id	sid	tid	token	lemma	pos
ma-88	1	1	2022	2022	NUM
ma-88	1	2	ada	ada	PROPN
ma-88	1	3	academica	academica	PROPN
ma-88	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-88	1	5	.	.	PUNCT
ma-88	2	1	j.	j.	PROPN
ma-88	2	2	math	math	PROPN
ma-88	2	3	.	.	PUNCT
ma-88	3	1	anal	anal	ADJ
ma-88	3	2	.	.	PUNCT
ma-88	3	3	2	2	NUM
ma-88	3	4	(	(	PUNCT
ma-88	3	5	2022	2022	NUM
ma-88	3	6	)	)	PUNCT
ma-88	3	7	16doi	16doi	NUM
ma-88	3	8	:	:	PUNCT
ma-88	3	9	10.28924	10.28924	NUM
ma-88	3	10	/	/	SYM
ma-88	3	11	ada	ada	PROPN
ma-88	3	12	/	/	SYM
ma-88	3	13	ma.2.16	ma.2.16	PROPN
ma-88	4	1	a	a	DET
ma-88	4	2	new	new	ADJ
ma-88	4	3	approximate	approximate	ADJ
ma-88	4	4	birkhoff	birkhoff	NOUN
ma-88	4	5	orthogonality	orthogonality	NOUN
ma-88	4	6	type	type	NOUN
ma-88	4	7	chuanjiang	chuanjiang	PROPN
ma-88	4	8	zhou	zhou	PROPN
ma-88	4	9	,	,	PUNCT
ma-88	4	10	qi	qi	PROPN
ma-88	4	11	liu	liu	PROPN
ma-88	4	12	,	,	PUNCT
ma-88	4	13	yongjin	yongjin	PROPN
ma-88	4	14	li∗	li∗	PROPN
ma-88	4	15	department	department	PROPN
ma-88	4	16	of	of	ADP
ma-88	4	17	mathematics	mathematics	PROPN
ma-88	4	18	,	,	PUNCT
ma-88	4	19	sun	sun	PROPN
ma-88	4	20	yat	yat	PROPN
ma-88	4	21	-	-	PUNCT
ma-88	4	22	sen	sen	PROPN
ma-88	4	23	university	university	PROPN
ma-88	4	24	,	,	PUNCT
ma-88	4	25	guangzhou	guangzhou	PROPN
ma-88	4	26	,	,	PUNCT
ma-88	4	27	510275	510275	NUM
ma-88	4	28	,	,	PUNCT
ma-88	4	29	p.	p.	PROPN
ma-88	4	30	r.	r.	PROPN
ma-88	4	31	china	china	PROPN
ma-88	4	32	1090871744@qq.com	1090871744@qq.com	NUM
ma-88	4	33	,	,	PUNCT
ma-88	4	34	liuq325@mail2.sysu.edu.cn	liuq325@mail2.sysu.edu.cn	PROPN
ma-88	4	35	,	,	PUNCT
ma-88	4	36	stslyj@mail.sysu.edu.cn	stslyj@mail.sysu.edu.cn	NOUN
ma-88	4	37	∗correspondence	∗correspondence	NOUN
ma-88	4	38	:	:	PUNCT
ma-88	4	39	stslyj@mail.sysu.edu.cn	stslyj@mail.sysu.edu.cn	NOUN
ma-88	4	40	abstract	abstract	NOUN
ma-88	4	41	.	.	PUNCT
ma-88	5	1	in	in	ADP
ma-88	5	2	this	this	DET
ma-88	5	3	note	note	NOUN
ma-88	5	4	,	,	PUNCT
ma-88	5	5	we	we	PRON
ma-88	5	6	introduce	introduce	VERB
ma-88	5	7	a	a	DET
ma-88	5	8	new	new	ADJ
ma-88	5	9	approximate	approximate	ADJ
ma-88	5	10	birkhoff	birkhoff	NOUN
ma-88	5	11	orthogonality	orthogonality	NOUN
ma-88	5	12	type	type	NOUN
ma-88	5	13	and	and	CCONJ
ma-88	5	14	give	give	VERB
ma-88	5	15	a	a	DET
ma-88	5	16	char	char	NOUN
ma-88	5	17	-	-	PUNCT
ma-88	5	18	acterization	acterization	NOUN
ma-88	5	19	for	for	ADP
ma-88	5	20	inner	inner	ADJ
ma-88	5	21	product	product	NOUN
ma-88	5	22	spaces	space	NOUN
ma-88	5	23	using	use	VERB
ma-88	5	24	the	the	DET
ma-88	5	25	approximate	approximate	ADJ
ma-88	5	26	orthogonality	orthogonality	NOUN
ma-88	5	27	.	.	PUNCT
ma-88	6	1	we	we	PRON
ma-88	6	2	show	show	VERB
ma-88	6	3	some	some	DET
ma-88	6	4	generalproperties	generalpropertie	NOUN
ma-88	6	5	of	of	ADP
ma-88	6	6	the	the	DET
ma-88	6	7	approximate	approximate	ADJ
ma-88	6	8	birkhoff	birkhoff	NOUN
ma-88	6	9	orthogonality	orthogonality	NOUN
ma-88	6	10	type	type	NOUN
ma-88	6	11	as	as	ADV
ma-88	6	12	well	well	ADV
ma-88	6	13	as	as	ADP
ma-88	6	14	applications	application	NOUN
ma-88	6	15	.	.	PUNCT
ma-88	7	1	in	in	ADP
ma-88	7	2	particular	particular	ADJ
ma-88	7	3	,	,	PUNCT
ma-88	7	4	westudy	westudy	VERB
ma-88	7	5	the	the	DET
ma-88	7	6	relationship	relationship	NOUN
ma-88	7	7	between	between	ADP
ma-88	7	8	the	the	DET
ma-88	7	9	new	new	ADJ
ma-88	7	10	approximate	approximate	ADJ
ma-88	7	11	birkhoff	birkhoff	NOUN
ma-88	7	12	orthogonality	orthogonality	NOUN
ma-88	7	13	type	type	NOUN
ma-88	7	14	and	and	CCONJ
ma-88	7	15	other	other	ADJ
ma-88	7	16	approx	approx	PROPN
ma-88	7	17	-	-	PUNCT
ma-88	7	18	imate	imate	PROPN
ma-88	7	19	orthogonality	orthogonality	NOUN
ma-88	7	20	types	type	NOUN
ma-88	7	21	that	that	PRON
ma-88	7	22	have	have	AUX
ma-88	7	23	been	be	AUX
ma-88	7	24	defined	define	VERB
ma-88	7	25	before	before	ADV
ma-88	7	26	.	.	PUNCT
ma-88	8	1	furthermore	furthermore	ADV
ma-88	8	2	we	we	PRON
ma-88	8	3	study	study	VERB
ma-88	8	4	the	the	DET
ma-88	8	5	approximatepreserving	approximatepreserve	VERB
ma-88	8	6	mapping	mapping	NOUN
ma-88	8	7	and	and	CCONJ
ma-88	8	8	give	give	VERB
ma-88	8	9	some	some	DET
ma-88	8	10	properties	property	NOUN
ma-88	8	11	.	.	PUNCT
ma-88	9	1	1	1	X
ma-88	9	2	.	.	X
ma-88	9	3	introduction	introduction	NOUN
ma-88	9	4	one	one	NUM
ma-88	9	5	of	of	ADP
ma-88	9	6	the	the	DET
ma-88	9	7	important	important	ADJ
ma-88	9	8	ideas	idea	NOUN
ma-88	9	9	playing	play	VERB
ma-88	9	10	a	a	DET
ma-88	9	11	fundamental	fundamental	ADJ
ma-88	9	12	role	role	NOUN
ma-88	9	13	in	in	ADP
ma-88	9	14	geometry	geometry	NOUN
ma-88	9	15	of	of	ADP
ma-88	9	16	normed	normed	ADJ
ma-88	9	17	spaces	space	NOUN
ma-88	9	18	is	be	AUX
ma-88	9	19	the	the	DET
ma-88	9	20	con	con	NOUN
ma-88	9	21	-	-	PUNCT
ma-88	9	22	cept	cept	NOUN
ma-88	9	23	of	of	ADP
ma-88	9	24	orthogonality	orthogonality	NOUN
ma-88	9	25	.	.	PUNCT
ma-88	10	1	many	many	ADJ
ma-88	10	2	mathematicians	mathematician	NOUN
ma-88	10	3	have	have	AUX
ma-88	10	4	introduced	introduce	VERB
ma-88	10	5	different	different	ADJ
ma-88	10	6	types	type	NOUN
ma-88	10	7	of	of	ADP
ma-88	10	8	orthogonality	orthogonality	NOUN
ma-88	10	9	for	for	ADP
ma-88	10	10	thenormed	thenormed	ADJ
ma-88	10	11	linear	linear	ADJ
ma-88	10	12	spaces	space	NOUN
ma-88	10	13	,	,	PUNCT
ma-88	10	14	cf	cf	NOUN
ma-88	10	15	.	.	PUNCT
ma-88	11	1	[	[	X
ma-88	11	2	2	2	NUM
ma-88	11	3	,	,	PUNCT
ma-88	11	4	20,24	20,24	PROPN
ma-88	11	5	]	]	PUNCT
ma-88	11	6	.	.	PUNCT
ma-88	12	1	in	in	ADP
ma-88	12	2	1934	1934	NUM
ma-88	12	3	[	[	X
ma-88	12	4	23	23	NUM
ma-88	12	5	]	]	PUNCT
ma-88	12	6	,	,	PUNCT
ma-88	12	7	the	the	DET
ma-88	12	8	first	first	ADJ
ma-88	12	9	orthogonality	orthogonality	NOUN
ma-88	12	10	type	type	NOUN
ma-88	12	11	:	:	PUNCT
ma-88	12	12	roberts	roberts	PROPN
ma-88	12	13	orthogonalitywas	orthogonalitywas	AUX
ma-88	12	14	introduced	introduce	VERB
ma-88	12	15	by	by	ADP
ma-88	12	16	roberts	roberts	PROPN
ma-88	12	17	.	.	PUNCT
ma-88	13	1	after	after	ADP
ma-88	13	2	that	that	PRON
ma-88	13	3	in	in	ADP
ma-88	13	4	1935	1935	NUM
ma-88	13	5	[	[	X
ma-88	13	6	5	5	NUM
ma-88	13	7	]	]	PUNCT
ma-88	13	8	,	,	PUNCT
ma-88	13	9	birkhoff	birkhoff	NOUN
ma-88	13	10	introduced	introduce	VERB
ma-88	13	11	one	one	NUM
ma-88	13	12	of	of	ADP
ma-88	13	13	the	the	DET
ma-88	13	14	most	most	ADJ
ma-88	13	15	importantorthogonality	importantorthogonality	NOUN
ma-88	13	16	types	type	NOUN
ma-88	13	17	:	:	PUNCT
ma-88	13	18	x	x	X
ma-88	13	19	is	be	AUX
ma-88	13	20	said	say	VERB
ma-88	13	21	to	to	PART
ma-88	13	22	be	be	AUX
ma-88	13	23	birhoff	birhoff	ADJ
ma-88	13	24	orthogonal	orthogonal	NOUN
ma-88	13	25	to	to	ADP
ma-88	13	26	y	y	PROPN
ma-88	13	27	(	(	PUNCT
ma-88	13	28	x	x	X
ma-88	13	29	⊥b	⊥b	PRON
ma-88	13	30	y	y	PROPN
ma-88	13	31	)	)	PUNCT
ma-88	13	32	if	if	SCONJ
ma-88	13	33	‖x+ty‖	‖x+ty‖	PROPN
ma-88	13	34	≥	≥	AUX
ma-88	13	35	‖x‖	‖x‖	VERB
ma-88	13	36	for	for	ADP
ma-88	13	37	all	all	DET
ma-88	13	38	t	t	NOUN
ma-88	13	39	∈	∈	PROPN
ma-88	13	40	r.then	r.then	ADV
ma-88	13	41	james	james	PROPN
ma-88	13	42	in	in	ADP
ma-88	13	43	1945	1945	NUM
ma-88	14	1	[	[	X
ma-88	14	2	15	15	NUM
ma-88	14	3	]	]	PUNCT
ma-88	14	4	introduced	introduce	VERB
ma-88	14	5	the	the	DET
ma-88	14	6	pythagorean	pythagorean	PROPN
ma-88	14	7	orthogonality	orthogonality	NOUN
ma-88	14	8	and	and	CCONJ
ma-88	14	9	isosceles	isoscele	NOUN
ma-88	14	10	orthogonality	orthogonality	NOUN
ma-88	14	11	:	:	PUNCT
ma-88	14	12	x	x	X
ma-88	14	13	is	be	AUX
ma-88	14	14	said	say	VERB
ma-88	14	15	to	to	PART
ma-88	14	16	be	be	AUX
ma-88	14	17	isosceles	isoscele	NOUN
ma-88	14	18	orthogonal	orthogonal	ADJ
ma-88	14	19	to	to	ADP
ma-88	14	20	y	y	PROPN
ma-88	14	21	(	(	PUNCT
ma-88	14	22	x	x	PROPN
ma-88	14	23	⊥i	⊥i	PROPN
ma-88	14	24	y	y	PROPN
ma-88	14	25	)	)	PUNCT
ma-88	14	26	if	if	SCONJ
ma-88	14	27	‖x	‖x	PRON
ma-88	14	28	+	+	NUM
ma-88	14	29	y‖	y‖	X
ma-88	15	1	=	=	PUNCT
ma-88	15	2	‖x	‖x	NUM
ma-88	16	1	−	−	PROPN
ma-88	16	2	y‖.	y‖.	NOUN
ma-88	16	3	there	there	PRON
ma-88	16	4	are	be	VERB
ma-88	16	5	also	also	ADV
ma-88	16	6	otherorthognality	otherorthognality	NOUN
ma-88	16	7	types	type	NOUN
ma-88	16	8	related	relate	VERB
ma-88	16	9	to	to	ADP
ma-88	16	10	norm	norm	NOUN
ma-88	16	11	limit	limit	NOUN
ma-88	16	12	such	such	ADJ
ma-88	16	13	as	as	ADP
ma-88	16	14	ρ	ρ	NOUN
ma-88	16	15	-	-	PUNCT
ma-88	16	16	orthogonality	orthogonality	NOUN
ma-88	16	17	and	and	CCONJ
ma-88	16	18	g	g	NOUN
ma-88	16	19	-	-	PUNCT
ma-88	16	20	orthogonality	orthogonality	NOUN
ma-88	16	21	[	[	X
ma-88	16	22	10,18].let	10,18].let	NUM
ma-88	16	23	x	x	PUNCT
ma-88	16	24	be	be	AUX
ma-88	16	25	inner	inner	ADJ
ma-88	16	26	product	product	NOUN
ma-88	16	27	spaces	space	NOUN
ma-88	16	28	(	(	PUNCT
ma-88	16	29	x	x	NOUN
ma-88	16	30	,	,	PUNCT
ma-88	16	31	〈	〈	PROPN
ma-88	16	32	·	·	SYM
ma-88	16	33	|	|	ADJ
ma-88	16	34	·	·	SYM
ma-88	16	35	〉	〉	NOUN
ma-88	16	36	)	)	PUNCT
ma-88	16	37	,	,	PUNCT
ma-88	16	38	all	all	DET
ma-88	16	39	the	the	DET
ma-88	16	40	orthogonality	orthogonality	NOUN
ma-88	16	41	types	type	NOUN
ma-88	16	42	are	be	AUX
ma-88	16	43	equivalent	equivalent	ADJ
ma-88	16	44	to	to	ADP
ma-88	16	45	x	x	PROPN
ma-88	16	46	⊥	⊥	PROPN
ma-88	16	47	y	y	PROPN
ma-88	16	48	orequivalently	orequivalently	ADV
ma-88	16	49	,	,	PUNCT
ma-88	16	50	〈	〈	PROPN
ma-88	16	51	x	x	PART
ma-88	16	52	|y	|y	NOUN
ma-88	17	1	〉	〉	NUM
ma-88	17	2	=	=	SYM
ma-88	17	3	0	0	X
ma-88	17	4	.	.	PUNCT
ma-88	18	1	in	in	ADP
ma-88	18	2	inner	inner	ADJ
ma-88	18	3	product	product	NOUN
ma-88	18	4	spaces	space	VERB
ma-88	18	5	a	a	DET
ma-88	18	6	natural	natural	ADJ
ma-88	18	7	way	way	NOUN
ma-88	18	8	to	to	PART
ma-88	18	9	generalize	generalize	VERB
ma-88	18	10	orthogonality	orthogonality	NOUN
ma-88	18	11	is	be	AUX
ma-88	18	12	todefine	todefine	NOUN
ma-88	18	13	the	the	DET
ma-88	18	14	approximate	approximate	ADJ
ma-88	18	15	orthogonality	orthogonality	NOUN
ma-88	18	16	by	by	ADP
ma-88	18	17	:	:	PUNCT
ma-88	18	18	x	x	X
ma-88	18	19	⊥ε	⊥ε	NUM
ma-88	19	1	y	y	NOUN
ma-88	19	2	if	if	SCONJ
ma-88	19	3	and	and	CCONJ
ma-88	19	4	only	only	ADV
ma-88	19	5	if	if	SCONJ
ma-88	19	6	|〈x	|〈x	VERB
ma-88	19	7	|y〉|	|y〉|	NOUN
ma-88	19	8	≤	≤	NOUN
ma-88	19	9	ε‖x‖‖y‖	ε‖x‖‖y‖	NOUN
ma-88	19	10	,	,	PUNCT
ma-88	19	11	x	x	PRON
ma-88	19	12	,	,	PUNCT
ma-88	19	13	y	y	PROPN
ma-88	19	14	∈	∈	PROPN
ma-88	19	15	x	x	PUNCT
ma-88	20	1	[	[	X
ma-88	20	2	9	9	NUM
ma-88	20	3	,	,	PUNCT
ma-88	20	4	26].inspired	26].inspired	NUM
ma-88	20	5	by	by	ADP
ma-88	20	6	the	the	DET
ma-88	20	7	approximate	approximate	ADJ
ma-88	20	8	orthogonality	orthogonality	NOUN
ma-88	20	9	,	,	PUNCT
ma-88	20	10	dragomir	dragomir	NOUN
ma-88	20	11	[	[	X
ma-88	20	12	13	13	NUM
ma-88	20	13	]	]	PUNCT
ma-88	20	14	gave	give	VERB
ma-88	20	15	the	the	DET
ma-88	20	16	definition	definition	NOUN
ma-88	20	17	of	of	ADP
ma-88	20	18	the	the	DET
ma-88	20	19	approximatebirkhoff	approximatebirkhoff	PROPN
ma-88	20	20	orthogonality	orthogonality	NOUN
ma-88	20	21	xε	xε	PUNCT
ma-88	21	1	⊥b	⊥b	PRON
ma-88	21	2	y	y	NOUN
ma-88	21	3	:	:	PUNCT
ma-88	21	4	‖x	‖x	NOUN
ma-88	22	1	+	+	CCONJ
ma-88	22	2	ty‖	ty‖	PRON
ma-88	22	3	≥	≥	NOUN
ma-88	22	4	(	(	PUNCT
ma-88	22	5	1	1	NUM
ma-88	22	6	−	−	PROPN
ma-88	22	7	ε)‖x‖	ε)‖x‖	PROPN
ma-88	22	8	for	for	ADP
ma-88	22	9	all	all	DET
ma-88	22	10	t	t	PROPN
ma-88	22	11	∈	∈	PROPN
ma-88	22	12	r.	r.	PROPN
ma-88	22	13	it	it	PRON
ma-88	22	14	is	be	AUX
ma-88	22	15	easy	easy	ADJ
ma-88	22	16	to	to	PART
ma-88	22	17	see	see	VERB
ma-88	22	18	thatthis	thatthis	DET
ma-88	22	19	type	type	NOUN
ma-88	22	20	of	of	ADP
ma-88	22	21	approximate	approximate	ADJ
ma-88	22	22	orthogonality	orthogonality	NOUN
ma-88	22	23	is	be	AUX
ma-88	22	24	equivalent	equivalent	ADJ
ma-88	22	25	to	to	ADP
ma-88	22	26	⊥ε	⊥ε	NUM
ma-88	22	27	in	in	ADP
ma-88	22	28	inner	inner	ADJ
ma-88	22	29	product	product	NOUN
ma-88	22	30	spaces	space	VERB
ma-88	22	31	[	[	X
ma-88	22	32	13	13	NUM
ma-88	22	33	]	]	PUNCT
ma-88	22	34	.	.	PUNCT
ma-88	23	1	after	after	SCONJ
ma-88	23	2	thatjacek	thatjacek	PROPN
ma-88	23	3	chmieliński	chmieliński	PROPN
ma-88	23	4	[	[	X
ma-88	23	5	21	21	NUM
ma-88	23	6	]	]	PUNCT
ma-88	23	7	introduced	introduce	VERB
ma-88	23	8	the	the	DET
ma-88	23	9	approximate	approximate	ADJ
ma-88	23	10	birkhoff	birkhoff	NOUN
ma-88	23	11	orthogonality	orthogonality	NOUN
ma-88	23	12	x	x	X
ma-88	23	13	⊥εb	⊥εb	NOUN
ma-88	23	14	y	y	PROPN
ma-88	23	15	:	:	PUNCT
ma-88	23	16	‖x	‖x	X
ma-88	24	1	+	+	CCONJ
ma-88	24	2	ty‖2	ty‖2	PROPN
ma-88	24	3	≥	≥	NOUN
ma-88	24	4	‖x‖2	‖x‖2	VERB
ma-88	24	5	−	−	ADP
ma-88	25	1	2ε‖x‖‖ty‖	2ε‖x‖‖ty‖	NUM
ma-88	25	2	for	for	ADP
ma-88	25	3	all	all	DET
ma-88	25	4	t	t	NOUN
ma-88	25	5	∈	∈	NOUN
ma-88	25	6	r	r	NOUN
ma-88	25	7	,	,	PUNCT
ma-88	25	8	the	the	DET
ma-88	25	9	approximate	approximate	ADJ
ma-88	25	10	isosceles	isoscele	NOUN
ma-88	25	11	orthogonality	orthogonality	NOUN
ma-88	25	12	[	[	X
ma-88	25	13	11	11	NUM
ma-88	25	14	]	]	X
ma-88	25	15	x	x	X
ma-88	25	16	⊥εi	⊥εi	PROPN
ma-88	25	17	y	y	NOUN
ma-88	25	18	:	:	PUNCT
ma-88	25	19	|‖x	|‖x	PROPN
ma-88	25	20	+	+	CCONJ
ma-88	25	21	y‖2	y‖2	PROPN
ma-88	25	22	−	−	PROPN
ma-88	25	23	received	receive	VERB
ma-88	25	24	:	:	PUNCT
ma-88	25	25	20	20	NUM
ma-88	25	26	feb	feb	PROPN
ma-88	25	27	2022	2022	NUM
ma-88	25	28	.	.	PUNCT
ma-88	26	1	key	key	ADJ
ma-88	26	2	words	word	NOUN
ma-88	26	3	and	and	CCONJ
ma-88	26	4	phrases	phrase	NOUN
ma-88	26	5	.	.	PUNCT
ma-88	27	1	birkhoff	birkhoff	NOUN
ma-88	27	2	orthogonality	orthogonality	NOUN
ma-88	27	3	;	;	PUNCT
ma-88	27	4	isosceles	isoscele	NOUN
ma-88	27	5	-	-	PUNCT
ma-88	27	6	orthogonality	orthogonality	NOUN
ma-88	27	7	;	;	PUNCT
ma-88	27	8	approximate	approximate	ADJ
ma-88	27	9	orthogonality	orthogonality	NOUN
ma-88	27	10	;	;	PUNCT
ma-88	27	11	orthogonalitypreserving	orthogonalitypreserving	NOUN
ma-88	27	12	mappings	mapping	NOUN
ma-88	27	13	.	.	PUNCT
ma-88	28	1	1	1	NUM
ma-88	28	2	https://adac.ee	https://adac.ee	PROPN
ma-88	28	3	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	28	4	eur	eur	PROPN
ma-88	28	5	.	.	PUNCT
ma-88	29	1	j.	j.	PROPN
ma-88	29	2	math	math	PROPN
ma-88	29	3	.	.	PUNCT
ma-88	30	1	anal	anal	PROPN
ma-88	30	2	.	.	PUNCT
ma-88	31	1	10.28924	10.28924	NUM
ma-88	31	2	/	/	SYM
ma-88	31	3	ada	ada	PROPN
ma-88	31	4	/	/	SYM
ma-88	31	5	ma.2.16	ma.2.16	PROPN
ma-88	31	6	2	2	NUM
ma-88	31	7	‖x	‖x	NOUN
ma-88	31	8	−	−	PROPN
ma-88	31	9	y‖2|	y‖2|	NOUN
ma-88	31	10	≤	≤	X
ma-88	31	11	4ε‖x‖‖y‖	4ε‖x‖‖y‖	NUM
ma-88	31	12	for	for	ADP
ma-88	31	13	all	all	DET
ma-88	31	14	t	t	NOUN
ma-88	31	15	∈	∈	NOUN
ma-88	31	16	r	r	NOUN
ma-88	31	17	,	,	PUNCT
ma-88	31	18	and	and	CCONJ
ma-88	31	19	xε	xε	X
ma-88	31	20	⊥i	⊥i	PROPN
ma-88	31	21	y	y	PROPN
ma-88	31	22	:	:	PUNCT
ma-88	31	23	|‖x	|‖x	PROPN
ma-88	31	24	+	+	NUM
ma-88	31	25	y‖	y‖	NOUN
ma-88	32	1	−	−	NOUN
ma-88	32	2	‖x	‖x	NOUN
ma-88	33	1	−	−	PROPN
ma-88	33	2	y‖|	y‖|	NOUN
ma-88	33	3	≤	≤	PUNCT
ma-88	34	1	ε‖x	ε‖x	INTJ
ma-88	35	1	+	+	NUM
ma-88	35	2	y‖‖x	y‖‖x	INTJ
ma-88	35	3	−	−	PROPN
ma-88	35	4	y‖	y‖	PROPN
ma-88	35	5	forall	forall	PROPN
ma-88	35	6	t	t	PROPN
ma-88	35	7	∈	∈	PROPN
ma-88	35	8	r.	r.	NOUN
ma-88	35	9	many	many	ADJ
ma-88	35	10	meaningful	meaningful	ADJ
ma-88	35	11	results	result	NOUN
ma-88	35	12	have	have	AUX
ma-88	35	13	been	be	AUX
ma-88	35	14	found	find	VERB
ma-88	35	15	about	about	ADP
ma-88	35	16	approximate	approximate	ADJ
ma-88	35	17	orthogonality	orthogonality	NOUN
ma-88	35	18	through	through	ADP
ma-88	35	19	thetireless	thetireless	ADJ
ma-88	35	20	efforts	effort	NOUN
ma-88	35	21	of	of	ADP
ma-88	35	22	mathematicians	mathematician	NOUN
ma-88	35	23	,	,	PUNCT
ma-88	35	24	see	see	VERB
ma-88	35	25	[	[	X
ma-88	35	26	12,14].in	12,14].in	NUM
ma-88	35	27	this	this	DET
ma-88	35	28	paper	paper	NOUN
ma-88	35	29	we	we	PRON
ma-88	35	30	will	will	AUX
ma-88	35	31	introduce	introduce	VERB
ma-88	35	32	a	a	DET
ma-88	35	33	new	new	ADJ
ma-88	35	34	approximate	approximate	ADJ
ma-88	35	35	birkhoff	birkhoff	NOUN
ma-88	35	36	othogonality	othogonality	NOUN
ma-88	35	37	type	type	NOUN
ma-88	35	38	and	and	CCONJ
ma-88	35	39	investigateits	investigateit	VERB
ma-88	35	40	properties	property	NOUN
ma-88	35	41	and	and	CCONJ
ma-88	35	42	its	its	PRON
ma-88	35	43	relationship	relationship	NOUN
ma-88	35	44	with	with	ADP
ma-88	35	45	other	other	ADJ
ma-88	35	46	approximate	approximate	ADJ
ma-88	35	47	orthogonality	orthogonality	NOUN
ma-88	35	48	types	type	NOUN
ma-88	35	49	.	.	PUNCT
ma-88	36	1	moreover	moreover	ADV
ma-88	36	2	we	we	PRON
ma-88	36	3	give	give	VERB
ma-88	36	4	acharacterization	acharacterization	NOUN
ma-88	36	5	of	of	ADP
ma-88	36	6	inner	inner	ADJ
ma-88	36	7	product	product	NOUN
ma-88	36	8	spaces	space	NOUN
ma-88	36	9	by	by	ADP
ma-88	36	10	approximate	approximate	ADJ
ma-88	36	11	orthogonality	orthogonality	NOUN
ma-88	36	12	types	type	NOUN
ma-88	36	13	and	and	CCONJ
ma-88	36	14	some	some	DET
ma-88	36	15	propertiesabout	propertiesabout	ADJ
ma-88	36	16	approximately	approximately	ADV
ma-88	36	17	orthogonality	orthogonality	NOUN
ma-88	36	18	preserving	preserve	VERB
ma-88	36	19	mapping.throughout	mapping.throughout	PROPN
ma-88	36	20	the	the	DET
ma-88	36	21	paper	paper	NOUN
ma-88	36	22	we	we	PRON
ma-88	36	23	will	will	AUX
ma-88	36	24	only	only	ADV
ma-88	36	25	consider	consider	VERB
ma-88	36	26	normed	normed	ADJ
ma-88	36	27	spaces	space	NOUN
ma-88	36	28	with	with	ADP
ma-88	36	29	dimx	dimx	PROPN
ma-88	36	30	≥	≥	NOUN
ma-88	36	31	2	2	NUM
ma-88	36	32	,	,	PUNCT
ma-88	36	33	we	we	PRON
ma-88	36	34	use	use	VERB
ma-88	36	35	〈	〈	PROPN
ma-88	36	36	·	·	SYM
ma-88	36	37	|	|	ADJ
ma-88	36	38	·	·	SYM
ma-88	36	39	〉	〉	NOUN
ma-88	36	40	denotingthe	denotingthe	ADJ
ma-88	36	41	inner	inner	ADJ
ma-88	36	42	product	product	NOUN
ma-88	36	43	and	and	CCONJ
ma-88	36	44	(	(	PUNCT
ma-88	36	45	·	·	PUNCT
ma-88	36	46	|	|	NOUN
ma-88	36	47	·	·	PUNCT
ma-88	36	48	)	)	PUNCT
ma-88	36	49	denoting	denote	VERB
ma-88	36	50	the	the	DET
ma-88	36	51	angle	angle	NOUN
ma-88	36	52	between	between	ADP
ma-88	36	53	x	x	PROPN
ma-88	36	54	and	and	CCONJ
ma-88	36	55	y	y	PROPN
ma-88	36	56	,	,	PUNCT
ma-88	36	57	i	i	PRON
ma-88	36	58	,	,	PUNCT
ma-88	36	59	e	e	NOUN
ma-88	36	60	,	,	PUNCT
ma-88	36	61	in	in	ADP
ma-88	36	62	inner	inner	ADJ
ma-88	36	63	product	product	NOUN
ma-88	36	64	spaces	space	NOUN
ma-88	36	65	(	(	PUNCT
ma-88	36	66	x	x	NOUN
ma-88	36	67	,	,	PUNCT
ma-88	36	68	y	y	NOUN
ma-88	36	69	)	)	PUNCT
ma-88	36	70	=	=	PUNCT
ma-88	37	1	‖x+y‖2−‖x‖2−‖y‖2	‖x+y‖2−‖x‖2−‖y‖2	NOUN
ma-88	37	2	2‖x‖‖y‖	2‖x‖‖y‖	NOUN
ma-88	37	3	.	.	PUNCT
ma-88	38	1	2	2	X
ma-88	38	2	.	.	NUM
ma-88	38	3	approximate	approximate	ADJ
ma-88	38	4	birkhoff	birkhoff	PROPN
ma-88	38	5	orthogonality	orthogonality	PROPN
ma-88	38	6	⊥bε	⊥bε	NOUN
ma-88	38	7	let	let	VERB
ma-88	38	8	ε	ε	PROPN
ma-88	38	9	∈	∈	PROPN
ma-88	39	1	[	[	X
ma-88	39	2	0	0	NUM
ma-88	39	3	,	,	PUNCT
ma-88	39	4	1	1	NUM
ma-88	39	5	)	)	PUNCT
ma-88	39	6	and	and	CCONJ
ma-88	39	7	x	x	X
ma-88	39	8	,	,	PUNCT
ma-88	39	9	y	y	PROPN
ma-88	39	10	be	be	VERB
ma-88	39	11	elements	element	NOUN
ma-88	39	12	of	of	ADP
ma-88	39	13	inner	inner	ADJ
ma-88	39	14	product	product	NOUN
ma-88	39	15	spaces	space	VERB
ma-88	39	16	x	x	X
ma-88	39	17	,	,	PUNCT
ma-88	39	18	we	we	PRON
ma-88	39	19	have	have	VERB
ma-88	39	20	the	the	DET
ma-88	39	21	vertical	vertical	ADJ
ma-88	39	22	relationship	relationship	NOUN
ma-88	39	23	:	:	PUNCT
ma-88	40	1	x	x	PROPN
ma-88	40	2	⊥	⊥	NOUN
ma-88	40	3	y	y	PROPN
ma-88	40	4	⇐	⇐	PROPN
ma-88	40	5	⇒	⇒	NOUN
ma-88	40	6	|〈x	|〈x	VERB
ma-88	40	7	|y〉|	|y〉|	NOUN
ma-88	40	8	=	=	SYM
ma-88	40	9	0	0	NUM
ma-88	40	10	.	.	PUNCT
ma-88	40	11	to	to	PART
ma-88	40	12	generalize	generalize	VERB
ma-88	40	13	the	the	DET
ma-88	40	14	orthogonality	orthogonality	NOUN
ma-88	40	15	,	,	PUNCT
ma-88	40	16	it	it	PRON
ma-88	40	17	is	be	AUX
ma-88	40	18	natural	natural	ADJ
ma-88	40	19	to	to	PART
ma-88	40	20	consider	consider	VERB
ma-88	40	21	the	the	DET
ma-88	40	22	approximateorthogonality	approximateorthogonality	NOUN
ma-88	40	23	(	(	PUNCT
ma-88	40	24	ε	ε	NOUN
ma-88	40	25	-	-	PUNCT
ma-88	40	26	orthogonality	orthogonality	NOUN
ma-88	40	27	:	:	PUNCT
ma-88	40	28	x	x	NUM
ma-88	40	29	⊥ε	⊥ε	NUM
ma-88	40	30	y	y	PROPN
ma-88	40	31	)	)	PUNCT
ma-88	40	32	defined	define	VERB
ma-88	40	33	by	by	ADP
ma-88	40	34	:	:	PUNCT
ma-88	40	35	x	x	X
ma-88	40	36	⊥ε	⊥ε	NUM
ma-88	40	37	y	y	PROPN
ma-88	40	38	⇐	⇐	PROPN
ma-88	40	39	⇒	⇒	PROPN
ma-88	40	40	|〈x	|〈x	VERB
ma-88	40	41	|y〉|	|y〉|	NOUN
ma-88	40	42	≤	≤	NOUN
ma-88	40	43	ε‖x‖‖y‖	ε‖x‖‖y‖	ADP
ma-88	40	44	⇐	⇐	ADJ
ma-88	40	45	⇒	⇒	NOUN
ma-88	40	46	|cos(x	|cos(x	VERB
ma-88	40	47	,	,	PUNCT
ma-88	40	48	y)|	y)|	PROPN
ma-88	40	49	≤	≤	PROPN
ma-88	40	50	ε	ε	PROPN
ma-88	40	51	.	.	PUNCT
ma-88	41	1	now	now	ADV
ma-88	41	2	we	we	PRON
ma-88	41	3	consider	consider	VERB
ma-88	41	4	normed	normed	ADJ
ma-88	41	5	spaces	space	NOUN
ma-88	41	6	,	,	PUNCT
ma-88	41	7	many	many	ADJ
ma-88	41	8	mathematicians	mathematician	NOUN
ma-88	41	9	have	have	AUX
ma-88	41	10	introduced	introduce	VERB
ma-88	41	11	different	different	ADJ
ma-88	41	12	types	type	NOUN
ma-88	41	13	of	of	ADP
ma-88	41	14	orthogo	orthogo	NOUN
ma-88	41	15	-	-	PUNCT
ma-88	41	16	nality	nality	NOUN
ma-88	41	17	to	to	PART
ma-88	41	18	represent	represent	VERB
ma-88	41	19	orthogonality	orthogonality	NOUN
ma-88	41	20	such	such	ADJ
ma-88	41	21	as	as	ADP
ma-88	41	22	birkhoff	birkhoff	NOUN
ma-88	41	23	orthogonality	orthogonality	NOUN
ma-88	41	24	[	[	X
ma-88	41	25	5	5	NUM
ma-88	41	26	]	]	PUNCT
ma-88	41	27	and	and	CCONJ
ma-88	41	28	isosceles	isoscele	NOUN
ma-88	41	29	orthogonality	orthogonality	NOUN
ma-88	41	30	[	[	X
ma-88	41	31	15].as	15].as	NUM
ma-88	41	32	an	an	DET
ma-88	41	33	extension	extension	NOUN
ma-88	41	34	for	for	ADP
ma-88	41	35	the	the	DET
ma-88	41	36	orthogonality	orthogonality	NOUN
ma-88	41	37	,	,	PUNCT
ma-88	41	38	approximately	approximately	ADV
ma-88	41	39	orthogonality	orthogonality	NOUN
ma-88	41	40	such	such	ADJ
ma-88	41	41	as	as	ADP
ma-88	41	42	approximate	approximate	ADJ
ma-88	41	43	birkhofforthogonalty	birkhofforthogonalty	NOUN
ma-88	41	44	[	[	X
