id	sid	tid	token	lemma	pos
ejpam-3237	1	1	european	european	PROPN
ejpam-3237	1	2	journal	journal	PROPN
ejpam-3237	1	3	of	of	ADP
ejpam-3237	1	4	pure	pure	ADJ
ejpam-3237	1	5	and	and	CCONJ
ejpam-3237	1	6	applied	apply	VERB
ejpam-3237	1	7	mathematics	mathematic	NOUN
ejpam-3237	1	8	vol	vol	NOUN
ejpam-3237	1	9	.	.	PUNCT
ejpam-3237	2	1	11	11	NUM
ejpam-3237	2	2	,	,	PUNCT
ejpam-3237	2	3	no	no	INTJ
ejpam-3237	2	4	.	.	NOUN
ejpam-3237	2	5	3	3	NUM
ejpam-3237	2	6	,	,	PUNCT
ejpam-3237	2	7	2018	2018	NUM
ejpam-3237	2	8	,	,	PUNCT
ejpam-3237	2	9	793	793	NUM
ejpam-3237	2	10	-	-	SYM
ejpam-3237	2	11	802	802	NUM
ejpam-3237	2	12	issn	issn	PROPN
ejpam-3237	2	13	1307	1307	NUM
ejpam-3237	2	14	-	-	SYM
ejpam-3237	2	15	5543	5543	NUM
ejpam-3237	2	16	–	–	PUNCT
ejpam-3237	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3237	2	18	published	publish	VERB
ejpam-3237	2	19	by	by	ADP
ejpam-3237	2	20	new	new	PROPN
ejpam-3237	2	21	york	york	PROPN
ejpam-3237	2	22	business	business	PROPN
ejpam-3237	2	23	global	global	PROPN
ejpam-3237	2	24	on	on	ADP
ejpam-3237	2	25	the	the	DET
ejpam-3237	2	26	continuity	continuity	NOUN
ejpam-3237	2	27	of	of	ADP
ejpam-3237	2	28	orthogonal	orthogonal	ADJ
ejpam-3237	2	29	sets	set	NOUN
ejpam-3237	2	30	in	in	ADP
ejpam-3237	2	31	the	the	DET
ejpam-3237	2	32	sense	sense	NOUN
ejpam-3237	2	33	of	of	ADP
ejpam-3237	2	34	operator	operator	NOUN
ejpam-3237	2	35	orthogonality	orthogonality	NOUN
ejpam-3237	2	36	m.	m.	PROPN
ejpam-3237	2	37	iranmanesh1,∗	iranmanesh1,∗	PROPN
ejpam-3237	2	38	,	,	PUNCT
ejpam-3237	2	39	m.	m.	NOUN
ejpam-3237	2	40	saeedi	saeedi	PROPN
ejpam-3237	2	41	khojasteh1	khojasteh1	PROPN
ejpam-3237	2	42	,	,	PUNCT
ejpam-3237	2	43	m.	m.	PROPN
ejpam-3237	2	44	k.	k.	PROPN
ejpam-3237	2	45	anwary2	anwary2	PROPN
ejpam-3237	3	1	1	1	NUM
ejpam-3237	3	2	department	department	NOUN
ejpam-3237	3	3	of	of	ADP
ejpam-3237	3	4	pure	pure	ADJ
ejpam-3237	3	5	mathematics	mathematic	NOUN
ejpam-3237	3	6	,	,	PUNCT
ejpam-3237	3	7	shahrood	shahrood	NOUN
ejpam-3237	3	8	university	university	PROPN
ejpam-3237	3	9	of	of	ADP
ejpam-3237	3	10	technology	technology	NOUN
ejpam-3237	3	11	,	,	PUNCT
ejpam-3237	3	12	iran	iran	PROPN
ejpam-3237	3	13	2	2	NUM
ejpam-3237	3	14	department	department	NOUN
ejpam-3237	3	15	of	of	ADP
ejpam-3237	3	16	pure	pure	ADJ
ejpam-3237	3	17	mathematics	mathematic	NOUN
ejpam-3237	3	18	,	,	PUNCT
ejpam-3237	3	19	ferdowsi	ferdowsi	NOUN
ejpam-3237	3	20	university	university	PROPN
ejpam-3237	3	21	of	of	ADP
ejpam-3237	3	22	mashhad	mashhad	PROPN
ejpam-3237	3	23	,	,	PUNCT
ejpam-3237	3	24	iran	iran	PROPN
ejpam-3237	3	25	abstract	abstract	NOUN
ejpam-3237	3	26	.	.	PUNCT
ejpam-3237	4	1	in	in	ADP
ejpam-3237	4	2	this	this	DET
ejpam-3237	4	3	paper	paper	NOUN
ejpam-3237	4	4	,	,	PUNCT
ejpam-3237	4	5	we	we	PRON
ejpam-3237	4	6	introduce	introduce	VERB
ejpam-3237	4	7	the	the	DET
ejpam-3237	4	8	operator	operator	NOUN
ejpam-3237	4	9	approach	approach	NOUN
ejpam-3237	4	10	for	for	ADP
ejpam-3237	4	11	orthogonality	orthogonality	NOUN
ejpam-3237	4	12	in	in	ADP
ejpam-3237	4	13	linear	linear	ADJ
ejpam-3237	4	14	spaces	space	NOUN
ejpam-3237	4	15	.	.	PUNCT
ejpam-3237	5	1	in	in	ADP
ejpam-3237	5	2	particular	particular	ADJ
ejpam-3237	5	3	,	,	PUNCT
ejpam-3237	5	4	we	we	PRON
ejpam-3237	5	5	represent	represent	VERB
ejpam-3237	5	6	the	the	DET
ejpam-3237	5	7	concept	concept	NOUN
ejpam-3237	5	8	of	of	ADP
ejpam-3237	5	9	orthogonal	orthogonal	ADJ
ejpam-3237	5	10	vectors	vector	NOUN
ejpam-3237	5	11	using	use	VERB
ejpam-3237	5	12	an	an	DET
ejpam-3237	5	13	operator	operator	NOUN
ejpam-3237	5	14	associated	associate	VERB
ejpam-3237	5	15	with	with	ADP
ejpam-3237	5	16	them	they	PRON
ejpam-3237	5	17	,	,	PUNCT
ejpam-3237	5	18	in	in	ADP
ejpam-3237	5	19	normed	normed	ADJ
ejpam-3237	5	20	spaces	space	NOUN
ejpam-3237	5	21	.	.	PUNCT
ejpam-3237	6	1	moreover	moreover	ADV
ejpam-3237	6	2	,	,	PUNCT
ejpam-3237	6	3	we	we	PRON
ejpam-3237	6	4	investigate	investigate	VERB
ejpam-3237	6	5	some	some	PRON
ejpam-3237	6	6	of	of	ADP
ejpam-3237	6	7	continuity	continuity	NOUN
ejpam-3237	6	8	properties	property	NOUN
ejpam-3237	6	9	of	of	ADP
ejpam-3237	6	10	this	this	DET
ejpam-3237	6	11	kind	kind	NOUN
ejpam-3237	6	12	of	of	ADP
ejpam-3237	6	13	orthogonality	orthogonality	NOUN
ejpam-3237	6	14	.	.	PUNCT
ejpam-3237	7	1	more	more	ADV
ejpam-3237	7	2	precisely	precisely	ADV
ejpam-3237	7	3	,	,	PUNCT
ejpam-3237	7	4	we	we	PRON
ejpam-3237	7	5	show	show	VERB
ejpam-3237	7	6	that	that	SCONJ
ejpam-3237	7	7	the	the	DET
ejpam-3237	7	8	set	set	NOUN
ejpam-3237	7	9	valued	value	VERB
ejpam-3237	7	10	function	function	NOUN
ejpam-3237	7	11	f	f	PROPN
ejpam-3237	7	12	(	(	PUNCT
ejpam-3237	7	13	x	x	NOUN
ejpam-3237	7	14	;	;	PUNCT
ejpam-3237	7	15	y	y	X
ejpam-3237	7	16	)	)	PUNCT
ejpam-3237	7	17	=	=	PRON
ejpam-3237	7	18	{	{	PUNCT
ejpam-3237	7	19	µ	µ	X
ejpam-3237	7	20	:	:	PUNCT
ejpam-3237	7	21	µ	µ	X
ejpam-3237	7	22	∈	∈	ADP
ejpam-3237	7	23	c	c	X
ejpam-3237	7	24	,	,	PUNCT
ejpam-3237	7	25	p(x−	p(x−	PROPN
ejpam-3237	7	26	µy	µy	PROPN
ejpam-3237	7	27	,	,	PUNCT
ejpam-3237	7	28	y	y	NOUN
ejpam-3237	7	29	)	)	PUNCT
ejpam-3237	7	30	=	=	SYM
ejpam-3237	7	31	1	1	X
ejpam-3237	7	32	}	}	PUNCT
ejpam-3237	7	33	is	be	AUX
ejpam-3237	7	34	upper	upper	ADJ
ejpam-3237	7	35	and	and	CCONJ
ejpam-3237	7	36	lower	low	ADJ
ejpam-3237	7	37	semi	semi	ADV
ejpam-3237	7	38	continuous	continuous	ADJ
ejpam-3237	7	39	,	,	PUNCT
ejpam-3237	7	40	where	where	SCONJ
ejpam-3237	7	41	p(x	p(x	PROPN
ejpam-3237	7	42	,	,	PUNCT
ejpam-3237	7	43	y	y	NOUN
ejpam-3237	7	44	)	)	PUNCT
ejpam-3237	7	45	=	=	SYM
ejpam-3237	7	46	sup{pz1,	sup{pz1,	PROPN
ejpam-3237	7	47	...	...	PUNCT
ejpam-3237	7	48	,zn−2	,zn−2	PUNCT
ejpam-3237	7	49	(	(	PUNCT
ejpam-3237	7	50	x	x	NOUN
ejpam-3237	7	51	,	,	PUNCT
ejpam-3237	7	52	y	y	PROPN
ejpam-3237	7	53	)	)	PUNCT
ejpam-3237	7	54	:	:	PUNCT
ejpam-3237	7	55	z1	z1	VERB
ejpam-3237	7	56	,	,	PUNCT
ejpam-3237	7	57	.	.	PUNCT
ejpam-3237	7	58	.	.	PUNCT
ejpam-3237	8	1	.	.	PUNCT
ejpam-3237	9	1	,	,	PUNCT
ejpam-3237	9	2	zn−2	zn−2	PROPN
ejpam-3237	9	3	∈	∈	PROPN
ejpam-3237	9	4	x	x	X
ejpam-3237	9	5	}	}	PUNCT
ejpam-3237	9	6	and	and	CCONJ
ejpam-3237	9	7	pz1,	pz1,	NOUN
ejpam-3237	9	8	...	...	PUNCT
ejpam-3237	9	9	,zn−2	,zn−2	PUNCT
ejpam-3237	9	10	(	(	PUNCT
ejpam-3237	9	11	x	x	NOUN
ejpam-3237	9	12	,	,	PUNCT
ejpam-3237	9	13	y	y	NOUN
ejpam-3237	9	14	)	)	PUNCT
ejpam-3237	9	15	=	=	SYM
ejpam-3237	9	16	‖px	‖px	PROPN
ejpam-3237	9	17	,	,	PUNCT
ejpam-3237	9	18	z1,	z1,	NOUN
ejpam-3237	9	19	...	...	PUNCT
ejpam-3237	9	20	,zn−2,y‖	,zn−2,y‖	PUNCT
ejpam-3237	9	21	−1	−1	ADV
ejpam-3237	9	22	where	where	SCONJ
ejpam-3237	9	23	px	px	NOUN
ejpam-3237	9	24	,	,	PUNCT
ejpam-3237	9	25	z1,	z1,	NOUN
ejpam-3237	9	26	...	...	PUNCT
ejpam-3237	9	27	,zn−2,y	,zn−2,y	PUNCT
ejpam-3237	9	28	denotes	denote	VERB
ejpam-3237	9	29	the	the	DET
ejpam-3237	9	30	projection	projection	NOUN
ejpam-3237	9	31	parallel	parallel	NOUN
ejpam-3237	9	32	to	to	ADP
ejpam-3237	9	33	y	y	PROPN
ejpam-3237	9	34	from	from	ADP
ejpam-3237	9	35	x	x	PUNCT
ejpam-3237	9	36	to	to	ADP
ejpam-3237	9	37	the	the	DET
ejpam-3237	9	38	subspace	subspace	NOUN
ejpam-3237	9	39	generated	generate	VERB
ejpam-3237	9	40	by	by	ADP
ejpam-3237	9	41	{	{	PUNCT
ejpam-3237	9	42	x	x	PROPN
ejpam-3237	9	43	,	,	PUNCT
ejpam-3237	9	44	z1	z1	NOUN
ejpam-3237	9	45	,	,	PUNCT
ejpam-3237	9	46	.	.	PUNCT
ejpam-3237	9	47	.	.	PUNCT
ejpam-3237	10	1	.	.	PUNCT
ejpam-3237	11	1	,	,	PUNCT
ejpam-3237	11	2	zn−2	zn−2	PROPN
ejpam-3237	11	3	}	}	PUNCT
ejpam-3237	11	4	.	.	PUNCT
ejpam-3237	12	1	this	this	PRON
ejpam-3237	12	2	can	can	AUX
ejpam-3237	12	3	be	be	AUX
ejpam-3237	12	4	considered	consider	VERB
ejpam-3237	12	5	as	as	ADP
ejpam-3237	12	6	an	an	DET
ejpam-3237	12	7	alternative	alternative	ADJ
ejpam-3237	12	8	definition	definition	NOUN
ejpam-3237	12	9	for	for	ADP
ejpam-3237	12	10	numerical	numerical	ADJ
ejpam-3237	12	11	range	range	NOUN
ejpam-3237	12	12	in	in	ADP
ejpam-3237	12	13	linear	linear	ADJ
ejpam-3237	12	14	spaces	space	NOUN
ejpam-3237	12	15	.	.	PUNCT
ejpam-3237	13	1	key	key	ADJ
ejpam-3237	13	2	words	word	NOUN
ejpam-3237	13	3	and	and	CCONJ
ejpam-3237	13	4	phrases	phrase	NOUN
ejpam-3237	13	5	:	:	PUNCT
ejpam-3237	13	6	birkhoff	birkhoff	NOUN
ejpam-3237	13	7	orthogonality	orthogonality	NOUN
ejpam-3237	13	8	,	,	PUNCT
ejpam-3237	13	9	minkowski	minkowski	ADJ
ejpam-3237	13	10	plane	plane	NOUN
ejpam-3237	13	11	,	,	PUNCT
ejpam-3237	13	12	set	set	VERB
ejpam-3237	13	13	valued	value	VERB
ejpam-3237	13	14	function	function	NOUN
ejpam-3237	13	15	,	,	PUNCT
ejpam-3237	13	16	upper	upper	ADJ
ejpam-3237	13	17	semi	semi	ADV
ejpam-3237	13	18	continuous	continuous	ADJ
ejpam-3237	13	19	,	,	PUNCT
ejpam-3237	13	20	lower	low	ADJ
ejpam-3237	13	21	semi	semi	ADV
ejpam-3237	13	22	continuous	continuous	ADJ
ejpam-3237	13	23	1	1	NUM
ejpam-3237	13	24	.	.	PUNCT
ejpam-3237	13	25	introduction	introduction	NOUN
ejpam-3237	13	26	orthogonality	orthogonality	NOUN
ejpam-3237	13	27	,	,	PUNCT
ejpam-3237	13	28	is	be	AUX
ejpam-3237	13	29	one	one	NUM
ejpam-3237	13	30	of	of	ADP
ejpam-3237	13	31	the	the	DET
ejpam-3237	13	32	important	important	ADJ
ejpam-3237	13	33	concepts	concept	NOUN
ejpam-3237	13	34	in	in	ADP
ejpam-3237	13	35	mathematical	mathematical	ADJ
ejpam-3237	13	36	and	and	CCONJ
ejpam-3237	13	37	numerical	numerical	ADJ
ejpam-3237	13	38	analysis	analysis	NOUN
ejpam-3237	13	39	.	.	PUNCT
ejpam-3237	14	1	perhaps	perhaps	ADV
ejpam-3237	14	2	,	,	PUNCT
ejpam-3237	14	3	it	it	PRON
ejpam-3237	14	4	is	be	AUX
ejpam-3237	14	5	the	the	DET
ejpam-3237	14	6	main	main	ADJ
ejpam-3237	14	7	property	property	NOUN
ejpam-3237	14	8	in	in	ADP
ejpam-3237	14	9	linear	linear	ADJ
ejpam-3237	14	10	spaces	space	NOUN
ejpam-3237	14	11	,	,	PUNCT
ejpam-3237	14	12	normed	normed	ADJ
ejpam-3237	14	13	spaces	space	NOUN
ejpam-3237	14	14	and	and	CCONJ
ejpam-3237	14	15	inner	inner	ADJ
ejpam-3237	14	16	product	product	NOUN
ejpam-3237	14	17	spaces	space	VERB
ejpam-3237	14	18	.	.	PUNCT
ejpam-3237	15	1	there	there	PRON
ejpam-3237	15	2	are	be	VERB
ejpam-3237	15	3	some	some	DET
ejpam-3237	15	4	various	various	ADJ
ejpam-3237	15	5	kinds	kind	NOUN
ejpam-3237	15	6	of	of	ADP
ejpam-3237	15	7	orthogonality	orthogonality	NOUN
ejpam-3237	15	8	.	.	PUNCT
ejpam-3237	16	1	in	in	ADP
ejpam-3237	16	2	fact	fact	NOUN
ejpam-3237	16	3	,	,	PUNCT
ejpam-3237	16	4	it	it	PRON
ejpam-3237	16	5	has	have	AUX
ejpam-3237	16	6	been	be	AUX
ejpam-3237	16	7	defined	define	VERB
ejpam-3237	16	8	different	different	ADJ
ejpam-3237	16	9	kinds	kind	NOUN
ejpam-3237	16	10	in	in	ADP
ejpam-3237	16	11	mathematical	mathematical	ADJ
ejpam-3237	16	12	spaces	space	NOUN
ejpam-3237	16	13	.	.	PUNCT
ejpam-3237	17	1	in	in	ADP
ejpam-3237	17	2	inner	inner	ADJ
ejpam-3237	17	3	product	product	NOUN
ejpam-3237	17	4	spaces	space	NOUN
ejpam-3237	17	5	,	,	PUNCT
ejpam-3237	17	6	it	it	PRON
ejpam-3237	17	7	is	be	AUX
ejpam-3237	17	8	easily	easily	ADV
ejpam-3237	17	9	said	say	VERB
ejpam-3237	17	10	that	that	SCONJ
ejpam-3237	17	11	two	two	NUM
ejpam-3237	17	12	vectors	vector	NOUN
ejpam-3237	17	13	x	x	PRON
ejpam-3237	17	14	,	,	PUNCT
ejpam-3237	17	15	y	y	PROPN
ejpam-3237	17	16	are	be	AUX
ejpam-3237	17	17	orthogonal	orthogonal	ADJ
ejpam-3237	17	18	if	if	SCONJ
ejpam-3237	17	19	〈	〈	PROPN
ejpam-3237	17	20	x	x	X
ejpam-3237	17	21	,	,	PUNCT
ejpam-3237	17	22	y	y	PROPN
ejpam-3237	17	23	〉	〉	NUM
ejpam-3237	17	24	=	=	SYM
ejpam-3237	17	25	0	0	X
ejpam-3237	17	26	.	.	PUNCT
ejpam-3237	17	27	∗corresponding	∗corresponde	VERB
ejpam-3237	17	28	author	author	NOUN
ejpam-3237	17	29	.	.	PUNCT
ejpam-3237	18	1	doi	doi	NOUN
ejpam-3237	18	2	:	:	PUNCT
ejpam-3237	18	3	https://doi.org/10.29020/nybg.ejpam.v11i3.3237	https://doi.org/10.29020/nybg.ejpam.v11i3.3237	PROPN
ejpam-3237	18	4	email	email	NOUN
ejpam-3237	18	5	addresses	address	NOUN
ejpam-3237	18	6	:	:	PUNCT
ejpam-3237	18	7	m.iranmanesh2012@gmail.com	m.iranmanesh2012@gmail.com	X
ejpam-3237	18	8	(	(	PUNCT
ejpam-3237	18	9	m.	m.	NOUN
ejpam-3237	18	10	iranmanesh	iranmanesh	PROPN
ejpam-3237	18	11	)	)	PUNCT
ejpam-3237	18	12	,	,	PUNCT
ejpam-3237	18	13	m.saeedi64@gmail.com	m.saeedi64@gmail.com	X
ejpam-3237	18	14	(	(	PUNCT
ejpam-3237	18	15	m.	m.	NOUN
ejpam-3237	18	16	saeedi	saeedi	PROPN
ejpam-3237	18	17	khojasteh	khojasteh	PROPN
ejpam-3237	18	18	)	)	PUNCT
ejpam-3237	18	19	,	,	PUNCT
ejpam-3237	18	20	abdh1248@gmail.com	abdh1248@gmail.com	PROPN
ejpam-3237	18	21	(	(	PUNCT
ejpam-3237	18	22	m.	m.	PROPN
ejpam-3237	18	23	k.	k.	PROPN
ejpam-3237	18	24	anwary	anwary	PROPN
ejpam-3237	18	25	)	)	PUNCT
ejpam-3237	18	26	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3237	19	1	793	793	NUM
ejpam-3237	19	2	c	c	NOUN
ejpam-3237	19	3	©	©	PROPN
ejpam-3237	19	4	2018	2018	NUM
ejpam-3237	19	5	ejpam	ejpam	VERB
ejpam-3237	19	6	all	all	DET
ejpam-3237	19	7	rights	right	NOUN
ejpam-3237	19	8	reserved	reserve	VERB
ejpam-3237	19	9	.	.	PUNCT
ejpam-3237	20	1	m.	m.	PROPN
ejpam-3237	20	2	iranmanesh	iranmanesh	PROPN
ejpam-3237	20	3	,	,	PUNCT
ejpam-3237	20	4	m.	m.	NOUN
ejpam-3237	20	5	saeedi	saeedi	PROPN
ejpam-3237	20	6	khojasteh	khojasteh	PROPN
ejpam-3237	20	7	,	,	PUNCT
ejpam-3237	20	8	m.	m.	NOUN
ejpam-3237	20	9	k.	k.	PROPN
ejpam-3237	20	10	anwary	anwary	PROPN
ejpam-3237	20	11	/	/	SYM
ejpam-3237	20	12	eur	eur	PROPN
ejpam-3237	20	13	.	.	PUNCT
ejpam-3237	21	1	j.	j.	PROPN
ejpam-3237	21	2	pure	pure	PROPN
ejpam-3237	21	3	appl	appl	PROPN
ejpam-3237	21	4	.	.	PROPN
ejpam-3237	21	5	math	math	PROPN
ejpam-3237	21	6	,	,	PUNCT
ejpam-3237	21	7	11	11	NUM
ejpam-3237	21	8	(	(	PUNCT
ejpam-3237	21	9	3	3	NUM
ejpam-3237	21	10	)	)	PUNCT
ejpam-3237	21	11	(	(	PUNCT
ejpam-3237	21	12	2018	2018	NUM
ejpam-3237	21	13	)	)	PUNCT
ejpam-3237	21	14	,	,	PUNCT
ejpam-3237	21	15	793	793	NUM
ejpam-3237	21	16	-	-	SYM
ejpam-3237	21	17	802	802	NUM
ejpam-3237	21	18	794	794	NUM
ejpam-3237	21	19	but	but	CCONJ
ejpam-3237	21	20	,	,	PUNCT
ejpam-3237	21	21	in	in	ADP
ejpam-3237	21	22	normed	normed	ADJ
ejpam-3237	21	23	spaces	space	NOUN
ejpam-3237	21	24	,	,	PUNCT
ejpam-3237	21	25	there	there	PRON
ejpam-3237	21	26	is	be	VERB
ejpam-3237	21	27	no	no	DET
ejpam-3237	21	28	simple	simple	ADJ
ejpam-3237	21	29	tool	tool	NOUN
ejpam-3237	21	30	for	for	ADP
ejpam-3237	21	31	define	define	VERB
ejpam-3237	21	32	orthogonal	orthogonal	ADJ
ejpam-3237	21	33	vectors	vector	NOUN
ejpam-3237	21	34	.	.	PUNCT
ejpam-3237	22	1	however	however	ADV
ejpam-3237	22	2	,	,	PUNCT
ejpam-3237	22	3	there	there	PRON
ejpam-3237	22	4	are	be	VERB
ejpam-3237	22	5	some	some	DET
ejpam-3237	22	6	good	good	ADJ
ejpam-3237	22	7	suggestions	suggestion	NOUN
ejpam-3237	22	8	.	.	PUNCT
ejpam-3237	23	1	one	one	NUM
ejpam-3237	23	2	of	of	ADP
ejpam-3237	23	3	them	they	PRON
ejpam-3237	23	4	,	,	PUNCT
ejpam-3237	23	5	is	be	AUX
ejpam-3237	23	6	the	the	DET
ejpam-3237	23	7	birkhoff	birkhoff	NOUN
ejpam-3237	23	8	james	james	PROPN
ejpam-3237	23	9	orthogonality	orthogonality	PROPN
ejpam-3237	23	10	[	[	X
ejpam-3237	23	11	2	2	NUM
ejpam-3237	23	12	]	]	PUNCT
ejpam-3237	23	13	.	.	PUNCT
ejpam-3237	24	1	let	let	VERB
ejpam-3237	24	2	x	x	PRON
ejpam-3237	24	3	be	be	AUX
ejpam-3237	24	4	a	a	DET
ejpam-3237	24	5	real	real	ADV
ejpam-3237	24	6	normed	normed	ADJ
ejpam-3237	24	7	space	space	NOUN
ejpam-3237	24	8	,	,	PUNCT
ejpam-3237	24	9	and	and	CCONJ
ejpam-3237	24	10	x	x	X
ejpam-3237	24	11	,	,	PUNCT
ejpam-3237	24	12	y	y	PROPN
ejpam-3237	24	13	be	be	VERB
ejpam-3237	24	14	in	in	ADP
ejpam-3237	24	15	x.	x.	NOUN
ejpam-3237	24	16	we	we	PRON
ejpam-3237	24	17	say	say	VERB
ejpam-3237	24	18	that	that	SCONJ
ejpam-3237	24	19	x	x	PRON
ejpam-3237	24	20	is	be	AUX
ejpam-3237	24	21	birkhoff	birkhoff	NOUN
ejpam-3237	24	22	orthogonal	orthogonal	NOUN
ejpam-3237	24	23	to	to	ADP
ejpam-3237	24	24	y	y	PRON
ejpam-3237	24	25	if	if	SCONJ
ejpam-3237	24	26	for	for	ADP
ejpam-3237	24	27	every	every	DET
ejpam-3237	24	28	constant	constant	ADJ
ejpam-3237	24	29	a	a	DET
ejpam-3237	24	30	,	,	PUNCT
ejpam-3237	24	31	‖x‖	‖x‖	PROPN
ejpam-3237	24	32	6	6	NUM
ejpam-3237	24	33	‖x+	‖x+	PROPN
ejpam-3237	24	34	ay‖.	ay‖.	PROPN
ejpam-3237	24	35	(	(	PUNCT
ejpam-3237	24	36	1	1	X
ejpam-3237	24	37	)	)	PUNCT
ejpam-3237	24	38	it	it	PRON
ejpam-3237	24	39	is	be	AUX
ejpam-3237	24	40	not	not	PART
ejpam-3237	24	41	difficult	difficult	ADJ
ejpam-3237	24	42	to	to	PART
ejpam-3237	24	43	show	show	VERB
ejpam-3237	24	44	that	that	SCONJ
ejpam-3237	24	45	this	this	DET
ejpam-3237	24	46	definition	definition	NOUN
ejpam-3237	24	47	is	be	AUX
ejpam-3237	24	48	the	the	DET
ejpam-3237	24	49	same	same	ADJ
ejpam-3237	24	50	in	in	ADP
ejpam-3237	24	51	inner	inner	ADJ
ejpam-3237	24	52	product	product	NOUN
ejpam-3237	24	53	spaces	space	VERB
ejpam-3237	24	54	[	[	X
ejpam-3237	24	55	6	6	NUM
ejpam-3237	24	56	]	]	PUNCT
ejpam-3237	24	57	.	.	PUNCT
ejpam-3237	25	1	in	in	ADP
