id	sid	tid	token	lemma	pos
ejpam-6613	1	1	european	european	PROPN
ejpam-6613	1	2	journal	journal	PROPN
ejpam-6613	1	3	of	of	ADP
ejpam-6613	1	4	pure	pure	ADJ
ejpam-6613	1	5	and	and	CCONJ
ejpam-6613	1	6	applied	applied	ADJ
ejpam-6613	1	7	mathematics	mathematic	NOUN
ejpam-6613	1	8	2025	2025	NUM
ejpam-6613	1	9	,	,	PUNCT
ejpam-6613	1	10	vol	vol	NOUN
ejpam-6613	1	11	.	.	PROPN
ejpam-6613	1	12	18	18	NUM
ejpam-6613	1	13	,	,	PUNCT
ejpam-6613	1	14	issue	issue	NOUN
ejpam-6613	1	15	4	4	NUM
ejpam-6613	1	16	,	,	PUNCT
ejpam-6613	1	17	article	article	NOUN
ejpam-6613	1	18	number	number	NOUN
ejpam-6613	1	19	6613	6613	NUM
ejpam-6613	1	20	issn	issn	PROPN
ejpam-6613	1	21	1307	1307	NUM
ejpam-6613	1	22	-	-	SYM
ejpam-6613	1	23	5543	5543	NUM
ejpam-6613	1	24	–	–	PUNCT
ejpam-6613	1	25	ejpam.com	ejpam.com	X
ejpam-6613	1	26	published	publish	VERB
ejpam-6613	1	27	by	by	ADP
ejpam-6613	1	28	new	new	PROPN
ejpam-6613	1	29	york	york	PROPN
ejpam-6613	1	30	business	business	PROPN
ejpam-6613	1	31	global	global	ADJ
ejpam-6613	1	32	tensor	tensor	NOUN
ejpam-6613	1	33	product	product	NOUN
ejpam-6613	1	34	of	of	ADP
ejpam-6613	1	35	spaces	space	NOUN
ejpam-6613	1	36	with	with	ADP
ejpam-6613	1	37	generalized	generalized	ADJ
ejpam-6613	1	38	2	2	NUM
ejpam-6613	1	39	-	-	PUNCT
ejpam-6613	1	40	inner	inner	ADJ
ejpam-6613	1	41	product	product	NOUN
ejpam-6613	1	42	moreno	moreno	PROPN
ejpam-6613	1	43	luis1	luis1	PROPN
ejpam-6613	1	44	,	,	PUNCT
ejpam-6613	1	45	ferrer	ferrer	PROPN
ejpam-6613	1	46	osmin1,∗	osmin1,∗	PROPN
ejpam-6613	1	47	,	,	PUNCT
ejpam-6613	1	48	sierra	sierra	PROPN
ejpam-6613	1	49	arley1,2	arley1,2	PROPN
ejpam-6613	1	50	1	1	NUM
ejpam-6613	1	51	university	university	NOUN
ejpam-6613	1	52	of	of	ADP
ejpam-6613	1	53	sucre	sucre	NOUN
ejpam-6613	1	54	,	,	PUNCT
ejpam-6613	1	55	faculty	faculty	NOUN
ejpam-6613	1	56	of	of	ADP
ejpam-6613	1	57	education	education	NOUN
ejpam-6613	1	58	and	and	CCONJ
ejpam-6613	1	59	sciences	science	NOUN
ejpam-6613	1	60	,	,	PUNCT
ejpam-6613	1	61	department	department	NOUN
ejpam-6613	1	62	of	of	ADP
ejpam-6613	1	63	mathematics	mathematic	NOUN
ejpam-6613	1	64	,	,	PUNCT
ejpam-6613	1	65	sincelejo	sincelejo	ADJ
ejpam-6613	1	66	,	,	PUNCT
ejpam-6613	1	67	colombia	colombia	PROPN
ejpam-6613	1	68	2	2	NUM
ejpam-6613	1	69	caribbean	caribbean	ADJ
ejpam-6613	1	70	university	university	NOUN
ejpam-6613	1	71	corporation	corporation	NOUN
ejpam-6613	1	72	,	,	PUNCT
ejpam-6613	1	73	department	department	NOUN
ejpam-6613	1	74	of	of	ADP
ejpam-6613	1	75	basic	basic	ADJ
ejpam-6613	1	76	sciences	science	NOUN
ejpam-6613	1	77	,	,	PUNCT
ejpam-6613	1	78	sincelejo	sincelejo	ADJ
ejpam-6613	1	79	,	,	PUNCT
ejpam-6613	1	80	colombia	colombia	PROPN
ejpam-6613	1	81	abstract	abstract	NOUN
ejpam-6613	1	82	.	.	PUNCT
ejpam-6613	2	1	in	in	ADP
ejpam-6613	2	2	this	this	DET
ejpam-6613	2	3	work	work	NOUN
ejpam-6613	2	4	,	,	PUNCT
ejpam-6613	2	5	we	we	PRON
ejpam-6613	2	6	introduce	introduce	VERB
ejpam-6613	2	7	the	the	DET
ejpam-6613	2	8	notion	notion	NOUN
ejpam-6613	2	9	of	of	ADP
ejpam-6613	2	10	tensor	tensor	NOUN
ejpam-6613	2	11	product	product	NOUN
ejpam-6613	2	12	of	of	ADP
ejpam-6613	2	13	spaces	space	NOUN
ejpam-6613	2	14	with	with	ADP
ejpam-6613	2	15	a	a	DET
ejpam-6613	2	16	generalized	generalized	ADJ
ejpam-6613	2	17	2	2	NUM
ejpam-6613	2	18	-	-	PUNCT
ejpam-6613	2	19	inner	inner	ADJ
ejpam-6613	2	20	product	product	NOUN
ejpam-6613	2	21	(	(	PUNCT
ejpam-6613	2	22	see	see	VERB
ejpam-6613	2	23	definition	definition	NOUN
ejpam-6613	2	24	5	5	NUM
ejpam-6613	2	25	)	)	PUNCT
ejpam-6613	2	26	,	,	PUNCT
ejpam-6613	2	27	and	and	CCONJ
ejpam-6613	2	28	we	we	PRON
ejpam-6613	2	29	establish	establish	VERB
ejpam-6613	2	30	several	several	ADJ
ejpam-6613	2	31	interesting	interesting	ADJ
ejpam-6613	2	32	properties	property	NOUN
ejpam-6613	2	33	(	(	PUNCT
ejpam-6613	2	34	see	see	VERB
ejpam-6613	2	35	proposition	proposition	NOUN
ejpam-6613	2	36	5	5	NUM
ejpam-6613	2	37	)	)	PUNCT
ejpam-6613	2	38	,	,	PUNCT
ejpam-6613	2	39	thereby	thereby	ADV
ejpam-6613	2	40	generalizing	generalize	VERB
ejpam-6613	2	41	the	the	DET
ejpam-6613	2	42	classical	classical	ADJ
ejpam-6613	2	43	properties	property	NOUN
ejpam-6613	2	44	of	of	ADP
ejpam-6613	2	45	the	the	DET
ejpam-6613	2	46	tensor	tensor	NOUN
ejpam-6613	2	47	product	product	NOUN
ejpam-6613	2	48	of	of	ADP
ejpam-6613	2	49	inner	inner	ADJ
ejpam-6613	2	50	product	product	NOUN
ejpam-6613	2	51	spaces	space	VERB
ejpam-6613	2	52	.	.	PUNCT
ejpam-6613	3	1	moreover	moreover	ADV
ejpam-6613	3	2	,	,	PUNCT
ejpam-6613	3	3	we	we	PRON
ejpam-6613	3	4	equip	equip	VERB
ejpam-6613	3	5	this	this	DET
ejpam-6613	3	6	tensor	tensor	NOUN
ejpam-6613	3	7	product	product	NOUN
ejpam-6613	3	8	with	with	ADP
ejpam-6613	3	9	a	a	DET
ejpam-6613	3	10	mapping	mapping	NOUN
ejpam-6613	3	11	that	that	PRON
ejpam-6613	3	12	defines	define	VERB
ejpam-6613	3	13	a	a	DET
ejpam-6613	3	14	generalized	generalized	ADJ
ejpam-6613	3	15	2	2	NUM
ejpam-6613	3	16	-	-	PUNCT
ejpam-6613	3	17	inner	inner	ADJ
ejpam-6613	3	18	product	product	NOUN
ejpam-6613	3	19	(	(	PUNCT
ejpam-6613	3	20	see	see	VERB
ejpam-6613	3	21	theorem	theorem	NOUN
ejpam-6613	3	22	3	3	NUM
ejpam-6613	3	23	)	)	PUNCT
ejpam-6613	3	24	and	and	CCONJ
ejpam-6613	3	25	,	,	PUNCT
ejpam-6613	3	26	consequently	consequently	ADV
ejpam-6613	3	27	,	,	PUNCT
ejpam-6613	3	28	endow	endow	VERB
ejpam-6613	3	29	it	it	PRON
ejpam-6613	3	30	with	with	ADP
ejpam-6613	3	31	a	a	DET
ejpam-6613	3	32	generalized	generalized	ADJ
ejpam-6613	3	33	2	2	NUM
ejpam-6613	3	34	-	-	PUNCT
ejpam-6613	3	35	norm	norm	NOUN
ejpam-6613	3	36	(	(	PUNCT
ejpam-6613	3	37	see	see	VERB
ejpam-6613	3	38	theorem	theorem	NOUN
ejpam-6613	3	39	1	1	NUM
ejpam-6613	3	40	)	)	PUNCT
ejpam-6613	3	41	.	.	PUNCT
ejpam-6613	4	1	in	in	ADP
ejpam-6613	4	2	this	this	DET
ejpam-6613	4	3	context	context	NOUN
ejpam-6613	4	4	,	,	PUNCT
ejpam-6613	4	5	we	we	PRON
ejpam-6613	4	6	also	also	ADV
ejpam-6613	4	7	define	define	VERB
ejpam-6613	4	8	the	the	DET
ejpam-6613	4	9	tensor	tensor	NOUN
ejpam-6613	4	10	product	product	NOUN
ejpam-6613	4	11	of	of	ADP
ejpam-6613	4	12	linear	linear	PROPN
ejpam-6613	4	13	operators	operator	NOUN
ejpam-6613	4	14	(	(	PUNCT
ejpam-6613	4	15	see	see	VERB
ejpam-6613	4	16	definition	definition	NOUN
ejpam-6613	4	17	9	9	NUM
ejpam-6613	4	18	)	)	PUNCT
ejpam-6613	4	19	and	and	CCONJ
ejpam-6613	4	20	prove	prove	VERB
ejpam-6613	4	21	a	a	DET
ejpam-6613	4	22	series	series	NOUN
ejpam-6613	4	23	of	of	ADP
ejpam-6613	4	24	results	result	NOUN
ejpam-6613	4	25	for	for	ADP
ejpam-6613	4	26	example	example	NOUN
ejpam-6613	4	27	,	,	PUNCT
ejpam-6613	4	28	that	that	SCONJ
ejpam-6613	4	29	the	the	DET
ejpam-6613	4	30	tensor	tensor	NOUN
ejpam-6613	4	31	product	product	NOUN
ejpam-6613	4	32	of	of	ADP
ejpam-6613	4	33	two	two	NUM
ejpam-6613	4	34	2	2	NUM
ejpam-6613	4	35	-	-	PUNCT
ejpam-6613	4	36	bounded	bound	VERB
ejpam-6613	4	37	linear	linear	PROPN
ejpam-6613	4	38	operators	operator	NOUN
ejpam-6613	4	39	is	be	AUX
ejpam-6613	4	40	again	again	ADV
ejpam-6613	4	41	2	2	NUM
ejpam-6613	4	42	-	-	PUNCT
ejpam-6613	4	43	bounded	bound	VERB
ejpam-6613	4	44	under	under	ADP
ejpam-6613	4	45	the	the	DET
ejpam-6613	4	46	tensor	tensor	NOUN
ejpam-6613	4	47	product	product	NOUN
ejpam-6613	4	48	(	(	PUNCT
ejpam-6613	4	49	see	see	VERB
ejpam-6613	4	50	proposition	proposition	NOUN
ejpam-6613	4	51	10	10	NUM
ejpam-6613	4	52	)	)	PUNCT
ejpam-6613	4	53	.	.	PUNCT
ejpam-6613	5	1	2020	2020	NUM
ejpam-6613	5	2	mathematics	mathematic	NOUN
ejpam-6613	5	3	subject	subject	NOUN
ejpam-6613	5	4	classifications	classification	NOUN
ejpam-6613	5	5	:	:	PUNCT
ejpam-6613	5	6	46m05	46m05	NUM
ejpam-6613	5	7	,	,	PUNCT
ejpam-6613	5	8	47a80	47a80	NUM
ejpam-6613	5	9	,	,	PUNCT
ejpam-6613	5	10	46b28	46b28	NUM
ejpam-6613	5	11	,	,	PUNCT
ejpam-6613	5	12	46c50	46c50	DET
ejpam-6613	5	13	key	key	ADJ
ejpam-6613	5	14	words	word	NOUN
ejpam-6613	5	15	and	and	CCONJ
ejpam-6613	5	16	phrases	phrase	NOUN
ejpam-6613	5	17	:	:	PUNCT
ejpam-6613	5	18	tensor	tensor	NOUN
ejpam-6613	5	19	product	product	NOUN
ejpam-6613	5	20	,	,	PUNCT
ejpam-6613	5	21	generalized	generalize	VERB
ejpam-6613	5	22	2	2	NUM
ejpam-6613	5	23	-	-	PUNCT
ejpam-6613	5	24	inner	inner	ADJ
ejpam-6613	5	25	product	product	NOUN
ejpam-6613	5	26	,	,	PUNCT
ejpam-6613	5	27	2	2	NUM
ejpam-6613	5	28	-	-	PUNCT
ejpam-6613	5	29	norm	norm	NOUN
ejpam-6613	5	30	1	1	NUM
ejpam-6613	5	31	.	.	PUNCT
ejpam-6613	5	32	introduction	introduction	NOUN
ejpam-6613	5	33	the	the	DET
ejpam-6613	5	34	ideas	idea	NOUN
ejpam-6613	5	35	that	that	PRON
ejpam-6613	5	36	gave	give	VERB
ejpam-6613	5	37	rise	rise	NOUN
ejpam-6613	5	38	to	to	ADP
ejpam-6613	5	39	the	the	DET
ejpam-6613	5	40	concept	concept	NOUN
ejpam-6613	5	41	of	of	ADP
ejpam-6613	5	42	the	the	DET
ejpam-6613	5	43	tensor	tensor	NOUN
ejpam-6613	5	44	product	product	NOUN
ejpam-6613	5	45	of	of	ADP
ejpam-6613	5	46	vector	vector	NOUN
ejpam-6613	5	47	spaces	space	NOUN
ejpam-6613	5	48	were	be	AUX
ejpam-6613	5	49	developed	develop	VERB
ejpam-6613	5	50	by	by	ADP
ejpam-6613	5	51	various	various	ADJ
ejpam-6613	5	52	researchers	researcher	NOUN
ejpam-6613	5	53	throughout	throughout	ADP
ejpam-6613	5	54	the	the	DET
ejpam-6613	5	55	nineteenth	nineteenth	ADJ
ejpam-6613	5	56	century	century	NOUN
ejpam-6613	5	57	;	;	PUNCT
ejpam-6613	5	58	however	however	ADV
ejpam-6613	5	59	,	,	PUNCT
ejpam-6613	5	60	it	it	PRON
ejpam-6613	5	61	was	be	AUX
ejpam-6613	5	62	not	not	PART
ejpam-6613	5	63	until	until	SCONJ
ejpam-6613	5	64	the	the	DET
ejpam-6613	5	65	work	work	NOUN
ejpam-6613	5	66	carried	carry	VERB
ejpam-6613	5	67	out	out	ADP
ejpam-6613	5	68	by	by	ADP
ejpam-6613	5	69	the	the	DET
ejpam-6613	5	70	mathematician	mathematician	ADJ
ejpam-6613	5	71	hassler	hassler	PROPN
ejpam-6613	5	72	whitney	whitney	PROPN
ejpam-6613	5	73	(	(	PUNCT
ejpam-6613	5	74	1938	1938	NUM
ejpam-6613	5	75	)	)	PUNCT
ejpam-6613	5	76	that	that	SCONJ
ejpam-6613	5	77	the	the	DET
ejpam-6613	5	78	notion	notion	NOUN
ejpam-6613	5	79	of	of	ADP
ejpam-6613	5	80	the	the	DET
ejpam-6613	5	81	tensor	tensor	NOUN
ejpam-6613	5	82	product	product	NOUN
ejpam-6613	5	83	of	of	ADP
ejpam-6613	5	84	abelian	abelian	ADJ
ejpam-6613	5	85	groups	group	NOUN
ejpam-6613	5	86	and	and	CCONJ
ejpam-6613	5	87	more	more	ADV
ejpam-6613	5	88	generally	generally	ADV
ejpam-6613	5	89	of	of	ADP
ejpam-6613	5	90	modules	module	NOUN
ejpam-6613	5	91	was	be	AUX
ejpam-6613	5	92	firmly	firmly	ADV
ejpam-6613	5	93	established	establish	VERB
ejpam-6613	5	94	[	[	X
ejpam-6613	5	95	1	1	NUM
ejpam-6613	5	96	]	]	PUNCT
ejpam-6613	5	97	.	.	PUNCT
ejpam-6613	6	1	consequently	consequently	ADV
ejpam-6613	6	2	,	,	PUNCT
ejpam-6613	6	3	the	the	DET
ejpam-6613	6	4	tensor	tensor	NOUN
ejpam-6613	6	5	product	product	NOUN
ejpam-6613	6	6	of	of	ADP
ejpam-6613	6	7	vector	vector	NOUN
ejpam-6613	6	8	spaces	space	NOUN
ejpam-6613	6	9	was	be	AUX
ejpam-6613	6	10	cast	cast	VERB
ejpam-6613	6	11	in	in	ADP
ejpam-6613	6	12	the	the	DET
ejpam-6613	6	13	language	language	NOUN
ejpam-6613	6	14	of	of	ADP
ejpam-6613	6	15	universal	universal	ADJ
ejpam-6613	6	16	properties	property	NOUN
ejpam-6613	6	17	in	in	ADP
ejpam-6613	6	18	works	work	NOUN
ejpam-6613	6	19	such	such	ADJ
ejpam-6613	6	20	as	as	ADP
ejpam-6613	6	21	those	those	PRON
ejpam-6613	6	22	of	of	ADP
ejpam-6613	6	23	bourbaki	bourbaki	NOUN
ejpam-6613	6	24	(	(	PUNCT
ejpam-6613	6	25	1943	1943	NUM
ejpam-6613	6	26	)	)	PUNCT
ejpam-6613	6	27	in	in	ADP
ejpam-6613	6	28	the	the	DET
ejpam-6613	6	29	article	article	NOUN
ejpam-6613	6	30	[	[	X
ejpam-6613	6	31	2	2	X
ejpam-6613	6	32	]	]	PUNCT
ejpam-6613	6	33	and	and	CCONJ
ejpam-6613	6	34	in	in	ADP
ejpam-6613	6	35	the	the	DET
ejpam-6613	6	36	book	book	NOUN
ejpam-6613	6	37	by	by	ADP
ejpam-6613	6	38	artin	artin	PROPN
ejpam-6613	6	39	,	,	PUNCT
ejpam-6613	6	40	nesbitt	nesbitt	PROPN
ejpam-6613	6	41	,	,	PUNCT
ejpam-6613	6	42	and	and	CCONJ
ejpam-6613	6	43	thrall	thrall	NOUN
ejpam-6613	6	44	(	(	PUNCT
ejpam-6613	6	45	1944	1944	NUM
ejpam-6613	6	46	)	)	PUNCT
ejpam-6613	7	1	[	[	X
ejpam-6613	7	2	3	3	NUM
ejpam-6613	7	3	]	]	PUNCT
ejpam-6613	7	4	.	.	PUNCT
ejpam-6613	8	1	these	these	DET
ejpam-6613	8	2	developments	development	NOUN
ejpam-6613	8	3	,	,	PUNCT
ejpam-6613	8	4	in	in	ADP
ejpam-6613	8	5	turn	turn	NOUN
ejpam-6613	8	6	,	,	PUNCT
ejpam-6613	8	7	spurred	spur	VERB
ejpam-6613	8	8	rapid	rapid	ADJ
ejpam-6613	8	9	advancement	advancement	NOUN
ejpam-6613	8	10	and	and	CCONJ
ejpam-6613	8	11	further	further	ADJ
ejpam-6613	8	12	elaboration	elaboration	NOUN
ejpam-6613	8	13	of	of	ADP
ejpam-6613	8	14	the	the	DET
ejpam-6613	8	15	concept	concept	NOUN
ejpam-6613	8	16	in	in	ADP
ejpam-6613	8	17	other	other	ADJ
ejpam-6613	8	18	mathematical	mathematical	ADJ
ejpam-6613	8	19	contexts	contexts	NOUN
ejpam-6613	8	20	.	.	PUNCT
ejpam-6613	9	1	the	the	DET
ejpam-6613	9	2	notion	notion	NOUN
ejpam-6613	9	3	of	of	ADP
ejpam-6613	9	4	the	the	DET
ejpam-6613	9	5	tensor	tensor	NOUN
ejpam-6613	9	6	product	product	NOUN
ejpam-6613	9	7	of	of	ADP
ejpam-6613	9	8	modules	module	NOUN
ejpam-6613	9	9	has	have	AUX
ejpam-6613	9	10	been	be	AUX
ejpam-6613	9	11	extensively	extensively	ADV
ejpam-6613	9	12	developed	develop	VERB
ejpam-6613	9	13	in	in	ADP
ejpam-6613	9	14	various	various	ADJ
ejpam-6613	9	15	mathematical	mathematical	ADJ
ejpam-6613	9	16	contexts	contexts	NOUN
ejpam-6613	9	17	for	for	ADP
ejpam-6613	9	18	example	example	NOUN
ejpam-6613	9	19	,	,	PUNCT
ejpam-6613	9	20	within	within	ADP
ejpam-6613	9	21	homological	homological	ADJ
ejpam-6613	9	22	algebra	algebra	NOUN
ejpam-6613	9	23	and	and	CCONJ
ejpam-6613	9	24	differential	differential	ADJ
ejpam-6613	9	25	geometry	geometry	NOUN
ejpam-6613	9	26	(	(	PUNCT
ejpam-6613	9	27	see	see	VERB
ejpam-6613	9	28	[	[	X
ejpam-6613	9	29	4–8	4–8	NOUN
ejpam-6613	9	30	]	]	X
ejpam-6613	9	31	)	)	PUNCT
ejpam-6613	9	32	.	.	PUNCT
ejpam-6613	10	1	it	it	PRON
ejpam-6613	10	2	is	be	AUX
ejpam-6613	10	3	a	a	DET
ejpam-6613	10	4	concept	concept	NOUN
ejpam-6613	10	5	of	of	ADP
ejpam-6613	10	6	great	great	ADJ
ejpam-6613	10	7	significance	significance	NOUN
ejpam-6613	10	8	,	,	PUNCT
ejpam-6613	10	9	since	since	SCONJ
ejpam-6613	10	10	,	,	PUNCT
ejpam-6613	10	11	broadly	broadly	ADV
ejpam-6613	10	12	speaking	speak	VERB
ejpam-6613	10	13	,	,	PUNCT
ejpam-6613	10	14	it	it	PRON
ejpam-6613	10	15	provides	provide	VERB
ejpam-6613	10	16	a	a	DET
ejpam-6613	10	17	∗corresponding	∗corresponding	NOUN
ejpam-6613	10	18	author	author	NOUN
ejpam-6613	10	19	.	.	PUNCT
ejpam-6613	11	1	doi	doi	NOUN
ejpam-6613	11	2	:	:	PUNCT
ejpam-6613	11	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6613	https://doi.org/10.29020/nybg.ejpam.v18i4.6613	PRON
ejpam-6613	11	4	email	email	NOUN
ejpam-6613	11	5	addresses	address	VERB
ejpam-6613	11	6	:	:	PUNCT
ejpam-6613	12	1	morenoarroyoluismario@gmail.com	morenoarroyoluismario@gmail.com	X
ejpam-6613	12	2	(	(	PUNCT
ejpam-6613	12	3	m.	m.	PROPN
ejpam-6613	12	4	luis	luis	PROPN
ejpam-6613	12	5	)	)	PUNCT
ejpam-6613	12	6	,	,	PUNCT
ejpam-6613	12	7	osmin.ferrer@unisucre.edu.co	osmin.ferrer@unisucre.edu.co	INTJ
ejpam-6613	12	8	(	(	PUNCT
ejpam-6613	12	9	f.	f.	PROPN
ejpam-6613	12	10	osmin	osmin	PROPN
ejpam-6613	12	11	)	)	PUNCT
ejpam-6613	12	12	,	,	PUNCT
ejpam-6613	12	13	arleysierra23@gmail.com	arleysierra23@gmail.com	X
ejpam-6613	12	14	(	(	PUNCT
ejpam-6613	12	15	s.	s.	PROPN
ejpam-6613	12	16	arley	arley	PROPN
ejpam-6613	12	17	)	)	PUNCT
ejpam-6613	12	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6613	13	1	1	1	NUM
ejpam-6613	13	2	copyright	copyright	NOUN
ejpam-6613	13	3	:	:	PUNCT
ejpam-6613	13	4	©	©	PROPN
ejpam-6613	13	5	2025	2025	NUM
ejpam-6613	13	6	the	the	DET
ejpam-6613	13	7	author(s	author(s	NOUN
ejpam-6613	13	8	)	)	PUNCT
ejpam-6613	13	9	.	.	PUNCT
ejpam-6613	14	1	(	(	PUNCT
ejpam-6613	14	2	cc	cc	NOUN
ejpam-6613	14	3	by	by	ADP
ejpam-6613	14	4	-	-	PUNCT
ejpam-6613	14	5	nc	nc	PROPN
ejpam-6613	14	6	4.0	4.0	NUM
ejpam-6613	14	7	)	)	PUNCT
ejpam-6613	14	8	m.	m.	NOUN
ejpam-6613	14	9	luis	luis	PROPN
ejpam-6613	14	10	,	,	PUNCT
ejpam-6613	14	11	f.	f.	PROPN
ejpam-6613	14	12	osmin	osmin	PROPN
ejpam-6613	14	13	,	,	PUNCT
ejpam-6613	14	14	s.	s.	PROPN
ejpam-6613	14	15	arley	arley	PROPN
ejpam-6613	14	16	/	/	SYM
ejpam-6613	14	17	eur	eur	PROPN
ejpam-6613	14	18	.	.	PUNCT
ejpam-6613	15	1	j.	j.	PROPN
ejpam-6613	15	2	pure	pure	PROPN
ejpam-6613	15	3	appl	appl	PROPN
ejpam-6613	15	4	.	.	PROPN
ejpam-6613	15	5	math	math	PROPN
ejpam-6613	15	6	,	,	PUNCT
ejpam-6613	15	7	18	18	NUM
ejpam-6613	15	8	(	(	PUNCT
ejpam-6613	15	9	4	4	NUM
ejpam-6613	15	10	)	)	PUNCT
ejpam-6613	15	11	(	(	PUNCT
ejpam-6613	15	12	2025	2025	NUM
ejpam-6613	15	13	)	)	PUNCT
ejpam-6613	15	14	,	,	PUNCT
ejpam-6613	15	15	6613	6613	NUM
ejpam-6613	15	16	2	2	NUM
ejpam-6613	15	17	of	of	ADP
ejpam-6613	15	18	17	17	NUM
ejpam-6613	15	19	method	method	NOUN
ejpam-6613	15	20	for	for	ADP
ejpam-6613	15	21	constructing	construct	VERB
ejpam-6613	15	22	new	new	ADJ
ejpam-6613	15	23	spaces	space	NOUN
ejpam-6613	15	24	with	with	ADP
ejpam-6613	15	25	the	the	DET
ejpam-6613	15	26	desired	desire	VERB
ejpam-6613	15	27	structure	structure	NOUN
ejpam-6613	15	28	and	and	CCONJ
ejpam-6613	15	29	,	,	PUNCT
ejpam-6613	15	30	in	in	ADP
ejpam-6613	15	31	the	the	DET
ejpam-6613	15	32	case	case	NOUN
ejpam-6613	15	33	of	of	ADP
ejpam-6613	15	34	vector	vector	NOUN
ejpam-6613	15	35	spaces	space	NOUN
ejpam-6613	15	36	,	,	PUNCT
ejpam-6613	15	37	offers	offer	VERB
ejpam-6613	15	38	a	a	DET
ejpam-6613	15	39	way	way	NOUN
ejpam-6613	15	40	to	to	PART
ejpam-6613	15	41	pass	pass	VERB
ejpam-6613	15	42	from	from	ADP
ejpam-6613	15	43	bilinear	bilinear	PROPN
ejpam-6613	15	44	operators	operator	NOUN
ejpam-6613	15	45	to	to	PART
ejpam-6613	15	46	linear	linear	VERB
ejpam-6613	15	47	operators	operator	NOUN
ejpam-6613	15	48	.	.	PUNCT
ejpam-6613	16	1	moreover	moreover	ADV
ejpam-6613	16	2	,	,	PUNCT
ejpam-6613	16	3	it	it	PRON
ejpam-6613	16	4	arises	arise	VERB
ejpam-6613	16	5	naturally	naturally	ADV
ejpam-6613	16	6	in	in	ADP
ejpam-6613	16	7	areas	area	NOUN
ejpam-6613	16	8	of	of	ADP
ejpam-6613	16	9	physics	physics	NOUN
ejpam-6613	16	10	such	such	ADJ
ejpam-6613	16	11	as	as	ADP
ejpam-6613	16	12	quantum	quantum	NOUN
ejpam-6613	16	13	physics	physics	NOUN
ejpam-6613	16	14	and	and	CCONJ
ejpam-6613	16	15	quantum	quantum	NOUN
ejpam-6613	16	16	computing	computing	NOUN
ejpam-6613	16	17	(	(	PUNCT
ejpam-6613	16	18	see	see	VERB
ejpam-6613	16	19	[	[	X
ejpam-6613	16	20	9	9	NUM
ejpam-6613	16	21	,	,	PUNCT
ejpam-6613	16	22	10	10	NUM
ejpam-6613	16	23	]	]	PUNCT
ejpam-6613	16	24	)	)	PUNCT
ejpam-6613	16	25	.	.	PUNCT
ejpam-6613	17	1	in	in	ADP
ejpam-6613	17	2	recent	recent	ADJ
ejpam-6613	17	3	years	year	NOUN
ejpam-6613	17	4	,	,	PUNCT
ejpam-6613	17	5	the	the	DET
ejpam-6613	17	6	concept	concept	NOUN
ejpam-6613	17	7	of	of	ADP
ejpam-6613	17	8	the	the	DET
ejpam-6613	17	9	tensor	tensor	NOUN
ejpam-6613	17	10	product	product	NOUN
ejpam-6613	17	11	of	of	ADP
ejpam-6613	17	12	complex	complex	ADJ
ejpam-6613	17	13	vector	vector	NOUN
ejpam-6613	17	14	spaces	space	NOUN
ejpam-6613	17	15	equipped	equip	VERB
ejpam-6613	17	16	with	with	ADP
ejpam-6613	17	17	a	a	DET
ejpam-6613	17	18	positive	positive	ADJ
ejpam-6613	17	19	definite	definite	ADJ
ejpam-6613	17	20	sesquilinear	sesquilinear	NOUN
ejpam-6613	17	21	form	form	NOUN
ejpam-6613	17	22	[	[	X
ejpam-6613	17	23	11	11	NUM
ejpam-6613	17	24	]	]	PUNCT
ejpam-6613	17	25	has	have	AUX
ejpam-6613	17	26	been	be	AUX
ejpam-6613	17	27	extended	extend	VERB
ejpam-6613	17	28	to	to	ADP
ejpam-6613	17	29	the	the	DET
ejpam-6613	17	30	tensor	tensor	NOUN
ejpam-6613	17	31	product	product	NOUN
ejpam-6613	17	32	of	of	ADP
ejpam-6613	17	33	spaces	space	NOUN
ejpam-6613	17	34	endowed	endow	VERB
ejpam-6613	17	35	with	with	ADP
ejpam-6613	17	36	an	an	DET
ejpam-6613	17	37	arbitrary	arbitrary	ADJ
ejpam-6613	17	38	sesquilinear	sesquilinear	NOUN
ejpam-6613	17	39	form	form	NOUN
ejpam-6613	17	40	[	[	X
ejpam-6613	17	41	12	12	NUM
ejpam-6613	17	42	]	]	PUNCT
ejpam-6613	17	43	,	,	PUNCT
ejpam-6613	17	44	defining	define	VERB
ejpam-6613	17	45	simple	simple	ADJ
ejpam-6613	17	46	tensors	tensor	NOUN
ejpam-6613	17	47	using	use	VERB
ejpam-6613	17	48	such	such	ADJ
ejpam-6613	17	49	sesquilinear	sesquilinear	ADJ
ejpam-6613	17	50	forms	form	NOUN
ejpam-6613	17	51	in	in	ADP
ejpam-6613	17	52	an	an	DET
ejpam-6613	17	53	appropriate	appropriate	ADJ
ejpam-6613	17	54	manner	manner	NOUN
ejpam-6613	17	55	.	.	PUNCT
ejpam-6613	18	1	moreover	moreover	ADV
ejpam-6613	18	2	,	,	PUNCT
ejpam-6613	18	3	since	since	SCONJ
ejpam-6613	18	4	the	the	DET
ejpam-6613	18	5	concept	concept	NOUN
ejpam-6613	18	6	of	of	ADP
ejpam-6613	18	7	the	the	DET
ejpam-6613	18	8	classical	classical	ADJ
ejpam-6613	18	9	inner	inner	ADJ
ejpam-6613	18	10	product	product	NOUN
ejpam-6613	18	11	can	can	AUX
ejpam-6613	18	12	be	be	AUX
ejpam-6613	18	13	extended	extend	VERB
ejpam-6613	18	14	to	to	ADP
ejpam-6613	18	15	what	what	PRON
ejpam-6613	18	16	is	be	AUX
ejpam-6613	18	17	known	know	VERB
ejpam-6613	18	18	as	as	ADP
ejpam-6613	18	19	the	the	DET
ejpam-6613	18	20	2	2	NUM
ejpam-6613	18	21	-	-	PUNCT
ejpam-6613	18	22	inner	inner	ADJ
ejpam-6613	18	23	product	product	NOUN
ejpam-6613	18	24	and	and	CCONJ
ejpam-6613	18	25	the	the	DET
ejpam-6613	18	26	generalized	generalized	ADJ
ejpam-6613	18	27	2	2	NUM
ejpam-6613	18	28	-	-	PUNCT
ejpam-6613	18	29	inner	inner	ADJ
ejpam-6613	18	30	product	product	NOUN
ejpam-6613	18	31	in	in	ADP
ejpam-6613	18	32	the	the	DET
ejpam-6613	18	33	senses	sense	NOUN
ejpam-6613	18	34	of	of	ADP
ejpam-6613	18	35	gähler	gähler	NOUN
ejpam-6613	18	36	[	[	X
ejpam-6613	18	37	13	13	NUM
ejpam-6613	18	38	]	]	PUNCT
ejpam-6613	18	39	and	and	CCONJ
ejpam-6613	18	40	lewandoska	lewandoska	VERB
ejpam-6613	19	1	[	[	X
ejpam-6613	19	2	14	14	NUM
ejpam-6613	19	3	]	]	PUNCT
ejpam-6613	19	4	,	,	PUNCT
ejpam-6613	19	5	respectively	respectively	ADV
ejpam-6613	19	6	,	,	PUNCT
ejpam-6613	19	7	it	it	PRON
ejpam-6613	19	8	is	be	AUX
ejpam-6613	19	9	natural	natural	ADJ
ejpam-6613	19	10	to	to	PART
ejpam-6613	19	11	consider	consider	VERB
ejpam-6613	19	12	defining	define	VERB
ejpam-6613	19	13	the	the	DET
ejpam-6613	19	14	tensor	tensor	NOUN
ejpam-6613	19	15	product	product	NOUN
ejpam-6613	19	16	of	of	ADP
ejpam-6613	19	17	spaces	space	NOUN
ejpam-6613	19	18	endowed	endow	VERB
ejpam-6613	19	19	with	with	ADP
ejpam-6613	19	20	a	a	DET
ejpam-6613	19	21	2	2	NUM
ejpam-6613	19	22	-	-	PUNCT
ejpam-6613	19	23	inner	inner	ADJ
ejpam-6613	19	24	product	product	NOUN
ejpam-6613	19	25	of	of	ADP
ejpam-6613	19	26	this	this	DET
ejpam-6613	19	27	kind	kind	NOUN
ejpam-6613	19	28	,	,	PUNCT
ejpam-6613	19	29	which	which	PRON
ejpam-6613	19	30	in	in	ADP
ejpam-6613	19	31	this	this	DET
ejpam-6613	19	32	work	work	NOUN
ejpam-6613	19	33	will	will	AUX
ejpam-6613	19	34	be	be	AUX
ejpam-6613	19	35	referred	refer	VERB
ejpam-6613	19	36	to	to	ADP
ejpam-6613	19	37	as	as	ADP
ejpam-6613	19	38	the	the	DET
ejpam-6613	19	39	2	2	NUM
ejpam-6613	19	40	-	-	PUNCT
ejpam-6613	19	41	tensor	tensor	NOUN
ejpam-6613	19	42	product	product	NOUN
ejpam-6613	19	43	of	of	ADP
ejpam-6613	19	44	spaces	space	NOUN
ejpam-6613	19	45	with	with	ADP
ejpam-6613	19	46	a	a	DET
ejpam-6613	19	47	generalized	generalized	ADJ
ejpam-6613	19	48	2	2	NUM
ejpam-6613	19	49	-	-	PUNCT
ejpam-6613	19	50	inner	inner	ADJ
ejpam-6613	19	51	product	product	NOUN
ejpam-6613	19	52	.	.	PUNCT
ejpam-6613	20	1	the	the	DET
ejpam-6613	20	2	theory	theory	NOUN
ejpam-6613	20	3	of	of	ADP
ejpam-6613	20	4	spaces	space	NOUN
ejpam-6613	20	5	with	with	ADP
ejpam-6613	20	6	a	a	DET
ejpam-6613	20	7	2	2	NUM
ejpam-6613	20	8	-	-	PUNCT
ejpam-6613	20	9	inner	inner	ADJ
ejpam-6613	20	10	product	product	NOUN
ejpam-6613	20	11	has	have	AUX
ejpam-6613	20	12	been	be	AUX
ejpam-6613	20	13	extensively	extensively	ADV
ejpam-6613	20	14	studied	study	VERB
ejpam-6613	20	15	in	in	ADP
ejpam-6613	20	16	articles	article	NOUN
ejpam-6613	20	17	by	by	ADP
ejpam-6613	20	18	diminnie	diminnie	NOUN
ejpam-6613	20	19	,	,	PUNCT
ejpam-6613	20	20	gähler	gähler	NOUN
ejpam-6613	20	21	and	and	CCONJ
ejpam-6613	20	22	white	white	ADJ
ejpam-6613	21	1	[	[	X
ejpam-6613	21	2	15–18	15–18	NUM
ejpam-6613	21	3	]	]	PUNCT
ejpam-6613	21	4	.	.	PUNCT
ejpam-6613	22	1	thus	thus	ADV
ejpam-6613	22	2	,	,	PUNCT
ejpam-6613	22	3	the	the	DET
ejpam-6613	22	4	present	present	ADJ
ejpam-6613	22	5	work	work	NOUN
ejpam-6613	22	6	is	be	AUX
ejpam-6613	22	7	organized	organize	VERB
ejpam-6613	22	8	as	as	SCONJ
ejpam-6613	22	9	follows	follow	VERB
ejpam-6613	22	10	:	:	PUNCT
ejpam-6613	22	11	in	in	ADP
ejpam-6613	22	12	section	section	NOUN
ejpam-6613	22	13	2	2	NUM
ejpam-6613	22	14	,	,	PUNCT
ejpam-6613	22	15	we	we	PRON
ejpam-6613	22	16	include	include	VERB
ejpam-6613	22	17	the	the	DET
ejpam-6613	22	18	necessary	necessary	ADJ
ejpam-6613	22	19	preliminaries	preliminary	NOUN
ejpam-6613	22	20	to	to	PART
ejpam-6613	22	21	introduce	introduce	VERB
ejpam-6613	22	22	the	the	DET
ejpam-6613	22	23	notion	notion	NOUN
ejpam-6613	22	24	of	of	ADP
ejpam-6613	22	25	the	the	DET
ejpam-6613	22	26	algebraic	algebraic	ADJ
ejpam-6613	22	27	2	2	NUM
ejpam-6613	22	28	-	-	PUNCT
ejpam-6613	22	29	tensor	tensor	NOUN
ejpam-6613	22	30	product	product	NOUN
ejpam-6613	22	31	of	of	ADP
ejpam-6613	22	32	vector	vector	NOUN
ejpam-6613	22	33	spaces	space	NOUN
ejpam-6613	22	34	endowed	endow	VERB
ejpam-6613	22	35	with	with	ADP
ejpam-6613	22	36	a	a	DET
ejpam-6613	22	37	generalized	generalized	ADJ
ejpam-6613	22	38	2	2	NUM
ejpam-6613	22	39	-	-	PUNCT
ejpam-6613	22	40	inner	inner	ADJ
ejpam-6613	22	41	product	product	NOUN
ejpam-6613	22	42	,	,	PUNCT
ejpam-6613	22	43	such	such	ADJ
ejpam-6613	22	44	as	as	ADP
ejpam-6613	22	45	the	the	DET
ejpam-6613	22	46	concept	concept	NOUN
ejpam-6613	22	47	of	of	ADP
ejpam-6613	22	48	a	a	DET
ejpam-6613	22	49	generalized	generalized	ADJ
ejpam-6613	22	50	2	2	NUM
ejpam-6613	22	51	-	-	PUNCT
ejpam-6613	22	52	inner	inner	ADJ
ejpam-6613	22	53	product	product	NOUN
ejpam-6613	22	54	and	and	CCONJ
ejpam-6613	22	55	some	some	PRON
ejpam-6613	22	56	of	of	ADP
ejpam-6613	22	57	its	its	PRON
ejpam-6613	22	58	properties	property	NOUN
ejpam-6613	22	59	.	.	PUNCT
ejpam-6613	23	1	in	in	ADP
ejpam-6613	23	2	section	section	NOUN
ejpam-6613	23	3	3	3	NUM
ejpam-6613	23	4	,	,	PUNCT
ejpam-6613	23	5	we	we	PRON
ejpam-6613	23	6	introduce	introduce	VERB
ejpam-6613	23	7	the	the	DET
ejpam-6613	23	8	notion	notion	NOUN
ejpam-6613	23	9	of	of	ADP
ejpam-6613	23	10	the	the	DET
ejpam-6613	23	11	2	2	NUM
ejpam-6613	23	12	-	-	PUNCT
ejpam-6613	23	13	tensor	tensor	NOUN
ejpam-6613	23	14	product	product	NOUN
ejpam-6613	23	15	of	of	ADP
ejpam-6613	23	16	elements	element	NOUN
ejpam-6613	23	17	in	in	ADP
ejpam-6613	23	18	spaces	space	NOUN
ejpam-6613	23	19	with	with	ADP
ejpam-6613	23	20	a	a	DET
ejpam-6613	23	21	generalized	generalized	ADJ
ejpam-6613	23	22	2	2	NUM
ejpam-6613	23	23	-	-	PUNCT
ejpam-6613	23	24	inner	inner	ADJ
ejpam-6613	23	25	product	product	NOUN
ejpam-6613	23	26	,	,	PUNCT
ejpam-6613	23	27	based	base	VERB
ejpam-6613	23	28	on	on	ADP
ejpam-6613	23	29	the	the	DET
ejpam-6613	23	30	ideas	idea	NOUN
ejpam-6613	23	31	of	of	ADP
ejpam-6613	23	32	the	the	DET
ejpam-6613	23	33	simple	simple	ADJ
ejpam-6613	23	34	tensor	tensor	NOUN
ejpam-6613	23	35	product	product	NOUN
ejpam-6613	23	36	of	of	ADP
ejpam-6613	23	37	elements	element	NOUN
ejpam-6613	23	38	in	in	ADP
ejpam-6613	23	39	spaces	space	NOUN
ejpam-6613	23	40	with	with	ADP
ejpam-6613	23	41	a	a	DET
ejpam-6613	23	42	classical	classical	ADJ
ejpam-6613	23	43	inner	inner	ADJ
ejpam-6613	23	44	product	product	NOUN
ejpam-6613	23	45	;	;	PUNCT
ejpam-6613	23	46	consequently	consequently	ADV
ejpam-6613	23	47	,	,	PUNCT
ejpam-6613	23	48	we	we	PRON
ejpam-6613	23	49	establish	establish	VERB
ejpam-6613	23	50	the	the	DET
ejpam-6613	23	51	notion	notion	NOUN
ejpam-6613	23	52	of	of	ADP
ejpam-6613	23	53	the	the	DET
ejpam-6613	23	54	2	2	NUM
ejpam-6613	23	55	-	-	PUNCT
ejpam-6613	23	56	tensor	tensor	NOUN
ejpam-6613	23	57	product	product	NOUN
ejpam-6613	23	58	of	of	ADP
ejpam-6613	23	59	spaces	space	NOUN
ejpam-6613	23	60	with	with	ADP
ejpam-6613	23	61	a	a	DET
ejpam-6613	23	62	generalized	generalized	ADJ
ejpam-6613	23	63	2	2	NUM
ejpam-6613	23	64	-	-	PUNCT
ejpam-6613	23	65	inner	inner	ADJ
ejpam-6613	23	66	product	product	NOUN
ejpam-6613	23	67	,	,	PUNCT
ejpam-6613	23	68	define	define	VERB
ejpam-6613	23	69	a	a	DET
ejpam-6613	23	70	generalized	generalized	ADJ
ejpam-6613	23	71	2	2	NUM
ejpam-6613	23	72	-	-	PUNCT
ejpam-6613	23	73	inner	inner	ADJ
ejpam-6613	23	74	product	product	NOUN
ejpam-6613	23	75	on	on	ADP
ejpam-6613	23	76	it	it	PRON
ejpam-6613	23	77	,	,	PUNCT
ejpam-6613	23	78	and	and	CCONJ
ejpam-6613	23	79	prove	prove	VERB
ejpam-6613	23	80	some	some	DET
ejpam-6613	23	81	interesting	interesting	ADJ
ejpam-6613	23	82	properties	property	NOUN
ejpam-6613	23	83	.	.	PUNCT
ejpam-6613	24	1	section	section	NOUN
ejpam-6613	24	2	4	4	NUM
ejpam-6613	24	3	contains	contain	VERB
ejpam-6613	24	4	the	the	DET
ejpam-6613	24	5	notion	notion	NOUN
ejpam-6613	24	6	of	of	ADP
ejpam-6613	24	7	the	the	DET
ejpam-6613	24	8	2	2	NUM
ejpam-6613	24	9	-	-	PUNCT
ejpam-6613	24	10	tensor	tensor	NOUN
ejpam-6613	24	11	product	product	NOUN
ejpam-6613	24	12	of	of	ADP
ejpam-6613	24	13	linear	linear	PROPN
ejpam-6613	24	14	operators	operator	NOUN
ejpam-6613	24	15	and	and	CCONJ
ejpam-6613	24	16	some	some	PRON
ejpam-6613	24	17	of	of	ADP
ejpam-6613	24	18	their	their	PRON
ejpam-6613	24	19	satisfied	satisfied	ADJ
ejpam-6613	24	20	properties	property	NOUN
ejpam-6613	24	21	.	.	PUNCT
ejpam-6613	25	1	finally	finally	ADV
ejpam-6613	25	2	,	,	PUNCT
ejpam-6613	25	3	section	section	NOUN
ejpam-6613	25	4	5	5	NUM
ejpam-6613	25	5	presents	present	VERB
ejpam-6613	25	6	the	the	DET
ejpam-6613	25	7	conclusions	conclusion	NOUN
ejpam-6613	25	8	of	of	ADP
ejpam-6613	25	9	this	this	DET
ejpam-6613	25	10	research	research	NOUN
ejpam-6613	25	11	.	.	PUNCT
ejpam-6613	26	1	2	2	X
ejpam-6613	26	2	.	.	X
ejpam-6613	26	3	preliminaries	preliminary	NOUN
ejpam-6613	26	4	in	in	ADP
ejpam-6613	26	5	this	this	DET
ejpam-6613	26	6	section	section	NOUN
