id	sid	tid	token	lemma	pos
ap-4612	1	1	acta	acta	PROPN
ap-4612	1	2	polytechnica	polytechnica	PROPN
ap-4612	1	3	doi:10.14311	doi:10.14311	PROPN
ap-4612	1	4	/	/	SYM
ap-4612	1	5	ap.2017.57.0379	ap.2017.57.0379	PROPN
ap-4612	1	6	acta	acta	PROPN
ap-4612	1	7	polytechnica	polytechnica	PROPN
ap-4612	1	8	57(6):379–384	57(6):379–384	PROPN
ap-4612	1	9	,	,	PUNCT
ap-4612	1	10	2017	2017	NUM
ap-4612	1	11	©	©	PROPN
ap-4612	1	12	czech	czech	PROPN
ap-4612	1	13	technical	technical	PROPN
ap-4612	1	14	university	university	PROPN
ap-4612	1	15	in	in	ADP
ap-4612	1	16	prague	prague	PROPN
ap-4612	1	17	,	,	PUNCT
ap-4612	1	18	2017	2017	NUM
ap-4612	1	19	available	available	ADJ
ap-4612	1	20	online	online	ADV
ap-4612	1	21	at	at	ADP
ap-4612	1	22	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-4612	1	23	lie	lie	NOUN
ap-4612	1	24	algebra	algebra	NOUN
ap-4612	1	25	representations	representation	NOUN
ap-4612	1	26	and	and	CCONJ
ap-4612	1	27	rigged	rig	VERB
ap-4612	1	28	hilbert	hilbert	NOUN
ap-4612	1	29	spaces	space	VERB
ap-4612	1	30	:	:	PUNCT
ap-4612	1	31	the	the	DET
ap-4612	1	32	so(2	so(2	NOUN
ap-4612	1	33	)	)	PUNCT
ap-4612	1	34	case	case	NOUN
ap-4612	1	35	enrico	enrico	PROPN
ap-4612	1	36	celeghinia	celeghinia	PROPN
ap-4612	1	37	,	,	PUNCT
ap-4612	1	38	b	b	PROPN
ap-4612	1	39	,	,	PUNCT
ap-4612	1	40	manuel	manuel	PROPN
ap-4612	1	41	gadellaa	gadellaa	PROPN
ap-4612	1	42	,	,	PUNCT
ap-4612	1	43	mariano	mariano	PROPN
ap-4612	1	44	a	a	DET
ap-4612	1	45	del	del	X
ap-4612	1	46	olmoa,∗	olmoa,∗	PROPN
ap-4612	1	47	a	a	DET
ap-4612	1	48	departamento	departamento	NOUN
ap-4612	1	49	de	de	PROPN
ap-4612	1	50	física	física	PROPN
ap-4612	1	51	teórica	teórica	PROPN
ap-4612	1	52	and	and	CCONJ
ap-4612	1	53	imuva	imuva	PROPN
ap-4612	1	54	,	,	PUNCT
ap-4612	1	55	universidad	universidad	PROPN
ap-4612	1	56	de	de	PROPN
ap-4612	1	57	valladolid	valladolid	PROPN
ap-4612	1	58	,	,	PUNCT
ap-4612	1	59	47011	47011	NUM
ap-4612	1	60	valladolid	valladolid	PROPN
ap-4612	1	61	,	,	PUNCT
ap-4612	1	62	spain	spain	PROPN
ap-4612	1	63	b	b	PROPN
ap-4612	1	64	dipartimento	dipartimento	PROPN
ap-4612	1	65	di	di	X
ap-4612	1	66	fisica	fisica	PROPN
ap-4612	1	67	,	,	PUNCT
ap-4612	1	68	università	università	PROPN
ap-4612	1	69	di	di	PROPN
ap-4612	1	70	firenze	firenze	PROPN
ap-4612	1	71	and	and	CCONJ
ap-4612	1	72	infn	infn	PROPN
ap-4612	1	73	-	-	PUNCT
ap-4612	1	74	sezione	sezione	NOUN
ap-4612	1	75	di	di	NOUN
ap-4612	1	76	firenze	firenze	NOUN
ap-4612	1	77	,	,	PUNCT
ap-4612	1	78	i50019	i50019	NUM
ap-4612	1	79	sesto	sesto	PROPN
ap-4612	1	80	fiorentino	fiorentino	PROPN
ap-4612	1	81	,	,	PUNCT
ap-4612	1	82	firenze	firenze	PROPN
ap-4612	1	83	,	,	PUNCT
ap-4612	1	84	italy	italy	PROPN
ap-4612	1	85	∗	∗	VERB
ap-4612	1	86	corresponding	correspond	VERB
ap-4612	1	87	author	author	NOUN
ap-4612	1	88	:	:	PUNCT
ap-4612	1	89	marianoantonio.olmo@uva.es	marianoantonio.olmo@uva.es	PROPN
ap-4612	1	90	abstract	abstract	NOUN
ap-4612	1	91	.	.	PUNCT
ap-4612	2	1	it	it	PRON
ap-4612	2	2	is	be	AUX
ap-4612	2	3	well	well	ADV
ap-4612	2	4	known	know	VERB
ap-4612	2	5	that	that	SCONJ
ap-4612	2	6	related	relate	VERB
ap-4612	2	7	with	with	ADP
ap-4612	2	8	the	the	DET
ap-4612	2	9	representations	representation	NOUN
ap-4612	2	10	of	of	ADP
ap-4612	2	11	the	the	DET
ap-4612	2	12	lie	lie	NOUN
ap-4612	2	13	group	group	NOUN
ap-4612	2	14	so(2	so(2	NOUN
ap-4612	2	15	)	)	PUNCT
ap-4612	2	16	we	we	PRON
ap-4612	2	17	find	find	VERB
ap-4612	2	18	a	a	DET
ap-4612	2	19	discrete	discrete	ADJ
ap-4612	2	20	basis	basis	NOUN
ap-4612	2	21	as	as	ADP
ap-4612	2	22	well	well	ADV
ap-4612	2	23	a	a	DET
ap-4612	2	24	continuous	continuous	ADJ
ap-4612	2	25	one	one	NUM
ap-4612	2	26	.	.	PUNCT
ap-4612	3	1	in	in	ADP
ap-4612	3	2	this	this	DET
ap-4612	3	3	paper	paper	NOUN
ap-4612	3	4	we	we	PRON
ap-4612	3	5	revisited	revisit	VERB
ap-4612	3	6	this	this	DET
ap-4612	3	7	situation	situation	NOUN
ap-4612	3	8	under	under	ADP
ap-4612	3	9	the	the	DET
ap-4612	3	10	light	light	NOUN
ap-4612	3	11	of	of	ADP
ap-4612	3	12	rigged	rig	VERB
ap-4612	3	13	hilbert	hilbert	NOUN
ap-4612	3	14	spaces	space	NOUN
ap-4612	3	15	,	,	PUNCT
ap-4612	3	16	which	which	PRON
ap-4612	3	17	are	be	AUX
ap-4612	3	18	the	the	DET
ap-4612	3	19	suitable	suitable	ADJ
ap-4612	3	20	framework	framework	NOUN
ap-4612	3	21	to	to	PART
ap-4612	3	22	deal	deal	VERB
ap-4612	3	23	with	with	ADP
ap-4612	3	24	both	both	CCONJ
ap-4612	3	25	discrete	discrete	ADJ
ap-4612	3	26	and	and	CCONJ
ap-4612	3	27	continuous	continuous	ADJ
ap-4612	3	28	bases	basis	NOUN
ap-4612	3	29	in	in	ADP
ap-4612	3	30	the	the	DET
ap-4612	3	31	same	same	ADJ
ap-4612	3	32	context	context	NOUN
ap-4612	3	33	and	and	CCONJ
ap-4612	3	34	in	in	ADP
ap-4612	3	35	relation	relation	NOUN
ap-4612	3	36	with	with	ADP
ap-4612	3	37	physical	physical	ADJ
ap-4612	3	38	applications	application	NOUN
ap-4612	3	39	.	.	PUNCT
ap-4612	4	1	keywords	keyword	NOUN
ap-4612	4	2	:	:	PUNCT
ap-4612	4	3	lie	lie	VERB
ap-4612	4	4	groups	group	NOUN
ap-4612	4	5	representations	representation	NOUN
ap-4612	4	6	;	;	PUNCT
ap-4612	4	7	special	special	ADJ
ap-4612	4	8	functions	function	NOUN
ap-4612	4	9	;	;	PUNCT
ap-4612	4	10	rigged	rig	VERB
ap-4612	4	11	hilbert	hilbert	NOUN
ap-4612	4	12	spaces	space	NOUN
ap-4612	4	13	.	.	PUNCT
ap-4612	5	1	1	1	X
ap-4612	5	2	.	.	X
ap-4612	5	3	introduction	introduction	NOUN
ap-4612	5	4	in	in	ADP
ap-4612	5	5	the	the	DET
ap-4612	5	6	last	last	ADJ
ap-4612	5	7	years	year	NOUN
ap-4612	5	8	we	we	PRON
ap-4612	5	9	have	have	AUX
ap-4612	5	10	been	be	AUX
ap-4612	5	11	involved	involve	VERB
ap-4612	5	12	in	in	ADP
ap-4612	5	13	a	a	DET
ap-4612	5	14	program	program	NOUN
ap-4612	5	15	of	of	ADP
ap-4612	5	16	revision	revision	NOUN
ap-4612	5	17	of	of	ADP
ap-4612	5	18	the	the	DET
ap-4612	5	19	connection	connection	NOUN
ap-4612	5	20	between	between	ADP
ap-4612	5	21	special	special	ADJ
ap-4612	5	22	functions	function	NOUN
ap-4612	5	23	(	(	PUNCT
ap-4612	5	24	in	in	ADP
ap-4612	5	25	particular	particular	ADJ
ap-4612	5	26	,	,	PUNCT
ap-4612	5	27	classical	classical	ADJ
ap-4612	5	28	orthogonal	orthogonal	ADJ
ap-4612	5	29	polynomials	polynomial	NOUN
ap-4612	5	30	)	)	PUNCT
ap-4612	5	31	,	,	PUNCT
ap-4612	5	32	lie	lie	NOUN
ap-4612	5	33	groups	group	NOUN
ap-4612	5	34	,	,	PUNCT
ap-4612	5	35	differential	differential	ADJ
ap-4612	5	36	equations	equation	NOUN
ap-4612	5	37	and	and	CCONJ
ap-4612	5	38	physical	physical	ADJ
ap-4612	5	39	spaces	space	NOUN
ap-4612	5	40	.	.	PUNCT
ap-4612	6	1	we	we	PRON
ap-4612	6	2	have	have	AUX
ap-4612	6	3	obtained	obtain	VERB
ap-4612	6	4	the	the	DET
ap-4612	6	5	ladder	ladder	NOUN
ap-4612	6	6	algebraic	algebraic	ADJ
ap-4612	6	7	structure	structure	NOUN
ap-4612	6	8	for	for	ADP
ap-4612	6	9	different	different	ADJ
ap-4612	6	10	orthogonal	orthogonal	ADJ
ap-4612	6	11	polynomials	polynomial	NOUN
ap-4612	6	12	,	,	PUNCT
ap-4612	6	13	like	like	ADP
ap-4612	6	14	hermite	hermite	PROPN
ap-4612	6	15	,	,	PUNCT
ap-4612	6	16	legendre	legendre	PROPN
ap-4612	6	17	,	,	PUNCT
ap-4612	6	18	laguerre	laguerre	NOUN
ap-4612	7	1	[	[	X
ap-4612	7	2	1	1	NUM
ap-4612	7	3	]	]	PUNCT
ap-4612	7	4	,	,	PUNCT
ap-4612	7	5	associated	associate	VERB
ap-4612	7	6	laguerre	laguerre	NOUN
ap-4612	7	7	polynomials	polynomial	NOUN
ap-4612	7	8	,	,	PUNCT
ap-4612	7	9	spherical	spherical	ADJ
ap-4612	7	10	harmonics	harmonic	NOUN
ap-4612	7	11	,	,	PUNCT
ap-4612	7	12	etc	etc	X
ap-4612	7	13	.	.	X
ap-4612	8	1	[	[	X
ap-4612	8	2	2	2	NUM
ap-4612	8	3	,	,	PUNCT
ap-4612	8	4	3	3	NUM
ap-4612	8	5	]	]	PUNCT
ap-4612	8	6	.	.	PUNCT
ap-4612	9	1	in	in	ADP
ap-4612	9	2	all	all	DET
ap-4612	9	3	cases	case	NOUN
ap-4612	9	4	,	,	PUNCT
ap-4612	9	5	we	we	PRON
ap-4612	9	6	have	have	AUX
ap-4612	9	7	obtained	obtain	VERB
ap-4612	9	8	a	a	DET
ap-4612	9	9	symmetry	symmetry	NOUN
ap-4612	9	10	group	group	NOUN
ap-4612	9	11	.	.	PUNCT
ap-4612	10	1	the	the	DET
ap-4612	10	2	corresponding	corresponding	ADJ
ap-4612	10	3	orthogonal	orthogonal	ADJ
ap-4612	10	4	polynomial	polynomial	NOUN
ap-4612	10	5	is	be	AUX
ap-4612	10	6	associated	associate	VERB
ap-4612	10	7	to	to	ADP
ap-4612	10	8	a	a	DET
ap-4612	10	9	particular	particular	ADJ
ap-4612	10	10	representation	representation	NOUN
ap-4612	10	11	of	of	ADP
ap-4612	10	12	its	its	PRON
ap-4612	10	13	lie	lie	NOUN
ap-4612	10	14	group	group	NOUN
ap-4612	10	15	.	.	PUNCT
ap-4612	11	1	for	for	ADP
ap-4612	11	2	instance	instance	NOUN
ap-4612	11	3	,	,	PUNCT
ap-4612	11	4	for	for	ADP
ap-4612	11	5	the	the	DET
ap-4612	11	6	associated	associated	ADJ
ap-4612	11	7	laguerre	laguerre	NOUN
ap-4612	11	8	polynomials	polynomial	NOUN
ap-4612	11	9	and	and	CCONJ
ap-4612	11	10	the	the	DET
ap-4612	11	11	spherical	spherical	ADJ
ap-4612	11	12	harmonics	harmonic	NOUN
ap-4612	11	13	,	,	PUNCT
ap-4612	11	14	we	we	PRON
ap-4612	11	15	obtain	obtain	VERB
ap-4612	11	16	the	the	DET
ap-4612	11	17	symmetry	symmetry	NOUN
ap-4612	11	18	group	group	NOUN
ap-4612	11	19	so(3	so(3	NOUN
ap-4612	11	20	,	,	PUNCT
ap-4612	11	21	2	2	NUM
ap-4612	11	22	)	)	PUNCT
ap-4612	11	23	and	and	CCONJ
ap-4612	11	24	in	in	ADP
ap-4612	11	25	both	both	DET
ap-4612	11	26	cases	case	NOUN
ap-4612	11	27	they	they	PRON
ap-4612	11	28	support	support	VERB
ap-4612	11	29	a	a	DET
ap-4612	11	30	unitary	unitary	ADJ
ap-4612	11	31	irreducible	irreducible	ADJ
ap-4612	11	32	representation	representation	NOUN
ap-4612	11	33	(	(	PUNCT
ap-4612	11	34	uir	uir	PROPN
ap-4612	11	35	)	)	PUNCT
ap-4612	11	36	with	with	ADP
ap-4612	11	37	quadratic	quadratic	ADJ
ap-4612	11	38	casimir	casimir	NOUN
ap-4612	11	39	−5/4	−5/4	NOUN
ap-4612	11	40	.	.	PUNCT
ap-4612	12	1	both	both	PRON
ap-4612	12	2	are	be	AUX
ap-4612	12	3	bases	basis	NOUN
ap-4612	12	4	of	of	ADP
ap-4612	12	5	square	square	ADJ
ap-4612	12	6	integrable	integrable	ADJ
ap-4612	12	7	functions	function	NOUN
ap-4612	12	8	defined	define	VERB
ap-4612	12	9	on	on	ADP
ap-4612	12	10	(	(	PUNCT
ap-4612	12	11	−1	−1	NOUN
ap-4612	12	12	,	,	PUNCT
ap-4612	12	13	1)×	1)×	NUM
ap-4612	12	14	z	z	NOUN
ap-4612	12	15	and	and	CCONJ
ap-4612	12	16	on	on	ADP
ap-4612	12	17	the	the	DET
ap-4612	12	18	sphere	sphere	NOUN
ap-4612	12	19	s2	s2	PROPN
ap-4612	12	20	,	,	PUNCT
ap-4612	12	21	respectively	respectively	ADV
ap-4612	12	22	.	.	PUNCT
ap-4612	13	1	in	in	ADP
ap-4612	13	2	any	any	DET
ap-4612	13	3	case	case	NOUN
ap-4612	13	4	we	we	PRON
ap-4612	13	5	get	get	VERB
ap-4612	13	6	discrete	discrete	ADJ
ap-4612	13	7	and	and	CCONJ
ap-4612	13	8	continuous	continuous	ADJ
ap-4612	13	9	bases	basis	NOUN
ap-4612	13	10	.	.	PUNCT
ap-4612	14	1	on	on	ADP
ap-4612	14	2	the	the	DET
ap-4612	14	3	other	other	ADJ
ap-4612	14	4	hand	hand	NOUN
ap-4612	14	5	,	,	PUNCT
ap-4612	14	6	the	the	DET
ap-4612	14	7	rigged	rig	VERB
ap-4612	14	8	hilbert	hilbert	NOUN
ap-4612	14	9	space	space	NOUN
ap-4612	14	10	(	(	PUNCT
ap-4612	14	11	rhs	rhs	PROPN
ap-4612	14	12	)	)	PUNCT
ap-4612	14	13	is	be	AUX
ap-4612	14	14	a	a	DET
ap-4612	14	15	suitable	suitable	ADJ
ap-4612	14	16	framework	framework	NOUN
ap-4612	14	17	for	for	ADP
ap-4612	14	18	a	a	DET
ap-4612	14	19	description	description	NOUN
ap-4612	14	20	of	of	ADP
ap-4612	14	21	quantum	quantum	ADJ
ap-4612	14	22	states	state	NOUN
ap-4612	14	23	,	,	PUNCT
ap-4612	14	24	when	when	SCONJ
ap-4612	14	25	the	the	DET
ap-4612	14	26	use	use	NOUN
ap-4612	14	27	of	of	ADP
ap-4612	14	28	both	both	CCONJ
ap-4612	14	29	discrete	discrete	ADJ
ap-4612	14	30	bases	basis	NOUN
ap-4612	14	31	,	,	PUNCT
ap-4612	14	32	i.e.	i.e.	X
ap-4612	14	33	,	,	PUNCT
ap-4612	14	34	complete	complete	ADJ
ap-4612	14	35	orthonormal	orthonormal	ADJ
ap-4612	14	36	sets	set	NOUN
ap-4612	14	37	,	,	PUNCT
ap-4612	14	38	and	and	CCONJ
ap-4612	14	39	generalized	generalize	VERB
ap-4612	14	40	continuous	continuous	ADJ
ap-4612	14	41	bases	basis	NOUN
ap-4612	14	42	like	like	ADP
ap-4612	14	43	those	those	PRON
ap-4612	14	44	used	use	VERB
ap-4612	14	45	in	in	ADP
ap-4612	14	46	the	the	DET
ap-4612	14	47	dirac	dirac	NOUN
ap-4612	14	48	formalism	formalism	NOUN
ap-4612	14	49	are	be	AUX
ap-4612	14	50	necessary	necessary	ADJ
ap-4612	14	51	[	[	X
ap-4612	14	52	4]–[15	4]–[15	X
ap-4612	14	53	]	]	X
ap-4612	14	54	.	.	PUNCT
ap-4612	15	1	as	as	SCONJ
ap-4612	15	2	mentioned	mention	VERB
ap-4612	15	3	above	above	ADV
ap-4612	15	4	,	,	PUNCT
ap-4612	15	5	this	this	PRON
ap-4612	15	6	is	be	AUX
ap-4612	15	7	a	a	DET
ap-4612	15	8	typical	typical	ADJ
ap-4612	15	9	situation	situation	NOUN
ap-4612	15	10	arisen	arise	VERB
ap-4612	15	11	when	when	SCONJ
ap-4612	15	12	we	we	PRON
ap-4612	15	13	deal	deal	VERB
ap-4612	15	14	with	with	ADP
ap-4612	15	15	special	special	ADJ
ap-4612	15	16	functions	function	NOUN
ap-4612	15	17	,	,	PUNCT
ap-4612	15	18	which	which	PRON
ap-4612	15	19	hold	hold	VERB
ap-4612	15	20	discrete	discrete	ADJ
ap-4612	15	21	labels	label	NOUN
ap-4612	15	22	and	and	CCONJ
ap-4612	15	23	depend	depend	VERB
ap-4612	15	24	on	on	ADP
ap-4612	15	25	continuous	continuous	ADJ
ap-4612	15	26	variables	variable	NOUN
ap-4612	15	27	.	.	PUNCT
ap-4612	16	1	we	we	PRON
ap-4612	16	2	have	have	AUX
ap-4612	16	3	analysed	analyse	VERB
ap-4612	16	4	this	this	DET
ap-4612	16	5	situation	situation	NOUN
ap-4612	16	6	for	for	ADP
ap-4612	16	7	the	the	DET
ap-4612	16	8	hermite	hermite	ADJ
ap-4612	16	9	and	and	CCONJ
ap-4612	16	10	laguerre	laguerre	NOUN
ap-4612	16	11	functions	function	NOUN
ap-4612	16	12	motivated	motivate	VERB
ap-4612	16	13	also	also	ADV
ap-4612	16	14	by	by	ADP
ap-4612	16	15	possible	possible	ADJ
ap-4612	16	16	applications	application	NOUN
ap-4612	16	17	on	on	ADP
ap-4612	16	18	signal	signal	ADJ
ap-4612	16	19	theory	theory	NOUN
ap-4612	16	20	in	in	ADP
ap-4612	16	21	recent	recent	ADJ
ap-4612	16	22	papers	paper	NOUN
ap-4612	16	23	[	[	X
ap-4612	16	24	11	11	NUM
ap-4612	16	25	,	,	PUNCT
ap-4612	16	26	12	12	NUM
ap-4612	16	27	]	]	PUNCT
ap-4612	16	28	.	.	PUNCT
ap-4612	17	1	moreover	moreover	ADV
ap-4612	17	2	,	,	PUNCT
ap-4612	17	3	the	the	DET
ap-4612	17	4	rhs	rhs	PROPN
ap-4612	17	5	fit	fit	VERB
ap-4612	17	6	very	very	ADV
ap-4612	17	7	well	well	ADV
ap-4612	17	8	with	with	ADP
ap-4612	17	9	lie	lie	NOUN
ap-4612	17	10	groups	group	NOUN
ap-4612	17	11	[	[	X
ap-4612	17	12	16	16	NUM
ap-4612	17	13	]	]	PUNCT
ap-4612	17	14	and	and	CCONJ
ap-4612	17	15	also	also	ADV
ap-4612	17	16	with	with	ADP
ap-4612	17	17	semigroups	semigroup	NOUN
ap-4612	17	18	,	,	PUNCT
ap-4612	17	19	see	see	VERB
ap-4612	17	20	[	[	X
ap-4612	17	21	17	17	NUM
ap-4612	17	22	]	]	PUNCT
ap-4612	17	23	and	and	CCONJ
ap-4612	17	24	references	reference	NOUN
ap-4612	17	25	therein	therein	ADV
ap-4612	17	26	.	.	PUNCT
ap-4612	18	1	in	in	ADP
ap-4612	18	2	this	this	DET
ap-4612	18	3	paper	paper	NOUN
ap-4612	18	4	,	,	PUNCT
ap-4612	18	5	we	we	PRON
ap-4612	18	6	continue	continue	VERB
ap-4612	18	7	the	the	DET
ap-4612	18	8	study	study	NOUN
ap-4612	18	9	of	of	ADP
ap-4612	18	10	the	the	DET
ap-4612	18	11	relation	relation	NOUN
ap-4612	18	12	between	between	ADP
ap-4612	18	13	lie	lie	NOUN
ap-4612	18	14	algebras	algebra	NOUN
ap-4612	18	15	,	,	PUNCT
ap-4612	18	16	special	special	ADJ
ap-4612	18	17	functions	function	NOUN
ap-4612	18	18	and	and	CCONJ
ap-4612	18	19	rhs	rhs	PROPN
ap-4612	18	20	.	.	PUNCT
ap-4612	19	1	here	here	ADV
ap-4612	19	2	we	we	PRON
ap-4612	19	3	propose	propose	VERB
ap-4612	19	4	a	a	DET
ap-4612	19	5	revision	revision	NOUN
ap-4612	19	6	of	of	ADP
ap-4612	19	7	the	the	DET
ap-4612	19	8	elementary	elementary	ADJ
ap-4612	19	9	case	case	NOUN
ap-4612	19	10	associated	associate	VERB
ap-4612	19	11	to	to	ADP
ap-4612	19	12	the	the	DET
ap-4612	19	13	lie	lie	NOUN
ap-4612	19	14	group	group	NOUN
ap-4612	19	15	so(2	so(2	NOUN
ap-4612	19	16	)	)	PUNCT
ap-4612	19	17	.	.	PUNCT
ap-4612	20	1	since	since	SCONJ
ap-4612	20	2	the	the	DET
ap-4612	20	3	representations	representation	NOUN
ap-4612	20	4	of	of	ADP
ap-4612	20	5	so(2	so(2	NOUN
ap-4612	20	6	)	)	PUNCT
ap-4612	20	7	admit	admit	VERB
ap-4612	20	8	continuous	continuous	ADJ
ap-4612	20	9	and	and	CCONJ
ap-4612	20	10	discrete	discrete	ADJ
ap-4612	20	11	bases	basis	NOUN
ap-4612	20	12	[	[	X
ap-4612	20	13	13	13	NUM
ap-4612	20	14	]	]	PUNCT
ap-4612	20	15	it	it	PRON
ap-4612	20	16	is	be	AUX
ap-4612	20	17	necessary	necessary	ADJ
ap-4612	20	18	a	a	DET
ap-4612	20	19	rhs	rhs	PROPN
ap-4612	20	20	for	for	ADP
ap-4612	20	21	a	a	DET
ap-4612	20	22	mathematical	mathematical	ADJ
ap-4612	20	23	rigorous	rigorous	ADJ
ap-4612	20	24	description	description	NOUN
ap-4612	20	25	.	.	PUNCT
ap-4612	21	1	the	the	DET
ap-4612	21	2	relation	relation	NOUN
ap-4612	21	3	with	with	ADP
ap-4612	21	4	the	the	DET
ap-4612	21	5	fourier	fourier	ADJ
ap-4612	21	6	series	series	NOUN
ap-4612	21	7	and	and	CCONJ
ap-4612	21	8	its	its	PRON
ap-4612	21	9	interest	interest	NOUN
ap-4612	21	10	in	in	ADP
ap-4612	21	11	quantum	quantum	ADJ
ap-4612	21	12	physics	physics	NOUN
ap-4612	21	13	make	make	VERB
ap-4612	21	14	that	that	SCONJ
ap-4612	21	15	this	this	DET
ap-4612	21	16	case	case	NOUN
ap-4612	21	17	provides	provide	VERB
ap-4612	21	18	a	a	DET
ap-4612	21	19	relevant	relevant	ADJ
ap-4612	21	20	example	example	NOUN
ap-4612	21	21	of	of	ADP
ap-4612	21	22	these	these	DET
ap-4612	21	23	mathematical	mathematical	ADJ
ap-4612	21	24	objects	object	NOUN
ap-4612	21	25	.	.	PUNCT
ap-4612	22	1	2	2	X
ap-4612	22	2	.	.	NUM
ap-4612	22	3	rigged	rig	VERB
ap-4612	22	4	hilbert	hilbert	NOUN
ap-4612	22	5	spaces	space	NOUN
ap-4612	22	6	there	there	PRON
ap-4612	22	7	are	be	VERB
ap-4612	22	8	several	several	ADJ
ap-4612	22	9	reasons	reason	NOUN
ap-4612	22	10	to	to	PART
ap-4612	22	11	assert	assert	VERB
ap-4612	22	12	that	that	SCONJ
ap-4612	22	13	hilbert	hilbert	NOUN
ap-4612	22	14	spaces	space	NOUN
ap-4612	22	15	are	be	AUX
ap-4612	22	16	not	not	PART
ap-4612	22	17	sufficient	sufficient	ADJ
ap-4612	22	18	for	for	ADP
ap-4612	22	19	a	a	DET
ap-4612	22	20	thoroughly	thoroughly	ADV
ap-4612	22	21	formulation	formulation	NOUN
ap-4612	22	22	of	of	ADP
ap-4612	22	23	quantum	quantum	ADJ
ap-4612	22	24	mechanics	mechanic	NOUN
ap-4612	22	25	even	even	ADV
ap-4612	22	26	within	within	ADP
