id	sid	tid	token	lemma	pos
ma-233	1	1	2024	2024	NUM
ma-233	1	2	ada	ada	PROPN
ma-233	1	3	academica	academica	PROPN
ma-233	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-233	1	5	.	.	PUNCT
ma-233	2	1	j.	j.	PROPN
ma-233	2	2	math	math	PROPN
ma-233	2	3	.	.	PUNCT
ma-233	3	1	anal	anal	ADJ
ma-233	3	2	.	.	PUNCT
ma-233	4	1	4	4	NUM
ma-233	4	2	(	(	PUNCT
ma-233	4	3	2024	2024	NUM
ma-233	4	4	)	)	PUNCT
ma-233	4	5	11doi	11doi	NUM
ma-233	4	6	:	:	PUNCT
ma-233	4	7	10.28924	10.28924	NUM
ma-233	4	8	/	/	SYM
ma-233	4	9	ada	ada	PROPN
ma-233	4	10	/	/	SYM
ma-233	4	11	ma.4.11	ma.4.11	ADJ
ma-233	4	12	on	on	ADP
ma-233	4	13	the	the	DET
ma-233	4	14	stability	stability	NOUN
ma-233	4	15	of	of	ADP
ma-233	4	16	hyers	hyer	NOUN
ma-233	4	17	orthogonality	orthogonality	VERB
ma-233	4	18	functional	functional	ADJ
ma-233	4	19	equations	equation	NOUN
ma-233	4	20	in	in	ADP
ma-233	4	21	non	non	ADJ
ma-233	4	22	-	-	ADJ
ma-233	4	23	archimedean	archimedean	ADJ
ma-233	4	24	spaces	space	NOUN
ma-233	4	25	wenhui	wenhui	PROPN
ma-233	4	26	xu	xu	PROPN
ma-233	4	27	,	,	PUNCT
ma-233	4	28	qi	qi	PROPN
ma-233	4	29	liu	liu	PROPN
ma-233	4	30	,	,	PUNCT
ma-233	4	31	jinyu	jinyu	ADJ
ma-233	4	32	xia∗	xia∗	ADJ
ma-233	4	33	school	school	NOUN
ma-233	4	34	of	of	ADP
ma-233	4	35	mathematics	mathematic	NOUN
ma-233	4	36	and	and	CCONJ
ma-233	4	37	physics	physics	NOUN
ma-233	4	38	,	,	PUNCT
ma-233	4	39	anqing	anqe	VERB
ma-233	4	40	normal	normal	ADJ
ma-233	4	41	university	university	NOUN
ma-233	4	42	,	,	PUNCT
ma-233	4	43	anqing	anqe	VERB
ma-233	4	44	246133	246133	NUM
ma-233	4	45	,	,	PUNCT
ma-233	5	1	p.	p.	PROPN
ma-233	5	2	r.	r.	PROPN
ma-233	5	3	chinaxuwenhuiwww@163.com	chinaxuwenhuiwww@163.com	PROPN
ma-233	5	4	,	,	PUNCT
ma-233	5	5	liuq67@aqnu.edu.cn	liuq67@aqnu.edu.cn	PROPN
ma-233	5	6	,	,	PUNCT
ma-233	5	7	y23060036@stu.aqnu.edu.cn	y23060036@stu.aqnu.edu.cn	NOUN
ma-233	5	8	∗correspondence	∗correspondence	NOUN
ma-233	5	9	:	:	PUNCT
ma-233	5	10	y23060036@stu.aqnu.edu.cn	y23060036@stu.aqnu.edu.cn	NOUN
ma-233	5	11	abstract	abstract	NOUN
ma-233	5	12	.	.	PUNCT
ma-233	6	1	in	in	ADP
ma-233	6	2	this	this	DET
ma-233	6	3	paper	paper	NOUN
ma-233	6	4	,	,	PUNCT
ma-233	6	5	we	we	PRON
ma-233	6	6	investigate	investigate	VERB
ma-233	6	7	the	the	DET
ma-233	6	8	stability	stability	NOUN
ma-233	6	9	of	of	ADP
ma-233	6	10	specially	specially	ADV
ma-233	6	11	orthogonally	orthogonally	ADV
ma-233	6	12	functional	functional	ADJ
ma-233	6	13	equationsderiving	equationsderiving	NOUN
ma-233	6	14	from	from	ADP
ma-233	6	15	additive	additive	ADJ
ma-233	6	16	and	and	CCONJ
ma-233	6	17	quadratic	quadratic	ADJ
ma-233	6	18	functions	function	NOUN
ma-233	6	19	4f	4f	NUM
ma-233	6	20	(	(	PUNCT
ma-233	6	21	x	x	SYM
ma-233	6	22	+	+	NUM
ma-233	6	23	y	y	NOUN
ma-233	6	24	)	)	PUNCT
ma-233	7	1	+	+	CCONJ
ma-233	7	2	4f	4f	NUM
ma-233	7	3	(	(	PUNCT
ma-233	7	4	x	x	SYM
ma-233	7	5	−	−	PROPN
ma-233	7	6	y	y	PROPN
ma-233	7	7	)	)	PUNCT
ma-233	7	8	+	+	CCONJ
ma-233	7	9	10f	10f	NOUN
ma-233	7	10	(	(	PUNCT
ma-233	7	11	x	x	X
ma-233	7	12	)	)	PUNCT
ma-233	7	13	+	+	NUM
ma-233	7	14	14f	14f	NUM
ma-233	7	15	(	(	PUNCT
ma-233	7	16	−x)−	−x)−	PROPN
ma-233	7	17	3f	3f	PROPN
ma-233	7	18	(	(	PUNCT
ma-233	7	19	y)−	y)−	PROPN
ma-233	7	20	3f	3f	X
ma-233	7	21	(	(	PUNCT
ma-233	7	22	−y	−y	PROPN
ma-233	7	23	)	)	PUNCT
ma-233	7	24	=	=	SYM
ma-233	7	25	f	f	PROPN
ma-233	7	26	(	(	PUNCT
ma-233	7	27	2x	2x	NUM
ma-233	7	28	+	+	CCONJ
ma-233	7	29	y	y	X
ma-233	7	30	)	)	PUNCT
ma-233	8	1	+	+	CCONJ
ma-233	8	2	f	f	X
ma-233	8	3	(	(	PUNCT
ma-233	8	4	2x	2x	NUM
ma-233	8	5	−	−	PROPN
ma-233	8	6	y	y	NOUN
ma-233	8	7	)	)	PUNCT
ma-233	8	8	and	and	CCONJ
ma-233	8	9	f	f	PROPN
ma-233	8	10	(	(	PUNCT
ma-233	8	11	x	x	PROPN
ma-233	8	12	+	+	NUM
ma-233	8	13	y	y	PROPN
ma-233	8	14	+	+	CCONJ
ma-233	8	15	z	z	NOUN
ma-233	8	16	2	2	NUM
ma-233	8	17	)	)	PUNCT
ma-233	9	1	+	+	CCONJ
ma-233	9	2	f	f	X
ma-233	9	3	(	(	PUNCT
ma-233	9	4	x	x	X
ma-233	9	5	+	+	NUM
ma-233	9	6	y	y	PROPN
ma-233	9	7	−	−	PROPN
ma-233	9	8	z	z	NOUN
ma-233	9	9	2	2	NUM
ma-233	9	10	)	)	PUNCT
ma-233	10	1	+	+	CCONJ
ma-233	10	2	f	f	X
ma-233	10	3	(	(	PUNCT
ma-233	10	4	x	x	SYM
ma-233	10	5	−	−	PROPN
ma-233	10	6	y	y	PROPN
ma-233	10	7	+	+	CCONJ
ma-233	10	8	z	z	NOUN
ma-233	10	9	2	2	NUM
ma-233	10	10	)	)	PUNCT
ma-233	11	1	+	+	CCONJ
ma-233	11	2	f	f	X
ma-233	11	3	(	(	PUNCT
ma-233	11	4	y	y	PROPN
ma-233	11	5	+	+	NOUN
ma-233	11	6	z	z	NOUN
ma-233	12	1	−	−	NOUN
ma-233	12	2	x	x	SYM
ma-233	12	3	2	2	X
ma-233	12	4	)	)	PUNCT
ma-233	12	5	=	=	SYM
ma-233	12	6	f	f	X
ma-233	12	7	(	(	PUNCT
ma-233	12	8	x	x	X
ma-233	12	9	)	)	PUNCT
ma-233	13	1	+	+	NUM
ma-233	13	2	f	f	X
ma-233	13	3	(	(	PUNCT
ma-233	13	4	y	y	NOUN
ma-233	13	5	)	)	PUNCT
ma-233	14	1	+	+	NOUN
ma-233	14	2	f	f	X
ma-233	14	3	(	(	PUNCT
ma-233	14	4	z	z	NOUN
ma-233	14	5	)	)	PUNCT
ma-233	14	6	where	where	SCONJ
ma-233	14	7	f	f	PROPN
ma-233	14	8	is	be	AUX
ma-233	14	9	a	a	DET
ma-233	14	10	mapping	mapping	NOUN
ma-233	14	11	from	from	ADP
ma-233	14	12	abelian	abelian	PROPN
ma-233	14	13	group	group	NOUN
ma-233	14	14	to	to	ADP
ma-233	14	15	a	a	DET
ma-233	14	16	non	non	ADJ
ma-233	14	17	-	-	ADJ
ma-233	14	18	archimedean	archimedean	ADJ
ma-233	14	19	space	space	NOUN
ma-233	14	20	.	.	PUNCT
ma-233	15	1	by	by	ADP
ma-233	15	2	adopting	adopt	VERB
ma-233	15	3	a	a	DET
ma-233	15	4	new	new	ADJ
ma-233	15	5	method	method	NOUN
ma-233	15	6	,	,	PUNCT
ma-233	15	7	we	we	PRON
ma-233	15	8	have	have	AUX
ma-233	15	9	made	make	VERB
ma-233	15	10	an	an	DET
ma-233	15	11	attempt	attempt	NOUN
ma-233	15	12	to	to	PART
ma-233	15	13	prove	prove	VERB
ma-233	15	14	the	the	DET
ma-233	15	15	hyers	hyers	PROPN
ma-233	15	16	-	-	PUNCT
ma-233	15	17	ulam	ulam	PROPN
ma-233	15	18	stability	stability	NOUN
ma-233	15	19	in	in	ADP
ma-233	15	20	non	non	ADJ
ma-233	15	21	-	-	ADJ
ma-233	15	22	archimedean	archimedean	ADJ
ma-233	15	23	spaces	space	NOUN
ma-233	15	24	.	.	PUNCT
ma-233	16	1	1	1	X
ma-233	16	2	.	.	X
ma-233	16	3	introduction	introduction	NOUN
ma-233	16	4	and	and	CCONJ
ma-233	16	5	preliminaries	preliminary	NOUN
ma-233	16	6	the	the	DET
ma-233	16	7	stability	stability	NOUN
ma-233	16	8	problem	problem	NOUN
ma-233	16	9	of	of	ADP
ma-233	16	10	functional	functional	ADJ
ma-233	16	11	equations	equation	NOUN
ma-233	16	12	originated	originate	VERB
ma-233	16	13	from	from	ADP
ma-233	16	14	ulam	ulam	PROPN
ma-233	16	15	in	in	ADP
ma-233	16	16	1940	1940	NUM
ma-233	16	17	when	when	SCONJ
ma-233	16	18	he	he	PRON
ma-233	16	19	posedthe	posedthe	PROPN
ma-233	16	20	group	group	NOUN
ma-233	16	21	homomorphism	homomorphism	PROPN
ma-233	16	22	problem	problem	NOUN
ma-233	16	23	"	"	PUNCT
ma-233	16	24	given	give	VERB
ma-233	16	25	an	an	DET
ma-233	16	26	approximately	approximately	ADV
ma-233	16	27	linear	linear	ADJ
ma-233	16	28	mapping	mapping	NOUN
ma-233	16	29	f	f	NOUN
ma-233	16	30	,	,	PUNCT
ma-233	16	31	when	when	SCONJ
ma-233	16	32	does	do	AUX
ma-233	16	33	a	a	DET
ma-233	16	34	linearmapping	linearmappe	VERB
ma-233	16	35	t	t	NOUN
ma-233	16	36	exist	exist	VERB
ma-233	16	37	that	that	PRON
ma-233	16	38	approximates	approximate	VERB
ma-233	16	39	f	f	PROPN
ma-233	16	40	?	?	PUNCT
ma-233	16	41	"	"	PUNCT
ma-233	16	42	in	in	ADP
ma-233	16	43	1941	1941	NUM
ma-233	16	44	,	,	PUNCT
ma-233	16	45	hyers	hyer	NOUN
ma-233	16	46	[	[	X
ma-233	16	47	1	1	X
ma-233	16	48	]	]	PUNCT
ma-233	16	49	explored	explore	VERB
ma-233	16	50	the	the	DET
ma-233	16	51	scenario	scenario	NOUN
ma-233	16	52	of	of	ADP
ma-233	16	53	approximatelyadditive	approximatelyadditive	ADJ
ma-233	16	54	mapping	mapping	NOUN
ma-233	16	55	f	f	NOUN
ma-233	16	56	:	:	PUNCT
ma-233	16	57	x	x	X
ma-233	16	58	→	→	PUNCT
ma-233	16	59	y	y	PROPN
ma-233	16	60	where	where	SCONJ
ma-233	16	61	x	x	PRON
ma-233	16	62	and	and	CCONJ
ma-233	16	63	y	y	PROPN
ma-233	16	64	are	be	AUX
ma-233	16	65	banach	banach	NOUN
ma-233	16	66	spaces	space	NOUN
ma-233	16	67	and	and	CCONJ
ma-233	16	68	f	f	NOUN
ma-233	16	69	satisfies	satisfie	NOUN
ma-233	16	70	‖f	‖f	PRON
ma-233	16	71	(	(	PUNCT
ma-233	16	72	x	x	X
ma-233	17	1	+	+	PUNCT
ma-233	17	2	y)−	y)−	PROPN
ma-233	17	3	f	f	NOUN
ma-233	17	4	(	(	PUNCT
ma-233	17	5	x)−	x)−	PROPN
ma-233	17	6	f	f	PROPN
ma-233	17	7	(	(	PUNCT
ma-233	17	8	y)‖	y)‖	PROPN
ma-233	17	9	6	6	NUM
ma-233	17	10	ε	ε	PROPN
ma-233	17	11	for	for	ADP
ma-233	17	12	all	all	DET
ma-233	17	13	x	x	NOUN
ma-233	17	14	,	,	PUNCT
ma-233	17	15	y	y	PROPN
ma-233	17	16	∈	∈	PROPN
ma-233	17	17	x	x	X
ma-233	17	18	.	.	PUNCT
ma-233	18	1	then	then	ADV
ma-233	18	2	there	there	PRON
ma-233	18	3	is	be	VERB
ma-233	18	4	a	a	DET
ma-233	18	5	unique	unique	ADJ
ma-233	18	6	mapping	mapping	NOUN
ma-233	18	7	additive	additive	ADJ
ma-233	18	8	l	l	NOUN
ma-233	18	9	:	:	PUNCT
ma-233	18	10	x	x	X
ma-233	18	11	→	→	SYM
ma-233	18	12	y	y	PROPN
ma-233	18	13	satisfying	satisfy	VERB
ma-233	18	14	‖f	‖f	ADP
ma-233	18	15	(	(	PUNCT
ma-233	18	16	x)−	x)−	PROPN
ma-233	18	17	l(x)‖	l(x)‖	PROPN
ma-233	18	18	6	6	NUM
ma-233	18	19	ε	ε	PROPN
ma-233	18	20	with	with	ADP
ma-233	18	21	the	the	DET
ma-233	18	22	limit	limit	NOUN
ma-233	18	23	l(x	l(x	PROPN
ma-233	18	24	)	)	PUNCT
ma-233	18	25	=	=	SYM
ma-233	18	26	lim	lim	PROPN
ma-233	18	27	n→∞	n→∞	X
ma-233	18	28	f	f	PROPN
ma-233	18	29	(	(	PUNCT
ma-233	18	30	2nx	2nx	ADJ
ma-233	18	31	)	)	PUNCT
ma-233	18	32	2n	2n	NUM
ma-233	18	33	.	.	PUNCT
ma-233	19	1	rassias	rassias	PROPN
ma-233	20	1	[	[	X
ma-233	20	2	14	14	NUM
ma-233	20	3	]	]	PUNCT
ma-233	20	4	weakened	weaken	VERB
ma-233	20	5	the	the	DET
ma-233	20	6	bounded	bounded	ADJ
ma-233	20	7	cauchy	cauchy	NOUN
ma-233	20	8	difference	difference	NOUN
ma-233	20	9	proposed	propose	VERB
ma-233	20	10	by	by	ADP
ma-233	20	11	hyers	hyer	NOUN
ma-233	20	12	in	in	ADP
ma-233	20	13	the	the	DET
ma-233	20	14	map	map	NOUN
ma-233	20	15	and	and	CCONJ
ma-233	20	16	ex	ex	NOUN
ma-233	20	17	-	-	VERB
ma-233	20	18	tended	tend	VERB
ma-233	20	19	it	it	PRON
ma-233	20	20	to	to	ADP
ma-233	20	21	the	the	DET
ma-233	20	22	unbounded	unbounded	ADJ
ma-233	20	23	cauchy	cauchy	NOUN
ma-233	20	24	difference	difference	NOUN
ma-233	20	25	‖f	‖f	PRON
ma-233	20	26	(	(	PUNCT
ma-233	20	27	x	x	X
ma-233	21	1	+	+	PUNCT
ma-233	21	2	y)−	y)−	PROPN
ma-233	21	3	f	f	NOUN
ma-233	21	4	(	(	PUNCT
ma-233	21	5	x)−	x)−	PROPN
ma-233	21	6	f	f	PROPN
ma-233	21	7	(	(	PUNCT
ma-233	21	8	y)‖	y)‖	PROPN
ma-233	21	9	6	6	NUM
ma-233	21	10	ε(‖x‖p	ε(‖x‖p	PROPN
ma-233	21	11	+	+	CCONJ
ma-233	21	12	‖y‖p	‖y‖p	NOUN
ma-233	21	13	)	)	PUNCT
ma-233	21	14	received	receive	VERB
ma-233	21	15	:	:	PUNCT
ma-233	21	16	5	5	NUM
ma-233	21	17	mar	mar	PROPN
ma-233	21	18	2024.key	2024.key	NOUN
ma-233	21	19	words	word	NOUN
ma-233	21	20	and	and	CCONJ
ma-233	21	21	phrases	phrase	NOUN
ma-233	21	22	.	.	PUNCT
ma-233	22	1	orthogonality	orthogonality	NOUN
ma-233	22	2	;	;	PUNCT
ma-233	22	3	stability	stability	NOUN
ma-233	22	4	;	;	PUNCT
ma-233	22	5	non	non	ADJ
ma-233	22	6	-	-	ADJ
ma-233	22	7	archimedean	archimedean	ADJ
ma-233	22	8	space	space	NOUN
ma-233	22	9	;	;	PUNCT
ma-233	22	10	functional	functional	ADJ
ma-233	22	11	equations.1	equations.1	PROPN
ma-233	22	12	https://adac.ee	https://adac.ee	PROPN
ma-233	22	13	https://doi.org/10.28924/ada/ma.4.11	https://doi.org/10.28924/ada/ma.4.11	VERB
ma-233	22	14	eur	eur	NOUN
ma-233	22	15	.	.	PUNCT
ma-233	23	1	j.	j.	PROPN
ma-233	23	2	math	math	PROPN
ma-233	23	3	.	.	PUNCT
ma-233	24	1	anal	anal	PROPN
ma-233	24	2	.	.	PUNCT
ma-233	25	1	10.28924	10.28924	NUM
ma-233	25	2	/	/	SYM
ma-233	25	3	ada	ada	PROPN
ma-233	25	4	/	/	SYM
ma-233	25	5	ma.4.11	ma.4.11	PROPN
ma-233	25	6	2where	2where	NUM
ma-233	25	7	ε	ε	PROPN
ma-233	25	8	>	>	PUNCT
ma-233	25	9	0	0	PUNCT
ma-233	26	1	and	and	CCONJ
ma-233	26	2	p	p	NOUN
ma-233	26	3	∈	∈	PROPN
ma-233	27	1	[	[	X
ma-233	27	2	0	0	NUM
ma-233	27	3	,	,	PUNCT
ma-233	27	4	1	1	NUM
ma-233	27	5	)	)	PUNCT
ma-233	27	6	,	,	PUNCT
ma-233	27	7	hyers	hyer	NOUN
ma-233	27	8	’	'	PUNCT
ma-233	27	9	theorem	theorem	NOUN
ma-233	27	10	was	be	AUX
ma-233	27	11	extended	extend	VERB
ma-233	27	12	to	to	ADP
ma-233	27	13	approximately	approximately	ADV
ma-233	27	14	linear	linear	ADJ
ma-233	27	15	maps	map	NOUN
ma-233	27	16	.	.	PUNCT
ma-233	28	1	r.	r.	PROPN
ma-233	28	2	gerand	gerand	PROPN
ma-233	28	3	j.	j.	PROPN
ma-233	28	4	sikorska	sikorska	PROPN
ma-233	29	1	[	[	X
ma-233	29	2	7	7	NUM
ma-233	29	3	]	]	PUNCT
ma-233	29	4	restricted	restrict	VERB
ma-233	29	5	the	the	DET
ma-233	29	6	conditions	condition	NOUN
ma-233	29	7	with	with	ADP
ma-233	29	8	(	(	PUNCT
ma-233	29	9	x	x	NOUN
ma-233	29	10	,	,	PUNCT
ma-233	29	11	y	y	NOUN
ma-233	29	12	)	)	PUNCT
ma-233	29	13	=	=	SYM
ma-233	29	14	0	0	PUNCT
ma-233	29	15	and	and	CCONJ
ma-233	29	16	investigated	investigate	VERB
ma-233	29	17	the	the	DET
ma-233	29	18	stability	stability	NOUN
ma-233	29	19	of	of	ADP
ma-233	29	20	thecauchy	thecauchy	ADJ
ma-233	29	21	functional	functional	ADJ
ma-233	29	22	f	f	X
ma-233	29	23	(	(	PUNCT
ma-233	29	24	x	x	PROPN
ma-233	29	25	+	+	NUM
ma-233	29	26	y	y	NOUN
ma-233	29	27	)	)	PUNCT
ma-233	30	1	=	=	SYM
ma-233	30	2	f	f	X
ma-233	30	3	(	(	PUNCT
ma-233	30	4	x	x	X
ma-233	30	5	)	)	PUNCT
ma-233	31	1	+	+	NUM
ma-233	31	2	f	f	X
ma-233	31	3	(	(	PUNCT
ma-233	31	4	y	y	NOUN
ma-233	31	5	)	)	PUNCT
ma-233	31	6	(	(	PUNCT
ma-233	31	7	1.1	1.1	NUM
ma-233	31	8	)	)	PUNCT
ma-233	31	9	of	of	ADP
ma-233	31	10	course	course	NOUN
ma-233	31	11	it	it	PRON
ma-233	31	12	is	be	AUX
ma-233	31	13	easy	easy	ADJ
ma-233	31	14	to	to	PART
ma-233	31	15	spot	spot	VERB
ma-233	31	16	that	that	SCONJ
ma-233	31	17	the	the	DET
ma-233	31	18	function	function	NOUN
ma-233	31	19	f	f	X
ma-233	31	20	(	(	PUNCT
ma-233	31	21	x	x	X
ma-233	31	22	)	)	PUNCT
ma-233	31	23	=	=	SYM
ma-233	32	1	‖x‖2	‖x‖2	ADJ
ma-233	32	2	satisfies	satisfy	VERB
ma-233	32	3	the	the	DET
ma-233	32	4	functional	functional	ADJ
ma-233	32	5	equations	equation	NOUN
ma-233	32	6	(	(	PUNCT
ma-233	32	7	1.1)by	1.1)by	NUM
ma-233	32	8	the	the	DET
ma-233	32	9	pythagorean	pythagorean	PROPN
ma-233	32	10	theorem	theorem	PROPN
ma-233	32	11	.	.	PUNCT
ma-233	33	1	they	they	PRON
ma-233	33	2	founded	found	VERB
ma-233	33	3	that	that	SCONJ
ma-233	33	4	there	there	PRON
ma-233	33	5	exists	exist	VERB
ma-233	33	6	a	a	DET
ma-233	33	7	orthogonality	orthogonality	NOUN
ma-233	33	8	additive	additive	ADJ
ma-233	33	9	mapping	mapping	NOUN
ma-233	33	10	g	g	NOUN
ma-233	33	11	:	:	PUNCT
ma-233	33	12	x	x	X
ma-233	33	13	→	→	PUNCT
ma-233	33	14	y	y	PRON
ma-233	33	15	such	such	ADJ
ma-233	33	16	that	that	SCONJ
ma-233	33	17	‖f	‖f	ADP
ma-233	33	18	(	(	PUNCT
ma-233	33	19	x)−	x)−	PROPN
ma-233	33	20	g(x)‖	g(x)‖	PROPN
ma-233	33	21	6	6	NUM
ma-233	33	22	16	16	NUM
ma-233	33	23	3	3	NUM
ma-233	33	24	ε	ε	NOUN
ma-233	33	25	for	for	ADP
ma-233	33	26	all	all	DET
ma-233	33	27	x	x	SYM
ma-233	33	28	∈	∈	PROPN
ma-233	33	29	x	x	PUNCT
ma-233	33	30	with	with	SCONJ
ma-233	33	31	restriction	restriction	NOUN
ma-233	33	32	on	on	ADP
ma-233	33	33	definition	definition	NOUN
ma-233	33	34	domain	domain	NOUN
ma-233	33	35	(	(	PUNCT
ma-233	33	36	1.1	1.1	NUM
ma-233	33	37	)	)	PUNCT
ma-233	33	38	was	be	AUX
ma-233	33	39	denoted	denote	VERB
ma-233	33	40	as	as	ADP
ma-233	33	41	a	a	DET
ma-233	33	42	additive	additive	NOUN
ma-233	33	43	equations.similarly	equations.similarly	ADV
ma-233	33	44	,	,	PUNCT
ma-233	33	45	the	the	DET
ma-233	33	46	equation	equation	NOUN
ma-233	33	47	was	be	AUX
ma-233	33	48	called	call	VERB
ma-233	33	49	as	as	ADP
ma-233	33	50	a	a	DET
ma-233	33	51	quadratic	quadratic	ADJ
ma-233	33	52	equation	equation	NOUN
ma-233	33	53	which	which	PRON
ma-233	33	54	satisfies	satisfy	VERB
ma-233	33	55	f	f	X
ma-233	33	56	(	(	PUNCT
ma-233	33	57	x	x	PROPN
ma-233	34	1	+	+	NUM
ma-233	34	2	y	y	NOUN
ma-233	34	3	)	)	PUNCT
ma-233	35	1	+	+	NOUN
ma-233	35	2	f	f	X
ma-233	35	3	(	(	PUNCT
ma-233	35	4	x	x	INTJ
ma-233	35	5	−	−	PROPN
ma-233	35	6	y	y	NOUN
ma-233	35	7	)	)	PUNCT
ma-233	35	8	=	=	SYM
ma-233	36	1	2f	2f	NOUN
ma-233	36	2	(	(	PUNCT
ma-233	36	3	x	x	X
ma-233	36	4	)	)	PUNCT
ma-233	37	1	+	+	NUM
ma-233	37	2	2f	2f	NUM
ma-233	37	3	(	(	PUNCT
ma-233	37	4	y	y	NOUN
ma-233	37	5	)	)	PUNCT
ma-233	37	6	.	.	PUNCT
ma-233	38	1	(	(	PUNCT
ma-233	38	2	1.2	1.2	NUM
ma-233	38	3	)	)	PUNCT
ma-233	38	4	during	during	ADP
ma-233	38	5	several	several	ADJ
ma-233	38	6	decades	decade	NOUN
ma-233	38	7	,	,	PUNCT
ma-233	38	8	mathematicians	mathematician	NOUN