ejpam-6613	26	7	,	,	PUNCT
ejpam-6613	26	8	we	we	PRON
ejpam-6613	26	9	study	study	VERB
ejpam-6613	26	10	the	the	DET
ejpam-6613	26	11	concepts	concept	NOUN
ejpam-6613	26	12	of	of	ADP
ejpam-6613	26	13	the	the	DET
ejpam-6613	26	14	generalized	generalized	ADJ
ejpam-6613	26	15	2	2	NUM
ejpam-6613	26	16	-	-	PUNCT
ejpam-6613	26	17	norm	norm	NOUN
ejpam-6613	26	18	,	,	PUNCT
ejpam-6613	26	19	the	the	DET
ejpam-6613	26	20	generalized	generalized	ADJ
ejpam-6613	26	21	2	2	NUM
ejpam-6613	26	22	-	-	PUNCT
ejpam-6613	26	23	inner	inner	ADJ
ejpam-6613	26	24	product	product	NOUN
ejpam-6613	26	25	,	,	PUNCT
ejpam-6613	26	26	and	and	CCONJ
ejpam-6613	26	27	some	some	PRON
ejpam-6613	26	28	of	of	ADP
ejpam-6613	26	29	the	the	DET
ejpam-6613	26	30	most	most	ADV
ejpam-6613	26	31	important	important	ADJ
ejpam-6613	26	32	properties	property	NOUN
ejpam-6613	26	33	that	that	PRON
ejpam-6613	26	34	are	be	AUX
ejpam-6613	26	35	satisfied	satisfied	ADJ
ejpam-6613	26	36	in	in	ADP
ejpam-6613	26	37	spaces	space	NOUN
ejpam-6613	26	38	endowed	endow	VERB
ejpam-6613	26	39	with	with	ADP
ejpam-6613	26	40	these	these	DET
ejpam-6613	26	41	structures	structure	NOUN
ejpam-6613	26	42	.	.	PUNCT
ejpam-6613	27	1	definition	definition	NOUN
ejpam-6613	27	2	1	1	NUM
ejpam-6613	27	3	.	.	PUNCT
ejpam-6613	28	1	[	[	X
ejpam-6613	28	2	14	14	NUM
ejpam-6613	28	3	]	]	PUNCT
ejpam-6613	28	4	a	a	DET
ejpam-6613	28	5	generalized	generalized	ADJ
ejpam-6613	28	6	2	2	NUM
ejpam-6613	28	7	-	-	PUNCT
ejpam-6613	28	8	norm	norm	NOUN
ejpam-6613	28	9	on	on	ADP
ejpam-6613	28	10	x	x	X
ejpam-6613	28	11	is	be	AUX
ejpam-6613	28	12	a	a	DET
ejpam-6613	28	13	map	map	NOUN
ejpam-6613	28	14	∥	∥	NOUN
ejpam-6613	28	15	·	·	PUNCT
ejpam-6613	28	16	,	,	PUNCT
ejpam-6613	28	17	·	·	PUNCT
ejpam-6613	28	18	∥	∥	X
ejpam-6613	28	19	:	:	PUNCT
ejpam-6613	29	1	x	x	PUNCT
ejpam-6613	29	2	×	×	NOUN
ejpam-6613	29	3	x	x	INTJ
ejpam-6613	29	4	→	→	PUNCT
ejpam-6613	29	5	r	r	NOUN
ejpam-6613	29	6	such	such	ADJ
ejpam-6613	29	7	that	that	PRON
ejpam-6613	29	8	for	for	ADP
ejpam-6613	29	9	every	every	DET
ejpam-6613	29	10	w	w	PROPN
ejpam-6613	29	11	,	,	PUNCT
ejpam-6613	29	12	y	y	PROPN
ejpam-6613	29	13	,	,	PUNCT
ejpam-6613	29	14	z	z	NOUN
ejpam-6613	29	15	∈	∈	PROPN
ejpam-6613	29	16	x	x	X
ejpam-6613	29	17	is	be	AUX
ejpam-6613	29	18	true	true	ADJ
ejpam-6613	29	19	i	i	NOUN
ejpam-6613	29	20	)	)	PUNCT
ejpam-6613	29	21	∥w	∥w	PROPN
ejpam-6613	29	22	,	,	PUNCT
ejpam-6613	29	23	z∥	z∥	NOUN
ejpam-6613	29	24	=	=	SYM
ejpam-6613	29	25	∥z	∥z	NOUN
ejpam-6613	29	26	,	,	PUNCT
ejpam-6613	29	27	w∥	w∥	NOUN
ejpam-6613	29	28	;	;	PUNCT
ejpam-6613	29	29	ii	ii	X
ejpam-6613	29	30	)	)	PUNCT
ejpam-6613	29	31	∥w	∥w	PROPN
ejpam-6613	29	32	,	,	PUNCT
ejpam-6613	29	33	αz∥	αz∥	PROPN
ejpam-6613	29	34	=	=	SYM
ejpam-6613	29	35	|α|∥w	|α|∥w	PROPN
ejpam-6613	29	36	,	,	PUNCT
ejpam-6613	29	37	z∥	z∥	VERB
ejpam-6613	29	38	for	for	ADP
ejpam-6613	29	39	all	all	DET
ejpam-6613	29	40	α	α	PRON
ejpam-6613	29	41	∈	∈	PROPN
ejpam-6613	29	42	c	c	X
ejpam-6613	29	43	;	;	PUNCT
ejpam-6613	29	44	iii	iii	X
ejpam-6613	29	45	)	)	PUNCT
ejpam-6613	29	46	∥w	∥w	PROPN
ejpam-6613	29	47	,	,	PUNCT
ejpam-6613	29	48	y	y	PROPN
ejpam-6613	29	49	+	+	PROPN
ejpam-6613	29	50	z∥	z∥	PROPN
ejpam-6613	29	51	≤	≤	NUM
ejpam-6613	29	52	∥w	∥w	PROPN
ejpam-6613	29	53	,	,	PUNCT
ejpam-6613	29	54	y∥+	y∥+	PROPN
ejpam-6613	29	55	∥w	∥w	PROPN
ejpam-6613	29	56	,	,	PUNCT
ejpam-6613	29	57	z∥.	z∥.	NOUN
ejpam-6613	29	58	the	the	DET
ejpam-6613	29	59	pair	pair	NOUN
ejpam-6613	29	60	(	(	PUNCT
ejpam-6613	29	61	x	x	NOUN
ejpam-6613	29	62	,	,	PUNCT
ejpam-6613	29	63	∥	∥	X
ejpam-6613	29	64	·	·	PUNCT
ejpam-6613	29	65	,	,	PUNCT
ejpam-6613	29	66	·	·	PUNCT
ejpam-6613	29	67	∥	∥	X
ejpam-6613	29	68	)	)	PUNCT
ejpam-6613	29	69	is	be	AUX
ejpam-6613	29	70	called	call	VERB
ejpam-6613	29	71	a	a	DET
ejpam-6613	29	72	space	space	NOUN
ejpam-6613	29	73	with	with	ADP
ejpam-6613	29	74	generalized	generalized	ADJ
ejpam-6613	29	75	2	2	NUM
ejpam-6613	29	76	-	-	PUNCT
ejpam-6613	29	77	norm	norm	NOUN
ejpam-6613	29	78	.	.	PUNCT
ejpam-6613	30	1	in	in	ADP
ejpam-6613	30	2	the	the	DET
ejpam-6613	30	3	following	following	ADJ
ejpam-6613	30	4	example	example	NOUN
ejpam-6613	30	5	we	we	PRON
ejpam-6613	30	6	show	show	VERB
ejpam-6613	30	7	how	how	SCONJ
ejpam-6613	30	8	to	to	PART
ejpam-6613	30	9	define	define	VERB
ejpam-6613	30	10	a	a	DET
ejpam-6613	30	11	generalized	generalized	ADJ
ejpam-6613	30	12	2	2	NUM
ejpam-6613	30	13	-	-	PUNCT
ejpam-6613	30	14	norm	norm	NOUN
ejpam-6613	30	15	from	from	ADP
ejpam-6613	30	16	an	an	DET
ejpam-6613	30	17	indefinite	indefinite	ADJ
ejpam-6613	30	18	sesquilinear	sesquilinear	NOUN
ejpam-6613	30	19	form	form	NOUN
ejpam-6613	30	20	.	.	PUNCT
ejpam-6613	31	1	m.	m.	PROPN
ejpam-6613	31	2	luis	luis	PROPN
ejpam-6613	31	3	,	,	PUNCT
ejpam-6613	31	4	f.	f.	PROPN
ejpam-6613	31	5	osmin	osmin	PROPN
ejpam-6613	31	6	,	,	PUNCT
ejpam-6613	31	7	s.	s.	PROPN
ejpam-6613	31	8	arley	arley	PROPN
ejpam-6613	31	9	/	/	SYM
ejpam-6613	31	10	eur	eur	PROPN
ejpam-6613	31	11	.	.	PUNCT
ejpam-6613	32	1	j.	j.	PROPN
ejpam-6613	32	2	pure	pure	PROPN
ejpam-6613	32	3	appl	appl	PROPN
ejpam-6613	32	4	.	.	PROPN
ejpam-6613	32	5	math	math	PROPN
ejpam-6613	32	6	,	,	PUNCT
ejpam-6613	32	7	18	18	NUM
ejpam-6613	32	8	(	(	PUNCT
ejpam-6613	32	9	4	4	NUM
ejpam-6613	32	10	)	)	PUNCT
ejpam-6613	32	11	(	(	PUNCT
ejpam-6613	32	12	2025	2025	NUM
ejpam-6613	32	13	)	)	PUNCT
ejpam-6613	32	14	,	,	PUNCT
ejpam-6613	32	15	6613	6613	NUM
ejpam-6613	32	16	3	3	NUM
ejpam-6613	32	17	of	of	ADP
ejpam-6613	32	18	17	17	NUM
ejpam-6613	32	19	example	example	NOUN
ejpam-6613	32	20	1	1	NUM
ejpam-6613	32	21	.	.	PUNCT
ejpam-6613	33	1	[	[	X
ejpam-6613	33	2	19	19	NUM
ejpam-6613	33	3	]	]	X
ejpam-6613	33	4	we	we	PRON
ejpam-6613	33	5	define	define	VERB
ejpam-6613	33	6	on	on	ADP
ejpam-6613	33	7	the	the	DET
ejpam-6613	33	8	vector	vector	NOUN
ejpam-6613	33	9	space	space	NOUN
ejpam-6613	33	10	cn	cn	PROPN
ejpam-6613	33	11	the	the	DET
ejpam-6613	33	12	function	function	NOUN
ejpam-6613	33	13	∥	∥	NOUN
ejpam-6613	33	14	·	·	PUNCT
ejpam-6613	33	15	,	,	PUNCT
ejpam-6613	33	16	·	·	PUNCT
ejpam-6613	33	17	∥	∥	X
ejpam-6613	33	18	:	:	PUNCT
ejpam-6613	33	19	cn	cn	PROPN
ejpam-6613	33	20	×cn	×cn	PROPN
ejpam-6613	33	21	→	→	SYM
ejpam-6613	33	22	r	r	NOUN
ejpam-6613	33	23	by	by	ADP
ejpam-6613	33	24	means	mean	NOUN
ejpam-6613	33	25	of	of	ADP
ejpam-6613	33	26	the	the	DET
ejpam-6613	33	27	rule	rule	NOUN
ejpam-6613	33	28	∥x	∥x	PROPN
ejpam-6613	33	29	,	,	PUNCT
ejpam-6613	33	30	y∥	y∥	VERB
ejpam-6613	33	31	:	:	PUNCT
ejpam-6613	33	32	=	=	PRON
ejpam-6613	33	33	∣∣∑n	∣∣∑n	PUNCT
ejpam-6613	33	34	i=1(−1)ixiyi	i=1(−1)ixiyi	PROPN
ejpam-6613	33	35	∣∣	∣∣	ADJ
ejpam-6613	33	36	for	for	ADP
ejpam-6613	33	37	all	all	PRON
ejpam-6613	33	38	x	x	X
ejpam-6613	33	39	=	=	SYM
ejpam-6613	33	40	(	(	PUNCT
ejpam-6613	33	41	x1	x1	PROPN
ejpam-6613	33	42	,	,	PUNCT
ejpam-6613	33	43	·	·	PUNCT
ejpam-6613	33	44	·	·	PUNCT
ejpam-6613	33	45	·	·	PUNCT
ejpam-6613	33	46	,	,	PUNCT
ejpam-6613	33	47	xn	xn	PROPN
ejpam-6613	33	48	)	)	PUNCT
ejpam-6613	33	49	,	,	PUNCT
ejpam-6613	33	50	y	y	PROPN
ejpam-6613	33	51	=	=	SYM
ejpam-6613	33	52	(	(	PUNCT
ejpam-6613	33	53	y1	y1	PROPN
ejpam-6613	33	54	,	,	PUNCT
ejpam-6613	33	55	·	·	PUNCT
ejpam-6613	33	56	·	·	PUNCT
ejpam-6613	33	57	·	·	PUNCT
ejpam-6613	33	58	,	,	PUNCT
ejpam-6613	33	59	yn	yn	X
ejpam-6613	33	60	)	)	PUNCT
ejpam-6613	33	61	∈	∈	PROPN
ejpam-6613	34	1	cn	cn	PROPN
ejpam-6613	34	2	.	.	PUNCT
ejpam-6613	35	1	what	what	PRON
ejpam-6613	35	2	defines	define	VERB
ejpam-6613	35	3	a	a	DET
ejpam-6613	35	4	generalized	generalized	ADJ
ejpam-6613	35	5	2	2	NUM
ejpam-6613	35	6	-	-	PUNCT
ejpam-6613	35	7	norm	norm	NOUN
ejpam-6613	35	8	.	.	PUNCT
ejpam-6613	36	1	in	in	ADP
ejpam-6613	36	2	general	general	ADJ
ejpam-6613	36	3	,	,	PUNCT
ejpam-6613	36	4	it	it	PRON
ejpam-6613	36	5	can	can	AUX
ejpam-6613	36	6	be	be	AUX
ejpam-6613	36	7	shown	show	VERB
ejpam-6613	36	8	that	that	SCONJ
ejpam-6613	36	9	,	,	PUNCT
ejpam-6613	36	10	given	give	VERB
ejpam-6613	36	11	a	a	DET
ejpam-6613	36	12	classical	classical	ADJ
ejpam-6613	36	13	inner	inner	ADJ
ejpam-6613	36	14	product	product	NOUN
ejpam-6613	36	15	space	space	NOUN
ejpam-6613	36	16	(	(	PUNCT
ejpam-6613	36	17	x	x	X
ejpam-6613	36	18	,	,	PUNCT
ejpam-6613	36	19	⟨	⟨	NOUN
ejpam-6613	36	20	·	·	SYM
ejpam-6613	36	21	,	,	PUNCT
ejpam-6613	36	22	·	·	PUNCT
ejpam-6613	36	23	⟩	⟩	NOUN
ejpam-6613	36	24	)	)	PUNCT
ejpam-6613	36	25	,	,	PUNCT
ejpam-6613	36	26	the	the	DET
ejpam-6613	36	27	mapping	mapping	NOUN
ejpam-6613	36	28	∥	∥	NOUN
ejpam-6613	36	29	·	·	PUNCT
ejpam-6613	36	30	,	,	PUNCT
ejpam-6613	36	31	·	·	PUNCT
ejpam-6613	36	32	∥	∥	X
ejpam-6613	36	33	:	:	PUNCT
ejpam-6613	36	34	x	x	PUNCT
ejpam-6613	36	35	×	×	NOUN
ejpam-6613	36	36	x	x	INTJ
ejpam-6613	36	37	→	→	SYM
ejpam-6613	36	38	r	r	NOUN
ejpam-6613	36	39	(	(	PUNCT
ejpam-6613	36	40	w	w	PROPN
ejpam-6613	36	41	,	,	PUNCT
ejpam-6613	36	42	z	z	NOUN
ejpam-6613	36	43	)	)	PUNCT
ejpam-6613	36	44	7→	7→	X
ejpam-6613	37	1	∥w	∥w	PROPN
ejpam-6613	37	2	,	,	PUNCT
ejpam-6613	37	3	z∥	z∥	ADJ
ejpam-6613	37	4	=	=	SYM
ejpam-6613	37	5	|⟨w	|⟨w	NOUN
ejpam-6613	37	6	,	,	PUNCT
ejpam-6613	37	7	z⟩|	z⟩|	PROPN
ejpam-6613	37	8	,	,	PUNCT
ejpam-6613	37	9	defines	define	VERB
ejpam-6613	37	10	a	a	DET
ejpam-6613	37	11	generalized	generalized	ADJ
ejpam-6613	37	12	2	2	NUM
ejpam-6613	37	13	-	-	PUNCT
ejpam-6613	37	14	norm	norm	NOUN
ejpam-6613	37	15	on	on	ADP
ejpam-6613	37	16	x	x	X
ejpam-6613	37	17	.	.	PUNCT
ejpam-6613	38	1	below	below	ADP
ejpam-6613	38	2	we	we	PRON
ejpam-6613	38	3	present	present	VERB
ejpam-6613	38	4	the	the	DET
ejpam-6613	38	5	concept	concept	NOUN
ejpam-6613	38	6	of	of	ADP
ejpam-6613	38	7	generalized	generalized	ADJ
ejpam-6613	38	8	2	2	NUM
ejpam-6613	38	9	-	-	PUNCT
ejpam-6613	38	10	inner	inner	ADJ
ejpam-6613	38	11	product	product	NOUN
ejpam-6613	38	12	,	,	PUNCT
ejpam-6613	38	13	a	a	DET
ejpam-6613	38	14	notion	notion	NOUN
ejpam-6613	38	15	of	of	ADP
ejpam-6613	38	16	great	great	ADJ
ejpam-6613	38	17	importance	importance	NOUN
ejpam-6613	38	18	for	for	ADP
ejpam-6613	38	19	this	this	DET
ejpam-6613	38	20	work	work	NOUN
ejpam-6613	38	21	.	.	PUNCT
ejpam-6613	39	1	definition	definition	NOUN
ejpam-6613	39	2	2	2	NUM
ejpam-6613	39	3	.	.	PUNCT
ejpam-6613	40	1	[	[	X
ejpam-6613	40	2	14	14	NUM
ejpam-6613	40	3	]	]	PUNCT
ejpam-6613	40	4	a	a	DET
ejpam-6613	40	5	generalized	generalized	ADJ
ejpam-6613	40	6	2	2	NUM
ejpam-6613	40	7	-	-	PUNCT
ejpam-6613	40	8	inner	inner	ADJ
ejpam-6613	40	9	product	product	NOUN
ejpam-6613	40	10	is	be	AUX
ejpam-6613	40	11	a	a	DET
ejpam-6613	40	12	map	map	NOUN
ejpam-6613	40	13	⟨	⟨	VERB
ejpam-6613	40	14	·	·	PUNCT
ejpam-6613	40	15	,	,	PUNCT
ejpam-6613	40	16	·	·	PUNCT
ejpam-6613	40	17	|·⟩	|·⟩	X
ejpam-6613	40	18	:	:	PUNCT
ejpam-6613	40	19	x	x	X
ejpam-6613	40	20	×x	×x	X
ejpam-6613	40	21	×x	×x	X
ejpam-6613	40	22	−→	−→	NOUN
ejpam-6613	40	23	c	c	NOUN
ejpam-6613	40	24	such	such	ADJ
ejpam-6613	40	25	that	that	PRON
ejpam-6613	40	26	for	for	ADP
ejpam-6613	40	27	all	all	DET
ejpam-6613	40	28	w	w	PROPN
ejpam-6613	40	29	,	,	PUNCT
ejpam-6613	40	30	y	y	PROPN
ejpam-6613	40	31	,	,	PUNCT
ejpam-6613	40	32	z	z	PROPN
ejpam-6613	40	33	,	,	PUNCT
ejpam-6613	40	34	w1	w1	NOUN
ejpam-6613	40	35	,	,	PUNCT
ejpam-6613	40	36	w2	w2	NOUN
ejpam-6613	40	37	∈	∈	PROPN
ejpam-6613	40	38	x	x	X
ejpam-6613	40	39	and	and	CCONJ
ejpam-6613	40	40	for	for	ADP
ejpam-6613	40	41	all	all	DET
ejpam-6613	40	42	α	α	NOUN
ejpam-6613	40	43	∈	∈	NOUN
ejpam-6613	40	44	c	c	X
ejpam-6613	40	45	i	i	NOUN
ejpam-6613	40	46	)	)	PUNCT
ejpam-6613	40	47	⟨y	⟨y	X
ejpam-6613	40	48	,	,	PUNCT
ejpam-6613	40	49	w|z⟩	w|z⟩	PROPN
ejpam-6613	40	50	=	=	PUNCT
ejpam-6613	40	51	⟨w	⟨w	PROPN
ejpam-6613	40	52	,	,	PUNCT
ejpam-6613	40	53	y|z⟩	y|z⟩	PROPN
ejpam-6613	40	54	;	;	PUNCT
ejpam-6613	40	55	ii	ii	NOUN
ejpam-6613	40	56	)	)	PUNCT
ejpam-6613	40	57	⟨w1	⟨w1	PROPN
ejpam-6613	40	58	+	+	PROPN
ejpam-6613	40	59	w2	w2	NOUN
ejpam-6613	40	60	,	,	PUNCT
ejpam-6613	40	61	y|z⟩	y|z⟩	PROPN
ejpam-6613	40	62	=	=	SYM
ejpam-6613	40	63	⟨w1	⟨w1	PROPN
ejpam-6613	40	64	,	,	PUNCT
ejpam-6613	40	65	y|z⟩+	y|z⟩+	PROPN
ejpam-6613	40	66	⟨w2	⟨w2	PROPN
ejpam-6613	40	67	,	,	PUNCT
ejpam-6613	40	68	y|z⟩	y|z⟩	PROPN
ejpam-6613	40	69	;	;	PUNCT
ejpam-6613	40	70	iii	iii	X
ejpam-6613	40	71	)	)	PUNCT
ejpam-6613	40	72	⟨αw	⟨αw	PROPN
ejpam-6613	40	73	,	,	PUNCT
ejpam-6613	40	74	y|z⟩	y|z⟩	PROPN
ejpam-6613	40	75	=	=	SYM
ejpam-6613	40	76	α⟨w	α⟨w	PROPN
ejpam-6613	40	77	,	,	PUNCT
ejpam-6613	40	78	y|z⟩	y|z⟩	PROPN
ejpam-6613	40	79	;	;	PUNCT
ejpam-6613	40	80	iv	iv	NUM
ejpam-6613	40	81	)	)	PUNCT
ejpam-6613	40	82	⟨w	⟨w	X
ejpam-6613	40	83	,	,	PUNCT
ejpam-6613	40	84	w|z⟩	w|z⟩	PROPN
ejpam-6613	40	85	=	=	SYM
ejpam-6613	40	86	⟨z	⟨z	PROPN
ejpam-6613	40	87	,	,	PUNCT
ejpam-6613	40	88	z|w⟩	z|w⟩	NOUN
ejpam-6613	40	89	;	;	PUNCT
ejpam-6613	40	90	v	v	NOUN
ejpam-6613	40	91	)	)	PUNCT
ejpam-6613	40	92	⟨w	⟨w	X
ejpam-6613	40	93	,	,	PUNCT
ejpam-6613	40	94	w|z⟩	w|z⟩	PROPN
ejpam-6613	40	95	≥	≥	NUM
ejpam-6613	40	96	0	0	NUM
ejpam-6613	40	97	.	.	PUNCT
ejpam-6613	41	1	a	a	DET
ejpam-6613	41	2	complex	complex	ADJ
ejpam-6613	41	3	vector	vector	NOUN
ejpam-6613	41	4	space	space	NOUN
ejpam-6613	41	5	x	x	PUNCT
ejpam-6613	41	6	together	together	ADV
ejpam-6613	41	7	a	a	DET
ejpam-6613	41	8	generalized	generalized	ADJ
ejpam-6613	41	9	2	2	NUM
ejpam-6613	41	10	-	-	PUNCT
ejpam-6613	41	11	inner	inner	ADJ
ejpam-6613	41	12	product	product	NOUN
ejpam-6613	41	13	⟨	⟨	VERB
ejpam-6613	41	14	·	·	PUNCT
ejpam-6613	41	15	,	,	PUNCT
ejpam-6613	41	16	·	·	PUNCT
ejpam-6613	41	17	|·⟩	|·⟩	X
ejpam-6613	41	18	,	,	PUNCT
ejpam-6613	41	19	denoted	denote	VERB
ejpam-6613	41	20	by	by	ADP
ejpam-6613	41	21	(	(	PUNCT
ejpam-6613	41	22	x	x	INTJ
ejpam-6613	41	23	,	,	PUNCT
ejpam-6613	41	24	⟨	⟨	NOUN
ejpam-6613	41	25	·	·	SYM
ejpam-6613	41	26	,	,	PUNCT
ejpam-6613	41	27	·	·	PUNCT
ejpam-6613	41	28	|·⟩	|·⟩	X
ejpam-6613	41	29	)	)	PUNCT
ejpam-6613	41	30	,	,	PUNCT
ejpam-6613	41	31	it	it	PRON
ejpam-6613	41	32	is	be	AUX
ejpam-6613	41	33	said	say	VERB
ejpam-6613	41	34	to	to	PART
ejpam-6613	41	35	be	be	AUX
ejpam-6613	41	36	a	a	DET
ejpam-6613	41	37	generalized	generalized	ADJ
ejpam-6613	41	38	2	2	NUM
ejpam-6613	41	39	-	-	PUNCT
ejpam-6613	41	40	inner	inner	ADJ
ejpam-6613	41	41	product	product	NOUN
ejpam-6613	41	42	space	space	NOUN
ejpam-6613	41	43	.	.	PUNCT
ejpam-6613	42	1	proposition	proposition	NOUN
ejpam-6613	42	2	1	1	NUM
ejpam-6613	42	3	.	.	PUNCT
ejpam-6613	43	1	[	[	X
ejpam-6613	43	2	20]it	20]it	NUM
ejpam-6613	43	3	is	be	AUX
ejpam-6613	43	4	easy	easy	ADJ
ejpam-6613	43	5	to	to	PART
ejpam-6613	43	6	check	check	VERB
ejpam-6613	43	7	that	that	SCONJ
ejpam-6613	43	8	the	the	DET
ejpam-6613	43	9	following	follow	VERB
ejpam-6613	43	10	function	function	NOUN
ejpam-6613	43	11	is	be	AUX
ejpam-6613	43	12	a	a	DET
ejpam-6613	43	13	generalized	generalized	ADJ
ejpam-6613	43	14	2	2	NUM
ejpam-6613	43	15	-	-	PUNCT
ejpam-6613	43	16	inner	inner	ADJ
ejpam-6613	43	17	product	product	NOUN
ejpam-6613	43	18	on	on	ADP
ejpam-6613	43	19	x	x	X
ejpam-6613	43	20	⟨w	⟨w	NOUN
ejpam-6613	43	21	,	,	PUNCT
ejpam-6613	43	22	y|z⟩	y|z⟩	PROPN
ejpam-6613	43	23	=	=	SYM
ejpam-6613	43	24	∣∣∣∣⟨w	∣∣∣∣⟨w	PROPN
ejpam-6613	43	25	,	,	PUNCT
ejpam-6613	43	26	y⟩	y⟩	NOUN
ejpam-6613	43	27	⟨w	⟨w	X
ejpam-6613	43	28	,	,	PUNCT
ejpam-6613	43	29	z⟩	z⟩	PROPN
ejpam-6613	43	30	⟨z	⟨z	PROPN
ejpam-6613	43	31	,	,	PUNCT
ejpam-6613	43	32	y⟩	y⟩	NOUN
ejpam-6613	43	33	⟨z	⟨z	PROPN
ejpam-6613	43	34	,	,	PUNCT
ejpam-6613	43	35	z⟩	z⟩	PROPN
ejpam-6613	43	36	∣∣∣∣	∣∣∣∣	PROPN
ejpam-6613	43	37	=	=	PUNCT
ejpam-6613	43	38	⟨w	⟨w	NOUN
ejpam-6613	43	39	,	,	PUNCT
ejpam-6613	43	40	y⟩	y⟩	NOUN
ejpam-6613	43	41	⟨z	⟨z	PROPN
ejpam-6613	43	42	,	,	PUNCT
ejpam-6613	43	43	z⟩	z⟩	PUNCT
ejpam-6613	43	44	−	−	PROPN
ejpam-6613	43	45	⟨w	⟨w	X
ejpam-6613	43	46	,	,	PUNCT
ejpam-6613	43	47	z⟩	z⟩	PROPN
ejpam-6613	43	48	⟨z	⟨z	PROPN
ejpam-6613	43	49	,	,	PUNCT
ejpam-6613	43	50	y⟩	y⟩	NOUN
ejpam-6613	43	51	for	for	ADP
ejpam-6613	43	52	all	all	DET
ejpam-6613	43	53	w	w	PROPN
ejpam-6613	43	54	,	,	PUNCT
ejpam-6613	43	55	y	y	PROPN
ejpam-6613	43	56	,	,	PUNCT
ejpam-6613	43	57	z	z	NOUN
ejpam-6613	43	58	∈	∈	PROPN
ejpam-6613	43	59	x	x	X
ejpam-6613	43	60	.	.	PUNCT
ejpam-6613	44	1	this	this	DET
ejpam-6613	44	2	2	2	NUM
ejpam-6613	44	3	-	-	PUNCT
ejpam-6613	44	4	inner	inner	ADJ
ejpam-6613	44	5	product	product	NOUN
ejpam-6613	44	6	is	be	AUX
ejpam-6613	44	7	called	call	VERB
ejpam-6613	44	8	generalized	generalized	ADJ
ejpam-6613	44	9	standard	standard	ADJ
ejpam-6613	44	10	2	2	NUM
ejpam-6613	44	11	-	-	PUNCT
ejpam-6613	44	12	inner	inner	ADJ
ejpam-6613	44	13	product	product	NOUN
ejpam-6613	44	14	and	and	CCONJ
ejpam-6613	44	15	it	it	PRON
ejpam-6613	44	16	is	be	AUX
ejpam-6613	44	17	denoted	denote	VERB
ejpam-6613	44	18	by	by	ADP
ejpam-6613	44	19	⟨	⟨	NOUN
ejpam-6613	44	20	·	·	PUNCT
ejpam-6613	44	21	,	,	PUNCT
ejpam-6613	44	22	·	·	PUNCT
ejpam-6613	45	1	|·⟩stand	|·⟩stand	PROPN
ejpam-6613	45	2	.	.	PUNCT
ejpam-6613	46	1	a	a	DET
ejpam-6613	46	2	consequence	consequence	NOUN
ejpam-6613	46	3	of	of	ADP
ejpam-6613	46	4	[	[	X
ejpam-6613	46	5	21	21	NUM
ejpam-6613	46	6	,	,	PUNCT
ejpam-6613	46	7	prop	prop	NOUN
ejpam-6613	46	8	3.1	3.1	NUM
ejpam-6613	46	9	]	]	PUNCT
ejpam-6613	46	10	when	when	SCONJ
ejpam-6613	46	11	(	(	PUNCT
ejpam-6613	46	12	x	x	X
ejpam-6613	46	13	,	,	PUNCT
ejpam-6613	46	14	⟨	⟨	NOUN
ejpam-6613	46	15	·	·	SYM
ejpam-6613	46	16	,	,	PUNCT
ejpam-6613	46	17	·	·	PUNCT
ejpam-6613	46	18	⟩	⟩	NOUN
ejpam-6613	46	19	)	)	PUNCT
ejpam-6613	46	20	is	be	AUX
ejpam-6613	46	21	a	a	DET
ejpam-6613	46	22	classical	classical	ADJ
ejpam-6613	46	23	inner	inner	ADJ
ejpam-6613	46	24	product	product	NOUN
ejpam-6613	46	25	space	space	NOUN
ejpam-6613	46	26	is	be	AUX
ejpam-6613	46	27	:	:	PUNCT
ejpam-6613	46	28	proposition	proposition	NOUN
ejpam-6613	46	29	2	2	NUM
ejpam-6613	46	30	.	.	PUNCT
ejpam-6613	47	1	[	[	X
ejpam-6613	47	2	21	21	NUM
ejpam-6613	47	3	]	]	X
ejpam-6613	47	4	let	let	VERB
ejpam-6613	47	5	(	(	PUNCT
ejpam-6613	47	6	x	x	X
ejpam-6613	47	7	,	,	PUNCT
ejpam-6613	47	8	⟨	⟨	NOUN
ejpam-6613	47	9	·	·	SYM
ejpam-6613	47	10	,	,	PUNCT
ejpam-6613	47	11	·	·	PUNCT
ejpam-6613	47	12	⟩	⟩	NOUN
ejpam-6613	47	13	)	)	PUNCT
ejpam-6613	47	14	be	be	AUX
ejpam-6613	47	15	a	a	DET
ejpam-6613	47	16	classical	classical	ADJ
ejpam-6613	47	17	inner	inner	ADJ
ejpam-6613	47	18	product	product	NOUN
ejpam-6613	47	19	space	space	NOUN
ejpam-6613	47	20	.	.	PUNCT
ejpam-6613	48	1	then	then	ADV
ejpam-6613	48	2	the	the	DET
ejpam-6613	48	3	mapping	mapping	NOUN
ejpam-6613	48	4	⟨	⟨	VERB
ejpam-6613	48	5	·	·	PUNCT
ejpam-6613	48	6	,	,	PUNCT
ejpam-6613	48	7	·	·	PUNCT
ejpam-6613	48	8	|·⟩	|·⟩	X
ejpam-6613	48	9	:	:	PUNCT
ejpam-6613	48	10	x	x	SYM
ejpam-6613	49	1	×	×	NOUN
ejpam-6613	49	2	x	x	SYM
ejpam-6613	49	3	×	×	NOUN
ejpam-6613	49	4	x	x	SYM
ejpam-6613	49	5	−→	−→	NOUN
ejpam-6613	49	6	c	c	NOUN
ejpam-6613	49	7	defined	define	VERB
ejpam-6613	49	8	by	by	ADP
ejpam-6613	49	9	⟨x	⟨x	NUM
ejpam-6613	49	10	,	,	PUNCT
ejpam-6613	49	11	y|z⟩	y|z⟩	PROPN
ejpam-6613	49	12	=	=	PUNCT
ejpam-6613	49	13	⟨x	⟨x	VERB
ejpam-6613	49	14	,	,	PUNCT
ejpam-6613	49	15	y⟩	y⟩	NOUN
ejpam-6613	49	16	∥z∥2	∥z∥2	VERB
ejpam-6613	49	17	for	for	ADP
ejpam-6613	49	18	all	all	DET
ejpam-6613	49	19	x	x	NOUN
ejpam-6613	49	20	,	,	PUNCT
ejpam-6613	49	21	y	y	PROPN
ejpam-6613	49	22	,	,	PUNCT
ejpam-6613	49	23	z	z	NOUN
ejpam-6613	49	24	∈	∈	PROPN
ejpam-6613	49	25	x	x	PUNCT
ejpam-6613	49	26	,	,	PUNCT
ejpam-6613	49	27	is	be	AUX
ejpam-6613	49	28	a	a	DET
ejpam-6613	49	29	generalized	generalized	ADJ
ejpam-6613	49	30	2	2	NUM
ejpam-6613	49	31	-	-	PUNCT
ejpam-6613	49	32	inner	inner	ADJ
ejpam-6613	49	33	product	product	NOUN
ejpam-6613	49	34	on	on	ADP
ejpam-6613	49	35	x	x	PROPN
ejpam-6613	49	36	.	.	PUNCT
ejpam-6613	49	37	m.	m.	PROPN
ejpam-6613	49	38	luis	luis	PROPN
ejpam-6613	49	39	,	,	PUNCT
ejpam-6613	49	40	f.	f.	PROPN
ejpam-6613	49	41	osmin	osmin	PROPN
ejpam-6613	49	42	,	,	PUNCT
ejpam-6613	49	43	s.	s.	PROPN
ejpam-6613	49	44	arley	arley	PROPN
ejpam-6613	49	45	/	/	SYM
ejpam-6613	49	46	eur	eur	PROPN
ejpam-6613	49	47	.	.	PUNCT
ejpam-6613	50	1	j.	j.	PROPN
ejpam-6613	50	2	pure	pure	PROPN
ejpam-6613	50	3	appl	appl	PROPN
ejpam-6613	50	4	.	.	PROPN
ejpam-6613	50	5	math	math	PROPN
ejpam-6613	50	6	,	,	PUNCT
ejpam-6613	50	7	18	18	NUM
ejpam-6613	50	8	(	(	PUNCT
ejpam-6613	50	9	4	4	NUM
ejpam-6613	50	10	)	)	PUNCT
ejpam-6613	50	11	(	(	PUNCT
ejpam-6613	50	12	2025	2025	NUM
ejpam-6613	50	13	)	)	PUNCT
ejpam-6613	50	14	,	,	PUNCT
ejpam-6613	50	15	6613	6613	NUM
ejpam-6613	50	16	4	4	NUM
ejpam-6613	50	17	of	of	ADP
ejpam-6613	50	18	17	17	NUM
ejpam-6613	50	19	proposition	proposition	NOUN
ejpam-6613	50	20	3	3	NUM
ejpam-6613	50	21	.	.	PUNCT
ejpam-6613	51	1	[	[	X
ejpam-6613	51	2	19	19	NUM
ejpam-6613	51	3	]	]	X
ejpam-6613	51	4	let	let	VERB
ejpam-6613	51	5	(	(	PUNCT
ejpam-6613	51	6	x	x	X
ejpam-6613	51	7	,	,	PUNCT
ejpam-6613	51	8	⟨	⟨	NOUN
ejpam-6613	51	9	·	·	SYM
ejpam-6613	51	10	,	,	PUNCT
ejpam-6613	51	11	·	·	PUNCT
ejpam-6613	51	12	|·⟩	|·⟩	X
ejpam-6613	51	13	)	)	PUNCT
ejpam-6613	51	14	be	be	VERB
ejpam-6613	51	15	a	a	DET
ejpam-6613	51	16	generalized	generalized	ADJ
ejpam-6613	51	17	2	2	NUM
ejpam-6613	51	18	-	-	PUNCT
ejpam-6613	51	19	inner	inner	ADJ
ejpam-6613	51	20	product	product	NOUN
ejpam-6613	51	21	space	space	NOUN
ejpam-6613	51	22	.	.	PUNCT
ejpam-6613	52	1	then	then	ADV
ejpam-6613	52	2	,	,	PUNCT
ejpam-6613	52	3	for	for	ADP
ejpam-6613	52	4	all	all	DET
ejpam-6613	52	5	w	w	PROPN
ejpam-6613	52	6	,	,	PUNCT
ejpam-6613	52	7	y	y	PROPN
ejpam-6613	52	8	,	,	PUNCT
ejpam-6613	52	9	z	z	NOUN
ejpam-6613	52	10	∈	∈	PROPN
ejpam-6613	52	11	x	x	PUNCT
ejpam-6613	52	12	it	it	PRON
ejpam-6613	52	13	holds	hold	VERB
ejpam-6613	52	14	that	that	SCONJ
ejpam-6613	52	15	:	:	PUNCT
ejpam-6613	52	16	|⟨w	|⟨w	NOUN
ejpam-6613	52	17	,	,	PUNCT
ejpam-6613	52	18	y|z⟩|2	y|z⟩|2	PROPN
ejpam-6613	52	19	≤	≤	ADV
ejpam-6613	52	20	⟨w	⟨w	X
ejpam-6613	52	21	,	,	PUNCT
ejpam-6613	52	22	w|z⟩⟨y	w|z⟩⟨y	PROPN
ejpam-6613	52	23	,	,	PUNCT
ejpam-6613	52	24	y|z⟩.	y|z⟩.	PROPN
ejpam-6613	52	25	remark	remark	VERB
ejpam-6613	52	26	1	1	NUM
ejpam-6613	52	27	.	.	PUNCT
ejpam-6613	53	1	the	the	DET
ejpam-6613	53	2	function	function	NOUN
ejpam-6613	53	3	∥w	∥w	PROPN
ejpam-6613	53	4	,	,	PUNCT
ejpam-6613	53	5	z∥	z∥	ADJ
ejpam-6613	53	6	:	:	PUNCT
ejpam-6613	53	7	=	=	SYM
ejpam-6613	53	8	√	√	NUM
ejpam-6613	53	9	⟨w	⟨w	NOUN
ejpam-6613	53	10	,	,	PUNCT
ejpam-6613	53	11	w|z⟩	w|z⟩	PROPN
ejpam-6613	53	12	,	,	PUNCT
ejpam-6613	53	13	w	w	PROPN
ejpam-6613	53	14	,	,	PUNCT
ejpam-6613	53	15	z	z	NOUN
ejpam-6613	53	16	∈	∈	PROPN
ejpam-6613	53	17	x	x	INTJ
ejpam-6613	53	18	,	,	PUNCT
ejpam-6613	53	19	sometimes	sometimes	ADV
ejpam-6613	53	20	it	it	PRON
ejpam-6613	53	21	is	be	AUX
ejpam-6613	53	22	called	call	VERB
ejpam-6613	53	23	induced	induced	ADJ
ejpam-6613	53	24	generalized	generalized	ADJ
ejpam-6613	53	25	2	2	NUM
ejpam-6613	53	26	-	-	PUNCT
ejpam-6613	53	27	norm	norm	NOUN
ejpam-6613	53	28	by	by	ADP
ejpam-6613	53	29	the	the	DET
ejpam-6613	53	30	generalized	generalized	ADJ
ejpam-6613	53	31	2	2	NUM
ejpam-6613	53	32	-	-	PUNCT
ejpam-6613	53	33	inner	inner	ADJ
ejpam-6613	53	34	product	product	NOUN
ejpam-6613	53	35	.	.	PUNCT
ejpam-6613	54	1	definition	definition	NOUN
ejpam-6613	54	2	3	3	NUM
ejpam-6613	54	3	(	(	PUNCT
ejpam-6613	54	4	2	2	NUM
ejpam-6613	54	5	-	-	PUNCT
ejpam-6613	54	6	bounded	bound	VERB
ejpam-6613	54	7	operator	operator	NOUN
ejpam-6613	54	8	)	)	PUNCT
ejpam-6613	54	9	.	.	PUNCT
ejpam-6613	55	1	[	[	X
ejpam-6613	55	2	14	14	NUM
ejpam-6613	55	3	]	]	PUNCT
ejpam-6613	55	4	let	let	VERB
ejpam-6613	55	5	x	x	PRON
ejpam-6613	55	6	be	be	AUX
ejpam-6613	55	7	a	a	DET
ejpam-6613	55	8	vector	vector	NOUN
ejpam-6613	55	9	space	space	NOUN
ejpam-6613	55	10	endowed	endow	VERB
ejpam-6613	55	11	with	with	ADP
ejpam-6613	55	12	two	two	NUM
ejpam-6613	55	13	generalized	generalized	ADJ
ejpam-6613	55	14	2	2	NUM
ejpam-6613	55	15	-	-	PUNCT
ejpam-6613	55	16	norms	norm	NOUN
ejpam-6613	55	17	∥	∥	ADJ
ejpam-6613	55	18	·	·	SYM
ejpam-6613	55	19	,	,	PUNCT
ejpam-6613	55	20	·	·	PUNCT
ejpam-6613	55	21	∥1	∥1	PRON
ejpam-6613	55	22	and	and	CCONJ
ejpam-6613	55	23	∥	∥	PROPN
ejpam-6613	55	24	·	·	PUNCT
ejpam-6613	55	25	,	,	PUNCT
ejpam-6613	55	26	·	·	PUNCT
ejpam-6613	55	27	∥2	∥2	X
ejpam-6613	55	28	.	.	PUNCT
ejpam-6613	56	1	an	an	DET
ejpam-6613	56	2	operator	operator	NOUN
ejpam-6613	56	3	t	t	NOUN
ejpam-6613	56	4	:	:	PUNCT
ejpam-6613	56	5	(	(	PUNCT
ejpam-6613	56	6	x	x	X
ejpam-6613	56	7	,	,	PUNCT
ejpam-6613	56	8	∥	∥	NOUN
ejpam-6613	56	9	·	·	SYM
ejpam-6613	56	10	,	,	PUNCT
ejpam-6613	56	11	·	·	PUNCT
ejpam-6613	56	12	∥1	∥1	X
ejpam-6613	56	13	)	)	PUNCT
ejpam-6613	56	14	→	→	SYM
ejpam-6613	56	15	(	(	PUNCT
ejpam-6613	56	16	x	x	X
ejpam-6613	56	17	,	,	PUNCT
ejpam-6613	56	18	∥	∥	NOUN
ejpam-6613	56	19	·	·	SYM
ejpam-6613	56	20	,	,	PUNCT
ejpam-6613	56	21	·	·	PUNCT
ejpam-6613	56	22	∥2	∥2	X
ejpam-6613	56	23	)	)	PUNCT
ejpam-6613	56	24	is	be	AUX
ejpam-6613	56	25	said	say	VERB
ejpam-6613	56	26	to	to	PART
ejpam-6613	56	27	be	be	AUX
ejpam-6613	56	28	2	2	NUM
ejpam-6613	56	29	-	-	PUNCT
ejpam-6613	56	30	bounded	bound	VERB
ejpam-6613	56	31	if	if	SCONJ
ejpam-6613	56	32	there	there	ADV
ejpam-6613	56	33	α	α	PRON
ejpam-6613	56	34	≥	≥	NOUN
ejpam-6613	56	35	0	0	NUM
ejpam-6613	57	1	such	such	ADJ
ejpam-6613	57	2	that	that	PRON
ejpam-6613	57	3	∥t	∥t	PROPN
ejpam-6613	57	4	(	(	PUNCT
ejpam-6613	57	5	x	x	NOUN
ejpam-6613	57	6	)	)	PUNCT
ejpam-6613	57	7	,	,	PUNCT
ejpam-6613	57	8	y∥2	y∥2	NOUN
ejpam-6613	57	9	+	+	CCONJ
ejpam-6613	57	10	∥x	∥x	PROPN
ejpam-6613	57	11	,	,	PUNCT
ejpam-6613	57	12	t	t	PROPN
ejpam-6613	57	13	(	(	PUNCT
ejpam-6613	57	14	y)∥2	y)∥2	PROPN
ejpam-6613	57	15	≤	≤	PROPN
ejpam-6613	57	16	α∥x	α∥x	PROPN
ejpam-6613	57	17	,	,	PUNCT
ejpam-6613	57	18	y∥1	y∥1	VERB
ejpam-6613	57	19	for	for	ADP
ejpam-6613	57	20	all	all	DET
ejpam-6613	57	21	x	x	NOUN
ejpam-6613	57	22	,	,	PUNCT
ejpam-6613	57	23	y	y	PROPN
ejpam-6613	57	24	∈	∈	PROPN
ejpam-6613	57	25	x	x	X
ejpam-6613	57	26	.	.	PUNCT
ejpam-6613	58	1	the	the	DET
ejpam-6613	58	2	symbol	symbol	NOUN
ejpam-6613	58	3	2	2	NUM
ejpam-6613	58	4	b(x	b(x	NOUN
ejpam-6613	58	5	)	)	PUNCT
ejpam-6613	58	6	will	will	AUX
ejpam-6613	58	7	denote	denote	VERB
ejpam-6613	58	8	the	the	DET
ejpam-6613	58	9	set	set	NOUN
ejpam-6613	58	10	of	of	ADP
ejpam-6613	58	11	2	2	NUM
ejpam-6613	58	12	-	-	PUNCT
ejpam-6613	58	13	bounded	bound	VERB
ejpam-6613	58	14	linear	linear	ADJ
ejpam-6613	58	15	operator	operator	NOUN
ejpam-6613	58	16	on	on	ADP
ejpam-6613	58	17	x	x	SYM
ejpam-6613	58	18	,	,	PUNCT
ejpam-6613	58	19	that	that	ADV
ejpam-6613	58	20	is	is	ADV
ejpam-6613	58	21	,	,	PUNCT
ejpam-6613	58	22	2	2	NUM
ejpam-6613	58	23	b(x	b(x	NOUN
ejpam-6613	58	24	)	)	PUNCT
ejpam-6613	58	25	:	:	PUNCT
ejpam-6613	59	1	=	=	SYM
ejpam-6613	59	2	{	{	PUNCT
ejpam-6613	59	3	t	t	NOUN
ejpam-6613	59	4	:	:	PUNCT
ejpam-6613	59	5	x	x	X
ejpam-6613	59	6	→	→	PUNCT
ejpam-6613	59	7	x	x	X
ejpam-6613	59	8	:	:	PUNCT
ejpam-6613	59	9	t	t	PROPN
ejpam-6613	59	10	is	be	AUX
ejpam-6613	59	11	linear	linear	ADJ
ejpam-6613	59	12	and	and	CCONJ
ejpam-6613	59	13	2	2	NUM
ejpam-6613	59	14	-	-	PUNCT
ejpam-6613	59	15	bounded	bound	VERB
ejpam-6613	59	16	}	}	PUNCT
ejpam-6613	59	17	.	.	PUNCT
ejpam-6613	60	1	remark	remark	NOUN
ejpam-6613	60	2	2	2	NUM
ejpam-6613	60	3	.	.	PUNCT
ejpam-6613	61	1	it	it	PRON
ejpam-6613	61	2	is	be	AUX
ejpam-6613	61	3	clear	clear	ADJ
ejpam-6613	61	4	that	that	SCONJ
ejpam-6613	61	5	the	the	DET
ejpam-6613	61	6	set	set	NOUN
ejpam-6613	61	7	2	2	NUM
ejpam-6613	61	8	b(x	b(x	NOUN
ejpam-6613	61	9	)	)	PUNCT
ejpam-6613	61	10	can	can	AUX
ejpam-6613	61	11	be	be	AUX
ejpam-6613	61	12	endowed	endow	VERB
ejpam-6613	61	13	with	with	ADP
ejpam-6613	61	14	a	a	DET
ejpam-6613	61	15	vector	vector	NOUN
ejpam-6613	61	16	space	space	NOUN
ejpam-6613	61	17	structure	structure	NOUN
ejpam-6613	61	18	over	over	ADP
ejpam-6613	61	19	c	c	PROPN
ejpam-6613	61	20	,	,	PUNCT
ejpam-6613	61	21	by	by	ADP
ejpam-6613	61	22	means	mean	NOUN
ejpam-6613	61	23	of	of	ADP
ejpam-6613	61	24	the	the	DET
ejpam-6613	61	25	pointwise	pointwise	ADJ
ejpam-6613	61	26	operations	operation	NOUN
ejpam-6613	61	27	of	of	ADP
ejpam-6613	61	28	operators	operator	NOUN
ejpam-6613	61	29	.	.	PUNCT
ejpam-6613	62	1	by	by	ADP
ejpam-6613	62	2	virtue	virtue	NOUN
ejpam-6613	62	3	of	of	ADP
ejpam-6613	62	4	the	the	DET
ejpam-6613	62	5	properties	property	NOUN
ejpam-6613	62	6	of	of	ADP
ejpam-6613	62	7	generalized	generalized	ADJ
ejpam-6613	62	8	2	2	NUM
ejpam-6613	62	9	-	-	PUNCT
ejpam-6613	62	10	normed	norme	VERB
ejpam-6613	62	11	spaces	space	NOUN
ejpam-6613	62	12	,	,	PUNCT
ejpam-6613	62	13	in	in	ADP
ejpam-6613	62	14	the	the	DET
ejpam-6613	62	15	following	following	NOUN
ejpam-6613	62	16	theorem	theorem	NOUN
ejpam-6613	62	17	we	we	PRON
ejpam-6613	62	18	establish	establish	VERB
ejpam-6613	62	19	an	an	DET
ejpam-6613	62	20	equivalent	equivalent	ADJ
ejpam-6613	62	21	result	result	NOUN
ejpam-6613	62	22	for	for	ADP
ejpam-6613	62	23	the	the	DET
ejpam-6613	62	24	definition	definition	NOUN
ejpam-6613	62	25	3	3	NUM
ejpam-6613	62	26	.	.	PUNCT
ejpam-6613	62	27	theorem	theorem	NOUN
ejpam-6613	62	28	1	1	NUM
ejpam-6613	62	29	.	.	PUNCT
ejpam-6613	62	30	note	note	VERB
ejpam-6613	62	31	that	that	SCONJ
ejpam-6613	62	32	a	a	DET
ejpam-6613	62	33	linear	linear	ADJ
ejpam-6613	62	34	operator	operator	NOUN
ejpam-6613	62	35	t	t	NOUN
ejpam-6613	62	36	:	:	PUNCT
ejpam-6613	62	37	(	(	PUNCT
ejpam-6613	62	38	x	x	X
ejpam-6613	62	39	,	,	PUNCT
ejpam-6613	62	40	∥	∥	X
ejpam-6613	62	41	·	·	SYM
ejpam-6613	62	42	,	,	PUNCT
ejpam-6613	62	43	·	·	PUNCT
ejpam-6613	62	44	∥1	∥1	X
ejpam-6613	62	45	)	)	PUNCT
ejpam-6613	62	46	→	→	SYM
ejpam-6613	62	47	(	(	PUNCT
ejpam-6613	62	48	x	x	INTJ
ejpam-6613	62	49	,	,	PUNCT
ejpam-6613	62	50	∥	∥	X
ejpam-6613	62	51	·	·	PUNCT
ejpam-6613	62	52	,	,	PUNCT
ejpam-6613	62	53	·	·	PUNCT
ejpam-6613	62	54	∥2	∥2	X
ejpam-6613	62	55	)	)	PUNCT
ejpam-6613	62	56	is	be	AUX
ejpam-6613	62	57	2	2	NUM
ejpam-6613	62	58	-	-	PUNCT
ejpam-6613	62	59	bounded	bound	VERB
ejpam-6613	62	60	according	accord	VERB
