id	sid	tid	token	lemma	pos
bibechana-10399	1	1	169	169	NUM
bibechana-10399	1	2	-	-	SYM
bibechana-10399	1	3	174	174	NUM
bibechana-10399	1	4	mahendra	mahendra	PROPN
bibechana-10399	1	5	sahi	sahi	PROPN
bibechana-10399	1	6	r	r	NOUN
bibechana-10399	1	7	c	c	NOUN
bibechana-10399	1	8	o	o	NOUN
bibechana-10399	1	9	s	s	PROPN
bibechana-10399	1	10	t	t	PROPN
bibechana-10399	1	11	n	n	NOUN
bibechana-10399	1	12	m.	m.	NOUN
bibechana-10399	1	13	sahi	sahi	PROPN
bibechana-10399	1	14	/	/	SYM
bibechana-10399	1	15	bibechana	bibechana	PROPN
bibechana-10399	1	16	11(1	11(1	NUM
bibechana-10399	1	17	)	)	PUNCT
bibechana-10399	1	18	(	(	PUNCT
bibechana-10399	1	19	2014	2014	NUM
bibechana-10399	1	20	)	)	PUNCT
bibechana-10399	1	21	169	169	NUM
bibechana-10399	1	22	-	-	SYM
bibechana-10399	1	23	174	174	NUM
bibechana-10399	1	24	:	:	PUNCT
bibechana-10399	1	25	(	(	PUNCT
bibechana-10399	1	26	online	online	ADJ
bibechana-10399	1	27	publication	publication	NOUN
bibechana-10399	1	28	:	:	PUNCT
bibechana-10399	1	29	march	march	PROPN
bibechana-10399	1	30	,	,	PUNCT
bibechana-10399	1	31	2014	2014	NUM
bibechana-10399	1	32	)	)	PUNCT
bibechana-10399	1	33	p.169	p.169	NOUN
bibechana-10399	1	34	bibechana	bibechana	NOUN
bibechana-10399	1	35	a	a	DET
bibechana-10399	1	36	multidisciplinary	multidisciplinary	ADJ
bibechana-10399	1	37	journal	journal	NOUN
bibechana-10399	1	38	of	of	ADP
bibechana-10399	1	39	science	science	NOUN
bibechana-10399	1	40	,	,	PUNCT
bibechana-10399	1	41	technology	technology	NOUN
bibechana-10399	1	42	and	and	CCONJ
bibechana-10399	1	43	mathematics	mathematic	NOUN
bibechana-10399	1	44	issn	issn	VERB
bibechana-10399	1	45	2091	2091	NUM
bibechana-10399	1	46	-	-	SYM
bibechana-10399	1	47	0762	0762	NUM
bibechana-10399	1	48	(	(	PUNCT
bibechana-10399	1	49	online	online	ADJ
bibechana-10399	1	50	)	)	PUNCT
bibechana-10399	1	51	journal	journal	NOUN
bibechana-10399	1	52	homepage	homepage	NOUN
bibechana-10399	1	53	:	:	PUNCT
bibechana-10399	1	54	http://nepjol.info/index.php/bibechana	http://nepjol.info/index.php/bibechana	PROPN
bibechana-10399	1	55	some	some	DET
bibechana-10399	1	56	special	special	ADJ
bibechana-10399	1	57	characterisations	characterisation	NOUN
bibechana-10399	1	58	of	of	ADP
bibechana-10399	1	59	fredholm	fredholm	NOUN
bibechana-10399	1	60	operators	operator	NOUN
bibechana-10399	1	61	in	in	ADP
bibechana-10399	1	62	banach	banach	NOUN
bibechana-10399	1	63	space	space	NOUN
bibechana-10399	1	64	mahendra	mahendra	PROPN
bibechana-10399	1	65	shahi	shahi	PROPN
bibechana-10399	1	66	department	department	PROPN
bibechana-10399	1	67	.	.	PUNCT
bibechana-10399	2	1	of	of	ADP
bibechana-10399	2	2	mathematics	mathematics	PROPN
bibechana-10399	2	3	,	,	PUNCT
bibechana-10399	2	4	m.m.a.m	m.m.a.m	PROPN
bibechana-10399	2	5	.	.	PUNCT
bibechana-10399	2	6	campus	campus	PROPN
bibechana-10399	2	7	,	,	PUNCT
bibechana-10399	2	8	biratnagar	biratnagar	PROPN
bibechana-10399	2	9	tribhuvan	tribhuvan	PROPN
bibechana-10399	2	10	university	university	PROPN
bibechana-10399	2	11	,	,	PUNCT
bibechana-10399	2	12	nepal	nepal	ADJ
bibechana-10399	2	13	e	e	NOUN
bibechana-10399	2	14	-	-	NOUN
bibechana-10399	2	15	mail	mail	NOUN
bibechana-10399	2	16	:	:	PUNCT
bibechana-10399	2	17	mshahi11@hotmail.com	mshahi11@hotmail.com	AUX
bibechana-10399	2	18	accepted	accept	VERB
bibechana-10399	2	19	for	for	ADP
bibechana-10399	2	20	publication	publication	NOUN
bibechana-10399	2	21	:	:	PUNCT
bibechana-10399	2	22	february	february	PROPN
bibechana-10399	2	23	06	06	NUM
bibechana-10399	2	24	,	,	PUNCT
bibechana-10399	2	25	2014	2014	NUM
bibechana-10399	2	26	abstract	abstract	ADV
bibechana-10399	2	27	a	a	DET
bibechana-10399	2	28	bounded	bounded	ADJ
bibechana-10399	2	29	linear	linear	ADJ
bibechana-10399	2	30	operator	operator	NOUN
bibechana-10399	2	31	which	which	PRON
bibechana-10399	2	32	has	have	VERB
bibechana-10399	2	33	a	a	DET
bibechana-10399	2	34	finite	finite	ADJ
bibechana-10399	2	35	index	index	NOUN
bibechana-10399	2	36	and	and	CCONJ
bibechana-10399	2	37	which	which	PRON
bibechana-10399	2	38	is	be	AUX
bibechana-10399	2	39	defined	define	VERB
bibechana-10399	2	40	on	on	ADP
bibechana-10399	2	41	a	a	DET
bibechana-10399	2	42	banach	banach	NOUN
bibechana-10399	2	43	space	space	NOUN
bibechana-10399	2	44	is	be	AUX
bibechana-10399	2	45	often	often	ADV
bibechana-10399	2	46	referred	refer	VERB
bibechana-10399	2	47	to	to	ADP
bibechana-10399	2	48	in	in	ADP
bibechana-10399	2	49	the	the	DET
bibechana-10399	2	50	literature	literature	NOUN
bibechana-10399	2	51	as	as	ADP
bibechana-10399	2	52	a	a	DET
bibechana-10399	2	53	fredholm	fredholm	NOUN
bibechana-10399	2	54	operator	operator	NOUN
bibechana-10399	2	55	.	.	PUNCT
bibechana-10399	3	1	fredholm	fredholm	NOUN
bibechana-10399	3	2	operators	operator	NOUN
bibechana-10399	3	3	are	be	AUX
bibechana-10399	3	4	important	important	ADJ
bibechana-10399	3	5	for	for	ADP
bibechana-10399	3	6	a	a	DET
bibechana-10399	3	7	variety	variety	NOUN
bibechana-10399	3	8	of	of	ADP
bibechana-10399	3	9	reasons	reason	NOUN
bibechana-10399	3	10	,	,	PUNCT
bibechana-10399	3	11	one	one	NUM
bibechana-10399	3	12	being	be	AUX
bibechana-10399	3	13	the	the	DET
bibechana-10399	3	14	role	role	NOUN
bibechana-10399	3	15	that	that	PRON
bibechana-10399	3	16	their	their	PRON
bibechana-10399	3	17	index	index	NOUN
bibechana-10399	3	18	plays	play	VERB
bibechana-10399	3	19	in	in	ADP
bibechana-10399	3	20	global	global	ADJ
bibechana-10399	3	21	analysis	analysis	NOUN
bibechana-10399	3	22	.	.	PUNCT
bibechana-10399	4	1	the	the	DET
bibechana-10399	4	2	aim	aim	NOUN
bibechana-10399	4	3	of	of	ADP
bibechana-10399	4	4	this	this	DET
bibechana-10399	4	5	paper	paper	NOUN
bibechana-10399	4	6	is	be	AUX
bibechana-10399	4	7	to	to	PART
bibechana-10399	4	8	prove	prove	VERB
bibechana-10399	4	9	the	the	DET
bibechana-10399	4	10	spectral	spectral	ADJ
bibechana-10399	4	11	theorem	theorem	NOUN
bibechana-10399	4	12	for	for	ADP
bibechana-10399	4	13	compact	compact	ADJ
bibechana-10399	4	14	operators	operator	NOUN
bibechana-10399	4	15	in	in	ADP
bibechana-10399	4	16	refined	refined	ADJ
bibechana-10399	4	17	form	form	NOUN
bibechana-10399	4	18	and	and	CCONJ
bibechana-10399	4	19	to	to	PART
bibechana-10399	4	20	describe	describe	VERB
bibechana-10399	4	21	some	some	DET
bibechana-10399	4	22	properties	property	NOUN
bibechana-10399	4	23	of	of	ADP
bibechana-10399	4	24	the	the	DET
bibechana-10399	4	25	essential	essential	ADJ
bibechana-10399	4	26	spectrum	spectrum	NOUN
bibechana-10399	4	27	of	of	ADP
bibechana-10399	4	28	general	general	PROPN
bibechana-10399	4	29	bounded	bound	VERB
bibechana-10399	4	30	operators	operator	NOUN
bibechana-10399	4	31	by	by	ADP
bibechana-10399	4	32	the	the	DET
bibechana-10399	4	33	use	use	NOUN
bibechana-10399	4	34	of	of	ADP
bibechana-10399	4	35	the	the	DET
bibechana-10399	4	36	theorem	theorem	NOUN
bibechana-10399	4	37	of	of	ADP
bibechana-10399	4	38	fredholm	fredholm	NOUN
bibechana-10399	4	39	operators	operator	NOUN
bibechana-10399	4	40	.	.	PUNCT
bibechana-10399	5	1	for	for	ADP
bibechana-10399	5	2	this	this	PRON
bibechana-10399	5	3	,	,	PUNCT
bibechana-10399	5	4	we	we	PRON
bibechana-10399	5	5	have	have	AUX
bibechana-10399	5	6	analysed	analyse	VERB
bibechana-10399	5	7	the	the	DET
bibechana-10399	5	8	fredholm	fredholm	NOUN
bibechana-10399	5	9	operator	operator	NOUN
bibechana-10399	5	10	which	which	PRON
bibechana-10399	5	11	is	be	AUX
bibechana-10399	5	12	defined	define	VERB
bibechana-10399	5	13	in	in	ADP
bibechana-10399	5	14	a	a	DET
bibechana-10399	5	15	banach	banach	NOUN
bibechana-10399	5	16	space	space	NOUN
bibechana-10399	5	17	for	for	ADP
bibechana-10399	5	18	some	some	DET
bibechana-10399	5	19	special	special	ADJ
bibechana-10399	5	20	characterisations	characterisation	NOUN
bibechana-10399	5	21	.	.	PUNCT
bibechana-10399	6	1	©	©	ADP
bibechana-10399	6	2	2014	2014	NUM
bibechana-10399	6	3	rcost	rcost	NOUN
bibechana-10399	6	4	:	:	PUNCT
bibechana-10399	6	5	all	all	DET
bibechana-10399	6	6	rights	right	NOUN
bibechana-10399	6	7	reserved	reserve	VERB
bibechana-10399	6	8	.	.	PUNCT
bibechana-10399	7	1	keywords	keyword	NOUN
bibechana-10399	7	2	:	:	PUNCT
bibechana-10399	7	3	bounded	bounded	ADJ
bibechana-10399	7	4	linear	linear	ADJ
bibechana-10399	7	5	operator	operator	NOUN
bibechana-10399	7	6	;	;	PUNCT
bibechana-10399	7	7	compact	compact	ADJ
bibechana-10399	7	8	operator	operator	NOUN
bibechana-10399	7	9	;	;	PUNCT
bibechana-10399	7	10	fredholm	fredholm	NOUN
bibechana-10399	7	11	operator	operator	NOUN
bibechana-10399	7	12	;	;	PUNCT
bibechana-10399	7	13	banach	banach	NOUN
bibechana-10399	7	14	space	space	NOUN
bibechana-10399	7	15	.	.	PUNCT
bibechana-10399	8	1	1	1	X
bibechana-10399	8	2	.	.	X
bibechana-10399	8	3	introduction	introduction	NOUN
bibechana-10399	8	4	an	an	DET
bibechana-10399	8	5	operator	operator	NOUN
bibechana-10399	8	6	k	k	PROPN
bibechana-10399	8	7	defined	define	VERB
bibechana-10399	8	8	by	by	ADP
bibechana-10399	8	9	a	a	DET
bibechana-10399	8	10	kernel	kernel	NOUN
bibechana-10399	8	11	k	k	PROPN
bibechana-10399	8	12	is	be	AUX
bibechana-10399	8	13	called	call	VERB
bibechana-10399	8	14	a	a	DET
bibechana-10399	8	15	fredholm	fredholm	NOUN
bibechana-10399	8	16	type	type	NOUN
bibechana-10399	8	17	operator	operator	NOUN
bibechana-10399	8	18	.	.	PUNCT
bibechana-10399	9	1	the	the	DET
bibechana-10399	9	2	name	name	NOUN
bibechana-10399	9	3	goes	go	VERB
bibechana-10399	9	4	back	back	ADV
bibechana-10399	9	5	to	to	ADP
bibechana-10399	9	6	swede	swede	NOUN
bibechana-10399	9	7	,	,	PUNCT
bibechana-10399	9	8	e.	e.	PROPN
bibechana-10399	9	9	ivar	ivar	PROPN
bibechana-10399	9	10	fredholm	fredholm	NOUN
bibechana-10399	9	11	who	who	PRON
bibechana-10399	9	12	developed	develop	VERB
bibechana-10399	9	13	a	a	DET
bibechana-10399	9	14	comprehensive	comprehensive	ADJ
bibechana-10399	9	15	theory	theory	NOUN
bibechana-10399	9	16	for	for	ADP
bibechana-10399	9	17	integral	integral	ADJ
bibechana-10399	9	18	equations	equation	NOUN
bibechana-10399	9	19	of	of	ADP
bibechana-10399	9	20	second	second	ADJ
bibechana-10399	9	21	kind	kind	NOUN
bibechana-10399	9	22	at	at	ADP
bibechana-10399	9	23	the	the	DET
bibechana-10399	9	24	beginning	beginning	NOUN
bibechana-10399	9	25	of	of	ADP
bibechana-10399	9	26	the	the	DET
bibechana-10399	9	27	twentieth	twentieth	ADJ
bibechana-10399	9	28	century	century	NOUN
bibechana-10399	9	29	[	[	X
bibechana-10399	9	30	4	4	NUM
bibechana-10399	9	31	]	]	PUNCT
bibechana-10399	9	32	.	.	PUNCT
bibechana-10399	10	1	let	let	VERB
bibechana-10399	10	2	x	x	SYM
bibechana-10399	10	3	=	=	PUNCT
bibechana-10399	10	4	y	y	PROPN
bibechana-10399	10	5	=	=	SYM
bibechana-10399	10	6	c[a	c[a	PROPN
bibechana-10399	10	7	,	,	PUNCT
bibechana-10399	10	8	b	b	AUX
bibechana-10399	10	9	]	]	PUNCT
bibechana-10399	10	10	be	be	AUX
bibechana-10399	10	11	a	a	DET
bibechana-10399	10	12	banach	banach	NOUN
bibechana-10399	10	13	space.let	space.let	X
bibechana-10399	10	14	k(s	k(s	PROPN
bibechana-10399	10	15	,	,	PUNCT
bibechana-10399	10	16	t	t	PROPN
bibechana-10399	10	17	)	)	PUNCT
bibechana-10399	10	18	be	be	AUX
bibechana-10399	10	19	defined	define	VERB
bibechana-10399	10	20	for	for	ADP
bibechana-10399	10	21	a≤s≤band	a≤s≤band	PROPN
bibechana-10399	10	22	a≤t≤b	a≤t≤b	PROPN
bibechana-10399	10	23	.	.	PUNCT
bibechana-10399	11	1	then	then	ADV
bibechana-10399	11	2	for	for	ADP
bibechana-10399	11	3	each	each	DET
bibechana-10399	11	4	xϵx	xϵx	NOUN
bibechana-10399	12	1	the	the	DET
bibechana-10399	12	2	riemann	riemann	PROPN
bibechana-10399	12	3	integral	integral	PROPN
bibechana-10399	12	4	�	�	PROPN
bibechana-10399	12	5	k	k	PROPN
bibechana-10399	12	6	�	�	PROPN
bibechana-10399	12	7	s	s	PART
bibechana-10399	12	8	,	,	PUNCT
bibechana-10399	12	9	t	t	PROPN
bibechana-10399	12	10	�	�	PROPN
bibechana-10399	12	11	x	x	PROPN
bibechana-10399	12	12	�	�	PROPN
bibechana-10399	12	13	t	t	PROPN
bibechana-10399	12	14	�	�	PROPN
bibechana-10399	12	15	dt	dt	NOUN
bibechana-10399	12	16	�	�	PROPN
bibechana-10399	12	17	1	1	NUM
bibechana-10399	12	18	�	�	PROPN
bibechana-10399	12	19	�	�	PROPN
bibechana-10399	12	20	exists	exist	VERB
bibechana-10399	12	21	and	and	CCONJ
bibechana-10399	12	22	defines	define	VERB
bibechana-10399	12	23	a	a	DET
bibechana-10399	12	24	continuous	continuous	ADJ
bibechana-10399	12	25	function	function	NOUN
bibechana-10399	12	26	of	of	ADP
bibechana-10399	12	27	s	s	PRON
bibechana-10399	12	28	on	on	ADP
bibechana-10399	12	29	[	[	X
bibechana-10399	12	30	a	a	DET
bibechana-10399	12	31	,	,	PUNCT
bibechana-10399	12	32	b	b	NOUN
bibechana-10399	12	33	]	]	X
bibechana-10399	12	34	.	.	PUNCT
bibechana-10399	13	1	the	the	DET
bibechana-10399	13	2	integral	integral	ADJ
bibechana-10399	13	3	(	(	PUNCT
