP
ejpam-4800	200	5	https://deanship	https://deanship	PROPN
ejpam-4800	200	6	of	of	ADP
ejpam-4800	200	7	research@zu.edu.jo	research@zu.edu.jo	NOUN
ejpam-4800	200	8	.	.	PUNCT
ejpam-4800	201	1	references	reference	NOUN
ejpam-4800	201	2	[	[	X
ejpam-4800	201	3	1	1	X
ejpam-4800	201	4	]	]	PUNCT
ejpam-4800	201	5	sh	sh	PROPN
ejpam-4800	201	6	al	al	PROPN
ejpam-4800	201	7	-	-	PUNCT
ejpam-4800	201	8	sharif	sharif	PROPN
ejpam-4800	201	9	,	,	PUNCT
ejpam-4800	201	10	a	a	DET
ejpam-4800	201	11	ababneh	ababneh	NOUN
ejpam-4800	201	12	,	,	PUNCT
ejpam-4800	201	13	and	and	CCONJ
ejpam-4800	201	14	m	m	PROPN
ejpam-4800	201	15	al	al	PROPN
ejpam-4800	201	16	-	-	PUNCT
ejpam-4800	201	17	qahtani	qahtani	PROPN
ejpam-4800	201	18	.	.	PUNCT
ejpam-4800	202	1	best	good	ADJ
ejpam-4800	202	2	coapproximation	coapproximation	NOUN
ejpam-4800	202	3	in	in	ADP
ejpam-4800	202	4	certain	certain	ADJ
ejpam-4800	202	5	metric	metric	ADJ
ejpam-4800	202	6	spaces	space	NOUN
ejpam-4800	202	7	.	.	PUNCT
ejpam-4800	203	1	jaen	jaen	PROPN
ejpam-4800	203	2	j.	j.	PROPN
ejpam-4800	203	3	approx	approx	PROPN
ejpam-4800	203	4	.	.	PROPN
ejpam-4800	203	5	,	,	PUNCT
ejpam-4800	203	6	11:91–100	11:91–100	NUM
ejpam-4800	203	7	,	,	PUNCT
ejpam-4800	203	8	2019	2019	NUM
ejpam-4800	203	9	.	.	PUNCT
ejpam-4800	204	1	[	[	X
ejpam-4800	204	2	2	2	NUM
ejpam-4800	204	3	]	]	X
ejpam-4800	204	4	c	c	PROPN
ejpam-4800	204	5	franchetti	franchetti	PROPN
ejpam-4800	204	6	and	and	CCONJ
ejpam-4800	204	7	m	m	PROPN
ejpam-4800	204	8	furi	furi	NOUN
ejpam-4800	204	9	.	.	PUNCT
ejpam-4800	205	1	some	some	DET
ejpam-4800	205	2	characteristic	characteristic	ADJ
ejpam-4800	205	3	properties	property	NOUN
ejpam-4800	205	4	of	of	ADP
ejpam-4800	205	5	real	real	ADJ
ejpam-4800	205	6	hilbert	hilbert	NOUN
ejpam-4800	205	7	spaces	space	NOUN
ejpam-4800	205	8	.	.	PUNCT
ejpam-4800	206	1	rev	rev	PROPN
ejpam-4800	206	2	.	.	PROPN
ejpam-4800	206	3	roumaine	roumaine	PROPN
ejpam-4800	206	4	math	math	NOUN
ejpam-4800	206	5	.	.	PUNCT
ejpam-4800	207	1	pure	pure	ADJ
ejpam-4800	207	2	appl	appl	PROPN
ejpam-4800	207	3	.	.	PROPN
ejpam-4800	207	4	,	,	PUNCT
ejpam-4800	207	5	17:1045–1048	17:1045–1048	NUM
ejpam-4800	207	6	,	,	PUNCT
ejpam-4800	207	7	1972	1972	NUM
ejpam-4800	207	8	.	.	PUNCT
ejpam-4800	208	1	[	[	X