ejpam-6613	62	61	to	to	ADP
ejpam-6613	62	62	definition	definition	NOUN
ejpam-6613	62	63	3	3	NUM
ejpam-6613	63	1	if	if	SCONJ
ejpam-6613	63	2	and	and	CCONJ
ejpam-6613	63	3	only	only	ADV
ejpam-6613	63	4	if	if	SCONJ
ejpam-6613	63	5	there	there	PRON
ejpam-6613	63	6	exists	exist	VERB
ejpam-6613	63	7	a	a	DET
ejpam-6613	63	8	constant	constant	ADJ
ejpam-6613	63	9	β	β	X
ejpam-6613	63	10	>	>	X
ejpam-6613	63	11	0	0	NUM
ejpam-6613	63	12	such	such	ADJ
ejpam-6613	63	13	that	that	DET
ejpam-6613	63	14	∥tz	∥tz	NOUN
ejpam-6613	63	15	,	,	PUNCT
ejpam-6613	63	16	w∥2	w∥2	ADJ
ejpam-6613	63	17	≤	≤	NOUN
ejpam-6613	63	18	β	β	X
ejpam-6613	63	19	∥z	∥z	NOUN
ejpam-6613	63	20	,	,	PUNCT
ejpam-6613	63	21	w∥1	w∥1	NOUN
ejpam-6613	63	22	for	for	ADP
ejpam-6613	63	23	all	all	DET
ejpam-6613	63	24	z	z	NOUN
ejpam-6613	63	25	,	,	PUNCT
ejpam-6613	63	26	w	w	PROPN
ejpam-6613	63	27	∈	∈	PROPN
ejpam-6613	63	28	x	x	X
ejpam-6613	63	29	.	.	PUNCT
ejpam-6613	64	1	proof	proof	NOUN
ejpam-6613	64	2	.	.	PUNCT
ejpam-6613	65	1	suppose	suppose	VERB
ejpam-6613	65	2	first	first	ADV
ejpam-6613	65	3	that	that	SCONJ
ejpam-6613	65	4	there	there	PRON
ejpam-6613	65	5	exists	exist	VERB
ejpam-6613	65	6	α	α	PROPN
ejpam-6613	65	7	>	>	X
ejpam-6613	65	8	0	0	NUM
ejpam-6613	66	1	such	such	ADJ
ejpam-6613	66	2	that	that	SCONJ
ejpam-6613	66	3	∥tx	∥tx	PROPN
ejpam-6613	66	4	,	,	PUNCT
ejpam-6613	66	5	y∥2	y∥2	NOUN
ejpam-6613	66	6	+	+	CCONJ
ejpam-6613	66	7	∥x	∥x	PROPN
ejpam-6613	66	8	,	,	PUNCT
ejpam-6613	66	9	ty∥2	ty∥2	NOUN
ejpam-6613	66	10	≤	≤	NUM
ejpam-6613	66	11	α	α	X
ejpam-6613	66	12	∥x	∥x	PROPN
ejpam-6613	66	13	,	,	PUNCT
ejpam-6613	66	14	y∥1	y∥1	VERB
ejpam-6613	66	15	for	for	ADP
ejpam-6613	66	16	all	all	DET
ejpam-6613	66	17	x	x	NOUN
ejpam-6613	66	18	,	,	PUNCT
ejpam-6613	66	19	y	y	PROPN
ejpam-6613	66	20	∈	∈	PROPN
ejpam-6613	66	21	x	x	X
ejpam-6613	66	22	.	.	PUNCT
ejpam-6613	67	1	since	since	SCONJ
ejpam-6613	67	2	each	each	DET
ejpam-6613	67	3	2	2	NUM
ejpam-6613	67	4	-	-	PUNCT
ejpam-6613	67	5	norm	norm	NOUN
ejpam-6613	67	6	is	be	AUX
ejpam-6613	67	7	nonnegative	nonnegative	ADJ
ejpam-6613	67	8	,	,	PUNCT
ejpam-6613	67	9	it	it	PRON
ejpam-6613	67	10	follows	follow	VERB
ejpam-6613	67	11	that	that	SCONJ
ejpam-6613	67	12	∥tx	∥tx	PROPN
ejpam-6613	67	13	,	,	PUNCT
ejpam-6613	67	14	y∥2	y∥2	NOUN
ejpam-6613	67	15	≤	≤	PROPN
ejpam-6613	67	16	∥tx	∥tx	PROPN
ejpam-6613	67	17	,	,	PUNCT
ejpam-6613	67	18	y∥2	y∥2	NOUN
ejpam-6613	67	19	+	+	CCONJ
ejpam-6613	67	20	∥x	∥x	PROPN
ejpam-6613	67	21	,	,	PUNCT
ejpam-6613	67	22	ty∥2	ty∥2	NOUN
ejpam-6613	67	23	≤	≤	NUM
ejpam-6613	68	1	α	α	X
ejpam-6613	68	2	∥x	∥x	PROPN
ejpam-6613	68	3	,	,	PUNCT
ejpam-6613	68	4	y∥1	y∥1	NOUN
ejpam-6613	68	5	,	,	PUNCT
ejpam-6613	68	6	hence	hence	ADV
ejpam-6613	68	7	t	t	PROPN
ejpam-6613	68	8	satisfies	satisfy	VERB
ejpam-6613	68	9	the	the	DET
ejpam-6613	68	10	above	above	ADJ
ejpam-6613	68	11	inequality	inequality	NOUN
ejpam-6613	68	12	with	with	ADP
ejpam-6613	68	13	β	β	X
ejpam-6613	68	14	=	=	SYM
ejpam-6613	68	15	α	α	X
ejpam-6613	68	16	.	.	PUNCT
ejpam-6613	69	1	conversely	conversely	ADV
ejpam-6613	69	2	,	,	PUNCT
ejpam-6613	69	3	assume	assume	VERB
ejpam-6613	69	4	there	there	PRON
ejpam-6613	69	5	exists	exist	VERB
ejpam-6613	69	6	β	β	X
ejpam-6613	69	7	>	>	X
ejpam-6613	69	8	0	0	NUM
ejpam-6613	69	9	such	such	ADJ
ejpam-6613	69	10	that	that	DET
ejpam-6613	69	11	∥tz	∥tz	NOUN
ejpam-6613	69	12	,	,	PUNCT
ejpam-6613	69	13	w∥2	w∥2	ADJ
ejpam-6613	69	14	≤	≤	NOUN
ejpam-6613	69	15	β	β	X
ejpam-6613	69	16	∥z	∥z	NOUN
ejpam-6613	69	17	,	,	PUNCT
ejpam-6613	69	18	w∥1	w∥1	NOUN
ejpam-6613	69	19	for	for	ADP
ejpam-6613	69	20	all	all	DET
ejpam-6613	69	21	z	z	NOUN
ejpam-6613	69	22	,	,	PUNCT
ejpam-6613	69	23	w	w	PROPN
ejpam-6613	69	24	∈	∈	PROPN
ejpam-6613	69	25	x	x	X
ejpam-6613	69	26	.	.	PUNCT
ejpam-6613	70	1	then	then	ADV
ejpam-6613	70	2	for	for	ADP
ejpam-6613	70	3	any	any	DET
ejpam-6613	70	4	x	x	NOUN
ejpam-6613	70	5	,	,	PUNCT
ejpam-6613	70	6	y	y	PROPN
ejpam-6613	70	7	∈	∈	PROPN
ejpam-6613	70	8	x	x	INTJ
ejpam-6613	70	9	we	we	PRON
ejpam-6613	70	10	have	have	VERB
ejpam-6613	70	11	∥tx	∥tx	PROPN
ejpam-6613	70	12	,	,	PUNCT
ejpam-6613	70	13	y∥2	y∥2	NOUN
ejpam-6613	71	1	+	+	CCONJ
ejpam-6613	72	1	∥x	∥x	PROPN
ejpam-6613	72	2	,	,	PUNCT
ejpam-6613	72	3	ty∥2	ty∥2	NOUN
ejpam-6613	72	4	=	=	SYM
ejpam-6613	72	5	∥tx	∥tx	PROPN
ejpam-6613	72	6	,	,	PUNCT
ejpam-6613	72	7	y∥2	y∥2	NOUN
ejpam-6613	72	8	+	+	CCONJ
ejpam-6613	72	9	∥ty	∥ty	VERB
ejpam-6613	72	10	,	,	PUNCT
ejpam-6613	72	11	x∥2	x∥2	NOUN
ejpam-6613	72	12	≤	≤	NUM
ejpam-6613	72	13	β	β	X
ejpam-6613	73	1	∥x	∥x	PROPN
ejpam-6613	73	2	,	,	PUNCT
ejpam-6613	73	3	y∥1	y∥1	NOUN
ejpam-6613	73	4	+	+	CCONJ
ejpam-6613	73	5	β	β	X
ejpam-6613	73	6	∥y	∥y	PROPN
ejpam-6613	73	7	,	,	PUNCT
ejpam-6613	73	8	x∥1	x∥1	PUNCT
ejpam-6613	73	9	=	=	PUNCT
ejpam-6613	74	1	2β	2β	NOUN
ejpam-6613	74	2	∥x	∥x	PROPN
ejpam-6613	74	3	,	,	PUNCT
ejpam-6613	74	4	y∥1	y∥1	NOUN
ejpam-6613	74	5	.	.	PUNCT
ejpam-6613	75	1	if	if	SCONJ
ejpam-6613	75	2	we	we	PRON
ejpam-6613	75	3	set	set	VERB
ejpam-6613	75	4	α	α	NOUN
ejpam-6613	75	5	=	=	NOUN
ejpam-6613	75	6	2β	2β	NOUN
ejpam-6613	75	7	,	,	PUNCT
ejpam-6613	75	8	the	the	DET
ejpam-6613	75	9	desired	desire	VERB
ejpam-6613	75	10	inequality	inequality	NOUN
ejpam-6613	75	11	follows	follow	VERB
ejpam-6613	75	12	.	.	PUNCT
ejpam-6613	76	1	m.	m.	PROPN
ejpam-6613	76	2	luis	luis	PROPN
ejpam-6613	76	3	,	,	PUNCT
ejpam-6613	76	4	f.	f.	PROPN
ejpam-6613	76	5	osmin	osmin	PROPN
ejpam-6613	76	6	,	,	PUNCT
ejpam-6613	76	7	s.	s.	PROPN
ejpam-6613	76	8	arley	arley	PROPN
ejpam-6613	76	9	/	/	SYM
ejpam-6613	76	10	eur	eur	PROPN
ejpam-6613	76	11	.	.	PUNCT
ejpam-6613	77	1	j.	j.	PROPN
ejpam-6613	77	2	pure	pure	PROPN
ejpam-6613	77	3	appl	appl	PROPN
ejpam-6613	77	4	.	.	PROPN
ejpam-6613	77	5	math	math	PROPN
ejpam-6613	77	6	,	,	PUNCT
ejpam-6613	77	7	18	18	NUM
ejpam-6613	77	8	(	(	PUNCT
ejpam-6613	77	9	4	4	NUM
ejpam-6613	77	10	)	)	PUNCT
ejpam-6613	77	11	(	(	PUNCT
ejpam-6613	77	12	2025	2025	NUM
ejpam-6613	77	13	)	)	PUNCT
ejpam-6613	77	14	,	,	PUNCT
ejpam-6613	77	15	6613	6613	NUM
ejpam-6613	77	16	5	5	NUM
ejpam-6613	77	17	of	of	ADP
ejpam-6613	77	18	17	17	NUM
ejpam-6613	77	19	definition	definition	NOUN
ejpam-6613	77	20	4	4	NUM
ejpam-6613	77	21	.	.	PUNCT
ejpam-6613	78	1	if	if	SCONJ
ejpam-6613	78	2	t	t	PROPN
ejpam-6613	78	3	∈	∈	PROPN
ejpam-6613	78	4	2	2	NUM
ejpam-6613	78	5	b(x	b(x	NOUN
ejpam-6613	78	6	)	)	PUNCT
ejpam-6613	78	7	,	,	PUNCT
ejpam-6613	78	8	we	we	PRON
ejpam-6613	78	9	define	define	VERB
ejpam-6613	78	10	∥t∥	∥t∥	ADP
ejpam-6613	78	11	,	,	PUNCT
ejpam-6613	78	12	by	by	ADP
ejpam-6613	78	13	∥t∥	∥t∥	ADV
ejpam-6613	78	14	=	=	PUNCT
ejpam-6613	78	15	inf{a	inf{a	VERB
ejpam-6613	78	16	≥	≥	NOUN
ejpam-6613	78	17	0	0	NUM
ejpam-6613	78	18	:	:	PUNCT
ejpam-6613	79	1	∥t	∥t	INTJ
ejpam-6613	79	2	(	(	PUNCT
ejpam-6613	79	3	w	w	NOUN
ejpam-6613	79	4	)	)	PUNCT
ejpam-6613	79	5	,	,	PUNCT
ejpam-6613	79	6	z∥	z∥	ADJ
ejpam-6613	79	7	≤	≤	NUM
ejpam-6613	79	8	a∥w	a∥w	PROPN
ejpam-6613	79	9	,	,	PUNCT
ejpam-6613	79	10	z∥	z∥	VERB
ejpam-6613	79	11	for	for	ADP
ejpam-6613	79	12	all	all	DET
ejpam-6613	79	13	w	w	NOUN
ejpam-6613	79	14	,	,	PUNCT
ejpam-6613	79	15	z	z	NOUN
ejpam-6613	79	16	∈	∈	PROPN
ejpam-6613	79	17	x	x	X
ejpam-6613	79	18	}	}	PUNCT
ejpam-6613	79	19	.	.	PUNCT
ejpam-6613	80	1	note	note	VERB
ejpam-6613	80	2	that	that	SCONJ
ejpam-6613	80	3	the	the	DET
ejpam-6613	80	4	identity	identity	NOUN
ejpam-6613	80	5	operator	operator	NOUN
ejpam-6613	80	6	in	in	ADP
ejpam-6613	80	7	x	x	SYM
ejpam-6613	80	8	,	,	PUNCT
ejpam-6613	80	9	i	i	PRON
ejpam-6613	80	10	:	:	PUNCT
ejpam-6613	80	11	x	x	X
ejpam-6613	80	12	→	→	SYM
ejpam-6613	80	13	x	x	SYM
ejpam-6613	80	14	,	,	PUNCT
ejpam-6613	80	15	fulfills	fulfill	VERB
ejpam-6613	80	16	∥i∥	∥i∥	PROPN
ejpam-6613	80	17	=	=	SYM
ejpam-6613	80	18	1	1	X
ejpam-6613	80	19	.	.	PUNCT
ejpam-6613	80	20	theorem	theorem	NOUN
ejpam-6613	80	21	2	2	NUM
ejpam-6613	80	22	.	.	PUNCT
ejpam-6613	80	23	in	in	ADP
ejpam-6613	80	24	the	the	DET
ejpam-6613	80	25	context	context	NOUN
ejpam-6613	80	26	of	of	ADP
ejpam-6613	80	27	the	the	DET
ejpam-6613	80	28	definition	definition	NOUN
ejpam-6613	80	29	4	4	NUM
ejpam-6613	80	30	,	,	PUNCT
ejpam-6613	80	31	for	for	ADP
ejpam-6613	80	32	all	all	DET
ejpam-6613	80	33	t	t	NOUN
ejpam-6613	80	34	∈	∈	NOUN
ejpam-6613	80	35	2	2	NUM
ejpam-6613	80	36	b(x	b(x	NOUN
ejpam-6613	80	37	)	)	PUNCT
ejpam-6613	80	38	it	it	PRON
ejpam-6613	80	39	is	be	AUX
ejpam-6613	80	40	true	true	ADJ
ejpam-6613	81	1	that	that	SCONJ
ejpam-6613	81	2	∥t∥	∥t∥	ADV
ejpam-6613	81	3	=	=	SYM
ejpam-6613	81	4	sup{∥tw	sup{∥tw	PROPN
ejpam-6613	81	5	,	,	PUNCT
ejpam-6613	81	6	z∥	z∥	NUM
ejpam-6613	81	7	:	:	PUNCT
ejpam-6613	81	8	w	w	X
ejpam-6613	81	9	,	,	PUNCT
ejpam-6613	81	10	z	z	NOUN
ejpam-6613	81	11	∈	∈	PROPN
ejpam-6613	81	12	x	x	X
ejpam-6613	81	13	and	and	CCONJ
ejpam-6613	81	14	∥w	∥w	PROPN
ejpam-6613	81	15	,	,	PUNCT
ejpam-6613	81	16	z∥	z∥	NOUN
ejpam-6613	81	17	=	=	SYM
ejpam-6613	81	18	1	1	NUM
ejpam-6613	81	19	}	}	PUNCT
ejpam-6613	81	20	∥t∥	∥t∥	ADV
ejpam-6613	81	21	=	=	SYM
ejpam-6613	81	22	sup{∥tw	sup{∥tw	PROPN
ejpam-6613	81	23	,	,	PUNCT
ejpam-6613	81	24	z∥	z∥	NUM
ejpam-6613	81	25	:	:	PUNCT
ejpam-6613	81	26	w	w	X
ejpam-6613	81	27	,	,	PUNCT
ejpam-6613	81	28	z	z	NOUN
ejpam-6613	81	29	∈	∈	PROPN
ejpam-6613	81	30	x	x	X
ejpam-6613	81	31	and	and	CCONJ
ejpam-6613	81	32	∥w	∥w	PROPN
ejpam-6613	81	33	,	,	PUNCT
ejpam-6613	81	34	z∥	z∥	ADJ
ejpam-6613	81	35	≤	≤	NUM
ejpam-6613	81	36	1	1	NUM
ejpam-6613	81	37	}	}	PUNCT
ejpam-6613	81	38	∥t∥	∥t∥	ADV
ejpam-6613	81	39	=	=	SYM
ejpam-6613	81	40	sup	sup	NOUN
ejpam-6613	81	41	{	{	PUNCT
ejpam-6613	81	42	∥tw	∥tw	PROPN
ejpam-6613	81	43	,	,	PUNCT
ejpam-6613	81	44	z∥	z∥	PROPN
ejpam-6613	81	45	∥w	∥w	PROPN
ejpam-6613	81	46	,	,	PUNCT
ejpam-6613	81	47	z∥	z∥	NUM
ejpam-6613	81	48	:	:	PUNCT
ejpam-6613	81	49	w	w	X
ejpam-6613	81	50	,	,	PUNCT
ejpam-6613	81	51	z	z	NOUN
ejpam-6613	81	52	∈	∈	PROPN
ejpam-6613	81	53	x	x	X
ejpam-6613	81	54	and	and	CCONJ
ejpam-6613	81	55	∥w	∥w	PROPN
ejpam-6613	81	56	,	,	PUNCT
ejpam-6613	81	57	z∥	z∥	NOUN
ejpam-6613	81	58	=	=	NOUN
ejpam-6613	81	59	̸	̸	NUM
ejpam-6613	81	60	0	0	NUM
ejpam-6613	81	61	}	}	PUNCT
ejpam-6613	81	62	moreover	moreover	ADV
ejpam-6613	81	63	,	,	PUNCT
ejpam-6613	81	64	thanks	thank	NOUN
ejpam-6613	81	65	to	to	ADP
ejpam-6613	81	66	the	the	DET
ejpam-6613	81	67	definition	definition	NOUN
ejpam-6613	81	68	4	4	NUM
ejpam-6613	81	69	we	we	PRON
ejpam-6613	81	70	prove	prove	VERB
ejpam-6613	81	71	that	that	SCONJ
ejpam-6613	81	72	given	give	VERB
ejpam-6613	81	73	a	a	DET
ejpam-6613	81	74	generalized	generalized	ADJ
ejpam-6613	81	75	2	2	NUM
ejpam-6613	81	76	-	-	PUNCT
ejpam-6613	81	77	normed	norme	VERB
ejpam-6613	81	78	space	space	NOUN
ejpam-6613	81	79	x	x	X
ejpam-6613	81	80	,	,	PUNCT
ejpam-6613	81	81	it	it	PRON
ejpam-6613	81	82	is	be	AUX
ejpam-6613	81	83	possible	possible	ADJ
ejpam-6613	81	84	to	to	PART
ejpam-6613	81	85	endow	endow	VERB
ejpam-6613	81	86	the	the	DET
ejpam-6613	81	87	vector	vector	NOUN
ejpam-6613	81	88	space	space	NOUN
ejpam-6613	81	89	2	2	NUM
ejpam-6613	81	90	b(x	b(x	NOUN
ejpam-6613	81	91	)	)	PUNCT
ejpam-6613	81	92	with	with	ADP
ejpam-6613	81	93	the	the	DET
ejpam-6613	81	94	structure	structure	NOUN
ejpam-6613	81	95	of	of	ADP
ejpam-6613	81	96	a	a	DET
ejpam-6613	81	97	semi	semi	ADJ
ejpam-6613	81	98	-	-	ADJ
ejpam-6613	81	99	normed	normed	ADJ
ejpam-6613	81	100	space	space	NOUN
ejpam-6613	81	101	.	.	PUNCT
ejpam-6613	82	1	proposition	proposition	NOUN
ejpam-6613	82	2	4	4	NUM
ejpam-6613	82	3	.	.	PUNCT
ejpam-6613	83	1	[	[	X
ejpam-6613	83	2	14	14	NUM
ejpam-6613	83	3	]	]	PUNCT
ejpam-6613	83	4	for	for	ADP
ejpam-6613	83	5	all	all	DET
ejpam-6613	83	6	t	t	NOUN
ejpam-6613	83	7	∈	∈	NOUN
ejpam-6613	83	8	2	2	NUM
ejpam-6613	83	9	b(x	b(x	NOUN
ejpam-6613	83	10	)	)	PUNCT
ejpam-6613	83	11	it	it	PRON
ejpam-6613	83	12	is	be	AUX
ejpam-6613	83	13	true	true	ADJ
ejpam-6613	83	14	that	that	SCONJ
ejpam-6613	83	15	∥t	∥t	PROPN
ejpam-6613	83	16	(	(	PUNCT
ejpam-6613	83	17	w	w	NOUN
ejpam-6613	83	18	)	)	PUNCT
ejpam-6613	83	19	,	,	PUNCT
ejpam-6613	83	20	y∥	y∥	VERB
ejpam-6613	83	21	≤	≤	NUM
ejpam-6613	83	22	∥t∥∥w	∥t∥∥w	ADJ
ejpam-6613	83	23	,	,	PUNCT
ejpam-6613	83	24	y∥	y∥	VERB
ejpam-6613	83	25	for	for	ADP
ejpam-6613	83	26	all	all	DET
ejpam-6613	83	27	w	w	PROPN
ejpam-6613	83	28	,	,	PUNCT
ejpam-6613	83	29	y	y	PROPN
ejpam-6613	83	30	,	,	PUNCT
ejpam-6613	83	31	z	z	NOUN
ejpam-6613	83	32	∈	∈	PROPN
ejpam-6613	83	33	x	x	X
ejpam-6613	83	34	.	.	PUNCT
ejpam-6613	84	1	3	3	X
ejpam-6613	84	2	.	.	X
ejpam-6613	84	3	main	main	ADJ
ejpam-6613	84	4	results	result	NOUN
ejpam-6613	84	5	3.1	3.1	NUM
ejpam-6613	84	6	.	.	NOUN
ejpam-6613	84	7	2	2	NUM
ejpam-6613	84	8	-	-	PUNCT
ejpam-6613	84	9	tensor	tensor	NOUN
ejpam-6613	84	10	product	product	NOUN
ejpam-6613	84	11	in	in	ADP
ejpam-6613	84	12	this	this	DET
ejpam-6613	84	13	section	section	NOUN
ejpam-6613	84	14	,	,	PUNCT
ejpam-6613	84	15	we	we	PRON
ejpam-6613	84	16	introduce	introduce	VERB
ejpam-6613	84	17	the	the	DET
ejpam-6613	84	18	notion	notion	NOUN
ejpam-6613	84	19	of	of	ADP
ejpam-6613	84	20	the	the	DET
ejpam-6613	84	21	2	2	NUM
ejpam-6613	84	22	-	-	PUNCT
ejpam-6613	84	23	tensor	tensor	NOUN
ejpam-6613	84	24	product	product	NOUN
ejpam-6613	84	25	of	of	ADP
ejpam-6613	84	26	elements	element	NOUN
ejpam-6613	84	27	in	in	ADP
ejpam-6613	84	28	vector	vector	NOUN
ejpam-6613	84	29	spaces	space	NOUN
ejpam-6613	84	30	equipped	equip	VERB
ejpam-6613	84	31	with	with	ADP
ejpam-6613	84	32	a	a	DET
ejpam-6613	84	33	generalized	generalized	ADJ
ejpam-6613	84	34	2	2	NUM
ejpam-6613	84	35	-	-	PUNCT
ejpam-6613	84	36	inner	inner	ADJ
ejpam-6613	84	37	product	product	NOUN
ejpam-6613	84	38	.	.	PUNCT
ejpam-6613	85	1	definition	definition	NOUN
ejpam-6613	85	2	5	5	NUM
ejpam-6613	85	3	.	.	PUNCT
ejpam-6613	86	1	let	let	VERB
ejpam-6613	86	2	(	(	PUNCT
ejpam-6613	86	3	x1	x1	ADJ
ejpam-6613	86	4	,	,	PUNCT
ejpam-6613	86	5	⟨	⟨	NOUN
ejpam-6613	86	6	·	·	SYM
ejpam-6613	86	7	,	,	PUNCT
ejpam-6613	86	8	·	·	PUNCT
ejpam-6613	86	9	|·⟩1	|·⟩1	X
ejpam-6613	86	10	)	)	PUNCT
ejpam-6613	86	11	and	and	CCONJ
ejpam-6613	86	12	(	(	PUNCT
ejpam-6613	86	13	x2	x2	INTJ
ejpam-6613	86	14	,	,	PUNCT
ejpam-6613	86	15	⟨	⟨	NOUN
ejpam-6613	86	16	·	·	SYM
ejpam-6613	86	17	,	,	PUNCT
ejpam-6613	86	18	·	·	PUNCT
ejpam-6613	86	19	|·⟩2	|·⟩2	X
ejpam-6613	86	20	)	)	PUNCT
ejpam-6613	86	21	be	be	AUX
ejpam-6613	86	22	spaces	space	NOUN
ejpam-6613	86	23	with	with	ADP
ejpam-6613	86	24	a	a	DET
ejpam-6613	86	25	generalized	generalized	ADJ
ejpam-6613	86	26	2	2	NUM
ejpam-6613	86	27	-	-	PUNCT
ejpam-6613	86	28	inner	inner	ADJ
ejpam-6613	86	29	product	product	NOUN
ejpam-6613	86	30	.	.	PUNCT
ejpam-6613	87	1	given	give	VERB
ejpam-6613	87	2	x1	x1	PROPN
ejpam-6613	87	3	∈	∈	PROPN
ejpam-6613	87	4	x1	x1	PROPN
ejpam-6613	88	1	and	and	CCONJ
ejpam-6613	88	2	x2	x2	PROPN
ejpam-6613	88	3	∈	∈	PROPN
ejpam-6613	88	4	x2	x2	PROPN
ejpam-6613	88	5	,	,	PUNCT
ejpam-6613	88	6	the	the	DET
ejpam-6613	88	7	2	2	NUM
ejpam-6613	88	8	-	-	PUNCT
ejpam-6613	88	9	tensor	tensor	NOUN
ejpam-6613	88	10	product	product	NOUN
ejpam-6613	88	11	of	of	ADP
ejpam-6613	88	12	x1	x1	PROPN
ejpam-6613	88	13	and	and	CCONJ
ejpam-6613	88	14	x2	x2	PROPN
ejpam-6613	88	15	,	,	PUNCT
ejpam-6613	88	16	denoted	denote	VERB
ejpam-6613	88	17	by	by	ADP
ejpam-6613	88	18	x1	x1	PROPN
ejpam-6613	88	19	2	2	NUM
ejpam-6613	88	20	⊙	⊙	X
ejpam-6613	88	21	x2	x2	PROPN
ejpam-6613	88	22	,	,	PUNCT
ejpam-6613	88	23	is	be	AUX
ejpam-6613	88	24	the	the	DET
ejpam-6613	88	25	function	function	NOUN
ejpam-6613	88	26	x1	x1	NOUN
ejpam-6613	88	27	2	2	NUM
ejpam-6613	88	28	⊙	⊙	NOUN
ejpam-6613	88	29	x2	x2	INTJ
ejpam-6613	88	30	:	:	PUNCT
ejpam-6613	88	31	(	(	PUNCT
ejpam-6613	88	32	x1	x1	PROPN
ejpam-6613	88	33	×x2)×	×x2)×	NOUN
ejpam-6613	88	34	(	(	PUNCT
ejpam-6613	88	35	x1	x1	ADJ
ejpam-6613	88	36	×x2	×x2	NOUN
ejpam-6613	88	37	)	)	PUNCT
ejpam-6613	88	38	−→	−→	NOUN
ejpam-6613	88	39	c	c	NOUN
ejpam-6613	88	40	defined	define	VERB
ejpam-6613	88	41	by	by	ADP
ejpam-6613	88	42	(	(	PUNCT
ejpam-6613	88	43	x1	x1	PROPN
ejpam-6613	88	44	2	2	NUM
ejpam-6613	88	45	⊙	⊙	X
ejpam-6613	88	46	x2	x2	PROPN
ejpam-6613	88	47	)	)	PUNCT
ejpam-6613	88	48	(	(	PUNCT
ejpam-6613	88	49	(	(	PUNCT
ejpam-6613	88	50	t1	t1	NOUN
ejpam-6613	88	51	,	,	PUNCT
ejpam-6613	88	52	t2	t2	NOUN
ejpam-6613	88	53	)	)	PUNCT
ejpam-6613	88	54	,	,	PUNCT
ejpam-6613	88	55	(	(	PUNCT
ejpam-6613	88	56	r1	r1	NOUN
ejpam-6613	88	57	,	,	PUNCT
ejpam-6613	88	58	r2	r2	PROPN
ejpam-6613	88	59	)	)	PUNCT
ejpam-6613	88	60	)	)	PUNCT
ejpam-6613	89	1	=	=	PUNCT
ejpam-6613	89	2	⟨x1	⟨x1	NOUN
ejpam-6613	89	3	,	,	PUNCT
ejpam-6613	89	4	t1	t1	NOUN
ejpam-6613	89	5	|	|	ADV
ejpam-6613	89	6	r1⟩1	r1⟩1	PROPN
ejpam-6613	89	7	⟨x2	⟨x2	PROPN
ejpam-6613	89	8	,	,	PUNCT
ejpam-6613	89	9	t2	t2	NOUN
ejpam-6613	89	10	|	|	NOUN
ejpam-6613	89	11	r2⟩2	r2⟩2	PROPN
ejpam-6613	89	12	.	.	PUNCT
ejpam-6613	90	1	proposition	proposition	NOUN
ejpam-6613	90	2	5	5	NUM
ejpam-6613	90	3	.	.	PUNCT
ejpam-6613	91	1	let	let	VERB
ejpam-6613	91	2	(	(	PUNCT
ejpam-6613	91	3	x1	x1	ADJ
ejpam-6613	91	4	,	,	PUNCT
ejpam-6613	91	5	⟨	⟨	NOUN
ejpam-6613	91	6	·	·	SYM
ejpam-6613	91	7	,	,	PUNCT
ejpam-6613	91	8	·	·	PUNCT
ejpam-6613	91	9	|·⟩1	|·⟩1	X
ejpam-6613	91	10	)	)	PUNCT
ejpam-6613	91	11	and	and	CCONJ
ejpam-6613	91	12	(	(	PUNCT
ejpam-6613	91	13	x2	x2	INTJ
ejpam-6613	91	14	,	,	PUNCT
ejpam-6613	91	15	⟨	⟨	NOUN
ejpam-6613	91	16	·	·	SYM
ejpam-6613	91	17	,	,	PUNCT
ejpam-6613	91	18	·	·	PUNCT
ejpam-6613	91	19	|·⟩2	|·⟩2	X
ejpam-6613	91	20	)	)	PUNCT
ejpam-6613	91	21	be	be	AUX
ejpam-6613	91	22	spaces	space	NOUN
ejpam-6613	91	23	with	with	ADP
ejpam-6613	91	24	a	a	DET
ejpam-6613	91	25	generalized	generalized	ADJ
ejpam-6613	91	26	2	2	NUM
ejpam-6613	91	27	-	-	PUNCT
ejpam-6613	91	28	inner	inner	ADJ
ejpam-6613	91	29	product	product	NOUN
ejpam-6613	91	30	.	.	PUNCT
ejpam-6613	92	1	for	for	ADP
ejpam-6613	92	2	all	all	DET
ejpam-6613	92	3	x	x	NOUN
ejpam-6613	92	4	,	,	PUNCT
ejpam-6613	92	5	x1	x1	PROPN
ejpam-6613	92	6	,	,	PUNCT
ejpam-6613	92	7	y1	y1	PROPN
ejpam-6613	92	8	∈	∈	PROPN
ejpam-6613	92	9	x1	x1	PROPN
ejpam-6613	92	10	,	,	PUNCT
ejpam-6613	92	11	x2	x2	PROPN
ejpam-6613	92	12	,	,	PUNCT
ejpam-6613	92	13	y2	y2	PROPN
ejpam-6613	92	14	,	,	PUNCT
ejpam-6613	92	15	y	y	PROPN
ejpam-6613	92	16	∈	∈	PROPN
ejpam-6613	92	17	x2	x2	PROPN
ejpam-6613	92	18	,	,	PUNCT
ejpam-6613	92	19	and	and	CCONJ
ejpam-6613	92	20	all	all	DET
ejpam-6613	92	21	α	α	NOUN
ejpam-6613	92	22	,	,	PUNCT
ejpam-6613	92	23	β	β	X
ejpam-6613	92	24	∈	∈	PROPN
ejpam-6613	92	25	c	c	X
ejpam-6613	92	26	,	,	PUNCT
ejpam-6613	92	27	the	the	DET
ejpam-6613	92	28	following	follow	VERB
ejpam-6613	92	29	properties	property	NOUN
ejpam-6613	92	30	hold	hold	VERB
ejpam-6613	92	31	:	:	PUNCT
ejpam-6613	92	32	m.	m.	PROPN
ejpam-6613	92	33	luis	luis	PROPN
ejpam-6613	92	34	,	,	PUNCT
ejpam-6613	92	35	f.	f.	PROPN
ejpam-6613	92	36	osmin	osmin	PROPN
ejpam-6613	92	37	,	,	PUNCT
ejpam-6613	92	38	s.	s.	PROPN
ejpam-6613	92	39	arley	arley	PROPN
ejpam-6613	92	40	/	/	SYM
ejpam-6613	92	41	eur	eur	PROPN
ejpam-6613	92	42	.	.	PUNCT
ejpam-6613	93	1	j.	j.	PROPN
ejpam-6613	93	2	pure	pure	PROPN
ejpam-6613	93	3	appl	appl	PROPN
ejpam-6613	93	4	.	.	PROPN
ejpam-6613	93	5	math	math	PROPN
ejpam-6613	93	6	,	,	PUNCT
ejpam-6613	93	7	18	18	NUM
ejpam-6613	93	8	(	(	PUNCT
ejpam-6613	93	9	4	4	NUM
ejpam-6613	93	10	)	)	PUNCT
ejpam-6613	93	11	(	(	PUNCT
ejpam-6613	93	12	2025	2025	NUM
ejpam-6613	93	13	)	)	PUNCT
ejpam-6613	93	14	,	,	PUNCT
ejpam-6613	93	15	6613	6613	NUM
ejpam-6613	93	16	6	6	NUM
ejpam-6613	93	17	of	of	ADP
ejpam-6613	93	18	17	17	NUM
ejpam-6613	93	19	i	i	NOUN
ejpam-6613	93	20	)	)	PUNCT
ejpam-6613	93	21	x	x	SYM
ejpam-6613	93	22	2	2	NUM
ejpam-6613	93	23	⊙	⊙	NOUN
ejpam-6613	93	24	02	02	NUM
ejpam-6613	93	25	=	=	SYM
ejpam-6613	93	26	0	0	NUM
ejpam-6613	93	27	ii	ii	NOUN
ejpam-6613	93	28	)	)	PUNCT
ejpam-6613	93	29	01	01	NUM
ejpam-6613	93	30	2	2	NUM
ejpam-6613	93	31	⊙	⊙	NOUN
ejpam-6613	93	32	y	y	PROPN
ejpam-6613	93	33	=	=	SYM
ejpam-6613	93	34	0	0	NUM
ejpam-6613	93	35	iii	iii	NOUN
ejpam-6613	93	36	)	)	PUNCT
ejpam-6613	93	37	(	(	PUNCT
ejpam-6613	93	38	αx1	αx1	NOUN
ejpam-6613	93	39	)	)	PUNCT
ejpam-6613	93	40	2	2	NUM
ejpam-6613	93	41	⊙	⊙	X
ejpam-6613	93	42	x2	x2	PROPN
ejpam-6613	94	1	=	=	PUNCT
ejpam-6613	94	2	α(x1	α(x1	PROPN
ejpam-6613	94	3	2	2	NUM
ejpam-6613	94	4	⊙	⊙	NOUN
ejpam-6613	94	5	x2	x2	PROPN
ejpam-6613	94	6	)	)	PUNCT
ejpam-6613	95	1	=	=	SYM
ejpam-6613	95	2	x1	x1	PROPN
ejpam-6613	95	3	2	2	NUM
ejpam-6613	95	4	⊙	⊙	X
ejpam-6613	95	5	(	(	PUNCT
ejpam-6613	95	6	αx2	αx2	NOUN
ejpam-6613	95	7	)	)	PUNCT
ejpam-6613	95	8	iv	iv	NUM
ejpam-6613	95	9	)	)	PUNCT
ejpam-6613	95	10	αβ(x1	αβ(x1	NOUN
ejpam-6613	95	11	2	2	NUM
ejpam-6613	95	12	⊙	⊙	X
ejpam-6613	95	13	x2	x2	PROPN
ejpam-6613	95	14	)	)	PUNCT
ejpam-6613	95	15	=	=	SYM
ejpam-6613	95	16	(	(	PUNCT
ejpam-6613	95	17	αx1	αx1	NOUN
ejpam-6613	95	18	2	2	NUM
ejpam-6613	95	19	⊙	⊙	NOUN
ejpam-6613	95	20	βx2	βx2	PROPN
ejpam-6613	95	21	)	)	PUNCT
ejpam-6613	95	22	v	v	NOUN
ejpam-6613	95	23	)	)	PUNCT
ejpam-6613	95	24	(	(	PUNCT
ejpam-6613	95	25	x1	x1	PROPN
ejpam-6613	95	26	+	+	NUM
ejpam-6613	95	27	y1	y1	NOUN
ejpam-6613	95	28	)	)	PUNCT
ejpam-6613	95	29	2	2	NUM
ejpam-6613	95	30	⊙	⊙	NOUN
ejpam-6613	95	31	x2	x2	PROPN
ejpam-6613	96	1	=	=	PRON
ejpam-6613	97	1	(	(	PUNCT
ejpam-6613	97	2	x1	x1	PROPN
ejpam-6613	97	3	2	2	NUM
ejpam-6613	97	4	⊙	⊙	X
ejpam-6613	97	5	x2	x2	PROPN
ejpam-6613	97	6	)	)	PUNCT
ejpam-6613	97	7	+	+	CCONJ
ejpam-6613	97	8	(	(	PUNCT
ejpam-6613	97	9	y1	y1	INTJ
ejpam-6613	97	10	2	2	NUM
ejpam-6613	97	11	⊙	⊙	NOUN
ejpam-6613	97	12	x2	x2	PROPN
ejpam-6613	97	13	)	)	PUNCT
ejpam-6613	97	14	vi	vi	PROPN
ejpam-6613	97	15	)	)	PUNCT
ejpam-6613	97	16	x1	x1	PROPN
ejpam-6613	97	17	2	2	NUM
ejpam-6613	97	18	⊙	⊙	NOUN
ejpam-6613	97	19	(	(	PUNCT
ejpam-6613	97	20	x2	x2	PROPN
ejpam-6613	97	21	+	+	CCONJ
ejpam-6613	97	22	y2	y2	NOUN
ejpam-6613	97	23	)	)	PUNCT
ejpam-6613	97	24	=	=	SYM
ejpam-6613	97	25	(	(	PUNCT
ejpam-6613	97	26	x1	x1	PROPN
ejpam-6613	97	27	2	2	NUM
ejpam-6613	97	28	⊙	⊙	X
ejpam-6613	97	29	x2	x2	PROPN
ejpam-6613	97	30	)	)	PUNCT
ejpam-6613	97	31	+	+	CCONJ
ejpam-6613	97	32	(	(	PUNCT
ejpam-6613	97	33	x1	x1	PROPN
ejpam-6613	97	34	2	2	NUM
ejpam-6613	97	35	⊙	⊙	NOUN
ejpam-6613	97	36	y2	y2	NOUN
ejpam-6613	97	37	)	)	PUNCT
ejpam-6613	97	38	proof	proof	NOUN
ejpam-6613	97	39	.	.	PUNCT
ejpam-6613	98	1	let	let	AUX
ejpam-6613	98	2	t1	t1	NOUN
ejpam-6613	98	3	,	,	PUNCT
ejpam-6613	98	4	r1	r1	PROPN
ejpam-6613	98	5	∈	∈	PROPN
ejpam-6613	98	6	x1	x1	PROPN
ejpam-6613	98	7	be	be	VERB
ejpam-6613	98	8	and	and	CCONJ
ejpam-6613	98	9	t2	t2	NOUN
ejpam-6613	98	10	,	,	PUNCT
ejpam-6613	98	11	r2	r2	PROPN
ejpam-6613	98	12	∈	∈	PROPN
ejpam-6613	99	1	x2	x2	PRON
ejpam-6613	99	2	be	be	VERB
ejpam-6613	99	3	.	.	PUNCT
ejpam-6613	100	1	i	i	PRON
ejpam-6613	100	2	)	)	PUNCT
ejpam-6613	101	1	(	(	PUNCT
ejpam-6613	101	2	x1	x1	PROPN
ejpam-6613	101	3	2	2	NUM
ejpam-6613	101	4	⊙	⊙	PROPN
ejpam-6613	101	5	02)((t1	02)((t1	NOUN
ejpam-6613	101	6	,	,	PUNCT
ejpam-6613	101	7	t2	t2	NOUN
ejpam-6613	101	8	)	)	PUNCT
ejpam-6613	101	9	,	,	PUNCT
ejpam-6613	101	10	(	(	PUNCT
ejpam-6613	101	11	r1	r1	NOUN
ejpam-6613	101	12	,	,	PUNCT
ejpam-6613	101	13	r2	r2	PROPN
ejpam-6613	101	14	)	)	PUNCT
ejpam-6613	101	15	)	)	PUNCT
ejpam-6613	102	1	=	=	PUNCT
ejpam-6613	102	2	⟨x1	⟨x1	NOUN
ejpam-6613	102	3	,	,	PUNCT
ejpam-6613	102	4	t1|r1⟩1⟨02	t1|r1⟩1⟨02	NUM
ejpam-6613	102	5	,	,	PUNCT
ejpam-6613	102	6	t2|r2⟩2	t2|r2⟩2	X
ejpam-6613	102	7	=	=	SYM
ejpam-6613	102	8	⟨x1	⟨x1	NOUN
ejpam-6613	102	9	,	,	PUNCT
ejpam-6613	102	10	t1|r1⟩1	t1|r1⟩1	X
ejpam-6613	102	11	·	·	PUNCT
ejpam-6613	102	12	0	0	PUNCT
ejpam-6613	102	13	=	=	SYM
ejpam-6613	102	14	0	0	NUM
ejpam-6613	102	15	ii	ii	NOUN
ejpam-6613	102	16	)	)	PUNCT
ejpam-6613	102	17	(	(	PUNCT
ejpam-6613	102	18	01	01	NUM
ejpam-6613	102	19	2	2	NUM
ejpam-6613	102	20	⊙	⊙	NOUN
ejpam-6613	102	21	x2)((t1	x2)((t1	PROPN
ejpam-6613	102	22	,	,	PUNCT
ejpam-6613	102	23	t2	t2	NOUN
ejpam-6613	102	24	)	)	PUNCT
ejpam-6613	102	25	,	,	PUNCT
ejpam-6613	102	26	(	(	PUNCT
ejpam-6613	102	27	r1	r1	NOUN
ejpam-6613	102	28	,	,	PUNCT
ejpam-6613	102	29	r2	r2	PROPN
ejpam-6613	102	30	)	)	PUNCT
ejpam-6613	102	31	)	)	PUNCT
ejpam-6613	102	32	=	=	SYM
ejpam-6613	103	1	⟨01	⟨01	PROPN
ejpam-6613	103	2	,	,	PUNCT
ejpam-6613	103	3	t1|r1⟩1⟨x2	t1|r1⟩1⟨x2	PROPN
ejpam-6613	103	4	,	,	PUNCT
ejpam-6613	103	5	t2|r2⟩2	t2|r2⟩2	PUNCT
ejpam-6613	103	6	=	=	SYM
ejpam-6613	103	7	0	0	X
ejpam-6613	103	8	·	·	PUNCT
ejpam-6613	103	9	⟨x2	⟨x2	PROPN
ejpam-6613	103	10	,	,	PUNCT
ejpam-6613	103	11	t2|r2⟩2	t2|r2⟩2	NUM
ejpam-6613	103	12	=	=	SYM
ejpam-6613	103	13	0	0	NUM
ejpam-6613	103	14	iii	iii	NOUN
ejpam-6613	103	15	)	)	PUNCT
ejpam-6613	103	16	(	(	PUNCT
ejpam-6613	103	17	(	(	PUNCT
ejpam-6613	103	18	αx1	αx1	NOUN
ejpam-6613	103	19	)	)	PUNCT
ejpam-6613	103	20	2	2	NUM
ejpam-6613	103	21	⊙	⊙	NOUN
ejpam-6613	103	22	x2)((t1	x2)((t1	PROPN
ejpam-6613	103	23	,	,	PUNCT
ejpam-6613	103	24	t2	t2	NOUN
ejpam-6613	103	25	)	)	PUNCT
ejpam-6613	103	26	,	,	PUNCT
ejpam-6613	103	27	(	(	PUNCT
ejpam-6613	103	28	r1	r1	NOUN
ejpam-6613	103	29	,	,	PUNCT
ejpam-6613	103	30	r2	r2	PROPN
ejpam-6613	103	31	)	)	PUNCT
ejpam-6613	103	32	)	)	PUNCT
ejpam-6613	104	1	=	=	SYM
ejpam-6613	104	2	⟨αx1	⟨αx1	PROPN
ejpam-6613	104	3	,	,	PUNCT
ejpam-6613	104	4	t1|r1⟩1⟨x2	t1|r1⟩1⟨x2	NOUN
ejpam-6613	104	5	,	,	PUNCT
ejpam-6613	104	6	t2|r2⟩2	t2|r2⟩2	PUNCT
ejpam-6613	104	7	=	=	SYM
ejpam-6613	104	8	α⟨x1	α⟨x1	NOUN
ejpam-6613	104	9	,	,	PUNCT
ejpam-6613	104	10	t1|r1⟩1⟨x2	t1|r1⟩1⟨x2	NOUN
ejpam-6613	104	11	,	,	PUNCT
ejpam-6613	104	12	t2|r2⟩2	t2|r2⟩2	X
ejpam-6613	104	13	=	=	SYM
ejpam-6613	104	14	⟨x1	⟨x1	NOUN
ejpam-6613	104	15	,	,	PUNCT
ejpam-6613	104	16	t1|r1⟩1⟨(αx2	t1|r1⟩1⟨(αx2	NOUN
ejpam-6613	104	17	)	)	PUNCT
ejpam-6613	104	18	,	,	PUNCT
ejpam-6613	104	19	t2|r2⟩2	t2|r2⟩2	NUM
ejpam-6613	104	20	=	=	SYM
ejpam-6613	104	21	(	(	PUNCT
ejpam-6613	104	22	x1	x1	PROPN
ejpam-6613	104	23	2	2	NUM
ejpam-6613	104	24	⊙	⊙	X
ejpam-6613	104	25	(	(	PUNCT
ejpam-6613	104	26	αx2))((t1	αx2))((t1	ADV
ejpam-6613	104	27	,	,	PUNCT
ejpam-6613	104	28	t2	t2	NOUN
ejpam-6613	104	29	)	)	PUNCT
ejpam-6613	104	30	,	,	PUNCT
ejpam-6613	104	31	(	(	PUNCT
ejpam-6613	104	32	r1	r1	NOUN
ejpam-6613	104	33	,	,	PUNCT
ejpam-6613	104	34	r2	r2	PROPN
ejpam-6613	104	35	)	)	PUNCT
ejpam-6613	104	36	)	)	PUNCT
ejpam-6613	105	1	=	=	SYM
ejpam-6613	105	2	α(x1	α(x1	ADJ
ejpam-6613	105	3	2	2	NUM
ejpam-6613	105	4	⊙	⊙	PROPN
ejpam-6613	105	5	x2)((t1	x2)((t1	PROPN
ejpam-6613	105	6	,	,	PUNCT
ejpam-6613	105	7	t2	t2	NOUN
ejpam-6613	105	8	)	)	PUNCT
ejpam-6613	105	9	,	,	PUNCT
ejpam-6613	105	10	(	(	PUNCT
ejpam-6613	105	11	r1	r1	NOUN
ejpam-6613	105	12	,	,	PUNCT
ejpam-6613	105	13	r2	r2	PROPN
ejpam-6613	105	14	)	)	PUNCT
ejpam-6613	105	15	)	)	PUNCT
ejpam-6613	105	16	iv	iv	X
ejpam-6613	105	17	)	)	PUNCT
ejpam-6613	105	18	(	(	PUNCT
ejpam-6613	105	19	αx1	αx1	NOUN
ejpam-6613	105	20	2	2	NUM
ejpam-6613	105	21	⊙	⊙	NOUN
ejpam-6613	105	22	βx2)((t1	βx2)((t1	PROPN
ejpam-6613	105	23	,	,	PUNCT
ejpam-6613	105	24	t2	t2	NOUN
ejpam-6613	105	25	)	)	PUNCT
ejpam-6613	105	26	,	,	PUNCT
ejpam-6613	105	27	(	(	PUNCT
ejpam-6613	105	28	r1	r1	NOUN
ejpam-6613	105	29	,	,	PUNCT
ejpam-6613	105	30	r2	r2	PROPN
ejpam-6613	105	31	)	)	PUNCT
ejpam-6613	105	32	)	)	PUNCT
ejpam-6613	106	1	=	=	SYM
ejpam-6613	107	1	⟨(αx1	⟨(αx1	PROPN
ejpam-6613	107	2	)	)	PUNCT
ejpam-6613	107	3	,	,	PUNCT
ejpam-6613	107	4	t1|r1⟩1⟨(βx2	t1|r1⟩1⟨(βx2	PROPN
ejpam-6613	107	5	)	)	PUNCT
ejpam-6613	107	6	,	,	PUNCT
ejpam-6613	107	7	t2|r2⟩2	t2|r2⟩2	NUM
ejpam-6613	107	8	=	=	SYM
ejpam-6613	107	9	α⟨x1	α⟨x1	NOUN
ejpam-6613	107	10	,	,	PUNCT
ejpam-6613	107	11	t1|r1⟩1β⟨x2	t1|r1⟩1β⟨x2	PRON
ejpam-6613	107	12	,	,	PUNCT
ejpam-6613	107	13	t2|r2⟩2	t2|r2⟩2	PUNCT
ejpam-6613	107	14	=	=	SYM
ejpam-6613	107	15	αβ⟨x1	αβ⟨x1	NOUN
ejpam-6613	107	16	,	,	PUNCT
ejpam-6613	107	17	t1|r1⟩1⟨x2	t1|r1⟩1⟨x2	NOUN
ejpam-6613	107	18	,	,	PUNCT
ejpam-6613	107	19	t2|r2⟩2	t2|r2⟩2	PUNCT
ejpam-6613	107	20	=	=	SYM
ejpam-6613	107	21	αβ(x1	αβ(x1	NOUN
ejpam-6613	107	22	2	2	NUM
ejpam-6613	107	23	⊙	⊙	NOUN
ejpam-6613	107	24	x2)((t1	x2)((t1	PROPN
ejpam-6613	107	25	,	,	PUNCT
ejpam-6613	107	26	t2	t2	NOUN
ejpam-6613	107	27	)	)	PUNCT
ejpam-6613	107	28	,	,	PUNCT
ejpam-6613	107	29	(	(	PUNCT
ejpam-6613	107	30	r1	r1	NOUN
ejpam-6613	107	31	,	,	PUNCT
ejpam-6613	107	32	r2	r2	PROPN
ejpam-6613	107	33	)	)	PUNCT
ejpam-6613	107	34	)	)	PUNCT
ejpam-6613	107	35	v	v	NOUN
ejpam-6613	107	36	)	)	PUNCT
ejpam-6613	107	37	(	(	PUNCT
ejpam-6613	107	38	(	(	PUNCT
ejpam-6613	107	39	x1	x1	PROPN
ejpam-6613	107	40	+	+	NUM
ejpam-6613	107	41	y1	y1	NOUN
ejpam-6613	107	42	)	)	PUNCT
ejpam-6613	107	43	2	2	NUM
ejpam-6613	107	44	⊙	⊙	NOUN
ejpam-6613	107	45	x2)((t1	x2)((t1	PROPN
ejpam-6613	107	46	,	,	PUNCT
ejpam-6613	107	47	t2	t2	NOUN
ejpam-6613	107	48	)	)	PUNCT
ejpam-6613	107	49	,	,	PUNCT
ejpam-6613	107	50	(	(	PUNCT
ejpam-6613	107	51	r1	r1	NOUN
ejpam-6613	107	52	,	,	PUNCT
ejpam-6613	107	53	r2	r2	PROPN
ejpam-6613	107	54	)	)	PUNCT
ejpam-6613	107	55	)	)	PUNCT
ejpam-6613	108	1	=	=	PUNCT
ejpam-6613	109	1	⟨(x1	⟨(x1	NOUN
ejpam-6613	109	2	+	+	NUM
ejpam-6613	109	3	y1	y1	NOUN
ejpam-6613	109	4	)	)	PUNCT
ejpam-6613	109	5	,	,	PUNCT
ejpam-6613	109	6	t1|r1⟩1⟨x2	t1|r1⟩1⟨x2	NOUN
ejpam-6613	109	7	,	,	PUNCT
ejpam-6613	109	8	t2|r2⟩2	t2|r2⟩2	NUM
ejpam-6613	109	9	=	=	SYM
ejpam-6613	109	10	(	(	PUNCT
ejpam-6613	109	11	⟨x1	⟨x1	NOUN
ejpam-6613	109	12	,	,	PUNCT
ejpam-6613	109	13	t1|r1⟩1	t1|r1⟩1	X
ejpam-6613	109	14	+	+	CCONJ
ejpam-6613	109	15	⟨y1	⟨y1	ADJ
ejpam-6613	109	16	,	,	PUNCT
ejpam-6613	109	17	t1|r1⟩1)⟨x2	t1|r1⟩1)⟨x2	NOUN
ejpam-6613	109	18	,	,	PUNCT
ejpam-6613	109	19	t2|r2⟩2	t2|r2⟩2	X
ejpam-6613	109	20	=	=	SYM
ejpam-6613	109	21	⟨x1	⟨x1	NOUN
ejpam-6613	109	22	,	,	PUNCT
ejpam-6613	109	23	t1|r1⟩1⟨x2	t1|r1⟩1⟨x2	NOUN
ejpam-6613	109	24	,	,	PUNCT
ejpam-6613	109	25	t2|r2⟩2	t2|r2⟩2	X
ejpam-6613	109	26	+	+	PUNCT
ejpam-6613	109	27	⟨y1	⟨y1	ADJ
ejpam-6613	109	28	,	,	PUNCT
ejpam-6613	109	29	t1|r1⟩1⟨x2	t1|r1⟩1⟨x2	NOUN
ejpam-6613	109	30	,	,	PUNCT
ejpam-6613	109	31	t2|r2⟩2	t2|r2⟩2	NUM
