id	sid	tid	token	lemma	pos
ma-134	1	1	2023	2023	NUM
ma-134	1	2	ada	ada	PROPN
ma-134	1	3	academica	academica	PROPN
ma-134	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-134	1	5	.	.	PUNCT
ma-134	2	1	j.	j.	PROPN
ma-134	2	2	math	math	PROPN
ma-134	2	3	.	.	PUNCT
ma-134	3	1	anal	anal	ADJ
ma-134	3	2	.	.	PUNCT
ma-134	4	1	3	3	NUM
ma-134	4	2	(	(	PUNCT
ma-134	4	3	2023	2023	NUM
ma-134	4	4	)	)	PUNCT
ma-134	5	1	7doi	7doi	NOUN
ma-134	5	2	:	:	PUNCT
ma-134	5	3	10.28924	10.28924	NUM
ma-134	5	4	/	/	SYM
ma-134	5	5	ada	ada	PROPN
ma-134	5	6	/	/	SYM
ma-134	5	7	ma.3.7	ma.3.7	NOUN
ma-134	5	8	a	a	DET
ma-134	5	9	note	note	NOUN
ma-134	5	10	on	on	ADP
ma-134	5	11	the	the	DET
ma-134	5	12	stability	stability	NOUN
ma-134	5	13	of	of	ADP
ma-134	5	14	functional	functional	ADJ
ma-134	5	15	equations	equation	NOUN
ma-134	5	16	via	via	ADP
ma-134	5	17	a	a	DET
ma-134	5	18	celebrated	celebrate	VERB
ma-134	5	19	direct	direct	ADJ
ma-134	5	20	method	method	NOUN
ma-134	5	21	dongwen	dongwen	NOUN
ma-134	5	22	zhang1	zhang1	PROPN
ma-134	5	23	,	,	PUNCT
ma-134	5	24	john	john	PROPN
ma-134	5	25	michael	michael	PROPN
ma-134	5	26	rassias2	rassias2	PROPN
ma-134	5	27	,	,	PUNCT
ma-134	5	28	qi	qi	PROPN
ma-134	5	29	liu3	liu3	PROPN
ma-134	5	30	,	,	PUNCT
ma-134	5	31	yongjin	yongjin	PROPN
ma-134	5	32	li4,∗	li4,∗	PROPN
ma-134	5	33	1school	1school	NUM
ma-134	5	34	of	of	ADP
ma-134	5	35	mathematics	mathematic	NOUN
ma-134	5	36	(	(	PUNCT
ma-134	5	37	zhuhai	zhuhai	PROPN
ma-134	5	38	)	)	PUNCT
ma-134	5	39	,	,	PUNCT
ma-134	5	40	sun	sun	PROPN
ma-134	5	41	yat	yat	PROPN
ma-134	5	42	-	-	PUNCT
ma-134	5	43	sen	sen	PROPN
ma-134	5	44	university	university	PROPN
ma-134	5	45	,	,	PUNCT
ma-134	5	46	zhuhai	zhuhai	PROPN
ma-134	5	47	519082	519082	NUM
ma-134	5	48	,	,	PUNCT
ma-134	5	49	p.r	p.r	PROPN
ma-134	5	50	.	.	PROPN
ma-134	5	51	china	china	PROPN
ma-134	5	52	zhangdw25@mail2.sysu.edu.cn	zhangdw25@mail2.sysu.edu.cn	PROPN
ma-134	5	53	2national	2national	PROPN
ma-134	5	54	and	and	CCONJ
ma-134	5	55	kapodistrian	kapodistrian	ADJ
ma-134	5	56	university	university	PROPN
ma-134	5	57	of	of	ADP
ma-134	5	58	athens	athens	PROPN
ma-134	5	59	,	,	PUNCT
ma-134	5	60	department	department	NOUN
ma-134	5	61	of	of	ADP
ma-134	5	62	mathematics	mathematics	PROPN
ma-134	5	63	and	and	CCONJ
ma-134	5	64	informatics	informatic	NOUN
ma-134	5	65	,	,	PUNCT
ma-134	5	66	attikis	attikis	PROPN
ma-134	5	67	15342	15342	NUM
ma-134	5	68	,	,	PUNCT
ma-134	5	69	greece	greece	PROPN
ma-134	5	70	jrassias@primedu.uoa.gr	jrassias@primedu.uoa.gr	VERB
ma-134	5	71	3school	3school	NUM
ma-134	5	72	of	of	ADP
ma-134	5	73	mathematics	mathematic	NOUN
ma-134	5	74	and	and	CCONJ
ma-134	5	75	physics	physics	NOUN
ma-134	5	76	,	,	PUNCT
ma-134	5	77	anqing	anqe	VERB
ma-134	5	78	normal	normal	ADJ
ma-134	5	79	university	university	NOUN
ma-134	5	80	,	,	PUNCT
ma-134	5	81	anqing	anqe	VERB
ma-134	5	82	246133	246133	NUM
ma-134	5	83	,	,	PUNCT
ma-134	5	84	p.r	p.r	PROPN
ma-134	5	85	.	.	PROPN
ma-134	5	86	china	china	PROPN
ma-134	5	87	liuq325@mail2.sysu.edu.cn	liuq325@mail2.sysu.edu.cn	PROPN
ma-134	5	88	4department	4department	NUM
ma-134	5	89	of	of	ADP
ma-134	5	90	mathematics	mathematic	NOUN
ma-134	5	91	,	,	PUNCT
ma-134	5	92	sun	sun	PROPN
ma-134	5	93	yat	yat	PROPN
ma-134	5	94	-	-	PUNCT
ma-134	5	95	sen	sen	PROPN
ma-134	5	96	university	university	PROPN
ma-134	5	97	,	,	PUNCT
ma-134	5	98	guangzhou	guangzhou	PROPN
ma-134	5	99	,	,	PUNCT
ma-134	5	100	510275	510275	NUM
ma-134	5	101	,	,	PUNCT
ma-134	6	1	p.r	p.r	PROPN
ma-134	6	2	.	.	PROPN
ma-134	6	3	china	china	PROPN
ma-134	6	4	stslyj@mail.sysu.edu.cn	stslyj@mail.sysu.edu.cn	PROPN
ma-134	6	5	∗correspondence	∗correspondence	NOUN
ma-134	6	6	:	:	PUNCT
ma-134	6	7	stslyj@mail.sysu.edu.cn	stslyj@mail.sysu.edu.cn	NOUN
ma-134	6	8	abstract	abstract	NOUN
ma-134	6	9	.	.	PUNCT
ma-134	7	1	more	more	ADJ
ma-134	7	2	than	than	ADP
ma-134	7	3	ten	ten	NUM
ma-134	7	4	years	year	NOUN
ma-134	7	5	after	after	SCONJ
ma-134	7	6	justyna	justyna	PROPN
ma-134	7	7	sikorska	sikorska	PROPN
ma-134	7	8	[	[	X
ma-134	7	9	8	8	NUM
ma-134	7	10	]	]	PUNCT
ma-134	7	11	attempted	attempt	VERB
ma-134	7	12	to	to	PART
ma-134	7	13	solve	solve	VERB
ma-134	7	14	the	the	DET
ma-134	7	15	heyers	heyer	NOUN
ma-134	7	16	-	-	PUNCT
ma-134	7	17	ulam	ulam	X
ma-134	7	18	sta	sta	NOUN
ma-134	7	19	-	-	PUNCT
ma-134	7	20	bility	bility	NOUN
ma-134	7	21	of	of	ADP
ma-134	7	22	a	a	DET
ma-134	7	23	single	single	ADJ
ma-134	7	24	variable	variable	ADJ
ma-134	7	25	equation	equation	NOUN
ma-134	7	26	by	by	ADP
ma-134	7	27	using	use	VERB
ma-134	7	28	direct	direct	ADJ
ma-134	7	29	method	method	NOUN
ma-134	7	30	.	.	PUNCT
ma-134	8	1	in	in	ADP
ma-134	8	2	this	this	DET
ma-134	8	3	paper	paper	NOUN
ma-134	8	4	,	,	PUNCT
ma-134	8	5	we	we	PRON
ma-134	8	6	will	will	AUX
ma-134	8	7	improve	improve	VERB
ma-134	8	8	the	the	DET
ma-134	8	9	resultsof	resultsof	PROPN
ma-134	8	10	justyna	justyna	PROPN
ma-134	8	11	sikorska	sikorska	PROPN
ma-134	8	12	by	by	ADP
ma-134	8	13	using	use	VERB
ma-134	8	14	a	a	DET
ma-134	8	15	more	more	ADV
ma-134	8	16	efficient	efficient	ADJ
ma-134	8	17	approach	approach	NOUN
ma-134	8	18	.	.	PUNCT
ma-134	9	1	relations	relation	NOUN
ma-134	9	2	between	between	ADP
ma-134	9	3	the	the	DET
ma-134	9	4	generalized	generalized	ADJ
ma-134	9	5	functionalequation	functionalequation	NOUN
ma-134	9	6	,	,	PUNCT
ma-134	9	7	the	the	DET
ma-134	9	8	dependence	dependence	NOUN
ma-134	9	9	of	of	ADP
ma-134	9	10	their	their	PRON
ma-134	9	11	different	different	ADJ
ma-134	9	12	parameters	parameter	NOUN
ma-134	9	13	and	and	CCONJ
ma-134	9	14	several	several	ADJ
ma-134	9	15	properties	property	NOUN
ma-134	9	16	are	be	AUX
ma-134	9	17	also	also	ADV
ma-134	9	18	further	further	ADJ
ma-134	9	19	ex	ex	VERB
ma-134	9	20	-	-	VERB
ma-134	9	21	plored	plored	ADJ
ma-134	9	22	.	.	PUNCT
ma-134	10	1	to	to	PART
ma-134	10	2	achieve	achieve	VERB
ma-134	10	3	the	the	DET
ma-134	10	4	problem	problem	NOUN
ma-134	10	5	,	,	PUNCT
ma-134	10	6	we	we	PRON
ma-134	10	7	try	try	VERB
ma-134	10	8	to	to	PART
ma-134	10	9	develop	develop	VERB
ma-134	10	10	some	some	DET
ma-134	10	11	new	new	ADJ
ma-134	10	12	techniques	technique	NOUN
ma-134	10	13	to	to	PART
ma-134	10	14	overcome	overcome	VERB
ma-134	10	15	the	the	DET
ma-134	10	16	fundamentaldifficulties	fundamentaldifficultie	NOUN
ma-134	10	17	caused	cause	VERB
ma-134	10	18	by	by	ADP
ma-134	10	19	the	the	DET
ma-134	10	20	different	different	ADJ
ma-134	10	21	properties	property	NOUN
ma-134	10	22	of	of	ADP
ma-134	10	23	the	the	DET
ma-134	10	24	function	function	NOUN
ma-134	10	25	and	and	CCONJ
ma-134	10	26	the	the	DET
ma-134	10	27	presence	presence	NOUN
ma-134	10	28	of	of	ADP
ma-134	10	29	several	several	ADJ
ma-134	10	30	variables	variable	NOUN
ma-134	10	31	inthe	inthe	DET
ma-134	10	32	equation	equation	NOUN
ma-134	10	33	.	.	PUNCT
ma-134	11	1	furthermore	furthermore	ADV
ma-134	11	2	,	,	PUNCT
ma-134	11	3	we	we	PRON
ma-134	11	4	continue	continue	VERB
ma-134	11	5	to	to	PART
ma-134	11	6	construct	construct	VERB
ma-134	11	7	and	and	CCONJ
ma-134	11	8	study	study	VERB
ma-134	11	9	a	a	DET
ma-134	11	10	couple	couple	NOUN
ma-134	11	11	of	of	ADP
ma-134	11	12	functional	functional	ADJ
ma-134	11	13	equations	equation	NOUN
ma-134	11	14	bymaking	bymake	VERB
ma-134	11	15	a	a	DET
ma-134	11	16	new	new	ADJ
ma-134	11	17	direct	direct	ADJ
ma-134	11	18	method	method	NOUN
ma-134	11	19	.	.	PUNCT
ma-134	12	1	1	1	X
ma-134	12	2	.	.	X
ma-134	12	3	introduction	introduction	NOUN
ma-134	12	4	the	the	DET
ma-134	12	5	core	core	ADJ
ma-134	12	6	idea	idea	NOUN
ma-134	12	7	of	of	ADP
ma-134	12	8	the	the	DET
ma-134	12	9	hyers	hyers	PROPN
ma-134	12	10	-	-	PUNCT
ma-134	12	11	ulam	ulam	PROPN
ma-134	12	12	stability	stability	NOUN
ma-134	12	13	for	for	ADP
ma-134	12	14	functional	functional	ADJ
ma-134	12	15	equations	equation	NOUN
ma-134	12	16	has	have	AUX
ma-134	12	17	been	be	AUX
ma-134	12	18	dated	date	VERB
ma-134	12	19	back	back	ADV
ma-134	12	20	to	to	ADP
ma-134	12	21	awell	awell	NOUN
ma-134	12	22	-	-	PUNCT
ma-134	12	23	known	know	VERB
ma-134	12	24	problem	problem	NOUN
ma-134	12	25	concerning	concern	VERB
ma-134	12	26	about	about	ADP
ma-134	12	27	group	group	NOUN
ma-134	12	28	homomorphisms	homomorphism	NOUN
ma-134	12	29	solved	solve	VERB
ma-134	12	30	by	by	ADP
ma-134	12	31	s.m	s.m	PROPN
ma-134	12	32	.	.	PROPN
ma-134	12	33	ulam	ulam	PROPN
ma-134	12	34	and	and	CCONJ
ma-134	12	35	d.h	d.h	PROPN
ma-134	12	36	.	.	PROPN
ma-134	12	37	hyers(see	hyers(see	PROPN
ma-134	13	1	[	[	X
ma-134	13	2	1–3	1–3	NOUN
ma-134	13	3	]	]	X
ma-134	13	4	)	)	PUNCT
ma-134	13	5	.	.	PUNCT
ma-134	14	1	in	in	ADP
ma-134	14	2	the	the	DET
ma-134	14	3	last	last	ADJ
ma-134	14	4	decades	decade	NOUN
ma-134	14	5	,	,	PUNCT
ma-134	14	6	a	a	DET
ma-134	14	7	great	great	ADJ
ma-134	14	8	number	number	NOUN
ma-134	14	9	of	of	ADP
ma-134	14	10	papers	paper	NOUN
ma-134	14	11	treating	treat	VERB
ma-134	14	12	the	the	DET
ma-134	14	13	stability	stability	NOUN
ma-134	14	14	problem	problem	NOUN
ma-134	14	15	aboutfunctional	aboutfunctional	ADJ
ma-134	14	16	equations	equation	NOUN
ma-134	14	17	has	have	AUX
ma-134	14	18	already	already	ADV
ma-134	14	19	been	be	AUX
ma-134	14	20	achieved	achieve	VERB
ma-134	14	21	and	and	CCONJ
ma-134	14	22	a	a	DET
ma-134	14	23	great	great	ADJ
ma-134	14	24	deal	deal	NOUN
ma-134	14	25	of	of	ADP
ma-134	14	26	important	important	ADJ
ma-134	14	27	problems	problem	NOUN
ma-134	14	28	about	about	ADP
ma-134	14	29	thisfield	thisfield	NOUN
ma-134	14	30	has	have	AUX
ma-134	14	31	been	be	AUX
ma-134	14	32	studied	study	VERB
ma-134	14	33	(	(	PUNCT
ma-134	14	34	[	[	X
ma-134	14	35	4–7	4–7	NOUN
ma-134	14	36	]	]	X
ma-134	14	37	)	)	PUNCT
ma-134	14	38	.	.	PUNCT
ma-134	15	1	it	it	PRON
ma-134	15	2	follows	follow	VERB
ma-134	15	3	that	that	SCONJ
ma-134	15	4	the	the	DET
ma-134	15	5	most	most	ADV
ma-134	15	6	efficient	efficient	ADJ
ma-134	15	7	methods	method	NOUN
ma-134	15	8	have	have	AUX
ma-134	15	9	been	be	AUX
ma-134	15	10	stated	state	VERB
ma-134	15	11	in	in	ADP
ma-134	15	12	manypapers	manypaper	NOUN
ma-134	15	13	(	(	PUNCT
ma-134	15	14	[	[	X
ma-134	15	15	10	10	NUM
ma-134	15	16	,	,	PUNCT
ma-134	15	17	18–24,27	18–24,27	NUM
ma-134	15	18	]	]	PUNCT
ma-134	15	19	)	)	PUNCT
ma-134	15	20	such	such	ADJ
ma-134	15	21	as	as	ADP
ma-134	15	22	the	the	DET
ma-134	15	23	direct	direct	ADJ
ma-134	15	24	approach	approach	NOUN
ma-134	15	25	,	,	PUNCT
ma-134	15	26	the	the	DET
ma-134	15	27	shadowing	shadow	VERB
ma-134	15	28	approach	approach	NOUN
ma-134	15	29	,	,	PUNCT
ma-134	15	30	and	and	CCONJ
ma-134	15	31	invariant	invariant	VERB
ma-134	15	32	meanapproach	meanapproach	NOUN
ma-134	15	33	and	and	CCONJ
ma-134	15	34	so	so	ADV
ma-134	15	35	on	on	ADV
ma-134	15	36	.	.	PUNCT
ma-134	16	1	in	in	ADP
ma-134	16	2	particular	particular	ADJ
ma-134	16	3	,	,	PUNCT
ma-134	16	4	the	the	DET
ma-134	16	5	direct	direct	ADJ
ma-134	16	6	method	method	NOUN
ma-134	16	7	is	be	AUX
ma-134	16	8	always	always	ADV
ma-134	16	9	the	the	DET
ma-134	16	10	main	main	ADJ
ma-134	16	11	studying	study	VERB
ma-134	16	12	tool	tool	NOUN
ma-134	16	13	on	on	ADP
ma-134	16	14	theinvestigation	theinvestigation	NOUN
ma-134	16	15	of	of	ADP
ma-134	16	16	functional	functional	ADJ
ma-134	16	17	equations	equation	NOUN
ma-134	16	18	of	of	ADP
ma-134	16	19	different	different	ADJ
ma-134	16	20	types	type	NOUN
ma-134	16	21	.	.	PUNCT
ma-134	17	1	received	receive	VERB
ma-134	17	2	:	:	PUNCT
ma-134	17	3	15	15	NUM
ma-134	17	4	sep	sep	NOUN
ma-134	17	5	.	.	PUNCT
ma-134	18	1	2022	2022	NUM
ma-134	18	2	.	.	PUNCT
ma-134	19	1	key	key	ADJ
ma-134	19	2	words	word	NOUN
ma-134	19	3	and	and	CCONJ
ma-134	19	4	phrases	phrase	NOUN
ma-134	19	5	.	.	PUNCT
ma-134	20	1	stability	stability	NOUN
ma-134	20	2	;	;	PUNCT
ma-134	20	3	several	several	ADJ
ma-134	20	4	functional	functional	ADJ
ma-134	20	5	equations	equation	NOUN
ma-134	20	6	;	;	PUNCT
ma-134	20	7	approximations	approximation	NOUN
ma-134	20	8	;	;	PUNCT
ma-134	20	9	odd	odd	ADJ
ma-134	20	10	function	function	NOUN
ma-134	20	11	;	;	PUNCT
ma-134	20	12	even	even	ADV
ma-134	20	13	function.1	function.1	PROPN
ma-134	20	14	https://adac.ee	https://adac.ee	PROPN
ma-134	20	15	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	20	16	eur	eur	PROPN
ma-134	20	17	.	.	PUNCT
ma-134	21	1	j.	j.	PROPN
ma-134	21	2	math	math	PROPN
ma-134	21	3	.	.	PUNCT
ma-134	22	1	anal	anal	PROPN
ma-134	22	2	.	.	PUNCT
ma-134	23	1	10.28924	10.28924	NUM
ma-134	23	2	/	/	SYM
ma-134	23	3	ada	ada	PROPN
ma-134	23	4	/	/	SYM
ma-134	23	5	ma.3.7	ma.3.7	NOUN
ma-134	23	6	2the	2the	NUM
ma-134	23	7	stability	stability	NOUN
ma-134	23	8	problems	problem	NOUN
ma-134	23	9	for	for	ADP
ma-134	23	10	an	an	DET
ma-134	23	11	appropriate	appropriate	ADJ
ma-134	23	12	simple	simple	ADJ
ma-134	23	13	variable	variable	ADJ
ma-134	23	14	functional	functional	ADJ
ma-134	23	15	equations	equation	NOUN
ma-134	23	16	have	have	AUX
ma-134	23	17	earlier	early	ADV
ma-134	23	18	beeninvestigated	beeninvestigate	VERB
ma-134	23	19	by	by	ADP
ma-134	23	20	direct	direct	ADJ
ma-134	23	21	method	method	NOUN
ma-134	23	22	.	.	PUNCT
ma-134	24	1	the	the	DET
ma-134	24	2	direct	direct	ADJ
ma-134	24	3	method	method	NOUN
ma-134	24	4	is	be	AUX
ma-134	24	5	familiar	familiar	ADJ
ma-134	24	6	with	with	ADP
ma-134	24	7	many	many	ADJ
ma-134	24	8	readers	reader	NOUN
ma-134	24	9	to	to	PART
ma-134	24	10	derive	derive	VERB
ma-134	24	11	thesolutions	thesolution	NOUN
ma-134	24	12	of	of	ADP
ma-134	24	13	the	the	DET
ma-134	24	14	equations	equation	NOUN
ma-134	24	15	.	.	PUNCT
ma-134	25	1	the	the	DET
ma-134	25	2	author	author	NOUN
ma-134	25	3	in	in	ADP
ma-134	25	4	[	[	X
ma-134	25	5	8	8	NUM
ma-134	25	6	]	]	PUNCT
ma-134	25	7	have	have	AUX
ma-134	25	8	made	make	VERB
ma-134	25	9	full	full	ADJ
ma-134	25	10	use	use	NOUN
ma-134	25	11	of	of	ADP
ma-134	25	12	quite	quite	DET
ma-134	25	13	a	a	DET
ma-134	25	14	general	general	ADJ
ma-134	25	15	way	way	NOUN
ma-134	25	16	to	to	PART
ma-134	25	17	solve	solve	VERB
ma-134	25	18	thehyers	thehyer	NOUN
ma-134	25	19	-	-	PUNCT
ma-134	25	20	ulam	ulam	ADJ
ma-134	25	21	stability	stability	NOUN
ma-134	25	22	problems	problem	NOUN
ma-134	25	23	on	on	ADP
ma-134	25	24	the	the	DET
ma-134	25	25	functional	functional	ADJ
ma-134	25	26	equations	equation	NOUN
ma-134	25	27	under	under	ADP
ma-134	25	28	which	which	PRON
ma-134	25	29	many	many	ADJ
ma-134	25	30	excellent	excellent	ADJ
ma-134	25	31	outcomeshave	outcomeshave	NOUN
ma-134	25	32	been	be	AUX
ma-134	25	33	achieved	achieve	VERB
ma-134	25	34	without	without	ADP
ma-134	25	35	reduplicating	reduplicate	VERB
ma-134	25	36	the	the	DET
ma-134	25	37	similar	similar	ADJ
ma-134	25	38	procedure	procedure	NOUN
ma-134	25	39	in	in	ADP
ma-134	25	40	the	the	DET
ma-134	25	41	whole	whole	ADJ
ma-134	25	42	process	process	NOUN
ma-134	25	43	of	of	ADP
ma-134	25	44	computation.however	computation.however	ADV
ma-134	25	45	,	,	PUNCT
ma-134	25	46	her	her	PRON
ma-134	25	47	results	result	NOUN
ma-134	25	48	can	can	AUX
ma-134	25	49	only	only	ADV
ma-134	25	50	be	be	AUX
ma-134	25	51	used	use	VERB
ma-134	25	52	to	to	PART
ma-134	25	53	derive	derive	VERB
ma-134	25	54	the	the	DET
ma-134	25	55	solutions	solution	NOUN
ma-134	25	56	of	of	ADP
ma-134	25	57	the	the	DET
ma-134	25	58	equation	equation	NOUN
ma-134	25	59	where	where	SCONJ
ma-134	25	60	the	the	DET
ma-134	25	61	mediatefunction	mediatefunction	NOUN
ma-134	25	62	is	be	AUX
ma-134	25	63	odd	odd	ADJ
ma-134	25	64	.	.	PUNCT
ma-134	26	1	this	this	PRON
ma-134	26	2	is	be	AUX
ma-134	26	3	exactly	exactly	ADV
ma-134	26	4	our	our	PRON
ma-134	26	5	contribution	contribution	NOUN
ma-134	26	6	to	to	ADP
ma-134	26	7	the	the	DET
ma-134	26	8	paper	paper	NOUN
ma-134	26	9	.	.	PUNCT
ma-134	27	1	in	in	ADP
ma-134	27	2	fact	fact	NOUN
ma-134	27	3	,	,	PUNCT
ma-134	27	4	a	a	DET
ma-134	27	5	straightforward	straightforward	ADJ
ma-134	27	6	observationis	observationis	NOUN
ma-134	27	7	that	that	PRON
ma-134	27	8	the	the	DET
ma-134	27	9	inequality	inequality	NOUN
ma-134	27	10	‖f	‖f	PRON
ma-134	27	11	(	(	PUNCT
ma-134	27	12	x)−	x)−	PROPN
ma-134	27	13	uf	uf	PROPN
ma-134	27	14	(	(	PUNCT
ma-134	27	15	e(x))−	e(x))−	NOUN
ma-134	27	16	vf	vf	X
ma-134	27	17	(	(	PUNCT
ma-134	27	18	−e(x))‖	−e(x))‖	PROPN
ma-134	27	19	6	6	NUM
ma-134	27	20	δ(x	δ(x	NOUN
ma-134	27	21	)	)	PUNCT
ma-134	27	22	can	can	AUX
ma-134	27	23	be	be	AUX
ma-134	27	24	solved	solve	VERB
ma-134	27	25	if	if	SCONJ
ma-134	27	26	the	the	DET
ma-134	27	27	function	function	NOUN
ma-134	27	28	h	h	NOUN
ma-134	27	29	is	be	AUX
ma-134	27	30	even	even	ADV
ma-134	27	31	.	.	PUNCT
ma-134	28	1	next	next	ADV
ma-134	28	2	,	,	PUNCT
ma-134	28	3	the	the	DET
ma-134	28	4	present	present	ADJ
ma-134	28	5	studying	study	VERB
ma-134	28	6	approach	approach	NOUN
ma-134	28	7	calls	call	VERB
ma-134	28	8	us	we	PRON
ma-134	28	9	to	to	ADP
ma-134	28	10	investigatethe	investigatethe	DET
ma-134	28	11	following	follow	VERB
ma-134	28	12	functional	functional	ADJ
ma-134	28	13	inequality	inequality	NOUN
ma-134	28	14	,	,	PUNCT
ma-134	28	15	by	by	ADP
ma-134	28	16	using	use	VERB
ma-134	28	17	a	a	DET
ma-134	28	18	direct	direct	ADJ
ma-134	28	19	method	method	NOUN
ma-134	28	20	,	,	PUNCT
ma-134	28	21	under	under	ADP
ma-134	28	22	which	which	PRON
ma-134	28	23	the	the	DET
ma-134	28	24	result	result	NOUN
ma-134	28	25	can	can	AUX
ma-134	28	26	not	not	PART
ma-134	28	27	becovered	becovere	VERB
ma-134	28	28	by	by	ADP
ma-134	28	29	earlier	early	ADJ
ma-134	28	30	works	work	NOUN
ma-134	28	31	‖f	‖f	PUNCT
ma-134	28	32	(	(	PUNCT
ma-134	28	33	x	x	X
ma-134	29	1	+	+	NUM
ma-134	29	2	y	y	PROPN
ma-134	29	3	+	+	PROPN
ma-134	29	4	z	z	NOUN
ma-134	29	5	)	)	PUNCT
ma-134	30	1	+	+	CCONJ
ma-134	30	2	f	f	X
ma-134	30	3	(	(	PUNCT
ma-134	30	4	x	x	X
ma-134	30	5	)	)	PUNCT
ma-134	31	1	+	+	NUM
ma-134	31	2	f	f	X
ma-134	31	3	(	(	PUNCT
ma-134	31	4	y	y	NOUN
ma-134	31	5	)	)	PUNCT
ma-134	32	1	+	+	NUM
ma-134	32	2	f	f	X
ma-134	32	3	(	(	PUNCT
ma-134	32	4	z)−	z)−	PROPN
ma-134	32	5	f	f	X
ma-134	32	6	(	(	PUNCT
ma-134	32	7	x	x	PROPN
ma-134	32	8	+	+	PUNCT
ma-134	32	9	y)−	y)−	PROPN
ma-134	32	10	f	f	NOUN
ma-134	32	11	(	(	PUNCT
ma-134	32	12	z	z	PROPN
ma-134	32	13	+	+	NOUN
ma-134	32	14	y)−	y)−	PROPN
ma-134	32	15	f	f	NOUN
ma-134	32	16	(	(	PUNCT
ma-134	32	17	x	x	PROPN
ma-134	32	18	+	+	PRON
ma-134	32	19	z)‖	z)‖	NUM
ma-134	32	20	6	6	NUM
ma-134	32	21	k	k	NOUN
ma-134	32	22	(	(	PUNCT
ma-134	32	23	‖x‖r	‖x‖r	NOUN
ma-134	32	24	+	+	CCONJ
ma-134	32	25	‖y‖r	‖y‖r	ADJ
ma-134	32	26	+	+	CCONJ
ma-134	32	27	‖z‖r	‖z‖r	NOUN
ma-134	32	28	)	)	PUNCT
ma-134	32	29	.	.	PUNCT
ma-134	33	1	(	(	PUNCT
ma-134	33	2	1.1	1.1	NUM
ma-134	33	3	)	)	PUNCT
ma-134	33	4	in	in	ADP
ma-134	33	5	fact	fact	NOUN
ma-134	33	6	,	,	PUNCT
ma-134	33	7	the	the	DET
ma-134	33	8	functional	functional	ADJ
ma-134	33	9	inequality	inequality	NOUN
ma-134	33	10	(	(	PUNCT
ma-134	33	11	1.1	1.1	NUM
ma-134	33	12	)	)	PUNCT
ma-134	33	13	comes	come	VERB
ma-134	33	14	from	from	ADP
ma-134	33	15	some	some	DET
ma-134	33	16	equivalent	equivalent	ADJ
ma-134	33	17	characterizations	characterization	NOUN
ma-134	33	18	of	of	ADP
ma-134	33	19	hilbert	hilbert	PROPN
ma-134	33	20	spacein	spacein	PROPN
ma-134	33	21	[	[	X
ma-134	33	22	15	15	NUM
ma-134	33	23	]	]	PUNCT
ma-134	33	24	.	.	PUNCT
ma-134	34	1	the	the	DET
ma-134	34	2	investigator	investigator	NOUN
ma-134	34	3	described	describe	VERB
ma-134	34	4	several	several	ADJ
ma-134	34	5	properties	property	NOUN
ma-134	34	6	of	of	ADP
ma-134	34	7	an	an	DET
ma-134	34	8	inner	inner	ADJ
ma-134	34	9	product	product	NOUN
ma-134	34	10	space	space	NOUN
ma-134	34	11	and	and	CCONJ
ma-134	34	12	applies	apply	VERB
ma-134	34	13	theseresults	theseresult	NOUN
ma-134	34	14	to	to	PART
ma-134	34	15	solve	solve	VERB
ma-134	34	16	many	many	ADJ
ma-134	34	17	interesting	interesting	ADJ
ma-134	34	18	functional	functional	ADJ
ma-134	34	19	inequalities	inequality	NOUN
ma-134	34	20	such	such	ADJ
ma-134	34	21	as	as	ADP
ma-134	34	22	:	:	PUNCT
ma-134	34	23	zarantone	zarantone	PROPN
ma-134	34	24	’s	’s	PART
ma-134	34	25	inequality	inequality	NOUN
ma-134	34	26	,	,	PUNCT
ma-134	34	27	hayashi’sinequality	hayashi’sinequality	NOUN
ma-134	34	28	and	and	CCONJ
ma-134	34	29	so	so	ADV
ma-134	34	30	on	on	ADV
ma-134	34	31	.	.	PUNCT
ma-134	35	1	however	however	ADV
ma-134	35	2	,	,	PUNCT
ma-134	35	3	the	the	DET
ma-134	35	4	more	more	ADV
ma-134	35	5	far	far	ADV
ma-134	35	6	reaching	reach	VERB
ma-134	35	7	work	work	NOUN
ma-134	35	8	can	can	AUX
ma-134	35	9	be	be	AUX
ma-134	35	10	done	do	VERB
ma-134	35	11	m.	m.	NOUN
ma-134	35	12	fréchet	fréchet	VERB
ma-134	35	13	in	in	ADP
ma-134	35	14	[	[	X
ma-134	35	15	16	16	NUM
ma-134	35	16	]	]	PUNCT
ma-134	35	17	underwhich	underwhich	PRON
ma-134	35	18	he	he	PRON
ma-134	35	19	ascertained	ascertain	VERB
ma-134	35	20	that	that	SCONJ
ma-134	35	21	the	the	DET
ma-134	35	22	corresponding	corresponding	ADJ
ma-134	35	23	equation	equation	NOUN
ma-134	35	24	is	be	AUX
ma-134	35	25	a	a	DET
ma-134	35	26	necessary	necessary	ADJ
ma-134	35	27	prerequisite	prerequisite	NOUN
ma-134	35	28	condition	condition	NOUN
ma-134	35	29	whencomplex	whencomplex	NOUN
ma-134	35	30	or	or	CCONJ
ma-134	35	31	real	real	ADV
ma-134	35	32	normed	norme	VERB
ma-134	35	33	completed	complete	VERB
ma-134	35	34	spaces	space	NOUN
ma-134	35	35	become	become	VERB
ma-134	35	36	hilbert	hilbert	NOUN
ma-134	35	37	spaces	space	NOUN
ma-134	35	38	.	.	PUNCT
ma-134	36	1	investigator	investigator	NOUN
ma-134	36	2	in	in	ADP
ma-134	36	3	[	[	X
ma-134	36	4	17	17	NUM
ma-134	36	5	]	]	PUNCT
ma-134	36	6	studied	study	VERB
ma-134	36	7	thestability	thestability	NOUN
ma-134	36	8	of	of	ADP
ma-134	36	9	fréchet	fréchet	ADJ
ma-134	36	10	functional	functional	ADJ
ma-134	36	11	equation	equation	NOUN
ma-134	36	12	from	from	ADP
ma-134	36	13	which	which	PRON
ma-134	36	14	a	a	DET
ma-134	36	15	characterization	characterization	NOUN
ma-134	36	16	of	of	ADP
ma-134	36	17	inner	inner	ADJ
ma-134	36	18	product	product	NOUN
ma-134	36	19	spaces	space	NOUN
ma-134	36	20	hadbeen	hadbeen	AUX
ma-134	36	21	achieved	achieve	VERB
ma-134	36	22	by	by	ADP
ma-134	36	23	using	use	VERB
ma-134	36	24	a	a	DET
ma-134	36	25	stationary	stationary	ADJ
ma-134	36	26	point	point	NOUN
ma-134	36	27	theorem	theorem	VERB
ma-134	36	28	in	in	ADP
ma-134	36	29	banach	banach	NOUN
ma-134	36	30	spaces	space	NOUN
ma-134	36	31	.	.	PUNCT
ma-134	37	1	compared	compare	VERB
ma-134	37	2	with	with	ADP
ma-134	37	3	the	the	DET
ma-134	37	4	beforestudying	beforestudying	NOUN
ma-134	37	5	approaches	approach	NOUN
ma-134	37	6	,	,	PUNCT
ma-134	37	7	we	we	PRON
ma-134	37	8	further	far	ADV
ma-134	37	9	explored	explore	VERB
ma-134	37	10	solutions	solution	NOUN
ma-134	37	11	of	of	ADP
ma-134	37	12	the	the	DET
ma-134	37	13	equation	equation	NOUN
ma-134	37	14	(	(	PUNCT
ma-134	37	15	1.1	1.1	NUM
ma-134	37	16	)	)	PUNCT
ma-134	37	17	in	in	ADP
ma-134	37	18	this	this	DET
ma-134	37	19	literature	literature	NOUN
ma-134	37	20	.	.	PUNCT
ma-134	38	1	of	of	ADP
ma-134	38	2	course	course	NOUN
ma-134	38	3	,	,	PUNCT
ma-134	38	4	to	to	ADP
ma-134	38	5	the	the	DET
ma-134	38	6	best	good	ADJ
ma-134	38	7	of	of	ADP
ma-134	38	8	our	our	PRON
ma-134	38	9	knowledge	knowledge	NOUN
ma-134	38	10	,	,	PUNCT
ma-134	38	11	it	it	PRON
ma-134	38	12	has	have	AUX
ma-134	38	13	also	also	ADV
ma-134	38	14	already	already	ADV
ma-134	38	15	been	be	AUX
ma-134	38	16	solved	solve	VERB
ma-134	38	17	by	by	ADP
ma-134	38	18	[	[	X
ma-134	38	19	8	8	NUM
ma-134	38	20	]	]	PUNCT
ma-134	38	21	under	under	ADP
ma-134	38	22	which	which	PRON
ma-134	38	23	a	a	DET
ma-134	38	24	direct	direct	ADJ
ma-134	38	25	methodwas	methodwas	NOUN
ma-134	38	26	to	to	PART
ma-134	38	27	derive	derive	VERB
ma-134	38	28	solutions	solution	NOUN
ma-134	38	29	of	of	ADP
ma-134	38	30	the	the	DET
ma-134	38	31	equation	equation	NOUN
ma-134	38	32	(	(	PUNCT
ma-134	38	33	1.1	1.1	NUM
ma-134	38	34	)	)	PUNCT
ma-134	38	35	and	and	CCONJ
ma-134	38	36	to	to	PART
ma-134	38	37	look	look	VERB
ma-134	38	38	for	for	ADP
ma-134	38	39	some	some	DET
ma-134	38	40	improvement	improvement	NOUN
ma-134	38	41	approximations.however	approximations.however	NOUN
ma-134	38	42	,	,	PUNCT
ma-134	38	43	in	in	ADP
ma-134	38	44	this	this	DET
ma-134	38	45	literature	literature	NOUN
ma-134	38	46	we	we	PRON
ma-134	38	47	make	make	VERB
ma-134	38	48	a	a	DET
ma-134	38	49	new	new	ADJ
ma-134	38	50	direct	direct	ADJ
ma-134	38	51	method	method	NOUN
ma-134	38	52	to	to	PART
ma-134	38	53	achieve	achieve	VERB
ma-134	38	54	the	the	DET
ma-134	38	55	solution	solution	NOUN
ma-134	38	56	of	of	ADP
ma-134	38	57	equation	equation	NOUN
ma-134	38	58	(	(	PUNCT
ma-134	38	59	1.1)must	1.1)must	NUM
ma-134	38	60	be	be	AUX
ma-134	38	61	close	close	ADJ
ma-134	38	62	to	to	ADP
ma-134	38	63	the	the	DET
ma-134	38	64	approximate	approximate	ADJ
ma-134	38	65	solution	solution	NOUN
ma-134	38	66	,	,	PUNCT
ma-134	38	67	approximately	approximately	ADV
ma-134	38	68	satisfying	satisfy	VERB
ma-134	38	69	the	the	DET
ma-134	38	70	corresponding	correspond	VERB
ma-134	38	71	equation.besides	equation.beside	NOUN
ma-134	38	72	this	this	PRON
ma-134	38	73	,	,	PUNCT
ma-134	38	74	we	we	PRON
ma-134	38	75	will	will	AUX
ma-134	38	76	consider	consider	VERB
ma-134	38	77	that	that	SCONJ
ma-134	38	78	the	the	DET
ma-134	38	79	functions	function	NOUN
ma-134	38	80	on	on	ADP
ma-134	38	81	the	the	DET
ma-134	38	82	functional	functional	ADJ
ma-134	38	83	equation	equation	NOUN
ma-134	38	84	of	of	ADP
ma-134	38	85	different	different	ADJ
ma-134	38	86	typeshave	typeshave	NOUN
ma-134	38	87	been	be	AUX
ma-134	38	88	defined	define	VERB
ma-134	38	89	in	in	ADP
ma-134	38	90	a	a	DET
ma-134	38	91	more	more	ADV
ma-134	38	92	general	general	ADJ
ma-134	38	93	domain	domain	NOUN
ma-134	38	94	.	.	PUNCT
ma-134	39	1	for	for	ADP
ma-134	39	2	instance	instance	NOUN
ma-134	39	3	,	,	PUNCT
ma-134	39	4	the	the	DET
ma-134	39	5	papers	paper	NOUN
ma-134	39	6	[	[	X
ma-134	39	7	11	11	NUM
ma-134	39	8	,	,	PUNCT
ma-134	39	9	12	12	NUM
ma-134	39	10	]	]	PUNCT
ma-134	39	11	have	have	AUX
ma-134	39	12	defined	define	VERB
ma-134	39	13	anadditive	anadditive	ADJ
ma-134	39	14	ρ	ρ	ADJ
ma-134	39	15	-	-	ADJ
ma-134	39	16	functional	functional	ADJ
ma-134	39	17	inequalities	inequality	NOUN
ma-134	39	18	in	in	ADP
ma-134	39	19	nonarchimedean	nonarchimedean	ADJ
ma-134	39	20	normed	normed	ADJ
ma-134	39	21	spaces	space	NOUN
ma-134	39	22	and	and	CCONJ
ma-134	39	23	banach	banach	NOUN
ma-134	39	24	spaces	space	NOUN
ma-134	39	25	.	.	PUNCT
ma-134	40	1	however	however	ADV
ma-134	40	2	,	,	PUNCT
ma-134	40	3	this	this	DET
ma-134	40	4	phenomenon	phenomenon	NOUN
ma-134	40	5	can	can	AUX
ma-134	40	6	not	not	PART
ma-134	40	7	attract	attract	VERB
ma-134	40	8	enough	enough	ADJ
ma-134	40	9	attention	attention	NOUN
ma-134	40	10	to	to	ADP
ma-134	40	11	the	the	DET
ma-134	40	12	study	study	NOUN
ma-134	40	13	of	of	ADP
ma-134	40	14	functional	functional	ADJ
ma-134	40	15	equations	equation	NOUN
ma-134	40	16	in	in	ADP
ma-134	40	17	themore	themore	NOUN
ma-134	40	18	general	general	ADJ
ma-134	40	19	and	and	CCONJ
ma-134	40	20	complex	complex	ADJ
ma-134	40	21	nonlinear	nonlinear	ADJ
ma-134	40	22	structure	structure	NOUN
ma-134	40	23	of	of	ADP
ma-134	40	24	f	f	NOUN
ma-134	40	25	-	-	PUNCT
ma-134	40	26	spaces	space	NOUN
ma-134	40	27	(	(	PUNCT
ma-134	40	28	see	see	VERB
ma-134	40	29	the	the	DET
ma-134	40	30	definition	definition	NOUN
ma-134	40	31	in	in	ADP
ma-134	40	32	[	[	X
ma-134	40	33	13	13	NUM
ma-134	40	34	,	,	PUNCT
ma-134	40	35	14	14	NUM
ma-134	40	36	]	]	PUNCT
ma-134	40	37	)	)	PUNCT
ma-134	40	38	.	.	PUNCT
ma-134	41	1	but	but	CCONJ
ma-134	41	2	,	,	PUNCT
ma-134	41	3	the	the	DET
ma-134	41	4	nonlinear	nonlinear	ADJ
ma-134	41	5	structure	structure	NOUN
ma-134	41	6	of	of	ADP
ma-134	41	7	space	space	NOUN
ma-134	41	8	has	have	AUX
ma-134	41	9	always	always	ADV
ma-134	41	10	stood	stand	VERB
ma-134	41	11	in	in	ADP
ma-134	41	12	a	a	DET
ma-134	41	13	very	very	ADV
ma-134	41	14	important	important	ADJ
ma-134	41	15	position	position	NOUN
ma-134	41	16	of	of	ADP
ma-134	41	17	leadership	leadership	NOUN
ma-134	41	18	infunctional	infunctional	ADJ
ma-134	41	19	analysis	analysis	NOUN
ma-134	41	20	.	.	PUNCT
ma-134	42	1	based	base	VERB
ma-134	42	2	on	on	ADP
ma-134	42	3	the	the	DET
ma-134	42	4	above	above	ADJ
ma-134	42	5	analysis	analysis	NOUN
ma-134	42	6	,	,	PUNCT
ma-134	42	7	it	it	PRON
ma-134	42	8	is	be	AUX
ma-134	42	9	of	of	ADP
ma-134	42	10	great	great	ADJ
ma-134	42	11	significance	significance	NOUN
ma-134	42	12	that	that	SCONJ
ma-134	42	13	the	the	DET
ma-134	42	14	functionalinequality	functionalinequality	NOUN
ma-134	42	15	is	be	AUX
ma-134	42	16	considered	consider	VERB
ma-134	42	17	in	in	ADP
ma-134	42	18	β	β	ADJ
ma-134	42	19	-	-	ADJ
ma-134	42	20	homogeneous	homogeneous	ADJ
ma-134	42	21	f	f	NOUN
ma-134	42	22	-	-	NOUN
ma-134	42	23	space	space	NOUN
ma-134	42	24	.	.	PUNCT
ma-134	43	1	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	43	2	eur	eur	PROPN
ma-134	43	3	.	.	PUNCT
ma-134	44	1	j.	j.	PROPN
ma-134	44	2	math	math	PROPN
ma-134	44	3	.	.	PUNCT
ma-134	45	1	anal	anal	PROPN
ma-134	45	2	.	.	PUNCT
ma-134	46	1	10.28924	10.28924	NUM
ma-134	46	2	/	/	SYM
ma-134	46	3	ada	ada	PROPN
ma-134	46	4	/	/	SYM
ma-134	46	5	ma.3.7	ma.3.7	NOUN
ma-134	46	6	3	3	NUM
ma-134	46	7	in	in	ADP
ma-134	46	8	section	section	NOUN
ma-134	46	9	2	2	NUM
ma-134	46	10	,	,	PUNCT
ma-134	46	11	the	the	DET
ma-134	46	12	counterpart	counterpart	NOUN
ma-134	46	13	of	of	ADP
ma-134	46	14	theorem	theorem	NOUN
ma-134	46	15	2.1	2.1	NUM
ma-134	46	16	from	from	ADP
ma-134	46	17	[	[	X
ma-134	46	18	8	8	NUM
ma-134	46	19	]	]	PUNCT
ma-134	46	20	where	where	SCONJ
ma-134	46	21	the	the	DET
ma-134	46	22	mediate	mediate	ADJ
ma-134	46	23	function	function	NOUN
ma-134	46	24	is	be	AUX
ma-134	46	25	odd	odd	ADJ
ma-134	46	26	willbe	willbe	NOUN
ma-134	46	27	considered	consider	VERB
ma-134	46	28	.	.	PUNCT
ma-134	47	1	in	in	ADP
ma-134	47	2	the	the	DET
ma-134	47	3	subsequent	subsequent	ADJ
ma-134	47	4	part	part	NOUN
ma-134	47	5	,	,	PUNCT
ma-134	47	6	a	a	DET
ma-134	47	7	new	new	ADJ
ma-134	47	8	direct	direct	ADJ
ma-134	47	9	method	method	NOUN
ma-134	47	10	for	for	ADP
ma-134	47	11	solving	solve	VERB
ma-134	47	12	(	(	PUNCT
ma-134	47	13	1.1	1.1	NUM
ma-134	47	14	)	)	PUNCT
ma-134	47	15	in	in	ADP
ma-134	47	16	f	f	NOUN
ma-134	47	17	-	-	PUNCT
ma-134	47	18	space	space	NOUN
ma-134	47	19	will	will	AUX
ma-134	47	20	bedescribed	bedescribe	VERB
ma-134	47	21	and	and	CCONJ
ma-134	47	22	some	some	DET
ma-134	47	23	new	new	ADJ
ma-134	47	24	extended	extended	ADJ
ma-134	47	25	results	result	NOUN
ma-134	47	26	of	of	ADP
ma-134	47	27	theorem	theorem	NOUN
ma-134	47	28	2.1	2.1	NUM
ma-134	47	29	from	from	ADP
ma-134	47	30	[	[	X
ma-134	47	31	8	8	NUM
ma-134	47	32	]	]	PUNCT
ma-134	47	33	will	will	AUX
ma-134	47	34	be	be	AUX
ma-134	47	35	presented	present	VERB
ma-134	47	36	.	.	PUNCT
ma-134	48	1	with	with	ADP
ma-134	48	2	it	it	PRON
ma-134	48	3	,	,	PUNCT
ma-134	48	4	twonew	twonew	VERB
ma-134	48	5	different	different	ADJ
ma-134	48	6	applications	application	NOUN
ma-134	48	7	of	of	ADP
ma-134	48	8	the	the	DET
ma-134	48	9	results	result	NOUN
ma-134	48	10	will	will	AUX
ma-134	48	11	be	be	AUX
ma-134	48	12	described	describe	VERB
ma-134	48	13	in	in	ADP
ma-134	48	14	the	the	DET
ma-134	48	15	final	final	ADJ
ma-134	48	16	part	part	NOUN
ma-134	48	17	.	.	PUNCT
ma-134	49	1	2	2	X
ma-134	49	2	.	.	X
ma-134	49	3	a	a	DET
ma-134	49	4	simple	simple	ADJ
ma-134	49	5	variable	variable	NOUN
ma-134	49	6	of	of	ADP
ma-134	49	7	abstract	abstract	ADJ
ma-134	49	8	equation	equation	NOUN
ma-134	49	9	in	in	ADP
ma-134	49	10	theorem	theorem	NOUN
ma-134	49	11	2.1	2.1	NUM
ma-134	49	12	from	from	ADP
ma-134	49	13	[	[	X
ma-134	49	14	8	8	NUM
ma-134	49	15	]	]	PUNCT
ma-134	49	16	,	,	PUNCT
ma-134	49	17	sikorska	sikorska	PROPN
ma-134	49	18	solved	solve	VERB
ma-134	49	19	the	the	DET
ma-134	49	20	equation	equation	NOUN
ma-134	49	21	(	(	PUNCT
ma-134	49	22	2.1	2.1	NUM
ma-134	49	23	)	)	PUNCT
ma-134	49	24	where	where	SCONJ
ma-134	49	25	the	the	DET
ma-134	49	26	mediate	mediate	ADJ
ma-134	49	27	function	function	NOUN
ma-134	49	28	e	e	NOUN
ma-134	49	29	is	be	AUX
ma-134	49	30	oddand	oddand	VERB
ma-134	49	31	the	the	DET
ma-134	49	32	related	relate	VERB
ma-134	49	33	parameters	parameter	NOUN
ma-134	49	34	u	u	NOUN
ma-134	49	35	,	,	PUNCT
ma-134	49	36	v	v	NUM
ma-134	49	37	are	be	AUX
ma-134	49	38	restricted	restrict	VERB
ma-134	49	39	on	on	ADP
ma-134	49	40	the	the	DET
ma-134	49	41	real	real	ADJ
ma-134	49	42	field	field	NOUN
ma-134	49	43	.	.	PUNCT