bibechana-10399	13	4	1	1	NUM
bibechana-10399	13	5	)	)	PUNCT
bibechana-10399	13	6	defines	define	VERB
bibechana-10399	13	7	a	a	DET
bibechana-10399	13	8	linear	linear	ADJ
bibechana-10399	13	9	operator	operator	NOUN
bibechana-10399	13	10	k	k	X
bibechana-10399	13	11	on	on	ADP
bibechana-10399	13	12	x	x	PUNCT
bibechana-10399	13	13	into	into	ADP
bibechana-10399	13	14	x.	x.	NOUN
bibechana-10399	13	15	if	if	SCONJ
bibechana-10399	13	16	we	we	PRON
bibechana-10399	13	17	take	take	VERB
bibechana-10399	13	18	kx	kx	NOUN
bibechana-10399	13	19	=	=	PUNCT
bibechana-10399	13	20	y	y	PROPN
bibechana-10399	13	21	to	to	PART
bibechana-10399	13	22	mean	mean	VERB
bibechana-10399	14	1	y	y	PROPN
bibechana-10399	14	2	�	�	PROPN
bibechana-10399	14	3	s	s	PART
bibechana-10399	14	4	�	�	PROPN
bibechana-10399	14	5	=	=	SYM
bibechana-10399	14	6	�	�	PROPN
bibechana-10399	14	7	k	k	PROPN
bibechana-10399	14	8	�	�	PROPN
bibechana-10399	14	9	s	s	PART
bibechana-10399	14	10	,	,	PUNCT
bibechana-10399	14	11	t	t	PROPN
bibechana-10399	14	12	�	�	PROPN
bibechana-10399	14	13	x	x	PROPN
bibechana-10399	14	14	�	�	PROPN
bibechana-10399	14	15	t	t	PROPN
bibechana-10399	14	16	�	�	PROPN
bibechana-10399	14	17	dt	dt	NOUN
bibechana-10399	14	18	�	�	PROPN
bibechana-10399	14	19	2	2	NUM
bibechana-10399	14	20	�	�	PROPN
bibechana-10399	14	21	�	�	PROPN
bibechana-10399	14	22	the	the	DET
bibechana-10399	14	23	equation(2	equation(2	PROPN
bibechana-10399	14	24	)	)	PUNCT
bibechana-10399	14	25	is	be	AUX
bibechana-10399	14	26	known	know	VERB
bibechana-10399	14	27	as	as	ADP
bibechana-10399	14	28	fredholm	fredholm	NOUN
bibechana-10399	14	29	type	type	NOUN
bibechana-10399	14	30	integral	integral	ADJ
bibechana-10399	14	31	equation	equation	NOUN
bibechana-10399	14	32	of	of	ADP
bibechana-10399	14	33	the	the	DET
bibechana-10399	14	34	first	first	ADJ
bibechana-10399	14	35	kind	kind	NOUN
bibechana-10399	14	36	[	[	X
bibechana-10399	14	37	7	7	NUM
bibechana-10399	14	38	]	]	PUNCT
bibechana-10399	14	39	.	.	PUNCT
bibechana-10399	15	1	another	another	DET
bibechana-10399	15	2	operator	operator	NOUN
bibechana-10399	15	3	t	t	NOUN
bibechana-10399	15	4	is	be	AUX
bibechana-10399	15	5	obtained	obtain	VERB
bibechana-10399	15	6	by	by	ADP
bibechana-10399	15	7	defining	define	VERB
bibechana-10399	15	8	tx	tx	PROPN
bibechana-10399	15	9	=	=	SYM
bibechana-10399	15	10	y	y	PROPN
bibechana-10399	15	11	to	to	PART
bibechana-10399	15	12	mean	mean	VERB
bibechana-10399	15	13	m.	m.	NOUN
bibechana-10399	15	14	sahi	sahi	PROPN
bibechana-10399	15	15	/	/	SYM
bibechana-10399	15	16	bibechana	bibechana	PROPN
bibechana-10399	15	17	11(1	11(1	NUM
bibechana-10399	15	18	)	)	PUNCT
bibechana-10399	15	19	(	(	PUNCT
bibechana-10399	15	20	2014	2014	NUM
bibechana-10399	15	21	)	)	PUNCT
bibechana-10399	15	22	169	169	NUM
bibechana-10399	15	23	-	-	SYM
bibechana-10399	15	24	174	174	NUM
bibechana-10399	15	25	:	:	PUNCT
bibechana-10399	15	26	(	(	PUNCT
bibechana-10399	15	27	online	online	ADJ
bibechana-10399	15	28	publication	publication	NOUN
bibechana-10399	15	29	:	:	PUNCT
bibechana-10399	15	30	march	march	PROPN
bibechana-10399	15	31	,	,	PUNCT
bibechana-10399	15	32	2014	2014	NUM
bibechana-10399	15	33	)	)	PUNCT
bibechana-10399	15	34	p.170	p.170	NOUN
bibechana-10399	15	35	y	y	PROPN
bibechana-10399	15	36	�	�	PROPN
bibechana-10399	15	37	s	s	PART
bibechana-10399	15	38	�	�	NOUN
bibechana-10399	15	39	=	=	SYM
bibechana-10399	15	40	x	x	SYM
bibechana-10399	15	41	�	�	PROPN
bibechana-10399	15	42	s	s	PART
bibechana-10399	15	43	�	�	PROPN
bibechana-10399	15	44	−	−	PROPN
bibechana-10399	15	45	�	�	PROPN
bibechana-10399	15	46	k	k	PROPN
bibechana-10399	15	47	�	�	PROPN
bibechana-10399	15	48	s	s	PART
bibechana-10399	15	49	,	,	PUNCT
bibechana-10399	15	50	t	t	PROPN
bibechana-10399	15	51	�	�	PROPN
bibechana-10399	15	52	x	x	PROPN
bibechana-10399	15	53	�	�	PROPN
bibechana-10399	15	54	t	t	PROPN
bibechana-10399	15	55	�	�	PROPN
bibechana-10399	15	56	dt	dt	NOUN
bibechana-10399	15	57	�	�	PROPN
bibechana-10399	15	58	3	3	NUM
bibechana-10399	15	59	�	�	PROPN
bibechana-10399	15	60	�	�	PROPN
bibechana-10399	15	61	here	here	ADV
bibechana-10399	15	62	k(s	k(s	PROPN
bibechana-10399	15	63	,	,	PUNCT
bibechana-10399	15	64	t	t	PROPN
bibechana-10399	15	65	)	)	PUNCT
bibechana-10399	15	66	is	be	AUX
bibechana-10399	15	67	a	a	DET
bibechana-10399	15	68	continuous	continuous	ADJ
bibechana-10399	15	69	function	function	NOUN
bibechana-10399	15	70	on	on	ADP
bibechana-10399	15	71	[	[	X
bibechana-10399	15	72	a	a	X
bibechana-10399	15	73	,	,	PUNCT
bibechana-10399	15	74	b]×[a	b]×[a	PROPN
bibechana-10399	15	75	,	,	PUNCT
bibechana-10399	15	76	b	b	NOUN
bibechana-10399	15	77	]	]	PUNCT
bibechana-10399	15	78	and	and	CCONJ
bibechana-10399	15	79	is	be	AUX
bibechana-10399	15	80	called	call	VERB
bibechana-10399	15	81	the	the	DET
bibechana-10399	15	82	kernel	kernel	NOUN
bibechana-10399	15	83	of	of	ADP
bibechana-10399	15	84	the	the	DET
bibechana-10399	15	85	integral	integral	ADJ
bibechana-10399	15	86	equation	equation	NOUN
bibechana-10399	15	87	.	.	PUNCT
bibechana-10399	16	1	y(s	y(s	PROPN
bibechana-10399	16	2	)	)	PUNCT
bibechana-10399	16	3	is	be	AUX
bibechana-10399	16	4	continuous	continuous	ADJ
bibechana-10399	16	5	on	on	ADP
bibechana-10399	16	6	[	[	X
bibechana-10399	16	7	a	a	DET
bibechana-10399	16	8	,	,	PUNCT
bibechana-10399	16	9	b	b	NOUN
bibechana-10399	16	10	]	]	PUNCT
bibechana-10399	16	11	and	and	CCONJ
bibechana-10399	16	12	therefore	therefore	ADV
bibechana-10399	16	13	yϵc[a	yϵc[a	PROPN
bibechana-10399	16	14	,	,	PUNCT
bibechana-10399	16	15	b	b	NOUN
bibechana-10399	16	16	]	]	X
bibechana-10399	16	17	.	.	PUNCT
bibechana-10399	17	1	the	the	DET
bibechana-10399	17	2	equation(3	equation(3	PROPN
bibechana-10399	17	3	)	)	PUNCT
bibechana-10399	17	4	is	be	AUX
bibechana-10399	17	5	known	know	VERB
bibechana-10399	17	6	as	as	ADP
bibechana-10399	17	7	fredholm	fredholm	NOUN
bibechana-10399	17	8	type	type	NOUN
bibechana-10399	17	9	integral	integral	ADJ
bibechana-10399	17	10	equation	equation	NOUN
bibechana-10399	17	11	of	of	ADP
bibechana-10399	17	12	the	the	DET
bibechana-10399	17	13	second	second	ADJ
bibechana-10399	17	14	kind	kind	NOUN
bibechana-10399	17	15	[	[	X
bibechana-10399	17	16	5	5	NUM
bibechana-10399	17	17	]	]	PUNCT
bibechana-10399	17	18	.	.	PUNCT
bibechana-10399	18	1	equations	equation	NOUN
bibechana-10399	18	2	of	of	ADP
bibechana-10399	18	3	this	this	DET
bibechana-10399	18	4	sort	sort	NOUN
bibechana-10399	18	5	are	be	AUX
bibechana-10399	18	6	of	of	ADP
bibechana-10399	18	7	great	great	ADJ
bibechana-10399	18	8	importance	importance	NOUN
bibechana-10399	18	9	.	.	PUNCT
bibechana-10399	19	1	the	the	DET
bibechana-10399	19	2	application	application	NOUN
bibechana-10399	19	3	of	of	ADP
bibechana-10399	19	4	this	this	DET
bibechana-10399	19	5	operator	operator	NOUN
bibechana-10399	19	6	plays	play	VERB
bibechana-10399	19	7	a	a	DET
bibechana-10399	19	8	vital	vital	ADJ
bibechana-10399	19	9	role	role	NOUN
bibechana-10399	19	10	in	in	ADP
bibechana-10399	19	11	the	the	DET
bibechana-10399	19	12	theory	theory	NOUN
bibechana-10399	19	13	of	of	ADP
bibechana-10399	19	14	boundary	boundary	ADJ
bibechana-10399	19	15	value	value	NOUN
bibechana-10399	19	16	problems	problem	NOUN
bibechana-10399	19	17	in	in	ADP
bibechana-10399	19	18	differential	differential	ADJ
bibechana-10399	19	19	equations	equation	NOUN
bibechana-10399	19	20	.	.	PUNCT
bibechana-10399	20	1	fredholm	fredholm	PROPN
bibechana-10399	20	2	studied	study	VERB
bibechana-10399	20	3	fredholm	fredholm	NOUN
bibechana-10399	20	4	type	type	NOUN
bibechana-10399	20	5	integral	integral	ADJ
bibechana-10399	20	6	equations	equation	NOUN
bibechana-10399	20	7	of	of	ADP
bibechana-10399	20	8	the	the	DET
bibechana-10399	20	9	second	second	ADJ
bibechana-10399	20	10	kind	kind	NOUN
bibechana-10399	20	11	,	,	PUNCT
bibechana-10399	20	12	which	which	PRON
bibechana-10399	20	13	gave	give	VERB
bibechana-10399	20	14	rise	rise	NOUN
bibechana-10399	20	15	to	to	ADP
bibechana-10399	20	16	such	such	ADJ
bibechana-10399	20	17	operators	operator	NOUN
bibechana-10399	20	18	.	.	PUNCT
bibechana-10399	21	1	in	in	ADP
bibechana-10399	21	2	view	view	NOUN
bibechana-10399	21	3	of	of	ADP
bibechana-10399	21	4	the	the	DET
bibechana-10399	21	5	development	development	NOUN
bibechana-10399	21	6	of	of	ADP
bibechana-10399	21	7	the	the	DET
bibechana-10399	21	8	theory	theory	NOUN
bibechana-10399	21	9	of	of	ADP
bibechana-10399	21	10	fredholm	fredholm	NOUN
bibechana-10399	21	11	operators	operator	NOUN
bibechana-10399	21	12	the	the	DET
bibechana-10399	21	13	following	follow	VERB
bibechana-10399	21	14	definitions	definition	NOUN
bibechana-10399	21	15	are	be	AUX
bibechana-10399	21	16	frequently	frequently	ADV
bibechana-10399	21	17	used	use	VERB
bibechana-10399	21	18	.	.	PUNCT
bibechana-10399	22	1	definitions	definition	NOUN
bibechana-10399	22	2	fredholm	fredholm	NOUN
bibechana-10399	22	3	operator	operator	NOUN
bibechana-10399	22	4	(	(	PUNCT
bibechana-10399	22	5	i	i	NOUN
bibechana-10399	22	6	)	)	PUNCT
bibechana-10399	22	7	a	a	DET
bibechana-10399	22	8	closed	close	VERB
bibechana-10399	22	9	linear	linear	NOUN
bibechana-10399	22	10	operator	operator	NOUN
bibechana-10399	22	11	which	which	PRON
bibechana-10399	22	12	has	have	VERB
bibechana-10399	22	13	a	a	DET
bibechana-10399	22	14	finite	finite	ADJ
bibechana-10399	22	15	index	index	NOUN
bibechana-10399	22	16	is	be	AUX
bibechana-10399	22	17	called	call	VERB
bibechana-10399	22	18	a	a	DET
bibechana-10399	22	19	fredholm	fredholm	NOUN
bibechana-10399	22	20	operator	operator	NOUN
bibechana-10399	22	21	.	.	PUNCT
bibechana-10399	23	1	(	(	PUNCT
bibechana-10399	23	2	ii	ii	NOUN
bibechana-10399	23	3	)	)	PUNCT
bibechana-10399	23	4	let	let	VERB
bibechana-10399	23	5	x	x	PRON
bibechana-10399	23	6	and	and	CCONJ
bibechana-10399	23	7	y	y	PROPN
bibechana-10399	23	8	are	be	AUX
bibechana-10399	23	9	banach	banach	ADV
bibechana-10399	23	10	spaces	space	NOUN
bibechana-10399	23	11	.	.	PUNCT
bibechana-10399	24	1	a	a	DET
bibechana-10399	24	2	linear	linear	ADJ
bibechana-10399	24	3	operator	operator	NOUN
bibechana-10399	24	4	t	t	NOUN
bibechana-10399	24	5	from	from	ADP
bibechana-10399	24	6	x	x	PUNCT
bibechana-10399	24	7	to	to	ADP
bibechana-10399	24	8	y	y	PROPN
bibechana-10399	24	9	is	be	AUX
bibechana-10399	24	10	called	call	VERB
bibechana-10399	24	11	a	a	DET
bibechana-10399	24	12	fredholmoperator	fredholmoperator	NOUN
bibechana-10399	24	13	if	if	SCONJ
bibechana-10399	24	14	i.	i.	PROPN
bibechana-10399	24	15	t	t	PROPN
bibechana-10399	24	16	is	be	AUX
bibechana-10399	24	17	closed	close	VERB
bibechana-10399	24	18	.	.	PUNCT
bibechana-10399	25	1	ii	ii	X
bibechana-10399	25	2	.	.	PUNCT
bibechana-10399	26	1	the	the	DET
bibechana-10399	26	2	domain	domain	NOUN
bibechana-10399	26	3	of	of	ADP
bibechana-10399	26	4	t	t	PROPN
bibechana-10399	26	5	is	be	AUX
bibechana-10399	26	6	dense	dense	ADJ
bibechana-10399	26	7	in	in	ADP
bibechana-10399	26	8	x.	x.	PROPN
bibechana-10399	26	9	iii	iii	PROPN
bibechana-10399	26	10	.	.	PUNCT
bibechana-10399	26	11	α(t	α(t	PROPN
bibechana-10399	26	12	)	)	PUNCT
bibechana-10399	26	13	,	,	PUNCT
bibechana-10399	26	14	the	the	DET
bibechana-10399	26	15	dimension	dimension	NOUN
bibechana-10399	26	16	of	of	ADP
bibechana-10399	26	17	the	the	DET
bibechana-10399	26	18	null	null	ADJ
bibechana-10399	26	19	space	space	NOUN
bibechana-10399	26	20	n(t	n(t	PROPN
bibechana-10399	26	21	)	)	PUNCT
bibechana-10399	26	22	of	of	ADP
bibechana-10399	26	23	t	t	PROPN
bibechana-10399	26	24	is	be	AUX
bibechana-10399	26	25	finite	finite	ADJ
bibechana-10399	26	26	.	.	PUNCT
bibechana-10399	27	1	iv	iv	X
bibechana-10399	27	2	.	.	PUNCT
bibechana-10399	28	1	the	the	DET
bibechana-10399	28	2	range	range	NOUN
bibechana-10399	28	3	of	of	ADP
bibechana-10399	28	4	t	t	PROPN
bibechana-10399	28	5	is	be	AUX
bibechana-10399	28	6	closed	close	VERB
bibechana-10399	28	7	in	in	ADP
bibechana-10399	28	8	y	y	PROPN
bibechana-10399	28	9	v.	v.	ADP
bibechana-10399	28	10	b(t	b(t	PROPN
bibechana-10399	28	11	)	)	PUNCT
bibechana-10399	28	12	,	,	PUNCT
bibechana-10399	28	13	the	the	DET
bibechana-10399	28	14	co	co	NOUN
bibechana-10399	28	15	-	-	NOUN
bibechana-10399	28	16	dimension	dimension	NOUN
bibechana-10399	28	17	of	of	ADP
bibechana-10399	28	18	r(t	r(t	NOUN
bibechana-10399	28	19	)	)	PUNCT
bibechana-10399	28	20	in	in	ADP
bibechana-10399	28	21	y	y	PROPN
bibechana-10399	28	22	is	be	AUX
bibechana-10399	28	23	finite	finite	ADJ
bibechana-10399	28	24	.	.	PUNCT
bibechana-10399	29	1	the	the	DET
bibechana-10399	29	2	terminology	terminology	NOUN
bibechana-10399	29	3	stems	stem	VERB
bibechana-10399	29	4	from	from	ADP
bibechana-10399	29	5	the	the	DET
