id	sid	tid	token	lemma	pos
ejpam-4919	1	1	european	european	PROPN
ejpam-4919	1	2	journal	journal	PROPN
ejpam-4919	1	3	of	of	ADP
ejpam-4919	1	4	pure	pure	ADJ
ejpam-4919	1	5	and	and	CCONJ
ejpam-4919	1	6	applied	apply	VERB
ejpam-4919	1	7	mathematics	mathematic	NOUN
ejpam-4919	1	8	vol	vol	NOUN
ejpam-4919	1	9	.	.	PUNCT
ejpam-4919	2	1	16	16	NUM
ejpam-4919	2	2	,	,	PUNCT
ejpam-4919	2	3	no	no	INTJ
ejpam-4919	2	4	.	.	NOUN
ejpam-4919	2	5	4	4	NUM
ejpam-4919	2	6	,	,	PUNCT
ejpam-4919	2	7	2023	2023	NUM
ejpam-4919	2	8	,	,	PUNCT
ejpam-4919	2	9	2397	2397	NUM
ejpam-4919	2	10	-	-	SYM
ejpam-4919	2	11	2404	2404	NUM
ejpam-4919	2	12	issn	issn	PROPN
ejpam-4919	2	13	1307	1307	NUM
ejpam-4919	2	14	-	-	SYM
ejpam-4919	2	15	5543	5543	NUM
ejpam-4919	2	16	–	–	PUNCT
ejpam-4919	2	17	ejpam.com	ejpam.com	X
ejpam-4919	2	18	published	publish	VERB
ejpam-4919	2	19	by	by	ADP
ejpam-4919	2	20	new	new	PROPN
ejpam-4919	2	21	york	york	PROPN
ejpam-4919	2	22	business	business	PROPN
ejpam-4919	2	23	global	global	PROPN
ejpam-4919	2	24	on	on	ADP
ejpam-4919	2	25	the	the	DET
ejpam-4919	2	26	farthest	farth	ADJ
ejpam-4919	2	27	point	point	NOUN
ejpam-4919	2	28	problem	problem	NOUN
ejpam-4919	2	29	in	in	ADP
ejpam-4919	2	30	hilbert	hilbert	PROPN
ejpam-4919	2	31	spaces	space	NOUN
ejpam-4919	2	32	a.	a.	PROPN
ejpam-4919	2	33	yousef1,2,∗	yousef1,2,∗	PROPN
ejpam-4919	2	34	,	,	PUNCT
ejpam-4919	2	35	r.	r.	PROPN
ejpam-4919	2	36	khalil1	khalil1	PROPN
ejpam-4919	2	37	,	,	PUNCT
ejpam-4919	2	38	a.	a.	PROPN
ejpam-4919	2	39	talafha1	talafha1	PROPN
ejpam-4919	2	40	,	,	PUNCT
ejpam-4919	2	41	b.	b.	PROPN
ejpam-4919	2	42	mutabagani3	mutabagani3	PROPN
ejpam-4919	2	43	1	1	NUM
ejpam-4919	2	44	department	department	NOUN
ejpam-4919	2	45	of	of	ADP
ejpam-4919	2	46	mathematics	mathematic	NOUN
ejpam-4919	2	47	,	,	PUNCT
ejpam-4919	2	48	school	school	NOUN
ejpam-4919	2	49	of	of	ADP
ejpam-4919	2	50	science	science	NOUN
ejpam-4919	2	51	,	,	PUNCT
ejpam-4919	2	52	the	the	DET
ejpam-4919	2	53	university	university	PROPN
ejpam-4919	2	54	of	of	ADP
ejpam-4919	2	55	jordan	jordan	PROPN
ejpam-4919	2	56	,	,	PUNCT
ejpam-4919	2	57	amman	amman	PROPN
ejpam-4919	2	58	,	,	PUNCT
ejpam-4919	2	59	jordan	jordan	PROPN
ejpam-4919	2	60	2	2	NUM
ejpam-4919	2	61	department	department	NOUN
ejpam-4919	2	62	of	of	ADP
ejpam-4919	2	63	mathematics	mathematic	NOUN
ejpam-4919	2	64	and	and	CCONJ
ejpam-4919	2	65	statistics	statistic	NOUN
ejpam-4919	2	66	,	,	PUNCT
ejpam-4919	2	67	college	college	NOUN
ejpam-4919	2	68	of	of	ADP
ejpam-4919	2	69	arts	art	NOUN
ejpam-4919	2	70	and	and	CCONJ
ejpam-4919	2	71	sciences	science	NOUN
ejpam-4919	2	72	,	,	PUNCT
ejpam-4919	2	73	american	american	PROPN
ejpam-4919	2	74	university	university	PROPN
ejpam-4919	2	75	of	of	ADP
ejpam-4919	2	76	sharjah	sharjah	PROPN
ejpam-4919	2	77	,	,	PUNCT
ejpam-4919	2	78	sharjah	sharjah	PROPN
ejpam-4919	2	79	,	,	PUNCT
ejpam-4919	2	80	uae	uae	PROPN
ejpam-4919	2	81	2	2	NUM
ejpam-4919	2	82	department	department	NOUN
ejpam-4919	2	83	of	of	ADP
ejpam-4919	2	84	mathematics	mathematic	NOUN
ejpam-4919	2	85	,	,	PUNCT
ejpam-4919	2	86	western	western	ADJ
ejpam-4919	2	87	illinois	illinois	PROPN
ejpam-4919	2	88	university	university	PROPN
ejpam-4919	2	89	,	,	PUNCT
ejpam-4919	2	90	illinois	illinois	PROPN
ejpam-4919	2	91	,	,	PUNCT
ejpam-4919	2	92	usa	usa	PROPN
ejpam-4919	2	93	abstract	abstract	PROPN
ejpam-4919	2	94	.	.	PUNCT
ejpam-4919	3	1	in	in	ADP
ejpam-4919	3	2	an	an	DET
ejpam-4919	3	3	attempt	attempt	NOUN
ejpam-4919	3	4	to	to	PART
ejpam-4919	3	5	solve	solve	VERB
ejpam-4919	3	6	the	the	DET
ejpam-4919	3	7	farthest	farth	ADJ
ejpam-4919	3	8	point	point	NOUN
ejpam-4919	3	9	problem	problem	NOUN
ejpam-4919	3	10	,	,	PUNCT
ejpam-4919	3	11	we	we	PRON
ejpam-4919	3	12	introduce	introduce	VERB
ejpam-4919	3	13	a	a	DET
ejpam-4919	3	14	new	new	ADJ
ejpam-4919	3	15	class	class	NOUN
ejpam-4919	3	16	of	of	ADP
ejpam-4919	3	17	uniquely	uniquely	ADV
ejpam-4919	3	18	remotal	remotal	ADJ
ejpam-4919	3	19	sets	set	NOUN
ejpam-4919	3	20	.	.	PUNCT
ejpam-4919	4	1	namely	namely	ADV
ejpam-4919	4	2	,	,	PUNCT
ejpam-4919	4	3	the	the	DET
ejpam-4919	4	4	class	class	NOUN
ejpam-4919	4	5	of	of	ADP
ejpam-4919	4	6	uniquely	uniquely	ADV
ejpam-4919	4	7	distant	distant	ADJ
ejpam-4919	4	8	sets	set	NOUN
ejpam-4919	4	9	.	.	PUNCT
ejpam-4919	5	1	then	then	ADV
ejpam-4919	5	2	,	,	PUNCT
ejpam-4919	5	3	we	we	PRON
ejpam-4919	5	4	prove	prove	VERB
ejpam-4919	5	5	that	that	SCONJ
ejpam-4919	5	6	in	in	ADP
ejpam-4919	5	7	a	a	DET
ejpam-4919	5	8	separable	separable	ADJ
ejpam-4919	5	9	hilbert	hilbert	NOUN
ejpam-4919	5	10	space	space	NOUN
ejpam-4919	5	11	,	,	PUNCT
ejpam-4919	5	12	every	every	DET
ejpam-4919	5	13	uniquely	uniquely	ADV
ejpam-4919	5	14	distant	distant	ADJ
ejpam-4919	5	15	set	set	NOUN
ejpam-4919	5	16	is	be	AUX
ejpam-4919	5	17	a	a	DET
ejpam-4919	5	18	singleton	singleton	NOUN
ejpam-4919	5	19	.	.	PUNCT
ejpam-4919	6	1	2020	2020	NUM
ejpam-4919	6	2	mathematics	mathematic	NOUN
ejpam-4919	6	3	subject	subject	NOUN
ejpam-4919	6	4	classifications	classification	NOUN
ejpam-4919	6	5	:	:	PUNCT
ejpam-4919	6	6	46b20	46b20	NUM
ejpam-4919	6	7	,	,	PUNCT
ejpam-4919	6	8	41a50	41a50	NUM
ejpam-4919	6	9	,	,	PUNCT
ejpam-4919	6	10	41a65	41a65	NUM
ejpam-4919	6	11	key	key	ADJ
ejpam-4919	6	12	words	word	NOUN
ejpam-4919	6	13	and	and	CCONJ
ejpam-4919	6	14	phrases	phrase	NOUN
ejpam-4919	6	15	:	:	PUNCT
ejpam-4919	6	16	uniquely	uniquely	ADV
ejpam-4919	6	17	remotal	remotal	ADJ
ejpam-4919	6	18	sets	set	NOUN
ejpam-4919	6	19	;	;	PUNCT
ejpam-4919	6	20	farthest	farth	ADJ
ejpam-4919	6	21	points	point	NOUN
ejpam-4919	6	22	;	;	PUNCT
ejpam-4919	6	23	uniquely	uniquely	ADV
ejpam-4919	6	24	distant	distant	ADJ
ejpam-4919	6	25	sets	set	NOUN
ejpam-4919	6	26	.	.	PUNCT
ejpam-4919	7	1	1	1	X
ejpam-4919	7	2	.	.	X
ejpam-4919	7	3	introduction	introduction	NOUN
ejpam-4919	7	4	let	let	VERB
ejpam-4919	7	5	x	x	PRON
ejpam-4919	7	6	be	be	AUX
ejpam-4919	7	7	a	a	DET
ejpam-4919	7	8	normed	normed	ADJ
ejpam-4919	7	9	space	space	NOUN
ejpam-4919	7	10	,	,	PUNCT
ejpam-4919	7	11	and	and	CCONJ
ejpam-4919	7	12	e	e	NOUN
ejpam-4919	7	13	be	be	AUX
ejpam-4919	7	14	a	a	DET
ejpam-4919	7	15	closed	closed	ADJ
ejpam-4919	7	16	bounded	bounded	ADJ
ejpam-4919	7	17	subset	subset	NOUN
ejpam-4919	7	18	of	of	ADP
ejpam-4919	7	19	x.	x.	NOUN
ejpam-4919	7	20	define	define	VERB
ejpam-4919	7	21	the	the	DET
ejpam-4919	7	22	real	real	ADV
ejpam-4919	7	23	valued	value	VERB
ejpam-4919	7	24	function	function	NOUN
ejpam-4919	7	25	r	r	NOUN
ejpam-4919	7	26	(	(	PUNCT
ejpam-4919	7	27	.	.	NUM
ejpam-4919	7	28	,	,	PUNCT
ejpam-4919	7	29	e	e	X
ejpam-4919	7	30	)	)	PUNCT
ejpam-4919	7	31	:	:	PUNCT
ejpam-4919	7	32	x	x	X
ejpam-4919	7	33	→	→	SYM
ejpam-4919	7	34	r	r	NOUN
ejpam-4919	7	35	by	by	ADP
ejpam-4919	7	36	r(x	r(x	PROPN
ejpam-4919	7	37	,	,	PUNCT
ejpam-4919	7	38	e	e	NOUN
ejpam-4919	7	39	)	)	PUNCT
ejpam-4919	7	40	=	=	SYM
ejpam-4919	8	1	sup{∥x−	sup{∥x−	NUM
ejpam-4919	8	2	e∥	e∥	NOUN
ejpam-4919	8	3	:	:	PUNCT
ejpam-4919	8	4	e	e	X
ejpam-4919	8	5	∈	∈	PROPN
ejpam-4919	8	6	e	e	PROPN
ejpam-4919	8	7	}	}	PUNCT
ejpam-4919	8	8	,	,	PUNCT
ejpam-4919	8	9	which	which	PRON
ejpam-4919	8	10	is	be	AUX
ejpam-4919	8	11	known	know	VERB
ejpam-4919	8	12	as	as	ADP
ejpam-4919	8	13	the	the	DET
ejpam-4919	8	14	farthest	farth	ADJ
ejpam-4919	8	15	distance	distance	NOUN
ejpam-4919	8	16	function	function	NOUN
ejpam-4919	8	17	.	.	PUNCT
ejpam-4919	9	1	the	the	DET
ejpam-4919	9	2	set	set	PROPN
ejpam-4919	9	3	e	e	NOUN
ejpam-4919	9	4	is	be	AUX
ejpam-4919	9	5	said	say	VERB
ejpam-4919	9	6	to	to	PART
ejpam-4919	9	7	be	be	AUX
ejpam-4919	9	8	remotal	remotal	ADJ
ejpam-4919	9	9	if	if	SCONJ
ejpam-4919	9	10	for	for	ADP
ejpam-4919	9	11	every	every	DET
ejpam-4919	9	12	x	x	SYM
ejpam-4919	9	13	∈	∈	PROPN
ejpam-4919	9	14	x	x	NOUN
ejpam-4919	9	15	,	,	PUNCT
ejpam-4919	9	16	there	there	PRON
ejpam-4919	9	17	exists	exist	VERB
ejpam-4919	9	18	e	e	X
ejpam-4919	9	19	∈	∈	PROPN
ejpam-4919	9	20	e	e	NOUN
ejpam-4919	9	21	such	such	ADJ
ejpam-4919	9	22	that	that	SCONJ
ejpam-4919	9	23	r(x	r(x	PROPN
ejpam-4919	9	24	,	,	PUNCT
ejpam-4919	9	25	e	e	NOUN
ejpam-4919	9	26	)	)	PUNCT
ejpam-4919	9	27	=	=	PUNCT
ejpam-4919	10	1	∥x	∥x	PROPN
ejpam-4919	10	2	−	−	NOUN
ejpam-4919	11	1	e∥.	e∥.	NOUN
ejpam-4919	11	2	in	in	ADP
ejpam-4919	11	3	this	this	DET
ejpam-4919	11	4	case	case	NOUN
ejpam-4919	11	5	,	,	PUNCT
ejpam-4919	11	6	we	we	PRON
ejpam-4919	11	7	denote	denote	VERB
ejpam-4919	11	8	the	the	DET
ejpam-4919	11	9	set	set	NOUN
ejpam-4919	11	10	{	{	PUNCT
ejpam-4919	11	11	e	e	NOUN
ejpam-4919	11	12	∈	∈	PROPN
ejpam-4919	11	13	e	e	NOUN
ejpam-4919	11	14	:	:	PUNCT
ejpam-4919	11	15	r(x	r(x	PROPN
ejpam-4919	11	16	,	,	PUNCT
ejpam-4919	11	17	e	e	NOUN
ejpam-4919	11	18	)	)	PUNCT
ejpam-4919	11	19	=	=	PUNCT
ejpam-4919	12	1	∥x	∥x	PROPN
ejpam-4919	12	2	−	−	NUM
ejpam-4919	12	3	e∥	e∥	NOUN
ejpam-4919	12	4	}	}	PUNCT
ejpam-4919	12	5	by	by	ADP
ejpam-4919	12	6	f	f	PROPN
ejpam-4919	12	7	(	(	PUNCT
ejpam-4919	12	8	x	x	X
ejpam-4919	12	9	,	,	PUNCT
ejpam-4919	12	10	e	e	NOUN
ejpam-4919	12	11	)	)	PUNCT
ejpam-4919	12	12	.	.	PUNCT
ejpam-4919	13	1	it	it	PRON
ejpam-4919	13	2	is	be	AUX
ejpam-4919	13	3	clear	clear	ADJ
ejpam-4919	13	4	that	that	SCONJ
ejpam-4919	13	5	f	f	PROPN
ejpam-4919	13	6	(	(	PUNCT
ejpam-4919	13	7	.	.	NUM
ejpam-4919	13	8	,	,	PUNCT
ejpam-4919	13	9	e	e	X
ejpam-4919	13	10	)	)	PUNCT
ejpam-4919	13	11	:	:	PUNCT
ejpam-4919	13	12	x	x	X
ejpam-4919	13	13	→	→	PUNCT
ejpam-4919	13	14	e	e	X
ejpam-4919	13	15	is	be	AUX
ejpam-4919	13	16	a	a	DET
ejpam-4919	13	17	multi	multi	ADJ
ejpam-4919	13	18	-	-	ADJ
ejpam-4919	13	19	valued	value	VERB
ejpam-4919	13	20	function	function	NOUN
ejpam-4919	13	21	.	.	PUNCT
ejpam-4919	14	1	however	however	ADV
ejpam-4919	14	2	,	,	PUNCT
ejpam-4919	14	3	if	if	SCONJ
ejpam-4919	14	4	f	f	PROPN
ejpam-4919	14	5	(	(	PUNCT
ejpam-4919	14	6	.	.	NUM
ejpam-4919	14	7	,	,	PUNCT
ejpam-4919	14	8	e	e	X
ejpam-4919	14	9	)	)	PUNCT
ejpam-4919	14	10	:	:	PUNCT
ejpam-4919	14	11	x	x	X
ejpam-4919	14	12	→	→	PUNCT
ejpam-4919	14	13	e	e	X
ejpam-4919	14	14	is	be	AUX
ejpam-4919	14	15	a	a	DET
ejpam-4919	14	16	single	single	ADV
ejpam-4919	14	17	-	-	PUNCT
ejpam-4919	14	18	valued	value	VERB
ejpam-4919	14	19	function	function	NOUN
ejpam-4919	14	20	,	,	PUNCT
ejpam-4919	14	21	then	then	ADV
ejpam-4919	14	22	e	e	PROPN
ejpam-4919	14	23	is	be	AUX
ejpam-4919	14	24	called	call	VERB
ejpam-4919	14	25	uniquely	uniquely	ADV
ejpam-4919	14	26	remotal	remotal	ADJ
ejpam-4919	14	27	,	,	PUNCT
ejpam-4919	14	28	also	also	ADV
ejpam-4919	14	29	known	know	VERB
ejpam-4919	14	30	as	as	ADP
ejpam-4919	14	31	max	max	PROPN
ejpam-4919	14	32	-	-	PUNCT
ejpam-4919	14	33	chebyshev	chebyshev	PROPN
ejpam-4919	14	34	set	set	NOUN
ejpam-4919	14	35	.	.	PUNCT
ejpam-4919	15	1	in	in	ADP
ejpam-4919	15	2	this	this	DET
ejpam-4919	15	3	case	case	NOUN
ejpam-4919	15	4	,	,	PUNCT
ejpam-4919	15	5	we	we	PRON
ejpam-4919	15	6	denote	denote	VERB
ejpam-4919	15	7	f	f	PROPN
ejpam-4919	15	8	(	(	PUNCT
ejpam-4919	15	9	x	x	X
ejpam-4919	15	10	,	,	PUNCT
ejpam-4919	15	11	e	e	NOUN
ejpam-4919	15	12	)	)	PUNCT
ejpam-4919	15	13	by	by	ADP
ejpam-4919	15	14	f	f	PROPN
ejpam-4919	15	15	(	(	PUNCT
ejpam-4919	15	16	x	x	NOUN
ejpam-4919	15	17	)	)	PUNCT
ejpam-4919	15	18	,	,	PUNCT
ejpam-4919	15	19	if	if	SCONJ
ejpam-4919	15	20	no	no	DET
ejpam-4919	15	21	confusion	confusion	NOUN
ejpam-4919	15	22	arises	arise	VERB
ejpam-4919	15	23	.	.	PUNCT
ejpam-4919	16	1	the	the	DET
ejpam-4919	16	2	study	study	NOUN
ejpam-4919	16	3	of	of	ADP
ejpam-4919	16	4	remotal	remotal	ADJ
ejpam-4919	16	5	and	and	CCONJ
ejpam-4919	16	6	uniquely	uniquely	ADV
ejpam-4919	16	7	remotal	remotal	ADJ
ejpam-4919	16	8	sets	set	NOUN
ejpam-4919	16	9	has	have	AUX
ejpam-4919	16	10	attracted	attract	VERB
ejpam-4919	16	11	many	many	ADJ
ejpam-4919	16	12	mathematicians	mathematician	NOUN
ejpam-4919	16	13	in	in	ADP
ejpam-4919	16	14	the	the	DET
ejpam-4919	16	15	last	last	ADJ
ejpam-4919	16	16	decades	decade	NOUN
ejpam-4919	16	17	,	,	PUNCT
ejpam-4919	16	18	due	due	ADP
ejpam-4919	16	19	to	to	ADP
ejpam-4919	16	20	its	its	PRON
ejpam-4919	16	21	connection	connection	NOUN
ejpam-4919	16	22	to	to	ADP
ejpam-4919	16	23	the	the	DET
ejpam-4919	16	24	geometry	geometry	NOUN
ejpam-4919	16	25	of	of	ADP
ejpam-4919	16	26	banach	banach	NOUN
ejpam-4919	16	27	spaces	space	VERB
ejpam-4919	16	28	.	.	PUNCT
ejpam-4919	17	1	we	we	PRON
ejpam-4919	17	2	refer	refer	VERB
ejpam-4919	17	3	the	the	DET
ejpam-4919	17	4	reader	reader	NOUN
ejpam-4919	17	5	to	to	ADP
ejpam-4919	17	6	[	[	X
ejpam-4919	17	7	1	1	NUM
ejpam-4919	17	8	]	]	PUNCT
ejpam-4919	17	9	,	,	PUNCT
ejpam-4919	17	10	[	[	X
ejpam-4919	17	11	10	10	NUM
ejpam-4919	17	12	]	]	PUNCT
ejpam-4919	17	13	,	,	PUNCT
ejpam-4919	17	14	[	[	X
ejpam-4919	17	15	11	11	NUM
ejpam-4919	17	16	]	]	PUNCT
ejpam-4919	17	17	,	,	PUNCT
ejpam-4919	17	18	[	[	X
ejpam-4919	17	19	12	12	NUM
ejpam-4919	17	20	]	]	PUNCT
ejpam-4919	17	21	,	,	PUNCT
ejpam-4919	17	22	[	[	X
ejpam-4919	17	23	13	13	NUM
ejpam-4919	17	24	]	]	PUNCT
ejpam-4919	17	25	,	,	PUNCT
ejpam-4919	17	26	[	[	X
ejpam-4919	17	27	15	15	NUM
ejpam-4919	17	28	]	]	PUNCT
ejpam-4919	17	29	,	,	PUNCT
ejpam-4919	17	30	[	[	X
ejpam-4919	17	31	8	8	NUM