ma-134	50	1	for	for	ADP
ma-134	50	2	simplicity	simplicity	NOUN
ma-134	50	3	in	in	ADP
ma-134	50	4	notation	notation	NOUN
ma-134	50	5	,	,	PUNCT
ma-134	50	6	weprovide	weprovide	ADP
ma-134	50	7	traditionally	traditionally	ADV
ma-134	50	8	our	our	PRON
ma-134	50	9	first	first	ADJ
ma-134	50	10	result	result	NOUN
ma-134	50	11	with	with	ADP
ma-134	50	12	the	the	DET
ma-134	50	13	studying	study	VERB
ma-134	50	14	mapping	mapping	NOUN
ma-134	50	15	defined	define	VERB
ma-134	50	16	in	in	ADP
ma-134	50	17	banach	banach	NOUN
ma-134	50	18	space	space	NOUN
ma-134	50	19	.	.	PUNCT
ma-134	51	1	by	by	ADP
ma-134	51	2	makinguse	makinguse	NOUN
ma-134	51	3	of	of	ADP
ma-134	51	4	small	small	ADJ
ma-134	51	5	conjectures	conjecture	VERB
ma-134	51	6	the	the	DET
ma-134	51	7	more	more	ADV
ma-134	51	8	general	general	ADJ
ma-134	51	9	form	form	NOUN
ma-134	51	10	of	of	ADP
ma-134	51	11	the	the	DET
ma-134	51	12	results	result	NOUN
ma-134	51	13	will	will	AUX
ma-134	51	14	be	be	AUX
ma-134	51	15	provided	provide	VERB
ma-134	51	16	in	in	ADP
ma-134	51	17	β	β	ADJ
ma-134	51	18	-	-	ADJ
ma-134	51	19	homogeneous	homogeneous	ADJ
ma-134	51	20	f	f	PROPN
ma-134	51	21	-space	-space	NOUN
ma-134	51	22	in	in	ADP
ma-134	51	23	section	section	NOUN
ma-134	51	24	3	3	NUM
ma-134	51	25	.	.	PUNCT
ma-134	52	1	therefore	therefore	ADV
ma-134	52	2	,	,	PUNCT
ma-134	52	3	our	our	PRON
ma-134	52	4	first	first	ADJ
ma-134	52	5	result	result	NOUN
ma-134	52	6	is	be	AUX
ma-134	52	7	simply	simply	ADV
ma-134	52	8	considered	consider	VERB
ma-134	52	9	in	in	ADP
ma-134	52	10	banach	banach	NOUN
ma-134	52	11	space	space	NOUN
ma-134	52	12	.	.	PUNCT
ma-134	53	1	theorem	theorem	VERB
ma-134	53	2	2.1	2.1	NUM
ma-134	53	3	let	let	NOUN
ma-134	53	4	(	(	PUNCT
ma-134	53	5	x,+	x,+	NUM
ma-134	53	6	)	)	PUNCT
ma-134	53	7	be	be	VERB
ma-134	53	8	a	a	DET
ma-134	53	9	group	group	NOUN
ma-134	53	10	,	,	PUNCT
ma-134	53	11	and	and	CCONJ
ma-134	53	12	(	(	PUNCT
ma-134	53	13	y	y	PROPN
ma-134	53	14	,	,	PUNCT
ma-134	53	15	‖	‖	PROPN
ma-134	53	16	·	·	PUNCT
ma-134	53	17	‖	‖	NUM
ma-134	53	18	)	)	PUNCT
ma-134	53	19	be	be	AUX
ma-134	53	20	a	a	DET
ma-134	53	21	banach	banach	NOUN
ma-134	53	22	space	space	NOUN
ma-134	53	23	,	,	PUNCT
ma-134	53	24	and	and	CCONJ
ma-134	53	25	assume	assume	VERB
ma-134	53	26	the	the	DET
ma-134	53	27	mapping	mapping	NOUN
ma-134	53	28	f	f	X
ma-134	53	29	:	:	PUNCT
ma-134	53	30	x	x	X
ma-134	53	31	→	→	SYM
ma-134	53	32	y	y	PROPN
ma-134	53	33	satisfying	satisfy	VERB
ma-134	53	34	the	the	DET
ma-134	53	35	inequality	inequality	NOUN
ma-134	53	36	‖f	‖f	ADP
ma-134	53	37	(	(	PUNCT
ma-134	53	38	x)−	x)−	PROPN
ma-134	53	39	uf	uf	PROPN
ma-134	53	40	(	(	PUNCT
ma-134	53	41	e(x))−	e(x))−	NOUN
ma-134	53	42	vf	vf	X
ma-134	53	43	(	(	PUNCT
ma-134	53	44	−e(x))‖	−e(x))‖	PROPN
ma-134	53	45	6	6	NUM
ma-134	53	46	δ(x	δ(x	NOUN
ma-134	53	47	)	)	PUNCT
ma-134	53	48	,	,	PUNCT
ma-134	53	49	x	x	PUNCT
ma-134	53	50	∈	∈	NOUN
ma-134	53	51	x	x	X
ma-134	53	52	,	,	PUNCT
ma-134	53	53	(	(	PUNCT
ma-134	53	54	2.1	2.1	NUM
ma-134	53	55	)	)	PUNCT
ma-134	53	56	where	where	SCONJ
ma-134	53	57	u	u	NOUN
ma-134	53	58	,	,	PUNCT
ma-134	53	59	v	v	PROPN
ma-134	53	60	∈	∈	PROPN
ma-134	53	61	(	(	PUNCT
ma-134	53	62	−∞,+∞	−∞,+∞	NUM
ma-134	53	63	)	)	PUNCT
ma-134	53	64	,	,	PUNCT
ma-134	53	65	and	and	CCONJ
ma-134	53	66	the	the	DET
ma-134	53	67	mappings	mapping	NOUN
ma-134	53	68	e	e	NOUN
ma-134	53	69	:	:	PUNCT
ma-134	53	70	x	x	X
ma-134	53	71	→	→	SYM
ma-134	53	72	x	x	PROPN
ma-134	53	73	,	,	PUNCT
ma-134	53	74	δ	δ	PROPN
ma-134	53	75	:	:	PUNCT
ma-134	53	76	x	x	X
ma-134	53	77	→	→	PUNCT
ma-134	53	78	[	[	X
ma-134	53	79	0,∞	0,∞	X
ma-134	53	80	)	)	PUNCT
ma-134	53	81	satisfy	satisfy	NOUN
ma-134	53	82	that	that	SCONJ
ma-134	53	83	e	e	NOUN
ma-134	53	84	is	be	AUX
ma-134	53	85	even	even	ADV
ma-134	53	86	(	(	PUNCT
ma-134	53	87	i.e.	i.e.	X
ma-134	53	88	,	,	PUNCT
ma-134	53	89	e(−x	e(−x	NOUN
ma-134	53	90	)	)	PUNCT
ma-134	53	91	=	=	SYM
ma-134	53	92	e(x	e(x	NUM
ma-134	53	93	)	)	PUNCT
ma-134	53	94	for	for	ADP
ma-134	53	95	every	every	DET
ma-134	53	96	x	x	SYM
ma-134	53	97	∈	∈	PROPN
ma-134	53	98	x	x	NOUN
ma-134	53	99	)	)	PUNCT
ma-134	53	100	.	.	PUNCT
ma-134	54	1	let	let	VERB
ma-134	54	2	the	the	DET
ma-134	54	3	infinite	infinite	ADJ
ma-134	54	4	progression	progression	NOUN
ma-134	54	5	∑∞n=0	∑∞n=0	PROPN
ma-134	55	1	[	[	X
ma-134	55	2	|un|	|un|	NOUN
ma-134	55	3	δ	δ	PROPN
ma-134	55	4	(	(	PUNCT
ma-134	55	5	en(x	en(x	X
ma-134	55	6	)	)	PUNCT
ma-134	55	7	)	)	PUNCT
ma-134	56	1	+	+	CCONJ
ma-134	56	2	|vn|δ	|vn|δ	NOUN
ma-134	56	3	(	(	PUNCT
ma-134	56	4	−en(x))]with	−en(x))]with	ADP
ma-134	56	5	u0	u0	ADJ
ma-134	56	6	:	:	PUNCT
ma-134	56	7	=	=	SYM
ma-134	56	8	1	1	NUM
ma-134	56	9	,	,	PUNCT
ma-134	56	10	un	un	PROPN
ma-134	56	11	:	:	PUNCT
ma-134	56	12	=	=	X
ma-134	56	13	[	[	PUNCT
ma-134	56	14	u(u	u(u	SYM
ma-134	56	15	+	+	CCONJ
ma-134	56	16	v)n−1	v)n−1	X
ma-134	56	17	]	]	PUNCT
ma-134	56	18	,	,	PUNCT
ma-134	56	19	v0	v0	NOUN
ma-134	56	20	:	:	PUNCT
ma-134	56	21	=	=	SYM
ma-134	56	22	0	0	NUM
ma-134	56	23	,	,	PUNCT
ma-134	56	24	vn	vn	X
ma-134	56	25	:	:	PUNCT
ma-134	56	26	=	=	X
ma-134	56	27	[	[	PUNCT
ma-134	57	1	v(u	v(u	ADP
ma-134	57	2	+	+	CCONJ
ma-134	57	3	v)n−1	v)n−1	VERB
ma-134	57	4	]	]	PUNCT
ma-134	57	5	,	,	PUNCT
ma-134	57	6	n	n	PROPN
ma-134	57	7	∈	∈	PROPN
ma-134	57	8	n(em	n(em	PROPN
ma-134	57	9	states	state	VERB
ma-134	57	10	the	the	DET
ma-134	57	11	m	m	PROPN
ma-134	57	12	-	-	PUNCT
ma-134	57	13	th	th	VERB
ma-134	57	14	composition	composition	NOUN
ma-134	57	15	of	of	ADP
ma-134	57	16	function	function	NOUN
ma-134	57	17	e	e	NOUN
ma-134	57	18	)	)	PUNCT
ma-134	57	19	,	,	PUNCT
ma-134	57	20	be	be	AUX
ma-134	57	21	assumed	assume	VERB
ma-134	57	22	convergence	convergence	NOUN
ma-134	57	23	for	for	ADP
ma-134	57	24	every	every	DET
ma-134	57	25	x	x	SYM
ma-134	57	26	∈	∈	PROPN
ma-134	57	27	x	x	X
ma-134	57	28	.	.	PUNCT
ma-134	58	1	thenthere	thenthere	PRON
ma-134	58	2	has	have	VERB
ma-134	58	3	a	a	DET
ma-134	58	4	unique	unique	ADJ
ma-134	58	5	even	even	ADV
ma-134	58	6	mapping	map	VERB
ma-134	58	7	g	g	NOUN
ma-134	58	8	:	:	PUNCT
ma-134	58	9	x	x	X
ma-134	58	10	→	→	SYM
ma-134	58	11	y	y	PROPN
ma-134	58	12	satisfying	satisfy	VERB
ma-134	58	13	g(x	g(x	NOUN
ma-134	58	14	)	)	PUNCT
ma-134	58	15	=	=	SYM
ma-134	58	16	ung(en(x	ung(en(x	PROPN
ma-134	58	17	)	)	PUNCT
ma-134	58	18	)	)	PUNCT
ma-134	59	1	+	+	CCONJ
ma-134	59	2	vng(−en(x	vng(−en(x	X
ma-134	59	3	)	)	PUNCT
ma-134	59	4	)	)	PUNCT
ma-134	59	5	,	,	PUNCT
ma-134	59	6	and	and	CCONJ
ma-134	59	7	‖f	‖f	ADP
ma-134	59	8	(	(	PUNCT
ma-134	59	9	x)−	x)−	PROPN
ma-134	59	10	g(x)‖	g(x)‖	PROPN
ma-134	59	11	6	6	NUM
ma-134	59	12	∞∑	∞∑	PROPN
ma-134	59	13	i=0	i=0	PROPN
ma-134	59	14	[	[	PUNCT
ma-134	59	15	|ui	|ui	X
ma-134	59	16	|	|	ADV
ma-134	59	17	δ	δ	PROPN
ma-134	59	18	(	(	PUNCT
ma-134	59	19	e	e	NOUN
ma-134	59	20	i(x	i(x	PROPN
ma-134	59	21	)	)	PUNCT
ma-134	59	22	)	)	PUNCT
ma-134	60	1	+	+	CCONJ
ma-134	60	2	|vi	|vi	NUM
ma-134	60	3	|	|	ADV
ma-134	60	4	δ	δ	PROPN
ma-134	60	5	(	(	PUNCT
ma-134	60	6	−e	−e	NOUN
ma-134	60	7	i(x	i(x	PROPN
ma-134	60	8	)	)	PUNCT
ma-134	60	9	)	)	PUNCT
ma-134	60	10	]	]	PUNCT
ma-134	60	11	,	,	PUNCT
ma-134	60	12	x	x	PUNCT
ma-134	60	13	∈	∈	PROPN
ma-134	60	14	x	x	X
ma-134	60	15	and	and	CCONJ
ma-134	60	16	n	n	PRON
ma-134	60	17	∈	∈	PROPN
ma-134	60	18	n.	n.	NOUN
ma-134	60	19	(	(	PUNCT
ma-134	60	20	2.2	2.2	NUM
ma-134	60	21	)	)	PUNCT
ma-134	60	22	proof	proof	NOUN
ma-134	60	23	.	.	PUNCT
ma-134	61	1	we	we	PRON
ma-134	61	2	will	will	AUX
ma-134	61	3	prove	prove	VERB
ma-134	61	4	that	that	SCONJ
ma-134	61	5	‖f	‖f	ADP
ma-134	61	6	(	(	PUNCT
ma-134	61	7	x)−	x)−	PROPN
ma-134	61	8	unf	unf	PROPN
ma-134	61	9	(	(	PUNCT
ma-134	61	10	en(x))−	en(x))−	NUM
ma-134	61	11	vnf	vnf	NOUN
ma-134	61	12	(	(	PUNCT
ma-134	61	13	−en(x))‖	−en(x))‖	PROPN
ma-134	61	14	6	6	NUM
ma-134	61	15	γn(x	γn(x	NUM
ma-134	61	16	)	)	PUNCT
ma-134	61	17	,	,	PUNCT
ma-134	61	18	x	x	PUNCT
ma-134	61	19	∈	∈	NOUN
ma-134	61	20	x	x	X
ma-134	61	21	,	,	PUNCT
ma-134	61	22	(	(	PUNCT
ma-134	61	23	2.3	2.3	NUM
ma-134	61	24	)	)	PUNCT
ma-134	61	25	where	where	SCONJ
ma-134	61	26	γn(x	γn(x	NUM
ma-134	61	27	)	)	PUNCT
ma-134	61	28	:	:	PUNCT
ma-134	62	1	=	=	PUNCT
ma-134	62	2	n−1∑	n−1∑	PROPN
ma-134	62	3	i=0	i=0	PROPN
ma-134	62	4	[	[	PUNCT
ma-134	62	5	|ui	|ui	X
ma-134	62	6	|	|	ADV
ma-134	62	7	δ	δ	PROPN
ma-134	62	8	(	(	PUNCT
ma-134	62	9	e	e	NOUN
ma-134	62	10	i(x	i(x	PROPN
ma-134	62	11	)	)	PUNCT
ma-134	62	12	)	)	PUNCT
ma-134	63	1	+	+	CCONJ
ma-134	63	2	|vi	|vi	NUM
ma-134	63	3	|	|	ADV
ma-134	63	4	δ	δ	PROPN
ma-134	63	5	(	(	PUNCT
ma-134	63	6	−e	−e	NOUN
ma-134	63	7	i(x	i(x	PROPN
ma-134	63	8	)	)	PUNCT
ma-134	63	9	)	)	PUNCT
ma-134	63	10	]	]	PUNCT
ma-134	63	11	,	,	PUNCT
ma-134	63	12	x	x	PUNCT
ma-134	63	13	∈	∈	PROPN
ma-134	63	14	x	x	NOUN
ma-134	63	15	,	,	PUNCT
ma-134	63	16	n	n	PROPN
ma-134	63	17	∈	∈	PROPN
ma-134	63	18	n.	n.	NOUN
ma-134	63	19	first	first	ADV
ma-134	63	20	of	of	ADP
ma-134	63	21	all	all	PRON
ma-134	63	22	,	,	PUNCT
ma-134	63	23	consider	consider	VERB
ma-134	63	24	with	with	ADP
ma-134	63	25	every	every	DET
ma-134	63	26	m	m	NOUN
ma-134	63	27	,	,	PUNCT
ma-134	63	28	n	n	PROPN
ma-134	63	29	∈	∈	PROPN
ma-134	63	30	n	n	NOUN
ma-134	63	31	and	and	CCONJ
ma-134	63	32	it	it	PRON
ma-134	63	33	is	be	AUX
ma-134	63	34	easy	easy	ADJ
ma-134	63	35	to	to	PART
ma-134	63	36	observe	observe	VERB
ma-134	63	37	that	that	PRON
ma-134	63	38	un+1	un+1	NOUN
ma-134	63	39	=	=	SYM
ma-134	63	40	uun	uun	PROPN
ma-134	63	41	+	+	CCONJ
ma-134	63	42	uvn	uvn	PROPN
ma-134	63	43	,	,	PUNCT
ma-134	63	44	vn+1	vn+1	PROPN
ma-134	63	45	=	=	SYM
ma-134	63	46	vvn	vvn	PROPN
ma-134	63	47	+	+	CCONJ
ma-134	63	48	vun	vun	PROPN
ma-134	63	49	,	,	PUNCT
ma-134	63	50	and	and	CCONJ
ma-134	63	51	un+m	un+m	PROPN
ma-134	63	52	=	=	PUNCT
ma-134	64	1	umun	umun	PROPN
ma-134	64	2	+	+	CCONJ
ma-134	64	3	vmun	vmun	PROPN
ma-134	64	4	,	,	PUNCT
ma-134	64	5	vn+m	vn+m	PROPN
ma-134	64	6	=	=	PUNCT
ma-134	64	7	umvn	umvn	ADJ
ma-134	64	8	+	+	CCONJ
ma-134	64	9	vmvn	vmvn	ADJ
ma-134	64	10	.	.	PUNCT
ma-134	65	1	(	(	PUNCT
ma-134	65	2	2.4	2.4	NUM
ma-134	65	3	)	)	PUNCT
ma-134	65	4	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	65	5	eur	eur	PROPN
ma-134	65	6	.	.	PUNCT
ma-134	66	1	j.	j.	PROPN
ma-134	66	2	math	math	PROPN
ma-134	66	3	.	.	PUNCT
ma-134	67	1	anal	anal	PROPN
ma-134	67	2	.	.	PUNCT
ma-134	68	1	10.28924	10.28924	NUM
ma-134	68	2	/	/	SYM
ma-134	68	3	ada	ada	PROPN
ma-134	68	4	/	/	SYM
ma-134	68	5	ma.3.7	ma.3.7	NOUN
ma-134	68	6	4from	4from	NUM
ma-134	68	7	the	the	DET
ma-134	68	8	definition	definition	NOUN
ma-134	68	9	of	of	ADP
ma-134	68	10	sequences	sequence	NOUN
ma-134	68	11	(	(	PUNCT
ma-134	68	12	un	un	PROPN
ma-134	68	13	)	)	PUNCT
ma-134	68	14	and	and	CCONJ
ma-134	68	15	(	(	PUNCT
ma-134	68	16	vn	vn	NOUN
ma-134	68	17	)	)	PUNCT
ma-134	68	18	we	we	PRON
ma-134	68	19	also	also	ADV
ma-134	68	20	have	have	VERB
ma-134	68	21	uvn	uvn	PROPN
ma-134	68	22	=	=	PROPN
ma-134	68	23	vun	vun	PROPN
ma-134	68	24	,	,	PUNCT
ma-134	68	25	vmun	vmun	NOUN
ma-134	68	26	=	=	SYM
ma-134	68	27	umvn	umvn	ADJ
ma-134	68	28	.	.	PUNCT
ma-134	69	1	first	first	ADV
ma-134	69	2	,	,	PUNCT
ma-134	69	3	(	(	PUNCT
ma-134	69	4	2.1	2.1	NUM
ma-134	69	5	)	)	PUNCT
ma-134	69	6	gives	give	VERB
ma-134	69	7	(	(	PUNCT
ma-134	69	8	2.3	2.3	NUM
ma-134	69	9	)	)	PUNCT
ma-134	69	10	with	with	ADP
ma-134	69	11	setting	set	VERB
ma-134	69	12	n	n	X
ma-134	69	13	=	=	SYM
ma-134	69	14	1	1	NUM
ma-134	69	15	,	,	PUNCT
ma-134	69	16	and	and	CCONJ
ma-134	69	17	by	by	ADP
ma-134	69	18	mathematical	mathematical	ADJ
ma-134	69	19	induction	induction	NOUN
ma-134	69	20	,	,	PUNCT
ma-134	69	21	later	later	ADV
ma-134	69	22	we	we	PRON
ma-134	69	23	suppose	suppose	VERB
ma-134	69	24	that	that	SCONJ
ma-134	69	25	(	(	PUNCT
ma-134	69	26	2.3	2.3	NUM
ma-134	69	27	)	)	PUNCT
ma-134	69	28	establishes	establish	VERB
ma-134	69	29	for	for	ADP
ma-134	69	30	some	some	DET
ma-134	69	31	n	n	PRON
ma-134	69	32	∈	∈	NOUN
ma-134	69	33	n	n	NOUN
ma-134	69	34	.	.	PUNCT
ma-134	70	1	we	we	PRON
ma-134	70	2	prove	prove	VERB
ma-134	70	3	that	that	SCONJ
ma-134	70	4	in	in	ADP
ma-134	70	5	the	the	DET
ma-134	70	6	case	case	NOUN
ma-134	70	7	for	for	ADP
ma-134	70	8	n	n	PROPN
ma-134	70	9	+	+	CCONJ
ma-134	70	10	1	1	NUM
ma-134	70	11	by	by	ADP
ma-134	70	12	virtue	virtue	NOUN
ma-134	70	13	of	of	ADP
ma-134	70	14	(	(	PUNCT
ma-134	70	15	2.1	2.1	NUM
ma-134	70	16	)	)	PUNCT
ma-134	70	17	‖f	‖f	ADP
ma-134	70	18	(	(	PUNCT
ma-134	70	19	x)−	x)−	PROPN
ma-134	70	20	un+1f	un+1f	PROPN
ma-134	70	21	(	(	PUNCT
ma-134	70	22	en+1(x	en+1(x	NOUN
ma-134	70	23	)	)	PUNCT
ma-134	70	24	)	)	PUNCT
ma-134	70	25	−	−	ADP
ma-134	70	26	vn+1f	vn+1f	NOUN
ma-134	70	27	(	(	PUNCT
ma-134	70	28	−en+1(x	−en+1(x	NOUN
ma-134	70	29	)	)	PUNCT
ma-134	70	30	)	)	PUNCT
ma-134	71	1	‖	‖	PROPN
ma-134	71	2	6	6	NUM
ma-134	71	3	‖f	‖f	ADP
ma-134	71	4	(	(	PUNCT
ma-134	71	5	x)−	x)−	PROPN
ma-134	71	6	unf	unf	PROPN
ma-134	71	7	(	(	PUNCT
ma-134	71	8	en(x))−	en(x))−	NUM
ma-134	71	9	vnf	vnf	NOUN
ma-134	71	10	(	(	PUNCT
ma-134	71	11	−en(x))‖	−en(x))‖	NOUN
ma-134	71	12	+	+	NUM
ma-134	71	13	|un|	|un|	NOUN
ma-134	71	14	∥∥f	∥∥f	NOUN
ma-134	71	15	(	(	PUNCT
ma-134	71	16	en(x))−	en(x))−	ADJ
ma-134	71	17	uf	uf	PROPN
ma-134	71	18	(	(	PUNCT
ma-134	71	19	en+1(x	en+1(x	NOUN
ma-134	71	20	)	)	PUNCT
ma-134	71	21	)	)	PUNCT
ma-134	72	1	−	−	PROPN
ma-134	73	1	vf	vf	X
ma-134	73	2	(	(	PUNCT
ma-134	73	3	−en+1(x	−en+1(x	NOUN
ma-134	73	4	)	)	PUNCT
ma-134	73	5	)	)	PUNCT
ma-134	73	6	∥∥	∥∥	PROPN
ma-134	74	1	+	+	CCONJ
ma-134	74	2	|vn|	|vn|	VERB
ma-134	74	3	∥∥f	∥∥f	NOUN
ma-134	74	4	(	(	PUNCT
ma-134	74	5	−en(x))−	−en(x))−	INTJ
ma-134	74	6	uf	uf	PROPN
ma-134	74	7	(	(	PUNCT
ma-134	74	8	en+1(x	en+1(x	NOUN
ma-134	74	9	)	)	PUNCT
ma-134	74	10	)	)	PUNCT
ma-134	75	1	−	−	PROPN
ma-134	75	2	vf	vf	X
ma-134	75	3	(	(	PUNCT
ma-134	75	4	−en+1(x	−en+1(x	NOUN
ma-134	75	5	)	)	PUNCT
ma-134	75	6	)	)	PUNCT
ma-134	75	7	∥∥	∥∥	PROPN
ma-134	75	8	6	6	NUM
ma-134	75	9	n−1∑	n−1∑	PROPN
ma-134	75	10	i=0	i=0	PROPN
ma-134	75	11	[	[	PUNCT
ma-134	75	12	|ui	|ui	X
ma-134	75	13	|	|	ADV
ma-134	75	14	δ	δ	PROPN
ma-134	75	15	(	(	PUNCT
ma-134	75	16	e	e	NOUN
ma-134	75	17	i(x	i(x	PROPN
ma-134	75	18	)	)	PUNCT
ma-134	75	19	)	)	PUNCT
ma-134	76	1	+	+	CCONJ
ma-134	76	2	|vi	|vi	NUM
ma-134	76	3	|	|	ADV
ma-134	76	4	δ	δ	PROPN
ma-134	76	5	(	(	PUNCT
ma-134	76	6	−e	−e	NOUN
ma-134	76	7	i(x	i(x	PROPN
ma-134	76	8	)	)	PUNCT
ma-134	76	9	)	)	PUNCT
ma-134	76	10	]	]	PUNCT
ma-134	77	1	+	+	CCONJ
ma-134	77	2	|un|	|un|	NOUN
ma-134	77	3	δ	δ	PROPN
ma-134	77	4	(	(	PUNCT
ma-134	77	5	en(x	en(x	X
ma-134	77	6	)	)	PUNCT
ma-134	77	7	)	)	PUNCT
ma-134	78	1	+	+	CCONJ
ma-134	78	2	|vn|	|vn|	PROPN
ma-134	78	3	δ	δ	NOUN
ma-134	78	4	(	(	PUNCT
ma-134	78	5	−en(x	−en(x	NOUN
ma-134	78	6	)	)	PUNCT
ma-134	78	7	)	)	PUNCT
ma-134	79	1	=	=	PUNCT
ma-134	79	2	n∑	n∑	PROPN
ma-134	79	3	i=0	i=0	PROPN
ma-134	80	1	[	[	PUNCT
ma-134	80	2	|ui	|ui	X
ma-134	80	3	|	|	ADV
ma-134	80	4	δ	δ	PROPN
ma-134	80	5	(	(	PUNCT
ma-134	80	6	e	e	NOUN
ma-134	80	7	i(x	i(x	PROPN
ma-134	80	8	)	)	PUNCT
ma-134	80	9	)	)	PUNCT
ma-134	81	1	+	+	CCONJ
ma-134	81	2	|vi	|vi	NUM
ma-134	81	3	|	|	ADV
ma-134	81	4	δ	δ	PROPN
ma-134	81	5	(	(	PUNCT
ma-134	81	6	−e	−e	NOUN
ma-134	81	7	i(x	i(x	PROPN
ma-134	81	8	)	)	PUNCT
ma-134	81	9	)	)	PUNCT
ma-134	81	10	]	]	PUNCT
ma-134	81	11	.	.	PUNCT
ma-134	82	1	since	since	SCONJ
ma-134	82	2	the	the	DET
ma-134	82	3	series	series	NOUN
ma-134	82	4	∑∞i=0	∑∞i=0	PROPN
ma-134	82	5	[	[	X
ma-134	82	6	|ui	|ui	X
ma-134	82	7	|	|	ADV
ma-134	82	8	δ	δ	PROPN
ma-134	82	9	(	(	PUNCT
ma-134	82	10	e	e	NOUN
ma-134	82	11	i(x	i(x	PROPN
ma-134	82	12	)	)	PUNCT
ma-134	82	13	)	)	PUNCT
ma-134	83	1	+	+	CCONJ
ma-134	83	2	|vi	|vi	NUM
ma-134	83	3	|	|	ADV
ma-134	83	4	δ	δ	PROPN
ma-134	83	5	(	(	PUNCT
ma-134	83	6	−e	−e	NOUN
ma-134	83	7	i(x	i(x	PROPN
ma-134	83	8	)	)	PUNCT
ma-134	83	9	)	)	PUNCT
ma-134	83	10	]	]	PUNCT
ma-134	83	11	is	be	AUX
ma-134	83	12	convergent	convergent	ADJ
ma-134	83	13	for	for	ADP
ma-134	83	14	every	every	DET
ma-134	83	15	x	x	SYM
ma-134	83	16	∈	∈	PROPN
ma-134	83	17	x	x	X
ma-134	83	18	,	,	PUNCT
ma-134	83	19	combinedwith	combinedwith	PROPN
ma-134	83	20	(	(	PUNCT
ma-134	83	21	2.3	2.3	NUM
ma-134	83	22	)	)	PUNCT
ma-134	83	23	and	and	CCONJ
ma-134	83	24	by	by	ADP
ma-134	83	25	virtue	virtue	NOUN
ma-134	83	26	of	of	ADP
ma-134	83	27	the	the	DET
ma-134	83	28	completeness	completeness	NOUN
ma-134	83	29	of	of	ADP
ma-134	83	30	y	y	PROPN
ma-134	83	31	,	,	PUNCT
ma-134	83	32	the	the	DET
ma-134	83	33	mapping	mapping	NOUN
ma-134	83	34	can	can	AUX
ma-134	83	35	be	be	AUX
ma-134	83	36	well	well	ADV
ma-134	83	37	defined	define	VERB
ma-134	83	38	as	as	ADP
ma-134	83	39	in	in	ADP
ma-134	83	40	thefollowing	thefollowing	NOUN
ma-134	83	41	:	:	PUNCT
ma-134	83	42	g(x	g(x	NOUN
ma-134	83	43	)	)	PUNCT
ma-134	83	44	:	:	PUNCT
ma-134	84	1	=	=	PUNCT
ma-134	84	2	lim	lim	PROPN
ma-134	84	3	n→∞	n→∞	X
ma-134	85	1	[	[	X
ma-134	85	2	unf	unf	INTJ
ma-134	85	3	(	(	PUNCT
ma-134	85	4	en(x	en(x	X
ma-134	85	5	)	)	PUNCT
ma-134	85	6	)	)	PUNCT
ma-134	86	1	+	+	CCONJ
ma-134	86	2	vnf	vnf	NOUN
ma-134	86	3	(	(	PUNCT
ma-134	86	4	−en(x	−en(x	NOUN
ma-134	86	5	)	)	PUNCT
ma-134	86	6	)	)	PUNCT
ma-134	86	7	]	]	PUNCT
ma-134	86	8	,	,	PUNCT
ma-134	86	9	x	x	PUNCT
ma-134	86	10	∈	∈	NOUN
ma-134	86	11	x	x	X
ma-134	86	12	,	,	PUNCT
ma-134	86	13	(	(	PUNCT
ma-134	86	14	2.5	2.5	NUM
ma-134	86	15	)	)	PUNCT
ma-134	86	16	and	and	CCONJ
ma-134	86	17	we	we	PRON
ma-134	86	18	prove	prove	VERB
ma-134	86	19	the	the	DET
ma-134	86	20	following	follow	VERB
ma-134	86	21	properties	property	NOUN
ma-134	86	22	of	of	ADP
ma-134	86	23	the	the	DET
ma-134	86	24	function	function	NOUN
ma-134	86	25	g.an	g.an	PROPN
ma-134	86	26	easy	easy	ADJ
ma-134	86	27	computation	computation	NOUN
ma-134	86	28	is	be	AUX
ma-134	86	29	to	to	PART
ma-134	86	30	prove	prove	VERB
ma-134	86	31	that	that	SCONJ
ma-134	86	32	ug(e(x	ug(e(x	NOUN
ma-134	86	33	)	)	PUNCT
ma-134	86	34	)	)	PUNCT
ma-134	87	1	+	+	CCONJ
ma-134	87	2	vg(−e(x	vg(−e(x	NOUN
ma-134	87	3	)	)	PUNCT
ma-134	87	4	)	)	PUNCT
ma-134	88	1	=	=	PUNCT
ma-134	89	1	u	u	PROPN
ma-134	89	2	lim	lim	PROPN
ma-134	89	3	n→∞	n→∞	X
ma-134	90	1	[	[	PUNCT
ma-134	90	2	unf	unf	INTJ
ma-134	90	3	(	(	PUNCT
ma-134	90	4	en+1(x	en+1(x	NOUN
ma-134	90	5	)	)	PUNCT
ma-134	90	6	)	)	PUNCT
ma-134	91	1	+	+	CCONJ
ma-134	91	2	vnf	vnf	NOUN
ma-134	91	3	(	(	PUNCT
ma-134	91	4	−en+1(x	−en+1(x	NOUN
ma-134	91	5	)	)	PUNCT
ma-134	91	6	)	)	PUNCT
ma-134	91	7	]	]	PUNCT
ma-134	92	1	+	+	CCONJ
ma-134	92	2	v	v	X
ma-134	92	3	lim	lim	PROPN
ma-134	92	4	n→∞	n→∞	X
ma-134	92	5	[	[	PUNCT
ma-134	92	6	unf	unf	INTJ
ma-134	92	7	(	(	PUNCT
ma-134	92	8	en+1(x	en+1(x	NOUN
ma-134	92	9	)	)	PUNCT
ma-134	92	10	)	)	PUNCT
ma-134	93	1	+	+	CCONJ
ma-134	93	2	vnf	vnf	NOUN
ma-134	93	3	(	(	PUNCT
ma-134	93	4	−en+1(x	−en+1(x	NOUN
ma-134	93	5	)	)	PUNCT
ma-134	93	6	)	)	PUNCT
ma-134	93	7	]	]	PUNCT
ma-134	94	1	=	=	PUNCT
ma-134	94	2	lim	lim	PROPN
ma-134	94	3	n→∞	n→∞	X
ma-134	95	1	[	[	PUNCT
ma-134	95	2	(	(	PUNCT
ma-134	95	3	uun	uun	PROPN
ma-134	95	4	+	+	CCONJ
ma-134	95	5	vun	vun	PROPN
ma-134	95	6	)	)	PUNCT
ma-134	95	7	f	f	PROPN
ma-134	95	8	(	(	PUNCT
ma-134	95	9	en+1(x	en+1(x	NOUN
ma-134	95	10	)	)	PUNCT
ma-134	95	11	)	)	PUNCT
ma-134	96	1	+	+	CCONJ
ma-134	96	2	(	(	PUNCT
ma-134	96	3	uvn	uvn	VERB
ma-134	96	4	+	+	CCONJ
ma-134	96	5	vvn	vvn	NOUN
ma-134	96	6	)	)	PUNCT
ma-134	96	7	f	f	PROPN
ma-134	96	8	(	(	PUNCT
ma-134	96	9	−en+1(x	−en+1(x	NOUN
ma-134	96	10	)	)	PUNCT
ma-134	96	11	)	)	PUNCT
ma-134	96	12	]	]	PUNCT
ma-134	97	1	=	=	PUNCT
ma-134	97	2	g(x	g(x	NOUN
ma-134	97	3	)	)	PUNCT
ma-134	97	4	.	.	PUNCT
ma-134	98	1	furthermore	furthermore	ADV
ma-134	98	2	,	,	PUNCT
ma-134	98	3	we	we	PRON
ma-134	98	4	will	will	AUX
ma-134	98	5	prove	prove	VERB
ma-134	98	6	the	the	DET
ma-134	98	7	more	more	ADV
ma-134	98	8	general	general	ADJ
ma-134	98	9	property	property	NOUN
ma-134	98	10	of	of	ADP
ma-134	98	11	g	g	PROPN
ma-134	98	12	g(x	g(x	NOUN
ma-134	98	13	)	)	PUNCT
ma-134	99	1	=	=	SYM
ma-134	99	2	ung	ung	PROPN
ma-134	99	3	(	(	PUNCT
ma-134	99	4	en(x	en(x	X
ma-134	99	5	)	)	PUNCT
ma-134	99	6	)	)	PUNCT
ma-134	100	1	+	+	CCONJ
ma-134	100	2	vng	vng	X
ma-134	100	3	(	(	PUNCT
ma-134	100	4	−en(x	−en(x	NOUN
ma-134	100	5	)	)	PUNCT
ma-134	100	6	)	)	PUNCT
ma-134	100	7	,	,	PUNCT
ma-134	100	8	f	f	PROPN
ma-134	100	9	or	or	CCONJ
ma-134	100	10	al	al	PROPN
ma-134	100	11	l	l	NOUN
ma-134	100	12	x	x	PUNCT
ma-134	100	13	∈	∈	PROPN
ma-134	100	14	x	x	X
ma-134	100	15	and	and	CCONJ
ma-134	100	16	n	n	PRON
ma-134	100	17	∈	∈	PROPN
ma-134	100	18	n.	n.	NOUN
ma-134	100	19	(	(	PUNCT
ma-134	100	20	2.6	2.6	NUM
ma-134	100	21	)	)	PUNCT
ma-134	100	22	by	by	ADP
ma-134	100	23	induction	induction	NOUN
ma-134	100	24	,	,	PUNCT
ma-134	100	25	we	we	PRON
ma-134	100	26	assume	assume	VERB
ma-134	100	27	that	that	SCONJ
ma-134	100	28	the	the	DET
ma-134	100	29	equation	equation	NOUN
ma-134	100	30	is	be	AUX
ma-134	100	31	true	true	ADJ
ma-134	100	32	for	for	ADP
ma-134	100	33	all	all	DET
ma-134	100	34	natural	natural	ADJ
ma-134	100	35	number	number	NOUN
ma-134	100	36	k	k	PROPN
ma-134	100	37	with	with	ADP
ma-134	100	38	k	k	PROPN
ma-134	100	39	≤	≤	PROPN
ma-134	100	40	n	n	CCONJ
ma-134	100	41	for	for	ADP
ma-134	100	42	some	some	DET
ma-134	100	43	n	n	PRON
ma-134	100	44	∈	∈	PROPN
ma-134	100	45	n.	n.	NOUN
ma-134	100	46	let	let	VERB
ma-134	100	47	us	we	PRON
ma-134	100	48	calculate	calculate	VERB
ma-134	100	49	with	with	ADP
ma-134	100	50	k	k	PROPN
ma-134	100	51	=	=	PUNCT
ma-134	100	52	n	n	PROPN
ma-134	100	53	+	+	CCONJ
ma-134	100	54	1	1	NUM
ma-134	100	55	g(x	g(x	NOUN
ma-134	100	56	)	)	PUNCT
ma-134	100	57	=	=	SYM
ma-134	100	58	ung	ung	PROPN
ma-134	100	59	(	(	PUNCT
ma-134	100	60	en(x	en(x	X
ma-134	100	61	)	)	PUNCT
ma-134	100	62	)	)	PUNCT
ma-134	101	1	+	+	CCONJ
ma-134	101	2	vng	vng	X
ma-134	101	3	(	(	PUNCT
ma-134	101	4	−en(x	−en(x	NOUN
ma-134	101	5	)	)	PUNCT
ma-134	101	6	)	)	PUNCT
ma-134	102	1	=	=	SYM
ma-134	102	2	un(ug	un(ug	NOUN
ma-134	102	3	(	(	PUNCT
ma-134	102	4	en+1(x	en+1(x	NOUN
ma-134	102	5	)	)	PUNCT
ma-134	102	6	)	)	PUNCT
ma-134	103	1	+	+	CCONJ
ma-134	103	2	vg	vg	X
ma-134	103	3	(	(	PUNCT
ma-134	103	4	−en+1(x	−en+1(x	NOUN
ma-134	103	5	)	)	PUNCT
ma-134	103	6	)	)	PUNCT
ma-134	103	7	)	)	PUNCT
ma-134	104	1	+	+	CCONJ
ma-134	104	2	vn(ug	vn(ug	NOUN
ma-134	104	3	(	(	PUNCT
ma-134	104	4	en+1(x	en+1(x	NOUN
ma-134	104	5	)	)	PUNCT
ma-134	104	6	)	)	PUNCT
ma-134	105	1	+	+	CCONJ
ma-134	105	2	vg	vg	X
ma-134	105	3	(	(	PUNCT
ma-134	105	4	−en+1(x	−en+1(x	NOUN
ma-134	105	5	)	)	PUNCT
ma-134	105	6	)	)	PUNCT
ma-134	105	7	)	)	PUNCT
ma-134	106	1	=	=	PUNCT
ma-134	106	2	un+1	un+1	PRON
ma-134	106	3	g	g	PROPN
ma-134	106	4	(	(	PUNCT
ma-134	106	5	en+1(x	en+1(x	NOUN
ma-134	106	6	)	)	PUNCT
ma-134	106	7	)	)	PUNCT
ma-134	107	1	+	+	CCONJ
ma-134	107	2	vn+1	vn+1	PRON
ma-134	107	3	g	g	NOUN
ma-134	107	4	(	(	PUNCT
ma-134	107	5	−en+1(x	−en+1(x	NOUN
ma-134	107	6	)	)	PUNCT
ma-134	107	7	)	)	PUNCT
ma-134	107	8	)	)	PUNCT
ma-134	107	9	.	.	PUNCT
ma-134	108	1	in	in	ADP
ma-134	108	2	particular	particular	ADJ
ma-134	108	3	,	,	PUNCT
ma-134	108	4	we	we	PRON
ma-134	108	5	also	also	ADV
ma-134	108	6	have	have	VERB
ma-134	108	7	that	that	SCONJ
ma-134	108	8	the	the	DET
ma-134	108	9	function	function	NOUN
ma-134	108	10	g	g	NOUN
ma-134	108	11	is	be	AUX
ma-134	108	12	even	even	ADV
ma-134	108	13	.	.	PUNCT
ma-134	109	1	an	an	DET
ma-134	109	2	easy	easy	ADJ
ma-134	109	3	computation	computation	NOUN
ma-134	109	4	is	be	AUX
ma-134	109	5	to	to	PART
ma-134	109	6	state	state	VERB
ma-134	109	7	that	that	DET
ma-134	109	8	g(−x	g(−x	NOUN
ma-134	109	9	)	)	PUNCT
ma-134	109	10	=	=	SYM
ma-134	109	11	ung	ung	PROPN
ma-134	109	12	(	(	PUNCT
ma-134	109	13	en(x	en(x	X
ma-134	109	14	)	)	PUNCT
ma-134	109	15	)	)	PUNCT
ma-134	110	1	+	+	CCONJ
ma-134	110	2	vng	vng	X
ma-134	110	3	(	(	PUNCT
ma-134	110	4	−en(x	−en(x	NOUN
ma-134	110	5	)	)	PUNCT
ma-134	110	6	)	)	PUNCT
ma-134	111	1	=	=	SYM
ma-134	111	2	g(x	g(x	NOUN
ma-134	111	3	)	)	PUNCT
ma-134	111	4	,	,	PUNCT
ma-134	111	5	f	f	PROPN
ma-134	111	6	or	or	CCONJ
ma-134	111	7	every	every	DET
ma-134	111	8	x	x	SYM
ma-134	111	9	∈	∈	PROPN
ma-134	111	10	x	x	X
ma-134	111	11	and	and	CCONJ
ma-134	111	12	n	n	CCONJ
ma-134	111	13	∈	∈	PROPN
ma-134	111	14	n.	n.	NOUN
ma-134	111	15	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	111	16	eur	eur	PROPN
ma-134	111	17	.	.	PUNCT
ma-134	112	1	j.	j.	PROPN
ma-134	112	2	math	math	PROPN
ma-134	112	3	.	.	PUNCT
ma-134	113	1	anal	anal	PROPN
ma-134	113	2	.	.	PUNCT
ma-134	114	1	10.28924	10.28924	NUM
ma-134	114	2	/	/	SYM
ma-134	114	3	ada	ada	PROPN
ma-134	114	4	/	/	SYM
ma-134	114	5	ma.3.7	ma.3.7	NOUN
ma-134	114	6	5	5	NUM
ma-134	114	7	in	in	ADP
ma-134	114	8	order	order	NOUN
ma-134	114	9	to	to	PART
ma-134	114	10	achieve	achieve	VERB
ma-134	114	11	the	the	DET
ma-134	114	12	uniqueness	uniqueness	NOUN
ma-134	114	13	of	of	ADP
ma-134	114	14	g	g	NOUN
ma-134	114	15	,	,	PUNCT
ma-134	114	16	suppose	suppose	VERB
ma-134	114	17	further	far	ADV
ma-134	114	18	that	that	SCONJ
ma-134	114	19	ḡ	ḡ	VERB
ma-134	114	20	:	:	PUNCT
ma-134	114	21	x	x	X
ma-134	114	22	→	→	SYM
ma-134	114	23	y	y	PROPN
ma-134	114	24	is	be	AUX
ma-134	114	25	the	the	DET
ma-134	114	26	another	another	DET
ma-134	114	27	mappingsuch	mappingsuch	NOUN
ma-134	114	28	that	that	SCONJ
ma-134	114	29	(	(	PUNCT
ma-134	114	30	2.2	2.2	NUM
ma-134	114	31	)	)	PUNCT
ma-134	114	32	and	and	CCONJ
ma-134	114	33	(	(	PUNCT
ma-134	114	34	2.6	2.6	NUM
ma-134	114	35	)	)	PUNCT
ma-134	114	36	hold	hold	NOUN
ma-134	114	37	.	.	PUNCT
ma-134	115	1	then	then	ADV
ma-134	115	2	‖g(x)−	‖g(x)−	PROPN
ma-134	115	3	ḡ(x)‖	ḡ(x)‖	VERB
ma-134	115	4	6	6	NUM
ma-134	115	5	2	2	NUM
ma-134	115	6	∞∑	∞∑	PRON
ma-134	115	7	i=0	i=0	PROPN
ma-134	115	8	[	[	PUNCT
ma-134	115	9	|ui	|ui	X
ma-134	115	10	|	|	ADV
ma-134	115	11	δ	δ	PROPN
ma-134	115	12	(	(	PUNCT
ma-134	115	13	e	e	NOUN
ma-134	115	14	i(x	i(x	PROPN
ma-134	115	15	)	)	PUNCT
ma-134	115	16	)	)	PUNCT
ma-134	116	1	+	+	CCONJ
ma-134	116	2	|vi	|vi	NUM
ma-134	116	3	|	|	ADV
ma-134	116	4	δ	δ	PROPN
ma-134	116	5	(	(	PUNCT
ma-134	116	6	−e	−e	NOUN
ma-134	116	7	i(x	i(x	PROPN
ma-134	116	8	)	)	PUNCT
ma-134	116	9	)	)	PUNCT
ma-134	116	10	]	]	PUNCT
ma-134	116	11	,	,	PUNCT
ma-134	116	12	x	x	PUNCT
ma-134	116	13	∈	∈	PROPN
ma-134	116	14	x.	x.	NOUN
ma-134	116	15	moreover	moreover	ADV
ma-134	116	16	,	,	PUNCT
ma-134	116	17	we	we	PRON
ma-134	116	18	have	have	VERB
ma-134	116	19	g(x)−	g(x)−	PROPN
ma-134	116	20	ḡ(x	ḡ(x	PROPN
ma-134	116	21	)	)	PUNCT
ma-134	117	1	=	=	PRON
ma-134	117	2	un	un	PROPN
ma-134	118	1	[	[	X
ma-134	118	2	g	g	X
ma-134	118	3	(	(	PUNCT
ma-134	118	4	en(x))−	en(x))−	PROPN
ma-134	118	5	ḡ	ḡ	VERB
ma-134	118	6	(	(	PUNCT
ma-134	118	7	en(x	en(x	X
ma-134	118	8	)	)	PUNCT
ma-134	118	9	)	)	PUNCT
ma-134	118	10	]	]	PUNCT
ma-134	119	1	+	+	CCONJ
ma-134	119	2	vn	vn	X
ma-134	120	1	[	[	X
ma-134	120	2	g	g	X
ma-134	120	3	(	(	PUNCT
ma-134	120	4	−en(x))−	−en(x))−	ADV
ma-134	120	5	ḡ	ḡ	VERB
ma-134	120	6	(	(	PUNCT
ma-134	120	7	−en(x	−en(x	NOUN
ma-134	120	8	)	)	PUNCT
ma-134	120	9	)	)	PUNCT
ma-134	120	10	]	]	PUNCT
ma-134	121	1	,	,	PUNCT
ma-134	121	2	x	x	PUNCT
ma-134	121	3	∈	∈	NOUN
ma-134	121	4	x	x	X
ma-134	121	5	,	,	PUNCT
ma-134	121	6	and	and	CCONJ
ma-134	121	7	on	on	ADP
ma-134	121	8	account	account	NOUN
ma-134	121	9	of	of	ADP
ma-134	121	10	(	(	PUNCT
ma-134	121	11	2.4	2.4	NUM
ma-134	121	12	)	)	PUNCT
ma-134	121	13	and	and	CCONJ
ma-134	121	14	(	(	PUNCT
ma-134	121	15	2.6	2.6	NUM
ma-134	121	16	)	)	PUNCT
ma-134	121	17	we	we	PRON
ma-134	121	18	can	can	AUX
ma-134	121	19	rewrite	rewrite	VERB
ma-134	121	20	‖g(x)−	‖g(x)−	PROPN
ma-134	121	21	ḡ(x)‖	ḡ(x)‖	NOUN
ma-134	121	22	6	6	NUM
ma-134	121	23	|un	|un	ADP
ma-134	121	24	+	+	CCONJ
ma-134	121	25	vn|	vn|	INTJ
ma-134	121	26	‖g	‖g	PROPN
ma-134	121	27	(	(	PUNCT
ma-134	121	28	en(x))−	en(x))−	PROPN
ma-134	121	29	ḡ	ḡ	VERB
ma-134	121	30	(	(	PUNCT
ma-134	121	31	en(x))‖	en(x))‖	VERB
ma-134	121	32	6	6	NUM
ma-134	121	33	2|un	2|un	NUM
ma-134	121	34	+	+	CCONJ
ma-134	121	35	vn|	vn|	VERB
ma-134	121	36	∞∑	∞∑	PRON
ma-134	121	37	i=0	i=0	PROPN
ma-134	121	38	[	[	PUNCT
ma-134	121	39	|ui	|ui	X
ma-134	121	40	|	|	ADV
ma-134	121	41	δ	δ	PROPN
ma-134	121	42	(	(	PUNCT
ma-134	121	43	e	e	NOUN
ma-134	121	44	i+n(x	i+n(x	NOUN
ma-134	121	45	)	)	PUNCT
ma-134	121	46	)	)	PUNCT
ma-134	122	1	+	+	CCONJ
ma-134	122	2	|vi	|vi	NUM
ma-134	122	3	|	|	ADV
ma-134	122	4	δ	δ	PROPN
ma-134	122	5	(	(	PUNCT
ma-134	122	6	−e	−e	NOUN
ma-134	122	7	i+n(x	i+n(x	NOUN
ma-134	122	8	)	)	PUNCT
ma-134	122	9	)	)	PUNCT
ma-134	122	10	]	]	PUNCT
ma-134	123	1	=	=	SYM
ma-134	123	2	2	2	NUM
ma-134	123	3	∞∑	∞∑	NUM
ma-134	123	4	i=0	i=0	PROPN
ma-134	123	5	[	[	X
ma-134	123	6	(	(	PUNCT
ma-134	123	7	|ui(un	|ui(un	NOUN
ma-134	123	8	+	+	CCONJ
ma-134	123	9	vn)|	vn)|	PROPN
ma-134	123	10	)	)	PUNCT
ma-134	123	11	δ	δ	NOUN
ma-134	123	12	(	(	PUNCT
ma-134	123	13	e	e	NOUN
ma-134	123	14	i+n(x	i+n(x	NOUN
ma-134	123	15	)	)	PUNCT
ma-134	123	16	)	)	PUNCT
ma-134	124	1	+	+	CCONJ
ma-134	124	2	|vi(un	|vi(un	NOUN
ma-134	124	3	+	+	CCONJ
ma-134	124	4	vn)|δ	vn)|δ	X
ma-134	124	5	(	(	PUNCT
ma-134	124	6	−e	−e	NOUN
ma-134	124	7	i+n(x	i+n(x	NOUN
ma-134	124	8	)	)	PUNCT
ma-134	124	9	)	)	PUNCT
ma-134	124	10	]	]	PUNCT
ma-134	125	1	=	=	SYM
ma-134	125	2	2	2	NUM
ma-134	125	3	∞∑	∞∑	NUM
ma-134	125	4	i=0	i=0	PROPN
ma-134	125	5	[	[	PUNCT
ma-134	125	6	|ui+n|	|ui+n|	PROPN
ma-134	125	7	δ	δ	PROPN
ma-134	125	8	(	(	PUNCT
ma-134	125	9	e	e	NOUN
ma-134	125	10	i+n(x	i+n(x	NOUN
ma-134	125	11	)	)	PUNCT
ma-134	125	12	)	)	PUNCT
ma-134	126	1	+	+	CCONJ
ma-134	126	2	|vi+n|	|vi+n|	PROPN
ma-134	126	3	δ	δ	PROPN
ma-134	126	4	(	(	PUNCT
ma-134	126	5	−e	−e	NOUN
ma-134	126	6	i+n(x	i+n(x	NOUN
ma-134	126	7	)	)	PUNCT
ma-134	126	8	)	)	PUNCT
ma-134	126	9	]	]	PUNCT
ma-134	127	1	=	=	SYM
ma-134	127	2	2	2	NUM
ma-134	127	3	∞∑	∞∑	NUM
ma-134	127	4	j	j	NOUN
ma-134	127	5	=	=	NOUN
ma-134	127	6	n	n	PRON
ma-134	127	7	[	[	X
ma-134	127	8	∣∣uj	∣∣uj	ADJ
ma-134	127	9	∣∣	∣∣	PROPN
ma-134	127	10	δ	δ	PROPN
ma-134	127	11	(	(	PUNCT
ma-134	127	12	e	e	PROPN
ma-134	127	13	j(x	j(x	PROPN
ma-134	127	14	)	)	PUNCT
ma-134	127	15	)	)	PUNCT
ma-134	127	16	+	+	CCONJ
ma-134	127	17	∣∣vj	∣∣vj	NOUN
ma-134	127	18	∣∣	∣∣	NUM
ma-134	127	19	δ	δ	PROPN
ma-134	127	20	(	(	PUNCT
ma-134	127	21	−e	−e	NOUN
ma-134	127	22	j(x	j(x	PROPN
ma-134	127	23	)	)	PUNCT
ma-134	127	24	)	)	PUNCT
ma-134	127	25	]	]	PUNCT
ma-134	127	26	for	for	ADP
ma-134	127	27	every	every	DET
ma-134	127	28	x	x	SYM
ma-134	127	29	∈	∈	PROPN
ma-134	127	30	x	x	X
ma-134	127	31	and	and	CCONJ
ma-134	127	32	n	n	CCONJ
ma-134	127	33	∈	∈	PROPN
ma-134	127	34	n	n	CCONJ
ma-134	127	35	,	,	PUNCT
ma-134	127	36	where	where	SCONJ
ma-134	127	37	it	it	PRON
ma-134	127	38	states	state	VERB
ma-134	127	39	that	that	SCONJ
ma-134	127	40	g	g	PROPN
ma-134	127	41	=	=	PUNCT
ma-134	127	42	ḡ	ḡ	VERB
ma-134	127	43	as	as	ADP
ma-134	127	44	n	n	X
ma-134	127	45	→∞.	→∞.	PROPN
ma-134	127	46	this	this	PRON
ma-134	127	47	proves	prove	VERB
ma-134	127	48	the	the	DET
ma-134	127	49	theorem	theorem	NOUN
ma-134	127	50	.	.	PUNCT
ma-134	127	51	�	�	PROPN
ma-134	127	52	the	the	DET
ma-134	127	53	purpose	purpose	NOUN
ma-134	127	54	of	of	ADP
ma-134	127	55	stating	state	VERB
ma-134	127	56	and	and	CCONJ
ma-134	127	57	proving	prove	VERB
ma-134	127	58	this	this	DET
ma-134	127	59	results	result	NOUN
ma-134	127	60	is	be	AUX
ma-134	127	61	of	of	ADP
ma-134	127	62	particular	particular	ADJ
ma-134	127	63	interest	interest	NOUN
ma-134	127	64	and	and	CCONJ
ma-134	127	65	give	give	VERB
ma-134	127	66	out	out	ADP
ma-134	127	67	a	a	DET
ma-134	127	68	solutionof	solutionof	NOUN
ma-134	127	69	a	a	DET
ma-134	127	70	simple	simple	ADJ
ma-134	127	71	variable	variable	ADJ
ma-134	127	72	functional	functional	ADJ
ma-134	127	73	equation	equation	NOUN
ma-134	127	74	(	(	PUNCT
ma-134	127	75	2.1	2.1	NUM
ma-134	127	76	)	)	PUNCT
ma-134	127	77	at	at	ADP
ma-134	127	78	least	least	ADJ
ma-134	127	79	.	.	PUNCT
ma-134	128	1	in	in	ADP
ma-134	128	2	section	section	NOUN
ma-134	128	3	3	3	NUM
ma-134	128	4	,	,	PUNCT
ma-134	128	5	we	we	PRON
ma-134	128	6	will	will	AUX
ma-134	128	7	extend	extend	VERB
ma-134	128	8	the	the	DET
ma-134	128	9	results	result	NOUN
ma-134	128	10	oftheorem	oftheorem	VERB
ma-134	128	11	2.1	2.1	NUM
ma-134	128	12	form	form	NOUN
ma-134	128	13	[	[	X
ma-134	128	14	8	8	NUM
ma-134	128	15	]	]	PUNCT
ma-134	128	16	to	to	ADP
ma-134	128	17	a	a	DET
ma-134	128	18	more	more	ADV
ma-134	128	19	general	general	ADJ
ma-134	128	20	setting	setting	NOUN
ma-134	128	21	.	.	PUNCT
ma-134	129	1	in	in	ADP
ma-134	129	2	particular	particular	ADJ
ma-134	129	3	,	,	PUNCT
