id	sid	tid	token	lemma	pos
ma-232	1	1	2024	2024	NUM
ma-232	1	2	ada	ada	PROPN
ma-232	1	3	academica	academica	PROPN
ma-232	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-232	1	5	.	.	PUNCT
ma-232	2	1	j.	j.	PROPN
ma-232	2	2	math	math	PROPN
ma-232	2	3	.	.	PUNCT
ma-232	3	1	anal	anal	ADJ
ma-232	3	2	.	.	PUNCT
ma-232	4	1	4	4	NUM
ma-232	4	2	(	(	PUNCT
ma-232	4	3	2024	2024	NUM
ma-232	4	4	)	)	PUNCT
ma-232	4	5	14doi	14doi	NOUN
ma-232	4	6	:	:	PUNCT
ma-232	4	7	10.28924	10.28924	NUM
ma-232	4	8	/	/	SYM
ma-232	4	9	ada	ada	PROPN
ma-232	4	10	/	/	SYM
ma-232	4	11	ma.4.14	ma.4.14	PROPN
ma-232	4	12	duality	duality	NOUN
ma-232	4	13	of	of	ADP
ma-232	4	14	the	the	DET
ma-232	4	15	nonreflexive	nonreflexive	ADJ
ma-232	4	16	bergman	bergman	PROPN
ma-232	4	17	space	space	NOUN
ma-232	4	18	of	of	ADP
ma-232	4	19	the	the	DET
ma-232	4	20	upper	upper	ADJ
ma-232	4	21	half	half	NOUN
ma-232	4	22	plane	plane	NOUN
ma-232	4	23	and	and	CCONJ
ma-232	4	24	composition	composition	NOUN
ma-232	4	25	groups	group	NOUN
ma-232	4	26	e.	e.	PROPN
ma-232	4	27	o.	o.	PROPN
ma-232	4	28	gori1	gori1	PROPN
ma-232	4	29	,	,	PUNCT
ma-232	4	30	j.	j.	PROPN
ma-232	4	31	o.	o.	PROPN
ma-232	4	32	bonyo2,∗	bonyo2,∗	PROPN
ma-232	4	33	1department	1department	NUM
ma-232	4	34	of	of	ADP
ma-232	4	35	pure	pure	ADJ
ma-232	4	36	and	and	CCONJ
ma-232	4	37	applied	applied	ADJ
ma-232	4	38	mathematics	mathematic	NOUN
ma-232	4	39	,	,	PUNCT
ma-232	4	40	maseno	maseno	NOUN
ma-232	4	41	university	university	NOUN
ma-232	4	42	,	,	PUNCT
ma-232	4	43	p.o	p.o	PROPN
ma-232	4	44	.	.	PROPN
ma-232	4	45	box	box	PROPN
ma-232	4	46	333	333	NUM
ma-232	4	47	-	-	SYM
ma-232	4	48	40105	40105	NUM
ma-232	4	49	,	,	PUNCT
ma-232	4	50	maseno	maseno	NOUN
ma-232	4	51	,	,	PUNCT
ma-232	4	52	kenya	kenya	PROPN
ma-232	5	1	omondierick75@gmail.com1	omondierick75@gmail.com1	PROPN
ma-232	5	2	2department	2department	NUM
ma-232	5	3	of	of	ADP
ma-232	5	4	mathematics	mathematic	NOUN
ma-232	5	5	,	,	PUNCT
ma-232	5	6	multimedia	multimedia	NOUN
ma-232	5	7	university	university	PROPN
ma-232	5	8	of	of	ADP
ma-232	5	9	kenya	kenya	PROPN
ma-232	5	10	,	,	PUNCT
ma-232	5	11	p.o	p.o	PROPN
ma-232	5	12	.	.	PROPN
ma-232	5	13	box	box	PROPN
ma-232	5	14	15653	15653	NUM
ma-232	5	15	-	-	SYM
ma-232	5	16	00503	00503	NUM
ma-232	5	17	,	,	PUNCT
ma-232	5	18	nairobi	nairobi	PROPN
ma-232	5	19	,	,	PUNCT
ma-232	5	20	kenya	kenya	PROPN
ma-232	5	21	jbonyo@mmu.ac.ke	jbonyo@mmu.ac.ke	NOUN
ma-232	5	22	correspondence	correspondence	NOUN
ma-232	5	23	:	:	PUNCT
ma-232	5	24	jbonyo@mmu.ac.ke	jbonyo@mmu.ac.ke	NOUN
ma-232	5	25	abstract	abstract	ADJ
ma-232	5	26	.	.	PUNCT
ma-232	6	1	we	we	PRON
ma-232	6	2	identify	identify	VERB
ma-232	6	3	the	the	DET
ma-232	6	4	predual	predual	ADJ
ma-232	6	5	of	of	ADP
ma-232	6	6	the	the	DET
ma-232	6	7	nonreflexive	nonreflexive	ADJ
ma-232	6	8	bergman	bergman	PROPN
ma-232	6	9	space	space	NOUN
ma-232	6	10	of	of	ADP
ma-232	6	11	the	the	DET
ma-232	6	12	upper	upper	ADJ
ma-232	6	13	half	half	ADJ
ma-232	6	14	plane	plane	NOUN
ma-232	6	15	withthe	withthe	PRON
ma-232	6	16	little	little	ADJ
ma-232	6	17	bloch	bloch	PROPN
ma-232	6	18	space	space	NOUN
ma-232	6	19	of	of	ADP
ma-232	6	20	the	the	DET
ma-232	6	21	upper	upper	ADJ
ma-232	6	22	half	half	ADJ
ma-232	6	23	plane	plane	NOUN
ma-232	6	24	consisting	consist	VERB
ma-232	6	25	of	of	ADP
ma-232	6	26	those	those	DET
ma-232	6	27	functions	function	NOUN
ma-232	6	28	vanishing	vanish	VERB
ma-232	6	29	at	at	ADP
ma-232	6	30	point	point	NOUN
ma-232	6	31	i	i	PRON
ma-232	6	32	.	.	PUNCT
ma-232	7	1	usingthe	usingthe	DET
ma-232	7	2	duality	duality	NOUN
ma-232	7	3	pairing	pair	VERB
ma-232	7	4	as	as	ADV
ma-232	7	5	well	well	ADV
ma-232	7	6	as	as	ADP
ma-232	7	7	the	the	DET
ma-232	7	8	composition	composition	NOUN
ma-232	7	9	groups	group	NOUN
ma-232	7	10	on	on	ADP
ma-232	7	11	the	the	DET
ma-232	7	12	nonreflexive	nonreflexive	ADJ
ma-232	7	13	bergman	bergman	PROPN
ma-232	7	14	space	space	NOUN
ma-232	7	15	,	,	PUNCT
ma-232	7	16	we	we	PRON
ma-232	7	17	obtainthe	obtainthe	VERB
ma-232	7	18	groups	group	NOUN
ma-232	7	19	of	of	ADP
ma-232	7	20	composition	composition	NOUN
ma-232	7	21	operators	operator	NOUN
ma-232	7	22	defined	define	VERB
ma-232	7	23	on	on	ADP
ma-232	7	24	the	the	DET
ma-232	7	25	identified	identify	VERB
ma-232	7	26	predual	predual	ADJ
ma-232	7	27	.	.	PUNCT
ma-232	8	1	we	we	PRON
ma-232	8	2	identify	identify	VERB
ma-232	8	3	the	the	DET
ma-232	8	4	infinitesimalgenerator	infinitesimalgenerator	NOUN
ma-232	8	5	of	of	ADP
ma-232	8	6	each	each	DET
ma-232	8	7	group	group	NOUN
ma-232	8	8	and	and	CCONJ
ma-232	8	9	prove	prove	VERB
ma-232	8	10	the	the	DET
ma-232	8	11	strong	strong	ADJ
ma-232	8	12	continuity	continuity	NOUN
ma-232	8	13	property	property	NOUN
ma-232	8	14	.	.	PUNCT
ma-232	9	1	we	we	PRON
ma-232	9	2	then	then	ADV
ma-232	9	3	obtain	obtain	VERB
ma-232	9	4	the	the	DET
ma-232	9	5	spectra	spectra	NOUN
ma-232	9	6	of	of	ADP
ma-232	9	7	thegenerator	thegenerator	NOUN
ma-232	9	8	γ	γ	PROPN
ma-232	9	9	,	,	PUNCT
ma-232	9	10	determine	determine	VERB
ma-232	9	11	the	the	DET
ma-232	9	12	resolvents	resolvent	NOUN
ma-232	9	13	and	and	CCONJ
ma-232	9	14	further	far	ADV
ma-232	9	15	obtain	obtain	VERB
ma-232	9	16	the	the	DET
ma-232	9	17	spectra	spectra	NOUN
ma-232	9	18	and	and	CCONJ
ma-232	9	19	the	the	DET
ma-232	9	20	norms	norm	NOUN
ma-232	9	21	of	of	ADP
ma-232	9	22	the	the	DET
ma-232	9	23	resultingresolvents	resultingresolvent	NOUN
ma-232	9	24	.	.	PUNCT
ma-232	10	1	1	1	X
ma-232	10	2	.	.	X
ma-232	10	3	introduction	introduction	NOUN
ma-232	10	4	let	let	VERB
ma-232	10	5	c	c	NOUN
ma-232	10	6	be	be	AUX
ma-232	10	7	the	the	DET
ma-232	10	8	complex	complex	ADJ
ma-232	10	9	plane	plane	NOUN
ma-232	10	10	.	.	PUNCT
ma-232	11	1	the	the	DET
ma-232	11	2	set	set	NOUN
ma-232	11	3	d	d	NOUN
ma-232	11	4	:	:	PUNCT
ma-232	11	5	=	=	SYM
ma-232	11	6	{	{	PUNCT
ma-232	11	7	z	z	NOUN
ma-232	11	8	∈	∈	PROPN
ma-232	11	9	c	c	NOUN
ma-232	11	10	:	:	PUNCT
ma-232	11	11	|z	|z	PROPN
ma-232	12	1	|	|	ADV
ma-232	12	2	<	<	X
ma-232	12	3	1	1	NUM
ma-232	12	4	}	}	PUNCT
ma-232	12	5	is	be	AUX
ma-232	12	6	called	call	VERB
ma-232	12	7	the	the	DET
ma-232	12	8	open	open	ADJ
ma-232	12	9	unit	unit	NOUN
ma-232	12	10	disc	disc	NOUN
ma-232	12	11	.	.	PUNCT
ma-232	13	1	let	let	VERB
ma-232	13	2	da	da	PROPN
ma-232	13	3	denote	denote	VERB
ma-232	13	4	the	the	DET
ma-232	13	5	area	area	NOUN
ma-232	13	6	measure	measure	NOUN
ma-232	13	7	on	on	ADP
ma-232	13	8	d	d	PROPN
ma-232	13	9	,	,	PUNCT
ma-232	13	10	normalized	normalize	VERB
ma-232	13	11	so	so	SCONJ
ma-232	13	12	that	that	SCONJ
ma-232	13	13	the	the	DET
ma-232	13	14	area	area	NOUN
ma-232	13	15	of	of	ADP
ma-232	13	16	d	d	PROPN
ma-232	13	17	is	be	AUX
ma-232	13	18	1	1	NUM
ma-232	13	19	.	.	PUNCT
ma-232	14	1	in	in	ADP
ma-232	14	2	terms	term	NOUN
ma-232	14	3	of	of	ADP
ma-232	14	4	rectangularand	rectangularand	NOUN
ma-232	14	5	polar	polar	ADJ
ma-232	14	6	coordinates	coordinate	NOUN
ma-232	14	7	,	,	PUNCT
ma-232	14	8	we	we	PRON
ma-232	14	9	have	have	VERB
ma-232	14	10	:	:	PUNCT
ma-232	14	11	da(z	da(z	X
ma-232	14	12	)	)	PUNCT
ma-232	14	13	=	=	SYM
ma-232	14	14	1	1	NUM
ma-232	14	15	πdxdy	πdxdy	NOUN
ma-232	14	16	=	=	SYM
ma-232	14	17	r	r	NOUN
ma-232	14	18	πdrdθ	πdrdθ	NOUN
ma-232	14	19	,	,	PUNCT
ma-232	14	20	where	where	SCONJ
ma-232	14	21	z	z	NOUN
ma-232	14	22	=	=	PUNCT
ma-232	14	23	x	x	PUNCT
ma-232	15	1	+	+	NUM
ma-232	15	2	iy	iy	X
ma-232	15	3	=	=	PUNCT
ma-232	15	4	re	re	X
ma-232	15	5	iθ	iθ	NOUN
ma-232	15	6	∈	∈	PROPN
ma-232	15	7	d.	d.	NOUN
ma-232	15	8	for	for	ADP
ma-232	15	9	α	α	PROPN
ma-232	15	10	∈	∈	PROPN
ma-232	15	11	r	r	PROPN
ma-232	15	12	,	,	PUNCT
ma-232	15	13	α	α	INTJ
ma-232	15	14	>	>	X
ma-232	15	15	−1	−1	NOUN
ma-232	15	16	,	,	PUNCT
ma-232	15	17	we	we	PRON
ma-232	15	18	define	define	VERB
ma-232	15	19	a	a	DET
ma-232	15	20	positive	positive	ADJ
ma-232	15	21	borel	borel	NOUN
ma-232	15	22	measure	measure	NOUN
ma-232	15	23	dmα	dmα	VERB
ma-232	15	24	on	on	ADP
ma-232	15	25	d	d	PROPN
ma-232	15	26	by	by	ADP
ma-232	15	27	dmα(z	dmα(z	PROPN
ma-232	15	28	)	)	PUNCT
ma-232	15	29	=	=	PUNCT
ma-232	15	30	(	(	PUNCT
ma-232	15	31	1−|z	1−|z	NUM
ma-232	15	32	|2)αda(z	|2)αda(z	NOUN
ma-232	15	33	)	)	PUNCT
ma-232	15	34	,	,	PUNCT
ma-232	15	35	andthus	andthus	PROPN
ma-232	15	36	dmα	dmα	PROPN
ma-232	15	37	is	be	AUX
ma-232	15	38	a	a	DET
ma-232	15	39	probability	probability	NOUN
ma-232	15	40	measure	measure	NOUN
ma-232	15	41	.	.	PUNCT
ma-232	16	1	moreover	moreover	ADV
ma-232	16	2	,	,	PUNCT
ma-232	16	3	if	if	SCONJ
ma-232	16	4	α	α	NOUN
ma-232	16	5	=	=	SYM
ma-232	16	6	0	0	NUM
ma-232	16	7	,	,	PUNCT
ma-232	16	8	then	then	ADV
ma-232	16	9	dmo	dmo	PROPN
ma-232	16	10	=	=	SYM
ma-232	16	11	da	da	PROPN
ma-232	16	12	.	.	PUNCT
ma-232	17	1	we	we	PRON
ma-232	17	2	consider	consider	VERB
ma-232	17	3	dmα	dmα	NOUN
ma-232	17	4	as	as	ADP
ma-232	17	5	aweighted	aweighte	VERB
ma-232	17	6	measure	measure	NOUN
ma-232	17	7	and	and	CCONJ
ma-232	17	8	a	a	DET
ma-232	17	9	generalization	generalization	NOUN
ma-232	17	10	of	of	ADP
ma-232	17	11	da	da	PROPN
ma-232	17	12	.	.	PUNCT
ma-232	18	1	on	on	ADP
ma-232	18	2	the	the	DET
ma-232	18	3	other	other	ADJ
ma-232	18	4	hand	hand	NOUN
ma-232	18	5	,	,	PUNCT
ma-232	18	6	the	the	DET
ma-232	18	7	set	set	ADJ
ma-232	18	8	u	u	NOUN
ma-232	18	9	:	:	PUNCT
ma-232	18	10	=	=	SYM
ma-232	18	11	{	{	PUNCT
ma-232	18	12	ω	ω	NUM
ma-232	18	13	∈	∈	PROPN
ma-232	18	14	c	c	NOUN
ma-232	18	15	:	:	PUNCT
ma-232	18	16	=(	=(	PROPN
ma-232	18	17	ω	ω	PROPN
ma-232	18	18	)	)	PUNCT
ma-232	18	19	>	>	X
ma-232	19	1	0}denotes	0}denote	NOUN
ma-232	19	2	the	the	DET
ma-232	19	3	upper	upper	ADJ
ma-232	19	4	half	half	NOUN
ma-232	19	5	of	of	ADP
ma-232	19	6	the	the	DET
ma-232	19	7	complex	complex	ADJ
ma-232	19	8	plane	plane	NOUN
ma-232	19	9	c	c	NOUN
ma-232	19	10	,	,	PUNCT
ma-232	19	11	with	with	ADP
ma-232	19	12	=(	=(	PROPN
ma-232	19	13	ω	ω	NOUN
ma-232	19	14	)	)	PUNCT
ma-232	19	15	being	be	AUX
ma-232	19	16	the	the	DET
ma-232	19	17	imaginary	imaginary	ADJ
ma-232	19	18	part	part	NOUN
ma-232	19	19	of	of	ADP
ma-232	19	20	ω	ω	PROPN
ma-232	19	21	∈	∈	PROPN
ma-232	19	22	c.	c.	NOUN
ma-232	19	23	for	for	ADP
ma-232	19	24	α	α	PROPN
ma-232	19	25	>	>	X
ma-232	19	26	−1	−1	NOUN
ma-232	19	27	,	,	PUNCT
ma-232	19	28	we	we	PRON
ma-232	19	29	define	define	VERB
ma-232	19	30	a	a	DET
ma-232	19	31	weighted	weighted	ADJ
ma-232	19	32	measure	measure	NOUN
ma-232	19	33	on	on	ADP
ma-232	19	34	u	u	NOUN
ma-232	19	35	by	by	ADP
ma-232	19	36	dµα(ω	dµα(ω	NOUN
ma-232	19	37	)	)	PUNCT
ma-232	19	38	=	=	SYM
ma-232	20	1	(	(	PUNCT
ma-232	20	2	=	=	NOUN
ma-232	20	3	(	(	PUNCT
ma-232	20	4	ω))αda(ω	ω))αda(ω	NOUN
ma-232	20	5	)	)	PUNCT
ma-232	20	6	,	,	PUNCT
ma-232	20	7	where	where	SCONJ
ma-232	20	8	ω	ω	PROPN
ma-232	20	9	∈	∈	PROPN
ma-232	20	10	u.	u.	VERB
ma-232	20	11	again	again	ADV
ma-232	20	12	itcan	itcan	AUX
ma-232	20	13	easily	easily	ADV
ma-232	20	14	be	be	AUX
ma-232	20	15	seen	see	VERB
ma-232	20	16	that	that	SCONJ
ma-232	20	17	α	α	PRON
ma-232	20	18	=	=	SYM
ma-232	20	19	0	0	PROPN
ma-232	20	20	coincides	coincide	VERB
ma-232	20	21	with	with	ADP
ma-232	20	22	the	the	DET
ma-232	20	23	unweighted	unweighted	ADJ
ma-232	20	24	measure	measure	NOUN
ma-232	20	25	.	.	PUNCT
ma-232	21	1	the	the	DET
ma-232	21	2	function	function	NOUN
ma-232	21	3	ψ(z	ψ(z	PROPN
ma-232	21	4	)	)	PUNCT
ma-232	21	5	=	=	SYM
ma-232	21	6	i(1+z	i(1+z	PROPN
ma-232	21	7	)	)	PUNCT
ma-232	21	8	1−zis	1−zi	NOUN
ma-232	21	9	referred	refer	VERB
ma-232	21	10	to	to	ADP
ma-232	21	11	as	as	SCONJ
ma-232	21	12	the	the	DET
ma-232	21	13	cayley	cayley	ADJ
ma-232	21	14	transform	transform	NOUN
ma-232	21	15	and	and	CCONJ
ma-232	21	16	maps	map	VERB
ma-232	21	17	the	the	DET
ma-232	21	18	unit	unit	NOUN
ma-232	21	19	disc	disc	VERB
ma-232	21	20	d	d	PROPN
ma-232	21	21	conformally	conformally	ADV
ma-232	21	22	onto	onto	ADP
ma-232	21	23	the	the	DET
ma-232	21	24	upper	upper	ADJ
ma-232	21	25	half	half	ADJ
ma-232	21	26	-	-	PUNCT
ma-232	21	27	plane	plane	NOUN
ma-232	21	28	u	u	NOUN
ma-232	21	29	with	with	ADP
ma-232	21	30	the	the	DET
ma-232	21	31	inverse	inverse	NOUN
ma-232	21	32	ψ−1(ω	ψ−1(ω	NOUN
ma-232	21	33	)	)	PUNCT
ma-232	21	34	=	=	PUNCT
ma-232	21	35	ω−i	ω−i	NUM
ma-232	21	36	ω+i	ω+i	NUM
ma-232	21	37	.for	.for	ADP
ma-232	21	38	an	an	DET
ma-232	21	39	open	open	ADJ
ma-232	21	40	subset	subset	NOUN
ma-232	21	41	ω	ω	PROPN
ma-232	21	42	of	of	ADP
ma-232	21	43	c	c	PROPN
ma-232	21	44	,	,	PUNCT
ma-232	21	45	let	let	VERB
ma-232	21	46	h(ω	h(ω	PROPN
ma-232	21	47	)	)	PUNCT
ma-232	21	48	denote	denote	VERB
ma-232	21	49	the	the	DET
ma-232	21	50	space	space	NOUN
ma-232	21	51	of	of	ADP
ma-232	21	52	analytic	analytic	ADJ
ma-232	21	53	functions	function	NOUN
ma-232	21	54	on	on	ADP
ma-232	21	55	ω	ω	NUM
ma-232	21	56	.	.	PUNCT
ma-232	22	1	for	for	ADP
ma-232	22	2	1	1	NUM
ma-232	22	3	≤	≤	NOUN
ma-232	22	4	p	p	NOUN
ma-232	22	5	<	<	X
ma-232	22	6	∞	∞	PROPN
ma-232	22	7	,	,	PUNCT
ma-232	22	8	received	receive	VERB
ma-232	22	9	:	:	PUNCT
ma-232	22	10	29	29	NUM
ma-232	22	11	feb	feb	NOUN
ma-232	22	12	2024	2024	NUM
ma-232	22	13	.	.	PUNCT
ma-232	23	1	key	key	ADJ
ma-232	23	2	words	word	NOUN
ma-232	23	3	and	and	CCONJ
ma-232	23	4	phrases	phrase	NOUN
ma-232	23	5	.	.	PUNCT
ma-232	24	1	duality	duality	NOUN
ma-232	24	2	,	,	PUNCT
ma-232	24	3	nonreflexive	nonreflexive	ADJ
ma-232	24	4	bergman	bergman	PROPN
ma-232	24	5	space	space	PROPN
ma-232	24	6	,	,	PUNCT
ma-232	24	7	bloch	bloch	PROPN
ma-232	24	8	space	space	NOUN
ma-232	24	9	,	,	PUNCT
ma-232	24	10	composition	composition	NOUN
ma-232	24	11	semigroups	semigroup	NOUN
ma-232	24	12	,	,	PUNCT
ma-232	24	13	infinitesimalgenerator	infinitesimalgenerator	NOUN
ma-232	24	14	,	,	PUNCT
ma-232	24	15	spectrum	spectrum	NOUN
ma-232	24	16	,	,	PUNCT
ma-232	24	17	resolvent	resolvent	ADJ
ma-232	24	18	.	.	PUNCT
ma-232	24	19	1	1	NUM
ma-232	24	20	https://adac.ee	https://adac.ee	PROPN
ma-232	24	21	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	24	22	https://orcid.org/0000-0003-1785-2202	https://orcid.org/0000-0003-1785-2202	NOUN
ma-232	24	23	https://orcid.org/0000-0002-6442-4211	https://orcid.org/0000-0002-6442-4211	PROPN
ma-232	24	24	eur	eur	PROPN
ma-232	24	25	.	.	PUNCT
ma-232	25	1	j.	j.	PROPN
ma-232	25	2	math	math	PROPN
ma-232	25	3	.	.	PUNCT
ma-232	26	1	anal	anal	PROPN
ma-232	26	2	.	.	PUNCT
ma-232	27	1	10.28924	10.28924	NUM
ma-232	27	2	/	/	SYM
ma-232	27	3	ada	ada	PROPN
ma-232	27	4	/	/	SYM
ma-232	27	5	ma.4.14	ma.4.14	PROPN
ma-232	27	6	2	2	NUM
ma-232	27	7	α	α	NOUN
ma-232	27	8	>	>	X
ma-232	27	9	−1	−1	NOUN
ma-232	27	10	,	,	PUNCT
ma-232	27	11	the	the	DET
ma-232	27	12	weighted	weight	VERB
ma-232	27	13	bergman	bergman	PROPN
ma-232	27	14	space	space	NOUN
ma-232	27	15	of	of	ADP
ma-232	27	16	the	the	DET
ma-232	27	17	upper	upper	ADJ
ma-232	27	18	half	half	ADJ
ma-232	27	19	-	-	PUNCT
ma-232	27	20	plane	plane	NOUN
ma-232	27	21	u	u	NOUN
ma-232	27	22	is	be	AUX
ma-232	27	23	defined	define	VERB
ma-232	27	24	by	by	ADP
ma-232	27	25	lpa(u	lpa(u	PROPN
ma-232	27	26	,	,	PUNCT
ma-232	27	27	µα	µα	ADP
ma-232	27	28	)	)	PUNCT
ma-232	27	29	:	:	PUNCT
ma-232	28	1	=	=	X
ma-232	28	2	{	{	PUNCT
ma-232	28	3	f	f	PROPN
ma-232	28	4	∈	∈	PROPN
ma-232	28	5	h(u	h(u	PROPN
ma-232	28	6	)	)	PUNCT
ma-232	28	7	:	:	PUNCT
ma-232	28	8	‖f	‖f	ADP
ma-232	28	9	‖lpa(u,µα	‖lpa(u,µα	NUM
ma-232	28	10	)	)	PUNCT
ma-232	28	11	=	=	SYM
ma-232	28	12	(	(	PUNCT
ma-232	28	13	∫	∫	PROPN
ma-232	28	14	u	u	PROPN
ma-232	28	15	|f	|f	PROPN
ma-232	28	16	(	(	PUNCT
ma-232	28	17	z)|pdµα(z	z)|pdµα(z	PROPN
ma-232	28	18	)	)	PUNCT
ma-232	28	19	)	)	PUNCT
ma-232	29	1	1	1	NUM
ma-232	30	1	p	p	NOUN
ma-232	30	2	<	<	X
ma-232	30	3	∞	∞	NUM
ma-232	30	4	}	}	PUNCT
ma-232	30	5	.	.	PUNCT
ma-232	31	1	in	in	ADP
ma-232	31	2	particular	particular	ADJ
ma-232	31	3	,	,	PUNCT
ma-232	31	4	lpa(u	lpa(u	PROPN
ma-232	31	5	,	,	PUNCT
ma-232	31	6	µα	µα	NOUN
ma-232	31	7	)	)	PUNCT
ma-232	31	8	=	=	SYM
ma-232	31	9	lp(u	lp(u	X
ma-232	31	10	,	,	PUNCT
ma-232	31	11	µα	µα	NOUN
ma-232	31	12	)	)	PUNCT
ma-232	31	13	∩	∩	NOUN
ma-232	31	14	h(u	h(u	PROPN
ma-232	31	15	)	)	PUNCT
ma-232	31	16	,	,	PUNCT
ma-232	31	17	where	where	SCONJ
ma-232	31	18	lp(u	lp(u	X
ma-232	31	19	,	,	PUNCT
ma-232	31	20	µα	µα	ADP
ma-232	31	21	)	)	PUNCT
ma-232	31	22	or	or	CCONJ
ma-232	31	23	simply	simply	ADV
ma-232	31	24	lp(µα	lp(µα	ADJ
ma-232	31	25	)	)	PUNCT
ma-232	31	26	denotes	denote	VERB
ma-232	31	27	theclassical	theclassical	ADJ
ma-232	31	28	lebesque	lebesque	NOUN
ma-232	31	29	spaces	space	NOUN
ma-232	31	30	with	with	ADP
ma-232	31	31	respect	respect	NOUN
ma-232	31	32	to	to	ADP
ma-232	31	33	the	the	DET
ma-232	31	34	weighted	weight	VERB
ma-232	31	35	measure	measure	NOUN
ma-232	31	36	dµα	dµα	PROPN
ma-232	31	37	.	.	PUNCT
ma-232	32	1	it	it	PRON
ma-232	32	2	is	be	AUX
ma-232	32	3	important	important	ADJ
ma-232	32	4	to	to	PART
ma-232	32	5	note	note	VERB
ma-232	32	6	thatthe	thatthe	NOUN
ma-232	32	7	case	case	NOUN
ma-232	32	8	α	α	X
ma-232	32	9	=	=	SYM
ma-232	32	10	0	0	NUM
ma-232	32	11	yields	yield	VERB
ma-232	32	12	the	the	DET
ma-232	32	13	unweighted	unweighted	ADJ
ma-232	32	14	bergman	bergman	PROPN
ma-232	32	15	space	space	NOUN
ma-232	32	16	.	.	PUNCT
ma-232	33	1	lpa(u	lpa(u	X
ma-232	33	2	,	,	PUNCT
ma-232	33	3	µα	µα	NOUN
ma-232	33	4	)	)	PUNCT
ma-232	33	5	is	be	AUX
ma-232	33	6	a	a	DET
ma-232	33	7	banach	banach	NOUN
ma-232	33	8	space	space	NOUN
ma-232	33	9	with	with	ADP
ma-232	33	10	respectto	respectto	ADJ
ma-232	33	11	the	the	DET
ma-232	33	12	norm	norm	NOUN
ma-232	33	13	‖f	‖f	PUNCT
ma-232	33	14	‖lpa(u,µα	‖lpa(u,µα	SYM
ma-232	33	15	)	)	PUNCT
ma-232	33	16	=	=	SYM
ma-232	34	1	(	(	PUNCT
ma-232	34	2	∫	∫	PROPN
ma-232	34	3	u	u	PROPN
ma-232	34	4	|f	|f	PROPN
ma-232	34	5	(	(	PUNCT
ma-232	34	6	ω)|pdµα(ω	ω)|pdµα(ω	PROPN
ma-232	34	7	)	)	PUNCT
ma-232	34	8	)	)	PUNCT
ma-232	34	9	1	1	NUM
ma-232	34	10	p	p	NOUN
ma-232	34	11	.	.	PUNCT
ma-232	35	1	for	for	ADP
ma-232	35	2	p	p	NOUN
ma-232	35	3	=	=	SYM
ma-232	35	4	2	2	NUM
ma-232	35	5	,	,	PUNCT
ma-232	35	6	l2	l2	NOUN
ma-232	35	7	a(u	a(u	NOUN
ma-232	35	8	,	,	PUNCT
ma-232	35	9	µα	µα	ADP
ma-232	35	10	)	)	PUNCT
ma-232	35	11	is	be	AUX
ma-232	35	12	a	a	DET
ma-232	35	13	hilbert	hilbert	NOUN
ma-232	35	14	space	space	NOUN
ma-232	35	15	.	.	PUNCT
ma-232	36	1	the	the	DET
ma-232	36	2	growth	growth	NOUN
ma-232	36	3	condition	condition	NOUN
ma-232	36	4	for	for	ADP
ma-232	36	5	the	the	DET
ma-232	36	6	weighted	weight	VERB
ma-232	36	7	bergman	bergman	PROPN
ma-232	36	8	spacefunctions	spacefunction	NOUN
ma-232	36	9	on	on	ADP
ma-232	36	10	u	u	NOUN
ma-232	36	11	is	be	AUX
ma-232	36	12	given	give	VERB
ma-232	36	13	by	by	ADP
ma-232	36	14	:	:	PUNCT
ma-232	36	15	for	for	ADP
ma-232	36	16	every	every	DET
ma-232	36	17	f	f	PROPN
ma-232	36	18	∈	∈	PROPN
ma-232	36	19	lpa(u	lpa(u	PROPN
ma-232	36	20	,	,	PUNCT
ma-232	36	21	µα	µα	ADJ
ma-232	36	22	)	)	PUNCT
ma-232	36	23	,	,	PUNCT
ma-232	36	24	γ	γ	X
ma-232	36	25	=	=	SYM
ma-232	36	26	α+2	α+2	PROPN
ma-232	36	27	p	p	NOUN
ma-232	36	28	and	and	CCONJ
ma-232	36	29	ω	ω	NUM
ma-232	36	30	∈	∈	PROPN
ma-232	36	31	u	u	NOUN
ma-232	36	32	,	,	PUNCT
ma-232	36	33	there	there	PRON
ma-232	36	34	exists	exist	VERB
ma-232	36	35	a	a	DET
ma-232	36	36	constant	constant	ADJ
ma-232	36	37	k	k	NOUN
ma-232	36	38	such	such	ADJ
ma-232	36	39	that	that	PRON
ma-232	36	40	,	,	PUNCT
ma-232	36	41	|f	|f	PROPN
ma-232	36	42	(	(	PUNCT
ma-232	36	43	ω)|	ω)|	ADJ
ma-232	36	44	≤	≤	NUM
ma-232	36	45	k‖f	k‖f	NOUN
ma-232	36	46	‖	‖	PROPN
ma-232	36	47	(=	(=	X
ma-232	36	48	(	(	PUNCT
ma-232	36	49	ω))γ	ω))γ	NOUN
ma-232	36	50	.	.	PUNCT
ma-232	37	1	for	for	ADP
ma-232	37	2	a	a	DET
ma-232	37	3	detailed	detailed	ADJ
ma-232	37	4	account	account	NOUN
ma-232	37	5	of	of	ADP
ma-232	37	6	the	the	DET
ma-232	37	7	theory	theory	NOUN
ma-232	37	8	of	of	ADP
ma-232	37	9	bergman	bergman	PROPN
ma-232	37	10	spaces	space	VERB
ma-232	37	11	,	,	PUNCT
ma-232	37	12	we	we	PRON
ma-232	37	13	refer	refer	VERB
ma-232	37	14	to	to	ADP
ma-232	37	15	[	[	X
ma-232	37	16	7	7	NUM
ma-232	37	17	,	,	PUNCT
ma-232	37	18	11,13].on	11,13].on	NUM
ma-232	37	19	the	the	DET
ma-232	37	20	other	other	ADJ
ma-232	37	21	hand	hand	NOUN
ma-232	37	22	,	,	PUNCT
ma-232	37	23	the	the	DET
ma-232	37	24	bloch	bloch	PROPN
ma-232	37	25	space	space	NOUN
ma-232	37	26	of	of	ADP
ma-232	37	27	the	the	DET
ma-232	37	28	unit	unit	NOUN
ma-232	37	29	disk	disk	NOUN
ma-232	37	30	,	,	PUNCT
ma-232	37	31	denoted	denote	VERB
ma-232	37	32	by	by	ADP
ma-232	37	33	b∞(d	b∞(d	PROPN
ma-232	37	34	)	)	PUNCT
ma-232	37	35	,	,	PUNCT
ma-232	37	36	is	be	AUX
ma-232	37	37	defined	define	VERB
ma-232	37	38	by	by	ADP
ma-232	37	39	b∞(d	b∞(d	PROPN
ma-232	37	40	)	)	PUNCT
ma-232	37	41	:	:	PUNCT
ma-232	38	1	=	=	PUNCT
ma-232	38	2	{	{	PUNCT
ma-232	38	3	f	f	PROPN
ma-232	38	4	∈	∈	PROPN
ma-232	38	5	h(d	h(d	PROPN
ma-232	38	6	)	)	PUNCT
ma-232	38	7	:	:	PUNCT
ma-232	39	1	‖f	‖f	DET
ma-232	39	2	‖b∞,1(d	‖b∞,1(d	NOUN
ma-232	39	3	)	)	PUNCT
ma-232	39	4	=	=	SYM
ma-232	39	5	sup	sup	NOUN
ma-232	39	6	z∈d	z∈d	NOUN
ma-232	39	7	(	(	PUNCT
ma-232	39	8	1−	1−	NUM
ma-232	39	9	|z	|z	PROPN
ma-232	39	10	|2)|f	|2)|f	PROPN
ma-232	39	11	′(z)|	′(z)|	PROPN
ma-232	39	12	<	<	X
ma-232	39	13	∞	∞	NUM
ma-232	39	14	}	}	PUNCT
ma-232	39	15	,	,	PUNCT
ma-232	39	16	with	with	ADP
ma-232	39	17	the	the	DET
ma-232	39	18	norm	norm	NOUN
ma-232	39	19	on	on	ADP
ma-232	39	20	b∞(d	b∞(d	NOUN
ma-232	39	21	)	)	PUNCT
ma-232	39	22	given	give	VERB
ma-232	39	23	by	by	ADP
ma-232	39	24	‖f	‖f	PRON
ma-232	39	25	‖b∞(d	‖b∞(d	CCONJ
ma-232	39	26	)	)	PUNCT
ma-232	39	27	:	:	PUNCT
ma-232	40	1	=	=	SYM
ma-232	40	2	|f	|f	PROPN
ma-232	40	3	(	(	PUNCT
ma-232	40	4	0)|+‖f	0)|+‖f	NOUN
ma-232	40	5	‖b∞,1(d	‖b∞,1(d	ADJ
ma-232	40	6	)	)	PUNCT
ma-232	40	7	,	,	PUNCT
ma-232	40	8	while	while	SCONJ
ma-232	40	9	‖.‖b∞,1(d	‖.‖b∞,1(d	NUM
ma-232	40	10	)	)	PUNCT
ma-232	40	11	is	be	AUX
ma-232	40	12	a	a	DET
ma-232	40	13	seminorm.the	seminorm.the	X
ma-232	40	14	bloch	bloch	NOUN
ma-232	40	15	space	space	NOUN
ma-232	40	16	of	of	ADP
ma-232	40	17	the	the	DET
ma-232	40	18	upper	upper	ADJ
ma-232	40	19	half	half	ADJ
ma-232	40	20	plane	plane	NOUN
ma-232	40	21	denoted	denote	VERB
ma-232	40	22	by	by	ADP
ma-232	40	23	b∞(u	b∞(u	NOUN
ma-232	40	24	)	)	PUNCT
ma-232	40	25	is	be	AUX
ma-232	40	26	defined	define	VERB
ma-232	40	27	by	by	ADP
ma-232	40	28	b∞(u	b∞(u	NOUN
ma-232	40	29	)	)	PUNCT
ma-232	40	30	:	:	PUNCT
ma-232	41	1	=	=	PUNCT
ma-232	41	2	{	{	PUNCT
ma-232	41	3	f	f	PROPN
ma-232	41	4	∈	∈	PROPN
ma-232	41	5	h(u	h(u	PROPN
ma-232	41	6	)	)	PUNCT
ma-232	41	7	:	:	PUNCT
ma-232	41	8	‖f	‖f	ADP
ma-232	41	9	‖b∞,1(u	‖b∞,1(u	NOUN
ma-232	41	10	)	)	PUNCT
ma-232	41	11	=	=	SYM
ma-232	41	12	sup	sup	NOUN
ma-232	41	13	ω∈u	ω∈u	NOUN
ma-232	41	14	=(	=(	NOUN
ma-232	41	15	ω)|f	ω)|f	NOUN
ma-232	41	16	′(ω)|	′(ω)|	X
ma-232	41	17	<	<	X
ma-232	41	18	∞	∞	NUM
ma-232	41	19	}	}	PUNCT
ma-232	41	20	,	,	PUNCT
ma-232	41	21	with	with	ADP
ma-232	41	22	the	the	DET
ma-232	41	23	norm	norm	NOUN
ma-232	41	24	given	give	VERB
ma-232	41	25	by	by	ADP
ma-232	41	26	‖f	‖f	DET
ma-232	41	27	‖b∞(u	‖b∞(u	NOUN
ma-232	41	28	)	)	PUNCT
ma-232	41	29	=	=	SYM
ma-232	41	30	|f	|f	PROPN
ma-232	41	31	(	(	PUNCT
ma-232	41	32	i)|	i)|	INTJ
ma-232	41	33	+	+	CCONJ
ma-232	41	34	‖f	‖f	ADJ
ma-232	41	35	‖b∞,1(u	‖b∞,1(u	NOUN
ma-232	41	36	)	)	PUNCT
ma-232	41	37	.	.	PUNCT
ma-232	42	1	the	the	DET
ma-232	42	2	little	little	ADJ
ma-232	42	3	bloch	bloch	PROPN
ma-232	42	4	space	space	NOUN
ma-232	42	5	of	of	ADP
ma-232	42	6	the	the	DET
ma-232	42	7	unit	unit	NOUN
ma-232	42	8	diskdenoted	diskdenote	VERB
ma-232	42	9	by	by	ADP
ma-232	42	10	b∞,	b∞,	NOUN
ma-232	42	11	◦	◦	NOUN
ma-232	42	12	(d	(d	NOUN
ma-232	42	13	)	)	PUNCT
ma-232	42	14	is	be	AUX
ma-232	42	15	defined	define	VERB
ma-232	42	16	as	as	ADP
ma-232	42	17	b∞,	b∞,	ADP
ma-232	42	18	◦	◦	NOUN
ma-232	42	19	(d	(d	NOUN
ma-232	42	20	)	)	PUNCT
ma-232	42	21	:	:	PUNCT
ma-232	43	1	=	=	SYM
ma-232	43	2	{	{	PUNCT
ma-232	43	3	f	f	PROPN
ma-232	43	4	∈	∈	PROPN
ma-232	43	5	h(d	h(d	PROPN
ma-232	43	6	)	)	PUNCT
ma-232	43	7	:	:	PUNCT
ma-232	43	8	lim	lim	PROPN
ma-232	43	9	|z	|z	PROPN
ma-232	43	10	|→1	|→1	PROPN
ma-232	43	11	(	(	PUNCT
ma-232	43	12	1−	1−	NUM
ma-232	43	13	|z	|z	PROPN
ma-232	43	14	|2)|f	|2)|f	PROPN
ma-232	43	15	′(z)|	′(z)|	PROPN
ma-232	43	16	=	=	PUNCT
ma-232	43	17	0	0	NUM
ma-232	43	18	}	}	PUNCT
ma-232	43	19	but	but	CCONJ
ma-232	43	20	with	with	ADP
ma-232	43	21	the	the	DET
ma-232	43	22	same	same	ADJ
ma-232	43	23	norm	norm	NOUN
ma-232	43	24	as	as	ADP
ma-232	43	25	b∞(d	b∞(d	NOUN
ma-232	43	26	)	)	PUNCT
ma-232	43	27	,	,	PUNCT
ma-232	43	28	while	while	SCONJ
ma-232	43	29	for	for	ADP
ma-232	43	30	the	the	DET
ma-232	43	31	upper	upper	ADJ
ma-232	43	32	half	half	ADJ
ma-232	43	33	-	-	PUNCT
ma-232	43	34	plane	plane	NOUN
ma-232	43	35	,	,	PUNCT
ma-232	43	36	the	the	DET
ma-232	43	37	little	little	ADJ
ma-232	43	38	bloch	bloch	NOUN
ma-232	43	39	space	space	NOUN
ma-232	43	40	is	be	AUX
ma-232	43	41	denotedby	denotedby	ADJ
ma-232	43	42	b∞,	b∞,	ADP
ma-232	43	43	◦	◦	NOUN
ma-232	43	44	(u	(u	NOUN
ma-232	43	45	)	)	PUNCT
ma-232	43	46	and	and	CCONJ
ma-232	43	47	is	be	AUX
ma-232	43	48	defined	define	VERB
ma-232	43	49	by	by	ADP
ma-232	43	50	b∞,	b∞,	NOUN
ma-232	43	51	◦	◦	NOUN
ma-232	43	52	(u	(u	NOUN
ma-232	43	53	)	)	PUNCT
ma-232	43	54	:	:	PUNCT
ma-232	43	55	=	=	PUNCT
ma-232	43	56	{	{	PUNCT
ma-232	43	57	f	f	PROPN
ma-232	43	58	∈	∈	PROPN
ma-232	43	59	h(u	h(u	PROPN
ma-232	43	60	)	)	PUNCT
ma-232	43	61	:	:	PUNCT
ma-232	44	1	lim	lim	PROPN
ma-232	44	2	=(	=(	PROPN
ma-232	44	3	ω)→0	ω)→0	NOUN
ma-232	44	4	=(	=(	NOUN
ma-232	44	5	ω)|f	ω)|f	NOUN
ma-232	44	6	′(ω)|	′(ω)|	PROPN
ma-232	45	1	=	=	SYM
ma-232	45	2	0	0	NUM
ma-232	45	3	}	}	PUNCT
ma-232	45	4	with	with	ADP
ma-232	45	5	the	the	DET
ma-232	45	6	same	same	ADJ
ma-232	45	7	norm	norm	NOUN
ma-232	45	8	as	as	ADP
ma-232	45	9	b∞(u	b∞(u	ADJ
ma-232	45	10	)	)	PUNCT
ma-232	45	11	.	.	PUNCT
ma-232	46	1	for	for	ADP
ma-232	46	2	a	a	DET
ma-232	46	3	comprehensive	comprehensive	ADJ
ma-232	46	4	theory	theory	NOUN
ma-232	46	5	of	of	ADP
ma-232	46	6	bloch	bloch	PROPN
ma-232	46	7	spaces	space	VERB
ma-232	46	8	,	,	PUNCT
ma-232	46	9	see	see	VERB
ma-232	46	10	[	[	PUNCT
ma-232	46	11	13,14].the	13,14].the	DET
ma-232	46	12	duality	duality	NOUN
ma-232	46	13	properties	property	NOUN
ma-232	46	14	of	of	ADP
ma-232	46	15	bergman	bergman	PROPN
ma-232	46	16	spaces	space	NOUN
ma-232	46	17	are	be	AUX
ma-232	46	18	well	well	ADV
ma-232	46	19	known	know	VERB
ma-232	46	20	in	in	ADP
ma-232	46	21	literature	literature	NOUN
ma-232	46	22	.	.	PUNCT
ma-232	47	1	for	for	ADP
ma-232	47	2	instance	instance	NOUN
ma-232	47	3	in	in	ADP
ma-232	47	4	[	[	X
ma-232	47	5	13	13	NUM
ma-232	47	6	,	,	PUNCT
ma-232	47	7	theorem4.2.9	theorem4.2.9	NOUN
ma-232	47	8	]	]	X
ma-232	47	9	,	,	PUNCT
ma-232	47	10	it	it	PRON
ma-232	47	11	is	be	AUX
ma-232	47	12	proved	prove	VERB
ma-232	47	13	that	that	SCONJ
ma-232	47	14	for	for	ADP
ma-232	47	15	1	1	NUM
ma-232	47	16	<	<	X
ma-232	47	17	p	p	X
ma-232	47	18	<	<	X
ma-232	47	19	∞	∞	PROPN
ma-232	47	20	,	,	PUNCT
ma-232	47	21	1	1	NUM
ma-232	47	22	p	p	NOUN
ma-232	47	23	+	+	NOUN
ma-232	47	24	1	1	NUM
ma-232	47	25	q	q	NOUN
ma-232	47	26	=	=	SYM
ma-232	47	27	1	1	NUM
ma-232	47	28	and	and	CCONJ
ma-232	47	29	α	α	NOUN
ma-232	47	30	>	>	X
ma-232	47	31	−1	−1	NOUN
ma-232	47	32	,	,	PUNCT
ma-232	47	33	the	the	DET
ma-232	47	34	dual	dual	ADJ
ma-232	47	35	of	of	ADP
ma-232	47	36	the	the	DET
ma-232	47	37	bergman	bergman	PROPN
ma-232	47	38	space	space	PROPN
ma-232	47	39	lpa(d	lpa(d	PROPN
ma-232	47	40	,	,	PUNCT
ma-232	47	41	mα	mα	PROPN
ma-232	47	42	)	)	PUNCT
ma-232	47	43	is	be	AUX
ma-232	47	44	given	give	VERB
ma-232	47	45	by	by	ADP
ma-232	47	46	(	(	PUNCT
ma-232	47	47	lpa(d	lpa(d	PROPN
ma-232	47	48	,	,	PUNCT
ma-232	47	49	mα))∗	mα))∗	NOUN
ma-232	48	1	≈	≈	PROPN
ma-232	48	2	lqa(d	lqa(d	PROPN
ma-232	48	3	,	,	PUNCT
ma-232	48	4	mα	mα	PROPN
ma-232	48	5	)	)	PUNCT
ma-232	48	6	under	under	ADP
ma-232	48	7	the	the	DET
ma-232	48	8	duality	duality	NOUN
ma-232	48	9	pairing	pairing	NOUN
ma-232	48	10	,	,	PUNCT
ma-232	48	11	〈	〈	PROPN
ma-232	48	12	g	g	PROPN
ma-232	48	13	,	,	PUNCT
ma-232	48	14	f	f	PROPN
ma-232	48	15	〉	〉	PROPN
ma-232	48	16	=	=	SYM
ma-232	49	1	∫	∫	PROPN
ma-232	50	1	d	d	X
ma-232	50	2	g(z)f	g(z)f	PROPN
ma-232	50	3	(	(	PUNCT
ma-232	50	4	z)dmα	z)dmα	PROPN
ma-232	50	5	(	(	PUNCT
ma-232	50	6	g	g	PROPN