ma-233	38	9	have	have	AUX
ma-233	38	10	achieved	achieve	VERB
ma-233	38	11	various	various	ADJ
ma-233	38	12	fruits	fruit	NOUN
ma-233	38	13	in	in	ADP
ma-233	38	14	studying	study	VERB
ma-233	38	15	the	the	DET
ma-233	38	16	stability	stability	NOUN
ma-233	38	17	offunctional	offunctional	ADJ
ma-233	38	18	equations	equation	NOUN
ma-233	38	19	based	base	VERB
ma-233	38	20	one	one	NUM
ma-233	38	21	these	these	DET
ma-233	38	22	two	two	NUM
ma-233	38	23	equations	equation	NOUN
ma-233	38	24	in	in	ADP
ma-233	38	25	the	the	DET
ma-233	38	26	spirit	spirit	NOUN
ma-233	38	27	of	of	ADP
ma-233	38	28	hyers-ulam-rassias.now	hyers-ulam-rassias.now	PROPN
ma-233	38	29	let	let	VERB
ma-233	38	30	us	we	PRON
ma-233	38	31	introduce	introduce	VERB
ma-233	38	32	the	the	DET
ma-233	38	33	concept	concept	NOUN
ma-233	38	34	of	of	ADP
ma-233	38	35	orthogonality	orthogonality	NOUN
ma-233	38	36	⊥	⊥	NOUN
ma-233	38	37	defined	define	VERB
ma-233	38	38	by	by	ADP
ma-233	38	39	rätz	rätz	NOUN
ma-233	39	1	[	[	X
ma-233	39	2	16	16	NUM
ma-233	39	3	]	]	PUNCT
ma-233	39	4	.	.	PUNCT
ma-233	40	1	suppose	suppose	VERB
ma-233	40	2	x	x	PRON
ma-233	40	3	is	be	AUX
ma-233	40	4	a	a	DET
ma-233	40	5	realvector	realvector	NOUN
ma-233	40	6	space	space	NOUN
ma-233	40	7	with	with	ADP
ma-233	40	8	dimx	dimx	NOUN
ma-233	40	9	>	>	X
ma-233	40	10	2	2	NUM
ma-233	40	11	and	and	CCONJ
ma-233	40	12	⊥	⊥	PROPN
ma-233	40	13	is	be	AUX
ma-233	40	14	a	a	DET
ma-233	40	15	binary	binary	ADJ
ma-233	40	16	relation	relation	NOUN
ma-233	40	17	on	on	ADP
ma-233	40	18	x	x	SYM
ma-233	40	19	are	be	AUX
ma-233	40	20	characterized	characterize	VERB
ma-233	40	21	by	by	ADP
ma-233	40	22	the	the	DET
ma-233	40	23	followingproperties:(i	followingproperties:(i	NOUN
ma-233	40	24	)	)	PUNCT
ma-233	40	25	totality	totality	NOUN
ma-233	40	26	of	of	ADP
ma-233	40	27	⊥	⊥	PROPN
ma-233	40	28	for	for	ADP
ma-233	40	29	zero	zero	NUM
ma-233	40	30	:	:	PUNCT
ma-233	40	31	x	x	PROPN
ma-233	40	32	⊥	⊥	NOUN
ma-233	40	33	0	0	NUM
ma-233	40	34	,	,	PUNCT
ma-233	40	35	0	0	NUM
ma-233	40	36	⊥	⊥	NOUN
ma-233	40	37	x	x	PUNCT
ma-233	40	38	for	for	ADP
ma-233	40	39	all	all	DET
ma-233	40	40	x	x	SYM
ma-233	40	41	∈	∈	NOUN
ma-233	40	42	x;(ii	x;(ii	PROPN
ma-233	40	43	)	)	PUNCT
ma-233	40	44	homogeneity	homogeneity	NOUN
ma-233	40	45	:	:	PUNCT
ma-233	40	46	if	if	SCONJ
ma-233	40	47	x	x	X
ma-233	40	48	,	,	PUNCT
ma-233	40	49	y	y	PROPN
ma-233	40	50	∈	∈	PROPN
ma-233	40	51	x	x	X
ma-233	40	52	,	,	PUNCT
ma-233	40	53	x	x	PROPN
ma-233	40	54	⊥	⊥	NOUN
ma-233	40	55	y	y	PROPN
ma-233	40	56	,	,	PUNCT
ma-233	40	57	then	then	ADV
ma-233	40	58	λx	λx	PROPN
ma-233	40	59	⊥	⊥	X
ma-233	40	60	µy	µy	VERB
ma-233	40	61	for	for	ADP
ma-233	40	62	all	all	DET
ma-233	40	63	λ	λ	PROPN
ma-233	40	64	,	,	PUNCT
ma-233	40	65	µ	µ	X
ma-233	40	66	∈	∈	NOUN
ma-233	40	67	r	r	NOUN
ma-233	40	68	;	;	PUNCT
ma-233	40	69	(	(	PUNCT
ma-233	40	70	iii	iii	NOUN
ma-233	40	71	)	)	PUNCT
ma-233	40	72	independence	independence	NOUN
ma-233	40	73	:	:	PUNCT
ma-233	40	74	if	if	SCONJ
ma-233	40	75	x	x	X
ma-233	40	76	,	,	PUNCT
ma-233	40	77	y	y	PROPN
ma-233	40	78	∈	∈	PROPN
ma-233	40	79	x	x	SYM
ma-233	40	80	\	\	X
ma-233	40	81	{	{	PUNCT
ma-233	40	82	0	0	NUM
ma-233	40	83	}	}	PUNCT
ma-233	40	84	,	,	PUNCT
ma-233	40	85	x	x	PROPN
ma-233	40	86	⊥	⊥	NOUN
ma-233	40	87	y	y	PROPN
ma-233	40	88	,	,	PUNCT
ma-233	40	89	if	if	SCONJ
ma-233	40	90	and	and	CCONJ
ma-233	40	91	only	only	ADV
ma-233	40	92	if	if	SCONJ
ma-233	40	93	x	x	NOUN
ma-233	40	94	,	,	PUNCT
ma-233	40	95	y	y	PROPN
ma-233	40	96	are	be	AUX
ma-233	40	97	linearly	linearly	ADV
ma-233	40	98	independent;(iv	independent;(iv	NOUN
ma-233	40	99	)	)	PUNCT
ma-233	40	100	for	for	ADP
ma-233	40	101	any	any	DET
ma-233	40	102	two	two	NUM
ma-233	40	103	-	-	PUNCT
ma-233	40	104	dimensional	dimensional	ADJ
ma-233	40	105	subspace	subspace	NOUN
ma-233	40	106	p	p	NOUN
ma-233	40	107	of	of	ADP
ma-233	40	108	x	x	X
ma-233	40	109	and	and	CCONJ
ma-233	40	110	for	for	ADP
ma-233	40	111	every	every	DET
ma-233	40	112	x	x	SYM
ma-233	40	113	∈	∈	PROPN
ma-233	40	114	p	p	NOUN
ma-233	40	115	,	,	PUNCT
ma-233	40	116	there	there	PRON
ma-233	40	117	exists	exist	VERB
ma-233	40	118	λ	λ	PROPN
ma-233	40	119	,	,	PUNCT
ma-233	40	120	y	y	PROPN
ma-233	40	121	∈	∈	PROPN
ma-233	41	1	p	p	NOUN
ma-233	41	2	such	such	ADJ
ma-233	41	3	that	that	SCONJ
ma-233	41	4	x	x	PROPN
ma-233	41	5	⊥	⊥	PROPN
ma-233	41	6	y	y	PROPN
ma-233	41	7	and	and	CCONJ
ma-233	41	8	x	x	PUNCT
ma-233	42	1	+	+	CCONJ
ma-233	42	2	y	y	PROPN
ma-233	42	3	⊥	⊥	PROPN
ma-233	42	4	λx	λx	PROPN
ma-233	43	1	−	−	PROPN
ma-233	43	2	y	y	PROPN
ma-233	43	3	.the	.the	PRON
ma-233	43	4	pair	pair	NOUN
ma-233	43	5	(	(	PUNCT
ma-233	43	6	x	x	X
ma-233	43	7	,	,	PUNCT
ma-233	43	8	⊥	⊥	NOUN
ma-233	43	9	)	)	PUNCT
ma-233	43	10	is	be	AUX
ma-233	43	11	called	call	VERB
ma-233	43	12	an	an	DET
ma-233	43	13	orthogonality	orthogonality	NOUN
ma-233	43	14	space	space	NOUN
ma-233	43	15	,	,	PUNCT
ma-233	43	16	which	which	PRON
ma-233	43	17	means	mean	VERB
ma-233	43	18	an	an	DET
ma-233	43	19	orthogonality	orthogonality	NOUN
ma-233	43	20	space	space	NOUN
ma-233	43	21	havinga	havinga	PROPN
ma-233	43	22	normed	norme	VERB
ma-233	43	23	structure	structure	NOUN
ma-233	43	24	.	.	PUNCT
ma-233	44	1	various	various	ADJ
ma-233	44	2	notions	notion	NOUN
ma-233	44	3	of	of	ADP
ma-233	44	4	othogonlity	othogonlity	NOUN
ma-233	44	5	on	on	ADP
ma-233	44	6	a	a	DET
ma-233	44	7	real	real	ADV
ma-233	44	8	normed	normed	ADJ
ma-233	44	9	space	space	NOUN
ma-233	44	10	such	such	ADJ
ma-233	44	11	as	as	ADP
ma-233	44	12	roberts	roberts	PROPN
ma-233	44	13	,	,	PUNCT
ma-233	44	14	pythagorean	pythagorean	PROPN
ma-233	44	15	,	,	PUNCT
ma-233	44	16	isosceles	isoscele	NOUN
ma-233	44	17	,	,	PUNCT
ma-233	44	18	birkhoff	birkhoff	NOUN
ma-233	44	19	-	-	PUNCT
ma-233	44	20	james	james	PROPN
ma-233	44	21	,	,	PUNCT
ma-233	44	22	carlsson	carlsson	PROPN
ma-233	44	23	,	,	PUNCT
ma-233	44	24	hermite	hermite	ADJ
ma-233	44	25	–	–	PUNCT
ma-233	44	26	hadamard	hadamard	NOUN
ma-233	44	27	(	(	PUNCT
ma-233	44	28	hh	hh	NOUN
ma-233	44	29	)	)	PUNCT
ma-233	44	30	type	type	NOUN
ma-233	44	31	orthogonalitieson	orthogonalitieson	NOUN
ma-233	44	32	the	the	DET
ma-233	44	33	basis	basis	NOUN
ma-233	44	34	of	of	ADP
ma-233	44	35	the	the	DET
ma-233	44	36	fundamental	fundamental	ADJ
ma-233	44	37	properties	property	NOUN
ma-233	44	38	.	.	PUNCT
ma-233	45	1	definition	definition	NOUN
ma-233	45	2	1.1	1.1	NUM
ma-233	45	3	.	.	PUNCT
ma-233	46	1	[	[	X
ma-233	46	2	16	16	NUM
ma-233	46	3	]	]	PUNCT
ma-233	46	4	a	a	DET
ma-233	46	5	function	function	NOUN
ma-233	46	6	‖·‖	‖·‖	PUNCT
ma-233	46	7	:x	:x	PUNCT
ma-233	47	1	→	→	PUNCT
ma-233	47	2	[	[	X
ma-233	47	3	0,∞	0,∞	NOUN
ma-233	47	4	)	)	PUNCT
ma-233	47	5	on	on	ADP
ma-233	47	6	a	a	DET
ma-233	47	7	vector	vector	NOUN
ma-233	47	8	space	space	NOUN
ma-233	47	9	over	over	ADP
ma-233	47	10	x	x	DET
ma-233	47	11	a	a	DET
ma-233	47	12	scalar	scalar	ADJ
ma-233	47	13	field	field	NOUN
ma-233	47	14	k	k	X
ma-233	47	15	with	with	ADP
ma-233	47	16	anon	anon	ADJ
ma-233	47	17	-	-	ADJ
ma-233	47	18	archimedean	archimedean	ADJ
ma-233	47	19	valuation	valuation	NOUN
ma-233	47	20	|	|	ADV
ma-233	47	21	·	·	PUNCT
ma-233	47	22	|	|	ADV
ma-233	47	23	,	,	PUNCT
ma-233	47	24	is	be	AUX
ma-233	47	25	classified	classify	VERB
ma-233	47	26	as	as	ADP
ma-233	47	27	a	a	DET
ma-233	47	28	non	non	ADJ
ma-233	47	29	-	-	ADJ
ma-233	47	30	archimedean	archimedean	ADJ
ma-233	47	31	norm	norm	NOUN
ma-233	47	32	if	if	SCONJ
ma-233	47	33	it	it	PRON
ma-233	47	34	meets	meet	VERB
ma-233	47	35	the	the	DET
ma-233	47	36	followingconditions:(i	followingconditions:(i	NOUN
ma-233	47	37	)	)	PUNCT
ma-233	47	38	nonnegativity	nonnegativity	NOUN
ma-233	47	39	:	:	PUNCT
ma-233	47	40	‖x‖	‖x‖	X
ma-233	47	41	>	>	X
ma-233	47	42	0	0	PUNCT
ma-233	47	43	and	and	CCONJ
ma-233	47	44	‖x‖	‖x‖	PROPN
ma-233	47	45	=	=	SYM
ma-233	47	46	0	0	PUNCT
ma-233	48	1	if	if	SCONJ
ma-233	48	2	and	and	CCONJ
ma-233	48	3	only	only	ADV
ma-233	48	4	if	if	SCONJ
ma-233	48	5	x	x	X
ma-233	48	6	=	=	PUNCT
ma-233	48	7	0;(ii	0;(ii	NOUN
ma-233	48	8	)	)	PUNCT
ma-233	48	9	homogeneity	homogeneity	NOUN
ma-233	48	10	:	:	PUNCT
ma-233	48	11	‖λx‖	‖λx‖	NOUN
ma-233	48	12	=	=	SYM
ma-233	48	13	|λ|	|λ|	PROPN
ma-233	48	14	‖x‖	‖x‖	PROPN
ma-233	48	15	∀λ	∀λ	NUM
ma-233	48	16	∈	∈	PROPN
ma-233	48	17	k,∀x	k,∀x	NOUN
ma-233	48	18	,	,	PUNCT
ma-233	48	19	y	y	PROPN
ma-233	48	20	∈	∈	PROPN
ma-233	48	21	x;(iii	x;(iii	NUM
ma-233	48	22	)	)	PUNCT
ma-233	48	23	the	the	DET
ma-233	48	24	strong	strong	ADJ
ma-233	48	25	triangle	triangle	NOUN
ma-233	48	26	inequality	inequality	NOUN
ma-233	48	27	‖x	‖x	NOUN
ma-233	48	28	+	+	CCONJ
ma-233	48	29	y‖	y‖	PROPN
ma-233	48	30	6	6	NUM
ma-233	48	31	max	max	PROPN
ma-233	48	32	{	{	PUNCT
ma-233	48	33	‖x‖	‖x‖	PROPN
ma-233	48	34	,	,	PUNCT
ma-233	48	35	‖y‖	‖y‖	PROPN
ma-233	48	36	}	}	PUNCT
ma-233	48	37	∀x	∀x	NUM
ma-233	48	38	,	,	PUNCT
ma-233	48	39	y	y	PROPN
ma-233	48	40	∈	∈	PROPN
ma-233	48	41	x	x	X
ma-233	48	42	then	then	ADV
ma-233	48	43	(	(	PUNCT
ma-233	48	44	x,‖·‖	x,‖·‖	PROPN
ma-233	48	45	)	)	PUNCT
ma-233	48	46	is	be	AUX
ma-233	48	47	called	call	VERB
ma-233	48	48	a	a	DET
ma-233	48	49	non	non	ADJ
ma-233	48	50	-	-	ADJ
ma-233	48	51	archimedean	archimedean	ADJ
ma-233	48	52	normed	normed	ADJ
ma-233	48	53	space	space	NOUN
ma-233	48	54	.	.	PUNCT
ma-233	49	1	https://doi.org/10.28924/ada/ma.4.11	https://doi.org/10.28924/ada/ma.4.11	PROPN
ma-233	49	2	eur	eur	PROPN
ma-233	49	3	.	.	PUNCT
ma-233	50	1	j.	j.	PROPN
ma-233	50	2	math	math	PROPN
ma-233	50	3	.	.	PUNCT
ma-233	51	1	anal	anal	PROPN
ma-233	51	2	.	.	PUNCT
ma-233	52	1	10.28924	10.28924	NUM
ma-233	52	2	/	/	SYM
ma-233	52	3	ada	ada	PROPN
ma-233	52	4	/	/	SYM
ma-233	52	5	ma.4.11	ma.4.11	ADJ
ma-233	53	1	3gordji	3gordji	NUM
ma-233	53	2	[	[	X
ma-233	53	3	9	9	NUM
ma-233	53	4	]	]	PUNCT
ma-233	53	5	investigated	investigate	VERB
ma-233	53	6	the	the	DET
ma-233	53	7	stability	stability	NOUN
ma-233	53	8	of	of	ADP
ma-233	53	9	the	the	DET
ma-233	53	10	traditionally	traditionally	ADV
ma-233	53	11	functional	functional	ADJ
ma-233	53	12	equations	equation	NOUN
ma-233	53	13	d(x	d(x	PROPN
ma-233	53	14	,	,	PUNCT
ma-233	53	15	y	y	NOUN
ma-233	53	16	)	)	PUNCT
ma-233	54	1	=	=	SYM
ma-233	54	2	f	f	X
ma-233	54	3	(	(	PUNCT
ma-233	54	4	x	x	PROPN
ma-233	54	5	+	+	PUNCT
ma-233	54	6	y)−	y)−	PROPN
ma-233	54	7	f	f	NOUN
ma-233	54	8	(	(	PUNCT
ma-233	54	9	x)−	x)−	PROPN
ma-233	54	10	f	f	PROPN
ma-233	54	11	(	(	PUNCT
ma-233	54	12	y	y	PROPN
ma-233	54	13	)	)	PUNCT
ma-233	54	14	where	where	SCONJ
ma-233	54	15	f	f	X
ma-233	54	16	:	:	PUNCT
ma-233	54	17	x	x	X
ma-233	54	18	→	→	SYM
ma-233	54	19	y	y	PROPN
ma-233	54	20	x	x	PROPN
ma-233	54	21	,	,	PUNCT
ma-233	54	22	y	y	PROPN
ma-233	54	23	are	be	AUX
ma-233	54	24	both	both	PRON
ma-233	54	25	non	non	ADJ
ma-233	54	26	-	-	ADJ
ma-233	54	27	arohimedean	arohimedean	ADJ
ma-233	54	28	banach	banach	NOUN
ma-233	54	29	spaces	space	VERB
ma-233	54	30	.	.	PUNCT
ma-233	55	1	they	they	PRON
ma-233	55	2	established	establish	VERB
ma-233	55	3	the	the	DET
ma-233	55	4	existenceof	existenceof	ADJ
ma-233	55	5	functions	function	NOUN
ma-233	55	6	ϕ,ψ	ϕ,ψ	NOUN
ma-233	55	7	:	:	PUNCT
ma-233	55	8	a×	a×	PROPN
ma-233	55	9	a→	a→	VERB
ma-233	55	10	[	[	X
ma-233	55	11	0,∞	0,∞	NOUN
ma-233	55	12	)	)	PUNCT
ma-233	55	13	such	such	ADJ
ma-233	55	14	that	that	SCONJ
ma-233	55	15	‖d(x	‖d(x	AUX
ma-233	55	16	,	,	PUNCT
ma-233	55	17	y)‖	y)‖	NOUN
ma-233	55	18	6	6	NUM
ma-233	55	19	ϕ(x	ϕ(x	NOUN
ma-233	55	20	,	,	PUNCT
ma-233	55	21	y	y	NOUN
ma-233	55	22	)	)	PUNCT
ma-233	55	23	‖f	‖f	PUNCT
ma-233	55	24	(	(	PUNCT
ma-233	55	25	xy)−	xy)−	X
ma-233	55	26	f	f	PROPN
ma-233	55	27	(	(	PUNCT
ma-233	55	28	x)f	x)f	X
ma-233	55	29	(	(	PUNCT
ma-233	55	30	y)‖	y)‖	X
ma-233	55	31	6	6	NUM
ma-233	55	32	ψ(x	ψ(x	NOUN
ma-233	55	33	,	,	PUNCT
ma-233	55	34	y	y	NOUN
ma-233	55	35	)	)	PUNCT
ma-233	55	36	for	for	ADP
ma-233	55	37	all	all	DET
ma-233	55	38	x	x	NOUN
ma-233	55	39	,	,	PUNCT
ma-233	55	40	y	y	PROPN
ma-233	55	41	∈	∈	PROPN
ma-233	55	42	x	x	X
ma-233	55	43	,	,	PUNCT
ma-233	55	44	and	and	CCONJ
ma-233	55	45	they	they	PRON
ma-233	55	46	considered	consider	VERB
ma-233	55	47	the	the	DET
ma-233	55	48	case	case	NOUN
ma-233	55	49	if	if	SCONJ
ma-233	55	50	there	there	PRON
ma-233	55	51	exists	exist	VERB
ma-233	55	52	a	a	DET
ma-233	55	53	constant	constant	ADJ
ma-233	55	54	0	0	NUM
ma-233	55	55	<	<	X
ma-233	55	56	l	l	X
ma-233	55	57	<	<	X
ma-233	55	58	1	1	NUM
ma-233	55	59	such	such	ADJ
ma-233	55	60	that	that	DET
ma-233	55	61	ϕ(2x	ϕ(2x	PROPN
ma-233	55	62	,	,	PUNCT
ma-233	55	63	2y	2y	NUM
ma-233	55	64	)	)	PUNCT
ma-233	55	65	6	6	NUM
ma-233	55	66	|2|lϕ(x	|2|lϕ(x	SYM
ma-233	55	67	,	,	PUNCT
ma-233	55	68	y	y	NOUN
ma-233	55	69	)	)	PUNCT
ma-233	55	70	ϕ(2x	ϕ(2x	PROPN
ma-233	55	71	,	,	PUNCT
ma-233	55	72	2y	2y	NUM
ma-233	55	73	)	)	PUNCT
ma-233	55	74	6	6	NUM
ma-233	55	75	|2|2lψ(x	|2|2lψ(x	NOUN
ma-233	55	76	,	,	PUNCT
ma-233	55	77	y	y	NOUN
ma-233	55	78	)	)	PUNCT
ma-233	55	79	then	then	ADV
ma-233	55	80	there	there	PRON
ma-233	55	81	exist	exist	VERB
ma-233	55	82	a	a	DET
ma-233	55	83	unique	unique	ADJ
ma-233	55	84	ring	ring	NOUN
ma-233	55	85	homomorphis	homomorphis	PRON
ma-233	55	86	h	h	NOUN
ma-233	55	87	:	:	PUNCT
ma-233	55	88	x	x	X
ma-233	55	89	→	→	PUNCT
ma-233	55	90	y	y	PRON
ma-233	55	91	such	such	ADJ
ma-233	55	92	that	that	SCONJ
ma-233	55	93	‖f	‖f	ADP
ma-233	55	94	(	(	PUNCT
ma-233	55	95	x)−h(x)‖	x)−h(x)‖	PROPN
ma-233	55	96	6	6	NUM
ma-233	55	97	1	1	NUM
ma-233	55	98	|2|(1−	|2|(1−	ADJ
ma-233	55	99	l)ϕ(x	l)ϕ(x	PROPN
ma-233	55	100	,	,	PUNCT
ma-233	55	101	x	x	X
ma-233	55	102	)	)	PUNCT
ma-233	55	103	kang	kang	PROPN
ma-233	56	1	[	[	X
ma-233	56	2	10	10	NUM
ma-233	56	3	]	]	PUNCT
ma-233	56	4	explored	explore	VERB
ma-233	56	5	the	the	DET
ma-233	56	6	stability	stability	NOUN
ma-233	56	7	of	of	ADP
ma-233	56	8	the	the	DET
ma-233	56	9	orthogonally	orthogonally	ADV
ma-233	56	10	functional	functional	ADJ
ma-233	56	11	equation(1.3	equation(1.3	NOUN
ma-233	56	12	)	)	PUNCT
ma-233	56	13	through	through	ADP
ma-233	56	14	the	the	DET
ma-233	56	15	classi	classi	NOUN
ma-233	56	16	-	-	NOUN
ma-233	56	17	fication	fication	NOUN
ma-233	56	18	of	of	ADP
ma-233	56	19	the	the	DET
ma-233	56	20	oddness	oddness	NOUN
ma-233	56	21	and	and	CCONJ
ma-233	56	22	evenness	evenness	NOUN
ma-233	56	23	of	of	ADP
ma-233	56	24	f	f	PROPN
ma-233	56	25	within	within	ADP
ma-233	56	26	the	the	DET
ma-233	56	27	same	same	ADJ
ma-233	56	28	spaces	space	NOUN
ma-233	56	29	4f	4f	NUM
ma-233	56	30	(	(	PUNCT
ma-233	56	31	x	x	SYM
ma-233	56	32	+	+	NUM
ma-233	56	33	y	y	NOUN
ma-233	56	34	)	)	PUNCT
ma-233	57	1	+	+	CCONJ
ma-233	57	2	4f	4f	NUM
ma-233	57	3	(	(	PUNCT
ma-233	57	4	x	x	SYM
ma-233	57	5	−	−	PROPN
ma-233	57	6	y	y	PROPN
ma-233	57	7	)	)	PUNCT
ma-233	57	8	+	+	CCONJ
ma-233	57	9	10f	10f	NOUN
ma-233	57	10	(	(	PUNCT
ma-233	57	11	x	x	X
ma-233	57	12	)	)	PUNCT
ma-233	57	13	+	+	NUM
ma-233	57	14	14f	14f	NUM
ma-233	57	15	(	(	PUNCT
ma-233	57	16	−x)−	−x)−	PROPN
ma-233	57	17	3f	3f	PROPN
ma-233	57	18	(	(	PUNCT
ma-233	57	19	y)−	y)−	PROPN
ma-233	57	20	3f	3f	X
ma-233	57	21	(	(	PUNCT
ma-233	57	22	−y	−y	PROPN
ma-233	57	23	)	)	PUNCT
ma-233	57	24	=	=	SYM
ma-233	57	25	f	f	PROPN
ma-233	57	26	(	(	PUNCT
ma-233	57	27	2x	2x	NUM
ma-233	57	28	+	+	CCONJ
ma-233	57	29	y	y	X
ma-233	57	30	)	)	PUNCT
ma-233	58	1	+	+	CCONJ
ma-233	58	2	f	f	X
ma-233	58	3	(	(	PUNCT
ma-233	58	4	2x	2x	NUM
ma-233	58	5	−	−	PROPN
ma-233	58	6	y	y	NOUN
ma-233	58	7	)	)	PUNCT
ma-233	58	8	(	(	PUNCT
ma-233	58	9	1.3	1.3	NUM
ma-233	58	10	)	)	PUNCT
ma-233	58	11	park	park	NOUN
ma-233	59	1	[	[	X
ma-233	59	2	12	12	NUM
ma-233	59	3	]	]	PUNCT
ma-233	59	4	investigated	investigate	VERB
ma-233	59	5	the	the	DET
ma-233	59	6	stability	stability	NOUN
ma-233	59	7	of	of	ADP
ma-233	59	8	the	the	DET
ma-233	59	9	orthogonally	orthogonally	ADV
ma-233	59	10	additive	additive	ADJ
ma-233	59	11	-	-	PUNCT
ma-233	59	12	additive	additive	ADJ
ma-233	59	13	and	and	CCONJ
ma-233	59	14	orthogonallyquadratic	orthogonallyquadratic	ADJ
ma-233	59	15	-	-	PUNCT
ma-233	59	16	quadratic	quadratic	ADJ
ma-233	59	17	functional	functional	ADJ
ma-233	59	18	equation(1.4	equation(1.4	NOUN
ma-233	59	19	)	)	PUNCT
ma-233	59	20	in	in	ADP
