id	sid	tid	token	lemma	pos
iajs-1822	1	1	microsoft	microsoft	PROPN
iajs-1822	1	2	word	word	NOUN
iajs-1822	1	3	500	500	NUM
iajs-1822	1	4	-	-	SYM
iajs-1822	1	5	509	509	NUM
iajs-1822	1	6	ihsciconf	ihsciconf	NOUN
iajs-1822	1	7	2017	2017	NUM
iajs-1822	1	8	special	special	ADJ
iajs-1822	1	9	issue	issue	NOUN
iajs-1822	1	10	ibn	ibn	PROPN
iajs-1822	1	11	al	al	PROPN
iajs-1822	1	12	-	-	PUNCT
iajs-1822	1	13	haitham	haitham	PROPN
iajs-1822	1	14	journal	journal	PROPN
iajs-1822	1	15	for	for	ADP
iajs-1822	1	16	pure	pure	ADJ
iajs-1822	1	17	and	and	CCONJ
iajs-1822	1	18	applied	apply	VERB
iajs-1822	1	19	science	science	NOUN
iajs-1822	1	20	https://doi.org/	https://doi.org/	NOUN
iajs-1822	1	21	10.30526/2017.ihsciconf.1822	10.30526/2017.ihsciconf.1822	NOUN
iajs-1822	1	22	for	for	ADP
iajs-1822	1	23	more	more	ADJ
iajs-1822	1	24	information	information	NOUN
iajs-1822	1	25	about	about	ADP
iajs-1822	1	26	the	the	DET
iajs-1822	1	27	conference	conference	NOUN
iajs-1822	1	28	please	please	INTJ
iajs-1822	1	29	visit	visit	VERB
iajs-1822	1	30	the	the	DET
iajs-1822	1	31	websites	website	NOUN
iajs-1822	1	32	:	:	PUNCT
iajs-1822	1	33	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1822	1	34	  	  	SPACE
iajs-1822	1	35	www.ihsciconf.org	www.ihsciconf.org	ADJ
iajs-1822	1	36	   	   	SPACE
iajs-1822	1	37	mathematics|500	mathematics|500	ADJ
iajs-1822	1	38	    	    	SPACE
iajs-1822	1	39	common	common	ADJ
iajs-1822	1	40	fixed	fix	VERB
iajs-1822	1	41	points	point	NOUN
iajs-1822	1	42	in	in	ADP
iajs-1822	1	43	modular	modular	ADJ
iajs-1822	1	44	spaces	space	NOUN
iajs-1822	1	45	salwa	salwa	PROPN
iajs-1822	1	46	salman	salman	PROPN
iajs-1822	1	47	abed	abe	VERB
iajs-1822	1	48	  	  	SPACE
iajs-1822	1	49	salwaalbundi@yahoo.com	salwaalbundi@yahoo.com	PROPN
iajs-1822	1	50	dept	dept	PROPN
iajs-1822	1	51	.	.	PROPN
iajs-1822	2	1	of	of	ADP
iajs-1822	2	2	mathematics	mathematics	PROPN
iajs-1822	2	3	/	/	SYM
iajs-1822	2	4	college	college	NOUN
iajs-1822	2	5	of	of	ADP
iajs-1822	2	6	education	education	NOUN
iajs-1822	2	7	forpure	forpure	NOUN
iajs-1822	2	8	science	science	NOUN
iajs-1822	2	9	(	(	PUNCT
iajs-1822	2	10	ibn	ibn	PROPN
iajs-1822	2	11	al	al	PROPN
iajs-1822	2	12	-	-	PUNCT
iajs-1822	2	13	haitham	haitham	PROPN
iajs-1822	2	14	)	)	PUNCT
iajs-1822	2	15	university	university	PROPN
iajs-1822	2	16	of	of	ADP
iajs-1822	2	17	baghdad	baghdad	PROPN
iajs-1822	2	18	sada	sada	PROPN
iajs-1822	2	19	emad	emad	PROPN
iajs-1822	2	20	abdulkarrar	abdulkarrar	PROPN
iajs-1822	2	21	kararemad1982@gmail.com	kararemad1982@gmail.com	PROPN
iajs-1822	2	22	dept	dept	PROPN
iajs-1822	2	23	.	.	PROPN
iajs-1822	3	1	of	of	ADP
iajs-1822	3	2	mathematics/	mathematics/	NUM
iajs-1822	3	3	college	college	NOUN
iajs-1822	3	4	of	of	ADP
iajs-1822	3	5	education	education	NOUN
iajs-1822	3	6	for	for	ADP
iajs-1822	3	7	pure	pure	ADJ
iajs-1822	3	8	science	science	NOUN
iajs-1822	3	9	(	(	PUNCT
iajs-1822	3	10	ibn	ibn	PROPN
iajs-1822	3	11	alhaitham	alhaitham	PROPN
iajs-1822	3	12	)	)	PUNCT
iajs-1822	3	13	university	university	NOUN
iajs-1822	3	14	of	of	ADP
iajs-1822	3	15	baghdad	baghdad	PROPN
iajs-1822	3	16	abstract	abstract	ADV
iajs-1822	3	17	in	in	ADP
iajs-1822	3	18	this	this	DET
iajs-1822	3	19	paper	paper	NOUN
iajs-1822	3	20	,	,	PUNCT
iajs-1822	3	21	there	there	PRON
iajs-1822	3	22	are	be	VERB
iajs-1822	3	23	   	   	SPACE
iajs-1822	3	24	new	new	ADJ
iajs-1822	3	25	considerations	consideration	NOUN
iajs-1822	3	26	about	about	ADP
iajs-1822	3	27	the	the	DET
iajs-1822	3	28	dual	dual	ADJ
iajs-1822	3	29	of	of	ADP
iajs-1822	3	30	a	a	DET
iajs-1822	3	31	modular	modular	ADJ
iajs-1822	3	32	spaces	space	NOUN
iajs-1822	3	33	and	and	CCONJ
iajs-1822	3	34	weak	weak	ADJ
iajs-1822	3	35	convergence	convergence	NOUN
iajs-1822	3	36	.	.	PUNCT
iajs-1822	4	1	two	two	NUM
iajs-1822	4	2	common	common	ADJ
iajs-1822	4	3	fixed	fix	VERB
iajs-1822	4	4	point	point	NOUN
iajs-1822	4	5	theorems	theorem	NOUN
iajs-1822	4	6	for	for	SCONJ
iajs-1822	4	7	a	a	DET
iajs-1822	4	8	𝑃-non	𝑃-non	ADJ
iajs-1822	4	9	-	-	ADJ
iajs-1822	4	10	expansive	expansive	ADJ
iajs-1822	4	11	mapping	mapping	NOUN
iajs-1822	4	12	defined	define	VERB
iajs-1822	4	13	on	on	ADP
iajs-1822	4	14	a	a	DET
iajs-1822	4	15	star	star	NOUN
iajs-1822	4	16	-	-	PUNCT
iajs-1822	4	17	shaped	shape	VERB
iajs-1822	4	18	weakly	weakly	ADJ
iajs-1822	4	19	compact	compact	ADJ
iajs-1822	4	20	subset	subset	NOUN
iajs-1822	4	21	are	be	AUX
iajs-1822	4	22	proved	prove	VERB
iajs-1822	4	23	,	,	PUNCT
iajs-1822	4	24	here	here	ADV
iajs-1822	4	25	the	the	DET
iajs-1822	4	26	conditions	condition	NOUN
iajs-1822	4	27	of	of	ADP
iajs-1822	4	28	affineness	affineness	NOUN
iajs-1822	4	29	,	,	PUNCT
iajs-1822	4	30	demi	demi	NOUN
iajs-1822	4	31	-	-	PUNCT
iajs-1822	4	32	closedness	closedness	ADJ
iajs-1822	4	33	and	and	CCONJ
iajs-1822	4	34	opial	opial	NOUN
iajs-1822	4	35	's	's	PART
iajs-1822	4	36	property	property	NOUN
iajs-1822	4	37	play	play	VERB
iajs-1822	4	38	an	an	DET
iajs-1822	4	39	active	active	ADJ
iajs-1822	4	40	role	role	NOUN
iajs-1822	4	41	in	in	ADP
iajs-1822	4	42	the	the	DET
iajs-1822	4	43	proving	prove	VERB
iajs-1822	4	44	our	our	PRON
iajs-1822	4	45	results	result	NOUN
iajs-1822	4	46	.	.	PUNCT
iajs-1822	5	1	keywords	keyword	NOUN
iajs-1822	5	2	:	:	PUNCT
iajs-1822	5	3	modular	modular	ADJ
iajs-1822	5	4	spaces	space	NOUN
iajs-1822	5	5	,	,	PUNCT
iajs-1822	5	6	fixed	fix	VERB
iajs-1822	5	7	points	point	NOUN
iajs-1822	5	8	,	,	PUNCT
iajs-1822	5	9	best	good	ADJ
iajs-1822	5	10	approximations	approximation	NOUN
iajs-1822	5	11	.	.	PUNCT
iajs-1822	6	1	ihsciconf	ihsciconf	PROPN
iajs-1822	6	2	2017	2017	NUM
iajs-1822	6	3	special	special	ADJ
iajs-1822	6	4	issue	issue	NOUN
iajs-1822	6	5	ibn	ibn	PROPN
iajs-1822	6	6	al	al	PROPN
iajs-1822	6	7	-	-	PUNCT
iajs-1822	6	8	haitham	haitham	PROPN
iajs-1822	6	9	journal	journal	PROPN
iajs-1822	6	10	for	for	ADP
iajs-1822	6	11	pure	pure	ADJ
iajs-1822	6	12	and	and	CCONJ
iajs-1822	6	13	applied	apply	VERB
iajs-1822	6	14	science	science	NOUN
iajs-1822	6	15	https://doi.org/	https://doi.org/	NOUN
iajs-1822	6	16	10.30526/2017.ihsciconf.1822	10.30526/2017.ihsciconf.1822	NOUN
iajs-1822	6	17	for	for	ADP
iajs-1822	6	18	more	more	ADJ
iajs-1822	6	19	information	information	NOUN
iajs-1822	6	20	about	about	ADP
iajs-1822	6	21	the	the	DET
iajs-1822	6	22	conference	conference	NOUN
iajs-1822	6	23	please	please	INTJ
iajs-1822	6	24	visit	visit	VERB
iajs-1822	6	25	the	the	DET
iajs-1822	6	26	websites	website	NOUN
iajs-1822	6	27	:	:	PUNCT
iajs-1822	6	28	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1822	6	29	  	  	SPACE
iajs-1822	6	30	www.ihsciconf.org	www.ihsciconf.org	ADJ
iajs-1822	6	31	   	   	SPACE
iajs-1822	6	32	mathematics|501	mathematics|501	NOUN
iajs-1822	6	33	    	    	SPACE
iajs-1822	6	34	1	1	NUM
iajs-1822	6	35	.	.	PUNCT
iajs-1822	7	1	introduction	introduction	NOUN
iajs-1822	7	2	and	and	CCONJ
iajs-1822	7	3	preliminaries	preliminary	NOUN
iajs-1822	7	4	dotson	dotson	PROPN
iajs-1822	7	5	1	1	NUM
iajs-1822	7	6	proved	prove	VERB
iajs-1822	7	7	existence	existence	NOUN
iajs-1822	7	8	of	of	ADP
iajs-1822	7	9	fixed	fix	VERB
iajs-1822	7	10	points	point	NOUN
iajs-1822	7	11	for	for	ADP
iajs-1822	7	12	non	non	ADJ
iajs-1822	7	13	-	-	ADJ
iajs-1822	7	14	expansive	expansive	ADJ
iajs-1822	7	15	self	self	NOUN
iajs-1822	7	16	-	-	PUNCT
iajs-1822	7	17	mappings	mapping	NOUN
iajs-1822	7	18	of	of	ADP
iajs-1822	7	19	starshaped	starshape	VERB
iajs-1822	7	20	subsets	subset	NOUN
iajs-1822	7	21	of	of	ADP
iajs-1822	7	22	banach	banach	NOUN
iajs-1822	7	23	spaces(under	spaces(under	NOUN
iajs-1822	7	24	appropriate	appropriate	ADJ
iajs-1822	7	25	conditions	condition	NOUN
iajs-1822	7	26	)	)	PUNCT
iajs-1822	7	27	.	.	PUNCT
iajs-1822	8	1	subrahmanyam	subrahmanyam	NOUN
iajs-1822	8	2	2	2	NUM
iajs-1822	8	3	and	and	CCONJ
iajs-1822	8	4	habinak	habinak	PROPN
iajs-1822	8	5	3	3	NUM
iajs-1822	8	6	used	use	VERB
iajs-1822	8	7	the	the	DET
iajs-1822	8	8	concept	concept	NOUN
iajs-1822	8	9	of	of	ADP
iajs-1822	8	10	banach	banach	NOUN
iajs-1822	8	11	operator	operator	NOUN
iajs-1822	8	12	to	to	PART
iajs-1822	8	13	generalize	generalize	VERB
iajs-1822	8	14	dotson	dotson	PROPN
iajs-1822	8	15	's	's	PART
iajs-1822	8	16	theorem	theorem	NOUN
iajs-1822	8	17	and	and	CCONJ
iajs-1822	8	18	its	its	PRON
iajs-1822	8	19	application	application	NOUN
iajs-1822	8	20	to	to	ADP
iajs-1822	8	21	invariant	invariant	ADJ
iajs-1822	8	22	approximation	approximation	NOUN
iajs-1822	8	23	.	.	PUNCT
iajs-1822	9	1	recently	recently	ADV
iajs-1822	9	2	,	,	PUNCT
iajs-1822	9	3	abed	abe	VERB
iajs-1822	9	4	[	[	PUNCT
iajs-1822	9	5	4	4	X
iajs-1822	9	6	]	]	PUNCT
iajs-1822	9	7	introduced	introduce	VERB
iajs-1822	9	8	the	the	DET
iajs-1822	9	9	notion	notion	NOUN
iajs-1822	9	10	of	of	ADP
iajs-1822	9	11	best	good	ADJ
iajs-1822	9	12	approximation	approximation	NOUN
iajs-1822	9	13	in	in	ADP
iajs-1822	9	14	modular	modular	ADJ
iajs-1822	9	15	spaces	space	NOUN
iajs-1822	9	16	and	and	CCONJ
iajs-1822	9	17	gave	give	VERB
iajs-1822	9	18	conditions	condition	NOUN
iajs-1822	9	19	to	to	ADP
iajs-1822	9	20	existences	existence	NOUN
iajs-1822	9	21	of	of	ADP
iajs-1822	9	22	proximinal	proximinal	ADJ
iajs-1822	9	23	and	and	CCONJ
iajs-1822	9	24	chebysev	chebysev	ADJ
iajs-1822	9	25	sets	set	NOUN
iajs-1822	9	26	in	in	ADP
iajs-1822	9	27	finite	finite	ADJ
iajs-1822	9	28	dimension	dimension	NOUN
iajs-1822	9	29	modular	modular	ADJ
iajs-1822	9	30	spaces	space	NOUN
iajs-1822	9	31	.	.	PUNCT
iajs-1822	10	1	also	also	ADV
iajs-1822	10	2	,	,	PUNCT
iajs-1822	10	3	abed	abe	VERB
iajs-1822	10	4	and	and	CCONJ
iajs-1822	10	5	  	  	SPACE
iajs-1822	10	6	abdul	abdul	PROPN
iajs-1822	10	7	sada	sada	PROPN
iajs-1822	11	1	[	[	X
iajs-1822	11	2	5	5	NUM
iajs-1822	11	3	-	-	SYM
iajs-1822	11	4	7	7	NUM
iajs-1822	11	5	]	]	PUNCT
iajs-1822	11	6	  	  	SPACE
iajs-1822	11	7	proved	prove	VERB
iajs-1822	11	8	a	a	DET
iajs-1822	11	9	theorem	theorem	NOUN
iajs-1822	11	10	of	of	ADP
iajs-1822	11	11	brosowski	brosowski	ADJ
iajs-1822	11	12	-	-	PUNCT
iajs-1822	11	13	meinaraus	meinaraus	NOUN
iajs-1822	11	14	type	type	NOUN
iajs-1822	11	15	on	on	ADP
iajs-1822	11	16	invariant	invariant	ADJ
iajs-1822	11	17	approximation	approximation	NOUN
iajs-1822	11	18	,	,	PUNCT
iajs-1822	11	19	proved	prove	VERB
iajs-1822	11	20	that	that	SCONJ
iajs-1822	11	21	two	two	NUM
iajs-1822	11	22	fixed	fix	VERB
iajs-1822	11	23	point	point	NOUN
iajs-1822	11	24	theorems	theorem	NOUN
iajs-1822	11	25	for	for	ADP
iajs-1822	11	26	compact	compact	ADJ
iajs-1822	11	27	set	set	NOUN
iajs-1822	11	28	-	-	PUNCT
iajs-1822	11	29	valued	value	VERB
iajs-1822	11	30	mappings	mapping	NOUN
iajs-1822	11	31	in	in	ADP
iajs-1822	11	32	modular	modular	ADJ
iajs-1822	11	33	spaces	space	NOUN
iajs-1822	11	34	with	with	ADP
iajs-1822	11	35	an	an	DET
iajs-1822	11	36	application	application	NOUN
iajs-1822	11	37	on	on	ADP
iajs-1822	11	38	invariant	invariant	ADJ
iajs-1822	11	39	best	good	ADJ
iajs-1822	11	40	approximation	approximation	NOUN
iajs-1822	11	41	.	.	PUNCT
iajs-1822	12	1	the	the	DET
iajs-1822	12	2	object	object	NOUN
iajs-1822	12	3	of	of	ADP
iajs-1822	12	4	the	the	DET
iajs-1822	12	5	present	present	ADJ
iajs-1822	12	6	paper	paper	NOUN
iajs-1822	12	7	is	be	AUX
iajs-1822	12	8	to	to	PART
iajs-1822	12	9	extend	extend	VERB
iajs-1822	12	10	and	and	CCONJ
iajs-1822	12	11	unified	unify	VERB
iajs-1822	12	12	the	the	DET
iajs-1822	12	13	above	above	ADJ
iajs-1822	12	14	results	result	NOUN
iajs-1822	12	15	[	[	X
iajs-1822	12	16	2	2	NUM
iajs-1822	12	17	]	]	PUNCT
iajs-1822	12	18	,	,	PUNCT
iajs-1822	12	19	[	[	X
iajs-1822	12	20	3	3	NUM
iajs-1822	12	21	]	]	PUNCT
iajs-1822	12	22	,	,	PUNCT
iajs-1822	12	23	[	[	X
iajs-1822	12	24	4	4	X
iajs-1822	12	25	]	]	PUNCT
iajs-1822	12	26	and	and	CCONJ
iajs-1822	12	27	others	other	NOUN
iajs-1822	12	28	to	to	ADP
iajs-1822	12	29	modular	modular	ADJ
iajs-1822	12	30	spaces	space	NOUN
iajs-1822	12	31	.	.	PUNCT
iajs-1822	13	1	for	for	ADP
iajs-1822	13	2	other	other	ADJ
iajs-1822	13	3	results	result	NOUN
iajs-1822	13	4	in	in	ADP
iajs-1822	13	5	this	this	DET
iajs-1822	13	6	field	field	NOUN
iajs-1822	13	7	see	see	VERB
iajs-1822	13	8	[	[	X
iajs-1822	13	9	8][10	8][10	NOUN
iajs-1822	13	10	]	]	X
iajs-1822	13	11	definition	definition	NOUN
iajs-1822	13	12	(	(	PUNCT
iajs-1822	13	13	1.1)[5	1.1)[5	NUM
iajs-1822	13	14	]	]	X
iajs-1822	13	15	:	:	PUNCT
iajs-1822	13	16	let𝑀	let𝑀	NOUN
iajs-1822	13	17	be	be	AUX
iajs-1822	13	18	a	a	DET
iajs-1822	13	19	linear	linear	ADJ
iajs-1822	13	20	space	space	NOUN
iajs-1822	13	21	over𝐹	over𝐹	PROPN
iajs-1822	13	22	𝑅	𝑅	PROPN
iajs-1822	13	23	𝑜𝑟	𝑜𝑟	NOUN
iajs-1822	13	24	₵	₵	NOUN
iajs-1822	13	25	.	.	PUNCT
iajs-1822	14	1	a	a	DET
iajs-1822	14	2	function𝛾:𝑀	function𝛾:𝑀	NOUN
iajs-1822	14	3	→	→	SYM
iajs-1822	14	4	0	0	NUM
iajs-1822	14	5	,	,	PUNCT
iajs-1822	14	6	∞	∞	PROPN
iajs-1822	14	7	is	be	AUX
iajs-1822	14	8	called	call	VERB
iajs-1822	14	9	modular	modular	ADJ
iajs-1822	14	10	if	if	SCONJ
iajs-1822	14	11	i.𝛾	i.𝛾	PROPN
iajs-1822	14	12	𝑣	𝑣	ADP
iajs-1822	14	13	0if	0if	NOUN
iajs-1822	14	14	and	and	CCONJ
iajs-1822	14	15	only	only	ADV
iajs-1822	14	16	if𝑣	if𝑣	ADJ
iajs-1822	14	17	0	0	NUM
iajs-1822	14	18	;	;	PUNCT
iajs-1822	14	19	ii.𝛾	ii.𝛾	NOUN
iajs-1822	14	20	𝛼𝑣	𝛼𝑣	X
iajs-1822	14	21	=	=	SYM
iajs-1822	14	22	𝛼	𝛼	PART
iajs-1822	14	23	𝑣	𝑣	NOUN
iajs-1822	14	24	for	for	ADP
iajs-1822	14	25	𝛼	𝛼	PROPN
iajs-1822	14	26	∈𝐹	∈𝐹	NOUN
iajs-1822	14	27	with	with	ADP
iajs-1822	14	28	|𝛼|	|𝛼|	PROPN
iajs-1822	14	29	1	1	NUM
iajs-1822	14	30	,	,	PUNCT
iajs-1822	14	31	for	for	ADP
iajs-1822	14	32	all	all	DET
iajs-1822	14	33	𝛼	𝛼	PRON
iajs-1822	14	34	∈	∈	NOUN
iajs-1822	14	35	𝐹	𝐹	PROPN
iajs-1822	14	36	;	;	PUNCT
iajs-1822	14	37	iii.𝛾	iii.𝛾	NUM
iajs-1822	14	38	𝛼𝑣	𝛼𝑣	NUM
iajs-1822	14	39	𝛽𝑢	𝛽𝑢	ADP
iajs-1822	14	40	𝛾	𝛾	NOUN
iajs-1822	14	41	𝑣	𝑣	ADP
iajs-1822	14	42	+	+	CCONJ
iajs-1822	14	43	𝛾	𝛾	NUM
iajs-1822	14	44	𝑢	𝑢	NUM
iajs-1822	14	45	iff	iff	NOUN
iajs-1822	14	46	𝛼	𝛼	PROPN
iajs-1822	14	47	,	,	PUNCT
iajs-1822	14	48	𝛽	𝛽	PROPN
iajs-1822	14	49	0	0	NUM
iajs-1822	14	50	,	,	PUNCT
iajs-1822	14	51	for	for	ADP
iajs-1822	14	52	all	all	PRON
iajs-1822	14	53	,	,	PUNCT
iajs-1822	14	54	∈	∈	PROPN
iajs-1822	14	55	𝑀.	𝑀.	NOUN
iajs-1822	14	56	if	if	SCONJ
iajs-1822	14	57	(	(	PUNCT
iajs-1822	14	58	iii	iii	NOUN
iajs-1822	14	59	)	)	PUNCT
iajs-1822	14	60	replaced	replace	VERB
iajs-1822	14	61	by	by	ADP
iajs-1822	14	62	(	(	PUNCT
iajs-1822	14	63	iii´	iii´	PROPN
iajs-1822	14	64	)	)	PUNCT
iajs-1822	14	65	𝛾	𝛾	ADP
iajs-1822	14	66	𝛼𝑣	𝛼𝑣	ADP
iajs-1822	14	67	𝛽𝑢	𝛽𝑢	ADP
iajs-1822	14	68	𝛼𝛾	𝛼𝛾	NOUN
iajs-1822	14	69	𝑣	𝑣	ADP
iajs-1822	14	70	+	+	NOUN
iajs-1822	14	71	𝛽𝛾	𝛽𝛾	X
iajs-1822	14	72	𝑢	𝑢	X
iajs-1822	14	73	,	,	PUNCT
iajs-1822	14	74	for	for	ADP
iajs-1822	14	75	𝛼	𝛼	X
iajs-1822	14	76	,	,	PUNCT
iajs-1822	14	77	𝛽	𝛽	PROPN
iajs-1822	14	78	0	0	NUM
iajs-1822	14	79	,	,	PUNCT
iajs-1822	14	80	𝛼	𝛼	PROPN
iajs-1822	14	81	𝛽	𝛽	NOUN
iajs-1822	14	82	1	1	NUM
iajs-1822	14	83	,	,	PUNCT
iajs-1822	14	84	for	for	ADP
iajs-1822	14	85	all	all	DET
iajs-1822	14	86	𝑣,u	𝑣,u	NOUN
iajs-1822	14	87	∈𝑀	∈𝑀	NOUN
iajs-1822	14	88	then	then	ADV
iajs-1822	14	89	𝑀	𝑀	PROPN
iajs-1822	14	90	modular	modular	NOUN
iajs-1822	14	91	𝛾	𝛾	PROPN
iajs-1822	14	92	is	be	AUX
iajs-1822	14	93	called	call	VERB
iajs-1822	14	94	con𝑣ex	con𝑣ex	PROPN
iajs-1822	14	95	modular	modular	NOUN
iajs-1822	14	96	.	.	PUNCT
iajs-1822	15	1	definition	definition	NOUN