ma-232	50	7	∈	∈	PROPN
ma-232	50	8	lpa(d	lpa(d	PROPN
ma-232	50	9	,	,	PUNCT
ma-232	50	10	mα	mα	PROPN
ma-232	50	11	)	)	PUNCT
ma-232	50	12	,	,	PUNCT
ma-232	50	13	f	f	PROPN
ma-232	50	14	∈	∈	PROPN
ma-232	50	15	lqa(d	lqa(d	PROPN
ma-232	50	16	,	,	PUNCT
ma-232	50	17	mα	mα	PROPN
ma-232	50	18	)	)	PUNCT
ma-232	50	19	)	)	PUNCT
ma-232	50	20	.	.	PUNCT
ma-232	51	1	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	51	2	eur	eur	PROPN
ma-232	51	3	.	.	PUNCT
ma-232	52	1	j.	j.	PROPN
ma-232	52	2	math	math	PROPN
ma-232	52	3	.	.	PUNCT
ma-232	53	1	anal	anal	PROPN
ma-232	53	2	.	.	PUNCT
ma-232	54	1	10.28924	10.28924	NUM
ma-232	54	2	/	/	SYM
ma-232	54	3	ada	ada	PROPN
ma-232	54	4	/	/	SYM
ma-232	54	5	ma.4.14	ma.4.14	PROPN
ma-232	55	1	3for	3for	ADP
ma-232	55	2	the	the	DET
ma-232	55	3	non	non	ADJ
ma-232	55	4	-	-	ADJ
ma-232	55	5	reflexive	reflexive	ADJ
ma-232	55	6	bergman	bergman	PROPN
ma-232	55	7	space	space	NOUN
ma-232	55	8	on	on	ADP
ma-232	55	9	the	the	DET
ma-232	55	10	unit	unit	NOUN
ma-232	55	11	disk	disk	NOUN
ma-232	55	12	,	,	PUNCT
ma-232	55	13	l1	l1	PROPN
ma-232	55	14	a(d	a(d	PROPN
ma-232	55	15	,	,	PUNCT
ma-232	55	16	mα	mα	PROPN
ma-232	55	17	)	)	PUNCT
ma-232	55	18	,	,	PUNCT
ma-232	55	19	it	it	PRON
ma-232	55	20	is	be	AUX
ma-232	55	21	shown	show	VERB
ma-232	55	22	in	in	ADP
ma-232	55	23	[	[	X
ma-232	55	24	13	13	NUM
ma-232	55	25	,	,	PUNCT
ma-232	55	26	theorems	theorem	VERB
ma-232	55	27	5.1.4and	5.1.4and	PROPN
ma-232	55	28	5.2.8	5.2.8	NUM
ma-232	55	29	]	]	PUNCT
ma-232	55	30	that	that	SCONJ
ma-232	55	31	the	the	DET
ma-232	55	32	dual	dual	ADJ
ma-232	55	33	and	and	CCONJ
ma-232	55	34	predual	predual	ADJ
ma-232	55	35	spaces	space	NOUN
ma-232	55	36	of	of	ADP
ma-232	55	37	l1	l1	PROPN
ma-232	55	38	a(d	a(d	PROPN
ma-232	55	39	,	,	PUNCT
ma-232	55	40	mα	mα	PROPN
ma-232	55	41	)	)	PUNCT
ma-232	55	42	are	be	AUX
ma-232	55	43	the	the	DET
ma-232	55	44	bloch	bloch	PROPN
ma-232	55	45	space	space	NOUN
ma-232	55	46	and	and	CCONJ
ma-232	55	47	the	the	DET
ma-232	55	48	little	little	ADJ
ma-232	55	49	blochspace	blochspace	NOUN
ma-232	55	50	respectively	respectively	ADV
ma-232	55	51	.	.	PUNCT
ma-232	56	1	in	in	ADP
ma-232	56	2	particular	particular	ADJ
ma-232	56	3	,	,	PUNCT
ma-232	56	4	(	(	PUNCT
ma-232	56	5	l1	l1	PROPN
ma-232	56	6	a(d	a(d	PROPN
ma-232	56	7	,	,	PUNCT
ma-232	56	8	mα))∗	mα))∗	NOUN
ma-232	57	1	≈	≈	PROPN
ma-232	57	2	b∞(d	b∞(d	PROPN
ma-232	57	3	)	)	PUNCT
ma-232	57	4	and	and	CCONJ
ma-232	57	5	(	(	PUNCT
ma-232	57	6	b∞,	b∞,	ADP
ma-232	57	7	◦	◦	NOUN
ma-232	57	8	(d))∗	(d))∗	X
ma-232	57	9	≈	≈	PROPN
ma-232	57	10	l1	l1	PROPN
ma-232	57	11	a(d	a(d	PROPN
ma-232	57	12	,	,	PUNCT
ma-232	57	13	mα	mα	PROPN
ma-232	57	14	)	)	PUNCT
ma-232	57	15	under	under	ADP
ma-232	57	16	theduality	theduality	NOUN
ma-232	57	17	pairings	pairing	NOUN
ma-232	57	18	given	give	VERB
ma-232	57	19	by	by	ADP
ma-232	57	20	respectively	respectively	ADV
ma-232	57	21	,	,	PUNCT
ma-232	57	22	〈	〈	PROPN
ma-232	57	23	g	g	PROPN
ma-232	57	24	,	,	PUNCT
ma-232	57	25	f	f	PROPN
ma-232	58	1	〉	〉	PROPN
ma-232	58	2	=	=	SYM
ma-232	58	3	∫	∫	PROPN
ma-232	59	1	d	d	X
ma-232	59	2	g(z)f	g(z)f	PROPN
ma-232	59	3	(	(	PUNCT
ma-232	59	4	z)dmα(z	z)dmα(z	PROPN
ma-232	59	5	)	)	PUNCT
ma-232	59	6	(	(	PUNCT
ma-232	59	7	g	g	PROPN
ma-232	59	8	∈	∈	PROPN
ma-232	59	9	l1	l1	PROPN
ma-232	59	10	a(d	a(d	PROPN
ma-232	59	11	,	,	PUNCT
ma-232	59	12	mα	mα	PROPN
ma-232	59	13	)	)	PUNCT
ma-232	59	14	,	,	PUNCT
ma-232	59	15	f	f	PROPN
ma-232	59	16	∈	∈	PROPN
ma-232	59	17	b∞(d	b∞(d	PROPN
ma-232	59	18	)	)	PUNCT
ma-232	59	19	)	)	PUNCT
ma-232	59	20	,	,	PUNCT
ma-232	59	21	and	and	CCONJ
ma-232	59	22	〈	〈	PROPN
ma-232	59	23	g	g	PROPN
ma-232	59	24	,	,	PUNCT
ma-232	59	25	f	f	PROPN
ma-232	59	26	〉	〉	PROPN
ma-232	59	27	=	=	SYM
ma-232	59	28	∫	∫	PROPN
ma-232	60	1	d	d	X
ma-232	60	2	g(z)f	g(z)f	PROPN
ma-232	60	3	(	(	PUNCT
ma-232	60	4	z)dmα(z	z)dmα(z	PROPN
ma-232	60	5	)	)	PUNCT
ma-232	60	6	(	(	PUNCT
ma-232	60	7	f	f	PROPN
ma-232	60	8	∈	∈	PROPN
ma-232	60	9	l1	l1	PROPN
ma-232	60	10	a(d	a(d	PROPN
ma-232	60	11	,	,	PUNCT
ma-232	60	12	mα	mα	PROPN
ma-232	60	13	)	)	PUNCT
ma-232	60	14	,	,	PUNCT
ma-232	60	15	g	g	PROPN
ma-232	60	16	∈	∈	PROPN
ma-232	60	17	b∞,	b∞,	ADP
ma-232	60	18	◦	◦	NOUN
ma-232	60	19	(d	(d	NOUN
ma-232	60	20	)	)	PUNCT
ma-232	60	21	)	)	PUNCT
ma-232	60	22	.	.	PUNCT
ma-232	61	1	for	for	ADP
ma-232	61	2	the	the	DET
ma-232	61	3	corresponding	corresponding	ADJ
ma-232	61	4	spaces	space	NOUN
ma-232	61	5	of	of	ADP
ma-232	61	6	the	the	DET
ma-232	61	7	upper	upper	ADJ
ma-232	61	8	half	half	NOUN
ma-232	61	9	plane	plane	NOUN
ma-232	61	10	,	,	PUNCT
ma-232	61	11	it	it	PRON
ma-232	61	12	has	have	AUX
ma-232	61	13	been	be	AUX
ma-232	61	14	proved	prove	VERB
ma-232	61	15	and	and	CCONJ
ma-232	61	16	noted	note	VERB
ma-232	61	17	that	that	SCONJ
ma-232	61	18	thedual	thedual	ADJ
ma-232	61	19	space	space	NOUN
ma-232	61	20	of	of	ADP
ma-232	61	21	the	the	DET
ma-232	61	22	reflexive	reflexive	ADJ
ma-232	61	23	bergman	bergman	PROPN
ma-232	61	24	space	space	NOUN
ma-232	61	25	of	of	ADP
ma-232	61	26	the	the	DET
ma-232	61	27	upper	upper	ADJ
ma-232	61	28	half	half	ADJ
ma-232	61	29	plane	plane	NOUN
ma-232	61	30	lpa(u	lpa(u	PROPN
ma-232	61	31	,	,	PUNCT
ma-232	61	32	µα	µα	NOUN
ma-232	61	33	)	)	PUNCT
ma-232	61	34	is	be	AUX
ma-232	61	35	lqa(u	lqa(u	PROPN
ma-232	61	36	,	,	PUNCT
ma-232	61	37	µα	µα	ADP
ma-232	61	38	)	)	PUNCT
ma-232	61	39	for	for	ADP
ma-232	61	40	1	1	NUM
ma-232	61	41	<	<	X
ma-232	61	42	p	p	X
ma-232	61	43	,	,	PUNCT
ma-232	61	44	q	q	X
ma-232	61	45	<	<	X
ma-232	61	46	∞	∞	NOUN
ma-232	61	47	with	with	ADP
ma-232	61	48	1	1	NUM
ma-232	61	49	p	p	NOUN
ma-232	62	1	+	+	NOUN
ma-232	62	2	1	1	NUM
ma-232	62	3	q	q	NOUN
ma-232	62	4	=	=	NOUN
ma-232	62	5	1	1	NUM
ma-232	62	6	under	under	ADP
ma-232	62	7	a	a	DET
ma-232	62	8	similar	similar	ADJ
ma-232	62	9	pairing	pairing	NOUN
ma-232	62	10	as	as	ADP
ma-232	62	11	above	above	ADV
ma-232	62	12	.	.	PUNCT
ma-232	63	1	see	see	VERB
ma-232	63	2	for	for	ADP
ma-232	63	3	instance	instance	NOUN
ma-232	63	4	,	,	PUNCT
ma-232	63	5	[	[	X
ma-232	63	6	2–4	2–4	X
ma-232	63	7	]	]	X
ma-232	63	8	or	or	CCONJ
ma-232	63	9	[	[	X
ma-232	63	10	11]for	11]for	NUM
ma-232	63	11	details	detail	NOUN
ma-232	63	12	.	.	PUNCT
ma-232	64	1	when	when	SCONJ
ma-232	64	2	p	p	NOUN
ma-232	64	3	=	=	NOUN
ma-232	64	4	1	1	NUM
ma-232	64	5	,	,	PUNCT
ma-232	64	6	the	the	DET
ma-232	64	7	space	space	NOUN
ma-232	64	8	l1	l1	PROPN
ma-232	64	9	a(u	a(u	PROPN
ma-232	64	10	,	,	PUNCT
ma-232	64	11	µα	µα	ADP
ma-232	64	12	)	)	PUNCT
ma-232	64	13	is	be	AUX
ma-232	64	14	non	non	ADJ
ma-232	64	15	-	-	ADJ
ma-232	64	16	reflexive	reflexive	ADJ
ma-232	64	17	,	,	PUNCT
ma-232	64	18	and	and	CCONJ
ma-232	64	19	it	it	PRON
ma-232	64	20	’s	’	VERB
ma-232	64	21	recently	recently	ADV
ma-232	64	22	that	that	SCONJ
ma-232	64	23	the	the	DET
ma-232	64	24	dual	dual	ADJ
ma-232	64	25	wasdetermined	wasdetermine	VERB
ma-232	64	26	by	by	ADP
ma-232	64	27	kang	kang	PROPN
ma-232	64	28	[	[	X
ma-232	64	29	8	8	NUM
ma-232	64	30	]	]	PUNCT
ma-232	64	31	as	as	SCONJ
ma-232	64	32	we	we	PRON
ma-232	64	33	give	give	VERB
ma-232	64	34	in	in	ADP
ma-232	64	35	theorem	theorem	ADJ
ma-232	64	36	2.1	2.1	NUM
ma-232	64	37	stated	state	VERB
ma-232	64	38	in	in	ADP
ma-232	64	39	the	the	DET
ma-232	64	40	next	next	ADJ
ma-232	64	41	section	section	NOUN
ma-232	64	42	.	.	PUNCT
ma-232	65	1	apparently	apparently	ADV
ma-232	65	2	,	,	PUNCT
ma-232	65	3	thepredual	thepredual	PROPN
ma-232	65	4	of	of	ADP
ma-232	65	5	l1	l1	PROPN
ma-232	65	6	a(u	a(u	PROPN
ma-232	65	7	,	,	PUNCT
ma-232	65	8	µα	µα	ADP
ma-232	65	9	)	)	PUNCT
ma-232	65	10	is	be	AUX
ma-232	65	11	not	not	PART
ma-232	65	12	explicitly	explicitly	ADV
ma-232	65	13	clear	clear	ADJ
ma-232	65	14	from	from	ADP
ma-232	65	15	the	the	DET
ma-232	65	16	literature	literature	NOUN
ma-232	65	17	.	.	PUNCT
ma-232	66	1	generally	generally	ADV
ma-232	66	2	,	,	PUNCT
ma-232	66	3	there	there	PRON
ma-232	66	4	’s	’	VERB
ma-232	66	5	no	no	DET
ma-232	66	6	unified	unified	ADJ
ma-232	66	7	andcomprehensive	andcomprehensive	ADJ
ma-232	66	8	exposition	exposition	NOUN
ma-232	66	9	of	of	ADP
ma-232	66	10	properties	property	NOUN
ma-232	66	11	of	of	ADP
ma-232	66	12	the	the	DET
ma-232	66	13	analytic	analytic	ADJ
ma-232	66	14	spaces	space	NOUN
ma-232	66	15	of	of	ADP
ma-232	66	16	upper	upper	ADJ
ma-232	66	17	half	half	ADJ
ma-232	66	18	plane	plane	NOUN
ma-232	66	19	u	u	NOUN
ma-232	66	20	as	as	SCONJ
ma-232	66	21	there	there	PRON
ma-232	66	22	are	be	VERB
ma-232	66	23	forthe	forthe	DET
ma-232	66	24	corresponding	corresponding	ADJ
ma-232	66	25	spaces	space	NOUN
ma-232	66	26	on	on	ADP
ma-232	66	27	the	the	DET
ma-232	66	28	unit	unit	NOUN
ma-232	66	29	disk	disk	NOUN
ma-232	66	30	d.	d.	PROPN
ma-232	66	31	therefore	therefore	ADV
ma-232	66	32	,	,	PUNCT
ma-232	66	33	the	the	DET
ma-232	66	34	main	main	ADJ
ma-232	66	35	focus	focus	NOUN
ma-232	66	36	of	of	ADP
ma-232	66	37	this	this	DET
ma-232	66	38	paper	paper	NOUN
ma-232	66	39	is	be	AUX
ma-232	66	40	to	to	ADP
ma-232	66	41	determinethe	determinethe	DET
ma-232	66	42	predual	predual	ADJ
ma-232	66	43	of	of	ADP
ma-232	66	44	l1	l1	PROPN
ma-232	66	45	a(u	a(u	PROPN
ma-232	66	46	,	,	PUNCT
ma-232	66	47	µα	µα	ADP
ma-232	66	48	)	)	PUNCT
ma-232	66	49	,	,	PUNCT
ma-232	66	50	that	that	ADV
ma-232	66	51	is	is	ADV
ma-232	66	52	,	,	PUNCT
ma-232	66	53	identifying	identify	VERB
ma-232	66	54	the	the	DET
ma-232	66	55	space	space	NOUN
ma-232	66	56	whose	whose	DET
ma-232	66	57	dual	dual	ADJ
ma-232	66	58	is	be	AUX
ma-232	66	59	l1	l1	PROPN
ma-232	66	60	a(u	a(u	PROPN
ma-232	66	61	,	,	PUNCT
ma-232	66	62	µα).let	µα).let	ADP
ma-232	66	63	aut(u	aut(u	NUM
ma-232	66	64	)	)	PUNCT
ma-232	66	65	denotes	denote	VERB
ma-232	66	66	the	the	DET
ma-232	66	67	collection	collection	NOUN
ma-232	66	68	of	of	ADP
ma-232	66	69	all	all	DET
ma-232	66	70	automorphisms	automorphism	NOUN
ma-232	66	71	of	of	ADP
ma-232	66	72	u.	u.	NOUN
ma-232	66	73	for	for	ADP
ma-232	66	74	ϕt	ϕt	PROPN
ma-232	66	75	∈	∈	PROPN
ma-232	66	76	aut(u	aut(u	PROPN
ma-232	66	77	)	)	PUNCT
ma-232	66	78	,	,	PUNCT
ma-232	66	79	t	t	PROPN
ma-232	66	80	≥	≥	NUM
ma-232	66	81	0	0	NUM
ma-232	66	82	,	,	PUNCT
ma-232	66	83	we	we	PRON
ma-232	66	84	define	define	VERB
ma-232	66	85	acomposition	acomposition	NOUN
ma-232	66	86	operator	operator	NOUN
ma-232	66	87	on	on	ADP
ma-232	66	88	h(u	h(u	PROPN
ma-232	66	89	)	)	PUNCT
ma-232	66	90	by	by	ADP
ma-232	66	91	cϕt	cϕt	PROPN
ma-232	66	92	f	f	X
ma-232	67	1	:	:	PUNCT
ma-232	67	2	=	=	SYM
ma-232	67	3	f	f	X
ma-232	67	4	◦	◦	NOUN
ma-232	67	5	ϕt	ϕt	ADV
ma-232	67	6	.	.	PUNCT
ma-232	68	1	the	the	DET
ma-232	68	2	corresponding	corresponding	ADJ
ma-232	68	3	group	group	NOUN
ma-232	68	4	of	of	ADP
ma-232	68	5	weighted	weight	VERB
ma-232	68	6	compositionoperator	compositionoperator	NOUN
ma-232	68	7	on	on	ADP
ma-232	68	8	h(u	h(u	PROPN
ma-232	68	9	)	)	PUNCT
ma-232	68	10	is	be	AUX
ma-232	68	11	therefore	therefore	ADV
ma-232	68	12	given	give	VERB
ma-232	68	13	by	by	ADP
ma-232	68	14	tt	tt	PROPN
ma-232	68	15	f	f	X
ma-232	68	16	:	:	PUNCT
ma-232	68	17	=	=	SYM
ma-232	68	18	sϕt	sϕt	VERB
ma-232	68	19	f	f	X
ma-232	68	20	=	=	SYM
ma-232	68	21	(	(	PUNCT
ma-232	68	22	ϕ′t	ϕ′t	INTJ
ma-232	68	23	)	)	PUNCT
ma-232	68	24	γf	γf	PROPN
ma-232	68	25	◦	◦	NOUN
ma-232	68	26	ϕt	ϕt	ADV
ma-232	68	27	for	for	ADP
ma-232	68	28	some	some	DET
ma-232	68	29	appropriate	appropriate	ADJ
ma-232	68	30	weight	weight	NOUN
ma-232	68	31	γ	γ	PROPN
ma-232	68	32	.	.	PROPN
ma-232	68	33	motivated	motivate	VERB
ma-232	68	34	by	by	ADP
ma-232	68	35	the	the	DET
ma-232	68	36	work	work	NOUN
ma-232	68	37	of	of	ADP
ma-232	68	38	arvanitidis	arvanitidi	NOUN
ma-232	68	39	and	and	CCONJ
ma-232	68	40	siskakis	siskaki	NOUN
ma-232	68	41	in	in	ADP
ma-232	68	42	[	[	X
ma-232	68	43	1	1	NUM
ma-232	68	44	]	]	PUNCT
ma-232	68	45	,	,	PUNCT
ma-232	68	46	the	the	DET
ma-232	68	47	current	current	ADJ
ma-232	68	48	second	second	ADJ
ma-232	68	49	author	author	NOUN
ma-232	68	50	and	and	CCONJ
ma-232	68	51	threeothers	threeother	NOUN
ma-232	68	52	in	in	ADP
ma-232	68	53	[	[	X
ma-232	68	54	3	3	NUM
ma-232	68	55	]	]	PUNCT
ma-232	68	56	classified	classify	VERB
ma-232	68	57	all	all	DET
ma-232	68	58	the	the	DET
ma-232	68	59	self	self	NOUN
ma-232	68	60	analytic	analytic	ADJ
ma-232	68	61	maps	map	NOUN
ma-232	68	62	of	of	ADP
ma-232	68	63	the	the	DET
ma-232	68	64	upper	upper	ADJ
ma-232	68	65	half	half	ADJ
ma-232	68	66	plane	plane	NOUN
ma-232	68	67	into	into	ADP
ma-232	68	68	three	three	NUM
ma-232	68	69	distinct	distinct	ADJ
ma-232	68	70	groups	group	NOUN
ma-232	68	71	,	,	PUNCT
ma-232	68	72	namely	namely	ADV
ma-232	68	73	:	:	PUNCT
ma-232	68	74	the	the	DET
ma-232	68	75	scaling	scaling	NOUN
ma-232	68	76	,	,	PUNCT
ma-232	68	77	the	the	DET
ma-232	68	78	translation	translation	NOUN
ma-232	68	79	and	and	CCONJ
ma-232	68	80	the	the	DET
ma-232	68	81	rotation	rotation	NOUN
ma-232	68	82	groups	group	NOUN
ma-232	68	83	.	.	PUNCT
ma-232	69	1	they	they	PRON
ma-232	69	2	then	then	ADV
ma-232	69	3	studied	study	VERB
ma-232	69	4	both	both	CCONJ
ma-232	69	5	the	the	DET
ma-232	69	6	semigroupand	semigroupand	NOUN
ma-232	69	7	spectral	spectral	ADJ
ma-232	69	8	properties	property	NOUN
ma-232	69	9	of	of	ADP
ma-232	69	10	the	the	DET
ma-232	69	11	corresponding	corresponding	ADJ
ma-232	69	12	groups	group	NOUN
ma-232	69	13	of	of	ADP
ma-232	69	14	weighted	weight	VERB
ma-232	69	15	composition	composition	NOUN
ma-232	69	16	operators	operator	NOUN
ma-232	69	17	.	.	PUNCT
ma-232	70	1	as	as	ADP
ma-232	70	2	for	for	ADP
ma-232	70	3	theproperties	thepropertie	NOUN
ma-232	70	4	of	of	ADP
ma-232	70	5	the	the	DET
ma-232	70	6	adjoint	adjoint	NOUN
ma-232	70	7	groups	group	NOUN
ma-232	70	8	on	on	ADP
ma-232	70	9	the	the	DET
ma-232	70	10	reflexive	reflexive	ADJ
ma-232	70	11	weighted	weight	VERB
ma-232	70	12	bergman	bergman	PROPN
ma-232	70	13	spaces	space	VERB
ma-232	70	14	lpa(u	lpa(u	PROPN
ma-232	70	15	,	,	PUNCT
ma-232	70	16	µα	µα	ADJ
ma-232	70	17	)	)	PUNCT
ma-232	70	18	,	,	PUNCT
ma-232	70	19	1	1	NUM
ma-232	70	20	<	<	X
ma-232	70	21	p	p	X
ma-232	70	22	<	<	X
ma-232	70	23	∞,only	∞,only	ADV
ma-232	70	24	the	the	DET
ma-232	70	25	scaling	scale	VERB
ma-232	70	26	group	group	NOUN
ma-232	70	27	was	be	AUX
ma-232	70	28	considered	consider	VERB
ma-232	70	29	in	in	ADP
ma-232	70	30	[	[	X
ma-232	70	31	3	3	NUM
ma-232	70	32	]	]	PUNCT
ma-232	70	33	and	and	CCONJ
ma-232	70	34	later	later	ADV
ma-232	70	35	completed	complete	VERB
ma-232	70	36	for	for	ADP
ma-232	70	37	the	the	DET
ma-232	70	38	other	other	ADJ
ma-232	70	39	two	two	NUM
ma-232	70	40	groups	group	NOUN
ma-232	70	41	by	by	ADP
ma-232	70	42	thesecond	thesecond	NOUN
ma-232	70	43	author	author	NOUN
ma-232	70	44	in	in	ADP
ma-232	70	45	[	[	X
ma-232	70	46	4	4	NUM
ma-232	70	47	]	]	PUNCT
ma-232	70	48	.	.	PUNCT
ma-232	71	1	in	in	ADP
ma-232	71	2	this	this	DET
ma-232	71	3	paper	paper	NOUN
ma-232	71	4	,	,	PUNCT
ma-232	71	5	we	we	PRON
ma-232	71	6	therefore	therefore	ADV
ma-232	71	7	determine	determine	VERB
ma-232	71	8	the	the	DET
ma-232	71	9	groups	group	NOUN
ma-232	71	10	of	of	ADP
ma-232	71	11	composition	composition	NOUN
ma-232	71	12	operators	operator	NOUN
ma-232	71	13	onthe	onthe	VERB
ma-232	71	14	predual	predual	ADJ
ma-232	71	15	of	of	ADP
ma-232	71	16	non	non	ADJ
ma-232	71	17	-	-	ADJ
ma-232	71	18	reflexive	reflexive	ADJ
ma-232	71	19	bergman	bergman	PROPN
ma-232	71	20	space	space	NOUN
ma-232	71	21	of	of	ADP
ma-232	71	22	the	the	DET
ma-232	71	23	upper	upper	ADJ
ma-232	71	24	half	half	ADJ
ma-232	71	25	-	-	PUNCT
ma-232	71	26	plane	plane	NOUN
ma-232	71	27	,	,	PUNCT
ma-232	71	28	l1	l1	PROPN
ma-232	71	29	a(u	a(u	PROPN
ma-232	71	30	,	,	PUNCT
ma-232	71	31	µα	µα	ADP
ma-232	71	32	)	)	PUNCT
ma-232	71	33	and	and	CCONJ
ma-232	71	34	investigate	investigate	VERB
ma-232	71	35	theadjoint	theadjoint	NOUN
ma-232	71	36	properties	property	NOUN
ma-232	71	37	of	of	ADP
ma-232	71	38	the	the	DET
ma-232	71	39	groups	group	NOUN
ma-232	71	40	of	of	ADP
ma-232	71	41	weighted	weight	VERB
ma-232	71	42	composition	composition	NOUN
ma-232	71	43	operators	operator	NOUN
ma-232	71	44	on	on	ADP
ma-232	71	45	nonreflexive	nonreflexive	ADJ
ma-232	71	46	bergman	bergman	PROPN
ma-232	71	47	space	space	PROPN
ma-232	71	48	l1	l1	PROPN
ma-232	71	49	a(u	a(u	PROPN
ma-232	71	50	,	,	PUNCT
ma-232	71	51	µα).let	µα).let	NOUN
ma-232	71	52	x	x	PUNCT
ma-232	71	53	and	and	CCONJ
ma-232	71	54	y	y	PROPN
ma-232	71	55	be	be	AUX
ma-232	71	56	banach	banach	ADV
ma-232	71	57	spaces	space	NOUN
ma-232	71	58	over	over	ADP
ma-232	71	59	c.	c.	PROPN
ma-232	71	60	the	the	DET
ma-232	71	61	space	space	NOUN
ma-232	71	62	l(x	l(x	PROPN
ma-232	71	63	,	,	PUNCT
ma-232	71	64	y	y	PROPN
ma-232	71	65	)	)	PUNCT
ma-232	71	66	=	=	PRON
ma-232	72	1	{	{	PUNCT
ma-232	72	2	t	t	NOUN
ma-232	72	3	:	:	PUNCT
ma-232	72	4	x	x	X
ma-232	72	5	→	→	SYM
ma-232	72	6	y	y	PROPN
ma-232	72	7	such	such	ADJ
ma-232	72	8	that	that	SCONJ
ma-232	72	9	t	t	PROPN
ma-232	72	10	is	be	AUX
ma-232	72	11	linearand	linearand	ADV
ma-232	72	12	continuous	continuous	ADJ
ma-232	72	13	}	}	PUNCT
ma-232	72	14	,	,	PUNCT
ma-232	72	15	endowed	endow	VERB
ma-232	72	16	with	with	ADP
ma-232	72	17	the	the	DET
ma-232	72	18	operator	operator	NOUN
ma-232	72	19	norm	norm	NOUN
ma-232	72	20	‖t‖	‖t‖	PROPN
ma-232	72	21	=	=	PROPN
ma-232	72	22	sup‖x‖≤1	sup‖x‖≤1	ADV
ma-232	72	23	‖tx‖	‖tx‖	PROPN
ma-232	72	24	,	,	PUNCT
ma-232	72	25	is	be	AUX
ma-232	72	26	a	a	DET
ma-232	72	27	banach	banach	NOUN
ma-232	72	28	space	space	NOUN
ma-232	72	29	[	[	X
ma-232	72	30	5].we	5].we	PRON
ma-232	72	31	write	write	VERB
ma-232	72	32	l(x	l(x	PROPN
ma-232	72	33	,	,	PUNCT
ma-232	72	34	x	x	NOUN
ma-232	72	35	)	)	PUNCT
ma-232	72	36	=	=	SYM
ma-232	72	37	l(x	l(x	PROPN
ma-232	72	38	)	)	PUNCT
ma-232	72	39	.	.	PUNCT
ma-232	73	1	t	t	PROPN
ma-232	73	2	is	be	AUX
ma-232	73	3	said	say	VERB
ma-232	73	4	to	to	PART
ma-232	73	5	be	be	AUX
ma-232	73	6	a	a	DET
ma-232	73	7	closed	closed	ADJ
ma-232	73	8	operator	operator	NOUN
ma-232	73	9	if	if	SCONJ
ma-232	73	10	its	its	PRON
ma-232	73	11	graph	graph	NOUN
ma-232	73	12	{	{	PUNCT
ma-232	73	13	(	(	PUNCT
ma-232	73	14	x	x	PROPN
ma-232	73	15	,	,	PUNCT
ma-232	73	16	t	t	PROPN
ma-232	73	17	x	x	NOUN
ma-232	73	18	)	)	PUNCT
ma-232	73	19	|	|	ADV
ma-232	73	20	x	x	SYM
ma-232	73	21	∈	∈	PROPN
ma-232	73	22	d(t	d(t	PROPN
ma-232	73	23	)	)	PUNCT
ma-232	73	24	}	}	PUNCT
ma-232	73	25	in	in	ADP
ma-232	73	26	x	x	SYM
ma-232	73	27	×	×	PROPN
ma-232	73	28	y	y	PROPN
ma-232	73	29	is	be	AUX
ma-232	73	30	closed	closed	ADJ
ma-232	73	31	.	.	PUNCT
ma-232	74	1	let	let	VERB
ma-232	74	2	t	t	NOUN
ma-232	74	3	be	be	AUX
ma-232	74	4	a	a	DET
ma-232	74	5	closed	closed	ADJ
ma-232	74	6	operator	operator	NOUN
ma-232	74	7	on	on	ADP
ma-232	74	8	x	x	X
ma-232	74	9	.	.	PUNCT
ma-232	75	1	the	the	DET
ma-232	75	2	resolvent	resolvent	ADJ
ma-232	75	3	set	set	NOUN
ma-232	75	4	of	of	ADP
ma-232	75	5	t	t	PROPN
ma-232	75	6	,	,	PUNCT
ma-232	75	7	ρ(t	ρ(t	PROPN
ma-232	75	8	)	)	PUNCT
ma-232	75	9	is	be	AUX
ma-232	75	10	given	give	VERB
ma-232	75	11	by	by	ADP
ma-232	75	12	ρ(t	ρ(t	PROPN
ma-232	75	13	)	)	PUNCT
ma-232	76	1	=	=	PUNCT
ma-232	76	2	{	{	PUNCT
ma-232	76	3	λ	λ	X
ma-232	76	4	∈	∈	NOUN
ma-232	76	5	c	c	NOUN
ma-232	76	6	:	:	PUNCT
ma-232	76	7	λi	λi	ADP
ma-232	76	8	−	−	PROPN
ma-232	76	9	t	t	PROPN
ma-232	76	10	is	be	AUX
ma-232	76	11	invertible	invertible	ADJ
ma-232	76	12	or	or	CCONJ
ma-232	76	13	bijective	bijective	ADJ
ma-232	76	14	}	}	PUNCT
ma-232	76	15	and	and	CCONJ
ma-232	76	16	its	its	PRON
ma-232	76	17	spectrum	spectrum	NOUN
ma-232	76	18	σ(t	σ(t	PROPN
ma-232	76	19	)	)	PUNCT
ma-232	77	1	=	=	PUNCT
ma-232	77	2	c	c	NOUN
ma-232	77	3	\	\	PROPN
ma-232	77	4	ρ(t	ρ(t	PROPN
ma-232	77	5	)	)	PUNCT
ma-232	77	6	.	.	PUNCT
ma-232	78	1	therefore	therefore	ADV
ma-232	78	2	σ(t	σ(t	PROPN
ma-232	78	3	)	)	PUNCT
ma-232	78	4	∪	∪	ADP
ma-232	78	5	ρ(t	ρ(t	PROPN
ma-232	78	6	)	)	PUNCT
ma-232	79	1	=	=	SYM
ma-232	79	2	c.	c.	PROPN
ma-232	79	3	the	the	DET
ma-232	79	4	spectral	spectral	ADJ
ma-232	79	5	radius	radius	NOUN
ma-232	79	6	of	of	ADP
ma-232	79	7	t	t	PROPN
ma-232	79	8	is	be	AUX
ma-232	79	9	defined	define	VERB
ma-232	79	10	by	by	ADP
ma-232	79	11	r(t	r(t	NOUN
ma-232	79	12	)	)	PUNCT
ma-232	80	1	=	=	NOUN
ma-232	80	2	sup{|λ|	sup{|λ|	NOUN
ma-232	80	3	:	:	PUNCT
ma-232	81	1	λ	λ	X
ma-232	81	2	∈	∈	PRON
ma-232	81	3	σ(t	σ(t	PROPN
ma-232	81	4	)	)	PUNCT
ma-232	81	5	}	}	PUNCT
ma-232	81	6	with	with	ADP
ma-232	81	7	the	the	DET
ma-232	81	8	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	81	9	eur	eur	PROPN
ma-232	81	10	.	.	PUNCT
ma-232	82	1	j.	j.	PROPN
ma-232	82	2	math	math	PROPN
ma-232	82	3	.	.	PUNCT
ma-232	83	1	anal	anal	PROPN
ma-232	83	2	.	.	PUNCT
ma-232	84	1	10.28924	10.28924	NUM
ma-232	84	2	/	/	SYM
ma-232	84	3	ada	ada	PROPN
ma-232	84	4	/	/	SYM
ma-232	84	5	ma.4.14	ma.4.14	NOUN
ma-232	84	6	4relation	4relation	NUM
ma-232	84	7	r(t	r(t	NOUN
ma-232	84	8	)	)	PUNCT
ma-232	85	1	≤	≤	NOUN
ma-232	85	2	‖t‖.	‖t‖.	VERB
ma-232	85	3	the	the	DET
ma-232	85	4	point	point	NOUN
ma-232	85	5	spectrum	spectrum	NOUN
ma-232	85	6	σp(t	σp(t	PUNCT
ma-232	85	7	)	)	PUNCT
ma-232	86	1	=	=	PUNCT
ma-232	86	2	{	{	PUNCT
ma-232	86	3	λ	λ	X
ma-232	86	4	∈	∈	NOUN
ma-232	86	5	c	c	NOUN
ma-232	86	6	:	:	PUNCT
ma-232	86	7	tx	tx	PROPN
ma-232	86	8	=	=	PUNCT
ma-232	86	9	λx	λx	PROPN
ma-232	86	10	for	for	ADP
ma-232	86	11	some	some	DET
ma-232	86	12	0	0	NUM
ma-232	86	13	6=	6=	NUM
ma-232	86	14	x	x	SYM
ma-232	86	15	∈	∈	PROPN
ma-232	86	16	dom(t	dom(t	PROPN
ma-232	86	17	)	)	PUNCT
ma-232	86	18	}	}	PUNCT
ma-232	86	19	.for	.for	PUNCT
ma-232	87	1	λ	λ	PROPN
ma-232	87	2	∈	∈	PROPN
ma-232	87	3	ρ(t	ρ(t	PROPN
ma-232	87	4	)	)	PUNCT
ma-232	87	5	,	,	PUNCT
ma-232	87	6	the	the	DET
ma-232	87	7	operator	operator	NOUN
ma-232	87	8	r(λ	r(λ	NOUN
ma-232	87	9	,	,	PUNCT
ma-232	87	10	t	t	NOUN
ma-232	87	11	)	)	PUNCT
ma-232	87	12	:	:	PUNCT
ma-232	87	13	=	=	SYM
ma-232	87	14	(	(	PUNCT
ma-232	87	15	λi	λi	ADP
ma-232	87	16	−	−	PROPN
ma-232	87	17	t	t	NOUN
ma-232	87	18	)	)	PUNCT
ma-232	87	19	−1	−1	NOUN
ma-232	87	20	is	be	AUX
ma-232	87	21	,	,	PUNCT
ma-232	87	22	by	by	ADP
ma-232	87	23	the	the	DET
ma-232	87	24	closed	closed	ADJ
ma-232	87	25	graph	graph	NOUN
ma-232	87	26	theorem	theorem	VERB
ma-232	87	27	a	a	DET
ma-232	87	28	boundedoperator	boundedoperator	NOUN
ma-232	87	29	on	on	ADP
ma-232	87	30	x	x	PUNCT
ma-232	87	31	and	and	CCONJ
ma-232	87	32	is	be	AUX
ma-232	87	33	called	call	VERB
ma-232	87	34	the	the	DET
ma-232	87	35	resolvent	resolvent	NOUN
ma-232	87	36	of	of	ADP
ma-232	87	37	t	t	PROPN
ma-232	87	38	at	at	ADP
ma-232	87	39	the	the	DET
ma-232	87	40	point	point	NOUN
ma-232	87	41	λ	λ	PROPN
ma-232	87	42	or	or	CCONJ
ma-232	87	43	simply	simply	ADV
ma-232	87	44	the	the	DET
ma-232	87	45	resolvent	resolvent	ADJ
ma-232	87	46	operator.in	operator.in	X
ma-232	87	47	fact	fact	NOUN
ma-232	87	48	,	,	PUNCT
ma-232	87	49	ρ(t	ρ(t	PROPN
ma-232	87	50	)	)	PUNCT
ma-232	87	51	is	be	AUX
ma-232	87	52	an	an	DET
ma-232	87	53	open	open	ADJ
ma-232	87	54	subset	subset	NOUN
ma-232	87	55	of	of	ADP
ma-232	87	56	c	c	PROPN
ma-232	87	57	and	and	CCONJ
ma-232	87	58	r(λ	r(λ	PROPN
ma-232	87	59	,	,	PUNCT
ma-232	87	60	t	t	PROPN
ma-232	87	61	)	)	PUNCT
ma-232	87	62	:	:	PUNCT
ma-232	87	63	ρ(t	ρ(t	PROPN
ma-232	87	64	)	)	PUNCT
ma-232	88	1	→	→	SYM
ma-232	88	2	l(x	l(x	PROPN
ma-232	88	3	)	)	PUNCT
ma-232	88	4	is	be	AUX
ma-232	88	5	an	an	DET
ma-232	88	6	analytic	analytic	ADJ
ma-232	88	7	function	function	NOUN
ma-232	88	8	.	.	PUNCT
ma-232	89	1	for	for	ADP
ma-232	89	2	adetailed	adetaile	VERB
ma-232	89	3	theory	theory	NOUN
ma-232	89	4	on	on	ADP
ma-232	89	5	spectra	spectra	PROPN
ma-232	89	6	,	,	PUNCT
ma-232	89	7	we	we	PRON
ma-232	89	8	refer	refer	VERB
ma-232	89	9	to	to	ADP
ma-232	89	10	[	[	X
ma-232	89	11	5	5	NUM
ma-232	89	12	,	,	PUNCT
ma-232	89	13	6	6	NUM
ma-232	89	14	,	,	PUNCT
ma-232	89	15	9	9	NUM
ma-232	89	16	,	,	PUNCT
ma-232	89	17	12	12	NUM
ma-232	89	18	]	]	PUNCT
ma-232	89	19	.	.	PUNCT
ma-232	90	1	2	2	X
ma-232	90	2	.	.	NUM
ma-232	90	3	predual	predual	ADJ
ma-232	90	4	of	of	ADP
ma-232	90	5	non	non	ADJ
ma-232	90	6	-	-	ADJ
ma-232	90	7	reflexive	reflexive	ADJ
ma-232	90	8	bergman	bergman	PROPN
ma-232	90	9	space	space	NOUN
ma-232	90	10	of	of	ADP
ma-232	90	11	the	the	DET
ma-232	90	12	upper	upper	ADJ
ma-232	90	13	half	half	ADJ
ma-232	90	14	-	-	PUNCT
ma-232	90	15	plane	plane	NOUN
ma-232	90	16	l1	l1	PROPN
ma-232	90	17	a(u	a(u	PROPN
ma-232	90	18	,	,	PUNCT
ma-232	90	19	µα	µα	ADP
ma-232	90	20	)	)	PUNCT
ma-232	90	21	let	let	VERB
ma-232	90	22	b∞(u	b∞(u	PROPN
ma-232	90	23	,	,	PUNCT
ma-232	90	24	i	i	NOUN
ma-232	90	25	)	)	PUNCT
ma-232	90	26	denote	denote	VERB
ma-232	90	27	the	the	DET
ma-232	90	28	subspace	subspace	NOUN
ma-232	90	29	of	of	ADP
ma-232	90	30	the	the	DET
ma-232	90	31	bloch	bloch	PROPN
ma-232	90	32	space	space	PROPN
ma-232	90	33	b∞(u	b∞(u	PROPN
ma-232	90	34	)	)	PUNCT
ma-232	90	35	consisting	consist	VERB
ma-232	90	36	of	of	ADP
ma-232	90	37	functions	function	NOUN
ma-232	90	38	vanishingat	vanishingat	ADJ
ma-232	90	39	point	point	VERB
ma-232	90	40	i	i	PRON
ma-232	90	41	.	.	PUNCT
ma-232	91	1	therefore	therefore	ADV
ma-232	91	2	b∞(u	b∞(u	PROPN
ma-232	91	3	,	,	PUNCT
ma-232	91	4	i	i	PROPN
ma-232	91	5	)	)	PUNCT
ma-232	91	6	is	be	AUX
ma-232	91	7	defined	define	VERB
ma-232	91	8	as	as	ADP
ma-232	91	9	b∞(u	b∞(u	PROPN
ma-232	91	10	,	,	PUNCT
ma-232	91	11	i	i	PROPN
ma-232	91	12	)	)	PUNCT
ma-232	91	13	:	:	PUNCT
ma-232	92	1	=	=	PUNCT
ma-232	92	2	{	{	PUNCT
ma-232	92	3	f	f	PROPN
ma-232	92	4	∈	∈	PROPN
ma-232	92	5	b∞(u	b∞(u	PROPN
ma-232	92	6	)	)	PUNCT
ma-232	92	7	:	:	PUNCT
ma-232	93	1	f	f	X
ma-232	93	2	(	(	PUNCT
ma-232	93	3	i	i	NOUN
ma-232	93	4	)	)	PUNCT
ma-232	93	5	=	=	PUNCT
ma-232	93	6	0	0	NUM
ma-232	93	7	}	}	PUNCT
ma-232	93	8	.	.	PUNCT
ma-232	94	1	then	then	ADV
ma-232	94	2	b∞(u	b∞(u	PROPN
ma-232	94	3	,	,	PUNCT
ma-232	94	4	i	i	PROPN
ma-232	94	5	)	)	PUNCT
ma-232	94	6	is	be	AUX
ma-232	94	7	a	a	DET
ma-232	94	8	closed	closed	ADJ
ma-232	94	9	subspace	subspace	NOUN
ma-232	94	10	of	of	ADP
ma-232	94	11	b∞(u	b∞(u	PROPN
ma-232	94	12	)	)	PUNCT
ma-232	94	13	and	and	CCONJ
ma-232	94	14	therefore	therefore	ADV
ma-232	94	15	is	be	AUX
ma-232	94	16	a	a	DET
ma-232	94	17	banach	banach	NOUN
ma-232	94	18	space	space	NOUN
ma-232	94	19	with	with	ADP
ma-232	94	20	respect	respect	NOUN
ma-232	94	21	to	to	ADP
ma-232	94	22	thenorm	thenorm	NOUN
ma-232	94	23	‖f	‖f	ADP
ma-232	94	24	‖b∞,i	‖b∞,i	NOUN
ma-232	94	25	:	:	PUNCT
ma-232	94	26	=	=	SYM
ma-232	94	27	‖f	‖f	X
ma-232	94	28	‖b∞(u	‖b∞(u	NOUN
ma-232	94	29	)	)	PUNCT
ma-232	94	30	=	=	SYM
ma-232	94	31	‖f	‖f	ADP
ma-232	94	32	‖b∞,1(u	‖b∞,1(u	NOUN
ma-232	94	33	)	)	PUNCT
ma-232	94	34	,	,	PUNCT
ma-232	94	35	see	see	VERB
ma-232	94	36	[	[	X
ma-232	94	37	8	8	NUM
ma-232	94	38	]	]	PUNCT
ma-232	94	39	.	.	PUNCT
ma-232	95	1	similarly	similarly	ADV
ma-232	95	2	,	,	PUNCT
ma-232	95	3	let	let	VERB
ma-232	95	4	b∞,	b∞,	PART
ma-232	95	5	◦	◦	VERB
ma-232	95	6	(u	(u	ADJ
ma-232	95	7	,	,	PUNCT
ma-232	95	8	i	i	NOUN
ma-232	95	9	)	)	PUNCT
ma-232	95	10	denotes	denote	VERB
ma-232	95	11	the	the	DET
ma-232	95	12	subspace	subspace	NOUN
ma-232	95	13	of	of	ADP
ma-232	95	14	b∞,	b∞,	ADP
ma-232	95	15	◦	◦	NOUN
ma-232	95	16	(u	(u	NOUN
ma-232	95	17	)	)	PUNCT
ma-232	95	18	consisting	consist	VERB
ma-232	95	19	of	of	ADP
ma-232	95	20	functions	function	NOUN
ma-232	95	21	vanishing	vanish	VERB
ma-232	95	22	at	at	ADP
ma-232	95	23	i	i	PRON
ma-232	95	24	.	.	PUNCT
ma-232	96	1	therefore	therefore	ADV
ma-232	96	2	b∞,	b∞,	ADP
ma-232	96	3	◦	◦	NOUN
ma-232	96	4	(u	(u	NOUN
ma-232	96	5	,	,	PUNCT
ma-232	96	6	i	i	NOUN
ma-232	96	7	)	)	PUNCT
ma-232	96	8	:	:	PUNCT
ma-232	96	9	=	=	X
ma-232	96	10	{	{	PUNCT
ma-232	96	11	f	f	PROPN
ma-232	96	12	∈	∈	PROPN
ma-232	96	13	b∞,	b∞,	ADP
ma-232	96	14	◦	◦	NOUN
ma-232	96	15	(u	(u	NOUN
ma-232	96	16	)	)	PUNCT
ma-232	96	17	:	:	PUNCT
ma-232	97	1	f	f	X
ma-232	97	2	(	(	PUNCT
ma-232	97	3	i	i	NOUN
ma-232	97	4	)	)	PUNCT
ma-232	97	5	=	=	PUNCT
ma-232	98	1	0	0	NUM
ma-232	98	2	}	}	PUNCT
ma-232	98	3	,	,	PUNCT
ma-232	98	4	with	with	ADP
ma-232	98	5	the	the	DET
ma-232	98	6	norm	norm	NOUN
ma-232	98	7	‖f	‖f	ADP
ma-232	98	8	‖b∞,i	‖b∞,i	NOUN
ma-232	98	9	:	:	PUNCT
ma-232	98	10	=	=	SYM
ma-232	98	11	‖f	‖f	X
ma-232	98	12	‖b∞(u	‖b∞(u	NOUN
ma-232	98	13	)	)	PUNCT
ma-232	98	14	=	=	SYM
ma-232	98	15	‖f	‖f	ADP