bibechana-10399	29	6	classical	classical	ADJ
bibechana-10399	29	7	theory	theory	NOUN
bibechana-10399	29	8	of	of	ADP
bibechana-10399	29	9	integral	integral	ADJ
bibechana-10399	29	10	equations	equation	NOUN
bibechana-10399	29	11	.	.	PUNCT
bibechana-10399	30	1	special	special	ADJ
bibechana-10399	30	2	types	type	NOUN
bibechana-10399	30	3	of	of	ADP
bibechana-10399	30	4	fredholm	fredholm	NOUN
bibechana-10399	30	5	operators	operator	NOUN
bibechana-10399	30	6	were	be	AUX
bibechana-10399	30	7	considered	consider	VERB
bibechana-10399	30	8	by	by	ADP
bibechana-10399	30	9	many	many	ADJ
bibechana-10399	30	10	authors	author	NOUN
bibechana-10399	30	11	since	since	SCONJ
bibechana-10399	30	12	that	that	DET
bibechana-10399	30	13	time	time	NOUN
bibechana-10399	30	14	but	but	CCONJ
bibechana-10399	30	15	systematic	systematic	ADJ
bibechana-10399	30	16	treatment	treatment	NOUN
bibechana-10399	30	17	were	be	AUX
bibechana-10399	30	18	not	not	PART
bibechana-10399	30	19	given	give	VERB
bibechana-10399	30	20	until	until	ADP
bibechana-10399	30	21	the	the	DET
bibechana-10399	30	22	work	work	NOUN
bibechana-10399	30	23	of	of	ADP
bibechana-10399	30	24	atkimon	atkimon	PROPN
bibechana-10399	31	1	[	[	X
bibechana-10399	31	2	1	1	NUM
bibechana-10399	31	3	]	]	PUNCT
bibechana-10399	31	4	,	,	PUNCT
bibechana-10399	31	5	gohberg	gohberg	PROPN
bibechana-10399	32	1	[	[	X
bibechana-10399	32	2	4	4	X
bibechana-10399	32	3	]	]	PUNCT
bibechana-10399	32	4	and	and	CCONJ
bibechana-10399	32	5	yood	yood	NOUN
bibechana-10399	33	1	[	[	X
bibechana-10399	33	2	9	9	NUM
bibechana-10399	33	3	]	]	PUNCT
bibechana-10399	33	4	.	.	PUNCT
bibechana-10399	34	1	a	a	DET
bibechana-10399	34	2	general	general	ADJ
bibechana-10399	34	3	account	account	NOUN
bibechana-10399	34	4	of	of	ADP
bibechana-10399	34	5	the	the	DET
bibechana-10399	34	6	history	history	NOUN
bibechana-10399	34	7	of	of	ADP
bibechana-10399	34	8	the	the	DET
bibechana-10399	34	9	theory	theory	NOUN
bibechana-10399	34	10	is	be	AUX
bibechana-10399	34	11	given	give	VERB
bibechana-10399	34	12	by	by	ADP
bibechana-10399	34	13	[	[	X
bibechana-10399	34	14	a]gohberg	a]gohberg	PROPN
bibechana-10399	34	15	krein	krein	NOUN
bibechana-10399	35	1	[	[	X
bibechana-10399	35	2	3	3	NUM
bibechana-10399	35	3	]	]	PUNCT
bibechana-10399	35	4	and	and	CCONJ
bibechana-10399	35	5	(	(	PUNCT
bibechana-10399	35	6	b	b	X
bibechana-10399	35	7	)	)	PUNCT
bibechana-10399	35	8	kato	kato	PROPN
bibechana-10399	36	1	[	[	X
bibechana-10399	36	2	5	5	NUM
bibechana-10399	36	3	]	]	PUNCT
bibechana-10399	36	4	.	.	PUNCT
bibechana-10399	37	1	for	for	ADP
bibechana-10399	37	2	a	a	DET
bibechana-10399	37	3	good	good	ADJ
bibechana-10399	37	4	general	general	ADJ
bibechana-10399	37	5	account	account	NOUN
bibechana-10399	37	6	of	of	ADP
bibechana-10399	37	7	the	the	DET
bibechana-10399	37	8	theory	theory	NOUN
bibechana-10399	37	9	can	can	AUX
bibechana-10399	37	10	be	be	AUX
bibechana-10399	37	11	found	find	VERB
bibechana-10399	37	12	in	in	ADP
bibechana-10399	37	13	the	the	DET
bibechana-10399	37	14	book	book	NOUN
bibechana-10399	37	15	written	write	VERB
bibechana-10399	37	16	by	by	ADP
bibechana-10399	37	17	gohberg	gohberg	PROPN
bibechana-10399	38	1	[	[	X
bibechana-10399	38	2	4	4	NUM
bibechana-10399	38	3	]	]	PUNCT
bibechana-10399	38	4	.	.	PUNCT
bibechana-10399	39	1	definition(3	definition(3	X
bibechana-10399	39	2	)	)	PUNCT
bibechana-10399	40	1	let	let	VERB
bibechana-10399	40	2	b	b	NOUN
bibechana-10399	40	3	&	&	CCONJ
bibechana-10399	40	4	c	c	PROPN
bibechana-10399	40	5	be	be	AUX
bibechana-10399	40	6	two	two	NUM
bibechana-10399	40	7	banach	banach	NOUN
bibechana-10399	40	8	spaces	space	VERB
bibechana-10399	40	9	.	.	PUNCT
bibechana-10399	41	1	a	a	DET
bibechana-10399	41	2	bounded	bounded	ADJ
bibechana-10399	41	3	linear	linear	ADJ
bibechana-10399	41	4	operator	operator	NOUN
bibechana-10399	41	5	a	a	DET
bibechana-10399	41	6	:	:	PUNCT
bibechana-10399	41	7	b	b	X
bibechana-10399	41	8	→	→	SYM
bibechana-10399	41	9	c	c	PROPN
bibechana-10399	41	10	is	be	AUX
bibechana-10399	41	11	defined	define	VERB
bibechana-10399	41	12	to	to	PART
bibechana-10399	41	13	be	be	AUX
bibechana-10399	41	14	a	a	DET
bibechana-10399	41	15	linear	linear	ADJ
bibechana-10399	41	16	map	map	NOUN
bibechana-10399	41	17	for	for	ADP
bibechana-10399	41	18	which	which	PRON
bibechana-10399	41	19	the	the	DET
bibechana-10399	41	20	norm	norm	NOUN
bibechana-10399	41	21	∥	∥	PUNCT
bibechana-10399	41	22	a	a	DET
bibechana-10399	41	23	∥∶=	∥∶=	NUM
bibechana-10399	41	24	sup	sup	NOUN
bibechana-10399	41	25	�	�	NOUN
bibechana-10399	41	26	∥	∥	NOUN
bibechana-10399	41	27	af	af	VERB
bibechana-10399	42	1	∥∶	∥∶	ADV
bibechana-10399	42	2	∥	∥	PUNCT
bibechana-10399	42	3	f	f	X
bibechana-10399	42	4	∥≤	∥≤	PROPN
bibechana-10399	42	5	1	1	NUM
bibechana-10399	42	6	�	�	PROPN
bibechana-10399	42	7	is	be	AUX
bibechana-10399	42	8	finite	finite	ADJ
bibechana-10399	42	9	.	.	PUNCT
bibechana-10399	43	1	definition(4	definition(4	PROPN
bibechana-10399	43	2	)	)	PUNCT
bibechana-10399	44	1	let	let	VERB
bibechana-10399	44	2	x	x	PRON
bibechana-10399	44	3	and	and	CCONJ
bibechana-10399	44	4	y	y	PROPN
bibechana-10399	44	5	be	be	AUX
bibechana-10399	44	6	normed	norme	VERB
bibechana-10399	44	7	linear	linear	ADJ
bibechana-10399	44	8	spaces	space	NOUN
bibechana-10399	44	9	.	.	PUNCT
bibechana-10399	45	1	suppose	suppose	VERB
bibechana-10399	45	2	t	t	PROPN
bibechana-10399	45	3	is	be	AUX
bibechana-10399	45	4	a	a	DET
bibechana-10399	45	5	linear	linear	ADJ
bibechana-10399	45	6	operator	operator	NOUN
bibechana-10399	45	7	with	with	ADP
bibechana-10399	45	8	domain	domain	NOUN
bibechana-10399	45	9	x	x	PUNCT
bibechana-10399	45	10	and	and	CCONJ
bibechana-10399	45	11	range	range	VERB
bibechana-10399	45	12	in	in	ADP
bibechana-10399	45	13	y.	y.	PROPN
bibechana-10399	45	14	we	we	PRON
bibechana-10399	45	15	say	say	VERB
bibechana-10399	45	16	that	that	SCONJ
bibechana-10399	45	17	t	t	PROPN
bibechana-10399	45	18	is	be	AUX
bibechana-10399	45	19	compact	compact	ADJ
bibechana-10399	45	20	if	if	SCONJ
bibechana-10399	45	21	for	for	ADP
bibechana-10399	45	22	each	each	DET
bibechana-10399	45	23	bounded	bound	VERB
bibechana-10399	45	24	sequence	sequence	NOUN
bibechana-10399	45	25	{	{	PUNCT
bibechana-10399	45	26	xn	xn	NOUN
bibechana-10399	45	27	}	}	PUNCT
bibechana-10399	45	28	in	in	ADP
bibechana-10399	45	29	x	x	PRON
bibechana-10399	45	30	,	,	PUNCT
bibechana-10399	45	31	the	the	DET
bibechana-10399	45	32	sequence	sequence	NOUN
bibechana-10399	45	33	{	{	PUNCT
bibechana-10399	45	34	t	t	NOUN
bibechana-10399	45	35	xn	xn	PROPN
bibechana-10399	45	36	}	}	PUNCT
bibechana-10399	45	37	contains	contain	VERB
bibechana-10399	45	38	a	a	DET
bibechana-10399	45	39	sub	sub	NOUN
bibechana-10399	45	40	sequence	sequence	NOUN
bibechana-10399	45	41	converging	converge	VERB
bibechana-10399	45	42	to	to	ADP
bibechana-10399	45	43	some	some	DET
bibechana-10399	45	44	limit	limit	NOUN
bibechana-10399	45	45	in	in	ADP
bibechana-10399	45	46	y.	y.	PROPN
bibechana-10399	45	47	a	a	DET
bibechana-10399	45	48	compact	compact	ADJ
bibechana-10399	45	49	operator	operator	NOUN
bibechana-10399	45	50	is	be	AUX
bibechana-10399	45	51	also	also	ADV
bibechana-10399	45	52	called	call	VERB
bibechana-10399	45	53	completely	completely	ADV
bibechana-10399	45	54	continuous	continuous	ADJ
bibechana-10399	45	55	.	.	PUNCT
bibechana-10399	46	1	lemma	lemma	PROPN
bibechana-10399	46	2	(	(	PUNCT
bibechana-10399	46	3	1	1	X
bibechana-10399	46	4	)	)	PUNCT
bibechana-10399	46	5	if	if	SCONJ
bibechana-10399	46	6	a	a	PRON
bibechana-10399	46	7	is	be	AUX
bibechana-10399	46	8	a	a	DET
bibechana-10399	46	9	compact	compact	ADJ
bibechana-10399	46	10	operator	operator	NOUN
bibechana-10399	46	11	on	on	ADP
bibechana-10399	46	12	b	b	NOUN
bibechana-10399	46	13	,	,	PUNCT
bibechana-10399	46	14	then	then	ADV
bibechana-10399	46	15	(	(	PUNCT
bibechana-10399	46	16	λia	λia	NOUN
bibechana-10399	46	17	)	)	PUNCT
bibechana-10399	46	18	is	be	AUX
bibechana-10399	46	19	fredholm	fredholm	NOUN
bibechana-10399	46	20	for	for	ADP
bibechana-10399	46	21	all	all	DET
bibechana-10399	46	22	λ	λ	PROPN
bibechana-10399	46	23	≠	≠	PROPN
bibechana-10399	46	24	0	0	NUM
bibechana-10399	46	25	.	.	PUNCT
bibechana-10399	47	1	proof	proof	NOUN
bibechana-10399	47	2	:	:	PUNCT
bibechana-10399	47	3	we	we	PRON
bibechana-10399	47	4	first	first	ADV
bibechana-10399	47	5	prove	prove	VERB
bibechana-10399	47	6	that	that	SCONJ
bibechana-10399	47	7	ℒ≔ker(λi	ℒ≔ker(λi	PRON
bibechana-10399	47	8	a	a	X
bibechana-10399	47	9	)	)	PUNCT
bibechana-10399	47	10	is	be	AUX
bibechana-10399	47	11	finite	finite	ADJ
bibechana-10399	47	12	dimensional	dimensional	ADJ
bibechana-10399	47	13	by	by	ADP
bibechana-10399	47	14	contradiction	contradiction	NOUN
bibechana-10399	47	15	.	.	PUNCT
bibechana-10399	48	1	if	if	SCONJ
bibechana-10399	48	2	this	this	PRON
bibechana-10399	48	3	were	be	AUX
bibechana-10399	48	4	not	not	PART
bibechana-10399	48	5	the	the	DET
bibechana-10399	48	6	case	case	NOUN
bibechana-10399	48	7	there	there	PRON
bibechana-10399	48	8	would	would	AUX
bibechana-10399	48	9	exist	exist	VERB
bibechana-10399	48	10	an	an	DET
bibechana-10399	48	11	infinite	infinite	ADJ
bibechana-10399	48	12	sequence	sequence	NOUN
bibechana-10399	48	13	xnєℒ	xnєℒ	SCONJ
bibechana-10399	48	14	such	such	ADJ
bibechana-10399	48	15	that	that	SCONJ
bibechana-10399	48	16	∥	∥	PROPN
bibechana-10399	48	17	x	x	SYM
bibechana-10399	48	18	�	�	NOUN
bibechana-10399	48	19	∥	∥	X
bibechana-10399	48	20	=	=	SYM
bibechana-10399	48	21	1	1	NUM
bibechana-10399	48	22	and	and	CCONJ
bibechana-10399	48	23	∥	∥	NUM
bibechana-10399	48	24	x	x	SYM
bibechana-10399	48	25	−	−	NOUN
bibechana-10399	48	26	x	x	X
bibechana-10399	48	27	�	�	PROPN
bibechana-10399	48	28	∥	∥	X
bibechana-10399	48	29	≥	≥	NOUN
bibechana-10399	48	30	1/2	1/2	NUM
bibechana-10399	48	31	for	for	ADP
bibechana-10399	48	32	all	all	DET
bibechana-10399	48	33	distinct	distinct	ADJ
bibechana-10399	48	34	m	m	NOUN
bibechana-10399	48	35	and	and	CCONJ
bibechana-10399	48	36	n.	n.	NOUN
bibechana-10399	48	37	since	since	SCONJ
bibechana-10399	48	38	axn	axn	PROPN
bibechana-10399	48	39	=	=	PUNCT
bibechana-10399	48	40	λxn	λxn	NOUN
bibechana-10399	48	41	and	and	CCONJ
bibechana-10399	48	42	λ	λ	X
bibechana-10399	48	43	≠	≠	PROPN
bibechana-10399	48	44	0	0	NUM
bibechana-10399	48	45	,	,	PUNCT
bibechana-10399	48	46	we	we	PRON
bibechana-10399	48	47	could	could	AUX
bibechana-10399	48	48	conclude	conclude	VERB
bibechana-10399	48	49	that	that	SCONJ
bibechana-10399	48	50	axn	axn	PROPN
bibechana-10399	48	51	has	have	VERB
bibechana-10399	48	52	no	no	DET
bibechana-10399	48	53	convergent	convergent	NOUN
bibechana-10399	48	54	subsequence	subsequence	NOUN
bibechana-10399	48	55	.	.	PUNCT
bibechana-10399	49	1	we	we	PRON
bibechana-10399	49	2	can	can	AUX
bibechana-10399	49	3	write	write	VERB
bibechana-10399	49	4	b=ℒ	b=ℒ	PROPN
bibechana-10399	50	1	+	+	CCONJ
bibechana-10399	50	2	m	m	PROPN
bibechana-10399	50	3	,	,	PUNCT
bibechana-10399	50	4	where	where	SCONJ
bibechana-10399	50	5	ℒ	ℒ	ADJ
bibechana-10399	50	6	∩	∩	NOUN
bibechana-10399	50	7	m	m	VERB
bibechana-10399	50	8	=	=	PUNCT
bibechana-10399	50	9	{	{	PUNCT
bibechana-10399	50	10	0}and	0}and	NUM
bibechana-10399	50	11	m	m	NOUN
bibechana-10399	50	12	is	be	AUX
bibechana-10399	50	13	a	a	DET
bibechana-10399	50	14	closed	closed	ADJ
bibechana-10399	50	15	linear	linear	ADJ
bibechana-10399	50	16	subspace	subspace	NOUN
bibechana-10399	50	17	on	on	ADP
bibechana-10399	50	18	which	which	PRON
bibechana-10399	50	19	(	(	PUNCT
bibechana-10399	50	20	λi	λi	ADP
bibechana-10399	50	21	a	a	NOUN
bibechana-10399	50	22	)	)	PUNCT
bibechana-10399	50	23	is	be	AUX
bibechana-10399	50	24	one	one	NUM
bibechana-10399	50	25	-	-	PUNCT
bibechana-10399	50	26	one	one	NUM
bibechana-10399	50	27	.	.	PUNCT
bibechana-10399	51	1	we	we	PRON
bibechana-10399	51	2	next	next	ADV
bibechana-10399	51	3	prove	prove	VERB
bibechana-10399	51	4	that	that	SCONJ
bibechana-10399	51	5	ℛ≔ran(λi	ℛ≔ran(λi	NOUN
bibechana-10399	51	6	-	-	PUNCT
bibechana-10399	51	7	a	a	NOUN
bibechana-10399	51	8	)	)	PUNCT
bibechana-10399	51	9	is	be	AUX
bibechana-10399	51	10	closed	closed	ADJ
bibechana-10399	51	11	.	.	PUNCT
bibechana-10399	52	1	if	if	SCONJ
bibechana-10399	52	2	gnє	gnє	VERB
bibechana-10399	52	3	ℛand	ℛand	PROPN
bibechana-10399	52	4	∥	∥	NOUN
bibechana-10399	52	5	#	#	SYM
bibechana-10399	52	6	$	$	SYM
bibechana-10399	52	7	−	−	NOUN
bibechana-10399	52	8	#	#	NOUN
bibechana-10399	52	9	∥	∥	NOUN
bibechana-10399	52	10	→0	→0	NUM
bibechana-10399	52	11	,	,	PUNCT
bibechana-10399	52	12	then	then	ADV
bibechana-10399	52	13	there	there	PRON
bibechana-10399	52	14	exist	exist	VERB
bibechana-10399	52	15	fn	fn	PROPN
bibechana-10399	52	16	є	є	ADV