ejpam-4919	17	32	]	]	PUNCT
ejpam-4919	17	33	,	,	PUNCT
ejpam-4919	17	34	[	[	X
ejpam-4919	17	35	2	2	NUM
ejpam-4919	17	36	]	]	PUNCT
ejpam-4919	17	37	and	and	CCONJ
ejpam-4919	17	38	[	[	X
ejpam-4919	17	39	9	9	NUM
ejpam-4919	17	40	]	]	PUNCT
ejpam-4919	17	41	for	for	ADP
ejpam-4919	17	42	samples	sample	NOUN
ejpam-4919	17	43	of	of	ADP
ejpam-4919	17	44	these	these	DET
ejpam-4919	17	45	studies	study	NOUN
ejpam-4919	17	46	.	.	PUNCT
ejpam-4919	18	1	however	however	ADV
ejpam-4919	18	2	,	,	PUNCT
ejpam-4919	18	3	uniquely	uniquely	ADV
ejpam-4919	18	4	remotal	remotal	ADJ
ejpam-4919	18	5	sets	set	NOUN
ejpam-4919	18	6	are	be	AUX
ejpam-4919	18	7	of	of	ADP
ejpam-4919	18	8	special	special	ADJ
ejpam-4919	18	9	interest	interest	NOUN
ejpam-4919	18	10	.	.	PUNCT
ejpam-4919	19	1	∗corresponding	∗corresponde	VERB
ejpam-4919	19	2	author	author	NOUN
ejpam-4919	19	3	.	.	PUNCT
ejpam-4919	20	1	doi	doi	NOUN
ejpam-4919	20	2	:	:	PUNCT
ejpam-4919	20	3	https://doi.org/10.29020/nybg.ejpam.v16i4.4919	https://doi.org/10.29020/nybg.ejpam.v16i4.4919	ADP
ejpam-4919	20	4	email	email	NOUN
ejpam-4919	20	5	addresses	address	VERB
ejpam-4919	20	6	:	:	PUNCT
ejpam-4919	20	7	abd.yousef@ju.edu.jo	abd.yousef@ju.edu.jo	NUM
ejpam-4919	20	8	;	;	PUNCT
ejpam-4919	20	9	afyousef@aus.edu	afyousef@aus.edu	NOUN
ejpam-4919	20	10	(	(	PUNCT
ejpam-4919	20	11	a.	a.	PROPN
ejpam-4919	20	12	yousef	yousef	PROPN
ejpam-4919	20	13	)	)	PUNCT
ejpam-4919	20	14	,	,	PUNCT
ejpam-4919	21	1	roshdi@ju.edu.jo	roshdi@ju.edu.jo	PROPN
ejpam-4919	21	2	(	(	PUNCT
ejpam-4919	21	3	r.	r.	PROPN
ejpam-4919	21	4	khalil	khalil	PROPN
ejpam-4919	21	5	)	)	PUNCT
ejpam-4919	21	6	,	,	PUNCT
ejpam-4919	21	7	a.tallafha@ju.edu.jo	a.tallafha@ju.edu.jo	PROPN
ejpam-4919	21	8	(	(	PUNCT
ejpam-4919	21	9	a.	a.	NOUN
ejpam-4919	21	10	talafha	talafha	PROPN
ejpam-4919	21	11	)	)	PUNCT
ejpam-4919	21	12	,	,	PUNCT
ejpam-4919	21	13	ba-mutabaganial-madani@wiu.edu	ba-mutabaganial-madani@wiu.edu	PROPN
ejpam-4919	21	14	(	(	PUNCT
ejpam-4919	21	15	b.	b.	PROPN
ejpam-4919	21	16	mutabagani	mutabagani	PROPN
ejpam-4919	21	17	)	)	PUNCT
ejpam-4919	21	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4919	21	19	2397	2397	NUM
ejpam-4919	22	1	©	©	PROPN
ejpam-4919	22	2	2023	2023	NUM
ejpam-4919	22	3	ejpam	ejpam	NOUN
ejpam-4919	22	4	all	all	DET
ejpam-4919	22	5	rights	right	NOUN
ejpam-4919	22	6	reserved	reserve	VERB
ejpam-4919	22	7	.	.	PUNCT
ejpam-4919	23	1	a.	a.	PROPN
ejpam-4919	23	2	yousef	yousef	PROPN
ejpam-4919	23	3	et	et	PROPN
ejpam-4919	23	4	al	al	PROPN
ejpam-4919	23	5	.	.	PUNCT
ejpam-4919	23	6	/	/	SYM
ejpam-4919	23	7	eur	eur	PROPN
ejpam-4919	23	8	.	.	PUNCT
ejpam-4919	24	1	j.	j.	PROPN
ejpam-4919	24	2	pure	pure	PROPN
ejpam-4919	24	3	appl	appl	PROPN
ejpam-4919	24	4	.	.	PROPN
ejpam-4919	24	5	math	math	PROPN
ejpam-4919	24	6	,	,	PUNCT
ejpam-4919	24	7	16	16	NUM
ejpam-4919	24	8	(	(	PUNCT
ejpam-4919	24	9	4	4	NUM
ejpam-4919	24	10	)	)	PUNCT
ejpam-4919	24	11	(	(	PUNCT
ejpam-4919	24	12	2023	2023	NUM
ejpam-4919	24	13	)	)	PUNCT
ejpam-4919	24	14	,	,	PUNCT
ejpam-4919	24	15	2397	2397	NUM
ejpam-4919	24	16	-	-	SYM
ejpam-4919	24	17	2404	2404	NUM
ejpam-4919	24	18	2398	2398	NUM
ejpam-4919	24	19	one	one	NUM
ejpam-4919	24	20	of	of	ADP
ejpam-4919	24	21	the	the	DET
ejpam-4919	24	22	most	most	ADV
ejpam-4919	24	23	interesting	interesting	ADJ
ejpam-4919	24	24	and	and	CCONJ
ejpam-4919	24	25	challenging	challenging	ADJ
ejpam-4919	24	26	problems	problem	NOUN
ejpam-4919	24	27	in	in	ADP
ejpam-4919	24	28	this	this	DET
ejpam-4919	24	29	field	field	NOUN
ejpam-4919	24	30	,	,	PUNCT
ejpam-4919	24	31	is	be	AUX
ejpam-4919	24	32	known	know	VERB
ejpam-4919	24	33	as	as	ADP
ejpam-4919	24	34	the	the	DET
ejpam-4919	24	35	farthest	farth	ADJ
ejpam-4919	24	36	point	point	NOUN
ejpam-4919	24	37	problem	problem	NOUN
ejpam-4919	24	38	,	,	PUNCT
ejpam-4919	24	39	which	which	PRON
ejpam-4919	24	40	is	be	AUX
ejpam-4919	24	41	stated	state	VERB
ejpam-4919	24	42	as	as	ADP
ejpam-4919	24	43	:	:	PUNCT
ejpam-4919	24	44	q	q	X
ejpam-4919	24	45	)	)	PUNCT
ejpam-4919	24	46	if	if	SCONJ
ejpam-4919	24	47	e	e	NOUN
ejpam-4919	24	48	is	be	AUX
ejpam-4919	24	49	a	a	DET
ejpam-4919	24	50	uniquely	uniquely	ADV
ejpam-4919	24	51	remotal	remotal	ADJ
ejpam-4919	24	52	set	set	NOUN
ejpam-4919	24	53	in	in	ADP
ejpam-4919	24	54	a	a	DET
ejpam-4919	24	55	normed	normed	ADJ
ejpam-4919	24	56	space	space	NOUN
ejpam-4919	24	57	x	x	NOUN
ejpam-4919	24	58	,	,	PUNCT
ejpam-4919	24	59	does	do	AUX
ejpam-4919	24	60	it	it	PRON
ejpam-4919	24	61	follow	follow	VERB
ejpam-4919	24	62	that	that	SCONJ
ejpam-4919	24	63	e	e	NOUN
ejpam-4919	24	64	is	be	AUX
ejpam-4919	24	65	a	a	DET
ejpam-4919	24	66	singleton	singleton	NOUN
ejpam-4919	24	67	?	?	PUNCT
ejpam-4919	25	1	this	this	DET
ejpam-4919	25	2	problem	problem	NOUN
ejpam-4919	25	3	became	become	VERB
ejpam-4919	25	4	more	more	ADV
ejpam-4919	25	5	important	important	ADJ
ejpam-4919	25	6	when	when	SCONJ
ejpam-4919	25	7	klee	klee	PROPN
ejpam-4919	25	8	[	[	X
ejpam-4919	25	9	14	14	NUM
ejpam-4919	25	10	]	]	PUNCT
ejpam-4919	25	11	proved	prove	VERB
ejpam-4919	25	12	that	that	SCONJ
ejpam-4919	25	13	singletoness	singletoness	NOUN
ejpam-4919	25	14	of	of	ADP
ejpam-4919	25	15	closed	closed	ADJ
ejpam-4919	25	16	uniquely	uniquely	ADV
ejpam-4919	25	17	remotal	remotal	ADJ
ejpam-4919	25	18	sets	set	NOUN
ejpam-4919	25	19	is	be	AUX
ejpam-4919	25	20	equivalent	equivalent	ADJ
ejpam-4919	25	21	to	to	ADP
ejpam-4919	25	22	convexity	convexity	NOUN
ejpam-4919	25	23	of	of	ADP
ejpam-4919	25	24	chebyshev	chebyshev	NOUN
ejpam-4919	25	25	sets	set	NOUN
ejpam-4919	25	26	in	in	ADP
ejpam-4919	25	27	hilbert	hilbert	NOUN
ejpam-4919	25	28	spaces	space	NOUN
ejpam-4919	25	29	(	(	PUNCT
ejpam-4919	25	30	which	which	PRON
ejpam-4919	25	31	is	be	AUX
ejpam-4919	25	32	an	an	DET
ejpam-4919	25	33	open	open	ADJ
ejpam-4919	25	34	problem	problem	NOUN
ejpam-4919	25	35	too	too	ADV
ejpam-4919	25	36	,	,	PUNCT
ejpam-4919	25	37	in	in	ADP
ejpam-4919	25	38	the	the	DET
ejpam-4919	25	39	theory	theory	NOUN
ejpam-4919	25	40	of	of	ADP
ejpam-4919	25	41	nearest	near	ADJ
ejpam-4919	25	42	points	point	NOUN
ejpam-4919	25	43	)	)	PUNCT
ejpam-4919	25	44	.	.	PUNCT
ejpam-4919	26	1	since	since	SCONJ
ejpam-4919	26	2	then	then	ADV
ejpam-4919	26	3	,	,	PUNCT
ejpam-4919	26	4	a	a	DET
ejpam-4919	26	5	considerable	considerable	ADJ
ejpam-4919	26	6	work	work	NOUN
ejpam-4919	26	7	has	have	AUX
ejpam-4919	26	8	been	be	AUX
ejpam-4919	26	9	done	do	VERB
ejpam-4919	26	10	to	to	PART
ejpam-4919	26	11	answer	answer	VERB
ejpam-4919	26	12	this	this	DET
ejpam-4919	26	13	question	question	NOUN
ejpam-4919	26	14	,	,	PUNCT
ejpam-4919	26	15	and	and	CCONJ
ejpam-4919	26	16	many	many	ADJ
ejpam-4919	26	17	partial	partial	ADJ
ejpam-4919	26	18	results	result	NOUN
ejpam-4919	26	19	have	have	AUX
ejpam-4919	26	20	been	be	AUX
ejpam-4919	26	21	obtained	obtain	VERB
ejpam-4919	26	22	in	in	ADP
ejpam-4919	26	23	the	the	DET
ejpam-4919	26	24	positive	positive	ADJ
ejpam-4919	26	25	direction	direction	NOUN
ejpam-4919	26	26	.	.	PUNCT
ejpam-4919	27	1	an	an	DET
ejpam-4919	27	2	element	element	NOUN
ejpam-4919	27	3	c	c	NOUN
ejpam-4919	27	4	in	in	ADP
ejpam-4919	27	5	a	a	DET
ejpam-4919	27	6	normed	normed	ADJ
ejpam-4919	27	7	space	space	NOUN
ejpam-4919	27	8	x	x	PUNCT
ejpam-4919	27	9	is	be	AUX
ejpam-4919	27	10	called	call	VERB
ejpam-4919	27	11	a	a	DET
ejpam-4919	27	12	chebyshev	chebyshev	NOUN
ejpam-4919	27	13	center	center	NOUN
ejpam-4919	27	14	of	of	ADP
ejpam-4919	27	15	e	e	PROPN
ejpam-4919	27	16	⊂	⊂	PROPN
ejpam-4919	27	17	x	x	PUNCT
ejpam-4919	27	18	if	if	SCONJ
ejpam-4919	27	19	r(c	r(c	ADJ
ejpam-4919	27	20	,	,	PUNCT
ejpam-4919	27	21	e	e	NOUN
ejpam-4919	27	22	)	)	PUNCT
ejpam-4919	27	23	=	=	SYM
ejpam-4919	27	24	inf	inf	NOUN
ejpam-4919	27	25	x∈x	x∈x	PROPN
ejpam-4919	27	26	r(x	r(x	PROPN
ejpam-4919	27	27	,	,	PUNCT
ejpam-4919	27	28	e	e	NOUN
ejpam-4919	27	29	)	)	PUNCT
ejpam-4919	27	30	.	.	PUNCT
ejpam-4919	28	1	whether	whether	SCONJ
ejpam-4919	28	2	a	a	DET
ejpam-4919	28	3	set	set	NOUN
ejpam-4919	28	4	has	have	VERB
ejpam-4919	28	5	a	a	DET
ejpam-4919	28	6	chebyshev	chebyshev	NOUN
ejpam-4919	28	7	center	center	NOUN
ejpam-4919	28	8	or	or	CCONJ
ejpam-4919	28	9	not	not	PART
ejpam-4919	28	10	is	be	AUX
ejpam-4919	28	11	still	still	ADV
ejpam-4919	28	12	an	an	DET
ejpam-4919	28	13	open	open	ADJ
ejpam-4919	28	14	question	question	NOUN
ejpam-4919	28	15	.	.	PUNCT
ejpam-4919	29	1	however	however	ADV
ejpam-4919	29	2	,	,	PUNCT
ejpam-4919	29	3	in	in	ADP
ejpam-4919	29	4	inner	inner	ADJ
ejpam-4919	29	5	product	product	NOUN
ejpam-4919	29	6	spaces	space	NOUN
ejpam-4919	29	7	,	,	PUNCT
ejpam-4919	29	8	any	any	DET
ejpam-4919	29	9	closed	closed	ADJ
ejpam-4919	29	10	bounded	bounded	ADJ
ejpam-4919	29	11	set	set	NOUN
ejpam-4919	29	12	does	do	AUX
ejpam-4919	29	13	have	have	VERB
ejpam-4919	29	14	a	a	DET
ejpam-4919	29	15	chebyshev	chebyshev	NOUN
ejpam-4919	29	16	center	center	NOUN
ejpam-4919	29	17	[	[	X
ejpam-4919	29	18	7	7	NUM
ejpam-4919	29	19	]	]	PUNCT
ejpam-4919	29	20	.	.	PUNCT
ejpam-4919	30	1	chebyshev	chebyshev	PROPN
ejpam-4919	30	2	centers	center	NOUN
ejpam-4919	30	3	of	of	ADP
ejpam-4919	30	4	sets	set	NOUN
ejpam-4919	30	5	have	have	AUX
ejpam-4919	30	6	played	play	VERB
ejpam-4919	30	7	a	a	DET
ejpam-4919	30	8	major	major	ADJ
ejpam-4919	30	9	role	role	NOUN
ejpam-4919	30	10	in	in	ADP
ejpam-4919	30	11	the	the	DET
ejpam-4919	30	12	study	study	NOUN
ejpam-4919	30	13	of	of	ADP
ejpam-4919	30	14	uniquely	uniquely	ADV
ejpam-4919	30	15	remotal	remotal	ADJ
ejpam-4919	30	16	sets	set	NOUN
ejpam-4919	30	17	.	.	PUNCT
ejpam-4919	31	1	we	we	PRON
ejpam-4919	31	2	refer	refer	VERB
ejpam-4919	31	3	the	the	DET
ejpam-4919	31	4	reader	reader	NOUN
ejpam-4919	31	5	to	to	ADP
ejpam-4919	31	6	[	[	X
ejpam-4919	31	7	4	4	NUM
ejpam-4919	31	8	]	]	PUNCT
ejpam-4919	31	9	,	,	PUNCT
ejpam-4919	31	10	[	[	X
ejpam-4919	31	11	5	5	NUM
ejpam-4919	31	12	]	]	PUNCT
ejpam-4919	31	13	,	,	PUNCT
ejpam-4919	31	14	[	[	X
ejpam-4919	31	15	3	3	NUM
ejpam-4919	31	16	]	]	PUNCT
ejpam-4919	31	17	and[6	and[6	NUM
ejpam-4919	31	18	]	]	PUNCT
ejpam-4919	31	19	.	.	PUNCT
ejpam-4919	32	1	for	for	ADP
ejpam-4919	32	2	x	x	SYM
ejpam-4919	32	3	,	,	PUNCT
ejpam-4919	32	4	y	y	PROPN
ejpam-4919	32	5	∈	∈	PROPN
ejpam-4919	32	6	x	x	X
ejpam-4919	32	7	,	,	PUNCT
ejpam-4919	32	8	[	[	X
ejpam-4919	32	9	x	x	X
ejpam-4919	32	10	,	,	PUNCT
ejpam-4919	32	11	y	y	PROPN
ejpam-4919	32	12	]	]	X
ejpam-4919	32	13	is	be	AUX
ejpam-4919	32	14	the	the	DET
ejpam-4919	32	15	line	line	NOUN
ejpam-4919	32	16	segment	segment	NOUN
ejpam-4919	32	17	joining	join	VERB
ejpam-4919	32	18	x	x	PUNCT
ejpam-4919	32	19	and	and	CCONJ
ejpam-4919	32	20	y.	y.	NOUN
ejpam-4919	32	21	in	in	ADP
ejpam-4919	32	22	[	[	X
ejpam-4919	32	23	16	16	NUM
ejpam-4919	32	24	]	]	X
ejpam-4919	32	25	it	it	PRON
ejpam-4919	32	26	was	be	AUX
ejpam-4919	32	27	proved	prove	VERB
ejpam-4919	32	28	that	that	SCONJ
ejpam-4919	32	29	if	if	SCONJ
ejpam-4919	32	30	e	e	NOUN
ejpam-4919	32	31	is	be	AUX
ejpam-4919	32	32	a	a	DET
ejpam-4919	32	33	uniquely	uniquely	ADV
ejpam-4919	32	34	remotal	remotal	ADJ
ejpam-4919	32	35	subset	subset	NOUN
ejpam-4919	32	36	of	of	ADP
ejpam-4919	32	37	a	a	DET
ejpam-4919	32	38	normed	normed	ADJ
ejpam-4919	32	39	space	space	NOUN
ejpam-4919	32	40	,	,	PUNCT
ejpam-4919	32	41	admitting	admit	VERB
ejpam-4919	32	42	a	a	DET
ejpam-4919	32	43	chebyshev	chebyshev	NOUN
ejpam-4919	32	44	center	center	NOUN
ejpam-4919	32	45	c	c	NOUN
ejpam-4919	32	46	,	,	PUNCT
ejpam-4919	32	47	and	and	CCONJ
ejpam-4919	32	48	if	if	SCONJ
ejpam-4919	32	49	f	f	PROPN
ejpam-4919	32	50	,	,	PUNCT
ejpam-4919	32	51	restricted	restrict	VERB
ejpam-4919	32	52	to	to	ADP
ejpam-4919	32	53	the	the	DET
ejpam-4919	32	54	line	line	NOUN
ejpam-4919	32	55	segment	segment	NOUN
ejpam-4919	33	1	[	[	X
ejpam-4919	33	2	c	c	X
ejpam-4919	33	3	,	,	PUNCT
ejpam-4919	33	4	f	f	PROPN
ejpam-4919	33	5	(	(	PUNCT
ejpam-4919	33	6	c	c	NOUN
ejpam-4919	33	7	)	)	PUNCT
ejpam-4919	33	8	]	]	PUNCT
ejpam-4919	33	9	is	be	AUX
ejpam-4919	33	10	continuous	continuous	ADJ
ejpam-4919	33	11	at	at	ADP
ejpam-4919	33	12	c	c	NOUN
ejpam-4919	33	13	,	,	PUNCT
ejpam-4919	33	14	then	then	ADV
ejpam-4919	33	15	e	e	PROPN
ejpam-4919	33	16	is	be	AUX
ejpam-4919	33	17	a	a	DET
ejpam-4919	33	18	singleton	singleton	NOUN
ejpam-4919	33	19	.	.	PUNCT
ejpam-4919	34	1	partially	partially	ADV
ejpam-4919	34	2	continuity	continuity	NOUN
ejpam-4919	34	3	was	be	AUX
ejpam-4919	34	4	introduced	introduce	VERB
ejpam-4919	34	5	in	in	ADP
ejpam-4919	34	6	[	[	X
ejpam-4919	34	7	17	17	NUM
ejpam-4919	34	8	]	]	PUNCT
ejpam-4919	34	9	as	as	SCONJ
ejpam-4919	34	10	follows	follow	VERB
ejpam-4919	34	11	:	:	PUNCT
ejpam-4919	34	12	let	let	VERB
ejpam-4919	34	13	g	g	NOUN
ejpam-4919	34	14	:	:	PUNCT
ejpam-4919	34	15	a	a	DET
ejpam-4919	34	16	⊂	⊂	X
ejpam-4919	34	17	x	x	PUNCT
ejpam-4919	34	18	→	→	PUNCT
ejpam-4919	34	19	x	x	PUNCT
ejpam-4919	34	20	be	be	AUX
ejpam-4919	34	21	a	a	DET
ejpam-4919	34	22	function	function	NOUN
ejpam-4919	34	23	,	,	PUNCT
ejpam-4919	34	24	and	and	CCONJ
ejpam-4919	34	25	let	let	VERB
ejpam-4919	34	26	a	a	DET
ejpam-4919	34	27	∈	∈	NOUN
ejpam-4919	34	28	a.	a.	NOUN
ejpam-4919	34	29	then	then	ADV
