id	sid	tid	token	lemma	pos
ma-119	1	1	2023	2023	NUM
ma-119	1	2	ada	ada	PROPN
ma-119	1	3	academica	academica	PROPN
ma-119	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-119	1	5	.	.	PUNCT
ma-119	2	1	j.	j.	PROPN
ma-119	2	2	math	math	PROPN
ma-119	2	3	.	.	PUNCT
ma-119	3	1	anal	anal	ADJ
ma-119	3	2	.	.	PUNCT
ma-119	4	1	3	3	NUM
ma-119	4	2	(	(	PUNCT
ma-119	4	3	2023	2023	NUM
ma-119	4	4	)	)	PUNCT
ma-119	5	1	9doi	9doi	NOUN
ma-119	5	2	:	:	PUNCT
ma-119	5	3	10.28924	10.28924	NUM
ma-119	5	4	/	/	SYM
ma-119	5	5	ada	ada	PROPN
ma-119	5	6	/	/	SYM
ma-119	5	7	ma.3.9	ma.3.9	PROPN
ma-119	5	8	on	on	ADP
ma-119	5	9	norm	norm	NOUN
ma-119	5	10	estimates	estimate	NOUN
ma-119	5	11	for	for	ADP
ma-119	5	12	derivations	derivation	NOUN
ma-119	5	13	in	in	ADP
ma-119	5	14	norm	norm	NOUN
ma-119	5	15	-	-	PUNCT
ma-119	5	16	attainable	attainable	ADJ
ma-119	5	17	classes	class	NOUN
ma-119	5	18	j.	j.	PROPN
ma-119	5	19	z.	z.	PROPN
ma-119	5	20	nyabonyi1,∗	nyabonyi1,∗	PROPN
ma-119	5	21	,	,	PUNCT
ma-119	5	22	n.	n.	PROPN
ma-119	5	23	b.	b.	PROPN
ma-119	5	24	okelo2	okelo2	PROPN
ma-119	5	25	,	,	PUNCT
ma-119	5	26	r.	r.	PROPN
ma-119	5	27	k.	k.	PROPN
ma-119	5	28	obogi1	obogi1	PROPN
ma-119	6	1	1department	1department	NUM
ma-119	6	2	of	of	ADP
ma-119	6	3	mathematics	mathematic	NOUN
ma-119	6	4	and	and	CCONJ
ma-119	6	5	actuarial	actuarial	ADJ
ma-119	6	6	science	science	NOUN
ma-119	6	7	,	,	PUNCT
ma-119	6	8	kisii	kisii	PROPN
ma-119	6	9	university	university	PROPN
ma-119	6	10	,	,	PUNCT
ma-119	6	11	kenya	kenya	PROPN
ma-119	6	12	nyabonyijanes@yahoo.com	nyabonyijanes@yahoo.com	PROPN
ma-119	6	13	,	,	PUNCT
ma-119	6	14	krbertobogi@yahoo.com	krbertobogi@yahoo.com	X
ma-119	7	1	2department	2department	NUM
ma-119	7	2	of	of	ADP
ma-119	7	3	pure	pure	ADJ
ma-119	7	4	and	and	CCONJ
ma-119	7	5	applied	applied	ADJ
ma-119	7	6	mathematics	mathematic	NOUN
ma-119	7	7	,	,	PUNCT
ma-119	7	8	jaramogi	jaramogi	PROPN
ma-119	7	9	oginga	oginga	PROPN
ma-119	7	10	odinga	odinga	PROPN
ma-119	7	11	university	university	PROPN
ma-119	7	12	of	of	ADP
ma-119	7	13	science	science	NOUN
ma-119	7	14	and	and	CCONJ
ma-119	7	15	technology	technology	NOUN
ma-119	7	16	,	,	PUNCT
ma-119	7	17	kenya	kenya	PROPN
ma-119	7	18	bnyaare@yahoo.com	bnyaare@yahoo.com	X
ma-119	8	1	∗correspondence	∗correspondence	NOUN
ma-119	8	2	:	:	PUNCT
ma-119	8	3	nyabonyijanes@yahoo.com	nyabonyijanes@yahoo.com	X
ma-119	9	1	abstract	abstract	ADJ
ma-119	9	2	.	.	PUNCT
ma-119	10	1	in	in	ADP
ma-119	10	2	this	this	DET
ma-119	10	3	note	note	NOUN
ma-119	10	4	,	,	PUNCT
ma-119	10	5	we	we	PRON
ma-119	10	6	provide	provide	VERB
ma-119	10	7	detailed	detailed	ADJ
ma-119	10	8	characterization	characterization	NOUN
ma-119	10	9	of	of	ADP
ma-119	10	10	operators	operator	NOUN
ma-119	10	11	in	in	ADP
ma-119	10	12	terms	term	NOUN
ma-119	10	13	of	of	ADP
ma-119	10	14	norm	norm	NOUN
ma-119	10	15	-	-	PUNCT
ma-119	10	16	attainabilityand	attainabilityand	NOUN
ma-119	10	17	norm	norm	NOUN
ma-119	10	18	estimates	estimate	NOUN
ma-119	10	19	in	in	ADP
ma-119	10	20	banach	banach	NOUN
ma-119	10	21	algebras	algebra	NOUN
ma-119	10	22	.	.	PUNCT
ma-119	11	1	in	in	ADP
ma-119	11	2	particular	particular	ADJ
ma-119	11	3	,	,	PUNCT
ma-119	11	4	we	we	PRON
ma-119	11	5	establish	establish	VERB
ma-119	11	6	the	the	DET
ma-119	11	7	necessary	necessary	ADJ
ma-119	11	8	and	and	CCONJ
ma-119	11	9	sufficientconditions	sufficientcondition	NOUN
ma-119	11	10	for	for	ADP
ma-119	11	11	norm	norm	NOUN
ma-119	11	12	-	-	PUNCT
ma-119	11	13	attainability	attainability	NOUN
ma-119	11	14	of	of	ADP
ma-119	11	15	the	the	DET
ma-119	11	16	derivations	derivation	NOUN
ma-119	11	17	and	and	CCONJ
ma-119	11	18	also	also	ADV
ma-119	11	19	give	give	VERB
ma-119	11	20	their	their	PRON
ma-119	11	21	norm	norm	NOUN
ma-119	11	22	bounds	bound	NOUN
ma-119	11	23	in	in	ADP
ma-119	11	24	the	the	DET
ma-119	11	25	norm	norm	NOUN
ma-119	11	26	-	-	PUNCT
ma-119	11	27	attainable	attainable	ADJ
ma-119	11	28	classes	class	NOUN
ma-119	11	29	.	.	PUNCT
ma-119	12	1	1	1	X
ma-119	12	2	.	.	X
ma-119	12	3	introduction	introduction	NOUN
ma-119	12	4	the	the	DET
ma-119	12	5	norm	norm	NOUN
ma-119	12	6	of	of	ADP
ma-119	12	7	a	a	DET
ma-119	12	8	derivation	derivation	NOUN
ma-119	12	9	was	be	AUX
ma-119	12	10	first	first	ADV
ma-119	12	11	introduced	introduce	VERB
ma-119	12	12	by	by	ADP
ma-119	12	13	stampfli	stampfli	NOUN
ma-119	13	1	[	[	X
ma-119	13	2	49	49	NUM
ma-119	13	3	]	]	PUNCT
ma-119	13	4	,	,	PUNCT
ma-119	13	5	who	who	PRON
ma-119	13	6	determined	determine	VERB
ma-119	13	7	the	the	DET
ma-119	13	8	inner	inner	ADJ
ma-119	13	9	derivation	derivation	NOUN
ma-119	13	10	δt0	δt0	NOUN
ma-119	13	11	:	:	PUNCT
ma-119	13	12	a0	a0	PROPN
ma-119	13	13	→	→	SYM
ma-119	13	14	t0a0	t0a0	PROPN
ma-119	13	15	−	−	PROPN
ma-119	13	16	a0t0	a0t0	PUNCT
ma-119	13	17	which	which	PRON
ma-119	13	18	acts	act	VERB
ma-119	13	19	on	on	ADP
ma-119	13	20	b(h	b(h	PROPN
ma-119	13	21	)	)	PUNCT
ma-119	13	22	,	,	PUNCT
ma-119	13	23	the	the	DET
ma-119	13	24	algebra	algebra	NOUN
ma-119	13	25	of	of	ADP
ma-119	13	26	all	all	DET
ma-119	13	27	bounded	bound	VERB
ma-119	13	28	linear	linear	PROPN
ma-119	13	29	operators	operator	NOUN
ma-119	13	30	on	on	ADP
ma-119	13	31	acomplex	acomplex	PROPN
ma-119	13	32	hilbert	hilbert	PROPN
ma-119	13	33	space	space	PROPN
ma-119	13	34	h.	h.	PROPN
ma-119	13	35	further	far	ADV
ma-119	13	36	,	,	PUNCT
ma-119	13	37	‖δt0‖	‖δt0‖	PROPN
ma-119	13	38	=	=	PROPN
ma-119	13	39	inf	inf	PROPN
ma-119	13	40	2‖t0	2‖t0	NUM
ma-119	13	41	−	−	NOUN
ma-119	13	42	λi0‖	λi0‖	PROPN
ma-119	13	43	,	,	PUNCT
ma-119	13	44	for	for	ADP
ma-119	13	45	every	every	DET
ma-119	13	46	complex	complex	ADJ
ma-119	13	47	λ	λ	NOUN
ma-119	13	48	was	be	AUX
ma-119	13	49	shown	show	VERB
ma-119	13	50	.	.	PUNCT
ma-119	14	1	fora	fora	ADJ
ma-119	14	2	normal	normal	ADJ
ma-119	14	3	operator	operator	NOUN
ma-119	14	4	t	t	PROPN
ma-119	14	5	,	,	PUNCT
ma-119	14	6	‖δt0‖	‖δt0‖	PROPN
ma-119	14	7	can	can	AUX
ma-119	14	8	be	be	AUX
ma-119	14	9	expressed	express	VERB
ma-119	14	10	as	as	ADP
ma-119	14	11	the	the	DET
ma-119	14	12	geometry	geometry	NOUN
ma-119	14	13	of	of	ADP
ma-119	14	14	the	the	DET
ma-119	14	15	spectrum	spectrum	NOUN
ma-119	14	16	of	of	ADP
ma-119	14	17	t0	t0	PROPN
ma-119	14	18	.	.	PUNCT
ma-119	15	1	johnson	johnson	PROPN
ma-119	16	1	[	[	X
ma-119	16	2	21]established	21]established	NUM
ma-119	16	3	methods	method	NOUN
ma-119	16	4	which	which	PRON
ma-119	16	5	apply	apply	VERB
ma-119	16	6	to	to	ADP
ma-119	16	7	a	a	DET
ma-119	16	8	uniformly	uniformly	ADV
ma-119	16	9	convex	convex	NOUN
ma-119	16	10	spaces	space	NOUN
ma-119	16	11	with	with	ADP
ma-119	16	12	a	a	DET
ma-119	16	13	large	large	ADJ
ma-119	16	14	class	class	NOUN
ma-119	16	15	,	,	PUNCT
ma-119	16	16	i.e	i.e	PRON
ma-119	16	17	the	the	DET
ma-119	16	18	formula	formula	NOUN
ma-119	16	19	‖δt	‖δt	PROPN
ma-119	16	20	‖	‖	PROPN
ma-119	16	21	is	be	AUX
ma-119	16	22	false	false	ADJ
ma-119	16	23	in	in	ADP
ma-119	16	24	lp	lp	NOUN
ma-119	16	25	and	and	CCONJ
ma-119	16	26	lp(0	lp(0	NOUN
ma-119	16	27	,	,	PUNCT
ma-119	16	28	1	1	NUM
ma-119	16	29	)	)	PUNCT
ma-119	16	30	1	1	NUM
ma-119	16	31	<	<	X
ma-119	16	32	p	p	X
ma-119	16	33	<	<	X
ma-119	16	34	∞	∞	PROPN
ma-119	16	35	,	,	PUNCT
ma-119	16	36	p	p	X
ma-119	16	37	6=	6=	PROPN
ma-119	16	38	2	2	NUM
ma-119	16	39	.	.	X
ma-119	16	40	for	for	ADP
ma-119	16	41	l1	l1	PROPN
ma-119	16	42	space	space	NOUN
ma-119	16	43	the	the	DET
ma-119	16	44	formula	formula	NOUN
ma-119	16	45	is	be	AUX
ma-119	16	46	true	true	ADJ
ma-119	16	47	for	for	ADP
ma-119	16	48	areal	areal	NOUN
ma-119	16	49	case	case	NOUN
ma-119	16	50	and	and	CCONJ
ma-119	16	51	not	not	PART
ma-119	16	52	for	for	ADP
ma-119	16	53	a	a	DET
ma-119	16	54	complex	complex	ADJ
ma-119	16	55	case	case	NOUN
ma-119	16	56	whose	whose	DET
ma-119	16	57	space	space	NOUN
ma-119	16	58	dimension	dimension	NOUN
ma-119	16	59	is	be	AUX
ma-119	16	60	3	3	NUM
ma-119	16	61	or	or	CCONJ
ma-119	16	62	more	more	ADJ
ma-119	16	63	.	.	PUNCT
ma-119	17	1	johnson	johnson	PROPN
ma-119	18	1	[	[	X
ma-119	18	2	20	20	NUM
ma-119	18	3	]	]	PUNCT
ma-119	18	4	foundthat	foundthat	PRON
ma-119	18	5	a	a	DET
ma-119	18	6	derivation	derivation	NOUN
ma-119	18	7	on	on	ADP
ma-119	18	8	b(h	b(h	PROPN
ma-119	18	9	)	)	PUNCT
ma-119	18	10	is	be	AUX
ma-119	18	11	a	a	DET
ma-119	18	12	mapping	mapping	NOUN
ma-119	18	13	∆	∆	PROPN
ma-119	18	14	:	:	PUNCT
ma-119	18	15	b(h)→	b(h)→	PUNCT
ma-119	18	16	b(h	b(h	PROPN
ma-119	18	17	)	)	PUNCT
ma-119	18	18	with	with	ADP
ma-119	18	19	∆(as	∆(as	PRON
ma-119	18	20	)	)	PUNCT
ma-119	18	21	=	=	PUNCT
ma-119	18	22	a∆(s	a∆(s	X
ma-119	18	23	)	)	PUNCT
ma-119	19	1	+	+	CCONJ
ma-119	19	2	∆(a)s	∆(a)s	PROPN
ma-119	19	3	,	,	PUNCT
ma-119	19	4	where	where	SCONJ
ma-119	19	5	a	a	PRON
ma-119	19	6	,	,	PUNCT
ma-119	19	7	s	s	NOUN
ma-119	19	8	∈	∈	PROPN
ma-119	19	9	b(h	b(h	PROPN
ma-119	19	10	)	)	PUNCT
ma-119	19	11	.	.	PUNCT
ma-119	20	1	such	such	ADJ
ma-119	20	2	derivations	derivation	NOUN
ma-119	20	3	are	be	AUX
ma-119	20	4	necessarily	necessarily	ADV
ma-119	20	5	continuous	continuous	ADJ
ma-119	20	6	and	and	CCONJ
ma-119	20	7	if	if	SCONJ
ma-119	20	8	s	s	X
ma-119	20	9	∈	∈	PROPN
ma-119	20	10	b(h	b(h	PROPN
ma-119	20	11	)	)	PUNCT
ma-119	20	12	then	then	ADV
ma-119	20	13	∆s(a	∆s(a	PROPN
ma-119	20	14	)	)	PUNCT
ma-119	21	1	=	=	PUNCT
ma-119	22	1	as−sais	as−sais	PRON
ma-119	22	2	a	a	DET
ma-119	22	3	derivation	derivation	NOUN
ma-119	22	4	on	on	ADP
ma-119	22	5	b(h	b(h	PROPN
ma-119	22	6	)	)	PUNCT
ma-119	22	7	.	.	PUNCT
ma-119	23	1	gajendragadka	gajendragadka	PROPN
ma-119	24	1	[	[	X
ma-119	24	2	18	18	NUM
ma-119	24	3	]	]	PUNCT
ma-119	24	4	was	be	AUX
ma-119	24	5	concerned	concern	VERB
ma-119	24	6	with	with	ADP
ma-119	24	7	the	the	DET
ma-119	24	8	von	von	PROPN
ma-119	24	9	neumann	neumann	PROPN
ma-119	24	10	algebra	algebra	PROPN
ma-119	24	11	andcomputed	andcompute	VERB
ma-119	24	12	the	the	DET
ma-119	24	13	norm	norm	NOUN
ma-119	24	14	of	of	ADP
ma-119	24	15	a	a	DET
ma-119	24	16	derivation	derivation	NOUN
ma-119	24	17	.	.	PUNCT
ma-119	25	1	specifically	specifically	ADV
ma-119	25	2	,	,	PUNCT
ma-119	25	3	it	it	PRON
ma-119	25	4	was	be	AUX
ma-119	25	5	proved	prove	VERB
ma-119	25	6	that	that	SCONJ
ma-119	25	7	the	the	DET
ma-119	25	8	von	von	PROPN
ma-119	25	9	neumann	neumann	PROPN
ma-119	25	10	algebra	algebra	PROPN
ma-119	25	11	actson	actson	PROPN
ma-119	25	12	a	a	DET
ma-119	25	13	separable	separable	ADJ
ma-119	25	14	hilbert	hilbert	NOUN
ma-119	25	15	space	space	NOUN
ma-119	25	16	h	h	NOUN
ma-119	25	17	,	,	PUNCT
ma-119	25	18	whereby	whereby	SCONJ
ma-119	25	19	if	if	SCONJ
ma-119	25	20	t	t	PROPN
ma-119	25	21	is	be	AUX
ma-119	25	22	in	in	ADP
ma-119	25	23	u	u	NOUN
ma-119	25	24	and	and	CCONJ
ma-119	25	25	δt	δt	PROPN
ma-119	25	26	is	be	AUX
ma-119	25	27	the	the	DET
ma-119	25	28	derivation	derivation	NOUN
ma-119	25	29	induced	induce	VERB
ma-119	25	30	by	by	ADP
ma-119	25	31	t	t	PROPN
ma-119	25	32	,	,	PUNCT
ma-119	25	33	then	then	ADV
ma-119	25	34	‖δt	‖δt	PROPN
ma-119	25	35	|u‖	|u‖	SYM
ma-119	25	36	=	=	SYM
ma-119	25	37	2	2	NUM
ma-119	25	38	inf	inf	NOUN
ma-119	25	39	‖t	‖t	PROPN
ma-119	25	40	−	−	PROPN
ma-119	25	41	z‖	z‖	NOUN
ma-119	25	42	,	,	PUNCT
ma-119	25	43	where	where	SCONJ
ma-119	25	44	z	z	NOUN
ma-119	25	45	is	be	AUX
ma-119	25	46	the	the	DET
ma-119	25	47	centre	centre	NOUN
ma-119	25	48	of	of	ADP
ma-119	25	49	u.	u.	PROPN
ma-119	25	50	therefore	therefore	ADV
ma-119	25	51	,	,	PUNCT
ma-119	25	52	anderson	anderson	PROPN
ma-119	26	1	[	[	X
ma-119	26	2	3	3	X
ma-119	26	3	]	]	PUNCT
ma-119	26	4	in	in	ADP
ma-119	26	5	his	his	PRON
ma-119	26	6	investigation	investigation	NOUN
ma-119	26	7	onnormal	onnormal	NOUN
ma-119	26	8	derivations	derivation	NOUN
ma-119	26	9	with	with	ADP
ma-119	26	10	the	the	DET
ma-119	26	11	operators	operator	NOUN
ma-119	26	12	a	a	PRON
ma-119	26	13	,	,	PUNCT
ma-119	26	14	c	c	PROPN
ma-119	26	15	∈	∈	PROPN
ma-119	26	16	b(h	b(h	PROPN
ma-119	26	17	)	)	PUNCT
ma-119	26	18	proved	prove	VERB
ma-119	26	19	if	if	SCONJ
ma-119	26	20	a	a	PRON
ma-119	26	21	is	be	AUX
ma-119	26	22	normal	normal	ADJ
ma-119	26	23	and	and	CCONJ
ma-119	26	24	ac	ac	PROPN
ma-119	26	25	commute	commute	PROPN
ma-119	26	26	,	,	PUNCT
ma-119	26	27	for	for	ADP
ma-119	26	28	every	every	DET
ma-119	26	29	x	x	PROPN
ma-119	26	30	∈	∈	PROPN
ma-119	26	31	b(h	b(h	PROPN
ma-119	26	32	)	)	PUNCT
ma-119	26	33	,	,	PUNCT
ma-119	26	34	‖δa(x	‖δa(x	PROPN
ma-119	26	35	)	)	PUNCT
ma-119	27	1	+	+	ADJ
ma-119	27	2	c‖	c‖	PROPN
ma-119	27	3	≥	≥	PRON
ma-119	27	4	‖c‖.	‖c‖.	X
ma-119	27	5	therefore	therefore	ADV
ma-119	27	6	,	,	PUNCT
ma-119	27	7	the	the	DET
ma-119	27	8	inequality	inequality	NOUN
ma-119	27	9	showed	show	VERB
ma-119	27	10	that	that	SCONJ
ma-119	27	11	the	the	DET
ma-119	27	12	kernel	kernel	NOUN
ma-119	27	13	and	and	CCONJ
ma-119	27	14	the	the	DET
ma-119	27	15	range	range	NOUN
ma-119	27	16	of	of	ADP
ma-119	27	17	δa	δa	PROPN
ma-119	27	18	are	be	AUX
ma-119	27	19	orthogonal	orthogonal	ADJ
ma-119	27	20	to	to	ADP
ma-119	27	21	δa	δa	PROPN
ma-119	27	22	which	which	PRON
ma-119	27	23	is	be	AUX
ma-119	27	24	the	the	DET
ma-119	27	25	commutation	commutation	NOUN
ma-119	27	26	of	of	ADP
ma-119	27	27	{	{	PUNCT
ma-119	27	28	a}′	a}′	NOUN
ma-119	27	29	of	of	ADP
ma-119	27	30	a.	a.	NOUN
ma-119	27	31	kyle	kyle	NOUN
ma-119	28	1	[	[	X
ma-119	28	2	24	24	NUM
ma-119	28	3	]	]	PUNCT
ma-119	28	4	examined	examine	VERB
ma-119	28	5	the	the	DET
ma-119	28	6	relationship	relationship	NOUN
ma-119	28	7	received	receive	VERB
ma-119	28	8	:	:	PUNCT
ma-119	28	9	22	22	NUM
ma-119	28	10	jun	jun	PROPN
ma-119	28	11	2022	2022	NUM
ma-119	28	12	.	.	PUNCT
ma-119	29	1	key	key	ADJ
ma-119	29	2	words	word	NOUN
ma-119	29	3	and	and	CCONJ
ma-119	29	4	phrases	phrase	NOUN
ma-119	29	5	.	.	PUNCT
ma-119	30	1	derivation	derivation	NOUN
ma-119	30	2	;	;	PUNCT
ma-119	30	3	norm	norm	NOUN
ma-119	30	4	;	;	PUNCT
ma-119	30	5	norm	norm	NOUN
ma-119	30	6	-	-	PUNCT
ma-119	30	7	attainability	attainability	NOUN
ma-119	30	8	;	;	PUNCT
ma-119	30	9	banach	banach	NOUN
ma-119	30	10	algebra.1	algebra.1	PROPN
ma-119	30	11	https://adac.ee	https://adac.ee	PROPN
ma-119	30	12	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	NUM
ma-119	30	13	eur	eur	NOUN
ma-119	30	14	.	.	PUNCT
ma-119	31	1	j.	j.	PROPN
ma-119	31	2	math	math	PROPN
ma-119	31	3	.	.	PUNCT
ma-119	32	1	anal	anal	PROPN
ma-119	32	2	.	.	PUNCT
ma-119	33	1	10.28924	10.28924	NUM
ma-119	33	2	/	/	SYM
ma-119	33	3	ada	ada	PROPN
ma-119	33	4	/	/	SYM
ma-119	33	5	ma.3.9	ma.3.9	PROPN
ma-119	33	6	2of	2of	NOUN
ma-119	33	7	the	the	DET
ma-119	33	8	numerical	numerical	ADJ
ma-119	33	9	range	range	NOUN
ma-119	33	10	of	of	ADP
ma-119	33	11	inner	inner	ADJ
ma-119	33	12	derivation	derivation	NOUN
ma-119	33	13	and	and	CCONJ
ma-119	33	14	that	that	PRON
ma-119	33	15	of	of	ADP
ma-119	33	16	the	the	DET
ma-119	33	17	implementing	implement	VERB
ma-119	33	18	element	element	NOUN
ma-119	33	19	.	.	PUNCT
ma-119	34	1	kyle	kyle	NOUN
ma-119	35	1	[	[	X
ma-119	35	2	25	25	NUM
ma-119	35	3	]	]	PUNCT
ma-119	35	4	studiednorms	studiednorm	NOUN
ma-119	35	5	of	of	ADP
ma-119	35	6	inner	inner	ADJ
ma-119	35	7	derivations	derivation	NOUN
ma-119	35	8	and	and	CCONJ
ma-119	35	9	used	use	VERB
ma-119	35	10	their	their	PRON
ma-119	35	11	properties	property	NOUN
ma-119	35	12	and	and	CCONJ
ma-119	35	13	concluded	conclude	VERB
ma-119	35	14	that	that	SCONJ
ma-119	35	15	a	a	DET
ma-119	35	16	closed	closed	ADJ
ma-119	35	17	subset	subset	NOUN
ma-119	35	18	of	of	ADP
ma-119	35	19	allderivations	allderivation	NOUN
ma-119	35	20	on	on	ADP
ma-119	35	21	a	a	DET
ma-119	35	22	c∗-algebra	c∗-algebra	NOUN
ma-119	35	23	,	,	PUNCT
ma-119	35	24	forms	form	VERB
ma-119	35	25	the	the	DET
ma-119	35	26	set	set	NOUN
ma-119	35	27	of	of	ADP
ma-119	35	28	inner	inner	ADJ
ma-119	35	29	derivations	derivation	NOUN
ma-119	35	30	and	and	CCONJ
ma-119	35	31	obtained	obtain	VERB
ma-119	35	32	the	the	DET
ma-119	35	33	result	result	NOUN
ma-119	35	34	which	which	PRON
ma-119	35	35	is	be	AUX
ma-119	35	36	aconverse	aconverse	ADJ
ma-119	35	37	of	of	ADP
ma-119	35	38	stampfli	stampfli	NOUN
ma-119	35	39	[	[	X
ma-119	35	40	49	49	NUM
ma-119	35	41	]	]	PUNCT
ma-119	35	42	.	.	PUNCT
ma-119	36	1	charles	charles	PROPN
ma-119	36	2	and	and	CCONJ
ma-119	36	3	steve	steve	PROPN
ma-119	37	1	[	[	X
ma-119	37	2	11	11	NUM
ma-119	37	3	]	]	PUNCT
ma-119	37	4	answered	answer	VERB
ma-119	37	5	the	the	DET
ma-119	37	6	question	question	NOUN
ma-119	37	7	when	when	SCONJ
ma-119	37	8	x	x	PRON
ma-119	37	9	=	=	SYM
ma-119	37	10	t	t	NOUN
ma-119	37	11	by	by	ADP
ma-119	37	12	structurecharacterization	structurecharacterization	NOUN
ma-119	37	13	of	of	ADP
ma-119	37	14	compact	compact	ADJ
ma-119	37	15	derivations	derivation	NOUN
ma-119	37	16	of	of	ADP
ma-119	37	17	c∗-algebras	c∗-algebra	NOUN
ma-119	37	18	.	.	PUNCT
ma-119	38	1	moreover	moreover	ADV
ma-119	38	2	,	,	PUNCT
ma-119	38	3	the	the	DET
ma-119	38	4	structure	structure	NOUN
ma-119	38	5	of	of	ADP
ma-119	38	6	weak	weak	ADJ
ma-119	38	7	compactderivations	compactderivation	NOUN
ma-119	38	8	of	of	ADP
ma-119	38	9	c∗-algebras	c∗-algebra	NOUN
ma-119	38	10	was	be	AUX
ma-119	38	11	determined	determine	VERB
ma-119	38	12	and	and	CCONJ
ma-119	38	13	as	as	ADP
ma-119	38	14	immediate	immediate	ADJ
ma-119	38	15	corollaries	corollary	NOUN
ma-119	38	16	of	of	ADP
ma-119	38	17	these	these	DET
ma-119	38	18	results	result	NOUN
ma-119	38	19	,	,	PUNCT
ma-119	38	20	conditionsthat	conditionsthat	PRON
ma-119	38	21	were	be	AUX
ma-119	38	22	necessary	necessary	ADJ
ma-119	38	23	and	and	CCONJ
ma-119	38	24	sufficient	sufficient	ADJ
ma-119	38	25	were	be	AUX
ma-119	38	26	obtained	obtain	VERB
ma-119	38	27	so	so	SCONJ
ma-119	38	28	that	that	SCONJ
ma-119	38	29	c∗-algebras	c∗-algebra	NOUN
ma-119	38	30	can	can	AUX
ma-119	38	31	admit	admit	VERB
ma-119	38	32	a	a	DET
ma-119	38	33	non	non	ADJ
ma-119	38	34	-	-	ADJ
ma-119	38	35	zero	zero	ADJ
ma-119	38	36	compactor	compactor	NOUN
ma-119	38	37	weakly	weakly	ADJ
ma-119	38	38	compact	compact	ADJ
ma-119	38	39	derivation	derivation	NOUN
ma-119	38	40	.	.	PUNCT
ma-119	39	1	stampfli	stampfli	NOUN
ma-119	40	1	[	[	X
ma-119	40	2	50	50	NUM
ma-119	40	3	]	]	PUNCT
ma-119	40	4	studied	study	VERB
ma-119	40	5	operators	operator	NOUN
ma-119	40	6	on	on	ADP
ma-119	40	7	hilbert	hilbert	NOUN
ma-119	40	8	spaces	space	NOUN
ma-119	40	9	and	and	CCONJ
ma-119	40	10	their	their	PRON
ma-119	40	11	propertiesinducing	propertiesinduce	VERB
ma-119	40	12	a	a	DET
ma-119	40	13	derivation	derivation	NOUN
ma-119	40	14	whose	whose	DET
ma-119	40	15	closure	closure	NOUN
ma-119	40	16	is	be	AUX
ma-119	40	17	self	self	NOUN
ma-119	40	18	-	-	PUNCT
ma-119	40	19	adjoint	adjoint	NOUN
ma-119	40	20	after	after	SCONJ
ma-119	40	21	the	the	DET
ma-119	40	22	range	range	NOUN
ma-119	40	23	of	of	ADP
ma-119	40	24	such	such	ADJ
ma-119	40	25	operators	operator	NOUN
ma-119	40	26	are	be	AUX
ma-119	40	27	termed	term	VERB
ma-119	40	28	d	d	ADJ
ma-119	40	29	-	-	PUNCT
ma-119	40	30	symmetric	symmetric	ADJ
ma-119	40	31	and	and	CCONJ
ma-119	40	32	then	then	ADV
ma-119	40	33	characterized	characterize	VERB
ma-119	40	34	compact	compact	ADJ
ma-119	40	35	d	d	ADJ
ma-119	40	36	-	-	PUNCT
ma-119	40	37	symmetric	symmetric	ADJ
ma-119	40	38	operators	operator	NOUN
ma-119	40	39	.	.	PUNCT
ma-119	41	1	erik	erik	PROPN
ma-119	42	1	[	[	X
ma-119	42	2	16	16	NUM
ma-119	42	3	]	]	PUNCT
ma-119	42	4	established	establish	VERB
ma-119	42	5	thatany	thatany	NOUN
ma-119	42	6	operator	operator	NOUN
ma-119	42	7	t	t	PROPN
ma-119	42	8	on	on	ADP
ma-119	42	9	a	a	DET
ma-119	42	10	hilbert	hilbert	NOUN
ma-119	42	11	space	space	NOUN
ma-119	42	12	h	h	NOUN
ma-119	42	13	with	with	ADP
ma-119	42	14	a	a	DET
ma-119	42	15	cyclic	cyclic	ADJ
ma-119	42	16	vector	vector	NOUN
ma-119	42	17	has	have	VERB
ma-119	42	18	a	a	DET
ma-119	42	19	property	property	NOUN
ma-119	42	20	with	with	ADP
ma-119	42	21	a	a	DET
ma-119	42	22	finite	finite	ADJ
ma-119	42	23	spectrum.mecheri	spectrum.mecheri	PROPN
ma-119	42	24	[	[	X
ma-119	42	25	31	31	NUM
ma-119	42	26	]	]	PUNCT
ma-119	42	27	established	establish	VERB
ma-119	42	28	that	that	SCONJ
ma-119	42	29	t	t	NOUN
ma-119	42	30	(	(	PUNCT
ma-119	42	31	x	x	X
ma-119	42	32	)	)	PUNCT
ma-119	42	33	is	be	AUX
ma-119	42	34	linear	linear	ADJ
ma-119	42	35	for	for	ADP
ma-119	42	36	any	any	DET
ma-119	42	37	m	m	ADJ
ma-119	42	38	-	-	ADJ
ma-119	42	39	linear	linear	ADJ
ma-119	42	40	derivation	derivation	NOUN
ma-119	42	41	and	and	CCONJ
ma-119	42	42	hence	hence	ADV
ma-119	42	43	,	,	PUNCT
ma-119	42	44	the	the	DET
ma-119	42	45	topologyof	topologyof	PROPN
ma-119	42	46	von	von	PROPN
ma-119	42	47	neumann	neumann	PROPN
ma-119	42	48	algebra	algebra	PROPN
ma-119	42	49	x	x	PUNCT
ma-119	42	50	of	of	ADP
ma-119	42	51	type	type	NOUN
ma-119	42	52	i	i	PRON
ma-119	42	53	is	be	AUX
ma-119	42	54	automatically	automatically	ADV
ma-119	42	55	continuous	continuous	ADJ
ma-119	42	56	in	in	ADP
ma-119	42	57	measure	measure	NOUN
ma-119	42	58	with	with	ADP
ma-119	42	59	center	center	NOUN
ma-119	42	60	m	m	NOUN
ma-119	42	61	and	and	CCONJ
ma-119	42	62	thesemi	thesemi	NOUN
ma-119	42	63	-	-	ADJ
ma-119	42	64	finite	finite	ADJ
ma-119	42	65	trace	trace	NOUN
ma-119	42	66	τ	τ	X
ma-119	42	67	which	which	PRON
ma-119	42	68	is	be	AUX
ma-119	42	69	normal	normal	ADJ
ma-119	42	70	is	be	AUX
ma-119	42	71	faithful	faithful	ADJ
ma-119	42	72	.	.	PUNCT
ma-119	43	1	therefore	therefore	ADV
ma-119	43	2	,	,	PUNCT
ma-119	43	3	t	t	PROPN
ma-119	43	4	(	(	PUNCT
ma-119	43	5	x	x	X
ma-119	43	6	)	)	PUNCT
ma-119	43	7	is	be	AUX
ma-119	43	8	the	the	DET
ma-119	43	9	algebra	algebra	NOUN
ma-119	43	10	of	of	ADP
ma-119	43	11	all	all	DET
ma-119	43	12	τ	τ	NOUN
ma-119	43	13	-	-	PUNCT
ma-119	43	14	measurableoperators	measurableoperator	NOUN
ma-119	43	15	affiliated	affiliate	VERB
ma-119	43	16	with	with	ADP
ma-119	43	17	x.	x.	PROPN
ma-119	43	18	mathieu	mathieu	PROPN
ma-119	44	1	[	[	X
ma-119	44	2	29	29	NUM
ma-119	44	3	]	]	PUNCT
ma-119	44	4	proved	prove	VERB
ma-119	44	5	that	that	SCONJ
ma-119	44	6	for	for	ADP
ma-119	44	7	non	non	ADJ
ma-119	44	8	-	-	ADJ
ma-119	44	9	zero	zero	NUM
ma-119	44	10	derivations	derivation	NOUN
ma-119	44	11	,	,	PUNCT
ma-119	44	12	the	the	DET
ma-119	44	13	product	product	NOUN
ma-119	44	14	of	of	ADP
ma-119	44	15	twoprime	twoprime	NOUN
ma-119	44	16	c∗-algebras	c∗-algebra	NOUN
ma-119	44	17	are	be	AUX
ma-119	44	18	bounded	bound	VERB
ma-119	44	19	if	if	SCONJ
ma-119	44	20	both	both	PRON
ma-119	44	21	of	of	ADP
ma-119	44	22	them	they	PRON
ma-119	44	23	are	be	AUX
ma-119	44	24	bounded	bound	VERB
ma-119	44	25	.	.	PUNCT
ma-119	45	1	in	in	ADP
ma-119	45	2	[	[	X
ma-119	45	3	51	51	NUM
ma-119	45	4	]	]	PUNCT
ma-119	45	5	,	,	PUNCT
ma-119	45	6	two	two	NUM
ma-119	45	7	automatic	automatic	ADJ
ma-119	45	8	continuityproblems	continuityproblem	NOUN
ma-119	45	9	for	for	ADP
ma-119	45	10	derivations	derivation	NOUN
ma-119	45	11	on	on	ADP
ma-119	45	12	commutating	commutate	VERB
ma-119	45	13	banach	banach	NOUN
ma-119	45	14	algebras	algebra	NOUN
ma-119	45	15	were	be	AUX
ma-119	45	16	discussed	discuss	VERB
ma-119	45	17	,	,	PUNCT
ma-119	45	18	that	that	ADV
ma-119	45	19	is	be	AUX
ma-119	45	20	,	,	PUNCT
ma-119	45	21	derivation	derivation	NOUN
ma-119	45	22	on	on	ADP
ma-119	45	23	acommutative	acommutative	ADJ
ma-119	45	24	algebra	algebra	NOUN
ma-119	45	25	is	be	AUX
ma-119	45	26	mapped	map	VERB
ma-119	45	27	onto	onto	ADP
ma-119	45	28	the	the	DET
ma-119	45	29	radical	radical	ADJ
ma-119	45	30	,	,	PUNCT
ma-119	45	31	and	and	CCONJ
ma-119	45	32	banach	banach	NOUN
ma-119	45	33	algebras	algebra	NOUN
ma-119	45	34	are	be	AUX
ma-119	45	35	continuous	continuous	ADJ
ma-119	45	36	on	on	ADP
ma-119	45	37	semiprimederivations	semiprimederivation	NOUN
ma-119	45	38	.	.	PUNCT
ma-119	46	1	bresar	bresar	VERB
ma-119	46	2	,	,	PUNCT
ma-119	46	3	zalar	zalar	ADJ
ma-119	46	4	[	[	X
ma-119	46	5	9	9	NUM
ma-119	46	6	]	]	PUNCT
ma-119	46	7	showed	show	VERB
ma-119	46	8	that	that	SCONJ
ma-119	46	9	a	a	DET
ma-119	46	10	jordan	jordan	PROPN
ma-119	46	11	∗-derivation	∗-derivation	NOUN
ma-119	46	12	is	be	AUX
ma-119	46	13	the	the	DET
ma-119	46	14	map	map	NOUN
ma-119	46	15	δa(x	δa(x	NOUN
ma-119	46	16	)	)	PUNCT
ma-119	46	17	=	=	NOUN
ma-119	46	18	ax	ax	NOUN
ma-119	46	19	−	−	PROPN
ma-119	46	20	x∗a	x∗a	PUNCT
ma-119	46	21	forfixed	forfixe	VERB
ma-119	46	22	a	a	DET
ma-119	46	23	∈	∈	PROPN
ma-119	46	24	u	u	NOUN
ma-119	46	25	;	;	PUNCT
ma-119	46	26	hence	hence	ADV
ma-119	46	27	,	,	PUNCT
ma-119	46	28	the	the	DET
ma-119	46	29	derivation	derivation	NOUN
ma-119	46	30	is	be	AUX
ma-119	46	31	inner	inner	ADJ
ma-119	46	32	.	.	PUNCT
ma-119	47	1	douglas	douglas	PROPN
ma-119	47	2	[	[	X
ma-119	47	3	15	15	NUM
ma-119	47	4	]	]	PUNCT
ma-119	47	5	continued	continue	VERB
ma-119	47	6	the	the	DET
ma-119	47	7	study	study	NOUN
ma-119	47	8	of	of	ADP
ma-119	47	9	ws(y	ws(y	PROPN
ma-119	47	10	)	)	PUNCT
ma-119	47	11	which	which	PRON
ma-119	47	12	wasconsiderably	wasconsiderably	VERB
ma-119	47	13	more	more	ADV
ma-119	47	14	amenable	amenable	ADJ
ma-119	47	15	where	where	SCONJ
ma-119	47	16	archbold	archbold	PROPN
ma-119	47	17	[	[	X
ma-119	47	18	1	1	X
ma-119	47	19	]	]	PUNCT
ma-119	47	20	defined	define	VERB
ma-119	47	21	the	the	DET
ma-119	47	22	smallest	small	ADJ
ma-119	47	23	numbers	number	NOUN
ma-119	47	24	to	to	PART
ma-119	47	25	be	be	AUX
ma-119	47	26	[	[	X
ma-119	47	27	0,∞	0,∞	X
ma-119	47	28	]	]	PUNCT
ma-119	47	29	andintroduced	andintroduce	VERB
ma-119	47	30	two	two	NUM
ma-119	47	31	constants	constant	NOUN
ma-119	47	32	w	w	ADP
ma-119	47	33	(	(	PUNCT
ma-119	47	34	y	y	PROPN
ma-119	47	35	)	)	PUNCT
ma-119	47	36	and	and	CCONJ
ma-119	47	37	wt(y	wt(y	X
ma-119	47	38	)	)	PUNCT
ma-119	47	39	such	such	ADJ
ma-119	47	40	that	that	DET
ma-119	47	41	d(y	d(y	NOUN
ma-119	47	42	,	,	PUNCT
ma-119	47	43	z(y	z(y	PROPN
ma-119	47	44	)	)	PUNCT
ma-119	47	45	)	)	PUNCT
ma-119	48	1	≤	≤	NUM
ma-119	48	2	w	w	ADP
ma-119	48	3	(	(	PUNCT
ma-119	48	4	y	y	PROPN
ma-119	48	5	)	)	PUNCT
ma-119	48	6	‖d(y	‖d(y	X
ma-119	48	7	,	,	PUNCT
ma-119	48	8	y	y	PROPN
ma-119	48	9	)	)	PUNCT
ma-119	48	10	‖	‖	PROPN
ma-119	48	11	,	,	PUNCT
ma-119	48	12	for	for	ADP
ma-119	48	13	all	all	DET
ma-119	48	14	y	y	PROPN
ma-119	48	15	∈	∈	PROPN
ma-119	48	16	yand	yand	PROPN
ma-119	48	17	d(y	d(y	PROPN
ma-119	48	18	,	,	PUNCT
ma-119	48	19	z(y	z(y	PROPN
ma-119	48	20	)	)	PUNCT
ma-119	48	21	)	)	PUNCT
ma-119	48	22	≤	≤	NOUN
ma-119	48	23	ws(y	ws(y	PUNCT
ma-119	48	24	)	)	PUNCT
ma-119	48	25	‖d(y	‖d(y	X
ma-119	48	26	,	,	PUNCT
ma-119	48	27	y	y	PROPN
ma-119	48	28	)	)	PUNCT
ma-119	48	29	‖	‖	PROPN
ma-119	48	30	,	,	PUNCT
ma-119	48	31	for	for	ADP
ma-119	48	32	all	all	PRON
ma-119	48	33	y	y	NOUN
ma-119	48	34	=	=	SYM
ma-119	48	35	y∗	y∗	PROPN
ma-119	48	36	∈	∈	PROPN
ma-119	48	37	y.	y.	NOUN
ma-119	48	38	the	the	DET
ma-119	48	39	author	author	NOUN
ma-119	48	40	in	in	ADP
ma-119	48	41	[	[	X
ma-119	48	42	26	26	NUM
ma-119	48	43	]	]	PUNCT
ma-119	48	44	showed	show	VERB
ma-119	48	45	that	that	SCONJ
ma-119	48	46	for	for	ADP
ma-119	48	47	the	the	DET
ma-119	48	48	nthorder	nthorder	NOUN
ma-119	48	49	commutator	commutator	NOUN
ma-119	49	1	[	[	X
ma-119	49	2	[	[	X
ma-119	49	3	[	[	X
ma-119	49	4	k(b	k(b	PROPN
ma-119	49	5	)	)	PUNCT
ma-119	49	6	,	,	PUNCT
ma-119	49	7	y	y	PROPN
ma-119	49	8	]	]	PUNCT
ma-119	49	9	,	,	PUNCT
ma-119	49	10	y	y	PROPN
ma-119	49	11	]	]	PUNCT
ma-119	49	12	,	,	PUNCT
ma-119	49	13	...	...	PUNCT
ma-119	49	14	,	,	PUNCT
ma-119	49	15	y	y	PROPN
ma-119	49	16	]	]	PUNCT
ma-119	49	17	,	,	PUNCT
ma-119	49	18	a	a	DET
ma-119	49	19	formula	formula	NOUN
ma-119	49	20	was	be	AUX
ma-119	49	21	obtained	obtain	VERB
ma-119	49	22	in	in	ADP
ma-119	49	23	terms	term	NOUN
ma-119	49	24	of	of	ADP
ma-119	49	25	the	the	DET
ma-119	49	26	frechet	frechet	PROPN
ma-119	49	27	derivatives	derivative	NOUN
ma-119	49	28	smk(b	smk(b	NOUN
ma-119	49	29	)	)	PUNCT
ma-119	49	30	in	in	ADP
ma-119	49	31	which	which	PRON
ma-119	49	32	the	the	DET
ma-119	49	33	formula	formula	NOUN
ma-119	49	34	illustrated	illustrate	VERB
ma-119	49	35	was	be	AUX
ma-119	49	36	used	use	VERB
ma-119	49	37	to	to	PART
ma-119	49	38	obtain	obtain	VERB
ma-119	49	39	bounds	bound	NOUN
ma-119	49	40	for	for	ADP
ma-119	49	41	norms	norm	NOUN
ma-119	49	42	of	of	ADP
ma-119	49	43	a	a	DET
ma-119	49	44	generalizedcommutator	generalizedcommutator	NOUN
ma-119	49	45	k(b)y	k(b)y	NOUN
ma-119	49	46	−	−	PROPN
ma-119	49	47	y	y	PROPN
ma-119	49	48	k(b	k(b	PROPN
ma-119	49	49	)	)	PUNCT
ma-119	49	50	and	and	CCONJ
ma-119	49	51	their	their	PRON
ma-119	49	52	higher	high	ADJ
ma-119	49	53	order	order	NOUN
ma-119	49	54	analogues	analogue	NOUN
ma-119	49	55	.	.	PUNCT
ma-119	50	1	in	in	ADP
ma-119	50	2	[	[	X
ma-119	50	3	17	17	NUM
ma-119	50	4	]	]	PUNCT
ma-119	50	5	,	,	PUNCT
ma-119	50	6	numerical	numerical	ADJ
ma-119	50	7	ranges	range	NOUN
ma-119	50	8	of	of	ADP
ma-119	50	9	2	2	NUM
ma-119	50	10	x	x	SYM
ma-119	50	11	2matrices	2matrices	NUM
ma-119	50	12	were	be	AUX
ma-119	50	13	determined	determine	VERB
ma-119	50	14	and	and	CCONJ
ma-119	50	15	the	the	DET
ma-119	50	16	convex	convex	NOUN
ma-119	50	17	of	of	ADP
ma-119	50	18	the	the	DET
ma-119	50	19	numerical	numerical	ADJ
ma-119	50	20	range	range	NOUN
ma-119	50	21	for	for	ADP
ma-119	50	22	any	any	DET
ma-119	50	23	hilbert	hilbert	NOUN
ma-119	50	24	space	space	NOUN
ma-119	50	25	operator	operator	NOUN
ma-119	50	26	wasestablished	wasestablishe	VERB
ma-119	50	27	in	in	ADP
ma-119	50	28	toeplitz	toeplitz	NOUN
ma-119	50	29	-	-	PUNCT
ma-119	50	30	hausdorff	hausdorff	NOUN
ma-119	50	31	theorem	theorem	NOUN
ma-119	50	32	and	and	CCONJ
ma-119	50	33	the	the	DET
ma-119	50	34	relation	relation	NOUN
ma-119	50	35	of	of	ADP
ma-119	50	36	the	the	DET
ma-119	50	37	numerical	numerical	ADJ
ma-119	50	38	range	range	NOUN
ma-119	50	39	to	to	ADP
ma-119	50	40	that	that	PRON
ma-119	50	41	of	of	ADP
ma-119	50	42	spectrumwas	spectrumwas	AUX
ma-119	50	43	discussed	discuss	VERB
ma-119	50	44	.	.	PUNCT
ma-119	51	1	further	far	ADV
ma-119	51	2	,	,	PUNCT
ma-119	51	3	the	the	DET
ma-119	51	4	closure	closure	NOUN
ma-119	51	5	of	of	ADP
ma-119	51	6	the	the	DET
ma-119	51	7	numerical	numerical	ADJ
ma-119	51	8	range	range	NOUN
ma-119	51	9	is	be	AUX
ma-119	51	10	contained	contain	VERB
ma-119	51	11	in	in	ADP
ma-119	51	12	the	the	DET
ma-119	51	13	spectrum	spectrum	NOUN
ma-119	51	14	and	and	CCONJ
ma-119	51	15	theintersection	theintersection	NOUN
ma-119	51	16	of	of	ADP
ma-119	51	17	closures	closure	NOUN
ma-119	51	18	of	of	ADP
ma-119	51	19	the	the	DET
ma-119	51	20	numerical	numerical	ADJ
ma-119	51	21	range	range	NOUN
ma-119	51	22	of	of	ADP
ma-119	51	23	all	all	DET
ma-119	51	24	operators	operator	NOUN
ma-119	51	25	were	be	AUX
ma-119	51	26	asserted	assert	VERB
ma-119	51	27	by	by	ADP
ma-119	51	28	hildebrandt’stheorem	hildebrandt’stheorem	NOUN
ma-119	51	29	.	.	PUNCT
ma-119	52	1	considering	consider	VERB
ma-119	52	2	results	result	NOUN
ma-119	52	3	on	on	ADP
ma-119	52	4	special	special	ADJ
ma-119	52	5	cases	case	NOUN
ma-119	52	6	[	[	X
ma-119	52	7	10	10	NUM
ma-119	52	8	]	]	PUNCT
ma-119	52	9	,	,	PUNCT
ma-119	52	10	established	establish	VERB
ma-119	52	11	that	that	SCONJ
ma-119	52	12	‖pxq	‖pxq	PROPN
ma-119	52	13	+	+	CCONJ
ma-119	52	14	qxp‖	qxp‖	PROPN
ma-119	52	15	≥	≥	NOUN
ma-119	52	16	‖p‖‖q‖.chi	‖p‖‖q‖.chi	PROPN
ma-119	52	17	-	-	PUNCT
ma-119	52	18	kwong	kwong	NOUN
ma-119	53	1	[	[	X
ma-119	53	2	13	13	NUM
ma-119	53	3	]	]	PUNCT
ma-119	53	4	established	establish	VERB
ma-119	53	5	that	that	SCONJ
ma-119	53	6	for	for	ADP
ma-119	53	7	an	an	DET
ma-119	53	8	n	n	NOUN
ma-119	53	9	x	x	SYM
ma-119	53	10	n	n	PRON
ma-119	53	11	matrix	matrix	NOUN
ma-119	53	12	x	x	NOUN
ma-119	53	13	,	,	PUNCT
ma-119	53	14	the	the	DET
ma-119	53	15	numerical	numerical	ADJ
ma-119	53	16	range	range	PROPN
ma-119	53	17	w	w	PROPN
ma-119	53	18	(	(	PUNCT
ma-119	53	19	x	x	X
ma-119	53	20	)	)	PUNCT
ma-119	53	21	has	have	VERB
ma-119	53	22	manyproperties	manypropertie	NOUN
ma-119	53	23	which	which	PRON
ma-119	53	24	can	can	AUX
ma-119	53	25	be	be	AUX
ma-119	53	26	used	use	VERB
ma-119	53	27	to	to	PART
ma-119	53	28	locate	locate	VERB
ma-119	53	29	eigenvalues	eigenvalue	NOUN
ma-119	53	30	to	to	PART
ma-119	53	31	obtain	obtain	VERB
ma-119	53	32	norm	norm	NOUN
ma-119	53	33	bounds	bound	NOUN
ma-119	53	34	.	.	PUNCT
ma-119	54	1	algebraic	algebraic	PROPN
ma-119	54	2	and	and	CCONJ
ma-119	54	3	analyticproperties	analyticpropertie	NOUN
ma-119	54	4	were	be	AUX
ma-119	54	5	deduced	deduce	VERB
ma-119	54	6	which	which	PRON
ma-119	54	7	help	help	NOUN
ma-119	54	8	in	in	ADP
ma-119	54	9	finding	find	VERB
ma-119	54	10	the	the	DET
ma-119	54	11	dilations	dilation	NOUN
ma-119	54	12	of	of	ADP
ma-119	54	13	simple	simple	ADJ
ma-119	54	14	structures	structure	NOUN
ma-119	54	15	.	.	PUNCT
ma-119	55	1	let	let	VERB
ma-119	55	2	the	the	DET
ma-119	55	3	linearoperators	linearoperator	NOUN
ma-119	55	4	xi	xi	X
ma-119	55	5	and	and	CCONJ
ma-119	55	6	yi	yi	PROPN
ma-119	55	7	,	,	PUNCT
ma-119	55	8	1	1	NUM
ma-119	55	9	≤	≤	NUM
ma-119	55	10	i	i	PRON
ma-119	55	11	≤	≤	NOUN
ma-119	55	12	n	n	PRON
ma-119	55	13	act	act	VERB
ma-119	55	14	on	on	ADP
ma-119	55	15	separate	separate	ADJ
ma-119	55	16	hilbert	hilbert	NOUN
ma-119	55	17	space	space	NOUN
ma-119	55	18	h	h	NOUN
ma-119	55	19	,	,	PUNCT
ma-119	55	20	therefore	therefore	ADV
ma-119	55	21	,	,	PUNCT
ma-119	55	22	hong	hong	PROPN
ma-119	55	23	-	-	PUNCT
ma-119	55	24	ke	ke	PROPN
ma-119	55	25	,	,	PUNCT
ma-119	55	26	yue	yue	PROPN
ma-119	55	27	-	-	PUNCT
ma-119	55	28	qing	qing	NOUN
ma-119	56	1	[	[	X
ma-119	56	2	19]proved	19]proved	NUM
ma-119	56	3	that	that	SCONJ
ma-119	56	4	sup{‖	sup{‖	NOUN
ma-119	57	1	∑n	∑n	PROPN
ma-119	57	2	i=1	i=1	PROPN
ma-119	58	1	pixqi‖	pixqi‖	PROPN
ma-119	58	2	:	:	PUNCT
ma-119	58	3	x	x	SYM
ma-119	58	4	∈	∈	PROPN