bibechana-10399	52	17	m	m	VERB
bibechana-10399	52	18	such	such	ADJ
bibechana-10399	52	19	that	that	DET
bibechana-10399	52	20	gn=	gn=	NOUN
bibechana-10399	52	21	(	(	PUNCT
bibechana-10399	52	22	λia)fn	λia)fn	VERB
bibechana-10399	52	23	.	.	PUNCT
bibechana-10399	53	1	if	if	SCONJ
bibechana-10399	53	2	∥	∥	PRON
bibechana-10399	53	3	f	f	X
bibechana-10399	53	4	�	�	PROPN
bibechana-10399	53	5	∥	∥	PROPN
bibechana-10399	53	6	is	be	AUX
bibechana-10399	53	7	not	not	PART
bibechana-10399	53	8	a	a	DET
bibechana-10399	53	9	bounded	bounded	ADJ
bibechana-10399	53	10	sequence	sequence	NOUN
bibechana-10399	53	11	then	then	ADV
bibechana-10399	53	12	by	by	ADP
bibechana-10399	53	13	passing	pass	VERB
bibechana-10399	53	14	to	to	ADP
bibechana-10399	53	15	a	a	DET
bibechana-10399	53	16	subsequence	subsequence	NOUN
bibechana-10399	53	17	(	(	PUNCT
bibechana-10399	53	18	without	without	ADP
bibechana-10399	53	19	m.	m.	NOUN
bibechana-10399	53	20	sahi	sahi	PROPN
bibechana-10399	53	21	/	/	SYM
bibechana-10399	53	22	bibechana	bibechana	PROPN
bibechana-10399	53	23	11(1	11(1	NUM
bibechana-10399	53	24	)	)	PUNCT
bibechana-10399	53	25	(	(	PUNCT
bibechana-10399	53	26	2014	2014	NUM
bibechana-10399	53	27	)	)	PUNCT
bibechana-10399	53	28	169	169	NUM
bibechana-10399	53	29	-	-	SYM
bibechana-10399	53	30	174	174	NUM
bibechana-10399	53	31	:	:	PUNCT
bibechana-10399	53	32	(	(	PUNCT
bibechana-10399	53	33	online	online	ADJ
bibechana-10399	53	34	publication	publication	NOUN
bibechana-10399	53	35	:	:	PUNCT
bibechana-10399	53	36	march	march	PROPN
bibechana-10399	53	37	,	,	PUNCT
bibechana-10399	53	38	2014	2014	NUM
bibechana-10399	53	39	)	)	PUNCT
bibechana-10399	53	40	p.171	p.171	NOUN
bibechana-10399	53	41	change	change	NOUN
bibechana-10399	53	42	of	of	ADP
bibechana-10399	53	43	notation	notation	NOUN
bibechana-10399	53	44	)	)	PUNCT
bibechana-10399	53	45	we	we	PRON
bibechana-10399	53	46	may	may	AUX
bibechana-10399	53	47	assume	assume	VERB
bibechana-10399	53	48	that	that	SCONJ
bibechana-10399	53	49	∥	∥	PROPN
bibechana-10399	53	50	f	f	X
bibechana-10399	53	51	�	�	NOUN
bibechana-10399	53	52	∥	∥	NUM
bibechana-10399	53	53	→	→	SYM
bibechana-10399	53	54	∞	∞	PROPN
bibechana-10399	53	55	as	as	ADP
bibechana-10399	53	56	n	n	PROPN
bibechana-10399	53	57	→	→	SYM
bibechana-10399	53	58	∞.	∞.	PROPN
bibechana-10399	53	59	putting	put	VERB
bibechana-10399	53	60	hn≔	hn≔	ADV
bibechana-10399	53	61	fn/∥	fn/∥	NOUN
bibechana-10399	53	62	%	%	INTJ
bibechana-10399	53	63	$	$	SYM
bibechana-10399	53	64	∥we	∥we	NUM
bibechana-10399	53	65	have∥	have∥	X
bibechana-10399	53	66	ℎ$	ℎ$	NOUN
bibechana-10399	53	67	∥	∥	PUNCT
bibechana-10399	54	1	=	=	SYM
bibechana-10399	54	2	1	1	NUM
bibechana-10399	54	3	and	and	CCONJ
bibechana-10399	54	4	kn≔	kn≔	X
bibechana-10399	54	5	(	(	PUNCT
bibechana-10399	54	6	λi	λi	ADP
bibechana-10399	54	7	a)hn	a)hn	PROPN
bibechana-10399	54	8	→	→	SYM
bibechana-10399	54	9	0	0	NUM
bibechana-10399	54	10	.	.	PUNCT
bibechana-10399	55	1	the	the	DET
bibechana-10399	55	2	compactness	compactness	NOUN
bibechana-10399	55	3	of	of	ADP
bibechana-10399	55	4	a	a	DET
bibechana-10399	55	5	implies	implie	NOUN
bibechana-10399	55	6	that	that	SCONJ
bibechana-10399	55	7	h	h	NOUN
bibechana-10399	55	8	n	n	NOUN
bibechana-10399	55	9	=	=	SYM
bibechana-10399	56	1	λ-1(ahn	λ-1(ahn	PROPN
bibechana-10399	56	2	+	+	NUM
bibechana-10399	56	3	kn	kn	PROPN
bibechana-10399	56	4	)	)	PUNCT
bibechana-10399	56	5	has	have	VERB
bibechana-10399	56	6	a	a	DET
bibechana-10399	56	7	convergent	convergent	NOUN
bibechana-10399	56	8	subsequence	subsequence	NOUN
bibechana-10399	56	9	.	.	PUNCT
bibechana-10399	57	1	passing	pass	VERB
bibechana-10399	57	2	to	to	ADP
bibechana-10399	57	3	this	this	DET
bibechana-10399	57	4	subsequence	subsequence	NOUN
bibechana-10399	57	5	we	we	PRON
bibechana-10399	57	6	have	have	VERB
bibechana-10399	57	7	hn	hn	PRON
bibechana-10399	57	8	→	→	PUNCT
bibechana-10399	57	9	hwhere	hwhere	ADJ
bibechana-10399	57	10	∥	∥	NOUN
bibechana-10399	57	11	ℎ	ℎ	X
bibechana-10399	57	12	∥	∥	PUNCT
bibechana-10399	57	13	=	=	SYM
bibechana-10399	57	14	1	1	NUM
bibechana-10399	57	15	,	,	PUNCT
bibechana-10399	57	16	h	h	NOUN
bibechana-10399	57	17	є	є	PROPN
bibechana-10399	57	18	m	m	NOUN
bibechana-10399	57	19	,	,	PUNCT
bibechana-10399	57	20	andh	andh	NOUN
bibechana-10399	57	21	=	=	SYM
bibechana-10399	57	22	λ-1ah	λ-1ah	X
bibechana-10399	57	23	.	.	PUNCT
bibechana-10399	58	1	we	we	PRON
bibechana-10399	58	2	conclude	conclude	VERB
bibechana-10399	58	3	that	that	PRON
bibechana-10399	58	4	h	h	PROPN
bibechana-10399	58	5	є	є	ADP
bibechana-10399	58	6	m	m	PROPN
bibechana-10399	58	7	∩	∩	ADJ
bibechana-10399	58	8	ℒ.	ℒ.	NOUN
bibechana-10399	58	9	the	the	DET
bibechana-10399	58	10	contradiction	contradiction	NOUN
bibechana-10399	58	11	implies	imply	VERB
bibechana-10399	58	12	that	that	SCONJ
bibechana-10399	58	13	∥	∥	NUM
bibechana-10399	58	14	%	%	NOUN
bibechana-10399	58	15	$	$	SYM
bibechana-10399	58	16	∥	∥	X
bibechana-10399	58	17	is	be	AUX
bibechana-10399	58	18	a	a	DET
bibechana-10399	58	19	bounded	bounded	ADJ
bibechana-10399	58	20	sequence	sequence	NOUN
bibechana-10399	58	21	.	.	PUNCT
bibechana-10399	59	1	given	give	VERB
bibechana-10399	59	2	this	this	DET
bibechana-10399	59	3	fact	fact	NOUN
bibechana-10399	59	4	the	the	DET
bibechana-10399	59	5	compactness	compactness	NOUN
bibechana-10399	59	6	of	of	ADP
bibechana-10399	59	7	a	a	DET
bibechana-10399	59	8	implies	implie	NOUN
bibechana-10399	59	9	that	that	SCONJ
bibechana-10399	59	10	the	the	DET
bibechana-10399	59	11	sequence	sequence	NOUN
bibechana-10399	59	12	fn	fn	NOUN
bibechana-10399	59	13	=	=	PUNCT
bibechana-10399	59	14	λ-1(afn	λ-1(afn	X
bibechana-10399	59	15	+	+	CCONJ
bibechana-10399	59	16	gn	gn	X
bibechana-10399	59	17	)	)	PUNCT
bibechana-10399	59	18	has	have	VERB
bibechana-10399	59	19	a	a	DET
bibechana-10399	59	20	convergent	convergent	NOUN
bibechana-10399	59	21	subsequence	subsequence	NOUN
bibechana-10399	59	22	.	.	PUNCT
bibechana-10399	60	1	passing	pass	VERB
bibechana-10399	60	2	to	to	ADP
bibechana-10399	60	3	this	this	DET
bibechana-10399	60	4	subsequence	subsequence	NOUN
bibechana-10399	60	5	we	we	PRON
bibechana-10399	60	6	obtain	obtain	VERB
bibechana-10399	60	7	fn→fas	fn→fa	NOUN
bibechana-10399	60	8	n	n	CCONJ
bibechana-10399	60	9	→	→	SYM
bibechana-10399	60	10	∞	∞	PROPN
bibechana-10399	60	11	,	,	PUNCT
bibechana-10399	60	12	sof	sof	VERB
bibechana-10399	60	13	=	=	SYM
bibechana-10399	60	14	λ-1(af	λ-1(af	PROPN
bibechana-10399	61	1	+	+	NUM
bibechana-10399	61	2	g	g	NOUN
bibechana-10399	61	3	)	)	PUNCT
bibechana-10399	61	4	,	,	PUNCT
bibechana-10399	61	5	and	and	CCONJ
bibechana-10399	61	6	g	g	PROPN
bibechana-10399	61	7	=	=	SYM
bibechana-10399	61	8	(	(	PUNCT
bibechana-10399	61	9	λi	λi	ADP
bibechana-10399	61	10	a)f	a)f	ADJ
bibechana-10399	61	11	.	.	PUNCT
bibechana-10399	62	1	therefore	therefore	ADV
bibechana-10399	62	2	ℛ	ℛ	PROPN
bibechana-10399	62	3	is	be	AUX
bibechana-10399	62	4	closed	close	VERB
bibechana-10399	62	5	.	.	PUNCT
bibechana-10399	63	1	since	since	SCONJ
bibechana-10399	63	2	ran(λi	ran(λi	NOUN
bibechana-10399	63	3	a	a	PRON
bibechana-10399	63	4	)	)	PUNCT
bibechana-10399	63	5	is	be	AUX
bibechana-10399	63	6	closed	close	VERB
bibechana-10399	63	7	,	,	PUNCT
bibechana-10399	63	8	an	an	DET
bibechana-10399	63	9	application	application	NOUN
bibechana-10399	63	10	of	of	ADP
bibechana-10399	63	11	the	the	DET
bibechana-10399	63	12	hahn	hahn	NOUN
bibechana-10399	63	13	-	-	PUNCT
bibechana-10399	63	14	banach	banach	NOUN
bibechana-10399	63	15	theorem	theorem	NOUN
bibechana-10399	63	16	implies	imply	VERB
bibechana-10399	63	17	that	that	SCONJ
bibechana-10399	63	18	its	its	PRON
bibechana-10399	63	19	codimension	codimension	NOUN
bibechana-10399	63	20	equals	equal	VERB
bibechana-10399	63	21	the	the	DET
bibechana-10399	63	22	dimension	dimension	NOUN
bibechana-10399	63	23	of	of	ADP
bibechana-10399	63	24	ker(λi	ker(λi	PROPN
bibechana-10399	63	25	a	a	PROPN
bibechana-10399	63	26	*	*	NOUN
bibechana-10399	63	27	)	)	PUNCT
bibechana-10399	63	28	in	in	ADP
bibechana-10399	63	29	b	b	NOUN
bibechana-10399	63	30	*	*	PROPN
bibechana-10399	63	31	.	.	PUNCT
bibechana-10399	64	1	but	but	CCONJ
bibechana-10399	64	2	a	a	PRON
bibechana-10399	64	3	*	*	PUNCT
bibechana-10399	64	4	is	be	AUX
bibechana-10399	64	5	compact	compact	ADJ
bibechana-10399	64	6	,	,	PUNCT
bibechana-10399	64	7	so	so	CCONJ
bibechana-10399	64	8	this	this	PRON
bibechana-10399	64	9	is	be	AUX
bibechana-10399	64	10	finite	finite	ADJ
bibechana-10399	64	11	by	by	ADP
bibechana-10399	64	12	the	the	DET
bibechana-10399	64	13	first	first	ADJ
bibechana-10399	64	14	paragraph	paragraph	NOUN
bibechana-10399	64	15	.	.	PUNCT
bibechana-10399	65	1	our	our	PRON
bibechana-10399	65	2	next	next	ADJ
bibechana-10399	65	3	theorem	theorem	NOUN
bibechana-10399	65	4	provides	provide	VERB
bibechana-10399	65	5	a	a	DET
bibechana-10399	65	6	second	second	ADJ
bibechana-10399	65	7	characterization	characterization	NOUN
bibechana-10399	65	8	of	of	ADP
bibechana-10399	65	9	fredholm	fredholm	NOUN
bibechana-10399	65	10	operators	operator	NOUN
bibechana-10399	65	11	.	.	PUNCT
bibechana-10399	66	1	theorem	theorem	NOUN
bibechana-10399	66	2	(	(	PUNCT
bibechana-10399	66	3	1	1	NUM
bibechana-10399	66	4	)	)	PUNCT
bibechana-10399	66	5	every	every	DET
bibechana-10399	66	6	fredholm	fredholm	NOUN
bibechana-10399	66	7	operator	operator	NOUN
bibechana-10399	66	8	has	have	AUX
bibechana-10399	66	9	closed	close	VERB
bibechana-10399	66	10	range	range	NOUN
bibechana-10399	66	11	.	.	PUNCT
bibechana-10399	67	1	the	the	DET
bibechana-10399	67	2	bounded	bounded	ADJ
bibechana-10399	67	3	operator	operator	NOUN
bibechana-10399	67	4	a	a	DET
bibechana-10399	67	5	:	:	PUNCT
bibechana-10399	67	6	b	b	X
bibechana-10399	67	7	→	→	SYM
bibechana-10399	67	8	c	c	PROPN
bibechana-10399	67	9	is	be	AUX
bibechana-10399	67	10	fredholm	fredholm	ADJ
bibechana-10399	67	11	if	if	SCONJ
bibechana-10399	67	12	and	and	CCONJ
bibechana-10399	67	13	only	only	ADV
bibechana-10399	67	14	if	if	SCONJ
bibechana-10399	67	15	there	there	PRON
bibechana-10399	67	16	is	be	VERB
bibechana-10399	67	17	a	a	DET
bibechana-10399	67	18	bounded	bounded	ADJ
bibechana-10399	67	19	operator	operator	NOUN
bibechana-10399	67	20	b	b	NOUN
bibechana-10399	67	21	:	:	PUNCT
bibechana-10399	67	22	c	c	X
bibechana-10399	67	23	→	→	SYM
bibechana-10399	67	24	b	b	X
bibechana-10399	67	25	such	such	ADJ
bibechana-10399	67	26	that	that	SCONJ
bibechana-10399	67	27	both	both	DET
bibechana-10399	67	28	(	(	PUNCT
bibechana-10399	67	29	ab	ab	PROPN
bibechana-10399	67	30	i	i	PROPN
bibechana-10399	67	31	)	)	PUNCT
bibechana-10399	67	32	and	and	CCONJ
bibechana-10399	67	33	(	(	PUNCT
bibechana-10399	67	34	ba	ba	PROPN
bibechana-10399	67	35	i	i	NOUN
bibechana-10399	67	36	)	)	PUNCT
bibechana-10399	67	37	are	be	AUX
bibechana-10399	67	38	compact	compact	ADJ
bibechana-10399	67	39	.	.	PUNCT
bibechana-10399	68	1	proof	proof	NOUN
bibechana-10399	68	2	:	:	PUNCT
bibechana-10399	68	3	if	if	SCONJ
bibechana-10399	68	4	a	a	PRON
bibechana-10399	68	5	is	be	AUX
bibechana-10399	68	6	fredholm	fredholm	NOUN
bibechana-10399	68	7	then	then	ADV
bibechana-10399	68	8	b1≔	b1≔	PROPN
bibechana-10399	68	9	ker(a	ker(a	PROPN
bibechana-10399	68	10	)	)	PUNCT
bibechana-10399	68	11	is	be	AUX
bibechana-10399	68	12	finite	finite	ADJ
bibechana-10399	68	13	-	-	ADJ
bibechana-10399	68	14	dimensional	dimensional	ADJ
bibechana-10399	68	15	and	and	CCONJ
bibechana-10399	68	16	so	so	ADV
bibechana-10399	68	17	has	have	VERB
bibechana-10399	68	18	a	a	DET
bibechana-10399	68	19	complementary	complementary	ADJ
bibechana-10399	68	20	closed	close	VERB
bibechana-10399	68	21	subspace	subspace	NOUN
bibechana-10399	68	22	b0	b0	NOUN
bibechana-10399	68	23	in	in	ADP
bibechana-10399	68	24	b.	b.	PROPN
bibechana-10399	68	25	moreover	moreover	ADV
bibechana-10399	68	26	a	a	DET
bibechana-10399	68	27	maps	map	NOUN
bibechana-10399	68	28	b0one	b0one	NOUN
bibechana-10399	68	29	-	-	NOUN
bibechana-10399	68	30	one	one	NUM
bibechana-10399	68	31	onto	onto	ADP
bibechana-10399	68	32	co	co	NOUN
bibechana-10399	68	33	≔	≔	NOUN
bibechana-10399	68	34	ran(a	ran(a	NOUN
bibechana-10399	68	35	)	)	PUNCT
bibechana-10399	68	36	.	.	PUNCT
bibechana-10399	69	1	if	if	SCONJ
bibechana-10399	69	2	c1	c1	PROPN
bibechana-10399	69	3	is	be	AUX
bibechana-10399	69	4	a	a	DET
bibechana-10399	69	5	complementary	complementary	ADJ
bibechana-10399	69	6	finite	finite	ADJ
bibechana-10399	69	7	-	-	ADJ
bibechana-10399	69	8	dimensional	dimensional	ADJ
bibechana-10399	69	9	subspace	subspace	NOUN
bibechana-10399	69	10	of	of	ADP
bibechana-10399	69	11	c0	c0	PROPN
bibechana-10399	69	12	in	in	ADP
bibechana-10399	69	13	c	c	PROPN