ejpam-6613	109	32	=	=	SYM
ejpam-6613	109	33	(	(	PUNCT
ejpam-6613	109	34	x1	x1	PROPN
ejpam-6613	109	35	2	2	NUM
ejpam-6613	109	36	⊙	⊙	NOUN
ejpam-6613	109	37	x2)((t1	x2)((t1	PROPN
ejpam-6613	109	38	,	,	PUNCT
ejpam-6613	109	39	t2	t2	NOUN
ejpam-6613	109	40	)	)	PUNCT
ejpam-6613	109	41	,	,	PUNCT
ejpam-6613	109	42	(	(	PUNCT
ejpam-6613	109	43	r1	r1	NOUN
ejpam-6613	109	44	,	,	PUNCT
ejpam-6613	109	45	r2	r2	PROPN
ejpam-6613	109	46	)	)	PUNCT
ejpam-6613	109	47	)	)	PUNCT
ejpam-6613	110	1	+	+	CCONJ
ejpam-6613	110	2	(	(	PUNCT
ejpam-6613	110	3	y1	y1	INTJ
ejpam-6613	110	4	2	2	NUM
ejpam-6613	110	5	⊙	⊙	NOUN
ejpam-6613	110	6	x2)((t1	x2)((t1	PROPN
ejpam-6613	110	7	,	,	PUNCT
ejpam-6613	110	8	t2	t2	NOUN
ejpam-6613	110	9	)	)	PUNCT
ejpam-6613	110	10	,	,	PUNCT
ejpam-6613	110	11	(	(	PUNCT
ejpam-6613	110	12	r1	r1	NOUN
ejpam-6613	110	13	,	,	PUNCT
ejpam-6613	110	14	r2	r2	PROPN
ejpam-6613	110	15	)	)	PUNCT
ejpam-6613	110	16	)	)	PUNCT
ejpam-6613	110	17	vi	vi	X
ejpam-6613	110	18	)	)	PUNCT
ejpam-6613	110	19	(	(	PUNCT
ejpam-6613	110	20	x1	x1	PROPN
ejpam-6613	110	21	2	2	NUM
ejpam-6613	110	22	⊙	⊙	NOUN
ejpam-6613	110	23	(	(	PUNCT
ejpam-6613	110	24	x2	x2	PROPN
ejpam-6613	110	25	+	+	NUM
ejpam-6613	110	26	y2))((t1	y2))((t1	PROPN
ejpam-6613	110	27	,	,	PUNCT
ejpam-6613	110	28	t2	t2	NOUN
ejpam-6613	110	29	)	)	PUNCT
ejpam-6613	110	30	,	,	PUNCT
ejpam-6613	110	31	(	(	PUNCT
ejpam-6613	110	32	r1	r1	NOUN
ejpam-6613	110	33	,	,	PUNCT
ejpam-6613	110	34	r2	r2	PROPN
ejpam-6613	110	35	)	)	PUNCT
ejpam-6613	110	36	)	)	PUNCT
ejpam-6613	111	1	=	=	PUNCT
ejpam-6613	112	1	⟨x1	⟨x1	NOUN
ejpam-6613	112	2	,	,	PUNCT
ejpam-6613	112	3	t1|r1⟩1⟨(x2	t1|r1⟩1⟨(x2	PROPN
ejpam-6613	112	4	+	+	NUM
ejpam-6613	112	5	y2	y2	PROPN
ejpam-6613	112	6	)	)	PUNCT
ejpam-6613	112	7	,	,	PUNCT
ejpam-6613	112	8	t2|r2⟩2	t2|r2⟩2	X
ejpam-6613	112	9	=	=	SYM
ejpam-6613	112	10	⟨x1	⟨x1	NOUN
ejpam-6613	112	11	,	,	PUNCT
ejpam-6613	112	12	t1|r1⟩1(⟨x2	t1|r1⟩1(⟨x2	NOUN
ejpam-6613	112	13	,	,	PUNCT
ejpam-6613	112	14	t2|r2⟩2	t2|r2⟩2	NUM
ejpam-6613	112	15	+	+	PUNCT
ejpam-6613	112	16	⟨y2	⟨y2	PROPN
ejpam-6613	112	17	,	,	PUNCT
ejpam-6613	112	18	t2|r2⟩2	t2|r2⟩2	NUM
ejpam-6613	112	19	)	)	PUNCT
ejpam-6613	112	20	=	=	SYM
ejpam-6613	113	1	⟨x1	⟨x1	NOUN
ejpam-6613	113	2	,	,	PUNCT
ejpam-6613	113	3	t1|r1⟩1⟨x2	t1|r1⟩1⟨x2	NOUN
ejpam-6613	113	4	,	,	PUNCT
ejpam-6613	113	5	t2|r2⟩2	t2|r2⟩2	X
ejpam-6613	113	6	+	+	CCONJ
ejpam-6613	113	7	⟨x1	⟨x1	NOUN
ejpam-6613	113	8	,	,	PUNCT
ejpam-6613	113	9	t1|r1⟩1⟨y2	t1|r1⟩1⟨y2	NOUN
ejpam-6613	113	10	,	,	PUNCT
ejpam-6613	113	11	t2|r2⟩2	t2|r2⟩2	NUM
ejpam-6613	113	12	=	=	SYM
ejpam-6613	113	13	(	(	PUNCT
ejpam-6613	113	14	x1	x1	PROPN
ejpam-6613	113	15	2	2	NUM
ejpam-6613	113	16	⊙	⊙	NOUN
ejpam-6613	113	17	x2)((t1	x2)((t1	PROPN
ejpam-6613	113	18	,	,	PUNCT
ejpam-6613	113	19	t2	t2	NOUN
ejpam-6613	113	20	)	)	PUNCT
ejpam-6613	113	21	,	,	PUNCT
ejpam-6613	113	22	(	(	PUNCT
ejpam-6613	113	23	r1	r1	NOUN
ejpam-6613	113	24	,	,	PUNCT
ejpam-6613	113	25	r2	r2	PROPN
ejpam-6613	113	26	)	)	PUNCT
ejpam-6613	113	27	)	)	PUNCT
ejpam-6613	114	1	+	+	CCONJ
ejpam-6613	115	1	(	(	PUNCT
ejpam-6613	115	2	x1	x1	PROPN
ejpam-6613	115	3	2	2	NUM
ejpam-6613	115	4	⊙	⊙	PROPN
ejpam-6613	115	5	y2)((t1	y2)((t1	PROPN
ejpam-6613	115	6	,	,	PUNCT
ejpam-6613	115	7	t2	t2	NOUN
ejpam-6613	115	8	)	)	PUNCT
ejpam-6613	115	9	,	,	PUNCT
ejpam-6613	115	10	(	(	PUNCT
ejpam-6613	115	11	r1	r1	NOUN
ejpam-6613	115	12	,	,	PUNCT
ejpam-6613	115	13	r2	r2	PROPN
ejpam-6613	115	14	)	)	PUNCT
ejpam-6613	115	15	)	)	PUNCT
ejpam-6613	115	16	m.	m.	PROPN
ejpam-6613	115	17	luis	luis	PROPN
ejpam-6613	115	18	,	,	PUNCT
ejpam-6613	115	19	f.	f.	PROPN
ejpam-6613	115	20	osmin	osmin	PROPN
ejpam-6613	115	21	,	,	PUNCT
ejpam-6613	115	22	s.	s.	PROPN
ejpam-6613	115	23	arley	arley	PROPN
ejpam-6613	115	24	/	/	SYM
ejpam-6613	115	25	eur	eur	PROPN
ejpam-6613	115	26	.	.	PUNCT
ejpam-6613	116	1	j.	j.	PROPN
ejpam-6613	116	2	pure	pure	PROPN
ejpam-6613	116	3	appl	appl	PROPN
ejpam-6613	116	4	.	.	PROPN
ejpam-6613	116	5	math	math	PROPN
ejpam-6613	116	6	,	,	PUNCT
ejpam-6613	116	7	18	18	NUM
ejpam-6613	116	8	(	(	PUNCT
ejpam-6613	116	9	4	4	NUM
ejpam-6613	116	10	)	)	PUNCT
ejpam-6613	116	11	(	(	PUNCT
ejpam-6613	116	12	2025	2025	NUM
ejpam-6613	116	13	)	)	PUNCT
ejpam-6613	116	14	,	,	PUNCT
ejpam-6613	116	15	6613	6613	NUM
ejpam-6613	116	16	7	7	NUM
ejpam-6613	116	17	of	of	ADP
ejpam-6613	116	18	17	17	NUM
ejpam-6613	116	19	3.2	3.2	NUM
ejpam-6613	116	20	.	.	PUNCT
ejpam-6613	117	1	algebraic	algebraic	ADJ
ejpam-6613	117	2	tensor	tensor	NOUN
ejpam-6613	117	3	product	product	NOUN
ejpam-6613	117	4	next	next	ADV
ejpam-6613	117	5	,	,	PUNCT
ejpam-6613	117	6	we	we	PRON
ejpam-6613	117	7	introduce	introduce	VERB
ejpam-6613	117	8	the	the	DET
ejpam-6613	117	9	notion	notion	NOUN
ejpam-6613	117	10	of	of	ADP
ejpam-6613	117	11	the	the	DET
ejpam-6613	117	12	algebraic	algebraic	ADJ
ejpam-6613	117	13	tensor	tensor	NOUN
ejpam-6613	117	14	product	product	NOUN
ejpam-6613	117	15	between	between	ADP
ejpam-6613	117	16	two	two	NUM
ejpam-6613	117	17	vector	vector	NOUN
ejpam-6613	117	18	spaces	space	NOUN
ejpam-6613	117	19	endowed	endow	VERB
ejpam-6613	117	20	with	with	ADP
ejpam-6613	117	21	a	a	DET
ejpam-6613	117	22	generalized	generalized	ADJ
ejpam-6613	117	23	2	2	NUM
ejpam-6613	117	24	-	-	PUNCT
ejpam-6613	117	25	inner	inner	ADJ
ejpam-6613	117	26	product	product	NOUN
ejpam-6613	117	27	.	.	PUNCT
ejpam-6613	118	1	definition	definition	NOUN
ejpam-6613	118	2	6	6	NUM
ejpam-6613	118	3	.	.	PUNCT
ejpam-6613	119	1	let	let	VERB
ejpam-6613	119	2	(	(	PUNCT
ejpam-6613	119	3	x1	x1	ADJ
ejpam-6613	119	4	,	,	PUNCT
ejpam-6613	119	5	⟨	⟨	NOUN
ejpam-6613	119	6	·	·	SYM
ejpam-6613	119	7	,	,	PUNCT
ejpam-6613	119	8	·	·	PUNCT
ejpam-6613	119	9	|·⟩1	|·⟩1	X
ejpam-6613	119	10	)	)	PUNCT
ejpam-6613	119	11	and	and	CCONJ
ejpam-6613	119	12	(	(	PUNCT
ejpam-6613	119	13	x2	x2	INTJ
ejpam-6613	119	14	,	,	PUNCT
ejpam-6613	119	15	⟨	⟨	NOUN
ejpam-6613	119	16	·	·	SYM
ejpam-6613	119	17	,	,	PUNCT
ejpam-6613	119	18	·	·	PUNCT
ejpam-6613	119	19	|·⟩2	|·⟩2	X
ejpam-6613	119	20	)	)	PUNCT
ejpam-6613	119	21	be	be	AUX
ejpam-6613	119	22	spaces	space	NOUN
ejpam-6613	119	23	with	with	ADP
ejpam-6613	119	24	a	a	DET
ejpam-6613	119	25	generalized	generalized	ADJ
ejpam-6613	119	26	2	2	NUM
ejpam-6613	119	27	-	-	PUNCT
ejpam-6613	119	28	inner	inner	ADJ
ejpam-6613	119	29	product	product	NOUN
ejpam-6613	119	30	.	.	PUNCT
ejpam-6613	120	1	we	we	PRON
ejpam-6613	120	2	define	define	VERB
ejpam-6613	120	3	their	their	PRON
ejpam-6613	120	4	algebraic	algebraic	ADJ
ejpam-6613	120	5	tensor	tensor	NOUN
ejpam-6613	120	6	product	product	NOUN
ejpam-6613	120	7	,	,	PUNCT
ejpam-6613	120	8	denoted	denote	VERB
ejpam-6613	120	9	by	by	ADP
ejpam-6613	120	10	x1	x1	PROPN
ejpam-6613	120	11	2	2	PROPN
ejpam-6613	120	12	⊙x2	⊙x2	PROPN
ejpam-6613	120	13	,	,	PUNCT
ejpam-6613	120	14	as	as	SCONJ
ejpam-6613	120	15	x1	x1	PROPN
ejpam-6613	120	16	2	2	NUM
ejpam-6613	120	17	⊙x2	⊙x2	PROPN
ejpam-6613	120	18	:	:	PUNCT
ejpam-6613	121	1	=	=	SYM
ejpam-6613	121	2	{	{	PUNCT
ejpam-6613	121	3	n∑	n∑	NOUN
ejpam-6613	121	4	i=1	i=1	X
ejpam-6613	121	5	βi	βi	PROPN
ejpam-6613	121	6	(	(	PUNCT
ejpam-6613	121	7	xi	xi	PROPN
ejpam-6613	121	8	2	2	NUM
ejpam-6613	121	9	⊙	⊙	PROPN
ejpam-6613	121	10	yi	yi	PROPN
ejpam-6613	121	11	)	)	PUNCT
ejpam-6613	121	12	∣∣	∣∣	PROPN
ejpam-6613	121	13	n	n	PROPN
ejpam-6613	121	14	∈	∈	PROPN
ejpam-6613	121	15	n∗	n∗	PROPN
ejpam-6613	121	16	,	,	PUNCT
ejpam-6613	121	17	βi	βi	PROPN
ejpam-6613	121	18	∈	∈	PROPN
ejpam-6613	121	19	c	c	NOUN
ejpam-6613	121	20	,	,	PUNCT
ejpam-6613	121	21	xi	xi	PROPN
ejpam-6613	121	22	∈	∈	PROPN
ejpam-6613	121	23	x1	x1	PROPN
ejpam-6613	121	24	,	,	PUNCT
ejpam-6613	121	25	yi	yi	PROPN
ejpam-6613	121	26	∈	∈	PROPN
ejpam-6613	121	27	x2	x2	PROPN
ejpam-6613	121	28	}	}	PUNCT
ejpam-6613	121	29	.	.	PUNCT
ejpam-6613	122	1	note	note	VERB
ejpam-6613	122	2	that	that	SCONJ
ejpam-6613	122	3	x1	x1	PROPN
ejpam-6613	122	4	2	2	NUM
ejpam-6613	122	5	⊙x2	⊙x2	PROPN
ejpam-6613	122	6	is	be	AUX
ejpam-6613	122	7	a	a	DET
ejpam-6613	122	8	complex	complex	ADJ
ejpam-6613	122	9	vector	vector	NOUN
ejpam-6613	122	10	space	space	NOUN
ejpam-6613	122	11	.	.	PUNCT
ejpam-6613	123	1	note	note	VERB
ejpam-6613	123	2	that	that	SCONJ
ejpam-6613	123	3	by	by	ADP
ejpam-6613	123	4	the	the	DET
ejpam-6613	123	5	properties	property	NOUN
ejpam-6613	123	6	of	of	ADP
ejpam-6613	123	7	the	the	DET
ejpam-6613	123	8	mapping	mapping	NOUN
ejpam-6613	123	9	2	2	NUM
ejpam-6613	123	10	⊙	⊙	NOUN
ejpam-6613	123	11	every	every	DET
ejpam-6613	123	12	element	element	NOUN
ejpam-6613	123	13	of	of	ADP
ejpam-6613	123	14	the	the	DET
ejpam-6613	123	15	vector	vector	NOUN
ejpam-6613	123	16	spacex1	spacex1	NOUN
ejpam-6613	123	17	2	2	NUM
ejpam-6613	123	18	⊙x2	⊙x2	PROPN
ejpam-6613	123	19	can	can	AUX
ejpam-6613	123	20	be	be	AUX
ejpam-6613	123	21	written	write	VERB
ejpam-6613	123	22	simply	simply	ADV
ejpam-6613	123	23	as	as	SCONJ
ejpam-6613	123	24	∑n	∑n	PROPN
ejpam-6613	123	25	i=1	i=1	PROPN
ejpam-6613	123	26	xi	xi	PART
ejpam-6613	123	27	2	2	NUM
ejpam-6613	123	28	⊙	⊙	PROPN
ejpam-6613	123	29	yi	yi	PROPN
ejpam-6613	123	30	with	with	ADP
ejpam-6613	123	31	xi	xi	PROPN
ejpam-6613	123	32	∈	∈	PROPN
ejpam-6613	123	33	x1	x1	PROPN
ejpam-6613	123	34	,	,	PUNCT
ejpam-6613	123	35	yi	yi	PROPN
ejpam-6613	123	36	∈	∈	PROPN
ejpam-6613	123	37	x2	x2	INTJ
ejpam-6613	123	38	and	and	CCONJ
ejpam-6613	123	39	n	n	PRON
ejpam-6613	123	40	∈	∈	PROPN
ejpam-6613	123	41	n.	n.	NOUN
ejpam-6613	123	42	in	in	ADP
ejpam-6613	123	43	the	the	DET
ejpam-6613	123	44	following	following	NOUN
ejpam-6613	123	45	theorem	theorem	NOUN
ejpam-6613	123	46	,	,	PUNCT
ejpam-6613	123	47	we	we	PRON
ejpam-6613	123	48	equip	equip	VERB
ejpam-6613	123	49	the	the	DET
ejpam-6613	123	50	vector	vector	NOUN
ejpam-6613	123	51	space	space	NOUN
ejpam-6613	123	52	from	from	ADP
ejpam-6613	123	53	definition	definition	NOUN
ejpam-6613	123	54	6	6	NUM
ejpam-6613	123	55	with	with	ADP
ejpam-6613	123	56	a	a	DET
ejpam-6613	123	57	generalized	generalized	ADJ
ejpam-6613	123	58	2	2	NUM
ejpam-6613	123	59	-	-	PUNCT
ejpam-6613	123	60	inner	inner	ADJ
ejpam-6613	123	61	product	product	NOUN
ejpam-6613	123	62	,	,	PUNCT
ejpam-6613	123	63	which	which	PRON
ejpam-6613	123	64	we	we	PRON
ejpam-6613	123	65	call	call	VERB
ejpam-6613	123	66	the	the	DET
ejpam-6613	123	67	generalized	generalized	ADJ
ejpam-6613	123	68	2	2	NUM
ejpam-6613	123	69	-	-	PUNCT
ejpam-6613	123	70	inner	inner	ADJ
ejpam-6613	123	71	tensor	tensor	NOUN
ejpam-6613	123	72	product	product	NOUN
ejpam-6613	123	73	.	.	PUNCT
ejpam-6613	124	1	theorem	theorem	NOUN
ejpam-6613	124	2	3	3	X
ejpam-6613	124	3	.	.	PUNCT
ejpam-6613	125	1	let	let	AUX
ejpam-6613	125	2	(	(	PUNCT
ejpam-6613	125	3	x1	x1	ADJ
ejpam-6613	125	4	,	,	PUNCT
ejpam-6613	125	5	⟨	⟨	NOUN
ejpam-6613	125	6	·	·	SYM
ejpam-6613	125	7	,	,	PUNCT
ejpam-6613	125	8	·	·	PUNCT
ejpam-6613	125	9	|·⟩1	|·⟩1	X
ejpam-6613	125	10	)	)	PUNCT
ejpam-6613	125	11	and	and	CCONJ
ejpam-6613	125	12	(	(	PUNCT
ejpam-6613	125	13	x2	x2	INTJ
ejpam-6613	125	14	,	,	PUNCT
ejpam-6613	125	15	⟨	⟨	NOUN
ejpam-6613	125	16	·	·	SYM
ejpam-6613	125	17	,	,	PUNCT
ejpam-6613	125	18	·	·	PUNCT
ejpam-6613	125	19	|·⟩2	|·⟩2	X
ejpam-6613	125	20	)	)	PUNCT
ejpam-6613	125	21	be	be	AUX
ejpam-6613	125	22	spaces	space	NOUN
ejpam-6613	125	23	with	with	ADP
ejpam-6613	125	24	a	a	DET
ejpam-6613	125	25	generalized	generalized	ADJ
ejpam-6613	125	26	2	2	NUM
ejpam-6613	125	27	-	-	PUNCT
ejpam-6613	125	28	inner	inner	ADJ
ejpam-6613	125	29	product	product	NOUN
ejpam-6613	125	30	.	.	PUNCT
ejpam-6613	126	1	then	then	ADV
ejpam-6613	126	2	the	the	DET
ejpam-6613	126	3	mapping	mapping	NOUN
ejpam-6613	126	4	⟨ξ	⟨ξ	NOUN
ejpam-6613	126	5	,	,	PUNCT
ejpam-6613	126	6	η	η	PROPN
ejpam-6613	126	7	|	|	NOUN
ejpam-6613	126	8	λ⟩	λ⟩	ADP
ejpam-6613	126	9	2	2	NUM
ejpam-6613	126	10	⊙	⊙	NOUN
ejpam-6613	126	11	:	:	PUNCT
ejpam-6613	126	12	(	(	PUNCT
ejpam-6613	126	13	x1	x1	NOUN
ejpam-6613	126	14	2	2	NUM
ejpam-6613	126	15	⊙x2)×	⊙x2)×	NUM
ejpam-6613	126	16	(	(	PUNCT
ejpam-6613	126	17	x1	x1	PROPN
ejpam-6613	126	18	2	2	NUM
ejpam-6613	126	19	⊙x2)×	⊙x2)×	NUM
ejpam-6613	126	20	(	(	PUNCT
ejpam-6613	126	21	x1	x1	PROPN
ejpam-6613	126	22	2	2	NUM
ejpam-6613	126	23	⊙x2	⊙x2	PROPN
ejpam-6613	126	24	)	)	PUNCT
ejpam-6613	126	25	−→	−→	NOUN
ejpam-6613	126	26	c	c	NOUN
ejpam-6613	126	27	defined	define	VERB
ejpam-6613	126	28	by	by	ADP
ejpam-6613	126	29	〈	〈	PROPN
ejpam-6613	126	30	ξ	ξ	PROPN
ejpam-6613	126	31	,	,	PUNCT
ejpam-6613	126	32	η	η	PROPN
ejpam-6613	126	33	|	|	NOUN
ejpam-6613	126	34	λ	λ	SYM
ejpam-6613	126	35	〉	〉	NOUN
ejpam-6613	126	36	2	2	NUM
ejpam-6613	126	37	⊙	⊙	NOUN
ejpam-6613	126	38	=	=	SYM
ejpam-6613	127	1	〈	〈	PROPN
ejpam-6613	127	2	n∑	n∑	NOUN
ejpam-6613	127	3	i=1	i=1	PROPN
ejpam-6613	127	4	xi	xi	PROPN
ejpam-6613	127	5	2	2	NUM
ejpam-6613	127	6	⊙	⊙	PROPN
ejpam-6613	127	7	yi	yi	PROPN
ejpam-6613	127	8	,	,	PUNCT
ejpam-6613	127	9	m∑	m∑	ADP
ejpam-6613	127	10	j=1	j=1	PROPN
ejpam-6613	127	11	zj	zj	PROPN
ejpam-6613	127	12	2	2	NUM
ejpam-6613	127	13	⊙wj	⊙wj	NOUN
ejpam-6613	127	14	∣∣∣	∣∣∣	NOUN
ejpam-6613	127	15	p∑	p∑	X
ejpam-6613	128	1	t=1	t=1	PROPN
ejpam-6613	128	2	rt	rt	PROPN
ejpam-6613	128	3	2	2	NUM
ejpam-6613	128	4	⊙	⊙	PROPN
ejpam-6613	128	5	st	st	PROPN
ejpam-6613	128	6	〉	〉	PROPN
ejpam-6613	128	7	:	:	PUNCT
ejpam-6613	129	1	=	=	SYM
ejpam-6613	129	2	n∑	n∑	NOUN
ejpam-6613	129	3	i=1	i=1	PROPN
ejpam-6613	130	1	m∑	m∑	ADV
ejpam-6613	131	1	j=1	j=1	PROPN
ejpam-6613	131	2	p∑	p∑	PROPN
ejpam-6613	132	1	t=1	t=1	PROPN
ejpam-6613	132	2	δi	δi	PROPN
ejpam-6613	132	3	,	,	PUNCT
ejpam-6613	132	4	j	j	PROPN
ejpam-6613	132	5	⟨xi	⟨xi	PROPN
ejpam-6613	132	6	,	,	PUNCT
ejpam-6613	132	7	zj	zj	PROPN
ejpam-6613	132	8	|	|	PROPN
ejpam-6613	132	9	rt⟩1	rt⟩1	PROPN
ejpam-6613	133	1	⟨yi	⟨yi	PRON
ejpam-6613	133	2	,	,	PUNCT
ejpam-6613	133	3	wj	wj	PROPN
ejpam-6613	133	4	|	|	ADV
ejpam-6613	133	5	st⟩2	st⟩2	PROPN
ejpam-6613	133	6	,	,	PUNCT
ejpam-6613	133	7	where	where	SCONJ
ejpam-6613	133	8	ξ	ξ	PROPN
ejpam-6613	133	9	=	=	SYM
ejpam-6613	133	10	n∑	n∑	NOUN
ejpam-6613	133	11	i=1	i=1	PROPN
ejpam-6613	133	12	xi	xi	PROPN
ejpam-6613	133	13	2	2	NUM
ejpam-6613	133	14	⊙	⊙	PROPN
ejpam-6613	133	15	yi	yi	PROPN
ejpam-6613	133	16	,	,	PUNCT
ejpam-6613	133	17	η	η	X
ejpam-6613	133	18	=	=	PROPN
ejpam-6613	133	19	m∑	m∑	PROPN
ejpam-6613	133	20	j=1	j=1	PROPN
ejpam-6613	133	21	zj	zj	PROPN
ejpam-6613	133	22	2	2	PROPN
ejpam-6613	133	23	⊙	⊙	X
ejpam-6613	133	24	wj	wj	PROPN
ejpam-6613	133	25	,	,	PUNCT
ejpam-6613	133	26	λ	λ	X
ejpam-6613	133	27	=	=	PUNCT
ejpam-6613	133	28	p∑	p∑	X
ejpam-6613	133	29	t=1	t=1	PROPN
ejpam-6613	133	30	rt	rt	PROPN
ejpam-6613	133	31	2	2	NUM
ejpam-6613	133	32	⊙	⊙	PROPN
ejpam-6613	133	33	st	st	PROPN
ejpam-6613	133	34	,	,	PUNCT
ejpam-6613	133	35	is	be	AUX
ejpam-6613	133	36	a	a	DET
ejpam-6613	133	37	generalized	generalized	ADJ
ejpam-6613	133	38	2	2	NUM
ejpam-6613	133	39	-	-	PUNCT
ejpam-6613	133	40	inner	inner	ADJ
ejpam-6613	133	41	product	product	NOUN
ejpam-6613	133	42	.	.	PUNCT
ejpam-6613	134	1	proof	proof	NOUN
ejpam-6613	134	2	.	.	PUNCT
ejpam-6613	135	1	let	let	VERB
ejpam-6613	135	2	ξ	ξ	X
ejpam-6613	135	3	=	=	PUNCT
ejpam-6613	135	4	n∑	n∑	NOUN
ejpam-6613	135	5	i=1	i=1	PROPN
ejpam-6613	135	6	xi	xi	PROPN
ejpam-6613	135	7	2	2	NUM
ejpam-6613	135	8	⊙	⊙	PROPN
ejpam-6613	135	9	yi	yi	PROPN
ejpam-6613	135	10	,	,	PUNCT
ejpam-6613	135	11	η	η	X
ejpam-6613	135	12	=	=	PROPN
ejpam-6613	135	13	m∑	m∑	PROPN
ejpam-6613	135	14	j=1	j=1	PROPN
ejpam-6613	135	15	zj	zj	PROPN
ejpam-6613	135	16	2	2	PROPN
ejpam-6613	135	17	⊙	⊙	X
ejpam-6613	135	18	wj	wj	PROPN
ejpam-6613	135	19	,	,	PUNCT
ejpam-6613	135	20	λ	λ	X
ejpam-6613	135	21	=	=	PUNCT
ejpam-6613	135	22	p∑	p∑	X
ejpam-6613	135	23	t=1	t=1	PROPN
ejpam-6613	135	24	rt	rt	PROPN
ejpam-6613	135	25	2	2	NUM
ejpam-6613	135	26	⊙	⊙	PROPN
ejpam-6613	135	27	st	st	PROPN
ejpam-6613	135	28	,	,	PUNCT
ejpam-6613	135	29	σ	σ	PROPN
ejpam-6613	135	30	=	=	SYM
ejpam-6613	135	31	g∑	g∑	PROPN
ejpam-6613	135	32	k=1	k=1	PUNCT
ejpam-6613	135	33	lk	lk	PROPN
ejpam-6613	135	34	2	2	NUM
ejpam-6613	135	35	⊙	⊙	X
ejpam-6613	135	36	vk	vk	AUX
ejpam-6613	135	37	be	be	AUX
ejpam-6613	135	38	elements	element	NOUN
ejpam-6613	135	39	of	of	ADP
ejpam-6613	135	40	x1	x1	PROPN
ejpam-6613	135	41	2	2	NUM
ejpam-6613	135	42	⊙	⊙	X
ejpam-6613	135	43	x2	x2	PROPN
ejpam-6613	135	44	and	and	CCONJ
ejpam-6613	135	45	let	let	VERB
ejpam-6613	135	46	α	α	PRON
ejpam-6613	135	47	∈	∈	PROPN
ejpam-6613	135	48	c.	c.	NOUN
ejpam-6613	135	49	to	to	PART
ejpam-6613	135	50	verify	verify	VERB
ejpam-6613	135	51	linearity	linearity	NOUN
ejpam-6613	135	52	in	in	ADP
ejpam-6613	135	53	the	the	DET
ejpam-6613	135	54	first	first	ADJ
ejpam-6613	135	55	slot	slot	NOUN
ejpam-6613	135	56	,	,	PUNCT
ejpam-6613	135	57	first	first	ADV
ejpam-6613	135	58	observe	observe	VERB
ejpam-6613	135	59	that	that	SCONJ
ejpam-6613	135	60	λ	λ	NOUN
ejpam-6613	135	61	=	=	PUNCT
ejpam-6613	135	62	p∑	p∑	X
ejpam-6613	135	63	t=1	t=1	PROPN
ejpam-6613	135	64	rt	rt	PROPN
ejpam-6613	135	65	2	2	NUM
ejpam-6613	135	66	⊙	⊙	PROPN
ejpam-6613	135	67	st	st	PROPN
ejpam-6613	136	1	=	=	PUNCT
ejpam-6613	136	2	n+p∑	n+p∑	PROPN
ejpam-6613	136	3	i	i	NOUN
ejpam-6613	136	4	=	=	NOUN
ejpam-6613	136	5	n+1	n+1	PROPN
ejpam-6613	136	6	xi	xi	ADP
ejpam-6613	136	7	2	2	NUM
ejpam-6613	136	8	⊙	⊙	PROPN
ejpam-6613	136	9	yi	yi	PROPN
ejpam-6613	136	10	,	,	PUNCT
ejpam-6613	136	11	m.	m.	PROPN
ejpam-6613	136	12	luis	luis	PROPN
ejpam-6613	136	13	,	,	PUNCT
ejpam-6613	136	14	f.	f.	PROPN
ejpam-6613	136	15	osmin	osmin	PROPN
ejpam-6613	136	16	,	,	PUNCT
ejpam-6613	136	17	s.	s.	PROPN
ejpam-6613	136	18	arley	arley	PROPN
ejpam-6613	136	19	/	/	SYM
ejpam-6613	136	20	eur	eur	PROPN
ejpam-6613	136	21	.	.	PUNCT
ejpam-6613	137	1	j.	j.	PROPN
ejpam-6613	137	2	pure	pure	PROPN
ejpam-6613	137	3	appl	appl	PROPN
ejpam-6613	137	4	.	.	PROPN
ejpam-6613	137	5	math	math	PROPN
ejpam-6613	137	6	,	,	PUNCT
ejpam-6613	137	7	18	18	NUM
ejpam-6613	137	8	(	(	PUNCT
ejpam-6613	137	9	4	4	NUM
ejpam-6613	137	10	)	)	PUNCT
ejpam-6613	137	11	(	(	PUNCT
ejpam-6613	137	12	2025	2025	NUM
ejpam-6613	137	13	)	)	PUNCT
ejpam-6613	137	14	,	,	PUNCT
ejpam-6613	137	15	6613	6613	NUM
ejpam-6613	137	16	8	8	NUM
ejpam-6613	137	17	of	of	ADP
ejpam-6613	137	18	17	17	NUM
ejpam-6613	137	19	where	where	SCONJ
ejpam-6613	137	20	we	we	PRON
ejpam-6613	137	21	set	set	VERB
ejpam-6613	137	22	xn+t	xn+t	PROPN
ejpam-6613	137	23	2	2	NUM
ejpam-6613	137	24	⊙	⊙	NOUN
ejpam-6613	137	25	yn+t	yn+t	PROPN
ejpam-6613	138	1	:	:	PUNCT
ejpam-6613	138	2	=	=	SYM
ejpam-6613	138	3	rt	rt	PROPN
ejpam-6613	138	4	2	2	NUM
ejpam-6613	138	5	⊙	⊙	PROPN
ejpam-6613	138	6	st	st	PROPN
ejpam-6613	138	7	for	for	ADP
ejpam-6613	138	8	each	each	DET
ejpam-6613	138	9	1	1	NUM
ejpam-6613	138	10	≤	≤	NUM
ejpam-6613	139	1	t	t	NOUN
ejpam-6613	139	2	≤	≤	NOUN
ejpam-6613	139	3	p.	p.	NOUN
ejpam-6613	139	4	hence	hence	NOUN
ejpam-6613	140	1	ξ	ξ	PROPN
ejpam-6613	141	1	+	+	PUNCT
ejpam-6613	141	2	λ	λ	X
ejpam-6613	141	3	=	=	VERB
ejpam-6613	141	4	n+p∑	n+p∑	NOUN
ejpam-6613	141	5	i=1	i=1	X
ejpam-6613	141	6	xi	xi	ADP
ejpam-6613	141	7	2	2	NUM
ejpam-6613	141	8	⊙	⊙	PROPN
ejpam-6613	141	9	yi	yi	PROPN
ejpam-6613	141	10	.	.	PUNCT
ejpam-6613	142	1	therefore	therefore	ADV
ejpam-6613	142	2	:	:	PUNCT
ejpam-6613	142	3	⟨ξ	⟨ξ	NOUN
ejpam-6613	142	4	+	+	SYM
ejpam-6613	142	5	λ	λ	PROPN
ejpam-6613	142	6	,	,	PUNCT
ejpam-6613	142	7	η|σ⟩	η|σ⟩	PROPN
ejpam-6613	142	8	2	2	NUM
ejpam-6613	142	9	⊙	⊙	NOUN
ejpam-6613	142	10	=	=	SYM
ejpam-6613	142	11	〈	〈	PROPN
ejpam-6613	142	12	n+p∑	n+p∑	NOUN
ejpam-6613	142	13	i=1	i=1	NOUN
ejpam-6613	142	14	xi	xi	ADP
ejpam-6613	142	15	2	2	NUM
ejpam-6613	142	16	⊙	⊙	PROPN
ejpam-6613	142	17	yi	yi	PROPN
ejpam-6613	142	18	,	,	PUNCT
ejpam-6613	142	19	m∑	m∑	ADP
ejpam-6613	142	20	j=1	j=1	PROPN
ejpam-6613	142	21	zj	zj	PROPN
ejpam-6613	142	22	2	2	NUM
ejpam-6613	142	23	⊙	⊙	X
ejpam-6613	142	24	wj	wj	PROPN
ejpam-6613	142	25	|	|	ADV
ejpam-6613	142	26	g∑	g∑	PROPN
ejpam-6613	142	27	k=1	k=1	PROPN
ejpam-6613	142	28	lk	lk	PROPN
ejpam-6613	142	29	2	2	NUM
ejpam-6613	142	30	⊙	⊙	X
ejpam-6613	142	31	vk	vk	ADP
ejpam-6613	142	32	〉	〉	NOUN
ejpam-6613	142	33	2	2	NUM
ejpam-6613	142	34	⊙	⊙	NOUN
ejpam-6613	142	35	=	=	PUNCT
ejpam-6613	143	1	n+p∑	n+p∑	NOUN
ejpam-6613	143	2	i=1	i=1	PROPN
ejpam-6613	143	3	m∑	m∑	ADV
ejpam-6613	143	4	j=1	j=1	ADJ
ejpam-6613	143	5	g∑	g∑	PROPN
ejpam-6613	143	6	k=1	k=1	X
ejpam-6613	143	7	δi	δi	PROPN
ejpam-6613	143	8	,	,	PUNCT
ejpam-6613	143	9	j⟨xi	j⟨xi	PROPN
ejpam-6613	143	10	,	,	PUNCT
ejpam-6613	143	11	zj	zj	PROPN
ejpam-6613	143	12	|lk⟩1⟨yi	|lk⟩1⟨yi	NUM
ejpam-6613	143	13	,	,	PUNCT
ejpam-6613	143	14	wj	wj	PROPN
ejpam-6613	143	15	|vk⟩2	|vk⟩2	NOUN
ejpam-6613	144	1	=	=	PUNCT
ejpam-6613	144	2	n∑	n∑	PROPN
ejpam-6613	144	3	i=1	i=1	PROPN
ejpam-6613	145	1	m∑	m∑	ADV
ejpam-6613	145	2	j=1	j=1	ADJ
ejpam-6613	145	3	g∑	g∑	PROPN
ejpam-6613	145	4	k=1	k=1	X
ejpam-6613	145	5	δi	δi	PROPN
ejpam-6613	145	6	,	,	PUNCT
ejpam-6613	145	7	j⟨xi	j⟨xi	PROPN
ejpam-6613	145	8	,	,	PUNCT
ejpam-6613	145	9	zj	zj	PROPN
ejpam-6613	145	10	|lk⟩1⟨yi	|lk⟩1⟨yi	NUM
ejpam-6613	145	11	,	,	PUNCT
ejpam-6613	145	12	wj	wj	PROPN
ejpam-6613	145	13	|vk⟩2	|vk⟩2	VERB
ejpam-6613	146	1	+	+	CCONJ
ejpam-6613	147	1	n+p∑	n+p∑	PROPN
ejpam-6613	147	2	i	i	NOUN
ejpam-6613	147	3	=	=	NOUN
ejpam-6613	147	4	n+1	n+1	ADJ
ejpam-6613	147	5	m∑	m∑	ADV
ejpam-6613	147	6	j=1	j=1	PROPN
ejpam-6613	147	7	g∑	g∑	PROPN
ejpam-6613	147	8	k=1	k=1	X
ejpam-6613	147	9	δi	δi	PROPN
ejpam-6613	147	10	,	,	PUNCT
ejpam-6613	147	11	j⟨xi	j⟨xi	PROPN
ejpam-6613	147	12	,	,	PUNCT
ejpam-6613	147	13	zj	zj	PROPN
ejpam-6613	147	14	|lk⟩1⟨yi	|lk⟩1⟨yi	NUM
ejpam-6613	147	15	,	,	PUNCT
ejpam-6613	147	16	wj	wj	PROPN
ejpam-6613	147	17	|vk⟩2	|vk⟩2	NOUN
ejpam-6613	148	1	=	=	PUNCT
ejpam-6613	148	2	n∑	n∑	PROPN
ejpam-6613	148	3	i=1	i=1	PROPN
ejpam-6613	149	1	m∑	m∑	ADV
ejpam-6613	149	2	j=1	j=1	ADJ
ejpam-6613	149	3	g∑	g∑	PROPN
ejpam-6613	149	4	k=1	k=1	X
ejpam-6613	149	5	δi	δi	PROPN
ejpam-6613	149	6	,	,	PUNCT
ejpam-6613	149	7	j⟨xi	j⟨xi	PROPN
ejpam-6613	149	8	,	,	PUNCT
ejpam-6613	149	9	zj	zj	PROPN
ejpam-6613	149	10	|lk⟩1⟨yi	|lk⟩1⟨yi	NUM
ejpam-6613	149	11	,	,	PUNCT
ejpam-6613	149	12	wj	wj	PROPN
ejpam-6613	149	13	|vk⟩2	|vk⟩2	NOUN
ejpam-6613	150	1	+	+	CCONJ
ejpam-6613	150	2	p∑	p∑	X
ejpam-6613	151	1	t=1	t=1	ADV
ejpam-6613	151	2	m∑	m∑	ADP
ejpam-6613	151	3	j=1	j=1	PROPN
ejpam-6613	151	4	g∑	g∑	PROPN
ejpam-6613	151	5	k=1	k=1	X
ejpam-6613	151	6	δn+t	δn+t	PROPN
ejpam-6613	151	7	,	,	PUNCT
ejpam-6613	151	8	j⟨xn+t	j⟨xn+t	PROPN
ejpam-6613	151	9	,	,	PUNCT
ejpam-6613	151	10	zj	zj	PROPN
ejpam-6613	151	11	|lk⟩1⟨yn+t	|lk⟩1⟨yn+t	PROPN
ejpam-6613	151	12	,	,	PUNCT
ejpam-6613	151	13	wj	wj	PROPN
ejpam-6613	151	14	|vk⟩2	|vk⟩2	NOUN
ejpam-6613	152	1	=	=	PUNCT
ejpam-6613	152	2	n∑	n∑	PROPN
ejpam-6613	152	3	i=1	i=1	PROPN
ejpam-6613	153	1	m∑	m∑	ADV
ejpam-6613	153	2	j=1	j=1	ADJ
ejpam-6613	153	3	g∑	g∑	PROPN
ejpam-6613	153	4	k=1	k=1	X
ejpam-6613	153	5	δi	δi	PROPN
ejpam-6613	153	6	,	,	PUNCT
ejpam-6613	153	7	j⟨xi	j⟨xi	PROPN
ejpam-6613	153	8	,	,	PUNCT
ejpam-6613	153	9	zj	zj	PROPN
ejpam-6613	153	10	|lk⟩1⟨yi	|lk⟩1⟨yi	NUM
ejpam-6613	153	11	,	,	PUNCT
ejpam-6613	153	12	wj	wj	PROPN
ejpam-6613	153	13	|vk⟩2	|vk⟩2	NOUN
ejpam-6613	154	1	+	+	CCONJ
ejpam-6613	154	2	p∑	p∑	X
ejpam-6613	155	1	t=1	t=1	ADV
ejpam-6613	155	2	m∑	m∑	ADP
ejpam-6613	155	3	j=1	j=1	PROPN
ejpam-6613	155	4	g∑	g∑	PROPN
ejpam-6613	155	5	k=1	k=1	X
ejpam-6613	155	6	δi	δi	PROPN
ejpam-6613	155	7	,	,	PUNCT
ejpam-6613	155	8	j⟨ri	j⟨ri	PROPN
ejpam-6613	155	9	,	,	PUNCT
ejpam-6613	155	10	zj	zj	PROPN
ejpam-6613	155	11	|lk⟩1⟨si	|lk⟩1⟨si	PROPN
ejpam-6613	155	12	,	,	PUNCT
ejpam-6613	155	13	wj	wj	PROPN
ejpam-6613	155	14	|vk⟩2	|vk⟩2	NOUN
ejpam-6613	155	15	=	=	SYM
ejpam-6613	155	16	⟨ξ	⟨ξ	PROPN
ejpam-6613	155	17	,	,	PUNCT
ejpam-6613	155	18	η|σ⟩	η|σ⟩	PROPN
ejpam-6613	155	19	2	2	NUM
ejpam-6613	155	20	⊙	⊙	NOUN
ejpam-6613	155	21	+	+	CCONJ
ejpam-6613	155	22	⟨λ	⟨λ	NUM
ejpam-6613	155	23	,	,	PUNCT
ejpam-6613	155	24	η|σ⟩	η|σ⟩	PROPN
ejpam-6613	155	25	2	2	NUM
ejpam-6613	155	26	⊙	⊙	NOUN
ejpam-6613	155	27	in	in	ADP
ejpam-6613	155	28	addition	addition	NOUN
ejpam-6613	155	29	,	,	PUNCT
ejpam-6613	155	30	⟨αξ	⟨αξ	NUM
ejpam-6613	155	31	,	,	PUNCT
ejpam-6613	155	32	η|λ⟩	η|λ⟩	PROPN
ejpam-6613	155	33	2	2	NUM
ejpam-6613	155	34	⊙	⊙	NOUN
ejpam-6613	155	35	=	=	SYM
ejpam-6613	156	1	〈	〈	PROPN
ejpam-6613	156	2	α	α	NOUN
ejpam-6613	156	3	n∑	n∑	NOUN
ejpam-6613	156	4	i=1	i=1	PROPN
ejpam-6613	156	5	xi	xi	PROPN
ejpam-6613	156	6	2	2	NUM
ejpam-6613	156	7	⊙	⊙	PROPN
ejpam-6613	156	8	yi	yi	PROPN
ejpam-6613	156	9	,	,	PUNCT
ejpam-6613	156	10	r∑	r∑	ADP
ejpam-6613	156	11	j=1	j=1	PROPN
ejpam-6613	156	12	zj	zj	PROPN
ejpam-6613	156	13	2	2	NUM
ejpam-6613	156	14	⊙	⊙	X
ejpam-6613	156	15	wj	wj	PROPN
ejpam-6613	157	1	|	|	ADV
ejpam-6613	157	2	p∑	p∑	PROPN
ejpam-6613	158	1	t=1	t=1	PROPN
ejpam-6613	158	2	rt	rt	PROPN
ejpam-6613	158	3	2	2	NUM
ejpam-6613	158	4	⊙	⊙	PROPN
ejpam-6613	158	5	st	st	PROPN
ejpam-6613	158	6	〉	〉	PROPN
ejpam-6613	158	7	2	2	NUM
ejpam-6613	158	8	⊙	⊙	NOUN
ejpam-6613	158	9	=	=	SYM
ejpam-6613	158	10	〈	〈	PROPN
ejpam-6613	158	11	n∑	n∑	PROPN
ejpam-6613	158	12	i=1	i=1	PROPN
ejpam-6613	159	1	α(xi	α(xi	ADJ
ejpam-6613	159	2	2	2	NUM
ejpam-6613	159	3	⊙	⊙	PROPN
ejpam-6613	159	4	yi	yi	PROPN
ejpam-6613	159	5	)	)	PUNCT
ejpam-6613	159	6	,	,	PUNCT
ejpam-6613	159	7	r∑	r∑	NOUN
ejpam-6613	159	8	j=1	j=1	PROPN
ejpam-6613	159	9	zj	zj	PROPN
ejpam-6613	159	10	2	2	NUM
ejpam-6613	159	11	⊙	⊙	X
ejpam-6613	159	12	wj	wj	PROPN
ejpam-6613	160	1	|	|	ADV
ejpam-6613	160	2	p∑	p∑	PROPN
ejpam-6613	161	1	t=1	t=1	PROPN
ejpam-6613	161	2	rt	rt	PROPN
ejpam-6613	161	3	2	2	NUM
ejpam-6613	161	4	⊙	⊙	PROPN
ejpam-6613	161	5	st	st	PROPN
ejpam-6613	161	6	〉	〉	PROPN
ejpam-6613	161	7	2	2	NUM
ejpam-6613	161	8	⊙	⊙	NOUN
ejpam-6613	161	9	=	=	SYM
ejpam-6613	161	10	〈	〈	PROPN
ejpam-6613	161	11	n∑	n∑	PROPN
ejpam-6613	161	12	i=1	i=1	PROPN
ejpam-6613	161	13	(	(	PUNCT
ejpam-6613	161	14	αxi	αxi	ADV
ejpam-6613	161	15	)	)	PUNCT
ejpam-6613	161	16	2	2	NUM
ejpam-6613	161	17	⊙	⊙	NOUN
ejpam-6613	161	18	yi	yi	PROPN
ejpam-6613	161	19	,	,	PUNCT
ejpam-6613	161	20	r∑	r∑	ADP
ejpam-6613	161	21	j=1	j=1	PROPN
ejpam-6613	161	22	zj	zj	PROPN
ejpam-6613	161	23	2	2	NUM
ejpam-6613	161	24	⊙	⊙	X
ejpam-6613	161	25	wj	wj	PROPN
ejpam-6613	162	1	|	|	ADV
ejpam-6613	162	2	p∑	p∑	PROPN
ejpam-6613	163	1	t=1	t=1	PROPN
ejpam-6613	163	2	rt	rt	PROPN
ejpam-6613	163	3	2	2	NUM
ejpam-6613	163	4	⊙	⊙	PROPN
ejpam-6613	163	5	st	st	PROPN
ejpam-6613	163	6	〉	〉	PROPN
ejpam-6613	163	7	2	2	NUM
ejpam-6613	163	8	⊙	⊙	NOUN
ejpam-6613	163	9	=	=	PROPN
ejpam-6613	164	1	n∑	n∑	PROPN
ejpam-6613	164	2	i=1	i=1	PROPN
ejpam-6613	165	1	m∑	m∑	ADV
ejpam-6613	166	1	j=1	j=1	PROPN
ejpam-6613	166	2	p∑	p∑	PROPN
ejpam-6613	167	1	t=1	t=1	X
ejpam-6613	167	2	δi	δi	PROPN
ejpam-6613	167	3	,	,	PUNCT
ejpam-6613	167	4	j⟨(αxi	j⟨(αxi	NOUN
ejpam-6613	167	5	)	)	PUNCT
ejpam-6613	167	6	,	,	PUNCT
ejpam-6613	167	7	zj	zj	PROPN
ejpam-6613	167	8	|rt⟩1⟨yi	|rt⟩1⟨yi	NUM
ejpam-6613	167	9	,	,	PUNCT
ejpam-6613	167	10	wj	wj	PROPN
ejpam-6613	167	11	|st⟩2	|st⟩2	PROPN
ejpam-6613	168	1	=	=	PUNCT
ejpam-6613	168	2	n∑	n∑	PROPN
ejpam-6613	168	3	i=1	i=1	PROPN
ejpam-6613	169	1	m∑	m∑	ADV
ejpam-6613	169	2	j=1	j=1	PROPN
ejpam-6613	169	3	p∑	p∑	PROPN
ejpam-6613	169	4	t=1	t=1	PROPN
ejpam-6613	169	5	αδi	αδi	PROPN
ejpam-6613	169	6	,	,	PUNCT
ejpam-6613	169	7	j⟨xi	j⟨xi	PROPN
ejpam-6613	169	8	,	,	PUNCT
ejpam-6613	169	9	zj	zj	PROPN
ejpam-6613	169	10	|rt⟩1⟨yi	|rt⟩1⟨yi	NUM
ejpam-6613	169	11	,	,	PUNCT
ejpam-6613	170	1	wj	wj	PROPN
ejpam-6613	170	2	|st⟩2	|st⟩2	PROPN
ejpam-6613	171	1	=	=	SYM
ejpam-6613	172	1	α	α	PROPN
ejpam-6613	172	2	n∑	n∑	NOUN
ejpam-6613	172	3	i=1	i=1	PROPN
ejpam-6613	173	1	m∑	m∑	ADV
ejpam-6613	174	1	j=1	j=1	PROPN
ejpam-6613	174	2	p∑	p∑	PROPN
ejpam-6613	174	3	t=1	t=1	PROPN
ejpam-6613	174	4	δi	δi	PROPN
ejpam-6613	174	5	,	,	PUNCT
ejpam-6613	174	6	j⟨xi	j⟨xi	PROPN
ejpam-6613	174	7	,	,	PUNCT
ejpam-6613	174	8	zj	zj	PROPN
ejpam-6613	174	9	|rt⟩1⟨yi	|rt⟩1⟨yi	NUM
ejpam-6613	174	10	,	,	PUNCT
ejpam-6613	174	11	wj	wj	PROPN
ejpam-6613	174	12	|st⟩2	|st⟩2	PROPN
ejpam-6613	174	13	=	=	SYM
ejpam-6613	175	1	α⟨ξ	α⟨ξ	PROPN
ejpam-6613	175	2	,	,	PUNCT
ejpam-6613	175	3	η|λ⟩	η|λ⟩	PROPN
ejpam-6613	175	4	2	2	NUM
ejpam-6613	175	5	⊙	⊙	NOUN
ejpam-6613	175	6	.	.	PUNCT
ejpam-6613	176	1	let	let	VERB
ejpam-6613	176	2	us	we	PRON
ejpam-6613	176	3	now	now	ADV
ejpam-6613	176	4	show	show	VERB
ejpam-6613	176	5	that	that	SCONJ
ejpam-6613	176	6	this	this	DET
ejpam-6613	176	7	mapping	mapping	NOUN
ejpam-6613	176	8	is	be	AUX
ejpam-6613	176	9	hermitian	hermitian	ADJ
ejpam-6613	176	10	.	.	PUNCT
ejpam-6613	177	1	⟨ξ	⟨ξ	PROPN
ejpam-6613	177	2	,	,	PUNCT
ejpam-6613	177	3	η|λ⟩	η|λ⟩	PROPN
ejpam-6613	177	4	2	2	NUM
ejpam-6613	177	5	⊙	⊙	NOUN
ejpam-6613	178	1	=	=	PROPN
ejpam-6613	178	2	n∑	n∑	PROPN
ejpam-6613	178	3	i=1	i=1	PRON
ejpam-6613	179	1	r∑	r∑	NOUN
ejpam-6613	179	2	j=1	j=1	ADJ
ejpam-6613	179	3	p∑	p∑	X
ejpam-6613	180	1	t=1	t=1	PROPN
ejpam-6613	180	2	δi	δi	PROPN
ejpam-6613	180	3	,	,	PUNCT
ejpam-6613	180	4	j	j	PROPN
ejpam-6613	180	5	⟨xi	⟨xi	PROPN
ejpam-6613	180	6	,	,	PUNCT
ejpam-6613	180	7	zj	zj	PROPN
ejpam-6613	180	8	|rt⟩1	|rt⟩1	X
ejpam-6613	180	9	⟨yi	⟨yi	PRON
ejpam-6613	180	10	,	,	PUNCT
ejpam-6613	180	11	wj	wj	PROPN
ejpam-6613	180	12	|st⟩2	|st⟩2	PROPN
ejpam-6613	181	1	=	=	PUNCT
ejpam-6613	181	2	n∑	n∑	PROPN
ejpam-6613	181	3	i=1	i=1	PRON
ejpam-6613	182	1	r∑	r∑	NOUN
ejpam-6613	182	2	j=1	j=1	ADJ
ejpam-6613	182	3	p∑	p∑	X
ejpam-6613	183	1	t=1	t=1	PROPN
ejpam-6613	183	2	δi	δi	PROPN
ejpam-6613	183	3	,	,	PUNCT
ejpam-6613	183	4	j	j	PROPN
ejpam-6613	183	5	⟨zj	⟨zj	NUM
ejpam-6613	183	6	,	,	PUNCT
ejpam-6613	183	7	xi|rt⟩1	xi|rt⟩1	PROPN
ejpam-6613	184	1	⟨wj	⟨wj	X
ejpam-6613	184	2	,	,	PUNCT
ejpam-6613	184	3	yi|st⟩2	yi|st⟩2	PROPN
ejpam-6613	184	4	=	=	PUNCT
ejpam-6613	185	1	r∑	r∑	NOUN
ejpam-6613	185	2	j=1	j=1	PROPN
ejpam-6613	185	3	n∑	n∑	PROPN
ejpam-6613	185	4	i=1	i=1	PROPN
ejpam-6613	186	1	p∑	p∑	X
ejpam-6613	186	2	t=1	t=1	X
ejpam-6613	186	3	δi	δi	PROPN
ejpam-6613	186	4	,	,	PUNCT
ejpam-6613	186	5	j	j	PROPN
ejpam-6613	186	6	⟨zj	⟨zj	NUM