ejpam-4800	208	2	3	3	X
ejpam-4800	208	3	]	]	X
ejpam-4800	208	4	g	g	NOUN
ejpam-4800	208	5	godefroy	godefroy	ADJ
ejpam-4800	208	6	and	and	CCONJ
ejpam-4800	208	7	v	v	ADP
ejpam-4800	208	8	indumathi	indumathi	NOUN
ejpam-4800	208	9	.	.	PUNCT
ejpam-4800	209	1	strong	strong	ADJ
ejpam-4800	209	2	proximinality	proximinality	NOUN
ejpam-4800	209	3	and	and	CCONJ
ejpam-4800	209	4	polyhedral	polyhedral	ADJ
ejpam-4800	209	5	spaces	space	NOUN
ejpam-4800	209	6	.	.	PUNCT
ejpam-4800	210	1	rev	rev	PROPN
ejpam-4800	210	2	.	.	PROPN
ejpam-4800	210	3	mat	mat	PROPN
ejpam-4800	210	4	.	.	PROPN
ejpam-4800	210	5	complut	complut	PROPN
ejpam-4800	210	6	.	.	PUNCT
ejpam-4800	210	7	,	,	PUNCT
ejpam-4800	210	8	14:105–125	14:105–125	NUM
ejpam-4800	210	9	,	,	PUNCT
ejpam-4800	210	10	2001	2001	NUM
ejpam-4800	210	11	.	.	PUNCT
ejpam-4800	211	1	[	[	X
ejpam-4800	211	2	4	4	NUM
ejpam-4800	211	3	]	]	PUNCT
ejpam-4800	211	4	m	m	VERB
ejpam-4800	211	5	haddadi	haddadi	NOUN
ejpam-4800	211	6	,	,	PUNCT
ejpam-4800	211	7	n	n	DET
ejpam-4800	211	8	hejazjpoor	hejazjpoor	NOUN
ejpam-4800	211	9	,	,	PUNCT
ejpam-4800	211	10	and	and	CCONJ
ejpam-4800	211	11	h	h	PROPN
ejpam-4800	211	12	mazaheri	mazaheri	NOUN
ejpam-4800	211	13	.	.	PUNCT
ejpam-4800	212	1	some	some	PRON
ejpam-4800	212	2	results	result	VERB
ejpam-4800	212	3	about	about	ADP
ejpam-4800	212	4	best	good	ADJ
ejpam-4800	212	5	coapproximation	coapproximation	NOUN
ejpam-4800	212	6	in	in	ADP
ejpam-4800	212	7	lp(s	lp(s	ADJ
ejpam-4800	212	8	,	,	PUNCT
ejpam-4800	212	9	x	x	X
ejpam-4800	212	10	)	)	PUNCT
ejpam-4800	212	11	.	.	PUNCT
ejpam-4800	213	1	anal	anal	PROPN
ejpam-4800	213	2	.	.	PUNCT
ejpam-4800	213	3	theory	theory	NOUN
ejpam-4800	213	4	appl	appl	PROPN
ejpam-4800	213	5	.	.	PROPN
ejpam-4800	213	6	,	,	PUNCT
ejpam-4800	213	7	26:69–75	26:69–75	NUM
ejpam-4800	213	8	,	,	PUNCT
ejpam-4800	213	9	2010	2010	NUM
ejpam-4800	213	10	.	.	PUNCT
ejpam-4800	214	1	[	[	X
ejpam-4800	214	2	5	5	NUM
ejpam-4800	214	3	]	]	SYM
ejpam-4800	214	4	j	j	PROPN
ejpam-4800	214	5	jawdat	jawdat	PROPN
ejpam-4800	214	6	.	.	PUNCT
ejpam-4800	215	1	strong	strong	ADJ
ejpam-4800	215	2	coproximinality	coproximinality	NOUN
ejpam-4800	215	3	in	in	ADP
ejpam-4800	215	4	bochner	bochner	NOUN
ejpam-4800	215	5	spaces	space	NOUN