ap-4612	22	27	the	the	DET
ap-4612	22	28	non	non	ADJ
ap-4612	22	29	-	-	ADJ
ap-4612	22	30	relativistic	relativistic	ADJ
ap-4612	22	31	context	context	NOUN
ap-4612	22	32	.	.	PUNCT
ap-4612	23	1	we	we	PRON
ap-4612	23	2	can	can	AUX
ap-4612	23	3	mention	mention	VERB
ap-4612	23	4	,	,	PUNCT
ap-4612	23	5	for	for	ADP
ap-4612	23	6	instance	instance	NOUN
ap-4612	23	7	,	,	PUNCT
ap-4612	23	8	the	the	DET
ap-4612	23	9	dirac	dirac	NOUN
ap-4612	23	10	formulation	formulation	NOUN
ap-4612	23	11	[	[	X
ap-4612	23	12	18	18	NUM
ap-4612	23	13	]	]	PUNCT
ap-4612	23	14	where	where	SCONJ
ap-4612	23	15	operators	operator	NOUN
ap-4612	23	16	with	with	ADP
ap-4612	23	17	continuous	continuous	ADJ
ap-4612	23	18	spectrum	spectrum	NOUN
ap-4612	23	19	play	play	VERB
ap-4612	23	20	a	a	DET
ap-4612	23	21	crucial	crucial	ADJ
ap-4612	23	22	role	role	NOUN
ap-4612	23	23	(	(	PUNCT
ap-4612	23	24	see	see	VERB
ap-4612	23	25	also	also	ADV
ap-4612	23	26	[	[	X
ap-4612	23	27	10	10	NUM
ap-4612	23	28	]	]	PUNCT
ap-4612	23	29	and	and	CCONJ
ap-4612	23	30	references	reference	NOUN
ap-4612	23	31	therein	therein	ADV
ap-4612	23	32	)	)	PUNCT
ap-4612	23	33	and	and	CCONJ
ap-4612	23	34	their	their	PRON
ap-4612	23	35	eigenvectors	eigenvector	NOUN
ap-4612	23	36	are	be	AUX
ap-4612	23	37	not	not	PART
ap-4612	23	38	in	in	ADP
ap-4612	23	39	the	the	DET
ap-4612	23	40	hilbert	hilbert	NOUN
ap-4612	23	41	space	space	NOUN
ap-4612	23	42	of	of	ADP
ap-4612	23	43	square	square	ADJ
ap-4612	23	44	integrable	integrable	ADJ
ap-4612	23	45	wave	wave	NOUN
ap-4612	23	46	functions	function	NOUN
ap-4612	23	47	.	.	PUNCT
ap-4612	24	1	another	another	DET
ap-4612	24	2	example	example	NOUN
ap-4612	24	3	is	be	AUX
ap-4612	24	4	related	relate	VERB
ap-4612	24	5	with	with	ADP
ap-4612	24	6	the	the	DET
ap-4612	24	7	proper	proper	ADJ
ap-4612	24	8	definition	definition	NOUN
ap-4612	24	9	of	of	ADP
ap-4612	24	10	gamow	gamow	NOUN
ap-4612	24	11	vectors	vector	NOUN
ap-4612	24	12	[	[	X
ap-4612	24	13	9	9	NUM
ap-4612	24	14	]	]	PUNCT
ap-4612	24	15	,	,	PUNCT
ap-4612	24	16	which	which	PRON
ap-4612	24	17	are	be	AUX
ap-4612	24	18	widely	widely	ADV
ap-4612	24	19	used	use	VERB
ap-4612	24	20	in	in	ADP
ap-4612	24	21	calculations	calculation	NOUN
ap-4612	24	22	including	include	VERB
ap-4612	24	23	unstable	unstable	ADJ
ap-4612	24	24	quantum	quantum	NOUN
ap-4612	24	25	systems	system	NOUN
ap-4612	24	26	and	and	CCONJ
ap-4612	24	27	are	be	AUX
ap-4612	24	28	non	non	ADJ
ap-4612	24	29	-	-	ADJ
ap-4612	24	30	normalizable	normalizable	ADJ
ap-4612	24	31	.	.	PUNCT
ap-4612	25	1	we	we	PRON
ap-4612	25	2	can	can	AUX
ap-4612	25	3	also	also	ADV
ap-4612	25	4	refer	refer	VERB
ap-4612	25	5	to	to	ADP
ap-4612	25	6	formulations	formulation	NOUN
ap-4612	25	7	of	of	ADP
ap-4612	25	8	time	time	NOUN
ap-4612	25	9	asymmetry	asymmetry	NOUN
ap-4612	25	10	in	in	ADP
ap-4612	25	11	quantum	quantum	ADJ
ap-4612	25	12	mechanics	mechanic	NOUN
ap-4612	25	13	that	that	PRON
ap-4612	25	14	may	may	AUX
ap-4612	25	15	require	require	VERB
ap-4612	25	16	the	the	DET
ap-4612	25	17	use	use	NOUN
ap-4612	25	18	of	of	ADP
ap-4612	25	19	tools	tool	NOUN
ap-4612	25	20	more	more	ADV
ap-4612	25	21	general	general	ADJ
ap-4612	25	22	than	than	ADP
ap-4612	25	23	hilbert	hilbert	NOUN
ap-4612	25	24	spaces	space	NOUN
ap-4612	25	25	[	[	X
ap-4612	25	26	19	19	NUM
ap-4612	25	27	]	]	PUNCT
ap-4612	25	28	.	.	PUNCT
ap-4612	26	1	the	the	DET
ap-4612	26	2	proper	proper	ADJ
ap-4612	26	3	framework	framework	NOUN
ap-4612	26	4	that	that	PRON
ap-4612	26	5	includes	include	VERB
ap-4612	26	6	naturally	naturally	ADV
ap-4612	26	7	the	the	DET
ap-4612	26	8	hilbert	hilbert	NOUN
ap-4612	26	9	space	space	NOUN
ap-4612	26	10	and	and	CCONJ
ap-4612	26	11	its	its	PRON
ap-4612	26	12	features	feature	NOUN
ap-4612	26	13	,	,	PUNCT
ap-4612	26	14	which	which	PRON
ap-4612	26	15	are	be	AUX
ap-4612	26	16	widely	widely	ADV
ap-4612	26	17	used	use	VERB
ap-4612	26	18	in	in	ADP
ap-4612	26	19	quantum	quantum	ADJ
ap-4612	26	20	mechanics	mechanic	NOUN
ap-4612	26	21	,	,	PUNCT
ap-4612	26	22	is	be	AUX
ap-4612	26	23	the	the	DET
ap-4612	26	24	rhs	rhs	PROPN
ap-4612	26	25	.	.	PUNCT
ap-4612	27	1	the	the	DET
ap-4612	27	2	rigged	rig	VERB
ap-4612	27	3	hilbert	hilbert	NOUN
ap-4612	27	4	spaces	space	NOUN
ap-4612	27	5	were	be	AUX
ap-4612	27	6	introduced	introduce	VERB
ap-4612	27	7	by	by	ADP
ap-4612	27	8	gelfand	gelfand	NOUN
ap-4612	27	9	and	and	CCONJ
ap-4612	27	10	collaborators	collaborator	NOUN
ap-4612	27	11	[	[	X
ap-4612	27	12	4	4	X
ap-4612	27	13	]	]	PUNCT
ap-4612	27	14	in	in	ADP
ap-4612	27	15	connection	connection	NOUN
ap-4612	27	16	with	with	ADP
ap-4612	27	17	the	the	DET
ap-4612	27	18	spectral	spectral	ADJ
ap-4612	27	19	theory	theory	NOUN
ap-4612	27	20	of	of	ADP
ap-4612	27	21	self	self	NOUN
ap-4612	27	22	-	-	PUNCT
ap-4612	27	23	adjoint	adjoint	NOUN
ap-4612	27	24	operators	operator	NOUN
ap-4612	27	25	.	.	PUNCT
ap-4612	28	1	they	they	PRON
ap-4612	28	2	also	also	ADV
ap-4612	28	3	proved	prove	VERB
ap-4612	28	4	,	,	PUNCT
ap-4612	28	5	together	together	ADV
ap-4612	28	6	with	with	ADP
ap-4612	28	7	maurin	maurin	NOUN
ap-4612	28	8	[	[	X
ap-4612	28	9	14	14	NUM
ap-4612	28	10	]	]	PUNCT
ap-4612	28	11	,	,	PUNCT
ap-4612	28	12	the	the	DET
ap-4612	28	13	nuclear	nuclear	ADJ
ap-4612	28	14	spectral	spectral	ADJ
ap-4612	28	15	theorem	theorem	NOUN
ap-4612	28	16	[	[	X
ap-4612	28	17	10	10	NUM
ap-4612	28	18	,	,	PUNCT
ap-4612	28	19	15	15	NUM
ap-4612	28	20	]	]	PUNCT
ap-4612	28	21	.	.	PUNCT
ap-4612	29	1	the	the	DET
ap-4612	29	2	rhs	rhs	PROPN
ap-4612	29	3	formulation	formulation	NOUN
ap-4612	29	4	of	of	ADP
ap-4612	29	5	quantum	quantum	ADJ
ap-4612	29	6	mechanics	mechanic	NOUN
ap-4612	29	7	was	be	AUX
ap-4612	29	8	introduced	introduce	VERB
ap-4612	29	9	by	by	ADP
ap-4612	29	10	bohm	bohm	PROPN
ap-4612	29	11	and	and	CCONJ
ap-4612	29	12	roberts	roberts	PROPN
ap-4612	29	13	around	around	ADP
ap-4612	29	14	1965	1965	NUM
ap-4612	29	15	[	[	X
ap-4612	29	16	5	5	NUM
ap-4612	29	17	,	,	PUNCT
ap-4612	29	18	8	8	NUM
ap-4612	29	19	]	]	PUNCT
ap-4612	29	20	.	.	PUNCT
ap-4612	30	1	a	a	DET
ap-4612	30	2	rigged	rig	VERB
ap-4612	30	3	hilbert	hilbert	NOUN
ap-4612	30	4	space	space	NOUN
ap-4612	30	5	(	(	PUNCT
ap-4612	30	6	also	also	ADV
ap-4612	30	7	called	call	VERB
ap-4612	30	8	gelfand	gelfand	PROPN
ap-4612	30	9	triplet	triplet	NOUN
ap-4612	30	10	)	)	PUNCT
ap-4612	30	11	is	be	AUX
ap-4612	30	12	a	a	DET
ap-4612	30	13	triplet	triplet	NOUN
ap-4612	30	14	of	of	ADP
ap-4612	30	15	spaces	space	NOUN
ap-4612	31	1	φ	φ	PROPN
ap-4612	31	2	⊂	⊂	PROPN
ap-4612	31	3	h	h	PROPN
ap-4612	31	4	⊂	⊂	PROPN
ap-4612	31	5	φ×	φ×	PROPN
ap-4612	31	6	,	,	PUNCT
ap-4612	31	7	withh	withh	VERB
ap-4612	31	8	an	an	DET
ap-4612	31	9	infinite	infinite	ADJ
ap-4612	31	10	dimensional	dimensional	ADJ
ap-4612	31	11	separable	separable	ADJ
ap-4612	31	12	hilbert	hilbert	NOUN
ap-4612	31	13	space	space	NOUN
ap-4612	31	14	,	,	PUNCT
ap-4612	31	15	φ	φ	PROPN
ap-4612	31	16	(	(	PUNCT
ap-4612	31	17	test	test	NOUN
ap-4612	31	18	vectors	vector	NOUN
ap-4612	31	19	space	space	NOUN
ap-4612	31	20	)	)	PUNCT
ap-4612	31	21	a	a	DET
ap-4612	31	22	dense	dense	ADJ
ap-4612	31	23	subspace	subspace	NOUN
ap-4612	31	24	of	of	ADP
ap-4612	31	25	h	h	NOUN
ap-4612	31	26	endowed	endow	VERB
ap-4612	31	27	with	with	ADP
ap-4612	31	28	its	its	PRON
ap-4612	31	29	own	own	ADJ
ap-4612	31	30	topology	topology	NOUN
ap-4612	31	31	,	,	PUNCT
ap-4612	31	32	and	and	CCONJ
ap-4612	31	33	φ×	φ×	NOUN
ap-4612	31	34	the	the	DET
ap-4612	31	35	dual	dual	ADJ
ap-4612	31	36	/antidual	/antidual	ADJ
ap-4612	31	37	space	space	NOUN
ap-4612	31	38	of	of	ADP
ap-4612	31	39	φ	φ	PROPN
ap-4612	31	40	.	.	PROPN
ap-4612	31	41	379	379	NUM
ap-4612	31	42	http://dx.doi.org/10.14311/ap.2017.57.0379	http://dx.doi.org/10.14311/ap.2017.57.0379	NOUN
ap-4612	31	43	http://ojs.cvut.cz/ojs/index.php/ap	http://ojs.cvut.cz/ojs/index.php/ap	PROPN
ap-4612	31	44	enrico	enrico	PROPN
ap-4612	31	45	celeghini	celeghini	PROPN
ap-4612	31	46	,	,	PUNCT
ap-4612	31	47	manuel	manuel	PROPN
ap-4612	31	48	gadella	gadella	PROPN
ap-4612	31	49	,	,	PUNCT
ap-4612	31	50	mariano	mariano	PROPN
ap-4612	31	51	a	a	DET
ap-4612	31	52	del	del	PROPN
ap-4612	31	53	olmo	olmo	PROPN
ap-4612	31	54	acta	acta	PROPN
ap-4612	31	55	polytechnica	polytechnica	PROPN
ap-4612	31	56	the	the	DET
ap-4612	31	57	topology	topology	NOUN
ap-4612	31	58	considered	consider	VERB
ap-4612	31	59	on	on	ADP
ap-4612	31	60	φ	φ	PROPN
ap-4612	31	61	is	be	AUX
ap-4612	31	62	finer	fine	ADJ
ap-4612	31	63	(	(	PUNCT
ap-4612	31	64	contains	contain	VERB
ap-4612	31	65	more	more	ADV
ap-4612	31	66	open	open	ADJ
ap-4612	31	67	sets	set	NOUN
ap-4612	31	68	)	)	PUNCT
ap-4612	31	69	than	than	ADP
ap-4612	31	70	the	the	DET
ap-4612	31	71	topology	topology	NOUN
ap-4612	31	72	that	that	PRON
ap-4612	31	73	φ	φ	PROPN
ap-4612	31	74	has	have	VERB
ap-4612	31	75	as	as	ADP
ap-4612	31	76	subspace	subspace	NOUN
ap-4612	31	77	of	of	ADP
ap-4612	31	78	h	h	NOUN
ap-4612	31	79	,	,	PUNCT
ap-4612	31	80	and	and	CCONJ
ap-4612	31	81	φ×	φ×	NOUN
ap-4612	31	82	is	be	AUX
ap-4612	31	83	equipped	equip	VERB
ap-4612	31	84	with	with	ADP
ap-4612	31	85	a	a	DET
ap-4612	31	86	topology	topology	NOUN
ap-4612	31	87	compatible	compatible	ADJ
ap-4612	31	88	with	with	ADP
ap-4612	31	89	the	the	DET
ap-4612	31	90	dual	dual	ADJ
ap-4612	31	91	pair	pair	NOUN
ap-4612	31	92	(	(	PUNCT
ap-4612	31	93	φ	φ	NOUN
ap-4612	31	94	,	,	PUNCT
ap-4612	31	95	φ×	φ×	NOUN
ap-4612	31	96	)	)	PUNCT
ap-4612	32	1	[	[	X
ap-4612	32	2	20	20	NUM
ap-4612	32	3	]	]	PUNCT
ap-4612	32	4	,	,	PUNCT
ap-4612	32	5	usually	usually	ADV
ap-4612	32	6	the	the	DET
ap-4612	32	7	weak	weak	ADJ
ap-4612	32	8	topology	topology	NOUN
ap-4612	32	9	.	.	PUNCT
ap-4612	33	1	one	one	NUM
ap-4612	33	2	consequence	consequence	NOUN
ap-4612	33	3	of	of	ADP
ap-4612	33	4	the	the	DET
ap-4612	33	5	topology	topology	NOUN
ap-4612	33	6	of	of	ADP
ap-4612	33	7	φ	φ	PROPN
ap-4612	33	8	[	[	X
ap-4612	33	9	10	10	NUM
ap-4612	33	10	,	,	PUNCT
ap-4612	33	11	21	21	NUM
ap-4612	33	12	]	]	PUNCT
ap-4612	33	13	is	be	AUX
ap-4612	33	14	that	that	SCONJ
ap-4612	33	15	all	all	DET
ap-4612	33	16	sequences	sequence	NOUN
ap-4612	33	17	which	which	PRON
ap-4612	33	18	converge	converge	VERB
ap-4612	33	19	on	on	ADP
ap-4612	33	20	φ	φ	NUM
ap-4612	33	21	,	,	PUNCT
ap-4612	33	22	also	also	ADV
ap-4612	33	23	converge	converge	VERB
ap-4612	33	24	on	on	ADP
ap-4612	33	25	h	h	NOUN
ap-4612	33	26	,	,	PUNCT
ap-4612	33	27	the	the	DET
ap-4612	33	28	converse	converse	NOUN
ap-4612	33	29	being	be	AUX
ap-4612	33	30	not	not	PART
ap-4612	33	31	true	true	ADJ
ap-4612	33	32	.	.	PUNCT
ap-4612	34	1	the	the	DET
ap-4612	34	2	difference	difference	NOUN
ap-4612	34	3	between	between	ADP
ap-4612	34	4	topologies	topology	NOUN
ap-4612	34	5	gives	give	VERB
ap-4612	34	6	rise	rise	NOUN
ap-4612	34	7	that	that	SCONJ
ap-4612	34	8	the	the	DET
ap-4612	34	9	dual	dual	ADJ
ap-4612	34	10	space	space	NOUN
ap-4612	34	11	of	of	ADP
ap-4612	34	12	φ	φ	PROPN
ap-4612	34	13	,	,	PUNCT
ap-4612	34	14	φ×	φ×	NOUN
ap-4612	34	15	,	,	PUNCT
ap-4612	34	16	is	be	AUX
ap-4612	34	17	bigger	big	ADJ
ap-4612	34	18	than	than	ADP
ap-4612	34	19	h	h	NOUN
ap-4612	34	20	,	,	PUNCT
ap-4612	34	21	which	which	PRON
ap-4612	34	22	is	be	AUX
ap-4612	34	23	self	self	NOUN
ap-4612	34	24	-	-	PUNCT
ap-4612	34	25	dual	dual	ADJ
ap-4612	34	26	.	.	PUNCT
ap-4612	35	1	here	here	ADV
ap-4612	35	2	,	,	PUNCT
ap-4612	35	3	the	the	DET
ap-4612	35	4	dual	dual	ADJ
ap-4612	35	5	φ×	φ×	NOUN
ap-4612	35	6	of	of	ADP
ap-4612	35	7	φ	φ	NUM
ap-4612	35	8	,	,	PUNCT
ap-4612	35	9	i.e.	i.e.	X
ap-4612	35	10	,	,	PUNCT
ap-4612	35	11	any	any	DET
ap-4612	35	12	f	f	PROPN
ap-4612	35	13	∈	∈	PROPN
ap-4612	35	14	φ×	φ×	NOUN
ap-4612	35	15	is	be	AUX
ap-4612	35	16	a	a	DET
ap-4612	35	17	continuous	continuous	ADJ
ap-4612	35	18	linear	linear	NOUN
ap-4612	35	19	mapping	mapping	NOUN
ap-4612	35	20	from	from	ADP
ap-4612	35	21	φ	φ	PROPN
ap-4612	35	22	into	into	ADP
ap-4612	35	23	c.	c.	PROPN
ap-4612	35	24	the	the	DET
ap-4612	35	25	linearity	linearity	NOUN
ap-4612	35	26	or	or	CCONJ
ap-4612	35	27	antilinearity	antilinearity	NOUN
ap-4612	35	28	of	of	ADP
ap-4612	35	29	f	f	PROPN
ap-4612	35	30	∈	∈	PROPN
ap-4612	35	31	φ×	φ×	NOUN
ap-4612	35	32	means	mean	NOUN
ap-4612	35	33	,	,	PUNCT
ap-4612	35	34	respectively	respectively	ADV
ap-4612	35	35	,	,	PUNCT
ap-4612	35	36	that	that	SCONJ
ap-4612	35	37	for	for	ADP
ap-4612	35	38	any	any	DET
ap-4612	35	39	pair	pair	NOUN
ap-4612	35	40	of	of	ADP
ap-4612	35	41	vectors	vector	NOUN
ap-4612	35	42	ψ,ϕ	ψ,ϕ	VERB
ap-4612	35	43	∈	∈	PROPN
ap-4612	35	44	φ	φ	PROPN
ap-4612	35	45	and	and	CCONJ
ap-4612	35	46	any	any	DET
ap-4612	35	47	pair	pair	NOUN
ap-4612	35	48	α	α	NOUN
ap-4612	35	49	,	,	PUNCT
ap-4612	35	50	β	β	X
ap-4612	35	51	∈	∈	PROPN
ap-4612	35	52	c	c	X
ap-4612	35	53	we	we	PRON
ap-4612	35	54	have	have	VERB
ap-4612	35	55	〈	〈	PROPN
ap-4612	35	56	f	f	PROPN
ap-4612	35	57	|αψ	|αψ	X
ap-4612	35	58	+	+	NUM
ap-4612	35	59	βϕ	βϕ	PROPN
ap-4612	35	60	〉	〉	NOUN
ap-4612	35	61	=	=	SYM
ap-4612	35	62	α〈f	α〈f	NUM
ap-4612	35	63	|ψ〉+	|ψ〉+	NOUN
ap-4612	35	64	β	β	X
ap-4612	35	65	〈	〈	PROPN
ap-4612	35	66	f	f	PROPN
ap-4612	35	67	|ϕ	|ϕ	PROPN
ap-4612	35	68	〉	〉	PROPN
ap-4612	35	69	,	,	PUNCT
ap-4612	35	70	〈	〈	PROPN
ap-4612	35	71	f	f	X
ap-4612	35	72	|αψ	|αψ	NUM
ap-4612	35	73	+	+	NUM
ap-4612	35	74	βϕ	βϕ	NOUN
ap-4612	35	75	〉	〉	NOUN
ap-4612	35	76	=	=	VERB
ap-4612	35	77	α∗〈f	α∗〈f	NOUN
ap-4612	35	78	|ψ〉+	|ψ〉+	NOUN
ap-4612	35	79	β∗	β∗	NOUN
ap-4612	35	80	〈	〈	PROPN
ap-4612	35	81	f	f	PROPN
ap-4612	35	82	|ϕ	|ϕ	PROPN
ap-4612	35	83	〉	〉	PROPN
ap-4612	35	84	,	,	PUNCT
ap-4612	35	85	where	where	SCONJ
ap-4612	35	86	we	we	PRON
ap-4612	35	87	have	have	AUX
ap-4612	35	88	followed	follow	VERB
ap-4612	35	89	the	the	DET
ap-4612	35	90	dirac	dirac	NOUN
ap-4612	35	91	bra	bra	NOUN
ap-4612	35	92	-	-	PUNCT
ap-4612	35	93	ket	ket	NOUN
ap-4612	35	94	notation	notation	NOUN
ap-4612	35	95	and	and	CCONJ
ap-4612	35	96	the	the	DET
ap-4612	35	97	star	star	NOUN
ap-4612	35	98	denotes	denote	VERB
ap-4612	35	99	complex	complex	ADJ
ap-4612	35	100	conjugation	conjugation	NOUN
ap-4612	35	101	.	.	PUNCT
ap-4612	36	1	a	a	DET
ap-4612	36	2	crucial	crucial	ADJ
ap-4612	36	3	property	property	NOUN
ap-4612	36	4	to	to	PART
ap-4612	36	5	be	be	AUX
ap-4612	36	6	taken	take	VERB
ap-4612	36	7	under	under	ADP
ap-4612	36	8	consideration	consideration	NOUN
ap-4612	36	9	is	be	AUX
ap-4612	36	10	that	that	SCONJ
ap-4612	36	11	if	if	SCONJ
ap-4612	36	12	a	a	PRON
ap-4612	36	13	is	be	AUX
ap-4612	36	14	a	a	DET
ap-4612	36	15	densely	densely	ADV
ap-4612	36	16	defined	define	VERB
ap-4612	36	17	operator	operator	NOUN
ap-4612	36	18	on	on	ADP
ap-4612	36	19	h	h	NOUN
ap-4612	36	20	,	,	PUNCT
ap-4612	36	21	such	such	ADJ
ap-4612	36	22	that	that	SCONJ
ap-4612	36	23	φ	φ	PROPN
ap-4612	36	24	be	be	VERB
ap-4612	36	25	a	a	DET
ap-4612	36	26	subspace	subspace	NOUN
ap-4612	36	27	of	of	ADP
ap-4612	36	28	its	its	PRON
ap-4612	36	29	domain	domain	NOUN
ap-4612	36	30	and	and	CCONJ
ap-4612	36	31	that	that	SCONJ
ap-4612	36	32	aϕ	aϕ	NUM
ap-4612	36	33	∈	∈	PROPN
ap-4612	36	34	φ	φ	PROPN
ap-4612	36	35	for	for	ADP
ap-4612	36	36	all	all	DET
ap-4612	36	37	ϕ	ϕ	PROPN
ap-4612	36	38	∈	∈	PROPN
ap-4612	36	39	φ	φ	NOUN
ap-4612	36	40	,	,	PUNCT
ap-4612	36	41	we	we	PRON
ap-4612	36	42	say	say	VERB
ap-4612	36	43	that	that	SCONJ
ap-4612	36	44	φ	φ	PROPN
ap-4612	36	45	reduces	reduce	VERB
ap-4612	36	46	a	a	PRON
ap-4612	36	47	or	or	CCONJ
ap-4612	36	48	that	that	SCONJ
ap-4612	36	49	φ	φ	PROPN
ap-4612	36	50	is	be	AUX
ap-4612	36	51	invariant	invariant	ADJ
ap-4612	36	52	under	under	ADP
ap-4612	36	53	the	the	DET
ap-4612	36	54	action	action	NOUN
ap-4612	36	55	of	of	ADP
ap-4612	36	56	a	a	PRON
ap-4612	36	57	,	,	PUNCT
ap-4612	36	58	(	(	PUNCT
ap-4612	36	59	i.e.	i.e.	X
ap-4612	36	60	,	,	PUNCT
ap-4612	36	61	aφ	aφ	ADP
ap-4612	36	62	⊂	⊂	PROPN
ap-4612	36	63	φ	φ	NUM
ap-4612	36	64	)	)	PUNCT
ap-4612	36	65	.	.	PUNCT
ap-4612	37	1	in	in	ADP
ap-4612	37	2	this	this	DET
ap-4612	37	3	case	case	NOUN
ap-4612	37	4	,	,	PUNCT
ap-4612	37	5	a	a	PRON
ap-4612	37	6	may	may	AUX
ap-4612	37	7	be	be	AUX
ap-4612	37	8	extended	extend	VERB
ap-4612	37	9	unambiguously	unambiguously	ADV
ap-4612	37	10	to	to	ADP
ap-4612	37	11	the	the	DET
ap-4612	37	12	dual	dual	ADJ
ap-4612	37	13	φ×	φ×	NOUN
ap-4612	37	14	by	by	ADP
ap-4612	37	15	making	make	VERB
ap-4612	37	16	use	use	NOUN
ap-4612	37	17	of	of	ADP
ap-4612	37	18	the	the	DET
ap-4612	37	19	duality	duality	NOUN
ap-4612	37	20	formula	formula	NOUN
ap-4612	37	21	〈	〈	PROPN
ap-4612	37	22	a×	a×	PROPN
ap-4612	37	23	f	f	SYM
ap-4612	37	24	|ϕ	|ϕ	ADJ
ap-4612	37	25	〉	〉	NOUN
ap-4612	37	26	:	:	PUNCT
ap-4612	37	27	=	=	SYM
ap-4612	37	28	〈	〈	PROPN
ap-4612	37	29	f	f	PROPN
ap-4612	37	30	|aϕ	|aϕ	NUM
ap-4612	37	31	〉	〉	NUM
ap-4612	37	32	∀ϕ	∀ϕ	PROPN
ap-4612	37	33	∈	∈	PROPN
ap-4612	37	34	φ	φ	PROPN
ap-4612	37	35	,	,	PUNCT
ap-4612	37	36	∀f	∀f	PROPN
ap-4612	37	37	∈	∈	PROPN
ap-4612	37	38	φ×.	φ×.	X
ap-4612	37	39	(	(	PUNCT
ap-4612	37	40	1	1	X
ap-4612	37	41	)	)	PUNCT
ap-4612	37	42	if	if	SCONJ
ap-4612	37	43	a	a	PRON
ap-4612	37	44	is	be	AUX
ap-4612	37	45	continuous	continuous	ADJ
ap-4612	37	46	on	on	ADP
ap-4612	37	47	φ	φ	NUM
ap-4612	37	48	,	,	PUNCT
ap-4612	37	49	then	then	ADV
ap-4612	37	50	a×	a×	PROPN
ap-4612	37	51	is	be	AUX
ap-4612	37	52	continuous	continuous	ADJ
ap-4612	37	53	on	on	ADP
ap-4612	37	54	φ×.	φ×.	X
ap-4612	37	55	the	the	DET
ap-4612	37	56	topology	topology	NOUN
ap-4612	37	57	on	on	ADP
ap-4612	37	58	φ	φ	PROPN
ap-4612	37	59	is	be	AUX
ap-4612	37	60	given	give	VERB
ap-4612	37	61	by	by	ADP
ap-4612	37	62	an	an	DET
ap-4612	37	63	infinite	infinite	ADJ
ap-4612	37	64	countable	countable	ADJ
ap-4612	37	65	set	set	NOUN
ap-4612	37	66	of	of	ADP
ap-4612	37	67	norms	norm	NOUN
ap-4612	37	68	{	{	PUNCT
ap-4612	37	69	‖·‖∞n=1	‖·‖∞n=1	NOUN
ap-4612	37	70	}	}	PUNCT
ap-4612	37	71	.	.	PUNCT
ap-4612	38	1	a	a	DET
ap-4612	38	2	linear	linear	ADJ
ap-4612	38	3	operator	operator	NOUN
ap-4612	38	4	a	a	PRON
ap-4612	38	5	on	on	ADP