ejpam-6613	186	7	,	,	PUNCT
ejpam-6613	186	8	xi|rt⟩1	xi|rt⟩1	PROPN
ejpam-6613	187	1	⟨wj	⟨wj	X
ejpam-6613	187	2	,	,	PUNCT
ejpam-6613	187	3	yi|st⟩2	yi|st⟩2	PROPN
ejpam-6613	187	4	=	=	SYM
ejpam-6613	187	5	⟨η	⟨η	PROPN
ejpam-6613	187	6	,	,	PUNCT
ejpam-6613	187	7	ξ|λ⟩	ξ|λ⟩	PROPN
ejpam-6613	187	8	2	2	NUM
ejpam-6613	187	9	⊙	⊙	NOUN
ejpam-6613	187	10	.	.	PUNCT
ejpam-6613	188	1	m.	m.	PROPN
ejpam-6613	188	2	luis	luis	PROPN
ejpam-6613	188	3	,	,	PUNCT
ejpam-6613	188	4	f.	f.	PROPN
ejpam-6613	188	5	osmin	osmin	PROPN
ejpam-6613	188	6	,	,	PUNCT
ejpam-6613	188	7	s.	s.	PROPN
ejpam-6613	188	8	arley	arley	PROPN
ejpam-6613	188	9	/	/	SYM
ejpam-6613	188	10	eur	eur	PROPN
ejpam-6613	188	11	.	.	PUNCT
ejpam-6613	189	1	j.	j.	PROPN
ejpam-6613	189	2	pure	pure	PROPN
ejpam-6613	189	3	appl	appl	PROPN
ejpam-6613	189	4	.	.	PROPN
ejpam-6613	189	5	math	math	PROPN
ejpam-6613	189	6	,	,	PUNCT
ejpam-6613	189	7	18	18	NUM
ejpam-6613	189	8	(	(	PUNCT
ejpam-6613	189	9	4	4	NUM
ejpam-6613	189	10	)	)	PUNCT
ejpam-6613	189	11	(	(	PUNCT
ejpam-6613	189	12	2025	2025	NUM
ejpam-6613	189	13	)	)	PUNCT
ejpam-6613	189	14	,	,	PUNCT
ejpam-6613	189	15	6613	6613	NUM
ejpam-6613	189	16	9	9	NUM
ejpam-6613	189	17	of	of	ADP
ejpam-6613	189	18	17	17	NUM
ejpam-6613	189	19	therefore	therefore	ADV
ejpam-6613	189	20	the	the	DET
ejpam-6613	189	21	mapping	mapping	NOUN
ejpam-6613	189	22	⟨	⟨	VERB
ejpam-6613	189	23	·	·	PUNCT
ejpam-6613	189	24	,	,	PUNCT
ejpam-6613	189	25	·	·	PUNCT
ejpam-6613	189	26	|·⟩	|·⟩	X
ejpam-6613	189	27	2	2	NUM
ejpam-6613	189	28	⊙	⊙	NOUN
ejpam-6613	189	29	is	be	AUX
ejpam-6613	189	30	hermitian	hermitian	ADJ
ejpam-6613	189	31	.	.	PUNCT
ejpam-6613	190	1	moreover	moreover	ADV
ejpam-6613	190	2	,	,	PUNCT
ejpam-6613	190	3	we	we	PRON
ejpam-6613	190	4	have	have	VERB
ejpam-6613	190	5	⟨ξ	⟨ξ	NOUN
ejpam-6613	190	6	,	,	PUNCT
ejpam-6613	190	7	ξ|λ⟩	ξ|λ⟩	PROPN
ejpam-6613	190	8	2	2	NUM
ejpam-6613	190	9	⊙	⊙	NOUN
ejpam-6613	190	10	=	=	SYM
ejpam-6613	190	11	〈	〈	PROPN
ejpam-6613	190	12	n∑	n∑	NOUN
ejpam-6613	190	13	i=1	i=1	PROPN
ejpam-6613	190	14	xi	xi	PROPN
ejpam-6613	190	15	2	2	NUM
ejpam-6613	190	16	⊙	⊙	PROPN
ejpam-6613	190	17	yi	yi	PROPN
ejpam-6613	190	18	,	,	PUNCT
ejpam-6613	190	19	n∑	n∑	PROPN
ejpam-6613	191	1	l=1	l=1	PROPN
ejpam-6613	191	2	xl	xl	PROPN
ejpam-6613	191	3	2	2	NUM
ejpam-6613	191	4	⊙	⊙	X
ejpam-6613	191	5	yl	yl	NOUN
ejpam-6613	191	6	∣∣∣	∣∣∣	NOUN
ejpam-6613	191	7	p∑	p∑	X
ejpam-6613	192	1	t=1	t=1	PROPN
ejpam-6613	192	2	rt	rt	PROPN
ejpam-6613	192	3	2	2	NUM
ejpam-6613	192	4	⊙	⊙	PROPN
ejpam-6613	192	5	st	st	PROPN
ejpam-6613	192	6	〉	〉	PROPN
ejpam-6613	192	7	2	2	NUM
ejpam-6613	192	8	⊙	⊙	NOUN
ejpam-6613	192	9	=	=	PROPN
ejpam-6613	193	1	n∑	n∑	PROPN
ejpam-6613	193	2	i=1	i=1	PROPN
ejpam-6613	194	1	n∑	n∑	PROPN
ejpam-6613	195	1	l=1	l=1	PROPN
ejpam-6613	195	2	p∑	p∑	X
ejpam-6613	196	1	t=1	t=1	X
ejpam-6613	196	2	δi	δi	PROPN
ejpam-6613	196	3	,	,	PUNCT
ejpam-6613	196	4	l	l	NOUN
ejpam-6613	196	5	⟨xi	⟨xi	PROPN
ejpam-6613	196	6	,	,	PUNCT
ejpam-6613	196	7	xl|rt⟩1	xl|rt⟩1	PROPN
ejpam-6613	196	8	⟨yi	⟨yi	PROPN
ejpam-6613	196	9	,	,	PUNCT
ejpam-6613	196	10	yl|st⟩2	yl|st⟩2	PROPN
ejpam-6613	196	11	=	=	SYM
ejpam-6613	196	12	n∑	n∑	PROPN
ejpam-6613	196	13	i=1	i=1	PROPN
ejpam-6613	197	1	p∑	p∑	X
ejpam-6613	197	2	t=1	t=1	PUNCT
ejpam-6613	197	3	⟨xi	⟨xi	PROPN
ejpam-6613	197	4	,	,	PUNCT
ejpam-6613	197	5	xi|rt⟩1	xi|rt⟩1	PROPN
ejpam-6613	197	6	⟨yi	⟨yi	PROPN
ejpam-6613	197	7	,	,	PUNCT
ejpam-6613	197	8	yi|st⟩2	yi|st⟩2	PROPN
ejpam-6613	197	9	=	=	SYM
ejpam-6613	197	10	n∑	n∑	PROPN
ejpam-6613	197	11	i=1	i=1	PROPN
ejpam-6613	198	1	p∑	p∑	X
ejpam-6613	198	2	t=1	t=1	SYM
ejpam-6613	198	3	⟨rt	⟨rt	PROPN
ejpam-6613	198	4	,	,	PUNCT
ejpam-6613	198	5	rt|xi⟩1	rt|xi⟩1	PROPN
ejpam-6613	198	6	⟨st	⟨st	ADJ
ejpam-6613	198	7	,	,	PUNCT
ejpam-6613	199	1	st|yi⟩2	st|yi⟩2	PROPN
ejpam-6613	199	2	=	=	PUNCT
ejpam-6613	199	3	p∑	p∑	X
ejpam-6613	200	1	j=1	j=1	NOUN
ejpam-6613	200	2	p∑	p∑	X
ejpam-6613	201	1	t=1	t=1	PROPN
ejpam-6613	201	2	n∑	n∑	PROPN
ejpam-6613	201	3	i=1	i=1	PROPN
ejpam-6613	201	4	δj	δj	PROPN
ejpam-6613	201	5	,	,	PUNCT
ejpam-6613	201	6	t	t	PROPN
ejpam-6613	201	7	⟨rj	⟨rj	ADJ
ejpam-6613	201	8	,	,	PUNCT
ejpam-6613	201	9	rt|xi⟩1	rt|xi⟩1	PROPN
ejpam-6613	201	10	⟨sj	⟨sj	PROPN
ejpam-6613	201	11	,	,	PUNCT
ejpam-6613	201	12	st|yi⟩2	st|yi⟩2	PROPN
ejpam-6613	201	13	=	=	SYM
ejpam-6613	201	14	〈	〈	PROPN
ejpam-6613	201	15	p∑	p∑	NOUN
ejpam-6613	201	16	j=1	j=1	PROPN
ejpam-6613	201	17	rj	rj	PROPN
ejpam-6613	201	18	2	2	NUM
ejpam-6613	201	19	⊙	⊙	X
ejpam-6613	201	20	sj	sj	INTJ
ejpam-6613	201	21	,	,	PUNCT
ejpam-6613	201	22	p∑	p∑	X
ejpam-6613	201	23	t=1	t=1	PROPN
ejpam-6613	201	24	rt	rt	PROPN
ejpam-6613	201	25	2	2	NUM
ejpam-6613	201	26	⊙	⊙	PROPN
ejpam-6613	201	27	st	st	PROPN
ejpam-6613	201	28	∣∣∣	∣∣∣	PROPN
ejpam-6613	201	29	n∑	n∑	PROPN
ejpam-6613	201	30	i=1	i=1	PROPN
ejpam-6613	201	31	xi	xi	ADP
ejpam-6613	201	32	2	2	NUM
ejpam-6613	201	33	⊙	⊙	PROPN
ejpam-6613	201	34	yi	yi	PROPN
ejpam-6613	201	35	〉	〉	NOUN
ejpam-6613	201	36	2	2	NUM
ejpam-6613	201	37	⊙	⊙	NOUN
ejpam-6613	201	38	=	=	PUNCT
ejpam-6613	201	39	⟨λ	⟨λ	X
ejpam-6613	201	40	,	,	PUNCT
ejpam-6613	201	41	λ|ξ⟩	λ|ξ⟩	PROPN
ejpam-6613	201	42	2	2	NUM
ejpam-6613	201	43	⊙	⊙	NOUN
ejpam-6613	201	44	.	.	PUNCT
ejpam-6613	202	1	finally	finally	ADV
ejpam-6613	202	2	,	,	PUNCT
ejpam-6613	202	3	it	it	PRON
ejpam-6613	202	4	is	be	AUX
ejpam-6613	202	5	clear	clear	ADJ
ejpam-6613	202	6	that	that	SCONJ
ejpam-6613	202	7	⟨ξ	⟨ξ	NOUN
ejpam-6613	202	8	,	,	PUNCT
ejpam-6613	202	9	ξ|λ⟩	ξ|λ⟩	PROPN
ejpam-6613	202	10	2	2	NUM
ejpam-6613	202	11	⊙	⊙	X
ejpam-6613	202	12	≥	≥	NOUN
ejpam-6613	202	13	0	0	NUM
ejpam-6613	202	14	for	for	ADP
ejpam-6613	202	15	all	all	DET
ejpam-6613	202	16	ξ	ξ	PROPN
ejpam-6613	202	17	,	,	PUNCT
ejpam-6613	202	18	λ	λ	PROPN
ejpam-6613	202	19	∈	∈	PROPN
ejpam-6613	202	20	x1	x1	PROPN
ejpam-6613	202	21	2	2	NUM
ejpam-6613	202	22	⊙	⊙	NOUN
ejpam-6613	202	23	x2	x2	PROPN
ejpam-6613	202	24	.	.	PUNCT
ejpam-6613	203	1	hence	hence	ADV
ejpam-6613	203	2	⟨	⟨	VERB
ejpam-6613	203	3	·	·	PUNCT
ejpam-6613	203	4	,	,	PUNCT
ejpam-6613	203	5	·	·	PUNCT
ejpam-6613	203	6	|·⟩	|·⟩	X
ejpam-6613	203	7	2	2	NUM
ejpam-6613	203	8	⊙	⊙	NOUN
ejpam-6613	203	9	defines	define	VERB
ejpam-6613	203	10	a	a	DET
ejpam-6613	203	11	generalized	generalized	ADJ
ejpam-6613	203	12	2	2	NUM
ejpam-6613	203	13	-	-	PUNCT
ejpam-6613	203	14	inner	inner	ADJ
ejpam-6613	203	15	product	product	NOUN
ejpam-6613	203	16	on	on	ADP
ejpam-6613	203	17	x1	x1	PROPN
ejpam-6613	203	18	2	2	NUM
ejpam-6613	203	19	⊙	⊙	NOUN
ejpam-6613	203	20	x2	x2	PROPN
ejpam-6613	203	21	.	.	PUNCT
ejpam-6613	203	22	example	example	NOUN
ejpam-6613	204	1	2	2	NUM
ejpam-6613	204	2	.	.	X
ejpam-6613	204	3	consider	consider	VERB
ejpam-6613	204	4	c2	c2	PROPN
ejpam-6613	204	5	equipped	equip	VERB
ejpam-6613	204	6	with	with	ADP
ejpam-6613	204	7	the	the	DET
ejpam-6613	204	8	application	application	NOUN
ejpam-6613	204	9	⟨	⟨	NOUN
ejpam-6613	204	10	·	·	PUNCT
ejpam-6613	204	11	,	,	PUNCT
ejpam-6613	204	12	·	·	PUNCT
ejpam-6613	204	13	|·⟩c2	|·⟩c2	PROPN
ejpam-6613	204	14	:	:	PUNCT
ejpam-6613	205	1	c2	c2	PROPN
ejpam-6613	205	2	×	×	PROPN
ejpam-6613	205	3	c2	c2	PROPN
ejpam-6613	205	4	×	×	PROPN
ejpam-6613	205	5	c2	c2	PROPN
ejpam-6613	205	6	−→	−→	NOUN
ejpam-6613	205	7	c	c	PROPN
ejpam-6613	205	8	given	give	VERB
ejpam-6613	205	9	by	by	ADP
ejpam-6613	205	10	⟨(x1	⟨(x1	PROPN
ejpam-6613	205	11	,	,	PUNCT
ejpam-6613	205	12	x2	x2	PROPN
ejpam-6613	205	13	)	)	PUNCT
ejpam-6613	205	14	,	,	PUNCT
ejpam-6613	205	15	(	(	PUNCT
ejpam-6613	205	16	y1	y1	INTJ
ejpam-6613	205	17	,	,	PUNCT
ejpam-6613	205	18	y2)|(z1	y2)|(z1	NOUN
ejpam-6613	205	19	,	,	PUNCT
ejpam-6613	205	20	z2)⟩c2	z2)⟩c2	NOUN
ejpam-6613	205	21	:	:	PUNCT
ejpam-6613	206	1	=	=	SYM
ejpam-6613	206	2	x1y1|z1|2	x1y1|z1|2	PUNCT
ejpam-6613	206	3	+	+	NUM
ejpam-6613	206	4	x2y2|z2|2	x2y2|z2|2	PROPN
ejpam-6613	206	5	,	,	PUNCT
ejpam-6613	206	6	(	(	PUNCT
ejpam-6613	206	7	x1	x1	PROPN
ejpam-6613	206	8	,	,	PUNCT
ejpam-6613	206	9	x2	x2	PROPN
ejpam-6613	206	10	)	)	PUNCT
ejpam-6613	206	11	,	,	PUNCT
ejpam-6613	206	12	(	(	PUNCT
ejpam-6613	206	13	y1	y1	INTJ
ejpam-6613	206	14	,	,	PUNCT
ejpam-6613	206	15	y2	y2	PROPN
ejpam-6613	206	16	)	)	PUNCT
ejpam-6613	206	17	,	,	PUNCT
ejpam-6613	206	18	(	(	PUNCT
ejpam-6613	206	19	z1	z1	PROPN
ejpam-6613	206	20	,	,	PUNCT
ejpam-6613	206	21	z2	z2	PROPN
ejpam-6613	206	22	)	)	PUNCT
ejpam-6613	206	23	∈	∈	PROPN
ejpam-6613	206	24	c2	c2	PROPN
ejpam-6613	206	25	.	.	PUNCT
ejpam-6613	207	1	in	in	ADP
ejpam-6613	207	2	addition	addition	NOUN
ejpam-6613	207	3	,	,	PUNCT
ejpam-6613	207	4	let	let	VERB
ejpam-6613	207	5	us	we	PRON
ejpam-6613	207	6	consider	consider	VERB
ejpam-6613	207	7	c	c	NOUN
ejpam-6613	207	8	with	with	ADP
ejpam-6613	207	9	the	the	DET
ejpam-6613	207	10	application	application	NOUN
ejpam-6613	207	11	⟨	⟨	NOUN
ejpam-6613	207	12	·	·	PUNCT
ejpam-6613	207	13	,	,	PUNCT
ejpam-6613	207	14	·	·	PUNCT
ejpam-6613	207	15	|·⟩c	|·⟩c	NOUN
ejpam-6613	207	16	:	:	PUNCT
ejpam-6613	208	1	c×	c×	PROPN
ejpam-6613	208	2	c×	c×	PROPN
ejpam-6613	208	3	c	c	PROPN
ejpam-6613	208	4	−→	−→	NOUN
ejpam-6613	208	5	c	c	NOUN
ejpam-6613	208	6	given	give	VERB
ejpam-6613	208	7	by	by	ADP
ejpam-6613	208	8	⟨x	⟨x	NUM
ejpam-6613	208	9	,	,	PUNCT
ejpam-6613	208	10	y|z⟩c	y|z⟩c	NOUN
ejpam-6613	208	11	:	:	PUNCT
ejpam-6613	208	12	=	=	SYM
ejpam-6613	209	1	xy|z|2	xy|z|2	NOUN
ejpam-6613	209	2	,	,	PUNCT
ejpam-6613	209	3	x	x	X
ejpam-6613	209	4	,	,	PUNCT
ejpam-6613	209	5	y	y	PROPN
ejpam-6613	209	6	,	,	PUNCT
ejpam-6613	209	7	z	z	PROPN
ejpam-6613	209	8	∈	∈	PROPN
ejpam-6613	209	9	c.	c.	PROPN
ejpam-6613	209	10	thus	thus	ADV
ejpam-6613	209	11	,	,	PUNCT
ejpam-6613	209	12	given	give	VERB
ejpam-6613	209	13	(	(	PUNCT
ejpam-6613	209	14	a	a	PRON
ejpam-6613	209	15	,	,	PUNCT
ejpam-6613	209	16	b	b	NOUN
ejpam-6613	209	17	)	)	PUNCT
ejpam-6613	209	18	∈	∈	PROPN
ejpam-6613	209	19	c2	c2	PROPN
ejpam-6613	209	20	y	y	PROPN
ejpam-6613	209	21	c	c	PROPN
ejpam-6613	209	22	∈	∈	PROPN
ejpam-6613	209	23	c	c	X
ejpam-6613	209	24	,	,	PUNCT
ejpam-6613	209	25	the	the	DET
ejpam-6613	209	26	function	function	NOUN
ejpam-6613	209	27	(	(	PUNCT
ejpam-6613	209	28	a	a	DET
ejpam-6613	209	29	,	,	PUNCT
ejpam-6613	209	30	b	b	NOUN
ejpam-6613	209	31	)	)	PUNCT
ejpam-6613	209	32	2	2	NUM
ejpam-6613	209	33	⊙	⊙	NOUN
ejpam-6613	209	34	c	c	NOUN
ejpam-6613	209	35	:	:	PUNCT
ejpam-6613	209	36	(	(	PUNCT
ejpam-6613	209	37	c2	c2	PROPN
ejpam-6613	209	38	×	×	PROPN
ejpam-6613	209	39	c)×	c)×	PROPN
ejpam-6613	209	40	(	(	PUNCT
ejpam-6613	209	41	c2	c2	PROPN
ejpam-6613	209	42	×	×	PROPN
ejpam-6613	209	43	c	c	NOUN
ejpam-6613	209	44	)	)	PUNCT
ejpam-6613	209	45	→	→	SYM
ejpam-6613	209	46	c	c	X
ejpam-6613	209	47	(	(	PUNCT
ejpam-6613	209	48	(	(	PUNCT
ejpam-6613	209	49	a	a	DET
ejpam-6613	209	50	,	,	PUNCT
ejpam-6613	209	51	b	b	NOUN
ejpam-6613	209	52	)	)	PUNCT
ejpam-6613	209	53	2	2	NUM
ejpam-6613	209	54	⊙	⊙	NOUN
ejpam-6613	209	55	c	c	PROPN
ejpam-6613	209	56	)	)	PUNCT
ejpam-6613	209	57	(	(	PUNCT
ejpam-6613	209	58	(	(	PUNCT
ejpam-6613	209	59	(	(	PUNCT
ejpam-6613	209	60	x1	x1	ADJ
ejpam-6613	209	61	,	,	PUNCT
ejpam-6613	209	62	x2	x2	PROPN
ejpam-6613	209	63	)	)	PUNCT
ejpam-6613	209	64	,	,	PUNCT
ejpam-6613	209	65	d	d	NOUN
ejpam-6613	209	66	)	)	PUNCT
ejpam-6613	209	67	,	,	PUNCT
ejpam-6613	209	68	(	(	PUNCT
ejpam-6613	209	69	(	(	PUNCT
ejpam-6613	209	70	y1	y1	INTJ
ejpam-6613	209	71	,	,	PUNCT
ejpam-6613	209	72	y2	y2	PROPN
ejpam-6613	209	73	)	)	PUNCT
ejpam-6613	209	74	,	,	PUNCT
ejpam-6613	209	75	e	e	NOUN
ejpam-6613	209	76	)	)	PUNCT
ejpam-6613	209	77	)	)	PUNCT
ejpam-6613	209	78	:	:	PUNCT
ejpam-6613	209	79	=	=	PUNCT
ejpam-6613	209	80	⟨(a	⟨(a	PROPN
ejpam-6613	209	81	,	,	PUNCT
ejpam-6613	209	82	b	b	NOUN
ejpam-6613	209	83	)	)	PUNCT
ejpam-6613	209	84	,	,	PUNCT
ejpam-6613	209	85	(	(	PUNCT
ejpam-6613	209	86	x1	x1	PROPN
ejpam-6613	209	87	,	,	PUNCT
ejpam-6613	209	88	x2)|(y1	x2)|(y1	PROPN
ejpam-6613	209	89	,	,	PUNCT
ejpam-6613	209	90	y2)⟩c2	y2)⟩c2	NOUN
ejpam-6613	209	91	·	·	PUNCT
ejpam-6613	209	92	⟨c	⟨c	NUM
ejpam-6613	209	93	,	,	PUNCT
ejpam-6613	209	94	d|e⟩c	d|e⟩c	NOUN
ejpam-6613	209	95	=	=	SYM
ejpam-6613	209	96	(	(	PUNCT
ejpam-6613	209	97	ax1|y1|2	ax1|y1|2	PROPN
ejpam-6613	209	98	+	+	CCONJ
ejpam-6613	209	99	bx2|y2|2	bx2|y2|2	PROPN
ejpam-6613	209	100	)	)	PUNCT
ejpam-6613	209	101	(	(	PUNCT
ejpam-6613	209	102	cd|e|2	cd|e|2	PROPN
ejpam-6613	209	103	)	)	PUNCT
ejpam-6613	209	104	=	=	PUNCT
ejpam-6613	210	1	acx1d|y1e|2	acx1d|y1e|2	NOUN
ejpam-6613	210	2	+	+	PRON
ejpam-6613	210	3	bcx2d|y2e|2	bcx2d|y2e|2	ADJ
ejpam-6613	210	4	=	=	SYM
ejpam-6613	210	5	⟨(ac	⟨(ac	PROPN
ejpam-6613	210	6	,	,	PUNCT
ejpam-6613	210	7	bc	bc	PROPN
ejpam-6613	210	8	)	)	PUNCT
ejpam-6613	210	9	,	,	PUNCT
ejpam-6613	210	10	(	(	PUNCT
ejpam-6613	210	11	x1d	x1d	X
ejpam-6613	210	12	,	,	PUNCT
ejpam-6613	210	13	x2d)|(y1e	x2d)|(y1e	PROPN
ejpam-6613	210	14	,	,	PUNCT
ejpam-6613	210	15	y2e)⟩c2	y2e)⟩c2	PROPN
ejpam-6613	210	16	,	,	PUNCT
ejpam-6613	210	17	for	for	ADP
ejpam-6613	210	18	all	all	PRON
ejpam-6613	210	19	(	(	PUNCT
ejpam-6613	210	20	(	(	PUNCT
ejpam-6613	210	21	x1	x1	PROPN
ejpam-6613	210	22	,	,	PUNCT
ejpam-6613	210	23	x2	x2	PROPN
ejpam-6613	210	24	)	)	PUNCT
ejpam-6613	210	25	,	,	PUNCT
ejpam-6613	210	26	d	d	NOUN
ejpam-6613	210	27	)	)	PUNCT
ejpam-6613	210	28	,	,	PUNCT
ejpam-6613	210	29	(	(	PUNCT
ejpam-6613	210	30	(	(	PUNCT
ejpam-6613	210	31	y1	y1	INTJ
ejpam-6613	210	32	,	,	PUNCT
ejpam-6613	210	33	y2	y2	PROPN
ejpam-6613	210	34	)	)	PUNCT
ejpam-6613	210	35	,	,	PUNCT
ejpam-6613	210	36	e	e	X
ejpam-6613	210	37	)	)	PUNCT
ejpam-6613	210	38	∈	∈	PROPN
ejpam-6613	210	39	c2	c2	PROPN
ejpam-6613	210	40	×	×	PROPN
ejpam-6613	210	41	c.	c.	PROPN
ejpam-6613	210	42	now	now	ADV
ejpam-6613	210	43	,	,	PUNCT
ejpam-6613	210	44	the	the	DET
ejpam-6613	210	45	tensor	tensor	NOUN
ejpam-6613	210	46	product	product	NOUN
ejpam-6613	210	47	of	of	ADP
ejpam-6613	210	48	the	the	DET
ejpam-6613	210	49	spaces	space	NOUN
ejpam-6613	210	50	with	with	ADP
ejpam-6613	210	51	generalized	generalized	ADJ
ejpam-6613	210	52	2	2	NUM
ejpam-6613	210	53	-	-	PUNCT
ejpam-6613	210	54	inner	inner	ADJ
ejpam-6613	210	55	product	product	NOUN
ejpam-6613	210	56	(	(	PUNCT
ejpam-6613	210	57	c2	c2	PROPN
ejpam-6613	210	58	,	,	PUNCT
ejpam-6613	210	59	⟨	⟨	NOUN
ejpam-6613	210	60	·	·	SYM
ejpam-6613	210	61	,	,	PUNCT
ejpam-6613	210	62	·	·	PUNCT
ejpam-6613	210	63	|·⟩c2	|·⟩c2	PUNCT
ejpam-6613	210	64	)	)	PUNCT
ejpam-6613	210	65	and	and	CCONJ
ejpam-6613	210	66	(	(	PUNCT
ejpam-6613	210	67	c	c	NOUN
ejpam-6613	210	68	,	,	PUNCT
ejpam-6613	210	69	⟨	⟨	NOUN
ejpam-6613	210	70	·	·	PUNCT
ejpam-6613	210	71	,	,	PUNCT
ejpam-6613	210	72	·	·	PUNCT
ejpam-6613	210	73	|·⟩c	|·⟩c	NOUN
ejpam-6613	210	74	)	)	PUNCT
ejpam-6613	210	75	,	,	PUNCT
ejpam-6613	210	76	in	in	ADP
ejpam-6613	210	77	accordance	accordance	NOUN
ejpam-6613	210	78	with	with	ADP
ejpam-6613	210	79	the	the	DET
ejpam-6613	210	80	definition	definition	NOUN
ejpam-6613	210	81	6	6	NUM
ejpam-6613	210	82	,	,	PUNCT
ejpam-6613	210	83	is	be	AUX
ejpam-6613	210	84	given	give	VERB
ejpam-6613	210	85	by	by	ADP
ejpam-6613	210	86	c2	c2	PROPN
ejpam-6613	210	87	2	2	NUM
ejpam-6613	210	88	⊙	⊙	NOUN
ejpam-6613	210	89	c	c	PROPN
ejpam-6613	211	1	=	=	PRON
ejpam-6613	211	2	{	{	PUNCT
ejpam-6613	211	3	∑n	∑n	PROPN
ejpam-6613	211	4	i=1(xi	i=1(xi	ADJ
ejpam-6613	211	5	,	,	PUNCT
ejpam-6613	211	6	yi	yi	PROPN
ejpam-6613	211	7	)	)	PUNCT
ejpam-6613	211	8	2	2	NUM
ejpam-6613	211	9	⊙	⊙	PROPN
ejpam-6613	211	10	ci	ci	PROPN
ejpam-6613	211	11	:	:	PUNCT
ejpam-6613	211	12	n	n	CCONJ
ejpam-6613	211	13	∈	∈	PROPN
ejpam-6613	211	14	n	n	CCONJ
ejpam-6613	211	15	,	,	PUNCT
ejpam-6613	211	16	(	(	PUNCT
ejpam-6613	211	17	xi	xi	PROPN
ejpam-6613	211	18	,	,	PUNCT
ejpam-6613	211	19	yi	yi	NOUN
ejpam-6613	211	20	)	)	PUNCT
ejpam-6613	211	21	∈	∈	PROPN
ejpam-6613	211	22	c2	c2	PROPN
ejpam-6613	211	23	,	,	PUNCT
ejpam-6613	211	24	ci	ci	PROPN
ejpam-6613	211	25	∈	∈	PROPN
ejpam-6613	211	26	c	c	PROPN
ejpam-6613	211	27	,	,	PUNCT
ejpam-6613	211	28	1	1	NUM
ejpam-6613	211	29	≤	≤	NUM
ejpam-6613	211	30	i	i	PRON
ejpam-6613	211	31	≤	≤	NOUN
ejpam-6613	211	32	n	n	CCONJ
ejpam-6613	211	33	}	}	PUNCT
ejpam-6613	211	34	,	,	PUNCT
ejpam-6613	211	35	and	and	CCONJ
ejpam-6613	211	36	the	the	DET
ejpam-6613	211	37	generalized	generalized	ADJ
ejpam-6613	211	38	2	2	NUM
ejpam-6613	211	39	-	-	PUNCT
ejpam-6613	211	40	inner	inner	ADJ
ejpam-6613	211	41	product	product	NOUN
ejpam-6613	211	42	of	of	ADP
ejpam-6613	211	43	theorem	theorem	ADJ
ejpam-6613	211	44	3	3	NUM
ejpam-6613	211	45	⟨	⟨	NOUN
ejpam-6613	211	46	·	·	NUM
ejpam-6613	211	47	,	,	PUNCT
ejpam-6613	211	48	·	·	PUNCT
ejpam-6613	211	49	|·⟩	|·⟩	X
ejpam-6613	211	50	:	:	PUNCT
ejpam-6613	211	51	(	(	PUNCT
ejpam-6613	211	52	c2	c2	PROPN
ejpam-6613	211	53	2	2	NUM
ejpam-6613	211	54	⊙c)×	⊙c)×	PROPN
ejpam-6613	211	55	(	(	PUNCT
ejpam-6613	211	56	c2	c2	PROPN
ejpam-6613	211	57	2	2	NUM
ejpam-6613	211	58	⊙c)×	⊙c)×	PROPN
ejpam-6613	211	59	(	(	PUNCT
ejpam-6613	211	60	c2	c2	PROPN
ejpam-6613	211	61	2	2	NUM
ejpam-6613	211	62	⊙c	⊙c	PROPN
ejpam-6613	211	63	)	)	PUNCT
ejpam-6613	211	64	→	→	SYM
ejpam-6613	211	65	c	c	X
ejpam-6613	211	66	,	,	PUNCT
ejpam-6613	211	67	has	have	VERB
ejpam-6613	211	68	the	the	DET
ejpam-6613	211	69	form	form	NOUN
ejpam-6613	211	70	m.	m.	PROPN
ejpam-6613	211	71	luis	luis	PROPN
ejpam-6613	211	72	,	,	PUNCT
ejpam-6613	211	73	f.	f.	PROPN
ejpam-6613	211	74	osmin	osmin	PROPN
ejpam-6613	211	75	,	,	PUNCT
ejpam-6613	211	76	s.	s.	PROPN
ejpam-6613	211	77	arley	arley	PROPN
ejpam-6613	211	78	/	/	SYM
ejpam-6613	211	79	eur	eur	PROPN
ejpam-6613	211	80	.	.	PUNCT
ejpam-6613	212	1	j.	j.	PROPN
ejpam-6613	212	2	pure	pure	PROPN
ejpam-6613	212	3	appl	appl	PROPN
ejpam-6613	212	4	.	.	PROPN
ejpam-6613	212	5	math	math	PROPN
ejpam-6613	212	6	,	,	PUNCT
ejpam-6613	212	7	18	18	NUM
ejpam-6613	212	8	(	(	PUNCT
ejpam-6613	212	9	4	4	NUM
ejpam-6613	212	10	)	)	PUNCT
ejpam-6613	212	11	(	(	PUNCT
ejpam-6613	212	12	2025	2025	NUM
ejpam-6613	212	13	)	)	PUNCT
ejpam-6613	212	14	,	,	PUNCT
ejpam-6613	212	15	6613	6613	NUM
ejpam-6613	212	16	10	10	NUM
ejpam-6613	212	17	of	of	ADP
ejpam-6613	212	18	17	17	NUM
ejpam-6613	212	19	⟨ξ	⟨ξ	NOUN
ejpam-6613	212	20	,	,	PUNCT
ejpam-6613	212	21	η|λ⟩	η|λ⟩	PROPN
ejpam-6613	212	22	2	2	NUM
ejpam-6613	212	23	⊙	⊙	NOUN
ejpam-6613	212	24	:	:	PUNCT
ejpam-6613	213	1	=	=	SYM
ejpam-6613	213	2	n∑	n∑	X
ejpam-6613	213	3	i=1	i=1	PROPN
ejpam-6613	214	1	m∑	m∑	ADV
ejpam-6613	215	1	j=1	j=1	PROPN
ejpam-6613	215	2	p∑	p∑	X
ejpam-6613	216	1	k=1	k=1	X
ejpam-6613	216	2	δi	δi	PROPN
ejpam-6613	216	3	,	,	PUNCT
ejpam-6613	216	4	j⟨(xi	j⟨(xi	NOUN
ejpam-6613	216	5	,	,	PUNCT
ejpam-6613	216	6	yi	yi	PROPN
ejpam-6613	216	7	)	)	PUNCT
ejpam-6613	216	8	,	,	PUNCT
ejpam-6613	216	9	(	(	PUNCT
ejpam-6613	216	10	zj	zj	INTJ
ejpam-6613	216	11	,	,	PUNCT
ejpam-6613	216	12	wj)|(rk	wj)|(rk	PROPN
ejpam-6613	216	13	,	,	PUNCT
ejpam-6613	216	14	sk)⟩c2⟨ci	sk)⟩c2⟨ci	PROPN
ejpam-6613	216	15	,	,	PUNCT
ejpam-6613	216	16	dj	dj	X
ejpam-6613	216	17	|ek⟩c	|ek⟩c	NOUN
ejpam-6613	216	18	=	=	PUNCT
ejpam-6613	217	1	n∑	n∑	NOUN
ejpam-6613	217	2	i=1	i=1	PROPN
ejpam-6613	218	1	m∑	m∑	ADV
ejpam-6613	219	1	j=1	j=1	PROPN
ejpam-6613	219	2	p∑	p∑	X
ejpam-6613	220	1	k=1	k=1	X
ejpam-6613	220	2	δi	δi	PROPN
ejpam-6613	220	3	,	,	PUNCT
ejpam-6613	220	4	j⟨(xici	j⟨(xici	NOUN
ejpam-6613	220	5	,	,	PUNCT
ejpam-6613	220	6	yici	yici	NOUN
ejpam-6613	220	7	)	)	PUNCT
ejpam-6613	220	8	,	,	PUNCT
ejpam-6613	220	9	(	(	PUNCT
ejpam-6613	220	10	zidi	zidi	PROPN
ejpam-6613	220	11	,	,	PUNCT
ejpam-6613	220	12	widi)|(rkek	widi)|(rkek	PROPN
ejpam-6613	220	13	,	,	PUNCT
ejpam-6613	220	14	skek)⟩c2	skek)⟩c2	PROPN
ejpam-6613	220	15	=	=	SYM
ejpam-6613	220	16	n∑	n∑	NOUN
ejpam-6613	220	17	i=1	i=1	PROPN
ejpam-6613	221	1	p∑	p∑	X
ejpam-6613	222	1	k=1	k=1	PROPN
ejpam-6613	222	2	⟨(xici	⟨(xici	NOUN
ejpam-6613	222	3	,	,	PUNCT
ejpam-6613	222	4	yici	yici	NOUN
ejpam-6613	222	5	)	)	PUNCT
ejpam-6613	222	6	,	,	PUNCT
ejpam-6613	222	7	(	(	PUNCT
ejpam-6613	222	8	zidi	zidi	PROPN
ejpam-6613	222	9	,	,	PUNCT
ejpam-6613	222	10	widi)|(rkek	widi)|(rkek	PROPN
ejpam-6613	222	11	,	,	PUNCT
ejpam-6613	222	12	skek)⟩c2	skek)⟩c2	PROPN
ejpam-6613	222	13	=	=	SYM
ejpam-6613	222	14	n∑	n∑	NOUN
ejpam-6613	222	15	i=1	i=1	PROPN
ejpam-6613	223	1	p∑	p∑	X
ejpam-6613	224	1	k=1	k=1	X
ejpam-6613	225	1	xicizidi|rkek|2	xicizidi|rkek|2	VERB
ejpam-6613	225	2	+	+	CCONJ
ejpam-6613	225	3	yiciwidi|skek|2	yiciwidi|skek|2	PROPN
ejpam-6613	225	4	for	for	ADP
ejpam-6613	225	5	all	all	PRON
ejpam-6613	225	6	ξ	ξ	PROPN
ejpam-6613	225	7	=	=	SYM
ejpam-6613	225	8	n∑	n∑	NOUN
ejpam-6613	225	9	i=1	i=1	PROPN
ejpam-6613	226	1	(	(	PUNCT
ejpam-6613	226	2	xi	xi	PROPN
ejpam-6613	226	3	,	,	PUNCT
ejpam-6613	226	4	yi	yi	PROPN
ejpam-6613	226	5	)	)	PUNCT
ejpam-6613	226	6	2	2	NUM
ejpam-6613	226	7	⊙	⊙	PROPN
ejpam-6613	226	8	ci	ci	PROPN
ejpam-6613	226	9	,	,	PUNCT
ejpam-6613	226	10	η	η	PROPN
ejpam-6613	226	11	=	=	PUNCT
ejpam-6613	226	12	m∑	m∑	ADP
ejpam-6613	226	13	j=1	j=1	PROPN
ejpam-6613	226	14	(	(	PUNCT
ejpam-6613	226	15	zj	zj	PROPN
ejpam-6613	226	16	,	,	PUNCT
ejpam-6613	226	17	wj	wj	PROPN
ejpam-6613	226	18	)	)	PUNCT
ejpam-6613	226	19	2	2	NUM
ejpam-6613	226	20	⊙	⊙	X
ejpam-6613	226	21	dj	dj	NOUN
ejpam-6613	226	22	,	,	PUNCT
ejpam-6613	226	23	λ	λ	X
ejpam-6613	226	24	=	=	PUNCT
ejpam-6613	226	25	p∑	p∑	X
ejpam-6613	226	26	k=1	k=1	X
ejpam-6613	226	27	(	(	PUNCT
ejpam-6613	226	28	rk	rk	NOUN
ejpam-6613	226	29	,	,	PUNCT
ejpam-6613	226	30	sk	sk	VERB
ejpam-6613	226	31	)	)	PUNCT
ejpam-6613	226	32	2	2	NUM
ejpam-6613	226	33	⊙	⊙	NOUN
ejpam-6613	226	34	ek	ek	PROPN
ejpam-6613	226	35	∈	∈	PROPN
ejpam-6613	226	36	c2	c2	PROPN
ejpam-6613	226	37	2	2	PROPN
ejpam-6613	226	38	⊙	⊙	PROPN
ejpam-6613	226	39	c.	c.	PROPN
ejpam-6613	226	40	next	next	ADV
ejpam-6613	226	41	,	,	PUNCT
ejpam-6613	226	42	we	we	PRON
ejpam-6613	226	43	equip	equip	VERB
ejpam-6613	226	44	the	the	DET
ejpam-6613	226	45	algebraic	algebraic	ADJ
ejpam-6613	226	46	tensor	tensor	NOUN
ejpam-6613	226	47	product	product	NOUN
ejpam-6613	226	48	space	space	NOUN
ejpam-6613	226	49	from	from	ADP
ejpam-6613	226	50	definition	definition	NOUN
ejpam-6613	226	51	6	6	NUM
ejpam-6613	226	52	with	with	ADP
ejpam-6613	226	53	a	a	DET
ejpam-6613	226	54	generalized	generalized	ADJ
ejpam-6613	226	55	2	2	NUM
ejpam-6613	226	56	-	-	PUNCT
ejpam-6613	226	57	norm	norm	NOUN
ejpam-6613	226	58	,	,	PUNCT
ejpam-6613	226	59	which	which	PRON
ejpam-6613	226	60	is	be	AUX
ejpam-6613	226	61	induced	induce	VERB
ejpam-6613	226	62	by	by	ADP
ejpam-6613	226	63	the	the	DET
ejpam-6613	226	64	mapping	mapping	NOUN
ejpam-6613	226	65	defined	define	VERB
ejpam-6613	226	66	in	in	ADP
ejpam-6613	226	67	theorem	theorem	ADJ
ejpam-6613	226	68	3	3	NUM
ejpam-6613	226	69	.	.	PUNCT
ejpam-6613	226	70	theorem	theorem	NOUN
ejpam-6613	226	71	4	4	NUM
ejpam-6613	226	72	.	.	PUNCT
ejpam-6613	227	1	let	let	AUX
ejpam-6613	227	2	(	(	PUNCT
ejpam-6613	227	3	x1	x1	ADJ
ejpam-6613	227	4	,	,	PUNCT
ejpam-6613	227	5	⟨	⟨	NOUN
ejpam-6613	227	6	·	·	SYM
ejpam-6613	227	7	,	,	PUNCT
ejpam-6613	227	8	·	·	PUNCT
ejpam-6613	227	9	|·⟩1	|·⟩1	X
ejpam-6613	227	10	)	)	PUNCT
ejpam-6613	227	11	and	and	CCONJ
ejpam-6613	227	12	(	(	PUNCT
ejpam-6613	227	13	x2	x2	INTJ
ejpam-6613	227	14	,	,	PUNCT
ejpam-6613	227	15	⟨	⟨	NOUN
ejpam-6613	227	16	·	·	SYM
ejpam-6613	227	17	,	,	PUNCT
ejpam-6613	227	18	·	·	PUNCT
ejpam-6613	227	19	|·⟩2	|·⟩2	X
ejpam-6613	227	20	)	)	PUNCT
ejpam-6613	227	21	be	be	AUX
ejpam-6613	227	22	spaces	space	NOUN
ejpam-6613	227	23	with	with	ADP
ejpam-6613	227	24	a	a	DET
ejpam-6613	227	25	generalized	generalized	ADJ
ejpam-6613	227	26	2	2	NUM
ejpam-6613	227	27	-	-	PUNCT
ejpam-6613	227	28	inner	inner	ADJ
ejpam-6613	227	29	product	product	NOUN
ejpam-6613	227	30	.	.	PUNCT
ejpam-6613	228	1	we	we	PRON
ejpam-6613	228	2	define	define	VERB
ejpam-6613	228	3	a	a	DET
ejpam-6613	228	4	generalized	generalized	ADJ
ejpam-6613	228	5	2	2	NUM
ejpam-6613	228	6	-	-	PUNCT
ejpam-6613	228	7	norm	norm	NOUN
ejpam-6613	228	8	on	on	ADP
ejpam-6613	228	9	x1	x1	PROPN
ejpam-6613	228	10	2	2	NUM
ejpam-6613	228	11	⊙	⊙	X
ejpam-6613	228	12	x2	x2	PROPN
ejpam-6613	228	13	,	,	PUNCT
ejpam-6613	228	14	called	call	VERB
ejpam-6613	228	15	the	the	DET
ejpam-6613	228	16	induced	induce	VERB
ejpam-6613	228	17	2	2	NUM
ejpam-6613	228	18	-	-	PUNCT
ejpam-6613	228	19	tensor	tensor	NOUN
ejpam-6613	228	20	norm	norm	NOUN
ejpam-6613	228	21	by	by	ADP
ejpam-6613	228	22	the	the	DET
ejpam-6613	228	23	generalized	generalized	ADJ
ejpam-6613	228	24	2	2	NUM
ejpam-6613	228	25	-	-	PUNCT
ejpam-6613	228	26	inner	inner	ADJ
ejpam-6613	228	27	product	product	NOUN
ejpam-6613	228	28	of	of	ADP
ejpam-6613	228	29	theorem	theorem	NOUN
ejpam-6613	228	30	3	3	NUM
ejpam-6613	228	31	,	,	PUNCT
ejpam-6613	228	32	as	as	ADP
ejpam-6613	228	33	the	the	DET
ejpam-6613	228	34	mapping	mapping	NOUN
ejpam-6613	228	35	∥	∥	NOUN
ejpam-6613	228	36	·	·	PUNCT
ejpam-6613	228	37	,	,	PUNCT
ejpam-6613	228	38	·	·	PUNCT
ejpam-6613	228	39	∥	∥	X
ejpam-6613	228	40	2	2	NUM
ejpam-6613	228	41	⊙	⊙	NOUN
ejpam-6613	228	42	:	:	PUNCT
ejpam-6613	228	43	(	(	PUNCT
ejpam-6613	228	44	x1	x1	NOUN
ejpam-6613	228	45	2	2	NUM
ejpam-6613	228	46	⊙x2)×	⊙x2)×	NUM
ejpam-6613	228	47	(	(	PUNCT
ejpam-6613	228	48	x1	x1	PROPN
ejpam-6613	228	49	2	2	NUM
ejpam-6613	228	50	⊙x2	⊙x2	PROPN
ejpam-6613	228	51	)	)	PUNCT
ejpam-6613	228	52	−→	−→	NOUN
ejpam-6613	228	53	r	r	NOUN
ejpam-6613	228	54	given	give	VERB
ejpam-6613	228	55	by	by	ADP
ejpam-6613	228	56	∥ξ	∥ξ	PROPN
ejpam-6613	228	57	,	,	PUNCT
ejpam-6613	228	58	λ∥	λ∥	X
ejpam-6613	228	59	2	2	NUM
ejpam-6613	228	60	⊙	⊙	NOUN
ejpam-6613	228	61	:	:	PUNCT
ejpam-6613	228	62	=	=	NOUN
ejpam-6613	228	63	√	√	NUM
ejpam-6613	228	64	⟨ξ	⟨ξ	NOUN
ejpam-6613	228	65	,	,	PUNCT
ejpam-6613	228	66	ξ|λ⟩	ξ|λ⟩	PROPN
ejpam-6613	228	67	2	2	NUM
ejpam-6613	228	68	⊙	⊙	NOUN
ejpam-6613	228	69	,	,	PUNCT
ejpam-6613	228	70	ξ	ξ	X
ejpam-6613	228	71	,	,	PUNCT
ejpam-6613	228	72	λ	λ	PROPN
ejpam-6613	228	73	∈	∈	PROPN
ejpam-6613	228	74	x1	x1	PROPN
ejpam-6613	228	75	2	2	NUM
ejpam-6613	228	76	⊙x2	⊙x2	PROPN
ejpam-6613	228	77	.	.	PUNCT
ejpam-6613	229	1	proof	proof	NOUN
ejpam-6613	229	2	.	.	PUNCT
ejpam-6613	230	1	since	since	SCONJ
ejpam-6613	230	2	⟨	⟨	NOUN
ejpam-6613	230	3	·	·	SYM
ejpam-6613	230	4	,	,	PUNCT
ejpam-6613	230	5	·	·	PUNCT
ejpam-6613	230	6	|·⟩	|·⟩	X
ejpam-6613	230	7	2	2	NUM
ejpam-6613	230	8	⊙	⊙	NOUN
ejpam-6613	230	9	is	be	AUX
ejpam-6613	230	10	a	a	DET
ejpam-6613	230	11	generalized	generalized	ADJ
ejpam-6613	230	12	2	2	NUM
ejpam-6613	230	13	-	-	PUNCT
ejpam-6613	230	14	inner	inner	ADJ
ejpam-6613	230	15	product	product	NOUN
ejpam-6613	230	16	,	,	PUNCT
ejpam-6613	230	17	it	it	PRON
ejpam-6613	230	18	follows	follow	VERB
ejpam-6613	230	19	that	that	SCONJ
ejpam-6613	230	20	∥	∥	PROPN
ejpam-6613	230	21	·	·	SYM
ejpam-6613	230	22	,	,	PUNCT
ejpam-6613	230	23	·	·	PUNCT
ejpam-6613	230	24	∥	∥	X
ejpam-6613	230	25	2	2	NUM
ejpam-6613	230	26	⊙	⊙	NOUN
ejpam-6613	230	27	defines	define	VERB
ejpam-6613	230	28	a	a	DET
ejpam-6613	230	29	generalized	generalized	ADJ
ejpam-6613	230	30	2	2	NUM
ejpam-6613	230	31	-	-	PUNCT
ejpam-6613	230	32	norm	norm	NOUN
ejpam-6613	230	33	.	.	PUNCT
ejpam-6613	231	1	the	the	DET
ejpam-6613	231	2	proof	proof	NOUN
ejpam-6613	231	3	is	be	AUX
ejpam-6613	231	4	straightforward	straightforward	ADJ
ejpam-6613	231	5	and	and	CCONJ
ejpam-6613	231	6	analogous	analogous	ADJ
ejpam-6613	231	7	to	to	ADP
ejpam-6613	231	8	the	the	DET
ejpam-6613	231	9	classical	classical	ADJ
ejpam-6613	231	10	case	case	NOUN
ejpam-6613	231	11	.	.	PUNCT
ejpam-6613	232	1	following	follow	VERB
ejpam-6613	232	2	the	the	DET
ejpam-6613	232	3	work	work	NOUN
ejpam-6613	232	4	of	of	ADP
ejpam-6613	232	5	lewandoska	lewandoska	NOUN
ejpam-6613	232	6	[	[	X
ejpam-6613	232	7	22	22	NUM
ejpam-6613	232	8	]	]	PUNCT
ejpam-6613	232	9	,	,	PUNCT
ejpam-6613	232	10	we	we	PRON
ejpam-6613	232	11	introduce	introduce	VERB
ejpam-6613	232	12	the	the	DET
ejpam-6613	232	13	notion	notion	NOUN
ejpam-6613	232	14	of	of	ADP
ejpam-6613	232	15	a	a	DET
ejpam-6613	232	16	2	2	NUM
ejpam-6613	232	17	-	-	PUNCT
ejpam-6613	232	18	bounded	bound	VERB
ejpam-6613	232	19	linear	linear	ADJ
ejpam-6613	232	20	operator	operator	NOUN
ejpam-6613	232	21	on	on	ADP
ejpam-6613	232	22	the	the	DET
ejpam-6613	232	23	2	2	NUM
ejpam-6613	232	24	-	-	PUNCT
ejpam-6613	232	25	tensor	tensor	NOUN
ejpam-6613	232	26	product	product	NOUN
ejpam-6613	232	27	of	of	ADP
ejpam-6613	232	28	spaces	space	NOUN
ejpam-6613	232	29	with	with	ADP
ejpam-6613	232	30	a	a	DET
ejpam-6613	232	31	generalized	generalized	ADJ
ejpam-6613	232	32	2	2	NUM
ejpam-6613	232	33	-	-	PUNCT
ejpam-6613	232	34	inner	inner	ADJ
ejpam-6613	232	35	product	product	NOUN
ejpam-6613	232	36	.	.	PUNCT
ejpam-6613	233	1	definition	definition	NOUN
ejpam-6613	233	2	7	7	NUM
ejpam-6613	233	3	(	(	PUNCT
ejpam-6613	233	4	2	2	NUM
ejpam-6613	233	5	-	-	PUNCT
ejpam-6613	233	6	bounded	bound	VERB
ejpam-6613	233	7	operator	operator	NOUN
ejpam-6613	233	8	)	)	PUNCT
ejpam-6613	233	9	.	.	PUNCT
ejpam-6613	234	1	let	let	VERB
ejpam-6613	234	2	(	(	PUNCT
ejpam-6613	234	3	x1	x1	PROPN
ejpam-6613	234	4	2	2	NUM
ejpam-6613	234	5	⊙	⊙	X
ejpam-6613	234	6	x2	x2	PROPN
ejpam-6613	234	7	,	,	PUNCT
ejpam-6613	234	8	∥	∥	X
ejpam-6613	234	9	·	·	PUNCT
ejpam-6613	234	10	,	,	PUNCT
ejpam-6613	234	11	·	·	PUNCT
ejpam-6613	234	12	∥	∥	X
ejpam-6613	234	13	2	2	NUM
ejpam-6613	234	14	⊙	⊙	NOUN
ejpam-6613	234	15	)	)	PUNCT
ejpam-6613	234	16	be	be	AUX
ejpam-6613	234	17	the	the	DET
ejpam-6613	234	18	generalized	generalized	ADJ
ejpam-6613	234	19	2	2	NUM
ejpam-6613	234	20	-	-	PUNCT
ejpam-6613	234	21	normed	norme	VERB
ejpam-6613	234	22	space	space	NOUN
ejpam-6613	234	23	from	from	ADP
ejpam-6613	234	24	theorem	theorem	ADJ
ejpam-6613	234	25	1	1	NUM
ejpam-6613	234	26	,	,	PUNCT
ejpam-6613	234	27	and	and	CCONJ
ejpam-6613	234	28	let	let	VERB
ejpam-6613	234	29	t	t	PROPN
ejpam-6613	234	30	be	be	AUX
ejpam-6613	234	31	a	a	DET
ejpam-6613	234	32	linear	linear	ADJ
ejpam-6613	234	33	operator	operator	NOUN
ejpam-6613	234	34	t	t	NOUN
ejpam-6613	234	35	a	a	DET
ejpam-6613	234	36	linear	linear	ADJ