ma-134	129	4	the	the	DET
ma-134	129	5	related	related	ADJ
ma-134	129	6	parameters	parameter	NOUN
ma-134	129	7	u	u	NOUN
ma-134	129	8	,	,	PUNCT
ma-134	129	9	v	v	PROPN
ma-134	129	10	can	can	AUX
ma-134	129	11	beextended	beextende	VERB
ma-134	129	12	to	to	PART
ma-134	129	13	complex	complex	VERB
ma-134	129	14	numbers.according	numbers.accorde	VERB
ma-134	129	15	to	to	ADP
ma-134	129	16	the	the	DET
ma-134	129	17	above	above	ADJ
ma-134	129	18	analysis	analysis	NOUN
ma-134	129	19	,	,	PUNCT
ma-134	129	20	we	we	PRON
ma-134	129	21	give	give	VERB
ma-134	129	22	out	out	ADP
ma-134	129	23	a	a	DET
ma-134	129	24	corollary	corollary	NOUN
ma-134	129	25	of	of	ADP
ma-134	129	26	theorem	theorem	ADJ
ma-134	129	27	2.1	2.1	NUM
ma-134	129	28	(	(	PUNCT
ma-134	129	29	still	still	ADV
ma-134	129	30	quite	quite	ADV
ma-134	129	31	general).first	general).first	NOUN
ma-134	129	32	of	of	ADP
ma-134	129	33	all	all	PRON
ma-134	129	34	,	,	PUNCT
ma-134	129	35	we	we	PRON
ma-134	129	36	must	must	AUX
ma-134	129	37	state	state	VERB
ma-134	129	38	that	that	SCONJ
ma-134	129	39	the	the	DET
ma-134	129	40	absolute	absolute	NOUN
ma-134	129	41	of	of	ADP
ma-134	129	42	an	an	DET
ma-134	129	43	element	element	NOUN
ma-134	129	44	x	x	SYM
ma-134	129	45	∈	∈	PROPN
ma-134	129	46	x	x	AUX
ma-134	129	47	can	can	AUX
ma-134	129	48	be	be	AUX
ma-134	129	49	given	give	VERB
ma-134	129	50	out	out	ADP
ma-134	129	51	in	in	ADP
ma-134	129	52	the	the	DET
ma-134	129	53	real	real	ADJ
ma-134	129	54	fieldconsidering	fieldconsidering	NOUN
ma-134	129	55	that	that	SCONJ
ma-134	129	56	the	the	DET
ma-134	129	57	function	function	NOUN
ma-134	129	58	h	h	NOUN
ma-134	129	59	is	be	AUX
ma-134	129	60	even	even	ADV
ma-134	129	61	for	for	ADP
ma-134	129	62	the	the	DET
ma-134	129	63	meaningful	meaningful	ADJ
ma-134	129	64	of	of	ADP
ma-134	129	65	the	the	DET
ma-134	129	66	results	result	NOUN
ma-134	129	67	,	,	PUNCT
ma-134	129	68	for	for	ADP
ma-134	129	69	example	example	NOUN
ma-134	129	70	h(x	h(x	PROPN
ma-134	129	71	)	)	PUNCT
ma-134	130	1	=	=	NOUN
ma-134	130	2	a|x	a|x	NOUN
ma-134	130	3	|.as	|.as	VERB
ma-134	130	4	a	a	DET
ma-134	130	5	matter	matter	NOUN
ma-134	130	6	of	of	ADP
ma-134	130	7	fact	fact	NOUN
ma-134	130	8	,	,	PUNCT
ma-134	130	9	we	we	PRON
ma-134	130	10	can	can	AUX
ma-134	130	11	also	also	ADV
ma-134	130	12	present	present	VERB
ma-134	130	13	the	the	DET
ma-134	130	14	absolute	absolute	NOUN
ma-134	130	15	of	of	ADP
ma-134	130	16	x	x	X
ma-134	130	17	=	=	SYM
ma-134	130	18	(	(	PUNCT
ma-134	130	19	x1	x1	PROPN
ma-134	130	20	,	,	PUNCT
ma-134	130	21	x2	x2	PROPN
ma-134	130	22	,	,	PUNCT
ma-134	130	23	·	·	PUNCT
ma-134	130	24	·	·	PUNCT
ma-134	130	25	·	·	PUNCT
ma-134	130	26	,	,	PUNCT
ma-134	130	27	xn	xn	X
ma-134	130	28	)	)	PUNCT
ma-134	130	29	∈	∈	PROPN
ma-134	130	30	rn	rn	PROPN
ma-134	130	31	by	by	ADP
ma-134	130	32	|x	|x	NOUN
ma-134	130	33	|	|	ADV
ma-134	130	34	=	=	SYM
ma-134	130	35	(	(	PUNCT
ma-134	130	36	|x1|	|x1|	PROPN
ma-134	130	37	,	,	PUNCT
ma-134	130	38	|x2|	|x2|	PROPN
ma-134	130	39	,	,	PUNCT
ma-134	130	40	·	·	PUNCT
ma-134	130	41	·	·	PUNCT
ma-134	130	42	·	·	PUNCT
ma-134	130	43	,	,	PUNCT
ma-134	130	44	|xn|	|xn|	PROPN
ma-134	130	45	)	)	PUNCT
ma-134	130	46	.	.	PUNCT
ma-134	131	1	thus	thus	ADV
ma-134	131	2	,	,	PUNCT
ma-134	131	3	it	it	PRON
ma-134	131	4	worth	worth	ADJ
ma-134	131	5	stating	state	VERB
ma-134	131	6	the	the	DET
ma-134	131	7	results	result	NOUN
ma-134	131	8	.	.	PUNCT
ma-134	132	1	in	in	ADP
ma-134	132	2	particular	particular	ADJ
ma-134	132	3	,	,	PUNCT
ma-134	132	4	we	we	PRON
ma-134	132	5	can	can	AUX
ma-134	132	6	present	present	VERB
ma-134	132	7	the	the	DET
ma-134	132	8	followingresults	followingresult	NOUN
ma-134	132	9	in	in	ADP
ma-134	132	10	the	the	DET
ma-134	132	11	euclidean	euclidean	ADJ
ma-134	132	12	space	space	NOUN
ma-134	132	13	if	if	SCONJ
ma-134	132	14	the	the	DET
ma-134	132	15	more	more	ADV
ma-134	132	16	general	general	ADJ
ma-134	132	17	setting	setting	NOUN
ma-134	132	18	can	can	AUX
ma-134	132	19	not	not	PART
ma-134	132	20	be	be	AUX
ma-134	132	21	judged	judge	VERB
ma-134	132	22	.	.	PUNCT
ma-134	133	1	corollary	corollary	ADJ
ma-134	133	2	2.1	2.1	NUM
ma-134	133	3	assume	assume	VERB
ma-134	133	4	that	that	SCONJ
ma-134	133	5	(	(	PUNCT
ma-134	133	6	x,+	x,+	NUM
ma-134	133	7	)	)	PUNCT
ma-134	133	8	is	be	AUX
ma-134	133	9	a	a	DET
ma-134	133	10	real	real	ADJ
ma-134	133	11	or	or	CCONJ
ma-134	133	12	complex	complex	ADJ
ma-134	133	13	normed	norme	VERB
ma-134	133	14	linear	linear	ADJ
ma-134	133	15	space	space	NOUN
ma-134	133	16	and	and	CCONJ
ma-134	133	17	set	set	NOUN
ma-134	133	18	(	(	PUNCT
ma-134	133	19	y	y	PROPN
ma-134	133	20	,	,	PUNCT
ma-134	133	21	‖	‖	PROPN
ma-134	133	22	·	·	PUNCT
ma-134	133	23	‖	‖	NUM
ma-134	133	24	)	)	PUNCT
ma-134	133	25	isa	isa	NOUN
ma-134	133	26	banach	banach	NOUN
ma-134	133	27	space	space	NOUN
ma-134	133	28	.	.	PUNCT
ma-134	134	1	suppose	suppose	VERB
ma-134	134	2	further	far	ADV
ma-134	134	3	that	that	SCONJ
ma-134	134	4	the	the	DET
ma-134	134	5	mapping	mapping	NOUN
ma-134	134	6	f	f	X
ma-134	134	7	:	:	PUNCT
ma-134	134	8	x	x	X
ma-134	134	9	→	→	SYM
ma-134	134	10	y	y	PROPN
ma-134	134	11	fulfils	fulfil	VERB
ma-134	134	12	the	the	DET
ma-134	134	13	inequality∥∥∥∥f	inequality∥∥∥∥f	NOUN
ma-134	134	14	(	(	PUNCT
ma-134	134	15	x)−	x)−	PROPN
ma-134	134	16	a	a	PRON
ma-134	134	17	+	+	NUM
ma-134	134	18	1	1	NUM
ma-134	134	19	2a2	2a2	NUM
ma-134	134	20	f	f	NOUN
ma-134	134	21	(	(	PUNCT
ma-134	134	22	a|x	a|x	NOUN
ma-134	134	23	|	|	ADV
ma-134	134	24	)	)	PUNCT
ma-134	135	1	+	+	CCONJ
ma-134	135	2	a	a	DET
ma-134	135	3	−	−	PROPN
ma-134	135	4	1	1	NUM
ma-134	135	5	2a2	2a2	NUM
ma-134	135	6	f	f	NOUN
ma-134	135	7	(	(	PUNCT
ma-134	135	8	−a|x	−a|x	PROPN
ma-134	135	9	|	|	ADV
ma-134	135	10	)	)	PUNCT
ma-134	135	11	∥∥∥∥	∥∥∥∥	NOUN
ma-134	135	12	6	6	NUM
ma-134	135	13	δ(x	δ(x	NOUN
ma-134	135	14	)	)	PUNCT
ma-134	135	15	,	,	PUNCT
ma-134	136	1	x	x	PUNCT
ma-134	136	2	∈	∈	NOUN
ma-134	136	3	x	x	X
ma-134	136	4	,	,	PUNCT
ma-134	136	5	(	(	PUNCT
ma-134	136	6	2.7	2.7	NUM
ma-134	136	7	)	)	PUNCT
ma-134	136	8	where	where	SCONJ
ma-134	136	9	a	a	DET
ma-134	136	10	∈	∈	NOUN
ma-134	136	11	r	r	NOUN
ma-134	136	12	with	with	ADP
ma-134	136	13	a	a	DET
ma-134	136	14	>	>	SYM
ma-134	136	15	1	1	NUM
ma-134	136	16	and	and	CCONJ
ma-134	136	17	mappings	mapping	NOUN
ma-134	136	18	e	e	NOUN
ma-134	136	19	:	:	PUNCT
ma-134	136	20	x	x	X
ma-134	136	21	→	→	SYM
ma-134	136	22	x	x	PROPN
ma-134	136	23	,	,	PUNCT
ma-134	136	24	δ	δ	PROPN
ma-134	136	25	:	:	PUNCT
ma-134	136	26	x	x	X
ma-134	136	27	→	→	PUNCT
ma-134	136	28	[	[	X
ma-134	136	29	0,∞	0,∞	X
ma-134	136	30	)	)	PUNCT
ma-134	136	31	make	make	VERB
ma-134	136	32	that	that	SCONJ
ma-134	136	33	e	e	NOUN
ma-134	136	34	is	be	AUX
ma-134	136	35	an	an	DET
ma-134	136	36	even	even	ADJ
ma-134	136	37	function	function	NOUN
ma-134	136	38	(	(	PUNCT
ma-134	136	39	i.e.	i.e.	X
ma-134	136	40	,	,	PUNCT
ma-134	136	41	e(−x	e(−x	NOUN
ma-134	136	42	)	)	PUNCT
ma-134	136	43	=	=	SYM
ma-134	136	44	e(x	e(x	NUM
ma-134	136	45	)	)	PUNCT
ma-134	136	46	for	for	ADP
ma-134	136	47	every	every	DET
ma-134	136	48	x	x	SYM
ma-134	136	49	∈	∈	PROPN
ma-134	136	50	x	x	NOUN
ma-134	136	51	)	)	PUNCT
ma-134	136	52	.	.	PUNCT
ma-134	137	1	the	the	DET
ma-134	137	2	infinite	infinite	ADJ
ma-134	137	3	progression	progression	NOUN
ma-134	137	4	∑∞i=0	∑∞i=0	PROPN
ma-134	137	5	1ai	1ai	PROPN
ma-134	137	6	δ	δ	PROPN
ma-134	137	7	(	(	PUNCT
ma-134	137	8	ai	ai	PROPN
ma-134	137	9	|x	|x	NOUN
ma-134	137	10	|	|	ADV
ma-134	137	11	)	)	PUNCT
ma-134	137	12	is	be	AUX
ma-134	137	13	convergence	convergence	NOUN
ma-134	137	14	for	for	ADP
ma-134	137	15	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	137	16	eur	eur	PROPN
ma-134	137	17	.	.	PUNCT
ma-134	138	1	j.	j.	PROPN
ma-134	138	2	math	math	PROPN
ma-134	138	3	.	.	PUNCT
ma-134	139	1	anal	anal	PROPN
ma-134	139	2	.	.	PUNCT
ma-134	140	1	10.28924	10.28924	NUM
ma-134	140	2	/	/	SYM
ma-134	140	3	ada	ada	NOUN
ma-134	140	4	/	/	SYM
ma-134	140	5	ma.3.7	ma.3.7	NOUN
ma-134	140	6	6every	6every	NUM
ma-134	140	7	x	x	SYM
ma-134	140	8	∈	∈	NOUN
ma-134	140	9	x	x	X
ma-134	140	10	.	.	PUNCT
ma-134	141	1	then	then	ADV
ma-134	141	2	there	there	PRON
ma-134	141	3	has	have	VERB
ma-134	141	4	a	a	DET
ma-134	141	5	unique	unique	ADJ
ma-134	141	6	mapping	mapping	NOUN
ma-134	141	7	g	g	NOUN
ma-134	141	8	:	:	PUNCT
ma-134	141	9	x	x	X
ma-134	141	10	→	→	SYM
ma-134	141	11	y	y	PROPN
ma-134	141	12	fulfilling	fulfil	VERB
ma-134	141	13	the	the	DET
ma-134	141	14	following	follow	VERB
ma-134	141	15	equations	equation	NOUN
ma-134	141	16	for	for	ADP
ma-134	141	17	all	all	DET
ma-134	141	18	x	x	SYM
ma-134	141	19	∈	∈	ADJ
ma-134	141	20	x	x	X
ma-134	141	21	g(x	g(x	NOUN
ma-134	141	22	)	)	PUNCT
ma-134	141	23	=	=	PUNCT
ma-134	142	1	a	a	DET
ma-134	142	2	+	+	NOUN
ma-134	142	3	1	1	NUM
ma-134	142	4	2a2	2a2	NUM
ma-134	142	5	g(a|x	g(a|x	VERB
ma-134	142	6	|)−	|)−	PROPN
ma-134	142	7	a	a	DET
ma-134	142	8	−	−	PROPN
ma-134	142	9	1	1	NUM
ma-134	142	10	2a2	2a2	NUM
ma-134	142	11	g(−a|x	g(−a|x	NOUN
ma-134	142	12	|	|	ADV
ma-134	142	13	)	)	PUNCT
ma-134	142	14	,	,	PUNCT
ma-134	142	15	and	and	CCONJ
ma-134	142	16	‖f	‖f	ADP
ma-134	142	17	(	(	PUNCT
ma-134	142	18	x)−	x)−	PROPN
ma-134	142	19	g(x)‖	g(x)‖	PROPN
ma-134	142	20	6	6	NUM
ma-134	142	21	∆(x	∆(x	NOUN
ma-134	142	22	)	)	PUNCT
ma-134	143	1	+	+	CCONJ
ma-134	144	1	λ(x	λ(x	PROPN
ma-134	144	2	)	)	PUNCT
ma-134	144	3	,	,	PUNCT
ma-134	144	4	where	where	SCONJ
ma-134	144	5	∆(x	∆(x	NOUN
ma-134	144	6	)	)	PUNCT
ma-134	144	7	:	:	PUNCT
ma-134	144	8	=	=	SYM
ma-134	144	9	1	1	NUM
ma-134	144	10	2	2	NUM
ma-134	144	11	∑∞	∑∞	NOUN
ma-134	144	12	i=0	i=0	PROPN
ma-134	144	13	1	1	NUM
ma-134	144	14	ai	ai	NOUN
ma-134	144	15	[	[	PUNCT
ma-134	144	16	δ	δ	PROPN
ma-134	144	17	(	(	PUNCT
ma-134	144	18	ai	ai	PROPN
ma-134	144	19	|x	|x	NOUN
ma-134	144	20	|	|	ADV
ma-134	144	21	)	)	PUNCT
ma-134	145	1	+	+	CCONJ
ma-134	145	2	δ	δ	PROPN
ma-134	145	3	(	(	PUNCT
ma-134	145	4	−ai	−ai	NOUN
ma-134	145	5	|x	|x	NOUN
ma-134	145	6	|	|	ADV
ma-134	145	7	)	)	PUNCT
ma-134	145	8	]	]	PUNCT
ma-134	145	9	,	,	PUNCT
ma-134	145	10	λ(x	λ(x	PROPN
ma-134	145	11	)	)	PUNCT
ma-134	145	12	:	:	PUNCT
ma-134	145	13	=	=	SYM
ma-134	145	14	1	1	NUM
ma-134	145	15	2	2	NUM
ma-134	145	16	∑∞	∑∞	NOUN
ma-134	145	17	i=0	i=0	PROPN
ma-134	145	18	1	1	NUM
ma-134	145	19	a2i	a2i	NOUN
ma-134	145	20	[	[	PUNCT
ma-134	145	21	δ	δ	PROPN
ma-134	145	22	(	(	PUNCT
ma-134	145	23	ai	ai	PROPN
ma-134	145	24	|x	|x	NOUN
ma-134	145	25	|	|	ADV
ma-134	145	26	)	)	PUNCT
ma-134	145	27	−	−	PROPN
ma-134	146	1	δ	δ	PROPN
ma-134	146	2	(	(	PUNCT
ma-134	146	3	−ai	−ai	NOUN
ma-134	146	4	|x	|x	NOUN
ma-134	146	5	|	|	ADV
ma-134	146	6	)	)	PUNCT
ma-134	146	7	]	]	PUNCT
ma-134	146	8	,	,	PUNCT
ma-134	146	9	x	x	PUNCT
ma-134	146	10	∈	∈	PROPN
ma-134	146	11	x.furthermore	x.furthermore	PROPN
ma-134	146	12	,	,	PUNCT
ma-134	146	13	g	g	PROPN
ma-134	146	14	can	can	AUX
ma-134	146	15	be	be	AUX
ma-134	146	16	obtained	obtain	VERB
ma-134	146	17	in	in	ADP
ma-134	146	18	the	the	DET
ma-134	146	19	following	follow	VERB
ma-134	146	20	limiting	limit	VERB
ma-134	146	21	equality	equality	NOUN
ma-134	146	22	g(x	g(x	NOUN
ma-134	146	23	)	)	PUNCT
ma-134	146	24	:	:	PUNCT
ma-134	146	25	=	=	PUNCT
ma-134	146	26	lim	lim	PROPN
ma-134	146	27	n→∞	n→∞	X
ma-134	146	28	(	(	PUNCT
ma-134	146	29	an	an	DET
ma-134	146	30	+	+	NUM
ma-134	146	31	1	1	NUM
ma-134	146	32	2a2n	2a2n	NUM
ma-134	146	33	f	f	NOUN
ma-134	146	34	(	(	PUNCT
ma-134	146	35	an|x	an|x	X
ma-134	146	36	|)−	|)−	PROPN
ma-134	146	37	an	an	DET
ma-134	146	38	−	−	PROPN
ma-134	146	39	1	1	NUM
ma-134	146	40	2a2n	2a2n	NOUN
ma-134	146	41	f	f	NOUN
ma-134	146	42	(	(	PUNCT
ma-134	146	43	−an|x	−an|x	PROPN
ma-134	146	44	|	|	NOUN
ma-134	146	45	)	)	PUNCT
ma-134	146	46	)	)	PUNCT
ma-134	146	47	,	,	PUNCT
ma-134	146	48	x	x	PUNCT
ma-134	146	49	∈	∈	NOUN
ma-134	146	50	x.	x.	NOUN
ma-134	146	51	proof	proof	NOUN
ma-134	146	52	.	.	PUNCT
ma-134	147	1	by	by	ADP
ma-134	147	2	using	use	VERB
ma-134	147	3	the	the	DET
ma-134	147	4	results	result	NOUN
ma-134	147	5	of	of	ADP
ma-134	147	6	theorem	theorem	ADJ
ma-134	147	7	2.1	2.1	NUM
ma-134	147	8	,	,	PUNCT
ma-134	147	9	u	u	NOUN
ma-134	147	10	:	:	PUNCT
ma-134	147	11	=	=	SYM
ma-134	147	12	1+a	1+a	NUM
ma-134	147	13	2a2	2a2	NUM
ma-134	147	14	,	,	PUNCT
ma-134	147	15	v	v	ADP
ma-134	147	16	:	:	PUNCT
ma-134	147	17	=	=	SYM
ma-134	147	18	1−a	1−a	NUM
ma-134	147	19	2a2	2a2	NUM
ma-134	147	20	and	and	CCONJ
ma-134	147	21	together	together	ADV
ma-134	147	22	e(x	e(x	NUM
ma-134	147	23	)	)	PUNCT
ma-134	147	24	:	:	PUNCT
ma-134	147	25	=	=	NOUN
ma-134	147	26	a|x	a|x	NOUN
ma-134	147	27	|	|	ADV
ma-134	147	28	,	,	PUNCT
ma-134	147	29	for	for	ADP
ma-134	147	30	all	all	DET
ma-134	147	31	x	x	SYM
ma-134	147	32	∈	∈	PROPN
ma-134	147	33	x	x	X
ma-134	147	34	,	,	PUNCT
ma-134	147	35	a	a	DET
ma-134	147	36	computation	computation	NOUN
ma-134	147	37	is	be	AUX
ma-134	147	38	to	to	PART
ma-134	147	39	prove	prove	VERB
ma-134	147	40	that	that	DET
ma-134	147	41	un	un	PROPN
ma-134	147	42	:	:	PUNCT
ma-134	147	43	=	=	SYM
ma-134	147	44	1	1	NUM
ma-134	147	45	+	+	CCONJ
ma-134	147	46	a	a	DET
ma-134	147	47	2a2n	2a2n	NOUN
ma-134	147	48	,	,	PUNCT
ma-134	147	49	vn	vn	X
ma-134	147	50	:	:	PUNCT
ma-134	147	51	=	=	SYM
ma-134	147	52	1−	1−	NUM
ma-134	147	53	a	a	DET
ma-134	147	54	2a2n	2a2n	NOUN
ma-134	147	55	,	,	PUNCT
ma-134	147	56	n	n	PROPN
ma-134	147	57	∈	∈	PROPN
ma-134	147	58	n.	n.	NOUN
ma-134	147	59	for	for	ADP
ma-134	147	60	the	the	DET
ma-134	147	61	convergent	convergent	NOUN
ma-134	147	62	series	series	PROPN
ma-134	147	63	∑∞i=0	∑∞i=0	PROPN
ma-134	147	64	1ai	1ai	PROPN
ma-134	147	65	δ	δ	PROPN
ma-134	147	66	(	(	PUNCT
ma-134	147	67	ai	ai	PROPN
ma-134	147	68	|x	|x	NOUN
ma-134	147	69	|	|	ADV
ma-134	147	70	)	)	PUNCT
ma-134	147	71	with	with	ADP
ma-134	147	72	x	x	PUNCT
ma-134	147	73	∈	∈	PROPN
ma-134	147	74	x	x	X
ma-134	147	75	,	,	PUNCT
ma-134	147	76	therefore	therefore	ADV
ma-134	147	77	∑∞i=0	∑∞i=0	PROPN
ma-134	147	78	1a2i	1a2i	PROPN
ma-134	147	79	δ	δ	PROPN
ma-134	147	80	(	(	PUNCT
ma-134	147	81	ai	ai	PROPN
ma-134	147	82	|x	|x	NOUN
ma-134	147	83	|	|	ADV
ma-134	147	84	)	)	PUNCT
ma-134	147	85	is	be	AUX
ma-134	147	86	convergence.applying	convergence.applye	VERB
ma-134	147	87	theorem	theorem	VERB
ma-134	147	88	2.1	2.1	NUM
ma-134	147	89	,	,	PUNCT
ma-134	147	90	there	there	PRON
ma-134	147	91	has	have	VERB
ma-134	147	92	a	a	DET
ma-134	147	93	unique	unique	ADJ
ma-134	147	94	limiting	limiting	NOUN
ma-134	147	95	function	function	NOUN
ma-134	147	96	g	g	NOUN
ma-134	147	97	:	:	PUNCT
ma-134	147	98	x	x	X
ma-134	147	99	→	→	SYM
ma-134	147	100	y	y	PROPN
ma-134	147	101	fulfilling	fulfil	VERB
ma-134	147	102	‖f	‖f	ADP
ma-134	147	103	(	(	PUNCT
ma-134	147	104	x)−	x)−	PROPN
ma-134	147	105	g(x)‖	g(x)‖	PROPN
ma-134	147	106	6	6	NUM
ma-134	147	107	∞∑	∞∑	PROPN
ma-134	147	108	i=0	i=0	PROPN
ma-134	147	109	[	[	X
ma-134	147	110	∣∣∣∣1	∣∣∣∣1	NOUN
ma-134	147	111	+	+	CCONJ
ma-134	147	112	ai	ai	VERB
ma-134	147	113	2a2i	2a2i	NOUN
ma-134	147	114	∣∣∣∣	∣∣∣∣	PROPN
ma-134	147	115	δ	δ	PROPN
ma-134	147	116	(	(	PUNCT
ma-134	147	117	ai	ai	PROPN
ma-134	147	118	|x	|x	NOUN
ma-134	147	119	|)+	|)+	ADP
ma-134	147	120	∣∣∣∣1−	∣∣∣∣1−	NUM
ma-134	147	121	ai2a2i	ai2a2i	PROPN
ma-134	147	122	∣∣∣∣	∣∣∣∣	PROPN
ma-134	147	123	δ	δ	PROPN
ma-134	147	124	(	(	PUNCT
ma-134	147	125	−ai	−ai	NOUN
ma-134	147	126	|x	|x	NOUN
ma-134	147	127	|	|	ADV
ma-134	147	128	)	)	PUNCT
ma-134	147	129	]	]	PUNCT
ma-134	148	1	=	=	PUNCT
ma-134	149	1	∞∑	∞∑	NUM
ma-134	149	2	i=0	i=0	PROPN
ma-134	149	3	1	1	NUM
ma-134	149	4	a2i	a2i	NOUN
ma-134	149	5	[	[	PUNCT
ma-134	149	6	δ	δ	PROPN
ma-134	149	7	(	(	PUNCT
ma-134	149	8	ai	ai	PROPN
ma-134	149	9	|x	|x	NOUN
ma-134	149	10	|	|	ADV
ma-134	149	11	)	)	PUNCT
ma-134	149	12	−	−	PROPN
ma-134	150	1	δ	δ	PROPN
ma-134	150	2	(	(	PUNCT
ma-134	150	3	−ai	−ai	NOUN
ma-134	150	4	|x	|x	NOUN
ma-134	150	5	|	|	ADV
ma-134	150	6	)	)	PUNCT
ma-134	150	7	]	]	PUNCT
ma-134	151	1	+	+	CCONJ
ma-134	151	2	∞∑	∞∑	NUM
ma-134	151	3	i=0	i=0	PROPN
ma-134	151	4	1	1	NUM
ma-134	151	5	ai	ai	NOUN
ma-134	151	6	[	[	PUNCT
ma-134	151	7	δ	δ	PROPN
ma-134	151	8	(	(	PUNCT
ma-134	151	9	ai	ai	PROPN
ma-134	151	10	|x	|x	NOUN
ma-134	151	11	|	|	ADV
ma-134	151	12	)	)	PUNCT
ma-134	152	1	+	+	CCONJ
ma-134	152	2	δ	δ	PROPN
ma-134	152	3	(	(	PUNCT
ma-134	152	4	−ai	−ai	NOUN
ma-134	152	5	|x	|x	NOUN
ma-134	152	6	|	|	ADV
ma-134	152	7	)	)	PUNCT
ma-134	152	8	]	]	PUNCT
ma-134	153	1	=	=	PUNCT
ma-134	154	1	λ(x	λ(x	X
ma-134	154	2	)	)	PUNCT
ma-134	155	1	+	+	CCONJ
ma-134	155	2	∆(x	∆(x	NOUN
ma-134	155	3	)	)	PUNCT
ma-134	155	4	.	.	PUNCT
ma-134	156	1	function	function	NOUN
ma-134	156	2	g	g	PROPN
ma-134	156	3	has	have	AUX
ma-134	156	4	been	be	AUX
ma-134	156	5	dated	date	VERB
ma-134	156	6	back	back	ADV
ma-134	156	7	to	to	ADP
ma-134	156	8	derived	derive	VERB
ma-134	156	9	from	from	ADP
ma-134	156	10	(	(	PUNCT
ma-134	156	11	2.5	2.5	NUM
ma-134	156	12	)	)	PUNCT
ma-134	156	13	.	.	PUNCT
ma-134	157	1	we	we	PRON
ma-134	157	2	complete	complete	VERB
ma-134	157	3	the	the	DET
ma-134	157	4	proof	proof	NOUN
ma-134	157	5	.	.	PUNCT
ma-134	158	1	�	�	PROPN
ma-134	158	2	remark	remark	VERB
ma-134	158	3	2.2	2.2	NUM
ma-134	158	4	if	if	SCONJ
ma-134	158	5	u	u	NOUN
ma-134	158	6	=	=	NOUN
ma-134	158	7	1	1	NUM
ma-134	158	8	,	,	PUNCT
ma-134	158	9	v	v	NOUN
ma-134	158	10	=	=	SYM
ma-134	158	11	0	0	NUM
ma-134	158	12	and	and	CCONJ
ma-134	158	13	e(x	e(x	NUM
ma-134	158	14	)	)	PUNCT
ma-134	159	1	=	=	SYM
ma-134	159	2	|x	|x	NOUN
ma-134	159	3	|	|	ADV
ma-134	159	4	,	,	PUNCT
ma-134	159	5	the	the	DET
ma-134	159	6	above	above	ADJ
ma-134	159	7	results	result	NOUN
ma-134	159	8	may	may	AUX
ma-134	159	9	be	be	AUX
ma-134	159	10	trivial	trivial	ADJ
ma-134	159	11	and	and	CCONJ
ma-134	159	12	meaningless.in	meaningless.in	PRON
ma-134	159	13	the	the	DET
ma-134	159	14	above	above	ADJ
ma-134	159	15	results	result	NOUN
ma-134	159	16	,	,	PUNCT
ma-134	159	17	suppose	suppose	VERB
ma-134	159	18	that	that	SCONJ
ma-134	159	19	a	a	DET
ma-134	159	20	∈	∈	PROPN
ma-134	159	21	(	(	PUNCT
ma-134	159	22	−∞,∞	−∞,∞	NOUN
ma-134	159	23	)	)	PUNCT
ma-134	159	24	which	which	PRON
ma-134	159	25	is	be	AUX
ma-134	159	26	not	not	PART
ma-134	159	27	equal	equal	ADJ
ma-134	159	28	to	to	ADP
ma-134	159	29	−1	−1	VERB
ma-134	159	30	,	,	PUNCT
ma-134	159	31	0	0	NUM
ma-134	159	32	,	,	PUNCT
ma-134	159	33	1	1	NUM
ma-134	159	34	.	.	X
ma-134	159	35	exchanging	exchange	VERB
ma-134	159	36	a	a	PRON
ma-134	159	37	with	with	ADP
ma-134	159	38	−a	−a	NOUN
ma-134	159	39	,	,	PUNCT
ma-134	159	40	this	this	DET
ma-134	159	41	transformation	transformation	NOUN
ma-134	159	42	may	may	AUX
ma-134	159	43	not	not	PART
ma-134	159	44	be	be	AUX
ma-134	159	45	different	different	ADJ
ma-134	159	46	from	from	ADP
ma-134	159	47	primary	primary	ADJ
ma-134	159	48	inequality	inequality	NOUN
ma-134	159	49	(	(	PUNCT
ma-134	159	50	2.7	2.7	NUM
ma-134	159	51	)	)	PUNCT
ma-134	159	52	.	.	PUNCT
ma-134	160	1	this	this	PRON
ma-134	160	2	is	be	AUX
ma-134	160	3	a	a	DET
ma-134	160	4	basic	basic	ADJ
ma-134	160	5	factleaving	factleaving	NOUN
ma-134	160	6	to	to	ADP
ma-134	160	7	the	the	DET
ma-134	160	8	reader	reader	NOUN
ma-134	160	9	to	to	PART
ma-134	160	10	check	check	VERB
ma-134	160	11	it	it	PRON
ma-134	160	12	.	.	PUNCT
ma-134	161	1	assume	assume	VERB
ma-134	161	2	that	that	SCONJ
ma-134	161	3	the	the	DET
ma-134	161	4	convergent	convergent	NOUN
ma-134	161	5	series	series	NOUN
ma-134	161	6	∑∞i=0	∑∞i=0	PROPN
ma-134	161	7	1|a|i	1|a|i	NUM
ma-134	161	8	δ	δ	PROPN
ma-134	161	9	(	(	PUNCT
ma-134	161	10	|a|i	|a|i	PROPN
ma-134	161	11	|x	|x	NOUN
ma-134	161	12	|	|	ADV
ma-134	161	13	)	)	PUNCT
ma-134	161	14	establishesfor	establishesfor	VERB
ma-134	161	15	every	every	DET
ma-134	161	16	x	x	SYM
ma-134	161	17	∈	∈	PROPN
ma-134	161	18	x	x	X
ma-134	161	19	.	.	PUNCT
ma-134	162	1	in	in	ADP
ma-134	162	2	fact	fact	NOUN
ma-134	162	3	,	,	PUNCT
ma-134	162	4	the	the	DET
ma-134	162	5	assertions	assertion	NOUN
ma-134	162	6	with	with	ADP
ma-134	162	7	|a|	|a|	NOUN
ma-134	162	8	exchanging	exchanging	NOUN
ma-134	162	9	for	for	ADP
ma-134	162	10	a	a	PRON
ma-134	162	11	has	have	AUX
ma-134	162	12	also	also	ADV
ma-134	162	13	been	be	AUX
ma-134	162	14	achieved	achieve	VERB
ma-134	162	15	by	by	ADP
ma-134	162	16	asimilar	asimilar	ADJ
ma-134	162	17	way	way	NOUN
ma-134	162	18	.	.	PUNCT
ma-134	163	1	corollary	corollary	ADJ
ma-134	163	2	2.2	2.2	NUM
ma-134	163	3	assume	assume	VERB
ma-134	163	4	that	that	SCONJ
ma-134	163	5	(	(	PUNCT
ma-134	163	6	x,+	x,+	NUM
ma-134	163	7	)	)	PUNCT
ma-134	163	8	is	be	AUX
ma-134	163	9	a	a	DET
ma-134	163	10	group	group	NOUN
ma-134	163	11	and	and	CCONJ
ma-134	163	12	set	set	NOUN
ma-134	163	13	(	(	PUNCT
ma-134	163	14	y	y	PROPN
ma-134	163	15	,	,	PUNCT
ma-134	163	16	‖	‖	PROPN
ma-134	163	17	·	·	PUNCT
ma-134	163	18	‖	‖	NUM
ma-134	163	19	)	)	PUNCT
ma-134	163	20	is	be	AUX
ma-134	163	21	a	a	DET
ma-134	163	22	banach	banach	NOUN
ma-134	163	23	space	space	NOUN
ma-134	163	24	.	.	PUNCT
ma-134	164	1	supposefurther	supposefurther	ADV
ma-134	164	2	that	that	SCONJ
ma-134	164	3	the	the	DET
ma-134	164	4	mapping	mapping	NOUN
ma-134	164	5	f	f	X
ma-134	164	6	:	:	PUNCT
ma-134	164	7	x	x	X
ma-134	164	8	→	→	SYM
ma-134	164	9	y	y	PROPN
ma-134	164	10	fulfils	fulfil	VERB
ma-134	164	11	the	the	DET
ma-134	164	12	inequality∥∥∥∥f	inequality∥∥∥∥f	NOUN
ma-134	164	13	(	(	PUNCT
ma-134	164	14	x)−	x)−	PROPN
ma-134	164	15	a	a	PRON
ma-134	164	16	+	+	NUM
ma-134	164	17	1	1	NUM
ma-134	164	18	2a2	2a2	NUM
ma-134	164	19	f	f	NOUN
ma-134	164	20	(	(	PUNCT
ma-134	164	21	a|x	a|x	NOUN
ma-134	164	22	|	|	ADV
ma-134	164	23	)	)	PUNCT
ma-134	165	1	+	+	CCONJ
ma-134	165	2	a	a	DET
ma-134	165	3	−	−	PROPN
ma-134	165	4	1	1	NUM
ma-134	165	5	2a2	2a2	NUM
ma-134	165	6	f	f	NOUN
ma-134	165	7	(	(	PUNCT
ma-134	165	8	−a|x	−a|x	PROPN
ma-134	165	9	|	|	ADV
ma-134	165	10	)	)	PUNCT
ma-134	165	11	∥∥∥∥	∥∥∥∥	PROPN
ma-134	165	12	6	6	NUM
ma-134	165	13	δ	δ	PROPN
ma-134	165	14	,	,	PUNCT
ma-134	165	15	x	x	X
ma-134	165	16	∈	∈	PROPN
ma-134	165	17	x	x	SYM
ma-134	165	18	,	,	PUNCT
ma-134	165	19	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	165	20	eur	eur	PROPN
ma-134	165	21	.	.	PUNCT
ma-134	166	1	j.	j.	PROPN
ma-134	166	2	math	math	PROPN
ma-134	166	3	.	.	PUNCT
ma-134	167	1	anal	anal	PROPN
ma-134	167	2	.	.	PUNCT
ma-134	168	1	10.28924	10.28924	NUM
ma-134	168	2	/	/	SYM
ma-134	168	3	ada	ada	PROPN
ma-134	168	4	/	/	SYM
ma-134	168	5	ma.3.7	ma.3.7	NOUN
ma-134	168	6	7where	7where	NUM
ma-134	168	7	a	a	DET
ma-134	168	8	∈	∈	NOUN
ma-134	168	9	(	(	PUNCT
ma-134	168	10	−∞,∞	−∞,∞	NOUN
ma-134	168	11	)	)	PUNCT
ma-134	168	12	with	with	ADP
ma-134	168	13	|a|	|a|	PROPN
ma-134	168	14	>	>	SYM
ma-134	168	15	1	1	NUM
ma-134	168	16	and	and	CCONJ
ma-134	168	17	δ	δ	PROPN
ma-134	168	18	>	>	X
ma-134	168	19	0	0	PUNCT
ma-134	168	20	is	be	AUX
ma-134	168	21	constant	constant	ADJ
ma-134	168	22	.	.	PUNCT
ma-134	169	1	then	then	ADV
ma-134	169	2	there	there	PRON
ma-134	169	3	is	be	VERB
ma-134	169	4	a	a	DET
ma-134	169	5	unique	unique	ADJ
ma-134	169	6	limiting	limiting	NOUN
ma-134	169	7	evenfunction	evenfunction	NOUN
ma-134	169	8	g	g	NOUN
ma-134	169	9	:	:	PUNCT
ma-134	169	10	x	x	X
ma-134	169	11	→	→	SYM
ma-134	169	12	y	y	PROPN
ma-134	169	13	fulfilling	fulfil	VERB
ma-134	169	14	‖f	‖f	ADP
ma-134	169	15	(	(	PUNCT
ma-134	169	16	x)−	x)−	PROPN
ma-134	169	17	g(x)‖	g(x)‖	PROPN
ma-134	169	18	6	6	NUM
ma-134	169	19	|a|δ	|a|δ	PROPN
ma-134	169	20	|a|	|a|	NOUN
ma-134	169	21	−	−	NOUN
ma-134	169	22	1	1	NUM
ma-134	169	23	.	.	PUNCT
ma-134	170	1	proof	proof	NOUN
ma-134	170	2	.	.	PUNCT
ma-134	171	1	since	since	SCONJ
ma-134	171	2	the	the	DET
ma-134	171	3	function	function	NOUN
ma-134	171	4	δ	δ	PROPN
ma-134	171	5	is	be	AUX
ma-134	171	6	a	a	DET
ma-134	171	7	positive	positive	ADJ
ma-134	171	8	constant	constant	ADJ
ma-134	171	9	,	,	PUNCT
ma-134	171	10	thus	thus	ADV
ma-134	171	11	∆(x	∆(x	ADJ
ma-134	171	12	)	)	PUNCT
ma-134	172	1	=	=	SYM
ma-134	172	2	|a|	|a|	NOUN
ma-134	172	3	|a|−1δ	|a|−1δ	PROPN
ma-134	172	4	and	and	CCONJ
ma-134	172	5	λ(x	λ(x	ADJ
ma-134	172	6	)	)	PUNCT
ma-134	173	1	=	=	SYM
ma-134	173	2	0	0	NUM
ma-134	174	1	for	for	ADP
ma-134	174	2	every	every	DET
ma-134	174	3	x	x	SYM
ma-134	174	4	∈	∈	PROPN
ma-134	174	5	x	x	X
ma-134	174	6	.	.	PUNCT
ma-134	175	1	the	the	DET
ma-134	175	2	mapping	mapping	NOUN
ma-134	175	3	g	g	NOUN
ma-134	175	4	has	have	AUX
ma-134	175	5	been	be	AUX
ma-134	175	6	stated	state	VERB
ma-134	175	7	in	in	ADP
ma-134	175	8	the	the	DET
ma-134	175	9	following	follow	VERB
ma-134	175	10	shape	shape	NOUN
ma-134	175	11	:	:	PUNCT
ma-134	175	12	g(x	g(x	NOUN
ma-134	175	13	)	)	PUNCT
ma-134	175	14	:	:	PUNCT
ma-134	176	1	=	=	PUNCT
ma-134	176	2	lim	lim	PROPN
ma-134	176	3	n→∞	n→∞	X
ma-134	176	4	(	(	PUNCT
ma-134	176	5	|a|n	|a|n	PROPN
ma-134	176	6	+	+	NUM
ma-134	176	7	1	1	NUM
ma-134	176	8	2a2n	2a2n	NUM
ma-134	176	9	f	f	NOUN
ma-134	176	10	(	(	PUNCT
ma-134	176	11	|a|n|x	|a|n|x	PROPN
ma-134	176	12	|)−	|)−	PROPN
ma-134	176	13	|a|n	|a|n	PROPN
ma-134	176	14	−	−	PROPN
ma-134	176	15	1	1	NUM
ma-134	176	16	2a2n	2a2n	NOUN
ma-134	176	17	f	f	X
ma-134	176	18	(	(	PUNCT
ma-134	176	19	−|a|n|x	−|a|n|x	X
ma-134	176	20	|	|	NOUN
ma-134	176	21	)	)	PUNCT
ma-134	176	22	)	)	PUNCT
ma-134	176	23	,	,	PUNCT
ma-134	176	24	x	x	PUNCT
ma-134	176	25	∈	∈	PROPN
ma-134	176	26	x.	x.	NOUN
ma-134	177	1	this	this	PRON
ma-134	177	2	proves	prove	VERB
ma-134	177	3	the	the	DET
ma-134	177	4	proof	proof	NOUN
ma-134	177	5	.	.	PUNCT
ma-134	178	1	�	�	PROPN
ma-134	178	2	remark	remark	VERB
ma-134	178	3	2.3	2.3	NUM
ma-134	178	4	the	the	DET
ma-134	178	5	above	above	ADJ
ma-134	178	6	corollaries	corollary	NOUN
ma-134	178	7	2.1	2.1	NUM
ma-134	178	8	and	and	CCONJ
ma-134	178	9	2.2	2.2	NUM
ma-134	178	10	will	will	AUX
ma-134	178	11	still	still	ADV
ma-134	178	12	establish	establish	VERB
ma-134	178	13	in	in	ADP
ma-134	178	14	β	β	ADJ
ma-134	178	15	-	-	ADJ
ma-134	178	16	homogeneous	homogeneous	ADJ
ma-134	178	17	f	f	PROPN
ma-134	178	18	-spacewith	-spacewith	NOUN
ma-134	178	19	a	a	DET
ma-134	178	20	∈	∈	NOUN
ma-134	178	21	(	(	PUNCT
ma-134	178	22	−∞,∞	−∞,∞	NOUN
ma-134	178	23	)	)	PUNCT
ma-134	178	24	and	and	CCONJ
ma-134	178	25	|a|	|a|	NOUN
ma-134	178	26	>	>	X
ma-134	178	27	1	1	NUM
ma-134	178	28	.	.	PUNCT
ma-134	179	1	if	if	SCONJ
ma-134	179	2	we	we	PRON
ma-134	179	3	exchange	exchange	VERB
ma-134	179	4	a	a	PRON
ma-134	179	5	for	for	ADP
ma-134	179	6	1a	1a	NOUN
ma-134	179	7	in	in	ADP
ma-134	179	8	the	the	DET
ma-134	179	9	equation	equation	NOUN
ma-134	179	10	f	f	X
ma-134	179	11	(	(	PUNCT
ma-134	179	12	x)−	x)−	PROPN
ma-134	179	13	a	a	DET
ma-134	179	14	+	+	NUM
ma-134	179	15	1	1	NUM
ma-134	179	16	2a2	2a2	NUM
ma-134	179	17	f	f	NOUN
ma-134	179	18	(	(	PUNCT
ma-134	179	19	a|x	a|x	NOUN
ma-134	179	20	|	|	ADV
ma-134	179	21	)	)	PUNCT
ma-134	180	1	+	+	CCONJ
ma-134	180	2	a	a	DET
ma-134	180	3	−	−	PROPN
ma-134	180	4	1	1	NUM
ma-134	180	5	2a2	2a2	NUM
ma-134	180	6	f	f	NOUN
ma-134	180	7	(	(	PUNCT
ma-134	180	8	−a|x	−a|x	PROPN
ma-134	180	9	|	|	ADV
ma-134	180	10	)	)	PUNCT
ma-134	180	11	from	from	ADP
ma-134	180	12	(	(	PUNCT
ma-134	180	13	2.7	2.7	NUM
ma-134	180	14	)	)	PUNCT
ma-134	180	15	,	,	PUNCT
ma-134	180	16	the	the	DET
ma-134	180	17	second	second	ADJ
ma-134	180	18	group	group	NOUN
ma-134	180	19	of	of	ADP
ma-134	180	20	results	result	NOUN
ma-134	180	21	will	will	AUX
ma-134	180	22	also	also	ADV
ma-134	180	23	be	be	AUX
ma-134	180	24	obtained	obtain	VERB
ma-134	180	25	with	with	ADP
ma-134	180	26	a	a	DET
ma-134	180	27	is	be	AUX
ma-134	180	28	a	a	DET
ma-134	180	29	positive	positive	ADJ
ma-134	180	30	constant	constant	NOUN
ma-134	180	31	stated	state	VERB
ma-134	180	32	inthe	inthe	ADJ
ma-134	180	33	following	follow	VERB
ma-134	180	34	results	result	NOUN
ma-134	180	35	.	.	PUNCT
ma-134	181	1	corollary	corollary	ADJ
ma-134	181	2	2.3	2.3	NUM
ma-134	181	3	assume	assume	VERB
ma-134	181	4	that	that	SCONJ
ma-134	181	5	(	(	PUNCT
ma-134	181	6	x,+	x,+	NUM
ma-134	181	7	)	)	PUNCT
ma-134	181	8	is	be	AUX
ma-134	181	9	a	a	DET
ma-134	181	10	group	group	NOUN
ma-134	181	11	divisible	divisible	ADJ
ma-134	181	12	by	by	ADP
ma-134	181	13	a	a	PRON
ma-134	181	14	with	with	ADP
ma-134	181	15	a	a	DET
ma-134	181	16	∈	∈	NOUN
ma-134	181	17	(	(	PUNCT
ma-134	181	18	−∞,∞	−∞,∞	NOUN
ma-134	181	19	)	)	PUNCT
ma-134	181	20	and	and	CCONJ
ma-134	181	21	|a|	|a|	NOUN
ma-134	181	22	>	>	SYM
ma-134	181	23	1	1	NUM
ma-134	181	24	andset	andset	NOUN
ma-134	181	25	(	(	PUNCT
ma-134	181	26	y	y	PROPN
ma-134	181	27	,	,	PUNCT
ma-134	181	28	‖	‖	PROPN
ma-134	181	29	·	·	PUNCT
ma-134	181	30	‖	‖	NUM
ma-134	181	31	)	)	PUNCT
ma-134	181	32	is	be	AUX
ma-134	181	33	a	a	DET
ma-134	181	34	banach	banach	NOUN
ma-134	181	35	space	space	NOUN
ma-134	181	36	.	.	PUNCT
ma-134	182	1	suppose	suppose	VERB
ma-134	182	2	further	far	ADV
ma-134	182	3	that	that	SCONJ
ma-134	182	4	the	the	DET
ma-134	182	5	mapping	mapping	NOUN
ma-134	182	6	f	f	X
ma-134	182	7	:	:	PUNCT
ma-134	182	8	x	x	X
ma-134	182	9	→	→	SYM
ma-134	182	10	y	y	PROPN
ma-134	182	11	fulfils	fulfil	VERB
ma-134	182	12	the	the	DET
ma-134	182	13	inequality∥∥∥∥f	inequality∥∥∥∥f	NOUN
ma-134	182	14	(	(	PUNCT
ma-134	182	15	x)−	x)−	PROPN
ma-134	182	16	a2	a2	PROPN
ma-134	182	17	+	+	CCONJ
ma-134	182	18	a	a	DET
ma-134	182	19	2	2	NUM
ma-134	182	20	f	f	NOUN
ma-134	182	21	(	(	PUNCT
ma-134	182	22	1	1	NUM
ma-134	182	23	a	a	DET
ma-134	182	24	|x	|x	NOUN
ma-134	182	25	|	|	ADV
ma-134	182	26	)	)	PUNCT
ma-134	182	27	−	−	PROPN
ma-134	182	28	a2	a2	PROPN
ma-134	182	29	−	−	PROPN
ma-134	182	30	a	a	DET
ma-134	182	31	2	2	NUM
ma-134	182	32	f	f	NOUN
ma-134	182	33	(	(	PUNCT
ma-134	182	34	−	−	PROPN
ma-134	182	35	1	1	NUM
ma-134	182	36	a	a	DET
ma-134	182	37	|x	|x	NOUN
ma-134	182	38	|	|	ADV
ma-134	182	39	)	)	PUNCT
ma-134	182	40	∥∥∥∥	∥∥∥∥	NOUN
ma-134	182	41	6	6	NUM
ma-134	182	42	δ(x	δ(x	NOUN
ma-134	182	43	)	)	PUNCT
ma-134	182	44	,	,	PUNCT
ma-134	182	45	x	x	PUNCT
ma-134	182	46	∈	∈	NOUN
ma-134	182	47	x	x	X
ma-134	182	48	,	,	PUNCT
ma-134	182	49	with	with	ADP
ma-134	182	50	δ	δ	PROPN
ma-134	182	51	:	:	PUNCT
ma-134	182	52	x	x	X
ma-134	182	53	→	→	PUNCT
ma-134	182	54	[	[	X
ma-134	182	55	0,∞	0,∞	NUM
ma-134	182	56	)	)	PUNCT
ma-134	182	57	is	be	AUX
ma-134	182	58	such	such	ADJ
ma-134	182	59	that	that	SCONJ
ma-134	182	60	the	the	DET
ma-134	182	61	convergent	convergent	NOUN
ma-134	182	62	series	series	NOUN
ma-134	182	63	∑∞i=0	∑∞i=0	PROPN
ma-134	182	64	a2iδ	a2iδ	PUNCT
ma-134	182	65	(	(	PUNCT
ma-134	182	66	1ai	1ai	ADJ
ma-134	182	67	|x	|x	NOUN
ma-134	182	68	|	|	ADV
ma-134	182	69	)	)	PUNCT
ma-134	182	70	holds	hold	VERB
ma-134	182	71	for	for	SCONJ
ma-134	182	72	every	every	DET
ma-134	182	73	x	x	SYM
ma-134	182	74	∈	∈	PROPN
ma-134	182	75	x	x	X
ma-134	182	76	.then	.then	VERB
ma-134	182	77	there	there	PRON
ma-134	182	78	has	have	VERB
ma-134	182	79	a	a	DET
ma-134	182	80	unique	unique	ADJ
ma-134	182	81	even	even	ADV
ma-134	182	82	limiting	limit	VERB
ma-134	182	83	mapping	mapping	NOUN
ma-134	182	84	g	g	NOUN
ma-134	182	85	:	:	PUNCT
ma-134	182	86	x	x	X
ma-134	182	87	→	→	SYM
ma-134	182	88	y	y	PROPN
ma-134	182	89	fulfilling	fulfil	VERB
ma-134	182	90	for	for	ADP
ma-134	182	91	every	every	DET
ma-134	182	92	x	x	SYM
ma-134	182	93	∈	∈	PROPN
ma-134	182	94	x	x	X
ma-134	182	95	,	,	PUNCT
ma-134	182	96	g(x	g(x	NOUN
ma-134	182	97	)	)	PUNCT
ma-134	182	98	=	=	SYM
ma-134	182	99	a2	a2	PROPN
ma-134	182	100	+	+	CCONJ
ma-134	182	101	a	a	DET
ma-134	182	102	2	2	NUM
ma-134	182	103	g	g	NOUN
ma-134	182	104	(	(	PUNCT
ma-134	182	105	1	1	NUM
ma-134	182	106	a	a	DET
ma-134	182	107	|x	|x	NOUN
ma-134	182	108	|	|	ADV
ma-134	182	109	)	)	PUNCT
ma-134	182	110	+	+	NUM
ma-134	182	111	a2	a2	PROPN
ma-134	182	112	−	−	PROPN
ma-134	182	113	a	a	DET
ma-134	182	114	2	2	NUM
ma-134	182	115	g	g	NOUN
ma-134	182	116	(	(	PUNCT
ma-134	182	117	−	−	PROPN
ma-134	182	118	1	1	NUM
ma-134	182	119	a	a	DET
ma-134	182	120	|x	|x	NOUN
ma-134	182	121	|	|	ADV
ma-134	182	122	)	)	PUNCT
ma-134	182	123	,	,	PUNCT
ma-134	182	124	and	and	CCONJ
ma-134	182	125	‖f	‖f	ADP
ma-134	182	126	(	(	PUNCT
ma-134	182	127	x)−	x)−	PROPN
ma-134	182	128	g(x)‖	g(x)‖	PROPN
ma-134	182	129	6	6	NUM
ma-134	182	130	∆̃(x	∆̃(x	NOUN
ma-134	182	131	)	)	PUNCT
ma-134	183	1	+	+	CCONJ
ma-134	184	1	λ̃(x	λ̃(x	NOUN
ma-134	184	2	)	)	PUNCT
ma-134	184	3	.	.	PUNCT
ma-134	185	1	furthermore	furthermore	ADV
ma-134	185	2	,	,	PUNCT
ma-134	185	3	the	the	DET
ma-134	185	4	mapping	mapping	NOUN
ma-134	185	5	g	g	NOUN
ma-134	185	6	can	can	AUX
ma-134	185	7	be	be	AUX
ma-134	185	8	stated	state	VERB
ma-134	185	9	in	in	ADP
ma-134	185	10	the	the	DET
ma-134	185	11	following	follow	VERB
