id	sid	tid	token	lemma	pos
ma-63	1	1	2022	2022	NUM
ma-63	1	2	ada	ada	PROPN
ma-63	1	3	academica	academica	PROPN
ma-63	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-63	1	5	.	.	PUNCT
ma-63	2	1	j.	j.	PROPN
ma-63	2	2	math	math	PROPN
ma-63	2	3	.	.	PUNCT
ma-63	3	1	anal	anal	ADJ
ma-63	3	2	.	.	PUNCT
ma-63	3	3	2	2	NUM
ma-63	3	4	(	(	PUNCT
ma-63	3	5	2022	2022	NUM
ma-63	3	6	)	)	PUNCT
ma-63	3	7	11doi	11doi	NUM
ma-63	3	8	:	:	PUNCT
ma-63	3	9	10.28924	10.28924	NUM
ma-63	3	10	/	/	SYM
ma-63	3	11	ada	ada	PROPN
ma-63	3	12	/	/	SYM
ma-63	3	13	ma.2.11	ma.2.11	PROPN
ma-63	3	14	on	on	ADP
ma-63	3	15	the	the	DET
ma-63	3	16	α−ψ−	α−ψ−	PROPN
ma-63	3	17	contractive	contractive	ADJ
ma-63	3	18	mappings	mapping	NOUN
ma-63	3	19	in	in	ADP
ma-63	3	20	c∗-algebra	c∗-algebra	PROPN
ma-63	3	21	valued	value	VERB
ma-63	3	22	b	b	X
ma-63	3	23	-	-	ADJ
ma-63	3	24	rectangular	rectangular	ADJ
ma-63	3	25	metric	metric	ADJ
ma-63	3	26	spaces	space	NOUN
ma-63	3	27	and	and	CCONJ
ma-63	3	28	fixed	fix	VERB
ma-63	3	29	point	point	NOUN
ma-63	3	30	theorems	theorems	PROPN
ma-63	3	31	mohamed	mohamed	PROPN
ma-63	3	32	rossafi1,∗	rossafi1,∗	PROPN
ma-63	3	33	,	,	PUNCT
ma-63	3	34	abdelkarim	abdelkarim	PROPN
ma-63	3	35	kari2	kari2	PROPN
ma-63	3	36	,	,	PUNCT
ma-63	3	37	hafida	hafida	PROPN
ma-63	3	38	massit3	massit3	PROPN
ma-63	3	39	1lasma	1lasma	NUM
ma-63	3	40	laboratory	laboratory	NOUN
ma-63	3	41	department	department	NOUN
ma-63	3	42	of	of	ADP
ma-63	3	43	mathematics	mathematic	NOUN
ma-63	3	44	,	,	PUNCT
ma-63	3	45	faculty	faculty	NOUN
ma-63	3	46	of	of	ADP
ma-63	3	47	sciences	sciences	PROPN
ma-63	3	48	dhar	dhar	PROPN
ma-63	3	49	el	el	PROPN
ma-63	3	50	mahraz	mahraz	PROPN
ma-63	3	51	,	,	PUNCT
ma-63	3	52	university	university	NOUN
ma-63	3	53	sidi	sidi	NOUN
ma-63	3	54	mohamed	mohamed	PROPN
ma-63	3	55	ben	ben	PROPN
ma-63	3	56	abdellah	abdellah	PROPN
ma-63	3	57	,	,	PUNCT
ma-63	3	58	b.	b.	PROPN
ma-63	3	59	p.	p.	NOUN
ma-63	3	60	1796	1796	NUM
ma-63	4	1	fes	fes	PROPN
ma-63	4	2	atlas	atlas	PROPN
ma-63	4	3	,	,	PUNCT
ma-63	4	4	morocco	morocco	PROPN
ma-63	4	5	mohamed.rossafi@usmba.ac.ma	mohamed.rossafi@usmba.ac.ma	NUM
ma-63	5	1	2ams	2am	NOUN
ma-63	5	2	laboratory	laboratory	NOUN
ma-63	5	3	faculty	faculty	NOUN
ma-63	5	4	of	of	ADP
ma-63	5	5	sciences	sciences	PROPN
ma-63	5	6	ben	ben	PROPN
ma-63	5	7	m’sik	m’sik	PROPN
ma-63	5	8	,	,	PUNCT
ma-63	5	9	hassan	hassan	PROPN
ma-63	5	10	ii	ii	PROPN
ma-63	5	11	university	university	PROPN
ma-63	5	12	,	,	PUNCT
ma-63	5	13	casablanca	casablanca	PROPN
ma-63	5	14	,	,	PUNCT
ma-63	5	15	morocco	morocco	PROPN
ma-63	5	16	abdkrimkariprofes@gmail.com	abdkrimkariprofes@gmail.com	X
ma-63	6	1	3laboratory	3laboratory	NUM
ma-63	6	2	of	of	ADP
ma-63	6	3	partial	partial	ADJ
ma-63	6	4	differential	differential	NOUN
ma-63	6	5	equations	equation	NOUN
ma-63	6	6	,	,	PUNCT
ma-63	6	7	spectral	spectral	ADJ
ma-63	6	8	algebra	algebra	NOUN
ma-63	6	9	and	and	CCONJ
ma-63	6	10	geometry	geometry	NOUN
ma-63	6	11	department	department	NOUN
ma-63	6	12	of	of	ADP
ma-63	6	13	mathematics	mathematic	NOUN
ma-63	6	14	,	,	PUNCT
ma-63	6	15	faculty	faculty	NOUN
ma-63	6	16	of	of	ADP
ma-63	6	17	sciences	science	NOUN
ma-63	6	18	,	,	PUNCT
ma-63	6	19	university	university	NOUN
ma-63	6	20	ibn	ibn	PROPN
ma-63	6	21	tofail	tofail	NOUN
ma-63	6	22	,	,	PUNCT
ma-63	6	23	kenitra	kenitra	PROPN
ma-63	6	24	,	,	PUNCT
ma-63	6	25	morocco	morocco	PROPN
ma-63	6	26	massithafida@yahoo.fr	massithafida@yahoo.fr	PROPN
ma-63	6	27	∗correspondence	∗correspondence	NOUN
ma-63	6	28	:	:	PUNCT
ma-63	6	29	mohamed.rossafi@usmba.ac.ma	mohamed.rossafi@usmba.ac.ma	NUM
ma-63	6	30	abstract	abstract	NOUN
ma-63	6	31	.	.	PUNCT
ma-63	7	1	this	this	DET
ma-63	7	2	present	present	ADJ
ma-63	7	3	paper	paper	NOUN
ma-63	7	4	extends	extend	VERB
ma-63	7	5	a	a	DET
ma-63	7	6	version	version	NOUN
ma-63	7	7	of	of	ADP
ma-63	7	8	α−ψ−contraction	α−ψ−contraction	NOUN
ma-63	7	9	in	in	ADP
ma-63	7	10	c∗-algebra	c∗-algebra	PROPN
ma-63	7	11	valued	value	VERB
ma-63	7	12	rectangu	rectangu	NOUN
ma-63	7	13	-	-	PUNCT
ma-63	7	14	lar	lar	NOUN
ma-63	7	15	b	b	NOUN
ma-63	7	16	-	-	PUNCT
ma-63	7	17	metric	metric	ADJ
ma-63	7	18	spaces	space	NOUN
ma-63	7	19	and	and	CCONJ
ma-63	7	20	establishing	establish	VERB
ma-63	7	21	the	the	DET
ma-63	7	22	existence	existence	NOUN
ma-63	7	23	and	and	CCONJ
ma-63	7	24	uniqueness	uniqueness	NOUN
ma-63	7	25	of	of	ADP
ma-63	7	26	fixed	fix	VERB
ma-63	7	27	point	point	NOUN
ma-63	7	28	for	for	ADP
ma-63	7	29	them	they	PRON
ma-63	7	30	.	.	PUNCT
ma-63	8	1	non	non	NOUN
ma-63	8	2	-	-	NOUN
ma-63	8	3	trivialexamples	trivialexample	NOUN
ma-63	8	4	are	be	AUX
ma-63	8	5	further	far	ADV
ma-63	8	6	provided	provide	VERB
ma-63	8	7	to	to	PART
ma-63	8	8	support	support	VERB
ma-63	8	9	the	the	DET
ma-63	8	10	hypotheses	hypothesis	NOUN
ma-63	8	11	of	of	ADP
ma-63	8	12	our	our	PRON
ma-63	8	13	results	result	NOUN
ma-63	8	14	.	.	PUNCT
ma-63	9	1	1	1	X
ma-63	9	2	.	.	X
ma-63	9	3	introduction	introduction	NOUN
ma-63	9	4	a	a	DET
ma-63	9	5	c∗-algebra	c∗-algebra	PROPN
ma-63	9	6	valued	value	VERB
ma-63	9	7	metric	metric	ADJ
ma-63	9	8	spaces	space	NOUN
ma-63	9	9	were	be	AUX
ma-63	9	10	introduced	introduce	VERB
ma-63	9	11	by	by	ADP
ma-63	9	12	ma	ma	PROPN
ma-63	9	13	et	et	PROPN
ma-63	9	14	al	al	PROPN
ma-63	9	15	.	.	PUNCT
ma-63	10	1	[	[	X
ma-63	10	2	6	6	NUM
ma-63	10	3	]	]	PUNCT
ma-63	10	4	as	as	ADP
ma-63	10	5	a	a	DET
ma-63	10	6	generalization	generalization	NOUN
ma-63	10	7	of	of	ADP
ma-63	10	8	metricspaces	metricspace	NOUN
ma-63	10	9	they	they	PRON
ma-63	10	10	proved	prove	VERB
ma-63	10	11	certain	certain	ADJ
ma-63	10	12	fixed	fixed	ADJ
ma-63	10	13	point	point	NOUN
ma-63	10	14	theorems	theorem	NOUN
ma-63	10	15	,	,	PUNCT
ma-63	10	16	by	by	ADP
ma-63	10	17	giving	give	VERB
ma-63	10	18	the	the	DET
ma-63	10	19	definition	definition	NOUN
ma-63	10	20	of	of	ADP
ma-63	10	21	c∗-algebra	c∗-algebra	NOUN
ma-63	10	22	valuedcontractive	valuedcontractive	ADJ
ma-63	10	23	mapping	mapping	NOUN
ma-63	10	24	analogous	analogous	ADJ
ma-63	10	25	to	to	PART
ma-63	10	26	banach	banach	NOUN
ma-63	10	27	contraction	contraction	NOUN
ma-63	10	28	principle	principle	NOUN
ma-63	10	29	.	.	PUNCT
ma-63	11	1	many	many	ADJ
ma-63	11	2	mathematicians	mathematician	NOUN
ma-63	11	3	worked	work	VERB
ma-63	11	4	onthis	onthis	PROPN
ma-63	11	5	interesting	interesting	ADJ
ma-63	11	6	space.various	space.various	ADJ
ma-63	11	7	fixed	fix	VERB
ma-63	11	8	point	point	NOUN
ma-63	11	9	results	result	NOUN
ma-63	11	10	were	be	AUX
ma-63	11	11	established	establish	VERB
ma-63	11	12	on	on	ADP
ma-63	11	13	such	such	ADJ
ma-63	11	14	spaces	space	NOUN
ma-63	11	15	,	,	PUNCT
ma-63	11	16	see	see	VERB
ma-63	11	17	[	[	X
ma-63	11	18	1–3	1–3	NOUN
ma-63	11	19	]	]	PUNCT
ma-63	11	20	and	and	CCONJ
ma-63	11	21	references	reference	NOUN
ma-63	11	22	therein.combining	therein.combine	VERB
ma-63	11	23	conditions	condition	NOUN
ma-63	11	24	used	use	VERB
ma-63	11	25	for	for	ADP
ma-63	11	26	definitions	definition	NOUN
ma-63	11	27	of	of	ADP
ma-63	11	28	c∗-algebra	c∗-algebra	NOUN
ma-63	11	29	valued	value	VERB
ma-63	11	30	metric	metric	ADJ
ma-63	11	31	and	and	CCONJ
ma-63	11	32	generalized	generalized	ADJ
ma-63	11	33	metricspaces	metricspace	NOUN
ma-63	11	34	,	,	PUNCT
ma-63	11	35	g	g	PROPN
ma-63	11	36	kalapana	kalapana	PROPN
ma-63	11	37	and	and	CCONJ
ma-63	11	38	tasneem	tasneem	X
ma-63	12	1	[	[	X
ma-63	12	2	4	4	X
ma-63	12	3	]	]	PUNCT
ma-63	12	4	announced	announce	VERB
ma-63	12	5	the	the	DET
ma-63	12	6	notions	notion	NOUN
ma-63	12	7	of	of	ADP
ma-63	12	8	c∗-algebra	c∗-algebra	PROPN
ma-63	12	9	valued	value	VERB
ma-63	12	10	metric	metric	ADJ
ma-63	12	11	space	space	NOUN
ma-63	12	12	andestablish	andestablish	VERB
ma-63	12	13	nice	nice	ADJ
ma-63	12	14	results	result	NOUN
ma-63	12	15	of	of	ADP
ma-63	12	16	fixed	fix	VERB
ma-63	12	17	point	point	NOUN
ma-63	12	18	on	on	ADP
ma-63	12	19	such	such	ADJ
ma-63	12	20	space.in	space.in	PROPN
ma-63	12	21	this	this	DET
ma-63	12	22	paper	paper	NOUN
ma-63	12	23	,	,	PUNCT
ma-63	12	24	inspired	inspire	VERB
ma-63	12	25	by	by	ADP
ma-63	12	26	the	the	DET
ma-63	12	27	work	work	NOUN
ma-63	12	28	done	do	VERB
ma-63	12	29	in	in	ADP
ma-63	12	30	[	[	X
ma-63	12	31	9	9	NUM
ma-63	12	32	]	]	PUNCT
ma-63	12	33	,	,	PUNCT
ma-63	12	34	we	we	PRON
ma-63	12	35	introduce	introduce	VERB
ma-63	12	36	the	the	DET
ma-63	12	37	notion	notion	NOUN
ma-63	12	38	of	of	ADP
ma-63	12	39	α	α	PROPN
ma-63	12	40	−	−	NOUN
ma-63	12	41	ψ−contractionand	ψ−contractionand	PUNCT
ma-63	12	42	establish	establish	VERB
ma-63	12	43	some	some	DET
ma-63	12	44	new	new	ADJ
ma-63	12	45	fixed	fix	VERB
ma-63	12	46	point	point	NOUN
ma-63	12	47	theorems	theorem	NOUN
ma-63	12	48	for	for	ADP
ma-63	12	49	mappings	mapping	NOUN
ma-63	12	50	in	in	ADP
ma-63	12	51	the	the	DET
ma-63	12	52	setting	setting	NOUN
ma-63	12	53	of	of	ADP
ma-63	12	54	complete	complete	ADJ
ma-63	12	55	c∗-algebravalued	c∗-algebravalue	VERB
ma-63	12	56	rectangular	rectangular	ADJ
ma-63	12	57	bmetric	bmetric	NOUN
ma-63	12	58	spaces.moreover	spaces.moreover	PROPN
ma-63	12	59	,	,	PUNCT
ma-63	12	60	an	an	DET
ma-63	12	61	illustrative	illustrative	ADJ
ma-63	12	62	examples	example	NOUN
ma-63	12	63	is	be	AUX
ma-63	12	64	presented	present	VERB
ma-63	12	65	to	to	PART
ma-63	12	66	support	support	VERB
ma-63	12	67	the	the	DET
ma-63	12	68	obtained	obtain	VERB
ma-63	12	69	results	result	NOUN
ma-63	12	70	.	.	PUNCT
ma-63	13	1	received	receive	VERB
ma-63	13	2	:	:	PUNCT
ma-63	13	3	18	18	NUM
ma-63	13	4	dec	dec	PROPN
ma-63	13	5	2021	2021	NUM
ma-63	13	6	.	.	PUNCT
ma-63	14	1	key	key	ADJ
ma-63	14	2	words	word	NOUN
ma-63	14	3	and	and	CCONJ
ma-63	14	4	phrases	phrase	NOUN
ma-63	14	5	.	.	PUNCT
ma-63	15	1	fixed	fix	VERB
ma-63	15	2	point	point	NOUN
ma-63	15	3	;	;	PUNCT
ma-63	15	4	c∗-algebra	c∗-algebra	NOUN
ma-63	15	5	valued	value	VERB
ma-63	15	6	metric	metric	ADJ
ma-63	15	7	spaces	space	NOUN
ma-63	15	8	;	;	PUNCT
ma-63	15	9	α	α	DET
ma-63	15	10	−	−	NOUN
ma-63	15	11	ψ−	ψ−	PROPN
ma-63	15	12	contraction	contraction	NOUN
ma-63	15	13	;	;	PUNCT
ma-63	15	14	α	α	X
ma-63	15	15	−	−	NOUN
ma-63	15	16	ψ	ψ	ADP
ma-63	15	17	−	−	PROPN
ma-63	15	18	c∗	c∗	ADJ
ma-63	15	19	valuedcontraction	valuedcontraction	NOUN
ma-63	15	20	.	.	PUNCT
ma-63	16	1	1	1	NUM
ma-63	16	2	https://adac.ee	https://adac.ee	PROPN
ma-63	16	3	https://doi.org/10.28924/ada/ma.2.11	https://doi.org/10.28924/ada/ma.2.11	PROPN
ma-63	16	4	eur	eur	NOUN
ma-63	16	5	.	.	PUNCT
ma-63	17	1	j.	j.	PROPN
ma-63	17	2	math	math	PROPN
ma-63	17	3	.	.	PUNCT
ma-63	18	1	anal	anal	PROPN
ma-63	18	2	.	.	PUNCT
ma-63	19	1	10.28924	10.28924	NUM
ma-63	19	2	/	/	SYM
ma-63	19	3	ada	ada	PROPN
ma-63	19	4	/	/	SYM
ma-63	19	5	ma.2.11	ma.2.11	PROPN
ma-63	19	6	22	22	NUM
ma-63	19	7	.	.	PUNCT
ma-63	20	1	preliminaries	preliminary	NOUN
ma-63	20	2	throughout	throughout	ADP
ma-63	20	3	this	this	DET
ma-63	20	4	paper	paper	NOUN
ma-63	20	5	,	,	PUNCT
ma-63	20	6	we	we	PRON
ma-63	20	7	denote	denote	VERB
ma-63	20	8	a	a	PRON
ma-63	20	9	by	by	ADP
ma-63	20	10	an	an	DET
ma-63	20	11	unital	unital	ADJ
ma-63	20	12	(	(	PUNCT
ma-63	20	13	i.e	i.e	INTJ
ma-63	20	14	,	,	PUNCT
ma-63	20	15	unity	unity	NOUN
ma-63	20	16	element	element	NOUN
ma-63	20	17	i	i	NOUN
ma-63	20	18	)	)	PUNCT
ma-63	21	1	c∗-algebra	c∗-algebra	VERB
ma-63	21	2	with	with	ADP
ma-63	21	3	linearinvolution	linearinvolution	NOUN
ma-63	21	4	∗	∗	NOUN
ma-63	21	5	,	,	PUNCT
ma-63	21	6	such	such	ADJ
ma-63	21	7	that	that	PRON
ma-63	21	8	for	for	ADP
ma-63	21	9	all	all	DET
ma-63	21	10	x	x	NOUN
ma-63	21	11	,	,	PUNCT
ma-63	21	12	y	y	PROPN
ma-63	21	13	∈	∈	PROPN
ma-63	21	14	a	a	PRON
ma-63	21	15	,	,	PUNCT
ma-63	21	16	(	(	PUNCT
ma-63	21	17	xy)∗	xy)∗	PUNCT
ma-63	21	18	=	=	SYM
ma-63	21	19	y∗x∗,and	y∗x∗,and	NOUN
ma-63	21	20	x∗∗	x∗∗	X
ma-63	22	1	=	=	PUNCT
ma-63	22	2	x	x	X
ma-63	22	3	.we	.we	PUNCT
ma-63	22	4	call	call	VERB
ma-63	22	5	an	an	DET
ma-63	22	6	element	element	NOUN
ma-63	22	7	x	x	SYM
ma-63	22	8	∈	∈	PROPN
ma-63	22	9	a	a	DET
ma-63	22	10	a	a	DET
ma-63	22	11	positive	positive	ADJ
ma-63	22	12	element	element	NOUN
ma-63	22	13	,	,	PUNCT
ma-63	22	14	denote	denote	VERB
ma-63	22	15	it	it	PRON
ma-63	22	16	by	by	ADP
ma-63	22	17	x	x	PROPN
ma-63	22	18	�	�	PROPN
ma-63	22	19	θif	θif	VERB