ejpam-4919	34	30	g	g	PROPN
ejpam-4919	34	31	is	be	AUX
ejpam-4919	34	32	partially	partially	ADV
ejpam-4919	34	33	continuous	continuous	ADJ
ejpam-4919	34	34	at	at	ADP
ejpam-4919	34	35	a	a	PRON
ejpam-4919	34	36	if	if	SCONJ
ejpam-4919	34	37	there	there	PRON
ejpam-4919	34	38	exists	exist	VERB
ejpam-4919	34	39	a	a	DET
ejpam-4919	34	40	nonconstant	nonconstant	ADJ
ejpam-4919	34	41	sequence	sequence	NOUN
ejpam-4919	34	42	(	(	PUNCT
ejpam-4919	34	43	an	an	PROPN
ejpam-4919	34	44	)	)	PUNCT
ejpam-4919	34	45	⊂	⊂	PROPN
ejpam-4919	34	46	a	a	X
ejpam-4919	34	47	,	,	PUNCT
ejpam-4919	34	48	such	such	ADJ
ejpam-4919	34	49	that	that	SCONJ
ejpam-4919	34	50	an	an	DET
ejpam-4919	34	51	→	→	SYM
ejpam-4919	34	52	a	a	PRON
ejpam-4919	34	53	and	and	CCONJ
ejpam-4919	34	54	g(an	g(an	PROPN
ejpam-4919	34	55	)	)	PUNCT
ejpam-4919	34	56	→	→	SYM
ejpam-4919	34	57	g(a	g(a	PROPN
ejpam-4919	34	58	)	)	PUNCT
ejpam-4919	34	59	.	.	PUNCT
ejpam-4919	35	1	in	in	ADP
ejpam-4919	35	2	[	[	X
ejpam-4919	35	3	17	17	NUM
ejpam-4919	35	4	]	]	PUNCT
ejpam-4919	35	5	,	,	PUNCT
ejpam-4919	35	6	the	the	DET
ejpam-4919	35	7	following	following	ADJ
ejpam-4919	35	8	result	result	NOUN
ejpam-4919	35	9	was	be	AUX
ejpam-4919	35	10	proved	prove	VERB
ejpam-4919	35	11	:	:	PUNCT
ejpam-4919	35	12	theorem	theorem	NOUN
ejpam-4919	35	13	1	1	NUM
ejpam-4919	35	14	.	.	PUNCT
ejpam-4919	36	1	let	let	VERB
ejpam-4919	36	2	e	e	PRON
ejpam-4919	36	3	be	be	AUX
ejpam-4919	36	4	a	a	DET
ejpam-4919	36	5	closed	closed	ADJ
ejpam-4919	36	6	subset	subset	NOUN
ejpam-4919	36	7	of	of	ADP
ejpam-4919	36	8	a	a	DET
ejpam-4919	36	9	normed	normed	ADJ
ejpam-4919	36	10	space	space	NOUN
ejpam-4919	36	11	x	x	NOUN
ejpam-4919	36	12	,	,	PUNCT
ejpam-4919	36	13	admitting	admit	VERB
ejpam-4919	36	14	a	a	DET
ejpam-4919	36	15	chebyshev	chebyshev	NOUN
ejpam-4919	36	16	center	center	NOUN
ejpam-4919	36	17	c.	c.	NOUN
ejpam-4919	36	18	if	if	SCONJ
ejpam-4919	36	19	e	e	NOUN
ejpam-4919	36	20	is	be	AUX
ejpam-4919	36	21	uniquely	uniquely	ADV
ejpam-4919	36	22	remotal	remotal	ADJ
ejpam-4919	36	23	,	,	PUNCT
ejpam-4919	36	24	and	and	CCONJ
ejpam-4919	36	25	if	if	SCONJ
ejpam-4919	36	26	f	f	X
ejpam-4919	36	27	:	:	PUNCT
ejpam-4919	37	1	[	[	X
ejpam-4919	37	2	c	c	X
ejpam-4919	37	3	,	,	PUNCT
ejpam-4919	37	4	f	f	PROPN
ejpam-4919	37	5	(	(	PUNCT
ejpam-4919	37	6	c	c	NOUN
ejpam-4919	37	7	)	)	PUNCT
ejpam-4919	37	8	]	]	PUNCT
ejpam-4919	37	9	→	→	PUNCT
ejpam-4919	37	10	e	e	X
ejpam-4919	37	11	is	be	AUX
ejpam-4919	37	12	partially	partially	ADV
ejpam-4919	37	13	continuous	continuous	ADJ
ejpam-4919	37	14	at	at	ADP
ejpam-4919	37	15	c	c	NOUN
ejpam-4919	37	16	,	,	PUNCT
ejpam-4919	37	17	then	then	ADV
ejpam-4919	37	18	e	e	PROPN
ejpam-4919	37	19	is	be	AUX
ejpam-4919	37	20	a	a	DET
ejpam-4919	37	21	singleton	singleton	NOUN
ejpam-4919	37	22	.	.	PUNCT
ejpam-4919	38	1	one	one	NUM
ejpam-4919	38	2	of	of	ADP
ejpam-4919	38	3	the	the	DET
ejpam-4919	38	4	important	important	ADJ
ejpam-4919	38	5	notions	notion	NOUN
ejpam-4919	38	6	that	that	PRON
ejpam-4919	38	7	appeared	appear	VERB
ejpam-4919	38	8	in	in	ADP
ejpam-4919	38	9	the	the	DET
ejpam-4919	38	10	literature	literature	NOUN
ejpam-4919	38	11	while	while	SCONJ
ejpam-4919	38	12	studying	study	VERB
ejpam-4919	38	13	remotal	remotal	ADJ
ejpam-4919	38	14	and	and	CCONJ
ejpam-4919	38	15	uniquely	uniquely	ADV
ejpam-4919	38	16	remotal	remotal	ADJ
ejpam-4919	38	17	sets	set	NOUN
ejpam-4919	38	18	is	be	AUX
ejpam-4919	38	19	the	the	DET
ejpam-4919	38	20	so	so	ADV
ejpam-4919	38	21	-	-	PUNCT
ejpam-4919	38	22	called	call	VERB
ejpam-4919	38	23	storngly	storngly	ADV
ejpam-4919	38	24	remotal	remotal	ADJ
ejpam-4919	38	25	set	set	NOUN
ejpam-4919	38	26	.	.	PUNCT
ejpam-4919	39	1	it	it	PRON
ejpam-4919	39	2	was	be	AUX
ejpam-4919	39	3	first	first	ADV
ejpam-4919	39	4	introduced	introduce	VERB
ejpam-4919	39	5	by	by	ADP
ejpam-4919	39	6	kalil	kalil	PROPN
ejpam-4919	39	7	et	et	PROPN
ejpam-4919	39	8	al	al	PROPN
ejpam-4919	39	9	in	in	ADP
ejpam-4919	39	10	[	[	X
ejpam-4919	39	11	8	8	NUM
ejpam-4919	39	12	]	]	PUNCT
ejpam-4919	39	13	for	for	ADP
ejpam-4919	39	14	general	general	ADJ
ejpam-4919	39	15	banach	banach	NOUN
ejpam-4919	39	16	spaces	space	NOUN
ejpam-4919	39	17	as	as	SCONJ
ejpam-4919	39	18	follows	follow	VERB
ejpam-4919	39	19	:	:	PUNCT
ejpam-4919	39	20	let	let	VERB
ejpam-4919	39	21	x	x	PRON
ejpam-4919	39	22	be	be	AUX
ejpam-4919	39	23	a	a	DET
ejpam-4919	39	24	banach	banach	NOUN
ejpam-4919	39	25	space	space	NOUN
ejpam-4919	39	26	and	and	CCONJ
ejpam-4919	39	27	e	e	NOUN
ejpam-4919	39	28	be	be	AUX
ejpam-4919	39	29	a	a	DET
ejpam-4919	39	30	nonempty	nonempty	ADV
ejpam-4919	39	31	closed	close	VERB
ejpam-4919	39	32	convex	convex	NOUN
ejpam-4919	39	33	bounded	bound	VERB
ejpam-4919	39	34	subset	subset	NOUN
ejpam-4919	39	35	of	of	ADP
ejpam-4919	39	36	x.	x.	PROPN
ejpam-4919	39	37	let	let	VERB
ejpam-4919	39	38	m	m	VERB
ejpam-4919	39	39	=	=	PRON
ejpam-4919	39	40	{	{	PUNCT
ejpam-4919	39	41	ϕ	ϕ	NOUN
ejpam-4919	39	42	:	:	PUNCT
ejpam-4919	39	43	ϕ	ϕ	NOUN
ejpam-4919	39	44	:	:	PUNCT
ejpam-4919	40	1	[	[	X
ejpam-4919	40	2	0,∞	0,∞	NOUN
ejpam-4919	40	3	)	)	PUNCT
ejpam-4919	40	4	→	→	PUNCT
ejpam-4919	41	1	[	[	X
ejpam-4919	41	2	0,∞	0,∞	X
ejpam-4919	41	3	)	)	PUNCT
ejpam-4919	41	4	be	be	AUX
ejpam-4919	41	5	strictly	strictly	ADV
ejpam-4919	41	6	increasing	increase	VERB
ejpam-4919	41	7	function	function	NOUN
ejpam-4919	41	8	,	,	PUNCT
ejpam-4919	41	9	ϕ(0	ϕ(0	PROPN
ejpam-4919	41	10	)	)	PUNCT
ejpam-4919	41	11	=	=	PUNCT
ejpam-4919	42	1	0	0	NUM
ejpam-4919	42	2	}	}	PUNCT
ejpam-4919	42	3	,	,	PUNCT
ejpam-4919	42	4	and	and	CCONJ
ejpam-4919	42	5	n	n	CCONJ
ejpam-4919	42	6	=	=	NOUN
ejpam-4919	42	7	{	{	PUNCT
ejpam-4919	42	8	ψ	ψ	X
ejpam-4919	42	9	:	:	PUNCT
ejpam-4919	42	10	ψ	ψ	X
ejpam-4919	42	11	:	:	PUNCT
ejpam-4919	42	12	x	x	SYM
ejpam-4919	42	13	→	→	SYM
ejpam-4919	42	14	(	(	PUNCT
ejpam-4919	42	15	0	0	NUM
ejpam-4919	42	16	,	,	PUNCT
ejpam-4919	42	17	1	1	NUM
ejpam-4919	42	18	)	)	PUNCT
ejpam-4919	42	19	such	such	ADJ
ejpam-4919	42	20	that	that	DET
ejpam-4919	42	21	ψ(x	ψ(x	NOUN
ejpam-4919	42	22	)	)	PUNCT
ejpam-4919	42	23	≤	≤	NUM
ejpam-4919	42	24	ψ(y	ψ(y	NOUN
ejpam-4919	42	25	)	)	PUNCT
ejpam-4919	42	26	whenever	whenever	SCONJ
ejpam-4919	42	27	||x||	||x||	ADV
ejpam-4919	42	28	<	<	X
ejpam-4919	42	29	||y||	||y||	X
ejpam-4919	42	30	}	}	PUNCT
ejpam-4919	42	31	.	.	PUNCT
ejpam-4919	43	1	e	e	NOUN
ejpam-4919	43	2	is	be	AUX
ejpam-4919	43	3	called	call	VERB
ejpam-4919	43	4	stongly	stongly	ADV
ejpam-4919	43	5	remotal	remotal	ADJ
ejpam-4919	43	6	in	in	ADP
ejpam-4919	43	7	x	x	SYM
ejpam-4919	43	8	if	if	SCONJ
ejpam-4919	43	9	there	there	PRON
ejpam-4919	43	10	exist	exist	VERB
ejpam-4919	43	11	ϕ	ϕ	PRON
ejpam-4919	43	12	∈	∈	PROPN
ejpam-4919	43	13	m	m	NOUN
ejpam-4919	43	14	and	and	CCONJ
ejpam-4919	43	15	ψ	ψ	X
ejpam-4919	43	16	∈	∈	NOUN
ejpam-4919	43	17	n	n	CCONJ
ejpam-4919	43	18	,	,	PUNCT
ejpam-4919	43	19	with	with	ADP
ejpam-4919	43	20	inf	inf	PROPN
ejpam-4919	43	21	y∈x	y∈x	PROPN
ejpam-4919	43	22	ψ(y	ψ(y	NOUN
ejpam-4919	43	23	)	)	PUNCT
ejpam-4919	43	24	>	>	X
ejpam-4919	43	25	0	0	NUM
ejpam-4919	43	26	,	,	PUNCT
ejpam-4919	43	27	such	such	ADJ
ejpam-4919	43	28	that	that	PRON
ejpam-4919	43	29	for	for	ADP
ejpam-4919	43	30	a.	a.	PROPN
ejpam-4919	43	31	yousef	yousef	PROPN
ejpam-4919	43	32	et	et	PROPN
ejpam-4919	43	33	al	al	PROPN
ejpam-4919	43	34	.	.	PUNCT
ejpam-4919	43	35	/	/	SYM
ejpam-4919	43	36	eur	eur	PROPN
ejpam-4919	43	37	.	.	PUNCT
ejpam-4919	44	1	j.	j.	PROPN
ejpam-4919	44	2	pure	pure	PROPN
ejpam-4919	44	3	appl	appl	PROPN
ejpam-4919	44	4	.	.	PROPN
ejpam-4919	44	5	math	math	PROPN
ejpam-4919	44	6	,	,	PUNCT
ejpam-4919	44	7	16	16	NUM
ejpam-4919	44	8	(	(	PUNCT
ejpam-4919	44	9	4	4	NUM
ejpam-4919	44	10	)	)	PUNCT
ejpam-4919	44	11	(	(	PUNCT
ejpam-4919	44	12	2023	2023	NUM
ejpam-4919	44	13	)	)	PUNCT
ejpam-4919	44	14	,	,	PUNCT
ejpam-4919	44	15	2397	2397	NUM
ejpam-4919	44	16	-	-	SYM
ejpam-4919	44	17	2404	2404	NUM
ejpam-4919	44	18	2399	2399	NUM
ejpam-4919	44	19	each	each	DET
ejpam-4919	44	20	x	x	SYM
ejpam-4919	44	21	∈	∈	PROPN
ejpam-4919	44	22	x	x	PUNCT
ejpam-4919	44	23	there	there	PRON
ejpam-4919	44	24	exists	exist	VERB
ejpam-4919	44	25	y	y	PROPN
ejpam-4919	44	26	∈	∈	PROPN
ejpam-4919	44	27	e	e	NOUN
ejpam-4919	44	28	such	such	ADJ
ejpam-4919	44	29	that	that	PRON
ejpam-4919	44	30	for	for	ADP
ejpam-4919	44	31	all	all	DET
ejpam-4919	44	32	z	z	NOUN
ejpam-4919	44	33	∈	∈	PROPN
ejpam-4919	44	34	e	e	NOUN
ejpam-4919	44	35	,	,	PUNCT
ejpam-4919	44	36	the	the	DET
ejpam-4919	44	37	following	follow	VERB
ejpam-4919	44	38	inequality	inequality	NOUN
ejpam-4919	44	39	holds	hold	VERB
ejpam-4919	44	40	ϕ(||x−	ϕ(||x−	PROPN
ejpam-4919	44	41	y||	y||	PROPN
ejpam-4919	44	42	)	)	PUNCT
ejpam-4919	44	43	≥	≥	NOUN
ejpam-4919	44	44	ϕ(||x−	ϕ(||x−	PROPN
ejpam-4919	44	45	z||	z||	PROPN
ejpam-4919	44	46	)	)	PUNCT
ejpam-4919	44	47	+	+	CCONJ
ejpam-4919	44	48	ψ(y)ϕ(||y	ψ(y)ϕ(||y	PROPN
ejpam-4919	44	49	−	−	PROPN
ejpam-4919	44	50	z||	z||	PROPN
ejpam-4919	44	51	)	)	PUNCT
ejpam-4919	44	52	(	(	PUNCT
ejpam-4919	44	53	1	1	X
ejpam-4919	44	54	)	)	PUNCT
ejpam-4919	44	55	it	it	PRON
ejpam-4919	44	56	was	be	AUX
ejpam-4919	44	57	also	also	ADV
ejpam-4919	44	58	proved	prove	VERB
ejpam-4919	44	59	in	in	ADP
ejpam-4919	44	60	[	[	X
ejpam-4919	44	61	8	8	NUM
ejpam-4919	44	62	]	]	PUNCT
ejpam-4919	44	63	that	that	SCONJ
ejpam-4919	44	64	strong	strong	ADJ
ejpam-4919	44	65	remotality	remotality	NOUN
ejpam-4919	44	66	of	of	ADP
ejpam-4919	44	67	e	e	PROPN
ejpam-4919	44	68	,	,	PUNCT
ejpam-4919	44	69	with	with	ADP
ejpam-4919	44	70	the	the	DET
ejpam-4919	44	71	associated	associated	PROPN
ejpam-4919	44	72	functions	function	NOUN
ejpam-4919	44	73	ϕ	ϕ	NOUN
ejpam-4919	44	74	and	and	CCONJ
ejpam-4919	44	75	ψ	ψ	NOUN
ejpam-4919	44	76	,	,	PUNCT
ejpam-4919	44	77	is	be	AUX
ejpam-4919	44	78	equivalent	equivalent	ADJ
ejpam-4919	44	79	to	to	ADP
ejpam-4919	44	80	saying	say	VERB
ejpam-4919	44	81	that	that	PRON
ejpam-4919	44	82	:	:	PUNCT
ejpam-4919	44	83	for	for	ADP
ejpam-4919	44	84	every	every	DET
ejpam-4919	44	85	x	x	SYM
ejpam-4919	44	86	∈	∈	PROPN
ejpam-4919	44	87	x	x	NOUN
ejpam-4919	44	88	,	,	PUNCT
ejpam-4919	44	89	there	there	PRON
ejpam-4919	44	90	exists	exist	VERB
ejpam-4919	44	91	y	y	PROPN
ejpam-4919	44	92	∈	∈	PROPN
ejpam-4919	44	93	e	e	NOUN
ejpam-4919	44	94	such	such	ADJ
ejpam-4919	44	95	that	that	DET
ejpam-4919	44	96	inf	inf	PROPN
ejpam-4919	44	97	z∈e\{y	z∈e\{y	NUM
ejpam-4919	44	98	}	}	PUNCT
ejpam-4919	44	99	{	{	PUNCT
ejpam-4919	44	100	ϕ(||x−	ϕ(||x−	PROPN
ejpam-4919	44	101	y||)−	y||)−	NOUN
ejpam-4919	44	102	ϕ(||x−	ϕ(||x−	PROPN
ejpam-4919	44	103	z||	z||	PROPN
ejpam-4919	44	104	)	)	PUNCT
ejpam-4919	44	105	ϕ(||y	ϕ(||y	NOUN
ejpam-4919	44	106	−	−	PROPN
ejpam-4919	44	107	z||	z||	PROPN
ejpam-4919	44	108	)	)	PUNCT
ejpam-4919	44	109	}	}	PUNCT
ejpam-4919	44	110	>	>	X
ejpam-4919	45	1	0	0	X
ejpam-4919	45	2	.	.	PUNCT
ejpam-4919	46	1	(	(	PUNCT
ejpam-4919	46	2	2	2	X
ejpam-4919	46	3	)	)	PUNCT
ejpam-4919	46	4	clearly	clearly	ADV
ejpam-4919	46	5	,	,	PUNCT
ejpam-4919	46	6	strong	strong	ADJ
ejpam-4919	46	7	remotal	remotal	ADJ
ejpam-4919	46	8	sets	set	NOUN
ejpam-4919	46	9	in	in	ADP
ejpam-4919	46	10	hilbert	hilbert	NOUN
ejpam-4919	46	11	space	space	NOUN
ejpam-4919	46	12	can	can	AUX
ejpam-4919	46	13	be	be	AUX
ejpam-4919	46	14	defined	define	VERB
ejpam-4919	46	15	similarly	similarly	ADV
ejpam-4919	46	16	.	.	PUNCT
ejpam-4919	47	1	we	we	PRON
ejpam-4919	47	2	provide	provide	VERB
ejpam-4919	47	3	the	the	DET
ejpam-4919	47	4	definition	definition	NOUN
ejpam-4919	47	5	below	below	ADP
ejpam-4919	47	6	with	with	ADP
ejpam-4919	47	7	ϕ(t	ϕ(t	NUM
ejpam-4919	47	8	)	)	PUNCT
ejpam-4919	48	1	=	=	SYM
ejpam-4919	48	2	t2	t2	NOUN
ejpam-4919	48	3	.	.	PUNCT
ejpam-4919	49	1	definition	definition	NOUN
ejpam-4919	49	2	1	1	NUM
ejpam-4919	49	3	.	.	PUNCT
ejpam-4919	50	1	let	let	VERB
ejpam-4919	50	2	e	e	PRON
ejpam-4919	50	3	be	be	AUX
ejpam-4919	50	4	a	a	DET
ejpam-4919	50	5	non	non	ADJ
ejpam-4919	50	6	-	-	ADJ
ejpam-4919	50	7	empty	empty	ADJ
ejpam-4919	50	8	closed	closed	ADJ
ejpam-4919	50	9	convex	convex	NOUN
ejpam-4919	50	10	bounded	bound	VERB
ejpam-4919	50	11	set	set	VERB
ejpam-4919	50	12	in	in	ADP
ejpam-4919	50	13	a	a	DET
ejpam-4919	50	14	hilbert	hilbert	NOUN
ejpam-4919	50	15	space	space	NOUN
ejpam-4919	50	16	h.	h.	PROPN
ejpam-4919	50	17	then	then	ADV
ejpam-4919	50	18	e	e	PROPN
ejpam-4919	50	19	is	be	AUX
ejpam-4919	50	20	called	call	VERB
ejpam-4919	50	21	strongly	strongly	ADV
ejpam-4919	50	22	remotal	remotal	ADJ
ejpam-4919	50	23	in	in	ADP
ejpam-4919	50	24	h	h	NOUN
ejpam-4919	50	25	if	if	SCONJ
ejpam-4919	50	26	for	for	ADP
ejpam-4919	50	27	every	every	DET
ejpam-4919	50	28	x	x	SYM
ejpam-4919	50	29	∈	∈	PROPN
ejpam-4919	50	30	h	h	NOUN
ejpam-4919	50	31	there	there	PRON
ejpam-4919	50	32	exists	exist	VERB
ejpam-4919	50	33	y	y	PROPN
ejpam-4919	50	34	∈	∈	PROPN
ejpam-4919	50	35	e	e	NOUN
ejpam-4919	50	36	such	such	ADJ
ejpam-4919	50	37	that	that	DET
ejpam-4919	50	38	inf	inf	PROPN
ejpam-4919	50	39	z∈e\{y	z∈e\{y	NUM
ejpam-4919	50	40	}	}	PUNCT
ejpam-4919	50	41	{	{	PUNCT
ejpam-4919	50	42	∥x−	∥x−	NUM
ejpam-4919	50	43	y∥2	y∥2	NOUN
ejpam-4919	51	1	−	−	PROPN