ejpam-4800	215	6	lp(µ,x	lp(µ,x	ADJ
ejpam-4800	215	7	)	)	PUNCT
ejpam-4800	215	8	,	,	PUNCT
ejpam-4800	215	9	1	1	NUM
ejpam-4800	215	10	≤	≤	NOUN
ejpam-4800	215	11	p	p	NOUN
ejpam-4800	215	12	<	<	X
ejpam-4800	215	13	∞.	∞.	PROPN
ejpam-4800	215	14	jmcs	jmcs	NOUN
ejpam-4800	215	15	,	,	PUNCT
ejpam-4800	215	16	accepted	accept	VERB
ejpam-4800	215	17	,	,	PUNCT
ejpam-4800	215	18	2022	2022	NUM
ejpam-4800	215	19	.	.	PUNCT
ejpam-4800	216	1	[	[	X
ejpam-4800	216	2	6	6	NUM
ejpam-4800	216	3	]	]	X
ejpam-4800	216	4	j	j	PROPN
ejpam-4800	216	5	jawdat	jawdat	PROPN
ejpam-4800	216	6	and	and	CCONJ
ejpam-4800	216	7	sh	sh	PROPN
ejpam-4800	216	8	al	al	PROPN
ejpam-4800	216	9	-	-	PUNCT
ejpam-4800	216	10	sharif	sharif	PROPN
ejpam-4800	216	11	.	.	PUNCT
ejpam-4800	217	1	coproximinality	coproximinality	NOUN
ejpam-4800	217	2	results	result	NOUN
ejpam-4800	217	3	in	in	ADP
ejpam-4800	217	4	köthe	köthe	DET
ejpam-4800	217	5	bochner	bochner	NOUN
ejpam-4800	217	6	spaces	space	NOUN
ejpam-4800	217	7	.	.	PUNCT
ejpam-4800	218	1	italian	italian	ADJ
ejpam-4800	218	2	j.	j.	PROPN
ejpam-4800	218	3	of	of	ADP
ejpam-4800	218	4	pure	pure	ADJ
ejpam-4800	218	5	and	and	CCONJ
ejpam-4800	218	6	applied	applied	ADJ
ejpam-4800	218	7	math	math	NOUN
ejpam-4800	218	8	.	.	PUNCT
ejpam-4800	218	9	,	,	PUNCT
ejpam-4800	218	10	36:783–790	36:783–790	NOUN
ejpam-4800	218	11	,	,	PUNCT
ejpam-4800	218	12	2016	2016	NUM
ejpam-4800	218	13	.	.	PUNCT
ejpam-4800	219	1	[	[	X
ejpam-4800	219	2	7	7	X
ejpam-4800	219	3	]	]	X
ejpam-4800	219	4	j	j	PROPN
ejpam-4800	219	5	jawdat	jawdat	NOUN
ejpam-4800	219	6	and	and	CCONJ
ejpam-4800	219	7	r	r	NOUN
ejpam-4800	219	8	shalabi	shalabi	NOUN
ejpam-4800	219	9	.	.	PUNCT
ejpam-4800	220	1	strong	strong	ADJ
ejpam-4800	220	2	proximinality	proximinality	NOUN
ejpam-4800	220	3	in	in	ADP
ejpam-4800	220	4	the	the	DET
ejpam-4800	220	5	function	function	NOUN
ejpam-4800	220	6	space	space	NOUN
ejpam-4800	220	7	lϕ(µ,x	lϕ(µ,x	NOUN
ejpam-4800	220	8	)	)	PUNCT
ejpam-4800	220	9	.	.	PUNCT
ejpam-4800	221	1	inter	inter	PROPN
ejpam-4800	221	2	.	.	PUNCT
ejpam-4800	222	1	math	math	PROPN
ejpam-4800	222	2	.	.	PUNCT
ejpam-4800	223	1	forum	forum	PROPN
ejpam-4800	223	2	,	,	PUNCT
ejpam-4800	223	3	15:103–111	15:103–111	NUM
ejpam-4800	223	4	,	,	PUNCT
ejpam-4800	223	5	2020	2020	NUM