ejpam-6613	234	37	operator	operator	NOUN
ejpam-6613	234	38	on	on	ADP
ejpam-6613	234	39	x1	x1	PROPN
ejpam-6613	234	40	2	2	NUM
ejpam-6613	234	41	⊙x2	⊙x2	PROPN
ejpam-6613	234	42	.	.	PUNCT
ejpam-6613	235	1	then	then	ADV
ejpam-6613	235	2	we	we	PRON
ejpam-6613	235	3	said	say	VERB
ejpam-6613	235	4	that	that	SCONJ
ejpam-6613	235	5	t	t	PROPN
ejpam-6613	235	6	is	be	AUX
ejpam-6613	235	7	a	a	DET
ejpam-6613	235	8	2	2	NUM
ejpam-6613	235	9	-	-	PUNCT
ejpam-6613	235	10	bounded	bound	VERB
ejpam-6613	235	11	linear	linear	ADJ
ejpam-6613	235	12	operator	operator	NOUN
ejpam-6613	235	13	if	if	SCONJ
ejpam-6613	235	14	there	there	PRON
ejpam-6613	235	15	exists	exist	VERB
ejpam-6613	235	16	a	a	DET
ejpam-6613	235	17	positive	positive	ADJ
ejpam-6613	235	18	number	number	NOUN
ejpam-6613	235	19	α	α	NOUN
ejpam-6613	235	20	>	>	X
ejpam-6613	235	21	0	0	NUM
ejpam-6613	236	1	such	such	ADJ
ejpam-6613	236	2	that	that	DET
ejpam-6613	236	3	∥tw	∥tw	PROPN
ejpam-6613	236	4	,	,	PUNCT
ejpam-6613	236	5	z∥	z∥	ADJ
ejpam-6613	236	6	2	2	NUM
ejpam-6613	236	7	⊙	⊙	NOUN
ejpam-6613	236	8	≤	≤	NOUN
ejpam-6613	237	1	α	α	X
ejpam-6613	237	2	∥w	∥w	PROPN
ejpam-6613	237	3	,	,	PUNCT
ejpam-6613	237	4	z∥	z∥	NOUN
ejpam-6613	237	5	2	2	NUM
ejpam-6613	237	6	⊙	⊙	NOUN
ejpam-6613	237	7	for	for	ADP
ejpam-6613	237	8	all	all	DET
ejpam-6613	237	9	w	w	NOUN
ejpam-6613	237	10	,	,	PUNCT
ejpam-6613	237	11	z	z	PROPN
ejpam-6613	237	12	∈	∈	PROPN
ejpam-6613	237	13	x1	x1	PROPN
ejpam-6613	237	14	2	2	NUM
ejpam-6613	237	15	⊙x2	⊙x2	PROPN
ejpam-6613	237	16	.	.	PUNCT
ejpam-6613	238	1	the	the	DET
ejpam-6613	238	2	symbol	symbol	NOUN
ejpam-6613	238	3	b(x1	b(x1	NOUN
ejpam-6613	238	4	2	2	NUM
ejpam-6613	238	5	⊙	⊙	X
ejpam-6613	238	6	x2	x2	PROPN
ejpam-6613	238	7	)	)	PUNCT
ejpam-6613	238	8	will	will	AUX
ejpam-6613	238	9	denote	denote	VERB
ejpam-6613	238	10	the	the	DET
ejpam-6613	238	11	set	set	NOUN
ejpam-6613	238	12	of	of	ADP
ejpam-6613	238	13	2	2	NUM
ejpam-6613	238	14	-	-	PUNCT
ejpam-6613	238	15	bounded	bound	VERB
ejpam-6613	238	16	linear	linear	ADJ
ejpam-6613	238	17	operator	operator	NOUN
ejpam-6613	238	18	on	on	ADP
ejpam-6613	238	19	x1	x1	PROPN
ejpam-6613	238	20	2	2	NUM
ejpam-6613	238	21	⊙	⊙	X
ejpam-6613	238	22	x2	x2	PROPN
ejpam-6613	238	23	,	,	PUNCT
ejpam-6613	238	24	that	that	ADV
ejpam-6613	238	25	is	is	ADV
ejpam-6613	238	26	,	,	PUNCT
ejpam-6613	238	27	b(x1	b(x1	NOUN
ejpam-6613	238	28	2	2	NUM
ejpam-6613	238	29	⊙x2	⊙x2	PROPN
ejpam-6613	238	30	)	)	PUNCT
ejpam-6613	238	31	:	:	PUNCT
ejpam-6613	239	1	=	=	SYM
ejpam-6613	239	2	{	{	PUNCT
ejpam-6613	239	3	t	t	X
ejpam-6613	239	4	:	:	PUNCT
ejpam-6613	239	5	x1	x1	PROPN
ejpam-6613	239	6	2	2	X
ejpam-6613	239	7	⊙x2	⊙x2	PROPN
ejpam-6613	239	8	→	→	SYM
ejpam-6613	239	9	x1	x1	PROPN
ejpam-6613	239	10	2	2	NUM
ejpam-6613	239	11	⊙x2	⊙x2	PROPN
ejpam-6613	239	12	:	:	PUNCT
ejpam-6613	239	13	t	t	PROPN
ejpam-6613	239	14	is	be	AUX
ejpam-6613	239	15	linear	linear	ADJ
ejpam-6613	239	16	and	and	CCONJ
ejpam-6613	239	17	2	2	NUM
ejpam-6613	239	18	-	-	PUNCT
ejpam-6613	239	19	bounded	bound	VERB
ejpam-6613	239	20	}	}	PUNCT
ejpam-6613	239	21	.	.	PUNCT
ejpam-6613	240	1	m.	m.	PROPN
ejpam-6613	240	2	luis	luis	PROPN
ejpam-6613	240	3	,	,	PUNCT
ejpam-6613	240	4	f.	f.	PROPN
ejpam-6613	240	5	osmin	osmin	PROPN
ejpam-6613	240	6	,	,	PUNCT
ejpam-6613	240	7	s.	s.	PROPN
ejpam-6613	240	8	arley	arley	PROPN
ejpam-6613	240	9	/	/	SYM
ejpam-6613	240	10	eur	eur	PROPN
ejpam-6613	240	11	.	.	PUNCT
ejpam-6613	241	1	j.	j.	PROPN
ejpam-6613	241	2	pure	pure	PROPN
ejpam-6613	241	3	appl	appl	PROPN
ejpam-6613	241	4	.	.	PROPN
ejpam-6613	241	5	math	math	PROPN
ejpam-6613	241	6	,	,	PUNCT
ejpam-6613	241	7	18	18	NUM
ejpam-6613	241	8	(	(	PUNCT
ejpam-6613	241	9	4	4	NUM
ejpam-6613	241	10	)	)	PUNCT
ejpam-6613	241	11	(	(	PUNCT
ejpam-6613	241	12	2025	2025	NUM
ejpam-6613	241	13	)	)	PUNCT
ejpam-6613	241	14	,	,	PUNCT
ejpam-6613	241	15	6613	6613	NUM
ejpam-6613	241	16	11	11	NUM
ejpam-6613	241	17	of	of	ADP
ejpam-6613	241	18	17	17	NUM
ejpam-6613	241	19	remark	remark	NOUN
ejpam-6613	241	20	3	3	NUM
ejpam-6613	241	21	.	.	PUNCT
ejpam-6613	242	1	it	it	PRON
ejpam-6613	242	2	is	be	AUX
ejpam-6613	242	3	clear	clear	ADJ
ejpam-6613	242	4	that	that	SCONJ
ejpam-6613	242	5	the	the	DET
ejpam-6613	242	6	set	set	ADJ
ejpam-6613	242	7	b(x1	b(x1	NOUN
ejpam-6613	242	8	2	2	NUM
ejpam-6613	242	9	⊙x2	⊙x2	PROPN
ejpam-6613	242	10	)	)	PUNCT
ejpam-6613	242	11	can	can	AUX
ejpam-6613	242	12	be	be	AUX
ejpam-6613	242	13	endowed	endow	VERB
ejpam-6613	242	14	with	with	ADP
ejpam-6613	242	15	a	a	DET
ejpam-6613	242	16	vector	vector	NOUN
ejpam-6613	242	17	space	space	NOUN
ejpam-6613	242	18	structure	structure	NOUN
ejpam-6613	242	19	over	over	ADP
ejpam-6613	242	20	c	c	PROPN
ejpam-6613	242	21	,	,	PUNCT
ejpam-6613	242	22	by	by	ADP
ejpam-6613	242	23	means	mean	NOUN
ejpam-6613	242	24	of	of	ADP
ejpam-6613	242	25	the	the	DET
ejpam-6613	242	26	pointwise	pointwise	ADJ
ejpam-6613	242	27	operations	operation	NOUN
ejpam-6613	242	28	of	of	ADP
ejpam-6613	242	29	operators	operator	NOUN
ejpam-6613	242	30	.	.	PUNCT
ejpam-6613	243	1	definition	definition	NOUN
ejpam-6613	243	2	8	8	NUM
ejpam-6613	243	3	.	.	PUNCT
ejpam-6613	244	1	[	[	X
ejpam-6613	244	2	14	14	NUM
ejpam-6613	244	3	]	]	X
ejpam-6613	244	4	if	if	SCONJ
ejpam-6613	244	5	t	t	PROPN
ejpam-6613	244	6	∈	∈	PROPN
ejpam-6613	244	7	b(x1	b(x1	NOUN
ejpam-6613	244	8	2	2	NUM
ejpam-6613	244	9	⊙x2	⊙x2	PROPN
ejpam-6613	244	10	)	)	PUNCT
ejpam-6613	244	11	,	,	PUNCT
ejpam-6613	244	12	we	we	PRON
ejpam-6613	244	13	define	define	VERB
ejpam-6613	244	14	∥t∥	∥t∥	ADP
ejpam-6613	244	15	2	2	NUM
ejpam-6613	244	16	⊙	⊙	NOUN
ejpam-6613	244	17	,	,	PUNCT
ejpam-6613	244	18	by	by	ADP
ejpam-6613	244	19	∥t∥	∥t∥	ADV
ejpam-6613	244	20	2	2	NUM
ejpam-6613	244	21	⊙	⊙	NOUN
ejpam-6613	244	22	=	=	PUNCT
ejpam-6613	244	23	inf{a	inf{a	PROPN
ejpam-6613	244	24	≥	≥	NOUN
ejpam-6613	244	25	0	0	NUM
ejpam-6613	244	26	:	:	PUNCT
ejpam-6613	245	1	∥t	∥t	INTJ
ejpam-6613	245	2	(	(	PUNCT
ejpam-6613	245	3	w	w	NOUN
ejpam-6613	245	4	)	)	PUNCT
ejpam-6613	245	5	,	,	PUNCT
ejpam-6613	245	6	z∥	z∥	NUM
ejpam-6613	245	7	2	2	NUM
ejpam-6613	245	8	⊙	⊙	PROPN
ejpam-6613	245	9	≤	≤	PROPN
ejpam-6613	245	10	a∥w	a∥w	PROPN
ejpam-6613	245	11	,	,	PUNCT
ejpam-6613	245	12	z∥	z∥	NOUN
ejpam-6613	245	13	2	2	NUM
ejpam-6613	245	14	⊙	⊙	NOUN
ejpam-6613	245	15	for	for	ADP
ejpam-6613	245	16	all	all	DET
ejpam-6613	245	17	w	w	NOUN
ejpam-6613	245	18	,	,	PUNCT
ejpam-6613	245	19	z	z	NOUN
ejpam-6613	245	20	∈	∈	PROPN
ejpam-6613	245	21	x	x	X
ejpam-6613	245	22	}	}	PUNCT
ejpam-6613	245	23	.	.	PUNCT
ejpam-6613	246	1	theorem	theorem	NOUN
ejpam-6613	246	2	5	5	NUM
ejpam-6613	246	3	.	.	PUNCT
ejpam-6613	247	1	[	[	X
ejpam-6613	247	2	14	14	NUM
ejpam-6613	247	3	]	]	PUNCT
ejpam-6613	247	4	in	in	ADP
ejpam-6613	247	5	the	the	DET
ejpam-6613	247	6	context	context	NOUN
ejpam-6613	247	7	of	of	ADP
ejpam-6613	247	8	the	the	DET
ejpam-6613	247	9	definition	definition	NOUN
ejpam-6613	247	10	8	8	NUM
ejpam-6613	247	11	,	,	PUNCT
ejpam-6613	247	12	for	for	ADP
ejpam-6613	247	13	all	all	DET
ejpam-6613	247	14	t	t	NOUN
ejpam-6613	247	15	∈	∈	NOUN
ejpam-6613	247	16	2	2	NUM
ejpam-6613	247	17	b(x	b(x	NOUN
ejpam-6613	247	18	)	)	PUNCT
ejpam-6613	247	19	it	it	PRON
ejpam-6613	247	20	is	be	AUX
ejpam-6613	247	21	true	true	ADJ
ejpam-6613	247	22	that	that	SCONJ
ejpam-6613	247	23	∥t∥	∥t∥	ADP
ejpam-6613	247	24	2	2	NUM
ejpam-6613	247	25	⊙	⊙	NOUN
ejpam-6613	247	26	=	=	SYM
ejpam-6613	247	27	sup{∥tw	sup{∥tw	PROPN
ejpam-6613	247	28	,	,	PUNCT
ejpam-6613	247	29	z∥	z∥	NOUN
ejpam-6613	247	30	2	2	NUM
ejpam-6613	247	31	⊙	⊙	NOUN
ejpam-6613	247	32	:	:	PUNCT
ejpam-6613	247	33	w	w	X
ejpam-6613	247	34	,	,	PUNCT
ejpam-6613	247	35	z	z	NOUN
ejpam-6613	247	36	∈	∈	PROPN
ejpam-6613	247	37	x	x	X
ejpam-6613	247	38	and	and	CCONJ
ejpam-6613	247	39	∥w	∥w	PROPN
ejpam-6613	247	40	,	,	PUNCT
ejpam-6613	247	41	z∥	z∥	NOUN
ejpam-6613	247	42	2	2	NUM
ejpam-6613	247	43	⊙	⊙	NOUN
ejpam-6613	247	44	=	=	SYM
ejpam-6613	247	45	1	1	NUM
ejpam-6613	247	46	}	}	PUNCT
ejpam-6613	247	47	∥t∥	∥t∥	CCONJ
ejpam-6613	247	48	2	2	NUM
ejpam-6613	247	49	⊙	⊙	NOUN
ejpam-6613	247	50	=	=	SYM
ejpam-6613	247	51	sup{∥tw	sup{∥tw	PROPN
ejpam-6613	247	52	,	,	PUNCT
ejpam-6613	247	53	z∥	z∥	NOUN
ejpam-6613	247	54	2	2	NUM
ejpam-6613	247	55	⊙	⊙	NOUN
ejpam-6613	247	56	:	:	PUNCT
ejpam-6613	247	57	w	w	X
ejpam-6613	247	58	,	,	PUNCT
ejpam-6613	247	59	z	z	NOUN
ejpam-6613	247	60	∈	∈	PROPN
ejpam-6613	247	61	x	x	X
ejpam-6613	247	62	and	and	CCONJ
ejpam-6613	247	63	∥w	∥w	PROPN
ejpam-6613	247	64	,	,	PUNCT
ejpam-6613	247	65	z∥	z∥	NUM
ejpam-6613	247	66	2	2	NUM
ejpam-6613	247	67	⊙	⊙	X
ejpam-6613	247	68	≤	≤	NOUN
ejpam-6613	247	69	1	1	NUM
ejpam-6613	247	70	}	}	PUNCT
ejpam-6613	247	71	∥t∥	∥t∥	CCONJ
ejpam-6613	247	72	2	2	NUM
ejpam-6613	247	73	⊙	⊙	NOUN
ejpam-6613	247	74	=	=	SYM
ejpam-6613	247	75	sup	sup	NOUN
ejpam-6613	247	76	∥tw	∥tw	NOUN
ejpam-6613	247	77	,	,	PUNCT
ejpam-6613	247	78	z∥	z∥	NOUN
ejpam-6613	247	79	2	2	NUM
ejpam-6613	247	80	⊙	⊙	PROPN
ejpam-6613	247	81	∥w	∥w	PROPN
ejpam-6613	247	82	,	,	PUNCT
ejpam-6613	247	83	z∥	z∥	PROPN
ejpam-6613	247	84	2	2	NUM
ejpam-6613	247	85	⊙	⊙	NOUN
ejpam-6613	247	86	:	:	PUNCT
ejpam-6613	247	87	w	w	X
ejpam-6613	247	88	,	,	PUNCT
ejpam-6613	247	89	z	z	NOUN
ejpam-6613	247	90	∈	∈	PROPN
ejpam-6613	247	91	x	x	X
ejpam-6613	247	92	and	and	CCONJ
ejpam-6613	247	93	∥w	∥w	PROPN
ejpam-6613	247	94	,	,	PUNCT
ejpam-6613	247	95	z∥	z∥	NUM
ejpam-6613	247	96	2	2	NUM
ejpam-6613	247	97	⊙	⊙	NOUN
ejpam-6613	247	98	̸=	̸=	PROPN
ejpam-6613	247	99	0	0	PUNCT
ejpam-6613	248	1			ADV
ejpam-6613	248	2	moreover	moreover	ADV
ejpam-6613	248	3	,	,	PUNCT
ejpam-6613	248	4	thanks	thank	NOUN
ejpam-6613	248	5	to	to	ADP
ejpam-6613	248	6	the	the	DET
ejpam-6613	248	7	definition	definition	NOUN
ejpam-6613	248	8	8	8	NUM
ejpam-6613	248	9	we	we	PRON
ejpam-6613	248	10	prove	prove	VERB
ejpam-6613	248	11	that	that	SCONJ
ejpam-6613	248	12	given	give	VERB
ejpam-6613	248	13	a	a	DET
ejpam-6613	248	14	generalized	generalized	ADJ
ejpam-6613	248	15	2	2	NUM
ejpam-6613	248	16	-	-	PUNCT
ejpam-6613	248	17	normed	norme	VERB
ejpam-6613	248	18	space	space	NOUN
ejpam-6613	248	19	x1	x1	PROPN
ejpam-6613	248	20	2	2	NUM
ejpam-6613	248	21	⊙	⊙	X
ejpam-6613	248	22	x2	x2	PROPN
ejpam-6613	248	23	,	,	PUNCT
ejpam-6613	248	24	it	it	PRON
ejpam-6613	248	25	is	be	AUX
ejpam-6613	248	26	possible	possible	ADJ
ejpam-6613	248	27	to	to	PART
ejpam-6613	248	28	endow	endow	VERB
ejpam-6613	248	29	the	the	DET
ejpam-6613	248	30	vector	vector	NOUN
ejpam-6613	248	31	space	space	NOUN
ejpam-6613	248	32	b(x1	b(x1	NOUN
ejpam-6613	248	33	2	2	NUM
ejpam-6613	248	34	⊙	⊙	X
ejpam-6613	248	35	x2	x2	PROPN
ejpam-6613	248	36	)	)	PUNCT
ejpam-6613	248	37	with	with	ADP
ejpam-6613	248	38	the	the	DET
ejpam-6613	248	39	structure	structure	NOUN
ejpam-6613	248	40	of	of	ADP
ejpam-6613	248	41	a	a	DET
ejpam-6613	248	42	semi	semi	ADJ
ejpam-6613	248	43	-	-	ADJ
ejpam-6613	248	44	normed	normed	ADJ
ejpam-6613	248	45	space	space	NOUN
ejpam-6613	248	46	.	.	PUNCT
ejpam-6613	249	1	proposition	proposition	NOUN
ejpam-6613	249	2	6	6	NUM
ejpam-6613	249	3	.	.	PUNCT
ejpam-6613	250	1	the	the	DET
ejpam-6613	250	2	mapping	mapping	NOUN
ejpam-6613	250	3	∥	∥	X
ejpam-6613	250	4	·	·	PUNCT
ejpam-6613	250	5	∥	∥	NUM
ejpam-6613	250	6	2	2	NUM
ejpam-6613	250	7	⊙	⊙	NOUN
ejpam-6613	250	8	:	:	PUNCT
ejpam-6613	250	9	b(x1	b(x1	NOUN
ejpam-6613	250	10	2	2	NUM
ejpam-6613	250	11	⊙x2	⊙x2	PROPN
ejpam-6613	250	12	)	)	PUNCT
ejpam-6613	250	13	→	→	SYM
ejpam-6613	250	14	r	r	NOUN
ejpam-6613	250	15	given	give	VERB
ejpam-6613	250	16	by	by	ADP
ejpam-6613	250	17	∥t∥	∥t∥	ADP
ejpam-6613	250	18	2	2	NUM
ejpam-6613	250	19	⊙	⊙	NOUN
ejpam-6613	250	20	=	=	SYM
ejpam-6613	250	21	sup	sup	NOUN
ejpam-6613	250	22	∥t	∥t	NOUN
ejpam-6613	250	23	(	(	PUNCT
ejpam-6613	250	24	w	w	NOUN
ejpam-6613	250	25	)	)	PUNCT
ejpam-6613	250	26	,	,	PUNCT
ejpam-6613	250	27	z∥	z∥	NUM
ejpam-6613	250	28	2	2	NUM
ejpam-6613	250	29	⊙	⊙	PROPN
ejpam-6613	250	30	∥w	∥w	PROPN
ejpam-6613	250	31	,	,	PUNCT
ejpam-6613	250	32	z∥	z∥	PROPN
ejpam-6613	250	33	2	2	NUM
ejpam-6613	250	34	⊙	⊙	NOUN
ejpam-6613	250	35	:	:	PUNCT
ejpam-6613	250	36	w	w	X
ejpam-6613	250	37	,	,	PUNCT
ejpam-6613	250	38	z	z	PROPN
ejpam-6613	250	39	∈	∈	PROPN
ejpam-6613	251	1	x1	x1	PROPN
ejpam-6613	251	2	2	2	NUM
ejpam-6613	251	3	⊙x2	⊙x2	PROPN
ejpam-6613	251	4	and	and	CCONJ
ejpam-6613	251	5	∥w	∥w	PROPN
ejpam-6613	251	6	,	,	PUNCT
ejpam-6613	251	7	z∥	z∥	NUM
ejpam-6613	251	8	2	2	NUM
ejpam-6613	251	9	⊙	⊙	NOUN
ejpam-6613	251	10	̸=	̸=	PROPN
ejpam-6613	251	11	0	0	NUM
ejpam-6613	252	1			NOUN
ejpam-6613	252	2	defines	define	VERB
ejpam-6613	252	3	a	a	DET
ejpam-6613	252	4	semi	semi	NOUN
ejpam-6613	252	5	-	-	NOUN
ejpam-6613	252	6	norm	norm	ADJ
ejpam-6613	252	7	in	in	ADP
ejpam-6613	252	8	b(x1	b(x1	NOUN
ejpam-6613	252	9	2	2	NUM
ejpam-6613	252	10	⊙x2	⊙x2	PROPN
ejpam-6613	252	11	)	)	PUNCT
ejpam-6613	252	12	.	.	PUNCT
ejpam-6613	253	1	the	the	DET
ejpam-6613	253	2	proofs	proof	NOUN
ejpam-6613	253	3	are	be	AUX
ejpam-6613	253	4	obtained	obtain	VERB
ejpam-6613	253	5	from	from	ADP
ejpam-6613	253	6	[	[	X
ejpam-6613	253	7	14	14	NUM
ejpam-6613	253	8	]	]	PUNCT
ejpam-6613	253	9	using	use	VERB
ejpam-6613	253	10	theorem	theorem	NOUN
ejpam-6613	253	11	1	1	NUM
ejpam-6613	253	12	.	.	PUNCT
ejpam-6613	253	13	proposition	proposition	NOUN
ejpam-6613	253	14	7	7	NUM
ejpam-6613	253	15	.	.	PUNCT
ejpam-6613	254	1	[	[	X
ejpam-6613	254	2	14	14	NUM
ejpam-6613	254	3	]	]	PUNCT
ejpam-6613	254	4	for	for	ADP
ejpam-6613	254	5	all	all	DET
ejpam-6613	254	6	t	t	NOUN
ejpam-6613	254	7	∈	∈	PROPN
ejpam-6613	254	8	b(x1	b(x1	NOUN
ejpam-6613	254	9	2	2	NUM
ejpam-6613	254	10	⊙x2	⊙x2	PROPN
ejpam-6613	254	11	)	)	PUNCT
ejpam-6613	254	12	it	it	PRON
ejpam-6613	254	13	is	be	AUX
ejpam-6613	254	14	true	true	ADJ
ejpam-6613	254	15	that	that	SCONJ
ejpam-6613	254	16	∥t	∥t	PROPN
ejpam-6613	254	17	(	(	PUNCT
ejpam-6613	254	18	w	w	NOUN
ejpam-6613	254	19	)	)	PUNCT
ejpam-6613	254	20	,	,	PUNCT
ejpam-6613	254	21	y∥	y∥	VERB
ejpam-6613	254	22	2	2	NUM
ejpam-6613	254	23	⊙	⊙	NOUN
ejpam-6613	254	24	≤	≤	NOUN
ejpam-6613	254	25	∥t∥∥w	∥t∥∥w	PROPN
ejpam-6613	254	26	,	,	PUNCT
ejpam-6613	254	27	y∥	y∥	VERB
ejpam-6613	254	28	2	2	NUM
ejpam-6613	254	29	⊙	⊙	NOUN
ejpam-6613	254	30	for	for	ADP
ejpam-6613	254	31	all	all	DET
ejpam-6613	254	32	w	w	PROPN
ejpam-6613	254	33	,	,	PUNCT
ejpam-6613	254	34	y	y	PROPN
ejpam-6613	254	35	,	,	PUNCT
ejpam-6613	254	36	z	z	PROPN
ejpam-6613	254	37	∈	∈	PROPN
ejpam-6613	255	1	x1	x1	PROPN
ejpam-6613	255	2	2	2	NUM
ejpam-6613	255	3	⊙x2	⊙x2	PROPN
ejpam-6613	255	4	.	.	PUNCT
ejpam-6613	256	1	4	4	NUM
ejpam-6613	256	2	.	.	NOUN
ejpam-6613	256	3	tensor	tensor	NOUN
ejpam-6613	256	4	product	product	NOUN
ejpam-6613	256	5	of	of	ADP
ejpam-6613	256	6	linear	linear	PROPN
ejpam-6613	256	7	operators	operator	NOUN
ejpam-6613	256	8	in	in	ADP
ejpam-6613	256	9	the	the	DET
ejpam-6613	256	10	following	follow	VERB
ejpam-6613	256	11	definition	definition	NOUN
ejpam-6613	256	12	,	,	PUNCT
ejpam-6613	256	13	we	we	PRON
ejpam-6613	256	14	establish	establish	VERB
ejpam-6613	256	15	the	the	DET
ejpam-6613	256	16	notion	notion	NOUN
ejpam-6613	256	17	of	of	ADP
ejpam-6613	256	18	the	the	DET
ejpam-6613	256	19	2	2	NUM
ejpam-6613	256	20	-	-	PUNCT
ejpam-6613	256	21	tensor	tensor	NOUN
ejpam-6613	256	22	product	product	NOUN
ejpam-6613	256	23	of	of	ADP
ejpam-6613	256	24	linear	linear	PROPN
ejpam-6613	256	25	operators	operator	NOUN
ejpam-6613	256	26	on	on	ADP
ejpam-6613	256	27	spaces	space	NOUN
ejpam-6613	256	28	with	with	ADP
ejpam-6613	256	29	a	a	DET
ejpam-6613	256	30	generalized	generalized	ADJ
ejpam-6613	256	31	2	2	NUM
ejpam-6613	256	32	-	-	PUNCT
ejpam-6613	256	33	inner	inner	ADJ
ejpam-6613	256	34	product	product	NOUN
ejpam-6613	256	35	.	.	PUNCT
ejpam-6613	257	1	definition	definition	NOUN
ejpam-6613	257	2	9	9	NUM
ejpam-6613	257	3	.	.	PUNCT
ejpam-6613	258	1	let	let	VERB
ejpam-6613	258	2	(	(	PUNCT
ejpam-6613	258	3	x1	x1	ADJ
ejpam-6613	258	4	,	,	PUNCT
ejpam-6613	258	5	⟨	⟨	NOUN
ejpam-6613	258	6	·	·	SYM
ejpam-6613	258	7	,	,	PUNCT
ejpam-6613	258	8	·	·	PUNCT
ejpam-6613	258	9	|·⟩1	|·⟩1	X
ejpam-6613	258	10	)	)	PUNCT
ejpam-6613	258	11	and	and	CCONJ
ejpam-6613	258	12	(	(	PUNCT
ejpam-6613	258	13	x2	x2	INTJ
ejpam-6613	258	14	,	,	PUNCT
ejpam-6613	258	15	⟨	⟨	NOUN
ejpam-6613	258	16	·	·	SYM
ejpam-6613	258	17	,	,	PUNCT
ejpam-6613	258	18	·	·	PUNCT
ejpam-6613	258	19	|·⟩2	|·⟩2	X
ejpam-6613	258	20	)	)	PUNCT
ejpam-6613	258	21	be	be	AUX
ejpam-6613	258	22	spaces	space	NOUN
ejpam-6613	258	23	with	with	ADP
ejpam-6613	258	24	a	a	DET
ejpam-6613	258	25	generalized	generalized	ADJ
ejpam-6613	258	26	2	2	NUM
ejpam-6613	258	27	-	-	PUNCT
ejpam-6613	258	28	inner	inner	ADJ
ejpam-6613	258	29	product	product	NOUN
ejpam-6613	258	30	,	,	PUNCT
ejpam-6613	258	31	and	and	CCONJ
ejpam-6613	258	32	let	let	VERB
ejpam-6613	258	33	t	t	NOUN
ejpam-6613	258	34	:	:	PUNCT
ejpam-6613	258	35	x1	x1	PROPN
ejpam-6613	258	36	→	→	SYM
ejpam-6613	258	37	x1	x1	PROPN
ejpam-6613	258	38	,	,	PUNCT
ejpam-6613	258	39	s	s	PART
ejpam-6613	258	40	:	:	PUNCT
ejpam-6613	258	41	x2	x2	PROPN
ejpam-6613	258	42	→	→	PUNCT
ejpam-6613	258	43	x2	x2	PROPN
ejpam-6613	258	44	be	be	AUX
ejpam-6613	258	45	linear	linear	ADJ
ejpam-6613	258	46	operators	operator	NOUN
ejpam-6613	258	47	on	on	ADP
ejpam-6613	258	48	x1	x1	PROPN
ejpam-6613	258	49	and	and	CCONJ
ejpam-6613	258	50	x2	x2	PROPN
ejpam-6613	258	51	,	,	PUNCT
ejpam-6613	258	52	respectively	respectively	ADV
ejpam-6613	258	53	.	.	PUNCT
ejpam-6613	259	1	then	then	ADV
ejpam-6613	259	2	the	the	DET
ejpam-6613	259	3	2	2	NUM
ejpam-6613	259	4	-	-	PUNCT
ejpam-6613	259	5	tensor	tensor	NOUN
ejpam-6613	259	6	product	product	NOUN
ejpam-6613	259	7	of	of	ADP
ejpam-6613	259	8	t	t	PROPN
ejpam-6613	259	9	and	and	CCONJ
ejpam-6613	259	10	s	s	VERB
ejpam-6613	259	11	on	on	ADP
ejpam-6613	259	12	x1	x1	PROPN
ejpam-6613	259	13	2	2	PROPN
ejpam-6613	259	14	⊙x2	⊙x2	PROPN
ejpam-6613	259	15	,	,	PUNCT
ejpam-6613	259	16	denoted	denote	VERB
ejpam-6613	259	17	by	by	ADP
ejpam-6613	259	18	t	t	PROPN
ejpam-6613	259	19	2	2	NUM
ejpam-6613	259	20	⊙s	⊙s	PROPN
ejpam-6613	259	21	,	,	PUNCT
ejpam-6613	259	22	is	be	AUX
ejpam-6613	259	23	the	the	DET
ejpam-6613	259	24	linear	linear	ADJ
ejpam-6613	259	25	operator	operator	NOUN
ejpam-6613	259	26	t	t	PROPN
ejpam-6613	259	27	2	2	NUM
ejpam-6613	259	28	⊙	⊙	NOUN
ejpam-6613	259	29	s	s	PART
ejpam-6613	259	30	:	:	PUNCT
ejpam-6613	259	31	x1	x1	PROPN
ejpam-6613	259	32	2	2	NUM
ejpam-6613	259	33	⊙x2	⊙x2	PROPN
ejpam-6613	259	34	−→	−→	ADJ
ejpam-6613	259	35	x1	x1	PROPN
ejpam-6613	259	36	2	2	NUM
ejpam-6613	259	37	⊙x2	⊙x2	PROPN
ejpam-6613	259	38	m.	m.	PROPN
ejpam-6613	259	39	luis	luis	PROPN
ejpam-6613	259	40	,	,	PUNCT
ejpam-6613	259	41	f.	f.	PROPN
ejpam-6613	259	42	osmin	osmin	PROPN
ejpam-6613	259	43	,	,	PUNCT
ejpam-6613	259	44	s.	s.	PROPN
ejpam-6613	259	45	arley	arley	PROPN
ejpam-6613	259	46	/	/	SYM
ejpam-6613	259	47	eur	eur	PROPN
ejpam-6613	259	48	.	.	PUNCT
ejpam-6613	260	1	j.	j.	PROPN
ejpam-6613	260	2	pure	pure	PROPN
ejpam-6613	260	3	appl	appl	PROPN
ejpam-6613	260	4	.	.	PROPN
ejpam-6613	260	5	math	math	PROPN
ejpam-6613	260	6	,	,	PUNCT
ejpam-6613	260	7	18	18	NUM
ejpam-6613	260	8	(	(	PUNCT
ejpam-6613	260	9	4	4	NUM
ejpam-6613	260	10	)	)	PUNCT
ejpam-6613	260	11	(	(	PUNCT
ejpam-6613	260	12	2025	2025	NUM
ejpam-6613	260	13	)	)	PUNCT
ejpam-6613	260	14	,	,	PUNCT
ejpam-6613	260	15	6613	6613	NUM
ejpam-6613	260	16	12	12	NUM
ejpam-6613	260	17	of	of	ADP
ejpam-6613	260	18	17	17	NUM
ejpam-6613	260	19	defined	define	VERB
ejpam-6613	260	20	by	by	ADP
ejpam-6613	260	21	(	(	PUNCT
ejpam-6613	260	22	t	t	PROPN
ejpam-6613	260	23	2	2	NUM
ejpam-6613	260	24	⊙	⊙	NOUN
ejpam-6613	260	25	s	s	PART
ejpam-6613	260	26	)	)	PUNCT
ejpam-6613	260	27	(	(	PUNCT
ejpam-6613	260	28	n∑	n∑	NOUN
ejpam-6613	260	29	i=1	i=1	PROPN
ejpam-6613	260	30	xi	xi	PROPN
ejpam-6613	260	31	2	2	NUM
ejpam-6613	260	32	⊙	⊙	PROPN
ejpam-6613	260	33	yi	yi	PROPN
ejpam-6613	260	34	)	)	PUNCT
ejpam-6613	260	35	:	:	PUNCT
ejpam-6613	261	1	=	=	PUNCT
ejpam-6613	261	2	n∑	n∑	NOUN
ejpam-6613	261	3	i=1	i=1	PROPN
ejpam-6613	262	1	(	(	PUNCT
ejpam-6613	262	2	txi	txi	PROPN
ejpam-6613	262	3	)	)	PUNCT
ejpam-6613	262	4	2	2	NUM
ejpam-6613	262	5	⊙	⊙	NOUN
ejpam-6613	262	6	(	(	PUNCT
ejpam-6613	262	7	syi	syi	NOUN
ejpam-6613	262	8	)	)	PUNCT
ejpam-6613	262	9	,	,	PUNCT
ejpam-6613	262	10	for	for	ADP
ejpam-6613	262	11	each	each	DET
ejpam-6613	262	12	∑n	∑n	PROPN
ejpam-6613	262	13	i=1	i=1	PROPN
ejpam-6613	262	14	xi	xi	ADP
ejpam-6613	262	15	2	2	NUM
ejpam-6613	262	16	⊙	⊙	PROPN
ejpam-6613	262	17	yi	yi	PROPN
ejpam-6613	262	18	∈	∈	PROPN
ejpam-6613	262	19	x1	x1	PROPN
ejpam-6613	262	20	2	2	NUM
ejpam-6613	262	21	⊙x2	⊙x2	PROPN
ejpam-6613	262	22	.	.	PUNCT
ejpam-6613	262	23	remark	remark	PROPN
ejpam-6613	262	24	4	4	NUM
ejpam-6613	262	25	.	.	PUNCT
ejpam-6613	263	1	let	let	VERB
ejpam-6613	263	2	(	(	PUNCT
ejpam-6613	263	3	x1	x1	ADJ
ejpam-6613	263	4	,	,	PUNCT
ejpam-6613	263	5	⟨	⟨	NOUN
ejpam-6613	263	6	·	·	SYM
ejpam-6613	263	7	,	,	PUNCT
ejpam-6613	263	8	·	·	PUNCT
ejpam-6613	263	9	|·⟩1	|·⟩1	X
ejpam-6613	263	10	)	)	PUNCT
ejpam-6613	263	11	and	and	CCONJ
ejpam-6613	263	12	(	(	PUNCT
ejpam-6613	263	13	x2	x2	INTJ
ejpam-6613	263	14	,	,	PUNCT
ejpam-6613	263	15	⟨	⟨	NOUN
ejpam-6613	263	16	·	·	SYM
ejpam-6613	263	17	,	,	PUNCT
ejpam-6613	263	18	·	·	PUNCT
ejpam-6613	263	19	|·⟩2	|·⟩2	X
ejpam-6613	263	20	)	)	PUNCT
ejpam-6613	263	21	be	be	AUX
ejpam-6613	263	22	spaces	space	NOUN
ejpam-6613	263	23	with	with	ADP
ejpam-6613	263	24	a	a	DET
ejpam-6613	263	25	generalized	generalized	ADJ
ejpam-6613	263	26	2	2	NUM
ejpam-6613	263	27	-	-	PUNCT
ejpam-6613	263	28	inner	inner	ADJ
ejpam-6613	263	29	product	product	NOUN
ejpam-6613	263	30	.	.	PUNCT
ejpam-6613	264	1	we	we	PRON
ejpam-6613	264	2	denote	denote	VERB
ejpam-6613	264	3	the	the	DET
ejpam-6613	264	4	tensor	tensor	NOUN
ejpam-6613	264	5	product	product	NOUN
ejpam-6613	264	6	(	(	PUNCT
ejpam-6613	264	7	i1	i1	PROPN
ejpam-6613	264	8	2	2	NUM
ejpam-6613	264	9	⊙	⊙	PROPN
ejpam-6613	264	10	i2	i2	PROPN
ejpam-6613	264	11	)	)	PUNCT
ejpam-6613	264	12	on	on	ADP
ejpam-6613	264	13	x1	x1	PROPN
ejpam-6613	264	14	2	2	NUM
ejpam-6613	264	15	⊙x2	⊙x2	PROPN
ejpam-6613	264	16	by	by	ADP
ejpam-6613	264	17	i⊙	i⊙	NOUN
ejpam-6613	264	18	;	;	PUNCT
ejpam-6613	264	19	this	this	PRON
ejpam-6613	264	20	is	be	AUX
ejpam-6613	264	21	the	the	DET
ejpam-6613	264	22	identity	identity	NOUN
ejpam-6613	264	23	mapping	mapping	NOUN
ejpam-6613	264	24	on	on	ADP
ejpam-6613	264	25	the	the	DET
ejpam-6613	264	26	vector	vector	NOUN
ejpam-6613	264	27	space	space	NOUN
ejpam-6613	264	28	x1	x1	PROPN
ejpam-6613	264	29	2	2	NUM
ejpam-6613	264	30	⊙x2	⊙x2	PROPN
ejpam-6613	264	31	.	.	PUNCT
ejpam-6613	265	1	indeed	indeed	ADV
ejpam-6613	265	2	,	,	PUNCT
ejpam-6613	265	3	for	for	ADP
ejpam-6613	265	4	every	every	DET
ejpam-6613	265	5	ξ	ξ	X
ejpam-6613	265	6	=	=	SYM
ejpam-6613	265	7	∑n	∑n	PROPN
ejpam-6613	265	8	i=1	i=1	PROPN
ejpam-6613	265	9	xi	xi	PART
ejpam-6613	265	10	2	2	NUM
ejpam-6613	265	11	⊙	⊙	PROPN
ejpam-6613	265	12	yi	yi	PROPN
ejpam-6613	266	1	∈	∈	PROPN
ejpam-6613	267	1	x1	x1	PROPN
ejpam-6613	267	2	2	2	NUM
ejpam-6613	267	3	⊙x2	⊙x2	PROPN
ejpam-6613	267	4	,	,	PUNCT
ejpam-6613	267	5	i⊙(ξ	i⊙(ξ	NUM
ejpam-6613	267	6	)	)	PUNCT
ejpam-6613	267	7	=	=	PRON
ejpam-6613	267	8	(	(	PUNCT
ejpam-6613	267	9	i1	i1	PROPN
ejpam-6613	267	10	2	2	NUM
ejpam-6613	267	11	⊙	⊙	PROPN
ejpam-6613	267	12	i2	i2	PROPN
ejpam-6613	267	13	)	)	PUNCT
ejpam-6613	267	14	(	(	PUNCT
ejpam-6613	267	15	n∑	n∑	NOUN
ejpam-6613	267	16	i=1	i=1	PROPN
ejpam-6613	267	17	xi	xi	PROPN
ejpam-6613	267	18	2	2	NUM
ejpam-6613	267	19	⊙	⊙	PROPN
ejpam-6613	267	20	yi	yi	PROPN
ejpam-6613	267	21	)	)	PUNCT
ejpam-6613	268	1	=	=	PUNCT
ejpam-6613	269	1	n∑	n∑	PROPN
ejpam-6613	269	2	i=1	i=1	PROPN
ejpam-6613	269	3	i1xi	i1xi	PROPN
ejpam-6613	269	4	2	2	NUM
ejpam-6613	269	5	⊙	⊙	X
ejpam-6613	269	6	i2yi	i2yi	PUNCT
ejpam-6613	270	1	=	=	PUNCT
ejpam-6613	270	2	n∑	n∑	ADJ
ejpam-6613	270	3	i=1	i=1	X
ejpam-6613	270	4	xi	xi	PROPN
ejpam-6613	270	5	2	2	NUM
ejpam-6613	270	6	⊙	⊙	X
ejpam-6613	270	7	yi	yi	PROPN
ejpam-6613	271	1	=	=	SYM
ejpam-6613	271	2	ξ	ξ	X
ejpam-6613	271	3	.	.	PUNCT
ejpam-6613	271	4	note	note	NOUN
ejpam-6613	271	5	also	also	ADV
ejpam-6613	271	6	that	that	SCONJ
ejpam-6613	271	7	∥i⊙∥	∥i⊙∥	PROPN
ejpam-6613	271	8	2	2	NUM
ejpam-6613	271	9	⊙	⊙	NOUN
ejpam-6613	271	10	=	=	SYM
ejpam-6613	271	11	1	1	X
ejpam-6613	271	12	.	.	X
ejpam-6613	271	13	proposition	proposition	NOUN
ejpam-6613	271	14	8	8	NUM
ejpam-6613	271	15	.	.	PUNCT
ejpam-6613	272	1	let	let	VERB
ejpam-6613	272	2	(	(	PUNCT
ejpam-6613	272	3	x1	x1	ADJ
ejpam-6613	272	4	,	,	PUNCT
ejpam-6613	272	5	⟨	⟨	NOUN
ejpam-6613	272	6	·	·	PUNCT
ejpam-6613	272	7	,	,	PUNCT
ejpam-6613	272	8	·	·	PUNCT
ejpam-6613	272	9	|	|	ADV
ejpam-6613	272	10	·	·	SYM
ejpam-6613	272	11	⟩1	⟩1	NOUN
ejpam-6613	272	12	)	)	PUNCT
ejpam-6613	272	13	and	and	CCONJ
ejpam-6613	272	14	(	(	PUNCT
ejpam-6613	272	15	x2	x2	INTJ
ejpam-6613	272	16	,	,	PUNCT
ejpam-6613	272	17	⟨	⟨	NOUN
ejpam-6613	272	18	·	·	PUNCT
ejpam-6613	272	19	,	,	PUNCT
ejpam-6613	272	20	·	·	PUNCT
ejpam-6613	272	21	|	|	ADV
ejpam-6613	272	22	·	·	SYM
ejpam-6613	272	23	⟩2	⟩2	NOUN
ejpam-6613	272	24	)	)	PUNCT
ejpam-6613	272	25	be	be	AUX
ejpam-6613	272	26	spaces	space	NOUN
ejpam-6613	272	27	with	with	ADP
ejpam-6613	272	28	a	a	DET
ejpam-6613	272	29	generalized	generalized	ADJ
ejpam-6613	272	30	2	2	NUM
ejpam-6613	272	31	-	-	PUNCT
ejpam-6613	272	32	inner	inner	ADJ
ejpam-6613	272	33	product	product	NOUN
ejpam-6613	272	34	,	,	PUNCT
ejpam-6613	272	35	let	let	VERB
ejpam-6613	272	36	t1	t1	NOUN
ejpam-6613	272	37	,	,	PUNCT
ejpam-6613	272	38	s1	s1	NOUN
ejpam-6613	272	39	:	:	PUNCT
ejpam-6613	272	40	x1	x1	PROPN
ejpam-6613	272	41	→	→	SYM
ejpam-6613	272	42	x1	x1	PROPN
ejpam-6613	272	43	,	,	PUNCT
ejpam-6613	272	44	t2	t2	NOUN
ejpam-6613	272	45	,	,	PUNCT
ejpam-6613	272	46	s2	s2	NOUN
ejpam-6613	272	47	:	:	PUNCT
ejpam-6613	272	48	x2	x2	PROPN
ejpam-6613	272	49	→	→	PUNCT
ejpam-6613	272	50	x2	x2	PROPN
ejpam-6613	272	51	be	be	AUX
ejpam-6613	272	52	linear	linear	PROPN
ejpam-6613	272	53	operators	operator	NOUN
ejpam-6613	272	54	,	,	PUNCT
ejpam-6613	272	55	and	and	CCONJ
ejpam-6613	272	56	let	let	VERB
ejpam-6613	272	57	α	α	PRON
ejpam-6613	272	58	,	,	PUNCT
ejpam-6613	272	59	β	β	PROPN
ejpam-6613	272	60	∈	∈	PROPN
ejpam-6613	272	61	c.	c.	NOUN
ejpam-6613	272	62	then	then	ADV
ejpam-6613	272	63	the	the	DET
ejpam-6613	272	64	following	follow	VERB
ejpam-6613	272	65	identities	identity	NOUN
ejpam-6613	272	66	hold	hold	VERB
ejpam-6613	272	67	:	:	PUNCT
ejpam-6613	272	68	i	i	NOUN
ejpam-6613	272	69	)	)	PUNCT
ejpam-6613	272	70	t1s1	t1s1	ADP
ejpam-6613	272	71	2	2	NUM
ejpam-6613	272	72	⊙	⊙	NOUN
ejpam-6613	272	73	t2s2	t2s2	PROPN
ejpam-6613	272	74	=	=	SYM
ejpam-6613	272	75	(	(	PUNCT
ejpam-6613	272	76	t1	t1	PROPN
ejpam-6613	272	77	2	2	NUM
ejpam-6613	272	78	⊙	⊙	NOUN
ejpam-6613	272	79	t2	t2	PROPN
ejpam-6613	272	80	)	)	PUNCT
ejpam-6613	273	1	◦	◦	NOUN
ejpam-6613	273	2	(	(	PUNCT
ejpam-6613	273	3	s1	s1	PROPN
ejpam-6613	273	4	2	2	NUM
ejpam-6613	273	5	⊙	⊙	X
ejpam-6613	273	6	s2	s2	PROPN
ejpam-6613	273	7	)	)	PUNCT
ejpam-6613	273	8	;	;	PUNCT
ejpam-6613	273	9	ii	ii	X
ejpam-6613	273	10	)	)	PUNCT
ejpam-6613	273	11	αβ	αβ	PROPN
ejpam-6613	273	12	(	(	PUNCT
ejpam-6613	273	13	t1	t1	PROPN
ejpam-6613	273	14	2	2	NUM
ejpam-6613	273	15	⊙	⊙	NOUN
ejpam-6613	273	16	t2	t2	NOUN
ejpam-6613	273	17	)	)	PUNCT
ejpam-6613	274	1	=	=	PRON
ejpam-6613	274	2	(	(	PUNCT
ejpam-6613	274	3	αt1	αt1	NOUN
ejpam-6613	274	4	)	)	PUNCT
ejpam-6613	274	5	2	2	NUM
ejpam-6613	274	6	⊙	⊙	NOUN
ejpam-6613	274	7	(	(	PUNCT
ejpam-6613	274	8	βt2	βt2	NOUN
ejpam-6613	274	9	)	)	PUNCT
ejpam-6613	274	10	;	;	PUNCT
ejpam-6613	274	11	iii	iii	X
ejpam-6613	274	12	)	)	PUNCT
ejpam-6613	274	13	t1	t1	NOUN
ejpam-6613	274	14	2	2	NUM
ejpam-6613	274	15	⊙	⊙	NOUN
ejpam-6613	274	16	(	(	PUNCT
ejpam-6613	274	17	t2	t2	PROPN
ejpam-6613	274	18	+	+	CCONJ
ejpam-6613	274	19	s2	s2	PROPN
ejpam-6613	274	20	)	)	PUNCT
ejpam-6613	274	21	=	=	PUNCT
ejpam-6613	274	22	(	(	PUNCT
ejpam-6613	274	23	t1	t1	PROPN
ejpam-6613	274	24	2	2	NUM
ejpam-6613	274	25	⊙	⊙	NOUN
ejpam-6613	274	26	t2	t2	PROPN
ejpam-6613	274	27	)	)	PUNCT
ejpam-6613	274	28	+	+	CCONJ
ejpam-6613	274	29	(	(	PUNCT
ejpam-6613	274	30	t1	t1	PROPN
ejpam-6613	274	31	2	2	NUM
ejpam-6613	274	32	⊙	⊙	PROPN
ejpam-6613	274	33	s2	s2	PROPN
ejpam-6613	274	34	)	)	PUNCT
ejpam-6613	274	35	;	;	PUNCT
ejpam-6613	274	36	iv	iv	X
ejpam-6613	274	37	)	)	PUNCT
ejpam-6613	274	38	(	(	PUNCT
ejpam-6613	274	39	t1	t1	NOUN
ejpam-6613	274	40	+	+	NUM
ejpam-6613	274	41	s1	s1	NOUN
ejpam-6613	274	42	)	)	PUNCT
ejpam-6613	274	43	2	2	NUM
ejpam-6613	274	44	⊙	⊙	NOUN
ejpam-6613	274	45	t2	t2	PROPN
ejpam-6613	274	46	=	=	SYM
ejpam-6613	274	47	(	(	PUNCT
ejpam-6613	274	48	t1	t1	PROPN
ejpam-6613	274	49	2	2	NUM
ejpam-6613	274	50	⊙	⊙	NOUN
ejpam-6613	274	51	t2	t2	PROPN
ejpam-6613	274	52	)	)	PUNCT
ejpam-6613	274	53	+	+	CCONJ
ejpam-6613	274	54	(	(	PUNCT
ejpam-6613	274	55	s1	s1	PROPN
ejpam-6613	274	56	2	2	NUM
ejpam-6613	274	57	⊙	⊙	NOUN
ejpam-6613	274	58	t2	t2	PROPN
ejpam-6613	274	59	)	)	PUNCT
ejpam-6613	274	60	;	;	PUNCT
ejpam-6613	274	61	v	v	X
ejpam-6613	274	62	)	)	PUNCT
ejpam-6613	274	63	(	(	PUNCT
ejpam-6613	274	64	t1	t1	NOUN
ejpam-6613	274	65	+	+	NUM
ejpam-6613	274	66	s1	s1	NOUN
ejpam-6613	274	67	)	)	PUNCT
ejpam-6613	274	68	2	2	NUM
ejpam-6613	274	69	⊙	⊙	NOUN
ejpam-6613	274	70	(	(	PUNCT
ejpam-6613	274	71	t2	t2	PROPN
ejpam-6613	274	72	+	+	CCONJ
ejpam-6613	274	73	s2	s2	PROPN
ejpam-6613	274	74	)	)	PUNCT
ejpam-6613	274	75	=	=	PUNCT
ejpam-6613	275	1	t1	t1	NOUN
ejpam-6613	275	2	2	2	NUM
ejpam-6613	275	3	⊙	⊙	NOUN
ejpam-6613	275	4	t2	t2	PROPN
ejpam-6613	275	5	+	+	CCONJ
ejpam-6613	275	6	t1	t1	PROPN
ejpam-6613	275	7	2	2	NUM
ejpam-6613	275	8	⊙	⊙	X
ejpam-6613	275	9	s2	s2	NOUN
ejpam-6613	275	10	+	+	CCONJ
ejpam-6613	275	11	s1	s1	PROPN
ejpam-6613	275	12	2	2	NUM
ejpam-6613	275	13	⊙	⊙	NOUN
ejpam-6613	275	14	t2	t2	PROPN
ejpam-6613	275	15	+	+	CCONJ
ejpam-6613	275	16	s1	s1	PROPN
ejpam-6613	275	17	2	2	NUM