ma-232	98	16	‖b∞,1(u	‖b∞,1(u	NOUN
ma-232	98	17	)	)	PUNCT
ma-232	98	18	.	.	PUNCT
ma-232	99	1	again	again	ADV
ma-232	99	2	,	,	PUNCT
ma-232	99	3	b∞,	b∞,	VERB
ma-232	99	4	◦	◦	NOUN
ma-232	99	5	(u	(u	ADJ
ma-232	99	6	,	,	PUNCT
ma-232	99	7	i	i	NOUN
ma-232	99	8	)	)	PUNCT
ma-232	99	9	is	be	AUX
ma-232	99	10	a	a	DET
ma-232	99	11	banach	banach	NOUN
ma-232	99	12	space	space	NOUN
ma-232	99	13	with	with	ADP
ma-232	99	14	respectto	respectto	ADJ
ma-232	99	15	the	the	DET
ma-232	99	16	norm	norm	NOUN
ma-232	99	17	given	give	VERB
ma-232	99	18	above.the	above.the	DET
ma-232	99	19	following	follow	VERB
ma-232	99	20	result	result	NOUN
ma-232	99	21	due	due	ADP
ma-232	99	22	to	to	ADP
ma-232	99	23	kang	kang	PROPN
ma-232	100	1	[	[	X
ma-232	100	2	8	8	NUM
ma-232	100	3	]	]	PUNCT
ma-232	100	4	gives	give	VERB
ma-232	100	5	the	the	DET
ma-232	100	6	dual	dual	ADJ
ma-232	100	7	of	of	ADP
ma-232	100	8	l1	l1	PROPN
ma-232	100	9	a(u	a(u	PROPN
ma-232	100	10	,	,	PUNCT
ma-232	100	11	µα	µα	ADP
ma-232	100	12	)	)	PUNCT
ma-232	100	13	;	;	PUNCT
ma-232	100	14	theorem	theorem	VERB
ma-232	100	15	2.1	2.1	NUM
ma-232	100	16	.	.	PUNCT
ma-232	101	1	for	for	ADP
ma-232	101	2	any	any	DET
ma-232	101	3	α	α	NOUN
ma-232	101	4	∈	∈	PROPN
ma-232	101	5	r	r	NOUN
ma-232	101	6	,	,	PUNCT
ma-232	101	7	α	α	INTJ
ma-232	101	8	>	>	X
ma-232	101	9	−1	−1	NOUN
ma-232	101	10	,	,	PUNCT
ma-232	101	11	we	we	PRON
ma-232	101	12	have	have	VERB
ma-232	101	13	(	(	PUNCT
ma-232	101	14	l1	l1	PROPN
ma-232	101	15	a(u	a(u	PROPN
ma-232	101	16	,	,	PUNCT
ma-232	101	17	µα))∗	µα))∗	VERB
ma-232	101	18	≈	≈	PROPN
ma-232	101	19	b∞(u	b∞(u	PROPN
ma-232	101	20	,	,	PUNCT
ma-232	101	21	i	i	PROPN
ma-232	101	22	)	)	PUNCT
ma-232	101	23	,	,	PUNCT
ma-232	101	24	under	under	ADP
ma-232	101	25	the	the	DET
ma-232	101	26	integral	integral	ADJ
ma-232	101	27	pairing	pairing	NOUN
ma-232	101	28	〈	〈	PROPN
ma-232	101	29	g	g	NOUN
ma-232	101	30	,	,	PUNCT
ma-232	102	1	f	f	PROPN
ma-232	102	2	〉	〉	PROPN
ma-232	102	3	=	=	SYM
ma-232	102	4	∫	∫	PROPN
ma-232	102	5	u	u	PROPN
ma-232	102	6	g(ω)f	g(ω)f	PROPN
ma-232	102	7	(	(	PUNCT
ma-232	102	8	ω)dµα(ω	ω)dµα(ω	X
ma-232	102	9	)	)	PUNCT
ma-232	102	10	(	(	PUNCT
ma-232	102	11	g	g	PROPN
ma-232	102	12	∈	∈	PROPN
ma-232	102	13	l1	l1	PROPN
ma-232	102	14	a(u	a(u	PROPN
ma-232	102	15	,	,	PUNCT
ma-232	102	16	µα	µα	ADP
ma-232	102	17	)	)	PUNCT
ma-232	102	18	,	,	PUNCT
ma-232	102	19	f	f	PROPN
ma-232	102	20	∈	∈	PROPN
ma-232	102	21	b∞(u	b∞(u	PROPN
ma-232	102	22	,	,	PUNCT
ma-232	102	23	i	i	NOUN
ma-232	102	24	)	)	PUNCT
ma-232	102	25	)	)	PUNCT
ma-232	102	26	.	.	PUNCT
ma-232	103	1	with	with	ADP
ma-232	103	2	the	the	DET
ma-232	103	3	help	help	NOUN
ma-232	103	4	of	of	ADP
ma-232	103	5	theorem	theorem	ADJ
ma-232	103	6	2.1	2.1	NUM
ma-232	103	7	above	above	ADV
ma-232	103	8	,	,	PUNCT
ma-232	103	9	we	we	PRON
ma-232	103	10	determine	determine	VERB
ma-232	103	11	the	the	DET
ma-232	103	12	predual	predual	ADJ
ma-232	103	13	space	space	NOUN
ma-232	103	14	of	of	ADP
ma-232	103	15	l1	l1	PROPN
ma-232	103	16	a(u	a(u	PROPN
ma-232	103	17	,	,	PUNCT
ma-232	103	18	µα	µα	ADP
ma-232	103	19	)	)	PUNCT
ma-232	103	20	,	,	PUNCT
ma-232	103	21	that	that	ADV
ma-232	103	22	is	is	ADV
ma-232	103	23	,	,	PUNCT
ma-232	103	24	a	a	DET
ma-232	103	25	setwhose	setwhose	NOUN
ma-232	103	26	dual	dual	ADV
ma-232	103	27	is	be	AUX
ma-232	103	28	l1	l1	PROPN
ma-232	103	29	a(u	a(u	PROPN
ma-232	103	30	,	,	PUNCT
ma-232	103	31	µα	µα	ADP
ma-232	103	32	)	)	PUNCT
ma-232	103	33	,	,	PUNCT
ma-232	103	34	but	but	CCONJ
ma-232	103	35	first	first	ADV
ma-232	103	36	we	we	PRON
ma-232	103	37	state	state	VERB
ma-232	103	38	some	some	DET
ma-232	103	39	results.let	results.let	X
ma-232	103	40	c(u	c(u	NOUN
ma-232	103	41	)	)	PUNCT
ma-232	103	42	be	be	VERB
ma-232	103	43	the	the	DET
ma-232	103	44	algebra	algebra	NOUN
ma-232	103	45	of	of	ADP
ma-232	103	46	complex	complex	ADJ
ma-232	103	47	valued	value	VERB
ma-232	103	48	continuous	continuous	ADJ
ma-232	103	49	functions	function	NOUN
ma-232	103	50	on	on	ADP
ma-232	103	51	u	u	NOUN
ma-232	103	52	=	=	PROPN
ma-232	103	53	u	u	PROPN
ma-232	103	54	⋃	⋃	PROPN
ma-232	103	55	∂u	∂u	PROPN
ma-232	103	56	,	,	PUNCT
ma-232	103	57	and	and	CCONJ
ma-232	103	58	c	c	NOUN
ma-232	103	59	◦	◦	NOUN
ma-232	103	60	(u	(u	NOUN
ma-232	103	61	)	)	PUNCT
ma-232	103	62	bethe	bethe	ADJ
ma-232	103	63	subalgebra	subalgebra	NOUN
ma-232	103	64	of	of	ADP
ma-232	103	65	c(u	c(u	PROPN
ma-232	103	66	)	)	PUNCT
ma-232	103	67	consisting	consist	VERB
ma-232	103	68	of	of	ADP
ma-232	103	69	functions	function	NOUN
ma-232	103	70	f	f	X
ma-232	103	71	such	such	ADJ
ma-232	103	72	that	that	SCONJ
ma-232	103	73	f	f	PROPN
ma-232	103	74	(	(	PUNCT
ma-232	103	75	ω)→	ω)→	PROPN
ma-232	103	76	0	0	NUM
ma-232	103	77	as	as	ADP
ma-232	103	78	=(	=(	ADJ
ma-232	103	79	ω)→	ω)→	PROPN
ma-232	103	80	0	0	NUM
ma-232	103	81	.	.	PUNCT
ma-232	104	1	proposition	proposition	NOUN
ma-232	104	2	2.2	2.2	NUM
ma-232	104	3	.	.	PUNCT
ma-232	105	1	let	let	VERB
ma-232	105	2	c	c	AUX
ma-232	105	3	◦	◦	VERB
ma-232	105	4	(u	(u	NOUN
ma-232	105	5	)	)	PUNCT
ma-232	105	6	be	be	AUX
ma-232	105	7	the	the	DET
ma-232	105	8	subalgebra	subalgebra	NOUN
ma-232	105	9	of	of	ADP
ma-232	105	10	c(u	c(u	PROPN
ma-232	105	11	)	)	PUNCT
ma-232	105	12	consisting	consist	VERB
ma-232	105	13	of	of	ADP
ma-232	105	14	functions	function	NOUN
ma-232	105	15	f	f	X
ma-232	105	16	such	such	ADJ
ma-232	105	17	that	that	SCONJ
ma-232	105	18	f	f	PROPN
ma-232	105	19	(	(	PUNCT
ma-232	105	20	ω	ω	NOUN
ma-232	105	21	)	)	PUNCT
ma-232	105	22	−→	−→	NOUN
ma-232	105	23	0	0	NUM
ma-232	105	24	as	as	ADP
ma-232	105	25	im(ω	im(ω	NOUN
ma-232	105	26	)	)	PUNCT
ma-232	105	27	−→	−→	NOUN
ma-232	105	28	0	0	NUM
ma-232	105	29	and	and	CCONJ
ma-232	105	30	c	c	NOUN
ma-232	105	31	◦	◦	NOUN
ma-232	105	32	(d	(d	NOUN
ma-232	105	33	)	)	PUNCT
ma-232	105	34	be	be	AUX
ma-232	105	35	the	the	DET
ma-232	105	36	subalgebra	subalgebra	NOUN
ma-232	105	37	of	of	ADP
ma-232	105	38	c(d	c(d	PROPN
ma-232	105	39	)	)	PUNCT
ma-232	105	40	consisting	consist	VERB
ma-232	105	41	of	of	ADP
ma-232	105	42	functions	function	NOUN
ma-232	105	43	f	f	PROPN
ma-232	105	44	with	with	ADP
ma-232	105	45	f	f	PROPN
ma-232	105	46	(	(	PUNCT
ma-232	105	47	z	z	NOUN
ma-232	105	48	)	)	PUNCT
ma-232	105	49	→	→	SYM
ma-232	105	50	0	0	PUNCT
ma-232	105	51	as	as	ADP
ma-232	105	52	|z	|z	PROPN
ma-232	105	53	|	|	PROPN
ma-232	105	54	→	→	SYM
ma-232	106	1	1−.	1−.	NUM
ma-232	106	2	then	then	ADV
ma-232	106	3	c	c	X
ma-232	106	4	◦	◦	NOUN
ma-232	106	5	(u	(u	NOUN
ma-232	106	6	)	)	PUNCT
ma-232	106	7	=	=	PRON
ma-232	107	1	{	{	PUNCT
ma-232	107	2	g	g	PROPN
ma-232	107	3	◦	◦	NOUN
ma-232	107	4	ψ−1	ψ−1	PROPN
ma-232	107	5	:	:	PUNCT
ma-232	107	6	g	g	PROPN
ma-232	107	7	∈	∈	PROPN
ma-232	107	8	c	c	X
ma-232	107	9	◦	◦	NOUN
ma-232	107	10	(d	(d	NOUN
ma-232	107	11	)	)	PUNCT
ma-232	107	12	}	}	PUNCT
ma-232	107	13	.	.	PUNCT
ma-232	108	1	proof	proof	NOUN
ma-232	108	2	.	.	PUNCT
ma-232	109	1	let	let	VERB
ma-232	109	2	k	k	PROPN
ma-232	109	3	⊂	⊂	PRON
ma-232	109	4	u	u	PRON
ma-232	109	5	be	be	VERB
ma-232	109	6	compact	compact	ADJ
ma-232	109	7	.	.	PUNCT
ma-232	110	1	since	since	SCONJ
ma-232	110	2	cayley	cayley	ADJ
ma-232	110	3	transform	transform	NOUN
ma-232	110	4	ψ	ψ	NOUN
ma-232	110	5	:	:	PUNCT
ma-232	110	6	d	d	X
ma-232	110	7	→	→	SYM
ma-232	110	8	u	u	NOUN
ma-232	110	9	is	be	AUX
ma-232	110	10	a	a	DET
ma-232	110	11	continuous	continuous	ADJ
ma-232	110	12	bijection	bijection	NOUN
ma-232	110	13	,	,	PUNCT
ma-232	110	14	itfollows	itfollow	VERB
ma-232	110	15	that	that	SCONJ
ma-232	110	16	k	k	PROPN
ma-232	110	17	⊂	⊂	PROPN
ma-232	110	18	u	u	PROPN
ma-232	110	19	is	be	AUX
ma-232	110	20	compact	compact	ADJ
ma-232	110	21	if	if	SCONJ
ma-232	110	22	and	and	CCONJ
ma-232	110	23	only	only	ADV
ma-232	110	24	if	if	SCONJ
ma-232	110	25	ψ−1(k	ψ−1(k	NOUN
ma-232	110	26	)	)	PUNCT
ma-232	110	27	is	be	AUX
ma-232	110	28	compact	compact	ADJ
ma-232	110	29	in	in	ADP
ma-232	110	30	d.	d.	PROPN
ma-232	110	31	if	if	SCONJ
ma-232	110	32	f	f	PROPN
ma-232	110	33	∈	∈	PROPN
ma-232	110	34	c	c	PROPN
ma-232	110	35	◦	◦	NOUN
ma-232	110	36	(u	(u	NOUN
ma-232	110	37	)	)	PUNCT
ma-232	110	38	and	and	CCONJ
ma-232	110	39	ε	ε	X
ma-232	110	40	>	>	X
ma-232	110	41	0	0	PROPN
ma-232	110	42	,	,	PUNCT
ma-232	110	43	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	110	44	eur	eur	PROPN
ma-232	110	45	.	.	PUNCT
ma-232	111	1	j.	j.	PROPN
ma-232	111	2	math	math	PROPN
ma-232	111	3	.	.	PUNCT
ma-232	112	1	anal	anal	PROPN
ma-232	112	2	.	.	PUNCT
ma-232	113	1	10.28924	10.28924	NUM
ma-232	113	2	/	/	SYM
ma-232	113	3	ada	ada	PROPN
ma-232	113	4	/	/	SYM
ma-232	113	5	ma.4.14	ma.4.14	X
ma-232	114	1	5then	5then	NOUN
ma-232	114	2	there	there	PRON
ma-232	114	3	exists	exist	VERB
ma-232	114	4	k	k	PROPN
ma-232	114	5	compact	compact	ADJ
ma-232	114	6	in	in	ADP
ma-232	114	7	u	u	PRON
ma-232	114	8	such	such	ADJ
ma-232	114	9	that	that	PRON
ma-232	114	10	supw∈u\k	supw∈u\k	NOUN
ma-232	114	11	|f	|f	PROPN
ma-232	115	1	(	(	PUNCT
ma-232	115	2	w)|	w)|	VERB
ma-232	115	3	<	<	X
ma-232	115	4	ε.now	ε.now	PROPN
ma-232	115	5	,	,	PUNCT
ma-232	115	6	g	g	PROPN
ma-232	115	7	=	=	SYM
ma-232	115	8	f	f	PROPN
ma-232	115	9	◦	◦	NOUN
ma-232	115	10	ψ	ψ	SYM
ma-232	115	11	is	be	AUX
ma-232	115	12	continuous	continuous	ADJ
ma-232	115	13	on	on	ADP
ma-232	115	14	d	d	PROPN
ma-232	115	15	with	with	ADP
ma-232	115	16	f	f	PROPN
ma-232	115	17	=	=	SYM
ma-232	115	18	g	g	PROPN
ma-232	115	19	◦	◦	PROPN
ma-232	115	20	ψ−1	ψ−1	PROPN
ma-232	115	21	,	,	PUNCT
ma-232	115	22	and	and	CCONJ
ma-232	115	23	sup	sup	PROPN
ma-232	115	24	z∈d\ψ−1(k	z∈d\ψ−1(k	NOUN
ma-232	115	25	)	)	PUNCT
ma-232	115	26	|g(z)|	|g(z)|	NOUN
ma-232	115	27	=	=	PUNCT
ma-232	115	28	sup	sup	X
ma-232	115	29	w∈u\k	w∈u\k	INTJ
ma-232	115	30	|f	|f	PROPN
ma-232	115	31	(	(	PUNCT
ma-232	115	32	w)|	w)|	VERB
ma-232	115	33	<	<	X
ma-232	115	34	ε	ε	PROPN
ma-232	115	35	.	.	PUNCT
ma-232	115	36	�	�	PROPN
ma-232	115	37	proposition	proposition	PROPN
ma-232	115	38	2.3	2.3	NUM
ma-232	115	39	.	.	PUNCT
ma-232	116	1	let	let	VERB
ma-232	116	2	cψ	cψ	NOUN
ma-232	116	3	be	be	AUX
ma-232	116	4	the	the	DET
ma-232	116	5	composition	composition	NOUN
ma-232	116	6	by	by	ADP
ma-232	116	7	ψ	ψ	NOUN
ma-232	116	8	operator	operator	NOUN
ma-232	116	9	.	.	PUNCT
ma-232	117	1	then	then	ADV
ma-232	117	2	(	(	PUNCT
ma-232	117	3	1	1	X
ma-232	117	4	)	)	PUNCT
ma-232	117	5	f	f	PROPN
ma-232	117	6	∈	∈	PROPN
ma-232	117	7	b∞(u	b∞(u	PROPN
ma-232	117	8	)	)	PUNCT
ma-232	117	9	if	if	SCONJ
ma-232	117	10	and	and	CCONJ
ma-232	117	11	only	only	ADV
ma-232	117	12	if	if	SCONJ
ma-232	117	13	cψf	cψf	PROPN
ma-232	117	14	∈	∈	PROPN
ma-232	117	15	b∞(d	b∞(d	PROPN
ma-232	117	16	)	)	PUNCT
ma-232	117	17	.	.	PUNCT
ma-232	118	1	in	in	ADP
ma-232	118	2	particular	particular	ADJ
ma-232	118	3	,	,	PUNCT
ma-232	118	4	‖f	‖f	ADJ
ma-232	118	5	‖b∞,1(u	‖b∞,1(u	NOUN
ma-232	118	6	)	)	PUNCT
ma-232	118	7	=	=	SYM
ma-232	118	8	1	1	NUM
ma-232	118	9	2‖cψf	2‖cψf	NOUN
ma-232	118	10	‖b∞,1(d	‖b∞,1(d	NOUN
ma-232	118	11	)	)	PUNCT
ma-232	118	12	.	.	PUNCT
ma-232	119	1	(	(	PUNCT
ma-232	119	2	2	2	X
ma-232	119	3	)	)	PUNCT
ma-232	119	4	f	f	NOUN
ma-232	119	5	∈	∈	PROPN
ma-232	119	6	b∞,	b∞,	ADP
ma-232	119	7	◦	◦	NOUN
ma-232	119	8	(u	(u	NOUN
ma-232	119	9	)	)	PUNCT
ma-232	119	10	if	if	SCONJ
ma-232	119	11	and	and	CCONJ
ma-232	119	12	only	only	ADV
ma-232	119	13	if	if	SCONJ
ma-232	119	14	cψf	cψf	NOUN
ma-232	119	15	∈	∈	NOUN
ma-232	119	16	b∞,	b∞,	ADP
ma-232	119	17	◦	◦	NOUN
ma-232	119	18	(d	(d	NOUN
ma-232	119	19	)	)	PUNCT
ma-232	119	20	.	.	PUNCT
ma-232	120	1	(	(	PUNCT
ma-232	120	2	3	3	X
ma-232	120	3	)	)	PUNCT
ma-232	120	4	f	f	PROPN
ma-232	120	5	∈	∈	PROPN
ma-232	120	6	l1(u	l1(u	PROPN
ma-232	120	7	,	,	PUNCT
ma-232	120	8	µα	µα	ADP
ma-232	120	9	)	)	PUNCT
ma-232	120	10	if	if	SCONJ
ma-232	120	11	and	and	CCONJ
ma-232	120	12	only	only	ADV
ma-232	120	13	if	if	SCONJ
ma-232	120	14	sψf	sψf	VERB
ma-232	120	15	∈	∈	PROPN
ma-232	120	16	l1(d	l1(d	PROPN
ma-232	120	17	,	,	PUNCT
ma-232	120	18	mα	mα	PROPN
ma-232	120	19	)	)	PUNCT
ma-232	120	20	.	.	PUNCT
ma-232	121	1	in	in	ADP
ma-232	121	2	particular	particular	ADJ
ma-232	121	3	,	,	PUNCT
ma-232	121	4	‖f	‖f	ADP
ma-232	121	5	‖l1	‖l1	PROPN
ma-232	121	6	a(u,µα	a(u,µα	NOUN
ma-232	121	7	)	)	PUNCT
ma-232	121	8	=	=	SYM
ma-232	121	9	1	1	NUM
ma-232	121	10	2α	2α	NOUN
ma-232	121	11	‖sψf	‖sψf	PROPN
ma-232	121	12	‖l1(d	‖l1(d	PROPN
ma-232	121	13	,	,	PUNCT
ma-232	121	14	mα	mα	NOUN
ma-232	121	15	)	)	PUNCT
ma-232	121	16	.	.	PUNCT
ma-232	122	1	(	(	PUNCT
ma-232	122	2	4	4	X
ma-232	122	3	)	)	PUNCT
ma-232	122	4	f	f	PROPN
ma-232	122	5	∈	∈	PROPN
ma-232	122	6	l∞(u	l∞(u	PROPN
ma-232	122	7	,	,	PUNCT
ma-232	122	8	µα	µα	ADP
ma-232	122	9	)	)	PUNCT
ma-232	122	10	if	if	SCONJ
ma-232	122	11	and	and	CCONJ
ma-232	122	12	only	only	ADV
ma-232	122	13	if	if	SCONJ
ma-232	122	14	cψf	cψf	PROPN
ma-232	122	15	∈	∈	PROPN
ma-232	122	16	l∞(d	l∞(d	NOUN
ma-232	122	17	,	,	PUNCT
ma-232	122	18	mα	mα	PROPN
ma-232	122	19	)	)	PUNCT
ma-232	122	20	.	.	PUNCT
ma-232	123	1	proof	proof	NOUN
ma-232	123	2	.	.	PUNCT
ma-232	124	1	for	for	ADP
ma-232	124	2	(	(	PUNCT
ma-232	124	3	1	1	NUM
ma-232	124	4	)	)	PUNCT
ma-232	124	5	,	,	PUNCT
ma-232	124	6	if	if	SCONJ
ma-232	124	7	f	f	PROPN
ma-232	124	8	∈	∈	PROPN
ma-232	124	9	b∞(u	b∞(u	PROPN
ma-232	124	10	)	)	PUNCT
ma-232	124	11	,	,	PUNCT
ma-232	124	12	then	then	ADV
ma-232	124	13	by	by	ADP
ma-232	124	14	definition	definition	NOUN
ma-232	124	15	,	,	PUNCT
ma-232	124	16	‖f	‖f	ADP
ma-232	124	17	‖b∞,1(u	‖b∞,1(u	NOUN
ma-232	124	18	)	)	PUNCT
ma-232	124	19	=	=	SYM
ma-232	124	20	sup	sup	NOUN
ma-232	124	21	ω∈u	ω∈u	NOUN
ma-232	124	22	(=	(=	X
ma-232	124	23	(	(	PUNCT
ma-232	124	24	ω))|f	ω))|f	NUM
ma-232	124	25	′(ω)|	′(ω)|	NOUN
ma-232	124	26	=	=	PUNCT
ma-232	124	27	sup	sup	NUM
ma-232	124	28	z∈d	z∈d	NUM
ma-232	124	29	1−	1−	PROPN
ma-232	124	30	|z	|z	PROPN
ma-232	124	31	|2	|2	NUM
ma-232	124	32	|1−	|1−	PROPN
ma-232	124	33	z	z	NOUN
ma-232	124	34	|2	|2	NUM
ma-232	124	35	∣∣f	∣∣f	NOUN
ma-232	124	36	′(ψ(z	′(ψ(z	NOUN
ma-232	124	37	)	)	PUNCT
ma-232	124	38	)	)	PUNCT
ma-232	124	39	∣∣	∣∣	PUNCT
ma-232	125	1	=	=	SYM
ma-232	125	2	1	1	NUM
ma-232	125	3	2	2	NUM
ma-232	125	4	sup	sup	NOUN
ma-232	125	5	z∈d	z∈d	NUM
ma-232	125	6	(	(	PUNCT
ma-232	125	7	1−	1−	NUM
ma-232	125	8	|z	|z	PROPN
ma-232	125	9	|2)|ψ′(z)||f	|2)|ψ′(z)||f	NUM
ma-232	125	10	′(ψ(z))|	′(ψ(z))|	PROPN
ma-232	125	11	=	=	SYM
ma-232	125	12	1	1	NUM
ma-232	125	13	2	2	NUM
ma-232	125	14	sup	sup	NOUN
ma-232	125	15	z∈d	z∈d	NUM
ma-232	125	16	(	(	PUNCT
ma-232	125	17	1−	1−	NUM
ma-232	125	18	|z	|z	PROPN
ma-232	125	19	|2)|(f	|2)|(f	ADP
ma-232	125	20	◦	◦	NOUN
ma-232	125	21	ψ)′(z)|	ψ)′(z)|	NOUN
ma-232	125	22	=	=	NOUN
ma-232	125	23	1	1	NUM
ma-232	125	24	2	2	NUM
ma-232	125	25	‖f	‖f	ADP
ma-232	125	26	◦	◦	NOUN
ma-232	125	27	ψ‖b∞,1(d	ψ‖b∞,1(d	NUM
ma-232	125	28	)	)	PUNCT
ma-232	125	29	.	.	PUNCT
ma-232	126	1	for	for	ADP
ma-232	126	2	(	(	PUNCT
ma-232	126	3	2	2	NUM
ma-232	126	4	)	)	PUNCT
ma-232	126	5	,	,	PUNCT
ma-232	126	6	we	we	PRON
ma-232	126	7	have	have	VERB
ma-232	126	8	f	f	PROPN
ma-232	126	9	∈	∈	PROPN
ma-232	126	10	b∞,0(u	b∞,0(u	PROPN
ma-232	126	11	)	)	PUNCT
ma-232	126	12	is	be	AUX
ma-232	126	13	equivalent	equivalent	ADJ
ma-232	126	14	to	to	ADP
ma-232	126	15	lim	lim	PROPN
ma-232	126	16	=(	=(	PROPN
ma-232	126	17	ω)→0	ω)→0	NOUN
ma-232	126	18	(=	(=	NOUN
ma-232	126	19	(	(	PUNCT
ma-232	126	20	ω))|f	ω))|f	NUM
ma-232	126	21	′(ω)|	′(ω)|	PROPN
ma-232	126	22	=	=	PROPN
ma-232	126	23	lim	lim	PROPN
ma-232	126	24	=(	=(	PROPN
ma-232	126	25	ψ(z))→0	ψ(z))→0	PUNCT
ma-232	126	26	1−	1−	NUM
ma-232	126	27	|z	|z	PROPN
ma-232	126	28	|2	|2	NUM
ma-232	126	29	|1−	|1−	PROPN
ma-232	126	30	z	z	PROPN
ma-232	126	31	|2	|2	X
ma-232	126	32	|f	|f	X
ma-232	127	1	′(ψ(z))|	′(ψ(z))|	PROPN
ma-232	127	2	=	=	SYM
ma-232	127	3	1	1	NUM
ma-232	127	4	2	2	NUM
ma-232	127	5	sup	sup	NOUN
ma-232	127	6	|z	|z	NOUN
ma-232	127	7	|→1	|→1	PROPN
ma-232	127	8	(	(	PUNCT
ma-232	127	9	1−	1−	NUM
ma-232	127	10	|z	|z	PROPN
ma-232	127	11	|2)|(f	|2)|(f	ADP
ma-232	127	12	◦	◦	NOUN
ma-232	127	13	ψ)′(z)|	ψ)′(z)|	PUNCT
ma-232	127	14	=	=	NOUN
ma-232	127	15	0	0	NUM
ma-232	127	16	,	,	PUNCT
ma-232	127	17	which	which	PRON
ma-232	127	18	in	in	ADP
ma-232	127	19	turn	turn	NOUN
ma-232	127	20	is	be	AUX
ma-232	127	21	equivalent	equivalent	ADJ
ma-232	127	22	to	to	ADP
ma-232	127	23	f	f	PROPN
ma-232	127	24	◦	◦	NOUN
ma-232	127	25	ψ	ψ	X
ma-232	127	26	∈	∈	PROPN
ma-232	127	27	b∞,0(d	b∞,0(d	PROPN
ma-232	127	28	)	)	PUNCT
ma-232	127	29	,	,	PUNCT
ma-232	127	30	as	as	SCONJ
ma-232	127	31	desired	desire	VERB
ma-232	127	32	.	.	PUNCT
ma-232	128	1	for	for	ADP
ma-232	128	2	f	f	PROPN
ma-232	128	3	∈	∈	PROPN
ma-232	128	4	l1	l1	PROPN
ma-232	128	5	a(u	a(u	PROPN
ma-232	128	6	,	,	PUNCT
ma-232	128	7	µα	µα	ADP
ma-232	128	8	)	)	PUNCT
ma-232	128	9	,	,	PUNCT
ma-232	128	10	we	we	PRON
ma-232	128	11	have	have	VERB
ma-232	128	12	‖f	‖f	PRON
ma-232	128	13	‖l1	‖l1	PROPN
ma-232	128	14	a(u,µα	a(u,µα	NOUN
ma-232	128	15	)	)	PUNCT
ma-232	129	1	=	=	SYM
ma-232	129	2	∫	∫	PROPN
ma-232	129	3	u	u	PROPN
ma-232	129	4	|f	|f	PROPN
ma-232	129	5	(	(	PUNCT
ma-232	129	6	ω)|	ω)|	ADJ
ma-232	129	7	dµα(ω	dµα(ω	NOUN
ma-232	129	8	)	)	PUNCT
ma-232	129	9	=	=	SYM
ma-232	129	10	∫	∫	PROPN
ma-232	129	11	u	u	PROPN
ma-232	129	12	|f	|f	PROPN
ma-232	129	13	(	(	PUNCT
ma-232	129	14	ω)|(=(ω))α	ω)|(=(ω))α	PROPN
ma-232	129	15	da(ω	da(ω	NOUN
ma-232	129	16	)	)	PUNCT
ma-232	130	1	=	=	SYM
ma-232	130	2	∫	∫	PROPN
ma-232	131	1	d	d	X
ma-232	131	2	|f	|f	PROPN
ma-232	131	3	(	(	PUNCT
ma-232	131	4	ψ(z))|	ψ(z))|	NOUN
ma-232	131	5	(	(	PUNCT
ma-232	131	6	1−	1−	NUM
ma-232	131	7	|z	|z	PROPN
ma-232	131	8	|2	|2	NUM
ma-232	131	9	|1−	|1−	PROPN
ma-232	131	10	z	z	PROPN
ma-232	131	11	|2	|2	NUM
ma-232	131	12	)	)	PUNCT
ma-232	131	13	α	α	PRON
ma-232	131	14	|ψ′(z)|2	|ψ′(z)|2	NOUN
ma-232	131	15	da(z	da(z	X
ma-232	131	16	)	)	PUNCT
ma-232	131	17	=	=	SYM
ma-232	131	18	1	1	NUM
ma-232	131	19	2α	2α	NOUN
ma-232	131	20	∫	∫	PROPN
ma-232	131	21	d	d	X
ma-232	131	22	|f	|f	PROPN
ma-232	131	23	(	(	PUNCT
ma-232	131	24	ψ(z))||ψ′(z)|α+2	ψ(z))||ψ′(z)|α+2	X
ma-232	131	25	(	(	PUNCT
ma-232	131	26	1−	1−	NUM
ma-232	131	27	|z	|z	NOUN
ma-232	131	28	|2	|2	NUM
ma-232	131	29	)	)	PUNCT
ma-232	131	30	da(z	da(z	PROPN
ma-232	131	31	)	)	PUNCT
ma-232	131	32	=	=	SYM
ma-232	131	33	1	1	NUM
ma-232	131	34	2α	2α	NOUN
ma-232	131	35	∫	∫	PROPN
ma-232	131	36	d	d	X
ma-232	131	37	|(ψ′(z))γ(f	|(ψ′(z))γ(f	NOUN
ma-232	131	38	◦	◦	VERB
ma-232	131	39	ψ)(z)|	ψ)(z)|	ADJ
ma-232	131	40	dmα(z	dmα(z	PROPN
ma-232	131	41	)	)	PUNCT
ma-232	131	42	=	=	SYM
ma-232	131	43	1	1	NUM
ma-232	131	44	2α	2α	NOUN
ma-232	131	45	‖sψf	‖sψf	PROPN
ma-232	131	46	‖l1(d	‖l1(d	PROPN
ma-232	131	47	,	,	PUNCT
ma-232	131	48	mα	mα	PROPN
ma-232	131	49	)	)	PUNCT
ma-232	131	50	,	,	PUNCT
ma-232	131	51	which	which	PRON
ma-232	131	52	proves	prove	VERB
ma-232	131	53	(	(	PUNCT
ma-232	131	54	3	3	NUM
ma-232	131	55	)	)	PUNCT
ma-232	131	56	.	.	PUNCT
ma-232	132	1	now	now	ADV
ma-232	132	2	,	,	PUNCT
ma-232	132	3	f	f	PROPN
ma-232	132	4	∈	∈	PROPN
ma-232	132	5	l∞(u	l∞(u	PROPN
ma-232	132	6	,	,	PUNCT
ma-232	132	7	µα	µα	NOUN
ma-232	132	8	)	)	PUNCT
ma-232	132	9	means	mean	VERB
ma-232	132	10	that	that	SCONJ
ma-232	132	11	f	f	PROPN
ma-232	132	12	is	be	AUX
ma-232	132	13	essentially	essentially	ADV
ma-232	132	14	bounded	bound	VERB
ma-232	132	15	which	which	PRON
ma-232	132	16	implies	imply	VERB
ma-232	132	17	that	that	SCONJ
ma-232	132	18	f	f	PROPN
ma-232	132	19	◦	◦	NOUN
ma-232	132	20	ψ	ψ	SYM
ma-232	132	21	is	be	AUX
ma-232	132	22	essentially	essentially	ADV
ma-232	132	23	bounded	bound	VERB
ma-232	132	24	as	as	ADV
ma-232	132	25	well	well	ADV
ma-232	132	26	.	.	PUNCT
ma-232	133	1	since	since	SCONJ
ma-232	133	2	ψ	ψ	NOUN
ma-232	133	3	is	be	AUX
ma-232	133	4	an	an	DET
ma-232	133	5	invertible	invertible	ADJ
ma-232	133	6	mapping	mapping	NOUN
ma-232	133	7	from	from	ADP
ma-232	133	8	d	d	PROPN
ma-232	133	9	onto	onto	ADP
ma-232	133	10	u	u	NOUN
ma-232	133	11	,	,	PUNCT
ma-232	133	12	it	it	PRON
ma-232	133	13	followsthat	followsthat	VERB
ma-232	133	14	f	f	INTJ
ma-232	133	15	◦	◦	NOUN
ma-232	133	16	ψ	ψ	X
ma-232	133	17	∈	∈	PROPN
ma-232	133	18	l∞(d	l∞(d	NOUN
ma-232	133	19	,	,	PUNCT
ma-232	133	20	mα	mα	PROPN
ma-232	133	21	)	)	PUNCT
ma-232	133	22	.	.	PUNCT
ma-232	134	1	the	the	DET
ma-232	134	2	converse	converse	NOUN
ma-232	134	3	follows	follow	VERB
ma-232	134	4	similarly	similarly	ADV
ma-232	134	5	.	.	PUNCT
ma-232	135	1	this	this	PRON
ma-232	135	2	completes	complete	VERB
ma-232	135	3	the	the	DET
ma-232	135	4	proof	proof	NOUN
ma-232	135	5	.	.	PUNCT
ma-232	136	1	�	�	PROPN
ma-232	136	2	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	136	3	eur	eur	PROPN
ma-232	136	4	.	.	PUNCT
ma-232	137	1	j.	j.	PROPN
ma-232	137	2	math	math	PROPN
ma-232	137	3	.	.	PUNCT
ma-232	138	1	anal	anal	PROPN
ma-232	138	2	.	.	PUNCT
ma-232	139	1	10.28924	10.28924	NUM
ma-232	139	2	/	/	SYM
ma-232	139	3	ada	ada	PROPN
ma-232	139	4	/	/	SYM
ma-232	139	5	ma.4.14	ma.4.14	PROPN
ma-232	139	6	6	6	NUM
ma-232	139	7	remark	remark	NOUN
ma-232	139	8	1	1	NUM
ma-232	139	9	.	.	PUNCT
ma-232	140	1	it	it	PRON
ma-232	140	2	is	be	AUX
ma-232	140	3	easy	easy	ADJ
ma-232	140	4	to	to	PART
ma-232	140	5	verify	verify	VERB
ma-232	140	6	that	that	DET
ma-232	140	7	cψ−1	cψ−1	PROPN
ma-232	140	8	=	=	SYM
ma-232	140	9	c−1	c−1	PROPN
ma-232	140	10	ψ	ψ	NOUN
ma-232	140	11	.	.	PUNCT
ma-232	141	1	proposition	proposition	NOUN
ma-232	141	2	2.3	2.3	NUM
ma-232	141	3	above	above	ADV
ma-232	141	4	therefore	therefore	ADV
ma-232	141	5	implies	imply	VERB
ma-232	141	6	that	that	SCONJ
ma-232	141	7	cψ	cψ	PROPN
ma-232	141	8	is	be	AUX
ma-232	141	9	an	an	DET
ma-232	141	10	is	be	AUX
ma-232	141	11	an	an	DET
ma-232	141	12	isometry	isometry	NOUN
ma-232	141	13	up	up	ADP
ma-232	141	14	to	to	ADP
ma-232	141	15	a	a	DET
ma-232	141	16	constant	constant	ADJ
ma-232	141	17	and	and	CCONJ
ma-232	141	18	at	at	ADP
ma-232	141	19	the	the	DET
ma-232	141	20	same	same	ADJ
ma-232	141	21	time	time	NOUN
ma-232	141	22	invertible	invertible	ADJ
ma-232	141	23	on	on	ADP
ma-232	141	24	the	the	DET
ma-232	141	25	respective	respective	ADJ
ma-232	141	26	spaces	space	NOUN
ma-232	141	27	with	with	ADP
ma-232	141	28	the	the	DET
ma-232	141	29	inverse	inverse	NOUN
ma-232	141	30	also	also	ADV
ma-232	141	31	acting	act	VERB
ma-232	141	32	on	on	ADP
ma-232	141	33	the	the	DET
ma-232	141	34	appropriate	appropriate	ADJ
ma-232	141	35	spaces	space	NOUN
ma-232	141	36	.	.	PUNCT
ma-232	142	1	more	more	ADV
ma-232	142	2	generally	generally	ADV
ma-232	142	3	,	,	PUNCT
ma-232	142	4	let	let	VERB
ma-232	142	5	{	{	PUNCT
ma-232	142	6	v1	v1	VERB
ma-232	142	7	,	,	PUNCT
ma-232	142	8	v2	v2	NOUN
ma-232	142	9	}	}	PUNCT
ma-232	142	10	=	=	PUNCT
ma-232	142	11	{	{	PUNCT
ma-232	142	12	d	d	PROPN
ma-232	142	13	,	,	PUNCT
ma-232	142	14	u	u	NOUN
ma-232	142	15	}	}	PUNCT
ma-232	142	16	,	,	PUNCT
ma-232	142	17	and	and	CCONJ
ma-232	142	18	let	let	VERB
ma-232	142	19	lf	lf	INTJ
ma-232	142	20	(	(	PUNCT
ma-232	142	21	vi	vi	PROPN
ma-232	142	22	,	,	PUNCT
ma-232	142	23	vj	vj	PROPN
ma-232	142	24	)	)	PUNCT
ma-232	142	25	denote	denote	VERB
ma-232	142	26	the	the	DET
ma-232	142	27	collection	collection	NOUN
ma-232	142	28	of	of	ADP
ma-232	142	29	conformal	conformal	ADJ
ma-232	142	30	mappings	mapping	NOUN
ma-232	142	31	from	from	ADP
ma-232	142	32	vi	vi	NOUN
ma-232	142	33	onto	onto	ADP
ma-232	142	34	vj	vj	PROPN
ma-232	142	35	.	.	PUNCT
ma-232	143	1	then	then	ADV
ma-232	143	2	lf	lf	INTJ
ma-232	143	3	(	(	PUNCT
ma-232	143	4	vi	vi	PROPN
ma-232	143	5	,	,	PUNCT
ma-232	143	6	vi	vi	NOUN
ma-232	143	7	)	)	PUNCT
ma-232	143	8	=	=	SYM
ma-232	143	9	aut(vi	aut(vi	NOUN
ma-232	143	10	)	)	PUNCT
ma-232	143	11	,	,	PUNCT
ma-232	143	12	and	and	CCONJ
ma-232	143	13	if	if	SCONJ
ma-232	143	14	h	h	PRON
ma-232	143	15	∈	∈	PROPN
ma-232	143	16	lf	lf	INTJ
ma-232	143	17	(	(	PUNCT
ma-232	143	18	vi	vi	PROPN
ma-232	143	19	,	,	PUNCT
ma-232	143	20	vj	vj	PROPN
ma-232	143	21	)	)	PUNCT
ma-232	143	22	,	,	PUNCT
ma-232	143	23	then	then	ADV
ma-232	143	24	g	g	PROPN
ma-232	143	25	∈	∈	PROPN
ma-232	143	26	aut(vj	aut(vj	NOUN
ma-232	143	27	)	)	PUNCT
ma-232	143	28	7→	7→	NUM
ma-232	144	1	h−1	h−1	PROPN
ma-232	144	2	◦	◦	NOUN
ma-232	144	3	g	g	NOUN
ma-232	144	4	◦	◦	NOUN
ma-232	145	1	h	h	NOUN
ma-232	145	2	∈	∈	NOUN
ma-232	145	3	aut(vi	aut(vi	X
ma-232	145	4	)	)	PUNCT
ma-232	145	5	is	be	AUX
ma-232	145	6	an	an	DET
ma-232	145	7	isomorphism	isomorphism	NOUN
ma-232	145	8	from	from	ADP
ma-232	145	9	aut(vi	aut(vi	NOUN
ma-232	145	10	)	)	PUNCT
ma-232	145	11	onto	onto	ADP
ma-232	145	12	aut(vj	aut(vj	NOUN
ma-232	145	13	)	)	PUNCT
ma-232	145	14	.	.	PUNCT
ma-232	146	1	for	for	ADP
ma-232	146	2	each	each	DET
ma-232	146	3	g	g	PROPN
ma-232	146	4	∈	∈	PROPN
ma-232	146	5	lf	lf	PROPN
ma-232	146	6	(	(	PUNCT
ma-232	146	7	vi	vi	PROPN
ma-232	146	8	,	,	PUNCT
ma-232	146	9	vj	vj	PROPN
ma-232	146	10	)	)	PUNCT
ma-232	146	11	,	,	PUNCT
ma-232	146	12	we	we	PRON
ma-232	146	13	define	define	VERB
ma-232	146	14	a	a	DET
ma-232	146	15	weighted	weighted	ADJ
ma-232	146	16	composition	composition	NOUN
ma-232	146	17	operator	operator	NOUN
ma-232	146	18	sg	sg	NOUN
ma-232	146	19	:	:	PUNCT
ma-232	146	20	h(vj	h(vj	PROPN
ma-232	146	21	)	)	PUNCT
ma-232	146	22	→	→	SYM
ma-232	146	23	h(vi	h(vi	PROPN
ma-232	146	24	)	)	PUNCT
ma-232	146	25	,	,	PUNCT
ma-232	146	26	by	by	ADP
ma-232	146	27	sgf	sgf	PROPN
ma-232	146	28	(	(	PUNCT
ma-232	146	29	z	z	NOUN
ma-232	146	30	)	)	PUNCT
ma-232	146	31	=	=	SYM
ma-232	146	32	(	(	PUNCT
ma-232	146	33	g′(z))γf	g′(z))γf	PROPN
ma-232	146	34	(	(	PUNCT
ma-232	146	35	g(z	g(z	PROPN
ma-232	146	36	)	)	PUNCT
ma-232	146	37	)	)	PUNCT
ma-232	146	38	,	,	PUNCT
ma-232	146	39	for	for	ADP
ma-232	146	40	all	all	DET
ma-232	146	41	z	z	NOUN
ma-232	146	42	∈	∈	PROPN
ma-232	146	43	vi	vi	PROPN
ma-232	146	44	.	.	PUNCT
ma-232	147	1	(	(	PUNCT
ma-232	147	2	2.1	2.1	NUM
ma-232	147	3	)	)	PUNCT
ma-232	147	4	we	we	PRON
ma-232	147	5	note	note	VERB
ma-232	147	6	that	that	SCONJ
ma-232	147	7	if	if	SCONJ
ma-232	147	8	g	g	PROPN
ma-232	147	9	∈	∈	PROPN
ma-232	147	10	lf	lf	ADP
ma-232	147	11	(	(	PUNCT
ma-232	147	12	vi	vi	PROPN
ma-232	147	13	,	,	PUNCT
ma-232	147	14	vj	vj	PROPN
ma-232	147	15	)	)	PUNCT
ma-232	147	16	and	and	CCONJ
ma-232	147	17	h	h	NOUN
ma-232	147	18	∈	∈	PROPN
ma-232	147	19	lf	lf	INTJ
ma-232	147	20	(	(	PUNCT
ma-232	147	21	vj	vj	INTJ
ma-232	147	22	,	,	PUNCT
ma-232	147	23	vi	vi	PROPN
ma-232	147	24	)	)	PUNCT
ma-232	147	25	,	,	PUNCT
ma-232	147	26	then	then	ADV
ma-232	147	27	it	it	PRON
ma-232	147	28	is	be	AUX
ma-232	147	29	clear	clear	ADJ
ma-232	147	30	by	by	ADP
ma-232	147	31	chain	chain	NOUN
ma-232	147	32	rule	rule	NOUN
ma-232	147	33	that	that	SCONJ
ma-232	147	34	sh	sh	PROPN
ma-232	147	35	◦	◦	VERB
ma-232	147	36	sg	sg	X
ma-232	147	37	=	=	PUNCT
ma-232	147	38	sg	sg	PROPN
ma-232	147	39	◦	◦	NOUN
ma-232	147	40	h	h	NOUN
ma-232	147	41	and	and	CCONJ
ma-232	147	42	s−1	s−1	PROPN
ma-232	147	43	g	g	NOUN
ma-232	147	44	=	=	PUNCT
ma-232	147	45	sg−1	sg−1	PROPN
ma-232	147	46	.	.	PUNCT
ma-232	148	1	now	now	ADV
ma-232	148	2	,	,	PUNCT
ma-232	148	3	using	use	VERB
ma-232	148	4	propositions	proposition	NOUN
ma-232	148	5	2.2	2.2	NUM
ma-232	148	6	and	and	CCONJ
ma-232	148	7	2.3	2.3	NUM
ma-232	148	8	above	above	ADV
ma-232	148	9	,	,	PUNCT
ma-232	148	10	we	we	PRON
ma-232	148	11	obtain	obtain	VERB
ma-232	148	12	the	the	DET
ma-232	148	13	following	following	ADJ
ma-232	148	14	result	result	NOUN
ma-232	148	15	which	which	PRON
ma-232	148	16	is	be	AUX
ma-232	148	17	the	the	DET
ma-232	148	18	upperhalf	upperhalf	ADJ
ma-232	148	19	-	-	PUNCT
ma-232	148	20	plane	plane	NOUN
ma-232	148	21	analogue	analogue	NOUN
ma-232	148	22	of	of	ADP
ma-232	148	23	[	[	X
ma-232	148	24	13	13	NUM
ma-232	148	25	,	,	PUNCT
ma-232	148	26	lemma	lemma	PROPN
ma-232	148	27	5.14	5.14	NUM
ma-232	148	28	]	]	PUNCT
ma-232	148	29	.	.	PUNCT
ma-232	149	1	proposition	proposition	NOUN
ma-232	149	2	2.4	2.4	NUM
ma-232	149	3	.	.	PUNCT
ma-232	150	1	for	for	ADP
ma-232	150	2	t	t	PROPN
ma-232	150	3	>	>	X
ma-232	150	4	0	0	PROPN
ma-232	150	5	,	,	PUNCT
ma-232	150	6	α	α	X
ma-232	150	7	>	>	X
ma-232	150	8	−1	−1	NOUN
ma-232	150	9	,	,	PUNCT
ma-232	150	10	let	let	VERB
ma-232	150	11	the	the	DET
ma-232	150	12	integral	integral	ADJ
ma-232	150	13	operator	operator	NOUN
ma-232	150	14	t	t	PROPN
ma-232	150	15	on	on	ADP
ma-232	150	16	h(d	h(d	PROPN