ejpam-4800	223	6	.	.	PUNCT
ejpam-4800	224	1	[	[	X
ejpam-4800	224	2	8	8	NUM
ejpam-4800	224	3	]	]	X
ejpam-4800	224	4	r	r	NOUN
ejpam-4800	224	5	khalil	khalil	PROPN
ejpam-4800	224	6	.	.	PUNCT
ejpam-4800	225	1	best	good	ADJ
ejpam-4800	225	2	approximation	approximation	NOUN
ejpam-4800	225	3	in	in	ADP
ejpam-4800	225	4	lp(i	lp(i	NOUN
ejpam-4800	225	5	,	,	PUNCT
ejpam-4800	225	6	x	x	NOUN
ejpam-4800	225	7	)	)	PUNCT
ejpam-4800	225	8	.	.	PUNCT
ejpam-4800	226	1	math	math	NOUN
ejpam-4800	226	2	.	.	PUNCT
ejpam-4800	227	1	proc	proc	PROPN
ejpam-4800	227	2	.	.	PUNCT
ejpam-4800	228	1	camb	camb	PROPN
ejpam-4800	228	2	.	.	PUNCT
ejpam-4800	229	1	phil	phil	PROPN
ejpam-4800	229	2	.	.	PUNCT
ejpam-4800	230	1	soc	soc	PROPN
ejpam-4800	230	2	.	.	PUNCT
ejpam-4800	230	3	,	,	PUNCT
ejpam-4800	231	1	94:177–289	94:177–289	PROPN
ejpam-4800	231	2	,	,	PUNCT
ejpam-4800	231	3	1983	1983	NUM
ejpam-4800	231	4	.	.	PUNCT
ejpam-4800	232	1	[	[	X
ejpam-4800	232	2	9	9	NUM
ejpam-4800	232	3	]	]	X
ejpam-4800	232	4	m	m	VERB
ejpam-4800	232	5	khandaqji	khandaqji	ADJ
ejpam-4800	232	6	,	,	PUNCT
ejpam-4800	232	7	w	w	NOUN
ejpam-4800	232	8	shatanawi	shatanawi	ADJ
ejpam-4800	232	9	,	,	PUNCT
ejpam-4800	232	10	and	and	CCONJ
ejpam-4800	232	11	z	z	PROPN
ejpam-4800	232	12	mustafa	mustafa	PROPN
ejpam-4800	232	13	.	.	PUNCT
ejpam-4800	233	1	approximation	approximation	NOUN
ejpam-4800	233	2	in	in	ADP
ejpam-4800	233	3	kothe	kothe	PROPN
ejpam-4800	233	4	bochner	bochner	NOUN
ejpam-4800	233	5	function	function	NOUN
ejpam-4800	233	6	space	space	NOUN
ejpam-4800	233	7	.	.	PUNCT
ejpam-4800	234	1	international	international	ADJ
ejpam-4800	234	2	journal	journal	PROPN
ejpam-4800	234	3	of	of	ADP
ejpam-4800	234	4	applied	apply	VERB
ejpam-4800	234	5	mathematics	mathematic	NOUN
ejpam-4800	234	6	,	,	PUNCT
ejpam-4800	234	7	20:937–942	20:937–942	NUM
ejpam-4800	234	8	,	,	PUNCT
ejpam-4800	234	9	2007	2007	NUM
ejpam-4800	234	10	.	.	PUNCT
ejpam-4800	235	1	references	reference	NOUN
ejpam-4800	235	2	1551	1551	NUM
ejpam-4800	236	1	[	[	X
ejpam-4800	236	2	10	10	NUM
ejpam-4800	236	3	]	]	X
ejpam-4800	236	4	w	w	NOUN
ejpam-4800	236	5	light	light	NOUN
ejpam-4800	236	6	and	and	CCONJ
ejpam-4800	236	7	e	e	NOUN
ejpam-4800	236	8	cheney	cheney	PROPN
ejpam-4800	236	9	.	.	PUNCT