ma-134	185	12	shape	shape	NOUN
ma-134	185	13	:	:	PUNCT
ma-134	185	14	g(x	g(x	NOUN
ma-134	185	15	)	)	PUNCT
ma-134	185	16	:	:	PUNCT
ma-134	186	1	=	=	PUNCT
ma-134	186	2	lim	lim	PROPN
ma-134	186	3	n→∞	n→∞	X
ma-134	186	4	[	[	PUNCT
ma-134	186	5	a2n	a2n	PROPN
ma-134	186	6	+	+	CCONJ
ma-134	186	7	an	an	DET
ma-134	186	8	2	2	NUM
ma-134	186	9	f	f	NOUN
ma-134	186	10	(	(	PUNCT
ma-134	186	11	1	1	NUM
ma-134	186	12	an	an	DET
ma-134	186	13	|x	|x	NOUN
ma-134	186	14	|	|	ADV
ma-134	186	15	)	)	PUNCT
ma-134	186	16	+	+	CCONJ
ma-134	186	17	a2n	a2n	VERB
ma-134	186	18	−	−	PUNCT
ma-134	186	19	an	an	DET
ma-134	186	20	2	2	NUM
ma-134	186	21	f	f	NOUN
ma-134	186	22	(	(	PUNCT
ma-134	186	23	−	−	PROPN
ma-134	186	24	1	1	NUM
ma-134	186	25	an	an	DET
ma-134	186	26	|x	|x	NOUN
ma-134	186	27	|	|	ADV
ma-134	186	28	)	)	PUNCT
ma-134	186	29	]	]	PUNCT
ma-134	186	30	,	,	PUNCT
ma-134	186	31	x	x	PUNCT
ma-134	186	32	∈	∈	NOUN
ma-134	186	33	x.	x.	NOUN
ma-134	186	34	proof	proof	NOUN
ma-134	186	35	.	.	PUNCT
ma-134	187	1	applying	apply	VERB
ma-134	187	2	for	for	ADP
ma-134	187	3	theorem	theorem	ADJ
ma-134	187	4	2.1	2.1	NUM
ma-134	187	5	for	for	ADP
ma-134	187	6	u	u	NOUN
ma-134	187	7	:	:	PUNCT
ma-134	187	8	=	=	SYM
ma-134	187	9	a2+a	a2+a	PROPN
ma-134	187	10	2	2	NUM
ma-134	187	11	,	,	PUNCT
ma-134	187	12	v	v	NOUN
ma-134	187	13	:	:	PUNCT
ma-134	187	14	=	=	SYM
ma-134	187	15	a2−a	a2−a	PROPN
ma-134	187	16	2	2	NUM
ma-134	187	17	and	and	CCONJ
ma-134	187	18	e(x	e(x	NUM
ma-134	187	19	)	)	PUNCT
ma-134	187	20	:	:	PUNCT
ma-134	188	1	=	=	SYM
ma-134	188	2	1	1	NUM
ma-134	188	3	a	a	DET
ma-134	188	4	|x	|x	NOUN
ma-134	188	5	|	|	ADV
ma-134	188	6	,	,	PUNCT
ma-134	188	7	for	for	ADP
ma-134	188	8	all	all	DET
ma-134	188	9	x	x	SYM
ma-134	188	10	∈	∈	PROPN
ma-134	188	11	x	x	X
ma-134	188	12	,	,	PUNCT
ma-134	188	13	an	an	DET
ma-134	188	14	easycomputation	easycomputation	NOUN
ma-134	188	15	is	be	AUX
ma-134	188	16	to	to	PART
ma-134	188	17	show	show	VERB
ma-134	188	18	that	that	DET
ma-134	188	19	un	un	PROPN
ma-134	188	20	:	:	PUNCT
ma-134	188	21	=	=	SYM
ma-134	188	22	a2n	a2n	PROPN
ma-134	188	23	+	+	CCONJ
ma-134	188	24	a2n−1	a2n−1	PROPN
ma-134	188	25	2	2	NUM
ma-134	188	26	,	,	PUNCT
ma-134	188	27	vn	vn	X
ma-134	188	28	:	:	PUNCT
ma-134	188	29	=	=	PUNCT
ma-134	188	30	a2n	a2n	PROPN
ma-134	188	31	−	−	X
ma-134	188	32	a2n−1	a2n−1	PROPN
ma-134	188	33	2	2	NUM
ma-134	188	34	,	,	PUNCT
ma-134	188	35	n	n	PRON
ma-134	188	36	∈	∈	PROPN
ma-134	188	37	n.	n.	NOUN
ma-134	188	38	according	accord	VERB
ma-134	188	39	to	to	ADP
ma-134	188	40	the	the	DET
ma-134	188	41	convergent	convergent	NOUN
ma-134	188	42	series	series	NOUN
ma-134	188	43	∑∞	∑∞	NOUN
ma-134	188	44	i=0	i=0	PROPN
ma-134	188	45	a	a	DET
ma-134	188	46	2iδ	2iδ	NOUN
ma-134	188	47	(	(	PUNCT
ma-134	188	48	1	1	NUM
ma-134	188	49	ai	ai	NOUN
ma-134	188	50	|x	|x	NOUN
ma-134	188	51	|	|	ADV
ma-134	188	52	)	)	PUNCT
ma-134	188	53	for	for	ADP
ma-134	188	54	all	all	PRON
ma-134	188	55	x	x	SYM
ma-134	188	56	∈	∈	PROPN
ma-134	188	57	x	x	X
ma-134	188	58	,	,	PUNCT
ma-134	188	59	hence	hence	ADV
ma-134	189	1	the	the	DET
ma-134	189	2	series∑∞	series∑∞	PROPN
ma-134	189	3	i=0	i=0	PROPN
ma-134	189	4	a	a	DET
ma-134	189	5	2i−1δ	2i−1δ	NUM
ma-134	189	6	(	(	PUNCT
ma-134	189	7	1	1	NUM
ma-134	189	8	ai	ai	NOUN
ma-134	189	9	|x	|x	NOUN
ma-134	189	10	|	|	ADV
ma-134	189	11	)	)	PUNCT
ma-134	189	12	is	be	AUX
ma-134	189	13	convergence	convergence	NOUN
ma-134	189	14	,	,	PUNCT
ma-134	189	15	and	and	CCONJ
ma-134	189	16	there	there	PRON
ma-134	189	17	has	have	VERB
ma-134	189	18	a	a	DET
ma-134	189	19	unique	unique	ADJ
ma-134	189	20	even	even	ADV
ma-134	189	21	limiting	limit	VERB
ma-134	189	22	mapping	mapping	NOUN
ma-134	189	23	g	g	NOUN
ma-134	189	24	:	:	PUNCT
ma-134	189	25	x	x	X
ma-134	189	26	→	→	SYM
ma-134	189	27	y	y	PROPN
ma-134	189	28	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	189	29	eur	eur	PROPN
ma-134	189	30	.	.	PUNCT
ma-134	190	1	j.	j.	PROPN
ma-134	190	2	math	math	PROPN
ma-134	190	3	.	.	PUNCT
ma-134	191	1	anal	anal	PROPN
ma-134	191	2	.	.	PUNCT
ma-134	192	1	10.28924	10.28924	NUM
ma-134	192	2	/	/	SYM
ma-134	192	3	ada	ada	NOUN
ma-134	192	4	/	/	SYM
ma-134	192	5	ma.3.7	ma.3.7	NOUN
ma-134	192	6	8fulfilling	8fulfilling	NUM
ma-134	192	7	‖f	‖f	PRON
ma-134	192	8	(	(	PUNCT
ma-134	192	9	x)−	x)−	PROPN
ma-134	192	10	g(x)‖	g(x)‖	PROPN
ma-134	192	11	6	6	NUM
ma-134	193	1	∞∑	∞∑	PROPN
ma-134	193	2	i=0	i=0	PROPN
ma-134	193	3	[	[	X
ma-134	193	4	∣∣∣∣a2i	∣∣∣∣a2i	NOUN
ma-134	193	5	+	+	CCONJ
ma-134	193	6	ai	ai	VERB
ma-134	193	7	2	2	NUM
ma-134	193	8	∣∣∣∣	∣∣∣∣	PROPN
ma-134	193	9	δ	δ	PROPN
ma-134	193	10	(	(	PUNCT
ma-134	193	11	1	1	NUM
ma-134	193	12	ai	ai	NOUN
ma-134	193	13	|x	|x	NOUN
ma-134	193	14	|	|	ADV
ma-134	193	15	)	)	PUNCT
ma-134	193	16	+	+	CCONJ
ma-134	193	17	∣∣∣∣a2i	∣∣∣∣a2i	NOUN
ma-134	193	18	−	−	NOUN
ma-134	193	19	ai2	ai2	NOUN
ma-134	193	20	∣∣∣∣	∣∣∣∣	NOUN
ma-134	193	21	δ(−	δ(−	NOUN
ma-134	193	22	1	1	NUM
ma-134	193	23	ai	ai	NOUN
ma-134	193	24	|x	|x	NOUN
ma-134	193	25	|	|	ADV
ma-134	193	26	)	)	PUNCT
ma-134	193	27	]	]	PUNCT
ma-134	194	1	=	=	PUNCT
ma-134	194	2	∞∑	∞∑	NUM
ma-134	194	3	i=0	i=0	PROPN
ma-134	194	4	a2i	a2i	NOUN
ma-134	194	5	2	2	NUM
ma-134	194	6	[	[	PUNCT
ma-134	194	7	δ	δ	PROPN
ma-134	194	8	(	(	PUNCT
ma-134	194	9	1	1	NUM
ma-134	194	10	ai	ai	NOUN
ma-134	194	11	|x	|x	NOUN
ma-134	194	12	|	|	ADV
ma-134	194	13	)	)	PUNCT
ma-134	195	1	+	+	CCONJ
ma-134	195	2	δ	δ	PROPN
ma-134	195	3	(	(	PUNCT
ma-134	195	4	−	−	PROPN
ma-134	195	5	1	1	NUM
ma-134	195	6	ai	ai	NOUN
ma-134	195	7	|x	|x	NOUN
ma-134	195	8	|	|	ADV
ma-134	195	9	)	)	PUNCT
ma-134	195	10	]	]	PUNCT
ma-134	196	1	+	+	CCONJ
ma-134	196	2	∞∑	∞∑	NUM
ma-134	196	3	i=0	i=0	PROPN
ma-134	196	4	ai	ai	VERB
ma-134	196	5	2	2	NUM
ma-134	196	6	[	[	PUNCT
ma-134	196	7	δ	δ	PROPN
ma-134	196	8	(	(	PUNCT
ma-134	196	9	1	1	NUM
ma-134	196	10	ai	ai	NOUN
ma-134	196	11	|x	|x	NOUN
ma-134	196	12	|	|	ADV
ma-134	196	13	)	)	PUNCT
ma-134	197	1	−	−	PROPN
ma-134	197	2	δ	δ	PROPN
ma-134	197	3	(	(	PUNCT
ma-134	197	4	−	−	PROPN
ma-134	197	5	1	1	NUM
ma-134	197	6	ai	ai	NOUN
ma-134	197	7	|x	|x	NOUN
ma-134	197	8	|	|	ADV
ma-134	197	9	)	)	PUNCT
ma-134	197	10	]	]	PUNCT
ma-134	198	1	=	=	PUNCT
ma-134	198	2	∆̃(x	∆̃(x	NOUN
ma-134	198	3	)	)	PUNCT
ma-134	199	1	+	+	CCONJ
ma-134	200	1	λ̃(x	λ̃(x	NOUN
ma-134	200	2	)	)	PUNCT
ma-134	200	3	.	.	PUNCT
ma-134	201	1	the	the	DET
ma-134	201	2	definition	definition	NOUN
ma-134	201	3	of	of	ADP
ma-134	201	4	g	g	PROPN
ma-134	201	5	is	be	AUX
ma-134	201	6	derived	derive	VERB
ma-134	201	7	from	from	ADP
ma-134	201	8	(	(	PUNCT
ma-134	201	9	2.5	2.5	NUM
ma-134	201	10	)	)	PUNCT
ma-134	201	11	.	.	PUNCT
ma-134	202	1	we	we	PRON
ma-134	202	2	complete	complete	VERB
ma-134	202	3	the	the	DET
ma-134	202	4	proof	proof	NOUN
ma-134	202	5	.	.	PUNCT
ma-134	203	1	�	�	PROPN
ma-134	203	2	remark	remark	VERB
ma-134	203	3	2.4	2.4	NUM
ma-134	203	4	corollary	corollary	NOUN
ma-134	203	5	2.3	2.3	NUM
ma-134	203	6	can	can	AUX
ma-134	203	7	be	be	AUX
ma-134	203	8	used	use	VERB
ma-134	203	9	to	to	PART
ma-134	203	10	investigate	investigate	VERB
ma-134	203	11	the	the	DET
ma-134	203	12	function	function	NOUN
ma-134	203	13	from	from	ADP
ma-134	203	14	which	which	PRON
ma-134	203	15	it	it	PRON
ma-134	203	16	could	could	AUX
ma-134	203	17	be	be	AUX
ma-134	203	18	splitinto	splitinto	NOUN
ma-134	203	19	even	even	ADV
ma-134	203	20	and	and	CCONJ
ma-134	203	21	odd	odd	ADJ
ma-134	203	22	parts	part	NOUN
ma-134	203	23	.	.	PUNCT
ma-134	204	1	there	there	PRON
ma-134	204	2	is	be	VERB
ma-134	204	3	a	a	DET
ma-134	204	4	good	good	ADJ
ma-134	204	5	point	point	NOUN
ma-134	204	6	of	of	ADP
ma-134	204	7	the	the	DET
ma-134	204	8	approach	approach	NOUN
ma-134	204	9	achieved	achieve	VERB
ma-134	204	10	here	here	ADV
ma-134	204	11	where	where	SCONJ
ma-134	204	12	the	the	DET
ma-134	204	13	functionssplit	functionssplit	NOUN
ma-134	204	14	into	into	ADP
ma-134	204	15	two	two	NUM
ma-134	204	16	two	two	NUM
ma-134	204	17	parts	part	NOUN
ma-134	204	18	of	of	ADP
ma-134	204	19	odd	odd	ADJ
ma-134	204	20	and	and	CCONJ
ma-134	204	21	even	even	ADV
ma-134	204	22	functions	function	NOUN
ma-134	204	23	can	can	AUX
ma-134	204	24	give	give	VERB
ma-134	204	25	more	more	ADJ
ma-134	204	26	concise	concise	ADJ
ma-134	204	27	approximations	approximation	NOUN
ma-134	204	28	than	than	ADP
ma-134	204	29	thebefore	thebefore	NOUN
ma-134	204	30	approximations	approximation	NOUN
ma-134	204	31	in	in	ADP
ma-134	204	32	theorem	theorem	ADJ
ma-134	204	33	2.1	2.1	NUM
ma-134	204	34	.	.	PUNCT
ma-134	205	1	the	the	DET
ma-134	205	2	above	above	ADJ
ma-134	205	3	results	result	NOUN
ma-134	205	4	is	be	AUX
ma-134	205	5	the	the	DET
ma-134	205	6	counterpart	counterpart	NOUN
ma-134	205	7	of	of	ADP
ma-134	205	8	the	the	DET
ma-134	205	9	correspondingresults	correspondingresult	NOUN
ma-134	205	10	of	of	ADP
ma-134	205	11	sikorska	sikorska	PROPN
ma-134	205	12	’s	’s	PART
ma-134	205	13	paper	paper	NOUN
ma-134	205	14	.	.	PUNCT
ma-134	206	1	however	however	ADV
ma-134	206	2	,	,	PUNCT
ma-134	206	3	it	it	PRON
ma-134	206	4	is	be	AUX
ma-134	206	5	not	not	PART
ma-134	206	6	copied	copy	VERB
ma-134	206	7	word	word	NOUN
ma-134	206	8	by	by	ADP
ma-134	206	9	word	word	NOUN
ma-134	206	10	.	.	PUNCT
ma-134	207	1	it	it	PRON
ma-134	207	2	is	be	AUX
ma-134	207	3	the	the	DET
ma-134	207	4	counterpart	counterpart	NOUN
ma-134	207	5	of	of	ADP
ma-134	207	6	evenfunction	evenfunction	NOUN
ma-134	207	7	.	.	PUNCT
ma-134	208	1	3	3	X
ma-134	208	2	.	.	X
ma-134	208	3	the	the	DET
ma-134	208	4	stability	stability	NOUN
ma-134	208	5	of	of	ADP
ma-134	208	6	functional	functional	ADJ
ma-134	208	7	equations	equation	NOUN
ma-134	208	8	in	in	ADP
ma-134	208	9	f	f	NOUN
ma-134	208	10	-	-	PUNCT
ma-134	208	11	space	space	NOUN
ma-134	208	12	an	an	DET
ma-134	208	13	f	f	PROPN
ma-134	208	14	-space	-space	NOUN
ma-134	208	15	is	be	AUX
ma-134	208	16	called	call	VERB
ma-134	208	17	β	β	NOUN
ma-134	208	18	-	-	ADJ
ma-134	208	19	homogeneous	homogeneous	ADJ
ma-134	208	20	if	if	SCONJ
ma-134	208	21	it	it	PRON
ma-134	208	22	satisfies	satisfy	VERB
ma-134	208	23	‖tx‖	‖tx‖	PROPN
ma-134	208	24	=	=	PUNCT
ma-134	208	25	|t|β‖x‖	|t|β‖x‖	VERB
ma-134	208	26	for	for	ADP
ma-134	208	27	every	every	DET
ma-134	208	28	x	x	SYM
ma-134	208	29	∈	∈	PROPN
ma-134	208	30	x	x	X
ma-134	208	31	,	,	PUNCT
ma-134	208	32	t	t	PROPN
ma-134	208	33	∈	∈	PROPN
ma-134	208	34	c.	c.	PROPN
ma-134	208	35	inthis	inthis	PROPN
ma-134	208	36	section	section	NOUN
ma-134	208	37	of	of	ADP
ma-134	208	38	the	the	DET
ma-134	208	39	first	first	ADJ
ma-134	208	40	two	two	NUM
ma-134	208	41	theorems	theorem	NOUN
ma-134	208	42	,	,	PUNCT
ma-134	208	43	β1	β1	PROPN
ma-134	208	44	,	,	PUNCT
ma-134	208	45	β2	β2	NOUN
ma-134	208	46	are	be	AUX
ma-134	208	47	to	to	PART
ma-134	208	48	be	be	AUX
ma-134	208	49	0	0	NUM
ma-134	208	50	<	<	X
ma-134	208	51	β1	β1	PROPN
ma-134	208	52	≤	≤	NUM
ma-134	208	53	1	1	NUM
ma-134	208	54	and	and	CCONJ
ma-134	208	55	0	0	NUM
ma-134	208	56	<	<	X
ma-134	208	57	β2	β2	VERB
ma-134	208	58	≤	≤	NOUN
ma-134	208	59	1	1	NUM
ma-134	208	60	.	.	PUNCT
ma-134	209	1	furthermore	furthermore	ADV
ma-134	209	2	,	,	PUNCT
ma-134	209	3	we	we	PRON
ma-134	209	4	suppose	suppose	VERB
ma-134	209	5	x	x	PRON
ma-134	209	6	is	be	AUX
ma-134	209	7	β1	β1	NOUN
ma-134	209	8	-	-	PUNCT
ma-134	209	9	homogeneous	homogeneous	ADJ
ma-134	209	10	f	f	NOUN
ma-134	209	11	-	-	PUNCT
ma-134	209	12	space	space	NOUN
ma-134	209	13	and	and	CCONJ
ma-134	209	14	y	y	PROPN
ma-134	209	15	is	be	AUX
ma-134	209	16	β2	β2	VERB
ma-134	209	17	-	-	PUNCT
ma-134	209	18	homogeneous	homogeneous	ADJ
ma-134	209	19	f	f	NOUN
ma-134	209	20	-	-	NOUN
ma-134	209	21	space	space	NOUN
ma-134	209	22	.	.	PUNCT
ma-134	210	1	before	before	ADP
ma-134	210	2	applyingtheorem	applyingtheorem	NOUN
ma-134	210	3	2.1	2.1	NUM
ma-134	210	4	we	we	PRON
ma-134	210	5	would	would	AUX
ma-134	210	6	like	like	VERB
ma-134	210	7	to	to	PART
ma-134	210	8	make	make	VERB
ma-134	210	9	an	an	DET
ma-134	210	10	answer	answer	NOUN
ma-134	210	11	that	that	SCONJ
ma-134	210	12	all	all	DET
ma-134	210	13	roads	road	NOUN
ma-134	210	14	lead	lead	VERB
ma-134	210	15	to	to	ADP
ma-134	210	16	rome	rome	PROPN
ma-134	210	17	.	.	PUNCT
ma-134	211	1	therefore	therefore	ADV
ma-134	211	2	anotherapproach	anotherapproach	VERB
ma-134	211	3	to	to	PART
ma-134	211	4	prove	prove	VERB
ma-134	211	5	the	the	DET
ma-134	211	6	following	follow	VERB
ma-134	211	7	functional	functional	ADJ
ma-134	211	8	inequality	inequality	NOUN
ma-134	211	9	has	have	AUX
ma-134	211	10	been	be	AUX
ma-134	211	11	stated	state	VERB
ma-134	211	12	in	in	ADP
ma-134	211	13	the	the	DET
ma-134	211	14	following	following	NOUN
ma-134	211	15	.	.	PUNCT
ma-134	212	1	in	in	ADP
ma-134	212	2	fact	fact	NOUN
ma-134	212	3	,	,	PUNCT
ma-134	212	4	there	there	PRON
ma-134	212	5	is	be	VERB
ma-134	212	6	also	also	ADV
ma-134	212	7	a	a	DET
ma-134	212	8	similar	similar	ADJ
ma-134	212	9	solution	solution	NOUN
ma-134	212	10	about	about	ADP
ma-134	212	11	functional	functional	ADJ
ma-134	212	12	equation	equation	NOUN
ma-134	212	13	being	be	AUX
ma-134	212	14	stated	state	VERB
ma-134	212	15	in	in	ADP
ma-134	212	16	[	[	X
ma-134	212	17	8	8	NUM
ma-134	212	18	]	]	PUNCT
ma-134	212	19	.	.	PUNCT
ma-134	213	1	theorem	theorem	ADJ
ma-134	213	2	3.1	3.1	NUM
ma-134	213	3	assume	assume	VERB
ma-134	213	4	the	the	DET
ma-134	213	5	mapping	mapping	NOUN
ma-134	213	6	f	f	X
ma-134	213	7	:	:	PUNCT
ma-134	214	1	x	x	X
ma-134	214	2	→	→	SYM
ma-134	214	3	y	y	PROPN
ma-134	214	4	fulfilling	fulfil	VERB
ma-134	214	5	for	for	ADP
ma-134	214	6	some	some	DET
ma-134	214	7	k	k	PROPN
ma-134	214	8	≥	≥	NOUN
ma-134	214	9	0	0	NUM
ma-134	214	10	and	and	CCONJ
ma-134	214	11	r	r	NOUN
ma-134	214	12	<	<	X
ma-134	214	13	β2	β2	NOUN
ma-134	214	14	β1	β1	PROPN
ma-134	214	15	‖f	‖f	PRON
ma-134	214	16	(	(	PUNCT
ma-134	214	17	x	x	X
ma-134	215	1	+	+	NUM
ma-134	215	2	y	y	PROPN
ma-134	215	3	+	+	PROPN
ma-134	215	4	z	z	NOUN
ma-134	215	5	)	)	PUNCT
ma-134	216	1	+	+	CCONJ
ma-134	216	2	f	f	X
ma-134	216	3	(	(	PUNCT
ma-134	216	4	x	x	X
ma-134	216	5	)	)	PUNCT
ma-134	217	1	+	+	NUM
ma-134	217	2	f	f	X
ma-134	217	3	(	(	PUNCT
ma-134	217	4	y	y	NOUN
ma-134	217	5	)	)	PUNCT
ma-134	218	1	+	+	NUM
ma-134	218	2	f	f	X
ma-134	218	3	(	(	PUNCT
ma-134	218	4	z)−	z)−	PROPN
ma-134	218	5	f	f	X
ma-134	218	6	(	(	PUNCT
ma-134	218	7	x	x	PROPN
ma-134	218	8	+	+	PUNCT
ma-134	218	9	y)−	y)−	PROPN
ma-134	218	10	f	f	NOUN
ma-134	218	11	(	(	PUNCT
ma-134	218	12	z	z	PROPN
ma-134	218	13	+	+	NOUN
ma-134	218	14	y)−	y)−	PROPN
ma-134	218	15	f	f	NOUN
ma-134	218	16	(	(	PUNCT
ma-134	218	17	x	x	PROPN
ma-134	218	18	+	+	PRON
ma-134	218	19	z)‖	z)‖	NUM
ma-134	218	20	6	6	NUM
ma-134	218	21	k	k	NOUN
ma-134	218	22	(	(	PUNCT
ma-134	218	23	‖x‖r	‖x‖r	NOUN
ma-134	218	24	+	+	CCONJ
ma-134	218	25	‖y‖r	‖y‖r	ADJ
ma-134	218	26	+	+	CCONJ
ma-134	218	27	‖z‖r	‖z‖r	NOUN
ma-134	218	28	)	)	PUNCT
ma-134	218	29	(	(	PUNCT
ma-134	218	30	3.1	3.1	NUM
ma-134	218	31	)	)	PUNCT
ma-134	218	32	for	for	ADP
ma-134	218	33	x	x	PROPN
ma-134	218	34	,	,	PUNCT
ma-134	218	35	y	y	PROPN
ma-134	218	36	,	,	PUNCT
ma-134	218	37	z	z	NOUN
ma-134	218	38	∈	∈	PROPN
ma-134	218	39	x	x	X
ma-134	218	40	.	.	PUNCT
ma-134	219	1	then	then	ADV
ma-134	219	2	there	there	PRON
ma-134	219	3	has	have	VERB
ma-134	219	4	a	a	DET
ma-134	219	5	unique	unique	ADJ
ma-134	219	6	limiting	limit	VERB
ma-134	219	7	mapping	mapping	NOUN
ma-134	219	8	ψ1	ψ1	NOUN
ma-134	219	9	:	:	PUNCT
ma-134	219	10	x	x	X
ma-134	219	11	→	→	PUNCT
ma-134	219	12	y	y	PRON
ma-134	219	13	such	such	ADJ
ma-134	219	14	that	that	SCONJ
ma-134	219	15	‖f	‖f	ADP
ma-134	219	16	(	(	PUNCT
ma-134	219	17	x)−	x)−	PROPN
ma-134	219	18	ψ1(x)‖	ψ1(x)‖	NOUN
ma-134	219	19	6	6	NUM
ma-134	219	20	(	(	PUNCT
ma-134	219	21	2	2	NUM
ma-134	219	22	+	+	NUM
ma-134	219	23	2rβ1	2rβ1	NUM
ma-134	219	24	+	+	CCONJ
ma-134	219	25	3	3	NUM
ma-134	219	26	·	·	SYM
ma-134	219	27	2β2)k	2β2)k	NUM
ma-134	219	28	(	(	PUNCT
ma-134	219	29	2β1r	2β1r	ADJ
ma-134	219	30	−	−	NOUN
ma-134	219	31	22β2)(2β1r	22β2)(2β1r	NUM
ma-134	219	32	−	−	NOUN
ma-134	219	33	2β2	2β2	NUM
ma-134	219	34	)	)	PUNCT
ma-134	219	35	‖x‖r	‖x‖r	NOUN
ma-134	219	36	for	for	ADP
ma-134	219	37	x	x	SYM
ma-134	219	38	∈	∈	PROPN
ma-134	219	39	x	x	X
ma-134	219	40	.	.	PUNCT
ma-134	220	1	moreover	moreover	ADV
ma-134	220	2	,	,	PUNCT
ma-134	220	3	ψ1	ψ1	ADV
ma-134	220	4	satisfying	satisfy	VERB
ma-134	220	5	the	the	DET
ma-134	220	6	above	above	ADJ
ma-134	220	7	inequality	inequality	NOUN
ma-134	220	8	is	be	AUX
ma-134	220	9	also	also	ADV
ma-134	220	10	satisfying	satisfy	VERB
ma-134	220	11	the	the	DET
ma-134	220	12	following	follow	VERB
ma-134	220	13	equation	equation	NOUN
ma-134	220	14	ψ1(x	ψ1(x	PROPN
ma-134	221	1	+	+	CCONJ
ma-134	221	2	y	y	PROPN
ma-134	221	3	+	+	PROPN
ma-134	221	4	z	z	NOUN
ma-134	221	5	)	)	PUNCT
ma-134	221	6	+	+	PUNCT
ma-134	221	7	ψ1(x	ψ1(x	NOUN
ma-134	221	8	)	)	PUNCT
ma-134	221	9	+	+	PUNCT
ma-134	221	10	ψ1(z	ψ1(z	X
ma-134	221	11	)	)	PUNCT
ma-134	221	12	+	+	PUNCT
ma-134	221	13	ψ1(y	ψ1(y	NOUN
ma-134	221	14	)	)	PUNCT
ma-134	221	15	=	=	SYM
ma-134	221	16	ψ1(x	ψ1(x	PROPN
ma-134	222	1	+	+	NUM
ma-134	222	2	y	y	NOUN
ma-134	222	3	)	)	PUNCT
ma-134	223	1	+	+	PUNCT
ma-134	223	2	ψ1(z	ψ1(z	PUNCT
ma-134	224	1	+	+	NUM
ma-134	224	2	y	y	NOUN
ma-134	224	3	)	)	PUNCT
ma-134	225	1	+	+	PUNCT
ma-134	225	2	ψ1(x	ψ1(x	PROPN
ma-134	226	1	+	+	ADJ
ma-134	226	2	z	z	NOUN
ma-134	226	3	)	)	PUNCT
ma-134	226	4	(	(	PUNCT
ma-134	226	5	3.2	3.2	NUM
ma-134	226	6	)	)	PUNCT
ma-134	226	7	for	for	ADP
ma-134	226	8	all	all	DET
ma-134	226	9	x	x	NOUN
ma-134	226	10	,	,	PUNCT
ma-134	226	11	y	y	PROPN
ma-134	226	12	,	,	PUNCT
ma-134	226	13	z	z	NOUN
ma-134	226	14	∈	∈	PROPN
ma-134	226	15	x	x	X
ma-134	226	16	.	.	PUNCT
ma-134	227	1	proof	proof	NOUN
ma-134	227	2	.	.	PUNCT
ma-134	228	1	from	from	ADP
ma-134	228	2	(	(	PUNCT
ma-134	228	3	x	x	NOUN
ma-134	228	4	,	,	PUNCT
ma-134	228	5	x	x	X
ma-134	228	6	,	,	PUNCT
ma-134	228	7	x	x	X
ma-134	228	8	)	)	PUNCT
ma-134	228	9	in	in	ADP
ma-134	228	10	place	place	NOUN
ma-134	228	11	of	of	ADP
ma-134	228	12	(	(	PUNCT
ma-134	228	13	x	x	X
ma-134	228	14	,	,	PUNCT
ma-134	228	15	y	y	PROPN
ma-134	228	16	,	,	PUNCT
ma-134	228	17	z	z	NOUN
ma-134	228	18	)	)	PUNCT
ma-134	228	19	in	in	ADP
ma-134	228	20	(	(	PUNCT
ma-134	228	21	3.1	3.1	NUM
ma-134	228	22	)	)	PUNCT
ma-134	228	23	we	we	PRON
ma-134	228	24	have	have	AUX
ma-134	228	25	‖f	‖f	PRON
ma-134	228	26	(	(	PUNCT
ma-134	228	27	3x	3x	NUM
ma-134	228	28	)	)	PUNCT
ma-134	229	1	+	+	CCONJ
ma-134	229	2	3f	3f	PROPN
ma-134	229	3	(	(	PUNCT
ma-134	229	4	x)−	x)−	PROPN
ma-134	229	5	3f	3f	PROPN
ma-134	229	6	(	(	PUNCT
ma-134	229	7	2x)‖	2x)‖	PROPN
ma-134	229	8	6	6	NUM
ma-134	229	9	3k	3k	NUM
ma-134	229	10	(	(	PUNCT
ma-134	229	11	‖x‖r	‖x‖r	NOUN
ma-134	229	12	)	)	PUNCT
ma-134	229	13	.	.	PUNCT
ma-134	230	1	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	230	2	eur	eur	PROPN
ma-134	230	3	.	.	PUNCT
ma-134	231	1	j.	j.	PROPN
ma-134	231	2	math	math	PROPN
ma-134	231	3	.	.	PUNCT
ma-134	232	1	anal	anal	PROPN
ma-134	232	2	.	.	PUNCT
ma-134	233	1	10.28924	10.28924	NUM
ma-134	233	2	/	/	SYM
ma-134	233	3	ada	ada	PROPN
ma-134	233	4	/	/	SYM
ma-134	233	5	ma.3.7	ma.3.7	NOUN
ma-134	233	6	9hence	9hence	NUM
ma-134	233	7	‖2f	‖2f	NOUN
ma-134	233	8	(	(	PUNCT
ma-134	233	9	3x	3x	NUM
ma-134	233	10	)	)	PUNCT
ma-134	234	1	+	+	CCONJ
ma-134	234	2	6f	6f	NUM
ma-134	234	3	(	(	PUNCT
ma-134	234	4	x)−	x)−	PROPN
ma-134	234	5	6f	6f	PROPN
ma-134	234	6	(	(	PUNCT
ma-134	234	7	2x)‖	2x)‖	NUM
ma-134	234	8	6	6	NUM
ma-134	234	9	3	3	NUM
ma-134	234	10	·	·	SYM
ma-134	234	11	2β2k	2β2k	NUM
ma-134	234	12	(	(	PUNCT
ma-134	234	13	‖x‖r	‖x‖r	NOUN
ma-134	234	14	)	)	PUNCT
ma-134	234	15	.	.	PUNCT
ma-134	235	1	substitute	substitute	NOUN
ma-134	235	2	(	(	PUNCT
ma-134	235	3	x	x	X
ma-134	235	4	,	,	PUNCT
ma-134	235	5	x	x	X
ma-134	235	6	,	,	PUNCT
ma-134	235	7	2x	2x	NUM
ma-134	235	8	)	)	PUNCT
ma-134	235	9	in	in	ADP
ma-134	235	10	place	place	NOUN
ma-134	235	11	of	of	ADP
ma-134	235	12	(	(	PUNCT
ma-134	235	13	x	x	X
ma-134	235	14	,	,	PUNCT
ma-134	235	15	y	y	PROPN
ma-134	235	16	,	,	PUNCT
ma-134	235	17	z	z	NOUN
ma-134	235	18	)	)	PUNCT
ma-134	235	19	in	in	ADP
ma-134	235	20	(	(	PUNCT
ma-134	235	21	3.1	3.1	NUM
ma-134	235	22	)	)	PUNCT
ma-134	235	23	,	,	PUNCT
ma-134	235	24	yielding	yield	VERB
ma-134	235	25	that	that	SCONJ
ma-134	235	26	‖f	‖f	ADP
ma-134	235	27	(	(	PUNCT
ma-134	235	28	4x	4x	NUM
ma-134	235	29	)	)	PUNCT
ma-134	236	1	+	+	NUM
ma-134	236	2	2f	2f	NUM
ma-134	236	3	(	(	PUNCT
ma-134	236	4	x)−	x)−	PROPN
ma-134	236	5	2f	2f	NUM
ma-134	236	6	(	(	PUNCT
ma-134	236	7	3x)‖	3x)‖	NUM
ma-134	236	8	6	6	NUM
ma-134	236	9	(	(	PUNCT
ma-134	236	10	2	2	NUM
ma-134	236	11	+	+	SYM
ma-134	236	12	2rβ1)k	2rβ1)k	NUM
ma-134	236	13	(	(	PUNCT
ma-134	236	14	‖x‖r	‖x‖r	NOUN
ma-134	236	15	)	)	PUNCT
ma-134	236	16	.	.	PUNCT
ma-134	237	1	and	and	CCONJ
ma-134	237	2	combining	combine	VERB
ma-134	237	3	the	the	DET
ma-134	237	4	above	above	ADJ
ma-134	237	5	two	two	NUM
ma-134	237	6	inequalities	inequality	NOUN
ma-134	237	7	,	,	PUNCT
ma-134	237	8	we	we	PRON
ma-134	237	9	get	get	VERB
ma-134	237	10	‖f	‖f	ADP
ma-134	237	11	(	(	PUNCT
ma-134	237	12	4x	4x	NOUN
ma-134	237	13	)	)	PUNCT
ma-134	238	1	+	+	NUM
ma-134	238	2	8f	8f	NOUN
ma-134	238	3	(	(	PUNCT
ma-134	238	4	x)−	x)−	PROPN
ma-134	238	5	6f	6f	PROPN
ma-134	238	6	(	(	PUNCT
ma-134	238	7	2x)‖	2x)‖	NUM
ma-134	238	8	6	6	NUM
ma-134	238	9	(	(	PUNCT
ma-134	238	10	2	2	NUM
ma-134	238	11	+	+	NUM
ma-134	238	12	2rβ1	2rβ1	NUM
ma-134	238	13	+	+	CCONJ
ma-134	238	14	3	3	NUM
ma-134	238	15	·	·	SYM
ma-134	238	16	2β2)k	2β2)k	NUM
ma-134	238	17	(	(	PUNCT
ma-134	238	18	‖x‖r	‖x‖r	NOUN
ma-134	238	19	)	)	PUNCT
ma-134	238	20	.	.	PUNCT
ma-134	239	1	(	(	PUNCT
ma-134	239	2	3.3	3.3	NUM
ma-134	239	3	)	)	PUNCT
ma-134	239	4	let	let	VERB
ma-134	239	5	us	we	PRON
ma-134	239	6	define	define	VERB
ma-134	239	7	g(x	g(x	NOUN
ma-134	239	8	)	)	PUNCT
ma-134	240	1	=	=	SYM
ma-134	240	2	f	f	X
ma-134	240	3	(	(	PUNCT
ma-134	240	4	2x)−	2x)−	NUM
ma-134	240	5	4f	4f	NUM
ma-134	240	6	(	(	PUNCT
ma-134	240	7	x	x	NOUN
ma-134	240	8	)	)	PUNCT
ma-134	240	9	for	for	ADP
ma-134	240	10	all	all	PRON
ma-134	240	11	x	x	SYM
ma-134	240	12	∈	∈	NOUN
ma-134	240	13	x	x	X
ma-134	240	14	.	.	PUNCT
ma-134	241	1	hence	hence	ADV
ma-134	241	2	‖g(2x)/2−	‖g(2x)/2−	ADV
ma-134	241	3	g(x)‖	g(x)‖	PROPN
ma-134	241	4	6	6	NUM
ma-134	241	5	(	(	PUNCT
ma-134	241	6	2	2	NUM
ma-134	241	7	+	+	NUM
ma-134	241	8	2rβ1	2rβ1	NUM
ma-134	241	9	+	+	CCONJ
ma-134	241	10	3	3	NUM
ma-134	241	11	·	·	SYM
ma-134	241	12	2β2)k	2β2)k	NUM
ma-134	241	13	(	(	PUNCT
ma-134	241	14	‖x‖r	‖x‖r	NOUN
ma-134	241	15	)	)	PUNCT
ma-134	241	16	/2β2	/2β2	PUNCT
ma-134	242	1	(	(	PUNCT
ma-134	242	2	3.4	3.4	NUM
ma-134	242	3	)	)	PUNCT
ma-134	242	4	for	for	ADP
ma-134	242	5	all	all	PRON
ma-134	242	6	x	x	SYM
ma-134	242	7	∈	∈	NOUN
ma-134	242	8	x	x	X
ma-134	242	9	.	.	PUNCT
ma-134	243	1	therefore	therefore	ADV
ma-134	243	2	‖g(2nx)/2n	‖g(2nx)/2n	NUM
ma-134	243	3	−	−	PUNCT
ma-134	244	1	g(2mx)/2m‖	g(2mx)/2m‖	NOUN
ma-134	244	2	6	6	NUM
ma-134	244	3	n−1∑	n−1∑	PROPN
ma-134	244	4	j	j	NOUN
ma-134	244	5	=	=	NOUN
ma-134	244	6	m	m	PROPN
ma-134	244	7	(	(	PUNCT
ma-134	244	8	2	2	NUM
ma-134	244	9	+	+	NUM
ma-134	244	10	2rβ1	2rβ1	NUM
ma-134	244	11	+	+	CCONJ
ma-134	244	12	3	3	NUM
ma-134	244	13	·	·	SYM
ma-134	244	14	2β2)k	2β2)k	NUM
ma-134	244	15	2jβ1r	2jβ1r	NUM
ma-134	244	16	2β22jβ2	2β22jβ2	NUM
ma-134	244	17	(	(	PUNCT
ma-134	244	18	‖x‖r	‖x‖r	NOUN
ma-134	244	19	)	)	PUNCT
ma-134	244	20	(	(	PUNCT
ma-134	244	21	3.5	3.5	NUM
ma-134	244	22	)	)	PUNCT
ma-134	244	23	for	for	ADP
ma-134	244	24	m	m	PROPN
ma-134	244	25	,	,	PUNCT
ma-134	244	26	n	n	PROPN
ma-134	244	27	∈	∈	PROPN
ma-134	244	28	n	n	NOUN
ma-134	244	29	with	with	ADP
ma-134	244	30	n	n	NOUN
ma-134	244	31	>	>	SYM
ma-134	244	32	m	m	PROPN
ma-134	244	33	and	and	CCONJ
ma-134	244	34	all	all	DET
ma-134	244	35	x	x	SYM
ma-134	244	36	∈	∈	NOUN
ma-134	244	37	x	x	X
ma-134	244	38	.	.	PUNCT
ma-134	245	1	since	since	SCONJ
ma-134	245	2	the	the	DET
ma-134	245	3	sequence	sequence	NOUN
ma-134	245	4	{	{	PUNCT
ma-134	245	5	g(2nx)/2n	g(2nx)/2n	PROPN
ma-134	245	6	}	}	PUNCT
ma-134	245	7	is	be	AUX
ma-134	245	8	a	a	DET
ma-134	245	9	cauchy	cauchy	ADJ
ma-134	245	10	sequencein	sequencein	NOUN
ma-134	245	11	y	y	PROPN
ma-134	245	12	for	for	ADP
ma-134	245	13	all	all	PRON
ma-134	245	14	x	x	SYM
ma-134	245	15	∈	∈	PROPN
ma-134	245	16	x	x	X
ma-134	245	17	and	and	CCONJ
ma-134	245	18	y	y	PROPN
ma-134	245	19	is	be	AUX
ma-134	245	20	complete	complete	ADJ
ma-134	245	21	,	,	PUNCT
ma-134	245	22	the	the	DET
ma-134	245	23	mapping	mapping	NOUN
ma-134	245	24	can	can	AUX
ma-134	245	25	be	be	AUX
ma-134	245	26	well	well	ADV
ma-134	245	27	defined	define	VERB
ma-134	245	28	as	as	ADP
ma-134	245	29	:	:	PUNCT
ma-134	245	30	φ(x	φ(x	NOUN
ma-134	245	31	)	)	PUNCT
ma-134	245	32	=	=	VERB
ma-134	245	33	lim	lim	PROPN
ma-134	245	34	n→∞	n→∞	NUM
ma-134	245	35	g(2nx)/2n	g(2nx)/2n	NOUN
ma-134	245	36	for	for	ADP
ma-134	245	37	all	all	DET
ma-134	245	38	x	x	SYM
ma-134	245	39	∈	∈	NOUN
ma-134	245	40	x	x	X
ma-134	245	41	.	.	PUNCT
ma-134	246	1	in	in	ADP
ma-134	246	2	particular	particular	ADJ
ma-134	246	3	,	,	PUNCT
ma-134	246	4	letting	let	VERB
ma-134	246	5	m	m	PROPN
ma-134	246	6	=	=	SYM
ma-134	246	7	0	0	PUNCT
ma-134	246	8	and	and	CCONJ
ma-134	246	9	setting	set	VERB
ma-134	246	10	n	n	PRON
ma-134	246	11	→∞	→∞	X
ma-134	246	12	in	in	ADP
ma-134	246	13	(	(	PUNCT
ma-134	246	14	3.4	3.4	NUM
ma-134	246	15	)	)	PUNCT
ma-134	246	16	,	,	PUNCT
ma-134	246	17	we	we	PRON
ma-134	246	18	have	have	VERB
ma-134	246	19	‖φ(x)−	‖φ(x)−	NOUN
ma-134	246	20	g(x)‖	g(x)‖	PROPN
ma-134	246	21	6	6	NUM
ma-134	246	22	(	(	PUNCT
ma-134	246	23	2	2	NUM
ma-134	246	24	+	+	NOUN
ma-134	246	25	2rβ1	2rβ1	NUM
ma-134	246	26	+	+	NOUN
ma-134	246	27	3·2β2)k	3·2β2)k	NUM
ma-134	246	28	2β1r−2β2	2β1r−2β2	NUM
ma-134	246	29	(	(	PUNCT
ma-134	246	30	‖x‖r	‖x‖r	NOUN
ma-134	246	31	)	)	PUNCT
ma-134	246	32	.	.	PUNCT
ma-134	247	1	(	(	PUNCT
ma-134	247	2	3.6	3.6	NUM
ma-134	247	3	)	)	PUNCT
ma-134	247	4	now	now	ADV
ma-134	247	5	,	,	PUNCT
ma-134	247	6	we	we	PRON
ma-134	247	7	prove	prove	VERB
ma-134	247	8	the	the	DET
ma-134	247	9	mapping	mapping	NOUN
ma-134	247	10	φ	φ	PROPN
ma-134	247	11	is	be	AUX
ma-134	247	12	additive	additive	ADJ
ma-134	247	13	and	and	CCONJ
ma-134	247	14	is	be	AUX
ma-134	247	15	unique	unique	ADJ
ma-134	247	16	.	.	PUNCT
ma-134	248	1	from	from	ADP
ma-134	248	2	(	(	PUNCT
ma-134	248	3	x	x	X
ma-134	248	4	,	,	PUNCT
ma-134	248	5	y	y	PROPN
ma-134	248	6	,	,	PUNCT
ma-134	248	7	y	y	PROPN
ma-134	248	8	+	+	NUM
ma-134	248	9	x	x	X
ma-134	248	10	)	)	PUNCT
ma-134	248	11	in	in	ADP
ma-134	248	12	(	(	PUNCT
ma-134	248	13	3.1	3.1	NUM
ma-134	248	14	)	)	PUNCT
ma-134	248	15	yields	yield	NOUN
ma-134	248	16	that	that	SCONJ
ma-134	248	17	‖f	‖f	ADP
ma-134	248	18	(	(	PUNCT
ma-134	248	19	2x	2x	NUM
ma-134	248	20	+	+	X
ma-134	248	21	2y	2y	NUM
ma-134	248	22	)	)	PUNCT
ma-134	248	23	+	+	CCONJ
ma-134	248	24	f	f	X
ma-134	248	25	(	(	PUNCT
ma-134	248	26	x	x	X
ma-134	248	27	)	)	PUNCT
ma-134	248	28	+	+	NUM
ma-134	248	29	f	f	X
ma-134	248	30	(	(	PUNCT
ma-134	248	31	y)−	y)−	PROPN
ma-134	248	32	f	f	X
ma-134	248	33	(	(	PUNCT
ma-134	248	34	x	x	PROPN
ma-134	248	35	+	+	NUM
ma-134	248	36	2y)−	2y)−	NUM
ma-134	248	37	f	f	X
ma-134	248	38	(	(	PUNCT
ma-134	248	39	2x	2x	NUM
ma-134	248	40	+	+	CCONJ
ma-134	248	41	y)‖	y)‖	PRON
ma-134	248	42	6	6	NUM
ma-134	248	43	k	k	NOUN
ma-134	248	44	(	(	PUNCT
ma-134	248	45	‖x‖r	‖x‖r	NOUN
ma-134	248	46	+	+	CCONJ
ma-134	248	47	‖y‖r	‖y‖r	ADJ
ma-134	248	48	+	+	CCONJ
ma-134	248	49	‖x	‖x	NOUN
ma-134	248	50	+	+	CCONJ
ma-134	248	51	y‖r	y‖r	PROPN
ma-134	248	52	)	)	PUNCT
ma-134	248	53	.	.	PUNCT
ma-134	249	1	from	from	ADP
ma-134	249	2	(	(	PUNCT
ma-134	249	3	x	x	NOUN
ma-134	249	4	,	,	PUNCT
ma-134	249	5	x	x	NOUN
ma-134	249	6	,	,	PUNCT
ma-134	249	7	y	y	PROPN
ma-134	249	8	)	)	PUNCT
ma-134	249	9	in	in	ADP
ma-134	249	10	equation	equation	NOUN
ma-134	249	11	(	(	PUNCT
ma-134	249	12	3.1	3.1	NUM
ma-134	249	13	)	)	PUNCT
ma-134	249	14	yields	yield	NOUN
ma-134	249	15	that	that	SCONJ
ma-134	249	16	‖f	‖f	ADP
ma-134	249	17	(	(	PUNCT
ma-134	249	18	2x	2x	NUM
ma-134	249	19	+	+	CCONJ
ma-134	249	20	y	y	X
ma-134	249	21	)	)	PUNCT
ma-134	250	1	+	+	CCONJ
ma-134	250	2	2f	2f	NUM
ma-134	250	3	(	(	PUNCT
ma-134	250	4	x	x	X
ma-134	250	5	)	)	PUNCT
ma-134	251	1	+	+	NUM
ma-134	251	2	f	f	X
ma-134	251	3	(	(	PUNCT
ma-134	251	4	y)−	y)−	PROPN
ma-134	251	5	f	f	X
ma-134	251	6	(	(	PUNCT
ma-134	251	7	2x)−	2x)−	NUM
ma-134	251	8	2f	2f	NUM
ma-134	251	9	(	(	PUNCT
ma-134	251	10	x	x	X
ma-134	251	11	+	+	CCONJ
ma-134	251	12	y)‖	y)‖	PRON
ma-134	251	13	6	6	NUM
ma-134	251	14	k	k	PROPN
ma-134	251	15	(	(	PUNCT
ma-134	251	16	2‖x‖r	2‖x‖r	NOUN
ma-134	251	17	+	+	CCONJ
ma-134	251	18	‖y‖r	‖y‖r	ADJ
ma-134	251	19	)	)	PUNCT
ma-134	251	20	.	.	PUNCT
ma-134	252	1	from	from	ADP
ma-134	252	2	(	(	PUNCT
ma-134	252	3	x	x	X
ma-134	252	4	,	,	PUNCT
ma-134	252	5	y	y	PROPN
ma-134	252	6	,	,	PUNCT
ma-134	252	7	y	y	PROPN
ma-134	252	8	)	)	PUNCT
ma-134	252	9	in	in	ADP
ma-134	252	10	(	(	PUNCT
ma-134	252	11	3.1	3.1	NUM
ma-134	252	12	)	)	PUNCT
ma-134	252	13	we	we	PRON
ma-134	252	14	have	have	VERB
ma-134	252	15	‖f	‖f	PRON
ma-134	252	16	(	(	PUNCT
ma-134	252	17	x	x	X
ma-134	252	18	+	+	NOUN
ma-134	252	19	2y	2y	NUM
ma-134	252	20	)	)	PUNCT
ma-134	253	1	+	+	CCONJ
ma-134	253	2	f	f	X
ma-134	253	3	(	(	PUNCT
ma-134	253	4	x	x	X
ma-134	253	5	)	)	PUNCT
ma-134	253	6	+	+	NUM
ma-134	253	7	2f	2f	NOUN
ma-134	253	8	(	(	PUNCT
ma-134	253	9	y)−	y)−	PROPN
ma-134	253	10	f	f	X
ma-134	253	11	(	(	PUNCT
ma-134	253	12	2y)−	2y)−	NUM
ma-134	253	13	2f	2f	NOUN
ma-134	253	14	(	(	PUNCT
ma-134	253	15	x	x	X
ma-134	253	16	+	+	CCONJ
ma-134	253	17	y)‖	y)‖	PRON
ma-134	253	18	6	6	NUM
ma-134	253	19	k	k	NOUN
ma-134	253	20	(	(	PUNCT
ma-134	253	21	‖x‖r	‖x‖r	NOUN
ma-134	253	22	+	+	CCONJ
ma-134	253	23	2‖y‖r	2‖y‖r	NUM
ma-134	253	24	)	)	PUNCT
ma-134	253	25	.	.	PUNCT
ma-134	254	1	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	254	2	eur	eur	PROPN
ma-134	254	3	.	.	PUNCT
ma-134	255	1	j.	j.	PROPN
ma-134	255	2	math	math	PROPN
ma-134	255	3	.	.	PUNCT
ma-134	256	1	anal	anal	PROPN
ma-134	256	2	.	.	PUNCT
ma-134	257	1	10.28924	10.28924	NUM
ma-134	257	2	/	/	SYM
ma-134	257	3	ada	ada	PROPN
ma-134	257	4	/	/	SYM
ma-134	257	5	ma.3.7	ma.3.7	NOUN
ma-134	257	6	10combining	10combine	VERB
ma-134	257	7	the	the	DET
ma-134	257	8	above	above	ADJ
ma-134	257	9	three	three	NUM
ma-134	257	10	inequalities	inequality	NOUN
ma-134	257	11	,	,	PUNCT
ma-134	257	12	we	we	PRON
ma-134	257	13	have	have	VERB
ma-134	257	14	that	that	PRON
ma-134	257	15	for	for	ADP
ma-134	257	16	x	x	PROPN
ma-134	257	17	,	,	PUNCT
ma-134	257	18	y	y	PROPN
ma-134	257	19	,	,	PUNCT
ma-134	257	20	z	z	NOUN
ma-134	257	21	∈	∈	PROPN
ma-134	257	22	x	x	PUNCT
ma-134	258	1	‖φ(x	‖φ(x	PUNCT
ma-134	259	1	+	+	CCONJ
ma-134	259	2	y)−	y)−	PROPN
ma-134	259	3	φ(x)−	φ(x)−	PROPN
ma-134	259	4	φ(y)‖	φ(y)‖	PROPN
ma-134	260	1	=	=	PROPN
ma-134	260	2	lim	lim	PROPN
ma-134	260	3	n→∞	n→∞	NUM
ma-134	260	4	1	1	NUM
ma-134	260	5	2β2n	2β2n	NUM
ma-134	260	6	‖f	‖f	ADP
ma-134	260	7	(	(	PUNCT
ma-134	260	8	2n+1x	2n+1x	NUM
ma-134	260	9	+	+	NOUN
ma-134	260	10	2n+1y	2n+1y	NUM
ma-134	260	11	)	)	PUNCT
ma-134	261	1	+	+	NUM
ma-134	261	2	4f	4f	NUM
ma-134	261	3	(	(	PUNCT
ma-134	261	4	2nx	2nx	NOUN
ma-134	261	5	)	)	PUNCT
ma-134	262	1	+	+	CCONJ
ma-134	262	2	4f	4f	NUM
ma-134	262	3	(	(	PUNCT
ma-134	262	4	2ny)−	2ny)−	NUM
ma-134	262	5	f	f	NOUN
ma-134	262	6	(	(	PUNCT
ma-134	262	7	2n+1x)−	2n+1x)−	PROPN
ma-134	262	8	f	f	PROPN
ma-134	262	9	(	(	PUNCT
ma-134	262	10	2n+1y)−	2n+1y)−	NUM
ma-134	262	11	4f	4f	NOUN
ma-134	262	12	(	(	PUNCT
ma-134	262	13	2nx	2nx	NOUN
ma-134	262	14	+	+	CCONJ
ma-134	262	15	2ny)‖	2ny)‖	NUM
ma-134	262	16	6	6	NUM
ma-134	262	17	lim	lim	NOUN
ma-134	262	18	n→∞	n→∞	NUM
ma-134	262	19	1	1	NUM
ma-134	262	20	2β2n	2β2n	NUM
ma-134	262	21	∥∥f	∥∥f	NOUN
ma-134	262	22	(	(	PUNCT
ma-134	262	23	2n+1x	2n+1x	NUM
ma-134	262	24	+	+	CCONJ
ma-134	262	25	2n+1y	2n+1y	NUM
ma-134	262	26	)	)	PUNCT
ma-134	263	1	+	+	NUM
ma-134	263	2	f	f	X
ma-134	263	3	(	(	PUNCT
ma-134	263	4	2nx	2nx	ADJ
ma-134	263	5	)	)	PUNCT
ma-134	264	1	+	+	CCONJ
ma-134	264	2	f	f	X
ma-134	264	3	(	(	PUNCT
ma-134	264	4	2ny)−	2ny)−	NUM
ma-134	264	5	f	f	PROPN
ma-134	264	6	(	(	PUNCT
ma-134	264	7	2n+1x	2n+1x	NUM
ma-134	264	8	+	+	CCONJ
ma-134	264	9	2ny)−	2ny)−	NUM
ma-134	264	10	f	f	NOUN
ma-134	264	11	(	(	PUNCT
ma-134	264	12	2n+1y	2n+1y	NUM
ma-134	264	13	+	+	CCONJ
ma-134	264	14	2nx	2nx	ADJ
ma-134	264	15	)	)	PUNCT
ma-134	264	16	∥∥	∥∥	X
ma-134	265	1	+	+	NUM
ma-134	265	2	lim	lim	NOUN