ma-63	22	20	x	x	SYM
ma-63	22	21	∈	∈	PROPN
ma-63	23	1	ah	ah	INTJ
ma-63	23	2	=	=	SYM
ma-63	23	3	{	{	PUNCT
ma-63	23	4	x	x	PUNCT
ma-63	23	5	∈	∈	PROPN
ma-63	23	6	a	a	DET
ma-63	23	7	:	:	PUNCT
ma-63	23	8	x	x	X
ma-63	23	9	=	=	SYM
ma-63	23	10	x∗	x∗	X
ma-63	23	11	}	}	PUNCT
ma-63	23	12	and	and	CCONJ
ma-63	23	13	σ(x	σ(x	NOUN
ma-63	23	14	)	)	PUNCT
ma-63	23	15	⊂	⊂	PROPN
ma-63	23	16	r+,where	r+,where	PROPN
ma-63	23	17	σ(x	σ(x	PROPN
ma-63	23	18	)	)	PUNCT
ma-63	23	19	is	be	AUX
ma-63	23	20	the	the	DET
ma-63	23	21	spectrum	spectrum	NOUN
ma-63	23	22	of	of	ADP
ma-63	23	23	x	x	SYM
ma-63	23	24	.using	.use	VERB
ma-63	23	25	positiveelement	positiveelement	NOUN
ma-63	23	26	,	,	PUNCT
ma-63	23	27	we	we	PRON
ma-63	23	28	can	can	AUX
ma-63	23	29	define	define	VERB
ma-63	23	30	a	a	DET
ma-63	23	31	partial	partial	ADJ
ma-63	23	32	ordering	order	VERB
ma-63	23	33	�	�	PROPN
ma-63	23	34	on	on	ADP
ma-63	23	35	ah	ah	INTJ
ma-63	23	36	as	as	SCONJ
ma-63	23	37	follows	follow	VERB
ma-63	23	38	:	:	PUNCT
ma-63	23	39	x	x	PUNCT
ma-63	23	40	�	�	PROPN
ma-63	23	41	y	y	PROPN
ma-63	23	42	if	if	SCONJ
ma-63	23	43	and	and	CCONJ
ma-63	23	44	only	only	ADV
ma-63	23	45	if	if	SCONJ
ma-63	23	46	y	y	PROPN
ma-63	23	47	−	−	PROPN
ma-63	23	48	x	x	SYM
ma-63	23	49	�	�	PROPN
ma-63	23	50	θwhere	θwhere	ADV
ma-63	23	51	θ	θ	PROPN
ma-63	23	52	means	mean	VERB
ma-63	23	53	the	the	DET
ma-63	23	54	zero	zero	NUM
ma-63	23	55	element	element	NOUN
ma-63	23	56	in	in	ADP
ma-63	23	57	a.	a.	NOUN
ma-63	23	58	we	we	PRON
ma-63	23	59	denote	denote	VERB
ma-63	23	60	the	the	DET
ma-63	23	61	set	set	NOUN
ma-63	23	62	x	x	X
ma-63	23	63	∈	∈	PROPN
ma-63	23	64	a	a	DET
ma-63	23	65	:	:	PUNCT
ma-63	23	66	x	x	SYM
ma-63	23	67	�	�	PROPN
ma-63	23	68	θ	θ	PROPN
ma-63	23	69	by	by	ADP
ma-63	23	70	a+	a+	PUNCT
ma-63	23	71	and	and	CCONJ
ma-63	23	72	|x	|x	NOUN
ma-63	23	73	|	|	ADV
ma-63	23	74	=	=	SYM
ma-63	23	75	(	(	PUNCT
ma-63	23	76	x∗x	x∗x	ADJ
ma-63	23	77	)	)	PUNCT
ma-63	23	78	1	1	NUM
ma-63	23	79	2	2	NUM
ma-63	23	80	.and	.and	NOUN
ma-63	23	81	a′	a′	PROPN
ma-63	23	82	will	will	AUX
ma-63	23	83	denote	denote	VERB
ma-63	23	84	the	the	DET
ma-63	23	85	set	set	NOUN
ma-63	23	86	{	{	PUNCT
ma-63	23	87	a	a	DET
ma-63	23	88	∈	∈	PROPN
ma-63	23	89	a+	a+	PUNCT
ma-63	23	90	;	;	PUNCT
ma-63	23	91	ab	ab	PROPN
ma-63	23	92	=	=	SYM
ma-63	23	93	ba	ba	PROPN
ma-63	23	94	,	,	PUNCT
ma-63	23	95	∀b	∀b	NOUN
ma-63	23	96	∈	∈	PROPN
ma-63	23	97	a	a	DET
ma-63	23	98	}	}	PUNCT
ma-63	23	99	lemma	lemma	PROPN
ma-63	23	100	2.1	2.1	NUM
ma-63	23	101	.	.	PUNCT
ma-63	24	1	[	[	X
ma-63	24	2	8	8	NUM
ma-63	24	3	]	]	PUNCT
ma-63	24	4	suppose	suppose	VERB
ma-63	24	5	that	that	SCONJ
ma-63	24	6	a	a	PRON
ma-63	24	7	is	be	AUX
ma-63	24	8	a	a	DET
ma-63	24	9	unital	unital	ADJ
ma-63	24	10	c∗-algebra	c∗-algebra	NOUN
ma-63	24	11	with	with	ADP
ma-63	24	12	a	a	DET
ma-63	24	13	unit	unit	NOUN
ma-63	24	14	i,(1	i,(1	PROPN
ma-63	24	15	)	)	PUNCT
ma-63	24	16	for	for	ADP
ma-63	24	17	any	any	DET
ma-63	24	18	x	x	SYM
ma-63	24	19	∈	∈	PROPN
ma-63	24	20	a+	a+	PUNCT
ma-63	24	21	we	we	PRON
ma-63	24	22	have	have	AUX
ma-63	24	23	x	x	PART
ma-63	24	24	�	�	PROPN
ma-63	24	25	i	i	PRON
ma-63	24	26	⇐	⇐	ADJ
ma-63	24	27	⇒	⇒	PROPN
ma-63	24	28	‖x‖	‖x‖	VERB
ma-63	24	29	≤	≤	PROPN
ma-63	24	30	1(2	1(2	NUM
ma-63	24	31	)	)	PUNCT
ma-63	24	32	if	if	SCONJ
ma-63	24	33	a	a	DET
ma-63	24	34	∈	∈	PROPN
ma-63	24	35	a+	a+	PUNCT
ma-63	24	36	with	with	ADP
ma-63	24	37	‖a‖	‖a‖	PROPN
ma-63	24	38	<	<	X
ma-63	24	39	1	1	NUM
ma-63	24	40	2	2	NUM
ma-63	24	41	then	then	ADV
ma-63	24	42	i	i	PRON
ma-63	24	43	−	−	VERB
ma-63	24	44	a	a	PRON
ma-63	24	45	is	be	AUX
ma-63	24	46	unvertible	unvertible	ADJ
ma-63	24	47	and	and	CCONJ
ma-63	24	48	‖a(1−	‖a(1−	PROPN
ma-63	24	49	a)−1‖	a)−1‖	VERB
ma-63	24	50	<	<	X
ma-63	24	51	1(3	1(3	NUM
ma-63	24	52	)	)	PUNCT
ma-63	24	53	suppose	suppose	VERB
ma-63	24	54	that	that	SCONJ
ma-63	24	55	a	a	DET
ma-63	24	56	,	,	PUNCT
ma-63	24	57	b	b	X
ma-63	24	58	∈	∈	PROPN
ma-63	24	59	a+	a+	PUNCT
ma-63	24	60	and	and	CCONJ
ma-63	24	61	ab	ab	PROPN
ma-63	24	62	=	=	PROPN
ma-63	24	63	ba	ba	PROPN
ma-63	24	64	,	,	PUNCT
ma-63	24	65	then	then	ADV
ma-63	24	66	ab	ab	PROPN
ma-63	24	67	�	�	PROPN
ma-63	24	68	θ(4	θ(4	PROPN
ma-63	24	69	)	)	PUNCT
ma-63	24	70	let	let	VERB
ma-63	24	71	a	a	DET
ma-63	24	72	∈	∈	NOUN
ma-63	24	73	a′	a′	PROPN
ma-63	24	74	,	,	PUNCT
ma-63	24	75	if	if	SCONJ
ma-63	24	76	b	b	X
ma-63	24	77	,	,	PUNCT
ma-63	24	78	c	c	PROPN
ma-63	24	79	∈	∈	PROPN
ma-63	24	80	a	a	PRON
ma-63	24	81	,	,	PUNCT
ma-63	24	82	with	with	ADP
ma-63	24	83	b	b	PROPN
ma-63	24	84	�	�	PROPN
ma-63	24	85	c	c	PROPN
ma-63	24	86	�	�	PROPN
ma-63	24	87	θ	θ	PROPN
ma-63	24	88	,	,	PUNCT
ma-63	24	89	and	and	CCONJ
ma-63	24	90	i	i	PRON
ma-63	24	91	−	−	VERB
ma-63	24	92	a	a	DET
ma-63	24	93	∈	∈	NOUN
ma-63	24	94	a′+	a′+	PRON
ma-63	24	95	is	be	AUX
ma-63	24	96	invertible	invertible	ADJ
ma-63	24	97	operator	operator	NOUN
ma-63	24	98	,	,	PUNCT
ma-63	24	99	then	then	ADV
ma-63	24	100	(	(	PUNCT
ma-63	24	101	i	i	PRON
ma-63	24	102	−	−	PROPN
ma-63	24	103	a)−1b	a)−1b	PROPN
ma-63	24	104	�	�	PROPN
ma-63	24	105	(	(	PUNCT
ma-63	24	106	i	i	PRON
ma-63	24	107	−	−	PROPN
ma-63	24	108	a)−1c	a)−1c	PROPN
ma-63	24	109	definition	definition	NOUN
ma-63	24	110	2.2	2.2	NUM
ma-63	24	111	.	.	PUNCT
ma-63	25	1	[	[	X
ma-63	25	2	4	4	X
ma-63	25	3	]	]	PUNCT
ma-63	25	4	let	let	VERB
ma-63	25	5	x	x	PRON
ma-63	25	6	be	be	AUX
ma-63	25	7	a	a	DET
ma-63	25	8	non	non	ADJ
ma-63	25	9	-	-	ADJ
ma-63	25	10	empty	empty	ADJ
ma-63	25	11	set	set	NOUN
ma-63	25	12	and	and	CCONJ
ma-63	25	13	b	b	NOUN
ma-63	25	14	∈	∈	PROPN
ma-63	25	15	a	a	DET
ma-63	25	16	such	such	ADJ
ma-63	25	17	that	that	DET
ma-63	25	18	b	b	PROPN
ma-63	25	19	�	�	PROPN
ma-63	25	20	i	i	PRON
ma-63	25	21	.	.	PUNCT
ma-63	26	1	supposa	supposa	VERB
ma-63	26	2	the	the	DET
ma-63	26	3	mapping	mapping	NOUN
ma-63	27	1	d	d	NOUN
ma-63	27	2	:	:	PUNCT
ma-63	27	3	x	x	PROPN
ma-63	27	4	×x	×x	ADP
ma-63	27	5	→	→	SYM
ma-63	27	6	a+	a+	PRON
ma-63	27	7	satisfies:(i	satisfies:(i	NOUN
ma-63	27	8	)	)	PUNCT
ma-63	27	9	d(x	d(x	PROPN
ma-63	27	10	,	,	PUNCT
ma-63	27	11	y	y	NOUN
ma-63	27	12	)	)	PUNCT
ma-63	27	13	=	=	SYM
ma-63	28	1	θ	θ	NOUN
ma-63	28	2	if	if	SCONJ
ma-63	28	3	and	and	CCONJ
ma-63	28	4	only	only	ADV
ma-63	28	5	if	if	SCONJ
ma-63	28	6	x	x	X
ma-63	28	7	=	=	SYM
ma-63	28	8	y	y	PROPN
ma-63	28	9	;	;	PUNCT
ma-63	28	10	(	(	PUNCT
ma-63	28	11	ii	ii	NOUN
ma-63	28	12	)	)	PUNCT
ma-63	28	13	d(x	d(x	PROPN
ma-63	28	14	,	,	PUNCT
ma-63	28	15	y	y	NOUN
ma-63	28	16	)	)	PUNCT
ma-63	28	17	=	=	NOUN
ma-63	29	1	d(y	d(y	NOUN
ma-63	29	2	,	,	PUNCT
ma-63	29	3	x	x	X
ma-63	29	4	)	)	PUNCT
ma-63	29	5	for	for	ADP
ma-63	29	6	all	all	DET
ma-63	29	7	distinct	distinct	ADJ
ma-63	29	8	points	point	NOUN
ma-63	29	9	x	x	X
ma-63	29	10	,	,	PUNCT
ma-63	29	11	y	y	PROPN
ma-63	29	12	∈	∈	PROPN
ma-63	29	13	x;(iii	x;(iii	NOUN
ma-63	29	14	)	)	PUNCT
ma-63	29	15	d(x	d(x	PROPN
ma-63	29	16	,	,	PUNCT
ma-63	29	17	y	y	NOUN
ma-63	29	18	)	)	PUNCT
ma-63	29	19	�	�	PROPN
ma-63	29	20	b[d(x	b[d(x	PROPN
ma-63	29	21	,	,	PUNCT
ma-63	29	22	u	u	NOUN
ma-63	29	23	)	)	PUNCT
ma-63	29	24	+	+	CCONJ
ma-63	29	25	d(u	d(u	PROPN
ma-63	29	26	,	,	PUNCT
ma-63	29	27	v	v	NOUN
ma-63	29	28	)	)	PUNCT
ma-63	29	29	+	+	CCONJ
ma-63	29	30	d(v	d(v	PROPN
ma-63	29	31	,	,	PUNCT
ma-63	29	32	y	y	PROPN
ma-63	29	33	)	)	PUNCT
ma-63	29	34	]	]	PUNCT
ma-63	29	35	for	for	ADP
ma-63	29	36	all	all	DET
ma-63	29	37	x	x	NOUN
ma-63	29	38	,	,	PUNCT
ma-63	29	39	y	y	PROPN
ma-63	29	40	∈	∈	PROPN
ma-63	29	41	x	x	X
ma-63	29	42	and	and	CCONJ
ma-63	29	43	for	for	ADP
ma-63	29	44	all	all	DET
ma-63	29	45	distinct	distinct	ADJ
ma-63	29	46	points	point	NOUN
ma-63	29	47	u	u	NOUN
ma-63	29	48	,	,	PUNCT
ma-63	29	49	v	v	NOUN
ma-63	29	50	∈	∈	PROPN
ma-63	29	51	x	x	PUNCT
ma-63	29	52	−	−	X
ma-63	29	53	{	{	PUNCT
ma-63	29	54	x	x	NOUN
ma-63	29	55	,	,	PUNCT
ma-63	29	56	y}.then	y}.then	ADV
ma-63	29	57	(	(	PUNCT
ma-63	29	58	x	x	NOUN
ma-63	29	59	,	,	PUNCT
ma-63	29	60	a+	a+	ADJ
ma-63	29	61	,	,	PUNCT
ma-63	29	62	d	d	X
ma-63	29	63	)	)	PUNCT
ma-63	29	64	is	be	AUX
ma-63	29	65	called	call	VERB
ma-63	29	66	a	a	DET
ma-63	29	67	c∗-algebra	c∗-algebra	PROPN
ma-63	29	68	valued	value	VERB
ma-63	29	69	rectangular	rectangular	ADJ
ma-63	29	70	b−metric	b−metric	ADJ
ma-63	29	71	space	space	NOUN
ma-63	29	72	.	.	PUNCT
ma-63	29	73	example	example	NOUN
ma-63	30	1	2.3	2.3	NUM
ma-63	30	2	.	.	PUNCT
ma-63	31	1	let	let	VERB
ma-63	31	2	x	x	PUNCT
ma-63	31	3	=	=	PUNCT
ma-63	31	4	r	r	NOUN
ma-63	31	5	and	and	CCONJ
ma-63	31	6	a	a	DET
ma-63	31	7	=	=	X
ma-63	31	8	m2(r	m2(r	PROPN
ma-63	31	9	)	)	PUNCT
ma-63	31	10	.	.	PUNCT
ma-63	32	1	define	define	VERB
ma-63	32	2	d(x	d(x	PROPN
ma-63	32	3	,	,	PUNCT
ma-63	32	4	y	y	NOUN
ma-63	32	5	)	)	PUNCT
ma-63	32	6	=	=	PUNCT
ma-63	33	1	diag(|x	diag(|x	NOUN
ma-63	34	1	−	−	NOUN
ma-63	34	2	y	y	PROPN
ma-63	34	3	|	|	ADV
ma-63	34	4	,	,	PUNCT
ma-63	34	5	2|x	2|x	NUM
ma-63	34	6	−	−	NOUN
ma-63	34	7	y	y	PROPN
ma-63	34	8	|	|	NOUN
ma-63	34	9	)	)	PUNCT
ma-63	34	10	where	where	SCONJ
ma-63	34	11	x	x	X
ma-63	34	12	,	,	PUNCT
ma-63	34	13	y	y	PROPN
ma-63	34	14	∈	∈	PROPN
ma-63	34	15	r.	r.	PROPN
ma-63	34	16	it	it	PRON
ma-63	34	17	is	be	AUX
ma-63	34	18	easy	easy	ADJ
ma-63	34	19	to	to	PART
ma-63	34	20	verify	verify	VERB
ma-63	34	21	d	d	NOUN
ma-63	34	22	is	be	AUX
ma-63	34	23	a	a	DET
ma-63	34	24	c∗−	c∗−	PROPN
ma-63	34	25	algebra	algebra	NOUN
ma-63	34	26	-	-	PUNCT
ma-63	34	27	valued	value	VERB
ma-63	34	28	rectangular	rectangular	ADJ
ma-63	34	29	b−	b−	NOUN
ma-63	34	30	metric	metric	NOUN
ma-63	34	31	and	and	CCONJ
ma-63	34	32	(	(	PUNCT
ma-63	34	33	x	x	X
ma-63	34	34	,	,	PUNCT
ma-63	34	35	m2(r	m2(r	NOUN
ma-63	34	36	)	)	PUNCT
ma-63	34	37	,	,	PUNCT
ma-63	34	38	d)is	d)is	PROPN
ma-63	34	39	a	a	DET
ma-63	34	40	copmlete	copmlete	NOUN
ma-63	34	41	c∗-algebra	c∗-algebra	PROPN
ma-63	34	42	valued	value	VERB
ma-63	34	43	rectangular	rectangular	ADJ
ma-63	34	44	b−metric	b−metric	ADJ
ma-63	34	45	space	space	NOUN
ma-63	34	46	.	.	PUNCT
ma-63	35	1	definition	definition	NOUN
ma-63	35	2	2.4	2.4	NUM
ma-63	35	3	.	.	PUNCT
ma-63	36	1	[	[	X
ma-63	36	2	9	9	NUM
ma-63	36	3	]	]	X
ma-63	36	4	if	if	SCONJ
ma-63	36	5	ψ	ψ	X
ma-63	36	6	:	:	PUNCT
ma-63	36	7	a	a	DET
ma-63	36	8	→	→	SYM
ma-63	36	9	b	b	NOUN
ma-63	36	10	is	be	AUX
ma-63	36	11	a	a	DET
ma-63	36	12	linear	linear	ADJ
ma-63	36	13	mapping	mapping	NOUN
ma-63	36	14	in	in	ADP
ma-63	36	15	c∗-algebra	c∗-algebra	PROPN
ma-63	36	16	,	,	PUNCT
ma-63	36	17	it	it	PRON
ma-63	36	18	is	be	AUX
ma-63	36	19	said	say	VERB
ma-63	36	20	to	to	PART
ma-63	36	21	be	be	AUX
ma-63	36	22	positive	positive	ADJ
ma-63	36	23	if	if	SCONJ
ma-63	36	24	ψ(a+	ψ(a+	NOUN
ma-63	36	25	)	)	PUNCT
ma-63	36	26	⊆	⊆	NUM
ma-63	36	27	b+	b+	NOUN
ma-63	36	28	.	.	PUNCT
ma-63	37	1	in	in	ADP
ma-63	37	2	this	this	DET
ma-63	37	3	case	case	NOUN
ma-63	37	4	ψ(ah	ψ(ah	NOUN
ma-63	37	5	)	)	PUNCT
ma-63	37	6	⊆	⊆	NUM
ma-63	37	7	bh	bh	NOUN
ma-63	37	8	,	,	PUNCT
ma-63	37	9	and	and	CCONJ
ma-63	37	10	the	the	DET
ma-63	37	11	restriction	restriction	NOUN
ma-63	37	12	map	map	NOUN
ma-63	37	13	ψ	ψ	X
ma-63	37	14	:	:	PUNCT
ma-63	37	15	ah	ah	INTJ
ma-63	37	16	→	→	PUNCT
ma-63	37	17	bh	bh	NOUN
ma-63	37	18	is	be	AUX
ma-63	37	19	increasing	increase	VERB
ma-63	37	20	.	.	PUNCT
ma-63	38	1	definition	definition	NOUN
ma-63	38	2	2.5	2.5	NUM
ma-63	38	3	.	.	PUNCT
ma-63	39	1	[	[	X
ma-63	39	2	9	9	X
ma-63	39	3	]	]	PUNCT
ma-63	39	4	suppose	suppose	VERB
ma-63	39	5	that	that	SCONJ
ma-63	39	6	a	a	PRON
ma-63	39	7	and	and	CCONJ
ma-63	39	8	b	b	NOUN
ma-63	39	9	are	be	AUX
ma-63	39	10	c∗-algebra	c∗-algebra	NOUN
ma-63	39	11	.a	.a	ADJ
ma-63	39	12	mapping	mapping	NOUN
ma-63	39	13	ψ	ψ	X
ma-63	39	14	:	:	PUNCT
ma-63	39	15	a→	a→	PROPN
ma-63	39	16	b	b	NOUN
ma-63	39	17	is	be	AUX
ma-63	39	18	said	say	VERB
ma-63	39	19	to	to	PART
ma-63	39	20	be	be	AUX
ma-63	39	21	c∗homomorphism	c∗homomorphism	NOUN
ma-63	39	22	if	if	SCONJ
ma-63	39	23	:(	:(	PROPN
ma-63	39	24	i	i	NOUN
ma-63	39	25	)	)	PUNCT
ma-63	39	26	ψ(ax	ψ(ax	PROPN
ma-63	39	27	+	+	CCONJ
ma-63	39	28	by	by	ADP
ma-63	39	29	)	)	PUNCT
ma-63	39	30	=	=	SYM
ma-63	39	31	aψ(x	aψ(x	NUM
ma-63	39	32	)	)	PUNCT
ma-63	39	33	+	+	CCONJ
ma-63	39	34	bψ(y	bψ(y	NOUN
ma-63	39	35	)	)	PUNCT
ma-63	39	36	for	for	ADP
ma-63	39	37	all	all	DET
ma-63	39	38	a	a	DET
ma-63	39	39	,	,	PUNCT