ma-88	41	45	13,21	13,21	PROPN
ma-88	41	46	]	]	X
ma-88	41	47	:	:	PUNCT
ma-88	41	48	xε	xε	NUM
ma-88	41	49	⊥b	⊥b	PRON
ma-88	41	50	y	y	NOUN
ma-88	41	51	⇐	⇐	ADJ
ma-88	41	52	⇒	⇒	NOUN
ma-88	41	53	‖x	‖x	NOUN
ma-88	42	1	+	+	CCONJ
ma-88	42	2	ty‖	ty‖	PRON
ma-88	42	3	≥	≥	NOUN
ma-88	42	4	(	(	PUNCT
ma-88	42	5	1−	1−	NUM
ma-88	42	6	ε)‖x‖	ε)‖x‖	PROPN
ma-88	42	7	t	t	PROPN
ma-88	42	8	∈	∈	PROPN
ma-88	42	9	r.	r.	PROPN
ma-88	42	10	x	x	PROPN
ma-88	42	11	⊥εb	⊥εb	PROPN
ma-88	42	12	y	y	PROPN
ma-88	42	13	⇐	⇐	PROPN
ma-88	42	14	⇒	⇒	PROPN
ma-88	42	15	‖x	‖x	NOUN
ma-88	43	1	+	+	CCONJ
ma-88	43	2	ty‖2	ty‖2	PROPN
ma-88	43	3	≥	≥	NOUN
ma-88	43	4	‖x‖2	‖x‖2	VERB
ma-88	43	5	−	−	PROPN
ma-88	43	6	2ε‖x‖‖ty‖	2ε‖x‖‖ty‖	NUM
ma-88	43	7	t	t	NOUN
ma-88	43	8	∈	∈	NOUN
ma-88	43	9	r	r	NOUN
ma-88	43	10	,	,	PUNCT
ma-88	43	11	and	and	CCONJ
ma-88	43	12	approximate	approximate	ADJ
ma-88	43	13	isosceles	isoscele	NOUN
ma-88	43	14	orthogonality	orthogonality	NOUN
ma-88	43	15	[	[	X
ma-88	43	16	11	11	NUM
ma-88	43	17	]	]	SYM
ma-88	43	18	:	:	PUNCT
ma-88	43	19	x	x	SYM
ma-88	43	20	⊥εi	⊥εi	ADP
ma-88	43	21	y	y	PROPN
ma-88	43	22	⇐	⇐	PROPN
ma-88	43	23	⇒	⇒	PROPN
ma-88	43	24	|‖x	|‖x	PROPN
ma-88	43	25	+	+	CCONJ
ma-88	43	26	y‖2	y‖2	X
ma-88	43	27	−	−	NOUN
ma-88	43	28	‖x	‖x	NOUN
ma-88	44	1	−	−	PROPN
ma-88	44	2	y‖2|	y‖2|	NOUN
ma-88	44	3	≤	≤	NOUN
ma-88	44	4	4ε‖x‖‖y‖	4ε‖x‖‖y‖	NUM
ma-88	44	5	t	t	PROPN
ma-88	44	6	∈	∈	PROPN
ma-88	44	7	r.	r.	PROPN
ma-88	44	8	xε	xε	PUNCT
ma-88	45	1	⊥i	⊥i	PROPN
ma-88	45	2	y	y	PROPN
ma-88	45	3	⇐	⇐	PROPN
ma-88	45	4	⇒	⇒	PROPN
ma-88	45	5	|‖x	|‖x	X
ma-88	45	6	+	+	CCONJ
ma-88	45	7	y‖	y‖	NOUN
ma-88	45	8	−	−	NOUN
ma-88	45	9	‖x	‖x	NOUN
ma-88	46	1	−	−	PROPN
ma-88	46	2	y‖|	y‖|	NOUN
ma-88	46	3	≤	≤	PUNCT
ma-88	47	1	ε‖x	ε‖x	INTJ
ma-88	48	1	+	+	NUM
ma-88	48	2	y‖‖x	y‖‖x	INTJ
ma-88	48	3	−	−	PROPN
ma-88	48	4	y‖	y‖	PROPN
ma-88	48	5	t	t	PROPN
ma-88	48	6	∈	∈	PROPN
ma-88	48	7	r	r	NOUN
ma-88	48	8	,	,	PUNCT
ma-88	48	9	have	have	AUX
ma-88	48	10	been	be	AUX
ma-88	48	11	defined	define	VERB
ma-88	48	12	and	and	CCONJ
ma-88	48	13	studied	study	VERB
ma-88	48	14	.	.	PUNCT
ma-88	49	1	notice	notice	VERB
ma-88	49	2	that	that	SCONJ
ma-88	49	3	the	the	DET
ma-88	49	4	definition	definition	NOUN
ma-88	49	5	of	of	ADP
ma-88	49	6	ε	ε	PROPN
ma-88	49	7	⊥b	⊥b	PRON
ma-88	49	8	is	be	AUX
ma-88	49	9	quadratic	quadratic	ADJ
ma-88	49	10	while	while	SCONJ
ma-88	49	11	the	the	DET
ma-88	49	12	definitionof	definitionof	PROPN
ma-88	49	13	⊥εb	⊥εb	NOUN
ma-88	49	14	is	be	AUX
ma-88	49	15	of	of	ADP
ma-88	49	16	first	first	ADJ
ma-88	49	17	order	order	NOUN
ma-88	49	18	,	,	PUNCT
ma-88	49	19	we	we	PRON
ma-88	49	20	give	give	VERB
ma-88	49	21	a	a	DET
ma-88	49	22	new	new	ADJ
ma-88	49	23	approximate	approximate	ADJ
ma-88	49	24	birkhoff	birkhoff	NOUN
ma-88	49	25	orthogonality	orthogonality	NOUN
ma-88	49	26	type	type	NOUN
ma-88	49	27	:	:	PUNCT
ma-88	49	28	x	x	PART
ma-88	49	29	⊥bε	⊥bε	NOUN
ma-88	49	30	y	y	PROPN
ma-88	49	31	⇐	⇐	PROPN
ma-88	49	32	⇒	⇒	NOUN
ma-88	49	33	‖x	‖x	NOUN
ma-88	50	1	+	+	CCONJ
ma-88	50	2	ty‖	ty‖	PRON
ma-88	50	3	≥	≥	NOUN
ma-88	50	4	‖x‖	‖x‖	VERB
ma-88	50	5	−	−	PROPN
ma-88	50	6	ε‖ty‖	ε‖ty‖	NOUN
ma-88	50	7	,	,	PUNCT
ma-88	50	8	which	which	PRON
ma-88	50	9	is	be	AUX
ma-88	50	10	also	also	ADV
ma-88	50	11	of	of	ADP
ma-88	50	12	first	first	ADJ
ma-88	50	13	order	order	NOUN
ma-88	50	14	but	but	CCONJ
ma-88	50	15	different	different	ADJ
ma-88	50	16	from	from	ADP
ma-88	50	17	ε	ε	PROPN
ma-88	50	18	⊥b	⊥b	X
ma-88	50	19	.	.	PUNCT
ma-88	51	1	it	it	PRON
ma-88	51	2	is	be	AUX
ma-88	51	3	easy	easy	ADJ
ma-88	51	4	to	to	PART
ma-88	51	5	see	see	VERB
ma-88	51	6	that	that	SCONJ
ma-88	51	7	the	the	DET
ma-88	51	8	inequality	inequality	NOUN
ma-88	51	9	is	be	AUX
ma-88	51	10	alwayscorrect	alwayscorrect	ADJ
ma-88	51	11	if	if	SCONJ
ma-88	51	12	t	t	PROPN
ma-88	51	13	≥	≥	NOUN
ma-88	51	14	‖x‖	‖x‖	PROPN
ma-88	51	15	ε‖y‖	ε‖y‖	PROPN
ma-88	51	16	.	.	PUNCT
ma-88	52	1	example	example	NOUN
ma-88	52	2	2.1	2.1	NUM
ma-88	52	3	.	.	PUNCT
ma-88	53	1	let	let	VERB
ma-88	53	2	x	x	PUNCT
ma-88	53	3	=	=	PRON
ma-88	53	4	(	(	PUNCT
ma-88	53	5	r2	r2	PROPN
ma-88	53	6	,	,	PUNCT
ma-88	53	7	‖	‖	PROPN
ma-88	53	8	·	·	SYM
ma-88	53	9	‖1	‖1	PROPN
ma-88	53	10	)	)	PUNCT
ma-88	53	11	,	,	PUNCT
ma-88	53	12	assume	assume	VERB
ma-88	53	13	that	that	SCONJ
ma-88	53	14	x	x	X
ma-88	53	15	=	=	SYM
ma-88	53	16	(	(	PUNCT
ma-88	53	17	1	1	NUM
ma-88	53	18	,	,	PUNCT
ma-88	53	19	0	0	NUM
ma-88	53	20	)	)	PUNCT
ma-88	53	21	,	,	PUNCT
ma-88	53	22	y	y	PROPN
ma-88	53	23	=	=	SYM
ma-88	53	24	(	(	PUNCT
ma-88	53	25	z	z	NOUN
ma-88	53	26	,	,	PUNCT
ma-88	53	27	1	1	NUM
ma-88	53	28	−	−	PROPN
ma-88	53	29	z	z	NOUN
ma-88	53	30	)	)	PUNCT
ma-88	53	31	,	,	PUNCT
ma-88	53	32	z	z	NOUN
ma-88	53	33	∈	∈	PROPN
ma-88	54	1	[	[	X
ma-88	54	2	0	0	NUM
ma-88	54	3	,	,	PUNCT
ma-88	54	4	1	1	NUM
ma-88	54	5	)	)	PUNCT
ma-88	54	6	.	.	PUNCT
ma-88	55	1	if	if	SCONJ
ma-88	55	2	we	we	PRON
ma-88	55	3	want	want	VERB
ma-88	55	4	xε	xε	VERB
ma-88	55	5	⊥b	⊥b	PRON
ma-88	55	6	y	y	PROPN
ma-88	55	7	,	,	PUNCT
ma-88	55	8	then	then	ADV
ma-88	55	9	the	the	DET
ma-88	55	10	inequality	inequality	NOUN
ma-88	55	11	‖x	‖x	NOUN
ma-88	55	12	+	+	CCONJ
ma-88	56	1	ty‖	ty‖	PRON
ma-88	56	2	≥	≥	NOUN
ma-88	56	3	(	(	PUNCT
ma-88	56	4	1−	1−	NUM
ma-88	56	5	ε)‖x‖	ε)‖x‖	PROPN
ma-88	56	6	should	should	AUX
ma-88	56	7	hold	hold	VERB
ma-88	56	8	for	for	ADP
ma-88	56	9	all	all	DET
ma-88	56	10	t	t	NOUN
ma-88	56	11	∈	∈	NOUN
ma-88	56	12	r	r	NOUN
ma-88	56	13	,	,	PUNCT
ma-88	56	14	thus	thus	ADV
ma-88	56	15	we	we	PRON
ma-88	56	16	have	have	VERB
ma-88	56	17	:	:	PUNCT
ma-88	56	18	‖(1	‖(1	NUM
ma-88	56	19	+	+	NUM
ma-88	56	20	tz	tz	NOUN
ma-88	56	21	,	,	PUNCT
ma-88	56	22	t(1−	t(1−	PROPN
ma-88	56	23	z))‖	z))‖	ADJ
ma-88	56	24	≥	≥	NOUN
ma-88	56	25	1−	1−	NUM
ma-88	56	26	ε	ε	PROPN
ma-88	56	27	t	t	PROPN
ma-88	56	28	∈	∈	PROPN
ma-88	56	29	r.	r.	PROPN
ma-88	56	30	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	56	31	eur	eur	PROPN
ma-88	56	32	.	.	PUNCT
ma-88	57	1	j.	j.	PROPN
ma-88	57	2	math	math	PROPN
ma-88	57	3	.	.	PUNCT
ma-88	58	1	anal	anal	PROPN
ma-88	58	2	.	.	PUNCT
ma-88	59	1	10.28924	10.28924	NUM
ma-88	59	2	/	/	SYM
ma-88	59	3	ada	ada	PROPN
ma-88	59	4	/	/	SYM
ma-88	59	5	ma.2.16	ma.2.16	PROPN
ma-88	59	6	3	3	NUM
ma-88	59	7	if	if	SCONJ
ma-88	59	8	t	t	PROPN
ma-88	59	9	≥	≥	NOUN
ma-88	59	10	0	0	NUM
ma-88	59	11	,	,	PUNCT
ma-88	59	12	the	the	DET
ma-88	59	13	inequality	inequality	NOUN
ma-88	59	14	is	be	AUX
ma-88	59	15	always	always	ADV
ma-88	59	16	correct	correct	ADJ
ma-88	59	17	.	.	PUNCT
ma-88	60	1	for	for	ADP
ma-88	60	2	t	t	PROPN
ma-88	60	3	<	<	X
ma-88	60	4	0	0	NUM
ma-88	60	5	,	,	PUNCT
ma-88	60	6	if	if	SCONJ
ma-88	60	7	1	1	NUM
ma-88	60	8	+	+	NUM
ma-88	60	9	tz	tz	PROPN
ma-88	60	10	≥	≥	NOUN
ma-88	60	11	0	0	NUM
ma-88	60	12	,	,	PUNCT
ma-88	60	13	we	we	PRON
ma-88	60	14	have	have	VERB
ma-88	60	15	:	:	PUNCT
ma-88	60	16	1	1	NUM
ma-88	60	17	+	+	NUM
ma-88	60	18	tz	tz	PROPN
ma-88	60	19	−	−	PROPN
ma-88	60	20	t	t	NOUN
ma-88	60	21	+	+	CCONJ
ma-88	60	22	tz	tz	PROPN
ma-88	60	23	≥	≥	NUM
ma-88	61	1	1−	1−	NUM
ma-88	61	2	ε	ε	PROPN
ma-88	61	3	.	.	PUNCT
ma-88	62	1	thus	thus	ADV
ma-88	62	2	z	z	NOUN
ma-88	62	3	≤	≤	NUM
ma-88	62	4	1	1	NUM
ma-88	62	5	2	2	NUM
ma-88	62	6	−	−	NOUN
ma-88	62	7	ε	ε	PROPN
ma-88	62	8	2	2	NUM
ma-88	62	9	t	t	NOUN
ma-88	62	10	.	.	PUNCT
ma-88	63	1	by	by	ADP
ma-88	63	2	1	1	NUM
ma-88	63	3	+	+	CCONJ
ma-88	63	4	tz	tz	PROPN
ma-88	63	5	≥	≥	NOUN
ma-88	63	6	0	0	NUM
ma-88	63	7	,	,	PUNCT
ma-88	63	8	we	we	PRON
ma-88	63	9	get	get	VERB
ma-88	63	10	z	z	NOUN
ma-88	63	11	≤	≤	NUM
ma-88	63	12	1	1	NUM
ma-88	63	13	2−ε	2−ε	NOUN
ma-88	63	14	.	.	PUNCT
ma-88	64	1	if	if	SCONJ
ma-88	64	2	1	1	NUM
ma-88	64	3	+	+	NUM
ma-88	64	4	tz	tz	X
ma-88	64	5	<	<	X
ma-88	64	6	0	0	NUM
ma-88	64	7	,	,	PUNCT
ma-88	64	8	similarly	similarly	ADV
ma-88	64	9	we	we	PRON
ma-88	64	10	need	need	VERB
ma-88	64	11	t	t	NOUN
ma-88	64	12	≤	≤	NOUN
ma-88	64	13	ε−	ε−	PROPN
ma-88	64	14	2	2	NUM
ma-88	64	15	.	.	PUNCT
ma-88	65	1	since	since	SCONJ
ma-88	65	2	z	z	NOUN
ma-88	65	3	≤	≤	NUM
ma-88	65	4	1	1	NUM
ma-88	65	5	2−ε	2−ε	NOUN
ma-88	65	6	,	,	PUNCT
ma-88	65	7	from	from	ADP
ma-88	65	8	1	1	NUM
ma-88	65	9	+	+	CCONJ
ma-88	65	10	tz	tz	PROPN
ma-88	65	11	<	<	X
ma-88	65	12	0	0	NUM
ma-88	65	13	,	,	PUNCT
ma-88	65	14	we	we	PRON
ma-88	65	15	get	get	VERB
ma-88	65	16	t	t	NOUN
ma-88	65	17	≤	≤	NOUN
ma-88	65	18	ε−	ε−	PROPN
ma-88	65	19	2	2	NUM
ma-88	65	20	.	.	PUNCT
ma-88	65	21	thus	thus	ADV
ma-88	65	22	xε	xε	ADP
ma-88	65	23	⊥b	⊥b	PRON
ma-88	65	24	y	y	PROPN
ma-88	65	25	iff	iff	PROPN
ma-88	65	26	z	z	PROPN
ma-88	65	27	≤	≤	PROPN
ma-88	65	28	1	1	NUM
ma-88	65	29	2−ε	2−ε	NOUN
ma-88	65	30	.	.	PUNCT
ma-88	66	1	on	on	ADP
ma-88	66	2	the	the	DET
ma-88	66	3	other	other	ADJ
ma-88	66	4	hand	hand	NOUN
ma-88	66	5	,	,	PUNCT
ma-88	66	6	if	if	SCONJ
ma-88	66	7	we	we	PRON
ma-88	66	8	want	want	VERB
ma-88	66	9	x	x	X
ma-88	66	10	⊥bε	⊥bε	NOUN
ma-88	66	11	y	y	PROPN
ma-88	66	12	,	,	PUNCT
ma-88	66	13	the	the	DET
ma-88	66	14	inequality	inequality	NOUN
ma-88	66	15	‖x	‖x	NOUN
ma-88	67	1	+	+	CCONJ
ma-88	67	2	ty‖	ty‖	DET
ma-88	67	3	≥	≥	NOUN
ma-88	67	4	‖x‖	‖x‖	PART
ma-88	67	5	−	−	PROPN
ma-88	67	6	ε‖ty‖	ε‖ty‖	NOUN
ma-88	67	7	should	should	AUX
ma-88	67	8	hold	hold	VERB
ma-88	67	9	for	for	ADP
ma-88	67	10	all	all	DET
ma-88	67	11	t	t	NOUN
ma-88	67	12	∈	∈	NOUN
ma-88	67	13	r	r	NOUN
ma-88	67	14	,	,	PUNCT
ma-88	67	15	thus	thus	ADV
ma-88	67	16	we	we	PRON
ma-88	67	17	have	have	VERB
ma-88	67	18	:	:	PUNCT
ma-88	67	19	‖(1	‖(1	NUM
ma-88	67	20	+	+	NUM
ma-88	67	21	tz	tz	NOUN
ma-88	67	22	,	,	PUNCT
ma-88	67	23	t(1−	t(1−	PROPN
ma-88	67	24	z))‖	z))‖	ADJ
ma-88	67	25	≥	≥	NOUN
ma-88	67	26	1−	1−	NUM
ma-88	67	27	ε|t|	ε|t|	PROPN
ma-88	67	28	t	t	PROPN
ma-88	67	29	∈	∈	PROPN
ma-88	67	30	r.	r.	PROPN
ma-88	67	31	if	if	SCONJ
ma-88	67	32	t	t	PROPN
ma-88	67	33	≥	≥	PROPN
ma-88	67	34	0	0	NUM
ma-88	67	35	,	,	PUNCT
ma-88	67	36	the	the	DET
ma-88	67	37	inequality	inequality	NOUN
ma-88	67	38	is	be	AUX
ma-88	67	39	also	also	ADV
ma-88	67	40	always	always	ADV
ma-88	67	41	correct	correct	ADJ
ma-88	67	42	.	.	PUNCT
ma-88	68	1	for	for	ADP
ma-88	68	2	t	t	PROPN
ma-88	68	3	<	<	X
ma-88	68	4	0	0	NUM
ma-88	68	5	,	,	PUNCT
ma-88	68	6	if	if	SCONJ
ma-88	68	7	1	1	NUM
ma-88	68	8	+	+	NUM
ma-88	68	9	tz	tz	PROPN
ma-88	68	10	≥	≥	NOUN
ma-88	68	11	0	0	NUM
ma-88	68	12	,	,	PUNCT
ma-88	68	13	we	we	PRON
ma-88	68	14	have	have	VERB
ma-88	68	15	:	:	PUNCT
ma-88	68	16	1	1	NUM
ma-88	68	17	+	+	NUM
ma-88	68	18	tz	tz	PROPN
ma-88	68	19	−	−	PROPN
ma-88	68	20	t	t	NOUN
ma-88	68	21	+	+	CCONJ
ma-88	68	22	tz	tz	PROPN
ma-88	68	23	≥	≥	NUM
ma-88	68	24	1	1	NUM
ma-88	69	1	+	+	CCONJ
ma-88	69	2	εt	εt	PROPN
ma-88	69	3	.	.	PUNCT
ma-88	70	1	thus	thus	ADV
ma-88	70	2	z	z	NOUN
ma-88	70	3	≤	≤	NOUN
ma-88	70	4	1+ε	1+ε	NUM
ma-88	70	5	2	2	NUM
ma-88	70	6	.	.	PUNCT
ma-88	71	1	similarly	similarly	ADV
ma-88	71	2	we	we	PRON
ma-88	71	3	can	can	AUX
ma-88	71	4	get	get	VERB
ma-88	71	5	‖(1	‖(1	NUM
ma-88	71	6	+	+	NUM
ma-88	71	7	tz	tz	NOUN
ma-88	71	8	,	,	PUNCT
ma-88	71	9	t(1−	t(1−	PROPN
ma-88	71	10	z))‖	z))‖	ADJ
ma-88	71	11	≥	≥	NOUN
ma-88	71	12	1−	1−	NUM
ma-88	71	13	ε|t|	ε|t|	ADV
ma-88	71	14	for	for	ADP
ma-88	71	15	1	1	NUM
ma-88	71	16	+	+	CCONJ
ma-88	71	17	tz	tz	X
ma-88	71	18	<	<	X
ma-88	71	19	0	0	PUNCT
ma-88	72	1	if	if	SCONJ
ma-88	72	2	z	z	NOUN
ma-88	72	3	≤	≤	NUM
ma-88	72	4	1+ε	1+ε	NUM
ma-88	72	5	2	2	NUM
ma-88	72	6	.	.	PUNCT
ma-88	73	1	thus	thus	ADV
ma-88	73	2	x	x	SYM
ma-88	73	3	⊥bε	⊥bε	NOUN
ma-88	73	4	y	y	PROPN
ma-88	73	5	iff	iff	PROPN
ma-88	73	6	z	z	PROPN
ma-88	73	7	≤	≤	PROPN
ma-88	73	8	1+ε	1+ε	NUM
ma-88	73	9	2	2	NUM
ma-88	73	10	.	.	PUNCT
ma-88	74	1	we	we	PRON
ma-88	74	2	have	have	VERB
ma-88	74	3	the	the	DET
ma-88	74	4	result	result	NOUN
ma-88	74	5	that	that	SCONJ
ma-88	74	6	⊥bε	⊥bε	NOUN
ma-88	74	7	is	be	AUX
ma-88	74	8	not	not	PART
ma-88	74	9	always	always	ADV
ma-88	74	10	equivalent	equivalent	ADJ
ma-88	74	11	to	to	PART
ma-88	74	12	ε	ε	PROPN
ma-88	74	13	⊥b	⊥b	PRON
ma-88	74	14	in	in	ADP
ma-88	74	15	x	x	X
ma-88	74	16	.	.	PUNCT
ma-88	75	1	since	since	SCONJ
ma-88	75	2	the	the	DET
ma-88	75	3	definition	definition	NOUN
ma-88	75	4	of	of	ADP
ma-88	75	5	approximate	approximate	ADJ
ma-88	75	6	birkhoff	birkhoff	NOUN
ma-88	75	7	orthogonality	orthogonality	NOUN
ma-88	75	8	comes	come	VERB
ma-88	75	9	from	from	ADP
ma-88	75	10	the	the	DET
ma-88	75	11	notion	notion	NOUN
ma-88	75	12	of	of	ADP
ma-88	75	13	approximateorthogonality	approximateorthogonality	NOUN
ma-88	75	14	⊥ε	⊥ε	VERB
ma-88	75	15	in	in	ADP
ma-88	75	16	inner	inner	ADJ
ma-88	75	17	product	product	NOUN
ma-88	75	18	spaces	space	NOUN
ma-88	75	19	,	,	PUNCT
ma-88	75	20	it	it	PRON
ma-88	75	21	is	be	AUX
ma-88	75	22	natural	natural	ADJ
ma-88	75	23	to	to	PART
ma-88	75	24	require	require	VERB
ma-88	75	25	the	the	DET
ma-88	75	26	equivalence	equivalence	NOUN
ma-88	75	27	:	:	PUNCT
ma-88	75	28	x	x	X
ma-88	75	29	⊥bε	⊥bε	NOUN
ma-88	75	30	y	y	PROPN
ma-88	75	31	⇐	⇐	PROPN
ma-88	75	32	⇒	⇒	PROPN
ma-88	75	33	x	x	X
ma-88	76	1	⊥ε	⊥ε	NUM
ma-88	76	2	y	y	NOUN
ma-88	76	3	in	in	ADP
ma-88	76	4	inner	inner	ADJ
ma-88	76	5	product	product	NOUN
ma-88	76	6	spaces	space	VERB
ma-88	76	7	.	.	PUNCT
ma-88	77	1	now	now	ADV
ma-88	77	2	we	we	PRON
ma-88	77	3	give	give	VERB
ma-88	77	4	some	some	DET
ma-88	77	5	basic	basic	ADJ
ma-88	77	6	properties	property	NOUN
ma-88	77	7	about	about	ADP
ma-88	77	8	⊥bε	⊥bε	NOUN
ma-88	77	9	before	before	ADP
ma-88	77	10	prove	prove	VERB
ma-88	77	11	theequivalence	theequivalence	NOUN
ma-88	77	12	.	.	PUNCT
ma-88	78	1	proposition	proposition	NOUN
ma-88	78	2	2.2	2.2	NUM
ma-88	78	3	.	.	PUNCT
ma-88	79	1	let	let	VERB
ma-88	79	2	x	x	PRON
ma-88	79	3	be	be	AUX
ma-88	79	4	normed	normed	ADJ
ma-88	79	5	spaces	space	NOUN
ma-88	79	6	,	,	PUNCT
ma-88	79	7	then	then	ADV
ma-88	79	8	⊥bε	⊥bε	NOUN
ma-88	79	9	is	be	AUX
ma-88	79	10	homogeneous	homogeneous	ADJ
ma-88	79	11	.	.	PUNCT
ma-88	79	12	,	,	PUNCT
ma-88	79	13	this	this	PRON
ma-88	79	14	is	be	AUX
ma-88	79	15	x	x	PART
ma-88	79	16	⊥bε	⊥bε	NOUN
ma-88	79	17	y	y	PROPN
ma-88	79	18	implies	imply	VERB
ma-88	79	19	αx	αx	PRON
ma-88	79	20	⊥bε	⊥bε	VERB
ma-88	79	21	βy	βy	X
ma-88	80	1	(	(	PUNCT
ma-88	80	2	x	x	X
ma-88	80	3	,	,	PUNCT
ma-88	80	4	y	y	PROPN
ma-88	80	5	∈	∈	PROPN
ma-88	80	6	x	x	PROPN
ma-88	80	7	,	,	PUNCT
ma-88	80	8	α	α	X
ma-88	80	9	,	,	PUNCT
ma-88	80	10	β	β	X
ma-88	80	11	∈	∈	NOUN
ma-88	80	12	r	r	NOUN
ma-88	80	13	)	)	PUNCT
ma-88	80	14	.	.	PUNCT
ma-88	81	1	proof	proof	NOUN
ma-88	81	2	.	.	PUNCT
ma-88	82	1	since	since	SCONJ
ma-88	82	2	x	x	PROPN
ma-88	82	3	⊥bε	⊥bε	NOUN
ma-88	82	4	y	y	PROPN
ma-88	82	5	,	,	PUNCT
ma-88	82	6	we	we	PRON
ma-88	82	7	have	have	VERB
ma-88	82	8	‖x	‖x	NOUN
ma-88	83	1	+	+	CCONJ
ma-88	83	2	ty‖	ty‖	PRON
ma-88	83	3	≥	≥	NOUN
ma-88	83	4	‖x‖	‖x‖	PART
ma-88	83	5	−	−	PROPN
ma-88	84	1	ε‖ty‖	ε‖ty‖	NOUN
ma-88	84	2	for	for	ADP
ma-88	84	3	any	any	DET
ma-88	84	4	t	t	PROPN
ma-88	84	5	∈	∈	PROPN
ma-88	84	6	r.	r.	PROPN
ma-88	84	7	if	if	SCONJ
ma-88	84	8	α	α	PROPN
ma-88	84	9	=	=	NOUN
ma-88	84	10	0	0	NUM
ma-88	84	11	,	,	PUNCT
ma-88	84	12	αx	αx	PRON
ma-88	84	13	⊥bε	⊥bε	NOUN
ma-88	84	14	βy	βy	PRON
ma-88	84	15	is	be	AUX
ma-88	84	16	always	always	ADV
ma-88	84	17	correct	correct	ADJ
ma-88	84	18	;	;	PUNCT
ma-88	84	19	if	if	SCONJ
ma-88	84	20	α	α	PROPN
ma-88	84	21	6=	6=	NOUN
ma-88	84	22	0	0	NUM
ma-88	84	23	,	,	PUNCT
ma-88	84	24	we	we	PRON
ma-88	84	25	have	have	VERB
ma-88	84	26	‖αx	‖αx	NUM
ma-88	84	27	+	+	NOUN
ma-88	84	28	tβy‖	tβy‖	NOUN
ma-88	84	29	=	=	PUNCT
ma-88	84	30	|α|‖x	|α|‖x	NOUN
ma-88	84	31	+	+	CCONJ
ma-88	84	32	β	β	X
ma-88	84	33	α	α	NOUN
ma-88	84	34	ty‖	ty‖	PROPN
ma-88	84	35	≥	≥	NOUN
ma-88	84	36	|α|{‖x‖	|α|{‖x‖	VERB
ma-88	84	37	−	−	PROPN
ma-88	85	1	ε‖	ε‖	NOUN
ma-88	85	2	β	β	X
ma-88	85	3	α	α	NOUN
ma-88	85	4	ty‖	ty‖	PROPN
ma-88	85	5	}	}	PUNCT
ma-88	85	6	=	=	PUNCT
ma-88	86	1	‖ax‖	‖ax‖	ADJ
ma-88	86	2	−	−	PROPN
ma-88	86	3	ε‖tβy‖.	ε‖tβy‖.	PROPN
ma-88	86	4	thus	thus	ADV
ma-88	86	5	αx	αx	ADV
ma-88	86	6	⊥bε	⊥bε	VERB
ma-88	86	7	βy	βy	PRON
ma-88	86	8	.	.	PUNCT
ma-88	87	1	�	�	PROPN
ma-88	87	2	recall	recall	VERB
ma-88	87	3	that	that	SCONJ
ma-88	87	4	the	the	DET
ma-88	87	5	limits	limit	NOUN
ma-88	87	6	[	[	X
ma-88	87	7	16	16	NUM
ma-88	87	8	]	]	PUNCT
ma-88	87	9	:	:	PUNCT
ma-88	87	10	n±(x	n±(x	PROPN
ma-88	87	11	;	;	PUNCT
ma-88	87	12	y	y	X
ma-88	87	13	)	)	PUNCT
ma-88	88	1	=	=	VERB
ma-88	88	2	lim	lim	PROPN
ma-88	88	3	n→±∞	n→±∞	NOUN
ma-88	88	4	‖nx	‖nx	PROPN
ma-88	88	5	+	+	NUM
ma-88	88	6	y‖	y‖	PROPN
ma-88	88	7	−	−	PROPN
ma-88	88	8	|nx‖	|nx‖	PROPN
ma-88	88	9	=	=	PROPN
ma-88	88	10	lim	lim	PROPN
ma-88	88	11	h→0±	h→0±	PROPN
ma-88	88	12	‖x	‖x	PROPN
ma-88	88	13	+	+	CCONJ
ma-88	88	14	hy‖	hy‖	PROPN
ma-88	88	15	−	−	PROPN
ma-88	88	16	‖x‖	‖x‖	PROPN
ma-88	88	17	h	h	NOUN
ma-88	88	18	,	,	PUNCT
ma-88	88	19	exist	exist	VERB
ma-88	88	20	and	and	CCONJ
ma-88	88	21	satisfy	satisfy	VERB
ma-88	88	22	the	the	DET
ma-88	88	23	weakened	weaken	VERB
ma-88	88	24	linearity	linearity	NOUN
ma-88	88	25	condition	condition	NOUN
ma-88	88	26	[	[	X
ma-88	88	27	4	4	NUM
ma-88	88	28	]	]	PUNCT
ma-88	88	29	.	.	PUNCT
ma-88	89	1	x	x	X
ma-88	89	2	,	,	PUNCT
ma-88	89	3	y	y	PROPN
ma-88	89	4	are	be	AUX
ma-88	89	5	said	say	VERB
ma-88	89	6	to	to	PART
ma-88	89	7	be	be	AUX
ma-88	89	8	gateaux	gateaux	ADV
ma-88	89	9	differentiable	differentiable	ADJ
ma-88	89	10	[	[	PUNCT
ma-88	89	11	1]at	1]at	NUM
ma-88	89	12	0	0	NUM
ma-88	90	1	i	i	PRON
ma-88	90	2	f	f	PROPN
ma-88	90	3	n−(x	n−(x	PROPN
ma-88	90	4	,	,	PUNCT
ma-88	90	5	y	y	PROPN
ma-88	90	6	)	)	PUNCT
ma-88	90	7	=	=	SYM
ma-88	90	8	n+(x	n+(x	PROPN
ma-88	90	9	,	,	PUNCT
ma-88	90	10	y	y	PROPN
ma-88	90	11	)	)	PUNCT
ma-88	90	12	.	.	PUNCT
ma-88	91	1	moreover	moreover	ADV
ma-88	91	2	we	we	PRON
ma-88	91	3	have	have	VERB
ma-88	91	4	[	[	X
ma-88	91	5	16	16	NUM
ma-88	91	6	]	]	X
ma-88	91	7	:	:	PUNCT
ma-88	91	8	n±(x	n±(x	PROPN
ma-88	91	9	;	;	PUNCT
ma-88	91	10	rx	rx	VERB
ma-88	91	11	+	+	CCONJ
ma-88	91	12	sy	sy	X
ma-88	91	13	)	)	PUNCT
ma-88	91	14	=	=	PUNCT
ma-88	92	1	r‖x‖+	r‖x‖+	NOUN
ma-88	92	2	s	s	PART
ma-88	92	3	·	·	PUNCT
ma-88	92	4	n±(x	n±(x	PROPN
ma-88	92	5	;	;	PUNCT
ma-88	92	6	y	y	PROPN
ma-88	92	7	)	)	PUNCT
ma-88	92	8	,	,	PUNCT
ma-88	92	9	f	f	PROPN
ma-88	92	10	or	or	CCONJ
ma-88	92	11	s	s	PROPN
ma-88	92	12	≥	≥	NOUN
ma-88	92	13	0	0	NUM
ma-88	92	14	and	and	CCONJ
ma-88	92	15	al	al	PROPN
ma-88	92	16	l	l	PROPN
ma-88	92	17	r.	r.	PROPN
ma-88	92	18	we	we	PRON
ma-88	92	19	then	then	ADV
ma-88	92	20	give	give	VERB
ma-88	92	21	a	a	DET
ma-88	92	22	characterization	characterization	NOUN
ma-88	92	23	of	of	ADP
ma-88	92	24	x	x	PART
ma-88	92	25	⊥bε	⊥bε	NOUN
ma-88	92	26	y	y	PROPN
ma-88	92	27	using	use	VERB
ma-88	92	28	the	the	DET
ma-88	92	29	definition	definition	NOUN
ma-88	92	30	of	of	ADP
ma-88	92	31	n±(x	n±(x	PROPN
ma-88	92	32	,	,	PUNCT
ma-88	92	33	y	y	PROPN
ma-88	92	34	)	)	PUNCT
ma-88	92	35	.	.	PUNCT
ma-88	93	1	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	93	2	eur	eur	PROPN
ma-88	93	3	.	.	PUNCT
ma-88	94	1	j.	j.	PROPN
ma-88	94	2	math	math	PROPN
ma-88	94	3	.	.	PUNCT
ma-88	95	1	anal	anal	PROPN
ma-88	95	2	.	.	PUNCT
ma-88	96	1	10.28924	10.28924	NUM
ma-88	96	2	/	/	SYM
ma-88	96	3	ada	ada	PROPN
ma-88	96	4	/	/	SYM
ma-88	96	5	ma.2.16	ma.2.16	PROPN
ma-88	96	6	4	4	NUM
ma-88	96	7	proposition	proposition	NOUN
ma-88	96	8	2.3	2.3	NUM
ma-88	96	9	.	.	PUNCT
ma-88	97	1	let	let	VERB
ma-88	97	2	x	x	PRON
ma-88	97	3	be	be	AUX
ma-88	97	4	normed	normed	ADJ
ma-88	97	5	spaces	space	NOUN
ma-88	97	6	,	,	PUNCT
ma-88	97	7	then	then	ADV
ma-88	97	8	x	x	PUNCT
ma-88	97	9	⊥bε	⊥bε	NOUN
ma-88	97	10	y	y	PROPN
ma-88	98	1	if	if	SCONJ
ma-88	98	2	and	and	CCONJ
ma-88	98	3	only	only	ADV
ma-88	98	4	if	if	SCONJ
ma-88	98	5	n+(x	n+(x	PROPN
ma-88	98	6	,	,	PUNCT
ma-88	98	7	y	y	PROPN
ma-88	98	8	)	)	PUNCT
ma-88	99	1	+	+	CCONJ
ma-88	99	2	ε‖y‖	ε‖y‖	NOUN
ma-88	99	3	≥	≥	NOUN
ma-88	99	4	0	0	NUM
ma-88	99	5	≥	≥	PROPN
ma-88	99	6	n−(x	n−(x	PROPN
ma-88	99	7	,	,	PUNCT
ma-88	99	8	y)−	y)−	PROPN
ma-88	99	9	ε‖y‖.	ε‖y‖.	PROPN
ma-88	99	10	proof	proof	NOUN
ma-88	99	11	.	.	PUNCT
ma-88	100	1	let	let	VERB
ma-88	100	2	x	x	NOUN
ma-88	100	3	⊥bε	⊥bε	VERB
ma-88	100	4	y	y	PROPN
ma-88	100	5	and	and	CCONJ
ma-88	100	6	t	t	PROPN
ma-88	100	7	∈	∈	PROPN
ma-88	100	8	r\{0	r\{0	PROPN
ma-88	100	9	}	}	PUNCT
ma-88	100	10	then	then	ADV
ma-88	100	11	‖x	‖x	NOUN
ma-88	101	1	+	+	CCONJ
ma-88	101	2	ty‖	ty‖	ADP
ma-88	101	3	−	−	NOUN
ma-88	101	4	‖x‖	‖x‖	PROPN
ma-88	101	5	|t|	|t|	PROPN
ma-88	101	6	≥	≥	NOUN
ma-88	101	7	ε‖y‖.	ε‖y‖.	PROPN
ma-88	101	8	let	let	VERB
ma-88	101	9	t	t	NOUN
ma-88	101	10	→	→	SYM
ma-88	101	11	0	0	NUM
ma-88	101	12	+	+	NUM
ma-88	101	13	,	,	PUNCT
ma-88	101	14	we	we	PRON
ma-88	101	15	have	have	VERB
ma-88	101	16	n+(x	n+(x	PROPN
ma-88	101	17	,	,	PUNCT
ma-88	101	18	y	y	PROPN
ma-88	101	19	)	)	PUNCT
ma-88	101	20	≥	≥	NOUN
ma-88	101	21	−ε‖y‖.	−ε‖y‖.	PRON
ma-88	101	22	similarly	similarly	ADV
ma-88	101	23	,	,	PUNCT
ma-88	101	24	let	let	VERB
ma-88	101	25	t	t	PROPN
ma-88	101	26	→	→	SYM
ma-88	101	27	0−	0−	NUM
ma-88	101	28	,	,	PUNCT
ma-88	101	29	we	we	PRON
ma-88	101	30	have	have	VERB
ma-88	101	31	n−(x	n−(x	PROPN
ma-88	101	32	,	,	PUNCT
ma-88	101	33	y	y	NOUN
ma-88	101	34	)	)	PUNCT
ma-88	101	35	≤	≤	NOUN
ma-88	102	1	ε‖y‖.	ε‖y‖.	PROPN
ma-88	102	2	to	to	PART
ma-88	102	3	sum	sum	VERB
ma-88	102	4	up	up	ADP
ma-88	102	5	,	,	PUNCT
ma-88	102	6	n+(x	n+(x	PROPN
ma-88	102	7	,	,	PUNCT
ma-88	102	8	y	y	PROPN
ma-88	102	9	)	)	PUNCT
ma-88	103	1	+	+	CCONJ
ma-88	103	2	ε‖y‖	ε‖y‖	NOUN
ma-88	103	3	≥	≥	NOUN
ma-88	103	4	0	0	NUM
ma-88	103	5	≥	≥	PROPN
ma-88	103	6	n−(x	n−(x	PROPN
ma-88	103	7	,	,	PUNCT
ma-88	103	8	y)−	y)−	PROPN
ma-88	103	9	ε‖y‖.	ε‖y‖.	PROPN
ma-88	103	10	conversely	conversely	ADV
ma-88	103	11	,	,	PUNCT
ma-88	103	12	if	if	SCONJ
ma-88	103	13	n+(x	n+(x	PROPN
ma-88	103	14	,	,	PUNCT
ma-88	103	15	y	y	PROPN
ma-88	103	16	)	)	PUNCT
ma-88	103	17	≥	≥	NOUN
ma-88	103	18	−ε‖y‖	−ε‖y‖	PROPN
ma-88	103	19	,	,	PUNCT
ma-88	103	20	for	for	ADP
ma-88	103	21	∀η	∀η	X
ma-88	103	22	>	>	X
ma-88	103	23	0	0	PROPN
ma-88	103	24	,	,	PUNCT
ma-88	103	25	there	there	PRON