iajs-1822	15	2	1.2	1.2	NUM
iajs-1822	16	1	[	[	X
iajs-1822	16	2	6	6	NUM
iajs-1822	16	3	]	]	PUNCT
iajs-1822	16	4	a	a	DET
iajs-1822	16	5	modular	modular	NOUN
iajs-1822	16	6	𝛾	𝛾	NOUN
iajs-1822	16	7	defines	define	VERB
iajs-1822	16	8	a	a	DET
iajs-1822	16	9	corresponding	corresponding	ADJ
iajs-1822	16	10	modular	modular	NOUN
iajs-1822	16	11	space,𝑡ℎ𝑒𝑛,the	space,𝑡ℎ𝑒𝑛,the	DET
iajs-1822	16	12	space𝑀	space𝑀	PROPN
iajs-1822	16	13	given	give	VERB
iajs-1822	16	14	by	by	ADP
iajs-1822	16	15	𝑀	𝑀	PROPN
iajs-1822	16	16	𝑣	𝑣	PROPN
iajs-1822	16	17	∈	∈	PROPN
iajs-1822	16	18	𝑀	𝑀	PROPN
iajs-1822	16	19	:	:	PUNCT
iajs-1822	16	20	𝛾	𝛾	NOUN
iajs-1822	16	21	𝛼𝑣	𝛼𝑣	NOUN
iajs-1822	16	22	→	→	SYM
iajs-1822	16	23	0	0	NUM
iajs-1822	16	24	𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟	𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟	NOUN
iajs-1822	16	25	𝛼	𝛼	NOUN
iajs-1822	16	26	→	→	SYM
iajs-1822	16	27	0	0	NUM
iajs-1822	16	28	.	.	PUNCT
iajs-1822	17	1	remark	remark	PROPN
iajs-1822	17	2	1.1[6	1.1[6	PROPN
iajs-1822	17	3	]	]	PUNCT
iajs-1822	17	4	by	by	ADP
iajs-1822	17	5	condition	condition	NOUN
iajs-1822	17	6	(	(	PUNCT
iajs-1822	17	7	iii	iii	NOUN
iajs-1822	17	8	)	)	PUNCT
iajs-1822	17	9	above	above	ADV
iajs-1822	17	10	,	,	PUNCT
iajs-1822	17	11	if	if	SCONJ
iajs-1822	17	12	u	u	NOUN
iajs-1822	17	13	0	0	VERB
iajs-1822	17	14	then	then	ADV
iajs-1822	17	15	𝛾	𝛾	ADP
iajs-1822	17	16	𝛼𝑣	𝛼𝑣	NOUN
iajs-1822	18	1	=	=	NOUN
iajs-1822	18	2	𝛾	𝛾	NOUN
iajs-1822	18	3	𝛽𝑣	𝛽𝑣	NOUN
iajs-1822	18	4	𝛾	𝛾	NOUN
iajs-1822	18	5	𝛽𝑣	𝛽𝑣	ADV
iajs-1822	18	6	,	,	PUNCT
iajs-1822	18	7	for	for	ADP
iajs-1822	18	8	all	all	DET
iajs-1822	18	9	𝛼	𝛼	PROPN
iajs-1822	18	10	,	,	PUNCT
iajs-1822	18	11	𝛽	𝛽	NOUN
iajs-1822	18	12	𝑖𝑛	𝑖𝑛	PRON
iajs-1822	18	13	𝐹	𝐹	PROPN
iajs-1822	18	14	,	,	PUNCT
iajs-1822	18	15	0	0	NUM
iajs-1822	18	16	𝛼	𝛼	NOUN
iajs-1822	18	17	𝛽.this	𝛽.this	PRON
iajs-1822	18	18	shows	show	VERB
iajs-1822	18	19	that	that	SCONJ
iajs-1822	18	20	𝛾	𝛾	NOUN
iajs-1822	18	21	is	be	AUX
iajs-1822	18	22	increasing	increase	VERB
iajs-1822	18	23	function	function	NOUN
iajs-1822	18	24	.	.	PUNCT
iajs-1822	19	1	definition	definition	NOUN
iajs-1822	19	2	1.3[6	1.3[6	NUM
iajs-1822	19	3	]	]	X
iajs-1822	19	4	the	the	DET
iajs-1822	19	5	𝛾-ball	𝛾-ball	NOUN
iajs-1822	19	6	,	,	PUNCT
iajs-1822	19	7	𝐵	𝐵	NOUN
iajs-1822	19	8	𝑢	𝑢	NOUN
iajs-1822	19	9	centered	center	VERB
iajs-1822	19	10	at	at	ADP
iajs-1822	19	11	𝑢	𝑢	PROPN
iajs-1822	19	12	∈	∈	PROPN
iajs-1822	19	13	𝑀	𝑀	PROPN
iajs-1822	19	14	with	with	ADP
iajs-1822	19	15	radius	radius	NOUN
iajs-1822	19	16	𝑟	𝑟	NOUN
iajs-1822	19	17	0	0	PUNCT
iajs-1822	19	18	as	as	SCONJ
iajs-1822	19	19	𝐵	𝐵	NOUN
iajs-1822	19	20	𝑢	𝑢	ADP
iajs-1822	19	21	𝒗	𝒗	PROPN
iajs-1822	19	22	∈	∈	PROPN
iajs-1822	19	23	𝑀	𝑀	PROPN
iajs-1822	19	24	;	;	PUNCT
iajs-1822	19	25	𝛾	𝛾	ADP
iajs-1822	19	26	𝑢	𝑢	X
iajs-1822	19	27	𝑣	𝑣	ADP
iajs-1822	19	28	𝑟	𝑟	NOUN
iajs-1822	19	29	.	.	PUNCT
iajs-1822	20	1	the	the	DET
iajs-1822	20	2	class	class	NOUN
iajs-1822	20	3	of	of	ADP
iajs-1822	20	4	all	all	DET
iajs-1822	20	5	𝛾-balls	𝛾-ball	NOUN
iajs-1822	20	6	in	in	ADP
iajs-1822	20	7	a	a	DET
iajs-1822	20	8	modular	modular	NOUN
iajs-1822	20	9	space𝑀	space𝑀	PROPN
iajs-1822	20	10	generates	generate	VERB
iajs-1822	20	11	a	a	DET
iajs-1822	20	12	topology	topology	NOUN
iajs-1822	20	13	which	which	PRON
iajs-1822	20	14	makes	make	VERB
iajs-1822	20	15	𝑀	𝑀	PROPN
iajs-1822	20	16	hausdorff	hausdorff	VERB
iajs-1822	20	17	topological	topological	PROPN
iajs-1822	20	18	linear	linear	PROPN
iajs-1822	20	19	space	space	NOUN
iajs-1822	20	20	.	.	PUNCT
iajs-1822	21	1	every	every	DET
iajs-1822	21	2	𝛾-ball	𝛾-ball	NOUN
iajs-1822	21	3	is	be	AUX
iajs-1822	21	4	convex	convex	NOUN
iajs-1822	21	5	set	set	NOUN
iajs-1822	21	6	,	,	PUNCT
iajs-1822	21	7	therefore	therefore	ADV
iajs-1822	21	8	every	every	DET
iajs-1822	21	9	modular	modular	ADJ
iajs-1822	21	10	space	space	NOUN
iajs-1822	21	11	locally	locally	ADV
iajs-1822	21	12	convex	convex	VERB
iajs-1822	21	13	hausdorff	hausdorff	PROPN
iajs-1822	21	14	topological	topological	ADJ
iajs-1822	21	15	vector	vector	NOUN
iajs-1822	21	16	space	space	NOUN
iajs-1822	21	17	[	[	X
iajs-1822	21	18	4	4	NUM
iajs-1822	21	19	]	]	PUNCT
iajs-1822	21	20	.	.	PUNCT
iajs-1822	22	1	definition	definition	NOUN
iajs-1822	22	2	1.5[6	1.5[6	NUM
iajs-1822	22	3	]	]	PUNCT
iajs-1822	22	4	let	let	VERB
iajs-1822	22	5	𝑀	𝑀	PRON
iajs-1822	22	6	be	be	AUX
iajs-1822	22	7	a	a	DET
iajs-1822	22	8	modular	modular	ADJ
iajs-1822	22	9	spase	spase	NOUN
iajs-1822	22	10	.	.	PUNCT
iajs-1822	23	1	a	a	PRON
iajs-1822	23	2	)	)	PUNCT
iajs-1822	23	3	a	a	DET
iajs-1822	23	4	sequence	sequence	NOUN
iajs-1822	23	5	𝑣	𝑣	ADP
iajs-1822	23	6	⊂	⊂	PROPN
iajs-1822	23	7	𝑀	𝑀	PROPN
iajs-1822	23	8	is	be	AUX
iajs-1822	23	9	said	say	VERB
iajs-1822	23	10	to	to	PART
iajs-1822	23	11	be	be	AUX
iajs-1822	23	12	𝛾	𝛾	ADP
iajs-1822	23	13	-convergent	-convergent	ADJ
iajs-1822	23	14	to	to	ADP
iajs-1822	23	15	𝑣	𝑣	DET
iajs-1822	23	16	∈	∈	PROPN
iajs-1822	23	17	𝑀	𝑀	PROPN
iajs-1822	23	18	and	and	CCONJ
iajs-1822	23	19	write	write	VERB
iajs-1822	23	20	𝑣	𝑣	PRON
iajs-1822	23	21	→	→	SYM
iajs-1822	23	22	𝑣	𝑣	X
iajs-1822	23	23	if	if	SCONJ
iajs-1822	23	24	𝛾	𝛾	ADP
iajs-1822	23	25	𝑣	𝑣	DET
iajs-1822	23	26	𝑣	𝑣	X
iajs-1822	23	27	→	→	SYM
iajs-1822	23	28	0	0	PROPN
iajs-1822	23	29	as	as	SCONJ
iajs-1822	23	30	n→	n→	PROPN
iajs-1822	23	31	∞.	∞.	PROPN
iajs-1822	23	32	b	b	NUM
iajs-1822	23	33	)	)	PUNCT
iajs-1822	23	34	a	a	DET
iajs-1822	23	35	sequence	sequence	NOUN
iajs-1822	23	36	{	{	PUNCT
iajs-1822	23	37	𝑣	𝑣	NOUN
iajs-1822	23	38	}	}	PUNCT
iajs-1822	23	39	is	be	AUX
iajs-1822	23	40	called	call	VERB
iajs-1822	23	41	𝛾ــ	𝛾ــ	NOUN
iajs-1822	23	42	cauchy	cauchy	PROPN
iajs-1822	23	43	whenever	whenever	SCONJ
iajs-1822	23	44	𝛾(𝑣	𝛾(𝑣	PROPN
iajs-1822	23	45	-𝑣	-𝑣	PROPN
iajs-1822	23	46	)	)	PUNCT
iajs-1822	23	47	→	→	SYM
iajs-1822	23	48	0	0	PUNCT
iajs-1822	23	49	as	as	SCONJ
iajs-1822	23	50	,	,	PUNCT
iajs-1822	23	51	𝑚	𝑚	X
iajs-1822	23	52	→	→	SYM
iajs-1822	23	53	∞.	∞.	PROPN
iajs-1822	23	54	c	c	PROPN
iajs-1822	23	55	)	)	PUNCT
iajs-1822	23	56	𝑀	𝑀	PROPN
iajs-1822	23	57	is	be	AUX
iajs-1822	23	58	called	call	VERB
iajs-1822	23	59	𝛾ــ	𝛾ــ	NOUN
iajs-1822	23	60	complete	complete	ADJ
iajs-1822	23	61	if	if	SCONJ
iajs-1822	23	62	any	any	DET
iajs-1822	23	63	𝛾ــ	𝛾ــ	NOUN
iajs-1822	23	64	cauchy	cauchy	NOUN
iajs-1822	23	65	sequence	sequence	NOUN
iajs-1822	23	66	in	in	ADP
iajs-1822	23	67	𝑀	𝑀	PROPN
iajs-1822	23	68	is	be	AUX
iajs-1822	23	69	𝛾ــ	𝛾ــ	NOUN
iajs-1822	23	70	convergent	convergent	NOUN
iajs-1822	23	71	.	.	PUNCT
iajs-1822	24	1	d	d	X
iajs-1822	24	2	)	)	PUNCT
iajs-1822	24	3	a	a	DET
iajs-1822	24	4	subset	subset	NOUN
iajs-1822	24	5	𝐵⊂𝑀	𝐵⊂𝑀	NOUN
iajs-1822	24	6	is	be	AUX
iajs-1822	24	7	called	call	VERB
iajs-1822	24	8	𝛾ــ	𝛾ــ	NOUN
iajs-1822	24	9	closed	close	VERB
iajs-1822	24	10	if	if	SCONJ
iajs-1822	24	11	for	for	ADP
iajs-1822	24	12	any	any	DET
iajs-1822	24	13	sequence	sequence	NOUN
iajs-1822	24	14	𝑣	𝑣	ADP
iajs-1822	24	15	⊂𝐵𝛾ــ	⊂𝐵𝛾ــ	NOUN
iajs-1822	24	16	convergent	convergent	NOUN
iajs-1822	24	17	to	to	ADP
iajs-1822	24	18	∈	∈	PROPN
iajs-1822	24	19	𝑀	𝑀	PROPN
iajs-1822	24	20	,	,	PUNCT
iajs-1822	24	21	we	we	PRON
iajs-1822	24	22	ha𝑣e	ha𝑣e	VERB
iajs-1822	24	23	𝑣	𝑣	ADP
iajs-1822	24	24	∈	∈	PROPN
iajs-1822	24	25	𝐵.	𝐵.	PROPN
iajs-1822	24	26	e	e	PROPN
iajs-1822	24	27	)	)	PUNCT
iajs-1822	24	28	a	a	DET
iajs-1822	24	29	𝛾ــ	𝛾ــ	NOUN
iajs-1822	24	30	closed	close	VERB
iajs-1822	24	31	subset	subset	NOUN
iajs-1822	24	32	𝐵⊂𝑀	𝐵⊂𝑀	NOUN
iajs-1822	24	33	is	be	AUX
iajs-1822	24	34	called	call	VERB
iajs-1822	24	35	𝛾ــ	𝛾ــ	NOUN
iajs-1822	24	36	compact	compact	ADJ
iajs-1822	24	37	if	if	SCONJ
iajs-1822	24	38	any	any	DET
iajs-1822	24	39	sequence	sequence	NOUN
iajs-1822	24	40	{	{	PUNCT
iajs-1822	24	41	𝑣	𝑣	PART
iajs-1822	24	42	}	}	PUNCT
iajs-1822	24	43	⊂𝐵	⊂𝐵	PROPN
iajs-1822	24	44	has	have	VERB
iajs-1822	24	45	a	a	DET
iajs-1822	24	46	𝛾ــ	𝛾ــ	NOUN
iajs-1822	24	47	convergent	convergent	NOUN
iajs-1822	24	48	subsequence	subsequence	NOUN
iajs-1822	24	49	.	.	PUNCT
iajs-1822	25	1	f	f	X
iajs-1822	25	2	)	)	PUNCT
iajs-1822	25	3	a	a	DET
iajs-1822	25	4	subset	subset	ADJ
iajs-1822	25	5	𝐵⊂	𝐵⊂	PROPN
iajs-1822	25	6	𝑀	𝑀	PROPN
iajs-1822	25	7	is	be	AUX
iajs-1822	25	8	said	say	VERB
iajs-1822	25	9	to	to	PART
iajs-1822	25	10	be	be	AUX
iajs-1822	25	11	𝛾ــ	𝛾ــ	NOUN
iajs-1822	25	12	bounded	bound	VERB
iajs-1822	25	13	if	if	SCONJ
iajs-1822	25	14	𝑑𝑎𝑖𝑚	𝑑𝑎𝑖𝑚	NOUN
iajs-1822	25	15	𝐵	𝐵	NOUN
iajs-1822	25	16	∞	∞	PROPN
iajs-1822	25	17	,	,	PUNCT
iajs-1822	25	18	where	where	SCONJ
iajs-1822	25	19	𝑑𝑎𝑖𝑚	𝑑𝑎𝑖𝑚	NOUN
iajs-1822	25	20	𝐵	𝐵	NOUN
iajs-1822	25	21	sup	sup	NOUN
iajs-1822	25	22	𝛾	𝛾	PROPN
iajs-1822	25	23	𝑣	𝑣	PRON
iajs-1822	25	24	𝑢	𝑢	NOUN
iajs-1822	25	25	;	;	PUNCT
iajs-1822	25	26	𝑣	𝑣	X
iajs-1822	25	27	,	,	PUNCT
iajs-1822	25	28	𝑢	𝑢	PROPN
iajs-1822	25	29	∈	∈	PROPN
iajs-1822	25	30	𝐵	𝐵	NOUN
iajs-1822	25	31	is	be	AUX
iajs-1822	25	32	called	call	VERB
iajs-1822	25	33	the	the	DET
iajs-1822	25	34	𝛾ــ	𝛾ــ	NOUN
iajs-1822	25	35	diameter	diameter	NOUN
iajs-1822	25	36	of	of	ADP
iajs-1822	25	37	𝐵.	𝐵.	PROPN
iajs-1822	25	38	ihsciconf	ihsciconf	PROPN
iajs-1822	25	39	2017	2017	NUM
iajs-1822	25	40	special	special	ADJ
iajs-1822	25	41	issue	issue	NOUN
iajs-1822	25	42	ibn	ibn	PROPN
iajs-1822	25	43	al	al	PROPN
iajs-1822	25	44	-	-	PUNCT
iajs-1822	25	45	haitham	haitham	PROPN
iajs-1822	25	46	journal	journal	PROPN
iajs-1822	25	47	for	for	ADP
iajs-1822	25	48	pure	pure	ADJ
iajs-1822	25	49	and	and	CCONJ
iajs-1822	25	50	applied	apply	VERB
iajs-1822	25	51	science	science	NOUN
iajs-1822	25	52	https://doi.org/	https://doi.org/	NOUN
iajs-1822	25	53	10.30526/2017.ihsciconf.1822	10.30526/2017.ihsciconf.1822	NOUN
iajs-1822	25	54	for	for	ADP
iajs-1822	25	55	more	more	ADJ
iajs-1822	25	56	information	information	NOUN
iajs-1822	25	57	about	about	ADP
iajs-1822	25	58	the	the	DET
iajs-1822	25	59	conference	conference	NOUN
iajs-1822	25	60	please	please	INTJ
iajs-1822	25	61	visit	visit	VERB
iajs-1822	25	62	the	the	DET
iajs-1822	25	63	websites	website	NOUN
iajs-1822	25	64	:	:	PUNCT
iajs-1822	25	65	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1822	25	66	  	  	SPACE
iajs-1822	25	67	www.ihsciconf.org	www.ihsciconf.org	ADJ
iajs-1822	25	68	   	   	SPACE
iajs-1822	25	69	mathematics|502	mathematics|502	NOUN
iajs-1822	25	70	    	    	SPACE
iajs-1822	25	71	definition	definition	NOUN
iajs-1822	25	72	(	(	PUNCT
iajs-1822	25	73	1.6	1.6	NUM
iajs-1822	25	74	)	)	PUNCT
iajs-1822	26	1	[	[	X
iajs-1822	26	2	7	7	X
iajs-1822	26	3	]	]	PUNCT
iajs-1822	26	4	let	let	VERB
iajs-1822	26	5	𝑀	𝑀	PROPN
iajs-1822	26	6	be	be	AUX
iajs-1822	26	7	a	a	DET
iajs-1822	26	8	modular	modular	ADJ
iajs-1822	26	9	space	space	NOUN
iajs-1822	26	10	and	and	CCONJ
iajs-1822	26	11	𝐴	𝐴	PROPN
iajs-1822	26	12	⊆	⊆	NUM
iajs-1822	26	13	𝑀	𝑀	PROPN
iajs-1822	26	14	𝑆:𝐴	𝑆:𝐴	PROPN
iajs-1822	26	15	→	→	SYM
iajs-1822	26	16	𝐴	𝐴	PROPN
iajs-1822	26	17	,	,	PUNCT
iajs-1822	26	18	𝑆	𝑆	PROPN
iajs-1822	26	19	is	be	AUX
iajs-1822	26	20	called	call	VERB
iajs-1822	26	21	contraction	contraction	NOUN
iajs-1822	26	22	mapping	mapping	NOUN
iajs-1822	26	23	if	if	SCONJ
iajs-1822	26	24	∃	∃	PROPN
iajs-1822	26	25	h	h	PROPN
iajs-1822	26	26	∈	∈	PROPN
iajs-1822	26	27	0	0	PUNCT
iajs-1822	26	28	,	,	PUNCT
iajs-1822	26	29	1	1	NUM
iajs-1822	26	30	for	for	ADP
iajs-1822	26	31	all	all	DET
iajs-1822	26	32	𝑣	𝑣	NOUN
iajs-1822	26	33	,	,	PUNCT
iajs-1822	26	34	𝑢	𝑢	NOUN
iajs-1822	26	35	in	in	ADP
iajs-1822	26	36	𝑀	𝑀	PROPN
iajs-1822	26	37	.	.	PUNCT
iajs-1822	27	1	such	such	ADJ
iajs-1822	27	2	that	that	PRON
iajs-1822	27	3	𝛾	𝛾	PROPN
iajs-1822	27	4	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	27	5	𝑆𝑢	𝑆𝑢	PROPN
iajs-1822	27	6	ℎ	ℎ	X
iajs-1822	27	7	𝑣	𝑣	DET
iajs-1822	27	8	𝑢	𝑢	NOUN
iajs-1822	27	9	and	and	CCONJ
iajs-1822	27	10	if	if	SCONJ
iajs-1822	27	11	h	h	NOUN
iajs-1822	27	12	1	1	NUM
iajs-1822	27	13	then	then	ADV
iajs-1822	27	14	𝑆	𝑆	PROPN
iajs-1822	27	15	is	be	AUX
iajs-1822	27	16	called	call	VERB
iajs-1822	27	17	a	a	DET
iajs-1822	27	18	non	non	ADJ
iajs-1822	27	19	–	–	PUNCT
iajs-1822	27	20	expansive	expansive	ADJ
iajs-1822	27	21	mapping	mapping	NOUN
iajs-1822	27	22	.	.	PUNCT
iajs-1822	28	1	definition	definition	NOUN
iajs-1822	28	2	(	(	PUNCT
iajs-1822	28	3	1.7	1.7	NUM
iajs-1822	28	4	):	):	PUNCT
iajs-1822	28	5	let	let	VERB
iajs-1822	28	6	𝑀	𝑀	PRON
iajs-1822	28	7	be	be	AUX
iajs-1822	28	8	a	a	DET
iajs-1822	28	9	modular	modular	ADJ
iajs-1822	28	10	space	space	NOUN
iajs-1822	28	11	and	and	CCONJ
iajs-1822	28	12	𝑃	𝑃	NOUN
iajs-1822	28	13	,	,	PUNCT
iajs-1822	28	14	𝑆	𝑆	PROPN
iajs-1822	28	15	:	:	PUNCT
iajs-1822	28	16	𝑀	𝑀	PROPN
iajs-1822	28	17	→	→	SYM
iajs-1822	28	18	𝑀	𝑀	PROPN
iajs-1822	28	19	be	be	VERB
iajs-1822	28	20	a	a	DET
iajs-1822	28	21	mapping	mapping	NOUN
iajs-1822	28	22	then	then	ADV
iajs-1822	28	23	𝑆	𝑆	PROPN
iajs-1822	28	24	is	be	AUX
iajs-1822	28	25	said	say	VERB
iajs-1822	28	26	to	to	PART
iajs-1822	28	27	be	be	AUX
iajs-1822	28	28	𝑃	𝑃	PROPN
iajs-1822	28	29	ــ	ــ	NOUN
iajs-1822	28	30	contraction	contraction	NOUN
iajs-1822	28	31	if	if	SCONJ
iajs-1822	28	32	there	there	PRON
iajs-1822	28	33	exists	exist	VERB
iajs-1822	28	34	h∈	h∈	ADV
iajs-1822	28	35	𝟶	𝟶	NUM
iajs-1822	28	36	,	,	PUNCT
iajs-1822	28	37	1	1	NUM
iajs-1822	28	38	such	such	ADJ
iajs-1822	28	39	that	that	PRON
iajs-1822	28	40	𝛾	𝛾	PROPN
iajs-1822	28	41	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	28	42	𝑆𝑢	𝑆𝑢	PROPN
iajs-1822	28	43	ℎ	ℎ	NOUN
iajs-1822	28	44	𝛾	𝛾	ADP
iajs-1822	28	45	𝑃𝑣	𝑃𝑣	NOUN
iajs-1822	28	46	𝑃𝑢	𝑃𝑢	NOUN
iajs-1822	28	47	∀	∀	NOUN
iajs-1822	28	48	𝑣	𝑣	NOUN
iajs-1822	28	49	,	,	PUNCT
iajs-1822	28	50	𝑢	𝑢	NOUN
iajs-1822	28	51	in	in	ADP
iajs-1822	28	52	𝑀	𝑀	PROPN
iajs-1822	28	53	.	.	PUNCT
iajs-1822	29	1	if	if	SCONJ
iajs-1822	29	2	h	h	NOUN
iajs-1822	29	3	1	1	NUM
iajs-1822	29	4	in	in	ADP
iajs-1822	29	5	(	(	PUNCT
iajs-1822	29	6	1.7	1.7	NUM
iajs-1822	29	7	)	)	PUNCT
iajs-1822	29	8	,	,	PUNCT
iajs-1822	29	9	then	then	ADV
iajs-1822	29	10	𝑆	𝑆	PROPN
iajs-1822	29	11	is	be	AUX
iajs-1822	29	12	called	call	VERB
iajs-1822	29	13	𝑃ــ	𝑃ــ	PROPN
iajs-1822	29	14	non	non	ADJ
iajs-1822	29	15	–	–	PUNCT
iajs-1822	29	16	expansive	expansive	ADJ
iajs-1822	29	17	mapping	mapping	NOUN
iajs-1822	29	18	.	.	PUNCT
iajs-1822	30	1	definition	definition	NOUN
iajs-1822	30	2	(	(	PUNCT
iajs-1822	30	3	1.8	1.8	NUM
iajs-1822	30	4	)	)	PUNCT
iajs-1822	30	5	a	a	NOUN
iajs-1822	30	6	)	)	PUNCT
iajs-1822	30	7	a	a	DET
iajs-1822	30	8	function	function	NOUN
iajs-1822	30	9	𝑆	𝑆	PROPN
iajs-1822	30	10	:	:	PUNCT
iajs-1822	30	11	𝑀	𝑀	PROPN
iajs-1822	30	12	→	→	SYM
iajs-1822	30	13	𝑁	𝑁	PROPN
iajs-1822	30	14	(	(	PUNCT
iajs-1822	30	15	where𝑀	where𝑀	PROPN
iajs-1822	30	16	,	,	PUNCT
iajs-1822	30	17	𝑁	𝑁	PROPN
iajs-1822	30	18	are	be	AUX
iajs-1822	30	19	modular	modular	ADJ
iajs-1822	30	20	spaces	space	NOUN
iajs-1822	30	21	)	)	PUNCT
iajs-1822	30	22	is	be	AUX
iajs-1822	30	23	said	say	VERB
iajs-1822	30	24	to	to	PART
iajs-1822	30	25	be	be	AUX
iajs-1822	30	26	continuous	continuous	ADJ
iajs-1822	30	27	at	at	ADP
iajs-1822	30	28	a	a	DET
iajs-1822	30	29	point	point	NOUN
iajs-1822	30	30	𝑣	𝑣	ADP
iajs-1822	30	31	∈	∈	PROPN
iajs-1822	30	32	𝑀	𝑀	PROPN
iajs-1822	30	33	if	if	SCONJ
iajs-1822	30	34	𝛾	𝛾	ADP
iajs-1822	30	35	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	30	36	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	30	37	→	→	SYM