ma-63	39	40	b	b	X
ma-63	39	41	∈	∈	PROPN
ma-63	39	42	c	c	NOUN
ma-63	39	43	and	and	CCONJ
ma-63	39	44	x	x	NOUN
ma-63	39	45	,	,	PUNCT
ma-63	39	46	y	y	PROPN
ma-63	39	47	∈	∈	PROPN
ma-63	39	48	a(ii	a(ii	PROPN
ma-63	39	49	)	)	PUNCT
ma-63	39	50	ψ(xy	ψ(xy	NUM
ma-63	39	51	)	)	PUNCT
ma-63	39	52	=	=	SYM
ma-63	39	53	ψ(x)ψ(y	ψ(x)ψ(y	NOUN
ma-63	39	54	)	)	PUNCT
ma-63	39	55	for	for	ADP
ma-63	39	56	all	all	DET
ma-63	39	57	x	x	NOUN
ma-63	39	58	,	,	PUNCT
ma-63	39	59	y	y	PROPN
ma-63	39	60	∈	∈	PROPN
ma-63	39	61	a	a	DET
ma-63	39	62	https://doi.org/10.28924/ada/ma.2.11	https://doi.org/10.28924/ada/ma.2.11	PROPN
ma-63	39	63	eur	eur	NOUN
ma-63	39	64	.	.	PUNCT
ma-63	40	1	j.	j.	PROPN
ma-63	40	2	math	math	PROPN
ma-63	40	3	.	.	PUNCT
ma-63	41	1	anal	anal	PROPN
ma-63	41	2	.	.	PUNCT
ma-63	42	1	10.28924	10.28924	NUM
ma-63	42	2	/	/	SYM
ma-63	42	3	ada	ada	PROPN
ma-63	42	4	/	/	SYM
ma-63	42	5	ma.2.11	ma.2.11	ADJ
ma-63	42	6	3(iii	3(iii	NUM
ma-63	42	7	)	)	PUNCT
ma-63	42	8	ψ(x∗	ψ(x∗	NOUN
ma-63	42	9	)	)	PUNCT
ma-63	42	10	=	=	SYM
ma-63	43	1	ψ(x)∗	ψ(x)∗	NOUN
ma-63	43	2	for	for	ADP
ma-63	43	3	all	all	DET
ma-63	43	4	x	x	SYM
ma-63	43	5	∈	∈	PROPN
ma-63	43	6	a(iv	a(iv	NOUN
ma-63	43	7	)	)	PUNCT
ma-63	43	8	ψ	ψ	PROPN
ma-63	43	9	maps	map	VERB
ma-63	43	10	the	the	DET
ma-63	43	11	unit	unit	NOUN
ma-63	43	12	in	in	ADP
ma-63	43	13	a	a	PRON
ma-63	43	14	to	to	ADP
ma-63	43	15	the	the	DET
ma-63	43	16	unit	unit	NOUN
ma-63	43	17	in	in	ADP
ma-63	43	18	b.	b.	PROPN
ma-63	43	19	definition	definition	NOUN
ma-63	43	20	2.6	2.6	NUM
ma-63	43	21	.	.	PUNCT
ma-63	44	1	[	[	X
ma-63	44	2	9	9	NUM
ma-63	44	3	]	]	PUNCT
ma-63	44	4	let	let	VERB
ma-63	44	5	a	a	PRON
ma-63	44	6	and	and	CCONJ
ma-63	44	7	b	b	NOUN
ma-63	44	8	be	be	AUX
ma-63	44	9	c∗-algebra	c∗-algebra	NOUN
ma-63	44	10	spaces	space	NOUN
ma-63	44	11	and	and	CCONJ
ma-63	44	12	let	let	VERB
ma-63	44	13	ψ	ψ	X
ma-63	44	14	:	:	PUNCT
ma-63	44	15	a→	a→	PUNCT
ma-63	44	16	b	b	X
ma-63	44	17	be	be	AUX
ma-63	44	18	a	a	DET
ma-63	44	19	homomorphismthen	homomorphismthen	NOUN
ma-63	44	20	ψ	ψ	NOUN
ma-63	44	21	is	be	AUX
ma-63	44	22	called	call	VERB
ma-63	44	23	an	an	DET
ma-63	44	24	∗−	∗−	ADJ
ma-63	44	25	homomorphism	homomorphism	NOUN
ma-63	44	26	if	if	SCONJ
ma-63	44	27	it	it	PRON
ma-63	44	28	is	be	AUX
ma-63	44	29	one	one	NUM
ma-63	44	30	to	to	ADP
ma-63	44	31	one	one	NUM
ma-63	44	32	∗−	∗−	NOUN
ma-63	44	33	homomorphism.a	homomorphism.a	NOUN
ma-63	44	34	c∗-algebra	c∗-algebra	NOUN
ma-63	44	35	a	a	PRON
ma-63	44	36	is	be	AUX
ma-63	44	37	∗−isomorphic	∗−isomorphic	ADJ
ma-63	44	38	to	to	ADP
ma-63	44	39	a	a	DET
ma-63	44	40	c∗-algebra	c∗-algebra	PROPN
ma-63	44	41	b	b	PROPN
ma-63	44	42	if	if	SCONJ
ma-63	44	43	there	there	PRON
ma-63	44	44	exists	exist	VERB
ma-63	44	45	∗−	∗−	ADJ
ma-63	44	46	isomorphism	isomorphism	NOUN
ma-63	44	47	of	of	ADP
ma-63	44	48	a	a	PRON
ma-63	44	49	onto	onto	ADP
ma-63	44	50	b.	b.	PROPN
ma-63	44	51	definition	definition	NOUN
ma-63	44	52	2.7	2.7	NUM
ma-63	44	53	.	.	PUNCT
ma-63	45	1	[	[	X
ma-63	45	2	9	9	NUM
ma-63	45	3	]	]	PUNCT
ma-63	45	4	let	let	VERB
ma-63	45	5	ψ	ψ	PART
ma-63	45	6	be	be	AUX
ma-63	45	7	the	the	DET
ma-63	45	8	set	set	NOUN
ma-63	45	9	of	of	ADP
ma-63	45	10	positive	positive	ADJ
ma-63	45	11	functions	function	NOUN
ma-63	45	12	ψ	ψ	NOUN
ma-63	45	13	:	:	PUNCT
ma-63	45	14	a+	a+	PUNCT
ma-63	45	15	→	→	PUNCT
ma-63	45	16	a+	a+	PUNCT
ma-63	45	17	satisfying	satisfy	VERB
ma-63	45	18	the	the	DET
ma-63	45	19	followingconditions	followingcondition	NOUN
ma-63	45	20	:(	:(	PUNCT
ma-63	46	1	i	i	PRON
ma-63	46	2	)	)	PUNCT
ma-63	46	3	ψ	ψ	X
ma-63	46	4	is	be	AUX
ma-63	46	5	continous	continous	ADJ
ma-63	46	6	and	and	CCONJ
ma-63	46	7	nondecrasing(ii	nondecrasing(ii	ADJ
ma-63	46	8	)	)	PUNCT
ma-63	46	9	ψ(a	ψ(a	PROPN
ma-63	46	10	)	)	PUNCT
ma-63	47	1	=	=	SYM
ma-63	47	2	θ	θ	NOUN
ma-63	47	3	if	if	SCONJ
ma-63	47	4	and	and	CCONJ
ma-63	47	5	only	only	ADV
ma-63	47	6	if	if	SCONJ
ma-63	47	7	a	a	DET
ma-63	47	8	=	=	PUNCT
ma-63	47	9	θ(iii	θ(iii	PROPN
ma-63	47	10	)	)	PUNCT
ma-63	47	11	l	l	PROPN
ma-63	47	12	imn−→∞ψn(a	imn−→∞ψn(a	PROPN
ma-63	47	13	)	)	PUNCT
ma-63	47	14	=	=	SYM
ma-63	47	15	θ	θ	PROPN
ma-63	47	16	,	,	PUNCT
ma-63	47	17	(	(	PUNCT
ma-63	47	18	a	a	DET
ma-63	47	19	�	�	PROPN
ma-63	47	20	θ	θ	PROPN
ma-63	47	21	)	)	PUNCT
ma-63	47	22	,	,	PUNCT
ma-63	47	23	∑∞n=1	∑∞n=1	PROPN
ma-63	47	24	ψn(a	ψn(a	PUNCT
ma-63	47	25	)	)	PUNCT
ma-63	47	26	<	<	X
ma-63	47	27	∞(iv	∞(iv	NOUN
ma-63	47	28	)	)	PUNCT
ma-63	47	29	the	the	DET
ma-63	47	30	series	series	PROPN
ma-63	47	31	∑∞k=1	∑∞k=1	PROPN
ma-63	47	32	bkψk(a	bkψk(a	PROPN
ma-63	47	33	)	)	PUNCT
ma-63	48	1	<	<	X
ma-63	48	2	∞	∞	NUM
ma-63	48	3	for	for	SCONJ
ma-63	48	4	a	a	DET
ma-63	48	5	�	�	PROPN
ma-63	48	6	θ	θ	PROPN
ma-63	48	7	is	be	AUX
ma-63	48	8	increasing	increase	VERB
ma-63	48	9	and	and	CCONJ
ma-63	48	10	continuous	continuous	ADJ
ma-63	48	11	at	at	ADP
ma-63	48	12	θ	θ	PROPN
ma-63	48	13	.	.	PUNCT
ma-63	48	14	corollary	corollary	ADJ
ma-63	48	15	2.8	2.8	NUM
ma-63	48	16	.	.	PUNCT
ma-63	49	1	[	[	X
ma-63	49	2	9	9	NUM
ma-63	49	3	]	]	PUNCT
ma-63	49	4	every	every	DET
ma-63	49	5	c∗−	c∗−	PROPN
ma-63	49	6	homomorphism	homomorphism	NOUN
ma-63	49	7	is	be	AUX
ma-63	49	8	contractive	contractive	ADJ
ma-63	49	9	and	and	CCONJ
ma-63	49	10	hence	hence	ADV
ma-63	49	11	bounded	bound	VERB
ma-63	49	12	.	.	PUNCT
ma-63	50	1	lemma	lemma	PROPN
ma-63	50	2	2.9	2.9	NUM
ma-63	50	3	.	.	PUNCT
ma-63	51	1	every	every	DET
ma-63	51	2	∗−	∗−	ADJ
ma-63	51	3	homomorphism	homomorphism	NOUN
ma-63	51	4	is	be	AUX
ma-63	51	5	positive	positive	ADJ
ma-63	51	6	.	.	PUNCT
ma-63	52	1	definition	definition	NOUN
ma-63	52	2	2.10	2.10	NUM
ma-63	52	3	.	.	PUNCT
ma-63	53	1	[	[	X
ma-63	53	2	9	9	NUM
ma-63	53	3	]	]	PUNCT
ma-63	53	4	let	let	VERB
ma-63	53	5	x	x	PRON
ma-63	53	6	be	be	AUX
ma-63	53	7	a	a	DET
ma-63	53	8	nonempty	nonempty	ADV
ma-63	53	9	set	set	VERB
ma-63	53	10	and	and	CCONJ
ma-63	53	11	α	α	NOUN
ma-63	53	12	:	:	PUNCT
ma-63	53	13	x	x	PROPN
ma-63	53	14	×x	×x	ADP
ma-63	53	15	→	→	PUNCT
ma-63	53	16	a′+	a′+	VERB
ma-63	53	17	be	be	AUX
ma-63	53	18	a	a	DET
ma-63	53	19	function	function	NOUN
ma-63	53	20	,	,	PUNCT
ma-63	53	21	wesay	wesay	VERB
ma-63	53	22	that	that	SCONJ
ma-63	53	23	theself	theself	PRON
ma-63	53	24	map	map	NOUN
ma-63	53	25	t	t	NOUN
ma-63	53	26	is	be	AUX
ma-63	53	27	α−	α−	ADP
ma-63	53	28	admissible	admissible	ADJ
ma-63	53	29	if	if	SCONJ
ma-63	53	30	(	(	PUNCT
ma-63	53	31	x	x	NOUN
ma-63	53	32	,	,	PUNCT
ma-63	53	33	y	y	NOUN
ma-63	53	34	)	)	PUNCT
ma-63	53	35	∈	∈	PROPN
ma-63	53	36	x	x	SYM
ma-63	53	37	×x	×x	PROPN
ma-63	53	38	,	,	PUNCT
ma-63	53	39	α(x	α(x	PROPN
ma-63	53	40	,	,	PUNCT
ma-63	53	41	y	y	PROPN
ma-63	53	42	)	)	PUNCT
ma-63	53	43	�	�	PROPN
ma-63	53	44	i	i	PRON
ma-63	53	45	⇒	⇒	VERB
ma-63	53	46	α(tx	α(tx	PROPN
ma-63	53	47	,	,	PUNCT
ma-63	53	48	t	t	PROPN
ma-63	53	49	y	y	PROPN
ma-63	53	50	)	)	PUNCT
ma-63	53	51	�	�	PROPN
ma-63	54	1	i	i	PRON
ma-63	54	2	,	,	PUNCT
ma-63	54	3	where	where	SCONJ
ma-63	54	4	i	i	PRON
ma-63	54	5	the	the	DET
ma-63	54	6	unit	unit	NOUN
ma-63	54	7	of	of	ADP
ma-63	54	8	a.	a.	NOUN
ma-63	54	9	definition	definition	NOUN
ma-63	54	10	2.11	2.11	NUM
ma-63	54	11	.	.	PUNCT
ma-63	55	1	[	[	X
ma-63	55	2	9	9	NUM
ma-63	55	3	]	]	X
ma-63	55	4	let	let	VERB
ma-63	55	5	(	(	PUNCT
ma-63	55	6	x	x	X
ma-63	55	7	,	,	PUNCT
ma-63	55	8	a	a	PRON
ma-63	55	9	,	,	PUNCT
ma-63	55	10	d	d	NOUN
ma-63	55	11	)	)	PUNCT
ma-63	55	12	be	be	AUX
ma-63	55	13	a	a	DET
ma-63	55	14	c∗-algebra	c∗-algebra	PROPN
ma-63	55	15	valued	value	VERB
ma-63	55	16	b−	b−	NOUN
ma-63	55	17	metric	metric	ADJ
ma-63	55	18	space	space	NOUN
ma-63	55	19	and	and	CCONJ
ma-63	55	20	t	t	NOUN
ma-63	55	21	:	:	PUNCT
ma-63	55	22	x	x	X
ma-63	55	23	→	→	SYM
ma-63	55	24	x	x	SYM
ma-63	55	25	ismapping	ismappe	VERB
ma-63	55	26	,	,	PUNCT
ma-63	55	27	we	we	PRON
ma-63	55	28	say	say	VERB
ma-63	55	29	that	that	SCONJ
ma-63	55	30	t	t	PROPN
ma-63	55	31	is	be	AUX
ma-63	55	32	an	an	DET
ma-63	55	33	α−ψ−	α−ψ−	PROPN
ma-63	55	34	contractive	contractive	ADJ
ma-63	55	35	mapping	mapping	NOUN
ma-63	55	36	if	if	SCONJ
ma-63	55	37	there	there	PRON
ma-63	55	38	exist	exist	VERB
ma-63	55	39	two	two	NUM
ma-63	55	40	functions	function	NOUN
ma-63	55	41	α	α	NOUN
ma-63	55	42	:	:	PUNCT
ma-63	56	1	x×x	x×x	PROPN
ma-63	56	2	→	→	SYM
ma-63	56	3	a+	a+	PUNCT
ma-63	56	4	and	and	CCONJ
ma-63	56	5	ψ	ψ	X
ma-63	56	6	∈	∈	NOUN
ma-63	56	7	ψ	ψ	ADP
ma-63	56	8	such	such	ADJ
ma-63	56	9	that	that	DET
ma-63	56	10	α(x	α(x	NOUN
ma-63	56	11	,	,	PUNCT
ma-63	56	12	y)d(tx	y)d(tx	NUM
ma-63	56	13	,	,	PUNCT
ma-63	56	14	t	t	PROPN
ma-63	56	15	y	y	PROPN
ma-63	56	16	)	)	PUNCT
ma-63	56	17	�	�	PROPN
ma-63	56	18	ψ(d(x	ψ(d(x	PROPN
ma-63	56	19	,	,	PUNCT
ma-63	56	20	y	y	NOUN
ma-63	56	21	)	)	PUNCT
ma-63	56	22	)	)	PUNCT
ma-63	56	23	,	,	PUNCT
ma-63	56	24	for	for	ADP
ma-63	56	25	all	all	DET
ma-63	56	26	x	x	NOUN
ma-63	56	27	,	,	PUNCT
ma-63	56	28	y	y	PROPN
ma-63	56	29	∈	∈	PROPN
ma-63	56	30	x	x	PUNCT
ma-63	56	31	3	3	X
ma-63	56	32	.	.	NOUN
ma-63	56	33	main	main	ADJ
ma-63	56	34	result	result	NOUN
ma-63	56	35	in	in	ADP
ma-63	56	36	[	[	X
ma-63	56	37	9	9	NUM
ma-63	56	38	]	]	PUNCT
ma-63	56	39	introduced	introduce	VERB
ma-63	56	40	the	the	DET
ma-63	56	41	concept	concept	NOUN
ma-63	56	42	of	of	ADP
ma-63	56	43	α−ψ−	α−ψ−	NUM
ma-63	56	44	contractive	contractive	ADJ
ma-63	56	45	mappings	mapping	NOUN
ma-63	56	46	in	in	ADP
ma-63	56	47	a	a	DET
ma-63	56	48	unital	unital	ADJ
ma-63	56	49	c∗-algebra	c∗-algebra	NOUN
ma-63	56	50	valued	value	VERB
ma-63	56	51	b−metric	b−metric	ADJ
ma-63	56	52	space	space	NOUN
ma-63	56	53	.	.	PUNCT
ma-63	57	1	in	in	ADP
ma-63	57	2	this	this	DET
ma-63	57	3	paper	paper	NOUN
ma-63	57	4	we	we	PRON
ma-63	57	5	will	will	AUX
ma-63	57	6	develop	develop	VERB
ma-63	57	7	the	the	DET
ma-63	57	8	definitions	definition	NOUN
ma-63	57	9	in	in	ADP
ma-63	57	10	case	case	NOUN
ma-63	57	11	of	of	ADP
ma-63	57	12	unital	unital	ADJ
ma-63	57	13	c∗-algebra	c∗-algebra	NOUN
ma-63	57	14	valuedrectangular	valuedrectangular	ADJ
ma-63	57	15	b−	b−	NOUN
ma-63	57	16	metric	metric	ADJ
ma-63	57	17	space	space	NOUN
ma-63	57	18	and	and	CCONJ
ma-63	57	19	give	give	VERB
ma-63	57	20	some	some	DET
ma-63	57	21	banach	banach	ADV
ma-63	57	22	fixed	fix	VERB
ma-63	57	23	point	point	NOUN
ma-63	57	24	theorems	theorem	NOUN
ma-63	57	25	.	.	PUNCT
ma-63	58	1	definition	definition	NOUN
ma-63	58	2	3.1	3.1	NUM
ma-63	58	3	.	.	PUNCT
ma-63	59	1	let	let	VERB
ma-63	59	2	(	(	PUNCT
ma-63	59	3	x	x	X
ma-63	59	4	,	,	PUNCT
ma-63	59	5	a	a	PRON
ma-63	59	6	,	,	PUNCT
ma-63	59	7	d	d	NOUN
ma-63	59	8	)	)	PUNCT
ma-63	59	9	be	be	AUX
ma-63	59	10	a	a	DET
ma-63	59	11	c∗-algebra	c∗-algebra	PROPN
ma-63	59	12	valued	value	VERB
ma-63	59	13	b−	b−	PRON
ma-63	59	14	rectangular	rectangular	VERB
ma-63	59	15	metric	metric	ADJ
ma-63	59	16	space	space	NOUN
ma-63	59	17	and	and	CCONJ
ma-63	59	18	t	t	NOUN
ma-63	59	19	:	:	PUNCT
ma-63	59	20	x	x	X
ma-63	59	21	→	→	SYM
ma-63	59	22	xis	xis	PROPN
ma-63	59	23	mapping	mapping	PROPN
ma-63	59	24	,	,	PUNCT
ma-63	59	25	we	we	PRON
ma-63	59	26	say	say	VERB
ma-63	59	27	that	that	SCONJ
ma-63	59	28	t	t	PROPN
ma-63	59	29	is	be	AUX
ma-63	59	30	an	an	DET
ma-63	59	31	α	α	NOUN
ma-63	59	32	−	−	PUNCT
ma-63	59	33	ψ−	ψ−	VERB
ma-63	59	34	contractive	contractive	ADJ
ma-63	59	35	mapping	mapping	NOUN
ma-63	59	36	if	if	SCONJ
ma-63	59	37	there	there	PRON
ma-63	59	38	exist	exist	VERB
ma-63	59	39	two	two	NUM
ma-63	59	40	functions	function	NOUN
ma-63	59	41	α	α	NOUN
ma-63	59	42	:	:	PUNCT
ma-63	59	43	x	x	PROPN
ma-63	59	44	×x	×x	ADP
ma-63	59	45	→	→	SYM
ma-63	59	46	a+	a+	PUNCT
ma-63	59	47	and	and	CCONJ
ma-63	59	48	ψ	ψ	X
ma-63	59	49	∈	∈	NOUN
ma-63	59	50	ψ	ψ	ADP
ma-63	59	51	such	such	ADJ
ma-63	59	52	that	that	DET
ma-63	59	53	α(x	α(x	NOUN
ma-63	59	54	,	,	PUNCT
ma-63	59	55	y)d(tx	y)d(tx	NUM
ma-63	59	56	,	,	PUNCT
ma-63	59	57	t	t	PROPN
ma-63	59	58	y	y	PROPN
ma-63	59	59	)	)	PUNCT
ma-63	59	60	�	�	PROPN
ma-63	59	61	ψ(d(x	ψ(d(x	PROPN
ma-63	59	62	,	,	PUNCT
ma-63	59	63	y	y	NOUN
ma-63	59	64	)	)	PUNCT
ma-63	59	65	)	)	PUNCT
ma-63	59	66	,	,	PUNCT
ma-63	59	67	f	f	PROPN
ma-63	59	68	oral	oral	ADJ
ma-63	59	69	lx	lx	NOUN
ma-63	59	70	,	,	PUNCT
ma-63	59	71	y	y	PROPN
ma-63	59	72	∈	∈	PROPN
ma-63	59	73	x	x	X
ma-63	59	74	(	(	PUNCT
ma-63	59	75	3.1	3.1	NUM
ma-63	59	76	)	)	PUNCT