ma-88	103	26	∃δ	∃δ	NUM
ma-88	103	27	such	such	ADJ
ma-88	103	28	that	that	SCONJ
ma-88	103	29	if	if	SCONJ
ma-88	103	30	0	0	NUM
ma-88	103	31	<	<	X
ma-88	103	32	t	t	X
ma-88	103	33	≤	≤	PROPN
ma-88	103	34	δ	δ	PROPN
ma-88	103	35	,	,	PUNCT
ma-88	103	36	we	we	PRON
ma-88	103	37	have	have	VERB
ma-88	103	38	:	:	PUNCT
ma-88	103	39	‖x	‖x	NOUN
ma-88	103	40	+	+	CCONJ
ma-88	103	41	ty‖	ty‖	PRON
ma-88	103	42	−	−	PROPN
ma-88	103	43	‖x‖	‖x‖	PROPN
ma-88	103	44	t	t	PROPN
ma-88	103	45	≥	≥	NOUN
ma-88	103	46	−(ε+	−(ε+	ADP
ma-88	103	47	η)‖y‖	η)‖y‖	PROPN
ma-88	103	48	,	,	PUNCT
ma-88	103	49	or	or	CCONJ
ma-88	103	50	equivalently	equivalently	ADV
ma-88	103	51	‖x	‖x	NOUN
ma-88	104	1	+	+	CCONJ
ma-88	104	2	ty‖	ty‖	ADP
ma-88	104	3	−	−	PUNCT
ma-88	104	4	‖x‖	‖x‖	PROPN
ma-88	104	5	≥	≥	PROPN
ma-88	104	6	−t(ε+	−t(ε+	PROPN
ma-88	104	7	η)‖y‖	η)‖y‖	PROPN
ma-88	104	8	f	f	PROPN
ma-88	104	9	or	or	CCONJ
ma-88	104	10	t	t	PROPN
ma-88	104	11	∈	∈	PROPN
ma-88	104	12	(	(	PUNCT
ma-88	104	13	0	0	NUM
ma-88	104	14	,	,	PUNCT
ma-88	104	15	δ	δ	PROPN
ma-88	104	16	]	]	X
ma-88	104	17	.	.	PUNCT
ma-88	105	1	because	because	SCONJ
ma-88	105	2	of	of	ADP
ma-88	105	3	the	the	DET
ma-88	105	4	convexity	convexity	NOUN
ma-88	105	5	of	of	ADP
ma-88	105	6	‖x	‖x	NOUN
ma-88	105	7	+	+	CCONJ
ma-88	105	8	ty‖	ty‖	PROPN
ma-88	105	9	,	,	PUNCT
ma-88	105	10	we	we	PRON
ma-88	105	11	have	have	VERB
ma-88	105	12	‖x	‖x	NOUN
ma-88	106	1	+	+	CCONJ
ma-88	106	2	ty‖	ty‖	PROPN
ma-88	106	3	−	−	PUNCT
ma-88	106	4	‖x‖	‖x‖	PROPN
ma-88	106	5	≥	≥	PROPN
ma-88	106	6	−t(ε+	−t(ε+	PROPN
ma-88	106	7	η)‖y‖	η)‖y‖	PROPN
ma-88	106	8	f	f	PROPN
ma-88	106	9	or	or	CCONJ
ma-88	106	10	t	t	PROPN
ma-88	106	11	>	>	X
ma-88	106	12	0	0	X
ma-88	106	13	.	.	PUNCT
ma-88	107	1	let	let	VERB
ma-88	107	2	δ	δ	PROPN
ma-88	107	3	→	→	X
ma-88	107	4	0	0	PROPN
ma-88	107	5	,	,	PUNCT
ma-88	107	6	we	we	PRON
ma-88	107	7	have	have	VERB
ma-88	107	8	‖x	‖x	NOUN
ma-88	108	1	+	+	CCONJ
ma-88	108	2	ty‖	ty‖	PROPN
ma-88	108	3	−	−	PUNCT
ma-88	108	4	‖x‖	‖x‖	PROPN
ma-88	108	5	≥	≥	PROPN
ma-88	108	6	−ε‖ty‖	−ε‖ty‖	VERB
ma-88	108	7	f	f	NOUN
ma-88	108	8	or	or	CCONJ
ma-88	108	9	t	t	PROPN
ma-88	108	10	>	>	X
ma-88	108	11	0	0	X
ma-88	108	12	.	.	PUNCT
ma-88	109	1	similarly	similarly	ADV
ma-88	109	2	,	,	PUNCT
ma-88	109	3	using	use	VERB
ma-88	109	4	n−(x	n−(x	PROPN
ma-88	109	5	,	,	PUNCT
ma-88	109	6	y	y	NOUN
ma-88	109	7	)	)	PUNCT
ma-88	109	8	≤	≤	NOUN
ma-88	109	9	ε‖y‖	ε‖y‖	NOUN
ma-88	109	10	,	,	PUNCT
ma-88	109	11	we	we	PRON
ma-88	109	12	have:‖x	have:‖x	VERB
ma-88	109	13	+	+	CCONJ
ma-88	109	14	ty‖	ty‖	PROPN
ma-88	109	15	−	−	PUNCT
ma-88	109	16	‖x‖	‖x‖	PROPN
ma-88	109	17	≥	≥	PRON
ma-88	110	1	t(ε)‖y‖	t(ε)‖y‖	NUM
ma-88	110	2	f	f	PROPN
ma-88	110	3	or	or	CCONJ
ma-88	110	4	t	t	X
ma-88	110	5	<	<	X
ma-88	110	6	0	0	NUM
ma-88	110	7	.	.	PUNCT
ma-88	111	1	if	if	SCONJ
ma-88	111	2	t	t	NOUN
ma-88	111	3	=	=	SYM
ma-88	111	4	0	0	PROPN
ma-88	111	5	,	,	PUNCT
ma-88	111	6	‖x	‖x	NOUN
ma-88	111	7	+	+	CCONJ
ma-88	111	8	ty‖	ty‖	ADP
ma-88	111	9	−	−	PUNCT
ma-88	111	10	‖x‖	‖x‖	PROPN
ma-88	111	11	≥	≥	NOUN
ma-88	111	12	ε‖ty‖	ε‖ty‖	NOUN
ma-88	111	13	is	be	AUX
ma-88	111	14	obvious	obvious	ADJ
ma-88	111	15	.	.	PUNCT
ma-88	112	1	to	to	PART
ma-88	112	2	conclude	conclude	VERB
ma-88	112	3	,	,	PUNCT
ma-88	112	4	we	we	PRON
ma-88	112	5	have	have	VERB
ma-88	112	6	:	:	PUNCT
ma-88	112	7	‖x	‖x	NOUN
ma-88	113	1	+	+	CCONJ
ma-88	113	2	ty‖	ty‖	ADP
ma-88	113	3	−	−	NOUN
ma-88	113	4	‖x‖	‖x‖	PROPN
ma-88	113	5	≥	≥	PROPN
ma-88	113	6	ε‖ty‖	ε‖ty‖	NOUN
ma-88	113	7	f	f	PROPN
ma-88	113	8	or	or	CCONJ
ma-88	113	9	t	t	PROPN
ma-88	113	10	∈	∈	PROPN
ma-88	113	11	r.	r.	PROPN
ma-88	113	12	thus	thus	ADV
ma-88	113	13	x	x	ADP
ma-88	113	14	⊥bε	⊥bε	NOUN
ma-88	113	15	y	y	PROPN
ma-88	113	16	.	.	PUNCT
ma-88	114	1	�	�	PROPN
ma-88	114	2	to	to	PART
ma-88	114	3	verify	verify	VERB
ma-88	114	4	the	the	DET
ma-88	114	5	validity	validity	NOUN
ma-88	114	6	of	of	ADP
ma-88	114	7	the	the	DET
ma-88	114	8	new	new	ADJ
ma-88	114	9	approximate	approximate	ADJ
ma-88	114	10	birkhoff	birkhoff	NOUN
ma-88	114	11	orthogonality	orthogonality	NOUN
ma-88	114	12	,	,	PUNCT
ma-88	114	13	we	we	PRON
ma-88	114	14	have	have	VERB
ma-88	114	15	the	the	DET
ma-88	114	16	followingproposition	followingproposition	NOUN
ma-88	114	17	:	:	PUNCT
ma-88	114	18	proposition	proposition	NOUN
ma-88	114	19	2.4	2.4	NUM
ma-88	114	20	.	.	PUNCT
ma-88	115	1	let	let	VERB
ma-88	115	2	x	x	PRON
ma-88	115	3	be	be	AUX
ma-88	115	4	normed	normed	ADJ
ma-88	115	5	spaces	space	NOUN
ma-88	115	6	,	,	PUNCT
ma-88	115	7	we	we	PRON
ma-88	115	8	have	have	VERB
ma-88	115	9	:	:	PUNCT
ma-88	116	1	x	x	PART
ma-88	116	2	⊥bε	⊥bε	NOUN
ma-88	116	3	y	y	PROPN
ma-88	116	4	if	if	SCONJ
ma-88	116	5	and	and	CCONJ
ma-88	116	6	only	only	ADV
ma-88	116	7	if	if	SCONJ
ma-88	116	8	x	x	PROPN
ma-88	116	9	⊥ε	⊥ε	ADJ
ma-88	116	10	y	y	NOUN
ma-88	116	11	.	.	PUNCT
ma-88	117	1	proof	proof	NOUN
ma-88	117	2	.	.	PUNCT
ma-88	118	1	since	since	SCONJ
ma-88	118	2	x	x	PROPN
ma-88	118	3	⊥bε	⊥bε	VERB
ma-88	118	4	y	y	PROPN
ma-88	118	5	,	,	PUNCT
ma-88	118	6	for	for	ADP
ma-88	118	7	0	0	NUM
ma-88	118	8	<	<	X
ma-88	118	9	t	t	X
ma-88	118	10	≤	≤	PUNCT
ma-88	118	11	‖x‖	‖x‖	PROPN
ma-88	118	12	ε‖y‖	ε‖y‖	NOUN
ma-88	118	13	we	we	PRON
ma-88	118	14	have	have	VERB
ma-88	118	15	‖x	‖x	NOUN
ma-88	119	1	+	+	CCONJ
ma-88	119	2	ty‖	ty‖	PRON
ma-88	119	3	≥	≥	NOUN
ma-88	120	1	‖x‖	‖x‖	PROPN
ma-88	120	2	−	−	PROPN
ma-88	120	3	ε‖ty‖.	ε‖ty‖.	PROPN
ma-88	120	4	square	square	NOUN
ma-88	120	5	both	both	DET
ma-88	120	6	sides	side	NOUN
ma-88	120	7	we	we	PRON
ma-88	120	8	get	get	VERB
ma-88	120	9	‖x‖2	‖x‖2	VERB
ma-88	121	1	+	+	CCONJ
ma-88	121	2	t2‖y‖2	t2‖y‖2	NUM
ma-88	121	3	+	+	CCONJ
ma-88	121	4	2t(x	2t(x	NUM
ma-88	121	5	,	,	PUNCT
ma-88	121	6	y	y	PROPN
ma-88	121	7	)	)	PUNCT
ma-88	121	8	≥	≥	PRON
ma-88	121	9	‖x‖2	‖x‖2	VERB
ma-88	122	1	+	+	CCONJ
ma-88	122	2	ε2t2‖y‖2	ε2t2‖y‖2	PROPN
ma-88	122	3	−	−	PROPN
ma-88	122	4	2ε‖x‖‖ty‖.	2ε‖x‖‖ty‖.	NUM
ma-88	122	5	thus	thus	ADV
ma-88	122	6	(	(	PUNCT
ma-88	122	7	1−	1−	NUM
ma-88	122	8	ε2)t‖y‖	ε2)t‖y‖	PROPN
ma-88	122	9	≥	≥	PROPN
ma-88	122	10	−2‖x‖(ε+	−2‖x‖(ε+	PROPN
ma-88	122	11	cos(x	cos(x	PROPN
ma-88	122	12	,	,	PUNCT
ma-88	122	13	y	y	NOUN
ma-88	122	14	)	)	PUNCT
ma-88	122	15	)	)	PUNCT
ma-88	122	16	when	when	SCONJ
ma-88	122	17	t	t	PROPN
ma-88	122	18	tends	tend	VERB
ma-88	122	19	to	to	ADP
ma-88	122	20	0	0	NUM
ma-88	122	21	,	,	PUNCT
ma-88	122	22	(	(	PUNCT
ma-88	122	23	1−	1−	NUM
ma-88	122	24	ε2)t‖y‖	ε2)t‖y‖	NOUN
ma-88	122	25	tends	tend	VERB
ma-88	122	26	to	to	ADP
ma-88	122	27	0	0	NUM
ma-88	122	28	,	,	PUNCT
ma-88	122	29	so	so	SCONJ
ma-88	122	30	we	we	PRON
ma-88	122	31	have	have	VERB
ma-88	122	32	ε+	ε+	NOUN
ma-88	122	33	cos(x	cos(x	PROPN
ma-88	122	34	,	,	PUNCT
ma-88	122	35	y	y	PROPN
ma-88	122	36	)	)	PUNCT
ma-88	122	37	≥	≥	NOUN
ma-88	122	38	0	0	NUM
ma-88	122	39	⇐	⇐	ADJ
ma-88	122	40	⇒	⇒	PROPN
ma-88	122	41	cos(x	cos(x	PROPN
ma-88	122	42	,	,	PUNCT
ma-88	122	43	y	y	PROPN
ma-88	122	44	)	)	PUNCT
ma-88	122	45	≥	≥	NOUN
ma-88	122	46	−ε	−ε	PROPN
ma-88	122	47	.	.	PUNCT
ma-88	123	1	similarly	similarly	ADV
ma-88	123	2	for	for	ADP
ma-88	123	3	0	0	NUM
ma-88	123	4	>	>	X
ma-88	123	5	t	t	PROPN
ma-88	123	6	≥	≥	NOUN
ma-88	123	7	−	−	PROPN
ma-88	123	8	‖x‖ε‖y‖	‖x‖ε‖y‖	NUM
ma-88	123	9	,	,	PUNCT
ma-88	123	10	we	we	PRON
ma-88	123	11	have	have	VERB
ma-88	123	12	cos(x	cos(x	PROPN
ma-88	123	13	,	,	PUNCT
ma-88	123	14	y	y	NOUN
ma-88	123	15	)	)	PUNCT
ma-88	123	16	≤	≤	PROPN
ma-88	123	17	ε	ε	PROPN
ma-88	123	18	,	,	PUNCT
ma-88	123	19	thus	thus	ADV
ma-88	123	20	x	x	NUM
ma-88	123	21	⊥ε	⊥ε	NUM
ma-88	123	22	y	y	PROPN
ma-88	123	23	.	.	PUNCT
ma-88	124	1	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	124	2	eur	eur	PROPN
ma-88	124	3	.	.	PUNCT
ma-88	125	1	j.	j.	PROPN
ma-88	125	2	math	math	PROPN
ma-88	125	3	.	.	PUNCT
ma-88	126	1	anal	anal	PROPN
ma-88	126	2	.	.	PUNCT
ma-88	127	1	10.28924	10.28924	NUM
ma-88	127	2	/	/	SYM
ma-88	127	3	ada	ada	PROPN
ma-88	127	4	/	/	SYM
ma-88	127	5	ma.2.16	ma.2.16	PROPN
ma-88	127	6	5	5	NUM
ma-88	127	7	conversely	conversely	ADV
ma-88	127	8	,	,	PUNCT
ma-88	127	9	if	if	SCONJ
ma-88	127	10	|cos(x	|cos(x	PROPN
ma-88	127	11	,	,	PUNCT
ma-88	127	12	y)|	y)|	PROPN
ma-88	127	13	≤	≤	PROPN
ma-88	127	14	ε	ε	PROPN
ma-88	127	15	,	,	PUNCT
ma-88	127	16	we	we	PRON
ma-88	127	17	have	have	VERB
ma-88	127	18	‖x	‖x	NOUN
ma-88	128	1	+	+	CCONJ
ma-88	128	2	ty‖	ty‖	PROPN
ma-88	128	3	−	−	ADP
ma-88	128	4	‖x‖	‖x‖	PROPN
ma-88	128	5	≥	≥	PROPN
ma-88	128	6	ε‖ty‖	ε‖ty‖	NOUN
ma-88	128	7	f	f	NOUN
ma-88	128	8	or	or	CCONJ
ma-88	128	9	|t|	|t|	ADP
ma-88	128	10	∈	∈	PROPN
ma-88	129	1	[	[	X
ma-88	129	2	0	0	NUM
ma-88	129	3	,	,	PUNCT
ma-88	129	4	‖x‖	‖x‖	VERB
ma-88	129	5	ε‖y‖	ε‖y‖	NOUN
ma-88	129	6	]	]	PUNCT
ma-88	129	7	.	.	PUNCT
ma-88	130	1	on	on	ADP
ma-88	130	2	the	the	DET
ma-88	130	3	other	other	ADJ
ma-88	130	4	hand	hand	NOUN
ma-88	130	5	,	,	PUNCT
ma-88	130	6	‖x	‖x	NOUN
ma-88	130	7	+	+	CCONJ
ma-88	130	8	ty‖	ty‖	ADP
ma-88	130	9	−	−	PUNCT
ma-88	130	10	‖x‖	‖x‖	PROPN
ma-88	130	11	≥	≥	NOUN
ma-88	130	12	ε‖ty‖	ε‖ty‖	NOUN
ma-88	130	13	is	be	AUX
ma-88	130	14	always	always	ADV
ma-88	130	15	correct	correct	ADJ
ma-88	130	16	for	for	ADP
ma-88	130	17	|t|	|t|	ADJ
ma-88	130	18	≥	≥	NOUN
ma-88	131	1	‖x‖	‖x‖	PROPN
ma-88	131	2	ε‖y‖	ε‖y‖	NOUN
ma-88	131	3	.	.	PUNCT
ma-88	132	1	to	to	PART
ma-88	132	2	conclude	conclude	VERB
ma-88	132	3	,	,	PUNCT
ma-88	132	4	‖x	‖x	NOUN
ma-88	133	1	+	+	CCONJ
ma-88	133	2	ty‖	ty‖	PROPN
ma-88	133	3	−	−	NOUN
ma-88	133	4	‖x‖	‖x‖	PROPN
ma-88	133	5	≥	≥	PROPN
ma-88	133	6	ε‖ty‖	ε‖ty‖	NOUN
ma-88	133	7	f	f	PROPN
ma-88	133	8	or	or	CCONJ
ma-88	133	9	t	t	PROPN
ma-88	133	10	∈	∈	PROPN
ma-88	133	11	r.	r.	PROPN
ma-88	133	12	thus	thus	ADV
ma-88	133	13	x	x	ADP
ma-88	133	14	⊥bε	⊥bε	NOUN
ma-88	133	15	y	y	PROPN
ma-88	133	16	.	.	PUNCT
ma-88	134	1	�	�	PROPN
ma-88	134	2	from	from	ADP
ma-88	134	3	n±(x	n±(x	NOUN
ma-88	134	4	;	;	PUNCT
ma-88	134	5	rx	rx	VERB
ma-88	134	6	+	+	CCONJ
ma-88	134	7	sy	sy	X
ma-88	134	8	)	)	PUNCT
ma-88	134	9	=	=	PUNCT
ma-88	135	1	r‖x‖+	r‖x‖+	NOUN
ma-88	135	2	s	s	PART
ma-88	135	3	·	·	PUNCT
ma-88	135	4	n±(x	n±(x	PROPN
ma-88	135	5	;	;	PUNCT
ma-88	135	6	y	y	PROPN
ma-88	135	7	)	)	PUNCT
ma-88	135	8	,	,	PUNCT
ma-88	135	9	f	f	PROPN
ma-88	135	10	or	or	CCONJ
ma-88	135	11	s	s	PROPN
ma-88	135	12	≥	≥	NOUN
ma-88	135	13	0	0	NUM
ma-88	135	14	and	and	CCONJ
ma-88	136	1	al	al	PROPN
ma-88	136	2	l	l	NOUN
ma-88	136	3	r	r	PROPN
ma-88	136	4	,	,	PUNCT
ma-88	136	5	we	we	PRON
ma-88	136	6	have	have	VERB
ma-88	136	7	the	the	DET
ma-88	136	8	following	following	NOUN
ma-88	136	9	:	:	PUNCT
ma-88	136	10	proposition	proposition	NOUN
ma-88	136	11	2.5	2.5	NUM
ma-88	136	12	.	.	PUNCT
ma-88	137	1	in	in	ADP
ma-88	137	2	the	the	DET
ma-88	137	3	normed	normed	ADJ
ma-88	137	4	space	space	NOUN
ma-88	137	5	x	x	PART
ma-88	137	6	,	,	PUNCT
ma-88	137	7	if	if	SCONJ
ma-88	137	8	x	x	PROPN
ma-88	137	9	⊥bε	⊥bε	NOUN
ma-88	137	10	y	y	PROPN
ma-88	137	11	,	,	PUNCT
ma-88	137	12	then	then	ADV
ma-88	137	13	we	we	PRON
ma-88	137	14	have	have	AUX
ma-88	137	15	x	x	PART
ma-88	137	16	⊥bε	⊥bε	VERB
ma-88	137	17	rx	rx	VERB
ma-88	137	18	+	+	CCONJ
ma-88	137	19	sy	sy	ADV
ma-88	137	20	for	for	ADP
ma-88	137	21	s	s	PRON
ma-88	137	22	≥	≥	NOUN
ma-88	137	23	0	0	NUM
ma-88	137	24	,	,	PUNCT
ma-88	137	25	r	r	NOUN
ma-88	137	26	satisfying	satisfy	VERB
ma-88	137	27	ε(‖rx	ε(‖rx	PROPN
ma-88	137	28	+	+	CCONJ
ma-88	137	29	sy‖	sy‖	PROPN
ma-88	137	30	−	−	PROPN
ma-88	137	31	s‖y‖	s‖y‖	PROPN
ma-88	137	32	)	)	PUNCT
ma-88	137	33	≥	≥	NOUN
ma-88	137	34	r‖x‖	r‖x‖	VERB
ma-88	137	35	≥	≥	NUM
ma-88	137	36	ε(s‖y‖	ε(s‖y‖	NOUN
ma-88	137	37	−	−	PROPN
ma-88	138	1	‖rx	‖rx	PUNCT
ma-88	138	2	+	+	CCONJ
ma-88	138	3	sy‖	sy‖	PROPN
ma-88	138	4	)	)	PUNCT
ma-88	138	5	.	.	PUNCT
ma-88	139	1	proof	proof	NOUN
ma-88	139	2	.	.	PUNCT
ma-88	140	1	since	since	SCONJ
ma-88	140	2	x	x	PROPN
ma-88	140	3	⊥bε	⊥bε	NOUN
ma-88	140	4	y	y	PROPN
ma-88	140	5	,	,	PUNCT
ma-88	140	6	we	we	PRON
ma-88	140	7	have	have	VERB
ma-88	140	8	n+(x	n+(x	PROPN
ma-88	140	9	,	,	PUNCT
ma-88	140	10	y	y	PROPN
ma-88	140	11	)	)	PUNCT
ma-88	140	12	≥	≥	NOUN
ma-88	140	13	−ε‖y‖	−ε‖y‖	PROPN
ma-88	140	14	and	and	CCONJ
ma-88	140	15	n−(x	n−(x	PROPN
ma-88	140	16	,	,	PUNCT
ma-88	140	17	y	y	NOUN
ma-88	140	18	)	)	PUNCT
ma-88	140	19	≤	≤	NOUN
ma-88	140	20	ε‖y‖	ε‖y‖	NOUN
ma-88	140	21	,	,	PUNCT
ma-88	141	1	so	so	ADV
ma-88	141	2	n+(x	n+(x	PROPN
ma-88	141	3	,	,	PUNCT
ma-88	141	4	rx	rx	VERB
ma-88	141	5	+	+	CCONJ
ma-88	141	6	sy	sy	NOUN
ma-88	141	7	)	)	PUNCT
ma-88	141	8	≥	≥	NOUN
ma-88	141	9	r‖x‖+	r‖x‖+	NOUN
ma-88	141	10	(	(	PUNCT
ma-88	141	11	−sε‖y‖	−sε‖y‖	NOUN
ma-88	141	12	)	)	PUNCT
ma-88	141	13	≥	≥	NOUN
ma-88	141	14	−ε‖rx	−ε‖rx	PROPN
ma-88	142	1	+	+	CCONJ
ma-88	142	2	sy‖.	sy‖.	PRON
ma-88	142	3	similarly	similarly	ADV
ma-88	142	4	we	we	PRON
ma-88	142	5	have	have	AUX
ma-88	142	6	n−(x	n−(x	ADV
ma-88	142	7	,	,	PUNCT
ma-88	142	8	rx	rx	VERB
ma-88	142	9	+	+	CCONJ
ma-88	142	10	sy	sy	NOUN
ma-88	142	11	)	)	PUNCT
ma-88	142	12	≤	≤	NOUN
ma-88	143	1	ε‖rx	ε‖rx	PROPN
ma-88	143	2	+	+	CCONJ
ma-88	143	3	sy‖	sy‖	PROPN
ma-88	143	4	,	,	PUNCT
ma-88	143	5	thus	thus	ADV
ma-88	143	6	x	x	PART
ma-88	143	7	⊥bε	⊥bε	NOUN
ma-88	143	8	rx	rx	VERB
ma-88	143	9	+	+	CCONJ
ma-88	143	10	sy	sy	INTJ
ma-88	143	11	.	.	PUNCT
ma-88	144	1	�	�	PROPN
ma-88	144	2	let	let	VERB
ma-88	144	3	x	x	PRON
ma-88	144	4	be	be	AUX
ma-88	144	5	normed	normed	ADJ
ma-88	144	6	spaces	space	NOUN
ma-88	144	7	,	,	PUNCT
ma-88	144	8	it	it	PRON
ma-88	144	9	is	be	AUX
ma-88	144	10	known	know	VERB
ma-88	144	11	that	that	SCONJ
ma-88	144	12	[	[	X
ma-88	144	13	16	16	NUM
ma-88	144	14	]	]	PUNCT
ma-88	144	15	for	for	ADP
ma-88	144	16	any	any	DET
ma-88	144	17	x	x	NOUN
ma-88	144	18	,	,	PUNCT
ma-88	144	19	y	y	PROPN
ma-88	144	20	∈	∈	PROPN
ma-88	144	21	x	x	PUNCT
ma-88	144	22	there	there	PRON
ma-88	144	23	exists	exist	VERB
ma-88	144	24	a	a	DET
ma-88	144	25	real	real	ADJ
ma-88	144	26	number	number	NOUN
ma-88	144	27	asuch	asuch	NOUN
ma-88	144	28	that	that	SCONJ
ma-88	144	29	x	x	PUNCT
ma-88	144	30	⊥b	⊥b	PRON
ma-88	144	31	ax	ax	NOUN
ma-88	144	32	+	+	CCONJ
ma-88	144	33	y	y	PROPN
ma-88	144	34	,	,	PUNCT
ma-88	144	35	moreover	moreover	ADV
ma-88	144	36	,	,	PUNCT
ma-88	144	37	such	such	DET
ma-88	144	38	a	a	DET
ma-88	144	39	number	number	NOUN
ma-88	144	40	satisfies	satisfy	VERB
ma-88	144	41	|a|	|a|	NOUN
ma-88	144	42	≤	≤	ADJ
ma-88	144	43	‖y‖‖x‖	‖y‖‖x‖	NOUN
ma-88	144	44	.	.	PUNCT
ma-88	145	1	on	on	ADP
ma-88	145	2	this	this	DET
ma-88	145	3	basis	basis	NOUN
ma-88	145	4	,	,	PUNCT
ma-88	145	5	chmieliński	chmieliński	NOUN
ma-88	146	1	[	[	X
ma-88	146	2	9]discovered	9]discovere	VERB
ma-88	146	3	that	that	SCONJ
ma-88	146	4	x	x	PROPN
ma-88	146	5	⊥εb	⊥εb	NOUN
ma-88	146	6	y	y	PROPN
ma-88	146	7	if	if	SCONJ
ma-88	146	8	and	and	CCONJ
ma-88	146	9	only	only	ADV
ma-88	146	10	if	if	SCONJ
ma-88	146	11	there	there	PRON
ma-88	146	12	exists	exist	VERB
ma-88	146	13	a	a	DET
ma-88	146	14	real	real	ADJ
ma-88	146	15	number	number	NOUN
ma-88	146	16	|a|	|a|	NOUN
ma-88	146	17	≤	≤	NUM
ma-88	146	18	‖y‖‖x‖ε	‖y‖‖x‖ε	NOUN
ma-88	146	19	such	such	ADJ
ma-88	146	20	that	that	SCONJ
ma-88	146	21	x	x	PRON
ma-88	146	22	⊥b	⊥b	PRON
ma-88	146	23	ax	ax	NOUN
ma-88	146	24	+	+	CCONJ
ma-88	146	25	y	y	PROPN
ma-88	146	26	.in	.in	PUNCT
ma-88	146	27	fact	fact	NOUN
ma-88	146	28	,	,	PUNCT
ma-88	146	29	in	in	ADP
ma-88	146	30	inner	inner	ADJ
ma-88	146	31	product	product	NOUN
ma-88	146	32	spaces	space	NOUN
ma-88	146	33	,	,	PUNCT
ma-88	146	34	it	it	PRON
ma-88	146	35	is	be	AUX
ma-88	146	36	easy	easy	ADJ
ma-88	146	37	to	to	PART
ma-88	146	38	see	see	VERB
ma-88	146	39	that	that	SCONJ
ma-88	146	40	x	x	X
ma-88	146	41	⊥ε	⊥ε	NUM
ma-88	147	1	y	y	NOUN
ma-88	147	2	if	if	SCONJ
ma-88	147	3	and	and	CCONJ
ma-88	147	4	only	only	ADV
ma-88	147	5	if	if	SCONJ
ma-88	147	6	there	there	PRON
ma-88	147	7	exists	exist	VERB
ma-88	147	8	|a|	|a|	PROPN
ma-88	147	9	≤	≤	NOUN
ma-88	147	10	‖y‖‖x‖εsuch	‖y‖‖x‖εsuch	ADJ
ma-88	148	1	that	that	PRON
ma-88	148	2	x	x	PUNCT
ma-88	148	3	⊥b	⊥b	PRON
ma-88	148	4	ax	ax	NOUN
ma-88	148	5	+	+	CCONJ
ma-88	148	6	y	y	NOUN
ma-88	148	7	by	by	ADP
ma-88	148	8	taking	take	VERB
ma-88	148	9	a	a	PRON
ma-88	148	10	=	=	NOUN
ma-88	148	11	−	−	PROPN
ma-88	148	12	〈	〈	PROPN
ma-88	148	13	x	x	X
ma-88	148	14	|y〉‖x‖2	|y〉‖x‖2	ADJ
ma-88	148	15	x	x	PUNCT
ma-88	149	1	+	+	CCONJ
ma-88	149	2	y	y	PROPN
ma-88	149	3	for	for	ADP
ma-88	149	4	x	x	SYM
ma-88	149	5	6=	6=	ADP
ma-88	149	6	0.in	0.in	NUM
ma-88	149	7	the	the	DET
ma-88	149	8	following	following	NOUN
ma-88	149	9	we	we	PRON
ma-88	149	10	will	will	AUX
ma-88	149	11	prove	prove	VERB
ma-88	149	12	that	that	SCONJ
ma-88	149	13	it	it	PRON
ma-88	149	14	is	be	AUX
ma-88	149	15	also	also	ADV
ma-88	149	16	true	true	ADJ
ma-88	149	17	for	for	ADP
ma-88	149	18	⊥bε	⊥bε	VERB
ma-88	149	19	,	,	PUNCT
ma-88	149	20	that	that	ADV
ma-88	149	21	is	is	ADV
ma-88	149	22	,	,	PUNCT
ma-88	149	23	in	in	ADP
ma-88	149	24	normed	normed	ADJ
ma-88	149	25	spaces	space	NOUN
ma-88	149	26	,	,	PUNCT
ma-88	149	27	x	x	SYM
ma-88	149	28	⊥bε	⊥bε	NOUN
ma-88	149	29	y	y	PROPN
ma-88	150	1	if	if	SCONJ
ma-88	150	2	and	and	CCONJ
ma-88	150	3	only	only	ADV
ma-88	150	4	if	if	SCONJ
ma-88	150	5	there	there	PRON
ma-88	150	6	exists	exist	VERB
ma-88	150	7	|a|	|a|	PROPN
ma-88	150	8	≤	≤	ADJ
ma-88	150	9	‖y‖‖x‖ε	‖y‖‖x‖ε	NOUN
ma-88	150	10	such	such	ADJ
ma-88	150	11	that	that	SCONJ
ma-88	150	12	x	x	PRON
ma-88	150	13	⊥b	⊥b	PRON
ma-88	150	14	ax	ax	NOUN
ma-88	150	15	+	+	X
ma-88	150	16	y	y	PROPN
ma-88	150	17	.	.	PUNCT
ma-88	151	1	before	before	ADP
ma-88	151	2	the	the	DET
ma-88	151	3	proof	proof	NOUN
ma-88	151	4	,	,	PUNCT
ma-88	151	5	we	we	PRON
ma-88	151	6	need	need	VERB
ma-88	151	7	some	some	DET
ma-88	151	8	lemma	lemma	PROPN
ma-88	151	9	.	.	PUNCT
ma-88	152	1	lemma	lemma	PROPN
ma-88	152	2	2.6	2.6	NUM
ma-88	152	3	.	.	PUNCT
ma-88	153	1	[	[	X
ma-88	153	2	16	16	NUM
ma-88	153	3	]	]	PUNCT
ma-88	153	4	let	let	VERB
ma-88	153	5	x	x	PRON
ma-88	153	6	be	be	AUX
ma-88	153	7	normed	normed	ADJ
ma-88	153	8	spaces	space	NOUN
ma-88	153	9	,	,	PUNCT
ma-88	153	10	n−(x	n−(x	PROPN
ma-88	153	11	,	,	PUNCT
ma-88	153	12	y	y	PROPN
ma-88	153	13	)	)	PUNCT
ma-88	153	14	≤	≤	PROPN
ma-88	153	15	n+(x	n+(x	PROPN
ma-88	153	16	,	,	PUNCT
ma-88	153	17	y	y	PROPN
ma-88	153	18	)	)	PUNCT
ma-88	153	19	.	.	PUNCT
ma-88	154	1	lemma	lemma	PROPN
ma-88	154	2	2.7	2.7	NUM
ma-88	154	3	.	.	PUNCT
ma-88	155	1	[	[	X
ma-88	155	2	16	16	NUM
ma-88	155	3	]	]	PUNCT
ma-88	155	4	let	let	VERB
ma-88	155	5	x	x	PRON
ma-88	155	6	be	be	AUX
ma-88	155	7	normed	normed	ADJ
ma-88	155	8	spaces	space	NOUN
ma-88	155	9	,	,	PUNCT
ma-88	155	10	a	a	DET
ma-88	155	11	≤	≤	PROPN
ma-88	155	12	b	b	NOUN
ma-88	155	13	,	,	PUNCT
ma-88	155	14	a	a	PRON
ma-88	155	15	,	,	PUNCT
ma-88	155	16	b	b	X
ma-88	155	17	∈	∈	PROPN
ma-88	155	18	x	x	X
ma-88	155	19	,	,	PUNCT
ma-88	155	20	if	if	SCONJ
ma-88	155	21	x	x	PRON
ma-88	155	22	⊥b	⊥b	PRON
ma-88	155	23	ax	ax	NOUN
ma-88	155	24	+	+	NOUN
ma-88	155	25	y	y	PROPN
ma-88	155	26	,	,	PUNCT
ma-88	155	27	x	x	PROPN
ma-88	155	28	⊥b	⊥b	X
ma-88	155	29	bx	bx	NOUN
ma-88	155	30	+	+	PROPN
ma-88	155	31	y	y	PROPN
ma-88	155	32	,	,	PUNCT
ma-88	155	33	then	then	ADV
ma-88	155	34	x	x	PUNCT
ma-88	155	35	⊥b	⊥b	PRON
ma-88	155	36	cx	cx	NOUN
ma-88	156	1	+	+	CCONJ
ma-88	156	2	y	y	PROPN
ma-88	156	3	f	f	PROPN
ma-88	156	4	or	or	CCONJ
ma-88	156	5	c	c	NOUN
ma-88	156	6	∈	∈	PROPN
ma-88	157	1	[	[	X
ma-88	157	2	a	a	X
ma-88	157	3	,	,	PUNCT
ma-88	157	4	b	b	NOUN
ma-88	157	5	]	]	PUNCT
ma-88	157	6	.	.	PUNCT
ma-88	158	1	lemma	lemma	PROPN
ma-88	158	2	2.8	2.8	NUM
ma-88	158	3	.	.	PUNCT
ma-88	159	1	[	[	X
ma-88	159	2	16	16	NUM
ma-88	159	3	]	]	PUNCT
ma-88	159	4	let	let	VERB
ma-88	159	5	x	x	PRON
ma-88	159	6	be	be	AUX
ma-88	159	7	normed	normed	ADJ
ma-88	159	8	spaces	space	NOUN
ma-88	159	9	,	,	PUNCT
ma-88	159	10	x	x	PRON
ma-88	159	11	⊥b	⊥b	PRON
ma-88	159	12	ax	ax	NOUN
ma-88	159	13	+	+	CCONJ
ma-88	159	14	y	y	PROPN
ma-88	159	15	⇐	⇐	PROPN
ma-88	159	16	⇒	⇒	PROPN
ma-88	159	17	n−(x	n−(x	PROPN
ma-88	159	18	,	,	PUNCT
ma-88	159	19	y	y	NOUN
ma-88	159	20	)	)	PUNCT
ma-88	159	21	≤	≤	NOUN
ma-88	159	22	−a‖x‖	−a‖x‖	VERB
ma-88	159	23	≤	≤	PUNCT
ma-88	159	24	n+(x	n+(x	PROPN
ma-88	159	25	,	,	PUNCT
ma-88	159	26	y	y	PROPN
ma-88	159	27	)	)	PUNCT
ma-88	159	28	.	.	PUNCT
ma-88	160	1	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	160	2	eur	eur	PROPN
ma-88	160	3	.	.	PUNCT
ma-88	161	1	j.	j.	PROPN
ma-88	161	2	math	math	PROPN
ma-88	161	3	.	.	PUNCT
ma-88	162	1	anal	anal	PROPN
ma-88	162	2	.	.	PUNCT
ma-88	163	1	10.28924	10.28924	NUM
ma-88	163	2	/	/	SYM
ma-88	163	3	ada	ada	PROPN
ma-88	163	4	/	/	SYM
ma-88	163	5	ma.2.16	ma.2.16	PROPN
ma-88	163	6	6	6	NUM
ma-88	163	7	theorem	theorem	VERB
ma-88	163	8	2.9	2.9	NUM
ma-88	163	9	.	.	PUNCT
ma-88	164	1	let	let	VERB
ma-88	164	2	x	x	PRON
ma-88	164	3	be	be	AUX
ma-88	164	4	normed	normed	ADJ
ma-88	164	5	spaces	space	NOUN
ma-88	164	6	,	,	PUNCT
ma-88	164	7	x	x	X
ma-88	164	8	⊥bε	⊥bε	NOUN
ma-88	164	9	y	y	PROPN
ma-88	165	1	if	if	SCONJ
ma-88	165	2	and	and	CCONJ
ma-88	165	3	only	only	ADV
ma-88	165	4	if	if	SCONJ
ma-88	165	5	there	there	PRON
ma-88	165	6	exists	exist	VERB
ma-88	165	7	|a|	|a|	PROPN
ma-88	165	8	≤	≤	ADJ
ma-88	165	9	‖y‖‖x‖ε	‖y‖‖x‖ε	NOUN
ma-88	165	10	such	such	ADJ
ma-88	165	11	that	that	SCONJ
ma-88	165	12	x	x	PRON
ma-88	165	13	⊥b	⊥b	PRON
ma-88	165	14	ax	ax	NOUN
ma-88	165	15	+	+	X
ma-88	165	16	y	y	PROPN
ma-88	165	17	.	.	PUNCT
ma-88	166	1	proof	proof	NOUN
ma-88	166	2	.	.	PUNCT
ma-88	167	1	if	if	SCONJ
ma-88	167	2	x	x	PROPN
ma-88	167	3	⊥bε	⊥bε	VERB
ma-88	167	4	y	y	PROPN
ma-88	167	5	,	,	PUNCT
ma-88	167	6	first	first	ADV
ma-88	167	7	we	we	PRON
ma-88	167	8	have	have	VERB
ma-88	167	9	n−(x	n−(x	PROPN
ma-88	167	10	,	,	PUNCT
ma-88	167	11	y	y	NOUN
ma-88	167	12	)	)	PUNCT
ma-88	167	13	≤	≤	NOUN
ma-88	167	14	ε‖y‖	ε‖y‖	NOUN
ma-88	167	15	,	,	PUNCT
ma-88	167	16	n+(x	n+(x	PROPN
ma-88	167	17	,	,	PUNCT
ma-88	167	18	y	y	PROPN
ma-88	167	19	)	)	PUNCT
ma-88	167	20	≥	≥	NOUN
ma-88	167	21	−ε‖y‖.	−ε‖y‖.	NUM
ma-88	167	22	on	on	ADP