ma-119	58	5	b(h	b(h	PROPN
ma-119	58	6	)	)	PUNCT
ma-119	58	7	,	,	PUNCT
ma-119	58	8	‖x‖	‖x‖	VERB
ma-119	58	9	≤	≤	NUM
ma-119	58	10	1	1	NUM
ma-119	58	11	}	}	PUNCT
ma-119	58	12	=	=	PUNCT
ma-119	58	13	sup{‖	sup{‖	NOUN
ma-119	58	14	∑n	∑n	NUM
ma-119	58	15	i=1	i=1	PROPN
ma-119	59	1	pitqi‖	pitqi‖	PROPN
ma-119	59	2	:	:	PUNCT
ma-119	59	3	uu∗	uu∗	X
ma-119	59	4	=	=	SYM
ma-119	59	5	t	t	PROPN
ma-119	59	6	∗u	∗u	NOUN
ma-119	59	7	=	=	PROPN
ma-119	59	8	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	NUM
ma-119	59	9	eur	eur	NOUN
ma-119	59	10	.	.	PUNCT
ma-119	60	1	j.	j.	PROPN
ma-119	60	2	math	math	PROPN
ma-119	60	3	.	.	PUNCT
ma-119	61	1	anal	anal	PROPN
ma-119	61	2	.	.	PUNCT
ma-119	62	1	10.28924	10.28924	NUM
ma-119	62	2	/	/	SYM
ma-119	62	3	ada	ada	PROPN
ma-119	62	4	/	/	SYM
ma-119	62	5	ma.3.9	ma.3.9	PROPN
ma-119	62	6	3	3	NUM
ma-119	62	7	i	i	NOUN
ma-119	62	8	,	,	PUNCT
ma-119	62	9	u	u	PROPN
ma-119	62	10	∈	∈	PROPN
ma-119	62	11	b(h	b(h	PROPN
ma-119	62	12	)	)	PUNCT
ma-119	62	13	}	}	PUNCT
ma-119	62	14	.	.	PUNCT
ma-119	63	1	in	in	ADP
ma-119	63	2	addition	addition	NOUN
ma-119	63	3	,	,	PUNCT
ma-119	63	4	okelo	okelo	NOUN
ma-119	63	5	,	,	PUNCT
ma-119	63	6	agure	agure	VERB
ma-119	63	7	and	and	CCONJ
ma-119	63	8	ambogo	ambogo	NOUN
ma-119	63	9	[	[	X
ma-119	63	10	35	35	NUM
ma-119	63	11	]	]	PUNCT
ma-119	63	12	established	establish	VERB
ma-119	63	13	the	the	DET
ma-119	63	14	norm	norm	NOUN
ma-119	63	15	of	of	ADP
ma-119	63	16	jordan	jordan	PROPN
ma-119	63	17	elementaryoperator	elementaryoperator	PROPN
ma-119	63	18	ua	ua	PROPN
ma-119	63	19	,	,	PUNCT
ma-119	63	20	b	b	PROPN
ma-119	63	21	:	:	PUNCT
ma-119	63	22	b(h)→	b(h)→	PROPN
ma-119	63	23	b(h	b(h	PROPN
ma-119	63	24	)	)	PUNCT
ma-119	63	25	which	which	PRON
ma-119	63	26	is	be	AUX
ma-119	63	27	given	give	VERB
ma-119	63	28	by	by	ADP
ma-119	63	29	ua	ua	PROPN
ma-119	63	30	,	,	PUNCT
ma-119	63	31	b	b	PROPN
ma-119	63	32	=	=	SYM
ma-119	63	33	ay	ay	PROPN
ma-119	63	34	b+by	b+by	NOUN
ma-119	63	35	a	a	PRON
ma-119	63	36	,	,	PUNCT
ma-119	63	37	∀y	∀y	PROPN
ma-119	63	38	∈	∈	PROPN
ma-119	63	39	b(h	b(h	PROPN
ma-119	63	40	)	)	PUNCT
ma-119	63	41	and	and	CCONJ
ma-119	63	42	a	a	PRON
ma-119	63	43	,	,	PUNCT
ma-119	63	44	b	b	NOUN
ma-119	63	45	fixedin	fixedin	ADJ
ma-119	63	46	b(h	b(h	PROPN
ma-119	63	47	)	)	PUNCT
ma-119	63	48	and	and	CCONJ
ma-119	63	49	showed	show	VERB
ma-119	63	50	that	that	SCONJ
ma-119	63	51	‖ua	‖ua	PROPN
ma-119	63	52	,	,	PUNCT
ma-119	63	53	b	b	PROPN
ma-119	63	54	‖	‖	ADJ
ma-119	63	55	≥	≥	NOUN
ma-119	63	56	‖a‖‖b‖	‖a‖‖b‖	ADJ
ma-119	63	57	and	and	CCONJ
ma-119	63	58	then	then	ADV
ma-119	63	59	characterized	characterize	VERB
ma-119	63	60	the	the	DET
ma-119	63	61	norm	norm	NOUN
ma-119	63	62	-	-	PUNCT
ma-119	63	63	attainable	attainable	ADJ
ma-119	63	64	operatorsusing	operatorsuse	VERB
ma-119	63	65	this	this	DET
ma-119	63	66	norm	norm	NOUN
ma-119	63	67	.	.	PUNCT
ma-119	64	1	inner	inner	ADJ
ma-119	64	2	derivations	derivation	NOUN
ma-119	64	3	implemented	implement	VERB
ma-119	64	4	by	by	ADP
ma-119	64	5	norm	norm	NOUN
ma-119	64	6	-	-	PUNCT
ma-119	64	7	attainable	attainable	ADJ
ma-119	64	8	elements	element	NOUN
ma-119	64	9	of	of	ADP
ma-119	64	10	a	a	DET
ma-119	64	11	c∗-algebra	c∗-algebra	PROPN
ma-119	64	12	hasrelation	hasrelation	NOUN
ma-119	64	13	to	to	ADP
ma-119	64	14	those	those	PRON
ma-119	64	15	of	of	ADP
ma-119	64	16	ideals	ideal	NOUN
ma-119	64	17	and	and	CCONJ
ma-119	64	18	primitive	primitive	ADJ
ma-119	64	19	ideals	ideal	NOUN
ma-119	64	20	.	.	PUNCT
ma-119	65	1	since	since	SCONJ
ma-119	65	2	there	there	PRON
ma-119	65	3	is	be	VERB
ma-119	65	4	a	a	DET
ma-119	65	5	relationship	relationship	NOUN
ma-119	65	6	between	between	ADP
ma-119	65	7	the	the	DET
ma-119	65	8	constants	constant	NOUN
ma-119	65	9	a(ξ	a(ξ	PROPN
ma-119	65	10	)	)	PUNCT
ma-119	65	11	and	and	CCONJ
ma-119	65	12	asξ	asξ	NOUN
ma-119	65	13	of	of	ADP
ma-119	65	14	c∗-algebras	c∗-algebra	NOUN
ma-119	65	15	to	to	ADP
ma-119	65	16	the	the	DET
ma-119	65	17	ideals	ideal	NOUN
ma-119	65	18	and	and	CCONJ
ma-119	65	19	primitive	primitive	ADJ
ma-119	65	20	ideals	ideal	NOUN
ma-119	65	21	then	then	ADV
ma-119	65	22	related	relate	VERB
ma-119	65	23	results	result	NOUN
ma-119	65	24	have	have	AUX
ma-119	65	25	beengiven	beengiven	VERB
ma-119	65	26	in	in	ADP
ma-119	65	27	general	general	ADJ
ma-119	65	28	banch	banch	PROPN
ma-119	65	29	settins	settin	NOUN
ma-119	65	30	.	.	PUNCT
ma-119	66	1	okelo	okelo	PROPN
ma-119	66	2	,	,	PUNCT
ma-119	66	3	agure	agure	NOUN
ma-119	66	4	and	and	CCONJ
ma-119	66	5	oleche	oleche	NOUN
ma-119	67	1	[	[	X
ma-119	67	2	38	38	NUM
ma-119	67	3	]	]	PUNCT
ma-119	67	4	gave	give	VERB
ma-119	67	5	results	result	NOUN
ma-119	67	6	on	on	ADP
ma-119	67	7	necessary	necessary	ADJ
ma-119	67	8	andsufficient	andsufficient	NOUN
ma-119	67	9	conditions	condition	NOUN
ma-119	67	10	for	for	ADP
ma-119	67	11	norm	norm	NOUN
ma-119	67	12	-	-	PUNCT
ma-119	67	13	attainable	attainable	ADJ
ma-119	67	14	operators	operator	NOUN
ma-119	67	15	and	and	CCONJ
ma-119	67	16	also	also	ADV
ma-119	67	17	studied	study	VERB
ma-119	67	18	norm	norm	NOUN
ma-119	67	19	-	-	PUNCT
ma-119	67	20	attainable	attainable	ADJ
ma-119	67	21	operators	operator	NOUN
ma-119	67	22	andgeneralized	andgeneralize	VERB
ma-119	67	23	derivations	derivation	NOUN
ma-119	67	24	.	.	PUNCT
ma-119	68	1	okelo	okelo	PROPN
ma-119	68	2	[	[	X
ma-119	68	3	37	37	NUM
ma-119	68	4	]	]	PUNCT
ma-119	68	5	extended	extend	VERB
ma-119	68	6	the	the	DET
ma-119	68	7	work	work	NOUN
ma-119	68	8	by	by	ADP
ma-119	68	9	presenting	present	VERB
ma-119	68	10	new	new	ADJ
ma-119	68	11	results	result	NOUN
ma-119	68	12	on	on	ADP
ma-119	68	13	conditionsthat	conditionsthat	PRON
ma-119	68	14	are	be	AUX
ma-119	68	15	necessary	necessary	ADJ
ma-119	68	16	and	and	CCONJ
ma-119	68	17	sufficient	sufficient	ADJ
ma-119	68	18	for	for	ADP
ma-119	68	19	norm	norm	NOUN
ma-119	68	20	-	-	PUNCT
ma-119	68	21	attainability	attainability	NOUN
ma-119	68	22	for	for	ADP
ma-119	68	23	operators	operator	NOUN
ma-119	68	24	in	in	ADP
ma-119	68	25	hilbert	hilbert	NOUN
ma-119	68	26	space	space	NOUN
ma-119	68	27	,	,	PUNCT
ma-119	68	28	elementaryoperators	elementaryoperator	NOUN
ma-119	68	29	and	and	CCONJ
ma-119	68	30	generalized	generalized	ADJ
ma-119	68	31	derivations	derivation	NOUN
ma-119	68	32	.	.	PUNCT
ma-119	69	1	further	far	ADV
ma-119	69	2	,	,	PUNCT
ma-119	69	3	okelo	okelo	X
ma-119	69	4	[	[	X
ma-119	69	5	37	37	NUM
ma-119	69	6	]	]	PUNCT
ma-119	69	7	established	establish	VERB
ma-119	69	8	that	that	SCONJ
ma-119	69	9	a	a	DET
ma-119	69	10	unit	unit	NOUN
ma-119	69	11	vector	vector	NOUN
ma-119	69	12	exists	exist	VERB
ma-119	69	13	λ	λ	PROPN
ma-119	69	14	∈	∈	PROPN
ma-119	69	15	h	h	NOUN
ma-119	69	16	,	,	PUNCT
ma-119	69	17	‖λ‖	‖λ‖	PROPN
ma-119	69	18	=	=	PUNCT
ma-119	70	1	1	1	NUM
ma-119	70	2	such	such	ADJ
ma-119	70	3	that	that	SCONJ
ma-119	70	4	‖sλ‖	‖sλ‖	PROPN
ma-119	70	5	=	=	SYM
ma-119	70	6	‖s‖	‖s‖	PROPN
ma-119	70	7	with	with	ADP
ma-119	70	8	〈	〈	PROPN
ma-119	70	9	sλ	sλ	NOUN
ma-119	70	10	,	,	PUNCT
ma-119	70	11	λ	λ	NOUN
ma-119	70	12	〉	〉	NOUN
ma-119	70	13	=	=	SYM
ma-119	70	14	η	η	PROPN
ma-119	70	15	.	.	PROPN
ma-119	70	16	results	result	NOUN
ma-119	70	17	from	from	ADP
ma-119	70	18	[	[	X
ma-119	70	19	23	23	NUM
ma-119	70	20	]	]	PUNCT
ma-119	70	21	showed	show	VERB
ma-119	70	22	that	that	SCONJ
ma-119	70	23	every	every	DET
ma-119	70	24	jordanderivation	jordanderivation	NOUN
ma-119	70	25	of	of	ADP
ma-119	70	26	the	the	DET
ma-119	70	27	trivial	trivial	ADJ
ma-119	70	28	extension	extension	NOUN
ma-119	70	29	of	of	ADP
ma-119	70	30	a	a	PRON
ma-119	70	31	by	by	ADP
ma-119	70	32	m	m	PROPN
ma-119	70	33	,	,	PUNCT
ma-119	70	34	under	under	ADP
ma-119	70	35	certain	certain	ADJ
ma-119	70	36	conditions	condition	NOUN
ma-119	70	37	,	,	PUNCT
ma-119	70	38	is	be	AUX
ma-119	70	39	the	the	DET
ma-119	70	40	sum	sum	NOUN
ma-119	70	41	of	of	ADP
ma-119	70	42	a	a	DET
ma-119	70	43	derivationand	derivationand	NOUN
ma-119	70	44	antiderivation	antiderivation	NOUN
ma-119	70	45	.	.	PUNCT
ma-119	71	1	in	in	ADP
ma-119	71	2	[	[	X
ma-119	71	3	10	10	NUM
ma-119	71	4	]	]	PUNCT
ma-119	71	5	,	,	PUNCT
ma-119	71	6	the	the	DET
ma-119	71	7	author	author	NOUN
ma-119	71	8	studied	study	VERB
ma-119	71	9	norm	norm	NOUN
ma-119	71	10	-	-	PUNCT
ma-119	71	11	attainable	attainable	ADJ
ma-119	71	12	operators	operator	NOUN
ma-119	71	13	that	that	PRON
ma-119	71	14	are	be	AUX
ma-119	71	15	convergent	convergent	ADJ
ma-119	71	16	andestablished	andestablishe	VERB
ma-119	71	17	norm	norm	NOUN
ma-119	71	18	-	-	PUNCT
ma-119	71	19	attainability	attainability	NOUN
ma-119	71	20	of	of	ADP
ma-119	71	21	operators	operator	NOUN
ma-119	71	22	via	via	ADP
ma-119	71	23	projective	projective	ADJ
ma-119	71	24	tensor	tensor	NOUN
ma-119	71	25	norm	norm	NOUN
ma-119	71	26	.	.	PUNCT
ma-119	72	1	wickstead	wickstead	PROPN
ma-119	73	1	[	[	X
ma-119	73	2	52	52	NUM
ma-119	73	3	]	]	PUNCT
ma-119	73	4	showed	show	VERB
ma-119	73	5	thatif	thatif	NOUN
ma-119	73	6	an	an	DET
ma-119	73	7	atomic	atomic	ADJ
ma-119	73	8	banach	banach	NOUN
ma-119	73	9	lattice	lattice	PROPN
ma-119	73	10	z	z	NOUN
ma-119	73	11	with	with	ADP
ma-119	73	12	a	a	DET
ma-119	73	13	continuous	continuous	ADJ
ma-119	73	14	norm	norm	NOUN
ma-119	73	15	order	order	NOUN
ma-119	73	16	,	,	PUNCT
ma-119	73	17	x	x	PRON
ma-119	73	18	,	,	PUNCT
ma-119	73	19	y	y	PROPN
ma-119	73	20	∈	∈	PROPN
ma-119	73	21	t	t	PROPN
ma-119	73	22	r	r	NOUN
ma-119	73	23	and	and	CCONJ
ma-119	73	24	mx	mx	PROPN
ma-119	73	25	,	,	PUNCT
ma-119	73	26	y	y	PROPN
ma-119	73	27	is	be	AUX
ma-119	73	28	the	the	DET
ma-119	73	29	operatoron	operatoron	NOUN
ma-119	73	30	t	t	PROPN
ma-119	73	31	r	r	NOUN
ma-119	73	32	(	(	PUNCT
ma-119	73	33	z	z	NOUN
ma-119	73	34	)	)	PUNCT
ma-119	73	35	defined	define	VERB
ma-119	73	36	by	by	ADP
ma-119	73	37	mx	mx	PROPN
ma-119	73	38	,	,	PUNCT
ma-119	73	39	y	y	PROPN
ma-119	73	40	(	(	PUNCT
ma-119	73	41	a	a	NOUN
ma-119	73	42	)	)	PUNCT
ma-119	73	43	=	=	SYM
ma-119	73	44	xay	xay	PROPN
ma-119	73	45	,	,	PUNCT
ma-119	73	46	then	then	ADV
ma-119	73	47	‖mx	‖mx	PROPN
ma-119	73	48	,	,	PUNCT
ma-119	73	49	y	y	PROPN
ma-119	73	50	‖r	‖r	NOUN
ma-119	73	51	=	=	PUNCT
ma-119	73	52	‖x‖r‖y	‖x‖r‖y	ADJ
ma-119	73	53	‖r	‖r	NOUN
ma-119	73	54	but	but	CCONJ
ma-119	73	55	there	there	PRON
ma-119	73	56	is	be	VERB
ma-119	73	57	no	no	DET
ma-119	73	58	real	real	ADJ
ma-119	73	59	β	β	X
ma-119	73	60	>	>	X
ma-119	73	61	0such	0such	PROPN
ma-119	73	62	that	that	SCONJ
ma-119	73	63	‖mx	‖mx	PROPN
ma-119	73	64	,	,	PUNCT
ma-119	73	65	y	y	PROPN
ma-119	73	66	‖r	‖r	NOUN
ma-119	73	67	=	=	PUNCT
ma-119	73	68	β‖x‖r‖y	β‖x‖r‖y	ADJ
ma-119	73	69	‖r	‖r	NOUN
ma-119	73	70	.	.	PUNCT
ma-119	74	1	okelo	okelo	PROPN
ma-119	75	1	[	[	X
ma-119	75	2	36	36	NUM
ma-119	75	3	]	]	PUNCT
ma-119	75	4	outlined	outline	VERB
ma-119	75	5	the	the	DET
ma-119	75	6	theory	theory	NOUN
ma-119	75	7	of	of	ADP
ma-119	75	8	normal	normal	ADJ
ma-119	75	9	,	,	PUNCT
ma-119	75	10	self	self	NOUN
ma-119	75	11	-	-	PUNCT
ma-119	75	12	adjoint	adjoint	NOUN
ma-119	75	13	and	and	CCONJ
ma-119	75	14	norm	norm	NOUN
ma-119	75	15	-	-	PUNCT
ma-119	75	16	attainable	attainable	ADJ
ma-119	75	17	operators	operator	NOUN
ma-119	75	18	then	then	ADV
ma-119	75	19	presented	present	VERB
ma-119	75	20	norms	norm	NOUN
ma-119	75	21	of	of	ADP
ma-119	75	22	operators	operator	NOUN
ma-119	75	23	in	in	ADP
ma-119	75	24	hilbert	hilbert	PROPN
ma-119	75	25	spaces	space	NOUN
ma-119	75	26	.	.	PUNCT
ma-119	76	1	in	in	ADP
ma-119	76	2	[	[	X
ma-119	76	3	8	8	NUM
ma-119	76	4	]	]	PUNCT
ma-119	76	5	the	the	DET
ma-119	76	6	author	author	NOUN
ma-119	76	7	provedthat	provedthat	NOUN
ma-119	76	8	for	for	ADP
ma-119	76	9	a	a	DET
ma-119	76	10	linear	linear	ADJ
ma-119	76	11	map	map	NOUN
ma-119	76	12	∆	∆	PROPN
ma-119	76	13	:	:	PUNCT
ma-119	76	14	u	u	X
ma-119	76	15	→	→	SYM
ma-119	76	16	u	u	PROPN
ma-119	76	17	,	,	PUNCT
ma-119	76	18	∆(xy	∆(xy	NUM
ma-119	76	19	)	)	PUNCT
ma-119	76	20	=	=	PUNCT
ma-119	77	1	∆(x)y	∆(x)y	X
ma-119	78	1	+	+	X
ma-119	78	2	∆x(y	∆x(y	PROPN
ma-119	78	3	)	)	PUNCT
ma-119	78	4	for	for	ADP
ma-119	78	5	each	each	DET
ma-119	78	6	x	x	NOUN
ma-119	78	7	,	,	PUNCT
ma-119	78	8	y	y	PROPN
ma-119	78	9	∈	∈	PROPN
ma-119	78	10	u	u	NOUN
ma-119	78	11	is	be	AUX
ma-119	78	12	a	a	DET
ma-119	78	13	derivation	derivation	NOUN
ma-119	78	14	,	,	PUNCT
ma-119	78	15	and	and	CCONJ
ma-119	78	16	for	for	ADP
ma-119	78	17	any	any	DET
ma-119	78	18	two	two	NUM
ma-119	78	19	derivations	derivation	NOUN
ma-119	78	20	∆	∆	PROPN
ma-119	78	21	and	and	CCONJ
ma-119	78	22	∆′	∆′	PROPN
ma-119	78	23	on	on	ADP
ma-119	78	24	a	a	DET
ma-119	78	25	c∗-algebra	c∗-algebra	PROPN
ma-119	78	26	u	u	NOUN
ma-119	78	27	there	there	ADV
ma-119	78	28	exists	exist	VERB
ma-119	78	29	a	a	DET
ma-119	78	30	derivation	derivation	NOUN
ma-119	78	31	δ	δ	NOUN
ma-119	78	32	∈	∈	NOUN
ma-119	79	1	u	u	NOUN
ma-119	79	2	suchthat	suchthat	VERB
ma-119	79	3	∆∆′	∆∆′	VERB
ma-119	79	4	=	=	X
ma-119	79	5	δ2	δ2	VERB
ma-119	79	6	if	if	SCONJ
ma-119	79	7	and	and	CCONJ
ma-119	79	8	only	only	ADV
ma-119	79	9	if	if	SCONJ
ma-119	79	10	either	either	PRON
ma-119	79	11	∆′	∆′	PROPN
ma-119	79	12	=	=	SYM
ma-119	79	13	0	0	NUM
ma-119	79	14	or	or	CCONJ
ma-119	79	15	∆	∆	PROPN
ma-119	79	16	=	=	SYM
ma-119	79	17	f	f	PROPN
ma-119	79	18	∆′	∆′	PROPN
ma-119	79	19	for	for	ADP
ma-119	79	20	any	any	DET
ma-119	79	21	f	f	PROPN
ma-119	79	22	∈	∈	PROPN
ma-119	79	23	c.	c.	PROPN
ma-119	79	24	clifford	clifford	PROPN
ma-119	80	1	[	[	X
ma-119	80	2	12	12	NUM
ma-119	80	3	]	]	PUNCT
ma-119	80	4	studiedhypercyclic	studiedhypercyclic	ADJ
ma-119	80	5	generalized	generalize	VERB
ma-119	80	6	derivations	derivation	NOUN
ma-119	80	7	acting	act	VERB
ma-119	80	8	on	on	ADP
ma-119	80	9	separable	separable	ADJ
ma-119	80	10	ideals	ideal	NOUN
ma-119	80	11	of	of	ADP
ma-119	80	12	operators	operator	NOUN
ma-119	80	13	and	and	CCONJ
ma-119	80	14	also	also	ADV
ma-119	80	15	identifiedconcrete	identifiedconcrete	ADJ
ma-119	80	16	examples	example	NOUN
ma-119	80	17	and	and	CCONJ
ma-119	80	18	established	establish	VERB
ma-119	80	19	some	some	DET
ma-119	80	20	conditions	condition	NOUN
ma-119	80	21	that	that	PRON
ma-119	80	22	are	be	AUX
ma-119	80	23	necessary	necessary	ADJ
ma-119	80	24	and	and	CCONJ
ma-119	80	25	sufficient	sufficient	ADJ
ma-119	80	26	for	for	ADP
ma-119	80	27	theirhypercyclicity	theirhypercyclicity	NOUN
ma-119	80	28	.	.	PUNCT
ma-119	81	1	okelo	okelo	PROPN
ma-119	82	1	[	[	X
ma-119	82	2	36	36	NUM
ma-119	82	3	]	]	PUNCT
ma-119	82	4	considered	consider	VERB
ma-119	82	5	orthogonal	orthogonal	ADJ
ma-119	82	6	and	and	CCONJ
ma-119	82	7	norm	norm	NOUN
ma-119	82	8	-	-	PUNCT
ma-119	82	9	attainable	attainable	ADJ
ma-119	82	10	operators	operator	NOUN
ma-119	82	11	in	in	ADP
ma-119	82	12	banach	banach	NOUN
ma-119	82	13	spaces	space	NOUN
ma-119	82	14	,	,	PUNCT
ma-119	82	15	gave	give	VERB
ma-119	82	16	in	in	ADP
ma-119	82	17	details	detail	NOUN
ma-119	82	18	the	the	DET
ma-119	82	19	characterization	characterization	NOUN
ma-119	82	20	and	and	CCONJ
ma-119	82	21	generalizations	generalization	NOUN
ma-119	82	22	of	of	ADP
ma-119	82	23	norm	norm	NOUN
ma-119	82	24	-	-	PUNCT
ma-119	82	25	attainability	attainability	NOUN
ma-119	82	26	and	and	CCONJ
ma-119	82	27	orthogonality.the	orthogonality.the	DET
ma-119	82	28	conditions	condition	NOUN
ma-119	82	29	that	that	PRON
ma-119	82	30	are	be	AUX
ma-119	82	31	sufficient	sufficient	ADJ
ma-119	82	32	and	and	CCONJ
ma-119	82	33	necessary	necessary	ADJ
ma-119	82	34	for	for	ADP
ma-119	82	35	norm	norm	NOUN
ma-119	82	36	-	-	PUNCT
ma-119	82	37	attainability	attainability	NOUN
ma-119	82	38	of	of	ADP
ma-119	82	39	operators	operator	NOUN
ma-119	82	40	on	on	ADP
ma-119	82	41	a	a	DET
ma-119	82	42	hilbertspace	hilbertspace	NOUN
ma-119	82	43	,	,	PUNCT
ma-119	82	44	the	the	DET
ma-119	82	45	result	result	NOUN
ma-119	82	46	on	on	ADP
ma-119	82	47	orthogonal	orthogonal	ADJ
ma-119	82	48	range	range	NOUN
ma-119	82	49	and	and	CCONJ
ma-119	82	50	the	the	DET
ma-119	82	51	kernel	kernel	NOUN
ma-119	82	52	of	of	ADP
ma-119	82	53	elementary	elementary	ADJ
ma-119	82	54	operators	operator	NOUN
ma-119	82	55	implemented	implement	VERB
ma-119	82	56	by	by	ADP
ma-119	82	57	norm	norm	NOUN
ma-119	82	58	-	-	PUNCT
ma-119	82	59	attainable	attainable	ADJ
ma-119	82	60	operators	operator	NOUN
ma-119	82	61	in	in	ADP
ma-119	82	62	banach	banach	NOUN
ma-119	82	63	spaces	space	NOUN
ma-119	82	64	were	be	AUX
ma-119	82	65	also	also	ADV
ma-119	82	66	given	give	VERB
ma-119	82	67	.	.	PUNCT
ma-119	83	1	okelo	okelo	PROPN
ma-119	83	2	[	[	X
ma-119	83	3	34	34	NUM
ma-119	83	4	]	]	PUNCT
ma-119	83	5	characterized	characterize	VERB
ma-119	83	6	norm	norm	NOUN
ma-119	83	7	-	-	PUNCT
ma-119	83	8	attainableclasses	attainableclasse	NOUN
ma-119	83	9	in	in	ADP
ma-119	83	10	terms	term	NOUN
ma-119	83	11	of	of	ADP
ma-119	83	12	orthogonality	orthogonality	NOUN
ma-119	83	13	by	by	ADP
ma-119	83	14	giving	give	VERB
ma-119	83	15	norm	norm	NOUN
ma-119	83	16	-	-	PUNCT
ma-119	83	17	attainability	attainability	NOUN
ma-119	83	18	conditions	condition	NOUN
ma-119	83	19	that	that	PRON
ma-119	83	20	were	be	AUX
ma-119	83	21	necessary	necessary	ADJ
ma-119	83	22	andsufficient	andsufficient	NOUN
ma-119	83	23	for	for	ADP
ma-119	83	24	hilbert	hilbert	NOUN
ma-119	83	25	space	space	NOUN
ma-119	83	26	operators	operator	NOUN
ma-119	83	27	first	first	ADV
ma-119	83	28	and	and	CCONJ
ma-119	83	29	the	the	DET
ma-119	83	30	orthogonality	orthogonality	NOUN
ma-119	83	31	result	result	NOUN
ma-119	83	32	on	on	ADP
ma-119	83	33	the	the	DET
ma-119	83	34	range	range	NOUN
ma-119	83	35	and	and	CCONJ
ma-119	83	36	kernel	kernel	PROPN
ma-119	83	37	ofelementary	ofelementary	ADJ
ma-119	83	38	operators	operator	NOUN
ma-119	83	39	when	when	SCONJ
ma-119	83	40	implemented	implement	VERB
ma-119	83	41	by	by	ADP
ma-119	83	42	norm	norm	NOUN
ma-119	83	43	-	-	PUNCT
ma-119	83	44	attainable	attainable	ADJ
ma-119	83	45	operators	operator	NOUN
ma-119	83	46	in	in	ADP
ma-119	83	47	norm	norm	NOUN
ma-119	83	48	-	-	PUNCT
ma-119	83	49	attainable	attainable	ADJ
ma-119	83	50	classeswere	classeswere	NOUN
ma-119	83	51	also	also	ADV
ma-119	83	52	given	give	VERB
ma-119	83	53	.	.	PUNCT
ma-119	84	1	okelo	okelo	PROPN
ma-119	85	1	[	[	X
ma-119	85	2	38	38	NUM
ma-119	85	3	]	]	PUNCT
ma-119	85	4	gave	give	VERB
ma-119	85	5	conditions	condition	NOUN
ma-119	85	6	for	for	ADP
ma-119	85	7	norm	norm	NOUN
ma-119	85	8	-	-	PUNCT
ma-119	85	9	attainability	attainability	NOUN
ma-119	85	10	for	for	ADP
ma-119	85	11	linear	linear	ADJ
ma-119	85	12	functionals	functional	NOUN
ma-119	85	13	in	in	ADP
ma-119	85	14	banachspaces	banachspace	NOUN
ma-119	85	15	,	,	PUNCT
ma-119	85	16	non	non	ADJ
ma-119	85	17	-	-	ADJ
ma-119	85	18	power	power	ADJ
ma-119	85	19	operators	operator	NOUN
ma-119	85	20	on	on	ADP
ma-119	85	21	h	h	PROPN
ma-119	85	22	and	and	CCONJ
ma-119	85	23	elementary	elementary	ADJ
ma-119	85	24	operators	operator	NOUN
ma-119	85	25	and	and	CCONJ
ma-119	85	26	also	also	ADV
ma-119	85	27	gave	give	VERB
ma-119	85	28	a	a	DET
ma-119	85	29	new	new	ADJ
ma-119	85	30	notion	notion	NOUN
ma-119	85	31	of	of	ADP
ma-119	85	32	norm	norm	NOUN
ma-119	85	33	-	-	PUNCT
ma-119	85	34	attainability	attainability	NOUN
ma-119	85	35	for	for	ADP
ma-119	85	36	power	power	NOUN
ma-119	85	37	operators	operator	NOUN
ma-119	85	38	then	then	ADV
ma-119	85	39	characterized	characterize	VERB
ma-119	85	40	norm	norm	NOUN
ma-119	85	41	-	-	PUNCT
ma-119	85	42	attainable	attainable	ADJ
ma-119	85	43	operators	operator	NOUN
ma-119	85	44	in	in	ADP
ma-119	85	45	normed	normed	PROPN
ma-119	85	46	spaces.in	spaces.in	X
ma-119	86	1	[	[	X
ma-119	86	2	51	51	NUM
ma-119	86	3	]	]	PUNCT
ma-119	86	4	determined	determine	VERB
ma-119	86	5	the	the	DET
ma-119	86	6	norm	norm	NOUN
ma-119	86	7	of	of	ADP
ma-119	86	8	the	the	DET
ma-119	86	9	inner	inner	ADJ
ma-119	86	10	jordan	jordan	PROPN
ma-119	86	11	∗-derivation	∗-derivation	NOUN
ma-119	86	12	δs	δs	NOUN
ma-119	86	13	:	:	PUNCT
ma-119	86	14	x	x	X
ma-119	86	15	→	→	SYM
ma-119	86	16	sx	sx	PROPN
ma-119	86	17	−	−	PROPN
ma-119	86	18	x∗s	x∗s	PUNCT
ma-119	86	19	acting	act	VERB
ma-119	86	20	on	on	ADP
ma-119	86	21	the	the	DET
ma-119	86	22	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	PROPN
ma-119	86	23	eur	eur	NOUN
ma-119	86	24	.	.	PUNCT
ma-119	87	1	j.	j.	PROPN
ma-119	87	2	math	math	PROPN
ma-119	87	3	.	.	PUNCT
ma-119	88	1	anal	anal	PROPN
ma-119	88	2	.	.	PUNCT
ma-119	89	1	10.28924	10.28924	NUM
ma-119	89	2	/	/	SYM
ma-119	89	3	ada	ada	PROPN
ma-119	89	4	/	/	SYM
ma-119	89	5	ma.3.9	ma.3.9	PROPN
ma-119	89	6	4banach	4banach	PROPN
ma-119	89	7	algebra	algebra	PROPN
ma-119	89	8	b(h	b(h	PROPN
ma-119	89	9	)	)	PUNCT
ma-119	89	10	.	.	PUNCT
ma-119	90	1	it	it	PRON
ma-119	90	2	was	be	AUX
ma-119	90	3	shown	show	VERB
ma-119	90	4	that	that	SCONJ
ma-119	90	5	‖δs‖	‖δs‖	ADJ
ma-119	90	6	≥	≥	NUM
ma-119	90	7	2	2	NUM
ma-119	90	8	supλ∈w0(s	supλ∈w0(s	PROPN
ma-119	90	9	)	)	PUNCT
ma-119	90	10	|=λ|	|=λ|	NOUN
ma-119	90	11	in	in	ADP
ma-119	90	12	which	which	PRON
ma-119	90	13	w0(s	w0(s	PROPN
ma-119	90	14	)	)	PUNCT
ma-119	90	15	is	be	AUX
ma-119	90	16	the	the	DET
ma-119	90	17	maximalnumerical	maximalnumerical	ADJ
ma-119	90	18	range	range	NOUN
ma-119	90	19	of	of	ADP
ma-119	90	20	operator	operator	NOUN
ma-119	90	21	s.	s.	PROPN
ma-119	90	22	the	the	DET
ma-119	90	23	work	work	NOUN
ma-119	90	24	of	of	ADP
ma-119	90	25	[	[	X
ma-119	90	26	1	1	NUM
ma-119	90	27	]	]	PUNCT
ma-119	90	28	obtained	obtain	VERB
ma-119	90	29	precisely	precisely	ADV
ma-119	90	30	when	when	SCONJ
ma-119	90	31	zero	zero	NUM
ma-119	90	32	belongs	belong	VERB
ma-119	90	33	to	to	ADP
ma-119	90	34	maximalnumerical	maximalnumerical	ADJ
ma-119	90	35	range	range	NOUN
ma-119	90	36	of	of	ADP
ma-119	90	37	composition	composition	NOUN
ma-119	90	38	operators	operator	NOUN
ma-119	90	39	on	on	ADP
ma-119	90	40	h	h	NOUN
ma-119	90	41	and	and	CCONJ
ma-119	90	42	then	then	ADV
ma-119	90	43	characterized	characterize	VERB
ma-119	90	44	the	the	DET
ma-119	90	45	norm	norm	NOUN
ma-119	90	46	-	-	PUNCT
ma-119	90	47	attainability	attainability	NOUN
ma-119	90	48	ofderivations	ofderivation	NOUN
ma-119	90	49	on	on	ADP
ma-119	90	50	b(h	b(h	PROPN
ma-119	90	51	)	)	PUNCT
ma-119	90	52	.	.	PUNCT
ma-119	91	1	in	in	ADP
ma-119	91	2	okelo	okelo	NOUN
ma-119	91	3	[	[	X
ma-119	91	4	41	41	NUM
ma-119	91	5	]	]	PUNCT
ma-119	91	6	norm	norm	NOUN
ma-119	91	7	-	-	PUNCT
ma-119	91	8	attainability	attainability	NOUN
ma-119	91	9	for	for	ADP
ma-119	91	10	hyponormal	hyponormal	ADJ
ma-119	91	11	operators	operator	NOUN
ma-119	91	12	that	that	PRON
ma-119	91	13	are	be	AUX
ma-119	91	14	compactwere	compactwere	ADV
ma-119	91	15	characterized	characterize	VERB
ma-119	91	16	,	,	PUNCT
ma-119	91	17	sufficient	sufficient	ADJ
ma-119	91	18	conditions	condition	NOUN
ma-119	91	19	for	for	ADP
ma-119	91	20	a	a	DET
ma-119	91	21	compact	compact	ADJ
ma-119	91	22	hyponormal	hyponormal	ADJ
ma-119	91	23	operator	operator	NOUN
ma-119	91	24	that	that	PRON
ma-119	91	25	is	be	AUX
ma-119	91	26	linear	linear	PROPN
ma-119	91	27	andbounded	andbounde	VERB
ma-119	91	28	on	on	ADP
ma-119	91	29	an	an	DET
ma-119	91	30	infinite	infinite	ADJ
ma-119	91	31	dimension	dimension	NOUN
ma-119	91	32	for	for	ADP
ma-119	91	33	a	a	DET
ma-119	91	34	complex	complex	ADJ
ma-119	91	35	hilbert	hilbert	NOUN
ma-119	91	36	space	space	NOUN
ma-119	91	37	to	to	PART
ma-119	91	38	be	be	AUX
ma-119	91	39	norm	norm	NOUN
ma-119	91	40	attainable	attainable	ADJ
ma-119	91	41	were	be	AUX
ma-119	91	42	given.further	given.further	PRON
ma-119	91	43	,	,	PUNCT
ma-119	91	44	the	the	DET
ma-119	91	45	structure	structure	NOUN
ma-119	91	46	and	and	CCONJ
ma-119	91	47	other	other	ADJ
ma-119	91	48	properties	property	NOUN
ma-119	91	49	of	of	ADP
ma-119	91	50	compact	compact	ADJ
ma-119	91	51	hyponormal	hyponormal	ADJ
ma-119	91	52	operators	operator	NOUN
ma-119	91	53	when	when	SCONJ
ma-119	91	54	they	they	PRON
ma-119	91	55	are	be	AUX
ma-119	91	56	self	self	NOUN
ma-119	91	57	-	-	PUNCT
ma-119	91	58	adjoint	adjoint	NOUN
ma-119	91	59	,	,	PUNCT
ma-119	91	60	normal	normal	ADJ
ma-119	91	61	and	and	CCONJ
ma-119	91	62	norm	norm	VERB
ma-119	91	63	attainable	attainable	ADJ
ma-119	91	64	with	with	ADP
ma-119	91	65	their	their	PRON
ma-119	91	66	commutators	commutator	NOUN
ma-119	91	67	were	be	AUX
ma-119	91	68	discussed	discuss	VERB
ma-119	91	69	in	in	ADP
ma-119	91	70	general	general	ADJ
ma-119	91	71	.	.	PUNCT
ma-119	92	1	lumer	lumer	PROPN
ma-119	93	1	[	[	X
ma-119	93	2	27]obtained	27]obtained	NUM
ma-119	93	3	a	a	DET
ma-119	93	4	sharp	sharp	ADJ
ma-119	93	5	estimate	estimate	NOUN
ma-119	93	6	not	not	PART
ma-119	93	7	only	only	ADV
ma-119	93	8	from	from	ADP
ma-119	93	9	|sp(r)|	|sp(r)|	PROPN
ma-119	93	10	equal	equal	ADJ
ma-119	93	11	to	to	ADP
ma-119	93	12	spectral	spectral	ADJ
ma-119	93	13	radius	radius	NOUN
ma-119	93	14	of	of	ADP
ma-119	93	15	r	r	NOUN
ma-119	93	16	but	but	CCONJ
ma-119	93	17	indeed	indeed	ADV
ma-119	93	18	for	for	ADP
ma-119	93	19	|sp(r)|in	|sp(r)|in	PROPN
ma-119	93	20	terms	term	NOUN
ma-119	93	21	of	of	ADP
ma-119	93	22	sup(|x(r)|	sup(|x(r)|	ADJ
ma-119	93	23	,	,	PUNCT
ma-119	93	24	|x(rn)|1	|x(rn)|1	NOUN
ma-119	93	25	/	/	SYM
ma-119	93	26	n	n	CCONJ
ma-119	93	27	)	)	PUNCT
ma-119	93	28	,	,	PUNCT
ma-119	93	29	n	n	PRON
ma-119	93	30	being	be	AUX
ma-119	93	31	any	any	PRON
ma-119	93	32	positive	positive	ADJ
ma-119	93	33	even	even	ADV
ma-119	93	34	integer	integer	NOUN
ma-119	93	35	.	.	PUNCT
ma-119	94	1	in	in	ADP
ma-119	94	2	[	[	X
ma-119	94	3	18	18	NUM
ma-119	94	4	]	]	PUNCT
ma-119	94	5	the	the	DET
ma-119	94	6	author	author	NOUN
ma-119	94	7	studiedthe	studiedthe	PROPN
ma-119	94	8	algebra	algebra	PROPN
ma-119	94	9	of	of	ADP
ma-119	94	10	functions	function	NOUN
ma-119	94	11	that	that	PRON
ma-119	94	12	are	be	AUX
ma-119	94	13	continuous	continuous	ADJ
ma-119	94	14	on	on	ADP
ma-119	94	15	[	[	X
ma-119	94	16	0	0	NUM
ma-119	94	17	,	,	PUNCT
ma-119	94	18	1	1	NUM
ma-119	94	19	]	]	PUNCT
ma-119	94	20	and	and	CCONJ
ma-119	94	21	are	be	AUX
ma-119	94	22	‖.‖w	‖.‖w	X
ma-119	94	23	-approximate	-approximate	ADJ
ma-119	94	24	polynomial	polynomial	ADJ
ma-119	94	25	;	;	PUNCT
ma-119	94	26	i.epoint	i.epoint	NOUN
ma-119	94	27	-	-	PUNCT
ma-119	94	28	wise	wise	ADJ
ma-119	94	29	functions	function	NOUN
ma-119	94	30	of	of	ADP
ma-119	94	31	limits	limit	NOUN
ma-119	94	32	of	of	ADP
ma-119	94	33	‖.‖w	‖.‖w	X
ma-119	94	34	-cauchy	-cauchy	ADJ
ma-119	94	35	sequence	sequence	NOUN
ma-119	94	36	of	of	ADP
ma-119	94	37	polynomial	polynomial	ADJ
ma-119	94	38	.	.	PUNCT
ma-119	95	1	archbold	archbold	PROPN
ma-119	96	1	[	[	X
ma-119	96	2	1	1	X
ma-119	96	3	]	]	X
ma-119	96	4	investigatedwhether	investigatedwhether	VERB
ma-119	96	5	the	the	DET
ma-119	96	6	simple	simple	ADJ
ma-119	96	7	triangle	triangle	NOUN
ma-119	96	8	inequality	inequality	NOUN
ma-119	96	9	‖t	‖t	NOUN
ma-119	96	10	(	(	PUNCT
ma-119	96	11	a	a	PRON
ma-119	96	12	,	,	PUNCT
ma-119	96	13	a)‖	a)‖	ADJ
ma-119	96	14	≤	≤	NUM
ma-119	96	15	2t(a	2t(a	NOUN
ma-119	96	16	,	,	PUNCT
ma-119	96	17	z	z	NOUN
ma-119	96	18	)	)	PUNCT
ma-119	96	19	if	if	SCONJ
ma-119	96	20	applied	apply	VERB
ma-119	96	21	holds	hold	NOUN
ma-119	96	22	.	.	PUNCT
ma-119	97	1	d(a	d(a	PROPN
ma-119	97	2	)	)	PUNCT
ma-119	97	3	was	be	AUX
ma-119	97	4	definedto	definedto	PROPN
ma-119	97	5	be	be	AUX
ma-119	97	6	a	a	DET
ma-119	97	7	minimum	minimum	NOUN
ma-119	97	8	value	value	NOUN
ma-119	97	9	d	d	NOUN
ma-119	97	10	in	in	ADP
ma-119	97	11	[	[	X
ma-119	97	12	0,∞	0,∞	X
ma-119	97	13	]	]	PUNCT
ma-119	97	14	such	such	ADJ
ma-119	97	15	that	that	DET
ma-119	97	16	t(a	t(a	NOUN
ma-119	97	17	,	,	PUNCT
ma-119	97	18	z	z	NOUN
ma-119	97	19	)	)	PUNCT
ma-119	97	20	≤	≤	NOUN
ma-119	97	21	d‖t	d‖t	VERB
ma-119	97	22	(	(	PUNCT
ma-119	97	23	a	a	PRON
ma-119	97	24	,	,	PUNCT
ma-119	97	25	a)‖.	a)‖.	VERB
ma-119	97	26	the	the	DET
ma-119	97	27	behaviour	behaviour	NOUN
ma-119	97	28	of	of	ADP
ma-119	97	29	d	d	PROPN
ma-119	97	30	in	in	ADP
ma-119	97	31	idealsand	idealsand	NOUN
ma-119	97	32	quotients	quotient	NOUN
ma-119	97	33	were	be	AUX
ma-119	97	34	discussed	discuss	VERB
ma-119	97	35	which	which	PRON
ma-119	97	36	proved	prove	VERB
ma-119	97	37	that	that	SCONJ
ma-119	97	38	ds(a	ds(a	VERB
ma-119	97	39	)	)	PUNCT
ma-119	97	40	≤	≤	NOUN
ma-119	97	41	1	1	NUM
ma-119	97	42	for	for	ADP
ma-119	97	43	a	a	DET
ma-119	97	44	weakly	weakly	ADJ
ma-119	97	45	central	central	ADJ
ma-119	97	46	c∗-algebra	c∗-algebra	NOUN
ma-119	97	47	a	a	DET
ma-119	97	48	andconsidered	andconsidere	VERB
ma-119	97	49	a	a	DET
ma-119	97	50	class	class	NOUN
ma-119	97	51	of	of	ADP
ma-119	97	52	n	n	CCONJ
ma-119	97	53	-	-	PUNCT
ma-119	97	54	homogeneous	homogeneous	ADJ
ma-119	97	55	c∗-algebras	c∗-algebra	NOUN
ma-119	97	56	that	that	PRON
ma-119	97	57	are	be	AUX
ma-119	97	58	special	special	ADJ
ma-119	97	59	.	.	PUNCT
ma-119	98	1	d	d	NOUN
ma-119	98	2	and	and	CCONJ
ma-119	98	3	ds	ds	PROPN
ma-119	98	4	were	be	AUX
ma-119	98	5	investigatedand	investigatedand	NOUN
ma-119	98	6	approximated	approximate	VERB
ma-119	98	7	finite	finite	ADJ
ma-119	98	8	-	-	NOUN
ma-119	98	9	dimension	dimension	NOUN
ma-119	98	10	(	(	PUNCT
ma-119	98	11	af	af	NOUN
ma-119	98	12	)	)	PUNCT
ma-119	99	1	c∗-algebra	c∗-algebra	NOUN
ma-119	99	2	in	in	ADP
ma-119	99	3	that	that	DET
ma-119	99	4	context	context	NOUN
ma-119	99	5	and	and	CCONJ
ma-119	99	6	an	an	DET
ma-119	99	7	example	example	NOUN
ma-119	99	8	was	be	AUX
ma-119	99	9	given	give	VERB
ma-119	99	10	toshow	toshow	NOUN
ma-119	99	11	certain	certain	ADJ
ma-119	99	12	estimates	estimate	NOUN
ma-119	99	13	.	.	PUNCT
ma-119	100	1	the	the	DET
ma-119	100	2	results	result	NOUN
ma-119	100	3	of	of	ADP
ma-119	100	4	[	[	X
ma-119	100	5	44	44	NUM
ma-119	100	6	]	]	PUNCT
ma-119	100	7	showed	show	VERB
ma-119	100	8	that	that	SCONJ
ma-119	100	9	for	for	ADP
ma-119	100	10	a	a	DET
ma-119	100	11	certain	certain	ADJ
ma-119	100	12	von	von	PROPN
ma-119	100	13	neumann	neumann	PROPN
ma-119	100	14	algebra	algebra	PROPN
ma-119	100	15	u	u	PROPN
ma-119	100	16	,	,	PUNCT
ma-119	100	17	a	a	DET
ma-119	100	18	constant	constant	ADJ
ma-119	100	19	f	f	NOUN
ma-119	100	20	existed	exist	VERB
ma-119	100	21	such	such	ADJ
ma-119	100	22	that	that	DET
ma-119	100	23	dist(t	dist(t	NOUN
ma-119	100	24	,	,	PUNCT
ma-119	100	25	u	u	NOUN
ma-119	100	26	)	)	PUNCT
ma-119	100	27	≤	≤	NUM
ma-119	100	28	f	f	PROPN
ma-119	100	29	supp∈latu	supp∈latu	NUM
ma-119	100	30	‖p⊥tp‖	‖p⊥tp‖	PROPN
ma-119	100	31	∀t	∀t	PROPN
ma-119	100	32	∈	∈	PROPN
ma-119	100	33	b(h	b(h	PROPN
ma-119	100	34	)	)	PUNCT
ma-119	100	35	.	.	PUNCT
ma-119	101	1	the	the	DET
ma-119	101	2	work	work	NOUN
ma-119	101	3	wasextended	wasextende	VERB
ma-119	101	4	to	to	ADP
ma-119	101	5	a	a	DET
ma-119	101	6	von	von	PROPN
ma-119	101	7	neumann	neumann	PROPN
ma-119	101	8	algebra	algebra	PROPN
ma-119	101	9	u	u	PROPN
ma-119	101	10	and	and	CCONJ
ma-119	101	11	showed	show	VERB
ma-119	101	12	that	that	SCONJ
ma-119	101	13	there	there	PRON
ma-119	101	14	exists	exist	VERB
ma-119	101	15	a	a	DET
ma-119	101	16	constant	constant	ADJ
ma-119	101	17	g	g	NOUN
ma-119	101	18	∈	∈	PROPN
ma-119	101	19	b(h),dist(t	b(h),dist(t	NUM
ma-119	101	20	,	,	PUNCT
ma-119	101	21	u	u	NOUN
ma-119	101	22	)	)	PUNCT
ma-119	101	23	≤	≤	NOUN
ma-119	101	24	g‖∆t	g‖∆t	VERB
ma-119	101	25	|u′‖	|u′‖	PROPN
ma-119	101	26	where	where	SCONJ
ma-119	101	27	δt	δt	PROPN
ma-119	101	28	is	be	AUX
ma-119	101	29	the	the	DET
ma-119	101	30	derivation	derivation	NOUN
ma-119	101	31	δt	δt	X
ma-119	101	32	(	(	PUNCT
ma-119	101	33	s	s	X
ma-119	101	34	)	)	PUNCT
ma-119	102	1	=	=	SYM
ma-119	102	2	st	st	PROPN
ma-119	103	1	−	−	NOUN
ma-119	103	2	ts	ts	AUX
ma-119	103	3	thus	thus	ADV
ma-119	103	4	proving	prove	VERB
ma-119	103	5	that	that	SCONJ
ma-119	103	6	theinequality	theinequality	NOUN
ma-119	103	7	holds	hold	VERB
ma-119	103	8	for	for	ADP
ma-119	103	9	large	large	ADJ
ma-119	103	10	classes	class	NOUN
ma-119	103	11	of	of	ADP
ma-119	103	12	von	von	PROPN
ma-119	103	13	neumann	neumann	PROPN
ma-119	103	14	algebras	algebras	PROPN
ma-119	103	15	.	.	PUNCT
ma-119	104	1	in	in	ADP
ma-119	104	2	[	[	X
ma-119	104	3	14	14	NUM
ma-119	104	4	]	]	PUNCT
ma-119	104	5	the	the	DET
ma-119	104	6	researcher	researcher	NOUN
ma-119	104	7	considered	consider	VERB
ma-119	104	8	λ(m	λ(m	NOUN
ma-119	104	9	)	)	PUNCT
ma-119	104	10	defined	define	VERB
ma-119	104	11	as	as	ADP
ma-119	104	12	the	the	DET
ma-119	104	13	smallest	small	ADJ
ma-119	104	14	number	number	NOUN
ma-119	104	15	‖z‖2	‖z‖2	NOUN
ma-119	104	16	of	of	ADP
ma-119	104	17	z	z	NOUN
ma-119	104	18	that	that	DET
ma-119	104	19	satisfy	satisfy	VERB
ma-119	104	20	[	[	X
ma-119	104	21	z∗	z∗	PROPN
ma-119	104	22	,	,	PUNCT
ma-119	104	23	z	z	X
ma-119	104	24	]	]	X
ma-119	104	25	=	=	PUNCT
ma-119	104	26	m	m	NOUN
ma-119	104	27	and	and	CCONJ
ma-119	104	28	showed	show	VERB
ma-119	104	29	that	that	SCONJ
ma-119	104	30	1	1	NUM
ma-119	104	31	≤	≤	NUM
ma-119	104	32	λ(m	λ(m	PROPN
ma-119	104	33	)	)	PUNCT
ma-119	104	34	≤	≤	NUM
ma-119	104	35	2	2	NUM
ma-119	104	36	.	.	PUNCT
ma-119	105	1	matej	matej	PROPN
ma-119	105	2	[	[	X
ma-119	105	3	28	28	NUM
ma-119	105	4	]	]	PUNCT
ma-119	105	5	estimated	estimate	VERB
ma-119	105	6	the	the	DET
ma-119	105	7	distance	distance	NOUN
ma-119	105	8	of	of	ADP
ma-119	105	9	d1	d1	PROPN
ma-119	105	10	and	and	CCONJ
ma-119	105	11	d2	d2	PROPN
ma-119	105	12	to	to	ADP
ma-119	105	13	the	the	DET
ma-119	105	14	generalized	generalized	ADJ
ma-119	105	15	derivations	derivation	NOUN
ma-119	105	16	andthe	andthe	X
ma-119	105	17	normed	norme	VERB
ma-119	105	18	algebra	algebra	NOUN
ma-119	105	19	of	of	ADP
ma-119	105	20	p	p	NOUN
ma-119	105	21	and	and	CCONJ
ma-119	105	22	considered	consider	VERB
ma-119	105	23	the	the	DET