ma-63	59	77	theorem	theorem	NOUN
ma-63	59	78	3.2	3.2	NUM
ma-63	59	79	.	.	PUNCT
ma-63	60	1	let	let	AUX
ma-63	60	2	(	(	PUNCT
ma-63	60	3	x	x	X
ma-63	60	4	,	,	PUNCT
ma-63	60	5	a	a	PRON
ma-63	60	6	,	,	PUNCT
ma-63	60	7	d	d	NOUN
ma-63	60	8	)	)	PUNCT
ma-63	60	9	be	be	AUX
ma-63	60	10	a	a	DET
ma-63	60	11	complete	complete	ADJ
ma-63	60	12	c∗-algebra	c∗-algebra	NOUN
ma-63	60	13	valued	value	VERB
ma-63	60	14	rectangular	rectangular	ADJ
ma-63	60	15	b−	b−	NOUN
ma-63	60	16	metric	metric	ADJ
ma-63	60	17	space	space	NOUN
ma-63	60	18	and	and	CCONJ
ma-63	60	19	let	let	VERB
ma-63	60	20	t	t	NOUN
ma-63	60	21	:	:	PUNCT
ma-63	60	22	x	x	X
ma-63	60	23	→	→	PUNCT
ma-63	60	24	x	x	PUNCT
ma-63	60	25	be	be	AUX
ma-63	60	26	a	a	DET
ma-63	60	27	α	α	NOUN
ma-63	60	28	,	,	PUNCT
ma-63	60	29	ψ−	ψ−	VERB
ma-63	60	30	contractive	contractive	ADJ
ma-63	60	31	mapping	mapping	NOUN
ma-63	60	32	satisfying	satisfy	VERB
ma-63	60	33	the	the	DET
ma-63	60	34	following	follow	VERB
ma-63	60	35	conditions:(i	conditions:(i	NOUN
ma-63	60	36	)	)	PUNCT
ma-63	61	1	t	t	PROPN
ma-63	61	2	is	be	AUX
ma-63	61	3	α−	α−	ADP
ma-63	61	4	admissible	admissible	ADJ
ma-63	61	5	https://doi.org/10.28924/ada/ma.2.11	https://doi.org/10.28924/ada/ma.2.11	NOUN
ma-63	61	6	eur	eur	NOUN
ma-63	61	7	.	.	PUNCT
ma-63	62	1	j.	j.	PROPN
ma-63	62	2	math	math	PROPN
ma-63	62	3	.	.	PUNCT
ma-63	63	1	anal	anal	PROPN
ma-63	63	2	.	.	PUNCT
ma-63	64	1	10.28924	10.28924	NUM
ma-63	64	2	/	/	SYM
ma-63	64	3	ada	ada	PROPN
ma-63	64	4	/	/	SYM
ma-63	64	5	ma.2.11	ma.2.11	PROPN
ma-63	64	6	4(ii	4(ii	NUM
ma-63	64	7	)	)	PUNCT
ma-63	64	8	there	there	PRON
ma-63	64	9	exists	exist	VERB
ma-63	64	10	x0	x0	PROPN
ma-63	64	11	∈	∈	PROPN
ma-63	64	12	x	x	PUNCT
ma-63	64	13	such	such	ADJ
ma-63	64	14	that	that	DET
ma-63	64	15	α(x0	α(x0	NOUN
ma-63	64	16	,	,	PUNCT
ma-63	64	17	t	t	PROPN
ma-63	64	18	x0	x0	PROPN
ma-63	64	19	)	)	PUNCT
ma-63	64	20	�	�	PROPN
ma-63	64	21	i(iii	i(iii	PROPN
ma-63	64	22	)	)	PUNCT
ma-63	64	23	for	for	ADP
ma-63	64	24	all	all	DET
ma-63	64	25	x	x	NOUN
ma-63	64	26	,	,	PUNCT
ma-63	64	27	y	y	PROPN
ma-63	64	28	∈	∈	PROPN
ma-63	64	29	x	x	INTJ
ma-63	64	30	,	,	PUNCT
ma-63	64	31	there	there	PRON
ma-63	64	32	exists	exist	VERB
ma-63	64	33	z	z	NOUN
ma-63	64	34	∈	∈	PROPN
ma-63	64	35	x	x	PUNCT
ma-63	64	36	such	such	ADJ
ma-63	64	37	that	that	DET
ma-63	64	38	α(x	α(x	NOUN
ma-63	64	39	,	,	PUNCT
ma-63	64	40	z	z	NOUN
ma-63	64	41	)	)	PUNCT
ma-63	64	42	�	�	PROPN
ma-63	64	43	i	i	PRON
ma-63	64	44	and	and	CCONJ
ma-63	64	45	α(y	α(y	NOUN
ma-63	64	46	,	,	PUNCT
ma-63	64	47	z	z	X
ma-63	64	48	)	)	PUNCT
ma-63	64	49	�	�	PROPN
ma-63	64	50	i(iv	i(iv	NOUN
ma-63	64	51	)	)	PUNCT
ma-63	64	52	t	t	PROPN
ma-63	64	53	is	be	AUX
ma-63	64	54	continuous	continuous	ADJ
ma-63	64	55	then	then	ADV
ma-63	64	56	,	,	PUNCT
ma-63	64	57	t	t	PROPN
ma-63	64	58	has	have	VERB
ma-63	64	59	a	a	DET
ma-63	64	60	unique	unique	ADJ
ma-63	64	61	fixed	fix	VERB
ma-63	64	62	point	point	NOUN
ma-63	64	63	in	in	ADP
ma-63	64	64	x	x	X
ma-63	64	65	.	.	PUNCT
ma-63	65	1	proof	proof	NOUN
ma-63	65	2	.	.	PUNCT
ma-63	66	1	let	let	VERB
ma-63	66	2	x0	x0	PROPN
ma-63	66	3	∈	∈	PROPN
ma-63	66	4	x	x	PUNCT
ma-63	66	5	such	such	ADJ
ma-63	66	6	that	that	DET
ma-63	66	7	α(x0	α(x0	NOUN
ma-63	66	8	,	,	PUNCT
ma-63	66	9	t	t	PROPN
ma-63	66	10	x0	x0	PROPN
ma-63	66	11	)	)	PUNCT
ma-63	66	12	�	�	PROPN
ma-63	67	1	i	i	PRON
ma-63	67	2	and	and	CCONJ
ma-63	67	3	define	define	VERB
ma-63	67	4	a	a	DET
ma-63	67	5	sequence	sequence	NOUN
ma-63	67	6	{	{	PUNCT
ma-63	67	7	xn	xn	NOUN
ma-63	67	8	}	}	PUNCT
ma-63	67	9	∈	∈	PROPN
ma-63	67	10	x	x	PUNCT
ma-63	67	11	such	such	ADJ
ma-63	67	12	that	that	PRON
ma-63	67	13	xn+1	xn+1	PROPN
ma-63	67	14	=	=	SYM
ma-63	67	15	txn	txn	NOUN
ma-63	67	16	,	,	PUNCT
ma-63	67	17	∀n	∀n	X
ma-63	67	18	∈	∈	PROPN
ma-63	67	19	n.	n.	NOUN
ma-63	67	20	suppose	suppose	VERB
ma-63	67	21	that	that	SCONJ
ma-63	67	22	there	there	PRON
ma-63	67	23	exists	exist	VERB
ma-63	67	24	n	n	PRON
ma-63	67	25	∈	∈	PROPN
ma-63	67	26	n	n	PRON
ma-63	67	27	such	such	ADJ
ma-63	67	28	that	that	PRON
ma-63	67	29	xn	xn	PUNCT
ma-63	67	30	=	=	SYM
ma-63	67	31	txn	txn	PROPN
ma-63	67	32	.	.	PUNCT
ma-63	68	1	then	then	ADV
ma-63	68	2	xn	xn	PROPN
ma-63	68	3	is	be	AUX
ma-63	68	4	a	a	DET
ma-63	68	5	fixed	fix	VERB
ma-63	68	6	point	point	NOUN
ma-63	68	7	of	of	ADP
ma-63	68	8	t	t	PROPN
ma-63	68	9	andthe	andthe	NOUN
ma-63	68	10	proof	proof	NOUN
ma-63	68	11	is	be	AUX
ma-63	68	12	finished.hence	finished.hence	NOUN
ma-63	68	13	,	,	PUNCT
ma-63	68	14	we	we	PRON
ma-63	68	15	assume	assume	VERB
ma-63	68	16	that	that	SCONJ
ma-63	68	17	xn	xn	PROPN
ma-63	68	18	6=	6=	NUM
ma-63	68	19	txn+1	txn+1	PROPN
ma-63	68	20	,	,	PUNCT
ma-63	68	21	∀n	∀n	SYM
ma-63	68	22	∈	∈	PROPN
ma-63	68	23	n	n	CCONJ
ma-63	68	24	,	,	PUNCT
ma-63	68	25	since	since	SCONJ
ma-63	68	26	t	t	PROPN
ma-63	68	27	is	be	AUX
ma-63	68	28	α−admissible	α−admissible	PROPN
ma-63	68	29	,	,	PUNCT
ma-63	68	30	we	we	PRON
ma-63	68	31	get	get	VERB
ma-63	68	32	α(x0	α(x0	ADJ
ma-63	68	33	,	,	PUNCT
ma-63	68	34	x1	x1	PROPN
ma-63	68	35	)	)	PUNCT
ma-63	69	1	=	=	SYM
ma-63	69	2	α(x0	α(x0	NOUN
ma-63	69	3	,	,	PUNCT
ma-63	69	4	t	t	PROPN
ma-63	69	5	x0	x0	PROPN
ma-63	69	6	)	)	PUNCT
ma-63	69	7	�	�	PROPN
ma-63	70	1	i	i	PRON
ma-63	70	2	⇒	⇒	VERB
ma-63	70	3	α(tx0	α(tx0	PROPN
ma-63	70	4	,	,	PUNCT
ma-63	70	5	t	t	PROPN
ma-63	70	6	2x0	2x0	NUM
ma-63	70	7	)	)	PUNCT
ma-63	70	8	=	=	SYM
ma-63	70	9	α(x1	α(x1	ADJ
ma-63	70	10	,	,	PUNCT
ma-63	70	11	x2	x2	PROPN
ma-63	70	12	)	)	PUNCT
ma-63	70	13	�	�	PROPN
ma-63	70	14	i	i	PRON
ma-63	70	15	.	.	PUNCT
ma-63	71	1	continuing	continue	VERB
ma-63	71	2	this	this	DET
ma-63	71	3	process	process	NOUN
ma-63	71	4	,	,	PUNCT
ma-63	71	5	we	we	PRON
ma-63	71	6	have	have	VERB
ma-63	71	7	α(xn	α(xn	NOUN
ma-63	71	8	,	,	PUNCT
ma-63	71	9	xn+1	xn+1	NUM
ma-63	71	10	)	)	PUNCT
ma-63	71	11	�	�	PROPN
ma-63	72	1	i	i	PRON
ma-63	72	2	∀n	∀n	PROPN
ma-63	72	3	∈	∈	PROPN
ma-63	72	4	n.	n.	NOUN
ma-63	72	5	(	(	PUNCT
ma-63	72	6	3.2	3.2	NUM
ma-63	72	7	)	)	PUNCT
ma-63	72	8	by	by	ADP
ma-63	72	9	3.1	3.1	NUM
ma-63	72	10	and	and	CCONJ
ma-63	72	11	3.2	3.2	NUM
ma-63	72	12	,	,	PUNCT
ma-63	72	13	we	we	PRON
ma-63	72	14	get	get	VERB
ma-63	72	15	d(xn	d(xn	NOUN
ma-63	72	16	,	,	PUNCT
ma-63	72	17	xn+1	xn+1	NUM
ma-63	72	18	)	)	PUNCT
ma-63	72	19	=	=	SYM
ma-63	72	20	d(txn−1	d(txn−1	PROPN
ma-63	72	21	,	,	PUNCT
ma-63	72	22	t	t	PROPN
ma-63	72	23	xn	xn	PROPN
ma-63	72	24	)	)	PUNCT
ma-63	72	25	�	�	PROPN
ma-63	72	26	α(xn−1	α(xn−1	NUM
ma-63	72	27	,	,	PUNCT
ma-63	72	28	xn)d(txn−1	xn)d(txn−1	PROPN
ma-63	72	29	,	,	PUNCT
ma-63	72	30	t	t	PROPN
ma-63	72	31	xn	xn	PROPN
ma-63	72	32	)	)	PUNCT
ma-63	72	33	�	�	PROPN
ma-63	72	34	ψ(d(xn−1	ψ(d(xn−1	PROPN
ma-63	72	35	,	,	PUNCT
ma-63	72	36	xn	xn	NUM
ma-63	72	37	)	)	PUNCT
ma-63	72	38	)	)	PUNCT
ma-63	72	39	�	�	PROPN
ma-63	72	40	�	�	PROPN
ma-63	72	41	ψn(d(x0	ψn(d(x0	PROPN
ma-63	72	42	,	,	PUNCT
ma-63	72	43	x1	x1	PROPN
ma-63	72	44	)	)	PUNCT
ma-63	72	45	)	)	PUNCT
ma-63	72	46	.	.	PUNCT
ma-63	73	1	for	for	ADP
ma-63	73	2	m	m	PROPN
ma-63	73	3	≥	≥	NOUN
ma-63	73	4	1	1	NUM
ma-63	73	5	and	and	CCONJ
ma-63	73	6	p	p	PRON
ma-63	73	7	≥	≥	NUM
ma-63	73	8	1	1	NUM
ma-63	73	9	,	,	PUNCT
ma-63	73	10	it	it	PRON
ma-63	73	11	follows	follow	VERB
ma-63	73	12	that	that	SCONJ
ma-63	73	13	d(xm+p	d(xm+p	PROPN
ma-63	73	14	,	,	PUNCT
ma-63	73	15	xm	xm	PROPN
ma-63	73	16	)	)	PUNCT
ma-63	73	17	�	�	PROPN
ma-63	73	18	b[d(xm+p	b[d(xm+p	PROPN
ma-63	73	19	,	,	PUNCT
ma-63	73	20	xm+p−1	xm+p−1	PROPN
ma-63	73	21	)	)	PUNCT
ma-63	74	1	+	+	SYM
ma-63	75	1	d(xm+p−1	d(xm+p−1	PROPN
ma-63	75	2	,	,	PUNCT
ma-63	75	3	xm+p−2	xm+p−2	PROPN
ma-63	75	4	)	)	PUNCT
ma-63	76	1	+	+	X
ma-63	77	1	d(xm+p−2	d(xm+p−2	ADJ
ma-63	77	2	,	,	PUNCT
ma-63	77	3	xm	xm	PROPN
ma-63	77	4	)	)	PUNCT
ma-63	77	5	]	]	PUNCT
ma-63	77	6	�	�	PROPN
ma-63	77	7	bd(xm+p	bd(xm+p	PROPN
ma-63	77	8	,	,	PUNCT
ma-63	77	9	xm+p−1	xm+p−1	PROPN
ma-63	77	10	)	)	PUNCT
ma-63	78	1	+	+	CCONJ
ma-63	78	2	bd(xm+p−1	bd(xm+p−1	PROPN
ma-63	78	3	,	,	PUNCT
ma-63	78	4	xm+p−2	xm+p−2	PROPN
ma-63	78	5	)	)	PUNCT
ma-63	79	1	+	+	CCONJ
ma-63	80	1	b[b[d(xm+p−2	b[b[d(xm+p−2	NOUN
ma-63	80	2	,	,	PUNCT
ma-63	80	3	xm+p−3	xm+p−3	PROPN
ma-63	80	4	)	)	PUNCT
ma-63	80	5	+	+	NUM
ma-63	80	6	d(xm+p−3	d(xm+p−3	PROPN
ma-63	80	7	,	,	PUNCT
ma-63	80	8	xm+p−4	xm+p−4	PROPN
ma-63	80	9	)	)	PUNCT
ma-63	80	10	+	+	X
ma-63	80	11	d(xm+p−4	d(xm+p−4	ADJ
ma-63	80	12	,	,	PUNCT
ma-63	80	13	xm	xm	PROPN
ma-63	80	14	)	)	PUNCT
ma-63	80	15	]	]	PUNCT
ma-63	80	16	]	]	X
ma-63	80	17	=	=	SYM
ma-63	80	18	bd(xm+p	bd(xm+p	PROPN
ma-63	80	19	,	,	PUNCT
ma-63	80	20	xm+p−1	xm+p−1	PROPN
ma-63	80	21	)	)	PUNCT
ma-63	81	1	+	+	CCONJ
ma-63	81	2	bd(xm+p−1	bd(xm+p−1	PROPN
ma-63	81	3	,	,	PUNCT
ma-63	81	4	xm+p−2	xm+p−2	PROPN
ma-63	81	5	)	)	PUNCT
ma-63	82	1	+	+	CCONJ
ma-63	82	2	b2d(xm+p−2	b2d(xm+p−2	ADJ
ma-63	82	3	,	,	PUNCT
ma-63	82	4	xm+p−3	xm+p−3	PROPN
ma-63	82	5	)	)	PUNCT
ma-63	82	6	+	+	NUM
ma-63	82	7	b2d(xm+p−3	b2d(xm+p−3	PROPN
ma-63	82	8	,	,	PUNCT
ma-63	82	9	xm+p−4	xm+p−4	PROPN
ma-63	82	10	)	)	PUNCT
ma-63	82	11	+	+	NUM
ma-63	82	12	b2d(xm+p−4	b2d(xm+p−4	NOUN
ma-63	82	13	,	,	PUNCT
ma-63	82	14	xm	xm	PROPN
ma-63	82	15	)	)	PUNCT
ma-63	82	16	�	�	PROPN
ma-63	82	17	bd(xm+p	bd(xm+p	PROPN
ma-63	82	18	,	,	PUNCT
ma-63	82	19	xm+p−1	xm+p−1	PROPN
ma-63	82	20	)	)	PUNCT
ma-63	83	1	+	+	CCONJ
ma-63	83	2	bd(xm+p−1	bd(xm+p−1	PROPN
ma-63	83	3	,	,	PUNCT
ma-63	83	4	xm+p−2	xm+p−2	PROPN
ma-63	83	5	)	)	PUNCT
ma-63	84	1	+	+	CCONJ
ma-63	84	2	b2d(xm+p−2	b2d(xm+p−2	ADJ
ma-63	84	3	,	,	PUNCT
ma-63	84	4	xm+p−3	xm+p−3	PROPN
ma-63	84	5	)	)	PUNCT
ma-63	84	6	+	+	NUM
ma-63	84	7	b2d(xm+p−3	b2d(xm+p−3	PROPN
ma-63	84	8	,	,	PUNCT
ma-63	84	9	xm+p−4	xm+p−4	PROPN
ma-63	84	10	)	)	PUNCT
ma-63	84	11	+	+	CCONJ
ma-63	84	12	....	....	PUNCT
ma-63	85	1	+	+	SYM
ma-63	85	2	b	b	X
ma-63	85	3	p−1	p−1	PROPN
ma-63	85	4	2	2	NUM
ma-63	85	5	d(xm+3	d(xm+3	NOUN
ma-63	85	6	,	,	PUNCT
ma-63	85	7	xm+2	xm+2	NUM
ma-63	85	8	)	)	PUNCT
ma-63	86	1	+	+	SYM
ma-63	86	2	b	b	X
ma-63	86	3	p−1	p−1	PROPN
ma-63	86	4	2	2	NUM
ma-63	86	5	d(xm+2	d(xm+2	PROPN
ma-63	86	6	,	,	PUNCT
ma-63	86	7	xm+1	xm+1	NUM
ma-63	86	8	)	)	PUNCT
ma-63	87	1	+	+	SYM
ma-63	87	2	b	b	X
ma-63	87	3	p−1	p−1	PROPN
ma-63	87	4	2	2	NUM
ma-63	87	5	d(xm+1	d(xm+1	PROPN
ma-63	87	6	,	,	PUNCT
ma-63	87	7	xm	xm	PROPN
ma-63	87	8	)	)	PUNCT
ma-63	87	9	�	�	PROPN
ma-63	87	10	bψm+p−1(d(x0	bψm+p−1(d(x0	PROPN
ma-63	87	11	,	,	PUNCT
ma-63	87	12	x1	x1	PROPN
ma-63	87	13	)	)	PUNCT
ma-63	87	14	)	)	PUNCT
ma-63	88	1	+	+	CCONJ
ma-63	88	2	bψm+p−2(d(x0	bψm+p−2(d(x0	X
ma-63	88	3	,	,	PUNCT
ma-63	88	4	x1	x1	PROPN
ma-63	88	5	)	)	PUNCT
ma-63	88	6	)	)	PUNCT
ma-63	89	1	+	+	CCONJ
ma-63	89	2	...	...	PUNCT
ma-63	90	1	+	+	NUM
ma-63	90	2	b	b	X
ma-63	90	3	p−1	p−1	PROPN
ma-63	90	4	2	2	NUM
ma-63	90	5	d(x0	d(x0	NOUN
ma-63	90	6	,	,	PUNCT
ma-63	90	7	x1)since	x1)since	PROPN
ma-63	90	8	b	b	PROPN
ma-63	90	9	�	�	PROPN
ma-63	90	10	i	i	PROPN
ma-63	90	11	,	,	PUNCT
ma-63	90	12	using	use	VERB
ma-63	90	13	definition	definition	NOUN
ma-63	90	14	2.6	2.6	NUM
ma-63	90	15	we	we	PRON
ma-63	90	16	have	have	VERB
ma-63	90	17	d(xm	d(xm	NOUN
ma-63	90	18	,	,	PUNCT
ma-63	90	19	xm+p	xm+p	NUM
ma-63	90	20	)	)	PUNCT
ma-63	90	21	�	�	PROPN
ma-63	90	22	bψm+p−1(d(x0	bψm+p−1(d(x0	PROPN
ma-63	90	23	,	,	PUNCT
ma-63	90	24	x1))+bψm+p−2(d(x0	x1))+bψm+p−2(d(x0	PROPN
ma-63	90	25	,	,	PUNCT
ma-63	90	26	x1))+	x1))+	PUNCT
ma-63	90	27	...	...	PUNCT
ma-63	91	1	+	+	PUNCT
ma-63	91	2	b	b	PROPN
ma-63	91	3	p−1	p−1	PROPN
ma-63	91	4	2	2	NUM
ma-63	91	5	d(x0	d(x0	NOUN
ma-63	91	6	,	,	PUNCT
ma-63	91	7	x1)→	x1)→	PROPN
ma-63	91	8	θ	θ	PROPN
ma-63	91	9	as	as	ADP
ma-63	91	10	n	n	PROPN
ma-63	91	11	→	→	SYM
ma-63	91	12	+	+	NOUN
ma-63	91	13	∞therefore	∞therefore	NOUN
ma-63	91	14	{	{	PUNCT
ma-63	91	15	xn	xn	PUNCT
ma-63	91	16	}	}	PUNCT
ma-63	91	17	is	be	AUX
ma-63	91	18	a	a	DET
ma-63	91	19	cauchy	cauchy	ADJ
ma-63	91	20	sequence	sequence	NOUN
ma-63	91	21	in	in	ADP
ma-63	91	22	x	x	X
ma-63	91	23	.	.	PUNCT
ma-63	92	1	by	by	ADP
ma-63	92	2	the	the	DET
ma-63	92	3	completeness	completeness	NOUN
ma-63	92	4	of	of	ADP
ma-63	92	5	(	(	PUNCT
ma-63	92	6	x	x	X
ma-63	92	7	,	,	PUNCT
ma-63	92	8	a	a	PRON
ma-63	92	9	,	,	PUNCT
ma-63	92	10	d	d	NOUN
ma-63	92	11	)	)	PUNCT
ma-63	92	12	there	there	PRON
ma-63	92	13	exists	exist	VERB