ma-134	265	3	n→∞	n→∞	NUM
ma-134	265	4	1	1	NUM
ma-134	265	5	2β2n	2β2n	NUM
ma-134	265	6	∥∥f	∥∥f	NOUN
ma-134	265	7	(	(	PUNCT
ma-134	265	8	2n+1x	2n+1x	NUM
ma-134	265	9	+	+	CCONJ
ma-134	265	10	2ny	2ny	ADJ
ma-134	265	11	)	)	PUNCT
ma-134	266	1	+	+	CCONJ
ma-134	266	2	2f	2f	NUM
ma-134	266	3	(	(	PUNCT
ma-134	266	4	2nx	2nx	NOUN
ma-134	266	5	)	)	PUNCT
ma-134	267	1	+	+	CCONJ
ma-134	267	2	f	f	X
ma-134	267	3	(	(	PUNCT
ma-134	267	4	2ny)−	2ny)−	NUM
ma-134	267	5	f	f	X
ma-134	267	6	(	(	PUNCT
ma-134	267	7	2n+1x)−	2n+1x)−	PROPN
ma-134	267	8	2f	2f	NUM
ma-134	267	9	(	(	PUNCT
ma-134	267	10	2nx	2nx	ADJ
ma-134	267	11	+	+	CCONJ
ma-134	267	12	2ny	2ny	ADJ
ma-134	267	13	)	)	PUNCT
ma-134	267	14	∥∥	∥∥	X
ma-134	268	1	+	+	NUM
ma-134	268	2	lim	lim	NOUN
ma-134	268	3	n→∞	n→∞	NUM
ma-134	268	4	1	1	NUM
ma-134	268	5	2β2n	2β2n	NUM
ma-134	268	6	∥∥f	∥∥f	NOUN
ma-134	268	7	(	(	PUNCT
ma-134	268	8	2nx	2nx	NOUN
ma-134	268	9	+	+	PUNCT
ma-134	268	10	2n+1y	2n+1y	NUM
ma-134	268	11	)	)	PUNCT
ma-134	269	1	+	+	NUM
ma-134	269	2	f	f	X
ma-134	269	3	(	(	PUNCT
ma-134	269	4	2nx	2nx	ADJ
ma-134	269	5	)	)	PUNCT
ma-134	270	1	+	+	CCONJ
ma-134	270	2	2f	2f	NUM
ma-134	270	3	(	(	PUNCT
ma-134	270	4	2ny)−	2ny)−	NUM
ma-134	270	5	f	f	X
ma-134	270	6	(	(	PUNCT
ma-134	270	7	2n+1y)−	2n+1y)−	NUM
ma-134	270	8	2f	2f	NUM
ma-134	270	9	(	(	PUNCT
ma-134	270	10	2nx	2nx	ADJ
ma-134	270	11	+	+	CCONJ
ma-134	270	12	2ny	2ny	ADJ
ma-134	270	13	)	)	PUNCT
ma-134	270	14	∥∥	∥∥	PROPN
ma-134	270	15	6	6	NUM
ma-134	270	16	lim	lim	NOUN
ma-134	270	17	n→∞	n→∞	X
ma-134	270	18	2β1rn	2β1rn	NUM
ma-134	270	19	2β2n	2β2n	NOUN
ma-134	270	20	k	k	PROPN
ma-134	270	21	(	(	PUNCT
ma-134	270	22	4‖x‖r	4‖x‖r	PROPN
ma-134	270	23	+	+	CCONJ
ma-134	270	24	4‖y‖r	4‖y‖r	NOUN
ma-134	270	25	+	+	CCONJ
ma-134	270	26	‖x	‖x	NOUN
ma-134	271	1	+	+	CCONJ
ma-134	271	2	y‖r	y‖r	PROPN
ma-134	271	3	)	)	PUNCT
ma-134	271	4	.	.	PUNCT
ma-134	272	1	so	so	ADV
ma-134	272	2	we	we	PRON
ma-134	272	3	have	have	VERB
ma-134	272	4	φ(x	φ(x	PROPN
ma-134	272	5	+	+	NOUN
ma-134	272	6	y	y	NOUN
ma-134	272	7	)	)	PUNCT
ma-134	272	8	=	=	SYM
ma-134	272	9	φ(x	φ(x	NOUN
ma-134	272	10	)	)	PUNCT
ma-134	272	11	+	+	X
ma-134	272	12	φ(y	φ(y	NOUN
ma-134	272	13	)	)	PUNCT
ma-134	272	14	for	for	ADP
ma-134	272	15	all	all	DET
ma-134	272	16	x	x	NOUN
ma-134	272	17	,	,	PUNCT
ma-134	272	18	y	y	PROPN
ma-134	272	19	∈	∈	PROPN
ma-134	272	20	x	x	PUNCT
ma-134	272	21	.next	.next	PROPN
ma-134	272	22	,	,	PUNCT
ma-134	272	23	the	the	DET
ma-134	272	24	uniqueness	uniqueness	NOUN
ma-134	272	25	of	of	ADP
ma-134	272	26	the	the	DET
ma-134	272	27	mapping	mapping	NOUN
ma-134	272	28	φ	φ	PROPN
ma-134	272	29	will	will	AUX
ma-134	272	30	be	be	AUX
ma-134	272	31	proved	prove	VERB
ma-134	272	32	.	.	PUNCT
ma-134	273	1	let	let	AUX
ma-134	273	2	u(x	u(x	NOUN
ma-134	273	3	)	)	PUNCT
ma-134	273	4	be	be	AUX
ma-134	273	5	another	another	DET
ma-134	273	6	additive	additive	ADJ
ma-134	273	7	mappingsuch	mappingsuch	NOUN
ma-134	273	8	that	that	SCONJ
ma-134	273	9	for	for	ADP
ma-134	273	10	some	some	DET
ma-134	273	11	k2	k2	ADJ
ma-134	273	12	≥	≥	NOUN
ma-134	273	13	0	0	NUM
ma-134	273	14	and	and	CCONJ
ma-134	273	15	r	r	NOUN
ma-134	273	16	<	<	X
ma-134	273	17	β2	β2	NOUN
ma-134	273	18	β1	β1	PROPN
ma-134	273	19	,	,	PUNCT
ma-134	273	20	‖g(x)−	‖g(x)−	PROPN
ma-134	273	21	u(x)‖	u(x)‖	PROPN
ma-134	273	22	6	6	NUM
ma-134	273	23	k2‖x‖r2	k2‖x‖r2	NOUN
ma-134	273	24	.	.	PUNCT
ma-134	274	1	hence	hence	ADV
ma-134	274	2	‖φ(x)−	‖φ(x)−	PROPN
ma-134	274	3	u(x)‖	u(x)‖	NOUN
ma-134	275	1	=	=	NOUN
ma-134	275	2	‖φ(nx)−	‖φ(nx)−	PROPN
ma-134	275	3	u(nx)‖/nβ2	u(nx)‖/nβ2	PROPN
ma-134	275	4	6‖φ(nx)−	6‖φ(nx)−	NOUN
ma-134	275	5	g(nx)‖/nβ2	g(nx)‖/nβ2	PRON
ma-134	276	1	+	+	CCONJ
ma-134	276	2	‖g(nx)−	‖g(nx)−	PROPN
ma-134	276	3	u(nx)‖/nβ2	u(nx)‖/nβ2	PROPN
ma-134	276	4	6	6	NUM
ma-134	276	5	(	(	PUNCT
ma-134	276	6	2	2	NUM
ma-134	276	7	+	+	NUM
ma-134	276	8	2rβ1	2rβ1	NUM
ma-134	276	9	+	+	CCONJ
ma-134	276	10	3	3	NUM
ma-134	276	11	·	·	SYM
ma-134	276	12	2β2)k	2β2)k	NUM
ma-134	277	1	2β1r	2β1r	ADJ
ma-134	277	2	−	−	NUM
ma-134	277	3	2β2	2β2	NUM
ma-134	277	4	‖x‖rnrβ1−β2	‖x‖rnrβ1−β2	NOUN
ma-134	277	5	+	+	NOUN
ma-134	277	6	k2‖x‖r2nr2β1−β2	k2‖x‖r2nr2β1−β2	NOUN
ma-134	277	7	for	for	ADP
ma-134	277	8	all	all	DET
ma-134	277	9	x	x	SYM
ma-134	277	10	∈	∈	NOUN
ma-134	277	11	x.	x.	NOUN
ma-134	277	12	therefore	therefore	ADV
ma-134	277	13	φ(x	φ(x	PROPN
ma-134	277	14	)	)	PUNCT
ma-134	277	15	=	=	SYM
ma-134	277	16	u(x	u(x	PROPN
ma-134	277	17	)	)	PUNCT
ma-134	277	18	for	for	ADP
ma-134	277	19	all	all	PRON
ma-134	277	20	x	x	SYM
ma-134	277	21	∈	∈	NOUN
ma-134	277	22	x.	x.	NOUN
ma-134	277	23	by	by	ADP
ma-134	277	24	the	the	DET
ma-134	277	25	condition	condition	NOUN
ma-134	277	26	r	r	NOUN
ma-134	277	27	<	<	X
ma-134	277	28	β2	β2	NOUN
ma-134	277	29	β1	β1	PROPN
ma-134	277	30	.	.	PUNCT
ma-134	278	1	so	so	ADV
ma-134	278	2	there	there	PRON
ma-134	278	3	has	have	VERB
ma-134	278	4	a	a	DET
ma-134	278	5	uniqueadditive	uniqueadditive	ADJ
ma-134	278	6	limiting	limit	VERB
ma-134	278	7	mapping	mapping	NOUN
ma-134	278	8	φ	φ	PROPN
ma-134	278	9	fulfilling	fulfil	VERB
ma-134	278	10	‖(f	‖(f	NOUN
ma-134	278	11	(	(	PUNCT
ma-134	278	12	x)−	x)−	NOUN
ma-134	278	13	1	1	NUM
ma-134	278	14	2	2	NUM
ma-134	278	15	φ(x))−	φ(x))−	NOUN
ma-134	278	16	(	(	PUNCT
ma-134	278	17	f	f	PROPN
ma-134	278	18	(	(	PUNCT
ma-134	278	19	2x)−	2x)−	NUM
ma-134	278	20	1	1	NUM
ma-134	278	21	2	2	NUM
ma-134	278	22	φ(2x))/4‖	φ(2x))/4‖	NOUN
ma-134	278	23	≤	≤	NUM
ma-134	278	24	(	(	PUNCT
ma-134	278	25	2	2	NUM
ma-134	278	26	+	+	NUM
ma-134	278	27	2rβ1	2rβ1	NUM
ma-134	278	28	+	+	CCONJ
ma-134	278	29	3	3	NUM
ma-134	278	30	·	·	SYM
ma-134	278	31	2β2)k	2β2)k	NUM
ma-134	278	32	2β1r	2β1r	ADJ
ma-134	278	33	−	−	NUM
ma-134	278	34	2β2	2β2	NUM
ma-134	278	35	‖x‖r/22β2	‖x‖r/22β2	NOUN
ma-134	278	36	.	.	PUNCT
ma-134	279	1	hence	hence	ADV
ma-134	279	2	‖(f	‖(f	NOUN
ma-134	279	3	(	(	PUNCT
ma-134	279	4	x)−	x)−	NOUN
ma-134	279	5	1	1	NUM
ma-134	279	6	2	2	NUM
ma-134	279	7	φ(x))−	φ(x))−	NOUN
ma-134	279	8	(	(	PUNCT
ma-134	279	9	f	f	PROPN
ma-134	279	10	(	(	PUNCT
ma-134	279	11	2nx)−	2nx)−	NUM
ma-134	279	12	1	1	NUM
ma-134	279	13	2	2	NUM
ma-134	279	14	φ(2nx))/4n‖	φ(2nx))/4n‖	PROPN
ma-134	279	15	≤	≤	NOUN
ma-134	279	16	n−1∑	n−1∑	NUM
ma-134	279	17	j=0	j=0	PROPN
ma-134	279	18	2β1r	2β1r	PROPN
ma-134	279	19	j	j	PROPN
ma-134	279	20	22β2j	22β2j	NUM
ma-134	279	21	(	(	PUNCT
ma-134	279	22	2	2	NUM
ma-134	279	23	+	+	NUM
ma-134	279	24	2rβ1	2rβ1	NUM
ma-134	279	25	+	+	CCONJ
ma-134	279	26	3	3	NUM
ma-134	279	27	·	·	SYM
ma-134	279	28	2β2)k	2β2)k	NUM
ma-134	280	1	22β2(2β1r	22β2(2β1r	NUM
ma-134	280	2	−	−	NOUN
ma-134	280	3	2β2	2β2	NUM
ma-134	280	4	)	)	PUNCT
ma-134	280	5	‖x‖r	‖x‖r	NOUN
ma-134	280	6	.	.	PUNCT
ma-134	281	1	then	then	ADV
ma-134	281	2	the	the	DET
ma-134	281	3	mapping	mapping	NOUN
ma-134	281	4	can	can	AUX
ma-134	281	5	be	be	AUX
ma-134	281	6	well	well	ADV
ma-134	281	7	defined	define	VERB
ma-134	281	8	as	as	ADP
ma-134	281	9	ψ(x	ψ(x	NOUN
ma-134	281	10	)	)	PUNCT
ma-134	282	1	=	=	SYM
ma-134	282	2	lim	lim	PROPN
ma-134	282	3	n→∞	n→∞	X
ma-134	282	4	(	(	PUNCT
ma-134	282	5	f	f	PROPN
ma-134	282	6	(	(	PUNCT
ma-134	282	7	2nx)−	2nx)−	NUM
ma-134	282	8	1	1	NUM
ma-134	282	9	2	2	NUM
ma-134	282	10	φ(2nx))/4n	φ(2nx))/4n	PROPN
ma-134	282	11	for	for	ADP
ma-134	282	12	all	all	DET
ma-134	282	13	x	x	SYM
ma-134	282	14	∈	∈	PROPN
ma-134	282	15	x	x	X
ma-134	282	16	,	,	PUNCT
ma-134	282	17	by	by	ADP
ma-134	282	18	the	the	DET
ma-134	282	19	completeness	completeness	NOUN
ma-134	282	20	of	of	ADP
ma-134	282	21	the	the	DET
ma-134	282	22	space	space	NOUN
ma-134	282	23	y	y	PROPN
ma-134	282	24	.	.	PUNCT
ma-134	283	1	thus	thus	ADV
ma-134	283	2	‖ψ(x)−	‖ψ(x)−	PROPN
ma-134	283	3	f	f	X
ma-134	283	4	(	(	PUNCT
ma-134	283	5	x	x	X
ma-134	283	6	)	)	PUNCT
ma-134	283	7	+	+	CCONJ
ma-134	283	8	φ(x)/2‖	φ(x)/2‖	PRON
ma-134	283	9	≤	≤	NUM
ma-134	283	10	(	(	PUNCT
ma-134	283	11	2	2	NUM
ma-134	283	12	+	+	NUM
ma-134	283	13	2rβ1	2rβ1	NUM
ma-134	283	14	+	+	CCONJ
ma-134	283	15	3	3	NUM
ma-134	283	16	·	·	SYM
ma-134	283	17	2β2)k	2β2)k	NUM
ma-134	283	18	(	(	PUNCT
ma-134	283	19	2β1r	2β1r	ADJ
ma-134	283	20	−	−	PROPN
ma-134	283	21	2β2)(2β1r	2β2)(2β1r	NUM
ma-134	283	22	−	−	NOUN
ma-134	283	23	22β2	22β2	NUM
ma-134	283	24	)	)	PUNCT
ma-134	283	25	‖x‖r	‖x‖r	NOUN
ma-134	283	26	.	.	PUNCT
ma-134	284	1	let	let	VERB
ma-134	284	2	u(x	u(x	NOUN
ma-134	284	3	)	)	PUNCT
ma-134	284	4	be	be	AUX
ma-134	284	5	another	another	DET
ma-134	284	6	limiting	limit	VERB
ma-134	284	7	mapping	mapping	NOUN
ma-134	284	8	which	which	PRON
ma-134	284	9	has	have	VERB
ma-134	284	10	the	the	DET
ma-134	284	11	same	same	ADJ
ma-134	284	12	property	property	NOUN
ma-134	284	13	to	to	ADP
ma-134	284	14	the	the	DET
ma-134	284	15	function	function	NOUN
ma-134	284	16	ψ(x	ψ(x	NOUN
ma-134	284	17	)	)	PUNCT
ma-134	284	18	such	such	ADJ
ma-134	284	19	that	that	SCONJ
ma-134	284	20	,	,	PUNCT
ma-134	284	21	‖u(x)−	‖u(x)−	PROPN
ma-134	284	22	φ(x)‖	φ(x)‖	NOUN
ma-134	284	23	6	6	NUM
ma-134	284	24	‖u(x)−	‖u(x)−	PROPN
ma-134	284	25	(	(	PUNCT
ma-134	284	26	f	f	PROPN
ma-134	284	27	(	(	PUNCT
ma-134	284	28	2nx)−	2nx)−	NUM
ma-134	284	29	1	1	NUM
ma-134	284	30	2	2	NUM
ma-134	284	31	φ(2nx))/4n‖+	φ(2nx))/4n‖+	PROPN
ma-134	284	32	‖(f	‖(f	NOUN
ma-134	284	33	(	(	PUNCT
ma-134	284	34	2nx)−	2nx)−	NOUN
ma-134	284	35	1	1	NUM
ma-134	284	36	2	2	NUM
ma-134	284	37	φ(2nx))/4n	φ(2nx))/4n	PROPN
ma-134	284	38	−	−	PROPN
ma-134	284	39	φ(x)‖	φ(x)‖	NOUN
ma-134	284	40	6	6	NUM
ma-134	284	41	2	2	NUM
ma-134	284	42	∞∑	∞∑	NUM
ma-134	284	43	j	j	NOUN
ma-134	284	44	=	=	NOUN
ma-134	284	45	n	n	PRON
ma-134	284	46	2β1r	2β1r	ADJ
ma-134	284	47	j	j	PROPN
ma-134	284	48	22β2j	22β2j	NUM
ma-134	284	49	(	(	PUNCT
ma-134	284	50	2	2	NUM
ma-134	284	51	+	+	NUM
ma-134	284	52	2rβ1	2rβ1	NUM
ma-134	284	53	+	+	CCONJ
ma-134	284	54	3	3	NUM
ma-134	284	55	·	·	SYM
ma-134	284	56	2β2)k	2β2)k	NUM
ma-134	284	57	22β2(2β1r	22β2(2β1r	NUM
ma-134	284	58	−	−	NOUN
ma-134	284	59	2β2	2β2	NUM
ma-134	284	60	)	)	PUNCT
ma-134	284	61	‖x‖r	‖x‖r	NOUN
ma-134	284	62	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	284	63	eur	eur	PROPN
ma-134	284	64	.	.	PUNCT
ma-134	285	1	j.	j.	PROPN
ma-134	285	2	math	math	PROPN
ma-134	285	3	.	.	PUNCT
ma-134	286	1	anal	anal	PROPN
ma-134	286	2	.	.	PUNCT
ma-134	287	1	10.28924	10.28924	NUM
ma-134	287	2	/	/	SYM
ma-134	287	3	ada	ada	PROPN
ma-134	287	4	/	/	SYM
ma-134	287	5	ma.3.7	ma.3.7	NOUN
ma-134	287	6	11which	11which	X
ma-134	287	7	shows	show	VERB
ma-134	287	8	that	that	SCONJ
ma-134	287	9	the	the	DET
ma-134	287	10	approximation	approximation	NOUN
ma-134	287	11	function	function	NOUN
ma-134	287	12	φ(x	φ(x	NOUN
ma-134	287	13	)	)	PUNCT
ma-134	287	14	is	be	AUX
ma-134	287	15	unique	unique	ADJ
ma-134	287	16	.	.	PUNCT
ma-134	288	1	finally	finally	ADV
ma-134	288	2	,	,	PUNCT
ma-134	288	3	it	it	PRON
ma-134	288	4	remains	remain	VERB
ma-134	288	5	to	to	PART
ma-134	288	6	prove	prove	VERB
ma-134	288	7	that	that	SCONJ
ma-134	288	8	φ(x)satisfies	φ(x)satisfie	NOUN
ma-134	288	9	(	(	PUNCT
ma-134	288	10	3.2	3.2	NUM
ma-134	288	11	)	)	PUNCT
ma-134	288	12	and	and	CCONJ
ma-134	288	13	we	we	PRON
ma-134	288	14	obtain	obtain	VERB
ma-134	288	15	1	1	NUM
ma-134	288	16	4n	4n	NOUN
ma-134	288	17	‖f	‖f	PUNCT
ma-134	288	18	(	(	PUNCT
ma-134	288	19	2nx	2nx	ADJ
ma-134	288	20	+	+	CCONJ
ma-134	288	21	2ny	2ny	ADJ
ma-134	288	22	+	+	CCONJ
ma-134	288	23	2nz	2nz	NOUN
ma-134	288	24	)	)	PUNCT
ma-134	289	1	+	+	NUM
ma-134	289	2	f	f	X
ma-134	289	3	(	(	PUNCT
ma-134	289	4	2nx	2nx	ADJ
ma-134	289	5	)	)	PUNCT
ma-134	290	1	+	+	CCONJ
ma-134	290	2	f	f	X
ma-134	290	3	(	(	PUNCT
ma-134	290	4	2ny	2ny	ADJ
ma-134	290	5	)	)	PUNCT
ma-134	291	1	+	+	CCONJ
ma-134	291	2	f	f	X
ma-134	291	3	(	(	PUNCT
ma-134	291	4	2nz)−	2nz)−	NUM
ma-134	291	5	f	f	X
ma-134	291	6	(	(	PUNCT
ma-134	291	7	2nx	2nx	NOUN
ma-134	291	8	+	+	CCONJ
ma-134	291	9	2ny)−	2ny)−	NUM
ma-134	291	10	f	f	NOUN
ma-134	291	11	(	(	PUNCT
ma-134	291	12	2nz	2nz	NOUN
ma-134	291	13	+	+	CCONJ
ma-134	291	14	2ny)−	2ny)−	NUM
ma-134	291	15	f	f	NOUN
ma-134	291	16	(	(	PUNCT
ma-134	291	17	2nx	2nx	ADJ
ma-134	291	18	+	+	CCONJ
ma-134	291	19	2nz)‖	2nz)‖	NUM
ma-134	291	20	6	6	NUM
ma-134	291	21	2β1n	2β1n	ADJ
ma-134	291	22	4n	4n	X
ma-134	291	23	k	k	NOUN
ma-134	291	24	(	(	PUNCT
ma-134	291	25	‖x‖r	‖x‖r	NOUN
ma-134	291	26	+	+	CCONJ
ma-134	291	27	‖y‖r	‖y‖r	ADJ
ma-134	291	28	+	+	CCONJ
ma-134	291	29	‖z‖r	‖z‖r	NOUN
ma-134	291	30	)	)	PUNCT
ma-134	291	31	.	.	PUNCT
ma-134	292	1	(	(	PUNCT
ma-134	292	2	3.7	3.7	NUM
ma-134	292	3	)	)	PUNCT
ma-134	292	4	letting	let	VERB
ma-134	292	5	n	n	DET
ma-134	292	6	→∞	→∞	NOUN
ma-134	292	7	,	,	PUNCT
ma-134	292	8	and	and	CCONJ
ma-134	292	9	we	we	PRON
ma-134	292	10	get	get	VERB
ma-134	292	11	our	our	PRON
ma-134	292	12	assertion	assertion	NOUN
ma-134	292	13	by	by	ADP
ma-134	292	14	using	use	VERB
ma-134	292	15	the	the	DET
ma-134	292	16	additivity	additivity	NOUN
ma-134	292	17	of	of	ADP
ma-134	292	18	φ(x	φ(x	PROPN
ma-134	292	19	)	)	PUNCT
ma-134	292	20	.	.	PUNCT
ma-134	293	1	�	�	PROPN
ma-134	293	2	in	in	ADP
ma-134	293	3	another	another	DET
ma-134	293	4	direction	direction	NOUN
ma-134	293	5	,	,	PUNCT
ma-134	293	6	we	we	PRON
ma-134	293	7	will	will	AUX
ma-134	293	8	describe	describe	VERB
ma-134	293	9	the	the	DET
ma-134	293	10	similar	similar	ADJ
ma-134	293	11	stability	stability	NOUN
ma-134	293	12	results	result	NOUN
ma-134	293	13	of	of	ADP
ma-134	293	14	the	the	DET
ma-134	293	15	above	above	ADJ
ma-134	293	16	theorem	theorem	ADJ
ma-134	293	17	3.1	3.1	NUM
ma-134	293	18	.	.	PUNCT
ma-134	293	19	theorem	theorem	ADJ
ma-134	293	20	3.2	3.2	NUM
ma-134	293	21	let	let	VERB
ma-134	293	22	r	r	NOUN
ma-134	293	23	>	>	SYM
ma-134	293	24	β2	β2	NOUN
ma-134	293	25	β1	β1	PROPN
ma-134	293	26	and	and	CCONJ
ma-134	293	27	assume	assume	VERB
ma-134	293	28	that	that	SCONJ
ma-134	293	29	f	f	X
ma-134	293	30	:	:	PUNCT
ma-134	293	31	x	x	X
ma-134	293	32	→	→	SYM
ma-134	293	33	y	y	PROPN
ma-134	293	34	is	be	AUX
ma-134	293	35	a	a	DET
ma-134	293	36	mapping	mapping	NOUN
ma-134	293	37	satisfying	satisfy	VERB
ma-134	293	38	the	the	DET
ma-134	293	39	equation	equation	NOUN
ma-134	293	40	(	(	PUNCT
ma-134	293	41	3.1).then	3.1).then	ADJ
ma-134	293	42	there	there	PRON
ma-134	293	43	has	have	VERB
ma-134	293	44	a	a	DET
ma-134	293	45	unique	unique	ADJ
ma-134	293	46	limiting	limit	VERB
ma-134	293	47	mapping	mapping	NOUN
ma-134	293	48	ψ1	ψ1	NOUN
ma-134	293	49	:	:	PUNCT
ma-134	293	50	x	x	X
ma-134	293	51	→	→	SYM
ma-134	293	52	y	y	PROPN
ma-134	293	53	satisfying	satisfy	VERB
ma-134	293	54	‖f	‖f	ADP
ma-134	293	55	(	(	PUNCT
ma-134	293	56	x)−	x)−	PROPN
ma-134	293	57	ψ1(x)‖	ψ1(x)‖	NOUN
ma-134	293	58	6	6	NUM
ma-134	293	59	(	(	PUNCT
ma-134	293	60	2	2	NUM
ma-134	293	61	+	+	NUM
ma-134	293	62	2rβ1	2rβ1	NUM
ma-134	293	63	+	+	CCONJ
ma-134	293	64	3	3	NUM
ma-134	293	65	·	·	SYM
ma-134	293	66	2β2)k	2β2)k	NUM
ma-134	293	67	(	(	PUNCT
ma-134	293	68	2β1r	2β1r	ADJ
ma-134	293	69	−	−	NOUN
ma-134	293	70	22β2)(2β1r	22β2)(2β1r	NUM
ma-134	293	71	−	−	NOUN
ma-134	293	72	2β2	2β2	NUM
ma-134	293	73	)	)	PUNCT
ma-134	293	74	‖x‖r	‖x‖r	NOUN
ma-134	293	75	for	for	ADP
ma-134	293	76	all	all	DET
ma-134	293	77	x	x	SYM
ma-134	293	78	∈	∈	PROPN
ma-134	293	79	x	x	X
ma-134	293	80	.	.	PUNCT
ma-134	294	1	moreover	moreover	ADV
ma-134	294	2	,	,	PUNCT
ma-134	294	3	ψ1	ψ1	ADJ
ma-134	294	4	solves	solve	NOUN
ma-134	294	5	also	also	ADV
ma-134	294	6	the	the	DET
ma-134	294	7	following	follow	VERB
ma-134	294	8	equation	equation	NOUN
ma-134	294	9	ψ1(x	ψ1(x	PROPN
ma-134	295	1	+	+	CCONJ
ma-134	295	2	y	y	PROPN
ma-134	295	3	+	+	PROPN
ma-134	295	4	z	z	NOUN
ma-134	295	5	)	)	PUNCT
ma-134	295	6	+	+	PUNCT
ma-134	295	7	ψ1(x	ψ1(x	NOUN
ma-134	295	8	)	)	PUNCT
ma-134	295	9	+	+	PUNCT
ma-134	295	10	ψ1(z	ψ1(z	X
ma-134	295	11	)	)	PUNCT
ma-134	295	12	+	+	PUNCT
ma-134	295	13	ψ1(y	ψ1(y	NOUN
ma-134	295	14	)	)	PUNCT
ma-134	295	15	=	=	SYM
ma-134	295	16	ψ1(x	ψ1(x	PROPN
ma-134	296	1	+	+	NUM
ma-134	296	2	y	y	NOUN
ma-134	296	3	)	)	PUNCT
ma-134	297	1	+	+	PUNCT
ma-134	297	2	ψ1(z	ψ1(z	PUNCT
ma-134	298	1	+	+	NUM
ma-134	298	2	y	y	NOUN
ma-134	298	3	)	)	PUNCT
ma-134	299	1	+	+	PUNCT
ma-134	299	2	ψ1(x	ψ1(x	PROPN
ma-134	300	1	+	+	ADJ
ma-134	300	2	z	z	NOUN
ma-134	300	3	)	)	PUNCT
ma-134	300	4	(	(	PUNCT
ma-134	300	5	3.8	3.8	NUM
ma-134	300	6	)	)	PUNCT
ma-134	300	7	for	for	ADP
ma-134	300	8	all	all	DET
ma-134	300	9	x	x	NOUN
ma-134	300	10	,	,	PUNCT
ma-134	300	11	y	y	PROPN
ma-134	300	12	,	,	PUNCT
ma-134	300	13	z	z	NOUN
ma-134	300	14	∈	∈	PROPN
ma-134	300	15	x	x	X
ma-134	300	16	.	.	PUNCT
ma-134	301	1	proof	proof	NOUN
ma-134	301	2	.	.	PUNCT
ma-134	302	1	according	accord	VERB
ma-134	302	2	to	to	ADP
ma-134	302	3	the	the	DET
ma-134	302	4	equation	equation	NOUN
ma-134	302	5	(	(	PUNCT
ma-134	302	6	3.3	3.3	NUM
ma-134	302	7	)	)	PUNCT
ma-134	302	8	,	,	PUNCT
ma-134	302	9	we	we	PRON
ma-134	302	10	obtain	obtain	VERB
ma-134	302	11	‖g(x)−	‖g(x)−	NOUN
ma-134	302	12	2	2	NUM
ma-134	302	13	g	g	NOUN
ma-134	302	14	(	(	PUNCT
ma-134	302	15	x	x	NOUN
ma-134	302	16	2	2	X
ma-134	302	17	)	)	PUNCT
ma-134	302	18	‖	‖	PROPN
ma-134	302	19	6	6	NUM
ma-134	302	20	(	(	PUNCT
ma-134	302	21	2	2	NUM
ma-134	302	22	+	+	NUM
ma-134	302	23	2rβ1	2rβ1	NUM
ma-134	302	24	+	+	CCONJ
ma-134	302	25	3	3	NUM
ma-134	302	26	·	·	SYM
ma-134	302	27	2β2)k	2β2)k	NUM
ma-134	302	28	(	(	PUNCT
ma-134	302	29	‖x‖r	‖x‖r	NOUN
ma-134	302	30	)	)	PUNCT
ma-134	302	31	/2β1r	/2β1r	PROPN
ma-134	302	32	.	.	PUNCT
ma-134	303	1	therefore	therefore	ADV
ma-134	303	2	‖2ng	‖2ng	PROPN
ma-134	303	3	(	(	PUNCT
ma-134	303	4	x	x	X
ma-134	303	5	2n	2n	NUM
ma-134	303	6	)	)	PUNCT
ma-134	303	7	−	−	PROPN
ma-134	303	8	2	2	NUM
ma-134	303	9	mg	mg	PROPN
ma-134	303	10	(	(	PUNCT
ma-134	303	11	x	x	PROPN
ma-134	303	12	2	2	NUM
ma-134	303	13	m	m	NOUN
ma-134	303	14	)	)	PUNCT
ma-134	303	15	‖	‖	PROPN
ma-134	303	16	6	6	NUM
ma-134	303	17	n−1∑	n−1∑	PROPN
ma-134	303	18	j	j	NOUN
ma-134	304	1	=	=	NOUN
ma-134	304	2	m	m	PROPN
ma-134	304	3	(	(	PUNCT
ma-134	304	4	2	2	NUM
ma-134	304	5	+	+	NUM
ma-134	304	6	2rβ1	2rβ1	NUM
ma-134	304	7	+	+	CCONJ
ma-134	304	8	3	3	NUM
ma-134	304	9	·	·	SYM
ma-134	304	10	2β2)k	2β2)k	NUM
ma-134	304	11	2jβ2	2jβ2	NUM
ma-134	304	12	2jβ1r2β1r	2jβ1r2β1r	NUM
ma-134	304	13	(	(	PUNCT
ma-134	304	14	‖x‖r	‖x‖r	NOUN
ma-134	304	15	)	)	PUNCT
ma-134	304	16	for	for	ADP
ma-134	304	17	m	m	PROPN
ma-134	304	18	,	,	PUNCT
ma-134	304	19	n	n	PROPN
ma-134	304	20	∈	∈	PROPN
ma-134	304	21	n	n	NOUN
ma-134	304	22	with	with	ADP
ma-134	304	23	n	n	NOUN
ma-134	304	24	>	>	SYM
ma-134	304	25	m	m	PROPN
ma-134	304	26	and	and	CCONJ
ma-134	304	27	x	x	SYM
ma-134	304	28	∈	∈	PROPN
ma-134	304	29	x	x	X
ma-134	304	30	.	.	PUNCT
ma-134	305	1	since	since	SCONJ
ma-134	305	2	the	the	DET
ma-134	305	3	sequence	sequence	NOUN
ma-134	305	4	{	{	PUNCT
ma-134	305	5	2ng	2ng	ADJ
ma-134	305	6	(	(	PUNCT
ma-134	305	7	x2n	x2n	PROPN
ma-134	305	8	)	)	PUNCT
ma-134	305	9	}	}	PUNCT
ma-134	305	10	is	be	AUX
ma-134	305	11	a	a	DET
ma-134	305	12	cauchy	cauchy	ADJ
ma-134	305	13	sequence	sequence	NOUN
ma-134	305	14	in	in	ADP
ma-134	305	15	yfor	yfor	PROPN
ma-134	305	16	all	all	DET
ma-134	306	1	x	x	SYM
ma-134	306	2	∈	∈	PROPN
ma-134	306	3	x	x	X
ma-134	306	4	and	and	CCONJ
ma-134	306	5	y	y	PROPN
ma-134	306	6	is	be	AUX
ma-134	306	7	complete	complete	ADJ
ma-134	306	8	,	,	PUNCT
ma-134	306	9	the	the	DET
ma-134	306	10	mapping	mapping	NOUN
ma-134	306	11	can	can	AUX
ma-134	306	12	be	be	AUX
ma-134	306	13	well	well	ADV
ma-134	306	14	defined	define	VERB
ma-134	306	15	as	as	ADP
ma-134	306	16	:	:	PUNCT
ma-134	306	17	φ(x	φ(x	NOUN
ma-134	306	18	)	)	PUNCT
ma-134	307	1	=	=	SYM
ma-134	307	2	lim	lim	PROPN
ma-134	307	3	n→∞	n→∞	NUM
ma-134	307	4	2ng	2ng	PROPN
ma-134	307	5	(	(	PUNCT
ma-134	307	6	x	x	PROPN
ma-134	307	7	2n	2n	NUM
ma-134	307	8	)	)	PUNCT
ma-134	307	9	for	for	ADP
ma-134	307	10	all	all	DET
ma-134	307	11	x	x	SYM
ma-134	307	12	∈	∈	NOUN
ma-134	307	13	x	x	X
ma-134	307	14	.	.	PUNCT
ma-134	308	1	using	use	VERB
ma-134	308	2	a	a	DET
ma-134	308	3	similar	similar	ADJ
ma-134	308	4	manner	manner	NOUN
ma-134	308	5	,	,	PUNCT
ma-134	308	6	we	we	PRON
ma-134	308	7	can	can	AUX
ma-134	308	8	complete	complete	VERB
ma-134	308	9	the	the	DET
ma-134	308	10	rest	rest	NOUN
ma-134	308	11	part	part	NOUN
ma-134	308	12	.	.	PUNCT
ma-134	309	1	�	�	PROPN
ma-134	310	1	if	if	SCONJ
ma-134	310	2	f	f	PROPN
ma-134	310	3	(	(	PUNCT
ma-134	310	4	x	x	X
ma-134	310	5	)	)	PUNCT
ma-134	310	6	is	be	AUX
ma-134	310	7	odd	odd	ADJ
ma-134	310	8	,	,	PUNCT
ma-134	310	9	then	then	ADV
ma-134	310	10	(	(	PUNCT
ma-134	310	11	x	x	X
ma-134	310	12	,	,	PUNCT
ma-134	310	13	y	y	PROPN
ma-134	310	14	,	,	PUNCT
ma-134	310	15	−x	−x	PROPN
ma-134	310	16	−	−	PROPN
ma-134	310	17	y	y	NOUN
ma-134	310	18	)	)	PUNCT
ma-134	310	19	in	in	ADP
ma-134	310	20	(	(	PUNCT
ma-134	310	21	3.2	3.2	NUM
ma-134	310	22	)	)	PUNCT
ma-134	310	23	can	can	AUX
ma-134	310	24	give	give	VERB
ma-134	310	25	a	a	DET
ma-134	310	26	precise	precise	ADJ
ma-134	310	27	condition	condition	NOUN
ma-134	310	28	to	to	PART
ma-134	310	29	ascertain	ascertain	VERB
ma-134	310	30	the	the	DET
ma-134	310	31	additiveproperty	additiveproperty	NOUN
ma-134	310	32	of	of	ADP
ma-134	310	33	the	the	DET
ma-134	310	34	function	function	NOUN
ma-134	310	35	f	f	PROPN
ma-134	310	36	(	(	PUNCT
ma-134	310	37	x	x	X
ma-134	310	38	)	)	PUNCT
ma-134	310	39	(	(	PUNCT
ma-134	310	40	see	see	VERB
ma-134	310	41	[	[	X
ma-134	310	42	9	9	NUM
ma-134	310	43	]	]	NUM
ma-134	310	44	)	)	PUNCT
ma-134	310	45	.	.	PUNCT
ma-134	311	1	obviously	obviously	ADV
ma-134	311	2	,	,	PUNCT
ma-134	311	3	the	the	DET
ma-134	311	4	additive	additive	ADJ
ma-134	311	5	property	property	NOUN
ma-134	311	6	is	be	AUX
ma-134	311	7	stronger	strong	ADJ
ma-134	311	8	than	than	ADP
ma-134	311	9	theproperty	theproperty	NOUN
ma-134	311	10	of	of	ADP
ma-134	311	11	the	the	DET
ma-134	311	12	equation	equation	NOUN
ma-134	311	13	(	(	PUNCT
ma-134	311	14	3.2	3.2	NUM
ma-134	311	15	)	)	PUNCT
ma-134	311	16	,	,	PUNCT
ma-134	311	17	but	but	CCONJ
ma-134	311	18	vice	vice	NOUN
ma-134	311	19	versa	versa	NOUN
ma-134	311	20	is	be	AUX
ma-134	311	21	not	not	PART
ma-134	311	22	true	true	ADJ
ma-134	311	23	.	.	PUNCT
ma-134	312	1	in	in	ADP
ma-134	312	2	contrast	contrast	NOUN
ma-134	312	3	with	with	ADP
ma-134	312	4	the	the	DET
ma-134	312	5	subadditive	subadditive	ADJ
ma-134	312	6	property	property	NOUN
ma-134	312	7	,	,	PUNCT
ma-134	312	8	we	we	PRON
ma-134	312	9	can	can	AUX
ma-134	312	10	not	not	PART
ma-134	312	11	get	get	VERB
ma-134	312	12	obvious	obvious	ADJ
ma-134	312	13	strong	strong	ADJ
ma-134	312	14	or	or	CCONJ
ma-134	312	15	weak	weak	ADJ
ma-134	312	16	property	property	NOUN
ma-134	312	17	temporarily	temporarily	ADV
ma-134	312	18	.	.	PUNCT
ma-134	313	1	by	by	ADP
ma-134	313	2	using	use	VERB
ma-134	313	3	another	another	DET
ma-134	313	4	approach	approach	NOUN
ma-134	313	5	to	to	ADP
ma-134	313	6	solvethe	solvethe	NOUN
ma-134	313	7	theorem	theorem	VERB
ma-134	313	8	3.1	3.1	NUM
ma-134	313	9	,	,	PUNCT
ma-134	313	10	according	accord	VERB
ma-134	313	11	to	to	ADP
ma-134	313	12	(	(	PUNCT
ma-134	313	13	3.6	3.6	NUM
ma-134	313	14	)	)	PUNCT
ma-134	313	15	,	,	PUNCT
ma-134	313	16	we	we	PRON
ma-134	313	17	have	have	VERB
ma-134	313	18	‖f	‖f	ADP
ma-134	313	19	(	(	PUNCT
ma-134	313	20	2nx)/22n	2nx)/22n	NUM
ma-134	313	21	−	−	PROPN
ma-134	313	22	f	f	NOUN
ma-134	313	23	(	(	PUNCT
ma-134	313	24	x)−	x)−	PROPN
ma-134	313	25	n−1∑	n−1∑	PROPN
ma-134	313	26	j=0	j=0	PROPN
ma-134	313	27	φ(2jx)/22(j+1)‖	φ(2jx)/22(j+1)‖	PROPN
ma-134	313	28	≤	≤	PUNCT
ma-134	313	29	n−1∑	n−1∑	NUM
ma-134	313	30	j=0	j=0	PROPN
ma-134	313	31	2β1r	2β1r	PROPN
ma-134	313	32	j	j	PROPN
ma-134	313	33	22β2j	22β2j	NUM
ma-134	313	34	(	(	PUNCT
ma-134	313	35	2	2	NUM
ma-134	313	36	+	+	NUM
ma-134	313	37	2rβ1	2rβ1	NUM
ma-134	313	38	+	+	CCONJ
ma-134	313	39	3	3	NUM
ma-134	313	40	·	·	SYM
ma-134	313	41	2β2)k	2β2)k	NUM
ma-134	313	42	22β2(2β1r	22β2(2β1r	NUM
ma-134	313	43	−	−	NOUN
ma-134	313	44	2β2	2β2	NUM
ma-134	313	45	)	)	PUNCT
ma-134	313	46	‖x‖r	‖x‖r	NOUN
ma-134	313	47	.	.	PUNCT
ma-134	314	1	then	then	ADV
ma-134	314	2	the	the	DET
ma-134	314	3	mapping	mapping	NOUN
ma-134	314	4	can	can	AUX
ma-134	314	5	be	be	AUX
ma-134	314	6	well	well	ADV
ma-134	314	7	defined	define	VERB
ma-134	314	8	as	as	ADP
ma-134	314	9	ψ(x	ψ(x	NOUN
ma-134	314	10	)	)	PUNCT
ma-134	315	1	=	=	VERB
ma-134	315	2	lim	lim	PROPN
ma-134	315	3	n→∞	n→∞	X
ma-134	316	1	f	f	PROPN
ma-134	316	2	(	(	PUNCT
ma-134	316	3	2nx)/22n	2nx)/22n	PROPN
ma-134	316	4	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	316	5	eur	eur	PROPN
ma-134	316	6	.	.	PUNCT
ma-134	317	1	j.	j.	PROPN
ma-134	317	2	math	math	PROPN
ma-134	317	3	.	.	PUNCT
ma-134	318	1	anal	anal	PROPN
ma-134	318	2	.	.	PUNCT
ma-134	319	1	10.28924	10.28924	NUM
ma-134	319	2	/	/	SYM
ma-134	319	3	ada	ada	PROPN
ma-134	319	4	/	/	SYM
ma-134	319	5	ma.3.7	ma.3.7	NOUN
ma-134	319	6	12for	12for	ADP
ma-134	319	7	all	all	DET
ma-134	319	8	x	x	SYM
ma-134	319	9	∈	∈	PROPN
ma-134	319	10	x	x	X
ma-134	319	11	,	,	PUNCT
ma-134	319	12	by	by	ADP
ma-134	319	13	the	the	DET
ma-134	319	14	completeness	completeness	NOUN
ma-134	319	15	of	of	ADP
ma-134	319	16	the	the	DET
ma-134	319	17	space	space	NOUN
ma-134	319	18	y	y	PROPN
ma-134	319	19	.	.	PUNCT
ma-134	320	1	thus	thus	ADV
ma-134	320	2	‖ψ(x)−	‖ψ(x)−	PROPN
ma-134	320	3	f	f	X
ma-134	320	4	(	(	PUNCT
ma-134	320	5	x)−	x)−	PROPN
ma-134	320	6	φ(x)/2‖	φ(x)/2‖	ADP
ma-134	320	7	≤	≤	NUM
ma-134	320	8	(	(	PUNCT
ma-134	320	9	2	2	NUM
ma-134	320	10	+	+	NUM
ma-134	320	11	2rβ1	2rβ1	NUM
ma-134	320	12	+	+	CCONJ
ma-134	320	13	3	3	NUM
ma-134	320	14	·	·	SYM
ma-134	320	15	2β2)k	2β2)k	NUM
ma-134	320	16	(	(	PUNCT
ma-134	320	17	2β1r	2β1r	ADJ
ma-134	320	18	−	−	PROPN
ma-134	320	19	2β2)(2β1r	2β2)(2β1r	NUM
ma-134	320	20	−	−	NOUN
ma-134	320	21	22β2	22β2	NUM
ma-134	320	22	)	)	PUNCT
ma-134	320	23	‖x‖r	‖x‖r	NOUN
ma-134	320	24	.	.	PUNCT
ma-134	321	1	in	in	ADP
ma-134	321	2	a	a	DET
ma-134	321	3	similar	similar	ADJ
ma-134	321	4	way	way	NOUN
ma-134	321	5	,	,	PUNCT
ma-134	321	6	we	we	PRON
ma-134	321	7	can	can	AUX
ma-134	321	8	use	use	VERB
ma-134	321	9	two	two	NUM
ma-134	321	10	steps	step	NOUN
ma-134	321	11	to	to	PART
ma-134	321	12	prove	prove	VERB
ma-134	321	13	that	that	SCONJ
ma-134	321	14	the	the	DET
ma-134	321	15	mapping	mapping	NOUN
ma-134	321	16	ψ(x	ψ(x	NOUN
ma-134	321	17	)	)	PUNCT
ma-134	321	18	is	be	AUX
ma-134	321	19	unique	unique	ADJ
ma-134	321	20	.	.	PUNCT
ma-134	322	1	the	the	DET
ma-134	322	2	first	first	ADJ
ma-134	322	3	stepwe	stepwe	PROPN
ma-134	322	4	show	show	VERB
ma-134	322	5	that	that	SCONJ
ma-134	322	6	the	the	DET
ma-134	322	7	mapping	mapping	NOUN
ma-134	322	8	satisfies	satisfy	VERB
ma-134	322	9	the	the	DET
ma-134	322	10	property	property	NOUN
ma-134	322	11	:	:	PUNCT
ma-134	322	12	ψ(kx	ψ(kx	PROPN
ma-134	322	13	)	)	PUNCT
ma-134	322	14	=	=	SYM
ma-134	322	15	k2ψ(x	k2ψ(x	PROPN
ma-134	322	16	)	)	PUNCT
ma-134	322	17	for	for	ADP
ma-134	322	18	all	all	DET
ma-134	322	19	k	k	PROPN
ma-134	322	20	∈	∈	PROPN
ma-134	322	21	n	n	NOUN
ma-134	322	22	,	,	PUNCT
ma-134	322	23	x	x	PUNCT
ma-134	322	24	∈	∈	NOUN
ma-134	322	25	x	x	X
ma-134	322	26	.	.	PUNCT
ma-134	323	1	we	we	PRON
ma-134	323	2	provethis	provethis	VERB
ma-134	323	3	by	by	ADP
ma-134	323	4	mathematical	mathematical	ADJ
ma-134	323	5	induction	induction	NOUN
ma-134	323	6	,	,	PUNCT
ma-134	323	7	for	for	ADP
ma-134	323	8	a	a	DET
ma-134	323	9	fixed	fix	VERB
ma-134	323	10	element	element	NOUN
ma-134	323	11	x	x	SYM
ma-134	323	12	∈	∈	PROPN
ma-134	323	13	x.	x.	NOUN
ma-134	324	1	we	we	PRON
ma-134	324	2	will	will	AUX
ma-134	324	3	prove	prove	VERB
ma-134	324	4	that	that	SCONJ
ma-134	324	5	the	the	DET
ma-134	324	6	property	property	NOUN
ma-134	324	7	is	be	AUX
ma-134	324	8	truefor	truefor	ADP
ma-134	324	9	k	k	NOUN
ma-134	325	1	=	=	SYM
ma-134	325	2	2	2	X
ma-134	325	3	.	.	X
ma-134	326	1	from	from	ADP
ma-134	326	2	(	(	PUNCT
ma-134	326	3	x,−x	x,−x	PROPN
ma-134	326	4	,	,	PUNCT
ma-134	326	5	x	x	NOUN
ma-134	326	6	)	)	PUNCT
ma-134	326	7	in	in	ADP
ma-134	326	8	equation	equation	NOUN
ma-134	326	9	(	(	PUNCT
ma-134	326	10	3.1	3.1	NUM
ma-134	326	11	)	)	PUNCT
ma-134	326	12	,	,	PUNCT
ma-134	326	13	we	we	PRON
ma-134	326	14	can	can	AUX
ma-134	326	15	get	get	VERB
ma-134	326	16	that	that	DET
ma-134	326	17	‖3f	‖3f	PROPN
ma-134	326	18	(	(	PUNCT
ma-134	326	19	x	x	NOUN
ma-134	326	20	)	)	PUNCT
ma-134	327	1	+	+	NUM
ma-134	327	2	f	f	X
ma-134	327	3	(	(	PUNCT
ma-134	327	4	−x)−	−x)−	PROPN
ma-134	327	5	f	f	PROPN
ma-134	327	6	(	(	PUNCT
ma-134	327	7	2x)‖	2x)‖	PROPN
ma-134	327	8	6	6	NUM
ma-134	327	9	3k	3k	NUM
ma-134	327	10	(	(	PUNCT
ma-134	327	11	‖x‖r	‖x‖r	NOUN
ma-134	327	12	)	)	PUNCT
ma-134	327	13	for	for	ADP
ma-134	327	14	all	all	DET
ma-134	327	15	x	x	SYM
ma-134	327	16	∈	∈	PROPN
ma-134	327	17	x.thus	x.thus	PROPN
ma-134	327	18	‖f	‖f	PRON
ma-134	327	19	(	(	PUNCT
ma-134	327	20	−x)−	−x)−	PROPN
ma-134	327	21	f	f	X
ma-134	327	22	(	(	PUNCT
ma-134	327	23	x)−	x)−	NOUN
ma-134	327	24	φ(x)‖	φ(x)‖	NOUN
ma-134	327	25	≤	≤	NOUN
ma-134	327	26	‖f	‖f	PRON
ma-134	327	27	(	(	PUNCT
ma-134	327	28	2x)−	2x)−	NUM
ma-134	327	29	4f	4f	NUM
ma-134	327	30	(	(	PUNCT
ma-134	327	31	x)−	x)−	PROPN
ma-134	327	32	φ(x)‖+	φ(x)‖+	PROPN
ma-134	327	33	‖3f	‖3f	PROPN
ma-134	327	34	(	(	PUNCT
ma-134	327	35	x	x	NOUN
ma-134	327	36	)	)	PUNCT
ma-134	328	1	+	+	NUM
ma-134	328	2	f	f	X
ma-134	328	3	(	(	PUNCT
ma-134	328	4	−x)−	−x)−	PROPN
ma-134	328	5	f	f	PROPN
ma-134	328	6	(	(	PUNCT
ma-134	328	7	2x)‖	2x)‖	NUM
ma-134	328	8	6	6	NUM
ma-134	328	9	(	(	PUNCT
ma-134	328	10	(	(	PUNCT
ma-134	328	11	2	2	NUM
ma-134	328	12	+	+	NUM
ma-134	328	13	2rβ1	2rβ1	NUM
ma-134	328	14	+	+	CCONJ
ma-134	328	15	3	3	NUM
ma-134	328	16	·	·	SYM
ma-134	328	17	2β2)k	2β2)k	NUM
ma-134	328	18	2β1r	2β1r	ADJ
ma-134	328	19	−	−	NUM
ma-134	328	20	2β2	2β2	NUM
ma-134	328	21	+	+	CCONJ
ma-134	328	22	3k	3k	NUM
ma-134	328	23	)	)	PUNCT
ma-134	328	24	(	(	PUNCT
ma-134	328	25	‖x‖r	‖x‖r	NOUN
ma-134	328	26	)	)	PUNCT
ma-134	328	27	for	for	ADP
ma-134	328	28	all	all	PRON
ma-134	328	29	x	x	SYM
ma-134	328	30	∈	∈	NOUN
ma-134	328	31	x.	x.	NOUN
ma-134	328	32	using	use	VERB
ma-134	328	33	the	the	DET
ma-134	328	34	similar	similar	ADJ
ma-134	328	35	above	above	ADJ
ma-134	328	36	argumentation	argumentation	NOUN
ma-134	328	37	together	together	ADV
ma-134	328	38	the	the	DET
ma-134	328	39	above	above	ADJ
ma-134	328	40	inequality	inequality	NOUN
ma-134	328	41	and	and	CCONJ
ma-134	328	42	equation	equation	NOUN
ma-134	328	43	(	(	PUNCT
ma-134	328	44	3.1	3.1	NUM
ma-134	328	45	)	)	PUNCT
ma-134	328	46	,	,	PUNCT
ma-134	328	47	yields	yield	NOUN
ma-134	328	48	ψ(−x	ψ(−x	PUNCT
ma-134	328	49	)	)	PUNCT
ma-134	328	50	=	=	SYM
ma-134	328	51	ψ(x	ψ(x	NOUN
ma-134	328	52	)	)	PUNCT
ma-134	329	1	+	+	CCONJ
ma-134	329	2	lim	lim	PROPN
ma-134	329	3	n→∞	n→∞	NUM
ma-134	329	4	φ(x	φ(x	PROPN
ma-134	329	5	)	)	PUNCT
ma-134	329	6	2nand	2nand	NUM
ma-134	329	7	ψ(x	ψ(x	NOUN
ma-134	329	8	+	+	CCONJ
ma-134	329	9	y	y	PROPN
ma-134	330	1	+	+	PROPN
ma-134	330	2	z	z	NOUN
ma-134	330	3	)	)	PUNCT
ma-134	330	4	+	+	NUM
ma-134	330	5	ψ(x	ψ(x	NOUN
ma-134	330	6	)	)	PUNCT
ma-134	331	1	+	+	NUM
ma-134	331	2	ψ(z	ψ(z	NOUN
ma-134	331	3	)	)	PUNCT
ma-134	331	4	+	+	NUM
ma-134	331	5	ψ(y	ψ(y	NOUN
ma-134	331	6	)	)	PUNCT
ma-134	331	7	=	=	SYM
ma-134	332	1	ψ(x	ψ(x	PROPN
ma-134	332	2	+	+	CCONJ
ma-134	332	3	y	y	NOUN
ma-134	332	4	)	)	PUNCT
ma-134	333	1	+	+	CCONJ
ma-134	333	2	ψ(z	ψ(z	PROPN
ma-134	333	3	+	+	CCONJ
ma-134	333	4	y	y	NOUN
ma-134	333	5	)	)	PUNCT
ma-134	334	1	+	+	NUM
ma-134	334	2	ψ(x	ψ(x	NOUN
ma-134	334	3	+	+	CCONJ
ma-134	334	4	z	z	NOUN
ma-134	334	5	)	)	PUNCT
ma-134	334	6	(	(	PUNCT
ma-134	334	7	3.9	3.9	NUM
ma-134	334	8	)	)	PUNCT
ma-134	334	9	for	for	ADP
ma-134	334	10	all	all	DET
ma-134	334	11	x	x	NOUN
ma-134	334	12	,	,	PUNCT
ma-134	334	13	y	y	PROPN
ma-134	334	14	,	,	PUNCT
ma-134	334	15	z	z	PROPN
ma-134	334	16	∈	∈	PROPN
ma-134	334	17	x.	x.	NOUN
ma-134	334	18	from	from	ADP
ma-134	334	19	(	(	PUNCT
ma-134	334	20	x,−x	x,−x	PROPN
ma-134	334	21	,	,	PUNCT
ma-134	334	22	x	x	NOUN
ma-134	334	23	)	)	PUNCT
ma-134	334	24	in	in	ADP
ma-134	334	25	equation	equation	NOUN
ma-134	334	26	(	(	PUNCT
ma-134	334	27	3.7	3.7	NUM
ma-134	334	28	)	)	PUNCT
ma-134	334	29	,	,	PUNCT
ma-134	334	30	we	we	PRON
ma-134	334	31	achieve	achieve	VERB
ma-134	334	32	ψ(2x	ψ(2x	NOUN
ma-134	334	33	)	)	PUNCT
ma-134	334	34	=	=	SYM
ma-134	334	35	3ψ(x	3ψ(x	NUM
ma-134	334	36	)	)	PUNCT
ma-134	335	1	+	+	NUM
ma-134	335	2	ψ(−x	ψ(−x	X
ma-134	335	3	)	)	PUNCT
ma-134	335	4	=	=	SYM
ma-134	335	5	4ψ(x	4ψ(x	NOUN
ma-134	335	6	)	)	PUNCT
ma-134	335	7	.	.	PUNCT
ma-134	336	1	fixed	fix	VERB
ma-134	336	2	x	x	SYM