ma-232	150	17	)	)	PUNCT
ma-232	150	18	be	be	AUX
ma-232	150	19	defined	define	VERB
ma-232	150	20	by	by	ADP
ma-232	150	21	t	t	PROPN
ma-232	150	22	f	f	PROPN
ma-232	150	23	(	(	PUNCT
ma-232	150	24	z	z	NOUN
ma-232	150	25	)	)	PUNCT
ma-232	150	26	=	=	SYM
ma-232	150	27	(	(	PUNCT
ma-232	150	28	1−	1−	NUM
ma-232	150	29	|z	|z	PROPN
ma-232	150	30	|2)t	|2)t	PROPN
ma-232	150	31	∫	∫	PROPN
ma-232	150	32	d	d	X
ma-232	150	33	f	f	PROPN
ma-232	150	34	(	(	PUNCT
ma-232	150	35	w	w	NOUN
ma-232	150	36	)	)	PUNCT
ma-232	150	37	(	(	PUNCT
ma-232	150	38	1−	1−	NUM
ma-232	150	39	zw)2+t+α	zw)2+t+α	NUM
ma-232	150	40	dmα(w	dmα(w	NOUN
ma-232	150	41	)	)	PUNCT
ma-232	150	42	.	.	PUNCT
ma-232	151	1	let	let	VERB
ma-232	151	2	s	s	PRON
ma-232	151	3	be	be	AUX
ma-232	151	4	the	the	DET
ma-232	151	5	corresponding	corresponding	ADJ
ma-232	151	6	integral	integral	ADJ
ma-232	151	7	operator	operator	NOUN
ma-232	151	8	on	on	ADP
ma-232	151	9	h(u	h(u	PROPN
ma-232	151	10	)	)	PUNCT
ma-232	151	11	defined	define	VERB
ma-232	151	12	by	by	ADP
ma-232	151	13	s	s	PRON
ma-232	151	14	:	:	PUNCT
ma-232	151	15	=	=	SYM
ma-232	151	16	cψ−1tcψ	cψ−1tcψ	PROPN
ma-232	151	17	.	.	PUNCT
ma-232	152	1	then	then	ADV
ma-232	152	2	the	the	DET
ma-232	152	3	following	follow	VERB
ma-232	152	4	properties	property	NOUN
ma-232	152	5	hold:(a	hold:(a	NOUN
ma-232	152	6	)	)	PUNCT
ma-232	153	1	s	s	PART
ma-232	153	2	=	=	PUNCT
ma-232	153	3	(	(	PUNCT
ma-232	153	4	α+	α+	PROPN
ma-232	153	5	t	t	NOUN
ma-232	153	6	+	+	CCONJ
ma-232	153	7	1)s2,(b	1)s2,(b	NUM
ma-232	153	8	)	)	PUNCT
ma-232	154	1	s	s	VERB
ma-232	154	2	is	be	AUX
ma-232	154	3	a	a	DET
ma-232	154	4	bounded	bounded	ADJ
ma-232	154	5	embedding	embedding	NOUN
ma-232	154	6	of	of	ADP
ma-232	154	7	b∞(u	b∞(u	ADJ
ma-232	154	8	)	)	PUNCT
ma-232	154	9	into	into	ADP
ma-232	154	10	l∞(u	l∞(u	NOUN
ma-232	154	11	)	)	PUNCT
ma-232	154	12	and(c	and(c	PROPN
ma-232	154	13	)	)	PUNCT
ma-232	154	14	s	s	VERB
ma-232	154	15	is	be	AUX
ma-232	154	16	an	an	DET
ma-232	154	17	embedding	embedding	NOUN
ma-232	154	18	of	of	ADP
ma-232	154	19	b∞,	b∞,	ADP
ma-232	154	20	◦	◦	NOUN
ma-232	154	21	(u	(u	NOUN
ma-232	154	22	)	)	PUNCT
ma-232	154	23	into	into	ADP
ma-232	154	24	c	c	NOUN
ma-232	154	25	◦	◦	NOUN
ma-232	154	26	(u	(u	NOUN
ma-232	154	27	)	)	PUNCT
ma-232	154	28	.	.	PUNCT
ma-232	155	1	proof	proof	NOUN
ma-232	155	2	.	.	PUNCT
ma-232	156	1	from	from	ADP
ma-232	156	2	[	[	X
ma-232	156	3	13	13	NUM
ma-232	156	4	,	,	PUNCT
ma-232	156	5	lemma	lemma	PROPN
ma-232	156	6	5.14	5.14	NUM
ma-232	156	7	]	]	PUNCT
ma-232	156	8	,	,	PUNCT
ma-232	156	9	we	we	PRON
ma-232	156	10	have	have	VERB
ma-232	156	11	,	,	PUNCT
ma-232	156	12	s	s	PART
ma-232	157	1	=	=	NOUN
ma-232	157	2	cψ−1tcψ	cψ−1tcψ	PROPN
ma-232	157	3	=	=	SYM
ma-232	157	4	cψ−1	cψ−1	PROPN
ma-232	157	5	(	(	PUNCT
ma-232	157	6	α+	α+	PROPN
ma-232	157	7	t	t	NOUN
ma-232	157	8	+	+	CCONJ
ma-232	157	9	1)t	1)t	PROPN
ma-232	157	10	2cψ	2cψ	NOUN
ma-232	157	11	=	=	PUNCT
ma-232	157	12	(	(	PUNCT
ma-232	157	13	α+	α+	PROPN
ma-232	157	14	t	t	NOUN
ma-232	157	15	+	+	CCONJ
ma-232	157	16	1)cψ−1	1)cψ−1	PROPN
ma-232	157	17	t	t	NOUN
ma-232	157	18	2cψ	2cψ	NOUN
ma-232	157	19	=	=	PUNCT
ma-232	157	20	(	(	PUNCT
ma-232	157	21	α+	α+	PROPN
ma-232	157	22	t	t	NOUN
ma-232	157	23	+	+	CCONJ
ma-232	157	24	1)s2	1)s2	NUM
ma-232	157	25	,	,	PUNCT
ma-232	157	26	which	which	PRON
ma-232	157	27	proves	prove	VERB
ma-232	157	28	(	(	PUNCT
ma-232	157	29	a).for	a).for	NOUN
ma-232	157	30	(	(	PUNCT
ma-232	157	31	b	b	NOUN
ma-232	157	32	)	)	PUNCT
ma-232	157	33	,	,	PUNCT
ma-232	157	34	we	we	PRON
ma-232	157	35	have	have	VERB
ma-232	157	36	b∞(u	b∞(u	VERB
ma-232	157	37	)	)	PUNCT
ma-232	157	38	cψ−−→	cψ−−→	NOUN
ma-232	157	39	b∞(d	b∞(d	PROPN
ma-232	157	40	)	)	PUNCT
ma-232	157	41	t−→	t−→	NOUN
ma-232	157	42	l∞(d	l∞(d	NOUN
ma-232	157	43	)	)	PUNCT
ma-232	157	44	cψ−1	cψ−1	PROPN
ma-232	157	45	−−−→	−−−→	VERB
ma-232	157	46	l∞(u	l∞(u	NOUN
ma-232	157	47	)	)	PUNCT
ma-232	157	48	.	.	PUNCT
ma-232	158	1	now	now	ADV
ma-232	158	2	,	,	PUNCT
ma-232	158	3	cψ	cψ	PROPN
ma-232	158	4	is	be	AUX
ma-232	158	5	an	an	DET
ma-232	158	6	isometry	isometry	NOUN
ma-232	158	7	of	of	ADP
ma-232	158	8	b∞(u	b∞(u	NOUN
ma-232	158	9	)	)	PUNCT
ma-232	158	10	onto	onto	ADP
ma-232	158	11	b∞(d	b∞(d	NOUN
ma-232	158	12	)	)	PUNCT
ma-232	158	13	up	up	ADP
ma-232	158	14	to	to	PART
ma-232	158	15	constant	constant	ADJ
ma-232	158	16	,	,	PUNCT
ma-232	158	17	t	t	PROPN
ma-232	158	18	is	be	AUX
ma-232	158	19	a	a	DET
ma-232	158	20	bounded	bounded	ADJ
ma-232	158	21	embedding	embedding	NOUN
ma-232	158	22	of	of	ADP
ma-232	158	23	b∞(d	b∞(d	NOUN
ma-232	158	24	)	)	PUNCT
ma-232	158	25	into	into	ADP
ma-232	158	26	l∞(d	l∞(d	PROPN
ma-232	158	27	)	)	PUNCT
ma-232	159	1	[	[	X
ma-232	159	2	13	13	NUM
ma-232	159	3	,	,	PUNCT
ma-232	159	4	lemma	lemma	PROPN
ma-232	159	5	5.14	5.14	NUM
ma-232	159	6	]	]	PUNCT
ma-232	159	7	,	,	PUNCT
ma-232	159	8	cψ−1	cψ−1	PROPN
ma-232	159	9	is	be	AUX
ma-232	159	10	also	also	ADV
ma-232	159	11	an	an	DET
ma-232	159	12	isometry	isometry	NOUN
ma-232	159	13	of	of	ADP
ma-232	159	14	l∞(d	l∞(d	NOUN
ma-232	159	15	)	)	PUNCT
ma-232	159	16	onto	onto	ADP
ma-232	159	17	l∞(u	l∞(u	NOUN
ma-232	159	18	)	)	PUNCT
ma-232	159	19	,	,	PUNCT
ma-232	159	20	it	it	PRON
ma-232	159	21	thereforefollows	thereforefollow	VERB
ma-232	159	22	that	that	SCONJ
ma-232	159	23	s	s	VERB
ma-232	159	24	=	=	SYM
ma-232	159	25	cψ−1tcψ	cψ−1tcψ	PROPN
ma-232	159	26	is	be	AUX
ma-232	159	27	a	a	DET
ma-232	159	28	bounded	bounded	ADJ
ma-232	159	29	embedding	embedding	NOUN
ma-232	159	30	of	of	ADP
ma-232	159	31	b∞(u	b∞(u	ADJ
ma-232	159	32	)	)	PUNCT
ma-232	159	33	into	into	ADP
ma-232	159	34	l∞(u).for	l∞(u).for	ADP
ma-232	159	35	(	(	PUNCT
ma-232	159	36	c	c	NOUN
ma-232	159	37	)	)	PUNCT
ma-232	159	38	,	,	PUNCT
ma-232	159	39	we	we	PRON
ma-232	159	40	have	have	VERB
ma-232	159	41	b∞,	b∞,	ADP
ma-232	159	42	◦	◦	NOUN
ma-232	159	43	(u	(u	NOUN
ma-232	159	44	)	)	PUNCT
ma-232	159	45	cψ−−→	cψ−−→	NOUN
ma-232	159	46	b∞,	b∞,	ADP
ma-232	159	47	◦	◦	NOUN
ma-232	159	48	(d	(d	NOUN
ma-232	159	49	)	)	PUNCT
ma-232	159	50	t−→	t−→	NOUN
ma-232	159	51	c	c	NOUN
ma-232	159	52	◦	◦	NOUN
ma-232	159	53	(d	(d	NOUN
ma-232	159	54	)	)	PUNCT
ma-232	159	55	cψ−1	cψ−1	NOUN
ma-232	159	56	−−−→	−−−→	PROPN
ma-232	159	57	c	c	NOUN
ma-232	159	58	◦	◦	NOUN
ma-232	159	59	(u	(u	NOUN
ma-232	159	60	)	)	PUNCT
ma-232	159	61	.	.	PUNCT
ma-232	160	1	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	160	2	eur	eur	PROPN
ma-232	160	3	.	.	PUNCT
ma-232	161	1	j.	j.	PROPN
ma-232	161	2	math	math	PROPN
ma-232	161	3	.	.	PUNCT
ma-232	162	1	anal	anal	PROPN
ma-232	162	2	.	.	PUNCT
ma-232	163	1	10.28924	10.28924	NUM
ma-232	163	2	/	/	SYM
ma-232	163	3	ada	ada	PROPN
ma-232	163	4	/	/	SYM
ma-232	163	5	ma.4.14	ma.4.14	PROPN
ma-232	163	6	7	7	NUM
ma-232	163	7	cψ	cψ	NOUN
ma-232	163	8	is	be	AUX
ma-232	163	9	a	a	DET
ma-232	163	10	bijection	bijection	NOUN
ma-232	163	11	of	of	ADP
ma-232	163	12	b∞,	b∞,	ADP
ma-232	163	13	◦	◦	NOUN
ma-232	163	14	(u	(u	NOUN
ma-232	163	15	)	)	PUNCT
ma-232	163	16	into	into	ADP
ma-232	163	17	b∞,	b∞,	ADP
ma-232	163	18	◦	◦	NOUN
ma-232	163	19	(d	(d	NOUN
ma-232	163	20	)	)	PUNCT
ma-232	163	21	,	,	PUNCT
ma-232	163	22	t	t	PROPN
ma-232	163	23	is	be	AUX
ma-232	163	24	an	an	DET
ma-232	163	25	embedding	embedding	NOUN
ma-232	163	26	of	of	ADP
ma-232	163	27	b∞,	b∞,	ADP
ma-232	163	28	◦	◦	NOUN
ma-232	163	29	(d	(d	NOUN
ma-232	163	30	)	)	PUNCT
ma-232	163	31	into	into	ADP
ma-232	163	32	c	c	NOUN
ma-232	163	33	◦	◦	NOUN
ma-232	163	34	(d	(d	NOUN
ma-232	163	35	)	)	PUNCT
ma-232	164	1	[	[	X
ma-232	164	2	13	13	NUM
ma-232	164	3	,	,	PUNCT
ma-232	164	4	lemma5.14	lemma5.14	NOUN
ma-232	164	5	]	]	X
ma-232	164	6	,	,	PUNCT
ma-232	164	7	while	while	SCONJ
ma-232	164	8	on	on	ADP
ma-232	164	9	the	the	DET
ma-232	164	10	other	other	ADJ
ma-232	164	11	hand	hand	NOUN
ma-232	164	12	,	,	PUNCT
ma-232	164	13	cψ−1	cψ−1	PROPN
ma-232	164	14	is	be	AUX
ma-232	164	15	also	also	ADV
ma-232	164	16	a	a	DET
ma-232	164	17	bijection	bijection	NOUN
ma-232	164	18	of	of	ADP
ma-232	164	19	c	c	NOUN
ma-232	164	20	◦	◦	NOUN
ma-232	164	21	(d	(d	NOUN
ma-232	164	22	)	)	PUNCT
ma-232	164	23	into	into	ADP
ma-232	164	24	c	c	NOUN
ma-232	164	25	◦	◦	NOUN
ma-232	164	26	(u	(u	NOUN
ma-232	164	27	)	)	PUNCT
ma-232	164	28	.	.	PUNCT
ma-232	165	1	therefore	therefore	ADV
ma-232	165	2	s	s	VERB
ma-232	165	3	=	=	SYM
ma-232	165	4	cψ−1tcψ	cψ−1tcψ	PROPN
ma-232	165	5	is	be	AUX
ma-232	165	6	an	an	DET
ma-232	165	7	embedding	embedding	NOUN
ma-232	165	8	of	of	ADP
ma-232	165	9	b∞,	b∞,	ADP
ma-232	165	10	◦	◦	NOUN
ma-232	165	11	(u	(u	NOUN
ma-232	165	12	)	)	PUNCT
ma-232	165	13	into	into	ADP
ma-232	165	14	c	c	NOUN
ma-232	165	15	◦	◦	NOUN
ma-232	165	16	(u	(u	NOUN
ma-232	165	17	)	)	PUNCT
ma-232	165	18	,	,	PUNCT
ma-232	165	19	which	which	PRON
ma-232	165	20	completes	complete	VERB
ma-232	165	21	the	the	DET
ma-232	165	22	proof	proof	NOUN
ma-232	165	23	.	.	PUNCT
ma-232	166	1	�	�	PROPN
ma-232	166	2	we	we	PRON
ma-232	166	3	now	now	ADV
ma-232	166	4	establish	establish	VERB
ma-232	166	5	the	the	DET
ma-232	166	6	predual	predual	ADJ
ma-232	166	7	space	space	NOUN
ma-232	166	8	of	of	ADP
ma-232	166	9	l1	l1	PROPN
ma-232	166	10	a(u	a(u	PROPN
ma-232	166	11	,	,	PUNCT
ma-232	166	12	µα	µα	ADP
ma-232	166	13	)	)	PUNCT
ma-232	166	14	as	as	SCONJ
ma-232	166	15	we	we	PRON
ma-232	166	16	give	give	VERB
ma-232	166	17	in	in	ADP
ma-232	166	18	the	the	DET
ma-232	166	19	following	following	NOUN
ma-232	166	20	theorem	theorem	NOUN
ma-232	166	21	:	:	PUNCT
ma-232	166	22	theorem	theorem	VERB
ma-232	166	23	2.5	2.5	NUM
ma-232	166	24	.	.	PUNCT
ma-232	167	1	for	for	ADP
ma-232	167	2	any	any	DET
ma-232	167	3	α	α	NOUN
ma-232	167	4	>	>	X
ma-232	167	5	−1	−1	NOUN
ma-232	167	6	,	,	PUNCT
ma-232	167	7	we	we	PRON
ma-232	167	8	have	have	VERB
ma-232	167	9	;	;	PUNCT
ma-232	167	10	(	(	PUNCT
ma-232	167	11	b∞,	b∞,	VERB
ma-232	167	12	◦	◦	NOUN
ma-232	167	13	(u	(u	ADJ
ma-232	167	14	,	,	PUNCT
ma-232	167	15	i))∗	i))∗	PROPN
ma-232	167	16	≈	≈	PROPN
ma-232	167	17	l1	l1	PROPN
ma-232	167	18	a(u	a(u	PROPN
ma-232	167	19	,	,	PUNCT
ma-232	167	20	µα	µα	ADP
ma-232	167	21	)	)	PUNCT
ma-232	167	22	,	,	PUNCT
ma-232	167	23	under	under	ADP
ma-232	167	24	the	the	DET
ma-232	167	25	pairing	pair	VERB
ma-232	167	26	〈	〈	PROPN
ma-232	167	27	g	g	PROPN
ma-232	167	28	,	,	PUNCT
ma-232	167	29	f	f	PROPN
ma-232	167	30	〉	〉	PROPN
ma-232	167	31	=	=	SYM
ma-232	167	32	∫	∫	PROPN
ma-232	167	33	u	u	PROPN
ma-232	167	34	g(ω)f	g(ω)f	PROPN
ma-232	167	35	(	(	PUNCT
ma-232	167	36	ω)dµα(ω	ω)dµα(ω	NUM
ma-232	167	37	)	)	PUNCT
ma-232	167	38	,	,	PUNCT
ma-232	167	39	where	where	SCONJ
ma-232	167	40	g	g	PROPN
ma-232	167	41	∈	∈	PROPN
ma-232	167	42	b∞,	b∞,	ADP
ma-232	167	43	◦	◦	NOUN
ma-232	167	44	(u	(u	NOUN
ma-232	167	45	,	,	PUNCT
ma-232	167	46	i	i	PROPN
ma-232	167	47	)	)	PUNCT
ma-232	167	48	and	and	CCONJ
ma-232	167	49	f	f	PROPN
ma-232	167	50	∈	∈	PROPN
ma-232	167	51	l1	l1	PROPN
ma-232	167	52	a(u	a(u	PROPN
ma-232	167	53	,	,	PUNCT
ma-232	167	54	µα	µα	NOUN
ma-232	167	55	)	)	PUNCT
ma-232	167	56	.	.	PUNCT
ma-232	168	1	here	here	ADV
ma-232	168	2	,	,	PUNCT
ma-232	168	3	b∞,	b∞,	VERB
ma-232	168	4	◦	◦	NOUN
ma-232	168	5	(u	(u	ADJ
ma-232	168	6	,	,	PUNCT
ma-232	168	7	i	i	PRON
ma-232	168	8	)	)	PUNCT
ma-232	168	9	is	be	AUX
ma-232	168	10	equipped	equip	VERB
ma-232	168	11	with	with	ADP
ma-232	168	12	the	the	DET
ma-232	168	13	same	same	ADJ
ma-232	168	14	norm	norm	NOUN
ma-232	168	15	as	as	ADP
ma-232	168	16	b∞(u	b∞(u	PROPN
ma-232	168	17	,	,	PUNCT
ma-232	168	18	i	i	NOUN
ma-232	168	19	)	)	PUNCT
ma-232	168	20	.	.	PUNCT
ma-232	169	1	proof	proof	NOUN
ma-232	169	2	.	.	PUNCT
ma-232	170	1	if	if	SCONJ
ma-232	170	2	f	f	PROPN
ma-232	170	3	∈	∈	PROPN
ma-232	170	4	l1	l1	PROPN
ma-232	170	5	a(u	a(u	PROPN
ma-232	170	6	,	,	PUNCT
ma-232	170	7	µα	µα	ADP
ma-232	170	8	)	)	PUNCT
ma-232	170	9	,	,	PUNCT
ma-232	170	10	then	then	ADV
ma-232	170	11	by	by	ADP
ma-232	170	12	theorem	theorem	NOUN
ma-232	170	13	2.1	2.1	NUM
ma-232	170	14	above	above	ADV
ma-232	170	15	,	,	PUNCT
ma-232	170	16	g	g	PROPN
ma-232	170	17	7−→	7−→	PROPN
ma-232	170	18	∫	∫	NOUN
ma-232	170	19	u	u	PROPN
ma-232	170	20	g(ω)f	g(ω)f	PROPN
ma-232	170	21	(	(	PUNCT
ma-232	170	22	ω)dµα(ω	ω)dµα(ω	NOUN
ma-232	170	23	)	)	PUNCT
ma-232	170	24	defines	define	VERB
ma-232	170	25	a	a	DET
ma-232	170	26	boundedlinear	boundedlinear	NOUN
ma-232	170	27	functional	functional	ADJ
ma-232	170	28	on	on	ADP
ma-232	170	29	b∞,	b∞,	ADP
ma-232	170	30	◦	◦	NOUN
ma-232	170	31	(u	(u	ADJ
ma-232	170	32	,	,	PUNCT
ma-232	170	33	i	i	NOUN
ma-232	170	34	)	)	PUNCT
ma-232	170	35	.	.	PUNCT
ma-232	171	1	conversely	conversely	ADV
ma-232	171	2	,	,	PUNCT
ma-232	171	3	if	if	SCONJ
ma-232	171	4	f	f	PROPN
ma-232	171	5	is	be	AUX
ma-232	171	6	a	a	DET
ma-232	171	7	bounded	bounded	ADJ
ma-232	171	8	linear	linear	ADJ
ma-232	171	9	functional	functional	NOUN
ma-232	171	10	on	on	ADP
ma-232	171	11	b∞,	b∞,	ADP
ma-232	171	12	◦	◦	NOUN
ma-232	171	13	(u	(u	ADJ
ma-232	171	14	,	,	PUNCT
ma-232	171	15	i	i	PROPN
ma-232	171	16	)	)	PUNCT
ma-232	171	17	,	,	PUNCT
ma-232	171	18	wewant	wewant	ADJ
ma-232	171	19	to	to	PART
ma-232	171	20	show	show	VERB
ma-232	171	21	that	that	SCONJ
ma-232	171	22	there	there	PRON
ma-232	171	23	exists	exist	VERB
ma-232	171	24	a	a	DET
ma-232	171	25	function	function	NOUN
ma-232	171	26	f	f	PROPN
ma-232	171	27	∈	∈	PROPN
ma-232	171	28	l1	l1	PROPN
ma-232	171	29	a(u	a(u	PROPN
ma-232	171	30	,	,	PUNCT
ma-232	171	31	µα	µα	ADP
ma-232	171	32	)	)	PUNCT
ma-232	172	1	such	such	ADJ
ma-232	172	2	that	that	SCONJ
ma-232	172	3	f	f	PROPN
ma-232	172	4	(	(	PUNCT
ma-232	172	5	g	g	NOUN
ma-232	172	6	)	)	PUNCT
ma-232	172	7	=	=	SYM
ma-232	172	8	∫	∫	PROPN
ma-232	172	9	u	u	PROPN
ma-232	172	10	g(ω)f	g(ω)f	PROPN
ma-232	172	11	(	(	PUNCT
ma-232	172	12	ω)dµα(ω	ω)dµα(ω	X
ma-232	172	13	)	)	PUNCT
ma-232	172	14	for	for	ADP
ma-232	172	15	g	g	NOUN
ma-232	172	16	in	in	ADP
ma-232	172	17	a	a	DET
ma-232	172	18	dense	dense	ADJ
ma-232	172	19	set	set	NOUN
ma-232	172	20	of	of	ADP
ma-232	172	21	b∞,	b∞,	ADP
ma-232	172	22	◦	◦	NOUN
ma-232	172	23	(u	(u	ADJ
ma-232	172	24	,	,	PUNCT
ma-232	172	25	i).now	i).now	PROPN
ma-232	172	26	we	we	PRON
ma-232	172	27	fix	fix	VERB
ma-232	172	28	any	any	DET
ma-232	172	29	positive	positive	ADJ
ma-232	172	30	parameter	parameter	NOUN
ma-232	172	31	t	t	PROPN
ma-232	172	32	and	and	CCONJ
ma-232	172	33	consider	consider	VERB
ma-232	172	34	the	the	DET
ma-232	172	35	embedding	embed	VERB
ma-232	172	36	s	s	PRON
ma-232	172	37	of	of	ADP
ma-232	172	38	b∞,	b∞,	ADP
ma-232	172	39	◦	◦	NOUN
ma-232	172	40	(u	(u	NOUN
ma-232	172	41	,	,	PUNCT
ma-232	172	42	i	i	NOUN
ma-232	172	43	)	)	PUNCT
ma-232	172	44	into	into	ADP
ma-232	172	45	c	c	NOUN
ma-232	172	46	◦	◦	NOUN
ma-232	172	47	(u)as	(u)as	PUNCT
ma-232	172	48	given	give	VERB
ma-232	172	49	by	by	ADP
ma-232	172	50	proposition	proposition	NOUN
ma-232	172	51	2.4	2.4	NUM
ma-232	172	52	.	.	PUNCT
ma-232	173	1	the	the	DET
ma-232	173	2	space	space	NOUN
ma-232	173	3	x	x	NOUN
ma-232	173	4	=	=	SYM
ma-232	173	5	s(b∞,	s(b∞,	NOUN
ma-232	173	6	◦	◦	NOUN
ma-232	173	7	(u	(u	NOUN
ma-232	173	8	,	,	PUNCT
ma-232	173	9	i	i	NOUN
ma-232	173	10	)	)	PUNCT
ma-232	173	11	)	)	PUNCT
ma-232	173	12	is	be	AUX
ma-232	173	13	a	a	DET
ma-232	173	14	closed	closed	ADJ
ma-232	173	15	subspace	subspace	NOUN
ma-232	173	16	of	of	ADP
ma-232	173	17	c	c	NOUN
ma-232	173	18	◦	◦	NOUN
ma-232	173	19	(u	(u	NOUN
ma-232	173	20	)	)	PUNCT
ma-232	173	21	and	and	CCONJ
ma-232	173	22	f	f	X
ma-232	173	23	◦	◦	NOUN
ma-232	173	24	s−1	s−1	PROPN
ma-232	173	25	:	:	PUNCT
ma-232	173	26	x	x	X
ma-232	173	27	→	→	SYM
ma-232	173	28	c	c	NOUN
ma-232	173	29	is	be	AUX
ma-232	173	30	a	a	DET
ma-232	173	31	bounded	bounded	ADJ
ma-232	173	32	linear	linear	ADJ
ma-232	173	33	functional	functional	NOUN
ma-232	173	34	on	on	ADP
ma-232	173	35	x	x	PUNCT
ma-232	173	36	since	since	SCONJ
ma-232	173	37	f	f	PROPN
ma-232	173	38	and	and	CCONJ
ma-232	173	39	s−1	s−1	PROPN
ma-232	173	40	are	be	AUX
ma-232	173	41	both	both	PRON
ma-232	173	42	bounded	bound	VERB
ma-232	173	43	.	.	PUNCT
ma-232	174	1	bythe	bythe	ADP
ma-232	174	2	hahn	hahn	NOUN
ma-232	174	3	-	-	PUNCT
ma-232	174	4	banach	banach	NOUN
ma-232	174	5	extension	extension	NOUN
ma-232	174	6	theorem	theorem	NOUN
ma-232	174	7	,	,	PUNCT
ma-232	174	8	f	f	SYM
ma-232	174	9	◦	◦	NOUN
ma-232	174	10	s−1	s−1	PROPN
ma-232	174	11	extends	extend	VERB
ma-232	174	12	to	to	ADP
ma-232	174	13	a	a	DET
ma-232	174	14	bounded	bounded	ADJ
ma-232	174	15	linear	linear	ADJ
ma-232	174	16	functional	functional	NOUN
ma-232	174	17	on	on	ADP
ma-232	174	18	c	c	PART
ma-232	174	19	◦	◦	NOUN
ma-232	174	20	(u).by	(u).by	PUNCT
ma-232	174	21	the	the	DET
ma-232	174	22	riesz	riesz	PROPN
ma-232	174	23	representation	representation	NOUN
ma-232	174	24	theorem	theorem	NOUN
ma-232	174	25	,	,	PUNCT
ma-232	174	26	there	there	PRON
ma-232	174	27	exists	exist	VERB
ma-232	174	28	a	a	DET
ma-232	174	29	finite	finite	NOUN
ma-232	174	30	weighted	weight	VERB
ma-232	174	31	measure	measure	NOUN
ma-232	174	32	µα	µα	ADP
ma-232	174	33	on	on	ADP
ma-232	174	34	u	u	PRON
ma-232	174	35	such	such	ADJ
ma-232	174	36	that	that	DET
ma-232	174	37	‖µα‖	‖µα‖	PROPN
ma-232	175	1	=	=	SYM
ma-232	175	2	‖f	‖f	ADP
ma-232	175	3	◦	◦	NOUN
ma-232	175	4	s−1‖	s−1‖	PROPN
ma-232	175	5	and	and	CCONJ
ma-232	175	6	f	f	PROPN
ma-232	175	7	◦	◦	PROPN
ma-232	175	8	s−1(h	s−1(h	PROPN
ma-232	175	9	)	)	PUNCT
ma-232	175	10	=	=	SYM
ma-232	175	11	∫	∫	PROPN
ma-232	175	12	u	u	NOUN
ma-232	175	13	h(z)dµα(z	h(z)dµα(z	PROPN
ma-232	175	14	)	)	PUNCT
ma-232	175	15	,	,	PUNCT
ma-232	175	16	h	h	NOUN
ma-232	175	17	∈	∈	PROPN
ma-232	175	18	c	c	X
ma-232	175	19	◦	◦	NOUN
ma-232	175	20	(u	(u	NOUN
ma-232	175	21	)	)	PUNCT
ma-232	175	22	.	.	PUNCT
ma-232	176	1	in	in	ADP
ma-232	176	2	particular	particular	ADJ
ma-232	176	3	,	,	PUNCT
ma-232	176	4	if	if	SCONJ
ma-232	176	5	g	g	PROPN
ma-232	176	6	is	be	AUX
ma-232	176	7	a	a	DET
ma-232	176	8	polynomial(polynomials	polynomial(polynomial	NOUN
ma-232	176	9	are	be	AUX
ma-232	176	10	dense	dense	ADJ
ma-232	176	11	in	in	ADP
ma-232	176	12	b∞,	b∞,	ADP
ma-232	176	13	◦	◦	NOUN
ma-232	176	14	(u	(u	NOUN
ma-232	176	15	,	,	PUNCT
ma-232	176	16	i	i	NOUN
ma-232	176	17	)	)	PUNCT
ma-232	176	18	)	)	PUNCT
ma-232	176	19	,	,	PUNCT
ma-232	176	20	then	then	ADV
ma-232	176	21	f	f	PROPN
ma-232	176	22	(	(	PUNCT
ma-232	176	23	g	g	NOUN
ma-232	176	24	)	)	PUNCT
ma-232	176	25	=	=	SYM
ma-232	177	1	f	f	X
ma-232	177	2	◦	◦	ADJ
ma-232	177	3	s−1	s−1	PROPN
ma-232	177	4	◦	◦	NOUN
ma-232	177	5	s(g	s(g	PROPN
ma-232	177	6	)	)	PUNCT
ma-232	178	1	=	=	SYM
ma-232	178	2	∫	∫	PROPN
ma-232	178	3	u	u	NOUN
ma-232	178	4	sg(z)dµα(z	sg(z)dµα(z	PROPN
ma-232	178	5	)	)	PUNCT
ma-232	178	6	.	.	PUNCT
ma-232	179	1	by	by	ADP
ma-232	179	2	fubini’stheorem	fubini’stheorem	VERB
ma-232	179	3	,	,	PUNCT
ma-232	179	4	we	we	PRON
ma-232	179	5	have	have	VERB
ma-232	179	6	f	f	PROPN
ma-232	179	7	(	(	PUNCT
ma-232	179	8	g	g	NOUN
ma-232	179	9	)	)	PUNCT
ma-232	179	10	=	=	SYM
ma-232	180	1	∫	∫	PROPN
ma-232	180	2	u	u	PROPN
ma-232	180	3	g(ω)f	g(ω)f	PROPN
ma-232	180	4	(	(	PUNCT
ma-232	180	5	ω)dµα(ω	ω)dµα(ω	NUM
ma-232	180	6	)	)	PUNCT
ma-232	180	7	,	,	PUNCT
ma-232	180	8	where	where	SCONJ
ma-232	180	9	f	f	PROPN
ma-232	180	10	=	=	SYM
ma-232	180	11	cψ−1tcψ	cψ−1tcψ	PROPN
ma-232	180	12	which	which	PRON
ma-232	180	13	is	be	AUX
ma-232	180	14	bounded	bound	VERB
ma-232	180	15	since	since	SCONJ
ma-232	180	16	t	t	PROPN
ma-232	180	17	isbounded	isbounde	VERB
ma-232	180	18	.	.	PUNCT
ma-232	181	1	�	�	PROPN
ma-232	181	2	3	3	NUM
ma-232	181	3	.	.	PUNCT
ma-232	182	1	groups	group	NOUN
ma-232	182	2	of	of	ADP
ma-232	182	3	weighted	weight	VERB
ma-232	182	4	composition	composition	NOUN
ma-232	182	5	operators	operator	NOUN
ma-232	182	6	on	on	ADP
ma-232	182	7	predual	predual	ADJ
ma-232	182	8	of	of	ADP
ma-232	182	9	l1	l1	PROPN
ma-232	182	10	a(u	a(u	PROPN
ma-232	182	11	,	,	PUNCT
ma-232	182	12	µα	µα	ADP
ma-232	182	13	)	)	PUNCT
ma-232	182	14	as	as	SCONJ
ma-232	182	15	remarked	remark	VERB
ma-232	182	16	in	in	ADP
ma-232	182	17	the	the	DET
ma-232	182	18	section	section	NOUN
ma-232	182	19	1	1	NUM
ma-232	182	20	,	,	PUNCT
ma-232	182	21	the	the	DET
ma-232	182	22	automorphisms	automorphism	NOUN
ma-232	182	23	of	of	ADP
ma-232	182	24	the	the	DET
ma-232	182	25	upper	upper	ADJ
ma-232	182	26	half	half	ADJ
ma-232	182	27	plane	plane	NOUN
ma-232	182	28	u	u	NOUN
ma-232	182	29	were	be	AUX
ma-232	182	30	identified	identify	VERB
ma-232	182	31	andclassified	andclassifie	VERB
ma-232	182	32	into	into	ADP
ma-232	182	33	three	three	NUM
ma-232	182	34	distinct	distinct	ADJ
ma-232	182	35	groups	group	NOUN
ma-232	182	36	according	accord	VERB
ma-232	182	37	to	to	ADP
ma-232	182	38	the	the	DET
ma-232	182	39	location	location	NOUN
ma-232	182	40	of	of	ADP
ma-232	182	41	their	their	PRON
ma-232	182	42	fixed	fix	VERB
ma-232	182	43	points	point	NOUN
ma-232	182	44	in	in	ADP
ma-232	182	45	[	[	X
ma-232	182	46	3	3	NUM
ma-232	182	47	]	]	PUNCT
ma-232	182	48	,	,	PUNCT
ma-232	182	49	namely	namely	ADV
ma-232	182	50	:	:	PUNCT
ma-232	182	51	the	the	DET
ma-232	182	52	scaling	scaling	NOUN
ma-232	182	53	,	,	PUNCT
ma-232	182	54	the	the	DET
ma-232	182	55	translation	translation	NOUN
ma-232	182	56	and	and	CCONJ
ma-232	182	57	the	the	DET
ma-232	182	58	rotation	rotation	NOUN
ma-232	182	59	groups	group	NOUN
ma-232	182	60	.	.	PUNCT
ma-232	183	1	since	since	SCONJ
ma-232	183	2	the	the	DET
ma-232	183	3	corresponding	correspond	VERB
ma-232	183	4	groups	group	NOUN
ma-232	183	5	of	of	ADP
ma-232	183	6	compositionoperators	compositionoperator	NOUN
ma-232	183	7	for	for	ADP
ma-232	183	8	the	the	DET
ma-232	183	9	rotation	rotation	NOUN
ma-232	183	10	group	group	NOUN
ma-232	183	11	are	be	AUX
ma-232	183	12	defined	define	VERB
ma-232	183	13	on	on	ADP
ma-232	183	14	the	the	DET
ma-232	183	15	analytic	analytic	ADJ
ma-232	183	16	spaces	space	NOUN
ma-232	183	17	of	of	ADP
ma-232	183	18	the	the	DET
ma-232	183	19	unit	unit	NOUN
ma-232	183	20	disk	disk	NOUN
ma-232	183	21	,	,	PUNCT
ma-232	183	22	we	we	PRON
ma-232	183	23	shall	shall	AUX
ma-232	183	24	onlyconsider	onlyconsider	NOUN
ma-232	183	25	groups	group	NOUN
ma-232	183	26	of	of	ADP
ma-232	183	27	composition	composition	NOUN
ma-232	183	28	operators	operator	NOUN
ma-232	183	29	associated	associate	VERB
ma-232	183	30	with	with	ADP
ma-232	183	31	the	the	DET
ma-232	183	32	scaling	scaling	NOUN
ma-232	183	33	and	and	CCONJ
ma-232	183	34	the	the	DET
ma-232	183	35	translation	translation	NOUN
ma-232	183	36	groups	group	NOUN
ma-232	183	37	inthis	inthis	PROPN
ma-232	183	38	paper	paper	NOUN
ma-232	183	39	.	.	PUNCT
ma-232	184	1	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	184	2	eur	eur	PROPN
ma-232	184	3	.	.	PUNCT
ma-232	185	1	j.	j.	PROPN
ma-232	185	2	math	math	PROPN
ma-232	185	3	.	.	PUNCT
ma-232	186	1	anal	anal	PROPN
ma-232	186	2	.	.	PUNCT
ma-232	187	1	10.28924	10.28924	NUM
ma-232	187	2	/	/	SYM
ma-232	187	3	ada	ada	PROPN
ma-232	187	4	/	/	SYM
ma-232	187	5	ma.4.14	ma.4.14	PROPN
ma-232	187	6	83.1	83.1	NUM
ma-232	187	7	.	.	PUNCT
ma-232	188	1	scaling	scale	VERB
ma-232	188	2	group	group	NOUN
ma-232	188	3	.	.	PUNCT
ma-232	189	1	the	the	DET
ma-232	189	2	automorphisms	automorphism	NOUN
ma-232	189	3	of	of	ADP
ma-232	189	4	this	this	DET
ma-232	189	5	group	group	NOUN
ma-232	189	6	are	be	AUX
ma-232	189	7	of	of	ADP
ma-232	189	8	the	the	DET
ma-232	189	9	form	form	NOUN
ma-232	189	10	ϕt(z	ϕt(z	NUM
ma-232	189	11	)	)	PUNCT
ma-232	190	1	=	=	SYM
ma-232	191	1	k	k	PROPN
ma-232	191	2	tz	tz	NOUN
ma-232	191	3	,	,	PUNCT
ma-232	191	4	where	where	SCONJ
ma-232	191	5	z	z	PROPN
ma-232	191	6	∈	∈	PROPN
ma-232	191	7	uand	uand	NOUN
ma-232	191	8	k	k	PROPN
ma-232	191	9	,	,	PUNCT
ma-232	191	10	t	t	PROPN
ma-232	191	11	∈	∈	PROPN
ma-232	191	12	r	r	NOUN
ma-232	191	13	with	with	ADP
ma-232	191	14	k	k	PROPN
ma-232	191	15	6=	6=	PROPN
ma-232	191	16	0	0	NUM
ma-232	191	17	.	.	PUNCT
ma-232	192	1	as	as	SCONJ
ma-232	192	2	noted	note	VERB
ma-232	192	3	in	in	ADP
ma-232	192	4	[	[	X
ma-232	192	5	3	3	NUM
ma-232	192	6	]	]	PUNCT
ma-232	192	7	and	and	CCONJ
ma-232	192	8	without	without	ADP
ma-232	192	9	loss	loss	NOUN
ma-232	192	10	of	of	ADP
ma-232	192	11	generality	generality	NOUN
ma-232	192	12	,	,	PUNCT
ma-232	192	13	we	we	PRON
ma-232	192	14	consider	consider	VERB
ma-232	192	15	the	the	DET
ma-232	192	16	analyticself	analyticself	NOUN
ma-232	192	17	maps	map	VERB
ma-232	192	18	ϕt	ϕt	ADV
ma-232	192	19	:	:	PUNCT
ma-232	192	20	u	u	X
ma-232	192	21	→	→	SYM
ma-232	192	22	u	u	NOUN
ma-232	192	23	of	of	ADP
ma-232	192	24	the	the	DET
ma-232	192	25	form	form	NOUN
ma-232	192	26	ϕt(z	ϕt(z	NUM
ma-232	192	27	)	)	PUNCT
ma-232	192	28	=	=	PUNCT
ma-232	192	29	e−tz	e−tz	VERB
ma-232	192	30	for	for	ADP
ma-232	192	31	z	z	PROPN
ma-232	192	32	∈	∈	PROPN
ma-232	192	33	u.	u.	VERB
ma-232	192	34	the	the	DET
ma-232	192	35	corresponding	corresponding	ADJ
ma-232	192	36	group	group	NOUN
ma-232	192	37	of	of	ADP
ma-232	192	38	weightedcomposition	weightedcomposition	NOUN
ma-232	192	39	operators	operator	NOUN
ma-232	192	40	on	on	ADP
ma-232	192	41	lpa(u	lpa(u	PROPN
ma-232	192	42	,	,	PUNCT
ma-232	192	43	µα	µα	NOUN
ma-232	192	44	)	)	PUNCT
ma-232	192	45	is	be	AUX
ma-232	192	46	given	give	VERB
ma-232	192	47	by	by	ADP
ma-232	192	48	tt	tt	PROPN
ma-232	192	49	f	f	PROPN
ma-232	192	50	(	(	PUNCT
ma-232	192	51	z	z	NOUN
ma-232	192	52	)	)	PUNCT
ma-232	192	53	=	=	NOUN
ma-232	192	54	e−tγf	e−tγf	NOUN
ma-232	192	55	(	(	PUNCT
ma-232	192	56	e−tz	e−tz	NUM
ma-232	192	57	)	)	PUNCT
ma-232	192	58	,	,	PUNCT
ma-232	192	59	for	for	ADP
ma-232	192	60	all	all	DET
ma-232	192	61	f	f	PROPN
ma-232	192	62	∈	∈	PROPN
ma-232	192	63	lpa(u	lpa(u	PROPN
ma-232	192	64	,	,	PUNCT
ma-232	192	65	µα),where	µα),where	ADV
ma-232	192	66	γ	γ	X
ma-232	192	67	=	=	NUM
ma-232	192	68	α+2	α+2	NUM
ma-232	192	69	p	p	NOUN
ma-232	192	70	and	and	CCONJ
ma-232	192	71	1	1	NUM
ma-232	192	72	≤	≤	NOUN
ma-232	193	1	p	p	NOUN
ma-232	193	2	<	<	X
ma-232	193	3	∞.	∞.	PROPN
ma-232	193	4	for	for	ADP
ma-232	193	5	p	p	NOUN
ma-232	193	6	=	=	PROPN
ma-232	193	7	1	1	NUM
ma-232	193	8	,	,	PUNCT
ma-232	193	9	(	(	PUNCT
ma-232	193	10	tt)t≥0	tt)t≥0	NOUN
ma-232	193	11	is	be	AUX
ma-232	193	12	defined	define	VERB
ma-232	193	13	on	on	ADP
ma-232	193	14	l1	l1	PROPN
ma-232	193	15	a(u	a(u	PROPN
ma-232	193	16	,	,	PUNCT
ma-232	193	17	µα	µα	ADP
ma-232	193	18	)	)	PUNCT
ma-232	193	19	with	with	ADP
ma-232	193	20	γ	γ	X
ma-232	193	21	=	=	SYM
ma-232	193	22	α+	α+	X
ma-232	193	23	2.following	2.following	NUM
ma-232	193	24	theorem	theorem	VERB
ma-232	193	25	2.5	2.5	NUM
ma-232	193	26	,	,	PUNCT
ma-232	193	27	the	the	DET
ma-232	193	28	predual	predual	ADJ
ma-232	193	29	of	of	ADP
ma-232	193	30	l1	l1	PROPN
ma-232	193	31	a(u	a(u	PROPN
ma-232	193	32	,	,	PUNCT
ma-232	193	33	µα	µα	ADP
ma-232	193	34	)	)	PUNCT
ma-232	193	35	is	be	AUX
ma-232	193	36	given	give	VERB
ma-232	193	37	by	by	ADP
ma-232	193	38	the	the	DET
ma-232	193	39	duality	duality	NOUN
ma-232	193	40	relation	relation	NOUN
ma-232	193	41	(	(	PUNCT
ma-232	193	42	b∞,	b∞,	VERB
ma-232	193	43	◦	◦	NOUN
ma-232	193	44	(u	(u	ADJ
ma-232	193	45	,	,	PUNCT
ma-232	193	46	i))∗	i))∗	PROPN
ma-232	193	47	≈	≈	PROPN
ma-232	193	48	l1	l1	PROPN
ma-232	193	49	a(u	a(u	PROPN
ma-232	193	50	,	,	PUNCT
ma-232	193	51	µα	µα	ADP
ma-232	193	52	)	)	PUNCT
ma-232	193	53	(	(	PUNCT
ma-232	193	54	3.1	3.1	NUM
ma-232	193	55	)	)	PUNCT
ma-232	193	56	under	under	ADP
ma-232	193	57	the	the	DET
ma-232	193	58	integral	integral	ADJ
ma-232	193	59	pairing	pairing	NOUN
ma-232	193	60	〈	〈	PROPN
ma-232	193	61	g	g	NOUN
ma-232	193	62	,	,	PUNCT
ma-232	193	63	f	f	PROPN
ma-232	193	64	〉	〉	PROPN
ma-232	193	65	=	=	SYM
ma-232	193	66	∫	∫	PROPN
ma-232	193	67	u	u	NOUN
ma-232	193	68	g(w)f	g(w)f	PROPN
ma-232	193	69	(	(	PUNCT
ma-232	193	70	w)dµα(w	w)dµα(w	NOUN
ma-232	193	71	)	)	PUNCT
ma-232	193	72	,	,	PUNCT
ma-232	193	73	(	(	PUNCT
ma-232	193	74	3.2	3.2	NUM
ma-232	193	75	)	)	PUNCT
ma-232	193	76	where	where	SCONJ
ma-232	193	77	g	g	PROPN
ma-232	193	78	∈	∈	PROPN
ma-232	193	79	b∞,	b∞,	ADP
ma-232	193	80	◦	◦	NOUN
ma-232	193	81	(u	(u	NOUN
ma-232	193	82	,	,	PUNCT
ma-232	193	83	i	i	PROPN
ma-232	193	84	)	)	PUNCT
ma-232	193	85	and	and	CCONJ
ma-232	193	86	f	f	PROPN
ma-232	193	87	∈	∈	PROPN
ma-232	193	88	l1	l1	PROPN
ma-232	193	89	a(u	a(u	PROPN
ma-232	193	90	,	,	PUNCT