ejpam-3237	25	2	1993	1993	NUM
ejpam-3237	25	3	,	,	PUNCT
ejpam-3237	25	4	milicic	milicic	NOUN
ejpam-3237	26	1	[	[	X
ejpam-3237	26	2	7	7	X
ejpam-3237	26	3	]	]	PUNCT
ejpam-3237	26	4	introduced	introduce	VERB
ejpam-3237	26	5	g	g	NOUN
ejpam-3237	26	6	-	-	PUNCT
ejpam-3237	26	7	orthogonality	orthogonality	NOUN
ejpam-3237	26	8	in	in	ADP
ejpam-3237	26	9	normed	normed	ADJ
ejpam-3237	26	10	spaces	space	NOUN
ejpam-3237	26	11	via	via	ADP
ejpam-3237	26	12	gateaux	gateaux	VERB
ejpam-3237	26	13	derivatives	derivative	NOUN
ejpam-3237	26	14	.	.	PUNCT
ejpam-3237	27	1	in	in	ADP
ejpam-3237	27	2	fact	fact	NOUN
ejpam-3237	27	3	,	,	PUNCT
ejpam-3237	27	4	one	one	PRON
ejpam-3237	27	5	has	have	VERB
ejpam-3237	27	6	the	the	DET
ejpam-3237	27	7	notion	notion	NOUN
ejpam-3237	27	8	of	of	ADP
ejpam-3237	27	9	g	g	NOUN
ejpam-3237	27	10	-	-	PUNCT
ejpam-3237	27	11	angle	angle	NOUN
ejpam-3237	27	12	related	relate	VERB
ejpam-3237	27	13	to	to	ADP
ejpam-3237	27	14	g	g	NOUN
ejpam-3237	27	15	-	-	PUNCT
ejpam-3237	27	16	orthogonality	orthogonality	NOUN
ejpam-3237	27	17	.	.	PUNCT
ejpam-3237	28	1	in	in	ADP
ejpam-3237	28	2	this	this	DET
ejpam-3237	28	3	paper	paper	NOUN
ejpam-3237	28	4	,	,	PUNCT
ejpam-3237	28	5	the	the	DET
ejpam-3237	28	6	authors	author	NOUN
ejpam-3237	28	7	define	define	VERB
ejpam-3237	28	8	a	a	DET
ejpam-3237	28	9	new	new	ADJ
ejpam-3237	28	10	type	type	NOUN
ejpam-3237	28	11	of	of	ADP
ejpam-3237	28	12	orthogonality	orthogonality	NOUN
ejpam-3237	28	13	in	in	ADP
ejpam-3237	28	14	a	a	DET
ejpam-3237	28	15	linear	linear	ADJ
ejpam-3237	28	16	space	space	NOUN
ejpam-3237	28	17	by	by	ADP
ejpam-3237	28	18	using	use	VERB
ejpam-3237	28	19	projection	projection	NOUN
ejpam-3237	28	20	operators	operator	NOUN
ejpam-3237	28	21	.	.	PUNCT
ejpam-3237	29	1	let	let	VERB
ejpam-3237	29	2	x	x	PRON
ejpam-3237	29	3	be	be	AUX
ejpam-3237	29	4	a	a	DET
ejpam-3237	29	5	minkowski	minkowski	ADJ
ejpam-3237	29	6	plane	plane	NOUN
ejpam-3237	29	7	.	.	PUNCT
ejpam-3237	30	1	denote	denote	VERB
ejpam-3237	30	2	by	by	ADP
ejpam-3237	30	3	‖.‖	‖.‖	NOUN
ejpam-3237	30	4	the	the	DET
ejpam-3237	30	5	norm	norm	NOUN
ejpam-3237	30	6	of	of	ADP
ejpam-3237	30	7	x.	x.	NOUN
ejpam-3237	30	8	fix	fix	VERB
ejpam-3237	30	9	a	a	DET
ejpam-3237	30	10	basis	basis	NOUN
ejpam-3237	30	11	{	{	PUNCT
ejpam-3237	30	12	e1	e1	NOUN
ejpam-3237	30	13	,	,	PUNCT
ejpam-3237	30	14	e2	e2	PROPN
ejpam-3237	30	15	}	}	PUNCT
ejpam-3237	30	16	of	of	ADP
ejpam-3237	30	17	x.	x.	NOUN
ejpam-3237	30	18	then	then	ADV
ejpam-3237	30	19	we	we	PRON
ejpam-3237	30	20	can	can	AUX
ejpam-3237	30	21	write	write	VERB
ejpam-3237	30	22	each	each	DET
ejpam-3237	30	23	x	x	SYM
ejpam-3237	30	24	∈	∈	PROPN
ejpam-3237	30	25	x	x	PUNCT
ejpam-3237	30	26	as	as	ADP
ejpam-3237	30	27	x	x	X
ejpam-3237	30	28	=	=	SYM
ejpam-3237	30	29	(	(	PUNCT
ejpam-3237	30	30	x1	x1	PROPN
ejpam-3237	30	31	,	,	PUNCT
ejpam-3237	30	32	x2	x2	PROPN
ejpam-3237	30	33	)	)	PUNCT
ejpam-3237	30	34	under	under	ADP
ejpam-3237	30	35	this	this	DET
ejpam-3237	30	36	basis	basis	NOUN
ejpam-3237	30	37	,	,	PUNCT
ejpam-3237	30	38	where	where	SCONJ
ejpam-3237	30	39	x1	x1	X
ejpam-3237	30	40	,	,	PUNCT
ejpam-3237	30	41	x2	x2	PROPN
ejpam-3237	30	42	∈	∈	PROPN
ejpam-3237	30	43	r.	r.	PROPN
ejpam-3237	30	44	moreover	moreover	ADV
ejpam-3237	30	45	,	,	PUNCT
ejpam-3237	30	46	{	{	PUNCT
ejpam-3237	30	47	δe1	δe1	PROPN
ejpam-3237	30	48	,	,	PUNCT
ejpam-3237	30	49	δe2	δe2	PROPN
ejpam-3237	30	50	}	}	PUNCT
ejpam-3237	30	51	is	be	AUX
ejpam-3237	30	52	a	a	DET
ejpam-3237	30	53	basis	basis	NOUN
ejpam-3237	30	54	of	of	ADP
ejpam-3237	30	55	the	the	DET
ejpam-3237	30	56	dual	dual	ADJ
ejpam-3237	30	57	space	space	NOUN
ejpam-3237	30	58	x∗	x∗	PROPN
ejpam-3237	30	59	,	,	PUNCT
ejpam-3237	30	60	where	where	SCONJ
ejpam-3237	30	61	δei	δei	VERB
ejpam-3237	30	62	for	for	ADP
ejpam-3237	30	63	i	i	PRON
ejpam-3237	30	64	=	=	SYM
ejpam-3237	30	65	1	1	NUM
ejpam-3237	30	66	,	,	PUNCT
ejpam-3237	30	67	2	2	NUM
ejpam-3237	30	68	is	be	AUX
ejpam-3237	30	69	a	a	DET
ejpam-3237	30	70	bounded	bounded	ADJ
ejpam-3237	30	71	linear	linear	ADJ
ejpam-3237	30	72	function	function	NOUN
ejpam-3237	30	73	on	on	ADP
ejpam-3237	30	74	x	x	PUNCT
ejpam-3237	30	75	with	with	ADP
ejpam-3237	30	76	δei(ej	δei(ej	NOUN
ejpam-3237	30	77	)	)	PUNCT
ejpam-3237	30	78	=	=	PRON
ejpam-3237	30	79	{	{	PUNCT
ejpam-3237	30	80	0	0	NUM
ejpam-3237	30	81	i	i	PRON
ejpam-3237	30	82	6=	6=	PROPN
ejpam-3237	31	1	j	j	NOUN
ejpam-3237	31	2	;	;	PUNCT
ejpam-3237	31	3	1	1	NUM
ejpam-3237	31	4	i	i	NOUN
ejpam-3237	31	5	=	=	PUNCT
ejpam-3237	31	6	j.	j.	PROPN
ejpam-3237	31	7	denote	denote	VERB
ejpam-3237	31	8	by	by	ADP
ejpam-3237	31	9	l(x	l(x	PROPN
ejpam-3237	31	10	)	)	PUNCT
ejpam-3237	31	11	the	the	DET
ejpam-3237	31	12	set	set	NOUN
ejpam-3237	31	13	of	of	ADP
ejpam-3237	31	14	all	all	DET
ejpam-3237	31	15	bounded	bound	VERB
ejpam-3237	31	16	linear	linear	PROPN
ejpam-3237	31	17	operators	operator	NOUN
ejpam-3237	31	18	from	from	ADP
ejpam-3237	31	19	x	x	PRON
ejpam-3237	31	20	to	to	ADP
ejpam-3237	31	21	x.	x.	NOUN
ejpam-3237	31	22	for	for	ADP
ejpam-3237	31	23	t	t	PROPN
ejpam-3237	31	24	∈	∈	PROPN
ejpam-3237	31	25	l(x	l(x	PROPN
ejpam-3237	31	26	)	)	PUNCT
ejpam-3237	31	27	,	,	PUNCT
ejpam-3237	31	28	the	the	DET
ejpam-3237	31	29	operator	operator	NOUN
ejpam-3237	31	30	t	t	PROPN
ejpam-3237	31	31	∗	∗	NOUN
ejpam-3237	31	32	∈	∈	PROPN
ejpam-3237	31	33	l(x∗	l(x∗	NOUN
ejpam-3237	31	34	)	)	PUNCT
ejpam-3237	31	35	is	be	AUX
ejpam-3237	31	36	said	say	VERB
ejpam-3237	31	37	to	to	PART
ejpam-3237	31	38	be	be	AUX
ejpam-3237	31	39	the	the	DET
ejpam-3237	31	40	banach	banach	ADV
ejpam-3237	31	41	conjugate	conjugate	ADJ
ejpam-3237	31	42	operator	operator	NOUN
ejpam-3237	31	43	of	of	ADP
ejpam-3237	31	44	t	t	PROPN
ejpam-3237	31	45	if	if	SCONJ
ejpam-3237	31	46	for	for	ADP
ejpam-3237	31	47	any	any	DET
ejpam-3237	31	48	z	z	NOUN
ejpam-3237	31	49	∈	∈	PROPN
ejpam-3237	31	50	x	x	X
ejpam-3237	31	51	and	and	CCONJ
ejpam-3237	31	52	any	any	DET
ejpam-3237	31	53	z∗	z∗	NOUN
ejpam-3237	31	54	∈	∈	PROPN
ejpam-3237	31	55	x∗	x∗	NOUN
ejpam-3237	31	56	,	,	PUNCT
ejpam-3237	31	57	there	there	PRON
ejpam-3237	31	58	must	must	AUX
ejpam-3237	31	59	be	be	AUX
ejpam-3237	31	60	(	(	PUNCT
ejpam-3237	31	61	t	t	NOUN
ejpam-3237	31	62	∗z∗)(z	∗z∗)(z	PROPN
ejpam-3237	31	63	)	)	PUNCT
ejpam-3237	32	1	=	=	SYM
ejpam-3237	32	2	z∗(tz	z∗(tz	NOUN
ejpam-3237	32	3	)	)	PUNCT
ejpam-3237	32	4	.	.	PUNCT
ejpam-3237	33	1	note	note	VERB
ejpam-3237	33	2	that	that	SCONJ
ejpam-3237	33	3	if	if	SCONJ
ejpam-3237	33	4	we	we	PRON
ejpam-3237	33	5	use	use	VERB
ejpam-3237	33	6	the	the	DET
ejpam-3237	33	7	following	following	ADJ
ejpam-3237	33	8	notation	notation	NOUN
ejpam-3237	33	9	f(x	f(x	PROPN
ejpam-3237	33	10	)	)	PUNCT
ejpam-3237	34	1	=	=	PUNCT
ejpam-3237	34	2	〈	〈	PROPN
ejpam-3237	34	3	x	x	X
ejpam-3237	34	4	,	,	PUNCT
ejpam-3237	34	5	f	f	PROPN
ejpam-3237	34	6	〉	〉	PROPN
ejpam-3237	34	7	then	then	ADV
ejpam-3237	34	8	the	the	DET
ejpam-3237	34	9	property	property	NOUN
ejpam-3237	34	10	of	of	ADP
ejpam-3237	34	11	conjugate	conjugate	NOUN
ejpam-3237	34	12	can	can	AUX
ejpam-3237	34	13	be	be	AUX
ejpam-3237	34	14	rewritten	rewrite	VERB
ejpam-3237	34	15	as	as	ADP
ejpam-3237	34	16	the	the	DET
ejpam-3237	34	17	following	following	ADJ
ejpam-3237	34	18	way	way	NOUN
ejpam-3237	34	19	〈	〈	PROPN
ejpam-3237	34	20	x	x	X
ejpam-3237	34	21	,	,	PUNCT
ejpam-3237	34	22	t	t	PROPN
ejpam-3237	34	23	∗f	∗f	PROPN
ejpam-3237	34	24	〉	〉	PROPN
ejpam-3237	35	1	=	=	SYM
ejpam-3237	36	1	〈	〈	PROPN
ejpam-3237	36	2	tx	tx	PROPN
ejpam-3237	36	3	,	,	PUNCT
ejpam-3237	36	4	f	f	PROPN
ejpam-3237	36	5	〉	〉	NOUN
ejpam-3237	36	6	as	as	ADP
ejpam-3237	36	7	usual	usual	ADJ
ejpam-3237	36	8	in	in	ADP
ejpam-3237	36	9	inner	inner	ADJ
ejpam-3237	36	10	product	product	NOUN
ejpam-3237	36	11	spaces	space	NOUN
ejpam-3237	36	12	.	.	PUNCT
ejpam-3237	37	1	recall	recall	VERB
ejpam-3237	37	2	that	that	SCONJ
ejpam-3237	37	3	an	an	DET
ejpam-3237	37	4	operator	operator	NOUN
ejpam-3237	37	5	p	p	NOUN
ejpam-3237	37	6	is	be	AUX
ejpam-3237	37	7	an	an	DET
ejpam-3237	37	8	orthogonal	orthogonal	ADJ
ejpam-3237	37	9	projection	projection	NOUN
ejpam-3237	37	10	if	if	SCONJ
ejpam-3237	37	11	it	it	PRON
ejpam-3237	37	12	is	be	AUX
ejpam-3237	37	13	idempotent	idempotent	ADJ
ejpam-3237	37	14	and	and	CCONJ
ejpam-3237	37	15	selfadjoint	selfadjoint	VERB
ejpam-3237	37	16	,	,	PUNCT
ejpam-3237	37	17	i.e.	i.e.	X
ejpam-3237	37	18	p	p	DET
ejpam-3237	37	19	2	2	NUM
ejpam-3237	37	20	=	=	SYM
ejpam-3237	37	21	p	p	NOUN
ejpam-3237	37	22	and	and	CCONJ
ejpam-3237	37	23	p	p	NOUN
ejpam-3237	37	24	∗	∗	NOUN
ejpam-3237	37	25	=	=	SYM
ejpam-3237	37	26	p	p	NOUN
ejpam-3237	37	27	.	.	PUNCT
ejpam-3237	38	1	in	in	ADP
ejpam-3237	38	2	an	an	DET
ejpam-3237	38	3	inner	inner	ADJ
ejpam-3237	38	4	product	product	NOUN
ejpam-3237	38	5	space	space	NOUN
ejpam-3237	38	6	it	it	PRON
ejpam-3237	38	7	is	be	AUX
ejpam-3237	38	8	equivalent	equivalent	ADJ
ejpam-3237	38	9	to	to	ADP
ejpam-3237	38	10	〈	〈	PROPN
ejpam-3237	38	11	px	px	PROPN
ejpam-3237	38	12	,	,	PUNCT
ejpam-3237	38	13	x	x	NOUN
ejpam-3237	38	14	〉	〉	NOUN
ejpam-3237	38	15	=	=	SYM
ejpam-3237	38	16	〈	〈	PROPN
ejpam-3237	38	17	px	px	PROPN
ejpam-3237	38	18	,	,	PUNCT
ejpam-3237	38	19	px	px	PROPN
ejpam-3237	38	20	〉	〉	NOUN
ejpam-3237	38	21	=	=	SYM
ejpam-3237	38	22	〈	〈	PROPN
ejpam-3237	38	23	x	x	X
ejpam-3237	38	24	,	,	PUNCT
ejpam-3237	38	25	px	px	PROPN
ejpam-3237	38	26	〉	〉	PROPN
ejpam-3237	38	27	.	.	PUNCT
ejpam-3237	39	1	suppose	suppose	VERB
ejpam-3237	39	2	that	that	SCONJ
ejpam-3237	39	3	x	x	X
ejpam-3237	39	4	=	=	PRON
ejpam-3237	39	5	(	(	PUNCT
ejpam-3237	39	6	x1	x1	PROPN
ejpam-3237	39	7	,	,	PUNCT
ejpam-3237	39	8	x2)t	x2)t	PROPN
ejpam-3237	39	9	and	and	CCONJ
ejpam-3237	39	10	y	y	PROPN
ejpam-3237	39	11	=	=	SYM
ejpam-3237	39	12	(	(	PUNCT
ejpam-3237	39	13	y1	y1	INTJ
ejpam-3237	39	14	,	,	PUNCT
ejpam-3237	39	15	y2)t	y2)t	ADV
ejpam-3237	39	16	are	be	AUX
ejpam-3237	39	17	two	two	NUM
ejpam-3237	39	18	linearly	linearly	ADV
ejpam-3237	39	19	independent	independent	ADJ
ejpam-3237	39	20	vectors	vector	NOUN
ejpam-3237	39	21	in	in	ADP
ejpam-3237	39	22	x	x	PUNCT
ejpam-3237	39	23	under	under	ADP
ejpam-3237	39	24	the	the	DET
ejpam-3237	39	25	basis	basis	NOUN
ejpam-3237	39	26	{	{	PUNCT
ejpam-3237	39	27	e1	e1	NOUN
ejpam-3237	39	28	,	,	PUNCT
ejpam-3237	39	29	e2	e2	PROPN
ejpam-3237	39	30	}	}	PUNCT
ejpam-3237	39	31	.	.	PUNCT
ejpam-3237	40	1	put	put	VERB
ejpam-3237	40	2	dxy	dxy	PROPN
ejpam-3237	40	3	=	=	PRON
ejpam-3237	41	1	[	[	PUNCT
ejpam-3237	41	2	x1	x1	PROPN
ejpam-3237	41	3	y1	y1	INTJ
ejpam-3237	41	4	x2	x2	PROPN
ejpam-3237	41	5	y2	y2	PROPN
ejpam-3237	41	6	]	]	PUNCT
ejpam-3237	41	7	notice	notice	NOUN
ejpam-3237	41	8	|dxy|	|dxy|	X
ejpam-3237	41	9	=	=	X
ejpam-3237	41	10	x1y2	x1y2	PROPN
ejpam-3237	41	11	−	−	PROPN
ejpam-3237	42	1	x2y1	x2y1	PROPN
ejpam-3237	42	2	6=	6=	ADP
ejpam-3237	42	3	0	0	NUM
ejpam-3237	42	4	since	since	SCONJ
ejpam-3237	42	5	x	x	PROPN
ejpam-3237	42	6	and	and	CCONJ
ejpam-3237	42	7	y	y	PROPN
ejpam-3237	42	8	are	be	AUX
ejpam-3237	42	9	linearly	linearly	ADV
ejpam-3237	42	10	independent	independent	ADJ
ejpam-3237	42	11	.	.	PUNCT
ejpam-3237	43	1	define	define	VERB
ejpam-3237	43	2	by	by	ADP
ejpam-3237	43	3	pxy	pxy	NOUN
ejpam-3237	43	4	the	the	DET
ejpam-3237	43	5	projection	projection	NOUN
ejpam-3237	43	6	parallel	parallel	NOUN
ejpam-3237	43	7	to	to	ADP
ejpam-3237	43	8	y	y	PROPN
ejpam-3237	43	9	from	from	ADP
ejpam-3237	43	10	x	x	X
ejpam-3237	43	11	to	to	ADP
ejpam-3237	43	12	the	the	DET
ejpam-3237	43	13	subspace	subspace	NOUN
ejpam-3237	43	14	{	{	PUNCT
ejpam-3237	43	15	λx;λ	λx;λ	PUNCT
ejpam-3237	43	16	∈	∈	PROPN
ejpam-3237	43	17	r	r	NOUN
ejpam-3237	43	18	}	}	PUNCT
ejpam-3237	43	19	.	.	PUNCT
ejpam-3237	44	1	then	then	ADV
ejpam-3237	44	2	pxy	pxy	NOUN
ejpam-3237	44	3	depends	depend	VERB
ejpam-3237	44	4	only	only	ADV
ejpam-3237	44	5	on	on	ADP
ejpam-3237	44	6	the	the	DET
ejpam-3237	44	7	vectors	vector	NOUN
ejpam-3237	44	8	x	x	PUNCT
ejpam-3237	44	9	and	and	CCONJ
ejpam-3237	44	10	y	y	PROPN
ejpam-3237	44	11	,	,	PUNCT
ejpam-3237	44	12	and	and	CCONJ
ejpam-3237	44	13	has	have	VERB
ejpam-3237	44	14	the	the	DET
ejpam-3237	44	15	following	following	ADJ
ejpam-3237	44	16	presentation	presentation	NOUN
ejpam-3237	44	17	under	under	ADP
ejpam-3237	44	18	the	the	DET
ejpam-3237	44	19	basis	basis	NOUN
ejpam-3237	44	20	{	{	PUNCT
ejpam-3237	44	21	e1	e1	NOUN
ejpam-3237	44	22	,	,	PUNCT
ejpam-3237	44	23	e2	e2	PROPN
ejpam-3237	44	24	}	}	PUNCT
ejpam-3237	44	25	:	:	PUNCT
ejpam-3237	44	26	pxy	pxy	PROPN
ejpam-3237	44	27	=	=	SYM
ejpam-3237	44	28	dxy	dxy	PROPN
ejpam-3237	44	29	.	.	PUNCT
ejpam-3237	45	1	[	[	PUNCT
ejpam-3237	45	2	1	1	NUM
ejpam-3237	45	3	0	0	NUM
ejpam-3237	45	4	0	0	NUM
ejpam-3237	45	5	0	0	NUM
ejpam-3237	45	6	]	]	PUNCT
ejpam-3237	45	7	.	.	PUNCT
ejpam-3237	46	1	dxy	dxy	PROPN
ejpam-3237	46	2	−1	−1	NOUN
ejpam-3237	46	3	=	=	SYM
ejpam-3237	46	4	1	1	NUM
ejpam-3237	46	5	|dxy|	|dxy|	PROPN
ejpam-3237	46	6	[	[	PUNCT
ejpam-3237	46	7	x1y2	x1y2	X
ejpam-3237	46	8	−x1y1	−x1y1	X
ejpam-3237	46	9	x2y2	x2y2	X
ejpam-3237	47	1	−x2y1	−x2y1	PROPN
ejpam-3237	47	2	]	]	PUNCT
ejpam-3237	47	3	.	.	PUNCT
ejpam-3237	48	1	m.	m.	PROPN
ejpam-3237	48	2	iranmanesh	iranmanesh	PROPN
ejpam-3237	48	3	,	,	PUNCT
ejpam-3237	48	4	m.	m.	NOUN
ejpam-3237	48	5	saeedi	saeedi	PROPN
ejpam-3237	48	6	khojasteh	khojasteh	PROPN
ejpam-3237	48	7	,	,	PUNCT
ejpam-3237	48	8	m.	m.	NOUN
ejpam-3237	48	9	k.	k.	PROPN
ejpam-3237	48	10	anwary	anwary	PROPN
ejpam-3237	48	11	/	/	SYM
ejpam-3237	48	12	eur	eur	PROPN
ejpam-3237	48	13	.	.	PUNCT
ejpam-3237	49	1	j.	j.	PROPN
ejpam-3237	49	2	pure	pure	PROPN
ejpam-3237	49	3	appl	appl	PROPN
ejpam-3237	49	4	.	.	PROPN
ejpam-3237	49	5	math	math	PROPN
ejpam-3237	49	6	,	,	PUNCT
ejpam-3237	49	7	11	11	NUM
ejpam-3237	49	8	(	(	PUNCT
ejpam-3237	49	9	3	3	NUM
ejpam-3237	49	10	)	)	PUNCT
ejpam-3237	49	11	(	(	PUNCT
ejpam-3237	49	12	2018	2018	NUM
ejpam-3237	49	13	)	)	PUNCT
ejpam-3237	49	14	,	,	PUNCT
ejpam-3237	49	15	793	793	NUM
ejpam-3237	49	16	-	-	SYM
ejpam-3237	49	17	802	802	NUM
ejpam-3237	49	18	795	795	NUM
ejpam-3237	49	19	it	it	PRON
ejpam-3237	49	20	is	be	AUX
ejpam-3237	49	21	clear	clear	ADJ
ejpam-3237	49	22	for	for	ADP
ejpam-3237	49	23	any	any	DET
ejpam-3237	49	24	two	two	NUM
ejpam-3237	49	25	linearly	linearly	ADV
ejpam-3237	49	26	independent	independent	ADJ
ejpam-3237	49	27	vectors	vector	NOUN
ejpam-3237	49	28	x	x	PUNCT
ejpam-3237	49	29	and	and	CCONJ
ejpam-3237	49	30	y	y	PROPN
ejpam-3237	49	31	in	in	ADP
ejpam-3237	49	32	x	x	PROPN
ejpam-3237	49	33	,	,	PUNCT
ejpam-3237	49	34	1	1	NUM
ejpam-3237	49	35	≤	≤	NOUN
ejpam-3237	49	36	‖pxy‖	‖pxy‖	PUNCT
ejpam-3237	49	37	<	<	X
ejpam-3237	49	38	+	+	NUM
ejpam-3237	49	39	∞.	∞.	PROPN
ejpam-3237	49	40	note	note	VERB
ejpam-3237	49	41	that	that	SCONJ
ejpam-3237	49	42	if	if	SCONJ
ejpam-3237	49	43	x	x	NOUN
ejpam-3237	49	44	,	,	PUNCT
ejpam-3237	49	45	y	y	PROPN
ejpam-3237	49	46	are	be	AUX
ejpam-3237	49	47	orthogonal	orthogonal	ADJ
ejpam-3237	49	48	,	,	PUNCT
ejpam-3237	49	49	in	in	ADP
ejpam-3237	49	50	the	the	DET
ejpam-3237	49	51	sense	sense	NOUN
ejpam-3237	49	52	of	of	ADP
ejpam-3237	49	53	inner	inner	ADJ
ejpam-3237	49	54	product	product	NOUN
ejpam-3237	49	55	space	space	NOUN
ejpam-3237	49	56	,	,	PUNCT
ejpam-3237	49	57	then	then	ADV
ejpam-3237	49	58	pxy	pxy	NOUN
ejpam-3237	49	59	is	be	AUX
ejpam-3237	49	60	an	an	DET
ejpam-3237	49	61	orthogonal	orthogonal	ADJ
ejpam-3237	49	62	projection	projection	NOUN
ejpam-3237	49	63	.	.	PUNCT
ejpam-3237	50	1	furthermore	furthermore	ADV
ejpam-3237	50	2	,	,	PUNCT
ejpam-3237	50	3	denote	denote	VERB
ejpam-3237	50	4	p(x	p(x	PROPN
ejpam-3237	50	5	,	,	PUNCT
ejpam-3237	50	6	y	y	NOUN
ejpam-3237	50	7	)	)	PUNCT
ejpam-3237	50	8	=	=	PRON
ejpam-3237	50	9	{	{	PUNCT
ejpam-3237	50	10	0	0	NUM
ejpam-3237	50	11	x	x	NOUN
ejpam-3237	50	12	and	and	CCONJ
ejpam-3237	50	13	y	y	PROPN
ejpam-3237	50	14	are	be	AUX
ejpam-3237	50	15	linearly	linearly	ADV
ejpam-3237	50	16	dependent	dependent	ADJ
ejpam-3237	50	17	;	;	PUNCT
ejpam-3237	50	18	‖pxy‖−1	‖pxy‖−1	NOUN
ejpam-3237	50	19	x	x	X
ejpam-3237	50	20	and	and	CCONJ
ejpam-3237	50	21	y	y	PROPN
ejpam-3237	50	22	are	be	AUX
ejpam-3237	50	23	linearly	linearly	ADV
ejpam-3237	50	24	independent	independent	ADJ
ejpam-3237	50	25	.	.	PUNCT
ejpam-3237	51	1	for	for	ADP
ejpam-3237	51	2	any	any	DET
ejpam-3237	51	3	x	x	NOUN
ejpam-3237	51	4	,	,	PUNCT
ejpam-3237	51	5	y	y	PROPN
ejpam-3237	51	6	∈	∈	PROPN
ejpam-3237	51	7	x	x	PROPN
ejpam-3237	51	8	,	,	PUNCT
ejpam-3237	51	9	the	the	DET
ejpam-3237	51	10	p	p	NOUN
ejpam-3237	51	11	-	-	PUNCT
ejpam-3237	51	12	angle	angle	NOUN
ejpam-3237	51	13	between	between	ADP
ejpam-3237	51	14	x	x	PUNCT
ejpam-3237	51	15	and	and	CCONJ
ejpam-3237	51	16	y	y	PROPN