ma-233	59	21	non	non	ADJ
ma-233	59	22	-	-	ADJ
ma-233	59	23	archimedean	archimedean	ADJ
ma-233	59	24	orthogonality	orthogonality	NOUN
ma-233	59	25	spaces	space	NOUN
ma-233	59	26	using	use	VERB
ma-233	59	27	con	con	ADJ
ma-233	59	28	-	-	PUNCT
ma-233	59	29	ventional	ventional	ADJ
ma-233	59	30	methods	method	NOUN
ma-233	59	31	f	f	PROPN
ma-233	59	32	(	(	PUNCT
ma-233	59	33	x	x	PROPN
ma-233	60	1	+	+	NUM
ma-233	60	2	y	y	PROPN
ma-233	61	1	+	+	CCONJ
ma-233	61	2	z	z	NOUN
ma-233	61	3	2	2	NUM
ma-233	61	4	)	)	PUNCT
ma-233	62	1	+	+	CCONJ
ma-233	62	2	f	f	X
ma-233	62	3	(	(	PUNCT
ma-233	62	4	x	x	X
ma-233	62	5	+	+	NUM
ma-233	62	6	y	y	PROPN
ma-233	62	7	−	−	PROPN
ma-233	62	8	z	z	NOUN
ma-233	62	9	2	2	NUM
ma-233	62	10	)	)	PUNCT
ma-233	63	1	+	+	CCONJ
ma-233	63	2	f	f	X
ma-233	63	3	(	(	PUNCT
ma-233	63	4	x	x	SYM
ma-233	63	5	−	−	PROPN
ma-233	63	6	y	y	PROPN
ma-233	63	7	+	+	CCONJ
ma-233	63	8	z	z	NOUN
ma-233	63	9	2	2	NUM
ma-233	63	10	)	)	PUNCT
ma-233	64	1	+	+	CCONJ
ma-233	64	2	f	f	X
ma-233	64	3	(	(	PUNCT
ma-233	64	4	y	y	PROPN
ma-233	64	5	+	+	NOUN
ma-233	64	6	z	z	NOUN
ma-233	65	1	−	−	NOUN
ma-233	65	2	x	x	SYM
ma-233	65	3	2	2	X
ma-233	65	4	)	)	PUNCT
ma-233	65	5	=	=	SYM
ma-233	65	6	f	f	X
ma-233	65	7	(	(	PUNCT
ma-233	65	8	x	x	X
ma-233	65	9	)	)	PUNCT
ma-233	66	1	+	+	NUM
ma-233	66	2	f	f	X
ma-233	66	3	(	(	PUNCT
ma-233	66	4	y	y	NOUN
ma-233	66	5	)	)	PUNCT
ma-233	67	1	+	+	NOUN
ma-233	67	2	f	f	AUX
ma-233	67	3	(	(	PUNCT
ma-233	67	4	z)(1.4	z)(1.4	NOUN
ma-233	67	5	)	)	PUNCT
ma-233	67	6	drawing	draw	VERB
ma-233	67	7	inspiration	inspiration	NOUN
ma-233	67	8	from	from	ADP
ma-233	67	9	[	[	X
ma-233	67	10	14	14	NUM
ma-233	67	11	]	]	PUNCT
ma-233	67	12	,	,	PUNCT
ma-233	67	13	this	this	DET
ma-233	67	14	paper	paper	NOUN
ma-233	67	15	we	we	PRON
ma-233	67	16	explore	explore	VERB
ma-233	67	17	different	different	ADJ
ma-233	67	18	spaces	space	NOUN
ma-233	67	19	and	and	CCONJ
ma-233	67	20	employ	employ	VERB
ma-233	67	21	new	new	ADJ
ma-233	67	22	methodsto	methodsto	NOUN
ma-233	67	23	investigete	investigete	NOUN
ma-233	67	24	the	the	DET
ma-233	67	25	stability	stability	NOUN
ma-233	67	26	of	of	ADP
ma-233	67	27	the	the	DET
ma-233	67	28	aforementioned	aforementioned	ADJ
ma-233	67	29	equation(1.4	equation(1.4	NOUN
ma-233	67	30	)	)	PUNCT
ma-233	67	31	and	and	CCONJ
ma-233	67	32	(	(	PUNCT
ma-233	67	33	1.3	1.3	NUM
ma-233	67	34	)	)	PUNCT
ma-233	67	35	.	.	PUNCT
ma-233	68	1	2	2	X
ma-233	68	2	.	.	X
ma-233	68	3	stability	stability	NOUN
ma-233	68	4	of	of	ADP
ma-233	68	5	the	the	DET
ma-233	68	6	orthogonally	orthogonally	ADV
ma-233	68	7	additive	additive	ADJ
ma-233	68	8	-	-	PUNCT
ma-233	68	9	quadratic	quadratic	ADJ
ma-233	68	10	functional	functional	ADJ
ma-233	68	11	equation	equation	NOUN
ma-233	68	12	in	in	ADP
ma-233	68	13	this	this	DET
ma-233	68	14	section	section	NOUN
ma-233	68	15	,	,	PUNCT
ma-233	68	16	we	we	PRON
ma-233	68	17	will	will	AUX
ma-233	68	18	use	use	VERB
ma-233	68	19	the	the	DET
ma-233	68	20	following	follow	VERB
ma-233	68	21	symbol	symbol	NOUN
ma-233	68	22	d1f	d1f	NOUN
ma-233	68	23	(	(	PUNCT
ma-233	68	24	x	x	PROPN
ma-233	68	25	,	,	PUNCT
ma-233	68	26	y	y	NOUN
ma-233	68	27	)	)	PUNCT
ma-233	69	1	=	=	SYM
ma-233	69	2	f	f	PROPN
ma-233	69	3	(	(	PUNCT
ma-233	69	4	2x	2x	NUM
ma-233	69	5	+	+	CCONJ
ma-233	69	6	y	y	X
ma-233	69	7	)	)	PUNCT
ma-233	70	1	+	+	CCONJ
ma-233	70	2	f	f	X
ma-233	70	3	(	(	PUNCT
ma-233	70	4	2x	2x	NUM
ma-233	70	5	−	−	PROPN
ma-233	70	6	y)−	y)−	NUM
ma-233	70	7	4f	4f	NOUN
ma-233	70	8	(	(	PUNCT
ma-233	70	9	x	x	SYM
ma-233	70	10	+	+	PUNCT
ma-233	70	11	y)−	y)−	NUM
ma-233	70	12	4f	4f	NOUN
ma-233	70	13	(	(	PUNCT
ma-233	70	14	x	x	SYM
ma-233	70	15	−	−	PROPN
ma-233	70	16	y	y	PROPN
ma-233	70	17	)	)	PUNCT
ma-233	70	18	−10f	−10f	PROPN
ma-233	70	19	(	(	PUNCT
ma-233	70	20	x)−	x)−	PROPN
ma-233	70	21	14f	14f	PROPN
ma-233	70	22	(	(	PUNCT
ma-233	70	23	−x	−x	NOUN
ma-233	70	24	)	)	PUNCT
ma-233	70	25	+	+	CCONJ
ma-233	70	26	3f	3f	PROPN
ma-233	70	27	(	(	PUNCT
ma-233	70	28	y	y	NOUN
ma-233	70	29	)	)	PUNCT
ma-233	70	30	+	+	CCONJ
ma-233	70	31	3f	3f	PROPN
ma-233	70	32	(	(	PUNCT
ma-233	70	33	−y	−y	PROPN
ma-233	70	34	)	)	PUNCT
ma-233	70	35	(	(	PUNCT
ma-233	70	36	2.1	2.1	NUM
ma-233	70	37	)	)	PUNCT
ma-233	70	38	we	we	PRON
ma-233	70	39	deal	deal	VERB
ma-233	70	40	with	with	ADP
ma-233	70	41	the	the	DET
ma-233	70	42	stability	stability	NOUN
ma-233	70	43	problem	problem	NOUN
ma-233	70	44	for	for	ADP
ma-233	70	45	the	the	DET
ma-233	70	46	orthogonally	orthogonally	ADV
ma-233	70	47	additive	additive	ADJ
ma-233	70	48	-	-	PUNCT
ma-233	70	49	quartic	quartic	ADJ
ma-233	70	50	functional	functional	ADJ
ma-233	70	51	equation	equation	NOUN
ma-233	70	52	for	for	ADP
ma-233	70	53	d1f	d1f	NOUN
ma-233	70	54	(	(	PUNCT
ma-233	70	55	x	x	PROPN
ma-233	70	56	,	,	PUNCT
ma-233	70	57	y	y	NOUN
ma-233	70	58	)	)	PUNCT
ma-233	70	59	=	=	SYM
ma-233	70	60	0	0	NUM
ma-233	70	61	by	by	ADP
ma-233	70	62	referring	refer	VERB
ma-233	70	63	to	to	ADP
ma-233	70	64	the	the	DET
ma-233	70	65	stability	stability	NOUN
ma-233	70	66	proof	proof	NOUN
ma-233	70	67	of	of	ADP
ma-233	70	68	[	[	X
ma-233	70	69	13	13	NUM
ma-233	70	70	,	,	PUNCT
ma-233	70	71	14	14	NUM
ma-233	70	72	]	]	PUNCT
ma-233	70	73	.	.	PUNCT
ma-233	71	1	https://doi.org/10.28924/ada/ma.4.11	https://doi.org/10.28924/ada/ma.4.11	PROPN
ma-233	71	2	eur	eur	PROPN
ma-233	71	3	.	.	PUNCT
ma-233	72	1	j.	j.	PROPN
ma-233	72	2	math	math	PROPN
ma-233	72	3	.	.	PUNCT
ma-233	73	1	anal	anal	PROPN
ma-233	73	2	.	.	PUNCT
ma-233	74	1	10.28924	10.28924	NUM
ma-233	74	2	/	/	SYM
ma-233	74	3	ada	ada	PROPN
ma-233	74	4	/	/	SYM
ma-233	74	5	ma.4.11	ma.4.11	ADJ
ma-233	74	6	4	4	NUM
ma-233	74	7	lemma	lemma	PROPN
ma-233	74	8	2.1	2.1	NUM
ma-233	74	9	.	.	PUNCT
ma-233	75	1	assume	assume	VERB
ma-233	75	2	f	f	X
ma-233	75	3	:	:	PUNCT
ma-233	75	4	g	g	PROPN
ma-233	75	5	→	→	SYM
ma-233	75	6	x	x	PART
ma-233	75	7	be	be	AUX
ma-233	75	8	a	a	DET
ma-233	75	9	mapping	mapping	NOUN
ma-233	75	10	with	with	ADP
ma-233	75	11	g	g	NOUN
ma-233	75	12	be	be	AUX
ma-233	75	13	an	an	DET
ma-233	75	14	abelian	abelian	ADJ
ma-233	75	15	group	group	NOUN
ma-233	75	16	and	and	CCONJ
ma-233	75	17	(	(	PUNCT
ma-233	75	18	x	x	NOUN
ma-233	75	19	,	,	PUNCT
ma-233	75	20	‖	‖	PROPN
ma-233	75	21	·	·	PUNCT
ma-233	75	22	‖	‖	NUM
ma-233	75	23	)	)	PUNCT
ma-233	75	24	be	be	AUX
ma-233	75	25	acomplete	acomplete	ADJ
ma-233	75	26	non	non	ADJ
ma-233	75	27	-archimedean	-archimedean	ADJ
ma-233	75	28	normed	normed	ADJ
ma-233	75	29	space	space	NOUN
ma-233	75	30	.	.	PUNCT
ma-233	76	1	for	for	ADP
ma-233	76	2	all	all	DET
ma-233	76	3	x	x	NOUN
ma-233	76	4	,	,	PUNCT
ma-233	76	5	y	y	PROPN
ma-233	76	6	∈	∈	PROPN
ma-233	76	7	g	g	PROPN
ma-233	77	1	and	and	CCONJ
ma-233	77	2	there	there	PRON
ma-233	77	3	is	be	VERB
ma-233	77	4	a	a	DET
ma-233	77	5	constant	constant	ADJ
ma-233	77	6	c	c	NOUN
ma-233	77	7	>	>	X
ma-233	77	8	0	0	PUNCT
ma-233	78	1	suchthat	suchthat	PROPN
ma-233	78	2	∥∥∥∥f	∥∥∥∥f	PUNCT
ma-233	79	1	(	(	PUNCT
ma-233	79	2	2x)−	2x)−	NUM
ma-233	79	3	38	38	NUM
ma-233	79	4	f	f	NOUN
ma-233	79	5	(	(	PUNCT
ma-233	79	6	4x	4x	NUM
ma-233	79	7	)	)	PUNCT
ma-233	79	8	+	+	CCONJ
ma-233	79	9	18	18	NUM
ma-233	79	10	f	f	NOUN
ma-233	79	11	(	(	PUNCT
ma-233	79	12	−4x	−4x	PROPN
ma-233	79	13	)	)	PUNCT
ma-233	79	14	∥∥∥∥	∥∥∥∥	NUM
ma-233	79	15	6	6	NUM
ma-233	79	16	c	c	NOUN
ma-233	79	17	(	(	PUNCT
ma-233	79	18	2.2	2.2	NUM
ma-233	79	19	)	)	PUNCT
ma-233	79	20	then	then	ADV
ma-233	79	21	we	we	PRON
ma-233	79	22	define	define	VERB
ma-233	79	23	h(x	h(x	PROPN
ma-233	79	24	,	,	PUNCT
ma-233	79	25	n	n	CCONJ
ma-233	79	26	)	)	PUNCT
ma-233	79	27	=	=	PUNCT
ma-233	79	28	∥∥∥∥f	∥∥∥∥f	NOUN
ma-233	80	1	(	(	PUNCT
ma-233	80	2	2x)−	2x)−	NUM
ma-233	80	3	2n	2n	NUM
ma-233	81	1	+	+	CCONJ
ma-233	82	1	12	12	NUM
ma-233	82	2	·	·	PUNCT
ma-233	82	3	4n	4n	X
ma-233	82	4	f	f	X
ma-233	82	5	(	(	PUNCT
ma-233	82	6	2n+1x	2n+1x	NUM
ma-233	82	7	)	)	PUNCT
ma-233	83	1	+	+	CCONJ
ma-233	83	2	2n	2n	NUM
ma-233	83	3	−	−	NOUN
ma-233	83	4	1	1	NUM
ma-233	83	5	2	2	NUM
ma-233	83	6	·	·	PUNCT
ma-233	83	7	4n	4n	X
ma-233	83	8	f	f	X
ma-233	83	9	(	(	PUNCT
ma-233	83	10	−2n+1x	−2n+1x	PROPN
ma-233	83	11	)	)	PUNCT
ma-233	83	12	∥∥∥∥	∥∥∥∥	NUM
ma-233	83	13	and	and	CCONJ
ma-233	83	14	gn(x	gn(x	NUM
ma-233	83	15	)	)	PUNCT
ma-233	83	16	=	=	SYM
ma-233	83	17	2n	2n	NUM
ma-233	83	18	+	+	CCONJ
ma-233	83	19	1	1	NUM
ma-233	83	20	2	2	NUM
ma-233	83	21	·	·	PUNCT
ma-233	83	22	4n	4n	X
ma-233	83	23	f	f	X
ma-233	83	24	(	(	PUNCT
ma-233	83	25	2	2	NUM
ma-233	83	26	nx)−	nx)−	NOUN
ma-233	83	27	2n	2n	NUM
ma-233	84	1	−	−	NOUN
ma-233	84	2	1	1	NUM
ma-233	84	3	2	2	NUM
ma-233	84	4	·	·	PUNCT
ma-233	84	5	4n	4n	X
ma-233	84	6	f	f	X
ma-233	84	7	(	(	PUNCT
ma-233	84	8	−2	−2	PROPN
ma-233	84	9	nx	nx	PROPN
ma-233	84	10	)	)	PUNCT
ma-233	84	11	.	.	PUNCT
ma-233	85	1	n	n	CCONJ
ma-233	85	2	∈	∈	PROPN
ma-233	85	3	n	n	CCONJ
ma-233	85	4	(	(	PUNCT
ma-233	85	5	1)then	1)then	ADV
ma-233	85	6	we	we	PRON
ma-233	85	7	have	have	AUX
ma-233	85	8	|h(x	|h(x	PROPN
ma-233	85	9	,	,	PUNCT
ma-233	85	10	n	n	PROPN
ma-233	85	11	+	+	PROPN
ma-233	85	12	1)−	1)−	PROPN
ma-233	85	13	h(x	h(x	PROPN
ma-233	85	14	,	,	PUNCT
ma-233	85	15	n)|	n)|	NOUN
ma-233	85	16	6	6	NUM
ma-233	85	17	2n	2n	NUM
ma-233	85	18	+	+	CCONJ
ma-233	85	19	1	1	NUM
ma-233	85	20	2	2	NUM
ma-233	85	21	·	·	PUNCT
ma-233	85	22	4n	4n	X
ma-233	85	23	c	c	NOUN
ma-233	85	24	(	(	PUNCT
ma-233	85	25	2.3	2.3	NUM
ma-233	85	26	)	)	PUNCT
ma-233	85	27	h(x	h(x	PROPN
ma-233	85	28	,	,	PUNCT
ma-233	85	29	n	n	CCONJ
ma-233	85	30	)	)	PUNCT
ma-233	85	31	6	6	NUM
ma-233	85	32	c	c	NOUN
ma-233	85	33	(	(	PUNCT
ma-233	85	34	2.4	2.4	NUM
ma-233	85	35	)	)	PUNCT
ma-233	85	36	(	(	PUNCT
ma-233	85	37	2)and	2)and	NUM
ma-233	85	38	{	{	PUNCT
ma-233	85	39	gn(x	gn(x	NUM
ma-233	85	40	)	)	PUNCT
ma-233	85	41	}	}	PUNCT
ma-233	85	42	is	be	AUX
ma-233	85	43	a	a	DET
ma-233	85	44	cauchy	cauchy	ADJ
ma-233	85	45	sequence	sequence	NOUN
ma-233	85	46	,	,	PUNCT
ma-233	85	47	for	for	ADP
ma-233	85	48	every	every	DET
ma-233	85	49	x	x	SYM
ma-233	85	50	∈	∈	PROPN
ma-233	85	51	g.	g.	NOUN
ma-233	85	52	hence	hence	ADV
ma-233	85	53	,	,	PUNCT
ma-233	85	54	the	the	DET
ma-233	85	55	mapping	mapping	NOUN
ma-233	85	56	g	g	NOUN
ma-233	85	57	:	:	PUNCT
ma-233	85	58	g	g	NOUN
ma-233	85	59	→	→	SYM
ma-233	85	60	x	x	X
ma-233	85	61	can	can	AUX
ma-233	85	62	bedefined	bedefine	VERB
ma-233	85	63	as	as	ADP
ma-233	85	64	g(x	g(x	NOUN
ma-233	85	65	)	)	PUNCT
ma-233	86	1	=	=	SYM
ma-233	86	2	lim	lim	PROPN
ma-233	86	3	n→∞	n→∞	NUM
ma-233	86	4	gn(x	gn(x	PUNCT
ma-233	86	5	)	)	PUNCT
ma-233	86	6	and	and	CCONJ
ma-233	86	7	then	then	ADV
ma-233	86	8	we	we	PRON
ma-233	86	9	get	get	VERB
ma-233	86	10	‖f	‖f	PRON
ma-233	86	11	(	(	PUNCT
ma-233	86	12	2x)−	2x)−	NUM
ma-233	86	13	g(2x)‖	g(2x)‖	PROPN
ma-233	86	14	6	6	NUM
ma-233	86	15	c	c	NOUN
ma-233	86	16	(	(	PUNCT
ma-233	86	17	2.5	2.5	NUM
ma-233	86	18	)	)	PUNCT
ma-233	86	19	proof	proof	NOUN
ma-233	86	20	:	:	PUNCT
ma-233	86	21	adding	add	VERB
ma-233	86	22	one	one	NUM
ma-233	86	23	and	and	CCONJ
ma-233	86	24	subtracting	subtract	VERB
ma-233	86	25	one	one	NUM
ma-233	86	26	with	with	ADP
ma-233	86	27	h(x	h(x	PROPN
ma-233	86	28	,	,	PUNCT
ma-233	86	29	n	n	PROPN
ma-233	86	30	+	+	NOUN
ma-233	86	31	1	1	NUM
ma-233	86	32	)	)	PUNCT
ma-233	86	33	for	for	ADP
ma-233	86	34	matching	match	VERB
ma-233	86	35	and	and	CCONJ
ma-233	86	36	then	then	ADV
ma-233	86	37	using	use	VERB
ma-233	86	38	the	the	DET
ma-233	86	39	in	in	ADP
ma-233	86	40	-	-	PUNCT
ma-233	86	41	equality	equality	NOUN
ma-233	86	42	,	,	PUNCT
ma-233	86	43	we	we	PRON
ma-233	86	44	obtain∥∥∥∥f	obtain∥∥∥∥f	VERB
ma-233	86	45	(	(	PUNCT
ma-233	86	46	2x)−	2x)−	NUM
ma-233	86	47	2n+1	2n+1	PROPN
ma-233	86	48	+	+	CCONJ
ma-233	86	49	12	12	NUM
ma-233	86	50	·	·	SYM
ma-233	86	51	4n+1	4n+1	PROPN
ma-233	86	52	f	f	PROPN
ma-233	86	53	(	(	PUNCT
ma-233	86	54	2n+2x	2n+2x	NUM
ma-233	86	55	)	)	PUNCT
ma-233	87	1	+	+	CCONJ
ma-233	87	2	2n+1	2n+1	NOUN
ma-233	87	3	−	−	NOUN
ma-233	87	4	1	1	NUM
ma-233	87	5	2	2	NUM
ma-233	87	6	·	·	SYM
ma-233	87	7	4n+1	4n+1	PROPN
ma-233	87	8	f	f	PROPN
ma-233	87	9	(	(	PUNCT
ma-233	87	10	−2n+2x	−2n+2x	NOUN
ma-233	87	11	)	)	PUNCT
ma-233	87	12	∥∥∥∥	∥∥∥∥	PROPN
ma-233	87	13	6	6	NUM
ma-233	87	14	∥∥∥∥f	∥∥∥∥f	SYM
ma-233	87	15	(	(	PUNCT
ma-233	87	16	2x)−	2x)−	NUM
ma-233	87	17	2n	2n	NUM
ma-233	87	18	+	+	CCONJ
ma-233	87	19	12	12	NUM
ma-233	87	20	·	·	PUNCT
ma-233	87	21	4n	4n	X
ma-233	87	22	f	f	X
ma-233	87	23	(	(	PUNCT
ma-233	87	24	2n+1x	2n+1x	NUM
ma-233	87	25	)	)	PUNCT
ma-233	88	1	+	+	CCONJ
ma-233	88	2	2n	2n	NUM
ma-233	88	3	−	−	NOUN
ma-233	88	4	1	1	NUM
ma-233	88	5	2	2	NUM
ma-233	88	6	·	·	PUNCT
ma-233	88	7	4n	4n	X
ma-233	88	8	f	f	X
ma-233	88	9	(	(	PUNCT
ma-233	88	10	−2n+1x	−2n+1x	PROPN
ma-233	88	11	)	)	PUNCT
ma-233	88	12	∥∥∥∥	∥∥∥∥	PUNCT
ma-233	89	1	+	+	NUM
ma-233	89	2	2n	2n	NUM
ma-233	89	3	+	+	CCONJ
ma-233	89	4	1	1	NUM
ma-233	89	5	2	2	NUM
ma-233	89	6	·	·	PUNCT
ma-233	89	7	4n	4n	X
ma-233	89	8	∥∥∥∥f	∥∥∥∥f	PUNCT
ma-233	90	1	(	(	PUNCT
ma-233	90	2	2n+1x)−	2n+1x)−	NUM
ma-233	90	3	38	38	NUM
ma-233	90	4	f	f	NOUN
ma-233	90	5	(	(	PUNCT
ma-233	90	6	2n+2x	2n+2x	NUM
ma-233	90	7	)	)	PUNCT
ma-233	90	8	+18	+18	ADJ
ma-233	90	9	f	f	PROPN
ma-233	90	10	(	(	PUNCT
ma-233	90	11	−2n+2x	−2n+2x	NOUN
ma-233	90	12	)	)	PUNCT
ma-233	90	13	∥∥∥∥	∥∥∥∥	PUNCT
ma-233	91	1	+	+	CCONJ
ma-233	91	2	2n	2n	NUM
ma-233	91	3	−	−	NOUN
ma-233	91	4	1	1	NUM
ma-233	91	5	2	2	NUM
ma-233	91	6	·	·	PUNCT
ma-233	91	7	4n	4n	X
ma-233	91	8	∥∥∥∥f	∥∥∥∥f	PUNCT
ma-233	91	9	(	(	PUNCT
ma-233	91	10	−2n+1	−2n+1	X
ma-233	91	11	·	·	PUNCT
ma-233	91	12	x)+	x)+	NUM
ma-233	91	13	18	18	NUM
ma-233	91	14	f	f	X
ma-233	91	15	(	(	PUNCT
ma-233	91	16	2n+2x)−	2n+2x)−	NUM
ma-233	91	17	38	38	NUM
ma-233	91	18	f	f	NOUN
ma-233	91	19	(	(	PUNCT
ma-233	91	20	−2n+2x	−2n+2x	NOUN
ma-233	91	21	)	)	PUNCT
ma-233	91	22	∥∥∥∥	∥∥∥∥	NUM
ma-233	91	23	6	6	NUM
ma-233	91	24	∥∥∥∥f	∥∥∥∥f	SYM
ma-233	91	25	(	(	PUNCT
ma-233	91	26	2x)−	2x)−	NUM
ma-233	91	27	2n	2n	NUM
ma-233	91	28	+	+	CCONJ
ma-233	91	29	12	12	NUM
ma-233	91	30	·	·	PUNCT
ma-233	91	31	4n	4n	X
ma-233	91	32	f	f	X
ma-233	91	33	(	(	PUNCT
ma-233	91	34	2n+1x	2n+1x	NUM
ma-233	91	35	)	)	PUNCT
ma-233	92	1	+	+	CCONJ
ma-233	92	2	2n	2n	NUM
ma-233	92	3	−	−	NOUN
ma-233	92	4	1	1	NUM
ma-233	92	5	2	2	NUM
ma-233	92	6	·	·	PUNCT
ma-233	92	7	4n	4n	X
ma-233	92	8	f	f	X
ma-233	92	9	(	(	PUNCT
ma-233	92	10	−2n+1x	−2n+1x	PROPN
ma-233	92	11	)	)	PUNCT
ma-233	92	12	∥∥∥∥+	∥∥∥∥+	PROPN
ma-233	92	13	c	c	NOUN
ma-233	92	14	·	·	PUNCT
ma-233	92	15	max{2n	max{2n	X
ma-233	93	1	+	+	CCONJ
ma-233	93	2	12	12	NUM
ma-233	93	3	·	·	SYM
ma-233	93	4	4n	4n	NOUN
ma-233	93	5	,	,	PUNCT
ma-233	93	6	2n	2n	NUM
ma-233	93	7	−	−	NOUN
ma-233	93	8	1	1	NUM
ma-233	93	9	2	2	NUM
ma-233	93	10	·	·	PUNCT
ma-233	93	11	4n	4n	NOUN
ma-233	93	12	}	}	PUNCT
ma-233	93	13	next	next	ADV
ma-233	93	14	,	,	PUNCT
ma-233	93	15	it	it	PRON
ma-233	93	16	is	be	AUX
ma-233	93	17	easy	easy	ADJ
ma-233	93	18	to	to	PART
ma-233	93	19	get	get	VERB
ma-233	93	20	|h(x	|h(x	PROPN
ma-233	93	21	,	,	PUNCT
ma-233	93	22	n	n	PROPN
ma-233	93	23	+	+	PROPN
ma-233	93	24	1)−	1)−	PROPN
ma-233	93	25	h(x	h(x	PROPN
ma-233	93	26	,	,	PUNCT
ma-233	93	27	n)|	n)|	NOUN
ma-233	93	28	6	6	NUM
ma-233	93	29	2n	2n	NUM
ma-233	93	30	+	+	CCONJ
ma-233	93	31	1	1	NUM
ma-233	93	32	2	2	NUM
ma-233	93	33	·	·	PUNCT
ma-233	93	34	4n	4n	NOUN
ma-233	93	35	c	c	PROPN
ma-233	93	36	https://doi.org/10.28924/ada/ma.4.11	https://doi.org/10.28924/ada/ma.4.11	PROPN
ma-233	93	37	eur	eur	PROPN
ma-233	93	38	.	.	PUNCT
ma-233	94	1	j.	j.	PROPN
ma-233	94	2	math	math	PROPN
ma-233	94	3	.	.	PUNCT
ma-233	95	1	anal	anal	PROPN
ma-233	95	2	.	.	PUNCT
ma-233	96	1	10.28924	10.28924	NUM
ma-233	96	2	/	/	SYM
ma-233	96	3	ada	ada	PROPN
ma-233	96	4	/	/	SYM
ma-233	96	5	ma.4.11	ma.4.11	PROPN
ma-233	96	6	5then	5then	PROPN
ma-233	96	7	h(x	h(x	PROPN
ma-233	96	8	,	,	PUNCT
ma-233	96	9	n	n	CCONJ
ma-233	96	10	)	)	PUNCT
ma-233	96	11	=	=	SYM
ma-233	97	1	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-233	97	2	(	(	PUNCT
ma-233	97	3	n∑	n∑	NOUN
ma-233	97	4	i=2	i=2	PROPN
ma-233	97	5	h(x	h(x	PROPN
ma-233	97	6	,	,	PUNCT
ma-233	97	7	i)−	i)−	PROPN
ma-233	97	8	h(x	h(x	PROPN
ma-233	97	9	,	,	PUNCT
ma-233	97	10	i	i	PRON
ma-233	97	11	−	−	PROPN
ma-233	97	12	1	1	NUM