ap-4612	38	6	φ	φ	PROPN
ap-4612	38	7	is	be	AUX
ap-4612	38	8	continuous	continuous	ADJ
ap-4612	38	9	if	if	SCONJ
ap-4612	38	10	and	and	CCONJ
ap-4612	38	11	only	only	ADV
ap-4612	38	12	if	if	SCONJ
ap-4612	38	13	for	for	ADP
ap-4612	38	14	each	each	DET
ap-4612	38	15	norm	norm	NOUN
ap-4612	38	16	‖·‖n	‖·‖n	ADV
ap-4612	38	17	there	there	PRON
ap-4612	38	18	is	be	VERB
ap-4612	38	19	a	a	DET
ap-4612	38	20	kn	kn	PROPN
ap-4612	38	21	>	>	X
ap-4612	38	22	0	0	PUNCT
ap-4612	38	23	and	and	CCONJ
ap-4612	38	24	a	a	DET
ap-4612	38	25	finite	finite	ADJ
ap-4612	38	26	sequence	sequence	NOUN
ap-4612	38	27	of	of	ADP
ap-4612	38	28	norms	norm	NOUN
ap-4612	38	29	‖·‖p1	‖·‖p1	PROPN
ap-4612	38	30	,	,	PUNCT
ap-4612	38	31	‖·‖p2	‖·‖p2	INTJ
ap-4612	38	32	,	,	PUNCT
ap-4612	38	33	.	.	PUNCT
ap-4612	38	34	.	.	PUNCT
ap-4612	39	1	.	.	PUNCT
ap-4612	40	1	,	,	PUNCT
ap-4612	40	2	‖·‖pr	‖·‖pr	VERB
ap-4612	40	3	such	such	ADJ
ap-4612	40	4	that	that	PRON
ap-4612	40	5	for	for	ADP
ap-4612	40	6	any	any	DET
ap-4612	40	7	ϕ	ϕ	PROPN
ap-4612	40	8	∈	∈	PROPN
ap-4612	40	9	φ	φ	PROPN
ap-4612	40	10	,	,	PUNCT
ap-4612	40	11	one	one	NUM
ap-4612	40	12	has	have	VERB
ap-4612	40	13	[	[	X
ap-4612	40	14	22	22	NUM
ap-4612	40	15	]	]	PUNCT
ap-4612	40	16	‖aϕ‖n	‖aϕ‖n	PUNCT
ap-4612	41	1	≤	≤	ADJ
ap-4612	41	2	kn	kn	PROPN
ap-4612	41	3	(	(	PUNCT
ap-4612	41	4	‖ϕ‖p1	‖ϕ‖p1	PROPN
ap-4612	41	5	+	+	SYM
ap-4612	41	6	‖ϕ‖p2	‖ϕ‖p2	NUM
ap-4612	41	7	+	+	NUM
ap-4612	41	8	·	·	PUNCT
ap-4612	41	9	·	·	PUNCT
ap-4612	41	10	·	·	PUNCT
ap-4612	41	11	+	+	NUM
ap-4612	41	12	‖ϕ‖pr	‖ϕ‖pr	NUM
ap-4612	41	13	)	)	PUNCT
ap-4612	41	14	.	.	PUNCT
ap-4612	42	1	(	(	PUNCT
ap-4612	42	2	2	2	X
ap-4612	42	3	)	)	PUNCT
ap-4612	42	4	the	the	DET
ap-4612	42	5	same	same	ADJ
ap-4612	42	6	result	result	NOUN
ap-4612	42	7	applies	apply	VERB
ap-4612	42	8	to	to	PART
ap-4612	42	9	check	check	VERB
ap-4612	42	10	the	the	DET
ap-4612	42	11	continuity	continuity	NOUN
ap-4612	42	12	of	of	ADP
ap-4612	42	13	any	any	DET
ap-4612	42	14	linear	linear	NOUN
ap-4612	42	15	or	or	CCONJ
ap-4612	42	16	antilinear	antilinear	ADJ
ap-4612	42	17	mapping	mapping	NOUN
ap-4612	42	18	f	f	NOUN
ap-4612	42	19	:	:	PUNCT
ap-4612	42	20	φ	φ	PROPN
ap-4612	42	21	7−→	7−→	PROPN
ap-4612	42	22	c.	c.	NOUN
ap-4612	42	23	in	in	ADP
ap-4612	42	24	this	this	DET
ap-4612	42	25	case	case	NOUN
ap-4612	42	26	,	,	PUNCT
ap-4612	42	27	the	the	DET
ap-4612	42	28	norm	norm	NOUN
ap-4612	42	29	‖aϕ‖p	‖aϕ‖p	PROPN
ap-4612	42	30	should	should	AUX
ap-4612	42	31	be	be	AUX
ap-4612	42	32	replaced	replace	VERB
ap-4612	42	33	by	by	ADP
ap-4612	42	34	the	the	DET
ap-4612	42	35	modulus	modulus	NOUN
ap-4612	42	36	|f	|f	PROPN
ap-4612	42	37	(	(	PUNCT
ap-4612	42	38	ϕ)|	ϕ)|	PROPN
ap-4612	42	39	.	.	PROPN
ap-4612	43	1	3	3	X
ap-4612	43	2	.	.	X
ap-4612	43	3	a	a	DET
ap-4612	43	4	paradigmatic	paradigmatic	ADJ
ap-4612	43	5	case	case	NOUN
ap-4612	43	6	:	:	PUNCT
ap-4612	43	7	rhs	rhs	PROPN
ap-4612	43	8	for	for	ADP
ap-4612	43	9	so(2	so(2	NOUN
ap-4612	43	10	)	)	PUNCT
ap-4612	43	11	as	as	SCONJ
ap-4612	43	12	mentioned	mention	VERB
ap-4612	43	13	before	before	ADV
ap-4612	43	14	,	,	PUNCT
ap-4612	43	15	we	we	PRON
ap-4612	43	16	have	have	AUX
ap-4612	43	17	considered	consider	VERB
ap-4612	43	18	the	the	DET
ap-4612	43	19	most	most	ADV
ap-4612	43	20	elementary	elementary	ADJ
ap-4612	43	21	situation	situation	NOUN
ap-4612	43	22	provided	provide	VERB
ap-4612	43	23	by	by	ADP
ap-4612	43	24	so(2	so(2	NOUN
ap-4612	43	25	)	)	PUNCT
ap-4612	43	26	,	,	PUNCT
ap-4612	43	27	where	where	SCONJ
ap-4612	43	28	we	we	PRON
ap-4612	43	29	have	have	VERB
ap-4612	43	30	two	two	NUM
ap-4612	43	31	rhs	rh	NOUN
ap-4612	43	32	serving	serve	VERB
ap-4612	43	33	as	as	ADP
ap-4612	43	34	support	support	NOUN
ap-4612	43	35	of	of	ADP
ap-4612	43	36	unitary	unitary	ADJ
ap-4612	43	37	equivalent	equivalent	ADJ
ap-4612	43	38	representations	representation	NOUN
ap-4612	43	39	of	of	ADP
ap-4612	43	40	so(2	so(2	NOUN
ap-4612	43	41	)	)	PUNCT
ap-4612	43	42	.	.	PUNCT
ap-4612	44	1	one	one	NUM
ap-4612	44	2	of	of	ADP
ap-4612	44	3	these	these	DET
ap-4612	44	4	rhs	rhs	PROPN
ap-4612	44	5	is	be	AUX
ap-4612	44	6	a	a	DET
ap-4612	44	7	concrete	concrete	ADJ
ap-4612	44	8	rhs	rh	NOUN
ap-4612	44	9	constructed	construct	VERB
ap-4612	44	10	with	with	ADP
ap-4612	44	11	functions	function	NOUN
ap-4612	44	12	or	or	CCONJ
ap-4612	44	13	generalised	generalise	VERB
ap-4612	44	14	functions	function	NOUN
ap-4612	44	15	and	and	CCONJ
ap-4612	44	16	the	the	DET
ap-4612	44	17	other	other	ADJ
ap-4612	44	18	one	one	NOUN
ap-4612	44	19	is	be	AUX
ap-4612	44	20	an	an	DET
ap-4612	44	21	abstract	abstract	ADJ
ap-4612	44	22	rhs	rhs	PROPN
ap-4612	44	23	.	.	PUNCT
ap-4612	45	1	a	a	DET
ap-4612	45	2	mapping	mapping	NOUN
ap-4612	45	3	of	of	ADP
ap-4612	45	4	the	the	DET
ap-4612	45	5	test	test	NOUN
ap-4612	45	6	vectors	vector	NOUN
ap-4612	45	7	of	of	ADP
ap-4612	45	8	the	the	DET
ap-4612	45	9	abstract	abstract	ADJ
ap-4612	45	10	rhs	rhs	PROPN
ap-4612	45	11	gives	give	VERB
ap-4612	45	12	the	the	DET
ap-4612	45	13	test	test	NOUN
ap-4612	45	14	functions	function	NOUN
ap-4612	45	15	of	of	ADP
ap-4612	45	16	the	the	DET
ap-4612	45	17	concrete	concrete	ADJ
ap-4612	45	18	one	one	NUM
ap-4612	45	19	.	.	PUNCT
ap-4612	46	1	also	also	ADV
ap-4612	46	2	we	we	PRON
ap-4612	46	3	have	have	VERB
ap-4612	46	4	to	to	PART
ap-4612	46	5	adjust	adjust	VERB
ap-4612	46	6	the	the	DET
ap-4612	46	7	topologies	topology	NOUN
ap-4612	46	8	so	so	SCONJ
ap-4612	46	9	that	that	SCONJ
ap-4612	46	10	the	the	DET
ap-4612	46	11	elements	element	NOUN
ap-4612	46	12	of	of	ADP
ap-4612	46	13	the	the	DET
ap-4612	46	14	lie	lie	NOUN
ap-4612	46	15	algebra	algebra	NOUN
ap-4612	46	16	be	be	VERB
ap-4612	46	17	continuous	continuous	ADJ
ap-4612	46	18	operators	operator	NOUN
ap-4612	46	19	on	on	ADP
ap-4612	46	20	both	both	DET
ap-4612	46	21	test	test	NOUN
ap-4612	46	22	spaces	space	NOUN
ap-4612	46	23	and	and	CCONJ
ap-4612	46	24	their	their	PRON
ap-4612	46	25	corresponding	corresponding	ADJ
ap-4612	46	26	duals	dual	NOUN
ap-4612	46	27	.	.	PUNCT
ap-4612	47	1	let	let	VERB
ap-4612	47	2	us	we	PRON
ap-4612	47	3	remember	remember	VERB
ap-4612	47	4	that	that	SCONJ
ap-4612	47	5	so(2	so(2	NOUN
ap-4612	47	6	)	)	PUNCT
ap-4612	47	7	is	be	AUX
ap-4612	47	8	the	the	DET
ap-4612	47	9	group	group	NOUN
ap-4612	47	10	of	of	ADP
ap-4612	47	11	rotations	rotation	NOUN
ap-4612	47	12	on	on	ADP
ap-4612	47	13	the	the	DET
ap-4612	47	14	euclidean	euclidean	ADJ
ap-4612	47	15	plane	plane	NOUN
ap-4612	47	16	.	.	PUNCT
ap-4612	48	1	it	it	PRON
ap-4612	48	2	is	be	AUX
ap-4612	48	3	a	a	DET
ap-4612	48	4	one	one	NUM
ap-4612	48	5	-	-	PUNCT
ap-4612	48	6	dimensional	dimensional	ADJ
ap-4612	48	7	abelian	abelian	ADJ
ap-4612	48	8	lie	lie	NOUN
ap-4612	48	9	group	group	NOUN
ap-4612	48	10	,	,	PUNCT
ap-4612	48	11	parametrized	parametrize	VERB
ap-4612	48	12	by	by	ADP
ap-4612	48	13	φ	φ	PROPN
ap-4612	48	14	∈	∈	PROPN
ap-4612	49	1	[	[	X
ap-4612	49	2	0	0	NUM
ap-4612	49	3	,	,	PUNCT
ap-4612	49	4	2π	2π	NOUN
ap-4612	49	5	)	)	PUNCT
ap-4612	49	6	.	.	PUNCT
ap-4612	50	1	the	the	DET
ap-4612	50	2	elements	element	NOUN
ap-4612	50	3	r(φ	r(φ	PROPN
ap-4612	50	4	)	)	PUNCT
ap-4612	50	5	of	of	ADP
ap-4612	50	6	so(2	so(2	NOUN
ap-4612	50	7	)	)	PUNCT
ap-4612	50	8	satisfy	satisfy	VERB
ap-4612	50	9	the	the	DET
ap-4612	50	10	product	product	NOUN
ap-4612	50	11	law	law	NOUN
ap-4612	50	12	r(φ1	r(φ1	NOUN
ap-4612	50	13	)	)	PUNCT
ap-4612	50	14	·	·	PUNCT
ap-4612	50	15	r(φ2	r(φ2	NOUN
ap-4612	50	16	)	)	PUNCT
ap-4612	51	1	=	=	SYM
ap-4612	51	2	r(φ1	r(φ1	PROPN
ap-4612	51	3	+	+	CCONJ
ap-4612	51	4	φ2	φ2	PROPN
ap-4612	51	5	)	)	PUNCT
ap-4612	51	6	(	(	PUNCT
ap-4612	51	7	mod	mod	ADJ
ap-4612	51	8	2π	2π	NOUN
ap-4612	51	9	)	)	PUNCT
ap-4612	51	10	.	.	PUNCT
ap-4612	52	1	here	here	ADV
ap-4612	52	2	,	,	PUNCT
ap-4612	52	3	we	we	PRON
ap-4612	52	4	are	be	AUX
ap-4612	52	5	considering	consider	VERB
ap-4612	52	6	two	two	NUM
ap-4612	52	7	equivalent	equivalent	ADJ
ap-4612	52	8	families	family	NOUN
ap-4612	52	9	of	of	ADP
ap-4612	52	10	uir	uir	NOUN
ap-4612	52	11	of	of	ADP
ap-4612	52	12	so(2	so(2	NOUN
ap-4612	52	13	):	):	PUNCT
ap-4612	52	14	one	one	NUM
ap-4612	52	15	of	of	ADP
ap-4612	52	16	them	they	PRON
ap-4612	52	17	supported	support	VERB
ap-4612	52	18	by	by	ADP
ap-4612	52	19	the	the	DET
ap-4612	52	20	hilbert	hilbert	PROPN
ap-4612	52	21	space	space	PROPN
ap-4612	52	22	l2[0	l2[0	PROPN
ap-4612	52	23	,	,	PUNCT
ap-4612	52	24	2π	2π	NOUN
ap-4612	52	25	]	]	PUNCT
ap-4612	52	26	(	(	PUNCT
ap-4612	52	27	via	via	ADP
ap-4612	52	28	the	the	DET
ap-4612	52	29	regular	regular	ADJ
ap-4612	52	30	representation	representation	NOUN
ap-4612	52	31	,	,	PUNCT
ap-4612	52	32	that	that	PRON
ap-4612	52	33	contains	contain	VERB
ap-4612	52	34	once	once	ADV
ap-4612	52	35	all	all	DET
ap-4612	52	36	the	the	DET
ap-4612	52	37	uir	uir	NOUN
ap-4612	52	38	,	,	PUNCT
ap-4612	52	39	each	each	DET
ap-4612	52	40	one	one	NUM
ap-4612	52	41	related	relate	VERB
ap-4612	52	42	to	to	ADP
ap-4612	52	43	an	an	DET
ap-4612	52	44	integer	integer	NOUN
ap-4612	52	45	number	number	NOUN
ap-4612	52	46	)	)	PUNCT
ap-4612	52	47	and	and	CCONJ
ap-4612	52	48	another	another	DET
ap-4612	52	49	set	set	NOUN
ap-4612	52	50	of	of	ADP
ap-4612	52	51	uir	uir	NOUN
ap-4612	52	52	(	(	PUNCT
ap-4612	52	53	also	also	ADV
ap-4612	52	54	labelled	label	VERB
ap-4612	52	55	by	by	ADP
ap-4612	52	56	z	z	PROPN
ap-4612	52	57	)	)	PUNCT
ap-4612	52	58	supported	support	VERB
ap-4612	52	59	by	by	ADP
ap-4612	52	60	an	an	DET
ap-4612	52	61	abstract	abstract	ADJ
ap-4612	52	62	infinite	infinite	ADJ
ap-4612	52	63	dimensional	dimensional	ADJ
ap-4612	52	64	separable	separable	ADJ
ap-4612	52	65	hilbert	hilbert	PROPN
ap-4612	52	66	space	space	PROPN
ap-4612	52	67	h.	h.	PROPN
ap-4612	52	68	3.1	3.1	NUM
ap-4612	52	69	.	.	PUNCT
ap-4612	53	1	uir	uir	NOUN
ap-4612	53	2	supported	support	VERB
ap-4612	53	3	by	by	ADP
ap-4612	53	4	the	the	DET
ap-4612	53	5	hs	hs	PROPN
ap-4612	53	6	l2[0	l2[0	PROPN
ap-4612	53	7	,	,	PUNCT
ap-4612	53	8	2π	2π	NOUN
ap-4612	53	9	]	]	PUNCT
ap-4612	54	1	we	we	PRON
ap-4612	54	2	consider	consider	VERB
ap-4612	54	3	the	the	DET
ap-4612	54	4	uir	uir	NOUN
ap-4612	54	5	characterised	characterise	VERB
ap-4612	54	6	by	by	ADP
ap-4612	54	7	the	the	DET
ap-4612	54	8	unitary	unitary	ADJ
ap-4612	54	9	operator	operator	NOUN
ap-4612	54	10	on	on	ADP
ap-4612	54	11	l2[0	l2[0	PROPN
ap-4612	54	12	,	,	PUNCT
ap-4612	54	13	2π	2π	NOUN
ap-4612	54	14	]	]	PUNCT
ap-4612	54	15	um(φ	um(φ	PRON
ap-4612	54	16	)	)	PUNCT
ap-4612	54	17	:	:	PUNCT
ap-4612	55	1	=	=	SYM
ap-4612	55	2	e−imφ	e−imφ	NOUN
ap-4612	55	3	∀φ	∀φ	X
ap-4612	55	4	∈	∈	PROPN
ap-4612	55	5	[	[	X
ap-4612	55	6	0	0	NUM
ap-4612	55	7	,	,	PUNCT
ap-4612	55	8	2π	2π	NOUN
ap-4612	55	9	)	)	PUNCT
ap-4612	55	10	,	,	PUNCT
ap-4612	55	11	m	m	PROPN
ap-4612	55	12	∈	∈	PROPN
ap-4612	55	13	z	z	NOUN
ap-4612	55	14	(	(	PUNCT
ap-4612	55	15	fixed	fix	VERB
ap-4612	55	16	)	)	PUNCT
ap-4612	55	17	.	.	PUNCT
ap-4612	56	1	(	(	PUNCT
ap-4612	56	2	3	3	X
ap-4612	56	3	)	)	PUNCT
ap-4612	56	4	an	an	DET
ap-4612	56	5	orthonormal	orthonormal	ADJ
ap-4612	56	6	basis	basis	NOUN
ap-4612	56	7	for	for	ADP
ap-4612	56	8	l2[0	l2[0	PROPN
ap-4612	56	9	,	,	PUNCT
ap-4612	56	10	2π	2π	NOUN
ap-4612	56	11	]	]	PUNCT
ap-4612	56	12	is	be	AUX
ap-4612	56	13	given	give	VERB
ap-4612	56	14	by	by	ADP
ap-4612	56	15	the	the	DET
ap-4612	56	16	sequence	sequence	NOUN
ap-4612	56	17	of	of	ADP
ap-4612	56	18	functions	function	NOUN
ap-4612	56	19	φm	φm	AUX
ap-4612	56	20	labelled	label	VERB
ap-4612	56	21	by	by	ADP
ap-4612	56	22	m	m	PROPN
ap-4612	56	23	∈	∈	PROPN
ap-4612	56	24	z	z	X
ap-4612	56	25	φm	φm	ADP
ap-4612	56	26	≡	≡	PROPN
ap-4612	56	27	1√	1√	PROPN
ap-4612	56	28	2π	2π	PROPN
ap-4612	56	29	e−imφ	e−imφ	NOUN
ap-4612	56	30	,	,	PUNCT
ap-4612	56	31	m	m	VERB
ap-4612	56	32	∈	∈	PROPN
ap-4612	56	33	z.	z.	PROPN
ap-4612	57	1	thus	thus	ADV
ap-4612	57	2	,	,	PUNCT
ap-4612	57	3	any	any	DET
ap-4612	57	4	lebesgue	lebesgue	NOUN
ap-4612	57	5	square	square	PROPN
ap-4612	57	6	integrable	integrable	ADJ
ap-4612	57	7	function	function	NOUN
ap-4612	57	8	f(φ	f(φ	PROPN
ap-4612	57	9	)	)	PUNCT
ap-4612	57	10	of	of	ADP
ap-4612	57	11	l2[0	l2[0	PROPN
ap-4612	57	12	,	,	PUNCT
ap-4612	57	13	2π	2π	NOUN
ap-4612	57	14	]	]	PUNCT
ap-4612	57	15	can	can	AUX
ap-4612	57	16	be	be	AUX
ap-4612	57	17	written	write	VERB
ap-4612	57	18	as	as	ADP
ap-4612	57	19	f(φ	f(φ	PROPN
ap-4612	57	20	)	)	PUNCT
ap-4612	57	21	=	=	PUNCT
ap-4612	58	1	∞∑	∞∑	NUM
ap-4612	58	2	m=−∞	m=−∞	ADP
ap-4612	58	3	fm	fm	PROPN
ap-4612	58	4	φm	φm	VERB
ap-4612	58	5	,	,	PUNCT
ap-4612	58	6	(	(	PUNCT
ap-4612	58	7	4	4	NUM
ap-4612	58	8	)	)	PUNCT
ap-4612	58	9	with	with	ADP
ap-4612	58	10	fm	fm	NOUN
ap-4612	58	11	=	=	SYM
ap-4612	58	12	1√	1√	PROPN
ap-4612	58	13	2π	2π	NUM
ap-4612	58	14	∫	∫	PROPN
ap-4612	58	15	2π	2π	NOUN
ap-4612	58	16	0	0	NUM
ap-4612	58	17	eimφ	eimφ	VERB
ap-4612	58	18	f(φ	f(φ	PROPN
ap-4612	58	19	)	)	PUNCT
ap-4612	58	20	dφ	dφ	ADP
ap-4612	58	21	,	,	PUNCT
ap-4612	58	22	(	(	PUNCT
ap-4612	58	23	5	5	NUM
ap-4612	58	24	)	)	PUNCT
ap-4612	58	25	under	under	ADP
ap-4612	58	26	the	the	DET
ap-4612	58	27	condition	condition	NOUN
ap-4612	58	28	that	that	SCONJ
ap-4612	58	29	∞∑	∞∑	NUM
ap-4612	58	30	m=−∞	m=−∞	NOUN
ap-4612	58	31	|fm|2	|fm|2	PUNCT
ap-4612	58	32	=	=	SYM
ap-4612	59	1	∫	∫	PROPN
ap-4612	59	2	2π	2π	PROPN
ap-4612	59	3	0	0	NUM
ap-4612	59	4	|f(φ)|2	|f(φ)|2	PRON
ap-4612	59	5	dφ	dφ	ADP
ap-4612	59	6	<	<	X
ap-4612	59	7	+	+	PRON
ap-4612	59	8	∞.	∞.	PROPN
ap-4612	59	9	note	note	VERB
ap-4612	59	10	that	that	SCONJ
ap-4612	59	11	the	the	DET
ap-4612	59	12	complex	complex	ADJ
ap-4612	59	13	numbers	number	NOUN
ap-4612	59	14	fm	fm	X
ap-4612	59	15	are	be	AUX
ap-4612	59	16	the	the	DET
ap-4612	59	17	fourier	fourier	ADJ
ap-4612	59	18	coefficients	coefficient	NOUN
ap-4612	59	19	of	of	ADP
ap-4612	59	20	f(φ	f(φ	PROPN
ap-4612	59	21	)	)	PUNCT
ap-4612	59	22	.	.	PUNCT
ap-4612	60	1	the	the	DET
ap-4612	60	2	functions	function	NOUN
ap-4612	60	3	um	um	INTJ
ap-4612	60	4	=	=	PUNCT
ap-4612	60	5	e−imφ	e−imφ	NOUN
ap-4612	60	6	satisfy	satisfy	VERB
ap-4612	60	7	the	the	DET
ap-4612	60	8	following	follow	VERB
ap-4612	60	9	orthogonality	orthogonality	NOUN
ap-4612	60	10	and	and	CCONJ
ap-4612	60	11	completeness	completeness	NOUN
ap-4612	60	12	relations	relation	NOUN
ap-4612	60	13	:	:	PUNCT
ap-4612	60	14	1	1	NUM
ap-4612	60	15	2π	2π	NUM
ap-4612	60	16	∫	∫	PROPN
ap-4612	60	17	2π	2π	NOUN
ap-4612	60	18	0	0	PUNCT
ap-4612	61	1	u†m(φ)un(φ	u†m(φ)un(φ	NOUN
ap-4612	61	2	)	)	PUNCT
ap-4612	61	3	dφ	dφ	ADP
ap-4612	61	4	=	=	PUNCT
ap-4612	61	5	δm	δm	PROPN
ap-4612	61	6	,	,	PUNCT
ap-4612	61	7	n	n	CCONJ
ap-4612	61	8	,	,	PUNCT
ap-4612	61	9	1	1	NUM
ap-4612	61	10	2π	2π	NOUN
ap-4612	61	11	∞∑	∞∑	NUM
ap-4612	61	12	m=−∞	m=−∞	X
ap-4612	61	13	u†m(φ)um(φ′	u†m(φ)um(φ′	PROPN
ap-4612	61	14	)	)	PUNCT
ap-4612	61	15	=	=	SYM
ap-4612	62	1	δ(φ−	δ(φ−	NOUN
ap-4612	62	2	φ′	φ′	NUM
ap-4612	62	3	)	)	PUNCT
ap-4612	62	4	.	.	PUNCT
ap-4612	63	1	3.2	3.2	NUM
ap-4612	63	2	.	.	PUNCT
ap-4612	64	1	uir	uir	NOUN
ap-4612	64	2	on	on	ADP
ap-4612	64	3	an	an	DET
ap-4612	64	4	infinite	infinite	ADJ
ap-4612	64	5	-	-	PUNCT
ap-4612	64	6	d	d	NOUN
ap-4612	64	7	separable	separable	NOUN
ap-4612	64	8	hs	hs	PROPN
ap-4612	64	9	equivalently	equivalently	PROPN
ap-4612	64	10	,	,	PUNCT
ap-4612	64	11	we	we	PRON
ap-4612	64	12	may	may	AUX
ap-4612	64	13	construct	construct	VERB
ap-4612	64	14	another	another	DET
ap-4612	64	15	set	set	NOUN
ap-4612	64	16	of	of	ADP
ap-4612	64	17	uir	uir	NOUN
ap-4612	64	18	’s	’s	X
ap-4612	64	19	of	of	ADP
ap-4612	64	20	so(2	so(2	NOUN
ap-4612	64	21	)	)	PUNCT
ap-4612	64	22	labelled	label	VERB
ap-4612	64	23	by	by	ADP
ap-4612	64	24	z	z	PROPN
ap-4612	64	25	and	and	CCONJ
ap-4612	64	26	supported	support	VERB
ap-4612	64	27	on	on	ADP
ap-4612	64	28	an	an	DET
ap-4612	64	29	abstract	abstract	ADJ
ap-4612	64	30	infinite	infinite	ADJ
ap-4612	64	31	dimensional	dimensional	ADJ
ap-4612	64	32	separable	separable	ADJ
ap-4612	64	33	hilbert	hilbert	PROPN
ap-4612	64	34	space	space	NOUN
ap-4612	64	35	h.	h.	PROPN
ap-4612	64	36	let	let	VERB
ap-4612	64	37	{	{	PUNCT
ap-4612	64	38	|m〉}m∈z	|m〉}m∈z	AUX
ap-4612	64	39	be	be	AUX
ap-4612	64	40	an	an	DET
ap-4612	64	41	orthonormal	orthonormal	ADJ
ap-4612	64	42	basis	basis	NOUN
ap-4612	64	43	of	of	ADP
ap-4612	64	44	h.	h.	PROPN
ap-4612	64	45	there	there	PRON
ap-4612	64	46	is	be	VERB
ap-4612	64	47	a	a	DET
ap-4612	64	48	unique	unique	ADJ
ap-4612	64	49	natural	natural	ADJ
ap-4612	64	50	unitary	unitary	ADJ
ap-4612	64	51	mapping	mapping	NOUN
ap-4612	64	52	s	s	VERB
ap-4612	64	53	such	such	ADJ
ap-4612	64	54	that	that	SCONJ
ap-4612	64	55	h	h	PROPN
ap-4612	64	56	s−→	s−→	PROPN
ap-4612	64	57	l2[0	l2[0	PROPN
ap-4612	64	58	,	,	PUNCT
ap-4612	64	59	2π	2π	NOUN
ap-4612	64	60	]	]	X
ap-4612	64	61	,	,	PUNCT
ap-4612	64	62	|m	|m	NOUN
ap-4612	64	63	〉	〉	PROPN
ap-4612	64	64	7−→	7−→	PROPN
ap-4612	64	65	s|m	s|m	NOUN
ap-4612	64	66	〉	〉	NOUN
ap-4612	64	67	=	=	SYM
ap-4612	64	68	φm	φm	PROPN
ap-4612	64	69	,	,	PUNCT
ap-4612	64	70	∀m	∀m	PROPN
ap-4612	64	71	∈	∈	PROPN
ap-4612	64	72	z.	z.	PROPN
ap-4612	64	73	380	380	NUM
ap-4612	64	74	vol	vol	NOUN
ap-4612	64	75	.	.	PUNCT
ap-4612	65	1	57	57	NUM
ap-4612	66	1	no	no	NOUN
ap-4612	66	2	.	.	PUNCT
ap-4612	67	1	6/2017	6/2017	PRON
ap-4612	67	2	lie	lie	VERB
ap-4612	67	3	algebra	algebra	NOUN
ap-4612	67	4	representations	representation	NOUN
ap-4612	67	5	and	and	CCONJ
ap-4612	67	6	rigged	rig	VERB
ap-4612	67	7	hilbert	hilbert	NOUN
ap-4612	67	8	spaces	space	VERB
ap-4612	67	9	:	:	PUNCT
ap-4612	67	10	the	the	DET
ap-4612	67	11	so(2	so(2	NOUN
ap-4612	67	12	)	)	PUNCT
ap-4612	67	13	case	case	NOUN
ap-4612	67	14	let	let	VERB
ap-4612	67	15	us	we	PRON
ap-4612	67	16	consider	consider	VERB
ap-4612	67	17	the	the	DET
ap-4612	67	18	subspace	subspace	NOUN
ap-4612	67	19	φ	φ	PROPN
ap-4612	67	20	of	of	ADP
ap-4612	67	21	h	h	PROPN
ap-4612	67	22	of	of	ADP
ap-4612	67	23	vectors	vector	NOUN
ap-4612	67	24	|f	|f	PROPN
ap-4612	67	25	〉	〉	NUM
ap-4612	67	26	=	=	PUNCT
ap-4612	68	1	∞∑	∞∑	NOUN