ma-119	105	24	cases	case	NOUN
ma-119	105	25	when	when	SCONJ
ma-119	105	26	p	p	NOUN
ma-119	105	27	is	be	AUX
ma-119	105	28	an	an	DET
ma-119	105	29	ultraprime	ultraprime	NOUN
ma-119	105	30	,	,	PUNCT
ma-119	105	31	when	when	SCONJ
ma-119	105	32	d1	d1	PROPN
ma-119	105	33	=	=	SYM
ma-119	105	34	d2	d2	PROPN
ma-119	105	35	and	and	CCONJ
ma-119	105	36	p	p	NOUN
ma-119	105	37	are	be	AUX
ma-119	105	38	ultrasemiprime	ultrasemiprime	ADJ
ma-119	105	39	and	and	CCONJ
ma-119	105	40	when	when	SCONJ
ma-119	105	41	p	p	NOUN
ma-119	105	42	is	be	AUX
ma-119	105	43	a	a	DET
ma-119	105	44	von	von	PROPN
ma-119	105	45	neumann	neumann	PROPN
ma-119	105	46	algebra	algebra	PROPN
ma-119	105	47	we	we	PRON
ma-119	105	48	have	have	VERB
ma-119	105	49	the	the	DET
ma-119	105	50	equation	equation	NOUN
ma-119	105	51	‖p	‖p	NOUN
ma-119	106	1	+	+	CCONJ
ma-119	106	2	q‖	q‖	NOUN
ma-119	106	3	=	=	SYM
ma-119	106	4	‖p‖	‖p‖	PROPN
ma-119	106	5	+	+	CCONJ
ma-119	106	6	‖q‖	‖q‖	PROPN
ma-119	106	7	,	,	PUNCT
ma-119	106	8	p	p	X
ma-119	106	9	,	,	PUNCT
ma-119	106	10	q	q	PROPN
ma-119	106	11	∈	∈	PROPN
ma-119	106	12	b(h	b(h	PROPN
ma-119	106	13	)	)	PUNCT
ma-119	106	14	.	.	PUNCT
ma-119	107	1	further	far	ADV
ma-119	107	2	,	,	PUNCT
ma-119	107	3	a	a	DET
ma-119	107	4	constructive	constructive	ADJ
ma-119	107	5	proof	proof	NOUN
ma-119	107	6	was	be	AUX
ma-119	107	7	provided	provide	VERB
ma-119	107	8	that	that	SCONJ
ma-119	107	9	a	a	DET
ma-119	107	10	minimum	minimum	NOUN
ma-119	107	11	bound	bind	VERB
ma-119	107	12	isnot	isnot	ADV
ma-119	107	13	valid	valid	ADJ
ma-119	107	14	and	and	CCONJ
ma-119	107	15	a	a	DET
ma-119	107	16	relevant	relevant	ADJ
ma-119	107	17	method	method	NOUN
ma-119	107	18	to	to	PART
ma-119	107	19	analyze	analyze	VERB
ma-119	107	20	the	the	DET
ma-119	107	21	problem	problem	NOUN
ma-119	107	22	on	on	ADP
ma-119	107	23	estimation	estimation	NOUN
ma-119	107	24	of	of	ADP
ma-119	107	25	eigenvalues	eigenvalue	NOUN
ma-119	107	26	such	such	ADJ
ma-119	107	27	aninterpolation	aninterpolation	NOUN
ma-119	107	28	matrix	matrix	NOUN
ma-119	107	29	was	be	AUX
ma-119	107	30	commented	comment	VERB
ma-119	107	31	on	on	ADP
ma-119	107	32	.	.	PUNCT
ma-119	108	1	the	the	DET
ma-119	108	2	norm	norm	NOUN
ma-119	108	3	property	property	NOUN
ma-119	108	4	was	be	AUX
ma-119	108	5	done	do	VERB
ma-119	108	6	by	by	ADP
ma-119	108	7	cabrera	cabrera	PROPN
ma-119	108	8	,	,	PUNCT
ma-119	108	9	rodriguez	rodriguez	PROPN
ma-119	109	1	[	[	X
ma-119	109	2	10]for	10]for	PROPN
ma-119	109	3	basic	basic	ADJ
ma-119	109	4	elementary	elementary	ADJ
ma-119	109	5	operators	operator	NOUN
ma-119	109	6	and	and	CCONJ
ma-119	109	7	obtained	obtain	VERB
ma-119	109	8	‖ma	‖ma	NOUN
ma-119	109	9	,	,	PUNCT
ma-119	109	10	b‖	b‖	NOUN
ma-119	109	11	≤	≤	PUNCT
ma-119	109	12	2‖a‖‖b‖	2‖a‖‖b‖	NOUN
ma-119	109	13	,	,	PUNCT
ma-119	109	14	for	for	ADP
ma-119	109	15	jordan	jordan	PROPN
ma-119	109	16	elementary	elementary	PROPN
ma-119	109	17	operator	operator	NOUN
ma-119	109	18	‖u‖	‖u‖	PROPN
ma-119	109	19	=	=	SYM
ma-119	109	20	‖ma	‖ma	PROPN
ma-119	109	21	,	,	PUNCT
ma-119	109	22	b‖	b‖	NOUN
ma-119	109	23	+	+	CCONJ
ma-119	109	24	‖ma	‖ma	NOUN
ma-119	109	25	,	,	PUNCT
ma-119	109	26	b‖	b‖	NOUN
ma-119	109	27	,	,	PUNCT
ma-119	109	28	‖ma	‖ma	PROPN
ma-119	109	29	,	,	PUNCT
ma-119	109	30	b‖	b‖	NOUN
ma-119	109	31	+	+	CCONJ
ma-119	109	32	‖ma	‖ma	NOUN
ma-119	109	33	,	,	PUNCT
ma-119	109	34	b‖	b‖	NOUN
ma-119	109	35	≤	≤	NUM
ma-119	109	36	2‖a‖‖b‖	2‖a‖‖b‖	NOUN
ma-119	109	37	for	for	ADP
ma-119	109	38	the	the	DET
ma-119	109	39	upper	upper	ADJ
ma-119	109	40	estimates	estimate	NOUN
ma-119	109	41	.	.	PUNCT
ma-119	110	1	in	in	ADP
ma-119	110	2	fact	fact	NOUN
ma-119	110	3	,	,	PUNCT
ma-119	110	4	[	[	X
ma-119	110	5	30	30	NUM
ma-119	110	6	]	]	X
ma-119	110	7	gavean	gavean	ADJ
ma-119	110	8	estimate	estimate	NOUN
ma-119	110	9	on	on	ADP
ma-119	110	10	matrix	matrix	NOUN
ma-119	110	11	-	-	PUNCT
ma-119	110	12	valued	value	VERB
ma-119	110	13	function	function	NOUN
ma-119	110	14	that	that	PRON
ma-119	110	15	is	be	AUX
ma-119	110	16	regular	regular	ADJ
ma-119	110	17	and	and	CCONJ
ma-119	110	18	showed	show	VERB
ma-119	110	19	that	that	SCONJ
ma-119	110	20	for	for	ADP
ma-119	110	21	normal	normal	ADJ
ma-119	110	22	matrices	matrix	NOUN
ma-119	110	23	it	it	PRON
ma-119	110	24	isattainable	isattainable	VERB
ma-119	110	25	and	and	CCONJ
ma-119	110	26	investigated	investigate	VERB
ma-119	110	27	their	their	PRON
ma-119	110	28	stability	stability	NOUN
ma-119	110	29	.	.	PUNCT
ma-119	111	1	kittaneh	kittaneh	PROPN
ma-119	112	1	[	[	X
ma-119	112	2	26	26	NUM
ma-119	112	3	]	]	PUNCT
ma-119	112	4	established	establish	VERB
ma-119	112	5	the	the	DET
ma-119	112	6	orthogonality	orthogonality	NOUN
ma-119	112	7	,	,	PUNCT
ma-119	112	8	kerneland	kerneland	VERB
ma-119	112	9	the	the	DET
ma-119	112	10	range	range	NOUN
ma-119	112	11	of	of	ADP
ma-119	112	12	a	a	DET
ma-119	112	13	normal	normal	ADJ
ma-119	112	14	derivation	derivation	NOUN
ma-119	112	15	associated	associate	VERB
ma-119	112	16	with	with	ADP
ma-119	112	17	norm	norm	NOUN
ma-119	112	18	ideals	ideal	NOUN
ma-119	112	19	of	of	ADP
ma-119	112	20	operators	operator	NOUN
ma-119	112	21	with	with	ADP
ma-119	112	22	respect	respect	NOUN
ma-119	112	23	tothe	tothe	PRON
ma-119	112	24	unitarily	unitarily	ADV
ma-119	112	25	invariant	invariant	ADJ
ma-119	112	26	norms	norm	NOUN
ma-119	112	27	.	.	PUNCT
ma-119	113	1	results	result	NOUN
ma-119	113	2	related	relate	VERB
ma-119	113	3	to	to	ADP
ma-119	113	4	orthorgonality	orthorgonality	NOUN
ma-119	113	5	of	of	ADP
ma-119	113	6	some	some	DET
ma-119	113	7	derivation	derivation	NOUN
ma-119	113	8	that	that	PRON
ma-119	113	9	are	be	AUX
ma-119	113	10	notnormal	notnormal	ADJ
ma-119	113	11	were	be	AUX
ma-119	113	12	also	also	ADV
ma-119	113	13	obtained	obtain	VERB
ma-119	113	14	.	.	PUNCT
ma-119	114	1	stacho	stacho	NOUN
ma-119	114	2	and	and	CCONJ
ma-119	114	3	zalar	zalar	ADJ
ma-119	115	1	[	[	X
ma-119	115	2	48	48	NUM
ma-119	115	3	]	]	PUNCT
ma-119	115	4	established	establish	VERB
ma-119	115	5	the	the	DET
ma-119	115	6	lower	low	ADJ
ma-119	115	7	estimates	estimate	NOUN
ma-119	115	8	for	for	ADP
ma-119	115	9	elementary	elementary	ADJ
ma-119	115	10	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	PROPN
ma-119	115	11	eur	eur	NOUN
ma-119	115	12	.	.	PUNCT
ma-119	116	1	j.	j.	PROPN
ma-119	116	2	math	math	PROPN
ma-119	116	3	.	.	PUNCT
ma-119	117	1	anal	anal	PROPN
ma-119	117	2	.	.	PUNCT
ma-119	118	1	10.28924	10.28924	NUM
ma-119	118	2	/	/	SYM
ma-119	118	3	ada	ada	PROPN
ma-119	118	4	/	/	SYM
ma-119	118	5	ma.3.9	ma.3.9	PROPN
ma-119	118	6	5operators	5operators	PROPN
ma-119	118	7	of	of	ADP
ma-119	118	8	jordan	jordan	PROPN
ma-119	118	9	type	type	PROPN
ma-119	118	10	in	in	ADP
ma-119	118	11	standard	standard	ADJ
ma-119	118	12	banach	banach	NOUN
ma-119	118	13	algebras	algebra	VERB
ma-119	118	14	.	.	PUNCT
ma-119	119	1	danko	danko	PROPN
ma-119	120	1	[	[	X
ma-119	120	2	14	14	NUM
ma-119	120	3	]	]	PUNCT
ma-119	120	4	established	establish	VERB
ma-119	120	5	that	that	SCONJ
ma-119	120	6	for	for	ADP
ma-119	120	7	all	all	DET
ma-119	120	8	unitarilyinvariant	unitarilyinvariant	ADJ
ma-119	120	9	norms	norm	NOUN
ma-119	120	10	and	and	CCONJ
ma-119	120	11	for	for	ADP
ma-119	120	12	bounded	bounded	ADJ
ma-119	120	13	hilbert	hilbert	PROPN
ma-119	120	14	space	space	NOUN
ma-119	120	15	operators	operator	NOUN
ma-119	120	16	there	there	ADV
ma-119	120	17	exist	exist	VERB
ma-119	120	18	{	{	PUNCT
ma-119	120	19	xn}n	xn}n	PROPN
ma-119	120	20	⊆	⊆	NUM
ma-119	120	21	h	h	NOUN
ma-119	120	22	which	which	PRON
ma-119	120	23	is	be	AUX
ma-119	120	24	a	a	DET
ma-119	120	25	unitsequence	unitsequence	NOUN
ma-119	120	26	such	such	ADJ
ma-119	120	27	that	that	DET
ma-119	120	28	limn	limn	PROPN
ma-119	121	1	‖c	‖c	NOUN
ma-119	122	1	−	−	PUNCT
ma-119	122	2	ω‖xn	ω‖xn	PROPN
ma-119	122	3	=	=	SYM
ma-119	122	4	0	0	X
ma-119	122	5	.	.	PUNCT
ma-119	123	1	from	from	ADP
ma-119	123	2	[	[	X
ma-119	123	3	11	11	NUM
ma-119	123	4	]	]	PUNCT
ma-119	123	5	,	,	PUNCT
ma-119	123	6	‖a‖	‖a‖	PROPN
ma-119	123	7	∈	∈	PROPN
ma-119	123	8	σ(a	σ(a	PROPN
ma-119	123	9	)	)	PUNCT
ma-119	124	1	if	if	SCONJ
ma-119	124	2	and	and	CCONJ
ma-119	124	3	only	only	ADV
ma-119	124	4	if	if	SCONJ
ma-119	124	5	‖a‖	‖a‖	PROPN
ma-119	124	6	∈	∈	PROPN
ma-119	124	7	σap(a	σap(a	NOUN
ma-119	124	8	)	)	PUNCT
ma-119	124	9	also	also	ADV
ma-119	124	10	σ(a	σ(a	PROPN
ma-119	124	11	)	)	PUNCT
ma-119	125	1	⊆	⊆	NUM
ma-119	125	2	w	w	NOUN
ma-119	125	3	(	(	PUNCT
ma-119	125	4	a	a	NOUN
ma-119	125	5	)	)	PUNCT
ma-119	125	6	(	(	PUNCT
ma-119	125	7	spectral	spectral	ADJ
ma-119	125	8	inclusion	inclusion	NOUN
ma-119	125	9	)	)	PUNCT
ma-119	125	10	and	and	CCONJ
ma-119	125	11	if	if	SCONJ
ma-119	125	12	ω(a	ω(a	NUM
ma-119	125	13	)	)	PUNCT
ma-119	125	14	=	=	SYM
ma-119	125	15	‖a‖	‖a‖	PROPN
ma-119	125	16	,	,	PUNCT
ma-119	125	17	then	then	ADV
ma-119	125	18	γ(a	γ(a	NOUN
ma-119	125	19	)	)	PUNCT
ma-119	125	20	=	=	SYM
ma-119	126	1	‖a‖.	‖a‖.	PROPN
ma-119	126	2	therefore	therefore	ADV
ma-119	126	3	,	,	PUNCT
ma-119	126	4	the	the	PRON
ma-119	126	5	resultimplied	resultimplie	VERB
ma-119	126	6	that	that	SCONJ
ma-119	126	7	‖a‖	‖a‖	PROPN
ma-119	126	8	⊆	⊆	NUM
ma-119	126	9	w	w	NOUN
ma-119	126	10	(	(	PUNCT
ma-119	126	11	a	a	NOUN
ma-119	126	12	)	)	PUNCT
ma-119	126	13	if	if	SCONJ
ma-119	126	14	and	and	CCONJ
ma-119	126	15	only	only	ADV
ma-119	126	16	if	if	SCONJ
ma-119	126	17	‖a‖	‖a‖	PROPN
ma-119	126	18	∈	∈	PROPN
ma-119	126	19	σ(a	σ(a	PROPN
ma-119	126	20	)	)	PUNCT
ma-119	126	21	.	.	PUNCT
ma-119	127	1	in	in	ADP
ma-119	127	2	fact	fact	NOUN
ma-119	127	3	,	,	PUNCT
ma-119	127	4	megginson	megginson	PROPN
ma-119	128	1	[	[	X
ma-119	128	2	32	32	NUM
ma-119	128	3	]	]	PUNCT
ma-119	128	4	established	establish	VERB
ma-119	128	5	that	that	DET
ma-119	128	6	forall	forall	NOUN
ma-119	128	7	y	y	PROPN
ma-119	128	8	∈	∈	PROPN
ma-119	128	9	k	k	NOUN
ma-119	128	10	,	,	PUNCT
ma-119	128	11	then	then	ADV
ma-119	128	12	δb(y	δb(y	PUNCT
ma-119	128	13	)	)	PUNCT
ma-119	128	14	∈	∈	PROPN
ma-119	128	15	j	j	PROPN
ma-119	128	16	and	and	CCONJ
ma-119	128	17	‖by	‖by	PROPN
ma-119	128	18	−	−	PROPN
ma-119	128	19	y	y	PROPN
ma-119	128	20	b‖k	b‖k	PROPN
ma-119	128	21	=	=	PUNCT
ma-119	128	22	‖(b	‖(b	NOUN
ma-119	129	1	−	−	PROPN
ma-119	129	2	λ)y	λ)y	NOUN
ma-119	129	3	−	−	PROPN
ma-119	129	4	y	y	PROPN
ma-119	129	5	(	(	PUNCT
ma-119	129	6	b	b	X
ma-119	129	7	−	−	NOUN
ma-119	129	8	α)‖j	α)‖j	ADJ
ma-119	129	9	≤	≤	ADJ
ma-119	129	10	2‖b	2‖b	NUM
ma-119	129	11	−	−	NOUN
ma-119	129	12	α‖‖y	α‖‖y	NUM
ma-119	129	13	‖kfor	‖kfor	ADP
ma-119	129	14	all	all	DET
ma-119	129	15	α	α	PROPN
ma-119	129	16	∈	∈	PROPN
ma-119	129	17	c.	c.	NOUN
ma-119	129	18	hence	hence	ADV
ma-119	129	19	,	,	PUNCT
ma-119	129	20	‖δb(y	‖δb(y	PROPN
ma-119	129	21	)	)	PUNCT
ma-119	129	22	‖k	‖k	VERB
ma-119	129	23	≤	≤	NUM
ma-119	129	24	2d(b)‖y	2d(b)‖y	NUM
ma-119	129	25	‖k	‖k	NOUN
ma-119	129	26	,	,	PUNCT
ma-119	129	27	implying	imply	VERB
ma-119	129	28	that	that	SCONJ
ma-119	129	29	‖δb|k‖	‖δb|k‖	DET
ma-119	129	30	≤	≤	NOUN
ma-119	129	31	2d(b	2d(b	NUM
ma-119	129	32	)	)	PUNCT
ma-119	129	33	.	.	PUNCT
ma-119	130	1	further	far	ADV
ma-119	130	2	,	,	PUNCT
ma-119	130	3	thenotion	thenotion	NOUN
ma-119	130	4	of	of	ADP
ma-119	130	5	r	r	NOUN
ma-119	130	6	-	-	PUNCT
ma-119	130	7	universal	universal	ADJ
ma-119	130	8	operators	operator	NOUN
ma-119	130	9	was	be	AUX
ma-119	130	10	introduced	introduce	VERB
ma-119	130	11	and	and	CCONJ
ma-119	130	12	that	that	SCONJ
ma-119	130	13	r	r	NOUN
ma-119	130	14	-	-	ADJ
ma-119	130	15	universal	universal	ADJ
ma-119	130	16	is	be	AUX
ma-119	130	17	an	an	DET
ma-119	130	18	operator	operator	NOUN
ma-119	130	19	a	a	DET
ma-119	130	20	∈	∈	PROPN
ma-119	130	21	b(h	b(h	NOUN
ma-119	130	22	)	)	PUNCT
ma-119	130	23	if	if	SCONJ
ma-119	130	24	‖δb|k‖	‖δb|k‖	PRON
ma-119	130	25	=	=	SYM
ma-119	130	26	2d(b	2d(b	NUM
ma-119	130	27	)	)	PUNCT
ma-119	130	28	for	for	ADP
ma-119	130	29	every	every	DET
ma-119	130	30	norm	norm	NOUN
ma-119	130	31	ideal	ideal	NOUN
ma-119	130	32	k	k	PROPN
ma-119	130	33	∈	∈	PROPN
ma-119	130	34	b(h	b(h	PROPN
ma-119	130	35	)	)	PUNCT
ma-119	130	36	.	.	PUNCT
ma-119	131	1	landsman	landsman	NOUN
ma-119	132	1	[	[	X
ma-119	132	2	23	23	NUM
ma-119	132	3	]	]	PUNCT
ma-119	132	4	proved	prove	VERB
ma-119	132	5	that	that	SCONJ
ma-119	132	6	for	for	ADP
ma-119	132	7	a	a	DET
ma-119	132	8	standard	standard	ADJ
ma-119	132	9	operatoralgebra	operatoralgebra	NOUN
ma-119	132	10	on	on	ADP
ma-119	132	11	h	h	PROPN
ma-119	132	12	‖ma	‖ma	PROPN
ma-119	132	13	,	,	PUNCT
ma-119	132	14	b‖+‖ma	b‖+‖ma	NOUN
ma-119	132	15	,	,	PUNCT
ma-119	132	16	b‖	b‖	PROPN
ma-119	132	17	≥	≥	NUM
ma-119	132	18	2	2	NUM
ma-119	132	19	(	(	PUNCT
ma-119	132	20	√	√	NUM
ma-119	132	21	2−1)‖a‖‖b‖.	2−1)‖a‖‖b‖.	NUM
ma-119	132	22	therefore	therefore	ADV
ma-119	132	23	,	,	PUNCT
ma-119	132	24	both	both	CCONJ
ma-119	132	25	the	the	DET
ma-119	132	26	lower	low	ADJ
ma-119	132	27	norm	norm	NOUN
ma-119	132	28	and	and	CCONJ
ma-119	132	29	upper	upper	ADJ
ma-119	132	30	normbounds	normbound	NOUN
ma-119	132	31	have	have	AUX
ma-119	132	32	been	be	AUX
ma-119	132	33	established	establish	VERB
ma-119	132	34	for	for	ADP
ma-119	132	35	normally	normally	ADV
ma-119	132	36	represented	represent	VERB
ma-119	132	37	elementary	elementary	ADJ
ma-119	132	38	operators	operator	NOUN
ma-119	132	39	.	.	PUNCT
ma-119	133	1	the	the	DET
ma-119	133	2	work	work	NOUN
ma-119	133	3	of	of	ADP
ma-119	133	4	[	[	X
ma-119	133	5	3	3	NUM
ma-119	133	6	]	]	PUNCT
ma-119	133	7	had	have	AUX
ma-119	133	8	anestimate	anestimate	NOUN
ma-119	133	9	on	on	ADP
ma-119	133	10	transfer	transfer	NOUN
ma-119	133	11	functions	function	NOUN
ma-119	133	12	of	of	ADP
ma-119	133	13	stable	stable	ADJ
ma-119	133	14	linear	linear	ADJ
ma-119	133	15	time	time	NOUN
ma-119	133	16	-	-	PUNCT
ma-119	133	17	invariant	invariant	ADJ
ma-119	133	18	systems	system	NOUN
ma-119	133	19	on	on	ADP
ma-119	133	20	stochastic	stochastic	ADJ
ma-119	133	21	assumptions	assumption	NOUN
ma-119	133	22	.	.	PUNCT
ma-119	134	1	theapproach	theapproach	NOUN
ma-119	134	2	of	of	ADP
ma-119	134	3	nonparametric	nonparametric	NOUN
ma-119	134	4	minimax	minimax	NOUN
ma-119	134	5	was	be	AUX
ma-119	134	6	adopted	adopt	VERB
ma-119	134	7	to	to	PART
ma-119	134	8	measure	measure	VERB
ma-119	134	9	the	the	DET
ma-119	134	10	estimate	estimate	NOUN
ma-119	134	11	accurately	accurately	ADV
ma-119	134	12	,	,	PUNCT
ma-119	134	13	an	an	DET
ma-119	134	14	estimator	estimator	NOUN
ma-119	134	15	ofquality	ofquality	NOUN
ma-119	134	16	was	be	AUX
ma-119	134	17	measured	measure	VERB
ma-119	134	18	over	over	ADP
ma-119	134	19	a	a	DET
ma-119	134	20	family	family	NOUN
ma-119	134	21	of	of	ADP
ma-119	134	22	transfer	transfer	NOUN
ma-119	134	23	functions	function	NOUN
ma-119	134	24	by	by	ADP
ma-119	134	25	its	its	PRON
ma-119	134	26	worst	bad	ADJ
ma-119	134	27	case	case	NOUN
ma-119	134	28	error	error	NOUN
ma-119	134	29	.	.	PUNCT
ma-119	135	1	in	in	ADP
ma-119	135	2	[	[	X
ma-119	135	3	32	32	NUM
ma-119	135	4	]	]	PUNCT
ma-119	135	5	the	the	PRON
ma-119	135	6	authorestablished	authorestablishe	VERB
ma-119	135	7	that	that	PRON
ma-119	135	8	for	for	ADP
ma-119	135	9	a	a	DET
ma-119	135	10	holomorphic	holomorphic	ADJ
ma-119	135	11	functions	function	NOUN
ma-119	135	12	f	f	PROPN
ma-119	135	13	with	with	ADP
ma-119	135	14	re{gf	re{gf	PROPN
ma-119	135	15	′(g	′(g	NOUN
ma-119	135	16	)	)	PUNCT
ma-119	135	17	}	}	PUNCT
ma-119	135	18	>	>	X
ma-119	135	19	α	α	PROPN
ma-119	135	20	and	and	CCONJ
ma-119	135	21	re{gf	re{gf	PROPN
ma-119	135	22	′′(g)/f	′′(g)/f	PROPN
ma-119	135	23	′(g	′(g	NOUN
ma-119	135	24	)	)	PUNCT
ma-119	135	25	}	}	PUNCT
ma-119	135	26	>	>	PUNCT
ma-119	136	1	α−1	α−1	PROPN
ma-119	136	2	,	,	PUNCT
ma-119	136	3	(	(	PUNCT
ma-119	136	4	0	0	NUM
ma-119	136	5	≤	≤	NUM
ma-119	136	6	α	α	NOUN
ma-119	136	7	<	<	X
ma-119	136	8	1	1	NUM
ma-119	136	9	)	)	PUNCT
ma-119	136	10	respectively	respectively	ADV
ma-119	136	11	in	in	ADP
ma-119	136	12	{	{	PUNCT
ma-119	136	13	|g|	|g|	PROPN
ma-119	136	14	<	<	X
ma-119	136	15	1	1	NUM
ma-119	136	16	}	}	PUNCT
ma-119	136	17	,	,	PUNCT
ma-119	136	18	estimates	estimate	NOUN
ma-119	136	19	of	of	ADP
ma-119	136	20	sup|g|<1(1−|g|2)|f	sup|g|<1(1−|g|2)|f	PROPN
ma-119	136	21	′′(g)/f	′′(g)/f	PROPN
ma-119	136	22	′(g)|	′(g)|	NUM
ma-119	136	23	were	be	AUX
ma-119	136	24	givenand	givenand	NOUN
ma-119	136	25	functions	function	NOUN
ma-119	136	26	gelfer	gelfer	NOUN
ma-119	136	27	-	-	PUNCT
ma-119	136	28	convex	convex	NOUN
ma-119	136	29	of	of	ADP
ma-119	136	30	exponential	exponential	ADJ
ma-119	136	31	order	order	NOUN
ma-119	136	32	α	α	NOUN
ma-119	136	33	,	,	PUNCT
ma-119	136	34	β	β	PROPN
ma-119	136	35	was	be	AUX
ma-119	136	36	also	also	ADV
ma-119	136	37	considered	consider	VERB
ma-119	136	38	.	.	PUNCT
ma-119	137	1	milos	milo	NOUN
ma-119	137	2	,	,	PUNCT
ma-119	137	3	dragoljub	dragoljub	PROPN
ma-119	138	1	[	[	X
ma-119	138	2	33]considered	33]considered	NUM
ma-119	138	3	elementary	elementary	ADJ
ma-119	138	4	operators	operator	NOUN
ma-119	138	5	x	x	PRON
ma-119	138	6	→	→	SYM
ma-119	138	7	∑n	∑n	PROPN
ma-119	138	8	j=1	j=1	PROPN
ma-119	138	9	vjxwj	vjxwj	ADJ
ma-119	138	10	that	that	PRON
ma-119	138	11	acts	act	VERB
ma-119	138	12	on	on	ADP
ma-119	138	13	a	a	DET
ma-119	138	14	banach	banach	NOUN
ma-119	138	15	algebra	algebra	NOUN
ma-119	138	16	.	.	PUNCT
ma-119	139	1	the	the	DET
ma-119	139	2	ascentestimation	ascentestimation	NOUN
ma-119	139	3	and	and	CCONJ
ma-119	139	4	lower	low	ADJ
ma-119	139	5	bound	bind	VERB
ma-119	139	6	estimation	estimation	NOUN
ma-119	139	7	of	of	ADP
ma-119	139	8	an	an	DET
ma-119	139	9	operator	operator	NOUN
ma-119	139	10	was	be	AUX
ma-119	139	11	given	give	VERB
ma-119	139	12	.	.	PUNCT
ma-119	140	1	barraa	barraa	NOUN
ma-119	140	2	and	and	CCONJ
ma-119	140	3	boumazgour	boumazgour	NOUN
ma-119	141	1	[	[	X
ma-119	141	2	4]showed	4]showe	VERB
ma-119	141	3	that	that	SCONJ
ma-119	141	4	the	the	DET
ma-119	141	5	norm	norm	NOUN
ma-119	141	6	of	of	ADP
ma-119	141	7	bounded	bounded	ADJ
ma-119	141	8	operators	operator	NOUN
ma-119	141	9	more	more	ADJ
ma-119	141	10	than	than	ADP
ma-119	141	11	one	one	NUM
ma-119	141	12	on	on	ADP
ma-119	141	13	a	a	DET
ma-119	141	14	hilbert	hilbert	NOUN
ma-119	141	15	space	space	NOUN
ma-119	141	16	is	be	AUX
ma-119	141	17	the	the	DET
ma-119	141	18	same	same	ADJ
ma-119	141	19	asthe	asthe	ADJ
ma-119	141	20	sum	sum	NOUN
ma-119	141	21	of	of	ADP
ma-119	141	22	the	the	DET
ma-119	141	23	norms	norm	NOUN
ma-119	141	24	and	and	CCONJ
ma-119	141	25	showed	show	VERB
ma-119	141	26	that	that	SCONJ
ma-119	141	27	δs	δs	NOUN
ma-119	141	28	,	,	PUNCT
ma-119	141	29	a	a	DET
ma-119	141	30	,	,	PUNCT
ma-119	141	31	b	b	NOUN
ma-119	141	32	is	be	AUX
ma-119	141	33	convexoid	convexoid	ADJ
ma-119	141	34	with	with	ADP
ma-119	141	35	the	the	DET
ma-119	141	36	convex	convex	PROPN
ma-119	141	37	hull	hull	NOUN
ma-119	141	38	of	of	ADP
ma-119	141	39	its	its	PRON
ma-119	141	40	spectrumif	spectrumif	NOUN
ma-119	141	41	and	and	CCONJ
ma-119	141	42	only	only	ADV
ma-119	141	43	if	if	SCONJ
ma-119	141	44	a	a	PRON
ma-119	141	45	and	and	CCONJ
ma-119	141	46	b	b	NOUN
ma-119	141	47	are	be	AUX
ma-119	141	48	convexoid	convexoid	ADJ
ma-119	141	49	.	.	PUNCT
ma-119	142	1	richard	richard	PROPN
ma-119	143	1	[	[	X
ma-119	143	2	44	44	NUM
ma-119	143	3	]	]	PUNCT
ma-119	143	4	established	establish	VERB
ma-119	143	5	the	the	DET
ma-119	143	6	cb	cb	NOUN
ma-119	143	7	-	-	PUNCT
ma-119	143	8	norms	norm	NOUN
ma-119	143	9	of	of	ADP
ma-119	143	10	elementaryoperators	elementaryoperator	NOUN
ma-119	143	11	and	and	CCONJ
ma-119	143	12	the	the	DET
ma-119	143	13	lower	low	ADJ
ma-119	143	14	bounds	bound	NOUN
ma-119	143	15	for	for	ADP
ma-119	143	16	norms	norm	NOUN
ma-119	143	17	on	on	ADP
ma-119	143	18	b(h	b(h	NOUN
ma-119	143	19	)	)	PUNCT
ma-119	143	20	.	.	PUNCT
ma-119	144	1	the	the	DET
ma-119	144	2	result	result	NOUN
ma-119	144	3	was	be	AUX
ma-119	144	4	concerned	concern	VERB
ma-119	144	5	with	with	ADP
ma-119	144	6	the	the	DET
ma-119	144	7	operator	operator	NOUN
ma-119	144	8	ua	ua	PROPN
ma-119	144	9	,	,	PUNCT
ma-119	144	10	bx	bx	PROPN
ma-119	144	11	=	=	SYM
ma-119	144	12	axb+bxa	axb+bxa	PROPN
ma-119	144	13	which	which	PRON
ma-119	144	14	showed	show	VERB
ma-119	144	15	that	that	SCONJ
ma-119	144	16	‖ua	‖ua	PROPN
ma-119	144	17	,	,	PUNCT
ma-119	144	18	b‖	b‖	PROPN
ma-119	144	19	≥	≥	PRON
ma-119	144	20	‖a‖‖b‖	‖a‖‖b‖	ADJ
ma-119	144	21	which	which	PRON
ma-119	144	22	proved	prove	VERB
ma-119	144	23	a	a	DET
ma-119	144	24	conjecture	conjecture	NOUN
ma-119	144	25	of	of	ADP
ma-119	144	26	mathieu	mathieu	PROPN
ma-119	144	27	,	,	PUNCT
ma-119	144	28	other	other	ADJ
ma-119	144	29	results	result	NOUN
ma-119	144	30	and	and	CCONJ
ma-119	144	31	formula	formula	NOUN
ma-119	144	32	of	of	ADP
ma-119	144	33	‖ua	‖ua	NUM
ma-119	144	34	,	,	PUNCT
ma-119	144	35	b‖cb	b‖cb	PROPN
ma-119	144	36	and	and	CCONJ
ma-119	144	37	‖ua	‖ua	PROPN
ma-119	144	38	,	,	PUNCT
ma-119	144	39	b‖	b‖	NOUN
ma-119	144	40	were	be	AUX
ma-119	144	41	established	establish	VERB
ma-119	144	42	.	.	PUNCT
ma-119	145	1	richard	richard	PROPN
ma-119	146	1	[	[	X
ma-119	146	2	45	45	NUM
ma-119	146	3	]	]	PUNCT
ma-119	146	4	provided	provide	VERB
ma-119	146	5	thehaagerup	thehaagerup	PROPN
ma-119	146	6	estimation	estimation	NOUN
ma-119	146	7	on	on	ADP
ma-119	146	8	the	the	DET
ma-119	146	9	norm	norm	NOUN
ma-119	146	10	of	of	ADP
ma-119	146	11	elementary	elementary	ADJ
ma-119	146	12	operators	operator	NOUN
ma-119	146	13	that	that	PRON
ma-119	146	14	are	be	AUX
ma-119	146	15	completely	completely	ADV
ma-119	146	16	bounded	bound	VERB
ma-119	146	17	.	.	PUNCT
ma-119	147	1	seddik	seddik	PROPN
ma-119	148	1	[	[	X
ma-119	148	2	46]proved	46]proved	X
ma-119	148	3	that	that	DET
ma-119	148	4	lower	low	ADJ
ma-119	148	5	estimate	estimate	NOUN
ma-119	148	6	bound	bind	VERB
ma-119	148	7	‖tm	‖tm	PROPN
ma-119	148	8	,	,	PUNCT
ma-119	148	9	n‖	n‖	NOUN
ma-119	148	10	≥	≥	NUM
ma-119	148	11	2	2	NUM
ma-119	148	12	(	(	PUNCT
ma-119	148	13	√	√	NUM
ma-119	148	14	2	2	NUM
ma-119	148	15	−	−	PROPN
ma-119	148	16	1)‖m‖‖n‖	1)‖m‖‖n‖	NUM
ma-119	148	17	holds	hold	VERB
ma-119	148	18	,	,	PUNCT
ma-119	148	19	if	if	SCONJ
ma-119	148	20	it	it	PRON
ma-119	148	21	is	be	AUX
ma-119	148	22	either	either	CCONJ
ma-119	148	23	a	a	DET
ma-119	148	24	standardoperator	standardoperator	NOUN
ma-119	148	25	algebra	algebra	NOUN
ma-119	148	26	or	or	CCONJ
ma-119	148	27	a	a	DET
ma-119	148	28	norm	norm	NOUN
ma-119	148	29	ideal	ideal	NOUN
ma-119	148	30	on	on	ADP
ma-119	148	31	b(h	b(h	PROPN
ma-119	148	32	)	)	PUNCT
ma-119	148	33	and	and	CCONJ
ma-119	148	34	m	m	PROPN
ma-119	148	35	,	,	PUNCT
ma-119	148	36	n	n	PROPN
ma-119	148	37	∈	∈	PROPN
ma-119	148	38	b(h	b(h	PROPN
ma-119	148	39	)	)	PUNCT
ma-119	148	40	.	.	PUNCT
ma-119	149	1	florin	florin	PROPN
ma-119	149	2	,	,	PUNCT
ma-119	149	3	alexandra	alexandra	PROPN
ma-119	150	1	[	[	X
ma-119	150	2	17	17	NUM
ma-119	150	3	]	]	X
ma-119	150	4	estimatedthe	estimatedthe	DET
ma-119	150	5	norm	norm	NOUN
ma-119	150	6	of	of	ADP
ma-119	150	7	operator	operator	NOUN
ma-119	150	8	hθ	hθ	NOUN
ma-119	150	9	,	,	PUNCT
ma-119	150	10	λ	λ	X
ma-119	150	11	=	=	PRON
ma-119	150	12	uθ	uθ	PROPN
ma-119	150	13	+	+	NOUN
ma-119	150	14	u∗θ	u∗θ	NUM
ma-119	150	15	+	+	CCONJ
ma-119	150	16	(	(	PUNCT
ma-119	150	17	λ/2)(vθ	λ/2)(vθ	X
ma-119	150	18	+	+	NOUN
ma-119	150	19	v	v	ADJ
ma-119	150	20	∗θ	∗θ	NOUN
ma-119	150	21	)	)	PUNCT
ma-119	150	22	which	which	PRON
ma-119	150	23	is	be	AUX
ma-119	150	24	an	an	DET
ma-119	150	25	element	element	NOUN
ma-119	150	26	on	on	ADP
ma-119	150	27	a	a	DET
ma-119	150	28	c∗-algebra	c∗-algebra	PROPN
ma-119	150	29	aθ	aθ	NOUN
ma-119	150	30	=	=	PUNCT
ma-119	150	31	c∗(uθ	c∗(uθ	NOUN
ma-119	150	32	,	,	PUNCT
ma-119	150	33	vθ	vθ	VERB
ma-119	150	34	unitaries	unitarie	NOUN
ma-119	150	35	:	:	PUNCT
ma-119	150	36	uθvθ	uθvθ	ADJ
ma-119	150	37	=	=	NOUN
ma-119	150	38	e2πiθvθuθ	e2πiθvθuθ	PROPN
ma-119	150	39	)	)	PUNCT
ma-119	150	40	,	,	PUNCT
ma-119	150	41	and	and	CCONJ
ma-119	150	42	proved	prove	VERB
ma-119	150	43	that	that	SCONJ
ma-119	150	44	for	for	ADP
ma-119	150	45	every	every	DET
ma-119	150	46	λ	λ	PROPN
ma-119	150	47	∈	∈	PROPN
ma-119	150	48	c	c	NOUN
ma-119	150	49	and	and	CCONJ
ma-119	150	50	θ	θ	PROPN
ma-119	150	51	∈	∈	PROPN
ma-119	151	1	[	[	X
ma-119	151	2	14	14	NUM
ma-119	151	3	,	,	PUNCT
ma-119	151	4	1	1	NUM
ma-119	151	5	2	2	NUM
ma-119	151	6	]	]	PUNCT
ma-119	151	7	the	the	DET
ma-119	151	8	inequality	inequality	NOUN
ma-119	151	9	‖hθ	‖hθ	PROPN
ma-119	151	10	,	,	PUNCT
ma-119	151	11	λ‖	λ‖	NUM
ma-119	151	12	≤√	≤√	NUM
ma-119	151	13	4	4	NUM
ma-119	151	14	+	+	NUM
ma-119	151	15	λ2	λ2	NOUN
ma-119	151	16	−	−	PROPN
ma-119	151	17	(	(	PUNCT
ma-119	151	18	1−	1−	NUM
ma-119	151	19	1	1	NUM
ma-119	151	20	tan	tan	PROPN
ma-119	151	21	θ	θ	PROPN
ma-119	151	22	,	,	PUNCT
ma-119	151	23	λ)(1−	λ)(1−	PROPN
ma-119	151	24	√	√	ADP
ma-119	151	25	1+cos2	1+cos2	NUM
ma-119	151	26	4πθ	4πθ	NOUN
ma-119	151	27	2	2	NUM
ma-119	151	28	)	)	PUNCT
ma-119	151	29	min{4	min{4	PROPN
ma-119	151	30	,	,	PUNCT
ma-119	151	31	λ2	λ2	NOUN
ma-119	151	32	}	}	PUNCT
ma-119	151	33	holds	hold	VERB
ma-119	151	34	.	.	PUNCT
ma-119	152	1	this	this	PRON
ma-119	152	2	significantlyimproved	significantlyimprove	VERB
ma-119	152	3	the	the	DET
ma-119	152	4	inequality	inequality	NOUN
ma-119	152	5	‖hθ,2‖	‖hθ,2‖	NOUN
ma-119	152	6	≤	≤	ADV
ma-119	152	7	2	2	NUM
ma-119	152	8	√	√	NUM
ma-119	152	9	2	2	NUM
ma-119	152	10	,	,	PUNCT
ma-119	152	11	θ	θ	PROPN
ma-119	152	12	∈	∈	PROPN
ma-119	153	1	[	[	X
ma-119	153	2	14	14	NUM
ma-119	153	3	,	,	PUNCT
ma-119	153	4	1	1	NUM
ma-119	153	5	2	2	NUM
ma-119	153	6	]	]	PUNCT
ma-119	153	7	,	,	PUNCT
ma-119	153	8	conjectured	conjecture	VERB
ma-119	153	9	by	by	ADP
ma-119	153	10	[	[	X
ma-119	153	11	18	18	NUM
ma-119	153	12	]	]	PUNCT
ma-119	153	13	.	.	PUNCT
ma-119	154	1	the	the	DET
ma-119	154	2	author	author	NOUN
ma-119	154	3	in	in	ADP
ma-119	154	4	[	[	X
ma-119	154	5	31	31	NUM
ma-119	154	6	]	]	PUNCT
ma-119	154	7	consideredcommuting	consideredcommute	VERB
ma-119	154	8	matrices	matrix	NOUN
ma-119	154	9	of	of	ADP
ma-119	154	10	matrix	matrix	NOUN
ma-119	154	11	valued	value	VERB
ma-119	154	12	analytic	analytic	ADJ
ma-119	154	13	function	function	NOUN
ma-119	154	14	and	and	CCONJ
ma-119	154	15	established	establish	VERB
ma-119	154	16	a	a	DET
ma-119	154	17	norm	norm	NOUN
ma-119	154	18	estimate	estimate	NOUN
ma-119	154	19	,	,	PUNCT
ma-119	154	20	in	in	ADP
ma-119	154	21	particular	particular	ADJ
ma-119	154	22	,	,	PUNCT
ma-119	154	23	two	two	NUM
ma-119	154	24	matrices	matrix	NOUN
ma-119	154	25	of	of	ADP
ma-119	154	26	matrix	matrix	NOUN
ma-119	154	27	valued	value	VERB
ma-119	154	28	functions	function	NOUN
ma-119	154	29	on	on	ADP
ma-119	154	30	a	a	DET
ma-119	154	31	tensor	tensor	NOUN
ma-119	154	32	product	product	NOUN
ma-119	154	33	in	in	ADP
ma-119	154	34	a	a	DET
ma-119	154	35	euclidean	euclidean	ADJ
ma-119	154	36	space	space	NOUN
ma-119	154	37	were	be	AUX
ma-119	154	38	explored	explore	VERB
ma-119	154	39	.	.	PUNCT
ma-119	155	1	in	in	ADP
ma-119	155	2	[	[	X
ma-119	155	3	5]the	5]the	DET
ma-119	155	4	research	research	NOUN
ma-119	155	5	communicated	communicate	VERB
ma-119	155	6	results	result	NOUN
ma-119	155	7	on	on	ADP
ma-119	155	8	complex	complex	ADJ
ma-119	155	9	symmetric	symmetric	ADJ
ma-119	155	10	operator	operator	NOUN
ma-119	155	11	theory	theory	NOUN
ma-119	155	12	and	and	CCONJ
ma-119	155	13	showed	show	VERB
ma-119	155	14	that	that	SCONJ
ma-119	155	15	two	two	NUM
ma-119	155	16	non	non	ADJ
ma-119	155	17	-	-	ADJ
ma-119	155	18	trivial	trivial	ADJ
ma-119	155	19	examples	example	NOUN
ma-119	155	20	were	be	AUX
ma-119	155	21	of	of	ADP
ma-119	155	22	great	great	ADJ
ma-119	155	23	use	use	NOUN
ma-119	155	24	in	in	ADP
ma-119	155	25	studying	study	VERB
ma-119	155	26	schrödinger	schrödinger	NOUN
ma-119	155	27	operators	operator	NOUN
ma-119	155	28	.	.	PUNCT
ma-119	156	1	the	the	DET
ma-119	156	2	work	work	NOUN
ma-119	156	3	of	of	ADP
ma-119	156	4	[	[	X
ma-119	156	5	43	43	NUM
ma-119	156	6	]	]	PUNCT
ma-119	156	7	showed	show	VERB
ma-119	156	8	that	that	SCONJ
ma-119	156	9	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	NUM
ma-119	156	10	eur	eur	NOUN
ma-119	156	11	.	.	PUNCT
ma-119	157	1	j.	j.	PROPN
ma-119	157	2	math	math	PROPN
ma-119	157	3	.	.	PUNCT
ma-119	158	1	anal	anal	PROPN
ma-119	158	2	.	.	PUNCT
ma-119	159	1	10.28924	10.28924	NUM
ma-119	159	2	/	/	SYM
ma-119	159	3	ada	ada	PROPN
ma-119	159	4	/	/	SYM
ma-119	159	5	ma.3.9	ma.3.9	PROPN
ma-119	159	6	6triangle	6triangle	NUM
ma-119	159	7	inequality	inequality	NOUN
ma-119	159	8	served	serve	VERB
ma-119	159	9	an	an	DET
ma-119	159	10	upper	upper	ADJ
ma-119	159	11	norm	norm	NOUN
ma-119	159	12	bound	bind	VERB
ma-119	159	13	for	for	ADP
ma-119	159	14	the	the	DET
ma-119	159	15	sum	sum	NOUN
ma-119	159	16	operators	operator	NOUN
ma-119	159	17	that	that	PRON
ma-119	159	18	is	be	AUX
ma-119	159	19	sup{‖t	sup{‖t	NOUN
ma-119	159	20	∗rt+v	∗rt+v	NUM
ma-119	159	21	∗sv	∗sv	PUNCT
ma-119	159	22	‖	‖	ADJ
ma-119	159	23	:	:	PUNCT
ma-119	159	24	tandv	tandv	NOUN
ma-119	159	25	}	}	PUNCT
ma-119	159	26	are	be	AUX
ma-119	159	27	unitaries	unitarie	NOUN
ma-119	159	28	.	.	PUNCT
ma-119	160	1	the	the	DET
ma-119	160	2	result	result	NOUN
ma-119	160	3	discussed	discuss	VERB
ma-119	160	4	had	have	VERB
ma-119	160	5	relationship	relationship	NOUN
ma-119	160	6	to	to	ADP
ma-119	160	7	normal	normal	ADJ
ma-119	160	8	dilations	dilation	NOUN
ma-119	160	9	,	,	PUNCT
ma-119	160	10	spectral	spectral	ADJ
ma-119	160	11	setsand	setsand	NOUN
ma-119	160	12	the	the	DET
ma-119	160	13	von	von	PROPN
ma-119	160	14	neumann	neumann	PROPN
ma-119	160	15	inequality	inequality	PROPN
ma-119	160	16	.	.	PUNCT
ma-119	161	1	yong	yong	PROPN
ma-119	161	2	,	,	PUNCT
ma-119	161	3	toshiyuki	toshiyuki	VERB
ma-119	162	1	[	[	X
ma-119	162	2	53	53	NUM
ma-119	162	3	]	]	PUNCT
ma-119	162	4	gave	give	VERB
ma-119	162	5	a	a	DET
ma-119	162	6	norm	norm	NOUN
ma-119	162	7	estimate	estimate	NOUN
ma-119	162	8	on	on	ADP
ma-119	162	9	pre	pre	NOUN
ma-119	162	10	-	-	NOUN
ma-119	162	11	schwarzianderivatives	schwarzianderivative	NOUN
ma-119	162	12	of	of	ADP
ma-119	162	13	a	a	DET
ma-119	162	14	specific	specific	ADJ
ma-119	162	15	type	type	NOUN
ma-119	162	16	of	of	ADP
ma-119	162	17	convex	convex	NOUN
ma-119	162	18	functions	function	NOUN
ma-119	162	19	by	by	ADP
ma-119	162	20	introducing	introduce	VERB
ma-119	162	21	a	a	DET
ma-119	162	22	maximal	maximal	ADJ
ma-119	162	23	operator	operator	NOUN
ma-119	162	24	of	of	ADP
ma-119	162	25	independentinterest	independentinter	ADJ
ma-119	162	26	of	of	ADP
ma-119	162	27	a	a	DET
ma-119	162	28	given	give	VERB
ma-119	162	29	kind	kind	NOUN
ma-119	162	30	.	.	PUNCT
ma-119	163	1	the	the	DET
ma-119	163	2	relationship	relationship	NOUN
ma-119	163	3	between	between	ADP
ma-119	163	4	the	the	DET
ma-119	163	5	convex	convex	NOUN
ma-119	163	6	functions	function	NOUN
ma-119	163	7	and	and	CCONJ
ma-119	163	8	the	the	DET
ma-119	163	9	hardy	hardy	ADJ
ma-119	163	10	spaces	space	NOUN
ma-119	163	11	wasdiscussed	wasdiscusse	VERB
ma-119	163	12	.	.	PUNCT
ma-119	164	1	in	in	ADP
ma-119	164	2	[	[	X
ma-119	164	3	16	16	NUM
ma-119	164	4	]	]	PUNCT
ma-119	164	5	the	the	DET
ma-119	164	6	author	author	NOUN
ma-119	164	7	analyzed	analyze	VERB
ma-119	164	8	the	the	DET
ma-119	164	9	structure	structure	NOUN
ma-119	164	10	of	of	ADP
ma-119	164	11	the	the	DET
ma-119	164	12	set	set	NOUN
ma-119	164	13	d	d	PROPN
ma-119	164	14	=	=	SYM
ma-119	164	15	{	{	PUNCT
ma-119	164	16	y	y	PROPN
ma-119	164	17	∈	∈	PROPN
ma-119	164	18	d(δ	d(δ	PROPN
ma-119	164	19	)	)	PUNCT
ma-119	164	20	:	:	PUNCT
ma-119	165	1	limn→∞	limn→∞	PROPN
ma-119	165	2	∆n(y	∆n(y	X
ma-119	165	3	)	)	PUNCT
ma-119	165	4	=	=	SYM
ma-119	165	5	∆(y	∆(y	NOUN
ma-119	165	6	)	)	PUNCT
ma-119	165	7	}	}	PUNCT
ma-119	165	8	for	for	ADP
ma-119	165	9	convergence	convergence	NOUN
ma-119	165	10	of	of	ADP
ma-119	165	11	the	the	DET
ma-119	165	12	generators	generator	NOUN
ma-119	165	13	that	that	PRON
ma-119	165	14	are	be	AUX
ma-119	165	15	pointwise	pointwise	VERB
ma-119	165	16	where	where	SCONJ
ma-119	165	17	α	α	NOUN
ma-119	165	18	is	be	AUX
ma-119	165	19	an	an	DET
ma-119	165	20	approximate	approximate	ADJ
ma-119	165	21	innerflow	innerflow	NOUN
ma-119	165	22	on	on	ADP
ma-119	165	23	a	a	DET
ma-119	165	24	c∗-algebra	c∗-algebra	PROPN
ma-119	165	25	t	t	PROPN
ma-119	165	26	with	with	ADP
ma-119	165	27	generator	generator	NOUN
ma-119	165	28	∆	∆	PROPN
ma-119	165	29	and	and	CCONJ
ma-119	165	30	∆n	∆n	PROPN
ma-119	165	31	for	for	ADP
ma-119	165	32	bounded	bounded	ADJ
ma-119	165	33	generators	generator	NOUN
ma-119	165	34	of	of	ADP
ma-119	165	35	the	the	DET
ma-119	165	36	approximateflows	approximateflow	NOUN
ma-119	165	37	αn	αn	VERB
ma-119	165	38	.	.	PUNCT
ma-119	166	1	in	in	ADP
ma-119	166	2	fact	fact	NOUN
ma-119	166	3	,	,	PUNCT
ma-119	166	4	the	the	DET
ma-119	166	5	relationship	relationship	NOUN
ma-119	166	6	of	of	ADP
ma-119	166	7	d	d	PROPN
ma-119	166	8	and	and	CCONJ
ma-119	166	9	various	various	ADJ
ma-119	166	10	cores	core	NOUN
ma-119	166	11	related	relate	VERB
ma-119	166	12	to	to	ADP
ma-119	166	13	spectral	spectral	ADJ
ma-119	166	14	subspaces	subspace	NOUN
ma-119	166	15	wereexamined	wereexamine	VERB
ma-119	166	16	.	.	PUNCT
ma-119	167	1	seddik	seddik	PROPN
ma-119	168	1	[	[	X
ma-119	168	2	47	47	NUM
ma-119	168	3	]	]	PUNCT
ma-119	168	4	showed	show	VERB
ma-119	168	5	that	that	SCONJ
ma-119	168	6	q	q	NOUN
ma-119	168	7	is	be	AUX
ma-119	168	8	a	a	DET
ma-119	168	9	normal	normal	ADJ
ma-119	168	10	operator	operator	NOUN
ma-119	168	11	which	which	PRON
ma-119	168	12	is	be	AUX
ma-119	168	13	invertible	invertible	ADJ
ma-119	168	14	in	in	ADP
ma-119	168	15	b(h	b(h	NOUN