bibechana-10399	69	14	then	then	ADV
bibechana-10399	69	15	the	the	DET
bibechana-10399	69	16	operatorx	operatorx	NOUN
bibechana-10399	69	17	:	:	PUNCT
bibechana-10399	69	18	b0⨁	b0⨁	PROPN
bibechana-10399	69	19	c1	c1	PROPN
bibechana-10399	69	20	→	→	SYM
bibechana-10399	69	21	c	c	PROPN
bibechana-10399	69	22	defined	define	VERB
bibechana-10399	69	23	by	by	ADP
bibechana-10399	69	24	(	(	PUNCT
bibechana-10399	69	25	�	�	PROPN
bibechana-10399	69	26	%	%	NOUN
bibechana-10399	69	27	⨁	⨁	PROPN
bibechana-10399	69	28	)	)	PUNCT
bibechana-10399	69	29	�	�	PROPN
bibechana-10399	69	30	≔	≔	NOUN
bibechana-10399	69	31	*	*	NOUN
bibechana-10399	69	32	%	%	NOUN
bibechana-10399	69	33	+	+	CCONJ
bibechana-10399	69	34	)	)	PUNCT
bibechana-10399	69	35	is	be	AUX
bibechana-10399	69	36	bounded	bound	VERB
bibechana-10399	69	37	and	and	CCONJ
bibechana-10399	69	38	invertible	invertible	ADJ
bibechana-10399	69	39	.	.	PUNCT
bibechana-10399	70	1	we	we	PRON
bibechana-10399	70	2	deduce	deduce	VERB
bibechana-10399	70	3	by	by	ADP
bibechana-10399	70	4	the	the	DET
bibechana-10399	70	5	inverse	inverse	NOUN
bibechana-10399	70	6	mapping	mapping	NOUN
bibechana-10399	70	7	theorem	theorem	VERB
bibechana-10399	70	8	that	that	PRON
bibechana-10399	70	9	c0≔x(b0	c0≔x(b0	NOUN
bibechana-10399	70	10	)	)	PUNCT
bibechana-10399	70	11	is	be	AUX
bibechana-10399	70	12	closed	closed	ADJ
bibechana-10399	70	13	.	.	PUNCT
bibechana-10399	71	1	this	this	PRON
bibechana-10399	71	2	completes	complete	VERB
bibechana-10399	71	3	the	the	DET
bibechana-10399	71	4	proof	proof	NOUN
bibechana-10399	71	5	of	of	ADP
bibechana-10399	71	6	the	the	DET
bibechana-10399	71	7	first	first	ADJ
bibechana-10399	71	8	statement	statement	NOUN
bibechana-10399	71	9	of	of	ADP
bibechana-10399	71	10	the	the	DET
bibechana-10399	71	11	theorem	theorem	NOUN
bibechana-10399	71	12	.	.	PUNCT
bibechana-10399	72	1	still	still	ADV
bibechana-10399	72	2	assuming	assume	VERB
bibechana-10399	72	3	that	that	SCONJ
bibechana-10399	72	4	a	a	PRON
bibechana-10399	72	5	is	be	AUX
bibechana-10399	72	6	fredholm	fredholm	NOUN
bibechana-10399	72	7	,	,	PUNCT
bibechana-10399	72	8	put	put	VERB
bibechana-10399	72	9	b(g⨁v	b(g⨁v	NOUN
bibechana-10399	72	10	)	)	PUNCT
bibechana-10399	72	11	≔(a	≔(a	NOUN
bibechana-10399	72	12	0	0	NUM
bibechana-10399	72	13	)	)	PUNCT
bibechana-10399	72	14	-1	-1	NOUN
bibechana-10399	72	15	g	g	NOUN
bibechana-10399	72	16	for	for	ADP
bibechana-10399	72	17	all	all	DET
bibechana-10399	72	18	g	g	NOUN
bibechana-10399	72	19	є	є	PROPN
bibechana-10399	72	20	c0	c0	NOUN
bibechana-10399	72	21	and	and	CCONJ
bibechana-10399	72	22	v	v	ADP
bibechana-10399	72	23	є	є	PROPN
bibechana-10399	72	24	c1	c1	NOUN
bibechana-10399	72	25	,	,	PUNCT
bibechana-10399	72	26	where	where	SCONJ
bibechana-10399	72	27	a0	a0	PROPN
bibechana-10399	72	28	:	:	PUNCT
bibechana-10399	72	29	b0	b0	PROPN
bibechana-10399	72	30	→	→	SYM
bibechana-10399	72	31	c0is	c0is	NOUN
bibechana-10399	72	32	the	the	DET
bibechana-10399	72	33	restriction	restriction	NOUN
bibechana-10399	72	34	of	of	ADP
bibechana-10399	72	35	a	a	PRON
bibechana-10399	72	36	to	to	PART
bibechana-10399	72	37	b0	b0	NOUN
bibechana-10399	72	38	.	.	PUNCT
bibechana-10399	73	1	then	then	ADV
bibechana-10399	73	2	bis	bis	VERB
bibechana-10399	73	3	a	a	DET
bibechana-10399	73	4	bounded	bounded	ADJ
bibechana-10399	73	5	operator	operator	NOUN
bibechana-10399	73	6	from	from	ADP
bibechana-10399	73	7	c	c	NOUN
bibechana-10399	73	8	to	to	PART
bibechana-10399	73	9	band	band	VERB
bibechana-10399	73	10	both	both	PRON
bibechana-10399	73	11	of	of	ADP
bibechana-10399	73	12	,	,	PUNCT
bibechana-10399	73	13	≔	≔	NOUN
bibechana-10399	73	14	*	*	NOUN
bibechana-10399	73	15	.	.	PUNCT
bibechana-10399	74	1	−	−	PROPN
bibechana-10399	74	2	/	/	SYM
bibechana-10399	74	3	,	,	PUNCT
bibechana-10399	74	4	,	,	PUNCT
bibechana-10399	74	5	0	0	NUM
bibechana-10399	74	6	≔	≔	NOUN
bibechana-10399	74	7	.	.	PUNCT
bibechana-10399	75	1	*	*	PUNCT
bibechana-10399	75	2	−	−	PROPN
bibechana-10399	75	3	/	/	SYM
bibechana-10399	75	4	�	�	PROPN
bibechana-10399	75	5	1	1	NUM
bibechana-10399	75	6	�	�	NOUN
bibechana-10399	75	7	are	be	AUX
bibechana-10399	75	8	finite	finite	ADJ
bibechana-10399	75	9	rank	rank	NOUN
bibechana-10399	75	10	and	and	CCONJ
bibechana-10399	75	11	hence	hence	ADV
bibechana-10399	75	12	compact	compact	ADJ
bibechana-10399	75	13	.	.	PUNCT
bibechana-10399	76	1	conversely	conversely	ADV
bibechana-10399	76	2	suppose	suppose	VERB
bibechana-10399	76	3	that	that	SCONJ
bibechana-10399	76	4	a	a	DET
bibechana-10399	76	5	,	,	PUNCT
bibechana-10399	76	6	bare	bare	ADJ
bibechana-10399	76	7	bounded	bound	VERB
bibechana-10399	76	8	,	,	PUNCT
bibechana-10399	76	9	k1	k1	PROPN
bibechana-10399	76	10	,	,	PUNCT
bibechana-10399	76	11	k2are	k2are	NOUN
bibechana-10399	76	12	compact	compact	ADJ
bibechana-10399	76	13	and	and	CCONJ
bibechana-10399	76	14	(	(	PUNCT
bibechana-10399	76	15	1	1	X
bibechana-10399	76	16	)	)	PUNCT
bibechana-10399	76	17	hold	hold	NOUN
bibechana-10399	76	18	.	.	PUNCT
bibechana-10399	77	1	then	then	ADV
bibechana-10399	77	2	,	,	PUNCT
bibechana-10399	77	3	12	12	NUM
bibechana-10399	77	4	�	�	PROPN
bibechana-10399	77	5	*	*	PUNCT
bibechana-10399	77	6	�	�	PROPN
bibechana-10399	77	7	⊆	⊆	NUM
bibechana-10399	77	8	,	,	PUNCT
bibechana-10399	77	9	12	12	NUM
bibechana-10399	77	10	�	�	PROPN
bibechana-10399	77	11	/	/	SYM
bibechana-10399	77	12	+	+	NUM
bibechana-10399	77	13	,	,	PUNCT
bibechana-10399	77	14	0	0	NUM
bibechana-10399	77	15	�	�	PROPN
bibechana-10399	77	16	,	,	PUNCT
bibechana-10399	77	17	456	456	NUM
bibechana-10399	77	18	�	�	PROPN
bibechana-10399	77	19	*	*	PUNCT
bibechana-10399	77	20	�	�	PROPN
bibechana-10399	77	21	⊇	⊇	PROPN
bibechana-10399	77	22	456	456	NUM
bibechana-10399	77	23	�	�	PROPN
bibechana-10399	77	24	/	/	SYM
bibechana-10399	77	25	+	+	NUM
bibechana-10399	77	26	,	,	PUNCT
bibechana-10399	77	27	-	-	PUNCT
bibechana-10399	77	28	�	�	NOUN
bibechana-10399	77	29	.	.	PUNCT
bibechana-10399	78	1	since	since	SCONJ
bibechana-10399	78	2	(	(	PUNCT
bibechana-10399	78	3	i	i	PROPN
bibechana-10399	78	4	+	+	NUM
bibechana-10399	78	5	k1	k1	NOUN
bibechana-10399	78	6	)	)	PUNCT
bibechana-10399	78	7	and	and	CCONJ
bibechana-10399	78	8	(	(	PUNCT
bibechana-10399	78	9	i	i	PROPN
bibechana-10399	78	10	+	+	CCONJ
bibechana-10399	78	11	k2	k2	ADJ
bibechana-10399	78	12	)	)	PUNCT
bibechana-10399	78	13	are	be	AUX
bibechana-10399	78	14	both	both	PRON
bibechana-10399	78	15	fredholm	fredholm	NOUN
bibechana-10399	78	16	by	by	ADP
bibechana-10399	78	17	lemma	lemma	PROPN
bibechana-10399	78	18	(	(	PUNCT
bibechana-10399	78	19	7.1	7.1	NUM
bibechana-10399	78	20	)	)	PUNCT
bibechana-10399	78	21	,	,	PUNCT
bibechana-10399	78	22	it	it	PRON
bibechana-10399	78	23	follows	follow	VERB
bibechana-10399	78	24	that	that	PRON
bibechana-10399	78	25	amust	amust	AUX
bibechana-10399	78	26	be	be	AUX
bibechana-10399	78	27	fredholm	fredholm	NOUN
bibechana-10399	78	28	.	.	PUNCT
bibechana-10399	79	1	the	the	DET
bibechana-10399	79	2	proof	proof	NOUN
bibechana-10399	79	3	of	of	ADP
bibechana-10399	79	4	theorem	theorem	NOUN
bibechana-10399	79	5	(	(	PUNCT
bibechana-10399	79	6	1	1	NUM
bibechana-10399	79	7	)	)	PUNCT
bibechana-10399	79	8	provides	provide	VERB
bibechana-10399	79	9	an	an	DET
bibechana-10399	79	10	important	important	ADJ
bibechana-10399	79	11	structure	structure	NOUN
bibechana-10399	79	12	theorem	theorem	NOUN
bibechana-10399	79	13	for	for	ADP
bibechana-10399	79	14	fredholm	fredholm	NOUN
bibechana-10399	79	15	operators	operator	NOUN
bibechana-10399	79	16	.	.	PUNCT
bibechana-10399	80	1	theorem	theorem	NOUN
bibechana-10399	80	2	(	(	PUNCT
bibechana-10399	80	3	2	2	NUM
bibechana-10399	80	4	)	)	PUNCT
bibechana-10399	80	5	if	if	SCONJ
bibechana-10399	80	6	a	a	PRON
bibechana-10399	80	7	is	be	AUX
bibechana-10399	80	8	a	a	DET
bibechana-10399	80	9	fredholm	fredholm	NOUN
bibechana-10399	80	10	operator	operator	NOUN
bibechana-10399	80	11	then	then	ADV
bibechana-10399	80	12	there	there	PRON
bibechana-10399	80	13	exist	exist	VERB
bibechana-10399	80	14	decompositions	decomposition	NOUN
bibechana-10399	80	15	b	b	NOUN
bibechana-10399	80	16	=	=	SYM
bibechana-10399	80	17	b0⨁	b0⨁	PROPN
bibechana-10399	80	18	b1	b1	NOUN
bibechana-10399	80	19	and	and	CCONJ
bibechana-10399	80	20	c	c	NOUN
bibechana-10399	80	21	=	=	SYM
bibechana-10399	80	22	c0⨁	c0⨁	PROPN
bibechana-10399	80	23	c1	c1	NOUN
bibechana-10399	80	24	such	such	ADJ
bibechana-10399	80	25	that	that	SCONJ
bibechana-10399	80	26	(	(	PUNCT
bibechana-10399	80	27	i	i	NOUN
bibechana-10399	80	28	)	)	PUNCT
bibechana-10399	80	29	b0and	b0and	NUM
bibechana-10399	81	1	c0	c0	NOUN
bibechana-10399	81	2	are	be	AUX
bibechana-10399	81	3	closed	closed	ADJ
bibechana-10399	81	4	subspaces	subspace	NOUN
bibechana-10399	81	5	;	;	PUNCT
bibechana-10399	81	6	m.	m.	NOUN
bibechana-10399	81	7	sahi	sahi	PROPN
bibechana-10399	81	8	/	/	SYM
bibechana-10399	81	9	bibechana	bibechana	PROPN
bibechana-10399	81	10	11(1	11(1	NUM
bibechana-10399	81	11	)	)	PUNCT
bibechana-10399	81	12	(	(	PUNCT
bibechana-10399	81	13	2014	2014	NUM
bibechana-10399	81	14	)	)	PUNCT
bibechana-10399	81	15	169	169	NUM
bibechana-10399	81	16	-	-	SYM
bibechana-10399	81	17	174	174	NUM
bibechana-10399	81	18	:	:	PUNCT
bibechana-10399	81	19	(	(	PUNCT
bibechana-10399	81	20	online	online	ADJ
bibechana-10399	81	21	publication	publication	NOUN
bibechana-10399	81	22	:	:	PUNCT
bibechana-10399	81	23	march	march	PROPN
bibechana-10399	81	24	,	,	PUNCT
bibechana-10399	81	25	2014	2014	NUM
bibechana-10399	81	26	)	)	PUNCT
bibechana-10399	81	27	p.172	p.172	NOUN
bibechana-10399	81	28	(	(	PUNCT
bibechana-10399	81	29	ii	ii	PROPN
bibechana-10399	81	30	)	)	PUNCT
bibechana-10399	82	1	b1and	b1and	ADP
bibechana-10399	82	2	c1	c1	PROPN
bibechana-10399	82	3	are	be	AUX
bibechana-10399	82	4	finite	finite	ADJ
bibechana-10399	82	5	-	-	ADJ
bibechana-10399	82	6	dimensional	dimensional	ADJ
bibechana-10399	82	7	subspaces	subspace	NOUN
bibechana-10399	82	8	;	;	PUNCT
bibechana-10399	82	9	(	(	PUNCT
bibechana-10399	82	10	iii	iii	X
bibechana-10399	82	11	)	)	PUNCT
bibechana-10399	82	12	b1	b1	NOUN
bibechana-10399	82	13	=	=	SYM
bibechana-10399	82	14	ker(a	ker(a	PROPN
bibechana-10399	82	15	)	)	PUNCT
bibechana-10399	82	16	and	and	CCONJ
bibechana-10399	82	17	c0=	c0=	NOUN
bibechana-10399	82	18	ran(a	ran(a	NOUN
bibechana-10399	82	19	)	)	PUNCT
bibechana-10399	82	20	;	;	PUNCT
bibechana-10399	82	21	(	(	PUNCT
bibechana-10399	82	22	iv	iv	X
bibechana-10399	82	23	)	)	PUNCT
bibechana-10399	82	24	index(a	index(a	NOUN
bibechana-10399	82	25	)	)	PUNCT
bibechana-10399	82	26	=	=	SYM
bibechana-10399	82	27	dim(b1	dim(b1	NOUN
bibechana-10399	82	28	)	)	PUNCT
bibechana-10399	82	29	dim(c1	dim(c1	NOUN
bibechana-10399	82	30	)	)	PUNCT
bibechana-10399	82	31	;	;	PUNCT
bibechana-10399	82	32	(	(	PUNCT
bibechana-10399	82	33	v	v	NOUN
bibechana-10399	82	34	)	)	PUNCT
bibechana-10399	82	35	a	a	PRON
bibechana-10399	82	36	has	have	VERB
bibechana-10399	82	37	the	the	DET
bibechana-10399	82	38	matrix	matrix	NOUN
bibechana-10399	82	39	representation	representation	NOUN
bibechana-10399	82	40	*	*	PUNCT
bibechana-10399	83	1	=	=	SYM
bibechana-10399	83	2	9	9	NUM
bibechana-10399	83	3	*	*	PUNCT
bibechana-10399	83	4	:	:	PUNCT
bibechana-10399	83	5	0	0	NUM
bibechana-10399	83	6	0	0	NUM
bibechana-10399	83	7	0	0	NUM
bibechana-10399	83	8	<	<	X
bibechana-10399	83	9	�	�	PROPN
bibechana-10399	83	10	2	2	NUM
bibechana-10399	83	11	�	�	PROPN
bibechana-10399	83	12	where	where	SCONJ
bibechana-10399	83	13	a0	a0	PROPN
bibechana-10399	83	14	:	:	PUNCT
bibechana-10399	83	15	b0	b0	PROPN
bibechana-10399	83	16	→	→	SYM
bibechana-10399	83	17	c0	c0	PROPN
bibechana-10399	83	18	is	be	AUX
bibechana-10399	83	19	one	one	NUM
bibechana-10399	83	20	-	-	PUNCT
bibechana-10399	83	21	one	one	NUM
bibechana-10399	83	22	onto	onto	ADP
bibechana-10399	83	23	.	.	PUNCT
bibechana-10399	84	1	example	example	NOUN
bibechana-10399	84	2	(	(	PUNCT
bibechana-10399	84	3	1	1	X
bibechana-10399	84	4	)	)	PUNCT
bibechana-10399	84	5	let	let	VERB
bibechana-10399	84	6	a	a	PRON
bibechana-10399	84	7	be	be	AUX
bibechana-10399	84	8	a	a	DET
bibechana-10399	84	9	fredholm	fredholm	NOUN
bibechana-10399	84	10	operator	operator	NOUN
bibechana-10399	84	11	on	on	ADP
bibechana-10399	84	12	the	the	DET
bibechana-10399	84	13	banach	banach	NOUN
bibechana-10399	84	14	space	space	NOUN
bibechana-10399	84	15	b.	b.	PROPN
bibechana-10399	84	16	prove	prove	VERB
bibechana-10399	84	17	that	that	SCONJ
bibechana-10399	84	18	if	if	SCONJ
bibechana-10399	84	19	ker(a	ker(a	PROPN
bibechana-10399	84	20	)	)	PUNCT
bibechana-10399	84	21	=	=	PRON
bibechana-10399	85	1	{	{	PUNCT
bibechana-10399	85	2	0	0	NUM
bibechana-10399	85	3	}	}	PUNCT