ejpam-4919	52	1	∥x−	∥x−	PROPN
ejpam-4919	52	2	z∥2	z∥2	VERB
ejpam-4919	52	3	∥y	∥y	PROPN
ejpam-4919	52	4	−	−	NOUN
ejpam-4919	52	5	z∥2	z∥2	NOUN
ejpam-4919	52	6	}	}	PUNCT
ejpam-4919	52	7	>	>	X
ejpam-4919	52	8	0	0	PUNCT
ejpam-4919	52	9	(	(	PUNCT
ejpam-4919	52	10	3	3	X
ejpam-4919	52	11	)	)	PUNCT
ejpam-4919	52	12	it	it	PRON
ejpam-4919	52	13	is	be	AUX
ejpam-4919	52	14	easy	easy	ADJ
ejpam-4919	52	15	to	to	PART
ejpam-4919	52	16	see	see	VERB
ejpam-4919	52	17	that	that	DET
ejpam-4919	52	18	inequality	inequality	NOUN
ejpam-4919	52	19	1	1	NUM
ejpam-4919	52	20	,	,	PUNCT
ejpam-4919	52	21	with	with	ADP
ejpam-4919	52	22	ϕ(t	ϕ(t	NUM
ejpam-4919	52	23	)	)	PUNCT
ejpam-4919	53	1	=	=	SYM
ejpam-4919	53	2	t2	t2	NOUN
ejpam-4919	53	3	,	,	PUNCT
ejpam-4919	53	4	can	can	AUX
ejpam-4919	53	5	be	be	AUX
ejpam-4919	53	6	written	write	VERB
ejpam-4919	53	7	as	as	ADP
ejpam-4919	53	8	||x−	||x−	PROPN
ejpam-4919	53	9	y||2	y||2	PROPN
ejpam-4919	53	10	≥	≥	NOUN
ejpam-4919	53	11	||x−	||x−	PROPN
ejpam-4919	53	12	z||2	z||2	NOUN
ejpam-4919	53	13	+	+	CCONJ
ejpam-4919	53	14	ψ(y)||y	ψ(y)||y	PROPN
ejpam-4919	53	15	−	−	PROPN
ejpam-4919	53	16	z||2	z||2	PROPN
ejpam-4919	53	17	.	.	PUNCT
ejpam-4919	54	1	(	(	PUNCT
ejpam-4919	54	2	4	4	X
ejpam-4919	54	3	)	)	PUNCT
ejpam-4919	54	4	the	the	DET
ejpam-4919	54	5	main	main	ADJ
ejpam-4919	54	6	purpose	purpose	NOUN
ejpam-4919	54	7	of	of	ADP
ejpam-4919	54	8	this	this	DET
ejpam-4919	54	9	article	article	NOUN
ejpam-4919	54	10	is	be	AUX
ejpam-4919	54	11	to	to	PART
ejpam-4919	54	12	give	give	VERB
ejpam-4919	54	13	an	an	DET
ejpam-4919	54	14	answer	answer	NOUN
ejpam-4919	54	15	for	for	ADP
ejpam-4919	54	16	the	the	DET
ejpam-4919	54	17	farthest	farth	ADJ
ejpam-4919	54	18	point	point	NOUN
ejpam-4919	54	19	problem	problem	NOUN
ejpam-4919	54	20	in	in	ADP
ejpam-4919	54	21	separable	separable	ADJ
ejpam-4919	54	22	hilbert	hilbert	PROPN
ejpam-4919	54	23	spaces	space	NOUN
ejpam-4919	54	24	,	,	PUNCT
ejpam-4919	54	25	considering	consider	VERB
ejpam-4919	54	26	a	a	DET
ejpam-4919	54	27	special	special	ADJ
ejpam-4919	54	28	class	class	NOUN
ejpam-4919	54	29	of	of	ADP
ejpam-4919	54	30	uniquely	uniquely	ADV
ejpam-4919	54	31	remotal	remotal	ADJ
ejpam-4919	54	32	sets	set	NOUN
ejpam-4919	54	33	that	that	PRON
ejpam-4919	54	34	satisfy	satisfy	VERB
ejpam-4919	54	35	a	a	DET
ejpam-4919	54	36	certain	certain	ADJ
ejpam-4919	54	37	condition	condition	NOUN
ejpam-4919	54	38	.	.	PUNCT
ejpam-4919	55	1	2	2	X
ejpam-4919	55	2	.	.	X
ejpam-4919	55	3	main	main	ADJ
ejpam-4919	55	4	result	result	NOUN
ejpam-4919	55	5	in	in	ADP
ejpam-4919	55	6	this	this	DET
ejpam-4919	55	7	section	section	NOUN
ejpam-4919	55	8	,	,	PUNCT
ejpam-4919	55	9	we	we	PRON
ejpam-4919	55	10	introduce	introduce	VERB
ejpam-4919	55	11	the	the	DET
ejpam-4919	55	12	so	so	ADV
ejpam-4919	55	13	-	-	PUNCT
ejpam-4919	55	14	called	call	VERB
ejpam-4919	55	15	uniquely	uniquely	ADV
ejpam-4919	55	16	distant	distant	ADJ
ejpam-4919	55	17	sets	set	NOUN
ejpam-4919	55	18	,	,	PUNCT
ejpam-4919	55	19	and	and	CCONJ
ejpam-4919	55	20	prove	prove	VERB
ejpam-4919	55	21	that	that	SCONJ
ejpam-4919	55	22	every	every	DET
ejpam-4919	55	23	uniquely	uniquely	ADV
ejpam-4919	55	24	distant	distant	ADJ
ejpam-4919	55	25	set	set	NOUN
ejpam-4919	55	26	in	in	ADP
ejpam-4919	55	27	a	a	DET
ejpam-4919	55	28	separable	separable	ADJ
ejpam-4919	55	29	hilbert	hilbert	NOUN
ejpam-4919	55	30	space	space	NOUN
ejpam-4919	55	31	is	be	AUX
ejpam-4919	55	32	a	a	DET
ejpam-4919	55	33	singleton	singleton	NOUN
ejpam-4919	55	34	.	.	PUNCT
ejpam-4919	56	1	first	first	ADV
ejpam-4919	56	2	we	we	PRON
ejpam-4919	56	3	prove	prove	VERB
ejpam-4919	56	4	the	the	DET
ejpam-4919	56	5	following	following	ADJ
ejpam-4919	56	6	result	result	NOUN
ejpam-4919	56	7	that	that	PRON
ejpam-4919	56	8	plays	play	VERB
ejpam-4919	56	9	an	an	DET
ejpam-4919	56	10	important	important	ADJ
ejpam-4919	56	11	role	role	NOUN
ejpam-4919	56	12	in	in	ADP
ejpam-4919	56	13	the	the	DET
ejpam-4919	56	14	proof	proof	NOUN
ejpam-4919	56	15	of	of	ADP
ejpam-4919	56	16	our	our	PRON
ejpam-4919	56	17	main	main	ADJ
ejpam-4919	56	18	result	result	NOUN
ejpam-4919	56	19	.	.	PUNCT
ejpam-4919	57	1	theorem	theorem	NOUN
ejpam-4919	57	2	2	2	NUM
ejpam-4919	57	3	.	.	PUNCT
ejpam-4919	58	1	let	let	VERB
ejpam-4919	58	2	e	e	PRON
ejpam-4919	58	3	be	be	AUX
ejpam-4919	58	4	a	a	DET
ejpam-4919	58	5	non	non	ADJ
ejpam-4919	58	6	-	-	ADJ
ejpam-4919	58	7	empty	empty	ADJ
ejpam-4919	58	8	strongly	strongly	ADV
ejpam-4919	58	9	remotal	remotal	ADJ
ejpam-4919	58	10	subset	subset	NOUN
ejpam-4919	58	11	of	of	ADP
ejpam-4919	58	12	a	a	DET
ejpam-4919	58	13	separable	separable	ADJ
ejpam-4919	58	14	hilbert	hilbert	PROPN
ejpam-4919	58	15	space	space	PROPN
ejpam-4919	58	16	h.	h.	PROPN
ejpam-4919	59	1	then	then	ADV
ejpam-4919	59	2	the	the	DET
ejpam-4919	59	3	mapping	mapping	NOUN
ejpam-4919	59	4	f	f	X
ejpam-4919	59	5	:	:	PUNCT
ejpam-4919	59	6	h	h	NOUN
ejpam-4919	59	7	→	→	SYM
ejpam-4919	59	8	e	e	NOUN
ejpam-4919	59	9	,	,	PUNCT
ejpam-4919	59	10	defined	define	VERB
ejpam-4919	59	11	by	by	ADP
ejpam-4919	59	12	f	f	PROPN
ejpam-4919	59	13	(	(	PUNCT
ejpam-4919	59	14	x	x	NOUN
ejpam-4919	59	15	)	)	PUNCT
ejpam-4919	59	16	=	=	SYM
ejpam-4919	59	17	f	f	X
ejpam-4919	59	18	(	(	PUNCT
ejpam-4919	59	19	x	x	X
ejpam-4919	59	20	,	,	PUNCT
ejpam-4919	59	21	e	e	NOUN
ejpam-4919	59	22	)	)	PUNCT
ejpam-4919	59	23	is	be	AUX
ejpam-4919	59	24	continuous	continuous	ADJ
ejpam-4919	59	25	on	on	ADP
ejpam-4919	59	26	h.	h.	NOUN
ejpam-4919	59	27	proof	proof	NOUN
ejpam-4919	59	28	.	.	PUNCT
ejpam-4919	60	1	let	let	VERB
ejpam-4919	60	2	x	x	SYM
ejpam-4919	60	3	∈	∈	NOUN
ejpam-4919	60	4	h	h	NOUN
ejpam-4919	60	5	be	be	AUX
ejpam-4919	60	6	such	such	ADJ
ejpam-4919	60	7	that	that	SCONJ
ejpam-4919	60	8	r(x	r(x	PROPN
ejpam-4919	60	9	,	,	PUNCT
ejpam-4919	60	10	e	e	NOUN
ejpam-4919	60	11	)	)	PUNCT
ejpam-4919	60	12	=	=	SYM
ejpam-4919	60	13	r0	r0	NOUN
ejpam-4919	60	14	.	.	PUNCT
ejpam-4919	61	1	let	let	AUX
ejpam-4919	61	2	(	(	PUNCT
ejpam-4919	61	3	xn	xn	X
ejpam-4919	61	4	)	)	PUNCT
ejpam-4919	61	5	be	be	VERB
ejpam-4919	61	6	a	a	DET
ejpam-4919	61	7	sequence	sequence	NOUN
ejpam-4919	61	8	in	in	ADP
ejpam-4919	61	9	h	h	NOUN
ejpam-4919	61	10	such	such	ADJ
ejpam-4919	61	11	that	that	PRON
ejpam-4919	61	12	xn	xn	PUNCT
ejpam-4919	62	1	→	→	PUNCT
ejpam-4919	62	2	x.	x.	NOUN
ejpam-4919	62	3	we	we	PRON
ejpam-4919	62	4	claim	claim	VERB
ejpam-4919	62	5	that	that	SCONJ
ejpam-4919	62	6	f	f	PROPN
ejpam-4919	62	7	(	(	PUNCT
ejpam-4919	62	8	xn	xn	PROPN
ejpam-4919	62	9	)	)	PUNCT
ejpam-4919	62	10	→	→	SYM
ejpam-4919	62	11	f	f	X
ejpam-4919	62	12	(	(	PUNCT
ejpam-4919	62	13	x	x	NOUN
ejpam-4919	62	14	)	)	PUNCT
ejpam-4919	62	15	.	.	PUNCT
ejpam-4919	63	1	e	e	NOUN
ejpam-4919	63	2	is	be	AUX
ejpam-4919	63	3	strongly	strongly	ADV
ejpam-4919	63	4	remotal	remotal	ADJ
ejpam-4919	63	5	,	,	PUNCT
ejpam-4919	63	6	so	so	ADV
ejpam-4919	64	1	∥x−	∥x−	PROPN
ejpam-4919	64	2	f	f	X
ejpam-4919	64	3	(	(	PUNCT
ejpam-4919	64	4	x	x	X
ejpam-4919	64	5	)	)	PUNCT
ejpam-4919	64	6	∥2	∥2	NOUN
ejpam-4919	64	7	≥	≥	NOUN
ejpam-4919	65	1	∥x−	∥x−	NUM
ejpam-4919	65	2	z∥2	z∥2	NOUN
ejpam-4919	65	3	+	+	PUNCT
ejpam-4919	65	4	ψ	ψ	X
ejpam-4919	65	5	(	(	PUNCT
ejpam-4919	65	6	z	z	X
ejpam-4919	65	7	)	)	PUNCT
ejpam-4919	65	8	∥f	∥f	PROPN
ejpam-4919	65	9	(	(	PUNCT
ejpam-4919	65	10	x)−	x)−	PROPN
ejpam-4919	65	11	z∥2	z∥2	PROPN
ejpam-4919	65	12	,	,	PUNCT
ejpam-4919	65	13	(	(	PUNCT
ejpam-4919	65	14	5	5	NUM
ejpam-4919	65	15	)	)	PUNCT
ejpam-4919	65	16	for	for	ADP
ejpam-4919	65	17	all	all	DET
ejpam-4919	65	18	z	z	PROPN
ejpam-4919	65	19	∈	∈	PROPN
ejpam-4919	65	20	e.	e.	PROPN
ejpam-4919	65	21	a.	a.	PROPN
ejpam-4919	65	22	yousef	yousef	PROPN
ejpam-4919	65	23	et	et	PROPN
ejpam-4919	65	24	al	al	PROPN
ejpam-4919	65	25	.	.	PUNCT
ejpam-4919	65	26	/	/	SYM
ejpam-4919	65	27	eur	eur	PROPN
ejpam-4919	65	28	.	.	PUNCT
ejpam-4919	66	1	j.	j.	PROPN
ejpam-4919	66	2	pure	pure	PROPN
ejpam-4919	66	3	appl	appl	PROPN
ejpam-4919	66	4	.	.	PROPN
ejpam-4919	66	5	math	math	PROPN
ejpam-4919	66	6	,	,	PUNCT
ejpam-4919	66	7	16	16	NUM
ejpam-4919	66	8	(	(	PUNCT
ejpam-4919	66	9	4	4	NUM
ejpam-4919	66	10	)	)	PUNCT
ejpam-4919	66	11	(	(	PUNCT
ejpam-4919	66	12	2023	2023	NUM
ejpam-4919	66	13	)	)	PUNCT
ejpam-4919	66	14	,	,	PUNCT
ejpam-4919	66	15	2397	2397	NUM
ejpam-4919	66	16	-	-	SYM
ejpam-4919	66	17	2404	2404	NUM
ejpam-4919	66	18	2400	2400	NUM
ejpam-4919	66	19	let	let	VERB
ejpam-4919	66	20	zn	zn	NOUN
ejpam-4919	66	21	=	=	SYM
ejpam-4919	66	22	f	f	PROPN
ejpam-4919	66	23	(	(	PUNCT
ejpam-4919	66	24	xn	xn	PROPN
ejpam-4919	66	25	)	)	PUNCT
ejpam-4919	66	26	.	.	PUNCT
ejpam-4919	67	1	then	then	ADV
ejpam-4919	67	2	,	,	PUNCT
ejpam-4919	67	3	∥x−	∥x−	PROPN
ejpam-4919	67	4	f	f	X
ejpam-4919	67	5	(	(	PUNCT
ejpam-4919	67	6	x	x	X
ejpam-4919	67	7	)	)	PUNCT
ejpam-4919	67	8	∥2	∥2	NOUN
ejpam-4919	67	9	≥	≥	NOUN
ejpam-4919	68	1	∥x−	∥x−	NUM
ejpam-4919	68	2	f	f	PROPN
ejpam-4919	68	3	(	(	PUNCT
ejpam-4919	68	4	xn	xn	PROPN
ejpam-4919	68	5	)	)	PUNCT
ejpam-4919	68	6	∥2	∥2	NOUN
ejpam-4919	69	1	+	+	NUM
ejpam-4919	69	2	ψ	ψ	X
ejpam-4919	69	3	(	(	PUNCT
ejpam-4919	69	4	y	y	NOUN
ejpam-4919	69	5	)	)	PUNCT
ejpam-4919	69	6	∥f	∥f	PROPN
ejpam-4919	69	7	(	(	PUNCT
ejpam-4919	69	8	x)−	x)−	PROPN
ejpam-4919	69	9	f	f	PROPN
ejpam-4919	69	10	(	(	PUNCT
ejpam-4919	69	11	xn	xn	PROPN
ejpam-4919	69	12	)	)	PUNCT
ejpam-4919	69	13	∥2	∥2	NOUN
ejpam-4919	69	14	for	for	ADP
ejpam-4919	69	15	all	all	DET
ejpam-4919	69	16	n.	n.	NOUN
ejpam-4919	69	17	(	(	PUNCT
ejpam-4919	69	18	6	6	NUM
ejpam-4919	69	19	)	)	PUNCT
ejpam-4919	69	20	this	this	PRON
ejpam-4919	69	21	implies	imply	VERB
ejpam-4919	69	22	∥f	∥f	PROPN
ejpam-4919	69	23	(	(	PUNCT
ejpam-4919	69	24	x)−	x)−	PROPN
ejpam-4919	69	25	f	f	PROPN
ejpam-4919	69	26	(	(	PUNCT
ejpam-4919	69	27	xn	xn	PROPN
ejpam-4919	69	28	)	)	PUNCT
ejpam-4919	69	29	∥2	∥2	NOUN
ejpam-4919	70	1	≤	≤	NUM
ejpam-4919	70	2	1	1	NUM
ejpam-4919	70	3	ψ	ψ	X
ejpam-4919	70	4	(	(	PUNCT
ejpam-4919	70	5	y	y	NOUN
ejpam-4919	70	6	)	)	PUNCT
ejpam-4919	70	7	[	[	PUNCT
ejpam-4919	71	1	∥x−	∥x−	PROPN
ejpam-4919	71	2	f	f	X
ejpam-4919	71	3	(	(	PUNCT
ejpam-4919	71	4	x	x	X
ejpam-4919	71	5	)	)	PUNCT
ejpam-4919	71	6	∥2	∥2	NOUN
ejpam-4919	71	7	−	−	NOUN
ejpam-4919	72	1	∥x−	∥x−	PROPN
ejpam-4919	72	2	f	f	PROPN
ejpam-4919	72	3	(	(	PUNCT
ejpam-4919	72	4	xn	xn	PROPN
ejpam-4919	72	5	)	)	PUNCT
ejpam-4919	72	6	∥2	∥2	NOUN
ejpam-4919	72	7	]	]	PUNCT
ejpam-4919	72	8	(	(	PUNCT
ejpam-4919	72	9	7	7	NUM
ejpam-4919	72	10	)	)	PUNCT
ejpam-4919	72	11	since	since	SCONJ
ejpam-4919	72	12	f	f	PROPN
ejpam-4919	72	13	(	(	PUNCT
ejpam-4919	72	14	xn	xn	PROPN
ejpam-4919	72	15	)	)	PUNCT
ejpam-4919	72	16	is	be	AUX
ejpam-4919	72	17	the	the	DET
ejpam-4919	72	18	farthest	farth	ADJ
ejpam-4919	72	19	point	point	NOUN
ejpam-4919	72	20	in	in	ADP
ejpam-4919	72	21	e	e	NOUN
ejpam-4919	72	22	from	from	ADP
ejpam-4919	72	23	xn	xn	PROPN
ejpam-4919	72	24	we	we	PRON
ejpam-4919	72	25	have	have	VERB
ejpam-4919	72	26	∥xn	∥xn	PRON
ejpam-4919	72	27	−	−	PROPN
ejpam-4919	72	28	f	f	NOUN
ejpam-4919	72	29	(	(	PUNCT
ejpam-4919	72	30	xn)∥	xn)∥	PROPN
ejpam-4919	72	31	≥	≥	NOUN
ejpam-4919	72	32	∥xn	∥xn	PART
ejpam-4919	72	33	−	−	PROPN
ejpam-4919	72	34	f	f	NOUN
ejpam-4919	72	35	(	(	PUNCT
ejpam-4919	72	36	x)∥	x)∥	PUNCT
ejpam-4919	72	37	,	,	PUNCT
ejpam-4919	72	38	it	it	PRON
ejpam-4919	72	39	follows	follow	VERB
ejpam-4919	72	40	that	that	SCONJ
ejpam-4919	72	41	∥xn	∥xn	VERB
ejpam-4919	72	42	−	−	PROPN
ejpam-4919	72	43	f	f	NOUN
ejpam-4919	72	44	(	(	PUNCT
ejpam-4919	72	45	x)∥	x)∥	PUNCT
ejpam-4919	72	46	≤	≤	NUM
ejpam-4919	72	47	∥xn	∥xn	PROPN
ejpam-4919	72	48	−	−	PROPN
ejpam-4919	72	49	f	f	PROPN
ejpam-4919	72	50	(	(	PUNCT
ejpam-4919	72	51	xn)∥	xn)∥	PROPN
ejpam-4919	72	52	≤	≤	NOUN
ejpam-4919	72	53	∥xn	∥xn	PROPN
ejpam-4919	73	1	−	−	NOUN
ejpam-4919	73	2	x∥+	x∥+	PROPN
ejpam-4919	74	1	∥x−	∥x−	PROPN
ejpam-4919	74	2	f	f	PROPN
ejpam-4919	74	3	(	(	PUNCT
ejpam-4919	74	4	xn)∥	xn)∥	PROPN
ejpam-4919	74	5	,	,	PUNCT
ejpam-4919	74	6	for	for	ADP
ejpam-4919	74	7	all	all	DET
ejpam-4919	74	8	n	n	PRON
ejpam-4919	74	9	∈	∈	PROPN
ejpam-4919	74	10	n	n	CCONJ
ejpam-4919	74	11	(	(	PUNCT
ejpam-4919	74	12	8)	8)	NUM
ejpam-4919	74	13	taking	take	VERB
ejpam-4919	74	14	the	the	DET
ejpam-4919	74	15	limits	limit	NOUN
ejpam-4919	74	16	on	on	ADP
ejpam-4919	74	17	both	both	DET
ejpam-4919	74	18	sides	side	NOUN
ejpam-4919	74	19	of	of	ADP
ejpam-4919	74	20	8	8	NUM
ejpam-4919	74	21	as	as	ADP
ejpam-4919	74	22	n→	n→	PROPN
ejpam-4919	74	23	∞	∞	PROPN
ejpam-4919	74	24	,	,	PUNCT
ejpam-4919	74	25	we	we	PRON
ejpam-4919	74	26	obtain	obtain	VERB
ejpam-4919	74	27	∥x−	∥x−	PROPN
ejpam-4919	74	28	f	f	PROPN
ejpam-4919	74	29	(	(	PUNCT
ejpam-4919	74	30	x)∥	x)∥	PUNCT
ejpam-4919	74	31	≤	≤	PROPN
ejpam-4919	74	32	lim	lim	PROPN
ejpam-4919	74	33	n→∞	n→∞	PRON
ejpam-4919	75	1	∥x−	∥x−	PROPN
ejpam-4919	75	2	f	f	PROPN
ejpam-4919	75	3	(	(	PUNCT
ejpam-4919	75	4	xn)∥	xn)∥	PROPN
ejpam-4919	75	5	(	(	PUNCT
ejpam-4919	75	6	9	9	NUM
ejpam-4919	75	7	)	)	PUNCT
ejpam-4919	75	8	inequality	inequality	NOUN
ejpam-4919	75	9	7	7	NUM
ejpam-4919	75	10	is	be	AUX
ejpam-4919	75	11	true	true	ADJ
ejpam-4919	75	12	for	for	ADP
ejpam-4919	75	13	all	all	DET