ejpam-3237	51	17	is	be	AUX
ejpam-3237	51	18	defined	define	VERB
ejpam-3237	51	19	by	by	ADP
ejpam-3237	51	20	ap(x	ap(x	PROPN
ejpam-3237	51	21	,	,	PUNCT
ejpam-3237	51	22	y	y	PROPN
ejpam-3237	51	23	)	)	PUNCT
ejpam-3237	52	1	=	=	SYM
ejpam-3237	52	2	arcsin(p(x	arcsin(p(x	PROPN
ejpam-3237	52	3	,	,	PUNCT
ejpam-3237	52	4	y	y	NOUN
ejpam-3237	52	5	)	)	PUNCT
ejpam-3237	52	6	)	)	PUNCT
ejpam-3237	52	7	.	.	PUNCT
ejpam-3237	53	1	in	in	ADP
ejpam-3237	53	2	an	an	DET
ejpam-3237	53	3	inner	inner	ADJ
ejpam-3237	53	4	product	product	NOUN
ejpam-3237	53	5	space	space	NOUN
ejpam-3237	53	6	(	(	PUNCT
ejpam-3237	53	7	x	x	NOUN
ejpam-3237	53	8	,	,	PUNCT
ejpam-3237	53	9	〈	〈	PROPN
ejpam-3237	53	10	.	.	PROPN
ejpam-3237	53	11	,	,	PUNCT
ejpam-3237	53	12	.	.	PUNCT
ejpam-3237	53	13	〉	〉	NOUN
ejpam-3237	53	14	)	)	PUNCT
ejpam-3237	53	15	,	,	PUNCT
ejpam-3237	53	16	obviously	obviously	ADV
ejpam-3237	53	17	p(x	p(x	PROPN
ejpam-3237	53	18	,	,	PUNCT
ejpam-3237	53	19	y	y	NOUN
ejpam-3237	53	20	)	)	PUNCT
ejpam-3237	53	21	=	=	PUNCT
ejpam-3237	53	22	〈	〈	PROPN
ejpam-3237	53	23	x	x	X
ejpam-3237	53	24	,	,	PUNCT
ejpam-3237	53	25	y	y	PROPN
ejpam-3237	53	26	〉	〉	NUM
ejpam-3237	53	27	‖x‖‖y‖	‖x‖‖y‖	NOUN
ejpam-3237	53	28	,	,	PUNCT
ejpam-3237	53	29	and	and	CCONJ
ejpam-3237	53	30	consequently	consequently	ADV
ejpam-3237	53	31	,	,	PUNCT
ejpam-3237	53	32	the	the	DET
ejpam-3237	53	33	p	p	NOUN
ejpam-3237	53	34	-	-	PUNCT
ejpam-3237	53	35	angle	angle	NOUN
ejpam-3237	53	36	is	be	AUX
ejpam-3237	53	37	identical	identical	ADJ
ejpam-3237	53	38	with	with	ADP
ejpam-3237	53	39	the	the	DET
ejpam-3237	53	40	usual	usual	ADJ
ejpam-3237	53	41	angle	angle	NOUN
ejpam-3237	53	42	.	.	PUNCT
ejpam-3237	54	1	let	let	AUX
ejpam-3237	54	2	(	(	PUNCT
ejpam-3237	54	3	x	x	NOUN
ejpam-3237	54	4	,	,	PUNCT
ejpam-3237	54	5	‖.‖	‖.‖	NOUN
ejpam-3237	54	6	)	)	PUNCT
ejpam-3237	54	7	be	be	VERB
ejpam-3237	54	8	a	a	DET
ejpam-3237	54	9	complex	complex	ADJ
ejpam-3237	54	10	banach	banach	NOUN
ejpam-3237	54	11	space	space	NOUN
ejpam-3237	54	12	,	,	PUNCT
ejpam-3237	54	13	(	(	PUNCT
ejpam-3237	54	14	x∗	x∗	X
ejpam-3237	54	15	,	,	PUNCT
ejpam-3237	54	16	‖.‖	‖.‖	NOUN
ejpam-3237	54	17	)	)	PUNCT
ejpam-3237	54	18	be	be	AUX
ejpam-3237	54	19	its	its	PRON
ejpam-3237	54	20	dual	dual	ADJ
ejpam-3237	54	21	space	space	NOUN
ejpam-3237	54	22	,	,	PUNCT
ejpam-3237	54	23	and	and	CCONJ
ejpam-3237	54	24	b(x	b(x	NOUN
ejpam-3237	54	25	)	)	PUNCT
ejpam-3237	54	26	be	be	VERB
ejpam-3237	54	27	the	the	DET
ejpam-3237	54	28	algebra	algebra	NOUN
ejpam-3237	54	29	of	of	ADP
ejpam-3237	54	30	all	all	DET
ejpam-3237	54	31	bounded	bound	VERB
ejpam-3237	54	32	linear	linear	PROPN
ejpam-3237	54	33	operators	operator	NOUN
ejpam-3237	54	34	acting	act	VERB
ejpam-3237	54	35	on	on	ADP
ejpam-3237	55	1	x.	x.	NOUN
ejpam-3237	55	2	define	define	VERB
ejpam-3237	55	3	the	the	DET
ejpam-3237	55	4	set	set	NOUN
ejpam-3237	55	5	of	of	ADP
ejpam-3237	55	6	normalized	normalize	VERB
ejpam-3237	55	7	states	states	PROPN
ejpam-3237	55	8	ω	ω	PROPN
ejpam-3237	55	9	=	=	SYM
ejpam-3237	55	10	{	{	PUNCT
ejpam-3237	55	11	ω	ω	NUM
ejpam-3237	55	12	∈	∈	PROPN
ejpam-3237	55	13	b(x)∗	b(x)∗	NOUN
ejpam-3237	55	14	:	:	PUNCT
ejpam-3237	56	1	ω(i	ω(i	X
ejpam-3237	56	2	)	)	PUNCT
ejpam-3237	56	3	=	=	PUNCT
ejpam-3237	57	1	‖ω‖	‖ω‖	NOUN
ejpam-3237	57	2	=	=	NOUN
ejpam-3237	57	3	1	1	NUM
ejpam-3237	57	4	}	}	PUNCT
ejpam-3237	57	5	,	,	PUNCT
ejpam-3237	57	6	where	where	SCONJ
ejpam-3237	57	7	i	i	PRON
ejpam-3237	57	8	denotes	denote	VERB
ejpam-3237	57	9	the	the	DET
ejpam-3237	57	10	identity	identity	NOUN
ejpam-3237	57	11	operator	operator	NOUN
ejpam-3237	57	12	.	.	PUNCT
ejpam-3237	58	1	for	for	ADP
ejpam-3237	58	2	any	any	DET
ejpam-3237	58	3	operator	operator	NOUN
ejpam-3237	58	4	a	a	DET
ejpam-3237	58	5	∈	∈	NOUN
ejpam-3237	58	6	b(x	b(x	NOUN
ejpam-3237	58	7	)	)	PUNCT
ejpam-3237	58	8	,	,	PUNCT
ejpam-3237	58	9	the	the	DET
ejpam-3237	58	10	(	(	PUNCT
ejpam-3237	58	11	algebraic	algebraic	ADJ
ejpam-3237	58	12	)	)	PUNCT
ejpam-3237	58	13	numerical	numerical	ADJ
ejpam-3237	58	14	range	range	NOUN
ejpam-3237	58	15	(	(	PUNCT
ejpam-3237	58	16	also	also	ADV
ejpam-3237	58	17	known	know	VERB
ejpam-3237	58	18	as	as	ADP
ejpam-3237	58	19	field	field	NOUN
ejpam-3237	58	20	of	of	ADP
ejpam-3237	58	21	values	value	NOUN
ejpam-3237	58	22	)	)	PUNCT
ejpam-3237	58	23	of	of	ADP
ejpam-3237	58	24	a	a	PRON
ejpam-3237	58	25	is	be	AUX
ejpam-3237	58	26	defined	define	VERB
ejpam-3237	58	27	by	by	ADP
ejpam-3237	58	28	f	f	PROPN
ejpam-3237	58	29	(	(	PUNCT
ejpam-3237	58	30	a	a	X
ejpam-3237	58	31	)	)	PUNCT
ejpam-3237	58	32	=	=	SYM
ejpam-3237	58	33	{	{	PUNCT
ejpam-3237	58	34	ω(a	ω(a	PROPN
ejpam-3237	58	35	)	)	PUNCT
ejpam-3237	58	36	:	:	PUNCT
ejpam-3237	59	1	ω	ω	NUM
ejpam-3237	59	2	∈	∈	PROPN
ejpam-3237	59	3	ω	ω	NOUN
ejpam-3237	59	4	}	}	PUNCT
ejpam-3237	59	5	.	.	PUNCT
ejpam-3237	60	1	in	in	ADP
ejpam-3237	60	2	the	the	DET
ejpam-3237	60	3	finite	finite	ADJ
ejpam-3237	60	4	-	-	ADJ
ejpam-3237	60	5	dimensional	dimensional	ADJ
ejpam-3237	60	6	case	case	NOUN
ejpam-3237	60	7	(	(	PUNCT
ejpam-3237	60	8	x	x	X
ejpam-3237	60	9	,	,	PUNCT
ejpam-3237	60	10	‖.‖	‖.‖	NOUN
ejpam-3237	60	11	)	)	PUNCT
ejpam-3237	60	12	=	=	SYM
ejpam-3237	60	13	(	(	PUNCT
ejpam-3237	60	14	cn	cn	INTJ
ejpam-3237	60	15	,	,	PUNCT
ejpam-3237	60	16	‖.‖2	‖.‖2	PROPN
ejpam-3237	60	17	)	)	PUNCT
ejpam-3237	60	18	,	,	PUNCT
ejpam-3237	60	19	where	where	SCONJ
ejpam-3237	60	20	‖.‖2	‖.‖2	NOUN
ejpam-3237	60	21	is	be	AUX
ejpam-3237	60	22	the	the	DET
ejpam-3237	60	23	spectral	spectral	ADJ
ejpam-3237	60	24	norm	norm	NOUN
ejpam-3237	60	25	,	,	PUNCT
ejpam-3237	60	26	the	the	DET
ejpam-3237	60	27	numerical	numerical	ADJ
ejpam-3237	60	28	range	range	NOUN
ejpam-3237	60	29	of	of	ADP
ejpam-3237	60	30	a	a	DET
ejpam-3237	60	31	square	square	ADJ
ejpam-3237	60	32	matrix	matrix	NOUN
ejpam-3237	60	33	a	a	DET
ejpam-3237	60	34	∈	∈	PROPN
ejpam-3237	60	35	cn×n	cn×n	NOUN
ejpam-3237	60	36	is	be	AUX
ejpam-3237	60	37	also	also	ADV
ejpam-3237	60	38	written	write	VERB
ejpam-3237	60	39	f	f	PROPN
ejpam-3237	60	40	(	(	PUNCT
ejpam-3237	60	41	a	a	X
ejpam-3237	60	42	)	)	PUNCT
ejpam-3237	60	43	=	=	SYM
ejpam-3237	60	44	{	{	PUNCT
ejpam-3237	61	1	x∗ax	x∗ax	PROPN
ejpam-3237	61	2	∈	∈	PROPN
ejpam-3237	61	3	c	c	NOUN
ejpam-3237	61	4	:	:	PUNCT
ejpam-3237	61	5	x	x	SYM
ejpam-3237	61	6	∈	∈	PROPN
ejpam-3237	61	7	cn	cn	PROPN
ejpam-3237	61	8	,	,	PUNCT
ejpam-3237	61	9	x∗x	x∗x	PUNCT
ejpam-3237	61	10	=	=	NOUN
ejpam-3237	61	11	1	1	NUM
ejpam-3237	61	12	}	}	PUNCT
ejpam-3237	61	13	.	.	PUNCT
ejpam-3237	62	1	the	the	DET
ejpam-3237	62	2	suggested	suggest	VERB
ejpam-3237	62	3	references	reference	NOUN
ejpam-3237	62	4	on	on	ADP
ejpam-3237	62	5	numerical	numerical	ADJ
ejpam-3237	62	6	ranges	range	NOUN
ejpam-3237	62	7	of	of	ADP
ejpam-3237	62	8	operators	operator	NOUN
ejpam-3237	62	9	and	and	CCONJ
ejpam-3237	62	10	matrices	matrix	NOUN
ejpam-3237	62	11	are	be	AUX
ejpam-3237	62	12	[	[	X
ejpam-3237	62	13	3	3	NUM
ejpam-3237	62	14	]	]	PUNCT
ejpam-3237	62	15	and	and	CCONJ
ejpam-3237	62	16	[	[	X
ejpam-3237	62	17	5	5	NUM
ejpam-3237	62	18	]	]	PUNCT
ejpam-3237	62	19	.	.	PUNCT
ejpam-3237	63	1	we	we	PRON
ejpam-3237	63	2	recall	recall	VERB
ejpam-3237	63	3	that	that	PRON
ejpam-3237	63	4	for	for	ADP
ejpam-3237	63	5	two	two	NUM
ejpam-3237	63	6	compact	compact	ADJ
ejpam-3237	63	7	subsets	subset	NOUN
ejpam-3237	63	8	ω1	ω1	PROPN
ejpam-3237	63	9	and	and	CCONJ
ejpam-3237	63	10	ω2	ω2	ADJ
ejpam-3237	63	11	of	of	ADP
ejpam-3237	63	12	a	a	DET
ejpam-3237	63	13	metric	metric	ADJ
ejpam-3237	63	14	space	space	NOUN
ejpam-3237	63	15	(	(	PUNCT
ejpam-3237	63	16	x	x	X
ejpam-3237	63	17	,	,	PUNCT
ejpam-3237	63	18	ρ	ρ	PROPN
ejpam-3237	63	19	)	)	PUNCT
ejpam-3237	63	20	,	,	PUNCT
ejpam-3237	63	21	the	the	DET
ejpam-3237	63	22	hausdorff	hausdorff	NOUN
ejpam-3237	63	23	distance	distance	NOUN
ejpam-3237	63	24	between	between	ADP
ejpam-3237	63	25	ω1	ω1	PROPN
ejpam-3237	63	26	and	and	CCONJ
ejpam-3237	63	27	ω2	ω2	NOUN
ejpam-3237	63	28	is	be	AUX
ejpam-3237	63	29	defined	define	VERB
ejpam-3237	63	30	by	by	ADP
ejpam-3237	63	31	dh(ω1,ω2	dh(ω1,ω2	NOUN
ejpam-3237	63	32	)	)	PUNCT
ejpam-3237	63	33	=	=	PUNCT
ejpam-3237	63	34	max{max	max{max	NOUN
ejpam-3237	63	35	x1∈ω1	x1∈ω1	PROPN
ejpam-3237	63	36	min	min	PROPN
ejpam-3237	63	37	x2∈ω2	x2∈ω2	PROPN
ejpam-3237	63	38	ρ(x1	ρ(x1	PROPN
ejpam-3237	63	39	,	,	PUNCT
ejpam-3237	63	40	x2	x2	PROPN
ejpam-3237	63	41	)	)	PUNCT
ejpam-3237	63	42	,	,	PUNCT
ejpam-3237	63	43	max	max	PROPN
ejpam-3237	63	44	x2∈ω2	x2∈ω2	PROPN
ejpam-3237	63	45	min	min	PROPN
ejpam-3237	63	46	x1∈ω1	x1∈ω1	PROPN
ejpam-3237	63	47	ρ(x1	ρ(x1	PROPN
ejpam-3237	63	48	,	,	PUNCT
ejpam-3237	63	49	x2	x2	PROPN
ejpam-3237	63	50	)	)	PUNCT
ejpam-3237	63	51	}	}	PUNCT
ejpam-3237	63	52	for	for	ADP
ejpam-3237	63	53	any	any	DET
ejpam-3237	63	54	x0	x0	PROPN
ejpam-3237	63	55	∈	∈	PROPN
ejpam-3237	63	56	x	x	X
ejpam-3237	63	57	and	and	CCONJ
ejpam-3237	63	58	δ	δ	PROPN
ejpam-3237	63	59	>	>	X
ejpam-3237	63	60	0	0	PROPN
ejpam-3237	63	61	,	,	PUNCT
ejpam-3237	63	62	we	we	PRON
ejpam-3237	63	63	define	define	VERB
ejpam-3237	63	64	the	the	DET
ejpam-3237	63	65	closed	closed	ADJ
ejpam-3237	63	66	ball	ball	NOUN
ejpam-3237	63	67	b(x0	b(x0	NOUN
ejpam-3237	63	68	,	,	PUNCT
ejpam-3237	63	69	δ	δ	PROPN
ejpam-3237	63	70	)	)	PUNCT
ejpam-3237	63	71	=	=	PRON
ejpam-3237	64	1	{	{	PUNCT
ejpam-3237	64	2	x	x	PUNCT
ejpam-3237	64	3	∈	∈	PROPN
ejpam-3237	64	4	x	x	X
ejpam-3237	64	5	:	:	PUNCT
ejpam-3237	64	6	ρ(x0	ρ(x0	NUM
ejpam-3237	64	7	,	,	PUNCT
ejpam-3237	64	8	x	x	X
ejpam-3237	64	9	)	)	PUNCT
ejpam-3237	64	10	≤	≤	NUM
ejpam-3237	64	11	δ	δ	PROPN
ejpam-3237	64	12	}	}	PUNCT
ejpam-3237	64	13	.	.	PUNCT
ejpam-3237	65	1	definition	definition	NOUN
ejpam-3237	65	2	1	1	NUM
ejpam-3237	65	3	.	.	PUNCT
ejpam-3237	66	1	[	[	X
ejpam-3237	66	2	1	1	X
ejpam-3237	66	3	]	]	PUNCT
ejpam-3237	66	4	suppose	suppose	VERB
ejpam-3237	66	5	(	(	PUNCT
ejpam-3237	66	6	x	x	X
ejpam-3237	66	7	,	,	PUNCT
ejpam-3237	66	8	ρx	ρx	VERB
ejpam-3237	66	9	)	)	PUNCT
ejpam-3237	66	10	is	be	AUX
ejpam-3237	66	11	a	a	DET
ejpam-3237	66	12	metric	metric	ADJ
ejpam-3237	66	13	space	space	NOUN
ejpam-3237	66	14	and	and	CCONJ
ejpam-3237	66	15	(	(	PUNCT
ejpam-3237	66	16	y	y	PROPN
ejpam-3237	66	17	,	,	PUNCT
ejpam-3237	66	18	ρy	ρy	ADP
ejpam-3237	66	19	)	)	PUNCT
ejpam-3237	66	20	is	be	AUX
ejpam-3237	66	21	a	a	DET
ejpam-3237	66	22	complete	complete	ADJ
ejpam-3237	66	23	metric	metric	ADJ
ejpam-3237	66	24	space	space	NOUN
ejpam-3237	66	25	.	.	PUNCT
ejpam-3237	67	1	consider	consider	VERB
ejpam-3237	67	2	a	a	DET
ejpam-3237	67	3	multi	multi	ADJ
ejpam-3237	67	4	-	-	ADJ
ejpam-3237	67	5	valued	value	VERB
ejpam-3237	67	6	mapping	mapping	NOUN
ejpam-3237	67	7	f	f	X
ejpam-3237	68	1	:	:	PUNCT
ejpam-3237	68	2	x	x	X
ejpam-3237	68	3	→	→	SYM
ejpam-3237	68	4	y	y	PROPN
ejpam-3237	68	5	,	,	PUNCT
ejpam-3237	68	6	and	and	CCONJ
ejpam-3237	68	7	let	let	VERB
ejpam-3237	68	8	x0	x0	PROPN
ejpam-3237	68	9	∈	∈	PROPN
ejpam-3237	69	1	x.	x.	NOUN
ejpam-3237	69	2	(	(	PUNCT
ejpam-3237	70	1	i	i	NOUN
ejpam-3237	70	2	)	)	PUNCT
ejpam-3237	70	3	f	f	PROPN
ejpam-3237	70	4	is	be	AUX
ejpam-3237	70	5	called	call	VERB
ejpam-3237	70	6	upper	upper	ADJ
ejpam-3237	70	7	semi	semi	ADJ
ejpam-3237	70	8	-	-	ADJ
ejpam-3237	70	9	continuous	continuous	ADJ
ejpam-3237	70	10	at	at	ADP
ejpam-3237	70	11	x0	x0	PROPN
ejpam-3237	70	12	if	if	SCONJ
ejpam-3237	70	13	for	for	ADP
ejpam-3237	70	14	every	every	DET
ejpam-3237	70	15	neighborhood	neighborhood	NOUN
ejpam-3237	70	16	n(f	n(f	PROPN
ejpam-3237	70	17	(	(	PUNCT
ejpam-3237	70	18	x0	x0	PROPN
ejpam-3237	70	19	)	)	PUNCT
ejpam-3237	70	20	)	)	PUNCT
ejpam-3237	71	1	⊂	⊂	PROPN
ejpam-3237	72	1	y	y	PROPN
ejpam-3237	72	2	of	of	ADP
ejpam-3237	72	3	the	the	DET
ejpam-3237	72	4	set	set	ADJ
ejpam-3237	72	5	f	f	PROPN
ejpam-3237	72	6	(	(	PUNCT
ejpam-3237	72	7	x0	x0	PROPN
ejpam-3237	72	8	)	)	PUNCT
ejpam-3237	72	9	,	,	PUNCT
ejpam-3237	72	10	there	there	PRON
ejpam-3237	72	11	is	be	VERB
ejpam-3237	72	12	a	a	DET
ejpam-3237	72	13	neighborhood	neighborhood	NOUN
ejpam-3237	72	14	n(x0	n(x0	NOUN
ejpam-3237	72	15	)	)	PUNCT
ejpam-3237	73	1	⊂	⊂	PROPN
ejpam-3237	73	2	x	x	PUNCT
ejpam-3237	74	1	of	of	ADP
ejpam-3237	74	2	x0	x0	PROPN
ejpam-3237	75	1	such	such	ADJ
ejpam-3237	75	2	that	that	SCONJ
ejpam-3237	75	3	f	f	PROPN
ejpam-3237	75	4	(	(	PUNCT
ejpam-3237	75	5	x	x	X
ejpam-3237	75	6	)	)	PUNCT
ejpam-3237	75	7	⊂	⊂	PROPN
ejpam-3237	75	8	n(f	n(f	PROPN
ejpam-3237	75	9	(	(	PUNCT
ejpam-3237	75	10	x0	x0	PROPN
ejpam-3237	75	11	)	)	PUNCT
ejpam-3237	75	12	)	)	PUNCT
ejpam-3237	75	13	,	,	PUNCT
ejpam-3237	75	14	∀x	∀x	X
ejpam-3237	75	15	∈	∈	PROPN
ejpam-3237	75	16	n(x0	n(x0	NOUN
ejpam-3237	75	17	)	)	PUNCT
ejpam-3237	75	18	.	.	PUNCT
ejpam-3237	76	1	m.	m.	PROPN
ejpam-3237	76	2	iranmanesh	iranmanesh	PROPN
ejpam-3237	76	3	,	,	PUNCT
ejpam-3237	76	4	m.	m.	NOUN
ejpam-3237	76	5	saeedi	saeedi	PROPN
ejpam-3237	76	6	khojasteh	khojasteh	PROPN
ejpam-3237	76	7	,	,	PUNCT
ejpam-3237	76	8	m.	m.	NOUN
ejpam-3237	76	9	k.	k.	PROPN
ejpam-3237	76	10	anwary	anwary	PROPN
ejpam-3237	76	11	/	/	SYM
ejpam-3237	76	12	eur	eur	PROPN
ejpam-3237	76	13	.	.	PUNCT
ejpam-3237	77	1	j.	j.	PROPN
ejpam-3237	77	2	pure	pure	PROPN
ejpam-3237	77	3	appl	appl	PROPN
ejpam-3237	77	4	.	.	PROPN
ejpam-3237	77	5	math	math	PROPN
ejpam-3237	77	6	,	,	PUNCT
ejpam-3237	77	7	11	11	NUM
ejpam-3237	77	8	(	(	PUNCT
ejpam-3237	77	9	3	3	NUM
ejpam-3237	77	10	)	)	PUNCT
ejpam-3237	77	11	(	(	PUNCT
ejpam-3237	77	12	2018	2018	NUM
ejpam-3237	77	13	)	)	PUNCT
ejpam-3237	77	14	,	,	PUNCT
ejpam-3237	77	15	793	793	NUM
ejpam-3237	77	16	-	-	SYM
ejpam-3237	77	17	802	802	NUM
ejpam-3237	77	18	796	796	NUM
ejpam-3237	77	19	(	(	PUNCT
ejpam-3237	77	20	ii	ii	NOUN
ejpam-3237	77	21	)	)	PUNCT
ejpam-3237	77	22	f	f	PROPN
ejpam-3237	77	23	is	be	AUX
ejpam-3237	77	24	called	call	VERB
ejpam-3237	77	25	lower	low	ADJ
ejpam-3237	77	26	semi	semi	ADJ
ejpam-3237	77	27	-	-	ADJ
ejpam-3237	77	28	continuous	continuous	ADJ
ejpam-3237	77	29	at	at	ADP
ejpam-3237	77	30	x0	x0	PROPN
ejpam-3237	77	31	if	if	SCONJ
ejpam-3237	77	32	for	for	ADP
ejpam-3237	77	33	every	every	DET
ejpam-3237	77	34	y0	y0	PROPN
ejpam-3237	77	35	∈	∈	ADJ
ejpam-3237	77	36	f	f	X
ejpam-3237	77	37	(	(	PUNCT
ejpam-3237	77	38	x0	x0	PROPN
ejpam-3237	77	39	)	)	PUNCT
ejpam-3237	77	40	and	and	CCONJ
ejpam-3237	77	41	every	every	DET
ejpam-3237	77	42	neighborhood	neighborhood	NOUN
ejpam-3237	77	43	n(y0	n(y0	NOUN
ejpam-3237	77	44	)	)	PUNCT
ejpam-3237	78	1	⊂	⊂	PROPN
ejpam-3237	79	1	y	y	PROPN
ejpam-3237	79	2	of	of	ADP
ejpam-3237	79	3	y0	y0	PROPN
ejpam-3237	79	4	,	,	PUNCT
ejpam-3237	79	5	there	there	PRON
ejpam-3237	79	6	exist	exist	VERB
ejpam-3237	79	7	a	a	DET
ejpam-3237	79	8	neighborhood	neighborhood	NOUN
ejpam-3237	79	9	n(x0	n(x0	NOUN
ejpam-3237	79	10	)	)	PUNCT
ejpam-3237	80	1	⊂	⊂	PROPN
ejpam-3237	80	2	x	x	PUNCT
ejpam-3237	81	1	of	of	ADP
ejpam-3237	81	2	x0	x0	PROPN
ejpam-3237	81	3	such	such	ADJ
ejpam-3237	81	4	that	that	SCONJ
ejpam-3237	81	5	f	f	PROPN
ejpam-3237	81	6	(	(	PUNCT
ejpam-3237	81	7	x	x	NOUN
ejpam-3237	81	8	)	)	PUNCT
ejpam-3237	81	9	∩n(y0	∩n(y0	NOUN
ejpam-3237	81	10	)	)	PUNCT
ejpam-3237	81	11	6=	6=	ADP
ejpam-3237	81	12	∅	∅	NOUN
ejpam-3237	81	13	,	,	PUNCT
ejpam-3237	81	14	∀x	∀x	X
ejpam-3237	81	15	∈	∈	PROPN
ejpam-3237	81	16	n(x0	n(x0	NOUN
ejpam-3237	81	17	)	)	PUNCT
ejpam-3237	81	18	.	.	PUNCT
ejpam-3237	82	1	(	(	PUNCT
ejpam-3237	82	2	iii	iii	X
ejpam-3237	82	3	)	)	PUNCT
ejpam-3237	82	4	f	f	PROPN
ejpam-3237	82	5	is	be	AUX
ejpam-3237	82	6	said	say	VERB
ejpam-3237	82	7	to	to	PART
ejpam-3237	82	8	be	be	AUX
ejpam-3237	82	9	semi	semi	ADV
ejpam-3237	82	10	continuous	continuous	ADJ
ejpam-3237	82	11	at	at	ADP
ejpam-3237	82	12	x0	x0	PROPN
ejpam-3237	82	13	if	if	SCONJ
ejpam-3237	82	14	it	it	PRON
ejpam-3237	82	15	is	be	AUX
ejpam-3237	82	16	upper	upper	ADJ
ejpam-3237	82	17	and	and	CCONJ
ejpam-3237	82	18	lower	low	ADJ
ejpam-3237	82	19	semi	semi	ADJ
ejpam-3237	82	20	-	-	ADJ
ejpam-3237	82	21	continuous	continuous	ADJ
ejpam-3237	82	22	.	.	PUNCT
ejpam-3237	83	1	the	the	DET
ejpam-3237	83	2	following	following	ADJ
ejpam-3237	83	3	example	example	NOUN
ejpam-3237	83	4	is	be	AUX
ejpam-3237	83	5	showing	show	VERB
ejpam-3237	83	6	some	some	PRON
ejpam-3237	83	7	of	of	ADP
ejpam-3237	83	8	the	the	DET
ejpam-3237	83	9	common	common	ADJ
ejpam-3237	83	10	behavior	behavior	NOUN
ejpam-3237	83	11	of	of	ADP
ejpam-3237	83	12	upper	upper	ADJ
ejpam-3237	83	13	semi	semi	ADJ
ejpam-3237	83	14	continuous	continuous	ADJ
ejpam-3237	83	15	functions	function	NOUN
ejpam-3237	83	16	.	.	PUNCT
ejpam-3237	84	1	example	example	NOUN
ejpam-3237	85	1	1	1	NUM
ejpam-3237	85	2	.	.	PUNCT
ejpam-3237	86	1	the	the	DET
ejpam-3237	86	2	following	follow	VERB
ejpam-3237	86	3	functions	function	NOUN
ejpam-3237	86	4	is	be	AUX