ma-233	97	13	)	)	PUNCT
ma-233	97	14	)	)	PUNCT
ma-233	98	1	+	+	CCONJ
ma-233	98	2	h(x	h(x	PROPN
ma-233	98	3	,	,	PUNCT
ma-233	98	4	1	1	NUM
ma-233	98	5	)	)	PUNCT
ma-233	98	6	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-233	98	7	6	6	NUM
ma-233	98	8	c	c	NOUN
ma-233	98	9	·	·	X
ma-233	98	10	max	max	X
ma-233	98	11	{	{	PUNCT
ma-233	98	12	2	2	NUM
ma-233	98	13	+	+	CCONJ
ma-233	99	1	1	1	NUM
ma-233	99	2	2	2	NUM
ma-233	99	3	·	·	SYM
ma-233	99	4	4	4	NUM
ma-233	99	5	,	,	PUNCT
ma-233	99	6	22	22	NUM
ma-233	99	7	+	+	CCONJ
ma-233	99	8	1	1	NUM
ma-233	99	9	2	2	NUM
ma-233	99	10	·	·	SYM
ma-233	99	11	42	42	NUM
ma-233	99	12	,	,	PUNCT
ma-233	99	13	·	·	PUNCT
ma-233	99	14	·	·	PUNCT
ma-233	99	15	·	·	PUNCT
ma-233	99	16	,	,	PUNCT
ma-233	99	17	2n	2n	NUM
ma-233	99	18	+	+	CCONJ
ma-233	99	19	1	1	NUM
ma-233	99	20	2	2	NUM
ma-233	99	21	·	·	PUNCT
ma-233	99	22	4n	4n	NOUN
ma-233	99	23	,	,	PUNCT
ma-233	99	24	1	1	NUM
ma-233	99	25	}	}	PUNCT
ma-233	99	26	=	=	SYM
ma-233	99	27	cnext	cnext	NOUN
ma-233	99	28	,	,	PUNCT
ma-233	99	29	we	we	PRON
ma-233	99	30	have	have	VERB
ma-233	99	31	to	to	PART
ma-233	99	32	prove	prove	VERB
ma-233	99	33	that	that	SCONJ
ma-233	99	34	for	for	ADP
ma-233	99	35	every	every	DET
ma-233	99	36	x	x	SYM
ma-233	99	37	∈	∈	PROPN
ma-233	99	38	g	g	PROPN
ma-233	99	39	,	,	PUNCT
ma-233	99	40	the	the	DET
ma-233	99	41	sequence	sequence	NOUN
ma-233	99	42	gn(x	gn(x	PUNCT
ma-233	99	43	)	)	PUNCT
ma-233	99	44	=	=	SYM
ma-233	100	1	2n	2n	NUM
ma-233	101	1	+	+	CCONJ
ma-233	102	1	1	1	NUM
ma-233	102	2	2	2	NUM
ma-233	102	3	·	·	PUNCT
ma-233	102	4	4n	4n	X
ma-233	102	5	f	f	X
ma-233	102	6	(	(	PUNCT
ma-233	102	7	2	2	NUM
ma-233	102	8	nx)−	nx)−	NOUN
ma-233	102	9	2n	2n	NUM
ma-233	102	10	−	−	NOUN
ma-233	102	11	1	1	NUM
ma-233	102	12	2	2	NUM
ma-233	102	13	·	·	PUNCT
ma-233	102	14	4n	4n	X
ma-233	102	15	f	f	X
ma-233	102	16	(	(	PUNCT
ma-233	102	17	−2	−2	PROPN
ma-233	102	18	nx	nx	PROPN
ma-233	102	19	)	)	PUNCT
ma-233	102	20	n	n	PRON
ma-233	102	21	∈	∈	NOUN
ma-233	102	22	n	n	VERB
ma-233	102	23	is	be	AUX
ma-233	102	24	convergent	convergent	ADJ
ma-233	102	25	in	in	ADP
ma-233	102	26	g.	g.	PROPN
ma-233	102	27	since	since	SCONJ
ma-233	102	28	x	x	PROPN
ma-233	102	29	is	be	AUX
ma-233	102	30	complete	complete	ADJ
ma-233	102	31	,	,	PUNCT
ma-233	102	32	it	it	PRON
ma-233	102	33	is	be	AUX
ma-233	102	34	sufficient	sufficient	ADJ
ma-233	102	35	to	to	PART
ma-233	102	36	show	show	VERB
ma-233	102	37	that	that	SCONJ
ma-233	102	38	(	(	PUNCT
ma-233	102	39	gn(x))n∈n	gn(x))n∈n	X
ma-233	102	40	is	be	AUX
ma-233	102	41	a	a	DET
ma-233	102	42	cauchysequence	cauchysequence	NOUN
ma-233	102	43	for	for	ADP
ma-233	102	44	all	all	DET
ma-233	102	45	x	x	SYM
ma-233	102	46	∈	∈	PROPN
ma-233	102	47	g.	g.	NOUN
ma-233	102	48	by	by	ADP
ma-233	102	49	matching	match	VERB
ma-233	102	50	‖gn+1(x)−	‖gn+1(x)−	PROPN
ma-233	102	51	gn(x)‖twice	gn(x)‖twice	NOUN
ma-233	102	52	then	then	ADV
ma-233	102	53	we	we	PRON
ma-233	102	54	have	have	VERB
ma-233	102	55	‖gn+1(x)−	‖gn+1(x)−	PROPN
ma-233	102	56	gn(x)‖	gn(x)‖	PROPN
ma-233	102	57	6	6	NUM
ma-233	102	58	2n	2n	NUM
ma-233	103	1	+	+	CCONJ
ma-233	103	2	1	1	NUM
ma-233	103	3	2	2	NUM
ma-233	103	4	·	·	PUNCT
ma-233	103	5	4n	4n	X
ma-233	103	6	∥∥∥∥f	∥∥∥∥f	PUNCT
ma-233	104	1	(	(	PUNCT
ma-233	104	2	2nx)−	2nx)−	NUM
ma-233	104	3	38	38	NUM
ma-233	104	4	f	f	NOUN
ma-233	104	5	(	(	PUNCT
ma-233	104	6	2n+1x)+	2n+1x)+	NUM
ma-233	104	7	18	18	NUM
ma-233	104	8	f	f	NOUN
ma-233	104	9	(	(	PUNCT
ma-233	104	10	−2n+1x	−2n+1x	ADJ
ma-233	104	11	)	)	PUNCT
ma-233	104	12	∥∥∥∥	∥∥∥∥	PUNCT
ma-233	105	1	+	+	CCONJ
ma-233	105	2	2n	2n	NUM
ma-233	105	3	−	−	NOUN
ma-233	105	4	1	1	NUM
ma-233	105	5	2	2	NUM
ma-233	105	6	·	·	PUNCT
ma-233	105	7	4n	4n	X
ma-233	105	8	∥∥∥∥f	∥∥∥∥f	PUNCT
ma-233	105	9	(	(	PUNCT
ma-233	105	10	−2nx)−	−2nx)−	PROPN
ma-233	105	11	38	38	NUM
ma-233	105	12	f	f	NOUN
ma-233	105	13	(	(	PUNCT
ma-233	105	14	−2n+1x)+	−2n+1x)+	NUM
ma-233	105	15	18	18	NUM
ma-233	105	16	f	f	NOUN
ma-233	105	17	(	(	PUNCT
ma-233	105	18	2n+1x	2n+1x	NUM
ma-233	105	19	)	)	PUNCT
ma-233	105	20	∥∥∥∥	∥∥∥∥	NUM
ma-233	105	21	6c	6c	NUM
ma-233	105	22	·	·	SYM
ma-233	105	23	max	max	PROPN
ma-233	105	24	{	{	PUNCT
ma-233	105	25	2n	2n	X
ma-233	105	26	+	+	CCONJ
ma-233	105	27	1	1	NUM
ma-233	105	28	2	2	NUM
ma-233	105	29	·	·	PUNCT
ma-233	105	30	4n	4n	NOUN
ma-233	105	31	,	,	PUNCT
ma-233	105	32	2n	2n	NUM
ma-233	105	33	−	−	NOUN
ma-233	105	34	1	1	NUM
ma-233	105	35	2	2	NUM
ma-233	105	36	·	·	PUNCT
ma-233	105	37	4n	4n	X
ma-233	105	38	}	}	PUNCT
ma-233	105	39	=	=	SYM
ma-233	105	40	2n	2n	NUM
ma-233	105	41	+	+	CCONJ
ma-233	105	42	1	1	NUM
ma-233	105	43	2	2	NUM
ma-233	105	44	·	·	PUNCT
ma-233	105	45	4n	4n	X
ma-233	105	46	cfor	cfor	ADP
ma-233	105	47	each	each	DET
ma-233	105	48	n	n	PRON
ma-233	105	49	∈	∈	PROPN
ma-233	105	50	n	n	NOUN
ma-233	105	51	.	.	PUNCT
ma-233	106	1	this	this	PRON
ma-233	106	2	easily	easily	ADV
ma-233	106	3	implies	imply	VERB
ma-233	106	4	that	that	SCONJ
ma-233	106	5	{	{	PUNCT
ma-233	106	6	gn(x	gn(x	PUNCT
ma-233	106	7	)	)	PUNCT
ma-233	106	8	}	}	PUNCT
ma-233	106	9	is	be	AUX
ma-233	106	10	a	a	DET
ma-233	106	11	cauchy	cauchy	ADJ
ma-233	106	12	sequence	sequence	NOUN
ma-233	106	13	.	.	PUNCT
ma-233	107	1	the	the	DET
ma-233	107	2	mapping	mapping	NOUN
ma-233	107	3	g	g	NOUN
ma-233	107	4	:	:	PUNCT
ma-233	107	5	g	g	PROPN
ma-233	107	6	→	→	SYM
ma-233	107	7	xcan	xcan	ADJ
ma-233	107	8	be	be	AUX
ma-233	107	9	defined	define	VERB
ma-233	107	10	as	as	ADP
ma-233	107	11	g(x	g(x	NOUN
ma-233	107	12	)	)	PUNCT
ma-233	108	1	=	=	SYM
ma-233	108	2	lim	lim	PROPN
ma-233	108	3	n→∞	n→∞	PRON
ma-233	108	4	gn(x)through	gn(x)through	NOUN
ma-233	108	5	the	the	DET
ma-233	108	6	above	above	ADJ
ma-233	108	7	results	result	NOUN
ma-233	108	8	,	,	PUNCT
ma-233	108	9	we	we	PRON
ma-233	108	10	can	can	AUX
ma-233	108	11	obtain	obtain	VERB
ma-233	108	12	‖f	‖f	PRON
ma-233	108	13	(	(	PUNCT
ma-233	108	14	2x)−	2x)−	NUM
ma-233	108	15	g(2x)‖	g(2x)‖	PROPN
ma-233	108	16	=	=	SYM
ma-233	108	17	‖h(x	‖h(x	PROPN
ma-233	108	18	,	,	PUNCT
ma-233	108	19	n	n	CCONJ
ma-233	108	20	)	)	PUNCT
ma-233	109	1	+	+	CCONJ
ma-233	109	2	gn(2x)−	gn(2x)−	X
ma-233	109	3	g(2x)‖	g(2x)‖	PROPN
ma-233	109	4	6	6	NUM
ma-233	109	5	c	c	NOUN
ma-233	109	6	in	in	ADP
ma-233	109	7	this	this	DET
ma-233	109	8	section	section	NOUN
ma-233	109	9	,	,	PUNCT
ma-233	109	10	let	let	VERB
ma-233	109	11	g	g	PRON
ma-233	109	12	be	be	AUX
ma-233	109	13	an	an	DET
ma-233	109	14	abelian	abelian	ADJ
ma-233	109	15	group	group	NOUN
ma-233	109	16	and	and	CCONJ
ma-233	109	17	let	let	VERB
ma-233	109	18	⊥	⊥	NOUN
ma-233	109	19	be	be	AUX
ma-233	109	20	a	a	DET
ma-233	109	21	binary	binary	ADJ
ma-233	109	22	relation	relation	NOUN
ma-233	109	23	defined	define	VERB
ma-233	109	24	on	on	ADP
ma-233	109	25	g	g	NOUN
ma-233	109	26	with	with	ADP
ma-233	109	27	theproperties:(i	theproperties:(i	NOUN
ma-233	109	28	)	)	PUNCT
ma-233	109	29	x	x	X
ma-233	110	1	⊥	⊥	NOUN
ma-233	110	2	0	0	NUM
ma-233	110	3	,	,	PUNCT
ma-233	110	4	0	0	NUM
ma-233	110	5	⊥	⊥	NOUN
ma-233	110	6	x	x	X
ma-233	110	7	,	,	PUNCT
ma-233	110	8	for	for	ADP
ma-233	110	9	all	all	DET
ma-233	110	10	x	x	SYM
ma-233	110	11	∈	∈	PROPN
ma-233	110	12	x;(ii	x;(ii	PROPN
ma-233	110	13	)	)	PUNCT
ma-233	110	14	if	if	SCONJ
ma-233	110	15	x	x	X
ma-233	110	16	,	,	PUNCT
ma-233	110	17	y	y	PROPN
ma-233	110	18	∈	∈	PROPN
ma-233	110	19	x	x	X
ma-233	110	20	and	and	CCONJ
ma-233	110	21	x	x	SYM
ma-233	110	22	⊥	⊥	NOUN
ma-233	110	23	y	y	PROPN
ma-233	110	24	,	,	PUNCT
ma-233	110	25	then	then	ADV
ma-233	110	26	x2	x2	PROPN
ma-233	110	27	⊥	⊥	PROPN
ma-233	110	28	y	y	PROPN
ma-233	110	29	2	2	NUM
ma-233	110	30	,	,	PUNCT
ma-233	110	31	2x	2x	NUM
ma-233	110	32	⊥	⊥	NUM
ma-233	110	33	2y	2y	NUM
ma-233	110	34	,	,	PUNCT
ma-233	110	35	4x	4x	NUM
ma-233	110	36	⊥	⊥	NUM
ma-233	110	37	4y	4y	NUM
ma-233	110	38	and	and	CCONJ
ma-233	110	39	−x	−x	PRON
ma-233	110	40	⊥	⊥	PROPN
ma-233	110	41	−y	−y	NOUN
ma-233	110	42	.	.	PUNCT
ma-233	111	1	theorem	theorem	VERB
ma-233	111	2	2.1	2.1	NUM
ma-233	111	3	.	.	PUNCT
ma-233	112	1	suppose	suppose	VERB
ma-233	112	2	f	f	X
ma-233	112	3	:	:	PUNCT
ma-233	112	4	g	g	PROPN
ma-233	112	5	→	→	SYM
ma-233	112	6	x	x	X
ma-233	112	7	where	where	SCONJ
ma-233	112	8	f	f	PROPN
ma-233	112	9	is	be	AUX
ma-233	112	10	a	a	DET
ma-233	112	11	mapping	mapping	NOUN
ma-233	112	12	from	from	ADP
ma-233	112	13	an	an	DET
ma-233	112	14	abelian	abelian	ADJ
ma-233	112	15	group	group	NOUN
ma-233	112	16	to	to	ADP
ma-233	112	17	a	a	DET
ma-233	112	18	completenon	completenon	NOUN
ma-233	112	19	-	-	PUNCT
ma-233	112	20	archimedean	archimedean	ADJ
ma-233	112	21	normed	normed	ADJ
ma-233	112	22	space	space	NOUN
ma-233	112	23	.	.	PUNCT
ma-233	113	1	for	for	ADP
ma-233	113	2	ε	ε	PROPN
ma-233	113	3	>	>	X
ma-233	113	4	0	0	PROPN
ma-233	113	5	,	,	PUNCT
ma-233	113	6	when	when	SCONJ
ma-233	113	7	x	x	PROPN
ma-233	113	8	⊥	⊥	PROPN
ma-233	113	9	y	y	PROPN
ma-233	113	10	for	for	ADP
ma-233	113	11	all	all	DET
ma-233	113	12	x	x	NOUN
ma-233	113	13	,	,	PUNCT
ma-233	113	14	y	y	PROPN
ma-233	113	15	∈	∈	PROPN
ma-233	113	16	g	g	PROPN
ma-233	113	17	,	,	PUNCT
ma-233	113	18	we	we	PRON
ma-233	113	19	obtain	obtain	VERB
ma-233	113	20	‖d1f	‖d1f	NOUN
ma-233	113	21	(	(	PUNCT
ma-233	113	22	x	x	X
ma-233	113	23	,	,	PUNCT
ma-233	113	24	y)‖	y)‖	PROPN
ma-233	113	25	6	6	NUM
ma-233	113	26	ε	ε	PROPN
ma-233	113	27	(	(	PUNCT
ma-233	113	28	2.6	2.6	NUM
ma-233	113	29	)	)	PUNCT
ma-233	113	30	and	and	CCONJ
ma-233	113	31	‖f	‖f	ADP
ma-233	113	32	(	(	PUNCT
ma-233	113	33	x	x	X
ma-233	113	34	)	)	PUNCT
ma-233	113	35	+	+	NUM
ma-233	113	36	f	f	X
ma-233	113	37	(	(	PUNCT
ma-233	113	38	−x)‖	−x)‖	PROPN
ma-233	113	39	6	6	NUM
ma-233	113	40	ε	ε	PROPN
ma-233	113	41	(	(	PUNCT
ma-233	113	42	2.7)then	2.7)then	NUM
ma-233	113	43	there	there	PRON
ma-233	113	44	exists	exist	VERB
ma-233	113	45	a	a	DET
ma-233	113	46	unique	unique	ADJ
ma-233	113	47	mapping	mapping	NOUN
ma-233	113	48	g	g	NOUN
ma-233	113	49	:	:	PUNCT
ma-233	113	50	g	g	PROPN
ma-233	113	51	→	→	SYM
ma-233	113	52	x	x	X
ma-233	113	53	such	such	ADJ
ma-233	113	54	that	that	SCONJ
ma-233	113	55	x	x	PROPN
ma-233	113	56	⊥	⊥	NOUN
ma-233	113	57	y	y	PROPN
ma-233	113	58	implies	imply	VERB
ma-233	113	59	4g(x	4g(x	PROPN
ma-233	114	1	+	+	CCONJ
ma-233	114	2	y	y	X
ma-233	114	3	)	)	PUNCT
ma-233	115	1	+	+	CCONJ
ma-233	115	2	4g(x	4g(x	NUM
ma-233	116	1	−	−	PROPN
ma-233	116	2	y	y	NOUN
ma-233	116	3	)	)	PUNCT
ma-233	117	1	+	+	CCONJ
ma-233	117	2	10g(x	10g(x	X
ma-233	117	3	)	)	PUNCT
ma-233	117	4	+	+	CCONJ
ma-233	117	5	14g(−x)−	14g(−x)−	NUM
ma-233	117	6	3g(y)−	3g(y)−	NUM
ma-233	117	7	3g(−y	3g(−y	NOUN
ma-233	117	8	)	)	PUNCT
ma-233	117	9	=	=	VERB
ma-233	118	1	g(2x	g(2x	VERB
ma-233	118	2	+	+	CCONJ
ma-233	118	3	y	y	NOUN
ma-233	118	4	)	)	PUNCT
ma-233	119	1	+	+	CCONJ
ma-233	119	2	g(2x	g(2x	VERB
ma-233	119	3	−	−	PROPN
ma-233	119	4	y	y	NOUN
ma-233	119	5	)	)	PUNCT
ma-233	119	6	(	(	PUNCT
ma-233	119	7	2.8	2.8	NUM
ma-233	119	8	)	)	PUNCT
ma-233	119	9	https://doi.org/10.28924/ada/ma.4.11	https://doi.org/10.28924/ada/ma.4.11	PROPN
ma-233	119	10	eur	eur	NOUN
ma-233	119	11	.	.	PUNCT
ma-233	120	1	j.	j.	PROPN
ma-233	120	2	math	math	PROPN
ma-233	120	3	.	.	PUNCT
ma-233	121	1	anal	anal	PROPN
ma-233	121	2	.	.	PUNCT
ma-233	122	1	10.28924	10.28924	NUM
ma-233	122	2	/	/	SYM
ma-233	122	3	ada	ada	PROPN
ma-233	122	4	/	/	SYM
ma-233	122	5	ma.4.11	ma.4.11	ADJ
ma-233	122	6	6and	6and	NUM
ma-233	122	7	‖f	‖f	ADP
ma-233	122	8	(	(	PUNCT
ma-233	122	9	x)−	x)−	PROPN
ma-233	122	10	g(x)‖	g(x)‖	PROPN
ma-233	122	11	6	6	NUM
ma-233	122	12	7	7	NUM
ma-233	122	13	2	2	NUM
ma-233	122	14	ε	ε	X
ma-233	122	15	(	(	PUNCT
ma-233	122	16	2.9	2.9	NUM
ma-233	122	17	)	)	PUNCT
ma-233	122	18	for	for	ADP
ma-233	122	19	all	all	PRON
ma-233	122	20	x	x	SYM
ma-233	122	21	∈	∈	PROPN
ma-233	122	22	2	2	NUM
ma-233	122	23	g	g	NOUN
ma-233	122	24	=	=	PUNCT
ma-233	122	25	{	{	PUNCT
ma-233	122	26	2x	2x	NUM
ma-233	122	27	:	:	PUNCT
ma-233	122	28	x	x	SYM
ma-233	122	29	∈	∈	NOUN
ma-233	122	30	g	g	NOUN
ma-233	122	31	}	}	PUNCT
ma-233	122	32	.	.	PUNCT
ma-233	123	1	proof	proof	NOUN
ma-233	123	2	.	.	PUNCT
ma-233	124	1	for	for	ADP
ma-233	124	2	all	all	DET
ma-233	124	3	x	x	SYM
ma-233	124	4	∈	∈	NOUN
ma-233	124	5	x	x	X
ma-233	124	6	,	,	PUNCT
ma-233	124	7	since	since	SCONJ
ma-233	124	8	0	0	NUM
ma-233	124	9	⊥	⊥	NOUN
ma-233	124	10	x	x	SYM
ma-233	124	11	,	,	PUNCT
ma-233	124	12	x	x	PROPN
ma-233	124	13	⊥	⊥	NOUN
ma-233	124	14	0	0	NUM
ma-233	124	15	and	and	CCONJ
ma-233	124	16	0	0	NUM
ma-233	124	17	⊥	⊥	NOUN
ma-233	124	18	0	0	NUM
ma-233	124	19	,	,	PUNCT
ma-233	124	20	setting	set	VERB
ma-233	124	21	x	x	PUNCT
ma-233	124	22	=	=	SYM
ma-233	124	23	0	0	NUM
ma-233	124	24	,	,	PUNCT
ma-233	124	25	y	y	PROPN
ma-233	124	26	=	=	PUNCT
ma-233	124	27	0	0	NUM
ma-233	124	28	in	in	ADP
ma-233	124	29	(	(	PUNCT
ma-233	124	30	2.6	2.6	NUM
ma-233	124	31	)	)	PUNCT
ma-233	124	32	,	,	PUNCT
ma-233	124	33	we	we	PRON
ma-233	124	34	obtain	obtain	VERB
ma-233	124	35	‖24f	‖24f	ADP
ma-233	124	36	(	(	PUNCT
ma-233	124	37	0)‖	0)‖	PROPN
ma-233	124	38	6	6	NUM
ma-233	124	39	ε	ε	PROPN
ma-233	124	40	,	,	PUNCT
ma-233	124	41	respectively	respectively	ADV
ma-233	124	42	,	,	PUNCT
ma-233	124	43	setting	set	VERB
ma-233	124	44	y	y	NOUN
ma-233	124	45	=	=	PUNCT
ma-233	124	46	0	0	NUM
ma-233	125	1	in	in	ADP
ma-233	125	2	(	(	PUNCT
ma-233	125	3	2.6),we	2.6),we	NUM
ma-233	125	4	obtain	obtain	VERB
ma-233	125	5	the	the	DET
ma-233	125	6	following	follow	VERB
ma-233	125	7	inequality	inequality	NOUN
ma-233	125	8	:	:	PUNCT
ma-233	125	9	‖2f	‖2f	PROPN
ma-233	125	10	(	(	PUNCT
ma-233	125	11	2x)−	2x)−	PROPN
ma-233	125	12	18f	18f	X
ma-233	125	13	(	(	PUNCT
ma-233	125	14	x)−	x)−	PROPN
ma-233	125	15	14f	14f	PROPN
ma-233	125	16	(	(	PUNCT
ma-233	125	17	−x	−x	NOUN
ma-233	125	18	)	)	PUNCT
ma-233	125	19	+	+	NUM
ma-233	125	20	6f	6f	NUM
ma-233	125	21	(	(	PUNCT
ma-233	125	22	0)‖	0)‖	PROPN
ma-233	125	23	6	6	NUM
ma-233	125	24	ε	ε	PROPN
ma-233	125	25	(	(	PUNCT
ma-233	125	26	2.10	2.10	NUM
ma-233	125	27	)	)	PUNCT
ma-233	125	28	by	by	ADP
ma-233	125	29	using	use	VERB
ma-233	125	30	the	the	DET
ma-233	125	31	strong	strong	ADJ
ma-233	125	32	triangle	triangle	NOUN
ma-233	125	33	inequality	inequality	NOUN
ma-233	125	34	,	,	PUNCT
ma-233	125	35	we	we	PRON
ma-233	125	36	obtain	obtain	VERB
ma-233	125	37	‖2f	‖2f	PROPN
ma-233	125	38	(	(	PUNCT
ma-233	125	39	2x)−18f	2x)−18f	NUM
ma-233	125	40	(	(	PUNCT
ma-233	125	41	x)−14f	x)−14f	PUNCT
ma-233	125	42	(	(	PUNCT
ma-233	125	43	−x)‖	−x)‖	PROPN
ma-233	125	44	6	6	NUM
ma-233	125	45	max{‖2f	max{‖2f	NOUN
ma-233	125	46	(	(	PUNCT
ma-233	125	47	2x)−18f	2x)−18f	NOUN
ma-233	125	48	(	(	PUNCT
ma-233	125	49	x)−14f	x)−14f	X
ma-233	125	50	(	(	PUNCT
ma-233	125	51	−x)+6f	−x)+6f	X
ma-233	125	52	(	(	PUNCT
ma-233	125	53	0)‖	0)‖	PROPN
ma-233	125	54	,	,	PUNCT
ma-233	125	55	‖6f	‖6f	PROPN
ma-233	125	56	(	(	PUNCT
ma-233	125	57	0)‖	0)‖	PROPN
ma-233	125	58	}	}	PUNCT
ma-233	125	59	6	6	NUM
ma-233	125	60	ε	ε	PROPN
ma-233	125	61	(	(	PUNCT
ma-233	125	62	2.11	2.11	NUM
ma-233	125	63	)	)	PUNCT
ma-233	125	64	by	by	ADP
ma-233	125	65	replacing	replace	VERB
ma-233	125	66	x	x	PUNCT
ma-233	125	67	with	with	ADP
ma-233	125	68	4x	4x	NUM
ma-233	125	69	in	in	ADP
ma-233	125	70	(	(	PUNCT
ma-233	125	71	2.7	2.7	NUM
ma-233	125	72	)	)	PUNCT
ma-233	125	73	and	and	CCONJ
ma-233	125	74	applying	apply	VERB
ma-233	125	75	the	the	DET
ma-233	125	76	triangle	triangle	NOUN
ma-233	125	77	inequality	inequality	NOUN
ma-233	125	78	twice	twice	ADV
ma-233	125	79	,	,	PUNCT
ma-233	125	80	we	we	PRON
ma-233	125	81	obtain	obtain	VERB
ma-233	125	82	‖2f	‖2f	PROPN
ma-233	125	83	(	(	PUNCT
ma-233	125	84	2x)−	2x)−	NUM
ma-233	125	85	4f	4f	NUM
ma-233	125	86	(	(	PUNCT
ma-233	125	87	x)‖	x)‖	PROPN
ma-233	125	88	6	6	NUM
ma-233	125	89	max{‖2f	max{‖2f	NOUN
ma-233	125	90	(	(	PUNCT
ma-233	125	91	2x)−	2x)−	PROPN
ma-233	125	92	18f	18f	NUM
ma-233	125	93	(	(	PUNCT
ma-233	125	94	x)−	x)−	PROPN
ma-233	125	95	14f	14f	PROPN
ma-233	125	96	(	(	PUNCT
ma-233	125	97	−x)‖	−x)‖	PROPN
ma-233	125	98	,	,	PUNCT
ma-233	125	99	14‖f	14‖f	NUM
ma-233	125	100	(	(	PUNCT
ma-233	125	101	x	x	X
ma-233	125	102	)	)	PUNCT
ma-233	126	1	+	+	NUM
ma-233	126	2	f	f	X
ma-233	126	3	(	(	PUNCT
ma-233	126	4	−x)‖	−x)‖	NOUN