ma-88	167	23	the	the	DET
ma-88	167	24	other	other	ADJ
ma-88	167	25	hand	hand	NOUN
ma-88	167	26	,	,	PUNCT
ma-88	167	27	from	from	ADP
ma-88	167	28	lemma	lemma	PROPN
ma-88	167	29	2.8	2.8	NUM
ma-88	167	30	,	,	PUNCT
ma-88	167	31	we	we	PRON
ma-88	167	32	have	have	VERB
ma-88	167	33	if	if	SCONJ
ma-88	167	34	−	−	PROPN
ma-88	167	35	n+(x	n+(x	PROPN
ma-88	167	36	,	,	PUNCT
ma-88	167	37	y	y	NOUN
ma-88	167	38	)	)	PUNCT
ma-88	167	39	‖x‖	‖x‖	VERB
ma-88	167	40	≤	≤	NOUN
ma-88	167	41	a	a	DET
ma-88	167	42	≤	≤	NUM
ma-88	167	43	−	−	PROPN
ma-88	168	1	n−(x	n−(x	PROPN
ma-88	168	2	,	,	PUNCT
ma-88	168	3	y	y	NOUN
ma-88	168	4	)	)	PUNCT
ma-88	168	5	‖x‖	‖x‖	PROPN
ma-88	168	6	,	,	PUNCT
ma-88	168	7	then	then	ADV
ma-88	168	8	x	x	PUNCT
ma-88	168	9	⊥b	⊥b	PRON
ma-88	168	10	ax	ax	NOUN
ma-88	168	11	+	+	X
ma-88	168	12	y	y	PROPN
ma-88	168	13	.	.	PUNCT
ma-88	169	1	if	if	SCONJ
ma-88	169	2	there	there	PRON
ma-88	169	3	exists	exist	VERB
ma-88	169	4	no	no	DET
ma-88	169	5	|a|	|a|	PROPN
ma-88	169	6	≤	≤	ADJ
ma-88	169	7	‖y‖‖x‖ε	‖y‖‖x‖ε	NOUN
ma-88	169	8	such	such	ADJ
ma-88	169	9	that	that	SCONJ
ma-88	169	10	x	x	PRON
ma-88	169	11	⊥b	⊥b	PRON
ma-88	169	12	ax	ax	NOUN
ma-88	169	13	+	+	CCONJ
ma-88	169	14	y	y	PROPN
ma-88	169	15	,	,	PUNCT
ma-88	169	16	then	then	ADV
ma-88	169	17	−	−	PROPN
ma-88	169	18	n+(x	n+(x	PROPN
ma-88	169	19	,	,	PUNCT
ma-88	169	20	y	y	PROPN
ma-88	169	21	)	)	PUNCT
ma-88	169	22	‖x‖	‖x‖	VERB
ma-88	169	23	>	>	X
ma-88	169	24	‖y‖	‖y‖	PROPN
ma-88	169	25	‖x‖ε	‖x‖ε	NOUN
ma-88	169	26	or	or	CCONJ
ma-88	169	27	−	−	PROPN
ma-88	170	1	n−(x	n−(x	PROPN
ma-88	170	2	,	,	PUNCT
ma-88	170	3	y	y	NOUN
ma-88	170	4	)	)	PUNCT
ma-88	170	5	‖x‖	‖x‖	VERB
ma-88	170	6	<	<	X
ma-88	170	7	−	−	PROPN
ma-88	170	8	‖y‖	‖y‖	PROPN
ma-88	170	9	‖x‖ε	‖x‖ε	NOUN
ma-88	170	10	.	.	PUNCT
ma-88	171	1	thus	thus	ADV
ma-88	171	2	n+(x	n+(x	PROPN
ma-88	171	3	,	,	PUNCT
ma-88	171	4	y	y	PROPN
ma-88	171	5	)	)	PUNCT
ma-88	171	6	<	<	X
ma-88	171	7	‖y‖	‖y‖	PROPN
ma-88	171	8	or	or	CCONJ
ma-88	171	9	n(x	n(x	PROPN
ma-88	171	10	,	,	PUNCT
ma-88	171	11	y	y	PROPN
ma-88	171	12	)	)	PUNCT
ma-88	171	13	>	>	X
ma-88	171	14	‖y‖ε	‖y‖ε	PROPN
ma-88	171	15	.	.	PUNCT
ma-88	172	1	contradict	contradict	VERB
ma-88	172	2	to	to	ADP
ma-88	172	3	x	x	PROPN
ma-88	172	4	⊥bε	⊥bε	NOUN
ma-88	172	5	y	y	PROPN
ma-88	172	6	,	,	PUNCT
ma-88	172	7	so	so	CCONJ
ma-88	172	8	there	there	PRON
ma-88	172	9	must	must	AUX
ma-88	172	10	exists	exist	VERB
ma-88	172	11	|a|	|a|	PROPN
ma-88	172	12	≤	≤	PROPN
ma-88	172	13	‖y‖‖x‖ε	‖y‖‖x‖ε	NOUN
ma-88	172	14	,	,	PUNCT
ma-88	172	15	such	such	ADJ
ma-88	172	16	that	that	SCONJ
ma-88	172	17	x	x	PRON
ma-88	172	18	⊥b	⊥b	PRON
ma-88	172	19	ax	ax	NOUN
ma-88	172	20	+	+	X
ma-88	172	21	y	y	PROPN
ma-88	172	22	.	.	PUNCT
ma-88	173	1	conversely	conversely	ADV
ma-88	173	2	,	,	PUNCT
ma-88	173	3	if	if	SCONJ
ma-88	173	4	there	there	PRON
ma-88	173	5	exists	exist	VERB
ma-88	173	6	|a|	|a|	PROPN
ma-88	173	7	≤	≤	ADJ
ma-88	173	8	ε‖y‖	ε‖y‖	NOUN
ma-88	173	9	‖x‖	‖x‖	PROPN
ma-88	173	10	such	such	ADJ
ma-88	173	11	that	that	SCONJ
ma-88	173	12	x	x	PRON
ma-88	173	13	⊥b	⊥b	PRON
ma-88	173	14	ax	ax	NOUN
ma-88	173	15	+	+	CCONJ
ma-88	173	16	y	y	PROPN
ma-88	173	17	,	,	PUNCT
ma-88	173	18	we	we	PRON
ma-88	173	19	have	have	VERB
ma-88	173	20	:	:	PUNCT
ma-88	173	21	x	x	X
ma-88	173	22	⊥b	⊥b	PRON
ma-88	173	23	ax	ax	NOUN
ma-88	173	24	+	+	CCONJ
ma-88	173	25	y	y	PROPN
ma-88	173	26	=	=	NOUN
ma-88	173	27	⇒	⇒	PROPN
ma-88	173	28	−n+(x	−n+(x	PROPN
ma-88	173	29	,	,	PUNCT
ma-88	173	30	y	y	NOUN
ma-88	173	31	)	)	PUNCT
ma-88	173	32	≤	≤	NUM
ma-88	173	33	a‖x‖	a‖x‖	PROPN
ma-88	173	34	≤	≤	NUM
ma-88	173	35	−n−(x	−n−(x	PROPN
ma-88	173	36	,	,	PUNCT
ma-88	173	37	y	y	NOUN
ma-88	173	38	)	)	PUNCT
ma-88	174	1	=	=	VERB
ma-88	174	2	⇒	⇒	PROPN
ma-88	174	3	n+(x	n+(x	PROPN
ma-88	174	4	,	,	PUNCT
ma-88	174	5	y	y	PROPN
ma-88	174	6	)	)	PUNCT
ma-88	175	1	+	+	CCONJ
ma-88	175	2	ε‖y‖	ε‖y‖	NOUN
ma-88	175	3	≥	≥	NOUN
ma-88	175	4	0	0	NUM
ma-88	175	5	≥	≥	PROPN
ma-88	175	6	n−(x	n−(x	PROPN
ma-88	175	7	,	,	PUNCT
ma-88	175	8	y)−	y)−	PROPN
ma-88	175	9	ε‖y‖	ε‖y‖	NOUN
ma-88	175	10	=	=	NOUN
ma-88	175	11	⇒	⇒	NOUN
ma-88	175	12	x	x	PROPN
ma-88	175	13	⊥bε	⊥bε	NOUN
ma-88	175	14	y	y	PROPN
ma-88	175	15	.	.	PUNCT
ma-88	176	1	to	to	PART
ma-88	176	2	conclude	conclude	VERB
ma-88	176	3	,	,	PUNCT
ma-88	176	4	in	in	ADP
ma-88	176	5	normed	normed	ADJ
ma-88	176	6	spaces	space	NOUN
ma-88	176	7	,	,	PUNCT
ma-88	176	8	x	x	SYM
ma-88	176	9	⊥bε	⊥bε	NOUN
ma-88	176	10	y	y	PROPN
ma-88	177	1	if	if	SCONJ
ma-88	177	2	and	and	CCONJ
ma-88	177	3	only	only	ADV
ma-88	177	4	if	if	SCONJ
ma-88	177	5	there	there	PRON
ma-88	177	6	exists	exist	VERB
ma-88	177	7	|a|	|a|	PROPN
ma-88	177	8	≤	≤	ADJ
ma-88	177	9	‖y‖‖x‖ε	‖y‖‖x‖ε	NOUN
ma-88	177	10	such	such	ADJ
ma-88	177	11	that	that	SCONJ
ma-88	177	12	x	x	PUNCT
ma-88	177	13	⊥b	⊥b	PRON
ma-88	177	14	ax+y	ax+y	PROPN
ma-88	177	15	.	.	PUNCT
ma-88	178	1	�	�	PROPN
ma-88	178	2	since	since	SCONJ
ma-88	178	3	both	both	PRON
ma-88	178	4	x	x	PROPN
ma-88	178	5	⊥bε	⊥bε	NOUN
ma-88	178	6	y	y	PROPN
ma-88	178	7	and	and	CCONJ
ma-88	178	8	x	x	PROPN
ma-88	178	9	⊥bε	⊥bε	NOUN
ma-88	178	10	y	y	PROPN
ma-88	178	11	are	be	AUX
ma-88	178	12	equivalent	equivalent	ADJ
ma-88	178	13	to	to	ADP
ma-88	178	14	there	there	PRON
ma-88	178	15	exists	exist	VERB
ma-88	178	16	|a|	|a|	PROPN
ma-88	178	17	≤	≤	ADJ
ma-88	178	18	‖y‖‖x‖ε	‖y‖‖x‖ε	NOUN
ma-88	178	19	such	such	ADJ
ma-88	178	20	that	that	SCONJ
ma-88	178	21	x	x	PUNCT
ma-88	178	22	⊥b	⊥b	PRON
ma-88	178	23	ax+y	ax+y	PROPN
ma-88	178	24	.we	.we	PUNCT
ma-88	178	25	have	have	VERB
ma-88	179	1	x	x	NOUN
ma-88	179	2	⊥bε	⊥bε	NOUN
ma-88	179	3	y	y	PROPN
ma-88	179	4	if	if	SCONJ
ma-88	179	5	and	and	CCONJ
ma-88	179	6	only	only	ADV
ma-88	179	7	if	if	SCONJ
ma-88	179	8	x	x	X
ma-88	179	9	⊥εb	⊥εb	NOUN
ma-88	179	10	y	y	PROPN
ma-88	179	11	in	in	ADP
ma-88	179	12	normed	normed	ADJ
ma-88	179	13	spaces	space	NOUN
ma-88	179	14	.	.	PUNCT
ma-88	180	1	now	now	ADV
ma-88	180	2	we	we	PRON
ma-88	180	3	give	give	VERB
ma-88	180	4	a	a	DET
ma-88	180	5	direct	direct	ADJ
ma-88	180	6	proof	proof	NOUN
ma-88	180	7	for	for	ADP
ma-88	180	8	this	this	PRON
ma-88	180	9	.	.	PUNCT
ma-88	181	1	theorem	theorem	VERB
ma-88	181	2	2.10	2.10	NUM
ma-88	181	3	.	.	PUNCT
ma-88	182	1	let	let	VERB
ma-88	182	2	x	x	PRON
ma-88	182	3	be	be	AUX
ma-88	182	4	normed	normed	ADJ
ma-88	182	5	spaces	space	NOUN
ma-88	182	6	,	,	PUNCT
ma-88	182	7	x	x	X
ma-88	182	8	⊥bε	⊥bε	NOUN
ma-88	182	9	y	y	PROPN
ma-88	183	1	if	if	SCONJ
ma-88	183	2	and	and	CCONJ
ma-88	183	3	only	only	ADV
ma-88	183	4	if	if	SCONJ
ma-88	183	5	x	x	X
ma-88	183	6	⊥εb	⊥εb	NOUN
ma-88	183	7	y	y	PROPN
ma-88	183	8	.	.	PUNCT
ma-88	184	1	proof.we	proof.we	NOUN
ma-88	184	2	can	can	AUX
ma-88	184	3	assume	assume	VERB
ma-88	184	4	that	that	PRON
ma-88	184	5	|t|	|t|	VERB
ma-88	184	6	≤	≤	NOUN
ma-88	184	7	‖x‖	‖x‖	PROPN
ma-88	184	8	ε‖y‖	ε‖y‖	NOUN
ma-88	184	9	and	and	CCONJ
ma-88	184	10	x	x	SYM
ma-88	184	11	6=	6=	ADP
ma-88	184	12	0	0	NUM
ma-88	184	13	.	.	PUNCT
ma-88	185	1	if	if	SCONJ
ma-88	185	2	x	x	PROPN
ma-88	185	3	⊥bε	⊥bε	NOUN
ma-88	185	4	y	y	PROPN
ma-88	185	5	,	,	PUNCT
ma-88	185	6	we	we	PRON
ma-88	185	7	have	have	VERB
ma-88	185	8	:	:	PUNCT
ma-88	185	9	‖x	‖x	NOUN
ma-88	186	1	+	+	CCONJ
ma-88	186	2	ty‖	ty‖	DET
ma-88	186	3	≥	≥	NOUN
ma-88	187	1	‖x‖	‖x‖	PROPN
ma-88	187	2	−	−	PROPN
ma-88	187	3	ε‖ty‖	ε‖ty‖	NOUN
ma-88	187	4	≥	≥	NOUN
ma-88	187	5	0	0	NUM
ma-88	187	6	.	.	PUNCT
ma-88	188	1	take	take	VERB
ma-88	188	2	square	square	NOUN
ma-88	188	3	on	on	ADP
ma-88	188	4	both	both	DET
ma-88	188	5	sides	side	NOUN
ma-88	188	6	,	,	PUNCT
ma-88	188	7	we	we	PRON
ma-88	188	8	get	get	VERB
ma-88	188	9	‖x	‖x	NOUN
ma-88	189	1	+	+	CCONJ
ma-88	189	2	ty‖2	ty‖2	PROPN
ma-88	189	3	≥	≥	NOUN
ma-88	189	4	‖x‖2	‖x‖2	VERB
ma-88	189	5	−	−	PROPN
ma-88	189	6	ε2‖ty‖2	ε2‖ty‖2	NOUN
ma-88	189	7	−	−	PROPN
ma-88	189	8	2ε‖x‖‖ty‖.	2ε‖x‖‖ty‖.	NUM
ma-88	189	9	thus	thus	ADV
ma-88	189	10	‖x	‖x	PRON
ma-88	190	1	+	+	CCONJ
ma-88	190	2	ty‖2	ty‖2	PROPN
ma-88	190	3	≥	≥	NOUN
ma-88	190	4	‖x‖2	‖x‖2	VERB
ma-88	190	5	−	−	ADP
ma-88	190	6	2ε‖x‖‖ty‖	2ε‖x‖‖ty‖	NUM
ma-88	190	7	,	,	PUNCT
ma-88	190	8	which	which	PRON
ma-88	190	9	means	mean	VERB
ma-88	190	10	that	that	SCONJ
ma-88	190	11	x	x	PROPN
ma-88	190	12	⊥εb	⊥εb	PROPN
ma-88	190	13	y	y	PROPN
ma-88	190	14	.	.	PUNCT
ma-88	191	1	conversely	conversely	ADV
ma-88	191	2	,	,	PUNCT
ma-88	191	3	if	if	SCONJ
ma-88	191	4	x	x	PROPN
ma-88	191	5	⊥εb	⊥εb	NOUN
ma-88	191	6	y	y	PROPN
ma-88	191	7	,	,	PUNCT
ma-88	191	8	we	we	PRON
ma-88	191	9	have	have	VERB
ma-88	191	10	‖x	‖x	NOUN
ma-88	192	1	+	+	CCONJ
ma-88	192	2	ty‖2	ty‖2	PROPN
ma-88	192	3	≥	≥	NOUN
ma-88	192	4	‖x‖2	‖x‖2	VERB
ma-88	192	5	−	−	PUNCT
ma-88	192	6	2ε‖x‖‖ty‖	2ε‖x‖‖ty‖	NUM
ma-88	192	7	≥	≥	NOUN
ma-88	192	8	0	0	NUM
ma-88	192	9	.	.	PUNCT
ma-88	193	1	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	193	2	eur	eur	PROPN
ma-88	193	3	.	.	PUNCT
ma-88	194	1	j.	j.	PROPN
ma-88	194	2	math	math	PROPN
ma-88	194	3	.	.	PUNCT
ma-88	195	1	anal	anal	PROPN
ma-88	195	2	.	.	PUNCT
ma-88	196	1	10.28924	10.28924	NUM
ma-88	196	2	/	/	SYM
ma-88	196	3	ada	ada	PROPN
ma-88	196	4	/	/	SYM
ma-88	196	5	ma.2.16	ma.2.16	PROPN
ma-88	196	6	7	7	NUM
ma-88	196	7	let	let	VERB
ma-88	196	8	both	both	DET
ma-88	196	9	sides	side	NOUN
ma-88	196	10	be	be	AUX
ma-88	196	11	divided	divide	VERB
ma-88	196	12	by	by	ADP
ma-88	196	13	‖x	‖x	NOUN
ma-88	196	14	+	+	CCONJ
ma-88	196	15	ty‖+	ty‖+	VERB
ma-88	196	16	‖x‖	‖x‖	PROPN
ma-88	196	17	,	,	PUNCT
ma-88	196	18	we	we	PRON
ma-88	196	19	get	get	VERB
ma-88	196	20	:	:	PUNCT
ma-88	196	21	‖x	‖x	NOUN
ma-88	197	1	+	+	CCONJ
ma-88	197	2	ty‖	ty‖	ADP
ma-88	197	3	−	−	PUNCT
ma-88	197	4	‖x‖	‖x‖	PROPN
ma-88	197	5	≥	≥	NOUN
ma-88	197	6	−2ε‖x‖‖ty‖	−2ε‖x‖‖ty‖	VERB
ma-88	197	7	‖x	‖x	PUNCT
ma-88	198	1	+	+	CCONJ
ma-88	198	2	ty‖+	ty‖+	PROPN
ma-88	198	3	‖x‖	‖x‖	PROPN
ma-88	198	4	.	.	PUNCT
ma-88	199	1	since	since	SCONJ
ma-88	199	2	‖x	‖x	PRON
ma-88	199	3	+	+	CCONJ
ma-88	199	4	ty‖	ty‖	PRON
ma-88	199	5	tends	tend	VERB
ma-88	199	6	to	to	PART
ma-88	199	7	‖x‖	‖x‖	VERB
ma-88	199	8	when	when	SCONJ
ma-88	199	9	t	t	PROPN
ma-88	199	10	→	→	SYM
ma-88	199	11	0	0	NUM
ma-88	199	12	,	,	PUNCT
ma-88	199	13	for	for	ADP
ma-88	199	14	every	every	DET
ma-88	199	15	2‖x‖	2‖x‖	NUM
ma-88	199	16	>	>	X
ma-88	199	17	δ	δ	PROPN
ma-88	199	18	>	>	X
ma-88	199	19	0	0	PROPN
ma-88	199	20	,	,	PUNCT
ma-88	199	21	we	we	PRON
ma-88	199	22	can	can	AUX
ma-88	199	23	find	find	VERB
ma-88	199	24	η	η	PROPN
ma-88	199	25	>	>	X
ma-88	199	26	0	0	PROPN
ma-88	199	27	,	,	PUNCT
ma-88	199	28	such	such	ADJ
ma-88	199	29	that	that	SCONJ
ma-88	199	30	‖x	‖x	PROPN
ma-88	199	31	+	+	CCONJ
ma-88	199	32	ty‖+	ty‖+	PROPN
ma-88	199	33	‖x‖	‖x‖	PROPN
ma-88	199	34	≥	≥	NUM
ma-88	199	35	2‖x‖	2‖x‖	NUM
ma-88	199	36	−	−	NOUN
ma-88	199	37	δ	δ	X
ma-88	199	38	i	i	PRON
ma-88	199	39	f	f	PROPN
ma-88	199	40	|t|	|t|	VERB
ma-88	199	41	≤	≤	PROPN
ma-88	199	42	η	η	PROPN
ma-88	199	43	.	.	PUNCT
ma-88	200	1	we	we	PRON
ma-88	200	2	then	then	ADV
ma-88	200	3	have	have	VERB
ma-88	200	4	‖x	‖x	NOUN
ma-88	200	5	+	+	CCONJ
ma-88	200	6	ty‖	ty‖	PROPN
ma-88	200	7	−	−	PUNCT
ma-88	200	8	‖x‖	‖x‖	PROPN
ma-88	200	9	≥	≥	PROPN
ma-88	200	10	−2‖x‖‖ty‖	−2‖x‖‖ty‖	NOUN
ma-88	200	11	2‖x‖	2‖x‖	NUM
ma-88	200	12	−	−	PROPN
ma-88	200	13	δ	δ	PROPN
ma-88	200	14	.	.	PUNCT
ma-88	201	1	let	let	VERB
ma-88	201	2	δ	δ	PRON
ma-88	201	3	→	→	SYM
ma-88	201	4	0	0	NUM
ma-88	201	5	we	we	PRON
ma-88	201	6	have	have	VERB
ma-88	201	7	‖x	‖x	NOUN
ma-88	202	1	+	+	CCONJ
ma-88	202	2	ty‖	ty‖	PROPN
ma-88	202	3	−	−	PUNCT
ma-88	202	4	‖x‖	‖x‖	PROPN
ma-88	202	5	≥	≥	PROPN
ma-88	202	6	−2‖x‖‖ty‖	−2‖x‖‖ty‖	NOUN
ma-88	202	7	2‖x‖	2‖x‖	NUM
ma-88	202	8	when	when	SCONJ
ma-88	202	9	t	t	NOUN
ma-88	202	10	→	→	SYM
ma-88	202	11	0	0	NUM
ma-88	202	12	.	.	PUNCT
ma-88	203	1	thus	thus	ADV
ma-88	203	2	n+(x	n+(x	PROPN
ma-88	203	3	,	,	PUNCT
ma-88	203	4	y	y	PROPN
ma-88	203	5	)	)	PUNCT
ma-88	204	1	+	+	CCONJ
ma-88	204	2	ε‖y‖	ε‖y‖	NOUN
ma-88	204	3	≥	≥	NOUN
ma-88	204	4	0	0	NUM
ma-88	204	5	≥	≥	PROPN
ma-88	204	6	n−(x	n−(x	PROPN
ma-88	204	7	,	,	PUNCT
ma-88	204	8	y)−	y)−	PROPN
ma-88	204	9	ε‖y‖	ε‖y‖	NOUN
ma-88	204	10	=	=	NOUN
ma-88	204	11	⇒	⇒	NOUN
ma-88	204	12	x	x	PROPN
ma-88	204	13	⊥bε	⊥bε	NOUN
ma-88	204	14	y	y	PROPN
ma-88	204	15	.	.	PUNCT
ma-88	205	1	�	�	PROPN
ma-88	205	2	recall	recall	VERB
ma-88	205	3	that	that	SCONJ
ma-88	205	4	dragomir	dragomir	PROPN
ma-88	205	5	gave	give	VERB
ma-88	205	6	the	the	DET
ma-88	205	7	following	follow	VERB
ma-88	205	8	definition	definition	NOUN
ma-88	205	9	about	about	ADP
ma-88	205	10	approximate	approximate	ADJ
ma-88	205	11	birkhoff	birkhoff	NOUN
ma-88	205	12	orthogonality	orthogonality	NOUN
ma-88	205	13	:	:	PUNCT
ma-88	205	14	xε	xε	NUM
ma-88	205	15	⊥b	⊥b	PRON
ma-88	205	16	y	y	NOUN
ma-88	205	17	⇐	⇐	ADJ
ma-88	205	18	⇒	⇒	NOUN
ma-88	205	19	‖x	‖x	NOUN
ma-88	206	1	+	+	CCONJ
ma-88	206	2	ty‖	ty‖	PRON
ma-88	206	3	≥	≥	NOUN
ma-88	206	4	(	(	PUNCT
ma-88	206	5	1−	1−	NUM
ma-88	206	6	ε)‖x‖.	ε)‖x‖.	NOUN
ma-88	206	7	it	it	PRON
ma-88	206	8	is	be	AUX
ma-88	206	9	known	know	VERB
ma-88	206	10	that	that	SCONJ
ma-88	206	11	[	[	X
ma-88	206	12	19	19	NUM
ma-88	206	13	]	]	PUNCT
ma-88	206	14	in	in	ADP
ma-88	206	15	normed	normed	ADJ
ma-88	206	16	spaces	space	NOUN
ma-88	206	17	,	,	PUNCT
ma-88	206	18	x	x	X
ma-88	206	19	⊥εb	⊥εb	NOUN
ma-88	206	20	y	y	PROPN
ma-88	206	21	implies	imply	VERB
ma-88	206	22	xδ	xδ	PROPN
ma-88	206	23	⊥b	⊥b	NOUN
ma-88	206	24	,	,	PUNCT
ma-88	206	25	where	where	SCONJ
ma-88	206	26	δ	δ	PROPN
ma-88	206	27	=	=	SYM
ma-88	206	28	1−	1−	NUM
ma-88	206	29	√	√	NUM
ma-88	206	30	1−	1−	NUM
ma-88	206	31	4ε	4ε	NUM
ma-88	206	32	.	.	PUNCT
ma-88	207	1	now	now	ADV
ma-88	207	2	wegive	wegive	ADJ
ma-88	207	3	a	a	DET
ma-88	207	4	more	more	ADV
ma-88	207	5	accurate	accurate	ADJ
ma-88	207	6	estimate	estimate	NOUN
ma-88	207	7	of	of	ADP
ma-88	207	8	δ	δ	PROPN
ma-88	207	9	as	as	ADP
ma-88	207	10	an	an	DET
ma-88	207	11	application	application	NOUN
ma-88	207	12	of	of	ADP
ma-88	207	13	the	the	DET
ma-88	207	14	above	above	ADJ
ma-88	207	15	proposition	proposition	NOUN
ma-88	207	16	.	.	PUNCT
ma-88	208	1	proposition	proposition	NOUN
ma-88	208	2	2.11	2.11	NUM
ma-88	208	3	.	.	PUNCT
ma-88	209	1	let	let	VERB
ma-88	209	2	x	x	PRON
ma-88	209	3	be	be	AUX
ma-88	209	4	normed	normed	ADJ
ma-88	209	5	spaces	space	NOUN
ma-88	209	6	,	,	PUNCT
ma-88	209	7	let	let	VERB
ma-88	209	8	x	x	PRON
ma-88	209	9	,	,	PUNCT
ma-88	209	10	y	y	PROPN
ma-88	209	11	∈	∈	PROPN
ma-88	209	12	x	x	NOUN
ma-88	209	13	,	,	PUNCT
ma-88	209	14	then	then	ADV
ma-88	209	15	:	:	PUNCT
ma-88	209	16	x	x	X
ma-88	210	1	⊥εb	⊥εb	NOUN
ma-88	210	2	y	y	PROPN
ma-88	210	3	=	=	PROPN
ma-88	210	4	⇒	⇒	PROPN
ma-88	210	5	xδ	xδ	PROPN
ma-88	210	6	⊥b	⊥b	PRON
ma-88	210	7	y	y	PROPN
ma-88	210	8	where	where	SCONJ
ma-88	210	9	δ	δ	PROPN
ma-88	210	10	=	=	SYM
ma-88	210	11	2ε	2ε	NUM
ma-88	210	12	.	.	PUNCT
ma-88	211	1	proof	proof	NOUN
ma-88	211	2	.	.	PUNCT
ma-88	212	1	let	let	VERB
ma-88	212	2	f	f	PROPN
ma-88	212	3	(	(	PUNCT
ma-88	212	4	t	t	PROPN
ma-88	212	5	)	)	PUNCT
ma-88	212	6	=	=	PUNCT
ma-88	213	1	‖x	‖x	NOUN
ma-88	213	2	+	+	CCONJ
ma-88	213	3	ty‖	ty‖	NOUN
ma-88	213	4	and	and	CCONJ
ma-88	213	5	assume	assume	VERB
ma-88	213	6	that	that	SCONJ
ma-88	213	7	f	f	PROPN
ma-88	213	8	(	(	PUNCT
ma-88	213	9	t	t	PROPN
ma-88	213	10	)	)	PUNCT
ma-88	213	11	attains	attain	VERB
ma-88	213	12	its	its	PRON
ma-88	213	13	minimum	minimum	NOUN
ma-88	213	14	at	at	ADP
ma-88	213	15	t0	t0	NOUN
ma-88	213	16	,	,	PUNCT
ma-88	213	17	hence	hence	ADV
ma-88	213	18	‖x	‖x	PUNCT
ma-88	214	1	+	+	PUNCT
ma-88	215	1	t0y	t0y	ADJ
ma-88	215	2	+	+	CCONJ
ma-88	215	3	ty‖	ty‖	PRON
ma-88	215	4	≥	≥	NUM
ma-88	215	5	‖x	‖x	PUNCT
ma-88	216	1	+	+	CCONJ
ma-88	216	2	t0y‖	t0y‖	X
ma-88	216	3	f	f	PROPN
ma-88	216	4	or	or	CCONJ
ma-88	216	5	al	al	PROPN
ma-88	216	6	l	l	PROPN
ma-88	216	7	t	t	PROPN
ma-88	217	1	∈	∈	PROPN
ma-88	217	2	r.	r.	PROPN
ma-88	217	3	choose	choose	PROPN
ma-88	217	4	t	t	PROPN
ma-88	217	5	=	=	PUNCT
ma-88	217	6	−t0	−t0	VERB
ma-88	217	7	we	we	PRON
ma-88	217	8	have	have	VERB
ma-88	217	9	‖x‖	‖x‖	PROPN
ma-88	217	10	≥	≥	NUM
ma-88	217	11	‖x	‖x	PUNCT
ma-88	218	1	+	+	CCONJ
ma-88	218	2	t0y‖	t0y‖	PROPN
ma-88	218	3	≥	≥	PROPN
ma-88	218	4	|‖x‖	|‖x‖	PROPN
ma-88	218	5	−	−	PROPN
ma-88	218	6	|t0|‖y‖|	|t0|‖y‖|	NOUN
ma-88	218	7	,	,	PUNCT
ma-88	218	8	thus	thus	ADV
ma-88	218	9	we	we	PRON
ma-88	218	10	get	get	VERB
ma-88	218	11	|t0|	|t0|	NOUN
ma-88	218	12	≤	≤	NOUN
ma-88	218	13	2‖x‖	2‖x‖	NUM
ma-88	218	14	‖y‖	‖y‖	PROPN
ma-88	218	15	,	,	PUNCT
ma-88	218	16	then	then	ADV
ma-88	218	17	‖x	‖x	PUNCT
ma-88	219	1	+	+	CCONJ
ma-88	219	2	ty‖	ty‖	PRON
ma-88	219	3	≥	≥	NUM
ma-88	219	4	‖x	‖x	PUNCT
ma-88	220	1	+	+	CCONJ
ma-88	220	2	t0y‖	t0y‖	X
ma-88	220	3	≥	≥	NOUN
ma-88	220	4	‖x‖	‖x‖	PROPN
ma-88	220	5	−	−	PROPN
ma-88	220	6	ε|t0|‖y‖	ε|t0|‖y‖	PROPN
ma-88	220	7	≥	≥	X
ma-88	220	8	(	(	PUNCT
ma-88	220	9	1−	1−	NUM
ma-88	220	10	2ε)‖x‖	2ε)‖x‖	NUM
ma-88	220	11	f	f	NOUN
ma-88	220	12	or	or	CCONJ
ma-88	220	13	al	al	PROPN
ma-88	220	14	l	l	PROPN
ma-88	220	15	t	t	PROPN
ma-88	220	16	∈	∈	PROPN
ma-88	220	17	r.	r.	PROPN
ma-88	220	18	thus	thus	ADV
ma-88	220	19	xδ	xδ	PROPN
ma-88	220	20	⊥b	⊥b	PRON
ma-88	220	21	y	y	PROPN
ma-88	220	22	,	,	PUNCT
ma-88	220	23	where	where	SCONJ
ma-88	220	24	δ	δ	PROPN
ma-88	220	25	=	=	SYM
ma-88	220	26	2ε	2ε	PROPN
ma-88	220	27	.	.	PUNCT
ma-88	221	1	by	by	ADP
ma-88	221	2	the	the	DET
ma-88	221	3	equivalence	equivalence	NOUN
ma-88	221	4	between	between	ADP
ma-88	221	5	⊥bε	⊥bε	NOUN
ma-88	221	6	and	and	CCONJ
ma-88	221	7	⊥εb	⊥εb	NOUN
ma-88	221	8	,	,	PUNCT
ma-88	221	9	we	we	PRON
ma-88	221	10	have	have	VERB
ma-88	221	11	the	the	DET
ma-88	221	12	result	result	NOUN
ma-88	221	13	that	that	SCONJ
ma-88	221	14	x	x	PROPN
ma-88	221	15	⊥εb	⊥εb	PROPN
ma-88	221	16	y	y	PROPN
ma-88	221	17	implies	imply	VERB
ma-88	221	18	xδ	xδ	PROPN
ma-88	221	19	⊥b	⊥b	PRON
ma-88	221	20	y	y	PROPN
ma-88	221	21	.	.	PUNCT
ma-88	222	1	since	since	SCONJ
ma-88	222	2	2ε	2ε	NOUN
ma-88	222	3	≤	≤	NUM
ma-88	222	4	1−	1−	NUM
ma-88	223	1	√	√	NUM
ma-88	223	2	1−	1−	NUM
ma-88	223	3	4ε	4ε	NUM
ma-88	223	4	,	,	PUNCT
ma-88	223	5	2ε	2ε	PROPN
ma-88	223	6	can	can	AUX
ma-88	223	7	be	be	AUX
ma-88	223	8	seen	see	VERB
ma-88	223	9	as	as	ADP
ma-88	223	10	a	a	DET
ma-88	223	11	more	more	ADV
ma-88	223	12	accurate	accurate	ADJ
ma-88	223	13	estimate	estimate	NOUN
ma-88	223	14	.	.	PUNCT
ma-88	224	1	�	�	PROPN
ma-88	224	2	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	224	3	eur	eur	PROPN
ma-88	224	4	.	.	PUNCT
ma-88	225	1	j.	j.	PROPN
ma-88	225	2	math	math	PROPN
ma-88	225	3	.	.	PUNCT
ma-88	226	1	anal	anal	PROPN
ma-88	226	2	.	.	PUNCT
ma-88	227	1	10.28924	10.28924	NUM
ma-88	227	2	/	/	SYM
ma-88	227	3	ada	ada	PROPN
ma-88	227	4	/	/	SYM
ma-88	227	5	ma.2.16	ma.2.16	PROPN
ma-88	227	6	83	83	NUM
ma-88	227	7	.	.	PUNCT
ma-88	228	1	approximate	approximate	ADJ
ma-88	228	2	isosceles	isoscele	NOUN
ma-88	228	3	orthogonality	orthogonality	NOUN
ma-88	228	4	and	and	CCONJ
ma-88	228	5	approximate	approximate	ADJ
ma-88	228	6	birkhoff	birkhoff	NOUN
ma-88	228	7	orthogonality	orthogonality	NOUN
ma-88	228	8	in	in	ADP
ma-88	228	9	the	the	DET
ma-88	228	10	following	following	NOUN
ma-88	228	11	we	we	PRON
ma-88	228	12	will	will	AUX
ma-88	228	13	use	use	VERB
ma-88	228	14	the	the	DET
ma-88	228	15	notion	notion	NOUN
ma-88	228	16	of	of	ADP
ma-88	228	17	approximate	approximate	ADJ
ma-88	228	18	isosceles	isoscele	NOUN
ma-88	228	19	orthogonality	orthogonality	NOUN
ma-88	228	20	[	[	X
ma-88	228	21	11	11	NUM
ma-88	228	22	]	]	PUNCT
ma-88	228	23	,	,	PUNCT
ma-88	228	24	recall	recall	VERB
ma-88	228	25	thatthe	thatthe	NOUN
ma-88	228	26	approximate	approximate	ADJ
ma-88	228	27	isosceles	isoscele	NOUN
ma-88	228	28	orthogonality	orthogonality	NOUN
ma-88	228	29	is	be	AUX
ma-88	228	30	defined	define	VERB
ma-88	228	31	by	by	ADP
ma-88	228	32	:	:	PUNCT
ma-88	228	33	x	x	X
ma-88	228	34	⊥εi	⊥εi	ADP
ma-88	228	35	y	y	NOUN
ma-88	228	36	:	:	PUNCT
ma-88	228	37	|‖x	|‖x	PROPN
ma-88	228	38	+	+	CCONJ
ma-88	228	39	y‖2	y‖2	X
ma-88	228	40	−	−	NOUN
ma-88	228	41	‖x	‖x	NOUN
ma-88	229	1	−	−	PROPN
ma-88	229	2	y‖2|	y‖2|	NOUN
ma-88	229	3	≤	≤	NOUN
ma-88	229	4	4ε‖x‖‖y‖.	4ε‖x‖‖y‖.	NUM
ma-88	229	5	xε	xε	NOUN
ma-88	229	6	⊥i	⊥i	PROPN
ma-88	229	7	y	y	PROPN
ma-88	229	8	:	:	PUNCT
ma-88	229	9	|‖x	|‖x	PROPN
ma-88	229	10	+	+	NUM
ma-88	229	11	y‖	y‖	NOUN
ma-88	229	12	−	−	NOUN
ma-88	229	13	‖x	‖x	NOUN
ma-88	230	1	−	−	PROPN
ma-88	230	2	y‖|	y‖|	NOUN
ma-88	230	3	≤	≤	NOUN
ma-88	230	4	ε(‖x	ε(‖x	PUNCT
ma-88	231	1	+	+	CCONJ
ma-88	231	2	y‖+	y‖+	ADJ
ma-88	231	3	‖x	‖x	NOUN
ma-88	231	4	−	−	PROPN
ma-88	231	5	y‖	y‖	PROPN
ma-88	231	6	)	)	PUNCT
ma-88	231	7	.	.	PUNCT
ma-88	232	1	it	it	PRON
ma-88	232	2	is	be	AUX
ma-88	232	3	easy	easy	ADJ
ma-88	232	4	to	to	PART
ma-88	232	5	see	see	VERB
ma-88	232	6	that	that	SCONJ
ma-88	232	7	in	in	ADP
ma-88	232	8	inner	inner	ADJ
ma-88	232	9	product	product	NOUN
ma-88	232	10	spaces	space	NOUN
ma-88	232	11	we	we	PRON
ma-88	232	12	have	have	VERB
ma-88	232	13	:	:	PUNCT
ma-88	232	14	x	x	SYM
ma-88	232	15	⊥εi	⊥εi	ADP
ma-88	232	16	y	y	PROPN
ma-88	232	17	⇐	⇐	PROPN
ma-88	232	18	⇒	⇒	PROPN
ma-88	232	19	|cos(x	|cos(x	VERB
ma-88	232	20	,	,	PUNCT
ma-88	232	21	y)|	y)|	PROPN
ma-88	232	22	≤	≤	NOUN
ma-88	232	23	ε	ε	VERB
ma-88	232	24	⇐	⇐	ADJ
ma-88	232	25	⇒	⇒	PROPN
ma-88	232	26	x	x	PROPN
ma-88	232	27	⊥bε	⊥bε	NOUN
ma-88	232	28	y	y	PROPN
ma-88	232	29	,	,	PUNCT
ma-88	232	30	and	and	CCONJ
ma-88	233	1	[	[	X
ma-88	233	2	11	11	NUM
ma-88	233	3	]	]	PUNCT
ma-88	233	4	xε	xε	PUNCT
ma-88	233	5	⊥i	⊥i	PROPN
ma-88	233	6	y	y	PROPN
ma-88	233	7	⇐	⇐	PROPN
ma-88	233	8	⇒	⇒	PROPN
ma-88	233	9	|cos(x	|cos(x	VERB
ma-88	233	10	,	,	PUNCT
ma-88	233	11	y)|	y)|	PROPN
ma-88	233	12	≤	≤	NOUN
ma-88	233	13	ε	ε	PROPN
ma-88	233	14	1	1	NUM
ma-88	233	15	+	+	CCONJ
ma-88	233	16	ε2	ε2	ADJ
ma-88	233	17	(	(	PUNCT
ma-88	233	18	‖x‖2	‖x‖2	PROPN
ma-88	233	19	+	+	X
ma-88	233	20	‖y‖2	‖y‖2	PROPN
ma-88	233	21	)	)	PUNCT
ma-88	233	22	.	.	PUNCT
ma-88	234	1	in	in	ADP
ma-88	234	2	the	the	DET
ma-88	234	3	following	following	NOUN