ejpam-3237	86	5	upper	upper	ADJ
ejpam-3237	86	6	semi	semi	ADJ
ejpam-3237	86	7	continuous	continuous	ADJ
ejpam-3237	86	8	f(α	f(α	NOUN
ejpam-3237	86	9	)	)	PUNCT
ejpam-3237	86	10	=	=	PRON
ejpam-3237	87	1	{	{	PUNCT
ejpam-3237	87	2	x	x	X
ejpam-3237	87	3	:	:	PUNCT
ejpam-3237	87	4	x	x	SYM
ejpam-3237	87	5	∈	∈	NOUN
ejpam-3237	87	6	x	x	X
ejpam-3237	87	7	,	,	PUNCT
ejpam-3237	87	8	‖x‖	‖x‖	VERB
ejpam-3237	87	9	≤	≤	NUM
ejpam-3237	87	10	|α|	|α|	PROPN
ejpam-3237	87	11	}	}	PUNCT
ejpam-3237	87	12	.	.	PUNCT
ejpam-3237	88	1	note	note	VERB
ejpam-3237	88	2	that	that	SCONJ
ejpam-3237	88	3	this	this	DET
ejpam-3237	88	4	function	function	NOUN
ejpam-3237	88	5	is	be	AUX
ejpam-3237	88	6	increasing	increase	VERB
ejpam-3237	88	7	in	in	ADP
ejpam-3237	88	8	the	the	DET
ejpam-3237	88	9	mean	mean	NOUN
ejpam-3237	88	10	that	that	SCONJ
ejpam-3237	88	11	if	if	SCONJ
ejpam-3237	88	12	|α|	|α|	PROPN
ejpam-3237	88	13	≤	≤	X
ejpam-3237	88	14	|β|	|β|	NOUN
ejpam-3237	88	15	then	then	ADV
ejpam-3237	88	16	f(α	f(α	NOUN
ejpam-3237	88	17	)	)	PUNCT
ejpam-3237	88	18	is	be	AUX
ejpam-3237	88	19	contained	contain	VERB
ejpam-3237	88	20	in	in	ADP
ejpam-3237	88	21	f(β	f(β	NOUN
ejpam-3237	88	22	)	)	PUNCT
ejpam-3237	88	23	.	.	PUNCT
ejpam-3237	89	1	we	we	PRON
ejpam-3237	89	2	will	will	AUX
ejpam-3237	89	3	investigate	investigate	VERB
ejpam-3237	89	4	the	the	DET
ejpam-3237	89	5	upper	upper	ADJ
ejpam-3237	89	6	semi	semi	ADJ
ejpam-3237	89	7	continuity	continuity	NOUN
ejpam-3237	89	8	at	at	ADP
ejpam-3237	89	9	α0	α0	ADJ
ejpam-3237	89	10	=	=	SYM
ejpam-3237	89	11	1	1	X
ejpam-3237	89	12	.	.	X
ejpam-3237	89	13	investigating	investigate	VERB
ejpam-3237	89	14	other	other	ADJ
ejpam-3237	89	15	points	point	NOUN
ejpam-3237	89	16	are	be	AUX
ejpam-3237	89	17	similar	similar	ADJ
ejpam-3237	89	18	.	.	PUNCT
ejpam-3237	90	1	assume	assume	VERB
ejpam-3237	90	2	that	that	SCONJ
ejpam-3237	90	3	n(f(1	n(f(1	NOUN
ejpam-3237	90	4	)	)	PUNCT
ejpam-3237	90	5	)	)	PUNCT
ejpam-3237	90	6	is	be	AUX
ejpam-3237	90	7	an	an	DET
ejpam-3237	90	8	open	open	ADJ
ejpam-3237	90	9	set	set	NOUN
ejpam-3237	90	10	containing	contain	VERB
ejpam-3237	90	11	f(1	f(1	PROPN
ejpam-3237	90	12	)	)	PUNCT
ejpam-3237	90	13	.	.	PUNCT
ejpam-3237	91	1	since	since	SCONJ
ejpam-3237	91	2	f(1	f(1	PROPN
ejpam-3237	91	3	)	)	PUNCT
ejpam-3237	91	4	is	be	AUX
ejpam-3237	91	5	closed	close	VERB
ejpam-3237	91	6	,	,	PUNCT
ejpam-3237	91	7	there	there	PRON
ejpam-3237	91	8	is	be	VERB
ejpam-3237	91	9	a	a	DET
ejpam-3237	91	10	scalar	scalar	NOUN
ejpam-3237	91	11	β	β	NOUN
ejpam-3237	91	12	such	such	ADJ
ejpam-3237	91	13	that	that	DET
ejpam-3237	91	14	f(1	f(1	PROPN
ejpam-3237	91	15	)	)	PUNCT
ejpam-3237	91	16	⊆	⊆	NUM
ejpam-3237	91	17	f(β	f(β	NOUN
ejpam-3237	91	18	)	)	PUNCT
ejpam-3237	91	19	=	=	PRON
ejpam-3237	92	1	{	{	PUNCT
ejpam-3237	92	2	x	x	X
ejpam-3237	92	3	:	:	PUNCT
ejpam-3237	92	4	x	x	SYM
ejpam-3237	92	5	∈	∈	NOUN
ejpam-3237	92	6	x	x	X
ejpam-3237	92	7	,	,	PUNCT
ejpam-3237	92	8	‖x‖	‖x‖	VERB
ejpam-3237	92	9	≤	≤	NOUN
ejpam-3237	92	10	|β|	|β|	NOUN
ejpam-3237	92	11	}	}	PUNCT
ejpam-3237	92	12	⊆	⊆	NUM
ejpam-3237	92	13	n(f(1	n(f(1	NOUN
ejpam-3237	92	14	)	)	PUNCT
ejpam-3237	92	15	)	)	PUNCT
ejpam-3237	92	16	.	.	PUNCT
ejpam-3237	93	1	it	it	PRON
ejpam-3237	93	2	is	be	AUX
ejpam-3237	93	3	clear	clear	ADJ
ejpam-3237	93	4	that	that	SCONJ
ejpam-3237	93	5	1	1	X
ejpam-3237	93	6	<	<	X
ejpam-3237	93	7	|β|	|β|	PROPN
ejpam-3237	93	8	.	.	PUNCT
ejpam-3237	93	9	now	now	ADV
ejpam-3237	93	10	consider	consider	VERB
ejpam-3237	93	11	the	the	DET
ejpam-3237	93	12	following	follow	VERB
ejpam-3237	93	13	open	open	ADJ
ejpam-3237	93	14	set	set	VERB
ejpam-3237	93	15	n(1	n(1	NOUN
ejpam-3237	93	16	)	)	PUNCT
ejpam-3237	93	17	=	=	PRON
ejpam-3237	93	18	{	{	PUNCT
ejpam-3237	93	19	α	α	NOUN
ejpam-3237	93	20	:	:	PUNCT
ejpam-3237	93	21	α	α	PROPN
ejpam-3237	93	22	∈	∈	PROPN
ejpam-3237	93	23	c	c	X
ejpam-3237	93	24	,	,	PUNCT
ejpam-3237	93	25	|α|	|α|	PROPN
ejpam-3237	93	26	<	<	X
ejpam-3237	93	27	|β|	|β|	NOUN
ejpam-3237	93	28	}	}	PUNCT
ejpam-3237	93	29	,	,	PUNCT
ejpam-3237	93	30	it	it	PRON
ejpam-3237	93	31	is	be	AUX
ejpam-3237	93	32	clear	clear	ADJ
ejpam-3237	93	33	that	that	SCONJ
ejpam-3237	93	34	for	for	ADP
ejpam-3237	93	35	any	any	DET
ejpam-3237	93	36	α	α	NOUN
ejpam-3237	93	37	in	in	ADP
ejpam-3237	93	38	n(1	n(1	NOUN
ejpam-3237	93	39	)	)	PUNCT
ejpam-3237	93	40	,	,	PUNCT
ejpam-3237	93	41	f(α	f(α	NOUN
ejpam-3237	93	42	)	)	PUNCT
ejpam-3237	93	43	is	be	AUX
ejpam-3237	93	44	contained	contain	VERB
ejpam-3237	93	45	in	in	ADP
ejpam-3237	93	46	f(β	f(β	NOUN
ejpam-3237	93	47	)	)	PUNCT
ejpam-3237	93	48	,	,	PUNCT
ejpam-3237	93	49	since	since	SCONJ
ejpam-3237	93	50	f	f	PROPN
ejpam-3237	93	51	is	be	AUX
ejpam-3237	93	52	increasing	increase	VERB
ejpam-3237	93	53	.	.	PUNCT
ejpam-3237	94	1	so	so	ADV
ejpam-3237	94	2	f(α	f(α	NOUN
ejpam-3237	94	3	)	)	PUNCT
ejpam-3237	94	4	⊆	⊆	NUM
ejpam-3237	94	5	n(f(1	n(f(1	NOUN
ejpam-3237	94	6	)	)	PUNCT
ejpam-3237	94	7	)	)	PUNCT
ejpam-3237	94	8	.	.	PUNCT
ejpam-3237	95	1	2	2	X
ejpam-3237	95	2	.	.	X
ejpam-3237	95	3	main	main	ADJ
ejpam-3237	95	4	results	result	NOUN
ejpam-3237	95	5	definition	definition	NOUN
ejpam-3237	95	6	2	2	NUM
ejpam-3237	95	7	.	.	PUNCT
ejpam-3237	96	1	let	let	VERB
ejpam-3237	96	2	x	x	PRON
ejpam-3237	96	3	be	be	AUX
ejpam-3237	96	4	a	a	DET
ejpam-3237	96	5	linear	linear	ADJ
ejpam-3237	96	6	space	space	NOUN
ejpam-3237	96	7	with	with	ADP
ejpam-3237	96	8	dimension	dimension	NOUN
ejpam-3237	96	9	n.	n.	PROPN
ejpam-3237	96	10	suppose	suppose	VERB
ejpam-3237	96	11	that	that	SCONJ
ejpam-3237	96	12	xk	xk	PROPN
ejpam-3237	96	13	=	=	PRON
ejpam-3237	96	14	(	(	PUNCT
ejpam-3237	96	15	xk1	xk1	PROPN
ejpam-3237	96	16	,	,	PUNCT
ejpam-3237	96	17	.	.	PUNCT
ejpam-3237	96	18	.	.	PUNCT
ejpam-3237	97	1	.	.	PUNCT
ejpam-3237	98	1	,	,	PUNCT
ejpam-3237	98	2	xkn)t	xkn)t	PROPN
ejpam-3237	98	3	,	,	PUNCT
ejpam-3237	98	4	k	k	PROPN
ejpam-3237	98	5	=	=	SYM
ejpam-3237	98	6	1	1	NUM
ejpam-3237	98	7	,	,	PUNCT
ejpam-3237	98	8	.	.	PUNCT
ejpam-3237	98	9	.	.	PUNCT
ejpam-3237	99	1	.	.	PUNCT
ejpam-3237	100	1	,	,	PUNCT
ejpam-3237	100	2	n	n	PRON
ejpam-3237	100	3	are	be	AUX
ejpam-3237	100	4	n	n	ADV
ejpam-3237	100	5	linearly	linearly	ADV
ejpam-3237	100	6	independent	independent	ADJ
ejpam-3237	100	7	vectors	vector	NOUN
ejpam-3237	100	8	in	in	ADP
ejpam-3237	100	9	x.	x.	NOUN
ejpam-3237	100	10	put	put	VERB
ejpam-3237	100	11	dx1,	dx1,	NOUN
ejpam-3237	100	12	...	...	PUNCT
ejpam-3237	100	13	,x2	,x2	PUNCT
ejpam-3237	101	1	=	=	PUNCT
ejpam-3237	101	2	x11	x11	PROPN
ejpam-3237	101	3	.	.	PUNCT
ejpam-3237	101	4	.	.	PUNCT
ejpam-3237	101	5	.	.	PUNCT
ejpam-3237	102	1	xn1	xn1	PRON
ejpam-3237	102	2	...	...	PUNCT
ejpam-3237	102	3	.	.	PUNCT
ejpam-3237	102	4	.	.	PUNCT
ejpam-3237	102	5	.	.	PUNCT
ejpam-3237	102	6	...	...	PUNCT
ejpam-3237	103	1	x1n	x1n	X
ejpam-3237	103	2	.	.	PUNCT
ejpam-3237	103	3	.	.	PUNCT
ejpam-3237	103	4	.	.	PUNCT
ejpam-3237	104	1	xnn	xnn	PROPN
ejpam-3237	104	2			NUM
ejpam-3237	104	3	since	since	SCONJ
ejpam-3237	104	4	x1	x1	PROPN
ejpam-3237	104	5	,	,	PUNCT
ejpam-3237	104	6	.	.	PUNCT
ejpam-3237	104	7	.	.	PUNCT
ejpam-3237	105	1	.	.	PUNCT
ejpam-3237	106	1	,	,	PUNCT
ejpam-3237	106	2	x2	x2	PRON
ejpam-3237	106	3	are	be	AUX
ejpam-3237	106	4	linearly	linearly	ADV
ejpam-3237	106	5	independent	independent	ADJ
ejpam-3237	106	6	,	,	PUNCT
ejpam-3237	106	7	we	we	PRON
ejpam-3237	106	8	have	have	VERB
ejpam-3237	106	9	|dx1,	|dx1,	NOUN
ejpam-3237	106	10	...	...	PUNCT
ejpam-3237	106	11	,x2	,x2	PUNCT
ejpam-3237	107	1	|	|	ADV
ejpam-3237	107	2	6=	6=	PROPN
ejpam-3237	107	3	0	0	NUM
ejpam-3237	107	4	.	.	PUNCT
ejpam-3237	107	5	m.	m.	PROPN
ejpam-3237	107	6	iranmanesh	iranmanesh	PROPN
ejpam-3237	107	7	,	,	PUNCT
ejpam-3237	107	8	m.	m.	NOUN
ejpam-3237	107	9	saeedi	saeedi	PROPN
ejpam-3237	107	10	khojasteh	khojasteh	PROPN
ejpam-3237	107	11	,	,	PUNCT
ejpam-3237	107	12	m.	m.	NOUN
ejpam-3237	107	13	k.	k.	PROPN
ejpam-3237	107	14	anwary	anwary	PROPN
ejpam-3237	107	15	/	/	SYM
ejpam-3237	107	16	eur	eur	PROPN
ejpam-3237	107	17	.	.	PUNCT
ejpam-3237	108	1	j.	j.	PROPN
ejpam-3237	108	2	pure	pure	PROPN
ejpam-3237	108	3	appl	appl	PROPN
ejpam-3237	108	4	.	.	PROPN
ejpam-3237	108	5	math	math	PROPN
ejpam-3237	108	6	,	,	PUNCT
ejpam-3237	108	7	11	11	NUM
ejpam-3237	108	8	(	(	PUNCT
ejpam-3237	108	9	3	3	NUM
ejpam-3237	108	10	)	)	PUNCT
ejpam-3237	108	11	(	(	PUNCT
ejpam-3237	108	12	2018	2018	NUM
ejpam-3237	108	13	)	)	PUNCT
ejpam-3237	108	14	,	,	PUNCT
ejpam-3237	108	15	793	793	NUM
ejpam-3237	108	16	-	-	SYM
ejpam-3237	108	17	802	802	NUM
ejpam-3237	108	18	797	797	NUM
ejpam-3237	108	19	definition	definition	NOUN
ejpam-3237	108	20	3	3	NUM
ejpam-3237	108	21	.	.	PUNCT
ejpam-3237	109	1	let	let	VERB
ejpam-3237	109	2	x	x	PRON
ejpam-3237	109	3	be	be	AUX
ejpam-3237	109	4	a	a	DET
ejpam-3237	109	5	linear	linear	ADJ
ejpam-3237	109	6	space	space	NOUN
ejpam-3237	109	7	with	with	ADP
ejpam-3237	109	8	dimension	dimension	NOUN
ejpam-3237	109	9	n.	n.	PROPN
ejpam-3237	109	10	suppose	suppose	VERB
ejpam-3237	109	11	that	that	SCONJ
ejpam-3237	109	12	x	x	X
ejpam-3237	109	13	=	=	PRON
ejpam-3237	109	14	(	(	PUNCT
ejpam-3237	109	15	x1	x1	PROPN
ejpam-3237	109	16	,	,	PUNCT
ejpam-3237	109	17	.	.	PUNCT
ejpam-3237	109	18	.	.	PUNCT
ejpam-3237	110	1	.	.	PUNCT
ejpam-3237	111	1	,	,	PUNCT
ejpam-3237	111	2	xn)t	xn)t	PROPN
ejpam-3237	111	3	,	,	PUNCT
ejpam-3237	111	4	y	y	PROPN
ejpam-3237	111	5	=	=	SYM
ejpam-3237	111	6	(	(	PUNCT
ejpam-3237	111	7	y1	y1	INTJ
ejpam-3237	111	8	,	,	PUNCT
ejpam-3237	111	9	.	.	PUNCT
ejpam-3237	111	10	.	.	PUNCT
ejpam-3237	112	1	.	.	PUNCT
ejpam-3237	113	1	,	,	PUNCT
ejpam-3237	113	2	yn)t	yn)t	PROPN
ejpam-3237	113	3	are	be	AUX
ejpam-3237	113	4	two	two	NUM
ejpam-3237	113	5	linearly	linearly	ADV
ejpam-3237	113	6	independent	independent	ADJ
ejpam-3237	113	7	vectors	vector	NOUN
ejpam-3237	113	8	in	in	ADP
ejpam-3237	113	9	x.	x.	NOUN
ejpam-3237	113	10	extend	extend	NOUN
ejpam-3237	113	11	x	x	PROPN
ejpam-3237	113	12	,	,	PUNCT
ejpam-3237	113	13	y	y	PROPN
ejpam-3237	113	14	to	to	ADP
ejpam-3237	113	15	a	a	DET
ejpam-3237	113	16	basis	basis	NOUN
ejpam-3237	113	17	for	for	ADP
ejpam-3237	113	18	x	x	SYM
ejpam-3237	113	19	by	by	ADP
ejpam-3237	113	20	adding	add	VERB
ejpam-3237	113	21	n	n	CCONJ
ejpam-3237	113	22	−	−	NUM
ejpam-3237	113	23	2	2	NUM
ejpam-3237	113	24	vector	vector	NOUN
ejpam-3237	113	25	as	as	ADP
ejpam-3237	113	26	zk	zk	PROPN
ejpam-3237	113	27	=	=	PUNCT
ejpam-3237	113	28	(	(	PUNCT
ejpam-3237	113	29	zk1	zk1	INTJ
ejpam-3237	113	30	,	,	PUNCT
ejpam-3237	113	31	.	.	PUNCT
ejpam-3237	113	32	.	.	PUNCT
ejpam-3237	114	1	.	.	PUNCT
ejpam-3237	115	1	,	,	PUNCT
ejpam-3237	115	2	zkn)t	zkn)t	PROPN
ejpam-3237	115	3	,	,	PUNCT
ejpam-3237	115	4	k	k	X
ejpam-3237	115	5	=	=	SYM
ejpam-3237	115	6	1	1	NUM
ejpam-3237	115	7	,	,	PUNCT
ejpam-3237	115	8	.	.	PUNCT
ejpam-3237	115	9	.	.	PUNCT
ejpam-3237	115	10	.	.	PUNCT
ejpam-3237	116	1	,	,	PUNCT
ejpam-3237	116	2	n−	n−	NOUN
ejpam-3237	116	3	2	2	NUM
ejpam-3237	116	4	.	.	PUNCT
ejpam-3237	116	5	denote	denote	VERB
ejpam-3237	116	6	by	by	ADP
ejpam-3237	116	7	px	px	NOUN
ejpam-3237	116	8	,	,	PUNCT
ejpam-3237	116	9	z1,	z1,	NOUN
ejpam-3237	116	10	...	...	PUNCT
ejpam-3237	116	11	,zn−2,y	,zn−2,y	PUNCT
ejpam-3237	116	12	the	the	DET
ejpam-3237	116	13	projection	projection	NOUN
ejpam-3237	116	14	parallel	parallel	NOUN
ejpam-3237	116	15	to	to	ADP
ejpam-3237	116	16	y	y	PROPN
ejpam-3237	116	17	from	from	ADP
ejpam-3237	116	18	x	x	PUNCT
ejpam-3237	116	19	to	to	ADP
ejpam-3237	116	20	the	the	DET
ejpam-3237	116	21	subspace	subspace	NOUN
ejpam-3237	116	22	generated	generate	VERB
ejpam-3237	116	23	by	by	ADP
ejpam-3237	116	24	x	x	PROPN
ejpam-3237	116	25	,	,	PUNCT
ejpam-3237	116	26	z1	z1	PROPN
ejpam-3237	116	27	,	,	PUNCT
ejpam-3237	116	28	.	.	PUNCT
ejpam-3237	116	29	.	.	PUNCT
ejpam-3237	117	1	.	.	PUNCT
ejpam-3237	118	1	,	,	PUNCT
ejpam-3237	118	2	zn−2	zn−2	PROPN
ejpam-3237	118	3	.	.	PROPN
ejpam-3237	118	4	since	since	SCONJ
ejpam-3237	118	5	the	the	DET
ejpam-3237	118	6	vectors	vector	NOUN
ejpam-3237	118	7	x	x	SYM
ejpam-3237	118	8	,	,	PUNCT
ejpam-3237	118	9	z1	z1	NOUN
ejpam-3237	118	10	,	,	PUNCT
ejpam-3237	118	11	.	.	PUNCT
ejpam-3237	118	12	.	.	PUNCT
ejpam-3237	119	1	.	.	PUNCT
ejpam-3237	120	1	,	,	PUNCT
ejpam-3237	120	2	zn−2	zn−2	PROPN
ejpam-3237	120	3	,	,	PUNCT
ejpam-3237	120	4	y	y	PROPN
ejpam-3237	120	5	are	be	AUX
ejpam-3237	120	6	the	the	DET
ejpam-3237	120	7	eigenvectors	eigenvector	NOUN
ejpam-3237	120	8	of	of	ADP
ejpam-3237	120	9	px	px	NOUN
ejpam-3237	120	10	,	,	PUNCT
ejpam-3237	120	11	z1,	z1,	NOUN
ejpam-3237	120	12	...	...	PUNCT
ejpam-3237	120	13	,zn−2,y	,zn−2,y	PUNCT
ejpam-3237	120	14	,	,	PUNCT
ejpam-3237	120	15	it	it	PRON
ejpam-3237	120	16	turn	turn	VERB
ejpam-3237	120	17	implies	imply	VERB
ejpam-3237	120	18	that	that	SCONJ
ejpam-3237	120	19	px	px	NOUN
ejpam-3237	120	20	,	,	PUNCT
ejpam-3237	120	21	z1,	z1,	NOUN
ejpam-3237	120	22	...	...	PUNCT
ejpam-3237	120	23	,zn−2,y	,zn−2,y	PUNCT
ejpam-3237	120	24	is	be	AUX
ejpam-3237	120	25	similar	similar	ADJ
ejpam-3237	120	26	to	to	ADP
ejpam-3237	120	27	the	the	DET
ejpam-3237	120	28	following	following	PROPN
ejpam-3237	120	29	1	1	NUM
ejpam-3237	120	30	.	.	PUNCT
ejpam-3237	120	31	.	.	PUNCT
ejpam-3237	121	1	.	.	PUNCT
ejpam-3237	122	1	0	0	NUM
ejpam-3237	122	2	0	0	NUM
ejpam-3237	122	3	...	...	PUNCT
ejpam-3237	122	4	.	.	PUNCT
ejpam-3237	122	5	.	.	PUNCT
ejpam-3237	122	6	.	.	PUNCT
ejpam-3237	123	1	...	...	PUNCT
ejpam-3237	124	1	...	...	PUNCT
ejpam-3237	124	2	0	0	X
ejpam-3237	124	3	.	.	PUNCT
ejpam-3237	124	4	.	.	PUNCT
ejpam-3237	124	5	.	.	PUNCT
ejpam-3237	125	1	1	1	NUM
ejpam-3237	125	2	0	0	NUM
ejpam-3237	125	3	0	0	NUM
ejpam-3237	125	4	.	.	PUNCT
ejpam-3237	125	5	.	.	PUNCT
ejpam-3237	125	6	.	.	PUNCT
ejpam-3237	126	1	0	0	NUM
ejpam-3237	126	2	0	0	NUM
ejpam-3237	126	3			ADJ
ejpam-3237	126	4	.	.	PUNCT
ejpam-3237	127	1	in	in	ADP
ejpam-3237	127	2	fact	fact	NOUN
ejpam-3237	127	3	,	,	PUNCT
ejpam-3237	127	4	px	px	NOUN
ejpam-3237	127	5	,	,	PUNCT
ejpam-3237	127	6	z1,	z1,	NOUN
ejpam-3237	127	7	...	...	PUNCT
ejpam-3237	127	8	,zn−2,y	,zn−2,y	PUNCT
ejpam-3237	127	9	has	have	VERB
ejpam-3237	127	10	a	a	DET
ejpam-3237	127	11	representation	representation	NOUN
ejpam-3237	127	12	as	as	SCONJ
ejpam-3237	127	13	follows	follow	VERB
ejpam-3237	127	14	px	px	NOUN
ejpam-3237	127	15	,	,	PUNCT
ejpam-3237	127	16	z1,	z1,	NOUN
ejpam-3237	127	17	...	...	PUNCT
ejpam-3237	127	18	,zn−2,y	,zn−2,y	PUNCT
ejpam-3237	127	19	=	=	SYM
ejpam-3237	127	20	dx	dx	PROPN
ejpam-3237	127	21	,	,	PUNCT
ejpam-3237	127	22	z1,	z1,	NOUN
ejpam-3237	127	23	...	...	PUNCT
ejpam-3237	127	24	,zn−2,y	,zn−2,y	PUNCT
ejpam-3237	127	25	.	.	PUNCT
ejpam-3237	128	1			NOUN
ejpam-3237	128	2	1	1	NUM
ejpam-3237	128	3	.	.	PUNCT
ejpam-3237	128	4	.	.	PUNCT
ejpam-3237	128	5	.	.	PUNCT
ejpam-3237	129	1	0	0	NUM
ejpam-3237	129	2	0	0	NUM
ejpam-3237	129	3	...	...	PUNCT
ejpam-3237	129	4	.	.	PUNCT
ejpam-3237	129	5	.	.	PUNCT
ejpam-3237	129	6	.	.	PUNCT
ejpam-3237	130	1	...	...	PUNCT
ejpam-3237	131	1	...	...	PUNCT
ejpam-3237	131	2	0	0	X
ejpam-3237	131	3	.	.	PUNCT
ejpam-3237	131	4	.	.	PUNCT
ejpam-3237	131	5	.	.	PUNCT
ejpam-3237	132	1	1	1	NUM
ejpam-3237	132	2	0	0	NUM
ejpam-3237	132	3	0	0	NUM
ejpam-3237	132	4	.	.	PUNCT
ejpam-3237	132	5	.	.	PUNCT
ejpam-3237	132	6	.	.	PUNCT
ejpam-3237	133	1	0	0	NUM
ejpam-3237	133	2	0	0	NUM
ejpam-3237	133	3			ADJ
ejpam-3237	133	4	.	.	PUNCT
ejpam-3237	134	1	d−1	d−1	PROPN
ejpam-3237	134	2	x	x	SYM
ejpam-3237	134	3	,	,	PUNCT
ejpam-3237	134	4	z1,	z1,	NOUN
ejpam-3237	134	5	...	...	PUNCT
ejpam-3237	134	6	,zn−2,y	,zn−2,y	X
ejpam-3237	134	7	.	.	PUNCT
ejpam-3237	135	1	proposition	proposition	NOUN
ejpam-3237	135	2	1	1	NUM
ejpam-3237	135	3	.	.	PUNCT
ejpam-3237	136	1	for	for	ADP
ejpam-3237	136	2	any	any	DET
ejpam-3237	136	3	two	two	NUM
ejpam-3237	136	4	linearly	linearly	ADV
ejpam-3237	136	5	independent	independent	ADJ
ejpam-3237	136	6	vectors	vector	NOUN
ejpam-3237	136	7	x	x	PUNCT
ejpam-3237	136	8	and	and	CCONJ
ejpam-3237	136	9	y	y	PROPN
ejpam-3237	136	10	in	in	ADP
ejpam-3237	136	11	x	x	PROPN
ejpam-3237	136	12	,	,	PUNCT
ejpam-3237	136	13	1	1	NUM
ejpam-3237	136	14	≤	≤	PROPN
ejpam-3237	136	15	‖px	‖px	PROPN
ejpam-3237	136	16	,	,	PUNCT
ejpam-3237	136	17	z1,	z1,	NOUN
ejpam-3237	136	18	...	...	PUNCT
ejpam-3237	136	19	,zn−2,y‖	,zn−2,y‖	PUNCT
ejpam-3237	136	20	<	<	X
ejpam-3237	137	1	+	+	ADJ
ejpam-3237	137	2	∞	∞	NUM
ejpam-3237	137	3	in	in	ADP
ejpam-3237	137	4	other	other	ADJ
ejpam-3237	137	5	words	word	NOUN
ejpam-3237	137	6	,	,	PUNCT
ejpam-3237	137	7	px	px	NOUN
ejpam-3237	137	8	,	,	PUNCT
ejpam-3237	137	9	z1,	z1,	NOUN
ejpam-3237	137	10	...	...	PUNCT
ejpam-3237	137	11	,zn−2,y	,zn−2,y	PUNCT
ejpam-3237	137	12	is	be	AUX
ejpam-3237	137	13	a	a	DET