ma-233	126	5	}	}	PUNCT
ma-233	126	6	=	=	SYM
ma-233	126	7	14ε	14ε	NOUN
ma-233	126	8	(	(	PUNCT
ma-233	126	9	2.12	2.12	NUM
ma-233	126	10	)	)	PUNCT
ma-233	126	11	applying	apply	VERB
ma-233	126	12	(	(	PUNCT
ma-233	126	13	2.7	2.7	NUM
ma-233	126	14	)	)	PUNCT
ma-233	126	15	and	and	CCONJ
ma-233	126	16	(	(	PUNCT
ma-233	126	17	2.12	2.12	NUM
ma-233	126	18	)	)	PUNCT
ma-233	126	19	to	to	ADP
ma-233	126	20	‖3f	‖3f	PROPN
ma-233	126	21	(	(	PUNCT
ma-233	126	22	4x)−	4x)−	PROPN
ma-233	126	23	8f	8f	NOUN
ma-233	126	24	(	(	PUNCT
ma-233	126	25	2x)−	2x)−	NUM
ma-233	126	26	f	f	X
ma-233	126	27	(	(	PUNCT
ma-233	126	28	−4x)‖	−4x)‖	PROPN
ma-233	126	29	,	,	PUNCT
ma-233	126	30	we	we	PRON
ma-233	126	31	can	can	AUX
ma-233	126	32	conclude	conclude	VERB
ma-233	126	33	that	that	SCONJ
ma-233	127	1	‖3f	‖3f	PROPN
ma-233	127	2	(	(	PUNCT
ma-233	127	3	4x)−	4x)−	PROPN
ma-233	127	4	8f	8f	NOUN
ma-233	127	5	(	(	PUNCT
ma-233	127	6	2x)−	2x)−	NUM
ma-233	127	7	f	f	X
ma-233	127	8	(	(	PUNCT
ma-233	127	9	−4x)‖	−4x)‖	PROPN
ma-233	127	10	=	=	PUNCT
ma-233	127	11	‖4[f	‖4[f	PROPN
ma-233	127	12	(	(	PUNCT
ma-233	127	13	4x)−	4x)−	PROPN
ma-233	127	14	2f	2f	NUM
ma-233	127	15	(	(	PUNCT
ma-233	127	16	2x)]−	2x)]−	PROPN
ma-233	128	1	[	[	X
ma-233	128	2	f	f	X
ma-233	128	3	(	(	PUNCT
ma-233	128	4	4x	4x	NUM
ma-233	128	5	)	)	PUNCT
ma-233	128	6	+	+	NUM
ma-233	128	7	f	f	X
ma-233	128	8	(	(	PUNCT
ma-233	128	9	−4x)]‖	−4x)]‖	X
ma-233	128	10	6	6	NUM
ma-233	128	11	max	max	PROPN
ma-233	128	12	{	{	PUNCT
ma-233	128	13	28ε	28ε	NUM
ma-233	128	14	,	,	PUNCT
ma-233	128	15	ε	ε	PROPN
ma-233	128	16	}	}	PUNCT
ma-233	128	17	=	=	NOUN
ma-233	128	18	28ε	28ε	NOUN
ma-233	128	19	(	(	PUNCT
ma-233	128	20	2.13	2.13	NUM
ma-233	128	21	)	)	PUNCT
ma-233	128	22	this	this	PRON
ma-233	128	23	means	mean	VERB
ma-233	128	24	that	that	SCONJ
ma-233	128	25	∥∥∥∥f	∥∥∥∥f	PUNCT
ma-233	129	1	(	(	PUNCT
ma-233	129	2	2x)−	2x)−	NUM
ma-233	129	3	38	38	NUM
ma-233	129	4	f	f	NOUN
ma-233	129	5	(	(	PUNCT
ma-233	129	6	4x	4x	NUM
ma-233	129	7	)	)	PUNCT
ma-233	129	8	+	+	CCONJ
ma-233	129	9	18	18	NUM
ma-233	129	10	f	f	NOUN
ma-233	129	11	(	(	PUNCT
ma-233	129	12	−4x	−4x	PROPN
ma-233	129	13	)	)	PUNCT
ma-233	129	14	∥∥∥∥	∥∥∥∥	NUM
ma-233	129	15	6	6	NUM
ma-233	129	16	72ε	72ε	NOUN
ma-233	129	17	(	(	PUNCT
ma-233	129	18	2.14	2.14	NUM
ma-233	129	19	)	)	PUNCT
ma-233	129	20	the	the	DET
ma-233	129	21	next	next	ADJ
ma-233	129	22	step	step	NOUN
ma-233	129	23	resembles	resemble	VERB
ma-233	129	24	lemma2.1	lemma2.1	NUM
ma-233	129	25	,	,	PUNCT
ma-233	129	26	let	let	VERB
ma-233	129	27	gn(x	gn(x	PUNCT
ma-233	129	28	)	)	PUNCT
ma-233	130	1	=	=	SYM
ma-233	130	2	2n	2n	NUM
ma-233	131	1	+	+	CCONJ
ma-233	132	1	1	1	NUM
ma-233	132	2	2	2	NUM
ma-233	132	3	·	·	PUNCT
ma-233	132	4	4n	4n	X
ma-233	132	5	f	f	X
ma-233	132	6	(	(	PUNCT
ma-233	132	7	2	2	NUM
ma-233	132	8	nx)−	nx)−	NOUN
ma-233	132	9	2n	2n	NUM
ma-233	132	10	−	−	NOUN
ma-233	132	11	1	1	NUM
ma-233	132	12	2	2	NUM
ma-233	132	13	·	·	PUNCT
ma-233	132	14	4n	4n	X
ma-233	132	15	f	f	X
ma-233	132	16	(	(	PUNCT
ma-233	132	17	−2	−2	PROPN
ma-233	132	18	nx	nx	PROPN
ma-233	132	19	)	)	PUNCT
ma-233	132	20	(	(	PUNCT
ma-233	132	21	2.15	2.15	NUM
ma-233	132	22	)	)	PUNCT
ma-233	132	23	then	then	ADV
ma-233	132	24	we	we	PRON
ma-233	132	25	can	can	AUX
ma-233	132	26	define	define	VERB
ma-233	132	27	a	a	DET
ma-233	132	28	mapping	mapping	NOUN
ma-233	132	29	g	g	NOUN
ma-233	132	30	g	g	PROPN
ma-233	132	31	:	:	PUNCT
ma-233	132	32	g	g	PROPN
ma-233	132	33	→	→	SYM
ma-233	132	34	x	x	SYM
ma-233	132	35	g(x	g(x	NOUN
ma-233	132	36	)	)	PUNCT
ma-233	133	1	=	=	SYM
ma-233	133	2	lim	lim	PROPN
ma-233	133	3	n→∞	n→∞	NUM
ma-233	133	4	gn(x	gn(x	PUNCT
ma-233	133	5	)	)	PUNCT
ma-233	133	6	.	.	PUNCT
ma-233	134	1	according	accord	VERB
ma-233	134	2	to	to	ADP
ma-233	134	3	lemma2.1	lemma2.1	PROPN
ma-233	134	4	,	,	PUNCT
ma-233	134	5	we	we	PRON
ma-233	134	6	obtain	obtain	VERB
ma-233	134	7	‖f	‖f	PRON
ma-233	134	8	(	(	PUNCT
ma-233	134	9	2x)−	2x)−	NUM
ma-233	134	10	g(2x)‖	g(2x)‖	NOUN
ma-233	134	11	6	6	NUM
ma-233	134	12	7	7	NUM
ma-233	134	13	2	2	NUM
ma-233	134	14	ε	ε	X
ma-233	134	15	(	(	PUNCT
ma-233	134	16	2.16	2.16	NUM
ma-233	134	17	)	)	PUNCT
ma-233	134	18	we	we	PRON
ma-233	134	19	consider	consider	VERB
ma-233	134	20	the	the	DET
ma-233	134	21	following	follow	VERB
ma-233	134	22	inequality	inequality	NOUN
ma-233	134	23	‖d1gn(x	‖d1gn(x	PROPN
ma-233	134	24	,	,	PUNCT
ma-233	134	25	y)‖	y)‖	PROPN
ma-233	134	26	6	6	NUM
ma-233	134	27	∥∥∥∥2n	∥∥∥∥2n	PROPN
ma-233	134	28	+	+	CCONJ
ma-233	134	29	12	12	NUM
ma-233	134	30	·	·	SYM
ma-233	134	31	4n	4n	ADJ
ma-233	134	32	d1f	d1f	NOUN
ma-233	134	33	(	(	PUNCT
ma-233	134	34	2	2	NUM
ma-233	134	35	nx	nx	NOUN
ma-233	134	36	,	,	PUNCT
ma-233	134	37	2ny	2ny	ADJ
ma-233	134	38	)	)	PUNCT
ma-233	135	1	+	+	NUM
ma-233	135	2	2n	2n	NUM
ma-233	135	3	−	−	NOUN
ma-233	135	4	1	1	NUM
ma-233	135	5	2	2	NUM
ma-233	135	6	·	·	PUNCT
ma-233	135	7	4n	4n	ADJ
ma-233	135	8	d1f	d1f	NOUN
ma-233	135	9	(	(	PUNCT
ma-233	135	10	2	2	NUM
ma-233	135	11	nx	nx	NOUN
ma-233	135	12	,	,	PUNCT
ma-233	135	13	2ny	2ny	ADJ
ma-233	135	14	)	)	PUNCT
ma-233	135	15	∥∥∥∥	∥∥∥∥	NUM
ma-233	135	16	6	6	NUM
ma-233	135	17	2n	2n	NUM
ma-233	135	18	+	+	CCONJ
ma-233	135	19	1	1	NUM
ma-233	135	20	2	2	NUM
ma-233	135	21	·	·	PUNCT
ma-233	135	22	4n	4n	X
ma-233	135	23	ε	ε	PROPN
ma-233	135	24	(	(	PUNCT
ma-233	135	25	2.17	2.17	NUM
ma-233	135	26	)	)	PUNCT
ma-233	135	27	https://doi.org/10.28924/ada/ma.4.11	https://doi.org/10.28924/ada/ma.4.11	PROPN
ma-233	135	28	eur	eur	NOUN
ma-233	135	29	.	.	PUNCT
ma-233	136	1	j.	j.	PROPN
ma-233	136	2	math	math	PROPN
ma-233	136	3	.	.	PUNCT
ma-233	137	1	anal	anal	PROPN
ma-233	137	2	.	.	PUNCT
ma-233	138	1	10.28924	10.28924	NUM
ma-233	138	2	/	/	SYM
ma-233	138	3	ada	ada	PROPN
ma-233	138	4	/	/	SYM
ma-233	138	5	ma.4.11	ma.4.11	PROPN
ma-233	138	6	7for	7for	NUM
ma-233	138	7	all	all	DET
ma-233	138	8	x	x	NOUN
ma-233	138	9	,	,	PUNCT
ma-233	139	1	y	y	PROPN
ma-233	139	2	∈	∈	PROPN
ma-233	139	3	g.	g.	NOUN
ma-233	140	1	then	then	ADV
ma-233	140	2	we	we	PRON
ma-233	140	3	let	let	VERB
ma-233	140	4	n	n	PRON
ma-233	140	5	→∞	→∞	PROPN
ma-233	140	6	,	,	PUNCT
ma-233	140	7	we	we	PRON
ma-233	140	8	get	get	VERB
ma-233	140	9	(	(	PUNCT
ma-233	140	10	2.8	2.8	NUM
ma-233	140	11	)	)	PUNCT
ma-233	140	12	.	.	PUNCT
ma-233	141	1	now	now	ADV
ma-233	141	2	,	,	PUNCT
ma-233	141	3	in	in	ADP
ma-233	141	4	order	order	NOUN
ma-233	141	5	to	to	PART
ma-233	141	6	prove	prove	VERB
ma-233	141	7	g	g	PROPN
ma-233	141	8	is	be	AUX
ma-233	141	9	unique	unique	ADJ
ma-233	141	10	,	,	PUNCT
ma-233	141	11	we	we	PRON
ma-233	141	12	assume	assume	VERB
ma-233	141	13	g′	g′	NOUN
ma-233	141	14	as	as	ADP
ma-233	141	15	another	another	DET
ma-233	141	16	mapping	mapping	NOUN
ma-233	141	17	satisfying	satisfy	VERB
ma-233	141	18	(	(	PUNCT
ma-233	141	19	2.8	2.8	NUM
ma-233	141	20	)	)	PUNCT
ma-233	141	21	and	and	CCONJ
ma-233	141	22	(	(	PUNCT
ma-233	141	23	2.9	2.9	NUM
ma-233	141	24	)	)	PUNCT
ma-233	141	25	that∥∥g(x)−	that∥∥g(x)−	PROPN
ma-233	141	26	g′(x)∥∥	g′(x)∥∥	NOUN
ma-233	141	27	=	=	PUNCT
ma-233	141	28	∥∥g(x)−	∥∥g(x)−	PROPN
ma-233	141	29	f	f	X
ma-233	141	30	(	(	PUNCT
ma-233	141	31	x	x	X
ma-233	141	32	)	)	PUNCT
ma-233	142	1	+	+	NUM
ma-233	142	2	f	f	X
ma-233	142	3	(	(	PUNCT
ma-233	142	4	x)−	x)−	PROPN
ma-233	142	5	g′(x)∥∥	g′(x)∥∥	PROPN
ma-233	142	6	6	6	NUM
ma-233	142	7	max	max	PROPN
ma-233	142	8	{	{	PUNCT
ma-233	142	9	‖g(x)−	‖g(x)−	PROPN
ma-233	142	10	f	f	PROPN
ma-233	142	11	(	(	PUNCT
ma-233	142	12	x)‖	x)‖	ADJ
ma-233	142	13	,	,	PUNCT
ma-233	142	14	∥∥f	∥∥f	PROPN
ma-233	142	15	(	(	PUNCT
ma-233	142	16	x)−	x)−	PROPN
ma-233	142	17	g′(x)∥∥	g′(x)∥∥	PROPN
ma-233	142	18	}	}	PUNCT
ma-233	142	19	=	=	SYM
ma-233	142	20	ε	ε	PROPN
ma-233	142	21	(	(	PUNCT
ma-233	142	22	2.18	2.18	NUM
ma-233	142	23	)	)	PUNCT
ma-233	142	24	for	for	ADP
ma-233	142	25	all	all	PRON
ma-233	142	26	x	x	SYM
ma-233	142	27	∈	∈	PROPN
ma-233	142	28	2	2	NUM
ma-233	142	29	g	g	NOUN
ma-233	142	30	=	=	PUNCT
ma-233	142	31	{	{	PUNCT
ma-233	142	32	2x	2x	NUM
ma-233	142	33	:	:	PUNCT
ma-233	142	34	x	x	X
ma-233	142	35	∈	∈	X
ma-233	142	36	g}on	g}on	PROPN
ma-233	142	37	the	the	DET
ma-233	142	38	other	other	ADJ
ma-233	142	39	hand	hand	NOUN
ma-233	142	40	,	,	PUNCT
ma-233	142	41	the	the	DET
ma-233	142	42	mapping	mapping	NOUN
ma-233	142	43	g	g	NOUN
ma-233	142	44	−	−	PROPN
ma-233	142	45	g′	g′	NOUN
ma-233	142	46	satisfy	satisfy	NOUN
ma-233	142	47	(	(	PUNCT
ma-233	142	48	2.6	2.6	NUM
ma-233	142	49	)	)	PUNCT
ma-233	142	50	and(2.8	and(2.8	NUM
ma-233	142	51	)	)	PUNCT
ma-233	142	52	g(2x)−	g(2x)−	PROPN
ma-233	142	53	g′(2x	g′(2x	NOUN
ma-233	142	54	)	)	PUNCT
ma-233	142	55	=	=	PUNCT
ma-233	143	1	2n	2n	NUM
ma-233	144	1	+	+	CCONJ
ma-233	144	2	1	1	NUM
ma-233	144	3	2	2	NUM
ma-233	144	4	·	·	PUNCT
ma-233	144	5	4n	4n	X
ma-233	144	6	[	[	PUNCT
ma-233	144	7	g	g	PROPN
ma-233	144	8	(	(	PUNCT
ma-233	144	9	2n+1x	2n+1x	NUM
ma-233	144	10	)	)	PUNCT
ma-233	144	11	−	−	PROPN
ma-233	145	1	g′	g′	NOUN
ma-233	145	2	(	(	PUNCT
ma-233	145	3	2n+1x	2n+1x	NUM
ma-233	145	4	)	)	PUNCT
ma-233	145	5	]	]	PUNCT
ma-233	146	1	−	−	PROPN
ma-233	146	2	2n	2n	NUM
ma-233	146	3	−	−	NOUN
ma-233	146	4	1	1	NUM
ma-233	146	5	2	2	NUM
ma-233	146	6	·	·	PUNCT
ma-233	146	7	4n	4n	X
ma-233	146	8	[	[	PUNCT
ma-233	146	9	g	g	NOUN
ma-233	146	10	(	(	PUNCT
ma-233	146	11	−2n+1x	−2n+1x	PROPN
ma-233	146	12	)	)	PUNCT
ma-233	146	13	−	−	PROPN
ma-233	146	14	g′	g′	NOUN
ma-233	146	15	(	(	PUNCT
ma-233	146	16	−2n+1x	−2n+1x	PROPN
ma-233	146	17	)	)	PUNCT
ma-233	146	18	]	]	PUNCT
ma-233	146	19	(	(	PUNCT
ma-233	146	20	2.19	2.19	NUM
ma-233	146	21	)	)	PUNCT
ma-233	146	22	and	and	CCONJ
ma-233	146	23	therefore∥∥g(2x)−	therefore∥∥g(2x)−	PROPN
ma-233	146	24	g′(2x)∥∥	g′(2x)∥∥	PROPN
ma-233	146	25	6	6	NUM
ma-233	146	26	max	max	NOUN
ma-233	146	27	{	{	PUNCT
ma-233	146	28	(	(	PUNCT
ma-233	146	29	2n	2n	NUM
ma-233	146	30	+	+	CCONJ
ma-233	146	31	1	1	NUM
ma-233	146	32	2	2	NUM
ma-233	146	33	·	·	SYM
ma-233	146	34	4n	4n	X
ma-233	146	35	)	)	PUNCT
ma-233	146	36	∥∥g	∥∥g	PROPN
ma-233	146	37	(	(	PUNCT
ma-233	146	38	2n+xx)−	2n+xx)−	NUM
ma-233	146	39	g′	g′	NOUN
ma-233	146	40	(	(	PUNCT
ma-233	146	41	2n+1	2n+1	PROPN
ma-233	146	42	·	·	PUNCT
ma-233	146	43	x)∥∥	x)∥∥	PUNCT
ma-233	147	1	,	,	PUNCT
ma-233	147	2	(	(	PUNCT
ma-233	147	3	2n	2n	NUM
ma-233	147	4	−	−	NOUN
ma-233	147	5	1	1	NUM
ma-233	147	6	2	2	NUM
ma-233	147	7	·	·	SYM
ma-233	147	8	4n	4n	X
ma-233	147	9	)	)	PUNCT
ma-233	147	10	∥∥g	∥∥g	PROPN
ma-233	147	11	(	(	PUNCT
ma-233	147	12	−2n+1x)−	−2n+1x)−	PROPN
ma-233	147	13	g′	g′	NOUN
ma-233	147	14	(	(	PUNCT
ma-233	147	15	−2n+1x)∥∥	−2n+1x)∥∥	PROPN
ma-233	147	16	6	6	NUM
ma-233	147	17	max	max	NOUN
ma-233	147	18	{	{	PUNCT
ma-233	147	19	(	(	PUNCT
ma-233	147	20	2n	2n	NUM
ma-233	148	1	+	+	CCONJ
ma-233	148	2	1	1	NUM
ma-233	148	3	2	2	NUM
ma-233	148	4	·	·	PUNCT
ma-233	148	5	4n	4n	X
ma-233	148	6	)	)	PUNCT
ma-233	148	7	ε	ε	PROPN
ma-233	148	8	,	,	PUNCT
ma-233	148	9	(	(	PUNCT
ma-233	148	10	2n	2n	NUM
ma-233	148	11	−	−	NOUN
ma-233	148	12	1	1	NUM
ma-233	148	13	2	2	NUM
ma-233	148	14	·	·	PUNCT
ma-233	148	15	4n	4n	X
ma-233	148	16	)	)	PUNCT
ma-233	148	17	ε	ε	PROPN
ma-233	148	18	}	}	PUNCT
ma-233	148	19	=	=	SYM
ma-233	148	20	2n	2n	NUM
ma-233	149	1	+	+	CCONJ
ma-233	149	2	1	1	NUM
ma-233	149	3	2	2	NUM
ma-233	149	4	·	·	PUNCT
ma-233	149	5	4n	4n	X
ma-233	149	6	ε	ε	PROPN
ma-233	149	7	(	(	PUNCT
ma-233	149	8	2.20	2.20	NUM
ma-233	149	9	)	)	PUNCT
ma-233	149	10	for	for	ADP
ma-233	149	11	x	x	PROPN
ma-233	149	12	∈	∈	PROPN
ma-233	149	13	g.	g.	NOUN
ma-233	149	14	by	by	ADP
ma-233	149	15	using	use	VERB
ma-233	149	16	the	the	DET
ma-233	149	17	nonnegativity	nonnegativity	NOUN
ma-233	149	18	of	of	ADP
ma-233	149	19	norm	norm	NOUN
ma-233	149	20	and	and	CCONJ
ma-233	149	21	the	the	DET
ma-233	149	22	forced	force	VERB
ma-233	149	23	convergence	convergence	NOUN
ma-233	149	24	we	we	PRON
ma-233	149	25	can	can	AUX
ma-233	149	26	get	get	VERB
ma-233	149	27	that	that	PRON
ma-233	149	28	themapping	themappe	VERB
ma-233	149	29	g	g	NOUN
ma-233	149	30	is	be	AUX
ma-233	149	31	unique	unique	ADJ
ma-233	149	32	on	on	ADP
ma-233	149	33	the	the	DET
ma-233	149	34	set	set	ADJ
ma-233	149	35	2	2	NUM
ma-233	149	36	g.	g.	NOUN
ma-233	149	37	�	�	PROPN
ma-233	149	38	3	3	NUM
ma-233	149	39	.	.	PUNCT
ma-233	149	40	stability	stability	NOUN
ma-233	149	41	of	of	ADP
ma-233	149	42	additive	additive	NOUN
ma-233	149	43	-	-	PUNCT
ma-233	149	44	additive	additive	ADJ
ma-233	149	45	and	and	CCONJ
ma-233	149	46	orthogonally	orthogonally	ADV
ma-233	149	47	quadratic	quadratic	ADJ
ma-233	149	48	-	-	PUNCT
ma-233	149	49	quadratic	quadratic	ADJ
ma-233	149	50	functional	functional	ADJ
ma-233	149	51	equation	equation	NOUN
ma-233	149	52	in	in	ADP
ma-233	149	53	this	this	DET
ma-233	149	54	section	section	NOUN
ma-233	149	55	,	,	PUNCT
ma-233	149	56	we	we	PRON
ma-233	149	57	substituted	substitute	VERB
ma-233	149	58	the	the	DET
ma-233	149	59	equations	equation	NOUN
ma-233	149	60	with	with	ADP
ma-233	149	61	the	the	DET
ma-233	149	62	orthogonally	orthogonally	ADV
ma-233	149	63	additive	additive	ADJ
ma-233	149	64	-	-	PUNCT
ma-233	149	65	additive	additive	NOUN
ma-233	149	66	and	and	CCONJ
ma-233	149	67	orthog	orthog	NOUN
ma-233	149	68	-	-	PUNCT
ma-233	149	69	onally	onally	ADV
ma-233	149	70	quadratic	quadratic	ADJ
ma-233	149	71	-	-	PUNCT
ma-233	149	72	quadratic	quadratic	ADJ
ma-233	149	73	functional	functional	ADJ
ma-233	149	74	equation	equation	NOUN
ma-233	149	75	concerning	concern	VERB
ma-233	149	76	[	[	X
ma-233	149	77	12	12	NUM
ma-233	149	78	]	]	PUNCT
ma-233	149	79	in	in	ADP
ma-233	149	80	the	the	DET
ma-233	149	81	same	same	ADJ
ma-233	149	82	method	method	NOUN
ma-233	149	83	and	and	CCONJ
ma-233	149	84	by	by	ADP
ma-233	149	85	referringto	referringto	NOUN
ma-233	149	86	the	the	DET
ma-233	149	87	stability	stability	NOUN
ma-233	149	88	proof	proof	NOUN
ma-233	149	89	of	of	ADP
ma-233	149	90	[	[	X
ma-233	149	91	13	13	NUM
ma-233	149	92	,	,	PUNCT
ma-233	149	93	14	14	NUM
ma-233	149	94	]	]	PUNCT
ma-233	149	95	,	,	PUNCT
ma-233	149	96	we	we	PRON
ma-233	149	97	define	define	VERB
ma-233	149	98	d2(x	d2(x	PROPN
ma-233	149	99	,	,	PUNCT
ma-233	149	100	y	y	PROPN
ma-233	149	101	,	,	PUNCT
ma-233	149	102	z	z	PROPN
ma-233	149	103	)	)	PUNCT
ma-233	149	104	as	as	ADP
ma-233	149	105	the	the	DET
ma-233	149	106	followig	followig	PROPN
ma-233	149	107	d2f	d2f	PROPN
ma-233	149	108	(	(	PUNCT
ma-233	149	109	x	x	X
ma-233	149	110	,	,	PUNCT
ma-233	149	111	y	y	PROPN
ma-233	149	112	,	,	PUNCT
ma-233	149	113	z	z	PROPN
ma-233	149	114	)	)	PUNCT
ma-233	150	1	=	=	SYM
ma-233	150	2	f	f	X
ma-233	150	3	(	(	PUNCT
ma-233	150	4	x	x	PROPN
ma-233	150	5	+	+	NUM
ma-233	150	6	y	y	PROPN
ma-233	151	1	+	+	CCONJ
ma-233	151	2	z	z	NOUN
ma-233	151	3	2	2	NUM
ma-233	151	4	)	)	PUNCT
ma-233	152	1	+	+	CCONJ
ma-233	152	2	f	f	X
ma-233	152	3	(	(	PUNCT
ma-233	152	4	x	x	X
ma-233	152	5	+	+	NUM
ma-233	152	6	y	y	PROPN
ma-233	152	7	−	−	PROPN
ma-233	152	8	z	z	NOUN
ma-233	152	9	2	2	NUM
ma-233	152	10	)	)	PUNCT
ma-233	153	1	+	+	CCONJ
ma-233	153	2	f	f	X
ma-233	153	3	(	(	PUNCT
ma-233	153	4	x	x	SYM
ma-233	153	5	−	−	PROPN
ma-233	153	6	y	y	PROPN
ma-233	153	7	+	+	CCONJ
ma-233	153	8	z	z	NOUN
ma-233	153	9	2	2	NUM
ma-233	153	10	)	)	PUNCT
ma-233	154	1	+	+	CCONJ
ma-233	154	2	f	f	X
ma-233	154	3	(	(	PUNCT
ma-233	154	4	y	y	PROPN
ma-233	154	5	+	+	NOUN
ma-233	154	6	z	z	NOUN
ma-233	154	7	−	−	NOUN
ma-233	154	8	x	x	SYM
ma-233	154	9	2	2	X
ma-233	154	10	)	)	PUNCT
ma-233	155	1	−	−	PROPN
ma-233	155	2	f	f	X
ma-233	155	3	(	(	PUNCT
ma-233	155	4	x)−	x)−	PROPN
ma-233	155	5	f	f	PROPN
ma-233	155	6	(	(	PUNCT
ma-233	155	7	y)−	y)−	PROPN
ma-233	155	8	f	f	X
ma-233	155	9	(	(	PUNCT
ma-233	155	10	z	z	NOUN
ma-233	155	11	)	)	PUNCT
ma-233	155	12	theorem	theorem	NOUN
ma-233	155	13	3.1	3.1	NUM
ma-233	155	14	.	.	PUNCT
ma-233	155	15	suppose	suppose	VERB
ma-233	155	16	f	f	X
ma-233	155	17	:	:	PUNCT
ma-233	155	18	g	g	PROPN
ma-233	155	19	→	→	SYM
ma-233	155	20	x	x	X
ma-233	155	21	where	where	SCONJ
ma-233	155	22	f	f	PROPN
ma-233	155	23	is	be	AUX
ma-233	155	24	a	a	DET
ma-233	155	25	mapping	mapping	NOUN
ma-233	155	26	from	from	ADP
ma-233	155	27	an	an	DET
ma-233	155	28	abelian	abelian	ADJ
ma-233	155	29	group	group	NOUN