ma-119	168	16	)	)	PUNCT
ma-119	168	17	if	if	SCONJ
ma-119	168	18	theestimate	theestimate	VERB
ma-119	168	19	‖q	‖q	PUNCT
ma-119	169	1	⊗	⊗	PROPN
ma-119	169	2	q−1	q−1	PROPN
ma-119	170	1	+	+	CCONJ
ma-119	170	2	q−1	q−1	PROPN
ma-119	171	1	⊗	⊗	NUM
ma-119	171	2	q‖λ	q‖λ	ADV
ma-119	171	3	≤	≤	PUNCT
ma-119	171	4	‖q‖‖q−1‖	‖q‖‖q−1‖	PROPN
ma-119	171	5	+	+	CCONJ
ma-119	171	6	1	1	NUM
ma-119	171	7	‖q‖‖q−1‖	‖q‖‖q−1‖	PROPN
ma-119	171	8	holds	hold	VERB
ma-119	171	9	,	,	PUNCT
ma-119	171	10	such	such	ADJ
ma-119	171	11	that	that	SCONJ
ma-119	171	12	‖.‖λ	‖.‖λ	ADJ
ma-119	171	13	is	be	AUX
ma-119	171	14	the	the	DET
ma-119	171	15	injectivenorm	injectivenorm	NOUN
ma-119	171	16	on	on	ADP
ma-119	171	17	the	the	DET
ma-119	171	18	tensor	tensor	NOUN
ma-119	171	19	product	product	NOUN
ma-119	171	20	b(h	b(h	PROPN
ma-119	171	21	)	)	PUNCT
ma-119	171	22	⊗	⊗	PROPN
ma-119	171	23	b(h	b(h	PROPN
ma-119	171	24	)	)	PUNCT
ma-119	171	25	,	,	PUNCT
ma-119	171	26	when	when	SCONJ
ma-119	171	27	q	q	NOUN
ma-119	171	28	is	be	AUX
ma-119	171	29	invertible	invertible	ADJ
ma-119	171	30	self	self	NOUN
ma-119	171	31	-	-	PUNCT
ma-119	171	32	adjoint	adjoint	NOUN
ma-119	171	33	then	then	ADV
ma-119	171	34	the	the	DET
ma-119	171	35	equationbecomes	equationbecome	VERB
ma-119	171	36	an	an	DET
ma-119	171	37	equality	equality	NOUN
ma-119	171	38	.	.	PUNCT
ma-119	172	1	bonyo	bonyo	PROPN
ma-119	172	2	and	and	CCONJ
ma-119	172	3	agure	agure	VERB
ma-119	172	4	[	[	X
ma-119	172	5	7	7	NUM
ma-119	172	6	]	]	PUNCT
ma-119	172	7	characterized	characterize	VERB
ma-119	172	8	the	the	DET
ma-119	172	9	norm	norm	NOUN
ma-119	172	10	of	of	ADP
ma-119	172	11	inner	inner	ADJ
ma-119	172	12	derivation	derivation	NOUN
ma-119	172	13	on	on	ADP
ma-119	172	14	normideal	normideal	NOUN
ma-119	172	15	to	to	PART
ma-119	172	16	be	be	AUX
ma-119	172	17	equal	equal	ADJ
ma-119	172	18	to	to	ADP
ma-119	172	19	the	the	DET
ma-119	172	20	quotient	quotient	NOUN
ma-119	172	21	algebra	algebra	NOUN
ma-119	172	22	and	and	CCONJ
ma-119	172	23	investigated	investigate	VERB
ma-119	172	24	them	they	PRON
ma-119	172	25	when	when	SCONJ
ma-119	172	26	they	they	PRON
ma-119	172	27	are	be	AUX
ma-119	172	28	implemented	implement	VERB
ma-119	172	29	bynormal	bynormal	ADJ
ma-119	172	30	and	and	CCONJ
ma-119	172	31	hyponormal	hyponormal	ADJ
ma-119	172	32	operators	operator	NOUN
ma-119	172	33	on	on	ADP
ma-119	172	34	norm	norm	NOUN
ma-119	172	35	ideals	ideal	NOUN
ma-119	172	36	.	.	PUNCT
ma-119	173	1	a	a	DET
ma-119	173	2	hyponormal	hyponormal	ADJ
ma-119	173	3	x	x	PUNCT
ma-119	173	4	is	be	AUX
ma-119	173	5	a	a	DET
ma-119	173	6	bounded	bounded	ADJ
ma-119	173	7	linear	linear	ADJ
ma-119	173	8	operatoron	operatoron	NOUN
ma-119	173	9	a	a	DET
ma-119	173	10	hilbert	hilbert	NOUN
ma-119	173	11	space	space	NOUN
ma-119	173	12	h	h	NOUN
ma-119	173	13	if	if	SCONJ
ma-119	173	14	x∗x	x∗x	NUM
ma-119	173	15	−	−	PROPN
ma-119	173	16	xx∗	xx∗	VERB
ma-119	173	17	≥	≥	NOUN
ma-119	173	18	0	0	PUNCT
ma-119	173	19	and	and	CCONJ
ma-119	173	20	is	be	AUX
ma-119	173	21	normal	normal	ADJ
ma-119	173	22	if	if	SCONJ
ma-119	173	23	x∗x	x∗x	NUM
ma-119	173	24	=	=	SYM
ma-119	173	25	xx∗.	xx∗.	PROPN
ma-119	173	26	bonyo	bonyo	NOUN
ma-119	173	27	and	and	CCONJ
ma-119	173	28	agure	agure	VERB
ma-119	173	29	[	[	X
ma-119	173	30	8]investigated	8]investigate	VERB
ma-119	173	31	the	the	DET
ma-119	173	32	relation	relation	NOUN
ma-119	173	33	of	of	ADP
ma-119	173	34	the	the	DET
ma-119	173	35	diameter	diameter	NOUN
ma-119	173	36	of	of	ADP
ma-119	173	37	the	the	DET
ma-119	173	38	numerical	numerical	ADJ
ma-119	173	39	range	range	NOUN
ma-119	173	40	of	of	ADP
ma-119	173	41	an	an	DET
ma-119	173	42	operator	operator	NOUN
ma-119	173	43	b	b	PROPN
ma-119	173	44	∈	∈	PROPN
ma-119	173	45	b(h	b(h	PROPN
ma-119	173	46	)	)	PUNCT
ma-119	173	47	and	and	CCONJ
ma-119	173	48	thenorm	thenorm	NOUN
ma-119	173	49	of	of	ADP
ma-119	173	50	inner	inner	ADJ
ma-119	173	51	derivation	derivation	NOUN
ma-119	173	52	implemented	implement	VERB
ma-119	173	53	by	by	ADP
ma-119	173	54	b	b	PROPN
ma-119	173	55	on	on	ADP
ma-119	173	56	a	a	DET
ma-119	173	57	norm	norm	NOUN
ma-119	173	58	ideal	ideal	NOUN
ma-119	173	59	j	j	PROPN
ma-119	173	60	and	and	CCONJ
ma-119	173	61	considered	consider	VERB
ma-119	173	62	the	the	DET
ma-119	173	63	application	application	NOUN
ma-119	173	64	of	of	ADP
ma-119	173	65	s	s	NOUN
ma-119	173	66	-	-	PUNCT
ma-119	173	67	universality	universality	NOUN
ma-119	173	68	to	to	ADP
ma-119	173	69	the	the	DET
ma-119	173	70	relation	relation	NOUN
ma-119	173	71	.	.	PUNCT
ma-119	174	1	bonyo	bonyo	PROPN
ma-119	174	2	and	and	CCONJ
ma-119	174	3	agure	agure	VERB
ma-119	174	4	[	[	X
ma-119	174	5	6	6	NUM
ma-119	174	6	]	]	PUNCT
ma-119	174	7	defined	define	VERB
ma-119	174	8	inner	inner	ADJ
ma-119	174	9	derivations	derivation	NOUN
ma-119	174	10	implemented	implement	VERB
ma-119	174	11	by	by	ADP
ma-119	174	12	a	a	PRON
ma-119	174	13	,	,	PUNCT
ma-119	174	14	brespectively	brespectively	ADV
ma-119	174	15	on	on	ADP
ma-119	174	16	b(h	b(h	NOUN
ma-119	174	17	)	)	PUNCT
ma-119	174	18	by	by	ADP
ma-119	174	19	δa(y	δa(y	NUM
ma-119	174	20	)	)	PUNCT
ma-119	175	1	=	=	SYM
ma-119	175	2	ay	ay	NOUN
ma-119	176	1	−	−	PROPN
ma-119	176	2	y	y	PROPN
ma-119	176	3	a	a	X
ma-119	176	4	,	,	PUNCT
ma-119	176	5	δb(y	δb(y	PUNCT
ma-119	176	6	)	)	PUNCT
ma-119	176	7	=	=	PUNCT
ma-119	176	8	by	by	ADP
ma-119	176	9	−	−	PROPN
ma-119	176	10	y	y	PROPN
ma-119	176	11	b	b	PROPN
ma-119	176	12	and	and	CCONJ
ma-119	176	13	generalized	generalized	ADJ
ma-119	176	14	derivation	derivation	NOUN
ma-119	176	15	by	by	ADP
ma-119	176	16	δa	δa	PROPN
ma-119	176	17	,	,	PUNCT
ma-119	176	18	b	b	PROPN
ma-119	176	19	(	(	PUNCT
ma-119	176	20	y	y	PROPN
ma-119	176	21	)	)	PUNCT
ma-119	177	1	=	=	SYM
ma-119	177	2	ay	ay	NOUN
ma-119	177	3	−	−	PROPN
ma-119	177	4	y	y	PROPN
ma-119	177	5	b	b	PROPN
ma-119	177	6	∀	∀	X
ma-119	177	7	y	y	PROPN
ma-119	177	8	∈	∈	PROPN
ma-119	177	9	b(h	b(h	PROPN
ma-119	177	10	)	)	PUNCT
ma-119	177	11	.	.	PUNCT
ma-119	178	1	further	far	ADV
ma-119	178	2	,	,	PUNCT
ma-119	178	3	a	a	DET
ma-119	178	4	relationship	relationship	NOUN
ma-119	178	5	between	between	ADP
ma-119	178	6	the	the	DET
ma-119	178	7	norms	norm	NOUN
ma-119	178	8	of	of	ADP
ma-119	178	9	δa	δa	NOUN
ma-119	178	10	,	,	PUNCT
ma-119	178	11	δb	δb	NOUN
ma-119	178	12	and	and	CCONJ
ma-119	178	13	δa	δa	NOUN
ma-119	178	14	,	,	PUNCT
ma-119	178	15	bon	bon	PROPN
ma-119	178	16	b(h	b(h	PROPN
ma-119	178	17	)	)	PUNCT
ma-119	178	18	was	be	AUX
ma-119	178	19	established	establish	VERB
ma-119	178	20	,	,	PUNCT
ma-119	178	21	specifically	specifically	ADV
ma-119	178	22	when	when	SCONJ
ma-119	178	23	the	the	DET
ma-119	178	24	operators	operator	NOUN
ma-119	178	25	a	a	PRON
ma-119	178	26	,	,	PUNCT
ma-119	178	27	b	b	NOUN
ma-119	178	28	are	be	AUX
ma-119	178	29	s	s	NOUN
ma-119	178	30	-	-	ADJ
ma-119	178	31	universal	universal	ADJ
ma-119	178	32	.	.	PUNCT
ma-119	179	1	ber	ber	PROPN
ma-119	179	2	,	,	PUNCT
ma-119	179	3	sukochev	sukochev	VERB
ma-119	179	4	[	[	X
ma-119	179	5	5]showed	5]showed	NUM
ma-119	179	6	that	that	SCONJ
ma-119	179	7	for	for	ADP
ma-119	179	8	every	every	DET
ma-119	179	9	self	self	NOUN
ma-119	179	10	-	-	PUNCT
ma-119	179	11	adjoint	adjoint	NOUN
ma-119	179	12	element	element	NOUN
ma-119	179	13	b	b	PROPN
ma-119	179	14	∈	∈	PROPN
ma-119	179	15	s(n	s(n	PROPN
ma-119	179	16	)	)	PUNCT
ma-119	179	17	a	a	DET
ma-119	179	18	scalar	scalar	ADJ
ma-119	179	19	λ0	λ0	NOUN
ma-119	179	20	∈	∈	NOUN
ma-119	179	21	r	r	NOUN
ma-119	179	22	exists	exist	VERB
ma-119	179	23	such	such	ADJ
ma-119	179	24	that	that	SCONJ
ma-119	179	25	∀	∀	NOUN
ma-119	180	1	ε	ε	X
ma-119	180	2	>	>	X
ma-119	180	3	0,then	0,then	NOUN
ma-119	180	4	there	there	PRON
ma-119	180	5	exists	exist	VERB
ma-119	180	6	a	a	DET
ma-119	180	7	unital	unital	ADJ
ma-119	180	8	element	element	NOUN
ma-119	180	9	uε	uε	NOUN
ma-119	180	10	from	from	ADP
ma-119	180	11	n	n	CCONJ
ma-119	180	12	satisfy	satisfy	PROPN
ma-119	180	13	|[b	|[b	PROPN
ma-119	180	14	,	,	PUNCT
ma-119	180	15	uε]|	uε]|	X
ma-119	180	16	≥	≥	X
ma-119	180	17	(	(	PUNCT
ma-119	180	18	1−	1−	NUM
ma-119	180	19	ε)|b−	ε)|b−	NOUN
ma-119	180	20	λ01|	λ01|	NOUN
ma-119	180	21	.	.	PUNCT
ma-119	181	1	from	from	ADP
ma-119	181	2	this	this	DET
ma-119	181	3	result	result	NOUN
ma-119	181	4	aconsequence	aconsequence	NOUN
ma-119	181	5	is	be	AUX
ma-119	181	6	that	that	SCONJ
ma-119	181	7	for	for	ADP
ma-119	181	8	any	any	DET
ma-119	181	9	derivation	derivation	NOUN
ma-119	181	10	δ	δ	NOUN
ma-119	181	11	on	on	ADP
ma-119	181	12	n	n	CCONJ
ma-119	181	13	with	with	ADP
ma-119	181	14	the	the	DET
ma-119	181	15	range	range	NOUN
ma-119	181	16	on	on	ADP
ma-119	181	17	an	an	DET
ma-119	181	18	ideal	ideal	NOUN
ma-119	181	19	i	i	PRON
ma-119	181	20	⊆	⊆	PROPN
ma-119	181	21	n	n	ADP
ma-119	181	22	the	the	DET
ma-119	181	23	derivation	derivation	NOUN
ma-119	181	24	δis	δis	NOUN
ma-119	181	25	inner	inner	ADV
ma-119	181	26	i.e	i.e	PROPN
ma-119	181	27	δ	δ	PROPN
ma-119	181	28	(	(	PUNCT
ma-119	181	29	.	.	PUNCT
ma-119	181	30	)	)	PUNCT
ma-119	182	1	=	=	SYM
ma-119	182	2	δa	δa	PROPN
ma-119	182	3	(	(	PUNCT
ma-119	182	4	.	.	PUNCT
ma-119	182	5	)	)	PUNCT
ma-119	183	1	=	=	PUNCT
ma-119	184	1	[	[	X
ma-119	184	2	a	a	X
ma-119	184	3	,	,	PUNCT
ma-119	184	4	.	.	PUNCT
ma-119	184	5	]	]	PUNCT
ma-119	185	1	and	and	CCONJ
ma-119	185	2	a	a	DET
ma-119	185	3	∈	∈	PROPN
ma-119	185	4	i.	i.	NOUN
ma-119	185	5	pablo	pablo	PROPN
ma-119	185	6	,	,	PUNCT
ma-119	185	7	jussi	jussi	PROPN
ma-119	185	8	,	,	PUNCT
ma-119	185	9	mikael	mikael	PROPN
ma-119	186	1	[	[	X
ma-119	186	2	42	42	NUM
ma-119	186	3	]	]	PUNCT
ma-119	186	4	provided	provide	VERB
ma-119	186	5	theoretic	theoretic	ADJ
ma-119	186	6	estimate	estimate	NOUN
ma-119	186	7	oftwo	oftwo	NOUN
ma-119	186	8	functions	function	NOUN
ma-119	186	9	for	for	ADP
ma-119	186	10	the	the	DET
ma-119	186	11	essential	essential	ADJ
ma-119	186	12	norm	norm	NOUN
ma-119	186	13	as	as	ADP
ma-119	186	14	a	a	DET
ma-119	186	15	composition	composition	NOUN
ma-119	186	16	operator	operator	NOUN
ma-119	186	17	cϕ	cϕ	ADP
ma-119	186	18	that	that	DET
ma-119	186	19	acts	act	VERB
ma-119	186	20	on	on	ADP
ma-119	186	21	the	the	DET
ma-119	186	22	space	space	NOUN
ma-119	186	23	bmoa;one	bmoa;one	NOUN
ma-119	186	24	in	in	ADP
ma-119	186	25	terms	term	NOUN
ma-119	186	26	of	of	ADP
ma-119	186	27	the	the	DET
ma-119	186	28	n	n	ADV
ma-119	186	29	-	-	PUNCT
ma-119	186	30	th	th	VERB
ma-119	186	31	power	power	NOUN
ma-119	186	32	ϕn	ϕn	PRON
ma-119	186	33	denoted	denote	VERB
ma-119	186	34	by	by	ADP
ma-119	186	35	ϕ	ϕ	PROPN
ma-119	186	36	and	and	CCONJ
ma-119	186	37	the	the	DET
ma-119	186	38	other	other	ADJ
ma-119	186	39	involved	involve	VERB
ma-119	186	40	the	the	DET
ma-119	186	41	nevanlinna	nevanlinna	NOUN
ma-119	186	42	countingfunction	countingfunction	NOUN
ma-119	186	43	.	.	PUNCT
ma-119	187	1	the	the	DET
ma-119	187	2	research	research	NOUN
ma-119	187	3	of	of	ADP
ma-119	187	4	[	[	X
ma-119	187	5	20	20	NUM
ma-119	187	6	]	]	PUNCT
ma-119	187	7	introduced	introduce	VERB
ma-119	187	8	a	a	DET
ma-119	187	9	new	new	ADJ
ma-119	187	10	type	type	NOUN
ma-119	187	11	of	of	ADP
ma-119	187	12	norm	norm	NOUN
ma-119	187	13	for	for	ADP
ma-119	187	14	semimartangles	semimartangle	NOUN
ma-119	187	15	,	,	PUNCT
ma-119	187	16	the	the	DET
ma-119	187	17	defined	define	VERB
ma-119	187	18	norm	norm	NOUN
ma-119	187	19	ofquasimartangales	ofquasimartangale	NOUN
ma-119	187	20	and	and	CCONJ
ma-119	187	21	then	then	ADV
ma-119	187	22	characterized	characterize	VERB
ma-119	187	23	the	the	DET
ma-119	187	24	square	square	ADJ
ma-119	187	25	integrable	integrable	ADJ
ma-119	187	26	semimartangales	semimartangale	NOUN
ma-119	187	27	.	.	PUNCT
ma-119	188	1	in	in	ADP
ma-119	188	2	[	[	X
ma-119	188	3	4	4	X
ma-119	188	4	]	]	PUNCT
ma-119	188	5	the	the	DET
ma-119	188	6	authorgave	authorgave	NOUN
ma-119	188	7	the	the	DET
ma-119	188	8	result	result	NOUN
ma-119	188	9	on	on	ADP
ma-119	188	10	lower	low	ADJ
ma-119	188	11	bound	bind	VERB
ma-119	188	12	of	of	ADP
ma-119	188	13	the	the	DET
ma-119	188	14	norms	norm	NOUN
ma-119	188	15	for	for	ADP
ma-119	188	16	finite	finite	ADJ
ma-119	188	17	dimensional	dimensional	ADJ
ma-119	188	18	operators	operator	NOUN
ma-119	188	19	.	.	PUNCT
ma-119	189	1	the	the	DET
ma-119	189	2	work	work	NOUN
ma-119	189	3	of	of	ADP
ma-119	189	4	[	[	X
ma-119	189	5	14]determined	14]determined	NUM
ma-119	189	6	the	the	DET
ma-119	189	7	norm	norm	NOUN
ma-119	189	8	of	of	ADP
ma-119	189	9	two	two	NUM
ma-119	189	10	-	-	PUNCT
ma-119	189	11	sided	sided	ADJ
ma-119	189	12	symmetric	symmetric	ADJ
ma-119	189	13	operator	operator	NOUN
ma-119	189	14	in	in	ADP
ma-119	189	15	an	an	DET
ma-119	189	16	algebra	algebra	NOUN
ma-119	189	17	.	.	PUNCT
ma-119	190	1	more	more	ADV
ma-119	190	2	precisely	precisely	ADV
ma-119	190	3	,	,	PUNCT
ma-119	190	4	the	the	DET
ma-119	190	5	lowerbound	lowerbound	NOUN
ma-119	190	6	of	of	ADP
ma-119	190	7	the	the	DET
ma-119	190	8	operator	operator	NOUN
ma-119	190	9	using	use	VERB
ma-119	190	10	injective	injective	ADJ
ma-119	190	11	tensor	tensor	NOUN
ma-119	190	12	norm	norm	NOUN
ma-119	190	13	was	be	AUX
ma-119	190	14	investigated	investigate	VERB
ma-119	190	15	.	.	PUNCT
ma-119	191	1	further	far	ADV
ma-119	191	2	,	,	PUNCT
ma-119	191	3	the	the	DET
ma-119	191	4	inner	inner	ADJ
ma-119	191	5	derivationnorm	derivationnorm	NOUN
ma-119	191	6	on	on	ADP
ma-119	191	7	irreducible	irreducible	ADJ
ma-119	191	8	c∗-algebra	c∗-algebra	NOUN
ma-119	191	9	was	be	AUX
ma-119	191	10	determined	determine	VERB
ma-119	191	11	and	and	CCONJ
ma-119	191	12	stampfli	stampfli	NOUN
ma-119	191	13	’s	’s	PART
ma-119	191	14	[	[	X
ma-119	191	15	49	49	NUM
ma-119	191	16	]	]	PUNCT
ma-119	191	17	result	result	NOUN
ma-119	191	18	for	for	ADP
ma-119	191	19	these	these	DET
ma-119	191	20	algebras	algebra	NOUN
ma-119	191	21	wasconfirmed	wasconfirmed	PROPN
ma-119	191	22	.	.	PUNCT
ma-119	192	1	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	NUM
ma-119	192	2	eur	eur	PROPN
ma-119	192	3	.	.	PUNCT
ma-119	193	1	j.	j.	PROPN
ma-119	193	2	math	math	PROPN
ma-119	193	3	.	.	PUNCT
ma-119	194	1	anal	anal	PROPN
ma-119	194	2	.	.	PUNCT
ma-119	195	1	10.28924	10.28924	NUM
ma-119	195	2	/	/	SYM
ma-119	195	3	ada	ada	PROPN
ma-119	195	4	/	/	SYM
ma-119	195	5	ma.3.9	ma.3.9	PROPN
ma-119	195	6	72	72	NUM
ma-119	195	7	.	.	PUNCT
ma-119	196	1	preliminaries	preliminary	NOUN
ma-119	196	2	this	this	DET
ma-119	196	3	section	section	NOUN
ma-119	196	4	provides	provide	VERB
ma-119	196	5	the	the	DET
ma-119	196	6	basic	basic	ADJ
ma-119	196	7	concepts	concept	NOUN
ma-119	196	8	which	which	PRON
ma-119	196	9	are	be	AUX
ma-119	196	10	useful	useful	ADJ
ma-119	196	11	in	in	ADP
ma-119	196	12	the	the	DET
ma-119	196	13	sequel	sequel	NOUN
ma-119	196	14	.	.	PUNCT
ma-119	197	1	definition	definition	NOUN
ma-119	197	2	1	1	NUM
ma-119	197	3	(	(	PUNCT
ma-119	197	4	[	[	X
ma-119	197	5	1	1	NUM
ma-119	197	6	]	]	PUNCT
ma-119	197	7	,	,	PUNCT
ma-119	197	8	definition	definition	NOUN
ma-119	197	9	1.5	1.5	NUM
ma-119	197	10	)	)	PUNCT
ma-119	197	11	.	.	PUNCT
ma-119	198	1	a	a	DET
ma-119	198	2	banach	banach	NOUN
ma-119	198	3	∗-algebra	∗-algebra	NOUN
ma-119	198	4	t	t	PROPN
ma-119	198	5	is	be	AUX
ma-119	198	6	called	call	VERB
ma-119	198	7	c∗-algebra	c∗-algebra	PROPN
ma-119	198	8	if	if	SCONJ
ma-119	198	9	‖tt∗‖	‖tt∗‖	PROPN
ma-119	198	10	=	=	SYM
ma-119	198	11	‖t‖2	‖t‖2	NOUN
ma-119	198	12	,	,	PUNCT
ma-119	198	13	∀	∀	X
ma-119	198	14	t	t	NOUN
ma-119	198	15	∈	∈	PROPN
ma-119	198	16	t	t	PROPN
ma-119	198	17	.	.	PUNCT
ma-119	199	1	definition	definition	NOUN
ma-119	199	2	2	2	NUM
ma-119	199	3	(	(	PUNCT
ma-119	199	4	[	[	X
ma-119	199	5	37	37	NUM
ma-119	199	6	]	]	PUNCT
ma-119	199	7	,	,	PUNCT
ma-119	199	8	definition	definition	NOUN
ma-119	199	9	2.1	2.1	NUM
ma-119	199	10	)	)	PUNCT
ma-119	199	11	.	.	PUNCT
ma-119	200	1	elementary	elementary	ADJ
ma-119	200	2	operator	operator	NOUN
ma-119	200	3	t	t	PROPN
ma-119	200	4	:	:	PUNCT
ma-119	200	5	b(h	b(h	PROPN
ma-119	200	6	)	)	PUNCT
ma-119	200	7	→	→	SYM
ma-119	200	8	b(h	b(h	PROPN
ma-119	200	9	)	)	PUNCT
ma-119	200	10	is	be	AUX
ma-119	200	11	defined	define	VERB
ma-119	200	12	by	by	ADP
ma-119	200	13	tdi	tdi	PROPN
ma-119	200	14	,	,	PUNCT
ma-119	200	15	ei	ei	X
ma-119	200	16	(	(	PUNCT
ma-119	200	17	x	x	X
ma-119	200	18	)	)	PUNCT
ma-119	200	19	=	=	SYM
ma-119	200	20	∑n	∑n	NUM
ma-119	200	21	i=1di	i=1di	NOUN
ma-119	200	22	x	x	NOUN
ma-119	200	23	ei	ei	X
ma-119	200	24	∀	∀	NOUN
ma-119	200	25	x	x	SYM
ma-119	200	26	∈	∈	PROPN
ma-119	200	27	b(h	b(h	PROPN
ma-119	200	28	)	)	PUNCT
ma-119	200	29	and	and	CCONJ
ma-119	200	30	∀	∀	X
ma-119	200	31	di	di	NOUN
ma-119	200	32	,	,	PUNCT
ma-119	200	33	ei	ei	SCONJ
ma-119	200	34	fixed	fix	VERB
ma-119	200	35	in	in	ADP
ma-119	200	36	b(h	b(h	PROPN
ma-119	200	37	)	)	PUNCT
ma-119	200	38	where	where	SCONJ
ma-119	200	39	i	i	PRON
ma-119	200	40	=	=	NOUN
ma-119	200	41	1	1	NUM
ma-119	200	42	,	,	PUNCT
ma-119	200	43	...	...	PUNCT
ma-119	200	44	,	,	PUNCT
ma-119	200	45	n.	n.	NOUN
ma-119	200	46	for	for	ADP
ma-119	200	47	b(h	b(h	PROPN
ma-119	200	48	)	)	PUNCT
ma-119	200	49	,	,	PUNCT
ma-119	200	50	we	we	PRON
ma-119	200	51	define	define	VERB
ma-119	200	52	the	the	DET
ma-119	200	53	particular	particular	ADJ
ma-119	200	54	elementary	elementary	ADJ
ma-119	200	55	operators	operator	NOUN
ma-119	200	56	as	as	ADP
ma-119	200	57	below:(i	below:(i	NOUN
ma-119	200	58	)	)	PUNCT
ma-119	200	59	.	.	PUNCT
ma-119	201	1	left	leave	VERB
ma-119	201	2	multiplication	multiplication	NOUN
ma-119	201	3	operator	operator	NOUN
ma-119	201	4	ld	ld	PROPN
ma-119	201	5	:	:	PUNCT
ma-119	201	6	b(h)→	b(h)→	PROPN
ma-119	201	7	b(h	b(h	PROPN
ma-119	201	8	)	)	PUNCT
ma-119	201	9	by	by	ADP
ma-119	201	10	ld(x	ld(x	NOUN
ma-119	201	11	)	)	PUNCT
ma-119	201	12	=	=	SYM
ma-119	201	13	dx	dx	PROPN
ma-119	201	14	,	,	PUNCT
ma-119	201	15	∀	∀	X
ma-119	201	16	x	x	SYM
ma-119	201	17	∈	∈	PROPN
ma-119	201	18	b(h).(ii	b(h).(ii	PROPN
ma-119	201	19	)	)	PUNCT
ma-119	201	20	.	.	PUNCT
ma-119	202	1	right	right	ADJ
ma-119	202	2	multiplication	multiplication	NOUN
ma-119	202	3	operator	operator	NOUN
ma-119	202	4	re	re	ADP
ma-119	202	5	:	:	PUNCT
ma-119	202	6	b(h)→	b(h)→	PUNCT
ma-119	202	7	b(h	b(h	PROPN
ma-119	202	8	)	)	PUNCT
ma-119	202	9	by	by	ADP
ma-119	202	10	re(x	re(x	NOUN
ma-119	202	11	)	)	PUNCT
ma-119	202	12	=	=	SYM
ma-119	202	13	xe	xe	PROPN
ma-119	202	14	,	,	PUNCT
ma-119	202	15	∀	∀	X
ma-119	202	16	x	x	SYM
ma-119	202	17	∈	∈	NOUN
ma-119	202	18	b(h).(iii	b(h).(iii	NOUN
ma-119	202	19	)	)	PUNCT
ma-119	202	20	.	.	PUNCT
ma-119	203	1	generalized	generalize	VERB
ma-119	203	2	derivation	derivation	NOUN
ma-119	203	3	(	(	PUNCT
ma-119	203	4	implemented	implement	VERB
ma-119	203	5	by	by	ADP
ma-119	203	6	d	d	PROPN
ma-119	203	7	,	,	PUNCT
ma-119	203	8	e	e	NOUN
ma-119	203	9	)	)	PUNCT
ma-119	203	10	by	by	ADP
ma-119	203	11	δd	δd	X
ma-119	203	12	,	,	PUNCT
ma-119	203	13	e	e	PROPN
ma-119	203	14	=	=	SYM
ma-119	203	15	ld	ld	PROPN
ma-119	203	16	−	−	PROPN
ma-119	203	17	re	re	X
ma-119	203	18	.(iv	.(iv	PROPN
ma-119	203	19	)	)	PUNCT
ma-119	203	20	.	.	PUNCT
ma-119	204	1	inner	inner	ADJ
ma-119	204	2	derivation	derivation	NOUN
ma-119	204	3	(	(	PUNCT
ma-119	204	4	implemented	implement	VERB
ma-119	204	5	by	by	ADP
ma-119	204	6	d	d	PROPN
ma-119	204	7	)	)	PUNCT
ma-119	204	8	by	by	ADP
ma-119	204	9	δd(x	δd(x	NOUN
ma-119	204	10	)	)	PUNCT
ma-119	204	11	=	=	PRON
ma-119	204	12	dx	dx	PROPN
ma-119	204	13	−xd.(v	−xd.(v	NUM
ma-119	204	14	)	)	PUNCT
ma-119	204	15	.	.	PUNCT
ma-119	205	1	basic	basic	ADJ
ma-119	205	2	elementary	elementary	ADJ
ma-119	205	3	operator	operator	NOUN
ma-119	205	4	(	(	PUNCT
ma-119	205	5	implemented	implement	VERB
ma-119	205	6	by	by	ADP
ma-119	205	7	d	d	PROPN
ma-119	205	8	,	,	PUNCT
ma-119	205	9	e	e	NOUN
ma-119	205	10	)	)	PUNCT
ma-119	205	11	by	by	ADP
ma-119	205	12	md	md	PROPN
ma-119	205	13	,	,	PUNCT
ma-119	205	14	e(x	e(x	NUM
ma-119	205	15	)	)	PUNCT
ma-119	206	1	=	=	SYM
ma-119	206	2	dxe	dxe	ADJ
ma-119	206	3	,	,	PUNCT
ma-119	206	4	∀	∀	PUNCT
ma-119	206	5	x	x	SYM
ma-119	206	6	∈	∈	PROPN
ma-119	206	7	b(h).(vi	b(h).(vi	PROPN
ma-119	206	8	)	)	PUNCT
ma-119	206	9	.	.	PUNCT
ma-119	207	1	jordan	jordan	PROPN
ma-119	207	2	elementary	elementary	PROPN
ma-119	207	3	operator	operator	NOUN
ma-119	207	4	(	(	PUNCT
ma-119	207	5	implemented	implement	VERB
ma-119	207	6	by	by	ADP
ma-119	207	7	d	d	PROPN
ma-119	207	8	,	,	PUNCT
ma-119	207	9	e	e	NOUN
ma-119	207	10	)	)	PUNCT
ma-119	207	11	by	by	ADP
ma-119	207	12	ud	ud	INTJ
ma-119	207	13	,	,	PUNCT
ma-119	207	14	e(x	e(x	NUM
ma-119	207	15	)	)	PUNCT
ma-119	207	16	=	=	SYM
ma-119	207	17	dxe	dxe	PROPN
ma-119	207	18	+	+	NUM
ma-119	207	19	exd	exd	NOUN
ma-119	207	20	,	,	PUNCT
ma-119	207	21	∀	∀	NOUN
ma-119	207	22	x	x	SYM
ma-119	207	23	∈	∈	PROPN
ma-119	207	24	b(h	b(h	PROPN
ma-119	207	25	)	)	PUNCT
ma-119	207	26	.	.	PUNCT
ma-119	208	1	definition	definition	NOUN
ma-119	208	2	3	3	NUM
ma-119	208	3	(	(	PUNCT
ma-119	208	4	[	[	X
ma-119	208	5	49	49	NUM
ma-119	208	6	]	]	PUNCT
ma-119	208	7	,	,	PUNCT
ma-119	208	8	definition	definition	NOUN
ma-119	208	9	2.3	2.3	NUM
ma-119	208	10	)	)	PUNCT
ma-119	208	11	.	.	PUNCT
ma-119	209	1	a	a	DET
ma-119	209	2	derivation	derivation	NOUN
ma-119	209	3	is	be	AUX
ma-119	209	4	a	a	DET
ma-119	209	5	map	map	NOUN
ma-119	210	1	d	d	NOUN
ma-119	210	2	:	:	PUNCT
ma-119	210	3	u	u	X
ma-119	210	4	→	→	SYM
ma-119	210	5	u	u	SYM
ma-119	210	6	satisfying	satisfy	VERB
ma-119	210	7	d(f	d(f	NOUN
ma-119	210	8	g	g	NOUN
ma-119	210	9	)	)	PUNCT
ma-119	210	10	=	=	SYM
ma-119	211	1	f	f	PROPN
ma-119	211	2	d(g	d(g	PROPN
ma-119	211	3	)	)	PUNCT
ma-119	212	1	+	+	NUM
ma-119	212	2	d(f	d(f	NOUN
ma-119	212	3	)	)	PUNCT
ma-119	212	4	g	g	NOUN
ma-119	212	5	for	for	ADP
ma-119	212	6	all	all	DET
ma-119	212	7	f	f	PROPN
ma-119	212	8	,	,	PUNCT
ma-119	212	9	g	g	PROPN
ma-119	212	10	∈	∈	PROPN
ma-119	212	11	u.	u.	NOUN
ma-119	212	12	definition	definition	NOUN
ma-119	212	13	4	4	NUM
ma-119	212	14	(	(	PUNCT
ma-119	212	15	[	[	X
ma-119	212	16	39	39	NUM
ma-119	212	17	]	]	PUNCT
ma-119	212	18	,	,	PUNCT
ma-119	212	19	definition	definition	NOUN
ma-119	212	20	1.2	1.2	NUM
ma-119	212	21	)	)	PUNCT
ma-119	212	22	.	.	PUNCT
ma-119	213	1	the	the	DET
ma-119	213	2	maximal	maximal	ADJ
ma-119	213	3	numerical	numerical	ADJ
ma-119	213	4	range	range	NOUN
ma-119	213	5	of	of	ADP
ma-119	213	6	an	an	DET
ma-119	213	7	operator	operator	NOUN
ma-119	213	8	s	s	VERB
ma-119	213	9	is	be	AUX
ma-119	213	10	defined	define	VERB
ma-119	213	11	by	by	ADP
ma-119	213	12	:	:	PUNCT
ma-119	213	13	w0(s	w0(s	X
ma-119	213	14	)	)	PUNCT
ma-119	213	15	=	=	SYM
ma-119	213	16	{	{	PUNCT
ma-119	213	17	β	β	X
ma-119	213	18	:	:	PUNCT
ma-119	213	19	〈	〈	PROPN
ma-119	213	20	st	st	PROPN
ma-119	213	21	,	,	PUNCT
ma-119	213	22	t	t	PROPN
ma-119	213	23	〉	〉	NUM
ma-119	213	24	→	→	SYM
ma-119	213	25	β	β	NOUN
ma-119	213	26	,	,	PUNCT
ma-119	213	27	where	where	SCONJ
ma-119	213	28	‖t‖	‖t‖	ADP
ma-119	213	29	=	=	SYM
ma-119	213	30	1	1	NUM
ma-119	213	31	and	and	CCONJ
ma-119	213	32	‖st‖	‖st‖	PROPN
ma-119	213	33	→	→	SYM
ma-119	213	34	‖s‖	‖s‖	PROPN
ma-119	213	35	}	}	PUNCT
ma-119	213	36	.	.	PUNCT
ma-119	214	1	definition	definition	NOUN
ma-119	214	2	5	5	NUM
ma-119	214	3	(	(	PUNCT
ma-119	214	4	[	[	X
ma-119	214	5	35	35	NUM
ma-119	214	6	]	]	PUNCT
ma-119	214	7	,	,	PUNCT
ma-119	214	8	definition	definition	NOUN
ma-119	214	9	2.1	2.1	NUM
ma-119	214	10	)	)	PUNCT
ma-119	214	11	.	.	PUNCT
ma-119	215	1	an	an	DET
ma-119	215	2	operator	operator	NOUN
ma-119	215	3	k	k	PROPN
ma-119	215	4	is	be	AUX
ma-119	215	5	norm	norm	NOUN
ma-119	215	6	-	-	PUNCT
ma-119	215	7	attainable	attainable	ADJ
ma-119	215	8	if	if	SCONJ
ma-119	215	9	t	t	PROPN
ma-119	215	10	∈	∈	PROPN
ma-119	215	11	h	h	NOUN
ma-119	215	12	exists	exist	VERB
ma-119	215	13	which	which	PRON
ma-119	215	14	is	be	AUX
ma-119	215	15	a	a	DET
ma-119	215	16	unit	unit	NOUN
ma-119	215	17	vector	vector	NOUN
ma-119	215	18	such	such	ADJ
ma-119	215	19	that	that	DET
ma-119	215	20	‖kt‖	‖kt‖	PROPN
ma-119	215	21	=	=	SYM
ma-119	215	22	‖k‖.	‖k‖.	PROPN
ma-119	215	23	moreover	moreover	ADV
ma-119	215	24	,	,	PUNCT
ma-119	215	25	it	it	PRON
ma-119	215	26	is	be	AUX
ma-119	215	27	self	self	NOUN
ma-119	215	28	-	-	PUNCT
ma-119	215	29	adjoint	adjoint	NOUN
ma-119	215	30	if	if	SCONJ
ma-119	215	31	k	k	PROPN
ma-119	215	32	=	=	SYM
ma-119	215	33	k∗.	k∗.	PROPN
ma-119	215	34	3	3	X
ma-119	215	35	.	.	PUNCT
ma-119	215	36	main	main	ADJ
ma-119	215	37	results	result	NOUN
ma-119	215	38	in	in	ADP
ma-119	215	39	this	this	DET
ma-119	215	40	section	section	NOUN
ma-119	215	41	,	,	PUNCT
ma-119	215	42	we	we	PRON
ma-119	215	43	give	give	VERB
ma-119	215	44	results	result	NOUN
ma-119	215	45	on	on	ADP
ma-119	215	46	norm	norm	NOUN
ma-119	215	47	-	-	PUNCT
ma-119	215	48	attainability	attainability	NOUN
ma-119	215	49	conditions	condition	NOUN
ma-119	215	50	an	an	DET
ma-119	215	51	norm	norm	NOUN
ma-119	215	52	estimates	estimate	NOUN
ma-119	215	53	for	for	ADP
ma-119	215	54	derivations.we	derivations.we	NOUN
ma-119	215	55	begin	begin	VERB
ma-119	215	56	with	with	ADP
ma-119	215	57	the	the	DET
ma-119	215	58	following	follow	VERB
ma-119	215	59	proposition	proposition	NOUN
ma-119	215	60	.	.	PUNCT
ma-119	216	1	proposition	proposition	NOUN
ma-119	216	2	6	6	NUM
ma-119	216	3	.	.	PUNCT
ma-119	217	1	let	let	VERB
ma-119	217	2	h	h	PRON
ma-119	217	3	be	be	AUX
ma-119	217	4	a	a	DET
ma-119	217	5	complex	complex	ADJ
ma-119	217	6	hilbert	hilbert	NOUN
ma-119	217	7	space	space	NOUN
ma-119	217	8	and	and	CCONJ
ma-119	217	9	b(h	b(h	NOUN
ma-119	217	10	)	)	PUNCT
ma-119	217	11	the	the	DET
ma-119	217	12	algebra	algebra	NOUN
ma-119	217	13	of	of	ADP
ma-119	217	14	all	all	DET
ma-119	217	15	bounded	bound	VERB
ma-119	217	16	linear	linear	PROPN
ma-119	217	17	operators	operator	NOUN
ma-119	217	18	on	on	ADP
ma-119	217	19	h.	h.	PROPN
ma-119	217	20	a	a	DET
ma-119	217	21	∈	∈	PROPN
ma-119	217	22	b(h	b(h	PROPN
ma-119	217	23	)	)	PUNCT
ma-119	217	24	is	be	AUX
ma-119	217	25	norm	norm	NOUN
ma-119	217	26	-	-	PUNCT
ma-119	217	27	attainable	attainable	ADJ
ma-119	217	28	if	if	SCONJ
ma-119	217	29	and	and	CCONJ
ma-119	217	30	only	only	ADV
ma-119	217	31	if	if	SCONJ
ma-119	217	32	its	its	PRON
ma-119	217	33	adjoint	adjoint	NOUN
ma-119	217	34	a∗	a∗	PROPN
ma-119	217	35	∈	∈	PROPN
ma-119	217	36	b(h	b(h	PROPN
ma-119	217	37	)	)	PUNCT
ma-119	217	38	is	be	AUX
ma-119	217	39	normattainable	normattainable	ADJ
ma-119	217	40	.	.	PUNCT
ma-119	218	1	proof	proof	NOUN
ma-119	218	2	.	.	PUNCT
ma-119	219	1	given	give	VERB
ma-119	219	2	a	a	DET
ma-119	219	3	∈	∈	PROPN
ma-119	219	4	b(h	b(h	PROPN
ma-119	219	5	)	)	PUNCT
ma-119	219	6	is	be	AUX
ma-119	219	7	norm	norm	NOUN
ma-119	219	8	-	-	PUNCT
ma-119	219	9	attainable	attainable	ADJ
ma-119	219	10	then	then	ADV
ma-119	219	11	we	we	PRON
ma-119	219	12	need	need	VERB
ma-119	219	13	to	to	PART
ma-119	219	14	show	show	VERB
ma-119	219	15	that	that	SCONJ
ma-119	219	16	a∗	a∗	PROPN
ma-119	219	17	∈	∈	PROPN
ma-119	219	18	b(h	b(h	PROPN
ma-119	219	19	)	)	PUNCT
ma-119	219	20	is	be	AUX
ma-119	219	21	norm	norm	NOUN
ma-119	219	22	-	-	PUNCT
ma-119	219	23	attainable	attainable	ADJ
ma-119	219	24	.	.	PUNCT
ma-119	220	1	if	if	SCONJ
ma-119	220	2	a	a	DET
ma-119	220	3	∈	∈	PROPN
ma-119	220	4	b(h	b(h	PROPN
ma-119	220	5	)	)	PUNCT
ma-119	220	6	is	be	AUX
ma-119	220	7	norm	norm	NOUN
ma-119	220	8	-	-	PUNCT
ma-119	220	9	attainable	attainable	ADJ
ma-119	220	10	then	then	ADV
ma-119	220	11	by	by	ADP
ma-119	220	12	definition	definition	NOUN
ma-119	220	13	of	of	ADP
ma-119	220	14	norm	norm	NOUN
ma-119	220	15	-	-	PUNCT
ma-119	220	16	attainability	attainability	NOUN
ma-119	220	17	there	there	PRON
ma-119	220	18	exists	exist	VERB
ma-119	220	19	aunit	aunit	PROPN
ma-119	220	20	vector	vector	NOUN
ma-119	220	21	x	x	PROPN
ma-119	220	22	∈	∈	PROPN
ma-119	220	23	h	h	NOUN
ma-119	220	24	with	with	ADP
ma-119	220	25	‖x‖	‖x‖	PROPN
ma-119	220	26	=	=	SYM
ma-119	220	27	1	1	NUM
ma-119	220	28	such	such	ADJ
ma-119	220	29	that	that	SCONJ
ma-119	220	30	‖ax‖	‖ax‖	PROPN
ma-119	220	31	=	=	SYM
ma-119	221	1	‖a‖.	‖a‖.	PROPN
ma-119	221	2	that	that	ADV
ma-119	221	3	is	be	AUX
ma-119	221	4	,	,	PUNCT
ma-119	221	5	‖aa∗x‖	‖aa∗x‖	ADJ
ma-119	221	6	=	=	PUNCT
ma-119	221	7	‖a2x‖.	‖a2x‖.	PUNCT
ma-119	221	8	let	let	VERB
ma-119	221	9	η	η	PROPN
ma-119	221	10	=	=	PROPN
ma-119	221	11	ax	ax	NOUN
ma-119	221	12	‖a‖	‖a‖	PROPN
ma-119	221	13	,	,	PUNCT
ma-119	221	14	then	then	ADV
ma-119	221	15	η	η	PROPN
ma-119	221	16	is	be	AUX
ma-119	221	17	a	a	DET
ma-119	221	18	unit	unit	NOUN
ma-119	221	19	vector	vector	NOUN
ma-119	221	20	such	such	ADJ
ma-119	221	21	that	that	SCONJ
ma-119	221	22	‖η‖	‖η‖	NOUN
ma-119	222	1	=	=	PUNCT
ma-119	222	2	1	1	NUM
ma-119	222	3	this	this	PRON
ma-119	222	4	implies	imply	VERB
ma-119	222	5	that	that	SCONJ
ma-119	222	6	‖a∗η‖	‖a∗η‖	NOUN
ma-119	222	7	=	=	SYM
ma-119	222	8	‖a‖	‖a‖	PROPN
ma-119	222	9	=	=	PUNCT
ma-119	223	1	‖a∗‖.	‖a∗‖.	PRON
ma-119	223	2	hence	hence	ADV
ma-119	223	3	,	,	PUNCT
ma-119	223	4	a∗	a∗	PROPN
ma-119	223	5	isnorm	isnorm	NOUN
ma-119	223	6	-	-	PUNCT
ma-119	223	7	attainable	attainable	ADJ
ma-119	223	8	.	.	PUNCT
ma-119	224	1	�	�	PROPN
ma-119	224	2	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	NUM
ma-119	224	3	eur	eur	NOUN
ma-119	224	4	.	.	PUNCT
ma-119	225	1	j.	j.	PROPN
ma-119	225	2	math	math	PROPN
ma-119	225	3	.	.	PUNCT
ma-119	226	1	anal	anal	PROPN
ma-119	226	2	.	.	PUNCT
ma-119	227	1	10.28924	10.28924	NUM
ma-119	227	2	/	/	SYM
ma-119	227	3	ada	ada	PROPN
ma-119	227	4	/	/	SYM
ma-119	227	5	ma.3.9	ma.3.9	PROPN
ma-119	227	6	8the	8the	DET
ma-119	227	7	next	next	ADJ
ma-119	227	8	result	result	NOUN
ma-119	227	9	gives	give	VERB
ma-119	227	10	norm	norm	NOUN
ma-119	227	11	-	-	PUNCT
ma-119	227	12	attainability	attainability	NOUN
ma-119	227	13	conditions	condition	NOUN
ma-119	227	14	for	for	ADP
ma-119	227	15	operators	operator	NOUN
ma-119	227	16	via	via	ADP
ma-119	227	17	the	the	DET
ma-119	227	18	essential	essential	ADJ
ma-119	227	19	numerical	numerical	ADJ
ma-119	227	20	range.an	range.an	ADJ
ma-119	227	21	analogy	analogy	NOUN
ma-119	227	22	of	of	ADP
ma-119	227	23	the	the	DET
ma-119	227	24	same	same	ADJ
ma-119	227	25	can	can	AUX
ma-119	227	26	be	be	AUX
ma-119	227	27	found	find	VERB
ma-119	227	28	in	in	ADP
ma-119	227	29	[	[	X
ma-119	227	30	37	37	NUM
ma-119	227	31	]	]	PUNCT
ma-119	227	32	.	.	PUNCT
ma-119	228	1	proposition	proposition	NOUN
ma-119	228	2	7	7	NUM
ma-119	228	3	.	.	PUNCT
ma-119	228	4	let	let	VERB
ma-119	228	5	a	a	DET
ma-119	228	6	∈	∈	PROPN
ma-119	228	7	b(h	b(h	PROPN
ma-119	228	8	)	)	PUNCT
ma-119	228	9	,	,	PUNCT
ma-119	228	10	λ	λ	PROPN
ma-119	228	11	∈	∈	PROPN
ma-119	228	12	wess(a	wess(a	NOUN
ma-119	228	13	)	)	PUNCT
ma-119	228	14	and	and	CCONJ
ma-119	228	15	η	η	PROPN
ma-119	228	16	>	>	X
ma-119	228	17	0	0	PROPN
ma-119	228	18	.	.	PUNCT
ma-119	229	1	then	then	ADV
ma-119	229	2	there	there	PRON
ma-119	229	3	exists	exist	VERB
ma-119	229	4	a0	a0	PROPN
ma-119	229	5	∈	∈	PROPN
ma-119	229	6	b(h	b(h	PROPN
ma-119	229	7	)	)	PUNCT
ma-119	230	1	such	such	ADJ
ma-119	230	2	that	that	SCONJ
ma-119	230	3	‖a‖	‖a‖	PROPN
ma-119	230	4	=	=	PUNCT
ma-119	230	5	‖a0‖	‖a0‖	ADP
ma-119	230	6	with	with	ADP
ma-119	230	7	‖a−	‖a−	PROPN
ma-119	230	8	a0‖	a0‖	PROPN
ma-119	230	9	>	>	X
ma-119	230	10	η	η	PROPN
ma-119	230	11	.	.	PROPN
ma-119	230	12	proof	proof	NOUN
ma-119	230	13	.	.	PUNCT
ma-119	231	1	see	see	VERB
ma-119	231	2	[	[	X
ma-119	231	3	37	37	NUM
ma-119	231	4	]	]	PUNCT
ma-119	231	5	for	for	ADP
ma-119	231	6	the	the	DET
ma-119	231	7	proof	proof	NOUN
ma-119	231	8	.	.	PUNCT
ma-119	232	1	�	�	PROPN
ma-119	232	2	remark	remark	VERB
ma-119	232	3	8	8	NUM
ma-119	232	4	.	.	PUNCT
ma-119	233	1	the	the	DET
ma-119	233	2	set	set	NOUN
ma-119	233	3	of	of	ADP
ma-119	233	4	all	all	DET
ma-119	233	5	norm	norm	NOUN
ma-119	233	6	-	-	PUNCT
ma-119	233	7	attainable	attainable	ADJ
ma-119	233	8	operators	operator	NOUN
ma-119	233	9	is	be	AUX
ma-119	233	10	denoted	denote	VERB
ma-119	233	11	by	by	ADP
ma-119	233	12	na(h	na(h	NOUN
ma-119	233	13	)	)	PUNCT
ma-119	233	14	,	,	PUNCT
ma-119	233	15	the	the	DET
ma-119	233	16	set	set	NOUN
ma-119	233	17	of	of	ADP
ma-119	233	18	all	all	DET
ma-119	233	19	normattainable	normattainable	ADJ
ma-119	233	20	self	self	NOUN
ma-119	233	21	adjoint	adjoint	PROPN
ma-119	233	22	operators	operator	NOUN
ma-119	233	23	is	be	AUX
ma-119	233	24	denoted	denote	VERB
ma-119	233	25	by	by	ADP
ma-119	233	26	na∗(h	na∗(h	PROPN
ma-119	233	27	)	)	PUNCT
ma-119	233	28	and	and	CCONJ
ma-119	233	29	the	the	DET
ma-119	233	30	set	set	NOUN
ma-119	233	31	of	of	ADP
ma-119	233	32	all	all	DET
ma-119	233	33	norm	norm	NOUN
ma-119	233	34	-	-	PUNCT
ma-119	233	35	attainable	attainable	ADJ
ma-119	233	36	elementary	elementary	ADJ
ma-119	233	37	operators	operator	NOUN
ma-119	233	38	is	be	AUX
ma-119	233	39	denoted	denote	VERB
ma-119	233	40	by	by	ADP
ma-119	233	41	ena[b(h	ena[b(h	NOUN
ma-119	233	42	)	)	PUNCT
ma-119	233	43	]	]	PUNCT
ma-119	233	44	.	.	PUNCT
ma-119	234	1	at	at	ADP
ma-119	234	2	this	this	DET
ma-119	234	3	point	point	NOUN
ma-119	234	4	,	,	PUNCT
ma-119	234	5	we	we	PRON
ma-119	234	6	consider	consider	VERB
ma-119	234	7	norm	norm	NOUN
ma-119	234	8	-	-	PUNCT
ma-119	234	9	attainability	attainability	NOUN
ma-119	234	10	in	in	ADP
ma-119	234	11	a	a	DET
ma-119	234	12	general	general	NOUN
ma-119	234	13	set	set	VERB
ma-119	234	14	up	up	ADP
ma-119	234	15	.	.	PUNCT
ma-119	235	1	we	we	PRON
ma-119	235	2	begin	begin	VERB
ma-119	235	3	with	with	ADP
ma-119	235	4	the	the	DET
ma-119	235	5	followingproposition	followingproposition	NOUN
ma-119	235	6	.	.	PUNCT
ma-119	236	1	proposition	proposition	NOUN
ma-119	236	2	9	9	NUM
ma-119	236	3	.	.	PUNCT
ma-119	237	1	let	let	VERB
ma-119	237	2	d	d	PRON
ma-119	237	3	be	be	AUX