ejpam-4919	75	14	n	n	CCONJ
ejpam-4919	75	15	,	,	PUNCT
ejpam-4919	75	16	which	which	PRON
ejpam-4919	75	17	implies	imply	VERB
ejpam-4919	75	18	lim	lim	PROPN
ejpam-4919	75	19	n→∞	n→∞	X
ejpam-4919	76	1	∥f	∥f	PROPN
ejpam-4919	76	2	(	(	PUNCT
ejpam-4919	76	3	x)−	x)−	PROPN
ejpam-4919	76	4	f	f	PROPN
ejpam-4919	76	5	(	(	PUNCT
ejpam-4919	76	6	xn	xn	PROPN
ejpam-4919	76	7	)	)	PUNCT
ejpam-4919	76	8	∥2	∥2	NOUN
ejpam-4919	77	1	≤	≤	NUM
ejpam-4919	77	2	1	1	NUM
ejpam-4919	77	3	ψ	ψ	X
ejpam-4919	77	4	(	(	PUNCT
ejpam-4919	77	5	y	y	NOUN
ejpam-4919	77	6	)	)	PUNCT
ejpam-4919	78	1	[	[	X
ejpam-4919	78	2	∥x−	∥x−	X
ejpam-4919	78	3	f	f	X
ejpam-4919	78	4	(	(	PUNCT
ejpam-4919	78	5	x	x	X
ejpam-4919	78	6	)	)	PUNCT
ejpam-4919	78	7	∥2	∥2	NOUN
ejpam-4919	78	8	−	−	PROPN
ejpam-4919	79	1	lim	lim	PROPN
ejpam-4919	79	2	n→∞	n→∞	PRON
ejpam-4919	80	1	∥x−	∥x−	PROPN
ejpam-4919	80	2	f	f	PROPN
ejpam-4919	80	3	(	(	PUNCT
ejpam-4919	80	4	xn	xn	PROPN
ejpam-4919	80	5	)	)	PUNCT
ejpam-4919	80	6	∥2	∥2	NOUN
ejpam-4919	80	7	]	]	PUNCT
ejpam-4919	80	8	(	(	PUNCT
ejpam-4919	80	9	10	10	NUM
ejpam-4919	80	10	)	)	PUNCT
ejpam-4919	80	11	substitute	substitute	NOUN
ejpam-4919	80	12	9	9	NUM
ejpam-4919	80	13	in	in	ADP
ejpam-4919	80	14	inequality	inequality	NOUN
ejpam-4919	80	15	10	10	NUM
ejpam-4919	80	16	to	to	PART
ejpam-4919	80	17	get	get	VERB
ejpam-4919	80	18	lim	lim	PROPN
ejpam-4919	80	19	n→∞	n→∞	X
ejpam-4919	81	1	∥f	∥f	PROPN
ejpam-4919	81	2	(	(	PUNCT
ejpam-4919	81	3	x)−	x)−	PROPN
ejpam-4919	81	4	f	f	PROPN
ejpam-4919	81	5	(	(	PUNCT
ejpam-4919	81	6	xn	xn	PROPN
ejpam-4919	81	7	)	)	PUNCT
ejpam-4919	81	8	∥2	∥2	NOUN
ejpam-4919	82	1	=	=	SYM
ejpam-4919	82	2	0	0	PUNCT
ejpam-4919	83	1	so	so	ADV
ejpam-4919	83	2	f	f	X
ejpam-4919	83	3	(	(	PUNCT
ejpam-4919	83	4	xn	xn	PROPN
ejpam-4919	83	5	)	)	PUNCT
ejpam-4919	83	6	→	→	SYM
ejpam-4919	83	7	f	f	X
ejpam-4919	83	8	(	(	PUNCT
ejpam-4919	83	9	x	x	X
ejpam-4919	83	10	)	)	PUNCT
ejpam-4919	83	11	,	,	PUNCT
ejpam-4919	83	12	which	which	PRON
ejpam-4919	83	13	completes	complete	VERB
ejpam-4919	83	14	the	the	DET
ejpam-4919	83	15	proof	proof	NOUN
ejpam-4919	83	16	.	.	PUNCT
ejpam-4919	84	1	theorems	theorems	PROPN
ejpam-4919	84	2	2	2	NUM
ejpam-4919	84	3	and	and	CCONJ
ejpam-4919	84	4	1	1	NUM
ejpam-4919	84	5	imply	imply	VERB
ejpam-4919	84	6	the	the	DET
ejpam-4919	84	7	following	follow	VERB
ejpam-4919	84	8	important	important	ADJ
ejpam-4919	84	9	result	result	NOUN
ejpam-4919	84	10	:	:	PUNCT
ejpam-4919	84	11	theorem	theorem	NOUN
ejpam-4919	84	12	3	3	NUM
ejpam-4919	84	13	.	.	PUNCT
ejpam-4919	85	1	every	every	DET
ejpam-4919	85	2	strongly	strongly	ADV
ejpam-4919	85	3	remotal	remotal	ADJ
ejpam-4919	85	4	set	set	NOUN
ejpam-4919	85	5	in	in	ADP
ejpam-4919	85	6	a	a	DET
ejpam-4919	85	7	hilbert	hilbert	NOUN
ejpam-4919	85	8	space	space	NOUN
ejpam-4919	85	9	is	be	AUX
ejpam-4919	85	10	a	a	DET
ejpam-4919	85	11	singleton	singleton	NOUN
ejpam-4919	85	12	.	.	PUNCT
ejpam-4919	86	1	remark	remark	PROPN
ejpam-4919	86	2	1	1	NUM
ejpam-4919	86	3	.	.	PUNCT
ejpam-4919	87	1	theorems	theorems	PROPN
ejpam-4919	87	2	1	1	NUM
ejpam-4919	87	3	and	and	CCONJ
ejpam-4919	87	4	2	2	NUM
ejpam-4919	87	5	imply	imply	VERB
ejpam-4919	87	6	garkavi	garkavi	NOUN
ejpam-4919	87	7	’s	’s	PART
ejpam-4919	87	8	result	result	NOUN
ejpam-4919	87	9	in	in	ADP
ejpam-4919	87	10	[	[	X
ejpam-4919	87	11	7	7	NUM
ejpam-4919	87	12	]	]	PUNCT
ejpam-4919	87	13	,	,	PUNCT
ejpam-4919	87	14	which	which	PRON
ejpam-4919	87	15	states	state	VERB
ejpam-4919	87	16	that	that	SCONJ
ejpam-4919	87	17	a	a	DET
ejpam-4919	87	18	chebyshev	chebyshev	NOUN
ejpam-4919	87	19	center	center	NOUN
ejpam-4919	87	20	in	in	ADP
ejpam-4919	87	21	a	a	DET
ejpam-4919	87	22	hilbert	hilbert	NOUN
ejpam-4919	87	23	space	space	NOUN
ejpam-4919	87	24	exists	exist	VERB
ejpam-4919	87	25	and	and	CCONJ
ejpam-4919	87	26	is	be	AUX
ejpam-4919	87	27	unique	unique	ADJ
ejpam-4919	87	28	.	.	PUNCT
ejpam-4919	88	1	next	next	ADV
ejpam-4919	88	2	,	,	PUNCT
ejpam-4919	88	3	we	we	PRON
ejpam-4919	88	4	introduce	introduce	VERB
ejpam-4919	88	5	the	the	DET
ejpam-4919	88	6	so	so	ADV
ejpam-4919	88	7	-	-	PUNCT
ejpam-4919	88	8	called	call	VERB
ejpam-4919	88	9	uniquely	uniquely	ADV
ejpam-4919	88	10	distant	distant	ADJ
ejpam-4919	88	11	sets	set	NOUN
ejpam-4919	88	12	.	.	PUNCT
ejpam-4919	89	1	definition	definition	NOUN
ejpam-4919	89	2	2	2	NUM
ejpam-4919	89	3	.	.	PUNCT
ejpam-4919	90	1	let	let	VERB
ejpam-4919	90	2	h	h	PRON
ejpam-4919	90	3	be	be	AUX
ejpam-4919	90	4	a	a	DET
ejpam-4919	90	5	hilbert	hilbert	NOUN
ejpam-4919	90	6	space	space	NOUN
ejpam-4919	90	7	and	and	CCONJ
ejpam-4919	90	8	e	e	PROPN
ejpam-4919	90	9	⊂	⊂	PROPN
ejpam-4919	90	10	h	h	PROPN
ejpam-4919	90	11	be	be	AUX
ejpam-4919	90	12	a	a	DET
ejpam-4919	90	13	closed	closed	ADJ
ejpam-4919	90	14	bounded	bound	VERB
ejpam-4919	90	15	subset	subset	NOUN
ejpam-4919	90	16	.	.	PUNCT
ejpam-4919	91	1	then	then	ADV
ejpam-4919	91	2	e	e	PROPN
ejpam-4919	91	3	is	be	AUX
ejpam-4919	91	4	said	say	VERB
ejpam-4919	91	5	to	to	PART
ejpam-4919	91	6	be	be	AUX
ejpam-4919	91	7	uniquely	uniquely	ADV
ejpam-4919	91	8	distant	distant	ADJ
ejpam-4919	91	9	set	set	NOUN
ejpam-4919	91	10	in	in	ADP
ejpam-4919	91	11	h	h	NOUN
ejpam-4919	91	12	if	if	SCONJ
ejpam-4919	91	13	the	the	DET
ejpam-4919	91	14	following	follow	VERB
ejpam-4919	91	15	two	two	NUM
ejpam-4919	91	16	conditions	condition	NOUN
ejpam-4919	91	17	are	be	AUX
ejpam-4919	91	18	satisfied	satisfied	ADJ
ejpam-4919	91	19	:	:	PUNCT
ejpam-4919	91	20	(	(	PUNCT
ejpam-4919	91	21	i	i	NOUN
ejpam-4919	91	22	)	)	PUNCT
ejpam-4919	91	23	e	e	NOUN
ejpam-4919	91	24	is	be	AUX
ejpam-4919	91	25	uniquely	uniquely	ADV
ejpam-4919	91	26	remotal	remotal	ADJ
ejpam-4919	91	27	a.	a.	NOUN
ejpam-4919	91	28	yousef	yousef	PROPN
ejpam-4919	91	29	et	et	PROPN
ejpam-4919	91	30	al	al	PROPN
ejpam-4919	91	31	.	.	PUNCT
ejpam-4919	91	32	/	/	SYM
ejpam-4919	91	33	eur	eur	PROPN
ejpam-4919	91	34	.	.	PUNCT
ejpam-4919	92	1	j.	j.	PROPN
ejpam-4919	92	2	pure	pure	PROPN
ejpam-4919	92	3	appl	appl	PROPN
ejpam-4919	92	4	.	.	PROPN
ejpam-4919	92	5	math	math	PROPN
ejpam-4919	92	6	,	,	PUNCT
ejpam-4919	92	7	16	16	NUM
ejpam-4919	92	8	(	(	PUNCT
ejpam-4919	92	9	4	4	NUM
ejpam-4919	92	10	)	)	PUNCT
ejpam-4919	92	11	(	(	PUNCT
ejpam-4919	92	12	2023	2023	NUM
ejpam-4919	92	13	)	)	PUNCT
ejpam-4919	92	14	,	,	PUNCT
ejpam-4919	92	15	2397	2397	NUM
ejpam-4919	92	16	-	-	SYM
ejpam-4919	92	17	2404	2404	NUM
ejpam-4919	92	18	2401	2401	NUM
ejpam-4919	92	19	(	(	PUNCT
ejpam-4919	92	20	ii	ii	NOUN
ejpam-4919	92	21	)	)	PUNCT
ejpam-4919	92	22	if	if	SCONJ
ejpam-4919	92	23	x	x	SYM
ejpam-4919	92	24	∈	∈	PROPN
ejpam-4919	92	25	h	h	NOUN
ejpam-4919	92	26	,	,	PUNCT
ejpam-4919	92	27	and	and	CCONJ
ejpam-4919	92	28	y	y	PROPN
ejpam-4919	92	29	is	be	AUX
ejpam-4919	92	30	the	the	DET
ejpam-4919	92	31	farthest	farth	ADJ
ejpam-4919	92	32	point	point	NOUN
ejpam-4919	92	33	from	from	ADP
ejpam-4919	92	34	x	x	PUNCT
ejpam-4919	92	35	in	in	ADP
ejpam-4919	92	36	e	e	NOUN
ejpam-4919	92	37	,	,	PUNCT
ejpam-4919	92	38	then	then	ADV
ejpam-4919	92	39	for	for	ADP
ejpam-4919	92	40	every	every	DET
ejpam-4919	92	41	ε	ε	PROPN
ejpam-4919	92	42	>	>	X
ejpam-4919	92	43	0	0	PROPN
ejpam-4919	92	44	,	,	PUNCT
ejpam-4919	92	45	there	there	PRON
ejpam-4919	92	46	exists	exist	VERB
ejpam-4919	92	47	δ	δ	PROPN
ejpam-4919	92	48	>	>	X
ejpam-4919	92	49	0	0	NUM
ejpam-4919	93	1	such	such	ADJ
ejpam-4919	93	2	that	that	SCONJ
ejpam-4919	93	3	r(x	r(x	PROPN
ejpam-4919	93	4	,	,	PUNCT
ejpam-4919	93	5	e\b(y	e\b(y	ADJ
ejpam-4919	93	6	,	,	PUNCT
ejpam-4919	93	7	δ	δ	PROPN
ejpam-4919	93	8	)	)	PUNCT
ejpam-4919	93	9	)	)	PUNCT
ejpam-4919	93	10	≤	≤	PROPN
ejpam-4919	93	11	r(x	r(x	PROPN
ejpam-4919	93	12	,	,	PUNCT
ejpam-4919	93	13	e)−	e)−	PROPN
ejpam-4919	93	14	ε	ε	PROPN
ejpam-4919	93	15	.	.	PUNCT
ejpam-4919	94	1	now	now	ADV
ejpam-4919	94	2	,	,	PUNCT
ejpam-4919	94	3	we	we	PRON
ejpam-4919	94	4	are	be	AUX
ejpam-4919	94	5	ready	ready	ADJ
ejpam-4919	94	6	to	to	PART
ejpam-4919	94	7	prove	prove	VERB
ejpam-4919	94	8	the	the	DET
ejpam-4919	94	9	main	main	ADJ
ejpam-4919	94	10	result	result	NOUN
ejpam-4919	94	11	of	of	ADP
ejpam-4919	94	12	this	this	DET
ejpam-4919	94	13	paper	paper	NOUN
ejpam-4919	94	14	.	.	PUNCT
ejpam-4919	95	1	theorem	theorem	ADJ
ejpam-4919	95	2	4	4	NUM
ejpam-4919	95	3	.	.	PUNCT
ejpam-4919	96	1	every	every	DET
ejpam-4919	96	2	uniquely	uniquely	ADV
ejpam-4919	96	3	distant	distant	ADJ
ejpam-4919	96	4	subset	subset	NOUN
ejpam-4919	96	5	of	of	ADP
ejpam-4919	96	6	a	a	DET
ejpam-4919	96	7	separable	separable	ADJ
ejpam-4919	96	8	hilbert	hilbert	NOUN
ejpam-4919	96	9	space	space	NOUN
ejpam-4919	96	10	is	be	AUX
ejpam-4919	96	11	a	a	DET
ejpam-4919	96	12	singleton	singleton	NOUN
ejpam-4919	96	13	.	.	PUNCT
ejpam-4919	97	1	proof	proof	NOUN
ejpam-4919	97	2	.	.	PUNCT
ejpam-4919	98	1	let	let	VERB
ejpam-4919	98	2	e	e	PRON
ejpam-4919	98	3	be	be	AUX
ejpam-4919	98	4	a	a	DET
ejpam-4919	98	5	nonsingleton	nonsingleton	NOUN
ejpam-4919	98	6	uniquely	uniquely	ADV
ejpam-4919	98	7	distant	distant	ADJ
ejpam-4919	98	8	subset	subset	NOUN
ejpam-4919	98	9	in	in	ADP
ejpam-4919	98	10	a	a	DET
ejpam-4919	98	11	hilbert	hilbert	NOUN
ejpam-4919	98	12	space	space	NOUN
ejpam-4919	98	13	h.	h.	PROPN
ejpam-4919	98	14	using	use	VERB
ejpam-4919	98	15	theorem	theorem	NOUN
ejpam-4919	98	16	2.6	2.6	NUM
ejpam-4919	98	17	in	in	ADP
ejpam-4919	98	18	[	[	X
ejpam-4919	98	19	10	10	NUM
ejpam-4919	98	20	]	]	PUNCT
ejpam-4919	98	21	,	,	PUNCT
ejpam-4919	98	22	we	we	PRON
ejpam-4919	98	23	can	can	AUX
ejpam-4919	98	24	assume	assume	VERB
ejpam-4919	98	25	,	,	PUNCT
ejpam-4919	98	26	without	without	ADP
ejpam-4919	98	27	loss	loss	NOUN
ejpam-4919	98	28	of	of	ADP
ejpam-4919	98	29	generality	generality	NOUN
ejpam-4919	98	30	,	,	PUNCT
ejpam-4919	98	31	that	that	SCONJ
ejpam-4919	98	32	e	e	NOUN
ejpam-4919	98	33	is	be	AUX
ejpam-4919	98	34	convex	convex	ADJ
ejpam-4919	98	35	.	.	PUNCT
ejpam-4919	99	1	let	let	VERB
ejpam-4919	99	2	x	x	SYM
ejpam-4919	99	3	∈	∈	NOUN
ejpam-4919	99	4	h	h	NOUN
ejpam-4919	99	5	be	be	AUX
ejpam-4919	99	6	any	any	DET
ejpam-4919	99	7	element	element	NOUN
ejpam-4919	99	8	,	,	PUNCT
ejpam-4919	99	9	and	and	CCONJ
ejpam-4919	99	10	y	y	PROPN
ejpam-4919	99	11	is	be	AUX
ejpam-4919	99	12	a	a	DET
ejpam-4919	99	13	farthest	farth	ADJ
ejpam-4919	99	14	element	element	NOUN
ejpam-4919	99	15	of	of	ADP
ejpam-4919	99	16	x	x	PROPN
ejpam-4919	99	17	in	in	ADP
ejpam-4919	99	18	e.	e.	PROPN
ejpam-4919	99	19	now	now	ADV
ejpam-4919	99	20	,	,	PUNCT
ejpam-4919	99	21	assume	assume	VERB
ejpam-4919	99	22	with	with	ADP
ejpam-4919	99	23	no	no	DET
ejpam-4919	99	24	loss	loss	NOUN
ejpam-4919	99	25	of	of	ADP
ejpam-4919	99	26	generality	generality	NOUN
ejpam-4919	99	27	that	that	PRON
ejpam-4919	99	28	∥x−	∥x−	PROPN
ejpam-4919	99	29	y∥	y∥	NOUN
ejpam-4919	99	30	=	=	PUNCT
ejpam-4919	99	31	r(x	r(x	PROPN
ejpam-4919	99	32	,	,	PUNCT
ejpam-4919	99	33	e	e	NOUN
ejpam-4919	99	34	)	)	PUNCT
ejpam-4919	99	35	=	=	SYM
ejpam-4919	100	1	1	1	X
ejpam-4919	100	2	.	.	X
ejpam-4919	100	3	we	we	PRON
ejpam-4919	100	4	claim	claim	VERB
ejpam-4919	100	5	that	that	SCONJ
ejpam-4919	100	6	inf{∥x−	inf{∥x−	PRON
ejpam-4919	100	7	y∥2	y∥2	ADJ
ejpam-4919	100	8	−	−	PROPN
ejpam-4919	100	9	∥x−	∥x−	PROPN
ejpam-4919	100	10	z∥2	z∥2	VERB
ejpam-4919	100	11	∥y	∥y	PROPN
ejpam-4919	100	12	−	−	NOUN
ejpam-4919	100	13	z∥2	z∥2	NOUN
ejpam-4919	100	14	:	:	PUNCT
ejpam-4919	100	15	z	z	PROPN
ejpam-4919	100	16	∈	∈	PROPN
ejpam-4919	100	17	e\{y	e\{y	PROPN
ejpam-4919	100	18	}	}	PUNCT
ejpam-4919	100	19	}	}	PUNCT
ejpam-4919	100	20	>	>	X
ejpam-4919	100	21	0	0	X
ejpam-4919	100	22	.	.	PUNCT
ejpam-4919	100	23	assume	assume	VERB
ejpam-4919	100	24	on	on	ADP
ejpam-4919	100	25	the	the	DET
ejpam-4919	100	26	contrary	contrary	NOUN
ejpam-4919	101	1	that	that	SCONJ
ejpam-4919	101	2	inf{∥x−	inf{∥x−	PRON
ejpam-4919	101	3	y∥2	y∥2	ADJ
ejpam-4919	101	4	−	−	PROPN
ejpam-4919	101	5	∥x−	∥x−	PROPN
ejpam-4919	101	6	z∥2	z∥2	VERB
ejpam-4919	101	7	∥y	∥y	PROPN
ejpam-4919	101	8	−	−	NOUN
ejpam-4919	101	9	z∥2	z∥2	NOUN
ejpam-4919	101	10	:	:	PUNCT
ejpam-4919	101	11	z	z	PROPN
ejpam-4919	101	12	∈	∈	PROPN
ejpam-4919	101	13	e\{y	e\{y	PROPN
ejpam-4919	101	14	}	}	PUNCT
ejpam-4919	101	15	}	}	PUNCT
ejpam-4919	102	1	=	=	SYM
ejpam-4919	102	2	0	0	PUNCT
ejpam-4919	102	3	then	then	ADV
ejpam-4919	102	4	,	,	PUNCT
ejpam-4919	102	5	there	there	PRON
ejpam-4919	102	6	exists	exist	VERB
ejpam-4919	102	7	a	a	DET
ejpam-4919	102	8	sequence	sequence	NOUN
ejpam-4919	102	9	zn	zn	PROPN
ejpam-4919	102	10	∈	∈	PROPN
ejpam-4919	102	11	e	e	NOUN
ejpam-4919	102	12	such	such	ADJ
ejpam-4919	102	13	that	that	SCONJ
ejpam-4919	102	14	lim	lim	PROPN
ejpam-4919	102	15	n→∞	n→∞	X
ejpam-4919	102	16	∥x−	∥x−	PROPN
ejpam-4919	102	17	y∥2	y∥2	NOUN
ejpam-4919	102	18	−	−	PROPN
ejpam-4919	103	1	∥x−	∥x−	PROPN
ejpam-4919	103	2	zn∥2	zn∥2	NUM
ejpam-4919	103	3	∥y	∥y	PROPN
ejpam-4919	103	4	−	−	NOUN
ejpam-4919	103	5	zn∥2	zn∥2	NOUN
ejpam-4919	103	6	=	=	SYM
ejpam-4919	103	7	0	0	NUM
ejpam-4919	104	1	(	(	PUNCT
ejpam-4919	104	2	11	11	NUM
ejpam-4919	104	3	)	)	PUNCT
ejpam-4919	104	4	since	since	SCONJ
ejpam-4919	104	5	e	e	PROPN
ejpam-4919	104	6	is	be	AUX
ejpam-4919	104	7	bounded	bound	VERB