ap-4612	68	2	m=−∞	m=−∞	AUX
ap-4612	68	3	am	be	AUX
ap-4612	68	4	|m	|m	NOUN
ap-4612	68	5	〉	〉	PROPN
ap-4612	68	6	∈	∈	PROPN
ap-4612	68	7	h	h	NOUN
ap-4612	68	8	,	,	PUNCT
ap-4612	68	9	am	be	AUX
ap-4612	68	10	∈	∈	PROPN
ap-4612	68	11	c	c	NOUN
ap-4612	68	12	,	,	PUNCT
ap-4612	68	13	(	(	PUNCT
ap-4612	68	14	6	6	NUM
ap-4612	68	15	)	)	PUNCT
ap-4612	68	16	such	such	ADJ
ap-4612	68	17	that	that	SCONJ
ap-4612	68	18	〈	〈	PROPN
ap-4612	68	19	f	f	PROPN
ap-4612	68	20	|f〉p	|f〉p	NUM
ap-4612	68	21	≡	≡	PROPN
ap-4612	68	22	‖f‖2	‖f‖2	PUNCT
ap-4612	69	1	p	p	X
ap-4612	69	2	:	:	PUNCT
ap-4612	69	3	=	=	SYM
ap-4612	69	4	∞∑	∞∑	NUM
ap-4612	69	5	m=−∞	m=−∞	NOUN
ap-4612	69	6	|am|2|m+	|am|2|m+	ADJ
ap-4612	69	7	i|2p	i|2p	ADJ
ap-4612	69	8	<	<	NOUN
ap-4612	69	9	∞	∞	PROPN
ap-4612	69	10	,	,	PUNCT
ap-4612	69	11	(	(	PUNCT
ap-4612	69	12	7	7	X
ap-4612	69	13	)	)	PUNCT
ap-4612	69	14	for	for	ADP
ap-4612	69	15	any	any	DET
ap-4612	69	16	p	p	NOUN
ap-4612	69	17	=	=	SYM
ap-4612	69	18	0	0	NUM
ap-4612	69	19	,	,	PUNCT
ap-4612	69	20	1	1	NUM
ap-4612	69	21	,	,	PUNCT
ap-4612	69	22	2	2	NUM
ap-4612	69	23	,	,	PUNCT
ap-4612	69	24	.	.	PUNCT
ap-4612	69	25	.	.	PUNCT
ap-4612	69	26	.	.	PUNCT
ap-4612	70	1	the	the	DET
ap-4612	70	2	imaginary	imaginary	ADJ
ap-4612	70	3	unit	unit	NOUN
ap-4612	70	4	i	i	PRON
ap-4612	70	5	has	have	AUX
ap-4612	70	6	been	be	AUX
ap-4612	70	7	introduced	introduce	VERB
ap-4612	70	8	to	to	PART
ap-4612	70	9	have	have	VERB
ap-4612	70	10	|m	|m	NOUN
ap-4612	70	11	+	+	CCONJ
ap-4612	70	12	i|	i|	PROPN
ap-4612	70	13	6=	6=	ADP
ap-4612	70	14	0	0	NUM
ap-4612	70	15	for	for	ADP
ap-4612	70	16	all	all	DET
ap-4612	70	17	m	m	PROPN
ap-4612	70	18	∈	∈	PROPN
ap-4612	70	19	z.	z.	PROPN
ap-4612	70	20	since	since	SCONJ
ap-4612	70	21	φ	φ	PROPN
ap-4612	70	22	contains	contain	VERB
ap-4612	70	23	all	all	DET
ap-4612	70	24	finite	finite	PROPN
ap-4612	70	25	linear	linear	ADJ
ap-4612	70	26	combinations	combination	NOUN
ap-4612	70	27	of	of	ADP
ap-4612	70	28	the	the	DET
ap-4612	70	29	basis	basis	NOUN
ap-4612	70	30	vectors	vector	NOUN
ap-4612	70	31	|m	|m	NOUN
ap-4612	70	32	〉	〉	PROPN
ap-4612	70	33	is	be	AUX
ap-4612	70	34	dense	dense	ADJ
ap-4612	70	35	on	on	ADP
ap-4612	70	36	h.	h.	PROPN
ap-4612	70	37	we	we	PRON
ap-4612	70	38	endow	endow	VERB
ap-4612	70	39	φ	φ	PROPN
ap-4612	70	40	with	with	ADP
ap-4612	70	41	the	the	DET
ap-4612	70	42	metrizable	metrizable	ADJ
ap-4612	70	43	topology	topology	NOUN
ap-4612	70	44	generated	generate	VERB
ap-4612	70	45	by	by	ADP
ap-4612	70	46	the	the	DET
ap-4612	70	47	norms	norm	NOUN
ap-4612	70	48	‖f‖p	‖f‖p	NOUN
ap-4612	70	49	,	,	PUNCT
ap-4612	70	50	(	(	PUNCT
ap-4612	70	51	p	p	NOUN
ap-4612	70	52	=	=	NOUN
ap-4612	70	53	0	0	NUM
ap-4612	70	54	,	,	PUNCT
ap-4612	70	55	1	1	NUM
ap-4612	70	56	,	,	PUNCT
ap-4612	70	57	2	2	NUM
ap-4612	70	58	,	,	PUNCT
ap-4612	70	59	.	.	PUNCT
ap-4612	70	60	.	.	PUNCT
ap-4612	70	61	.	.	PUNCT
ap-4612	70	62	)	)	PUNCT
ap-4612	70	63	.	.	PUNCT
ap-4612	71	1	in	in	ADP
ap-4612	71	2	this	this	DET
ap-4612	71	3	way	way	NOUN
ap-4612	71	4	we	we	PRON
ap-4612	71	5	have	have	AUX
ap-4612	71	6	constructed	construct	VERB
ap-4612	71	7	a	a	DET
ap-4612	71	8	rhs	rhs	PROPN
ap-4612	71	9	:	:	PUNCT
ap-4612	71	10	φ	φ	PROPN
ap-4612	71	11	⊂	⊂	PROPN
ap-4612	71	12	h	h	PROPN
ap-4612	71	13	⊂	⊂	PROPN
ap-4612	71	14	φ×.	φ×.	ADJ
ap-4612	71	15	considering	consider	VERB
ap-4612	71	16	that	that	SCONJ
ap-4612	71	17	the	the	DET
ap-4612	71	18	unitary	unitary	ADJ
ap-4612	71	19	mapping	mapping	NOUN
ap-4612	71	20	s	s	PART
ap-4612	71	21	transports	transport	NOUN
ap-4612	71	22	the	the	DET
ap-4612	71	23	topologies	topology	NOUN
ap-4612	71	24	,	,	PUNCT
ap-4612	71	25	we	we	PRON
ap-4612	71	26	get	get	VERB
ap-4612	71	27	two	two	NUM
ap-4612	71	28	rhs	rhs	PROPN
ap-4612	71	29	φ	φ	PROPN
ap-4612	71	30	⊂	⊂	PROPN
ap-4612	71	31	h	h	PROPN
ap-4612	71	32	⊂	⊂	PROPN
ap-4612	71	33	φ×	φ×	PROPN
ap-4612	71	34	,	,	PUNCT
ap-4612	71	35	sφ⊂	sφ⊂	PROPN
ap-4612	71	36	l2[0	l2[0	PROPN
ap-4612	71	37	,	,	PUNCT
ap-4612	71	38	2π]⊂	2π]⊂	NUM
ap-4612	71	39	(	(	PUNCT
ap-4612	71	40	sφ)×.	sφ)×.	NOUN
ap-4612	71	41	such	such	ADJ
ap-4612	71	42	that	that	SCONJ
ap-4612	71	43	φ	φ	PROPN
ap-4612	71	44	and	and	CCONJ
ap-4612	71	45	h	h	NOUN
ap-4612	71	46	have	have	VERB
ap-4612	71	47	the	the	DET
ap-4612	71	48	discrete	discrete	ADJ
ap-4612	71	49	basis	basis	NOUN
ap-4612	71	50	{	{	PUNCT
ap-4612	71	51	|m〉}m∈z	|m〉}m∈z	NOUN
ap-4612	71	52	and	and	CCONJ
ap-4612	71	53	sφ	sφ	PROPN
ap-4612	71	54	and	and	CCONJ
ap-4612	71	55	l2[0	l2[0	PROPN
ap-4612	71	56	,	,	PUNCT
ap-4612	71	57	2π	2π	NOUN
ap-4612	71	58	]	]	PUNCT
ap-4612	71	59	have	have	VERB
ap-4612	71	60	its	its	PRON
ap-4612	71	61	equivalent	equivalent	ADJ
ap-4612	71	62	discrete	discrete	ADJ
ap-4612	71	63	basis	basis	NOUN
ap-4612	71	64	{	{	PUNCT
ap-4612	71	65	φm}m∈z	φm}m∈z	NOUN
ap-4612	71	66	.	.	PUNCT
ap-4612	72	1	now	now	ADV
ap-4612	72	2	,	,	PUNCT
ap-4612	72	3	we	we	PRON
ap-4612	72	4	may	may	AUX
ap-4612	72	5	define	define	VERB
ap-4612	72	6	continuous	continuous	ADJ
ap-4612	72	7	bases	basis	NOUN
ap-4612	72	8	in	in	ADP
ap-4612	72	9	both	both	DET
ap-4612	72	10	rhs	rhs	PROPN
ap-4612	72	11	as	as	SCONJ
ap-4612	72	12	follows	follow	VERB
ap-4612	72	13	.	.	PUNCT
ap-4612	73	1	since	since	SCONJ
ap-4612	73	2	these	these	DET
ap-4612	73	3	two	two	NUM
ap-4612	73	4	rhs	rh	NOUN
ap-4612	73	5	are	be	AUX
ap-4612	73	6	unitarily	unitarily	ADV
ap-4612	73	7	equivalent	equivalent	ADJ
ap-4612	73	8	,	,	PUNCT
ap-4612	73	9	it	it	PRON
ap-4612	73	10	is	be	AUX
ap-4612	73	11	enough	enough	ADJ
ap-4612	73	12	to	to	PART
ap-4612	73	13	construct	construct	VERB
ap-4612	73	14	the	the	DET
ap-4612	73	15	continuous	continuous	ADJ
ap-4612	73	16	basis	basis	NOUN
ap-4612	73	17	on	on	ADP
ap-4612	73	18	the	the	DET
ap-4612	73	19	abstract	abstract	ADJ
ap-4612	73	20	rhs	rhs	PROPN
ap-4612	73	21	and	and	CCONJ
ap-4612	73	22	to	to	PART
ap-4612	73	23	induce	induce	VERB
ap-4612	73	24	the	the	DET
ap-4612	73	25	equivalent	equivalent	ADJ
ap-4612	73	26	one	one	NUM
ap-4612	73	27	in	in	ADP
ap-4612	73	28	the	the	DET
ap-4612	73	29	other	other	ADJ
ap-4612	73	30	rhs	rhs	PROPN
ap-4612	73	31	.	.	PUNCT
ap-4612	74	1	let	let	VERB
ap-4612	74	2	us	we	PRON
ap-4612	74	3	consider	consider	VERB
ap-4612	74	4	the	the	DET
ap-4612	74	5	abstract	abstract	ADJ
ap-4612	74	6	rhs	rhs	PROPN
ap-4612	75	1	φ	φ	PROPN
ap-4612	75	2	⊂	⊂	PROPN
ap-4612	75	3	h	h	PROPN
ap-4612	75	4	⊂	⊂	PROPN
ap-4612	75	5	φ×.	φ×.	VERB
ap-4612	75	6	since	since	SCONJ
ap-4612	75	7	|m	|m	NOUN
ap-4612	75	8	〉	〉	PROPN
ap-4612	75	9	∈	∈	PROPN
ap-4612	75	10	h	h	NOUN
ap-4612	75	11	we	we	PRON
ap-4612	75	12	can	can	AUX
ap-4612	75	13	consider	consider	VERB
ap-4612	75	14	〈	〈	PROPN
ap-4612	75	15	m|	m|	NOUN
ap-4612	75	16	∈	∈	NOUN
ap-4612	75	17	h×	h×	PRON
ap-4612	75	18	=	=	SYM
ap-4612	75	19	h.	h.	PROPN
ap-4612	75	20	then	then	ADV
ap-4612	75	21	,	,	PUNCT
ap-4612	75	22	for	for	ADP
ap-4612	75	23	any	any	DET
ap-4612	75	24	φ	φ	PROPN
ap-4612	75	25	∈	∈	PROPN
ap-4612	76	1	[	[	X
ap-4612	76	2	0	0	NUM
ap-4612	76	3	,	,	PUNCT
ap-4612	76	4	2π	2π	NOUN
ap-4612	76	5	)	)	PUNCT
ap-4612	76	6	,	,	PUNCT
ap-4612	76	7	we	we	PRON
ap-4612	76	8	can	can	AUX
ap-4612	76	9	define	define	VERB
ap-4612	76	10	a	a	DET
ap-4612	76	11	ket	ket	NOUN
ap-4612	76	12	|φ	|φ	PROPN
ap-4612	76	13	〉	〉	PROPN
ap-4612	76	14	such	such	ADJ
ap-4612	76	15	that	that	SCONJ
ap-4612	76	16	〈	〈	PROPN
ap-4612	76	17	m|φ	m|φ	PROPN
ap-4612	76	18	〉	〉	NOUN
ap-4612	76	19	:	:	PUNCT
ap-4612	76	20	=	=	SYM
ap-4612	76	21	1√	1√	ADJ
ap-4612	76	22	2π	2π	NOUN
ap-4612	76	23	eimφ	eimφ	VERB
ap-4612	76	24	.	.	PUNCT
ap-4612	77	1	from	from	ADP
ap-4612	77	2	the	the	DET
ap-4612	77	3	duality	duality	NOUN
ap-4612	77	4	relation	relation	NOUN
ap-4612	77	5	〈	〈	PROPN
ap-4612	77	6	φ|m	φ|m	NOUN
ap-4612	77	7	〉	〉	PROPN
ap-4612	77	8	=	=	SYM
ap-4612	77	9	〈	〈	PROPN
ap-4612	77	10	m|φ〉∗	m|φ〉∗	PROPN
ap-4612	77	11	and	and	CCONJ
ap-4612	77	12	for	for	ADP
ap-4612	77	13	any	any	DET
ap-4612	77	14	|f	|f	PROPN
ap-4612	77	15	〉	〉	NOUN
ap-4612	77	16	=	=	PUNCT
ap-4612	77	17	∑∞	∑∞	NOUN
ap-4612	77	18	m=−∞	m=−∞	X
ap-4612	77	19	am	be	AUX
ap-4612	77	20	|m	|m	NOUN
ap-4612	77	21	〉	〉	PROPN
ap-4612	77	22	∈	∈	PROPN
ap-4612	77	23	φ	φ	NOUN
ap-4612	77	24	we	we	PRON
ap-4612	77	25	get	get	VERB
ap-4612	77	26	〈	〈	PROPN
ap-4612	77	27	φ|f	φ|f	NOUN
ap-4612	77	28	〉	〉	NOUN
ap-4612	77	29	=	=	NOUN
ap-4612	78	1	∞∑	∞∑	NOUN
ap-4612	78	2	m=−∞	m=−∞	AUX
ap-4612	78	3	am	be	AUX
ap-4612	78	4	〈	〈	NOUN
ap-4612	78	5	φ|m	φ|m	NOUN
ap-4612	78	6	〉	〉	NOUN
ap-4612	78	7	=	=	SYM
ap-4612	78	8	1√	1√	PROPN
ap-4612	78	9	2π	2π	NOUN
ap-4612	78	10	∞∑	∞∑	NUM
ap-4612	78	11	m=−∞	m=−∞	ADP
ap-4612	78	12	ame	ame	PROPN
ap-4612	78	13	−imφ	−imφ	NOUN
ap-4612	78	14	,	,	PUNCT
ap-4612	78	15	where	where	SCONJ
ap-4612	78	16	am	be	AUX
ap-4612	78	17	=	=	SYM
ap-4612	78	18	fm	fm	NOUN
ap-4612	78	19	as	as	ADP
ap-4612	78	20	in	in	ADP
ap-4612	78	21	(	(	PUNCT
ap-4612	78	22	4	4	NUM
ap-4612	78	23	)	)	PUNCT
ap-4612	78	24	.	.	PUNCT
ap-4612	79	1	the	the	DET
ap-4612	79	2	action	action	NOUN
ap-4612	79	3	of	of	ADP
ap-4612	79	4	〈	〈	PROPN
ap-4612	79	5	φ|	φ|	PROPN
ap-4612	79	6	on	on	ADP
ap-4612	79	7	φ	φ	NUM
ap-4612	79	8	,	,	PUNCT
ap-4612	79	9	〈	〈	PROPN
ap-4612	79	10	φ|f	φ|f	NOUN
ap-4612	79	11	〉	〉	PROPN
ap-4612	79	12	,	,	PUNCT
ap-4612	79	13	is	be	AUX
ap-4612	79	14	well	well	ADV
ap-4612	79	15	defined	define	VERB
ap-4612	79	16	since	since	SCONJ
ap-4612	79	17	the	the	DET
ap-4612	79	18	following	follow	VERB
ap-4612	79	19	series	series	NOUN
ap-4612	79	20	is	be	AUX
ap-4612	79	21	absolutely	absolutely	ADV
ap-4612	79	22	convergent	convergent	ADJ
ap-4612	79	23	∞∑	∞∑	NUM
ap-4612	79	24	m=−∞	m=−∞	NOUN
ap-4612	79	25	|am|	|am|	NOUN
ap-4612	79	26	=	=	NOUN
ap-4612	79	27	∞∑	∞∑	NUM
ap-4612	79	28	m=−∞	m=−∞	X
ap-4612	79	29	|am||m+	|am||m+	ADJ
ap-4612	79	30	i|	i|	PROPN
ap-4612	79	31	|m+	|m+	PROPN
ap-4612	79	32	i|	i|	PROPN
ap-4612	79	33	≤	≤	NOUN
ap-4612	79	34	√√√√	√√√√	ADP
ap-4612	79	35	∞∑	∞∑	NUM
ap-4612	79	36	m=−∞	m=−∞	NOUN
ap-4612	79	37	|am|2|m+	|am|2|m+	VERB
ap-4612	79	38	i|2	i|2	VERB
ap-4612	79	39	√√√√	√√√√	NOUN
ap-4612	79	40	∞∑	∞∑	NUM
ap-4612	79	41	m=−∞	m=−∞	ADP
ap-4612	79	42	1	1	NUM
ap-4612	79	43	|m+	|m+	PROPN
ap-4612	79	44	i|2	i|2	PROPN
ap-4612	79	45	.	.	PUNCT
ap-4612	80	1	(	(	PUNCT
ap-4612	80	2	8)	8)	NUM
ap-4612	80	3	note	note	NOUN
ap-4612	80	4	that	that	SCONJ
ap-4612	80	5	both	both	DET
ap-4612	80	6	series	serie	NOUN
ap-4612	80	7	on	on	ADP
ap-4612	80	8	the	the	DET
ap-4612	80	9	right	right	ADJ
ap-4612	80	10	converge	converge	NOUN
ap-4612	80	11	:	:	PUNCT
ap-4612	80	12	the	the	DET
ap-4612	80	13	first	first	ADJ
ap-4612	80	14	one	one	NUM
ap-4612	80	15	because	because	SCONJ
ap-4612	80	16	it	it	PRON
ap-4612	80	17	verifies	verify	VERB
ap-4612	80	18	(	(	PUNCT
ap-4612	80	19	7	7	NUM
ap-4612	80	20	)	)	PUNCT
ap-4612	80	21	for	for	ADP
ap-4612	80	22	p	p	NOUN
ap-4612	80	23	=	=	SYM
ap-4612	80	24	1	1	NUM
ap-4612	80	25	,	,	PUNCT
ap-4612	80	26	and	and	CCONJ
ap-4612	80	27	it	it	PRON
ap-4612	80	28	is	be	AUX
ap-4612	80	29	obvious	obvious	ADJ
ap-4612	80	30	for	for	ADP
ap-4612	80	31	the	the	DET
ap-4612	80	32	second	second	ADJ
ap-4612	80	33	series	series	NOUN
ap-4612	80	34	.	.	PUNCT
ap-4612	81	1	since	since	SCONJ
ap-4612	81	2	|〈φ|f〉|	|〈φ|f〉|	ADJ
ap-4612	81	3	≤	≤	PROPN
ap-4612	81	4	c	c	PROPN
ap-4612	81	5	‖f‖1	‖f‖1	NOUN
ap-4612	81	6	with	with	ADP
ap-4612	81	7	‖f‖1	‖f‖1	NOUN
ap-4612	81	8	=	=	SYM
ap-4612	81	9	√√√√	√√√√	VERB
ap-4612	81	10	∞∑	∞∑	NUM
ap-4612	81	11	m=−∞	m=−∞	X
ap-4612	81	12	|am|2	|am|2	NUM
ap-4612	81	13	|m+	|m+	PROPN
ap-4612	81	14	i|2	i|2	PROPN
ap-4612	81	15	,	,	PUNCT
ap-4612	81	16	c	c	NOUN
ap-4612	81	17	=	=	PUNCT
ap-4612	81	18	√√√√	√√√√	VERB
ap-4612	81	19	∞∑	∞∑	NUM
ap-4612	81	20	m=−∞	m=−∞	NOUN
ap-4612	81	21	1	1	NUM
ap-4612	81	22	|m+	|m+	PROPN
ap-4612	81	23	i|2	i|2	PROPN
ap-4612	81	24	,	,	PUNCT
ap-4612	81	25	then	then	ADV
ap-4612	81	26	〈	〈	PROPN
ap-4612	81	27	φ|	φ|	PROPN
ap-4612	81	28	∈	∈	PROPN
ap-4612	81	29	φ×.	φ×.	NOUN
ap-4612	81	30	note	note	VERB
ap-4612	81	31	that	that	SCONJ
ap-4612	81	32	〈	〈	PROPN
ap-4612	81	33	φ|f	φ|f	NOUN
ap-4612	81	34	〉	〉	NOUN
ap-4612	81	35	=	=	SYM
ap-4612	81	36	〈	〈	PROPN
ap-4612	81	37	f	f	PROPN
ap-4612	81	38	|φ〉∗	|φ〉∗	PROPN
ap-4612	81	39	and	and	CCONJ
ap-4612	81	40	since	since	SCONJ
ap-4612	81	41	〈	〈	PROPN
ap-4612	81	42	φ|	φ|	PROPN
ap-4612	81	43	is	be	AUX
ap-4612	81	44	a	a	DET
ap-4612	81	45	linear	linear	ADJ
ap-4612	81	46	map	map	NOUN
ap-4612	81	47	on	on	ADP
ap-4612	81	48	φ	φ	PROPN
ap-4612	81	49	then	then	ADV
ap-4612	81	50	|φ	|φ	PROPN
ap-4612	81	51	〉	〉	PROPN
ap-4612	81	52	is	be	AUX
ap-4612	81	53	antilinear	antilinear	ADJ
ap-4612	81	54	.	.	PUNCT
ap-4612	82	1	on	on	ADP
ap-4612	82	2	the	the	DET
ap-4612	82	3	other	other	ADJ
ap-4612	82	4	hand	hand	NOUN
ap-4612	82	5	,	,	PUNCT
ap-4612	82	6	{	{	PUNCT
ap-4612	82	7	|φ〉}φ∈[0,2π	|φ〉}φ∈[0,2π	ADJ
ap-4612	82	8	)	)	PUNCT
ap-4612	82	9	,	,	PUNCT
ap-4612	82	10	is	be	AUX
ap-4612	82	11	a	a	DET
ap-4612	82	12	continuous	continuous	ADJ
ap-4612	82	13	basis	basis	NOUN
ap-4612	82	14	.	.	PUNCT
ap-4612	83	1	in	in	ADP
ap-4612	83	2	fact	fact	NOUN
ap-4612	83	3	,	,	PUNCT
ap-4612	83	4	if	if	SCONJ
ap-4612	83	5	we	we	PRON
ap-4612	83	6	apply	apply	VERB
ap-4612	83	7	the	the	DET
ap-4612	83	8	map	map	NOUN
ap-4612	83	9	s	s	VERB
ap-4612	83	10	to	to	ADP
ap-4612	83	11	an	an	DET
ap-4612	83	12	arbitrary	arbitrary	ADJ
ap-4612	83	13	|f	|f	PROPN
ap-4612	83	14	〉	〉	PROPN
ap-4612	83	15	∈	∈	PROPN
ap-4612	83	16	φ	φ	NOUN
ap-4612	83	17	as	as	ADP
ap-4612	83	18	in	in	ADP
ap-4612	83	19	(	(	PUNCT
ap-4612	83	20	6	6	NUM
ap-4612	83	21	)	)	PUNCT
ap-4612	83	22	,	,	PUNCT
ap-4612	83	23	we	we	PRON
ap-4612	83	24	obtain	obtain	VERB
ap-4612	83	25	that	that	DET
ap-4612	83	26	s|f	s|f	PROPN
ap-4612	83	27	〉	〉	PROPN
ap-4612	83	28	∈	∈	PROPN
ap-4612	83	29	sφ	sφ	PROPN
ap-4612	83	30	⊂	⊂	PROPN
ap-4612	83	31	l2[0	l2[0	PROPN
ap-4612	83	32	,	,	PUNCT
ap-4612	83	33	2π	2π	NOUN
ap-4612	83	34	]	]	PUNCT
ap-4612	83	35	and	and	CCONJ
ap-4612	83	36	s|f	s|f	PROPN
ap-4612	83	37	〉	〉	NOUN
ap-4612	83	38	=	=	NOUN
ap-4612	84	1	∞∑	∞∑	NOUN
ap-4612	84	2	m=−∞	m=−∞	AUX
ap-4612	84	3	am	be	AUX
ap-4612	84	4	s|m	s|m	NOUN
ap-4612	84	5	〉	〉	NOUN
ap-4612	84	6	=	=	NOUN
ap-4612	85	1	∞∑	∞∑	NOUN
ap-4612	85	2	m=−∞	m=−∞	X
ap-4612	85	3	am	be	AUX
ap-4612	85	4	e−imφ√	e−imφ√	X
ap-4612	85	5	2π	2π	PROPN
ap-4612	85	6	=	=	SYM
ap-4612	85	7	〈	〈	PROPN
ap-4612	85	8	φ|f	φ|f	NOUN
ap-4612	85	9	〉	〉	NOUN
ap-4612	85	10	=	=	SYM
ap-4612	85	11	f(φ	f(φ	PROPN
ap-4612	85	12	)	)	PUNCT
ap-4612	85	13	.	.	PUNCT
ap-4612	86	1	(	(	PUNCT
ap-4612	86	2	9	9	X
ap-4612	86	3	)	)	PUNCT
ap-4612	86	4	if	if	SCONJ
ap-4612	86	5	|f	|f	PROPN
ap-4612	86	6	〉	〉	PROPN
ap-4612	86	7	,	,	PUNCT
ap-4612	86	8	|g	|g	PROPN
ap-4612	86	9	〉	〉	PROPN
ap-4612	86	10	∈	∈	PROPN
ap-4612	86	11	φ	φ	NOUN
ap-4612	86	12	,	,	PUNCT
ap-4612	86	13	then	then	ADV
ap-4612	86	14	f(φ	f(φ	PROPN
ap-4612	86	15	)	)	PUNCT
ap-4612	86	16	=	=	PUNCT
ap-4612	87	1	s|f	s|f	PROPN
ap-4612	87	2	〉	〉	PROPN
ap-4612	87	3	and	and	CCONJ
ap-4612	87	4	g(φ	g(φ	PROPN
ap-4612	87	5	)	)	PUNCT
ap-4612	87	6	=	=	SYM
ap-4612	87	7	s|g	s|g	PROPN
ap-4612	87	8	〉	〉	PROPN
ap-4612	87	9	belong	belong	VERB
ap-4612	87	10	to	to	ADP
ap-4612	87	11	(	(	PUNCT
ap-4612	87	12	sφ	sφ	PROPN
ap-4612	87	13	)	)	PUNCT
ap-4612	87	14	⊂	⊂	PROPN
ap-4612	87	15	l2[0	l2[0	PROPN
ap-4612	87	16	,	,	PUNCT
ap-4612	87	17	2π	2π	NOUN
ap-4612	87	18	]	]	PUNCT
ap-4612	87	19	.	.	PUNCT
ap-4612	88	1	thus	thus	ADV
ap-4612	88	2	,	,	PUNCT
ap-4612	88	3	and	and	CCONJ
ap-4612	88	4	due	due	ADP
ap-4612	88	5	to	to	ADP
ap-4612	88	6	the	the	DET
ap-4612	88	7	unitarity	unitarity	NOUN
ap-4612	88	8	of	of	ADP
ap-4612	88	9	s	s	PROPN
ap-4612	88	10	,	,	PUNCT
ap-4612	88	11	we	we	PRON
ap-4612	88	12	get	get	VERB
ap-4612	88	13	〈	〈	PROPN
ap-4612	88	14	f	f	PROPN
ap-4612	88	15	|g	|g	PROPN
ap-4612	89	1	〉	〉	PROPN
ap-4612	89	2	=	=	SYM
ap-4612	89	3	∫	∫	PROPN
ap-4612	89	4	2π	2π	NOUN
ap-4612	89	5	0	0	NUM
ap-4612	89	6	f∗(φ)g(φ	f∗(φ)g(φ	NOUN
ap-4612	89	7	)	)	PUNCT
ap-4612	89	8	dφ	dφ	ADP
ap-4612	89	9	=	=	PUNCT
ap-4612	89	10	∫	∫	PROPN
ap-4612	89	11	2π	2π	PROPN
ap-4612	89	12	0	0	PUNCT
ap-4612	90	1	〈	〈	PROPN
ap-4612	90	2	f	f	PROPN
ap-4612	90	3	|φ〉〈φ|g	|φ〉〈φ|g	PROPN
ap-4612	90	4	〉	〉	PROPN
ap-4612	90	5	dφ	dφ	ADP
ap-4612	90	6	,	,	PUNCT
ap-4612	90	7	(	(	PUNCT
ap-4612	90	8	10	10	NUM
ap-4612	90	9	)	)	PUNCT
ap-4612	90	10	and	and	CCONJ
ap-4612	90	11	thus	thus	ADV
ap-4612	90	12	i	i	PRON
ap-4612	90	13	=	=	SYM
ap-4612	90	14	∫	∫	PROPN
ap-4612	90	15	2π	2π	PROPN
ap-4612	90	16	0	0	NUM
ap-4612	90	17	|φ〉〈φ|	|φ〉〈φ|	NOUN
ap-4612	90	18	dφ	dφ	ADP
ap-4612	90	19	.	.	PUNCT
ap-4612	91	1	(	(	PUNCT
ap-4612	91	2	11	11	X
ap-4612	91	3	)	)	PUNCT
ap-4612	91	4	applying	apply	VERB
ap-4612	91	5	this	this	DET
ap-4612	91	6	identity	identity	NOUN
ap-4612	91	7	to	to	ADP
ap-4612	91	8	|f	|f	PROPN
ap-4612	91	9	〉	〉	PROPN
ap-4612	91	10	∈	∈	PROPN
ap-4612	91	11	φ	φ	NOUN
ap-4612	91	12	,	,	PUNCT
ap-4612	91	13	we	we	PRON
ap-4612	91	14	have	have	VERB
ap-4612	91	15	i|f	i|f	PROPN
ap-4612	92	1	〉	〉	PROPN
ap-4612	92	2	=	=	SYM
ap-4612	92	3	∫	∫	PROPN
ap-4612	92	4	2π	2π	NOUN
ap-4612	92	5	0	0	PUNCT
ap-4612	93	1	|φ〉〈φ|f	|φ〉〈φ|f	NOUN
ap-4612	93	2	〉	〉	NOUN
ap-4612	93	3	dφ	dφ	ADP
ap-4612	93	4	=	=	SYM
ap-4612	93	5	∫	∫	PROPN
ap-4612	93	6	2π	2π	PROPN
ap-4612	93	7	0	0	NUM
ap-4612	93	8	f(φ	f(φ	PROPN
ap-4612	93	9	)	)	PUNCT
ap-4612	93	10	|φ	|φ	NOUN
ap-4612	93	11	〉	〉	NOUN
ap-4612	93	12	dφ	dφ	ADP
ap-4612	93	13	.	.	PUNCT
ap-4612	94	1	(	(	PUNCT
ap-4612	94	2	12	12	NUM
ap-4612	94	3	)	)	PUNCT
ap-4612	94	4	this	this	PRON
ap-4612	94	5	gives	give	VERB
ap-4612	94	6	a	a	DET
ap-4612	94	7	span	span	NOUN
ap-4612	94	8	of	of	ADP
ap-4612	94	9	|f	|f	PROPN
ap-4612	94	10	〉	〉	PROPN
ap-4612	94	11	in	in	ADP
ap-4612	94	12	terms	term	NOUN
ap-4612	94	13	of	of	ADP
ap-4612	94	14	|φ	|φ	NOUN
ap-4612	94	15	〉	〉	NOUN
ap-4612	94	16	with	with	ADP
ap-4612	94	17	coefficients	coefficient	NOUN
ap-4612	94	18	f(φ	f(φ	PROPN
ap-4612	94	19	)	)	PUNCT
ap-4612	94	20	for	for	ADP
ap-4612	94	21	all	all	DET
ap-4612	94	22	φ	φ	PROPN