ejpam-3237	137	14	bounded	bounded	ADJ
ejpam-3237	137	15	operator	operator	NOUN
ejpam-3237	137	16	.	.	PUNCT
ejpam-3237	138	1	furthermore	furthermore	ADV
ejpam-3237	138	2	,	,	PUNCT
ejpam-3237	138	3	denote	denote	VERB
ejpam-3237	138	4	pz1,	pz1,	NOUN
ejpam-3237	138	5	...	...	PUNCT
ejpam-3237	138	6	,zn−2(x	,zn−2(x	PROPN
ejpam-3237	138	7	,	,	PUNCT
ejpam-3237	138	8	y	y	NOUN
ejpam-3237	138	9	)	)	PUNCT
ejpam-3237	138	10	=	=	SYM
ejpam-3237	138	11	‖px	‖px	PROPN
ejpam-3237	138	12	,	,	PUNCT
ejpam-3237	138	13	z1,	z1,	NOUN
ejpam-3237	138	14	...	...	PUNCT
ejpam-3237	138	15	,zn−2,y‖	,zn−2,y‖	PUNCT
ejpam-3237	138	16	−1	−1	NOUN
ejpam-3237	138	17	and	and	CCONJ
ejpam-3237	138	18	let	let	VERB
ejpam-3237	138	19	p(x	p(x	PROPN
ejpam-3237	138	20	,	,	PUNCT
ejpam-3237	138	21	y	y	NOUN
ejpam-3237	138	22	)	)	PUNCT
ejpam-3237	138	23	=	=	SYM
ejpam-3237	139	1	sup{pz1,	sup{pz1,	ADJ
ejpam-3237	139	2	...	...	PUNCT
ejpam-3237	139	3	,zn−2(x	,zn−2(x	NOUN
ejpam-3237	139	4	,	,	PUNCT
ejpam-3237	139	5	y	y	PROPN
ejpam-3237	139	6	)	)	PUNCT
ejpam-3237	139	7	:	:	PUNCT
ejpam-3237	139	8	z1	z1	VERB
ejpam-3237	139	9	,	,	PUNCT
ejpam-3237	139	10	.	.	PUNCT
ejpam-3237	139	11	.	.	PUNCT
ejpam-3237	139	12	.	.	PUNCT
ejpam-3237	140	1	,	,	PUNCT
ejpam-3237	140	2	zn−2	zn−2	PROPN
ejpam-3237	140	3	∈	∈	PROPN
ejpam-3237	140	4	x	x	X
ejpam-3237	140	5	}	}	PUNCT
ejpam-3237	140	6	.	.	PUNCT
ejpam-3237	141	1	it	it	PRON
ejpam-3237	141	2	is	be	AUX
ejpam-3237	141	3	obvious	obvious	ADJ
ejpam-3237	141	4	that	that	SCONJ
ejpam-3237	141	5	p(x	p(x	PROPN
ejpam-3237	141	6	,	,	PUNCT
ejpam-3237	141	7	y	y	NOUN
ejpam-3237	141	8	)	)	PUNCT
ejpam-3237	141	9	=	=	SYM
ejpam-3237	141	10	max{‖px	max{‖px	NOUN
ejpam-3237	141	11	,	,	PUNCT
ejpam-3237	141	12	z1,	z1,	NOUN
ejpam-3237	141	13	...	...	PUNCT
ejpam-3237	141	14	,zn−2,y‖	,zn−2,y‖	NOUN
ejpam-3237	141	15	−1	−1	NOUN
ejpam-3237	141	16	:	:	PUNCT
ejpam-3237	141	17	z1	z1	VERB
ejpam-3237	141	18	,	,	PUNCT
ejpam-3237	141	19	.	.	PUNCT
ejpam-3237	141	20	.	.	PUNCT
ejpam-3237	141	21	.	.	PUNCT
ejpam-3237	142	1	,	,	PUNCT
ejpam-3237	142	2	zn−2	zn−2	PROPN
ejpam-3237	142	3	∈	∈	PROPN
ejpam-3237	142	4	x	x	X
ejpam-3237	142	5	,	,	PUNCT
ejpam-3237	142	6	‖z1‖	‖z1‖	X
ejpam-3237	142	7	=	=	NOUN
ejpam-3237	142	8	1	1	NUM
ejpam-3237	142	9	,	,	PUNCT
ejpam-3237	142	10	.	.	PUNCT
ejpam-3237	142	11	.	.	PUNCT
ejpam-3237	142	12	.	.	PUNCT
ejpam-3237	143	1	,	,	PUNCT
ejpam-3237	143	2	‖zn−2‖	‖zn−2‖	NOUN
ejpam-3237	143	3	=	=	NOUN
ejpam-3237	143	4	1	1	NUM
ejpam-3237	143	5	}	}	PUNCT
ejpam-3237	143	6	.	.	PUNCT
ejpam-3237	144	1	definition	definition	NOUN
ejpam-3237	144	2	4	4	NUM
ejpam-3237	144	3	.	.	PUNCT
ejpam-3237	145	1	for	for	ADP
ejpam-3237	145	2	any	any	DET
ejpam-3237	145	3	linearly	linearly	ADV
ejpam-3237	145	4	independent	independent	ADJ
ejpam-3237	145	5	x	x	NOUN
ejpam-3237	145	6	,	,	PUNCT
ejpam-3237	145	7	y	y	PROPN
ejpam-3237	145	8	in	in	ADP
ejpam-3237	145	9	x	x	X
ejpam-3237	145	10	,	,	PUNCT
ejpam-3237	145	11	the	the	DET
ejpam-3237	145	12	p	p	NOUN
ejpam-3237	145	13	-	-	PUNCT
ejpam-3237	145	14	angle	angle	NOUN
ejpam-3237	145	15	between	between	ADP
ejpam-3237	145	16	x	x	PROPN
ejpam-3237	145	17	,	,	PUNCT
ejpam-3237	145	18	y	y	PROPN
ejpam-3237	145	19	is	be	AUX
ejpam-3237	145	20	defined	define	VERB
ejpam-3237	145	21	by	by	ADP
ejpam-3237	145	22	ap(x	ap(x	PROPN
ejpam-3237	145	23	,	,	PUNCT
ejpam-3237	145	24	y	y	PROPN
ejpam-3237	145	25	)	)	PUNCT
ejpam-3237	146	1	=	=	SYM
ejpam-3237	146	2	arcsin(p(x	arcsin(p(x	PROPN
ejpam-3237	146	3	,	,	PUNCT
ejpam-3237	146	4	y	y	NOUN
ejpam-3237	146	5	)	)	PUNCT
ejpam-3237	146	6	)	)	PUNCT
ejpam-3237	146	7	.	.	PUNCT
ejpam-3237	147	1	note	note	VERB
ejpam-3237	147	2	that	that	SCONJ
ejpam-3237	147	3	p	p	X
ejpam-3237	147	4	-	-	PUNCT
ejpam-3237	147	5	angle	angle	NOUN
ejpam-3237	147	6	is	be	AUX
ejpam-3237	147	7	not	not	PART
ejpam-3237	147	8	depending	depend	VERB
ejpam-3237	147	9	on	on	ADP
ejpam-3237	147	10	selected	select	VERB
ejpam-3237	147	11	vectors	vector	NOUN
ejpam-3237	147	12	z1	z1	VERB
ejpam-3237	147	13	,	,	PUNCT
ejpam-3237	147	14	.	.	PUNCT
ejpam-3237	147	15	.	.	PUNCT
ejpam-3237	148	1	.	.	PUNCT
ejpam-3237	149	1	,	,	PUNCT
ejpam-3237	149	2	zn−2	zn−2	PROPN
ejpam-3237	149	3	.	.	PROPN
ejpam-3237	149	4	m.	m.	NOUN
ejpam-3237	149	5	iranmanesh	iranmanesh	PROPN
ejpam-3237	149	6	,	,	PUNCT
ejpam-3237	149	7	m.	m.	NOUN
ejpam-3237	149	8	saeedi	saeedi	PROPN
ejpam-3237	149	9	khojasteh	khojasteh	PROPN
ejpam-3237	149	10	,	,	PUNCT
ejpam-3237	149	11	m.	m.	NOUN
ejpam-3237	149	12	k.	k.	PROPN
ejpam-3237	149	13	anwary	anwary	PROPN
ejpam-3237	149	14	/	/	SYM
ejpam-3237	149	15	eur	eur	PROPN
ejpam-3237	149	16	.	.	PUNCT
ejpam-3237	150	1	j.	j.	PROPN
ejpam-3237	150	2	pure	pure	PROPN
ejpam-3237	150	3	appl	appl	PROPN
ejpam-3237	150	4	.	.	PROPN
ejpam-3237	150	5	math	math	PROPN
ejpam-3237	150	6	,	,	PUNCT
ejpam-3237	150	7	11	11	NUM
ejpam-3237	150	8	(	(	PUNCT
ejpam-3237	150	9	3	3	NUM
ejpam-3237	150	10	)	)	PUNCT
ejpam-3237	150	11	(	(	PUNCT
ejpam-3237	150	12	2018	2018	NUM
ejpam-3237	150	13	)	)	PUNCT
ejpam-3237	150	14	,	,	PUNCT
ejpam-3237	150	15	793	793	NUM
ejpam-3237	150	16	-	-	SYM
ejpam-3237	150	17	802	802	NUM
ejpam-3237	150	18	798	798	NUM
ejpam-3237	150	19	definition	definition	NOUN
ejpam-3237	150	20	5	5	NUM
ejpam-3237	150	21	.	.	PUNCT
ejpam-3237	151	1	for	for	ADP
ejpam-3237	151	2	linearly	linearly	ADV
ejpam-3237	151	3	independent	independent	ADJ
ejpam-3237	151	4	vectors	vector	NOUN
ejpam-3237	151	5	x	x	PRON
ejpam-3237	151	6	,	,	PUNCT
ejpam-3237	151	7	y	y	PROPN
ejpam-3237	151	8	in	in	ADP
ejpam-3237	151	9	x	x	PROPN
ejpam-3237	151	10	,	,	PUNCT
ejpam-3237	151	11	we	we	PRON
ejpam-3237	151	12	say	say	VERB
ejpam-3237	151	13	that	that	SCONJ
ejpam-3237	151	14	x	x	PRON
ejpam-3237	151	15	is	be	AUX
ejpam-3237	151	16	p	p	NOUN
ejpam-3237	151	17	-	-	PUNCT
ejpam-3237	151	18	orthogonal	orthogonal	ADJ
ejpam-3237	151	19	to	to	ADP
ejpam-3237	151	20	y	y	PROPN
ejpam-3237	151	21	if	if	SCONJ
ejpam-3237	151	22	ap(x	ap(x	PROPN
ejpam-3237	151	23	,	,	PUNCT
ejpam-3237	151	24	y	y	PROPN
ejpam-3237	151	25	)	)	PUNCT
ejpam-3237	151	26	=	=	PUNCT
ejpam-3237	152	1	π	π	NOUN
ejpam-3237	152	2	2	2	NUM
ejpam-3237	152	3	.	.	PUNCT
ejpam-3237	153	1	it	it	PRON
ejpam-3237	153	2	is	be	AUX
ejpam-3237	153	3	clear	clear	ADJ
ejpam-3237	153	4	that	that	SCONJ
ejpam-3237	153	5	x	x	X
ejpam-3237	153	6	,	,	PUNCT
ejpam-3237	153	7	y	y	PROPN
ejpam-3237	153	8	are	be	AUX
ejpam-3237	153	9	p	p	NOUN
ejpam-3237	153	10	-	-	PUNCT
ejpam-3237	153	11	orthogonal	orthogonal	ADJ
ejpam-3237	153	12	if	if	SCONJ
ejpam-3237	153	13	there	there	PRON
ejpam-3237	153	14	exist	exist	VERB
ejpam-3237	153	15	suitable	suitable	ADJ
ejpam-3237	153	16	vectors	vector	NOUN
ejpam-3237	153	17	z1	z1	VERB
ejpam-3237	153	18	,	,	PUNCT
ejpam-3237	153	19	.	.	PUNCT
ejpam-3237	153	20	.	.	PUNCT
ejpam-3237	154	1	.	.	PUNCT
ejpam-3237	155	1	,	,	PUNCT
ejpam-3237	155	2	zn−2	zn−2	ADP
ejpam-3237	155	3	such	such	ADJ
ejpam-3237	155	4	that	that	SCONJ
ejpam-3237	155	5	‖px	‖px	PROPN
ejpam-3237	155	6	,	,	PUNCT
ejpam-3237	155	7	z1,	z1,	NOUN
ejpam-3237	155	8	...	...	PUNCT
ejpam-3237	155	9	,zn−2,y‖	,zn−2,y‖	PUNCT
ejpam-3237	155	10	=	=	NOUN
ejpam-3237	155	11	1	1	X
ejpam-3237	155	12	.	.	PUNCT
ejpam-3237	155	13	theorem	theorem	NOUN
ejpam-3237	155	14	1	1	NUM
ejpam-3237	155	15	.	.	PUNCT
ejpam-3237	156	1	the	the	DET
ejpam-3237	156	2	concept	concept	NOUN
ejpam-3237	156	3	of	of	ADP
ejpam-3237	156	4	p	p	NOUN
ejpam-3237	156	5	-	-	PUNCT
ejpam-3237	156	6	orthogonality	orthogonality	NOUN
ejpam-3237	156	7	is	be	AUX
ejpam-3237	156	8	compatible	compatible	ADJ
ejpam-3237	156	9	with	with	ADP
ejpam-3237	156	10	the	the	DET
ejpam-3237	156	11	usual	usual	ADJ
ejpam-3237	156	12	orthogonality	orthogonality	NOUN
ejpam-3237	156	13	in	in	ADP
ejpam-3237	156	14	the	the	DET
ejpam-3237	156	15	inner	inner	ADJ
ejpam-3237	156	16	product	product	NOUN
ejpam-3237	156	17	spaces	space	VERB
ejpam-3237	156	18	.	.	PUNCT
ejpam-3237	157	1	proof	proof	NOUN
ejpam-3237	157	2	.	.	PUNCT
ejpam-3237	158	1	let	let	VERB
ejpam-3237	158	2	x	x	PRON
ejpam-3237	158	3	be	be	AUX
ejpam-3237	158	4	an	an	DET
ejpam-3237	158	5	inner	inner	ADJ
ejpam-3237	158	6	product	product	NOUN
ejpam-3237	158	7	space	space	NOUN
ejpam-3237	158	8	.	.	PUNCT
ejpam-3237	159	1	first	first	ADV
ejpam-3237	159	2	,	,	PUNCT
ejpam-3237	159	3	assume	assume	VERB
ejpam-3237	159	4	that	that	SCONJ
ejpam-3237	159	5	x	x	X
ejpam-3237	159	6	,	,	PUNCT
ejpam-3237	159	7	y	y	PROPN
ejpam-3237	159	8	are	be	AUX
ejpam-3237	159	9	orthogonal	orthogonal	ADJ
ejpam-3237	159	10	.	.	PUNCT
ejpam-3237	160	1	we	we	PRON
ejpam-3237	160	2	shall	shall	AUX
ejpam-3237	160	3	show	show	VERB
ejpam-3237	160	4	that	that	SCONJ
ejpam-3237	160	5	‖px	‖px	NOUN
ejpam-3237	160	6	,	,	PUNCT
ejpam-3237	160	7	z1,	z1,	NOUN
ejpam-3237	160	8	...	...	PUNCT
ejpam-3237	160	9	,zn−2,y‖	,zn−2,y‖	SYM
ejpam-3237	160	10	=	=	NOUN
ejpam-3237	160	11	1	1	NUM
ejpam-3237	160	12	for	for	ADP
ejpam-3237	160	13	suitable	suitable	ADJ
ejpam-3237	160	14	choice	choice	NOUN
ejpam-3237	160	15	of	of	ADP
ejpam-3237	160	16	z1	z1	NUM
ejpam-3237	160	17	,	,	PUNCT
ejpam-3237	160	18	.	.	PUNCT
ejpam-3237	160	19	.	.	PUNCT
ejpam-3237	160	20	.	.	PUNCT
ejpam-3237	161	1	,	,	PUNCT
ejpam-3237	161	2	zn−2	zn−2	PROPN
ejpam-3237	161	3	.	.	PUNCT
ejpam-3237	162	1	to	to	ADP
ejpam-3237	162	2	this	this	DET
ejpam-3237	162	3	end	end	NOUN
ejpam-3237	162	4	,	,	PUNCT
ejpam-3237	162	5	extending	extend	VERB
ejpam-3237	162	6	x	x	SYM
ejpam-3237	162	7	,	,	PUNCT
ejpam-3237	162	8	y	y	PROPN
ejpam-3237	162	9	to	to	ADP
ejpam-3237	162	10	a	a	DET
ejpam-3237	162	11	basis	basis	NOUN
ejpam-3237	162	12	as	as	ADP
ejpam-3237	162	13	{	{	PUNCT
ejpam-3237	162	14	x	x	NOUN
ejpam-3237	162	15	,	,	PUNCT
ejpam-3237	162	16	z1	z1	NOUN
ejpam-3237	162	17	,	,	PUNCT
ejpam-3237	162	18	.	.	PUNCT
ejpam-3237	162	19	.	.	PUNCT
ejpam-3237	162	20	.	.	PUNCT
ejpam-3237	163	1	,	,	PUNCT
ejpam-3237	163	2	zn−2	zn−2	PROPN
ejpam-3237	163	3	,	,	PUNCT
ejpam-3237	163	4	y	y	NOUN
ejpam-3237	163	5	}	}	PUNCT
ejpam-3237	163	6	to	to	ADP
ejpam-3237	163	7	an	an	DET
ejpam-3237	163	8	orthogonal	orthogonal	ADJ
ejpam-3237	163	9	basis	basis	NOUN
ejpam-3237	163	10	for	for	ADP
ejpam-3237	163	11	x	x	X
ejpam-3237	163	12	,	,	PUNCT
ejpam-3237	163	13	we	we	PRON
ejpam-3237	163	14	show	show	VERB
ejpam-3237	163	15	that	that	SCONJ
ejpam-3237	163	16	‖p‖	‖p‖	PROPN
ejpam-3237	163	17	=	=	NOUN
ejpam-3237	163	18	1	1	NUM
ejpam-3237	163	19	where	where	SCONJ
ejpam-3237	163	20	p	p	NOUN
ejpam-3237	163	21	=	=	PROPN
ejpam-3237	163	22	px	px	PROPN
ejpam-3237	163	23	,	,	PUNCT
ejpam-3237	163	24	z1,	z1,	NOUN
ejpam-3237	163	25	...	...	PUNCT
ejpam-3237	163	26	,zn−2,y	,zn−2,y	PUNCT
ejpam-3237	163	27	is	be	AUX
ejpam-3237	163	28	the	the	DET
ejpam-3237	163	29	orthogonal	orthogonal	ADJ
ejpam-3237	163	30	projection	projection	NOUN
ejpam-3237	163	31	associated	associate	VERB
ejpam-3237	163	32	with	with	ADP
ejpam-3237	163	33	the	the	DET
ejpam-3237	163	34	subspace	subspace	NOUN
ejpam-3237	163	35	generated	generate	VERB
ejpam-3237	163	36	by	by	ADP
ejpam-3237	163	37	{	{	PUNCT
ejpam-3237	163	38	x	x	PROPN
ejpam-3237	163	39	,	,	PUNCT
ejpam-3237	163	40	z1	z1	NOUN
ejpam-3237	163	41	,	,	PUNCT
ejpam-3237	163	42	.	.	PUNCT
ejpam-3237	163	43	.	.	PUNCT
ejpam-3237	164	1	.	.	PUNCT
ejpam-3237	165	1	,	,	PUNCT
ejpam-3237	165	2	zn−2	zn−2	PROPN
ejpam-3237	165	3	,	,	PUNCT
ejpam-3237	165	4	y	y	NOUN
ejpam-3237	165	5	}	}	PUNCT
ejpam-3237	165	6	.	.	PUNCT
ejpam-3237	166	1	since	since	SCONJ
ejpam-3237	166	2	y	y	PROPN
ejpam-3237	166	3	∈	∈	PROPN
ejpam-3237	166	4	[	[	X
ejpam-3237	166	5	span{x	span{x	ADJ
ejpam-3237	166	6	,	,	PUNCT
ejpam-3237	166	7	z1	z1	NOUN
ejpam-3237	166	8	,	,	PUNCT
ejpam-3237	166	9	.	.	PUNCT
ejpam-3237	166	10	.	.	PUNCT
ejpam-3237	166	11	.	.	PUNCT
ejpam-3237	167	1	,	,	PUNCT
ejpam-3237	167	2	zn−2}]⊥	zn−2}]⊥	PROPN
ejpam-3237	167	3	and	and	CCONJ
ejpam-3237	167	4	for	for	ADP
ejpam-3237	167	5	any	any	DET
ejpam-3237	167	6	z	z	NOUN
ejpam-3237	167	7	in	in	ADP
ejpam-3237	167	8	x	x	SYM
ejpam-3237	167	9	,	,	PUNCT
ejpam-3237	167	10	we	we	PRON
ejpam-3237	167	11	have	have	VERB
ejpam-3237	167	12	pz	pz	PROPN
ejpam-3237	167	13	∈	∈	PROPN
ejpam-3237	167	14	span{x	span{x	PROPN
ejpam-3237	167	15	,	,	PUNCT
ejpam-3237	167	16	z1	z1	NOUN
ejpam-3237	167	17	,	,	PUNCT
ejpam-3237	167	18	.	.	PUNCT
ejpam-3237	167	19	.	.	PUNCT
ejpam-3237	168	1	.	.	PUNCT
ejpam-3237	169	1	,	,	PUNCT
ejpam-3237	169	2	zn−2	zn−2	PROPN
ejpam-3237	169	3	}	}	PUNCT
ejpam-3237	169	4	we	we	PRON
ejpam-3237	169	5	conclude	conclude	VERB
ejpam-3237	169	6	that	that	SCONJ
ejpam-3237	169	7	y	y	PROPN
ejpam-3237	169	8	⊥	⊥	PROPN
ejpam-3237	169	9	pz	pz	PROPN
ejpam-3237	169	10	and	and	CCONJ
ejpam-3237	169	11	we	we	PRON
ejpam-3237	169	12	have	have	VERB
ejpam-3237	169	13	‖z‖2	‖z‖2	NOUN
ejpam-3237	169	14	=	=	SYM
ejpam-3237	169	15	‖z	‖z	NOUN
ejpam-3237	169	16	−	−	PROPN
ejpam-3237	169	17	pz	pz	NOUN
ejpam-3237	169	18	+	+	CCONJ
ejpam-3237	169	19	pz‖2	pz‖2	NOUN
ejpam-3237	169	20	=	=	PUNCT
ejpam-3237	169	21	‖z	‖z	NOUN
ejpam-3237	170	1	−	−	NOUN
ejpam-3237	170	2	pz‖2	pz‖2	NOUN
ejpam-3237	170	3	+	+	CCONJ
ejpam-3237	170	4	‖pz‖2	‖pz‖2	NOUN
ejpam-3237	170	5	≥	≥	NUM
ejpam-3237	170	6	‖pz‖2	‖pz‖2	NOUN
ejpam-3237	170	7	therefore	therefore	ADV
ejpam-3237	170	8	‖p‖	‖p‖	PROPN
ejpam-3237	170	9	≤	≤	NOUN
ejpam-3237	170	10	1	1	NUM
ejpam-3237	170	11	,	,	PUNCT
ejpam-3237	170	12	now	now	ADV
ejpam-3237	170	13	,	,	PUNCT
ejpam-3237	170	14	taking	take	VERB
ejpam-3237	170	15	z	z	NOUN
ejpam-3237	170	16	=	=	SYM
ejpam-3237	170	17	x	x	NOUN
ejpam-3237	170	18	,	,	PUNCT
ejpam-3237	170	19	we	we	PRON
ejpam-3237	170	20	have	have	VERB
ejpam-3237	170	21	px	px	NOUN
ejpam-3237	170	22	=	=	PUNCT
ejpam-3237	170	23	x	x	PUNCT
ejpam-3237	170	24	so	so	ADV
ejpam-3237	171	1	‖px‖	‖px‖	PROPN
ejpam-3237	171	2	=	=	PUNCT
ejpam-3237	171	3	‖x‖	‖x‖	PROPN
ejpam-3237	171	4	hence	hence	ADV
ejpam-3237	171	5	‖p‖	‖p‖	PROPN
ejpam-3237	171	6	=	=	PUNCT
ejpam-3237	171	7	1	1	X
ejpam-3237	171	8	.	.	PUNCT
ejpam-3237	171	9	m.	m.	NOUN
ejpam-3237	171	10	iranmanesh	iranmanesh	PROPN
ejpam-3237	171	11	,	,	PUNCT
ejpam-3237	171	12	m.	m.	NOUN
ejpam-3237	171	13	saeedi	saeedi	PROPN
ejpam-3237	171	14	khojasteh	khojasteh	PROPN
ejpam-3237	171	15	,	,	PUNCT
ejpam-3237	171	16	m.	m.	NOUN
ejpam-3237	171	17	k.	k.	PROPN
ejpam-3237	171	18	anwary	anwary	PROPN
ejpam-3237	171	19	/	/	SYM
ejpam-3237	171	20	eur	eur	PROPN
ejpam-3237	171	21	.	.	PUNCT
ejpam-3237	172	1	j.	j.	PROPN
ejpam-3237	172	2	pure	pure	PROPN
ejpam-3237	172	3	appl	appl	PROPN
ejpam-3237	172	4	.	.	PROPN
ejpam-3237	172	5	math	math	PROPN
ejpam-3237	172	6	,	,	PUNCT
ejpam-3237	172	7	11	11	NUM
ejpam-3237	172	8	(	(	PUNCT
ejpam-3237	172	9	3	3	NUM
ejpam-3237	172	10	)	)	PUNCT
ejpam-3237	172	11	(	(	PUNCT
ejpam-3237	172	12	2018	2018	NUM
ejpam-3237	172	13	)	)	PUNCT
ejpam-3237	172	14	,	,	PUNCT
ejpam-3237	172	15	793	793	NUM
ejpam-3237	172	16	-	-	SYM
ejpam-3237	172	17	802	802	NUM
ejpam-3237	172	18	799	799	NUM
ejpam-3237	172	19	next	next	ADV
ejpam-3237	172	20	,	,	PUNCT
ejpam-3237	172	21	assume	assume	VERB
ejpam-3237	172	22	that	that	SCONJ
ejpam-3237	172	23	x	x	X
ejpam-3237	172	24	,	,	PUNCT
ejpam-3237	172	25	y	y	PROPN
ejpam-3237	172	26	are	be	AUX
ejpam-3237	172	27	not	not	PART
ejpam-3237	172	28	orthogonal	orthogonal	ADJ
ejpam-3237	172	29	.	.	PUNCT
ejpam-3237	173	1	we	we	PRON
ejpam-3237	173	2	shall	shall	AUX
ejpam-3237	173	3	show	show	VERB
ejpam-3237	173	4	that	that	SCONJ
ejpam-3237	173	5	‖px	‖px	NOUN
ejpam-3237	173	6	,	,	PUNCT
ejpam-3237	173	7	z1,	z1,	NOUN
ejpam-3237	173	8	...	...	PUNCT
ejpam-3237	173	9	,zn−2,y‖	,zn−2,y‖	X
ejpam-3237	173	10	>	>	X
ejpam-3237	173	11	1	1	NUM
ejpam-3237	173	12	for	for	ADP
ejpam-3237	173	13	all	all	DET
ejpam-3237	173	14	choices	choice	NOUN
ejpam-3237	173	15	of	of	ADP
ejpam-3237	173	16	z1	z1	NOUN
ejpam-3237	173	17	,	,	PUNCT
ejpam-3237	173	18	.	.	PUNCT
ejpam-3237	173	19	.	.	PUNCT
ejpam-3237	173	20	.	.	PUNCT
ejpam-3237	174	1	,	,	PUNCT
ejpam-3237	174	2	zn−2	zn−2	PROPN
ejpam-3237	174	3	.	.	PUNCT
ejpam-3237	175	1	since	since	SCONJ
ejpam-3237	175	2	{	{	PUNCT
ejpam-3237	175	3	y}⊥	y}⊥	PROPN
ejpam-3237	175	4	6=	6=	NUM
ejpam-3237	175	5	span{x	span{x	PROPN
ejpam-3237	175	6	,	,	PUNCT
ejpam-3237	175	7	z1	z1	NOUN
ejpam-3237	175	8	,	,	PUNCT
ejpam-3237	175	9	.	.	PUNCT
ejpam-3237	175	10	.	.	PUNCT
ejpam-3237	175	11	.	.	PUNCT
ejpam-3237	176	1	,	,	PUNCT
ejpam-3237	176	2	zn−2	zn−2	PROPN
ejpam-3237	176	3	}	}	PUNCT
ejpam-3237	176	4	there	there	ADV
ejpam-3237	176	5	exists	exist	VERB
ejpam-3237	176	6	a	a	DET
ejpam-3237	176	7	nonzero	nonzero	PROPN
ejpam-3237	176	8	vector	vector	NOUN
ejpam-3237	176	9	z	z	NOUN
ejpam-3237	176	10	in	in	ADP
ejpam-3237	176	11	{	{	PUNCT
ejpam-3237	176	12	y}⊥	y}⊥	X
ejpam-3237	176	13	that	that	PRON
ejpam-3237	176	14	does	do	AUX