ma-119	237	4	the	the	DET
ma-119	237	5	unit	unit	NOUN
ma-119	237	6	disc	disc	NOUN
ma-119	237	7	of	of	ADP
ma-119	237	8	a	a	DET
ma-119	237	9	complex	complex	ADJ
ma-119	237	10	hilbert	hilbert	NOUN
ma-119	237	11	space	space	NOUN
ma-119	237	12	h	h	NOUN
ma-119	237	13	and	and	CCONJ
ma-119	237	14	a	a	DET
ma-119	237	15	:	:	PUNCT
ma-119	237	16	h	h	NOUN
ma-119	237	17	→	→	SYM
ma-119	237	18	h	h	NOUN
ma-119	237	19	be	be	AUX
ma-119	237	20	compact	compact	ADJ
ma-119	237	21	and	and	CCONJ
ma-119	237	22	self	self	NOUN
ma-119	237	23	adjoint	adjoint	NOUN
ma-119	237	24	.	.	PUNCT
ma-119	238	1	then	then	ADV
ma-119	238	2	there	there	PRON
ma-119	238	3	exists	exist	VERB
ma-119	238	4	x	x	X
ma-119	238	5	∈	∈	PROPN
ma-119	238	6	d	d	X
ma-119	238	7	such	such	ADJ
ma-119	238	8	that	that	DET
ma-119	238	9	‖ax‖	‖ax‖	PROPN
ma-119	238	10	=	=	SYM
ma-119	238	11	‖a‖.	‖a‖.	PROPN
ma-119	238	12	proof	proof	NOUN
ma-119	238	13	.	.	PUNCT
ma-119	239	1	by	by	ADP
ma-119	239	2	the	the	DET
ma-119	239	3	definition	definition	NOUN
ma-119	239	4	of	of	ADP
ma-119	239	5	usual	usual	ADJ
ma-119	239	6	norm	norm	NOUN
ma-119	239	7	,	,	PUNCT
ma-119	239	8	we	we	PRON
ma-119	239	9	have	have	VERB
ma-119	239	10	‖a‖	‖a‖	NOUN
ma-119	239	11	=	=	SYM
ma-119	239	12	supx∈d	supx∈d	ADP
ma-119	239	13	‖ax‖.	‖ax‖.	VERB
ma-119	239	14	so	so	ADV
ma-119	239	15	,	,	PUNCT
ma-119	239	16	there	there	PRON
ma-119	239	17	exists	exist	VERB
ma-119	239	18	a	a	DET
ma-119	239	19	sequence	sequence	NOUN
ma-119	239	20	x1	x1	PROPN
ma-119	239	21	,	,	PUNCT
ma-119	239	22	x2	x2	PROPN
ma-119	239	23	,	,	PUNCT
ma-119	239	24	...	...	PUNCT
ma-119	239	25	,	,	PUNCT
ma-119	239	26	xn	xn	PROPN
ma-119	240	1	in	in	ADP
ma-119	240	2	d	d	PROPN
ma-119	240	3	such	such	ADJ
ma-119	240	4	that	that	DET
ma-119	240	5	‖axn‖	‖axn‖	NOUN
ma-119	240	6	=	=	SYM
ma-119	240	7	‖a‖.	‖a‖.	PROPN
ma-119	240	8	but	but	CCONJ
ma-119	240	9	a	a	PRON
ma-119	240	10	is	be	AUX
ma-119	240	11	compact	compact	ADJ
ma-119	240	12	so	so	ADV
ma-119	240	13	let	let	VERB
ma-119	240	14	y0	y0	PROPN
ma-119	240	15	=	=	SYM
ma-119	240	16	limn→∞	limn→∞	PRON
ma-119	240	17	axn	axn	PROPN
ma-119	240	18	exist	exist	VERB
ma-119	240	19	in	in	ADP
ma-119	240	20	h.	h.	PROPN
ma-119	240	21	suppose	suppose	VERB
ma-119	240	22	y	y	PROPN
ma-119	240	23	=	=	PUNCT
ma-119	240	24	span{x1	span{x1	PROPN
ma-119	240	25	,	,	PUNCT
ma-119	240	26	x2	x2	PROPN
ma-119	240	27	}	}	PUNCT
ma-119	240	28	,	,	PUNCT
ma-119	240	29	then	then	ADV
ma-119	240	30	it	it	PRON
ma-119	240	31	is	be	AUX
ma-119	240	32	a	a	DET
ma-119	240	33	closed	closed	ADJ
ma-119	240	34	subspace	subspace	NOUN
ma-119	240	35	of	of	ADP
ma-119	240	36	h.	h.	PROPN
ma-119	240	37	if	if	SCONJ
ma-119	240	38	we	we	PRON
ma-119	240	39	pick	pick	VERB
ma-119	240	40	a	a	DET
ma-119	240	41	subsequence	subsequence	NOUN
ma-119	240	42	xnkof	xnkof	PROPN
ma-119	240	43	xn	xn	PROPN
ma-119	240	44	,	,	PUNCT
ma-119	240	45	then	then	ADV
ma-119	240	46	it	it	PRON
ma-119	240	47	converges	converge	VERB
ma-119	240	48	weakly	weakly	ADV
ma-119	240	49	to	to	ADP
ma-119	240	50	x	x	PUNCT
ma-119	240	51	and	and	CCONJ
ma-119	240	52	we	we	PRON
ma-119	240	53	have	have	AUX
ma-119	240	54	done	do	VERB
ma-119	240	55	〈	〈	PROPN
ma-119	240	56	x	x	X
ma-119	240	57	,	,	PUNCT
ma-119	240	58	x	x	NOUN
ma-119	240	59	〉	〉	NOUN
ma-119	240	60	=	=	SYM
ma-119	240	61	limk→∞〈xnk	limk→∞〈xnk	NOUN
ma-119	240	62	,	,	PUNCT
ma-119	240	63	x	x	X
ma-119	240	64	〉	〉	NUM
ma-119	240	65	and	and	CCONJ
ma-119	240	66	|〈xnk	|〈xnk	PROPN
ma-119	240	67	,	,	PUNCT
ma-119	240	68	x〉|	x〉|	PROPN
ma-119	240	69	≤	≤	NUM
ma-119	240	70	‖xnk‖‖x‖	‖xnk‖‖x‖	PUNCT
ma-119	240	71	=	=	PUNCT
ma-119	240	72	1	1	NUM
ma-119	240	73	for	for	ADP
ma-119	240	74	all	all	DET
ma-119	240	75	k	k	PROPN
ma-119	240	76	.	.	PUNCT
ma-119	241	1	therefore	therefore	ADV
ma-119	241	2	,	,	PUNCT
ma-119	241	3	‖x‖	‖x‖	VERB
ma-119	241	4	≤	≤	ADV
ma-119	241	5	1	1	NUM
ma-119	242	1	but	but	CCONJ
ma-119	242	2	we	we	PRON
ma-119	242	3	can	can	AUX
ma-119	242	4	not	not	PART
ma-119	242	5	have	have	VERB
ma-119	242	6	‖x‖	‖x‖	PROPN
ma-119	242	7	<	<	X
ma-119	242	8	1	1	NUM
ma-119	242	9	since	since	SCONJ
ma-119	242	10	then	then	ADV
ma-119	242	11	‖ax‖	‖ax‖	ADJ
ma-119	242	12	=	=	PUNCT
ma-119	242	13	‖a‖‖x‖	‖a‖‖x‖	PROPN
ma-119	242	14	<	<	X
ma-119	242	15	‖t‖	‖t‖	NOUN
ma-119	242	16	which	which	PRON
ma-119	242	17	is	be	AUX
ma-119	242	18	a	a	DET
ma-119	242	19	contradiction	contradiction	NOUN
ma-119	242	20	.	.	PUNCT
ma-119	243	1	thus	thus	ADV
ma-119	243	2	,	,	PUNCT
ma-119	243	3	‖x‖	‖x‖	PROPN
ma-119	243	4	=	=	SYM
ma-119	243	5	1	1	NUM
ma-119	243	6	i.e	i.e	PROPN
ma-119	243	7	x	x	SYM
ma-119	243	8	∈	∈	PROPN
ma-119	243	9	d.	d.	PROPN
ma-119	243	10	hence	hence	ADV
ma-119	243	11	,	,	PUNCT
ma-119	243	12	the	the	DET
ma-119	243	13	existence	existence	NOUN
ma-119	243	14	of	of	ADP
ma-119	243	15	x	x	SYM
ma-119	243	16	isshown	isshown	ADJ
ma-119	243	17	and	and	CCONJ
ma-119	243	18	thus	thus	ADV
ma-119	243	19	completes	complete	VERB
ma-119	243	20	the	the	DET
ma-119	243	21	proof	proof	NOUN
ma-119	243	22	.	.	PUNCT
ma-119	244	1	�	�	PROPN
ma-119	244	2	at	at	ADP
ma-119	244	3	this	this	DET
ma-119	244	4	point	point	NOUN
ma-119	244	5	,	,	PUNCT
ma-119	244	6	we	we	PRON
ma-119	244	7	consider	consider	VERB
ma-119	244	8	q	q	ADJ
ma-119	244	9	-	-	PUNCT
ma-119	244	10	normality	normality	NOUN
ma-119	244	11	and	and	CCONJ
ma-119	244	12	q	q	ADJ
ma-119	244	13	-	-	PUNCT
ma-119	244	14	norm	norm	NOUN
ma-119	244	15	-	-	PUNCT
ma-119	244	16	attainability	attainability	NOUN
ma-119	244	17	.	.	PUNCT
ma-119	245	1	lemma	lemma	PROPN
ma-119	245	2	10	10	NUM
ma-119	245	3	.	.	PUNCT
ma-119	246	1	let	let	VERB
ma-119	246	2	a	a	DET
ma-119	246	3	∈	∈	PROPN
ma-119	246	4	na(h	na(h	NOUN
ma-119	246	5	)	)	PUNCT
ma-119	246	6	then	then	ADV
ma-119	246	7	a	a	PRON
ma-119	246	8	is	be	AUX
ma-119	246	9	q	q	ADJ
ma-119	246	10	-	-	PUNCT
ma-119	246	11	norm	norm	NOUN
ma-119	246	12	-	-	PUNCT
ma-119	246	13	attainable	attainable	ADJ
ma-119	246	14	if	if	SCONJ
ma-119	246	15	it	it	PRON
ma-119	246	16	is	be	AUX
ma-119	246	17	q	q	ADJ
ma-119	246	18	-	-	ADJ
ma-119	246	19	normal	normal	ADJ
ma-119	246	20	.	.	PUNCT
ma-119	247	1	proof	proof	NOUN
ma-119	247	2	.	.	PUNCT
ma-119	248	1	let	let	VERB
ma-119	248	2	a	a	DET
ma-119	248	3	∈	∈	PROPN
ma-119	248	4	na(h	na(h	NOUN
ma-119	248	5	)	)	PUNCT
ma-119	248	6	be	be	AUX
ma-119	248	7	q	q	ADJ
ma-119	248	8	-	-	ADJ
ma-119	248	9	normal	normal	ADJ
ma-119	248	10	i.e	i.e	PRON
ma-119	248	11	aqa∗	aqa∗	PROPN
ma-119	248	12	=	=	SYM
ma-119	248	13	a∗aq	a∗aq	X
ma-119	248	14	.	.	PUNCT
ma-119	249	1	raising	raise	VERB
ma-119	249	2	a∗	a∗	NOUN
ma-119	249	3	to	to	ADP
ma-119	249	4	power	power	NOUN
ma-119	249	5	q	q	PROPN
ma-119	249	6	and	and	CCONJ
ma-119	249	7	using	use	VERB
ma-119	249	8	it	it	PRON
ma-119	249	9	toreplace	toreplace	ADJ
ma-119	249	10	a∗	a∗	NOUN
ma-119	249	11	we	we	PRON
ma-119	249	12	have	have	VERB
ma-119	249	13	aq(a∗)q	aq(a∗)q	NUM
ma-119	249	14	=	=	SYM
ma-119	249	15	(	(	PUNCT
ma-119	249	16	a∗)qaq	a∗)qaq	PROPN
ma-119	249	17	.	.	PUNCT
ma-119	250	1	this	this	PRON
ma-119	250	2	shows	show	VERB
ma-119	250	3	that	that	SCONJ
ma-119	250	4	aq	aq	PROPN
ma-119	250	5	is	be	AUX
ma-119	250	6	normal	normal	ADJ
ma-119	250	7	.	.	PUNCT
ma-119	251	1	now	now	ADV
ma-119	251	2	aqa∗	aqa∗	PROPN
ma-119	251	3	=	=	SYM
ma-119	251	4	a∗aq	a∗aq	ADJ
ma-119	251	5	byfuglede	byfuglede	NOUN
ma-119	251	6	property	property	NOUN
ma-119	251	7	.	.	PUNCT
ma-119	252	1	therefore	therefore	ADV
ma-119	252	2	,	,	PUNCT
ma-119	252	3	a	a	PRON
ma-119	252	4	is	be	AUX
ma-119	252	5	q	q	ADJ
ma-119	252	6	-	-	ADJ
ma-119	252	7	normal	normal	ADJ
ma-119	252	8	.	.	PUNCT
ma-119	253	1	however	however	ADV
ma-119	253	2	,	,	PUNCT
ma-119	253	3	a	a	DET
ma-119	253	4	∈	∈	PROPN
ma-119	253	5	na(h	na(h	NOUN
ma-119	253	6	)	)	PUNCT
ma-119	253	7	and	and	CCONJ
ma-119	253	8	aq	aq	PROPN
ma-119	253	9	is	be	AUX
ma-119	253	10	normal	normal	ADJ
ma-119	253	11	so	so	SCONJ
ma-119	253	12	it	it	PRON
ma-119	253	13	followsthat	followsthat	VERB
ma-119	253	14	there	there	PRON
ma-119	253	15	exists	exist	VERB
ma-119	253	16	a	a	DET
ma-119	253	17	unit	unit	NOUN
ma-119	253	18	vector	vector	NOUN
ma-119	253	19	x	x	PROPN
ma-119	253	20	∈	∈	PROPN
ma-119	253	21	h	h	NOUN
ma-119	253	22	such	such	ADJ
ma-119	253	23	that	that	SCONJ
ma-119	253	24	‖aqx‖	‖aqx‖	PROPN
ma-119	253	25	=	=	SYM
ma-119	253	26	‖aq‖	‖aq‖	PROPN
ma-119	253	27	,	,	PUNCT
ma-119	253	28	for	for	ADP
ma-119	253	29	any	any	DET
ma-119	253	30	q	q	PROPN
ma-119	253	31	∈	∈	PROPN
ma-119	253	32	n.	n.	NOUN
ma-119	253	33	hence	hence	ADV
ma-119	253	34	,	,	PUNCT
ma-119	253	35	aq	aq	X
ma-119	253	36	isnorm	isnorm	NOUN
ma-119	253	37	-	-	PUNCT
ma-119	253	38	attainable	attainable	ADJ
ma-119	253	39	.	.	PUNCT
ma-119	254	1	�	�	PROPN
ma-119	254	2	remark	remark	VERB
ma-119	254	3	11	11	NUM
ma-119	254	4	.	.	PUNCT
ma-119	255	1	every	every	DET
ma-119	255	2	norm	norm	NOUN
ma-119	255	3	-	-	PUNCT
ma-119	255	4	attainable	attainable	ADJ
ma-119	255	5	operator	operator	NOUN
ma-119	255	6	and	and	CCONJ
ma-119	255	7	every	every	DET
ma-119	255	8	self	self	NOUN
ma-119	255	9	adjoint	adjoint	NOUN
ma-119	255	10	operator	operator	NOUN
ma-119	255	11	is	be	AUX
ma-119	255	12	q	q	ADJ
ma-119	255	13	-	-	PUNCT
ma-119	255	14	norm	norm	NOUN
ma-119	255	15	-	-	PUNCT
ma-119	255	16	attainable	attainable	ADJ
ma-119	255	17	and	and	CCONJ
ma-119	255	18	q	q	NOUN
ma-119	255	19	-	-	ADJ
ma-119	255	20	normal	normal	ADJ
ma-119	255	21	for	for	ADP
ma-119	255	22	any	any	DET
ma-119	255	23	q	q	PROPN
ma-119	255	24	∈	∈	PROPN
ma-119	255	25	n.	n.	NOUN
ma-119	255	26	however	however	ADV
ma-119	255	27	,	,	PUNCT
ma-119	255	28	the	the	DET
ma-119	255	29	converse	converse	NOUN
ma-119	255	30	need	need	AUX
ma-119	255	31	not	not	PART
ma-119	255	32	be	be	AUX
ma-119	255	33	true	true	ADJ
ma-119	255	34	in	in	ADP
ma-119	255	35	general	general	ADJ
ma-119	255	36	see	see	VERB
ma-119	255	37	[	[	X
ma-119	255	38	66	66	NUM
ma-119	255	39	]	]	PUNCT
ma-119	255	40	.	.	PUNCT
ma-119	256	1	lemma	lemma	PROPN
ma-119	256	2	12	12	NUM
ma-119	256	3	.	.	PUNCT
ma-119	257	1	let	let	AUX
ma-119	257	2	naq(h	naq(h	PROPN
ma-119	257	3	)	)	PUNCT
ma-119	257	4	be	be	VERB
ma-119	257	5	the	the	DET
ma-119	257	6	set	set	NOUN
ma-119	257	7	of	of	ADP
ma-119	257	8	all	all	DET
ma-119	257	9	q	q	ADJ
ma-119	257	10	-	-	PUNCT
ma-119	257	11	norm	norm	NOUN
ma-119	257	12	-	-	PUNCT
ma-119	257	13	attainable	attainable	ADJ
ma-119	257	14	operators	operator	NOUN
ma-119	257	15	on	on	ADP
ma-119	257	16	h.	h.	PROPN
ma-119	257	17	then	then	ADV
ma-119	257	18	naq(h)is	naq(h)is	PROPN
ma-119	257	19	a	a	DET
ma-119	257	20	closed	closed	ADJ
ma-119	257	21	subset	subset	NOUN
ma-119	257	22	of	of	ADP
ma-119	257	23	na(h	na(h	NOUN
ma-119	257	24	)	)	PUNCT
ma-119	257	25	which	which	PRON
ma-119	257	26	is	be	AUX
ma-119	257	27	algebraic	algebraic	ADJ
ma-119	257	28	if	if	SCONJ
ma-119	257	29	and	and	CCONJ
ma-119	257	30	only	only	ADV
ma-119	257	31	if	if	SCONJ
ma-119	257	32	for	for	ADP
ma-119	257	33	any	any	DET
ma-119	257	34	a	a	DET
ma-119	257	35	∈	∈	PROPN
ma-119	257	36	na(h	na(h	NOUN
ma-119	257	37	)	)	PUNCT
ma-119	257	38	,	,	PUNCT
ma-119	257	39	a	a	PRON
ma-119	257	40	is	be	AUX
ma-119	257	41	q	q	ADJ
ma-119	257	42	-	-	ADJ
ma-119	257	43	normal	normal	ADJ
ma-119	257	44	.	.	PUNCT
ma-119	258	1	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	NUM
ma-119	258	2	eur	eur	NOUN
ma-119	258	3	.	.	PUNCT
ma-119	259	1	j.	j.	PROPN
ma-119	259	2	math	math	PROPN
ma-119	259	3	.	.	PUNCT
ma-119	260	1	anal	anal	PROPN
ma-119	260	2	.	.	PUNCT
ma-119	261	1	10.28924	10.28924	NUM
ma-119	261	2	/	/	SYM
ma-119	261	3	ada	ada	PROPN
ma-119	261	4	/	/	SYM
ma-119	261	5	ma.3.9	ma.3.9	PROPN
ma-119	261	6	9	9	NUM
ma-119	261	7	proof	proof	NOUN
ma-119	261	8	.	.	PUNCT
ma-119	262	1	let	let	VERB
ma-119	262	2	a	a	DET
ma-119	262	3	be	be	AUX
ma-119	262	4	q	q	ADJ
ma-119	262	5	-	-	ADJ
ma-119	262	6	normal	normal	ADJ
ma-119	262	7	and	and	CCONJ
ma-119	262	8	pick	pick	VERB
ma-119	262	9	λ	λ	PROPN
ma-119	262	10	∈	∈	PROPN
ma-119	262	11	k.	k.	NOUN
ma-119	262	12	by	by	ADP
ma-119	262	13	premultiplying	premultiplye	VERB
ma-119	262	14	by	by	ADP
ma-119	262	15	λ	λ	PROPN
ma-119	262	16	and	and	CCONJ
ma-119	262	17	postmultiplying	postmultiplye	VERB
ma-119	262	18	by	by	ADP
ma-119	262	19	q	q	PROPN
ma-119	262	20	asa	asa	PROPN
ma-119	262	21	power	power	NOUN
ma-119	262	22	on	on	ADP
ma-119	262	23	the	the	DET
ma-119	262	24	normal	normal	ADJ
ma-119	262	25	a	a	PRON
ma-119	262	26	we	we	PRON
ma-119	262	27	have	have	VERB
ma-119	262	28	(	(	PUNCT
ma-119	262	29	λa)q(λa)∗	λa)q(λa)∗	NOUN
ma-119	262	30	=	=	SYM
ma-119	262	31	(	(	PUNCT
ma-119	262	32	λa)∗(λa)q	λa)∗(λa)q	X
ma-119	262	33	.	.	PUNCT
ma-119	263	1	this	this	PRON
ma-119	263	2	proves	prove	VERB
ma-119	263	3	the	the	DET
ma-119	263	4	normality	normality	NOUN
ma-119	263	5	of	of	ADP
ma-119	263	6	λa.now	λa.now	PROPN
ma-119	263	7	if	if	SCONJ
ma-119	263	8	a	a	DET
ma-119	263	9	∈	∈	PROPN
ma-119	263	10	na(h	na(h	NOUN
ma-119	263	11	)	)	PUNCT
ma-119	263	12	then	then	ADV
ma-119	263	13	the	the	DET
ma-119	263	14	converse	converse	NOUN
ma-119	263	15	is	be	AUX
ma-119	263	16	true	true	ADJ
ma-119	263	17	if	if	SCONJ
ma-119	263	18	we	we	PRON
ma-119	263	19	take	take	VERB
ma-119	263	20	limits	limit	NOUN
ma-119	263	21	over	over	ADP
ma-119	263	22	a	a	DET
ma-119	263	23	sequence	sequence	NOUN
ma-119	263	24	of	of	ADP
ma-119	263	25	vectors	vector	NOUN
ma-119	263	26	in	in	ADP
ma-119	263	27	h	h	NOUN
ma-119	263	28	andalso	andalso	ADV
ma-119	263	29	by	by	ADP
ma-119	263	30	proposition	proposition	NOUN
ma-119	263	31	9	9	NUM
ma-119	263	32	.	.	PUNCT
ma-119	264	1	therefore	therefore	ADV
ma-119	264	2	,	,	PUNCT
ma-119	264	3	a	a	PRON
ma-119	264	4	is	be	AUX
ma-119	264	5	a	a	DET
ma-119	264	6	q	q	NOUN
ma-119	264	7	-	-	ADJ
ma-119	264	8	normal	normal	ADJ
ma-119	264	9	.	.	PUNCT
ma-119	265	1	�	�	PROPN
ma-119	265	2	theorem	theorem	VERB
ma-119	265	3	13	13	NUM
ma-119	265	4	.	.	PUNCT
ma-119	266	1	let	let	VERB
ma-119	266	2	a	a	DET
ma-119	266	3	∈	∈	PROPN
ma-119	266	4	naq(h	naq(h	PROPN
ma-119	266	5	)	)	PUNCT
ma-119	266	6	.	.	PUNCT
ma-119	267	1	then	then	ADV
ma-119	267	2	the	the	DET
ma-119	267	3	following	follow	VERB
ma-119	267	4	conditions	condition	NOUN
ma-119	267	5	are	be	AUX
ma-119	267	6	true	true	ADJ
ma-119	267	7	.	.	PUNCT
ma-119	268	1	(	(	PUNCT
ma-119	268	2	i	i	NOUN
ma-119	268	3	)	)	PUNCT
ma-119	268	4	.	.	PUNCT
ma-119	269	1	a∗	a∗	PROPN
ma-119	269	2	is	be	AUX
ma-119	269	3	q	q	ADJ
ma-119	269	4	-	-	PUNCT
ma-119	269	5	norm	norm	NOUN
ma-119	269	6	-	-	PUNCT
ma-119	269	7	attainable.(ii	attainable.(ii	NOUN
ma-119	269	8	)	)	PUNCT
ma-119	269	9	.	.	PUNCT
ma-119	270	1	v	v	X
ma-119	270	2	av	av	PROPN
ma-119	270	3	∗	∗	NOUN
ma-119	270	4	is	be	AUX
ma-119	270	5	q	q	ADJ
ma-119	270	6	-	-	ADJ
ma-119	270	7	normal	normal	ADJ
ma-119	270	8	,	,	PUNCT
ma-119	270	9	for	for	ADP
ma-119	270	10	a	a	DET
ma-119	270	11	unitary	unitary	ADJ
ma-119	270	12	operator	operator	NOUN
ma-119	270	13	v	v	ADP
ma-119	270	14	∈	∈	PROPN
ma-119	270	15	naq(h).(iii	naq(h).(iii	NOUN
ma-119	270	16	)	)	PUNCT
ma-119	270	17	.	.	PUNCT
ma-119	271	1	a−1	a−1	PROPN
ma-119	271	2	is	be	AUX
ma-119	271	3	q	q	ADJ
ma-119	271	4	-	-	PUNCT
ma-119	271	5	norm	norm	NOUN
ma-119	271	6	-	-	PUNCT
ma-119	271	7	attainable	attainable	ADJ
ma-119	271	8	if	if	SCONJ
ma-119	271	9	it	it	PRON
ma-119	271	10	exists.(iv	exists.(iv	NOUN
ma-119	271	11	)	)	PUNCT
ma-119	271	12	.	.	PUNCT
ma-119	272	1	a0	a0	NOUN
ma-119	272	2	=	=	PUNCT
ma-119	272	3	a	a	DET
ma-119	272	4	/	/	SYM
ma-119	272	5	g	g	NOUN
ma-119	272	6	is	be	AUX
ma-119	272	7	q	q	ADJ
ma-119	272	8	-	-	PUNCT
ma-119	272	9	norm	norm	NOUN
ma-119	272	10	-	-	PUNCT
ma-119	272	11	attainable	attainable	ADJ
ma-119	272	12	for	for	ADP
ma-119	272	13	some	some	DET
ma-119	272	14	g	g	NOUN
ma-119	272	15	which	which	PRON
ma-119	272	16	is	be	AUX
ma-119	272	17	a	a	DET
ma-119	272	18	uniformly	uniformly	ADV
ma-119	272	19	invariable	invariable	ADJ
ma-119	272	20	subspace	subspace	NOUN
ma-119	272	21	of	of	ADP
ma-119	272	22	hwhich	hwhich	PROPN
ma-119	272	23	reduces	reduce	VERB
ma-119	272	24	to	to	ADP
ma-119	272	25	a.(v	a.(v	NUM
ma-119	272	26	)	)	PUNCT
ma-119	272	27	.	.	PUNCT
ma-119	273	1	a0	a0	PROPN
ma-119	273	2	is	be	AUX
ma-119	273	3	uniformly	uniformly	ADV
ma-119	273	4	equivalent	equivalent	ADJ
ma-119	273	5	to	to	ADP
ma-119	273	6	a	a	DET
ma-119	273	7	implies	implie	NOUN
ma-119	273	8	a0	a0	NOUN
ma-119	273	9	is	be	AUX
ma-119	273	10	norm	norm	NOUN
ma-119	273	11	-	-	PUNCT
ma-119	273	12	attainable	attainable	ADJ
ma-119	273	13	.	.	PUNCT
ma-119	274	1	proof	proof	NOUN
ma-119	274	2	.	.	PUNCT
ma-119	275	1	(	(	PUNCT
ma-119	275	2	i	i	NOUN
ma-119	275	3	)	)	PUNCT
ma-119	275	4	.	.	PUNCT
ma-119	276	1	since	since	SCONJ
ma-119	276	2	a	a	DET
ma-119	276	3	∈	∈	PROPN
ma-119	276	4	naq(h	naq(h	PROPN
ma-119	276	5	)	)	PUNCT
ma-119	276	6	,	,	PUNCT
ma-119	276	7	then	then	ADV
ma-119	276	8	from	from	ADP
ma-119	276	9	lemma	lemma	PROPN
ma-119	276	10	10	10	NUM
ma-119	276	11	,	,	PUNCT
ma-119	276	12	aq	aq	X
ma-119	276	13	is	be	AUX
ma-119	276	14	q	q	ADJ
ma-119	276	15	-	-	PUNCT
ma-119	276	16	norm	norm	NOUN
ma-119	276	17	-	-	PUNCT
ma-119	276	18	attainable	attainable	ADJ
ma-119	276	19	and	and	CCONJ
ma-119	276	20	so	so	ADV
ma-119	276	21	(	(	PUNCT
ma-119	276	22	a∗)q	a∗)q	PROPN
ma-119	276	23	isnorm	isnorm	NOUN
ma-119	276	24	-	-	PUNCT
ma-119	276	25	attainable	attainable	ADJ
ma-119	276	26	.	.	PUNCT
ma-119	277	1	consequently	consequently	ADV
ma-119	277	2	,	,	PUNCT
ma-119	277	3	a∗	a∗	PROPN
ma-119	277	4	is	be	AUX
ma-119	277	5	q	q	ADJ
ma-119	277	6	-	-	PUNCT
ma-119	277	7	norm	norm	NOUN
ma-119	277	8	-	-	PUNCT
ma-119	277	9	attainable.(ii	attainable.(ii	NOUN
ma-119	277	10	)	)	PUNCT
ma-119	277	11	.	.	PUNCT
ma-119	278	1	since	since	SCONJ
ma-119	278	2	v	v	NOUN
ma-119	278	3	is	be	AUX
ma-119	278	4	unitary	unitary	ADJ
ma-119	278	5	then	then	ADV
ma-119	278	6	v	v	NOUN
ma-119	278	7	v	v	NOUN
ma-119	278	8	∗	∗	NOUN
ma-119	278	9	=	=	SYM
ma-119	278	10	v	v	ADP
ma-119	278	11	∗v	∗v	NOUN
ma-119	278	12	=	=	SYM
ma-119	278	13	i	i	PROPN
ma-119	278	14	,	,	PUNCT
ma-119	278	15	where	where	SCONJ
ma-119	278	16	i	i	PRON
ma-119	278	17	is	be	AUX
ma-119	278	18	the	the	DET
ma-119	278	19	identity	identity	NOUN
ma-119	278	20	operator	operator	NOUN
ma-119	278	21	.	.	PUNCT
ma-119	279	1	by	by	ADP
ma-119	279	2	definition	definition	NOUN
ma-119	279	3	ofnorm	ofnorm	ADJ
ma-119	279	4	-	-	PUNCT
ma-119	279	5	attainability	attainability	NOUN
ma-119	279	6	and	and	CCONJ
ma-119	279	7	lemma	lemma	PROPN
ma-119	279	8	10	10	NUM
ma-119	279	9	we	we	PRON
ma-119	279	10	obtain	obtain	VERB
ma-119	279	11	the	the	DET
ma-119	279	12	desired	desire	VERB
ma-119	279	13	results.(iii	results.(iii	NOUN
ma-119	279	14	)	)	PUNCT
ma-119	279	15	.	.	PUNCT
ma-119	280	1	if	if	SCONJ
ma-119	280	2	a−1	a−1	PROPN
ma-119	280	3	exists	exist	VERB
ma-119	280	4	then	then	ADV
ma-119	280	5	since	since	SCONJ
ma-119	280	6	a	a	PRON
ma-119	280	7	is	be	AUX
ma-119	280	8	q	q	ADJ
ma-119	280	9	-	-	PUNCT
ma-119	280	10	norm	norm	NOUN
ma-119	280	11	-	-	PUNCT
ma-119	280	12	attainable	attainable	ADJ
ma-119	280	13	,	,	PUNCT
ma-119	280	14	aq	aq	X
ma-119	280	15	is	be	AUX
ma-119	280	16	q	q	ADJ
ma-119	280	17	-	-	PUNCT
ma-119	280	18	norm	norm	NOUN
ma-119	280	19	-	-	PUNCT
ma-119	280	20	attainable	attainable	ADJ
ma-119	280	21	.	.	PUNCT
ma-119	281	1	now	now	ADV
ma-119	281	2	since	since	SCONJ
ma-119	281	3	a	a	PRON
ma-119	281	4	is	be	AUX
ma-119	281	5	q	q	ADJ
ma-119	281	6	-	-	PUNCT
ma-119	281	7	norm	norm	NOUN
ma-119	281	8	-	-	PUNCT
ma-119	281	9	attainable	attainable	ADJ
ma-119	281	10	then	then	ADV
ma-119	281	11	by	by	ADP
ma-119	281	12	lemma	lemma	PROPN
ma-119	281	13	10	10	NUM
ma-119	281	14	aq	aq	NOUN
ma-119	281	15	is	be	AUX
ma-119	281	16	q	q	ADJ
ma-119	281	17	-	-	PUNCT
ma-119	281	18	norm	norm	NOUN
ma-119	281	19	-	-	PUNCT
ma-119	281	20	attainable	attainable	ADJ
ma-119	281	21	.	.	PUNCT
ma-119	282	1	but	but	CCONJ
ma-119	282	2	(	(	PUNCT
ma-119	282	3	aq)−1	aq)−1	X
ma-119	282	4	=	=	X
ma-119	282	5	(	(	PUNCT
ma-119	282	6	a−1)q	a−1)q	NOUN
ma-119	282	7	is	be	AUX
ma-119	282	8	q	q	ADJ
ma-119	282	9	-	-	PUNCT
ma-119	282	10	norm	norm	NOUN
ma-119	282	11	-	-	PUNCT
ma-119	282	12	attainable	attainable	ADJ
ma-119	282	13	.	.	PUNCT
ma-119	283	1	so	so	ADV
ma-119	283	2	a−1	a−1	PROPN
ma-119	283	3	is	be	AUX
ma-119	283	4	q	q	ADJ
ma-119	283	5	-	-	PUNCT
ma-119	283	6	norm	norm	NOUN
ma-119	283	7	-	-	PUNCT
ma-119	283	8	attainable.(iv	attainable.(iv	NOUN
ma-119	283	9	)	)	PUNCT
ma-119	283	10	.	.	PUNCT
ma-119	284	1	follows	follow	VERB
ma-119	284	2	from	from	ADP
ma-119	284	3	the	the	DET
ma-119	284	4	fact	fact	NOUN
ma-119	284	5	that	that	SCONJ
ma-119	284	6	g	g	PROPN
ma-119	284	7	invariant	invariant	VERB
ma-119	284	8	under	under	ADP
ma-119	284	9	a.(v	a.(v	NUM
ma-119	284	10	)	)	PUNCT
ma-119	284	11	.	.	PUNCT
ma-119	285	1	follows	follow	VERB
ma-119	285	2	from	from	ADP
ma-119	285	3	(	(	PUNCT
ma-119	285	4	iii	iii	NOUN
ma-119	285	5	)	)	PUNCT
ma-119	285	6	since	since	SCONJ
ma-119	285	7	v	v	NOUN
ma-119	285	8	is	be	AUX
ma-119	285	9	unitary	unitary	ADJ
ma-119	285	10	.	.	PUNCT
ma-119	286	1	�	�	PROPN
ma-119	286	2	corollary	corollary	ADJ
ma-119	286	3	14	14	NUM
ma-119	286	4	.	.	PUNCT
ma-119	287	1	let	let	VERB
ma-119	287	2	aq	aq	VERB
ma-119	287	3	,	,	PUNCT
ma-119	287	4	aq0	aq0	PROPN
ma-119	287	5	∈	∈	PROPN
ma-119	287	6	naq(h	naq(h	PROPN
ma-119	287	7	)	)	PUNCT
ma-119	287	8	be	be	AUX
ma-119	287	9	commuting	commute	VERB
ma-119	287	10	operators	operator	NOUN
ma-119	287	11	,	,	PUNCT
ma-119	287	12	then	then	ADV
ma-119	287	13	a	a	PRON
ma-119	287	14	,	,	PUNCT
ma-119	287	15	a0	a0	PROPN
ma-119	287	16	∈	∈	PROPN
ma-119	287	17	naq(h	naq(h	PROPN
ma-119	287	18	)	)	PUNCT
ma-119	287	19	.	.	PUNCT
ma-119	288	1	proof	proof	NOUN
ma-119	288	2	.	.	PUNCT
ma-119	289	1	since	since	SCONJ
ma-119	289	2	aq	aq	SYM
ma-119	289	3	,	,	PUNCT
ma-119	289	4	aq0	aq0	PROPN
ma-119	289	5	∈	∈	PROPN
ma-119	289	6	naq(h	naq(h	PROPN
ma-119	289	7	)	)	PUNCT
ma-119	289	8	are	be	AUX
ma-119	289	9	commuting	commute	VERB
ma-119	289	10	then	then	ADV
ma-119	289	11	a	a	PRON
ma-119	289	12	,	,	PUNCT
ma-119	289	13	a0	a0	PROPN
ma-119	289	14	are	be	AUX
ma-119	289	15	commuting	commute	VERB
ma-119	289	16	normal	normal	ADJ
ma-119	289	17	operators	operator	NOUN
ma-119	289	18	.	.	PUNCT
ma-119	290	1	bysupraposinormality	bysupraposinormality	NOUN
ma-119	290	2	of	of	ADP
ma-119	290	3	operators	operator	NOUN
ma-119	290	4	in	in	ADP
ma-119	290	5	dense	dense	ADJ
ma-119	290	6	classes	class	NOUN
ma-119	290	7	we	we	PRON
ma-119	290	8	have	have	VERB
ma-119	290	9	a	a	DET
ma-119	290	10	,	,	PUNCT
ma-119	290	11	a0	a0	PROPN
ma-119	290	12	∈	∈	PROPN
ma-119	290	13	naq(h	naq(h	PROPN
ma-119	290	14	)	)	PUNCT
ma-119	290	15	and	and	CCONJ
ma-119	290	16	hence	hence	ADV
ma-119	290	17	are	be	AUX
ma-119	290	18	norm	norm	NOUN
ma-119	290	19	-	-	PUNCT
ma-119	290	20	attainable	attainable	ADJ
ma-119	290	21	.	.	PUNCT
ma-119	291	1	indeed	indeed	ADV
ma-119	291	2	,	,	PUNCT
ma-119	291	3	aqaq0	aqaq0	PROPN
ma-119	292	1	=	=	SYM
ma-119	292	2	(	(	PUNCT
ma-119	292	3	aa0	aa0	NOUN
ma-119	292	4	)	)	PUNCT
ma-119	292	5	q	q	NOUN
ma-119	293	1	=	=	PUNCT
ma-119	293	2	(	(	PUNCT
ma-119	293	3	a0a)q	a0a)q	X
ma-119	293	4	which	which	PRON
ma-119	293	5	is	be	AUX
ma-119	293	6	normal	normal	ADJ
ma-119	293	7	and	and	CCONJ
ma-119	293	8	norm	norm	NOUN
ma-119	293	9	-	-	PUNCT
ma-119	293	10	attainable	attainable	ADJ
ma-119	293	11	.	.	PUNCT
ma-119	294	1	hence	hence	ADV
ma-119	294	2	,	,	PUNCT
ma-119	294	3	a	a	DET
ma-119	294	4	,	,	PUNCT
ma-119	294	5	a0	a0	PROPN
ma-119	294	6	∈	∈	PROPN
ma-119	294	7	naq(h	naq(h	PROPN
ma-119	294	8	)	)	PUNCT
ma-119	294	9	.	.	PUNCT
ma-119	295	1	�	�	PROPN
ma-119	295	2	remark	remark	VERB
ma-119	295	3	15	15	NUM
ma-119	295	4	.	.	PUNCT
ma-119	296	1	not	not	PART
ma-119	296	2	all	all	PRON
ma-119	296	3	q	q	ADJ
ma-119	296	4	-	-	PUNCT
ma-119	296	5	norm	norm	NOUN
ma-119	296	6	-	-	PUNCT
ma-119	296	7	attainable	attainable	ADJ
ma-119	296	8	operators	operator	NOUN
ma-119	296	9	are	be	AUX
ma-119	296	10	q	q	ADJ
ma-119	296	11	-	-	ADJ
ma-119	296	12	normal	normal	ADJ
ma-119	296	13	.	.	PUNCT
ma-119	297	1	thus	thus	ADV
ma-119	297	2	,	,	PUNCT
ma-119	297	3	the	the	DET
ma-119	297	4	following	follow	VERB
ma-119	297	5	example	example	NOUN
ma-119	297	6	shows	show	VERB
ma-119	297	7	that	that	SCONJ
ma-119	297	8	the	the	DET
ma-119	297	9	two	two	NUM
ma-119	297	10	commuting	commute	VERB
ma-119	297	11	q	q	ADJ
ma-119	297	12	-	-	ADJ
ma-119	297	13	normal	normal	ADJ
ma-119	297	14	operators	operator	NOUN
ma-119	297	15	need	need	AUX
ma-119	297	16	not	not	PART
ma-119	297	17	be	be	AUX
ma-119	297	18	q	q	ADJ
ma-119	297	19	-	-	ADJ
ma-119	297	20	normal	normal	ADJ
ma-119	297	21	.	.	PUNCT
ma-119	297	22	example	example	NOUN
ma-119	298	1	16	16	NUM
ma-119	298	2	.	.	PUNCT
ma-119	299	1	let	let	VERB
ma-119	299	2	a	a	DET
ma-119	299	3	=	=	X
ma-119	299	4	[	[	PUNCT
ma-119	299	5	1	1	NUM
ma-119	299	6	0	0	NUM
ma-119	299	7	0	0	NUM
ma-119	299	8	1	1	NUM
ma-119	299	9	]	]	PUNCT
ma-119	299	10	and	and	CCONJ
ma-119	299	11	a0	a0	NOUN
ma-119	299	12	=	=	PUNCT
ma-119	300	1	[	[	PUNCT
ma-119	300	2	0	0	NUM
ma-119	300	3	1	1	NUM
ma-119	300	4	0	0	NUM
ma-119	300	5	0	0	NUM
ma-119	300	6	]	]	PUNCT
ma-119	300	7	.	.	PUNCT
ma-119	301	1	now	now	ADV
ma-119	301	2	a+	a+	PUNCT
ma-119	301	3	a0	a0	PROPN
ma-119	301	4	=	=	PUNCT
ma-119	302	1	[	[	PUNCT
ma-119	302	2	1	1	NUM
ma-119	302	3	1	1	NUM
ma-119	302	4	0	0	NUM
ma-119	302	5	1	1	NUM
ma-119	302	6	]	]	PUNCT
ma-119	302	7	and	and	CCONJ
ma-119	302	8	(	(	PUNCT
ma-119	302	9	a+	a+	X
ma-119	302	10	a0	a0	NOUN
ma-119	302	11	)	)	PUNCT
ma-119	302	12	2	2	NUM
ma-119	302	13	=[	=[	NOUN
ma-119	302	14	1	1	NUM
ma-119	302	15	2	2	NUM
ma-119	302	16	0	0	NUM
ma-119	302	17	1	1	NUM
ma-119	302	18	]	]	PUNCT
ma-119	302	19	are	be	AUX
ma-119	302	20	not	not	PART
ma-119	302	21	normal	normal	ADJ
ma-119	302	22	.	.	PUNCT
ma-119	303	1	so	so	ADV
ma-119	303	2	a+	a+	PUNCT
ma-119	303	3	a0	a0	PROPN
ma-119	303	4	is	be	AUX
ma-119	303	5	not	not	PART
ma-119	303	6	2	2	NUM
ma-119	303	7	-	-	PUNCT
ma-119	303	8	normal	normal	ADJ
ma-119	303	9	.	.	PUNCT
ma-119	304	1	we	we	PRON
ma-119	304	2	note	note	VERB
ma-119	304	3	that	that	SCONJ
ma-119	304	4	a0	a0	PROPN
ma-119	304	5	is	be	AUX
ma-119	304	6	self	self	NOUN
ma-119	304	7	-	-	PUNCT
ma-119	304	8	adjoint	adjoint	NOUN
ma-119	304	9	.	.	PUNCT
ma-119	305	1	lemma	lemma	PROPN
ma-119	305	2	17	17	NUM
ma-119	305	3	.	.	PUNCT
ma-119	306	1	the	the	DET
ma-119	306	2	sum	sum	NOUN
ma-119	306	3	of	of	ADP
ma-119	306	4	norm	norm	NOUN
ma-119	306	5	-	-	PUNCT
ma-119	306	6	attainable	attainable	ADJ
ma-119	306	7	operators	operator	NOUN
ma-119	306	8	is	be	AUX
ma-119	306	9	norm	norm	NOUN
ma-119	306	10	-	-	PUNCT
ma-119	306	11	attainable	attainable	ADJ
ma-119	306	12	.	.	PUNCT
ma-119	307	1	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	NUM
ma-119	307	2	eur	eur	NOUN
ma-119	307	3	.	.	PUNCT
ma-119	308	1	j.	j.	PROPN
ma-119	308	2	math	math	PROPN
ma-119	308	3	.	.	PUNCT
ma-119	309	1	anal	anal	PROPN
ma-119	309	2	.	.	PUNCT
ma-119	310	1	10.28924	10.28924	NUM
ma-119	310	2	/	/	SYM
ma-119	310	3	ada	ada	PROPN
ma-119	310	4	/	/	SYM
ma-119	310	5	ma.3.9	ma.3.9	PROPN
ma-119	310	6	10	10	NUM
ma-119	310	7	proof	proof	NOUN
ma-119	310	8	.	.	PUNCT
ma-119	311	1	consider	consider	VERB
ma-119	311	2	a	a	DET
ma-119	311	3	,	,	PUNCT
ma-119	311	4	b	b	PROPN
ma-119	311	5	∈	∈	PROPN
ma-119	311	6	b(h	b(h	PROPN
ma-119	311	7	)	)	PUNCT
ma-119	311	8	.	.	PUNCT
ma-119	312	1	we	we	PRON
ma-119	312	2	need	need	VERB
ma-119	312	3	to	to	PART
ma-119	312	4	show	show	VERB
ma-119	312	5	that	that	SCONJ
ma-119	312	6	the	the	DET
ma-119	312	7	sum	sum	NOUN
ma-119	312	8	of	of	ADP
ma-119	312	9	a	a	PRON
ma-119	312	10	and	and	CCONJ
ma-119	312	11	b	b	NOUN
ma-119	312	12	is	be	AUX
ma-119	312	13	norm	norm	NOUN
ma-119	312	14	-	-	PUNCT
ma-119	312	15	attainable	attainable	ADJ
ma-119	312	16	.	.	PUNCT
ma-119	313	1	for	for	ADP
ma-119	313	2	a	a	DET
ma-119	313	3	,	,	PUNCT
ma-119	313	4	b	b	NOUN
ma-119	313	5	to	to	PART
ma-119	313	6	be	be	AUX
ma-119	313	7	norm	norm	NOUN
ma-119	313	8	-	-	PUNCT
ma-119	313	9	attainable	attainable	ADJ
ma-119	313	10	then	then	ADV
ma-119	313	11	there	there	PRON
ma-119	313	12	exists	exist	VERB
ma-119	313	13	a	a	DET
ma-119	313	14	unit	unit	NOUN
ma-119	313	15	vector	vector	NOUN
ma-119	313	16	x	x	PROPN
ma-119	313	17	∈	∈	NOUN
ma-119	313	18	h	h	NOUN
ma-119	313	19	such	such	ADJ
ma-119	314	1	that	that	SCONJ
ma-119	314	2	‖x‖	‖x‖	PROPN
ma-119	314	3	=	=	SYM
ma-119	314	4	1	1	NUM
ma-119	314	5	,	,	PUNCT
ma-119	314	6	‖(a+b)x‖	‖(a+b)x‖	PROPN
ma-119	314	7	=	=	SYM
ma-119	314	8	‖ax+bx‖	‖ax+bx‖	NOUN
ma-119	314	9	=	=	SYM
ma-119	314	10	‖a+b‖	‖a+b‖	NOUN
ma-119	314	11	=	=	SYM
ma-119	314	12	‖a‖+‖b‖.	‖a‖+‖b‖.	PROPN
ma-119	314	13	since	since	SCONJ
ma-119	314	14	‖ax+bx‖	‖ax+bx‖	NOUN
ma-119	314	15	≤	≤	X
ma-119	314	16	‖ax‖+‖bx‖	‖ax‖+‖bx‖	NOUN
ma-119	314	17	≤	≤	ADJ
ma-119	314	18	‖a‖+‖bx‖	‖a‖+‖bx‖	NOUN
ma-119	314	19	≤	≤	NOUN
ma-119	314	20	‖a‖+‖b‖then	‖a‖+‖b‖then	PROPN
ma-119	314	21	for	for	ADP
ma-119	314	22	an	an	DET
ma-119	314	23	orthonormal	orthonormal	ADJ
ma-119	314	24	sequence	sequence	NOUN
ma-119	314	25	xn	xn	PROPN
ma-119	314	26	∈	∈	PROPN
ma-119	314	27	h	h	NOUN
ma-119	314	28	we	we	PRON
ma-119	314	29	have	have	VERB
ma-119	314	30	limn→∞(‖axn	limn→∞(‖axn	ADJ
ma-119	314	31	+	+	NOUN
ma-119	314	32	bxn‖	bxn‖	ADJ
ma-119	314	33	)	)	PUNCT
ma-119	314	34	=	=	PUNCT
ma-119	315	1	‖ax	‖ax	PUNCT
ma-119	316	1	+	+	PROPN
ma-119	316	2	bx‖.	bx‖.	PROPN
ma-119	316	3	but	but	CCONJ
ma-119	316	4	since	since	SCONJ
ma-119	316	5	a	a	PRON
ma-119	316	6	and	and	CCONJ
ma-119	316	7	b	b	NOUN
ma-119	316	8	are	be	AUX
ma-119	316	9	norm	norm	ADV
ma-119	316	10	-	-	PUNCT
ma-119	316	11	attainable	attainable	ADJ
ma-119	316	12	we	we	PRON
ma-119	316	13	have	have	VERB
ma-119	316	14	‖ax+bx‖	‖ax+bx‖	NOUN
ma-119	316	15	=	=	SYM
ma-119	317	1	‖(a+b)x‖	‖(a+b)x‖	VERB
ma-119	317	2	=	=	NOUN
ma-119	317	3	‖a+b‖	‖a+b‖	NOUN
ma-119	317	4	is	be	AUX
ma-119	317	5	norm	norm	NOUN
ma-119	317	6	-	-	PUNCT
ma-119	317	7	attainable	attainable	ADJ
ma-119	317	8	.	.	PUNCT
ma-119	318	1	�	�	PROPN
ma-119	318	2	theorem	theorem	VERB
ma-119	318	3	18	18	NUM
ma-119	318	4	.	.	PUNCT
ma-119	319	1	a	a	DET
ma-119	319	2	norm	norm	NOUN
ma-119	319	3	-	-	PUNCT
ma-119	319	4	attainable	attainable	ADJ
ma-119	319	5	operator	operator	NOUN
ma-119	319	6	perturbed	perturb	VERB
ma-119	319	7	by	by	ADP
ma-119	319	8	an	an	DET
ma-119	319	9	identity	identity	NOUN
ma-119	319	10	operators	operator	NOUN
ma-119	319	11	is	be	AUX
ma-119	319	12	norm	norm	NOUN
ma-119	319	13	-	-	PUNCT
ma-119	319	14	attainable	attainable	ADJ
ma-119	319	15	.	.	PUNCT
ma-119	320	1	proof	proof	NOUN
ma-119	320	2	.	.	PUNCT
ma-119	321	1	let	let	VERB
ma-119	321	2	b	b	PROPN
ma-119	321	3	∈	∈	PROPN
ma-119	321	4	b(h	b(h	PROPN
ma-119	321	5	)	)	PUNCT
ma-119	321	6	be	be	VERB
ma-119	321	7	norm	norm	NOUN
ma-119	321	8	-	-	PUNCT
ma-119	321	9	attainable	attainable	ADJ
ma-119	321	10	.	.	PUNCT
ma-119	322	1	since	since	SCONJ
ma-119	322	2	b	b	PROPN
ma-119	322	3	is	be	AUX
ma-119	322	4	norm	norm	NOUN
ma-119	322	5	-	-	PUNCT
ma-119	322	6	attainable	attainable	ADJ
ma-119	322	7	then	then	ADV
ma-119	322	8	there	there	PRON
ma-119	322	9	exists	exist	VERB
ma-119	322	10	a	a	DET
ma-119	322	11	unitvector	unitvector	NOUN
ma-119	322	12	x0	x0	PROPN
ma-119	322	13	∈	∈	PROPN
ma-119	322	14	h	h	NOUN
ma-119	322	15	,	,	PUNCT
ma-119	322	16	an	an	DET
ma-119	322	17	identity	identity	NOUN
ma-119	322	18	i	i	PROPN
ma-119	322	19	∈	∈	PROPN
ma-119	322	20	b(h	b(h	PROPN
ma-119	322	21	)	)	PUNCT
ma-119	322	22	and	and	CCONJ
ma-119	322	23	for	for	ADP
ma-119	322	24	every	every	DET
ma-119	322	25	ε	ε	PROPN
ma-119	322	26	>	>	X
ma-119	322	27	0	0	NUM
ma-119	322	28	we	we	PRON
ma-119	322	29	have	have	VERB
ma-119	322	30	‖(bi)x0‖	‖(bi)x0‖	NOUN
ma-119	322	31	≤	≤	NOUN
ma-119	322	32	‖bix0‖	‖bix0‖	PUNCT
ma-119	323	1	+	+	CCONJ
ma-119	323	2	ε	ε	PROPN
ma-119	323	3	≤	≤	NOUN
ma-119	323	4	‖b‖‖i‖‖x0‖+	‖b‖‖i‖‖x0‖+	NUM
ma-119	323	5	ε	ε	PROPN
ma-119	323	6	.	.	PUNCT
ma-119	324	1	since	since	SCONJ
ma-119	324	2	ε	ε	PROPN
ma-119	324	3	is	be	AUX
ma-119	324	4	arbitrary	arbitrary	ADJ
ma-119	324	5	then	then	ADV
ma-119	324	6	it	it	PRON
ma-119	324	7	follows	follow	VERB
ma-119	324	8	that	that	SCONJ
ma-119	324	9	‖(bi)x0‖	‖(bi)x0‖	PROPN
ma-119	324	10	≤	≤	ADV
ma-119	324	11	‖b‖‖i‖‖x0‖	‖b‖‖i‖‖x0‖	X
ma-119	324	12	=	=	X
ma-119	324	13	‖b‖.	‖b‖.	PUNCT
ma-119	324	14	hence	hence	ADV
ma-119	324	15	,	,	PUNCT
ma-119	324	16	‖(bi)x0‖	‖(bi)x0‖	PROPN
ma-119	324	17	=	=	SYM
ma-119	324	18	‖b‖.	‖b‖.	X
ma-119	324	19	�	�	PROPN
ma-119	324	20	at	at	ADP
ma-119	324	21	this	this	DET
ma-119	324	22	point	point	NOUN
ma-119	324	23	,	,	PUNCT
ma-119	324	24	we	we	PRON
ma-119	324	25	consider	consider	VERB
ma-119	324	26	norm	norm	NOUN
ma-119	324	27	-	-	PUNCT
ma-119	324	28	attainability	attainability	NOUN
ma-119	324	29	for	for	ADP
ma-119	324	30	elementary	elementary	ADJ
ma-119	324	31	operators	operator	NOUN
ma-119	324	32	.	.	PUNCT
ma-119	325	1	we	we	PRON
ma-119	325	2	begin	begin	VERB
ma-119	325	3	with	with	ADP
ma-119	325	4	inner	inner	ADJ
ma-119	325	5	deriva	deriva	NOUN
ma-119	325	6	-	-	PUNCT
ma-119	325	7	tions	tion	NOUN
ma-119	325	8	.	.	PUNCT
ma-119	326	1	lemma	lemma	PROPN
ma-119	326	2	19	19	NUM
ma-119	326	3	.	.	PUNCT
ma-119	327	1	let	let	VERB
ma-119	327	2	δa	δa	PRON
ma-119	327	3	∈	∈	PROPN
ma-119	327	4	e	e	X
ma-119	328	1	[	[	X
ma-119	328	2	b(h	b(h	PROPN
ma-119	328	3	)	)	PUNCT
ma-119	328	4	]	]	PUNCT
ma-119	328	5	,	,	PUNCT
ma-119	328	6	then	then	ADV
ma-119	328	7	δa	δa	PROPN
ma-119	328	8	is	be	AUX
ma-119	328	9	norm	norm	NOUN
ma-119	328	10	-	-	PUNCT
ma-119	328	11	attainable	attainable	ADJ
ma-119	328	12	if	if	SCONJ