ap-4612	94	23	∈	∈	PROPN
ap-4612	95	1	[	[	X
ap-4612	95	2	0	0	NUM
ap-4612	95	3	,	,	PUNCT
ap-4612	95	4	2π	2π	NOUN
ap-4612	95	5	)	)	PUNCT
ap-4612	95	6	,	,	PUNCT
ap-4612	95	7	which	which	PRON
ap-4612	95	8	shows	show	VERB
ap-4612	95	9	that	that	SCONJ
ap-4612	95	10	{	{	PUNCT
ap-4612	95	11	|φ	|φ	NUM
ap-4612	95	12	〉	〉	NOUN
ap-4612	95	13	}	}	PUNCT
ap-4612	95	14	is	be	AUX
ap-4612	95	15	a	a	DET
ap-4612	95	16	continuous	continuous	ADJ
ap-4612	95	17	basis	basis	NOUN
ap-4612	95	18	on	on	ADP
ap-4612	95	19	φ	φ	NUM
ap-4612	95	20	,	,	PUNCT
ap-4612	95	21	although	although	SCONJ
ap-4612	95	22	its	its	PRON
ap-4612	95	23	elements	element	NOUN
ap-4612	95	24	are	be	AUX
ap-4612	95	25	not	not	PART
ap-4612	95	26	in	in	ADP
ap-4612	95	27	φ	φ	PROPN
ap-4612	95	28	but	but	CCONJ
ap-4612	95	29	instead	instead	ADV
ap-4612	95	30	in	in	ADP
ap-4612	95	31	φ×.	φ×.	NUM
ap-4612	95	32	since	since	SCONJ
ap-4612	95	33	〈	〈	PROPN
ap-4612	95	34	φ|	φ|	PROPN
ap-4612	95	35	acts	act	NOUN
ap-4612	95	36	on	on	ADP
ap-4612	95	37	φ	φ	PROPN
ap-4612	95	38	only	only	ADV
ap-4612	95	39	(	(	PUNCT
ap-4612	95	40	not	not	PART
ap-4612	95	41	on	on	ADP
ap-4612	95	42	all	all	DET
ap-4612	95	43	h	h	NOUN
ap-4612	95	44	)	)	PUNCT
ap-4612	95	45	,	,	PUNCT
ap-4612	95	46	then	then	ADV
ap-4612	95	47	for	for	ADP
ap-4612	95	48	an	an	DET
ap-4612	95	49	arbitrary	arbitrary	ADJ
ap-4612	95	50	|g	|g	PROPN
ap-4612	95	51	〉	〉	PROPN
ap-4612	95	52	∈	∈	PROPN
ap-4612	95	53	φ	φ	NOUN
ap-4612	95	54	we	we	PRON
ap-4612	95	55	have	have	VERB
ap-4612	95	56	〈	〈	PROPN
ap-4612	95	57	g|if	g|if	NOUN
ap-4612	96	1	〉	〉	NOUN
ap-4612	96	2	=	=	SYM
ap-4612	96	3	∫	∫	PROPN
ap-4612	96	4	2π	2π	NOUN
ap-4612	96	5	0	0	PUNCT
ap-4612	97	1	〈	〈	NOUN
ap-4612	97	2	g|φ〉〈φ|f	g|φ〉〈φ|f	NOUN
ap-4612	97	3	〉	〉	NOUN
ap-4612	97	4	dφ	dφ	ADP
ap-4612	97	5	=	=	SYM
ap-4612	97	6	〈	〈	PROPN
ap-4612	97	7	g|f	g|f	NOUN
ap-4612	97	8	〉	〉	PROPN
ap-4612	97	9	.	.	PUNCT
ap-4612	98	1	because	because	SCONJ
ap-4612	98	2	of	of	ADP
ap-4612	98	3	the	the	DET
ap-4612	98	4	definition	definition	NOUN
ap-4612	98	5	of	of	ADP
ap-4612	98	6	rhs	rhs	PROPN
ap-4612	98	7	to	to	ADP
ap-4612	98	8	any	any	DET
ap-4612	98	9	|f	|f	PROPN
ap-4612	98	10	〉	〉	PROPN
ap-4612	98	11	∈	∈	PROPN
ap-4612	98	12	φ	φ	PROPN
ap-4612	98	13	corresponds	correspond	VERB
ap-4612	98	14	a	a	DET
ap-4612	98	15	〈	〈	PROPN
ap-4612	98	16	f	f	X
ap-4612	98	17	|	|	NOUN
ap-4612	98	18	∈	∈	PROPN
ap-4612	98	19	φ×	φ×	NOUN
ap-4612	98	20	and	and	CCONJ
ap-4612	98	21	the	the	DET
ap-4612	98	22	action	action	NOUN
ap-4612	98	23	of	of	ADP
ap-4612	98	24	〈	〈	PROPN
ap-4612	98	25	f	f	PROPN
ap-4612	98	26	|	|	ADV
ap-4612	98	27	on	on	ADP
ap-4612	98	28	any	any	DET
ap-4612	98	29	|g	|g	PROPN
ap-4612	98	30	〉	〉	PROPN
ap-4612	98	31	∈	∈	PROPN
ap-4612	98	32	φ	φ	PROPN
ap-4612	98	33	is	be	AUX
ap-4612	98	34	given	give	VERB
ap-4612	98	35	by	by	ADP
ap-4612	98	36	the	the	DET
ap-4612	98	37	scalar	scalar	ADJ
ap-4612	98	38	product	product	NOUN
ap-4612	98	39	〈	〈	PROPN
ap-4612	98	40	f	f	PROPN
ap-4612	98	41	|g	|g	PROPN
ap-4612	98	42	〉	〉	PROPN
ap-4612	98	43	(	(	PUNCT
ap-4612	98	44	10	10	NUM
ap-4612	98	45	)	)	PUNCT
ap-4612	98	46	from	from	ADP
ap-4612	98	47	l2[0	l2[0	PROPN
ap-4612	98	48	,	,	PUNCT
ap-4612	98	49	2π	2π	NOUN
ap-4612	98	50	]	]	PUNCT
ap-4612	98	51	.	.	PUNCT
ap-4612	99	1	thus	thus	ADV
ap-4612	99	2	,	,	PUNCT
ap-4612	99	3	i	i	PRON
ap-4612	99	4	is	be	AUX
ap-4612	99	5	the	the	DET
ap-4612	99	6	canonical	canonical	ADJ
ap-4612	99	7	injection	injection	NOUN
ap-4612	100	1	i	i	PRON
ap-4612	100	2	:	:	PUNCT
ap-4612	100	3	φ	φ	PROPN
ap-4612	100	4	7−→	7−→	NOUN
ap-4612	100	5	φ×	φ×	NOUN
ap-4612	100	6	,	,	PUNCT
ap-4612	100	7	and	and	CCONJ
ap-4612	100	8	it	it	PRON
ap-4612	100	9	is	be	AUX
ap-4612	100	10	continuous	continuous	ADJ
ap-4612	100	11	.	.	PUNCT
ap-4612	101	1	consequently	consequently	ADV
ap-4612	101	2	,	,	PUNCT
ap-4612	101	3	f(φ	f(φ	PROPN
ap-4612	101	4	)	)	PUNCT
ap-4612	101	5	=	=	PUNCT
ap-4612	101	6	〈	〈	PROPN
ap-4612	101	7	φ|f	φ|f	NOUN
ap-4612	101	8	〉	〉	NOUN
ap-4612	101	9	=	=	SYM
ap-4612	101	10	∫	∫	PROPN
ap-4612	101	11	2π	2π	NOUN
ap-4612	101	12	0	0	PUNCT
ap-4612	102	1	〈	〈	NOUN
ap-4612	102	2	φ|φ′〉〈φ′|f	φ|φ′〉〈φ′|f	ADJ
ap-4612	102	3	〉	〉	PROPN
ap-4612	102	4	dφ′	dφ′	NOUN
ap-4612	102	5	,	,	PUNCT
ap-4612	102	6	and	and	CCONJ
ap-4612	102	7	then	then	ADV
ap-4612	102	8	,	,	PUNCT
ap-4612	102	9	〈	〈	PROPN
ap-4612	102	10	φ|φ′	φ|φ′	PROPN
ap-4612	102	11	〉	〉	NOUN
ap-4612	102	12	=	=	SYM
ap-4612	102	13	δ(φ−	δ(φ−	NOUN
ap-4612	102	14	φ′	φ′	NUM
ap-4612	102	15	)	)	PUNCT
ap-4612	102	16	.	.	PUNCT
ap-4612	103	1	(	(	PUNCT
ap-4612	103	2	13	13	NUM
ap-4612	103	3	)	)	PUNCT
ap-4612	103	4	therefore	therefore	ADV
ap-4612	103	5	,	,	PUNCT
ap-4612	103	6	the	the	DET
ap-4612	103	7	set	set	NOUN
ap-4612	103	8	{	{	PUNCT
ap-4612	103	9	|φ	|φ	PROPN
ap-4612	103	10	〉	〉	NOUN
ap-4612	103	11	}	}	PUNCT
ap-4612	103	12	satisfies	satisfy	VERB
ap-4612	103	13	the	the	DET
ap-4612	103	14	relations	relation	NOUN
ap-4612	103	15	of	of	ADP
ap-4612	103	16	orthogonality	orthogonality	NOUN
ap-4612	103	17	(	(	PUNCT
ap-4612	103	18	13	13	NUM
ap-4612	103	19	)	)	PUNCT
ap-4612	103	20	and	and	CCONJ
ap-4612	103	21	completeness	completeness	NOUN
ap-4612	103	22	(	(	PUNCT
ap-4612	103	23	11	11	NUM
ap-4612	103	24	)	)	PUNCT
ap-4612	103	25	that	that	PRON
ap-4612	103	26	allow	allow	VERB
ap-4612	103	27	us	we	PRON
ap-4612	103	28	,	,	PUNCT
ap-4612	103	29	once	once	ADV
ap-4612	103	30	more	more	ADV
ap-4612	103	31	,	,	PUNCT
ap-4612	103	32	to	to	PART
ap-4612	103	33	write	write	VERB
ap-4612	103	34	(	(	PUNCT
ap-4612	103	35	12	12	NUM
ap-4612	103	36	)	)	PUNCT
ap-4612	103	37	and	and	CCONJ
ap-4612	103	38	to	to	PART
ap-4612	103	39	show	show	VERB
ap-4612	103	40	that	that	SCONJ
ap-4612	103	41	{	{	PUNCT
ap-4612	103	42	|φ	|φ	NUM
ap-4612	103	43	〉	〉	NOUN
ap-4612	103	44	}	}	PUNCT
ap-4612	103	45	is	be	AUX
ap-4612	103	46	a	a	DET
ap-4612	103	47	continuous	continuous	ADJ
ap-4612	103	48	basis	basis	NOUN
ap-4612	103	49	.	.	PUNCT
ap-4612	104	1	however	however	ADV
ap-4612	104	2	the	the	DET
ap-4612	104	3	above	above	ADJ
ap-4612	104	4	formulae	formulae	NOUN
ap-4612	104	5	are	be	AUX
ap-4612	104	6	not	not	PART
ap-4612	104	7	rigurously	rigurously	ADV
ap-4612	104	8	correct	correct	ADJ
ap-4612	104	9	for	for	ADP
ap-4612	104	10	elements	element	NOUN
ap-4612	104	11	of	of	ADP
ap-4612	104	12	h.	h.	PROPN
ap-4612	104	13	effectively	effectively	ADV
ap-4612	104	14	,	,	PUNCT
ap-4612	104	15	formula	formula	NOUN
ap-4612	104	16	(	(	PUNCT
ap-4612	104	17	10	10	NUM
ap-4612	104	18	)	)	PUNCT
ap-4612	104	19	,	,	PUNCT
ap-4612	104	20	and	and	CCONJ
ap-4612	104	21	hence	hence	ADV
ap-4612	104	22	all	all	PRON
ap-4612	104	23	formulae	formulae	ADJ
ap-4612	104	24	derived	derive	VERB
ap-4612	104	25	thereof	thereof	ADV
ap-4612	104	26	including	include	VERB
ap-4612	104	27	(	(	PUNCT
ap-4612	104	28	12	12	NUM
ap-4612	104	29	)	)	PUNCT
ap-4612	104	30	,	,	PUNCT
ap-4612	104	31	is	be	AUX
ap-4612	104	32	a	a	DET
ap-4612	104	33	consequence	consequence	NOUN
ap-4612	104	34	of	of	ADP
ap-4612	104	35	the	the	DET
ap-4612	104	36	gelfand	gelfand	ADJ
ap-4612	104	37	–	–	PUNCT
ap-4612	104	38	maurin	maurin	NOUN
ap-4612	104	39	theorem	theorem	NOUN
ap-4612	104	40	[	[	X
ap-4612	104	41	4	4	NUM
ap-4612	104	42	,	,	PUNCT
ap-4612	104	43	16	16	NUM
ap-4612	104	44	]	]	PUNCT
ap-4612	104	45	.	.	PUNCT
ap-4612	105	1	since	since	SCONJ
ap-4612	105	2	this	this	DET
ap-4612	105	3	theorem	theorem	NOUN
ap-4612	105	4	is	be	AUX
ap-4612	105	5	only	only	ADV
ap-4612	105	6	valid	valid	ADJ
ap-4612	105	7	for	for	ADP
ap-4612	105	8	|f	|f	PROPN
ap-4612	105	9	〉	〉	PROPN
ap-4612	105	10	,	,	PUNCT
ap-4612	105	11	|g	|g	PROPN
ap-4612	105	12	〉	〉	PROPN
ap-4612	105	13	∈	∈	PROPN
ap-4612	105	14	φ	φ	NOUN
ap-4612	105	15	,	,	PUNCT
ap-4612	105	16	we	we	PRON
ap-4612	105	17	conclude	conclude	VERB
ap-4612	105	18	that	that	SCONJ
ap-4612	105	19	(	(	PUNCT
ap-4612	105	20	12	12	NUM
ap-4612	105	21	)	)	PUNCT
ap-4612	105	22	is	be	AUX
ap-4612	105	23	only	only	ADV
ap-4612	105	24	valid	valid	ADJ
ap-4612	105	25	for	for	ADP
ap-4612	105	26	|f	|f	PROPN
ap-4612	105	27	〉	〉	PROPN
ap-4612	105	28	∈	∈	PROPN
ap-4612	105	29	φ	φ	NOUN
ap-4612	105	30	from	from	ADP
ap-4612	105	31	a	a	DET
ap-4612	105	32	strictly	strictly	ADV
ap-4612	105	33	rigorous	rigorous	ADJ
ap-4612	105	34	point	point	NOUN
ap-4612	105	35	of	of	ADP
ap-4612	105	36	view	view	NOUN
ap-4612	105	37	.	.	PUNCT
ap-4612	106	1	however	however	ADV
ap-4612	106	2	,	,	PUNCT
ap-4612	106	3	we	we	PRON
ap-4612	106	4	may	may	AUX
ap-4612	106	5	write	write	VERB
ap-4612	106	6	formal	formal	ADJ
ap-4612	106	7	expressions	expression	NOUN
ap-4612	106	8	like	like	ADP
ap-4612	106	9	|h	|h	PRON
ap-4612	107	1	〉	〉	NOUN
ap-4612	107	2	=	=	SYM
ap-4612	107	3	∫	∫	PROPN
ap-4612	107	4	2π	2π	PROPN
ap-4612	107	5	0	0	NUM
ap-4612	108	1	φ	φ	NUM
ap-4612	108	2	|φ	|φ	PROPN
ap-4612	108	3	〉	〉	PROPN
ap-4612	108	4	dφ	dφ	ADP
ap-4612	109	1	,	,	PUNCT
ap-4612	109	2	381	381	NUM
ap-4612	109	3	enrico	enrico	PROPN
ap-4612	109	4	celeghini	celeghini	PROPN
ap-4612	109	5	,	,	PUNCT
ap-4612	109	6	manuel	manuel	PROPN
ap-4612	109	7	gadella	gadella	PROPN
ap-4612	109	8	,	,	PUNCT
ap-4612	109	9	mariano	mariano	PROPN
ap-4612	109	10	a	a	DET
ap-4612	109	11	del	del	PROPN
ap-4612	109	12	olmo	olmo	PROPN
ap-4612	109	13	acta	acta	PROPN
ap-4612	109	14	polytechnica	polytechnica	PROPN
ap-4612	109	15	for	for	ADP
ap-4612	109	16	the	the	DET
ap-4612	109	17	function	function	NOUN
ap-4612	109	18	h(φ	h(φ	PROPN
ap-4612	109	19	)	)	PUNCT
ap-4612	109	20	=	=	PUNCT
ap-4612	109	21	φ	φ	PROPN
ap-4612	109	22	∈	∈	PROPN
ap-4612	109	23	l2[0	l2[0	PROPN
ap-4612	109	24	,	,	PUNCT
ap-4612	109	25	2π	2π	NOUN
ap-4612	109	26	]	]	X
ap-4612	109	27	,	,	PUNCT
ap-4612	109	28	i.e.	i.e.	X
ap-4612	109	29	|h	|h	VERB
ap-4612	109	30	〉	〉	PROPN
ap-4612	109	31	∈	∈	PROPN
ap-4612	109	32	h.	h.	NOUN
ap-4612	110	1	but	but	CCONJ
ap-4612	110	2	this	this	DET
ap-4612	110	3	expression	expression	NOUN
ap-4612	110	4	is	be	AUX
ap-4612	110	5	meaningless	meaningless	ADJ
ap-4612	110	6	from	from	ADP
ap-4612	110	7	the	the	DET
ap-4612	110	8	point	point	NOUN
ap-4612	110	9	of	of	ADP
ap-4612	110	10	view	view	NOUN
ap-4612	110	11	of	of	ADP
ap-4612	110	12	the	the	DET
ap-4612	110	13	gelfand	gelfand	NOUN
ap-4612	110	14	-	-	PUNCT
ap-4612	110	15	maurin	maurin	NOUN
ap-4612	110	16	theorem	theorem	NOUN
ap-4612	110	17	,	,	PUNCT
ap-4612	110	18	since	since	SCONJ
ap-4612	110	19	|h	|h	PROPN
ap-4612	110	20	〉	〉	PROPN
ap-4612	110	21	/∈	/∈	PUNCT
ap-4612	111	1	φ	φ	PROPN
ap-4612	111	2	,	,	PUNCT
ap-4612	111	3	as	as	SCONJ
ap-4612	111	4	one	one	NUM
ap-4612	111	5	easily	easily	ADV
ap-4612	111	6	checks	check	VERB
ap-4612	111	7	in	in	ADP
ap-4612	111	8	the	the	DET
ap-4612	111	9	following	following	NOUN
ap-4612	111	10	.	.	PUNCT
ap-4612	112	1	taking	take	VERB
ap-4612	112	2	into	into	ADP
ap-4612	112	3	account	account	NOUN
ap-4612	112	4	formulae	formulae	NOUN
ap-4612	112	5	(	(	PUNCT
ap-4612	112	6	4	4	NUM
ap-4612	112	7	)	)	PUNCT
ap-4612	112	8	and	and	CCONJ
ap-4612	112	9	(	(	PUNCT
ap-4612	112	10	5	5	X
ap-4612	112	11	)	)	PUNCT
ap-4612	112	12	we	we	PRON
ap-4612	112	13	have	have	VERB
ap-4612	112	14	hm	hm	NOUN
ap-4612	112	15	=	=	SYM
ap-4612	112	16	1√	1√	PROPN
ap-4612	112	17	2π	2π	NUM
ap-4612	112	18	∫	∫	PROPN
ap-4612	112	19	2π	2π	PROPN
ap-4612	112	20	0	0	PUNCT
ap-4612	112	21	φeimφ	φeimφ	NOUN
ap-4612	112	22	dφ	dφ	ADP
ap-4612	112	23	=	=	PUNCT
ap-4612	112	24			PUNCT
ap-4612	112	25	i√	i√	PROPN
ap-4612	112	26	2π	2π	NOUN
ap-4612	112	27	1	1	NUM
ap-4612	112	28	m	m	NOUN
ap-4612	112	29	,	,	PUNCT
ap-4612	112	30	m	m	VERB
ap-4612	112	31	6=	6=	NUM
ap-4612	112	32	0	0	NUM
ap-4612	112	33	,	,	PUNCT
ap-4612	112	34	√	√	ADP
ap-4612	112	35	2π	2π	NOUN
ap-4612	112	36	,	,	PUNCT
ap-4612	112	37	m	m	VERB
ap-4612	112	38	=	=	NOUN
ap-4612	112	39	0	0	NUM
ap-4612	112	40	,	,	PUNCT
ap-4612	112	41	then	then	ADV
ap-4612	112	42	,	,	PUNCT
ap-4612	112	43	∑	∑	PROPN
ap-4612	112	44	m	m	PROPN
ap-4612	112	45	|hm|2	|hm|2	PUNCT
ap-4612	112	46	=	=	NOUN
ap-4612	112	47	13π	13π	NUM
ap-4612	112	48	6	6	NUM
ap-4612	112	49	<	<	X
ap-4612	112	50	∞.	∞.	PROPN
ap-4612	112	51	however	however	ADV
ap-4612	112	52	,	,	PUNCT
ap-4612	112	53	∑	∑	PROPN
ap-4612	112	54	m	m	VERB
ap-4612	112	55	|hm|2|m+	|hm|2|m+	VERB
ap-4612	112	56	i|2p	i|2p	ADJ
ap-4612	112	57	diverges	diverge	VERB
ap-4612	112	58	for	for	ADP
ap-4612	112	59	p	p	PRON
ap-4612	112	60	≥	≥	NOUN
ap-4612	112	61	1	1	NUM
ap-4612	112	62	.	.	PUNCT
ap-4612	113	1	this	this	PRON
ap-4612	113	2	proves	prove	VERB
ap-4612	113	3	that	that	SCONJ
ap-4612	113	4	h(φ	h(φ	ADJ
ap-4612	113	5	)	)	PUNCT
ap-4612	113	6	≡	≡	PROPN
ap-4612	113	7	φ	φ	PROPN
ap-4612	113	8	is	be	AUX
ap-4612	113	9	not	not	PART
ap-4612	113	10	in	in	ADP
ap-4612	113	11	sφ	sφ	PROPN
ap-4612	113	12	and	and	CCONJ
ap-4612	113	13	,	,	PUNCT
ap-4612	113	14	hence	hence	ADV
ap-4612	113	15	,	,	PUNCT
ap-4612	113	16	|h	|h	PROPN
ap-4612	113	17	〉	〉	PROPN
ap-4612	113	18	/∈	/∈	PUNCT
ap-4612	114	1	φ	φ	X
ap-4612	114	2	.	.	PUNCT
ap-4612	115	1	there	there	PRON
ap-4612	115	2	are	be	VERB
ap-4612	115	3	some	some	DET
ap-4612	115	4	formal	formal	ADJ
ap-4612	115	5	relations	relation	NOUN
ap-4612	115	6	between	between	ADP
ap-4612	115	7	both	both	DET
ap-4612	115	8	bases	basis	NOUN
ap-4612	115	9	,	,	PUNCT
ap-4612	115	10	{	{	PUNCT
ap-4612	115	11	|m	|m	NOUN
ap-4612	115	12	〉	〉	NOUN
ap-4612	115	13	}	}	PUNCT
ap-4612	115	14	and	and	CCONJ
ap-4612	115	15	{	{	PUNCT
ap-4612	115	16	|φ	|φ	NOUN
ap-4612	115	17	〉	〉	NOUN
ap-4612	115	18	}	}	PUNCT
ap-4612	115	19	.	.	PUNCT
ap-4612	116	1	for	for	ADP
ap-4612	116	2	instance	instance	NOUN
ap-4612	116	3	,	,	PUNCT
ap-4612	116	4	replacing	replace	VERB
ap-4612	116	5	|f	|f	PRON
ap-4612	116	6	〉	〉	NUM
ap-4612	116	7	by	by	ADP
ap-4612	116	8	|m	|m	NOUN
ap-4612	116	9	〉	〉	PROPN
ap-4612	116	10	in	in	ADP
ap-4612	116	11	(	(	PUNCT
ap-4612	116	12	12	12	NUM
ap-4612	116	13	)	)	PUNCT
ap-4612	116	14	we	we	PRON
ap-4612	116	15	have	have	VERB
ap-4612	116	16	that	that	SCONJ
ap-4612	116	17	|m	|m	NOUN
ap-4612	117	1	〉	〉	PROPN
ap-4612	117	2	=	=	SYM
ap-4612	117	3	∫	∫	PROPN
ap-4612	117	4	2π	2π	NOUN
ap-4612	117	5	0	0	PUNCT
ap-4612	118	1	〈	〈	NOUN
ap-4612	118	2	φ|m	φ|m	NOUN
ap-4612	118	3	〉	〉	PROPN
ap-4612	118	4	|φ	|φ	NOUN
ap-4612	118	5	〉	〉	PROPN
ap-4612	118	6	dφ	dφ	ADP
ap-4612	118	7	=	=	SYM
ap-4612	118	8	1√	1√	PROPN
ap-4612	118	9	2π	2π	NUM
ap-4612	118	10	∫	∫	PROPN
ap-4612	118	11	2π	2π	PROPN
ap-4612	118	12	0	0	NUM
ap-4612	118	13	e−imφ	e−imφ	NOUN
ap-4612	118	14	|φ	|φ	NOUN
ap-4612	118	15	〉	〉	NOUN
ap-4612	118	16	dφ	dφ	ADP
ap-4612	118	17	.	.	PUNCT
ap-4612	119	1	since	since	SCONJ
ap-4612	119	2	{	{	PUNCT
ap-4612	119	3	|m	|m	NOUN
ap-4612	119	4	〉	〉	NOUN
ap-4612	119	5	}	}	PUNCT
ap-4612	119	6	is	be	AUX
ap-4612	119	7	a	a	DET
ap-4612	119	8	basis	basis	NOUN
ap-4612	119	9	in	in	ADP
ap-4612	119	10	h	h	NOUN
ap-4612	119	11	,	,	PUNCT
ap-4612	119	12	the	the	DET
ap-4612	119	13	following	follow	VERB
ap-4612	119	14	completeness	completeness	NOUN
ap-4612	119	15	relation	relation	NOUN
ap-4612	119	16	holds	hold	VERB
ap-4612	119	17	∞∑	∞∑	NOUN
ap-4612	119	18	m=−∞	m=−∞	NOUN
ap-4612	119	19	|m〉〈m|	|m〉〈m|	X
ap-4612	120	1	=	=	PUNCT
ap-4612	120	2	i	i	PROPN
ap-4612	120	3	,	,	PUNCT
ap-4612	120	4	(	(	PUNCT
ap-4612	120	5	14	14	NUM
ap-4612	120	6	)	)	PUNCT
ap-4612	120	7	where	where	SCONJ
ap-4612	120	8	i	i	PRON
ap-4612	120	9	is	be	AUX
ap-4612	120	10	the	the	DET
ap-4612	120	11	identity	identity	NOUN
ap-4612	120	12	on	on	ADP
ap-4612	120	13	h	h	NOUN
ap-4612	120	14	(	(	PUNCT
ap-4612	120	15	and	and	CCONJ
ap-4612	120	16	also	also	ADV
ap-4612	120	17	on	on	ADP
ap-4612	120	18	φ	φ	NUM
ap-4612	120	19	)	)	PUNCT
ap-4612	120	20	.	.	PUNCT
ap-4612	121	1	do	do	AUX
ap-4612	121	2	not	not	PART
ap-4612	121	3	confuse	confuse	VERB
ap-4612	121	4	this	this	DET
ap-4612	121	5	identity	identity	NOUN
ap-4612	121	6	with	with	ADP
ap-4612	121	7	i	i	PRON
ap-4612	121	8	previously	previously	ADV
ap-4612	121	9	defined	define	VERB
ap-4612	121	10	(	(	PUNCT
ap-4612	121	11	11	11	NUM
ap-4612	121	12	)	)	PUNCT
ap-4612	121	13	that	that	PRON
ap-4612	121	14	is	be	AUX
ap-4612	121	15	the	the	DET
ap-4612	121	16	canonical	canonical	ADJ
ap-4612	121	17	injection	injection	NOUN
ap-4612	121	18	from	from	ADP
ap-4612	121	19	φ	φ	PROPN
ap-4612	121	20	to	to	ADP
ap-4612	121	21	φ×.	φ×.	NOUN
ap-4612	121	22	because	because	SCONJ
ap-4612	121	23	|m	|m	NOUN
ap-4612	121	24	〉	〉	PROPN
ap-4612	121	25	∈	∈	PROPN
ap-4612	121	26	φ	φ	NOUN
ap-4612	121	27	,	,	PUNCT
ap-4612	121	28	we	we	PRON
ap-4612	121	29	may	may	AUX
ap-4612	121	30	apply	apply	VERB
ap-4612	121	31	to	to	ADP
ap-4612	121	32	it	it	PRON
ap-4612	121	33	any	any	DET
ap-4612	121	34	element	element	NOUN
ap-4612	121	35	of	of	ADP
ap-4612	121	36	φ×	φ×	NOUN
ap-4612	121	37	so	so	SCONJ
ap-4612	121	38	that	that	SCONJ
ap-4612	121	39	i	i	PRON
ap-4612	121	40	becomes	become	VERB
ap-4612	121	41	a	a	DET
ap-4612	121	42	well	well	ADV
ap-4612	121	43	defined	define	VERB
ap-4612	121	44	identity	identity	NOUN
ap-4612	121	45	on	on	ADP
ap-4612	121	46	the	the	DET
ap-4612	121	47	dual	dual	ADJ
ap-4612	121	48	φ×	φ×	NOUN
ap-4612	121	49	〈	〈	PROPN
ap-4612	121	50	φ|	φ|	PROPN
ap-4612	121	51	i	i	NOUN
ap-4612	122	1	=	=	PUNCT
ap-4612	122	2	〈	〈	PROPN
ap-4612	122	3	φ|=	φ|=	NOUN
ap-4612	122	4	∞∑	∞∑	NUM
ap-4612	123	1	m=−∞	m=−∞	ADP
ap-4612	123	2	〈	〈	PROPN
ap-4612	123	3	φ|m〉〈m|=	φ|m〉〈m|=	X
ap-4612	123	4	1√	1√	ADJ
ap-4612	123	5	2π	2π	NOUN
ap-4612	123	6	∞∑	∞∑	NUM
ap-4612	123	7	m=−∞	m=−∞	NOUN
ap-4612	123	8	e−imφ	e−imφ	NOUN
ap-4612	123	9	〈	〈	PROPN
ap-4612	123	10	m|	m|	PROPN
ap-4612	123	11	,	,	PUNCT
ap-4612	123	12	which	which	PRON
ap-4612	123	13	gives	give	VERB
ap-4612	123	14	the	the	DET
ap-4612	123	15	second	second	ADJ
ap-4612	123	16	formal	formal	ADJ
ap-4612	123	17	identity	identity	NOUN
ap-4612	123	18	(	(	PUNCT
ap-4612	123	19	14	14	NUM
ap-4612	123	20	)	)	PUNCT
ap-4612	123	21	.	.	PUNCT
ap-4612	124	1	nevertheless	nevertheless	ADV
ap-4612	124	2	and	and	CCONJ
ap-4612	124	3	due	due	ADP
ap-4612	124	4	to	to	ADP