ejpam-4919	104	8	,	,	PUNCT
ejpam-4919	104	9	this	this	PRON
ejpam-4919	104	10	implies	imply	VERB
ejpam-4919	104	11	that	that	SCONJ
ejpam-4919	104	12	lim	lim	PROPN
ejpam-4919	104	13	n→∞	n→∞	X
ejpam-4919	105	1	∥x−	∥x−	PROPN
ejpam-4919	105	2	zn∥	zn∥	NOUN
ejpam-4919	105	3	=	=	SYM
ejpam-4919	106	1	∥x−	∥x−	PROPN
ejpam-4919	106	2	y∥	y∥	NOUN
ejpam-4919	106	3	(	(	PUNCT
ejpam-4919	106	4	12	12	NUM
ejpam-4919	106	5	)	)	PUNCT
ejpam-4919	106	6	we	we	PRON
ejpam-4919	106	7	claim	claim	VERB
ejpam-4919	106	8	that	that	SCONJ
ejpam-4919	106	9	zn	zn	PROPN
ejpam-4919	106	10	→	→	PUNCT
ejpam-4919	106	11	y.	y.	PROPN
ejpam-4919	106	12	if	if	SCONJ
ejpam-4919	106	13	not	not	PART
ejpam-4919	106	14	,	,	PUNCT
ejpam-4919	106	15	then	then	ADV
ejpam-4919	106	16	there	there	PRON
ejpam-4919	106	17	exists	exist	VERB
ejpam-4919	106	18	an	an	DET
ejpam-4919	106	19	open	open	ADJ
ejpam-4919	106	20	ball	ball	NOUN
ejpam-4919	106	21	b(y	b(y	PROPN
ejpam-4919	106	22	,	,	PUNCT
ejpam-4919	106	23	δ	δ	PROPN
ejpam-4919	106	24	)	)	PUNCT
ejpam-4919	107	1	such	such	ADJ
ejpam-4919	107	2	that	that	SCONJ
ejpam-4919	107	3	zn	zn	PROPN
ejpam-4919	107	4	/∈	/∈	PUNCT
ejpam-4919	108	1	b(y	b(y	PROPN
ejpam-4919	108	2	,	,	PUNCT
ejpam-4919	108	3	δ	δ	NOUN
ejpam-4919	108	4	)	)	PUNCT
ejpam-4919	108	5	∩	∩	NOUN
ejpam-4919	108	6	e	e	NOUN
ejpam-4919	108	7	for	for	ADP
ejpam-4919	108	8	all	all	DET
ejpam-4919	108	9	n.	n.	NOUN
ejpam-4919	108	10	the	the	DET
ejpam-4919	108	11	set	set	NOUN
ejpam-4919	108	12	e\b(y	e\b(y	NOUN
ejpam-4919	108	13	,	,	PUNCT
ejpam-4919	108	14	δ	δ	PROPN
ejpam-4919	108	15	)	)	PUNCT
ejpam-4919	108	16	⊊	⊊	VERB
ejpam-4919	108	17	e	e	NOUN
ejpam-4919	108	18	is	be	AUX
ejpam-4919	108	19	a	a	DET
ejpam-4919	108	20	closed	closed	ADJ
ejpam-4919	108	21	set	set	NOUN
ejpam-4919	108	22	.	.	PUNCT
ejpam-4919	109	1	since	since	SCONJ
ejpam-4919	109	2	e	e	PROPN
ejpam-4919	109	3	is	be	AUX
ejpam-4919	109	4	uniquely	uniquely	ADV
ejpam-4919	109	5	distant	distant	ADJ
ejpam-4919	109	6	then	then	ADV
ejpam-4919	109	7	for	for	ADP
ejpam-4919	109	8	any	any	DET
ejpam-4919	109	9	ε	ε	PROPN
ejpam-4919	109	10	>	>	X
ejpam-4919	109	11	0	0	PROPN
ejpam-4919	109	12	,	,	PUNCT
ejpam-4919	109	13	we	we	PRON
ejpam-4919	109	14	haveb(y	haveb(y	VERB
ejpam-4919	109	15	,	,	PUNCT
ejpam-4919	109	16	δ	δ	PROPN
ejpam-4919	109	17	)	)	PUNCT
ejpam-4919	109	18	such	such	ADJ
ejpam-4919	109	19	that	that	SCONJ
ejpam-4919	109	20	r(x	r(x	PROPN
ejpam-4919	109	21	,	,	PUNCT
ejpam-4919	109	22	e\b(y	e\b(y	ADJ
ejpam-4919	109	23	,	,	PUNCT
ejpam-4919	109	24	δ	δ	PROPN
ejpam-4919	109	25	)	)	PUNCT
ejpam-4919	109	26	)	)	PUNCT
ejpam-4919	109	27	≤	≤	NOUN
ejpam-4919	109	28	1−	1−	NUM
ejpam-4919	110	1	ε	ε	PROPN
ejpam-4919	110	2	.	.	PUNCT
ejpam-4919	111	1	in	in	ADP
ejpam-4919	111	2	this	this	DET
ejpam-4919	111	3	case	case	NOUN
ejpam-4919	111	4	equation	equation	NOUN
ejpam-4919	111	5	12	12	NUM
ejpam-4919	111	6	is	be	AUX
ejpam-4919	111	7	not	not	PART
ejpam-4919	111	8	true	true	ADJ
ejpam-4919	111	9	.	.	PUNCT
ejpam-4919	112	1	thus	thus	ADV
ejpam-4919	112	2	,	,	PUNCT
ejpam-4919	112	3	we	we	PRON
ejpam-4919	112	4	must	must	AUX
ejpam-4919	112	5	have	have	VERB
ejpam-4919	112	6	limn→∞	limn→∞	PROPN
ejpam-4919	112	7	zn	zn	NOUN
ejpam-4919	112	8	=	=	SYM
ejpam-4919	112	9	y.	y.	PROPN
ejpam-4919	112	10	now	now	ADV
ejpam-4919	112	11	,	,	PUNCT
ejpam-4919	112	12	let	let	VERB
ejpam-4919	112	13	[	[	X
ejpam-4919	112	14	p	p	X
ejpam-4919	112	15	,	,	PUNCT
ejpam-4919	112	16	q	q	X
ejpam-4919	112	17	]	]	PUNCT
ejpam-4919	112	18	be	be	AUX
ejpam-4919	112	19	the	the	DET
ejpam-4919	112	20	line	line	NOUN
ejpam-4919	112	21	segment	segment	NOUN
ejpam-4919	112	22	joining	join	VERB
ejpam-4919	112	23	p	p	PROPN
ejpam-4919	112	24	and	and	CCONJ
ejpam-4919	112	25	q.	q.	PROPN
ejpam-4919	112	26	since	since	SCONJ
ejpam-4919	112	27	h	h	NOUN
ejpam-4919	112	28	is	be	AUX
ejpam-4919	112	29	a	a	DET
ejpam-4919	112	30	hilbert	hilbert	NOUN
ejpam-4919	112	31	space	space	NOUN
ejpam-4919	112	32	,	,	PUNCT
ejpam-4919	112	33	it	it	PRON
ejpam-4919	112	34	follows	follow	VERB
ejpam-4919	112	35	that	that	SCONJ
ejpam-4919	112	36	for	for	ADP
ejpam-4919	112	37	each	each	DET
ejpam-4919	112	38	zn	zn	NOUN
ejpam-4919	112	39	there	there	PRON
ejpam-4919	112	40	exists	exist	VERB
ejpam-4919	112	41	wn	wn	INTJ
ejpam-4919	112	42	in	in	ADP
ejpam-4919	112	43	[	[	X
ejpam-4919	112	44	x	x	NOUN
ejpam-4919	112	45	,	,	PUNCT
ejpam-4919	112	46	y	y	PROPN
ejpam-4919	112	47	]	]	X
ejpam-4919	112	48	such	such	ADJ
ejpam-4919	112	49	that	that	SCONJ
ejpam-4919	112	50	[	[	X
ejpam-4919	112	51	zn	zn	X
ejpam-4919	112	52	,	,	PUNCT
ejpam-4919	112	53	wn	wn	PROPN
ejpam-4919	112	54	]	]	PUNCT
ejpam-4919	112	55	is	be	AUX
ejpam-4919	112	56	orthogonal	orthogonal	ADJ
ejpam-4919	112	57	to	to	ADP
ejpam-4919	112	58	[	[	X
ejpam-4919	112	59	x	x	X
ejpam-4919	112	60	,	,	PUNCT
ejpam-4919	112	61	y	y	PROPN
ejpam-4919	112	62	]	]	PUNCT
ejpam-4919	112	63	.	.	PUNCT
ejpam-4919	113	1	assume	assume	VERB
ejpam-4919	113	2	that	that	SCONJ
ejpam-4919	113	3	∥zn	∥zn	PROPN
ejpam-4919	114	1	−	−	PROPN
ejpam-4919	114	2	wn∥	wn∥	PROPN
ejpam-4919	114	3	=	=	SYM
ejpam-4919	114	4	dn	dn	PROPN
ejpam-4919	114	5	,	,	PUNCT
ejpam-4919	114	6	and	and	CCONJ
ejpam-4919	114	7	∥y	∥y	PROPN
ejpam-4919	114	8	−	−	PROPN
ejpam-4919	115	1	wn∥	wn∥	PROPN
ejpam-4919	115	2	=	=	PUNCT
ejpam-4919	115	3	an	an	PROPN
ejpam-4919	115	4	and	and	CCONJ
ejpam-4919	115	5	∥x−	∥x−	NUM
ejpam-4919	116	1	wn∥	wn∥	PROPN
ejpam-4919	116	2	=	=	SYM
ejpam-4919	116	3	bn	bn	PROPN
ejpam-4919	116	4	.	.	PROPN
ejpam-4919	117	1	from	from	ADP
ejpam-4919	117	2	basic	basic	ADJ
ejpam-4919	117	3	geometry	geometry	NOUN
ejpam-4919	117	4	in	in	ADP
ejpam-4919	117	5	hilbert	hilbert	PROPN
ejpam-4919	117	6	spaces	space	NOUN
ejpam-4919	117	7	we	we	PRON
ejpam-4919	117	8	have	have	VERB
ejpam-4919	117	9	:	:	PUNCT
ejpam-4919	117	10	a.	a.	PROPN
ejpam-4919	117	11	yousef	yousef	PROPN
ejpam-4919	117	12	et	et	PROPN
ejpam-4919	117	13	al	al	PROPN
ejpam-4919	117	14	.	.	PUNCT
ejpam-4919	117	15	/	/	SYM
ejpam-4919	117	16	eur	eur	PROPN
ejpam-4919	117	17	.	.	PUNCT
ejpam-4919	118	1	j.	j.	PROPN
ejpam-4919	118	2	pure	pure	PROPN
ejpam-4919	118	3	appl	appl	PROPN
ejpam-4919	118	4	.	.	PROPN
ejpam-4919	118	5	math	math	PROPN
ejpam-4919	118	6	,	,	PUNCT
ejpam-4919	118	7	16	16	NUM
ejpam-4919	118	8	(	(	PUNCT
ejpam-4919	118	9	4	4	NUM
ejpam-4919	118	10	)	)	PUNCT
ejpam-4919	118	11	(	(	PUNCT
ejpam-4919	118	12	2023	2023	NUM
ejpam-4919	118	13	)	)	PUNCT
ejpam-4919	118	14	,	,	PUNCT
ejpam-4919	118	15	2397	2397	NUM
ejpam-4919	118	16	-	-	SYM
ejpam-4919	118	17	2404	2404	NUM
ejpam-4919	118	18	2402	2402	NUM
ejpam-4919	118	19	∥x−	∥x−	NUM
ejpam-4919	118	20	y∥2	y∥2	NOUN
ejpam-4919	118	21	=	=	SYM
ejpam-4919	118	22	1	1	NUM
ejpam-4919	118	23	=	=	SYM
ejpam-4919	118	24	(	(	PUNCT
ejpam-4919	118	25	an	an	DET
ejpam-4919	118	26	+	+	NOUN
ejpam-4919	118	27	bn	bn	NOUN
ejpam-4919	118	28	)	)	PUNCT
ejpam-4919	118	29	2	2	NUM
ejpam-4919	118	30	,	,	PUNCT
ejpam-4919	118	31	∥x−	∥x−	PROPN
ejpam-4919	118	32	zn∥2	zn∥2	NUM
ejpam-4919	118	33	=	=	SYM
ejpam-4919	118	34	d2n	d2n	NOUN
ejpam-4919	118	35	+	+	CCONJ
ejpam-4919	118	36	b2n	b2n	NUM
ejpam-4919	118	37	,	,	PUNCT
ejpam-4919	118	38	and	and	CCONJ
ejpam-4919	118	39	∥y	∥y	PROPN
ejpam-4919	118	40	−	−	NOUN
ejpam-4919	118	41	zn∥2	zn∥2	NOUN
ejpam-4919	118	42	=	=	SYM
ejpam-4919	118	43	d2n	d2n	NOUN
ejpam-4919	118	44	+	+	CCONJ
ejpam-4919	118	45	a2n	a2n	PROPN
ejpam-4919	118	46	.	.	PUNCT
ejpam-4919	119	1	thus	thus	ADV
ejpam-4919	119	2	∥x−	∥x−	NUM
ejpam-4919	119	3	y∥2	y∥2	NOUN
ejpam-4919	119	4	−	−	PROPN
ejpam-4919	120	1	∥x−	∥x−	PROPN
ejpam-4919	120	2	zn∥2	zn∥2	NUM
ejpam-4919	120	3	∥y	∥y	PROPN
ejpam-4919	120	4	−	−	NOUN
ejpam-4919	120	5	zn∥2	zn∥2	NOUN
ejpam-4919	120	6	=	=	SYM
ejpam-4919	120	7	(	(	PUNCT
ejpam-4919	120	8	an	an	DET
ejpam-4919	120	9	+	+	NOUN
ejpam-4919	120	10	bn	bn	NOUN
ejpam-4919	120	11	)	)	PUNCT
ejpam-4919	120	12	2	2	NUM
ejpam-4919	120	13	−	−	PROPN
ejpam-4919	120	14	(	(	PUNCT
ejpam-4919	120	15	d2n	d2n	PROPN
ejpam-4919	120	16	+	+	CCONJ
ejpam-4919	120	17	b2n	b2n	NUM
ejpam-4919	120	18	)	)	PUNCT
ejpam-4919	120	19	d2n	d2n	PROPN
ejpam-4919	121	1	+	+	CCONJ
ejpam-4919	121	2	a2n	a2n	PROPN
ejpam-4919	121	3	.	.	PUNCT
ejpam-4919	122	1	(	(	PUNCT
ejpam-4919	122	2	13	13	NUM
ejpam-4919	122	3	)	)	PUNCT
ejpam-4919	122	4	but	but	CCONJ
ejpam-4919	122	5	(	(	PUNCT
ejpam-4919	122	6	an	an	DET
ejpam-4919	122	7	+	+	NOUN
ejpam-4919	122	8	bn	bn	NOUN
ejpam-4919	122	9	)	)	PUNCT
ejpam-4919	122	10	2	2	NUM
ejpam-4919	122	11	−	−	PROPN
ejpam-4919	122	12	(	(	PUNCT
ejpam-4919	122	13	d2n	d2n	PROPN
ejpam-4919	122	14	+	+	CCONJ
ejpam-4919	122	15	b2n	b2n	NUM
ejpam-4919	122	16	)	)	PUNCT
ejpam-4919	122	17	=	=	PRON
ejpam-4919	123	1	a2n	a2n	PROPN
ejpam-4919	124	1	+	+	CCONJ
ejpam-4919	124	2	b2n	b2n	NUM
ejpam-4919	124	3	+	+	SYM
ejpam-4919	124	4	2anbn	2anbn	NUM
ejpam-4919	124	5	−	−	NOUN
ejpam-4919	124	6	(	(	PUNCT
ejpam-4919	124	7	d2n	d2n	PROPN
ejpam-4919	124	8	+	+	CCONJ
ejpam-4919	124	9	b2n	b2n	NUM
ejpam-4919	124	10	)	)	PUNCT
ejpam-4919	124	11	=	=	SYM
ejpam-4919	125	1	2a2n	2a2n	NOUN
ejpam-4919	126	1	+	+	SYM
ejpam-4919	127	1	2anbn	2anbn	NUM
ejpam-4919	127	2	−	−	NOUN
ejpam-4919	127	3	(	(	PUNCT
ejpam-4919	127	4	d2n	d2n	PROPN
ejpam-4919	127	5	+	+	CCONJ
ejpam-4919	127	6	a2n	a2n	NOUN
ejpam-4919	127	7	)	)	PUNCT
ejpam-4919	127	8	.	.	PUNCT
ejpam-4919	128	1	since	since	SCONJ
ejpam-4919	128	2	e	e	PROPN
ejpam-4919	128	3	is	be	AUX
ejpam-4919	128	4	uniquely	uniquely	ADV
ejpam-4919	128	5	remotal	remotal	ADJ
ejpam-4919	128	6	and	and	CCONJ
ejpam-4919	128	7	r(x	r(x	PROPN
ejpam-4919	128	8	,	,	PUNCT
ejpam-4919	128	9	e	e	NOUN
ejpam-4919	128	10	)	)	PUNCT
ejpam-4919	128	11	=	=	SYM
ejpam-4919	129	1	∥x−	∥x−	PROPN
ejpam-4919	129	2	y∥	y∥	NOUN
ejpam-4919	129	3	,	,	PUNCT
ejpam-4919	129	4	so	so	ADV
ejpam-4919	129	5	∥x−	∥x−	PROPN
ejpam-4919	129	6	y∥	y∥	VERB
ejpam-4919	129	7	>	>	PUNCT
ejpam-4919	129	8	∥x−	∥x−	PROPN
ejpam-4919	129	9	zn∥	zn∥	PROPN
ejpam-4919	129	10	for	for	ADP
ejpam-4919	129	11	all	all	DET
ejpam-4919	129	12	n	n	CCONJ
ejpam-4919	129	13	,	,	PUNCT
ejpam-4919	129	14	it	it	PRON
ejpam-4919	129	15	follows	follow	VERB
ejpam-4919	129	16	that	that	SCONJ
ejpam-4919	129	17	2a2n	2a2n	PROPN
ejpam-4919	130	1	+	+	PROPN
ejpam-4919	130	2	2anbn	2anbn	NUM
ejpam-4919	130	3	−	−	NOUN
ejpam-4919	130	4	(	(	PUNCT
ejpam-4919	130	5	d2n	d2n	PROPN
ejpam-4919	130	6	+	+	CCONJ
ejpam-4919	130	7	a2n	a2n	NOUN
ejpam-4919	130	8	)	)	PUNCT
ejpam-4919	130	9	>	>	X
ejpam-4919	130	10	0	0	PUNCT
ejpam-4919	131	1	but	but	CCONJ
ejpam-4919	131	2	2a2n	2a2n	NUM
ejpam-4919	132	1	+	+	CCONJ
ejpam-4919	132	2	2anbn	2anbn	NUM
ejpam-4919	132	3	=	=	SYM
ejpam-4919	132	4	2an(an	2an(an	PROPN
ejpam-4919	132	5	+	+	CCONJ
ejpam-4919	132	6	bn	bn	NUM
ejpam-4919	132	7	)	)	PUNCT
ejpam-4919	132	8	=	=	SYM
ejpam-4919	132	9	2an	2an	NOUN
ejpam-4919	132	10	.	.	PUNCT
ejpam-4919	133	1	thus	thus	ADV
ejpam-4919	133	2	2an	2an	ADJ
ejpam-4919	133	3	−	−	PROPN
ejpam-4919	133	4	(	(	PUNCT
ejpam-4919	133	5	d2n	d2n	PROPN
ejpam-4919	133	6	+	+	CCONJ
ejpam-4919	133	7	a2n	a2n	NOUN
ejpam-4919	133	8	)	)	PUNCT
ejpam-4919	133	9	>	>	X
ejpam-4919	133	10	0	0	X
ejpam-4919	133	11	.	.	PUNCT
ejpam-4919	134	1	(	(	PUNCT
ejpam-4919	134	2	14	14	NUM
ejpam-4919	134	3	)	)	PUNCT
ejpam-4919	134	4	equations	equation	NOUN
ejpam-4919	134	5	13	13	NUM
ejpam-4919	134	6	and	and	CCONJ
ejpam-4919	134	7	14	14	NUM
ejpam-4919	134	8	gives	give	VERB
ejpam-4919	134	9	∥x−	∥x−	PROPN
ejpam-4919	134	10	y∥2	y∥2	NOUN
ejpam-4919	134	11	−	−	PROPN
ejpam-4919	135	1	∥x−	∥x−	PROPN
ejpam-4919	135	2	zn∥2	zn∥2	NUM
ejpam-4919	135	3	∥y	∥y	PROPN
ejpam-4919	135	4	−	−	NOUN
ejpam-4919	135	5	zn∥2	zn∥2	NOUN
ejpam-4919	135	6	=	=	SYM
ejpam-4919	135	7	2an	2an	NOUN
ejpam-4919	135	8	−	−	PROPN
ejpam-4919	135	9	(	(	PUNCT
ejpam-4919	135	10	d2n	d2n	PROPN
ejpam-4919	135	11	+	+	CCONJ
ejpam-4919	135	12	a2n	a2n	NOUN
ejpam-4919	135	13	)	)	PUNCT
ejpam-4919	135	14	d2n	d2n	PROPN
ejpam-4919	136	1	+	+	CCONJ
ejpam-4919	136	2	a2n	a2n	PROPN
ejpam-4919	136	3	=	=	SYM
ejpam-4919	136	4	2an	2an	ADJ
ejpam-4919	136	5	d2n	d2n	PROPN
ejpam-4919	136	6	+	+	CCONJ
ejpam-4919	136	7	a2n	a2n	PROPN
ejpam-4919	136	8	−	−	ADP
ejpam-4919	136	9	1	1	NUM
ejpam-4919	136	10	.	.	PUNCT
ejpam-4919	136	11	(	(	PUNCT
ejpam-4919	136	12	15	15	NUM
ejpam-4919	136	13	)	)	PUNCT
ejpam-4919	136	14	from	from	ADP
ejpam-4919	136	15	14	14	NUM
ejpam-4919	136	16	we	we	PRON
ejpam-4919	136	17	have	have	VERB
ejpam-4919	136	18	2an	2an	ADJ
ejpam-4919	136	19	d2n	d2n	PROPN
ejpam-4919	136	20	+	+	CCONJ
ejpam-4919	136	21	a2n	a2n	PROPN
ejpam-4919	136	22	−	−	ADP
ejpam-4919	136	23	1	1	NUM
ejpam-4919	136	24	>	>	SYM
ejpam-4919	136	25	0	0	NUM
ejpam-4919	136	26	,	,	PUNCT
ejpam-4919	136	27	which	which	PRON
ejpam-4919	136	28	is	be	AUX
ejpam-4919	136	29	equivalent	equivalent	ADJ
ejpam-4919	136	30	to	to	ADP
ejpam-4919	136	31	2an	2an	ADJ
ejpam-4919	136	32	d2n	d2n	PROPN
ejpam-4919	136	33	+	+	CCONJ
ejpam-4919	136	34	a2n	a2n	PROPN
ejpam-4919	136	35	>	>	X
ejpam-4919	136	36	1	1	X
ejpam-4919	136	37	.	.	PUNCT
ejpam-4919	137	1	(	(	PUNCT
ejpam-4919	137	2	16	16	NUM
ejpam-4919	137	3	)	)	PUNCT
ejpam-4919	137	4	now	now	ADV