ma-119	328	13	there	there	PRON
ma-119	328	14	exists	exist	VERB
ma-119	328	15	a	a	DET
ma-119	328	16	unit	unit	NOUN
ma-119	328	17	vector	vector	NOUN
ma-119	328	18	x0	x0	PROPN
ma-119	328	19	∈	∈	PROPN
ma-119	328	20	h	h	NOUN
ma-119	328	21	,	,	PUNCT
ma-119	328	22	a	a	DET
ma-119	328	23	∈	∈	PROPN
ma-119	328	24	na(h	na(h	NOUN
ma-119	328	25	)	)	PUNCT
ma-119	328	26	and	and	CCONJ
ma-119	328	27	〈	〈	PROPN
ma-119	328	28	ax0	ax0	NOUN
ma-119	328	29	,	,	PUNCT
ma-119	328	30	x0	x0	PROPN
ma-119	328	31	〉	〉	PROPN
ma-119	328	32	∈	∈	PROPN
ma-119	328	33	wess(a	wess(a	NOUN
ma-119	328	34	)	)	PUNCT
ma-119	328	35	.	.	PUNCT
ma-119	329	1	proof	proof	NOUN
ma-119	329	2	.	.	PUNCT
ma-119	330	1	for	for	ADP
ma-119	330	2	an	an	DET
ma-119	330	3	operator	operator	NOUN
ma-119	330	4	a	a	DET
ma-119	330	5	∈	∈	PROPN
ma-119	330	6	na(h	na(h	NOUN
ma-119	330	7	)	)	PUNCT
ma-119	330	8	we	we	PRON
ma-119	330	9	know	know	VERB
ma-119	330	10	that	that	SCONJ
ma-119	330	11	an	an	DET
ma-119	330	12	operator	operator	NOUN
ma-119	330	13	is	be	AUX
ma-119	330	14	norm	norm	NOUN
ma-119	330	15	-	-	PUNCT
ma-119	330	16	attainable	attainable	ADJ
ma-119	330	17	via	via	ADP
ma-119	330	18	essentialnumerical	essentialnumerical	ADJ
ma-119	330	19	range	range	NOUN
ma-119	330	20	from	from	ADP
ma-119	330	21	proposition	proposition	NOUN
ma-119	330	22	4.2	4.2	NUM
ma-119	330	23	.	.	PUNCT
ma-119	331	1	now	now	ADV
ma-119	331	2	,	,	PUNCT
ma-119	331	3	we	we	PRON
ma-119	331	4	need	need	VERB
ma-119	331	5	to	to	PART
ma-119	331	6	show	show	VERB
ma-119	331	7	that	that	SCONJ
ma-119	331	8	δa	δa	PROPN
ma-119	331	9	∈	∈	PROPN
ma-119	331	10	e	e	X
ma-119	331	11	[	[	X
ma-119	331	12	b(h	b(h	PROPN
ma-119	331	13	)	)	PUNCT
ma-119	331	14	]	]	PUNCT
ma-119	331	15	is	be	AUX
ma-119	331	16	norm-attainable.by	norm-attainable.by	PROPN
ma-119	331	17	the	the	DET
ma-119	331	18	definition	definition	NOUN
ma-119	331	19	of	of	ADP
ma-119	331	20	inner	inner	ADJ
ma-119	331	21	derivation	derivation	NOUN
ma-119	331	22	,	,	PUNCT
ma-119	331	23	δa	δa	PROPN
ma-119	331	24	=	=	PUNCT
ma-119	331	25	ay0−y0a	ay0−y0a	NOUN
ma-119	331	26	.	.	PUNCT
ma-119	332	1	since	since	SCONJ
ma-119	332	2	a	a	PRON
ma-119	332	3	is	be	AUX
ma-119	332	4	norm	norm	NOUN
ma-119	332	5	-	-	PUNCT
ma-119	332	6	attainable	attainable	ADJ
ma-119	332	7	then	then	ADV
ma-119	332	8	there	there	PRON
ma-119	332	9	exists	exist	VERB
ma-119	332	10	aunit	aunit	PROPN
ma-119	332	11	vector	vector	NOUN
ma-119	332	12	x0	x0	PROPN
ma-119	332	13	∈	∈	PROPN
ma-119	332	14	h	h	NOUN
ma-119	333	1	such	such	ADJ
ma-119	333	2	that	that	SCONJ
ma-119	333	3	‖x0‖	‖x0‖	NOUN
ma-119	333	4	=	=	SYM
ma-119	333	5	1	1	NUM
ma-119	333	6	,	,	PUNCT
ma-119	333	7	‖ax0‖	‖ax0‖	PROPN
ma-119	333	8	=	=	SYM
ma-119	333	9	‖a‖.	‖a‖.	PROPN
ma-119	333	10	by	by	ADP
ma-119	333	11	orthogonality	orthogonality	NOUN
ma-119	333	12	let	let	VERB
ma-119	333	13	y0	y0	PRON
ma-119	333	14	satisfy	satisfy	VERB
ma-119	333	15	y0⊥{ax0	y0⊥{ax0	PROPN
ma-119	333	16	,	,	PUNCT
ma-119	333	17	x0}and	x0}and	PROPN
ma-119	334	1	a	a	DET
ma-119	334	2	contractive	contractive	ADJ
ma-119	334	3	y0	y0	NOUN
ma-119	334	4	be	be	AUX
ma-119	334	5	defined	define	VERB
ma-119	334	6	as	as	ADP
ma-119	334	7	a	a	DET
ma-119	334	8	linear	linear	ADJ
ma-119	334	9	transformation	transformation	NOUN
ma-119	334	10	y0	y0	NOUN
ma-119	334	11	:	:	PUNCT
ma-119	334	12	x0	x0	PROPN
ma-119	334	13	→	→	PUNCT
ma-119	334	14	x0	x0	PROPN
ma-119	334	15	with	with	ADP
ma-119	334	16	ax0	ax0	NOUN
ma-119	334	17	→	→	SYM
ma-119	334	18	−ax0	−ax0	NUM
ma-119	334	19	as	as	ADP
ma-119	334	20	y0	y0	PROPN
ma-119	334	21	→	→	SYM
ma-119	334	22	0	0	NUM
ma-119	334	23	.	.	PUNCT
ma-119	335	1	since	since	SCONJ
ma-119	335	2	y0	y0	PROPN
ma-119	335	3	is	be	AUX
ma-119	335	4	a	a	DET
ma-119	335	5	bounded	bounded	ADJ
ma-119	335	6	linear	linear	ADJ
ma-119	335	7	operator	operator	NOUN
ma-119	335	8	on	on	ADP
ma-119	335	9	h	h	NOUN
ma-119	335	10	,	,	PUNCT
ma-119	335	11	then	then	ADV
ma-119	335	12	by	by	ADP
ma-119	335	13	norm	norm	NOUN
ma-119	335	14	-	-	PUNCT
ma-119	335	15	attainability	attainability	NOUN
ma-119	335	16	‖y0x0‖	‖y0x0‖	X
ma-119	335	17	=	=	SYM
ma-119	335	18	‖y0‖	‖y0‖	PROPN
ma-119	335	19	=	=	SYM
ma-119	335	20	1and	1and	NUM
ma-119	335	21	‖ay0x0	‖ay0x0	NOUN
ma-119	335	22	−	−	PROPN
ma-119	335	23	y0ax0‖	y0ax0‖	PROPN
ma-119	335	24	=	=	PUNCT
ma-119	335	25	‖ax0	‖ax0	PROPN
ma-119	335	26	−	−	PROPN
ma-119	335	27	(	(	PUNCT
ma-119	335	28	−ax0)‖	−ax0)‖	NOUN
ma-119	335	29	=	=	SYM
ma-119	335	30	2‖a‖.it	2‖a‖.it	PROPN
ma-119	335	31	follows	follow	VERB
ma-119	335	32	from	from	ADP
ma-119	335	33	lemma	lemma	PROPN
ma-119	335	34	3.1	3.1	NUM
ma-119	335	35	in	in	ADP
ma-119	335	36	[	[	X
ma-119	335	37	49	49	NUM
ma-119	335	38	]	]	PUNCT
ma-119	335	39	that	that	PRON
ma-119	335	40	‖δa‖	‖δa‖	AUX
ma-119	335	41	=	=	SYM
ma-119	335	42	2‖a‖.	2‖a‖.	NUM
ma-119	335	43	by	by	ADP
ma-119	335	44	the	the	DET
ma-119	335	45	inner	inner	ADJ
ma-119	335	46	product	product	NOUN
ma-119	335	47	〈	〈	PROPN
ma-119	335	48	ax0	ax0	NOUN
ma-119	335	49	,	,	PUNCT
ma-119	335	50	x0	x0	PROPN
ma-119	335	51	〉	〉	NOUN
ma-119	335	52	=	=	SYM
ma-119	335	53	0	0	NUM
ma-119	335	54	∈	∈	PROPN
ma-119	335	55	wess(a),it	wess(a),it	PROPN
ma-119	335	56	follows	follow	VERB
ma-119	335	57	that	that	SCONJ
ma-119	335	58	‖δa‖	‖δa‖	PROPN
ma-119	335	59	=	=	SYM
ma-119	335	60	2‖a‖.	2‖a‖.	PROPN
ma-119	335	61	therefore	therefore	ADV
ma-119	335	62	,	,	PUNCT
ma-119	335	63	‖ay0	‖ay0	NOUN
ma-119	335	64	−	−	PROPN
ma-119	335	65	y0a‖	y0a‖	NOUN
ma-119	335	66	=	=	SYM
ma-119	335	67	2‖a‖	2‖a‖	NUM
ma-119	335	68	=	=	PUNCT
ma-119	335	69	‖δa‖.	‖δa‖.	ADV
ma-119	335	70	hence	hence	ADV
ma-119	335	71	,	,	PUNCT
ma-119	335	72	δa	δa	PROPN
ma-119	335	73	is	be	AUX
ma-119	335	74	norm	norm	NOUN
ma-119	335	75	-	-	PUNCT
ma-119	335	76	attainable	attainable	ADJ
ma-119	335	77	.	.	PUNCT
ma-119	336	1	�	�	PROPN
ma-119	336	2	lemma	lemma	PROPN
ma-119	336	3	20	20	NUM
ma-119	336	4	.	.	PUNCT
ma-119	337	1	let	let	VERB
ma-119	337	2	a	a	DET
ma-119	337	3	,	,	PUNCT
ma-119	337	4	a0	a0	PROPN
ma-119	337	5	∈	∈	PROPN
ma-119	337	6	b(h	b(h	PROPN
ma-119	337	7	)	)	PUNCT
ma-119	337	8	.	.	PUNCT
ma-119	338	1	if	if	SCONJ
ma-119	338	2	there	there	PRON
ma-119	338	3	exists	exist	VERB
ma-119	338	4	unit	unit	NOUN
ma-119	338	5	vectors	vector	NOUN
ma-119	338	6	y	y	PROPN
ma-119	338	7	and	and	CCONJ
ma-119	338	8	y0	y0	PROPN
ma-119	338	9	on	on	ADP
ma-119	338	10	h	h	NOUN
ma-119	338	11	such	such	ADJ
ma-119	338	12	that	that	SCONJ
ma-119	338	13	a	a	PRON
ma-119	338	14	,	,	PUNCT
ma-119	338	15	a0	a0	PROPN
ma-119	338	16	are	be	AUX
ma-119	338	17	norm	norm	NOUN
ma-119	338	18	-	-	PUNCT
ma-119	338	19	attainable	attainable	ADJ
ma-119	338	20	then	then	ADV
ma-119	338	21	δa	δa	PROPN
ma-119	338	22	,	,	PUNCT
ma-119	338	23	a0	a0	PROPN
ma-119	338	24	is	be	AUX
ma-119	338	25	also	also	ADV
ma-119	338	26	norm	norm	NOUN
ma-119	338	27	-	-	PUNCT
ma-119	338	28	attainable	attainable	ADJ
ma-119	338	29	.	.	PUNCT
ma-119	339	1	proof	proof	NOUN
ma-119	339	2	.	.	PUNCT
ma-119	340	1	given	give	VERB
ma-119	340	2	the	the	DET
ma-119	340	3	operators	operator	NOUN
ma-119	340	4	a	a	DET
ma-119	340	5	,	,	PUNCT
ma-119	340	6	a0	a0	PROPN
ma-119	340	7	∈	∈	PROPN
ma-119	340	8	b(h	b(h	PROPN
ma-119	340	9	)	)	PUNCT
ma-119	340	10	are	be	AUX
ma-119	340	11	norm	norm	NOUN
ma-119	340	12	-	-	PUNCT
ma-119	340	13	attainable	attainable	ADJ
ma-119	340	14	then	then	ADV
ma-119	340	15	we	we	PRON
ma-119	340	16	need	need	VERB
ma-119	340	17	to	to	PART
ma-119	340	18	show	show	VERB
ma-119	340	19	that	that	SCONJ
ma-119	340	20	δa	δa	PROPN
ma-119	340	21	,	,	PUNCT
ma-119	340	22	a0	a0	PROPN
ma-119	340	23	isalso	isalso	ADV
ma-119	340	24	norm	norm	NOUN
ma-119	340	25	-	-	PUNCT
ma-119	340	26	attainable	attainable	ADJ
ma-119	340	27	.	.	PUNCT
ma-119	341	1	we	we	PRON
ma-119	341	2	define	define	VERB
ma-119	341	3	the	the	DET
ma-119	341	4	generalized	generalized	ADJ
ma-119	341	5	derivation	derivation	NOUN
ma-119	341	6	by	by	ADP
ma-119	341	7	δa	δa	PROPN
ma-119	341	8	,	,	PUNCT
ma-119	341	9	a0	a0	PROPN
ma-119	341	10	=	=	SYM
ma-119	341	11	ay	ay	PROPN
ma-119	341	12	−	−	PROPN
ma-119	341	13	y	y	PROPN
ma-119	341	14	a0	a0	PROPN
ma-119	341	15	.	.	PUNCT
ma-119	342	1	since	since	SCONJ
ma-119	342	2	a	a	DET
ma-119	342	3	,	,	PUNCT
ma-119	342	4	a0	a0	PROPN
ma-119	342	5	arenorm	arenorm	NOUN
ma-119	342	6	-	-	PUNCT
ma-119	342	7	attainable	attainable	ADJ
ma-119	342	8	then	then	ADV
ma-119	342	9	there	there	PRON
ma-119	342	10	exists	exist	VERB
ma-119	342	11	unit	unit	NOUN
ma-119	342	12	vectors	vector	NOUN
ma-119	342	13	y	y	PROPN
ma-119	342	14	and	and	CCONJ
ma-119	342	15	y0	y0	PROPN
ma-119	342	16	on	on	ADP
ma-119	342	17	h	h	NOUN
ma-119	342	18	such	such	ADJ
ma-119	342	19	that	that	SCONJ
ma-119	342	20	‖y‖	‖y‖	PROPN
ma-119	342	21	=	=	SYM
ma-119	342	22	‖y0‖	‖y0‖	PROPN
ma-119	342	23	=	=	SYM
ma-119	342	24	1	1	NUM
ma-119	342	25	,	,	PUNCT
ma-119	342	26	‖ay‖	‖ay‖	PROPN
ma-119	342	27	=	=	PUNCT
ma-119	342	28	‖a‖and	‖a‖and	CCONJ
ma-119	342	29	‖a0y0‖	‖a0y0‖	PUNCT
ma-119	343	1	=	=	SYM
ma-119	343	2	‖a0‖.	‖a0‖.	PROPN
ma-119	343	3	by	by	ADP
ma-119	343	4	linear	linear	ADJ
ma-119	343	5	dependence	dependence	NOUN
ma-119	343	6	of	of	ADP
ma-119	343	7	vectors	vector	NOUN
ma-119	343	8	,	,	PUNCT
ma-119	343	9	if	if	SCONJ
ma-119	343	10	y	y	PROPN
ma-119	343	11	and	and	CCONJ
ma-119	343	12	ay	ay	PROPN
ma-119	343	13	are	be	AUX
ma-119	343	14	linearly	linearly	ADV
ma-119	343	15	dependent	dependent	ADJ
ma-119	343	16	then	then	ADV
ma-119	343	17	wehave	wehave	VERB
ma-119	343	18	‖ay‖	‖ay‖	PROPN
ma-119	343	19	=	=	PUNCT
ma-119	343	20	η‖a‖y	η‖a‖y	PROPN
ma-119	343	21	where	where	SCONJ
ma-119	343	22	|η|	|η|	PROPN
ma-119	343	23	=	=	SYM
ma-119	343	24	1	1	NUM
ma-119	343	25	and	and	CCONJ
ma-119	343	26	|〈ay	|〈ay	NOUN
ma-119	343	27	,	,	PUNCT
ma-119	343	28	y〉|	y〉|	X
ma-119	344	1	=	=	SYM
ma-119	344	2	‖a‖.	‖a‖.	PROPN
ma-119	344	3	it	it	PRON
ma-119	344	4	follows	follow	VERB
ma-119	344	5	that	that	SCONJ
ma-119	344	6	|〈a0y0	|〈a0y0	NOUN
ma-119	344	7	,	,	PUNCT
ma-119	344	8	y0〉|	y0〉|	PROPN
ma-119	344	9	=	=	PUNCT
ma-119	345	1	‖a0‖	‖a0‖	ADP
ma-119	345	2	whichimplies	whichimplie	NOUN
ma-119	345	3	that	that	PRON
ma-119	345	4	‖a0y0‖	‖a0y0‖	PUNCT
ma-119	345	5	=	=	NOUN
ma-119	345	6	φ‖a0‖y0	φ‖a0‖y0	PUNCT
ma-119	345	7	and	and	CCONJ
ma-119	345	8	|φ|	|φ|	PROPN
ma-119	345	9	=	=	SYM
ma-119	345	10	1	1	NUM
ma-119	345	11	.	.	PUNCT
ma-119	345	12	therefore	therefore	ADV
ma-119	345	13	,	,	PUNCT
ma-119	345	14	〈	〈	PROPN
ma-119	345	15	a0y0‖a0‖	a0y0‖a0‖	NUM
ma-119	345	16	,	,	PUNCT
ma-119	345	17	y0	y0	NOUN
ma-119	345	18	〉	〉	NOUN
ma-119	345	19	=	=	SYM
ma-119	345	20	φ	φ	PROPN
ma-119	345	21	=	=	PUNCT
ma-119	346	1	−	−	ADP
ma-119	346	2	〈	〈	PROPN
ma-119	346	3	ay‖a‖	ay‖a‖	PROPN
ma-119	346	4	,	,	PUNCT
ma-119	346	5	y	y	PROPN
ma-119	346	6	〉	〉	NOUN
ma-119	346	7	=	=	SYM
ma-119	346	8	−η	−η	NOUN
ma-119	346	9	.	.	PUNCT
ma-119	347	1	if	if	SCONJ
ma-119	347	2	y	y	PROPN
ma-119	347	3	is	be	AUX
ma-119	347	4	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	NUM
ma-119	347	5	eur	eur	NOUN
ma-119	347	6	.	.	PUNCT
ma-119	348	1	j.	j.	PROPN
ma-119	348	2	math	math	PROPN
ma-119	348	3	.	.	PUNCT
ma-119	349	1	anal	anal	PROPN
ma-119	349	2	.	.	PUNCT
ma-119	350	1	10.28924	10.28924	NUM
ma-119	350	2	/	/	SYM
ma-119	350	3	ada	ada	PROPN
ma-119	350	4	/	/	SYM
ma-119	350	5	ma.3.9	ma.3.9	PROPN
ma-119	350	6	11defined	11defined	NUM
ma-119	350	7	as	as	ADP
ma-119	350	8	y	y	PROPN
ma-119	350	9	:	:	PUNCT
ma-119	350	10	y	y	PROPN
ma-119	350	11	→	→	SYM
ma-119	350	12	y0	y0	PROPN
ma-119	350	13	and	and	CCONJ
ma-119	350	14	y0	y0	PROPN
ma-119	350	15	→	→	SYM
ma-119	350	16	0	0	NUM
ma-119	350	17	,	,	PUNCT
ma-119	350	18	‖y	‖y	PUNCT
ma-119	351	1	‖	‖	PROPN
ma-119	351	2	=	=	NOUN
ma-119	351	3	1	1	NUM
ma-119	351	4	then	then	ADV
ma-119	351	5	(	(	PUNCT
ma-119	351	6	ay	ay	AUX
ma-119	351	7	−y	−y	PROPN
ma-119	351	8	a0)y0	a0)y0	PROPN
ma-119	352	1	=	=	SYM
ma-119	352	2	φ(‖a‖+‖a0‖)y0	φ(‖a‖+‖a0‖)y0	PRON
ma-119	353	1	which	which	PRON
ma-119	353	2	implies	imply	VERB
ma-119	353	3	‖ay	‖ay	NUM
ma-119	353	4	−	−	PROPN
ma-119	353	5	y	y	PROPN
ma-119	353	6	a0‖	a0‖	PROPN
ma-119	353	7	=	=	SYM
ma-119	353	8	‖(ay	‖(ay	PROPN
ma-119	353	9	−	−	PROPN
ma-119	353	10	y	y	PROPN
ma-119	353	11	a0)y0‖	a0)y0‖	PROPN
ma-119	353	12	=	=	SYM
ma-119	353	13	‖a‖+	‖a‖+	NUM
ma-119	353	14	‖a0‖	‖a0‖	ADP
ma-119	353	15	=	=	SYM
ma-119	353	16	‖δa	‖δa	PROPN
ma-119	353	17	,	,	PUNCT
ma-119	353	18	a0‖.	a0‖.	NOUN
ma-119	353	19	hence	hence	ADV
ma-119	353	20	,	,	PUNCT
ma-119	353	21	δa	δa	PROPN
ma-119	353	22	,	,	PUNCT
ma-119	353	23	a0	a0	PROPN
ma-119	353	24	is	be	AUX
ma-119	353	25	norm	norm	NOUN
ma-119	353	26	-	-	PUNCT
ma-119	353	27	attainable	attainable	ADJ
ma-119	353	28	.	.	PUNCT
ma-119	354	1	�	�	PROPN
ma-119	354	2	lemma	lemma	PROPN
ma-119	354	3	21	21	NUM
ma-119	354	4	.	.	PUNCT
ma-119	355	1	every	every	DET
ma-119	355	2	inner	inner	ADJ
ma-119	355	3	derivation	derivation	NOUN
ma-119	355	4	is	be	AUX
ma-119	355	5	norm	norm	NOUN
ma-119	355	6	-	-	PUNCT
ma-119	355	7	attainable	attainable	ADJ
ma-119	355	8	if	if	SCONJ
ma-119	355	9	and	and	CCONJ
ma-119	355	10	only	only	ADV
ma-119	355	11	if	if	SCONJ
ma-119	355	12	it	it	PRON
ma-119	355	13	is	be	AUX
ma-119	355	14	self	self	NOUN
ma-119	355	15	-	-	PUNCT
ma-119	355	16	adjoint	adjoint	NOUN
ma-119	355	17	.	.	PUNCT
ma-119	356	1	proof	proof	NOUN
ma-119	356	2	.	.	PUNCT
ma-119	357	1	let	let	VERB
ma-119	357	2	δa	δa	PROPN
ma-119	357	3	∈	∈	PROPN
ma-119	357	4	b(h	b(h	PROPN
ma-119	357	5	)	)	PUNCT
ma-119	357	6	be	be	VERB
ma-119	357	7	norm	norm	NOUN
ma-119	357	8	-	-	PUNCT
ma-119	357	9	attainable	attainable	ADJ
ma-119	357	10	then	then	ADV
ma-119	357	11	we	we	PRON
ma-119	357	12	show	show	VERB
ma-119	357	13	that	that	SCONJ
ma-119	357	14	δa	δa	NOUN
ma-119	357	15	=	=	PUNCT
ma-119	357	16	δ∗a	δ∗a	PROPN
ma-119	357	17	.	.	PUNCT
ma-119	357	18	now	now	ADV
ma-119	357	19	since	since	SCONJ
ma-119	357	20	δa	δa	PROPN
ma-119	357	21	∈	∈	PROPN
ma-119	357	22	b(h)is	b(h)is	PROPN
ma-119	357	23	norm	norm	NOUN
ma-119	357	24	-	-	PUNCT
ma-119	357	25	attainable	attainable	ADJ
ma-119	357	26	then	then	ADV
ma-119	357	27	there	there	PRON
ma-119	357	28	exists	exist	VERB
ma-119	357	29	a	a	DET
ma-119	357	30	contraction	contraction	NOUN
ma-119	357	31	y	y	PROPN
ma-119	357	32	∈	∈	PROPN
ma-119	357	33	b(h	b(h	PROPN
ma-119	357	34	)	)	PUNCT
ma-119	357	35	such	such	ADJ
ma-119	357	36	that	that	SCONJ
ma-119	357	37	‖δay	‖δay	ADJ
ma-119	357	38	‖	‖	PROPN
ma-119	357	39	=	=	PRON
ma-119	358	1	‖δa‖.	‖δa‖.	INTJ
ma-119	358	2	that	that	PRON
ma-119	358	3	is	be	AUX
ma-119	358	4	,	,	PUNCT
ma-119	358	5	‖δ∗aδay	‖δ∗aδay	PUNCT
ma-119	359	1	‖	‖	PROPN
ma-119	359	2	=	=	SYM
ma-119	360	1	‖δ2ay	‖δ2ay	NUM
ma-119	360	2	‖.	‖.	NOUN
ma-119	360	3	let	let	VERB
ma-119	360	4	η	η	PROPN
ma-119	360	5	∈	∈	PROPN
ma-119	360	6	h	h	NOUN
ma-119	360	7	be	be	AUX
ma-119	360	8	defined	define	VERB
ma-119	360	9	as	as	ADP
ma-119	360	10	η	η	PROPN
ma-119	360	11	=	=	PROPN
ma-119	360	12	δa	δa	PROPN
ma-119	360	13	‖δa‖	‖δa‖	PROPN
ma-119	360	14	then	then	ADV
ma-119	360	15	η	η	PROPN
ma-119	360	16	is	be	AUX
ma-119	360	17	contractive	contractive	ADJ
ma-119	360	18	such	such	ADJ
ma-119	360	19	that	that	SCONJ
ma-119	360	20	‖δ∗aη‖	‖δ∗aη‖	DET
ma-119	360	21	=	=	SYM
ma-119	360	22	‖δa‖	‖δa‖	PROPN
ma-119	360	23	=	=	SYM
ma-119	360	24	‖δ∗a‖.	‖δ∗a‖.	NOUN
ma-119	360	25	hence	hence	ADV
ma-119	360	26	,	,	PUNCT
ma-119	360	27	δa	δa	PROPN
ma-119	360	28	is	be	AUX
ma-119	360	29	self	self	NOUN
ma-119	360	30	-	-	PUNCT
ma-119	360	31	adjoint	adjoint	NOUN
ma-119	360	32	.	.	PUNCT
ma-119	361	1	conversely	conversely	ADV
ma-119	361	2	,	,	PUNCT
ma-119	361	3	let	let	VERB
ma-119	361	4	δa	δa	PART
ma-119	361	5	be	be	AUX
ma-119	361	6	self	self	NOUN
ma-119	361	7	-	-	PUNCT
ma-119	361	8	adjoint	adjoint	NOUN
ma-119	361	9	.	.	PUNCT
ma-119	362	1	now	now	ADV
ma-119	362	2	since	since	SCONJ
ma-119	362	3	δ∗a	δ∗a	PROPN
ma-119	362	4	is	be	AUX
ma-119	362	5	norm	norm	NOUN
ma-119	362	6	-	-	PUNCT
ma-119	362	7	attainable	attainable	ADJ
ma-119	362	8	from	from	ADP
ma-119	362	9	the	the	DET
ma-119	362	10	first	first	ADJ
ma-119	362	11	part	part	NOUN
ma-119	362	12	,	,	PUNCT
ma-119	362	13	then	then	ADV
ma-119	362	14	there	there	PRON
ma-119	362	15	exists	exist	VERB
ma-119	362	16	a	a	DET
ma-119	362	17	contractivem	contractivem	ADJ
ma-119	362	18	∈	∈	PROPN
ma-119	362	19	b(h	b(h	PROPN
ma-119	362	20	)	)	PUNCT
ma-119	362	21	such	such	ADJ
ma-119	362	22	that	that	SCONJ
ma-119	362	23	‖δ∗am‖	‖δ∗am‖	PROPN
ma-119	362	24	=	=	SYM
ma-119	362	25	‖δ∗a‖	‖δ∗a‖	PROPN
ma-119	362	26	,	,	PUNCT
ma-119	362	27	i.e	i.e	PROPN
ma-119	362	28	‖δaδ∗am‖	‖δaδ∗am‖	PROPN
ma-119	362	29	=	=	SYM
ma-119	362	30	‖δ2am‖.	‖δ2am‖.	PROPN
ma-119	362	31	let	let	VERB
ma-119	362	32	ζ	ζ	NOUN
ma-119	362	33	be	be	AUX
ma-119	362	34	denoted	denote	VERB
ma-119	362	35	by	by	ADP
ma-119	362	36	ζ	ζ	NOUN
ma-119	362	37	=	=	SYM
ma-119	362	38	δ∗a	δ∗a	PROPN
ma-119	362	39	‖δ∗a‖	‖δ∗a‖	PROPN
ma-119	362	40	where	where	SCONJ
ma-119	362	41	‖ζ‖	‖ζ‖	PROPN
ma-119	362	42	=	=	SYM
ma-119	362	43	1	1	NUM
ma-119	362	44	such	such	ADJ
ma-119	362	45	that	that	PRON
ma-119	362	46	‖δaζ‖	‖δaζ‖	PUNCT
ma-119	362	47	=	=	SYM
ma-119	362	48	‖δ∗a‖	‖δ∗a‖	PROPN
ma-119	362	49	=	=	SYM
ma-119	363	1	‖δa‖.hence	‖δa‖.hence	NUM
ma-119	363	2	,	,	PUNCT
ma-119	363	3	δa	δa	PROPN
ma-119	363	4	is	be	AUX
ma-119	363	5	norm	norm	NOUN
ma-119	363	6	-	-	PUNCT
ma-119	363	7	attainable	attainable	ADJ
ma-119	363	8	.	.	PUNCT
ma-119	364	1	�	�	PROPN
ma-119	364	2	lemma	lemma	PROPN
ma-119	364	3	22	22	NUM
ma-119	364	4	.	.	PUNCT
ma-119	365	1	every	every	DET
ma-119	365	2	generalized	generalized	ADJ
ma-119	365	3	derivation	derivation	NOUN
ma-119	365	4	is	be	AUX
ma-119	365	5	norm	norm	NOUN
ma-119	365	6	-	-	PUNCT
ma-119	365	7	attainable	attainable	ADJ
ma-119	365	8	if	if	SCONJ
ma-119	365	9	and	and	CCONJ
ma-119	365	10	only	only	ADV
ma-119	365	11	if	if	SCONJ
ma-119	365	12	it	it	PRON
ma-119	365	13	is	be	AUX
ma-119	365	14	implemented	implement	VERB
ma-119	365	15	by	by	ADP
ma-119	365	16	orthogonal	orthogonal	ADJ
ma-119	365	17	projections	projection	NOUN
ma-119	365	18	.	.	PUNCT
ma-119	366	1	proof	proof	NOUN
ma-119	366	2	.	.	PUNCT
ma-119	367	1	let	let	VERB
ma-119	367	2	a	a	DET
ma-119	367	3	,	,	PUNCT
ma-119	367	4	a0	a0	PROPN
ma-119	367	5	∈	∈	PROPN
ma-119	367	6	b(h	b(h	PROPN
ma-119	367	7	)	)	PUNCT
ma-119	367	8	be	be	AUX
ma-119	367	9	orthogonal	orthogonal	ADJ
ma-119	367	10	projections	projection	NOUN
ma-119	367	11	.	.	PUNCT
ma-119	368	1	indeed	indeed	ADV
ma-119	368	2	,	,	PUNCT
ma-119	368	3	to	to	PART
ma-119	368	4	show	show	VERB
ma-119	368	5	that	that	SCONJ
ma-119	368	6	a	a	DET
ma-119	368	7	generalized	generalized	ADJ
ma-119	368	8	derivationis	derivationis	NOUN
ma-119	368	9	implemented	implement	VERB
ma-119	368	10	by	by	ADP
ma-119	368	11	orthogonal	orthogonal	ADJ
ma-119	368	12	projections	projection	NOUN
ma-119	368	13	a	a	PRON
ma-119	368	14	and	and	CCONJ
ma-119	368	15	a0	a0	NOUN
ma-119	368	16	,	,	PUNCT
ma-119	368	17	it	it	PRON
ma-119	368	18	is	be	AUX
ma-119	368	19	enough	enough	ADJ
ma-119	368	20	to	to	PART
ma-119	368	21	show	show	VERB
ma-119	368	22	that	that	SCONJ
ma-119	368	23	it	it	PRON
ma-119	368	24	is	be	AUX
ma-119	368	25	self	self	NOUN
ma-119	368	26	-	-	PUNCT
ma-119	368	27	adjoint	adjoint	NOUN
ma-119	368	28	ifand	ifand	NOUN
ma-119	368	29	only	only	ADV
ma-119	368	30	if	if	SCONJ
ma-119	368	31	it	it	PRON
ma-119	368	32	is	be	AUX
ma-119	368	33	normal	normal	ADJ
ma-119	368	34	as	as	SCONJ
ma-119	368	35	proved	prove	VERB
ma-119	368	36	in	in	ADP
ma-119	368	37	[	[	X
ma-119	368	38	22	22	NUM
ma-119	368	39	]	]	PUNCT
ma-119	368	40	.	.	PUNCT
ma-119	369	1	let	let	VERB
ma-119	369	2	δa	δa	NOUN
ma-119	369	3	,	,	PUNCT
ma-119	369	4	a0	a0	PROPN
ma-119	369	5	:	:	PUNCT
ma-119	369	6	b(h)→	b(h)→	NUM
ma-119	369	7	b(h	b(h	PROPN
ma-119	369	8	)	)	PUNCT
ma-119	369	9	be	be	AUX
ma-119	369	10	bounded	bound	VERB
ma-119	369	11	linear	linear	ADJ
ma-119	369	12	operator	operator	NOUN
ma-119	369	13	on	on	ADP
ma-119	369	14	b(h	b(h	PROPN
ma-119	369	15	)	)	PUNCT
ma-119	369	16	.	.	PUNCT
ma-119	370	1	then	then	ADV
ma-119	370	2	exists	exist	VERB
ma-119	370	3	a	a	DET
ma-119	370	4	unique	unique	ADJ
ma-119	370	5	bounded	bounded	ADJ
ma-119	370	6	linear	linear	ADJ
ma-119	370	7	operator	operator	NOUN
ma-119	370	8	δ∗a	δ∗a	NOUN
ma-119	370	9	,	,	PUNCT
ma-119	370	10	a0	a0	NOUN
ma-119	370	11	:	:	PUNCT
ma-119	370	12	b(h)→	b(h)→	NUM
ma-119	370	13	b(h	b(h	PROPN
ma-119	370	14	)	)	PUNCT
ma-119	370	15	such	such	ADJ
ma-119	370	16	that	that	SCONJ
ma-119	370	17	〈	〈	PROPN
ma-119	370	18	δa	δa	NOUN
ma-119	370	19	,	,	PUNCT
ma-119	370	20	a0x	a0x	PROPN
ma-119	370	21	,	,	PUNCT
ma-119	370	22	y	y	PROPN
ma-119	370	23	〉	〉	PROPN
ma-119	370	24	=	=	SYM
ma-119	370	25	〈	〈	PROPN
ma-119	370	26	x	x	X
ma-119	370	27	,	,	PUNCT
ma-119	370	28	δ∗a	δ∗a	PROPN
ma-119	370	29	,	,	PUNCT
ma-119	370	30	a0y	a0y	PROPN
ma-119	370	31	〉	〉	PROPN
ma-119	370	32	,	,	PUNCT
ma-119	370	33	for	for	ADP
ma-119	370	34	all	all	DET
ma-119	370	35	x	x	NOUN
ma-119	370	36	,	,	PUNCT
ma-119	370	37	y	y	PROPN
ma-119	370	38	∈	∈	PROPN
ma-119	370	39	b(h	b(h	PROPN
ma-119	370	40	)	)	PUNCT
ma-119	370	41	.	.	PUNCT
ma-119	371	1	now	now	ADV
ma-119	371	2	,	,	PUNCT
ma-119	371	3	‖δ∗a	‖δ∗a	PROPN
ma-119	371	4	,	,	PUNCT
ma-119	371	5	a0y	a0y	PROPN
ma-119	371	6	‖	‖	PROPN
ma-119	371	7	=	=	PROPN
ma-119	371	8	sup	sup	PROPN
ma-119	371	9	‖x‖=1	‖x‖=1	PROPN
ma-119	371	10	〈	〈	PROPN
ma-119	371	11	δa	δa	NOUN
ma-119	371	12	,	,	PUNCT
ma-119	371	13	a0x	a0x	PROPN
ma-119	371	14	,	,	PUNCT
ma-119	371	15	y	y	PROPN
ma-119	371	16	〉	〉	PROPN
ma-119	371	17	≤	≤	NUM
ma-119	371	18	sup	sup	NOUN
ma-119	371	19	‖x‖=‖y	‖x‖=‖y	NOUN
ma-119	371	20	‖=1	‖=1	X
ma-119	371	21	‖δa	‖δa	NUM
ma-119	371	22	,	,	PUNCT
ma-119	371	23	a0‖‖x‖‖y	a0‖‖x‖‖y	PROPN
ma-119	371	24	‖	‖	PROPN
ma-119	371	25	=	=	SYM
ma-119	371	26	‖δa	‖δa	PROPN
ma-119	371	27	,	,	PUNCT
ma-119	371	28	a0‖so	a0‖so	NOUN
ma-119	371	29	,	,	PUNCT
ma-119	371	30	we	we	PRON
ma-119	371	31	conclude	conclude	VERB
ma-119	371	32	that	that	SCONJ
ma-119	371	33	δ∗a	δ∗a	PUNCT
ma-119	371	34	,	,	PUNCT
ma-119	371	35	a0	a0	NOUN
ma-119	371	36	is	be	AUX
ma-119	371	37	norm	norm	NOUN
ma-119	371	38	-	-	PUNCT
ma-119	371	39	attainable	attainable	ADJ
ma-119	371	40	.	.	PUNCT
ma-119	372	1	conversely	conversely	ADV
ma-119	372	2	,	,	PUNCT
ma-119	372	3	let	let	VERB
ma-119	372	4	δa	δa	PROPN
ma-119	372	5	,	,	PUNCT
ma-119	372	6	a0	a0	PROPN
ma-119	372	7	be	be	VERB
ma-119	372	8	norm	norm	NOUN
ma-119	372	9	-	-	PUNCT
ma-119	372	10	attainable	attainable	ADJ
ma-119	372	11	.	.	PUNCT
ma-119	373	1	we	we	PRON
ma-119	373	2	needto	needto	AUX
ma-119	373	3	show	show	VERB
ma-119	373	4	that	that	SCONJ
ma-119	373	5	it	it	PRON
ma-119	373	6	is	be	AUX
ma-119	373	7	implemented	implement	VERB
ma-119	373	8	by	by	ADP
ma-119	373	9	orthogonal	orthogonal	ADJ
ma-119	373	10	projections	projection	NOUN
ma-119	373	11	.	.	PUNCT
ma-119	374	1	this	this	PRON
ma-119	374	2	follows	follow	VERB
ma-119	374	3	immediately	immediately	ADV
ma-119	374	4	from	from	ADP
ma-119	374	5	[	[	X
ma-119	374	6	22	22	NUM
ma-119	374	7	]	]	PUNCT
ma-119	374	8	andthis	andthis	NOUN
ma-119	374	9	completes	complete	VERB
ma-119	374	10	the	the	DET
ma-119	374	11	proof	proof	NOUN
ma-119	374	12	.	.	PUNCT
ma-119	375	1	�	�	PROPN
ma-119	375	2	at	at	ADP
ma-119	375	3	this	this	DET
ma-119	375	4	point	point	NOUN
ma-119	375	5	,	,	PUNCT
ma-119	375	6	we	we	PRON
ma-119	375	7	give	give	VERB
ma-119	375	8	results	result	NOUN
ma-119	375	9	on	on	ADP
ma-119	375	10	upper	upper	ADJ
ma-119	375	11	norm	norm	NOUN
ma-119	375	12	estimates	estimate	NOUN
ma-119	375	13	for	for	ADP
ma-119	375	14	norm	norm	NOUN
ma-119	375	15	-	-	PUNCT
ma-119	375	16	attainable	attainable	ADJ
ma-119	375	17	derivations	derivation	NOUN
ma-119	375	18	.	.	PUNCT
ma-119	376	1	we	we	PRON
ma-119	376	2	con	con	NOUN
ma-119	376	3	-	-	PUNCT
ma-119	376	4	sider	sider	NOUN
ma-119	376	5	both	both	CCONJ
ma-119	376	6	inner	inner	ADJ
ma-119	376	7	derivations	derivation	NOUN
ma-119	376	8	and	and	CCONJ
ma-119	376	9	generalized	generalized	ADJ
ma-119	376	10	derivations	derivation	NOUN
ma-119	376	11	.	.	PUNCT
ma-119	377	1	we	we	PRON
ma-119	377	2	begin	begin	VERB
ma-119	377	3	with	with	ADP
ma-119	377	4	the	the	DET
ma-119	377	5	following	follow	VERB
ma-119	377	6	proposition	proposition	NOUN
ma-119	377	7	.	.	PUNCT
ma-119	378	1	proposition	proposition	NOUN
ma-119	378	2	23	23	NUM
ma-119	378	3	.	.	PUNCT
ma-119	379	1	let	let	VERB
ma-119	379	2	a	a	DET
ma-119	379	3	,	,	PUNCT
ma-119	379	4	b	b	PROPN
ma-119	379	5	∈	∈	PROPN
ma-119	379	6	na(h	na(h	NOUN
ma-119	379	7	)	)	PUNCT
ma-119	379	8	and	and	CCONJ
ma-119	379	9	δa	δa	PROPN
ma-119	379	10	,	,	PUNCT
ma-119	379	11	b	b	PROPN
ma-119	379	12	be	be	AUX
ma-119	379	13	bounded	bound	VERB
ma-119	379	14	then	then	ADV
ma-119	379	15	‖δa	‖δa	NUM
ma-119	379	16	,	,	PUNCT
ma-119	379	17	b‖	b‖	VERB
ma-119	379	18	≤	≤	NUM
ma-119	379	19	‖a‖+	‖a‖+	PRON
ma-119	379	20	‖b‖.	‖b‖.	NOUN
ma-119	379	21	proof	proof	NOUN
ma-119	379	22	.	.	PUNCT
ma-119	380	1	since	since	SCONJ
ma-119	380	2	δa	δa	PROPN
ma-119	380	3	,	,	PUNCT
ma-119	380	4	b	b	PROPN
ma-119	380	5	is	be	AUX
ma-119	380	6	bounded	bound	VERB
ma-119	380	7	then	then	ADV
ma-119	380	8	for	for	ADP
ma-119	380	9	fixed	fix	VERB
ma-119	380	10	a	a	DET
ma-119	380	11	,	,	PUNCT
ma-119	380	12	b	b	PROPN
ma-119	380	13	∈	∈	PROPN
ma-119	380	14	na(h	na(h	NOUN
ma-119	380	15	)	)	PUNCT
ma-119	380	16	we	we	PRON
ma-119	380	17	have	have	VERB
ma-119	380	18	‖δa	‖δa	NUM
ma-119	380	19	,	,	PUNCT
ma-119	380	20	b(x)‖	b(x)‖	ADP
ma-119	380	21	≤	≤	PROPN
ma-119	380	22	‖ax	‖ax	CCONJ
ma-119	381	1	−	−	PROPN
ma-119	381	2	xb‖	xb‖	PROPN
ma-119	381	3	≤	≤	PUNCT
ma-119	381	4	‖ax‖	‖ax‖	PROPN
ma-119	381	5	+	+	CCONJ
ma-119	381	6	‖xb‖	‖xb‖	PROPN
ma-119	381	7	≤	≤	ADV
ma-119	381	8	‖a‖‖x‖	‖a‖‖x‖	PROPN
ma-119	381	9	+	+	CCONJ
ma-119	381	10	‖x‖‖b‖.	‖x‖‖b‖.	NUM
ma-119	381	11	let	let	VERB
ma-119	381	12	x	x	PRON
ma-119	381	13	be	be	AUX
ma-119	381	14	of	of	ADP
ma-119	381	15	norm	norm	NOUN
ma-119	381	16	1	1	NUM
ma-119	381	17	and	and	CCONJ
ma-119	381	18	take	take	VERB
ma-119	381	19	supremum	supremum	ADV
ma-119	381	20	over	over	ADP
ma-119	381	21	x	x	SYM
ma-119	381	22	∈	∈	PROPN
ma-119	381	23	na(h)then	na(h)then	X
ma-119	381	24	‖δa	‖δa	NUM
ma-119	381	25	,	,	PUNCT
ma-119	381	26	b‖	b‖	VERB
ma-119	381	27	≤	≤	NUM
ma-119	381	28	‖a‖+	‖a‖+	NUM
ma-119	381	29	‖b‖.	‖b‖.	PUNCT
ma-119	381	30	�	�	PROPN
ma-119	381	31	remark	remark	NOUN
ma-119	381	32	24	24	NUM
ma-119	381	33	.	.	PUNCT
ma-119	382	1	if	if	SCONJ
ma-119	382	2	a	a	DET
ma-119	382	3	=	=	SYM
ma-119	382	4	b	b	NOUN
ma-119	382	5	then	then	ADV
ma-119	382	6	‖δa‖	‖δa‖	VERB
ma-119	382	7	≤	≤	NOUN
ma-119	382	8	2‖a‖.	2‖a‖.	NUM
ma-119	382	9	next	next	ADV
ma-119	382	10	,	,	PUNCT
ma-119	382	11	we	we	PRON
ma-119	382	12	consider	consider	VERB
ma-119	382	13	upper	upper	ADJ
ma-119	382	14	bounds	bound	NOUN
ma-119	382	15	in	in	ADP
ma-119	382	16	the	the	DET
ma-119	382	17	unit	unit	NOUN
ma-119	382	18	ball	ball	NOUN
ma-119	382	19	of	of	ADP
ma-119	382	20	na(h	na(h	NOUN
ma-119	382	21	)	)	PUNCT
ma-119	382	22	denoted	denote	VERB
ma-119	382	23	by	by	ADP
ma-119	382	24	[	[	X
ma-119	382	25	na(h)]0	na(h)]0	PROPN
ma-119	382	26	.	.	PUNCT
ma-119	382	27	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	NUM
ma-119	382	28	eur	eur	NOUN
ma-119	382	29	.	.	PUNCT
ma-119	383	1	j.	j.	PROPN
ma-119	383	2	math	math	PROPN
ma-119	383	3	.	.	PUNCT
ma-119	384	1	anal	anal	PROPN
ma-119	384	2	.	.	PUNCT
ma-119	385	1	10.28924	10.28924	NUM
ma-119	385	2	/	/	SYM
ma-119	385	3	ada	ada	PROPN
ma-119	385	4	/	/	SYM
ma-119	385	5	ma.3.9	ma.3.9	PROPN
ma-119	385	6	12	12	NUM
ma-119	385	7	lemma	lemma	PROPN
ma-119	385	8	25	25	NUM
ma-119	385	9	.	.	PUNCT
ma-119	386	1	let	let	VERB
ma-119	386	2	[	[	PUNCT
ma-119	386	3	na(h)]0	na(h)]0	PROPN
ma-119	386	4	be	be	AUX
ma-119	386	5	the	the	DET
ma-119	386	6	unit	unit	NOUN
ma-119	386	7	ball	ball	NOUN
ma-119	386	8	of	of	ADP
ma-119	386	9	na(h	na(h	NOUN
ma-119	386	10	)	)	PUNCT
ma-119	386	11	and	and	CCONJ
ma-119	386	12	s	s	AUX
ma-119	386	13	be	be	AUX
ma-119	386	14	a	a	DET
ma-119	386	15	fixed	fix	VERB
ma-119	386	16	element	element	NOUN
ma-119	386	17	of	of	ADP
ma-119	386	18	na(h	na(h	NOUN
ma-119	386	19	)	)	PUNCT
ma-119	386	20	.	.	PUNCT
ma-119	387	1	let	let	VERB
ma-119	387	2	x	x	X
ma-119	387	3	∈	∈	PROPN
ma-119	388	1	[	[	X
ma-119	388	2	na(h)]0	na(h)]0	PROPN
ma-119	388	3	then	then	ADV
ma-119	388	4	‖δs|[na(h)]0‖	‖δs|[na(h)]0‖	PUNCT
ma-119	388	5	≤	≤	ADJ
ma-119	388	6	2d(s	2d(s	NUM
ma-119	388	7	)	)	PUNCT
ma-119	388	8	.	.	PUNCT
ma-119	389	1	proof	proof	NOUN
ma-119	389	2	.	.	PUNCT
ma-119	390	1	since	since	SCONJ
ma-119	390	2	x	x	PROPN
ma-119	390	3	∈	∈	PROPN
ma-119	390	4	[	[	X
ma-119	390	5	na(h)]0	na(h)]0	PROPN
ma-119	390	6	has	have	AUX
ma-119	390	7	norm	norm	NOUN
ma-119	390	8	1	1	NUM
ma-119	390	9	then	then	ADV
ma-119	390	10	we	we	PRON
ma-119	390	11	have	have	VERB
ma-119	390	12	‖δs|[na(h)]0(x)‖	‖δs|[na(h)]0(x)‖	ADJ
ma-119	390	13	=	=	SYM
ma-119	390	14	‖sx	‖sx	NUM
ma-119	390	15	−	−	NUM
ma-119	390	16	xs‖[na(h)]0	xs‖[na(h)]0	X
ma-119	390	17	=	=	SYM
ma-119	390	18	‖(s−λ)x−x(s−λ)‖[na(h)]0	‖(s−λ)x−x(s−λ)‖[na(h)]0	PROPN
ma-119	390	19	≤	≤	NUM
ma-119	390	20	‖s−λ‖‖x‖[na(h)]0	‖s−λ‖‖x‖[na(h)]0	VERB
ma-119	390	21	+	+	PROPN
ma-119	390	22	‖x‖‖s−λ‖[na(h)]0	‖x‖‖s−λ‖[na(h)]0	PROPN
ma-119	390	23	.	.	PUNCT
ma-119	391	1	taking	take	VERB
ma-119	391	2	the	the	DET
ma-119	391	3	supremumover	supremumover	NOUN
ma-119	391	4	[	[	X
ma-119	391	5	na(h)]0	na(h)]0	PROPN
ma-119	391	6	,	,	PUNCT
ma-119	391	7	we	we	PRON
ma-119	391	8	obtain	obtain	VERB
ma-119	391	9	‖δs|[na(h)]0‖	‖δs|[na(h)]0‖	PUNCT
ma-119	391	10	≤	≤	NOUN
ma-119	391	11	2‖s	2‖s	NUM
ma-119	391	12	−	−	PROPN
ma-119	391	13	λ‖	λ‖	NOUN
ma-119	392	1	and	and	CCONJ
ma-119	392	2	considering	consider	VERB
ma-119	392	3	the	the	DET
ma-119	392	4	infimum	infimum	NOUN
ma-119	392	5	over	over	ADP
ma-119	392	6	λ	λ	PROPN
ma-119	392	7	∈	∈	PROPN
ma-119	392	8	c	c	NOUN
ma-119	392	9	weobtain	weobtain	NOUN
ma-119	392	10	‖δs|[na(h)]0‖	‖δs|[na(h)]0‖	PUNCT
ma-119	392	11	≤	≤	ADV
ma-119	392	12	2	2	NUM
ma-119	392	13	infλ∈c	infλ∈c	NOUN
ma-119	392	14	‖s	‖s	NOUN
ma-119	392	15	−	−	PROPN
ma-119	393	1	λ‖	λ‖	PROPN
ma-119	393	2	=	=	SYM
ma-119	393	3	2d(s	2d(s	NUM
ma-119	393	4	)	)	PUNCT
ma-119	393	5	.	.	PUNCT
ma-119	394	1	�	�	PROPN
ma-119	394	2	remark	remark	VERB
ma-119	394	3	26	26	NUM
ma-119	394	4	.	.	PUNCT
ma-119	395	1	the	the	DET
ma-119	395	2	restriction	restriction	NOUN
ma-119	395	3	of	of	ADP
ma-119	395	4	δa|[na(h)]0	δa|[na(h)]0	NOUN
ma-119	395	5	i.e	i.e	X
ma-119	395	6	δa	δa	NOUN
ma-119	395	7	to	to	ADP
ma-119	395	8	[	[	X
ma-119	395	9	na(h)]0	na(h)]0	PROPN
ma-119	395	10	is	be	AUX
ma-119	395	11	a	a	DET
ma-119	395	12	bounded	bounded	ADJ
ma-119	395	13	linear	linear	ADJ
ma-119	395	14	operator	operator	NOUN
ma-119	395	15	.	.	PUNCT
ma-119	396	1	next	next	ADV
ma-119	396	2	we	we	PRON
ma-119	396	3	give	give	VERB
ma-119	396	4	an	an	DET
ma-119	396	5	extension	extension	NOUN
ma-119	396	6	of	of	ADP
ma-119	396	7	lemma	lemma	PROPN
ma-119	396	8	25	25	NUM
ma-119	396	9	to	to	ADP
ma-119	396	10	a	a	DET
ma-119	396	11	generalized	generalized	ADJ
ma-119	396	12	derivation	derivation	NOUN
ma-119	396	13	in	in	ADP
ma-119	396	14	the	the	DET
ma-119	396	15	following	follow	VERB
ma-119	396	16	theorem	theorem	PROPN
ma-119	396	17	.	.	PUNCT
ma-119	396	18	theorem	theorem	PROPN
ma-119	396	19	27	27	NUM
ma-119	396	20	.	.	PUNCT
ma-119	397	1	let	let	VERB
ma-119	397	2	s	s	NOUN
ma-119	397	3	,	,	PUNCT
ma-119	397	4	s0	s0	PROPN
ma-119	397	5	be	be	AUX
ma-119	397	6	fixed	fix	VERB
ma-119	397	7	elements	element	NOUN
ma-119	397	8	of	of	ADP
ma-119	397	9	na(h	na(h	NOUN
ma-119	397	10	)	)	PUNCT
ma-119	397	11	then	then	ADV
ma-119	397	12	‖δs	‖δs	NUM
ma-119	397	13	,	,	PUNCT
ma-119	397	14	s0	s0	PROPN
ma-119	397	15	|[na(h)]0‖	|[na(h)]0‖	PROPN
ma-119	397	16	≤	≤	PROPN
ma-119	397	17	‖δs	‖δs	PROPN
ma-119	397	18	,	,	PUNCT
ma-119	397	19	s0‖.	s0‖.	NOUN
ma-119	397	20	proof	proof	NOUN
ma-119	397	21	.	.	PUNCT
ma-119	398	1	since	since	SCONJ
ma-119	398	2	x	x	PROPN
ma-119	398	3	∈	∈	PROPN
ma-119	398	4	[	[	X
ma-119	398	5	na(h)]0	na(h)]0	PROPN
ma-119	398	6	has	have	AUX
ma-119	398	7	norm	norm	NOUN
ma-119	398	8	1	1	NUM
ma-119	398	9	then	then	ADV
ma-119	398	10	we	we	PRON
ma-119	398	11	have	have	VERB
ma-119	398	12	‖δs	‖δs	NUM
ma-119	398	13	,	,	PUNCT
ma-119	398	14	s0	s0	PROPN
ma-119	398	15	|[na(h)]0(x)‖	|[na(h)]0(x)‖	PROPN
ma-119	398	16	=	=	SYM
ma-119	398	17	‖sx−xs0‖.	‖sx−xs0‖.	PROPN
ma-119	398	18	followingproof	followingproof	NOUN
ma-119	398	19	of	of	ADP
ma-119	398	20	lemma	lemma	PROPN
ma-119	398	21	25	25	NUM
ma-119	398	22	anologously	anologously	ADV
ma-119	398	23	we	we	PRON
ma-119	398	24	have	have	VERB
ma-119	398	25	‖δs	‖δs	NUM
ma-119	398	26	,	,	PUNCT
ma-119	398	27	s0	s0	PROPN
ma-119	398	28	|[na(h)]0(x)‖	|[na(h)]0(x)‖	PROPN
ma-119	398	29	≤	≤	ADV
ma-119	399	1	‖s	‖s	ADV
ma-119	399	2	−	−	PROPN
ma-119	400	1	λ‖‖x‖[na(h)]0	λ‖‖x‖[na(h)]0	X
ma-119	400	2	+	+	NUM
ma-119	400	3	‖x‖‖s0	‖x‖‖s0	NOUN
ma-119	400	4	−	−	PROPN
ma-119	400	5	λ‖[na(h)]0	λ‖[na(h)]0	PROPN
ma-119	400	6	.taking	.take	VERB
ma-119	400	7	the	the	DET
ma-119	400	8	supremum	supremum	ADJ
ma-119	400	9	over	over	ADP
ma-119	400	10	x	x	PUNCT
ma-119	400	11	∈	∈	PROPN
ma-119	400	12	[	[	X
ma-119	400	13	na(h)]0	na(h)]0	PROPN
ma-119	400	14	we	we	PRON
ma-119	400	15	obtain	obtain	VERB
ma-119	400	16	‖δs	‖δs	NUM
ma-119	400	17	,	,	PUNCT
ma-119	400	18	s0	s0	PROPN
ma-119	400	19	|[na(h)]0‖	|[na(h)]0‖	PROPN
ma-119	400	20	≤	≤	NUM
ma-119	400	21	infλ∈c(‖s	infλ∈c(‖s	PROPN
ma-119	400	22	−	−	PROPN
ma-119	400	23	λ‖+	λ‖+	VERB
ma-119	400	24	‖s0	‖s0	NOUN
ma-119	400	25	−	−	PROPN
ma-119	400	26	λ‖	λ‖	NUM