ma-232	193	91	µα).using	µα).use	VERB
ma-232	193	92	the	the	DET
ma-232	193	93	duality	duality	NOUN
ma-232	193	94	pairing	pair	VERB
ma-232	193	95	above	above	ADV
ma-232	193	96	,	,	PUNCT
ma-232	193	97	we	we	PRON
ma-232	193	98	obtain	obtain	VERB
ma-232	193	99	the	the	DET
ma-232	193	100	corresponding	corresponding	ADJ
ma-232	193	101	group	group	NOUN
ma-232	193	102	of	of	ADP
ma-232	193	103	weighted	weight	VERB
ma-232	193	104	composition	composition	NOUN
ma-232	193	105	op	op	NOUN
ma-232	193	106	-	-	PUNCT
ma-232	193	107	erators	erator	NOUN
ma-232	193	108	on	on	ADP
ma-232	193	109	b∞,	b∞,	ADP
ma-232	193	110	◦	◦	NOUN
ma-232	193	111	(u	(u	ADJ
ma-232	193	112	,	,	PUNCT
ma-232	193	113	i	i	NOUN
ma-232	193	114	)	)	PUNCT
ma-232	193	115	as	as	ADP
ma-232	193	116	below	below	ADV
ma-232	193	117	:	:	PUNCT
ma-232	193	118	let	let	VERB
ma-232	193	119	g	g	PROPN
ma-232	193	120	∈	∈	PROPN
ma-232	193	121	b∞,	b∞,	ADP
ma-232	193	122	◦	◦	NOUN
ma-232	193	123	(u	(u	NOUN
ma-232	193	124	,	,	PUNCT
ma-232	193	125	i	i	PROPN
ma-232	193	126	)	)	PUNCT
ma-232	193	127	and	and	CCONJ
ma-232	193	128	f	f	PROPN
ma-232	193	129	∈	∈	PROPN
ma-232	193	130	l1	l1	PROPN
ma-232	193	131	a(u	a(u	PROPN
ma-232	193	132	,	,	PUNCT
ma-232	193	133	µα	µα	ADP
ma-232	193	134	)	)	PUNCT
ma-232	193	135	,	,	PUNCT
ma-232	193	136	then	then	ADV
ma-232	193	137	,	,	PUNCT
ma-232	193	138	〈	〈	PROPN
ma-232	193	139	g	g	PROPN
ma-232	193	140	,	,	PUNCT
ma-232	193	141	tt	tt	PROPN
ma-232	193	142	f	f	PROPN
ma-232	193	143	〉	〉	PROPN
ma-232	193	144	=	=	SYM
ma-232	193	145	∫	∫	PROPN
ma-232	193	146	u	u	X
ma-232	193	147	g(z)e−tγf	g(z)e−tγf	X
ma-232	193	148	(	(	PUNCT
ma-232	193	149	e−tz)dµα(z	e−tz)dµα(z	PROPN
ma-232	193	150	)	)	PUNCT
ma-232	193	151	=	=	SYM
ma-232	194	1	∫	∫	PROPN
ma-232	194	2	u	u	X
ma-232	194	3	g(z)e−tγf	g(z)e−tγf	X
ma-232	194	4	(	(	PUNCT
ma-232	194	5	e−tz)(=(z))αda(z	e−tz)(=(z))αda(z	NOUN
ma-232	194	6	)	)	PUNCT
ma-232	194	7	.	.	PUNCT
ma-232	195	1	by	by	ADP
ma-232	195	2	change	change	NOUN
ma-232	195	3	of	of	ADP
ma-232	195	4	variables	variable	NOUN
ma-232	195	5	,	,	PUNCT
ma-232	195	6	let	let	VERB
ma-232	195	7	ω	ω	NOUN
ma-232	195	8	=	=	SYM
ma-232	195	9	e−tz	e−tz	PROPN
ma-232	195	10	,	,	PUNCT
ma-232	195	11	then	then	ADV
ma-232	195	12	z	z	NOUN
ma-232	195	13	=	=	SYM
ma-232	195	14	etω	etω	ADJ
ma-232	195	15	,	,	PUNCT
ma-232	195	16	da(ω	da(ω	NOUN
ma-232	195	17	)	)	PUNCT
ma-232	195	18	=	=	SYM
ma-232	195	19	e−2tda(z	e−2tda(z	NOUN
ma-232	195	20	)	)	PUNCT
ma-232	195	21	and	and	CCONJ
ma-232	195	22	=(	=(	NOUN
ma-232	195	23	z	z	NOUN
ma-232	195	24	)	)	PUNCT
ma-232	196	1	=	=	SYM
ma-232	196	2	et	et	NOUN
ma-232	196	3	im(ω).then	im(ω).then	NOUN
ma-232	196	4	,	,	PUNCT
ma-232	196	5	〈	〈	PROPN
ma-232	196	6	g	g	PROPN
ma-232	196	7	,	,	PUNCT
ma-232	196	8	tt	tt	PROPN
ma-232	196	9	f	f	PROPN
ma-232	196	10	〉	〉	PROPN
ma-232	196	11	=	=	SYM
ma-232	196	12	∫	∫	PROPN
ma-232	196	13	u	u	INTJ
ma-232	196	14	g(etω)e−tγf	g(etω)e−tγf	PROPN
ma-232	196	15	(	(	PUNCT
ma-232	196	16	ω)eαt(=(ω))αe2tda(ω	ω)eαt(=(ω))αe2tda(ω	NOUN
ma-232	196	17	)	)	PUNCT
ma-232	196	18	=	=	SYM
ma-232	197	1	∫	∫	PUNCT
ma-232	197	2	u	u	PROPN
ma-232	197	3	g(etω)e−tγetγf	g(etω)e−tγetγf	X
ma-232	197	4	(	(	PUNCT
ma-232	197	5	ω)dµα(ω	ω)dµα(ω	X
ma-232	197	6	)	)	PUNCT
ma-232	197	7	=	=	SYM
ma-232	197	8	∫	∫	PROPN
ma-232	197	9	u	u	PROPN
ma-232	197	10	g(etω)f	g(etω)f	X
ma-232	197	11	(	(	PUNCT
ma-232	197	12	ω)dµα(ω	ω)dµα(ω	X
ma-232	197	13	)	)	PUNCT
ma-232	197	14	=	=	PUNCT
ma-232	198	1	〈	〈	PROPN
ma-232	198	2	t	t	PROPN
ma-232	198	3	∗t	∗t	PROPN
ma-232	198	4	g	g	PROPN
ma-232	198	5	,	,	PUNCT
ma-232	198	6	f	f	PROPN
ma-232	198	7	〉	〉	PROPN
ma-232	198	8	.	.	PUNCT
ma-232	199	1	(	(	PUNCT
ma-232	199	2	3.3	3.3	NUM
ma-232	199	3	)	)	PUNCT
ma-232	199	4	where	where	SCONJ
ma-232	199	5	t	t	PROPN
ma-232	199	6	∗t	∗t	PROPN
ma-232	199	7	g(ω	g(ω	PROPN
ma-232	199	8	)	)	PUNCT
ma-232	200	1	=	=	SYM
ma-232	200	2	g(etω).now	g(etω).now	PROPN
ma-232	200	3	,	,	PUNCT
ma-232	200	4	st	st	PROPN
ma-232	200	5	:	:	PUNCT
ma-232	200	6	=	=	SYM
ma-232	200	7	t	t	PROPN
ma-232	200	8	∗t	∗t	PROPN
ma-232	200	9	is	be	AUX
ma-232	200	10	defined	define	VERB
ma-232	200	11	on	on	ADP
ma-232	200	12	b∞,	b∞,	ADP
ma-232	200	13	◦	◦	NOUN
ma-232	200	14	(u	(u	ADJ
ma-232	200	15	,	,	PUNCT
ma-232	200	16	i	i	PROPN
ma-232	200	17	)	)	PUNCT
ma-232	200	18	.	.	PUNCT
ma-232	201	1	but	but	CCONJ
ma-232	201	2	we	we	PRON
ma-232	201	3	see	see	VERB
ma-232	201	4	that	that	PRON
ma-232	201	5	stg(i	stg(i	NOUN
ma-232	201	6	)	)	PUNCT
ma-232	201	7	=	=	SYM
ma-232	201	8	g(et	g(et	NOUN
ma-232	201	9	i	i	NOUN
ma-232	201	10	)	)	PUNCT
ma-232	201	11	6=	6=	ADP
ma-232	201	12	0	0	NUM
ma-232	202	1	and	and	CCONJ
ma-232	202	2	therefore	therefore	ADV
ma-232	202	3	stgdoes	stgdoes	AUX
ma-232	202	4	not	not	PART
ma-232	202	5	vanish	vanish	VERB
ma-232	202	6	at	at	ADP
ma-232	202	7	i	i	PRON
ma-232	202	8	.	.	PUNCT
ma-232	203	1	this	this	PRON
ma-232	203	2	means	mean	VERB
ma-232	203	3	that	that	SCONJ
ma-232	203	4	st	st	PROPN
ma-232	203	5	does	do	AUX
ma-232	203	6	not	not	PART
ma-232	203	7	map	map	VERB
ma-232	203	8	b∞,	b∞,	ADP
ma-232	203	9	◦	◦	NOUN
ma-232	203	10	(u	(u	ADJ
ma-232	203	11	,	,	PUNCT
ma-232	203	12	i	i	NOUN
ma-232	203	13	)	)	PUNCT
ma-232	203	14	onto	onto	ADP
ma-232	203	15	itself	itself	PRON
ma-232	203	16	,	,	PUNCT
ma-232	203	17	and	and	CCONJ
ma-232	203	18	there	there	ADV
ma-232	203	19	(	(	PUNCT
ma-232	203	20	st)t≥0	st)t≥0	NOUN
ma-232	203	21	isnot	isnot	ADP
ma-232	203	22	a	a	DET
ma-232	203	23	good	good	ADJ
ma-232	203	24	semigroup	semigroup	NOUN
ma-232	203	25	.	.	PUNCT
ma-232	204	1	we	we	PRON
ma-232	204	2	therefore	therefore	ADV
ma-232	204	3	propose	propose	VERB
ma-232	204	4	two	two	NUM
ma-232	204	5	remedies	remedy	NOUN
ma-232	204	6	to	to	PART
ma-232	204	7	correct	correct	VERB
ma-232	204	8	the	the	DET
ma-232	204	9	defect	defect	NOUN
ma-232	204	10	.	.	PUNCT
ma-232	205	1	the	the	DET
ma-232	205	2	first	first	ADJ
ma-232	205	3	one	one	NOUN
ma-232	205	4	is	be	AUX
ma-232	205	5	toapply	toapply	NOUN
ma-232	205	6	a	a	DET
ma-232	205	7	correction	correction	NOUN
ma-232	205	8	factor	factor	NOUN
ma-232	205	9	by	by	ADP
ma-232	205	10	writing	write	VERB
ma-232	205	11	stg(ω	stg(ω	PROPN
ma-232	205	12	)	)	PUNCT
ma-232	205	13	=	=	NOUN
ma-232	205	14	g(etω)−	g(etω)−	NOUN
ma-232	205	15	g(et	g(et	NOUN
ma-232	205	16	i	i	PRON
ma-232	205	17	)	)	PUNCT
ma-232	205	18	for	for	ADP
ma-232	205	19	all	all	PRON
ma-232	205	20	g	g	PROPN
ma-232	205	21	∈	∈	NOUN
ma-232	205	22	b∞,	b∞,	ADP
ma-232	205	23	◦	◦	NOUN
ma-232	205	24	(u	(u	NOUN
ma-232	205	25	,	,	PUNCT
ma-232	205	26	i	i	PROPN
ma-232	205	27	)	)	PUNCT
ma-232	205	28	.	.	PUNCT
ma-232	206	1	this	this	DET
ma-232	206	2	meansthat	meansthat	PROPN
ma-232	206	3	stg(i	stg(i	PROPN
ma-232	206	4	)	)	PUNCT
ma-232	206	5	=	=	SYM
ma-232	206	6	0	0	NUM
ma-232	206	7	and	and	CCONJ
ma-232	206	8	therefore	therefore	ADV
ma-232	206	9	st	st	PROPN
ma-232	206	10	maps	maps	PROPN
ma-232	206	11	b∞,	b∞,	ADP
ma-232	206	12	◦	◦	NOUN
ma-232	206	13	(u	(u	PROPN
ma-232	206	14	,	,	PUNCT
ma-232	206	15	i	i	NOUN
ma-232	206	16	)	)	PUNCT
ma-232	206	17	onto	onto	ADP
ma-232	206	18	itself	itself	PRON
ma-232	206	19	,	,	PUNCT
ma-232	206	20	as	as	SCONJ
ma-232	206	21	desired.the	desired.the	DET
ma-232	206	22	second	second	ADJ
ma-232	206	23	remedy	remedy	NOUN
ma-232	206	24	is	be	AUX
ma-232	206	25	to	to	PART
ma-232	206	26	redefined	redefined	VERB
ma-232	206	27	st	st	PROPN
ma-232	206	28	to	to	PART
ma-232	206	29	act	act	VERB
ma-232	206	30	on	on	ADP
ma-232	206	31	b∞,	b∞,	ADP
ma-232	206	32	◦	◦	NOUN
ma-232	206	33	(u	(u	NOUN
ma-232	206	34	)	)	PUNCT
ma-232	206	35	instead	instead	ADV
ma-232	206	36	of	of	ADP
ma-232	206	37	b∞,	b∞,	ADP
ma-232	206	38	◦	◦	NOUN
ma-232	206	39	(u	(u	ADJ
ma-232	206	40	,	,	PUNCT
ma-232	206	41	i	i	PROPN
ma-232	206	42	)	)	PUNCT
ma-232	206	43	.	.	PUNCT
ma-232	207	1	this	this	PRON
ma-232	207	2	simply	simply	ADV
ma-232	207	3	meansthat	meansthat	VERB
ma-232	207	4	the	the	DET
ma-232	207	5	domain	domain	NOUN
ma-232	207	6	of	of	ADP
ma-232	207	7	st	st	PROPN
ma-232	207	8	has	have	AUX
ma-232	207	9	been	be	AUX
ma-232	207	10	enlarged	enlarge	VERB
ma-232	207	11	and	and	CCONJ
ma-232	207	12	therefore	therefore	ADV
ma-232	207	13	st	st	PROPN
ma-232	207	14	is	be	AUX
ma-232	207	15	well	well	ADV
ma-232	207	16	defined	define	VERB
ma-232	207	17	on	on	ADP
ma-232	207	18	b∞,	b∞,	ADP
ma-232	207	19	◦	◦	NOUN
ma-232	207	20	(u	(u	ADJ
ma-232	207	21	,	,	PUNCT
ma-232	207	22	i	i	PROPN
ma-232	207	23	)	)	PUNCT
ma-232	207	24	.	.	PUNCT
ma-232	208	1	we	we	PRON
ma-232	208	2	shall	shall	AUX
ma-232	208	3	now	now	ADV
ma-232	208	4	carry	carry	VERB
ma-232	208	5	out	out	ADP
ma-232	208	6	a	a	DET
ma-232	208	7	complete	complete	ADJ
ma-232	208	8	study	study	NOUN
ma-232	208	9	of	of	ADP
ma-232	208	10	both	both	CCONJ
ma-232	208	11	the	the	DET
ma-232	208	12	semigroup	semigroup	ADJ
ma-232	208	13	and	and	CCONJ
ma-232	208	14	spectral	spectral	ADJ
ma-232	208	15	properties	property	NOUN
ma-232	208	16	of	of	ADP
ma-232	208	17	thisgroup	thisgroup	NOUN
ma-232	208	18	on	on	ADP
ma-232	208	19	b∞,	b∞,	ADP
ma-232	208	20	◦	◦	NOUN
ma-232	208	21	(u	(u	NOUN
ma-232	208	22	)	)	PUNCT
ma-232	208	23	.	.	PUNCT
ma-232	209	1	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	209	2	eur	eur	PROPN
ma-232	209	3	.	.	PUNCT
ma-232	210	1	j.	j.	PROPN
ma-232	210	2	math	math	PROPN
ma-232	210	3	.	.	PUNCT
ma-232	211	1	anal	anal	PROPN
ma-232	211	2	.	.	PUNCT
ma-232	212	1	10.28924	10.28924	NUM
ma-232	212	2	/	/	SYM
ma-232	212	3	ada	ada	PROPN
ma-232	212	4	/	/	SYM
ma-232	212	5	ma.4.14	ma.4.14	PROPN
ma-232	212	6	93.1.1	93.1.1	NOUN
ma-232	212	7	.	.	PUNCT
ma-232	213	1	semigroup	semigroup	PROPN
ma-232	213	2	properties	property	NOUN
ma-232	213	3	.	.	PUNCT
ma-232	214	1	in	in	ADP
ma-232	214	2	this	this	DET
ma-232	214	3	section	section	NOUN
ma-232	214	4	,	,	PUNCT
ma-232	214	5	we	we	PRON
ma-232	214	6	investigate	investigate	VERB
ma-232	214	7	the	the	DET
ma-232	214	8	semigroup	semigroup	ADJ
ma-232	214	9	properties	property	NOUN
ma-232	214	10	and	and	CCONJ
ma-232	214	11	determinethe	determinethe	DET
ma-232	214	12	infinitesimal	infinitesimal	ADJ
ma-232	214	13	generator	generator	NOUN
ma-232	214	14	γ	γ	PROPN
ma-232	214	15	of	of	ADP
ma-232	214	16	(	(	PUNCT
ma-232	214	17	st)t≥0	st)t≥0	NOUN
ma-232	214	18	on	on	ADP
ma-232	214	19	b∞,	b∞,	NOUN
ma-232	214	20	◦	◦	NOUN
ma-232	214	21	(u	(u	NOUN
ma-232	214	22	)	)	PUNCT
ma-232	214	23	where	where	SCONJ
ma-232	214	24	,	,	PUNCT
ma-232	214	25	stg(w	stg(w	PROPN
ma-232	214	26	)	)	PUNCT
ma-232	214	27	:	:	PUNCT
ma-232	214	28	=	=	SYM
ma-232	214	29	g(etw	g(etw	PROPN
ma-232	214	30	)	)	PUNCT
ma-232	214	31	.	.	PUNCT
ma-232	215	1	we	we	PRON
ma-232	215	2	begin	begin	VERB
ma-232	215	3	byproving	byprove	VERB
ma-232	215	4	the	the	DET
ma-232	215	5	strong	strong	ADJ
ma-232	215	6	continuity	continuity	NOUN
ma-232	215	7	property	property	NOUN
ma-232	215	8	.	.	PUNCT
ma-232	216	1	theorem	theorem	VERB
ma-232	216	2	3.1	3.1	NUM
ma-232	216	3	.	.	PUNCT
ma-232	217	1	let	let	VERB
ma-232	217	2	stg(w	stg(w	NOUN
ma-232	217	3	)	)	PUNCT
ma-232	217	4	:	:	PUNCT
ma-232	217	5	=	=	SYM
ma-232	217	6	g(etw	g(etw	NOUN
ma-232	217	7	)	)	PUNCT
ma-232	217	8	be	be	VERB
ma-232	217	9	a	a	DET
ma-232	217	10	semigroup	semigroup	NOUN
ma-232	217	11	of	of	ADP
ma-232	217	12	composition	composition	NOUN
ma-232	217	13	operators	operator	NOUN
ma-232	217	14	defined	define	VERB
ma-232	217	15	on	on	ADP
ma-232	217	16	b∞,	b∞,	ADP
ma-232	217	17	◦	◦	NOUN
ma-232	217	18	(u	(u	NOUN
ma-232	217	19	)	)	PUNCT
ma-232	217	20	.	.	PUNCT
ma-232	218	1	then	then	ADV
ma-232	218	2	,	,	PUNCT
ma-232	218	3	(	(	PUNCT
ma-232	218	4	st)t∈r	st)t∈r	NOUN
ma-232	218	5	is	be	AUX
ma-232	218	6	a	a	DET
ma-232	218	7	strongly	strongly	ADV
ma-232	218	8	continuous	continuous	ADJ
ma-232	218	9	group	group	NOUN
ma-232	218	10	of	of	ADP
ma-232	218	11	isometries	isometry	NOUN
ma-232	218	12	on	on	ADP
ma-232	218	13	b∞,	b∞,	ADP
ma-232	218	14	◦	◦	NOUN
ma-232	218	15	(u	(u	NOUN
ma-232	218	16	)	)	PUNCT
ma-232	218	17	.	.	PUNCT
ma-232	219	1	proof	proof	NOUN
ma-232	219	2	.	.	PUNCT
ma-232	220	1	it	it	PRON
ma-232	220	2	is	be	AUX
ma-232	220	3	clear	clear	ADJ
ma-232	220	4	from	from	ADP
ma-232	220	5	the	the	DET
ma-232	220	6	definition	definition	NOUN
ma-232	220	7	that	that	SCONJ
ma-232	220	8	(	(	PUNCT
ma-232	220	9	st)t∈r	st)t∈r	NOUN
ma-232	220	10	is	be	AUX
ma-232	220	11	a	a	DET
ma-232	220	12	group	group	NOUN
ma-232	220	13	.	.	PUNCT
ma-232	221	1	to	to	PART
ma-232	221	2	prove	prove	VERB
ma-232	221	3	that	that	SCONJ
ma-232	221	4	(	(	PUNCT
ma-232	221	5	st)t∈r	st)t∈r	NOUN
ma-232	221	6	is	be	AUX
ma-232	221	7	an	an	DET
ma-232	221	8	isometryon	isometryon	NOUN
ma-232	221	9	b∞,	b∞,	ADP
ma-232	221	10	◦	◦	NOUN
ma-232	221	11	(u	(u	NOUN
ma-232	221	12	)	)	PUNCT
ma-232	221	13	,	,	PUNCT
ma-232	221	14	we	we	PRON
ma-232	221	15	have	have	VERB
ma-232	221	16	;	;	PUNCT
ma-232	221	17	‖stg‖b∞,	‖stg‖b∞,	VERB
ma-232	221	18	◦	◦	NOUN
ma-232	221	19	(u	(u	NOUN
ma-232	221	20	)	)	PUNCT
ma-232	222	1	=	=	SYM
ma-232	222	2	sup	sup	NOUN
ma-232	222	3	ω∈u	ω∈u	NOUN
ma-232	222	4	=(	=(	NOUN
ma-232	222	5	ω)|stg′(ω)|	ω)|stg′(ω)|	PROPN
ma-232	222	6	=	=	SYM
ma-232	222	7	sup	sup	NOUN
ma-232	222	8	ω∈u	ω∈u	NOUN
ma-232	222	9	=(	=(	ADJ
ma-232	222	10	ω)et	ω)et	NOUN
ma-232	222	11	|g′(etω)|	|g′(etω)|	PROPN
ma-232	222	12	.	.	PUNCT
ma-232	223	1	now	now	ADV
ma-232	223	2	by	by	ADP
ma-232	223	3	change	change	NOUN
ma-232	223	4	of	of	ADP
ma-232	223	5	variables	variable	NOUN
ma-232	223	6	,	,	PUNCT
ma-232	223	7	let	let	VERB
ma-232	223	8	z	z	NOUN
ma-232	223	9	=	=	SYM
ma-232	223	10	etω	etω	NOUN
ma-232	223	11	then	then	ADV
ma-232	223	12	ω	ω	NUM
ma-232	223	13	=	=	PROPN
ma-232	223	14	e−tz	e−tz	PROPN
ma-232	223	15	,	,	PUNCT
ma-232	223	16	and	and	CCONJ
ma-232	223	17	=(	=(	PROPN
ma-232	223	18	ω	ω	NOUN
ma-232	223	19	)	)	PUNCT
ma-232	223	20	=	=	SYM
ma-232	223	21	e−t=(z	e−t=(z	PROPN
ma-232	223	22	)	)	PUNCT
ma-232	223	23	.	.	PUNCT
ma-232	224	1	therefore	therefore	ADV
ma-232	224	2	,	,	PUNCT
ma-232	224	3	‖stg‖b∞,	‖stg‖b∞,	NOUN
ma-232	224	4	◦	◦	NOUN
ma-232	224	5	(u	(u	NOUN
ma-232	224	6	)	)	PUNCT
ma-232	224	7	=	=	SYM
ma-232	224	8	sup	sup	NOUN
ma-232	224	9	z∈u	z∈u	NOUN
ma-232	224	10	e−t=(z)et	e−t=(z)et	NUM
ma-232	224	11	|g′(z)|	|g′(z)|	PUNCT
ma-232	224	12	=	=	SYM
ma-232	224	13	sup	sup	NOUN
ma-232	224	14	z∈u	z∈u	VERB
ma-232	224	15	=(	=(	NOUN
ma-232	224	16	z)|g′(z)|	z)|g′(z)|	PROPN
ma-232	224	17	=	=	SYM
ma-232	224	18	‖g‖b∞,	‖g‖b∞,	PROPN
ma-232	224	19	◦	◦	NOUN
ma-232	224	20	(u	(u	NOUN
ma-232	224	21	)	)	PUNCT
ma-232	224	22	,	,	PUNCT
ma-232	224	23	as	as	SCONJ
ma-232	224	24	desired	desire	VERB
ma-232	224	25	.	.	PUNCT
ma-232	225	1	for	for	ADP
ma-232	225	2	strongly	strongly	ADV
ma-232	225	3	continuity	continuity	NOUN
ma-232	225	4	,	,	PUNCT
ma-232	225	5	we	we	PRON
ma-232	225	6	first	first	ADV
ma-232	225	7	take	take	VERB
ma-232	225	8	note	note	NOUN
ma-232	225	9	that	that	SCONJ
ma-232	225	10	st	st	PROPN
ma-232	225	11	=	=	NOUN
ma-232	225	12	cϕ−t	cϕ−t	PROPN
ma-232	225	13	since	since	SCONJ
ma-232	225	14	stg(ω	stg(ω	PROPN
ma-232	225	15	)	)	PUNCT
ma-232	225	16	=	=	PROPN
ma-232	225	17	g(ϕt(ω	g(ϕt(ω	NOUN
ma-232	225	18	)	)	PUNCT
ma-232	225	19	)	)	PUNCT
ma-232	225	20	.	.	PUNCT
ma-232	226	1	thenby	thenby	VERB
ma-232	226	2	proposition	proposition	NOUN
ma-232	226	3	2.3	2.3	NUM
ma-232	226	4	,	,	PUNCT
ma-232	226	5	it	it	PRON
ma-232	226	6	is	be	AUX
ma-232	226	7	easy	easy	ADJ
ma-232	226	8	to	to	PART
ma-232	226	9	see	see	VERB
ma-232	226	10	that	that	PRON
ma-232	226	11	cψ−t	cψ−t	NOUN
ma-232	226	12	is	be	AUX
ma-232	226	13	strongly	strongly	ADV
ma-232	226	14	continuous	continuous	ADJ
ma-232	226	15	on	on	ADP
ma-232	226	16	b∞,	b∞,	NOUN
ma-232	226	17	◦	◦	NOUN
ma-232	226	18	(u	(u	NOUN
ma-232	226	19	)	)	PUNCT
ma-232	226	20	if	if	SCONJ
ma-232	226	21	and	and	CCONJ
ma-232	226	22	only	only	ADV
ma-232	226	23	if	if	SCONJ
ma-232	226	24	(	(	PUNCT
ma-232	226	25	cψ−1	cψ−1	PROPN
ma-232	226	26	◦	◦	NOUN
ma-232	226	27	ϕ−t	ϕ−t	NOUN
ma-232	226	28	◦	◦	NOUN
ma-232	226	29	ψ)t∈r	ψ)t∈r	PROPN
ma-232	226	30	is	be	AUX
ma-232	226	31	strongly	strongly	ADV
ma-232	226	32	continuous	continuous	ADJ
ma-232	226	33	on	on	ADP
ma-232	226	34	b∞,	b∞,	ADP
ma-232	226	35	◦	◦	NOUN
ma-232	226	36	(d	(d	NOUN
ma-232	226	37	)	)	PUNCT
ma-232	226	38	.	.	PUNCT
ma-232	227	1	now	now	ADV
ma-232	227	2	by	by	ADP
ma-232	227	3	simple	simple	ADJ
ma-232	227	4	computation	computation	NOUN
ma-232	227	5	of	of	ADP
ma-232	227	6	ψ−1	ψ−1	PROPN
ma-232	227	7	◦	◦	NOUN
ma-232	227	8	ϕ−t	ϕ−t	CCONJ
ma-232	227	9	◦	◦	NOUN
ma-232	227	10	ψ(z),we	ψ(z),we	NOUN
ma-232	227	11	obtain	obtain	VERB
ma-232	227	12	;	;	PUNCT
ma-232	227	13	ψ−1	ψ−1	PROPN
ma-232	227	14	◦	◦	NOUN
ma-232	227	15	ϕ−t	ϕ−t	NOUN
ma-232	227	16	◦	◦	NOUN
ma-232	227	17	ψ(z	ψ(z	NOUN
ma-232	227	18	)	)	PUNCT
ma-232	227	19	=	=	SYM
ma-232	228	1	z	z	NOUN
ma-232	229	1	−	−	NOUN
ma-232	229	2	1−et	1−et	NUM
ma-232	229	3	1+et	1+et	NUM
ma-232	229	4	1−	1−	NUM
ma-232	229	5	1−et	1−et	NUM
ma-232	229	6	1+et	1+et	NUM
ma-232	229	7	z	z	X
ma-232	229	8	=	=	SYM
ma-232	229	9	z	z	NOUN
ma-232	229	10	−	−	NOUN
ma-232	229	11	at	at	ADP
ma-232	229	12	1−	1−	NUM
ma-232	229	13	atz	atz	NOUN
ma-232	229	14	,	,	PUNCT
ma-232	229	15	where	where	SCONJ
ma-232	229	16	at	at	ADP
ma-232	229	17	=	=	PROPN
ma-232	229	18	1−et	1−et	NUM
ma-232	229	19	1+et	1+et	NUM
ma-232	229	20	.	.	PUNCT
ma-232	230	1	as	as	SCONJ
ma-232	230	2	t	t	PROPN
ma-232	230	3	→	→	SYM
ma-232	230	4	0	0	NUM
ma-232	230	5	,	,	PUNCT
ma-232	230	6	at	at	ADP
ma-232	230	7	→	→	X
ma-232	230	8	0	0	X
ma-232	230	9	.	.	PUNCT
ma-232	230	10	let	let	VERB
ma-232	230	11	ha(z	ha(z	PRON
ma-232	230	12	)	)	PUNCT
ma-232	231	1	=	=	SYM
ma-232	232	1	z−at	z−at	NUM
ma-232	232	2	1−atz	1−atz	PUNCT
ma-232	232	3	=	=	PUNCT
ma-232	232	4	ψ−1	ψ−1	PROPN
ma-232	232	5	◦	◦	NOUN
ma-232	232	6	ϕ−t	ϕ−t	NOUN
ma-232	232	7	◦	◦	NOUN
ma-232	232	8	ψ(z	ψ(z	PROPN
ma-232	232	9	)	)	PUNCT
ma-232	232	10	,	,	PUNCT
ma-232	232	11	then	then	ADV
ma-232	232	12	for	for	ADP
ma-232	232	13	strongcontinuity	strongcontinuity	NOUN
ma-232	232	14	,	,	PUNCT
ma-232	232	15	it	it	PRON
ma-232	232	16	therefore	therefore	ADV
ma-232	232	17	suffices	suffice	VERB
ma-232	232	18	to	to	PART
ma-232	232	19	show	show	VERB
ma-232	232	20	that	that	DET
ma-232	232	21	‖cha	‖cha	PROPN
ma-232	233	1	f	f	NOUN
ma-232	234	1	−	−	PROPN
ma-232	234	2	f	f	PROPN
ma-232	234	3	‖b∞,	‖b∞,	PROPN
ma-232	234	4	◦	◦	NOUN
ma-232	234	5	(d	(d	NOUN
ma-232	234	6	)	)	PUNCT
ma-232	234	7	→	→	SYM
ma-232	234	8	0	0	NUM
ma-232	234	9	as	as	ADP
ma-232	234	10	a→	a→	X
ma-232	234	11	0	0	NUM
ma-232	234	12	(	(	PUNCT
ma-232	234	13	at	at	ADP
ma-232	234	14	→	→	SYM
ma-232	234	15	0	0	NUM
ma-232	234	16	)	)	PUNCT
ma-232	234	17	.	.	PUNCT
ma-232	235	1	using	use	VERB
ma-232	235	2	thedensity	thedensity	NOUN
ma-232	235	3	of	of	ADP
ma-232	235	4	polynomials	polynomial	NOUN
ma-232	235	5	in	in	ADP
ma-232	235	6	b∞,	b∞,	ADP
ma-232	235	7	◦	◦	NOUN
ma-232	235	8	(d	(d	NOUN
ma-232	235	9	)	)	PUNCT
ma-232	235	10	,	,	PUNCT
ma-232	235	11	let	let	VERB
ma-232	235	12	f	f	PROPN
ma-232	235	13	(	(	PUNCT
ma-232	235	14	z	z	NOUN
ma-232	235	15	)	)	PUNCT
ma-232	235	16	=	=	SYM
ma-232	236	1	zn	zn	X
ma-232	236	2	.	.	PUNCT
ma-232	236	3	then	then	ADV
ma-232	236	4	chazn	chazn	VERB
ma-232	236	5	−	−	PROPN
ma-232	236	6	zn	zn	PROPN
ma-232	236	7	=	=	SYM
ma-232	236	8	(	(	PUNCT
ma-232	236	9	ha(z))n	ha(z))n	INTJ
ma-232	236	10	−	−	NOUN
ma-232	236	11	zn	zn	NUM
ma-232	236	12	,	,	PUNCT
ma-232	236	13	n	n	PRON
ma-232	236	14	≥	≥	NOUN
ma-232	236	15	1	1	NUM
ma-232	236	16	,	,	PUNCT
ma-232	236	17	and	and	CCONJ
ma-232	236	18	(	(	PUNCT
ma-232	236	19	cha	cha	NOUN
ma-232	236	20	f	f	NOUN
ma-232	237	1	−	−	PROPN
ma-232	237	2	f	f	PROPN
ma-232	237	3	)	)	PUNCT
ma-232	237	4	′(z	′(z	NOUN
ma-232	237	5	)	)	PUNCT
ma-232	238	1	=	=	SYM
ma-232	238	2	n[(ha(z))n−1h′a(z)−	n[(ha(z))n−1h′a(z)−	NUM
ma-232	238	3	zn−1	zn−1	PROPN
ma-232	238	4	]	]	PUNCT
ma-232	238	5	.	.	PUNCT
ma-232	239	1	but	but	CCONJ
ma-232	239	2	ha(z	ha(z	PRON
ma-232	239	3	)	)	PUNCT
ma-232	239	4	=	=	SYM
ma-232	240	1	z−at	z−at	NUM
ma-232	240	2	1−atz	1−atz	PROPN
ma-232	240	3	,	,	PUNCT
ma-232	240	4	and	and	CCONJ
ma-232	240	5	hence	hence	ADV
ma-232	240	6	h′a(z	h′a(z	NOUN
ma-232	240	7	)	)	PUNCT
ma-232	240	8	=	=	SYM
ma-232	240	9	1−atat	1−atat	NUM
ma-232	240	10	(	(	PUNCT
ma-232	240	11	1−atz)2	1−atz)2	NOUN
ma-232	240	12	.	.	PUNCT
ma-232	241	1	therefore	therefore	ADV
ma-232	241	2	,	,	PUNCT
ma-232	241	3	(	(	PUNCT
ma-232	241	4	cha	cha	NOUN
ma-232	241	5	f	f	NOUN
ma-232	241	6	−	−	PROPN
ma-232	241	7	f	f	PROPN
ma-232	241	8	)	)	PUNCT
ma-232	241	9	′(z	′(z	NOUN
ma-232	241	10	)	)	PUNCT
ma-232	241	11	=	=	SYM
ma-232	241	12	n	n	CCONJ
ma-232	241	13	[	[	PUNCT
ma-232	241	14	(	(	PUNCT
ma-232	241	15	ha(z))n−1(1−	ha(z))n−1(1−	PROPN
ma-232	241	16	atat	atat	PROPN
ma-232	241	17	)	)	PUNCT
ma-232	241	18	(	(	PUNCT
ma-232	241	19	1−	1−	NUM
ma-232	241	20	atz)2	atz)2	PROPN
ma-232	241	21	−	−	PROPN
ma-232	241	22	zn−1	zn−1	PROPN
ma-232	241	23	]	]	PUNCT
ma-232	242	1	=	=	PUNCT
ma-232	242	2	n	n	CCONJ
ma-232	242	3	[	[	PUNCT
ma-232	242	4	(	(	PUNCT
ma-232	242	5	z−at1−atz	z−at1−atz	X
ma-232	242	6	)	)	PUNCT
ma-232	242	7	n−1(1−	n−1(1−	PROPN
ma-232	242	8	atat	atat	PROPN
ma-232	242	9	)	)	PUNCT
ma-232	242	10	(	(	PUNCT
ma-232	242	11	1−	1−	NUM
ma-232	242	12	atz)2	atz)2	PROPN
ma-232	242	13	−	−	PROPN
ma-232	242	14	zn−1	zn−1	PROPN
ma-232	242	15	]	]	PUNCT
ma-232	242	16	=	=	PUNCT
ma-232	243	1	n	n	PART
ma-232	243	2	[	[	PUNCT
ma-232	243	3	(	(	PUNCT
ma-232	243	4	z	z	NOUN
ma-232	243	5	−	−	NOUN
ma-232	243	6	at)n−1(1−	at)n−1(1−	PROPN
ma-232	243	7	atat)−	atat)−	PROPN
ma-232	243	8	zn−1((1−	zn−1((1−	X
ma-232	243	9	atz)n+1	atz)n+1	PROPN
ma-232	243	10	)	)	PUNCT
ma-232	243	11	(	(	PUNCT
ma-232	243	12	1−	1−	NUM
ma-232	243	13	atz)n+1	atz)n+1	NOUN
ma-232	243	14	]	]	PUNCT
ma-232	243	15	.	.	PUNCT
ma-232	244	1	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	244	2	eur	eur	PROPN
ma-232	244	3	.	.	PUNCT
ma-232	245	1	j.	j.	PROPN
ma-232	245	2	math	math	PROPN
ma-232	245	3	.	.	PUNCT
ma-232	246	1	anal	anal	PROPN
ma-232	246	2	.	.	PUNCT
ma-232	247	1	10.28924	10.28924	NUM
ma-232	247	2	/	/	SYM
ma-232	247	3	ada	ada	PROPN
ma-232	247	4	/	/	SYM
ma-232	247	5	ma.4.14	ma.4.14	NOUN
ma-232	247	6	10now	10now	ADV
ma-232	247	7	,	,	PUNCT
ma-232	247	8	lim	lim	PROPN
ma-232	247	9	a→0	a→0	PROPN
ma-232	247	10	‖cha	‖cha	PROPN
ma-232	248	1	f	f	PROPN
ma-232	248	2	−	−	PROPN
ma-232	248	3	f	f	PROPN
ma-232	248	4	‖b∞,	‖b∞,	PROPN
ma-232	248	5	◦	◦	NOUN
ma-232	248	6	(d	(d	NOUN
ma-232	248	7	)	)	PUNCT
ma-232	248	8	=	=	SYM
ma-232	248	9	lim	lim	PROPN
ma-232	248	10	a→0	a→0	PROPN
ma-232	249	1	(	(	PUNCT
ma-232	249	2	sup	sup	NOUN
ma-232	249	3	z∈d	z∈d	NUM
ma-232	249	4	(	(	PUNCT
ma-232	249	5	1−	1−	NUM
ma-232	249	6	|z	|z	PROPN
ma-232	250	1	|2)|(cha	|2)|(cha	PROPN
ma-232	250	2	f	f	PROPN
ma-232	250	3	−	−	PROPN
ma-232	250	4	f	f	PROPN
ma-232	250	5	)	)	PUNCT
ma-232	250	6	′|(z	′|(z	PROPN
ma-232	250	7	)	)	PUNCT
ma-232	250	8	)	)	PUNCT
ma-232	251	1	=	=	SYM
ma-232	251	2	lim	lim	PROPN
ma-232	251	3	t→0	t→0	PUNCT
ma-232	251	4	(	(	PUNCT
ma-232	251	5	sup	sup	NUM
ma-232	251	6	z∈d	z∈d	NUM
ma-232	251	7	(	(	PUNCT
ma-232	251	8	1−	1−	NUM
ma-232	251	9	|z	|z	NOUN
ma-232	251	10	|2	|2	NUM
ma-232	251	11	)	)	PUNCT
ma-232	251	12	∣∣∣∣n	∣∣∣∣n	NOUN
ma-232	252	1	[	[	X
ma-232	252	2	(	(	PUNCT
ma-232	252	3	zn−1)(1)−	zn−1)(1)−	NOUN
ma-232	252	4	zn−1(1	zn−1(1	NOUN
ma-232	252	5	)	)	PUNCT
ma-232	252	6	(	(	PUNCT
ma-232	252	7	1)n+1	1)n+1	NUM
ma-232	252	8	]	]	SYM
ma-232	252	9	∣∣∣∣	∣∣∣∣	NOUN
ma-232	252	10	)	)	PUNCT
ma-232	252	11	=	=	SYM
ma-232	252	12	lim	lim	PROPN
ma-232	252	13	t→0	t→0	PUNCT
ma-232	252	14	(	(	PUNCT
ma-232	252	15	sup	sup	NUM
ma-232	252	16	z∈d	z∈d	NUM
ma-232	252	17	(	(	PUNCT
ma-232	252	18	1−	1−	NUM
ma-232	252	19	|z	|z	NOUN
ma-232	252	20	|2	|2	NUM
ma-232	252	21	)	)	PUNCT
ma-232	252	22	∣∣n[zn−1	∣∣n[zn−1	PROPN
ma-232	252	23	−	−	PROPN
ma-232	252	24	zn−1	zn−1	PROPN
ma-232	252	25	]	]	X
ma-232	252	26	∣∣	∣∣	X
ma-232	252	27	)	)	PUNCT
ma-232	252	28	=	=	SYM
ma-232	252	29	0	0	X
ma-232	252	30	.	.	PUNCT
ma-232	253	1	hence	hence	ADV
ma-232	253	2	,	,	PUNCT
ma-232	253	3	(	(	PUNCT
ma-232	253	4	st)t∈r	st)t∈r	NOUN
ma-232	253	5	is	be	AUX
ma-232	253	6	strongly	strongly	ADV
ma-232	253	7	continuous	continuous	ADJ
ma-232	253	8	on	on	ADP
ma-232	253	9	b∞,	b∞,	NOUN
ma-232	253	10	◦	◦	NOUN
ma-232	253	11	(u	(u	NOUN
ma-232	253	12	)	)	PUNCT
ma-232	253	13	,	,	PUNCT
ma-232	253	14	as	as	SCONJ
ma-232	253	15	claimed	claim	VERB
ma-232	253	16	.	.	PUNCT
ma-232	254	1	�	�	PROPN
ma-232	254	2	theorem	theorem	VERB
ma-232	254	3	3.2	3.2	NUM
ma-232	254	4	.	.	PUNCT
ma-232	255	1	the	the	DET
ma-232	255	2	infinitesimal	infinitesimal	ADJ
ma-232	255	3	generator	generator	NOUN
ma-232	255	4	γ	γ	X
ma-232	255	5	of	of	ADP
ma-232	255	6	(	(	PUNCT
ma-232	255	7	st)t≥0	st)t≥0	NOUN
ma-232	255	8	on	on	ADP
ma-232	255	9	b∞,	b∞,	ADP
ma-232	255	10	◦	◦	NOUN
ma-232	255	11	(u	(u	NOUN
ma-232	255	12	)	)	PUNCT
ma-232	255	13	is	be	AUX
ma-232	255	14	given	give	VERB
ma-232	255	15	by	by	ADP
ma-232	255	16	γg(ω)=ωg′(ω	γg(ω)=ωg′(ω	NOUN
ma-232	255	17	)	)	PUNCT
ma-232	255	18	with	with	ADP
ma-232	255	19	the	the	DET
ma-232	255	20	domain	domain	NOUN
ma-232	255	21	d(γ	d(γ	PROPN
ma-232	255	22	)	)	PUNCT
ma-232	255	23	=	=	SYM
ma-232	255	24	{	{	PUNCT
ma-232	255	25	g	g	NOUN
ma-232	255	26	∈	∈	PROPN
ma-232	255	27	b∞,	b∞,	ADP
ma-232	255	28	◦	◦	NOUN
ma-232	255	29	(u	(u	NOUN
ma-232	255	30	)	)	PUNCT
ma-232	255	31	:	:	PUNCT
ma-232	255	32	ωg′(ω	ωg′(ω	NOUN
ma-232	255	33	)	)	PUNCT
ma-232	255	34	∈	∈	PROPN
ma-232	255	35	b∞,	b∞,	ADP
ma-232	255	36	◦	◦	NOUN
ma-232	255	37	(u	(u	NOUN
ma-232	255	38	)	)	PUNCT
ma-232	255	39	}	}	PUNCT
ma-232	255	40	.	.	PUNCT
ma-232	256	1	proof	proof	NOUN
ma-232	256	2	.	.	PUNCT
ma-232	257	1	by	by	ADP
ma-232	257	2	definition	definition	NOUN
ma-232	257	3	,	,	PUNCT
ma-232	257	4	the	the	DET
ma-232	257	5	infinitesimal	infinitesimal	ADJ
ma-232	257	6	generator	generator	NOUN
ma-232	257	7	denoted	denote	VERB
ma-232	257	8	by	by	ADP
ma-232	257	9	γ	γ	NOUN
ma-232	257	10	of	of	ADP
ma-232	257	11	(	(	PUNCT
ma-232	257	12	st)t≥0	st)t≥0	NOUN
ma-232	257	13	is	be	AUX
ma-232	257	14	given	give	VERB
ma-232	257	15	by	by	ADP
ma-232	257	16	;	;	PUNCT
ma-232	257	17	γg(ω	γg(ω	NUM
ma-232	257	18	)	)	PUNCT
ma-232	257	19	=	=	SYM
ma-232	257	20	lim	lim	PROPN
ma-232	257	21	t→0	t→0	PROPN
ma-232	257	22	+	+	NUM
ma-232	257	23	g(etω)−	g(etω)−	PROPN
ma-232	257	24	g(ω	g(ω	PROPN
ma-232	257	25	)	)	PUNCT
ma-232	257	26	t	t	NOUN
ma-232	257	27	=	=	SYM
ma-232	257	28	∂	∂	NUM
ma-232	257	29	∂t	∂t	PROPN
ma-232	257	30	g(etω	g(etω	PROPN
ma-232	257	31	)	)	PUNCT
ma-232	257	32	∣∣∣∣	∣∣∣∣	NOUN
ma-232	257	33	t=0	t=0	VERB
ma-232	257	34	=	=	SYM
ma-232	257	35	ωg′(ω	ωg′(ω	PROPN
ma-232	257	36	)	)	PUNCT
ma-232	257	37	.	.	PUNCT
ma-232	258	1	it	it	PRON
ma-232	258	2	therefore	therefore	ADV
ma-232	258	3	follows	follow	VERB
ma-232	258	4	that	that	SCONJ
ma-232	258	5	d(γ	d(γ	PROPN
ma-232	258	6	)	)	PUNCT
ma-232	258	7	⊆	⊆	NUM
ma-232	258	8	{	{	PUNCT
ma-232	258	9	g	g	NOUN
ma-232	258	10	∈	∈	PROPN
ma-232	258	11	b∞,	b∞,	ADP
ma-232	258	12	◦	◦	NOUN
ma-232	258	13	(u	(u	NOUN
ma-232	258	14	)	)	PUNCT
ma-232	258	15	:	:	PUNCT
ma-232	258	16	ωg′(ω	ωg′(ω	NOUN
ma-232	258	17	)	)	PUNCT
ma-232	258	18	∈	∈	PROPN
ma-232	258	19	b∞,	b∞,	ADP
ma-232	258	20	◦	◦	NOUN
ma-232	258	21	(u	(u	NOUN
ma-232	258	22	)	)	PUNCT
ma-232	258	23	}	}	PUNCT
ma-232	258	24	.	.	PUNCT
ma-232	259	1	to	to	PART
ma-232	259	2	prove	prove	VERB
ma-232	259	3	the	the	DET
ma-232	259	4	reverseinclusion	reverseinclusion	NOUN
ma-232	259	5	,	,	PUNCT
ma-232	259	6	we	we	PRON
ma-232	259	7	let	let	VERB
ma-232	259	8	g	g	PROPN
ma-232	259	9	∈	∈	PROPN
ma-232	259	10	b∞,	b∞,	ADP
ma-232	259	11	◦	◦	NOUN
ma-232	259	12	(u	(u	NOUN
ma-232	259	13	)	)	PUNCT
ma-232	259	14	be	be	AUX
ma-232	259	15	such	such	ADJ
ma-232	259	16	that	that	PRON
ma-232	259	17	ωg′(ω	ωg′(ω	NOUN
ma-232	259	18	)	)	PUNCT
ma-232	259	19	∈	∈	PROPN
ma-232	259	20	b∞,	b∞,	ADP
ma-232	259	21	◦	◦	NOUN
ma-232	259	22	(u	(u	NOUN
ma-232	259	23	)	)	PUNCT
ma-232	259	24	.	.	PUNCT
ma-232	260	1	then	then	ADV
ma-232	260	2	for	for	ADP
ma-232	260	3	ω	ω	PROPN
ma-232	260	4	∈	∈	PROPN
ma-232	260	5	u	u	NOUN
ma-232	260	6	,	,	PUNCT
ma-232	260	7	we	we	PRON
ma-232	260	8	have	have	VERB
ma-232	260	9	;	;	PUNCT