bibechana-10399	85	4	then	then	ADV
bibechana-10399	85	5	ker(a	ker(a	PROPN
bibechana-10399	85	6	i	i	PUNCT
bibechana-10399	85	7	)	)	PUNCT
bibechana-10399	86	1	=	=	PRON
bibechana-10399	86	2	{	{	PUNCT
bibechana-10399	86	3	0	0	NUM
bibechana-10399	86	4	}	}	PUNCT
bibechana-10399	86	5	for	for	ADP
bibechana-10399	86	6	all	all	PRON
bibechana-10399	86	7	small	small	ADJ
bibechana-10399	86	8	enough	enough	ADV
bibechana-10399	86	9	.	.	NUM
bibechana-10399	86	10	it	it	PRON
bibechana-10399	86	11	can	can	AUX
bibechana-10399	86	12	be	be	AUX
bibechana-10399	86	13	proved	prove	VERB
bibechana-10399	86	14	that	that	SCONJ
bibechana-10399	86	15	if	if	SCONJ
bibechana-10399	86	16	ran(a	ran(a	NOUN
bibechana-10399	86	17	)	)	PUNCT
bibechana-10399	86	18	=	=	PRON
bibechana-10399	86	19	bthen	bthen	VERB
bibechana-10399	86	20	ran(a	ran(a	ADV
bibechana-10399	86	21	i	i	X
bibechana-10399	86	22	)	)	PUNCT
bibechana-10399	87	1	=	=	SYM
bibechana-10399	87	2	b	b	PROPN
bibechana-10399	87	3	for	for	ADP
bibechana-10399	87	4	all	all	PRON
bibechana-10399	87	5	small	small	ADJ
bibechana-10399	87	6	enough	enough	ADV
bibechana-10399	87	7	.	.	NUM
bibechana-10399	87	8	before	before	ADP
bibechana-10399	87	9	stating	state	VERB
bibechana-10399	87	10	our	our	PRON
bibechana-10399	87	11	next	next	ADJ
bibechana-10399	87	12	theorem	theorem	NOUN
bibechana-10399	87	13	we	we	PRON
bibechana-10399	87	14	make	make	VERB
bibechana-10399	87	15	some	some	DET
bibechana-10399	87	16	definitions	definition	NOUN
bibechana-10399	87	17	.	.	PUNCT
bibechana-10399	88	1	we	we	PRON
bibechana-10399	88	2	say	say	VERB
bibechana-10399	88	3	that	that	SCONJ
bibechana-10399	88	4	λ	λ	PROPN
bibechana-10399	88	5	lies	lie	VERB
bibechana-10399	88	6	in	in	ADP
bibechana-10399	88	7	the	the	DET
bibechana-10399	88	8	essential	essential	ADJ
bibechana-10399	88	9	spectrum	spectrum	NOUN
bibechana-10399	88	10	essspec(a	essspec(a	NOUN
bibechana-10399	88	11	)	)	PUNCT
bibechana-10399	88	12	of	of	ADP
bibechana-10399	88	13	a	a	DET
bibechana-10399	88	14	bounded	bounded	ADJ
bibechana-10399	88	15	operator	operator	NOUN
bibechana-10399	88	16	a	a	DET
bibechana-10399	88	17	if	if	NOUN
bibechana-10399	88	18	(	(	PUNCT
bibechana-10399	88	19	λi	λi	NOUN
bibechana-10399	88	20	a	a	NOUN
bibechana-10399	88	21	)	)	PUNCT
bibechana-10399	88	22	is	be	AUX
bibechana-10399	88	23	not	not	PART
bibechana-10399	88	24	a	a	DET
bibechana-10399	88	25	fredholm	fredholm	NOUN
bibechana-10399	88	26	operator	operator	NOUN
bibechana-10399	88	27	.	.	PUNCT
bibechana-10399	89	1	since	since	SCONJ
bibechana-10399	89	2	the	the	DET
bibechana-10399	89	3	set	set	NOUN
bibechana-10399	89	4	κ	κ	X
bibechana-10399	89	5	(	(	PUNCT
bibechana-10399	89	6	b	b	NOUN
bibechana-10399	89	7	)	)	PUNCT
bibechana-10399	89	8	of	of	ADP
bibechana-10399	89	9	all	all	DET
bibechana-10399	89	10	compact	compact	ADJ
bibechana-10399	89	11	operators	operator	NOUN
bibechana-10399	89	12	is	be	AUX
bibechana-10399	89	13	a	a	DET
bibechana-10399	89	14	norm	norm	NOUN
bibechana-10399	89	15	closed	close	VERB
bibechana-10399	89	16	two	two	NUM
bibechana-10399	89	17	-	-	PUNCT
bibechana-10399	89	18	sided	sided	ADJ
bibechana-10399	89	19	ideal	ideal	NOUN
bibechana-10399	89	20	in	in	ADP
bibechana-10399	89	21	the	the	DET
bibechana-10399	89	22	banach	banach	NOUN
bibechana-10399	89	23	algebra	algebra	NOUN
bibechana-10399	89	24	ℒ(b	ℒ(b	NOUN
bibechana-10399	89	25	)	)	PUNCT
bibechana-10399	89	26	of	of	ADP
bibechana-10399	89	27	all	all	DET
bibechana-10399	89	28	bounded	bound	VERB
bibechana-10399	89	29	operators	operator	NOUN
bibechana-10399	89	30	on	on	ADP
bibechana-10399	89	31	b	b	NOUN
bibechana-10399	89	32	,	,	PUNCT
bibechana-10399	89	33	the	the	DET
bibechana-10399	89	34	quotient	quotient	NOUN
bibechana-10399	89	35	algebra	algebra	NOUN
bibechana-10399	89	36	c	c	NOUN
bibechana-10399	89	37	:	:	PUNCT
bibechana-10399	89	38	=	=	SYM
bibechana-10399	89	39	ℒ(b)/	ℒ(b)/	PROPN
bibechana-10399	89	40	κ(b	κ(b	PROPN
bibechana-10399	89	41	)	)	PUNCT
bibechana-10399	89	42	is	be	AUX
bibechana-10399	89	43	a	a	DET
bibechana-10399	89	44	banach	banach	NOUN
bibechana-10399	89	45	algebra	algebra	NOUN
bibechana-10399	89	46	with	with	ADP
bibechana-10399	89	47	respect	respect	NOUN
bibechana-10399	89	48	to	to	ADP
bibechana-10399	89	49	the	the	DET
bibechana-10399	89	50	quotient	quotient	NOUN
bibechana-10399	89	51	norm	norm	NOUN
bibechana-10399	89	52	∥	∥	PROPN
bibechana-10399	89	53	>	>	PUNCT
bibechana-10399	89	54	�	�	PROPN
bibechana-10399	89	55	*	*	PUNCT
bibechana-10399	89	56	�	�	PROPN
bibechana-10399	89	57	∥≔	∥≔	PROPN
bibechana-10399	89	58	inf	inf	PROPN
bibechana-10399	89	59	�	�	PROPN
bibechana-10399	89	60	∥	∥	X
bibechana-10399	89	61	*	*	PUNCT
bibechana-10399	90	1	+	+	CCONJ
bibechana-10399	90	2	,	,	PUNCT
bibechana-10399	90	3	∥∶	∥∶	ADV
bibechana-10399	90	4	,	,	PUNCT
bibechana-10399	90	5	∈	∈	PROPN
bibechana-10399	90	6	κ	κ	PROPN
bibechana-10399	90	7	�	�	PROPN
bibechana-10399	90	8	.	.	PUNCT
bibechana-10399	90	9	�	�	PROPN
bibechana-10399	90	10	�	�	PROPN
bibechana-10399	90	11	where	where	SCONJ
bibechana-10399	90	12	π	π	X
bibechana-10399	90	13	:	:	PUNCT
bibechana-10399	90	14	ℒ(b	ℒ(b	NUM
bibechana-10399	90	15	)	)	PUNCT
bibechana-10399	90	16	→	→	PUNCT
bibechana-10399	90	17	cis	cis	NOUN
bibechana-10399	90	18	the	the	DET
bibechana-10399	90	19	quotient	quotient	NOUN
bibechana-10399	90	20	map	map	NOUN
bibechana-10399	90	21	.	.	PUNCT
bibechana-10399	91	1	the	the	DET
bibechana-10399	91	2	calkin	calkin	ADJ
bibechana-10399	91	3	algebra	algebra	PROPN
bibechana-10399	91	4	c	c	NOUN
bibechana-10399	91	5	enables	enable	VERB
bibechana-10399	91	6	us	we	PRON
bibechana-10399	91	7	to	to	PART
bibechana-10399	91	8	rewrite	rewrite	VERB
bibechana-10399	91	9	theorem	theorem	NOUN
bibechana-10399	91	10	(	(	PUNCT
bibechana-10399	91	11	1	1	NUM
bibechana-10399	91	12	)	)	PUNCT
bibechana-10399	91	13	is	be	AUX
bibechana-10399	91	14	particularly	particularly	ADV
bibechana-10399	91	15	in	in	ADP
bibechana-10399	91	16	a	a	DET
bibechana-10399	91	17	simple	simple	ADJ
bibechana-10399	91	18	form	form	NOUN
bibechana-10399	91	19	.	.	PUNCT
bibechana-10399	92	1	theorem	theorem	NOUN
bibechana-10399	92	2	(	(	PUNCT
bibechana-10399	92	3	3	3	NUM
bibechana-10399	92	4	)	)	PUNCT
bibechana-10399	92	5	the	the	DET
bibechana-10399	92	6	bounded	bounded	ADJ
bibechana-10399	92	7	operator	operator	NOUN
bibechana-10399	92	8	a	a	PRON
bibechana-10399	92	9	on	on	ADP
bibechana-10399	92	10	b	b	PROPN
bibechana-10399	92	11	is	be	AUX
bibechana-10399	92	12	fredholm	fredholm	NOUN
bibechana-10399	92	13	if	if	SCONJ
bibechana-10399	92	14	and	and	CCONJ
bibechana-10399	92	15	only	only	ADV
bibechana-10399	92	16	if	if	SCONJ
bibechana-10399	92	17	π(a	π(a	PROPN
bibechana-10399	92	18	)	)	PUNCT
bibechana-10399	92	19	is	be	AUX
bibechana-10399	92	20	invertible	invertible	ADJ
bibechana-10399	92	21	in	in	ADP
bibechana-10399	92	22	the	the	DET
bibechana-10399	92	23	calkin	calkin	ADJ
bibechana-10399	92	24	algebra	algebra	PROPN
bibechana-10399	92	25	c.	c.	NOUN
bibechana-10399	92	26	if	if	SCONJ
bibechana-10399	92	27	a	a	DET
bibechana-10399	92	28	є	є	NOUN
bibechana-10399	92	29	ℒ(b	ℒ(b	NOUN
bibechana-10399	92	30	)	)	PUNCT
bibechana-10399	92	31	then	then	ADV
bibechana-10399	92	32	bccde1f	bccde1f	VERB
bibechana-10399	92	33	�	�	PROPN
bibechana-10399	92	34	*	*	PUNCT
bibechana-10399	92	35	�	�	PROPN
bibechana-10399	92	36	=	=	SYM
bibechana-10399	92	37	de1f	de1f	PROPN
bibechana-10399	92	38	�	�	PROPN
bibechana-10399	92	39	>	>	NOUN
bibechana-10399	92	40	�	�	PROPN
bibechana-10399	92	41	*	*	PUNCT
bibechana-10399	92	42	�	�	PROPN
bibechana-10399	92	43	�	�	PROPN
bibechana-10399	92	44	.	.	PUNCT
bibechana-10399	93	1	proof	proof	NOUN
bibechana-10399	93	2	:	:	PUNCT
bibechana-10399	93	3	both	both	DET
bibechana-10399	93	4	statements	statement	NOUN
bibechana-10399	93	5	of	of	ADP
bibechana-10399	93	6	the	the	DET
bibechana-10399	93	7	theorem	theorem	NOUN
bibechana-10399	93	8	are	be	AUX
bibechana-10399	93	9	elementary	elementary	ADJ
bibechana-10399	93	10	consequences	consequence	NOUN
bibechana-10399	93	11	of	of	ADP
bibechana-10399	93	12	theorem	theorem	NOUN
bibechana-10399	93	13	(	(	PUNCT
bibechana-10399	93	14	1	1	NUM
bibechana-10399	93	15	)	)	PUNCT
bibechana-10399	93	16	.	.	PUNCT
bibechana-10399	94	1	corollary	corollary	ADJ
bibechana-10399	94	2	:	:	PUNCT
bibechana-10399	94	3	if	if	SCONJ
bibechana-10399	94	4	a	a	DET
bibechana-10399	94	5	:	:	PUNCT
bibechana-10399	94	6	b	b	X
bibechana-10399	94	7	→	→	SYM
bibechana-10399	94	8	b	b	PROPN
bibechana-10399	94	9	is	be	AUX
bibechana-10399	94	10	a	a	DET
bibechana-10399	94	11	fredholm	fredholm	NOUN
bibechana-10399	94	12	operator	operator	NOUN
bibechana-10399	94	13	and	and	CCONJ
bibechana-10399	94	14	b	b	NOUN
bibechana-10399	94	15	≔	≔	VERB
bibechana-10399	94	16	a	a	PRON
bibechana-10399	95	1	+	+	X
bibechana-10399	95	2	k	k	NOUN
bibechana-10399	95	3	where	where	SCONJ
bibechana-10399	95	4	k	k	PROPN
bibechana-10399	95	5	is	be	AUX
bibechana-10399	95	6	compact	compact	ADJ
bibechana-10399	95	7	,	,	PUNCT
bibechana-10399	95	8	then	then	ADV
bibechana-10399	95	9	b	b	X
bibechana-10399	95	10	is	be	AUX
bibechana-10399	95	11	a	a	DET
bibechana-10399	95	12	fredholm	fredholm	NOUN
bibechana-10399	95	13	operator	operator	NOUN
bibechana-10399	95	14	and	and	CCONJ
bibechana-10399	95	15	bccde1f	bccde1f	VERB
bibechana-10399	95	16	�	�	PROPN
bibechana-10399	95	17	*	*	PUNCT
bibechana-10399	95	18	�	�	PROPN
bibechana-10399	95	19	=	=	PUNCT
bibechana-10399	95	20	bccde1f	bccde1f	PROPN
bibechana-10399	95	21	�	�	PROPN
bibechana-10399	95	22	.	.	PUNCT
bibechana-10399	95	23	�	�	PROPN
bibechana-10399	95	24	.	.	PUNCT
bibechana-10399	96	1	theorem	theorem	PROPN
bibechana-10399	96	2	(	(	PUNCT
bibechana-10399	96	3	4	4	NUM
bibechana-10399	96	4	)	)	PUNCT
bibechana-10399	96	5	if	if	SCONJ
bibechana-10399	96	6	a	a	PRON
bibechana-10399	96	7	is	be	AUX
bibechana-10399	96	8	a	a	DET
bibechana-10399	96	9	fredholm	fredholm	NOUN
bibechana-10399	96	10	operator	operator	NOUN
bibechana-10399	96	11	on	on	ADP
bibechana-10399	96	12	b	b	NOUN
bibechana-10399	96	13	then	then	ADV
bibechana-10399	96	14	a	a	PRON
bibechana-10399	96	15	*	*	PUNCT
bibechana-10399	96	16	is	be	AUX
bibechana-10399	96	17	fredholm	fredholm	NOUN
bibechana-10399	96	18	.	.	PUNCT
bibechana-10399	97	1	proof	proof	NOUN
bibechana-10399	97	2	:	:	PUNCT
bibechana-10399	97	3	suppose	suppose	VERB
bibechana-10399	97	4	that	that	SCONJ
bibechana-10399	97	5	ab	ab	PROPN
bibechana-10399	97	6	=	=	PUNCT
bibechana-10399	97	7	i	i	PROPN
bibechana-10399	97	8	+	+	CCONJ
bibechana-10399	97	9	k1	k1	PROPN
bibechana-10399	97	10	and	and	CCONJ
bibechana-10399	97	11	ba	ba	NOUN
bibechana-10399	98	1	=	=	PUNCT
bibechana-10399	98	2	i	i	PRON
bibechana-10399	99	1	+	+	CCONJ
bibechana-10399	99	2	k2	k2	PROPN
bibechana-10399	99	3	where	where	SCONJ
bibechana-10399	99	4	k1	k1	NOUN
bibechana-10399	99	5	,	,	PUNCT
bibechana-10399	99	6	k2	k2	PROPN
bibechana-10399	99	7	are	be	AUX
bibechana-10399	99	8	compact	compact	ADJ
bibechana-10399	99	9	.	.	PUNCT
bibechana-10399	100	1	then	then	ADV
bibechana-10399	100	2	b*a	b*a	NOUN
bibechana-10399	100	3	*	*	PUNCT
bibechana-10399	101	1	=	=	PUNCT
bibechana-10399	101	2	i	i	PRON
bibechana-10399	101	3	+	+	NUM
bibechana-10399	101	4	k1	k1	NOUN
bibechana-10399	101	5	*	*	PUNCT
bibechana-10399	101	6	and	and	CCONJ
bibechana-10399	101	7	a*b	a*b	PROPN
bibechana-10399	101	8	*	*	PUNCT
bibechana-10399	102	1	=	=	PUNCT
bibechana-10399	102	2	i	i	PRON
bibechana-10399	102	3	+	+	CCONJ
bibechana-10399	102	4	k2	k2	PROPN
bibechana-10399	102	5	*	*	NOUN
bibechana-10399	102	6	.	.	PUNCT
bibechana-10399	103	1	we	we	PRON
bibechana-10399	103	2	deduce	deduce	VERB
bibechana-10399	103	3	that	that	SCONJ
bibechana-10399	103	4	a	a	PRON
bibechana-10399	103	5	*	*	PUNCT
bibechana-10399	103	6	is	be	AUX
bibechana-10399	103	7	fredholm	fredholm	NOUN
bibechana-10399	103	8	by	by	ADP
bibechana-10399	103	9	applying	apply	VERB
bibechana-10399	103	10	theorem	theorem	NOUN
bibechana-10399	103	11	(	(	PUNCT
bibechana-10399	103	12	1	1	NUM
bibechana-10399	103	13	)	)	PUNCT
bibechana-10399	103	14	.	.	PUNCT
bibechana-10399	104	1	m.	m.	NOUN
bibechana-10399	104	2	sahi	sahi	PROPN
bibechana-10399	104	3	/	/	SYM
bibechana-10399	104	4	bibechana	bibechana	PROPN
bibechana-10399	104	5	11(1	11(1	NUM
bibechana-10399	104	6	)	)	PUNCT
bibechana-10399	104	7	(	(	PUNCT
bibechana-10399	104	8	2014	2014	NUM
bibechana-10399	104	9	)	)	PUNCT
bibechana-10399	104	10	169	169	NUM
bibechana-10399	104	11	-	-	SYM
bibechana-10399	104	12	174	174	NUM
bibechana-10399	104	13	:	:	PUNCT
bibechana-10399	104	14	(	(	PUNCT
bibechana-10399	104	15	online	online	ADJ