ma-88	234	4	we	we	PRON
ma-88	234	5	give	give	VERB
ma-88	234	6	some	some	DET
ma-88	234	7	simple	simple	ADJ
ma-88	234	8	properties	property	NOUN
ma-88	234	9	about	about	ADP
ma-88	234	10	approxiamte	approxiamte	NOUN
ma-88	234	11	isosceles	isoscele	NOUN
ma-88	234	12	orthogonality	orthogonality	NOUN
ma-88	234	13	.	.	PUNCT
ma-88	235	1	proposition	proposition	NOUN
ma-88	235	2	3.1	3.1	NUM
ma-88	235	3	.	.	PUNCT
ma-88	236	1	let	let	VERB
ma-88	236	2	x	x	PRON
ma-88	236	3	be	be	AUX
ma-88	236	4	normed	normed	ADJ
ma-88	236	5	spaces	space	NOUN
ma-88	236	6	,	,	PUNCT
ma-88	236	7	if	if	SCONJ
ma-88	236	8	there	there	PRON
ma-88	236	9	exists	exist	VERB
ma-88	236	10	|a|	|a|	PROPN
ma-88	236	11	≤	≤	ADJ
ma-88	236	12	‖y‖‖x‖ε	‖y‖‖x‖ε	NOUN
ma-88	236	13	such	such	ADJ
ma-88	236	14	that	that	SCONJ
ma-88	236	15	x	x	SYM
ma-88	236	16	⊥i	⊥i	ADJ
ma-88	236	17	ax	ax	NOUN
ma-88	236	18	+	+	CCONJ
ma-88	236	19	y	y	PROPN
ma-88	236	20	,	,	PUNCT
ma-88	236	21	then	then	ADV
ma-88	236	22	xε	xε	PROPN
ma-88	236	23	⊥i	⊥i	PROPN
ma-88	236	24	y	y	PROPN
ma-88	236	25	.	.	PUNCT
ma-88	237	1	proof	proof	NOUN
ma-88	237	2	.	.	PUNCT
ma-88	238	1	since	since	SCONJ
ma-88	238	2	x	x	SYM
ma-88	238	3	⊥i	⊥i	ADJ
ma-88	238	4	ax	ax	NOUN
ma-88	238	5	+	+	CCONJ
ma-88	238	6	y	y	NOUN
ma-88	238	7	we	we	PRON
ma-88	238	8	have	have	VERB
ma-88	238	9	‖x	‖x	NOUN
ma-88	238	10	+	+	CCONJ
ma-88	238	11	ax	ax	NOUN
ma-88	238	12	+	+	CCONJ
ma-88	238	13	y‖	y‖	NOUN
ma-88	238	14	=	=	PUNCT
ma-88	239	1	‖x	‖x	NOUN
ma-88	239	2	−	−	NOUN
ma-88	240	1	ax	ax	NOUN
ma-88	240	2	−	−	PROPN
ma-88	240	3	y‖	y‖	PROPN
ma-88	240	4	,	,	PUNCT
ma-88	240	5	then	then	ADV
ma-88	240	6	|‖x	|‖x	PROPN
ma-88	240	7	+	+	CCONJ
ma-88	240	8	y‖	y‖	PROPN
ma-88	240	9	−	−	NOUN
ma-88	240	10	‖x	‖x	NOUN
ma-88	241	1	−	−	PROPN
ma-88	241	2	y‖|	y‖|	NOUN
ma-88	241	3	=	=	PUNCT
ma-88	241	4	|‖x	|‖x	NOUN
ma-88	241	5	+	+	CCONJ
ma-88	241	6	ax	ax	NOUN
ma-88	241	7	+	+	CCONJ
ma-88	241	8	y	y	PROPN
ma-88	241	9	−	−	PROPN
ma-88	241	10	ax‖	ax‖	PROPN
ma-88	241	11	−	−	PROPN
ma-88	241	12	‖x	‖x	NOUN
ma-88	242	1	−	−	PROPN
ma-88	242	2	ax	ax	NOUN
ma-88	242	3	−	−	PROPN
ma-88	242	4	y	y	PROPN
ma-88	242	5	+	+	CCONJ
ma-88	242	6	ax‖|	ax‖|	PROPN
ma-88	242	7	.	.	PUNCT
ma-88	243	1	on	on	ADP
ma-88	243	2	the	the	DET
ma-88	243	3	other	other	ADJ
ma-88	243	4	hand	hand	NOUN
ma-88	243	5	,	,	PUNCT
ma-88	243	6	by	by	ADP
ma-88	243	7	trigonometric	trigonometric	ADJ
ma-88	243	8	inequality	inequality	NOUN
ma-88	243	9	we	we	PRON
ma-88	243	10	have	have	VERB
ma-88	243	11	:	:	PUNCT
ma-88	243	12	‖x	‖x	X
ma-88	243	13	+	+	CCONJ
ma-88	243	14	ax	ax	NOUN
ma-88	243	15	+	+	CCONJ
ma-88	243	16	y‖	y‖	NOUN
ma-88	244	1	−	−	PROPN
ma-88	244	2	‖ax‖	‖ax‖	ADJ
ma-88	244	3	−	−	PROPN
ma-88	245	1	(	(	PUNCT
ma-88	245	2	‖x	‖x	NOUN
ma-88	245	3	−	−	NOUN
ma-88	245	4	ax	ax	NOUN
ma-88	245	5	−	−	PROPN
ma-88	245	6	y‖+	y‖+	ADJ
ma-88	245	7	‖ax‖	‖ax‖	PROPN
ma-88	245	8	)	)	PUNCT
ma-88	245	9	≤	≤	NUM
ma-88	245	10	‖x	‖x	PUNCT
ma-88	246	1	+	+	CCONJ
ma-88	246	2	ax	ax	NOUN
ma-88	246	3	+	+	CCONJ
ma-88	246	4	y	y	PROPN
ma-88	246	5	−	−	PROPN
ma-88	246	6	ax‖	ax‖	PROPN
ma-88	246	7	−	−	PROPN
ma-88	246	8	‖x	‖x	NOUN
ma-88	247	1	−	−	PROPN
ma-88	247	2	ax	ax	NOUN
ma-88	247	3	−	−	PROPN
ma-88	247	4	y	y	PROPN
ma-88	247	5	+	+	CCONJ
ma-88	247	6	ax‖	ax‖	PROPN
ma-88	247	7	,	,	PUNCT
ma-88	247	8	and	and	CCONJ
ma-88	247	9	‖x	‖x	NOUN
ma-88	247	10	+	+	CCONJ
ma-88	247	11	ax	ax	NOUN
ma-88	248	1	+	+	CCONJ
ma-88	248	2	y	y	PROPN
ma-88	248	3	−	−	PROPN
ma-88	248	4	ax‖	ax‖	PROPN
ma-88	248	5	−	−	PROPN
ma-88	248	6	‖x	‖x	NOUN
ma-88	249	1	−	−	PROPN
ma-88	249	2	ax	ax	NOUN
ma-88	249	3	−	−	PROPN
ma-88	249	4	y	y	PROPN
ma-88	249	5	+	+	CCONJ
ma-88	249	6	ax‖	ax‖	VERB
ma-88	249	7	≤	≤	NUM
ma-88	249	8	‖x	‖x	PUNCT
ma-88	250	1	+	+	CCONJ
ma-88	250	2	ax	ax	NOUN
ma-88	250	3	+	+	CCONJ
ma-88	250	4	y‖+	y‖+	ADJ
ma-88	250	5	‖ax‖	‖ax‖	ADJ
ma-88	250	6	−	−	PROPN
ma-88	251	1	(	(	PUNCT
ma-88	251	2	‖x	‖x	NOUN
ma-88	251	3	−	−	NOUN
ma-88	251	4	ax	ax	NOUN
ma-88	251	5	−	−	PROPN
ma-88	251	6	y‖	y‖	PROPN
ma-88	251	7	−	−	PROPN
ma-88	251	8	‖ax‖	‖ax‖	ADJ
ma-88	251	9	)	)	PUNCT
ma-88	251	10	.	.	PUNCT
ma-88	252	1	thus	thus	ADV
ma-88	252	2	|‖x	|‖x	X
ma-88	252	3	+	+	CCONJ
ma-88	252	4	ax	ax	NOUN
ma-88	252	5	+	+	CCONJ
ma-88	252	6	y	y	PROPN
ma-88	252	7	−	−	PROPN
ma-88	252	8	ax‖	ax‖	PROPN
ma-88	252	9	−	−	PROPN
ma-88	252	10	‖x	‖x	NOUN
ma-88	253	1	−	−	PROPN
ma-88	253	2	ax	ax	NOUN
ma-88	253	3	−	−	PROPN
ma-88	253	4	y	y	PROPN
ma-88	253	5	+	+	CCONJ
ma-88	253	6	ax‖|	ax‖|	PROPN
ma-88	253	7	≤	≤	NOUN
ma-88	254	1	2‖ax‖	2‖ax‖	NUM
ma-88	254	2	,	,	PUNCT
ma-88	254	3	then	then	ADV
ma-88	254	4	|‖x	|‖x	PROPN
ma-88	254	5	+	+	CCONJ
ma-88	254	6	y‖	y‖	PROPN
ma-88	255	1	−	−	NOUN
ma-88	255	2	‖x	‖x	NOUN
ma-88	256	1	−	−	PROPN
ma-88	256	2	y‖|	y‖|	NOUN
ma-88	256	3	≤	≤	NOUN
ma-88	256	4	2‖ax‖	2‖ax‖	NUM
ma-88	256	5	≤	≤	NOUN
ma-88	256	6	ε(‖x	ε(‖x	PUNCT
ma-88	257	1	+	+	CCONJ
ma-88	257	2	y‖+	y‖+	ADJ
ma-88	257	3	‖x	‖x	NOUN
ma-88	257	4	−	−	PROPN
ma-88	257	5	y‖	y‖	PROPN
ma-88	257	6	)	)	PUNCT
ma-88	257	7	,	,	PUNCT
ma-88	257	8	thus	thus	ADV
ma-88	257	9	xε	xε	PUNCT
ma-88	257	10	⊥i	⊥i	PROPN
ma-88	257	11	y	y	PROPN
ma-88	257	12	.	.	PUNCT
ma-88	258	1	proposition	proposition	NOUN
ma-88	258	2	3.2	3.2	NUM
ma-88	258	3	.	.	PUNCT
ma-88	259	1	let	let	VERB
ma-88	259	2	x	x	PRON
ma-88	259	3	be	be	AUX
ma-88	259	4	normed	normed	ADJ
ma-88	259	5	spaces	space	NOUN
ma-88	259	6	,	,	PUNCT
ma-88	259	7	if	if	SCONJ
ma-88	259	8	for	for	ADP
ma-88	259	9	every	every	DET
ma-88	259	10	‖x‖	‖x‖	PROPN
ma-88	259	11	=	=	SYM
ma-88	259	12	‖y‖	‖y‖	PROPN
ma-88	259	13	=	=	SYM
ma-88	259	14	1	1	NUM
ma-88	259	15	,	,	PUNCT
ma-88	259	16	there	there	PRON
ma-88	259	17	is	be	VERB
ma-88	259	18	no	no	DET
ma-88	259	19	0	0	NUM
ma-88	259	20	≤	≤	NUM
ma-88	259	21	ε	ε	X
ma-88	259	22	<	<	X
ma-88	259	23	1	1	NUM
ma-88	259	24	such	such	ADJ
ma-88	259	25	that	that	SCONJ
ma-88	259	26	x	x	PROPN
ma-88	259	27	⊥εi	⊥εi	ADP
ma-88	259	28	y	y	PROPN
ma-88	259	29	,	,	PUNCT
ma-88	259	30	then	then	ADV
ma-88	259	31	x	x	PUNCT
ma-88	259	32	is	be	AUX
ma-88	259	33	a	a	DET
ma-88	259	34	strictly	strictly	ADV
ma-88	259	35	convex	convex	ADJ
ma-88	259	36	space	space	NOUN
ma-88	259	37	.	.	PUNCT
ma-88	260	1	proof	proof	NOUN
ma-88	260	2	.	.	PUNCT
ma-88	261	1	for	for	ADP
ma-88	261	2	any	any	DET
ma-88	261	3	‖x‖	‖x‖	PROPN
ma-88	261	4	=	=	SYM
ma-88	261	5	‖y‖	‖y‖	PROPN
ma-88	261	6	=	=	PUNCT
ma-88	261	7	‖x+y‖	‖x+y‖	ADV
ma-88	261	8	2	2	NUM
ma-88	261	9	=	=	SYM
ma-88	261	10	1	1	NUM
ma-88	261	11	,	,	PUNCT
ma-88	261	12	if	if	SCONJ
ma-88	261	13	x	x	PROPN
ma-88	261	14	6=	6=	NUM
ma-88	261	15	y	y	PROPN
ma-88	261	16	,	,	PUNCT
ma-88	261	17	we	we	PRON
ma-88	261	18	have	have	VERB
ma-88	261	19	|‖x	|‖x	NOUN
ma-88	261	20	+	+	NOUN
ma-88	261	21	y‖2	y‖2	X
ma-88	261	22	−	−	NOUN
ma-88	261	23	‖x	‖x	NOUN
ma-88	261	24	−	−	PROPN
ma-88	261	25	y‖2|	y‖2|	NOUN
ma-88	261	26	=	=	NOUN
ma-88	261	27	|4−	|4−	VERB
ma-88	261	28	‖x	‖x	PUNCT
ma-88	262	1	−	−	PROPN
ma-88	262	2	y‖|	y‖|	NOUN
ma-88	262	3	<	<	X
ma-88	262	4	4	4	NUM
ma-88	262	5	,	,	PUNCT
ma-88	262	6	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	262	7	eur	eur	PROPN
ma-88	262	8	.	.	PUNCT
ma-88	263	1	j.	j.	PROPN
ma-88	263	2	math	math	PROPN
ma-88	263	3	.	.	PUNCT
ma-88	264	1	anal	anal	PROPN
ma-88	264	2	.	.	PUNCT
ma-88	265	1	10.28924	10.28924	NUM
ma-88	265	2	/	/	SYM
ma-88	265	3	ada	ada	PROPN
ma-88	265	4	/	/	SYM
ma-88	265	5	ma.2.16	ma.2.16	PROPN
ma-88	265	6	9	9	NUM
ma-88	265	7	thus	thus	ADV
ma-88	265	8	there	there	PRON
ma-88	265	9	must	must	AUX
ma-88	265	10	exist	exist	VERB
ma-88	265	11	a	a	DET
ma-88	265	12	0	0	NUM
ma-88	265	13	≤	≤	NUM
ma-88	265	14	ε	ε	X
ma-88	265	15	<	<	X
ma-88	265	16	1	1	NUM
ma-88	265	17	such	such	ADJ
ma-88	265	18	that	that	SCONJ
ma-88	265	19	|4	|4	NUM
ma-88	266	1	−	−	PROPN
ma-88	266	2	‖x	‖x	NOUN
ma-88	267	1	−	−	PROPN
ma-88	267	2	y‖2|	y‖2|	NOUN
ma-88	267	3	≤	≤	NOUN
ma-88	267	4	4ε	4ε	NOUN
ma-88	267	5	which	which	PRON
ma-88	267	6	means	mean	VERB
ma-88	267	7	that	that	SCONJ
ma-88	267	8	x	x	PROPN
ma-88	267	9	⊥εi	⊥εi	ADP
ma-88	267	10	y	y	PROPN
ma-88	267	11	,	,	PUNCT
ma-88	267	12	contradict	contradict	VERB
ma-88	267	13	to	to	ADP
ma-88	267	14	the	the	DET
ma-88	267	15	condition	condition	NOUN
ma-88	267	16	.	.	PUNCT
ma-88	268	1	so	so	ADV
ma-88	268	2	there	there	PRON
ma-88	268	3	must	must	AUX
ma-88	268	4	be	be	AUX
ma-88	268	5	x	x	X
ma-88	268	6	=	=	PUNCT
ma-88	268	7	y	y	PROPN
ma-88	268	8	.	.	PUNCT
ma-88	269	1	from	from	ADP
ma-88	269	2	the	the	DET
ma-88	269	3	equivalent	equivalent	ADJ
ma-88	269	4	characterization	characterization	NOUN
ma-88	269	5	of	of	ADP
ma-88	269	6	strictly	strictly	ADV
ma-88	269	7	convex	convex	ADJ
ma-88	269	8	space	space	NOUN
ma-88	269	9	[	[	X
ma-88	269	10	23	23	NUM
ma-88	269	11	]	]	PUNCT
ma-88	269	12	.	.	PUNCT
ma-88	270	1	we	we	PRON
ma-88	270	2	get	get	VERB
ma-88	270	3	the	the	DET
ma-88	270	4	result	result	NOUN
ma-88	270	5	that	that	SCONJ
ma-88	270	6	x	x	PRON
ma-88	270	7	must	must	AUX
ma-88	270	8	be	be	AUX
ma-88	270	9	a	a	DET
ma-88	270	10	strictly	strictly	ADV
ma-88	270	11	convex	convex	ADJ
ma-88	270	12	space	space	NOUN
ma-88	270	13	.	.	PUNCT
ma-88	271	1	it	it	PRON
ma-88	271	2	is	be	AUX
ma-88	271	3	known	know	VERB
ma-88	271	4	that	that	SCONJ
ma-88	271	5	in	in	ADP
ma-88	271	6	inner	inner	ADJ
ma-88	271	7	product	product	NOUN
ma-88	271	8	spaces	space	NOUN
ma-88	271	9	,	,	PUNCT
ma-88	271	10	different	different	ADJ
ma-88	271	11	orthogonality	orthogonality	NOUN
ma-88	271	12	types	type	NOUN
ma-88	271	13	such	such	ADJ
ma-88	271	14	as	as	ADP
ma-88	271	15	isosceles	isoscele	NOUN
ma-88	271	16	,	,	PUNCT
ma-88	271	17	pythagorean	pythagorean	NOUN
ma-88	271	18	,	,	PUNCT
ma-88	271	19	and	and	CCONJ
ma-88	271	20	birkhoff	birkhoff	NOUN
ma-88	271	21	orthogonality	orthogonality	NOUN
ma-88	271	22	is	be	AUX
ma-88	271	23	equivalent	equivalent	ADJ
ma-88	272	1	[	[	X
ma-88	272	2	3	3	NUM
ma-88	272	3	]	]	PUNCT
ma-88	272	4	.	.	PUNCT
ma-88	273	1	using	use	VERB
ma-88	273	2	the	the	DET
ma-88	273	3	notions	notion	NOUN
ma-88	273	4	of	of	ADP
ma-88	273	5	orthogonality	orthogonality	NOUN
ma-88	273	6	innormed	innorme	VERB
ma-88	273	7	linear	linear	PROPN
ma-88	273	8	spaces	space	NOUN
ma-88	273	9	it	it	PRON
ma-88	273	10	is	be	AUX
ma-88	273	11	possible	possible	ADJ
ma-88	273	12	to	to	PART
ma-88	273	13	give	give	VERB
ma-88	273	14	different	different	ADJ
ma-88	273	15	characterizations	characterization	NOUN
ma-88	273	16	for	for	ADP
ma-88	273	17	inner	inner	ADJ
ma-88	273	18	product	product	NOUN
ma-88	273	19	spaces	space	NOUN
ma-88	273	20	.	.	PUNCT
ma-88	274	1	forinstance	forinstance	NOUN
ma-88	275	1	[	[	X
ma-88	275	2	17	17	NUM
ma-88	275	3	]	]	PUNCT
ma-88	275	4	,	,	PUNCT
ma-88	275	5	if	if	SCONJ
ma-88	275	6	x	x	X
ma-88	275	7	⊥i	⊥i	PROPN
ma-88	275	8	y	y	PROPN
ma-88	275	9	=	=	PROPN
ma-88	275	10	⇒	⇒	PROPN
ma-88	275	11	x	x	PUNCT
ma-88	275	12	⊥b	⊥b	PRON
ma-88	275	13	y	y	PROPN
ma-88	275	14	in	in	ADP
ma-88	275	15	a	a	DET
ma-88	275	16	normed	normed	ADJ
ma-88	275	17	space	space	NOUN
ma-88	275	18	x	x	X
ma-88	275	19	,	,	PUNCT
ma-88	275	20	then	then	ADV
ma-88	275	21	x	x	PRON
ma-88	275	22	must	must	AUX
ma-88	275	23	be	be	AUX
ma-88	275	24	inner	inner	ADJ
ma-88	275	25	product	product	NOUN
ma-88	275	26	spaces.inspired	spaces.inspire	VERB
ma-88	275	27	by	by	ADP
ma-88	275	28	this	this	PRON
ma-88	275	29	,	,	PUNCT
ma-88	275	30	now	now	ADV
ma-88	275	31	we	we	PRON
ma-88	275	32	give	give	VERB
ma-88	275	33	a	a	DET
ma-88	275	34	characterization	characterization	NOUN
ma-88	275	35	for	for	ADP
ma-88	275	36	inner	inner	ADJ
ma-88	275	37	product	product	NOUN
ma-88	275	38	spaces	space	NOUN
ma-88	275	39	using	use	VERB
ma-88	275	40	approximate	approximate	ADJ
ma-88	275	41	or	or	CCONJ
ma-88	275	42	-	-	PUNCT
ma-88	275	43	thogonality	thogonality	NOUN
ma-88	275	44	.	.	PUNCT
ma-88	276	1	theorem	theorem	VERB
ma-88	276	2	3.3	3.3	NUM
ma-88	276	3	.	.	PUNCT
ma-88	277	1	let	let	VERB
ma-88	277	2	x	x	PRON
ma-88	277	3	be	be	AUX
ma-88	277	4	normed	normed	ADJ
ma-88	277	5	spaces	space	NOUN
ma-88	277	6	,	,	PUNCT
ma-88	277	7	then	then	ADV
ma-88	277	8	x	x	PUNCT
ma-88	277	9	is	be	AUX
ma-88	277	10	inner	inner	ADJ
ma-88	277	11	product	product	NOUN
ma-88	277	12	spaces	space	VERB
ma-88	277	13	iff	iff	VERB
ma-88	277	14	the	the	DET
ma-88	277	15	following	follow	VERB
ma-88	277	16	two	two	NUM
ma-88	277	17	conditions	condition	NOUN
ma-88	277	18	are	be	AUX
ma-88	277	19	satisfied	satisfied	ADJ
ma-88	277	20	.	.	PUNCT
ma-88	278	1	(	(	PUNCT
ma-88	278	2	1	1	X
ma-88	278	3	)	)	PUNCT
ma-88	278	4	if	if	SCONJ
ma-88	278	5	there	there	PRON
ma-88	278	6	exists	exist	VERB
ma-88	278	7	|a|	|a|	PROPN
ma-88	278	8	≤	≤	ADJ
ma-88	278	9	‖y‖‖x‖ε	‖y‖‖x‖ε	NOUN
ma-88	278	10	such	such	ADJ
ma-88	278	11	that	that	SCONJ
ma-88	278	12	x	x	SYM
ma-88	278	13	⊥i	⊥i	ADJ
ma-88	278	14	ax	ax	NOUN
ma-88	278	15	+	+	CCONJ
ma-88	278	16	y	y	PROPN
ma-88	278	17	,	,	PUNCT
ma-88	278	18	then	then	ADV
ma-88	278	19	x	x	PUNCT
ma-88	278	20	⊥εi	⊥εi	PROPN
ma-88	278	21	y	y	PROPN
ma-88	278	22	.	.	PUNCT
ma-88	279	1	(	(	PUNCT
ma-88	279	2	2	2	X
ma-88	279	3	)	)	PUNCT
ma-88	279	4	x	x	PRON
ma-88	279	5	⊥εi	⊥εi	PROPN
ma-88	279	6	y	y	PROPN
ma-88	279	7	impliesx	impliesx	PROPN
ma-88	279	8	⊥bε	⊥bε	NOUN
ma-88	279	9	y	y	PROPN
ma-88	279	10	.	.	PUNCT
ma-88	280	1	proof	proof	NOUN
ma-88	280	2	.	.	PUNCT
ma-88	281	1	if	if	SCONJ
ma-88	281	2	x	x	PRON
ma-88	281	3	is	be	AUX
ma-88	281	4	an	an	DET
ma-88	281	5	inner	inner	ADJ
ma-88	281	6	product	product	NOUN
ma-88	281	7	space	space	NOUN
ma-88	281	8	,	,	PUNCT
ma-88	281	9	we	we	PRON
ma-88	281	10	have	have	VERB
ma-88	281	11	x	x	PROPN
ma-88	281	12	⊥i	⊥i	PROPN
ma-88	281	13	y	y	PROPN
ma-88	281	14	⇐	⇐	PROPN
ma-88	281	15	⇒	⇒	PROPN
ma-88	281	16	x	x	PUNCT
ma-88	281	17	⊥b	⊥b	PRON
ma-88	281	18	y	y	PROPN
ma-88	281	19	and	and	CCONJ
ma-88	281	20	x	x	PROPN
ma-88	281	21	⊥εi	⊥εi	PROPN
ma-88	281	22	y	y	PROPN
ma-88	281	23	⇐	⇐	PROPN
ma-88	281	24	⇒	⇒	PROPN
ma-88	281	25	x	x	PROPN
ma-88	281	26	⊥bε	⊥bε	NOUN
ma-88	281	27	y	y	PROPN
ma-88	281	28	.	.	PUNCT
ma-88	282	1	thus	thus	ADV
ma-88	282	2	(	(	PUNCT
ma-88	282	3	2	2	X
ma-88	282	4	)	)	PUNCT
ma-88	282	5	is	be	AUX
ma-88	282	6	satisfied	satisfied	ADJ
ma-88	282	7	.	.	PUNCT
ma-88	283	1	if	if	SCONJ
ma-88	283	2	there	there	PRON
ma-88	283	3	exists	exist	VERB
ma-88	283	4	|a|	|a|	PROPN
ma-88	283	5	≤	≤	ADJ
ma-88	283	6	‖y‖‖x‖ε	‖y‖‖x‖ε	NOUN
ma-88	283	7	such	such	ADJ
ma-88	283	8	that	that	SCONJ
ma-88	283	9	x	x	SYM
ma-88	283	10	⊥i	⊥i	ADJ
ma-88	283	11	ax	ax	NOUN
ma-88	283	12	+	+	CCONJ
ma-88	283	13	y	y	PROPN
ma-88	283	14	,	,	PUNCT
ma-88	283	15	we	we	PRON
ma-88	283	16	have	have	VERB
ma-88	283	17	:	:	PUNCT
ma-88	283	18	x	x	X
ma-88	283	19	⊥b	⊥b	PRON
ma-88	283	20	ax	ax	NOUN
ma-88	283	21	+	+	CCONJ
ma-88	283	22	y	y	PROPN
ma-88	283	23	,	,	PUNCT
ma-88	283	24	|a|	|a|	PROPN
ma-88	283	25	≤	≤	ADJ
ma-88	283	26	‖y‖	‖y‖	PROPN
ma-88	283	27	‖x‖ε	‖x‖ε	NOUN
ma-88	283	28	.	.	PUNCT
ma-88	284	1	thus	thus	ADV
ma-88	284	2	x	x	SYM
ma-88	284	3	⊥bε	⊥bε	X
ma-88	284	4	y	y	X
ma-88	284	5	which	which	PRON
ma-88	284	6	implies	imply	VERB
ma-88	284	7	x	x	PROPN
ma-88	284	8	⊥εi	⊥εi	PROPN
ma-88	284	9	y	y	PROPN
ma-88	284	10	.	.	PUNCT
ma-88	285	1	thus	thus	ADV
ma-88	285	2	both	both	DET
ma-88	285	3	(	(	PUNCT
ma-88	285	4	1	1	NUM
ma-88	285	5	)	)	PUNCT
ma-88	285	6	and	and	CCONJ
ma-88	285	7	(	(	PUNCT
ma-88	285	8	2	2	X
ma-88	285	9	)	)	PUNCT
ma-88	285	10	are	be	AUX
ma-88	285	11	satisfied	satisfied	ADJ
ma-88	285	12	.	.	PUNCT
ma-88	286	1	conversely	conversely	ADV
ma-88	286	2	,	,	PUNCT
ma-88	286	3	assume	assume	VERB
ma-88	286	4	that	that	SCONJ
ma-88	286	5	both	both	DET
ma-88	286	6	(	(	PUNCT
ma-88	286	7	1	1	NUM
ma-88	286	8	)	)	PUNCT
ma-88	286	9	and	and	CCONJ
ma-88	286	10	(	(	PUNCT
ma-88	286	11	2	2	X
ma-88	286	12	)	)	PUNCT
ma-88	286	13	are	be	AUX
ma-88	286	14	satisfied	satisfied	ADJ
ma-88	286	15	,	,	PUNCT
ma-88	286	16	let	let	VERB
ma-88	286	17	x	x	SYM
ma-88	286	18	⊥i	⊥i	PROPN
ma-88	286	19	y	y	PROPN
ma-88	286	20	,	,	PUNCT
ma-88	286	21	x	x	PROPN
ma-88	286	22	6=	6=	ADP
ma-88	286	23	0	0	NUM
ma-88	286	24	.	.	PUNCT
ma-88	287	1	if	if	SCONJ
ma-88	287	2	|a|	|a|	NOUN
ma-88	287	3	≤	≤	X
ma-88	287	4	ε	ε	VERB
ma-88	287	5	‖ax+y‖	‖ax+y‖	ADV
ma-88	287	6	‖x‖	‖x‖	PROPN
ma-88	287	7	,	,	PUNCT
ma-88	287	8	let	let	VERB
ma-88	287	9	b	b	NOUN
ma-88	287	10	=	=	SYM
ma-88	287	11	−a	−a	NOUN
ma-88	287	12	,	,	PUNCT
ma-88	287	13	then	then	ADV
ma-88	287	14	|b|	|b|	PROPN
ma-88	287	15	≤	≤	ADJ
ma-88	287	16	ε‖ax+y‖‖x‖	ε‖ax+y‖‖x‖	NOUN
ma-88	287	17	and	and	CCONJ
ma-88	287	18	x	x	ADP
ma-88	287	19	⊥i	⊥i	PROPN
ma-88	287	20	bx	bx	NOUN
ma-88	287	21	+	+	CCONJ
ma-88	287	22	ax	ax	NOUN
ma-88	287	23	+	+	CCONJ
ma-88	287	24	y	y	PROPN
ma-88	287	25	,	,	PUNCT
ma-88	287	26	thus	thus	ADV
ma-88	287	27	form	form	NOUN
ma-88	287	28	(	(	PUNCT
ma-88	287	29	1	1	X
ma-88	287	30	)	)	PUNCT
ma-88	287	31	we	we	PRON
ma-88	287	32	have	have	AUX
ma-88	287	33	x	x	NOUN
ma-88	287	34	⊥εi	⊥εi	ADJ
ma-88	287	35	ax	ax	NOUN
ma-88	287	36	+	+	CCONJ
ma-88	287	37	y	y	PROPN
ma-88	287	38	.	.	PUNCT
ma-88	288	1	to	to	PART
ma-88	288	2	conclude	conclude	VERB
ma-88	288	3	we	we	PRON
ma-88	288	4	have	have	VERB
ma-88	288	5	:	:	PUNCT
ma-88	288	6	x	x	NOUN
ma-88	288	7	⊥εi	⊥εi	ADV
ma-88	288	8	ax	ax	NOUN
ma-88	288	9	+	+	CCONJ
ma-88	289	1	y	y	PROPN
ma-88	290	1	i	i	PRON
ma-88	290	2	f	f	PROPN
ma-88	290	3	|a|	|a|	NOUN
ma-88	290	4	≤	≤	PUNCT
ma-88	290	5	ε	ε	PROPN
ma-88	290	6	‖ax	‖ax	CCONJ
ma-88	290	7	+	+	NUM
ma-88	290	8	y‖	y‖	PROPN
ma-88	290	9	‖x‖	‖x‖	PROPN
ma-88	290	10	.	.	PUNCT
ma-88	291	1	now	now	ADV
ma-88	291	2	define	define	VERB
ma-88	291	3	:	:	PUNCT
ma-88	291	4	f	f	PROPN
ma-88	291	5	(	(	PUNCT
ma-88	291	6	t	t	PROPN
ma-88	291	7	)	)	PUNCT
ma-88	291	8	=	=	PUNCT
ma-88	291	9	‖tx+y‖	‖tx+y‖	PROPN
ma-88	291	10	‖x‖	‖x‖	PROPN
ma-88	291	11	ε	ε	PROPN
ma-88	291	12	,	,	PUNCT
ma-88	291	13	we	we	PRON
ma-88	291	14	have	have	VERB
ma-88	291	15	f	f	PROPN
ma-88	291	16	(	(	PUNCT
ma-88	291	17	t	t	PROPN
ma-88	291	18	)	)	PUNCT
ma-88	291	19	=	=	PUNCT
ma-88	292	1	‖tx	‖tx	PUNCT
ma-88	292	2	+	+	NUM
ma-88	292	3	y‖	y‖	PROPN
ma-88	292	4	‖x‖	‖x‖	PROPN
ma-88	292	5	ε	ε	PROPN
ma-88	292	6	≤	≤	NUM
ma-88	292	7	‖tx‖+	‖tx‖+	ADV
ma-88	292	8	‖y‖	‖y‖	PROPN
ma-88	292	9	‖x‖	‖x‖	PROPN
ma-88	292	10	ε	ε	PROPN
ma-88	292	11	=	=	SYM
ma-88	292	12	ε|t|+	ε|t|+	PROPN
ma-88	292	13	‖y‖	‖y‖	PROPN
ma-88	292	14	‖x‖ε	‖x‖ε	NOUN
ma-88	292	15	.	.	PUNCT
ma-88	293	1	since	since	SCONJ
ma-88	293	2	0	0	NUM
ma-88	293	3	≤	≤	NUM
ma-88	293	4	ε	ε	X
ma-88	293	5	<	<	X
ma-88	293	6	1	1	NUM
ma-88	293	7	,	,	PUNCT
ma-88	293	8	when	when	SCONJ
ma-88	293	9	|t|	|t|	NOUN
ma-88	293	10	tends	tend	VERB
ma-88	293	11	to	to	PART
ma-88	293	12	infinite	infinite	VERB
ma-88	293	13	,	,	PUNCT
ma-88	293	14	f	f	PROPN
ma-88	293	15	(	(	PUNCT
ma-88	293	16	t	t	PROPN
ma-88	293	17	)	)	PUNCT
ma-88	293	18	<	<	X
ma-88	293	19	|t|	|t|	PROPN
ma-88	293	20	.	.	PUNCT
ma-88	294	1	when	when	SCONJ
ma-88	294	2	t	t	PROPN
ma-88	294	3	=	=	SYM
ma-88	294	4	0	0	PROPN
ma-88	294	5	,	,	PUNCT
ma-88	294	6	f	f	X
ma-88	294	7	(	(	PUNCT
ma-88	294	8	0	0	NUM
ma-88	294	9	)	)	PUNCT
ma-88	294	10	=	=	SYM
ma-88	295	1	‖y‖	‖y‖	PROPN
ma-88	295	2	‖x‖ε	‖x‖ε	VERB
ma-88	295	3	>	>	X
ma-88	295	4	0	0	X
ma-88	295	5	.	.	PUNCT
ma-88	296	1	by	by	ADP
ma-88	296	2	the	the	DET
ma-88	296	3	convexity	convexity	NOUN
ma-88	296	4	of	of	ADP
ma-88	296	5	f	f	PROPN
ma-88	296	6	(	(	PUNCT
ma-88	296	7	t	t	PROPN
ma-88	296	8	)	)	PUNCT
ma-88	296	9	we	we	PRON
ma-88	296	10	have	have	VERB
ma-88	296	11	x	x	NOUN
ma-88	296	12	⊥εi	⊥εi	ADV
ma-88	296	13	t1x	t1x	ADJ
ma-88	296	14	+	+	PROPN
ma-88	296	15	y	y	PROPN
ma-88	296	16	,	,	PUNCT
ma-88	296	17	t1	t1	NOUN
ma-88	296	18	<	<	X
ma-88	296	19	0	0	NUM
ma-88	296	20	,	,	PUNCT
ma-88	296	21	‖t1x	‖t1x	NOUN
ma-88	296	22	+	+	CCONJ
ma-88	296	23	y‖	y‖	NOUN
ma-88	296	24	‖x‖	‖x‖	PROPN
ma-88	296	25	ε	ε	PROPN
ma-88	296	26	=	=	SYM
ma-88	296	27	−t1	−t1	PROPN
ma-88	296	28	.	.	PUNCT
ma-88	297	1	x	x	X
ma-88	297	2	⊥εi	⊥εi	ADV
ma-88	297	3	t2x	t2x	ADJ
ma-88	297	4	+	+	CCONJ
ma-88	297	5	y	y	PROPN
ma-88	297	6	,	,	PUNCT
ma-88	297	7	t2	t2	PROPN
ma-88	297	8	>	>	X
ma-88	297	9	0	0	NUM
ma-88	297	10	,	,	PUNCT
ma-88	297	11	‖t2x	‖t2x	NOUN
ma-88	297	12	+	+	CCONJ
ma-88	297	13	y‖	y‖	PROPN
ma-88	297	14	‖x‖	‖x‖	PROPN
ma-88	297	15	ε	ε	PROPN
ma-88	297	16	=	=	SYM
ma-88	297	17	t2	t2	PROPN
ma-88	297	18	.	.	PUNCT
ma-88	298	1	from	from	ADP
ma-88	298	2	(	(	PUNCT
ma-88	298	3	2	2	X
ma-88	298	4	)	)	PUNCT
ma-88	298	5	we	we	PRON
ma-88	298	6	have	have	VERB
ma-88	298	7	x	x	NOUN
ma-88	298	8	⊥bε	⊥bε	NOUN
ma-88	298	9	t1x	t1x	PROPN
ma-88	298	10	+	+	PROPN
ma-88	298	11	y	y	PROPN
ma-88	298	12	and	and	CCONJ
ma-88	298	13	x	x	NOUN
ma-88	298	14	⊥bε	⊥bε	NOUN
ma-88	298	15	t2x	t2x	PROPN
ma-88	299	1	+	+	CCONJ
ma-88	299	2	y	y	PROPN
ma-88	299	3	.	.	PUNCT
ma-88	300	1	from	from	ADP
ma-88	300	2	proposition	proposition	NOUN
ma-88	300	3	2.9,there	2.9,there	NUM
ma-88	300	4	must	must	AUX
ma-88	300	5	exist	exist	VERB
ma-88	300	6	|a1|	|a1|	NOUN
ma-88	300	7	≤	≤	NUM
ma-88	300	8	‖t1x	‖t1x	NOUN
ma-88	300	9	+	+	CCONJ
ma-88	300	10	y‖	y‖	NOUN
ma-88	300	11	‖x‖	‖x‖	PROPN
ma-88	300	12	ε	ε	PROPN
ma-88	300	13	=	=	SYM
ma-88	300	14	−t1	−t1	PROPN
ma-88	300	15	and	and	CCONJ
ma-88	300	16	|a2|	|a2|	VERB
ma-88	300	17	≤	≤	ADJ
ma-88	300	18	‖t2x	‖t2x	NOUN
ma-88	300	19	+	+	CCONJ
ma-88	300	20	y‖	y‖	NOUN
ma-88	300	21	‖x‖	‖x‖	PROPN
ma-88	300	22	ε	ε	PROPN
ma-88	300	23	=	=	SYM
ma-88	300	24	t2	t2	PROPN
ma-88	300	25	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	300	26	eur	eur	PROPN
ma-88	300	27	.	.	PUNCT
ma-88	301	1	j.	j.	PROPN
ma-88	301	2	math	math	PROPN
ma-88	301	3	.	.	PUNCT
ma-88	302	1	anal	anal	PROPN
ma-88	302	2	.	.	PUNCT
ma-88	303	1	10.28924	10.28924	NUM
ma-88	303	2	/	/	SYM
ma-88	303	3	ada	ada	PROPN
ma-88	303	4	/	/	SYM
ma-88	303	5	ma.2.16	ma.2.16	PROPN
ma-88	303	6	10	10	NUM
ma-88	304	1	such	such	ADJ
ma-88	304	2	that	that	SCONJ
ma-88	304	3	x	x	PUNCT
ma-88	304	4	⊥b	⊥b	PRON
ma-88	304	5	a1	a1	NOUN
ma-88	304	6	+	+	CCONJ
ma-88	304	7	t1	t1	NOUN
ma-88	304	8	+	+	X
ma-88	304	9	y	y	PROPN