ma-134	336	3	∈	∈	PROPN
ma-134	336	4	x	x	X
ma-134	336	5	,	,	PUNCT
ma-134	336	6	we	we	PRON
ma-134	336	7	prove	prove	VERB
ma-134	336	8	this	this	PRON
ma-134	336	9	by	by	ADP
ma-134	336	10	induction	induction	NOUN
ma-134	336	11	.	.	PUNCT
ma-134	337	1	we	we	PRON
ma-134	337	2	have	have	AUX
ma-134	337	3	already	already	ADV
ma-134	337	4	proved	prove	VERB
ma-134	337	5	that	that	SCONJ
ma-134	337	6	the	the	DET
ma-134	337	7	property	property	NOUN
ma-134	337	8	is	be	AUX
ma-134	337	9	true	true	ADJ
ma-134	337	10	for	for	ADP
ma-134	337	11	n	n	NOUN
ma-134	337	12	=	=	SYM
ma-134	337	13	2	2	NUM
ma-134	337	14	.	.	PUNCT
ma-134	337	15	supposing	suppose	VERB
ma-134	337	16	that	that	SCONJ
ma-134	337	17	ψ(nx	ψ(nx	NOUN
ma-134	337	18	)	)	PUNCT
ma-134	337	19	=	=	SYM
ma-134	337	20	n2ψ(x	n2ψ(x	PROPN
ma-134	337	21	)	)	PUNCT
ma-134	337	22	for	for	ADP
ma-134	337	23	all	all	DET
ma-134	337	24	natural	natural	ADJ
ma-134	337	25	n	n	CCONJ
ma-134	337	26	≤	≤	NOUN
ma-134	337	27	2k	2k	NOUN
ma-134	337	28	,	,	PUNCT
ma-134	337	29	with	with	ADP
ma-134	337	30	k	k	PROPN
ma-134	337	31	≥	≥	NUM
ma-134	337	32	1	1	NUM
ma-134	337	33	,	,	PUNCT
ma-134	337	34	let	let	VERB
ma-134	337	35	us	we	PRON
ma-134	337	36	calculate	calculate	VERB
ma-134	337	37	ψ((2k	ψ((2k	ADV
ma-134	337	38	+	+	NOUN
ma-134	337	39	1)x	1)x	NUM
ma-134	337	40	)	)	PUNCT
ma-134	337	41	.	.	PUNCT
ma-134	338	1	from	from	ADP
ma-134	338	2	(	(	PUNCT
ma-134	338	3	kx	kx	PROPN
ma-134	338	4	,	,	PUNCT
ma-134	338	5	kx	kx	PROPN
ma-134	338	6	,	,	PUNCT
ma-134	338	7	x	x	NOUN
ma-134	338	8	)	)	PUNCT
ma-134	338	9	in	in	ADP
ma-134	338	10	(	(	PUNCT
ma-134	338	11	3.7	3.7	NUM
ma-134	338	12	)	)	PUNCT
ma-134	338	13	,	,	PUNCT
ma-134	338	14	we	we	PRON
ma-134	338	15	know	know	VERB
ma-134	338	16	ψ((2k	ψ((2k	ADJ
ma-134	338	17	+	+	ADJ
ma-134	338	18	1)x	1)x	NUM
ma-134	338	19	)	)	PUNCT
ma-134	338	20	=	=	SYM
ma-134	338	21	ψ(2kx	ψ(2kx	NOUN
ma-134	338	22	)	)	PUNCT
ma-134	339	1	+	+	CCONJ
ma-134	339	2	2ψ((k	2ψ((k	NUM
ma-134	339	3	+	+	CCONJ
ma-134	339	4	1)x)−	1)x)−	NUM
ma-134	339	5	2ψ(kx)−	2ψ(kx)−	NUM
ma-134	339	6	ψ(x	ψ(x	NOUN
ma-134	339	7	)	)	PUNCT
ma-134	339	8	=	=	PUNCT
ma-134	339	9	(	(	PUNCT
ma-134	339	10	4k2	4k2	NUM
ma-134	339	11	+	+	CCONJ
ma-134	339	12	2(k	2(k	NUM
ma-134	339	13	+	+	CCONJ
ma-134	339	14	1)2	1)2	NUM
ma-134	339	15	−	−	NOUN
ma-134	339	16	2k2	2k2	NUM
ma-134	339	17	−	−	PROPN
ma-134	339	18	1)ψ(x	1)ψ(x	NUM
ma-134	339	19	)	)	PUNCT
ma-134	339	20	=	=	SYM
ma-134	339	21	(	(	PUNCT
ma-134	339	22	2k	2k	NOUN
ma-134	339	23	+	+	CCONJ
ma-134	339	24	1)2ψ(x	1)2ψ(x	NUM
ma-134	339	25	)	)	PUNCT
ma-134	339	26	.	.	PUNCT
ma-134	340	1	now	now	ADV
ma-134	340	2	,	,	PUNCT
ma-134	340	3	we	we	PRON
ma-134	340	4	show	show	VERB
ma-134	340	5	the	the	DET
ma-134	340	6	mapping	mapping	NOUN
ma-134	340	7	ψ	ψ	NOUN
ma-134	340	8	satisfies	satisfie	NOUN
ma-134	340	9	the	the	DET
ma-134	340	10	property	property	NOUN
ma-134	340	11	ψ(kx	ψ(kx	PROPN
ma-134	340	12	)	)	PUNCT
ma-134	340	13	=	=	SYM
ma-134	340	14	k2ψ(x	k2ψ(x	PROPN
ma-134	340	15	)	)	PUNCT
ma-134	340	16	for	for	ADP
ma-134	340	17	all	all	DET
ma-134	340	18	k	k	PROPN
ma-134	340	19	∈	∈	PROPN
ma-134	340	20	n	n	NOUN
ma-134	340	21	,	,	PUNCT
ma-134	340	22	x	x	PUNCT
ma-134	340	23	∈	∈	PROPN
ma-134	340	24	x	x	X
ma-134	340	25	.	.	PUNCT
ma-134	341	1	thesecond	thesecond	NOUN
ma-134	341	2	step	step	NOUN
ma-134	341	3	,	,	PUNCT
ma-134	341	4	we	we	PRON
ma-134	341	5	claim	claim	VERB
ma-134	341	6	that	that	SCONJ
ma-134	341	7	the	the	DET
ma-134	341	8	mapping	mapping	NOUN
ma-134	341	9	φ	φ	PROPN
ma-134	341	10	is	be	AUX
ma-134	341	11	unique	unique	ADJ
ma-134	341	12	.	.	PUNCT
ma-134	342	1	let	let	VERB
ma-134	342	2	u(x	u(x	NOUN
ma-134	342	3	)	)	PUNCT
ma-134	342	4	be	be	AUX
ma-134	342	5	another	another	DET
ma-134	342	6	limiting	limit	VERB
ma-134	342	7	mapping	mapping	NOUN
ma-134	342	8	suchthat	suchthat	NOUN
ma-134	342	9	for	for	ADP
ma-134	342	10	some	some	DET
ma-134	342	11	k2	k2	ADJ
ma-134	342	12	≥	≥	NOUN
ma-134	342	13	0	0	NUM
ma-134	343	1	and	and	CCONJ
ma-134	343	2	r	r	NOUN
ma-134	343	3	<	<	X
ma-134	343	4	β2	β2	NOUN
ma-134	343	5	β1	β1	PROPN
ma-134	343	6	,	,	PUNCT
ma-134	343	7	‖u(x)−	‖u(x)−	PROPN
ma-134	343	8	f	f	PROPN
ma-134	343	9	(	(	PUNCT
ma-134	343	10	x)−	x)−	NOUN
ma-134	343	11	φ(x)/2‖	φ(x)/2‖	ADP
ma-134	343	12	6	6	NUM
ma-134	343	13	k2‖x‖r2	k2‖x‖r2	NOUN
ma-134	343	14	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	343	15	eur	eur	PROPN
ma-134	343	16	.	.	PUNCT
ma-134	344	1	j.	j.	PROPN
ma-134	344	2	math	math	PROPN
ma-134	344	3	.	.	PUNCT
ma-134	345	1	anal	anal	PROPN
ma-134	345	2	.	.	PUNCT
ma-134	346	1	10.28924	10.28924	NUM
ma-134	346	2	/	/	SYM
ma-134	346	3	ada	ada	PROPN
ma-134	346	4	/	/	SYM
ma-134	346	5	ma.3.7	ma.3.7	NOUN
ma-134	346	6	13which	13which	NUM
ma-134	346	7	satisfies	satisfy	VERB
ma-134	346	8	the	the	DET
ma-134	346	9	property	property	NOUN
ma-134	346	10	u(kx	u(kx	PROPN
ma-134	346	11	)	)	PUNCT
ma-134	346	12	=	=	SYM
ma-134	346	13	k2u(x	k2u(x	PROPN
ma-134	346	14	)	)	PUNCT
ma-134	346	15	for	for	ADP
ma-134	346	16	all	all	DET
ma-134	346	17	k	k	PROPN
ma-134	346	18	∈	∈	PROPN
ma-134	346	19	n	n	NOUN
ma-134	346	20	and	and	CCONJ
ma-134	346	21	x	x	PUNCT
ma-134	346	22	∈	∈	PROPN
ma-134	346	23	x	x	X
ma-134	346	24	.	.	PUNCT
ma-134	347	1	therefore	therefore	ADV
ma-134	347	2	‖ψ(x)−	‖ψ(x)−	PROPN
ma-134	347	3	u(x)‖	u(x)‖	NOUN
ma-134	348	1	=	=	NOUN
ma-134	348	2	‖ψ1(kx)−	‖ψ1(kx)−	PROPN
ma-134	348	3	u(kx)‖/k2β2	u(kx)‖/k2β2	NUM
ma-134	349	1	6‖u(xk)−	6‖u(xk)−	NOUN
ma-134	349	2	f	f	X
ma-134	349	3	(	(	PUNCT
ma-134	349	4	kx)−	kx)−	PROPN
ma-134	349	5	φ(kx)/2‖/k2β2	φ(kx)/2‖/k2β2	PUNCT
ma-134	349	6	+	+	CCONJ
ma-134	349	7	|ψ(xk)−	|ψ(xk)−	ADJ
ma-134	349	8	f	f	NOUN
ma-134	349	9	(	(	PUNCT
ma-134	349	10	kx)−	kx)−	PROPN
ma-134	349	11	φ(kx)/2‖/k2β2	φ(kx)/2‖/k2β2	NUM
ma-134	349	12	6	6	NUM
ma-134	349	13	(	(	PUNCT
ma-134	349	14	2	2	NUM
ma-134	349	15	+	+	NUM
ma-134	349	16	2rβ1	2rβ1	NUM
ma-134	349	17	+	+	CCONJ
ma-134	349	18	3	3	NUM
ma-134	349	19	·	·	SYM
ma-134	349	20	2β2)k	2β2)k	NUM
ma-134	349	21	(	(	PUNCT
ma-134	349	22	2β1r	2β1r	ADJ
ma-134	349	23	−	−	NOUN
ma-134	349	24	22β2)(2β1r	22β2)(2β1r	NUM
ma-134	349	25	−	−	NOUN
ma-134	349	26	2β2	2β2	NUM
ma-134	349	27	)	)	PUNCT
ma-134	349	28	‖x‖rk	‖x‖rk	INTJ
ma-134	350	1	rβ1−2β2	rβ1−2β2	PROPN
ma-134	350	2	+	+	PUNCT
ma-134	350	3	|k2‖x‖r2k	|k2‖x‖r2k	ADJ
ma-134	350	4	r2β1−2β2	r2β1−2β2	NOUN
ma-134	350	5	.	.	PUNCT
ma-134	351	1	hence	hence	ADV
ma-134	351	2	φ(x	φ(x	NOUN
ma-134	351	3	)	)	PUNCT
ma-134	351	4	=	=	SYM
ma-134	351	5	u(x	u(x	PROPN
ma-134	351	6	)	)	PUNCT
ma-134	351	7	for	for	ADP
ma-134	351	8	all	all	DET
ma-134	351	9	x	x	SYM
ma-134	351	10	∈	∈	PROPN
ma-134	351	11	x	x	X
ma-134	351	12	.	.	PUNCT
ma-134	352	1	this	this	PRON
ma-134	352	2	shows	show	VERB
ma-134	352	3	that	that	SCONJ
ma-134	352	4	ψ	ψ	NOUN
ma-134	352	5	is	be	AUX
ma-134	352	6	unique	unique	ADJ
ma-134	352	7	.	.	PUNCT
ma-134	353	1	let	let	VERB
ma-134	353	2	ψ1(x	ψ1(x	PRON
ma-134	353	3	)	)	PUNCT
ma-134	354	1	=	=	SYM
ma-134	354	2	ψ(x)−	ψ(x)−	PROPN
ma-134	354	3	φ(x)/2	φ(x)/2	PROPN
ma-134	354	4	.	.	PUNCT
ma-134	354	5	thiscompletes	thiscomplete	VERB
ma-134	354	6	the	the	DET
ma-134	354	7	uniqueness	uniqueness	NOUN
ma-134	354	8	of	of	ADP
ma-134	354	9	ψ1(x	ψ1(x	PROPN
ma-134	354	10	)	)	PUNCT
ma-134	354	11	.	.	PUNCT
ma-134	355	1	we	we	PRON
ma-134	355	2	have	have	VERB
ma-134	355	3	‖ψ1(x)−	‖ψ1(x)−	PROPN
ma-134	355	4	f	f	PROPN
ma-134	355	5	(	(	PUNCT
ma-134	355	6	x)‖	x)‖	PROPN
ma-134	355	7	6	6	NUM
ma-134	355	8	(	(	PUNCT
ma-134	355	9	2	2	NUM
ma-134	355	10	+	+	NUM
ma-134	355	11	2rβ1	2rβ1	NUM
ma-134	355	12	+	+	CCONJ
ma-134	355	13	3	3	NUM
ma-134	355	14	·	·	SYM
ma-134	355	15	2β2)k	2β2)k	NUM
ma-134	355	16	(	(	PUNCT
ma-134	355	17	2β1r	2β1r	ADJ
ma-134	355	18	−	−	NOUN
ma-134	355	19	22β2)(2β1r	22β2)(2β1r	NUM
ma-134	355	20	−	−	NOUN
ma-134	355	21	2β2	2β2	NUM
ma-134	355	22	)	)	PUNCT
ma-134	355	23	‖x‖r	‖x‖r	NOUN
ma-134	355	24	for	for	ADP
ma-134	355	25	all	all	PRON
ma-134	355	26	x	x	SYM
ma-134	355	27	∈	∈	ADJ
ma-134	355	28	x	x	X
ma-134	355	29	and	and	CCONJ
ma-134	355	30	also	also	ADV
ma-134	355	31	the	the	DET
ma-134	355	32	equation	equation	NOUN
ma-134	355	33	(	(	PUNCT
ma-134	355	34	3.2	3.2	NUM
ma-134	355	35	)	)	PUNCT
ma-134	355	36	holds	hold	VERB
ma-134	355	37	by	by	ADP
ma-134	355	38	using	use	VERB
ma-134	355	39	the	the	DET
ma-134	355	40	additive	additive	ADJ
ma-134	355	41	property	property	NOUN
ma-134	355	42	of	of	ADP
ma-134	355	43	φ	φ	PROPN
ma-134	355	44	and	and	CCONJ
ma-134	355	45	equation	equation	NOUN
ma-134	355	46	(	(	PUNCT
ma-134	355	47	3.7	3.7	NUM
ma-134	355	48	)	)	PUNCT
ma-134	355	49	.	.	PUNCT
ma-134	356	1	we	we	PRON
ma-134	356	2	complete	complete	VERB
ma-134	356	3	the	the	DET
ma-134	356	4	proof	proof	NOUN
ma-134	356	5	.	.	PUNCT
ma-134	357	1	we	we	PRON
ma-134	357	2	may	may	AUX
ma-134	357	3	also	also	ADV
ma-134	357	4	assume	assume	VERB
ma-134	357	5	that	that	SCONJ
ma-134	357	6	limn→∞	limn→∞	PROPN
ma-134	357	7	φ(x	φ(x	NOUN
ma-134	357	8	)	)	PUNCT
ma-134	357	9	2n	2n	NUM
ma-134	357	10	=	=	SYM
ma-134	357	11	limn→∞	limn→∞	X
ma-134	357	12	g(2nx)/2n	g(2nx)/2n	NOUN
ma-134	357	13	2n	2n	NUM
ma-134	357	14	=	=	SYM
ma-134	357	15	0.otherwise	0.otherwise	NUM
ma-134	357	16	,	,	PUNCT
ma-134	357	17	this	this	DET
ma-134	357	18	limit	limit	NOUN
ma-134	357	19	may	may	AUX
ma-134	357	20	not	not	PART
ma-134	357	21	be	be	AUX
ma-134	357	22	convergence	convergence	NOUN
ma-134	357	23	to	to	ADP
ma-134	357	24	zero	zero	NUM
ma-134	357	25	.	.	PUNCT
ma-134	358	1	conversely	conversely	ADV
ma-134	358	2	,	,	PUNCT
ma-134	358	3	we	we	PRON
ma-134	358	4	may	may	AUX
ma-134	358	5	add	add	VERB
ma-134	358	6	some	some	DET
ma-134	358	7	similar	similar	ADJ
ma-134	358	8	smalladditional	smalladditional	ADJ
ma-134	358	9	assumptions	assumption	NOUN
ma-134	358	10	to	to	PART
ma-134	358	11	guarantee	guarantee	VERB
ma-134	358	12	the	the	DET
ma-134	358	13	convergence	convergence	NOUN
ma-134	358	14	in	in	ADP
ma-134	358	15	theorem	theorem	NOUN
ma-134	358	16	3.2	3.2	NUM
ma-134	358	17	.	.	PUNCT
ma-134	359	1	4	4	NUM
ma-134	359	2	.	.	X
ma-134	360	1	the	the	DET
ma-134	360	2	stability	stability	NOUN
ma-134	360	3	of	of	ADP
ma-134	360	4	functional	functional	ADJ
ma-134	360	5	equations	equation	NOUN
ma-134	360	6	in	in	ADP
ma-134	360	7	banach	banach	NOUN
ma-134	360	8	space	space	NOUN
ma-134	360	9	in	in	ADP
ma-134	360	10	this	this	DET
ma-134	360	11	section	section	NOUN
ma-134	360	12	,	,	PUNCT
ma-134	360	13	we	we	PRON
ma-134	360	14	will	will	AUX
ma-134	360	15	prove	prove	VERB
ma-134	360	16	the	the	DET
ma-134	360	17	counterpart	counterpart	NOUN
ma-134	360	18	of	of	ADP
ma-134	360	19	the	the	DET
ma-134	360	20	results	result	NOUN
ma-134	360	21	of	of	ADP
ma-134	360	22	theorem	theorem	NOUN
ma-134	360	23	2.1	2.1	NUM
ma-134	360	24	from	from	ADP
ma-134	360	25	[	[	X
ma-134	360	26	8	8	NUM
ma-134	360	27	]	]	PUNCT
ma-134	360	28	to	to	ADP
ma-134	360	29	moregeneral	moregeneral	ADJ
ma-134	360	30	case	case	NOUN
ma-134	360	31	.	.	PUNCT
ma-134	361	1	we	we	PRON
ma-134	361	2	generalize	generalize	VERB
ma-134	361	3	the	the	DET
ma-134	361	4	results	result	NOUN
ma-134	361	5	of	of	ADP
ma-134	361	6	sikorska	sikorska	NOUN
ma-134	361	7	in	in	ADP
ma-134	361	8	2010	2010	NUM
ma-134	361	9	.	.	PUNCT
ma-134	362	1	in	in	ADP
ma-134	362	2	particular	particular	ADJ
ma-134	362	3	,	,	PUNCT
ma-134	362	4	the	the	DET
ma-134	362	5	related	related	ADJ
ma-134	362	6	parameters	parameter	NOUN
ma-134	362	7	u	u	NOUN
ma-134	362	8	,	,	PUNCT
ma-134	362	9	v	v	NOUN
ma-134	362	10	can	can	AUX
ma-134	362	11	be	be	AUX
ma-134	362	12	extended	extend	VERB
ma-134	362	13	to	to	ADP
ma-134	362	14	complex	complex	ADJ
ma-134	362	15	numbers	number	NOUN
ma-134	362	16	by	by	ADP
ma-134	362	17	using	use	VERB
ma-134	362	18	a	a	DET
ma-134	362	19	more	more	ADV
ma-134	362	20	efficient	efficient	ADJ
ma-134	362	21	approach	approach	NOUN
ma-134	362	22	.	.	PUNCT
ma-134	363	1	beyond	beyond	ADP
ma-134	363	2	that	that	PRON
ma-134	363	3	,	,	PUNCT
ma-134	363	4	westate	westate	VERB
ma-134	363	5	that	that	SCONJ
ma-134	363	6	the	the	DET
ma-134	363	7	first	first	ADJ
ma-134	363	8	results	result	NOUN
ma-134	363	9	in	in	ADP
ma-134	363	10	section	section	NOUN
ma-134	363	11	2	2	NUM
ma-134	363	12	are	be	AUX
ma-134	363	13	presented	present	VERB
ma-134	363	14	and	and	CCONJ
ma-134	363	15	combined	combine	VERB
ma-134	363	16	the	the	DET
ma-134	363	17	first	first	ADJ
ma-134	363	18	results	result	NOUN
ma-134	363	19	in	in	ADP
ma-134	363	20	[	[	X
ma-134	363	21	8	8	NUM
ma-134	363	22	]	]	PUNCT
ma-134	363	23	.	.	PUNCT
ma-134	364	1	ourcontribution	ourcontribution	NOUN
ma-134	364	2	to	to	ADP
ma-134	364	3	the	the	DET
ma-134	364	4	parameters	parameter	NOUN
ma-134	364	5	u	u	NOUN
ma-134	364	6	,	,	PUNCT
ma-134	364	7	v	v	NOUN
ma-134	364	8	are	be	AUX
ma-134	364	9	complex	complex	ADJ
ma-134	364	10	numbers	number	NOUN
ma-134	364	11	.	.	PUNCT
ma-134	365	1	the	the	DET
ma-134	365	2	results	result	NOUN
ma-134	365	3	is	be	AUX
ma-134	365	4	stated	state	VERB
ma-134	365	5	in	in	ADP
ma-134	365	6	this	this	DET
ma-134	365	7	section	section	NOUN
ma-134	365	8	inmore	inmore	NOUN
ma-134	365	9	detail	detail	NOUN
ma-134	365	10	.	.	PUNCT
ma-134	366	1	theorem	theorem	VERB
ma-134	366	2	4.1	4.1	NUM
ma-134	366	3	suppose	suppose	VERB
ma-134	366	4	that	that	SCONJ
ma-134	366	5	(	(	PUNCT
ma-134	366	6	x,+	x,+	NUM
ma-134	366	7	)	)	PUNCT
ma-134	366	8	is	be	AUX
ma-134	366	9	a	a	DET
ma-134	366	10	group	group	NOUN
ma-134	366	11	,	,	PUNCT
ma-134	366	12	and	and	CCONJ
ma-134	366	13	(	(	PUNCT
ma-134	366	14	y	y	PROPN
ma-134	366	15	,	,	PUNCT
ma-134	366	16	‖	‖	PROPN
ma-134	366	17	·	·	PUNCT
ma-134	366	18	‖	‖	NUM
ma-134	366	19	)	)	PUNCT
ma-134	366	20	is	be	AUX
ma-134	366	21	a	a	DET
ma-134	366	22	banach	banach	NOUN
ma-134	366	23	space	space	NOUN
ma-134	366	24	,	,	PUNCT
ma-134	366	25	and	and	CCONJ
ma-134	366	26	let	let	VERB
ma-134	366	27	themapping	themappe	VERB
ma-134	366	28	f	f	X
ma-134	366	29	:	:	PUNCT
ma-134	366	30	x	x	X
ma-134	366	31	→	→	SYM
ma-134	366	32	y	y	PROPN
ma-134	366	33	satisfy	satisfy	VERB
ma-134	366	34	the	the	DET
ma-134	366	35	inequality	inequality	NOUN
ma-134	366	36	‖f	‖f	ADP
ma-134	366	37	(	(	PUNCT
ma-134	366	38	x)−	x)−	PROPN
ma-134	366	39	uf	uf	PROPN
ma-134	366	40	(	(	PUNCT
ma-134	366	41	e(x))−	e(x))−	NOUN
ma-134	366	42	vf	vf	X
ma-134	366	43	(	(	PUNCT
ma-134	366	44	−e(x))‖	−e(x))‖	PROPN
ma-134	366	45	6	6	NUM
ma-134	366	46	δ(x	δ(x	NOUN
ma-134	366	47	)	)	PUNCT
ma-134	366	48	,	,	PUNCT
ma-134	366	49	x	x	PUNCT
ma-134	366	50	∈	∈	NOUN
ma-134	366	51	x	x	NOUN
ma-134	366	52	,	,	PUNCT
ma-134	366	53	where	where	SCONJ
ma-134	366	54	u	u	NOUN
ma-134	366	55	,	,	PUNCT
ma-134	366	56	v	v	ADP
ma-134	366	57	∈	∈	X
ma-134	366	58	c	c	NOUN
ma-134	366	59	(	(	PUNCT
ma-134	366	60	c	c	PROPN
ma-134	366	61	denotes	denote	VERB
ma-134	366	62	the	the	DET
ma-134	366	63	complex	complex	ADJ
ma-134	366	64	field	field	NOUN
ma-134	366	65	.	.	PUNCT
ma-134	366	66	)	)	PUNCT
ma-134	366	67	,	,	PUNCT
ma-134	366	68	and	and	CCONJ
ma-134	366	69	e	e	NOUN
ma-134	366	70	:	:	PUNCT
ma-134	366	71	x	x	X
ma-134	366	72	→	→	SYM
ma-134	366	73	x	x	PROPN
ma-134	366	74	,	,	PUNCT
ma-134	366	75	δ	δ	PROPN
ma-134	366	76	:	:	PUNCT
ma-134	366	77	x	x	X
ma-134	366	78	→	→	PUNCT
ma-134	366	79	[	[	X
ma-134	366	80	0,∞	0,∞	NOUN
ma-134	366	81	)	)	PUNCT
ma-134	366	82	are	be	AUX
ma-134	366	83	arbitrary	arbitrary	ADJ
ma-134	366	84	givenfunctions.(1	givenfunctions.(1	PROPN
ma-134	366	85	):	):	PUNCT
ma-134	366	86	if	if	SCONJ
ma-134	366	87	e	e	PRON
ma-134	366	88	is	be	AUX
ma-134	366	89	a	a	DET
ma-134	366	90	even	even	ADJ
ma-134	366	91	function	function	NOUN
ma-134	366	92	(	(	PUNCT
ma-134	366	93	i.e.	i.e.	X
ma-134	366	94	,	,	PUNCT
ma-134	366	95	e(−x	e(−x	NOUN
ma-134	366	96	)	)	PUNCT
ma-134	366	97	=	=	SYM
ma-134	366	98	e(x	e(x	NUM
ma-134	366	99	)	)	PUNCT
ma-134	366	100	for	for	ADP
ma-134	366	101	x	x	PROPN
ma-134	366	102	∈	∈	PROPN
ma-134	366	103	x	x	NOUN
ma-134	366	104	)	)	PUNCT
ma-134	366	105	and	and	CCONJ
ma-134	366	106	the	the	DET
ma-134	366	107	convergent	convergent	NOUN
ma-134	366	108	series	series	NOUN
ma-134	366	109	∞∑	∞∑	PROPN
ma-134	366	110	n=0	n=0	PUNCT
ma-134	367	1	[	[	X
ma-134	367	2	|un|	|un|	NOUN
ma-134	367	3	δ	δ	PROPN
ma-134	367	4	(	(	PUNCT
ma-134	367	5	en(x	en(x	X
ma-134	367	6	)	)	PUNCT
ma-134	367	7	)	)	PUNCT
ma-134	367	8	+	+	CCONJ
ma-134	367	9	|vn|δ	|vn|δ	NOUN
ma-134	367	10	(	(	PUNCT
ma-134	367	11	−en(x	−en(x	NOUN
ma-134	367	12	)	)	PUNCT
ma-134	367	13	)	)	PUNCT
ma-134	367	14	]	]	PUNCT
ma-134	367	15	with	with	ADP
ma-134	367	16	u0	u0	ADJ
ma-134	367	17	:	:	PUNCT
ma-134	367	18	=	=	SYM
ma-134	367	19	1	1	NUM
ma-134	367	20	,	,	PUNCT
ma-134	367	21	un	un	PROPN
ma-134	367	22	:	:	PUNCT
ma-134	367	23	=	=	X
ma-134	367	24	[	[	PUNCT
ma-134	367	25	u(u	u(u	SYM
ma-134	367	26	+	+	CCONJ
ma-134	367	27	v)n−1	v)n−1	X
ma-134	367	28	]	]	PUNCT
ma-134	367	29	,	,	PUNCT
ma-134	367	30	n	n	PROPN
ma-134	367	31	∈	∈	PROPN
ma-134	367	32	n	n	CCONJ
ma-134	367	33	,	,	PUNCT
ma-134	367	34	v0	v0	NOUN
ma-134	367	35	:	:	PUNCT
ma-134	367	36	=	=	SYM
ma-134	367	37	0	0	NUM
ma-134	367	38	,	,	PUNCT
ma-134	367	39	vn	vn	X
ma-134	367	40	:	:	PUNCT
ma-134	367	41	=	=	X
ma-134	367	42	[	[	PUNCT
ma-134	368	1	v(u	v(u	ADP
ma-134	368	2	+	+	CCONJ
ma-134	368	3	v)n−1	v)n−1	VERB
ma-134	368	4	]	]	PUNCT
ma-134	368	5	,	,	PUNCT
ma-134	368	6	n	n	CCONJ
ma-134	368	7	∈	∈	PROPN
ma-134	368	8	n	n	CCONJ
ma-134	368	9	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	368	10	eur	eur	PROPN
ma-134	368	11	.	.	PUNCT
ma-134	369	1	j.	j.	PROPN
ma-134	369	2	math	math	PROPN
ma-134	369	3	.	.	PUNCT
ma-134	370	1	anal	anal	PROPN
ma-134	370	2	.	.	PUNCT
ma-134	371	1	10.28924	10.28924	NUM
ma-134	371	2	/	/	SYM
ma-134	371	3	ada	ada	PROPN
ma-134	371	4	/	/	SYM
ma-134	371	5	ma.3.7	ma.3.7	NOUN
ma-134	371	6	14(and	14(and	INTJ
ma-134	371	7	where	where	SCONJ
ma-134	371	8	en	en	ADV
ma-134	371	9	states	state	VERB
ma-134	371	10	the	the	DET
ma-134	371	11	n	n	ADV
ma-134	371	12	-	-	PUNCT
ma-134	371	13	th	th	NOUN
ma-134	371	14	composition	composition	NOUN
ma-134	371	15	of	of	ADP
ma-134	371	16	the	the	DET
ma-134	371	17	function	function	NOUN
ma-134	371	18	e	e	NOUN
ma-134	371	19	)	)	PUNCT
ma-134	372	1	,	,	PUNCT
ma-134	372	2	establishes	establish	VERB
ma-134	372	3	for	for	ADP
ma-134	372	4	every	every	DET
ma-134	372	5	x	x	SYM
ma-134	372	6	∈	∈	PROPN
ma-134	372	7	x	x	X
ma-134	372	8	.	.	PUNCT
ma-134	373	1	thenthere	thenthere	PRON
ma-134	373	2	has	have	VERB
ma-134	373	3	a	a	DET
ma-134	373	4	unique	unique	ADJ
ma-134	373	5	even	even	ADV
ma-134	373	6	limiting	limit	VERB
ma-134	373	7	function	function	NOUN
ma-134	373	8	g	g	NOUN
ma-134	373	9	:	:	PUNCT
ma-134	373	10	x	x	X
ma-134	373	11	→	→	SYM
ma-134	373	12	y	y	PROPN
ma-134	373	13	fulfilling	fulfil	VERB
ma-134	373	14	g(x	g(x	NOUN
ma-134	373	15	)	)	PUNCT
ma-134	373	16	=	=	SYM
ma-134	373	17	ung(en(x	ung(en(x	PROPN
ma-134	373	18	)	)	PUNCT
ma-134	373	19	)	)	PUNCT
ma-134	374	1	+	+	CCONJ
ma-134	374	2	vng(−en(x	vng(−en(x	X
ma-134	374	3	)	)	PUNCT
ma-134	374	4	)	)	PUNCT
ma-134	374	5	,	,	PUNCT
ma-134	374	6	x	x	PUNCT
ma-134	374	7	∈	∈	NOUN
ma-134	374	8	x	x	X
ma-134	374	9	and	and	CCONJ
ma-134	374	10	n	n	CCONJ
ma-134	374	11	∈	∈	PROPN
ma-134	374	12	n	n	CCONJ
ma-134	374	13	,	,	PUNCT
ma-134	374	14	(	(	PUNCT
ma-134	374	15	4.1	4.1	NUM
ma-134	374	16	)	)	PUNCT
ma-134	374	17	and	and	CCONJ
ma-134	374	18	‖f	‖f	ADP
ma-134	374	19	(	(	PUNCT
ma-134	374	20	x)−	x)−	PROPN
ma-134	374	21	g(x)‖	g(x)‖	PROPN
ma-134	374	22	6	6	NUM
ma-134	374	23	∞∑	∞∑	PROPN
ma-134	374	24	i=0	i=0	PROPN
ma-134	374	25	[	[	PUNCT
ma-134	374	26	|ui	|ui	X
ma-134	374	27	|	|	ADV
ma-134	374	28	δ	δ	PROPN
ma-134	374	29	(	(	PUNCT
ma-134	374	30	e	e	NOUN
ma-134	374	31	i(x	i(x	PROPN
ma-134	374	32	)	)	PUNCT
ma-134	374	33	)	)	PUNCT
ma-134	375	1	+	+	CCONJ
ma-134	375	2	|vi	|vi	NUM
ma-134	375	3	|	|	ADV
ma-134	375	4	δ	δ	PROPN
ma-134	375	5	(	(	PUNCT
ma-134	375	6	−e	−e	NOUN
ma-134	375	7	i(x	i(x	PROPN
ma-134	375	8	)	)	PUNCT
ma-134	375	9	)	)	PUNCT
ma-134	375	10	]	]	PUNCT
ma-134	375	11	,	,	PUNCT
ma-134	375	12	x	x	PUNCT
ma-134	375	13	∈	∈	NOUN
ma-134	375	14	x.	x.	NOUN
ma-134	375	15	(	(	PUNCT
ma-134	375	16	4.2	4.2	NUM
ma-134	375	17	)	)	PUNCT
ma-134	375	18	(	(	PUNCT
ma-134	375	19	2	2	NUM
ma-134	375	20	):	):	PUNCT
ma-134	375	21	if	if	SCONJ
ma-134	375	22	e	e	NOUN
ma-134	375	23	is	be	AUX
ma-134	375	24	odd	odd	ADJ
ma-134	375	25	(	(	PUNCT
ma-134	375	26	i·e	i·e	PROPN
ma-134	375	27	.	.	PROPN
ma-134	375	28	,	,	PUNCT
ma-134	375	29	e(−x	e(−x	NOUN
ma-134	375	30	)	)	PUNCT
ma-134	375	31	=	=	SYM
ma-134	375	32	−e(x	−e(x	ADJ
ma-134	375	33	)	)	PUNCT
ma-134	375	34	for	for	ADP
ma-134	375	35	al	al	PROPN
ma-134	375	36	l	l	NOUN
ma-134	375	37	x	x	PUNCT
ma-134	375	38	∈	∈	PROPN
ma-134	375	39	x	x	NOUN
ma-134	375	40	)	)	PUNCT
ma-134	375	41	and	and	CCONJ
ma-134	375	42	the	the	DET
ma-134	375	43	convergent	convergent	NOUN
ma-134	375	44	series	series	NOUN
ma-134	375	45	∞∑	∞∑	PROPN
ma-134	375	46	n=0	n=0	PUNCT
ma-134	376	1	[	[	X
ma-134	376	2	|un|	|un|	NOUN
ma-134	376	3	δ	δ	PROPN
ma-134	376	4	(	(	PUNCT
ma-134	376	5	en(x	en(x	X
ma-134	376	6	)	)	PUNCT
ma-134	376	7	)	)	PUNCT
ma-134	377	1	+	+	CCONJ
ma-134	377	2	|vn|	|vn|	PROPN
ma-134	377	3	δ	δ	NOUN
ma-134	377	4	(	(	PUNCT
ma-134	377	5	−en(x	−en(x	NOUN
ma-134	377	6	)	)	PUNCT
ma-134	377	7	)	)	PUNCT
ma-134	377	8	]	]	PUNCT
ma-134	377	9	.	.	PUNCT
ma-134	378	1	with	with	ADP
ma-134	378	2	u0	u0	ADJ
ma-134	378	3	:	:	PUNCT
ma-134	378	4	=	=	SYM
ma-134	378	5	1	1	NUM
ma-134	378	6	,	,	PUNCT
ma-134	378	7	un	un	PROPN
ma-134	378	8	:	:	PUNCT
ma-134	378	9	=	=	SYM
ma-134	378	10	1	1	NUM
ma-134	378	11	2	2	NUM
ma-134	378	12	[	[	X
ma-134	378	13	(	(	PUNCT
ma-134	378	14	u	u	NOUN
ma-134	378	15	+	+	X
ma-134	378	16	v)n	v)n	X
ma-134	378	17	+	+	CCONJ
ma-134	378	18	(	(	PUNCT
ma-134	378	19	u	u	NOUN
ma-134	378	20	−	−	PROPN
ma-134	378	21	v)n	v)n	NOUN
ma-134	378	22	]	]	PUNCT
ma-134	378	23	,	,	PUNCT
ma-134	378	24	n	n	PROPN
ma-134	378	25	∈	∈	PROPN
ma-134	378	26	n	n	CCONJ
ma-134	378	27	,	,	PUNCT
ma-134	378	28	v0	v0	NOUN
ma-134	378	29	:	:	PUNCT
ma-134	378	30	=	=	SYM
ma-134	378	31	0	0	NUM
ma-134	378	32	,	,	PUNCT
ma-134	378	33	vn	vn	X
ma-134	378	34	:	:	PUNCT
ma-134	378	35	=	=	SYM
ma-134	378	36	1	1	NUM
ma-134	378	37	2	2	NUM
ma-134	378	38	[	[	X
ma-134	378	39	(	(	PUNCT
ma-134	378	40	u	u	NOUN
ma-134	378	41	+	+	X
ma-134	378	42	v)n	v)n	NOUN
ma-134	378	43	−	−	PROPN
ma-134	378	44	(	(	PUNCT
ma-134	378	45	u	u	NOUN
ma-134	378	46	−	−	PROPN
ma-134	378	47	v)n	v)n	NOUN
ma-134	378	48	]	]	PUNCT
ma-134	378	49	,	,	PUNCT
ma-134	378	50	n	n	PROPN
ma-134	378	51	∈	∈	PROPN
ma-134	378	52	nestablishes	nestablishe	NOUN
ma-134	378	53	for	for	ADP
ma-134	378	54	all	all	PRON
ma-134	378	55	x	x	SYM
ma-134	378	56	∈	∈	NOUN
ma-134	378	57	x	x	X
ma-134	378	58	.	.	PUNCT
ma-134	379	1	then	then	ADV
ma-134	379	2	there	there	PRON
ma-134	379	3	has	have	VERB
ma-134	379	4	a	a	DET
ma-134	379	5	unique	unique	ADJ
ma-134	379	6	limiting	limiting	NOUN
ma-134	379	7	mapping	mapping	NOUN
ma-134	379	8	g	g	NOUN
ma-134	379	9	:	:	PUNCT
ma-134	379	10	x	x	X
ma-134	379	11	→	→	SYM
ma-134	379	12	y	y	PROPN
ma-134	379	13	fulfilling	fulfil	VERB
ma-134	379	14	(	(	PUNCT
ma-134	379	15	3.10)and	3.10)and	NUM
ma-134	379	16	(	(	PUNCT
ma-134	379	17	3.11	3.11	NUM
ma-134	379	18	)	)	PUNCT
ma-134	379	19	.	.	PUNCT
ma-134	380	1	proof	proof	NOUN
ma-134	380	2	.	.	PUNCT
ma-134	381	1	we	we	PRON
ma-134	381	2	only	only	ADV
ma-134	381	3	need	need	VERB
ma-134	381	4	to	to	PART
ma-134	381	5	prove	prove	VERB
ma-134	381	6	the	the	DET
ma-134	381	7	uniqueness	uniqueness	NOUN
ma-134	381	8	of	of	ADP
ma-134	381	9	the	the	DET
ma-134	381	10	approximation	approximation	NOUN
ma-134	381	11	function	function	NOUN
ma-134	381	12	.	.	PUNCT
ma-134	382	1	(	(	PUNCT
ma-134	382	2	1	1	NUM
ma-134	382	3	):	):	PUNCT
ma-134	382	4	let	let	VERB
ma-134	382	5	us	we	PRON
ma-134	382	6	supposethat	supposethat	VERB
ma-134	382	7	g̃	g̃	PROPN
ma-134	382	8	:	:	PUNCT
ma-134	382	9	x	x	X
ma-134	382	10	→	→	SYM
ma-134	382	11	y	y	PROPN
ma-134	382	12	is	be	AUX
ma-134	382	13	another	another	DET
ma-134	382	14	approximating	approximate	VERB
ma-134	382	15	mapping	mapping	NOUN
ma-134	382	16	.	.	PUNCT
ma-134	383	1	so	so	ADV
ma-134	383	2	let	let	VERB
ma-134	383	3	’s	’s	PRON
ma-134	383	4	first	first	ADV
ma-134	383	5	prove	prove	VERB
ma-134	383	6	the	the	DET
ma-134	383	7	inequality	inequality	NOUN
ma-134	383	8	together	together	ADV
ma-134	383	9	withthe	withthe	ADJ
ma-134	383	10	equation	equation	NOUN
ma-134	383	11	(	(	PUNCT
ma-134	383	12	2.5	2.5	NUM
ma-134	383	13	)	)	PUNCT
ma-134	383	14	and	and	CCONJ
ma-134	383	15	g(−x	g(−x	NOUN
ma-134	383	16	)	)	PUNCT
ma-134	383	17	=	=	SYM
ma-134	383	18	g(x	g(x	NOUN
ma-134	383	19	)	)	PUNCT
ma-134	383	20	‖f	‖f	PUNCT
ma-134	383	21	(	(	PUNCT
ma-134	383	22	em(x))−	em(x))−	X
ma-134	383	23	um(unf	um(unf	ADJ
ma-134	383	24	(	(	PUNCT
ma-134	383	25	en+m(x	en+m(x	PROPN
ma-134	383	26	)	)	PUNCT
ma-134	383	27	)	)	PUNCT
ma-134	384	1	+	+	CCONJ
ma-134	384	2	vnf	vnf	PROPN
ma-134	384	3	(	(	PUNCT
ma-134	384	4	−em+n(x	−em+n(x	PROPN
ma-134	384	5	)	)	PUNCT
ma-134	384	6	)	)	PUNCT
ma-134	384	7	)	)	PUNCT
ma-134	385	1	−	−	PROPN
ma-134	385	2	vm(unf	vm(unf	ADJ
ma-134	385	3	(	(	PUNCT
ma-134	385	4	en+m(x	en+m(x	PROPN
ma-134	385	5	)	)	PUNCT
ma-134	385	6	)	)	PUNCT
ma-134	386	1	+	+	CCONJ
ma-134	386	2	vnf	vnf	PROPN
ma-134	386	3	(	(	PUNCT
ma-134	386	4	−em+n(x	−em+n(x	PROPN
ma-134	386	5	)	)	PUNCT
ma-134	386	6	)	)	PUNCT
ma-134	386	7	)	)	PUNCT
ma-134	387	1	‖	‖	PROPN
ma-134	388	1	=	=	NOUN
ma-134	388	2	‖f	‖f	ADP
ma-134	388	3	(	(	PUNCT
ma-134	388	4	em(x))−	em(x))−	PRON
ma-134	388	5	un+mf	un+mf	ADJ
ma-134	388	6	(	(	PUNCT
ma-134	388	7	en+m(x	en+m(x	NUM
ma-134	388	8	)	)	PUNCT
ma-134	388	9	)	)	PUNCT
ma-134	389	1	−	−	ADP
ma-134	389	2	vn+mf	vn+mf	NOUN
ma-134	389	3	(	(	PUNCT
ma-134	389	4	−en+m(x	−en+m(x	PROPN
ma-134	389	5	)	)	PUNCT
ma-134	389	6	)	)	PUNCT
ma-134	390	1	‖	‖	PROPN
ma-134	390	2	6	6	NUM
ma-134	390	3	n+m−1∑	n+m−1∑	PROPN
ma-134	390	4	j	j	PROPN
ma-134	390	5	=	=	NOUN
ma-134	390	6	m	m	PROPN
ma-134	391	1	[	[	X
ma-134	391	2	∣∣uj	∣∣uj	ADJ
ma-134	391	3	∣∣	∣∣	NUM
ma-134	391	4	δ	δ	PROPN
ma-134	391	5	(	(	PUNCT
ma-134	391	6	e	e	PROPN
ma-134	391	7	j(x	j(x	PROPN
ma-134	391	8	)	)	PUNCT
ma-134	391	9	)	)	PUNCT
ma-134	392	1	+	+	CCONJ
ma-134	392	2	∣∣vj	∣∣vj	NOUN
ma-134	392	3	∣∣	∣∣	NUM
ma-134	392	4	δ	δ	PROPN
ma-134	392	5	(	(	PUNCT
ma-134	392	6	−e	−e	NOUN
ma-134	392	7	j(x	j(x	PROPN
ma-134	392	8	)	)	PUNCT
ma-134	392	9	)	)	PUNCT
ma-134	392	10	]	]	PUNCT
ma-134	392	11	,	,	PUNCT
ma-134	392	12	and	and	CCONJ
ma-134	392	13	letting	let	VERB
ma-134	392	14	n	n	PRON
ma-134	392	15	→∞	→∞	NOUN
ma-134	392	16	we	we	PRON
ma-134	392	17	have	have	VERB
ma-134	392	18	for	for	ADP
ma-134	392	19	any	any	DET
ma-134	392	20	m	m	NOUN
ma-134	392	21	∈	∈	NOUN
ma-134	392	22	n	n	NOUN
ma-134	392	23	‖f	‖f	ADJ
ma-134	392	24	(	(	PUNCT
ma-134	392	25	em(x))−	em(x))−	X
ma-134	392	26	umg	umg	X
ma-134	392	27	(	(	PUNCT
ma-134	392	28	em(x))−	em(x))−	X
ma-134	392	29	vmg	vmg	NOUN
ma-134	392	30	(	(	PUNCT
ma-134	392	31	−em(x	−em(x	NOUN
ma-134	392	32	)	)	PUNCT
ma-134	392	33	)	)	PUNCT
ma-134	392	34	‖	‖	PROPN
ma-134	392	35	6	6	NUM
ma-134	393	1	∞∑	∞∑	NUM
ma-134	393	2	j	j	X
ma-134	393	3	=	=	NOUN
ma-134	393	4	m	m	PROPN
ma-134	393	5	[	[	X
ma-134	393	6	∣∣uj	∣∣uj	ADJ
ma-134	393	7	∣∣	∣∣	NUM
ma-134	393	8	δ	δ	PROPN
ma-134	393	9	(	(	PUNCT
ma-134	393	10	e	e	PROPN
ma-134	393	11	j(x	j(x	PROPN
ma-134	393	12	)	)	PUNCT
ma-134	393	13	)	)	PUNCT
ma-134	393	14	+	+	CCONJ
ma-134	393	15	∣∣vj	∣∣vj	NOUN
ma-134	393	16	∣∣	∣∣	NUM
ma-134	393	17	δ	δ	PROPN
ma-134	393	18	(	(	PUNCT
ma-134	393	19	−e	−e	NOUN
ma-134	393	20	j(x	j(x	PROPN
ma-134	393	21	)	)	PUNCT
ma-134	393	22	)	)	PUNCT
ma-134	393	23	]	]	PUNCT
ma-134	393	24	,	,	PUNCT
ma-134	393	25	and	and	CCONJ
ma-134	393	26	we	we	PRON
ma-134	393	27	can	can	AUX
ma-134	393	28	rewrite	rewrite	VERB
ma-134	393	29	‖g(x)−	‖g(x)−	PROPN
ma-134	393	30	g̃(x)‖	g̃(x)‖	NOUN
ma-134	393	31	6‖f	6‖f	NUM
ma-134	393	32	(	(	PUNCT
ma-134	393	33	km(x))−	km(x))−	NOUN
ma-134	393	34	umg	umg	NOUN
ma-134	393	35	(	(	PUNCT
ma-134	393	36	em(x))−	em(x))−	X
ma-134	393	37	vmg	vmg	NOUN
ma-134	393	38	(	(	PUNCT
ma-134	393	39	−em(x	−em(x	NOUN
ma-134	393	40	)	)	PUNCT
ma-134	393	41	)	)	PUNCT
ma-134	394	1	‖	‖	PROPN
ma-134	395	1	+	+	CCONJ
ma-134	395	2	‖f	‖f	ADP
ma-134	395	3	(	(	PUNCT
ma-134	395	4	km(x))−	km(x))−	VERB
ma-134	395	5	umg̃	umg̃	X
ma-134	395	6	(	(	PUNCT
ma-134	395	7	em(x))−	em(x))−	PRON
ma-134	395	8	vmg̃	vmg̃	X
ma-134	395	9	(	(	PUNCT
ma-134	395	10	em(x	em(x	X
ma-134	395	11	)	)	PUNCT
ma-134	395	12	)	)	PUNCT
ma-134	396	1	‖	‖	PROPN
ma-134	396	2	62	62	NUM
ma-134	397	1	∞∑	∞∑	NUM
ma-134	397	2	j	j	PROPN
ma-134	397	3	=	=	NOUN
ma-134	397	4	m	m	PROPN
ma-134	397	5	[	[	X
ma-134	397	6	∣∣uj	∣∣uj	ADJ
ma-134	397	7	∣∣	∣∣	NUM
ma-134	397	8	δ	δ	PROPN
ma-134	397	9	(	(	PUNCT
ma-134	397	10	e	e	PROPN
ma-134	397	11	j(x	j(x	PROPN
ma-134	397	12	)	)	PUNCT
ma-134	397	13	)	)	PUNCT
ma-134	398	1	+	+	CCONJ
ma-134	398	2	∣∣vj	∣∣vj	NOUN
ma-134	398	3	∣∣	∣∣	NUM
ma-134	398	4	δ	δ	PROPN
ma-134	398	5	(	(	PUNCT
ma-134	398	6	−e	−e	NOUN
ma-134	398	7	j(x	j(x	PROPN
ma-134	398	8	)	)	PUNCT
ma-134	398	9	)	)	PUNCT
ma-134	398	10	]	]	PUNCT
ma-134	398	11	for	for	ADP
ma-134	398	12	any	any	DET
ma-134	398	13	x	x	SYM
ma-134	398	14	∈	∈	PROPN
ma-134	398	15	x	x	X
ma-134	398	16	and	and	CCONJ
ma-134	398	17	m	m	PROPN
ma-134	398	18	∈	∈	PROPN
ma-134	398	19	n	n	CCONJ
ma-134	398	20	,	,	PUNCT
ma-134	398	21	which	which	PRON
ma-134	398	22	yields	yield	VERB
ma-134	398	23	g	g	PROPN
ma-134	398	24	=	=	SYM
ma-134	398	25	g̃	g̃	PROPN
ma-134	398	26	in	in	ADP
ma-134	398	27	x	x	PUNCT
ma-134	398	28	as	as	ADP
ma-134	398	29	m	m	PRON
ma-134	398	30	→∞.(2	→∞.(2	VERB
ma-134	398	31	):	):	PUNCT
ma-134	398	32	combined	combine	VERB
ma-134	398	33	with	with	ADP
ma-134	398	34	the	the	DET
ma-134	398	35	results	result	NOUN
ma-134	398	36	of	of	ADP
ma-134	398	37	theorem	theorem	NOUN
ma-134	398	38	2.1	2.1	NUM
ma-134	398	39	from	from	ADP
ma-134	398	40	[	[	X
ma-134	398	41	8	8	NUM
ma-134	398	42	]	]	PUNCT
ma-134	398	43	where	where	SCONJ
ma-134	398	44	e	e	NOUN
ma-134	398	45	is	be	AUX
ma-134	398	46	odd	odd	ADJ
ma-134	398	47	,	,	PUNCT
ma-134	398	48	we	we	PRON
ma-134	398	49	only	only	ADV
ma-134	398	50	need	need	VERB
ma-134	398	51	to	to	PART
ma-134	398	52	prove	prove	VERB
ma-134	398	53	theuniqueness	theuniqueness	NOUN
ma-134	398	54	of	of	ADP
ma-134	398	55	the	the	DET
ma-134	398	56	approximation	approximation	NOUN
ma-134	398	57	function	function	NOUN
ma-134	398	58	.	.	PUNCT
ma-134	399	1	let	let	VERB
ma-134	399	2	us	we	PRON
ma-134	399	3	suppose	suppose	VERB
ma-134	399	4	that	that	SCONJ
ma-134	399	5	g̃	g̃	PROPN
ma-134	399	6	:	:	PUNCT
ma-134	399	7	x	x	SYM
ma-134	399	8	→	→	SYM
ma-134	399	9	y	y	PROPN
ma-134	399	10	is	be	AUX
ma-134	399	11	another	another	DET
ma-134	399	12	approximating	approximate	VERB
ma-134	399	13	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	NOUN
ma-134	399	14	eur	eur	PROPN
ma-134	399	15	.	.	PUNCT
ma-134	400	1	j.	j.	PROPN
ma-134	400	2	math	math	PROPN
ma-134	400	3	.	.	PUNCT
ma-134	401	1	anal	anal	PROPN
ma-134	401	2	.	.	PUNCT
ma-134	402	1	10.28924	10.28924	NUM
ma-134	402	2	/	/	SYM
ma-134	402	3	ada	ada	PROPN
ma-134	402	4	/	/	SYM
ma-134	402	5	ma.3.7	ma.3.7	NOUN
ma-134	402	6	15mapping	15mapping	NUM
ma-134	402	7	.	.	PUNCT
ma-134	403	1	so	so	ADV
ma-134	403	2	let	let	VERB
ma-134	403	3	’s	’s	PRON
ma-134	403	4	first	first	ADV
ma-134	403	5	prove	prove	VERB
ma-134	403	6	the	the	DET
ma-134	403	7	inequality	inequality	NOUN
ma-134	403	8	together	together	ADV
ma-134	403	9	with	with	ADP
ma-134	403	10	the	the	DET
ma-134	403	11	equation	equation	NOUN
ma-134	403	12	the	the	DET
ma-134	403	13	results	result	NOUN
ma-134	403	14	in	in	ADP
ma-134	403	15	theorem	theorem	NOUN
ma-134	403	16	2.1	2.1	NUM
ma-134	403	17	in	in	ADP
ma-134	403	18	[	[	X
ma-134	403	19	8	8	NUM
ma-134	403	20	]	]	PUNCT
ma-134	403	21	‖f	‖f	PRON
ma-134	403	22	(	(	PUNCT
ma-134	403	23	em(x))−	em(x))−	X
ma-134	403	24	um(unf	um(unf	ADJ
ma-134	403	25	(	(	PUNCT
ma-134	403	26	en+m(x	en+m(x	PROPN
ma-134	403	27	)	)	PUNCT
ma-134	403	28	)	)	PUNCT
ma-134	404	1	+	+	CCONJ
ma-134	404	2	vnf	vnf	PROPN
ma-134	404	3	(	(	PUNCT
ma-134	404	4	−em+n(x	−em+n(x	PROPN
ma-134	404	5	)	)	PUNCT
ma-134	404	6	)	)	PUNCT
ma-134	404	7	)	)	PUNCT
ma-134	405	1	−	−	PROPN
ma-134	405	2	vm(unf	vm(unf	ADJ
ma-134	405	3	(	(	PUNCT
ma-134	405	4	−en+m(x	−en+m(x	PROPN
ma-134	405	5	)	)	PUNCT
ma-134	405	6	)	)	PUNCT
ma-134	406	1	+	+	CCONJ
ma-134	406	2	vnf	vnf	NOUN
ma-134	406	3	(	(	PUNCT