bibechana-10399	104	16	publication	publication	NOUN
bibechana-10399	104	17	:	:	PUNCT
bibechana-10399	104	18	march	march	PROPN
bibechana-10399	104	19	,	,	PUNCT
bibechana-10399	104	20	2014	2014	NUM
bibechana-10399	104	21	)	)	PUNCT
bibechana-10399	104	22	p.173	p.173	NOUN
bibechana-10399	104	23	example	example	NOUN
bibechana-10399	104	24	(	(	PUNCT
bibechana-10399	104	25	2	2	NUM
bibechana-10399	104	26	):	):	PUNCT
bibechana-10399	104	27	prove	prove	VERB
bibechana-10399	104	28	directly	directly	ADV
bibechana-10399	104	29	from	from	ADP
bibechana-10399	104	30	the	the	DET
bibechana-10399	104	31	definition	definition	NOUN
bibechana-10399	104	32	that	that	SCONJ
bibechana-10399	104	33	if	if	SCONJ
bibechana-10399	104	34	a1	a1	NOUN
bibechana-10399	104	35	and	and	CCONJ
bibechana-10399	104	36	a2	a2	PROPN
bibechana-10399	104	37	are	be	AUX
bibechana-10399	104	38	both	both	PRON
bibechana-10399	104	39	fredholm	fredholm	NOUN
bibechana-10399	104	40	operators	operator	NOUN
bibechana-10399	104	41	then	then	ADV
bibechana-10399	104	42	so	so	ADV
bibechana-10399	104	43	is	be	AUX
bibechana-10399	104	44	a1a2	a1a2	PROPN
bibechana-10399	104	45	.	.	PUNCT
bibechana-10399	104	46	note	note	NOUN
bibechana-10399	104	47	:	:	PUNCT
bibechana-10399	105	1	if	if	SCONJ
bibechana-10399	105	2	b1	b1	NOUN
bibechana-10399	105	3	=	=	SYM
bibechana-10399	105	4	b2	b2	NOUN
bibechana-10399	105	5	then	then	ADV
bibechana-10399	105	6	this	this	PRON
bibechana-10399	105	7	is	be	AUX
bibechana-10399	105	8	an	an	DET
bibechana-10399	105	9	obvious	obvious	ADJ
bibechana-10399	105	10	consequence	consequence	NOUN
bibechana-10399	105	11	of	of	ADP
bibechana-10399	105	12	theorem	theorem	NOUN
bibechana-10399	105	13	(	(	PUNCT
bibechana-10399	105	14	7.3	7.3	NUM
bibechana-10399	105	15	)	)	PUNCT
bibechana-10399	105	16	,	,	PUNCT
bibechana-10399	105	17	but	but	CCONJ
bibechana-10399	105	18	there	there	PRON
bibechana-10399	105	19	is	be	VERB
bibechana-10399	105	20	an	an	DET
bibechana-10399	105	21	elementary	elementary	ADJ
bibechana-10399	105	22	direct	direct	ADJ
bibechana-10399	105	23	proof	proof	NOUN
bibechana-10399	105	24	.	.	PUNCT
bibechana-10399	106	1	theorem	theorem	NOUN
bibechana-10399	106	2	(	(	PUNCT
bibechana-10399	106	3	5	5	NUM
bibechana-10399	106	4	)	)	PUNCT
bibechana-10399	106	5	if	if	SCONJ
bibechana-10399	106	6	a	a	PRON
bibechana-10399	106	7	:	:	PUNCT
bibechana-10399	106	8	b	b	NOUN
bibechana-10399	106	9	→	→	SYM
bibechana-10399	106	10	c	c	PROPN
bibechana-10399	106	11	is	be	AUX
bibechana-10399	106	12	a	a	DET
bibechana-10399	106	13	fredholm	fredholm	NOUN
bibechana-10399	106	14	operator	operator	NOUN
bibechana-10399	106	15	,	,	PUNCT
bibechana-10399	106	16	then	then	ADV
bibechana-10399	106	17	there	there	PRON
bibechana-10399	106	18	exists	exist	VERB
bibechana-10399	106	19	є	є	ADP
bibechana-10399	106	20	>	>	X
bibechana-10399	106	21	0	0	NUM
bibechana-10399	106	22	such	such	ADJ
bibechana-10399	106	23	that	that	SCONJ
bibechana-10399	106	24	every	every	DET
bibechana-10399	106	25	bounded	bounded	ADJ
bibechana-10399	106	26	operator	operator	NOUN
bibechana-10399	106	27	x	x	PUNCT
bibechana-10399	106	28	satisfying	satisfy	VERB
bibechana-10399	106	29	∥	∥	NOUN
bibechana-10399	106	30	(	(	PUNCT
bibechana-10399	106	31	−	−	PROPN
bibechana-10399	106	32	*	*	PUNCT
bibechana-10399	106	33	∥<∈	∥<∈	PROPN
bibechana-10399	106	34	is	be	AUX
bibechana-10399	106	35	also	also	ADV
bibechana-10399	106	36	fredholm	fredholm	VERB
bibechana-10399	106	37	with	with	ADP
bibechana-10399	106	38	h6i1j	h6i1j	NUM
bibechana-10399	106	39	�	�	PROPN
bibechana-10399	106	40	(	(	PUNCT
bibechana-10399	106	41	�	�	PROPN
bibechana-10399	106	42	=	=	SYM
bibechana-10399	106	43	h6i1j	h6i1j	SYM
bibechana-10399	106	44	�	�	PROPN
bibechana-10399	106	45	*	*	PUNCT
bibechana-10399	106	46	�	�	PROPN
bibechana-10399	106	47	.	.	PUNCT
bibechana-10399	107	1	proof	proof	NOUN
bibechana-10399	107	2	:	:	PUNCT
bibechana-10399	107	3	we	we	PRON
bibechana-10399	107	4	make	make	VERB
bibechana-10399	107	5	use	use	NOUN
bibechana-10399	107	6	of	of	ADP
bibechana-10399	107	7	the	the	DET
bibechana-10399	107	8	matrix	matrix	NOUN
bibechana-10399	107	9	representation	representation	NOUN
bibechana-10399	107	10	of	of	ADP
bibechana-10399	107	11	theorem	theorem	NOUN
bibechana-10399	107	12	(	(	PUNCT
bibechana-10399	107	13	2	2	NUM
bibechana-10399	107	14	)	)	PUNCT
bibechana-10399	107	15	.	.	PUNCT
bibechana-10399	108	1	if	if	SCONJ
bibechana-10399	108	2	(	(	PUNCT
bibechana-10399	108	3	=	=	NOUN
bibechana-10399	108	4	9	9	X
bibechana-10399	108	5	.	.	PUNCT
bibechana-10399	109	1	k	k	PROPN
bibechana-10399	109	2	l	l	PROPN
bibechana-10399	110	1	b	b	X
bibechana-10399	110	2	<	<	X
bibechana-10399	110	3	and	and	CCONJ
bibechana-10399	110	4	∥	∥	NUM
bibechana-10399	110	5	(	(	PUNCT
bibechana-10399	110	6	−	−	PROPN
bibechana-10399	110	7	*	*	PUNCT
bibechana-10399	111	1	∥<є	∥<є	NOUN
bibechana-10399	111	2	then	then	ADV
bibechana-10399	111	3	∥	∥	NUM
bibechana-10399	111	4	.	.	PUNCT
bibechana-10399	112	1	−	−	PROPN
bibechana-10399	113	1	*	*	PUNCT
bibechana-10399	113	2	:	:	PUNCT
bibechana-10399	113	3	∥	∥	X
bibechana-10399	113	4	<	<	X
bibechana-10399	113	5	cϵ	cϵ	NOUN
bibechana-10399	113	6	,	,	PUNCT
bibechana-10399	113	7	so	so	ADV
bibechana-10399	113	8	b	b	PROPN
bibechana-10399	113	9	is	be	AUX
bibechana-10399	113	10	invertible	invertible	ADJ
bibechana-10399	113	11	provided	provide	VERB
bibechana-10399	113	12	ϵ	ϵ	ADP
bibechana-10399	113	13	>	>	X
bibechana-10399	113	14	0	0	NUM
bibechana-10399	113	15	is	be	AUX
bibechana-10399	113	16	small	small	ADJ
bibechana-10399	113	17	enough	enough	ADV
bibechana-10399	113	18	.	.	PUNCT
bibechana-10399	114	1	if	if	SCONJ
bibechana-10399	114	2	f	f	PROPN
bibechana-10399	114	3	ϵ	ϵ	X
bibechana-10399	114	4	boand	boand	NOUN
bibechana-10399	114	5	g	g	PROPN
bibechana-10399	114	6	ϵ	ϵ	X
bibechana-10399	114	7	b1then	b1then	PROPN
bibechana-10399	114	8	x(f	x(f	PROPN
bibechana-10399	114	9	⨁g	⨁g	PROPN
bibechana-10399	114	10	)	)	PUNCT
bibechana-10399	115	1	=	=	SYM
bibechana-10399	115	2	0	0	PUNCT
bibechana-10399	116	1	if	if	SCONJ
bibechana-10399	116	2	and	and	CCONJ
bibechana-10399	116	3	only	only	ADV
bibechana-10399	116	4	if	if	SCONJ
bibechana-10399	116	5	.%	.%	PROPN
bibechana-10399	117	1	+	+	CCONJ
bibechana-10399	117	2	k	k	X
bibechana-10399	117	3	#	#	NOUN
bibechana-10399	117	4	=	=	SYM
bibechana-10399	117	5	0	0	NUM
bibechana-10399	117	6	,	,	PUNCT
bibechana-10399	117	7	l%	l%	NOUN
bibechana-10399	117	8	+	+	CCONJ
bibechana-10399	117	9	b	b	X
bibechana-10399	117	10	#	#	NOUN
bibechana-10399	117	11	=	=	NOUN
bibechana-10399	117	12	0	0	NUM
bibechana-10399	117	13	.	.	PUNCT
bibechana-10399	118	1	this	this	PRON
bibechana-10399	118	2	reduces	reduce	VERB
bibechana-10399	118	3	to	to	ADP
bibechana-10399	118	4	�	�	PROPN
bibechana-10399	118	5	b	b	PROPN
bibechana-10399	118	6	–	–	PUNCT
bibechana-10399	118	7	l.n	l.n	PROPN
bibechana-10399	118	8	-	-	PUNCT
bibechana-10399	118	9	k	k	NOUN
bibechana-10399	118	10	�	�	NOUN
bibechana-10399	118	11	#	#	NOUN
bibechana-10399	118	12	=	=	SYM
bibechana-10399	118	13	0	0	NUM
bibechana-10399	118	14	,	,	PUNCT
bibechana-10399	118	15	where	where	SCONJ
bibechana-10399	118	16	(	(	PUNCT
bibechana-10399	118	17	e	e	X
bibechana-10399	118	18	–	–	PUNCT
bibechana-10399	118	19	db-1	db-1	ADV
bibechana-10399	118	20	c)g	c)g	PROPN
bibechana-10399	118	21	:	:	PUNCT
bibechana-10399	118	22	b1	b1	PROPN
bibechana-10399	118	23	→	→	SYM
bibechana-10399	118	24	c1	c1	PROPN
bibechana-10399	118	25	,	,	PUNCT
bibechana-10399	118	26	both	both	PRON
bibechana-10399	118	27	of	of	ADP
bibechana-10399	118	28	these	these	DET
bibechana-10399	118	29	spaces	space	NOUN
bibechana-10399	118	30	being	be	AUX
bibechana-10399	118	31	finite	finite	ADJ
bibechana-10399	118	32	-	-	ADJ
bibechana-10399	118	33	dimensional	dimensional	ADJ
bibechana-10399	118	34	.	.	PUNCT
bibechana-10399	119	1	we	we	PRON
bibechana-10399	119	2	deduce	deduce	VERB
bibechana-10399	119	3	that	that	SCONJ
bibechana-10399	119	4	iho	iho	PROPN
bibechana-10399	119	5	�	�	PROPN
bibechana-10399	119	6	,12	,12	SYM
bibechana-10399	119	7	�	�	PROPN
bibechana-10399	119	8	(	(	PUNCT
bibechana-10399	119	9	�	�	PROPN
bibechana-10399	119	10	�	�	PROPN
bibechana-10399	119	11	=	=	PUNCT
bibechana-10399	119	12	iho	iho	PROPN
bibechana-10399	119	13	�	�	PROPN
bibechana-10399	119	14	,12	,12	PUNCT
bibechana-10399	119	15	�	�	PROPN
bibechana-10399	119	16	b	b	PROPN
bibechana-10399	119	17	–	–	PUNCT
bibechana-10399	119	18	l.n	l.n	PROPN
bibechana-10399	119	19	-	-	PUNCT
bibechana-10399	119	20	k	k	NOUN
bibechana-10399	119	21	�	�	PROPN
bibechana-10399	119	22	�	�	PROPN
bibechana-10399	119	23	for	for	ADP
bibechana-10399	119	24	all	all	DET
bibechana-10399	119	25	small	small	ADJ
bibechana-10399	119	26	enough	enough	ADV
bibechana-10399	119	27	є	є	ADP
bibechana-10399	119	28	>	>	X
bibechana-10399	119	29	0	0	NUM
bibechana-10399	119	30	.	.	PUNCT
bibechana-10399	119	31	by	by	ADP
bibechana-10399	119	32	applying	apply	VERB
bibechana-10399	119	33	a	a	DET
bibechana-10399	119	34	similar	similar	ADJ
bibechana-10399	119	35	argument	argument	NOUN
bibechana-10399	119	36	to	to	ADP
bibechana-10399	119	37	(	(	PUNCT
bibechana-10399	119	38	∗	∗	X
bibechana-10399	119	39	=	=	SYM
bibechana-10399	119	40	9.∗	9.∗	NUM
bibechana-10399	119	41	l∗	l∗	PROPN
bibechana-10399	119	42	k∗	k∗	VERB
bibechana-10399	119	43	b∗	b∗	ADV
bibechana-10399	119	44	<	<	X
bibechana-10399	119	45	we	we	PRON
bibechana-10399	119	46	obtain	obtain	VERB
bibechana-10399	119	47	iho	iho	PROPN
bibechana-10399	119	48	�	�	PROPN
bibechana-10399	119	49	kqr12	kqr12	PROPN
bibechana-10399	119	50	�	�	PROPN
bibechana-10399	119	51	(	(	PUNCT
bibechana-10399	119	52	�	�	PROPN
bibechana-10399	119	53	�	�	PROPN
bibechana-10399	119	54	=	=	PUNCT
bibechana-10399	119	55	iho	iho	PROPN
bibechana-10399	119	56	�	�	PROPN
bibechana-10399	119	57	kqr12	kqr12	PROPN
bibechana-10399	119	58	�	�	PROPN
bibechana-10399	119	59	b	b	PROPN
bibechana-10399	119	60	−	−	PROPN
bibechana-10399	119	61	l.n	l.n	PROPN
bibechana-10399	119	62	-	-	PUNCT
bibechana-10399	119	63	k	k	NOUN
bibechana-10399	119	64	�	�	PROPN
bibechana-10399	119	65	�	�	PROPN
bibechana-10399	119	66	for	for	ADP
bibechana-10399	119	67	all	all	DET
bibechana-10399	119	68	small	small	ADJ
bibechana-10399	119	69	enough	enough	ADV
bibechana-10399	119	70	ε	ε	PROPN
bibechana-10399	119	71	>	>	X
bibechana-10399	119	72	0	0	PROPN
bibechana-10399	119	73	.	.	PUNCT
bibechana-10399	120	1	problem	problem	NOUN
bibechana-10399	120	2	now	now	ADV
bibechana-10399	120	3	implies	imply	VERB
bibechana-10399	120	4	that	that	SCONJ
bibechana-10399	120	5	h6i1j	h6i1j	NUM
bibechana-10399	120	6	�	�	PROPN
bibechana-10399	120	7	(	(	PUNCT
bibechana-10399	120	8	�	�	PROPN
bibechana-10399	120	9	=	=	SYM
bibechana-10399	120	10	h6i1j	h6i1j	X
bibechana-10399	120	11	�	�	PROPN
bibechana-10399	120	12	b	b	NOUN
bibechana-10399	120	13	−	−	PROPN
bibechana-10399	120	14	l.n	l.n	PROPN
bibechana-10399	120	15	-	-	PUNCT
bibechana-10399	120	16	k	k	PROPN
bibechana-10399	120	17	�	�	PROPN
bibechana-10399	120	18	�	�	PROPN
bibechana-10399	120	19	=	=	PUNCT
bibechana-10399	120	20	iho	iho	PROPN
bibechana-10399	120	21	�	�	PROPN
bibechana-10399	120	22	ℬ-	ℬ-	SYM
bibechana-10399	120	23	�	�	PROPN
bibechana-10399	120	24	−	−	PROPN
bibechana-10399	120	25	iho	iho	PROPN
bibechana-10399	120	26	�	�	PROPN
bibechana-10399	120	27	k-	k-	PROPN
bibechana-10399	120	28	�	�	PROPN
bibechana-10399	120	29	.	.	PUNCT
bibechana-10399	121	1	this	this	DET
bibechana-10399	121	2	formula	formula	NOUN
bibechana-10399	121	3	establishes	establish	VERB
bibechana-10399	121	4	that	that	SCONJ
bibechana-10399	121	5	index(x	index(x	NOUN
bibechana-10399	121	6	)	)	PUNCT
bibechana-10399	121	7	does	do	AUX
bibechana-10399	121	8	not	not	PART
bibechana-10399	121	9	depend	depend	VERB
bibechana-10399	121	10	on	on	ADP
bibechana-10399	121	11	x	x	PRON
bibechana-10399	121	12	,	,	PUNCT
bibechana-10399	121	13	provided	provide	VERB
bibechana-10399	121	14	∥	∥	PROPN
bibechana-10399	121	15	(	(	PUNCT
bibechana-10399	121	16	−	−	PROPN
bibechana-10399	121	17	*	*	PUNCT
bibechana-10399	121	18	∥is	∥is	X
bibechana-10399	121	19	small	small	ADJ
bibechana-10399	121	20	enough	enough	ADV
bibechana-10399	121	21	.	.	PUNCT
bibechana-10399	122	1	theorem	theorem	NOUN
bibechana-10399	122	2	(	(	PUNCT
bibechana-10399	122	3	5	5	NUM
bibechana-10399	122	4	)	)	PUNCT
bibechana-10399	122	5	establishes	establish	VERB
bibechana-10399	122	6	that	that	SCONJ
bibechana-10399	122	7	the	the	DET
bibechana-10399	122	8	index	index	NOUN
bibechana-10399	122	9	is	be	AUX
bibechana-10399	122	10	a	a	DET
bibechana-10399	122	11	homotopy	homotopy	NOUN
bibechana-10399	122	12	invariant	invariant	ADJ