ma-233	155	30	to	to	ADP
ma-233	155	31	a	a	DET
ma-233	155	32	completenon	completenon	NOUN
ma-233	155	33	-	-	PUNCT
ma-233	155	34	archimedean	archimedean	ADJ
ma-233	155	35	normed	normed	ADJ
ma-233	155	36	space	space	NOUN
ma-233	155	37	.	.	PUNCT
ma-233	156	1	for	for	ADP
ma-233	156	2	ε	ε	PROPN
ma-233	156	3	>	>	X
ma-233	156	4	0	0	PROPN
ma-233	156	5	,	,	PUNCT
ma-233	156	6	when	when	SCONJ
ma-233	156	7	x	x	PROPN
ma-233	156	8	⊥	⊥	PROPN
ma-233	156	9	y	y	PROPN
ma-233	156	10	for	for	ADP
ma-233	156	11	all	all	DET
ma-233	156	12	x	x	NOUN
ma-233	156	13	,	,	PUNCT
ma-233	156	14	y	y	PROPN
ma-233	156	15	,	,	PUNCT
ma-233	156	16	z	z	PROPN
ma-233	156	17	∈	∈	PROPN
ma-233	156	18	g	g	PROPN
ma-233	156	19	,	,	PUNCT
ma-233	156	20	we	we	PRON
ma-233	156	21	obtain	obtain	VERB
ma-233	156	22	‖d2f	‖d2f	PROPN
ma-233	156	23	(	(	PUNCT
ma-233	156	24	x	x	X
ma-233	156	25	,	,	PUNCT
ma-233	156	26	y	y	PROPN
ma-233	156	27	,	,	PUNCT
ma-233	156	28	z)‖	z)‖	PROPN
ma-233	156	29	6	6	NUM
ma-233	156	30	ε	ε	PROPN
ma-233	156	31	(	(	PUNCT
ma-233	156	32	3.1	3.1	NUM
ma-233	156	33	)	)	PUNCT
ma-233	156	34	and	and	CCONJ
ma-233	156	35	‖f	‖f	ADP
ma-233	156	36	(	(	PUNCT
ma-233	156	37	x	x	X
ma-233	156	38	)	)	PUNCT
ma-233	156	39	+	+	NUM
ma-233	156	40	f	f	X
ma-233	156	41	(	(	PUNCT
ma-233	156	42	−x)‖	−x)‖	PROPN
ma-233	156	43	6	6	NUM
ma-233	156	44	ε	ε	PROPN
ma-233	156	45	.	.	PUNCT
ma-233	157	1	(	(	PUNCT
ma-233	157	2	3.2	3.2	NUM
ma-233	157	3	)	)	PUNCT
ma-233	157	4	https://doi.org/10.28924/ada/ma.4.11	https://doi.org/10.28924/ada/ma.4.11	PROPN
ma-233	157	5	eur	eur	NOUN
ma-233	157	6	.	.	PUNCT
ma-233	158	1	j.	j.	PROPN
ma-233	158	2	math	math	PROPN
ma-233	158	3	.	.	PUNCT
ma-233	159	1	anal	anal	PROPN
ma-233	159	2	.	.	PUNCT
ma-233	160	1	10.28924	10.28924	NUM
ma-233	160	2	/	/	SYM
ma-233	160	3	ada	ada	PROPN
ma-233	160	4	/	/	SYM
ma-233	160	5	ma.4.11	ma.4.11	ADJ
ma-233	161	1	8then	8then	X
ma-233	161	2	there	there	PRON
ma-233	161	3	exists	exist	VERB
ma-233	161	4	a	a	DET
ma-233	161	5	unique	unique	ADJ
ma-233	161	6	mapping	mapping	NOUN
ma-233	161	7	g	g	NOUN
ma-233	161	8	:	:	PUNCT
ma-233	161	9	x	x	X
ma-233	161	10	→	→	PUNCT
ma-233	161	11	y	y	NUM
ma-233	161	12	such	such	ADJ
ma-233	161	13	that	that	SCONJ
ma-233	161	14	x	x	PROPN
ma-233	161	15	⊥	⊥	NOUN
ma-233	161	16	y	y	PROPN
ma-233	161	17	implies	imply	VERB
ma-233	161	18	g	g	PROPN
ma-233	161	19	(	(	PUNCT
ma-233	161	20	x	x	PROPN
ma-233	162	1	+	+	NUM
ma-233	162	2	y	y	PROPN
ma-233	163	1	+	+	CCONJ
ma-233	163	2	z	z	NOUN
ma-233	163	3	2	2	NUM
ma-233	163	4	)	)	PUNCT
ma-233	164	1	+	+	ADP
ma-233	164	2	g	g	NOUN
ma-233	164	3	(	(	PUNCT
ma-233	164	4	x	x	PROPN
ma-233	164	5	+	+	NUM
ma-233	164	6	y	y	PROPN
ma-233	164	7	−	−	PROPN
ma-233	164	8	z	z	NOUN
ma-233	164	9	2	2	NUM
ma-233	164	10	)	)	PUNCT
ma-233	165	1	+	+	ADP
ma-233	165	2	g	g	NOUN
ma-233	165	3	(	(	PUNCT
ma-233	165	4	x	x	SYM
ma-233	165	5	−	−	PROPN
ma-233	165	6	y	y	PROPN
ma-233	165	7	+	+	CCONJ
ma-233	165	8	z	z	NOUN
ma-233	165	9	2	2	NUM
ma-233	165	10	)	)	PUNCT
ma-233	166	1	+	+	ADP
ma-233	166	2	g	g	NOUN
ma-233	166	3	(	(	PUNCT
ma-233	166	4	y	y	PROPN
ma-233	166	5	+	+	NOUN
ma-233	166	6	z	z	NOUN
ma-233	166	7	−	−	NOUN
ma-233	166	8	x	x	SYM
ma-233	166	9	2	2	X
ma-233	166	10	)	)	PUNCT
ma-233	166	11	=	=	SYM
ma-233	166	12	g(x)+g(y)+g(z	g(x)+g(y)+g(z	NOUN
ma-233	166	13	)	)	PUNCT
ma-233	166	14	(	(	PUNCT
ma-233	166	15	3.3	3.3	NUM
ma-233	166	16	)	)	PUNCT
ma-233	166	17	and	and	CCONJ
ma-233	166	18	‖f	‖f	ADP
ma-233	166	19	(	(	PUNCT
ma-233	166	20	x)−	x)−	PROPN
ma-233	166	21	g(x)‖	g(x)‖	PROPN
ma-233	166	22	6	6	NUM
ma-233	166	23	ε	ε	PROPN
ma-233	166	24	(	(	PUNCT
ma-233	166	25	3.4)for	3.4)for	ADP
ma-233	166	26	all	all	PRON
ma-233	166	27	x	x	PUNCT
ma-233	166	28	∈	∈	PROPN
ma-233	166	29	2	2	NUM
ma-233	166	30	g	g	NOUN
ma-233	166	31	=	=	PUNCT
ma-233	166	32	{	{	PUNCT
ma-233	166	33	2x	2x	NUM
ma-233	166	34	:	:	PUNCT
ma-233	166	35	x	x	SYM
ma-233	166	36	∈	∈	NOUN
ma-233	166	37	g	g	NOUN
ma-233	166	38	}	}	PUNCT
ma-233	166	39	.	.	PUNCT
ma-233	167	1	proof	proof	NOUN
ma-233	167	2	.	.	PUNCT
ma-233	168	1	for	for	ADP
ma-233	168	2	all	all	DET
ma-233	168	3	x	x	SYM
ma-233	168	4	∈	∈	PROPN
ma-233	168	5	g	g	NOUN
ma-233	168	6	,	,	PUNCT
ma-233	168	7	since	since	SCONJ
ma-233	168	8	0	0	NUM
ma-233	168	9	⊥	⊥	NOUN
ma-233	168	10	x	x	SYM
ma-233	168	11	,	,	PUNCT
ma-233	168	12	x	x	PROPN
ma-233	168	13	⊥	⊥	NOUN
ma-233	168	14	0	0	NUM
ma-233	168	15	,	,	PUNCT
ma-233	168	16	and	and	CCONJ
ma-233	168	17	0	0	NUM
ma-233	168	18	⊥	⊥	NOUN
ma-233	168	19	0	0	NUM
ma-233	168	20	,	,	PUNCT
ma-233	168	21	setting	set	VERB
ma-233	168	22	x	x	PUNCT
ma-233	168	23	=	=	SYM
ma-233	168	24	0	0	NUM
ma-233	168	25	,	,	PUNCT
ma-233	168	26	y	y	PROPN
ma-233	168	27	=	=	SYM
ma-233	168	28	0	0	NUM
ma-233	168	29	,	,	PUNCT
ma-233	168	30	z	z	NOUN
ma-233	168	31	=	=	SYM
ma-233	168	32	0	0	NUM
ma-233	168	33	in	in	ADP
ma-233	168	34	inquality	inquality	NOUN
ma-233	168	35	(	(	PUNCT
ma-233	168	36	3.1),we	3.1),we	NUM
ma-233	168	37	obtain	obtain	VERB
ma-233	168	38	‖f	‖f	PRON
ma-233	168	39	(	(	PUNCT
ma-233	168	40	0)‖	0)‖	PROPN
ma-233	168	41	6	6	NUM
ma-233	168	42	ε	ε	PROPN
ma-233	168	43	,	,	PUNCT
ma-233	168	44	then	then	ADV
ma-233	168	45	similarily	similarily	ADV
ma-233	168	46	setting	set	VERB
ma-233	168	47	y	y	PROPN
ma-233	168	48	=	=	SYM
ma-233	168	49	0	0	NUM
ma-233	168	50	,	,	PUNCT
ma-233	168	51	z	z	NOUN
ma-233	168	52	=	=	SYM
ma-233	168	53	0	0	NUM
ma-233	168	54	in	in	ADP
ma-233	168	55	inequality	inequality	NOUN
ma-233	168	56	(	(	PUNCT
ma-233	168	57	3.1	3.1	NUM
ma-233	168	58	)	)	PUNCT
ma-233	168	59	,	,	PUNCT
ma-233	168	60	we	we	PRON
ma-233	168	61	obtain∥∥∥∥3f	obtain∥∥∥∥3f	VERB
ma-233	168	62	(	(	PUNCT
ma-233	168	63	x2)+	x2)+	PROPN
ma-233	168	64	f	f	PROPN
ma-233	168	65	(	(	PUNCT
ma-233	168	66	−x	−x	NOUN
ma-233	168	67	2	2	NUM
ma-233	168	68	)	)	PUNCT
ma-233	169	1	−	−	PROPN
ma-233	169	2	f	f	X
ma-233	169	3	(	(	PUNCT
ma-233	169	4	x)−	x)−	PROPN
ma-233	169	5	2f	2f	X
ma-233	169	6	(	(	PUNCT
ma-233	169	7	0	0	NUM
ma-233	169	8	)	)	PUNCT
ma-233	169	9	∥∥∥∥	∥∥∥∥	NUM
ma-233	169	10	6	6	NUM
ma-233	169	11	ε	ε	PROPN
ma-233	169	12	(	(	PUNCT
ma-233	169	13	3.5	3.5	NUM
ma-233	169	14	)	)	PUNCT
ma-233	169	15	then	then	ADV
ma-233	169	16	,	,	PUNCT
ma-233	169	17	by	by	ADP
ma-233	169	18	using	use	VERB
ma-233	169	19	the	the	DET
ma-233	169	20	strong	strong	ADJ
ma-233	169	21	triangle	triangle	NOUN
ma-233	169	22	inequality	inequality	NOUN
ma-233	169	23	,	,	PUNCT
ma-233	169	24	we	we	PRON
ma-233	169	25	obtain∥∥∥∥3f	obtain∥∥∥∥3f	VERB
ma-233	169	26	(	(	PUNCT
ma-233	169	27	x2)+	x2)+	PROPN
ma-233	169	28	f	f	PROPN
ma-233	169	29	(	(	PUNCT
ma-233	169	30	−x	−x	NOUN
ma-233	169	31	2	2	NUM
ma-233	169	32	)	)	PUNCT
ma-233	170	1	−	−	PROPN
ma-233	170	2	f	f	PROPN
ma-233	170	3	(	(	PUNCT
ma-233	170	4	x	x	NOUN
ma-233	170	5	)	)	PUNCT
ma-233	170	6	∥∥∥∥	∥∥∥∥	NUM
ma-233	170	7	6	6	NUM
ma-233	170	8	max	max	PROPN
ma-233	170	9	{	{	PUNCT
ma-233	170	10	‖2f	‖2f	PROPN
ma-233	170	11	(	(	PUNCT
ma-233	170	12	0)‖	0)‖	PROPN
ma-233	170	13	,	,	PUNCT
ma-233	170	14	∥∥∥3f	∥∥∥3f	PROPN
ma-233	170	15	(	(	PUNCT
ma-233	170	16	x	x	SYM
ma-233	170	17	2	2	NUM
ma-233	170	18	)	)	PUNCT
ma-233	170	19	+	+	CCONJ
ma-233	170	20	f	f	X
ma-233	170	21	(	(	PUNCT
ma-233	170	22	−	−	PROPN
ma-233	170	23	x	x	SYM
ma-233	170	24	2	2	X
ma-233	170	25	)	)	PUNCT
ma-233	170	26	−	−	PROPN
ma-233	171	1	f	f	X
ma-233	171	2	(	(	PUNCT
ma-233	171	3	x)−	x)−	PROPN
ma-233	171	4	2f	2f	X
ma-233	171	5	(	(	PUNCT
ma-233	171	6	0	0	NUM
ma-233	171	7	)	)	PUNCT
ma-233	171	8	∥∥∥	∥∥∥	NOUN
ma-233	171	9	}	}	PUNCT
ma-233	171	10	=	=	SYM
ma-233	171	11	2ε	2ε	NOUN
ma-233	171	12	(	(	PUNCT
ma-233	171	13	3.6	3.6	NUM
ma-233	171	14	)	)	PUNCT
ma-233	171	15	by	by	ADP
ma-233	171	16	replacing	replace	VERB
ma-233	171	17	x	x	PRON
ma-233	171	18	witn	witn	NOUN
ma-233	171	19	2x	2x	NUM
ma-233	171	20	in	in	ADP
ma-233	171	21	(	(	PUNCT
ma-233	171	22	3.6	3.6	NUM
ma-233	171	23	)	)	PUNCT
ma-233	171	24	,	,	PUNCT
ma-233	171	25	we	we	PRON
ma-233	171	26	obtain	obtain	VERB
ma-233	171	27	‖3f	‖3f	PROPN
ma-233	171	28	(	(	PUNCT
ma-233	171	29	x	x	NOUN
ma-233	171	30	)	)	PUNCT
ma-233	172	1	+	+	NUM
ma-233	172	2	f	f	X
ma-233	172	3	(	(	PUNCT
ma-233	172	4	−x)−	−x)−	PROPN
ma-233	172	5	f	f	PROPN
ma-233	172	6	(	(	PUNCT
ma-233	172	7	2x)‖	2x)‖	NUM
ma-233	172	8	6	6	NUM
ma-233	172	9	2ε	2ε	NOUN
ma-233	172	10	(	(	PUNCT
ma-233	172	11	3.7	3.7	NUM
ma-233	172	12	)	)	PUNCT
ma-233	172	13	by	by	ADP
ma-233	172	14	using	use	VERB
ma-233	172	15	the	the	DET
ma-233	172	16	strong	strong	ADJ
ma-233	172	17	triangle	triangle	NOUN
ma-233	172	18	inequality	inequality	NOUN
ma-233	172	19	twice	twice	ADV
ma-233	172	20	,	,	PUNCT
ma-233	172	21	we	we	PRON
ma-233	172	22	can	can	AUX
ma-233	172	23	easily	easily	ADV
ma-233	172	24	obtain	obtain	VERB
ma-233	172	25	‖2f	‖2f	PROPN
ma-233	172	26	(	(	PUNCT
ma-233	172	27	x)−	x)−	PROPN
ma-233	172	28	f	f	PROPN
ma-233	172	29	(	(	PUNCT
ma-233	172	30	2x)‖	2x)‖	NUM
ma-233	172	31	6	6	NUM
ma-233	172	32	max{‖f	max{‖f	PROPN
ma-233	172	33	(	(	PUNCT
ma-233	172	34	x	x	NOUN
ma-233	172	35	)	)	PUNCT
ma-233	173	1	+	+	NUM
ma-233	173	2	f	f	X
ma-233	173	3	(	(	PUNCT
ma-233	173	4	−x)‖	−x)‖	PROPN
ma-233	173	5	,	,	PUNCT
ma-233	173	6	‖3f	‖3f	PROPN
ma-233	173	7	(	(	PUNCT
ma-233	173	8	x	x	NOUN
ma-233	173	9	)	)	PUNCT
ma-233	174	1	+	+	NUM
ma-233	174	2	f	f	X
ma-233	174	3	(	(	PUNCT
ma-233	174	4	−x)−	−x)−	PROPN
ma-233	174	5	f	f	PROPN
ma-233	174	6	(	(	PUNCT
ma-233	174	7	2x)‖	2x)‖	NUM
ma-233	174	8	}	}	PUNCT
ma-233	174	9	=	=	SYM
ma-233	174	10	2ε	2ε	NOUN
ma-233	174	11	(	(	PUNCT
ma-233	174	12	3.8	3.8	NUM
ma-233	174	13	)	)	PUNCT
ma-233	174	14	by	by	ADP
ma-233	174	15	replacing	replace	VERB
ma-233	174	16	x	x	PRON
ma-233	174	17	witn	witn	NOUN
ma-233	174	18	4x	4x	NUM
ma-233	174	19	in	in	ADP
ma-233	174	20	(	(	PUNCT
ma-233	174	21	3.2),then	3.2),then	NUM
ma-233	174	22	combining	combine	VERB
ma-233	174	23	the	the	DET
ma-233	174	24	following	following	NOUN
ma-233	174	25	with	with	ADP
ma-233	174	26	(	(	PUNCT
ma-233	174	27	3.2)and(3.8	3.2)and(3.8	NUM
ma-233	174	28	)	)	PUNCT
ma-233	174	29	,	,	PUNCT
ma-233	174	30	we	we	PRON
ma-233	174	31	can	can	AUX
ma-233	174	32	concludethat	concludethat	VERB
ma-233	174	33	‖3f	‖3f	PROPN
ma-233	174	34	(	(	PUNCT
ma-233	174	35	4x)−	4x)−	PROPN
ma-233	174	36	8f	8f	NOUN
ma-233	174	37	(	(	PUNCT
ma-233	174	38	2x)−	2x)−	NUM
ma-233	174	39	f	f	X
ma-233	174	40	(	(	PUNCT
ma-233	174	41	−4x)‖	−4x)‖	PROPN
ma-233	174	42	=	=	PUNCT
ma-233	174	43	‖4[f	‖4[f	PROPN
ma-233	174	44	(	(	PUNCT
ma-233	174	45	4x)−	4x)−	PROPN
ma-233	174	46	2f	2f	NUM
ma-233	174	47	(	(	PUNCT
ma-233	174	48	2x)]−	2x)]−	PROPN
ma-233	175	1	[	[	X
ma-233	175	2	f	f	X
ma-233	175	3	(	(	PUNCT
ma-233	175	4	4ẋ	4ẋ	NUM
ma-233	175	5	)	)	PUNCT
ma-233	175	6	+	+	NUM
ma-233	175	7	f	f	X
ma-233	175	8	(	(	PUNCT
ma-233	175	9	−4x)]‖	−4x)]‖	X
ma-233	175	10	6	6	NUM
ma-233	175	11	max{8ε	max{8ε	PROPN
ma-233	175	12	,	,	PUNCT
ma-233	175	13	ε	ε	NOUN
ma-233	175	14	}	}	PUNCT
ma-233	175	15	=	=	NUM
ma-233	175	16	8ε	8ε	NUM
ma-233	175	17	(	(	PUNCT
ma-233	175	18	3.9	3.9	NUM
ma-233	175	19	)	)	PUNCT
ma-233	175	20	then	then	ADV
ma-233	175	21	dividing	divide	VERB
ma-233	175	22	both	both	DET
ma-233	175	23	side	side	NOUN
ma-233	175	24	of	of	ADP
ma-233	175	25	the	the	DET
ma-233	175	26	inequality	inequality	NOUN
ma-233	175	27	by	by	ADP
ma-233	175	28	8,we	8,we	PROPN
ma-233	175	29	obtain∥∥∥∥f	obtain∥∥∥∥f	PROPN
ma-233	175	30	(	(	PUNCT
ma-233	175	31	2x)−	2x)−	NUM
ma-233	175	32	38	38	NUM
ma-233	175	33	f	f	NOUN
ma-233	175	34	(	(	PUNCT
ma-233	175	35	4x	4x	NUM
ma-233	175	36	)	)	PUNCT
ma-233	175	37	+	+	CCONJ
ma-233	175	38	18	18	NUM
ma-233	175	39	f	f	NOUN
ma-233	175	40	(	(	PUNCT
ma-233	175	41	−4x	−4x	PROPN
ma-233	175	42	)	)	PUNCT
ma-233	175	43	∥∥∥∥	∥∥∥∥	NUM
ma-233	175	44	6	6	NUM
ma-233	175	45	ε	ε	PROPN
ma-233	175	46	(	(	PUNCT
ma-233	175	47	3.10	3.10	NUM
ma-233	175	48	)	)	PUNCT
ma-233	175	49	the	the	DET
ma-233	175	50	next	next	ADJ
ma-233	175	51	step	step	NOUN
ma-233	175	52	resembles	resemble	VERB
ma-233	175	53	lemma2.1	lemma2.1	PROPN
ma-233	175	54	,	,	PUNCT
ma-233	175	55	gn(x	gn(x	PUNCT
ma-233	175	56	)	)	PUNCT
ma-233	175	57	=	=	SYM
ma-233	175	58	2n	2n	NUM
ma-233	176	1	+	+	CCONJ
ma-233	177	1	1	1	NUM
ma-233	177	2	2	2	NUM
ma-233	177	3	·	·	PUNCT
ma-233	177	4	4n	4n	X
ma-233	177	5	f	f	X
ma-233	177	6	(	(	PUNCT
ma-233	177	7	2	2	NUM
ma-233	177	8	nx)−	nx)−	NOUN
ma-233	177	9	2n	2n	NUM
ma-233	177	10	−	−	NOUN
ma-233	177	11	1	1	NUM
ma-233	177	12	2	2	NUM
ma-233	177	13	·	·	PUNCT
ma-233	177	14	4n	4n	X
ma-233	177	15	f	f	X
ma-233	177	16	(	(	PUNCT
ma-233	177	17	−2	−2	PROPN
ma-233	177	18	nx	nx	PROPN
ma-233	177	19	)	)	PUNCT
ma-233	177	20	n	n	PRON
ma-233	177	21	∈	∈	PROPN
ma-233	177	22	n.	n.	NOUN
ma-233	177	23	let	let	VERB
ma-233	177	24	g	g	NOUN
ma-233	177	25	:	:	PUNCT
ma-233	177	26	g	g	PROPN
ma-233	177	27	→	→	SYM
ma-233	177	28	x	x	SYM
ma-233	177	29	g(x	g(x	NOUN
ma-233	177	30	)	)	PUNCT
ma-233	178	1	=	=	SYM
ma-233	178	2	lim	lim	PROPN
ma-233	178	3	n→∞	n→∞	NUM
ma-233	178	4	gn(x	gn(x	PUNCT
ma-233	178	5	)	)	PUNCT
ma-233	178	6	.	.	PUNCT
ma-233	179	1	https://doi.org/10.28924/ada/ma.4.11	https://doi.org/10.28924/ada/ma.4.11	PROPN
ma-233	179	2	eur	eur	PROPN
ma-233	179	3	.	.	PUNCT
ma-233	180	1	j.	j.	PROPN
ma-233	180	2	math	math	PROPN
ma-233	180	3	.	.	PUNCT
ma-233	181	1	anal	anal	PROPN
ma-233	181	2	.	.	PUNCT
ma-233	182	1	10.28924	10.28924	NUM
ma-233	182	2	/	/	SYM
ma-233	182	3	ada	ada	PROPN
ma-233	182	4	/	/	SYM
ma-233	182	5	ma.4.11	ma.4.11	ADJ
ma-233	182	6	9according	9accorde	VERB
ma-233	182	7	to	to	ADP
ma-233	182	8	lemma2.1	lemma2.1	PROPN
ma-233	182	9	,	,	PUNCT
ma-233	182	10	we	we	PRON
ma-233	182	11	obtain	obtain	VERB
ma-233	182	12	‖f	‖f	PRON
ma-233	182	13	(	(	PUNCT
ma-233	182	14	2x)−	2x)−	NUM
ma-233	182	15	g(2x)‖	g(2x)‖	PROPN
ma-233	182	16	=	=	SYM
ma-233	182	17	‖h(x	‖h(x	PROPN
ma-233	182	18	,	,	PUNCT
ma-233	182	19	n	n	CCONJ
ma-233	182	20	)	)	PUNCT
ma-233	183	1	+	+	CCONJ
ma-233	183	2	gn(2x)−	gn(2x)−	PROPN
ma-233	183	3	g(2x)‖	g(2x)‖	PROPN
ma-233	183	4	6	6	NUM
ma-233	183	5	ε	ε	PROPN
ma-233	183	6	(	(	PUNCT
ma-233	183	7	3.11	3.11	NUM
ma-233	183	8	)	)	PUNCT
ma-233	183	9	for	for	ADP
ma-233	183	10	the	the	DET
ma-233	183	11	purpose	purpose	NOUN
ma-233	183	12	of	of	ADP
ma-233	183	13	proving	prove	VERB
ma-233	183	14	that	that	SCONJ
ma-233	183	15	g	g	PROPN
ma-233	183	16	is	be	AUX
ma-233	183	17	orthogonally	orthogonally	ADV
ma-233	183	18	additive	additive	ADJ
ma-233	183	19	,	,	PUNCT
ma-233	183	20	firstiy	firstiy	NOUN
ma-233	183	21	,	,	PUNCT
ma-233	183	22	we	we	PRON
ma-233	183	23	apply	apply	VERB
ma-233	183	24	the	the	DET
ma-233	183	25	strong	strong	ADJ
ma-233	183	26	triangleinequality	triangleinequality	NOUN
ma-233	183	27	and	and	CCONJ
ma-233	183	28	the	the	DET
ma-233	183	29	nonnegativity	nonnegativity	NOUN
ma-233	183	30	property	property	NOUN
ma-233	183	31	for	for	ADP
ma-233	183	32	the	the	DET
ma-233	183	33	following	following	NOUN
ma-233	183	34	,	,	PUNCT
ma-233	183	35	we	we	PRON
ma-233	183	36	obtain	obtain	VERB
ma-233	183	37	‖d2g(x	‖d2g(x	X
ma-233	183	38	,	,	PUNCT
ma-233	183	39	y	y	PROPN
ma-233	183	40	,	,	PUNCT
ma-233	183	41	z)‖	z)‖	X
ma-233	183	42	=	=	SYM
ma-233	183	43	∥∥∥∥2n	∥∥∥∥2n	PROPN
ma-233	183	44	+	+	CCONJ
ma-233	183	45	12	12	NUM
ma-233	183	46	·	·	SYM
ma-233	183	47	4n	4n	X
ma-233	183	48	d2f	d2f	NOUN
ma-233	183	49	(	(	PUNCT
ma-233	183	50	2	2	NUM
ma-233	183	51	nx	nx	NOUN
ma-233	183	52	,	,	PUNCT
ma-233	183	53	2ny	2ny	ADJ
ma-233	183	54	,	,	PUNCT
ma-233	183	55	2nz	2nz	NOUN
ma-233	183	56	)	)	PUNCT
ma-233	184	1	+	+	CCONJ
ma-233	184	2	2n	2n	NUM
ma-233	184	3	−	−	NOUN
ma-233	184	4	1	1	NUM
ma-233	184	5	2	2	NUM
ma-233	184	6	·	·	PUNCT
ma-233	184	7	4n	4n	X
ma-233	184	8	d2f	d2f	NOUN
ma-233	184	9	(	(	PUNCT
ma-233	184	10	2	2	NUM
ma-233	184	11	nx	nx	NOUN
ma-233	184	12	,	,	PUNCT
ma-233	184	13	2ny	2ny	ADJ
ma-233	184	14	,	,	PUNCT
ma-233	184	15	2nz	2nz	NOUN
ma-233	184	16	)	)	PUNCT
ma-233	184	17	∥∥∥∥	∥∥∥∥	SYM
ma-233	185	1	6max	6max	NUM
ma-233	185	2	{	{	PUNCT
ma-233	185	3	2n	2n	NOUN
ma-233	185	4	+	+	CCONJ
ma-233	185	5	1	1	NUM
ma-233	185	6	2	2	NUM
ma-233	185	7	·	·	PUNCT
ma-233	185	8	4n	4n	X
ma-233	185	9	ε	ε	PROPN
ma-233	185	10	,	,	PUNCT
ma-233	185	11	2n	2n	NUM
ma-233	185	12	−	−	NOUN