ma-232	260	10	stg(ω)−	stg(ω)−	PROPN
ma-232	260	11	g(ω	g(ω	PROPN
ma-232	260	12	)	)	PUNCT
ma-232	261	1	=	=	SYM
ma-232	261	2	∫	∫	PROPN
ma-232	261	3	t	t	PROPN
ma-232	261	4	0	0	NUM
ma-232	261	5	∂	∂	NUM
ma-232	261	6	∂s	∂s	PROPN
ma-232	261	7	g(esω	g(esω	NOUN
ma-232	261	8	)	)	PUNCT
ma-232	261	9	ds	ds	PROPN
ma-232	261	10	=	=	SYM
ma-232	261	11	∫	∫	PROPN
ma-232	261	12	t	t	PROPN
ma-232	261	13	0	0	NUM
ma-232	261	14	esωg′(esω	esωg′(esω	PROPN
ma-232	261	15	)	)	PUNCT
ma-232	261	16	ds	ds	PROPN
ma-232	261	17	=	=	SYM
ma-232	261	18	∫	∫	PROPN
ma-232	261	19	t	t	PROPN
ma-232	261	20	0	0	NUM
ma-232	261	21	ssg(ω	ssg(ω	PROPN
ma-232	261	22	)	)	PUNCT
ma-232	261	23	ds	ds	X
ma-232	261	24	where	where	SCONJ
ma-232	261	25	g(ω	g(ω	NOUN
ma-232	261	26	)	)	PUNCT
ma-232	261	27	=	=	SYM
ma-232	261	28	ωg′(ω	ωg′(ω	PROPN
ma-232	261	29	)	)	PUNCT
ma-232	261	30	.	.	PUNCT
ma-232	262	1	thus	thus	ADV
ma-232	262	2	,	,	PUNCT
ma-232	262	3	lim	lim	PROPN
ma-232	262	4	t→0	t→0	PROPN
ma-232	262	5	+	+	PROPN
ma-232	262	6	stg	stg	PROPN
ma-232	262	7	−	−	PROPN
ma-232	262	8	g	g	PROPN
ma-232	262	9	t	t	PROPN
ma-232	262	10	=	=	SYM
ma-232	262	11	lim	lim	PROPN
ma-232	262	12	t→0	t→0	PROPN
ma-232	262	13	+	+	CCONJ
ma-232	262	14	1	1	NUM
ma-232	262	15	t	t	NOUN
ma-232	262	16	∫	∫	PROPN
ma-232	262	17	t	t	PROPN
ma-232	262	18	0	0	NUM
ma-232	262	19	ssg(ω	ssg(ω	PROPN
ma-232	262	20	)	)	PUNCT
ma-232	262	21	ds	ds	ADJ
ma-232	262	22	and	and	CCONJ
ma-232	262	23	strong	strong	ADJ
ma-232	262	24	continuity	continuity	NOUN
ma-232	262	25	of	of	ADP
ma-232	262	26	(	(	PUNCT
ma-232	262	27	ss)t≥0	ss)t≥0	PROPN
ma-232	262	28	implies	imply	VERB
ma-232	262	29	that	that	SCONJ
ma-232	262	30	1	1	NUM
ma-232	262	31	t	t	NOUN
ma-232	262	32	∫	∫	PROPN
ma-232	262	33	t	t	PROPN
ma-232	262	34	0	0	NUM
ma-232	262	35	‖ssg	‖ssg	PROPN
ma-232	263	1	−	−	PROPN
ma-232	264	1	g‖ds	g‖ds	PROPN
ma-232	264	2	→	→	SYM
ma-232	264	3	0	0	PROPN
ma-232	264	4	as	as	ADP
ma-232	264	5	t	t	PROPN
ma-232	264	6	→	→	SYM
ma-232	264	7	0	0	NUM
ma-232	264	8	+	+	NOUN
ma-232	264	9	.	.	PUNCT
ma-232	264	10	hence	hence	ADV
ma-232	264	11	d(γ	d(γ	PROPN
ma-232	264	12	)	)	PUNCT
ma-232	264	13	⊇	⊇	NOUN
ma-232	264	14	{	{	PUNCT
ma-232	264	15	g	g	PROPN
ma-232	264	16	∈	∈	PROPN
ma-232	264	17	b∞,	b∞,	ADP
ma-232	264	18	◦	◦	NOUN
ma-232	264	19	(u	(u	NOUN
ma-232	264	20	)	)	PUNCT
ma-232	264	21	:	:	PUNCT
ma-232	264	22	ωg′(ω	ωg′(ω	NOUN
ma-232	264	23	)	)	PUNCT
ma-232	264	24	∈	∈	PROPN
ma-232	264	25	b∞,	b∞,	ADP
ma-232	264	26	◦	◦	NOUN
ma-232	264	27	(u	(u	NOUN
ma-232	264	28	)	)	PUNCT
ma-232	264	29	}	}	PUNCT
ma-232	264	30	,	,	PUNCT
ma-232	264	31	which	which	PRON
ma-232	264	32	completes	complete	VERB
ma-232	264	33	the	the	DET
ma-232	264	34	proof	proof	NOUN
ma-232	264	35	.	.	PUNCT
ma-232	265	1	�	�	PROPN
ma-232	265	2	3.1.2	3.1.2	NUM
ma-232	265	3	.	.	PUNCT
ma-232	266	1	spectral	spectral	ADJ
ma-232	266	2	properties	property	NOUN
ma-232	266	3	.	.	PUNCT
ma-232	267	1	now	now	ADV
ma-232	267	2	for	for	ADP
ma-232	267	3	the	the	DET
ma-232	267	4	spectral	spectral	ADJ
ma-232	267	5	properties	property	NOUN
ma-232	267	6	,	,	PUNCT
ma-232	267	7	we	we	PRON
ma-232	267	8	obtain	obtain	VERB
ma-232	267	9	the	the	DET
ma-232	267	10	spectra	spectra	NOUN
ma-232	267	11	of	of	ADP
ma-232	267	12	the	the	DET
ma-232	267	13	generator	generator	NOUN
ma-232	267	14	γ	γ	NOUN
ma-232	267	15	,	,	PUNCT
ma-232	267	16	determine	determine	VERB
ma-232	267	17	the	the	DET
ma-232	267	18	resolvents	resolvent	NOUN
ma-232	267	19	and	and	CCONJ
ma-232	267	20	further	far	ADV
ma-232	267	21	obtain	obtain	VERB
ma-232	267	22	the	the	DET
ma-232	267	23	spectra	spectra	NOUN
ma-232	267	24	and	and	CCONJ
ma-232	267	25	the	the	DET
ma-232	267	26	norms	norm	NOUN
ma-232	267	27	of	of	ADP
ma-232	267	28	the	the	DET
ma-232	267	29	resulting	result	VERB
ma-232	267	30	resolvents	resolvent	NOUN
ma-232	267	31	.	.	PUNCT
ma-232	268	1	theorem	theorem	VERB
ma-232	268	2	3.3	3.3	NUM
ma-232	268	3	.	.	PUNCT
ma-232	269	1	let	let	VERB
ma-232	269	2	γ	γ	NOUN
ma-232	269	3	be	be	AUX
ma-232	269	4	the	the	DET
ma-232	269	5	infinitesimal	infinitesimal	ADJ
ma-232	269	6	generator	generator	NOUN
ma-232	269	7	of	of	ADP
ma-232	269	8	(	(	PUNCT
ma-232	269	9	st)t∈r	st)t∈r	NOUN
ma-232	269	10	on	on	ADP
ma-232	269	11	b∞,	b∞,	ADP
ma-232	269	12	◦	◦	NOUN
ma-232	269	13	(u	(u	NOUN
ma-232	269	14	)	)	PUNCT
ma-232	269	15	.	.	PUNCT
ma-232	270	1	then	then	ADV
ma-232	270	2	σp(γ	σp(γ	NOUN
ma-232	270	3	)	)	PUNCT
ma-232	270	4	=	=	SYM
ma-232	270	5	∅	∅	NOUN
ma-232	270	6	and	and	CCONJ
ma-232	270	7	σ(γ	σ(γ	PROPN
ma-232	270	8	)	)	PUNCT
ma-232	271	1	=	=	SYM
ma-232	271	2	ir	ir	PROPN
ma-232	271	3	.	.	PUNCT
ma-232	272	1	in	in	ADP
ma-232	272	2	particular	particular	ADJ
ma-232	272	3	,	,	PUNCT
ma-232	272	4	γ	γ	PROPN
ma-232	272	5	is	be	AUX
ma-232	272	6	an	an	DET
ma-232	272	7	unbounded	unbounded	ADJ
ma-232	272	8	operator	operator	NOUN
ma-232	272	9	on	on	ADP
ma-232	272	10	b∞,	b∞,	ADP
ma-232	272	11	◦	◦	NOUN
ma-232	272	12	(u	(u	NOUN
ma-232	272	13	)	)	PUNCT
ma-232	272	14	.	.	PUNCT
ma-232	273	1	before	before	SCONJ
ma-232	273	2	we	we	PRON
ma-232	273	3	prove	prove	VERB
ma-232	273	4	this	this	DET
ma-232	273	5	theorem	theorem	NOUN
ma-232	273	6	,	,	PUNCT
ma-232	273	7	we	we	PRON
ma-232	273	8	first	first	ADV
ma-232	273	9	give	give	VERB
ma-232	273	10	the	the	DET
ma-232	273	11	following	follow	VERB
ma-232	273	12	lemma	lemma	PROPN
ma-232	273	13	:	:	PUNCT
ma-232	273	14	lemma	lemma	PROPN
ma-232	273	15	3.4	3.4	NUM
ma-232	273	16	.	.	PUNCT
ma-232	274	1	if	if	SCONJ
ma-232	274	2	ν	ν	PROPN
ma-232	274	3	∈	∈	PROPN
ma-232	274	4	c	c	NOUN
ma-232	274	5	and	and	CCONJ
ma-232	274	6	c	c	NOUN
ma-232	274	7	∈	∈	PROPN
ma-232	275	1	r	r	NOUN
ma-232	275	2	,	,	PUNCT
ma-232	275	3	we	we	PRON
ma-232	275	4	have	have	VERB
ma-232	275	5	(	(	PUNCT
ma-232	275	6	1	1	X
ma-232	275	7	)	)	PUNCT
ma-232	275	8	g(ω	g(ω	NOUN
ma-232	275	9	)	)	PUNCT
ma-232	276	1	=	=	SYM
ma-232	276	2	cων	cων	PROPN
ma-232	276	3	/∈	/∈	PUNCT
ma-232	276	4	b∞,0(u	b∞,0(u	PROPN
ma-232	276	5	)	)	PUNCT
ma-232	276	6	for	for	ADP
ma-232	276	7	any	any	DET
ma-232	276	8	c	c	NOUN
ma-232	276	9	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	276	10	eur	eur	PROPN
ma-232	276	11	.	.	PUNCT
ma-232	277	1	j.	j.	PROPN
ma-232	277	2	math	math	PROPN
ma-232	277	3	.	.	PUNCT
ma-232	278	1	anal	anal	PROPN
ma-232	278	2	.	.	PUNCT
ma-232	279	1	10.28924	10.28924	NUM
ma-232	279	2	/	/	SYM
ma-232	279	3	ada	ada	PROPN
ma-232	279	4	/	/	SYM
ma-232	279	5	ma.4.14	ma.4.14	PROPN
ma-232	279	6	11	11	NUM
ma-232	279	7	(	(	PUNCT
ma-232	279	8	2	2	NUM
ma-232	279	9	)	)	PUNCT
ma-232	279	10	f	f	PROPN
ma-232	279	11	(	(	PUNCT
ma-232	279	12	ω	ω	NOUN
ma-232	279	13	)	)	PUNCT
ma-232	279	14	=	=	SYM
ma-232	280	1	(	(	PUNCT
ma-232	280	2	w	w	PROPN
ma-232	280	3	−	−	PROPN
ma-232	280	4	i)ν	i)ν	ADJ
ma-232	280	5	∈	∈	PROPN
ma-232	280	6	b∞,0(u	b∞,0(u	PROPN
ma-232	280	7	)	)	PUNCT
ma-232	281	1	if	if	SCONJ
ma-232	281	2	and	and	CCONJ
ma-232	281	3	only	only	ADV
ma-232	281	4	if	if	SCONJ
ma-232	281	5	<	<	X
ma-232	281	6	(	(	PUNCT
ma-232	281	7	ν	ν	NOUN
ma-232	281	8	)	)	PUNCT
ma-232	281	9	<	<	X
ma-232	281	10	0	0	X
ma-232	281	11	.	.	PUNCT
ma-232	282	1	in	in	ADP
ma-232	282	2	particular	particular	ADJ
ma-232	282	3	,	,	PUNCT
ma-232	282	4	g(ω	g(ω	PROPN
ma-232	282	5	)	)	PUNCT
ma-232	282	6	/∈	/∈	PUNCT
ma-232	283	1	b∞,0(u	b∞,0(u	PROPN
ma-232	283	2	)	)	PUNCT
ma-232	283	3	for	for	ADP
ma-232	283	4	any	any	DET
ma-232	283	5	c	c	PROPN
ma-232	283	6	and	and	CCONJ
ma-232	283	7	f	f	PROPN
ma-232	283	8	(	(	PUNCT
ma-232	283	9	ω	ω	NOUN
ma-232	283	10	)	)	PUNCT
ma-232	283	11	∈	∈	PROPN
ma-232	283	12	b∞,0(u	b∞,0(u	PROPN
ma-232	283	13	)	)	PUNCT
ma-232	283	14	if	if	SCONJ
ma-232	283	15	and	and	CCONJ
ma-232	283	16	only	only	ADV
ma-232	283	17	if	if	SCONJ
ma-232	283	18	<	<	X
ma-232	283	19	(	(	PUNCT
ma-232	283	20	ν	ν	NOUN
ma-232	283	21	)	)	PUNCT
ma-232	283	22	<	<	X
ma-232	283	23	0	0	X
ma-232	283	24	.	.	PUNCT
ma-232	284	1	proof	proof	NOUN
ma-232	284	2	.	.	PUNCT
ma-232	285	1	from	from	ADP
ma-232	285	2	proposition	proposition	NOUN
ma-232	285	3	2.3	2.3	NUM
ma-232	285	4	,	,	PUNCT
ma-232	285	5	we	we	PRON
ma-232	285	6	know	know	VERB
ma-232	285	7	that	that	SCONJ
ma-232	285	8	g	g	PROPN
ma-232	285	9	∈	∈	PROPN
ma-232	285	10	b∞,	b∞,	ADP
ma-232	285	11	◦	◦	NOUN
ma-232	285	12	(u	(u	NOUN
ma-232	285	13	)	)	PUNCT
ma-232	285	14	if	if	SCONJ
ma-232	285	15	and	and	CCONJ
ma-232	285	16	only	only	ADV
ma-232	285	17	if	if	SCONJ
ma-232	285	18	g	g	PROPN
ma-232	285	19	◦	◦	NOUN
ma-232	285	20	ψ	ψ	X
ma-232	285	21	∈	∈	NOUN
ma-232	285	22	b∞,	b∞,	ADP
ma-232	285	23	◦	◦	NOUN
ma-232	285	24	(d	(d	NOUN
ma-232	285	25	)	)	PUNCT
ma-232	285	26	.	.	PUNCT
ma-232	286	1	then	then	ADV
ma-232	286	2	for	for	ADP
ma-232	286	3	z	z	PROPN
ma-232	286	4	∈	∈	PROPN
ma-232	286	5	d	d	PROPN
ma-232	286	6	,	,	PUNCT
ma-232	286	7	(	(	PUNCT
ma-232	286	8	g	g	PROPN
ma-232	286	9	◦	◦	NOUN
ma-232	286	10	ψ)(z	ψ)(z	PUNCT
ma-232	286	11	)	)	PUNCT
ma-232	286	12	=	=	SYM
ma-232	286	13	g(ψ(z	g(ψ(z	NOUN
ma-232	286	14	)	)	PUNCT
ma-232	286	15	)	)	PUNCT
ma-232	287	1	=	=	SYM
ma-232	287	2	c(ψ(z))ν	c(ψ(z))ν	NOUN
ma-232	288	1	=	=	SYM
ma-232	288	2	c	c	X
ma-232	288	3	(	(	PUNCT
ma-232	288	4	i(1	i(1	PROPN
ma-232	288	5	+	+	PROPN
ma-232	288	6	z	z	NOUN
ma-232	288	7	)	)	PUNCT
ma-232	288	8	1−	1−	NUM
ma-232	288	9	z	z	NOUN
ma-232	288	10	)	)	PUNCT
ma-232	289	1	ν	ν	NOUN
ma-232	289	2	=	=	SYM
ma-232	289	3	ci(1	ci(1	PROPN
ma-232	289	4	+	+	NOUN
ma-232	289	5	z)ν(1−	z)ν(1−	PROPN
ma-232	289	6	z)−ν	z)−ν	PROPN
ma-232	289	7	.	.	PUNCT
ma-232	290	1	now	now	ADV
ma-232	290	2	g	g	ADP
ma-232	290	3	◦	◦	NOUN
ma-232	290	4	ψ	ψ	X
ma-232	290	5	∈	∈	PROPN
ma-232	290	6	h(d	h(d	PROPN
ma-232	290	7	)	)	PUNCT
ma-232	291	1	if	if	SCONJ
ma-232	291	2	and	and	CCONJ
ma-232	291	3	only	only	ADV
ma-232	291	4	if	if	SCONJ
ma-232	291	5	<	<	X
ma-232	291	6	(	(	PUNCT
ma-232	291	7	ν	ν	NOUN
ma-232	291	8	)	)	PUNCT
ma-232	291	9	>	>	X
ma-232	291	10	0	0	PUNCT
ma-232	291	11	and	and	CCONJ
ma-232	291	12	<	<	X
ma-232	291	13	(	(	PUNCT
ma-232	291	14	−ν	−ν	ADV
ma-232	291	15	)	)	PUNCT
ma-232	291	16	>	>	X
ma-232	291	17	0	0	NUM
ma-232	292	1	which	which	PRON
ma-232	292	2	is	be	AUX
ma-232	292	3	not	not	PART
ma-232	292	4	possible	possible	ADJ
ma-232	292	5	,	,	PUNCT
ma-232	292	6	and	and	CCONJ
ma-232	292	7	therefore	therefore	ADV
ma-232	292	8	g	g	PROPN
ma-232	292	9	◦	◦	NOUN
ma-232	292	10	ψ	ψ	X
ma-232	292	11	/∈	/∈	PUNCT
ma-232	292	12	h(d	h(d	PROPN
ma-232	292	13	)	)	PUNCT
ma-232	292	14	.	.	PUNCT
ma-232	293	1	hence	hence	ADV
ma-232	293	2	g	g	PROPN
ma-232	293	3	/∈	/∈	PUNCT
ma-232	293	4	b∞,0(u	b∞,0(u	PROPN
ma-232	293	5	)	)	PUNCT
ma-232	293	6	.	.	PUNCT
ma-232	294	1	this	this	PRON
ma-232	294	2	proves	prove	VERB
ma-232	294	3	(	(	PUNCT
ma-232	294	4	1	1	NUM
ma-232	294	5	)	)	PUNCT
ma-232	294	6	.	.	PUNCT
ma-232	295	1	for	for	ADP
ma-232	295	2	(	(	PUNCT
ma-232	295	3	2	2	NUM
ma-232	295	4	)	)	PUNCT
ma-232	295	5	,	,	PUNCT
ma-232	295	6	following	follow	VERB
ma-232	295	7	[	[	X
ma-232	295	8	3	3	NUM
ma-232	295	9	,	,	PUNCT
ma-232	295	10	lemma	lemma	PROPN
ma-232	295	11	3.2	3.2	NUM
ma-232	295	12	]	]	PUNCT
ma-232	295	13	,	,	PUNCT
ma-232	295	14	for	for	ADP
ma-232	295	15	any	any	DET
ma-232	295	16	ν	ν	NOUN
ma-232	295	17	∈	∈	PROPN
ma-232	295	18	c	c	X
ma-232	295	19	,	,	PUNCT
ma-232	295	20	(	(	PUNCT
ma-232	295	21	w	w	NOUN
ma-232	295	22	−	−	PROPN
ma-232	295	23	i)ν	i)ν	ADJ
ma-232	295	24	∈	∈	PROPN
ma-232	295	25	h(u	h(u	PROPN
ma-232	295	26	)	)	PUNCT
ma-232	295	27	if	if	SCONJ
ma-232	295	28	and	and	CCONJ
ma-232	295	29	only	only	ADV
ma-232	295	30	if	if	SCONJ
ma-232	295	31	<	<	X
ma-232	295	32	(	(	PUNCT
ma-232	295	33	ν	ν	NOUN
ma-232	295	34	)	)	PUNCT
ma-232	295	35	<	<	X
ma-232	295	36	0	0	PUNCT
ma-232	295	37	since	since	SCONJ
ma-232	295	38	γ	γ	X
ma-232	295	39	=	=	SYM
ma-232	295	40	0	0	NUM
ma-232	295	41	in	in	ADP
ma-232	295	42	this	this	PRON
ma-232	295	43	case.the	case.the	DET
ma-232	295	44	particular	particular	ADJ
ma-232	295	45	cases	case	NOUN
ma-232	295	46	follow	follow	VERB
ma-232	295	47	immediately	immediately	ADV
ma-232	295	48	since	since	SCONJ
ma-232	295	49	b∞,0(u	b∞,0(u	PROPN
ma-232	295	50	,	,	PUNCT
ma-232	295	51	i	i	NOUN
ma-232	295	52	)	)	PUNCT
ma-232	295	53	⊆	⊆	NUM
ma-232	295	54	b∞,0(u	b∞,0(u	PROPN
ma-232	295	55	)	)	PUNCT
ma-232	295	56	and	and	CCONJ
ma-232	295	57	g(i	g(i	NOUN
ma-232	295	58	)	)	PUNCT
ma-232	295	59	6=	6=	ADP
ma-232	295	60	0	0	NUM
ma-232	295	61	for	for	ADP
ma-232	295	62	(	(	PUNCT
ma-232	295	63	1	1	NUM
ma-232	295	64	)	)	PUNCT
ma-232	295	65	,	,	PUNCT
ma-232	295	66	while	while	SCONJ
ma-232	295	67	f	f	PROPN
ma-232	295	68	(	(	PUNCT
ma-232	295	69	i	i	NOUN
ma-232	295	70	)	)	PUNCT
ma-232	295	71	=	=	SYM
ma-232	295	72	0	0	NUM
ma-232	295	73	for	for	ADP
ma-232	295	74	(	(	PUNCT
ma-232	295	75	2	2	NUM
ma-232	295	76	)	)	PUNCT
ma-232	295	77	.	.	PUNCT
ma-232	296	1	�	�	PROPN
ma-232	296	2	proof	proof	NOUN
ma-232	296	3	of	of	ADP
ma-232	296	4	theorem	theorem	ADJ
ma-232	296	5	3.3	3.3	NUM
ma-232	296	6	.	.	PUNCT
ma-232	297	1	to	to	PART
ma-232	297	2	obtain	obtain	VERB
ma-232	297	3	the	the	DET
ma-232	297	4	point	point	NOUN
ma-232	297	5	spectrum	spectrum	NOUN
ma-232	297	6	of	of	ADP
ma-232	297	7	γ	γ	PROPN
ma-232	297	8	,	,	PUNCT
ma-232	297	9	let	let	VERB
ma-232	297	10	λ	λ	PRON
ma-232	297	11	be	be	AUX
ma-232	297	12	an	an	DET
ma-232	297	13	eigenvalue	eigenvalue	NOUN
ma-232	297	14	of	of	ADP
ma-232	297	15	γ	γ	NOUN
ma-232	297	16	and	and	CCONJ
ma-232	297	17	g	g	PROPN
ma-232	297	18	be	be	AUX
ma-232	297	19	thecorresponding	thecorresponde	VERB
ma-232	297	20	eigenvector	eigenvector	NOUN
ma-232	297	21	.	.	PUNCT
ma-232	298	1	then	then	ADV
ma-232	298	2	γg(ω	γg(ω	PUNCT
ma-232	298	3	)	)	PUNCT
ma-232	299	1	=	=	SYM
ma-232	299	2	λg(ω	λg(ω	X
ma-232	299	3	)	)	PUNCT
ma-232	299	4	is	be	AUX
ma-232	299	5	equivalent	equivalent	ADJ
ma-232	299	6	to	to	ADP
ma-232	299	7	ωg′(ω	ωg′(ω	PROPN
ma-232	299	8	)	)	PUNCT
ma-232	299	9	=	=	PUNCT
ma-232	299	10	λg(ω	λg(ω	X
ma-232	299	11	)	)	PUNCT
ma-232	299	12	which	which	PRON
ma-232	299	13	yields	yield	VERB
ma-232	299	14	ωg′(ω	ωg′(ω	PROPN
ma-232	299	15	)	)	PUNCT
ma-232	299	16	ω	ω	PROPN
ma-232	299	17	=	=	SYM
ma-232	299	18	λg(ω	λg(ω	CCONJ
ma-232	299	19	)	)	PUNCT
ma-232	299	20	ω	ω	NOUN
ma-232	299	21	by	by	ADP
ma-232	299	22	dividing	divide	VERB
ma-232	299	23	both	both	DET
ma-232	299	24	sides	side	NOUN
ma-232	299	25	by	by	ADP
ma-232	299	26	ω	ω	PROPN
ma-232	299	27	.	.	PUNCT
ma-232	300	1	by	by	ADP
ma-232	300	2	integrating	integrate	VERB
ma-232	300	3	both	both	DET
ma-232	300	4	sides	side	NOUN
ma-232	300	5	,	,	PUNCT
ma-232	300	6	we	we	PRON
ma-232	300	7	obtain	obtain	VERB
ma-232	300	8	g(ω	g(ω	X
ma-232	300	9	)	)	PUNCT
ma-232	300	10	=	=	SYM
ma-232	300	11	cωλ	cωλ	NOUN
ma-232	300	12	,	,	PUNCT
ma-232	300	13	which	which	PRON
ma-232	300	14	is	be	AUX
ma-232	300	15	not	not	PART
ma-232	300	16	in	in	ADP
ma-232	300	17	b∞,	b∞,	ADP
ma-232	300	18	◦	◦	NOUN
ma-232	300	19	(u	(u	NOUN
ma-232	300	20	)	)	PUNCT
ma-232	300	21	for	for	ADP
ma-232	300	22	any	any	DET
ma-232	300	23	c	c	NOUN
ma-232	300	24	.	.	PUNCT
ma-232	301	1	therefore	therefore	ADV
ma-232	301	2	σp(γ	σp(γ	NOUN
ma-232	301	3	)	)	PUNCT
ma-232	301	4	=	=	PUNCT
ma-232	301	5	∅.since	∅.since	NOUN
ma-232	301	6	each	each	DET
ma-232	301	7	st	st	PROPN
ma-232	301	8	is	be	AUX
ma-232	301	9	an	an	DET
ma-232	301	10	invertible	invertible	ADJ
ma-232	301	11	isometry	isometry	NOUN
ma-232	301	12	,	,	PUNCT
ma-232	301	13	its	its	PRON
ma-232	301	14	spectrum	spectrum	NOUN
ma-232	301	15	satisfies	satisfy	VERB
ma-232	301	16	σ(st	σ(st	PROPN
ma-232	301	17	)	)	PUNCT
ma-232	301	18	⊆	⊆	NUM
ma-232	301	19	∂d	∂d	PROPN
ma-232	301	20	.	.	PUNCT
ma-232	302	1	therefore	therefore	ADV
ma-232	302	2	the	the	DET
ma-232	302	3	spectralmapping	spectralmapping	NOUN
ma-232	302	4	theorem	theorem	NOUN
ma-232	302	5	for	for	ADP
ma-232	302	6	strongly	strongly	ADV
ma-232	302	7	continuous	continuous	ADJ
ma-232	302	8	groups	group	NOUN
ma-232	302	9	[	[	X
ma-232	302	10	10	10	NUM
ma-232	302	11	,	,	PUNCT
ma-232	302	12	theorem	theorem	VERB
ma-232	302	13	2.3	2.3	NUM
ma-232	302	14	]	]	PUNCT
ma-232	302	15	implies	imply	VERB
ma-232	302	16	that	that	SCONJ
ma-232	302	17	etσ(γ	etσ(γ	PROPN
ma-232	302	18	)	)	PUNCT
ma-232	302	19	⊆	⊆	NUM
ma-232	302	20	σ(st	σ(st	PROPN
ma-232	302	21	)	)	PUNCT
ma-232	302	22	⊆	⊆	NUM
ma-232	302	23	∂d	∂d	PROPN
ma-232	302	24	.	.	PUNCT
ma-232	303	1	now	now	ADV
ma-232	303	2	let	let	VERB
ma-232	303	3	λ	λ	X
ma-232	303	4	∈	∈	VERB
ma-232	303	5	σ(γ	σ(γ	PROPN
ma-232	303	6	)	)	PUNCT
ma-232	303	7	,	,	PUNCT
ma-232	303	8	then	then	ADV
ma-232	303	9	|etλ|	|etλ|	VERB
ma-232	303	10	=	=	SYM
ma-232	303	11	1	1	NUM
ma-232	303	12	which	which	PRON
ma-232	303	13	further	far	ADV
ma-232	303	14	implies	imply	VERB
ma-232	303	15	that	that	SCONJ
ma-232	303	16	<	<	X
ma-232	303	17	(	(	PUNCT
ma-232	303	18	λ	λ	X
ma-232	303	19	)	)	PUNCT
ma-232	303	20	=	=	SYM
ma-232	303	21	0	0	X
ma-232	303	22	.	.	PUNCT
ma-232	304	1	thus	thus	ADV
ma-232	304	2	λ	λ	X
ma-232	304	3	∈	∈	PROPN
ma-232	304	4	ir	ir	PROPN
ma-232	304	5	andtherefore	andtherefore	NOUN
ma-232	304	6	σ(γ	σ(γ	PROPN
ma-232	304	7	)	)	PUNCT
ma-232	304	8	⊆	⊆	NUM
ma-232	304	9	ir.we	ir.we	NOUN
ma-232	304	10	now	now	ADV
ma-232	304	11	need	need	VERB
ma-232	304	12	to	to	PART
ma-232	304	13	show	show	VERB
ma-232	304	14	the	the	DET
ma-232	304	15	reverse	reverse	ADJ
ma-232	304	16	inclusion	inclusion	NOUN
ma-232	304	17	,	,	PUNCT
ma-232	304	18	that	that	ADV
ma-232	304	19	is	is	ADV
ma-232	304	20	,	,	PUNCT
ma-232	304	21	ir	ir	PROPN
ma-232	304	22	⊆	⊆	NUM
ma-232	304	23	σ(γ	σ(γ	PROPN
ma-232	304	24	)	)	PUNCT
ma-232	304	25	.	.	PUNCT
ma-232	305	1	fix	fix	VERB
ma-232	305	2	λ	λ	PROPN
ma-232	305	3	∈	∈	PROPN
ma-232	305	4	ir	ir	PROPN
ma-232	305	5	and	and	CCONJ
ma-232	305	6	assume	assume	VERB
ma-232	305	7	λ	λ	X
ma-232	305	8	/∈	/∈	PROPN
ma-232	305	9	σ(γ)which	σ(γ)which	PROPN
ma-232	305	10	implies	imply	VERB
ma-232	305	11	that	that	SCONJ
ma-232	305	12	the	the	DET
ma-232	305	13	resolvent	resolvent	ADJ
ma-232	305	14	operator	operator	NOUN
ma-232	305	15	r(λ	r(λ	NOUN
ma-232	305	16	,	,	PUNCT
ma-232	305	17	γ	γ	NOUN
ma-232	305	18	)	)	PUNCT
ma-232	305	19	:	:	PUNCT
ma-232	305	20	b∞,	b∞,	PUNCT
ma-232	305	21	◦	◦	NOUN
ma-232	305	22	(u	(u	NOUN
ma-232	305	23	)	)	PUNCT
ma-232	305	24	→	→	SYM
ma-232	305	25	b∞,	b∞,	ADP
ma-232	305	26	◦	◦	NOUN
ma-232	305	27	(u	(u	NOUN
ma-232	305	28	)	)	PUNCT
ma-232	305	29	is	be	AUX
ma-232	305	30	bounded	bound	VERB
ma-232	305	31	.	.	PUNCT
ma-232	306	1	considerthe	considerthe	PROPN
ma-232	306	2	function	function	NOUN
ma-232	306	3	h(w	h(w	PROPN
ma-232	306	4	)	)	PUNCT
ma-232	307	1	=	=	PRON
ma-232	307	2	(	(	PUNCT
ma-232	307	3	w	w	NOUN
ma-232	307	4	−	−	NOUN
ma-232	307	5	i)−(λ+1	i)−(λ+1	NOUN
ma-232	307	6	)	)	PUNCT
ma-232	307	7	.	.	PUNCT
ma-232	308	1	then	then	ADV
ma-232	308	2	<	<	X
ma-232	308	3	(	(	PUNCT
ma-232	308	4	−(λ	−(λ	NOUN
ma-232	308	5	+	+	X
ma-232	308	6	1	1	NUM
ma-232	308	7	)	)	PUNCT
ma-232	308	8	)	)	PUNCT
ma-232	308	9	=	=	PUNCT
ma-232	309	1	−1	−1	NOUN
ma-232	309	2	<	<	X
ma-232	309	3	0	0	PUNCT
ma-232	310	1	and	and	CCONJ
ma-232	310	2	following	follow	VERB
ma-232	310	3	lemma	lemma	PROPN
ma-232	310	4	3.4	3.4	NUM
ma-232	310	5	,	,	PUNCT
ma-232	310	6	itis	itis	NOUN
ma-232	310	7	immediate	immediate	ADJ
ma-232	310	8	that	that	SCONJ
ma-232	310	9	h	h	NOUN
ma-232	310	10	∈	∈	NOUN
ma-232	310	11	b∞,	b∞,	ADP
ma-232	310	12	◦	◦	NOUN
ma-232	310	13	(u	(u	NOUN
ma-232	310	14	)	)	PUNCT
ma-232	310	15	.	.	PUNCT
ma-232	311	1	the	the	DET
ma-232	311	2	image	image	NOUN
ma-232	311	3	function	function	NOUN
ma-232	311	4	f	f	PROPN
ma-232	311	5	=	=	SYM
ma-232	311	6	r(λ	r(λ	PROPN
ma-232	311	7	,	,	PUNCT
ma-232	311	8	γ)h	γ)h	X
ma-232	311	9	is	be	AUX
ma-232	311	10	equivalent	equivalent	ADJ
ma-232	311	11	to	to	ADP
ma-232	311	12	(	(	PUNCT
ma-232	311	13	λ	λ	X
ma-232	311	14	−	−	PROPN
ma-232	311	15	γ)f	γ)f	PROPN
ma-232	311	16	=	=	PUNCT
ma-232	311	17	hwhich	hwhich	PRON
ma-232	311	18	yields	yield	VERB
ma-232	311	19	a	a	DET
ma-232	311	20	differential	differential	ADJ
ma-232	311	21	equation	equation	NOUN
ma-232	311	22	f	f	PROPN
ma-232	311	23	′(ω)−	′(ω)−	PROPN
ma-232	311	24	λ	λ	PROPN
ma-232	311	25	ω	ω	NUM
ma-232	311	26	f	f	PROPN
ma-232	311	27	(	(	PUNCT
ma-232	311	28	ω	ω	NOUN
ma-232	311	29	)	)	PUNCT
ma-232	311	30	=	=	SYM
ma-232	311	31	−	−	PROPN
ma-232	311	32	h(ω	h(ω	PROPN
ma-232	311	33	)	)	PUNCT
ma-232	311	34	ω	ω	PROPN
ma-232	311	35	,	,	PUNCT
ma-232	311	36	whose	whose	DET
ma-232	311	37	general	general	ADJ
ma-232	311	38	solution	solution	NOUN
ma-232	311	39	is	be	AUX
ma-232	311	40	f	f	PROPN
ma-232	311	41	(	(	PUNCT
ma-232	311	42	ω	ω	NOUN
ma-232	311	43	)	)	PUNCT
ma-232	311	44	=	=	SYM
ma-232	312	1	(	(	PUNCT
ma-232	312	2	ω	ω	NOUN
ma-232	312	3	−	−	X
ma-232	313	1	i)−λ	i)−λ	PUNCT
ma-232	313	2	+	+	NUM
ma-232	313	3	cωλwhich	cωλwhich	NOUN
ma-232	313	4	does	do	AUX
ma-232	313	5	not	not	PART
ma-232	313	6	belong	belong	VERB
ma-232	313	7	to	to	ADP
ma-232	313	8	b∞,	b∞,	ADP
ma-232	313	9	◦	◦	VERB
ma-232	313	10	(u	(u	NOUN
ma-232	313	11	)	)	PUNCT
ma-232	313	12	for	for	ADP
ma-232	313	13	any	any	DET
ma-232	313	14	c	c	NOUN
ma-232	313	15	,	,	PUNCT
ma-232	313	16	by	by	ADP
ma-232	313	17	lemma	lemma	PROPN
ma-232	313	18	3.4	3.4	NUM
ma-232	313	19	.	.	PUNCT
ma-232	314	1	thus	thus	ADV
ma-232	314	2	h	h	NOUN
ma-232	314	3	/∈	/∈	PUNCT
ma-232	315	1	r(λ	r(λ	NOUN
ma-232	315	2	−	−	PROPN
ma-232	315	3	γ	γ	NOUN
ma-232	315	4	)	)	PUNCT
ma-232	315	5	and	and	CCONJ
ma-232	315	6	so	so	ADV
ma-232	315	7	σ(γ	σ(γ	PROPN
ma-232	315	8	)	)	PUNCT
ma-232	315	9	=	=	SYM
ma-232	315	10	ir	ir	PROPN
ma-232	315	11	.	.	PROPN
ma-232	315	12	�	�	PROPN
ma-232	315	13	theorem	theorem	VERB
ma-232	315	14	3.5	3.5	NUM
ma-232	315	15	.	.	PUNCT
ma-232	316	1	let	let	VERB
ma-232	316	2	γ	γ	NOUN
ma-232	316	3	be	be	AUX
ma-232	316	4	the	the	DET
ma-232	316	5	infinitesimal	infinitesimal	ADJ
ma-232	316	6	generator	generator	NOUN
ma-232	316	7	of	of	ADP
ma-232	316	8	(	(	PUNCT
ma-232	316	9	st)t∈r	st)t∈r	NOUN
ma-232	316	10	.	.	PUNCT
ma-232	317	1	then	then	ADV
ma-232	317	2	the	the	DET
ma-232	317	3	following	follow	VERB
ma-232	317	4	hold	hold	NOUN
ma-232	317	5	;	;	PUNCT
ma-232	317	6	(	(	PUNCT
ma-232	317	7	1	1	X
ma-232	317	8	)	)	PUNCT
ma-232	317	9	for	for	ADP
ma-232	317	10	λ	λ	PROPN
ma-232	317	11	∈	∈	PROPN
ma-232	317	12	ρ(γ	ρ(γ	NOUN
ma-232	317	13	)	)	PUNCT
ma-232	317	14	,	,	PUNCT
ma-232	317	15	and	and	CCONJ
ma-232	317	16	h	h	NOUN
ma-232	317	17	∈	∈	PROPN
ma-232	317	18	b∞,	b∞,	ADP
ma-232	317	19	◦	◦	NOUN
ma-232	317	20	(u	(u	NOUN
ma-232	317	21	)	)	PUNCT
ma-232	317	22	then	then	ADV
ma-232	317	23	,	,	PUNCT
ma-232	317	24	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	317	25	eur	eur	PROPN
ma-232	317	26	.	.	PUNCT
ma-232	318	1	j.	j.	PROPN
ma-232	318	2	math	math	PROPN
ma-232	318	3	.	.	PUNCT
ma-232	319	1	anal	anal	PROPN
ma-232	319	2	.	.	PUNCT
ma-232	320	1	10.28924	10.28924	NUM
ma-232	320	2	/	/	SYM
ma-232	320	3	ada	ada	PROPN
ma-232	320	4	/	/	SYM
ma-232	320	5	ma.4.14	ma.4.14	NOUN
ma-232	320	6	12(i	12(i	NUM
ma-232	320	7	)	)	PUNCT
ma-232	320	8	r(λ	r(λ	NOUN
ma-232	320	9	,	,	PUNCT
ma-232	320	10	γ)h(ω	γ)h(ω	ADJ
ma-232	320	11	)	)	PUNCT
ma-232	320	12	=	=	VERB
ma-232	320	13	ωλ	ωλ	VERB
ma-232	320	14	∫∞	∫∞	NOUN
ma-232	320	15	ω	ω	NOUN
ma-232	320	16	1	1	NUM
ma-232	320	17	zλ+1	zλ+1	NUM
ma-232	320	18	h(z	h(z	NOUN
ma-232	320	19	)	)	PUNCT
ma-232	320	20	dz	dz	NOUN
ma-232	320	21	,	,	PUNCT
ma-232	320	22	if	if	SCONJ
ma-232	320	23	<	<	X
ma-232	320	24	(	(	PUNCT
ma-232	320	25	λ	λ	X
ma-232	320	26	)	)	PUNCT
ma-232	320	27	>	>	PUNCT
ma-232	320	28	0.(ii	0.(ii	X
ma-232	320	29	)	)	PUNCT
ma-232	320	30	r(λ	r(λ	NOUN
ma-232	320	31	,	,	PUNCT
ma-232	320	32	γ)h(ω	γ)h(ω	ADJ
ma-232	320	33	)	)	PUNCT
ma-232	320	34	=	=	SYM
ma-232	321	1	−ωλ	−ωλ	PROPN
ma-232	321	2	∫	∫	PROPN
ma-232	321	3	ω	ω	NOUN
ma-232	321	4	0	0	NUM
ma-232	321	5	1	1	NUM
ma-232	321	6	zλ+1	zλ+1	NUM
ma-232	321	7	h(z	h(z	NOUN
ma-232	321	8	)	)	PUNCT
ma-232	321	9	dz	dz	NOUN
ma-232	321	10	,	,	PUNCT
ma-232	321	11	if	if	SCONJ
ma-232	321	12	<	<	X
ma-232	321	13	(	(	PUNCT
ma-232	321	14	λ	λ	X
ma-232	321	15	)	)	PUNCT
ma-232	321	16	<	<	X
ma-232	321	17	0	0	NUM
ma-232	321	18	.	.	PUNCT
ma-232	322	1	(	(	PUNCT
ma-232	322	2	2	2	X
ma-232	322	3	)	)	PUNCT
ma-232	322	4	σ(r(λ	σ(r(λ	PROPN
ma-232	322	5	,	,	PUNCT
ma-232	322	6	γ	γ	NOUN
ma-232	322	7	)	)	PUNCT
ma-232	322	8	)	)	PUNCT
ma-232	323	1	=	=	PRON
ma-232	323	2	{	{	PUNCT
ma-232	323	3	ω	ω	NOUN
ma-232	323	4	:	:	PUNCT
ma-232	323	5	|ω	|ω	PROPN
ma-232	323	6	−	−	PROPN
ma-232	323	7	1	1	NUM
ma-232	323	8	2<(λ	2<(λ	NUM
ma-232	323	9	)	)	PUNCT
ma-232	324	1	|	|	ADV
ma-232	324	2	=	=	SYM
ma-232	324	3	1	1	NUM
ma-232	324	4	2<(λ	2<(λ	NUM
ma-232	324	5	)	)	PUNCT
ma-232	324	6	}	}	PUNCT
ma-232	324	7	.	.	PUNCT
ma-232	325	1	(	(	PUNCT
ma-232	325	2	3	3	X
ma-232	325	3	)	)	PUNCT
ma-232	325	4	r(r(λ	r(r(λ	PROPN
ma-232	325	5	,	,	PUNCT
ma-232	325	6	γ	γ	NOUN
ma-232	325	7	)	)	PUNCT
ma-232	325	8	)	)	PUNCT
ma-232	326	1	=	=	SYM
ma-232	326	2	‖r(λ	‖r(λ	X
ma-232	326	3	,	,	PUNCT
ma-232	326	4	γ)‖	γ)‖	NOUN
ma-232	326	5	=	=	SYM
ma-232	326	6	1	1	NUM
ma-232	326	7	|<(λ)|	|<(λ)|	NOUN
ma-232	326	8	.	.	PUNCT
ma-232	327	1	proof	proof	NOUN
ma-232	327	2	.	.	PUNCT
ma-232	328	1	to	to	PART
ma-232	328	2	prove	prove	VERB
ma-232	328	3	(	(	PUNCT
ma-232	328	4	1	1	NUM
ma-232	328	5	)	)	PUNCT
ma-232	328	6	,	,	PUNCT
ma-232	328	7	we	we	PRON
ma-232	328	8	take	take	VERB
ma-232	328	9	note	note	VERB
ma-232	328	10	the	the	DET
ma-232	328	11	resolvent	resolvent	ADJ
ma-232	328	12	set	set	NOUN
ma-232	328	13	is	be	AUX
ma-232	328	14	given	give	VERB
ma-232	328	15	as	as	ADP
ma-232	328	16	ρ(γ	ρ(γ	NOUN
ma-232	328	17	)	)	PUNCT
ma-232	328	18	=	=	SYM
ma-232	328	19	{	{	PUNCT
ma-232	328	20	λ	λ	X
ma-232	328	21	∈	∈	NOUN
ma-232	328	22	c	c	NOUN
ma-232	328	23	:	:	PUNCT
ma-232	329	1	<	<	X
ma-232	329	2	(	(	PUNCT
ma-232	329	3	λ	λ	X
ma-232	329	4	)	)	PUNCT
ma-232	329	5	6=	6=	ADP
ma-232	329	6	0	0	NUM
ma-232	329	7	}	}	PUNCT
ma-232	329	8	.	.	PUNCT
ma-232	330	1	wetherefore	wetherefore	PROPN
ma-232	330	2	consider	consider	VERB
ma-232	330	3	the	the	DET
ma-232	330	4	following	follow	VERB
ma-232	330	5	cases	case	NOUN
ma-232	330	6	:	:	PUNCT
ma-232	330	7	case	case	NOUN
ma-232	330	8	1	1	NUM
ma-232	330	9	:	:	PUNCT
ma-232	330	10	if	if	SCONJ
ma-232	330	11	<	<	X
ma-232	330	12	(	(	PUNCT
ma-232	330	13	λ	λ	X
ma-232	330	14	)	)	PUNCT
ma-232	330	15	>	>	X
ma-232	330	16	0	0	NUM
ma-232	330	17	,	,	PUNCT
ma-232	330	18	then	then	ADV
ma-232	330	19	the	the	DET
ma-232	330	20	resolvent	resolvent	ADJ
ma-232	330	21	operator	operator	NOUN
ma-232	330	22	is	be	AUX
ma-232	330	23	given	give	VERB
ma-232	330	24	by	by	ADP
ma-232	330	25	the	the	DET
ma-232	330	26	laplace	laplace	NOUN
ma-232	330	27	transform	transform	NOUN
ma-232	330	28	:	:	PUNCT
ma-232	330	29	for	for	ADP
ma-232	330	30	every	every	DET
ma-232	330	31	h	h	NOUN
ma-232	330	32	∈	∈	NOUN
ma-232	330	33	b∞,	b∞,	ADP
ma-232	330	34	◦	◦	NOUN
ma-232	330	35	(u	(u	NOUN
ma-232	330	36	)	)	PUNCT
ma-232	330	37	,	,	PUNCT
ma-232	330	38	we	we	PRON
ma-232	330	39	have	have	VERB
ma-232	330	40	r(λ	r(λ	NOUN
ma-232	330	41	,	,	PUNCT
ma-232	330	42	γ)h	γ)h	X
ma-232	330	43	=	=	SYM
ma-232	330	44	∫∞	∫∞	NOUN
ma-232	330	45	0	0	PUNCT
ma-232	330	46	e−λtsthdt	e−λtsthdt	X
ma-232	330	47	with	with	ADP
ma-232	330	48	convergence	convergence	NOUN
ma-232	330	49	in	in	ADP
ma-232	330	50	norm	norm	NOUN
ma-232	330	51	.	.	PUNCT
ma-232	331	1	therefore	therefore	ADV
ma-232	331	2	,	,	PUNCT
ma-232	331	3	r(λ	r(λ	PROPN
ma-232	331	4	,	,	PUNCT
ma-232	331	5	γ)h	γ)h	X
ma-232	331	6	=	=	SYM
ma-232	331	7	∫∞	∫∞	NOUN
ma-232	331	8	0	0	PUNCT
ma-232	331	9	e−λth(etω)dt	e−λth(etω)dt	NOUN
ma-232	331	10	.	.	PUNCT
ma-232	332	1	by	by	ADP
ma-232	332	2	change	change	NOUN
ma-232	332	3	of	of	ADP
ma-232	332	4	variables	variable	NOUN
ma-232	332	5	,	,	PUNCT
ma-232	332	6	let	let	VERB
ma-232	332	7	z	z	NOUN
ma-232	332	8	=	=	SYM
ma-232	332	9	etω	etω	ADJ
ma-232	332	10	,	,	PUNCT
ma-232	332	11	then	then	ADV
ma-232	332	12	ω	ω	X
ma-232	332	13	=	=	PROPN
ma-232	332	14	e−tz	e−tz	PROPN
ma-232	332	15	,	,	PUNCT
ma-232	332	16	dzdt	dzdt	PROPN
ma-232	332	17	=	=	PUNCT
ma-232	333	1	ωet	ωet	NOUN
ma-232	333	2	then	then	ADV
ma-232	333	3	dt	dt	X
ma-232	334	1	=	=	PUNCT
ma-232	334	2	dz	dz	PROPN
ma-232	334	3	ωet	ωet	NOUN
ma-232	334	4	=	=	PUNCT
ma-232	334	5	dz	dz	PROPN
ma-232	334	6	z	z	NOUN
ma-232	334	7	.	.	PUNCT