bibechana-10399	122	13	:	:	PUNCT
bibechana-10399	122	14	if	if	SCONJ
bibechana-10399	122	15	t	t	PROPN
bibechana-10399	122	16	→	→	PUNCT
bibechana-10399	122	17	a	a	DET
bibechana-10399	122	18	t	t	NOUN
bibechana-10399	122	19	is	be	AUX
bibechana-10399	122	20	a	a	DET
bibechana-10399	122	21	norm	norm	NOUN
bibechana-10399	122	22	continuous	continuous	ADJ
bibechana-10399	122	23	family	family	NOUN
bibechana-10399	122	24	of	of	ADP
bibechana-10399	122	25	fredholm	fredholm	NOUN
bibechana-10399	122	26	operators	operator	NOUN
bibechana-10399	122	27	then	then	ADV
bibechana-10399	122	28	index(at	index(at	NOUN
bibechana-10399	122	29	)	)	PUNCT
bibechana-10399	122	30	does	do	AUX
bibechana-10399	122	31	not	not	PART
bibechana-10399	122	32	depend	depend	VERB
bibechana-10399	122	33	on	on	ADP
bibechana-10399	122	34	t.	t.	PROPN
bibechana-10399	122	35	in	in	ADP
bibechana-10399	122	36	a	a	DET
bibechana-10399	122	37	hilbert	hilbert	NOUN
bibechana-10399	122	38	space	space	NOUN
bibechana-10399	122	39	context	context	NOUN
bibechana-10399	122	40	one	one	NUM
bibechana-10399	122	41	can	can	AUX
bibechana-10399	122	42	even	even	ADV
bibechana-10399	122	43	identify	identify	VERB
bibechana-10399	122	44	the	the	DET
bibechana-10399	122	45	homotopy	homotopy	NOUN
bibechana-10399	122	46	classes	class	NOUN
bibechana-10399	122	47	.	.	PUNCT
bibechana-10399	123	1	m.	m.	NOUN
bibechana-10399	123	2	sahi	sahi	PROPN
bibechana-10399	123	3	/	/	SYM
bibechana-10399	123	4	bibechana	bibechana	PROPN
bibechana-10399	123	5	11(1	11(1	NUM
bibechana-10399	123	6	)	)	PUNCT
bibechana-10399	123	7	(	(	PUNCT
bibechana-10399	123	8	2014	2014	NUM
bibechana-10399	123	9	)	)	PUNCT
bibechana-10399	123	10	169	169	NUM
bibechana-10399	123	11	-	-	SYM
bibechana-10399	123	12	174	174	NUM
bibechana-10399	123	13	:	:	PUNCT
bibechana-10399	123	14	(	(	PUNCT
bibechana-10399	123	15	online	online	ADJ
bibechana-10399	123	16	publication	publication	NOUN
bibechana-10399	123	17	:	:	PUNCT
bibechana-10399	123	18	march	march	PROPN
bibechana-10399	123	19	,	,	PUNCT
bibechana-10399	123	20	2014	2014	NUM
bibechana-10399	123	21	)	)	PUNCT
bibechana-10399	123	22	p.174	p.174	ADJ
bibechana-10399	123	23	conclusion	conclusion	NOUN
bibechana-10399	123	24	thus	thus	ADV
bibechana-10399	123	25	,	,	PUNCT
bibechana-10399	123	26	we	we	PRON
bibechana-10399	123	27	see	see	VERB
bibechana-10399	123	28	that	that	SCONJ
bibechana-10399	123	29	fredholm	fredholm	NOUN
bibechana-10399	123	30	operators	operator	NOUN
bibechana-10399	123	31	are	be	AUX
bibechana-10399	123	32	important	important	ADJ
bibechana-10399	123	33	for	for	ADP
bibechana-10399	123	34	variety	variety	NOUN
bibechana-10399	123	35	of	of	ADP
bibechana-10399	123	36	reasons	reason	NOUN
bibechana-10399	123	37	,	,	PUNCT
bibechana-10399	123	38	one	one	NUM
bibechana-10399	123	39	being	be	AUX
bibechana-10399	123	40	the	the	DET
bibechana-10399	123	41	role	role	NOUN
bibechana-10399	123	42	that	that	PRON
bibechana-10399	123	43	their	their	PRON
bibechana-10399	123	44	index	index	NOUN
bibechana-10399	123	45	plays	play	VERB
bibechana-10399	123	46	in	in	ADP
bibechana-10399	123	47	global	global	ADJ
bibechana-10399	123	48	analysis	analysis	NOUN
bibechana-10399	123	49	.	.	PUNCT
bibechana-10399	124	1	the	the	DET
bibechana-10399	124	2	dimension	dimension	NOUN
bibechana-10399	124	3	of	of	ADP
bibechana-10399	124	4	null	null	ADJ
bibechana-10399	124	5	space	space	NOUN
bibechana-10399	124	6	n(t	n(t	PROPN
bibechana-10399	124	7	)	)	PUNCT
bibechana-10399	124	8	and	and	CCONJ
bibechana-10399	124	9	the	the	DET
bibechana-10399	124	10	co	co	NOUN
bibechana-10399	124	11	-	-	NOUN
bibechana-10399	124	12	dimension	dimension	NOUN
bibechana-10399	124	13	of	of	ADP
bibechana-10399	124	14	r(t	r(t	NOUN
bibechana-10399	124	15	)	)	PUNCT
bibechana-10399	124	16	of	of	ADP
bibechana-10399	124	17	the	the	DET
bibechana-10399	124	18	operator	operator	NOUN
bibechana-10399	124	19	t	t	NOUN
bibechana-10399	124	20	are	be	AUX
bibechana-10399	124	21	finite	finite	ADJ
bibechana-10399	124	22	and	and	CCONJ
bibechana-10399	124	23	it	it	PRON
bibechana-10399	124	24	is	be	AUX
bibechana-10399	124	25	closed	closed	ADJ
bibechana-10399	124	26	and	and	CCONJ
bibechana-10399	124	27	range	range	NOUN
bibechana-10399	124	28	of	of	ADP
bibechana-10399	124	29	the	the	DET
bibechana-10399	124	30	operator	operator	NOUN
bibechana-10399	124	31	is	be	AUX
bibechana-10399	124	32	also	also	ADV
bibechana-10399	124	33	closed	close	VERB
bibechana-10399	124	34	.	.	PUNCT
bibechana-10399	125	1	the	the	DET
bibechana-10399	125	2	application	application	NOUN
bibechana-10399	125	3	of	of	ADP
bibechana-10399	125	4	this	this	DET
bibechana-10399	125	5	operator	operator	NOUN
bibechana-10399	125	6	plays	play	VERB
bibechana-10399	125	7	a	a	DET
bibechana-10399	125	8	vital	vital	ADJ
bibechana-10399	125	9	role	role	NOUN
bibechana-10399	125	10	in	in	ADP
bibechana-10399	125	11	the	the	DET
bibechana-10399	125	12	theory	theory	NOUN
bibechana-10399	125	13	of	of	ADP
bibechana-10399	125	14	boundary	boundary	ADJ
bibechana-10399	125	15	value	value	NOUN
bibechana-10399	125	16	problems	problem	NOUN
bibechana-10399	125	17	in	in	ADP
bibechana-10399	125	18	differential	differential	ADJ
bibechana-10399	125	19	equations	equation	NOUN
bibechana-10399	125	20	.	.	PUNCT
bibechana-10399	126	1	references	reference	NOUN
bibechana-10399	126	2	[	[	X
bibechana-10399	126	3	1	1	NUM
bibechana-10399	126	4	]	]	X
bibechana-10399	126	5	f	f	PROPN
bibechana-10399	126	6	-	-	PUNCT
bibechana-10399	126	7	v	v	NOUN
bibechana-10399	126	8	atkinson	atkinson	PROPN
bibechana-10399	126	9	,	,	PUNCT
bibechana-10399	126	10	acta	acta	PROPN
bibechana-10399	126	11	sci	sci	PROPN
bibechana-10399	126	12	.	.	PROPN
bibechana-10399	126	13	math	math	PROPN
bibechana-10399	126	14	.	.	PUNCT
bibechana-10399	127	1	,	,	PUNCT
bibechana-10399	127	2	15	15	NUM
bibechana-10399	127	3	(	(	PUNCT
bibechana-10399	127	4	1953	1953	NUM
bibechana-10399	127	5	)	)	PUNCT
bibechana-10399	127	6	38	38	NUM
bibechana-10399	127	7	.	.	PUNCT
bibechana-10399	128	1	[	[	X
bibechana-10399	128	2	2	2	NUM
bibechana-10399	128	3	]	]	PUNCT
bibechana-10399	128	4	bovenbek	bovenbek	ADJ
bibechana-10399	128	5	boss	boss	NOUN
bibechana-10399	128	6	,	,	PUNCT
bibechana-10399	128	7	j.	j.	PROPN
bibechana-10399	128	8	phillips	phillips	PROPN
bibechana-10399	128	9	,	,	PUNCT
bibechana-10399	128	10	canada	canada	PROPN
bibechana-10399	128	11	.	.	PUNCT
bibechana-10399	129	1	j.	j.	PROPN
bibechana-10399	129	2	math	math	PROPN
bibechana-10399	129	3	.	.	PUNCT
bibechana-10399	129	4	,	,	PUNCT
bibechana-10399	129	5	57	57	NUM
bibechana-10399	129	6	(	(	PUNCT
bibechana-10399	129	7	2005	2005	NUM
bibechana-10399	129	8	)	)	PUNCT
bibechana-10399	129	9	225	225	NUM
bibechana-10399	129	10	.	.	PUNCT
bibechana-10399	130	1	[	[	X
bibechana-10399	130	2	3	3	X
bibechana-10399	130	3	]	]	X
bibechana-10399	130	4	i.c	i.c	PROPN
bibechana-10399	130	5	.	.	PROPN
bibechana-10399	130	6	gohberb	gohberb	PROPN
bibechana-10399	130	7	,	,	PUNCT
bibechana-10399	130	8	m.	m.	NOUN
bibechana-10399	130	9	g.	g.	PROPN
bibechana-10399	130	10	krein	krein	PROPN
bibechana-10399	130	11	,	,	PUNCT
bibechana-10399	130	12	amer	amer	PROPN
bibechana-10399	130	13	.	.	PROPN
bibechana-10399	130	14	math	math	PROPN
bibechana-10399	130	15	.	.	PUNCT
bibechana-10399	131	1	soc	soc	PROPN
bibechana-10399	131	2	.	.	PUNCT
bibechana-10399	132	1	transl	transl	PROPN
bibechana-10399	132	2	.	.	PUNCT
bibechana-10399	133	1	,	,	PUNCT
bibechana-10399	133	2	13	13	NUM
bibechana-10399	133	3	(	(	PUNCT
bibechana-10399	133	4	1960)185	1960)185	NOUN
bibechana-10399	133	5	.	.	PUNCT
bibechana-10399	134	1	[	[	X
bibechana-10399	134	2	4	4	X
bibechana-10399	134	3	]	]	PUNCT
bibechana-10399	134	4	s.	s.	PROPN
bibechana-10399	134	5	goldberg	goldberg	PROPN
bibechana-10399	134	6	,	,	PUNCT
bibechana-10399	134	7	unbounded	unbounded	ADJ
bibechana-10399	134	8	linear	linear	PROPN
bibechana-10399	134	9	operators	operator	NOUN
bibechana-10399	134	10	,	,	PUNCT
bibechana-10399	134	11	mc	mc	PROPN
bibechana-10399	134	12	graw	graw	PROPN
bibechana-10399	134	13	hill	hill	PROPN
bibechana-10399	134	14	,	,	PUNCT
bibechana-10399	134	15	new	new	PROPN
bibechana-10399	134	16	york	york	PROPN
bibechana-10399	134	17	,	,	PUNCT
bibechana-10399	134	18	1966	1966	NUM
bibechana-10399	134	19	.	.	PUNCT
bibechana-10399	135	1	[	[	X
bibechana-10399	135	2	5	5	NUM
bibechana-10399	135	3	]	]	PUNCT
bibechana-10399	135	4	t	t	PROPN
bibechana-10399	135	5	kato	kato	PROPN
bibechana-10399	135	6	,	,	PUNCT
bibechana-10399	135	7	perturbation	perturbation	NOUN
bibechana-10399	135	8	theory	theory	NOUN
bibechana-10399	135	9	for	for	ADP
bibechana-10399	135	10	linear	linear	PROPN
bibechana-10399	135	11	operators	operator	NOUN
bibechana-10399	135	12	,	,	PUNCT
bibechana-10399	135	13	springer	springer	NOUN
bibechana-10399	135	14	,	,	PUNCT
bibechana-10399	135	15	1996	1996	NUM
bibechana-10399	135	16	.	.	PUNCT
bibechana-10399	136	1	[	[	X
bibechana-10399	136	2	6	6	NUM
bibechana-10399	136	3	]	]	PUNCT
bibechana-10399	136	4	g.	g.	NOUN
bibechana-10399	136	5	kothe	kothe	PROPN
bibechana-10399	136	6	,	,	PUNCT
bibechana-10399	136	7	topological	topological	ADJ
bibechana-10399	136	8	vector	vector	NOUN
bibechana-10399	136	9	spaces	space	NOUN
bibechana-10399	136	10	,	,	PUNCT
bibechana-10399	136	11	vol	vol	NOUN
bibechana-10399	136	12	.	.	PUNCT
bibechana-10399	137	1	1.and	1.and	NUM
bibechana-10399	137	2	vol	vol	NOUN
bibechana-10399	137	3	.	.	PUNCT
bibechana-10399	137	4	ii	ii	PROPN
bibechana-10399	137	5	,	,	PUNCT
bibechana-10399	137	6	1979	1979	NUM
bibechana-10399	137	7	.	.	PUNCT
bibechana-10399	138	1	[	[	X
bibechana-10399	138	2	7	7	X
bibechana-10399	138	3	]	]	X
bibechana-10399	138	4	a.e	a.e	PROPN
bibechana-10399	138	5	.	.	PROPN
bibechana-10399	138	6	taylor	taylor	PROPN
bibechana-10399	138	7	,	,	PUNCT
bibechana-10399	138	8	d.	d.	PROPN
bibechana-10399	138	9	c.	c.	PROPN
bibechana-10399	138	10	lay	lay	PROPN
bibechana-10399	138	11	,	,	PUNCT
bibechana-10399	138	12	introduction	introduction	NOUN
bibechana-10399	138	13	functional	functional	ADJ
bibechana-10399	138	14	analysis	analysis	NOUN
bibechana-10399	138	15	,	,	PUNCT
bibechana-10399	138	16	2nd	2nd	ADJ
bibechana-10399	138	17	ed	ed	NOUN
bibechana-10399	138	18	.	.	PUNCT
bibechana-10399	139	1	wiley	wiley	PROPN
bibechana-10399	139	2	,	,	PUNCT
bibechana-10399	139	3	new	new	PROPN
bibechana-10399	139	4	york	york	PROPN
bibechana-10399	139	5	,	,	PUNCT
bibechana-10399	139	6	1980	1980	NUM
bibechana-10399	139	7	.	.	PUNCT
bibechana-10399	140	1	[	[	X
bibechana-10399	140	2	8	8	NUM
bibechana-10399	140	3	]	]	X
bibechana-10399	140	4	a.e	a.e	PROPN
bibechana-10399	140	5	.	.	PROPN
bibechana-10399	140	6	taylor	taylor	PROPN
bibechana-10399	140	7	,	,	PUNCT
bibechana-10399	140	8	theorems	theorem	NOUN
bibechana-10399	140	9	on	on	ADP
bibechana-10399	140	10	ascent	ascent	NOUN
bibechana-10399	140	11	,	,	PUNCT
bibechana-10399	140	12	descent	descent	NOUN
bibechana-10399	140	13	,	,	PUNCT
bibechana-10399	140	14	nullity	nullity	NOUN
bibechana-10399	140	15	and	and	CCONJ
bibechana-10399	140	16	defect	defect	NOUN
bibechana-10399	140	17	of	of	ADP
bibechana-10399	140	18	linear	linear	PROPN
bibechana-10399	140	19	operators	operator	NOUN
bibechana-10399	140	20	,	,	PUNCT
bibechana-10399	140	21	math	math	NOUN
bibechana-10399	140	22	.	.	PUNCT
bibechana-10399	141	1	ann	ann	PROPN
bibechana-10399	141	2	.	.	PROPN
bibechana-10399	141	3	,	,	PUNCT
bibechana-10399	141	4	163	163	NUM
bibechana-10399	141	5	(	(	PUNCT
bibechana-10399	141	6	1966	1966	NUM
bibechana-10399	141	7	)	)	PUNCT
bibechana-10399	141	8	18	18	NUM
bibechana-10399	141	9	.	.	PUNCT
bibechana-10399	142	1	[	[	X
bibechana-10399	142	2	9	9	NUM
bibechana-10399	142	3	]	]	PUNCT
bibechana-10399	142	4	b.	b.	PROPN
bibechana-10399	142	5	yood	yood	PROPN
bibechana-10399	142	6	,	,	PUNCT
bibechana-10399	142	7	properties	property	NOUN
bibechana-10399	142	8	of	of	ADP
bibechana-10399	142	9	linear	linear	ADJ
bibechana-10399	142	10	transformations	transformation	NOUN
bibechana-10399	142	11	:	:	PUNCT
bibechana-10399	142	12	preserved	preserve	VERB
bibechana-10399	142	13	under	under	ADP
bibechana-10399	142	14	addition	addition	NOUN
bibechana-10399	142	15	of	of	ADP
bibechana-10399	142	16	a	a	DET
bibechana-10399	142	17	completely	completely	ADV
bibechana-10399	142	18	continuous	continuous	ADJ
bibechana-10399	142	19	trans	tran	NOUN
bibechana-10399	142	20	.	.	PUNCT
bibechana-10399	142	21	duke	duke	PROPN
bibechana-10399	142	22	math	math	PROPN
bibechana-10399	142	23	.	.	PUNCT
bibechana-10399	142	24	,	,	PUNCT
bibechana-10399	142	25	j.	j.	PROPN
bibechana-10399	142	26	18	18	NUM
bibechana-10399	142	27	.	.	PUNCT