ma-63	92	14	an	an	DET
ma-63	92	15	x	x	SYM
ma-63	92	16	∈	∈	PROPN
ma-63	92	17	x	x	X
ma-63	92	18	such	such	ADJ
ma-63	92	19	that	that	DET
ma-63	92	20	l	l	NOUN
ma-63	92	21	imn→∞xn	imn→∞xn	PROPN
ma-63	92	22	=	=	SYM
ma-63	92	23	l	l	NOUN
ma-63	92	24	imn→∞txn−1	imn→∞txn−1	X
ma-63	93	1	=	=	PUNCT
ma-63	93	2	x	x	X
ma-63	93	3	.	.	PUNCT
ma-63	94	1	from	from	ADP
ma-63	94	2	continuity	continuity	NOUN
ma-63	94	3	of	of	ADP
ma-63	94	4	t	t	NOUN
ma-63	94	5	and	and	CCONJ
ma-63	94	6	by	by	ADP
ma-63	94	7	uniqueness	uniqueness	NOUN
ma-63	94	8	of	of	ADP
ma-63	94	9	the	the	DET
ma-63	94	10	limit	limit	NOUN
ma-63	94	11	,	,	PUNCT
ma-63	94	12	we	we	PRON
ma-63	94	13	get	get	VERB
ma-63	94	14	tx	tx	ADJ
ma-63	94	15	=	=	SYM
ma-63	94	16	x	x	SYM
ma-63	94	17	,	,	PUNCT
ma-63	94	18	ie	ie	X
ma-63	94	19	.	.	PUNCT
ma-63	95	1	x	x	PUNCT
ma-63	95	2	is	be	AUX
ma-63	95	3	a	a	DET
ma-63	95	4	fixed	fix	VERB
ma-63	95	5	point	point	NOUN
ma-63	95	6	of	of	ADP
ma-63	95	7	t	t	PROPN
ma-63	95	8	.now	.now	PUNCT
ma-63	96	1	suppose	suppose	VERB
ma-63	96	2	that	that	SCONJ
ma-63	96	3	y	y	PROPN
ma-63	96	4	6=	6=	PROPN
ma-63	96	5	x	x	X
ma-63	96	6	is	be	AUX
ma-63	96	7	another	another	DET
ma-63	96	8	fixed	fix	VERB
ma-63	96	9	point	point	NOUN
ma-63	96	10	of	of	ADP
ma-63	96	11	t	t	PROPN
ma-63	96	12	.	.	PUNCT
ma-63	97	1	https://doi.org/10.28924/ada/ma.2.11	https://doi.org/10.28924/ada/ma.2.11	PROPN
ma-63	97	2	eur	eur	PROPN
ma-63	97	3	.	.	PUNCT
ma-63	98	1	j.	j.	PROPN
ma-63	98	2	math	math	PROPN
ma-63	98	3	.	.	PUNCT
ma-63	99	1	anal	anal	PROPN
ma-63	99	2	.	.	PUNCT
ma-63	100	1	10.28924	10.28924	NUM
ma-63	100	2	/	/	SYM
ma-63	100	3	ada	ada	PROPN
ma-63	100	4	/	/	SYM
ma-63	100	5	ma.2.11	ma.2.11	PROPN
ma-63	101	1	5from	5from	NUM
ma-63	101	2	(	(	PUNCT
ma-63	101	3	i	i	PRON
ma-63	101	4	i	i	PROPN
ma-63	101	5	i	i	PROPN
ma-63	101	6	)	)	PUNCT
ma-63	101	7	,	,	PUNCT
ma-63	101	8	there	there	PRON
ma-63	101	9	exists	exist	VERB
ma-63	101	10	z	z	NOUN
ma-63	101	11	∈	∈	PROPN
ma-63	101	12	x	x	PUNCT
ma-63	101	13	such	such	ADJ
ma-63	101	14	that	that	DET
ma-63	101	15	α(x	α(x	NOUN
ma-63	101	16	,	,	PUNCT
ma-63	101	17	z	z	NOUN
ma-63	101	18	)	)	PUNCT
ma-63	101	19	�	�	PROPN
ma-63	101	20	i	i	PRON
ma-63	101	21	and	and	CCONJ
ma-63	101	22	α(y	α(y	NOUN
ma-63	101	23	,	,	PUNCT
ma-63	101	24	z	z	NOUN
ma-63	101	25	)	)	PUNCT
ma-63	101	26	�	�	PROPN
ma-63	102	1	i	i	PRON
ma-63	102	2	.since	.since	NOUN
ma-63	102	3	t	t	PROPN
ma-63	102	4	is	be	AUX
ma-63	102	5	α−	α−	ADP
ma-63	102	6	admissible	admissible	ADJ
ma-63	102	7	,	,	PUNCT
ma-63	102	8	we	we	PRON
ma-63	102	9	have	have	VERB
ma-63	102	10	α(x	α(x	PROPN
ma-63	102	11	,	,	PUNCT
ma-63	102	12	t	t	PROPN
ma-63	102	13	nz	nz	PROPN
ma-63	102	14	)	)	PUNCT
ma-63	102	15	�	�	PROPN
ma-63	102	16	i	i	PROPN
ma-63	102	17	and	and	CCONJ
ma-63	102	18	α(y	α(y	NOUN
ma-63	102	19	,	,	PUNCT
ma-63	102	20	t	t	PROPN
ma-63	102	21	nz	nz	PROPN
ma-63	102	22	)	)	PUNCT
ma-63	102	23	�	�	PROPN
ma-63	103	1	i	i	PRON
ma-63	103	2	for	for	ADP
ma-63	103	3	all	all	DET
ma-63	103	4	n	n	PRON
ma-63	103	5	∈	∈	PRON
ma-63	103	6	n	n	CCONJ
ma-63	103	7	using	use	VERB
ma-63	103	8	(	(	PUNCT
ma-63	103	9	1	1	NUM
ma-63	103	10	)	)	PUNCT
ma-63	103	11	,	,	PUNCT
ma-63	103	12	we	we	PRON
ma-63	103	13	obtain	obtain	VERB
ma-63	103	14	d(x	d(x	PROPN
ma-63	103	15	,	,	PUNCT
ma-63	103	16	t	t	PROPN
ma-63	103	17	nz	nz	PROPN
ma-63	103	18	)	)	PUNCT
ma-63	103	19	=	=	PUNCT
ma-63	104	1	d(tx	d(tx	PROPN
ma-63	104	2	,	,	PUNCT
ma-63	104	3	t	t	PROPN
ma-63	104	4	(	(	PUNCT
ma-63	104	5	t	t	PROPN
ma-63	104	6	n−1z	n−1z	PROPN
ma-63	104	7	)	)	PUNCT
ma-63	104	8	)	)	PUNCT
ma-63	104	9	�	�	PROPN
ma-63	104	10	α(x	α(x	PROPN
ma-63	104	11	,	,	PUNCT
ma-63	104	12	t	t	PROPN
ma-63	104	13	n−1z)d(tx	n−1z)d(tx	PROPN
ma-63	104	14	,	,	PUNCT
ma-63	104	15	t	t	PROPN
ma-63	104	16	(	(	PUNCT
ma-63	104	17	t	t	PROPN
ma-63	104	18	n−1z	n−1z	PROPN
ma-63	104	19	)	)	PUNCT
ma-63	104	20	)	)	PUNCT
ma-63	104	21	�	�	PROPN
ma-63	104	22	ψn(d(x	ψn(d(x	PROPN
ma-63	104	23	,	,	PUNCT
ma-63	104	24	z))→	z))→	PROPN
ma-63	104	25	θ	θ	PROPN
ma-63	104	26	as	as	ADP
ma-63	104	27	n	n	PROPN
ma-63	104	28	→∞.	→∞.	PUNCT
ma-63	104	29	thus	thus	ADV
ma-63	104	30	,	,	PUNCT
ma-63	104	31	t	t	PROPN
ma-63	104	32	nz	nz	PROPN
ma-63	104	33	=	=	SYM
ma-63	105	1	x	x	X
ma-63	105	2	.	.	PUNCT
ma-63	106	1	similary	similary	PROPN
ma-63	106	2	t	t	PROPN
ma-63	106	3	nz	nz	PROPN
ma-63	106	4	=	=	SYM
ma-63	106	5	y	y	PROPN
ma-63	106	6	as	as	ADP
ma-63	106	7	n	n	PROPN
ma-63	106	8	→∞	→∞	PROPN
ma-63	106	9	so	so	ADV
ma-63	107	1	,	,	PUNCT
ma-63	107	2	the	the	DET
ma-63	107	3	uniqueness	uniqueness	NOUN
ma-63	107	4	of	of	ADP
ma-63	107	5	the	the	DET
ma-63	107	6	limit	limit	NOUN
ma-63	107	7	we	we	PRON
ma-63	107	8	obtain	obtain	VERB
ma-63	107	9	x	x	X
ma-63	107	10	=	=	SYM
ma-63	107	11	y	y	PROPN
ma-63	107	12	.	.	PUNCT
ma-63	107	13	�	�	PROPN
ma-63	107	14	example	example	NOUN
ma-63	107	15	3.3	3.3	NUM
ma-63	107	16	.	.	PUNCT
ma-63	108	1	let	let	VERB
ma-63	108	2	x	x	PUNCT
ma-63	108	3	=	=	PUNCT
ma-63	108	4	r	r	NOUN
ma-63	108	5	and	and	CCONJ
ma-63	108	6	a	a	DET
ma-63	108	7	=	=	X
ma-63	108	8	m2(r	m2(r	PROPN
ma-63	108	9	)	)	PUNCT
ma-63	108	10	as	as	SCONJ
ma-63	108	11	given	give	VERB
ma-63	108	12	in	in	ADP
ma-63	108	13	example	example	NOUN
ma-63	108	14	2.3	2.3	NUM
ma-63	108	15	,	,	PUNCT
ma-63	108	16	define	define	VERB
ma-63	108	17	t	t	NOUN
ma-63	108	18	:	:	PUNCT
ma-63	108	19	x	x	X
ma-63	108	20	→	→	SYM
ma-63	108	21	x	x	SYM
ma-63	108	22	,	,	PUNCT
ma-63	108	23	by	by	ADP
ma-63	108	24	tx	tx	PROPN
ma-63	108	25	=	=	PUNCT
ma-63	109	1	x	x	SYM
ma-63	109	2	3and	3and	NUM
ma-63	109	3	α	α	NOUN
ma-63	109	4	:	:	PUNCT
ma-63	109	5	x	x	PROPN
ma-63	109	6	×x	×x	X
ma-63	109	7	→	→	SYM
ma-63	109	8	m2(r	m2(r	ADJ
ma-63	109	9	)	)	PUNCT
ma-63	109	10	such	such	ADJ
ma-63	109	11	that	that	SCONJ
ma-63	109	12	α(x	α(x	PROPN
ma-63	109	13	,	,	PUNCT
ma-63	109	14	y	y	PROPN
ma-63	109	15	)	)	PUNCT
ma-63	109	16	=	=	PRON
ma-63	109	17	(	(	PUNCT
ma-63	109	18	|x	|x	X
ma-63	109	19	−	−	PROPN
ma-63	110	1	y	y	PROPN
ma-63	111	1	|	|	ADV
ma-63	111	2	0	0	NUM
ma-63	111	3	0	0	NUM
ma-63	111	4	0	0	NUM
ma-63	111	5	)	)	PUNCT
ma-63	112	1	thus	thus	ADV
ma-63	112	2	,	,	PUNCT
ma-63	112	3	t	t	PROPN
ma-63	112	4	is	be	AUX
ma-63	112	5	α−	α−	ADP
ma-63	112	6	admissible	admissible	ADJ
ma-63	112	7	,	,	PUNCT
ma-63	112	8	and	and	CCONJ
ma-63	112	9	ψ	ψ	X
ma-63	112	10	:	:	PUNCT
ma-63	112	11	m2(r)+	m2(r)+	PROPN
ma-63	112	12	→	→	SYM
ma-63	112	13	m2(r)+	m2(r)+	PROPN
ma-63	112	14	,	,	PUNCT
ma-63	112	15	ψ(a	ψ(a	PROPN
ma-63	112	16	)	)	PUNCT
ma-63	113	1	=	=	PRON
ma-63	113	2	(	(	PUNCT
ma-63	113	3	a2	a2	PROPN
ma-63	113	4	0	0	NUM
ma-63	113	5	0	0	NUM
ma-63	113	6	a2	a2	PROPN
ma-63	113	7	)	)	PUNCT
ma-63	113	8	∀a	∀a	NOUN
ma-63	114	1	∈	∈	NOUN
ma-63	114	2	(	(	PUNCT
ma-63	114	3	r)+	r)+	PROPN
ma-63	114	4	.	.	PUNCT
ma-63	115	1	this	this	PRON
ma-63	115	2	is	be	AUX
ma-63	115	3	clear	clear	ADJ
ma-63	115	4	that	that	SCONJ
ma-63	115	5	t	t	PROPN
ma-63	115	6	is	be	AUX
ma-63	115	7	α−	α−	AUX
ma-63	115	8	ψ−	ψ−	VERB
ma-63	115	9	contractive	contractive	ADJ
ma-63	115	10	mapping	mapping	NOUN
ma-63	115	11	and	and	CCONJ
ma-63	115	12	satisfies	satisfie	NOUN
ma-63	115	13	α(x	α(x	PROPN
ma-63	115	14	,	,	PUNCT
ma-63	115	15	y)d(tx	y)d(tx	NUM
ma-63	115	16	,	,	PUNCT
ma-63	115	17	t	t	PROPN
ma-63	115	18	y	y	PROPN
ma-63	115	19	)	)	PUNCT
ma-63	115	20	�	�	PROPN
ma-63	116	1	ψ(d(x	ψ(d(x	PROPN
ma-63	116	2	,	,	PUNCT
ma-63	116	3	y	y	NOUN
ma-63	116	4	)	)	PUNCT
ma-63	116	5	)	)	PUNCT
ma-63	116	6	,	,	PUNCT
ma-63	116	7	for	for	ADP
ma-63	116	8	all	all	DET
ma-63	116	9	x	x	NOUN
ma-63	116	10	,	,	PUNCT
ma-63	116	11	y	y	PROPN
ma-63	116	12	∈	∈	PROPN
ma-63	116	13	x	x	PUNCT
ma-63	116	14	theorem	theorem	VERB
ma-63	116	15	3.4	3.4	NUM
ma-63	116	16	.	.	PUNCT
ma-63	117	1	let	let	AUX
ma-63	117	2	(	(	PUNCT
ma-63	117	3	x	x	X
ma-63	117	4	,	,	PUNCT
ma-63	117	5	a	a	PRON
ma-63	117	6	,	,	PUNCT
ma-63	117	7	d	d	NOUN
ma-63	117	8	)	)	PUNCT
ma-63	117	9	be	be	AUX
ma-63	117	10	a	a	DET
ma-63	117	11	complete	complete	ADJ
ma-63	117	12	c∗-algebra	c∗-algebra	NOUN
ma-63	117	13	valued	value	VERB
ma-63	117	14	rectangular	rectangular	ADJ
ma-63	117	15	b−	b−	NOUN
ma-63	117	16	metric	metric	ADJ
ma-63	117	17	space	space	NOUN
ma-63	117	18	and	and	CCONJ
ma-63	117	19	let	let	VERB
ma-63	117	20	t	t	NOUN
ma-63	117	21	:	:	PUNCT
ma-63	117	22	x	x	X
ma-63	117	23	→	→	PUNCT
ma-63	117	24	x	x	PUNCT
ma-63	117	25	be	be	AUX
ma-63	117	26	a	a	DET
ma-63	117	27	α	α	NOUN
ma-63	117	28	,	,	PUNCT
ma-63	117	29	ψ−	ψ−	VERB
ma-63	117	30	contractive	contractive	ADJ
ma-63	117	31	mapping	mapping	NOUN
ma-63	117	32	of	of	ADP
ma-63	117	33	kannan	kannan	PROPN
ma-63	117	34	type	type	PROPN
ma-63	117	35	ie	ie	X
ma-63	117	36	,	,	PUNCT
ma-63	117	37	α(x	α(x	NOUN
ma-63	117	38	,	,	PUNCT
ma-63	117	39	y)d(tx	y)d(tx	NUM
ma-63	117	40	,	,	PUNCT
ma-63	117	41	t	t	PROPN
ma-63	117	42	y	y	PROPN
ma-63	117	43	)	)	PUNCT
ma-63	117	44	�	�	PROPN
ma-63	117	45	ψ(d(tx	ψ(d(tx	NOUN
ma-63	117	46	,	,	PUNCT
ma-63	117	47	x	x	PRON
ma-63	117	48	)	)	PUNCT
ma-63	118	1	+	+	CCONJ
ma-63	118	2	d(ty	d(ty	PROPN
ma-63	118	3	,	,	PUNCT
ma-63	118	4	y	y	NOUN
ma-63	118	5	)	)	PUNCT
ma-63	118	6	)	)	PUNCT
ma-63	118	7	(	(	PUNCT
ma-63	118	8	3.3	3.3	NUM
ma-63	118	9	)	)	PUNCT
ma-63	118	10	for	for	ADP
ma-63	118	11	all	all	DET
ma-63	118	12	x	x	NOUN
ma-63	118	13	,	,	PUNCT
ma-63	118	14	y	y	PROPN
ma-63	118	15	∈	∈	PROPN
ma-63	118	16	x	x	INTJ
ma-63	119	1	where	where	SCONJ
ma-63	119	2	ψ	ψ	VERB
ma-63	119	3	∈	∈	PROPN
ma-63	119	4	ψ	ψ	NOUN
ma-63	119	5	and	and	CCONJ
ma-63	119	6	α	α	NOUN
ma-63	119	7	:	:	PUNCT
ma-63	119	8	x	x	X
ma-63	119	9	×x	×x	ADP
ma-63	119	10	→	→	PUNCT
ma-63	119	11	a+	a+	PUNCT
ma-63	119	12	and	and	CCONJ
ma-63	119	13	the	the	DET
ma-63	119	14	following	follow	VERB
ma-63	119	15	conditions	condition	NOUN
ma-63	119	16	holds:(i	holds:(i	NOUN
ma-63	119	17	)	)	PUNCT
ma-63	119	18	t	t	PROPN
ma-63	119	19	is	be	AUX
ma-63	119	20	α−	α−	ADP
ma-63	119	21	admissible(ii	admissible(ii	NOUN
ma-63	119	22	)	)	PUNCT
ma-63	119	23	there	there	ADV
ma-63	119	24	exists	exist	VERB
ma-63	119	25	x0	x0	PROPN
ma-63	119	26	∈	∈	PROPN
ma-63	119	27	x	x	PUNCT
ma-63	119	28	such	such	ADJ
ma-63	119	29	that	that	DET
ma-63	119	30	α(x0	α(x0	NOUN
ma-63	119	31	,	,	PUNCT
ma-63	119	32	t	t	PROPN
ma-63	119	33	x0	x0	PROPN
ma-63	119	34	)	)	PUNCT
ma-63	119	35	�	�	PROPN
ma-63	119	36	i(iii	i(iii	PROPN
ma-63	119	37	)	)	PUNCT
ma-63	119	38	t	t	PROPN
ma-63	119	39	is	be	AUX
ma-63	119	40	continuous	continuous	ADJ
ma-63	119	41	then	then	ADV
ma-63	119	42	,	,	PUNCT
ma-63	119	43	t	t	PROPN
ma-63	119	44	has	have	VERB
ma-63	119	45	a	a	DET
ma-63	119	46	fixed	fix	VERB
ma-63	119	47	point	point	NOUN
ma-63	119	48	in	in	ADP
ma-63	119	49	x	x	X
ma-63	119	50	.	.	PUNCT
ma-63	120	1	proof	proof	NOUN
ma-63	120	2	.	.	PUNCT
ma-63	121	1	by	by	ADP
ma-63	121	2	(	(	PUNCT
ma-63	121	3	3.3	3.3	NUM
ma-63	121	4	)	)	PUNCT
ma-63	121	5	,	,	PUNCT
ma-63	121	6	we	we	PRON
ma-63	121	7	obtain	obtain	VERB
ma-63	121	8	d(xn	d(xn	NOUN
ma-63	121	9	,	,	PUNCT
ma-63	121	10	xn+1	xn+1	NUM
ma-63	121	11	)	)	PUNCT
ma-63	121	12	=	=	SYM
ma-63	121	13	d(txn−1	d(txn−1	PROPN
ma-63	121	14	,	,	PUNCT
ma-63	121	15	t	t	PROPN
ma-63	121	16	xn	xn	PROPN
ma-63	121	17	)	)	PUNCT
ma-63	121	18	�	�	PROPN
ma-63	121	19	α(xn−1	α(xn−1	NUM
ma-63	121	20	,	,	PUNCT
ma-63	121	21	xn)d(txn−1	xn)d(txn−1	PROPN
ma-63	121	22	,	,	PUNCT
ma-63	121	23	t	t	PROPN
ma-63	121	24	xn	xn	PROPN
ma-63	121	25	)	)	PUNCT
ma-63	121	26	�	�	PROPN
ma-63	121	27	ψ(d(txn−1	ψ(d(txn−1	PROPN
ma-63	121	28	,	,	PUNCT
ma-63	121	29	xn−1	xn−1	PROPN
ma-63	121	30	)	)	PUNCT
ma-63	122	1	+	+	CCONJ
ma-63	122	2	d(txn	d(txn	PROPN
ma-63	122	3	,	,	PUNCT
ma-63	122	4	xn	xn	PROPN
ma-63	122	5	)	)	PUNCT
ma-63	122	6	)	)	PUNCT
ma-63	123	1	=	=	PUNCT
ma-63	123	2	ψ(d(xn	ψ(d(xn	NOUN
ma-63	123	3	,	,	PUNCT
ma-63	123	4	xn−1	xn−1	PROPN
ma-63	123	5	)	)	PUNCT
ma-63	124	1	+	+	PUNCT
ma-63	125	1	d(xn+1	d(xn+1	ADJ
ma-63	125	2	,	,	PUNCT
ma-63	125	3	xn	xn	PROPN
ma-63	125	4	)	)	PUNCT
ma-63	125	5	)	)	PUNCT
ma-63	126	1	=	=	PUNCT
ma-63	126	2	ψ(d(xn	ψ(d(xn	NOUN
ma-63	126	3	,	,	PUNCT
ma-63	126	4	xn−1	xn−1	PROPN
ma-63	126	5	)	)	PUNCT
ma-63	126	6	)	)	PUNCT
ma-63	127	1	+	+	PUNCT
ma-63	128	1	ψ(d(xn+1	ψ(d(xn+1	ADJ
ma-63	128	2	,	,	PUNCT
ma-63	128	3	xn	xn	PROPN
ma-63	128	4	)	)	PUNCT
ma-63	128	5	)	)	PUNCT