ejpam-4919	137	5	,	,	PUNCT
ejpam-4919	137	6	one	one	PRON
ejpam-4919	137	7	should	should	AUX
ejpam-4919	137	8	remark	remark	VERB
ejpam-4919	137	9	that	that	SCONJ
ejpam-4919	137	10	as	as	SCONJ
ejpam-4919	137	11	zn	zn	PROPN
ejpam-4919	137	12	approaches	approach	VERB
ejpam-4919	137	13	y	y	PROPN
ejpam-4919	137	14	,	,	PUNCT
ejpam-4919	137	15	then	then	ADV
ejpam-4919	137	16	(	(	PUNCT
ejpam-4919	137	17	an	an	PRON
ejpam-4919	137	18	,	,	PUNCT
ejpam-4919	137	19	dn	dn	PROPN
ejpam-4919	137	20	)	)	PUNCT
ejpam-4919	137	21	→	→	SYM
ejpam-4919	137	22	(	(	PUNCT
ejpam-4919	137	23	0	0	NUM
ejpam-4919	137	24	,	,	PUNCT
ejpam-4919	137	25	0	0	NUM
ejpam-4919	137	26	)	)	PUNCT
ejpam-4919	137	27	.	.	PUNCT
ejpam-4919	138	1	equation	equation	NOUN
ejpam-4919	138	2	16	16	NUM
ejpam-4919	138	3	implies	imply	VERB
ejpam-4919	138	4	that	that	SCONJ
ejpam-4919	138	5	an	an	PRON
ejpam-4919	138	6	does	do	AUX
ejpam-4919	138	7	not	not	PART
ejpam-4919	138	8	converge	converge	VERB
ejpam-4919	138	9	to	to	ADP
ejpam-4919	138	10	zero	zero	NUM
ejpam-4919	138	11	before	before	ADP
ejpam-4919	138	12	dn	dn	NOUN
ejpam-4919	138	13	,	,	PUNCT
ejpam-4919	138	14	otherwise	otherwise	ADV
ejpam-4919	138	15	we	we	PRON
ejpam-4919	138	16	obtain	obtain	VERB
ejpam-4919	138	17	2an	2an	ADJ
ejpam-4919	138	18	d2n	d2n	NOUN
ejpam-4919	138	19	+	+	CCONJ
ejpam-4919	138	20	a2n	a2n	PROPN
ejpam-4919	138	21	→	→	SYM
ejpam-4919	138	22	0	0	NUM
ejpam-4919	138	23	,	,	PUNCT
ejpam-4919	138	24	which	which	PRON
ejpam-4919	138	25	contradicts	contradict	VERB
ejpam-4919	138	26	16	16	NUM
ejpam-4919	138	27	.	.	PUNCT
ejpam-4919	139	1	therefore	therefore	ADV
ejpam-4919	139	2	,	,	PUNCT
ejpam-4919	139	3	we	we	PRON
ejpam-4919	139	4	can	can	AUX
ejpam-4919	139	5	assume	assume	VERB
ejpam-4919	139	6	that	that	SCONJ
ejpam-4919	139	7	dn	dn	PROPN
ejpam-4919	139	8	converges	converge	VERB
ejpam-4919	139	9	to	to	ADP
ejpam-4919	139	10	0	0	NUM
ejpam-4919	139	11	as	as	ADP
ejpam-4919	139	12	a	a	DET
ejpam-4919	139	13	function	function	NOUN
ejpam-4919	139	14	of	of	ADP
ejpam-4919	139	15	an	an	PRON
ejpam-4919	139	16	,	,	PUNCT
ejpam-4919	139	17	since	since	SCONJ
ejpam-4919	139	18	∥y	∥y	PROPN
ejpam-4919	139	19	−	−	PROPN
ejpam-4919	139	20	zn∥2	zn∥2	NOUN
ejpam-4919	139	21	=	=	SYM
ejpam-4919	139	22	d2n	d2n	NOUN
ejpam-4919	139	23	+	+	CCONJ
ejpam-4919	139	24	a2n	a2n	PROPN
ejpam-4919	139	25	.	.	PUNCT
ejpam-4919	140	1	also	also	ADV
ejpam-4919	140	2	,	,	PUNCT
ejpam-4919	140	3	by	by	ADP
ejpam-4919	140	4	sequential	sequential	ADJ
ejpam-4919	140	5	criterion	criterion	NOUN
ejpam-4919	140	6	,	,	PUNCT
ejpam-4919	140	7	assume	assume	VERB
ejpam-4919	140	8	with	with	ADP
ejpam-4919	140	9	no	no	DET
ejpam-4919	140	10	loss	loss	NOUN
ejpam-4919	140	11	of	of	ADP
ejpam-4919	140	12	generality	generality	NOUN
ejpam-4919	140	13	that	that	PRON
ejpam-4919	140	14	dn	dn	PROPN
ejpam-4919	140	15	is	be	AUX
ejpam-4919	140	16	a	a	DET
ejpam-4919	140	17	differentiable	differentiable	ADJ
ejpam-4919	140	18	function	function	NOUN
ejpam-4919	140	19	of	of	ADP
ejpam-4919	140	20	an	an	PRON
ejpam-4919	140	21	.	.	PUNCT
ejpam-4919	141	1	now	now	ADV
ejpam-4919	141	2	,	,	PUNCT
ejpam-4919	141	3	we	we	PRON
ejpam-4919	141	4	can	can	AUX
ejpam-4919	141	5	think	think	VERB
ejpam-4919	141	6	of	of	ADP
ejpam-4919	141	7	the	the	DET
ejpam-4919	141	8	following	follow	VERB
ejpam-4919	141	9	limit	limit	NOUN
ejpam-4919	141	10	:	:	PUNCT
ejpam-4919	141	11	lim	lim	PROPN
ejpam-4919	141	12	z→y	z→y	NUM
ejpam-4919	141	13	∥x−	∥x−	PROPN
ejpam-4919	141	14	y∥2	y∥2	NOUN
ejpam-4919	141	15	−	−	PROPN
ejpam-4919	142	1	∥x−	∥x−	PROPN
ejpam-4919	142	2	zn∥2	zn∥2	NUM
ejpam-4919	142	3	∥y	∥y	PROPN
ejpam-4919	142	4	−	−	NOUN
ejpam-4919	142	5	zn∥2	zn∥2	NOUN
ejpam-4919	142	6	=	=	SYM
ejpam-4919	142	7	lim	lim	PROPN
ejpam-4919	142	8	(	(	PUNCT
ejpam-4919	142	9	an	an	PROPN
ejpam-4919	142	10	,	,	PUNCT
ejpam-4919	142	11	dn)→(0,0	dn)→(0,0	NOUN
ejpam-4919	142	12	)	)	PUNCT
ejpam-4919	142	13	(	(	PUNCT
ejpam-4919	142	14	2an	2an	ADJ
ejpam-4919	142	15	d2n	d2n	PROPN
ejpam-4919	142	16	+	+	CCONJ
ejpam-4919	142	17	a2n	a2n	PROPN
ejpam-4919	142	18	−	−	ADP
ejpam-4919	142	19	1	1	NUM
ejpam-4919	142	20	)	)	PUNCT
ejpam-4919	142	21	,	,	PUNCT
ejpam-4919	142	22	(	(	PUNCT
ejpam-4919	142	23	17	17	NUM
ejpam-4919	142	24	)	)	PUNCT
ejpam-4919	142	25	as	as	ADP
ejpam-4919	142	26	lim	lim	PROPN
ejpam-4919	142	27	w→0	w→0	NOUN
ejpam-4919	142	28	(	(	PUNCT
ejpam-4919	142	29	2w	2w	NUM
ejpam-4919	142	30	p	p	NOUN
ejpam-4919	142	31	2(w	2(w	NUM
ejpam-4919	142	32	)	)	PUNCT
ejpam-4919	142	33	+	+	NUM
ejpam-4919	142	34	w2	w2	NOUN
ejpam-4919	142	35	−	−	PROPN
ejpam-4919	142	36	1	1	NUM
ejpam-4919	142	37	)	)	PUNCT
ejpam-4919	142	38	.	.	PUNCT
ejpam-4919	143	1	using	use	VERB
ejpam-4919	143	2	l’hopital	l’hopital	PROPN
ejpam-4919	143	3	’s	’s	PART
ejpam-4919	143	4	rule	rule	NOUN
ejpam-4919	143	5	we	we	PRON
ejpam-4919	143	6	get	get	AUX
ejpam-4919	143	7	:	:	PUNCT
ejpam-4919	143	8	references	reference	NOUN
ejpam-4919	143	9	2403	2403	NUM
ejpam-4919	143	10	lim	lim	PROPN
ejpam-4919	143	11	(	(	PUNCT
ejpam-4919	143	12	an	an	PROPN
ejpam-4919	143	13	,	,	PUNCT
ejpam-4919	143	14	dn)→(0,0	dn)→(0,0	NOUN
ejpam-4919	143	15	)	)	PUNCT
ejpam-4919	143	16	(	(	PUNCT
ejpam-4919	143	17	2an	2an	ADJ
ejpam-4919	143	18	d2n	d2n	PROPN
ejpam-4919	143	19	+	+	CCONJ
ejpam-4919	144	1	a2n	a2n	PROPN
ejpam-4919	144	2	−	−	ADP
ejpam-4919	144	3	1	1	NUM
ejpam-4919	144	4	)	)	PUNCT
ejpam-4919	144	5	=	=	SYM
ejpam-4919	144	6	lim	lim	PROPN
ejpam-4919	144	7	(	(	PUNCT
ejpam-4919	144	8	an	an	PROPN
ejpam-4919	144	9	,	,	PUNCT
ejpam-4919	144	10	dn)→(0,0	dn)→(0,0	NOUN
ejpam-4919	144	11	)	)	PUNCT
ejpam-4919	144	12	(	(	PUNCT
ejpam-4919	144	13	2	2	NUM
ejpam-4919	144	14	2dn.d′n	2dn.d′n	NUM
ejpam-4919	144	15	+	+	NUM
ejpam-4919	144	16	2an	2an	ADJ
ejpam-4919	144	17	−	−	NOUN
ejpam-4919	144	18	1	1	NUM
ejpam-4919	144	19	)	)	PUNCT
ejpam-4919	144	20	.	.	PUNCT
ejpam-4919	145	1	(	(	PUNCT
ejpam-4919	145	2	18	18	NUM
ejpam-4919	145	3	)	)	PUNCT
ejpam-4919	145	4	clearly	clearly	ADV
ejpam-4919	145	5	,	,	PUNCT
ejpam-4919	145	6	the	the	DET
ejpam-4919	145	7	limit	limit	NOUN
ejpam-4919	145	8	in	in	ADP
ejpam-4919	145	9	18	18	NUM
ejpam-4919	145	10	is	be	AUX
ejpam-4919	145	11	certainly	certainly	ADV
ejpam-4919	145	12	bounded	bound	VERB
ejpam-4919	145	13	away	away	ADV
ejpam-4919	145	14	from	from	ADP
ejpam-4919	145	15	zero	zero	NUM
ejpam-4919	145	16	noting	note	VERB
ejpam-4919	145	17	that	that	SCONJ
ejpam-4919	145	18	by	by	ADP
ejpam-4919	145	19	unique	unique	ADJ
ejpam-4919	145	20	remotality	remotality	NOUN
ejpam-4919	145	21	,	,	PUNCT
ejpam-4919	145	22	d′n	d′n	VERB
ejpam-4919	145	23	can	can	AUX
ejpam-4919	145	24	not	not	PART
ejpam-4919	145	25	equal	equal	VERB
ejpam-4919	145	26	to	to	ADP
ejpam-4919	145	27	infinity	infinity	NOUN
ejpam-4919	145	28	.	.	PUNCT
ejpam-4919	146	1	otherwise	otherwise	ADV
ejpam-4919	146	2	,	,	PUNCT
ejpam-4919	146	3	∥x−	∥x−	PROPN
ejpam-4919	146	4	zn∥	zn∥	PROPN
ejpam-4919	146	5	will	will	AUX
ejpam-4919	146	6	reach	reach	VERB
ejpam-4919	146	7	∥x−	∥x−	PROPN
ejpam-4919	146	8	y∥	y∥	NOUN
ejpam-4919	146	9	before	before	SCONJ
ejpam-4919	146	10	zn	zn	PROPN
ejpam-4919	146	11	reaches	reach	VERB
ejpam-4919	146	12	y.	y.	PROPN
ejpam-4919	146	13	hence	hence	ADV
ejpam-4919	146	14	e	e	PROPN
ejpam-4919	146	15	is	be	AUX
ejpam-4919	146	16	strongly	strongly	ADV
ejpam-4919	146	17	remotal	remotal	ADJ
ejpam-4919	146	18	,	,	PUNCT
ejpam-4919	146	19	and	and	CCONJ
ejpam-4919	146	20	by	by	ADP
ejpam-4919	146	21	theorem	theorem	NOUN
ejpam-4919	146	22	3	3	NUM
ejpam-4919	146	23	,	,	PUNCT
ejpam-4919	146	24	e	e	X
ejpam-4919	146	25	is	be	AUX
ejpam-4919	146	26	a	a	DET
ejpam-4919	146	27	singleton	singleton	NOUN
ejpam-4919	146	28	.	.	PUNCT
ejpam-4919	147	1	this	this	PRON
ejpam-4919	147	2	completes	complete	VERB
ejpam-4919	147	3	the	the	DET
ejpam-4919	147	4	proof	proof	NOUN
ejpam-4919	147	5	.	.	PUNCT
ejpam-4919	148	1	clearly	clearly	ADV
ejpam-4919	148	2	,	,	PUNCT
ejpam-4919	148	3	one	one	PRON
ejpam-4919	148	4	can	can	AUX
ejpam-4919	148	5	see	see	VERB
ejpam-4919	148	6	that	that	SCONJ
ejpam-4919	148	7	every	every	DET
ejpam-4919	148	8	uniquely	uniquely	ADV
ejpam-4919	148	9	distant	distant	ADJ
ejpam-4919	148	10	set	set	NOUN
ejpam-4919	148	11	is	be	AUX
ejpam-4919	148	12	uniquely	uniquely	ADV
ejpam-4919	148	13	remotal	remotal	ADJ
ejpam-4919	148	14	.	.	PUNCT
ejpam-4919	149	1	however	however	ADV
ejpam-4919	149	2	,	,	PUNCT
ejpam-4919	149	3	we	we	PRON
ejpam-4919	149	4	could	could	AUX
ejpam-4919	149	5	not	not	PART
ejpam-4919	149	6	find	find	VERB
ejpam-4919	149	7	a	a	DET
ejpam-4919	149	8	uniquely	uniquely	ADV
ejpam-4919	149	9	remotal	remotal	ADJ
ejpam-4919	149	10	set	set	NOUN
ejpam-4919	149	11	that	that	PRON
ejpam-4919	149	12	is	be	AUX
ejpam-4919	149	13	not	not	PART
ejpam-4919	149	14	uniquely	uniquely	ADV
ejpam-4919	149	15	distant	distant	ADJ
ejpam-4919	149	16	.	.	PUNCT
ejpam-4919	150	1	so	so	ADV
ejpam-4919	150	2	,	,	PUNCT
ejpam-4919	150	3	we	we	PRON
ejpam-4919	150	4	believe	believe	VERB
ejpam-4919	150	5	that	that	SCONJ
ejpam-4919	150	6	they	they	PRON
ejpam-4919	150	7	are	be	AUX
ejpam-4919	150	8	equivalent	equivalent	ADJ
ejpam-4919	150	9	,	,	PUNCT
ejpam-4919	150	10	and	and	CCONJ
ejpam-4919	150	11	we	we	PRON
ejpam-4919	150	12	conjecture	conjecture	VERB
ejpam-4919	150	13	the	the	DET
ejpam-4919	150	14	following	following	NOUN
ejpam-4919	150	15	:	:	PUNCT
ejpam-4919	150	16	conjecture	conjecture	NOUN
ejpam-4919	150	17	:	:	PUNCT
ejpam-4919	150	18	every	every	DET
ejpam-4919	150	19	uniquely	uniquely	ADV
ejpam-4919	150	20	remotal	remotal	ADJ
ejpam-4919	150	21	set	set	NOUN
ejpam-4919	150	22	in	in	ADP
ejpam-4919	150	23	hilbert	hilbert	NOUN
ejpam-4919	150	24	space	space	NOUN
ejpam-4919	150	25	is	be	AUX
ejpam-4919	150	26	uniquely	uniquely	ADV
ejpam-4919	150	27	distant	distant	ADJ
ejpam-4919	150	28	.	.	PUNCT
ejpam-4919	151	1	acknowledgements	acknowledgement	NOUN
ejpam-4919	151	2	this	this	DET
ejpam-4919	151	3	work	work	NOUN
ejpam-4919	151	4	has	have	AUX
ejpam-4919	151	5	been	be	AUX
ejpam-4919	151	6	done	do	VERB
ejpam-4919	151	7	while	while	SCONJ
ejpam-4919	151	8	the	the	DET
ejpam-4919	151	9	first	first	ADJ
ejpam-4919	151	10	author	author	NOUN
ejpam-4919	151	11	is	be	AUX
ejpam-4919	151	12	on	on	ADP
ejpam-4919	151	13	sabbatical	sabbatical	ADJ
ejpam-4919	151	14	leave	leave	NOUN
ejpam-4919	151	15	from	from	ADP
ejpam-4919	151	16	the	the	DET
ejpam-4919	151	17	university	university	PROPN
ejpam-4919	151	18	of	of	ADP
ejpam-4919	151	19	jordan	jordan	PROPN
ejpam-4919	151	20	to	to	ADP
ejpam-4919	151	21	the	the	DET
ejpam-4919	151	22	american	american	PROPN
ejpam-4919	151	23	university	university	PROPN
ejpam-4919	151	24	of	of	ADP
ejpam-4919	151	25	sharjah	sharjah	PROPN
ejpam-4919	151	26	.	.	PUNCT
ejpam-4919	152	1	the	the	DET
ejpam-4919	152	2	author	author	NOUN
ejpam-4919	152	3	is	be	AUX
ejpam-4919	152	4	grateful	grateful	ADJ
ejpam-4919	152	5	to	to	ADP
ejpam-4919	152	6	both	both	DET
ejpam-4919	152	7	institutions	institution	NOUN
ejpam-4919	152	8	for	for	ADP
ejpam-4919	152	9	their	their	PRON
ejpam-4919	152	10	support	support	NOUN
ejpam-4919	152	11	.	.	PUNCT
ejpam-4919	153	1	references	reference	NOUN
ejpam-4919	153	2	[	[	X
ejpam-4919	153	3	1	1	NUM
ejpam-4919	153	4	]	]	X
ejpam-4919	153	5	sh	sh	PROPN
ejpam-4919	153	6	.	.	PUNCT
ejpam-4919	153	7	al	al	PROPN
ejpam-4919	153	8	-	-	PUNCT
ejpam-4919	153	9	sharif	sharif	PROPN
ejpam-4919	153	10	and	and	CCONJ
ejpam-4919	153	11	r.	r.	PROPN
ejpam-4919	153	12	khalil	khalil	PROPN
ejpam-4919	153	13	.	.	PUNCT
ejpam-4919	154	1	remotal	remotal	ADJ
ejpam-4919	154	2	sets	set	NOUN
ejpam-4919	154	3	in	in	ADP
ejpam-4919	154	4	valued	value	VERB
ejpam-4919	154	5	function	function	NOUN
ejpam-4919	154	6	spaces	space	NOUN
ejpam-4919	154	7	.	.	PUNCT
ejpam-4919	155	1	scientiae	scientiae	PROPN
ejpam-4919	155	2	mathematicae	mathematicae	PROPN
ejpam-4919	155	3	japonica	japonica	PROPN
ejpam-4919	155	4	,	,	PUNCT
ejpam-4919	155	5	3:433–441	3:433–441	NUM
ejpam-4919	155	6	,	,	PUNCT
ejpam-4919	155	7	2006	2006	NUM
ejpam-4919	155	8	.	.	PUNCT
ejpam-4919	156	1	[	[	X
ejpam-4919	156	2	2	2	NUM
ejpam-4919	156	3	]	]	PUNCT
ejpam-4919	156	4	a.	a.	PROPN
ejpam-4919	156	5	r.	r.	PROPN
ejpam-4919	156	6	alimov	alimov	PROPN
ejpam-4919	156	7	.	.	PUNCT
ejpam-4919	157	1	solarity	solarity	NOUN
ejpam-4919	157	2	of	of	ADP
ejpam-4919	157	3	chebyshev	chebyshev	NOUN
ejpam-4919	157	4	sets	set	NOUN
ejpam-4919	157	5	in	in	ADP
ejpam-4919	157	6	dual	dual	ADJ
ejpam-4919	157	7	spaces	space	NOUN
ejpam-4919	157	8	and	and	CCONJ
ejpam-4919	157	9	uniquely	uniquely	ADV
ejpam-4919	157	10	remotal	remotal	ADJ
ejpam-4919	157	11	sets	set	NOUN
ejpam-4919	157	12	.	.	PUNCT
ejpam-4919	158	1	lobachevskii	lobachevskii	PROPN
ejpam-4919	158	2	journal	journal	PROPN
ejpam-4919	158	3	of	of	ADP
ejpam-4919	158	4	mathematics	mathematic	NOUN
ejpam-4919	158	5	,	,	PUNCT
ejpam-4919	158	6	42(4):785–790	42(4):785–790	NOUN
ejpam-4919	158	7	,	,	PUNCT
ejpam-4919	158	8	2021	2021	NUM
ejpam-4919	158	9	.	.	PUNCT
ejpam-4919	159	1	[	[	X
ejpam-4919	159	2	3	3	NUM
ejpam-4919	159	3	]	]	PUNCT
ejpam-4919	159	4	a.	a.	PROPN
ejpam-4919	159	5	r.	r.	PROPN