ap-4612	124	5	the	the	DET
ap-4612	124	6	absolute	absolute	ADJ
ap-4612	124	7	convergence	convergence	NOUN
ap-4612	124	8	of	of	ADP
ap-4612	124	9	the	the	DET
ap-4612	124	10	series	series	NOUN
ap-4612	124	11	〈	〈	PROPN
ap-4612	124	12	φ|f	φ|f	NOUN
ap-4612	124	13	〉	〉	NOUN
ap-4612	124	14	=	=	NOUN
ap-4612	125	1	∞∑	∞∑	NOUN
ap-4612	125	2	m=−∞	m=−∞	AUX
ap-4612	125	3	am	be	AUX
ap-4612	125	4	〈	〈	NOUN
ap-4612	125	5	φ|m	φ|m	NOUN
ap-4612	125	6	〉	〉	NOUN
ap-4612	125	7	=	=	SYM
ap-4612	125	8	1√	1√	PROPN
ap-4612	125	9	2π	2π	NOUN
ap-4612	125	10	∞∑	∞∑	NUM
ap-4612	125	11	m=−∞	m=−∞	ADP
ap-4612	125	12	ame	ame	PROPN
ap-4612	125	13	−imφ	−imφ	PROPN
ap-4612	125	14	,	,	PUNCT
ap-4612	125	15	it	it	PRON
ap-4612	125	16	is	be	AUX
ap-4612	125	17	easy	easy	ADJ
ap-4612	125	18	to	to	PART
ap-4612	125	19	prove	prove	VERB
ap-4612	125	20	that	that	SCONJ
ap-4612	125	21	〈	〈	PROPN
ap-4612	125	22	φ|	φ|	PROPN
ap-4612	125	23	i	i	PRON
ap-4612	125	24	converges	converge	VERB
ap-4612	125	25	in	in	ADP
ap-4612	125	26	the	the	DET
ap-4612	125	27	weak	weak	ADJ
ap-4612	125	28	topology	topology	NOUN
ap-4612	125	29	on	on	ADP
ap-4612	125	30	φ×.	φ×.	X
ap-4612	125	31	3.3	3.3	NUM
ap-4612	125	32	.	.	PUNCT
ap-4612	126	1	action	action	NOUN
ap-4612	126	2	of	of	ADP
ap-4612	126	3	so(2	so(2	NOUN
ap-4612	126	4	)	)	PUNCT
ap-4612	126	5	on	on	ADP
ap-4612	126	6	the	the	DET
ap-4612	126	7	rhs	rhs	PROPN
ap-4612	126	8	the	the	DET
ap-4612	126	9	hilbert	hilbert	PROPN
ap-4612	126	10	space	space	PROPN
ap-4612	126	11	l2[0	l2[0	PROPN
ap-4612	126	12	,	,	PUNCT
ap-4612	126	13	2π	2π	NOUN
ap-4612	126	14	]	]	PUNCT
ap-4612	126	15	also	also	ADV
ap-4612	126	16	supports	support	VERB
ap-4612	126	17	the	the	DET
ap-4612	126	18	regular	regular	ADJ
ap-4612	126	19	representation	representation	NOUN
ap-4612	126	20	of	of	ADP
ap-4612	126	21	so(2	so(2	NOUN
ap-4612	126	22	)	)	PUNCT
ap-4612	126	23	,	,	PUNCT
ap-4612	126	24	r(θ	r(θ	NOUN
ap-4612	126	25	)	)	PUNCT
ap-4612	126	26	,	,	PUNCT
ap-4612	126	27	defined	define	VERB
ap-4612	126	28	by	by	ADP
ap-4612	126	29	[	[	X
ap-4612	126	30	r(θ)f	r(θ)f	NOUN
ap-4612	126	31	]	]	X
ap-4612	126	32	(	(	PUNCT
ap-4612	126	33	φ	φ	NOUN
ap-4612	126	34	)	)	PUNCT
ap-4612	126	35	:	:	PUNCT
ap-4612	127	1	=	=	NUM
ap-4612	127	2	f(φ−	f(φ−	PROPN
ap-4612	127	3	θ	θ	NOUN
ap-4612	127	4	)	)	PUNCT
ap-4612	127	5	(	(	PUNCT
ap-4612	127	6	mod	mod	ADJ
ap-4612	127	7	2π	2π	NOUN
ap-4612	127	8	)	)	PUNCT
ap-4612	127	9	,	,	PUNCT
ap-4612	127	10	∀f	∀f	PROPN
ap-4612	127	11	∈	∈	PROPN
ap-4612	127	12	l2[0	l2[0	PROPN
ap-4612	127	13	,	,	PUNCT
ap-4612	127	14	2π	2π	NOUN
ap-4612	127	15	]	]	PUNCT
ap-4612	127	16	,	,	PUNCT
ap-4612	127	17	for	for	ADP
ap-4612	127	18	any	any	DET
ap-4612	127	19	θ	θ	PROPN
ap-4612	127	20	∈	∈	PROPN
ap-4612	128	1	[	[	X
ap-4612	128	2	0	0	NUM
ap-4612	128	3	,	,	PUNCT
ap-4612	128	4	2π	2π	NOUN
ap-4612	128	5	)	)	PUNCT
ap-4612	128	6	.	.	PUNCT
ap-4612	129	1	the	the	DET
ap-4612	129	2	unitary	unitary	ADJ
ap-4612	129	3	map	map	NOUN
ap-4612	129	4	s	s	PART
ap-4612	129	5	:	:	PUNCT
ap-4612	129	6	h	h	PROPN
ap-4612	129	7	7−→	7−→	PROPN
ap-4612	129	8	l2[0	l2[0	PROPN
ap-4612	129	9	,	,	PUNCT
ap-4612	129	10	2π	2π	NOUN
ap-4612	129	11	]	]	PUNCT
ap-4612	129	12	also	also	ADV
ap-4612	129	13	allows	allow	VERB
ap-4612	129	14	us	we	PRON
ap-4612	129	15	to	to	PART
ap-4612	129	16	transport	transport	VERB
ap-4612	129	17	r	r	NOUN
ap-4612	129	18	to	to	ADP
ap-4612	129	19	an	an	DET
ap-4612	129	20	equivalent	equivalent	ADJ
ap-4612	129	21	representation	representation	NOUN
ap-4612	129	22	r	r	NOUN
ap-4612	129	23	supported	support	VERB
ap-4612	129	24	on	on	ADP
ap-4612	129	25	h	h	NOUN
ap-4612	129	26	by	by	ADP
ap-4612	129	27	r(θ	r(θ	NOUN
ap-4612	129	28	)	)	PUNCT
ap-4612	129	29	=	=	SYM
ap-4612	129	30	s−1r(θ)s	s−1r(θ)s	ADJ
ap-4612	129	31	,	,	PUNCT
ap-4612	129	32	such	such	ADJ
ap-4612	129	33	that	that	DET
ap-4612	129	34	r(θ)φ	r(θ)φ	PROPN
ap-4612	129	35	=	=	SYM
ap-4612	129	36	φ	φ	PROPN
ap-4612	129	37	,	,	PUNCT
ap-4612	129	38	∀	∀	X
ap-4612	129	39	θ	θ	NOUN
ap-4612	129	40	∈	∈	PROPN
ap-4612	130	1	[	[	X
ap-4612	130	2	0	0	NUM
ap-4612	130	3	,	,	PUNCT
ap-4612	130	4	2π	2π	NOUN
ap-4612	130	5	)	)	PUNCT
ap-4612	130	6	.	.	PUNCT
ap-4612	131	1	since	since	SCONJ
ap-4612	131	2	r(θ	r(θ	NOUN
ap-4612	131	3	)	)	PUNCT
ap-4612	131	4	is	be	AUX
ap-4612	131	5	unitary	unitary	ADJ
ap-4612	131	6	on	on	ADP
ap-4612	131	7	l2[0	l2[0	PROPN
ap-4612	131	8	,	,	PUNCT
ap-4612	131	9	2π	2π	NOUN
ap-4612	131	10	]	]	PUNCT
ap-4612	131	11	for	for	ADP
ap-4612	131	12	any	any	DET
ap-4612	131	13	value	value	NOUN
ap-4612	131	14	of	of	ADP
ap-4612	131	15	θ	θ	PROPN
ap-4612	131	16	then	then	ADV
ap-4612	131	17	r(θ	r(θ	VERB
ap-4612	131	18	)	)	PUNCT
ap-4612	132	1	is	be	AUX
ap-4612	132	2	also	also	ADV
ap-4612	132	3	unitary	unitary	ADJ
ap-4612	132	4	on	on	SCONJ
ap-4612	132	5	h	h	NOUN
ap-4612	132	6	due	due	ADP
ap-4612	132	7	to	to	ADP
ap-4612	132	8	the	the	DET
ap-4612	132	9	unitarity	unitarity	NOUN
ap-4612	132	10	of	of	ADP
ap-4612	132	11	s.	s.	PROPN
ap-4612	132	12	unitary	unitary	PROPN
ap-4612	132	13	operators	operator	NOUN
ap-4612	132	14	leaving	leave	VERB
ap-4612	132	15	φ	φ	PROPN
ap-4612	132	16	invariant	invariant	PROPN
ap-4612	132	17	can	can	AUX
ap-4612	132	18	be	be	AUX
ap-4612	132	19	extended	extend	VERB
ap-4612	132	20	to	to	ADP
ap-4612	132	21	φ×	φ×	NOUN
ap-4612	132	22	by	by	ADP
ap-4612	132	23	the	the	DET
ap-4612	132	24	duality	duality	NOUN
ap-4612	132	25	formula	formula	NOUN
ap-4612	132	26	(	(	PUNCT
ap-4612	132	27	1	1	NUM
ap-4612	132	28	)	)	PUNCT
ap-4612	132	29	,	,	PUNCT
ap-4612	133	1	i.e.	i.e.	X
ap-4612	133	2	,	,	PUNCT
ap-4612	133	3	〈	〈	PROPN
ap-4612	133	4	r(θ)f	r(θ)f	PROPN
ap-4612	133	5	|f	|f	PROPN
ap-4612	133	6	〉	〉	PROPN
ap-4612	133	7	=	=	SYM
ap-4612	133	8	〈	〈	PROPN
ap-4612	133	9	f	f	PROPN
ap-4612	133	10	|r(−θ)f	|r(−θ)f	PROPN
ap-4612	133	11	〉	〉	PROPN
ap-4612	133	12	,	,	PUNCT
ap-4612	133	13	∀	∀	PUNCT
ap-4612	133	14	|f	|f	PROPN
ap-4612	133	15	〉	〉	PROPN
ap-4612	133	16	∈	∈	PROPN
ap-4612	133	17	φ	φ	PROPN
ap-4612	133	18	,	,	PUNCT
ap-4612	133	19	∀f	∀f	PROPN
ap-4612	133	20	∈	∈	PROPN
ap-4612	133	21	φ×.	φ×.	PUNCT
ap-4612	133	22	therefore	therefore	ADV
ap-4612	133	23	,	,	PUNCT
ap-4612	133	24	〈	〈	PROPN
ap-4612	133	25	r(θ)φ|f	r(θ)φ|f	NOUN
ap-4612	133	26	〉	〉	NOUN
ap-4612	133	27	=	=	PUNCT
ap-4612	134	1	[	[	X
ap-4612	134	2	r(−θ)f	r(−θ)f	NOUN
ap-4612	134	3	]	]	X
ap-4612	134	4	(	(	PUNCT
ap-4612	134	5	φ	φ	NOUN
ap-4612	134	6	)	)	PUNCT
ap-4612	134	7	=	=	SYM
ap-4612	134	8	f(φ+	f(φ+	NOUN
ap-4612	134	9	θ	θ	NOUN
ap-4612	134	10	)	)	PUNCT
ap-4612	134	11	=	=	SYM
ap-4612	134	12	〈	〈	PROPN
ap-4612	134	13	φ+	φ+	X
ap-4612	134	14	θ|f	θ|f	NOUN
ap-4612	134	15	〉	〉	NOUN
ap-4612	134	16	.	.	PUNCT
ap-4612	135	1	combining	combine	VERB
ap-4612	135	2	both	both	DET
ap-4612	135	3	expressions	expression	NOUN
ap-4612	135	4	and	and	CCONJ
ap-4612	135	5	dropping	drop	VERB
ap-4612	135	6	the	the	DET
ap-4612	135	7	arbitrary	arbitrary	ADJ
ap-4612	135	8	|f	|f	PROPN
ap-4612	135	9	〉	〉	PROPN
ap-4612	135	10	∈	∈	PROPN
ap-4612	135	11	φ	φ	NOUN
ap-4612	135	12	we	we	PRON
ap-4612	135	13	arrive	arrive	VERB
ap-4612	135	14	to	to	ADP
ap-4612	135	15	〈	〈	PROPN
ap-4612	135	16	r(θ)φ|	r(θ)φ|	PART
ap-4612	135	17	≡	≡	PROPN
ap-4612	135	18	〈	〈	PROPN
ap-4612	135	19	φ|r(θ	φ|r(θ	NOUN
ap-4612	135	20	)	)	PUNCT
ap-4612	135	21	=	=	SYM
ap-4612	135	22	〈	〈	PROPN
ap-4612	135	23	φ+	φ+	X
ap-4612	135	24	θ|	θ|	PROPN
ap-4612	135	25	(	(	PUNCT
ap-4612	135	26	mod	mod	ADJ
ap-4612	135	27	2π	2π	NOUN
ap-4612	135	28	)	)	PUNCT
ap-4612	135	29	,	,	PUNCT
ap-4612	135	30	which	which	PRON
ap-4612	135	31	is	be	AUX
ap-4612	135	32	a	a	DET
ap-4612	135	33	rigorous	rigorous	ADJ
ap-4612	135	34	expression	expression	NOUN
ap-4612	135	35	in	in	ADP
ap-4612	135	36	φ×.	φ×.	NUM
ap-4612	135	37	in	in	ADP
ap-4612	135	38	fact	fact	NOUN
ap-4612	135	39	,	,	PUNCT
ap-4612	135	40	let	let	VERB
ap-4612	135	41	|f	|f	PRON
ap-4612	135	42	〉	〉	PROPN
ap-4612	135	43	∈	∈	PROPN
ap-4612	135	44	φ	φ	NOUN
ap-4612	135	45	as	as	ADP
ap-4612	135	46	in	in	ADP
ap-4612	135	47	(	(	PUNCT
ap-4612	135	48	6	6	NUM
ap-4612	135	49	)	)	PUNCT
ap-4612	135	50	.	.	PUNCT
ap-4612	136	1	then	then	ADV
ap-4612	136	2	,	,	PUNCT
ap-4612	136	3	we	we	PRON
ap-4612	136	4	have	have	VERB
ap-4612	136	5	r(θ	r(θ	NOUN
ap-4612	136	6	)	)	PUNCT
ap-4612	137	1	∞∑	∞∑	NOUN
ap-4612	137	2	m=−∞	m=−∞	AUX
ap-4612	137	3	am	be	AUX
ap-4612	137	4	|m	|m	NOUN
ap-4612	137	5	〉	〉	NOUN
ap-4612	137	6	=	=	PUNCT
ap-4612	138	1	s−1	s−1	NOUN
ap-4612	138	2	∞∑	∞∑	NUM
ap-4612	138	3	m=−∞	m=−∞	AUX
ap-4612	138	4	amsr(θ)s−1	amsr(θ)s−1	VERB
ap-4612	138	5	s|m	s|m	NOUN
ap-4612	138	6	〉	〉	NOUN
ap-4612	138	7	=	=	PUNCT
ap-4612	139	1	s−1	s−1	NOUN
ap-4612	139	2	∞∑	∞∑	NUM
ap-4612	139	3	m=−∞	m=−∞	X
ap-4612	139	4	amr(θ	amr(θ	PROPN
ap-4612	139	5	)	)	PUNCT
ap-4612	139	6	1√	1√	PROPN
ap-4612	139	7	2π	2π	NOUN
ap-4612	139	8	e−imφ	e−imφ	NOUN
ap-4612	139	9	=	=	PUNCT
ap-4612	140	1	s−1	s−1	NOUN
ap-4612	140	2	∞∑	∞∑	NUM
ap-4612	140	3	m=−∞	m=−∞	X
ap-4612	140	4	am	be	AUX
ap-4612	140	5	1√	1√	ADJ
ap-4612	140	6	2π	2π	NOUN
ap-4612	140	7	e−im(φ−θ	e−im(φ−θ	NOUN
ap-4612	140	8	)	)	PUNCT
ap-4612	141	1	=	=	PUNCT
ap-4612	142	1	s−1	s−1	NOUN
ap-4612	142	2	∞∑	∞∑	NUM
ap-4612	142	3	m=−∞	m=−∞	AUX
ap-4612	142	4	ame	ame	PROPN
ap-4612	142	5	imθ	imθ	PROPN
ap-4612	142	6	1√	1√	PROPN
ap-4612	142	7	2π	2π	PROPN
ap-4612	142	8	e−imφ	e−imφ	NOUN
ap-4612	142	9	=	=	NOUN
ap-4612	143	1	∞∑	∞∑	NOUN
ap-4612	143	2	m=−∞	m=−∞	ADP
ap-4612	143	3	ame	ame	PROPN
ap-4612	143	4	imθ	imθ	PROPN
ap-4612	143	5	|m	|m	NOUN
ap-4612	143	6	〉	〉	PROPN
ap-4612	143	7	∈	∈	PROPN
ap-4612	143	8	φ	φ	NOUN
ap-4612	143	9	.	.	PUNCT
ap-4612	144	1	hence	hence	ADV
ap-4612	144	2	,	,	PUNCT
ap-4612	144	3	we	we	PRON
ap-4612	144	4	see	see	VERB
ap-4612	144	5	that	that	SCONJ
ap-4612	144	6	r(θ)φ	r(θ)φ	PROPN
ap-4612	144	7	⊂	⊂	PROPN
ap-4612	144	8	φ	φ	PROPN
ap-4612	144	9	and	and	CCONJ
ap-4612	144	10	since	since	SCONJ
ap-4612	144	11	r−1(θ	r−1(θ	PROPN
ap-4612	144	12	)	)	PUNCT
ap-4612	144	13	=	=	SYM
ap-4612	144	14	r(−θ	r(−θ	NOUN
ap-4612	144	15	)	)	PUNCT
ap-4612	144	16	then	then	ADV
ap-4612	144	17	φ	φ	PROPN
ap-4612	144	18	⊂	⊂	PROPN
ap-4612	144	19	r(−θ)φ	r(−θ)φ	PROPN
ap-4612	144	20	.	.	PUNCT
ap-4612	145	1	so	so	ADV
ap-4612	145	2	,	,	PUNCT
ap-4612	145	3	φ	φ	PROPN
ap-4612	145	4	=	=	SYM
ap-4612	145	5	r(θ)φ	r(θ)φ	PROPN
ap-4612	145	6	,	,	PUNCT
ap-4612	145	7	∀θ	∀θ	PROPN
ap-4612	145	8	.	.	PUNCT
ap-4612	146	1	the	the	DET
ap-4612	146	2	uir	uir	PROPN
ap-4612	146	3	,	,	PUNCT
ap-4612	146	4	um	um	INTJ
ap-4612	146	5	,	,	PUNCT
ap-4612	146	6	on	on	ADP
ap-4612	146	7	h	h	NOUN
ap-4612	146	8	is	be	AUX
ap-4612	146	9	given	give	VERB
ap-4612	146	10	in	in	ADP
ap-4612	146	11	terms	term	NOUN
ap-4612	146	12	of	of	ADP
ap-4612	146	13	the	the	PRON
ap-4612	146	14	um	um	INTJ
ap-4612	146	15	,	,	PUNCT
ap-4612	146	16	defined	define	VERB
ap-4612	146	17	in	in	ADP
ap-4612	146	18	(	(	PUNCT
ap-4612	146	19	3	3	NUM
ap-4612	146	20	)	)	PUNCT
ap-4612	146	21	,	,	PUNCT
ap-4612	146	22	by	by	ADP
ap-4612	146	23	um(φ	um(φ	NOUN
ap-4612	146	24	)	)	PUNCT
ap-4612	146	25	:	:	PUNCT
ap-4612	146	26	=	=	SYM
ap-4612	146	27	s−1um(φ)s	s−1um(φ)s	PROPN
ap-4612	146	28	,	,	PUNCT
ap-4612	146	29	∀m	∀m	PROPN
ap-4612	146	30	∈	∈	PROPN
ap-4612	146	31	z	z	PROPN
ap-4612	146	32	,	,	PUNCT
ap-4612	146	33	∀φ	∀φ	X
ap-4612	146	34	∈	∈	PROPN
ap-4612	147	1	[	[	X
ap-4612	147	2	0	0	NUM
ap-4612	147	3	,	,	PUNCT
ap-4612	147	4	2π	2π	NOUN
ap-4612	147	5	)	)	PUNCT
ap-4612	147	6	.	.	PUNCT
ap-4612	148	1	let	let	VERB
ap-4612	148	2	j	j	PROPN
ap-4612	148	3	be	be	AUX
ap-4612	148	4	the	the	DET
ap-4612	148	5	infinitesimal	infinitesimal	ADJ
ap-4612	148	6	generator	generator	NOUN
ap-4612	148	7	associated	associate	VERB
ap-4612	148	8	to	to	ADP
ap-4612	148	9	this	this	DET
ap-4612	148	10	representation	representation	NOUN
ap-4612	148	11	,	,	PUNCT
ap-4612	148	12	i.e.	i.e.	X
ap-4612	148	13	um(φ	um(φ	NOUN
ap-4612	148	14	)	)	PUNCT
ap-4612	148	15	=	=	SYM
ap-4612	148	16	e−ijφ	e−ijφ	NOUN
ap-4612	148	17	.	.	PUNCT
ap-4612	149	1	since	since	SCONJ
ap-4612	149	2	um	um	INTJ
ap-4612	149	3	is	be	AUX
ap-4612	149	4	unitary	unitary	ADJ
ap-4612	149	5	then	then	ADV
ap-4612	149	6	j	j	PROPN
ap-4612	149	7	is	be	AUX
ap-4612	149	8	self	self	NOUN
ap-4612	149	9	-	-	PUNCT
ap-4612	149	10	adjoint	adjoint	NOUN
ap-4612	149	11	.	.	PUNCT
ap-4612	150	1	its	its	PRON
ap-4612	150	2	action	action	NOUN
ap-4612	150	3	on	on	ADP
ap-4612	150	4	the	the	DET
ap-4612	150	5	vectors	vector	NOUN
ap-4612	150	6	|m	|m	NOUN
ap-4612	150	7	〉	〉	PROPN
ap-4612	150	8	is	be	AUX
ap-4612	150	9	j	j	PROPN
ap-4612	150	10	|m	|m	NOUN
ap-4612	150	11	〉	〉	PROPN
ap-4612	150	12	=	=	SYM
ap-4612	150	13	m	m	VERB
ap-4612	150	14	|m	|m	NOUN
ap-4612	150	15	〉	〉	NUM
ap-4612	150	16	.	.	PUNCT
ap-4612	151	1	hence	hence	ADV
ap-4612	151	2	for	for	ADP
ap-4612	151	3	any	any	DET
ap-4612	151	4	|f	|f	PROPN
ap-4612	151	5	〉	〉	PROPN
ap-4612	151	6	∈	∈	PROPN
ap-4612	151	7	φ	φ	NOUN
ap-4612	151	8	as	as	ADP
ap-4612	151	9	in	in	ADP
ap-4612	151	10	(	(	PUNCT
ap-4612	151	11	6	6	NUM
ap-4612	151	12	)	)	PUNCT
ap-4612	151	13	,	,	PUNCT
ap-4612	151	14	we	we	PRON
ap-4612	151	15	have	have	VERB
ap-4612	151	16	that	that	DET
ap-4612	151	17	j	j	PROPN
ap-4612	151	18	|f	|f	PROPN
ap-4612	151	19	〉	〉	PROPN
ap-4612	151	20	=	=	PUNCT
ap-4612	152	1	∞∑	∞∑	NUM
ap-4612	152	2	m=−∞	m=−∞	ADP
ap-4612	152	3	amm	amm	PROPN
ap-4612	152	4	|m	|m	PROPN
ap-4612	152	5	〉	〉	PROPN
ap-4612	152	6	.	.	PUNCT
ap-4612	153	1	from	from	ADP
ap-4612	153	2	the	the	DET
ap-4612	153	3	set	set	NOUN
ap-4612	153	4	of	of	ADP
ap-4612	153	5	norms	norm	NOUN
ap-4612	153	6	‖jf‖2	‖jf‖2	VERB
ap-4612	153	7	p	p	X
ap-4612	153	8	,	,	PUNCT
ap-4612	153	9	p	p	X
ap-4612	153	10	=	=	NOUN
ap-4612	153	11	0	0	NUM
ap-4612	153	12	,	,	PUNCT
ap-4612	153	13	1	1	NUM
ap-4612	153	14	,	,	PUNCT
ap-4612	153	15	2	2	NUM
ap-4612	153	16	,	,	PUNCT
ap-4612	153	17	.	.	PUNCT
ap-4612	153	18	.	.	PUNCT
ap-4612	153	19	.	.	PUNCT
ap-4612	154	1	,	,	PUNCT
ap-4612	154	2	that	that	SCONJ
ap-4612	154	3	we	we	PRON
ap-4612	154	4	have	have	AUX
ap-4612	154	5	defined	define	VERB
ap-4612	154	6	in	in	ADP
ap-4612	154	7	(	(	PUNCT
ap-4612	154	8	7	7	NUM
ap-4612	154	9	)	)	PUNCT
ap-4612	154	10	,	,	PUNCT
ap-4612	154	11	we	we	PRON
ap-4612	154	12	obtain	obtain	VERB
ap-4612	154	13	the	the	DET
ap-4612	154	14	following	follow	VERB
ap-4612	154	15	inequality	inequality	NOUN
ap-4612	154	16	∞∑	∞∑	NUM
ap-4612	154	17	m=−∞	m=−∞	AUX
ap-4612	154	18	|am|2m2|m+	|am|2m2|m+	VERB
ap-4612	154	19	i|2p	i|2p	ADJ
ap-4612	154	20	≤	≤	NOUN
ap-4612	154	21	∞∑	∞∑	NUM
ap-4612	154	22	m=−∞	m=−∞	ADP
ap-4612	154	23	|am|2|m+	|am|2|m+	ADJ
ap-4612	154	24	i|2p+2	i|2p+2	PROPN
ap-4612	154	25	,	,	PUNCT
ap-4612	154	26	which	which	PRON
ap-4612	154	27	shows	show	VERB
ap-4612	154	28	that	that	SCONJ
ap-4612	154	29	jφ	jφ	PROPN
ap-4612	154	30	⊂	⊂	PROPN
ap-4612	154	31	φ	φ	PROPN
ap-4612	154	32	.	.	PUNCT
ap-4612	155	1	also	also	ADV
ap-4612	155	2	,	,	PUNCT
ap-4612	155	3	this	this	DET
ap-4612	155	4	inequality	inequality	NOUN
ap-4612	155	5	may	may	AUX
ap-4612	155	6	be	be	AUX
ap-4612	155	7	also	also	ADV
ap-4612	155	8	read	read	VERB
ap-4612	155	9	as	as	ADP
ap-4612	155	10	‖jf‖2	‖jf‖2	NOUN
ap-4612	155	11	p	p	PROPN
ap-4612	155	12	≤	≤	NUM
ap-4612	155	13	‖f‖p+1	‖f‖p+1	PROPN
ap-4612	155	14	,	,	PUNCT
ap-4612	155	15	∀	∀	PUNCT
ap-4612	155	16	|f	|f	PROPN
ap-4612	155	17	〉	〉	PROPN
ap-4612	155	18	∈	∈	PROPN
ap-4612	155	19	φ	φ	PROPN
ap-4612	155	20	,	,	PUNCT
ap-4612	155	21	∀	∀	PUNCT
ap-4612	155	22	p	p	NOUN
ap-4612	155	23	∈	∈	PROPN
ap-4612	155	24	n.	n.	NOUN
ap-4612	155	25	382	382	NUM
ap-4612	155	26	vol	vol	NOUN
ap-4612	155	27	.	.	PUNCT
ap-4612	156	1	57	57	NUM
ap-4612	157	1	no	no	NOUN
ap-4612	157	2	.	.	PUNCT
ap-4612	158	1	6/2017	6/2017	PRON
ap-4612	158	2	lie	lie	VERB
ap-4612	158	3	algebra	algebra	NOUN
ap-4612	158	4	representations	representation	NOUN
ap-4612	158	5	and	and	CCONJ
ap-4612	158	6	rigged	rig	VERB
ap-4612	158	7	hilbert	hilbert	NOUN
ap-4612	158	8	spaces	space	VERB
ap-4612	158	9	:	:	PUNCT
ap-4612	158	10	the	the	DET
ap-4612	158	11	so(2	so(2	NOUN
ap-4612	158	12	)	)	PUNCT
ap-4612	158	13	case	case	NOUN
ap-4612	158	14	thus	thus	ADV
ap-4612	158	15	,	,	PUNCT
ap-4612	158	16	we	we	PRON
ap-4612	158	17	have	have	AUX
ap-4612	158	18	proved	prove	VERB
ap-4612	158	19	that	that	SCONJ
ap-4612	158	20	j	j	PROPN
ap-4612	158	21	is	be	AUX
ap-4612	158	22	continuous	continuous	ADJ
ap-4612	158	23	on	on	ADP
ap-4612	158	24	φ	φ	NUM
ap-4612	158	25	.	.	PUNCT
ap-4612	159	1	moreover	moreover	ADV
ap-4612	159	2	,	,	PUNCT
ap-4612	159	3	since	since	SCONJ
ap-4612	159	4	the	the	DET
ap-4612	159	5	self	self	NOUN
ap-4612	159	6	-	-	PUNCT
ap-4612	159	7	adjoint	adjoint	NOUN
ap-4612	159	8	operator	operator	NOUN
ap-4612	159	9	j	j	PROPN
ap-4612	159	10	verifies	verifie	NOUN
ap-4612	159	11	jφ	jφ	PROPN
ap-4612	159	12	⊂	⊂	PROPN
ap-4612	159	13	φ	φ	PROPN
ap-4612	159	14	,	,	PUNCT
ap-4612	159	15	it	it	PRON
ap-4612	159	16	can	can	AUX
ap-4612	159	17	be	be	AUX
ap-4612	159	18	extended	extend	VERB
ap-4612	159	19	to	to	ADP
ap-4612	159	20	φ×	φ×	NOUN
ap-4612	159	21	using	use	VERB
ap-4612	159	22	the	the	DET
ap-4612	159	23	duality	duality	NOUN
ap-4612	159	24	form	form	NOUN
ap-4612	159	25	,	,	PUNCT
ap-4612	159	26	i.e.	i.e.	X
ap-4612	159	27	,	,	PUNCT
ap-4612	159	28	〈	〈	PROPN
ap-4612	159	29	jf	jf	PROPN
ap-4612	159	30	|f	|f	PROPN
ap-4612	159	31	〉	〉	PROPN
ap-4612	159	32	=	=	SYM
ap-4612	160	1	〈	〈	PROPN
ap-4612	160	2	f	f	PROPN
ap-4612	160	3	|jf	|jf	PROPN
ap-4612	160	4	〉	〉	PROPN
ap-4612	160	5	,	,	PUNCT
ap-4612	160	6	∀	∀	PUNCT
ap-4612	160	7	|f	|f	PROPN
ap-4612	160	8	〉	〉	PROPN
ap-4612	160	9	∈	∈	PROPN
ap-4612	160	10	φ	φ	PROPN
ap-4612	160	11	,	,	PUNCT
ap-4612	160	12	∀f	∀f	PROPN
ap-4612	160	13	∈	∈	PROPN
ap-4612	160	14	φ×.	φ×.	PUNCT
ap-4612	160	15	furthermore	furthermore	ADV
ap-4612	160	16	,	,	PUNCT
ap-4612	160	17	since	since	SCONJ
ap-4612	160	18	j	j	PROPN
ap-4612	160	19	is	be	AUX
ap-4612	160	20	continuous	continuous	ADJ
ap-4612	160	21	on	on	ADP
ap-4612	160	22	φ	φ	NUM
ap-4612	160	23	,	,	PUNCT