ma-134	406	4	em+n(x	em+n(x	PROPN
ma-134	406	5	)	)	PUNCT
ma-134	406	6	)	)	PUNCT
ma-134	406	7	)	)	PUNCT
ma-134	407	1	‖	‖	PROPN
ma-134	408	1	=	=	NOUN
ma-134	408	2	‖f	‖f	ADP
ma-134	408	3	(	(	PUNCT
ma-134	408	4	em(x))−	em(x))−	PRON
ma-134	408	5	un+mf	un+mf	ADJ
ma-134	408	6	(	(	PUNCT
ma-134	408	7	en+m(x	en+m(x	NUM
ma-134	408	8	)	)	PUNCT
ma-134	408	9	)	)	PUNCT
ma-134	409	1	−	−	ADP
ma-134	409	2	vn+mf	vn+mf	NOUN
ma-134	409	3	(	(	PUNCT
ma-134	409	4	−en+m(x	−en+m(x	PROPN
ma-134	409	5	)	)	PUNCT
ma-134	409	6	)	)	PUNCT
ma-134	410	1	‖	‖	PROPN
ma-134	410	2	6	6	NUM
ma-134	410	3	n+m−1∑	n+m−1∑	PROPN
ma-134	410	4	j	j	PROPN
ma-134	410	5	=	=	NOUN
ma-134	410	6	m	m	PROPN
ma-134	411	1	[	[	X
ma-134	411	2	∣∣uj	∣∣uj	ADJ
ma-134	411	3	∣∣	∣∣	NUM
ma-134	411	4	δ	δ	PROPN
ma-134	411	5	(	(	PUNCT
ma-134	411	6	e	e	PROPN
ma-134	411	7	j(x	j(x	PROPN
ma-134	411	8	)	)	PUNCT
ma-134	411	9	)	)	PUNCT
ma-134	412	1	+	+	CCONJ
ma-134	412	2	∣∣vj	∣∣vj	NOUN
ma-134	412	3	∣∣	∣∣	NUM
ma-134	412	4	δ	δ	PROPN
ma-134	412	5	(	(	PUNCT
ma-134	412	6	−e	−e	NOUN
ma-134	412	7	j(x	j(x	PROPN
ma-134	412	8	)	)	PUNCT
ma-134	412	9	)	)	PUNCT
ma-134	412	10	]	]	PUNCT
ma-134	412	11	,	,	PUNCT
ma-134	412	12	and	and	CCONJ
ma-134	412	13	letting	let	VERB
ma-134	412	14	n	n	PRON
ma-134	412	15	→∞	→∞	NOUN
ma-134	412	16	we	we	PRON
ma-134	412	17	have	have	VERB
ma-134	412	18	for	for	ADP
ma-134	412	19	any	any	DET
ma-134	412	20	m	m	NOUN
ma-134	412	21	∈	∈	NOUN
ma-134	412	22	n	n	NOUN
ma-134	412	23	‖f	‖f	ADJ
ma-134	412	24	(	(	PUNCT
ma-134	412	25	em(x))−	em(x))−	X
ma-134	412	26	umg	umg	X
ma-134	412	27	(	(	PUNCT
ma-134	412	28	em(x))−	em(x))−	X
ma-134	412	29	vmg	vmg	NOUN
ma-134	412	30	(	(	PUNCT
ma-134	412	31	−em(x	−em(x	NOUN
ma-134	412	32	)	)	PUNCT
ma-134	412	33	)	)	PUNCT
ma-134	412	34	‖	‖	PROPN
ma-134	412	35	6	6	NUM
ma-134	413	1	∞∑	∞∑	NUM
ma-134	413	2	j	j	X
ma-134	413	3	=	=	NOUN
ma-134	413	4	m	m	PROPN
ma-134	413	5	[	[	X
ma-134	413	6	∣∣uj	∣∣uj	ADJ
ma-134	413	7	∣∣	∣∣	NUM
ma-134	413	8	δ	δ	PROPN
ma-134	413	9	(	(	PUNCT
ma-134	413	10	e	e	PROPN
ma-134	413	11	j(x	j(x	PROPN
ma-134	413	12	)	)	PUNCT
ma-134	413	13	)	)	PUNCT
ma-134	413	14	+	+	CCONJ
ma-134	413	15	∣∣vj	∣∣vj	NOUN
ma-134	413	16	∣∣	∣∣	NUM
ma-134	413	17	δ	δ	PROPN
ma-134	413	18	(	(	PUNCT
ma-134	413	19	−e	−e	NOUN
ma-134	413	20	j(x	j(x	PROPN
ma-134	413	21	)	)	PUNCT
ma-134	413	22	)	)	PUNCT
ma-134	413	23	]	]	PUNCT
ma-134	413	24	,	,	PUNCT
ma-134	413	25	and	and	CCONJ
ma-134	413	26	we	we	PRON
ma-134	413	27	can	can	AUX
ma-134	413	28	rewrite	rewrite	VERB
ma-134	413	29	‖g(x)−	‖g(x)−	PROPN
ma-134	413	30	g̃(x)‖	g̃(x)‖	NOUN
ma-134	413	31	6‖f	6‖f	NUM
ma-134	413	32	(	(	PUNCT
ma-134	413	33	km(x))−	km(x))−	NOUN
ma-134	413	34	umg	umg	NOUN
ma-134	413	35	(	(	PUNCT
ma-134	413	36	em(x))−	em(x))−	X
ma-134	413	37	vmg	vmg	NOUN
ma-134	413	38	(	(	PUNCT
ma-134	413	39	−em(x	−em(x	NOUN
ma-134	413	40	)	)	PUNCT
ma-134	413	41	)	)	PUNCT
ma-134	414	1	‖	‖	PROPN
ma-134	415	1	+	+	CCONJ
ma-134	415	2	‖f	‖f	ADP
ma-134	415	3	(	(	PUNCT
ma-134	415	4	km(x))−	km(x))−	VERB
ma-134	415	5	umg̃	umg̃	X
ma-134	415	6	(	(	PUNCT
ma-134	415	7	em(x))−	em(x))−	PRON
ma-134	415	8	vmg̃	vmg̃	X
ma-134	415	9	(	(	PUNCT
ma-134	415	10	em(x	em(x	X
ma-134	415	11	)	)	PUNCT
ma-134	415	12	)	)	PUNCT
ma-134	416	1	‖	‖	PROPN
ma-134	416	2	62	62	NUM
ma-134	417	1	∞∑	∞∑	NUM
ma-134	417	2	j	j	PROPN
ma-134	417	3	=	=	NOUN
ma-134	417	4	m	m	PROPN
ma-134	417	5	[	[	X
ma-134	417	6	∣∣uj	∣∣uj	ADJ
ma-134	417	7	∣∣	∣∣	NUM
ma-134	417	8	δ	δ	PROPN
ma-134	417	9	(	(	PUNCT
ma-134	417	10	e	e	PROPN
ma-134	417	11	j(x	j(x	PROPN
ma-134	417	12	)	)	PUNCT
ma-134	417	13	)	)	PUNCT
ma-134	418	1	+	+	CCONJ
ma-134	418	2	∣∣vj	∣∣vj	NOUN
ma-134	418	3	∣∣	∣∣	NUM
ma-134	418	4	δ	δ	PROPN
ma-134	418	5	(	(	PUNCT
ma-134	418	6	−e	−e	NOUN
ma-134	418	7	j(x	j(x	PROPN
ma-134	418	8	)	)	PUNCT
ma-134	418	9	)	)	PUNCT
ma-134	418	10	]	]	PUNCT
ma-134	418	11	for	for	ADP
ma-134	418	12	any	any	DET
ma-134	418	13	x	x	SYM
ma-134	418	14	∈	∈	PROPN
ma-134	418	15	x	x	X
ma-134	418	16	and	and	CCONJ
ma-134	418	17	m	m	PROPN
ma-134	418	18	∈	∈	PROPN
ma-134	418	19	n	n	CCONJ
ma-134	418	20	,	,	PUNCT
ma-134	418	21	which	which	PRON
ma-134	418	22	yields	yield	VERB
ma-134	418	23	g	g	PROPN
ma-134	418	24	=	=	SYM
ma-134	418	25	g̃	g̃	PROPN
ma-134	418	26	in	in	ADP
ma-134	418	27	x	x	PUNCT
ma-134	418	28	as	as	ADP
ma-134	418	29	m	m	PROPN
ma-134	418	30	→∞.	→∞.	PROPN
ma-134	418	31	this	this	PRON
ma-134	418	32	completes	complete	VERB
ma-134	418	33	the	the	DET
ma-134	418	34	proof	proof	NOUN
ma-134	418	35	.	.	PUNCT
ma-134	419	1	�	�	PROPN
ma-134	419	2	for	for	ADP
ma-134	419	3	the	the	DET
ma-134	419	4	euler	euler	ADJ
ma-134	419	5	-	-	PUNCT
ma-134	419	6	lagrange	lagrange	NOUN
ma-134	419	7	equation	equation	NOUN
ma-134	419	8	,	,	PUNCT
ma-134	419	9	we	we	PRON
ma-134	419	10	provide	provide	VERB
ma-134	419	11	another	another	DET
ma-134	419	12	method	method	NOUN
ma-134	419	13	to	to	PART
ma-134	419	14	solve	solve	VERB
ma-134	419	15	it	it	PRON
ma-134	419	16	in	in	ADP
ma-134	419	17	contrast	contrast	NOUN
ma-134	419	18	with	with	ADP
ma-134	419	19	[	[	X
ma-134	419	20	10	10	NUM
ma-134	419	21	]	]	PUNCT
ma-134	419	22	.	.	PUNCT
ma-134	420	1	theorem	theorem	ADJ
ma-134	420	2	4.2	4.2	NUM
ma-134	420	3	suppose	suppose	VERB
ma-134	420	4	that	that	SCONJ
ma-134	420	5	(	(	PUNCT
ma-134	420	6	x,+	x,+	NUM
ma-134	420	7	)	)	PUNCT
ma-134	420	8	is	be	AUX
ma-134	420	9	a	a	DET
ma-134	420	10	group	group	NOUN
ma-134	420	11	,	,	PUNCT
ma-134	420	12	and	and	CCONJ
ma-134	420	13	(	(	PUNCT
ma-134	420	14	y	y	PROPN
ma-134	420	15	,	,	PUNCT
ma-134	420	16	‖	‖	PROPN
ma-134	420	17	·	·	PUNCT
ma-134	420	18	‖	‖	NUM
ma-134	420	19	)	)	PUNCT
ma-134	420	20	is	be	AUX
ma-134	420	21	a	a	DET
ma-134	420	22	banach	banach	NOUN
ma-134	420	23	space	space	NOUN
ma-134	420	24	and	and	CCONJ
ma-134	420	25	let	let	VERB
ma-134	420	26	themapping	themappe	VERB
ma-134	420	27	f	f	X
ma-134	420	28	:	:	PUNCT
ma-134	420	29	x	x	X
ma-134	420	30	→	→	SYM
ma-134	420	31	y	y	PROPN
ma-134	420	32	satisfy	satisfy	VERB
ma-134	420	33	the	the	DET
ma-134	420	34	inequality	inequality	NOUN
ma-134	420	35	for	for	ADP
ma-134	420	36	all	all	DET
ma-134	420	37	x	x	NOUN
ma-134	420	38	,	,	PUNCT
ma-134	420	39	y	y	PROPN
ma-134	420	40	,	,	PUNCT
ma-134	420	41	z	z	NOUN
ma-134	420	42	∈	∈	PROPN
ma-134	420	43	x	x	X
ma-134	420	44	and	and	CCONJ
ma-134	420	45	some	some	DET
ma-134	420	46	ε	ε	PROPN
ma-134	420	47	>	>	X
ma-134	420	48	0	0	NUM
ma-134	421	1	‖f	‖f	PRON
ma-134	421	2	(	(	PUNCT
ma-134	421	3	x	x	X
ma-134	421	4	+	+	NUM
ma-134	421	5	y	y	PROPN
ma-134	421	6	+	+	PROPN
ma-134	421	7	z	z	NOUN
ma-134	421	8	)	)	PUNCT
ma-134	422	1	+	+	NOUN
ma-134	422	2	f	f	X
ma-134	422	3	(	(	PUNCT
ma-134	422	4	x	x	X
ma-134	422	5	−	−	PROPN
ma-134	422	6	y	y	PROPN
ma-134	422	7	+	+	PROPN
ma-134	422	8	z	z	NOUN
ma-134	422	9	)	)	PUNCT
ma-134	423	1	+	+	NOUN
ma-134	423	2	f	f	X
ma-134	423	3	(	(	PUNCT
ma-134	423	4	x	x	X
ma-134	423	5	+	+	NUM
ma-134	423	6	y	y	PROPN
ma-134	423	7	−	−	PROPN
ma-134	423	8	z	z	NOUN
ma-134	423	9	)	)	PUNCT
ma-134	424	1	+	+	CCONJ
ma-134	424	2	f	f	X
ma-134	424	3	(	(	PUNCT
ma-134	424	4	x	x	SYM
ma-134	424	5	−	−	PROPN
ma-134	424	6	y	y	PROPN
ma-134	424	7	−	−	PROPN
ma-134	424	8	z)−	z)−	PROPN
ma-134	424	9	4f	4f	NUM
ma-134	424	10	(	(	PUNCT
ma-134	424	11	x)−	x)−	PROPN
ma-134	424	12	4f	4f	NUM
ma-134	424	13	(	(	PUNCT
ma-134	424	14	y)−	y)−	PROPN
ma-134	424	15	4f	4f	NOUN
ma-134	424	16	(	(	PUNCT
ma-134	424	17	z)‖	z)‖	NUM
ma-134	424	18	6	6	NUM
ma-134	424	19	ε	ε	PROPN
ma-134	424	20	.	.	PUNCT
ma-134	424	21	(	(	PUNCT
ma-134	424	22	4.3	4.3	NUM
ma-134	424	23	)	)	PUNCT
ma-134	424	24	then	then	ADV
ma-134	424	25	there	there	PRON
ma-134	424	26	has	have	VERB
ma-134	424	27	a	a	DET
ma-134	424	28	unique	unique	ADJ
ma-134	424	29	limiting	limiting	NOUN
ma-134	424	30	function	function	NOUN
ma-134	424	31	g	g	NOUN
ma-134	424	32	:	:	PUNCT
ma-134	424	33	x	x	X
ma-134	424	34	→	→	PUNCT
ma-134	424	35	y	y	NUM
ma-134	424	36	such	such	ADJ
ma-134	424	37	that	that	PRON
ma-134	424	38	g(x	g(x	NOUN
ma-134	424	39	)	)	PUNCT
ma-134	424	40	=	=	SYM
ma-134	425	1	2	2	NUM
ma-134	425	2	9	9	NUM
ma-134	425	3	g(3x)−	g(3x)−	PROPN
ma-134	425	4	1	1	NUM
ma-134	425	5	9	9	NUM
ma-134	425	6	g(−3x	g(−3x	NOUN
ma-134	425	7	)	)	PUNCT
ma-134	425	8	,	,	PUNCT
ma-134	425	9	x	x	PUNCT
ma-134	425	10	∈	∈	NOUN
ma-134	425	11	x	x	X
ma-134	425	12	and	and	CCONJ
ma-134	425	13	‖f	‖f	ADP
ma-134	425	14	(	(	PUNCT
ma-134	425	15	x)−	x)−	PROPN
ma-134	425	16	g(x)‖	g(x)‖	PROPN
ma-134	425	17	6	6	NUM
ma-134	425	18	3ε	3ε	NUM
ma-134	425	19	8	8	NUM
ma-134	425	20	x	x	SYM
ma-134	425	21	∈	∈	PROPN
ma-134	425	22	x.in	x.in	X
ma-134	425	23	particular	particular	ADJ
ma-134	425	24	,	,	PUNCT
ma-134	425	25	if	if	SCONJ
ma-134	425	26	x	x	PRON
ma-134	425	27	is	be	AUX
ma-134	425	28	abelian	abelian	ADJ
ma-134	425	29	,	,	PUNCT
ma-134	425	30	then	then	ADV
ma-134	425	31	g	g	PROPN
ma-134	425	32	is	be	AUX
ma-134	425	33	a	a	DET
ma-134	425	34	solution	solution	NOUN
ma-134	425	35	of	of	ADP
ma-134	425	36	the	the	DET
ma-134	425	37	equation	equation	NOUN
ma-134	425	38	in	in	ADP
ma-134	425	39	the	the	DET
ma-134	425	40	following	follow	VERB
ma-134	425	41	f	f	X
ma-134	425	42	(	(	PUNCT
ma-134	425	43	x	x	PROPN
ma-134	425	44	+	+	NUM
ma-134	425	45	y	y	PROPN
ma-134	425	46	+	+	PROPN
ma-134	425	47	z	z	NOUN
ma-134	425	48	)	)	PUNCT
ma-134	426	1	+	+	NOUN
ma-134	426	2	f	f	X
ma-134	426	3	(	(	PUNCT
ma-134	426	4	x	x	X
ma-134	426	5	−	−	PROPN
ma-134	426	6	y	y	PROPN
ma-134	426	7	+	+	PROPN
ma-134	426	8	z	z	NOUN
ma-134	426	9	)	)	PUNCT
ma-134	427	1	+	+	NOUN
ma-134	427	2	f	f	X
ma-134	427	3	(	(	PUNCT
ma-134	427	4	x	x	X
ma-134	427	5	+	+	NUM
ma-134	427	6	y	y	PROPN
ma-134	427	7	−	−	PROPN
ma-134	427	8	z	z	NOUN
ma-134	427	9	)	)	PUNCT
ma-134	428	1	+	+	CCONJ
ma-134	428	2	f	f	X
ma-134	428	3	(	(	PUNCT
ma-134	428	4	x	x	SYM
ma-134	428	5	−	−	PROPN
ma-134	428	6	y	y	PROPN
ma-134	428	7	−	−	PROPN
ma-134	428	8	z	z	NOUN
ma-134	428	9	)	)	PUNCT
ma-134	428	10	=	=	SYM
ma-134	428	11	4f	4f	NUM
ma-134	428	12	(	(	PUNCT
ma-134	428	13	x	x	X
ma-134	428	14	)	)	PUNCT
ma-134	429	1	+	+	NUM
ma-134	429	2	4f	4f	NUM
ma-134	429	3	(	(	PUNCT
ma-134	429	4	y	y	NOUN
ma-134	429	5	)	)	PUNCT
ma-134	430	1	+	+	CCONJ
ma-134	430	2	4f	4f	NUM
ma-134	430	3	(	(	PUNCT
ma-134	430	4	y	y	NOUN
ma-134	430	5	)	)	PUNCT
ma-134	430	6	,	,	PUNCT
ma-134	430	7	(	(	PUNCT
ma-134	430	8	4.4	4.4	NUM
ma-134	430	9	)	)	PUNCT
ma-134	430	10	for	for	ADP
ma-134	430	11	all	all	DET
ma-134	430	12	x	x	NOUN
ma-134	430	13	,	,	PUNCT
ma-134	430	14	y	y	PROPN
ma-134	430	15	∈	∈	PROPN
ma-134	430	16	x	x	X
ma-134	430	17	.	.	PUNCT
ma-134	431	1	proof	proof	NOUN
ma-134	431	2	.	.	PUNCT
ma-134	432	1	from	from	ADP
ma-134	432	2	(	(	PUNCT
ma-134	432	3	x	x	NOUN
ma-134	432	4	,	,	PUNCT
ma-134	432	5	x,−x	x,−x	NUM
ma-134	432	6	)	)	PUNCT
ma-134	432	7	in	in	ADP
ma-134	432	8	(	(	PUNCT
ma-134	432	9	4.3	4.3	NUM
ma-134	432	10	)	)	PUNCT
ma-134	432	11	,	,	PUNCT
ma-134	432	12	we	we	PRON
ma-134	432	13	obtain	obtain	VERB
ma-134	432	14	‖6f	‖6f	PROPN
ma-134	432	15	(	(	PUNCT
ma-134	432	16	x	x	X
ma-134	432	17	)	)	PUNCT
ma-134	433	1	+	+	NUM
ma-134	433	2	3f	3f	PROPN
ma-134	433	3	(	(	PUNCT
ma-134	433	4	−x)−	−x)−	PROPN
ma-134	433	5	f	f	PROPN
ma-134	433	6	(	(	PUNCT
ma-134	433	7	3x)‖	3x)‖	NUM
ma-134	433	8	6	6	NUM
ma-134	433	9	ε	ε	PROPN
ma-134	433	10	,	,	PUNCT
ma-134	433	11	x	x	SYM
ma-134	433	12	∈	∈	NOUN
ma-134	433	13	x.	x.	NOUN
ma-134	433	14	replacing	replace	VERB
ma-134	433	15	x	x	PUNCT
ma-134	433	16	by	by	ADP
ma-134	433	17	−x	−x	NOUN
ma-134	433	18	in	in	ADP
ma-134	433	19	the	the	DET
ma-134	433	20	above	above	ADJ
ma-134	433	21	inequality	inequality	NOUN
ma-134	433	22	we	we	PRON
ma-134	433	23	obtain	obtain	VERB
ma-134	433	24	‖6f	‖6f	PROPN
ma-134	433	25	(	(	PUNCT
ma-134	433	26	−x	−x	NOUN
ma-134	433	27	)	)	PUNCT
ma-134	433	28	+	+	CCONJ
ma-134	433	29	3f	3f	PROPN
ma-134	433	30	(	(	PUNCT
ma-134	433	31	x)−	x)−	PROPN
ma-134	433	32	f	f	PROPN
ma-134	433	33	(	(	PUNCT
ma-134	433	34	−3x)‖	−3x)‖	PROPN
ma-134	433	35	6	6	NUM
ma-134	433	36	ε	ε	PROPN
ma-134	433	37	,	,	PUNCT
ma-134	433	38	x	x	SYM
ma-134	433	39	∈	∈	ADJ
ma-134	433	40	x.	x.	NOUN
ma-134	433	41	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	433	42	eur	eur	PROPN
ma-134	433	43	.	.	PUNCT
ma-134	434	1	j.	j.	PROPN
ma-134	434	2	math	math	PROPN
ma-134	434	3	.	.	PUNCT
ma-134	435	1	anal	anal	PROPN
ma-134	435	2	.	.	PUNCT
ma-134	436	1	10.28924	10.28924	NUM
ma-134	436	2	/	/	SYM
ma-134	436	3	ada	ada	PROPN
ma-134	436	4	/	/	SYM
ma-134	436	5	ma.3.7	ma.3.7	NOUN
ma-134	436	6	16consequently	16consequently	ADV
ma-134	436	7	,	,	PUNCT
ma-134	436	8	combining	combine	VERB
ma-134	436	9	the	the	DET
ma-134	436	10	above	above	ADJ
ma-134	436	11	two	two	NUM
ma-134	436	12	inequalities	inequality	NOUN
ma-134	436	13	yield	yield	VERB
ma-134	436	14	that	that	SCONJ
ma-134	436	15	‖9f	‖9f	PROPN
ma-134	436	16	(	(	PUNCT
ma-134	436	17	x	x	X
ma-134	436	18	)	)	PUNCT
ma-134	437	1	+	+	NUM
ma-134	437	2	f	f	X
ma-134	437	3	(	(	PUNCT
ma-134	437	4	−3x)−	−3x)−	PROPN
ma-134	437	5	2f	2f	NUM
ma-134	437	6	(	(	PUNCT
ma-134	437	7	3x)‖	3x)‖	NUM
ma-134	437	8	6	6	NUM
ma-134	437	9	3ε	3ε	NUM
ma-134	437	10	,	,	PUNCT
ma-134	437	11	x	x	PUNCT
ma-134	437	12	∈	∈	NOUN
ma-134	437	13	x.	x.	NOUN
ma-134	437	14	by	by	ADP
ma-134	437	15	using	use	VERB
ma-134	437	16	the	the	DET
ma-134	437	17	results	result	NOUN
ma-134	437	18	second	second	ADJ
ma-134	437	19	part	part	NOUN
ma-134	437	20	of	of	ADP
ma-134	437	21	theorem	theorem	NOUN
ma-134	437	22	4.1	4.1	NUM
ma-134	437	23	,	,	PUNCT
ma-134	437	24	a	a	DET
ma-134	437	25	computation	computation	NOUN
ma-134	437	26	is	be	AUX
ma-134	437	27	to	to	PART
ma-134	437	28	prove	prove	VERB
ma-134	437	29	that	that	PRON
ma-134	437	30	un	un	PROPN
ma-134	437	31	:	:	PUNCT
ma-134	437	32	=	=	SYM
ma-134	437	33	3n	3n	NOUN
ma-134	437	34	+	+	CCONJ
ma-134	437	35	1	1	NUM
ma-134	437	36	2	2	NUM
ma-134	437	37	·	·	PUNCT
ma-134	437	38	9n	9n	NOUN
ma-134	437	39	,	,	PUNCT
ma-134	437	40	vn	vn	X
ma-134	437	41	:	:	PUNCT
ma-134	437	42	=	=	SYM
ma-134	437	43	1−	1−	NUM
ma-134	437	44	3n	3n	NUM
ma-134	437	45	2	2	NUM
ma-134	437	46	·	·	PUNCT
ma-134	437	47	9n	9n	NOUN
ma-134	437	48	,	,	PUNCT
ma-134	437	49	n	n	PROPN
ma-134	437	50	∈	∈	PROPN
ma-134	437	51	n.	n.	NOUN
ma-134	437	52	and	and	CCONJ
ma-134	437	53	the	the	DET
ma-134	437	54	convergent	convergent	NOUN
ma-134	437	55	series	series	NOUN
ma-134	437	56	can	can	AUX
ma-134	437	57	be	be	AUX
ma-134	437	58	described	describe	VERB
ma-134	437	59	as	as	ADP
ma-134	437	60	∞∑	∞∑	NUM
ma-134	437	61	n=0	n=0	PUNCT
ma-134	438	1	[	[	X
ma-134	438	2	|un|	|un|	NOUN
ma-134	438	3	δ	δ	PROPN
ma-134	438	4	(	(	PUNCT
ma-134	438	5	en(x	en(x	X
ma-134	438	6	)	)	PUNCT
ma-134	438	7	)	)	PUNCT
ma-134	438	8	+	+	CCONJ
ma-134	438	9	|vn|δ	|vn|δ	NOUN
ma-134	438	10	(	(	PUNCT
ma-134	438	11	−en(x	−en(x	NOUN
ma-134	438	12	)	)	PUNCT
ma-134	438	13	)	)	PUNCT
ma-134	438	14	]	]	PUNCT
ma-134	439	1	=	=	PUNCT
ma-134	439	2	3ε	3ε	NUM
ma-134	439	3	8	8	NUM
ma-134	439	4	.	.	PUNCT
ma-134	440	1	and	and	CCONJ
ma-134	440	2	we	we	PRON
ma-134	440	3	show	show	VERB
ma-134	440	4	that	that	SCONJ
ma-134	440	5	if	if	SCONJ
ma-134	440	6	x	x	PRON
ma-134	440	7	is	be	AUX
ma-134	440	8	commutative	commutative	ADJ
ma-134	440	9	,	,	PUNCT
ma-134	440	10	by	by	ADP
ma-134	440	11	using	use	VERB
ma-134	440	12	(	(	PUNCT
ma-134	440	13	x	x	NOUN
ma-134	440	14	,	,	PUNCT
ma-134	440	15	y	y	PROPN
ma-134	440	16	,	,	PUNCT
ma-134	440	17	z	z	PROPN
ma-134	440	18	)	)	PUNCT
ma-134	440	19	=	=	SYM
ma-134	440	20	(	(	PUNCT
ma-134	440	21	3nx	3nx	NOUN
ma-134	440	22	,	,	PUNCT
ma-134	440	23	3ny	3ny	ADJ
ma-134	440	24	,	,	PUNCT
ma-134	440	25	3nz	3nz	ADJ
ma-134	440	26	)	)	PUNCT
ma-134	440	27	,	,	PUNCT
ma-134	440	28	then	then	ADV
ma-134	440	29	‖un[f	‖un[f	PRON
ma-134	440	30	(	(	PUNCT
ma-134	440	31	3n(x	3n(x	NUM
ma-134	441	1	+	+	CCONJ
ma-134	441	2	y	y	PROPN
ma-134	441	3	+	+	PROPN
ma-134	441	4	z	z	NOUN
ma-134	441	5	)	)	PUNCT
ma-134	441	6	)	)	PUNCT
ma-134	442	1	+	+	CCONJ
ma-134	443	1	f	f	X
ma-134	443	2	(	(	PUNCT
ma-134	443	3	3n(x	3n(x	NUM
ma-134	443	4	−	−	NOUN
ma-134	444	1	y	y	PROPN
ma-134	445	1	+	+	PROPN
ma-134	446	1	z	z	NOUN
ma-134	446	2	)	)	PUNCT
ma-134	446	3	)	)	PUNCT
ma-134	447	1	+	+	CCONJ
ma-134	447	2	f	f	X
ma-134	447	3	(	(	PUNCT
ma-134	447	4	3n(x	3n(x	NUM
ma-134	447	5	+	+	CCONJ
ma-134	447	6	y	y	PROPN
ma-134	447	7	−	−	PROPN
ma-134	447	8	z	z	NOUN
ma-134	447	9	)	)	PUNCT
ma-134	447	10	)	)	PUNCT
ma-134	448	1	+	+	CCONJ
ma-134	449	1	f	f	X
ma-134	449	2	(	(	PUNCT
ma-134	449	3	3n(x	3n(x	NUM
ma-134	449	4	−	−	NOUN
ma-134	450	1	y	y	PROPN
ma-134	451	1	−	−	PROPN
ma-134	452	1	z	z	NOUN
ma-134	452	2	)	)	PUNCT
ma-134	452	3	)	)	PUNCT
ma-134	453	1	−	−	ADP
ma-134	453	2	4f	4f	NUM
ma-134	453	3	(	(	PUNCT
ma-134	453	4	3nx)−	3nx)−	NUM
ma-134	453	5	4f	4f	NUM
ma-134	453	6	(	(	PUNCT
ma-134	453	7	3ny)−	3ny)−	NUM
ma-134	453	8	4f	4f	NUM
ma-134	453	9	(	(	PUNCT
ma-134	453	10	3ny	3ny	NOUN
ma-134	453	11	)	)	PUNCT
ma-134	453	12	]	]	PUNCT
ma-134	454	1	+	+	CCONJ
ma-134	454	2	vn[f	vn[f	PROPN
ma-134	454	3	(	(	PUNCT
ma-134	454	4	3n(x	3n(x	NUM
ma-134	455	1	+	+	CCONJ
ma-134	455	2	y	y	PROPN
ma-134	455	3	+	+	PROPN
ma-134	455	4	z	z	NOUN
ma-134	455	5	)	)	PUNCT
ma-134	455	6	)	)	PUNCT
ma-134	456	1	+	+	CCONJ
ma-134	457	1	f	f	X
ma-134	457	2	(	(	PUNCT
ma-134	457	3	3n(x	3n(x	NUM
ma-134	457	4	−	−	NOUN
ma-134	458	1	y	y	PROPN
ma-134	459	1	+	+	PROPN
ma-134	460	1	z	z	NOUN
ma-134	460	2	)	)	PUNCT
ma-134	460	3	)	)	PUNCT
ma-134	461	1	+	+	CCONJ
ma-134	461	2	f	f	X
ma-134	461	3	(	(	PUNCT
ma-134	461	4	3n(x	3n(x	NUM
ma-134	461	5	+	+	CCONJ
ma-134	461	6	y	y	PROPN
ma-134	461	7	−	−	PROPN
ma-134	461	8	z	z	NOUN
ma-134	461	9	)	)	PUNCT
ma-134	461	10	)	)	PUNCT
ma-134	462	1	+	+	CCONJ
ma-134	462	2	f	f	X
ma-134	462	3	(	(	PUNCT
ma-134	462	4	3n(x	3n(x	NUM
ma-134	462	5	−	−	PROPN
ma-134	462	6	y	y	PROPN
ma-134	462	7	−	−	PROPN
ma-134	462	8	z))−	z))−	NUM
ma-134	462	9	4f	4f	NOUN
ma-134	462	10	(	(	PUNCT
ma-134	462	11	3nx)−	3nx)−	NUM
ma-134	462	12	4f	4f	NUM
ma-134	462	13	(	(	PUNCT
ma-134	462	14	3ny)−	3ny)−	NUM
ma-134	462	15	4f	4f	NUM
ma-134	462	16	(	(	PUNCT
ma-134	462	17	3ny)]‖	3ny)]‖	PROPN
ma-134	462	18	6	6	NUM
ma-134	462	19	ε	ε	PROPN
ma-134	462	20	9nwhich	9nwhich	NUM
ma-134	462	21	we	we	PRON
ma-134	462	22	achieve	achieve	VERB
ma-134	462	23	our	our	PRON
ma-134	462	24	result	result	NOUN
ma-134	462	25	(	(	PUNCT
ma-134	462	26	3.13	3.13	NUM
ma-134	462	27	)	)	PUNCT
ma-134	462	28	by	by	ADP
ma-134	462	29	letting	let	VERB
ma-134	462	30	n	n	X
ma-134	462	31	→∞.	→∞.	X
ma-134	462	32	�	�	PROPN
ma-134	462	33	theorem	theorem	VERB
ma-134	462	34	4.3	4.3	NUM
ma-134	462	35	suppose	suppose	VERB
ma-134	462	36	that	that	SCONJ
ma-134	462	37	x	x	PRON
ma-134	462	38	is	be	AUX
ma-134	462	39	a	a	DET
ma-134	462	40	group	group	NOUN
ma-134	462	41	,	,	PUNCT
ma-134	462	42	and	and	CCONJ
ma-134	462	43	(	(	PUNCT
ma-134	462	44	y	y	PROPN
ma-134	462	45	,	,	PUNCT
ma-134	462	46	‖	‖	PROPN
ma-134	462	47	·	·	PUNCT
ma-134	462	48	‖	‖	NUM
ma-134	462	49	)	)	PUNCT
ma-134	462	50	is	be	AUX
ma-134	462	51	a	a	DET
ma-134	462	52	banach	banach	NOUN
ma-134	462	53	space	space	NOUN
ma-134	462	54	and	and	CCONJ
ma-134	462	55	let	let	VERB
ma-134	462	56	the	the	DET
ma-134	462	57	mapping	mapping	NOUN
ma-134	462	58	f	f	X
ma-134	462	59	:	:	PUNCT
ma-134	462	60	x	x	X
ma-134	462	61	→	→	SYM
ma-134	462	62	y	y	PROPN
ma-134	462	63	satisfy	satisfy	VERB
ma-134	462	64	the	the	DET
ma-134	462	65	inequality	inequality	NOUN
ma-134	462	66	for	for	ADP
ma-134	462	67	all	all	DET
ma-134	462	68	x	x	NOUN
ma-134	462	69	,	,	PUNCT
ma-134	462	70	y	y	PROPN
ma-134	462	71	∈	∈	PROPN
ma-134	462	72	x	x	X
ma-134	462	73	and	and	CCONJ
ma-134	462	74	some	some	DET
ma-134	462	75	ε	ε	PROPN
ma-134	462	76	>	>	X
ma-134	462	77	0	0	NUM
ma-134	463	1	‖f	‖f	PRON
ma-134	463	2	(	(	PUNCT
ma-134	463	3	x	x	X
ma-134	463	4	+	+	NUM
ma-134	463	5	y	y	NOUN
ma-134	463	6	)	)	PUNCT
ma-134	464	1	+	+	NOUN
ma-134	464	2	f	f	X
ma-134	464	3	(	(	PUNCT
ma-134	464	4	x	x	X
ma-134	464	5	−	−	PUNCT
ma-134	464	6	y)−	y)−	PROPN
ma-134	464	7	2f	2f	NOUN
ma-134	464	8	(	(	PUNCT
ma-134	464	9	x)−	x)−	PROPN
ma-134	464	10	f	f	PROPN
ma-134	464	11	(	(	PUNCT
ma-134	464	12	y)−	y)−	PROPN
ma-134	464	13	f	f	X
ma-134	464	14	(	(	PUNCT
ma-134	464	15	−y)‖	−y)‖	PROPN
ma-134	464	16	6	6	NUM
ma-134	464	17	ε	ε	PROPN
ma-134	464	18	,	,	PUNCT
ma-134	464	19	x	x	PRON
ma-134	464	20	,	,	PUNCT
ma-134	464	21	y	y	PROPN
ma-134	464	22	∈	∈	PROPN
ma-134	464	23	x.	x.	NOUN
ma-134	464	24	(	(	PUNCT
ma-134	464	25	4.5	4.5	NUM
ma-134	464	26	)	)	PUNCT
ma-134	464	27	then	then	ADV
ma-134	464	28	there	there	PRON
ma-134	464	29	has	have	VERB
ma-134	464	30	a	a	DET
ma-134	464	31	unique	unique	ADJ
ma-134	464	32	limiting	limiting	NOUN
ma-134	464	33	function	function	NOUN
ma-134	464	34	g	g	NOUN
ma-134	464	35	:	:	PUNCT
ma-134	464	36	x	x	X
ma-134	464	37	→	→	SYM
ma-134	464	38	y	y	PROPN
ma-134	464	39	fulfilling	fulfil	VERB
ma-134	464	40	g(x	g(x	NOUN
ma-134	464	41	)	)	PUNCT
ma-134	465	1	=	=	SYM
ma-134	466	1	3	3	NUM
ma-134	466	2	8	8	NUM
ma-134	466	3	g(2x)−	g(2x)−	PROPN
ma-134	466	4	1	1	NUM
ma-134	466	5	8	8	NUM
ma-134	466	6	g(−2x	g(−2x	NOUN
ma-134	466	7	)	)	PUNCT
ma-134	466	8	,	,	PUNCT
ma-134	466	9	x	x	PUNCT
ma-134	466	10	∈	∈	NOUN
ma-134	466	11	x	x	X
ma-134	466	12	and	and	CCONJ
ma-134	466	13	‖f	‖f	ADP
ma-134	466	14	(	(	PUNCT
ma-134	466	15	x)−	x)−	PROPN
ma-134	466	16	g(x)‖	g(x)‖	PROPN
ma-134	466	17	6	6	NUM
ma-134	466	18	2ε	2ε	NOUN
ma-134	466	19	3	3	NUM
ma-134	466	20	x	x	SYM
ma-134	466	21	∈	∈	PROPN
ma-134	466	22	x.in	x.in	X
ma-134	466	23	particular	particular	ADJ
ma-134	466	24	,	,	PUNCT
ma-134	466	25	if	if	SCONJ
ma-134	466	26	x	x	PRON
ma-134	466	27	is	be	AUX
ma-134	466	28	commutative	commutative	ADJ
ma-134	466	29	,	,	PUNCT
ma-134	466	30	then	then	ADV
ma-134	466	31	g	g	PROPN
ma-134	466	32	also	also	ADV
ma-134	466	33	fulfils	fulfil	VERB
ma-134	466	34	g(x	g(x	PROPN
ma-134	466	35	+	+	CCONJ
ma-134	466	36	y	y	NOUN
ma-134	466	37	)	)	PUNCT
ma-134	467	1	+	+	CCONJ
ma-134	467	2	g(x	g(x	NUM
ma-134	467	3	−	−	NOUN
ma-134	467	4	y	y	NOUN
ma-134	467	5	)	)	PUNCT
ma-134	467	6	=	=	SYM
ma-134	467	7	2g(x	2g(x	NUM
ma-134	467	8	)	)	PUNCT
ma-134	468	1	+	+	CCONJ
ma-134	469	1	g(y	g(y	NOUN
ma-134	469	2	)	)	PUNCT
ma-134	469	3	+	+	NUM
ma-134	469	4	g(−y	g(−y	NOUN
ma-134	469	5	)	)	PUNCT
ma-134	469	6	,	,	PUNCT
ma-134	469	7	x	x	X
ma-134	469	8	,	,	PUNCT
ma-134	469	9	y	y	PROPN
ma-134	469	10	∈	∈	PROPN
ma-134	469	11	x.	x.	NOUN
ma-134	469	12	proof	proof	NOUN
ma-134	469	13	.	.	PUNCT
ma-134	470	1	substituting	substitute	VERB
ma-134	470	2	in	in	ADP
ma-134	470	3	the	the	DET
ma-134	470	4	sequel	sequel	NOUN
ma-134	470	5	(	(	PUNCT
ma-134	470	6	x	x	X
ma-134	470	7	,	,	PUNCT
ma-134	470	8	x	x	X
ma-134	470	9	)	)	PUNCT
ma-134	470	10	in	in	ADP
ma-134	470	11	(	(	PUNCT
ma-134	470	12	4.5	4.5	NUM
ma-134	470	13	)	)	PUNCT
ma-134	470	14	,	,	PUNCT
ma-134	470	15	we	we	PRON
ma-134	470	16	obtain	obtain	VERB
ma-134	470	17	‖f	‖f	PRON
ma-134	470	18	(	(	PUNCT
ma-134	470	19	2x	2x	NUM
ma-134	470	20	)	)	PUNCT
ma-134	470	21	+	+	CCONJ
ma-134	470	22	f	f	X
ma-134	470	23	(	(	PUNCT
ma-134	470	24	0)−	0)−	NUM
ma-134	470	25	3f	3f	PROPN
ma-134	470	26	(	(	PUNCT
ma-134	470	27	x)−	x)−	PROPN
ma-134	470	28	f	f	PROPN
ma-134	470	29	(	(	PUNCT
ma-134	470	30	−x)‖	−x)‖	PROPN
ma-134	470	31	6	6	NUM
ma-134	470	32	ε	ε	PROPN
ma-134	470	33	,	,	PUNCT
ma-134	470	34	x	x	SYM
ma-134	470	35	∈	∈	NOUN
ma-134	470	36	x.	x.	NOUN
ma-134	470	37	(	(	PUNCT
ma-134	470	38	4.6	4.6	NUM
ma-134	470	39	)	)	PUNCT
ma-134	470	40	replacing	replace	VERB
ma-134	470	41	x	x	PUNCT
ma-134	470	42	by	by	ADP
ma-134	470	43	−x	−x	NOUN
ma-134	470	44	in	in	ADP
ma-134	470	45	(	(	PUNCT
ma-134	470	46	4.6	4.6	NUM
ma-134	470	47	)	)	PUNCT
ma-134	470	48	we	we	PRON
ma-134	470	49	have	have	VERB
ma-134	470	50	‖f	‖f	ADP
ma-134	470	51	(	(	PUNCT
ma-134	470	52	−2x	−2x	NOUN
ma-134	470	53	)	)	PUNCT
ma-134	471	1	+	+	NUM
ma-134	471	2	f	f	X
ma-134	471	3	(	(	PUNCT
ma-134	471	4	0)−	0)−	NUM
ma-134	471	5	3f	3f	PROPN
ma-134	471	6	(	(	PUNCT
ma-134	471	7	−x)−	−x)−	PROPN
ma-134	471	8	f	f	PROPN
ma-134	471	9	(	(	PUNCT
ma-134	471	10	x)‖	x)‖	PROPN
ma-134	471	11	6	6	NUM
ma-134	471	12	ε	ε	PROPN
ma-134	471	13	,	,	PUNCT
ma-134	471	14	x	x	SYM
ma-134	471	15	∈	∈	NOUN
ma-134	471	16	x.	x.	NOUN
ma-134	471	17	(	(	PUNCT
ma-134	471	18	4.7	4.7	NUM
ma-134	471	19	)	)	PUNCT
ma-134	471	20	consequently	consequently	ADV
ma-134	471	21	,	,	PUNCT
ma-134	471	22	(	(	PUNCT
ma-134	471	23	4.6	4.6	NUM
ma-134	471	24	)	)	PUNCT
ma-134	471	25	and	and	CCONJ
ma-134	471	26	(	(	PUNCT
ma-134	471	27	4.7	4.7	NUM
ma-134	471	28	)	)	PUNCT
ma-134	471	29	yield	yield	NOUN
ma-134	471	30	that	that	PRON
ma-134	471	31	‖8f	‖8f	PROPN
ma-134	471	32	(	(	PUNCT
ma-134	471	33	x	x	NOUN
ma-134	471	34	)	)	PUNCT
ma-134	472	1	+	+	NUM
ma-134	472	2	f	f	X
ma-134	472	3	(	(	PUNCT
ma-134	472	4	−2x)−	−2x)−	PROPN
ma-134	472	5	3f	3f	PROPN
ma-134	472	6	(	(	PUNCT
ma-134	472	7	2x)‖	2x)‖	PROPN
ma-134	472	8	6	6	NUM
ma-134	472	9	4ε	4ε	NOUN
ma-134	472	10	,	,	PUNCT
ma-134	472	11	x	x	SYM
ma-134	473	1	∈	∈	NOUN
ma-134	473	2	x.	x.	NOUN
ma-134	473	3	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	473	4	eur	eur	PROPN
ma-134	473	5	.	.	PUNCT
ma-134	474	1	j.	j.	PROPN
ma-134	474	2	math	math	PROPN
ma-134	474	3	.	.	PUNCT
ma-134	475	1	anal	anal	PROPN
ma-134	475	2	.	.	PUNCT
ma-134	476	1	10.28924	10.28924	NUM
ma-134	476	2	/	/	SYM
ma-134	476	3	ada	ada	NOUN
ma-134	476	4	/	/	SYM
ma-134	476	5	ma.3.7	ma.3.7	NOUN
ma-134	476	6	17by	17by	NOUN
ma-134	476	7	using	use	VERB
ma-134	476	8	the	the	DET
ma-134	476	9	results	result	NOUN
ma-134	476	10	of	of	ADP
ma-134	476	11	theorem	theorem	ADJ
ma-134	476	12	3.3	3.3	NUM
ma-134	476	13	,	,	PUNCT
ma-134	476	14	a	a	DET
ma-134	476	15	computation	computation	NOUN
ma-134	476	16	is	be	AUX
ma-134	476	17	to	to	PART
ma-134	476	18	prove	prove	VERB
ma-134	476	19	that	that	DET
ma-134	476	20	un	un	PROPN
ma-134	476	21	:	:	PUNCT
ma-134	477	1	=	=	NOUN
ma-134	477	2	2n	2n	NUM
ma-134	478	1	+	+	CCONJ
ma-134	478	2	1	1	NUM
ma-134	478	3	2	2	NUM
ma-134	478	4	·	·	PUNCT
ma-134	478	5	4n	4n	NOUN
ma-134	478	6	,	,	PUNCT
ma-134	478	7	vn	vn	X
ma-134	478	8	:	:	PUNCT
ma-134	478	9	=	=	SYM
ma-134	478	10	1−	1−	NUM
ma-134	478	11	2n	2n	NUM
ma-134	478	12	2	2	NUM
ma-134	478	13	·	·	SYM
ma-134	478	14	4n	4n	NOUN
ma-134	478	15	,	,	PUNCT
ma-134	478	16	n	n	PROPN
ma-134	478	17	∈	∈	PROPN
ma-134	478	18	n.	n.	NOUN
ma-134	478	19	and	and	CCONJ
ma-134	478	20	the	the	DET
ma-134	478	21	convergent	convergent	NOUN
ma-134	478	22	series	series	NOUN
ma-134	479	1	∞∑	∞∑	PROPN
ma-134	479	2	n=0	n=0	PUNCT
ma-134	480	1	[	[	X
ma-134	480	2	|un|	|un|	NOUN
ma-134	480	3	δ	δ	PROPN
ma-134	480	4	(	(	PUNCT
ma-134	480	5	en(x	en(x	X
ma-134	480	6	)	)	PUNCT
ma-134	480	7	)	)	PUNCT
ma-134	480	8	+	+	CCONJ
ma-134	480	9	|vn|δ	|vn|δ	NOUN
ma-134	480	10	(	(	PUNCT
ma-134	480	11	−en(x	−en(x	NOUN
ma-134	480	12	)	)	PUNCT
ma-134	480	13	)	)	PUNCT
ma-134	480	14	]	]	PUNCT
ma-134	481	1	=	=	PUNCT
ma-134	481	2	2ε	2ε	NOUN
ma-134	481	3	3	3	X
ma-134	481	4	.	.	PUNCT
ma-134	482	1	and	and	CCONJ
ma-134	482	2	we	we	PRON
ma-134	482	3	show	show	VERB
ma-134	482	4	that	that	SCONJ
ma-134	482	5	if	if	SCONJ
ma-134	482	6	x	x	PRON
ma-134	482	7	is	be	AUX
ma-134	482	8	commutative	commutative	ADJ
ma-134	482	9	,	,	PUNCT
ma-134	482	10	by	by	ADP
ma-134	482	11	using	use	VERB
ma-134	482	12	(	(	PUNCT
ma-134	482	13	x	x	NOUN
ma-134	482	14	,	,	PUNCT
ma-134	482	15	y	y	NOUN
ma-134	482	16	)	)	PUNCT
ma-134	482	17	=	=	SYM
ma-134	482	18	(	(	PUNCT
ma-134	482	19	2nx	2nx	ADJ
ma-134	482	20	,	,	PUNCT
ma-134	482	21	2ny	2ny	ADJ
ma-134	482	22	)	)	PUNCT
ma-134	482	23	,	,	PUNCT
ma-134	482	24	then	then	ADV
ma-134	482	25	‖un[f	‖un[f	PRON
ma-134	482	26	(	(	PUNCT
ma-134	482	27	2nx	2nx	ADJ
ma-134	482	28	+	+	CCONJ
ma-134	482	29	2ny	2ny	ADJ
ma-134	482	30	)	)	PUNCT
ma-134	483	1	+	+	CCONJ
ma-134	483	2	f	f	X
ma-134	483	3	(	(	PUNCT
ma-134	483	4	2nx	2nx	ADJ
ma-134	483	5	−	−	ADP
ma-134	483	6	2ny)−	2ny)−	NUM
ma-134	483	7	2f	2f	NUM
ma-134	483	8	(	(	PUNCT
ma-134	483	9	2nx)−	2nx)−	PROPN
ma-134	483	10	f	f	X
ma-134	483	11	(	(	PUNCT
ma-134	483	12	2ny)−	2ny)−	NUM
ma-134	483	13	f	f	X
ma-134	483	14	(	(	PUNCT
ma-134	483	15	−2ny	−2ny	PROPN
ma-134	483	16	)	)	PUNCT
ma-134	483	17	]	]	PUNCT
ma-134	484	1	+	+	CCONJ
ma-134	484	2	vn[f	vn[f	PROPN
ma-134	484	3	(	(	PUNCT
ma-134	484	4	2nx	2nx	NOUN
ma-134	484	5	+	+	CCONJ
ma-134	484	6	2ny	2ny	ADJ
ma-134	484	7	)	)	PUNCT
ma-134	485	1	+	+	CCONJ
ma-134	485	2	f	f	X
ma-134	485	3	(	(	PUNCT
ma-134	485	4	2nx	2nx	ADJ
ma-134	485	5	−	−	ADP
ma-134	485	6	2ny)−	2ny)−	NUM
ma-134	485	7	2f	2f	NUM
ma-134	485	8	(	(	PUNCT
ma-134	485	9	2nx)−	2nx)−	PROPN
ma-134	485	10	f	f	X
ma-134	485	11	(	(	PUNCT
ma-134	485	12	2ny)−	2ny)−	NUM
ma-134	485	13	f	f	PROPN
ma-134	485	14	(	(	PUNCT
ma-134	485	15	−2ny)]‖	−2ny)]‖	PROPN
ma-134	485	16	6	6	NUM
ma-134	485	17	ε	ε	PROPN
ma-134	485	18	4nwhich	4nwhich	NUM
ma-134	485	19	we	we	PRON
ma-134	485	20	achieve	achieve	VERB
ma-134	485	21	our	our	PRON
ma-134	485	22	result	result	NOUN
ma-134	485	23	(	(	PUNCT
ma-134	485	24	∗	∗	NOUN
ma-134	485	25	)	)	PUNCT
ma-134	485	26	by	by	ADP
ma-134	485	27	letting	let	VERB
ma-134	485	28	n	n	X
ma-134	485	29	→∞.	→∞.	X
ma-134	485	30	�	�	PROPN
ma-134	485	31	if	if	SCONJ
ma-134	485	32	we	we	PRON
ma-134	485	33	can	can	AUX
ma-134	485	34	not	not	PART
ma-134	485	35	set	set	VERB
ma-134	485	36	f	f	PROPN
ma-134	485	37	(	(	PUNCT
ma-134	485	38	0	0	NUM
ma-134	485	39	)	)	PUNCT
ma-134	485	40	=	=	SYM
ma-134	485	41	0	0	NUM
ma-134	485	42	,	,	PUNCT
ma-134	485	43	then	then	ADV
ma-134	485	44	the	the	DET
ma-134	485	45	approximate	approximate	ADJ
ma-134	485	46	constat	constat	NOUN
ma-134	485	47	is	be	AUX
ma-134	485	48	56ε	56ε	NUM
ma-134	485	49	.	.	PUNCT
ma-134	486	1	acknowledgments	acknowledgment	NOUN
ma-134	486	2	the	the	DET
ma-134	486	3	authors	author	NOUN
ma-134	486	4	express	express	VERB
ma-134	486	5	their	their	PRON
ma-134	486	6	gratitude	gratitude	NOUN
ma-134	486	7	to	to	ADP
ma-134	486	8	the	the	DET
ma-134	486	9	anonymous	anonymous	ADJ
ma-134	486	10	reviewers	reviewer	NOUN
ma-134	486	11	and	and	CCONJ
ma-134	486	12	editor	editor	NOUN
ma-134	486	13	for	for	ADP
ma-134	486	14	their	their	PRON
ma-134	486	15	carefulreading	carefulreade	VERB
ma-134	486	16	the	the	DET
ma-134	486	17	manuscript	manuscript	NOUN
ma-134	486	18	and	and	CCONJ
ma-134	486	19	for	for	ADP
ma-134	486	20	many	many	ADJ
ma-134	486	21	valuable	valuable	ADJ
ma-134	486	22	remarks	remark	NOUN
ma-134	486	23	and	and	CCONJ
ma-134	486	24	suggestions	suggestion	NOUN
ma-134	486	25	.	.	PUNCT
ma-134	487	1	conflict	conflict	NOUN
ma-134	487	2	of	of	ADP
ma-134	487	3	interest	interest	NOUN
ma-134	487	4	the	the	DET
ma-134	487	5	author(s	author(s	NOUN
ma-134	487	6	)	)	PUNCT
ma-134	487	7	declare(s	declare(s	NOUN
ma-134	487	8	)	)	PUNCT
ma-134	487	9	that	that	SCONJ
ma-134	487	10	there	there	PRON
ma-134	487	11	is	be	VERB
ma-134	487	12	no	no	DET
ma-134	487	13	conflict	conflict	NOUN
ma-134	487	14	of	of	ADP
ma-134	487	15	interest	interest	NOUN
ma-134	487	16	regarding	regard	VERB
ma-134	487	17	this	this	DET
ma-134	487	18	manuscript	manuscript	NOUN
ma-134	487	19	.	.	PUNCT
ma-134	488	1	data	datum	NOUN
ma-134	488	2	availability	availability	NOUN
ma-134	488	3	data	datum	NOUN
ma-134	488	4	sharing	share	VERB
ma-134	488	5	not	not	PART
ma-134	488	6	applicable	applicable	ADJ
ma-134	488	7	to	to	ADP
ma-134	488	8	this	this	DET
ma-134	488	9	article	article	NOUN
ma-134	488	10	as	as	SCONJ
ma-134	488	11	no	no	DET
ma-134	488	12	datasets	dataset	NOUN
ma-134	488	13	were	be	AUX
ma-134	488	14	generated	generate	VERB
ma-134	488	15	or	or	CCONJ
ma-134	488	16	analysed	analyse	VERB
ma-134	488	17	duringthe	duringthe	DET
ma-134	488	18	current	current	ADJ
ma-134	488	19	study	study	NOUN
ma-134	488	20	.	.	PUNCT
ma-134	489	1	funding	funding	NOUN
ma-134	489	2	statement	statement	NOUN
ma-134	489	3	this	this	DET
ma-134	489	4	work	work	NOUN
ma-134	489	5	was	be	AUX
ma-134	489	6	supported	support	VERB
ma-134	489	7	by	by	ADP
ma-134	489	8	the	the	DET
ma-134	489	9	national	national	ADJ
ma-134	489	10	natural	natural	PROPN
ma-134	489	11	science	science	PROPN
ma-134	489	12	foundation	foundation	PROPN
ma-134	489	13	of	of	ADP
ma-134	489	14	china	china	PROPN
ma-134	489	15	(	(	PUNCT
ma-134	489	16	11971493	11971493	NUM
ma-134	489	17	)	)	PUNCT
ma-134	489	18	and(12071491	and(12071491	NOUN
ma-134	489	19	)	)	PUNCT
ma-134	489	20	.	.	PUNCT
ma-134	490	1	references	reference	NOUN
ma-134	490	2	[	[	X
ma-134	490	3	1	1	NUM
ma-134	490	4	]	]	X
ma-134	490	5	s.m	s.m	PROPN
ma-134	490	6	.	.	PROPN
ma-134	490	7	ulam	ulam	PROPN
ma-134	490	8	,	,	PUNCT