ma-63	129	1	(	(	PUNCT
ma-63	129	2	i	i	PRON
ma-63	129	3	−	−	PROPN
ma-63	129	4	ψ)(d(xn	ψ)(d(xn	NOUN
ma-63	129	5	,	,	PUNCT
ma-63	129	6	xn−1	xn−1	PROPN
ma-63	129	7	)	)	PUNCT
ma-63	129	8	)	)	PUNCT
ma-63	129	9	�	�	PROPN
ma-63	129	10	ψ(d(xn	ψ(d(xn	NOUN
ma-63	129	11	,	,	PUNCT
ma-63	129	12	xn−1	xn−1	PROPN
ma-63	129	13	)	)	PUNCT
ma-63	129	14	)	)	PUNCT
ma-63	130	1	from	from	ADP
ma-63	130	2	lemma	lemma	PROPN
ma-63	130	3	2.1	2.1	NUM
ma-63	130	4	and	and	CCONJ
ma-63	130	5	definition	definition	NOUN
ma-63	130	6	2.6	2.6	NUM
ma-63	130	7	,	,	PUNCT
ma-63	130	8	we	we	PRON
ma-63	130	9	obtain	obtain	VERB
ma-63	130	10	https://doi.org/10.28924/ada/ma.2.11	https://doi.org/10.28924/ada/ma.2.11	NOUN
ma-63	130	11	eur	eur	NOUN
ma-63	130	12	.	.	PUNCT
ma-63	131	1	j.	j.	PROPN
ma-63	131	2	math	math	PROPN
ma-63	131	3	.	.	PUNCT
ma-63	132	1	anal	anal	PROPN
ma-63	132	2	.	.	PUNCT
ma-63	133	1	10.28924	10.28924	NUM
ma-63	133	2	/	/	SYM
ma-63	133	3	ada	ada	PROPN
ma-63	133	4	/	/	SYM
ma-63	133	5	ma.2.11	ma.2.11	PROPN
ma-63	133	6	6	6	NUM
ma-63	133	7	d(xn	d(xn	PROPN
ma-63	133	8	,	,	PUNCT
ma-63	133	9	xn+1	xn+1	NUM
ma-63	133	10	)	)	PUNCT
ma-63	133	11	�	�	PROPN
ma-63	133	12	(	(	PUNCT
ma-63	133	13	i	i	PRON
ma-63	133	14	−	−	PROPN
ma-63	133	15	ψ)−1ψ(d(xn	ψ)−1ψ(d(xn	PROPN
ma-63	133	16	,	,	PUNCT
ma-63	133	17	xn−1	xn−1	PROPN
ma-63	133	18	)	)	PUNCT
ma-63	133	19	)	)	PUNCT
ma-63	134	1	=	=	SYM
ma-63	135	1	φ(d(xn	φ(d(xn	NOUN
ma-63	135	2	,	,	PUNCT
ma-63	135	3	xn−1	xn−1	PROPN
ma-63	135	4	)	)	PUNCT
ma-63	135	5	)	)	PUNCT
ma-63	135	6	where	where	SCONJ
ma-63	135	7	φ	φ	PROPN
ma-63	135	8	=	=	SYM
ma-63	135	9	(	(	PUNCT
ma-63	135	10	i	i	PRON
ma-63	135	11	−	−	PROPN
ma-63	135	12	ψ)−1ψ	ψ)−1ψ	VERB
ma-63	135	13	therefore	therefore	ADV
ma-63	135	14	d(xn	d(xn	PROPN
ma-63	135	15	,	,	PUNCT
ma-63	135	16	xn+1	xn+1	NUM
ma-63	135	17	)	)	PUNCT
ma-63	135	18	�	�	PROPN
ma-63	135	19	φn(d(x0	φn(d(x0	PROPN
ma-63	135	20	,	,	PUNCT
ma-63	135	21	x1))∀n	x1))∀n	PROPN
ma-63	135	22	∈	∈	PROPN
ma-63	135	23	n	n	PROPN
ma-63	135	24	for	for	ADP
ma-63	135	25	any	any	DET
ma-63	135	26	m	m	PROPN
ma-63	135	27	≥	≥	NOUN
ma-63	135	28	1	1	NUM
ma-63	135	29	and	and	CCONJ
ma-63	135	30	p	p	PRON
ma-63	135	31	≥	≥	NUM
ma-63	135	32	1	1	NUM
ma-63	135	33	similary	similary	ADJ
ma-63	135	34	in	in	ADP
ma-63	135	35	theorem	theorem	ADJ
ma-63	135	36	3.1	3.1	NUM
ma-63	135	37	we	we	PRON
ma-63	135	38	have	have	AUX
ma-63	135	39	d(xm	d(xm	NOUN
ma-63	135	40	,	,	PUNCT
ma-63	135	41	xm+p	xm+p	NUM
ma-63	135	42	)	)	PUNCT
ma-63	135	43	�	�	PROPN
ma-63	135	44	bψm+p−1(d(x0	bψm+p−1(d(x0	PROPN
ma-63	135	45	,	,	PUNCT
ma-63	135	46	x1))+bψm+p−2(d(x0	x1))+bψm+p−2(d(x0	PROPN
ma-63	135	47	,	,	PUNCT
ma-63	135	48	x1))+	x1))+	PROPN
ma-63	135	49	...	...	PUNCT
ma-63	135	50	+b	+b	X
ma-63	136	1	p−1	p−1	PROPN
ma-63	136	2	2	2	NUM
ma-63	136	3	d(x0	d(x0	NOUN
ma-63	136	4	,	,	PUNCT
ma-63	136	5	x1)→	x1)→	PROPN
ma-63	136	6	θ	θ	PROPN
ma-63	136	7	as	as	ADP
ma-63	136	8	n	n	NOUN
ma-63	136	9	→	→	SYM
ma-63	136	10	+	+	NOUN
ma-63	136	11	∞.thus	∞.thus	PART
ma-63	136	12	{	{	PUNCT
ma-63	136	13	xn	xn	X
ma-63	136	14	}	}	PUNCT
ma-63	136	15	is	be	AUX
ma-63	136	16	a	a	DET
ma-63	136	17	cauchy	cauchy	ADJ
ma-63	136	18	sequence	sequence	NOUN
ma-63	136	19	in	in	ADP
ma-63	136	20	x	x	X
ma-63	136	21	.	.	PUNCT
ma-63	137	1	by	by	ADP
ma-63	137	2	the	the	DET
ma-63	137	3	completeness	completeness	NOUN
ma-63	137	4	of	of	ADP
ma-63	137	5	(	(	PUNCT
ma-63	137	6	x	x	X
ma-63	137	7	,	,	PUNCT
ma-63	137	8	a	a	PRON
ma-63	137	9	,	,	PUNCT
ma-63	137	10	d	d	NOUN
ma-63	137	11	)	)	PUNCT
ma-63	137	12	,	,	PUNCT
ma-63	137	13	there	there	PRON
ma-63	137	14	exists	exist	VERB
ma-63	137	15	x	x	X
ma-63	137	16	∈	∈	PROPN
ma-63	137	17	xsuch	xsuch	PROPN
ma-63	137	18	that	that	SCONJ
ma-63	137	19	l	l	NOUN
ma-63	137	20	imn→∞xn	imn→∞xn	PROPN
ma-63	137	21	=	=	PUNCT
ma-63	137	22	l	l	NOUN
ma-63	137	23	imn→∞txn−1	imn→∞txn−1	X
ma-63	138	1	=	=	PUNCT
ma-63	138	2	x	x	X
ma-63	138	3	.	.	PUNCT
ma-63	139	1	the	the	DET
ma-63	139	2	continuity	continuity	NOUN
ma-63	139	3	of	of	ADP
ma-63	139	4	t	t	PROPN
ma-63	139	5	gives	give	VERB
ma-63	139	6	that	that	SCONJ
ma-63	139	7	x	x	PRON
ma-63	139	8	is	be	AUX
ma-63	139	9	a	a	DET
ma-63	139	10	fixed	fix	VERB
ma-63	139	11	point	point	NOUN
ma-63	139	12	of	of	ADP
ma-63	139	13	t	t	PROPN
ma-63	139	14	.to	.to	PUNCT
ma-63	140	1	prove	prove	VERB
ma-63	140	2	that	that	SCONJ
ma-63	140	3	x	x	PRON
ma-63	140	4	is	be	AUX
ma-63	140	5	the	the	DET
ma-63	140	6	unique	unique	ADJ
ma-63	140	7	fixed	fix	VERB
ma-63	140	8	point	point	NOUN
ma-63	140	9	,	,	PUNCT
ma-63	140	10	we	we	PRON
ma-63	140	11	suppose	suppose	VERB
ma-63	140	12	that	that	SCONJ
ma-63	140	13	y	y	PROPN
ma-63	140	14	∈	∈	PROPN
ma-63	140	15	x	x	PUNCT
ma-63	140	16	is	be	AUX
ma-63	140	17	another	another	DET
ma-63	140	18	fixed	fix	VERB
ma-63	140	19	point	point	NOUN
ma-63	140	20	of	of	ADP
ma-63	140	21	t	t	PROPN
ma-63	140	22	.then	.then	PUNCT
ma-63	140	23	θ	θ	PROPN
ma-63	140	24	�	�	PROPN
ma-63	140	25	d(x	d(x	PROPN
ma-63	140	26	,	,	PUNCT
ma-63	140	27	y	y	NOUN
ma-63	140	28	)	)	PUNCT
ma-63	140	29	=	=	PUNCT
ma-63	141	1	d(tx	d(tx	PROPN
ma-63	141	2	,	,	PUNCT
ma-63	141	3	t	t	PROPN
ma-63	141	4	y	y	PROPN
ma-63	141	5	)	)	PUNCT
ma-63	141	6	�	�	PROPN
ma-63	141	7	α(x	α(x	PROPN
ma-63	141	8	,	,	PUNCT
ma-63	141	9	y)d(tx	y)d(tx	NUM
ma-63	141	10	,	,	PUNCT
ma-63	141	11	t	t	PROPN
ma-63	141	12	y	y	PROPN
ma-63	141	13	)	)	PUNCT
ma-63	141	14	�	�	PROPN
ma-63	141	15	ψ(d(tx	ψ(d(tx	NOUN
ma-63	141	16	,	,	PUNCT
ma-63	141	17	x	x	PRON
ma-63	141	18	)	)	PUNCT
ma-63	141	19	+	+	CCONJ
ma-63	141	20	d(ty	d(ty	PROPN
ma-63	141	21	,	,	PUNCT
ma-63	141	22	y	y	NOUN
ma-63	141	23	)	)	PUNCT
ma-63	141	24	)	)	PUNCT
ma-63	142	1	=	=	PUNCT
ma-63	142	2	ψ(d(x	ψ(d(x	NOUN
ma-63	142	3	,	,	PUNCT
ma-63	142	4	x	x	PRON
ma-63	142	5	)	)	PUNCT
ma-63	143	1	+	+	CCONJ
ma-63	143	2	d(y	d(y	NOUN
ma-63	143	3	,	,	PUNCT
ma-63	143	4	y	y	NOUN
ma-63	143	5	)	)	PUNCT
ma-63	143	6	)	)	PUNCT
ma-63	144	1	=	=	PUNCT
ma-63	144	2	θ	θ	X
ma-63	144	3	hence	hence	ADV
ma-63	144	4	x	x	X
ma-63	144	5	=	=	SYM
ma-63	144	6	y	y	PROPN
ma-63	144	7	.therefore	.therefore	ADV
ma-63	145	1	the	the	DET
ma-63	145	2	fixed	fix	VERB
ma-63	145	3	point	point	NOUN
ma-63	145	4	is	be	AUX
ma-63	145	5	unique	unique	ADJ
ma-63	145	6	.	.	PUNCT
ma-63	146	1	�	�	PROPN
ma-63	146	2	theorem	theorem	VERB
ma-63	146	3	3.5	3.5	NUM
ma-63	146	4	.	.	PUNCT
ma-63	147	1	let	let	AUX
ma-63	147	2	(	(	PUNCT
ma-63	147	3	x	x	X
ma-63	147	4	,	,	PUNCT
ma-63	147	5	a	a	PRON
ma-63	147	6	,	,	PUNCT
ma-63	147	7	d	d	NOUN
ma-63	147	8	)	)	PUNCT
ma-63	147	9	be	be	AUX
ma-63	147	10	a	a	DET
ma-63	147	11	complete	complete	ADJ
ma-63	147	12	c∗-algebra	c∗-algebra	NOUN
ma-63	147	13	valued	value	VERB
ma-63	147	14	rectangular	rectangular	ADJ
ma-63	147	15	b−	b−	NOUN
ma-63	147	16	metric	metric	ADJ
ma-63	147	17	space	space	NOUN
ma-63	147	18	and	and	CCONJ
ma-63	147	19	let	let	VERB
ma-63	147	20	t	t	NOUN
ma-63	147	21	:	:	PUNCT
ma-63	147	22	x	x	X
ma-63	147	23	→	→	PUNCT
ma-63	147	24	x	x	PUNCT
ma-63	147	25	be	be	AUX
ma-63	147	26	a	a	DET
ma-63	147	27	α	α	NOUN
ma-63	147	28	,	,	PUNCT
ma-63	147	29	ψ−	ψ−	VERB
ma-63	147	30	contractive	contractive	ADJ
ma-63	147	31	mapping	mapping	NOUN
ma-63	147	32	of	of	ADP
ma-63	147	33	banach	banach	NOUN
ma-63	147	34	-	-	PUNCT
ma-63	147	35	kannan	kannan	PROPN
ma-63	147	36	type	type	NOUN
ma-63	147	37	ie	ie	X
ma-63	147	38	,	,	PUNCT
ma-63	147	39	α(x	α(x	NOUN
ma-63	147	40	,	,	PUNCT
ma-63	147	41	y)d(tx	y)d(tx	NUM
ma-63	147	42	,	,	PUNCT
ma-63	147	43	t	t	PROPN
ma-63	147	44	y	y	PROPN
ma-63	147	45	)	)	PUNCT
ma-63	147	46	�	�	PROPN
ma-63	147	47	ψ(d(x	ψ(d(x	PROPN
ma-63	147	48	,	,	PUNCT
ma-63	147	49	y	y	NOUN
ma-63	147	50	)	)	PUNCT
ma-63	148	1	+	+	CCONJ
ma-63	149	1	d(tx	d(tx	ADJ
ma-63	149	2	,	,	PUNCT
ma-63	149	3	x	x	PRON
ma-63	149	4	)	)	PUNCT
ma-63	149	5	+	+	CCONJ
ma-63	149	6	d(ty	d(ty	PROPN
ma-63	149	7	,	,	PUNCT
ma-63	149	8	y	y	NOUN
ma-63	149	9	)	)	PUNCT
ma-63	149	10	)	)	PUNCT
ma-63	149	11	(	(	PUNCT
ma-63	149	12	3.4	3.4	NUM
ma-63	149	13	)	)	PUNCT
ma-63	149	14	for	for	ADP
ma-63	149	15	all	all	DET
ma-63	149	16	x	x	NOUN
ma-63	149	17	,	,	PUNCT
ma-63	149	18	y	y	PROPN
ma-63	149	19	∈	∈	PROPN
ma-63	149	20	x	x	INTJ
ma-63	149	21	where	where	SCONJ
ma-63	149	22	ψ	ψ	VERB
ma-63	149	23	∈	∈	PROPN
ma-63	149	24	ψ	ψ	NOUN
ma-63	149	25	and	and	CCONJ
ma-63	149	26	α	α	NOUN
ma-63	149	27	:	:	PUNCT
ma-63	149	28	x	x	X
ma-63	149	29	×x	×x	ADP
ma-63	149	30	→	→	SYM
ma-63	149	31	a+	a+	PUNCT
ma-63	149	32	such	such	ADJ
ma-63	149	33	that	that	DET
ma-63	149	34	ψ(1−	ψ(1−	PROPN
ma-63	149	35	ψ)−1	ψ)−1	X
ma-63	149	36	�	�	PROPN
ma-63	149	37	1	1	NUM
ma-63	149	38	2i	2i	NUM
ma-63	149	39	,	,	PUNCT
ma-63	149	40	and	and	CCONJ
ma-63	149	41	the	the	DET
ma-63	149	42	following	follow	VERB
ma-63	149	43	conditions	condition	NOUN
ma-63	149	44	holds:(i	holds:(i	NOUN
ma-63	149	45	)	)	PUNCT
ma-63	149	46	t	t	PROPN
ma-63	149	47	is	be	AUX
ma-63	149	48	α−	α−	ADP
ma-63	149	49	admissible(ii	admissible(ii	NOUN
ma-63	149	50	)	)	PUNCT
ma-63	149	51	there	there	ADV
ma-63	149	52	exists	exist	VERB
ma-63	149	53	x0	x0	PROPN
ma-63	149	54	∈	∈	PROPN
ma-63	149	55	x	x	PUNCT
ma-63	149	56	such	such	ADJ
ma-63	149	57	that	that	DET
ma-63	149	58	α(x0	α(x0	NOUN
ma-63	149	59	,	,	PUNCT
ma-63	149	60	t	t	PROPN
ma-63	149	61	x0	x0	PROPN
ma-63	149	62	)	)	PUNCT
ma-63	149	63	�	�	PROPN
ma-63	149	64	i(iii	i(iii	PROPN
ma-63	149	65	)	)	PUNCT
ma-63	149	66	t	t	PROPN
ma-63	149	67	is	be	AUX
ma-63	149	68	continuous	continuous	ADJ
ma-63	149	69	then	then	ADV
ma-63	149	70	,	,	PUNCT
ma-63	149	71	t	t	PROPN
ma-63	149	72	has	have	VERB
ma-63	149	73	a	a	DET
ma-63	149	74	fixed	fix	VERB
ma-63	149	75	point	point	NOUN
ma-63	149	76	in	in	ADP
ma-63	149	77	x	x	NOUN
ma-63	149	78	proof	proof	NOUN
ma-63	149	79	.	.	PUNCT
ma-63	150	1	using	use	VERB
ma-63	150	2	(	(	PUNCT
ma-63	150	3	3.4	3.4	NUM
ma-63	150	4	)	)	PUNCT
ma-63	150	5	,	,	PUNCT
ma-63	150	6	we	we	PRON
ma-63	150	7	get	get	VERB
ma-63	150	8	d(xn	d(xn	NOUN
ma-63	150	9	,	,	PUNCT
ma-63	150	10	xn+1	xn+1	NUM
ma-63	150	11	)	)	PUNCT
ma-63	150	12	=	=	SYM
ma-63	150	13	d(txn−1	d(txn−1	PROPN
ma-63	150	14	,	,	PUNCT
ma-63	150	15	t	t	PROPN
ma-63	150	16	xn	xn	PROPN
ma-63	150	17	)	)	PUNCT
ma-63	150	18	�	�	PROPN
ma-63	150	19	α(xn−1	α(xn−1	NUM
ma-63	150	20	,	,	PUNCT
ma-63	150	21	xn)d(txn−1	xn)d(txn−1	PROPN
ma-63	150	22	,	,	PUNCT
ma-63	150	23	t	t	PROPN
ma-63	150	24	xn	xn	PROPN
ma-63	150	25	)	)	PUNCT
ma-63	150	26	�	�	PROPN
ma-63	150	27	ψ(d(xn−1	ψ(d(xn−1	PROPN
ma-63	150	28	,	,	PUNCT
ma-63	150	29	xn	xn	PUNCT
ma-63	150	30	)	)	PUNCT
ma-63	151	1	+	+	X
ma-63	151	2	d(txn−1	d(txn−1	ADJ
ma-63	151	3	,	,	PUNCT
ma-63	151	4	xn−1	xn−1	PROPN
ma-63	151	5	)	)	PUNCT
ma-63	151	6	+	+	CCONJ
ma-63	152	1	d(txn	d(txn	PROPN
ma-63	152	2	,	,	PUNCT
ma-63	152	3	xn	xn	PROPN
ma-63	152	4	)	)	PUNCT
ma-63	152	5	)	)	PUNCT
ma-63	153	1	=	=	SYM
ma-63	153	2	ψ(d(xn−1	ψ(d(xn−1	PROPN
ma-63	153	3	,	,	PUNCT
ma-63	153	4	xn)2i	xn)2i	NOUN
ma-63	154	1	+	+	CCONJ
ma-63	154	2	d(xn	d(xn	PROPN
ma-63	154	3	,	,	PUNCT
ma-63	154	4	xn+1	xn+1	NUM
ma-63	154	5	)	)	PUNCT
ma-63	154	6	)	)	PUNCT
ma-63	154	7	⇒	⇒	NOUN
ma-63	154	8	(	(	PUNCT
ma-63	154	9	i	i	PRON
ma-63	154	10	−	−	PROPN
ma-63	154	11	ψ)(d(xn	ψ)(d(xn	NOUN
ma-63	154	12	,	,	PUNCT
ma-63	154	13	xn+1	xn+1	NUM
ma-63	154	14	)	)	PUNCT
ma-63	154	15	)	)	PUNCT
ma-63	154	16	�	�	PROPN
ma-63	154	17	2iψ(d(xn	2iψ(d(xn	NUM
ma-63	154	18	,	,	PUNCT
ma-63	154	19	xn−1	xn−1	PROPN
ma-63	154	20	)	)	PUNCT
ma-63	154	21	)	)	PUNCT
ma-63	154	22	⇒	⇒	VERB
ma-63	154	23	d(xn	d(xn	PROPN
ma-63	154	24	,	,	PUNCT
ma-63	154	25	xn+1	xn+1	NUM
ma-63	154	26	)	)	PUNCT
ma-63	154	27	�	�	PROPN
ma-63	154	28	2i(i	2i(i	NUM
ma-63	155	1	−	−	NOUN
ma-63	155	2	ψ)−1ψ(d(xn	ψ)−1ψ(d(xn	PROPN
ma-63	155	3	,	,	PUNCT
ma-63	155	4	xn−1	xn−1	PROPN
ma-63	155	5	)	)	PUNCT
ma-63	155	6	)	)	PUNCT
ma-63	155	7	�	�	PROPN
ma-63	155	8	φ(d(xn	φ(d(xn	PROPN
ma-63	155	9	,	,	PUNCT
ma-63	155	10	xn−1)).where	xn−1)).where	PROPN
ma-63	155	11	https://doi.org/10.28924/ada/ma.2.11	https://doi.org/10.28924/ada/ma.2.11	PROPN
ma-63	155	12	eur	eur	PROPN
ma-63	155	13	.	.	PUNCT
ma-63	156	1	j.	j.	PROPN
ma-63	156	2	math	math	PROPN
ma-63	156	3	.	.	PUNCT
ma-63	157	1	anal	anal	PROPN
ma-63	157	2	.	.	PUNCT
ma-63	158	1	10.28924	10.28924	NUM
ma-63	158	2	/	/	SYM