ma-232	335	1	therefore	therefore	ADV
ma-232	336	1	when	when	SCONJ
ma-232	336	2	t	t	PROPN
ma-232	336	3	=	=	SYM
ma-232	336	4	0⇒	0⇒	PROPN
ma-232	336	5	z	z	NOUN
ma-232	336	6	=	=	SYM
ma-232	336	7	ω	ω	PROPN
ma-232	336	8	and	and	CCONJ
ma-232	336	9	t	t	PROPN
ma-232	336	10	=	=	NOUN
ma-232	336	11	∞⇒	∞⇒	PROPN
ma-232	336	12	z	z	PROPN
ma-232	336	13	=	=	SYM
ma-232	336	14	∞	∞	PROPN
ma-232	336	15	,	,	PUNCT
ma-232	336	16	and	and	CCONJ
ma-232	336	17	so	so	ADV
ma-232	336	18	;	;	PUNCT
ma-232	336	19	r(λ	r(λ	NOUN
ma-232	336	20	,	,	PUNCT
ma-232	336	21	γ)h(ω	γ)h(ω	ADJ
ma-232	336	22	)	)	PUNCT
ma-232	336	23	=	=	SYM
ma-232	337	1	∫	∫	PROPN
ma-232	337	2	∞	∞	PROPN
ma-232	337	3	ω	ω	PROPN
ma-232	337	4	e−λth(z	e−λth(z	NOUN
ma-232	337	5	)	)	PUNCT
ma-232	338	1	dz	dz	PROPN
ma-232	338	2	z	z	NOUN
ma-232	338	3	=	=	SYM
ma-232	338	4	∫	∫	PROPN
ma-232	339	1	∞	∞	PROPN
ma-232	339	2	ω	ω	PROPN
ma-232	339	3	(	(	PUNCT
ma-232	339	4	z	z	PROPN
ma-232	339	5	ω	ω	PROPN
ma-232	339	6	)	)	PUNCT
ma-232	339	7	−λ	−λ	PROPN
ma-232	339	8	1	1	NUM
ma-232	339	9	z	z	NOUN
ma-232	339	10	h(z)dz	h(z)dz	NOUN
ma-232	339	11	=	=	SYM
ma-232	339	12	ωλ	ωλ	VERB
ma-232	339	13	∫	∫	PROPN
ma-232	339	14	∞	∞	PROPN
ma-232	339	15	ω	ω	NUM
ma-232	339	16	1	1	NUM
ma-232	339	17	zλ+1	zλ+1	NUM
ma-232	339	18	h(z)dz	h(z)dz	NOUN
ma-232	339	19	.	.	PUNCT
ma-232	339	20	case	case	NOUN
ma-232	339	21	2	2	NUM
ma-232	339	22	:	:	PUNCT
ma-232	339	23	if	if	SCONJ
ma-232	339	24	<	<	X
ma-232	339	25	(	(	PUNCT
ma-232	339	26	λ	λ	X
ma-232	339	27	)	)	PUNCT
ma-232	339	28	<	<	X
ma-232	339	29	0	0	NUM
ma-232	339	30	,	,	PUNCT
ma-232	339	31	then	then	ADV
ma-232	339	32	r(λ	r(λ	PROPN
ma-232	339	33	,	,	PUNCT
ma-232	339	34	γ)h	γ)h	ADJ
ma-232	339	35	=	=	SYM
ma-232	339	36	−r(−λ,−γ)h	−r(−λ,−γ)h	NOUN
ma-232	339	37	=	=	SYM
ma-232	339	38	−	−	PROPN
ma-232	339	39	∫∞	∫∞	NOUN
ma-232	339	40	0	0	NUM
ma-232	339	41	eλth(e−tω)dt	eλth(e−tω)dt	PROPN
ma-232	339	42	.	.	PUNCT
ma-232	340	1	then	then	ADV
ma-232	340	2	again	again	ADV
ma-232	340	3	bychange	bychange	VERB
ma-232	340	4	of	of	ADP
ma-232	340	5	variables	variable	NOUN
ma-232	340	6	,	,	PUNCT
ma-232	340	7	let	let	VERB
ma-232	340	8	z	z	NOUN
ma-232	340	9	=	=	PUNCT
ma-232	340	10	e−tω	e−tω	NOUN
ma-232	340	11	,	,	PUNCT
ma-232	340	12	then	then	ADV
ma-232	340	13	et	et	NOUN
ma-232	340	14	=	=	SYM
ma-232	340	15	ω	ω	PROPN
ma-232	340	16	z	z	PROPN
ma-232	340	17	,	,	PUNCT
ma-232	340	18	dzdt	dzdt	PROPN
ma-232	340	19	=	=	SYM
ma-232	340	20	−ωe−t	−ωe−t	NOUN
ma-232	340	21	and	and	CCONJ
ma-232	340	22	dt	dt	NOUN
ma-232	341	1	=	=	SYM
ma-232	341	2	−dz	−dz	PROPN
ma-232	341	3	ωe−t	ωe−t	NOUN
ma-232	341	4	=	=	PUNCT
ma-232	341	5	−dzz	−dzz	PROPN
ma-232	341	6	.	.	PUNCT
ma-232	342	1	therefore	therefore	ADV
ma-232	342	2	t	t	PROPN
ma-232	342	3	=	=	PUNCT
ma-232	342	4	0⇒	0⇒	PROPN
ma-232	342	5	z	z	NOUN
ma-232	343	1	=	=	SYM
ma-232	343	2	w	w	PROPN
ma-232	343	3	and	and	CCONJ
ma-232	343	4	t	t	PROPN
ma-232	344	1	=	=	NOUN
ma-232	344	2	∞⇒	∞⇒	X
ma-232	344	3	z	z	NOUN
ma-232	344	4	=	=	SYM
ma-232	344	5	0	0	NUM
ma-232	344	6	and	and	CCONJ
ma-232	344	7	so	so	ADV
ma-232	345	1	;	;	PUNCT
ma-232	345	2	r(λ	r(λ	NOUN
ma-232	345	3	,	,	PUNCT
ma-232	345	4	γ)h(w	γ)h(w	ADJ
ma-232	345	5	)	)	PUNCT
ma-232	345	6	=	=	PUNCT
ma-232	346	1	−	−	PROPN
ma-232	346	2	∫	∫	NOUN
ma-232	346	3	0	0	NUM
ma-232	347	1	ω	ω	PROPN
ma-232	347	2	eλth(z).−	eλth(z).−	PROPN
ma-232	347	3	dz	dz	PROPN
ma-232	347	4	z	z	NOUN
ma-232	347	5	=	=	PUNCT
ma-232	348	1	−	−	PROPN
ma-232	348	2	∫	∫	PROPN
ma-232	348	3	ω	ω	PROPN
ma-232	348	4	0	0	NUM
ma-232	348	5	(	(	PUNCT
ma-232	348	6	ω	ω	NOUN
ma-232	348	7	z	z	NOUN
ma-232	348	8	)	)	PUNCT
ma-232	348	9	λ	λ	PROPN
ma-232	348	10	h(z	h(z	NOUN
ma-232	348	11	)	)	PUNCT
ma-232	348	12	.	.	PUNCT
ma-232	349	1	dz	dz	PROPN
ma-232	349	2	z	z	NOUN
ma-232	349	3	=	=	PUNCT
ma-232	349	4	−ωλ	−ωλ	PROPN
ma-232	349	5	∫	∫	PROPN
ma-232	349	6	ω	ω	X
ma-232	349	7	0	0	NUM
ma-232	350	1	(	(	PUNCT
ma-232	350	2	1	1	NUM
ma-232	350	3	z	z	NOUN
ma-232	350	4	)	)	PUNCT
ma-232	350	5	λ	λ	NOUN
ma-232	350	6	.	.	PUNCT
ma-232	351	1	1	1	NUM
ma-232	351	2	z	z	NOUN
ma-232	351	3	h(z)dz	h(z)dz	NOUN
ma-232	351	4	=	=	SYM
ma-232	351	5	−ωλ	−ωλ	PROPN
ma-232	351	6	∫	∫	PROPN
ma-232	351	7	ω	ω	NOUN
ma-232	351	8	0	0	NUM
ma-232	351	9	1	1	NUM
ma-232	351	10	zλ+1	zλ+1	NUM
ma-232	351	11	h(z)dz	h(z)dz	NOUN
ma-232	351	12	.	.	PUNCT
ma-232	352	1	to	to	PART
ma-232	352	2	prove	prove	VERB
ma-232	352	3	(	(	PUNCT
ma-232	352	4	2	2	NUM
ma-232	352	5	)	)	PUNCT
ma-232	352	6	,	,	PUNCT
ma-232	352	7	we	we	PRON
ma-232	352	8	use	use	VERB
ma-232	352	9	the	the	DET
ma-232	352	10	spectral	spectral	ADJ
ma-232	352	11	mapping	mapping	NOUN
ma-232	352	12	theorem	theorem	NOUN
ma-232	352	13	for	for	ADP
ma-232	352	14	the	the	DET
ma-232	352	15	resolvents	resolvent	NOUN
ma-232	352	16	which	which	PRON
ma-232	352	17	asserts	assert	VERB
ma-232	352	18	that	that	SCONJ
ma-232	352	19	σ(r(λ	σ(r(λ	PROPN
ma-232	352	20	,	,	PUNCT
ma-232	352	21	γ	γ	NOUN
ma-232	352	22	)	)	PUNCT
ma-232	352	23	)	)	PUNCT
ma-232	353	1	=	=	NOUN
ma-232	353	2	{	{	PUNCT
ma-232	353	3	1	1	NUM
ma-232	353	4	λ−µ	λ−µ	NOUN
ma-232	353	5	:	:	PUNCT
ma-232	353	6	µ	µ	X
ma-232	353	7	∈	∈	PROPN
ma-232	353	8	σ(γ	σ(γ	PROPN
ma-232	353	9	)	)	PUNCT
ma-232	353	10	}	}	PUNCT
ma-232	353	11	\	\	NOUN
ma-232	353	12	{	{	PUNCT
ma-232	353	13	0	0	NUM
ma-232	353	14	}	}	PUNCT
ma-232	353	15	for	for	ADP
ma-232	353	16	λ	λ	PROPN
ma-232	353	17	∈	∈	PROPN
ma-232	353	18	ρ(γ	ρ(γ	PROPN
ma-232	353	19	)	)	PUNCT
ma-232	353	20	.	.	PUNCT
ma-232	354	1	therefore	therefore	ADV
ma-232	354	2	,	,	PUNCT
ma-232	354	3	σ(r(λ	σ(r(λ	PROPN
ma-232	354	4	,	,	PUNCT
ma-232	354	5	γ	γ	NOUN
ma-232	354	6	)	)	PUNCT
ma-232	354	7	)	)	PUNCT
ma-232	355	1	=	=	PRON
ma-232	355	2	{	{	PUNCT
ma-232	355	3	1	1	NUM
ma-232	355	4	λ−	λ−	PROPN
ma-232	356	1	i	i	PRON
ma-232	356	2	r	r	VERB
ma-232	356	3	:	:	PUNCT
ma-232	356	4	r	r	NOUN
ma-232	356	5	∈	∈	NOUN
ma-232	356	6	r	r	NOUN
ma-232	356	7	}	}	PUNCT
ma-232	356	8	\	\	NOUN
ma-232	356	9	{	{	PUNCT
ma-232	356	10	0	0	NUM
ma-232	356	11	}	}	PUNCT
ma-232	356	12	=	=	SYM
ma-232	356	13	{	{	PUNCT
ma-232	356	14	1	1	NUM
ma-232	356	15	<	<	X
ma-232	356	16	(	(	PUNCT
ma-232	356	17	λ	λ	X
ma-232	356	18	)	)	PUNCT
ma-232	357	1	+	+	CCONJ
ma-232	357	2	i(im(λ)−	i(im(λ)−	ADJ
ma-232	357	3	r	r	NOUN
ma-232	357	4	)	)	PUNCT
ma-232	357	5	:	:	PUNCT
ma-232	357	6	r	r	NOUN
ma-232	357	7	∈	∈	PROPN
ma-232	357	8	r	r	NOUN
ma-232	357	9	}	}	PUNCT
ma-232	357	10	\	\	NOUN
ma-232	357	11	{	{	PUNCT
ma-232	357	12	0	0	NUM
ma-232	357	13	}	}	PUNCT
ma-232	357	14	.	.	PUNCT
ma-232	358	1	rationalizing	rationalize	VERB
ma-232	358	2	the	the	DET
ma-232	358	3	denominator	denominator	NOUN
ma-232	358	4	and	and	CCONJ
ma-232	358	5	simplifying	simplify	VERB
ma-232	358	6	we	we	PRON
ma-232	358	7	get	get	VERB
ma-232	358	8	σ(r(λ	σ(r(λ	PROPN
ma-232	358	9	,	,	PUNCT
ma-232	358	10	γ	γ	NOUN
ma-232	358	11	)	)	PUNCT
ma-232	358	12	)	)	PUNCT
ma-232	359	1	=	=	PRON
ma-232	359	2	{	{	PUNCT
ma-232	359	3	(	(	PUNCT
ma-232	359	4	<	<	X
ma-232	359	5	(	(	PUNCT
ma-232	359	6	λ)−i(=(λ)−r	λ)−i(=(λ)−r	PROPN
ma-232	359	7	)	)	PUNCT
ma-232	359	8	)	)	PUNCT
ma-232	360	1	(	(	PUNCT
ma-232	360	2	<	<	X
ma-232	360	3	(	(	PUNCT
ma-232	360	4	λ))2+(=(λ)−r)2	λ))2+(=(λ)−r)2	X
ma-232	360	5	:	:	PUNCT
ma-232	360	6	r	r	NOUN
ma-232	360	7	∈	∈	NOUN
ma-232	360	8	r	r	NOUN
ma-232	360	9	}	}	PUNCT
ma-232	360	10	.now	.now	PUNCT
ma-232	360	11	by	by	ADP
ma-232	360	12	letting	let	VERB
ma-232	360	13	w	w	ADP
ma-232	360	14	=	=	PUNCT
ma-232	360	15	(	(	PUNCT
ma-232	360	16	<	<	X
ma-232	360	17	(	(	PUNCT
ma-232	360	18	λ)−i(=(λ)−r	λ)−i(=(λ)−r	PROPN
ma-232	360	19	)	)	PUNCT
ma-232	360	20	)	)	PUNCT
ma-232	360	21	(	(	PUNCT
ma-232	360	22	<	<	X
ma-232	360	23	(	(	PUNCT
ma-232	360	24	λ))2+(=(λ)−r)2	λ))2+(=(λ)−r)2	INTJ
ma-232	360	25	,	,	PUNCT
ma-232	360	26	subtracting	subtract	VERB
ma-232	360	27	1	1	NUM
ma-232	360	28	2<(λ	2<(λ	NUM
ma-232	360	29	)	)	PUNCT
ma-232	360	30	and	and	CCONJ
ma-232	360	31	finding	find	VERB
ma-232	360	32	the	the	DET
ma-232	360	33	magnitude	magnitude	NOUN
ma-232	360	34	of	of	ADP
ma-232	360	35	both	both	DET
ma-232	360	36	sideswe	sideswe	NOUN
ma-232	360	37	get	get	VERB
ma-232	360	38	,	,	PUNCT
ma-232	360	39	∣∣∣∣w	∣∣∣∣w	ADV
ma-232	360	40	−	−	NOUN
ma-232	360	41	1	1	NUM
ma-232	360	42	2<(λ	2<(λ	NUM
ma-232	360	43	)	)	PUNCT
ma-232	360	44	∣∣∣∣2	∣∣∣∣2	NOUN
ma-232	360	45	=	=	SYM
ma-232	360	46	1	1	NUM
ma-232	360	47	(	(	PUNCT
ma-232	360	48	2<(λ))2	2<(λ))2	NUM
ma-232	360	49	,	,	PUNCT
ma-232	360	50	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	360	51	eur	eur	PROPN
ma-232	360	52	.	.	PUNCT
ma-232	361	1	j.	j.	PROPN
ma-232	361	2	math	math	PROPN
ma-232	361	3	.	.	PUNCT
ma-232	362	1	anal	anal	PROPN
ma-232	362	2	.	.	PUNCT
ma-232	363	1	10.28924	10.28924	NUM
ma-232	363	2	/	/	SYM
ma-232	363	3	ada	ada	PROPN
ma-232	363	4	/	/	SYM
ma-232	363	5	ma.4.14	ma.4.14	X
ma-232	364	1	13and	13and	VERB
ma-232	364	2	so	so	ADV
ma-232	364	3	∣∣∣∣w	∣∣∣∣w	VERB
ma-232	364	4	−	−	NOUN
ma-232	364	5	1	1	NUM
ma-232	364	6	2<(λ	2<(λ	NUM
ma-232	364	7	)	)	PUNCT
ma-232	364	8	∣∣∣∣	∣∣∣∣	NOUN
ma-232	364	9	=	=	NOUN
ma-232	364	10	1	1	NUM
ma-232	364	11	2<(λ	2<(λ	NUM
ma-232	364	12	)	)	PUNCT
ma-232	364	13	.	.	PUNCT
ma-232	365	1	therefore	therefore	ADV
ma-232	365	2	,	,	PUNCT
ma-232	365	3	σ(r(λ	σ(r(λ	PROPN
ma-232	365	4	,	,	PUNCT
ma-232	365	5	γ	γ	NOUN
ma-232	365	6	)	)	PUNCT
ma-232	365	7	)	)	PUNCT
ma-232	366	1	=	=	PRON
ma-232	366	2	{	{	PUNCT
ma-232	366	3	w	w	NOUN
ma-232	366	4	:	:	PUNCT
ma-232	366	5	|w	|w	ADJ
ma-232	366	6	−	−	PROPN
ma-232	366	7	1	1	NUM
ma-232	366	8	2<(λ	2<(λ	NUM
ma-232	366	9	)	)	PUNCT
ma-232	367	1	|	|	ADV
ma-232	367	2	=	=	SYM
ma-232	367	3	1	1	NUM
ma-232	367	4	2<(λ	2<(λ	NUM
ma-232	367	5	)	)	PUNCT
ma-232	367	6	}	}	PUNCT
ma-232	367	7	.	.	PUNCT
ma-232	368	1	for	for	ADP
ma-232	368	2	part	part	NOUN
ma-232	368	3	(	(	PUNCT
ma-232	368	4	3	3	NUM
ma-232	368	5	)	)	PUNCT
ma-232	368	6	,	,	PUNCT
ma-232	368	7	the	the	DET
ma-232	368	8	spectral	spectral	ADJ
ma-232	368	9	radius	radius	NOUN
ma-232	368	10	r(r(λ	r(r(λ	PROPN
ma-232	368	11	,	,	PUNCT
ma-232	368	12	γ))is	γ))is	PROPN
ma-232	368	13	given	give	VERB
ma-232	368	14	by	by	ADP
ma-232	368	15	;	;	PUNCT
ma-232	368	16	r(r(λ	r(r(λ	PROPN
ma-232	368	17	,	,	PUNCT
ma-232	368	18	γ	γ	NOUN
ma-232	368	19	)	)	PUNCT
ma-232	368	20	)	)	PUNCT
ma-232	369	1	=	=	PUNCT
ma-232	370	1	sup{|w	sup{|w	PROPN
ma-232	371	1	|	|	ADV
ma-232	371	2	:	:	PUNCT
ma-232	371	3	w	w	PROPN
ma-232	371	4	∈	∈	PROPN
ma-232	371	5	σ(r(λ	σ(r(λ	PROPN
ma-232	371	6	,	,	PUNCT
ma-232	371	7	γ	γ	NOUN
ma-232	371	8	)	)	PUNCT
ma-232	371	9	)	)	PUNCT
ma-232	371	10	}	}	PUNCT
ma-232	371	11	=	=	SYM
ma-232	371	12	sup	sup	X
ma-232	371	13	{	{	PUNCT
ma-232	371	14	|w	|w	NOUN
ma-232	371	15	|	|	NOUN
ma-232	371	16	:	:	PUNCT
ma-232	371	17	∣∣∣∣w	∣∣∣∣w	PROPN
ma-232	371	18	−	−	NOUN
ma-232	371	19	1	1	NUM
ma-232	371	20	2<(λ	2<(λ	NUM
ma-232	371	21	)	)	PUNCT
ma-232	371	22	∣∣∣∣	∣∣∣∣	NOUN
ma-232	371	23	=	=	NOUN
ma-232	371	24	1	1	NUM
ma-232	371	25	2<(λ	2<(λ	NUM
ma-232	371	26	)	)	PUNCT
ma-232	371	27	}	}	PUNCT
ma-232	371	28	=	=	SYM
ma-232	371	29	1	1	NUM
ma-232	371	30	|<(λ)|	|<(λ)|	VERB
ma-232	371	31	.finally	.finally	ADV
ma-232	371	32	to	to	PART
ma-232	371	33	determine	determine	VERB
ma-232	371	34	‖r(λ	‖r(λ	ADP
ma-232	371	35	,	,	PUNCT
ma-232	371	36	γ)‖	γ)‖	NOUN
ma-232	371	37	,	,	PUNCT
ma-232	371	38	we	we	PRON
ma-232	371	39	use	use	VERB
ma-232	371	40	the	the	DET
ma-232	371	41	hille	hille	PROPN
ma-232	371	42	yosida	yosida	PROPN
ma-232	371	43	theorem	theorem	VERB
ma-232	371	44	as	as	ADV
ma-232	371	45	well	well	ADV
ma-232	371	46	as	as	ADP
ma-232	371	47	the	the	DET
ma-232	371	48	fact	fact	NOUN
ma-232	371	49	that	that	SCONJ
ma-232	371	50	the	the	DET
ma-232	371	51	spectralradius	spectralradius	NOUN
ma-232	371	52	is	be	AUX
ma-232	371	53	always	always	ADV
ma-232	371	54	bounded	bound	VERB
ma-232	371	55	by	by	ADP
ma-232	371	56	the	the	DET
ma-232	371	57	norm.therefore	norm.therefore	PROPN
ma-232	371	58	,	,	PUNCT
ma-232	371	59	1	1	NUM
ma-232	371	60	|<(λ)|	|<(λ)|	NOUN
ma-232	371	61	=	=	SYM
ma-232	371	62	r(r(λ	r(r(λ	PROPN
ma-232	371	63	,	,	PUNCT
ma-232	371	64	γ	γ	NOUN
ma-232	371	65	)	)	PUNCT
ma-232	371	66	)	)	PUNCT
ma-232	371	67	≤	≤	NUM
ma-232	372	1	‖r(λ	‖r(λ	PRON
ma-232	372	2	,	,	PUNCT
ma-232	372	3	γ)‖	γ)‖	NOUN
ma-232	372	4	≤	≤	PROPN
ma-232	372	5	1	1	NUM
ma-232	372	6	|<(λ)|	|<(λ)|	PROPN
ma-232	372	7	.thus	.thus	PROPN
ma-232	372	8	,	,	PUNCT
ma-232	372	9	r(r(λ	r(r(λ	PROPN
ma-232	372	10	,	,	PUNCT
ma-232	372	11	γ	γ	NOUN
ma-232	372	12	)	)	PUNCT
ma-232	372	13	)	)	PUNCT
ma-232	373	1	=	=	SYM
ma-232	374	1	‖r(λ	‖r(λ	X
ma-232	374	2	,	,	PUNCT
ma-232	374	3	γ)‖	γ)‖	NOUN
ma-232	374	4	=	=	SYM
ma-232	374	5	1	1	NUM
ma-232	374	6	|<(λ)|	|<(λ)|	NOUN
ma-232	374	7	,	,	PUNCT
ma-232	374	8	as	as	SCONJ
ma-232	374	9	desired	desire	VERB
ma-232	374	10	.	.	PUNCT
ma-232	375	1	�	�	PROPN
ma-232	375	2	3.2	3.2	NUM
ma-232	375	3	.	.	PUNCT
ma-232	376	1	translation	translation	NOUN
ma-232	376	2	group	group	NOUN
ma-232	376	3	.	.	PUNCT
ma-232	377	1	in	in	ADP
ma-232	377	2	this	this	DET
ma-232	377	3	group	group	NOUN
ma-232	377	4	the	the	DET
ma-232	377	5	automorphisms	automorphism	NOUN
ma-232	377	6	are	be	AUX
ma-232	377	7	of	of	ADP
ma-232	377	8	the	the	DET
ma-232	377	9	form	form	NOUN
ma-232	377	10	ϕt(z	ϕt(z	NUM
ma-232	377	11	)	)	PUNCT
ma-232	378	1	=	=	SYM
ma-232	378	2	z	z	PROPN
ma-232	379	1	+	+	CCONJ
ma-232	379	2	kt	kt	PROPN
ma-232	379	3	,	,	PUNCT
ma-232	379	4	where	where	SCONJ
ma-232	379	5	z	z	PROPN
ma-232	379	6	∈	∈	PROPN
ma-232	379	7	u	u	NOUN
ma-232	379	8	and	and	CCONJ
ma-232	379	9	k	k	PROPN
ma-232	379	10	,	,	PUNCT
ma-232	379	11	t	t	PROPN
ma-232	379	12	∈	∈	PROPN
ma-232	379	13	r	r	NOUN
ma-232	379	14	with	with	ADP
ma-232	379	15	k	k	PROPN
ma-232	379	16	6=	6=	PROPN
ma-232	379	17	0	0	NUM
ma-232	379	18	.	.	PUNCT
ma-232	380	1	as	as	SCONJ
ma-232	380	2	noted	note	VERB
ma-232	380	3	earlier	early	ADV
ma-232	380	4	in	in	ADP
ma-232	380	5	subsection	subsection	NOUN
ma-232	380	6	3.1	3.1	NUM
ma-232	380	7	,	,	PUNCT
ma-232	380	8	without	without	ADP
ma-232	380	9	loss	loss	NOUN
ma-232	380	10	of	of	ADP
ma-232	380	11	generality	generality	NOUN
ma-232	380	12	welet	welet	NOUN
ma-232	380	13	k	k	PROPN
ma-232	380	14	=	=	SYM
ma-232	380	15	1	1	NUM
ma-232	380	16	and	and	CCONJ
ma-232	380	17	consider	consider	VERB
ma-232	380	18	the	the	DET
ma-232	380	19	self	self	NOUN
ma-232	380	20	analytic	analytic	ADJ
ma-232	380	21	maps	map	NOUN
ma-232	380	22	ϕt	ϕt	ADV
ma-232	381	1	:	:	PUNCT
ma-232	381	2	u	u	X
ma-232	381	3	→	→	SYM
ma-232	381	4	u	u	X
ma-232	381	5	given	give	VERB
ma-232	381	6	by	by	ADP
ma-232	381	7	ϕt(z	ϕt(z	NUM
ma-232	381	8	)	)	PUNCT
ma-232	381	9	=	=	SYM
ma-232	382	1	z	z	NOUN
ma-232	383	1	+	+	NUM
ma-232	383	2	t	t	PROPN
ma-232	383	3	for	for	ADP
ma-232	383	4	z	z	PROPN
ma-232	383	5	∈	∈	PROPN
ma-232	383	6	u.then	u.then	ADV
ma-232	383	7	the	the	DET
ma-232	383	8	corresponding	corresponding	ADJ
ma-232	383	9	group	group	NOUN
ma-232	383	10	of	of	ADP
ma-232	383	11	composition	composition	NOUN
ma-232	383	12	operators	operator	NOUN
ma-232	383	13	defined	define	VERB
ma-232	383	14	on	on	ADP
ma-232	383	15	l1	l1	PROPN
ma-232	383	16	a(u	a(u	PROPN
ma-232	383	17	,	,	PUNCT
ma-232	383	18	µα	µα	ADP
ma-232	383	19	)	)	PUNCT
ma-232	383	20	is	be	AUX
ma-232	383	21	therefore	therefore	ADV
ma-232	383	22	given	give	VERB
ma-232	383	23	by	by	ADP
ma-232	383	24	tt	tt	PROPN
ma-232	383	25	f	f	PROPN
ma-232	383	26	(	(	PUNCT
ma-232	383	27	z	z	NOUN
ma-232	383	28	)	)	PUNCT
ma-232	383	29	=	=	SYM
ma-232	383	30	f	f	X
ma-232	383	31	(	(	PUNCT
ma-232	383	32	z	z	PROPN
ma-232	383	33	+	+	NUM
ma-232	383	34	t	t	PROPN
ma-232	383	35	)	)	PUNCT
ma-232	383	36	,	,	PUNCT
ma-232	383	37	for	for	ADP
ma-232	383	38	all	all	DET
ma-232	383	39	f	f	PROPN
ma-232	383	40	∈	∈	PROPN
ma-232	383	41	lpa(u	lpa(u	PROPN
ma-232	383	42	,	,	PUNCT
ma-232	383	43	µα).now	µα).now	ADV
ma-232	383	44	using	use	VERB
ma-232	383	45	the	the	DET
ma-232	383	46	duality	duality	NOUN
ma-232	383	47	relation	relation	NOUN
ma-232	383	48	given	give	VERB
ma-232	383	49	by	by	ADP
ma-232	383	50	equation	equation	NOUN
ma-232	383	51	(	(	PUNCT
ma-232	383	52	3.1	3.1	NUM
ma-232	383	53	)	)	PUNCT
ma-232	383	54	and	and	CCONJ
ma-232	383	55	its	its	PRON
ma-232	383	56	sesquilinear	sesquilinear	NOUN
ma-232	383	57	pairing	pairing	NOUN
ma-232	383	58	given	give	VERB
ma-232	383	59	by	by	ADP
ma-232	383	60	equation(3.2	equation(3.2	NOUN
ma-232	383	61	)	)	PUNCT
ma-232	383	62	,	,	PUNCT
ma-232	383	63	we	we	PRON
ma-232	383	64	have	have	AUX
ma-232	383	65	:	:	PUNCT
ma-232	383	66	let	let	VERB
ma-232	383	67	g	g	PROPN
ma-232	383	68	∈	∈	PROPN
ma-232	383	69	b∞,	b∞,	ADP
ma-232	383	70	◦	◦	NOUN
ma-232	383	71	(u	(u	NOUN
ma-232	383	72	,	,	PUNCT
ma-232	383	73	i	i	PROPN
ma-232	383	74	)	)	PUNCT
ma-232	383	75	and	and	CCONJ
ma-232	383	76	f	f	PROPN
ma-232	383	77	∈	∈	PROPN
ma-232	383	78	l1	l1	PROPN
ma-232	383	79	a(u	a(u	PROPN
ma-232	383	80	,	,	PUNCT
ma-232	383	81	µα	µα	ADP
ma-232	383	82	)	)	PUNCT
ma-232	383	83	,	,	PUNCT
ma-232	383	84	then	then	ADV
ma-232	383	85	〈	〈	PROPN
ma-232	383	86	g	g	PROPN
ma-232	383	87	,	,	PUNCT
ma-232	383	88	tt	tt	PROPN
ma-232	383	89	f	f	PROPN
ma-232	383	90	〉	〉	PROPN
ma-232	383	91	=	=	SYM
ma-232	383	92	∫	∫	PROPN
ma-232	383	93	u	u	INTJ
ma-232	383	94	g(z)f	g(z)f	PROPN
ma-232	383	95	(	(	PUNCT
ma-232	383	96	z	z	NOUN
ma-232	383	97	+	+	NOUN
ma-232	383	98	t)dµα(z	t)dµα(z	NOUN
ma-232	383	99	)	)	PUNCT
ma-232	383	100	=	=	SYM
ma-232	384	1	∫	∫	PUNCT
ma-232	385	1	u	u	INTJ
ma-232	385	2	g(z)f	g(z)f	PROPN
ma-232	385	3	(	(	PUNCT
ma-232	385	4	z	z	NOUN
ma-232	385	5	+	+	NUM
ma-232	385	6	t)(=(z))αda(z	t)(=(z))αda(z	PROPN
ma-232	385	7	)	)	PUNCT
ma-232	385	8	.	.	PUNCT
ma-232	386	1	now	now	ADV
ma-232	386	2	by	by	ADP
ma-232	386	3	a	a	DET
ma-232	386	4	change	change	NOUN
ma-232	386	5	of	of	ADP
ma-232	386	6	variables	variable	NOUN
ma-232	386	7	,	,	PUNCT
ma-232	386	8	let	let	VERB
ma-232	386	9	ω	ω	NOUN
ma-232	386	10	=	=	PUNCT
ma-232	387	1	z	z	PROPN
ma-232	388	1	+	+	NUM
ma-232	388	2	t	t	PROPN
ma-232	388	3	,	,	PUNCT
ma-232	388	4	then	then	ADV
ma-232	388	5	z	z	X
ma-232	388	6	=	=	SYM
ma-232	388	7	ω	ω	PROPN
ma-232	388	8	−	−	PROPN
ma-232	389	1	t	t	NOUN
ma-232	389	2	and	and	CCONJ
ma-232	389	3	da(ω	da(ω	NOUN
ma-232	389	4	)	)	PUNCT
ma-232	389	5	=	=	SYM
ma-232	389	6	da(z	da(z	X
ma-232	389	7	)	)	PUNCT
ma-232	389	8	.	.	PUNCT
ma-232	390	1	therefore	therefore	ADV
ma-232	390	2	,	,	PUNCT
ma-232	390	3	〈	〈	PROPN
ma-232	390	4	g	g	PROPN
ma-232	390	5	,	,	PUNCT
ma-232	390	6	tt	tt	PROPN
ma-232	390	7	f	f	PROPN
ma-232	390	8	〉	〉	PROPN
ma-232	390	9	=	=	SYM
ma-232	390	10	∫	∫	PROPN
ma-232	390	11	u	u	NOUN
ma-232	390	12	g(ω	g(ω	PROPN
ma-232	390	13	−	−	PROPN
ma-232	390	14	t)f	t)f	SYM
ma-232	390	15	(	(	PUNCT
ma-232	390	16	ω)(=(ω))αda(ω	ω)(=(ω))αda(ω	NOUN
ma-232	390	17	)	)	PUNCT
ma-232	390	18	=	=	SYM
ma-232	391	1	∫	∫	PROPN
ma-232	391	2	u	u	X
ma-232	391	3	g(ω	g(ω	PROPN
ma-232	391	4	−	−	PROPN
ma-232	391	5	t)f	t)f	SYM
ma-232	391	6	(	(	PUNCT
ma-232	391	7	ω)dµα(ω	ω)dµα(ω	X
ma-232	391	8	)	)	PUNCT
ma-232	391	9	=	=	PUNCT
ma-232	392	1	〈	〈	PROPN
ma-232	392	2	t	t	PROPN
ma-232	392	3	∗t	∗t	PROPN
ma-232	392	4	g	g	PROPN
ma-232	392	5	,	,	PUNCT
ma-232	392	6	f	f	PROPN
ma-232	392	7	〉	〉	PROPN
ma-232	392	8	.	.	PUNCT
ma-232	393	1	(	(	PUNCT
ma-232	393	2	3.4	3.4	NUM
ma-232	393	3	)	)	PUNCT
ma-232	393	4	now	now	ADV
ma-232	393	5	,	,	PUNCT
ma-232	393	6	we	we	PRON
ma-232	393	7	define	define	VERB
ma-232	393	8	st	st	PROPN
ma-232	393	9	:	:	PUNCT
ma-232	393	10	=	=	SYM
ma-232	393	11	t	t	PROPN
ma-232	393	12	∗t	∗t	PROPN
ma-232	393	13	on	on	ADP
ma-232	393	14	b∞,	b∞,	ADP
ma-232	393	15	◦	◦	NOUN
ma-232	393	16	(u	(u	ADJ
ma-232	393	17	,	,	PUNCT
ma-232	393	18	i	i	PROPN
ma-232	393	19	)	)	PUNCT
ma-232	393	20	.	.	PUNCT
ma-232	394	1	but	but	CCONJ
ma-232	394	2	again	again	ADV
ma-232	394	3	we	we	PRON
ma-232	394	4	see	see	VERB
ma-232	394	5	that	that	SCONJ
ma-232	394	6	just	just	ADV
ma-232	394	7	as	as	ADP
ma-232	394	8	in	in	ADP
ma-232	394	9	the	the	DET
ma-232	394	10	case	case	NOUN
ma-232	394	11	of	of	ADP
ma-232	394	12	the	the	DET
ma-232	394	13	scalinggroup	scalinggroup	NOUN
ma-232	394	14	,	,	PUNCT
ma-232	394	15	stg(i	stg(i	PROPN
ma-232	394	16	)	)	PUNCT
ma-232	394	17	=	=	PUNCT
ma-232	394	18	g(i	g(i	PROPN
ma-232	394	19	−	−	PROPN
ma-232	394	20	t	t	PROPN
ma-232	394	21	)	)	PUNCT
ma-232	394	22	and	and	CCONJ
ma-232	394	23	therefore	therefore	ADV
ma-232	394	24	stg(i	stg(i	PROPN
ma-232	394	25	)	)	PUNCT
ma-232	394	26	does	do	AUX
ma-232	394	27	not	not	PART
ma-232	394	28	vanish	vanish	VERB
ma-232	394	29	at	at	ADP
ma-232	394	30	point	point	NOUN
ma-232	394	31	i	i	PRON
ma-232	394	32	.	.	PUNCT
ma-232	395	1	this	this	PRON
ma-232	395	2	means	mean	VERB
ma-232	395	3	that	that	SCONJ
ma-232	395	4	st	st	PROPN
ma-232	395	5	doesnot	doesnot	ADV
ma-232	395	6	map	map	NOUN
ma-232	395	7	b∞,	b∞,	ADP
ma-232	395	8	◦	◦	NOUN
ma-232	395	9	(u	(u	NOUN
ma-232	395	10	,	,	PUNCT
ma-232	395	11	i	i	NOUN
ma-232	395	12	)	)	PUNCT
ma-232	395	13	onto	onto	ADP
ma-232	395	14	itself	itself	PRON
ma-232	395	15	.	.	PUNCT
ma-232	396	1	we	we	PRON
ma-232	396	2	can	can	AUX
ma-232	396	3	therefore	therefore	ADV
ma-232	396	4	apply	apply	VERB
ma-232	396	5	similar	similar	ADJ
ma-232	396	6	remedies	remedy	NOUN
ma-232	396	7	proposed	propose	VERB
ma-232	396	8	in	in	ADP
ma-232	396	9	subsection3.1	subsection3.1	PROPN
ma-232	396	10	above	above	ADV
ma-232	396	11	.	.	PUNCT
ma-232	397	1	in	in	ADP
ma-232	397	2	the	the	DET
ma-232	397	3	next	next	ADJ
ma-232	397	4	sections	section	NOUN
ma-232	397	5	,	,	PUNCT
ma-232	397	6	we	we	PRON
ma-232	397	7	study	study	VERB
ma-232	397	8	the	the	DET
ma-232	397	9	semigroup	semigroup	ADJ
ma-232	397	10	properties	property	NOUN
ma-232	397	11	of	of	ADP
ma-232	397	12	(	(	PUNCT
ma-232	397	13	st)t≥0	st)t≥0	NOUN
ma-232	397	14	on	on	ADP
ma-232	397	15	b∞,	b∞,	NOUN
ma-232	397	16	◦	◦	NOUN
ma-232	397	17	(u	(u	NOUN
ma-232	397	18	)	)	PUNCT
ma-232	397	19	.	.	PUNCT
ma-232	398	1	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	398	2	eur	eur	PROPN
ma-232	398	3	.	.	PUNCT
ma-232	399	1	j.	j.	PROPN
ma-232	399	2	math	math	PROPN
ma-232	399	3	.	.	PUNCT
ma-232	400	1	anal	anal	PROPN
ma-232	400	2	.	.	PUNCT
ma-232	401	1	10.28924	10.28924	NUM
ma-232	401	2	/	/	SYM
ma-232	401	3	ada	ada	PROPN
ma-232	401	4	/	/	SYM
ma-232	401	5	ma.4.14	ma.4.14	NOUN
ma-232	401	6	143.2.1	143.2.1	NUM
ma-232	401	7	.	.	PUNCT
ma-232	402	1	semigroup	semigroup	PROPN
ma-232	402	2	properties	property	NOUN
ma-232	402	3	.	.	PUNCT
ma-232	403	1	in	in	ADP
ma-232	403	2	this	this	DET
ma-232	403	3	section	section	NOUN
ma-232	403	4	,	,	PUNCT
ma-232	403	5	we	we	PRON
ma-232	403	6	begin	begin	VERB
ma-232	403	7	by	by	ADP
ma-232	403	8	proving	prove	VERB
ma-232	403	9	the	the	DET
ma-232	403	10	strong	strong	ADJ
ma-232	403	11	continuity	continuity	NOUN
ma-232	403	12	property	property	NOUN
ma-232	403	13	.	.	PUNCT
ma-232	404	1	theorem	theorem	VERB
ma-232	404	2	3.6	3.6	NUM
ma-232	404	3	.	.	PUNCT
ma-232	405	1	let	let	VERB
ma-232	405	2	stg(ω	stg(ω	PROPN
ma-232	405	3	)	)	PUNCT
ma-232	405	4	:	:	PUNCT
ma-232	406	1	=	=	PUNCT
ma-232	406	2	g(ω−t	g(ω−t	X
ma-232	406	3	)	)	PUNCT
ma-232	406	4	be	be	VERB
ma-232	406	5	a	a	DET
ma-232	406	6	semigroup	semigroup	NOUN
ma-232	406	7	of	of	ADP
ma-232	406	8	composition	composition	NOUN
ma-232	406	9	operators	operator	NOUN
ma-232	406	10	defined	define	VERB
ma-232	406	11	on	on	ADP
ma-232	406	12	b∞,	b∞,	ADP
ma-232	406	13	◦	◦	NOUN
ma-232	406	14	(u	(u	NOUN
ma-232	406	15	)	)	PUNCT
ma-232	406	16	.	.	PUNCT
ma-232	407	1	then	then	ADV
ma-232	407	2	,	,	PUNCT
ma-232	407	3	(	(	PUNCT
ma-232	407	4	st)t∈r	st)t∈r	NOUN
ma-232	407	5	is	be	AUX
ma-232	407	6	a	a	DET
ma-232	407	7	strongly	strongly	ADV
ma-232	407	8	continuous	continuous	ADJ
ma-232	407	9	group	group	NOUN
ma-232	407	10	of	of	ADP
ma-232	407	11	isometries	isometry	NOUN
ma-232	407	12	on	on	ADP
ma-232	407	13	b∞,	b∞,	ADP
ma-232	407	14	◦	◦	NOUN
ma-232	407	15	(u	(u	NOUN
ma-232	407	16	)	)	PUNCT
ma-232	407	17	.	.	PUNCT
ma-232	408	1	proof	proof	NOUN
ma-232	408	2	.	.	PUNCT
ma-232	409	1	it	it	PRON
ma-232	409	2	is	be	AUX
ma-232	409	3	clear	clear	ADJ
ma-232	409	4	from	from	ADP
ma-232	409	5	the	the	DET
ma-232	409	6	definition	definition	NOUN
ma-232	409	7	that	that	SCONJ
ma-232	409	8	(	(	PUNCT
ma-232	409	9	st)t∈r	st)t∈r	NOUN
ma-232	409	10	is	be	AUX
ma-232	409	11	a	a	DET
ma-232	409	12	group	group	NOUN
ma-232	409	13	.	.	PUNCT
ma-232	410	1	to	to	PART
ma-232	410	2	prove	prove	VERB
ma-232	410	3	that	that	SCONJ
ma-232	410	4	(	(	PUNCT
ma-232	410	5	st)t∈r	st)t∈r	NOUN
ma-232	410	6	is	be	AUX
ma-232	410	7	an	an	DET
ma-232	410	8	isometry	isometry	NOUN
ma-232	410	9	,	,	PUNCT
ma-232	410	10	then	then	ADV
ma-232	410	11	by	by	ADP
ma-232	410	12	the	the	DET
ma-232	410	13	definition	definition	NOUN
ma-232	410	14	of	of	ADP
ma-232	410	15	isometry	isometry	NOUN
ma-232	410	16	,	,	PUNCT
ma-232	410	17	we	we	PRON
ma-232	410	18	have	have	VERB
ma-232	410	19	;	;	PUNCT
ma-232	410	20	‖stg‖b∞,	‖stg‖b∞,	VERB
ma-232	410	21	◦	◦	NOUN
ma-232	410	22	(u	(u	NOUN
ma-232	410	23	)	)	PUNCT
ma-232	410	24	=	=	SYM
ma-232	410	25	sup	sup	NOUN
ma-232	410	26	ω∈u	ω∈u	NOUN
ma-232	410	27	=(	=(	NOUN
ma-232	410	28	ω)|(stg)′(ω)|	ω)|(stg)′(ω)|	ADJ
ma-232	411	1	=	=	SYM
ma-232	411	2	sup	sup	NOUN
ma-232	411	3	ω∈u	ω∈u	NOUN
ma-232	411	4	=(	=(	NOUN
ma-232	411	5	ω)|g′(ω	ω)|g′(ω	NOUN
ma-232	411	6	−	−	PROPN
ma-232	411	7	t)|	t)|	NOUN
ma-232	411	8	.	.	PUNCT
ma-232	412	1	by	by	ADP
ma-232	412	2	change	change	NOUN
ma-232	412	3	of	of	ADP
ma-232	412	4	variables	variable	NOUN
ma-232	412	5	,	,	PUNCT
ma-232	412	6	let	let	VERB
ma-232	413	1	z	z	NOUN
ma-232	413	2	=	=	SYM
ma-232	413	3	ω	ω	PROPN
ma-232	413	4	−	−	PROPN
ma-232	414	1	t	t	NOUN
ma-232	414	2	then	then	ADV
ma-232	414	3	ω	ω	PROPN
ma-232	414	4	=	=	PUNCT
ma-232	415	1	z	z	PROPN
ma-232	415	2	+	+	NUM
ma-232	415	3	t	t	PROPN
ma-232	415	4	and	and	CCONJ
ma-232	415	5	=(	=(	PROPN
ma-232	415	6	ω	ω	NUM
ma-232	415	7	)	)	PUNCT
ma-232	416	1	=	=	SYM
ma-232	416	2	=(	=(	NOUN
ma-232	416	3	z	z	NOUN
ma-232	416	4	)	)	PUNCT
ma-232	416	5	.	.	PUNCT
ma-232	417	1	hence	hence	ADV
ma-232	417	2	,	,	PUNCT
ma-232	417	3	‖stg‖b∞,	‖stg‖b∞,	NOUN
ma-232	417	4	◦	◦	NOUN
ma-232	417	5	(u	(u	NOUN
ma-232	417	6	)	)	PUNCT
ma-232	417	7	=	=	SYM
ma-232	417	8	sup	sup	NOUN
ma-232	417	9	z∈u	z∈u	VERB
ma-232	417	10	=(	=(	NOUN
ma-232	417	11	z)|g′(z)|	z)|g′(z)|	PROPN
ma-232	417	12	=	=	SYM
ma-232	417	13	‖g‖b∞,	‖g‖b∞,	PROPN
ma-232	417	14	◦	◦	NOUN
ma-232	417	15	(u	(u	NOUN
ma-232	417	16	)	)	PUNCT
ma-232	417	17	,	,	PUNCT
ma-232	417	18	as	as	SCONJ
ma-232	417	19	desired	desire	VERB
ma-232	417	20	.	.	PUNCT
ma-232	418	1	for	for	ADP
ma-232	418	2	strongly	strongly	ADV
ma-232	418	3	continuity	continuity	NOUN
ma-232	418	4	property	property	NOUN
ma-232	418	5	,	,	PUNCT
ma-232	418	6	we	we	PRON
ma-232	418	7	argue	argue	VERB
ma-232	418	8	as	as	SCONJ
ma-232	418	9	we	we	PRON
ma-232	418	10	did	do	VERB
ma-232	418	11	in	in	ADP
ma-232	418	12	the	the	DET
ma-232	418	13	previous	previous	ADJ
ma-232	418	14	section	section	NOUN
ma-232	418	15	.	.	PUNCT
ma-232	419	1	we	we	PRON
ma-232	419	2	note	note	VERB
ma-232	419	3	that	that	SCONJ
ma-232	419	4	st	st	PROPN
ma-232	419	5	=	=	SYM
ma-232	419	6	cϕ−t	cϕ−t	NOUN
ma-232	419	7	which	which	PRON
ma-232	419	8	is	be	AUX
ma-232	419	9	strongly	strongly	ADV
ma-232	419	10	continuous	continuous	ADJ
ma-232	419	11	on	on	ADP
ma-232	419	12	b∞,	b∞,	NOUN
ma-232	419	13	◦	◦	NOUN
ma-232	419	14	(u	(u	NOUN
ma-232	419	15	)	)	PUNCT
ma-232	420	1	if	if	SCONJ
ma-232	420	2	and	and	CCONJ
ma-232	420	3	only	only	ADV
ma-232	420	4	if	if	SCONJ
ma-232	420	5	(	(	PUNCT
ma-232	420	6	cψ−1	cψ−1	PROPN
ma-232	420	7	◦	◦	NOUN
ma-232	420	8	ϕ−t	ϕ−t	NOUN
ma-232	420	9	◦	◦	NOUN
ma-232	420	10	ψ)t∈r	ψ)t∈r	PROPN
ma-232	420	11	is	be	AUX
ma-232	420	12	strongly	strongly	ADV
ma-232	420	13	continuouson	continuouson	NOUN
ma-232	420	14	b∞,	b∞,	ADP
ma-232	420	15	◦	◦	NOUN
ma-232	420	16	(d	(d	NOUN
ma-232	420	17	)	)	PUNCT
ma-232	420	18	,	,	PUNCT
ma-232	420	19	which	which	PRON
ma-232	420	20	consists	consist	VERB
ma-232	420	21	of	of	ADP
ma-232	420	22	functions	function	NOUN
ma-232	420	23	vanishing	vanish	VERB
ma-232	420	24	at	at	ADP
ma-232	420	25	point	point	NOUN
ma-232	420	26	0.we	0.we	NUM
ma-232	420	27	compute	compute	VERB
ma-232	420	28	ψ−1	ψ−1	PROPN
ma-232	420	29	◦	◦	NOUN
ma-232	420	30	ϕ−t	ϕ−t	NOUN