ma-119	400	27	)	)	PUNCT
ma-119	401	1	=	=	SYM
ma-119	401	2	‖δs	‖δs	PROPN
ma-119	401	3	,	,	PUNCT
ma-119	401	4	s0‖.	s0‖.	PROPN
ma-119	401	5	�	�	PROPN
ma-119	401	6	corollary	corollary	NOUN
ma-119	401	7	28	28	NUM
ma-119	401	8	.	.	PUNCT
ma-119	402	1	every	every	DET
ma-119	402	2	generalized	generalized	ADJ
ma-119	402	3	derivation	derivation	NOUN
ma-119	402	4	δs	δs	NOUN
ma-119	402	5	,	,	PUNCT
ma-119	402	6	s0	s0	PROPN
ma-119	402	7	is	be	AUX
ma-119	402	8	norm	norm	NOUN
ma-119	402	9	-	-	PUNCT
ma-119	402	10	bounded	bound	VERB
ma-119	402	11	.	.	PUNCT
ma-119	403	1	proof	proof	NOUN
ma-119	403	2	.	.	PUNCT
ma-119	404	1	this	this	PRON
ma-119	404	2	follows	follow	VERB
ma-119	404	3	immediately	immediately	ADV
ma-119	404	4	from	from	ADP
ma-119	404	5	[	[	X
ma-119	404	6	49	49	NUM
ma-119	404	7	]	]	PUNCT
ma-119	404	8	and	and	CCONJ
ma-119	404	9	from	from	ADP
ma-119	404	10	theorem	theorem	ADJ
ma-119	404	11	27	27	NUM
ma-119	404	12	.	.	PUNCT
ma-119	405	1	this	this	PRON
ma-119	405	2	completes	complete	VERB
ma-119	405	3	the	the	DET
ma-119	405	4	proof	proof	NOUN
ma-119	405	5	.	.	PUNCT
ma-119	406	1	�	�	PROPN
ma-119	406	2	now	now	ADV
ma-119	406	3	,	,	PUNCT
ma-119	406	4	we	we	PRON
ma-119	406	5	consider	consider	VERB
ma-119	406	6	lower	low	ADJ
ma-119	406	7	bounds	bound	NOUN
ma-119	406	8	for	for	ADP
ma-119	406	9	norms	norm	NOUN
ma-119	406	10	of	of	ADP
ma-119	406	11	derivations	derivation	NOUN
ma-119	406	12	.	.	PUNCT
ma-119	407	1	we	we	PRON
ma-119	407	2	begin	begin	VERB
ma-119	407	3	the	the	DET
ma-119	407	4	following	follow	VERB
ma-119	407	5	proposition	proposition	NOUN
ma-119	407	6	ongeneralized	ongeneralize	VERB
ma-119	407	7	derivation	derivation	NOUN
ma-119	407	8	.	.	PUNCT
ma-119	408	1	proposition	proposition	NOUN
ma-119	408	2	29	29	NUM
ma-119	408	3	.	.	PUNCT
ma-119	409	1	let	let	VERB
ma-119	409	2	s	s	NOUN
ma-119	409	3	,	,	PUNCT
ma-119	409	4	s0	s0	PROPN
ma-119	409	5	be	be	AUX
ma-119	409	6	fixed	fix	VERB
ma-119	409	7	elements	element	NOUN
ma-119	409	8	of	of	ADP
ma-119	409	9	na(h	na(h	NOUN
ma-119	409	10	)	)	PUNCT
ma-119	409	11	then	then	ADV
ma-119	409	12	‖δs	‖δs	NUM
ma-119	409	13	,	,	PUNCT
ma-119	409	14	s0	s0	PROPN
ma-119	409	15	|[na(h)]0‖	|[na(h)]0‖	PROPN
ma-119	409	16	≥	≥	PROPN
ma-119	409	17	‖s‖+	‖s‖+	NUM
ma-119	409	18	‖s0‖.	‖s0‖.	PROPN
ma-119	409	19	proof	proof	NOUN
ma-119	409	20	.	.	PUNCT
ma-119	410	1	let	let	VERB
ma-119	410	2	η	η	PROPN
ma-119	410	3	,	,	PUNCT
ma-119	410	4	ξ	ξ	PROPN
ma-119	410	5	and	and	CCONJ
ma-119	410	6	x	x	ADJ
ma-119	410	7	be	be	AUX
ma-119	410	8	unit	unit	NOUN
ma-119	410	9	vectors	vector	NOUN
ma-119	410	10	in	in	ADP
ma-119	410	11	h	h	NOUN
ma-119	410	12	and	and	CCONJ
ma-119	410	13	φ,ϕ	φ,ϕ	PRON
ma-119	410	14	be	be	VERB
ma-119	410	15	positive	positive	ADJ
ma-119	410	16	linear	linear	NOUN
ma-119	410	17	functionals	functional	NOUN
ma-119	410	18	such	such	ADJ
ma-119	410	19	that	that	SCONJ
ma-119	410	20	φ⊗	φ⊗	PROPN
ma-119	410	21	η	η	PROPN
ma-119	410	22	:	:	PUNCT
ma-119	410	23	h	h	PROPN
ma-119	410	24	→	→	SYM
ma-119	410	25	c	c	PROPN
ma-119	410	26	and	and	CCONJ
ma-119	410	27	ϕ	ϕ	PROPN
ma-119	410	28	⊗	⊗	PROPN
ma-119	410	29	ξ	ξ	PROPN
ma-119	410	30	:	:	PUNCT
ma-119	410	31	h	h	NOUN
ma-119	410	32	→	→	SYM
ma-119	410	33	c	c	X
ma-119	410	34	be	be	AUX
ma-119	410	35	of	of	ADP
ma-119	410	36	rank	rank	NOUN
ma-119	410	37	1	1	NUM
ma-119	410	38	defined	define	VERB
ma-119	410	39	as	as	ADP
ma-119	410	40	(	(	PUNCT
ma-119	410	41	φ	φ	PROPN
ma-119	410	42	⊗	⊗	NOUN
ma-119	410	43	η)x	η)x	X
ma-119	410	44	=	=	PUNCT
ma-119	411	1	φ(x)η	φ(x)η	PROPN
ma-119	411	2	and	and	CCONJ
ma-119	411	3	(	(	PUNCT
ma-119	411	4	ϕ	ϕ	PROPN
ma-119	411	5	⊗	⊗	PROPN
ma-119	411	6	ξ)x	ξ)x	NUM
ma-119	412	1	=	=	SYM
ma-119	412	2	ϕ(x)ξ	ϕ(x)ξ	PROPN
ma-119	412	3	,	,	PUNCT
ma-119	412	4	∀x	∀x	X
ma-119	412	5	∈	∈	PROPN
ma-119	412	6	h	h	NOUN
ma-119	412	7	,	,	PUNCT
ma-119	412	8	‖x‖	‖x‖	PROPN
ma-119	412	9	=	=	SYM
ma-119	412	10	1	1	X
ma-119	412	11	.	.	PUNCT
ma-119	413	1	now	now	ADV
ma-119	413	2	we	we	PRON
ma-119	413	3	have	have	VERB
ma-119	413	4	that	that	DET
ma-119	413	5	‖(φ⊗η)x‖	‖(φ⊗η)x‖	PROPN
ma-119	413	6	=	=	SYM
ma-119	413	7	sup{‖(φ⊗η)x‖	sup{‖(φ⊗η)x‖	PROPN
ma-119	413	8	,	,	PUNCT
ma-119	413	9	‖x‖	‖x‖	PROPN
ma-119	413	10	=	=	SYM
ma-119	413	11	1	1	X
ma-119	413	12	}	}	PUNCT
ma-119	413	13	=	=	PUNCT
ma-119	413	14	|φ(x)|	|φ(x)|	NOUN
ma-119	413	15	=	=	PUNCT
ma-119	413	16	|φ|.similarly	|φ|.similarly	ADV
ma-119	413	17	,	,	PUNCT
ma-119	413	18	we	we	PRON
ma-119	413	19	have	have	VERB
ma-119	413	20	‖(ϕ	‖(ϕ	PROPN
ma-119	413	21	⊗	⊗	PROPN
ma-119	413	22	ξ)x‖	ξ)x‖	NUM
ma-119	413	23	=	=	SYM
ma-119	413	24	‖ϕ‖.	‖ϕ‖.	NOUN
ma-119	413	25	letting	let	VERB
ma-119	413	26	s	s	X
ma-119	413	27	=	=	SYM
ma-119	413	28	φ	φ	PROPN
ma-119	413	29	⊗	⊗	PROPN
ma-119	413	30	η	η	PROPN
ma-119	413	31	and	and	CCONJ
ma-119	413	32	s0	s0	PROPN
ma-119	413	33	=	=	SYM
ma-119	413	34	ϕ	ϕ	PROPN
ma-119	414	1	⊗	⊗	PROPN
ma-119	414	2	ξ	ξ	PROPN
ma-119	414	3	then	then	ADV
ma-119	414	4	‖s‖	‖s‖	PROPN
ma-119	414	5	=	=	PUNCT
ma-119	414	6	‖φ‖and	‖φ‖and	NUM
ma-119	414	7	‖s0‖	‖s0‖	NOUN
ma-119	414	8	=	=	SYM
ma-119	414	9	‖ϕ‖.	‖ϕ‖.	PROPN
ma-119	414	10	now	now	ADV
ma-119	414	11	from	from	ADP
ma-119	414	12	corollary	corollary	ADJ
ma-119	414	13	28	28	NUM
ma-119	414	14	we	we	PRON
ma-119	414	15	have	have	VERB
ma-119	414	16	that	that	SCONJ
ma-119	414	17	every	every	DET
ma-119	414	18	generalized	generalized	ADJ
ma-119	414	19	derivation	derivation	NOUN
ma-119	414	20	is	be	AUX
ma-119	414	21	norm	norm	NOUN
ma-119	414	22	-	-	PUNCT
ma-119	414	23	bounded	bound	VERB
ma-119	414	24	this	this	PRON
ma-119	414	25	implies	imply	VERB
ma-119	414	26	that	that	SCONJ
ma-119	414	27	‖δs	‖δs	PROPN
ma-119	414	28	,	,	PUNCT
ma-119	414	29	s0	s0	PROPN
ma-119	414	30	|[na(h)]0(x)‖	|[na(h)]0(x)‖	PROPN
ma-119	414	31	≥	≥	PROPN
ma-119	414	32	‖δs	‖δs	PROPN
ma-119	414	33	,	,	PUNCT
ma-119	414	34	s0(x)‖	s0(x)‖	NOUN
ma-119	414	35	where	where	SCONJ
ma-119	414	36	x	x	X
ma-119	414	37	∈	∈	PROPN
ma-119	414	38	[	[	X
ma-119	414	39	na(h)]0	na(h)]0	PROPN
ma-119	414	40	.	.	PROPN
ma-119	414	41	therefore	therefore	ADV
ma-119	414	42	,	,	PUNCT
ma-119	414	43	‖δs	‖δs	PROPN
ma-119	414	44	,	,	PUNCT
ma-119	414	45	s0	s0	PROPN
ma-119	414	46	|[na(h)]0‖2	|[na(h)]0‖2	NUM
ma-119	414	47	≥	≥	NUM
ma-119	414	48	‖sx−xs0‖2	‖sx−xs0‖2	NUM
ma-119	414	49	implying	imply	VERB
ma-119	414	50	that	that	SCONJ
ma-119	414	51	‖δs	‖δs	PROPN
ma-119	414	52	,	,	PUNCT
ma-119	414	53	s0	s0	PROPN
ma-119	414	54	|[na(h)]0‖2	|[na(h)]0‖2	NUM
ma-119	414	55	≥	≥	VERB
ma-119	414	56	[	[	X
ma-119	414	57	‖s‖+	‖s‖+	NOUN
ma-119	414	58	‖s0‖]2	‖s0‖]2	NOUN
ma-119	414	59	.	.	PUNCT
ma-119	415	1	taking	take	VERB
ma-119	415	2	positivesquare	positivesquare	NOUN
ma-119	415	3	root	root	NOUN
ma-119	415	4	on	on	ADP
ma-119	415	5	both	both	DET
ma-119	415	6	sides	side	NOUN
ma-119	415	7	we	we	PRON
ma-119	415	8	obtain	obtain	VERB
ma-119	415	9	‖δs	‖δs	NUM
ma-119	415	10	,	,	PUNCT
ma-119	415	11	s0	s0	PROPN
ma-119	415	12	|[na(h)]0‖	|[na(h)]0‖	PROPN
ma-119	415	13	=	=	SYM
ma-119	415	14	‖δs	‖δs	PROPN
ma-119	415	15	,	,	PUNCT
ma-119	415	16	s0‖	s0‖	NOUN
ma-119	415	17	≥	≥	NOUN
ma-119	415	18	‖s‖+	‖s‖+	NUM
ma-119	415	19	‖s0‖.	‖s0‖.	PROPN
ma-119	415	20	�	�	PROPN
ma-119	415	21	remark	remark	VERB
ma-119	415	22	30	30	NUM
ma-119	415	23	.	.	PUNCT
ma-119	416	1	if	if	SCONJ
ma-119	416	2	s	s	PART
ma-119	416	3	=	=	SYM
ma-119	416	4	s0	s0	PROPN
ma-119	416	5	then	then	ADV
ma-119	416	6	‖δs	‖δs	NUM
ma-119	416	7	,	,	PUNCT
ma-119	416	8	s0‖	s0‖	PROPN
ma-119	416	9	=	=	SYM
ma-119	416	10	‖δs‖	‖δs‖	ADJ
ma-119	416	11	≥	≥	NOUN
ma-119	416	12	2‖s‖.	2‖s‖.	NOUN
ma-119	416	13	remark	remark	NOUN
ma-119	416	14	31	31	NUM
ma-119	416	15	.	.	PUNCT
ma-119	417	1	from	from	ADP
ma-119	417	2	theorem	theorem	ADJ
ma-119	417	3	27	27	NUM
ma-119	417	4	and	and	CCONJ
ma-119	417	5	proposition	proposition	NOUN
ma-119	417	6	3	3	NUM
ma-119	417	7	it	it	PRON
ma-119	417	8	is	be	AUX
ma-119	417	9	easy	easy	ADJ
ma-119	417	10	to	to	PART
ma-119	417	11	see	see	VERB
ma-119	417	12	that	that	SCONJ
ma-119	417	13	‖δs	‖δs	NUM
ma-119	417	14	,	,	PUNCT
ma-119	417	15	s0‖	s0‖	NOUN
ma-119	417	16	=	=	SYM
ma-119	417	17	‖s‖+	‖s‖+	NUM
ma-119	417	18	‖s0‖	‖s0‖	NOUN
ma-119	417	19	and	and	CCONJ
ma-119	417	20	hence	hence	ADV
ma-119	417	21	‖δs‖	‖δs‖	ADJ
ma-119	418	1	=	=	SYM
ma-119	418	2	2‖s‖.	2‖s‖.	PROPN
ma-119	418	3	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	NUM
ma-119	418	4	eur	eur	NOUN
ma-119	418	5	.	.	PUNCT
ma-119	419	1	j.	j.	PROPN
ma-119	419	2	math	math	PROPN
ma-119	419	3	.	.	PUNCT
ma-119	420	1	anal	anal	PROPN
ma-119	420	2	.	.	PUNCT
ma-119	421	1	10.28924	10.28924	NUM
ma-119	421	2	/	/	SYM
ma-119	421	3	ada	ada	PROPN
ma-119	421	4	/	/	SYM
ma-119	421	5	ma.3.9	ma.3.9	PROPN
ma-119	421	6	13	13	NUM
ma-119	421	7	theorem	theorem	NOUN
ma-119	421	8	32	32	NUM
ma-119	421	9	.	.	PUNCT
ma-119	422	1	let	let	VERB
ma-119	422	2	s	s	NOUN
ma-119	422	3	,	,	PUNCT
ma-119	422	4	s0	s0	PROPN
ma-119	422	5	∈	∈	PROPN
ma-119	422	6	na(h	na(h	NOUN
ma-119	422	7	)	)	PUNCT
ma-119	422	8	and	and	CCONJ
ma-119	422	9	α1	α1	PROPN
ma-119	422	10	∈	∈	PROPN
ma-119	422	11	w0(s	w0(s	PROPN
ma-119	422	12	)	)	PUNCT
ma-119	422	13	and	and	CCONJ
ma-119	422	14	α2	α2	PROPN
ma-119	422	15	∈	∈	PROPN
ma-119	422	16	w0(s0	w0(s0	ADV
ma-119	422	17	)	)	PUNCT
ma-119	422	18	.	.	PUNCT
ma-119	423	1	then	then	ADV
ma-119	423	2	‖δs	‖δs	NUM
ma-119	423	3	,	,	PUNCT
ma-119	423	4	s0‖	s0‖	PROPN
ma-119	423	5	≥	≥	PROPN
ma-119	423	6	(	(	PUNCT
ma-119	423	7	‖s‖2	‖s‖2	PROPN
ma-119	423	8	−	−	PROPN
ma-119	423	9	|α1|2)1/2	|α1|2)1/2	VERB
ma-119	424	1	+	+	CCONJ
ma-119	424	2	(	(	PUNCT
ma-119	424	3	‖s0‖2	‖s0‖2	NUM
ma-119	424	4	−	−	PROPN
ma-119	424	5	|α2|2)1/2	|α2|2)1/2	PROPN
ma-119	424	6	.	.	PUNCT
ma-119	425	1	proof	proof	NOUN
ma-119	425	2	.	.	PUNCT
ma-119	426	1	by	by	ADP
ma-119	426	2	definition	definition	NOUN
ma-119	426	3	of	of	ADP
ma-119	426	4	w0(s	w0(s	PROPN
ma-119	426	5	)	)	PUNCT
ma-119	426	6	we	we	PRON
ma-119	426	7	have	have	VERB
ma-119	426	8	xn	xn	PROPN
ma-119	426	9	∈	∈	PROPN
ma-119	426	10	h	h	NOUN
ma-119	426	11	such	such	ADJ
ma-119	426	12	that	that	SCONJ
ma-119	426	13	‖sxn‖	‖sxn‖	PUNCT
ma-119	426	14	=	=	SYM
ma-119	426	15	‖s‖	‖s‖	PROPN
ma-119	426	16	and	and	CCONJ
ma-119	426	17	〈	〈	PROPN
ma-119	426	18	sxn	sxn	PROPN
ma-119	426	19	,	,	PUNCT
ma-119	426	20	xn	xn	PROPN
ma-119	426	21	〉	〉	NUM
ma-119	426	22	→	→	SYM
ma-119	426	23	α1for	α1for	ADP
ma-119	426	24	α1	α1	PROPN
ma-119	426	25	∈	∈	PROPN
ma-119	426	26	w0(s	w0(s	PROPN
ma-119	426	27	)	)	PUNCT
ma-119	426	28	.	.	PUNCT
ma-119	427	1	this	this	DET
ma-119	427	2	argument	argument	NOUN
ma-119	427	3	follows	follow	VERB
ma-119	427	4	for	for	ADP
ma-119	427	5	w0(s0	w0(s0	ADV
ma-119	427	6	)	)	PUNCT
ma-119	427	7	and	and	CCONJ
ma-119	427	8	α2	α2	PROPN
ma-119	427	9	∈	∈	PROPN
ma-119	427	10	w0(s0	w0(s0	ADV
ma-119	427	11	)	)	PUNCT
ma-119	427	12	.	.	PUNCT
ma-119	428	1	let	let	VERB
ma-119	428	2	sxn	sxn	NOUN
ma-119	428	3	=	=	SYM
ma-119	428	4	δnxn	δnxn	NOUN
ma-119	428	5	+	+	NUM
ma-119	428	6	βnynso	βnynso	NOUN
ma-119	428	7	s0xn	s0xn	PUNCT
ma-119	429	1	=	=	SYM
ma-119	429	2	σnxn	σnxn	PROPN
ma-119	429	3	+	+	CCONJ
ma-119	429	4	λnyn	λnyn	PROPN
ma-119	429	5	where	where	SCONJ
ma-119	429	6	〈	〈	PROPN
ma-119	429	7	xn	xn	PROPN
ma-119	429	8	,	,	PUNCT
ma-119	429	9	yn	yn	PROPN
ma-119	429	10	〉	〉	PROPN
ma-119	429	11	=	=	SYM
ma-119	429	12	0	0	NUM
ma-119	429	13	,	,	PUNCT
ma-119	429	14	‖yn‖	‖yn‖	X
ma-119	429	15	=	=	PUNCT
ma-119	430	1	1	1	X
ma-119	430	2	.	.	X
ma-119	430	3	take	take	VERB
ma-119	430	4	unxn	unxn	NOUN
ma-119	430	5	=	=	PUNCT
ma-119	430	6	xn	xn	PROPN
ma-119	430	7	and	and	CCONJ
ma-119	430	8	unyn	unyn	NOUN
ma-119	430	9	=	=	PUNCT
ma-119	430	10	−yn	−yn	NOUN
ma-119	430	11	for	for	ADP
ma-119	430	12	un	un	PROPN
ma-119	430	13	=	=	PROPN
ma-119	430	14	0	0	PROPN
ma-119	430	15	in	in	ADP
ma-119	430	16	{	{	PUNCT
ma-119	430	17	xn	xn	PROPN
ma-119	430	18	,	,	PUNCT
ma-119	430	19	yn	yn	PROPN
ma-119	430	20	}	}	PUNCT
ma-119	430	21	.	.	PUNCT
ma-119	431	1	then	then	ADV
ma-119	431	2	‖sunxn	‖sunxn	NOUN
ma-119	431	3	−	−	NOUN
ma-119	431	4	uns0xn‖	uns0xn‖	ADJ
ma-119	431	5	=	=	PUNCT
ma-119	431	6	‖δn	‖δn	NUM
ma-119	431	7	+	+	NUM
ma-119	431	8	βn‖	βn‖	NOUN
ma-119	431	9	≤	≤	ADJ
ma-119	431	10	|δn|	|δn|	PROPN
ma-119	431	11	+	+	CCONJ
ma-119	431	12	|βn|	|βn|	PROPN
ma-119	431	13	.	.	PUNCT
ma-119	432	1	but	but	CCONJ
ma-119	432	2	|δn|	|δn|	PROPN
ma-119	432	3	+	+	CCONJ
ma-119	432	4	|βn|	|βn|	PROPN
ma-119	432	5	≥	≥	NUM
ma-119	432	6	(	(	PUNCT
ma-119	432	7	‖s‖2	‖s‖2	PROPN
ma-119	432	8	−	−	PROPN
ma-119	432	9	|δn|2)1/2	|δn|2)1/2	PROPN
ma-119	432	10	−	−	PROPN
ma-119	433	1	ξn	ξn	PROPN
ma-119	433	2	+	+	CCONJ
ma-119	433	3	(	(	PUNCT
ma-119	433	4	‖s0‖2	‖s0‖2	NUM
ma-119	433	5	−	−	NOUN
ma-119	433	6	|βn|2)1/2	|βn|2)1/2	VERB
ma-119	433	7	−	−	PROPN
ma-119	433	8	ξn	ξn	PROPN
ma-119	433	9	)	)	PUNCT
ma-119	433	10	.	.	PUNCT
ma-119	434	1	since	since	SCONJ
ma-119	434	2	ξn	ξn	PROPN
ma-119	434	3	is	be	AUX
ma-119	434	4	arbitrary	arbitrary	ADJ
ma-119	434	5	and	and	CCONJ
ma-119	434	6	letting	let	VERB
ma-119	434	7	n	n	PRON
ma-119	434	8	→∞	→∞	NOUN
ma-119	434	9	,	,	PUNCT
ma-119	434	10	so	so	SCONJ
ma-119	434	11	itfollows	itfollow	VERB
ma-119	434	12	that	that	SCONJ
ma-119	434	13	‖δs	‖δs	NUM
ma-119	434	14	,	,	PUNCT
ma-119	434	15	s0‖	s0‖	PROPN
ma-119	434	16	≥	≥	NOUN
ma-119	434	17	‖(sun−uns0)xn‖	‖(sun−uns0)xn‖	PUNCT
ma-119	434	18	=	=	SYM
ma-119	434	19	|δn|+|βn|	|δn|+|βn|	NOUN
ma-119	434	20	=	=	SYM
ma-119	434	21	(	(	PUNCT
ma-119	434	22	‖s‖2−|α1|2)1/2+(‖s0‖2−|α2|2)1/2	‖s‖2−|α1|2)1/2+(‖s0‖2−|α2|2)1/2	PROPN
ma-119	434	23	.	.	PUNCT
ma-119	434	24	�	�	PROPN
ma-119	434	25	corollary	corollary	PROPN
ma-119	434	26	33	33	NUM
ma-119	434	27	.	.	PUNCT
ma-119	435	1	let	let	VERB
ma-119	435	2	〈	〈	PROPN
ma-119	435	3	xn	xn	PROPN
ma-119	435	4	,	,	PUNCT
ma-119	435	5	yn	yn	PROPN
ma-119	435	6	〉	〉	NUM
ma-119	435	7	=	=	SYM
ma-119	435	8	0	0	PUNCT
ma-119	436	1	then	then	ADV
ma-119	436	2	0	0	NUM
ma-119	436	3	∈	∈	PROPN
ma-119	436	4	w0(s	w0(s	PROPN
ma-119	436	5	)	)	PUNCT
ma-119	436	6	and	and	CCONJ
ma-119	436	7	if	if	SCONJ
ma-119	436	8	0	0	NUM
ma-119	436	9	∈	∈	PROPN
ma-119	436	10	w0(s0	w0(s0	ADV
ma-119	436	11	)	)	PUNCT
ma-119	436	12	then	then	ADV
ma-119	436	13	‖δs	‖δs	NUM
ma-119	436	14	,	,	PUNCT
ma-119	436	15	s0‖	s0‖	NOUN
ma-119	436	16	≥	≥	NOUN
ma-119	436	17	‖s‖+	‖s‖+	NUM
ma-119	436	18	‖s0‖.	‖s0‖.	PROPN
ma-119	436	19	proof	proof	NOUN
ma-119	436	20	.	.	PUNCT
ma-119	436	21	follows	follow	VERB
ma-119	436	22	immediately	immediately	ADV
ma-119	436	23	from	from	ADP
ma-119	436	24	definition	definition	NOUN
ma-119	436	25	of	of	ADP
ma-119	436	26	w0(s	w0(s	PROPN
ma-119	436	27	)	)	PUNCT
ma-119	436	28	and	and	CCONJ
ma-119	436	29	the	the	DET
ma-119	436	30	theorem	theorem	ADJ
ma-119	436	31	32	32	NUM
ma-119	436	32	.	.	PUNCT
ma-119	437	1	�	�	PROPN
ma-119	437	2	4	4	NUM
ma-119	437	3	.	.	PUNCT
ma-119	437	4	conclusion	conclusion	NOUN
ma-119	437	5	in	in	ADP
ma-119	437	6	this	this	DET
ma-119	437	7	paper	paper	NOUN
ma-119	437	8	,	,	PUNCT
ma-119	437	9	we	we	PRON
ma-119	437	10	have	have	AUX
ma-119	437	11	given	give	VERB
ma-119	437	12	a	a	DET
ma-119	437	13	detailed	detailed	ADJ
ma-119	437	14	characterization	characterization	NOUN
ma-119	437	15	of	of	ADP
ma-119	437	16	operators	operator	NOUN
ma-119	437	17	in	in	ADP
ma-119	437	18	terms	term	NOUN
ma-119	437	19	of	of	ADP
ma-119	437	20	norm	norm	NOUN
ma-119	437	21	-	-	PUNCT
ma-119	437	22	attainabilityconditions	attainabilitycondition	NOUN
ma-119	437	23	and	and	CCONJ
ma-119	437	24	norm	norm	NOUN
ma-119	437	25	estimates	estimate	NOUN
ma-119	437	26	for	for	ADP
ma-119	437	27	in	in	ADP
ma-119	437	28	banach	banach	NOUN
ma-119	437	29	algebras	algebra	NOUN
ma-119	437	30	.	.	PUNCT
ma-119	438	1	in	in	ADP
ma-119	438	2	particular	particular	ADJ
ma-119	438	3	,	,	PUNCT
ma-119	438	4	we	we	PRON
ma-119	438	5	have	have	AUX
ma-119	438	6	established	establish	VERB
ma-119	438	7	norm	norm	NOUN
ma-119	438	8	-	-	PUNCT
ma-119	438	9	attainability	attainability	NOUN
ma-119	438	10	conditions	condition	NOUN
ma-119	438	11	for	for	ADP
ma-119	438	12	the	the	DET
ma-119	438	13	derivations	derivation	NOUN
ma-119	438	14	and	and	CCONJ
ma-119	438	15	also	also	ADV
ma-119	438	16	given	give	VERB
ma-119	438	17	the	the	DET
ma-119	438	18	norm	norm	NOUN
ma-119	438	19	bounds	bound	NOUN
ma-119	438	20	in	in	ADP
ma-119	438	21	the	the	DET
ma-119	438	22	norm	norm	NOUN
ma-119	438	23	-	-	PUNCT
ma-119	438	24	attainableclasses	attainableclasse	NOUN
ma-119	438	25	.	.	PUNCT
ma-119	439	1	references	reference	NOUN
ma-119	439	2	[	[	X
ma-119	439	3	1	1	NUM
ma-119	439	4	]	]	X
ma-119	439	5	r.j	r.j	PROPN
ma-119	439	6	.	.	PROPN
ma-119	439	7	archbold	archbold	PROPN
ma-119	439	8	,	,	PUNCT
ma-119	439	9	on	on	ADP
ma-119	439	10	the	the	DET
ma-119	439	11	norm	norm	NOUN
ma-119	439	12	of	of	ADP
ma-119	439	13	an	an	DET
ma-119	439	14	inner	inner	ADJ
ma-119	439	15	derivation	derivation	NOUN
ma-119	439	16	of	of	ADP
ma-119	439	17	a	a	DET
ma-119	439	18	c∗-algebra	c∗-algebra	PROPN
ma-119	439	19	,	,	PUNCT
ma-119	439	20	math	math	NOUN
ma-119	439	21	.	.	PUNCT
ma-119	440	1	proc	proc	PROPN
ma-119	440	2	.	.	PUNCT
ma-119	441	1	camb	camb	PROPN
ma-119	441	2	.	.	PUNCT
ma-119	442	1	phil	phil	PROPN
ma-119	442	2	.	.	PUNCT
ma-119	443	1	soc	soc	PROPN
ma-119	443	2	.	.	PUNCT
ma-119	444	1	84	84	NUM
ma-119	444	2	(	(	PUNCT
ma-119	444	3	1978	1978	NUM
ma-119	444	4	)	)	PUNCT
ma-119	445	1	273–291	273–291	NUM
ma-119	445	2	.	.	PUNCT
ma-119	446	1	https://doi.org/10.1017/s0305004100055109.[2	https://doi.org/10.1017/s0305004100055109.[2	NOUN
ma-119	446	2	]	]	X
ma-119	446	3	n.m	n.m	PROPN
ma-119	446	4	.	.	PROPN
ma-119	446	5	abolfazl	abolfazl	PROPN
ma-119	446	6	,	,	PUNCT
ma-119	446	7	on	on	ADP
ma-119	446	8	the	the	DET
ma-119	446	9	norm	norm	NOUN
ma-119	446	10	of	of	ADP
ma-119	446	11	jordan	jordan	PROPN
ma-119	446	12	∗-derivations	∗-derivations	PROPN
ma-119	446	13	,	,	PUNCT
ma-119	446	14	khaayyam	khaayyam	PROPN
ma-119	446	15	j.	j.	PROPN
ma-119	446	16	math	math	PROPN
ma-119	446	17	.	.	PUNCT
ma-119	447	1	6	6	NUM
ma-119	447	2	(	(	PUNCT
ma-119	447	3	2020	2020	NUM
ma-119	447	4	)	)	PUNCT
ma-119	447	5	104	104	NUM
ma-119	447	6	-	-	SYM
ma-119	447	7	107	107	NUM
ma-119	447	8	.	.	PUNCT
ma-119	448	1	https://doi.org/10	https://doi.org/10	PROPN
ma-119	448	2	.	.	PUNCT
ma-119	449	1	22034	22034	NUM
ma-119	449	2	/	/	SYM
ma-119	449	3	kjm.2019.97176.[3	kjm.2019.97176.[3	PROPN
ma-119	449	4	]	]	X
ma-119	449	5	j.	j.	PROPN
ma-119	449	6	anderson	anderson	PROPN
ma-119	449	7	,	,	PUNCT
ma-119	449	8	on	on	ADP
ma-119	449	9	normal	normal	ADJ
ma-119	449	10	derivations	derivation	NOUN
ma-119	449	11	,	,	PUNCT
ma-119	449	12	proc	proc	NOUN
ma-119	449	13	.	.	PUNCT
ma-119	450	1	amer	amer	PROPN
ma-119	450	2	.	.	PUNCT
ma-119	450	3	math	math	PROPN
ma-119	450	4	.	.	PUNCT
ma-119	451	1	soc	soc	PROPN
ma-119	451	2	.	.	PUNCT
ma-119	452	1	38	38	NUM
ma-119	452	2	(	(	PUNCT
ma-119	452	3	1973	1973	NUM
ma-119	452	4	)	)	PUNCT
ma-119	452	5	135–140	135–140	NUM
ma-119	452	6	.	.	PUNCT
ma-119	453	1	https://doi.org/10.1090/	https://doi.org/10.1090/	DET
ma-119	453	2	s0002	s0002	NOUN
ma-119	453	3	-	-	PUNCT
ma-119	453	4	9939	9939	NUM
ma-119	453	5	-	-	PUNCT
ma-119	453	6	1973	1973	NUM
ma-119	453	7	-	-	PUNCT
ma-119	453	8	0312313	0312313	NUM
ma-119	453	9	-	-	SYM
ma-119	453	10	6.[4	6.[4	PROPN
ma-119	453	11	]	]	X
ma-119	453	12	m.	m.	NOUN
ma-119	453	13	barraa	barraa	NOUN
ma-119	453	14	,	,	PUNCT
ma-119	453	15	m.	m.	NOUN
ma-119	453	16	boumazgour	boumazgour	NOUN
ma-119	453	17	,	,	PUNCT
ma-119	453	18	inner	inner	ADJ
ma-119	453	19	derivations	derivation	NOUN
ma-119	453	20	and	and	CCONJ
ma-119	453	21	norm	norm	NOUN
ma-119	453	22	equality	equality	NOUN
ma-119	453	23	,	,	PUNCT
ma-119	453	24	proc	proc	NOUN
ma-119	453	25	.	.	PUNCT
ma-119	454	1	amer	amer	PROPN
ma-119	454	2	.	.	PUNCT
ma-119	454	3	math	math	PROPN
ma-119	454	4	.	.	PUNCT
ma-119	455	1	soc	soc	PROPN
ma-119	455	2	.	.	PUNCT
ma-119	456	1	130	130	NUM
ma-119	456	2	(	(	PUNCT
ma-119	456	3	2001	2001	NUM
ma-119	456	4	)	)	PUNCT
ma-119	456	5	471	471	NUM
ma-119	456	6	-	-	SYM
ma-119	456	7	476	476	NUM
ma-119	456	8	.	.	PUNCT
ma-119	457	1	https://www.jstor.org/stable/2699643.[5	https://www.jstor.org/stable/2699643.[5	PRON
ma-119	457	2	]	]	X
ma-119	457	3	a.f	a.f	PROPN
ma-119	457	4	.	.	PROPN
ma-119	457	5	ber	ber	PROPN
ma-119	457	6	,	,	PUNCT
ma-119	457	7	f.a	f.a	PROPN
ma-119	457	8	.	.	PROPN
ma-119	457	9	sukochev	sukochev	PROPN
ma-119	457	10	,	,	PUNCT
ma-119	457	11	commutator	commutator	NOUN
ma-119	457	12	estimates	estimate	NOUN
ma-119	457	13	in	in	ADP
ma-119	457	14	w	w	PROPN
ma-119	457	15	∗-factors	∗-factor	NOUN
ma-119	457	16	,	,	PUNCT
ma-119	457	17	trans	trans	PROPN
ma-119	457	18	.	.	PROPN
ma-119	458	1	amer	amer	PROPN
ma-119	458	2	.	.	PUNCT
ma-119	458	3	math	math	PROPN
ma-119	458	4	.	.	PUNCT
ma-119	459	1	soc	soc	PROPN
ma-119	459	2	.	.	PUNCT
ma-119	460	1	364	364	NUM
ma-119	460	2	(	(	PUNCT
ma-119	460	3	2012	2012	NUM
ma-119	460	4	)	)	PUNCT
ma-119	460	5	5571–5587	5571–5587	NUM
ma-119	460	6	.	.	PUNCT
ma-119	461	1	https://doi.org/10.1090/s0002-9947-2012-05568-1.[6	https://doi.org/10.1090/s0002-9947-2012-05568-1.[6	PROPN
ma-119	461	2	]	]	X
ma-119	461	3	j.o	j.o	PROPN
ma-119	461	4	.	.	PROPN
ma-119	461	5	bonyo	bonyo	PROPN
ma-119	461	6	,	,	PUNCT
ma-119	461	7	j.o	j.o	PROPN
ma-119	461	8	.	.	PROPN
ma-119	461	9	agure	agure	PROPN
ma-119	461	10	,	,	PUNCT
ma-119	461	11	norms	norm	NOUN
ma-119	461	12	of	of	ADP
ma-119	461	13	derivations	derivation	NOUN
ma-119	461	14	implemented	implement	VERB
ma-119	461	15	by	by	ADP
ma-119	461	16	s	s	NOUN
ma-119	461	17	-	-	ADJ
ma-119	461	18	universal	universal	ADJ
ma-119	461	19	operators	operator	NOUN
ma-119	461	20	,	,	PUNCT
ma-119	461	21	int	int	PROPN
ma-119	461	22	.	.	PUNCT
ma-119	462	1	j.	j.	PROPN
ma-119	462	2	math	math	PROPN
ma-119	462	3	.	.	PUNCT
ma-119	463	1	anal	anal	ADJ
ma-119	463	2	.	.	PUNCT
ma-119	464	1	5	5	NUM
ma-119	464	2	(	(	PUNCT
ma-119	464	3	2011)215	2011)215	NOUN
ma-119	464	4	-	-	PUNCT
ma-119	464	5	222.[7	222.[7	NUM
ma-119	464	6	]	]	X
ma-119	464	7	j.o	j.o	PROPN
ma-119	464	8	.	.	PROPN
ma-119	464	9	bonyo	bonyo	PROPN
ma-119	464	10	,	,	PUNCT
ma-119	464	11	j.o	j.o	PROPN
ma-119	464	12	.	.	PROPN
ma-119	464	13	agure	agure	PROPN
ma-119	464	14	,	,	PUNCT
ma-119	464	15	norm	norm	NOUN
ma-119	464	16	of	of	ADP
ma-119	464	17	a	a	DET
ma-119	464	18	derivation	derivation	NOUN
ma-119	464	19	and	and	CCONJ
ma-119	464	20	hyponormal	hyponormal	ADJ
ma-119	464	21	operators	operator	NOUN
ma-119	464	22	,	,	PUNCT
ma-119	464	23	int	int	NOUN
ma-119	464	24	.	.	PUNCT
ma-119	465	1	j.	j.	PROPN
ma-119	465	2	math	math	PROPN
ma-119	465	3	.	.	PUNCT
ma-119	466	1	anal	anal	ADJ
ma-119	466	2	.	.	PUNCT
ma-119	467	1	4	4	NUM
ma-119	467	2	(	(	PUNCT
ma-119	467	3	2010	2010	NUM
ma-119	467	4	)	)	PUNCT
ma-119	467	5	687	687	NUM
ma-119	467	6	-	-	SYM
ma-119	467	7	693.[8	693.[8	PROPN
ma-119	467	8	]	]	X
ma-119	467	9	j.o	j.o	PROPN
ma-119	467	10	.	.	PROPN
ma-119	467	11	bonyo	bonyo	PROPN
ma-119	467	12	,	,	PUNCT
ma-119	467	13	j.o	j.o	PROPN
ma-119	467	14	.	.	PROPN
ma-119	467	15	agure	agure	PROPN
ma-119	467	16	,	,	PUNCT
ma-119	467	17	norms	norm	NOUN
ma-119	467	18	of	of	ADP
ma-119	467	19	inner	inner	ADJ
ma-119	467	20	derivations	derivation	NOUN
ma-119	467	21	on	on	ADP
ma-119	467	22	norm	norm	NOUN
ma-119	467	23	ideals	ideal	NOUN
ma-119	467	24	,	,	PUNCT
ma-119	467	25	int	int	NOUN
ma-119	467	26	.	.	PUNCT
ma-119	468	1	j.	j.	PROPN
ma-119	468	2	math	math	PROPN
ma-119	468	3	.	.	PUNCT
ma-119	469	1	anal	anal	ADJ
ma-119	469	2	.	.	PUNCT
ma-119	470	1	4	4	NUM
ma-119	470	2	(	(	PUNCT
ma-119	470	3	2010	2010	NUM
ma-119	470	4	)	)	PUNCT
ma-119	470	5	695	695	NUM
ma-119	470	6	-	-	SYM
ma-119	470	7	701.[9	701.[9	NUM
ma-119	470	8	]	]	PUNCT
ma-119	470	9	m.	m.	NOUN
ma-119	470	10	bresar	bresar	PROPN
ma-119	470	11	,	,	PUNCT
ma-119	470	12	b.	b.	PROPN
ma-119	470	13	zalar	zalar	PROPN
ma-119	470	14	,	,	PUNCT
ma-119	470	15	on	on	ADP
ma-119	470	16	the	the	DET
ma-119	470	17	structure	structure	NOUN
ma-119	470	18	of	of	ADP
ma-119	470	19	jordan	jordan	PROPN
ma-119	470	20	∗-derivations	∗-derivations	PROPN
ma-119	470	21	,	,	PUNCT
ma-119	470	22	colloq	colloq	PROPN
ma-119	470	23	.	.	PUNCT
ma-119	470	24	math	math	PROPN
ma-119	470	25	.	.	PUNCT
ma-119	471	1	63	63	NUM
ma-119	471	2	(	(	PUNCT
ma-119	471	3	1992	1992	NUM
ma-119	471	4	)	)	PUNCT
ma-119	471	5	163	163	NUM
ma-119	471	6	-	-	SYM
ma-119	471	7	171.[10	171.[10	NUM
ma-119	471	8	]	]	PUNCT
ma-119	471	9	m.	m.	NOUN
ma-119	471	10	cabrera	cabrera	PROPN
ma-119	471	11	,	,	PUNCT
ma-119	471	12	a.	a.	NOUN
ma-119	471	13	rodriguez	rodriguez	PROPN
ma-119	471	14	,	,	PUNCT
ma-119	471	15	nondegenerately	nondegenerately	ADV
ma-119	471	16	ultraprint	ultraprint	PROPN
ma-119	471	17	jordan	jordan	PROPN
ma-119	471	18	banach	banach	PROPN
ma-119	471	19	algebras	algebras	PROPN
ma-119	471	20	,	,	PUNCT
ma-119	471	21	proc	proc	NOUN
ma-119	471	22	.	.	PUNCT
ma-119	472	1	london	london	PROPN
ma-119	472	2	math	math	PROPN
ma-119	472	3	.	.	PUNCT
ma-119	473	1	soc	soc	PROPN
ma-119	473	2	.	.	PUNCT
ma-119	474	1	69	69	NUM
ma-119	474	2	(	(	PUNCT
ma-119	474	3	1994)576	1994)576	PROPN
ma-119	474	4	-	-	PUNCT
ma-119	474	5	604.[11	604.[11	PROPN
ma-119	474	6	]	]	PUNCT
ma-119	474	7	a.a	a.a	PROPN
ma-119	474	8	.	.	PROPN
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ma-119	474	20	c∗-algebras	c∗-algebra	NOUN
ma-119	474	21	,	,	PUNCT
ma-119	474	22	pac	pac	PROPN
ma-119	474	23	.	.	PUNCT
ma-119	475	1	j.	j.	PROPN
ma-119	475	2	math	math	PROPN
ma-119	475	3	.	.	PROPN
ma-119	476	1	85	85	NUM
ma-119	476	2	(	(	PUNCT
ma-119	476	3	1979	1979	NUM
ma-119	476	4	)	)	PUNCT
ma-119	476	5	79	79	NUM
ma-119	476	6	-	-	SYM
ma-119	476	7	96.[12	96.[12	NUM
ma-119	476	8	]	]	X
ma-119	476	9	g.	g.	PROPN
ma-119	476	10	clifford	clifford	PROPN
ma-119	476	11	,	,	PUNCT
ma-119	476	12	dynamics	dynamic	NOUN
ma-119	476	13	of	of	ADP
ma-119	476	14	generalized	generalized	ADJ
ma-119	476	15	derivations	derivation	NOUN
ma-119	476	16	and	and	CCONJ
ma-119	476	17	elementary	elementary	ADJ
ma-119	476	18	operators	operator	NOUN
ma-119	476	19	,	,	PUNCT
ma-119	476	20	(	(	PUNCT
ma-119	476	21	2017	2017	NUM
ma-119	476	22	)	)	PUNCT
ma-119	476	23	,	,	PUNCT
ma-119	476	24	arxiv:1605.07409v2	arxiv:1605.07409v2	VERB
ma-119	477	1	[	[	X
ma-119	477	2	math.fa	math.fa	X
ma-119	477	3	]	]	PUNCT
ma-119	477	4	.	.	PUNCT
ma-119	478	1	https://arxiv.org/abs/1605.07409v2.[13	https://arxiv.org/abs/1605.07409v2.[13	X
ma-119	479	1	]	]	X
ma-119	479	2	c.k	c.k	PROPN
ma-119	479	3	.	.	PUNCT
ma-119	479	4	li	li	PROPN
ma-119	479	5	,	,	PUNCT
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ma-119	479	10	range	range	NOUN
ma-119	479	11	,	,	PUNCT
ma-119	479	12	2005	2005	NUM
ma-119	479	13	.	.	PUNCT
ma-119	480	1	http://www.math.wm.edu/~ckli/nrnote	http://www.math.wm.edu/~ckli/nrnote	VERB
ma-119	480	2	.	.	PUNCT
ma-119	481	1	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	NUM
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ma-119	481	3	https://doi.org/10.22034/kjm.2019.97176	https://doi.org/10.22034/kjm.2019.97176	NOUN
ma-119	481	4	https://doi.org/10.22034/kjm.2019.97176	https://doi.org/10.22034/kjm.2019.97176	NOUN
ma-119	481	5	https://doi.org/10.1090/s0002-9939-1973-0312313-6	https://doi.org/10.1090/s0002-9939-1973-0312313-6	PROPN
ma-119	481	6	https://doi.org/10.1090/s0002-9939-1973-0312313-6	https://doi.org/10.1090/s0002-9939-1973-0312313-6	PROPN
ma-119	481	7	https://www.jstor.org/stable/2699643	https://www.jstor.org/stable/2699643	PROPN
ma-119	481	8	https://doi.org/10.1090/s0002-9947-2012-05568-1	https://doi.org/10.1090/s0002-9947-2012-05568-1	PROPN
ma-119	481	9	https://arxiv.org/abs/1605.07409v2	https://arxiv.org/abs/1605.07409v2	PUNCT
ma-119	481	10	http://www.math.wm.edu/~ckli/nrnote	http://www.math.wm.edu/~ckli/nrnote	PROPN
ma-119	481	11	eur	eur	NOUN
ma-119	481	12	.	.	PUNCT
ma-119	482	1	j.	j.	PROPN
ma-119	482	2	math	math	PROPN
ma-119	482	3	.	.	PUNCT
ma-119	483	1	anal	anal	PROPN
ma-119	483	2	.	.	PUNCT
ma-119	484	1	10.28924	10.28924	NUM
ma-119	484	2	/	/	SYM
ma-119	484	3	ada	ada	PROPN
ma-119	484	4	/	/	SYM
ma-119	484	5	ma.3.9	ma.3.9	PROPN
ma-119	484	6	14	14	NUM
ma-119	484	7	[	[	X
ma-119	484	8	14	14	NUM
ma-119	484	9	]	]	X
ma-119	484	10	d.r	d.r	PROPN
ma-119	484	11	.	.	PROPN
ma-119	484	12	jocić	jocić	PROPN
ma-119	484	13	,	,	PUNCT
ma-119	484	14	norm	norm	NOUN
ma-119	484	15	inequalities	inequality	NOUN
ma-119	484	16	for	for	ADP
ma-119	484	17	self	self	NOUN
ma-119	484	18	-	-	PUNCT
ma-119	484	19	adjoint	adjoint	NOUN
ma-119	484	20	derivations	derivation	NOUN
ma-119	484	21	,	,	PUNCT
ma-119	484	22	j.	j.	PROPN
ma-119	484	23	funct	funct	PROPN
ma-119	484	24	.	.	PUNCT
ma-119	485	1	anal	anal	PROPN
ma-119	485	2	.	.	PUNCT
ma-119	486	1	145	145	NUM
ma-119	486	2	(	(	PUNCT
ma-119	486	3	1997	1997	NUM
ma-119	486	4	)	)	PUNCT
ma-119	486	5	24–34	24–34	NUM
ma-119	486	6	.	.	PUNCT
ma-119	486	7	https://doi.org/10	https://doi.org/10	PROPN
ma-119	486	8	.	.	PUNCT
ma-119	487	1	1006	1006	NUM
ma-119	487	2	/	/	SYM
ma-119	487	3	jfan.1996.3004.[15	jfan.1996.3004.[15	PROPN
ma-119	487	4	]	]	PUNCT
ma-119	487	5	d.w.b	d.w.b	PROPN
ma-119	487	6	.	.	PROPN
ma-119	487	7	somerset	somerset	PROPN
ma-119	487	8	,	,	PUNCT
ma-119	487	9	the	the	DET
ma-119	487	10	inner	inner	ADJ
ma-119	487	11	derivations	derivation	NOUN
ma-119	487	12	and	and	CCONJ
ma-119	487	13	the	the	DET
ma-119	487	14	primitive	primitive	ADJ
ma-119	487	15	ideal	ideal	ADJ
ma-119	487	16	space	space	NOUN
ma-119	487	17	of	of	ADP
ma-119	487	18	a	a	DET
ma-119	487	19	c∗-algebra	c∗-algebra	PROPN
ma-119	487	20	,	,	PUNCT
ma-119	487	21	j.	j.	PROPN
ma-119	487	22	oper	oper	PROPN
ma-119	487	23	.	.	PROPN
ma-119	487	24	theory	theory	NOUN
ma-119	487	25	,	,	PUNCT
ma-119	487	26	29	29	NUM
ma-119	487	27	(	(	PUNCT
ma-119	487	28	1993)307	1993)307	NUM
ma-119	487	29	-	-	SYM
ma-119	487	30	321	321	NUM
ma-119	487	31	.	.	PUNCT
ma-119	487	32	https://www.jstor.org/stable/24714573.[16	https://www.jstor.org/stable/24714573.[16	NOUN
ma-119	487	33	]	]	PUNCT
ma-119	488	1	c.	c.	PROPN
ma-119	488	2	erik	erik	PROPN
ma-119	488	3	,	,	PUNCT
ma-119	488	4	extensions	extension	NOUN
ma-119	488	5	of	of	ADP
ma-119	488	6	derivations	derivations	PROPN
ma-119	488	7	ii	ii	PROPN
ma-119	488	8	,	,	PUNCT
ma-119	488	9	math	math	NOUN
ma-119	488	10	.	.	PUNCT
ma-119	489	1	scand	scand	PROPN
ma-119	489	2	.	.	PROPN
ma-119	490	1	50	50	NUM
ma-119	490	2	(	(	PUNCT
ma-119	490	3	1982	1982	NUM
ma-119	490	4	)	)	PUNCT
ma-119	490	5	111	111	NUM
ma-119	490	6	-	-	SYM
ma-119	490	7	122.[17	122.[17	NUM
ma-119	490	8	]	]	X
ma-119	490	9	f.p	f.p	PROPN
ma-119	490	10	.	.	PROPN
ma-119	490	11	boca	boca	PROPN
ma-119	490	12	,	,	PUNCT
ma-119	490	13	a.	a.	NOUN
ma-119	490	14	zaharescu	zaharescu	PROPN
ma-119	490	15	,	,	PUNCT
ma-119	490	16	norm	norm	NOUN
ma-119	490	17	estimates	estimate	NOUN
ma-119	490	18	of	of	ADP
ma-119	490	19	almost	almost	ADV
ma-119	490	20	mathieu	mathieu	PROPN
ma-119	490	21	operators	operators	PROPN
ma-119	490	22	,	,	PUNCT
ma-119	490	23	j.	j.	PROPN
ma-119	490	24	funct	funct	PROPN
ma-119	490	25	.	.	PUNCT
ma-119	491	1	anal	anal	PROPN
ma-119	491	2	.	.	PUNCT
ma-119	492	1	220	220	NUM
ma-119	492	2	(	(	PUNCT
ma-119	492	3	2005	2005	NUM
ma-119	492	4	)	)	PUNCT
ma-119	493	1	76–96	76–96	NUM
ma-119	493	2	.	.	PUNCT
ma-119	494	1	https	https	NOUN
ma-119	494	2	:	:	PUNCT
ma-119	495	1	//doi.org/10.1016	//doi.org/10.1016	PROPN
ma-119	495	2	/	/	SYM
ma-119	495	3	j.jfa.2004.09.013.[18	j.jfa.2004.09.013.[18	NUM
ma-119	495	4	]	]	PUNCT
ma-119	495	5	p.	p.	NOUN
ma-119	495	6	gajendragadkar	gajendragadkar	NOUN
ma-119	495	7	,	,	PUNCT
ma-119	495	8	norm	norm	NOUN
ma-119	495	9	of	of	ADP
ma-119	495	10	a	a	DET
ma-119	495	11	derivation	derivation	NOUN
ma-119	495	12	on	on	ADP
ma-119	495	13	a	a	DET
ma-119	495	14	von	von	PROPN
ma-119	495	15	neumann	neumann	PROPN
ma-119	495	16	algebra	algebra	PROPN
ma-119	495	17	,	,	PUNCT
ma-119	495	18	trans	trans	PROPN
ma-119	495	19	.	.	PROPN
ma-119	496	1	amer	amer	PROPN
ma-119	496	2	.	.	PUNCT
ma-119	496	3	math	math	PROPN
ma-119	496	4	.	.	PUNCT
ma-119	497	1	soc	soc	PROPN
ma-119	497	2	.	.	PUNCT
ma-119	498	1	170	170	NUM
ma-119	498	2	(	(	PUNCT
ma-119	498	3	1972	1972	NUM
ma-119	498	4	)	)	PUNCT
ma-119	499	1	165–165	165–165	NUM
ma-119	499	2	.	.	PUNCT
ma-119	500	1	https://doi.org/10.1090/s0002-9947-1972-0305090-x.[19	https://doi.org/10.1090/s0002-9947-1972-0305090-x.[19	PRON
ma-119	500	2	]	]	PUNCT
ma-119	500	3	h.k	h.k	PROPN
ma-119	500	4	.	.	PROPN
ma-119	500	5	du	du	PROPN
ma-119	500	6	,	,	PUNCT
ma-119	500	7	y.q	y.q	PROPN
ma-119	500	8	.	.	PROPN
ma-119	500	9	wang	wang	PROPN
ma-119	500	10	,	,	PUNCT
ma-119	501	1	g.b	g.b	PROPN
ma-119	501	2	.	.	PUNCT
ma-119	501	3	gao	gao	PROPN
ma-119	501	4	,	,	PUNCT
ma-119	501	5	norms	norm	NOUN
ma-119	501	6	of	of	ADP
ma-119	501	7	elementary	elementary	ADJ
ma-119	501	8	operators	operator	NOUN
ma-119	501	9	,	,	PUNCT
ma-119	501	10	proc	proc	PROPN
ma-119	501	11	.	.	PUNCT
ma-119	502	1	amer	amer	PROPN
ma-119	502	2	.	.	PUNCT
ma-119	502	3	math	math	PROPN
ma-119	502	4	.	.	PUNCT
ma-119	503	1	soc	soc	PROPN
ma-119	503	2	.	.	PUNCT
ma-119	504	1	136	136	NUM
ma-119	504	2	(	(	PUNCT
ma-119	504	3	2008	2008	NUM
ma-119	504	4	)	)	PUNCT
ma-119	504	5	1337	1337	NUM
ma-119	504	6	-	-	SYM
ma-119	504	7	1348	1348	NUM
ma-119	504	8	.	.	PUNCT
ma-119	505	1	https://doi.org/10.1090/s0002-9939-07-09112-5.[20	https://doi.org/10.1090/s0002-9939-07-09112-5.[20	PROPN
ma-119	505	2	]	]	PUNCT
ma-119	505	3	b.	b.	PROPN
ma-119	505	4	johnson	johnson	PROPN
ma-119	505	5	,	,	PUNCT
ma-119	505	6	characterization	characterization	NOUN
ma-119	505	7	and	and	CCONJ
ma-119	505	8	norms	norm	NOUN