ma-88	304	10	,	,	PUNCT
ma-88	304	11	x	x	PUNCT
ma-88	304	12	⊥b	⊥b	PRON
ma-88	304	13	a2	a2	NOUN
ma-88	304	14	+	+	CCONJ
ma-88	304	15	t2	t2	PROPN
ma-88	304	16	+	+	CCONJ
ma-88	304	17	y	y	PROPN
ma-88	304	18	.	.	PUNCT
ma-88	305	1	by	by	ADP
ma-88	305	2	a1	a1	NOUN
ma-88	305	3	+	+	CCONJ
ma-88	305	4	t1	t1	NOUN
ma-88	305	5	≤	≤	NUM
ma-88	305	6	0	0	NUM
ma-88	305	7	,	,	PUNCT
ma-88	305	8	a2	a2	PROPN
ma-88	305	9	+	+	CCONJ
ma-88	305	10	t2	t2	PROPN
ma-88	305	11	≥	≥	NOUN
ma-88	305	12	0	0	NUM
ma-88	305	13	and	and	CCONJ
ma-88	305	14	lemma	lemma	PROPN
ma-88	305	15	2.7	2.7	NUM
ma-88	305	16	,	,	PUNCT
ma-88	305	17	we	we	PRON
ma-88	305	18	have	have	VERB
ma-88	305	19	x	x	PUNCT
ma-88	305	20	⊥b	⊥b	PRON
ma-88	305	21	y	y	PROPN
ma-88	305	22	.	.	PUNCT
ma-88	306	1	thus	thus	ADV
ma-88	306	2	we	we	PRON
ma-88	306	3	have	have	VERB
ma-88	306	4	x	x	PROPN
ma-88	306	5	⊥i	⊥i	PROPN
ma-88	306	6	y	y	PROPN
ma-88	306	7	=	=	PROPN
ma-88	306	8	⇒	⇒	PROPN
ma-88	306	9	x	x	PUNCT
ma-88	306	10	⊥b	⊥b	PRON
ma-88	306	11	y	y	PROPN
ma-88	306	12	,	,	PUNCT
ma-88	306	13	which	which	PRON
ma-88	306	14	means	mean	VERB
ma-88	306	15	that	that	SCONJ
ma-88	306	16	x	x	PRON
ma-88	306	17	is	be	AUX
ma-88	306	18	inner	inner	ADJ
ma-88	306	19	product	product	NOUN
ma-88	306	20	spaces	space	NOUN
ma-88	306	21	.	.	PUNCT
ma-88	307	1	example	example	NOUN
ma-88	307	2	3.4	3.4	NUM
ma-88	307	3	.	.	PUNCT
ma-88	308	1	let	let	VERB
ma-88	308	2	x	x	PUNCT
ma-88	308	3	=	=	PRON
ma-88	308	4	(	(	PUNCT
ma-88	308	5	r2	r2	PROPN
ma-88	308	6	,	,	PUNCT
ma-88	308	7	‖	‖	PROPN
ma-88	308	8	·	·	SYM
ma-88	308	9	‖∞	‖∞	PROPN
ma-88	308	10	)	)	PUNCT
ma-88	308	11	,	,	PUNCT
ma-88	308	12	that	that	ADV
ma-88	308	13	is	is	ADV
ma-88	308	14	,	,	PUNCT
ma-88	308	15	‖(x1	‖(x1	PROPN
ma-88	308	16	,	,	PUNCT
ma-88	308	17	x2)‖	x2)‖	PROPN
ma-88	308	18	=	=	SYM
ma-88	308	19	max(|x1|	max(|x1|	NOUN
ma-88	308	20	,	,	PUNCT
ma-88	308	21	|x2|	|x2|	PROPN
ma-88	308	22	)	)	PUNCT
ma-88	308	23	,	,	PUNCT
ma-88	308	24	assume	assume	VERB
ma-88	308	25	that	that	SCONJ
ma-88	308	26	x	x	X
ma-88	308	27	=	=	SYM
ma-88	308	28	(	(	PUNCT
ma-88	308	29	1	1	NUM
ma-88	308	30	,	,	PUNCT
ma-88	308	31	0	0	NUM
ma-88	308	32	)	)	PUNCT
ma-88	308	33	,	,	PUNCT
ma-88	308	34	y	y	PROPN
ma-88	308	35	=	=	SYM
ma-88	308	36	(	(	PUNCT
ma-88	308	37	z	z	NOUN
ma-88	308	38	,	,	PUNCT
ma-88	308	39	1	1	NUM
ma-88	308	40	)	)	PUNCT
ma-88	308	41	,	,	PUNCT
ma-88	308	42	|z	|z	PROPN
ma-88	309	1	|	|	ADV
ma-88	309	2	<	<	X
ma-88	309	3	1	1	X
ma-88	309	4	.	.	PUNCT
ma-88	310	1	in	in	ADP
ma-88	310	2	order	order	NOUN
ma-88	310	3	to	to	PART
ma-88	310	4	satisfy	satisfy	VERB
ma-88	310	5	x	x	SYM
ma-88	310	6	⊥bε	⊥bε	NOUN
ma-88	310	7	y	y	PROPN
ma-88	310	8	or	or	CCONJ
ma-88	310	9	equivalently	equivalently	ADV
ma-88	310	10	‖x	‖x	NOUN
ma-88	311	1	+	+	CCONJ
ma-88	311	2	ty‖	ty‖	PRON
ma-88	311	3	≥	≥	NOUN
ma-88	311	4	‖x‖	‖x‖	VERB
ma-88	311	5	−	−	PROPN
ma-88	311	6	ε‖ty‖	ε‖ty‖	NOUN
ma-88	311	7	,	,	PUNCT
ma-88	311	8	the	the	DET
ma-88	311	9	following	follow	VERB
ma-88	311	10	inequality	inequality	NOUN
ma-88	311	11	shuold	shuold	PRON
ma-88	311	12	hold	hold	VERB
ma-88	311	13	for	for	ADP
ma-88	311	14	al	al	PROPN
ma-88	311	15	l	l	PROPN
ma-88	311	16	t	t	NOUN
ma-88	311	17	∈	∈	PROPN
ma-88	312	1	r	r	NOUN
ma-88	312	2	:	:	PUNCT
ma-88	312	3	‖(1	‖(1	NUM
ma-88	312	4	+	+	NUM
ma-88	312	5	tc	tc	NOUN
ma-88	312	6	,	,	PUNCT
ma-88	312	7	t)‖	t)‖	NOUN
ma-88	312	8	≥	≥	NOUN
ma-88	312	9	1−	1−	NUM
ma-88	312	10	ε|t|	ε|t|	X
ma-88	312	11	.	.	PUNCT
ma-88	313	1	since	since	SCONJ
ma-88	313	2	‖(1	‖(1	NUM
ma-88	313	3	+	+	NUM
ma-88	313	4	tc	tc	NOUN
ma-88	313	5	,	,	PUNCT
ma-88	313	6	t)‖	t)‖	NOUN
ma-88	313	7	≥	≥	NOUN
ma-88	313	8	‖t‖	‖t‖	NOUN
ma-88	313	9	,	,	PUNCT
ma-88	313	10	we	we	PRON
ma-88	313	11	know	know	VERB
ma-88	313	12	the	the	DET
ma-88	313	13	above	above	ADJ
ma-88	313	14	inequality	inequality	NOUN
ma-88	313	15	is	be	AUX
ma-88	313	16	always	always	ADV
ma-88	313	17	correct	correct	ADJ
ma-88	313	18	if	if	SCONJ
ma-88	313	19	|t|	|t|	VERB
ma-88	313	20	≥	≥	NOUN
ma-88	313	21	1	1	NUM
ma-88	313	22	1+ε	1+ε	NUM
ma-88	313	23	,	,	PUNCT
ma-88	313	24	then	then	ADV
ma-88	313	25	we	we	PRON
ma-88	313	26	may	may	AUX
ma-88	313	27	assume	assume	VERB
ma-88	313	28	that	that	SCONJ
ma-88	313	29	|t|	|t|	VERB
ma-88	313	30	<	<	X
ma-88	313	31	1	1	NUM
ma-88	313	32	1+ε	1+ε	NUM
ma-88	313	33	≤	≤	NUM
ma-88	313	34	1	1	NUM
ma-88	313	35	.	.	PUNCT
ma-88	314	1	when	when	SCONJ
ma-88	314	2	1	1	X
ma-88	314	3	>	>	X
ma-88	314	4	t	t	PROPN
ma-88	314	5	≥	≥	PROPN
ma-88	314	6	0	0	NUM
ma-88	314	7	,	,	PUNCT
ma-88	314	8	if	if	SCONJ
ma-88	314	9	1	1	NUM
ma-88	314	10	+	+	NUM
ma-88	314	11	tc	tc	NUM
ma-88	314	12	≥	≥	NUM
ma-88	314	13	t	t	NUM
ma-88	314	14	which	which	PRON
ma-88	314	15	means	mean	VERB
ma-88	314	16	that	that	SCONJ
ma-88	314	17	t	t	VERB
ma-88	314	18	≤	≤	NUM
ma-88	314	19	1	1	NUM
ma-88	314	20	1−c	1−c	NUM
ma-88	314	21	,	,	PUNCT
ma-88	314	22	we	we	PRON
ma-88	314	23	have	have	VERB
ma-88	314	24	1	1	NUM
ma-88	314	25	+	+	NUM
ma-88	314	26	tc	tc	NUM
ma-88	314	27	≥	≥	NOUN
ma-88	314	28	1−	1−	NUM
ma-88	314	29	εt	εt	NUM
ma-88	314	30	which	which	PRON
ma-88	314	31	implies	imply	VERB
ma-88	314	32	that	that	SCONJ
ma-88	314	33	c	c	PROPN
ma-88	314	34	≥	≥	PROPN
ma-88	314	35	−ε	−ε	NOUN
ma-88	314	36	.	.	PUNCT
ma-88	315	1	if	if	SCONJ
ma-88	315	2	1	1	NUM
ma-88	315	3	+	+	NUM
ma-88	315	4	tc	tc	NOUN
ma-88	315	5	<	<	X
ma-88	315	6	t	t	NOUN
ma-88	315	7	or	or	CCONJ
ma-88	315	8	equivalently	equivalently	ADV
ma-88	315	9	t	t	X
ma-88	315	10	>	>	X
ma-88	315	11	1	1	NUM
ma-88	315	12	1−c	1−c	NUM
ma-88	315	13	,	,	PUNCT
ma-88	315	14	we	we	PRON
ma-88	315	15	have	have	VERB
ma-88	315	16	t	t	PROPN
ma-88	315	17	≥	≥	NUM
ma-88	315	18	1	1	NUM
ma-88	315	19	−	−	PROPN
ma-88	316	1	εt	εt	INTJ
ma-88	316	2	that	that	PRON
ma-88	316	3	is	be	AUX
ma-88	316	4	t	t	PROPN
ma-88	316	5	≥	≥	NUM
ma-88	316	6	1	1	NUM
ma-88	316	7	1+ε	1+ε	NUM
ma-88	316	8	.	.	PUNCT
ma-88	317	1	so	so	ADV
ma-88	317	2	there	there	PRON
ma-88	317	3	must	must	AUX
ma-88	317	4	be	be	AUX
ma-88	317	5	1	1	NUM
ma-88	317	6	1−	1−	NUM
ma-88	317	7	c	c	X
ma-88	317	8	≥	≥	NUM
ma-88	317	9	1	1	NUM
ma-88	317	10	1	1	NUM
ma-88	317	11	+	+	NUM
ma-88	317	12	ε	ε	PROPN
ma-88	317	13	which	which	PRON
ma-88	317	14	implies	imply	VERB
ma-88	317	15	that	that	SCONJ
ma-88	317	16	c	c	PROPN
ma-88	317	17	≥	≥	PROPN
ma-88	317	18	−ε	−ε	PROPN
ma-88	317	19	.	.	PUNCT
ma-88	318	1	similarly	similarly	ADV
ma-88	318	2	when	when	SCONJ
ma-88	318	3	−1	−1	NOUN
ma-88	318	4	<	<	X
ma-88	318	5	t	t	X
ma-88	318	6	≤	≤	NUM
ma-88	318	7	0	0	NUM
ma-88	318	8	,	,	PUNCT
ma-88	318	9	we	we	PRON
ma-88	318	10	can	can	AUX
ma-88	318	11	get	get	VERB
ma-88	318	12	c	c	NOUN
ma-88	318	13	≤	≤	X
ma-88	318	14	ε	ε	PROPN
ma-88	318	15	.	.	PUNCT
ma-88	319	1	to	to	PART
ma-88	319	2	conclude	conclude	VERB
ma-88	319	3	we	we	PRON
ma-88	319	4	have	have	VERB
ma-88	319	5	:	:	PUNCT
ma-88	319	6	(	(	PUNCT
ma-88	319	7	1	1	NUM
ma-88	319	8	,	,	PUNCT
ma-88	319	9	0	0	NUM
ma-88	319	10	)	)	PUNCT
ma-88	319	11	⊥bε	⊥bε	NOUN
ma-88	320	1	(	(	PUNCT
ma-88	320	2	c	c	X
ma-88	320	3	,	,	PUNCT
ma-88	320	4	1	1	NUM
ma-88	320	5	)	)	PUNCT
ma-88	320	6	if	if	SCONJ
ma-88	320	7	|c	|c	VERB
ma-88	320	8	|	|	ADV
ma-88	320	9	≤	≤	X
ma-88	320	10	ε	ε	PROPN
ma-88	320	11	.	.	PUNCT
ma-88	321	1	in	in	ADP
ma-88	321	2	order	order	NOUN
ma-88	321	3	to	to	PART
ma-88	321	4	satisfy	satisfy	VERB
ma-88	321	5	x	x	PUNCT
ma-88	321	6	⊥iε	⊥iε	PROPN
ma-88	321	7	y	y	PROPN
ma-88	321	8	or	or	CCONJ
ma-88	321	9	equivalently	equivalently	ADV
ma-88	321	10	|‖x+y‖2−‖x−y‖2|	|‖x+y‖2−‖x−y‖2|	VERB
ma-88	321	11	≤	≤	ADJ
ma-88	321	12	4ε‖x‖‖y‖	4ε‖x‖‖y‖	NUM
ma-88	321	13	,	,	PUNCT
ma-88	321	14	the	the	DET
ma-88	321	15	following	follow	VERB
ma-88	321	16	inequality	inequality	NOUN
ma-88	321	17	shuold	shuold	PRON
ma-88	321	18	hold	hold	VERB
ma-88	321	19	:	:	PUNCT
ma-88	321	20	|‖(1	|‖(1	NUM
ma-88	322	1	+	+	CCONJ
ma-88	322	2	c	c	X
ma-88	322	3	,	,	PUNCT
ma-88	322	4	1)‖2	1)‖2	NUM
ma-88	322	5	−	−	PROPN
ma-88	322	6	‖(1−	‖(1−	PROPN
ma-88	322	7	c,−1)‖2|	c,−1)‖2|	NOUN
ma-88	322	8	≤	≤	NUM
ma-88	322	9	4ε	4ε	NUM
ma-88	322	10	.	.	PUNCT
ma-88	323	1	if	if	SCONJ
ma-88	323	2	c	c	PROPN
ma-88	323	3	≥	≥	NOUN
ma-88	323	4	0	0	NUM
ma-88	323	5	,	,	PUNCT
ma-88	323	6	we	we	PRON
ma-88	323	7	have	have	VERB
ma-88	323	8	(	(	PUNCT
ma-88	323	9	1	1	NUM
ma-88	323	10	+	+	NUM
ma-88	323	11	c)2	c)2	NOUN
ma-88	323	12	−	−	PROPN
ma-88	323	13	1	1	NUM
ma-88	323	14	≤	≤	NOUN
ma-88	323	15	4ε	4ε	NUM
ma-88	324	1	=	=	NOUN
ma-88	324	2	⇒	⇒	NOUN
ma-88	324	3	0	0	NUM
ma-88	325	1	≤	≤	NUM
ma-88	325	2	c	c	X
ma-88	325	3	<	<	X
ma-88	325	4	−1	−1	NOUN
ma-88	326	1	+	+	CCONJ
ma-88	326	2	√	√	NUM
ma-88	326	3	1	1	NUM
ma-88	326	4	+	+	NUM
ma-88	326	5	4ε	4ε	NOUN
ma-88	326	6	.	.	PUNCT
ma-88	327	1	if	if	SCONJ
ma-88	327	2	c	c	PROPN
ma-88	327	3	<	<	X
ma-88	327	4	0	0	NUM
ma-88	327	5	,	,	PUNCT
ma-88	327	6	we	we	PRON
ma-88	327	7	have	have	VERB
ma-88	327	8	(	(	PUNCT
ma-88	327	9	1−	1−	NUM
ma-88	327	10	c)2	c)2	PROPN
ma-88	327	11	−	−	PROPN
ma-88	327	12	1	1	NUM
ma-88	327	13	≤	≤	NUM
ma-88	328	1	4ε	4ε	NUM
ma-88	329	1	=	=	NOUN
ma-88	329	2	⇒	⇒	NOUN
ma-88	329	3	1−	1−	NUM
ma-88	329	4	√	√	NUM
ma-88	329	5	1	1	NUM
ma-88	329	6	+	+	NUM
ma-88	329	7	4ε	4ε	NUM
ma-88	330	1	≤	≤	NUM
ma-88	330	2	c	c	X
ma-88	330	3	<	<	X
ma-88	330	4	0	0	NUM
ma-88	330	5	.	.	PUNCT
ma-88	330	6	to	to	PART
ma-88	330	7	conclude	conclude	VERB
ma-88	330	8	we	we	PRON
ma-88	330	9	have	have	VERB
ma-88	330	10	(	(	PUNCT
ma-88	330	11	1	1	NUM
ma-88	330	12	,	,	PUNCT
ma-88	330	13	0	0	NUM
ma-88	330	14	)	)	PUNCT
ma-88	330	15	⊥εi	⊥εi	NOUN
ma-88	330	16	(	(	PUNCT
ma-88	330	17	c	c	X
ma-88	330	18	,	,	PUNCT
ma-88	330	19	1	1	NUM
ma-88	330	20	)	)	PUNCT
ma-88	330	21	if	if	SCONJ
ma-88	330	22	|c	|c	VERB
ma-88	330	23	|	|	NOUN
ma-88	330	24	≤	≤	NUM
ma-88	330	25	−1	−1	NOUN
ma-88	331	1	+	+	CCONJ
ma-88	331	2	√	√	NUM
ma-88	331	3	1	1	NUM
ma-88	331	4	+	+	NUM
ma-88	331	5	4ε	4ε	NOUN
ma-88	331	6	.	.	PUNCT
ma-88	332	1	since	since	SCONJ
ma-88	332	2	ε	ε	PROPN
ma-88	332	3	≤	≤	NOUN
ma-88	332	4	−1	−1	NOUN
ma-88	332	5	+	+	CCONJ
ma-88	332	6	√	√	NUM
ma-88	332	7	1	1	NUM
ma-88	332	8	+	+	NUM
ma-88	332	9	4ε	4ε	NOUN
ma-88	332	10	,	,	PUNCT
ma-88	332	11	it	it	PRON
ma-88	332	12	can	can	AUX
ma-88	332	13	be	be	AUX
ma-88	332	14	seen	see	VERB
ma-88	332	15	as	as	ADP
ma-88	332	16	an	an	DET
ma-88	332	17	example	example	NOUN
ma-88	332	18	that	that	SCONJ
ma-88	332	19	x	x	SYM
ma-88	332	20	⊥εi	⊥εi	PROPN
ma-88	332	21	y	y	PROPN
ma-88	332	22	does	do	AUX
ma-88	332	23	not	not	PART
ma-88	332	24	imply	imply	VERB
ma-88	332	25	x	x	NOUN
ma-88	332	26	⊥bε	⊥bε	NOUN
ma-88	332	27	y	y	PROPN
ma-88	332	28	.	.	PUNCT
ma-88	333	1	example	example	NOUN
ma-88	334	1	3.5	3.5	NUM
ma-88	334	2	.	.	PUNCT
ma-88	335	1	let	let	VERB
ma-88	335	2	x	x	PUNCT
ma-88	335	3	=	=	PRON
ma-88	335	4	(	(	PUNCT
ma-88	335	5	r2	r2	PROPN
ma-88	335	6	,	,	PUNCT
ma-88	335	7	‖	‖	PROPN
ma-88	335	8	·	·	SYM
ma-88	335	9	‖∞	‖∞	PROPN
ma-88	335	10	)	)	PUNCT
ma-88	335	11	,	,	PUNCT
ma-88	335	12	we	we	PRON
ma-88	335	13	assume	assume	VERB
ma-88	335	14	that	that	SCONJ
ma-88	335	15	x	x	X
ma-88	335	16	=	=	SYM
ma-88	335	17	(	(	PUNCT
ma-88	335	18	1	1	NUM
ma-88	335	19	,	,	PUNCT
ma-88	335	20	1	1	NUM
ma-88	335	21	)	)	PUNCT
ma-88	335	22	,	,	PUNCT
ma-88	336	1	y	y	PROPN
ma-88	336	2	=	=	PUNCT
ma-88	336	3	(	(	PUNCT
ma-88	336	4	−1−	−1−	NOUN
ma-88	336	5	√	√	ADV
ma-88	336	6	2	2	NUM
ma-88	336	7	2	2	NUM
ma-88	336	8	ε	ε	PROPN
ma-88	336	9	,	,	PUNCT
ma-88	336	10	1−	1−	NUM
ma-88	336	11	√	√	ADV
ma-88	336	12	2	2	NUM
ma-88	336	13	2	2	NUM
ma-88	336	14	ε	ε	PROPN
ma-88	336	15	)	)	PUNCT
ma-88	336	16	,	,	PUNCT
ma-88	336	17	z	z	NOUN
ma-88	336	18	=	=	SYM
ma-88	336	19	(	(	PUNCT
ma-88	336	20	−1	−1	NOUN
ma-88	336	21	,	,	PUNCT
ma-88	336	22	1	1	NUM
ma-88	336	23	)	)	PUNCT
ma-88	336	24	.	.	PUNCT
ma-88	337	1	we	we	PRON
ma-88	337	2	have	have	VERB
ma-88	337	3	x	x	PROPN
ma-88	337	4	⊥i	⊥i	PROPN
ma-88	337	5	z	z	NOUN
ma-88	337	6	,	,	PUNCT
ma-88	337	7	and	and	CCONJ
ma-88	337	8	z	z	NOUN
ma-88	337	9	=	=	PUNCT
ma-88	338	1	−	−	PROPN
ma-88	338	2	ε√	ε√	NOUN
ma-88	338	3	2	2	NUM
ma-88	338	4	x	x	SYM
ma-88	338	5	+	+	NUM
ma-88	338	6	y	y	PROPN
ma-88	338	7	,	,	PUNCT
ma-88	338	8	ε	ε	PROPN
ma-88	338	9	≤	≤	ADV
ma-88	338	10	‖y‖	‖y‖	PROPN
ma-88	338	11	‖x‖ε	‖x‖ε	NOUN
ma-88	338	12	.	.	PUNCT
ma-88	339	1	thus	thus	ADV
ma-88	339	2	|	|	ADV
ma-88	339	3	−	−	PROPN
ma-88	339	4	ε√	ε√	NOUN
ma-88	339	5	2	2	NUM
ma-88	340	1	|	|	ADV
ma-88	340	2	≤	≤	NUM
ma-88	340	3	‖y‖‖x‖ε	‖y‖‖x‖ε	NOUN
ma-88	340	4	,	,	PUNCT
ma-88	340	5	which	which	PRON
ma-88	340	6	implies	imply	VERB
ma-88	340	7	that	that	SCONJ
ma-88	340	8	there	there	PRON
ma-88	340	9	exists	exist	VERB
ma-88	340	10	|a|	|a|	PROPN
ma-88	340	11	≤	≤	ADJ
ma-88	340	12	‖y‖‖x‖ε	‖y‖‖x‖ε	NOUN
ma-88	340	13	such	such	ADJ
ma-88	340	14	that	that	SCONJ
ma-88	340	15	x	x	SYM
ma-88	340	16	⊥i	⊥i	ADJ
ma-88	340	17	ax	ax	NOUN
ma-88	340	18	+	+	X
ma-88	340	19	y	y	PROPN
ma-88	340	20	.	.	PUNCT
ma-88	341	1	on	on	ADP
ma-88	341	2	the	the	DET
ma-88	341	3	other	other	ADJ
ma-88	341	4	hand	hand	NOUN
ma-88	341	5	,	,	PUNCT
ma-88	341	6	since	since	SCONJ
ma-88	341	7	‖x	‖x	NOUN
ma-88	341	8	+	+	CCONJ
ma-88	341	9	y‖	y‖	X
ma-88	342	1	=	=	PUNCT
ma-88	342	2	‖(−	‖(−	NOUN
ma-88	342	3	√	√	NUM
ma-88	342	4	2	2	NUM
ma-88	342	5	2	2	NUM
ma-88	342	6	ε	ε	PROPN
ma-88	342	7	,	,	PUNCT
ma-88	342	8	2−	2−	NUM
ma-88	342	9	√	√	ADV
ma-88	342	10	2	2	NUM
ma-88	342	11	2	2	NUM
ma-88	342	12	ε)‖	ε)‖	ADV
ma-88	342	13	=	=	NOUN
ma-88	342	14	2−	2−	NUM
ma-88	342	15	√	√	NUM
ma-88	342	16	2	2	NUM
ma-88	342	17	2	2	NUM
ma-88	342	18	ε	ε	PROPN
ma-88	342	19	.	.	PUNCT
ma-88	342	20	‖x	‖x	NOUN
ma-88	343	1	−	−	PROPN
ma-88	343	2	y‖	y‖	NOUN
ma-88	343	3	=	=	PUNCT
ma-88	344	1	‖(2	‖(2	NOUN
ma-88	344	2	+	+	NUM
ma-88	344	3	√	√	NUM
ma-88	344	4	2	2	NUM
ma-88	344	5	2	2	NUM
ma-88	344	6	ε	ε	PROPN
ma-88	344	7	,	,	PUNCT
ma-88	344	8	√	√	ADV
ma-88	344	9	2	2	NUM
ma-88	344	10	2	2	NUM
ma-88	344	11	ε)‖	ε)‖	ADV
ma-88	344	12	=	=	X
ma-88	344	13	2	2	NUM
ma-88	344	14	+	+	CCONJ
ma-88	344	15	√	√	NUM
ma-88	344	16	2	2	NUM
ma-88	344	17	2	2	NUM
ma-88	344	18	ε	ε	PROPN
ma-88	344	19	.	.	PUNCT
ma-88	345	1	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	345	2	eur	eur	PROPN
ma-88	345	3	.	.	PUNCT
ma-88	346	1	j.	j.	PROPN
ma-88	346	2	math	math	PROPN
ma-88	346	3	.	.	PUNCT
ma-88	347	1	anal	anal	PROPN
ma-88	347	2	.	.	PUNCT
ma-88	348	1	10.28924	10.28924	NUM
ma-88	348	2	/	/	SYM
ma-88	348	3	ada	ada	PROPN
ma-88	348	4	/	/	SYM
ma-88	348	5	ma.2.16	ma.2.16	PROPN
ma-88	348	6	11	11	NUM
ma-88	348	7	we	we	PRON
ma-88	348	8	have	have	VERB
ma-88	348	9	|‖x	|‖x	NOUN
ma-88	348	10	+	+	NOUN
ma-88	348	11	y‖2	y‖2	X
ma-88	348	12	−	−	NOUN
ma-88	348	13	‖x	‖x	NOUN
ma-88	349	1	−	−	PROPN
ma-88	349	2	y‖2|	y‖2|	NOUN
ma-88	349	3	=	=	SYM
ma-88	349	4	4	4	NUM
ma-88	349	5	√	√	NOUN
ma-88	349	6	2ε	2ε	NUM
ma-88	349	7	≥	≥	NUM
ma-88	349	8	4	4	NUM
ma-88	349	9	√	√	NUM
ma-88	349	10	2	2	NUM
ma-88	349	11	.	.	PUNCT
ma-88	350	1	so	so	ADV
ma-88	350	2	x	x	SYM
ma-88	350	3	6⊥εi	6⊥εi	NUM
ma-88	350	4	y	y	NOUN
ma-88	350	5	,	,	PUNCT
ma-88	350	6	thus	thus	ADV
ma-88	350	7	it	it	PRON
ma-88	350	8	can	can	AUX
ma-88	350	9	be	be	AUX
ma-88	350	10	seen	see	VERB
ma-88	350	11	as	as	ADP
ma-88	350	12	an	an	DET
ma-88	350	13	example	example	NOUN
ma-88	350	14	that	that	SCONJ
ma-88	350	15	condition	condition	NOUN
ma-88	350	16	(	(	PUNCT
ma-88	350	17	1	1	X
ma-88	350	18	)	)	PUNCT
ma-88	350	19	is	be	AUX
ma-88	350	20	not	not	PART
ma-88	350	21	satisfied	satisfied	ADJ
ma-88	350	22	.	.	PUNCT
ma-88	351	1	a	a	DET
ma-88	351	2	mapping	mapping	NOUN
ma-88	351	3	t	t	NOUN
ma-88	351	4	:	:	PUNCT
ma-88	351	5	h→	h→	X
ma-88	351	6	k	k	NOUN
ma-88	351	7	which	which	PRON
ma-88	351	8	satisfies	satisfy	VERB
ma-88	351	9	the	the	DET
ma-88	351	10	condtion	condtion	NOUN
ma-88	351	11	x	x	PROPN
ma-88	352	1	⊥	⊥	NOUN
ma-88	352	2	y	y	PROPN
ma-88	352	3	=	=	AUX
ma-88	352	4	⇒	⇒	PROPN
ma-88	352	5	t	t	PROPN
ma-88	352	6	(	(	PUNCT
ma-88	352	7	x	x	X
ma-88	352	8	)	)	PUNCT
ma-88	352	9	⊥	⊥	PROPN
ma-88	352	10	t	t	PROPN
ma-88	352	11	(	(	PUNCT
ma-88	352	12	y	y	NOUN
ma-88	352	13	)	)	PUNCT
ma-88	352	14	.	.	PUNCT
ma-88	353	1	is	be	AUX
ma-88	353	2	called	call	VERB
ma-88	353	3	orthogonality	orthogonality	NOUN
ma-88	353	4	preserving(o.p	preserving(o.p	NOUN
ma-88	353	5	.	.	PUNCT
ma-88	353	6	)	)	PUNCT
ma-88	354	1	[	[	X
ma-88	354	2	7,8	7,8	NUM
ma-88	354	3	]	]	PUNCT
ma-88	354	4	,	,	PUNCT
ma-88	354	5	and	and	CCONJ
ma-88	354	6	t	t	PROPN
ma-88	354	7	is	be	AUX
ma-88	354	8	said	say	VERB
ma-88	354	9	to	to	PART
ma-88	354	10	be	be	AUX
ma-88	354	11	an	an	DET
ma-88	354	12	isometry	isometry	NOUN
ma-88	354	13	maping	maping	NOUN
ma-88	354	14	[	[	X
ma-88	354	15	22	22	NUM
ma-88	354	16	]	]	PUNCT
ma-88	354	17	if	if	SCONJ
ma-88	354	18	‖tx‖	‖tx‖	PROPN
ma-88	354	19	=	=	SYM
ma-88	354	20	‖x‖.	‖x‖.	PROPN
ma-88	354	21	to	to	PART
ma-88	354	22	promote	promote	VERB
ma-88	354	23	the	the	DET
ma-88	354	24	concept	concept	NOUN
ma-88	354	25	,	,	PUNCT
ma-88	354	26	jacek	jacek	PROPN
ma-88	354	27	chmielinski	chmielinski	PROPN
ma-88	355	1	[	[	X
ma-88	355	2	6	6	NUM
ma-88	355	3	]	]	PUNCT
ma-88	355	4	introduced	introduce	VERB
ma-88	355	5	the	the	DET
ma-88	355	6	notion	notion	NOUN
ma-88	355	7	of	of	ADP
ma-88	355	8	approximately	approximately	ADV
ma-88	355	9	orthogo	orthogo	NOUN
ma-88	355	10	-	-	PUNCT
ma-88	355	11	nality	nality	NOUN
ma-88	355	12	preserving	preserve	VERB
ma-88	355	13	(	(	PUNCT
ma-88	355	14	a.o.p	a.o.p	NOUN
ma-88	355	15	.	.	PUNCT
ma-88	355	16	)	)	PUNCT
ma-88	356	1	mapping	mapping	NOUN
ma-88	356	2	and	and	CCONJ
ma-88	356	3	have	have	AUX
ma-88	356	4	studied	study	VERB
ma-88	356	5	the	the	DET
ma-88	356	6	properties	property	NOUN
ma-88	356	7	of	of	ADP
ma-88	356	8	mapping	mapping	NOUN
ma-88	356	9	that	that	PRON
ma-88	356	10	is	be	AUX
ma-88	356	11	approximarelyisosceles	approximarelyisoscele	NOUN
ma-88	356	12	orthogonality	orthogonality	NOUN
ma-88	356	13	preserving(t	preserving(t	NOUN
ma-88	356	14	:	:	PUNCT
ma-88	356	15	x	x	PROPN
ma-88	356	16	⊥i	⊥i	PROPN
ma-88	356	17	y	y	PROPN
ma-88	356	18	=	=	PROPN
ma-88	356	19	⇒	⇒	PROPN
ma-88	356	20	t	t	PROPN
ma-88	356	21	(	(	PUNCT
ma-88	356	22	x	x	NOUN
ma-88	356	23	)	)	PUNCT
ma-88	356	24	⊥εi	⊥εi	ADP
ma-88	356	25	t	t	PROPN
ma-88	356	26	(	(	PUNCT
ma-88	356	27	y	y	NOUN
ma-88	356	28	)	)	PUNCT
ma-88	356	29	)	)	PUNCT
ma-88	356	30	.	.	PUNCT
ma-88	357	1	after	after	SCONJ
ma-88	357	2	that	that	DET
ma-88	357	3	many	many	ADJ
ma-88	357	4	mathematicianshave	mathematicianshave	NOUN
ma-88	357	5	show	show	VERB
ma-88	357	6	great	great	ADJ
ma-88	357	7	interest	interest	NOUN
ma-88	357	8	in	in	ADP
ma-88	357	9	the	the	DET
ma-88	357	10	a.o.p	a.o.p	NOUN
ma-88	357	11	mapping	mapping	NOUN
ma-88	357	12	[	[	X
ma-88	357	13	25	25	NUM
ma-88	357	14	]	]	PUNCT
ma-88	357	15	,	,	PUNCT
ma-88	357	16	and	and	CCONJ
ma-88	357	17	aleksej	aleksej	ADJ
ma-88	357	18	turnšek	turnšek	NOUN
ma-88	357	19	[	[	X
ma-88	357	20	19	19	NUM
ma-88	357	21	]	]	PUNCT
ma-88	357	22	studied	study	VERB
ma-88	357	23	the	the	PRON
ma-88	357	24	mappingthat	mappingthat	PRON
ma-88	357	25	is	be	AUX
ma-88	357	26	approximately	approximately	ADV
ma-88	357	27	birkhoff	birkhoff	NOUN
ma-88	357	28	orthogonality	orthogonality	NOUN
ma-88	357	29	preserving(t	preserving(t	NOUN
ma-88	357	30	:	:	PUNCT
ma-88	357	31	x	x	SYM
ma-88	357	32	⊥b	⊥b	PRON
ma-88	357	33	y	y	NOUN
ma-88	357	34	=	=	NOUN
ma-88	357	35	⇒	⇒	PROPN
ma-88	357	36	t	t	PROPN
ma-88	357	37	(	(	PUNCT
ma-88	357	38	x	x	X
ma-88	357	39	)	)	PUNCT
ma-88	357	40	⊥εb	⊥εb	PROPN
ma-88	357	41	t	t	PROPN
ma-88	357	42	(	(	PUNCT
ma-88	357	43	x))innormed	x))innormed	PROPN
ma-88	357	44	spaces	space	NOUN
ma-88	357	45	.	.	PUNCT
ma-88	358	1	now	now	ADV
ma-88	358	2	we	we	PRON
ma-88	358	3	try	try	VERB
ma-88	358	4	to	to	PART
ma-88	358	5	study	study	VERB
ma-88	358	6	the	the	DET
ma-88	358	7	approximarely	approximarely	ADV
ma-88	358	8	orthogonality	orthogonality	NOUN
ma-88	358	9	preserving	preserve	VERB
ma-88	358	10	mapping	mapping	NOUN
ma-88	358	11	types	type	NOUN
ma-88	358	12	:	:	PUNCT
ma-88	358	13	t	t	X
ma-88	358	14	:	:	PUNCT
ma-88	358	15	x	x	PROPN
ma-88	358	16	⊥i	⊥i	PROPN
ma-88	358	17	y	y	PROPN
ma-88	358	18	=	=	PROPN
ma-88	358	19	⇒	⇒	PROPN
ma-88	358	20	tx	tx	PROPN
ma-88	358	21	⊥εb	⊥εb	PROPN
ma-88	358	22	ty	ty	PROPN
ma-88	358	23	,	,	PUNCT
ma-88	358	24	and	and	CCONJ
ma-88	358	25	t	t	NOUN
ma-88	358	26	:	:	PUNCT
ma-88	358	27	x	x	PROPN
ma-88	358	28	⊥i	⊥i	PROPN
ma-88	358	29	y	y	PROPN
ma-88	358	30	=	=	PROPN
ma-88	358	31	⇒	⇒	X
ma-88	358	32	tx	tx	X
ma-88	358	33	εx	εx	ADP
ma-88	358	34	⊥b	⊥b	PRON
ma-88	358	35	ty	ty	PROPN
ma-88	358	36	.	.	PUNCT
ma-88	359	1	proposition	proposition	NOUN
ma-88	359	2	3.6	3.6	NUM
ma-88	359	3	.	.	PUNCT
ma-88	360	1	let	let	VERB
ma-88	360	2	t	t	NOUN
ma-88	360	3	:	:	PUNCT
ma-88	360	4	x	x	X
ma-88	360	5	→	→	SYM
ma-88	360	6	y	y	X
ma-88	360	7	be	be	AUX
ma-88	360	8	a	a	DET
ma-88	360	9	nontrivial	nontrivial	ADJ
ma-88	360	10	linear	linear	NOUN
ma-88	360	11	mapping	mapping	NOUN
ma-88	360	12	satisfying	satisfying	ADJ
ma-88	360	13	x	x	SYM
ma-88	360	14	⊥i	⊥i	PROPN
ma-88	360	15	y	y	PROPN
ma-88	360	16	=	=	PROPN
ma-88	360	17	⇒	⇒	PROPN
ma-88	360	18	tx	tx	PROPN
ma-88	360	19	⊥εb	⊥εb	PROPN
ma-88	360	20	ty	ty	PROPN
ma-88	360	21	,	,	PUNCT
ma-88	360	22	x	x	PRON
ma-88	360	23	,	,	PUNCT
ma-88	360	24	y	y	PROPN
ma-88	360	25	∈	∈	PROPN
ma-88	361	1	x.	x.	NOUN
ma-88	362	1	then	then	ADV
ma-88	362	2	t	t	PROPN
ma-88	362	3	is	be	AUX
ma-88	362	4	a	a	DET
ma-88	362	5	bounded	bound	VERB
ma-88	362	6	and	and	CCONJ
ma-88	362	7	bounded	bound	VERB
ma-88	362	8	from	from	ADP
ma-88	362	9	below	below	ADV
ma-88	362	10	,	,	PUNCT
ma-88	362	11	‖tx‖	‖tx‖	PROPN
ma-88	362	12	≥	≥	NUM
ma-88	362	13	(	(	PUNCT
ma-88	362	14	1−ε)2	1−ε)2	NUM
ma-88	362	15	3−ε2	3−ε2	NUM
ma-88	362	16	+	+	NOUN
ma-88	362	17	2	2	NUM
ma-88	362	18	√	√	NUM
ma-88	362	19	2−ε2	2−ε2	NUM
ma-88	362	20	.	.	PUNCT
ma-88	363	1	proof	proof	NOUN
ma-88	363	2	.	.	PUNCT
ma-88	364	1	take	take	VERB
ma-88	364	2	two	two	NUM
ma-88	364	3	arbitrary	arbitrary	ADJ
ma-88	364	4	unit	unit	NOUN
ma-88	364	5	vectors	vector	NOUN