iajs-1822	30	38	0	0	NUM
iajs-1822	30	39	as	as	ADP
iajs-1822	30	40	n→	n→	ADV
iajs-1822	30	41	∞	∞	PROPN
iajs-1822	30	42	whenever	whenever	SCONJ
iajs-1822	30	43	𝛿	𝛿	PRON
iajs-1822	30	44	𝑣	𝑣	ADP
iajs-1822	30	45	𝑣	𝑣	X
iajs-1822	30	46	→	→	SYM
iajs-1822	30	47	0	0	PROPN
iajs-1822	30	48	as	as	ADP
iajs-1822	30	49	n	n	PROPN
iajs-1822	30	50	→	→	SYM
iajs-1822	30	51	∞.	∞.	PROPN
iajs-1822	30	52	b	b	X
iajs-1822	30	53	)	)	PUNCT
iajs-1822	30	54	a	a	DET
iajs-1822	30	55	mapping	map	VERB
iajs-1822	30	56	𝑆	𝑆	PROPN
iajs-1822	30	57	:	:	PUNCT
iajs-1822	30	58	𝑀	𝑀	PROPN
iajs-1822	30	59	→	→	PUNCT
iajs-1822	30	60	𝑁	𝑁	PROPN
iajs-1822	30	61	is	be	AUX
iajs-1822	30	62	said	say	VERB
iajs-1822	30	63	to	to	PART
iajs-1822	30	64	be	be	AUX
iajs-1822	30	65	affine	affine	NOUN
iajs-1822	30	66	if	if	SCONJ
iajs-1822	30	67	∀𝑣	∀𝑣	PROPN
iajs-1822	30	68	,	,	PUNCT
iajs-1822	30	69	𝑢	𝑢	PROPN
iajs-1822	30	70	in	in	ADP
iajs-1822	30	71	𝑀	𝑀	PROPN
iajs-1822	30	72	and	and	CCONJ
iajs-1822	30	73	∀𝜆	∀𝜆	X
iajs-1822	30	74	,	,	PUNCT
iajs-1822	30	75	0	0	PUNCT
iajs-1822	30	76	𝜆	𝜆	DET
iajs-1822	30	77	1	1	NUM
iajs-1822	30	78	,	,	PUNCT
iajs-1822	30	79	𝑆	𝑆	PROPN
iajs-1822	30	80	𝜆𝑣	𝜆𝑣	NOUN
iajs-1822	30	81	1	1	NUM
iajs-1822	30	82	𝜆	𝜆	NOUN
iajs-1822	30	83	𝑢	𝑢	X
iajs-1822	30	84	𝜆𝑆	𝜆𝑆	VERB
iajs-1822	30	85	𝑣	𝑣	ADP
iajs-1822	30	86	1	1	NUM
iajs-1822	30	87	𝜆	𝜆	DET
iajs-1822	30	88	𝑆	𝑆	PROPN
iajs-1822	30	89	𝑢	𝑢	NOUN
iajs-1822	30	90	.	.	PUNCT
iajs-1822	31	1	definition	definition	NOUN
iajs-1822	31	2	(	(	PUNCT
iajs-1822	31	3	1.9	1.9	NUM
iajs-1822	31	4	):	):	PUNCT
iajs-1822	31	5	a	a	DET
iajs-1822	31	6	two	two	NUM
iajs-1822	31	7	mappings	mapping	NOUN
iajs-1822	31	8	𝑆	𝑆	PROPN
iajs-1822	31	9	and	and	CCONJ
iajs-1822	31	10	𝑃	𝑃	PROPN
iajs-1822	31	11	on	on	ADP
iajs-1822	31	12	𝑀	𝑀	PROPN
iajs-1822	31	13	are	be	AUX
iajs-1822	31	14	said	say	VERB
iajs-1822	31	15	to	to	PART
iajs-1822	31	16	be	be	AUX
iajs-1822	31	17	commute	commute	VERB
iajs-1822	31	18	if	if	SCONJ
iajs-1822	31	19	𝑆𝑃𝑣	𝑆𝑃𝑣	NOUN
iajs-1822	31	20	𝑃𝑆𝑣	𝑃𝑆𝑣	X
iajs-1822	31	21	∀	∀	X
iajs-1822	32	1	𝑣∈	𝑣∈	NUM
iajs-1822	32	2	𝑀	𝑀	PROPN
iajs-1822	32	3	.	.	PUNCT
iajs-1822	33	1	the	the	DET
iajs-1822	33	2	purpose	purpose	NOUN
iajs-1822	33	3	of	of	ADP
iajs-1822	33	4	this	this	DET
iajs-1822	33	5	article	article	NOUN
iajs-1822	33	6	is	be	AUX
iajs-1822	33	7	to	to	PART
iajs-1822	33	8	prove	prove	VERB
iajs-1822	33	9	the	the	DET
iajs-1822	33	10	completeness	completeness	NOUN
iajs-1822	33	11	of	of	ADP
iajs-1822	33	12	dual	dual	ADJ
iajs-1822	33	13	space	space	NOUN
iajs-1822	33	14	of	of	ADP
iajs-1822	33	15	a	a	DET
iajs-1822	33	16	modular	modular	ADJ
iajs-1822	33	17	space	space	NOUN
iajs-1822	33	18	and	and	CCONJ
iajs-1822	33	19	to	to	PART
iajs-1822	33	20	give	give	VERB
iajs-1822	33	21	some	some	DET
iajs-1822	33	22	related	relate	VERB
iajs-1822	33	23	concepts	concept	NOUN
iajs-1822	33	24	and	and	CCONJ
iajs-1822	33	25	properties	property	NOUN
iajs-1822	33	26	,	,	PUNCT
iajs-1822	33	27	also	also	ADV
iajs-1822	33	28	,	,	PUNCT
iajs-1822	33	29	to	to	PART
iajs-1822	33	30	prove	prove	VERB
iajs-1822	33	31	the	the	DET
iajs-1822	33	32	existence	existence	NOUN
iajs-1822	33	33	of	of	ADP
iajs-1822	33	34	common	common	ADJ
iajs-1822	33	35	fixed	fix	VERB
iajs-1822	33	36	points	point	NOUN
iajs-1822	33	37	for	for	ADP
iajs-1822	33	38	pair	pair	NOUN
iajs-1822	33	39	mapping	map	VERB
iajs-1822	33	40	𝑆	𝑆	PROPN
iajs-1822	33	41	,	,	PUNCT
iajs-1822	33	42	𝑃	𝑃	VERB
iajs-1822	33	43	where	where	SCONJ
iajs-1822	33	44	𝑆	𝑆	PROPN
iajs-1822	33	45	is	be	AUX
iajs-1822	33	46	𝑃	𝑃	X
iajs-1822	33	47	non	non	X
iajs-1822	33	48	expansive	expansive	ADJ
iajs-1822	33	49	.	.	PUNCT
iajs-1822	34	1	2	2	X
iajs-1822	34	2	.	.	X
iajs-1822	34	3	dual	dual	ADJ
iajs-1822	34	4	of	of	ADP
iajs-1822	34	5	a	a	DET
iajs-1822	34	6	modular	modular	ADJ
iajs-1822	34	7	space	space	NOUN
iajs-1822	34	8	let	let	VERB
iajs-1822	34	9	𝑃	𝑃	PRON
iajs-1822	34	10	be	be	AUX
iajs-1822	34	11	a	a	DET
iajs-1822	34	12	linear	linear	ADJ
iajs-1822	34	13	functional	functional	ADJ
iajs-1822	34	14	with	with	ADP
iajs-1822	34	15	domain	domain	NOUN
iajs-1822	34	16	in	in	ADP
iajs-1822	34	17	a	a	DET
iajs-1822	34	18	modular	modular	ADJ
iajs-1822	34	19	space	space	NOUN
iajs-1822	34	20	𝑀	𝑀	PROPN
iajs-1822	34	21	and	and	CCONJ
iajs-1822	34	22	range	range	VERB
iajs-1822	34	23	in	in	ADP
iajs-1822	34	24	the	the	DET
iajs-1822	34	25	scalar	scalar	ADJ
iajs-1822	34	26	field	field	NOUN
iajs-1822	34	27	𝐾	𝐾	PROPN
iajs-1822	34	28	𝑃:𝐷	𝑃:𝐷	PROPN
iajs-1822	34	29	𝑃	𝑃	PROPN
iajs-1822	34	30	→	→	SYM
iajs-1822	34	31	𝐾	𝐾	PROPN
iajs-1822	34	32	,	,	PUNCT
iajs-1822	34	33	𝑃	𝑃	PROPN
iajs-1822	34	34	is	be	AUX
iajs-1822	34	35	bounded	bound	VERB
iajs-1822	34	36	linear	linear	ADJ
iajs-1822	34	37	functional	functional	PROPN
iajs-1822	34	38	𝑐	𝑐	ADP
iajs-1822	34	39	such	such	ADJ
iajs-1822	34	40	that	that	PRON
iajs-1822	34	41	for	for	ADP
iajs-1822	34	42	all	all	PRON
iajs-1822	34	43	𝑣	𝑣	PRON
iajs-1822	34	44	∈	∈	NOUN
iajs-1822	34	45	𝐷	𝐷	NOUN
iajs-1822	34	46	𝑃	𝑃	NOUN
iajs-1822	34	47	,	,	PUNCT
iajs-1822	34	48	𝛾	𝛾	X
iajs-1822	35	1	𝑃𝑣	𝑃𝑣	NOUN
iajs-1822	35	2	𝑐𝛾	𝑐𝛾	NOUN
iajs-1822	35	3	𝑣	𝑣	PRON
iajs-1822	35	4	.	.	PUNCT
iajs-1822	36	1	the	the	DET
iajs-1822	36	2	set	set	NOUN
iajs-1822	36	3	of	of	ADP
iajs-1822	36	4	all	all	DET
iajs-1822	36	5	bounded	bounded	ADJ
iajs-1822	36	6	linear	linear	ADJ
iajs-1822	36	7	functional	functional	NOUN
iajs-1822	36	8	on	on	ADP
iajs-1822	36	9	𝑀	𝑀	PROPN
iajs-1822	36	10	,	,	PUNCT
iajs-1822	37	1	𝑴𝜸	𝑴𝜸	PROPN
iajs-1822	37	2			ADV
iajs-1822	37	3	is	be	AUX
iajs-1822	37	4	linear	linear	ADJ
iajs-1822	37	5	space	space	NOUN
iajs-1822	37	6	with	with	ADP
iajs-1822	37	7	point	point	NOUN
iajs-1822	37	8	-	-	PUNCT
iajs-1822	37	9	wise	wise	ADJ
iajs-1822	37	10	operations	operation	NOUN
iajs-1822	37	11	.	.	PUNCT
iajs-1822	38	1	in	in	ADP
iajs-1822	38	2	the	the	DET
iajs-1822	38	3	following	following	NOUN
iajs-1822	38	4	,	,	PUNCT
iajs-1822	38	5	we	we	PRON
iajs-1822	38	6	reform	reform	VERB
iajs-1822	38	7	some	some	DET
iajs-1822	38	8	concepts	concept	NOUN
iajs-1822	38	9	about	about	ADP
iajs-1822	38	10	dual	dual	ADJ
iajs-1822	38	11	space	space	NOUN
iajs-1822	38	12	in	in	ADP
iajs-1822	38	13	the	the	DET
iajs-1822	38	14	setting	setting	NOUN
iajs-1822	38	15	of	of	ADP
iajs-1822	38	16	modular	modular	ADJ
iajs-1822	38	17	spaces	space	NOUN
iajs-1822	38	18	,	,	PUNCT
iajs-1822	38	19	we	we	PRON
iajs-1822	38	20	begin	begin	VERB
iajs-1822	38	21	with	with	ADP
iajs-1822	38	22	following	follow	VERB
iajs-1822	38	23	:	:	PUNCT
iajs-1822	38	24	proposition	proposition	NOUN
iajs-1822	38	25	(	(	PUNCT
iajs-1822	38	26	2.1	2.1	NUM
iajs-1822	38	27	):	):	PUNCT
iajs-1822	38	28	let	let	VERB
iajs-1822	38	29	𝑃	𝑃	NOUN
iajs-1822	38	30	∈	∈	VERB
iajs-1822	38	31	𝑴𝜸	𝑴𝜸	PROPN
iajs-1822	38	32			ADV
iajs-1822	38	33	,	,	PUNCT
iajs-1822	38	34	define	define	VERB
iajs-1822	38	35	𝜸	𝜸	X
iajs-1822	38	36	:	:	PUNCT
iajs-1822	38	37	𝑴𝜸	𝑴𝜸	PROPN
iajs-1822	38	38			PROPN
iajs-1822	38	39	→	→	SYM
iajs-1822	38	40	𝑅	𝑅	PROPN
iajs-1822	38	41	∋	∋	NOUN
iajs-1822	38	42	𝜸	𝜸	X
iajs-1822	38	43	𝑃	𝑃	NOUN
iajs-1822	38	44	sup	sup	NOUN
iajs-1822	38	45	𝛾	𝛾	ADP
iajs-1822	38	46	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	38	47	∶	∶	NOUN
iajs-1822	38	48	𝛾	𝛾	ADP
iajs-1822	38	49	𝑣	𝑣	DET
iajs-1822	38	50	1	1	NUM
iajs-1822	38	51	then	then	ADV
iajs-1822	38	52	i.	i.	PROPN
iajs-1822	38	53	𝜸	𝜸	PROPN
iajs-1822	38	54	𝛼𝑃	𝛼𝑃	PROPN
iajs-1822	38	55	𝜸	𝜸	X
iajs-1822	38	56	𝑃	𝑃	NOUN
iajs-1822	38	57	,	,	PUNCT
iajs-1822	38	58	for	for	ADP
iajs-1822	38	59	𝛼	𝛼	DET
iajs-1822	38	60	∈	∈	PROPN
iajs-1822	38	61	k	k	NOUN
iajs-1822	38	62	with	with	ADP
iajs-1822	38	63	|𝛼|	|𝛼|	PROPN
iajs-1822	38	64	1	1	NUM
iajs-1822	38	65	ii	ii	NOUN
iajs-1822	38	66	.	.	PUNCT
iajs-1822	39	1	𝜸	𝜸	X
iajs-1822	39	2	𝛼𝑃	𝛼𝑃	NOUN
iajs-1822	39	3	𝛽𝑄	𝛽𝑄	VERB
iajs-1822	39	4	𝜸	𝜸	X
iajs-1822	39	5	𝑃	𝑃	NOUN
iajs-1822	39	6	𝜸	𝜸	X
iajs-1822	39	7	𝑄	𝑄	PROPN
iajs-1822	39	8	,	,	PUNCT
iajs-1822	39	9	iii	iii	PROPN
iajs-1822	39	10	.	.	PUNCT
iajs-1822	39	11	𝜸	𝜸	DET
iajs-1822	39	12	𝑃	𝑃	NOUN
iajs-1822	39	13	0	0	NUM
iajs-1822	39	14	iff	iff	PROPN
iajs-1822	39	15	𝑃	𝑃	PROPN
iajs-1822	39	16	0	0	NUM
iajs-1822	39	17	.	.	PUNCT
iajs-1822	40	1	proof	proof	NOUN
iajs-1822	40	2	:	:	PUNCT
iajs-1822	40	3	for	for	ADP
iajs-1822	40	4	𝑖	𝑖	NOUN
iajs-1822	40	5	𝜸	𝜸	X
iajs-1822	40	6	𝛼𝑃	𝛼𝑃	PROPN
iajs-1822	40	7	sup	sup	NOUN
iajs-1822	40	8	𝛾	𝛾	NOUN
iajs-1822	40	9	𝛼𝑃𝑣	𝛼𝑃𝑣	NUM
iajs-1822	40	10	sup	sup	NOUN
iajs-1822	40	11	𝛾	𝛾	ADP
iajs-1822	40	12	𝑃𝑣	𝑃𝑣	NOUN
iajs-1822	40	13	𝜸	𝜸	NOUN
iajs-1822	40	14	𝑃	𝑃	NOUN
iajs-1822	40	15	.	.	PUNCT
iajs-1822	41	1	for	for	ADP
iajs-1822	41	2	(	(	PUNCT
iajs-1822	41	3	ii	ii	NOUN
iajs-1822	41	4	)	)	PUNCT
iajs-1822	41	5	𝜸	𝜸	X
iajs-1822	41	6	𝛼𝑃	𝛼𝑃	PROPN
iajs-1822	41	7	𝛽𝑄	𝛽𝑄	PROPN
iajs-1822	41	8	sup	sup	VERB
iajs-1822	41	9	𝛾	𝛾	NOUN
iajs-1822	41	10	𝛼𝑃𝑣	𝛼𝑃𝑣	NUM
iajs-1822	41	11	𝛽𝑄𝑣	𝛽𝑄𝑣	NOUN
iajs-1822	41	12	sup	sup	NOUN
iajs-1822	41	13	𝛾	𝛾	ADP
iajs-1822	41	14	𝑃𝑣	𝑃𝑣	NOUN
iajs-1822	41	15	𝛾	𝛾	ADP
iajs-1822	41	16	𝑄𝑣	𝑄𝑣	PROPN
iajs-1822	41	17	sup	sup	NOUN
iajs-1822	41	18	𝛾	𝛾	X
iajs-1822	42	1	𝑃𝑣	𝑃𝑣	NOUN
iajs-1822	42	2	sup	sup	NOUN
iajs-1822	42	3	𝛾	𝛾	AUX
iajs-1822	42	4	𝑄𝑣	𝑄𝑣	PROPN
iajs-1822	42	5	ihsciconf	ihsciconf	NOUN
iajs-1822	42	6	2017	2017	NUM
iajs-1822	42	7	special	special	ADJ
iajs-1822	42	8	issue	issue	NOUN
iajs-1822	42	9	ibn	ibn	PROPN
iajs-1822	42	10	al	al	PROPN
iajs-1822	42	11	-	-	PUNCT
iajs-1822	42	12	haitham	haitham	PROPN
iajs-1822	42	13	journal	journal	PROPN
iajs-1822	42	14	for	for	ADP
iajs-1822	42	15	pure	pure	ADJ
iajs-1822	42	16	and	and	CCONJ
iajs-1822	42	17	applied	apply	VERB
iajs-1822	42	18	science	science	NOUN
iajs-1822	42	19	https://doi.org/	https://doi.org/	NOUN
iajs-1822	42	20	10.30526/2017.ihsciconf.1822	10.30526/2017.ihsciconf.1822	NOUN
iajs-1822	42	21	for	for	ADP
iajs-1822	42	22	more	more	ADJ
iajs-1822	42	23	information	information	NOUN
iajs-1822	42	24	about	about	ADP
iajs-1822	42	25	the	the	DET
iajs-1822	42	26	conference	conference	NOUN
iajs-1822	42	27	please	please	INTJ
iajs-1822	42	28	visit	visit	VERB
iajs-1822	42	29	the	the	DET
iajs-1822	42	30	websites	website	NOUN
iajs-1822	42	31	:	:	PUNCT
iajs-1822	42	32	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1822	42	33	  	  	SPACE
iajs-1822	42	34	www.ihsciconf.org	www.ihsciconf.org	ADJ
iajs-1822	42	35	   	   	SPACE
iajs-1822	42	36	mathematics|503	mathematics|503	ADJ
iajs-1822	42	37	    	    	SPACE
iajs-1822	42	38	𝜸	𝜸	ADP
iajs-1822	42	39	𝑃	𝑃	NOUN
iajs-1822	42	40	𝜸	𝜸	X
iajs-1822	42	41	𝑄	𝑄	PRON
iajs-1822	42	42	for	for	ADP
iajs-1822	42	43	(	(	PUNCT
iajs-1822	42	44	iii	iii	NOUN
iajs-1822	42	45	)	)	PUNCT
iajs-1822	42	46	𝜸	𝜸	NOUN
iajs-1822	42	47	𝑃	𝑃	PROPN
iajs-1822	42	48	0	0	NUM
iajs-1822	42	49	iff	iff	PROPN
iajs-1822	42	50	sup	sup	NOUN
iajs-1822	42	51	𝛾	𝛾	PROPN
iajs-1822	42	52	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	42	53	∶	∶	NOUN
iajs-1822	42	54	𝛾	𝛾	ADP
iajs-1822	42	55	𝑣	𝑣	ADP
iajs-1822	42	56	1	1	NUM
iajs-1822	42	57	iff	iff	NOUN
iajs-1822	42	58	𝛾	𝛾	PROPN
iajs-1822	42	59	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	42	60	0	0	NUM
iajs-1822	42	61	for	for	ADP
iajs-1822	42	62	all𝑣	all𝑣	NOUN
iajs-1822	42	63	iff	iff	PROPN
iajs-1822	42	64	𝑃	𝑃	PROPN
iajs-1822	42	65	0	0	NUM
iajs-1822	42	66	.	.	PUNCT
iajs-1822	43	1	a	a	DET
iajs-1822	43	2	modular	modular	NOUN
iajs-1822	43	3	𝜸	𝜸	ADP
iajs-1822	43	4	defines	define	NOUN
iajs-1822	43	5	a	a	DET
iajs-1822	43	6	corresponding	correspond	VERB
iajs-1822	43	7	modular	modular	ADJ
iajs-1822	43	8	space,𝑖.	space,𝑖.	PROPN
iajs-1822	43	9	𝑒.,the	𝑒.,the	PRON
iajs-1822	43	10	space𝑴𝜸	space𝑴𝜸	VERB
iajs-1822	43	11			ADV
iajs-1822	43	12	given	give	VERB
iajs-1822	43	13	by	by	ADP
iajs-1822	43	14	𝑴𝜸	𝑴𝜸	PROPN
iajs-1822	43	15			ADJ
iajs-1822	43	16	𝑣	𝑣	DET
iajs-1822	43	17	∈	∈	PROPN
iajs-1822	43	18	𝑀	𝑀	PROPN
iajs-1822	43	19	:	:	PUNCT
iajs-1822	43	20	𝜸	𝜸	X
iajs-1822	43	21	𝛼𝑃	𝛼𝑃	PROPN
iajs-1822	43	22	→	→	SYM
iajs-1822	43	23	0	0	NUM
iajs-1822	43	24	𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟	𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟	NOUN
iajs-1822	43	25	𝛼	𝛼	NOUN
iajs-1822	43	26	→	→	SYM
iajs-1822	43	27	0	0	NUM
iajs-1822	43	28	theorem	theorem	NOUN
iajs-1822	43	29	(	(	PUNCT
iajs-1822	43	30	2.2	2.2	NUM
iajs-1822	43	31	):	):	PUNCT
iajs-1822	43	32	𝑴𝜸	𝑴𝜸	PROPN
iajs-1822	43	33			ADV
iajs-1822	43	34	is	be	AUX
iajs-1822	43	35	complete	complete	ADJ
iajs-1822	43	36	modular	modular	ADJ
iajs-1822	43	37	space	space	NOUN
iajs-1822	43	38	.	.	PUNCT
iajs-1822	44	1	proof	proof	NOUN
iajs-1822	44	2	:	:	PUNCT
iajs-1822	44	3	we	we	PRON
iajs-1822	44	4	consider	consider	VERB
iajs-1822	44	5	an	an	DET
iajs-1822	44	6	arbitrary	arbitrary	ADJ
iajs-1822	44	7	cauchy	cauchy	NOUN
iajs-1822	44	8	sequence	sequence	NOUN
iajs-1822	44	9	𝑆	𝑆	PROPN
iajs-1822	44	10	in	in	ADP
iajs-1822	44	11	𝑴𝜸	𝑴𝜸	PROPN
iajs-1822	44	12			ADV
iajs-1822	44	13	and	and	CCONJ
iajs-1822	44	14	show	show	VERB
iajs-1822	44	15	that	that	SCONJ
iajs-1822	44	16	𝑆	𝑆	PROPN
iajs-1822	44	17	converges	converge	VERB
iajs-1822	44	18	to	to	ADP
iajs-1822	44	19	a	a	DET
iajs-1822	44	20	𝑆∈	𝑆∈	NOUN
iajs-1822	44	21	𝑴𝜸	𝑴𝜸	PROPN
iajs-1822	45	1			ADV
iajs-1822	45	2	since	since	SCONJ
iajs-1822	45	3	𝑆	𝑆	PROPN
iajs-1822	45	4	is	be	AUX
iajs-1822	45	5	cauchy	cauchy	NOUN
iajs-1822	45	6	,	,	PUNCT
iajs-1822	45	7	for	for	ADP
iajs-1822	45	8	every	every	DET
iajs-1822	45	9	ϵ	ϵ	NOUN
iajs-1822	45	10	𝟢	𝟢	NUM
iajs-1822	45	11	there	there	PRON
iajs-1822	45	12	is	be	VERB
iajs-1822	45	13	an	an	DET
iajs-1822	45	14	l	l	NOUN
iajs-1822	45	15	such	such	ADJ
iajs-1822	45	16	that	that	SCONJ
iajs-1822	45	17	𝜸	𝜸	X
iajs-1822	45	18	𝑆	𝑆	PROPN
iajs-1822	45	19	𝑆	𝑆	PROPN
iajs-1822	45	20	∈	∈	PROPN
iajs-1822	45	21	,	,	PUNCT
iajs-1822	45	22	𝑛	𝑛	PROPN
iajs-1822	45	23	,	,	PUNCT
iajs-1822	45	24	𝑚	𝑚	PROPN
iajs-1822	45	25	𝐿	𝐿	PROPN
iajs-1822	45	26	,	,	PUNCT
iajs-1822	45	27	for	for	ADP
iajs-1822	45	28	any	any	DET
iajs-1822	45	29	𝑣	𝑣	PROPN
iajs-1822	45	30	∈	∈	PROPN
iajs-1822	45	31	𝑀	𝑀	PROPN
iajs-1822	45	32	and	and	CCONJ
iajs-1822	45	33	𝑛	𝑛	PROPN
iajs-1822	45	34	,	,	PUNCT
iajs-1822	45	35	𝑚	𝑚	PROPN
iajs-1822	45	36	𝐿	𝐿	PROPN
iajs-1822	45	37	,	,	PUNCT
iajs-1822	45	38	this	this	PRON
iajs-1822	45	39	implies	imply	VERB
iajs-1822	45	40	that	that	SCONJ
iajs-1822	45	41	|𝑆	|𝑆	PROPN
iajs-1822	45	42	𝑣	𝑣	DET
iajs-1822	45	43	𝑆	𝑆	PROPN
iajs-1822	45	44	𝑣|	𝑣|	PROPN
iajs-1822	45	45	|	|	ADV
iajs-1822	45	46	𝑆	𝑆	PROPN
iajs-1822	45	47	𝑆	𝑆	PROPN
iajs-1822	45	48	𝑣|	𝑣|	VERB
iajs-1822	45	49	𝛾	𝛾	ADP
iajs-1822	45	50	𝑆	𝑆	PROPN
iajs-1822	45	51	𝑆	𝑆	PROPN
iajs-1822	45	52	𝛾	𝛾	PROPN
iajs-1822	45	53	𝑣	𝑣	PRON
iajs-1822	45	54	∈	∈	NOUN
iajs-1822	45	55	𝛾	𝛾	ADP
iajs-1822	45	56	𝑣	𝑣	NOUN
iajs-1822	45	57	.	.	PUNCT
iajs-1822	46	1	…	…	PUNCT
iajs-1822	46	2	(	(	PUNCT
iajs-1822	46	3	2.1	2.1	NUM
iajs-1822	46	4	)	)	PUNCT
iajs-1822	46	5	now	now	ADV
iajs-1822	46	6	,	,	PUNCT
iajs-1822	46	7	for	for	ADP
iajs-1822	46	8	any	any	DET
iajs-1822	46	9	fixed	fixed	ADJ
iajs-1822	46	10	point	point	NOUN
iajs-1822	46	11	𝑣	𝑣	ADP
iajs-1822	46	12	and	and	CCONJ
iajs-1822	46	13	given	give	VERB
iajs-1822	46	14	∈	∈	PROPN
iajs-1822	46	15	we	we	PRON
iajs-1822	46	16	may	may	AUX
iajs-1822	46	17	choose	choose	VERB
iajs-1822	46	18	∈	∈	PROPN
iajs-1822	46	19	∈	∈	NOUN
iajs-1822	46	20	so	so	SCONJ
iajs-1822	46	21	that	that	SCONJ
iajs-1822	46	22	∈	∈	PROPN
iajs-1822	46	23	𝛾	𝛾	VERB
iajs-1822	46	24	𝑣	𝑣	PRON