ejpam-6613	275	18	⊙	⊙	X
ejpam-6613	275	19	s2	s2	PROPN
ejpam-6613	275	20	;	;	PUNCT
ejpam-6613	275	21	vi	vi	X
ejpam-6613	275	22	)	)	PUNCT
ejpam-6613	275	23	t1	t1	NOUN
ejpam-6613	275	24	2	2	NUM
ejpam-6613	275	25	⊙	⊙	NOUN
ejpam-6613	275	26	t2	t2	PROPN
ejpam-6613	275	27	is	be	AUX
ejpam-6613	275	28	invertible	invertible	ADJ
ejpam-6613	275	29	if	if	SCONJ
ejpam-6613	275	30	and	and	CCONJ
ejpam-6613	275	31	only	only	ADV
ejpam-6613	275	32	if	if	SCONJ
ejpam-6613	275	33	both	both	PRON
ejpam-6613	275	34	t1	t1	NOUN
ejpam-6613	275	35	and	and	CCONJ
ejpam-6613	275	36	t2	t2	NOUN
ejpam-6613	275	37	are	be	AUX
ejpam-6613	275	38	invertible	invertible	ADJ
ejpam-6613	275	39	,	,	PUNCT
ejpam-6613	275	40	and	and	CCONJ
ejpam-6613	275	41	in	in	ADP
ejpam-6613	275	42	that	that	DET
ejpam-6613	275	43	case	case	NOUN
ejpam-6613	275	44	(	(	PUNCT
ejpam-6613	275	45	t1	t1	NOUN
ejpam-6613	275	46	2	2	NUM
ejpam-6613	275	47	⊙	⊙	NOUN
ejpam-6613	275	48	t2	t2	NOUN
ejpam-6613	275	49	)	)	PUNCT
ejpam-6613	275	50	−1	−1	NOUN
ejpam-6613	276	1	=	=	SYM
ejpam-6613	276	2	t−1	t−1	PROPN
ejpam-6613	276	3	1	1	NUM
ejpam-6613	276	4	2	2	NUM
ejpam-6613	276	5	⊙	⊙	NOUN
ejpam-6613	276	6	t−1	t−1	PROPN
ejpam-6613	276	7	2	2	NUM
ejpam-6613	276	8	.	.	PUNCT
ejpam-6613	277	1	proof	proof	NOUN
ejpam-6613	277	2	.	.	PUNCT
ejpam-6613	278	1	the	the	DET
ejpam-6613	278	2	proofs	proof	NOUN
ejpam-6613	278	3	follow	follow	VERB
ejpam-6613	278	4	directly	directly	ADV
ejpam-6613	278	5	by	by	ADP
ejpam-6613	278	6	applying	apply	VERB
ejpam-6613	278	7	each	each	DET
ejpam-6613	278	8	operator	operator	NOUN
ejpam-6613	278	9	definition	definition	NOUN
ejpam-6613	278	10	to	to	ADP
ejpam-6613	278	11	a	a	DET
ejpam-6613	278	12	simple	simple	ADJ
ejpam-6613	278	13	tensor	tensor	NOUN
ejpam-6613	278	14	sum	sum	NOUN
ejpam-6613	278	15	ξ	ξ	X
ejpam-6613	278	16	=	=	SYM
ejpam-6613	278	17	∑n	∑n	PROPN
ejpam-6613	278	18	i=1	i=1	PROPN
ejpam-6613	278	19	xi	xi	PART
ejpam-6613	278	20	2	2	NUM
ejpam-6613	278	21	⊙	⊙	PROPN
ejpam-6613	278	22	yi	yi	PROPN
ejpam-6613	278	23	and	and	CCONJ
ejpam-6613	278	24	grouping	group	VERB
ejpam-6613	278	25	like	like	ADP
ejpam-6613	278	26	terms	term	NOUN
ejpam-6613	278	27	,	,	PUNCT
ejpam-6613	278	28	using	use	VERB
ejpam-6613	278	29	associativity	associativity	NOUN
ejpam-6613	278	30	and	and	CCONJ
ejpam-6613	278	31	distributivity	distributivity	NOUN
ejpam-6613	278	32	of	of	ADP
ejpam-6613	278	33	sums	sum	NOUN
ejpam-6613	278	34	and	and	CCONJ
ejpam-6613	278	35	scalars	scalar	NOUN
ejpam-6613	278	36	,	,	PUNCT
ejpam-6613	278	37	as	as	ADV
ejpam-6613	278	38	well	well	ADV
ejpam-6613	278	39	as	as	ADP
ejpam-6613	278	40	the	the	DET
ejpam-6613	278	41	definition	definition	NOUN
ejpam-6613	278	42	of	of	ADP
ejpam-6613	278	43	the	the	DET
ejpam-6613	278	44	tensor	tensor	NOUN
ejpam-6613	278	45	-	-	PUNCT
ejpam-6613	278	46	product	product	NOUN
ejpam-6613	278	47	operator	operator	NOUN
ejpam-6613	278	48	.	.	PUNCT
ejpam-6613	279	1	the	the	DET
ejpam-6613	279	2	invertibility	invertibility	NOUN
ejpam-6613	279	3	statement	statement	NOUN
ejpam-6613	279	4	in	in	ADP
ejpam-6613	279	5	(	(	PUNCT
ejpam-6613	279	6	vi	vi	NOUN
ejpam-6613	279	7	)	)	PUNCT
ejpam-6613	279	8	uses	use	VERB
ejpam-6613	279	9	the	the	DET
ejpam-6613	279	10	fact	fact	NOUN
ejpam-6613	279	11	that	that	SCONJ
ejpam-6613	279	12	a	a	DET
ejpam-6613	279	13	tensor	tensor	NOUN
ejpam-6613	279	14	-	-	PUNCT
ejpam-6613	279	15	product	product	NOUN
ejpam-6613	279	16	of	of	ADP
ejpam-6613	279	17	invertible	invertible	ADJ
ejpam-6613	279	18	operators	operator	NOUN
ejpam-6613	279	19	is	be	AUX
ejpam-6613	279	20	itself	itself	PRON
ejpam-6613	279	21	invertible	invertible	ADJ
ejpam-6613	279	22	,	,	PUNCT
ejpam-6613	279	23	with	with	ADP
ejpam-6613	279	24	inverse	inverse	NOUN
ejpam-6613	279	25	given	give	VERB
ejpam-6613	279	26	by	by	ADP
ejpam-6613	279	27	the	the	DET
ejpam-6613	279	28	tensor	tensor	NOUN
ejpam-6613	279	29	product	product	NOUN
ejpam-6613	279	30	of	of	ADP
ejpam-6613	279	31	the	the	DET
ejpam-6613	279	32	individual	individual	ADJ
ejpam-6613	279	33	inverses	inverse	NOUN
ejpam-6613	279	34	.	.	PUNCT
ejpam-6613	280	1	example	example	NOUN
ejpam-6613	281	1	3	3	NUM
ejpam-6613	281	2	.	.	PUNCT
ejpam-6613	282	1	if	if	SCONJ
ejpam-6613	282	2	we	we	PRON
ejpam-6613	282	3	consider	consider	VERB
ejpam-6613	282	4	(	(	PUNCT
ejpam-6613	282	5	c2	c2	PROPN
ejpam-6613	282	6	,	,	PUNCT
ejpam-6613	282	7	⟨	⟨	NOUN
ejpam-6613	282	8	·	·	SYM
ejpam-6613	282	9	,	,	PUNCT
ejpam-6613	282	10	·	·	PUNCT
ejpam-6613	282	11	|·⟩c2	|·⟩c2	PUNCT
ejpam-6613	282	12	)	)	PUNCT
ejpam-6613	282	13	and	and	CCONJ
ejpam-6613	282	14	(	(	PUNCT
ejpam-6613	282	15	c	c	NOUN
ejpam-6613	282	16	,	,	PUNCT
ejpam-6613	282	17	⟨	⟨	NOUN
ejpam-6613	282	18	·	·	PUNCT
ejpam-6613	282	19	,	,	PUNCT
ejpam-6613	282	20	·	·	PUNCT
ejpam-6613	282	21	|·⟩c	|·⟩c	X
ejpam-6613	282	22	)	)	PUNCT
ejpam-6613	282	23	as	as	ADP
ejpam-6613	282	24	in	in	ADP
ejpam-6613	282	25	example	example	NOUN
ejpam-6613	282	26	2	2	NUM
ejpam-6613	282	27	,	,	PUNCT
ejpam-6613	282	28	then	then	ADV
ejpam-6613	282	29	the	the	DET
ejpam-6613	282	30	following	follow	VERB
ejpam-6613	282	31	mappings	mapping	NOUN
ejpam-6613	282	32	are	be	AUX
ejpam-6613	282	33	linear	linear	PROPN
ejpam-6613	282	34	operators	operator	NOUN
ejpam-6613	282	35	t	t	NOUN
ejpam-6613	282	36	:	:	PUNCT
ejpam-6613	282	37	c2	c2	PROPN
ejpam-6613	282	38	→	→	SYM
ejpam-6613	282	39	c2	c2	PROPN
ejpam-6613	282	40	,	,	PUNCT
ejpam-6613	282	41	t	t	PROPN
ejpam-6613	282	42	(	(	PUNCT
ejpam-6613	282	43	a	a	DET
ejpam-6613	282	44	,	,	PUNCT
ejpam-6613	282	45	b	b	NOUN
ejpam-6613	282	46	)	)	PUNCT
ejpam-6613	282	47	=	=	SYM
ejpam-6613	282	48	(	(	PUNCT
ejpam-6613	282	49	0	0	NUM
ejpam-6613	282	50	,	,	PUNCT
ejpam-6613	282	51	a	a	PRON
ejpam-6613	282	52	)	)	PUNCT
ejpam-6613	282	53	,	,	PUNCT
ejpam-6613	282	54	(	(	PUNCT
ejpam-6613	282	55	a	a	PRON
ejpam-6613	282	56	,	,	PUNCT
ejpam-6613	282	57	b	b	NOUN
ejpam-6613	282	58	)	)	PUNCT
ejpam-6613	282	59	∈	∈	PROPN
ejpam-6613	282	60	c2	c2	PROPN
ejpam-6613	282	61	.	.	PUNCT
ejpam-6613	282	62	m.	m.	PROPN
ejpam-6613	282	63	luis	luis	PROPN
ejpam-6613	282	64	,	,	PUNCT
ejpam-6613	282	65	f.	f.	PROPN
ejpam-6613	282	66	osmin	osmin	PROPN
ejpam-6613	282	67	,	,	PUNCT
ejpam-6613	282	68	s.	s.	PROPN
ejpam-6613	282	69	arley	arley	PROPN
ejpam-6613	282	70	/	/	SYM
ejpam-6613	282	71	eur	eur	PROPN
ejpam-6613	282	72	.	.	PUNCT
ejpam-6613	283	1	j.	j.	PROPN
ejpam-6613	283	2	pure	pure	PROPN
ejpam-6613	283	3	appl	appl	PROPN
ejpam-6613	283	4	.	.	PROPN
ejpam-6613	283	5	math	math	PROPN
ejpam-6613	283	6	,	,	PUNCT
ejpam-6613	283	7	18	18	NUM
ejpam-6613	283	8	(	(	PUNCT
ejpam-6613	283	9	4	4	NUM
ejpam-6613	283	10	)	)	PUNCT
ejpam-6613	283	11	(	(	PUNCT
ejpam-6613	283	12	2025	2025	NUM
ejpam-6613	283	13	)	)	PUNCT
ejpam-6613	283	14	,	,	PUNCT
ejpam-6613	283	15	6613	6613	NUM
ejpam-6613	283	16	13	13	NUM
ejpam-6613	283	17	of	of	ADP
ejpam-6613	283	18	17	17	NUM
ejpam-6613	283	19	s	s	PART
ejpam-6613	283	20	:	:	PUNCT
ejpam-6613	283	21	c	c	NOUN
ejpam-6613	283	22	→	→	SYM
ejpam-6613	283	23	c	c	PROPN
ejpam-6613	283	24	,	,	PUNCT
ejpam-6613	283	25	s(c	s(c	ADJ
ejpam-6613	283	26	)	)	PUNCT
ejpam-6613	283	27	=	=	SYM
ejpam-6613	283	28	ic	ic	PROPN
ejpam-6613	283	29	,	,	PUNCT
ejpam-6613	283	30	c	c	PROPN
ejpam-6613	283	31	∈	∈	PROPN
ejpam-6613	283	32	c	c	NOUN
ejpam-6613	283	33	,	,	PUNCT
ejpam-6613	283	34	and	and	CCONJ
ejpam-6613	283	35	consequently	consequently	ADV
ejpam-6613	283	36	,	,	PUNCT
ejpam-6613	283	37	for	for	SCONJ
ejpam-6613	283	38	each	each	DET
ejpam-6613	283	39	∑n	∑n	PROPN
ejpam-6613	283	40	k=1(ak	k=1(ak	PROPN
ejpam-6613	283	41	,	,	PUNCT
ejpam-6613	283	42	bk	bk	PROPN
ejpam-6613	283	43	)	)	PUNCT
ejpam-6613	283	44	2	2	NUM
ejpam-6613	283	45	⊙	⊙	NOUN
ejpam-6613	283	46	ck	ck	PROPN
ejpam-6613	283	47	∈	∈	PROPN
ejpam-6613	283	48	c2	c2	PROPN
ejpam-6613	283	49	2	2	NUM
ejpam-6613	283	50	⊙c	⊙c	PROPN
ejpam-6613	283	51	the	the	DET
ejpam-6613	283	52	map	map	NOUN
ejpam-6613	283	53	t	t	PROPN
ejpam-6613	283	54	2	2	NUM
ejpam-6613	283	55	⊙s	⊙s	PROPN
ejpam-6613	283	56	:	:	PUNCT
ejpam-6613	283	57	c2	c2	PROPN
ejpam-6613	283	58	2	2	NUM
ejpam-6613	283	59	⊙c	⊙c	PROPN
ejpam-6613	283	60	→	→	SYM
ejpam-6613	283	61	c2	c2	PROPN
ejpam-6613	283	62	2	2	NUM
ejpam-6613	283	63	⊙c	⊙c	PROPN
ejpam-6613	283	64	is	be	AUX
ejpam-6613	283	65	given	give	VERB
ejpam-6613	283	66	by	by	ADP
ejpam-6613	283	67	(	(	PUNCT
ejpam-6613	283	68	t	t	PROPN
ejpam-6613	283	69	2	2	NUM
ejpam-6613	283	70	⊙	⊙	NOUN
ejpam-6613	283	71	s	s	PART
ejpam-6613	283	72	)	)	PUNCT
ejpam-6613	283	73	(	(	PUNCT
ejpam-6613	283	74	n∑	n∑	INTJ
ejpam-6613	283	75	k=1	k=1	PROPN
ejpam-6613	283	76	(	(	PUNCT
ejpam-6613	283	77	ak	ak	PROPN
ejpam-6613	283	78	,	,	PUNCT
ejpam-6613	283	79	bk	bk	PROPN
ejpam-6613	283	80	)	)	PUNCT
ejpam-6613	283	81	2	2	NUM
ejpam-6613	283	82	⊙	⊙	NOUN
ejpam-6613	283	83	ck	ck	PUNCT
ejpam-6613	283	84	)	)	PUNCT
ejpam-6613	284	1	=	=	PUNCT
ejpam-6613	285	1	n∑	n∑	NOUN
ejpam-6613	285	2	k=1	k=1	PROPN
ejpam-6613	285	3	t	t	PROPN
ejpam-6613	285	4	(	(	PUNCT
ejpam-6613	285	5	ak	ak	PROPN
ejpam-6613	285	6	,	,	PUNCT
ejpam-6613	285	7	bk	bk	PROPN
ejpam-6613	285	8	)	)	PUNCT
ejpam-6613	285	9	2	2	NUM
ejpam-6613	285	10	⊙	⊙	X
ejpam-6613	285	11	sck	sck	PROPN
ejpam-6613	286	1	=	=	PUNCT
ejpam-6613	286	2	n∑	n∑	PROPN
ejpam-6613	286	3	i=1	i=1	PROPN
ejpam-6613	286	4	(	(	PUNCT
ejpam-6613	286	5	0	0	NUM
ejpam-6613	286	6	,	,	PUNCT
ejpam-6613	286	7	ak	ak	NOUN
ejpam-6613	286	8	)	)	PUNCT
ejpam-6613	286	9	2	2	NUM
ejpam-6613	286	10	⊙	⊙	NOUN
ejpam-6613	286	11	ick	ick	VERB
ejpam-6613	287	1	=	=	VERB
ejpam-6613	288	1	i	i	PRON
ejpam-6613	288	2	n∑	n∑	PROPN
ejpam-6613	288	3	i=1	i=1	PROPN
ejpam-6613	289	1	(	(	PUNCT
ejpam-6613	289	2	0	0	NUM
ejpam-6613	289	3	,	,	PUNCT
ejpam-6613	289	4	ak	ak	NOUN
ejpam-6613	289	5	)	)	PUNCT
ejpam-6613	289	6	2	2	NUM
ejpam-6613	289	7	⊙	⊙	NOUN
ejpam-6613	289	8	ck	ck	PROPN
ejpam-6613	289	9	.	.	PUNCT
ejpam-6613	290	1	now	now	ADV
ejpam-6613	290	2	,	,	PUNCT
ejpam-6613	290	3	for	for	ADP
ejpam-6613	290	4	all	all	PRON
ejpam-6613	290	5	(	(	PUNCT
ejpam-6613	290	6	(	(	PUNCT
ejpam-6613	290	7	x1	x1	PROPN
ejpam-6613	290	8	,	,	PUNCT
ejpam-6613	290	9	x2	x2	PROPN
ejpam-6613	290	10	)	)	PUNCT
ejpam-6613	290	11	,	,	PUNCT
ejpam-6613	290	12	d	d	NOUN
ejpam-6613	290	13	)	)	PUNCT
ejpam-6613	290	14	,	,	PUNCT
ejpam-6613	290	15	(	(	PUNCT
ejpam-6613	290	16	(	(	PUNCT
ejpam-6613	290	17	y1	y1	INTJ
ejpam-6613	290	18	,	,	PUNCT
ejpam-6613	290	19	y2	y2	PROPN
ejpam-6613	290	20	)	)	PUNCT
ejpam-6613	290	21	,	,	PUNCT
ejpam-6613	290	22	e	e	X
ejpam-6613	290	23	)	)	PUNCT
ejpam-6613	290	24	∈	∈	PROPN
ejpam-6613	290	25	c2	c2	PROPN
ejpam-6613	290	26	×	×	PROPN
ejpam-6613	290	27	c	c	NOUN
ejpam-6613	290	28	we	we	PRON
ejpam-6613	290	29	have	have	VERB
ejpam-6613	290	30	(	(	PUNCT
ejpam-6613	290	31	i	i	PRON
ejpam-6613	290	32	n∑	n∑	INTJ
ejpam-6613	290	33	i=1	i=1	PROPN
ejpam-6613	291	1	(	(	PUNCT
ejpam-6613	291	2	0	0	NUM
ejpam-6613	291	3	,	,	PUNCT
ejpam-6613	291	4	ak	ak	NOUN
ejpam-6613	291	5	)	)	PUNCT
ejpam-6613	291	6	2	2	NUM
ejpam-6613	291	7	⊙	⊙	NOUN
ejpam-6613	291	8	ck	ck	PROPN
ejpam-6613	291	9	)	)	PUNCT
ejpam-6613	292	1	(	(	PUNCT
ejpam-6613	292	2	(	(	PUNCT
ejpam-6613	292	3	(	(	PUNCT
ejpam-6613	292	4	x1	x1	ADJ
ejpam-6613	292	5	,	,	PUNCT
ejpam-6613	292	6	x2	x2	PROPN
ejpam-6613	292	7	)	)	PUNCT
ejpam-6613	292	8	,	,	PUNCT
ejpam-6613	292	9	d	d	NOUN
ejpam-6613	292	10	)	)	PUNCT
ejpam-6613	292	11	,	,	PUNCT
ejpam-6613	292	12	(	(	PUNCT
ejpam-6613	292	13	(	(	PUNCT
ejpam-6613	292	14	y1	y1	INTJ
ejpam-6613	292	15	,	,	PUNCT
ejpam-6613	292	16	y2	y2	PROPN
ejpam-6613	292	17	)	)	PUNCT
ejpam-6613	292	18	,	,	PUNCT
ejpam-6613	292	19	e	e	NOUN
ejpam-6613	292	20	)	)	PUNCT
ejpam-6613	292	21	)	)	PUNCT
ejpam-6613	293	1	=	=	PUNCT
ejpam-6613	294	1	i	i	PRON
ejpam-6613	294	2	n∑	n∑	INTJ
ejpam-6613	294	3	k=1	k=1	X
ejpam-6613	294	4	⟨(0	⟨(0	PROPN
ejpam-6613	294	5	,	,	PUNCT
ejpam-6613	294	6	ak	ak	PROPN
ejpam-6613	294	7	)	)	PUNCT
ejpam-6613	294	8	,	,	PUNCT
ejpam-6613	294	9	(	(	PUNCT
ejpam-6613	294	10	x1	x1	PROPN
ejpam-6613	294	11	,	,	PUNCT
ejpam-6613	294	12	x2)|(y1	x2)|(y1	PROPN
ejpam-6613	294	13	,	,	PUNCT
ejpam-6613	294	14	y2)⟩c2⟨ck	y2)⟩c2⟨ck	NUM
ejpam-6613	294	15	,	,	PUNCT
ejpam-6613	294	16	d|e⟩c	d|e⟩c	NOUN
ejpam-6613	294	17	=	=	PUNCT
ejpam-6613	295	1	i	i	PRON
ejpam-6613	295	2	n∑	n∑	INTJ
ejpam-6613	296	1	k=1	k=1	VERB
ejpam-6613	296	2	akckx2d|y2e|2	akckx2d|y2e|2	PROPN
ejpam-6613	297	1	=	=	SYM
ejpam-6613	297	2	ix2d|y2e|2	ix2d|y2e|2	PROPN
ejpam-6613	298	1	n∑	n∑	INTJ
ejpam-6613	298	2	k=1	k=1	PROPN
ejpam-6613	298	3	akck	akck	PROPN
ejpam-6613	298	4	=	=	PROPN
ejpam-6613	298	5	ix2d|y2e|2⟨a	ix2d|y2e|2⟨a	PROPN
ejpam-6613	298	6	,	,	PUNCT
ejpam-6613	298	7	c̄⟩cn	c̄⟩cn	NOUN
ejpam-6613	298	8	where	where	SCONJ
ejpam-6613	298	9	a	a	DET
ejpam-6613	298	10	=	=	X
ejpam-6613	298	11	(	(	PUNCT
ejpam-6613	298	12	a1	a1	PROPN
ejpam-6613	298	13	,	,	PUNCT
ejpam-6613	298	14	·	·	PUNCT
ejpam-6613	298	15	·	·	PUNCT
ejpam-6613	298	16	·	·	PUNCT
ejpam-6613	298	17	,	,	PUNCT
ejpam-6613	298	18	an	an	X
ejpam-6613	298	19	)	)	PUNCT
ejpam-6613	298	20	,	,	PUNCT
ejpam-6613	298	21	c̄	c̄	PROPN
ejpam-6613	298	22	=	=	SYM
ejpam-6613	298	23	(	(	PUNCT
ejpam-6613	298	24	c̄1	c̄1	X
ejpam-6613	298	25	,	,	PUNCT
ejpam-6613	298	26	·	·	PUNCT
ejpam-6613	298	27	·	·	PUNCT
ejpam-6613	298	28	·	·	PUNCT
ejpam-6613	298	29	,	,	PUNCT
ejpam-6613	298	30	c̄n	c̄n	PROPN
ejpam-6613	298	31	)	)	PUNCT
ejpam-6613	298	32	∈	∈	PROPN
ejpam-6613	298	33	cn	cn	PROPN
ejpam-6613	298	34	and	and	CCONJ
ejpam-6613	298	35	⟨	⟨	VERB
ejpam-6613	298	36	·	·	PUNCT
ejpam-6613	298	37	,	,	PUNCT
ejpam-6613	298	38	·	·	PUNCT
ejpam-6613	298	39	⟩cn	⟩cn	ADP
ejpam-6613	298	40	denotes	denote	VERB
ejpam-6613	298	41	the	the	DET
ejpam-6613	298	42	classical	classical	ADJ
ejpam-6613	298	43	inner	inner	ADJ
ejpam-6613	298	44	product	product	NOUN
ejpam-6613	298	45	on	on	ADP
ejpam-6613	298	46	cn	cn	PROPN
ejpam-6613	298	47	.	.	PROPN
ejpam-6613	298	48	remark	remark	PROPN
ejpam-6613	298	49	5	5	NUM
ejpam-6613	298	50	.	.	PUNCT
ejpam-6613	299	1	given	give	VERB
ejpam-6613	299	2	two	two	NUM
ejpam-6613	299	3	spaces	space	NOUN
ejpam-6613	299	4	with	with	ADP
ejpam-6613	299	5	classical	classical	ADJ
ejpam-6613	299	6	inner	inner	ADJ
ejpam-6613	299	7	product	product	NOUN
ejpam-6613	299	8	(	(	PUNCT
ejpam-6613	299	9	x1	x1	ADJ
ejpam-6613	299	10	,	,	PUNCT
ejpam-6613	299	11	⟨	⟨	NOUN
ejpam-6613	299	12	·	·	SYM
ejpam-6613	299	13	,	,	PUNCT
ejpam-6613	299	14	·	·	PUNCT
ejpam-6613	299	15	⟩x1	⟩x1	X
ejpam-6613	299	16	)	)	PUNCT
ejpam-6613	299	17	and	and	CCONJ
ejpam-6613	299	18	(	(	PUNCT
ejpam-6613	299	19	x2	x2	INTJ
ejpam-6613	299	20	,	,	PUNCT
ejpam-6613	299	21	⟨	⟨	NOUN
ejpam-6613	299	22	·	·	SYM
ejpam-6613	299	23	,	,	PUNCT
ejpam-6613	299	24	·	·	PUNCT
ejpam-6613	299	25	⟩x2	⟩x2	ADJ
ejpam-6613	299	26	)	)	PUNCT
ejpam-6613	299	27	,	,	PUNCT
ejpam-6613	299	28	and	and	CCONJ
ejpam-6613	299	29	x	x	PUNCT
ejpam-6613	299	30	∈	∈	PROPN
ejpam-6613	299	31	x1	x1	PROPN
ejpam-6613	299	32	,	,	PUNCT
ejpam-6613	299	33	y	y	PROPN
ejpam-6613	299	34	∈	∈	PROPN
ejpam-6613	299	35	x2	x2	PROPN
ejpam-6613	299	36	,	,	PUNCT
ejpam-6613	299	37	we	we	PRON
ejpam-6613	299	38	know	know	VERB
ejpam-6613	299	39	that	that	PRON
ejpam-6613	299	40	x⊙y	x⊙y	PROPN
ejpam-6613	299	41	is	be	AUX
ejpam-6613	299	42	defined	define	VERB
ejpam-6613	299	43	as	as	ADP
ejpam-6613	299	44	the	the	DET
ejpam-6613	299	45	mapping	mapping	NOUN
ejpam-6613	299	46	x⊙y	x⊙y	PUNCT
ejpam-6613	299	47	:	:	PUNCT
ejpam-6613	300	1	x1×x2	x1×x2	PROPN
ejpam-6613	300	2	→	→	SYM
ejpam-6613	300	3	c	c	X
ejpam-6613	300	4	,	,	PUNCT
ejpam-6613	300	5	given	give	VERB
ejpam-6613	300	6	by	by	ADP
ejpam-6613	300	7	(	(	PUNCT
ejpam-6613	300	8	x⊙	x⊙	PROPN
ejpam-6613	300	9	y)(r	y)(r	PROPN
ejpam-6613	300	10	,	,	PUNCT
ejpam-6613	300	11	s	s	X
ejpam-6613	300	12	)	)	PUNCT
ejpam-6613	300	13	=	=	PUNCT
ejpam-6613	300	14	⟨x	⟨x	VERB
ejpam-6613	300	15	,	,	PUNCT
ejpam-6613	300	16	r⟩x⟨y	r⟩x⟨y	PROPN
ejpam-6613	300	17	,	,	PUNCT
ejpam-6613	300	18	s⟩y	s⟩y	PROPN
ejpam-6613	300	19	for	for	ADP
ejpam-6613	300	20	all	all	DET
ejpam-6613	300	21	(	(	PUNCT
ejpam-6613	300	22	r	r	NOUN
ejpam-6613	300	23	,	,	PUNCT
ejpam-6613	300	24	s	s	PART
ejpam-6613	300	25	)	)	PUNCT
ejpam-6613	300	26	∈	∈	PROPN
ejpam-6613	300	27	x1	x1	NUM
ejpam-6613	300	28	×	×	NOUN
ejpam-6613	300	29	x2	x2	NOUN
ejpam-6613	300	30	(	(	PUNCT
ejpam-6613	300	31	see	see	VERB
ejpam-6613	300	32	[	[	X
ejpam-6613	300	33	11	11	NUM
ejpam-6613	300	34	]	]	NUM
ejpam-6613	300	35	)	)	PUNCT
ejpam-6613	300	36	.	.	PUNCT
ejpam-6613	301	1	now	now	ADV
ejpam-6613	301	2	,	,	PUNCT
ejpam-6613	301	3	we	we	PRON
ejpam-6613	301	4	can	can	AUX
ejpam-6613	301	5	consider	consider	VERB
ejpam-6613	301	6	the	the	DET
ejpam-6613	301	7	application	application	NOUN
ejpam-6613	301	8	x⊙̃y	x⊙̃y	NOUN
ejpam-6613	301	9	:	:	PUNCT
ejpam-6613	302	1	(	(	PUNCT
ejpam-6613	302	2	x1	x1	NUM
ejpam-6613	302	3	×	×	PROPN
ejpam-6613	302	4	x2	x2	PROPN
ejpam-6613	302	5	)	)	PUNCT
ejpam-6613	302	6	×	×	NOUN
ejpam-6613	302	7	(	(	PUNCT
ejpam-6613	302	8	x1	x1	PROPN
ejpam-6613	302	9	×	×	PROPN
ejpam-6613	302	10	x2	x2	PROPN
ejpam-6613	302	11	)	)	PUNCT
ejpam-6613	302	12	→	→	SYM
ejpam-6613	302	13	c	c	X
ejpam-6613	302	14	,	,	PUNCT
ejpam-6613	302	15	defined	define	VERB
ejpam-6613	302	16	by	by	ADP
ejpam-6613	302	17	(	(	PUNCT
ejpam-6613	302	18	x⊙̃y)((x1	x⊙̃y)((x1	PROPN
ejpam-6613	302	19	,	,	PUNCT
ejpam-6613	302	20	y1	y1	PROPN
ejpam-6613	302	21	)	)	PUNCT
ejpam-6613	302	22	,	,	PUNCT
ejpam-6613	302	23	(	(	PUNCT
ejpam-6613	302	24	x2	x2	PROPN
ejpam-6613	302	25	,	,	PUNCT
ejpam-6613	302	26	y2	y2	PROPN
ejpam-6613	302	27	)	)	PUNCT
ejpam-6613	302	28	)	)	PUNCT
ejpam-6613	302	29	:	:	PUNCT
ejpam-6613	303	1	=	=	SYM
ejpam-6613	303	2	(	(	PUNCT
ejpam-6613	303	3	x⊙	x⊙	PROPN
ejpam-6613	303	4	y)(x1	y)(x1	PROPN
ejpam-6613	303	5	,	,	PUNCT
ejpam-6613	303	6	y1	y1	NOUN
ejpam-6613	303	7	)	)	PUNCT
ejpam-6613	303	8	.	.	PUNCT
ejpam-6613	304	1	thus	thus	ADV
ejpam-6613	304	2	we	we	PRON
ejpam-6613	304	3	obtain	obtain	VERB
ejpam-6613	304	4	the	the	DET
ejpam-6613	304	5	following	follow	VERB
ejpam-6613	304	6	complex	complex	ADJ
ejpam-6613	304	7	vector	vector	NOUN
ejpam-6613	304	8	spaces	space	NOUN
ejpam-6613	304	9	x1	x1	PROPN
ejpam-6613	304	10	⊙x2	⊙x2	PROPN
ejpam-6613	305	1	=	=	PRON
ejpam-6613	306	1	{	{	PUNCT
ejpam-6613	306	2	n∑	n∑	NOUN
ejpam-6613	306	3	i=1	i=1	PROPN
ejpam-6613	307	1	αi(xi	αi(xi	PROPN
ejpam-6613	307	2	⊙	⊙	PROPN
ejpam-6613	307	3	yi	yi	PROPN
ejpam-6613	307	4	)	)	PUNCT
ejpam-6613	307	5	:	:	PUNCT
ejpam-6613	307	6	αi	αi	VERB
ejpam-6613	307	7	∈	∈	PROPN
ejpam-6613	308	1	c	c	X
ejpam-6613	308	2	,	,	PUNCT
ejpam-6613	308	3	xi	xi	PROPN
ejpam-6613	308	4	∈	∈	PROPN
ejpam-6613	308	5	x1	x1	PROPN
ejpam-6613	308	6	,	,	PUNCT
ejpam-6613	308	7	yi	yi	PROPN
ejpam-6613	308	8	∈	∈	PROPN
ejpam-6613	308	9	x2	x2	PROPN
ejpam-6613	308	10	,	,	PUNCT
ejpam-6613	308	11	n	n	PROPN
ejpam-6613	308	12	∈	∈	PROPN
ejpam-6613	308	13	n	n	CCONJ
ejpam-6613	308	14	}	}	PUNCT
ejpam-6613	308	15	x1⊙̃x2	x1⊙̃x2	PUNCT
ejpam-6613	308	16	=	=	PUNCT
ejpam-6613	308	17			PUNCT
ejpam-6613	309	1	m∑	m∑	ADV
ejpam-6613	309	2	j=1	j=1	NOUN
ejpam-6613	309	3	βj(xj⊙̃yj	βj(xj⊙̃yj	PUNCT
ejpam-6613	309	4	)	)	PUNCT
ejpam-6613	309	5	:	:	PUNCT
ejpam-6613	310	1	βj	βj	X
ejpam-6613	310	2	∈	∈	PROPN
ejpam-6613	310	3	c	c	X
ejpam-6613	310	4	,	,	PUNCT
ejpam-6613	310	5	xj	xj	PROPN
ejpam-6613	310	6	∈	∈	PROPN
ejpam-6613	310	7	x1	x1	PROPN
ejpam-6613	310	8	,	,	PUNCT
ejpam-6613	310	9	yj	yj	PROPN
ejpam-6613	310	10	∈	∈	PROPN
ejpam-6613	310	11	x2,m	x2,m	PROPN
ejpam-6613	310	12	∈	∈	PROPN
ejpam-6613	310	13	n	n	CCONJ
ejpam-6613	310	14			X
ejpam-6613	310	15	whose	whose	DET
ejpam-6613	310	16	relationship	relationship	NOUN
ejpam-6613	310	17	is	be	AUX
ejpam-6613	310	18	expressed	express	VERB
ejpam-6613	310	19	in	in	ADP
ejpam-6613	310	20	the	the	DET
ejpam-6613	310	21	following	follow	VERB
ejpam-6613	310	22	proposition	proposition	NOUN
ejpam-6613	310	23	:	:	PUNCT
ejpam-6613	310	24	proposition	proposition	NOUN
ejpam-6613	310	25	9	9	NUM
ejpam-6613	310	26	.	.	PUNCT
ejpam-6613	311	1	let	let	VERB
ejpam-6613	311	2	(	(	PUNCT
ejpam-6613	311	3	x1	x1	ADJ
ejpam-6613	311	4	,	,	PUNCT
ejpam-6613	311	5	⟨	⟨	NOUN
ejpam-6613	311	6	·	·	SYM
ejpam-6613	311	7	,	,	PUNCT
ejpam-6613	311	8	·	·	PUNCT
ejpam-6613	311	9	⟩1	⟩1	NOUN
ejpam-6613	311	10	)	)	PUNCT
ejpam-6613	311	11	and	and	CCONJ
ejpam-6613	311	12	(	(	PUNCT
ejpam-6613	311	13	x2	x2	INTJ
ejpam-6613	311	14	,	,	PUNCT
ejpam-6613	311	15	⟨	⟨	NOUN
ejpam-6613	311	16	·	·	SYM
ejpam-6613	311	17	,	,	PUNCT
ejpam-6613	311	18	·	·	PUNCT
ejpam-6613	311	19	⟩2	⟩2	NOUN
ejpam-6613	311	20	)	)	PUNCT
ejpam-6613	311	21	spaces	space	VERB
ejpam-6613	311	22	with	with	ADP
ejpam-6613	311	23	a	a	DET
ejpam-6613	311	24	classical	classical	ADJ
ejpam-6613	311	25	inner	inner	ADJ
ejpam-6613	311	26	product	product	NOUN
ejpam-6613	311	27	.	.	PUNCT
ejpam-6613	312	1	then	then	ADV
ejpam-6613	312	2	x1	x1	PROPN
ejpam-6613	312	3	⊙x2	⊙x2	PROPN
ejpam-6613	312	4	∼=	∼=	PART
ejpam-6613	312	5	x1⊙̃x2	x1⊙̃x2	NOUN
ejpam-6613	312	6	.	.	PUNCT
ejpam-6613	313	1	proof	proof	NOUN
ejpam-6613	313	2	.	.	PUNCT
ejpam-6613	314	1	consider	consider	VERB
ejpam-6613	314	2	the	the	DET
ejpam-6613	314	3	linear	linear	ADJ
ejpam-6613	314	4	transformation	transformation	NOUN
ejpam-6613	314	5	φ	φ	X
ejpam-6613	314	6	:	:	PUNCT
ejpam-6613	314	7	x1	x1	PROPN
ejpam-6613	314	8	⊙x2	⊙x2	PROPN
ejpam-6613	315	1	→	→	PUNCT
ejpam-6613	315	2	x1⊙̃x2	x1⊙̃x2	PROPN
ejpam-6613	315	3	given	give	VERB
ejpam-6613	315	4	by	by	ADP
ejpam-6613	315	5	φ	φ	PROPN
ejpam-6613	315	6	(	(	PUNCT
ejpam-6613	315	7	n∑	n∑	NOUN
ejpam-6613	315	8	i=1	i=1	PROPN
ejpam-6613	315	9	xi	xi	PROPN
ejpam-6613	315	10	⊙	⊙	PROPN
ejpam-6613	315	11	yi	yi	PROPN
ejpam-6613	315	12	)	)	PUNCT
ejpam-6613	316	1	=	=	PUNCT
ejpam-6613	317	1	n∑	n∑	NOUN
ejpam-6613	317	2	i=1	i=1	PROPN
ejpam-6613	318	1	xi⊙̃yi	xi⊙̃yi	PROPN
ejpam-6613	318	2	note	note	NOUN
ejpam-6613	318	3	that	that	SCONJ
ejpam-6613	318	4	clearly	clearly	ADV
ejpam-6613	318	5	φ	φ	PROPN
ejpam-6613	318	6	is	be	AUX
ejpam-6613	318	7	bijective	bijective	ADJ
ejpam-6613	319	1	.	.	PUNCT
ejpam-6613	319	2	m.	m.	PROPN
ejpam-6613	319	3	luis	luis	PROPN
ejpam-6613	319	4	,	,	PUNCT
ejpam-6613	319	5	f.	f.	PROPN
ejpam-6613	319	6	osmin	osmin	PROPN
ejpam-6613	319	7	,	,	PUNCT
ejpam-6613	319	8	s.	s.	PROPN
ejpam-6613	319	9	arley	arley	PROPN
ejpam-6613	319	10	/	/	SYM
ejpam-6613	319	11	eur	eur	PROPN
ejpam-6613	319	12	.	.	PUNCT
ejpam-6613	320	1	j.	j.	PROPN
ejpam-6613	320	2	pure	pure	PROPN
ejpam-6613	320	3	appl	appl	PROPN
ejpam-6613	320	4	.	.	PROPN
ejpam-6613	320	5	math	math	PROPN
ejpam-6613	320	6	,	,	PUNCT
ejpam-6613	320	7	18	18	NUM
ejpam-6613	320	8	(	(	PUNCT
ejpam-6613	320	9	4	4	NUM
ejpam-6613	320	10	)	)	PUNCT
ejpam-6613	320	11	(	(	PUNCT
ejpam-6613	320	12	2025	2025	NUM
ejpam-6613	320	13	)	)	PUNCT
ejpam-6613	320	14	,	,	PUNCT
ejpam-6613	320	15	6613	6613	NUM
ejpam-6613	320	16	14	14	NUM
ejpam-6613	320	17	of	of	ADP
ejpam-6613	320	18	17	17	NUM
ejpam-6613	320	19	theorem	theorem	NOUN
ejpam-6613	320	20	6	6	NUM
ejpam-6613	320	21	.	.	PUNCT
ejpam-6613	321	1	let	let	VERB
ejpam-6613	321	2	(	(	PUNCT
ejpam-6613	321	3	x1	x1	ADJ
ejpam-6613	321	4	,	,	PUNCT
ejpam-6613	321	5	⟨	⟨	NOUN
ejpam-6613	321	6	·	·	SYM
ejpam-6613	321	7	,	,	PUNCT
ejpam-6613	321	8	·	·	PUNCT
ejpam-6613	321	9	⟩1	⟩1	NOUN
ejpam-6613	321	10	)	)	PUNCT
ejpam-6613	321	11	and	and	CCONJ
ejpam-6613	321	12	(	(	PUNCT
ejpam-6613	321	13	x2	x2	INTJ
ejpam-6613	321	14	,	,	PUNCT
ejpam-6613	321	15	⟨	⟨	NOUN
ejpam-6613	321	16	·	·	SYM
ejpam-6613	321	17	,	,	PUNCT
ejpam-6613	321	18	·	·	PUNCT
ejpam-6613	321	19	⟩2	⟩2	NOUN
ejpam-6613	321	20	)	)	PUNCT
ejpam-6613	321	21	spaces	space	VERB
ejpam-6613	321	22	with	with	ADP
ejpam-6613	321	23	a	a	DET
ejpam-6613	321	24	classical	classical	ADJ
ejpam-6613	321	25	inner	inner	ADJ
ejpam-6613	321	26	product	product	NOUN
ejpam-6613	321	27	.	.	PUNCT
ejpam-6613	322	1	then	then	ADV
ejpam-6613	322	2	,	,	PUNCT
ejpam-6613	322	3	x1	x1	PROPN
ejpam-6613	322	4	2	2	NUM
ejpam-6613	322	5	⊙	⊙	X
ejpam-6613	322	6	x2	x2	PROPN
ejpam-6613	322	7	is	be	AUX
ejpam-6613	322	8	a	a	DET
ejpam-6613	322	9	vector	vector	NOUN
ejpam-6613	322	10	subspace	subspace	NOUN
ejpam-6613	322	11	of	of	ADP
ejpam-6613	322	12	x1⊙̃x2	x1⊙̃x2	PROPN
ejpam-6613	322	13	,	,	PUNCT
ejpam-6613	322	14	where	where	SCONJ
ejpam-6613	322	15	the	the	DET
ejpam-6613	322	16	vector	vector	NOUN
ejpam-6613	322	17	space	space	NOUN
ejpam-6613	322	18	x1	x1	PROPN
ejpam-6613	322	19	2	2	NUM
ejpam-6613	322	20	⊙	⊙	NOUN
ejpam-6613	322	21	x2	x2	PROPN
ejpam-6613	322	22	is	be	AUX
ejpam-6613	322	23	defined	define	VERB
ejpam-6613	322	24	as	as	ADP
ejpam-6613	322	25	in	in	ADP
ejpam-6613	322	26	the	the	DET
ejpam-6613	322	27	definition	definition	NOUN
ejpam-6613	322	28	5	5	NUM
ejpam-6613	322	29	.	.	PUNCT
ejpam-6613	323	1	proof	proof	NOUN
ejpam-6613	323	2	.	.	PUNCT
ejpam-6613	324	1	it	it	PRON
ejpam-6613	324	2	is	be	AUX
ejpam-6613	324	3	enough	enough	ADJ
ejpam-6613	324	4	to	to	PART
ejpam-6613	324	5	see	see	VERB
ejpam-6613	324	6	that	that	SCONJ
ejpam-6613	324	7	x1	x1	PROPN
ejpam-6613	324	8	2	2	NUM
ejpam-6613	324	9	⊙	⊙	X
ejpam-6613	324	10	x2	x2	NOUN
ejpam-6613	324	11	⊆	⊆	NUM
ejpam-6613	324	12	x1⊙̃x2	x1⊙̃x2	PROPN
ejpam-6613	324	13	.	.	PUNCT
ejpam-6613	325	1	in	in	ADP
ejpam-6613	325	2	fact	fact	NOUN
ejpam-6613	325	3	,	,	PUNCT
ejpam-6613	325	4	whether	whether	SCONJ
ejpam-6613	325	5	ξ	ξ	PROPN
ejpam-6613	325	6	∈	∈	NOUN
ejpam-6613	325	7	x1	x1	PROPN
ejpam-6613	325	8	2	2	NUM
ejpam-6613	325	9	⊙	⊙	NOUN
ejpam-6613	326	1	x2	x2	PROPN
ejpam-6613	326	2	then	then	ADV
ejpam-6613	326	3	there	there	PRON
ejpam-6613	326	4	is	be	VERB
ejpam-6613	326	5	a	a	DET
ejpam-6613	326	6	natural	natural	ADJ
ejpam-6613	326	7	number	number	NOUN
ejpam-6613	326	8	n	n	CCONJ
ejpam-6613	326	9	,	,	PUNCT
ejpam-6613	326	10	and	and	CCONJ
ejpam-6613	326	11	there	there	PRON
ejpam-6613	326	12	are	be	VERB
ejpam-6613	326	13	{	{	PUNCT
ejpam-6613	326	14	αi}ni=1	αi}ni=1	PROPN
ejpam-6613	326	15	⊆	⊆	NUM
ejpam-6613	326	16	c	c	X
ejpam-6613	326	17	,	,	PUNCT
ejpam-6613	326	18	{	{	PUNCT
ejpam-6613	326	19	xi}ni=1	xi}ni=1	PROPN
ejpam-6613	326	20	⊆	⊆	NUM
ejpam-6613	326	21	x1	x1	NUM
ejpam-6613	326	22	,	,	PUNCT
ejpam-6613	326	23	{	{	PUNCT
ejpam-6613	326	24	yi}ni=1	yi}ni=1	ADV
ejpam-6613	326	25	⊆	⊆	NUM
ejpam-6613	326	26	x2	x2	NOUN
ejpam-6613	326	27	such	such	ADJ
ejpam-6613	326	28	that	that	SCONJ
ejpam-6613	326	29	ξ	ξ	X
ejpam-6613	326	30	=	=	SYM
ejpam-6613	326	31	∑n	∑n	PROPN
ejpam-6613	326	32	i=1	i=1	PROPN
ejpam-6613	327	1	αi(xi	αi(xi	PROPN
ejpam-6613	327	2	2	2	NUM
ejpam-6613	327	3	⊙	⊙	X
ejpam-6613	327	4	yi	yi	PROPN
ejpam-6613	327	5	)	)	PUNCT
ejpam-6613	327	6	.	.	PUNCT
ejpam-6613	328	1	however	however	ADV
ejpam-6613	328	2	,	,	PUNCT
ejpam-6613	328	3	for	for	ADP
ejpam-6613	328	4	all	all	DET
ejpam-6613	328	5	(	(	PUNCT
ejpam-6613	328	6	x	x	NOUN
ejpam-6613	328	7	,	,	PUNCT
ejpam-6613	328	8	y	y	NOUN
ejpam-6613	328	9	)	)	PUNCT
ejpam-6613	328	10	∈	∈	PROPN
ejpam-6613	329	1	x1	x1	NUM
ejpam-6613	329	2	×	×	NOUN
ejpam-6613	329	3	x2	x2	NOUN
ejpam-6613	329	4	and	and	CCONJ
ejpam-6613	329	5	all	all	PRON
ejpam-6613	329	6	(	(	PUNCT
ejpam-6613	329	7	r	r	NOUN
ejpam-6613	329	8	,	,	PUNCT
ejpam-6613	329	9	s	s	PART
ejpam-6613	329	10	)	)	PUNCT
ejpam-6613	329	11	∈	∈	PROPN
ejpam-6613	330	1	x1	x1	NUM
ejpam-6613	330	2	×	×	NOUN
ejpam-6613	330	3	x2	x2	NOUN
ejpam-6613	330	4	with	with	ADP
ejpam-6613	330	5	∥r∥x	∥r∥x	NOUN
ejpam-6613	330	6	=	=	SYM
ejpam-6613	330	7	1	1	NUM
ejpam-6613	330	8	,	,	PUNCT
ejpam-6613	330	9	∥s∥y	∥s∥y	NOUN
ejpam-6613	330	10	=	=	SYM
ejpam-6613	331	1	1	1	NUM
ejpam-6613	331	2	it	it	PRON
ejpam-6613	331	3	is	be	AUX
ejpam-6613	331	4	necessary	necessary	ADJ
ejpam-6613	331	5	to	to	PART
ejpam-6613	331	6	,	,	PUNCT
ejpam-6613	331	7	ξ((x	ξ((x	PROPN
ejpam-6613	331	8	,	,	PUNCT
ejpam-6613	331	9	y	y	PROPN
ejpam-6613	331	10	)	)	PUNCT
ejpam-6613	331	11	,	,	PUNCT
ejpam-6613	331	12	(	(	PUNCT
ejpam-6613	331	13	r	r	NOUN
ejpam-6613	331	14	,	,	PUNCT
ejpam-6613	331	15	s	s	NOUN
ejpam-6613	331	16	)	)	PUNCT
ejpam-6613	331	17	)	)	PUNCT
ejpam-6613	332	1	=	=	PRON
ejpam-6613	332	2	(	(	PUNCT
ejpam-6613	332	3	n∑	n∑	NOUN
ejpam-6613	332	4	i=1	i=1	X
ejpam-6613	333	1	αi(xi	αi(xi	PROPN
ejpam-6613	333	2	2	2	NUM
ejpam-6613	333	3	⊙	⊙	NOUN
ejpam-6613	333	4	yi))((x	yi))((x	NOUN
ejpam-6613	333	5	,	,	PUNCT
ejpam-6613	333	6	y	y	PROPN
ejpam-6613	333	7	)	)	PUNCT
ejpam-6613	333	8	,	,	PUNCT
ejpam-6613	333	9	(	(	PUNCT
ejpam-6613	333	10	r	r	NOUN
ejpam-6613	333	11	,	,	PUNCT
ejpam-6613	333	12	s	s	NOUN
ejpam-6613	333	13	)	)	PUNCT
ejpam-6613	333	14	)	)	PUNCT
ejpam-6613	334	1	=	=	PUNCT
ejpam-6613	335	1	n∑	n∑	NOUN
ejpam-6613	335	2	i=1	i=1	PROPN
ejpam-6613	336	1	(	(	PUNCT
ejpam-6613	336	2	αi(xi	αi(xi	PROPN
ejpam-6613	336	3	2	2	NUM
ejpam-6613	336	4	⊙	⊙	NOUN
ejpam-6613	336	5	yi))((x	yi))((x	NOUN
ejpam-6613	336	6	,	,	PUNCT
ejpam-6613	336	7	y	y	PROPN
ejpam-6613	336	8	)	)	PUNCT
ejpam-6613	336	9	,	,	PUNCT
ejpam-6613	336	10	(	(	PUNCT
ejpam-6613	336	11	r	r	NOUN
ejpam-6613	336	12	,	,	PUNCT
ejpam-6613	336	13	s	s	NOUN
ejpam-6613	336	14	)	)	PUNCT
ejpam-6613	336	15	)	)	PUNCT
ejpam-6613	337	1	=	=	PUNCT
ejpam-6613	338	1	n∑	n∑	PROPN
ejpam-6613	338	2	i=1	i=1	PROPN
ejpam-6613	338	3	αi⟨xi	αi⟨xi	PROPN
ejpam-6613	338	4	,	,	PUNCT
ejpam-6613	338	5	x|r⟩x⟨yi	x|r⟩x⟨yi	PROPN
ejpam-6613	338	6	,	,	PUNCT
ejpam-6613	338	7	y|s⟩y	y|s⟩y	NOUN
ejpam-6613	338	8	=	=	SYM
ejpam-6613	338	9	n∑	n∑	PROPN
ejpam-6613	338	10	i=1	i=1	PROPN
ejpam-6613	338	11	αi⟨xi	αi⟨xi	PROPN
ejpam-6613	338	12	,	,	PUNCT
ejpam-6613	338	13	x⟩x∥r∥2x⟨yi	x⟩x∥r∥2x⟨yi	PROPN
ejpam-6613	338	14	,	,	PUNCT
ejpam-6613	338	15	y⟩y	y⟩y	PROPN
ejpam-6613	338	16	∥s∥2y	∥s∥2y	PROPN
ejpam-6613	338	17	=	=	SYM
ejpam-6613	338	18	n∑	n∑	PROPN
ejpam-6613	338	19	i=1	i=1	PROPN
ejpam-6613	338	20	αi⟨xi	αi⟨xi	PROPN
ejpam-6613	338	21	,	,	PUNCT
ejpam-6613	338	22	x⟩x⟨yi	x⟩x⟨yi	PRON
ejpam-6613	338	23	,	,	PUNCT
ejpam-6613	338	24	y⟩y	y⟩y	PROPN
ejpam-6613	338	25	=	=	PUNCT
ejpam-6613	338	26	n∑	n∑	PROPN
ejpam-6613	338	27	i=1	i=1	PROPN
ejpam-6613	338	28	αi(xi⊙̃yi)((x	αi(xi⊙̃yi)((x	NOUN