ap-4612	160	24	this	this	DET
ap-4612	160	25	extension	extension	NOUN
ap-4612	160	26	is	be	AUX
ap-4612	160	27	weakly	weakly	ADJ
ap-4612	160	28	(	(	PUNCT
ap-4612	160	29	with	with	ADP
ap-4612	160	30	the	the	DET
ap-4612	160	31	weak	weak	ADJ
ap-4612	160	32	topology	topology	NOUN
ap-4612	160	33	)	)	PUNCT
ap-4612	160	34	continuous	continuous	ADJ
ap-4612	160	35	on	on	ADP
ap-4612	160	36	φ×.	φ×.	NUM
ap-4612	160	37	in	in	ADP
ap-4612	160	38	fact	fact	NOUN
ap-4612	160	39	,	,	PUNCT
ap-4612	160	40	since	since	SCONJ
ap-4612	160	41	the	the	DET
ap-4612	160	42	series	series	NOUN
ap-4612	160	43	i|φ	i|φ	VERB
ap-4612	160	44	〉	〉	PROPN
ap-4612	160	45	=	=	SYM
ap-4612	160	46	|φ	|φ	PROPN
ap-4612	160	47	〉	〉	NOUN
ap-4612	160	48	=	=	NOUN
ap-4612	161	1	∞∑	∞∑	NUM
ap-4612	161	2	m=−∞	m=−∞	ADP
ap-4612	161	3	|m〉〈m|φ	|m〉〈m|φ	ADJ
ap-4612	161	4	〉	〉	PROPN
ap-4612	161	5	=	=	SYM
ap-4612	161	6	1√	1√	NUM
ap-4612	161	7	2π	2π	NOUN
ap-4612	161	8	∞∑	∞∑	NUM
ap-4612	161	9	m=−∞	m=−∞	NOUN
ap-4612	161	10	eimφ	eimφ	VERB
ap-4612	161	11	|m	|m	NOUN
ap-4612	161	12	〉	〉	PROPN
ap-4612	161	13	is	be	AUX
ap-4612	161	14	weakly	weakly	ADJ
ap-4612	161	15	convergent	convergent	NOUN
ap-4612	161	16	,	,	PUNCT
ap-4612	161	17	then	then	ADV
ap-4612	161	18	j	j	PROPN
ap-4612	161	19	|φ	|φ	PROPN
ap-4612	161	20	〉	〉	PROPN
ap-4612	161	21	=	=	SYM
ap-4612	161	22	1√	1√	NUM
ap-4612	161	23	2π	2π	NOUN
ap-4612	161	24	∞∑	∞∑	NUM
ap-4612	161	25	m=−∞	m=−∞	AUX
ap-4612	161	26	eimφ	eimφ	VERB
ap-4612	161	27	j	j	PROPN
ap-4612	161	28	|m	|m	NOUN
ap-4612	161	29	〉	〉	PROPN
ap-4612	161	30	=	=	SYM
ap-4612	161	31	1√	1√	PROPN
ap-4612	161	32	2π	2π	NOUN
ap-4612	161	33	∞∑	∞∑	NUM
ap-4612	161	34	m=−∞	m=−∞	X
ap-4612	161	35	eimφm	eimφm	VERB
ap-4612	161	36	|m	|m	NOUN
ap-4612	161	37	〉	〉	PROPN
ap-4612	161	38	=	=	SYM
ap-4612	161	39	−idφ	−idφ	PROPN
ap-4612	161	40	|φ	|φ	PROPN
ap-4612	161	41	〉	〉	PROPN
ap-4612	161	42	,	,	PUNCT
ap-4612	161	43	where	where	SCONJ
ap-4612	161	44	the	the	DET
ap-4612	161	45	operator	operator	NOUN
ap-4612	161	46	dφ	dφ	ADP
ap-4612	161	47	is	be	AUX
ap-4612	161	48	defined	define	VERB
ap-4612	161	49	as	as	SCONJ
ap-4612	161	50	follows	follow	VERB
ap-4612	161	51	:	:	PUNCT
ap-4612	161	52	for	for	ADP
ap-4612	161	53	any	any	DET
ap-4612	161	54	|f	|f	PROPN
ap-4612	161	55	〉	〉	PROPN
ap-4612	161	56	in	in	ADP
ap-4612	161	57	φ	φ	PROPN
ap-4612	161	58	we	we	PRON
ap-4612	161	59	know	know	VERB
ap-4612	161	60	that	that	SCONJ
ap-4612	161	61	s|f	s|f	PROPN
ap-4612	161	62	〉	〉	PROPN
ap-4612	161	63	=	=	SYM
ap-4612	161	64	f(φ	f(φ	PROPN
ap-4612	161	65	)	)	PUNCT
ap-4612	161	66	∈	∈	PROPN
ap-4612	161	67	(	(	PUNCT
ap-4612	161	68	sφ	sφ	NOUN
ap-4612	161	69	)	)	PUNCT
ap-4612	161	70	as	as	ADP
ap-4612	161	71	in	in	ADP
ap-4612	161	72	(	(	PUNCT
ap-4612	161	73	9	9	NUM
ap-4612	161	74	)	)	PUNCT
ap-4612	161	75	.	.	PUNCT
ap-4612	162	1	then	then	ADV
ap-4612	162	2	,	,	PUNCT
ap-4612	162	3	i	i	PRON
ap-4612	162	4	d	d	X
ap-4612	162	5	dφ	dφ	ADP
ap-4612	162	6	f(φ	f(φ	PROPN
ap-4612	162	7	)	)	PUNCT
ap-4612	163	1	=	=	PUNCT
ap-4612	164	1	i	i	PRON
ap-4612	165	1	d	d	X
ap-4612	165	2	dφ	dφ	ADP
ap-4612	165	3	∞∑	∞∑	NUM
ap-4612	165	4	m=−∞	m=−∞	AUX
ap-4612	165	5	am	be	AUX
ap-4612	165	6	e−imφ√	e−imφ√	NOUN
ap-4612	165	7	2π	2π	NOUN
ap-4612	165	8	=	=	PUNCT
ap-4612	166	1	∞∑	∞∑	NUM
ap-4612	166	2	m=−∞	m=−∞	ADP
ap-4612	166	3	amm	amm	PROPN
ap-4612	166	4	e−imφ√	e−imφ√	PROPN
ap-4612	166	5	2π	2π	PROPN
ap-4612	166	6	.	.	PUNCT
ap-4612	167	1	(	(	PUNCT
ap-4612	167	2	15	15	X
ap-4612	167	3	)	)	PUNCT
ap-4612	167	4	we	we	PRON
ap-4612	167	5	easily	easily	ADV
ap-4612	167	6	conclude	conclude	VERB
ap-4612	167	7	that	that	SCONJ
ap-4612	167	8	the	the	DET
ap-4612	167	9	operator	operator	NOUN
ap-4612	167	10	i	i	PRON
ap-4612	167	11	d	d	VERB
ap-4612	167	12	/	/	SYM
ap-4612	167	13	dφ	dφ	ADP
ap-4612	167	14	is	be	AUX
ap-4612	167	15	continuous	continuous	ADJ
ap-4612	167	16	on	on	ADP
ap-4612	167	17	sφ	sφ	PROPN
ap-4612	167	18	with	with	ADP
ap-4612	167	19	the	the	DET
ap-4612	167	20	topology	topology	NOUN
ap-4612	167	21	transported	transport	VERB
ap-4612	167	22	by	by	ADP
ap-4612	167	23	s	s	PROPN
ap-4612	167	24	from	from	ADP
ap-4612	167	25	φ	φ	PROPN
ap-4612	167	26	(	(	PUNCT
ap-4612	167	27	norms	norm	NOUN
ap-4612	167	28	on	on	ADP
ap-4612	167	29	sφ	sφ	PART
ap-4612	167	30	look	look	VERB
ap-4612	167	31	like	like	INTJ
ap-4612	167	32	exactly	exactly	ADV
ap-4612	167	33	as	as	ADP
ap-4612	167	34	the	the	DET
ap-4612	167	35	norms	norm	NOUN
ap-4612	167	36	on	on	ADP
ap-4612	167	37	φ	φ	NUM
ap-4612	167	38	)	)	PUNCT
ap-4612	167	39	.	.	PUNCT
ap-4612	168	1	hence	hence	ADV
ap-4612	168	2	,	,	PUNCT
ap-4612	168	3	−idφ	−idφ	ADJ
ap-4612	168	4	:	:	PUNCT
ap-4612	168	5	=	=	PUNCT
ap-4612	169	1	s−1	s−1	NOUN
ap-4612	170	1	i	i	NOUN
ap-4612	170	2	d	d	X
ap-4612	170	3	dφ	dφ	ADP
ap-4612	170	4	s.	s.	PROPN
ap-4612	170	5	this	this	DET
ap-4612	170	6	definition	definition	NOUN
ap-4612	170	7	implies	imply	VERB
ap-4612	170	8	that	that	SCONJ
ap-4612	170	9	−idφ	−idφ	PROPN
ap-4612	170	10	is	be	AUX
ap-4612	170	11	continuous	continuous	ADJ
ap-4612	170	12	on	on	ADP
ap-4612	170	13	φ	φ	NUM
ap-4612	170	14	.	.	PUNCT
ap-4612	171	1	moreover	moreover	ADV
ap-4612	171	2	,	,	PUNCT
ap-4612	171	3	it	it	PRON
ap-4612	171	4	is	be	AUX
ap-4612	171	5	self	self	NOUN
ap-4612	171	6	-	-	PUNCT
ap-4612	171	7	adjoint	adjoint	NOUN
ap-4612	171	8	on	on	ADP
ap-4612	171	9	h	h	NOUN
ap-4612	171	10	,	,	PUNCT
ap-4612	171	11	so	so	SCONJ
ap-4612	171	12	that	that	SCONJ
ap-4612	171	13	it	it	PRON
ap-4612	171	14	can	can	AUX
ap-4612	171	15	be	be	AUX
ap-4612	171	16	extended	extend	VERB
ap-4612	171	17	to	to	ADP
ap-4612	171	18	a	a	DET
ap-4612	171	19	weakly	weakly	ADJ
ap-4612	171	20	continuous	continuous	ADJ
ap-4612	171	21	operator	operator	NOUN
ap-4612	171	22	on	on	ADP
ap-4612	171	23	φ×	φ×	NOUN
ap-4612	171	24	as	as	ADP
ap-4612	171	25	the	the	DET
ap-4612	171	26	last	last	ADJ
ap-4612	171	27	identity	identity	NOUN
ap-4612	171	28	in	in	ADP
ap-4612	171	29	(	(	PUNCT
ap-4612	171	30	15	15	NUM
ap-4612	171	31	)	)	PUNCT
ap-4612	171	32	shows	show	NOUN
ap-4612	171	33	.	.	PUNCT
ap-4612	172	1	therefore	therefore	ADV
ap-4612	172	2	on	on	ADP
ap-4612	172	3	φ×	φ×	NUM
ap-4612	172	4	j	j	PROPN
ap-4612	172	5	≡	≡	PROPN
ap-4612	172	6	−idφ	−idφ	PROPN
ap-4612	172	7	.	.	PUNCT
ap-4612	173	1	4	4	X
ap-4612	173	2	.	.	X
ap-4612	173	3	conclusions	conclusion	NOUN
ap-4612	173	4	we	we	PRON
ap-4612	173	5	have	have	AUX
ap-4612	173	6	construct	construct	VERB
ap-4612	173	7	two	two	NUM
ap-4612	173	8	rhs	rh	NOUN
ap-4612	173	9	that	that	PRON
ap-4612	173	10	support	support	VERB
ap-4612	173	11	the	the	DET
ap-4612	173	12	uir	uir	NOUN
ap-4612	173	13	of	of	ADP
ap-4612	173	14	the	the	DET
ap-4612	173	15	lie	lie	NOUN
ap-4612	173	16	group	group	NOUN
ap-4612	173	17	so(2	so(2	PROPN
ap-4612	173	18	)	)	PUNCT
ap-4612	173	19	,	,	PUNCT
ap-4612	174	1	φ	φ	PROPN
ap-4612	174	2	⊂	⊂	PROPN
ap-4612	174	3	h	h	PROPN
ap-4612	174	4	⊂	⊂	X
ap-4612	174	5	φ×	φ×	PROPN
ap-4612	174	6	and	and	CCONJ
ap-4612	174	7	sφ	sφ	PROPN
ap-4612	174	8	⊂	⊂	PROPN
ap-4612	174	9	l2[0	l2[0	PROPN
ap-4612	174	10	,	,	PUNCT
ap-4612	174	11	2π	2π	NOUN
ap-4612	174	12	]	]	X
ap-4612	175	1	⊂	⊂	X
ap-4612	175	2	(	(	PUNCT
ap-4612	175	3	sφ)×.	sφ)×.	NOUN
ap-4612	175	4	the	the	DET
ap-4612	175	5	first	first	ADJ
ap-4612	175	6	one	one	NOUN
ap-4612	175	7	is	be	AUX
ap-4612	175	8	related	relate	VERB
ap-4612	175	9	with	with	ADP
ap-4612	175	10	the	the	DET
ap-4612	175	11	discrete	discrete	ADJ
ap-4612	175	12	basis	basis	NOUN
ap-4612	175	13	{	{	PUNCT
ap-4612	175	14	|m	|m	NOUN
ap-4612	175	15	〉	〉	NOUN
ap-4612	175	16	}	}	PUNCT
ap-4612	175	17	and	and	CCONJ
ap-4612	175	18	in	in	ADP
ap-4612	175	19	some	some	DET
ap-4612	175	20	sense	sense	NOUN
ap-4612	175	21	is	be	AUX
ap-4612	175	22	an	an	DET
ap-4612	175	23	abstract	abstract	ADJ
ap-4612	175	24	rhs	rhs	PROPN
ap-4612	175	25	,	,	PUNCT
ap-4612	175	26	but	but	CCONJ
ap-4612	175	27	the	the	DET
ap-4612	175	28	second	second	ADJ
ap-4612	175	29	one	one	NUM
ap-4612	175	30	,	,	PUNCT
ap-4612	175	31	related	relate	VERB
ap-4612	175	32	with	with	ADP
ap-4612	175	33	the	the	DET
ap-4612	175	34	continuous	continuous	ADJ
ap-4612	175	35	basis	basis	NOUN
ap-4612	175	36	{	{	PUNCT
ap-4612	175	37	|φ	|φ	NOUN
ap-4612	175	38	〉	〉	NOUN
ap-4612	175	39	}	}	PUNCT
ap-4612	175	40	,	,	PUNCT
ap-4612	175	41	is	be	AUX
ap-4612	175	42	obtained	obtain	VERB
ap-4612	175	43	by	by	ADP
ap-4612	175	44	means	mean	NOUN
ap-4612	175	45	of	of	ADP
ap-4612	175	46	a	a	DET
ap-4612	175	47	unitary	unitary	ADJ
ap-4612	175	48	map	map	NOUN
ap-4612	175	49	s	s	PART
ap-4612	175	50	:	:	PUNCT
ap-4612	175	51	|m	|m	NOUN
ap-4612	175	52	〉	〉	PROPN
ap-4612	175	53	→	→	SYM
ap-4612	175	54	e−imφ/	e−imφ/	NOUN
ap-4612	176	1	√	√	NUM
ap-4612	176	2	2π	2π	NOUN
ap-4612	176	3	that	that	PRON
ap-4612	176	4	allows	allow	VERB
ap-4612	176	5	to	to	PART
ap-4612	176	6	translate	translate	VERB
ap-4612	176	7	the	the	DET
ap-4612	176	8	topologies	topology	NOUN
ap-4612	176	9	of	of	ADP
ap-4612	176	10	the	the	DET
ap-4612	176	11	first	first	ADJ
ap-4612	176	12	rhs	rhs	PROPN
ap-4612	176	13	as	as	ADV
ap-4612	176	14	well	well	ADV
ap-4612	176	15	as	as	ADP
ap-4612	176	16	all	all	DET
ap-4612	176	17	its	its	PRON
ap-4612	176	18	properties	property	NOUN
ap-4612	176	19	to	to	ADP
ap-4612	176	20	the	the	DET
ap-4612	176	21	second	second	ADJ
ap-4612	176	22	one	one	NUM
ap-4612	176	23	.	.	PUNCT
ap-4612	177	1	another	another	DET
ap-4612	177	2	interesting	interesting	ADJ
ap-4612	177	3	point	point	NOUN
ap-4612	177	4	to	to	ADP
ap-4612	177	5	stress	stress	NOUN
ap-4612	177	6	is	be	AUX
ap-4612	177	7	the	the	DET
ap-4612	177	8	fact	fact	NOUN
ap-4612	177	9	that	that	SCONJ
ap-4612	177	10	rhs	rhs	PROPN
ap-4612	177	11	,	,	PUNCT
ap-4612	177	12	from	from	ADP
ap-4612	177	13	one	one	NUM
ap-4612	177	14	side	side	NOUN
ap-4612	177	15	,	,	PUNCT
ap-4612	177	16	and	and	CCONJ
ap-4612	177	17	lie	lie	VERB
ap-4612	177	18	algebras	algebra	NOUN
ap-4612	177	19	and	and	CCONJ
ap-4612	177	20	universal	universal	ADJ
ap-4612	177	21	enveloping	enveloping	NOUN
ap-4612	177	22	algebras	algebra	NOUN
ap-4612	177	23	,	,	PUNCT
ap-4612	177	24	from	from	ADP
ap-4612	177	25	the	the	DET
ap-4612	177	26	other	other	ADJ
ap-4612	177	27	,	,	PUNCT
ap-4612	177	28	are	be	AUX
ap-4612	177	29	closely	closely	ADV
ap-4612	177	30	related	relate	VERB
ap-4612	177	31	.	.	PUNCT
ap-4612	178	1	this	this	PRON
ap-4612	178	2	means	mean	VERB
ap-4612	178	3	that	that	SCONJ
ap-4612	178	4	starting	start	VERB
ap-4612	178	5	from	from	ADP
ap-4612	178	6	a	a	DET
ap-4612	178	7	lie	lie	NOUN
ap-4612	178	8	algebra	algebra	NOUN
ap-4612	178	9	we	we	PRON
ap-4612	178	10	can	can	AUX
ap-4612	178	11	construct	construct	VERB
ap-4612	178	12	a	a	DET
ap-4612	178	13	rhs	rhs	PROPN
ap-4612	178	14	that	that	PRON
ap-4612	178	15	supports	support	VERB
ap-4612	178	16	it	it	PRON
ap-4612	178	17	in	in	ADP
ap-4612	178	18	such	such	DET
ap-4612	178	19	a	a	DET
ap-4612	178	20	way	way	NOUN
ap-4612	178	21	that	that	PRON
ap-4612	178	22	generators	generator	NOUN
ap-4612	178	23	and	and	CCONJ
ap-4612	178	24	universal	universal	ADJ
ap-4612	178	25	enveloping	enveloping	NOUN
ap-4612	178	26	elements	element	NOUN
ap-4612	178	27	can	can	AUX
ap-4612	178	28	be	be	AUX
ap-4612	178	29	represented	represent	VERB
ap-4612	178	30	by	by	ADP
ap-4612	178	31	operators	operator	NOUN
ap-4612	178	32	in	in	ADP
ap-4612	178	33	the	the	DET
ap-4612	178	34	rhs	rhs	PROPN
ap-4612	178	35	avoiding	avoid	VERB
ap-4612	178	36	domain	domain	NOUN
ap-4612	178	37	difficulties	difficulty	NOUN
ap-4612	178	38	[	[	X
ap-4612	178	39	5	5	NUM
ap-4612	178	40	]	]	PUNCT
ap-4612	178	41	.	.	PUNCT
ap-4612	179	1	vice	vice	PROPN
ap-4612	179	2	versa	versa	ADV
ap-4612	179	3	a	a	DET
ap-4612	179	4	rhs	rhs	PROPN
ap-4612	179	5	contains	contain	VERB
ap-4612	179	6	itself	itself	PRON
ap-4612	179	7	the	the	DET
ap-4612	179	8	symmetries	symmetry	NOUN
ap-4612	179	9	that	that	PRON
ap-4612	179	10	allow	allow	VERB
ap-4612	179	11	to	to	PART
ap-4612	179	12	construct	construct	VERB
ap-4612	179	13	its	its	PRON
ap-4612	179	14	related	related	ADJ
ap-4612	179	15	algebraical	algebraical	ADJ
ap-4612	179	16	structures	structure	NOUN
ap-4612	179	17	.	.	PUNCT
ap-4612	180	1	acknowledgements	acknowledgement	NOUN
ap-4612	180	2	partial	partial	ADJ
ap-4612	180	3	financial	financial	ADJ
ap-4612	180	4	support	support	NOUN
ap-4612	180	5	is	be	AUX
ap-4612	180	6	acknowledged	acknowledge	VERB
ap-4612	180	7	to	to	ADP
ap-4612	180	8	the	the	DET
ap-4612	180	9	spanish	spanish	ADJ
ap-4612	180	10	junta	junta	PROPN
ap-4612	180	11	de	de	PROPN
ap-4612	180	12	castilla	castilla	PROPN
ap-4612	180	13	y	y	PROPN
ap-4612	180	14	león	león	PROPN
ap-4612	180	15	and	and	CCONJ
ap-4612	180	16	feder	feder	PROPN
ap-4612	180	17	(	(	PUNCT
ap-4612	180	18	project	project	NOUN
ap-4612	180	19	va057u16	va057u16	PROPN
ap-4612	180	20	)	)	PUNCT
ap-4612	180	21	and	and	CCONJ
ap-4612	180	22	mineco	mineco	PROPN
ap-4612	180	23	(	(	PUNCT
ap-4612	180	24	project	project	NOUN
ap-4612	180	25	mtm2014	mtm2014	NOUN
ap-4612	180	26	-	-	PUNCT
ap-4612	180	27	57129	57129	NUM
ap-4612	180	28	-	-	PUNCT
ap-4612	180	29	c2	c2	PROPN
ap-4612	180	30	-	-	PUNCT
ap-4612	180	31	1	1	NUM
ap-4612	180	32	-	-	PUNCT
ap-4612	180	33	p	p	NOUN
ap-4612	180	34	)	)	PUNCT
ap-4612	180	35	.	.	PUNCT
ap-4612	181	1	references	reference	NOUN
ap-4612	181	2	[	[	X
ap-4612	181	3	1	1	X
ap-4612	181	4	]	]	PUNCT
ap-4612	181	5	e.	e.	PROPN
ap-4612	181	6	celeghini	celeghini	PROPN
ap-4612	181	7	and	and	CCONJ
ap-4612	181	8	m.a	m.a	PROPN
ap-4612	181	9	.	.	PROPN
ap-4612	181	10	del	del	PROPN
ap-4612	181	11	olmo	olmo	PROPN
ap-4612	181	12	.	.	PUNCT
ap-4612	182	1	coherent	coherent	ADJ
ap-4612	182	2	orthogonal	orthogonal	ADJ
ap-4612	182	3	polynomials	polynomial	NOUN
ap-4612	182	4	,	,	PUNCT
ap-4612	182	5	ann	ann	PROPN
ap-4612	182	6	.	.	PUNCT
ap-4612	183	1	phys	phys	PROPN
ap-4612	183	2	.	.	PUNCT
ap-4612	184	1	(	(	PUNCT
ap-4612	184	2	ny	ny	NOUN
ap-4612	184	3	)	)	PUNCT
ap-4612	184	4	335	335	NUM
ap-4612	184	5	,	,	PUNCT
ap-4612	184	6	78–85	78–85	NUM
ap-4612	184	7	(	(	PUNCT
ap-4612	184	8	2013	2013	NUM
ap-4612	184	9	)	)	PUNCT
ap-4612	184	10	.	.	PUNCT
ap-4612	185	1	http://dx.doi.org/10.1016/j.aop.2013.04.017	http://dx.doi.org/10.1016/j.aop.2013.04.017	PROPN
ap-4612	186	1	[	[	X
ap-4612	186	2	2	2	X
ap-4612	186	3	]	]	PUNCT
ap-4612	186	4	e.	e.	PROPN
ap-4612	186	5	celeghini	celeghini	PROPN
ap-4612	186	6	and	and	CCONJ
ap-4612	186	7	m.a	m.a	PROPN
ap-4612	186	8	.	.	PROPN
ap-4612	186	9	del	del	PROPN
ap-4612	186	10	olmo	olmo	PROPN
ap-4612	186	11	.	.	PUNCT
ap-4612	187	1	algebraic	algebraic	ADJ
ap-4612	187	2	special	special	ADJ
ap-4612	187	3	functions	function	NOUN
ap-4612	187	4	and	and	CCONJ
ap-4612	187	5	so(3	so(3	NOUN
ap-4612	187	6	,	,	PUNCT
ap-4612	187	7	2	2	NUM
ap-4612	187	8	)	)	PUNCT
ap-4612	187	9	,	,	PUNCT
ap-4612	187	10	ann	ann	PROPN
ap-4612	187	11	.	.	PUNCT
ap-4612	187	12	phys	phys	PROPN
ap-4612	187	13	.	.	PUNCT
ap-4612	188	1	(	(	PUNCT
ap-4612	188	2	ny	ny	NOUN
ap-4612	188	3	)	)	PUNCT
ap-4612	188	4	333	333	NUM
ap-4612	188	5	,	,	PUNCT
ap-4612	188	6	90–103	90–103	NUM
ap-4612	188	7	(	(	PUNCT
ap-4612	188	8	2013	2013	NUM
ap-4612	188	9	)	)	PUNCT
ap-4612	188	10	.	.	PUNCT
ap-4612	189	1	http://dx.doi.org/10.1016/j.aop.2013.02.010	http://dx.doi.org/10.1016/j.aop.2013.02.010	NOUN
ap-4612	190	1	[	[	X
ap-4612	190	2	3	3	X
ap-4612	190	3	]	]	X
ap-4612	190	4	e.	e.	PROPN
ap-4612	190	5	celeghini	celeghini	PROPN
ap-4612	190	6	,	,	PUNCT
ap-4612	190	7	m.	m.	NOUN
ap-4612	190	8	a.	a.	PROPN
ap-4612	190	9	del	del	PROPN
ap-4612	190	10	olmo	olmo	PROPN
ap-4612	190	11	and	and	CCONJ
ap-4612	190	12	m.a	m.a	PROPN
ap-4612	190	13	.	.	PROPN
ap-4612	190	14	velasco	velasco	PROPN
ap-4612	190	15	.	.	PROPN
ap-4612	191	1	lie	lie	PROPN
ap-4612	191	2	groups	group	NOUN
ap-4612	191	3	,	,	PUNCT
ap-4612	191	4	algebraic	algebraic	ADJ
ap-4612	191	5	special	special	ADJ
ap-4612	191	6	functions	function	NOUN
ap-4612	191	7	and	and	CCONJ
ap-4612	191	8	jacobi	jacobi	PROPN
ap-4612	191	9	polynomials	polynomials	PROPN
ap-4612	191	10	,	,	PUNCT
ap-4612	191	11	j.	j.	PROPN
ap-4612	191	12	phys	phys	PROPN
ap-4612	191	13	.	.	PUNCT
ap-4612	191	14	:	:	PUNCT
ap-4612	192	1	conf	conf	PROPN
ap-4612	192	2	.	.	PUNCT
ap-4612	192	3	ser	ser	PROPN
ap-4612	192	4	.	.	PROPN
ap-4612	192	5	597	597	NUM
ap-4612	192	6	,	,	PUNCT
ap-4612	192	7	012023	012023	NUM
ap-4612	192	8	(	(	PUNCT
ap-4612	192	9	2015	2015	NUM
ap-4612	192	10	)	)	PUNCT
ap-4612	192	11	.	.	PUNCT
ap-4612	193	1	doi:10.1088/1742	doi:10.1088/1742	PROPN
ap-4612	193	2	-	-	PUNCT
ap-4612	193	3	6596/597/1/012023	6596/597/1/012023	NUM
ap-4612	193	4	[	[	X
ap-4612	193	5	4	4	NUM
ap-4612	193	6	]	]	X
ap-4612	193	7	i.m	i.m	PROPN
ap-4612	193	8	.	.	PROPN
ap-4612	193	9	gelfand	gelfand	PROPN
ap-4612	193	10	and	and	CCONJ
ap-4612	193	11	n.ya	n.ya	PROPN
ap-4612	193	12	.	.	PROPN
ap-4612	193	13	vilenkin	vilenkin	PROPN
ap-4612	193	14	.	.	PUNCT
ap-4612	194	1	generalized	generalized	ADJ
ap-4612	194	2	functions	function	NOUN
ap-4612	194	3	:	:	PUNCT
ap-4612	194	4	applications	application	NOUN
ap-4612	194	5	to	to	PART
ap-4612	194	6	harmonic	harmonic	VERB
ap-4612	194	7	analysis	analysis	NOUN
ap-4612	194	8	.	.	PUNCT
ap-4612	195	1	academic	academic	ADJ
ap-4612	195	2	,	,	PUNCT
ap-4612	195	3	new	new	PROPN
ap-4612	195	4	york	york	PROPN
ap-4612	195	5	,	,	PUNCT
ap-4612	195	6	1964	1964	NUM
ap-4612	195	7	.	.	PUNCT
ap-4612	196	1	[	[	X
ap-4612	196	2	5	5	NUM
ap-4612	196	3	]	]	X
ap-4612	196	4	j.e	j.e	PROPN
ap-4612	196	5	.	.	PROPN
ap-4612	196	6	roberts	roberts	PROPN
ap-4612	196	7	.	.	PROPN
ap-4612	197	1	rigged	rig	VERB
ap-4612	197	2	hilbert	hilbert	NOUN
ap-4612	197	3	spaces	space	NOUN
ap-4612	197	4	in	in	ADP
ap-4612	197	5	quantum	quantum	ADJ
ap-4612	197	6	mechanics	mechanic	NOUN
ap-4612	197	7	,	,	PUNCT
ap-4612	197	8	comm	comm	NOUN
ap-4612	197	9	.	.	PUNCT
ap-4612	198	1	math	math	NOUN
ap-4612	198	2	.	.	PUNCT
ap-4612	199	1	phys	phy	NOUN
ap-4612	199	2	.	.	PUNCT
ap-4612	199	3	,	,	PUNCT
ap-4612	199	4	3	3	X