ma-63	158	3	ada	ada	PROPN
ma-63	158	4	/	/	SYM
ma-63	158	5	ma.2.11	ma.2.11	PROPN
ma-63	158	6	7	7	NUM
ma-63	158	7	ϕ	ϕ	NOUN
ma-63	158	8	=	=	SYM
ma-63	158	9	2i(i	2i(i	NUM
ma-63	158	10	−	−	PROPN
ma-63	158	11	ψ)−1ψ	ψ)−1ψ	NOUN
ma-63	158	12	.	.	PUNCT
ma-63	159	1	then	then	ADV
ma-63	159	2	d(xn	d(xn	PROPN
ma-63	159	3	,	,	PUNCT
ma-63	159	4	xn+1	xn+1	NUM
ma-63	159	5	)	)	PUNCT
ma-63	159	6	�	�	PROPN
ma-63	159	7	φn(d(x0	φn(d(x0	PROPN
ma-63	159	8	,	,	PUNCT
ma-63	159	9	x1	x1	PROPN
ma-63	159	10	)	)	PUNCT
ma-63	159	11	.	.	PUNCT
ma-63	160	1	we	we	PRON
ma-63	160	2	refer	refer	VERB
ma-63	160	3	to	to	ADP
ma-63	160	4	the	the	DET
ma-63	160	5	proof	proof	NOUN
ma-63	160	6	of	of	ADP
ma-63	160	7	the	the	DET
ma-63	160	8	theorem	theorem	ADJ
ma-63	160	9	3.1	3.1	NUM
ma-63	160	10	we	we	PRON
ma-63	160	11	get	get	VERB
ma-63	160	12	that	that	PRON
ma-63	160	13	x	x	PRON
ma-63	160	14	is	be	AUX
ma-63	160	15	a	a	DET
ma-63	160	16	fixed	fix	VERB
ma-63	160	17	point	point	NOUN
ma-63	160	18	of	of	ADP
ma-63	160	19	t	t	PROPN
ma-63	160	20	.	.	PUNCT
ma-63	161	1	now	now	ADV
ma-63	161	2	,	,	PUNCT
ma-63	161	3	if	if	SCONJ
ma-63	161	4	y	y	PROPN
ma-63	161	5	6=	6=	PROPN
ma-63	161	6	x	x	SYM
ma-63	161	7	isanother	isanother	ADJ
ma-63	161	8	fixed	fixed	ADJ
ma-63	161	9	point	point	NOUN
ma-63	161	10	of	of	ADP
ma-63	161	11	t	t	PROPN
ma-63	161	12	,	,	PUNCT
ma-63	161	13	we	we	PRON
ma-63	161	14	have	have	VERB
ma-63	161	15	θ	θ	PROPN
ma-63	161	16	�	�	PROPN
ma-63	161	17	d(x	d(x	PROPN
ma-63	161	18	,	,	PUNCT
ma-63	161	19	y	y	NOUN
ma-63	161	20	)	)	PUNCT
ma-63	161	21	=	=	PUNCT
ma-63	162	1	d(tx	d(tx	PROPN
ma-63	162	2	,	,	PUNCT
ma-63	162	3	t	t	PROPN
ma-63	162	4	y	y	PROPN
ma-63	162	5	)	)	PUNCT
ma-63	162	6	�	�	PROPN
ma-63	162	7	α(x	α(x	PROPN
ma-63	162	8	,	,	PUNCT
ma-63	162	9	y)d(tx	y)d(tx	NUM
ma-63	162	10	,	,	PUNCT
ma-63	162	11	t	t	PROPN
ma-63	162	12	y	y	PROPN
ma-63	162	13	)	)	PUNCT
ma-63	162	14	�	�	PROPN
ma-63	162	15	ψ(d(x	ψ(d(x	PROPN
ma-63	162	16	,	,	PUNCT
ma-63	162	17	y	y	NOUN
ma-63	162	18	)	)	PUNCT
ma-63	162	19	+	+	CCONJ
ma-63	162	20	d(tx	d(tx	ADJ
ma-63	162	21	,	,	PUNCT
ma-63	162	22	x	x	PRON
ma-63	162	23	)	)	PUNCT
ma-63	162	24	+	+	CCONJ
ma-63	162	25	d(ty	d(ty	PROPN
ma-63	162	26	,	,	PUNCT
ma-63	162	27	y	y	NOUN
ma-63	162	28	)	)	PUNCT
ma-63	162	29	)	)	PUNCT
ma-63	163	1	=	=	PUNCT
ma-63	163	2	ψ(d(x	ψ(d(x	PROPN
ma-63	163	3	,	,	PUNCT
ma-63	163	4	y	y	NOUN
ma-63	163	5	)	)	PUNCT
ma-63	163	6	+	+	CCONJ
ma-63	163	7	d(x	d(x	PROPN
ma-63	163	8	,	,	PUNCT
ma-63	163	9	x	x	PRON
ma-63	163	10	)	)	PUNCT
ma-63	163	11	+	+	CCONJ
ma-63	163	12	d(y	d(y	NOUN
ma-63	163	13	,	,	PUNCT
ma-63	163	14	y	y	NOUN
ma-63	163	15	)	)	PUNCT
ma-63	163	16	)	)	PUNCT
ma-63	164	1	=	=	PUNCT
ma-63	164	2	ψ(d(x	ψ(d(x	PROPN
ma-63	164	3	,	,	PUNCT
ma-63	164	4	y	y	NOUN
ma-63	164	5	)	)	PUNCT
ma-63	164	6	.	.	PUNCT
ma-63	165	1	so	so	ADV
ma-63	165	2	d(x	d(x	PROPN
ma-63	165	3	,	,	PUNCT
ma-63	165	4	y	y	NOUN
ma-63	165	5	)	)	PUNCT
ma-63	165	6	=	=	SYM
ma-63	165	7	θ	θ	NOUN
ma-63	165	8	;	;	PUNCT
ma-63	165	9	ie	ie	ADV
ma-63	165	10	x	x	X
ma-63	165	11	=	=	SYM
ma-63	165	12	y	y	PROPN
ma-63	165	13	.	.	PUNCT
ma-63	166	1	�	�	PROPN
ma-63	166	2	4	4	NUM
ma-63	166	3	.	.	PUNCT
ma-63	166	4	applications	application	NOUN
ma-63	166	5	as	as	ADP
ma-63	166	6	application	application	NOUN
ma-63	166	7	of	of	ADP
ma-63	166	8	α	α	NOUN
ma-63	166	9	−	−	PROPN
ma-63	167	1	ψ	ψ	ADP
ma-63	167	2	contractive	contractive	ADJ
ma-63	167	3	in	in	ADP
ma-63	167	4	unital	unital	ADJ
ma-63	167	5	c∗-algebra	c∗-algebra	NOUN
ma-63	167	6	valued	value	VERB
ma-63	167	7	rectangular	rectangular	ADJ
ma-63	167	8	b−	b−	NOUN
ma-63	167	9	metric	metric	ADJ
ma-63	167	10	spaces	space	NOUN
ma-63	167	11	,	,	PUNCT
ma-63	167	12	existence	existence	NOUN
ma-63	167	13	and	and	CCONJ
ma-63	167	14	uniqueness	uniqueness	NOUN
ma-63	167	15	results	result	NOUN
ma-63	167	16	for	for	ADP
ma-63	167	17	a	a	DET
ma-63	167	18	type	type	NOUN
ma-63	167	19	of	of	ADP
ma-63	167	20	operator	operator	NOUN
ma-63	167	21	equation	equation	NOUN
ma-63	167	22	is	be	AUX
ma-63	167	23	given	give	VERB
ma-63	167	24	.	.	PUNCT
ma-63	168	1	example	example	NOUN
ma-63	168	2	4.1	4.1	NUM
ma-63	168	3	.	.	PUNCT
ma-63	168	4	suppose	suppose	VERB
ma-63	168	5	that	that	SCONJ
ma-63	168	6	h	h	NOUN
ma-63	168	7	is	be	AUX
ma-63	168	8	a	a	DET
ma-63	168	9	hilbert	hilbert	NOUN
ma-63	168	10	space	space	NOUN
ma-63	168	11	,	,	PUNCT
ma-63	168	12	b(h	b(h	PROPN
ma-63	168	13	)	)	PUNCT
ma-63	168	14	is	be	AUX
ma-63	168	15	the	the	DET
ma-63	168	16	set	set	NOUN
ma-63	168	17	of	of	ADP
ma-63	168	18	linear	linear	PROPN
ma-63	168	19	bounded	bounded	PROPN
ma-63	168	20	operators	operator	NOUN
ma-63	168	21	on	on	ADP
ma-63	168	22	h.	h.	PROPN
ma-63	168	23	let	let	VERB
ma-63	168	24	a1	a1	PROPN
ma-63	168	25	,	,	PUNCT
ma-63	168	26	a2	a2	PROPN
ma-63	168	27	,	,	PUNCT
ma-63	168	28	...	...	PUNCT
ma-63	168	29	,	,	PUNCT
ma-63	169	1	an	an	PRON
ma-63	169	2	,	,	PUNCT
ma-63	169	3	...	...	PUNCT
ma-63	169	4	∈	∈	NOUN
ma-63	169	5	b(h)which	b(h)which	PRON
ma-63	169	6	satisfy	satisfy	NOUN
ma-63	169	7	∑∞n=1	∑∞n=1	PROPN
ma-63	169	8	‖an‖	‖an‖	PROPN
ma-63	169	9	<	<	X
ma-63	169	10	1	1	NUM
ma-63	169	11	and	and	CCONJ
ma-63	169	12	q	q	NOUN
ma-63	169	13	∈	∈	PROPN
ma-63	169	14	b(h)+.then	b(h)+.then	SCONJ
ma-63	169	15	the	the	DET
ma-63	169	16	operator	operator	NOUN
ma-63	169	17	equation	equation	NOUN
ma-63	169	18	x	x	X
ma-63	169	19	−∑∞n=1	−∑∞n=1	NOUN
ma-63	169	20	a∗nxan	a∗nxan	NOUN
ma-63	169	21	=	=	X
ma-63	169	22	q	q	PROPN
ma-63	169	23	has	have	VERB
ma-63	169	24	a	a	DET
ma-63	169	25	unique	unique	ADJ
ma-63	169	26	solution	solution	NOUN
ma-63	169	27	in	in	ADP
ma-63	169	28	b(h	b(h	PROPN
ma-63	169	29	)	)	PUNCT
ma-63	169	30	.	.	PUNCT
ma-63	170	1	proof	proof	NOUN
ma-63	170	2	.	.	PUNCT
ma-63	171	1	set	set	VERB
ma-63	171	2	a	a	PRON
ma-63	171	3	=	=	SYM
ma-63	171	4	(	(	PUNCT
ma-63	171	5	∑∞	∑∞	NOUN
ma-63	171	6	n=1	n=1	PROPN
ma-63	171	7	‖an‖)p	‖an‖)p	PROPN
ma-63	171	8	with	with	ADP
ma-63	171	9	p	p	PRON
ma-63	171	10	≥	≥	NUM
ma-63	171	11	1	1	NUM
ma-63	171	12	,	,	PUNCT
ma-63	171	13	then	then	ADV
ma-63	171	14	‖a‖	‖a‖	PROPN
ma-63	171	15	<	<	X
ma-63	171	16	1.without	1.without	NUM
ma-63	171	17	loss	loss	NOUN
ma-63	171	18	of	of	ADP
ma-63	171	19	generality	generality	NOUN
ma-63	171	20	,	,	PUNCT
ma-63	171	21	one	one	PRON
ma-63	171	22	can	can	AUX
ma-63	171	23	supposethat	supposethat	VERB
ma-63	171	24	a	a	DET
ma-63	171	25	>	>	X
ma-63	171	26	0.choose	0.choose	NUM
ma-63	171	27	a	a	DET
ma-63	171	28	positive	positive	ADJ
ma-63	171	29	operator	operator	NOUN
ma-63	171	30	m	m	NOUN
ma-63	171	31	∈	∈	NOUN
ma-63	171	32	b(h).for	b(h).for	ADP
ma-63	171	33	x	x	SYM
ma-63	171	34	,	,	PUNCT
ma-63	171	35	y	y	PROPN
ma-63	171	36	∈	∈	PROPN
ma-63	171	37	b(h	b(h	PROPN
ma-63	171	38	)	)	PUNCT
ma-63	171	39	and	and	CCONJ
ma-63	171	40	p	p	X
ma-63	171	41	≥	≥	NUM
ma-63	171	42	1	1	NUM
ma-63	171	43	,	,	PUNCT
ma-63	171	44	set	set	VERB
ma-63	171	45	d(x	d(x	PROPN
ma-63	171	46	,	,	PUNCT
ma-63	171	47	y	y	PROPN
ma-63	171	48	)	)	PUNCT
ma-63	172	1	=	=	PUNCT
ma-63	172	2	‖x	‖x	NOUN
ma-63	173	1	−	−	PROPN
ma-63	173	2	y	y	PROPN
ma-63	173	3	‖pm	‖pm	NUM
ma-63	173	4	.then	.then	X
ma-63	173	5	d(x	d(x	PROPN
ma-63	173	6	,	,	PUNCT
ma-63	173	7	y	y	PROPN
ma-63	173	8	)	)	PUNCT
ma-63	173	9	is	be	AUX
ma-63	173	10	a	a	DET
ma-63	173	11	c∗-algebra	c∗-algebra	PROPN
ma-63	173	12	valued	value	VERB
ma-63	173	13	rectangular	rectangular	ADJ
ma-63	173	14	b−	b−	NOUN
ma-63	173	15	metric.suppose	metric.suppose	VERB
ma-63	173	16	that	that	PRON
ma-63	173	17	x	x	PROPN
ma-63	173	18	,	,	PUNCT
ma-63	173	19	y	y	PROPN
ma-63	173	20	,	,	PUNCT
ma-63	173	21	z	z	PROPN
ma-63	173	22	,	,	PUNCT
ma-63	173	23	w	w	PROPN
ma-63	173	24	∈	∈	PROPN
ma-63	173	25	b(h	b(h	PROPN
ma-63	173	26	)	)	PUNCT
ma-63	173	27	we	we	PRON
ma-63	173	28	have	have	VERB
ma-63	173	29	‖x	‖x	NOUN
ma-63	174	1	−	−	PROPN
ma-63	174	2	y	y	PROPN
ma-63	174	3	‖p	‖p	PROPN
ma-63	174	4	�	�	PROPN
ma-63	174	5	2p	2p	NOUN
ma-63	174	6	(	(	PUNCT
ma-63	174	7	‖x	‖x	NOUN
ma-63	174	8	−	−	PROPN
ma-63	175	1	z‖p	z‖p	NOUN
ma-63	175	2	+	+	CCONJ
ma-63	175	3	‖z	‖z	NOUN
ma-63	175	4	−w‖p	−w‖p	VERB
ma-63	175	5	+	+	CCONJ
ma-63	175	6	‖w	‖w	NOUN
ma-63	175	7	−	−	PROPN
ma-63	175	8	y	y	PROPN
ma-63	175	9	‖p).which	‖p).which	PRON
ma-63	175	10	implies	imply	VERB
ma-63	175	11	that	that	SCONJ
ma-63	175	12	d(x	d(x	PROPN
ma-63	175	13	,	,	PUNCT
ma-63	175	14	y	y	PROPN
ma-63	175	15	)	)	PUNCT
ma-63	175	16	�	�	PROPN
ma-63	175	17	a[d(x	a[d(x	PROPN
ma-63	175	18	,	,	PUNCT
ma-63	175	19	z	z	NOUN
ma-63	175	20	)	)	PUNCT
ma-63	176	1	+	+	CCONJ
ma-63	176	2	d(z	d(z	PROPN
ma-63	176	3	,	,	PUNCT
ma-63	176	4	w	w	NOUN
ma-63	176	5	)	)	PUNCT
ma-63	176	6	+	+	CCONJ
ma-63	176	7	d(w	d(w	PROPN
ma-63	176	8	,	,	PUNCT
ma-63	176	9	y	y	PROPN
ma-63	176	10	)	)	PUNCT
ma-63	176	11	]	]	X
ma-63	176	12	where	where	SCONJ
ma-63	176	13	a	a	DET
ma-63	176	14	=	=	X
ma-63	176	15	2pi	2pi	NOUN
ma-63	176	16	.	.	PUNCT
ma-63	177	1	consider	consider	VERB
ma-63	177	2	the	the	DET
ma-63	177	3	map	map	NOUN
ma-63	177	4	t	t	PROPN
ma-63	177	5	:	:	PUNCT
ma-63	177	6	b(h)→	b(h)→	PROPN
ma-63	177	7	b(h	b(h	PROPN
ma-63	177	8	)	)	PUNCT
ma-63	178	1	such	such	ADJ
ma-63	178	2	that	that	SCONJ
ma-63	178	3	t	t	PROPN
ma-63	178	4	(	(	PUNCT
ma-63	178	5	x	x	NOUN
ma-63	178	6	)	)	PUNCT
ma-63	178	7	=	=	NOUN
ma-63	178	8	∑∞	∑∞	NOUN
ma-63	178	9	n=1	n=1	PROPN
ma-63	178	10	a	a	DET
ma-63	178	11	∗	∗	NOUN
ma-63	178	12	nxan	nxan	NOUN
ma-63	178	13	+	+	PROPN
ma-63	178	14	q.then	q.then	PROPN
ma-63	178	15	d(t	d(t	PROPN
ma-63	178	16	(	(	PUNCT
ma-63	178	17	x	x	NOUN
ma-63	178	18	)	)	PUNCT
ma-63	178	19	,	,	PUNCT
ma-63	178	20	t	t	PROPN
ma-63	178	21	(	(	PUNCT
ma-63	178	22	y	y	PROPN
ma-63	178	23	)	)	PUNCT
ma-63	178	24	)	)	PUNCT
ma-63	179	1	=	=	SYM
ma-63	179	2	‖t	‖t	NOUN
ma-63	179	3	(	(	PUNCT
ma-63	179	4	x	x	NOUN
ma-63	179	5	)	)	PUNCT
ma-63	179	6	,	,	PUNCT
ma-63	179	7	t	t	PROPN
ma-63	179	8	(	(	PUNCT
ma-63	179	9	y	y	PROPN
ma-63	179	10	)	)	PUNCT
ma-63	179	11	‖pm	‖pm	PROPN
ma-63	179	12	=	=	SYM
ma-63	179	13	‖	‖	PROPN
ma-63	179	14	∑∞	∑∞	NOUN
ma-63	179	15	n=1	n=1	PUNCT
ma-63	179	16	a	a	DET
ma-63	179	17	∗	∗	NOUN
ma-63	179	18	n(x	n(x	ADJ
ma-63	179	19	−	−	PROPN
ma-63	179	20	y	y	PROPN
ma-63	179	21	)	)	PUNCT
ma-63	179	22	an‖pm	an‖pm	PART
ma-63	179	23	�	�	PROPN
ma-63	179	24	∑∞	∑∞	PROPN
ma-63	179	25	n=1	n=1	PROPN
ma-63	179	26	‖an‖2p‖x	‖an‖2p‖x	PROPN
ma-63	179	27	−	−	PROPN
ma-63	179	28	y	y	PROPN
ma-63	179	29	‖pm	‖pm	PROPN
ma-63	179	30	�	�	PROPN
ma-63	179	31	a2d(x	a2d(x	PROPN
ma-63	179	32	,	,	PUNCT
ma-63	179	33	y	y	PROPN
ma-63	179	34	)	)	PUNCT
ma-63	179	35	https://doi.org/10.28924/ada/ma.2.11	https://doi.org/10.28924/ada/ma.2.11	PROPN
ma-63	179	36	eur	eur	PROPN
ma-63	179	37	.	.	PUNCT
ma-63	180	1	j.	j.	PROPN
ma-63	180	2	math	math	PROPN
ma-63	180	3	.	.	PUNCT
ma-63	181	1	anal	anal	PROPN
ma-63	181	2	.	.	PUNCT
ma-63	182	1	10.28924	10.28924	NUM
ma-63	182	2	/	/	SYM
ma-63	182	3	ada	ada	PROPN
ma-63	182	4	/	/	SYM
ma-63	182	5	ma.2.11	ma.2.11	PROPN
ma-63	182	6	8let	8let	PROPN
ma-63	182	7	α	α	NOUN
ma-63	182	8	:	:	PUNCT
ma-63	182	9	b(h)×	b(h)×	NOUN
ma-63	182	10	b(h)→	b(h)→	NOUN
ma-63	182	11	b(h)+	b(h)+	AUX
ma-63	182	12	defined	define	VERB
ma-63	182	13	by	by	ADP
ma-63	182	14	α(x	α(x	PROPN
ma-63	182	15	,	,	PUNCT
ma-63	182	16	y	y	PROPN
ma-63	182	17	)	)	PUNCT
ma-63	182	18	=	=	PUNCT
ma-63	183	1	(	(	PUNCT
ma-63	183	2	x	x	X
ma-63	183	3	,	,	PUNCT
ma-63	183	4	y	y	PROPN
ma-63	183	5	)	)	PUNCT
ma-63	183	6	iand	iand	NOUN
ma-63	183	7	ψ	ψ	X
ma-63	183	8	:	:	PUNCT
ma-63	183	9	b(h)+	b(h)+	X
ma-63	183	10	→	→	SYM
ma-63	183	11	b(h)+	b(h)+	VERB
ma-63	183	12	defined	define	VERB
ma-63	183	13	by	by	ADP
ma-63	183	14	ψ(x	ψ(x	NOUN
ma-63	183	15	)	)	PUNCT
ma-63	184	1	=	=	SYM
ma-63	184	2	x	x	X
ma-63	184	3	.we	.we	PUNCT
ma-63	184	4	get	get	VERB
ma-63	184	5	α(x	α(x	PROPN
ma-63	184	6	,	,	PUNCT
ma-63	184	7	y	y	PROPN
ma-63	184	8	)	)	PUNCT
ma-63	184	9	d(tx	d(tx	PROPN
ma-63	184	10	,	,	PUNCT
ma-63	184	11	ty	ty	INTJ
ma-63	184	12	)	)	PUNCT
ma-63	184	13	�	�	PROPN
ma-63	184	14	ψ(d(x	ψ(d(x	PROPN
ma-63	184	15	,	,	PUNCT
ma-63	184	16	y	y	PROPN
ma-63	184	17	)	)	PUNCT
ma-63	184	18	)	)	PUNCT
ma-63	184	19	.using	.using	PUNCT
ma-63	184	20	theorem	theorem	VERB
ma-63	184	21	3.1	3.1	NUM
ma-63	184	22	,	,	PUNCT