iajs-1822	46	25	∈	∈	PROPN
iajs-1822	46	26	.	.	PUNCT
iajs-1822	47	1	then	then	ADV
iajs-1822	47	2	from	from	ADP
iajs-1822	47	3	(	(	PUNCT
iajs-1822	47	4	2.1	2.1	NUM
iajs-1822	47	5	)	)	PUNCT
iajs-1822	47	6	,	,	PUNCT
iajs-1822	47	7	we	we	PRON
iajs-1822	47	8	have	have	VERB
iajs-1822	47	9	|𝑆	|𝑆	ADV
iajs-1822	47	10	𝑣	𝑣	SCONJ
iajs-1822	47	11	𝑆	𝑆	PROPN
iajs-1822	47	12	𝑣|	𝑣|	PROPN
iajs-1822	47	13	∈	∈	PROPN
iajs-1822	47	14	and	and	CCONJ
iajs-1822	47	15	𝑆	𝑆	PROPN
iajs-1822	47	16	𝑣	𝑣	PROPN
iajs-1822	47	17	is	be	AUX
iajs-1822	47	18	cauchy	cauchy	NOUN
iajs-1822	47	19	in	in	ADP
iajs-1822	47	20	𝐾.	𝐾.	PROPN
iajs-1822	47	21	by	by	ADP
iajs-1822	47	22	completeness	completeness	NOUN
iajs-1822	47	23	of	of	ADP
iajs-1822	47	24	𝐾	𝐾	PROPN
iajs-1822	47	25	,	,	PUNCT
iajs-1822	47	26	𝑆	𝑆	PROPN
iajs-1822	47	27	𝑣	𝑣	PROPN
iajs-1822	47	28	converges	converge	NOUN
iajs-1822	47	29	,	,	PUNCT
iajs-1822	47	30	say	say	INTJ
iajs-1822	47	31	,	,	PUNCT
iajs-1822	47	32	𝑆	𝑆	PROPN
iajs-1822	47	33	𝑣	𝑣	PROPN
iajs-1822	47	34	→	→	PUNCT
iajs-1822	47	35	𝑟.	𝑟.	NOUN
iajs-1822	47	36	clearly	clearly	ADV
iajs-1822	47	37	,	,	PUNCT
iajs-1822	47	38	the	the	DET
iajs-1822	47	39	limit	limit	NOUN
iajs-1822	47	40	𝑟	𝑟	X
iajs-1822	47	41	∈	∈	PROPN
iajs-1822	47	42	𝐾	𝐾	PROPN
iajs-1822	47	43	depends	depend	VERB
iajs-1822	47	44	on	on	ADP
iajs-1822	47	45	the	the	DET
iajs-1822	47	46	choice	choice	NOUN
iajs-1822	47	47	of	of	ADP
iajs-1822	47	48	𝑣	𝑣	DET
iajs-1822	47	49	∈	∈	PROPN
iajs-1822	47	50	𝑀	𝑀	PROPN
iajs-1822	47	51	.	.	PUNCT
iajs-1822	48	1	this	this	PRON
iajs-1822	48	2	defines	define	VERB
iajs-1822	48	3	a	a	DET
iajs-1822	48	4	functional	functional	ADJ
iajs-1822	48	5	𝑆	𝑆	PROPN
iajs-1822	48	6	:	:	PUNCT
iajs-1822	48	7	𝑀	𝑀	PROPN
iajs-1822	48	8	→	→	SYM
iajs-1822	48	9	𝐾	𝐾	PROPN
iajs-1822	48	10	where	where	SCONJ
iajs-1822	48	11	𝑟	𝑟	X
iajs-1822	48	12	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	48	13	.the	.the	PROPN
iajs-1822	48	14	functional	functional	PROPN
iajs-1822	48	15	𝑆	𝑆	PROPN
iajs-1822	48	16	is	be	AUX
iajs-1822	48	17	linear	linear	ADJ
iajs-1822	48	18	since	since	SCONJ
iajs-1822	48	19	lim	lim	PROPN
iajs-1822	48	20	→	→	SYM
iajs-1822	48	21	𝑆	𝑆	PROPN
iajs-1822	48	22	𝛼𝑣	𝛼𝑣	INTJ
iajs-1822	48	23	𝛽𝑧	𝛽𝑧	ADP
iajs-1822	48	24	lim	lim	PROPN
iajs-1822	48	25	→	→	SYM
iajs-1822	48	26	𝛼𝑆	𝛼𝑆	PROPN
iajs-1822	48	27	𝑣	𝑣	ADP
iajs-1822	48	28	𝛽𝑆	𝛽𝑆	PROPN
iajs-1822	48	29	𝑧	𝑧	ADP
iajs-1822	48	30	𝛼	𝛼	PROPN
iajs-1822	48	31	lim	lim	PROPN
iajs-1822	48	32	→	→	SYM
iajs-1822	48	33	𝑆	𝑆	PROPN
iajs-1822	48	34	𝑣	𝑣	ADP
iajs-1822	48	35	𝛽	𝛽	PROPN
iajs-1822	48	36	lim	lim	PROPN
iajs-1822	48	37	→	→	SYM
iajs-1822	48	38	𝑆	𝑆	PROPN
iajs-1822	48	39	𝑧.	𝑧.	NOUN
iajs-1822	48	40	we	we	PRON
iajs-1822	48	41	prove	prove	VERB
iajs-1822	48	42	that	that	SCONJ
iajs-1822	48	43	𝑆	𝑆	PROPN
iajs-1822	48	44	is	be	AUX
iajs-1822	48	45	bounded	bound	VERB
iajs-1822	48	46	and	and	CCONJ
iajs-1822	48	47	𝑆	𝑆	PROPN
iajs-1822	48	48	→𝑆	→𝑆	NOUN
iajs-1822	48	49	,	,	PUNCT
iajs-1822	48	50	that	that	PRON
iajs-1822	48	51	is	be	AUX
iajs-1822	48	52	𝜸	𝜸	PRON
iajs-1822	48	53	𝑆	𝑆	PROPN
iajs-1822	48	54	𝑆	𝑆	PROPN
iajs-1822	48	55	→	→	SYM
iajs-1822	48	56	𝟢.	𝟢.	X
iajs-1822	48	57	since	since	SCONJ
iajs-1822	48	58	(	(	PUNCT
iajs-1822	48	59	2.1	2.1	NUM
iajs-1822	48	60	)	)	PUNCT
iajs-1822	48	61	holds	hold	VERB
iajs-1822	48	62	for	for	ADP
iajs-1822	48	63	every	every	DET
iajs-1822	48	64	𝑚	𝑚	PROPN
iajs-1822	48	65	𝐿	𝐿	PROPN
iajs-1822	48	66	and	and	CCONJ
iajs-1822	48	67	𝑆	𝑆	PROPN
iajs-1822	48	68	𝑣	𝑣	PROPN
iajs-1822	48	69	→	→	SYM
iajs-1822	48	70	𝑆	𝑆	PROPN
iajs-1822	48	71	,	,	PUNCT
iajs-1822	48	72	we	we	PRON
iajs-1822	48	73	may	may	AUX
iajs-1822	48	74	let	let	VERB
iajs-1822	48	75	𝑚→	𝑚→	VERB
iajs-1822	48	76	∞.	∞.	PROPN
iajs-1822	48	77	using	use	VERB
iajs-1822	48	78	the	the	DET
iajs-1822	48	79	continuity	continuity	NOUN
iajs-1822	48	80	of	of	ADP
iajs-1822	48	81	the	the	DET
iajs-1822	48	82	modular	modular	NOUN
iajs-1822	48	83	,	,	PUNCT
iajs-1822	48	84	then	then	ADV
iajs-1822	48	85	for	for	ADP
iajs-1822	48	86	every	every	DET
iajs-1822	48	87	𝑛	𝑛	PRON
iajs-1822	48	88	𝐿	𝐿	PROPN
iajs-1822	48	89	and	and	CCONJ
iajs-1822	48	90	all	all	PRON
iajs-1822	48	91	𝑣	𝑣	DET
iajs-1822	48	92	∈	∈	PROPN
iajs-1822	48	93	𝑀	𝑀	PROPN
iajs-1822	48	94	.	.	PUNCT
iajs-1822	49	1	|𝑆	|𝑆	PROPN
iajs-1822	49	2	𝑣	𝑣	ADP
iajs-1822	49	3	𝑆𝑣|	𝑆𝑣|	X
iajs-1822	49	4	𝑆	𝑆	PROPN
iajs-1822	49	5	𝑣	𝑣	PROPN
iajs-1822	49	6	lim	lim	PROPN
iajs-1822	49	7	→	→	SYM
iajs-1822	49	8	𝑆	𝑆	PROPN
iajs-1822	49	9	𝑣	𝑣	PROPN
iajs-1822	49	10	lim	lim	PROPN
iajs-1822	49	11	→	→	SYM
iajs-1822	49	12	|𝑆	|𝑆	PROPN
iajs-1822	49	13	𝑣	𝑣	ADP
iajs-1822	49	14	𝑆	𝑆	PROPN
iajs-1822	49	15	𝑣|	𝑣|	PROPN
iajs-1822	49	16	𝜖𝛾	𝜖𝛾	VERB
iajs-1822	49	17	𝑣	𝑣	PART
iajs-1822	49	18	…	…	PUNCT
iajs-1822	49	19	(	(	PUNCT
iajs-1822	49	20	2.2	2.2	NUM
iajs-1822	49	21	)	)	PUNCT
iajs-1822	49	22	this	this	PRON
iajs-1822	49	23	shows	show	VERB
iajs-1822	49	24	that	that	SCONJ
iajs-1822	49	25	𝑆	𝑆	PROPN
iajs-1822	49	26	𝑆	𝑆	PROPN
iajs-1822	49	27	with	with	ADP
iajs-1822	49	28	𝑛	𝑛	DET
iajs-1822	49	29	𝐿	𝐿	PROPN
iajs-1822	49	30	is	be	AUX
iajs-1822	49	31	a	a	DET
iajs-1822	49	32	bounded	bounded	ADJ
iajs-1822	49	33	linear	linear	ADJ
iajs-1822	49	34	functional	functional	NOUN
iajs-1822	49	35	.	.	PUNCT
iajs-1822	50	1	since	since	SCONJ
iajs-1822	50	2	𝑆	𝑆	PROPN
iajs-1822	50	3	is	be	AUX
iajs-1822	50	4	bounded	bound	VERB
iajs-1822	50	5	,	,	PUNCT
iajs-1822	50	6	𝑆	𝑆	PROPN
iajs-1822	50	7	𝑆	𝑆	PROPN
iajs-1822	50	8	𝑆	𝑆	PROPN
iajs-1822	50	9	𝑆	𝑆	PROPN
iajs-1822	50	10	is	be	AUX
iajs-1822	50	11	bounded	bound	VERB
iajs-1822	50	12	,	,	PUNCT
iajs-1822	50	13	that	that	ADV
iajs-1822	50	14	is	is	ADV
iajs-1822	50	15	,	,	PUNCT
iajs-1822	50	16	𝑆∈	𝑆∈	PROPN
iajs-1822	50	17	𝑴𝜸	𝑴𝜸	PROPN
iajs-1822	50	18			ADV
iajs-1822	50	19	.	.	PUNCT
iajs-1822	51	1	furthermore	furthermore	ADV
iajs-1822	51	2	,	,	PUNCT
iajs-1822	51	3	if	if	SCONJ
iajs-1822	51	4	in	in	ADP
iajs-1822	51	5	(	(	PUNCT
iajs-1822	51	6	2.2	2.2	NUM
iajs-1822	51	7	)	)	PUNCT
iajs-1822	51	8	we	we	PRON
iajs-1822	51	9	take	take	VERB
iajs-1822	51	10	the	the	DET
iajs-1822	51	11	supremum	supremum	ADJ
iajs-1822	51	12	over	over	ADP
iajs-1822	51	13	all	all	DET
iajs-1822	51	14	𝑣	𝑣	PRON
iajs-1822	51	15	of	of	ADP
iajs-1822	51	16	modular	modular	ADJ
iajs-1822	51	17	𝟣	𝟣	NUM
iajs-1822	51	18	,	,	PUNCT
iajs-1822	51	19	we	we	PRON
iajs-1822	51	20	obtain	obtain	VERB
iajs-1822	51	21	𝛾	𝛾	ADP
iajs-1822	51	22	𝑆	𝑆	PROPN
iajs-1822	51	23	𝑆	𝑆	PROPN
iajs-1822	51	24	𝜖	𝜖	PROPN
iajs-1822	51	25	,	,	PUNCT
iajs-1822	51	26	𝑛	𝑛	PRON
iajs-1822	51	27	𝐿.	𝐿.	VERB
iajs-1822	51	28	hence	hence	ADV
iajs-1822	51	29	𝜸	𝜸	ADP
iajs-1822	51	30	𝑆	𝑆	PROPN
iajs-1822	51	31	𝑆	𝑆	PROPN
iajs-1822	51	32	→	→	SYM
iajs-1822	51	33	𝟢.	𝟢.	NOUN
iajs-1822	51	34	this	this	PRON
iajs-1822	51	35	completes	complete	VERB
iajs-1822	51	36	proof	proof	NOUN
iajs-1822	51	37	.	.	PUNCT
iajs-1822	52	1	definition	definition	NOUN
iajs-1822	52	2	(	(	PUNCT
iajs-1822	52	3	2.3	2.3	NUM
iajs-1822	52	4	):	):	PUNCT
iajs-1822	52	5	a	a	DET
iajs-1822	52	6	sequence	sequence	NOUN
iajs-1822	52	7	𝑣	𝑣	X
iajs-1822	52	8	in	in	ADP
iajs-1822	52	9	a	a	DET
iajs-1822	52	10	modular	modular	ADJ
iajs-1822	52	11	space	space	NOUN
iajs-1822	52	12	𝑀	𝑀	PROPN
iajs-1822	52	13	is	be	AUX
iajs-1822	52	14	said	say	VERB
iajs-1822	52	15	to	to	PART
iajs-1822	52	16	be	be	AUX
iajs-1822	52	17	weakly	weakly	ADV
iajs-1822	52	18	convergent	convergent	ADJ
iajs-1822	52	19	if	if	SCONJ
iajs-1822	52	20	there	there	PRON
iajs-1822	52	21	is	be	VERB
iajs-1822	52	22	an	an	DET
iajs-1822	52	23	𝑣	𝑣	PROPN
iajs-1822	52	24			NOUN
iajs-1822	52	25	𝑀	𝑀	PROPN
iajs-1822	52	26	such	such	ADJ
iajs-1822	52	27	that	that	PRON
iajs-1822	52	28	for	for	ADP
iajs-1822	52	29	every	every	DET
iajs-1822	52	30	𝑃	𝑃	NOUN
iajs-1822	52	31	𝑴𝜸	𝑴𝜸	NOUN
iajs-1822	52	32			ADJ
iajs-1822	52	33	lim	lim	PROPN
iajs-1822	52	34	→	→	SYM
iajs-1822	52	35	𝛾	𝛾	PROPN
iajs-1822	52	36	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	52	37	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	52	38	0	0	PUNCT
iajs-1822	53	1	this	this	PRON
iajs-1822	53	2	denoted	denote	VERB
iajs-1822	53	3	by	by	ADP
iajs-1822	53	4	𝑣	𝑣	PROPN
iajs-1822	53	5	→	→	SYM
iajs-1822	53	6	𝑣.	𝑣.	NOUN
iajs-1822	53	7	ihsciconf	ihsciconf	PROPN
iajs-1822	53	8	2017	2017	NUM
iajs-1822	53	9	special	special	ADJ
iajs-1822	53	10	issue	issue	NOUN
iajs-1822	53	11	ibn	ibn	PROPN
iajs-1822	53	12	al	al	PROPN
iajs-1822	53	13	-	-	PUNCT
iajs-1822	53	14	haitham	haitham	PROPN
iajs-1822	53	15	journal	journal	PROPN
iajs-1822	53	16	for	for	ADP
iajs-1822	53	17	pure	pure	ADJ
iajs-1822	53	18	and	and	CCONJ
iajs-1822	53	19	applied	apply	VERB
iajs-1822	53	20	science	science	NOUN
iajs-1822	53	21	https://doi.org/	https://doi.org/	NOUN
iajs-1822	53	22	10.30526/2017.ihsciconf.1822	10.30526/2017.ihsciconf.1822	NOUN
iajs-1822	53	23	for	for	ADP
iajs-1822	53	24	more	more	ADJ
iajs-1822	53	25	information	information	NOUN
iajs-1822	53	26	about	about	ADP
iajs-1822	53	27	the	the	DET
iajs-1822	53	28	conference	conference	NOUN
iajs-1822	53	29	please	please	INTJ
iajs-1822	53	30	visit	visit	VERB
iajs-1822	53	31	the	the	DET
iajs-1822	53	32	websites	website	NOUN
iajs-1822	53	33	:	:	PUNCT
iajs-1822	53	34	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1822	53	35	  	  	SPACE
iajs-1822	53	36	www.ihsciconf.org	www.ihsciconf.org	ADV
iajs-1822	53	37	   	   	SPACE
iajs-1822	53	38	mathematics|504	mathematics|504	ADJ
iajs-1822	53	39	    	    	SPACE
iajs-1822	53	40	proposition	proposition	NOUN
iajs-1822	53	41	(	(	PUNCT
iajs-1822	53	42	2.4	2.4	NUM
iajs-1822	53	43	):	):	PUNCT
iajs-1822	53	44	in	in	ADP
iajs-1822	53	45	a	a	DET
iajs-1822	53	46	modular	modular	ADJ
iajs-1822	53	47	space	space	NOUN
iajs-1822	53	48	𝑀	𝑀	PROPN
iajs-1822	53	49	,	,	PUNCT
iajs-1822	53	50	every	every	DET
iajs-1822	53	51	convergent	convergent	NOUN
iajs-1822	53	52	sequence	sequence	NOUN
iajs-1822	53	53	is	be	AUX
iajs-1822	53	54	weakly	weakly	ADJ
iajs-1822	53	55	convergent	convergent	NOUN
iajs-1822	53	56	.	.	PUNCT
iajs-1822	54	1	proof	proof	NOUN
iajs-1822	54	2	:	:	PUNCT
iajs-1822	54	3	by	by	ADP
iajs-1822	54	4	definition	definition	NOUN
iajs-1822	54	5	,	,	PUNCT
iajs-1822	54	6	𝑣	𝑣	X
iajs-1822	54	7	→	→	SYM
iajs-1822	54	8	𝑣	𝑣	PRON
iajs-1822	54	9	means	mean	VERB
iajs-1822	54	10	𝛾	𝛾	ADP
iajs-1822	54	11	𝑣	𝑣	PRON
iajs-1822	54	12	𝑣	𝑣	X
iajs-1822	54	13	→	→	SYM
iajs-1822	54	14	0	0	NUM
iajs-1822	54	15	and	and	CCONJ
iajs-1822	54	16	implies	imply	VERB
iajs-1822	54	17	that	that	SCONJ
iajs-1822	54	18	for	for	ADP
iajs-1822	54	19	every	every	DET
iajs-1822	54	20	p∈	p∈	PROPN
iajs-1822	54	21	𝑴𝜸	𝑴𝜸	PROPN
iajs-1822	54	22	,	,	PUNCT
iajs-1822	54	23			ADJ
iajs-1822	54	24	|𝑃	|𝑃	NOUN
iajs-1822	54	25	𝑣	𝑣	DET
iajs-1822	54	26	𝑃	𝑃	NOUN
iajs-1822	54	27	𝑣	𝑣	NOUN
iajs-1822	54	28	|	|	NOUN
iajs-1822	54	29	|𝑃	|𝑃	X
iajs-1822	54	30	𝑣	𝑣	PRON
iajs-1822	54	31	𝑣	𝑣	X
iajs-1822	54	32	|	|	ADV
iajs-1822	54	33	𝛾	𝛾	ADP
iajs-1822	54	34	𝑃	𝑃	VERB
iajs-1822	54	35	𝛾	𝛾	NOUN
iajs-1822	54	36	𝑣	𝑣	PRON
iajs-1822	54	37	𝑣	𝑣	X
iajs-1822	54	38	→	→	SYM
iajs-1822	54	39	0	0	PROPN
iajs-1822	54	40	.	.	PUNCT
iajs-1822	55	1	this	this	PRON
iajs-1822	55	2	shows	show	VERB
iajs-1822	55	3	that	that	SCONJ
iajs-1822	55	4	𝑣	𝑣	X
iajs-1822	55	5	→	→	SYM
iajs-1822	55	6	𝑣.	𝑣.	NOUN
iajs-1822	55	7	note	note	NOUN
iajs-1822	55	8	that	that	SCONJ
iajs-1822	55	9	,	,	PUNCT
iajs-1822	55	10	the	the	DET
iajs-1822	55	11	converse	converse	NOUN
iajs-1822	55	12	of	of	ADP
iajs-1822	55	13	proposition	proposition	NOUN
iajs-1822	55	14	(	(	PUNCT
iajs-1822	55	15	2.4	2.4	NUM
iajs-1822	55	16	)	)	PUNCT
iajs-1822	55	17	is	be	AUX
iajs-1822	55	18	not	not	PART
iajs-1822	55	19	necessary	necessary	ADJ
iajs-1822	55	20	true	true	ADJ
iajs-1822	55	21	.	.	PUNCT
iajs-1822	56	1	to	to	PART
iajs-1822	56	2	show	show	VERB
iajs-1822	56	3	this	this	DET
iajs-1822	56	4	recall	recall	VERB
iajs-1822	56	5	the	the	DET
iajs-1822	56	6	usual	usual	ADJ
iajs-1822	56	7	case	case	NOUN
iajs-1822	56	8	is	be	AUX
iajs-1822	56	9	in	in	ADP
iajs-1822	56	10	a	a	DET
iajs-1822	56	11	normed	normed	ADJ
iajs-1822	56	12	space.in	space.in	PROPN
iajs-1822	56	13	the	the	DET
iajs-1822	56	14	following	follow	VERB
iajs-1822	56	15	some	some	DET
iajs-1822	56	16	other	other	ADJ
iajs-1822	56	17	needed	need	VERB
iajs-1822	56	18	properties	property	NOUN
iajs-1822	56	19	of	of	ADP
iajs-1822	56	20	weak	weak	ADJ
iajs-1822	56	21	convergence	convergence	NOUN
iajs-1822	56	22	are	be	AUX
iajs-1822	56	23	given	give	VERB
iajs-1822	56	24	:	:	PUNCT
iajs-1822	56	25	proposition	proposition	NOUN
iajs-1822	56	26	(	(	PUNCT
iajs-1822	56	27	2.5	2.5	NUM
iajs-1822	56	28	):	):	PUNCT
iajs-1822	56	29	let	let	VERB
iajs-1822	56	30	𝑣	𝑣	PART
iajs-1822	56	31	be	be	AUX
iajs-1822	56	32	weakly	weakly	ADJ
iajs-1822	56	33	convergent	convergent	ADJ
iajs-1822	56	34	sequence	sequence	NOUN
iajs-1822	56	35	in	in	ADP
iajs-1822	56	36	a	a	DET
iajs-1822	56	37	modular	modular	ADJ
iajs-1822	56	38	space	space	NOUN
iajs-1822	56	39	𝑀	𝑀	PROPN
iajs-1822	56	40	,	,	PUNCT
iajs-1822	56	41	say	say	VERB
iajs-1822	56	42	𝑣	𝑣	ADP
iajs-1822	56	43	→	→	SYM
iajs-1822	57	1	𝑣	𝑣	X
iajs-1822	57	2	then	then	ADV
iajs-1822	57	3	:	:	PUNCT
iajs-1822	57	4	i.	i.	NOUN
iajs-1822	57	5	the	the	DET
iajs-1822	57	6	weak	weak	ADJ
iajs-1822	57	7	limit	limit	NOUN
iajs-1822	57	8	𝑣	𝑣	ADP
iajs-1822	57	9	of	of	ADP
iajs-1822	57	10	𝑣	𝑣	PROPN
iajs-1822	57	11	is	be	AUX
iajs-1822	57	12	unique	unique	ADJ
iajs-1822	57	13	.	.	PUNCT
iajs-1822	58	1	ii	ii	X
iajs-1822	58	2	.	.	PUNCT
iajs-1822	59	1	every	every	DET
iajs-1822	59	2	subsequence	subsequence	NOUN
iajs-1822	59	3	of	of	ADP
iajs-1822	59	4	𝑣	𝑣	PRON
iajs-1822	59	5	converges	converge	VERB
iajs-1822	59	6	weakly	weakly	ADV
iajs-1822	59	7	to	to	ADP
iajs-1822	59	8	𝑣.	𝑣.	NOUN
iajs-1822	59	9	proof	proof	NOUN
iajs-1822	59	10	:	:	PUNCT
iajs-1822	59	11	for	for	ADP
iajs-1822	59	12	(	(	PUNCT
iajs-1822	59	13	i	i	NOUN
iajs-1822	59	14	)	)	PUNCT
iajs-1822	59	15	,	,	PUNCT
iajs-1822	59	16	suppose	suppose	VERB
iajs-1822	59	17	that	that	SCONJ
iajs-1822	59	18	𝑣	𝑣	ADP
iajs-1822	59	19	𝒘	𝒘	X
iajs-1822	59	20	→	→	SYM
iajs-1822	59	21	𝑣	𝑣	X
iajs-1822	59	22	as	as	ADV
iajs-1822	59	23	well	well	ADV
iajs-1822	59	24	as	as	ADP
iajs-1822	59	25	𝑣	𝑣	PRON
iajs-1822	59	26	𝒘	𝒘	X
iajs-1822	59	27	→	→	SYM
iajs-1822	59	28	𝑢.	𝑢.	NOUN
iajs-1822	59	29	then	then	ADV
iajs-1822	59	30	p	p	X
iajs-1822	59	31	𝑣	𝑣	X
iajs-1822	59	32	→	→	PUNCT
iajs-1822	59	33	𝑃	𝑃	NOUN
iajs-1822	59	34	𝑣	𝑣	PRON
iajs-1822	59	35	as	as	ADV
iajs-1822	59	36	well	well	ADV
iajs-1822	59	37	as	as	ADP
iajs-1822	59	38	p	p	NOUN
iajs-1822	59	39	𝑣	𝑣	X
iajs-1822	59	40	→	→	SYM
iajs-1822	59	41	𝑃	𝑃	NOUN
iajs-1822	59	42	𝑢	𝑢	NOUN
iajs-1822	59	43	.	.	PUNCT
iajs-1822	60	1	since	since	SCONJ
iajs-1822	60	2	p	p	NOUN
iajs-1822	60	3	𝑣	𝑣	PROPN
iajs-1822	60	4	is	be	AUX
iajs-1822	60	5	a	a	DET
iajs-1822	60	6	sequence	sequence	NOUN
iajs-1822	60	7	of	of	ADP
iajs-1822	60	8	numbers	number	NOUN
iajs-1822	60	9	,	,	PUNCT
iajs-1822	60	10	its	its	PRON
iajs-1822	60	11	limit	limit	NOUN
iajs-1822	60	12	is	be	AUX
iajs-1822	60	13	unique	unique	ADJ
iajs-1822	60	14	.	.	PUNCT
iajs-1822	61	1	hence	hence	ADV
iajs-1822	61	2	𝑃	𝑃	VERB
iajs-1822	61	3	𝑣	𝑣	DET
iajs-1822	61	4	𝑃	𝑃	NOUN
iajs-1822	61	5	𝑢	𝑢	NOUN
iajs-1822	61	6	,	,	PUNCT
iajs-1822	61	7	that	that	ADV
iajs-1822	61	8	is	is	ADV
iajs-1822	61	9	,	,	PUNCT
iajs-1822	61	10	for	for	ADP
iajs-1822	61	11	every	every	DET