ejpam-4800	237	1	approximation	approximation	NOUN
ejpam-4800	237	2	theory	theory	NOUN
ejpam-4800	237	3	in	in	ADP
ejpam-4800	237	4	tensor	tensor	NOUN
ejpam-4800	237	5	product	product	NOUN
ejpam-4800	237	6	spaces	space	VERB
ejpam-4800	237	7	.	.	PUNCT
ejpam-4800	238	1	lecture	lecture	NOUN
ejpam-4800	238	2	notes	note	NOUN
ejpam-4800	238	3	in	in	ADP
ejpam-4800	238	4	math	math	NOUN
ejpam-4800	238	5	.	.	PUNCT
ejpam-4800	239	1	springer	springer	NOUN
ejpam-4800	239	2	,	,	PUNCT
ejpam-4800	239	3	new	new	PROPN
ejpam-4800	239	4	york	york	PROPN
ejpam-4800	239	5	,	,	PUNCT
ejpam-4800	239	6	1980	1980	NUM
ejpam-4800	239	7	.	.	PUNCT
ejpam-4800	240	1	[	[	X
ejpam-4800	240	2	11	11	NUM
ejpam-4800	240	3	]	]	X
ejpam-4800	240	4	p	p	X
ejpam-4800	240	5	k	k	PROPN
ejpam-4800	240	6	lin	lin	PROPN
ejpam-4800	240	7	.	.	PUNCT
ejpam-4800	241	1	köthe	köthe	PRON
ejpam-4800	241	2	bochner	bochner	NOUN
ejpam-4800	241	3	function	function	NOUN
ejpam-4800	241	4	space	space	NOUN
ejpam-4800	241	5	.	.	PUNCT
ejpam-4800	242	1	springer	springer	NOUN
ejpam-4800	242	2	-	-	PUNCT
ejpam-4800	242	3	verlag	verlag	PROPN
ejpam-4800	242	4	,	,	PUNCT
ejpam-4800	242	5	new	new	PROPN
ejpam-4800	242	6	york	york	PROPN
ejpam-4800	242	7	,	,	PUNCT
ejpam-4800	242	8	2004	2004	NUM
ejpam-4800	242	9	.	.	PUNCT
ejpam-4800	243	1	[	[	X
ejpam-4800	243	2	12	12	NUM
ejpam-4800	243	3	]	]	X
ejpam-4800	243	4	j	j	PROPN
ejpam-4800	243	5	mendoza	mendoza	PROPN
ejpam-4800	243	6	.	.	PUNCT
ejpam-4800	244	1	proximinality	proximinality	NOUN
ejpam-4800	244	2	in	in	ADP
ejpam-4800	244	3	lp(i	lp(i	NUM
ejpam-4800	244	4	,	,	PUNCT
ejpam-4800	244	5	x	x	NOUN
ejpam-4800	244	6	)	)	PUNCT
ejpam-4800	244	7	.	.	PUNCT
ejpam-4800	245	1	journal	journal	PROPN
ejpam-4800	245	2	of	of	ADP
ejpam-4800	245	3	approximation	approximation	NOUN
ejpam-4800	245	4	theory	theory	NOUN
ejpam-4800	245	5	,	,	PUNCT
ejpam-4800	245	6	93:331–343	93:331–343	PROPN
ejpam-4800	245	7	,	,	PUNCT
ejpam-4800	245	8	1998	1998	NUM
ejpam-4800	245	9	.	.	PUNCT
ejpam-4800	246	1	[	[	X
ejpam-4800	246	2	13	13	NUM
ejpam-4800	246	3	]	]	PUNCT
ejpam-4800	246	4	p	p	NOUN
ejpam-4800	246	5	l	l	NOUN
ejpam-4800	246	6	papini	papini	NOUN
ejpam-4800	246	7	and	and	CCONJ
ejpam-4800	246	8	i	i	PROPN
ejpam-4800	246	9	singer	singer	NOUN