ma-88	364	6	x	x	PUNCT
ma-88	364	7	and	and	CCONJ
ma-88	364	8	y	y	PROPN
ma-88	364	9	and	and	CCONJ
ma-88	364	10	note	note	VERB
ma-88	364	11	that	that	SCONJ
ma-88	364	12	x+y	x+y	PROPN
ma-88	364	13	2	2	NUM
ma-88	364	14	⊥i	⊥i	NOUN
ma-88	364	15	x−y	x−y	PROPN
ma-88	364	16	2	2	NUM
ma-88	364	17	,	,	PUNCT
ma-88	364	18	it	it	PRON
ma-88	364	19	follows	follow	VERB
ma-88	364	20	that	that	SCONJ
ma-88	364	21	t	t	PROPN
ma-88	364	22	(	(	PUNCT
ma-88	364	23	x	x	PROPN
ma-88	364	24	+	+	NUM
ma-88	364	25	y	y	NOUN
ma-88	364	26	)	)	PUNCT
ma-88	364	27	⊥εb	⊥εb	NOUN
ma-88	364	28	t	t	PROPN
ma-88	364	29	(	(	PUNCT
ma-88	364	30	x	x	PROPN
ma-88	364	31	−	−	PROPN
ma-88	364	32	y	y	PROPN
ma-88	364	33	)	)	PUNCT
ma-88	364	34	,	,	PUNCT
ma-88	364	35	hence	hence	ADV
ma-88	364	36	for	for	SCONJ
ma-88	364	37	all	all	DET
ma-88	364	38	λ	λ	NOUN
ma-88	364	39	∈	∈	NOUN
ma-88	364	40	r	r	NOUN
ma-88	364	41	we	we	PRON
ma-88	364	42	have	have	VERB
ma-88	364	43	‖t	‖t	NOUN
ma-88	364	44	(	(	PUNCT
ma-88	364	45	x	x	SYM
ma-88	364	46	+	+	NUM
ma-88	364	47	y	y	NOUN
ma-88	364	48	)	)	PUNCT
ma-88	365	1	+	+	CCONJ
ma-88	365	2	λt	λt	ADP
ma-88	365	3	(	(	PUNCT
ma-88	365	4	x	x	NOUN
ma-88	365	5	−	−	PROPN
ma-88	365	6	y)‖2	y)‖2	PROPN
ma-88	365	7	≥	≥	NOUN
ma-88	365	8	‖t	‖t	NOUN
ma-88	365	9	(	(	PUNCT
ma-88	365	10	x	x	X
ma-88	365	11	+	+	NUM
ma-88	365	12	y)‖2	y)‖2	NOUN
ma-88	365	13	−	−	PROPN
ma-88	365	14	2ε‖t	2ε‖t	NOUN
ma-88	365	15	(	(	PUNCT
ma-88	365	16	x	x	X
ma-88	366	1	+	+	NUM
ma-88	366	2	y)‖‖λt	y)‖‖λt	PRON
ma-88	366	3	(	(	PUNCT
ma-88	366	4	x	x	NOUN
ma-88	366	5	−	−	X
ma-88	366	6	y)‖	y)‖	NOUN
ma-88	366	7	,	,	PUNCT
ma-88	366	8	by	by	ADP
ma-88	366	9	the	the	DET
ma-88	366	10	triangle	triangle	NOUN
ma-88	366	11	inequality	inequality	NOUN
ma-88	366	12	and	and	CCONJ
ma-88	366	13	the	the	DET
ma-88	366	14	linearity	linearity	NOUN
ma-88	366	15	of	of	ADP
ma-88	366	16	t	t	PROPN
ma-88	366	17	it	it	PRON
ma-88	366	18	follows	follow	VERB
ma-88	366	19	that	that	SCONJ
ma-88	366	20	‖t	‖t	NOUN
ma-88	366	21	(	(	PUNCT
ma-88	366	22	x	x	SYM
ma-88	367	1	+	+	NUM
ma-88	367	2	y)‖2	y)‖2	NOUN
ma-88	367	3	≤	≤	NOUN
ma-88	367	4	‖(1	‖(1	NUM
ma-88	368	1	+	+	PUNCT
ma-88	368	2	λ)tx	λ)tx	PROPN
ma-88	368	3	+	+	CCONJ
ma-88	368	4	(	(	PUNCT
ma-88	368	5	1−	1−	NUM
ma-88	368	6	λ)ty‖2	λ)ty‖2	NOUN
ma-88	368	7	+	+	CCONJ
ma-88	368	8	2ε|λ|‖tx	2ε|λ|‖tx	NUM
ma-88	368	9	+	+	NUM
ma-88	368	10	ty‖2	ty‖2	PROPN
ma-88	368	11	.	.	PROPN
ma-88	369	1	on	on	ADP
ma-88	369	2	the	the	DET
ma-88	369	3	other	other	ADJ
ma-88	369	4	hand	hand	NOUN
ma-88	369	5	we	we	PRON
ma-88	369	6	have	have	VERB
ma-88	369	7	‖t	‖t	NOUN
ma-88	369	8	(	(	PUNCT
ma-88	369	9	x	x	SYM
ma-88	369	10	+	+	NUM
ma-88	369	11	y)‖2	y)‖2	NOUN
ma-88	369	12	≥	≥	NOUN
ma-88	369	13	(	(	PUNCT
ma-88	369	14	‖tx‖	‖tx‖	PROPN
ma-88	369	15	−	−	PROPN
ma-88	369	16	‖ty‖)2	‖ty‖)2	PROPN
ma-88	369	17	,	,	PUNCT
ma-88	369	18	thus	thus	ADV
ma-88	369	19	we	we	PRON
ma-88	369	20	get	get	VERB
ma-88	369	21	:	:	PUNCT
ma-88	369	22	‖tx‖2+‖ty‖2−2‖tx‖‖ty‖	‖tx‖2+‖ty‖2−2‖tx‖‖ty‖	NOUN
ma-88	369	23	≤	≤	NOUN
ma-88	369	24	(	(	PUNCT
ma-88	369	25	1+λ)2‖tx‖2+(1−λ)2‖ty‖2	1+λ)2‖tx‖2+(1−λ)2‖ty‖2	NUM
ma-88	369	26	+	+	NOUN
ma-88	369	27	2(1−λ2)‖tx‖‖ty‖+2ε|λ|‖tx+ty‖2	2(1−λ2)‖tx‖‖ty‖+2ε|λ|‖tx+ty‖2	NUM
ma-88	369	28	.	.	PUNCT
ma-88	370	1	if	if	SCONJ
ma-88	370	2	tx	tx	VERB
ma-88	370	3	=	=	SYM
ma-88	370	4	0	0	PROPN
ma-88	370	5	,	,	PUNCT
ma-88	370	6	then	then	ADV
ma-88	370	7	let	let	VERB
ma-88	370	8	y	y	PROPN
ma-88	370	9	∈	∈	PROPN
ma-88	370	10	x	x	PUNCT
ma-88	370	11	such	such	ADJ
ma-88	370	12	that	that	PRON
ma-88	370	13	ty	ty	NUM
ma-88	370	14	6=	6=	ADP
ma-88	370	15	0	0	NUM
ma-88	370	16	,	,	PUNCT
ma-88	370	17	substitute	substitute	NOUN
ma-88	370	18	x	x	SYM
ma-88	370	19	,	,	PUNCT
ma-88	370	20	y	y	PROPN
ma-88	370	21	into	into	ADP
ma-88	370	22	the	the	DET
ma-88	370	23	above	above	ADJ
ma-88	370	24	formula	formula	NOUN
ma-88	370	25	,	,	PUNCT
ma-88	370	26	we	we	PRON
ma-88	370	27	have	have	VERB
ma-88	370	28	0	0	NUM
ma-88	370	29	≤	≤	NUM
ma-88	370	30	λ2	λ2	NOUN
ma-88	370	31	−	−	PROPN
ma-88	370	32	2λ+	2λ+	NUM
ma-88	371	1	2ε|λ|	2ε|λ|	NUM
ma-88	371	2	f	f	NOUN
ma-88	371	3	or	or	CCONJ
ma-88	371	4	al	al	PROPN
ma-88	371	5	l	l	PROPN
ma-88	371	6	λ	λ	PROPN
ma-88	371	7	∈	∈	PROPN
ma-88	371	8	r.	r.	PROPN
ma-88	371	9	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	371	10	eur	eur	PROPN
ma-88	371	11	.	.	PUNCT
ma-88	372	1	j.	j.	PROPN
ma-88	372	2	math	math	PROPN
ma-88	372	3	.	.	PUNCT
ma-88	373	1	anal	anal	PROPN
ma-88	373	2	.	.	PUNCT
ma-88	374	1	10.28924	10.28924	NUM
ma-88	374	2	/	/	SYM
ma-88	374	3	ada	ada	PROPN
ma-88	374	4	/	/	SYM
ma-88	374	5	ma.2.16	ma.2.16	PROPN
ma-88	374	6	12	12	NUM
ma-88	375	1	it	it	PRON
ma-88	375	2	is	be	AUX
ma-88	375	3	impossible	impossible	ADJ
ma-88	375	4	cause	cause	SCONJ
ma-88	375	5	0	0	NUM
ma-88	375	6	≤	≤	NUM
ma-88	375	7	ε	ε	PROPN
ma-88	375	8	<	<	X
ma-88	375	9	0	0	PROPN
ma-88	375	10	,	,	PUNCT
ma-88	375	11	so	so	SCONJ
ma-88	375	12	we	we	PRON
ma-88	375	13	can	can	AUX
ma-88	375	14	divide	divide	VERB
ma-88	375	15	both	both	DET
ma-88	375	16	sides	side	NOUN
ma-88	375	17	of	of	ADP
ma-88	375	18	the	the	DET
ma-88	375	19	inequality	inequality	NOUN
ma-88	375	20	by	by	ADP
ma-88	375	21	‖tx‖2	‖tx‖2	PROPN
ma-88	375	22	and	and	CCONJ
ma-88	375	23	denote	denote	VERB
ma-88	375	24	z	z	NOUN
ma-88	375	25	=	=	SYM
ma-88	375	26	‖ty‖	‖ty‖	PROPN
ma-88	375	27	‖tx‖	‖tx‖	PROPN
ma-88	375	28	.	.	PUNCT
ma-88	376	1	we	we	PRON
ma-88	376	2	get	get	VERB
ma-88	376	3	:	:	PUNCT
ma-88	376	4	(	(	PUNCT
ma-88	376	5	z	z	NOUN
ma-88	376	6	−	−	NOUN
ma-88	376	7	1)2λ2	1)2λ2	NUM
ma-88	376	8	+	+	CCONJ
ma-88	376	9	(	(	PUNCT
ma-88	376	10	2−	2−	NUM
ma-88	376	11	2z2	2z2	NUM
ma-88	376	12	+	+	CCONJ
ma-88	376	13	2ε(1	2ε(1	NUM
ma-88	376	14	+	+	SYM
ma-88	376	15	z)2	z)2	NOUN
ma-88	376	16	)	)	PUNCT
ma-88	376	17	λ+	λ+	PUNCT
ma-88	376	18	4z	4z	X
ma-88	376	19	≥	≥	NOUN
ma-88	376	20	0	0	NUM
ma-88	376	21	f	f	PROPN
ma-88	376	22	or	or	CCONJ
ma-88	376	23	al	al	PROPN
ma-88	376	24	l	l	PROPN
ma-88	376	25	λ	λ	PROPN
ma-88	376	26	≥	≥	PROPN
ma-88	376	27	0	0	NUM
ma-88	376	28	.	.	PUNCT
ma-88	377	1	the	the	DET
ma-88	377	2	inequality	inequality	NOUN
ma-88	377	3	is	be	AUX
ma-88	377	4	satisfied	satisfied	ADJ
ma-88	377	5	when	when	SCONJ
ma-88	377	6	−	−	PROPN
ma-88	377	7	b	b	PROPN
ma-88	377	8	2a	2a	NUM
ma-88	377	9	≤	≤	NOUN
ma-88	377	10	0	0	NUM
ma-88	377	11	or	or	CCONJ
ma-88	377	12	∆	∆	PROPN
ma-88	377	13	=	=	SYM
ma-88	377	14	b2	b2	PROPN
ma-88	377	15	−	−	PROPN
ma-88	377	16	4ac	4ac	NOUN
ma-88	377	17	≤	≤	NOUN
ma-88	377	18	0	0	NUM
ma-88	377	19	.	.	PUNCT
ma-88	378	1	∆	∆	PUNCT
ma-88	379	1	=	=	SYM
ma-88	379	2	4	4	NUM
ma-88	379	3	(	(	PUNCT
ma-88	379	4	(	(	PUNCT
ma-88	379	5	1−	1−	NUM
ma-88	379	6	z2)2	z2)2	NUM
ma-88	379	7	+	+	NUM
ma-88	379	8	ε2(1	ε2(1	NOUN
ma-88	379	9	+	+	CCONJ
ma-88	379	10	z)4	z)4	X
ma-88	379	11	+	+	CCONJ
ma-88	379	12	2ε(1	2ε(1	NUM
ma-88	379	13	+	+	CCONJ
ma-88	379	14	z)2(1−	z)2(1−	ADJ
ma-88	379	15	z2)−	z2)−	NOUN
ma-88	379	16	(	(	PUNCT
ma-88	379	17	4z)(z	4z)(z	NUM
ma-88	379	18	−	−	PROPN
ma-88	379	19	1)2	1)2	NUM
ma-88	379	20	)	)	PUNCT
ma-88	379	21	.	.	PUNCT
ma-88	380	1	(	(	PUNCT
ma-88	380	2	1−	1−	NUM
ma-88	380	3	z2)2	z2)2	NUM
ma-88	380	4	+	+	NUM
ma-88	380	5	ε2(1	ε2(1	NOUN
ma-88	380	6	+	+	CCONJ
ma-88	380	7	z)4	z)4	X
ma-88	380	8	+	+	CCONJ
ma-88	380	9	2ε(1	2ε(1	NUM
ma-88	380	10	+	+	CCONJ
ma-88	380	11	z)2(1−	z)2(1−	ADJ
ma-88	380	12	z2)−	z2)−	NOUN
ma-88	380	13	(	(	PUNCT
ma-88	380	14	4z)(z	4z)(z	NUM
ma-88	380	15	+	+	X
ma-88	380	16	1)2	1)2	NUM
ma-88	380	17	≤	≤	NOUN
ma-88	380	18	0	0	NUM
ma-88	381	1	=	=	NOUN
ma-88	381	2	⇒	⇒	NOUN
ma-88	381	3	∆	∆	VERB
ma-88	381	4	≤	≤	ADV
ma-88	381	5	0	0	NUM
ma-88	381	6	.	.	PUNCT
ma-88	382	1	thus	thus	ADV
ma-88	382	2	we	we	PRON
ma-88	382	3	get	get	VERB
ma-88	382	4	z	z	NOUN
ma-88	382	5	≤	≤	NOUN
ma-88	382	6	3−ε2	3−ε2	NUM
ma-88	382	7	+	+	NOUN
ma-88	382	8	2	2	NUM
ma-88	382	9	√	√	NUM
ma-88	382	10	2−ε2	2−ε2	NUM
ma-88	382	11	(	(	PUNCT
ma-88	382	12	1−ε)2	1−ε)2	NUM
ma-88	382	13	which	which	PRON
ma-88	382	14	implies	imply	VERB
ma-88	382	15	‖ty‖	‖ty‖	NOUN
ma-88	382	16	≤	≤	ADJ
ma-88	382	17	3−ε2	3−ε2	NUM
ma-88	382	18	+	+	NOUN
ma-88	382	19	2	2	NUM
ma-88	382	20	√	√	NUM
ma-88	382	21	2−ε2	2−ε2	NUM
ma-88	382	22	(	(	PUNCT
ma-88	382	23	1−ε)2	1−ε)2	NUM
ma-88	382	24	‖tx‖.	‖tx‖.	PROPN
ma-88	382	25	since	since	SCONJ
ma-88	382	26	x	x	PRON
ma-88	382	27	,	,	PUNCT
ma-88	382	28	y	y	PROPN
ma-88	382	29	are	be	AUX
ma-88	382	30	arbitrary	arbitrary	ADJ
ma-88	382	31	,	,	PUNCT
ma-88	382	32	t	t	PROPN
ma-88	382	33	is	be	AUX
ma-88	382	34	bounded	bound	VERB
ma-88	382	35	and	and	CCONJ
ma-88	382	36	‖tx‖	‖tx‖	PROPN
ma-88	382	37	≥	≥	NUM
ma-88	382	38	(	(	PUNCT
ma-88	382	39	1−	1−	NUM
ma-88	382	40	ε)2	ε)2	NOUN
ma-88	382	41	3−	3−	NUM
ma-88	382	42	ε2	ε2	ADJ
ma-88	382	43	+	+	CCONJ
ma-88	382	44	2	2	NUM
ma-88	382	45	√	√	NUM
ma-88	382	46	2−	2−	NUM
ma-88	382	47	ε2	ε2	ADJ
ma-88	382	48	‖t‖‖x‖.	‖t‖‖x‖.	PUNCT
ma-88	382	49	theorem	theorem	ADJ
ma-88	382	50	3.7	3.7	NUM
ma-88	382	51	.	.	PUNCT
ma-88	383	1	let	let	VERB
ma-88	383	2	t	t	NOUN
ma-88	383	3	:	:	PUNCT
ma-88	383	4	x	x	X
ma-88	383	5	→	→	SYM
ma-88	383	6	y	y	X
ma-88	383	7	be	be	AUX
ma-88	383	8	a	a	DET
ma-88	383	9	nontrivial	nontrivial	ADJ
ma-88	383	10	linear	linear	NOUN
ma-88	383	11	mapping	mapping	NOUN
ma-88	383	12	satisfying	satisfying	ADJ
ma-88	383	13	x	x	SYM
ma-88	383	14	⊥i	⊥i	PROPN
ma-88	383	15	y	y	PROPN
ma-88	383	16	=	=	PROPN
ma-88	383	17	⇒	⇒	PROPN
ma-88	383	18	tx	tx	PROPN
ma-88	383	19	ε	ε	PROPN
ma-88	383	20	⊥b	⊥b	PRON
ma-88	383	21	ty	ty	PROPN
ma-88	383	22	,	,	PUNCT
ma-88	383	23	x	x	PRON
ma-88	383	24	,	,	PUNCT
ma-88	383	25	y	y	PROPN
ma-88	383	26	∈	∈	PROPN
ma-88	384	1	x.	x.	NOUN
ma-88	384	2	then	then	ADV
ma-88	384	3	t	t	PROPN
ma-88	384	4	is	be	AUX
ma-88	384	5	a	a	DET
ma-88	384	6	scalar	scalar	ADJ
ma-88	384	7	mutiple	mutiple	NOUN
ma-88	384	8	of	of	ADP
ma-88	384	9	a	a	DET
ma-88	384	10	isometric	isometric	ADJ
ma-88	384	11	mapping	mapping	NOUN
ma-88	384	12	,	,	PUNCT
ma-88	384	13	i.e.	i.e.	X
ma-88	384	14	,	,	PUNCT
ma-88	384	15	for	for	ADP
ma-88	384	16	some	some	DET
ma-88	384	17	γ	γ	X
ma-88	384	18	>	>	X
ma-88	384	19	0	0	PROPN
ma-88	384	20	,	,	PUNCT
ma-88	384	21	‖tx‖	‖tx‖	PROPN
ma-88	384	22	=	=	SYM
ma-88	384	23	γ‖x‖.	γ‖x‖.	PROPN
ma-88	384	24	proof	proof	NOUN
ma-88	384	25	.	.	PUNCT
ma-88	385	1	take	take	VERB
ma-88	385	2	two	two	NUM
ma-88	385	3	arbitrary	arbitrary	ADJ
ma-88	385	4	unit	unit	NOUN
ma-88	385	5	vectors	vector	NOUN
ma-88	385	6	x	x	PUNCT
ma-88	385	7	and	and	CCONJ
ma-88	385	8	y	y	PROPN
ma-88	385	9	and	and	CCONJ
ma-88	385	10	note	note	VERB
ma-88	385	11	that	that	SCONJ
ma-88	385	12	x+y	x+y	PROPN
ma-88	385	13	2	2	NUM
ma-88	385	14	⊥i	⊥i	NOUN
ma-88	385	15	x−y	x−y	PROPN
ma-88	385	16	2	2	NUM
ma-88	385	17	,	,	PUNCT
ma-88	385	18	it	it	PRON
ma-88	385	19	follows	follow	VERB
ma-88	385	20	that	that	SCONJ
ma-88	385	21	t	t	PROPN
ma-88	385	22	(	(	PUNCT
ma-88	385	23	x	x	SYM
ma-88	385	24	+	+	NUM
ma-88	385	25	y)ε	y)ε	ADJ
ma-88	385	26	⊥b	⊥b	DET
ma-88	385	27	t	t	NOUN
ma-88	385	28	(	(	PUNCT
ma-88	385	29	x	x	PROPN
ma-88	385	30	−	−	PROPN
ma-88	385	31	y	y	PROPN
ma-88	385	32	)	)	PUNCT
ma-88	385	33	.	.	PUNCT
ma-88	386	1	hence	hence	ADV
ma-88	386	2	for	for	ADP
ma-88	386	3	all	all	DET
ma-88	386	4	λ	λ	NOUN
ma-88	386	5	∈	∈	NOUN
ma-88	386	6	r	r	NOUN
ma-88	386	7	we	we	PRON
ma-88	386	8	have	have	VERB
ma-88	386	9	‖t	‖t	NOUN
ma-88	386	10	(	(	PUNCT
ma-88	386	11	x	x	SYM
ma-88	386	12	+	+	NUM
ma-88	386	13	y	y	NOUN
ma-88	386	14	)	)	PUNCT
ma-88	387	1	+	+	CCONJ
ma-88	387	2	λt	λt	ADP
ma-88	387	3	(	(	PUNCT
ma-88	387	4	x	x	X
ma-88	387	5	−	−	X
ma-88	387	6	y)‖	y)‖	PRON
ma-88	387	7	≥	≥	NOUN
ma-88	387	8	(	(	PUNCT
ma-88	387	9	1−	1−	NUM
ma-88	387	10	ε)‖t	ε)‖t	NOUN
ma-88	387	11	(	(	PUNCT
ma-88	387	12	x	x	SYM
ma-88	387	13	+	+	CCONJ
ma-88	387	14	y)‖	y)‖	ADJ
ma-88	387	15	,	,	PUNCT
ma-88	387	16	thus	thus	ADV
ma-88	387	17	(	(	PUNCT
ma-88	387	18	‖(1	‖(1	NUM
ma-88	387	19	+	+	CCONJ
ma-88	387	20	λ)tx‖+	λ)tx‖+	ADJ
ma-88	387	21	‖(1−	‖(1−	NUM
ma-88	387	22	λ)ty‖	λ)ty‖	NOUN
ma-88	387	23	)	)	PUNCT
ma-88	387	24	2	2	NUM
ma-88	387	25	≥	≥	NOUN
ma-88	387	26	(	(	PUNCT
ma-88	387	27	1−	1−	NUM
ma-88	387	28	ε)2‖tx	ε)2‖tx	PROPN
ma-88	387	29	+	+	CCONJ
ma-88	387	30	ty‖2	ty‖2	PROPN
ma-88	387	31	≥	≥	X
ma-88	387	32	(	(	PUNCT
ma-88	387	33	1−	1−	NUM
ma-88	387	34	ε)2(‖tx‖	ε)2(‖tx‖	PROPN
ma-88	387	35	−	−	PROPN
ma-88	387	36	‖ty‖)2	‖ty‖)2	PROPN
ma-88	387	37	.	.	PUNCT
ma-88	388	1	if	if	SCONJ
ma-88	388	2	tx	tx	PROPN
ma-88	388	3	=	=	SYM
ma-88	388	4	0	0	PROPN
ma-88	388	5	,	,	PUNCT
ma-88	388	6	then	then	ADV
ma-88	388	7	let	let	VERB
ma-88	388	8	y	y	PROPN
ma-88	388	9	∈	∈	PROPN
ma-88	388	10	x	x	PUNCT
ma-88	388	11	such	such	ADJ
ma-88	388	12	that	that	PRON
ma-88	388	13	ty	ty	NUM
ma-88	388	14	6=	6=	ADP
ma-88	388	15	0	0	NUM
ma-88	388	16	,	,	PUNCT
ma-88	388	17	substitute	substitute	NOUN
ma-88	388	18	x	x	SYM
ma-88	388	19	,	,	PUNCT
ma-88	388	20	y	y	PROPN
ma-88	388	21	into	into	ADP
ma-88	388	22	the	the	DET
ma-88	388	23	above	above	ADJ
ma-88	388	24	formula	formula	NOUN
ma-88	388	25	,	,	PUNCT
ma-88	388	26	we	we	PRON
ma-88	388	27	have	have	VERB
ma-88	388	28	‖(1−	‖(1−	PROPN
ma-88	388	29	λ)ty‖2	λ)ty‖2	NUM
ma-88	388	30	≥	≥	NOUN
ma-88	388	31	(	(	PUNCT
ma-88	388	32	1−	1−	NUM
ma-88	388	33	ε)2‖ty‖2	ε)2‖ty‖2	PROPN
ma-88	388	34	=	=	SYM
ma-88	388	35	⇒	⇒	NOUN
ma-88	388	36	(	(	PUNCT
ma-88	388	37	1−	1−	NUM
ma-88	388	38	λ)2	λ)2	PROPN
ma-88	388	39	≥	≥	NUM
ma-88	388	40	(	(	PUNCT
ma-88	388	41	1−	1−	NUM
ma-88	388	42	ε)2	ε)2	PROPN
ma-88	388	43	f	f	PROPN
ma-88	388	44	or	or	CCONJ
ma-88	389	1	al	al	PROPN
ma-88	389	2	l	l	PROPN
ma-88	389	3	λ	λ	PROPN
ma-88	389	4	∈	∈	PROPN
ma-88	389	5	r	r	NOUN
ma-88	389	6	,	,	PUNCT
ma-88	389	7	it	it	PRON
ma-88	389	8	is	be	AUX
ma-88	389	9	impossible	impossible	ADJ
ma-88	389	10	,	,	PUNCT
ma-88	389	11	then	then	ADV
ma-88	389	12	we	we	PRON
ma-88	389	13	can	can	AUX
ma-88	389	14	divide	divide	VERB
ma-88	389	15	both	both	DET
ma-88	389	16	sides	side	NOUN
ma-88	389	17	by	by	ADP
ma-88	389	18	‖tx‖2	‖tx‖2	PROPN
ma-88	389	19	like	like	ADP
ma-88	389	20	before	before	ADV
ma-88	389	21	and	and	CCONJ
ma-88	389	22	denote	denote	VERB
ma-88	389	23	z	z	NOUN
ma-88	389	24	=	=	SYM
ma-88	389	25	‖ty‖	‖ty‖	PROPN
ma-88	389	26	‖tx‖	‖tx‖	PROPN
ma-88	389	27	,	,	PUNCT
ma-88	389	28	we	we	PRON
ma-88	389	29	get	get	VERB
ma-88	389	30	(	(	PUNCT
ma-88	389	31	1−	1−	NUM
ma-88	389	32	z)2λ2	z)2λ2	PROPN
ma-88	389	33	+	+	CCONJ
ma-88	390	1	2(1−	2(1−	X
ma-88	390	2	z2)λ+	z2)λ+	X
ma-88	390	3	(	(	PUNCT
ma-88	390	4	z	z	NOUN
ma-88	390	5	+	+	NOUN
ma-88	390	6	1)2	1)2	NUM
ma-88	390	7	−	−	NOUN
ma-88	390	8	(	(	PUNCT
ma-88	390	9	1−	1−	NUM
ma-88	390	10	ε)2(1−	ε)2(1−	PROPN
ma-88	390	11	z)2	z)2	PROPN
ma-88	390	12	≥	≥	NOUN
ma-88	390	13	0	0	NUM
ma-88	390	14	f	f	PROPN
ma-88	390	15	or	or	CCONJ
ma-88	390	16	al	al	PROPN
ma-88	390	17	l	l	PROPN
ma-88	390	18	λ	λ	PROPN
ma-88	390	19	∈	∈	PROPN
ma-88	390	20	r.	r.	PROPN
ma-88	390	21	if	if	SCONJ
ma-88	390	22	z	z	PROPN
ma-88	390	23	6=	6=	NUM
ma-88	390	24	1	1	NUM
ma-88	390	25	,	,	PUNCT
ma-88	390	26	then	then	ADV
ma-88	390	27	∆	∆	PROPN
ma-88	390	28	=	=	SYM
ma-88	390	29	4	4	X
ma-88	390	30	·	·	PUNCT
ma-88	390	31	(	(	PUNCT
ma-88	390	32	1	1	NUM
ma-88	390	33	−	−	PROPN
ma-88	390	34	z)4(1	z)4(1	PROPN
ma-88	390	35	−	−	PROPN
ma-88	390	36	ε)2	ε)2	PROPN
ma-88	390	37	>	>	X
ma-88	390	38	0	0	NUM
ma-88	390	39	.	.	PUNCT
ma-88	391	1	thus	thus	ADV
ma-88	391	2	the	the	DET
ma-88	391	3	inequality	inequality	NOUN
ma-88	391	4	is	be	AUX
ma-88	391	5	satisfied	satisfied	ADJ
ma-88	391	6	only	only	ADV
ma-88	391	7	when	when	SCONJ
ma-88	391	8	z	z	NOUN
ma-88	391	9	=	=	SYM
ma-88	391	10	1	1	NUM
ma-88	391	11	,	,	PUNCT
ma-88	391	12	so	so	ADV
ma-88	391	13	z	z	NOUN
ma-88	391	14	=	=	SYM
ma-88	391	15	1	1	NUM
ma-88	391	16	which	which	PRON
ma-88	391	17	means	mean	VERB
ma-88	391	18	that	that	SCONJ
ma-88	391	19	‖tx‖	‖tx‖	PROPN
ma-88	391	20	=	=	SYM
ma-88	391	21	‖ty‖	‖ty‖	NOUN
ma-88	391	22	,	,	PUNCT
ma-88	391	23	thus	thus	ADV
ma-88	391	24	t	t	PROPN
ma-88	391	25	must	must	AUX
ma-88	391	26	be	be	AUX
ma-88	391	27	a	a	DET
ma-88	391	28	scalar	scalar	ADJ
ma-88	391	29	mutiple	mutiple	NOUN
ma-88	391	30	of	of	ADP
ma-88	391	31	a	a	DET
ma-88	391	32	isometric	isometric	ADJ
ma-88	391	33	mapping	mapping	NOUN
ma-88	391	34	.	.	PUNCT
ma-88	392	1	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	392	2	eur	eur	PROPN
ma-88	392	3	.	.	PUNCT
ma-88	393	1	j.	j.	PROPN
ma-88	393	2	math	math	PROPN
ma-88	393	3	.	.	PUNCT
ma-88	394	1	anal	anal	PROPN
ma-88	394	2	.	.	PUNCT
ma-88	395	1	10.28924	10.28924	NUM
ma-88	395	2	/	/	SYM
ma-88	395	3	ada	ada	PROPN
ma-88	395	4	/	/	SYM
ma-88	395	5	ma.2.16	ma.2.16	PROPN
ma-88	395	6	13references	13references	NUM
ma-88	396	1	[	[	X
ma-88	396	2	1	1	NUM
ma-88	396	3	]	]	PUNCT
ma-88	396	4	m.	m.	PROPN
ma-88	396	5	abbasi	abbasi	PROPN
ma-88	396	6	,	,	PUNCT
ma-88	396	7	a.y	a.y	PROPN
ma-88	396	8	.	.	PROPN
ma-88	396	9	kruger	kruger	PROPN
ma-88	396	10	,	,	PUNCT
ma-88	396	11	m.	m.	NOUN
ma-88	396	12	théra	théra	NOUN
ma-88	396	13	,	,	PUNCT
ma-88	396	14	gateaux	gateaux	PRON
ma-88	396	15	differentiability	differentiability	NOUN
ma-88	396	16	revisited	revisit	VERB
ma-88	396	17	,	,	PUNCT
ma-88	396	18	appl	appl	PROPN
ma-88	396	19	.	.	PROPN
ma-88	396	20	math	math	PROPN
ma-88	396	21	.	.	PUNCT
ma-88	397	1	optim	optim	ADJ
ma-88	397	2	.	.	PUNCT
ma-88	398	1	84	84	NUM
ma-88	398	2	(	(	PUNCT
ma-88	398	3	2021	2021	NUM
ma-88	398	4	)	)	PUNCT
ma-88	398	5	3499–3516	3499–3516	NUM
ma-88	398	6	.	.	PUNCT
ma-88	399	1	https://doi.org/10.1007/s00245-021-09754-y.[2	https://doi.org/10.1007/s00245-021-09754-y.[2	PROPN
ma-88	399	2	]	]	PUNCT
ma-88	399	3	j.	j.	PROPN
ma-88	399	4	alonso	alonso	PROPN
ma-88	399	5	,	,	PUNCT
ma-88	399	6	h.	h.	PROPN
ma-88	399	7	martini	martini	PROPN
ma-88	399	8	,	,	PUNCT
ma-88	399	9	s.	s.	PROPN
ma-88	399	10	wu	wu	PROPN
ma-88	399	11	,	,	PUNCT
ma-88	399	12	on	on	ADP
ma-88	399	13	birkhoff	birkhoff	NOUN
ma-88	399	14	orthogonality	orthogonality	NOUN
ma-88	399	15	and	and	CCONJ
ma-88	399	16	isosceles	isoscele	NOUN
ma-88	399	17	orthogonality	orthogonality	NOUN
ma-88	399	18	in	in	ADP
ma-88	399	19	normed	normed	ADJ
ma-88	399	20	linear	linear	PROPN
ma-88	399	21	spaces	space	NOUN
ma-88	399	22	,	,	PUNCT
ma-88	399	23	aequat.math	aequat.math	PROPN
ma-88	399	24	.	.	PROPN
ma-88	399	25	83	83	NUM
ma-88	399	26	(	(	PUNCT
ma-88	399	27	2011	2011	NUM
ma-88	399	28	)	)	PUNCT
ma-88	399	29	153–189	153–189	NUM
ma-88	399	30	.	.	PUNCT
ma-88	400	1	https://doi.org/10.1007/s00010-011-0092-z.[3	https://doi.org/10.1007/s00010-011-0092-z.[3	NOUN
ma-88	400	2	]	]	X
ma-88	400	3	d.	d.	PROPN
ma-88	400	4	amir	amir	PROPN
ma-88	400	5	,	,	PUNCT
ma-88	400	6	characterizations	characterization	NOUN
ma-88	400	7	of	of	ADP
ma-88	400	8	inner	inner	ADJ
ma-88	400	9	product	product	NOUN
ma-88	400	10	spaces	space	NOUN
ma-88	400	11	,	,	PUNCT
ma-88	400	12	birkhäuser	birkhäuser	PROPN
ma-88	400	13	basel	basel	PROPN
ma-88	400	14	,	,	PUNCT
ma-88	400	15	1986	1986	NUM
ma-88	400	16	.	.	PUNCT
ma-88	401	1	https://doi.org/10.1007/	https://doi.org/10.1007/	PROPN
ma-88	401	2	978	978	NUM
ma-88	401	3	-	-	PUNCT
ma-88	401	4	3	3	NUM
ma-88	401	5	-	-	PUNCT
ma-88	401	6	0348	0348	NUM
ma-88	401	7	-	-	PUNCT
ma-88	401	8	5487	5487	NUM
ma-88	401	9	-	-	SYM
ma-88	401	10	0.[4	0.[4	NUM
ma-88	401	11	]	]	X
ma-88	401	12	g.	g.	PROPN
ma-88	401	13	ascoli	ascoli	PROPN
ma-88	401	14	,	,	PUNCT
ma-88	401	15	sugli	sugli	PROPN
ma-88	401	16	spazi	spazi	X
ma-88	401	17	lineari	lineari	PROPN
ma-88	401	18	metrici	metrici	PROPN
ma-88	401	19	e	e	PROPN
ma-88	401	20	le	le	X
ma-88	401	21	loro	loro	PROPN
ma-88	401	22	varietà	varietà	PROPN
ma-88	401	23	lineari	lineari	PROPN
ma-88	401	24	,	,	PUNCT
ma-88	401	25	ann	ann	PROPN
ma-88	401	26	.	.	PROPN
ma-88	401	27	mat	mat	PROPN
ma-88	401	28	.	.	PROPN
ma-88	401	29	10	10	NUM
ma-88	401	30	(	(	PUNCT
ma-88	401	31	1932	1932	NUM
ma-88	401	32	)	)	PUNCT
ma-88	401	33	203–232	203–232	NUM
ma-88	401	34	.	.	PUNCT
ma-88	402	1	https://doi.org/10	https://doi.org/10	PROPN
ma-88	402	2	.	.	PUNCT
ma-88	403	1	1007	1007	NUM
ma-88	403	2	/	/	SYM
ma-88	403	3	bf02417142.[5	bf02417142.[5	PROPN
ma-88	403	4	]	]	X
ma-88	403	5	g.	g.	PROPN
ma-88	403	6	birkhoff	birkhoff	PROPN
ma-88	403	7	,	,	PUNCT
ma-88	403	8	orthogonality	orthogonality	NOUN
ma-88	403	9	in	in	ADP
ma-88	403	10	linear	linear	ADJ
ma-88	403	11	metric	metric	ADJ
ma-88	403	12	spaces	space	NOUN
ma-88	403	13	,	,	PUNCT
ma-88	403	14	duke	duke	PROPN
ma-88	403	15	math	math	PROPN
ma-88	403	16	.	.	PUNCT
ma-88	404	1	j.	j.	PROPN
ma-88	404	2	1	1	NUM
ma-88	404	3	(	(	PUNCT
ma-88	404	4	1935	1935	NUM
ma-88	404	5	)	)	PUNCT
ma-88	404	6	169	169	NUM
ma-88	404	7	-	-	SYM
ma-88	404	8	172	172	NUM
ma-88	404	9	.	.	PUNCT
ma-88	405	1	https://doi.org/10.1215/	https://doi.org/10.1215/	PROPN
ma-88	405	2	s0012	s0012	PROPN
ma-88	405	3	-	-	PUNCT
ma-88	405	4	7094	7094	NUM
ma-88	405	5	-	-	SYM
ma-88	405	6	35	35	NUM
ma-88	405	7	-	-	PUNCT
ma-88	405	8	00115	00115	NUM
ma-88	405	9	-	-	PUNCT
ma-88	405	10	6.[6	6.[6	PROPN
ma-88	405	11	]	]	X
ma-88	405	12	j.	j.	PROPN
ma-88	405	13	chmieliński	chmieliński	PROPN
ma-88	405	14	,	,	PUNCT
ma-88	405	15	linear	linear	ADJ
ma-88	405	16	mappings	mapping	NOUN
ma-88	405	17	approximately	approximately	ADV
ma-88	405	18	preserving	preserve	VERB
ma-88	405	19	orthogonality	orthogonality	NOUN
ma-88	405	20	,	,	PUNCT
ma-88	405	21	j.	j.	PROPN
ma-88	405	22	math	math	PROPN
ma-88	405	23	.	.	PUNCT
ma-88	406	1	anal	anal	PROPN
ma-88	406	2	.	.	PUNCT
ma-88	407	1	appl	appl	PROPN
ma-88	407	2	.	.	PUNCT
ma-88	408	1	304	304	NUM
ma-88	408	2	(	(	PUNCT
ma-88	408	3	2005	2005	NUM
ma-88	408	4	)	)	PUNCT
ma-88	408	5	158–169	158–169	NUM
ma-88	408	6	.	.	PUNCT
ma-88	409	1	https://doi.org/10.1016/j.jmaa.2004.09.011.[7	https://doi.org/10.1016/j.jmaa.2004.09.011.[7	PROPN
ma-88	409	2	]	]	PUNCT
ma-88	409	3	j.	j.	PROPN
ma-88	409	4	chmieliński	chmieliński	PROPN
ma-88	409	5	,	,	PUNCT
ma-88	409	6	stability	stability	NOUN
ma-88	409	7	of	of	ADP