iajs-1822	61	12	p∈	p∈	PROPN
iajs-1822	61	13	𝑴𝜸.	𝑴𝜸.	PROPN
iajs-1822	62	1			CCONJ
iajs-1822	62	2	we	we	PRON
iajs-1822	62	3	have	have	VERB
iajs-1822	62	4	𝑃	𝑃	VERB
iajs-1822	62	5	𝑣	𝑣	DET
iajs-1822	62	6	𝑃	𝑃	NOUN
iajs-1822	62	7	𝑢	𝑢	X
iajs-1822	62	8	𝑃	𝑃	NOUN
iajs-1822	62	9	𝑣	𝑣	ADP
iajs-1822	62	10	𝑢	𝑢	PRON
iajs-1822	62	11	𝟢.this	𝟢.this	PRON
iajs-1822	62	12	implies	imply	VERB
iajs-1822	62	13	𝑣	𝑣	ADP
iajs-1822	62	14	𝑢	𝑢	X
iajs-1822	62	15	𝟢	𝟢	NUM
iajs-1822	62	16	and	and	CCONJ
iajs-1822	62	17	shows	show	VERB
iajs-1822	62	18	that	that	SCONJ
iajs-1822	62	19	the	the	DET
iajs-1822	62	20	weak	weak	ADJ
iajs-1822	62	21	limit	limit	NOUN
iajs-1822	62	22	is	be	AUX
iajs-1822	62	23	unique	unique	ADJ
iajs-1822	62	24	.	.	PUNCT
iajs-1822	63	1	part	part	NOUN
iajs-1822	63	2	(	(	PUNCT
iajs-1822	63	3	ii	ii	NOUN
iajs-1822	63	4	)	)	PUNCT
iajs-1822	63	5	follows	follow	VERB
iajs-1822	63	6	from	from	ADP
iajs-1822	63	7	the	the	DET
iajs-1822	63	8	fact	fact	NOUN
iajs-1822	63	9	that	that	SCONJ
iajs-1822	63	10	p	p	PROPN
iajs-1822	63	11	𝑣	𝑣	PROPN
iajs-1822	63	12	is	be	AUX
iajs-1822	63	13	convergent	convergent	ADJ
iajs-1822	63	14	sequence	sequence	NOUN
iajs-1822	63	15	of	of	ADP
iajs-1822	63	16	numbers	number	NOUN
iajs-1822	63	17	.	.	PUNCT
iajs-1822	64	1	so	so	SCONJ
iajs-1822	64	2	that	that	SCONJ
iajs-1822	64	3	every	every	DET
iajs-1822	64	4	subsequence	subsequence	NOUN
iajs-1822	64	5	of	of	ADP
iajs-1822	64	6	p	p	NOUN
iajs-1822	64	7	𝑣	𝑣	ADP
iajs-1822	64	8	converges	converge	NOUN
iajs-1822	64	9	and	and	CCONJ
iajs-1822	64	10	has	have	VERB
iajs-1822	64	11	same	same	ADJ
iajs-1822	64	12	limit	limit	NOUN
iajs-1822	64	13	as	as	ADP
iajs-1822	64	14	the	the	DET
iajs-1822	64	15	sequence	sequence	NOUN
iajs-1822	64	16	.	.	PUNCT
iajs-1822	65	1	definition	definition	NOUN
iajs-1822	65	2	(	(	PUNCT
iajs-1822	65	3	2.6	2.6	NUM
iajs-1822	65	4	):	):	PUNCT
iajs-1822	65	5	𝐴	𝐴	PROPN
iajs-1822	65	6	a	a	DET
iajs-1822	65	7	subset	subset	NOUN
iajs-1822	65	8	of	of	ADP
iajs-1822	65	9	a	a	DET
iajs-1822	65	10	modular	modular	ADJ
iajs-1822	65	11	space	space	NOUN
iajs-1822	65	12	𝑀	𝑀	PROPN
iajs-1822	65	13	is	be	AUX
iajs-1822	65	14	said	say	VERB
iajs-1822	65	15	to	to	PART
iajs-1822	65	16	be	be	AUX
iajs-1822	65	17	weakly	weakly	ADV
iajs-1822	65	18	compact	compact	ADJ
iajs-1822	65	19	if	if	SCONJ
iajs-1822	65	20	every	every	DET
iajs-1822	65	21	sequence	sequence	NOUN
iajs-1822	65	22	in	in	ADP
iajs-1822	65	23	𝑀	𝑀	PROPN
iajs-1822	65	24	has	have	VERB
iajs-1822	65	25	a	a	DET
iajs-1822	65	26	weak	weak	ADJ
iajs-1822	65	27	convergent	convergent	NOUN
iajs-1822	65	28	subsequence	subsequence	NOUN
iajs-1822	65	29	.	.	PUNCT
iajs-1822	66	1	definition	definition	NOUN
iajs-1822	66	2	(	(	PUNCT
iajs-1822	66	3	2.7	2.7	NUM
iajs-1822	66	4	):	):	PUNCT
iajs-1822	66	5	let	let	VERB
iajs-1822	66	6	𝑀	𝑀	PRON
iajs-1822	66	7	,	,	PUNCT
iajs-1822	66	8	𝑁	𝑁	PROPN
iajs-1822	66	9	be	be	VERB
iajs-1822	66	10	two	two	NUM
iajs-1822	66	11	modular	modular	ADJ
iajs-1822	66	12	spaces	space	NOUN
iajs-1822	66	13	and	and	CCONJ
iajs-1822	66	14	𝑆	𝑆	PROPN
iajs-1822	66	15	:	:	PUNCT
iajs-1822	67	1	𝑀	𝑀	PROPN
iajs-1822	67	2	𝑁	𝑁	NOUN
iajs-1822	67	3	be	be	AUX
iajs-1822	67	4	mappings	mapping	NOUN
iajs-1822	67	5	then	then	ADV
iajs-1822	67	6	:	:	PUNCT
iajs-1822	67	7	i.	i.	PROPN
iajs-1822	67	8	𝑆	𝑆	PROPN
iajs-1822	67	9	is	be	AUX
iajs-1822	67	10	continuous	continuous	ADJ
iajs-1822	67	11	if𝑣	if𝑣	PROPN
iajs-1822	67	12			PROPN
iajs-1822	67	13	𝑣	𝑣	ADP
iajs-1822	67	14			PROPN
iajs-1822	67	15	𝑆(𝑣	𝑆(𝑣	NOUN
iajs-1822	67	16	)	)	PUNCT
iajs-1822	67	17			PROPN
iajs-1822	67	18	𝑆	𝑆	PROPN
iajs-1822	67	19	(	(	PUNCT
iajs-1822	67	20	𝑣	𝑣	NOUN
iajs-1822	67	21	)	)	PUNCT
iajs-1822	67	22	.	.	PUNCT
iajs-1822	68	1	ii	ii	PROPN
iajs-1822	68	2	.	.	PUNCT
iajs-1822	69	1	𝑆	𝑆	PROPN
iajs-1822	69	2	is	be	AUX
iajs-1822	69	3	weakly	weakly	ADV
iajs-1822	69	4	continuous	continuous	ADJ
iajs-1822	69	5	if	if	SCONJ
iajs-1822	69	6	𝑣	𝑣	ADP
iajs-1822	69	7	→	→	SYM
iajs-1822	69	8	𝑣	𝑣	DET
iajs-1822	69	9			NOUN
iajs-1822	69	10	𝑆	𝑆	PROPN
iajs-1822	69	11	𝑣	𝑣	PROPN
iajs-1822	69	12	→	→	SYM
iajs-1822	69	13	𝑆	𝑆	PROPN
iajs-1822	69	14	(	(	PUNCT
iajs-1822	69	15	𝑣	𝑣	NOUN
iajs-1822	69	16	)	)	PUNCT
iajs-1822	69	17	.	.	PUNCT
iajs-1822	70	1	definition	definition	NOUN
iajs-1822	70	2	(	(	PUNCT
iajs-1822	70	3	2.8	2.8	NUM
iajs-1822	70	4	):	):	PUNCT
iajs-1822	70	5	let	let	VERB
iajs-1822	70	6	𝑀	𝑀	PRON
iajs-1822	70	7	be	be	AUX
iajs-1822	70	8	a	a	DET
iajs-1822	70	9	modular	modular	ADJ
iajs-1822	70	10	space	space	NOUN
iajs-1822	70	11	,	,	PUNCT
iajs-1822	70	12	𝐴⊆𝑀	𝐴⊆𝑀	PROPN
iajs-1822	70	13	and	and	CCONJ
iajs-1822	70	14	𝑆	𝑆	PROPN
iajs-1822	70	15	:	:	PUNCT
iajs-1822	70	16	𝐴→	𝐴→	PROPN
iajs-1822	70	17	𝑀	𝑀	PROPN
iajs-1822	70	18	be	be	VERB
iajs-1822	70	19	a	a	DET
iajs-1822	70	20	mapping	mapping	NOUN
iajs-1822	70	21	,	,	PUNCT
iajs-1822	70	22	𝑆	𝑆	PROPN
iajs-1822	70	23	is	be	AUX
iajs-1822	70	24	called	call	VERB
iajs-1822	70	25	demi	demi	NOUN
iajs-1822	70	26	-	-	PUNCT
iajs-1822	70	27	closed	closed	ADJ
iajs-1822	70	28	of	of	ADP
iajs-1822	70	29	𝑣	𝑣	DET
iajs-1822	70	30	∈	∈	PROPN
iajs-1822	70	31	𝐴	𝐴	PROPN
iajs-1822	70	32	,	,	PUNCT
iajs-1822	70	33	if	if	SCONJ
iajs-1822	70	34	for	for	ADP
iajs-1822	70	35	every	every	DET
iajs-1822	70	36	sequence	sequence	NOUN
iajs-1822	70	37	𝑣	𝑣	ADP
iajs-1822	70	38	in	in	ADP
iajs-1822	70	39	𝐴	𝐴	PROPN
iajs-1822	70	40	such	such	ADJ
iajs-1822	70	41	that	that	SCONJ
iajs-1822	70	42	𝑣	𝑣	X
iajs-1822	70	43	→	→	SYM
iajs-1822	70	44	𝑣	𝑣	PROPN
iajs-1822	70	45	and	and	CCONJ
iajs-1822	70	46	𝑣	𝑣	X
iajs-1822	70	47	→	→	SYM
iajs-1822	70	48	𝑢	𝑢	PROPN
iajs-1822	70	49	∈	∈	PROPN
iajs-1822	70	50	𝑀	𝑀	PROPN
iajs-1822	70	51	then	then	ADV
iajs-1822	70	52	𝑢	𝑢	PROPN
iajs-1822	70	53	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	70	54	and	and	CCONJ
iajs-1822	70	55	𝑆	𝑆	PROPN
iajs-1822	70	56	is	be	AUX
iajs-1822	70	57	demi	demi	NOUN
iajs-1822	70	58	closed	close	VERB
iajs-1822	70	59	on	on	ADP
iajs-1822	70	60	𝐴	𝐴	PROPN
iajs-1822	70	61	if	if	SCONJ
iajs-1822	70	62	it	it	PRON
iajs-1822	70	63	is	be	AUX
iajs-1822	70	64	demi	demi	NOUN
iajs-1822	70	65	-	-	PUNCT
iajs-1822	70	66	closed	closed	ADJ
iajs-1822	70	67	of	of	ADP
iajs-1822	70	68	each	each	DET
iajs-1822	70	69	𝑣	𝑣	NOUN
iajs-1822	70	70	in	in	ADP
iajs-1822	70	71	𝐴.	𝐴.	PROPN
iajs-1822	70	72	definition	definition	NOUN
iajs-1822	70	73	(	(	PUNCT
iajs-1822	70	74	2.9	2.9	NUM
iajs-1822	70	75	):	):	PUNCT
iajs-1822	70	76	let	let	VERB
iajs-1822	70	77	𝑀	𝑀	PRON
iajs-1822	70	78	be	be	AUX
iajs-1822	70	79	a	a	DET
iajs-1822	70	80	modular	modular	ADJ
iajs-1822	70	81	space	space	NOUN
iajs-1822	70	82	,	,	PUNCT
iajs-1822	70	83	𝑀	𝑀	PROPN
iajs-1822	70	84	is	be	AUX
iajs-1822	70	85	said	say	VERB
iajs-1822	70	86	to	to	PART
iajs-1822	70	87	be	be	AUX
iajs-1822	70	88	opial	opial	ADJ
iajs-1822	70	89	if	if	SCONJ
iajs-1822	70	90	for	for	SCONJ
iajs-1822	70	91	every	every	DET
iajs-1822	70	92	sequence	sequence	NOUN
iajs-1822	70	93	𝑣	𝑣	X
iajs-1822	70	94	in	in	ADP
iajs-1822	70	95	𝑀	𝑀	PROPN
iajs-1822	70	96	weakly	weakly	ADJ
iajs-1822	70	97	convergent	convergent	NOUN
iajs-1822	70	98	to	to	ADP
iajs-1822	70	99	𝑣	𝑣	DET
iajs-1822	70	100	∈	∈	PROPN
iajs-1822	70	101	𝑀	𝑀	PROPN
iajs-1822	70	102	the	the	DET
iajs-1822	70	103	inequality	inequality	NOUN
iajs-1822	70	104	lim	lim	PROPN
iajs-1822	70	105	→	→	PROPN
iajs-1822	70	106	𝑖𝑛𝑓	𝑖𝑛𝑓	PROPN
iajs-1822	70	107	𝛾	𝛾	ADP
iajs-1822	70	108	𝑣	𝑣	PRON
iajs-1822	70	109	𝑣	𝑣	PROPN
iajs-1822	70	110	lim	lim	PROPN
iajs-1822	70	111	→	→	PROPN
iajs-1822	70	112	𝑖𝑛𝑓𝛾	𝑖𝑛𝑓𝛾	PROPN
iajs-1822	70	113	𝑣	𝑣	ADP
iajs-1822	70	114	𝑢	𝑢	PROPN
iajs-1822	70	115	holds	hold	VERB
iajs-1822	70	116	for	for	ADP
iajs-1822	70	117	all	all	PRON
iajs-1822	70	118	𝑢	𝑢	DET
iajs-1822	70	119	𝑣.	𝑣.	NOUN
iajs-1822	70	120	ihsciconf	ihsciconf	PROPN
iajs-1822	70	121	2017	2017	NUM
iajs-1822	70	122	special	special	ADJ
iajs-1822	70	123	issue	issue	NOUN
iajs-1822	70	124	ibn	ibn	PROPN
iajs-1822	70	125	al	al	PROPN
iajs-1822	70	126	-	-	PUNCT
iajs-1822	70	127	haitham	haitham	PROPN
iajs-1822	70	128	journal	journal	PROPN
iajs-1822	70	129	for	for	ADP
iajs-1822	70	130	pure	pure	ADJ
iajs-1822	70	131	and	and	CCONJ
iajs-1822	70	132	applied	apply	VERB
iajs-1822	70	133	science	science	NOUN
iajs-1822	70	134	https://doi.org/	https://doi.org/	NOUN
iajs-1822	70	135	10.30526/2017.ihsciconf.1822	10.30526/2017.ihsciconf.1822	NOUN
iajs-1822	70	136	for	for	ADP
iajs-1822	70	137	more	more	ADJ
iajs-1822	70	138	information	information	NOUN
iajs-1822	70	139	about	about	ADP
iajs-1822	70	140	the	the	DET
iajs-1822	70	141	conference	conference	NOUN
iajs-1822	70	142	please	please	INTJ
iajs-1822	70	143	visit	visit	VERB
iajs-1822	70	144	the	the	DET
iajs-1822	70	145	websites	website	NOUN
iajs-1822	70	146	:	:	PUNCT
iajs-1822	70	147	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1822	70	148	  	  	SPACE
iajs-1822	70	149	www.ihsciconf.org	www.ihsciconf.org	ADV
iajs-1822	70	150	   	   	SPACE
iajs-1822	70	151	mathematics|505	mathematics|505	NOUN
iajs-1822	70	152	    	    	SPACE
iajs-1822	70	153	3	3	NUM
iajs-1822	70	154	.	.	PUNCT
iajs-1822	70	155	common	common	ADJ
iajs-1822	70	156	fixed	fix	VERB
iajs-1822	70	157	point	point	NOUN
iajs-1822	70	158	for	for	ADP
iajs-1822	70	159	commuting	commute	VERB
iajs-1822	70	160	mappings	mapping	NOUN
iajs-1822	70	161	mongkolkeha	mongkolkeha	ADJ
iajs-1822	70	162	,	,	PUNCT
iajs-1822	70	163	sintunavarat	sintunavarat	NOUN
iajs-1822	70	164	and	and	CCONJ
iajs-1822	70	165	kumamstudy[11]and	kumamstudy[11]and	X
iajs-1822	71	1	[	[	X
iajs-1822	71	2	12	12	NUM
iajs-1822	71	3	]	]	PUNCT
iajs-1822	71	4	proved	prove	VERB
iajs-1822	71	5	the	the	DET
iajs-1822	71	6	existence	existence	NOUN
iajs-1822	71	7	theorems	theorem	NOUN
iajs-1822	71	8	of	of	ADP
iajs-1822	71	9	fixed	fix	VERB
iajs-1822	71	10	points	point	NOUN
iajs-1822	71	11	for	for	ADP
iajs-1822	71	12	contraction	contraction	NOUN
iajs-1822	71	13	mappings	mapping	NOUN
iajs-1822	71	14	in	in	ADP
iajs-1822	71	15	modular	modular	ADJ
iajs-1822	71	16	metric	metric	ADJ
iajs-1822	71	17	spaces	space	NOUN
iajs-1822	71	18	with	with	ADP
iajs-1822	71	19	condition	condition	NOUN
iajs-1822	71	20	γ	γ	X
iajs-1822	71	21	𝑃	𝑃	ADP
iajs-1822	71	22	𝑣	𝑣	PRON
iajs-1822	71	23	∞	∞	NUM
iajs-1822	71	24	to	to	PART
iajs-1822	71	25	guarantee	guarantee	VERB
iajs-1822	71	26	the	the	DET
iajs-1822	71	27	existence	existence	NOUN
iajs-1822	71	28	and	and	CCONJ
iajs-1822	71	29	uniqueness	uniqueness	NOUN
iajs-1822	71	30	of	of	ADP
iajs-1822	71	31	the	the	DET
iajs-1822	71	32	fixed	fix	VERB
iajs-1822	71	33	points	point	NOUN
iajs-1822	71	34	.	.	PUNCT
iajs-1822	72	1	we	we	PRON
iajs-1822	72	2	start	start	VERB
iajs-1822	72	3	with	with	ADP
iajs-1822	72	4	following	follow	VERB
iajs-1822	72	5	proposition	proposition	NOUN
iajs-1822	72	6	(	(	PUNCT
iajs-1822	72	7	3.1	3.1	NUM
iajs-1822	72	8	):	):	PUNCT
iajs-1822	72	9	let	let	VERB
iajs-1822	72	10	𝑃	𝑃	PRON
iajs-1822	72	11	be	be	AUX
iajs-1822	72	12	a	a	DET
iajs-1822	72	13	continuous	continuous	ADJ
iajs-1822	72	14	self	self	NOUN
iajs-1822	72	15	-	-	PUNCT
iajs-1822	72	16	mapping	mapping	NOUN
iajs-1822	72	17	of	of	ADP
iajs-1822	72	18	a	a	DET
iajs-1822	72	19	complete	complete	ADJ
iajs-1822	72	20	modular	modular	ADJ
iajs-1822	72	21	space	space	NOUN
iajs-1822	72	22	𝑀	𝑀	PROPN
iajs-1822	72	23	,	,	PUNCT
iajs-1822	72	24	𝛾	𝛾	X
iajs-1822	72	25	if	if	SCONJ
iajs-1822	72	26	𝑆	𝑆	PROPN
iajs-1822	72	27	:	:	PUNCT
iajs-1822	72	28	𝑀	𝑀	PROPN
iajs-1822	72	29	→	→	SYM
iajs-1822	72	30	𝑀	𝑀	PROPN
iajs-1822	72	31	is	be	AUX
iajs-1822	72	32	𝑃contraction	𝑃contraction	PROPN
iajs-1822	72	33	mapping	mapping	NOUN
iajs-1822	72	34	which	which	PRON
iajs-1822	72	35	commutes	commute	VERB
iajs-1822	72	36	with	with	ADP
iajs-1822	72	37	𝑃	𝑃	NOUN
iajs-1822	72	38	and	and	CCONJ
iajs-1822	72	39	𝑆	𝑆	PROPN
iajs-1822	72	40	𝑀	𝑀	PROPN
iajs-1822	72	41	⊆	⊆	NUM
iajs-1822	72	42	𝑃	𝑃	PROPN
iajs-1822	72	43	𝑀	𝑀	PROPN
iajs-1822	72	44	and	and	CCONJ
iajs-1822	72	45	∃	∃	PROPN
iajs-1822	72	46	𝑣	𝑣	PROPN
iajs-1822	72	47	∈	∈	PROPN
iajs-1822	72	48	𝑀	𝑀	PROPN
iajs-1822	72	49	such	such	ADJ
iajs-1822	72	50	that	that	SCONJ
iajs-1822	72	51	γ	γ	PROPN
iajs-1822	72	52	𝑃	𝑃	VERB
iajs-1822	72	53	𝑣	𝑣	NOUN
iajs-1822	72	54	∞	∞	NUM
iajs-1822	72	55	then	then	ADV
iajs-1822	72	56	𝐹	𝐹	PRON
iajs-1822	72	57	𝑃	𝑃	PROPN
iajs-1822	72	58	∩	∩	NOUN
iajs-1822	72	59	𝐹	𝐹	PROPN
iajs-1822	72	60	𝑆	𝑆	PROPN
iajs-1822	72	61	singleton	singleton	NOUN
iajs-1822	72	62	.	.	PUNCT
iajs-1822	73	1	proof	proof	NOUN
iajs-1822	73	2	:	:	PUNCT
iajs-1822	73	3	suppose	suppose	VERB
iajs-1822	73	4	p	p	X
iajs-1822	73	5	𝑎	𝑎	NOUN
iajs-1822	73	6	𝑎	𝑎	NOUN
iajs-1822	73	7	for	for	ADP
iajs-1822	73	8	some	some	DET
iajs-1822	73	9	𝑎	𝑎	PROPN
iajs-1822	73	10	∈	∈	PROPN
iajs-1822	73	11	𝑀	𝑀	PROPN
iajs-1822	73	12	,	,	PUNCT
iajs-1822	73	13	define	define	VERB
iajs-1822	73	14	𝑆	𝑆	PROPN
iajs-1822	73	15	:	:	PUNCT
iajs-1822	73	16	𝑀	𝑀	PROPN
iajs-1822	73	17	→	→	SYM
iajs-1822	73	18	𝑀	𝑀	PROPN
iajs-1822	73	19	by	by	ADP
iajs-1822	73	20	𝑆	𝑆	PROPN
iajs-1822	73	21	𝑣	𝑣	ADP
iajs-1822	73	22	𝑎	𝑎	NOUN
iajs-1822	73	23	∀	∀	NOUN
iajs-1822	73	24	𝑣	𝑣	ADP
iajs-1822	73	25	∈	∈	PROPN
iajs-1822	73	26	𝑀	𝑀	PROPN
iajs-1822	73	27	then	then	ADV
iajs-1822	73	28	𝑆	𝑆	PROPN
iajs-1822	73	29	𝑃	𝑃	VERB
iajs-1822	73	30	𝑣	𝑣	DET
iajs-1822	73	31	𝑎	𝑎	NOUN
iajs-1822	73	32	and	and	CCONJ
iajs-1822	73	33	𝑃	𝑃	PROPN
iajs-1822	73	34	𝑆	𝑆	PROPN
iajs-1822	73	35	𝑣	𝑣	PART
iajs-1822	73	36	𝑃	𝑃	VERB
iajs-1822	73	37	𝑎	𝑎	NOUN
iajs-1822	73	38	for	for	ADP
iajs-1822	73	39	all	all	DET
iajs-1822	73	40	𝑣	𝑣	PRON
iajs-1822	73	41	∈	∈	NOUN
iajs-1822	73	42	𝑀	𝑀	PROPN
iajs-1822	74	1	so	so	ADV
iajs-1822	74	2	𝑆	𝑆	PROPN
iajs-1822	74	3	𝑃	𝑃	PROPN
iajs-1822	74	4	𝑣	𝑣	DET
iajs-1822	74	5	𝑃	𝑃	NOUN
iajs-1822	74	6	𝑆	𝑆	PROPN
iajs-1822	74	7	𝑣	𝑣	NOUN
iajs-1822	74	8	,	,	PUNCT
iajs-1822	74	9	∀	∀	VERB
iajs-1822	74	10	𝑣	𝑣	ADP
iajs-1822	74	11	∈	∈	PROPN
iajs-1822	74	12	𝑀	𝑀	PROPN
iajs-1822	74	13	and	and	CCONJ
iajs-1822	74	14	𝑆	𝑆	PROPN
iajs-1822	74	15	commutes	commute	NOUN
iajs-1822	74	16	with	with	ADP
iajs-1822	74	17	𝑃	𝑃	NOUN
iajs-1822	74	18	moreover	moreover	ADV
iajs-1822	74	19	𝑆	𝑆	PROPN
iajs-1822	74	20	𝑣	𝑣	PART
iajs-1822	74	21	𝑎	𝑎	PRON
iajs-1822	74	22	𝑃	𝑃	NOUN
iajs-1822	74	23	𝑎	𝑎	NOUN
iajs-1822	74	24	∀	∀	NOUN
iajs-1822	74	25	𝑣	𝑣	ADP
iajs-1822	74	26	∈	∈	PROPN
iajs-1822	74	27	𝑀	𝑀	PROPN
iajs-1822	74	28	so	so	SCONJ
iajs-1822	74	29	that	that	SCONJ
iajs-1822	74	30	𝑆	𝑆	PROPN
iajs-1822	74	31	𝑀	𝑀	PROPN
iajs-1822	74	32	⊆	⊆	NUM
iajs-1822	74	33	𝑃	𝑃	PROPN
iajs-1822	74	34	𝑀	𝑀	PROPN
iajs-1822	74	35	.	.	PUNCT
iajs-1822	75	1	finally	finally	ADV
iajs-1822	75	2	,	,	PUNCT
iajs-1822	75	3	∀𝑎	∀𝑎	PROPN
iajs-1822	75	4	∈	∈	PROPN
iajs-1822	75	5	0,1	0,1	NUM
iajs-1822	75	6	,	,	PUNCT
iajs-1822	75	7	∀	∀	X
iajs-1822	75	8	𝑣,𝑢	𝑣,𝑢	NOUN
iajs-1822	75	9	in	in	ADP
iajs-1822	75	10	𝑀	𝑀	PROPN
iajs-1822	75	11	we	we	PRON
iajs-1822	75	12	have	have	VERB
iajs-1822	75	13	𝛾	𝛾	ADP
iajs-1822	75	14	𝑆	𝑆	PROPN
iajs-1822	75	15	𝑣	𝑣	NOUN
iajs-1822	75	16	,	,	PUNCT
iajs-1822	75	17	𝑆	𝑆	PROPN
iajs-1822	75	18	𝑢	𝑢	X
iajs-1822	75	19	𝛾	𝛾	NOUN
iajs-1822	75	20	𝑎	𝑎	X
iajs-1822	75	21	,	,	PUNCT
iajs-1822	75	22	𝑎	𝑎	PRON
iajs-1822	75	23	𝟶	𝟶	NUM
iajs-1822	75	24	𝑎	𝑎	NOUN
iajs-1822	75	25	𝛾	𝛾	NOUN
iajs-1822	75	26	𝑃	𝑃	NOUN
iajs-1822	75	27	𝑣	𝑣	NOUN
iajs-1822	75	28	,	,	PUNCT
iajs-1822	75	29	𝑃	𝑃	VERB
iajs-1822	75	30	𝑢	𝑢	NOUN
iajs-1822	75	31	.	.	PUNCT
iajs-1822	76	1	this	this	PRON