ejpam-3237	176	15	not	not	PART
ejpam-3237	176	16	belong	belong	VERB
ejpam-3237	176	17	to	to	ADP
ejpam-3237	176	18	span{x	span{x	PROPN
ejpam-3237	176	19	,	,	PUNCT
ejpam-3237	176	20	z1	z1	NOUN
ejpam-3237	176	21	,	,	PUNCT
ejpam-3237	176	22	.	.	PUNCT
ejpam-3237	176	23	.	.	PUNCT
ejpam-3237	177	1	.	.	PUNCT
ejpam-3237	178	1	,	,	PUNCT
ejpam-3237	178	2	zn−2	zn−2	PROPN
ejpam-3237	178	3	}	}	PUNCT
ejpam-3237	178	4	.	.	PUNCT
ejpam-3237	179	1	for	for	ADP
ejpam-3237	179	2	this	this	PRON
ejpam-3237	179	3	z	z	NOUN
ejpam-3237	179	4	we	we	PRON
ejpam-3237	179	5	have	have	VERB
ejpam-3237	179	6	pz	pz	NOUN
ejpam-3237	179	7	−	−	PROPN
ejpam-3237	179	8	z	z	PROPN
ejpam-3237	180	1	⊥	⊥	PROPN
ejpam-3237	180	2	z.	z.	PROPN
ejpam-3237	180	3	we	we	PRON
ejpam-3237	180	4	conclude	conclude	VERB
ejpam-3237	180	5	that	that	SCONJ
ejpam-3237	180	6	‖pz‖2	‖pz‖2	NOUN
ejpam-3237	180	7	=	=	SYM
ejpam-3237	180	8	‖pz	‖pz	PROPN
ejpam-3237	180	9	−	−	PROPN
ejpam-3237	180	10	z	z	NOUN
ejpam-3237	180	11	+	+	NUM
ejpam-3237	180	12	z‖2	z‖2	NOUN
ejpam-3237	180	13	=	=	SYM
ejpam-3237	180	14	‖pz	‖pz	PROPN
ejpam-3237	180	15	−	−	PROPN
ejpam-3237	180	16	z‖2	z‖2	NOUN
ejpam-3237	180	17	+	+	CCONJ
ejpam-3237	180	18	‖z‖2	‖z‖2	NOUN
ejpam-3237	180	19	>	>	SYM
ejpam-3237	180	20	‖z‖2	‖z‖2	NOUN
ejpam-3237	180	21	therefore	therefore	ADV
ejpam-3237	180	22	‖p‖	‖p‖	PROPN
ejpam-3237	180	23	>	>	X
ejpam-3237	180	24	1	1	PROPN
ejpam-3237	180	25	as	as	SCONJ
ejpam-3237	180	26	claimed	claim	VERB
ejpam-3237	180	27	.	.	PUNCT
ejpam-3237	181	1	for	for	ADP
ejpam-3237	181	2	a	a	DET
ejpam-3237	181	3	complex	complex	ADJ
ejpam-3237	181	4	linear	linear	ADJ
ejpam-3237	181	5	space	space	NOUN
ejpam-3237	181	6	,	,	PUNCT
ejpam-3237	181	7	we	we	PRON
ejpam-3237	181	8	have	have	AUX
ejpam-3237	181	9	already	already	ADV
ejpam-3237	181	10	defined	define	VERB
ejpam-3237	181	11	the	the	DET
ejpam-3237	181	12	operator	operator	NOUN
ejpam-3237	181	13	orthogonality	orthogonality	NOUN
ejpam-3237	181	14	.	.	PUNCT
ejpam-3237	182	1	we	we	PRON
ejpam-3237	182	2	denote	denote	VERB
ejpam-3237	182	3	this	this	DET
ejpam-3237	182	4	kind	kind	NOUN
ejpam-3237	182	5	of	of	ADP
ejpam-3237	182	6	orthogonality	orthogonality	NOUN
ejpam-3237	182	7	by	by	ADP
ejpam-3237	182	8	notation	notation	NOUN
ejpam-3237	182	9	⊥p	⊥p	PROPN
ejpam-3237	182	10	.	.	PUNCT
ejpam-3237	183	1	let	let	VERB
ejpam-3237	183	2	x	x	PRON
ejpam-3237	183	3	,	,	PUNCT
ejpam-3237	183	4	y	y	PROPN
ejpam-3237	183	5	be	be	VERB
ejpam-3237	183	6	two	two	NUM
ejpam-3237	183	7	vectors	vector	NOUN
ejpam-3237	183	8	in	in	ADP
ejpam-3237	183	9	x.	x.	NOUN
ejpam-3237	183	10	similar	similar	ADJ
ejpam-3237	183	11	to	to	ADP
ejpam-3237	183	12	[	[	X
ejpam-3237	183	13	?	?	PUNCT
ejpam-3237	184	1	]	]	X
ejpam-3237	184	2	,	,	PUNCT
ejpam-3237	184	3	we	we	PRON
ejpam-3237	184	4	consider	consider	VERB
ejpam-3237	184	5	the	the	DET
ejpam-3237	184	6	following	following	NOUN
ejpam-3237	184	7	set	set	VERB
ejpam-3237	184	8	in	in	ADP
ejpam-3237	184	9	c	c	PROPN
ejpam-3237	184	10	as	as	ADP
ejpam-3237	184	11	the	the	DET
ejpam-3237	184	12	orthogonality	orthogonality	NOUN
ejpam-3237	184	13	set	set	NOUN
ejpam-3237	184	14	of	of	ADP
ejpam-3237	184	15	x	x	PUNCT
ejpam-3237	184	16	with	with	ADP
ejpam-3237	184	17	respect	respect	NOUN
ejpam-3237	184	18	to	to	ADP
ejpam-3237	184	19	y	y	PROPN
ejpam-3237	184	20	:	:	PUNCT
ejpam-3237	184	21	f	f	PROPN
ejpam-3237	184	22	(	(	PUNCT
ejpam-3237	184	23	x	x	X
ejpam-3237	184	24	;	;	PUNCT
ejpam-3237	184	25	y	y	X
ejpam-3237	184	26	)	)	PUNCT
ejpam-3237	184	27	=	=	PRON
ejpam-3237	184	28	{	{	PUNCT
ejpam-3237	184	29	µ	µ	X
ejpam-3237	184	30	:	:	PUNCT
ejpam-3237	184	31	µ	µ	X
ejpam-3237	184	32	∈	∈	NOUN
ejpam-3237	184	33	c	c	X
ejpam-3237	184	34	,	,	PUNCT
ejpam-3237	184	35	(	(	PUNCT
ejpam-3237	184	36	x−	x−	PROPN
ejpam-3237	184	37	µy	µy	PROPN
ejpam-3237	184	38	)	)	PUNCT
ejpam-3237	184	39	⊥p	⊥p	ADP
ejpam-3237	184	40	y	y	PROPN
ejpam-3237	184	41	}	}	PUNCT
ejpam-3237	184	42	or	or	CCONJ
ejpam-3237	184	43	equivalently	equivalently	ADV
ejpam-3237	184	44	f	f	X
ejpam-3237	184	45	(	(	PUNCT
ejpam-3237	184	46	x	x	X
ejpam-3237	184	47	;	;	PUNCT
ejpam-3237	184	48	y	y	X
ejpam-3237	184	49	)	)	PUNCT
ejpam-3237	184	50	=	=	PRON
ejpam-3237	184	51	{	{	PUNCT
ejpam-3237	184	52	µ	µ	X
ejpam-3237	184	53	:	:	PUNCT
ejpam-3237	184	54	µ	µ	X
ejpam-3237	184	55	∈	∈	PROPN
ejpam-3237	184	56	c	c	X
ejpam-3237	184	57	,	,	PUNCT
ejpam-3237	184	58	‖px−µy	‖px−µy	PROPN
ejpam-3237	184	59	,	,	PUNCT
ejpam-3237	184	60	z1,	z1,	NOUN
ejpam-3237	184	61	...	...	PUNCT
ejpam-3237	184	62	,zn−2,y‖	,zn−2,y‖	NOUN
ejpam-3237	184	63	=	=	NOUN
ejpam-3237	184	64	1	1	NUM
ejpam-3237	184	65	}	}	PUNCT
ejpam-3237	184	66	.	.	PUNCT
ejpam-3237	185	1	moreover	moreover	ADV
ejpam-3237	185	2	,	,	PUNCT
ejpam-3237	185	3	we	we	PRON
ejpam-3237	185	4	can	can	AUX
ejpam-3237	185	5	involve	involve	VERB
ejpam-3237	185	6	an	an	DET
ejpam-3237	185	7	other	other	ADJ
ejpam-3237	185	8	parameter	parameter	NOUN
ejpam-3237	185	9	α	α	NOUN
ejpam-3237	185	10	for	for	ADP
ejpam-3237	185	11	more	more	ADJ
ejpam-3237	185	12	benefits	benefit	NOUN
ejpam-3237	185	13	:	:	PUNCT
ejpam-3237	185	14	f	f	X
ejpam-3237	185	15	(	(	PUNCT
ejpam-3237	185	16	x	x	NOUN
ejpam-3237	185	17	;	;	PUNCT
ejpam-3237	185	18	y;α	y;α	NUM
ejpam-3237	185	19	)	)	PUNCT
ejpam-3237	185	20	=	=	PRON
ejpam-3237	185	21	{	{	PUNCT
ejpam-3237	185	22	µ	µ	X
ejpam-3237	185	23	:	:	PUNCT
ejpam-3237	185	24	µ	µ	X
ejpam-3237	185	25	∈	∈	NOUN
ejpam-3237	185	26	c	c	NOUN
ejpam-3237	185	27	,	,	PUNCT
ejpam-3237	185	28	(	(	PUNCT
ejpam-3237	185	29	αx−	αx−	NUM
ejpam-3237	185	30	µy	µy	NOUN
ejpam-3237	185	31	)	)	PUNCT
ejpam-3237	185	32	⊥p	⊥p	ADP
ejpam-3237	185	33	y	y	PROPN
ejpam-3237	185	34	}	}	PUNCT
ejpam-3237	185	35	or	or	CCONJ
ejpam-3237	185	36	equivalently	equivalently	ADV
ejpam-3237	185	37	f	f	X
ejpam-3237	185	38	(	(	PUNCT
ejpam-3237	185	39	x	x	NOUN
ejpam-3237	185	40	;	;	PUNCT
ejpam-3237	185	41	y;α	y;α	NUM
ejpam-3237	185	42	)	)	PUNCT
ejpam-3237	185	43	=	=	PRON
ejpam-3237	185	44	{	{	PUNCT
ejpam-3237	185	45	µ	µ	X
ejpam-3237	185	46	:	:	PUNCT
ejpam-3237	185	47	µ	µ	X
ejpam-3237	185	48	∈	∈	PROPN
ejpam-3237	185	49	c	c	X
ejpam-3237	185	50	,	,	PUNCT
ejpam-3237	185	51	‖pαx−µy	‖pαx−µy	PROPN
ejpam-3237	185	52	,	,	PUNCT
ejpam-3237	185	53	z1,	z1,	NOUN
ejpam-3237	185	54	...	...	PUNCT
ejpam-3237	185	55	,zn−2,y‖	,zn−2,y‖	NOUN
ejpam-3237	185	56	=	=	NOUN
ejpam-3237	185	57	1	1	NUM
ejpam-3237	185	58	}	}	PUNCT
ejpam-3237	185	59	.	.	PUNCT
ejpam-3237	186	1	lemma	lemma	PROPN
ejpam-3237	186	2	1	1	NUM
ejpam-3237	186	3	.	.	PUNCT
ejpam-3237	187	1	for	for	ADP
ejpam-3237	187	2	any	any	DET
ejpam-3237	187	3	non	non	ADJ
ejpam-3237	187	4	zero	zero	NUM
ejpam-3237	187	5	α	α	NOUN
ejpam-3237	187	6	,	,	PUNCT
ejpam-3237	187	7	we	we	PRON
ejpam-3237	187	8	have	have	VERB
ejpam-3237	187	9	the	the	DET
ejpam-3237	187	10	following	follow	VERB
ejpam-3237	187	11	f	f	X
ejpam-3237	187	12	(	(	PUNCT
ejpam-3237	187	13	x	x	NOUN
ejpam-3237	187	14	;	;	PUNCT
ejpam-3237	187	15	y;α	y;α	NUM
ejpam-3237	187	16	)	)	PUNCT
ejpam-3237	187	17	=	=	SYM
ejpam-3237	187	18	αf	αf	X
ejpam-3237	187	19	(	(	PUNCT
ejpam-3237	187	20	x	x	X
ejpam-3237	187	21	;	;	PUNCT
ejpam-3237	187	22	y	y	PROPN
ejpam-3237	187	23	;	;	PUNCT
ejpam-3237	187	24	1	1	NUM
ejpam-3237	187	25	)	)	PUNCT
ejpam-3237	187	26	.	.	PUNCT
ejpam-3237	188	1	proof	proof	NOUN
ejpam-3237	188	2	.	.	PUNCT
ejpam-3237	189	1	by	by	ADP
ejpam-3237	189	2	definition	definition	NOUN
ejpam-3237	189	3	,	,	PUNCT
ejpam-3237	189	4	we	we	PRON
ejpam-3237	189	5	have	have	VERB
ejpam-3237	189	6	f	f	PROPN
ejpam-3237	189	7	(	(	PUNCT
ejpam-3237	189	8	x	x	NOUN
ejpam-3237	189	9	;	;	PUNCT
ejpam-3237	189	10	y;α	y;α	NUM
ejpam-3237	189	11	)	)	PUNCT
ejpam-3237	189	12	=	=	PRON
ejpam-3237	189	13	{	{	PUNCT
ejpam-3237	189	14	µ	µ	X
ejpam-3237	189	15	:	:	PUNCT
ejpam-3237	189	16	µ	µ	X
ejpam-3237	189	17	∈	∈	PROPN
ejpam-3237	189	18	c	c	X
ejpam-3237	189	19	,	,	PUNCT
ejpam-3237	189	20	‖pαx−µy	‖pαx−µy	PROPN
ejpam-3237	189	21	,	,	PUNCT
ejpam-3237	189	22	z1,	z1,	NOUN
ejpam-3237	189	23	...	...	PUNCT
ejpam-3237	189	24	,zn−2,y‖	,zn−2,y‖	NOUN
ejpam-3237	189	25	=	=	NOUN
ejpam-3237	189	26	1	1	X
ejpam-3237	189	27	}	}	PUNCT
ejpam-3237	189	28	=	=	PRON
ejpam-3237	189	29	{	{	PUNCT
ejpam-3237	189	30	µ	µ	X
ejpam-3237	189	31	:	:	PUNCT
ejpam-3237	189	32	µ	µ	X
ejpam-3237	189	33	∈	∈	PROPN
ejpam-3237	189	34	c	c	X
ejpam-3237	189	35	,	,	PUNCT
ejpam-3237	189	36	‖pα(x−	‖pα(x−	PROPN
ejpam-3237	189	37	µ	µ	PROPN
ejpam-3237	189	38	α	α	PRON
ejpam-3237	189	39	y),z1,	y),z1,	NOUN
ejpam-3237	189	40	...	...	PUNCT
ejpam-3237	189	41	,zn−2,y‖	,zn−2,y‖	PUNCT
ejpam-3237	189	42	=	=	NOUN
ejpam-3237	189	43	1	1	X
ejpam-3237	189	44	}	}	PUNCT
ejpam-3237	189	45	=	=	PRON
ejpam-3237	189	46	{	{	PUNCT
ejpam-3237	189	47	µ	µ	X
ejpam-3237	189	48	:	:	PUNCT
ejpam-3237	189	49	µ	µ	X
ejpam-3237	189	50	∈	∈	PROPN
ejpam-3237	189	51	c	c	X
ejpam-3237	189	52	,	,	PUNCT
ejpam-3237	189	53	‖p(x−	‖p(x−	PROPN
ejpam-3237	189	54	µ	µ	PROPN
ejpam-3237	189	55	α	α	NOUN
ejpam-3237	189	56	y),z1,	y),z1,	NOUN
ejpam-3237	189	57	...	...	PUNCT
ejpam-3237	189	58	,zn−2,y‖	,zn−2,y‖	PUNCT
ejpam-3237	190	1	=	=	NOUN
ejpam-3237	190	2	1	1	NUM
ejpam-3237	190	3	}	}	PUNCT
ejpam-3237	190	4	m.	m.	NOUN
ejpam-3237	190	5	iranmanesh	iranmanesh	PROPN
ejpam-3237	190	6	,	,	PUNCT
ejpam-3237	190	7	m.	m.	NOUN
ejpam-3237	190	8	saeedi	saeedi	PROPN
ejpam-3237	190	9	khojasteh	khojasteh	PROPN
ejpam-3237	190	10	,	,	PUNCT
ejpam-3237	190	11	m.	m.	NOUN
ejpam-3237	190	12	k.	k.	PROPN
ejpam-3237	190	13	anwary	anwary	PROPN
ejpam-3237	190	14	/	/	SYM
ejpam-3237	190	15	eur	eur	PROPN
ejpam-3237	190	16	.	.	PUNCT
ejpam-3237	191	1	j.	j.	PROPN
ejpam-3237	191	2	pure	pure	PROPN
ejpam-3237	191	3	appl	appl	PROPN
ejpam-3237	191	4	.	.	PROPN
ejpam-3237	191	5	math	math	PROPN
ejpam-3237	191	6	,	,	PUNCT
ejpam-3237	191	7	11	11	NUM
ejpam-3237	191	8	(	(	PUNCT
ejpam-3237	191	9	3	3	NUM
ejpam-3237	191	10	)	)	PUNCT
ejpam-3237	191	11	(	(	PUNCT
ejpam-3237	191	12	2018	2018	NUM
ejpam-3237	191	13	)	)	PUNCT
ejpam-3237	191	14	,	,	PUNCT
ejpam-3237	191	15	793	793	NUM
ejpam-3237	191	16	-	-	SYM
ejpam-3237	191	17	802	802	NUM
ejpam-3237	191	18	800	800	NUM
ejpam-3237	191	19	since	since	SCONJ
ejpam-3237	191	20	p	p	NOUN
ejpam-3237	191	21	is	be	AUX
ejpam-3237	191	22	homogenized	homogenize	VERB
ejpam-3237	191	23	.	.	PUNCT
ejpam-3237	192	1	it	it	PRON
ejpam-3237	192	2	turn	turn	VERB
ejpam-3237	192	3	implies	imply	VERB
ejpam-3237	192	4	that	that	SCONJ
ejpam-3237	192	5	if	if	SCONJ
ejpam-3237	192	6	µ	µ	X
ejpam-3237	192	7	∈	∈	X
ejpam-3237	192	8	f	f	X
ejpam-3237	192	9	(	(	PUNCT
ejpam-3237	192	10	x	x	NOUN
ejpam-3237	192	11	;	;	PUNCT
ejpam-3237	192	12	y;α	y;α	NUM
ejpam-3237	192	13	)	)	PUNCT
ejpam-3237	192	14	,	,	PUNCT
ejpam-3237	192	15	then	then	ADV
ejpam-3237	192	16	µ	µ	VERB
ejpam-3237	192	17	α	α	NOUN
ejpam-3237	192	18	∈	∈	PROPN
ejpam-3237	192	19	f	f	X
ejpam-3237	192	20	(	(	PUNCT
ejpam-3237	192	21	x	x	X
ejpam-3237	192	22	;	;	PUNCT
ejpam-3237	192	23	y	y	PROPN
ejpam-3237	192	24	;	;	PUNCT
ejpam-3237	192	25	1	1	NUM
ejpam-3237	192	26	)	)	PUNCT
ejpam-3237	192	27	;	;	PUNCT
ejpam-3237	192	28	or	or	CCONJ
ejpam-3237	192	29	µ	µ	X
ejpam-3237	192	30	∈	∈	NOUN
ejpam-3237	192	31	αf	αf	X
ejpam-3237	192	32	(	(	PUNCT
ejpam-3237	192	33	x	x	X
ejpam-3237	192	34	;	;	PUNCT
ejpam-3237	192	35	y	y	PROPN
ejpam-3237	192	36	;	;	PUNCT
ejpam-3237	192	37	1	1	NUM
ejpam-3237	192	38	)	)	PUNCT
ejpam-3237	192	39	;	;	PUNCT
ejpam-3237	192	40	it	it	PRON
ejpam-3237	192	41	completes	complete	VERB
ejpam-3237	192	42	the	the	DET
ejpam-3237	192	43	proof	proof	NOUN
ejpam-3237	192	44	.	.	PUNCT
ejpam-3237	193	1	in	in	ADP
ejpam-3237	193	2	the	the	DET
ejpam-3237	193	3	following	follow	VERB
ejpam-3237	193	4	theorems	theorem	NOUN
ejpam-3237	193	5	we	we	PRON
ejpam-3237	193	6	will	will	AUX
ejpam-3237	193	7	see	see	VERB
ejpam-3237	193	8	upper	upper	ADJ
ejpam-3237	193	9	semi	semi	ADJ
ejpam-3237	193	10	continuity	continuity	NOUN
ejpam-3237	193	11	and	and	CCONJ
ejpam-3237	193	12	lower	low	ADJ
ejpam-3237	193	13	semi	semi	ADJ
ejpam-3237	193	14	continuity	continuity	NOUN
ejpam-3237	193	15	of	of	ADP
ejpam-3237	193	16	f	f	PROPN
ejpam-3237	193	17	(	(	PUNCT
ejpam-3237	193	18	x	x	NOUN
ejpam-3237	193	19	;	;	PUNCT
ejpam-3237	193	20	y;α	y;α	NUM
ejpam-3237	193	21	)	)	PUNCT
ejpam-3237	193	22	in	in	ADP
ejpam-3237	193	23	α	α	NUM
ejpam-3237	193	24	,	,	PUNCT
ejpam-3237	193	25	using	use	VERB
ejpam-3237	193	26	lemma	lemma	PROPN
ejpam-3237	193	27	1	1	NUM
ejpam-3237	193	28	.	.	PUNCT
ejpam-3237	193	29	theorem	theorem	NOUN
ejpam-3237	193	30	2	2	NUM
ejpam-3237	193	31	.	.	PUNCT
ejpam-3237	194	1	the	the	DET
ejpam-3237	194	2	set	set	NOUN
ejpam-3237	194	3	valued	value	VERB
ejpam-3237	194	4	function	function	NOUN
ejpam-3237	194	5	which	which	PRON
ejpam-3237	194	6	maps	map	VERB
ejpam-3237	194	7	α	α	PRON
ejpam-3237	194	8	to	to	ADP
ejpam-3237	194	9	f	f	PROPN
ejpam-3237	194	10	(	(	PUNCT
ejpam-3237	194	11	x	x	NOUN
ejpam-3237	194	12	;	;	PUNCT
ejpam-3237	194	13	y;α	y;α	NUM
ejpam-3237	194	14	)	)	PUNCT
ejpam-3237	194	15	,	,	PUNCT
ejpam-3237	194	16	is	be	AUX
ejpam-3237	194	17	upper	upper	ADJ
ejpam-3237	194	18	semi	semi	ADV
ejpam-3237	194	19	continuous	continuous	ADJ
ejpam-3237	194	20	.	.	PUNCT
ejpam-3237	195	1	proof	proof	NOUN
ejpam-3237	195	2	.	.	PUNCT
ejpam-3237	196	1	with	with	ADP
ejpam-3237	196	2	out	out	ADP
ejpam-3237	196	3	loss	loss	NOUN
ejpam-3237	196	4	of	of	ADP
ejpam-3237	196	5	generality	generality	NOUN
ejpam-3237	196	6	,	,	PUNCT
ejpam-3237	196	7	we	we	PRON
ejpam-3237	196	8	will	will	AUX
ejpam-3237	196	9	show	show	VERB
ejpam-3237	196	10	upper	upper	ADJ
ejpam-3237	196	11	continuity	continuity	NOUN
ejpam-3237	196	12	at	at	ADP
ejpam-3237	196	13	α0	α0	ADJ
ejpam-3237	196	14	=	=	SYM
ejpam-3237	196	15	1	1	X
ejpam-3237	196	16	.	.	X
ejpam-3237	196	17	continuity	continuity	NOUN
ejpam-3237	196	18	at	at	ADP
ejpam-3237	196	19	other	other	ADJ
ejpam-3237	196	20	points	point	NOUN
ejpam-3237	196	21	are	be	AUX
ejpam-3237	196	22	similar	similar	ADJ
ejpam-3237	196	23	.	.	PUNCT
ejpam-3237	197	1	for	for	ADP
ejpam-3237	197	2	more	more	ADJ
ejpam-3237	197	3	simplicity	simplicity	NOUN
ejpam-3237	197	4	,	,	PUNCT
ejpam-3237	197	5	fix	fix	NOUN
ejpam-3237	197	6	x	x	NOUN
ejpam-3237	197	7	,	,	PUNCT
ejpam-3237	197	8	y	y	PROPN
ejpam-3237	197	9	and	and	CCONJ
ejpam-3237	197	10	let	let	VERB
ejpam-3237	197	11	f(α	f(α	NOUN
ejpam-3237	197	12	)	)	PUNCT
ejpam-3237	197	13	=	=	SYM
ejpam-3237	197	14	f	f	X
ejpam-3237	197	15	(	(	PUNCT
ejpam-3237	197	16	x	x	NOUN
ejpam-3237	197	17	;	;	PUNCT
ejpam-3237	197	18	y;α	y;α	NUM
ejpam-3237	197	19	)	)	PUNCT
ejpam-3237	197	20	.	.	PUNCT
ejpam-3237	198	1	assume	assume	VERB
ejpam-3237	198	2	that	that	SCONJ
ejpam-3237	198	3	n(f(1	n(f(1	NOUN
ejpam-3237	198	4	)	)	PUNCT
ejpam-3237	198	5	)	)	PUNCT
ejpam-3237	198	6	is	be	AUX
ejpam-3237	198	7	an	an	DET
ejpam-3237	198	8	open	open	ADJ
ejpam-3237	198	9	set	set	NOUN
ejpam-3237	198	10	containing	contain	VERB
ejpam-3237	198	11	f(1	f(1	PROPN
ejpam-3237	198	12	)	)	PUNCT
ejpam-3237	198	13	.	.	PUNCT
ejpam-3237	199	1	the	the	DET
ejpam-3237	199	2	following	follow	VERB
ejpam-3237	199	3	scalars	scalar	NOUN
ejpam-3237	199	4	are	be	AUX
ejpam-3237	199	5	well	well	ADV
ejpam-3237	199	6	defined	define	VERB
ejpam-3237	199	7	β1	β1	NOUN
ejpam-3237	199	8	=	=	SYM
ejpam-3237	199	9	inf{β	inf{β	NOUN
ejpam-3237	199	10	:	:	PUNCT
ejpam-3237	199	11	f(β	f(β	NOUN
ejpam-3237	199	12	)	)	PUNCT
ejpam-3237	199	13	is	be	AUX
ejpam-3237	199	14	contained	contain	VERB
ejpam-3237	199	15	in	in	ADP
ejpam-3237	199	16	n(f(1	n(f(1	PROPN
ejpam-3237	199	17	)	)	PUNCT
ejpam-3237	199	18	)	)	PUNCT
ejpam-3237	199	19	}	}	PUNCT
ejpam-3237	199	20	β2	β2	NOUN
ejpam-3237	199	21	=	=	PUNCT
ejpam-3237	199	22	sup{β	sup{β	X
ejpam-3237	199	23	:	:	PUNCT
ejpam-3237	199	24	f(β	f(β	NUM
ejpam-3237	199	25	)	)	PUNCT
ejpam-3237	199	26	is	be	AUX
ejpam-3237	199	27	contained	contain	VERB
ejpam-3237	199	28	in	in	ADP
ejpam-3237	199	29	n(f(1	n(f(1	PROPN
ejpam-3237	199	30	)	)	PUNCT
ejpam-3237	199	31	)	)	PUNCT
ejpam-3237	199	32	}	}	PUNCT
ejpam-3237	199	33	.	.	PUNCT
ejpam-3237	200	1	on	on	ADP
ejpam-3237	200	2	the	the	DET
ejpam-3237	200	3	other	other	ADJ
ejpam-3237	200	4	hand	hand	NOUN
ejpam-3237	200	5	,	,	PUNCT
ejpam-3237	200	6	since	since	SCONJ
ejpam-3237	200	7	f(1	f(1	PROPN
ejpam-3237	200	8	)	)	PUNCT
ejpam-3237	200	9	is	be	AUX
ejpam-3237	200	10	closed	close	VERB
ejpam-3237	200	11	,	,	PUNCT
ejpam-3237	200	12	we	we	PRON
ejpam-3237	200	13	have	have	AUX
ejpam-3237	200	14	β1	β1	VERB
ejpam-3237	200	15	<	<	X
ejpam-3237	200	16	1	1	NUM
ejpam-3237	200	17	<	<	X
ejpam-3237	200	18	β2	β2	NOUN
ejpam-3237	200	19	.	.	PUNCT
ejpam-3237	201	1	now	now	ADV
ejpam-3237	201	2	consider	consider	VERB
ejpam-3237	201	3	the	the	DET