ejpam-4800	246	10	.	.	PUNCT
ejpam-4800	247	1	best	good	ADJ
ejpam-4800	247	2	coapproximation	coapproximation	NOUN
ejpam-4800	247	3	in	in	ADP
ejpam-4800	247	4	normed	normed	ADJ
ejpam-4800	247	5	linear	linear	PROPN
ejpam-4800	247	6	spaces	space	NOUN
ejpam-4800	247	7	.	.	PUNCT
ejpam-4800	248	1	mh	mh	PROPN
ejpam-4800	248	2	.	.	PUNCT
ejpam-4800	248	3	math	math	PROPN
ejpam-4800	248	4	.	.	PUNCT
ejpam-4800	248	5	,	,	PUNCT
ejpam-4800	248	6	88:27–44	88:27–44	NUM
ejpam-4800	248	7	,	,	PUNCT
ejpam-4800	248	8	1979	1979	NUM
ejpam-4800	248	9	.	.	PUNCT
ejpam-4800	249	1	[	[	X
ejpam-4800	249	2	14	14	NUM
ejpam-4800	249	3	]	]	X
ejpam-4800	249	4	i	i	PROPN
ejpam-4800	249	5	singer	singer	NOUN
ejpam-4800	249	6	.	.	PUNCT
ejpam-4800	250	1	best	good	ADJ
ejpam-4800	250	2	approximation	approximation	NOUN
ejpam-4800	250	3	in	in	ADP
ejpam-4800	250	4	normed	normed	ADJ
ejpam-4800	250	5	linear	linear	PROPN
ejpam-4800	250	6	spaces	space	NOUN
ejpam-4800	250	7	of	of	ADP
ejpam-4800	250	8	linear	linear	ADJ
ejpam-4800	250	9	subspaces	subspace	NOUN
ejpam-4800	250	10	.	.	PUNCT
ejpam-4800	251	1	springerverlag	springerverlag	NOUN
ejpam-4800	251	2	,	,	PUNCT
ejpam-4800	251	3	new	new	PROPN
ejpam-4800	251	4	york	york	PROPN
ejpam-4800	251	5	,	,	PUNCT
ejpam-4800	251	6	1970	1970	NUM
ejpam-4800	251	7	.	.	PUNCT
ejpam-4800	252	1	[	[	X
ejpam-4800	252	2	15	15	NUM
ejpam-4800	252	3	]	]	X
ejpam-4800	252	4	p	p	DET
ejpam-4800	252	5	tanmoy	tanmoy	NOUN
ejpam-4800	252	6	.	.	PUNCT
ejpam-4800	253	1	various	various	ADJ
ejpam-4800	253	2	notions	notion	NOUN
ejpam-4800	253	3	of	of	ADP
ejpam-4800	253	4	best	good	ADJ
ejpam-4800	253	5	approximation	approximation	NOUN
ejpam-4800	253	6	property	property	NOUN
ejpam-4800	253	7	in	in	ADP
ejpam-4800	253	8	spaces	space	NOUN
ejpam-4800	253	9	of	of	ADP
ejpam-4800	253	10	bochner	bochner	NOUN
ejpam-4800	253	11	integrable	integrable	ADJ
ejpam-4800	253	12	functions	function	NOUN
ejpam-4800	253	13	.	.	PUNCT
ejpam-4800	254	1	adv	adv	PROPN
ejpam-4800	254	2	.	.	PUNCT
ejpam-4800	255	1	oper	oper	PROPN
ejpam-4800	255	2	.	.	PROPN
ejpam-4800	255	3	theory	theory	PROPN
ejpam-4800	255	4	,	,	PUNCT
ejpam-4800	255	5	2:59–77	2:59–77	NUM
ejpam-4800	255	6	,	,	PUNCT
ejpam-4800	255	7	2017	2017	NUM
ejpam-4800	255	8	.	.	PUNCT