iajs-1822	76	2	completes	complete	VERB
iajs-1822	76	3	the	the	DET
iajs-1822	76	4	proof	proof	NOUN
iajs-1822	76	5	.	.	PUNCT
iajs-1822	77	1	now	now	ADV
iajs-1822	77	2	,	,	PUNCT
iajs-1822	77	3	it	it	PRON
iajs-1822	77	4	is	be	AUX
iajs-1822	77	5	easy	easy	ADJ
iajs-1822	77	6	to	to	PART
iajs-1822	77	7	show	show	VERB
iajs-1822	77	8	that	that	SCONJ
iajs-1822	77	9	the	the	DET
iajs-1822	77	10	following	follow	VERB
iajs-1822	77	11	needed	need	VERB
iajs-1822	77	12	lemma	lemma	PROPN
iajs-1822	77	13	.	.	PUNCT
iajs-1822	78	1	lemma	lemma	PROPN
iajs-1822	78	2	(	(	PUNCT
iajs-1822	78	3	3.2	3.2	NUM
iajs-1822	78	4	):	):	PUNCT
iajs-1822	78	5	let	let	VERB
iajs-1822	78	6	𝑀	𝑀	PRON
iajs-1822	78	7	be	be	AUX
iajs-1822	78	8	a	a	DET
iajs-1822	78	9	modular	modular	ADJ
iajs-1822	78	10	space	space	NOUN
iajs-1822	78	11	,	,	PUNCT
iajs-1822	78	12	𝑆	𝑆	PROPN
iajs-1822	78	13	:	:	PUNCT
iajs-1822	78	14	𝑀	𝑀	PROPN
iajs-1822	78	15	→	→	SYM
iajs-1822	78	16	𝑀	𝑀	PROPN
iajs-1822	78	17	be	be	VERB
iajs-1822	78	18	mapping	mapping	NOUN
iajs-1822	78	19	,	,	PUNCT
iajs-1822	78	20	and	and	CCONJ
iajs-1822	78	21	𝑢	𝑢	PROPN
iajs-1822	78	22	∈	∈	PROPN
iajs-1822	78	23	𝑀	𝑀	PROPN
iajs-1822	78	24	.	.	PUNCT
iajs-1822	79	1	if	if	SCONJ
iajs-1822	79	2	𝑆	𝑆	PROPN
iajs-1822	79	3	ℎ𝑢	ℎ𝑢	PROPN
iajs-1822	79	4	1	1	NUM
iajs-1822	79	5	ℎ	ℎ	PART
iajs-1822	79	6	𝑣	𝑣	PROPN
iajs-1822	79	7	ℎ𝑆𝑢	ℎ𝑆𝑢	PROPN
iajs-1822	79	8	1	1	NUM
iajs-1822	79	9	ℎ	ℎ	PART
iajs-1822	79	10	𝑣	𝑣	NOUN
iajs-1822	79	11	,	,	PUNCT
iajs-1822	79	12	∀𝑣	∀𝑣	PROPN
iajs-1822	79	13	∈	∈	PROPN
iajs-1822	79	14	𝑀	𝑀	PROPN
iajs-1822	79	15	and	and	CCONJ
iajs-1822	79	16	ℎ	ℎ	PART
iajs-1822	79	17	∈	∈	NOUN
iajs-1822	79	18	0,1	0,1	NUM
iajs-1822	79	19	,	,	PUNCT
iajs-1822	79	20	then	then	ADV
iajs-1822	79	21	𝑢	𝑢	PROPN
iajs-1822	79	22	is	be	AUX
iajs-1822	79	23	a	a	DET
iajs-1822	79	24	fixed	fix	VERB
iajs-1822	79	25	point	point	NOUN
iajs-1822	79	26	.	.	PUNCT
iajs-1822	80	1	theorem	theorem	NOUN
iajs-1822	80	2	(	(	PUNCT
iajs-1822	80	3	3.3	3.3	NUM
iajs-1822	80	4	):	):	PUNCT
iajs-1822	80	5	let	let	VERB
iajs-1822	80	6	∅	∅	NOUN
iajs-1822	80	7	𝐴	𝐴	PROPN
iajs-1822	80	8	weakly	weakly	ADJ
iajs-1822	80	9	compact	compact	ADJ
iajs-1822	80	10	subset	subset	NOUN
iajs-1822	80	11	of	of	ADP
iajs-1822	80	12	a	a	DET
iajs-1822	80	13	complete	complete	ADJ
iajs-1822	80	14	modular	modular	ADJ
iajs-1822	80	15	space	space	NOUN
iajs-1822	80	16	𝑀	𝑀	PROPN
iajs-1822	80	17	.	.	PUNCT
iajs-1822	81	1	let	let	VERB
iajs-1822	81	2	𝑝	𝑝	PART
iajs-1822	81	3	be	be	AUX
iajs-1822	81	4	a	a	DET
iajs-1822	81	5	continuous	continuous	ADJ
iajs-1822	81	6	and	and	CCONJ
iajs-1822	81	7	affine	affine	NOUN
iajs-1822	81	8	mapping	mapping	NOUN
iajs-1822	81	9	on	on	ADP
iajs-1822	81	10	𝑀	𝑀	PROPN
iajs-1822	81	11	with	with	ADP
iajs-1822	81	12	p	p	PROPN
iajs-1822	81	13	𝐴	𝐴	PROPN
iajs-1822	81	14	𝐴	𝐴	PROPN
iajs-1822	81	15	,	,	PUNCT
iajs-1822	81	16	𝑆	𝑆	PROPN
iajs-1822	81	17	:	:	PUNCT
iajs-1822	81	18	𝐴	𝐴	PROPN
iajs-1822	81	19	→𝐴	→𝐴	NOUN
iajs-1822	81	20	be	be	AUX
iajs-1822	81	21	an	an	DET
iajs-1822	81	22	𝑃non	𝑃non	PROPN
iajs-1822	81	23	–	–	PUNCT
iajs-1822	81	24	expansive	expansive	ADJ
iajs-1822	81	25	mapping	mapping	NOUN
iajs-1822	81	26	commutes	commute	NOUN
iajs-1822	81	27	with	with	ADP
iajs-1822	81	28	𝑃.	𝑃.	PROPN
iajs-1822	81	29	if	if	SCONJ
iajs-1822	81	30	𝐴	𝐴	PROPN
iajs-1822	81	31	is	be	AUX
iajs-1822	81	32	star	star	NOUN
iajs-1822	81	33	-	-	PUNCT
iajs-1822	81	34	shaped	shape	VERB
iajs-1822	81	35	with	with	ADP
iajs-1822	81	36	respect	respect	NOUN
iajs-1822	81	37	to	to	ADP
iajs-1822	81	38	𝑆,and	𝑆,and	NOUN
iajs-1822	81	39	there	there	PRON
iajs-1822	81	40	is	be	VERB
iajs-1822	81	41	some	some	DET
iajs-1822	81	42	𝑣	𝑣	ADP
iajs-1822	81	43	∈	∈	PROPN
iajs-1822	81	44	𝐴	𝐴	PROPN
iajs-1822	81	45	𝛾	𝛾	VERB
iajs-1822	81	46	𝑆	𝑆	PROPN
iajs-1822	81	47	𝑣	𝑣	ADP
iajs-1822	81	48	∞	∞	NUM
iajs-1822	81	49	and	and	CCONJ
iajs-1822	81	50	𝑃	𝑃	PROPN
iajs-1822	81	51	𝑆	𝑆	PROPN
iajs-1822	81	52	is	be	AUX
iajs-1822	81	53	demi	demi	NOUN
iajs-1822	81	54	-	-	PUNCT
iajs-1822	81	55	closed	closed	ADJ
iajs-1822	81	56	on	on	ADP
iajs-1822	81	57	𝑀	𝑀	PROPN
iajs-1822	81	58	,	,	PUNCT
iajs-1822	81	59	then	then	ADV
iajs-1822	81	60	𝐹	𝐹	PROPN
iajs-1822	81	61	𝑆	𝑆	PROPN
iajs-1822	81	62	∩	∩	NOUN
iajs-1822	81	63	𝐹	𝐹	PRON
iajs-1822	81	64	𝑃	𝑃	VERB
iajs-1822	81	65	∅.	∅.	NOUN
iajs-1822	81	66	proof	proof	NOUN
iajs-1822	81	67	:	:	PUNCT
iajs-1822	81	68	since	since	SCONJ
iajs-1822	81	69	𝐴	𝐴	PROPN
iajs-1822	81	70	is	be	AUX
iajs-1822	81	71	star	star	NOUN
iajs-1822	81	72	-	-	PUNCT
iajs-1822	81	73	shaped	shape	VERB
iajs-1822	81	74	with	with	ADP
iajs-1822	81	75	respect	respect	NOUN
iajs-1822	81	76	to	to	ADP
iajs-1822	81	77	𝑢∈	𝑢∈	PROPN
iajs-1822	81	78	𝐴	𝐴	PROPN
iajs-1822	81	79	,	,	PUNCT
iajs-1822	81	80	then	then	ADV
iajs-1822	81	81	𝑆	𝑆	PROPN
iajs-1822	81	82	:	:	PUNCT
iajs-1822	81	83	𝐴	𝐴	PROPN
iajs-1822	81	84	→	→	SYM
iajs-1822	81	85	𝐴	𝐴	PROPN
iajs-1822	81	86	,	,	PUNCT
iajs-1822	81	87	we	we	PRON
iajs-1822	81	88	define	define	VERB
iajs-1822	81	89	𝑆	𝑆	PROPN
iajs-1822	81	90	on	on	ADP
iajs-1822	81	91	𝐴	𝐴	PROPN
iajs-1822	81	92	for	for	ADP
iajs-1822	81	93	any	any	DET
iajs-1822	81	94	𝑣	𝑣	NOUN
iajs-1822	81	95	in	in	ADP
iajs-1822	81	96	𝐴	𝐴	PROPN
iajs-1822	81	97	by	by	ADP
iajs-1822	81	98	,	,	PUNCT
iajs-1822	81	99	𝑆	𝑆	PROPN
iajs-1822	81	100	𝑣	𝑣	ADP
iajs-1822	81	101	ℎ	ℎ	PART
iajs-1822	81	102	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	81	103	1	1	NUM
iajs-1822	81	104	ℎ	ℎ	PART
iajs-1822	81	105	𝑢	𝑢	NOUN
iajs-1822	81	106	and	and	CCONJ
iajs-1822	81	107	there	there	PRON
iajs-1822	81	108	is	be	VERB
iajs-1822	81	109	𝑢	𝑢	PRON
iajs-1822	81	110	∈	∈	PROPN
iajs-1822	81	111	𝐴	𝐴	PROPN
iajs-1822	81	112	,	,	PUNCT
iajs-1822	81	113	and	and	CCONJ
iajs-1822	81	114	the	the	DET
iajs-1822	81	115	sequence	sequence	NOUN
iajs-1822	81	116	ℎ	ℎ	X
iajs-1822	81	117	→	→	SYM
iajs-1822	81	118	1	1	NUM
iajs-1822	81	119	as	as	ADP
iajs-1822	81	120	𝑛	𝑛	PROPN
iajs-1822	81	121	→	→	SYM
iajs-1822	81	122	∞	∞	PROPN
iajs-1822	81	123	,	,	PUNCT
iajs-1822	81	124	𝟶	𝟶	NUM
iajs-1822	81	125	ℎ	ℎ	PART
iajs-1822	81	126	1	1	NUM
iajs-1822	81	127	such	such	ADJ
iajs-1822	81	128	that	that	SCONJ
iajs-1822	81	129	1	1	NUM
iajs-1822	81	130	ℎ	ℎ	PART
iajs-1822	81	131	𝑢	𝑢	NOUN
iajs-1822	81	132	ℎ	ℎ	ADP
iajs-1822	81	133	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	81	134	∈	∈	PROPN
iajs-1822	81	135	𝐴	𝐴	PROPN
iajs-1822	81	136	∀	∀	X
iajs-1822	82	1	𝑣,𝑢	𝑣,𝑢	NOUN
iajs-1822	82	2	∈	∈	PROPN
iajs-1822	82	3	𝐴.	𝐴.	NOUN
iajs-1822	82	4	it	it	PRON
iajs-1822	82	5	is	be	AUX
iajs-1822	82	6	clear	clear	ADJ
iajs-1822	82	7	that	that	SCONJ
iajs-1822	82	8	𝑆	𝑆	PROPN
iajs-1822	82	9	∶	∶	PROPN
iajs-1822	82	10	𝐴	𝐴	PROPN
iajs-1822	82	11	→	→	SYM
iajs-1822	82	12	𝐴.	𝐴.	PROPN
iajs-1822	82	13	note	note	NOUN
iajs-1822	82	14	that	that	SCONJ
iajs-1822	82	15	𝑆	𝑆	PROPN
iajs-1822	82	16	𝐴	𝐴	PROPN
iajs-1822	82	17	⊆	⊆	NUM
iajs-1822	82	18	𝐴	𝐴	PROPN
iajs-1822	82	19	and	and	CCONJ
iajs-1822	82	20	𝑆	𝑆	PROPN
iajs-1822	82	21	𝐴	𝐴	PROPN
iajs-1822	82	22	⊆	⊆	PROPN
iajs-1822	82	23	𝑝	𝑝	PROPN
iajs-1822	82	24	𝐴	𝐴	PROPN
iajs-1822	82	25	.	.	PUNCT
iajs-1822	83	1	since	since	SCONJ
iajs-1822	83	2	𝑆	𝑆	PROPN
iajs-1822	83	3	commutes	commute	NOUN
iajs-1822	83	4	with	with	ADP
iajs-1822	83	5	𝑃	𝑃	NOUN
iajs-1822	83	6	and	and	CCONJ
iajs-1822	83	7	𝑃	𝑃	PROPN
iajs-1822	83	8	is	be	AUX
iajs-1822	83	9	affine	affine	NOUN
iajs-1822	83	10	mapping	mapping	NOUN
iajs-1822	83	11	,	,	PUNCT
iajs-1822	83	12	for	for	ADP
iajs-1822	83	13	each	each	DET
iajs-1822	83	14	𝑣	𝑣	PRON
iajs-1822	83	15	∈	∈	PROPN
iajs-1822	84	1	𝐴.	𝐴.	PROPN
iajs-1822	84	2	𝑆	𝑆	PROPN
iajs-1822	84	3	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	84	4	ℎ	ℎ	NOUN
iajs-1822	84	5	𝑆𝑝𝑣	𝑆𝑝𝑣	PROPN
iajs-1822	84	6	1	1	NUM
iajs-1822	84	7	ℎ	ℎ	NOUN
iajs-1822	84	8	𝑃𝑢	𝑃𝑢	VERB
iajs-1822	84	9	ℎ	ℎ	NOUN
iajs-1822	84	10	𝑃𝑆𝑣	𝑃𝑆𝑣	ADJ
iajs-1822	84	11	1	1	NUM
iajs-1822	84	12	ℎ	ℎ	X
iajs-1822	85	1	𝑃𝑢	𝑃𝑢	VERB
iajs-1822	85	2	𝑃	𝑃	NOUN
iajs-1822	85	3	ℎ	ℎ	NOUN
iajs-1822	85	4	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	85	5	1	1	NUM
iajs-1822	85	6	ℎ	ℎ	NOUN
iajs-1822	85	7	𝑃𝑆	𝑃𝑆	NOUN
iajs-1822	85	8	𝑣	𝑣	PRON
iajs-1822	85	9	∋	∋	NOUN
iajs-1822	85	10	𝑆	𝑆	PROPN
iajs-1822	85	11	commutes	commute	VERB
iajs-1822	85	12	with	with	ADP
iajs-1822	85	13	𝑃.	𝑃.	PROPN
iajs-1822	85	14	further	far	ADV
iajs-1822	85	15	,	,	PUNCT
iajs-1822	85	16	we	we	PRON
iajs-1822	85	17	observe	observe	VERB
iajs-1822	85	18	that	that	SCONJ
iajs-1822	85	19	for	for	ADP
iajs-1822	85	20	each	each	DET
iajs-1822	85	21	𝑛	𝑛	DET
iajs-1822	85	22	1	1	NUM
iajs-1822	85	23	,	,	PUNCT
iajs-1822	85	24	𝑆	𝑆	PROPN
iajs-1822	85	25	is	be	AUX
iajs-1822	85	26	𝑃non	𝑃non	PROPN
iajs-1822	85	27	-	-	PUNCT
iajs-1822	85	28	expansive	expansive	ADJ
iajs-1822	85	29	mapping	mapping	NOUN
iajs-1822	85	30	,	,	PUNCT
iajs-1822	85	31	ihsciconf	ihsciconf	PROPN
iajs-1822	85	32	2017	2017	NUM
iajs-1822	85	33	special	special	ADJ
iajs-1822	85	34	issue	issue	NOUN
iajs-1822	85	35	ibn	ibn	PROPN
iajs-1822	85	36	al	al	PROPN
iajs-1822	85	37	-	-	PUNCT
iajs-1822	85	38	haitham	haitham	PROPN
iajs-1822	85	39	journal	journal	PROPN
iajs-1822	85	40	for	for	ADP
iajs-1822	85	41	pure	pure	ADJ
iajs-1822	85	42	and	and	CCONJ
iajs-1822	85	43	applied	apply	VERB
iajs-1822	85	44	science	science	NOUN
iajs-1822	85	45	https://doi.org/	https://doi.org/	NOUN
iajs-1822	85	46	10.30526/2017.ihsciconf.1822	10.30526/2017.ihsciconf.1822	NOUN
iajs-1822	85	47	for	for	ADP
iajs-1822	85	48	more	more	ADJ
iajs-1822	85	49	information	information	NOUN
iajs-1822	85	50	about	about	ADP
iajs-1822	85	51	the	the	DET
iajs-1822	85	52	conference	conference	NOUN
iajs-1822	85	53	please	please	INTJ
iajs-1822	85	54	visit	visit	VERB
iajs-1822	85	55	the	the	DET
iajs-1822	85	56	websites	website	NOUN
iajs-1822	85	57	:	:	PUNCT
iajs-1822	85	58	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1822	85	59	  	  	SPACE
iajs-1822	85	60	www.ihsciconf.org	www.ihsciconf.org	ADJ
iajs-1822	85	61	   	   	SPACE
iajs-1822	85	62	mathematics|506	mathematics|506	NOUN
iajs-1822	85	63	    	    	SPACE
iajs-1822	85	64	𝛾	𝛾	ADP
iajs-1822	85	65	𝑆	𝑆	PROPN
iajs-1822	85	66	𝑣	𝑣	ADP
iajs-1822	85	67	𝑆	𝑆	PROPN
iajs-1822	85	68	𝑢	𝑢	NOUN
iajs-1822	85	69	𝛾	𝛾	NOUN
iajs-1822	85	70	ℎ	ℎ	PART
iajs-1822	85	71	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	85	72	1	1	NUM
iajs-1822	85	73	ℎ	ℎ	VERB
iajs-1822	85	74	𝑢	𝑢	NOUN
iajs-1822	85	75	ℎ	ℎ	PART
iajs-1822	85	76	𝑆𝑢	𝑆𝑢	PROPN
iajs-1822	85	77	1	1	NUM
iajs-1822	85	78	ℎ	ℎ	PART
iajs-1822	85	79	𝑢	𝑢	PROPN
iajs-1822	85	80	ℎ	ℎ	ADP
iajs-1822	85	81	𝛾	𝛾	PROPN
iajs-1822	85	82	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	85	83	𝑆𝑢	𝑆𝑢	PROPN
iajs-1822	85	84	ℎ	ℎ	NOUN
iajs-1822	85	85	𝛾	𝛾	AUX
iajs-1822	85	86	𝑃𝑣	𝑃𝑣	NOUN
iajs-1822	85	87	𝑃𝑢	𝑃𝑢	NOUN
iajs-1822	85	88	∀	∀	NOUN
iajs-1822	85	89	𝑣	𝑣	NOUN
iajs-1822	85	90	,	,	PUNCT
iajs-1822	85	91	𝑢	𝑢	PROPN
iajs-1822	85	92	∈	∈	PROPN
iajs-1822	85	93	𝐴	𝐴	PROPN
iajs-1822	85	94	hence	hence	ADV
iajs-1822	85	95	𝑆	𝑆	PROPN
iajs-1822	85	96	is	be	AUX
iajs-1822	85	97	𝑃contraction	𝑃contraction	PROPN
iajs-1822	85	98	.	.	PUNCT
iajs-1822	86	1	thus	thus	ADV
iajs-1822	86	2	by	by	ADP
iajs-1822	86	3	proposition	proposition	NOUN
iajs-1822	86	4	(	(	PUNCT
iajs-1822	86	5	3.1	3.1	NUM
iajs-1822	86	6	)	)	PUNCT
iajs-1822	86	7	,	,	PUNCT
iajs-1822	86	8	there	there	PRON
iajs-1822	86	9	is	be	VERB
iajs-1822	86	10	a	a	DET
iajs-1822	86	11	unique	unique	ADJ
iajs-1822	86	12	𝑣	𝑣	ADP
iajs-1822	86	13	∈	∈	PROPN
iajs-1822	86	14	𝐴	𝐴	PROPN
iajs-1822	86	15	such	such	ADJ
iajs-1822	86	16	that	that	SCONJ
iajs-1822	86	17	𝑣	𝑣	DET
iajs-1822	86	18	𝑆	𝑆	PROPN
iajs-1822	86	19	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	86	20	for	for	ADP
iajs-1822	86	21	all	all	PRON
iajs-1822	86	22	𝑛	𝑛	DET
iajs-1822	86	23	1	1	NUM
iajs-1822	86	24	.	.	PUNCT
iajs-1822	87	1	since	since	SCONJ
iajs-1822	87	2	𝐴	𝐴	PROPN
iajs-1822	87	3	is	be	AUX
iajs-1822	87	4	weakly	weakly	ADV
iajs-1822	87	5	compact	compact	ADJ
iajs-1822	87	6	,	,	PUNCT
iajs-1822	87	7	there	there	PRON
iajs-1822	87	8	is	be	VERB
iajs-1822	87	9	a	a	DET
iajs-1822	87	10	subsequence	subsequence	NOUN
iajs-1822	87	11	𝑣	𝑣	ADP
iajs-1822	87	12	of	of	ADP
iajs-1822	87	13	sequence	sequence	NOUN
iajs-1822	87	14	𝑣	𝑣	X
iajs-1822	87	15	which	which	PRON
iajs-1822	87	16	converges	converge	VERB
iajs-1822	87	17	weakly	weakly	ADV
iajs-1822	87	18	to	to	ADP
iajs-1822	87	19	some	some	DET
iajs-1822	87	20	𝑣𝟶	𝑣𝟶	PROPN
iajs-1822	87	21	∈	∈	PROPN
iajs-1822	87	22	𝐴.	𝐴.	PROPN
iajs-1822	87	23	since	since	SCONJ
iajs-1822	87	24	𝑃	𝑃	NOUN
iajs-1822	87	25	is	be	AUX
iajs-1822	87	26	a	a	DET
iajs-1822	87	27	continuous	continuous	ADJ
iajs-1822	87	28	affine	affine	NOUN
iajs-1822	87	29	mapping	mapping	NOUN
iajs-1822	87	30	then	then	ADV
iajs-1822	87	31	𝑃	𝑃	PROPN
iajs-1822	87	32	is	be	AUX
iajs-1822	87	33	weakly	weakly	ADV
iajs-1822	87	34	continuous	continuous	ADJ
iajs-1822	87	35	and	and	CCONJ
iajs-1822	87	36	so	so	ADV
iajs-1822	87	37	,	,	PUNCT
iajs-1822	87	38	since	since	SCONJ
iajs-1822	87	39	s𝑣	s𝑣	ADV
iajs-1822	87	40	and	and	CCONJ
iajs-1822	87	41	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	87	42	𝑣	𝑣	PROPN
iajs-1822	87	43	.	.	PUNCT
iajs-1822	88	1	now	now	ADV
iajs-1822	88	2	,	,	PUNCT
iajs-1822	88	3	𝑃	𝑃	VERB
iajs-1822	88	4	𝑆	𝑆	PROPN
iajs-1822	88	5	𝑣	𝑣	ADP
iajs-1822	88	6	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	88	7	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	88	8	𝑣	𝑣	ADP
iajs-1822	88	9	𝑢	𝑢	PRON
iajs-1822	88	10	𝑣	𝑣	PROPN
iajs-1822	88	11	1	1	NUM
iajs-1822	88	12	𝑢	𝑢	NOUN
iajs-1822	88	13	𝑣	𝑣	X
iajs-1822	88	14	therefore	therefore	ADV
iajs-1822	88	15	𝑃	𝑃	VERB
iajs-1822	88	16	𝑆	𝑆	PROPN
iajs-1822	88	17	𝑣	𝑣	ADP
iajs-1822	88	18	1	1	NUM
iajs-1822	88	19	𝑢	𝑢	NOUN
iajs-1822	88	20	𝑣	𝑣	AUX
iajs-1822	88	21	thus	thus	ADV
iajs-1822	88	22	𝑃	𝑃	VERB
iajs-1822	88	23	𝑆	𝑆	PROPN
iajs-1822	88	24	𝑣	𝑣	ADP
iajs-1822	88	25	1	1	NUM
iajs-1822	88	26	𝛾	𝛾	NOUN
iajs-1822	88	27	𝑢	𝑢	PRON
iajs-1822	88	28	𝑣	𝑣	ADP
iajs-1822	88	29	1	1	NUM
iajs-1822	88	30	𝛾	𝛾	NOUN
iajs-1822	88	31	𝑣	𝑣	ADP
iajs-1822	88	32	𝛾	𝛾	NOUN
iajs-1822	88	33	𝑢	𝑢	NOUN
iajs-1822	88	34	.	.	PUNCT
iajs-1822	89	1	since	since	SCONJ
iajs-1822	89	2	𝐴	𝐴	PROPN
iajs-1822	89	3	is	be	AUX
iajs-1822	89	4	bounded	bound	VERB
iajs-1822	89	5	,	,	PUNCT
iajs-1822	89	6	𝑣	𝑣	PRON
iajs-1822	89	7	∈	∈	PROPN
iajs-1822	89	8	𝐴	𝐴	PROPN
iajs-1822	89	9	implies	imply	VERB
iajs-1822	89	10	𝛾	𝛾	AUX
iajs-1822	89	11	𝑣	𝑣	NOUN
iajs-1822	89	12	is	be	AUX
iajs-1822	89	13	bounded	bound	VERB
iajs-1822	89	14	and	and	CCONJ
iajs-1822	89	15	so	so	ADV
iajs-1822	89	16	by	by	ADP
iajs-1822	89	17	the	the	DET
iajs-1822	89	18	fact	fact	NOUN
iajs-1822	89	19	that	that	SCONJ
iajs-1822	89	20	ℎ	ℎ	PROPN
iajs-1822	89	21	→	→	SYM
iajs-1822	89	22	1	1	NUM
iajs-1822	89	23	,	,	PUNCT
iajs-1822	89	24	we	we	PRON
iajs-1822	89	25	have	have	AUX
iajs-1822	89	26	𝛾	𝛾	PART
iajs-1822	89	27	𝑃	𝑃	VERB
iajs-1822	89	28	𝑆	𝑆	PROPN
iajs-1822	89	29	𝑣	𝑣	NOUN
iajs-1822	89	30	→	→	SYM
iajs-1822	89	31	𝟶	𝟶	NUM
iajs-1822	89	32	now	now	ADV
iajs-1822	89	33	,	,	PUNCT
iajs-1822	89	34	since	since	SCONJ
iajs-1822	89	35	𝑃	𝑃	PROPN
iajs-1822	89	36	𝑆	𝑆	PROPN
iajs-1822	89	37	is	be	AUX
iajs-1822	89	38	demi	demi	NOUN
iajs-1822	89	39	-	-	PUNCT
iajs-1822	89	40	closed	closed	ADJ
iajs-1822	89	41	then	then	ADV
iajs-1822	89	42	𝑃	𝑃	PROPN
iajs-1822	89	43	𝑆	𝑆	PROPN
iajs-1822	89	44	𝑣𝟶	𝑣𝟶	PROPN
iajs-1822	89	45	𝟶	𝟶	NUM
iajs-1822	89	46	and	and	CCONJ