ma-233	185	13	1	1	NUM
ma-233	185	14	2	2	NUM
ma-233	185	15	·	·	PUNCT
ma-233	185	16	4n	4n	X
ma-233	185	17	ε	ε	NOUN
ma-233	185	18	}	}	PUNCT
ma-233	185	19	=	=	SYM
ma-233	185	20	2n	2n	NUM
ma-233	186	1	+	+	CCONJ
ma-233	186	2	1	1	NUM
ma-233	186	3	2	2	NUM
ma-233	186	4	·	·	PUNCT
ma-233	186	5	4n	4n	X
ma-233	186	6	ε	ε	PROPN
ma-233	186	7	(	(	PUNCT
ma-233	186	8	3.12	3.12	NUM
ma-233	186	9	)	)	PUNCT
ma-233	186	10	for	for	ADP
ma-233	186	11	all	all	DET
ma-233	186	12	x	x	PROPN
ma-233	186	13	,	,	PUNCT
ma-233	186	14	y	y	PROPN
ma-233	186	15	,	,	PUNCT
ma-233	186	16	z	z	PROPN
ma-233	186	17	∈	∈	PROPN
ma-233	186	18	g	g	NOUN
ma-233	186	19	with	with	ADP
ma-233	186	20	x	x	PROPN
ma-233	186	21	⊥	⊥	PROPN
ma-233	186	22	y	y	PROPN
ma-233	186	23	and	and	CCONJ
ma-233	186	24	n	n	CCONJ
ma-233	186	25	∈	∈	PROPN
ma-233	186	26	n	n	CCONJ
ma-233	186	27	,	,	PUNCT
ma-233	186	28	n	n	X
ma-233	186	29	>	>	X
ma-233	186	30	1	1	X
ma-233	186	31	.	.	PUNCT
ma-233	186	32	when	when	SCONJ
ma-233	186	33	we	we	PRON
ma-233	186	34	let	let	VERB
ma-233	186	35	n	n	PRON
ma-233	186	36	→	→	SYM
ma-233	186	37	∞	∞	PROPN
ma-233	186	38	,	,	PUNCT
ma-233	186	39	we	we	PRON
ma-233	186	40	get	get	VERB
ma-233	186	41	(	(	PUNCT
ma-233	186	42	3.3	3.3	NUM
ma-233	186	43	)	)	PUNCT
ma-233	186	44	.	.	PUNCT
ma-233	187	1	the	the	DET
ma-233	187	2	rest	rest	NOUN
ma-233	187	3	ofproof	ofproof	PROPN
ma-233	187	4	resembles	resemble	VERB
ma-233	187	5	theorem	theorem	VERB
ma-233	187	6	2.1	2.1	NUM
ma-233	187	7	,	,	PUNCT
ma-233	187	8	according	accord	VERB
ma-233	187	9	to	to	ADP
ma-233	187	10	(	(	PUNCT
ma-233	187	11	2.18)to	2.18)to	X
ma-233	187	12	(	(	PUNCT
ma-233	187	13	2.20	2.20	NUM
ma-233	187	14	)	)	PUNCT
ma-233	187	15	,	,	PUNCT
ma-233	187	16	we	we	PRON
ma-233	187	17	can	can	AUX
ma-233	187	18	get	get	VERB
ma-233	187	19	the	the	DET
ma-233	187	20	mapping	mapping	NOUN
ma-233	187	21	g	g	NOUN
ma-233	187	22	is	be	AUX
ma-233	187	23	unique	unique	ADJ
ma-233	187	24	onthe	onthe	NOUN
ma-233	187	25	set	set	VERB
ma-233	187	26	2	2	NUM
ma-233	187	27	g	g	NOUN
ma-233	187	28	similarly	similarly	ADV
ma-233	187	29	.	.	PUNCT
ma-233	188	1	�	�	PROPN
ma-233	188	2	theorem	theorem	VERB
ma-233	188	3	3.2	3.2	NUM
ma-233	188	4	.	.	PUNCT
ma-233	189	1	suppose	suppose	VERB
ma-233	189	2	f	f	X
ma-233	189	3	:	:	PUNCT
ma-233	189	4	g	g	PROPN
ma-233	189	5	→	→	SYM
ma-233	189	6	x	x	X
ma-233	189	7	where	where	SCONJ
ma-233	189	8	f	f	PROPN
ma-233	189	9	is	be	AUX
ma-233	189	10	a	a	DET
ma-233	189	11	mapping	mapping	NOUN
ma-233	189	12	from	from	ADP
ma-233	189	13	an	an	DET
ma-233	189	14	abelian	abelian	ADJ
ma-233	189	15	group	group	NOUN
ma-233	189	16	to	to	ADP
ma-233	189	17	a	a	DET
ma-233	189	18	completenon	completenon	NOUN
ma-233	189	19	-	-	PUNCT
ma-233	189	20	archimedean	archimedean	ADJ
ma-233	189	21	normed	normed	ADJ
ma-233	189	22	space	space	NOUN
ma-233	189	23	.	.	PUNCT
ma-233	190	1	for	for	ADP
ma-233	190	2	ε	ε	PROPN
ma-233	190	3	>	>	X
ma-233	190	4	0	0	PROPN
ma-233	190	5	,	,	PUNCT
ma-233	190	6	when	when	SCONJ
ma-233	190	7	x	x	PROPN
ma-233	190	8	⊥	⊥	PROPN
ma-233	190	9	y	y	PROPN
ma-233	190	10	for	for	ADP
ma-233	190	11	all	all	DET
ma-233	190	12	x	x	NOUN
ma-233	190	13	,	,	PUNCT
ma-233	190	14	y	y	PROPN
ma-233	190	15	,	,	PUNCT
ma-233	190	16	z	z	PROPN
ma-233	190	17	∈	∈	PROPN
ma-233	190	18	g	g	PROPN
ma-233	190	19	,	,	PUNCT
ma-233	190	20	we	we	PRON
ma-233	190	21	obtain	obtain	VERB
ma-233	190	22	‖d2f	‖d2f	PROPN
ma-233	190	23	(	(	PUNCT
ma-233	190	24	x	x	X
ma-233	190	25	,	,	PUNCT
ma-233	190	26	y	y	PROPN
ma-233	190	27	,	,	PUNCT
ma-233	190	28	z)‖	z)‖	PROPN
ma-233	190	29	6	6	NUM
ma-233	190	30	ε	ε	PROPN
ma-233	190	31	(	(	PUNCT
ma-233	190	32	3.13	3.13	NUM
ma-233	190	33	)	)	PUNCT
ma-233	190	34	and	and	CCONJ
ma-233	190	35	‖f	‖f	ADP
ma-233	190	36	(	(	PUNCT
ma-233	190	37	x)−	x)−	PROPN
ma-233	190	38	f	f	PROPN
ma-233	190	39	(	(	PUNCT
ma-233	190	40	−x)‖	−x)‖	PROPN
ma-233	190	41	6	6	NUM
ma-233	190	42	ε	ε	PROPN
ma-233	190	43	.	.	PUNCT
ma-233	191	1	(	(	PUNCT
ma-233	191	2	3.14	3.14	NUM
ma-233	191	3	)	)	PUNCT
ma-233	191	4	then	then	ADV
ma-233	191	5	there	there	PRON
ma-233	191	6	exists	exist	VERB
ma-233	191	7	a	a	DET
ma-233	191	8	unique	unique	ADJ
ma-233	191	9	mapping	mapping	NOUN
ma-233	191	10	g	g	NOUN
ma-233	191	11	:	:	PUNCT
ma-233	191	12	x	x	X
ma-233	191	13	→	→	PUNCT
ma-233	191	14	y	y	NUM
ma-233	191	15	such	such	ADJ
ma-233	191	16	that	that	SCONJ
ma-233	191	17	x	x	PROPN
ma-233	191	18	⊥	⊥	NOUN
ma-233	191	19	y	y	PROPN
ma-233	191	20	implies	imply	VERB
ma-233	191	21	g	g	PROPN
ma-233	191	22	(	(	PUNCT
ma-233	191	23	x	x	PROPN
ma-233	192	1	+	+	NUM
ma-233	192	2	y	y	PROPN
ma-233	193	1	+	+	CCONJ
ma-233	193	2	z	z	NOUN
ma-233	193	3	2	2	NUM
ma-233	193	4	)	)	PUNCT
ma-233	194	1	+	+	ADP
ma-233	194	2	g	g	NOUN
ma-233	194	3	(	(	PUNCT
ma-233	194	4	x	x	PROPN
ma-233	194	5	+	+	NUM
ma-233	194	6	y	y	PROPN
ma-233	194	7	−	−	PROPN
ma-233	194	8	z	z	NOUN
ma-233	194	9	2	2	NUM
ma-233	194	10	)	)	PUNCT
ma-233	195	1	+	+	ADP
ma-233	195	2	g	g	NOUN
ma-233	195	3	(	(	PUNCT
ma-233	195	4	x	x	SYM
ma-233	195	5	−	−	PROPN
ma-233	195	6	y	y	PROPN
ma-233	195	7	+	+	CCONJ
ma-233	195	8	z	z	NOUN
ma-233	195	9	2	2	NUM
ma-233	195	10	)	)	PUNCT
ma-233	196	1	+	+	ADP
ma-233	196	2	g	g	NOUN
ma-233	196	3	(	(	PUNCT
ma-233	196	4	y	y	PROPN
ma-233	196	5	+	+	NOUN
ma-233	196	6	z	z	NOUN
ma-233	196	7	−	−	NOUN
ma-233	196	8	x	x	SYM
ma-233	196	9	2	2	X
ma-233	196	10	)	)	PUNCT
ma-233	196	11	=	=	SYM
ma-233	196	12	g(x)+g(y)+g(z	g(x)+g(y)+g(z	NOUN
ma-233	196	13	)	)	PUNCT
ma-233	196	14	(	(	PUNCT
ma-233	196	15	3.15	3.15	NUM
ma-233	196	16	)	)	PUNCT
ma-233	196	17	and	and	CCONJ
ma-233	196	18	‖f	‖f	ADP
ma-233	196	19	(	(	PUNCT
ma-233	196	20	x)−	x)−	PROPN
ma-233	196	21	g(x)‖	g(x)‖	PROPN
ma-233	196	22	6	6	NUM
ma-233	196	23	1	1	NUM
ma-233	196	24	2	2	NUM
ma-233	196	25	ε	ε	X
ma-233	196	26	(	(	PUNCT
ma-233	196	27	3.16	3.16	NUM
ma-233	196	28	)	)	PUNCT
ma-233	196	29	for	for	ADP
ma-233	196	30	all	all	PRON
ma-233	196	31	x	x	SYM
ma-233	196	32	∈	∈	PROPN
ma-233	196	33	2	2	NUM
ma-233	196	34	g	g	NOUN
ma-233	196	35	=	=	PUNCT
ma-233	196	36	{	{	PUNCT
ma-233	196	37	2x	2x	NUM
ma-233	196	38	:	:	PUNCT
ma-233	196	39	x	x	SYM
ma-233	196	40	∈	∈	NOUN
ma-233	196	41	g	g	NOUN
ma-233	196	42	}	}	PUNCT
ma-233	196	43	.	.	PUNCT
ma-233	197	1	proof	proof	NOUN
ma-233	197	2	.	.	PUNCT
ma-233	198	1	our	our	PRON
ma-233	198	2	proof	proof	NOUN
ma-233	198	3	resembles	resemble	VERB
ma-233	198	4	theorem3.1	theorem3.1	PROPN
ma-233	198	5	,	,	PUNCT
ma-233	198	6	the	the	DET
ma-233	198	7	same	same	ADJ
ma-233	198	8	step	step	NOUN
ma-233	198	9	from	from	ADP
ma-233	198	10	(	(	PUNCT
ma-233	198	11	3.5	3.5	NUM
ma-233	198	12	)	)	PUNCT
ma-233	198	13	to	to	ADP
ma-233	198	14	(	(	PUNCT
ma-233	198	15	3.7	3.7	NUM
ma-233	198	16	)	)	PUNCT
ma-233	198	17	,	,	PUNCT
ma-233	198	18	we	we	PRON
ma-233	198	19	get	get	VERB
ma-233	198	20	that	that	DET
ma-233	198	21	‖3f	‖3f	PROPN
ma-233	198	22	(	(	PUNCT
ma-233	198	23	x	x	NOUN
ma-233	198	24	)	)	PUNCT
ma-233	199	1	+	+	NUM
ma-233	199	2	f	f	X
ma-233	199	3	(	(	PUNCT
ma-233	199	4	−x)−	−x)−	PROPN
ma-233	199	5	f	f	PROPN
ma-233	199	6	(	(	PUNCT
ma-233	199	7	2x)‖	2x)‖	NUM
ma-233	199	8	6	6	NUM
ma-233	199	9	2ε	2ε	NOUN
ma-233	199	10	(	(	PUNCT
ma-233	199	11	3.17	3.17	NUM
ma-233	199	12	)	)	PUNCT
ma-233	199	13	adding	add	VERB
ma-233	199	14	(	(	PUNCT
ma-233	199	15	3.14	3.14	NUM
ma-233	199	16	)	)	PUNCT
ma-233	199	17	to	to	ADP
ma-233	199	18	(	(	PUNCT
ma-233	199	19	3.17	3.17	NUM
ma-233	199	20	)	)	PUNCT
ma-233	199	21	and	and	CCONJ
ma-233	199	22	using	use	VERB
ma-233	199	23	the	the	DET
ma-233	199	24	triangle	triangle	NOUN
ma-233	199	25	inequality	inequality	NOUN
ma-233	199	26	,	,	PUNCT
ma-233	199	27	we	we	PRON
ma-233	199	28	can	can	AUX
ma-233	199	29	obtain	obtain	VERB
ma-233	199	30	‖f	‖f	PRON
ma-233	199	31	(	(	PUNCT
ma-233	199	32	2x)−	2x)−	NUM
ma-233	199	33	4f	4f	NUM
ma-233	199	34	(	(	PUNCT
ma-233	199	35	x)‖	x)‖	PROPN
ma-233	199	36	6	6	NUM
ma-233	199	37	max	max	PROPN
ma-233	199	38	{	{	PUNCT
ma-233	199	39	‖3f	‖3f	PROPN
ma-233	199	40	(	(	PUNCT
ma-233	199	41	x	x	NOUN
ma-233	199	42	)	)	PUNCT
ma-233	200	1	+	+	NUM
ma-233	200	2	f	f	X
ma-233	200	3	(	(	PUNCT
ma-233	200	4	−x)−	−x)−	PROPN
ma-233	200	5	f	f	PROPN
ma-233	200	6	(	(	PUNCT
ma-233	200	7	2x)‖	2x)‖	NUM
ma-233	200	8	,	,	PUNCT
ma-233	200	9	‖f	‖f	ADP
ma-233	200	10	(	(	PUNCT
ma-233	200	11	x)−	x)−	PROPN
ma-233	200	12	f	f	PROPN
ma-233	200	13	(	(	PUNCT
ma-233	200	14	−x)‖	−x)‖	NOUN
ma-233	200	15	}	}	PUNCT
ma-233	200	16	=	=	SYM
ma-233	200	17	2ε	2ε	NOUN
ma-233	200	18	(	(	PUNCT
ma-233	200	19	3.18	3.18	NUM
ma-233	200	20	)	)	PUNCT
ma-233	200	21	https://doi.org/10.28924/ada/ma.4.11	https://doi.org/10.28924/ada/ma.4.11	PROPN
ma-233	200	22	eur	eur	NOUN
ma-233	200	23	.	.	PUNCT
ma-233	201	1	j.	j.	PROPN
ma-233	201	2	math	math	PROPN
ma-233	201	3	.	.	PUNCT
ma-233	202	1	anal	anal	PROPN
ma-233	202	2	.	.	PUNCT
ma-233	203	1	10.28924	10.28924	NUM
ma-233	203	2	/	/	SYM
ma-233	203	3	ada	ada	PROPN
ma-233	203	4	/	/	SYM
ma-233	203	5	ma.4.11	ma.4.11	PROPN
ma-233	204	1	10hence	10hence	NUM
ma-233	204	2	,	,	PUNCT
ma-233	204	3	by	by	ADP
ma-233	204	4	using	use	VERB
ma-233	204	5	the	the	DET
ma-233	204	6	result	result	NOUN
ma-233	204	7	,	,	PUNCT
ma-233	204	8	there	there	PRON
ma-233	204	9	is	be	VERB
ma-233	204	10	‖3f	‖3f	PROPN
ma-233	204	11	(	(	PUNCT
ma-233	204	12	4x)−	4x)−	PROPN
ma-233	204	13	8f	8f	NOUN
ma-233	204	14	(	(	PUNCT
ma-233	204	15	2x)−	2x)−	NUM
ma-233	204	16	f	f	X
ma-233	204	17	(	(	PUNCT
ma-233	204	18	−4x)‖	−4x)‖	PROPN
ma-233	204	19	=	=	SYM
ma-233	204	20	‖2[f	‖2[f	PROPN
ma-233	204	21	(	(	PUNCT
ma-233	204	22	4x)−	4x)−	PROPN
ma-233	204	23	4f	4f	NOUN
ma-233	204	24	(	(	PUNCT
ma-233	204	25	2x	2x	NUM
ma-233	204	26	)	)	PUNCT
ma-233	204	27	]	]	PUNCT
ma-233	205	1	+	+	CCONJ
ma-233	205	2	f	f	X
ma-233	205	3	(	(	PUNCT
ma-233	205	4	4x)−	4x)−	PROPN
ma-233	205	5	f	f	PROPN
ma-233	205	6	(	(	PUNCT
ma-233	205	7	−4x)]‖	−4x)]‖	X
ma-233	205	8	6	6	NUM
ma-233	205	9	max{4ε	max{4ε	PROPN
ma-233	205	10	,	,	PUNCT
ma-233	205	11	ε	ε	PROPN
ma-233	205	12	}	}	PUNCT
ma-233	205	13	=	=	SYM
ma-233	205	14	4ε	4ε	NOUN
ma-233	205	15	(	(	PUNCT
ma-233	205	16	3.19	3.19	NUM
ma-233	205	17	)	)	PUNCT
ma-233	205	18	the	the	DET
ma-233	205	19	rest	rest	NOUN
ma-233	205	20	of	of	ADP
ma-233	205	21	proof	proof	NOUN
ma-233	205	22	is	be	AUX
ma-233	205	23	similar	similar	ADJ
ma-233	205	24	to	to	ADP
ma-233	205	25	the	the	DET
ma-233	205	26	theorem	theorem	ADJ
ma-233	205	27	3.1	3.1	NUM
ma-233	205	28	.	.	PUNCT
ma-233	205	29	�	�	PROPN
ma-233	205	30	acknowledgments	acknowledgment	NOUN
ma-233	205	31	thanks	thank	NOUN
ma-233	205	32	to	to	ADP
ma-233	205	33	all	all	DET
ma-233	205	34	the	the	DET
ma-233	205	35	members	member	NOUN
ma-233	205	36	of	of	ADP
ma-233	205	37	the	the	DET
ma-233	205	38	functional	functional	ADJ
ma-233	205	39	analysis	analysis	NOUN
ma-233	205	40	research	research	NOUN
ma-233	205	41	team	team	NOUN
ma-233	205	42	of	of	ADP
ma-233	205	43	the	the	DET
ma-233	205	44	college	college	PROPN
ma-233	205	45	of	of	ADP
ma-233	205	46	mathemat	mathemat	PROPN
ma-233	205	47	-	-	PUNCT
ma-233	205	48	ics	ics	PROPN
ma-233	205	49	and	and	CCONJ
ma-233	205	50	physics	physics	NOUN
ma-233	205	51	of	of	ADP
ma-233	205	52	anqing	anqe	VERB
ma-233	205	53	normal	normal	ADJ
ma-233	205	54	university	university	NOUN
ma-233	205	55	for	for	ADP
ma-233	205	56	their	their	PRON
ma-233	205	57	discussion	discussion	NOUN
ma-233	205	58	and	and	CCONJ
ma-233	205	59	correction	correction	NOUN
ma-233	205	60	of	of	ADP
ma-233	205	61	the	the	DET
ma-233	205	62	difficultiesand	difficultiesand	NOUN
ma-233	205	63	errors	error	NOUN
ma-233	205	64	encountered	encounter	VERB
ma-233	205	65	in	in	ADP
ma-233	205	66	this	this	DET
ma-233	205	67	paper	paper	NOUN
ma-233	205	68	.	.	PUNCT
ma-233	206	1	this	this	DET
ma-233	206	2	research	research	NOUN
ma-233	206	3	work	work	NOUN
ma-233	206	4	was	be	AUX
ma-233	206	5	funded	fund	VERB
ma-233	206	6	by	by	ADP
ma-233	206	7	anhui	anhui	PROPN
ma-233	206	8	province	province	PROPN
ma-233	206	9	highereducation	highereducation	PROPN
ma-233	206	10	science	science	PROPN
ma-233	206	11	research	research	NOUN
ma-233	206	12	project	project	NOUN
ma-233	206	13	(	(	PUNCT
ma-233	206	14	natural	natural	ADJ
ma-233	206	15	science	science	NOUN
ma-233	206	16	)	)	PUNCT
ma-233	206	17	,	,	PUNCT
ma-233	206	18	2023ah050487	2023ah050487	NUM
ma-233	206	19	.	.	PUNCT
ma-233	207	1	references	reference	NOUN
ma-233	207	2	[	[	X
ma-233	207	3	1	1	NUM
ma-233	207	4	]	]	X
ma-233	207	5	d.h	d.h	PROPN
ma-233	207	6	.	.	PROPN
ma-233	207	7	hyers	hyer	NOUN
ma-233	207	8	,	,	PUNCT
ma-233	207	9	on	on	ADP
ma-233	207	10	the	the	DET
ma-233	207	11	stability	stability	NOUN
ma-233	207	12	of	of	ADP
ma-233	207	13	the	the	DET
ma-233	207	14	linear	linear	ADJ
ma-233	207	15	functional	functional	ADJ
ma-233	207	16	equation	equation	NOUN
ma-233	207	17	,	,	PUNCT
ma-233	207	18	proc	proc	NOUN
ma-233	207	19	.	.	PUNCT
ma-233	208	1	natl	natl	PROPN
ma-233	208	2	.	.	PUNCT
ma-233	209	1	acad	acad	PROPN
ma-233	209	2	.	.	PUNCT
ma-233	210	1	sci	sci	PROPN
ma-233	210	2	.	.	PROPN
ma-233	210	3	27	27	NUM
ma-233	210	4	(	(	PUNCT
ma-233	210	5	1941	1941	NUM
ma-233	210	6	)	)	PUNCT
ma-233	211	1	222–224	222–224	NUM
ma-233	211	2	.	.	PUNCT
ma-233	212	1	https	https	NOUN
ma-233	212	2	:	:	PUNCT
ma-233	212	3	//doi.org/10.1073	//doi.org/10.1073	X
ma-233	212	4	/	/	SYM
ma-233	212	5	pnas.27.4.222.[2	pnas.27.4.222.[2	NUM
ma-233	212	6	]	]	X
ma-233	212	7	t.m	t.m	PROPN
ma-233	212	8	.	.	PROPN
ma-233	212	9	rassias	rassias	PROPN
ma-233	212	10	,	,	PUNCT
ma-233	212	11	on	on	ADP
ma-233	212	12	the	the	DET
ma-233	212	13	stability	stability	NOUN
ma-233	212	14	of	of	ADP
ma-233	212	15	the	the	DET
ma-233	212	16	linear	linear	ADJ
ma-233	212	17	mapping	mapping	NOUN
ma-233	212	18	in	in	ADP
ma-233	212	19	banach	banach	NOUN
ma-233	212	20	spaces	space	NOUN
ma-233	212	21	,	,	PUNCT
ma-233	212	22	proc	proc	NOUN
ma-233	212	23	.	.	PUNCT
ma-233	213	1	am	be	AUX
ma-233	213	2	.	.	PUNCT
ma-233	214	1	math	math	NOUN
ma-233	214	2	.	.	PUNCT
ma-233	215	1	soc	soc	PROPN
ma-233	215	2	.	.	PUNCT
ma-233	216	1	251	251	NUM
ma-233	216	2	(	(	PUNCT
ma-233	216	3	1978	1978	NUM
ma-233	216	4	)	)	PUNCT
ma-233	216	5	264–284	264–284	NUM
ma-233	216	6	.	.	PUNCT
ma-233	217	1	https://doi.org/10.1090/s0002-9939-1978-0507327-1.[3	https://doi.org/10.1090/s0002-9939-1978-0507327-1.[3	NOUN
ma-233	217	2	]	]	PUNCT
ma-233	217	3	katsaras	katsaras	PROPN
ma-233	217	4	,	,	PUNCT
ma-233	217	5	ak	ak	PROPN
ma-233	217	6	,	,	PUNCT
ma-233	217	7	beoyiannis	beoyiannis	PROPN
ma-233	217	8	,	,	PUNCT
ma-233	217	9	a	a	DET
ma-233	217	10	:	:	PUNCT
ma-233	217	11	tensor	tensor	NOUN
ma-233	217	12	products	product	NOUN
ma-233	217	13	of	of	ADP
ma-233	217	14	non	non	ADJ
ma-233	217	15	-	-	ADJ
ma-233	217	16	archimedean	archimedean	ADJ
ma-233	217	17	weighted	weight	VERB
ma-233	217	18	spaces	space	NOUN
ma-233	217	19	of	of	ADP
ma-233	217	20	continuous	continuous	ADJ
ma-233	217	21	functions	function	NOUN
ma-233	217	22	,	,	PUNCT
ma-233	217	23	georgianmath	georgianmath	NOUN
ma-233	217	24	j.	j.	PROPN
ma-233	217	25	6	6	NUM
ma-233	217	26	(	(	PUNCT
ma-233	217	27	1999	1999	NUM
ma-233	217	28	)	)	PUNCT
ma-233	217	29	33–44	33–44	NUM
ma-233	217	30	.	.	PUNCT
ma-233	218	1	https://doi.org/10.1515/gmj.1999.33.[4	https://doi.org/10.1515/gmj.1999.33.[4	PROPN
ma-233	218	2	]	]	X
ma-233	218	3	m.s	m.s	PROPN
ma-233	218	4	.	.	PROPN
ma-233	218	5	moslehian	moslehian	PROPN
ma-233	218	6	,	,	PUNCT
ma-233	218	7	on	on	ADP
ma-233	218	8	the	the	DET
ma-233	218	9	stability	stability	NOUN
ma-233	218	10	of	of	ADP
ma-233	218	11	the	the	DET
ma-233	218	12	orthogonal	orthogonal	ADJ
ma-233	218	13	pexiderized	pexiderize	VERB
ma-233	218	14	cauchy	cauchy	NOUN
ma-233	218	15	equation	equation	NOUN
ma-233	218	16	j.math	j.math	NOUN
ma-233	218	17	.	.	PUNCT
ma-233	219	1	anal	anal	PROPN
ma-233	219	2	.	.	PUNCT
ma-233	219	3	appl	appl	PROPN
ma-233	219	4	.	.	PUNCT
ma-233	220	1	318(1	318(1	NUM
ma-233	220	2	)	)	PUNCT
ma-233	220	3	(	(	PUNCT
ma-233	220	4	2006)211–223	2006)211–223	NUM
ma-233	220	5	.	.	PUNCT
ma-233	221	1	https://doi.org/10.1016/j.jmaa.2005.05.052.[5	https://doi.org/10.1016/j.jmaa.2005.05.052.[5	PROPN