ma-232	420	31	◦	◦	NOUN
ma-232	420	32	ψ(z	ψ(z	PROPN
ma-232	420	33	)	)	PUNCT
ma-232	420	34	.	.	PUNCT
ma-232	421	1	let	let	VERB
ma-232	421	2	at	at	ADP
ma-232	421	3	=	=	PROPN
ma-232	421	4	t	t	PROPN
ma-232	421	5	2i+t	2i+t	NUM
ma-232	421	6	and	and	CCONJ
ma-232	421	7	bt	bt	X
ma-232	421	8	=	=	SYM
ma-232	421	9	2i−t	2i−t	NUM
ma-232	421	10	2i+t	2i+t	NUM
ma-232	421	11	,	,	PUNCT
ma-232	421	12	then	then	ADV
ma-232	421	13	a	a	DET
ma-232	421	14	straight	straight	ADV
ma-232	421	15	forward	forward	ADV
ma-232	421	16	calculationyields	calculationyield	VERB
ma-232	421	17	ψ−1	ψ−1	PROPN
ma-232	421	18	◦	◦	NOUN
ma-232	421	19	ϕ−t	ϕ−t	NOUN
ma-232	421	20	◦	◦	NOUN
ma-232	421	21	ψ(z	ψ(z	NOUN
ma-232	421	22	)	)	PUNCT
ma-232	421	23	=	=	SYM
ma-232	422	1	z	z	NOUN
ma-232	423	1	−	−	NOUN
ma-232	423	2	at	at	ADP
ma-232	423	3	bt	bt	PROPN
ma-232	423	4	+	+	CCONJ
ma-232	423	5	atz	atz	PROPN
ma-232	423	6	=	=	SYM
ma-232	423	7	ha(z	ha(z	NOUN
ma-232	423	8	)	)	PUNCT
ma-232	423	9	,	,	PUNCT
ma-232	423	10	where	where	SCONJ
ma-232	423	11	we	we	PRON
ma-232	423	12	have	have	AUX
ma-232	423	13	let	let	VERB
ma-232	423	14	ha(z	ha(z	PRON
ma-232	423	15	)	)	PUNCT
ma-232	424	1	=	=	SYM
ma-232	424	2	z−at	z−at	NOUN
ma-232	424	3	bt+atz	bt+atz	NOUN
ma-232	424	4	.	.	PUNCT
ma-232	425	1	clearly	clearly	ADV
ma-232	425	2	,	,	PUNCT
ma-232	425	3	t	t	PROPN
ma-232	425	4	→	→	SYM
ma-232	425	5	0	0	PUNCT
ma-232	425	6	as	as	ADP
ma-232	425	7	at	at	ADP
ma-232	425	8	→	→	SYM
ma-232	425	9	0	0	NUM
ma-232	425	10	and	and	CCONJ
ma-232	425	11	bt	bt	NOUN
ma-232	425	12	→	→	SYM
ma-232	425	13	1	1	NUM
ma-232	425	14	.	.	PUNCT
ma-232	425	15	it	it	PRON
ma-232	425	16	therefore	therefore	ADV
ma-232	425	17	suffices	suffice	VERB
ma-232	425	18	toshow	toshow	ADJ
ma-232	425	19	that	that	SCONJ
ma-232	425	20	‖cha	‖cha	PROPN
ma-232	426	1	f	f	PROPN
ma-232	426	2	−f	−f	PROPN
ma-232	426	3	‖b∞,	‖b∞,	PROPN
ma-232	426	4	◦	◦	NOUN
ma-232	426	5	(d	(d	NOUN
ma-232	426	6	)	)	PUNCT
ma-232	426	7	→	→	SYM
ma-232	426	8	0	0	NUM
ma-232	426	9	as	as	ADP
ma-232	426	10	t	t	PROPN
ma-232	426	11	→	→	SYM
ma-232	426	12	0	0	X
ma-232	426	13	.	.	PUNCT
ma-232	427	1	using	use	VERB
ma-232	427	2	density	density	NOUN
ma-232	427	3	of	of	ADP
ma-232	427	4	polynomials	polynomial	NOUN
ma-232	427	5	in	in	ADP
ma-232	427	6	b∞,	b∞,	ADP
ma-232	427	7	◦	◦	NOUN
ma-232	427	8	(d	(d	NOUN
ma-232	427	9	)	)	PUNCT
ma-232	427	10	,	,	PUNCT
ma-232	427	11	we	we	PRON
ma-232	427	12	let	let	VERB
ma-232	427	13	f	f	PROPN
ma-232	427	14	(	(	PUNCT
ma-232	427	15	z	z	NOUN
ma-232	427	16	)	)	PUNCT
ma-232	427	17	=	=	SYM
ma-232	428	1	zn	zn	X
ma-232	428	2	.	.	PUNCT
ma-232	428	3	then	then	ADV
ma-232	428	4	chazn−zn	chazn−zn	NOUN
ma-232	428	5	=	=	SYM
ma-232	428	6	(	(	PUNCT
ma-232	428	7	ha(z))n−zn	ha(z))n−zn	PROPN
ma-232	428	8	,	,	PUNCT
ma-232	428	9	n	n	PRON
ma-232	428	10	≥	≥	NOUN
ma-232	428	11	1	1	NUM
ma-232	428	12	.	.	PUNCT
ma-232	429	1	therefore	therefore	ADV
ma-232	429	2	(	(	PUNCT
ma-232	429	3	cha	cha	NOUN
ma-232	429	4	f	f	PROPN
ma-232	429	5	−f	−f	PROPN
ma-232	429	6	)	)	PUNCT
ma-232	429	7	′(z	′(z	NOUN
ma-232	429	8	)	)	PUNCT
ma-232	429	9	=	=	SYM
ma-232	429	10	n[(ha(z))n−1h′a(z)−zn−1].but	n[(ha(z))n−1h′a(z)−zn−1].but	PROPN
ma-232	429	11	ha(z	ha(z	PRON
ma-232	429	12	)	)	PUNCT
ma-232	429	13	=	=	SYM
ma-232	430	1	z−at	z−at	NOUN
ma-232	430	2	bt+atz	bt+atz	PROPN
ma-232	430	3	⇒	⇒	VERB
ma-232	430	4	h′a(z	h′a(z	PROPN
ma-232	430	5	)	)	PUNCT
ma-232	431	1	=	=	PRON
ma-232	431	2	(	(	PUNCT
ma-232	431	3	bt+atz)(1)−(z−at)(at	bt+atz)(1)−(z−at)(at	NOUN
ma-232	431	4	)	)	PUNCT
ma-232	431	5	(	(	PUNCT
ma-232	431	6	bt+atz)2	bt+atz)2	NOUN
ma-232	431	7	.	.	PUNCT
ma-232	432	1	therefore	therefore	ADV
ma-232	432	2	by	by	ADP
ma-232	432	3	substituting	substitute	VERB
ma-232	432	4	,	,	PUNCT
ma-232	432	5	(	(	PUNCT
ma-232	432	6	cha	cha	NOUN
ma-232	432	7	f	f	NOUN
ma-232	433	1	−	−	PROPN
ma-232	433	2	f	f	PROPN
ma-232	433	3	)	)	PUNCT
ma-232	433	4	′(z	′(z	NOUN
ma-232	433	5	)	)	PUNCT
ma-232	434	1	=	=	SYM
ma-232	434	2	n[(ha(z))n−1h′a(z)−	n[(ha(z))n−1h′a(z)−	NUM
ma-232	434	3	zn−1	zn−1	PROPN
ma-232	434	4	]	]	X
ma-232	434	5	=	=	SYM
ma-232	434	6	n	n	PRON
ma-232	434	7	[	[	X
ma-232	434	8	(	(	PUNCT
ma-232	434	9	z	z	NOUN
ma-232	434	10	−	−	NOUN
ma-232	434	11	at	at	ADP
ma-232	434	12	bt	bt	PROPN
ma-232	434	13	+	+	CCONJ
ma-232	434	14	atz	atz	PROPN
ma-232	434	15	)	)	PUNCT
ma-232	434	16	n−1	n−1	PROPN
ma-232	434	17	(	(	PUNCT
ma-232	434	18	bt	bt	NOUN
ma-232	434	19	+	+	X
ma-232	434	20	atz)−	atz)−	PROPN
ma-232	434	21	(	(	PUNCT
ma-232	434	22	z	z	NOUN
ma-232	434	23	−	−	PROPN
ma-232	434	24	at)(at	at)(at	PROPN
ma-232	434	25	)	)	PUNCT
ma-232	434	26	(	(	PUNCT
ma-232	434	27	bt	bt	NOUN
ma-232	434	28	+	+	CCONJ
ma-232	434	29	atz)2	atz)2	PROPN
ma-232	434	30	−	−	PROPN
ma-232	434	31	zn−1	zn−1	PROPN
ma-232	434	32	]	]	PUNCT
ma-232	434	33	=	=	PUNCT
ma-232	435	1	n	n	PART
ma-232	435	2	[	[	PUNCT
ma-232	435	3	(	(	PUNCT
ma-232	435	4	z	z	NOUN
ma-232	435	5	−	−	PROPN
ma-232	435	6	at)n−1(bt	at)n−1(bt	NOUN
ma-232	435	7	+	+	CCONJ
ma-232	435	8	atz)−	atz)−	PROPN
ma-232	435	9	(	(	PUNCT
ma-232	435	10	z	z	NOUN
ma-232	435	11	−	−	PROPN
ma-232	435	12	at)(at	at)(at	PROPN
ma-232	435	13	)	)	PUNCT
ma-232	435	14	(	(	PUNCT
ma-232	435	15	bt	bt	X
ma-232	435	16	+	+	CCONJ
ma-232	435	17	atz)n+1	atz)n+1	PROPN
ma-232	435	18	−	−	PROPN
ma-232	435	19	zn−1	zn−1	PROPN
ma-232	435	20	]	]	PUNCT
ma-232	435	21	.	.	PUNCT
ma-232	436	1	now	now	ADV
ma-232	436	2	,	,	PUNCT
ma-232	436	3	lim	lim	PROPN
ma-232	436	4	t→0	t→0	PROPN
ma-232	436	5	+	+	CCONJ
ma-232	436	6	‖cha	‖cha	PROPN
ma-232	437	1	f	f	NOUN
ma-232	437	2	−	−	PROPN
ma-232	438	1	f	f	PROPN
ma-232	438	2	‖b∞,	‖b∞,	PROPN
ma-232	438	3	◦	◦	NOUN
ma-232	438	4	(d	(d	NOUN
ma-232	438	5	)	)	PUNCT
ma-232	438	6	=	=	SYM
ma-232	438	7	lim	lim	PROPN
ma-232	438	8	t→0	t→0	PROPN
ma-232	438	9	+	+	CCONJ
ma-232	438	10	(	(	PUNCT
ma-232	438	11	sup	sup	NUM
ma-232	438	12	z∈d	z∈d	NUM
ma-232	438	13	(	(	PUNCT
ma-232	438	14	1−	1−	NUM
ma-232	438	15	|z	|z	PROPN
ma-232	438	16	|2)|(cha	|2)|(cha	PROPN
ma-232	438	17	f	f	PROPN
ma-232	439	1	−	−	PROPN
ma-232	439	2	f	f	PROPN
ma-232	439	3	)	)	PUNCT
ma-232	439	4	′|(z	′|(z	PROPN
ma-232	439	5	)	)	PUNCT
ma-232	439	6	)	)	PUNCT
ma-232	440	1	=	=	SYM
ma-232	440	2	lim	lim	PROPN
ma-232	440	3	t→0	t→0	PROPN
ma-232	441	1	+	+	CCONJ
ma-232	441	2	(	(	PUNCT
ma-232	441	3	sup	sup	NUM
ma-232	441	4	z∈d	z∈d	NUM
ma-232	441	5	(	(	PUNCT
ma-232	441	6	1−	1−	NUM
ma-232	441	7	|z	|z	PROPN
ma-232	442	1	|2)∣∣∣∣n	|2)∣∣∣∣n	ADV
ma-232	443	1	[	[	X
ma-232	443	2	(	(	PUNCT
ma-232	443	3	z	z	NOUN
ma-232	443	4	−	−	PROPN
ma-232	443	5	at)n−1(bt	at)n−1(bt	NOUN
ma-232	443	6	+	+	CCONJ
ma-232	443	7	atz)−	atz)−	PROPN
ma-232	443	8	(	(	PUNCT
ma-232	443	9	z	z	NOUN
ma-232	443	10	−	−	PROPN
ma-232	443	11	at)(at	at)(at	PROPN
ma-232	443	12	)	)	PUNCT
ma-232	443	13	(	(	PUNCT
ma-232	443	14	bt	bt	X
ma-232	443	15	+	+	CCONJ
ma-232	443	16	atz)n+1	atz)n+1	PROPN
ma-232	443	17	−	−	PROPN
ma-232	443	18	zn−1	zn−1	PROPN
ma-232	443	19	]	]	PUNCT
ma-232	443	20	∣∣∣∣	∣∣∣∣	NOUN
ma-232	443	21	)	)	PUNCT
ma-232	443	22	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	443	23	eur	eur	PROPN
ma-232	443	24	.	.	PUNCT
ma-232	444	1	j.	j.	PROPN
ma-232	444	2	math	math	PROPN
ma-232	444	3	.	.	PUNCT
ma-232	445	1	anal	anal	PROPN
ma-232	445	2	.	.	PUNCT
ma-232	446	1	10.28924	10.28924	NUM
ma-232	446	2	/	/	SYM
ma-232	446	3	ada	ada	PROPN
ma-232	446	4	/	/	SYM
ma-232	446	5	ma.4.14	ma.4.14	PROPN
ma-232	447	1	15	15	NUM
ma-232	447	2	=	=	SYM
ma-232	447	3	lim	lim	PROPN
ma-232	447	4	t→0	t→0	PROPN
ma-232	448	1	+	+	CCONJ
ma-232	449	1	(	(	PUNCT
ma-232	449	2	sup	sup	NUM
ma-232	449	3	z∈d	z∈d	NUM
ma-232	449	4	(	(	PUNCT
ma-232	449	5	1−	1−	NUM
ma-232	449	6	|z	|z	NOUN
ma-232	449	7	|2	|2	NUM
ma-232	449	8	)	)	PUNCT
ma-232	449	9	∣∣∣∣n[zn−1	∣∣∣∣n[zn−1	PRON
ma-232	449	10	−	−	NOUN
ma-232	449	11	0−	0−	NUM
ma-232	450	1	zn−1	zn−1	PROPN
ma-232	450	2	]	]	X
ma-232	450	3	1	1	NUM
ma-232	450	4	∣∣∣∣	∣∣∣∣	PROPN
ma-232	450	5	)	)	PUNCT
ma-232	450	6	=	=	NOUN
ma-232	451	1	0	0	X
ma-232	451	2	.	.	PUNCT
ma-232	452	1	hence	hence	ADV
ma-232	452	2	(	(	PUNCT
ma-232	452	3	st)∈r	st)∈r	PRON
ma-232	452	4	is	be	AUX
ma-232	452	5	strongly	strongly	ADV
ma-232	452	6	continuous	continuous	ADJ
ma-232	452	7	,	,	PUNCT
ma-232	452	8	as	as	SCONJ
ma-232	452	9	claimed	claim	VERB
ma-232	452	10	.	.	PUNCT
ma-232	453	1	�	�	PROPN
ma-232	453	2	theorem	theorem	VERB
ma-232	453	3	3.7	3.7	NUM
ma-232	453	4	.	.	PUNCT
ma-232	454	1	the	the	DET
ma-232	454	2	infinitesimal	infinitesimal	ADJ
ma-232	454	3	generator	generator	NOUN
ma-232	454	4	γ	γ	X
ma-232	454	5	of	of	ADP
ma-232	454	6	(	(	PUNCT
ma-232	454	7	st)t≥0	st)t≥0	NOUN
ma-232	454	8	on	on	ADP
ma-232	454	9	b∞,	b∞,	ADP
ma-232	454	10	◦	◦	NOUN
ma-232	454	11	(u	(u	NOUN
ma-232	454	12	)	)	PUNCT
ma-232	454	13	is	be	AUX
ma-232	454	14	given	give	VERB
ma-232	454	15	by	by	ADP
ma-232	454	16	γg(ω	γg(ω	PRON
ma-232	454	17	)	)	PUNCT
ma-232	455	1	=	=	SYM
ma-232	455	2	−g′(ω	−g′(ω	PROPN
ma-232	455	3	)	)	PUNCT
ma-232	455	4	with	with	ADP
ma-232	455	5	the	the	DET
ma-232	455	6	domain	domain	NOUN
ma-232	455	7	d(γ	d(γ	PROPN
ma-232	455	8	)	)	PUNCT
ma-232	455	9	=	=	SYM
ma-232	455	10	{	{	PUNCT
ma-232	455	11	g	g	NOUN
ma-232	455	12	∈	∈	PROPN
ma-232	455	13	b∞,	b∞,	ADP
ma-232	455	14	◦	◦	NOUN
ma-232	455	15	(u	(u	NOUN
ma-232	455	16	)	)	PUNCT
ma-232	455	17	:	:	PUNCT
ma-232	455	18	g′	g′	NOUN
ma-232	455	19	∈	∈	PROPN
ma-232	455	20	b∞,	b∞,	ADP
ma-232	455	21	◦	◦	NOUN
ma-232	455	22	(u	(u	NOUN
ma-232	455	23	)	)	PUNCT
ma-232	455	24	}	}	PUNCT
ma-232	455	25	.	.	PUNCT
ma-232	456	1	proof	proof	NOUN
ma-232	456	2	.	.	PUNCT
ma-232	457	1	by	by	ADP
ma-232	457	2	definition	definition	NOUN
ma-232	457	3	,	,	PUNCT
ma-232	457	4	the	the	DET
ma-232	457	5	infinitesimal	infinitesimal	ADJ
ma-232	457	6	generator	generator	NOUN
ma-232	457	7	γ	γ	X
ma-232	457	8	on	on	ADP
ma-232	457	9	b∞,	b∞,	ADP
ma-232	457	10	◦	◦	NOUN
ma-232	457	11	(u	(u	NOUN
ma-232	457	12	)	)	PUNCT
ma-232	457	13	is	be	AUX
ma-232	457	14	given	give	VERB
ma-232	457	15	by	by	ADP
ma-232	457	16	;	;	PUNCT
ma-232	457	17	γg(ω	γg(ω	NUM
ma-232	457	18	)	)	PUNCT
ma-232	457	19	=	=	SYM
ma-232	457	20	lim	lim	PROPN
ma-232	457	21	t→0	t→0	PROPN
ma-232	457	22	+	+	CCONJ
ma-232	457	23	g(ω	g(ω	PROPN
ma-232	457	24	−	−	PROPN
ma-232	457	25	t)−	t)−	PROPN
ma-232	457	26	g(ω	g(ω	PROPN
ma-232	457	27	)	)	PUNCT
ma-232	457	28	t	t	PROPN
ma-232	457	29	=	=	SYM
ma-232	457	30	∂	∂	NUM
ma-232	457	31	∂t	∂t	PROPN
ma-232	457	32	g(ω	g(ω	PROPN
ma-232	457	33	−	−	PROPN
ma-232	457	34	t	t	PROPN
ma-232	457	35	)	)	PUNCT
ma-232	457	36	∣∣∣∣	∣∣∣∣	NOUN
ma-232	457	37	t=0	t=0	VERB
ma-232	457	38	=	=	SYM
ma-232	457	39	−g′(ω	−g′(ω	PROPN
ma-232	457	40	)	)	PUNCT
ma-232	457	41	.	.	PUNCT
ma-232	458	1	therefore	therefore	ADV
ma-232	458	2	d(γ	d(γ	PROPN
ma-232	458	3	)	)	PUNCT
ma-232	459	1	⊂	⊂	PROPN
ma-232	459	2	{	{	PUNCT
ma-232	459	3	g	g	PROPN
ma-232	459	4	∈	∈	PROPN
ma-232	459	5	b∞,	b∞,	ADP
ma-232	459	6	◦	◦	NOUN
ma-232	459	7	(u	(u	NOUN
ma-232	459	8	)	)	PUNCT
ma-232	459	9	:	:	PUNCT
ma-232	459	10	−g′	−g′	PROPN
ma-232	459	11	∈	∈	PROPN
ma-232	459	12	b∞,	b∞,	ADP
ma-232	459	13	◦	◦	NOUN
ma-232	459	14	(u	(u	NOUN
ma-232	459	15	)	)	PUNCT
ma-232	459	16	}	}	PUNCT
ma-232	459	17	.	.	PUNCT
ma-232	460	1	conversely	conversely	ADV
ma-232	460	2	,	,	PUNCT
ma-232	460	3	let	let	VERB
ma-232	460	4	g	g	PROPN
ma-232	460	5	∈	∈	PROPN
ma-232	460	6	b∞,	b∞,	ADP
ma-232	460	7	◦	◦	NOUN
ma-232	460	8	(u	(u	NOUN
ma-232	460	9	)	)	PUNCT
ma-232	460	10	be	be	AUX
ma-232	460	11	such	such	ADJ
ma-232	460	12	that	that	SCONJ
ma-232	460	13	−g′	−g′	PROPN
ma-232	460	14	∈	∈	PROPN
ma-232	460	15	b∞,	b∞,	ADP
ma-232	460	16	◦	◦	NOUN
ma-232	460	17	(u	(u	NOUN
ma-232	460	18	)	)	PUNCT
ma-232	460	19	.	.	PUNCT
ma-232	461	1	thus	thus	ADV
ma-232	461	2	we	we	PRON
ma-232	461	3	have	have	VERB
ma-232	461	4	;	;	PUNCT
ma-232	462	1	stg	stg	PROPN
ma-232	462	2	−	−	PROPN
ma-232	462	3	g	g	PROPN
ma-232	462	4	t	t	PROPN
ma-232	462	5	=	=	SYM
ma-232	462	6	1	1	NUM
ma-232	462	7	t	t	NOUN
ma-232	462	8	∫	∫	PROPN
ma-232	462	9	t	t	PROPN
ma-232	462	10	0	0	NUM
ma-232	462	11	∂	∂	NUM
ma-232	462	12	∂s	∂s	PROPN
ma-232	462	13	ssg	ssg	X
ma-232	462	14	ds	ds	NOUN
ma-232	462	15	and	and	CCONJ
ma-232	462	16	for	for	ADP
ma-232	462	17	every	every	DET
ma-232	462	18	ω	ω	PROPN
ma-232	462	19	∈	∈	PROPN
ma-232	462	20	u	u	NOUN
ma-232	462	21	,	,	PUNCT
ma-232	462	22	∂	∂	NUM
ma-232	462	23	∂sssg(ω	∂sssg(ω	NUM
ma-232	462	24	)	)	PUNCT
ma-232	463	1	=	=	SYM
ma-232	463	2	−g′(ω	−g′(ω	NOUN
ma-232	463	3	−	−	PROPN
ma-232	463	4	s	s	PART
ma-232	463	5	)	)	PUNCT
ma-232	463	6	=	=	SYM
ma-232	463	7	ssg	ssg	PROPN
ma-232	463	8	′(ω	′(ω	NOUN
ma-232	463	9	)	)	PUNCT
ma-232	463	10	.	.	PUNCT
ma-232	464	1	thus,∥∥∥∥ssg	thus,∥∥∥∥ssg	VERB
ma-232	464	2	−	−	PROPN
ma-232	465	1	gt	gt	INTJ
ma-232	466	1	−	−	PROPN
ma-232	466	2	f	f	NOUN
ma-232	467	1	′	′	NOUN
ma-232	467	2	∥∥∥∥	∥∥∥∥	NUM
ma-232	468	1	≤	≤	NUM
ma-232	468	2	1	1	NUM
ma-232	468	3	t	t	NOUN
ma-232	468	4	∫	∫	PROPN
ma-232	468	5	t	t	PROPN
ma-232	468	6	0	0	NUM
ma-232	468	7	∥∥ts	∥∥ts	SYM
ma-232	468	8	f	f	NOUN
ma-232	469	1	′	′	NOUN
ma-232	469	2	−	−	PROPN
ma-232	470	1	f	f	PROPN
ma-232	470	2	′∥∥	′∥∥	PROPN
ma-232	470	3	ds	ds	X
ma-232	470	4	→	→	SYM
ma-232	470	5	0	0	NUM
ma-232	470	6	as	as	ADP
ma-232	470	7	t	t	PROPN
ma-232	470	8	→	→	SYM
ma-232	470	9	0	0	NUM
ma-232	470	10	by	by	ADP
ma-232	470	11	strong	strong	ADJ
ma-232	470	12	continuity	continuity	NOUN
ma-232	470	13	,	,	PUNCT
ma-232	470	14	and	and	CCONJ
ma-232	470	15	therefore	therefore	ADV
ma-232	470	16	d(γ	d(γ	PROPN
ma-232	470	17	)	)	PUNCT
ma-232	470	18	⊇	⊇	NOUN
ma-232	470	19	{	{	PUNCT
ma-232	470	20	g	g	PROPN
ma-232	470	21	∈	∈	PROPN
ma-232	470	22	b∞,	b∞,	ADP
ma-232	470	23	◦	◦	NOUN
ma-232	470	24	(u	(u	NOUN
ma-232	470	25	)	)	PUNCT
ma-232	470	26	:	:	PUNCT
ma-232	471	1	−g′	−g′	PROPN
ma-232	471	2	∈	∈	PROPN
ma-232	471	3	b∞,	b∞,	ADP
ma-232	471	4	◦	◦	NOUN
ma-232	471	5	(u	(u	NOUN
ma-232	471	6	)	)	PUNCT
ma-232	471	7	}	}	PUNCT
ma-232	471	8	,	,	PUNCT
ma-232	471	9	which	which	PRON
ma-232	471	10	completes	complete	VERB
ma-232	471	11	theproof	theproof	NOUN
ma-232	471	12	.	.	PUNCT
ma-232	472	1	�	�	PROPN
ma-232	472	2	acknowledgement	acknowledgement	NOUN
ma-232	472	3	this	this	DET
ma-232	472	4	work	work	NOUN
ma-232	472	5	was	be	AUX
ma-232	472	6	completed	complete	VERB
ma-232	472	7	during	during	ADP
ma-232	472	8	the	the	DET
ma-232	472	9	period	period	NOUN
ma-232	472	10	when	when	SCONJ
ma-232	472	11	the	the	DET
ma-232	472	12	second	second	ADJ
ma-232	472	13	author	author	NOUN
ma-232	472	14	was	be	AUX
ma-232	472	15	visiting	visit	VERB
ma-232	472	16	the	the	DET
ma-232	472	17	aristotleuniversity	aristotleuniversity	NOUN
ma-232	472	18	of	of	ADP
ma-232	472	19	thessaloniki	thessaloniki	PROPN
ma-232	472	20	,	,	PUNCT
ma-232	472	21	greece	greece	PROPN
ma-232	472	22	.	.	PUNCT
ma-232	473	1	he	he	PRON
ma-232	473	2	would	would	AUX
ma-232	473	3	like	like	VERB
ma-232	473	4	to	to	PART
ma-232	473	5	sincerely	sincerely	ADV
ma-232	473	6	that	that	SCONJ
ma-232	473	7	the	the	DET
ma-232	473	8	simon	simon	PROPN
ma-232	473	9	’s	’s	PART
ma-232	473	10	foundation	foundation	NOUN
ma-232	473	11	forfunding	forfunde	VERB
ma-232	473	12	his	his	PRON
ma-232	473	13	visit	visit	NOUN
ma-232	473	14	.	.	PUNCT
ma-232	474	1	he	he	PRON
ma-232	474	2	would	would	AUX
ma-232	474	3	also	also	ADV
ma-232	474	4	wish	wish	VERB
ma-232	474	5	to	to	PART
ma-232	474	6	thank	thank	VERB
ma-232	474	7	his	his	PRON
ma-232	474	8	host	host	NOUN
ma-232	474	9	prof	prof	NOUN
ma-232	474	10	.	.	PROPN
ma-232	475	1	aristomenis	aristomenis	PROPN
ma-232	475	2	g.	g.	PROPN
ma-232	475	3	siskakis	siskakis	PROPN
ma-232	475	4	and	and	CCONJ
ma-232	475	5	thedepartment	thedepartment	NOUN
ma-232	475	6	of	of	ADP
ma-232	475	7	mathematics	mathematic	NOUN
ma-232	475	8	for	for	ADP
ma-232	475	9	the	the	DET
ma-232	475	10	unmatched	unmatched	ADJ
ma-232	475	11	hospitality	hospitality	NOUN
ma-232	475	12	references	reference	NOUN
ma-232	475	13	[	[	X
ma-232	475	14	1	1	NUM
ma-232	475	15	]	]	PUNCT
ma-232	475	16	a.	a.	NOUN
ma-232	475	17	g.	g.	PROPN
ma-232	475	18	arvanitidis	arvanitidis	PROPN
ma-232	475	19	,	,	PUNCT
ma-232	475	20	a.	a.	PROPN
ma-232	475	21	g.	g.	PROPN
ma-232	475	22	siskakis	siskakis	PROPN
ma-232	475	23	,	,	PUNCT
ma-232	475	24	cesàro	cesàro	PROPN
ma-232	475	25	operators	operator	NOUN
ma-232	475	26	on	on	ADP
ma-232	475	27	the	the	DET
ma-232	475	28	hardy	hardy	ADJ
ma-232	475	29	spaces	space	NOUN
ma-232	475	30	of	of	ADP
ma-232	475	31	the	the	DET
ma-232	475	32	half	half	ADJ
ma-232	475	33	plane	plane	NOUN
ma-232	475	34	.	.	PUNCT
ma-232	476	1	canadian	canadian	ADJ
ma-232	476	2	math	math	PROPN
ma-232	476	3	.	.	PUNCT
ma-232	477	1	bull	bull	NOUN
ma-232	477	2	.	.	PUNCT
ma-232	478	1	56(2013	56(2013	NUM
ma-232	478	2	)	)	PUNCT
ma-232	478	3	,	,	PUNCT
ma-232	478	4	229–240.[2	229–240.[2	NUM
ma-232	478	5	]	]	PUNCT
ma-232	478	6	s.	s.	PROPN
ma-232	478	7	axler	axler	PROPN
ma-232	478	8	,	,	PUNCT
ma-232	478	9	bergman	bergman	PROPN
ma-232	478	10	spaces	space	VERB
ma-232	478	11	and	and	CCONJ
ma-232	478	12	their	their	PRON
ma-232	478	13	operators	operator	NOUN
ma-232	478	14	,	,	PUNCT
ma-232	478	15	lecture	lecture	NOUN
ma-232	478	16	notes	note	NOUN
ma-232	478	17	at	at	ADP
ma-232	478	18	the	the	DET
ma-232	478	19	indiana	indiana	PROPN
ma-232	478	20	university	university	PROPN
ma-232	478	21	function	function	NOUN
ma-232	478	22	theoretic	theoretic	NOUN
ma-232	478	23	operatorstheory	operatorstheory	ADJ
ma-232	478	24	conference	conference	NOUN
ma-232	478	25	,	,	PUNCT
ma-232	478	26	1985.[3	1985.[3	NUM
ma-232	478	27	]	]	X
ma-232	478	28	s.	s.	PROPN
ma-232	478	29	ballamoole	ballamoole	PROPN
ma-232	478	30	,	,	PUNCT
ma-232	478	31	j.	j.	PROPN
ma-232	478	32	o.	o.	PROPN
ma-232	478	33	bonyo	bonyo	PROPN
ma-232	478	34	,	,	PUNCT
ma-232	478	35	t.	t.	PROPN
ma-232	478	36	l.	l.	PROPN
ma-232	478	37	miller	miller	PROPN
ma-232	478	38	,	,	PUNCT
ma-232	478	39	v.	v.	PROPN
ma-232	478	40	g.	g.	PROPN
ma-232	478	41	miller	miller	PROPN
ma-232	478	42	,	,	PUNCT
ma-232	478	43	cesaro	cesaro	NOUN
ma-232	478	44	-	-	PUNCT
ma-232	478	45	like	like	ADJ
ma-232	478	46	operators	operator	NOUN
ma-232	478	47	on	on	ADP
ma-232	478	48	the	the	DET
ma-232	478	49	hardy	hardy	ADJ
ma-232	478	50	and	and	CCONJ
ma-232	478	51	bergman	bergman	PROPN
ma-232	478	52	spaces	space	NOUN
ma-232	478	53	of	of	ADP
ma-232	478	54	thehalf	thehalf	NOUN
ma-232	478	55	plane	plane	NOUN
ma-232	478	56	,	,	PUNCT
ma-232	478	57	complex	complex	ADJ
ma-232	478	58	anal	anal	NOUN
ma-232	478	59	.	.	PUNCT
ma-232	479	1	oper	oper	PROPN
ma-232	479	2	.	.	PROPN
ma-232	479	3	theory	theory	NOUN
ma-232	479	4	,	,	PUNCT
ma-232	479	5	10	10	NUM
ma-232	479	6	(	(	PUNCT
ma-232	479	7	2016	2016	NUM
ma-232	479	8	)	)	PUNCT
ma-232	479	9	,	,	PUNCT
ma-232	479	10	187	187	NUM
ma-232	479	11	-	-	SYM
ma-232	479	12	203.[4	203.[4	NUM
ma-232	479	13	]	]	PUNCT
ma-232	479	14	j.	j.	PROPN
ma-232	479	15	o.	o.	PROPN
ma-232	479	16	bonyo	bonyo	PROPN
ma-232	479	17	,	,	PUNCT
ma-232	479	18	spectral	spectral	ADJ
ma-232	479	19	analysis	analysis	NOUN
ma-232	479	20	of	of	ADP
ma-232	479	21	certain	certain	ADJ
ma-232	479	22	groups	group	NOUN
ma-232	479	23	of	of	ADP
ma-232	479	24	isometries	isometry	NOUN
ma-232	479	25	on	on	ADP
ma-232	479	26	hardy	hardy	ADJ
ma-232	479	27	and	and	CCONJ
ma-232	479	28	bergman	bergman	PROPN
ma-232	479	29	spaces	space	VERB
ma-232	479	30	,	,	PUNCT
ma-232	479	31	j.	j.	PROPN
ma-232	479	32	math	math	PROPN
ma-232	479	33	.	.	PUNCT
ma-232	480	1	anal	anal	PROPN
ma-232	480	2	.	.	PUNCT
ma-232	481	1	appl.456	appl.456	NOUN
ma-232	481	2	(	(	PUNCT
ma-232	481	3	2017	2017	NUM
ma-232	481	4	)	)	PUNCT
ma-232	481	5	,	,	PUNCT
ma-232	481	6	1470–1481.[5	1470–1481.[5	NUM
ma-232	481	7	]	]	X
ma-232	481	8	j.	j.	PROPN
ma-232	481	9	b.	b.	PROPN
ma-232	481	10	conway	conway	PROPN
ma-232	481	11	,	,	PUNCT
ma-232	481	12	a	a	DET
ma-232	481	13	course	course	NOUN
ma-232	481	14	in	in	ADP
ma-232	481	15	functional	functional	ADJ
ma-232	481	16	analysis	analysis	NOUN
ma-232	481	17	,	,	PUNCT
ma-232	481	18	springer	springer	NOUN
ma-232	481	19	verlag	verlag	NOUN
ma-232	481	20	,	,	PUNCT
ma-232	481	21	new	new	PROPN
ma-232	481	22	york	york	PROPN
ma-232	481	23	,	,	PUNCT
ma-232	481	24	1985.[6	1985.[6	NUM
ma-232	481	25	]	]	X
ma-232	481	26	n.	n.	PROPN
ma-232	481	27	dunford	dunford	PROPN
ma-232	481	28	,	,	PUNCT
ma-232	481	29	j.	j.	PROPN
ma-232	481	30	t.	t.	PROPN
ma-232	481	31	schwartz	schwartz	PROPN
ma-232	481	32	,	,	PUNCT
ma-232	481	33	linear	linear	PROPN
ma-232	481	34	operators	operator	NOUN
ma-232	481	35	part	part	PROPN
ma-232	481	36	i.	i.	PROPN
ma-232	481	37	interscience	interscience	PROPN
ma-232	481	38	publishers	publisher	NOUN
ma-232	481	39	,	,	PUNCT
ma-232	481	40	new	new	PROPN
ma-232	481	41	york	york	PROPN
ma-232	481	42	,	,	PUNCT
ma-232	481	43	1958	1958	NUM
ma-232	481	44	.	.	PUNCT
ma-232	482	1	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	482	2	eur	eur	PROPN
ma-232	482	3	.	.	PUNCT
ma-232	483	1	j.	j.	PROPN
ma-232	483	2	math	math	PROPN
ma-232	483	3	.	.	PUNCT
ma-232	484	1	anal	anal	PROPN
ma-232	484	2	.	.	PUNCT
ma-232	485	1	10.28924	10.28924	NUM
ma-232	485	2	/	/	SYM
ma-232	485	3	ada	ada	PROPN
ma-232	485	4	/	/	SYM
ma-232	485	5	ma.4.14	ma.4.14	NOUN
ma-232	485	6	16	16	NUM
ma-232	486	1	[	[	X
ma-232	486	2	7	7	X
ma-232	486	3	]	]	X
ma-232	486	4	p.	p.	PROPN
ma-232	486	5	duren	duren	PROPN
ma-232	486	6	,	,	PUNCT
ma-232	486	7	a.	a.	NOUN
ma-232	486	8	schuster	schuster	PROPN
ma-232	486	9	,	,	PUNCT
ma-232	486	10	bergman	bergman	PROPN
ma-232	486	11	spaces	space	VERB
ma-232	486	12	,	,	PUNCT
ma-232	486	13	mathematical	mathematical	ADJ
ma-232	486	14	surveys	survey	NOUN
ma-232	486	15	and	and	CCONJ
ma-232	486	16	monographs	monograph	NOUN
ma-232	486	17	100	100	NUM
ma-232	486	18	,	,	PUNCT
ma-232	486	19	amer	amer	PROPN
ma-232	486	20	.	.	PROPN
ma-232	486	21	math	math	PROPN
ma-232	486	22	.	.	PUNCT
ma-232	487	1	soc	soc	PROPN
ma-232	487	2	.	.	PUNCT
ma-232	487	3	,	,	PUNCT
ma-232	487	4	providence	providence	NOUN
ma-232	487	5	,	,	PUNCT
ma-232	487	6	ri	ri	PROPN
ma-232	487	7	,	,	PUNCT
ma-232	487	8	2004.[8	2004.[8	NUM
ma-232	487	9	]	]	PUNCT
ma-232	487	10	s.	s.	PROPN
ma-232	487	11	h.	h.	PROPN
ma-232	487	12	kang	kang	PROPN
ma-232	487	13	,	,	PUNCT
ma-232	487	14	some	some	DET
ma-232	487	15	duality	duality	NOUN
ma-232	487	16	of	of	ADP
ma-232	487	17	weighted	weight	VERB
ma-232	487	18	bergman	bergman	PROPN
ma-232	487	19	spaces	space	NOUN
ma-232	487	20	of	of	ADP
ma-232	487	21	the	the	DET
ma-232	487	22	half	half	ADJ
ma-232	487	23	-	-	PUNCT
ma-232	487	24	plane	plane	NOUN
ma-232	487	25	,	,	PUNCT
ma-232	487	26	bull	bull	NOUN
ma-232	487	27	.	.	PUNCT
ma-232	488	1	korean	korean	ADJ
ma-232	488	2	math	math	PROPN
ma-232	488	3	.	.	PUNCT
ma-232	489	1	soc	soc	PROPN
ma-232	489	2	.	.	PUNCT
ma-232	490	1	42	42	NUM
ma-232	490	2	(	(	PUNCT
ma-232	490	3	2005	2005	NUM
ma-232	490	4	)	)	PUNCT
ma-232	490	5	,	,	PUNCT
ma-232	490	6	385	385	NUM
ma-232	490	7	-	-	SYM
ma-232	490	8	396.[9	396.[9	PROPN
ma-232	490	9	]	]	PUNCT
ma-232	490	10	k.	k.	PROPN
ma-232	490	11	b.	b.	PROPN
ma-232	491	1	laursen	laursen	PROPN
ma-232	491	2	,	,	PUNCT
ma-232	491	3	m.	m.	NOUN
ma-232	491	4	m.	m.	PROPN
ma-232	491	5	neumann	neumann	PROPN
ma-232	491	6	,	,	PUNCT
ma-232	491	7	an	an	DET
ma-232	491	8	introduction	introduction	NOUN
ma-232	491	9	to	to	ADP
ma-232	491	10	local	local	ADJ
ma-232	491	11	spectral	spectral	ADJ
ma-232	491	12	theory	theory	NOUN
ma-232	491	13	,	,	PUNCT
ma-232	491	14	clarendon	clarendon	PROPN
ma-232	491	15	press	press	PROPN
ma-232	491	16	,	,	PUNCT
ma-232	491	17	oxford	oxford	PROPN
ma-232	491	18	,	,	PUNCT
ma-232	491	19	2000.[10	2000.[10	NUM
ma-232	491	20	]	]	PUNCT
ma-232	491	21	a.	a.	NOUN
ma-232	491	22	pazy	pazy	NOUN
ma-232	491	23	,	,	PUNCT
ma-232	491	24	semigroups	semigroup	NOUN
ma-232	491	25	of	of	ADP
ma-232	491	26	linear	linear	PROPN
ma-232	491	27	operators	operator	NOUN
ma-232	491	28	and	and	CCONJ
ma-232	491	29	applications	application	NOUN
ma-232	491	30	to	to	ADP
ma-232	491	31	partial	partial	ADJ
ma-232	491	32	differential	differential	NOUN
ma-232	491	33	equations	equation	NOUN
ma-232	491	34	,	,	PUNCT
ma-232	491	35	applied	apply	VERB
ma-232	491	36	mathematicalsciences	mathematicalscience	NOUN
ma-232	491	37	40	40	NUM
ma-232	491	38	,	,	PUNCT
ma-232	491	39	springer	springer	NOUN
ma-232	491	40	,	,	PUNCT
ma-232	491	41	new	new	PROPN
ma-232	491	42	york	york	PROPN
ma-232	491	43	,	,	PUNCT
ma-232	491	44	1983.[11	1983.[11	NUM
ma-232	491	45	]	]	X
ma-232	491	46	m.	m.	NOUN
ma-232	491	47	m.	m.	NOUN
ma-232	491	48	peloso	peloso	NOUN
ma-232	491	49	,	,	PUNCT
ma-232	491	50	classical	classical	ADJ
ma-232	491	51	spaces	space	NOUN
ma-232	491	52	of	of	ADP
ma-232	491	53	holomorphic	holomorphic	ADJ
ma-232	491	54	functions	function	NOUN
ma-232	491	55	,	,	PUNCT
ma-232	491	56	technical	technical	ADJ
ma-232	491	57	report	report	NOUN
ma-232	491	58	,	,	PUNCT
ma-232	491	59	universìt	universìt	PROPN
ma-232	491	60	di	di	PROPN
ma-232	491	61	milano	milano	PROPN
ma-232	491	62	,	,	PUNCT
ma-232	491	63	2014.[12	2014.[12	NUM
ma-232	491	64	]	]	X
ma-232	491	65	w.	w.	PROPN
ma-232	491	66	rudin	rudin	PROPN
ma-232	491	67	,	,	PUNCT
ma-232	491	68	functional	functional	ADJ
ma-232	491	69	analysis	analysis	NOUN
ma-232	491	70	,	,	PUNCT
ma-232	491	71	mcgraw	mcgraw	PROPN
ma-232	491	72	-	-	PUNCT
ma-232	491	73	hill	hill	PROPN
ma-232	491	74	,	,	PUNCT
ma-232	491	75	inc	inc	PROPN
ma-232	491	76	.	.	PROPN
ma-232	491	77	new	new	PROPN
ma-232	491	78	york	york	PROPN
ma-232	491	79	(	(	PUNCT
ma-232	491	80	1991).[13	1991).[13	PROPN
ma-232	491	81	]	]	PUNCT
ma-232	491	82	k.	k.	PROPN
ma-232	491	83	zhu	zhu	PROPN
ma-232	491	84	,	,	PUNCT
ma-232	491	85	operator	operator	NOUN
ma-232	491	86	theory	theory	NOUN
ma-232	491	87	in	in	ADP
ma-232	491	88	function	function	NOUN
ma-232	491	89	spaces	space	NOUN
ma-232	491	90	,	,	PUNCT
ma-232	491	91	marcel	marcel	PROPN
ma-232	491	92	dekker	dekker	PROPN
ma-232	491	93	inc	inc	PROPN
ma-232	491	94	.	.	PROPN
ma-232	491	95	new	new	PROPN
ma-232	491	96	york	york	PROPN
ma-232	491	97	,	,	PUNCT
ma-232	491	98	basel	basel	PROPN
ma-232	491	99	(	(	PUNCT
ma-232	491	100	1990).[14	1990).[14	NUM
ma-232	491	101	]	]	PUNCT
ma-232	491	102	k.	k.	PROPN
ma-232	491	103	zhu	zhu	PROPN
ma-232	491	104	,	,	PUNCT
ma-232	491	105	bloch	bloch	PROPN
ma-232	491	106	type	type	NOUN
ma-232	491	107	spaces	space	NOUN
ma-232	491	108	of	of	ADP
ma-232	491	109	analytic	analytic	ADJ
ma-232	491	110	functions	function	NOUN
ma-232	491	111	,	,	PUNCT
ma-232	491	112	rocky	rocky	ADJ
ma-232	491	113	mountain	mountain	NOUN
ma-232	491	114	j.	j.	PROPN
ma-232	491	115	math	math	PROPN
ma-232	491	116	.	.	PUNCT
ma-232	492	1	23	23	NUM
ma-232	492	2	(	(	PUNCT
ma-232	492	3	1993	1993	NUM
ma-232	492	4	)	)	PUNCT
ma-232	492	5	,	,	PUNCT
ma-232	492	6	1143–1177	1143–1177	NUM
ma-232	492	7	.	.	PUNCT
ma-232	493	1	https://doi.org/10.28924/ada/ma.4.14	https://doi.org/10.28924/ada/ma.4.14	PROPN
ma-232	493	2	1	1	NUM
ma-232	493	3	.	.	PUNCT
ma-232	493	4	introduction	introduction	NOUN
ma-232	493	5	2	2	NUM
ma-232	493	6	.	.	PUNCT
ma-232	493	7	predual	predual	ADJ
ma-232	493	8	of	of	ADP
ma-232	493	9	non	non	ADJ
ma-232	493	10	-	-	ADJ
ma-232	493	11	reflexive	reflexive	ADJ
ma-232	493	12	bergman	bergman	PROPN
ma-232	493	13	space	space	NOUN
ma-232	493	14	of	of	ADP
ma-232	493	15	the	the	DET
ma-232	493	16	upper	upper	ADJ
ma-232	493	17	half	half	ADJ
ma-232	493	18	-	-	PUNCT
ma-232	493	19	plane	plane	NOUN
ma-232	493	20	l1a(u	l1a(u	NOUN
ma-232	493	21	,	,	PUNCT
ma-232	493	22	)	)	PUNCT
ma-232	493	23	3	3	X
ma-232	493	24	.	.	PUNCT
ma-232	493	25	groups	group	NOUN
ma-232	493	26	of	of	ADP
ma-232	493	27	weighted	weight	VERB
ma-232	493	28	composition	composition	NOUN
ma-232	493	29	operators	operator	NOUN
ma-232	493	30	on	on	ADP
ma-232	493	31	predual	predual	ADJ
ma-232	493	32	of	of	ADP
ma-232	493	33	l1a(u	l1a(u	PROPN
ma-232	493	34	,	,	PUNCT
ma-232	493	35	)	)	PUNCT
ma-232	493	36	3.1	3.1	NUM
ma-232	493	37	.	.	PUNCT
ma-232	493	38	scaling	scale	VERB
ma-232	493	39	group	group	NOUN
ma-232	493	40	3.2	3.2	NUM
ma-232	493	41	.	.	PUNCT
ma-232	494	1	translation	translation	NOUN
ma-232	494	2	group	group	NOUN
ma-232	494	3	acknowledgement	acknowledgement	NOUN
ma-232	494	4	references	reference	VERB