iajs-1822	89	47	thus	thus	ADV
iajs-1822	89	48	𝑃𝑣𝟶	𝑃𝑣𝟶	NOUN
iajs-1822	89	49	𝑣𝟶	𝑣𝟶	PROPN
iajs-1822	89	50	𝑆𝑣𝟶.	𝑆𝑣𝟶.	PROPN
iajs-1822	89	51	hence	hence	ADV
iajs-1822	89	52	,	,	PUNCT
iajs-1822	89	53	𝐹	𝐹	PROPN
iajs-1822	89	54	𝑆	𝑆	PROPN
iajs-1822	89	55	∩	∩	NOUN
iajs-1822	89	56	𝐹	𝐹	PROPN
iajs-1822	89	57	𝑃	𝑃	VERB
iajs-1822	89	58	∅.	∅.	VERB
iajs-1822	89	59	another	another	DET
iajs-1822	89	60	common	common	ADJ
iajs-1822	89	61	fixed	fix	VERB
iajs-1822	89	62	point	point	NOUN
iajs-1822	89	63	theorem	theorem	NOUN
iajs-1822	89	64	will	will	AUX
iajs-1822	89	65	be	be	AUX
iajs-1822	89	66	given	give	VERB
iajs-1822	89	67	for	for	ADP
iajs-1822	89	68	opial	opial	NOUN
iajs-1822	89	69	's	's	PART
iajs-1822	89	70	space	space	NOUN
iajs-1822	89	71	.	.	PUNCT
iajs-1822	90	1	theorem	theorem	NOUN
iajs-1822	90	2	(	(	PUNCT
iajs-1822	90	3	3.4	3.4	NUM
iajs-1822	90	4	):	):	PUNCT
iajs-1822	90	5	let∅	let∅	PROPN
iajs-1822	90	6	𝐴	𝐴	PROPN
iajs-1822	90	7	weakly	weakly	ADJ
iajs-1822	90	8	compact	compact	ADJ
iajs-1822	90	9	subset	subset	NOUN
iajs-1822	90	10	of	of	ADP
iajs-1822	90	11	opia	opia	NOUN
iajs-1822	90	12	's	's	PART
iajs-1822	90	13	complete	complete	ADJ
iajs-1822	90	14	modular	modular	ADJ
iajs-1822	90	15	space	space	NOUN
iajs-1822	90	16	𝑀	𝑀	PROPN
iajs-1822	90	17	.	.	PUNCT
iajs-1822	91	1	let	let	VERB
iajs-1822	91	2	𝑃	𝑃	PRON
iajs-1822	91	3	be	be	AUX
iajs-1822	91	4	a	a	DET
iajs-1822	91	5	continuous	continuous	ADJ
iajs-1822	91	6	and	and	CCONJ
iajs-1822	91	7	affine	affine	NOUN
iajs-1822	91	8	mapping	mapping	NOUN
iajs-1822	91	9	on	on	ADP
iajs-1822	91	10	𝑀	𝑀	PROPN
iajs-1822	91	11	with	with	ADP
iajs-1822	91	12	𝑃	𝑃	PROPN
iajs-1822	91	13	𝐴	𝐴	PROPN
iajs-1822	91	14	𝐴	𝐴	PROPN
iajs-1822	91	15	,	,	PUNCT
iajs-1822	91	16	𝑆	𝑆	PROPN
iajs-1822	91	17	:	:	PUNCT
iajs-1822	91	18	𝐴	𝐴	PROPN
iajs-1822	91	19	→	→	SYM
iajs-1822	91	20	𝐴	𝐴	PROPN
iajs-1822	91	21	be	be	AUX
iajs-1822	91	22	𝑃nonihsciconf	𝑃nonihsciconf	PROPN
iajs-1822	91	23	2017	2017	NUM
iajs-1822	91	24	special	special	ADJ
iajs-1822	91	25	issue	issue	NOUN
iajs-1822	91	26	ibn	ibn	PROPN
iajs-1822	91	27	al	al	PROPN
iajs-1822	91	28	-	-	PUNCT
iajs-1822	91	29	haitham	haitham	PROPN
iajs-1822	91	30	journal	journal	PROPN
iajs-1822	91	31	for	for	ADP
iajs-1822	91	32	pure	pure	ADJ
iajs-1822	91	33	and	and	CCONJ
iajs-1822	91	34	applied	apply	VERB
iajs-1822	91	35	science	science	NOUN
iajs-1822	91	36	https://doi.org/	https://doi.org/	NOUN
iajs-1822	91	37	10.30526/2017.ihsciconf.1822	10.30526/2017.ihsciconf.1822	NOUN
iajs-1822	91	38	for	for	ADP
iajs-1822	91	39	more	more	ADJ
iajs-1822	91	40	information	information	NOUN
iajs-1822	91	41	about	about	ADP
iajs-1822	91	42	the	the	DET
iajs-1822	91	43	conference	conference	NOUN
iajs-1822	91	44	please	please	INTJ
iajs-1822	91	45	visit	visit	VERB
iajs-1822	91	46	the	the	DET
iajs-1822	91	47	websites	website	NOUN
iajs-1822	91	48	:	:	PUNCT
iajs-1822	91	49	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1822	91	50	  	  	SPACE
iajs-1822	91	51	www.ihsciconf.org	www.ihsciconf.org	ADJ
iajs-1822	91	52	   	   	SPACE
iajs-1822	91	53	mathematics|507	mathematics|507	VERB
iajs-1822	91	54	    	    	SPACE
iajs-1822	91	55	expansive	expansive	ADJ
iajs-1822	91	56	mapping	mapping	NOUN
iajs-1822	91	57	commutes	commute	NOUN
iajs-1822	91	58	with	with	ADP
iajs-1822	91	59	𝑃.	𝑃.	PROPN
iajs-1822	91	60	if	if	SCONJ
iajs-1822	91	61	𝐴	𝐴	PROPN
iajs-1822	91	62	has	have	AUX
iajs-1822	91	63	star	star	NOUN
iajs-1822	91	64	-	-	PUNCT
iajs-1822	91	65	shaped	shape	VERB
iajs-1822	91	66	with	with	ADP
iajs-1822	91	67	respect	respect	NOUN
iajs-1822	91	68	to	to	ADP
iajs-1822	91	69	𝑆	𝑆	PROPN
iajs-1822	91	70	,	,	PUNCT
iajs-1822	91	71	then	then	ADV
iajs-1822	91	72	𝐹	𝐹	PROPN
iajs-1822	91	73	𝑆	𝑆	PROPN
iajs-1822	91	74	∩	∩	NOUN
iajs-1822	91	75	𝐹	𝐹	PRON
iajs-1822	91	76	𝑃	𝑃	VERB
iajs-1822	91	77	∅.	∅.	NOUN
iajs-1822	91	78	proof	proof	NOUN
iajs-1822	91	79	:	:	PUNCT
iajs-1822	91	80	since	since	SCONJ
iajs-1822	91	81	𝐴	𝐴	PROPN
iajs-1822	91	82	has	have	AUX
iajs-1822	91	83	star	star	NOUN
iajs-1822	91	84	-	-	PUNCT
iajs-1822	91	85	shaped	shape	VERB
iajs-1822	91	86	then	then	ADV
iajs-1822	91	87	𝑆:𝐴→	𝑆:𝐴→	PROPN
iajs-1822	91	88	𝐴	𝐴	PROPN
iajs-1822	91	89	and	and	CCONJ
iajs-1822	91	90	there	there	PRON
iajs-1822	91	91	is	be	VERB
iajs-1822	91	92	𝑢	𝑢	PRON
iajs-1822	91	93	∈	∈	PROPN
iajs-1822	91	94	𝐴	𝐴	PROPN
iajs-1822	91	95	and	and	CCONJ
iajs-1822	91	96	the	the	DET
iajs-1822	91	97	sequence	sequence	NOUN
iajs-1822	91	98	ℎ	ℎ	X
iajs-1822	91	99	→	→	SYM
iajs-1822	91	100	1	1	NUM
iajs-1822	91	101	,	,	PUNCT
iajs-1822	91	102	as	as	ADP
iajs-1822	91	103	𝑛→	𝑛→	PROPN
iajs-1822	91	104	∞	∞	PROPN
iajs-1822	91	105	,	,	PUNCT
iajs-1822	91	106	𝟶	𝟶	NUM
iajs-1822	91	107	ℎ	ℎ	PART
iajs-1822	91	108	1	1	NUM
iajs-1822	91	109	∋	∋	NOUN
iajs-1822	91	110	1	1	NUM
iajs-1822	91	111	ℎ	ℎ	PROPN
iajs-1822	91	112	𝑢	𝑢	NOUN
iajs-1822	91	113	ℎ	ℎ	PROPN
iajs-1822	91	114	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	91	115	∈	∈	PROPN
iajs-1822	91	116	𝐴	𝐴	PROPN
iajs-1822	91	117	for	for	ADP
iajs-1822	91	118	all	all	DET
iajs-1822	91	119	𝑣∈	𝑣∈	NUM
iajs-1822	91	120	𝐴.	𝐴.	ADV
iajs-1822	92	1	now	now	ADV
iajs-1822	92	2	,	,	PUNCT
iajs-1822	92	3	define	define	VERB
iajs-1822	92	4	𝑆	𝑆	PROPN
iajs-1822	92	5	on	on	ADP
iajs-1822	92	6	𝐴	𝐴	PROPN
iajs-1822	92	7	for	for	ADP
iajs-1822	92	8	any	any	DET
iajs-1822	92	9	𝑣	𝑣	NOUN
iajs-1822	92	10	in	in	ADP
iajs-1822	92	11	𝐴	𝐴	PROPN
iajs-1822	92	12	by	by	ADP
iajs-1822	92	13	,	,	PUNCT
iajs-1822	92	14	𝑆	𝑆	PROPN
iajs-1822	92	15	𝑣	𝑣	ADP
iajs-1822	92	16	ℎ	ℎ	PART
iajs-1822	92	17	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	92	18	1	1	NUM
iajs-1822	92	19	ℎ	ℎ	PART
iajs-1822	92	20	𝑢	𝑢	NOUN
iajs-1822	92	21	and	and	CCONJ
iajs-1822	92	22	there	there	PRON
iajs-1822	92	23	is	be	VERB
iajs-1822	92	24	𝑢∈	𝑢∈	PROPN
iajs-1822	92	25	𝐴	𝐴	PROPN
iajs-1822	92	26	,	,	PUNCT
iajs-1822	92	27	it	it	PRON
iajs-1822	92	28	is	be	AUX
iajs-1822	92	29	clear	clear	ADJ
iajs-1822	92	30	that	that	SCONJ
iajs-1822	92	31	𝑆	𝑆	PROPN
iajs-1822	92	32	:	:	PUNCT
iajs-1822	92	33	𝐴	𝐴	PROPN
iajs-1822	92	34	→	→	SYM
iajs-1822	92	35	𝐴.	𝐴.	PROPN
iajs-1822	92	36	note	note	NOUN
iajs-1822	92	37	that	that	SCONJ
iajs-1822	92	38	𝑆	𝑆	PROPN
iajs-1822	92	39	𝐴	𝐴	PROPN
iajs-1822	92	40	⊆	⊆	NUM
iajs-1822	92	41	𝐴	𝐴	PROPN
iajs-1822	92	42	and	and	CCONJ
iajs-1822	92	43	𝑆	𝑆	PROPN
iajs-1822	92	44	𝐴	𝐴	PROPN
iajs-1822	92	45	⊆	⊆	PROPN
iajs-1822	92	46	𝑝	𝑝	PROPN
iajs-1822	92	47	𝐴	𝐴	PROPN
iajs-1822	92	48	.	.	PUNCT
iajs-1822	93	1	since	since	SCONJ
iajs-1822	93	2	𝑆	𝑆	PROPN
iajs-1822	93	3	commutes	commute	NOUN
iajs-1822	93	4	with	with	ADP
iajs-1822	93	5	𝑝	𝑝	NOUN
iajs-1822	93	6	and	and	CCONJ
iajs-1822	93	7	𝑝	𝑝	PROPN
iajs-1822	93	8	is	be	AUX
iajs-1822	93	9	affine	affine	NOUN
iajs-1822	93	10	mapping	mapping	NOUN
iajs-1822	93	11	,	,	PUNCT
iajs-1822	93	12	for	for	ADP
iajs-1822	93	13	each	each	DET
iajs-1822	93	14	𝑣	𝑣	PRON
iajs-1822	93	15	∈	∈	PROPN
iajs-1822	93	16	𝐴.	𝐴.	PROPN
iajs-1822	94	1	𝑆	𝑆	PROPN
iajs-1822	95	1	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	95	2	ℎ	ℎ	NOUN
iajs-1822	95	3	𝑆𝑃𝑣	𝑆𝑃𝑣	NOUN
iajs-1822	95	4	1	1	NUM
iajs-1822	95	5	ℎ	ℎ	NOUN
iajs-1822	95	6	𝑃𝑢	𝑃𝑢	VERB
iajs-1822	95	7	ℎ	ℎ	NOUN
iajs-1822	95	8	𝑃𝑆𝑣	𝑃𝑆𝑣	ADJ
iajs-1822	95	9	1	1	NUM
iajs-1822	95	10	ℎ	ℎ	X
iajs-1822	96	1	𝑃𝑢	𝑃𝑢	VERB
iajs-1822	96	2	𝑃	𝑃	NOUN
iajs-1822	96	3	ℎ	ℎ	NOUN
iajs-1822	96	4	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	96	5	1	1	NUM
iajs-1822	96	6	ℎ	ℎ	PROPN
iajs-1822	96	7	𝑢	𝑢	PROPN
iajs-1822	96	8	𝑃𝑆	𝑃𝑆	NOUN
iajs-1822	96	9	𝑣	𝑣	ADP
iajs-1822	96	10	thus	thus	ADV
iajs-1822	96	11	each	each	DET
iajs-1822	96	12	ℎ	ℎ	NOUN
iajs-1822	96	13	commutes	commute	VERB
iajs-1822	96	14	with	with	ADP
iajs-1822	96	15	𝑃.	𝑃.	PROPN
iajs-1822	96	16	further	far	ADV
iajs-1822	96	17	observe	observe	VERB
iajs-1822	96	18	that	that	SCONJ
iajs-1822	96	19	for	for	ADP
iajs-1822	96	20	each	each	DET
iajs-1822	96	21	𝑛	𝑛	DET
iajs-1822	96	22	1	1	NUM
iajs-1822	96	23	,	,	PUNCT
iajs-1822	96	24	𝑆	𝑆	PROPN
iajs-1822	96	25	is	be	AUX
iajs-1822	96	26	𝑃	𝑃	NOUN
iajs-1822	96	27	–	–	PUNCT
iajs-1822	96	28	non	non	ADJ
iajs-1822	96	29	-	-	ADJ
iajs-1822	96	30	expansive	expansive	ADJ
iajs-1822	96	31	mapping	mapping	NOUN
iajs-1822	96	32	.	.	PUNCT
iajs-1822	97	1	𝛾	𝛾	X
iajs-1822	97	2	𝑆	𝑆	PROPN
iajs-1822	97	3	𝑣	𝑣	ADP
iajs-1822	97	4	𝑆	𝑆	PROPN
iajs-1822	97	5	𝑢	𝑢	NOUN
iajs-1822	97	6	𝛾	𝛾	NOUN
iajs-1822	97	7	ℎ	ℎ	PART
iajs-1822	97	8	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	97	9	1	1	NUM
iajs-1822	97	10	ℎ	ℎ	VERB
iajs-1822	97	11	𝑢	𝑢	NOUN
iajs-1822	97	12	ℎ	ℎ	PART
iajs-1822	97	13	𝑆𝑢	𝑆𝑢	PROPN
iajs-1822	97	14	1	1	NUM
iajs-1822	97	15	ℎ	ℎ	PART
iajs-1822	97	16	𝑢	𝑢	PROPN
iajs-1822	97	17	ℎ	ℎ	ADP
iajs-1822	97	18	𝛾	𝛾	PROPN
iajs-1822	97	19	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	97	20	𝑆𝑢	𝑆𝑢	PROPN
iajs-1822	97	21	ℎ	ℎ	NOUN
iajs-1822	97	22	𝛾	𝛾	ADP
iajs-1822	97	23	𝑃𝑣	𝑃𝑣	NOUN
iajs-1822	97	24	𝑃𝑢	𝑃𝑢	NOUN
iajs-1822	97	25	∀	∀	PUNCT
iajs-1822	97	26	𝑢∈	𝑢∈	NOUN
iajs-1822	97	27	𝐴	𝐴	PROPN
iajs-1822	97	28	,	,	PUNCT
iajs-1822	97	29	hence	hence	ADV
iajs-1822	97	30	𝑆	𝑆	PROPN
iajs-1822	97	31	is	be	AUX
iajs-1822	97	32	𝑃contraction	𝑃contraction	PROPN
iajs-1822	97	33	.	.	PUNCT
iajs-1822	98	1	thus	thus	ADV
iajs-1822	98	2	by	by	ADP
iajs-1822	98	3	proposition	proposition	NOUN
iajs-1822	98	4	(	(	PUNCT
iajs-1822	98	5	3.1	3.1	NUM
iajs-1822	98	6	)	)	PUNCT
iajs-1822	98	7	,	,	PUNCT
iajs-1822	98	8	there	there	PRON
iajs-1822	98	9	is	be	VERB
iajs-1822	98	10	a	a	DET
iajs-1822	98	11	unique	unique	ADJ
iajs-1822	98	12	𝑣	𝑣	ADP
iajs-1822	98	13	∈	∈	PROPN
iajs-1822	98	14	𝐴	𝐴	PROPN
iajs-1822	98	15	such	such	ADJ
iajs-1822	98	16	that	that	SCONJ
iajs-1822	98	17	𝑣	𝑣	PRON
iajs-1822	98	18	𝑆	𝑆	PROPN
iajs-1822	98	19	𝑣	𝑣	PART
iajs-1822	98	20	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	98	21	for	for	ADP
iajs-1822	98	22	all	all	PRON
iajs-1822	98	23	𝑛	𝑛	DET
iajs-1822	98	24	1	1	NUM
iajs-1822	98	25	.	.	PUNCT
iajs-1822	99	1	since	since	SCONJ
iajs-1822	99	2	𝐴	𝐴	PROPN
iajs-1822	99	3	is	be	AUX
iajs-1822	99	4	weakly	weakly	ADV
iajs-1822	99	5	compact	compact	ADJ
iajs-1822	99	6	,	,	PUNCT
iajs-1822	99	7	there	there	PRON
iajs-1822	99	8	is	be	VERB
iajs-1822	99	9	a	a	DET
iajs-1822	99	10	subsequence	subsequence	NOUN
iajs-1822	99	11	𝑣	𝑣	ADP
iajs-1822	99	12	of	of	ADP
iajs-1822	99	13	sequence	sequence	NOUN
iajs-1822	99	14	𝑣	𝑣	X
iajs-1822	99	15	which	which	PRON
iajs-1822	99	16	converges	converge	VERB
iajs-1822	99	17	weakly	weakly	ADV
iajs-1822	99	18	to	to	ADP
iajs-1822	99	19	some	some	DET
iajs-1822	99	20	𝑣𝟶	𝑣𝟶	PROPN
iajs-1822	99	21	∈	∈	PROPN
iajs-1822	99	22	𝐴.	𝐴.	PROPN
iajs-1822	99	23	since	since	SCONJ
iajs-1822	99	24	𝑃	𝑃	NOUN
iajs-1822	99	25	is	be	AUX
iajs-1822	99	26	a	a	DET
iajs-1822	99	27	continuous	continuous	ADJ
iajs-1822	99	28	affine	affine	NOUN
iajs-1822	99	29	mapping	mapping	NOUN
iajs-1822	99	30	then	then	ADV
iajs-1822	99	31	𝑃	𝑃	PROPN
iajs-1822	99	32	is	be	AUX
iajs-1822	99	33	weakly	weakly	ADV
iajs-1822	99	34	continuous	continuous	ADJ
iajs-1822	99	35	and	and	CCONJ
iajs-1822	99	36	so	so	ADV
iajs-1822	99	37	we	we	PRON
iajs-1822	99	38	have	have	VERB
iajs-1822	99	39	:	:	PUNCT
iajs-1822	99	40	𝑃𝑣𝟶	𝑃𝑣𝟶	NOUN
iajs-1822	99	41	lim	lim	PROPN
iajs-1822	99	42	→	→	PUNCT
iajs-1822	99	43	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	99	44	lim	lim	PROPN
iajs-1822	99	45	→	→	SYM
iajs-1822	99	46	𝑣	𝑣	ADP
iajs-1822	99	47	𝑣𝟶	𝑣𝟶	PROPN
iajs-1822	99	48	since	since	SCONJ
iajs-1822	99	49	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	99	50	and	and	CCONJ
iajs-1822	99	51	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	99	52	𝑣	𝑣	PROPN
iajs-1822	99	53	,	,	PUNCT
iajs-1822	99	54	we	we	PRON
iajs-1822	99	55	have	have	AUX
iajs-1822	99	56	:	:	PUNCT
iajs-1822	99	57	𝑃	𝑃	VERB
iajs-1822	99	58	𝑆	𝑆	PROPN
iajs-1822	99	59	𝑣	𝑣	ADP
iajs-1822	100	1	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	100	2	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	101	1	𝑣	𝑣	ADP
iajs-1822	101	2	𝑢	𝑢	NOUN
iajs-1822	101	3	𝑣	𝑣	PROPN
iajs-1822	101	4	ihsciconf	ihsciconf	ADJ
iajs-1822	101	5	2017	2017	NUM
iajs-1822	101	6	special	special	ADJ
iajs-1822	101	7	issue	issue	NOUN
iajs-1822	101	8	ibn	ibn	PROPN
iajs-1822	101	9	al	al	PROPN
iajs-1822	101	10	-	-	PUNCT
iajs-1822	101	11	haitham	haitham	PROPN
iajs-1822	101	12	journal	journal	PROPN
iajs-1822	101	13	for	for	ADP
iajs-1822	101	14	pure	pure	ADJ
iajs-1822	101	15	and	and	CCONJ
iajs-1822	101	16	applied	apply	VERB
iajs-1822	101	17	science	science	NOUN
iajs-1822	101	18	https://doi.org/	https://doi.org/	NOUN
iajs-1822	101	19	10.30526/2017.ihsciconf.1822	10.30526/2017.ihsciconf.1822	NOUN
iajs-1822	101	20	for	for	ADP
iajs-1822	101	21	more	more	ADJ
iajs-1822	101	22	information	information	NOUN
iajs-1822	101	23	about	about	ADP
iajs-1822	101	24	the	the	DET
iajs-1822	101	25	conference	conference	NOUN
iajs-1822	101	26	please	please	INTJ
iajs-1822	101	27	visit	visit	VERB
iajs-1822	101	28	the	the	DET
iajs-1822	101	29	websites	website	NOUN
iajs-1822	101	30	:	:	PUNCT
iajs-1822	101	31	http://www.ihsciconf.org/conf/	http://www.ihsciconf.org/conf/	VERB
iajs-1822	101	32	  	  	SPACE
iajs-1822	101	33	www.ihsciconf.org	www.ihsciconf.org	ADV
iajs-1822	101	34	   	   	SPACE
iajs-1822	101	35	mathematics|508	mathematics|508	NOUN
iajs-1822	101	36	    	    	SPACE
iajs-1822	101	37	𝑃	𝑃	PROPN
iajs-1822	101	38	𝑆	𝑆	PROPN
iajs-1822	101	39	𝑣	𝑣	ADP
iajs-1822	101	40	1	1	NUM
iajs-1822	101	41	𝑢	𝑢	NOUN
iajs-1822	101	42	𝑣	𝑣	AUX
iajs-1822	101	43	therefore	therefore	ADV
iajs-1822	101	44	𝑃	𝑃	VERB
iajs-1822	101	45	𝑆	𝑆	PROPN
iajs-1822	101	46	𝑣	𝑣	ADP
iajs-1822	101	47	1	1	NUM
iajs-1822	101	48	𝑢	𝑢	PROPN
iajs-1822	101	49	𝑣	𝑣	NOUN
iajs-1822	101	50	.	.	PUNCT
iajs-1822	102	1	thus	thus	ADV
iajs-1822	102	2	𝛾	𝛾	AUX
iajs-1822	102	3	𝑃	𝑃	VERB
iajs-1822	102	4	𝑆	𝑆	PROPN
iajs-1822	102	5	𝑣	𝑣	ADP
iajs-1822	102	6	1	1	NUM
iajs-1822	102	7	𝛾	𝛾	NOUN
iajs-1822	102	8	𝑢	𝑢	PRON
iajs-1822	102	9	𝑣	𝑣	ADP
iajs-1822	102	10	1	1	NUM
iajs-1822	102	11	𝛾	𝛾	NOUN
iajs-1822	102	12	𝑣	𝑣	ADP
iajs-1822	102	13	𝛾	𝛾	NOUN
iajs-1822	102	14	𝑢	𝑢	NOUN
iajs-1822	102	15	.	.	PUNCT
iajs-1822	103	1	since	since	SCONJ
iajs-1822	103	2	𝐴	𝐴	PROPN
iajs-1822	103	3	is	be	AUX
iajs-1822	103	4	bounded	bound	VERB
iajs-1822	103	5	by	by	ADP
iajs-1822	103	6	𝐴	𝐴	PROPN
iajs-1822	103	7	is	be	AUX
iajs-1822	103	8	weakly	weakly	ADV
iajs-1822	103	9	compact	compact	ADJ
iajs-1822	103	10	,	,	PUNCT
iajs-1822	103	11	𝑣	𝑣	DET
iajs-1822	103	12	∈	∈	PROPN
iajs-1822	103	13	𝐴	𝐴	PROPN
iajs-1822	103	14	implies	imply	VERB
iajs-1822	103	15	𝛾	𝛾	AUX
iajs-1822	103	16	𝑣	𝑣	NOUN
iajs-1822	103	17	is	be	AUX
iajs-1822	103	18	bounded	bound	VERB
iajs-1822	103	19	and	and	CCONJ
iajs-1822	103	20	so	so	ADV
iajs-1822	103	21	by	by	ADP
iajs-1822	103	22	the	the	DET
iajs-1822	103	23	fact	fact	NOUN
iajs-1822	103	24	that	that	SCONJ
iajs-1822	103	25	ℎ	ℎ	PROPN
iajs-1822	103	26	→	→	SYM
iajs-1822	103	27	1	1	NUM
iajs-1822	103	28	,	,	PUNCT
iajs-1822	103	29	we	we	PRON
iajs-1822	103	30	have	have	AUX
iajs-1822	103	31	𝛾	𝛾	PART
iajs-1822	103	32	𝑃	𝑃	VERB
iajs-1822	103	33	𝑆	𝑆	PROPN
iajs-1822	103	34	𝑣	𝑣	NOUN
iajs-1822	103	35	→	→	SYM
iajs-1822	103	36	0	0	NUM
iajs-1822	103	37	now	now	ADV
iajs-1822	103	38	,	,	PUNCT
iajs-1822	103	39	since	since	SCONJ
iajs-1822	103	40	𝑀	𝑀	PROPN
iajs-1822	103	41	is	be	AUX
iajs-1822	103	42	opial	opial	ADJ
iajs-1822	103	43	space	space	NOUN
iajs-1822	103	44	and	and	CCONJ