ma-88	409	8	the	the	DET
ma-88	409	9	orthogonality	orthogonality	NOUN
ma-88	409	10	preserving	preserve	VERB
ma-88	409	11	property	property	NOUN
ma-88	409	12	in	in	ADP
ma-88	409	13	finite	finite	ADJ
ma-88	409	14	-	-	ADJ
ma-88	409	15	dimensional	dimensional	ADJ
ma-88	409	16	inner	inner	ADJ
ma-88	409	17	product	product	NOUN
ma-88	409	18	spaces	space	NOUN
ma-88	409	19	,	,	PUNCT
ma-88	409	20	j.	j.	PROPN
ma-88	409	21	math.anal	math.anal	PROPN
ma-88	409	22	.	.	PUNCT
ma-88	409	23	appl	appl	PROPN
ma-88	409	24	.	.	PUNCT
ma-88	410	1	318	318	NUM
ma-88	410	2	(	(	PUNCT
ma-88	410	3	2006	2006	NUM
ma-88	410	4	)	)	PUNCT
ma-88	410	5	433–443	433–443	NUM
ma-88	410	6	.	.	PUNCT
ma-88	411	1	https://doi.org/10.1016/j.jmaa.2005.06.016.[8	https://doi.org/10.1016/j.jmaa.2005.06.016.[8	PROPN
ma-88	411	2	]	]	PUNCT
ma-88	411	3	j.	j.	PROPN
ma-88	411	4	chmieliński	chmieliński	PROPN
ma-88	411	5	,	,	PUNCT
ma-88	411	6	j.	j.	PROPN
ma-88	411	7	chmieliński	chmieliński	PROPN
ma-88	411	8	,	,	PUNCT
ma-88	411	9	orthogonality	orthogonality	NOUN
ma-88	411	10	preserving	preserve	VERB
ma-88	411	11	property	property	NOUN
ma-88	411	12	and	and	CCONJ
ma-88	411	13	its	its	PRON
ma-88	411	14	ulam	ulam	NOUN
ma-88	411	15	stability	stability	NOUN
ma-88	411	16	.	.	PUNCT
ma-88	412	1	in	in	ADP
ma-88	412	2	:	:	PUNCT
ma-88	412	3	rassias	rassias	PROPN
ma-88	412	4	,	,	PUNCT
ma-88	412	5	t.	t.	PROPN
ma-88	412	6	,	,	PUNCT
ma-88	412	7	brzdek	brzdek	PROPN
ma-88	412	8	,	,	PUNCT
ma-88	412	9	j.(eds	j.(ed	NOUN
ma-88	412	10	)	)	PUNCT
ma-88	412	11	functional	functional	ADJ
ma-88	412	12	equations	equation	NOUN
ma-88	412	13	in	in	ADP
ma-88	412	14	mathematical	mathematical	ADJ
ma-88	412	15	analysis	analysis	NOUN
ma-88	412	16	.	.	PUNCT
ma-88	413	1	springer	springer	NOUN
ma-88	413	2	optimization	optimization	NOUN
ma-88	413	3	and	and	CCONJ
ma-88	413	4	its	its	PRON
ma-88	413	5	applications	application	NOUN
ma-88	413	6	,	,	PUNCT
ma-88	413	7	vol	vol	NOUN
ma-88	413	8	52	52	NUM
ma-88	413	9	.	.	PUNCT
ma-88	413	10	springer	springer	NOUN
ma-88	413	11	,	,	PUNCT
ma-88	413	12	new	new	PROPN
ma-88	413	13	york	york	PROPN
ma-88	413	14	,	,	PUNCT
ma-88	413	15	ny	ny	PROPN
ma-88	413	16	.	.	PROPN
ma-88	413	17	(	(	PUNCT
ma-88	413	18	2011	2011	NUM
ma-88	413	19	)	)	PUNCT
ma-88	413	20	.	.	PUNCT
ma-88	414	1	https://doi.org/10.1007/978-1-4614-0055-4_4.[9	https://doi.org/10.1007/978-1-4614-0055-4_4.[9	X
ma-88	414	2	]	]	X
ma-88	414	3	j.	j.	PROPN
ma-88	414	4	chmieliński	chmieliński	PROPN
ma-88	414	5	,	,	PUNCT
ma-88	414	6	t.	t.	PROPN
ma-88	414	7	stypuła	stypuła	PROPN
ma-88	414	8	,	,	PUNCT
ma-88	414	9	p.	p.	PROPN
ma-88	414	10	wójcik	wójcik	PROPN
ma-88	414	11	,	,	PUNCT
ma-88	414	12	approximate	approximate	ADJ
ma-88	414	13	orthogonality	orthogonality	NOUN
ma-88	414	14	in	in	ADP
ma-88	414	15	normed	normed	ADJ
ma-88	414	16	spaces	space	NOUN
ma-88	414	17	and	and	CCONJ
ma-88	414	18	its	its	PRON
ma-88	414	19	applications	application	NOUN
ma-88	414	20	,	,	PUNCT
ma-88	414	21	linearalgebra	linearalgebra	PROPN
ma-88	414	22	appl	appl	NOUN
ma-88	414	23	.	.	PUNCT
ma-88	415	1	531	531	NUM
ma-88	415	2	(	(	PUNCT
ma-88	415	3	2017	2017	NUM
ma-88	415	4	)	)	PUNCT
ma-88	416	1	305–317	305–317	NUM
ma-88	416	2	.	.	PUNCT
ma-88	417	1	https://doi.org/10.1016/j.laa.2017.06.001.[10	https://doi.org/10.1016/j.laa.2017.06.001.[10	PROPN
ma-88	417	2	]	]	PUNCT
ma-88	417	3	j.	j.	PROPN
ma-88	417	4	chmieliński	chmieliński	PROPN
ma-88	417	5	,	,	PUNCT
ma-88	417	6	p.	p.	PROPN
ma-88	417	7	wójcik	wójcik	PROPN
ma-88	417	8	,	,	PUNCT
ma-88	417	9	on	on	ADP
ma-88	417	10	a	a	DET
ma-88	417	11	ρ	ρ	NOUN
ma-88	417	12	-	-	PUNCT
ma-88	417	13	orthogonality	orthogonality	NOUN
ma-88	417	14	,	,	PUNCT
ma-88	417	15	aequat	aequat	PROPN
ma-88	417	16	.	.	PUNCT
ma-88	418	1	math	math	NOUN
ma-88	418	2	.	.	PUNCT
ma-88	419	1	80	80	NUM
ma-88	419	2	(	(	PUNCT
ma-88	419	3	2010	2010	NUM
ma-88	419	4	)	)	PUNCT
ma-88	419	5	45–55	45–55	NOUN
ma-88	419	6	.	.	PUNCT
ma-88	420	1	https://doi.org/10.1007/	https://doi.org/10.1007/	PROPN
ma-88	420	2	s00010	s00010	PROPN
ma-88	420	3	-	-	PUNCT
ma-88	420	4	010	010	NUM
ma-88	420	5	-	-	PUNCT
ma-88	420	6	0042	0042	NUM
ma-88	420	7	-	-	SYM
ma-88	420	8	1.[11	1.[11	NUM
ma-88	420	9	]	]	PUNCT
ma-88	420	10	j.	j.	PROPN
ma-88	420	11	chmieliński	chmieliński	PROPN
ma-88	420	12	,	,	PUNCT
ma-88	420	13	p.	p.	PROPN
ma-88	420	14	wójcik	wójcik	PROPN
ma-88	420	15	,	,	PUNCT
ma-88	420	16	isosceles	isoscele	NOUN
ma-88	420	17	-	-	PUNCT
ma-88	420	18	orthogonality	orthogonality	NOUN
ma-88	420	19	preserving	preserve	VERB
ma-88	420	20	property	property	NOUN
ma-88	420	21	and	and	CCONJ
ma-88	420	22	its	its	PRON
ma-88	420	23	stability	stability	NOUN
ma-88	420	24	,	,	PUNCT
ma-88	420	25	nonlinear	nonlinear	ADJ
ma-88	420	26	anal	anal	NOUN
ma-88	420	27	.	.	PUNCT
ma-88	420	28	:	:	PUNCT
ma-88	421	1	theorymethods	theorymethod	NOUN
ma-88	421	2	appl	appl	NOUN
ma-88	421	3	.	.	PUNCT
ma-88	422	1	72	72	NUM
ma-88	422	2	(	(	PUNCT
ma-88	422	3	2010	2010	NUM
ma-88	422	4	)	)	PUNCT
ma-88	422	5	1445–1453	1445–1453	NUM
ma-88	422	6	.	.	PUNCT
ma-88	423	1	https://doi.org/10.1016/j.na.2009.08.028.[12	https://doi.org/10.1016/j.na.2009.08.028.[12	PROPN
ma-88	423	2	]	]	X
ma-88	423	3	c.	c.	PROPN
ma-88	423	4	chorianopoulos	chorianopoulos	PROPN
ma-88	423	5	,	,	PUNCT
ma-88	423	6	p.	p.	PROPN
ma-88	423	7	psarrakos	psarrakos	NOUN
ma-88	423	8	,	,	PUNCT
ma-88	423	9	birkhoff	birkhoff	NOUN
ma-88	423	10	–	–	PUNCT
ma-88	423	11	james	james	PROPN
ma-88	423	12	approximate	approximate	ADJ
ma-88	423	13	orthogonality	orthogonality	NOUN
ma-88	423	14	sets	set	NOUN
ma-88	423	15	and	and	CCONJ
ma-88	423	16	numerical	numerical	ADJ
ma-88	423	17	ranges	range	NOUN
ma-88	423	18	,	,	PUNCT
ma-88	423	19	linear	linear	PROPN
ma-88	423	20	al	al	PROPN
ma-88	423	21	-	-	PUNCT
ma-88	423	22	gebra	gebra	PROPN
ma-88	423	23	appl	appl	NOUN
ma-88	423	24	.	.	PROPN
ma-88	423	25	434	434	NUM
ma-88	423	26	(	(	PUNCT
ma-88	423	27	2011	2011	NUM
ma-88	423	28	)	)	PUNCT
ma-88	423	29	2089–2108	2089–2108	NUM
ma-88	423	30	.	.	PUNCT
ma-88	424	1	https://doi.org/10.1016/j.laa.2010.12.008.[13	https://doi.org/10.1016/j.laa.2010.12.008.[13	PROPN
ma-88	424	2	]	]	PUNCT
ma-88	424	3	s.	s.	PROPN
ma-88	424	4	s.	s.	PROPN
ma-88	424	5	dragomir	dragomir	PROPN
ma-88	424	6	,	,	PUNCT
ma-88	424	7	on	on	ADP
ma-88	424	8	approximation	approximation	NOUN
ma-88	424	9	of	of	ADP
ma-88	424	10	continuous	continuous	ADJ
ma-88	424	11	linear	linear	NOUN
ma-88	424	12	functionals	functional	NOUN
ma-88	424	13	in	in	ADP
ma-88	424	14	normed	normed	ADJ
ma-88	424	15	linear	linear	PROPN
ma-88	424	16	spaces	space	NOUN
ma-88	424	17	,	,	PUNCT
ma-88	424	18	an	an	PROPN
ma-88	424	19	.	.	PUNCT
ma-88	424	20	univ	univ	PROPN
ma-88	424	21	.	.	PUNCT
ma-88	425	1	timişoara	timişoara	PROPN
ma-88	425	2	ser.ştiinţ	ser.ştiinţ	PROPN
ma-88	425	3	.	.	PUNCT
ma-88	425	4	mat	mat	PROPN
ma-88	425	5	.	.	PROPN
ma-88	425	6	29	29	NUM
ma-88	425	7	(	(	PUNCT
ma-88	425	8	1991	1991	NUM
ma-88	425	9	)	)	PUNCT
ma-88	425	10	51–58.[14	51–58.[14	NUM
ma-88	425	11	]	]	PUNCT
ma-88	425	12	l.	l.	PROPN
ma-88	425	13	flaminio	flaminio	PROPN
ma-88	425	14	,	,	PUNCT
ma-88	425	15	k.	k.	PROPN
ma-88	425	16	frączek	frączek	PROPN
ma-88	425	17	,	,	PUNCT
ma-88	425	18	j.	j.	PROPN
ma-88	425	19	kułaga	kułaga	PROPN
ma-88	425	20	-	-	PUNCT
ma-88	425	21	przymus	przymus	PROPN
ma-88	425	22	,	,	PUNCT
ma-88	425	23	m.	m.	NOUN
ma-88	425	24	lemańczyk	lemańczyk	NOUN
ma-88	425	25	,	,	PUNCT
ma-88	425	26	approximate	approximate	ADJ
ma-88	425	27	orthogonality	orthogonality	NOUN
ma-88	425	28	of	of	ADP
ma-88	425	29	powers	power	NOUN
ma-88	425	30	for	for	ADP
ma-88	425	31	ergodicaffine	ergodicaffine	NOUN
ma-88	425	32	unipotent	unipotent	ADJ
ma-88	425	33	diffeomorphisms	diffeomorphism	NOUN
ma-88	425	34	on	on	ADP
ma-88	425	35	nilmanifolds	nilmanifold	NOUN
ma-88	425	36	,	,	PUNCT
ma-88	425	37	studia	studia	PROPN
ma-88	425	38	math	math	NOUN
ma-88	425	39	.	.	PUNCT
ma-88	426	1	244	244	NUM
ma-88	426	2	(	(	PUNCT
ma-88	426	3	2019	2019	NUM
ma-88	426	4	)	)	PUNCT
ma-88	426	5	43–97	43–97	NUM
ma-88	426	6	.	.	PUNCT
ma-88	427	1	https://doi.org/10.4064/	https://doi.org/10.4064/	PROPN
ma-88	427	2	sm170512	sm170512	PROPN
ma-88	427	3	-	-	PUNCT
ma-88	427	4	25	25	NUM
ma-88	427	5	-	-	PUNCT
ma-88	427	6	9.[15	9.[15	NUM
ma-88	427	7	]	]	X
ma-88	427	8	r.c	r.c	PROPN
ma-88	427	9	.	.	PROPN
ma-88	427	10	james	james	PROPN
ma-88	427	11	,	,	PUNCT
ma-88	427	12	orthogonality	orthogonality	NOUN
ma-88	427	13	in	in	ADP
ma-88	427	14	normed	normed	ADJ
ma-88	427	15	linear	linear	PROPN
ma-88	427	16	spaces	space	NOUN
ma-88	427	17	,	,	PUNCT
ma-88	427	18	duke	duke	PROPN
ma-88	427	19	math	math	PROPN
ma-88	427	20	.	.	PUNCT
ma-88	428	1	j.	j.	PROPN
ma-88	428	2	12	12	NUM
ma-88	428	3	(	(	PUNCT
ma-88	428	4	1945	1945	NUM
ma-88	428	5	)	)	PUNCT
ma-88	428	6	.	.	PUNCT
ma-88	429	1	https://doi.org/10.1215/	https://doi.org/10.1215/	PROPN
ma-88	429	2	s0012	s0012	PROPN
ma-88	429	3	-	-	PUNCT
ma-88	429	4	7094	7094	NUM
ma-88	429	5	-	-	PUNCT
ma-88	429	6	45	45	NUM
ma-88	429	7	-	-	PUNCT
ma-88	429	8	01223	01223	NUM
ma-88	429	9	-	-	PUNCT
ma-88	429	10	3.[16	3.[16	NUM
ma-88	429	11	]	]	X
ma-88	429	12	r.c	r.c	PROPN
ma-88	429	13	.	.	PROPN
ma-88	429	14	james	james	PROPN
ma-88	429	15	,	,	PUNCT
ma-88	429	16	orthogonality	orthogonality	NOUN
ma-88	429	17	and	and	CCONJ
ma-88	429	18	linear	linear	ADJ
ma-88	429	19	functionals	functional	NOUN
ma-88	429	20	in	in	ADP
ma-88	429	21	normed	normed	ADJ
ma-88	429	22	linear	linear	PROPN
ma-88	429	23	spaces	space	NOUN
ma-88	429	24	,	,	PUNCT
ma-88	429	25	trans	trans	PROPN
ma-88	429	26	.	.	PROPN
ma-88	430	1	amer	amer	PROPN
ma-88	430	2	.	.	PUNCT
ma-88	430	3	math	math	PROPN
ma-88	430	4	.	.	PUNCT
ma-88	431	1	soc	soc	PROPN
ma-88	431	2	.	.	PUNCT
ma-88	432	1	61	61	NUM
ma-88	432	2	(	(	PUNCT
ma-88	432	3	1947	1947	NUM
ma-88	432	4	)	)	PUNCT
ma-88	433	1	265–292	265–292	NUM
ma-88	433	2	.	.	PUNCT
ma-88	434	1	https://doi.org/10.1090/s0002-9947-1947-0021241-4.[17	https://doi.org/10.1090/s0002-9947-1947-0021241-4.[17	PROPN
ma-88	434	2	]	]	PUNCT
ma-88	434	3	o.p	o.p	PROPN
ma-88	434	4	.	.	PROPN
ma-88	434	5	kapoor	kapoor	PROPN
ma-88	434	6	,	,	PUNCT
ma-88	434	7	j.	j.	PROPN
ma-88	434	8	prasad	prasad	PROPN
ma-88	434	9	,	,	PUNCT
ma-88	434	10	orthogonality	orthogonality	NOUN
ma-88	434	11	and	and	CCONJ
ma-88	434	12	characterizations	characterization	NOUN
ma-88	434	13	of	of	ADP
ma-88	434	14	inner	inner	ADJ
ma-88	434	15	product	product	NOUN
ma-88	434	16	spaces	space	NOUN
ma-88	434	17	,	,	PUNCT
ma-88	434	18	bull	bull	NOUN
ma-88	434	19	.	.	PUNCT
ma-88	435	1	austral	austral	PROPN
ma-88	435	2	.	.	PUNCT
ma-88	436	1	math	math	NOUN
ma-88	436	2	.	.	PUNCT
ma-88	437	1	soc	soc	PROPN
ma-88	437	2	.	.	PUNCT
ma-88	438	1	19(1978	19(1978	X
ma-88	438	2	)	)	PUNCT
ma-88	439	1	403–416	403–416	NUM
ma-88	439	2	.	.	PUNCT
ma-88	440	1	https://doi.org/10.1017/s0004972700008947.[18	https://doi.org/10.1017/s0004972700008947.[18	X
ma-88	440	2	]	]	X
ma-88	440	3	p.m.	p.m.	NOUN
ma-88	440	4	miličić	miličić	PROPN
ma-88	440	5	,	,	PUNCT
ma-88	440	6	sur	sur	PROPN
ma-88	440	7	la	la	PRON
ma-88	440	8	g	g	NOUN
ma-88	440	9	-	-	PUNCT
ma-88	440	10	orthogonalité	orthogonalité	ADJ
ma-88	440	11	dans	dan	NOUN
ma-88	440	12	des	des	PROPN
ma-88	440	13	espaces	espaces	PROPN
ma-88	440	14	normés	normés	NOUN
ma-88	440	15	,	,	PUNCT
ma-88	440	16	mat	mat	PROPN
ma-88	440	17	.	.	PROPN
ma-88	440	18	vesnik	vesnik	PROPN
ma-88	440	19	39	39	NUM
ma-88	440	20	(	(	PUNCT
ma-88	440	21	1987	1987	NUM
ma-88	440	22	)	)	PUNCT
ma-88	441	1	325–334.[19	325–334.[19	NUM
ma-88	441	2	]	]	X
ma-88	441	3	b.	b.	PROPN
ma-88	441	4	mojškerc	mojškerc	PROPN
ma-88	441	5	,	,	PUNCT
ma-88	441	6	a.	a.	NOUN
ma-88	441	7	turnšek	turnšek	NOUN
ma-88	441	8	,	,	PUNCT
ma-88	441	9	mappings	mapping	NOUN
ma-88	441	10	approximately	approximately	ADV
ma-88	441	11	preserving	preserve	VERB
ma-88	441	12	orthogonality	orthogonality	NOUN
ma-88	441	13	in	in	ADP
ma-88	441	14	normed	normed	ADJ
ma-88	441	15	spaces	space	NOUN
ma-88	441	16	,	,	PUNCT
ma-88	441	17	nonlinear	nonlinear	ADJ
ma-88	441	18	anal.:theory	anal.:theory	PROPN
ma-88	441	19	methods	method	NOUN
ma-88	441	20	appl	appl	PROPN
ma-88	441	21	.	.	PUNCT
ma-88	442	1	73	73	NUM
ma-88	442	2	(	(	PUNCT
ma-88	442	3	2010	2010	NUM
ma-88	442	4	)	)	PUNCT
ma-88	442	5	3821–3831	3821–3831	NUM
ma-88	442	6	.	.	PUNCT
ma-88	443	1	https://doi.org/10.1016/j.na.2010.08.007.[20	https://doi.org/10.1016/j.na.2010.08.007.[20	PROPN
ma-88	443	2	]	]	PUNCT
ma-88	443	3	m.s	m.s	PROPN
ma-88	443	4	.	.	PROPN
ma-88	443	5	moslehian	moslehian	PROPN
ma-88	443	6	,	,	PUNCT
ma-88	443	7	on	on	ADP
ma-88	443	8	the	the	DET
ma-88	443	9	stability	stability	NOUN
ma-88	443	10	of	of	ADP
ma-88	443	11	the	the	DET
ma-88	443	12	orthogonal	orthogonal	ADJ
ma-88	443	13	pexiderized	pexiderize	VERB
ma-88	443	14	cauchy	cauchy	NOUN
ma-88	443	15	equation	equation	NOUN
ma-88	443	16	,	,	PUNCT
ma-88	443	17	j.	j.	PROPN
ma-88	443	18	math	math	PROPN
ma-88	443	19	.	.	PUNCT
ma-88	444	1	anal	anal	PROPN
ma-88	444	2	.	.	PUNCT
ma-88	444	3	appl	appl	PROPN
ma-88	444	4	.	.	PUNCT
ma-88	445	1	318	318	NUM
ma-88	445	2	(	(	PUNCT
ma-88	445	3	2006)211–223	2006)211–223	NUM
ma-88	445	4	.	.	PUNCT
ma-88	446	1	https://doi.org/10.1016/j.jmaa.2005.05.052.[21	https://doi.org/10.1016/j.jmaa.2005.05.052.[21	PROPN
ma-88	446	2	]	]	X
ma-88	447	1	j.	j.	PROPN
ma-88	447	2	chmieliński	chmieliński	PROPN
ma-88	447	3	,	,	PUNCT
ma-88	447	4	on	on	ADP
ma-88	447	5	an	an	DET
ma-88	447	6	ε	ε	PROPN
ma-88	447	7	-	-	PUNCT
ma-88	447	8	birkhoff	birkhoff	NOUN
ma-88	447	9	orthogonality	orthogonality	NOUN
ma-88	447	10	,	,	PUNCT
ma-88	447	11	j.	j.	PROPN
ma-88	447	12	inequal	inequal	PROPN
ma-88	447	13	.	.	PUNCT
ma-88	448	1	pure	pure	ADJ
ma-88	448	2	appl	appl	PROPN
ma-88	448	3	.	.	PUNCT
ma-88	448	4	math	math	NOUN
ma-88	448	5	.	.	PUNCT
ma-88	449	1	6	6	NUM
ma-88	449	2	(	(	PUNCT
ma-88	449	3	2005	2005	NUM
ma-88	449	4	)	)	PUNCT
ma-88	450	1	79.[22	79.[22	NUM
ma-88	450	2	]	]	X
ma-88	450	3	m.r	m.r	PROPN
ma-88	450	4	.	.	PROPN
ma-88	450	5	themistocles	themistocle	NOUN
ma-88	450	6	,	,	PUNCT
ma-88	450	7	some	some	DET
ma-88	450	8	remarks	remark	NOUN
ma-88	450	9	on	on	ADP
ma-88	450	10	isometric	isometric	ADJ
ma-88	450	11	mappings	mapping	NOUN
ma-88	450	12	,	,	PUNCT
ma-88	450	13	facta	facta	PROPN
ma-88	450	14	univ	univ	PROPN
ma-88	450	15	.	.	PUNCT
ma-88	451	1	ser	ser	PROPN
ma-88	451	2	.	.	PROPN
ma-88	451	3	math	math	PROPN
ma-88	451	4	.	.	PUNCT
ma-88	452	1	inform	inform	NOUN
ma-88	452	2	.	.	PUNCT
ma-88	453	1	2	2	NUM
ma-88	453	2	(	(	PUNCT
ma-88	453	3	1987	1987	NUM
ma-88	453	4	)	)	PUNCT
ma-88	453	5	49–52	49–52	NUM
ma-88	453	6	.	.	PUNCT
ma-88	454	1	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	454	2	https://doi.org/10.1007/s00245-021-09754-y	https://doi.org/10.1007/s00245-021-09754-y	PROPN
ma-88	454	3	https://doi.org/10.1007/s00010-011-0092-z	https://doi.org/10.1007/s00010-011-0092-z	ADV
ma-88	455	1	https://doi.org/10.1007/978-3-0348-5487-0	https://doi.org/10.1007/978-3-0348-5487-0	PROPN
ma-88	455	2	https://doi.org/10.1007/978-3-0348-5487-0	https://doi.org/10.1007/978-3-0348-5487-0	PROPN
ma-88	455	3	https://doi.org/10.1007/bf02417142	https://doi.org/10.1007/bf02417142	PROPN
ma-88	455	4	https://doi.org/10.1007/bf02417142	https://doi.org/10.1007/bf02417142	PROPN
ma-88	455	5	https://doi.org/10.1215/s0012-7094-35-00115-6	https://doi.org/10.1215/s0012-7094-35-00115-6	ADV
ma-88	455	6	https://doi.org/10.1215/s0012-7094-35-00115-6	https://doi.org/10.1215/s0012-7094-35-00115-6	ADV
ma-88	455	7	https://doi.org/10.1016/j.jmaa.2004.09.011	https://doi.org/10.1016/j.jmaa.2004.09.011	NOUN
ma-88	455	8	https://doi.org/10.1016/j.jmaa.2005.06.016	https://doi.org/10.1016/j.jmaa.2005.06.016	PROPN
ma-88	455	9	https://doi.org/10.1007/978-1-4614-0055-4_4	https://doi.org/10.1007/978-1-4614-0055-4_4	PROPN
ma-88	455	10	https://doi.org/10.1016/j.laa.2017.06.001	https://doi.org/10.1016/j.laa.2017.06.001	NOUN
ma-88	455	11	https://doi.org/10.1007/s00010-010-0042-1	https://doi.org/10.1007/s00010-010-0042-1	NOUN
ma-88	456	1	https://doi.org/10.1007/s00010-010-0042-1	https://doi.org/10.1007/s00010-010-0042-1	PROPN
ma-88	456	2	https://doi.org/10.1016/j.na.2009.08.028	https://doi.org/10.1016/j.na.2009.08.028	ADJ
ma-88	456	3	https://doi.org/10.1016/j.laa.2010.12.008	https://doi.org/10.1016/j.laa.2010.12.008	NOUN
ma-88	456	4	https://doi.org/10.4064/sm170512-25-9	https://doi.org/10.4064/sm170512-25-9	VERB
ma-88	456	5	https://doi.org/10.4064/sm170512-25-9	https://doi.org/10.4064/sm170512-25-9	VERB
ma-88	456	6	https://doi.org/10.1215/s0012-7094-45-01223-3	https://doi.org/10.1215/s0012-7094-45-01223-3	NUM
ma-88	456	7	https://doi.org/10.1215/s0012-7094-45-01223-3	https://doi.org/10.1215/s0012-7094-45-01223-3	NUM
ma-88	457	1	https://doi.org/10.1090/s0002-9947-1947-0021241-4	https://doi.org/10.1090/s0002-9947-1947-0021241-4	NOUN
ma-88	457	2	https://doi.org/10.1017/s0004972700008947	https://doi.org/10.1017/s0004972700008947	NOUN
ma-88	457	3	https://doi.org/10.1016/j.na.2010.08.007	https://doi.org/10.1016/j.na.2010.08.007	VERB
ma-88	457	4	https://doi.org/10.1016/j.jmaa.2005.05.052	https://doi.org/10.1016/j.jmaa.2005.05.052	DET
ma-88	457	5	eur	eur	NOUN
ma-88	457	6	.	.	PUNCT
ma-88	458	1	j.	j.	PROPN
ma-88	458	2	math	math	PROPN
ma-88	458	3	.	.	PUNCT
ma-88	459	1	anal	anal	PROPN
ma-88	459	2	.	.	PUNCT
ma-88	460	1	10.28924	10.28924	NUM
ma-88	460	2	/	/	SYM
ma-88	460	3	ada	ada	PROPN
ma-88	460	4	/	/	SYM
ma-88	460	5	ma.2.16	ma.2.16	PROPN
ma-88	460	6	14	14	NUM
ma-88	461	1	[	[	X
ma-88	461	2	23	23	NUM
ma-88	461	3	]	]	X
ma-88	461	4	b.d	b.d	PROPN
ma-88	461	5	.	.	PROPN
ma-88	461	6	roberts	roberts	PROPN
ma-88	461	7	,	,	PUNCT
ma-88	461	8	on	on	ADP
ma-88	461	9	the	the	DET
ma-88	461	10	geometry	geometry	NOUN
ma-88	461	11	of	of	ADP
ma-88	461	12	abstract	abstract	ADJ
ma-88	461	13	vector	vector	NOUN
ma-88	461	14	spaces	space	NOUN
ma-88	461	15	.	.	PUNCT
ma-88	462	1	tôhoku	tôhoku	PROPN
ma-88	462	2	math	math	PROPN
ma-88	462	3	.	.	PUNCT
ma-88	463	1	j.	j.	PROPN
ma-88	463	2	39	39	NUM
ma-88	463	3	(	(	PUNCT
ma-88	463	4	1934	1934	NUM
ma-88	463	5	)	)	PUNCT
ma-88	463	6	42–59.[24	42–59.[24	PUNCT
ma-88	463	7	]	]	PUNCT
ma-88	463	8	j.	j.	PROPN
ma-88	463	9	alonso	alonso	PROPN
ma-88	463	10	,	,	PUNCT
ma-88	463	11	b.	b.	PROPN
ma-88	463	12	javier	javier	PROPN
ma-88	463	13	,	,	PUNCT
ma-88	463	14	orthogonality	orthogonality	NOUN
ma-88	463	15	in	in	ADP
ma-88	463	16	normed	normed	ADJ
ma-88	463	17	linear	linear	PROPN
ma-88	463	18	spaces	space	NOUN
ma-88	463	19	:	:	PUNCT
ma-88	463	20	a	a	DET
ma-88	463	21	survey	survey	NOUN
ma-88	463	22	.	.	PUNCT
ma-88	464	1	i.	i.	NOUN
ma-88	464	2	main	main	ADJ
ma-88	464	3	properties	property	NOUN
ma-88	464	4	.	.	PUNCT
ma-88	465	1	extracta	extracta	PROPN
ma-88	465	2	math	math	PROPN
ma-88	465	3	.	.	PUNCT
ma-88	466	1	3	3	NUM
ma-88	466	2	(	(	PUNCT
ma-88	466	3	1988)1–15.[25	1988)1–15.[25	NUM
ma-88	466	4	]	]	X
ma-88	466	5	a.	a.	NOUN
ma-88	466	6	turnšek	turnšek	NOUN
ma-88	466	7	,	,	PUNCT
ma-88	466	8	on	on	ADP
ma-88	466	9	mappings	mapping	NOUN
ma-88	466	10	approximately	approximately	ADV
ma-88	466	11	preserving	preserve	VERB
ma-88	466	12	orthogonality	orthogonality	NOUN
ma-88	466	13	,	,	PUNCT
ma-88	466	14	j.	j.	PROPN
ma-88	466	15	math	math	PROPN
ma-88	466	16	.	.	PUNCT
ma-88	467	1	anal	anal	PROPN
ma-88	467	2	.	.	PUNCT
ma-88	468	1	appl	appl	PROPN
ma-88	468	2	.	.	PROPN
ma-88	469	1	336	336	NUM
ma-88	469	2	(	(	PUNCT
ma-88	469	3	2007	2007	NUM
ma-88	469	4	)	)	PUNCT
ma-88	470	1	625–631	625–631	NUM
ma-88	470	2	.	.	PUNCT
ma-88	470	3	https	https	NOUN
ma-88	470	4	:	:	PUNCT
ma-88	470	5	//doi.org/10.1016	//doi.org/10.1016	PROPN
ma-88	470	6	/	/	SYM
ma-88	470	7	j.jmaa.2007.03.016.[26	j.jmaa.2007.03.016.[26	PROPN
ma-88	470	8	]	]	X
ma-88	470	9	a.	a.	NOUN
ma-88	470	10	zamani	zamani	PROPN
ma-88	470	11	,	,	PUNCT
ma-88	470	12	m.s	m.s	PROPN
ma-88	470	13	.	.	PROPN
ma-88	470	14	moslehian	moslehian	PROPN
ma-88	470	15	,	,	PUNCT
ma-88	470	16	approximate	approximate	ADJ
ma-88	470	17	roberts	roberts	PROPN
ma-88	470	18	orthogonality	orthogonality	PROPN
ma-88	470	19	,	,	PUNCT
ma-88	470	20	aequat	aequat	PROPN
ma-88	470	21	.	.	PUNCT
ma-88	471	1	math	math	NOUN
ma-88	471	2	.	.	PUNCT
ma-88	472	1	89	89	NUM
ma-88	472	2	(	(	PUNCT
ma-88	472	3	2013	2013	NUM
ma-88	472	4	)	)	PUNCT
ma-88	473	1	529–541	529–541	NUM
ma-88	473	2	.	.	PUNCT
ma-88	474	1	https://doi	https://doi	X
ma-88	474	2	.	.	PUNCT
ma-88	474	3	org/10.1007	org/10.1007	PROPN
ma-88	474	4	/	/	SYM
ma-88	474	5	s00010	s00010	NOUN
ma-88	474	6	-	-	PUNCT
ma-88	474	7	013	013	NUM
ma-88	474	8	-	-	PUNCT
ma-88	474	9	0233	0233	NUM
ma-88	474	10	-	-	SYM
ma-88	474	11	7	7	NUM
ma-88	474	12	.	.	PUNCT
ma-88	475	1	https://doi.org/10.28924/ada/ma.2.16	https://doi.org/10.28924/ada/ma.2.16	PROPN
ma-88	475	2	https://doi.org/10.1016/j.jmaa.2007.03.016	https://doi.org/10.1016/j.jmaa.2007.03.016	PROPN
ma-88	475	3	https://doi.org/10.1016/j.jmaa.2007.03.016	https://doi.org/10.1016/j.jmaa.2007.03.016	PROPN
ma-88	475	4	https://doi.org/10.1007/s00010-013-0233-7	https://doi.org/10.1007/s00010-013-0233-7	NUM
ma-88	475	5	https://doi.org/10.1007/s00010-013-0233-7	https://doi.org/10.1007/s00010-013-0233-7	NUM
ma-88	475	6	1	1	NUM
ma-88	475	7	.	.	PUNCT
ma-88	475	8	introduction	introduction	NOUN
ma-88	475	9	2	2	NUM
ma-88	475	10	.	.	NUM
ma-88	475	11	approximate	approximate	ADJ
ma-88	475	12	birkhoff	birkhoff	PROPN
ma-88	475	13	orthogonality	orthogonality	PROPN
ma-88	475	14	b	b	PROPN
ma-88	475	15	3	3	NUM
ma-88	475	16	.	.	NOUN
ma-88	475	17	approximate	approximate	ADJ
ma-88	475	18	isosceles	isoscele	NOUN
ma-88	475	19	orthogonality	orthogonality	NOUN
ma-88	475	20	and	and	CCONJ
ma-88	475	21	approximate	approximate	ADJ
ma-88	475	22	birkhoff	birkhoff	NOUN
ma-88	475	23	orthogonality	orthogonality	NOUN
ma-88	475	24	references	reference	NOUN