ejpam-4919	159	6	alimov	alimov	PROPN
ejpam-4919	159	7	and	and	CCONJ
ejpam-4919	159	8	i.	i.	PROPN
ejpam-4919	159	9	g.tsarkov	g.tsarkov	PROPN
ejpam-4919	159	10	.	.	PUNCT
ejpam-4919	160	1	chebyshev	chebyshev	PROPN
ejpam-4919	160	2	centres	centre	NOUN
ejpam-4919	160	3	,	,	PUNCT
ejpam-4919	160	4	jung	jung	PROPN
ejpam-4919	160	5	constants	constant	NOUN
ejpam-4919	160	6	,	,	PUNCT
ejpam-4919	160	7	and	and	CCONJ
ejpam-4919	160	8	their	their	PRON
ejpam-4919	160	9	applications	application	NOUN
ejpam-4919	160	10	.	.	PUNCT
ejpam-4919	161	1	russian	russian	ADJ
ejpam-4919	161	2	math	math	PROPN
ejpam-4919	161	3	.	.	PUNCT
ejpam-4919	162	1	surveys	survey	NOUN
ejpam-4919	162	2	,	,	PUNCT
ejpam-4919	162	3	74(5):775–849	74(5):775–849	PROPN
ejpam-4919	162	4	,	,	PUNCT
ejpam-4919	162	5	2019	2019	NUM
ejpam-4919	162	6	.	.	PUNCT
ejpam-4919	163	1	[	[	X
ejpam-4919	163	2	4	4	NUM
ejpam-4919	163	3	]	]	PUNCT
ejpam-4919	163	4	a.	a.	NOUN
ejpam-4919	163	5	astaneh	astaneh	NOUN
ejpam-4919	163	6	.	.	PUNCT
ejpam-4919	164	1	on	on	ADP
ejpam-4919	164	2	uniquely	uniquely	ADV
ejpam-4919	164	3	remotal	remotal	ADJ
ejpam-4919	164	4	subsets	subset	NOUN
ejpam-4919	164	5	of	of	ADP
ejpam-4919	164	6	hilbert	hilbert	PROPN
ejpam-4919	164	7	spaces	space	NOUN
ejpam-4919	164	8	.	.	PUNCT
ejpam-4919	165	1	indian	indian	ADJ
ejpam-4919	165	2	journal	journal	PROPN
ejpam-4919	165	3	of	of	ADP
ejpam-4919	165	4	pure	pure	ADJ
ejpam-4919	165	5	and	and	CCONJ
ejpam-4919	165	6	applied	applied	ADJ
ejpam-4919	165	7	mathematics	mathematic	NOUN
ejpam-4919	165	8	,	,	PUNCT
ejpam-4919	165	9	14(10):1311–1317	14(10):1311–1317	NUM
ejpam-4919	165	10	,	,	PUNCT
ejpam-4919	165	11	1983	1983	NUM
ejpam-4919	165	12	.	.	PUNCT
ejpam-4919	166	1	[	[	X
ejpam-4919	166	2	5	5	NUM
ejpam-4919	166	3	]	]	PUNCT
ejpam-4919	166	4	a.	a.	NOUN
ejpam-4919	166	5	astaneh	astaneh	NOUN
ejpam-4919	166	6	.	.	PUNCT
ejpam-4919	167	1	on	on	ADP
ejpam-4919	167	2	singletoness	singletoness	NOUN
ejpam-4919	167	3	of	of	ADP
ejpam-4919	167	4	uniquely	uniquely	ADV
ejpam-4919	167	5	remotal	remotal	ADJ
ejpam-4919	167	6	sets	set	NOUN
ejpam-4919	167	7	.	.	PUNCT
ejpam-4919	168	1	indian	indian	ADJ
ejpam-4919	168	2	journal	journal	PROPN
ejpam-4919	168	3	of	of	ADP
ejpam-4919	168	4	pure	pure	ADJ
ejpam-4919	168	5	and	and	CCONJ
ejpam-4919	168	6	applied	applied	ADJ
ejpam-4919	168	7	mathematics	mathematic	NOUN
ejpam-4919	168	8	,	,	PUNCT
ejpam-4919	168	9	17(9):1137–1139	17(9):1137–1139	NUM
ejpam-4919	168	10	,	,	PUNCT
ejpam-4919	168	11	1986	1986	NUM
ejpam-4919	168	12	.	.	PUNCT
ejpam-4919	169	1	[	[	X
ejpam-4919	169	2	6	6	NUM
ejpam-4919	169	3	]	]	PUNCT
ejpam-4919	169	4	m.	m.	NOUN
ejpam-4919	169	5	baronti	baronti	PROPN
ejpam-4919	169	6	.	.	PUNCT
ejpam-4919	170	1	a	a	DET
ejpam-4919	170	2	note	note	NOUN
ejpam-4919	170	3	on	on	ADP
ejpam-4919	170	4	remotal	remotal	ADJ
ejpam-4919	170	5	sets	set	NOUN
ejpam-4919	170	6	in	in	ADP
ejpam-4919	170	7	banach	banach	NOUN
ejpam-4919	170	8	spaces	space	NOUN
ejpam-4919	170	9	.	.	PUNCT
ejpam-4919	171	1	publications	publication	NOUN
ejpam-4919	171	2	de	de	ADP
ejpam-4919	171	3	l’institute	l’institute	NOUN
ejpam-4919	171	4	mathematique	mathematique	NOUN
ejpam-4919	171	5	,	,	PUNCT
ejpam-4919	171	6	53(67):95–98	53(67):95–98	NUM
ejpam-4919	171	7	,	,	PUNCT
ejpam-4919	171	8	1993	1993	NUM
ejpam-4919	171	9	.	.	PUNCT
ejpam-4919	172	1	[	[	X
ejpam-4919	172	2	7	7	NUM
ejpam-4919	172	3	]	]	PUNCT
ejpam-4919	172	4	a.	a.	NOUN
ejpam-4919	172	5	l.	l.	PROPN
ejpam-4919	172	6	garkavi	garkavi	PROPN
ejpam-4919	172	7	.	.	PUNCT
ejpam-4919	173	1	on	on	ADP
ejpam-4919	173	2	the	the	DET
ejpam-4919	173	3	chebyshev	chebyshev	PROPN
ejpam-4919	173	4	centre	centre	NOUN
ejpam-4919	173	5	of	of	ADP
ejpam-4919	173	6	a	a	DET
ejpam-4919	173	7	set	set	NOUN
ejpam-4919	173	8	in	in	ADP
ejpam-4919	173	9	a	a	DET
ejpam-4919	173	10	normed	normed	ADJ
ejpam-4919	173	11	space	space	NOUN
ejpam-4919	173	12	.	.	PUNCT
ejpam-4919	174	1	studies	study	NOUN
ejpam-4919	174	2	of	of	ADP
ejpam-4919	174	3	modern	modern	ADJ
ejpam-4919	174	4	problems	problem	NOUN
ejpam-4919	174	5	of	of	ADP
ejpam-4919	174	6	the	the	DET
ejpam-4919	174	7	constructive	constructive	ADJ
ejpam-4919	174	8	theory	theory	NOUN
ejpam-4919	174	9	of	of	ADP
ejpam-4919	174	10	functions	function	NOUN
ejpam-4919	174	11	,	,	PUNCT
ejpam-4919	174	12	fizmatgiz	fizmatgiz	NOUN
ejpam-4919	174	13	,	,	PUNCT
ejpam-4919	174	14	moscow	moscow	PROPN
ejpam-4919	174	15	,	,	PUNCT
ejpam-4919	174	16	pages	page	VERB
ejpam-4919	174	17	328–331	328–331	NUM
ejpam-4919	174	18	,	,	PUNCT
ejpam-4919	174	19	1961	1961	NUM
ejpam-4919	174	20	.	.	PUNCT
ejpam-4919	175	1	references	reference	NOUN
ejpam-4919	175	2	2404	2404	NUM
ejpam-4919	176	1	[	[	X
ejpam-4919	176	2	8	8	NUM
ejpam-4919	176	3	]	]	X
ejpam-4919	176	4	f.	f.	PROPN
ejpam-4919	176	5	saidi	saidi	PROPN
ejpam-4919	176	6	;	;	PUNCT
ejpam-4919	176	7	d.	d.	PROPN
ejpam-4919	176	8	hussein	hussein	PROPN
ejpam-4919	176	9	and	and	CCONJ
ejpam-4919	176	10	r.	r.	PROPN
ejpam-4919	176	11	khalil	khalil	PROPN
ejpam-4919	176	12	.	.	PUNCT
ejpam-4919	177	1	strong	strong	ADJ
ejpam-4919	177	2	proximinality	proximinality	NOUN
ejpam-4919	177	3	in	in	ADP
ejpam-4919	177	4	banach	banach	NOUN
ejpam-4919	177	5	spaces	space	NOUN
ejpam-4919	177	6	.	.	PUNCT
ejpam-4919	178	1	math	math	NOUN
ejpam-4919	178	2	.	.	PUNCT
ejpam-4919	179	1	j.	j.	PROPN
ejpam-4919	179	2	of	of	ADP
ejpam-4919	179	3	toyama	toyama	PROPN
ejpam-4919	179	4	university	university	PROPN
ejpam-4919	179	5	,	,	PUNCT
ejpam-4919	179	6	19(2):67–95	19(2):67–95	NUM
ejpam-4919	179	7	,	,	PUNCT
ejpam-4919	179	8	1996	1996	NUM
ejpam-4919	179	9	.	.	PUNCT
ejpam-4919	180	1	[	[	X
ejpam-4919	180	2	9	9	NUM
ejpam-4919	180	3	]	]	PUNCT
ejpam-4919	180	4	a.	a.	NOUN
ejpam-4919	180	5	yousef	yousef	PROPN
ejpam-4919	180	6	;	;	PUNCT
ejpam-4919	180	7	r.	r.	PROPN
ejpam-4919	180	8	khalil	khalil	PROPN
ejpam-4919	180	9	and	and	CCONJ
ejpam-4919	180	10	b.	b.	PROPN
ejpam-4919	180	11	mutabagani	mutabagani	PROPN
ejpam-4919	180	12	.	.	PUNCT
ejpam-4919	181	1	on	on	ADP
ejpam-4919	181	2	the	the	DET
ejpam-4919	181	3	farthest	farth	ADJ
ejpam-4919	181	4	point	point	NOUN
ejpam-4919	181	5	problem	problem	NOUN
ejpam-4919	181	6	in	in	ADP
ejpam-4919	181	7	banach	banach	NOUN
ejpam-4919	181	8	spaces	space	NOUN
ejpam-4919	181	9	.	.	PUNCT
ejpam-4919	182	1	journal	journal	NOUN
ejpam-4919	182	2	of	of	ADP
ejpam-4919	182	3	computational	computational	ADJ
ejpam-4919	182	4	analysis	analysis	NOUN
ejpam-4919	182	5	and	and	CCONJ
ejpam-4919	182	6	applications	application	NOUN
ejpam-4919	182	7	,	,	PUNCT
ejpam-4919	182	8	29(1):123–128	29(1):123–128	NUM
ejpam-4919	182	9	,	,	PUNCT
ejpam-4919	182	10	2020	2020	NUM
ejpam-4919	182	11	.	.	PUNCT
ejpam-4919	183	1	[	[	X
ejpam-4919	183	2	10	10	NUM
ejpam-4919	183	3	]	]	X
ejpam-4919	183	4	r.	r.	PROPN
ejpam-4919	183	5	khalil	khalil	PROPN
ejpam-4919	183	6	and	and	CCONJ
ejpam-4919	183	7	m.	m.	NOUN
ejpam-4919	183	8	sababheh	sababheh	NOUN
ejpam-4919	183	9	.	.	PUNCT
ejpam-4919	184	1	remotality	remotality	NOUN
ejpam-4919	184	2	of	of	ADP
ejpam-4919	184	3	closed	closed	ADJ
ejpam-4919	184	4	bounded	bound	VERB
ejpam-4919	184	5	convex	convex	NOUN
ejpam-4919	184	6	sets	set	NOUN
ejpam-4919	184	7	in	in	ADP
ejpam-4919	184	8	reflexive	reflexive	ADJ
ejpam-4919	184	9	spaces	space	NOUN
ejpam-4919	184	10	.	.	PUNCT
ejpam-4919	185	1	numerical	numerical	ADJ
ejpam-4919	185	2	functional	functional	ADJ
ejpam-4919	185	3	analysis	analysis	NOUN
ejpam-4919	185	4	and	and	CCONJ
ejpam-4919	185	5	optimization	optimization	NOUN
ejpam-4919	185	6	,	,	PUNCT
ejpam-4919	185	7	29(10):1166–1170	29(10):1166–1170	NUM
ejpam-4919	185	8	,	,	PUNCT
ejpam-4919	185	9	2008	2008	NUM
ejpam-4919	185	10	.	.	PUNCT
ejpam-4919	186	1	[	[	X
ejpam-4919	186	2	11	11	NUM
ejpam-4919	186	3	]	]	PUNCT
ejpam-4919	186	4	r.	r.	PROPN
ejpam-4919	186	5	khalil	khalil	PROPN
ejpam-4919	186	6	and	and	CCONJ
ejpam-4919	186	7	m.	m.	NOUN
ejpam-4919	186	8	sababheh	sababheh	PROPN
ejpam-4919	186	9	.	.	PUNCT
ejpam-4919	187	1	a	a	DET
ejpam-4919	187	2	study	study	NOUN
ejpam-4919	187	3	of	of	ADP
ejpam-4919	187	4	uniquely	uniquely	ADV
ejpam-4919	187	5	remotal	remotal	ADJ
ejpam-4919	187	6	sets	set	NOUN
ejpam-4919	187	7	.	.	PUNCT
ejpam-4919	188	1	journal	journal	NOUN
ejpam-4919	188	2	of	of	ADP
ejpam-4919	188	3	computational	computational	ADJ
ejpam-4919	188	4	analysis	analysis	NOUN
ejpam-4919	188	5	,	,	PUNCT
ejpam-4919	188	6	13:1233–1239	13:1233–1239	NUM
ejpam-4919	188	7	,	,	PUNCT
ejpam-4919	188	8	2010	2010	NUM
ejpam-4919	188	9	.	.	PUNCT
ejpam-4919	189	1	[	[	X
ejpam-4919	189	2	12	12	NUM
ejpam-4919	189	3	]	]	X
ejpam-4919	189	4	r.	r.	PROPN
ejpam-4919	189	5	khalil	khalil	PROPN
ejpam-4919	189	6	and	and	CCONJ
ejpam-4919	189	7	m.	m.	NOUN
ejpam-4919	189	8	sababheh	sababheh	PROPN
ejpam-4919	189	9	.	.	PUNCT
ejpam-4919	190	1	remotal	remotal	ADJ
ejpam-4919	190	2	points	point	NOUN
ejpam-4919	190	3	and	and	CCONJ
ejpam-4919	190	4	krein	krein	NOUN
ejpam-4919	190	5	-	-	PUNCT
ejpam-4919	190	6	milman	milman	NOUN
ejpam-4919	190	7	type	type	NOUN
ejpam-4919	190	8	theorem	theorem	VERB
ejpam-4919	190	9	.	.	PROPN
ejpam-4919	190	10	journal	journal	PROPN
ejpam-4919	190	11	of	of	ADP
ejpam-4919	190	12	non	non	ADJ
ejpam-4919	190	13	-	-	ADJ
ejpam-4919	190	14	linear	linear	ADJ
ejpam-4919	190	15	convex	convex	NOUN
ejpam-4919	190	16	analysis	analysis	NOUN
ejpam-4919	190	17	,	,	PUNCT
ejpam-4919	190	18	12(2):5–15	12(2):5–15	NUM
ejpam-4919	190	19	,	,	PUNCT
ejpam-4919	190	20	2011	2011	NUM
ejpam-4919	190	21	.	.	PUNCT
ejpam-4919	191	1	[	[	X
ejpam-4919	191	2	13	13	NUM
ejpam-4919	191	3	]	]	PUNCT
ejpam-4919	191	4	r.	r.	PROPN
ejpam-4919	191	5	khalil	khalil	PROPN
ejpam-4919	191	6	and	and	CCONJ
ejpam-4919	191	7	m.	m.	NOUN
ejpam-4919	191	8	sababheh	sababheh	PROPN
ejpam-4919	191	9	.	.	PUNCT
ejpam-4919	192	1	new	new	ADJ
ejpam-4919	192	2	results	result	NOUN
ejpam-4919	192	3	on	on	ADP
ejpam-4919	192	4	remotality	remotality	NOUN
ejpam-4919	192	5	in	in	ADP
ejpam-4919	192	6	banach	banach	NOUN
ejpam-4919	192	7	spaces	space	NOUN
ejpam-4919	192	8	.	.	PUNCT
ejpam-4919	193	1	italian	italian	PROPN
ejpam-4919	193	2	j.p.a.m	j.p.a.m	PROPN
ejpam-4919	193	3	,	,	PUNCT
ejpam-4919	193	4	30:59–66	30:59–66	NUM
ejpam-4919	193	5	,	,	PUNCT
ejpam-4919	193	6	2013	2013	NUM
ejpam-4919	193	7	.	.	PUNCT
ejpam-4919	194	1	[	[	X
ejpam-4919	194	2	14	14	NUM
ejpam-4919	194	3	]	]	X
ejpam-4919	194	4	v.	v.	PROPN
ejpam-4919	194	5	l.	l.	PROPN
ejpam-4919	194	6	klee	klee	PROPN
ejpam-4919	194	7	.	.	PUNCT
ejpam-4919	195	1	convexity	convexity	NOUN
ejpam-4919	195	2	of	of	ADP
ejpam-4919	195	3	chebyshev	chebyshev	NOUN
ejpam-4919	195	4	sets	set	NOUN
ejpam-4919	195	5	.	.	PUNCT
ejpam-4919	196	1	math	math	NOUN
ejpam-4919	196	2	.	.	PUNCT
ejpam-4919	197	1	ann	ann	PROPN
ejpam-4919	197	2	.	.	PROPN
ejpam-4919	197	3	,	,	PUNCT
ejpam-4919	197	4	142:292–304	142:292–304	NUM
ejpam-4919	197	5	,	,	PUNCT
ejpam-4919	197	6	1961	1961	NUM
ejpam-4919	197	7	.	.	PUNCT
ejpam-4919	198	1	[	[	X
ejpam-4919	198	2	15	15	NUM
ejpam-4919	198	3	]	]	PUNCT
ejpam-4919	198	4	maaden	maaden	NOUN
ejpam-4919	198	5	.	.	PUNCT
ejpam-4919	199	1	on	on	ADP
ejpam-4919	199	2	the	the	DET
ejpam-4919	199	3	c	c	NOUN
ejpam-4919	199	4	-	-	PUNCT
ejpam-4919	199	5	farthest	farth	ADJ
ejpam-4919	199	6	points	point	NOUN
ejpam-4919	199	7	.	.	PUNCT
ejpam-4919	200	1	extracta	extracta	PROPN
ejpam-4919	200	2	mathematicae	mathematicae	PROPN
ejpam-4919	200	3	,	,	PUNCT
ejpam-4919	200	4	16(2):211–222	16(2):211–222	PROPN
ejpam-4919	200	5	,	,	PUNCT
ejpam-4919	200	6	2001	2001	NUM
ejpam-4919	200	7	.	.	PUNCT
ejpam-4919	201	1	[	[	X
ejpam-4919	201	2	16	16	NUM
ejpam-4919	201	3	]	]	PUNCT
ejpam-4919	201	4	a.	a.	NOUN
ejpam-4919	201	5	niknam	niknam	PROPN
ejpam-4919	201	6	.	.	PUNCT
ejpam-4919	202	1	on	on	ADP
ejpam-4919	202	2	uniquely	uniquely	ADV
ejpam-4919	202	3	remotal	remotal	ADJ
ejpam-4919	202	4	sets	set	NOUN
ejpam-4919	202	5	.	.	PUNCT
ejpam-4919	203	1	indian	indian	ADJ
ejpam-4919	203	2	journal	journal	PROPN
ejpam-4919	203	3	of	of	ADP
ejpam-4919	203	4	pure	pure	ADJ
ejpam-4919	203	5	and	and	CCONJ
ejpam-4919	203	6	applied	applied	ADJ
ejpam-4919	203	7	mathematics	mathematic	NOUN
ejpam-4919	203	8	,	,	PUNCT
ejpam-4919	203	9	15(10):1079–1083	15(10):1079–1083	NOUN
ejpam-4919	203	10	,	,	PUNCT
ejpam-4919	203	11	1984	1984	NUM
ejpam-4919	203	12	.	.	PUNCT
ejpam-4919	204	1	[	[	X
ejpam-4919	204	2	17	17	NUM
ejpam-4919	204	3	]	]	PUNCT
ejpam-4919	204	4	m.	m.	NOUN
ejpam-4919	204	5	sababheh	sababheh	NOUN
ejpam-4919	204	6	;	;	PUNCT
ejpam-4919	204	7	a.	a.	PROPN
ejpam-4919	204	8	yousef	yousef	PROPN
ejpam-4919	204	9	and	and	CCONJ
ejpam-4919	204	10	r.	r.	PROPN
ejpam-4919	204	11	khalil	khalil	PROPN
ejpam-4919	204	12	.	.	PUNCT
ejpam-4919	205	1	uniquely	uniquely	ADV
ejpam-4919	205	2	remotal	remotal	ADJ
ejpam-4919	205	3	sets	set	NOUN
ejpam-4919	205	4	in	in	ADP
ejpam-4919	205	5	banach	banach	NOUN
ejpam-4919	205	6	spaces	space	NOUN
ejpam-4919	205	7	.	.	PUNCT
ejpam-4919	206	1	filomat	filomat	NOUN
ejpam-4919	206	2	,	,	PUNCT
ejpam-4919	206	3	31:2773–2777	31:2773–2777	NUM
ejpam-4919	206	4	,	,	PUNCT
ejpam-4919	206	5	2017	2017	NUM
ejpam-4919	206	6	.	.	PUNCT