ma-63	184	23	there	there	PRON
ma-63	184	24	exists	exist	VERB
ma-63	184	25	a	a	DET
ma-63	184	26	unique	unique	ADJ
ma-63	184	27	fixed	fixed	ADJ
ma-63	184	28	point	point	NOUN
ma-63	184	29	x	x	PUNCT
ma-63	184	30	in	in	ADP
ma-63	184	31	b(h	b(h	NOUN
ma-63	184	32	)	)	PUNCT
ma-63	184	33	.	.	PUNCT
ma-63	185	1	�	�	PROPN
ma-63	185	2	5	5	NUM
ma-63	185	3	.	.	PUNCT
ma-63	185	4	acknowledgments	acknowledgment	NOUN
ma-63	185	5	it	it	PRON
ma-63	185	6	is	be	AUX
ma-63	185	7	our	our	PRON
ma-63	185	8	great	great	ADJ
ma-63	185	9	pleasure	pleasure	NOUN
ma-63	185	10	to	to	PART
ma-63	185	11	thank	thank	VERB
ma-63	185	12	the	the	DET
ma-63	185	13	referee	referee	NOUN
ma-63	185	14	for	for	ADP
ma-63	185	15	his	his	PRON
ma-63	185	16	careful	careful	ADJ
ma-63	185	17	reading	reading	NOUN
ma-63	185	18	of	of	ADP
ma-63	185	19	the	the	DET
ma-63	185	20	paper	paper	NOUN
ma-63	185	21	and	and	CCONJ
ma-63	185	22	for	for	ADP
ma-63	185	23	severalhelpful	severalhelpful	ADJ
ma-63	185	24	suggestions	suggestion	NOUN
ma-63	185	25	.	.	PUNCT
ma-63	186	1	references	reference	NOUN
ma-63	186	2	[	[	X
ma-63	186	3	1	1	NUM
ma-63	186	4	]	]	X
ma-63	186	5	h.h	h.h	PROPN
ma-63	186	6	.	.	PROPN
ma-63	186	7	alsulami	alsulami	PROPN
ma-63	186	8	,	,	PUNCT
ma-63	186	9	r.p	r.p	PROPN
ma-63	186	10	.	.	PROPN
ma-63	186	11	agarwal	agarwal	PROPN
ma-63	186	12	,	,	PUNCT
ma-63	186	13	e.	e.	PROPN
ma-63	186	14	karapınar	karapınar	PROPN
ma-63	186	15	,	,	PUNCT
ma-63	186	16	f.	f.	PROPN
ma-63	186	17	khojasteh	khojasteh	PROPN
ma-63	186	18	,	,	PUNCT
ma-63	186	19	a	a	DET
ma-63	186	20	short	short	ADJ
ma-63	186	21	note	note	NOUN
ma-63	186	22	on	on	ADP
ma-63	186	23	c∗-valued	c∗-value	VERB
ma-63	186	24	contraction	contraction	NOUN
ma-63	186	25	mappings	mapping	NOUN
ma-63	186	26	,	,	PUNCT
ma-63	186	27	j.	j.	PROPN
ma-63	186	28	inequal.appl	inequal.appl	PROPN
ma-63	186	29	.	.	PROPN
ma-63	186	30	2016	2016	NUM
ma-63	186	31	(	(	PUNCT
ma-63	186	32	2016	2016	NUM
ma-63	186	33	)	)	PUNCT
ma-63	186	34	50	50	NUM
ma-63	186	35	.	.	PUNCT
ma-63	187	1	https://doi.org/10.1186/s13660-016-0992-5.[2	https://doi.org/10.1186/s13660-016-0992-5.[2	ADV
ma-63	187	2	]	]	PUNCT
ma-63	187	3	s.	s.	PROPN
ma-63	187	4	chandok	chandok	PROPN
ma-63	187	5	,	,	PUNCT
ma-63	187	6	d.	d.	PROPN
ma-63	187	7	kumar	kumar	PROPN
ma-63	187	8	,	,	PUNCT
ma-63	187	9	c.	c.	PROPN
ma-63	187	10	park	park	PROPN
ma-63	187	11	,	,	PUNCT
ma-63	187	12	c∗−algebra	c∗−algebra	PROPN
ma-63	187	13	-	-	PUNCT
ma-63	187	14	valued	value	VERB
ma-63	187	15	partial	partial	ADJ
ma-63	187	16	metric	metric	ADJ
ma-63	187	17	space	space	NOUN
ma-63	187	18	and	and	CCONJ
ma-63	187	19	fixed	fix	VERB
ma-63	187	20	point	point	NOUN
ma-63	187	21	theorems	theorem	NOUN
ma-63	187	22	.	.	PUNCT
ma-63	188	1	proc	proc	PROPN
ma-63	188	2	.	.	PUNCT
ma-63	189	1	math	math	NOUN
ma-63	189	2	.	.	PUNCT
ma-63	190	1	sci.129	sci.129	PROPN
ma-63	190	2	(	(	PUNCT
ma-63	190	3	2019	2019	NUM
ma-63	190	4	)	)	PUNCT
ma-63	190	5	37	37	NUM
ma-63	190	6	.	.	PUNCT
ma-63	191	1	https://doi.org/10.1007/s12044-019-0481-0.[3	https://doi.org/10.1007/s12044-019-0481-0.[3	NOUN
ma-63	191	2	]	]	PUNCT
ma-63	191	3	m.	m.	NOUN
ma-63	191	4	jleli	jleli	PROPN
ma-63	191	5	,	,	PUNCT
ma-63	191	6	b.	b.	PROPN
ma-63	191	7	samet	samet	PROPN
ma-63	191	8	,	,	PUNCT
ma-63	191	9	a	a	DET
ma-63	191	10	new	new	ADJ
ma-63	191	11	generalization	generalization	NOUN
ma-63	191	12	of	of	ADP
ma-63	191	13	the	the	DET
ma-63	191	14	banach	banach	NOUN
ma-63	191	15	contraction	contraction	NOUN
ma-63	191	16	principle	principle	NOUN
ma-63	191	17	.	.	PUNCT
ma-63	192	1	j.	j.	PROPN
ma-63	192	2	inequal	inequal	PROPN
ma-63	192	3	.	.	PUNCT
ma-63	193	1	appl	appl	PROPN
ma-63	193	2	.	.	PROPN
ma-63	194	1	2014	2014	NUM
ma-63	194	2	(	(	PUNCT
ma-63	194	3	2014	2014	NUM
ma-63	194	4	)	)	PUNCT
ma-63	194	5	,	,	PUNCT
ma-63	194	6	38	38	NUM
ma-63	194	7	.	.	PUNCT
ma-63	195	1	https://doi.org/10.1186/1029-242x-2014-38.[4	https://doi.org/10.1186/1029-242x-2014-38.[4	PROPN
ma-63	195	2	]	]	X
ma-63	195	3	g.	g.	PROPN
ma-63	195	4	kalapana	kalapana	PROPN
ma-63	195	5	,	,	PUNCT
ma-63	195	6	z.s	z.s	PROPN
ma-63	195	7	.	.	PROPN
ma-63	195	8	tasneem	tasneem	PROPN
ma-63	195	9	c∗−algebra	c∗−algebra	PROPN
ma-63	195	10	-	-	PUNCT
ma-63	195	11	valued	value	VERB
ma-63	195	12	rectangular	rectangular	ADJ
ma-63	195	13	b	b	X
ma-63	195	14	-	-	ADJ
ma-63	195	15	metric	metric	ADJ
ma-63	195	16	spaces	space	NOUN
ma-63	195	17	and	and	CCONJ
ma-63	195	18	some	some	DET
ma-63	195	19	fixed	fix	VERB
ma-63	195	20	point	point	NOUN
ma-63	195	21	theorems	theorem	NOUN
ma-63	195	22	,	,	PUNCT
ma-63	195	23	commun.fac	commun.fac	PROPN
ma-63	195	24	.	.	PUNCT
ma-63	196	1	sci	sci	PROPN
ma-63	196	2	.	.	PROPN
ma-63	196	3	univ	univ	PROPN
ma-63	196	4	.	.	PUNCT
ma-63	197	1	ank	ank	PROPN
ma-63	197	2	.	.	PROPN
ma-63	197	3	ser	ser	PROPN
ma-63	197	4	.	.	PUNCT
ma-63	198	1	a1	a1	NOUN
ma-63	198	2	math	math	NOUN
ma-63	198	3	.	.	PUNCT
ma-63	199	1	stat	stat	PROPN
ma-63	199	2	.	.	PUNCT
ma-63	200	1	68	68	NUM
ma-63	200	2	(	(	PUNCT
ma-63	200	3	2019	2019	NUM
ma-63	200	4	)	)	PUNCT
ma-63	200	5	2198	2198	NUM
ma-63	200	6	-	-	PUNCT
ma-63	200	7	2208	2208	NUM
ma-63	200	8	.	.	PUNCT
ma-63	201	1	https://doi.org/10.31801/cfsuasmas.598146.[5	https://doi.org/10.31801/cfsuasmas.598146.[5	PROPN
ma-63	201	2	]	]	PUNCT
ma-63	201	3	w.	w.	PROPN
ma-63	201	4	a.	a.	PROPN
ma-63	201	5	kirk	kirk	PROPN
ma-63	201	6	,	,	PUNCT
ma-63	201	7	n.	n.	PROPN
ma-63	201	8	shahzad	shahzad	PROPN
ma-63	201	9	,	,	PUNCT
ma-63	201	10	generalized	generalized	ADJ
ma-63	201	11	metrics	metric	NOUN
ma-63	201	12	and	and	CCONJ
ma-63	201	13	caristi	caristi	NOUN
ma-63	201	14	’s	’s	PART
ma-63	201	15	theorem	theorem	ADJ
ma-63	201	16	,	,	PUNCT
ma-63	201	17	fixed	fix	VERB
ma-63	201	18	point	point	NOUN
ma-63	201	19	theory	theory	NOUN
ma-63	201	20	appl	appl	NOUN
ma-63	201	21	.	.	PUNCT
ma-63	202	1	2013	2013	NUM
ma-63	202	2	(	(	PUNCT
ma-63	202	3	2013	2013	NUM
ma-63	202	4	)	)	PUNCT
ma-63	202	5	129	129	NUM
ma-63	202	6	.	.	PUNCT
ma-63	203	1	https://doi.org/10.1186/1687-1812-2013-129.[6	https://doi.org/10.1186/1687-1812-2013-129.[6	NOUN
ma-63	203	2	]	]	PUNCT
ma-63	203	3	z.	z.	PROPN
ma-63	203	4	ma	ma	PROPN
ma-63	203	5	,	,	PUNCT
ma-63	203	6	l.	l.	PROPN
ma-63	203	7	jiang	jiang	PROPN
ma-63	203	8	,	,	PUNCT
ma-63	203	9	h.	h.	PROPN
ma-63	203	10	sun	sun	PROPN
ma-63	203	11	,	,	PUNCT
ma-63	203	12	c∗-algebra	c∗-algebra	PROPN
ma-63	203	13	-	-	PUNCT
ma-63	203	14	valued	value	VERB
ma-63	203	15	metric	metric	ADJ
ma-63	203	16	spaces	space	NOUN
ma-63	203	17	and	and	CCONJ
ma-63	203	18	related	relate	VERB
ma-63	203	19	fixed	fix	VERB
ma-63	203	20	point	point	NOUN
ma-63	203	21	theorems	theorem	NOUN
ma-63	203	22	,	,	PUNCT
ma-63	203	23	fixed	fixed	ADJ
ma-63	203	24	point	point	NOUN
ma-63	203	25	theoryappl	theoryappl	ADJ
ma-63	203	26	.	.	PUNCT
ma-63	204	1	(	(	PUNCT
ma-63	204	2	2014	2014	NUM
ma-63	204	3	)	)	PUNCT
ma-63	204	4	2014	2014	NUM
ma-63	204	5	,	,	PUNCT
ma-63	204	6	206	206	NUM
ma-63	204	7	.	.	PUNCT
ma-63	205	1	https://doi.org/10.1186/1687-1812-2014-206.[7	https://doi.org/10.1186/1687-1812-2014-206.[7	PROPN
ma-63	205	2	]	]	X
ma-63	205	3	h.	h.	PROPN
ma-63	205	4	massit	massit	PROPN
ma-63	205	5	,	,	PUNCT
ma-63	205	6	m.	m.	NOUN
ma-63	205	7	rossafi	rossafi	NOUN
ma-63	205	8	,	,	PUNCT
ma-63	205	9	fixed	fix	VERB
ma-63	205	10	point	point	NOUN
ma-63	205	11	for	for	ADP
ma-63	205	12	ψ−	ψ−	VERB
ma-63	205	13	contractive	contractive	ADJ
ma-63	205	14	mapping	mapping	NOUN
ma-63	205	15	in	in	ADP
ma-63	205	16	c∗−	c∗−	PROPN
ma-63	205	17	algebra	algebra	PROPN
ma-63	205	18	valued	value	VERB
ma-63	205	19	rectangular	rectangular	ADJ
ma-63	205	20	b	b	NOUN
ma-63	205	21	-	-	ADJ
ma-63	205	22	metric	metric	ADJ
ma-63	205	23	,	,	PUNCT
ma-63	205	24	j.	j.	PROPN
ma-63	205	25	math.comput	math.comput	PROPN
ma-63	205	26	.	.	PUNCT
ma-63	206	1	sci	sci	PROPN
ma-63	206	2	.	.	PUNCT
ma-63	206	3	11(2021	11(2021	NUM
ma-63	206	4	)	)	PUNCT
ma-63	206	5	6507	6507	NUM
ma-63	206	6	-	-	SYM
ma-63	206	7	6521	6521	NUM
ma-63	206	8	.	.	PUNCT
ma-63	207	1	https://doi.org/10.28919/jmcs/6363.[8	https://doi.org/10.28919/jmcs/6363.[8	PROPN
ma-63	207	2	]	]	X
ma-63	207	3	g.j	g.j	PROPN
ma-63	207	4	.	.	PROPN
ma-63	207	5	murphy	murphy	PROPN
ma-63	207	6	,	,	PUNCT
ma-63	207	7	c∗-algebras	c∗-algebras	PROPN
ma-63	207	8	and	and	CCONJ
ma-63	207	9	operator	operator	NOUN
ma-63	207	10	theory	theory	NOUN
ma-63	207	11	,	,	PUNCT
ma-63	207	12	academic	academic	ADJ
ma-63	207	13	press	press	NOUN
ma-63	207	14	,	,	PUNCT
ma-63	207	15	london	london	PROPN
ma-63	207	16	,	,	PUNCT
ma-63	207	17	uk	uk	PROPN
ma-63	207	18	,	,	PUNCT
ma-63	207	19	1990.[9	1990.[9	NUM
ma-63	207	20	]	]	X
ma-63	207	21	s.	s.	PROPN
ma-63	207	22	omran	omran	PROPN
ma-63	207	23	,	,	PUNCT
ma-63	207	24	i.	i.	PROPN
ma-63	207	25	masmali	masmali	PROPN
ma-63	207	26	,	,	PUNCT
ma-63	207	27	on	on	ADP
ma-63	207	28	the	the	DET
ma-63	207	29	(	(	PUNCT
ma-63	207	30	α−	α−	ADP
ma-63	207	31	ψ)-contractive	ψ)-contractive	ADJ
ma-63	207	32	mappings	mapping	NOUN
ma-63	207	33	in	in	ADP
ma-63	207	34	c∗-algebra	c∗-algebra	PROPN
ma-63	207	35	valued	value	VERB
ma-63	207	36	b	b	X
ma-63	207	37	-	-	PUNCT
ma-63	207	38	metric	metric	ADJ
ma-63	207	39	spaces	space	NOUN
ma-63	207	40	and	and	CCONJ
ma-63	207	41	fixed	fix	VERB
ma-63	207	42	pointtheorems	pointtheorem	NOUN
ma-63	207	43	,	,	PUNCT
ma-63	207	44	j.	j.	PROPN
ma-63	207	45	math	math	PROPN
ma-63	207	46	.	.	PUNCT
ma-63	208	1	2021	2021	NUM
ma-63	208	2	(	(	PUNCT
ma-63	208	3	2021	2021	NUM
ma-63	208	4	)	)	PUNCT
ma-63	208	5	7865976	7865976	NUM
ma-63	208	6	.	.	PUNCT
ma-63	209	1	https://doi.org/10.1155/2021/7865976.[10	https://doi.org/10.1155/2021/7865976.[10	PROPN
ma-63	209	2	]	]	PUNCT
ma-63	209	3	b.	b.	PROPN
ma-63	209	4	samet	samet	PROPN
ma-63	209	5	,	,	PUNCT
ma-63	209	6	c.	c.	PROPN
ma-63	209	7	vetro	vetro	PROPN
ma-63	209	8	,	,	PUNCT
ma-63	209	9	p.	p.	NOUN
ma-63	209	10	vetro	vetro	NOUN
ma-63	209	11	,	,	PUNCT
ma-63	209	12	fixed	fix	VERB
ma-63	209	13	point	point	NOUN
ma-63	209	14	theorems	theorem	NOUN
ma-63	209	15	for	for	ADP
ma-63	209	16	α−	α−	ADP
ma-63	209	17	ψ−contractive	ψ−contractive	NUM
ma-63	209	18	type	type	NOUN
ma-63	209	19	mappings	mapping	NOUN
ma-63	209	20	,	,	PUNCT
ma-63	209	21	nonlinear	nonlinear	ADJ
ma-63	209	22	anal	anal	NOUN
ma-63	209	23	.	.	PUNCT
ma-63	209	24	:	:	PUNCT
ma-63	210	1	theorymethods	theorymethod	NOUN
ma-63	210	2	appl	appl	NOUN
ma-63	210	3	.	.	PUNCT
ma-63	211	1	75	75	NUM
ma-63	211	2	(	(	PUNCT
ma-63	211	3	2012	2012	NUM
ma-63	211	4	)	)	PUNCT
ma-63	211	5	2154–2165	2154–2165	NUM
ma-63	211	6	.	.	PUNCT
ma-63	212	1	https://doi.org/10.1016/j.na.2011.10.014	https://doi.org/10.1016/j.na.2011.10.014	PROPN
ma-63	212	2	.	.	PUNCT
ma-63	213	1	https://doi.org/10.28924/ada/ma.2.11	https://doi.org/10.28924/ada/ma.2.11	PROPN
ma-63	213	2	https://doi.org/10.1186/s13660-016-0992-5	https://doi.org/10.1186/s13660-016-0992-5	PROPN
ma-63	213	3	https://doi.org/10.1007/s12044-019-0481-0	https://doi.org/10.1007/s12044-019-0481-0	ADJ
ma-63	213	4	https://doi.org/10.1186/1029-242x-2014-38	https://doi.org/10.1186/1029-242x-2014-38	PROPN
ma-63	213	5	https://doi.org/10.31801/cfsuasmas.598146	https://doi.org/10.31801/cfsuasmas.598146	PROPN
ma-63	213	6	https://doi.org/10.1186/1687-1812-2013-129	https://doi.org/10.1186/1687-1812-2013-129	PROPN
ma-63	213	7	https://doi.org/10.1186/1687-1812-2014-206	https://doi.org/10.1186/1687-1812-2014-206	PROPN
ma-63	213	8	https://doi.org/10.28919/jmcs/6363	https://doi.org/10.28919/jmcs/6363	PROPN
ma-63	213	9	https://doi.org/10.1155/2021/7865976	https://doi.org/10.1155/2021/7865976	PROPN
ma-63	213	10	https://doi.org/10.1016/j.na.2011.10.014	https://doi.org/10.1016/j.na.2011.10.014	PROPN
ma-63	213	11	1	1	NUM
ma-63	213	12	.	.	PUNCT
ma-63	213	13	introduction	introduction	NOUN
ma-63	213	14	2	2	NUM
ma-63	213	15	.	.	PUNCT
ma-63	213	16	preliminaries	preliminary	NOUN
ma-63	213	17	3	3	NUM
ma-63	213	18	.	.	X
ma-63	213	19	main	main	ADJ
ma-63	213	20	result	result	NOUN
ma-63	213	21	4	4	NUM
ma-63	213	22	.	.	PUNCT
ma-63	213	23	applications	application	NOUN
ma-63	213	24	5	5	NUM
ma-63	213	25	.	.	PUNCT
ma-63	214	1	acknowledgments	acknowledgment	NOUN
ma-63	214	2	references	reference	NOUN