ejpam-3237	201	4	open	open	ADJ
ejpam-3237	201	5	set	set	VERB
ejpam-3237	201	6	around	around	ADP
ejpam-3237	201	7	1	1	NUM
ejpam-3237	201	8	,	,	PUNCT
ejpam-3237	201	9	n(1	n(1	NOUN
ejpam-3237	201	10	)	)	PUNCT
ejpam-3237	201	11	=	=	SYM
ejpam-3237	201	12	{	{	PUNCT
ejpam-3237	201	13	α	α	NOUN
ejpam-3237	201	14	:	:	PUNCT
ejpam-3237	201	15	α	α	PROPN
ejpam-3237	201	16	∈	∈	PROPN
ejpam-3237	201	17	c	c	AUX
ejpam-3237	201	18	,	,	PUNCT
ejpam-3237	201	19	β1	β1	PROPN
ejpam-3237	201	20	<	<	X
ejpam-3237	201	21	|α|	|α|	PROPN
ejpam-3237	201	22	<	<	X
ejpam-3237	201	23	β2	β2	NOUN
ejpam-3237	201	24	}	}	PUNCT
ejpam-3237	201	25	.	.	PUNCT
ejpam-3237	202	1	it	it	PRON
ejpam-3237	202	2	is	be	AUX
ejpam-3237	202	3	clear	clear	ADJ
ejpam-3237	202	4	that	that	SCONJ
ejpam-3237	202	5	for	for	ADP
ejpam-3237	202	6	any	any	DET
ejpam-3237	202	7	α	α	NOUN
ejpam-3237	202	8	in	in	ADP
ejpam-3237	202	9	n(1	n(1	NOUN
ejpam-3237	202	10	)	)	PUNCT
ejpam-3237	202	11	,	,	PUNCT
ejpam-3237	202	12	f(α	f(α	NOUN
ejpam-3237	202	13	)	)	PUNCT
ejpam-3237	202	14	is	be	AUX
ejpam-3237	202	15	contained	contain	VERB
ejpam-3237	202	16	in	in	ADP
ejpam-3237	202	17	n(f(1	n(f(1	PROPN
ejpam-3237	202	18	)	)	PUNCT
ejpam-3237	202	19	)	)	PUNCT
ejpam-3237	202	20	.	.	PUNCT
ejpam-3237	203	1	theorem	theorem	NOUN
ejpam-3237	203	2	3	3	NUM
ejpam-3237	203	3	.	.	PUNCT
ejpam-3237	204	1	the	the	DET
ejpam-3237	204	2	set	set	NOUN
ejpam-3237	204	3	valued	value	VERB
ejpam-3237	204	4	function	function	NOUN
ejpam-3237	204	5	which	which	PRON
ejpam-3237	204	6	maps	map	VERB
ejpam-3237	204	7	α	α	PRON
ejpam-3237	204	8	to	to	ADP
ejpam-3237	204	9	f	f	PROPN
ejpam-3237	204	10	(	(	PUNCT
ejpam-3237	204	11	x	x	NOUN
ejpam-3237	204	12	;	;	PUNCT
ejpam-3237	204	13	y;α	y;α	NUM
ejpam-3237	204	14	)	)	PUNCT
ejpam-3237	204	15	,	,	PUNCT
ejpam-3237	204	16	is	be	AUX
ejpam-3237	204	17	lower	low	ADJ
ejpam-3237	204	18	semi	semi	ADJ
ejpam-3237	204	19	continuous	continuous	ADJ
ejpam-3237	204	20	proof	proof	NOUN
ejpam-3237	204	21	.	.	PUNCT
ejpam-3237	205	1	with	with	ADP
ejpam-3237	205	2	out	out	ADP
ejpam-3237	205	3	loss	loss	NOUN
ejpam-3237	205	4	of	of	ADP
ejpam-3237	205	5	generality	generality	NOUN
ejpam-3237	205	6	,	,	PUNCT
ejpam-3237	205	7	we	we	PRON
ejpam-3237	205	8	will	will	AUX
ejpam-3237	205	9	show	show	VERB
ejpam-3237	205	10	lower	low	ADJ
ejpam-3237	205	11	continuity	continuity	NOUN
ejpam-3237	205	12	at	at	ADP
ejpam-3237	205	13	α0	α0	ADJ
ejpam-3237	205	14	=	=	SYM
ejpam-3237	205	15	1	1	X
ejpam-3237	205	16	.	.	X
ejpam-3237	205	17	continuity	continuity	NOUN
ejpam-3237	205	18	at	at	ADP
ejpam-3237	205	19	other	other	ADJ
ejpam-3237	205	20	points	point	NOUN
ejpam-3237	205	21	are	be	AUX
ejpam-3237	205	22	similar	similar	ADJ
ejpam-3237	205	23	.	.	PUNCT
ejpam-3237	206	1	for	for	ADP
ejpam-3237	206	2	more	more	ADJ
ejpam-3237	206	3	simplicity	simplicity	NOUN
ejpam-3237	206	4	,	,	PUNCT
ejpam-3237	206	5	fix	fix	NOUN
ejpam-3237	206	6	x	x	NOUN
ejpam-3237	206	7	,	,	PUNCT
ejpam-3237	206	8	y	y	PROPN
ejpam-3237	206	9	and	and	CCONJ
ejpam-3237	206	10	let	let	VERB
ejpam-3237	206	11	f(α	f(α	NOUN
ejpam-3237	206	12	)	)	PUNCT
ejpam-3237	206	13	=	=	SYM
ejpam-3237	206	14	f	f	X
ejpam-3237	206	15	(	(	PUNCT
ejpam-3237	206	16	x	x	NOUN
ejpam-3237	206	17	;	;	PUNCT
ejpam-3237	206	18	y;α	y;α	NUM
ejpam-3237	206	19	)	)	PUNCT
ejpam-3237	206	20	.	.	PUNCT
ejpam-3237	207	1	assume	assume	VERB
ejpam-3237	207	2	that	that	SCONJ
ejpam-3237	207	3	β0	β0	PROPN
ejpam-3237	207	4	∈	∈	PROPN
ejpam-3237	207	5	f(1	f(1	PROPN
ejpam-3237	207	6	)	)	PUNCT
ejpam-3237	207	7	is	be	AUX
ejpam-3237	207	8	an	an	DET
ejpam-3237	207	9	arbitrary	arbitrary	ADJ
ejpam-3237	207	10	point	point	NOUN
ejpam-3237	207	11	.	.	PUNCT
ejpam-3237	208	1	moreover	moreover	ADV
ejpam-3237	208	2	,	,	PUNCT
ejpam-3237	208	3	assume	assume	VERB
ejpam-3237	208	4	that	that	SCONJ
ejpam-3237	208	5	n(β0	n(β0	NOUN
ejpam-3237	208	6	)	)	PUNCT
ejpam-3237	208	7	is	be	AUX
ejpam-3237	208	8	an	an	DET
ejpam-3237	208	9	open	open	ADJ
ejpam-3237	208	10	neighborhood	neighborhood	NOUN
ejpam-3237	208	11	of	of	ADP
ejpam-3237	208	12	β0	β0	PROPN
ejpam-3237	208	13	.	.	PUNCT
ejpam-3237	209	1	the	the	DET
ejpam-3237	209	2	following	follow	VERB
ejpam-3237	209	3	scalars	scalar	NOUN
ejpam-3237	209	4	are	be	AUX
ejpam-3237	209	5	well	well	ADV
ejpam-3237	209	6	defined	define	VERB
ejpam-3237	209	7	β1	β1	NOUN
ejpam-3237	209	8	=	=	SYM
ejpam-3237	209	9	inf{β	inf{β	NOUN
ejpam-3237	209	10	:	:	PUNCT
ejpam-3237	209	11	f(β	f(β	NOUN
ejpam-3237	209	12	)	)	PUNCT
ejpam-3237	209	13	intersects	intersect	NOUN
ejpam-3237	209	14	n(β0	n(β0	NOUN
ejpam-3237	209	15	)	)	PUNCT
ejpam-3237	209	16	}	}	PUNCT
ejpam-3237	209	17	β2	β2	NOUN
ejpam-3237	209	18	=	=	PUNCT
ejpam-3237	209	19	sup{β	sup{β	X
ejpam-3237	209	20	:	:	PUNCT
ejpam-3237	209	21	f(β	f(β	NUM
ejpam-3237	209	22	)	)	PUNCT
ejpam-3237	209	23	intersects	intersect	NOUN
ejpam-3237	209	24	n(β0	n(β0	NOUN
ejpam-3237	209	25	)	)	PUNCT
ejpam-3237	209	26	}	}	PUNCT
ejpam-3237	209	27	.	.	PUNCT
ejpam-3237	210	1	on	on	ADP
ejpam-3237	210	2	the	the	DET
ejpam-3237	210	3	other	other	ADJ
ejpam-3237	210	4	hand	hand	NOUN
ejpam-3237	210	5	,	,	PUNCT
ejpam-3237	210	6	since	since	SCONJ
ejpam-3237	210	7	f(1	f(1	PROPN
ejpam-3237	210	8	)	)	PUNCT
ejpam-3237	210	9	is	be	AUX
ejpam-3237	210	10	closed	close	VERB
ejpam-3237	210	11	,	,	PUNCT
ejpam-3237	210	12	we	we	PRON
ejpam-3237	210	13	have	have	AUX
ejpam-3237	210	14	β1	β1	VERB
ejpam-3237	210	15	<	<	X
ejpam-3237	210	16	1	1	NUM
ejpam-3237	210	17	<	<	X
ejpam-3237	210	18	β2	β2	NOUN
ejpam-3237	210	19	.	.	PUNCT
ejpam-3237	211	1	now	now	ADV
ejpam-3237	211	2	consider	consider	VERB
ejpam-3237	211	3	the	the	DET
ejpam-3237	211	4	open	open	ADJ
ejpam-3237	211	5	set	set	VERB
ejpam-3237	211	6	around	around	ADP
ejpam-3237	211	7	1	1	NUM
ejpam-3237	211	8	,	,	PUNCT
ejpam-3237	211	9	n(1	n(1	NOUN
ejpam-3237	211	10	)	)	PUNCT
ejpam-3237	211	11	=	=	SYM
ejpam-3237	211	12	{	{	PUNCT
ejpam-3237	211	13	α	α	NOUN
ejpam-3237	211	14	:	:	PUNCT
ejpam-3237	211	15	α	α	PROPN
ejpam-3237	211	16	∈	∈	PROPN
ejpam-3237	211	17	c	c	AUX
ejpam-3237	211	18	,	,	PUNCT
ejpam-3237	211	19	β1	β1	PROPN
ejpam-3237	211	20	<	<	X
ejpam-3237	211	21	|α|	|α|	PROPN
ejpam-3237	211	22	<	<	X
ejpam-3237	211	23	β2	β2	NOUN
ejpam-3237	211	24	}	}	PUNCT
ejpam-3237	211	25	.	.	PUNCT
ejpam-3237	212	1	it	it	PRON
ejpam-3237	212	2	is	be	AUX
ejpam-3237	212	3	clear	clear	ADJ
ejpam-3237	212	4	that	that	SCONJ
ejpam-3237	212	5	for	for	ADP
ejpam-3237	212	6	any	any	DET
ejpam-3237	212	7	α	α	NOUN
ejpam-3237	212	8	in	in	ADP
ejpam-3237	212	9	n(1	n(1	NOUN
ejpam-3237	212	10	)	)	PUNCT
ejpam-3237	212	11	,	,	PUNCT
ejpam-3237	212	12	f(α	f(α	NOUN
ejpam-3237	212	13	)	)	PUNCT
ejpam-3237	212	14	intersects	intersect	NOUN
ejpam-3237	212	15	n(β0	n(β0	NOUN
ejpam-3237	212	16	)	)	PUNCT
ejpam-3237	212	17	.	.	PUNCT
ejpam-3237	213	1	corollary	corollary	ADJ
ejpam-3237	213	2	1	1	NUM
ejpam-3237	213	3	.	.	PUNCT
ejpam-3237	214	1	the	the	DET
ejpam-3237	214	2	set	set	NOUN
ejpam-3237	214	3	valued	value	VERB
ejpam-3237	214	4	function	function	NOUN
ejpam-3237	214	5	which	which	PRON
ejpam-3237	214	6	maps	map	VERB
ejpam-3237	214	7	α	α	PRON
ejpam-3237	214	8	to	to	ADP
ejpam-3237	214	9	f	f	PROPN
ejpam-3237	214	10	(	(	PUNCT
ejpam-3237	214	11	x	x	NOUN
ejpam-3237	214	12	;	;	PUNCT
ejpam-3237	214	13	y;α	y;α	NUM
ejpam-3237	214	14	)	)	PUNCT
ejpam-3237	214	15	,	,	PUNCT
ejpam-3237	214	16	is	be	AUX
ejpam-3237	214	17	semi	semi	ADV
ejpam-3237	214	18	continuous	continuous	ADJ
ejpam-3237	214	19	as	as	SCONJ
ejpam-3237	214	20	we	we	PRON
ejpam-3237	214	21	proved	prove	VERB
ejpam-3237	214	22	,	,	PUNCT
ejpam-3237	214	23	this	this	DET
ejpam-3237	214	24	functions	function	NOUN
ejpam-3237	214	25	is	be	AUX
ejpam-3237	214	26	both	both	PRON
ejpam-3237	214	27	upper	upper	ADJ
ejpam-3237	214	28	and	and	CCONJ
ejpam-3237	214	29	lower	low	ADJ
ejpam-3237	214	30	semi	semi	ADV
ejpam-3237	214	31	continuous	continuous	ADJ
ejpam-3237	214	32	,	,	PUNCT
ejpam-3237	214	33	the	the	DET
ejpam-3237	214	34	corollary	corollary	NOUN
ejpam-3237	214	35	is	be	AUX
ejpam-3237	214	36	hold	hold	NOUN
ejpam-3237	214	37	.	.	PUNCT
ejpam-3237	215	1	m.	m.	NOUN
ejpam-3237	215	2	iranmanesh	iranmanesh	PROPN
ejpam-3237	215	3	,	,	PUNCT
ejpam-3237	215	4	m.	m.	NOUN
ejpam-3237	215	5	saeedi	saeedi	PROPN
ejpam-3237	215	6	khojasteh	khojasteh	PROPN
ejpam-3237	215	7	,	,	PUNCT
ejpam-3237	215	8	m.	m.	NOUN
ejpam-3237	215	9	k.	k.	PROPN
ejpam-3237	215	10	anwary	anwary	PROPN
ejpam-3237	215	11	/	/	SYM
ejpam-3237	215	12	eur	eur	PROPN
ejpam-3237	215	13	.	.	PUNCT
ejpam-3237	216	1	j.	j.	PROPN
ejpam-3237	216	2	pure	pure	PROPN
ejpam-3237	216	3	appl	appl	PROPN
ejpam-3237	216	4	.	.	PROPN
ejpam-3237	216	5	math	math	PROPN
ejpam-3237	216	6	,	,	PUNCT
ejpam-3237	216	7	11	11	NUM
ejpam-3237	216	8	(	(	PUNCT
ejpam-3237	216	9	3	3	NUM
ejpam-3237	216	10	)	)	PUNCT
ejpam-3237	216	11	(	(	PUNCT
ejpam-3237	216	12	2018	2018	NUM
ejpam-3237	216	13	)	)	PUNCT
ejpam-3237	216	14	,	,	PUNCT
ejpam-3237	216	15	793	793	NUM
ejpam-3237	216	16	-	-	SYM
ejpam-3237	216	17	802	802	NUM
ejpam-3237	216	18	801	801	NUM
ejpam-3237	216	19	as	as	SCONJ
ejpam-3237	216	20	we	we	PRON
ejpam-3237	216	21	saw	see	VERB
ejpam-3237	216	22	in	in	ADP
ejpam-3237	216	23	theorem	theorem	NOUN
ejpam-3237	216	24	1	1	NUM
ejpam-3237	216	25	,	,	PUNCT
ejpam-3237	216	26	the	the	DET
ejpam-3237	216	27	concept	concept	NOUN
ejpam-3237	216	28	of	of	ADP
ejpam-3237	216	29	operator	operator	NOUN
ejpam-3237	216	30	orthogonal	orthogonal	ADJ
ejpam-3237	216	31	vectors	vector	NOUN
ejpam-3237	216	32	in	in	ADP
ejpam-3237	216	33	the	the	DET
ejpam-3237	216	34	inner	inner	ADJ
ejpam-3237	216	35	product	product	NOUN
ejpam-3237	216	36	spaces	space	NOUN
ejpam-3237	216	37	,	,	PUNCT
ejpam-3237	216	38	is	be	AUX
ejpam-3237	216	39	the	the	DET
ejpam-3237	216	40	same	same	ADJ
ejpam-3237	216	41	of	of	ADP
ejpam-3237	216	42	usual	usual	ADJ
ejpam-3237	216	43	orthogonal	orthogonal	ADJ
ejpam-3237	216	44	vectors	vector	NOUN
ejpam-3237	216	45	,	,	PUNCT
ejpam-3237	216	46	i.e.	i.e.	X
ejpam-3237	216	47	for	for	ADP
ejpam-3237	216	48	vector	vector	NOUN
ejpam-3237	216	49	x	x	NOUN
ejpam-3237	216	50	,	,	PUNCT
ejpam-3237	216	51	y	y	PROPN
ejpam-3237	216	52	in	in	ADP
ejpam-3237	216	53	an	an	DET
ejpam-3237	216	54	inner	inner	ADJ
ejpam-3237	216	55	product	product	NOUN
ejpam-3237	216	56	space	space	NOUN
ejpam-3237	216	57	x	x	NOUN
ejpam-3237	216	58	,	,	PUNCT
ejpam-3237	216	59	we	we	PRON
ejpam-3237	216	60	have	have	AUX
ejpam-3237	216	61	p(x	p(x	PROPN
ejpam-3237	216	62	,	,	PUNCT
ejpam-3237	216	63	y	y	NOUN
ejpam-3237	216	64	)	)	PUNCT
ejpam-3237	216	65	=	=	SYM
ejpam-3237	216	66	1	1	NUM
ejpam-3237	217	1	if	if	SCONJ
ejpam-3237	217	2	and	and	CCONJ
ejpam-3237	217	3	only	only	ADV
ejpam-3237	217	4	if	if	SCONJ
ejpam-3237	217	5	〈	〈	PROPN
ejpam-3237	217	6	x	x	X
ejpam-3237	217	7	,	,	PUNCT
ejpam-3237	217	8	y	y	PROPN
ejpam-3237	217	9	〉	〉	NUM
ejpam-3237	217	10	=	=	SYM
ejpam-3237	217	11	0	0	PUNCT
ejpam-3237	217	12	this	this	PRON
ejpam-3237	217	13	leads	lead	VERB
ejpam-3237	217	14	us	we	PRON
ejpam-3237	217	15	to	to	ADP
ejpam-3237	217	16	a	a	DET
ejpam-3237	217	17	simple	simple	ADJ
ejpam-3237	217	18	computation	computation	NOUN
ejpam-3237	217	19	for	for	ADP
ejpam-3237	217	20	the	the	DET
ejpam-3237	217	21	set	set	NOUN
ejpam-3237	217	22	f	f	PROPN
ejpam-3237	217	23	(	(	PUNCT
ejpam-3237	217	24	x	x	NOUN
ejpam-3237	217	25	;	;	PUNCT
ejpam-3237	217	26	y;α	y;α	NUM
ejpam-3237	217	27	)	)	PUNCT
ejpam-3237	217	28	in	in	ADP
ejpam-3237	217	29	an	an	DET
ejpam-3237	217	30	inner	inner	ADJ
ejpam-3237	217	31	product	product	NOUN
ejpam-3237	217	32	space	space	NOUN
ejpam-3237	217	33	.	.	PUNCT
ejpam-3237	218	1	example	example	NOUN
ejpam-3237	219	1	2	2	NUM
ejpam-3237	219	2	.	.	X
ejpam-3237	219	3	in	in	ADP
ejpam-3237	219	4	an	an	DET
ejpam-3237	219	5	inner	inner	ADJ
ejpam-3237	219	6	product	product	NOUN
ejpam-3237	219	7	space	space	NOUN
ejpam-3237	219	8	we	we	PRON
ejpam-3237	219	9	have	have	VERB
ejpam-3237	219	10	f	f	PROPN
ejpam-3237	219	11	(	(	PUNCT
ejpam-3237	219	12	x	x	NOUN
ejpam-3237	219	13	;	;	PUNCT
ejpam-3237	219	14	y;α	y;α	NUM
ejpam-3237	219	15	)	)	PUNCT
ejpam-3237	219	16	=	=	PRON
ejpam-3237	219	17	{	{	PUNCT
ejpam-3237	219	18	µ	µ	X
ejpam-3237	219	19	:	:	PUNCT
ejpam-3237	219	20	µ	µ	X
ejpam-3237	219	21	∈	∈	NOUN
ejpam-3237	219	22	c	c	X
ejpam-3237	219	23	,	,	PUNCT
ejpam-3237	219	24	〈	〈	PROPN
ejpam-3237	219	25	αx−	αx−	NUM
ejpam-3237	219	26	µy	µy	PROPN
ejpam-3237	219	27	,	,	PUNCT
ejpam-3237	219	28	y	y	PROPN
ejpam-3237	219	29	〉	〉	NUM
ejpam-3237	219	30	=	=	SYM
ejpam-3237	219	31	0	0	NUM
ejpam-3237	219	32	}	}	PUNCT
ejpam-3237	219	33	.	.	PUNCT
ejpam-3237	220	1	it	it	PRON
ejpam-3237	220	2	turn	turn	VERB
ejpam-3237	220	3	implies	imply	VERB
ejpam-3237	221	1	that	that	SCONJ
ejpam-3237	221	2	f	f	PROPN
ejpam-3237	221	3	(	(	PUNCT
ejpam-3237	221	4	x	x	NOUN
ejpam-3237	221	5	;	;	PUNCT
ejpam-3237	221	6	y;α	y;α	NUM
ejpam-3237	221	7	)	)	PUNCT
ejpam-3237	221	8	=	=	PRON
ejpam-3237	221	9	{	{	PUNCT
ejpam-3237	221	10	µ	µ	X
ejpam-3237	221	11	:	:	PUNCT
ejpam-3237	221	12	µ	µ	X
ejpam-3237	221	13	∈	∈	ADJ
ejpam-3237	221	14	c	c	NOUN
ejpam-3237	221	15	,	,	PUNCT
ejpam-3237	221	16	α〈x	α〈x	PROPN
ejpam-3237	221	17	,	,	PUNCT
ejpam-3237	221	18	y	y	PROPN
ejpam-3237	221	19	〉	〉	NUM
ejpam-3237	221	20	=	=	SYM
ejpam-3237	221	21	µ〈y	µ〈y	PROPN
ejpam-3237	221	22	,	,	PUNCT
ejpam-3237	221	23	y	y	PROPN
ejpam-3237	221	24	〉	〉	PROPN
ejpam-3237	221	25	}	}	PUNCT
ejpam-3237	221	26	.	.	PUNCT
ejpam-3237	222	1	or	or	CCONJ
ejpam-3237	222	2	equivalently	equivalently	ADV
ejpam-3237	222	3	f	f	X
ejpam-3237	222	4	(	(	PUNCT
ejpam-3237	222	5	x	x	NOUN
ejpam-3237	222	6	;	;	PUNCT
ejpam-3237	222	7	y;α	y;α	NUM
ejpam-3237	222	8	)	)	PUNCT
ejpam-3237	222	9	=	=	PRON
ejpam-3237	222	10	{	{	PUNCT
ejpam-3237	222	11	α〈x	α〈x	PROPN
ejpam-3237	222	12	,	,	PUNCT
ejpam-3237	222	13	y	y	PROPN
ejpam-3237	222	14	〉	〉	NUM
ejpam-3237	222	15	〈	〈	PROPN
ejpam-3237	222	16	y	y	PROPN
ejpam-3237	222	17	,	,	PUNCT
ejpam-3237	222	18	y	y	PROPN
ejpam-3237	222	19	〉	〉	PROPN
ejpam-3237	222	20	}	}	PUNCT
ejpam-3237	222	21	.	.	PUNCT
ejpam-3237	223	1	this	this	PRON
ejpam-3237	223	2	means	mean	VERB
ejpam-3237	223	3	that	that	SCONJ
ejpam-3237	223	4	in	in	ADP
ejpam-3237	223	5	inner	inner	ADJ
ejpam-3237	223	6	product	product	NOUN
ejpam-3237	223	7	spaces	space	NOUN
ejpam-3237	223	8	,	,	PUNCT
ejpam-3237	223	9	f	f	PROPN
ejpam-3237	223	10	(	(	PUNCT
ejpam-3237	223	11	x	x	NOUN
ejpam-3237	223	12	;	;	PUNCT
ejpam-3237	223	13	y;α	y;α	NUM
ejpam-3237	223	14	)	)	PUNCT
ejpam-3237	223	15	is	be	AUX
ejpam-3237	223	16	a	a	DET
ejpam-3237	223	17	singleton	singleton	NOUN
ejpam-3237	223	18	set	set	NOUN
ejpam-3237	223	19	.	.	PUNCT
ejpam-3237	224	1	therefore	therefore	ADV
ejpam-3237	224	2	the	the	DET
ejpam-3237	224	3	concept	concept	NOUN
ejpam-3237	224	4	of	of	ADP
ejpam-3237	224	5	semi	semi	ADJ
ejpam-3237	224	6	continuity	continuity	NOUN
ejpam-3237	224	7	of	of	ADP
ejpam-3237	224	8	this	this	DET
ejpam-3237	224	9	set	set	NOUN
ejpam-3237	224	10	valued	value	VERB
ejpam-3237	224	11	function	function	NOUN
ejpam-3237	224	12	is	be	AUX
ejpam-3237	224	13	the	the	DET
ejpam-3237	224	14	same	same	ADJ
ejpam-3237	224	15	of	of	ADP
ejpam-3237	224	16	its	its	PRON
ejpam-3237	224	17	usual	usual	ADJ
ejpam-3237	224	18	continuity	continuity	NOUN
ejpam-3237	224	19	.	.	PUNCT
ejpam-3237	224	20	example	example	NOUN
ejpam-3237	225	1	3	3	X
ejpam-3237	225	2	.	.	PUNCT
ejpam-3237	225	3	let	let	VERB
ejpam-3237	225	4	1	1	NUM
ejpam-3237	225	5	≤	≤	NOUN
ejpam-3237	225	6	r	r	NOUN
ejpam-3237	225	7	≤	≤	PUNCT
ejpam-3237	225	8	∞.	∞.	PROPN
ejpam-3237	225	9	for	for	ADP
ejpam-3237	225	10	any	any	DET
ejpam-3237	225	11	two	two	NUM
ejpam-3237	225	12	vectors	vector	NOUN
ejpam-3237	225	13	x	x	PUNCT
ejpam-3237	225	14	=	=	SYM
ejpam-3237	225	15	(	(	PUNCT
ejpam-3237	225	16	x1	x1	PROPN
ejpam-3237	225	17	,	,	PUNCT
ejpam-3237	225	18	x2	x2	PROPN
ejpam-3237	225	19	)	)	PUNCT
ejpam-3237	225	20	and	and	CCONJ
ejpam-3237	225	21	y	y	PROPN
ejpam-3237	225	22	=	=	SYM
ejpam-3237	225	23	(	(	PUNCT
ejpam-3237	225	24	y1	y1	INTJ
ejpam-3237	225	25	,	,	PUNCT
ejpam-3237	225	26	y2	y2	PROPN
ejpam-3237	225	27	)	)	PUNCT
ejpam-3237	225	28	in	in	ADP
ejpam-3237	225	29	l2	l2	NOUN
ejpam-3237	225	30	r	r	NOUN
ejpam-3237	225	31	,	,	PUNCT
ejpam-3237	225	32	it	it	PRON
ejpam-3237	225	33	has	have	AUX
ejpam-3237	225	34	been	be	AUX
ejpam-3237	225	35	shown	show	VERB
ejpam-3237	225	36	in	in	ADP
ejpam-3237	225	37	[	[	X
ejpam-3237	225	38	8	8	NUM
ejpam-3237	225	39	]	]	PUNCT
ejpam-3237	225	40	that	that	SCONJ
ejpam-3237	225	41	the	the	DET
ejpam-3237	225	42	operator	operator	NOUN
ejpam-3237	225	43	angle	angle	NOUN
ejpam-3237	225	44	between	between	ADP
ejpam-3237	225	45	x	x	PROPN