iajs-1822	103	45	suppose	suppose	VERB
iajs-1822	103	46	that	that	SCONJ
iajs-1822	103	47	,	,	PUNCT
iajs-1822	103	48	𝑆𝑣𝟶	𝑆𝑣𝟶	NOUN
iajs-1822	103	49	𝑣𝟶	𝑣𝟶	PROPN
iajs-1822	103	50	we	we	PRON
iajs-1822	103	51	have	have	VERB
iajs-1822	103	52	:	:	PUNCT
iajs-1822	103	53	lim	lim	PROPN
iajs-1822	103	54	→	→	PROPN
iajs-1822	103	55	𝑖𝑛𝑓𝛾	𝑖𝑛𝑓𝛾	PROPN
iajs-1822	103	56	𝑣	𝑣	ADP
iajs-1822	103	57	𝑣𝟶	𝑣𝟶	PROPN
iajs-1822	103	58	lim	lim	PROPN
iajs-1822	103	59	→	→	SYM
iajs-1822	103	60	inf	inf	PROPN
iajs-1822	103	61	𝛾	𝛾	ADP
iajs-1822	103	62	𝑣	𝑣	X
iajs-1822	103	63	𝑆𝑣𝟶	𝑆𝑣𝟶	ADJ
iajs-1822	103	64	lim	lim	PROPN
iajs-1822	103	65	→	→	SYM
iajs-1822	103	66	inf	inf	PROPN
iajs-1822	103	67	𝛾	𝛾	ADP
iajs-1822	103	68	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	103	69	𝑃	𝑃	PROPN
iajs-1822	103	70	𝑆	𝑆	PROPN
iajs-1822	103	71	𝑣	𝑣	PROPN
iajs-1822	103	72	𝑆𝑣𝟶	𝑆𝑣𝟶	NOUN
iajs-1822	103	73	lim	lim	PROPN
iajs-1822	103	74	→	→	SYM
iajs-1822	103	75	inf	inf	PROPN
iajs-1822	103	76	𝛾	𝛾	PROPN
iajs-1822	103	77	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	103	78	𝑆𝑣𝟶	𝑆𝑣𝟶	NOUN
iajs-1822	103	79	lim	lim	PROPN
iajs-1822	103	80	→	→	PROPN
iajs-1822	103	81	𝑖𝑛𝑓𝛾	𝑖𝑛𝑓𝛾	PROPN
iajs-1822	103	82	𝑃	𝑃	PROPN
iajs-1822	103	83	𝑆	𝑆	PROPN
iajs-1822	103	84	𝑣	𝑣	NOUN
iajs-1822	103	85	,	,	PUNCT
iajs-1822	103	86	since	since	SCONJ
iajs-1822	103	87	𝑣	𝑣	PRON
iajs-1822	103	88	𝑃	𝑃	VERB
iajs-1822	103	89	𝑆	𝑆	PROPN
iajs-1822	103	90	𝑣	𝑣	PROPN
iajs-1822	103	91	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	103	92	.	.	PUNCT
iajs-1822	104	1	and	and	CCONJ
iajs-1822	104	2	thus	thus	ADV
iajs-1822	104	3	lim	lim	PROPN
iajs-1822	104	4	→	→	PROPN
iajs-1822	104	5	𝑖𝑛𝑓𝛾	𝑖𝑛𝑓𝛾	PROPN
iajs-1822	104	6	𝑣	𝑣	ADP
iajs-1822	104	7	𝑣𝟶	𝑣𝟶	PROPN
iajs-1822	104	8	lim	lim	PROPN
iajs-1822	104	9	→	→	SYM
iajs-1822	104	10	inf	inf	PROPN
iajs-1822	104	11	𝛾	𝛾	PROPN
iajs-1822	104	12	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	104	13	𝑆𝑣𝟶	𝑆𝑣𝟶	NOUN
iajs-1822	104	14	but	but	CCONJ
iajs-1822	104	15	on	on	ADP
iajs-1822	104	16	the	the	DET
iajs-1822	104	17	other	other	ADJ
iajs-1822	104	18	hand	hand	NOUN
iajs-1822	104	19	,	,	PUNCT
iajs-1822	104	20	we	we	PRON
iajs-1822	104	21	have	have	VERB
iajs-1822	104	22	lim	lim	PROPN
iajs-1822	104	23	→	→	SYM
iajs-1822	104	24	inf	inf	PROPN
iajs-1822	105	1	𝛾	𝛾	PROPN
iajs-1822	105	2	𝑆𝑣	𝑆𝑣	PROPN
iajs-1822	105	3	𝑆𝑣𝟶	𝑆𝑣𝟶	NOUN
iajs-1822	105	4	lim	lim	PROPN
iajs-1822	105	5	→	→	SYM
iajs-1822	105	6	inf	inf	PROPN
iajs-1822	105	7	𝛾	𝛾	PROPN
iajs-1822	105	8	𝑃𝑣	𝑃𝑣	PROPN
iajs-1822	105	9	𝑃𝑣𝟶	𝑃𝑣𝟶	NOUN
iajs-1822	105	10	lim	lim	PROPN
iajs-1822	105	11	→	→	SYM
iajs-1822	105	12	inf	inf	PROPN
iajs-1822	105	13	𝛾	𝛾	ADP
iajs-1822	105	14	𝑣	𝑣	ADP
iajs-1822	105	15	𝑣𝟶	𝑣𝟶	NOUN
iajs-1822	105	16	this	this	PRON
iajs-1822	105	17	is	be	AUX
iajs-1822	105	18	a	a	DET
iajs-1822	105	19	contradiction	contradiction	NOUN
iajs-1822	105	20	.	.	PUNCT
iajs-1822	106	1	hence	hence	ADV
iajs-1822	106	2	𝑣𝟶	𝑣𝟶	PROPN
iajs-1822	106	3	∈	∈	PROPN
iajs-1822	106	4	𝐹	𝐹	PROPN
iajs-1822	106	5	𝑆	𝑆	PROPN
iajs-1822	106	6	∩	∩	NOUN
iajs-1822	106	7	𝐹	𝐹	PRON
iajs-1822	106	8	𝑃	𝑃	NOUN
iajs-1822	106	9	⇒	⇒	NOUN
iajs-1822	106	10	𝐹	𝐹	PROPN
iajs-1822	106	11	𝑆	𝑆	PROPN
iajs-1822	106	12	∩	∩	NOUN
iajs-1822	106	13	𝐹	𝐹	PRON
iajs-1822	106	14	𝑃	𝑃	VERB
iajs-1822	106	15	∅.	∅.	NOUN
iajs-1822	106	16	acknowledgements	acknowledgement	NOUN
iajs-1822	106	17	:	:	PUNCT
iajs-1822	106	18	we	we	PRON
iajs-1822	106	19	would	would	AUX
iajs-1822	106	20	like	like	VERB
iajs-1822	106	21	to	to	PART
iajs-1822	106	22	acknowledge	acknowledge	VERB
iajs-1822	106	23	the	the	DET
iajs-1822	106	24	generous	generous	ADJ
iajs-1822	106	25	help	help	NOUN
iajs-1822	106	26	of	of	ADP
iajs-1822	106	27	editors	editor	NOUN
iajs-1822	106	28	and	and	CCONJ
iajs-1822	106	29	we	we	PRON
iajs-1822	106	30	are	be	AUX
iajs-1822	106	31	grateful	grateful	ADJ
iajs-1822	106	32	to	to	ADP
iajs-1822	106	33	the	the	DET
iajs-1822	106	34	referees	referee	NOUN
iajs-1822	106	35	for	for	ADP
iajs-1822	106	36	their	their	PRON
iajs-1822	106	37	constructive	constructive	ADJ
iajs-1822	106	38	input	input	NOUN
iajs-1822	106	39	.	.	PUNCT
iajs-1822	107	1	references	reference	NOUN
iajs-1822	107	2	[	[	X
iajs-1822	107	3	1	1	NUM
iajs-1822	107	4	]	]	SYM
iajs-1822	107	5	jr.dotson	jr.dotson	NOUN
iajs-1822	107	6	,	,	PUNCT
iajs-1822	107	7	,	,	PUNCT
iajs-1822	107	8	w.	w.	PROPN
iajs-1822	107	9	g.	g.	PROPN
iajs-1822	107	10	,	,	PUNCT
iajs-1822	107	11	"	"	PUNCT
iajs-1822	107	12	fixed	fix	VERB
iajs-1822	107	13	point	point	NOUN
iajs-1822	107	14	theorems	theorem	NOUN
iajs-1822	107	15	for	for	ADP
iajs-1822	107	16	non	non	ADJ
iajs-1822	107	17	-	-	ADJ
iajs-1822	107	18	expansive	expansive	ADJ
iajs-1822	107	19	mappings	mapping	NOUN
iajs-1822	107	20	on	on	ADP
iajs-1822	107	21	star	star	NOUN
iajs-1822	107	22	-	-	PUNCT
iajs-1822	107	23	shaped	shape	VERB
iajs-1822	107	24	subsets	subset	NOUN
iajs-1822	107	25	of	of	ADP
iajs-1822	107	26	banach	banach	NOUN
iajs-1822	107	27	spaces	space	NOUN
iajs-1822	107	28	"	"	PUNCT
iajs-1822	107	29	,	,	PUNCT
iajs-1822	107	30	j.	j.	PROPN
iajs-1822	107	31	london	london	PROPN
iajs-1822	107	32	math	math	PROPN
iajs-1822	107	33	.	.	PUNCT
iajs-1822	108	1	soc	soc	PROPN
iajs-1822	108	2	.	.	PROPN
iajs-1822	108	3	,	,	PUNCT
iajs-1822	108	4	4(2	4(2	NUM
iajs-1822	108	5	)	)	PUNCT
iajs-1822	108	6	,	,	PUNCT
iajs-1822	108	7	.	.	PUNCT
iajs-1822	109	1	408	408	NUM
iajs-1822	109	2	-	-	SYM
iajs-1822	109	3	410	410	NUM
iajs-1822	109	4	.	.	NUM
iajs-1822	109	5	1972	1972	NUM
iajs-1822	109	6	full/3.408.4	full/3.408.4	NOUN
iajs-1822	109	7	-	-	PUNCT
iajs-1822	109	8	http://onlinelibrary.wiley.com	http://onlinelibrary.wiley.com	X
iajs-1822	109	9	/	/	SYM
iajs-1822	109	10	doi/10.1112	doi/10.1112	PROPN
iajs-1822	109	11	/	/	SYM
iajs-1822	109	12	jlms	jlm	NOUN
iajs-1822	109	13	/	/	SYM
iajs-1822	109	14	s2	s2	PROPN
iajs-1822	109	15	[	[	X
iajs-1822	109	16	2	2	NUM
iajs-1822	109	17	]	]	PUNCT
iajs-1822	109	18	p.	p.	NOUN
iajs-1822	109	19	v.subrahmanyam	v.subrahmanyam	NOUN
iajs-1822	109	20	,	,	PUNCT
iajs-1822	109	21	,	,	PUNCT
iajs-1822	109	22	"	"	PUNCT
iajs-1822	109	23	remarks	remark	NOUN
iajs-1822	109	24	on	on	ADP
iajs-1822	109	25	some	some	DET
iajs-1822	109	26	fixed	fix	VERB
iajs-1822	109	27	point	point	NOUN
iajs-1822	109	28	theorems	theorem	NOUN
iajs-1822	109	29	related	relate	VERB
iajs-1822	109	30	to	to	ADP
iajs-1822	109	31	banach	banach	NOUN
iajs-1822	109	32	contraction	contraction	NOUN
iajs-1822	109	33	principle	principle	NOUN
iajs-1822	109	34	"	"	PUNCT
iajs-1822	109	35	,	,	PUNCT
iajs-1822	109	36	j.	j.	PROPN
iajs-1822	109	37	math	math	PROPN
iajs-1822	109	38	.	.	PUNCT
iajs-1822	110	1	phys	phy	NOUN
iajs-1822	110	2	.	.	PUNCT
iajs-1822	111	1	sci.8	sci.8	NOUN
iajs-1822	111	2	(	(	PUNCT
iajs-1822	111	3	1974	1974	NUM
iajs-1822	111	4	)	)	PUNCT
iajs-1822	111	5	,	,	PUNCT
iajs-1822	111	6	445457	445457	NUM
iajs-1822	111	7	;	;	PUNCT
iajs-1822	111	8	erratum	erratum	PROPN
iajs-1822	111	9	,	,	PUNCT
iajs-1822	111	10	j.	j.	PROPN
iajs-1822	111	11	math.phys	math.phys	PROPN
iajs-1822	111	12	.	.	PUNCT
iajs-1822	111	13	sci	sci	PROPN
iajs-1822	111	14	.	.	PROPN
iajs-1822	111	15	9	9	NUM
iajs-1822	111	16	,	,	PUNCT
iajs-1822	111	17	195,1975	195,1975	NUM
iajs-1822	112	1	[	[	X
iajs-1822	112	2	3	3	X
iajs-1822	112	3	]	]	X
iajs-1822	112	4	l.	l.	PROPN
iajs-1822	112	5	habiniak	habiniak	PROPN
iajs-1822	112	6	,	,	PUNCT
iajs-1822	112	7	"	"	PUNCT
iajs-1822	112	8	fixed	fixed	ADJ
iajs-1822	112	9	point	point	NOUN
iajs-1822	112	10	theorems	theorem	NOUN
iajs-1822	112	11	and	and	CCONJ
iajs-1822	112	12	invariant	invariant	ADJ
iajs-1822	112	13	approximation	approximation	NOUN
iajs-1822	112	14	"	"	PUNCT
iajs-1822	112	15	,	,	PUNCT
iajs-1822	112	16	j.	j.	PROPN
iajs-1822	112	17	approx	approx	PROPN
iajs-1822	112	18	.	.	PUNCT
iajs-1822	113	1	theory	theory	NOUN
iajs-1822	113	2	,	,	PUNCT
iajs-1822	113	3	56	56	NUM
iajs-1822	113	4	,	,	PUNCT
iajs-1822	113	5	pp	pp	ADJ
iajs-1822	113	6	.	.	PUNCT
iajs-1822	114	1	241	241	NUM
iajs-1822	114	2	-	-	NUM
iajs-1822	114	3	244,1989	244,1989	NUM
iajs-1822	114	4	.	.	PUNCT
iajs-1822	115	1	[	[	X
iajs-1822	115	2	4	4	NUM
iajs-1822	115	3	]	]	X
iajs-1822	115	4	s.s	s.s	PROPN
iajs-1822	115	5	.	.	PROPN
iajs-1822	115	6	abed	abed	PROPN
iajs-1822	115	7	,	,	PUNCT
iajs-1822	115	8	,	,	PUNCT
iajs-1822	115	9	"	"	PUNCT
iajs-1822	115	10	on	on	ADP
iajs-1822	115	11	invariant	invariant	ADJ
iajs-1822	115	12	best	good	ADJ
iajs-1822	115	13	approximation	approximation	NOUN
iajs-1822	115	14	in	in	ADP
iajs-1822	115	15	modular	modular	ADJ
iajs-1822	115	16	spaces	space	NOUN
iajs-1822	115	17	"	"	PUNCT
iajs-1822	115	18	,	,	PUNCT
iajs-1822	115	19	global	global	ADJ
iajs-1822	115	20	journal	journal	NOUN
iajs-1822	115	21	of	of	ADP
iajs-1822	115	22	pure	pure	ADJ
iajs-1822	115	23	and	and	CCONJ
iajs-1822	115	24	applied	applied	ADJ
iajs-1822	115	25	mathematics	mathematic	NOUN
iajs-1822	115	26	,	,	PUNCT
iajs-1822	115	27	13	13	NUM
iajs-1822	115	28	,	,	PUNCT
iajs-1822	115	29	.	.	PUNCT
iajs-1822	116	1	9	9	NUM
iajs-1822	116	2	,	,	PUNCT
iajs-1822	116	3	.	.	PUNCT
iajs-1822	117	1	5227	5227	NUM
iajs-1822	117	2	-	-	SYM
iajs-1822	117	3	5233	5233	NUM
iajs-1822	117	4	,	,	PUNCT
iajs-1822	117	5	2017	2017	NUM
iajs-1822	117	6	.	.	PUNCT
iajs-1822	118	1	http://www.ripublication.com/gjpam17/gjpamv13n9_102.pdf	http://www.ripublication.com/gjpam17/gjpamv13n9_102.pdf	NOUN
iajs-1822	118	2	[	[	X
iajs-1822	118	3	5	5	NUM
iajs-1822	118	4	]	]	PUNCT
iajs-1822	118	5	s.s.abed	s.s.abe	VERB
iajs-1822	118	6	,	,	PUNCT
iajs-1822	118	7	,	,	PUNCT
iajs-1822	118	8	k.a	k.a	PROPN
iajs-1822	118	9	.	.	PROPN
iajs-1822	118	10	abdul	abdul	PROPN
iajs-1822	118	11	sada	sada	PROPN
iajs-1822	118	12	,	,	PUNCT
iajs-1822	118	13	,	,	PUNCT
iajs-1822	118	14	"	"	PUNCT
iajs-1822	118	15	an	an	DET
iajs-1822	118	16	extension	extension	NOUN
iajs-1822	118	17	of	of	ADP
iajs-1822	118	18	brosowskimeinaraus	brosowskimeinaraus	NOUN
iajs-1822	118	19	theorem	theorem	VERB
iajs-1822	118	20	in	in	ADP
iajs-1822	118	21	modular	modular	ADJ
iajs-1822	118	22	spaces	space	NOUN
iajs-1822	118	23	"	"	PUNCT
iajs-1822	118	24	,	,	PUNCT
iajs-1822	118	25	inter	inter	PROPN
iajs-1822	118	26	.	.	PUNCT
iajs-1822	119	1	j.	j.	PROPN
iajs-1822	119	2	of	of	ADP
iajs-1822	119	3	math	math	PROPN
iajs-1822	119	4	.	.	PUNCT
iajs-1822	120	1	anal	anal	PROPN
iajs-1822	120	2	.	.	PUNCT
iajs-1822	120	3	,	,	PUNCT
iajs-1822	120	4	hikari	hikari	PROPN
iajs-1822	120	5	ltd	ltd	PROPN
iajs-1822	120	6	.	.	PROPN
iajs-1822	120	7	,	,	PUNCT
iajs-1822	120	8	11	11	NUM
iajs-1822	120	9	,	,	PUNCT
iajs-1822	120	10	18	18	NUM
iajs-1822	120	11	,	,	PUNCT
iajs-1822	120	12	877	877	NUM
iajs-1822	120	13	–	–	PUNCT
iajs-1822	120	14	882	882	NUM
iajs-1822	120	15	,	,	PUNCT
iajs-1822	120	16	2017	2017	NUM
iajs-1822	121	1	https://doi.org/10.12988/ijma.2017.77101	https://doi.org/10.12988/ijma.2017.77101	PROPN
iajs-1822	121	2	[	[	X
iajs-1822	121	3	6	6	NUM
iajs-1822	121	4	]	]	PUNCT
iajs-1822	121	5	abed	abe	VERB
iajs-1822	121	6	,	,	PUNCT
iajs-1822	121	7	s.s	s.s	PROPN
iajs-1822	121	8	.	.	PROPN
iajs-1822	121	9	abdul	abdul	PROPN
iajs-1822	121	10	sada	sada	PROPN
iajs-1822	121	11	,	,	PUNCT
iajs-1822	121	12	k,.a	k,.a	PROPN
iajs-1822	121	13	.	.	PUNCT
iajs-1822	122	1	"	"	PUNCT
iajs-1822	122	2	approximatively	approximatively	ADV
iajs-1822	122	3	compactness	compactness	NOUN
iajs-1822	122	4	and	and	CCONJ
iajs-1822	122	5	best	good	ADJ
iajs-1822	122	6	approximation	approximation	NOUN
iajs-1822	122	7	in	in	ADP
iajs-1822	122	8	modular	modular	ADJ
iajs-1822	122	9	spaces	space	NOUN
iajs-1822	122	10	"	"	PUNCT
iajs-1822	122	11	accepted	accept	VERB
iajs-1822	122	12	in	in	ADP
iajs-1822	122	13	conf	conf	NOUN
iajs-1822	122	14	.	.	PUNCT
iajs-1822	123	1	of	of	ADP
iajs-1822	123	2	scie	scie	PROPN
iajs-1822	123	3	.	.	PUNCT
iajs-1822	124	1	coll	coll	PROPN
iajs-1822	124	2	.	.	PROPN
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iajs-1822	124	6	.	.	PUNCT
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iajs-1822	125	12	and	and	CCONJ
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iajs-1822	125	14	science	science	NOUN
iajs-1822	125	15	https://doi.org/	https://doi.org/	NOUN
iajs-1822	125	16	10.30526/2017.ihsciconf.1822	10.30526/2017.ihsciconf.1822	NOUN
iajs-1822	125	17	for	for	ADP
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iajs-1822	125	29	  	  	SPACE
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iajs-1822	125	31	   	   	SPACE
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iajs-1822	125	33	    	    	SPACE
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iajs-1822	126	16	-	-	PUNCT
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iajs-1822	126	20	approximation	approximation	NOUN
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iajs-1822	126	25	"	"	PUNCT
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iajs-1822	126	29	in	in	ADP
iajs-1822	126	30	inter	inter	PROPN
iajs-1822	126	31	.	.	PUNCT
iajs-1822	127	1	j.	j.	PROPN
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iajs-1822	129	23	"	"	PUNCT
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iajs-1822	131	2	.	.	PUNCT
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iajs-1822	132	31	.	.	PUNCT
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iajs-1822	135	4	)	)	PUNCT
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iajs-1822	137	22	"	"	PUNCT
iajs-1822	137	23	.	.	PUNCT
iajs-1822	138	1	fixed	fix	VERB
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iajs-1822	138	5	.	.	PROPN
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iajs-1822	139	2	,	,	PUNCT
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iajs-1822	140	5	,	,	PUNCT
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iajs-1822	140	10	,	,	PUNCT
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iajs-1822	140	12	.	.	PUNCT
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iajs-1822	141	2	,	,	PUNCT
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iajs-1822	141	10	in	in	ADP
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iajs-1822	141	14	"	"	PUNCT
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iajs-1822	141	17	point	point	NOUN
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iajs-1822	141	21	,	,	PUNCT
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iajs-1822	141	23	2011	2011	NUM
iajs-1822	141	24	,	,	PUNCT
iajs-1822	141	25	2011:93	2011:93	NUM
iajs-1822	141	26	.	.	PUNCT
iajs-1822	141	27	/http://www.fixedpointtheoryandapplications.com	/http://www.fixedpointtheoryandapplications.com	PUNCT
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iajs-1822	142	18	for	for	ADP
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iajs-1822	142	20	in	in	ADP
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iajs-1822	142	22	metric	metric	ADJ
iajs-1822	142	23	spaces	space	NOUN
iajs-1822	142	24	fixed	fix	VERB
iajs-1822	142	25	point	point	NOUN
iajs-1822	142	26	theory	theory	NOUN
iajs-1822	142	27	appl	appl	NOUN
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iajs-1822	143	2	"	"	PUNCT
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iajs-1822	143	9	,	,	PUNCT
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iajs-1822	143	11	,	,	PUNCT
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iajs-1822	143	13	.	.	PUNCT
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