ma-134	490	9	problems	problem	NOUN
ma-134	490	10	in	in	ADP
ma-134	490	11	modern	modern	ADJ
ma-134	490	12	mathematics	mathematic	NOUN
ma-134	490	13	,	,	PUNCT
ma-134	490	14	science	science	NOUN
ma-134	490	15	editions	edition	NOUN
ma-134	490	16	john	john	PROPN
ma-134	490	17	wiley	wiley	PROPN
ma-134	490	18	and	and	CCONJ
ma-134	490	19	sons	son	NOUN
ma-134	490	20	.	.	PUNCT
ma-134	491	1	inc	inc	PROPN
ma-134	491	2	.	.	PROPN
ma-134	491	3	,	,	PUNCT
ma-134	491	4	new	new	PROPN
ma-134	491	5	york	york	PROPN
ma-134	491	6	(	(	PUNCT
ma-134	491	7	1964).[2	1964).[2	PROPN
ma-134	491	8	]	]	X
ma-134	491	9	s.m	s.m	PROPN
ma-134	491	10	.	.	PROPN
ma-134	491	11	ulam	ulam	PROPN
ma-134	491	12	,	,	PUNCT
ma-134	491	13	a	a	DET
ma-134	491	14	collection	collection	NOUN
ma-134	491	15	of	of	ADP
ma-134	491	16	mathematical	mathematical	ADJ
ma-134	491	17	problems	problem	NOUN
ma-134	491	18	,	,	PUNCT
ma-134	492	1	interscience	interscience	NOUN
ma-134	492	2	tracts	tract	NOUN
ma-134	492	3	in	in	ADP
ma-134	492	4	pure	pure	ADJ
ma-134	492	5	and	and	CCONJ
ma-134	492	6	applied	applied	ADJ
ma-134	492	7	mathematics	mathematic	NOUN
ma-134	492	8	.	.	PUNCT
ma-134	493	1	no	no	INTJ
ma-134	493	2	.	.	PUNCT
ma-134	494	1	8interscience	8interscience	NUM
ma-134	494	2	publishers	publisher	NOUN
ma-134	494	3	,	,	PUNCT
ma-134	494	4	new	new	PROPN
ma-134	494	5	york	york	PROPN
ma-134	494	6	-	-	PROPN
ma-134	494	7	london	london	PROPN
ma-134	494	8	(	(	PUNCT
ma-134	494	9	1960).[3	1960).[3	PROPN
ma-134	494	10	]	]	X
ma-134	494	11	d.h	d.h	PROPN
ma-134	494	12	.	.	PROPN
ma-134	494	13	hyers	hyer	NOUN
ma-134	494	14	,	,	PUNCT
ma-134	494	15	on	on	ADP
ma-134	494	16	the	the	DET
ma-134	494	17	stability	stability	NOUN
ma-134	494	18	of	of	ADP
ma-134	494	19	the	the	DET
ma-134	494	20	linear	linear	ADJ
ma-134	494	21	functional	functional	ADJ
ma-134	494	22	equation	equation	NOUN
ma-134	494	23	,	,	PUNCT
ma-134	494	24	proc	proc	NOUN
ma-134	494	25	.	.	PUNCT
ma-134	495	1	nat	nat	PROPN
ma-134	495	2	.	.	PUNCT
ma-134	496	1	acad	acad	PROPN
ma-134	496	2	.	.	PUNCT
ma-134	497	1	sci	sci	PROPN
ma-134	497	2	.	.	PROPN
ma-134	497	3	u.s.a	u.s.a	PROPN
ma-134	497	4	.	.	PROPN
ma-134	497	5	27	27	NUM
ma-134	497	6	(	(	PUNCT
ma-134	497	7	1941	1941	NUM
ma-134	497	8	)	)	PUNCT
ma-134	497	9	222	222	NUM
ma-134	497	10	-	-	SYM
ma-134	497	11	224	224	NUM
ma-134	497	12	.	.	PUNCT
ma-134	498	1	https://doi.org/10.1073/pnas.27.4.222.[4	https://doi.org/10.1073/pnas.27.4.222.[4	PROPN
ma-134	498	2	]	]	X
ma-134	498	3	d.h	d.h	PROPN
ma-134	498	4	.	.	PROPN
ma-134	498	5	hyers	hyers	PROPN
ma-134	498	6	,	,	PUNCT
ma-134	498	7	g.	g.	PROPN
ma-134	498	8	isac	isac	PROPN
ma-134	498	9	,	,	PUNCT
ma-134	498	10	t.m	t.m	PROPN
ma-134	498	11	.	.	PROPN
ma-134	498	12	rassias	rassias	PROPN
ma-134	498	13	,	,	PUNCT
ma-134	498	14	stability	stability	NOUN
ma-134	498	15	of	of	ADP
ma-134	498	16	functional	functional	ADJ
ma-134	498	17	equations	equation	NOUN
ma-134	498	18	in	in	ADP
ma-134	498	19	several	several	ADJ
ma-134	498	20	variables	variable	NOUN
ma-134	498	21	,	,	PUNCT
ma-134	498	22	birkhäuser	birkhäuser	PROPN
ma-134	498	23	boston	boston	PROPN
ma-134	498	24	,	,	PUNCT
ma-134	498	25	boston	boston	PROPN
ma-134	498	26	,	,	PUNCT
ma-134	498	27	ma	ma	PROPN
ma-134	498	28	,	,	PUNCT
ma-134	498	29	1998	1998	NUM
ma-134	498	30	.	.	PUNCT
ma-134	499	1	https://doi.org/10.1007/978-1-4612-1790-9	https://doi.org/10.1007/978-1-4612-1790-9	INTJ
ma-134	499	2	.	.	PUNCT
ma-134	500	1	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	NOUN
ma-134	501	1	https://doi.org/10.1073/pnas.27.4.222	https://doi.org/10.1073/pnas.27.4.222	PROPN
ma-134	501	2	https://doi.org/10.1007/978-1-4612-1790-9	https://doi.org/10.1007/978-1-4612-1790-9	PROPN
ma-134	501	3	eur	eur	NOUN
ma-134	501	4	.	.	PUNCT
ma-134	502	1	j.	j.	PROPN
ma-134	502	2	math	math	PROPN
ma-134	502	3	.	.	PUNCT
ma-134	503	1	anal	anal	PROPN
ma-134	503	2	.	.	PUNCT
ma-134	504	1	10.28924	10.28924	NUM
ma-134	504	2	/	/	SYM
ma-134	504	3	ada	ada	PROPN
ma-134	504	4	/	/	SYM
ma-134	504	5	ma.3.7	ma.3.7	NOUN
ma-134	504	6	18	18	NUM
ma-134	504	7	[	[	SYM
ma-134	504	8	5	5	NUM
ma-134	504	9	]	]	PUNCT
ma-134	504	10	m.	m.	NOUN
ma-134	504	11	zenon	zenon	NOUN
ma-134	504	12	,	,	PUNCT
ma-134	504	13	on	on	ADP
ma-134	504	14	the	the	DET
ma-134	504	15	stability	stability	NOUN
ma-134	504	16	of	of	ADP
ma-134	504	17	functional	functional	ADJ
ma-134	504	18	equations	equation	NOUN
ma-134	504	19	,	,	PUNCT
ma-134	504	20	aequat	aequat	PROPN
ma-134	504	21	.	.	PUNCT
ma-134	504	22	math	math	NOUN
ma-134	504	23	.	.	PUNCT
ma-134	505	1	77	77	NUM
ma-134	505	2	(	(	PUNCT
ma-134	505	3	2009	2009	NUM
ma-134	505	4	)	)	PUNCT
ma-134	505	5	33	33	NUM
ma-134	505	6	-	-	SYM
ma-134	505	7	88	88	NUM
ma-134	505	8	.	.	PUNCT
ma-134	506	1	https://doi.org/10.1007/	https://doi.org/10.1007/	PROPN
ma-134	506	2	s00010	s00010	PROPN
ma-134	506	3	-	-	PUNCT
ma-134	506	4	008	008	NUM
ma-134	506	5	-	-	PUNCT
ma-134	506	6	2945	2945	NUM
ma-134	506	7	-	-	PUNCT
ma-134	506	8	7.[6	7.[6	PROPN
ma-134	506	9	]	]	X
ma-134	506	10	g.l	g.l	PROPN
ma-134	506	11	.	.	PROPN
ma-134	506	12	forti	forti	PROPN
ma-134	506	13	,	,	PUNCT
ma-134	506	14	an	an	DET
ma-134	506	15	existence	existence	NOUN
ma-134	506	16	and	and	CCONJ
ma-134	506	17	stability	stability	NOUN
ma-134	506	18	theorem	theorem	VERB
ma-134	506	19	for	for	ADP
ma-134	506	20	a	a	DET
ma-134	506	21	class	class	NOUN
ma-134	506	22	of	of	ADP
ma-134	506	23	functional	functional	ADJ
ma-134	506	24	equations	equation	NOUN
ma-134	506	25	,	,	PUNCT
ma-134	506	26	stochastica	stochastica	PROPN
ma-134	506	27	.	.	PROPN
ma-134	506	28	1	1	NUM
ma-134	506	29	(	(	PUNCT
ma-134	506	30	1980	1980	NUM
ma-134	506	31	)	)	PUNCT
ma-134	506	32	23	23	NUM
ma-134	506	33	-	-	SYM
ma-134	506	34	30	30	NUM
ma-134	506	35	.	.	PUNCT
ma-134	507	1	https://eudml.org/doc/38832.[7	https://eudml.org/doc/38832.[7	ADP
ma-134	507	2	]	]	X
ma-134	507	3	g.l	g.l	PROPN
ma-134	507	4	.	.	PROPN
ma-134	507	5	forti	forti	PROPN
ma-134	507	6	,	,	PUNCT
ma-134	507	7	elementary	elementary	ADJ
ma-134	507	8	remarks	remark	NOUN
ma-134	507	9	on	on	ADP
ma-134	507	10	ulam	ulam	NOUN
ma-134	507	11	-	-	PUNCT
ma-134	507	12	hyers	hyer	NOUN
ma-134	507	13	stability	stability	NOUN
ma-134	507	14	of	of	ADP
ma-134	507	15	linear	linear	ADJ
ma-134	507	16	functional	functional	ADJ
ma-134	507	17	equations	equation	NOUN
ma-134	507	18	,	,	PUNCT
ma-134	507	19	j.	j.	PROPN
ma-134	507	20	math	math	PROPN
ma-134	507	21	.	.	PUNCT
ma-134	508	1	anal	anal	PROPN
ma-134	508	2	.	.	PUNCT
ma-134	509	1	appl	appl	PROPN
ma-134	509	2	.	.	PUNCT
ma-134	510	1	328(2007	328(2007	NUM
ma-134	510	2	)	)	PUNCT
ma-134	510	3	109	109	NUM
ma-134	510	4	-	-	SYM
ma-134	510	5	118	118	NUM
ma-134	510	6	.	.	PUNCT
ma-134	511	1	https://doi.org/10.1016/j.jmaa.2006.04.079.[8	https://doi.org/10.1016/j.jmaa.2006.04.079.[8	PROPN
ma-134	511	2	]	]	PUNCT
ma-134	511	3	j.	j.	PROPN
ma-134	511	4	sikorska	sikorska	PROPN
ma-134	511	5	,	,	PUNCT
ma-134	511	6	on	on	ADP
ma-134	511	7	a	a	DET
ma-134	511	8	direct	direct	ADJ
ma-134	511	9	method	method	NOUN
ma-134	511	10	for	for	ADP
ma-134	511	11	proving	prove	VERB
ma-134	511	12	the	the	DET
ma-134	511	13	hyers	hyer	NOUN
ma-134	511	14	-	-	PUNCT
ma-134	511	15	ulam	ulam	ADJ
ma-134	511	16	stability	stability	NOUN
ma-134	511	17	of	of	ADP
ma-134	511	18	functional	functional	ADJ
ma-134	511	19	equations	equation	NOUN
ma-134	511	20	,	,	PUNCT
ma-134	511	21	j.	j.	PROPN
ma-134	511	22	math	math	PROPN
ma-134	511	23	.	.	PUNCT
ma-134	512	1	anal	anal	PROPN
ma-134	512	2	.	.	PUNCT
ma-134	513	1	appl.372	appl.372	PROPN
ma-134	513	2	(	(	PUNCT
ma-134	513	3	2010	2010	NUM
ma-134	513	4	)	)	PUNCT
ma-134	513	5	99	99	NUM
ma-134	513	6	-	-	SYM
ma-134	513	7	109	109	NUM
ma-134	513	8	.	.	PUNCT
ma-134	514	1	https://doi.org/10.1016/j.jmaa.2010.06.056.[9	https://doi.org/10.1016/j.jmaa.2010.06.056.[9	NOUN
ma-134	514	2	]	]	PUNCT
ma-134	514	3	t.	t.	PROPN
ma-134	514	4	aoki	aoki	PROPN
ma-134	514	5	,	,	PUNCT
ma-134	514	6	on	on	ADP
ma-134	514	7	the	the	DET
ma-134	514	8	stability	stability	NOUN
ma-134	514	9	of	of	ADP
ma-134	514	10	the	the	DET
ma-134	514	11	linear	linear	ADJ
ma-134	514	12	transformation	transformation	NOUN
ma-134	514	13	in	in	ADP
ma-134	514	14	banach	banach	NOUN
ma-134	514	15	spaces	space	NOUN
ma-134	514	16	,	,	PUNCT
ma-134	514	17	j.	j.	PROPN
ma-134	514	18	math	math	PROPN
ma-134	514	19	.	.	PUNCT
ma-134	515	1	soc	soc	PROPN
ma-134	515	2	.	.	PUNCT
ma-134	516	1	japan	japan	PROPN
ma-134	516	2	.	.	PROPN
ma-134	517	1	2	2	NUM
ma-134	517	2	(	(	PUNCT
ma-134	517	3	1950	1950	NUM
ma-134	517	4	)	)	PUNCT
ma-134	517	5	64	64	NUM
ma-134	517	6	-	-	SYM
ma-134	517	7	66	66	NUM
ma-134	517	8	.	.	PUNCT
ma-134	518	1	https://doi.org/10.2969/jmsj/00210064.[10	https://doi.org/10.2969/jmsj/00210064.[10	X
ma-134	518	2	]	]	X
ma-134	518	3	g.l	g.l	PROPN
ma-134	518	4	.	.	PROPN
ma-134	518	5	forti	forti	PROPN
ma-134	518	6	,	,	PUNCT
ma-134	518	7	e.	e.	PROPN
ma-134	518	8	shulman	shulman	PROPN
ma-134	518	9	,	,	PUNCT
ma-134	518	10	a	a	DET
ma-134	518	11	comparison	comparison	NOUN
ma-134	518	12	among	among	ADP
ma-134	518	13	methods	method	NOUN
ma-134	518	14	for	for	ADP
ma-134	518	15	proving	prove	VERB
ma-134	518	16	stability	stability	NOUN
ma-134	518	17	,	,	PUNCT
ma-134	518	18	aequationes	aequatione	VERB
ma-134	518	19	math	math	NOUN
ma-134	518	20	.	.	PUNCT
ma-134	519	1	94	94	NUM
ma-134	519	2	(	(	PUNCT
ma-134	519	3	2020	2020	NUM
ma-134	519	4	)	)	PUNCT
ma-134	519	5	547	547	NUM
ma-134	519	6	-	-	SYM
ma-134	519	7	574	574	NUM
ma-134	519	8	.	.	PUNCT
ma-134	520	1	https://doi.org/10.1007/s00010-019-00679-0.[11	https://doi.org/10.1007/s00010-019-00679-0.[11	PROPN
ma-134	520	2	]	]	PUNCT
ma-134	520	3	c.	c.	PROPN
ma-134	520	4	park	park	PROPN
ma-134	520	5	,	,	PUNCT
ma-134	520	6	additive	additive	NOUN
ma-134	520	7	ρ−functional	ρ−functional	PROPN
ma-134	520	8	inequalities	inequality	NOUN
ma-134	520	9	and	and	CCONJ
ma-134	520	10	equations	equation	NOUN
ma-134	520	11	,	,	PUNCT
ma-134	520	12	j.	j.	PROPN
ma-134	520	13	math	math	PROPN
ma-134	520	14	.	.	PUNCT
ma-134	521	1	inequal	inequal	ADJ
ma-134	521	2	.	.	PUNCT
ma-134	522	1	9	9	NUM
ma-134	522	2	(	(	PUNCT
ma-134	522	3	2015	2015	NUM
ma-134	522	4	)	)	PUNCT
ma-134	522	5	17	17	NUM
ma-134	522	6	-	-	SYM
ma-134	522	7	26	26	NUM
ma-134	522	8	.	.	PUNCT
ma-134	523	1	https://doi.org/	https://doi.org/	VERB
ma-134	523	2	10.7153	10.7153	NUM
ma-134	523	3	/	/	SYM
ma-134	523	4	jmi-09	jmi-09	NOUN
ma-134	523	5	-	-	PUNCT
ma-134	523	6	02.[12	02.[12	PROPN
ma-134	523	7	]	]	X
ma-134	523	8	a.k	a.k	PROPN
ma-134	523	9	.	.	PUNCT
ma-134	524	1	hassan	hassan	PROPN
ma-134	524	2	,	,	PUNCT
ma-134	524	3	h.	h.	PROPN
ma-134	524	4	keshavarz	keshavarz	PROPN
ma-134	524	5	,	,	PUNCT
ma-134	524	6	c.	c.	PROPN
ma-134	524	7	park	park	PROPN
ma-134	524	8	,	,	PUNCT
ma-134	524	9	s.	s.	PROPN
ma-134	524	10	dong	dong	PROPN
ma-134	524	11	,	,	PUNCT
ma-134	524	12	on	on	ADP
ma-134	524	13	the	the	DET
ma-134	524	14	generalized	generalize	VERB
ma-134	524	15	hyers	hyers	PROPN
ma-134	524	16	-	-	PUNCT
ma-134	524	17	ulam	ulam	ADJ
ma-134	524	18	stability	stability	NOUN
ma-134	524	19	of	of	ADP
ma-134	524	20	quartic	quartic	ADJ
ma-134	524	21	mappings	mapping	NOUN
ma-134	524	22	innon	innon	NOUN
ma-134	524	23	-	-	PUNCT
ma-134	524	24	archimedean	archimedean	ADJ
ma-134	524	25	banach	banach	NOUN
ma-134	524	26	spaces	space	VERB
ma-134	524	27	,	,	PUNCT
ma-134	524	28	j.	j.	PROPN
ma-134	524	29	math	math	PROPN
ma-134	524	30	.	.	PUNCT
ma-134	525	1	inequal	inequal	ADJ
ma-134	525	2	.	.	PUNCT
ma-134	526	1	9	9	NUM
ma-134	526	2	(	(	PUNCT
ma-134	526	3	2015	2015	NUM
ma-134	526	4	)	)	PUNCT
ma-134	526	5	553	553	NUM
ma-134	526	6	-	-	SYM
ma-134	526	7	569	569	NUM
ma-134	526	8	.	.	PUNCT
ma-134	527	1	https://doi.org/10.7153/jmi-09-48.[13	https://doi.org/10.7153/jmi-09-48.[13	PROPN
ma-134	527	2	]	]	PUNCT
ma-134	527	3	l.	l.	PROPN
ma-134	527	4	qi	qi	PROPN
ma-134	527	5	„	„	PROPN
ma-134	527	6	z.	z.	PROPN
ma-134	527	7	shaomo	shaomo	PROPN
ma-134	527	8	,	,	PUNCT
ma-134	527	9	l.	l.	PROPN
ma-134	527	10	yongjin	yongjin	PROPN
ma-134	527	11	,	,	PUNCT
ma-134	527	12	additive	additive	ADJ
ma-134	527	13	double	double	ADJ
ma-134	527	14	ρfunctional	ρfunctional	ADJ
ma-134	527	15	inequalities	inequality	NOUN
ma-134	527	16	in	in	ADP
ma-134	527	17	ρhomogeneous	ρhomogeneous	ADJ
ma-134	527	18	f	f	PROPN
ma-134	527	19	-	-	PUNCT
ma-134	527	20	spaces	space	NOUN
ma-134	527	21	,	,	PUNCT
ma-134	527	22	j.	j.	PROPN
ma-134	527	23	math	math	PROPN
ma-134	527	24	.	.	PUNCT
ma-134	528	1	inequal.15	inequal.15	NOUN
ma-134	528	2	(	(	PUNCT
ma-134	528	3	2021	2021	NUM
ma-134	528	4	)	)	PUNCT
ma-134	528	5	605	605	NUM
ma-134	528	6	-	-	SYM
ma-134	528	7	613	613	NUM
ma-134	528	8	.	.	PUNCT
ma-134	528	9	https://doi.org/10.7153/jmi-2021-15-44.[14	https://doi.org/10.7153/jmi-2021-15-44.[14	PROPN
ma-134	528	10	]	]	X
ma-134	528	11	r.	r.	PROPN
ma-134	528	12	walter	walter	PROPN
ma-134	528	13	,	,	PUNCT
ma-134	528	14	functional	functional	ADJ
ma-134	528	15	analysis	analysis	NOUN
ma-134	528	16	.	.	PUNCT
ma-134	529	1	second	second	ADJ
ma-134	529	2	edition	edition	NOUN
ma-134	529	3	,	,	PUNCT
ma-134	529	4	international	international	ADJ
ma-134	529	5	series	series	NOUN
ma-134	529	6	in	in	ADP
ma-134	529	7	pure	pure	ADJ
ma-134	529	8	and	and	CCONJ
ma-134	529	9	applied	applied	ADJ
ma-134	529	10	mathematics	mathematic	NOUN
ma-134	529	11	,	,	PUNCT
ma-134	529	12	mcgraw	mcgraw	PROPN
ma-134	529	13	-	-	PUNCT
ma-134	529	14	hill	hill	PROPN
ma-134	529	15	,	,	PUNCT
ma-134	529	16	inc	inc	PROPN
ma-134	529	17	.	.	PROPN
ma-134	529	18	,	,	PUNCT
ma-134	529	19	new	new	PROPN
ma-134	529	20	york	york	PROPN
ma-134	529	21	,	,	PUNCT
ma-134	529	22	(	(	PUNCT
ma-134	529	23	1991).[15	1991).[15	NUM
ma-134	529	24	]	]	X
ma-134	529	25	d.	d.	PROPN
ma-134	529	26	marinescu	marinescu	PROPN
ma-134	529	27	,	,	PUNCT
ma-134	529	28	m.	m.	NOUN
ma-134	529	29	monea	monea	PROPN
ma-134	529	30	,	,	PUNCT
ma-134	529	31	m.	m.	NOUN
ma-134	529	32	opincariu	opincariu	NOUN
ma-134	529	33	,	,	PUNCT
ma-134	529	34	m.	m.	NOUN
ma-134	529	35	stroe	stroe	NOUN
ma-134	529	36	,	,	PUNCT
ma-134	529	37	some	some	DET
ma-134	529	38	equivalent	equivalent	ADJ
ma-134	529	39	characterizations	characterization	NOUN
ma-134	529	40	of	of	ADP
ma-134	529	41	inner	inner	ADJ
ma-134	529	42	product	product	NOUN
ma-134	529	43	spaces	space	VERB
ma-134	529	44	andtheir	andtheir	PRON
ma-134	529	45	consequences	consequence	NOUN
ma-134	529	46	.	.	PUNCT
ma-134	530	1	filomat	filomat	NOUN
ma-134	530	2	29	29	NUM
ma-134	530	3	(	(	PUNCT
ma-134	530	4	2015	2015	NUM
ma-134	530	5	)	)	PUNCT
ma-134	530	6	1587	1587	NUM
ma-134	530	7	-	-	SYM
ma-134	530	8	1599	1599	NUM
ma-134	530	9	.	.	PUNCT
ma-134	531	1	https://doi.org/10.2298/fil1507587m.[16	https://doi.org/10.2298/fil1507587m.[16	PROPN
ma-134	531	2	]	]	PUNCT
ma-134	531	3	m.	m.	PROPN
ma-134	531	4	frechet	frechet	PROPN
ma-134	531	5	,	,	PUNCT
ma-134	531	6	sur	sur	PROPN
ma-134	531	7	la	la	PROPN
ma-134	531	8	definition	definition	NOUN
ma-134	531	9	axiomatique	axiomatique	ADJ
ma-134	531	10	d’une	d’une	NOUN
ma-134	531	11	classe	classe	NOUN
ma-134	531	12	d’espaces	d’espace	NOUN
ma-134	531	13	vectoriels	vectoriel	NOUN
ma-134	531	14	distancies	distancie	NOUN
ma-134	531	15	applicables	applicable	NOUN
ma-134	531	16	vectoriellementsur	vectoriellementsur	PROPN
ma-134	531	17	l’espace	l’espace	PROPN
ma-134	531	18	de	de	PROPN
ma-134	531	19	hilbert	hilbert	PROPN
ma-134	531	20	,	,	PUNCT
ma-134	531	21	ann	ann	PROPN
ma-134	531	22	.	.	PROPN
ma-134	531	23	math	math	PROPN
ma-134	531	24	.	.	PUNCT
ma-134	532	1	36	36	NUM
ma-134	532	2	(	(	PUNCT
ma-134	532	3	1935	1935	NUM
ma-134	532	4	)	)	PUNCT
ma-134	532	5	705	705	NUM
ma-134	532	6	-	-	SYM
ma-134	532	7	718	718	NUM
ma-134	532	8	.	.	PUNCT
ma-134	533	1	https://doi.org/10.2307/1968652.[17	https://doi.org/10.2307/1968652.[17	PROPN
ma-134	533	2	]	]	X
ma-134	533	3	s.s	s.s	PROPN
ma-134	533	4	.	.	PROPN
ma-134	533	5	kim	kim	PROPN
ma-134	533	6	,	,	PUNCT
ma-134	533	7	stability	stability	NOUN
ma-134	533	8	of	of	ADP
ma-134	533	9	the	the	DET
ma-134	533	10	frechet	frechet	NOUN
ma-134	533	11	equation	equation	NOUN
ma-134	533	12	in	in	ADP
ma-134	533	13	quasi	quasi	ADJ
ma-134	533	14	-	-	ADJ
ma-134	533	15	banach	banach	ADJ
ma-134	533	16	spaces	space	NOUN
ma-134	533	17	,	,	PUNCT
ma-134	533	18	mathematics	mathematic	NOUN
ma-134	533	19	.	.	PUNCT
ma-134	533	20	8	8	NUM
ma-134	533	21	(	(	PUNCT
ma-134	533	22	2020	2020	NUM
ma-134	533	23	)	)	PUNCT
ma-134	533	24	490	490	NUM
ma-134	533	25	.	.	PUNCT
ma-134	534	1	https://doi	https://doi	NOUN
ma-134	534	2	.	.	PUNCT
ma-134	534	3	org/10.3390	org/10.3390	PROPN
ma-134	534	4	/	/	SYM
ma-134	534	5	math8040490.[18	math8040490.[18	PROPN
ma-134	534	6	]	]	PUNCT
ma-134	534	7	r.	r.	PROPN
ma-134	534	8	badora	badora	PROPN
ma-134	534	9	,	,	PUNCT
ma-134	534	10	j.	j.	PROPN
ma-134	534	11	brzdk	brzdk	PROPN
ma-134	534	12	,	,	PUNCT
ma-134	534	13	a	a	DET
ma-134	534	14	note	note	NOUN
ma-134	534	15	on	on	ADP
ma-134	534	16	a	a	DET
ma-134	534	17	fixed	fix	VERB
ma-134	534	18	point	point	NOUN
ma-134	534	19	theorem	theorem	NOUN
ma-134	534	20	and	and	CCONJ
ma-134	534	21	the	the	DET
ma-134	534	22	hyers	hyers	PROPN
ma-134	534	23	-	-	PUNCT
ma-134	534	24	ulam	ulam	PROPN
ma-134	534	25	stability	stability	PROPN
ma-134	534	26	,	,	PUNCT
ma-134	534	27	j.	j.	PROPN
ma-134	534	28	difference	difference	PROPN
ma-134	534	29	equ	equ	PROPN
ma-134	534	30	.	.	PUNCT
ma-134	535	1	appl	appl	PROPN
ma-134	535	2	.	.	PUNCT
ma-134	536	1	18(2012	18(2012	NUM
ma-134	536	2	)	)	PUNCT
ma-134	536	3	1115	1115	NUM
ma-134	536	4	-	-	SYM
ma-134	536	5	1119	1119	NUM
ma-134	536	6	.	.	PUNCT
ma-134	537	1	https://doi.org/10.1080/10236198.2011.559165.[19	https://doi.org/10.1080/10236198.2011.559165.[19	PROPN
ma-134	537	2	]	]	X
ma-134	537	3	j.a	j.a	PROPN
ma-134	537	4	.	.	PROPN
ma-134	537	5	baker	baker	PROPN
ma-134	537	6	,	,	PUNCT
ma-134	537	7	the	the	DET
ma-134	537	8	stability	stability	NOUN
ma-134	537	9	of	of	ADP
ma-134	537	10	certain	certain	ADJ
ma-134	537	11	functional	functional	ADJ
ma-134	537	12	equations	equation	NOUN
ma-134	537	13	.	.	PUNCT
ma-134	538	1	proc	proc	PROPN
ma-134	538	2	.	.	PUNCT
ma-134	539	1	amer	amer	PROPN
ma-134	539	2	.	.	PUNCT
ma-134	539	3	math	math	PROPN
ma-134	539	4	.	.	PUNCT
ma-134	540	1	soc	soc	PROPN
ma-134	540	2	.	.	PUNCT
ma-134	541	1	3	3	NUM
ma-134	541	2	(	(	PUNCT
ma-134	541	3	1991	1991	NUM
ma-134	541	4	)	)	PUNCT
ma-134	541	5	729	729	NUM
ma-134	541	6	-	-	SYM
ma-134	541	7	732	732	NUM
ma-134	541	8	.	.	PUNCT
ma-134	542	1	https://doi	https://doi	PROPN
ma-134	542	2	.	.	PUNCT
ma-134	542	3	org/10.1090	org/10.1090	NOUN
ma-134	542	4	/	/	SYM
ma-134	542	5	s0002	s0002	NOUN
ma-134	542	6	-	-	PUNCT
ma-134	542	7	9939	9939	NUM
ma-134	542	8	-	-	PUNCT
ma-134	542	9	1991	1991	NUM
ma-134	542	10	-	-	PUNCT
ma-134	542	11	1052568	1052568	NUM
ma-134	542	12	-	-	PUNCT
ma-134	542	13	7.[20	7.[20	PROPN
ma-134	542	14	]	]	PUNCT
ma-134	542	15	l.	l.	PROPN
ma-134	542	16	câdariu	câdariu	PROPN
ma-134	542	17	,	,	PUNCT
ma-134	542	18	v.	v.	PROPN
ma-134	542	19	radu	radu	PROPN
ma-134	542	20	,	,	PUNCT
ma-134	542	21	on	on	ADP
ma-134	542	22	the	the	DET
ma-134	542	23	stability	stability	NOUN
ma-134	542	24	of	of	ADP
ma-134	542	25	the	the	DET
ma-134	542	26	cauchy	cauchy	ADJ
ma-134	542	27	functional	functional	ADJ
ma-134	542	28	equation	equation	NOUN
ma-134	542	29	:	:	PUNCT
ma-134	542	30	a	a	DET
ma-134	542	31	fixed	fix	VERB
ma-134	542	32	point	point	NOUN
ma-134	542	33	approach	approach	NOUN
ma-134	542	34	,	,	PUNCT
ma-134	542	35	grazer	grazer	NOUN
ma-134	542	36	math	math	NOUN
ma-134	542	37	.	.	PUNCT
ma-134	543	1	ber.346	ber.346	PROPN
ma-134	543	2	(	(	PUNCT
ma-134	543	3	2004	2004	NUM
ma-134	543	4	)	)	PUNCT
ma-134	544	1	43–52.[21	43–52.[21	PUNCT
ma-134	544	2	]	]	X
ma-134	545	1	y.j	y.j	PROPN
ma-134	545	2	.	.	PUNCT
ma-134	545	3	cho	cho	PROPN
ma-134	545	4	,	,	PUNCT
ma-134	545	5	r.	r.	PROPN
ma-134	545	6	saadati	saadati	PROPN
ma-134	545	7	,	,	PUNCT
ma-134	545	8	j.	j.	PROPN
ma-134	545	9	vahidi	vahidi	PROPN
ma-134	545	10	,	,	PUNCT
ma-134	545	11	approximation	approximation	NOUN
ma-134	545	12	of	of	ADP
ma-134	545	13	homomorphisms	homomorphism	NOUN
ma-134	545	14	and	and	CCONJ
ma-134	545	15	derivations	derivation	NOUN
ma-134	545	16	on	on	ADP
ma-134	545	17	non	non	ADJ
ma-134	545	18	-	-	ADJ
ma-134	545	19	archimedean	archimedean	ADJ
ma-134	545	20	lie	lie	NOUN
ma-134	545	21	c*-algebrasvia	c*-algebrasvia	ADP
ma-134	545	22	fixed	fix	VERB
ma-134	545	23	point	point	NOUN
ma-134	545	24	method	method	NOUN
ma-134	545	25	,	,	PUNCT
ma-134	545	26	discrete	discrete	ADJ
ma-134	545	27	dyn	dyn	NOUN
ma-134	545	28	.	.	PUNCT
ma-134	546	1	nat	nat	PROPN
ma-134	546	2	.	.	PUNCT
ma-134	547	1	soc	soc	PROPN
ma-134	547	2	.	.	PUNCT
ma-134	548	1	2012	2012	NUM
ma-134	548	2	(	(	PUNCT
ma-134	548	3	2012	2012	NUM
ma-134	548	4	)	)	PUNCT
ma-134	548	5	373904	373904	NUM
ma-134	548	6	.	.	PUNCT
ma-134	549	1	https://doi.org/10.1155/2012/373904.[22	https://doi.org/10.1155/2012/373904.[22	NOUN
ma-134	549	2	]	]	X
ma-134	549	3	h.	h.	PROPN
ma-134	549	4	khodaei	khodaei	PROPN
ma-134	549	5	,	,	PUNCT
ma-134	549	6	m.	m.	PROPN
ma-134	549	7	eshaghi	eshaghi	PROPN
ma-134	549	8	gordji	gordji	PROPN
ma-134	549	9	,	,	PUNCT
ma-134	549	10	s.s	s.s	PROPN
ma-134	549	11	.	.	PROPN
ma-134	549	12	kim	kim	PROPN
ma-134	549	13	,	,	PUNCT
ma-134	549	14	y.j	y.j	PROPN
ma-134	549	15	.	.	PUNCT
ma-134	549	16	cho	cho	PROPN
ma-134	549	17	,	,	PUNCT
ma-134	549	18	approximation	approximation	NOUN
ma-134	549	19	of	of	ADP
ma-134	549	20	radical	radical	ADJ
ma-134	549	21	functional	functional	ADJ
ma-134	549	22	equations	equation	NOUN
ma-134	549	23	related	relate	VERB
ma-134	549	24	to	to	PART
ma-134	549	25	quadraticand	quadraticand	VERB
ma-134	549	26	quartic	quartic	ADJ
ma-134	549	27	mappings	mapping	NOUN
ma-134	549	28	,	,	PUNCT
ma-134	549	29	j.	j.	PROPN
ma-134	549	30	math	math	PROPN
ma-134	549	31	.	.	PUNCT
ma-134	550	1	anal	anal	PROPN
ma-134	550	2	.	.	PUNCT
ma-134	550	3	appl	appl	PROPN
ma-134	550	4	.	.	PUNCT
ma-134	551	1	395	395	NUM
ma-134	551	2	(	(	PUNCT
ma-134	551	3	2012	2012	NUM
ma-134	551	4	)	)	PUNCT
ma-134	551	5	284–297	284–297	NUM
ma-134	551	6	.	.	PUNCT
ma-134	551	7	https://doi.org/10.1016/j.jmaa.2012.04	https://doi.org/10.1016/j.jmaa.2012.04	NOUN
ma-134	551	8	.	.	PUNCT
ma-134	552	1	086.[23	086.[23	ADP
ma-134	552	2	]	]	X
ma-134	552	3	p.	p.	PROPN
ma-134	552	4	kaskasem	kaskasem	PROPN
ma-134	552	5	,	,	PUNCT
ma-134	552	6	c.	c.	PROPN
ma-134	552	7	klin	klin	PROPN
ma-134	552	8	-	-	PROPN
ma-134	552	9	eam	eam	PROPN
ma-134	552	10	,	,	PUNCT
ma-134	552	11	y.j	y.j	PROPN
ma-134	552	12	.	.	PUNCT
ma-134	552	13	cho	cho	PROPN
ma-134	552	14	,	,	PUNCT
ma-134	552	15	on	on	ADP
ma-134	552	16	the	the	DET
ma-134	552	17	stability	stability	NOUN
ma-134	552	18	of	of	ADP
ma-134	552	19	the	the	DET
ma-134	552	20	generalized	generalize	VERB
ma-134	552	21	cauchy	cauchy	PROPN
ma-134	552	22	–	–	PUNCT
ma-134	552	23	jensen	jensen	PROPN
ma-134	552	24	set	set	PROPN
ma-134	552	25	-	-	PUNCT
ma-134	552	26	valued	value	VERB
ma-134	552	27	functional	functional	ADJ
ma-134	552	28	equa	equa	NOUN
ma-134	552	29	-	-	PUNCT
ma-134	552	30	tions	tion	NOUN
ma-134	552	31	,	,	PUNCT
ma-134	552	32	j.	j.	PROPN
ma-134	552	33	fixed	fix	VERB
ma-134	552	34	point	point	PROPN
ma-134	552	35	theory	theory	NOUN
ma-134	552	36	appl	appl	PROPN
ma-134	552	37	.	.	PROPN
ma-134	553	1	20	20	NUM
ma-134	553	2	(	(	PUNCT
ma-134	553	3	2018	2018	NUM
ma-134	553	4	)	)	PUNCT
ma-134	553	5	76	76	NUM
ma-134	553	6	.	.	PUNCT
ma-134	554	1	https://doi.org/10.1007/s11784-018-0558-x.[24	https://doi.org/10.1007/s11784-018-0558-x.[24	X
ma-134	554	2	]	]	PUNCT
ma-134	555	1	s.	s.	PROPN
ma-134	555	2	kim	kim	PROPN
ma-134	555	3	sik	sik	PROPN
ma-134	555	4	,	,	PUNCT
ma-134	555	5	j.	j.	PROPN
ma-134	555	6	rassias	rassias	PROPN
ma-134	555	7	michael	michael	PROPN
ma-134	555	8	,	,	PUNCT
ma-134	555	9	n.	n.	PROPN
ma-134	555	10	hussain	hussain	PROPN
ma-134	555	11	,	,	PUNCT
ma-134	555	12	y.	y.	PROPN
ma-134	555	13	cho	cho	PROPN
ma-134	556	1	je	je	PROPN
ma-134	556	2	,	,	PUNCT
ma-134	556	3	generalized	generalize	VERB
ma-134	556	4	hyers	hyers	PROPN
ma-134	556	5	-	-	PUNCT
ma-134	556	6	ulam	ulam	PROPN
ma-134	556	7	stability	stability	NOUN
ma-134	556	8	of	of	ADP
ma-134	556	9	general	general	ADJ
ma-134	556	10	cubic	cubic	ADJ
ma-134	556	11	functionalequation	functionalequation	NOUN
ma-134	556	12	in	in	ADP
ma-134	556	13	random	random	ADJ
ma-134	556	14	normed	normed	ADJ
ma-134	556	15	spaces	space	NOUN
ma-134	556	16	,	,	PUNCT
ma-134	556	17	filomat	filomat	PROPN
ma-134	556	18	.	.	PROPN
ma-134	556	19	30	30	NUM
ma-134	556	20	(	(	PUNCT
ma-134	556	21	2016	2016	NUM
ma-134	556	22	)	)	PUNCT
ma-134	556	23	89–98	89–98	NUM
ma-134	556	24	.	.	PUNCT
ma-134	557	1	https://doi.org/10.2298/fil1601089k.[25	https://doi.org/10.2298/fil1601089k.[25	NOUN
ma-134	557	2	]	]	X
ma-134	557	3	y.w	y.w	PROPN
ma-134	557	4	.	.	PROPN
ma-134	557	5	lee	lee	PROPN
ma-134	557	6	,	,	PUNCT
ma-134	557	7	stability	stability	NOUN
ma-134	557	8	of	of	ADP
ma-134	557	9	a	a	DET
ma-134	557	10	quadratic	quadratic	ADJ
ma-134	557	11	jensen	jensen	PROPN
ma-134	557	12	type	type	NOUN
ma-134	557	13	functional	functional	ADJ
ma-134	557	14	equation	equation	NOUN
ma-134	557	15	.	.	PUNCT
ma-134	558	1	j.	j.	PROPN
ma-134	558	2	math	math	PROPN
ma-134	558	3	.	.	PUNCT
ma-134	559	1	anal	anal	PROPN
ma-134	559	2	.	.	PUNCT
ma-134	560	1	appl	appl	PROPN
ma-134	560	2	.	.	PROPN
ma-134	561	1	27	27	NUM
ma-134	561	2	(	(	PUNCT
ma-134	561	3	2002	2002	NUM
ma-134	561	4	)	)	PUNCT
ma-134	561	5	590	590	NUM
ma-134	561	6	-	-	SYM
ma-134	561	7	601	601	NUM
ma-134	561	8	.	.	PUNCT
ma-134	562	1	https	https	NOUN
ma-134	562	2	:	:	PUNCT
ma-134	562	3	//doi.org/10.1016	//doi.org/10.1016	PROPN
ma-134	562	4	/	/	SYM
ma-134	562	5	s0022	s0022	NOUN
ma-134	562	6	-	-	PUNCT
ma-134	562	7	247x(02)00093	247x(02)00093	NUM
ma-134	562	8	-	-	SYM
ma-134	562	9	8	8	NUM
ma-134	562	10	.	.	PUNCT
ma-134	563	1	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	563	2	https://doi.org/10.1007/s00010-008-2945-7	https://doi.org/10.1007/s00010-008-2945-7	NUM
ma-134	563	3	https://doi.org/10.1007/s00010-008-2945-7	https://doi.org/10.1007/s00010-008-2945-7	NUM
ma-134	563	4	https://eudml.org/doc/38832	https://eudml.org/doc/38832	VERB
ma-134	563	5	https://doi.org/10.1016/j.jmaa.2006.04.079	https://doi.org/10.1016/j.jmaa.2006.04.079	NOUN
ma-134	563	6	https://doi.org/10.1016/j.jmaa.2010.06.056	https://doi.org/10.1016/j.jmaa.2010.06.056	PROPN
ma-134	563	7	https://doi.org/10.2969/jmsj/00210064	https://doi.org/10.2969/jmsj/00210064	PROPN
ma-134	563	8	https://doi.org/10.1007/s00010-019-00679-0	https://doi.org/10.1007/s00010-019-00679-0	NUM
ma-134	563	9	https://doi.org/10.7153/jmi-09-02	https://doi.org/10.7153/jmi-09-02	PROPN
ma-134	563	10	https://doi.org/10.7153/jmi-09-02	https://doi.org/10.7153/jmi-09-02	PROPN
ma-134	563	11	https://doi.org/10.7153/jmi-09-48	https://doi.org/10.7153/jmi-09-48	X
ma-134	563	12	https://doi.org/10.7153/jmi-2021-15-44	https://doi.org/10.7153/jmi-2021-15-44	PROPN
ma-134	563	13	https://doi.org/10.2298/fil1507587	https://doi.org/10.2298/fil1507587	PROPN
ma-134	563	14	m	m	PROPN
ma-134	563	15	https://doi.org/10.2307/1968652	https://doi.org/10.2307/1968652	PROPN
ma-134	563	16	https://doi.org/10.3390/math8040490	https://doi.org/10.3390/math8040490	PROPN
ma-134	563	17	https://doi.org/10.3390/math8040490	https://doi.org/10.3390/math8040490	PROPN
ma-134	563	18	https://doi.org/10.1080/10236198.2011.559165	https://doi.org/10.1080/10236198.2011.559165	PUNCT
ma-134	564	1	https://doi.org/10.1090/s0002-9939-1991-1052568-7	https://doi.org/10.1090/s0002-9939-1991-1052568-7	PROPN
ma-134	564	2	https://doi.org/10.1090/s0002-9939-1991-1052568-7	https://doi.org/10.1090/s0002-9939-1991-1052568-7	PROPN
ma-134	564	3	https://doi.org/10.1155/2012/373904	https://doi.org/10.1155/2012/373904	PROPN
ma-134	564	4	https://doi.org/10.1016/j.jmaa.2012.04.086	https://doi.org/10.1016/j.jmaa.2012.04.086	NUM
ma-134	564	5	https://doi.org/10.1016/j.jmaa.2012.04.086	https://doi.org/10.1016/j.jmaa.2012.04.086	VERB
ma-134	564	6	https://doi.org/10.1007/s11784-018-0558-x	https://doi.org/10.1007/s11784-018-0558-x	ADJ
ma-134	564	7	https://doi.org/10.2298/fil1601089k	https://doi.org/10.2298/fil1601089k	NOUN
ma-134	564	8	https://doi.org/10.1016/s0022-247x(02)00093-8	https://doi.org/10.1016/s0022-247x(02)00093-8	VERB
ma-134	564	9	https://doi.org/10.1016/s0022-247x(02)00093-8	https://doi.org/10.1016/s0022-247x(02)00093-8	NOUN
ma-134	564	10	eur	eur	NOUN
ma-134	564	11	.	.	PUNCT
ma-134	565	1	j.	j.	PROPN
ma-134	565	2	math	math	PROPN
ma-134	565	3	.	.	PUNCT
ma-134	566	1	anal	anal	PROPN
ma-134	566	2	.	.	PUNCT
ma-134	567	1	10.28924	10.28924	NUM
ma-134	567	2	/	/	SYM
ma-134	567	3	ada	ada	PROPN
ma-134	567	4	/	/	SYM
ma-134	567	5	ma.3.7	ma.3.7	NOUN
ma-134	567	6	19	19	NUM
ma-134	567	7	[	[	SYM
ma-134	567	8	26	26	NUM
ma-134	567	9	]	]	SYM
ma-134	567	10	y.w	y.w	PROPN
ma-134	567	11	.	.	PROPN
ma-134	567	12	lee	lee	PROPN
ma-134	567	13	,	,	PUNCT
ma-134	567	14	stability	stability	NOUN
ma-134	567	15	of	of	ADP
ma-134	567	16	a	a	DET
ma-134	567	17	generalized	generalized	ADJ
ma-134	567	18	quadratic	quadratic	ADJ
ma-134	567	19	functional	functional	ADJ
ma-134	567	20	equation	equation	NOUN
ma-134	567	21	with	with	ADP
ma-134	567	22	jensen	jensen	PROPN
ma-134	567	23	type	type	PROPN
ma-134	567	24	,	,	PUNCT
ma-134	567	25	bull	bull	NOUN
ma-134	567	26	.	.	PUNCT
ma-134	568	1	korean	korean	ADJ
ma-134	568	2	math	math	PROPN
ma-134	568	3	.	.	PUNCT
ma-134	569	1	soc	soc	PROPN
ma-134	569	2	.	.	PUNCT
ma-134	570	1	42(2005	42(2005	NUM
ma-134	570	2	)	)	PUNCT
ma-134	570	3	57	57	NUM
ma-134	570	4	-	-	SYM
ma-134	570	5	73	73	NUM
ma-134	570	6	.	.	PUNCT
ma-134	571	1	https://doi.org/10.4134/bkms.2005.42.1.057.[27	https://doi.org/10.4134/bkms.2005.42.1.057.[27	X
ma-134	571	2	]	]	X
ma-134	571	3	d.	d.	PROPN
ma-134	571	4	zhang	zhang	PROPN
ma-134	571	5	,	,	PUNCT
ma-134	571	6	q.	q.	PROPN
ma-134	571	7	liu	liu	PROPN
ma-134	571	8	,	,	PUNCT
ma-134	571	9	j.m	j.m	PROPN
ma-134	571	10	.	.	PROPN
ma-134	571	11	rassias	rassias	PROPN
ma-134	571	12	,	,	PUNCT
ma-134	571	13	y.	y.	PROPN
ma-134	571	14	li	li	PROPN
ma-134	571	15	,	,	PUNCT
ma-134	571	16	the	the	DET
ma-134	571	17	stability	stability	NOUN
ma-134	571	18	of	of	ADP
ma-134	571	19	functional	functional	ADJ
ma-134	571	20	equations	equation	NOUN
ma-134	571	21	with	with	ADP
ma-134	571	22	a	a	DET
ma-134	571	23	new	new	ADJ
ma-134	571	24	direct	direct	ADJ
ma-134	571	25	method	method	NOUN
ma-134	571	26	,	,	PUNCT
ma-134	571	27	mathematics.10	mathematics.10	NOUN
ma-134	571	28	(	(	PUNCT
ma-134	571	29	2022	2022	NUM
ma-134	571	30	)	)	PUNCT
ma-134	571	31	1188	1188	NUM
ma-134	571	32	.	.	PUNCT
ma-134	572	1	https://doi.org/10.3390/math10071188	https://doi.org/10.3390/math10071188	PROPN
ma-134	572	2	.	.	PUNCT
ma-134	573	1	https://doi.org/10.28924/ada/ma.3.7	https://doi.org/10.28924/ada/ma.3.7	PROPN
ma-134	573	2	https://doi.org/10.4134/bkms.2005.42.1.057	https://doi.org/10.4134/bkms.2005.42.1.057	VERB
ma-134	573	3	https://doi.org/10.3390/math10071188	https://doi.org/10.3390/math10071188	NOUN
ma-134	573	4	1	1	NUM
ma-134	573	5	.	.	PUNCT
ma-134	574	1	introduction	introduction	NOUN
ma-134	574	2	2	2	NUM
ma-134	574	3	.	.	PUNCT
ma-134	575	1	a	a	DET
ma-134	575	2	simple	simple	ADJ
ma-134	575	3	variable	variable	NOUN
ma-134	575	4	of	of	ADP
ma-134	575	5	abstract	abstract	ADJ
ma-134	575	6	equation	equation	NOUN
ma-134	575	7	3	3	NUM
ma-134	575	8	.	.	PUNCT
ma-134	576	1	the	the	DET
ma-134	576	2	stability	stability	NOUN
ma-134	576	3	of	of	ADP
ma-134	576	4	functional	functional	ADJ
ma-134	576	5	equations	equation	NOUN
ma-134	576	6	in	in	ADP
ma-134	576	7	f	f	NOUN
ma-134	576	8	-	-	PUNCT
ma-134	576	9	space	space	NOUN
ma-134	576	10	4	4	NUM
ma-134	576	11	.	.	PUNCT
ma-134	577	1	the	the	DET
ma-134	577	2	stability	stability	NOUN
ma-134	577	3	of	of	ADP
ma-134	577	4	functional	functional	ADJ
ma-134	577	5	equations	equation	NOUN
ma-134	577	6	in	in	ADP
ma-134	577	7	banach	banach	NOUN
ma-134	577	8	space	space	NOUN
ma-134	577	9	acknowledgments	acknowledgment	NOUN
ma-134	577	10	conflict	conflict	NOUN
ma-134	577	11	of	of	ADP
ma-134	577	12	interest	interest	NOUN
ma-134	577	13	data	datum	NOUN
ma-134	577	14	availability	availability	NOUN
ma-134	577	15	funding	funding	NOUN
ma-134	577	16	statement	statement	NOUN
ma-134	577	17	references	reference	NOUN