ejpam-3237	225	46	,	,	PUNCT
ejpam-3237	225	47	y	y	PROPN
ejpam-3237	225	48	is	be	AUX
ejpam-3237	225	49	the	the	DET
ejpam-3237	225	50	following	follow	VERB
ejpam-3237	225	51	arcsin	arcsin	PROPN
ejpam-3237	225	52	(	(	PUNCT
ejpam-3237	225	53	|x1y2	|x1y2	PROPN
ejpam-3237	225	54	−	−	PROPN
ejpam-3237	225	55	x2y1|	x2y1|	PROPN
ejpam-3237	225	56	‖x‖r‖y‖	‖x‖r‖y‖	PROPN
ejpam-3237	226	1	r−1	r−1	PROPN
ejpam-3237	226	2	r	r	NOUN
ejpam-3237	226	3	)	)	PUNCT
ejpam-3237	226	4	.	.	PUNCT
ejpam-3237	227	1	it	it	PRON
ejpam-3237	227	2	implies	imply	VERB
ejpam-3237	227	3	that	that	SCONJ
ejpam-3237	227	4	x	x	X
ejpam-3237	227	5	,	,	PUNCT
ejpam-3237	227	6	y	y	PROPN
ejpam-3237	227	7	are	be	AUX
ejpam-3237	227	8	operator	operator	NOUN
ejpam-3237	227	9	orthogonal	orthogonal	ADJ
ejpam-3237	227	10	if	if	SCONJ
ejpam-3237	227	11	‖x‖r‖y‖	‖x‖r‖y‖	PROPN
ejpam-3237	227	12	r−1	r−1	PROPN
ejpam-3237	227	13	r	r	NOUN
ejpam-3237	227	14	=	=	PUNCT
ejpam-3237	227	15	|x1y2	|x1y2	NOUN
ejpam-3237	227	16	−	−	NOUN
ejpam-3237	227	17	x2y1|	x2y1|	PROPN
ejpam-3237	227	18	.	.	PUNCT
ejpam-3237	228	1	therefore	therefore	ADV
ejpam-3237	228	2	,	,	PUNCT
ejpam-3237	228	3	in	in	ADP
ejpam-3237	228	4	this	this	DET
ejpam-3237	228	5	case	case	NOUN
ejpam-3237	228	6	we	we	PRON
ejpam-3237	228	7	have	have	VERB
ejpam-3237	228	8	f	f	PROPN
ejpam-3237	228	9	(	(	PUNCT
ejpam-3237	228	10	x	x	NOUN
ejpam-3237	228	11	;	;	PUNCT
ejpam-3237	228	12	y;α	y;α	NUM
ejpam-3237	228	13	)	)	PUNCT
ejpam-3237	228	14	=	=	PRON
ejpam-3237	228	15	{	{	PUNCT
ejpam-3237	228	16	µ	µ	X
ejpam-3237	228	17	:	:	PUNCT
ejpam-3237	228	18	µ	µ	X
ejpam-3237	228	19	∈	∈	NOUN
ejpam-3237	228	20	c	c	NOUN
ejpam-3237	228	21	,	,	PUNCT
ejpam-3237	228	22	(	(	PUNCT
ejpam-3237	228	23	αx−	αx−	NUM
ejpam-3237	228	24	µy	µy	NOUN
ejpam-3237	228	25	)	)	PUNCT
ejpam-3237	228	26	⊥p	⊥p	ADP
ejpam-3237	228	27	y	y	X
ejpam-3237	228	28	}	}	PUNCT
ejpam-3237	228	29	=	=	PUNCT
ejpam-3237	228	30	{	{	PUNCT
ejpam-3237	228	31	µ	µ	X
ejpam-3237	228	32	:	:	PUNCT
ejpam-3237	228	33	µ	µ	X
ejpam-3237	228	34	∈	∈	ADJ
ejpam-3237	228	35	c	c	X
ejpam-3237	228	36	,	,	PUNCT
ejpam-3237	228	37	‖αx−	‖αx−	PROPN
ejpam-3237	228	38	µy‖r‖y‖	µy‖r‖y‖	PROPN
ejpam-3237	228	39	r−1	r−1	PROPN
ejpam-3237	228	40	r	r	NOUN
ejpam-3237	228	41	=	=	PUNCT
ejpam-3237	228	42	|(αx1	|(αx1	NOUN
ejpam-3237	228	43	−	−	PROPN
ejpam-3237	228	44	µy1)y2	µy1)y2	NOUN
ejpam-3237	228	45	−	−	PROPN
ejpam-3237	228	46	(	(	PUNCT
ejpam-3237	228	47	αx2	αx2	NOUN
ejpam-3237	228	48	−	−	NOUN
ejpam-3237	228	49	µy2)y1|	µy2)y1|	ADP
ejpam-3237	228	50	}	}	PUNCT
ejpam-3237	228	51	.	.	PUNCT
ejpam-3237	229	1	so	so	ADV
ejpam-3237	229	2	by	by	ADP
ejpam-3237	229	3	definition	definition	NOUN
ejpam-3237	229	4	of	of	ADP
ejpam-3237	229	5	‖x‖r	‖x‖r	NOUN
ejpam-3237	229	6	,	,	PUNCT
ejpam-3237	229	7	we	we	PRON
ejpam-3237	229	8	have	have	VERB
ejpam-3237	229	9	f	f	PROPN
ejpam-3237	229	10	(	(	PUNCT
ejpam-3237	229	11	x	x	NOUN
ejpam-3237	229	12	;	;	PUNCT
ejpam-3237	229	13	y;α	y;α	NUM
ejpam-3237	229	14	)	)	PUNCT
ejpam-3237	229	15	=	=	PRON
ejpam-3237	229	16	{	{	PUNCT
ejpam-3237	229	17	µ	µ	X
ejpam-3237	229	18	:	:	PUNCT
ejpam-3237	229	19	µ	µ	X
ejpam-3237	229	20	∈	∈	NOUN
ejpam-3237	229	21	c,(|αx1	c,(|αx1	PROPN
ejpam-3237	229	22	−	−	PROPN
ejpam-3237	229	23	µy1|r	µy1|r	PROPN
ejpam-3237	229	24	+	+	PROPN
ejpam-3237	230	1	|αx2	|αx2	ADP
ejpam-3237	230	2	−	−	NOUN
ejpam-3237	230	3	µy2|r	µy2|r	NUM
ejpam-3237	230	4	)	)	PUNCT
ejpam-3237	230	5	1	1	NUM
ejpam-3237	230	6	r	r	NOUN
ejpam-3237	230	7	(	(	PUNCT
ejpam-3237	230	8	|y1|	|y1|	ADP
ejpam-3237	230	9	r−1	r−1	PROPN
ejpam-3237	230	10	r	r	NOUN
ejpam-3237	230	11	+	+	NUM
ejpam-3237	230	12	|y2|	|y2|	NOUN
ejpam-3237	230	13	r−1	r−1	PROPN
ejpam-3237	230	14	r	r	NOUN
ejpam-3237	230	15	)	)	PUNCT
ejpam-3237	231	1	r	r	NOUN
ejpam-3237	231	2	r−1	r−1	NOUN
ejpam-3237	231	3	=	=	PUNCT
ejpam-3237	231	4	|(αx1	|(αx1	NOUN
ejpam-3237	231	5	−	−	PROPN
ejpam-3237	231	6	µy1)y2	µy1)y2	NOUN
ejpam-3237	231	7	−	−	PROPN
ejpam-3237	231	8	(	(	PUNCT
ejpam-3237	231	9	αx2	αx2	NOUN
ejpam-3237	231	10	−	−	NOUN
ejpam-3237	231	11	µy2)y1|	µy2)y1|	ADP
ejpam-3237	231	12	}	}	PUNCT
ejpam-3237	231	13	.	.	PUNCT
ejpam-3237	232	1	specially	specially	ADV
ejpam-3237	232	2	,	,	PUNCT
ejpam-3237	232	3	for	for	ADP
ejpam-3237	232	4	r	r	NOUN
ejpam-3237	232	5	=	=	SYM
ejpam-3237	232	6	2	2	NUM
ejpam-3237	232	7	,	,	PUNCT
ejpam-3237	232	8	f	f	PROPN
ejpam-3237	232	9	(	(	PUNCT
ejpam-3237	232	10	x	x	NOUN
ejpam-3237	232	11	;	;	PUNCT
ejpam-3237	232	12	y;α	y;α	NUM
ejpam-3237	232	13	)	)	PUNCT
ejpam-3237	232	14	=	=	PRON
ejpam-3237	232	15	{	{	PUNCT
ejpam-3237	232	16	µ	µ	X
ejpam-3237	232	17	:	:	PUNCT
ejpam-3237	232	18	µ	µ	X
ejpam-3237	232	19	∈	∈	NOUN
ejpam-3237	232	20	c,(|αx1	c,(|αx1	NOUN
ejpam-3237	232	21	−	−	PROPN
ejpam-3237	232	22	µy1|2	µy1|2	PROPN
ejpam-3237	232	23	+	+	NUM
ejpam-3237	233	1	|αx2	|αx2	PROPN
ejpam-3237	233	2	−	−	NOUN
ejpam-3237	233	3	µy2|2	µy2|2	ADJ
ejpam-3237	233	4	)	)	PUNCT
ejpam-3237	233	5	1	1	NUM
ejpam-3237	233	6	2	2	NUM
ejpam-3237	233	7	(	(	PUNCT
ejpam-3237	233	8	|y1|	|y1|	ADP
ejpam-3237	233	9	1	1	NUM
ejpam-3237	233	10	2	2	NUM
ejpam-3237	233	11	+	+	NUM
ejpam-3237	233	12	|y2|	|y2|	NOUN
ejpam-3237	233	13	1	1	NUM
ejpam-3237	233	14	2	2	NUM
ejpam-3237	233	15	)	)	PUNCT
ejpam-3237	233	16	2	2	NUM
ejpam-3237	233	17	=	=	SYM
ejpam-3237	233	18	|(αx1	|(αx1	NOUN
ejpam-3237	233	19	−	−	PROPN
ejpam-3237	233	20	µy1)y2	µy1)y2	NOUN
ejpam-3237	233	21	−	−	PROPN
ejpam-3237	233	22	(	(	PUNCT
ejpam-3237	233	23	αx2	αx2	NOUN
ejpam-3237	233	24	−	−	NOUN
ejpam-3237	233	25	µy2)y1|	µy2)y1|	VERB
ejpam-3237	233	26	}	}	PUNCT
ejpam-3237	233	27	.	.	PUNCT
ejpam-3237	234	1	references	reference	NOUN
ejpam-3237	234	2	802	802	NUM
ejpam-3237	234	3	giving	give	VERB
ejpam-3237	234	4	x1	x1	PROPN
ejpam-3237	234	5	,	,	PUNCT
ejpam-3237	234	6	x2	x2	PROPN
ejpam-3237	234	7	,	,	PUNCT
ejpam-3237	234	8	y1	y1	NOUN
ejpam-3237	234	9	,	,	PUNCT
ejpam-3237	234	10	y2	y2	PROPN
ejpam-3237	234	11	,	,	PUNCT
ejpam-3237	234	12	α	α	PROPN
ejpam-3237	234	13	,	,	PUNCT
ejpam-3237	234	14	this	this	PRON
ejpam-3237	234	15	is	be	AUX
ejpam-3237	234	16	an	an	DET
ejpam-3237	234	17	equation	equation	NOUN
ejpam-3237	234	18	on	on	ADP
ejpam-3237	234	19	µ	µ	NUM
ejpam-3237	234	20	;	;	PUNCT
ejpam-3237	234	21	in	in	ADP
ejpam-3237	234	22	fact	fact	NOUN
ejpam-3237	234	23	we	we	PRON
ejpam-3237	234	24	have	have	VERB
ejpam-3237	234	25	a((αx1	a((αx1	NOUN
ejpam-3237	234	26	−	−	PROPN
ejpam-3237	234	27	µy1)2	µy1)2	PROPN
ejpam-3237	234	28	+	+	NUM
ejpam-3237	234	29	(	(	PUNCT
ejpam-3237	234	30	αx2	αx2	NOUN
ejpam-3237	234	31	−	−	PROPN
ejpam-3237	234	32	µy2)2	µy2)2	PROPN
ejpam-3237	234	33	)	)	PUNCT
ejpam-3237	234	34	=	=	SYM
ejpam-3237	235	1	(	(	PUNCT
ejpam-3237	235	2	(	(	PUNCT
ejpam-3237	235	3	αx1	αx1	NOUN
ejpam-3237	235	4	−	−	NOUN
ejpam-3237	235	5	µy1)y2	µy1)y2	NOUN
ejpam-3237	235	6	−	−	PROPN
ejpam-3237	236	1	(	(	PUNCT
ejpam-3237	236	2	αx2	αx2	NOUN
ejpam-3237	236	3	−	−	PROPN
ejpam-3237	237	1	µy2)y1)2	µy2)y1)2	PROPN
ejpam-3237	237	2	where	where	SCONJ
ejpam-3237	237	3	a	a	PRON
ejpam-3237	237	4	=	=	X
ejpam-3237	237	5	(	(	PUNCT
ejpam-3237	237	6	|y1|	|y1|	ADP
ejpam-3237	237	7	1	1	NUM
ejpam-3237	237	8	2	2	NUM
ejpam-3237	237	9	+	+	NUM
ejpam-3237	237	10	|y2|	|y2|	NOUN
ejpam-3237	237	11	1	1	NUM
ejpam-3237	237	12	2	2	NUM
ejpam-3237	237	13	)	)	PUNCT
ejpam-3237	237	14	4	4	NUM
ejpam-3237	237	15	.	.	PUNCT
ejpam-3237	238	1	it	it	PRON
ejpam-3237	238	2	leads	lead	VERB
ejpam-3237	238	3	to	to	ADP
ejpam-3237	238	4	the	the	DET
ejpam-3237	238	5	following	follow	VERB
ejpam-3237	238	6	a((y1	a((y1	NOUN
ejpam-3237	238	7	2	2	NUM
ejpam-3237	238	8	+	+	CCONJ
ejpam-3237	239	1	y2	y2	NOUN
ejpam-3237	239	2	2)µ2	2)µ2	NUM
ejpam-3237	240	1	−	−	NOUN
ejpam-3237	240	2	2α(x1y1	2α(x1y1	NUM
ejpam-3237	241	1	+	+	CCONJ
ejpam-3237	241	2	x2y2)µ+	x2y2)µ+	PROPN
ejpam-3237	241	3	α2(x1	α2(x1	NUM
ejpam-3237	241	4	2	2	NUM
ejpam-3237	242	1	+	+	CCONJ
ejpam-3237	242	2	x2	x2	PROPN
ejpam-3237	242	3	2	2	NUM
ejpam-3237	242	4	)	)	PUNCT
ejpam-3237	242	5	)	)	PUNCT
ejpam-3237	243	1	=	=	PUNCT
ejpam-3237	243	2	α2(x1	α2(x1	NUM
ejpam-3237	243	3	2y2	2y2	NUM
ejpam-3237	243	4	2	2	NUM
ejpam-3237	243	5	−	−	NUM
ejpam-3237	243	6	2x1x2y1y2	2x1x2y1y2	NUM
ejpam-3237	244	1	+	+	CCONJ
ejpam-3237	244	2	x2	x2	PROPN
ejpam-3237	244	3	2y1	2y1	NUM
ejpam-3237	244	4	2	2	NUM
ejpam-3237	244	5	)	)	PUNCT
ejpam-3237	244	6	,	,	PUNCT
ejpam-3237	244	7	and	and	CCONJ
ejpam-3237	244	8	µ	µ	X
ejpam-3237	244	9	is	be	AUX
ejpam-3237	244	10	obtained	obtain	VERB
ejpam-3237	244	11	from	from	ADP
ejpam-3237	244	12	this	this	DET
ejpam-3237	244	13	equation	equation	NOUN
ejpam-3237	244	14	.	.	PUNCT
ejpam-3237	245	1	references	reference	NOUN
ejpam-3237	245	2	[	[	X
ejpam-3237	245	3	1	1	X
ejpam-3237	245	4	]	]	X
ejpam-3237	245	5	m.v	m.v	PROPN
ejpam-3237	245	6	.	.	PROPN
ejpam-3237	245	7	balashov	balashov	PROPN
ejpam-3237	245	8	,	,	PUNCT
ejpam-3237	245	9	geometric	geometric	ADJ
ejpam-3237	245	10	difference	difference	NOUN
ejpam-3237	245	11	of	of	ADP
ejpam-3237	245	12	multivalued	multivalued	ADJ
ejpam-3237	245	13	maps	map	NOUN
ejpam-3237	245	14	,	,	PUNCT
ejpam-3237	245	15	mathematical	mathematical	ADJ
ejpam-3237	245	16	notes	note	NOUN
ejpam-3237	245	17	70	70	NUM
ejpam-3237	245	18	(	(	PUNCT
ejpam-3237	245	19	2001	2001	NUM
ejpam-3237	245	20	)	)	PUNCT
ejpam-3237	245	21	,	,	PUNCT
ejpam-3237	245	22	147–153	147–153	NUM
ejpam-3237	245	23	.	.	PUNCT
ejpam-3237	246	1	[	[	X
ejpam-3237	246	2	2	2	NUM
ejpam-3237	246	3	]	]	X
ejpam-3237	246	4	g.	g.	NOUN
ejpam-3237	246	5	birkhoff	birkhoff	PROPN
ejpam-3237	246	6	,	,	PUNCT
ejpam-3237	246	7	orthogonality	orthogonality	NOUN
ejpam-3237	246	8	in	in	ADP
ejpam-3237	246	9	normed	normed	ADJ
ejpam-3237	246	10	linear	linear	PROPN
ejpam-3237	246	11	spaces	space	NOUN
ejpam-3237	246	12	,	,	PUNCT
ejpam-3237	246	13	duke	duke	PROPN
ejpam-3237	246	14	math	math	PROPN
ejpam-3237	246	15	.	.	PUNCT
ejpam-3237	247	1	j.	j.	PROPN
ejpam-3237	247	2	,	,	PUNCT
ejpam-3237	247	3	1	1	NUM
ejpam-3237	247	4	(	(	PUNCT
ejpam-3237	247	5	1935	1935	NUM
ejpam-3237	247	6	)	)	PUNCT
ejpam-3237	247	7	,	,	PUNCT
ejpam-3237	247	8	169–172	169–172	NUM
ejpam-3237	247	9	.	.	PUNCT
ejpam-3237	248	1	[	[	X
ejpam-3237	248	2	3	3	X
ejpam-3237	248	3	]	]	X
ejpam-3237	248	4	bonsall	bonsall	PROPN
ejpam-3237	248	5	f.f	f.f	PROPN
ejpam-3237	248	6	.	.	PROPN
ejpam-3237	248	7	and	and	CCONJ
ejpam-3237	248	8	duncan	duncan	PROPN
ejpam-3237	248	9	j.	j.	PROPN
ejpam-3237	248	10	,	,	PUNCT
ejpam-3237	248	11	numerical	numerical	PROPN
ejpam-3237	248	12	ranges	ranges	PROPN
ejpam-3237	248	13	ii	ii	PROPN
ejpam-3237	248	14	,	,	PUNCT
ejpam-3237	248	15	london	london	PROPN
ejpam-3237	248	16	mathematical	mathematical	ADJ
ejpam-3237	248	17	society	society	NOUN
ejpam-3237	248	18	lecture	lecture	NOUN
ejpam-3237	248	19	note	note	NOUN
ejpam-3237	248	20	series	series	PROPN
ejpam-3237	248	21	,	,	PUNCT
ejpam-3237	248	22	cambridge	cambridge	PROPN
ejpam-3237	248	23	university	university	PROPN
ejpam-3237	248	24	press	press	NOUN
ejpam-3237	248	25	,	,	PUNCT
ejpam-3237	248	26	new	new	PROPN
ejpam-3237	248	27	york	york	PROPN
ejpam-3237	248	28	(	(	PUNCT
ejpam-3237	248	29	1973	1973	NUM
ejpam-3237	248	30	)	)	PUNCT
ejpam-3237	248	31	.	.	PUNCT
ejpam-3237	249	1	[	[	X
ejpam-3237	249	2	4	4	X
ejpam-3237	249	3	]	]	PUNCT
ejpam-3237	249	4	christos	christos	PROPN
ejpam-3237	249	5	chorianopoulos	chorianopoulos	PROPN
ejpam-3237	249	6	and	and	CCONJ
ejpam-3237	249	7	panayiotis	panayiotis	PROPN
ejpam-3237	249	8	j.	j.	PROPN
ejpam-3237	249	9	psarrakos	psarrakos	PROPN
ejpam-3237	249	10	,	,	PUNCT
ejpam-3237	249	11	on	on	ADP
ejpam-3237	249	12	the	the	DET
ejpam-3237	249	13	continuity	continuity	NOUN
ejpam-3237	249	14	of	of	ADP
ejpam-3237	249	15	birkhoffjames	birkhoffjame	NOUN
ejpam-3237	249	16	ε	ε	PROPN
ejpam-3237	249	17	-orthogonality	-orthogonality	NOUN
ejpam-3237	249	18	sets	set	NOUN
ejpam-3237	249	19	,	,	PUNCT
ejpam-3237	249	20	linear	linear	PROPN
ejpam-3237	249	21	multilinear	multilinear	PROPN
ejpam-3237	249	22	algebra	algebra	PROPN
ejpam-3237	249	23	,	,	PUNCT
ejpam-3237	249	24	61	61	NUM
ejpam-3237	249	25	(	(	PUNCT
ejpam-3237	249	26	2013	2013	NUM
ejpam-3237	249	27	)	)	PUNCT
ejpam-3237	249	28	.	.	PUNCT
ejpam-3237	250	1	[	[	X
ejpam-3237	250	2	5	5	X
ejpam-3237	250	3	]	]	X
ejpam-3237	250	4	gustafson	gustafson	PROPN
ejpam-3237	250	5	k.e	k.e	PROPN
ejpam-3237	250	6	.	.	PROPN
ejpam-3237	250	7	and	and	CCONJ
ejpam-3237	250	8	rao	rao	PROPN
ejpam-3237	250	9	d.k.m	d.k.m	PROPN
ejpam-3237	250	10	.	.	PROPN
ejpam-3237	250	11	,	,	PUNCT
ejpam-3237	250	12	numerical	numerical	ADJ
ejpam-3237	250	13	range	range	NOUN
ejpam-3237	250	14	.	.	PUNCT
ejpam-3237	251	1	the	the	DET
ejpam-3237	251	2	field	field	NOUN
ejpam-3237	251	3	of	of	ADP
ejpam-3237	251	4	values	value	NOUN
ejpam-3237	251	5	of	of	ADP
ejpam-3237	251	6	linear	linear	PROPN
ejpam-3237	251	7	operators	operator	NOUN
ejpam-3237	251	8	and	and	CCONJ
ejpam-3237	251	9	matrices	matrix	NOUN
ejpam-3237	251	10	,	,	PUNCT
ejpam-3237	251	11	springer	springer	NOUN
ejpam-3237	251	12	-	-	PUNCT
ejpam-3237	251	13	verlag	verlag	PROPN
ejpam-3237	251	14	,	,	PUNCT
ejpam-3237	251	15	new	new	PROPN
ejpam-3237	251	16	york	york	PROPN
ejpam-3237	251	17	,	,	PUNCT
ejpam-3237	251	18	(	(	PUNCT
ejpam-3237	251	19	1997	1997	NUM
ejpam-3237	251	20	)	)	PUNCT
ejpam-3237	251	21	.	.	PUNCT
ejpam-3237	252	1	[	[	X
ejpam-3237	252	2	6	6	NUM
ejpam-3237	252	3	]	]	PUNCT
ejpam-3237	252	4	d.	d.	PROPN
ejpam-3237	252	5	h.	h.	PROPN
ejpam-3237	252	6	ji	ji	PROPN
ejpam-3237	252	7	,	,	PUNCT
ejpam-3237	252	8	and	and	CCONJ
ejpam-3237	252	9	s.	s.	PROPN
ejpam-3237	252	10	l.	l.	PROPN
ejpam-3237	252	11	wu	wu	PROPN
ejpam-3237	252	12	,	,	PUNCT
ejpam-3237	252	13	quantitative	quantitative	ADJ
ejpam-3237	252	14	characterization	characterization	NOUN
ejpam-3237	252	15	of	of	ADP
ejpam-3237	252	16	the	the	DET
ejpam-3237	252	17	difference	difference	NOUN
ejpam-3237	252	18	between	between	ADP
ejpam-3237	252	19	birkhoff	birkhoff	NOUN
ejpam-3237	252	20	orthogonality	orthogonality	NOUN
ejpam-3237	252	21	and	and	CCONJ
ejpam-3237	252	22	isosceles	isoscele	NOUN
ejpam-3237	252	23	orthogonality	orthogonality	NOUN
ejpam-3237	252	24	,	,	PUNCT
ejpam-3237	252	25	j.	j.	PROPN
ejpam-3237	252	26	math	math	PROPN
ejpam-3237	252	27	.	.	PUNCT
ejpam-3237	253	1	ana	ana	PROPN
ejpam-3237	253	2	.	.	PUNCT
ejpam-3237	253	3	appl	appl	PROPN
ejpam-3237	253	4	.	.	PROPN
ejpam-3237	254	1	,	,	PUNCT
ejpam-3237	254	2	323	323	NUM
ejpam-3237	254	3	(	(	PUNCT
ejpam-3237	254	4	2006	2006	NUM
ejpam-3237	254	5	)	)	PUNCT
ejpam-3237	254	6	,	,	PUNCT
ejpam-3237	254	7	17	17	NUM
ejpam-3237	254	8	.	.	PUNCT
ejpam-3237	255	1	[	[	X
ejpam-3237	255	2	7	7	X
ejpam-3237	255	3	]	]	X
ejpam-3237	255	4	p.	p.	NOUN
ejpam-3237	255	5	m.	m.	NOUN
ejpam-3237	255	6	milicic	milicic	PROPN
ejpam-3237	255	7	,	,	PUNCT
ejpam-3237	255	8	sur	sur	PROPN
ejpam-3237	255	9	le	le	X
ejpam-3237	255	10	g	g	PROPN
ejpam-3237	255	11	-	-	PUNCT
ejpam-3237	255	12	angle	angle	NOUN
ejpam-3237	255	13	dans	dans	PROPN
ejpam-3237	255	14	un	un	PROPN
ejpam-3237	255	15	espace	espace	PROPN
ejpam-3237	255	16	norme	norme	PROPN
ejpam-3237	255	17	,	,	PUNCT
ejpam-3237	255	18	mat	mat	PROPN
ejpam-3237	255	19	.	.	PROPN
ejpam-3237	255	20	vesnik	vesnik	PROPN
ejpam-3237	255	21	,	,	PUNCT
ejpam-3237	255	22	45	45	NUM
ejpam-3237	255	23	(	(	PUNCT
ejpam-3237	255	24	1993	1993	NUM
ejpam-3237	255	25	)	)	PUNCT
ejpam-3237	255	26	,	,	PUNCT
ejpam-3237	255	27	4348	4348	NUM
ejpam-3237	255	28	.	.	PUNCT
ejpam-3237	256	1	[	[	X
ejpam-3237	256	2	8	8	NUM
ejpam-3237	256	3	]	]	X
ejpam-3237	256	4	chen	chen	PROPN
ejpam-3237	256	5	zhi	zhi	PROPN
ejpam-3237	256	6	-	-	PUNCT
ejpam-3237	256	7	zhi	zhi	PROPN
ejpam-3237	256	8	,	,	PUNCT
ejpam-3237	256	9	lin	lin	PROPN
ejpam-3237	256	10	wei	wei	PROPN
ejpam-3237	256	11	and	and	CCONJ
ejpam-3237	256	12	luo	luo	PROPN
ejpam-3237	256	13	lu	lu	PROPN
ejpam-3237	256	14	-	-	PUNCT
ejpam-3237	256	15	lin	lin	PROPN
ejpam-3237	256	16	,	,	PUNCT
ejpam-3237	256	17	projections	projection	NOUN
ejpam-3237	256	18	,	,	PUNCT
ejpam-3237	256	19	birkhoff	birkhoff	NOUN
ejpam-3237	256	20	orthogonality	orthogonality	NOUN
ejpam-3237	256	21	and	and	CCONJ
ejpam-3237	256	22	angles	angle	NOUN
ejpam-3237	256	23	in	in	ADP
ejpam-3237	256	24	normed	normed	ADJ
ejpam-3237	256	25	spaces	space	NOUN
ejpam-3237	256	26	,	,	PUNCT
ejpam-3237	256	27	communications	communication	NOUN
ejpam-3237	256	28	in	in	ADP
ejpam-3237	256	29	mathematical	mathematical	ADJ
ejpam-3237	256	30	research	research	NOUN
ejpam-3237	256	31	27(4	27(4	PROPN
ejpam-3237	256	32	)	)	PUNCT
ejpam-3237	256	33	(	(	PUNCT
ejpam-3237	256	34	2011	2011	NUM
ejpam-3237	256	35	)	)	PUNCT
ejpam-3237	256	36	,	,	PUNCT
ejpam-3237	256	37	378–384	378–384	NUM
ejpam-3237	256	38	.	.	PUNCT