ap-4612	199	5	,	,	PUNCT
ap-4612	199	6	98–119	98–119	NUM
ap-4612	199	7	(	(	PUNCT
ap-4612	199	8	1966	1966	NUM
ap-4612	199	9	)	)	PUNCT
ap-4612	199	10	.	.	PUNCT
ap-4612	200	1	https://doi.org/10.1007/bf01645448	https://doi.org/10.1007/bf01645448	X
ap-4612	201	1	[	[	X
ap-4612	201	2	6	6	NUM
ap-4612	201	3	]	]	X
ap-4612	201	4	j.p	j.p	PROPN
ap-4612	201	5	.	.	PROPN
ap-4612	201	6	antoine	antoine	PROPN
ap-4612	201	7	.	.	PUNCT
ap-4612	201	8	dirac	dirac	NOUN
ap-4612	201	9	formalism	formalism	NOUN
ap-4612	201	10	and	and	CCONJ
ap-4612	201	11	symmetry	symmetry	NOUN
ap-4612	201	12	problems	problem	NOUN
ap-4612	201	13	in	in	ADP
ap-4612	201	14	quantum	quantum	ADJ
ap-4612	201	15	mechanics	mechanic	NOUN
ap-4612	201	16	.	.	PUNCT
ap-4612	201	17	i.	i.	PROPN
ap-4612	201	18	general	general	PROPN
ap-4612	201	19	dirac	dirac	PROPN
ap-4612	201	20	formalism	formalism	NOUN
ap-4612	201	21	,	,	PUNCT
ap-4612	201	22	j.	j.	PROPN
ap-4612	201	23	math	math	PROPN
ap-4612	201	24	.	.	PUNCT
ap-4612	202	1	phys	phy	NOUN
ap-4612	202	2	.	.	PUNCT
ap-4612	202	3	,	,	PUNCT
ap-4612	202	4	10	10	NUM
ap-4612	202	5	,	,	PUNCT
ap-4612	202	6	53–69	53–69	NUM
ap-4612	202	7	(	(	PUNCT
ap-4612	202	8	1969	1969	NUM
ap-4612	202	9	)	)	PUNCT
ap-4612	202	10	.	.	PUNCT
ap-4612	203	1	https://doi.org/10.1063/1.1664761	https://doi.org/10.1063/1.1664761	PROPN
ap-4612	204	1	[	[	X
ap-4612	204	2	7	7	NUM
ap-4612	204	3	]	]	X
ap-4612	204	4	o.	o.	PROPN
ap-4612	204	5	melsheimer	melsheimer	PROPN
ap-4612	204	6	.	.	PUNCT
ap-4612	205	1	rigged	rig	VERB
ap-4612	205	2	hilbert	hilbert	NOUN
ap-4612	205	3	space	space	NOUN
ap-4612	205	4	formalism	formalism	NOUN
ap-4612	205	5	as	as	ADP
ap-4612	205	6	an	an	DET
ap-4612	205	7	extended	extended	ADJ
ap-4612	205	8	mathematical	mathematical	ADJ
ap-4612	205	9	formalism	formalism	NOUN
ap-4612	205	10	for	for	ADP
ap-4612	205	11	quantum	quantum	NOUN
ap-4612	205	12	systems	system	NOUN
ap-4612	205	13	.	.	PUNCT
ap-4612	206	1	i.	i.	PROPN
ap-4612	206	2	general	general	PROPN
ap-4612	206	3	theory	theory	PROPN
ap-4612	206	4	,	,	PUNCT
ap-4612	206	5	j.	j.	PROPN
ap-4612	206	6	math	math	PROPN
ap-4612	206	7	.	.	PUNCT
ap-4612	207	1	phys	phy	NOUN
ap-4612	207	2	.	.	PUNCT
ap-4612	207	3	,	,	PUNCT
ap-4612	207	4	15	15	NUM
ap-4612	207	5	,	,	PUNCT
ap-4612	207	6	902–916	902–916	NUM
ap-4612	207	7	(	(	PUNCT
ap-4612	207	8	1974	1974	NUM
ap-4612	207	9	)	)	PUNCT
ap-4612	207	10	.	.	PUNCT
ap-4612	208	1	https://doi.org/10.1063/1.1666769	https://doi.org/10.1063/1.1666769	NOUN
ap-4612	209	1	[	[	X
ap-4612	209	2	8	8	NUM
ap-4612	209	3	]	]	X
ap-4612	209	4	a.	a.	PROPN
ap-4612	209	5	bohm	bohm	PROPN
ap-4612	209	6	.	.	PUNCT
ap-4612	210	1	the	the	DET
ap-4612	210	2	rigged	rig	VERB
ap-4612	210	3	hilbert	hilbert	NOUN
ap-4612	210	4	space	space	NOUN
ap-4612	210	5	and	and	CCONJ
ap-4612	210	6	quantum	quantum	NOUN
ap-4612	210	7	mechanics	mechanic	NOUN
ap-4612	210	8	,	,	PUNCT
ap-4612	210	9	springer	springer	NOUN
ap-4612	210	10	lecture	lecture	NOUN
ap-4612	210	11	notes	note	NOUN
ap-4612	210	12	in	in	ADP
ap-4612	210	13	physics	physics	NOUN
ap-4612	210	14	,	,	PUNCT
ap-4612	210	15	78	78	NUM
ap-4612	210	16	.	.	PUNCT
ap-4612	210	17	springer	springer	NOUN
ap-4612	210	18	,	,	PUNCT
ap-4612	210	19	berlin	berlin	PROPN
ap-4612	210	20	,	,	PUNCT
ap-4612	210	21	1978	1978	NUM
ap-4612	210	22	.	.	PUNCT
ap-4612	211	1	[	[	X
ap-4612	211	2	9	9	NUM
ap-4612	211	3	]	]	PUNCT
ap-4612	211	4	a.	a.	PROPN
ap-4612	211	5	bohm	bohm	PROPN
ap-4612	211	6	and	and	CCONJ
ap-4612	211	7	m.	m.	NOUN
ap-4612	211	8	gadella	gadella	PROPN
ap-4612	211	9	.	.	PUNCT
ap-4612	212	1	dirac	dirac	PROPN
ap-4612	212	2	kets	kets	PROPN
ap-4612	212	3	,	,	PUNCT
ap-4612	212	4	gamow	gamow	NOUN
ap-4612	212	5	vectors	vector	NOUN
ap-4612	212	6	and	and	CCONJ
ap-4612	212	7	gelfand	gelfand	ADJ
ap-4612	212	8	triplets	triplet	NOUN
ap-4612	212	9	,	,	PUNCT
ap-4612	212	10	springer	springer	NOUN
ap-4612	212	11	lecture	lecture	NOUN
ap-4612	212	12	notes	note	NOUN
ap-4612	212	13	in	in	ADP
ap-4612	212	14	physics	physics	NOUN
ap-4612	212	15	,	,	PUNCT
ap-4612	212	16	348	348	NUM
ap-4612	212	17	,	,	PUNCT
ap-4612	212	18	springer	springer	NOUN
ap-4612	212	19	,	,	PUNCT
ap-4612	212	20	berlin	berlin	PROPN
ap-4612	212	21	,	,	PUNCT
ap-4612	212	22	1989	1989	NUM
ap-4612	212	23	.	.	PUNCT
ap-4612	213	1	[	[	X
ap-4612	213	2	10	10	NUM
ap-4612	213	3	]	]	PUNCT
ap-4612	213	4	m.	m.	NOUN
ap-4612	213	5	gadella	gadella	PROPN
ap-4612	213	6	and	and	CCONJ
ap-4612	213	7	f.	f.	PROPN
ap-4612	213	8	gómez	gómez	PROPN
ap-4612	213	9	.	.	PUNCT
ap-4612	214	1	a	a	DET
ap-4612	214	2	unified	unified	ADJ
ap-4612	214	3	mathematical	mathematical	ADJ
ap-4612	214	4	formalism	formalism	NOUN
ap-4612	214	5	for	for	ADP
ap-4612	214	6	the	the	DET
ap-4612	214	7	dirac	dirac	NOUN
ap-4612	214	8	formulation	formulation	NOUN
ap-4612	214	9	of	of	ADP
ap-4612	214	10	quantum	quantum	ADJ
ap-4612	214	11	mechanics	mechanic	NOUN
ap-4612	214	12	,	,	PUNCT
ap-4612	214	13	found	find	VERB
ap-4612	214	14	.	.	PUNCT
ap-4612	215	1	phys	phy	NOUN
ap-4612	215	2	.	.	PUNCT
ap-4612	215	3	,	,	PUNCT
ap-4612	215	4	32	32	NUM
ap-4612	215	5	,	,	PUNCT
ap-4612	215	6	815–869	815–869	NUM
ap-4612	215	7	(	(	PUNCT
ap-4612	215	8	2002	2002	NUM
ap-4612	215	9	)	)	PUNCT
ap-4612	215	10	.	.	PUNCT
ap-4612	216	1	https://doi.org/10.1023/a:1016069311589	https://doi.org/10.1023/a:1016069311589	NOUN
ap-4612	217	1	[	[	X
ap-4612	217	2	11	11	NUM
ap-4612	217	3	]	]	X
ap-4612	217	4	e.	e.	PROPN
ap-4612	217	5	celeghini	celeghini	PROPN
ap-4612	217	6	and	and	CCONJ
ap-4612	217	7	m.a	m.a	PROPN
ap-4612	217	8	.	.	PROPN
ap-4612	217	9	del	del	PROPN
ap-4612	217	10	olmo	olmo	PROPN
ap-4612	217	11	.	.	PUNCT
ap-4612	217	12	quantum	quantum	PROPN
ap-4612	217	13	physics	physics	NOUN
ap-4612	217	14	and	and	CCONJ
ap-4612	217	15	signal	signal	NOUN
ap-4612	217	16	processing	processing	NOUN
ap-4612	217	17	in	in	ADP
ap-4612	217	18	rigged	rig	VERB
ap-4612	217	19	hilbert	hilbert	NOUN
ap-4612	217	20	spaces	space	NOUN
ap-4612	217	21	by	by	ADP
ap-4612	217	22	means	mean	NOUN
ap-4612	217	23	of	of	ADP
ap-4612	217	24	special	special	ADJ
ap-4612	217	25	functions	function	NOUN
ap-4612	217	26	,	,	PUNCT
ap-4612	217	27	lie	lie	NOUN
ap-4612	217	28	algebras	algebra	NOUN
ap-4612	217	29	and	and	CCONJ
ap-4612	217	30	fourier	fourier	NOUN
ap-4612	217	31	-	-	PUNCT
ap-4612	217	32	like	like	ADJ
ap-4612	217	33	transforms	transform	VERB
ap-4612	217	34	j.	j.	PROPN
ap-4612	217	35	phys	phys	PROPN
ap-4612	217	36	.	.	PUNCT
ap-4612	218	1	:	:	PUNCT
ap-4612	218	2	conf	conf	PROPN
ap-4612	218	3	.	.	PUNCT
ap-4612	219	1	ser	ser	PROPN
ap-4612	219	2	.	.	PROPN
ap-4612	220	1	597	597	NUM
ap-4612	220	2	,	,	PUNCT
ap-4612	220	3	012022	012022	NUM
ap-4612	220	4	(	(	PUNCT
ap-4612	220	5	2015	2015	NUM
ap-4612	220	6	)	)	PUNCT
ap-4612	220	7	.	.	PUNCT
ap-4612	221	1	https://doi.org/10.1088/1742-6596/597/1/012022	https://doi.org/10.1088/1742-6596/597/1/012022	X
ap-4612	222	1	[	[	X
ap-4612	222	2	12	12	NUM
ap-4612	222	3	]	]	X
ap-4612	222	4	e.	e.	PROPN
ap-4612	222	5	celeghini	celeghini	PROPN
ap-4612	222	6	,	,	PUNCT
ap-4612	222	7	m.	m.	NOUN
ap-4612	222	8	gadella	gadella	PROPN
ap-4612	222	9	and	and	CCONJ
ap-4612	222	10	m.a	m.a	PROPN
ap-4612	222	11	.	.	PROPN
ap-4612	222	12	del	del	PROPN
ap-4612	222	13	olmo	olmo	PROPN
ap-4612	222	14	.	.	PUNCT
ap-4612	222	15	applications	application	NOUN
ap-4612	222	16	of	of	ADP
ap-4612	222	17	rigged	rig	VERB
ap-4612	222	18	hilbert	hilbert	NOUN
ap-4612	222	19	spaces	space	NOUN
ap-4612	222	20	in	in	ADP
ap-4612	222	21	quantum	quantum	ADJ
ap-4612	222	22	mechanics	mechanic	NOUN
ap-4612	222	23	and	and	CCONJ
ap-4612	222	24	signal	signal	NOUN
ap-4612	222	25	processing	processing	NOUN
ap-4612	222	26	.	.	PUNCT
ap-4612	223	1	j.	j.	PROPN
ap-4612	223	2	math	math	PROPN
ap-4612	223	3	.	.	PUNCT
ap-4612	224	1	phys	phy	NOUN
ap-4612	224	2	.	.	PUNCT
ap-4612	225	1	57	57	NUM
ap-4612	225	2	(	(	PUNCT
ap-4612	225	3	2016	2016	NUM
ap-4612	225	4	)	)	PUNCT
ap-4612	225	5	072105	072105	NUM
ap-4612	225	6	.	.	PUNCT
ap-4612	226	1	http://dx.doi.org/10.1063/1.4958725	http://dx.doi.org/10.1063/1.4958725	PROPN
ap-4612	226	2	[	[	X
ap-4612	226	3	13	13	NUM
ap-4612	226	4	]	]	X
ap-4612	226	5	wu	wu	PROPN
ap-4612	226	6	-	-	PUNCT
ap-4612	226	7	ki	ki	PROPN
ap-4612	226	8	tung	tung	PROPN
ap-4612	226	9	.	.	PUNCT
ap-4612	227	1	group	group	NOUN
ap-4612	227	2	theory	theory	NOUN
ap-4612	227	3	in	in	ADP
ap-4612	227	4	physics	physics	PROPN
ap-4612	227	5	,	,	PUNCT
ap-4612	227	6	chap	chap	NOUN
ap-4612	227	7	.	.	PUNCT
ap-4612	228	1	6	6	NUM
ap-4612	228	2	,	,	PUNCT
ap-4612	228	3	world	world	NOUN
ap-4612	228	4	scientific	scientific	NOUN
ap-4612	228	5	,	,	PUNCT
ap-4612	228	6	singapore	singapore	PROPN
ap-4612	228	7	,	,	PUNCT
ap-4612	228	8	1985	1985	NUM
ap-4612	228	9	.	.	PUNCT
ap-4612	229	1	383	383	NUM
ap-4612	229	2	enrico	enrico	PROPN
ap-4612	229	3	celeghini	celeghini	PROPN
ap-4612	229	4	,	,	PUNCT
ap-4612	229	5	manuel	manuel	PROPN
ap-4612	229	6	gadella	gadella	PROPN
ap-4612	229	7	,	,	PUNCT
ap-4612	229	8	mariano	mariano	PROPN
ap-4612	229	9	a	a	DET
ap-4612	229	10	del	del	PROPN
ap-4612	229	11	olmo	olmo	PROPN
ap-4612	229	12	acta	acta	PROPN
ap-4612	229	13	polytechnica	polytechnica	PROPN
ap-4612	229	14	[	[	X
ap-4612	229	15	14	14	NUM
ap-4612	229	16	]	]	PUNCT
ap-4612	229	17	k.	k.	PROPN
ap-4612	229	18	maurin	maurin	PROPN
ap-4612	229	19	.	.	PUNCT
ap-4612	230	1	bull	bull	NOUN
ap-4612	230	2	.	.	PUNCT
ap-4612	231	1	acad	acad	PROPN
ap-4612	231	2	.	.	PUNCT
ap-4612	232	1	polon	polon	PROPN
ap-4612	232	2	.	.	PUNCT
ap-4612	233	1	sci	sci	PROPN
ap-4612	233	2	.	.	PUNCT
ap-4612	233	3	ser	ser	PROPN
ap-4612	233	4	.	.	PUNCT
ap-4612	234	1	sci	sci	PROPN
ap-4612	234	2	.	.	PROPN
ap-4612	234	3	math	math	PROPN
ap-4612	234	4	.	.	PUNCT
ap-4612	235	1	astronom	astronom	NOUN
ap-4612	235	2	.	.	PUNCT
ap-4612	236	1	phys	phy	NOUN
ap-4612	236	2	.	.	PUNCT
ap-4612	236	3	,	,	PUNCT
ap-4612	236	4	7	7	NUM
ap-4612	236	5	(	(	PUNCT
ap-4612	236	6	1959	1959	NUM
ap-4612	236	7	)	)	PUNCT
ap-4612	236	8	471	471	NUM
ap-4612	236	9	.	.	PUNCT
ap-4612	237	1	[	[	X
ap-4612	237	2	15	15	NUM
ap-4612	237	3	]	]	X
ap-4612	237	4	m.	m.	NOUN
ap-4612	237	5	gadella	gadella	PROPN
ap-4612	237	6	and	and	CCONJ
ap-4612	237	7	f.	f.	PROPN
ap-4612	237	8	gómez	gómez	PROPN
ap-4612	237	9	.	.	PUNCT
ap-4612	238	1	eigenfunction	eigenfunction	NOUN
ap-4612	238	2	expansions	expansion	NOUN
ap-4612	238	3	and	and	CCONJ
ap-4612	238	4	transformation	transformation	NOUN
ap-4612	238	5	theory	theory	NOUN
ap-4612	238	6	,	,	PUNCT
ap-4612	238	7	acta	acta	PROPN
ap-4612	238	8	appl	appl	PROPN
ap-4612	238	9	.	.	PROPN
ap-4612	238	10	math	math	PROPN
ap-4612	238	11	.	.	PUNCT
ap-4612	238	12	,	,	PUNCT
ap-4612	238	13	109	109	NUM
ap-4612	238	14	,	,	PUNCT
ap-4612	238	15	721–742	721–742	NUM
ap-4612	238	16	(	(	PUNCT
ap-4612	238	17	2010	2010	NUM
ap-4612	238	18	)	)	PUNCT
ap-4612	238	19	.	.	PUNCT
ap-4612	239	1	https://doi.org/10.1007/s10440-008-9342-z	https://doi.org/10.1007/s10440-008-9342-z	NOUN
ap-4612	240	1	[	[	X
ap-4612	240	2	16	16	NUM
ap-4612	240	3	]	]	PUNCT
ap-4612	240	4	k.	k.	PROPN
ap-4612	240	5	maurin	maurin	PROPN
ap-4612	240	6	.	.	PUNCT
ap-4612	241	1	general	general	ADJ
ap-4612	241	2	eigenfunction	eigenfunction	NOUN
ap-4612	241	3	expansions	expansion	NOUN
ap-4612	241	4	and	and	CCONJ
ap-4612	241	5	unitary	unitary	ADJ
ap-4612	241	6	representations	representation	NOUN
ap-4612	241	7	of	of	ADP
ap-4612	241	8	topological	topological	ADJ
ap-4612	241	9	groups	group	NOUN
ap-4612	241	10	,	,	PUNCT
ap-4612	241	11	monografie	monografie	ADJ
ap-4612	241	12	matematyczne	matematyczne	PROPN
ap-4612	241	13	,	,	PUNCT
ap-4612	241	14	48	48	NUM
ap-4612	241	15	,	,	PUNCT
ap-4612	241	16	pwn	pwn	NOUN
ap-4612	241	17	-	-	PUNCT
ap-4612	241	18	polish	polish	ADJ
ap-4612	241	19	scientific	scientific	ADJ
ap-4612	241	20	publishers	publisher	NOUN
ap-4612	241	21	,	,	PUNCT
ap-4612	241	22	warsaw	warsaw	PROPN
ap-4612	241	23	,	,	PUNCT
ap-4612	241	24	1968	1968	NUM
ap-4612	241	25	.	.	PUNCT
ap-4612	242	1	[	[	X
ap-4612	242	2	17	17	NUM
ap-4612	242	3	]	]	PUNCT
ap-4612	242	4	m.	m.	NOUN
ap-4612	242	5	gadella	gadella	PROPN
ap-4612	242	6	,	,	PUNCT
ap-4612	242	7	f.	f.	PROPN
ap-4612	242	8	gómez	gómez	PROPN
ap-4612	242	9	-	-	PUNCT
ap-4612	242	10	cubillo	cubillo	PROPN
ap-4612	242	11	,	,	PUNCT
ap-4612	242	12	l.	l.	PROPN
ap-4612	242	13	rodríguez	rodríguez	PROPN
ap-4612	242	14	and	and	CCONJ
ap-4612	242	15	s.	s.	PROPN
ap-4612	242	16	wickramasekara	wickramasekara	PROPN
ap-4612	242	17	.	.	PUNCT
ap-4612	243	1	point	point	NOUN
ap-4612	243	2	-	-	PUNCT
ap-4612	243	3	form	form	NOUN
ap-4612	243	4	dynamics	dynamic	NOUN
ap-4612	243	5	of	of	ADP
ap-4612	243	6	quasistable	quasistable	ADJ
ap-4612	243	7	states	state	NOUN
ap-4612	243	8	,	,	PUNCT
ap-4612	243	9	j.	j.	PROPN
ap-4612	243	10	math	math	PROPN
ap-4612	243	11	.	.	PUNCT
ap-4612	244	1	phys	phy	NOUN
ap-4612	244	2	.	.	PUNCT
ap-4612	244	3	,	,	PUNCT
ap-4612	244	4	54	54	NUM
ap-4612	244	5	,	,	PUNCT
ap-4612	244	6	072303	072303	NUM
ap-4612	244	7	(	(	PUNCT
ap-4612	244	8	2013	2013	NUM
ap-4612	244	9	)	)	PUNCT
ap-4612	244	10	.	.	PUNCT
ap-4612	245	1	https://doi.org/10.1063/1.4811563	https://doi.org/10.1063/1.4811563	PROPN
ap-4612	246	1	[	[	X
ap-4612	246	2	18	18	NUM
ap-4612	246	3	]	]	X
ap-4612	246	4	p.a.m	p.a.m	PROPN
ap-4612	246	5	.	.	PUNCT
ap-4612	246	6	dirac	dirac	PROPN
ap-4612	246	7	.	.	PUNCT
ap-4612	247	1	the	the	DET
ap-4612	247	2	principles	principle	NOUN
ap-4612	247	3	of	of	ADP
ap-4612	247	4	quantum	quantum	NOUN
ap-4612	247	5	mechanics	mechanic	NOUN
ap-4612	247	6	,	,	PUNCT
ap-4612	247	7	clarendon	clarendon	PROPN
ap-4612	247	8	press	press	NOUN
ap-4612	247	9	,	,	PUNCT
ap-4612	247	10	oxford	oxford	PROPN
ap-4612	247	11	,	,	PUNCT
ap-4612	247	12	1958	1958	NUM
ap-4612	247	13	.	.	PUNCT
ap-4612	248	1	[	[	X
ap-4612	248	2	19	19	NUM
ap-4612	248	3	]	]	X
ap-4612	248	4	a.r	a.r	PROPN
ap-4612	248	5	.	.	PROPN
ap-4612	248	6	bohm	bohm	PROPN
ap-4612	248	7	,	,	PUNCT
ap-4612	248	8	m.	m.	NOUN
ap-4612	248	9	gadella	gadella	PROPN
ap-4612	248	10	and	and	CCONJ
ap-4612	248	11	p.	p.	PROPN
ap-4612	248	12	kielanowski	kielanowski	PROPN
ap-4612	248	13	.	.	PUNCT
ap-4612	249	1	time	time	PROPN
ap-4612	249	2	asymmetric	asymmetric	ADJ
ap-4612	249	3	quantum	quantum	ADJ
ap-4612	249	4	mechanics	mechanic	NOUN
ap-4612	249	5	,	,	PUNCT
ap-4612	249	6	sigma	sigma	PROPN
ap-4612	249	7	,	,	PUNCT
ap-4612	249	8	7	7	NUM
ap-4612	249	9	,	,	PUNCT
ap-4612	249	10	086	086	NUM
ap-4612	249	11	(	(	PUNCT
ap-4612	249	12	2011	2011	NUM
ap-4612	249	13	)	)	PUNCT
ap-4612	249	14	.	.	PUNCT
ap-4612	250	1	https://doi.org/10.3842/sigma.2011.086	https://doi.org/10.3842/sigma.2011.086	PROPN
ap-4612	250	2	[	[	X
ap-4612	250	3	20	20	NUM
ap-4612	250	4	]	]	PUNCT
ap-4612	250	5	j.	j.	PROPN
ap-4612	250	6	horvath	horvath	PROPN
ap-4612	250	7	,	,	PUNCT
ap-4612	250	8	topological	topological	ADJ
ap-4612	250	9	vector	vector	NOUN
ap-4612	250	10	spaces	space	NOUN
ap-4612	250	11	and	and	CCONJ
ap-4612	250	12	distributions	distribution	NOUN
ap-4612	250	13	,	,	PUNCT
ap-4612	250	14	addison	addison	PROPN
ap-4612	250	15	-	-	PUNCT
ap-4612	250	16	wesley	wesley	PROPN
ap-4612	250	17	,	,	PUNCT
ap-4612	250	18	reading	read	VERB
ap-4612	250	19	ma	ma	PROPN
ap-4612	250	20	.	.	PROPN
ap-4612	250	21	,	,	PUNCT
ap-4612	250	22	1966	1966	NUM
ap-4612	250	23	.	.	PUNCT
ap-4612	251	1	[	[	X
ap-4612	251	2	21	21	NUM
ap-4612	251	3	]	]	PUNCT
ap-4612	251	4	a.	a.	NOUN
ap-4612	251	5	pietsch	pietsch	PROPN
ap-4612	251	6	,	,	PUNCT
ap-4612	251	7	nuclear	nuclear	ADJ
ap-4612	251	8	topological	topological	ADJ
ap-4612	251	9	vector	vector	NOUN
ap-4612	251	10	spaces	space	NOUN
ap-4612	251	11	,	,	PUNCT
ap-4612	251	12	springer	springer	NOUN
ap-4612	251	13	,	,	PUNCT
ap-4612	251	14	berlin	berlin	PROPN
ap-4612	251	15	,	,	PUNCT
ap-4612	251	16	1972	1972	NUM
ap-4612	251	17	.	.	PUNCT
ap-4612	252	1	[	[	X
ap-4612	252	2	22	22	NUM
ap-4612	252	3	]	]	PUNCT
ap-4612	252	4	m.	m.	NOUN
ap-4612	252	5	reed	reed	PROPN
ap-4612	252	6	and	and	CCONJ
ap-4612	252	7	b.	b.	PROPN
ap-4612	252	8	simon	simon	PROPN
ap-4612	252	9	,	,	PUNCT
ap-4612	252	10	functional	functional	ADJ
ap-4612	252	11	analysis	analysis	NOUN
ap-4612	252	12	,	,	PUNCT
ap-4612	252	13	academic	academic	ADJ
ap-4612	252	14	,	,	PUNCT
ap-4612	252	15	new	new	PROPN
ap-4612	252	16	york	york	PROPN
ap-4612	252	17	,	,	PUNCT
ap-4612	252	18	1972	1972	NUM
ap-4612	252	19	.	.	PUNCT
ap-4612	253	1	384	384	NUM
ap-4612	253	2	acta	acta	PROPN
ap-4612	253	3	polytechnica	polytechnica	PROPN
ap-4612	253	4	57(6):379–384	57(6):379–384	PROPN
ap-4612	253	5	,	,	PUNCT
ap-4612	253	6	2017	2017	NUM
ap-4612	253	7	1	1	NUM
ap-4612	253	8	introduction	introduction	NOUN
ap-4612	253	9	2	2	NUM
ap-4612	253	10	rigged	rig	VERB
ap-4612	253	11	hilbert	hilbert	NOUN
ap-4612	253	12	spaces	space	VERB
ap-4612	253	13	3	3	NUM
ap-4612	253	14	a	a	DET
ap-4612	253	15	paradigmatic	paradigmatic	ADJ
ap-4612	253	16	case	case	NOUN
ap-4612	253	17	:	:	PUNCT
ap-4612	253	18	rhs	rhs	PROPN
ap-4612	253	19	for	for	ADP
ap-4612	253	20	so(2	so(2	NOUN
ap-4612	253	21	)	)	PUNCT
ap-4612	253	22	3.1	3.1	NUM
ap-4612	253	23	uir	uir	NOUN
ap-4612	253	24	supported	support	VERB
ap-4612	253	25	by	by	ADP
ap-4612	253	26	the	the	DET
ap-4612	253	27	hs	hs	PROPN
ap-4612	253	28	l2[0,2	l2[0,2	PROPN
ap-4612	253	29	]	]	X
ap-4612	253	30	3.2	3.2	NUM
ap-4612	253	31	uir	uir	NOUN
ap-4612	253	32	on	on	ADP
ap-4612	253	33	an	an	DET
ap-4612	253	34	infinite	infinite	ADJ
ap-4612	253	35	-	-	PUNCT
ap-4612	253	36	d	d	NOUN
ap-4612	253	37	separable	separable	ADJ
ap-4612	253	38	hs	hs	PROPN
ap-4612	253	39	3.3	3.3	NUM
ap-4612	253	40	action	action	NOUN
ap-4612	253	41	of	of	ADP
ap-4612	253	42	so(2	so(2	NOUN
ap-4612	253	43	)	)	PUNCT
ap-4612	253	44	on	on	ADP
ap-4612	253	45	the	the	DET
ap-4612	253	46	rhs	rhs	PROPN
ap-4612	253	47	4	4	NUM
ap-4612	253	48	conclusions	conclusion	NOUN
ap-4612	253	49	acknowledgements	acknowledgement	NOUN
ap-4612	253	50	references	reference	NOUN