ma-119	505	9	of	of	ADP
ma-119	505	10	derivations	derivation	NOUN
ma-119	505	11	on	on	ADP
ma-119	505	12	von	von	PROPN
ma-119	505	13	neumann	neumann	PROPN
ma-119	505	14	algebras	algebras	PROPN
ma-119	505	15	,	,	PUNCT
ma-119	505	16	in	in	ADP
ma-119	505	17	:	:	PUNCT
ma-119	505	18	p.	p.	NOUN
ma-119	505	19	de	de	X
ma-119	505	20	la	la	PROPN
ma-119	505	21	harpe	harpe	PROPN
ma-119	505	22	(	(	PUNCT
ma-119	505	23	ed	ed	NOUN
ma-119	505	24	.	.	PUNCT
ma-119	505	25	)	)	PUNCT
ma-119	505	26	,	,	PUNCT
ma-119	505	27	algèbresd’opérateurs	algèbresd’opérateur	NOUN
ma-119	505	28	,	,	PUNCT
ma-119	505	29	springer	springer	NOUN
ma-119	505	30	berlin	berlin	PROPN
ma-119	505	31	heidelberg	heidelberg	PROPN
ma-119	505	32	,	,	PUNCT
ma-119	505	33	berlin	berlin	PROPN
ma-119	505	34	,	,	PUNCT
ma-119	505	35	heidelberg	heidelberg	PROPN
ma-119	505	36	,	,	PUNCT
ma-119	505	37	1979	1979	NUM
ma-119	505	38	:	:	PUNCT
ma-119	505	39	pp	pp	ADP
ma-119	505	40	.	.	PUNCT
ma-119	506	1	228–236	228–236	NUM
ma-119	506	2	.	.	PUNCT
ma-119	506	3	https://doi.org/10.1007/	https://doi.org/10.1007/	PROPN
ma-119	506	4	bfb0062619.[21	bfb0062619.[21	PROPN
ma-119	506	5	]	]	X
ma-119	506	6	b.e	b.e	PROPN
ma-119	506	7	.	.	PROPN
ma-119	506	8	johnson	johnson	PROPN
ma-119	506	9	,	,	PUNCT
ma-119	506	10	norms	norm	NOUN
ma-119	506	11	of	of	ADP
ma-119	506	12	derivations	derivation	NOUN
ma-119	506	13	on	on	ADP
ma-119	506	14	l(x	l(x	PROPN
ma-119	506	15	)	)	PUNCT
ma-119	506	16	,	,	PUNCT
ma-119	506	17	pac	pac	PROPN
ma-119	506	18	.	.	PUNCT
ma-119	506	19	j.	j.	PROPN
ma-119	506	20	math	math	PROPN
ma-119	506	21	.	.	PUNCT
ma-119	507	1	38	38	NUM
ma-119	507	2	(	(	PUNCT
ma-119	507	3	1971	1971	NUM
ma-119	507	4	)	)	PUNCT
ma-119	507	5	465	465	NUM
ma-119	507	6	-	-	SYM
ma-119	507	7	469	469	NUM
ma-119	507	8	.	.	PUNCT
ma-119	508	1	https://doi.org/10.2140/pjm	https://doi.org/10.2140/pjm	X
ma-119	508	2	.	.	PUNCT
ma-119	509	1	1971.38.465.[22	1971.38.465.[22	NUM
ma-119	509	2	]	]	X
ma-119	509	3	e.	e.	PROPN
ma-119	509	4	kreyszig	kreyszig	PROPN
ma-119	509	5	,	,	PUNCT
ma-119	509	6	introduction	introduction	NOUN
ma-119	509	7	functional	functional	ADJ
ma-119	509	8	analysis	analysis	NOUN
ma-119	509	9	with	with	ADP
ma-119	509	10	applications	application	NOUN
ma-119	509	11	,	,	PUNCT
ma-119	509	12	book.canada	book.canada	NOUN
ma-119	509	13	publications	publication	NOUN
ma-119	509	14	,	,	PUNCT
ma-119	509	15	toronto	toronto	PROPN
ma-119	509	16	,	,	PUNCT
ma-119	509	17	1978.[23	1978.[23	NUM
ma-119	509	18	]	]	X
ma-119	509	19	n.p	n.p	PROPN
ma-119	509	20	.	.	PROPN
ma-119	509	21	landsman	landsman	NOUN
ma-119	509	22	,	,	PUNCT
ma-119	509	23	c∗-algebras	c∗-algebra	NOUN
ma-119	509	24	and	and	CCONJ
ma-119	509	25	quantum	quantum	ADJ
ma-119	509	26	mechanics	mechanic	NOUN
ma-119	509	27	.	.	PUNCT
ma-119	510	1	lecture	lecture	NOUN
ma-119	510	2	notes	note	NOUN
ma-119	510	3	,	,	PUNCT
ma-119	510	4	1998.[24	1998.[24	NUM
ma-119	510	5	]	]	X
ma-119	510	6	j.	j.	PROPN
ma-119	510	7	kyle	kyle	PROPN
ma-119	510	8	,	,	PUNCT
ma-119	510	9	numerical	numerical	ADJ
ma-119	510	10	ranges	range	NOUN
ma-119	510	11	of	of	ADP
ma-119	510	12	derivations	derivation	NOUN
ma-119	510	13	,	,	PUNCT
ma-119	510	14	proc	proc	NOUN
ma-119	510	15	.	.	PUNCT
ma-119	511	1	edinburgh	edinburgh	PROPN
ma-119	511	2	math	math	PROPN
ma-119	511	3	.	.	PUNCT
ma-119	512	1	soc	soc	PROPN
ma-119	512	2	.	.	PUNCT
ma-119	513	1	21	21	NUM
ma-119	513	2	(	(	PUNCT
ma-119	513	3	1978	1978	NUM
ma-119	513	4	)	)	PUNCT
ma-119	513	5	33	33	NUM
ma-119	513	6	-	-	SYM
ma-119	513	7	39	39	NUM
ma-119	513	8	.	.	PUNCT
ma-119	514	1	https://doi.org/10	https://doi.org/10	PROPN
ma-119	514	2	.	.	PUNCT
ma-119	515	1	1017	1017	NUM
ma-119	515	2	/	/	SYM
ma-119	515	3	s0013091500015856.[25	s0013091500015856.[25	PROPN
ma-119	515	4	]	]	PUNCT
ma-119	515	5	j.	j.	PROPN
ma-119	515	6	kyle	kyle	PROPN
ma-119	515	7	,	,	PUNCT
ma-119	515	8	norms	norm	NOUN
ma-119	515	9	of	of	ADP
ma-119	515	10	derivations	derivation	NOUN
ma-119	515	11	,	,	PUNCT
ma-119	515	12	j.	j.	PROPN
ma-119	515	13	london	london	PROPN
ma-119	515	14	math	math	PROPN
ma-119	515	15	.	.	PUNCT
ma-119	516	1	soc	soc	PROPN
ma-119	516	2	.	.	PUNCT
ma-119	517	1	16	16	NUM
ma-119	517	2	(	(	PUNCT
ma-119	517	3	1977	1977	NUM
ma-119	517	4	)	)	PUNCT
ma-119	517	5	297	297	NUM
ma-119	517	6	-	-	SYM
ma-119	517	7	312	312	NUM
ma-119	517	8	.	.	PUNCT
ma-119	518	1	https://doi.org/10.1112/jlms/s2-16	https://doi.org/10.1112/jlms/s2-16	X
ma-119	518	2	.	.	PUNCT
ma-119	518	3	2.297.[26	2.297.[26	PROPN
ma-119	518	4	]	]	X
ma-119	518	5	f.	f.	PROPN
ma-119	518	6	kittaneh	kittaneh	PROPN
ma-119	518	7	,	,	PUNCT
ma-119	518	8	normal	normal	ADJ
ma-119	518	9	derivations	derivation	NOUN
ma-119	518	10	in	in	ADP
ma-119	518	11	norm	norm	NOUN
ma-119	518	12	ideals	ideal	NOUN
ma-119	518	13	,	,	PUNCT
ma-119	518	14	proc	proc	NOUN
ma-119	518	15	.	.	PUNCT
ma-119	519	1	amer	amer	PROPN
ma-119	519	2	.	.	PUNCT
ma-119	519	3	math	math	PROPN
ma-119	519	4	.	.	PUNCT
ma-119	520	1	soc	soc	PROPN
ma-119	520	2	.	.	PUNCT
ma-119	521	1	123	123	NUM
ma-119	521	2	(	(	PUNCT
ma-119	521	3	1995	1995	NUM
ma-119	521	4	)	)	PUNCT
ma-119	521	5	1779	1779	NUM
ma-119	521	6	-	-	SYM
ma-119	521	7	1785	1785	NUM
ma-119	521	8	.	.	PUNCT
ma-119	522	1	https://doi.org/	https://doi.org/	VERB
ma-119	522	2	10.2307/2160991.[27	10.2307/2160991.[27	NUM
ma-119	522	3	]	]	X
ma-119	522	4	g.	g.	PROPN
ma-119	522	5	lumer	lumer	PROPN
ma-119	522	6	,	,	PUNCT
ma-119	522	7	complex	complex	ADJ
ma-119	522	8	methods	method	NOUN
ma-119	522	9	and	and	CCONJ
ma-119	522	10	the	the	DET
ma-119	522	11	estimation	estimation	NOUN
ma-119	522	12	of	of	ADP
ma-119	522	13	operator	operator	NOUN
ma-119	522	14	norms	norm	NOUN
ma-119	522	15	and	and	CCONJ
ma-119	522	16	spectra	spectra	NOUN
ma-119	522	17	from	from	ADP
ma-119	522	18	real	real	ADJ
ma-119	522	19	numerical	numerical	ADJ
ma-119	522	20	ranges	range	NOUN
ma-119	522	21	,	,	PUNCT
ma-119	522	22	j.	j.	PROPN
ma-119	522	23	funct.anal	funct.anal	PROPN
ma-119	522	24	.	.	PROPN
ma-119	523	1	10	10	NUM
ma-119	523	2	(	(	PUNCT
ma-119	523	3	1972	1972	NUM
ma-119	523	4	)	)	PUNCT
ma-119	523	5	482	482	NUM
ma-119	523	6	-	-	SYM
ma-119	523	7	495	495	NUM
ma-119	523	8	.	.	PUNCT
ma-119	524	1	https://doi.org/10.1016/0022-1236(72)90043-2.[28	https://doi.org/10.1016/0022-1236(72)90043-2.[28	PROPN
ma-119	524	2	]	]	PUNCT
ma-119	524	3	b.	b.	PROPN
ma-119	524	4	matej	matej	PROPN
ma-119	524	5	,	,	PUNCT
ma-119	524	6	on	on	ADP
ma-119	524	7	distance	distance	NOUN
ma-119	524	8	of	of	ADP
ma-119	524	9	the	the	DET
ma-119	524	10	composition	composition	NOUN
ma-119	524	11	of	of	ADP
ma-119	524	12	two	two	NUM
ma-119	524	13	derivations	derivation	NOUN
ma-119	524	14	to	to	ADP
ma-119	524	15	the	the	DET
ma-119	524	16	generalized	generalized	ADJ
ma-119	524	17	derivations	derivation	NOUN
ma-119	524	18	,	,	PUNCT
ma-119	524	19	glasgow	glasgow	PROPN
ma-119	524	20	math	math	NOUN
ma-119	524	21	.	.	PUNCT
ma-119	525	1	j.	j.	PROPN
ma-119	525	2	33(1991	33(1991	NUM
ma-119	525	3	)	)	PUNCT
ma-119	525	4	89	89	NUM
ma-119	525	5	-	-	SYM
ma-119	525	6	93	93	NUM
ma-119	525	7	.	.	PUNCT
ma-119	526	1	https://doi.org/10.1017/s0017089500008077.[29	https://doi.org/10.1017/s0017089500008077.[29	X
ma-119	526	2	]	]	X
ma-119	526	3	m.	m.	PROPN
ma-119	526	4	mathieu	mathieu	PROPN
ma-119	526	5	,	,	PUNCT
ma-119	526	6	more	more	ADJ
ma-119	526	7	properties	property	NOUN
ma-119	526	8	of	of	ADP
ma-119	526	9	the	the	DET
ma-119	526	10	product	product	NOUN
ma-119	526	11	of	of	ADP
ma-119	526	12	two	two	NUM
ma-119	526	13	derivations	derivation	NOUN
ma-119	526	14	of	of	ADP
ma-119	526	15	a	a	DET
ma-119	526	16	c∗-algebra	c∗-algebra	PROPN
ma-119	526	17	,	,	PUNCT
ma-119	526	18	bull	bull	NOUN
ma-119	526	19	.	.	PUNCT
ma-119	527	1	austral	austral	PROPN
ma-119	527	2	.	.	PUNCT
ma-119	528	1	math	math	NOUN
ma-119	528	2	.	.	PUNCT
ma-119	529	1	soc	soc	PROPN
ma-119	529	2	.	.	PUNCT
ma-119	530	1	42	42	NUM
ma-119	530	2	(	(	PUNCT
ma-119	530	3	1990)115	1990)115	PROPN
ma-119	530	4	-	-	SYM
ma-119	530	5	120	120	NUM
ma-119	530	6	.	.	PUNCT
ma-119	531	1	https://doi.org/10.1017/s0004972700028203.[30	https://doi.org/10.1017/s0004972700028203.[30	X
ma-119	531	2	]	]	X
ma-119	531	3	m.	m.	NOUN
ma-119	531	4	mathieu	mathieu	PROPN
ma-119	531	5	,	,	PUNCT
ma-119	531	6	elementary	elementary	PROPN
ma-119	531	7	operators	operator	NOUN
ma-119	531	8	on	on	ADP
ma-119	531	9	calkin	calkin	ADJ
ma-119	531	10	algebras	algebra	NOUN
ma-119	531	11	,	,	PUNCT
ma-119	531	12	irish	irish	ADJ
ma-119	531	13	math	math	NOUN
ma-119	531	14	.	.	PUNCT
ma-119	532	1	soc	soc	PROPN
ma-119	532	2	.	.	PUNCT
ma-119	533	1	bull	bull	NOUN
ma-119	533	2	.	.	PUNCT
ma-119	534	1	46	46	NUM
ma-119	534	2	(	(	PUNCT
ma-119	534	3	2001	2001	NUM
ma-119	534	4	)	)	PUNCT
ma-119	534	5	33	33	NUM
ma-119	534	6	-	-	SYM
ma-119	534	7	44.[31	44.[31	PROPN
ma-119	534	8	]	]	X
ma-119	534	9	s.	s.	PROPN
ma-119	534	10	mecheri	mecheri	PROPN
ma-119	534	11	,	,	PUNCT
ma-119	534	12	the	the	DET
ma-119	534	13	gateaux	gateaux	ADV
ma-119	534	14	derivative	derivative	ADJ
ma-119	534	15	orthogonality	orthogonality	NOUN
ma-119	534	16	in	in	ADP
ma-119	534	17	c∞.	c∞.	PROPN
ma-119	534	18	lecture	lecture	NOUN
ma-119	534	19	notes	note	NOUN
ma-119	534	20	,	,	PUNCT
ma-119	534	21	1991.[32	1991.[32	PROPN
ma-119	534	22	]	]	X
ma-119	534	23	r.e	r.e	PROPN
ma-119	534	24	.	.	PROPN
ma-119	534	25	megginson	megginson	PROPN
ma-119	534	26	,	,	PUNCT
ma-119	534	27	an	an	DET
ma-119	534	28	introduction	introduction	NOUN
ma-119	534	29	to	to	ADP
ma-119	534	30	banach	banach	NOUN
ma-119	534	31	space	space	NOUN
ma-119	534	32	theory	theory	NOUN
ma-119	534	33	,	,	PUNCT
ma-119	534	34	springer	springer	NOUN
ma-119	534	35	-	-	PUNCT
ma-119	534	36	verlag	verlag	PROPN
ma-119	534	37	,	,	PUNCT
ma-119	534	38	new	new	PROPN
ma-119	534	39	york	york	PROPN
ma-119	534	40	,	,	PUNCT
ma-119	534	41	1998.[33	1998.[33	NUM
ma-119	534	42	]	]	PUNCT
ma-119	534	43	m.	m.	NOUN
ma-119	534	44	arsenovic	arsenovic	PROPN
ma-119	534	45	,	,	PUNCT
ma-119	534	46	d.	d.	PROPN
ma-119	534	47	keckic	keckic	PROPN
ma-119	534	48	,	,	PUNCT
ma-119	534	49	elementary	elementary	ADJ
ma-119	534	50	operators	operator	NOUN
ma-119	534	51	on	on	ADP
ma-119	534	52	banach	banach	NOUN
ma-119	534	53	algebras	algebra	NOUN
ma-119	534	54	and	and	CCONJ
ma-119	534	55	fourier	fourier	NOUN
ma-119	534	56	transform	transform	NOUN
ma-119	534	57	,	,	PUNCT
ma-119	534	58	stud	stud	NOUN
ma-119	534	59	.	.	PUNCT
ma-119	535	1	math	math	NOUN
ma-119	535	2	.	.	PUNCT
ma-119	536	1	173	173	NUM
ma-119	536	2	(	(	PUNCT
ma-119	536	3	2006)149	2006)149	PROPN
ma-119	536	4	-	-	PUNCT
ma-119	536	5	166.[34	166.[34	NUM
ma-119	536	6	]	]	PUNCT
ma-119	536	7	n.b	n.b	PROPN
ma-119	536	8	.	.	PROPN
ma-119	536	9	okelo	okelo	PROPN
ma-119	536	10	,	,	PUNCT
ma-119	536	11	on	on	ADP
ma-119	536	12	orthogonality	orthogonality	NOUN
ma-119	536	13	of	of	ADP
ma-119	536	14	elementary	elementary	ADJ
ma-119	536	15	operators	operator	NOUN
ma-119	536	16	in	in	ADP
ma-119	536	17	norm	norm	NOUN
ma-119	536	18	-	-	PUNCT
ma-119	536	19	attainable	attainable	ADJ
ma-119	536	20	classes	class	NOUN
ma-119	536	21	,	,	PUNCT
ma-119	536	22	taiwan	taiwan	PROPN
ma-119	536	23	.	.	PUNCT
ma-119	537	1	j.	j.	PROPN
ma-119	537	2	math	math	PROPN
ma-119	537	3	.	.	PUNCT
ma-119	538	1	24	24	NUM
ma-119	538	2	(	(	PUNCT
ma-119	538	3	2020	2020	NUM
ma-119	538	4	)	)	PUNCT
ma-119	538	5	119	119	NUM
ma-119	538	6	-	-	SYM
ma-119	538	7	130	130	NUM
ma-119	538	8	.	.	PUNCT
ma-119	539	1	https://doi.org/10.11650/tjm/190502.[35	https://doi.org/10.11650/tjm/190502.[35	PROPN
ma-119	539	2	]	]	PUNCT
ma-119	539	3	n.b	n.b	PROPN
ma-119	539	4	.	.	PROPN
ma-119	539	5	okelo	okelo	PROPN
ma-119	539	6	,	,	PUNCT
ma-119	539	7	j.o	j.o	PROPN
ma-119	539	8	.	.	PROPN
ma-119	539	9	agure	agure	PROPN
ma-119	539	10	,	,	PUNCT
ma-119	539	11	d.o	d.o	PROPN
ma-119	539	12	.	.	PROPN
ma-119	539	13	ambogo	ambogo	PROPN
ma-119	539	14	,	,	PUNCT
ma-119	539	15	norms	norm	NOUN
ma-119	539	16	of	of	ADP
ma-119	539	17	elementary	elementary	ADJ
ma-119	539	18	operators	operator	NOUN
ma-119	539	19	and	and	CCONJ
ma-119	539	20	characterization	characterization	NOUN
ma-119	539	21	of	of	ADP
ma-119	539	22	norm	norm	NOUN
ma-119	539	23	-	-	PUNCT
ma-119	539	24	attainableoperators	attainableoperator	NOUN
ma-119	539	25	,	,	PUNCT
ma-119	539	26	int	int	NOUN
ma-119	539	27	.	.	PUNCT
ma-119	540	1	j.	j.	PROPN
ma-119	540	2	math	math	PROPN
ma-119	540	3	.	.	PUNCT
ma-119	541	1	anal	anal	ADJ
ma-119	541	2	.	.	PUNCT
ma-119	542	1	4	4	NUM
ma-119	542	2	(	(	PUNCT
ma-119	542	3	2010	2010	NUM
ma-119	542	4	)	)	PUNCT
ma-119	542	5	1197	1197	NUM
ma-119	542	6	-	-	SYM
ma-119	542	7	1204.[36	1204.[36	NUM
ma-119	542	8	]	]	PUNCT
ma-119	542	9	n.b	n.b	PROPN
ma-119	542	10	.	.	PROPN
ma-119	542	11	okelo	okelo	PROPN
ma-119	542	12	,	,	PUNCT
ma-119	542	13	norm	norm	NOUN
ma-119	542	14	-	-	PUNCT
ma-119	542	15	attainability	attainability	NOUN
ma-119	542	16	and	and	CCONJ
ma-119	542	17	range	range	NOUN
ma-119	542	18	-	-	PUNCT
ma-119	542	19	kernel	kernel	NOUN
ma-119	542	20	orthogonality	orthogonality	NOUN
ma-119	542	21	of	of	ADP
ma-119	542	22	elementary	elementary	ADJ
ma-119	542	23	operators	operator	NOUN
ma-119	542	24	,	,	PUNCT
ma-119	542	25	commun	commun	PROPN
ma-119	542	26	.	.	PUNCT
ma-119	543	1	adv	adv	PROPN
ma-119	543	2	.	.	PUNCT
ma-119	543	3	math	math	NOUN
ma-119	543	4	.	.	PUNCT
ma-119	544	1	sci.1	sci.1	INTJ
ma-119	544	2	(	(	PUNCT
ma-119	544	3	2018	2018	NUM
ma-119	544	4	)	)	PUNCT
ma-119	544	5	91–98	91–98	NUM
ma-119	544	6	.	.	PUNCT
ma-119	545	1	https://doi.org/10.33434/cams.442556.[37	https://doi.org/10.33434/cams.442556.[37	PUNCT
ma-119	545	2	]	]	X
ma-119	546	1	n.b	n.b	PROPN
ma-119	546	2	.	.	PROPN
ma-119	546	3	okelo	okelo	PROPN
ma-119	546	4	,	,	PUNCT
ma-119	546	5	the	the	DET
ma-119	546	6	norm	norm	NOUN
ma-119	546	7	-	-	PUNCT
ma-119	546	8	attainability	attainability	NOUN
ma-119	546	9	of	of	ADP
ma-119	546	10	some	some	DET
ma-119	546	11	elementary	elementary	ADJ
ma-119	546	12	operators	operator	NOUN
ma-119	546	13	,	,	PUNCT
ma-119	546	14	appl	appl	PROPN
ma-119	546	15	.	.	PROPN
ma-119	546	16	math	math	NOUN
ma-119	546	17	.	.	PUNCT
ma-119	547	1	e	e	X
ma-119	547	2	-	-	NOUN
ma-119	547	3	notes	note	NOUN
ma-119	547	4	,	,	PUNCT
ma-119	547	5	13	13	NUM
ma-119	547	6	(	(	PUNCT
ma-119	547	7	2013	2013	NUM
ma-119	547	8	)	)	PUNCT
ma-119	547	9	1	1	NUM
ma-119	547	10	-	-	SYM
ma-119	547	11	7.[38	7.[38	NUM
ma-119	547	12	]	]	PUNCT
ma-119	547	13	n.b	n.b	PROPN
ma-119	547	14	.	.	PROPN
ma-119	547	15	okelo	okelo	PROPN
ma-119	547	16	,	,	PUNCT
ma-119	547	17	j.o	j.o	PROPN
ma-119	547	18	.	.	PROPN
ma-119	547	19	agure	agure	PROPN
ma-119	547	20	,	,	PUNCT
ma-119	547	21	p.o	p.o	PROPN
ma-119	547	22	.	.	PROPN
ma-119	547	23	oleche	oleche	PROPN
ma-119	547	24	,	,	PUNCT
ma-119	547	25	certain	certain	ADJ
ma-119	547	26	conditions	condition	NOUN
ma-119	547	27	for	for	ADP
ma-119	547	28	norm	norm	NOUN
ma-119	547	29	-	-	PUNCT
ma-119	547	30	attainability	attainability	NOUN
ma-119	547	31	of	of	ADP
ma-119	547	32	elementary	elementary	ADJ
ma-119	547	33	operators	operator	NOUN
ma-119	547	34	and	and	CCONJ
ma-119	547	35	deriva	deriva	NOUN
ma-119	547	36	-	-	PUNCT
ma-119	547	37	tions	tion	NOUN
ma-119	547	38	,	,	PUNCT
ma-119	547	39	int	int	NOUN
ma-119	547	40	.	.	PUNCT
ma-119	548	1	j.	j.	PROPN
ma-119	548	2	math	math	PROPN
ma-119	548	3	.	.	PUNCT
ma-119	549	1	soft	soft	ADJ
ma-119	549	2	comput	comput	NOUN
ma-119	549	3	.	.	PUNCT
ma-119	550	1	3	3	NUM
ma-119	550	2	(	(	PUNCT
ma-119	550	3	2013	2013	NUM
ma-119	550	4	)	)	PUNCT
ma-119	550	5	53	53	NUM
ma-119	550	6	-	-	SYM
ma-119	550	7	59	59	NUM
ma-119	550	8	.	.	PUNCT
ma-119	551	1	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	X
ma-119	551	2	https://doi.org/10.1006/jfan.1996.3004	https://doi.org/10.1006/jfan.1996.3004	PROPN
ma-119	551	3	https://doi.org/10.1006/jfan.1996.3004	https://doi.org/10.1006/jfan.1996.3004	PROPN
ma-119	551	4	https://www.jstor.org/stable/24714573	https://www.jstor.org/stable/24714573	NOUN
ma-119	551	5	https://doi.org/10.1016/j.jfa.2004.09.013	https://doi.org/10.1016/j.jfa.2004.09.013	NOUN
ma-119	551	6	https://doi.org/10.1016/j.jfa.2004.09.013	https://doi.org/10.1016/j.jfa.2004.09.013	NOUN
ma-119	551	7	https://doi.org/10.1090/s0002-9947-1972-0305090-x	https://doi.org/10.1090/s0002-9947-1972-0305090-x	NOUN
ma-119	551	8	https://doi.org/10.1090/s0002-9939-07-09112-5	https://doi.org/10.1090/s0002-9939-07-09112-5	NOUN
ma-119	551	9	https://doi.org/10.1007/bfb0062619	https://doi.org/10.1007/bfb0062619	NUM
ma-119	551	10	https://doi.org/10.1007/bfb0062619	https://doi.org/10.1007/bfb0062619	NUM
ma-119	551	11	https://doi.org/10.2140/pjm.1971.38.465	https://doi.org/10.2140/pjm.1971.38.465	PRON
ma-119	551	12	https://doi.org/10.2140/pjm.1971.38.465	https://doi.org/10.2140/pjm.1971.38.465	SYM
ma-119	551	13	https://doi.org/10.1017/s0013091500015856	https://doi.org/10.1017/s0013091500015856	NUM
ma-119	551	14	https://doi.org/10.1017/s0013091500015856	https://doi.org/10.1017/s0013091500015856	NUM
ma-119	551	15	https://doi.org/10.1112/jlms/s2-16.2.297	https://doi.org/10.1112/jlms/s2-16.2.297	PROPN
ma-119	551	16	https://doi.org/10.1112/jlms/s2-16.2.297	https://doi.org/10.1112/jlms/s2-16.2.297	PROPN
ma-119	551	17	https://doi.org/10.2307/2160991	https://doi.org/10.2307/2160991	PROPN
ma-119	551	18	https://doi.org/10.2307/2160991	https://doi.org/10.2307/2160991	X
ma-119	551	19	https://doi.org/10.1016/0022-1236(72)90043-2	https://doi.org/10.1016/0022-1236(72)90043-2	PROPN
ma-119	552	1	https://doi.org/10.1017/s0017089500008077	https://doi.org/10.1017/s0017089500008077	PRON
ma-119	552	2	https://doi.org/10.1017/s0004972700028203	https://doi.org/10.1017/s0004972700028203	NUM
ma-119	552	3	https://doi.org/10.11650/tjm/190502	https://doi.org/10.11650/tjm/190502	ADJ
ma-119	552	4	https://doi.org/10.33434/cams.442556	https://doi.org/10.33434/cams.442556	PROPN
ma-119	552	5	eur	eur	PROPN
ma-119	552	6	.	.	PUNCT
ma-119	553	1	j.	j.	PROPN
ma-119	553	2	math	math	PROPN
ma-119	553	3	.	.	PUNCT
ma-119	554	1	anal	anal	PROPN
ma-119	554	2	.	.	PUNCT
ma-119	555	1	10.28924	10.28924	NUM
ma-119	555	2	/	/	SYM
ma-119	555	3	ada	ada	PROPN
ma-119	555	4	/	/	SYM
ma-119	555	5	ma.3.9	ma.3.9	PROPN
ma-119	555	6	15	15	NUM
ma-119	555	7	[	[	X
ma-119	555	8	39	39	NUM
ma-119	555	9	]	]	X
ma-119	555	10	n.b	n.b	PROPN
ma-119	555	11	.	.	PROPN
ma-119	555	12	okelo	okelo	PROPN
ma-119	555	13	,	,	PUNCT
ma-119	555	14	j.o	j.o	PROPN
ma-119	555	15	.	.	PROPN
ma-119	555	16	agure	agure	PROPN
ma-119	555	17	,	,	PUNCT
ma-119	555	18	a	a	DET
ma-119	555	19	two	two	NUM
ma-119	555	20	-	-	PUNCT
ma-119	555	21	sided	sided	ADJ
ma-119	555	22	multiplication	multiplication	NOUN
ma-119	555	23	operator	operator	NOUN
ma-119	555	24	norm	norm	NOUN
ma-119	555	25	,	,	PUNCT
ma-119	555	26	gen	gen	PROPN
ma-119	555	27	.	.	PROPN
ma-119	555	28	math	math	PROPN
ma-119	555	29	.	.	PUNCT
ma-119	556	1	notes	note	NOUN
ma-119	556	2	,	,	PUNCT
ma-119	556	3	2	2	NUM
ma-119	556	4	(	(	PUNCT
ma-119	556	5	2011	2011	NUM
ma-119	556	6	)	)	PUNCT
ma-119	556	7	18	18	NUM
ma-119	556	8	-	-	SYM
ma-119	556	9	23.[40	23.[40	PROPN
ma-119	556	10	]	]	PUNCT
ma-119	556	11	n.b	n.b	PROPN
ma-119	556	12	.	.	PROPN
ma-119	556	13	okelo	okelo	PROPN
ma-119	556	14	,	,	PUNCT
ma-119	556	15	fixed	fix	VERB
ma-119	556	16	points	point	NOUN
ma-119	556	17	approximation	approximation	NOUN
ma-119	556	18	for	for	ADP
ma-119	556	19	nonexpansive	nonexpansive	ADJ
ma-119	556	20	operators	operator	NOUN
ma-119	556	21	in	in	ADP
ma-119	556	22	hilbert	hilbert	PROPN
ma-119	556	23	spaces	space	NOUN
ma-119	556	24	,	,	PUNCT
ma-119	556	25	int	int	NOUN
ma-119	556	26	.	.	PUNCT
ma-119	557	1	j.	j.	PROPN
ma-119	557	2	open	open	PROPN
ma-119	557	3	problems	problem	NOUN
ma-119	557	4	comput.math	comput.math	NUM
ma-119	557	5	.	.	PUNCT
ma-119	558	1	14	14	NUM
ma-119	558	2	(	(	PUNCT
ma-119	558	3	2021	2021	NUM
ma-119	558	4	)	)	PUNCT
ma-119	558	5	1	1	NUM
ma-119	558	6	-	-	SYM
ma-119	558	7	5.[41	5.[41	NUM
ma-119	558	8	]	]	X
ma-119	558	9	n.b	n.b	PROPN
ma-119	558	10	.	.	PROPN
ma-119	558	11	okelo	okelo	PROPN
ma-119	558	12	,	,	PUNCT
ma-119	558	13	characterization	characterization	NOUN
ma-119	558	14	of	of	ADP
ma-119	558	15	absolutely	absolutely	ADV
ma-119	558	16	norm	norm	NOUN
ma-119	558	17	attaining	attain	VERB
ma-119	558	18	compact	compact	ADJ
ma-119	558	19	hyponormal	hyponormal	ADJ
ma-119	558	20	operators	operator	NOUN
ma-119	558	21	,	,	PUNCT
ma-119	558	22	proc	proc	NOUN
ma-119	558	23	.	.	PUNCT
ma-119	559	1	int	int	NOUN
ma-119	559	2	.	.	PUNCT
ma-119	560	1	math	math	NOUN
ma-119	560	2	.	.	PUNCT
ma-119	561	1	sci	sci	PROPN
ma-119	561	2	.	.	PUNCT
ma-119	562	1	2(2020	2(2020	X
ma-119	562	2	)	)	PUNCT
ma-119	562	3	96	96	NUM
ma-119	562	4	-	-	SYM
ma-119	562	5	102	102	NUM
ma-119	562	6	.	.	PUNCT
ma-119	563	1	https://doi.org/10.47086/pims.689633.[42	https://doi.org/10.47086/pims.689633.[42	X
ma-119	563	2	]	]	PUNCT
ma-119	564	1	p.	p.	PROPN
ma-119	564	2	galindo	galindo	PROPN
ma-119	564	3	,	,	PUNCT
ma-119	564	4	j.	j.	PROPN
ma-119	564	5	laitila	laitila	PROPN
ma-119	564	6	,	,	PUNCT
ma-119	564	7	m.	m.	NOUN
ma-119	564	8	lindström	lindström	NOUN
ma-119	564	9	,	,	PUNCT
ma-119	564	10	essential	essential	ADJ
ma-119	564	11	norm	norm	NOUN
ma-119	564	12	estimates	estimate	NOUN
ma-119	564	13	for	for	ADP
ma-119	564	14	composition	composition	NOUN
ma-119	564	15	operators	operator	NOUN
ma-119	564	16	on	on	ADP
ma-119	564	17	bmoa	bmoa	NOUN
ma-119	564	18	,	,	PUNCT
ma-119	564	19	j.	j.	PROPN
ma-119	564	20	funct	funct	PROPN
ma-119	564	21	.	.	PUNCT
ma-119	565	1	anal.265	anal.265	NOUN
ma-119	565	2	(	(	PUNCT
ma-119	565	3	2013	2013	NUM
ma-119	565	4	)	)	PUNCT
ma-119	565	5	629–643	629–643	NUM
ma-119	565	6	.	.	PUNCT
ma-119	566	1	https://doi.org/10.1016/j.jfa.2013.05.002.[43	https://doi.org/10.1016/j.jfa.2013.05.002.[43	PROPN
ma-119	566	2	]	]	PUNCT
ma-119	566	3	a.	a.	NOUN
ma-119	566	4	pinchuck	pinchuck	NOUN
ma-119	566	5	,	,	PUNCT
ma-119	566	6	functional	functional	ADJ
ma-119	566	7	analysis	analysis	NOUN
ma-119	566	8	notes	note	NOUN
ma-119	566	9	,	,	PUNCT
ma-119	566	10	springer	springer	NOUN
ma-119	566	11	verlag	verlag	PROPN
ma-119	566	12	,	,	PUNCT
ma-119	566	13	new	new	PROPN
ma-119	566	14	york	york	PROPN
ma-119	566	15	,	,	PUNCT
ma-119	566	16	2011.[44	2011.[44	NUM
ma-119	566	17	]	]	X
ma-119	566	18	r.m	r.m	PROPN
ma-119	566	19	.	.	PROPN
ma-119	566	20	timoney	timoney	PROPN
ma-119	566	21	,	,	PUNCT
ma-119	566	22	norms	norm	NOUN
ma-119	566	23	and	and	CCONJ
ma-119	566	24	cb	cb	PROPN
ma-119	566	25	norms	norm	NOUN
ma-119	566	26	of	of	ADP
ma-119	566	27	jordan	jordan	PROPN
ma-119	566	28	elementary	elementary	PROPN
ma-119	566	29	operators	operators	PROPN
ma-119	566	30	,	,	PUNCT
ma-119	566	31	bull	bull	NOUN
ma-119	566	32	.	.	PUNCT
ma-119	567	1	sci	sci	PROPN
ma-119	567	2	.	.	PROPN
ma-119	567	3	math	math	PROPN
ma-119	567	4	.	.	PUNCT
ma-119	568	1	127	127	NUM
ma-119	568	2	(	(	PUNCT
ma-119	568	3	2003	2003	NUM
ma-119	568	4	)	)	PUNCT
ma-119	568	5	597	597	NUM
ma-119	568	6	-	-	SYM
ma-119	568	7	609	609	NUM
ma-119	568	8	.	.	PUNCT
ma-119	569	1	https	https	NOUN
ma-119	569	2	:	:	PUNCT
ma-119	570	1	//doi.org/10.1016	//doi.org/10.1016	PROPN
ma-119	570	2	/	/	SYM
ma-119	570	3	s0007	s0007	VERB
ma-119	570	4	-	-	PUNCT
ma-119	570	5	4497(03)00046	4497(03)00046	NUM
ma-119	570	6	-	-	PUNCT
ma-119	570	7	0.[45	0.[45	NUM
ma-119	570	8	]	]	X
ma-119	570	9	r.m	r.m	PROPN
ma-119	570	10	.	.	PROPN
ma-119	570	11	timoney	timoney	PROPN
ma-119	570	12	,	,	PUNCT
ma-119	570	13	computing	compute	VERB
ma-119	570	14	the	the	DET
ma-119	570	15	norms	norm	NOUN
ma-119	570	16	of	of	ADP
ma-119	570	17	elementary	elementary	ADJ
ma-119	570	18	operators	operator	NOUN
ma-119	570	19	,	,	PUNCT
ma-119	570	20	illinois	illinois	PROPN
ma-119	570	21	j.	j.	PROPN
ma-119	570	22	math	math	PROPN
ma-119	570	23	.	.	PUNCT
ma-119	571	1	47	47	NUM
ma-119	571	2	(	(	PUNCT
ma-119	571	3	2003	2003	NUM
ma-119	571	4	)	)	PUNCT
ma-119	571	5	1207	1207	NUM
ma-119	571	6	-	-	SYM
ma-119	571	7	1226	1226	NUM
ma-119	571	8	.	.	PUNCT
ma-119	572	1	https://doi	https://doi	X
ma-119	572	2	.	.	PUNCT
ma-119	572	3	org/10.1215	org/10.1215	PROPN
ma-119	572	4	/	/	SYM
ma-119	572	5	ijm/1258138100.[46	ijm/1258138100.[46	PROPN
ma-119	572	6	]	]	PUNCT
ma-119	572	7	a.	a.	NOUN
ma-119	572	8	seddik	seddik	PROPN
ma-119	572	9	,	,	PUNCT
ma-119	572	10	on	on	ADP
ma-119	572	11	the	the	DET
ma-119	572	12	numerical	numerical	ADJ
ma-119	572	13	range	range	NOUN
ma-119	572	14	and	and	CCONJ
ma-119	572	15	norm	norm	NOUN
ma-119	572	16	of	of	ADP
ma-119	572	17	elementary	elementary	ADJ
ma-119	572	18	operators	operator	NOUN
ma-119	572	19	,	,	PUNCT
ma-119	572	20	linear	linear	PROPN
ma-119	572	21	multilinear	multilinear	PROPN
ma-119	572	22	algebra	algebra	PROPN
ma-119	572	23	.	.	PUNCT
ma-119	573	1	52	52	NUM
ma-119	573	2	(	(	PUNCT
ma-119	573	3	2004	2004	NUM
ma-119	573	4	)	)	PUNCT
ma-119	573	5	293–302	293–302	NUM
ma-119	573	6	.	.	PUNCT
ma-119	574	1	https://doi.org/10.1080/0308108031000122515.[47	https://doi.org/10.1080/0308108031000122515.[47	PROPN
ma-119	574	2	]	]	PUNCT
ma-119	574	3	a.	a.	NOUN
ma-119	574	4	seddik	seddik	PROPN
ma-119	574	5	,	,	PUNCT
ma-119	574	6	on	on	ADP
ma-119	574	7	the	the	DET
ma-119	574	8	injective	injective	ADJ
ma-119	574	9	norm	norm	NOUN
ma-119	574	10	and	and	CCONJ
ma-119	574	11	characterization	characterization	NOUN
ma-119	574	12	of	of	ADP
ma-119	574	13	some	some	DET
ma-119	574	14	subclasses	subclass	NOUN
ma-119	574	15	of	of	ADP
ma-119	574	16	normal	normal	ADJ
ma-119	574	17	operators	operator	NOUN
ma-119	574	18	by	by	ADP
ma-119	574	19	inequalities	inequality	NOUN
ma-119	574	20	orequalities	orequalitie	NOUN
ma-119	574	21	,	,	PUNCT
ma-119	574	22	j.	j.	PROPN
ma-119	574	23	math	math	PROPN
ma-119	574	24	.	.	PUNCT
ma-119	575	1	anal	anal	PROPN
ma-119	575	2	.	.	PUNCT
ma-119	575	3	appl	appl	PROPN
ma-119	575	4	.	.	PUNCT
ma-119	576	1	351	351	NUM
ma-119	576	2	(	(	PUNCT
ma-119	576	3	2009	2009	NUM
ma-119	576	4	)	)	PUNCT
ma-119	577	1	277–284	277–284	NUM
ma-119	577	2	.	.	PUNCT
ma-119	577	3	https://doi.org/10.1016/j.jmaa.2008.10.008.[48	https://doi.org/10.1016/j.jmaa.2008.10.008.[48	PROPN
ma-119	577	4	]	]	X
ma-119	577	5	l.l	l.l	PROPN
ma-119	577	6	.	.	PROPN
ma-119	577	7	stacho	stacho	PROPN
ma-119	577	8	,	,	PUNCT
ma-119	577	9	b.	b.	PROPN
ma-119	577	10	zalar	zalar	PROPN
ma-119	577	11	,	,	PUNCT
ma-119	577	12	on	on	ADP
ma-119	577	13	the	the	DET
ma-119	577	14	norm	norm	NOUN
ma-119	577	15	of	of	ADP
ma-119	577	16	jordan	jordan	PROPN
ma-119	577	17	elementary	elementary	PROPN
ma-119	577	18	operators	operators	PROPN
ma-119	577	19	in	in	ADP
ma-119	577	20	standard	standard	ADJ
ma-119	577	21	operator	operator	NOUN
ma-119	577	22	algebra	algebra	NOUN
ma-119	577	23	,	,	PUNCT
ma-119	577	24	publ	publ	NOUN
ma-119	577	25	.	.	PUNCT
ma-119	578	1	math.debrecen	math.debrecen	PROPN
ma-119	578	2	,	,	PUNCT
ma-119	578	3	49	49	NUM
ma-119	578	4	(	(	PUNCT
ma-119	578	5	1996	1996	NUM
ma-119	578	6	)	)	PUNCT
ma-119	578	7	127	127	NUM
ma-119	578	8	-	-	SYM
ma-119	578	9	134.[49	134.[49	NUM
ma-119	578	10	]	]	X
ma-119	578	11	j.	j.	PROPN
ma-119	578	12	stampfli	stampfli	PROPN
ma-119	578	13	,	,	PUNCT
ma-119	578	14	the	the	DET
ma-119	578	15	norm	norm	NOUN
ma-119	578	16	of	of	ADP
ma-119	578	17	a	a	DET
ma-119	578	18	derivation	derivation	NOUN
ma-119	578	19	,	,	PUNCT
ma-119	578	20	pac	pac	PROPN
ma-119	578	21	.	.	PUNCT
ma-119	578	22	j.	j.	PROPN
ma-119	578	23	math	math	PROPN
ma-119	578	24	.	.	PUNCT
ma-119	579	1	33	33	NUM
ma-119	579	2	(	(	PUNCT
ma-119	579	3	1970	1970	NUM
ma-119	579	4	)	)	PUNCT
ma-119	580	1	737–747	737–747	NUM
ma-119	580	2	.	.	PUNCT
ma-119	581	1	https://doi.org/10.2140/pjm.1970	https://doi.org/10.2140/pjm.1970	PROPN
ma-119	581	2	.	.	PUNCT
ma-119	582	1	33.737.[50	33.737.[50	NUM
ma-119	582	2	]	]	X
ma-119	582	3	j.	j.	PROPN
ma-119	582	4	stampfli	stampfli	PROPN
ma-119	582	5	,	,	PUNCT
ma-119	582	6	on	on	ADP
ma-119	582	7	selfadjoint	selfadjoint	NOUN
ma-119	582	8	derivation	derivation	NOUN
ma-119	582	9	ranges	range	NOUN
ma-119	582	10	,	,	PUNCT
ma-119	582	11	pac	pac	PROPN
ma-119	582	12	.	.	PUNCT
ma-119	582	13	j.	j.	PROPN
ma-119	582	14	math	math	PROPN
ma-119	582	15	.	.	PUNCT
ma-119	583	1	82	82	NUM
ma-119	583	2	(	(	PUNCT
ma-119	583	3	1979	1979	NUM
ma-119	583	4	)	)	PUNCT
ma-119	583	5	257–277	257–277	NUM
ma-119	583	6	.	.	PUNCT
ma-119	584	1	https://doi.org/10.2140/pjm	https://doi.org/10.2140/pjm	X
ma-119	584	2	.	.	PUNCT
ma-119	585	1	1979.82.257.[51	1979.82.257.[51	NUM
ma-119	585	2	]	]	X
ma-119	586	1	v.	v.	CCONJ
ma-119	586	2	runde	runde	PROPN
ma-119	586	3	,	,	PUNCT
ma-119	586	4	automatic	automatic	ADJ
ma-119	586	5	continuity	continuity	NOUN
ma-119	586	6	of	of	ADP
ma-119	586	7	derivations	derivation	NOUN
ma-119	586	8	and	and	CCONJ
ma-119	586	9	epimorphisms	epimorphism	NOUN
ma-119	586	10	,	,	PUNCT
ma-119	586	11	pac	pac	PROPN
ma-119	586	12	.	.	PUNCT
ma-119	586	13	j.	j.	PROPN
ma-119	586	14	math	math	PROPN
ma-119	586	15	.	.	PUNCT
ma-119	587	1	147	147	NUM
ma-119	587	2	(	(	PUNCT
ma-119	587	3	1991	1991	NUM
ma-119	587	4	)	)	PUNCT
ma-119	588	1	365–374	365–374	NUM
ma-119	588	2	.	.	PUNCT
ma-119	589	1	https://doi	https://doi	PROPN
ma-119	589	2	.	.	PUNCT
ma-119	589	3	org/10.2140	org/10.2140	NOUN
ma-119	589	4	/	/	SYM
ma-119	589	5	pjm.1991.147.365.[52	pjm.1991.147.365.[52	PROPN
ma-119	589	6	]	]	X
ma-119	589	7	a.w	a.w	PROPN
ma-119	589	8	.	.	PROPN
ma-119	589	9	wickstead	wickstead	PROPN
ma-119	589	10	,	,	PUNCT
ma-119	589	11	norms	norm	NOUN
ma-119	589	12	of	of	ADP
ma-119	589	13	basic	basic	ADJ
ma-119	589	14	elementary	elementary	ADJ
ma-119	589	15	operators	operator	NOUN
ma-119	589	16	on	on	ADP
ma-119	589	17	algebras	algebra	NOUN
ma-119	589	18	of	of	ADP
ma-119	589	19	regular	regular	ADJ
ma-119	589	20	operators	operator	NOUN
ma-119	589	21	,	,	PUNCT
ma-119	589	22	proc	proc	PROPN
ma-119	589	23	.	.	PUNCT
ma-119	590	1	amer	amer	PROPN
ma-119	590	2	.	.	PUNCT
ma-119	590	3	math	math	PROPN
ma-119	590	4	.	.	PUNCT
ma-119	591	1	soc	soc	PROPN
ma-119	591	2	.	.	PUNCT
ma-119	592	1	143(2015	143(2015	NUM
ma-119	592	2	)	)	PUNCT
ma-119	592	3	5275–5280	5275–5280	NUM
ma-119	592	4	.	.	PUNCT
ma-119	593	1	https://doi.org/10.1090/proc/12664.[53	https://doi.org/10.1090/proc/12664.[53	PROPN
ma-119	593	2	]	]	X
ma-119	593	3	y.c	y.c	PROPN
ma-119	593	4	.	.	PROPN
ma-119	593	5	kim	kim	PROPN
ma-119	593	6	,	,	PUNCT
ma-119	593	7	t.	t.	PROPN
ma-119	593	8	sugawa	sugawa	PROPN
ma-119	593	9	,	,	PUNCT
ma-119	593	10	norm	norm	NOUN
ma-119	593	11	estimates	estimate	NOUN
ma-119	593	12	of	of	ADP
ma-119	593	13	the	the	DET
ma-119	593	14	pre	pre	ADJ
ma-119	593	15	-	-	ADJ
ma-119	593	16	schwarzian	schwarzian	ADJ
ma-119	593	17	derivatives	derivative	NOUN
ma-119	593	18	for	for	ADP
ma-119	593	19	certain	certain	ADJ
ma-119	593	20	classes	class	NOUN
ma-119	593	21	of	of	ADP
ma-119	593	22	univalent	univalent	ADJ
ma-119	593	23	functions	function	NOUN
ma-119	593	24	,	,	PUNCT
ma-119	593	25	proc	proc	NOUN
ma-119	593	26	.	.	PUNCT
ma-119	594	1	edinburgh	edinburgh	PROPN
ma-119	594	2	math	math	PROPN
ma-119	594	3	.	.	PUNCT
ma-119	595	1	soc	soc	PROPN
ma-119	595	2	.	.	PUNCT
ma-119	596	1	49	49	NUM
ma-119	596	2	(	(	PUNCT
ma-119	596	3	2006	2006	NUM
ma-119	596	4	)	)	PUNCT
ma-119	596	5	131	131	NUM
ma-119	596	6	-	-	SYM
ma-119	596	7	143	143	NUM
ma-119	596	8	.	.	PUNCT
ma-119	597	1	https://doi.org/10.1017/s0013091504000306	https://doi.org/10.1017/s0013091504000306	NOUN
ma-119	597	2	.	.	PUNCT
ma-119	598	1	https://doi.org/10.28924/ada/ma.3.9	https://doi.org/10.28924/ada/ma.3.9	NUM
ma-119	598	2	https://doi.org/10.47086/pims.689633	https://doi.org/10.47086/pims.689633	ADJ
ma-119	598	3	https://doi.org/10.1016/j.jfa.2013.05.002	https://doi.org/10.1016/j.jfa.2013.05.002	NOUN
ma-119	598	4	https://doi.org/10.1016/s0007-4497(03)00046-0	https://doi.org/10.1016/s0007-4497(03)00046-0	PROPN
ma-119	598	5	https://doi.org/10.1016/s0007-4497(03)00046-0	https://doi.org/10.1016/s0007-4497(03)00046-0	PROPN
ma-119	598	6	https://doi.org/10.1215/ijm/1258138100	https://doi.org/10.1215/ijm/1258138100	PROPN
ma-119	598	7	https://doi.org/10.1215/ijm/1258138100	https://doi.org/10.1215/ijm/1258138100	PROPN
ma-119	598	8	https://doi.org/10.1080/0308108031000122515	https://doi.org/10.1080/0308108031000122515	X
ma-119	598	9	https://doi.org/10.1016/j.jmaa.2008.10.008	https://doi.org/10.1016/j.jmaa.2008.10.008	PROPN
ma-119	598	10	https://doi.org/10.2140/pjm.1970.33.737	https://doi.org/10.2140/pjm.1970.33.737	NUM
ma-119	598	11	https://doi.org/10.2140/pjm.1970.33.737	https://doi.org/10.2140/pjm.1970.33.737	PROPN
ma-119	598	12	https://doi.org/10.2140/pjm.1979.82.257	https://doi.org/10.2140/pjm.1979.82.257	PROPN
ma-119	598	13	https://doi.org/10.2140/pjm.1979.82.257	https://doi.org/10.2140/pjm.1979.82.257	NOUN
ma-119	598	14	https://doi.org/10.2140/pjm.1991.147.365	https://doi.org/10.2140/pjm.1991.147.365	NOUN
ma-119	598	15	https://doi.org/10.2140/pjm.1991.147.365	https://doi.org/10.2140/pjm.1991.147.365	NOUN
ma-119	598	16	https://doi.org/10.1090/proc/12664	https://doi.org/10.1090/proc/12664	PROPN
ma-119	598	17	https://doi.org/10.1017/s0013091504000306	https://doi.org/10.1017/s0013091504000306	PROPN
ma-119	598	18	1	1	NUM
ma-119	598	19	.	.	PUNCT
ma-119	598	20	introduction	introduction	NOUN
ma-119	598	21	2	2	NUM
ma-119	598	22	.	.	PUNCT
ma-119	598	23	preliminaries	preliminary	NOUN
ma-119	598	24	3	3	NUM
ma-119	598	25	.	.	X
ma-119	598	26	main	main	ADJ
ma-119	598	27	results	result	NOUN
ma-119	598	28	4	4	NUM
ma-119	598	29	.	.	PUNCT
ma-119	599	1	conclusion	conclusion	NOUN
ma-119	599	2	references	reference	NOUN