ma-233	221	2	]	]	PUNCT
ma-233	221	3	m.s	m.s	PROPN
ma-233	221	4	.	.	PROPN
ma-233	221	5	moslehian	moslehian	PROPN
ma-233	221	6	and	and	CCONJ
ma-233	221	7	gh	gh	PROPN
ma-233	221	8	.	.	PROPN
ma-233	221	9	sadeghi	sadeghi	PROPN
ma-233	221	10	,	,	PUNCT
ma-233	221	11	a	a	DET
ma-233	221	12	mazur	mazur	PROPN
ma-233	221	13	-	-	PUNCT
ma-233	221	14	ulam	ulam	PROPN
ma-233	221	15	theorem	theorem	NOUN
ma-233	221	16	in	in	ADP
ma-233	221	17	non	non	ADJ
ma-233	221	18	-	-	ADJ
ma-233	221	19	archimedean	archimedean	ADJ
ma-233	221	20	normed	normed	ADJ
ma-233	221	21	spaces	space	NOUN
ma-233	221	22	,	,	PUNCT
ma-233	221	23	nonlinear	nonlinear	ADJ
ma-233	221	24	anal.–tma69	anal.–tma69	X
ma-233	221	25	(	(	PUNCT
ma-233	221	26	2008	2008	NUM
ma-233	221	27	)	)	PUNCT
ma-233	221	28	3405–3408	3405–3408	NUM
ma-233	221	29	.	.	PUNCT
ma-233	222	1	https://doi.org/10.1016/j.na.2007.09.023.[6	https://doi.org/10.1016/j.na.2007.09.023.[6	NOUN
ma-233	222	2	]	]	PUNCT
ma-233	222	3	a.	a.	NOUN
ma-233	222	4	najati	najati	PROPN
ma-233	222	5	,	,	PUNCT
ma-233	222	6	m.	m.	PROPN
ma-233	222	7	b.	b.	PROPN
ma-233	222	8	moghimi	moghimi	PROPN
ma-233	222	9	stability	stability	NOUN
ma-233	222	10	of	of	ADP
ma-233	222	11	a	a	DET
ma-233	222	12	functional	functional	ADJ
ma-233	222	13	equation	equation	NOUN
ma-233	222	14	deriving	derive	VERB
ma-233	222	15	from	from	ADP
ma-233	222	16	quadratic	quadratic	ADJ
ma-233	222	17	and	and	CCONJ
ma-233	222	18	additive	additive	ADJ
ma-233	222	19	functions	function	NOUN
ma-233	222	20	in	in	ADP
ma-233	222	21	quasi	quasi	ADJ
ma-233	222	22	-	-	ADJ
ma-233	222	23	banach	banach	ADJ
ma-233	222	24	spaces	space	NOUN
ma-233	222	25	,	,	PUNCT
ma-233	222	26	j.	j.	PROPN
ma-233	222	27	math	math	PROPN
ma-233	222	28	.	.	PUNCT
ma-233	223	1	anal	anal	PROPN
ma-233	223	2	.	.	PUNCT
ma-233	223	3	appl	appl	PROPN
ma-233	223	4	.	.	PUNCT
ma-233	224	1	337	337	NUM
ma-233	224	2	(	(	PUNCT
ma-233	224	3	2008	2008	NUM
ma-233	224	4	)	)	PUNCT
ma-233	224	5	399–415.https://doi.org/10.1016	399–415.https://doi.org/10.1016	NUM
ma-233	224	6	/	/	SYM
ma-233	224	7	j.jmaa.2007.03.104.[7	j.jmaa.2007.03.104.[7	PROPN
ma-233	224	8	]	]	PUNCT
ma-233	224	9	r.	r.	PROPN
ma-233	224	10	ger	ger	PROPN
ma-233	224	11	,	,	PUNCT
ma-233	224	12	j.	j.	PROPN
ma-233	224	13	sikorska	sikorska	PROPN
ma-233	224	14	,	,	PUNCT
ma-233	224	15	stability	stability	NOUN
ma-233	224	16	of	of	ADP
ma-233	224	17	the	the	DET
ma-233	224	18	orthogonal	orthogonal	ADJ
ma-233	224	19	additivity	additivity	NOUN
ma-233	224	20	,	,	PUNCT
ma-233	224	21	bull	bull	NOUN
ma-233	224	22	.	.	PUNCT
ma-233	225	1	pol	pol	PROPN
ma-233	225	2	.	.	PUNCT
ma-233	226	1	acad	acad	PROPN
ma-233	226	2	.	.	PUNCT
ma-233	227	1	sci	sci	PROPN
ma-233	227	2	.	.	PROPN
ma-233	227	3	,	,	PUNCT
ma-233	227	4	math	math	NOUN
ma-233	227	5	.	.	PUNCT
ma-233	228	1	43	43	NUM
ma-233	228	2	(	(	PUNCT
ma-233	228	3	1995	1995	NUM
ma-233	228	4	)	)	PUNCT
ma-233	229	1	143–151	143–151	NUM
ma-233	229	2	.	.	PUNCT
ma-233	230	1	https	https	NOUN
ma-233	230	2	:	:	PUNCT
ma-233	230	3	//doi.org/10.1007	//doi.org/10.1007	PROPN
ma-233	230	4	/	/	SYM
ma-233	230	5	s00010	s00010	NOUN
ma-233	230	6	-	-	PUNCT
ma-233	230	7	006	006	NUM
ma-233	230	8	-	-	NUM
ma-233	230	9	2868	2868	NUM
ma-233	230	10	-	-	SYM
ma-233	230	11	0.[8	0.[8	NUM
ma-233	230	12	]	]	X
ma-233	230	13	w.	w.	PROPN
ma-233	230	14	fechner	fechner	PROPN
ma-233	230	15	,	,	PUNCT
ma-233	230	16	j.	j.	PROPN
ma-233	230	17	sikorska	sikorska	PROPN
ma-233	230	18	,	,	PUNCT
ma-233	230	19	on	on	ADP
ma-233	230	20	the	the	DET
ma-233	230	21	stability	stability	NOUN
ma-233	230	22	of	of	ADP
ma-233	230	23	orthogonal	orthogonal	ADJ
ma-233	230	24	additivity	additivity	NOUN
ma-233	230	25	,	,	PUNCT
ma-233	230	26	bull	bull	NOUN
ma-233	230	27	.	.	PUNCT
ma-233	231	1	pol	pol	PROPN
ma-233	231	2	.	.	PUNCT
ma-233	232	1	acad	acad	PROPN
ma-233	232	2	.	.	PUNCT
ma-233	233	1	sci	sci	PROPN
ma-233	233	2	.	.	PROPN
ma-233	233	3	,	,	PUNCT
ma-233	233	4	math	math	NOUN
ma-233	233	5	.	.	PUNCT
ma-233	234	1	58	58	NUM
ma-233	234	2	(	(	PUNCT
ma-233	234	3	2010	2010	NUM
ma-233	234	4	)	)	PUNCT
ma-233	234	5	23–30	23–30	NUM
ma-233	234	6	.	.	PUNCT
ma-233	235	1	https://doi.org/10.4064/ba58-1-3.[9	https://doi.org/10.4064/ba58-1-3.[9	PROPN
ma-233	235	2	]	]	X
ma-233	235	3	m.e	m.e	PROPN
ma-233	235	4	.	.	PROPN
ma-233	235	5	gordji	gordji	PROPN
ma-233	235	6	,	,	PUNCT
ma-233	235	7	z.	z.	PROPN
ma-233	235	8	alizadeh	alizadeh	PROPN
ma-233	235	9	,	,	PUNCT
ma-233	235	10	stability	stability	NOUN
ma-233	235	11	and	and	CCONJ
ma-233	235	12	superstability	superstability	NOUN
ma-233	235	13	of	of	ADP
ma-233	235	14	ring	ring	NOUN
ma-233	235	15	homomorphisms	homomorphism	NOUN
ma-233	235	16	on	on	ADP
ma-233	235	17	non	non	ADJ
ma-233	235	18	-	-	ADJ
ma-233	235	19	archimedean	archimedean	ADJ
ma-233	235	20	banach	banach	NOUN
ma-233	235	21	algebras	algebras	PROPN
ma-233	235	22	,	,	PUNCT
ma-233	236	1	abstr	abstr	PROPN
ma-233	236	2	.	.	PUNCT
ma-233	236	3	appl	appl	PROPN
ma-233	236	4	.	.	PUNCT
ma-233	237	1	anal	anal	PROPN
ma-233	237	2	.	.	PUNCT
ma-233	238	1	2011	2011	NUM
ma-233	238	2	(	(	PUNCT
ma-233	238	3	2011	2011	NUM
ma-233	238	4	)	)	PUNCT
ma-233	238	5	123656	123656	NUM
ma-233	238	6	.	.	PUNCT
ma-233	239	1	https://doi.org/10.1155/2011/123656.[10	https://doi.org/10.1155/2011/123656.[10	ADJ
ma-233	239	2	]	]	PUNCT
ma-233	240	1	s.y	s.y	PROPN
ma-233	240	2	.	.	PROPN
ma-233	240	3	kang	kang	PROPN
ma-233	240	4	,	,	PUNCT
ma-233	240	5	s.w	s.w	PROPN
ma-233	240	6	.	.	PROPN
ma-233	240	7	kim	kim	PROPN
ma-233	240	8	,	,	PUNCT
ma-233	240	9	orthogonal	orthogonal	ADJ
ma-233	240	10	stability	stability	NOUN
ma-233	240	11	of	of	ADP
ma-233	240	12	an	an	DET
ma-233	240	13	additive	additive	ADJ
ma-233	240	14	-	-	PUNCT
ma-233	240	15	quartic	quartic	ADJ
ma-233	240	16	functional	functional	ADJ
ma-233	240	17	equation	equation	NOUN
ma-233	240	18	in	in	ADP
ma-233	240	19	non	non	ADJ
ma-233	240	20	-	-	ADJ
ma-233	240	21	archimedean	archimedean	ADJ
ma-233	240	22	spaces	space	NOUN
ma-233	240	23	,	,	PUNCT
ma-233	240	24	j.nonlinear	j.nonlinear	ADJ
ma-233	240	25	anal	anal	PROPN
ma-233	240	26	.	.	PUNCT
ma-233	241	1	appl	appl	PROPN
ma-233	241	2	.	.	PROPN
ma-233	242	1	2012	2012	NUM
ma-233	242	2	(	(	PUNCT
ma-233	242	3	2012	2012	NUM
ma-233	242	4	)	)	PUNCT
ma-233	242	5	jnaa-00123	jnaa-00123	NOUN
ma-233	242	6	.	.	PUNCT
ma-233	243	1	https://doi.org/10.5899/2012/jnaa-00123.[11	https://doi.org/10.5899/2012/jnaa-00123.[11	NOUN
ma-233	243	2	]	]	X
ma-233	243	3	s.g	s.g	PROPN
ma-233	243	4	.	.	PROPN
ma-233	243	5	ghaleh	ghaleh	PROPN
ma-233	243	6	,	,	PUNCT
ma-233	243	7	k.	k.	PROPN
ma-233	243	8	ghasem	ghasem	PROPN
ma-233	243	9	,	,	PUNCT
ma-233	243	10	stability	stability	NOUN
ma-233	243	11	of	of	ADP
ma-233	243	12	n	n	NOUN
ma-233	243	13	-	-	PUNCT
ma-233	243	14	jordan*-derivations	jordan*-derivation	NOUN
ma-233	243	15	in	in	ADP
ma-233	243	16	c*-algebras	c*-algebra	NOUN
ma-233	243	17	and	and	CCONJ
ma-233	243	18	jc*-algebras	jc*-algebras	PROPN
ma-233	243	19	,	,	PUNCT
ma-233	243	20	taiwan	taiwan	PROPN
ma-233	243	21	.	.	PUNCT
ma-233	244	1	j.	j.	PROPN
ma-233	244	2	math	math	PROPN
ma-233	244	3	.	.	PUNCT
ma-233	245	1	16(2012	16(2012	NUM
ma-233	245	2	)	)	PUNCT
ma-233	245	3	1791	1791	NUM
ma-233	245	4	-	-	SYM
ma-233	245	5	1802	1802	NUM
ma-233	245	6	.	.	PUNCT
ma-233	246	1	https://doi.org/10.11650/twjm/1500406797.[12	https://doi.org/10.11650/twjm/1500406797.[12	PROPN
ma-233	246	2	]	]	X
ma-233	246	3	c.	c.	PROPN
ma-233	246	4	park	park	PROPN
ma-233	246	5	,	,	PUNCT
ma-233	246	6	g.h	g.h	PROPN
ma-233	246	7	.	.	PROPN
ma-233	246	8	kim	kim	PROPN
ma-233	246	9	orthogonally	orthogonally	ADV
ma-233	246	10	additive	additive	ADJ
ma-233	246	11	-	-	PUNCT
ma-233	246	12	additive	additive	ADJ
ma-233	246	13	and	and	CCONJ
ma-233	246	14	orthogonally	orthogonally	ADV
ma-233	246	15	quadratic	quadratic	ADJ
ma-233	246	16	-	-	PUNCT
ma-233	246	17	quadratic	quadratic	ADJ
ma-233	246	18	functional	functional	ADJ
ma-233	246	19	equation	equation	NOUN
ma-233	246	20	inorthogonality	inorthogonality	NOUN
ma-233	246	21	spaces	space	VERB
ma-233	246	22	,	,	PUNCT
ma-233	246	23	j.	j.	PROPN
ma-233	246	24	ineq	ineq	PROPN
ma-233	246	25	.	.	PUNCT
ma-233	247	1	appl	appl	PROPN
ma-233	247	2	.	.	PROPN
ma-233	248	1	2012	2012	NUM
ma-233	248	2	(	(	PUNCT
ma-233	248	3	2012	2012	NUM
ma-233	248	4	)	)	PUNCT
ma-233	248	5	139	139	NUM
ma-233	248	6	.	.	PUNCT
ma-233	249	1	https://doi.org/10.1186/1029-242x-2012-139.[13	https://doi.org/10.1186/1029-242x-2012-139.[13	PROPN
ma-233	249	2	]	]	X
ma-233	249	3	a.	a.	NOUN
ma-233	249	4	thanyacharoen	thanyacharoen	PROPN
ma-233	249	5	,	,	PUNCT
ma-233	249	6	w.	w.	PROPN
ma-233	249	7	sintunavarat	sintunavarat	PROPN
ma-233	249	8	,	,	PUNCT
ma-233	249	9	the	the	DET
ma-233	249	10	new	new	ADJ
ma-233	249	11	investigation	investigation	NOUN
ma-233	249	12	of	of	ADP
ma-233	249	13	the	the	DET
ma-233	249	14	stability	stability	NOUN
ma-233	249	15	of	of	ADP
ma-233	249	16	mixed	mixed	ADJ
ma-233	249	17	type	type	NOUN
ma-233	249	18	additive	additive	NOUN
ma-233	249	19	-	-	PUNCT
ma-233	249	20	quartic	quartic	ADJ
ma-233	249	21	func	func	ADJ
ma-233	249	22	-	-	PUNCT
ma-233	249	23	tional	tional	ADJ
ma-233	249	24	equations	equation	NOUN
ma-233	249	25	in	in	ADP
ma-233	249	26	non	non	ADJ
ma-233	249	27	-	-	ADJ
ma-233	249	28	archimedean	archimedean	ADJ
ma-233	249	29	spaces	space	NOUN
ma-233	249	30	,	,	PUNCT
ma-233	249	31	demonstr	demonstr	NOUN
ma-233	249	32	.	.	PUNCT
ma-233	249	33	math	math	NOUN
ma-233	249	34	.	.	PUNCT
ma-233	250	1	53	53	NUM
ma-233	250	2	(	(	PUNCT
ma-233	250	3	2020	2020	NUM
ma-233	250	4	)	)	PUNCT
ma-233	250	5	174	174	NUM
ma-233	250	6	-	-	SYM
ma-233	250	7	192	192	NUM
ma-233	250	8	.	.	PUNCT
ma-233	251	1	https://doi.org/10.1515/	https://doi.org/10.1515/	PROPN
ma-233	251	2	dema-2020	dema-2020	NOUN
ma-233	251	3	-	-	PUNCT
ma-233	251	4	0009	0009	NUM
ma-233	251	5	.	.	PUNCT
ma-233	252	1	https://doi.org/10.28924/ada/ma.4.11	https://doi.org/10.28924/ada/ma.4.11	PROPN
ma-233	253	1	https://doi.org/10.1073/pnas.27.4.222	https://doi.org/10.1073/pnas.27.4.222	NOUN
ma-233	254	1	https://doi.org/10.1073/pnas.27.4.222	https://doi.org/10.1073/pnas.27.4.222	PROPN
ma-233	254	2	https://doi.org/10.1090/s0002-9939-1978-0507327-1	https://doi.org/10.1090/s0002-9939-1978-0507327-1	PROPN
ma-233	254	3	https://doi.org/10.1515/gmj.1999.33	https://doi.org/10.1515/gmj.1999.33	PROPN
ma-233	254	4	https://doi.org/10.1016/j.jmaa.2005.05.052	https://doi.org/10.1016/j.jmaa.2005.05.052	NUM
ma-233	254	5	https://doi.org/10.1016/j.na.2007.09.023	https://doi.org/10.1016/j.na.2007.09.023	PROPN
ma-233	254	6	https://doi.org/10.1016/j.jmaa.2007.03.104	https://doi.org/10.1016/j.jmaa.2007.03.104	PROPN
ma-233	254	7	https://doi.org/10.1007/s00010-006-2868-0	https://doi.org/10.1007/s00010-006-2868-0	NUM
ma-233	254	8	https://doi.org/10.1007/s00010-006-2868-0	https://doi.org/10.1007/s00010-006-2868-0	NUM
ma-233	254	9	https://doi.org/10.4064/ba58-1-3	https://doi.org/10.4064/ba58-1-3	NOUN
ma-233	254	10	https://doi.org/10.1155/2011/123656	https://doi.org/10.1155/2011/123656	VERB
ma-233	254	11	https://doi.org/10.5899/2012/jnaa-00123	https://doi.org/10.5899/2012/jnaa-00123	NUM
ma-233	254	12	https://doi.org/10.11650/twjm/1500406797	https://doi.org/10.11650/twjm/1500406797	PROPN
ma-233	254	13	https://doi.org/10.1186/1029-242x-2012-139	https://doi.org/10.1186/1029-242x-2012-139	NOUN
ma-233	254	14	https://doi.org/10.1515/dema-2020-0009	https://doi.org/10.1515/dema-2020-0009	PROPN
ma-233	254	15	https://doi.org/10.1515/dema-2020-0009	https://doi.org/10.1515/dema-2020-0009	PROPN
ma-233	254	16	eur	eur	PROPN
ma-233	254	17	.	.	PUNCT
ma-233	255	1	j.	j.	PROPN
ma-233	255	2	math	math	PROPN
ma-233	255	3	.	.	PUNCT
ma-233	256	1	anal	anal	PROPN
ma-233	256	2	.	.	PUNCT
ma-233	257	1	10.28924	10.28924	NUM
ma-233	257	2	/	/	SYM
ma-233	257	3	ada	ada	PROPN
ma-233	257	4	/	/	SYM
ma-233	257	5	ma.4.11	ma.4.11	ADJ
ma-233	258	1	11	11	NUM
ma-233	258	2	[	[	X
ma-233	258	3	14	14	NUM
ma-233	258	4	]	]	X
ma-233	258	5	l.	l.	PROPN
ma-233	258	6	fu	fu	PROPN
ma-233	258	7	,	,	PUNCT
ma-233	258	8	q.	q.	PROPN
ma-233	258	9	liu	liu	PROPN
ma-233	258	10	,	,	PUNCT
ma-233	258	11	y.	y.	PROPN
ma-233	258	12	li	li	PROPN
ma-233	258	13	,	,	PUNCT
ma-233	258	14	on	on	ADP
ma-233	258	15	the	the	DET
ma-233	258	16	stability	stability	NOUN
ma-233	258	17	of	of	ADP
ma-233	258	18	orthogonally	orthogonally	ADV
ma-233	258	19	jensen	jensen	PROPN
ma-233	258	20	additive	additive	NOUN
ma-233	258	21	and	and	CCONJ
ma-233	258	22	quadratic	quadratic	ADJ
ma-233	258	23	functional	functional	ADJ
ma-233	258	24	equation	equation	NOUN
ma-233	258	25	,	,	PUNCT
ma-233	258	26	j.	j.	PROPN
ma-233	258	27	math	math	PROPN
ma-233	258	28	.	.	PUNCT
ma-233	259	1	anal.appl	anal.appl	PROPN
ma-233	259	2	.	.	PROPN
ma-233	260	1	519	519	NUM
ma-233	260	2	(	(	PUNCT
ma-233	260	3	2023	2023	NUM
ma-233	260	4	)	)	PUNCT
ma-233	260	5	126744	126744	NUM
ma-233	260	6	.	.	PUNCT
ma-233	261	1	https://doi.org/10.1016/j.jmaa.2022.126744.[15	https://doi.org/10.1016/j.jmaa.2022.126744.[15	PROPN
ma-233	261	2	]	]	PUNCT
ma-233	261	3	k.	k.	PROPN
ma-233	261	4	hensel	hensel	PROPN
ma-233	261	5	,	,	PUNCT
ma-233	261	6	über	über	PROPN
ma-233	261	7	eine	eine	PROPN
ma-233	261	8	neue	neue	PROPN
ma-233	261	9	begründung	begründung	PROPN
ma-233	261	10	der	der	PROPN
ma-233	261	11	theorie	theorie	PROPN
ma-233	261	12	der	der	PROPN
ma-233	261	13	algebraischen	algebraischen	PROPN
ma-233	261	14	zahlen	zahlen	PROPN
ma-233	261	15	,	,	PUNCT
ma-233	261	16	jahresber	jahresber	PROPN
ma-233	261	17	.	.	PROPN
ma-233	261	18	dtsch	dtsch	PROPN
ma-233	261	19	.	.	PUNCT
ma-233	262	1	math.-ver	math.-ver	ADV
ma-233	262	2	.	.	PROPN
ma-233	263	1	6	6	NUM
ma-233	263	2	(	(	PUNCT
ma-233	263	3	1897)83	1897)83	NUM
ma-233	263	4	-	-	SYM
ma-233	263	5	88	88	NUM
ma-233	263	6	.	.	PUNCT
ma-233	264	1	http://eudml.org/doc/144593.[16	http://eudml.org/doc/144593.[16	PROPN
ma-233	264	2	]	]	X
ma-233	265	1	j.	j.	PROPN
ma-233	265	2	ratz	ratz	PROPN
ma-233	265	3	,	,	PUNCT
ma-233	265	4	on	on	ADP
ma-233	265	5	orthogonally	orthogonally	ADV
ma-233	265	6	additive	additive	ADJ
ma-233	265	7	mappings	mapping	NOUN
ma-233	265	8	,	,	PUNCT
ma-233	265	9	aequat	aequat	PROPN
ma-233	265	10	.	.	PUNCT
ma-233	266	1	math	math	NOUN
ma-233	266	2	.	.	PUNCT
ma-233	267	1	28	28	NUM
ma-233	267	2	(	(	PUNCT
ma-233	267	3	1985	1985	NUM
ma-233	267	4	)	)	PUNCT
ma-233	267	5	35–49	35–49	NUM
ma-233	267	6	.	.	PUNCT
ma-233	268	1	https://doi.org/10.1007/	https://doi.org/10.1007/	PROPN
ma-233	268	2	bf02189629	bf02189629	PROPN
ma-233	268	3	.	.	PUNCT
ma-233	268	4	https://doi.org/10.28924/ada/ma.4.11	https://doi.org/10.28924/ada/ma.4.11	PROPN
ma-233	268	5	https://doi.org/10.1016/j.jmaa.2022.126744	https://doi.org/10.1016/j.jmaa.2022.126744	VERB
ma-233	268	6	http://eudml.org/doc/144593	http://eudml.org/doc/144593	PROPN
ma-233	268	7	https://doi.org/10.1007/bf02189629	https://doi.org/10.1007/bf02189629	X
ma-233	268	8	https://doi.org/10.1007/bf02189629	https://doi.org/10.1007/bf02189629	PRON
ma-233	268	9	1	1	NUM
ma-233	268	10	.	.	PUNCT
ma-233	268	11	introduction	introduction	NOUN
ma-233	268	12	and	and	CCONJ
ma-233	268	13	preliminaries	preliminary	NOUN
ma-233	268	14	2	2	NUM
ma-233	268	15	.	.	PUNCT
ma-233	268	16	stability	stability	NOUN
ma-233	268	17	of	of	ADP
ma-233	268	18	the	the	DET
ma-233	268	19	orthogonally	orthogonally	ADV
ma-233	268	20	additive	additive	ADJ
ma-233	268	21	-	-	PUNCT
ma-233	268	22	quadratic	quadratic	ADJ
ma-233	268	23	functional	functional	ADJ
ma-233	268	24	equation	equation	NOUN
ma-233	268	25	3	3	NUM
ma-233	268	26	.	.	PUNCT
ma-233	269	1	stability	stability	NOUN
ma-233	269	2	of	of	ADP
ma-233	269	3	additive	additive	NOUN
ma-233	269	4	-	-	PUNCT
ma-233	269	5	additive	additive	ADJ
ma-233	269	6	and	and	CCONJ
ma-233	269	7	orthogonally	orthogonally	ADV
ma-233	269	8	quadratic	quadratic	ADJ
ma-233	269	9	-	-	PUNCT
ma-233	269	10	quadratic	quadratic	ADJ
ma-233	269	11	functional	functional	ADJ
ma-233	269	12	equation	equation	NOUN
ma-233	269	13	acknowledgments	acknowledgment	NOUN
ma-233	269	14	references	reference	NOUN