ejpam-6613	338	29	,	,	PUNCT
ejpam-6613	338	30	y	y	PROPN
ejpam-6613	338	31	)	)	PUNCT
ejpam-6613	338	32	,	,	PUNCT
ejpam-6613	338	33	(	(	PUNCT
ejpam-6613	338	34	r	r	NOUN
ejpam-6613	338	35	,	,	PUNCT
ejpam-6613	338	36	s	s	NOUN
ejpam-6613	338	37	)	)	PUNCT
ejpam-6613	338	38	)	)	PUNCT
ejpam-6613	339	1	=	=	PRON
ejpam-6613	339	2	(	(	PUNCT
ejpam-6613	339	3	n∑	n∑	NOUN
ejpam-6613	339	4	i=1	i=1	PROPN
ejpam-6613	339	5	αi(xi⊙̃yi))((x	αi(xi⊙̃yi))((x	PROPN
ejpam-6613	339	6	,	,	PUNCT
ejpam-6613	339	7	y	y	NOUN
ejpam-6613	339	8	)	)	PUNCT
ejpam-6613	339	9	,	,	PUNCT
ejpam-6613	339	10	(	(	PUNCT
ejpam-6613	339	11	r	r	NOUN
ejpam-6613	339	12	,	,	PUNCT
ejpam-6613	339	13	s	s	NOUN
ejpam-6613	339	14	)	)	PUNCT
ejpam-6613	339	15	)	)	PUNCT
ejpam-6613	339	16	then	then	ADV
ejpam-6613	339	17	,	,	PUNCT
ejpam-6613	339	18	ξ	ξ	X
ejpam-6613	339	19	=	=	SYM
ejpam-6613	339	20	∑n	∑n	PROPN
ejpam-6613	339	21	i=1	i=1	PROPN
ejpam-6613	339	22	αi(xi⊙̃yi	αi(xi⊙̃yi	PROPN
ejpam-6613	339	23	)	)	PUNCT
ejpam-6613	339	24	∈	∈	PROPN
ejpam-6613	339	25	x1⊙̃x2	x1⊙̃x2	PROPN
ejpam-6613	339	26	.	.	PUNCT
ejpam-6613	340	1	therefore	therefore	ADV
ejpam-6613	340	2	ξ	ξ	X
ejpam-6613	340	3	∈	∈	PROPN
ejpam-6613	340	4	x1⊙̃x2	x1⊙̃x2	PROPN
ejpam-6613	340	5	.	.	PUNCT
ejpam-6613	341	1	proposition	proposition	NOUN
ejpam-6613	341	2	10	10	NUM
ejpam-6613	341	3	.	.	PUNCT
ejpam-6613	342	1	let	let	VERB
ejpam-6613	342	2	(	(	PUNCT
ejpam-6613	342	3	x1	x1	ADJ
ejpam-6613	342	4	,	,	PUNCT
ejpam-6613	342	5	⟨	⟨	NOUN
ejpam-6613	342	6	·	·	PUNCT
ejpam-6613	342	7	,	,	PUNCT
ejpam-6613	342	8	·	·	PUNCT
ejpam-6613	342	9	|	|	ADV
ejpam-6613	342	10	·	·	SYM
ejpam-6613	342	11	⟩1	⟩1	NOUN
ejpam-6613	342	12	)	)	PUNCT
ejpam-6613	342	13	and	and	CCONJ
ejpam-6613	342	14	(	(	PUNCT
ejpam-6613	342	15	x2	x2	INTJ
ejpam-6613	342	16	,	,	PUNCT
ejpam-6613	342	17	⟨	⟨	NOUN
ejpam-6613	342	18	·	·	PUNCT
ejpam-6613	342	19	,	,	PUNCT
ejpam-6613	342	20	·	·	PUNCT
ejpam-6613	342	21	|	|	ADV
ejpam-6613	342	22	·	·	SYM
ejpam-6613	342	23	⟩2	⟩2	NOUN
ejpam-6613	342	24	)	)	PUNCT
ejpam-6613	342	25	be	be	AUX
ejpam-6613	342	26	spaces	space	NOUN
ejpam-6613	342	27	with	with	ADP
ejpam-6613	342	28	a	a	DET
ejpam-6613	342	29	generalized	generalized	ADJ
ejpam-6613	342	30	2	2	NUM
ejpam-6613	342	31	-	-	PUNCT
ejpam-6613	342	32	inner	inner	ADJ
ejpam-6613	342	33	product	product	NOUN
ejpam-6613	342	34	,	,	PUNCT
ejpam-6613	342	35	if	if	SCONJ
ejpam-6613	342	36	t1	t1	PROPN
ejpam-6613	342	37	∈	∈	PROPN
ejpam-6613	342	38	2	2	NUM
ejpam-6613	342	39	b(x1	b(x1	NOUN
ejpam-6613	342	40	)	)	PUNCT
ejpam-6613	342	41	,	,	PUNCT
ejpam-6613	342	42	t2	t2	NOUN
ejpam-6613	342	43	∈	∈	PROPN
ejpam-6613	342	44	2	2	NUM
ejpam-6613	342	45	b(x2	b(x2	NOUN
ejpam-6613	342	46	)	)	PUNCT
ejpam-6613	342	47	then	then	ADV
ejpam-6613	342	48	t1	t1	PROPN
ejpam-6613	342	49	2	2	NUM
ejpam-6613	342	50	⊙	⊙	PROPN
ejpam-6613	342	51	t2	t2	PROPN
ejpam-6613	342	52	∈	∈	PROPN
ejpam-6613	342	53	2	2	NUM
ejpam-6613	342	54	b(x1	b(x1	NOUN
ejpam-6613	342	55	2	2	NUM
ejpam-6613	342	56	⊙x2	⊙x2	PROPN
ejpam-6613	342	57	)	)	PUNCT
ejpam-6613	342	58	.	.	PUNCT
ejpam-6613	343	1	proof	proof	NOUN
ejpam-6613	343	2	.	.	PUNCT
ejpam-6613	344	1	let	let	VERB
ejpam-6613	344	2	ξ1	ξ1	NOUN
ejpam-6613	344	3	=	=	PUNCT
ejpam-6613	345	1	∑n	∑n	PROPN
ejpam-6613	345	2	i=1	i=1	PROPN
ejpam-6613	345	3	xi	xi	PROPN
ejpam-6613	345	4	2	2	NUM
ejpam-6613	345	5	⊙	⊙	PROPN
ejpam-6613	345	6	yi	yi	PROPN
ejpam-6613	345	7	,	,	PUNCT
ejpam-6613	345	8	ξ2	ξ2	NOUN
ejpam-6613	345	9	=	=	PUNCT
ejpam-6613	345	10	∑m	∑m	PROPN
ejpam-6613	346	1	k=1	k=1	ADJ
ejpam-6613	346	2	rk	rk	PROPN
ejpam-6613	346	3	2	2	NUM
ejpam-6613	346	4	⊙	⊙	NOUN
ejpam-6613	346	5	sk	sk	ADP
ejpam-6613	346	6	∈	∈	PROPN
ejpam-6613	346	7	x1	x1	PROPN
ejpam-6613	346	8	2	2	NUM
ejpam-6613	346	9	⊙	⊙	NOUN
ejpam-6613	346	10	x2	x2	NOUN
ejpam-6613	346	11	with	with	ADP
ejpam-6613	346	12	∥ξ1	∥ξ1	NOUN
ejpam-6613	346	13	,	,	PUNCT
ejpam-6613	346	14	ξ2∥	ξ2∥	PROPN
ejpam-6613	346	15	2	2	NUM
ejpam-6613	346	16	⊙	⊙	NOUN
ejpam-6613	346	17	=	=	SYM
ejpam-6613	346	18	1	1	X
ejpam-6613	346	19	.	.	PUNCT
ejpam-6613	347	1	thus	thus	ADV
ejpam-6613	347	2	,	,	PUNCT
ejpam-6613	347	3	we	we	PRON
ejpam-6613	347	4	have	have	VERB
ejpam-6613	347	5	∥∥∥∥(t1	∥∥∥∥(t1	PROPN
ejpam-6613	347	6	2	2	NUM
ejpam-6613	347	7	⊙	⊙	NOUN
ejpam-6613	347	8	t2)(ξ1	t2)(ξ1	NUM
ejpam-6613	347	9	)	)	PUNCT
ejpam-6613	347	10	,	,	PUNCT
ejpam-6613	347	11	ξ2	ξ2	PROPN
ejpam-6613	347	12	∥∥∥∥22	∥∥∥∥22	PROPN
ejpam-6613	347	13	⊙	⊙	NOUN
ejpam-6613	347	14	=	=	SYM
ejpam-6613	348	1	〈	〈	PROPN
ejpam-6613	348	2	(	(	PUNCT
ejpam-6613	348	3	t1	t1	NOUN
ejpam-6613	348	4	2	2	NUM
ejpam-6613	348	5	⊙	⊙	NOUN
ejpam-6613	348	6	t2)(ξ1	t2)(ξ1	NUM
ejpam-6613	348	7	)	)	PUNCT
ejpam-6613	348	8	,	,	PUNCT
ejpam-6613	348	9	(	(	PUNCT
ejpam-6613	348	10	t1	t1	NOUN
ejpam-6613	348	11	2	2	NUM
ejpam-6613	348	12	⊙	⊙	NOUN
ejpam-6613	348	13	t2)(ξ1)|ξ2	t2)(ξ1)|ξ2	PROPN
ejpam-6613	348	14	〉	〉	NOUN
ejpam-6613	348	15	2	2	NUM
ejpam-6613	348	16	⊙	⊙	NOUN
ejpam-6613	348	17	=	=	SYM
ejpam-6613	348	18	〈	〈	PROPN
ejpam-6613	348	19	n∑	n∑	NOUN
ejpam-6613	348	20	i=1	i=1	X
ejpam-6613	349	1	t1(xi	t1(xi	PROPN
ejpam-6613	349	2	)	)	PUNCT
ejpam-6613	349	3	2	2	NUM
ejpam-6613	349	4	⊙	⊙	NOUN
ejpam-6613	349	5	t2(yi	t2(yi	NUM
ejpam-6613	349	6	)	)	PUNCT
ejpam-6613	349	7	,	,	PUNCT
ejpam-6613	349	8	n∑	n∑	PROPN
ejpam-6613	349	9	j=1	j=1	PROPN
ejpam-6613	349	10	t1(xj	t1(xj	PROPN
ejpam-6613	349	11	)	)	PUNCT
ejpam-6613	349	12	2	2	NUM
ejpam-6613	349	13	⊙	⊙	NOUN
ejpam-6613	349	14	t2(yj)|	t2(yj)|	X
ejpam-6613	350	1	m∑	m∑	ADV
ejpam-6613	350	2	k=1	k=1	PROPN
ejpam-6613	350	3	rk	rk	PROPN
ejpam-6613	350	4	2	2	NUM
ejpam-6613	350	5	⊙	⊙	NOUN
ejpam-6613	350	6	sk	sk	VERB
ejpam-6613	350	7	〉	〉	NOUN
ejpam-6613	350	8	2	2	NUM
ejpam-6613	350	9	⊙	⊙	NOUN
ejpam-6613	351	1	=	=	PROPN
ejpam-6613	352	1	n∑	n∑	PROPN
ejpam-6613	352	2	i=1	i=1	PROPN
ejpam-6613	353	1	n∑	n∑	PROPN
ejpam-6613	353	2	j=1	j=1	NOUN
ejpam-6613	354	1	m∑	m∑	VERB
ejpam-6613	354	2	k=1	k=1	PROPN
ejpam-6613	354	3	δi	δi	PROPN
ejpam-6613	354	4	,	,	PUNCT
ejpam-6613	354	5	j	j	PROPN
ejpam-6613	354	6	⟨t1(xi	⟨t1(xi	PROPN
ejpam-6613	354	7	)	)	PUNCT
ejpam-6613	354	8	,	,	PUNCT
ejpam-6613	354	9	t1(xj)|rk⟩1	t1(xj)|rk⟩1	X
ejpam-6613	354	10	⟨t2(yi	⟨t2(yi	NUM
ejpam-6613	354	11	)	)	PUNCT
ejpam-6613	354	12	,	,	PUNCT
ejpam-6613	354	13	t2(yj)|sk⟩2	t2(yj)|sk⟩2	PUNCT
ejpam-6613	355	1	=	=	PUNCT
ejpam-6613	356	1	n∑	n∑	PROPN
ejpam-6613	356	2	i=1	i=1	PROPN
ejpam-6613	357	1	m∑	m∑	INTJ
ejpam-6613	357	2	k=1	k=1	PROPN
ejpam-6613	357	3	⟨t1(xi	⟨t1(xi	PROPN
ejpam-6613	357	4	)	)	PUNCT
ejpam-6613	357	5	,	,	PUNCT
ejpam-6613	357	6	t1(xj)|rk⟩1	t1(xj)|rk⟩1	X
ejpam-6613	357	7	⟨t2(yi	⟨t2(yi	NUM
ejpam-6613	357	8	)	)	PUNCT
ejpam-6613	357	9	,	,	PUNCT
ejpam-6613	357	10	t2(yj)|sk⟩2	t2(yj)|sk⟩2	PUNCT
ejpam-6613	358	1	=	=	PUNCT
ejpam-6613	359	1	n∑	n∑	PROPN
ejpam-6613	359	2	i=1	i=1	PROPN
ejpam-6613	360	1	m∑	m∑	INTJ
ejpam-6613	360	2	k=1	k=1	PROPN
ejpam-6613	361	1	∥t1(xi	∥t1(xi	NUM
ejpam-6613	361	2	)	)	PUNCT
ejpam-6613	361	3	,	,	PUNCT
ejpam-6613	362	1	rk∥21	rk∥21	NOUN
ejpam-6613	362	2	∥t2(yi	∥t2(yi	NUM
ejpam-6613	362	3	)	)	PUNCT
ejpam-6613	362	4	,	,	PUNCT
ejpam-6613	362	5	sk∥22	sk∥22	PROPN
ejpam-6613	362	6	m.	m.	PROPN
ejpam-6613	362	7	luis	luis	PROPN
ejpam-6613	362	8	,	,	PUNCT
ejpam-6613	362	9	f.	f.	PROPN
ejpam-6613	362	10	osmin	osmin	PROPN
ejpam-6613	362	11	,	,	PUNCT
ejpam-6613	362	12	s.	s.	PROPN
ejpam-6613	362	13	arley	arley	PROPN
ejpam-6613	362	14	/	/	SYM
ejpam-6613	362	15	eur	eur	PROPN
ejpam-6613	362	16	.	.	PUNCT
ejpam-6613	363	1	j.	j.	PROPN
ejpam-6613	363	2	pure	pure	PROPN
ejpam-6613	363	3	appl	appl	PROPN
ejpam-6613	363	4	.	.	PROPN
ejpam-6613	363	5	math	math	PROPN
ejpam-6613	363	6	,	,	PUNCT
ejpam-6613	363	7	18	18	NUM
ejpam-6613	363	8	(	(	PUNCT
ejpam-6613	363	9	4	4	NUM
ejpam-6613	363	10	)	)	PUNCT
ejpam-6613	363	11	(	(	PUNCT
ejpam-6613	363	12	2025	2025	NUM
ejpam-6613	363	13	)	)	PUNCT
ejpam-6613	363	14	,	,	PUNCT
ejpam-6613	363	15	6613	6613	NUM
ejpam-6613	363	16	15	15	NUM
ejpam-6613	363	17	of	of	ADP
ejpam-6613	363	18	17	17	NUM
ejpam-6613	363	19	≤	≤	NOUN
ejpam-6613	363	20	n∑	n∑	PART
ejpam-6613	363	21	i=1	i=1	PROPN
ejpam-6613	364	1	m∑	m∑	INTJ
ejpam-6613	364	2	k=1	k=1	PROPN
ejpam-6613	364	3	∥t1∥2	∥t1∥2	PROPN
ejpam-6613	364	4	∥xi	∥xi	PROPN
ejpam-6613	364	5	,	,	PUNCT
ejpam-6613	364	6	rk∥21	rk∥21	NOUN
ejpam-6613	364	7	∥t2∥2	∥t2∥2	PROPN
ejpam-6613	364	8	∥yi	∥yi	PROPN
ejpam-6613	364	9	,	,	PUNCT
ejpam-6613	364	10	sk∥22	sk∥22	NOUN
ejpam-6613	364	11	=	=	SYM
ejpam-6613	364	12	∥t1∥2	∥t1∥2	PROPN
ejpam-6613	365	1	∥t2∥2	∥t2∥2	VERB
ejpam-6613	365	2	n∑	n∑	PROPN
ejpam-6613	365	3	i=1	i=1	INTJ
ejpam-6613	366	1	m∑	m∑	INTJ
ejpam-6613	366	2	k=1	k=1	PROPN
ejpam-6613	366	3	∥xi	∥xi	PROPN
ejpam-6613	366	4	,	,	PUNCT
ejpam-6613	366	5	rk∥21	rk∥21	NOUN
ejpam-6613	366	6	∥yi	∥yi	PROPN
ejpam-6613	366	7	,	,	PUNCT
ejpam-6613	366	8	sk∥	sk∥	PROPN
ejpam-6613	366	9	2	2	NUM
ejpam-6613	366	10	2	2	NUM
ejpam-6613	366	11	=	=	SYM
ejpam-6613	366	12	∥t1∥2	∥t1∥2	NOUN
ejpam-6613	366	13	∥t2∥2	∥t2∥2	VERB
ejpam-6613	366	14	∥ξ1	∥ξ1	NOUN
ejpam-6613	366	15	,	,	PUNCT
ejpam-6613	366	16	ξ2∥22	ξ2∥22	PROPN
ejpam-6613	366	17	⊙	⊙	NOUN
ejpam-6613	366	18	=	=	SYM
ejpam-6613	366	19	∥t1∥2	∥t1∥2	PROPN
ejpam-6613	367	1	∥t2∥2	∥t2∥2	VERB
ejpam-6613	367	2	therefore	therefore	ADV
ejpam-6613	367	3	∥∥∥∥(t1	∥∥∥∥(t1	PROPN
ejpam-6613	367	4	2	2	NUM
ejpam-6613	367	5	⊙	⊙	NOUN
ejpam-6613	367	6	t2)(ξ1	t2)(ξ1	NUM
ejpam-6613	367	7	)	)	PUNCT
ejpam-6613	367	8	,	,	PUNCT
ejpam-6613	367	9	ξ2	ξ2	NOUN
ejpam-6613	367	10	∥∥∥∥	∥∥∥∥	NUM
ejpam-6613	367	11	2	2	NUM
ejpam-6613	367	12	⊙	⊙	NOUN
ejpam-6613	367	13	≤	≤	NOUN
ejpam-6613	367	14	∥t1∥	∥t1∥	PROPN
ejpam-6613	367	15	∥t2∥	∥t2∥	PROPN
ejpam-6613	367	16	.	.	PUNCT
ejpam-6613	368	1	proposition	proposition	NOUN
ejpam-6613	368	2	11	11	NUM
ejpam-6613	368	3	.	.	PUNCT
ejpam-6613	369	1	let	let	VERB
ejpam-6613	369	2	(	(	PUNCT
ejpam-6613	369	3	x1	x1	ADJ
ejpam-6613	369	4	,	,	PUNCT
ejpam-6613	369	5	⟨	⟨	NOUN
ejpam-6613	369	6	·	·	PUNCT
ejpam-6613	369	7	,	,	PUNCT
ejpam-6613	369	8	·	·	PUNCT
ejpam-6613	369	9	|·⟩x1	|·⟩x1	NOUN
ejpam-6613	369	10	)	)	PUNCT
ejpam-6613	369	11	,	,	PUNCT
ejpam-6613	369	12	(	(	PUNCT
ejpam-6613	369	13	x2	x2	INTJ
ejpam-6613	369	14	,	,	PUNCT
ejpam-6613	369	15	⟨	⟨	NOUN
ejpam-6613	369	16	·	·	SYM
ejpam-6613	369	17	,	,	PUNCT
ejpam-6613	369	18	·	·	PUNCT
ejpam-6613	369	19	|·⟩x2	|·⟩x2	NUM
ejpam-6613	369	20	)	)	PUNCT
ejpam-6613	369	21	,	,	PUNCT
ejpam-6613	369	22	(	(	PUNCT
ejpam-6613	369	23	y1	y1	INTJ
ejpam-6613	369	24	,	,	PUNCT
ejpam-6613	369	25	⟨	⟨	NOUN
ejpam-6613	369	26	·	·	SYM
ejpam-6613	369	27	,	,	PUNCT
ejpam-6613	369	28	·	·	PUNCT
ejpam-6613	369	29	|·⟩y1	|·⟩y1	NUM
ejpam-6613	369	30	)	)	PUNCT
ejpam-6613	369	31	,	,	PUNCT
ejpam-6613	369	32	(	(	PUNCT
ejpam-6613	369	33	y2	y2	INTJ
ejpam-6613	369	34	,	,	PUNCT
ejpam-6613	369	35	⟨	⟨	NOUN
ejpam-6613	369	36	·	·	SYM
ejpam-6613	369	37	,	,	PUNCT
ejpam-6613	369	38	·	·	PUNCT
ejpam-6613	369	39	|·⟩y2	|·⟩y2	ADV
ejpam-6613	369	40	)	)	PUNCT
ejpam-6613	369	41	spaces	space	NOUN
ejpam-6613	369	42	with	with	ADP
ejpam-6613	369	43	generalized	generalized	ADJ
ejpam-6613	369	44	2	2	NUM
ejpam-6613	369	45	-	-	PUNCT
ejpam-6613	369	46	inner	inner	ADJ
ejpam-6613	369	47	product	product	NOUN
ejpam-6613	369	48	and	and	CCONJ
ejpam-6613	369	49	t1	t1	NOUN
ejpam-6613	369	50	:	:	PUNCT
ejpam-6613	370	1	x1	x1	PROPN
ejpam-6613	370	2	→	→	SYM
ejpam-6613	370	3	x2	x2	PROPN
ejpam-6613	370	4	and	and	CCONJ
ejpam-6613	370	5	t2	t2	PROPN
ejpam-6613	370	6	:	:	PUNCT
ejpam-6613	370	7	y1	y1	INTJ
ejpam-6613	370	8	→	→	PUNCT
ejpam-6613	370	9	y2	y2	NOUN
ejpam-6613	370	10	linear	linear	PROPN
ejpam-6613	370	11	operators	operator	NOUN
ejpam-6613	370	12	,	,	PUNCT
ejpam-6613	370	13	then	then	ADV
ejpam-6613	370	14	t1	t1	PROPN
ejpam-6613	370	15	×	×	PROPN
ejpam-6613	370	16	t2	t2	NOUN
ejpam-6613	370	17	:	:	PUNCT
ejpam-6613	370	18	x1	x1	NUM
ejpam-6613	370	19	×	×	NOUN
ejpam-6613	370	20	y1	y1	INTJ
ejpam-6613	370	21	→	→	SYM
ejpam-6613	370	22	x2	x2	NUM
ejpam-6613	370	23	×	×	NOUN
ejpam-6613	370	24	y2	y2	PROPN
ejpam-6613	370	25	(	(	PUNCT
ejpam-6613	370	26	x1	x1	PROPN
ejpam-6613	370	27	,	,	PUNCT
ejpam-6613	370	28	y1	y1	PROPN
ejpam-6613	370	29	)	)	PUNCT
ejpam-6613	370	30	→	→	SYM
ejpam-6613	370	31	(	(	PUNCT
ejpam-6613	370	32	t1	t1	NUM
ejpam-6613	370	33	×	×	PROPN
ejpam-6613	370	34	t2)(x1	t2)(x1	NOUN
ejpam-6613	370	35	,	,	PUNCT
ejpam-6613	370	36	y1	y1	NOUN
ejpam-6613	370	37	)	)	PUNCT
ejpam-6613	370	38	=	=	PUNCT
ejpam-6613	370	39	t1(x1)⊙	t1(x1)⊙	X
ejpam-6613	370	40	t2(y2	t2(y2	NOUN
ejpam-6613	370	41	)	)	PUNCT
ejpam-6613	370	42	is	be	AUX
ejpam-6613	370	43	a	a	DET
ejpam-6613	370	44	bilinear	bilinear	NOUN
ejpam-6613	370	45	operator	operator	NOUN
ejpam-6613	370	46	.	.	PUNCT
ejpam-6613	371	1	proof	proof	NOUN
ejpam-6613	371	2	.	.	PUNCT
ejpam-6613	372	1	in	in	ADP
ejpam-6613	372	2	fact	fact	NOUN
ejpam-6613	372	3	,	,	PUNCT
ejpam-6613	372	4	note	note	VERB
ejpam-6613	372	5	that	that	SCONJ
ejpam-6613	372	6	for	for	ADP
ejpam-6613	372	7	all	all	DET
ejpam-6613	372	8	x1	x1	PROPN
ejpam-6613	372	9	,	,	PUNCT
ejpam-6613	372	10	x	x	SYM
ejpam-6613	372	11	′	′	NOUN
ejpam-6613	372	12	1	1	NUM
ejpam-6613	372	13	∈	∈	NOUN
ejpam-6613	372	14	x1	x1	NOUN
ejpam-6613	372	15	and	and	CCONJ
ejpam-6613	372	16	all	all	DET
ejpam-6613	372	17	y1	y1	NOUN
ejpam-6613	372	18	,	,	PUNCT
ejpam-6613	372	19	y	y	PROPN
ejpam-6613	372	20	′	′	NOUN
ejpam-6613	372	21	1	1	NUM
ejpam-6613	372	22	∈	∈	NOUN
ejpam-6613	372	23	y1	y1	NOUN
ejpam-6613	372	24	is	be	AUX
ejpam-6613	372	25	met	meet	VERB
ejpam-6613	372	26	(	(	PUNCT
ejpam-6613	372	27	t1	t1	NOUN
ejpam-6613	372	28	×	×	NOUN
ejpam-6613	372	29	t2)(αx1	t2)(αx1	PROPN
ejpam-6613	372	30	+	+	CCONJ
ejpam-6613	373	1	x′1	x′1	PROPN
ejpam-6613	373	2	,	,	PUNCT
ejpam-6613	373	3	y1	y1	ADJ
ejpam-6613	373	4	)	)	PUNCT
ejpam-6613	373	5	=	=	SYM
ejpam-6613	373	6	t1(αx1	t1(αx1	NOUN
ejpam-6613	374	1	+	+	CCONJ
ejpam-6613	374	2	x′1)⊙	x′1)⊙	PROPN
ejpam-6613	374	3	t2(y1	t2(y1	NUM
ejpam-6613	374	4	)	)	PUNCT
ejpam-6613	374	5	=	=	SYM
ejpam-6613	374	6	(	(	PUNCT
ejpam-6613	374	7	αt1(x1	αt1(x1	NUM
ejpam-6613	374	8	)	)	PUNCT
ejpam-6613	375	1	+	+	CCONJ
ejpam-6613	375	2	t1(x	t1(x	PUNCT
ejpam-6613	375	3	′	′	NUM
ejpam-6613	375	4	1))⊙	1))⊙	NUM
ejpam-6613	375	5	t2(y1	t2(y1	NUM
ejpam-6613	375	6	)	)	PUNCT
ejpam-6613	375	7	=	=	PUNCT
ejpam-6613	375	8	(	(	PUNCT
ejpam-6613	375	9	αt1(x1))⊙	αt1(x1))⊙	NOUN
ejpam-6613	375	10	t2(y1	t2(y1	NUM
ejpam-6613	375	11	)	)	PUNCT
ejpam-6613	375	12	+	+	NUM
ejpam-6613	375	13	t1(x	t1(x	NOUN
ejpam-6613	375	14	′	′	NUM
ejpam-6613	375	15	1)⊙	1)⊙	PROPN
ejpam-6613	375	16	t2(y1	t2(y1	NUM
ejpam-6613	375	17	)	)	PUNCT
ejpam-6613	375	18	=	=	SYM
ejpam-6613	375	19	α(t1(x1)⊙	α(t1(x1)⊙	PROPN
ejpam-6613	375	20	t2(y1	t2(y1	NUM
ejpam-6613	375	21	)	)	PUNCT
ejpam-6613	375	22	)	)	PUNCT
ejpam-6613	376	1	+	+	CCONJ
ejpam-6613	377	1	t1(x	t1(x	PUNCT
ejpam-6613	377	2	′	′	NUM
ejpam-6613	377	3	1)⊙	1)⊙	PROPN
ejpam-6613	377	4	t2(y1	t2(y1	NUM
ejpam-6613	377	5	)	)	PUNCT
ejpam-6613	377	6	=	=	SYM
ejpam-6613	377	7	α(t1	α(t1	NUM
ejpam-6613	377	8	×	×	PROPN
ejpam-6613	377	9	t2)(x1	t2)(x1	NOUN
ejpam-6613	377	10	,	,	PUNCT
ejpam-6613	377	11	y1	y1	NOUN
ejpam-6613	377	12	)	)	PUNCT
ejpam-6613	377	13	+	+	CCONJ
ejpam-6613	377	14	(	(	PUNCT
ejpam-6613	377	15	t1	t1	NUM
ejpam-6613	377	16	×	×	NOUN
ejpam-6613	378	1	t2)(x	t2)(x	NOUN
ejpam-6613	378	2	′	′	NUM
ejpam-6613	378	3	1	1	NUM
ejpam-6613	378	4	,	,	PUNCT
ejpam-6613	378	5	y1	y1	PROPN
ejpam-6613	378	6	)	)	PUNCT
ejpam-6613	378	7	(	(	PUNCT
ejpam-6613	378	8	t1	t1	NOUN
ejpam-6613	378	9	×	×	PROPN
ejpam-6613	378	10	t2)(x1	t2)(x1	NOUN
ejpam-6613	378	11	,	,	PUNCT
ejpam-6613	378	12	y1	y1	NOUN
ejpam-6613	378	13	+	+	NUM
ejpam-6613	378	14	y′1	y′1	X
ejpam-6613	378	15	)	)	PUNCT
ejpam-6613	378	16	=	=	PUNCT
ejpam-6613	378	17	t1(x1)⊙	t1(x1)⊙	X
ejpam-6613	378	18	t2(y1	t2(y1	NUM
ejpam-6613	378	19	+	+	CCONJ
ejpam-6613	378	20	y′1	y′1	X
ejpam-6613	378	21	)	)	PUNCT
ejpam-6613	378	22	=	=	SYM
ejpam-6613	378	23	t1(x1)⊙	t1(x1)⊙	X
ejpam-6613	378	24	(	(	PUNCT
ejpam-6613	378	25	(	(	PUNCT
ejpam-6613	378	26	t2(αy1	t2(αy1	NOUN
ejpam-6613	378	27	)	)	PUNCT
ejpam-6613	378	28	+	+	CCONJ
ejpam-6613	379	1	t2(y	t2(y	X
ejpam-6613	380	1	′	′	NUM
ejpam-6613	380	2	1	1	NUM
ejpam-6613	380	3	)	)	PUNCT
ejpam-6613	380	4	)	)	PUNCT
ejpam-6613	381	1	=	=	SYM
ejpam-6613	381	2	t1(x1)⊙	t1(x1)⊙	X
ejpam-6613	381	3	(	(	PUNCT
ejpam-6613	381	4	αt2(y1	αt2(y1	NUM
ejpam-6613	381	5	)	)	PUNCT
ejpam-6613	381	6	)	)	PUNCT
ejpam-6613	382	1	+	+	CCONJ
ejpam-6613	382	2	t1(x1)⊙	t1(x1)⊙	X
ejpam-6613	383	1	t2(y	t2(y	X
ejpam-6613	384	1	′	′	NUM
ejpam-6613	384	2	1	1	NUM
ejpam-6613	384	3	)	)	PUNCT
ejpam-6613	384	4	=	=	SYM
ejpam-6613	384	5	α(t1(x1)⊙	α(t1(x1)⊙	PROPN
ejpam-6613	384	6	t2(y1	t2(y1	NUM
ejpam-6613	384	7	)	)	PUNCT
ejpam-6613	384	8	)	)	PUNCT
ejpam-6613	385	1	+	+	CCONJ
ejpam-6613	385	2	t1(x1)⊙	t1(x1)⊙	X
ejpam-6613	386	1	t2(y	t2(y	X
ejpam-6613	387	1	′	′	NUM
ejpam-6613	387	2	1	1	X
ejpam-6613	387	3	)	)	PUNCT
ejpam-6613	387	4	=	=	PRON
ejpam-6613	387	5	α(t1	α(t1	NUM
ejpam-6613	387	6	×	×	PROPN
ejpam-6613	387	7	t2)(x1	t2)(x1	NOUN
ejpam-6613	387	8	,	,	PUNCT
ejpam-6613	387	9	y1	y1	NOUN
ejpam-6613	387	10	)	)	PUNCT
ejpam-6613	387	11	+	+	CCONJ
ejpam-6613	387	12	(	(	PUNCT
ejpam-6613	387	13	t1	t1	NUM
ejpam-6613	387	14	×	×	PROPN
ejpam-6613	387	15	t2)(x1	t2)(x1	NOUN
ejpam-6613	387	16	,	,	PUNCT
ejpam-6613	387	17	y	y	PROPN
ejpam-6613	387	18	′	′	NUM
ejpam-6613	387	19	1	1	NUM
ejpam-6613	387	20	)	)	PUNCT
ejpam-6613	387	21	5	5	NUM
ejpam-6613	387	22	.	.	PUNCT
ejpam-6613	387	23	conclusions	conclusion	NOUN
ejpam-6613	387	24	in	in	ADP
ejpam-6613	387	25	this	this	DET
ejpam-6613	387	26	research	research	NOUN
ejpam-6613	387	27	,	,	PUNCT
ejpam-6613	387	28	we	we	PRON
ejpam-6613	387	29	introduced	introduce	VERB
ejpam-6613	387	30	the	the	DET
ejpam-6613	387	31	notion	notion	NOUN
ejpam-6613	387	32	of	of	ADP
ejpam-6613	387	33	tensor	tensor	NOUN
ejpam-6613	387	34	product	product	NOUN
ejpam-6613	387	35	of	of	ADP
ejpam-6613	387	36	elements	element	NOUN
ejpam-6613	387	37	of	of	ADP
ejpam-6613	387	38	spaces	space	NOUN
ejpam-6613	387	39	equipped	equip	VERB
ejpam-6613	387	40	with	with	ADP
ejpam-6613	387	41	a	a	DET
ejpam-6613	387	42	generalized	generalized	ADJ
ejpam-6613	387	43	2	2	NUM
ejpam-6613	387	44	-	-	PUNCT
ejpam-6613	387	45	inner	inner	ADJ
ejpam-6613	387	46	product	product	NOUN
ejpam-6613	387	47	,	,	PUNCT
ejpam-6613	387	48	calling	call	VERB
ejpam-6613	387	49	it	it	PRON
ejpam-6613	387	50	the	the	DET
ejpam-6613	387	51	2	2	NUM
ejpam-6613	387	52	-	-	PUNCT
ejpam-6613	387	53	tensor	tensor	NOUN
ejpam-6613	387	54	product	product	NOUN
ejpam-6613	387	55	(	(	PUNCT
ejpam-6613	387	56	see	see	VERB
ejpam-6613	387	57	definition	definition	NOUN
ejpam-6613	387	58	5	5	NUM
ejpam-6613	387	59	)	)	PUNCT
ejpam-6613	387	60	,	,	PUNCT
ejpam-6613	387	61	which	which	PRON
ejpam-6613	387	62	turns	turn	VERB
ejpam-6613	387	63	out	out	ADP
ejpam-6613	387	64	to	to	PART
ejpam-6613	387	65	be	be	AUX
ejpam-6613	387	66	a	a	DET
ejpam-6613	387	67	bilinear	bilinear	NOUN
ejpam-6613	387	68	mapping	mapping	NOUN
ejpam-6613	387	69	in	in	ADP
ejpam-6613	387	70	the	the	DET
ejpam-6613	387	71	sense	sense	NOUN
ejpam-6613	387	72	of	of	ADP
ejpam-6613	387	73	proposition	proposition	NOUN
ejpam-6613	387	74	5	5	NUM
ejpam-6613	387	75	,	,	PUNCT
ejpam-6613	387	76	as	as	ADP
ejpam-6613	387	77	in	in	ADP
ejpam-6613	387	78	the	the	DET
ejpam-6613	387	79	classic	classic	ADJ
ejpam-6613	387	80	case	case	NOUN
ejpam-6613	387	81	.	.	PUNCT
ejpam-6613	388	1	we	we	PRON
ejpam-6613	388	2	also	also	ADV
ejpam-6613	388	3	define	define	VERB
ejpam-6613	388	4	the	the	DET
ejpam-6613	388	5	algebraic	algebraic	ADJ
ejpam-6613	388	6	tensor	tensor	NOUN
ejpam-6613	388	7	product	product	NOUN
ejpam-6613	388	8	of	of	ADP
ejpam-6613	388	9	spaces	space	NOUN
ejpam-6613	388	10	endowed	endow	VERB
ejpam-6613	388	11	with	with	ADP
ejpam-6613	388	12	a	a	DET
ejpam-6613	388	13	generalized	generalized	ADJ
ejpam-6613	388	14	2	2	NUM
ejpam-6613	388	15	-	-	PUNCT
ejpam-6613	388	16	inner	inner	ADJ
ejpam-6613	388	17	product	product	NOUN
ejpam-6613	388	18	and	and	CCONJ
ejpam-6613	388	19	prove	prove	VERB
ejpam-6613	388	20	that	that	SCONJ
ejpam-6613	388	21	it	it	PRON
ejpam-6613	388	22	satisfies	satisfy	VERB
ejpam-6613	388	23	all	all	DET
ejpam-6613	388	24	fundamental	fundamental	ADJ
ejpam-6613	388	25	properties	property	NOUN
ejpam-6613	388	26	(	(	PUNCT
ejpam-6613	388	27	see	see	VERB
ejpam-6613	388	28	proposition	proposition	NOUN
ejpam-6613	388	29	5	5	NUM
ejpam-6613	388	30	)	)	PUNCT
ejpam-6613	388	31	.	.	PUNCT
ejpam-6613	389	1	furthermore	furthermore	ADV
ejpam-6613	389	2	,	,	PUNCT
ejpam-6613	389	3	we	we	PRON
ejpam-6613	389	4	define	define	VERB
ejpam-6613	389	5	a	a	DET
ejpam-6613	389	6	generalized	generalized	ADJ
ejpam-6613	389	7	2	2	NUM
ejpam-6613	389	8	-	-	PUNCT
ejpam-6613	389	9	inner	inner	ADJ
ejpam-6613	389	10	product	product	NOUN
ejpam-6613	389	11	on	on	ADP
ejpam-6613	389	12	the	the	DET
ejpam-6613	389	13	algebraic	algebraic	ADJ
ejpam-6613	389	14	tensor	tensor	NOUN
ejpam-6613	389	15	product	product	NOUN
ejpam-6613	389	16	(	(	PUNCT
ejpam-6613	389	17	see	see	VERB
ejpam-6613	389	18	definition	definition	NOUN
ejpam-6613	389	19	6	6	NUM
ejpam-6613	389	20	)	)	PUNCT
ejpam-6613	389	21	and	and	CCONJ
ejpam-6613	389	22	induced	induce	VERB
ejpam-6613	389	23	a	a	DET
ejpam-6613	389	24	2	2	NUM
ejpam-6613	389	25	-	-	PUNCT
ejpam-6613	389	26	norm	norm	NOUN
ejpam-6613	389	27	,	,	PUNCT
ejpam-6613	389	28	which	which	PRON
ejpam-6613	389	29	we	we	PRON
ejpam-6613	389	30	call	call	VERB
ejpam-6613	389	31	the	the	DET
ejpam-6613	389	32	induced	induced	ADJ
ejpam-6613	389	33	2	2	NUM
ejpam-6613	389	34	-	-	PUNCT
ejpam-6613	389	35	tensor	tensor	NOUN
ejpam-6613	389	36	norm	norm	NOUN
ejpam-6613	389	37	(	(	PUNCT
ejpam-6613	389	38	see	see	VERB
ejpam-6613	389	39	m.	m.	PROPN
ejpam-6613	389	40	luis	luis	PROPN
ejpam-6613	389	41	,	,	PUNCT
ejpam-6613	389	42	f.	f.	PROPN
ejpam-6613	389	43	osmin	osmin	PROPN
ejpam-6613	389	44	,	,	PUNCT
ejpam-6613	389	45	s.	s.	PROPN
ejpam-6613	389	46	arley	arley	PROPN
ejpam-6613	389	47	/	/	SYM
ejpam-6613	389	48	eur	eur	PROPN
ejpam-6613	389	49	.	.	PUNCT
ejpam-6613	390	1	j.	j.	PROPN
ejpam-6613	390	2	pure	pure	PROPN
ejpam-6613	390	3	appl	appl	PROPN
ejpam-6613	390	4	.	.	PROPN
ejpam-6613	390	5	math	math	PROPN
ejpam-6613	390	6	,	,	PUNCT
ejpam-6613	390	7	18	18	NUM
ejpam-6613	390	8	(	(	PUNCT
ejpam-6613	390	9	4	4	NUM
ejpam-6613	390	10	)	)	PUNCT
ejpam-6613	390	11	(	(	PUNCT
ejpam-6613	390	12	2025	2025	NUM
ejpam-6613	390	13	)	)	PUNCT
ejpam-6613	390	14	,	,	PUNCT
ejpam-6613	390	15	6613	6613	NUM
ejpam-6613	390	16	16	16	NUM
ejpam-6613	390	17	of	of	ADP
ejpam-6613	390	18	17	17	NUM
ejpam-6613	390	19	theorem	theorem	NOUN
ejpam-6613	390	20	1	1	NUM
ejpam-6613	390	21	)	)	PUNCT
ejpam-6613	390	22	.	.	PUNCT
ejpam-6613	391	1	we	we	PRON
ejpam-6613	391	2	also	also	ADV
ejpam-6613	391	3	established	establish	VERB
ejpam-6613	391	4	the	the	DET
ejpam-6613	391	5	notion	notion	NOUN
ejpam-6613	391	6	of	of	ADP
ejpam-6613	391	7	the	the	DET
ejpam-6613	391	8	2	2	NUM
ejpam-6613	391	9	-	-	PUNCT
ejpam-6613	391	10	tensor	tensor	NOUN
ejpam-6613	391	11	product	product	NOUN
ejpam-6613	391	12	of	of	ADP
ejpam-6613	391	13	linear	linear	PROPN
ejpam-6613	391	14	operators	operator	NOUN
ejpam-6613	391	15	on	on	ADP
ejpam-6613	391	16	spaces	space	NOUN
ejpam-6613	391	17	with	with	ADP
ejpam-6613	391	18	a	a	DET
ejpam-6613	391	19	generalized	generalized	ADJ
ejpam-6613	391	20	2	2	NUM
ejpam-6613	391	21	-	-	PUNCT
ejpam-6613	391	22	inner	inner	ADJ
ejpam-6613	391	23	product	product	NOUN
ejpam-6613	391	24	(	(	PUNCT
ejpam-6613	391	25	see	see	VERB
ejpam-6613	391	26	definition	definition	NOUN
ejpam-6613	391	27	9	9	NUM
ejpam-6613	391	28	)	)	PUNCT
ejpam-6613	391	29	,	,	PUNCT
ejpam-6613	391	30	and	and	CCONJ
ejpam-6613	391	31	demonstrated	demonstrate	VERB
ejpam-6613	391	32	that	that	SCONJ
ejpam-6613	391	33	several	several	ADJ
ejpam-6613	391	34	well	well	ADV
ejpam-6613	391	35	-	-	PUNCT
ejpam-6613	391	36	known	know	VERB
ejpam-6613	391	37	identities	identity	NOUN
ejpam-6613	391	38	for	for	ADP
ejpam-6613	391	39	operators	operator	NOUN
ejpam-6613	391	40	in	in	ADP
ejpam-6613	391	41	inner	inner	ADJ
ejpam-6613	391	42	product	product	NOUN
ejpam-6613	391	43	spaces	space	NOUN
ejpam-6613	391	44	remain	remain	VERB
ejpam-6613	391	45	valid	valid	ADJ
ejpam-6613	391	46	in	in	ADP
ejpam-6613	391	47	this	this	DET
ejpam-6613	391	48	new	new	ADJ
ejpam-6613	391	49	structure	structure	NOUN
ejpam-6613	391	50	(	(	PUNCT
ejpam-6613	391	51	see	see	VERB
ejpam-6613	391	52	proposition	proposition	NOUN
ejpam-6613	391	53	8)	8)	NUM
ejpam-6613	391	54	.	.	PUNCT
ejpam-6613	392	1	finally	finally	ADV
ejpam-6613	392	2	,	,	PUNCT
ejpam-6613	392	3	the	the	DET
ejpam-6613	392	4	notions	notion	NOUN
ejpam-6613	392	5	established	establish	VERB
ejpam-6613	392	6	in	in	ADP
ejpam-6613	392	7	this	this	DET
ejpam-6613	392	8	work	work	NOUN
ejpam-6613	392	9	have	have	AUX
ejpam-6613	392	10	been	be	AUX
ejpam-6613	392	11	illustrated	illustrate	VERB
ejpam-6613	392	12	with	with	ADP
ejpam-6613	392	13	examples	example	NOUN
ejpam-6613	392	14	(	(	PUNCT
ejpam-6613	392	15	see	see	VERB
ejpam-6613	392	16	example	example	NOUN
ejpam-6613	392	17	2	2	NUM
ejpam-6613	392	18	and	and	CCONJ
ejpam-6613	392	19	example	example	NOUN
ejpam-6613	392	20	3	3	NUM
ejpam-6613	392	21	)	)	PUNCT
ejpam-6613	392	22	.	.	PUNCT
ejpam-6613	393	1	in	in	ADP
ejpam-6613	393	2	this	this	DET
ejpam-6613	393	3	context	context	NOUN
ejpam-6613	393	4	,	,	PUNCT
ejpam-6613	393	5	other	other	ADJ
ejpam-6613	393	6	theories	theory	NOUN
ejpam-6613	393	7	can	can	AUX
ejpam-6613	393	8	be	be	AUX
ejpam-6613	393	9	developed	develop	VERB
ejpam-6613	393	10	,	,	PUNCT
ejpam-6613	393	11	such	such	ADJ
ejpam-6613	393	12	as	as	ADP
ejpam-6613	393	13	frame	frame	NOUN
ejpam-6613	393	14	theory	theory	NOUN
ejpam-6613	393	15	,	,	PUNCT
ejpam-6613	393	16	soft	soft	ADJ
ejpam-6613	393	17	set	set	NOUN
ejpam-6613	393	18	theory	theory	NOUN
ejpam-6613	393	19	,	,	PUNCT
ejpam-6613	393	20	the	the	DET
ejpam-6613	393	21	theory	theory	NOUN
ejpam-6613	393	22	of	of	ADP
ejpam-6613	393	23	functions	function	NOUN
ejpam-6613	393	24	of	of	ADP
ejpam-6613	393	25	bounded	bounded	ADJ
ejpam-6613	393	26	variation	variation	NOUN
ejpam-6613	393	27	,	,	PUNCT
ejpam-6613	393	28	the	the	DET
ejpam-6613	393	29	study	study	NOUN
ejpam-6613	393	30	of	of	ADP
ejpam-6613	393	31	the	the	DET
ejpam-6613	393	32	numerical	numerical	ADJ
ejpam-6613	393	33	range	range	NOUN
ejpam-6613	393	34	of	of	ADP
ejpam-6613	393	35	operators	operator	NOUN
ejpam-6613	393	36	,	,	PUNCT
ejpam-6613	393	37	and	and	CCONJ
ejpam-6613	393	38	others	other	NOUN
ejpam-6613	393	39	.	.	PUNCT
ejpam-6613	394	1	conflict	conflict	NOUN
ejpam-6613	394	2	of	of	ADP
ejpam-6613	394	3	interest	interest	NOUN
ejpam-6613	394	4	the	the	DET
ejpam-6613	394	5	authors	author	NOUN
ejpam-6613	394	6	declare	declare	VERB
ejpam-6613	394	7	that	that	SCONJ
ejpam-6613	394	8	they	they	PRON
ejpam-6613	394	9	have	have	VERB
ejpam-6613	394	10	no	no	DET
ejpam-6613	394	11	conflict	conflict	NOUN
ejpam-6613	394	12	of	of	ADP
ejpam-6613	394	13	interest	interest	NOUN
ejpam-6613	394	14	in	in	ADP
ejpam-6613	394	15	this	this	DET
ejpam-6613	394	16	work	work	NOUN
ejpam-6613	394	17	.	.	PUNCT
ejpam-6613	395	1	references	reference	NOUN
ejpam-6613	395	2	[	[	X
ejpam-6613	395	3	1	1	NUM
ejpam-6613	395	4	]	]	X
ejpam-6613	395	5	h.	h.	PROPN
ejpam-6613	395	6	whitney	whitney	PROPN
ejpam-6613	395	7	.	.	PUNCT
ejpam-6613	396	1	tensor	tensor	NOUN
ejpam-6613	396	2	products	product	NOUN
ejpam-6613	396	3	of	of	ADP
ejpam-6613	396	4	abelian	abelian	ADJ
ejpam-6613	396	5	groups	group	NOUN
ejpam-6613	396	6	.	.	PUNCT
ejpam-6613	397	1	duke	duke	PROPN
ejpam-6613	397	2	mathematical	mathematical	PROPN
ejpam-6613	397	3	journal	journal	PROPN
ejpam-6613	397	4	,	,	PUNCT
ejpam-6613	397	5	4(3):495–528	4(3):495–528	NUM
ejpam-6613	397	6	,	,	PUNCT
ejpam-6613	397	7	1938	1938	NUM
ejpam-6613	397	8	.	.	PUNCT
ejpam-6613	398	1	[	[	X
ejpam-6613	398	2	2	2	NUM
ejpam-6613	398	3	]	]	X
ejpam-6613	398	4	n.	n.	NOUN
ejpam-6613	398	5	bourbaki	bourbaki	PROPN
ejpam-6613	398	6	.	.	PUNCT
ejpam-6613	399	1	elements	element	NOUN
ejpam-6613	399	2	de	de	X
ejpam-6613	399	3	mathematique	mathematique	NOUN
ejpam-6613	399	4	.	.	PUNCT
ejpam-6613	400	1	part	part	PROPN
ejpam-6613	400	2	i.	i.	PROPN
ejpam-6613	400	3	,	,	PUNCT
ejpam-6613	400	4	livre	livre	PROPN
ejpam-6613	400	5	ii	ii	PROPN
ejpam-6613	400	6	,	,	PUNCT
ejpam-6613	400	7	algebre	algebre	NOUN
ejpam-6613	400	8	ch	ch	NOUN
ejpam-6613	400	9	.	.	PROPN
ejpam-6613	401	1	2	2	NUM
ejpam-6613	401	2	.	.	X
ejpam-6613	401	3	hermann	hermann	PROPN
ejpam-6613	401	4	,	,	PUNCT
ejpam-6613	401	5	1942	1942	NUM
ejpam-6613	401	6	.	.	PUNCT
ejpam-6613	402	1	[	[	X
ejpam-6613	402	2	3	3	X
ejpam-6613	402	3	]	]	X
ejpam-6613	402	4	e.	e.	PROPN
ejpam-6613	402	5	artin	artin	PROPN
ejpam-6613	402	6	,	,	PUNCT
ejpam-6613	402	7	j.	j.	PROPN
ejpam-6613	402	8	nesbitt	nesbitt	PROPN
ejpam-6613	402	9	,	,	PUNCT
ejpam-6613	402	10	and	and	CCONJ
ejpam-6613	402	11	r.	r.	NOUN
ejpam-6613	402	12	thrall	thrall	NOUN
ejpam-6613	402	13	.	.	PUNCT
ejpam-6613	403	1	rings	ring	NOUN
ejpam-6613	403	2	with	with	ADP
ejpam-6613	403	3	minimum	minimum	ADJ
ejpam-6613	403	4	condition	condition	NOUN
ejpam-6613	403	5	.	.	PUNCT
ejpam-6613	404	1	university	university	NOUN
ejpam-6613	404	2	of	of	ADP
ejpam-6613	404	3	michigan	michigan	PROPN
ejpam-6613	404	4	press	press	PROPN
ejpam-6613	404	5	,	,	PUNCT
ejpam-6613	404	6	1944	1944	NUM
ejpam-6613	404	7	.	.	PUNCT
ejpam-6613	405	1	[	[	X
ejpam-6613	405	2	4	4	X
ejpam-6613	405	3	]	]	PUNCT
ejpam-6613	405	4	e.	e.	PROPN
ejpam-6613	405	5	h.	h.	PROPN
ejpam-6613	405	6	brown	brown	PROPN
ejpam-6613	405	7	.	.	PUNCT
ejpam-6613	406	1	twisted	twisted	ADJ
ejpam-6613	406	2	tensor	tensor	NOUN
ejpam-6613	406	3	products	product	NOUN
ejpam-6613	406	4	,	,	PUNCT
ejpam-6613	406	5	i.	i.	NOUN
ejpam-6613	406	6	annals	annals	NOUN
ejpam-6613	406	7	of	of	ADP
ejpam-6613	406	8	mathematics	mathematic	NOUN
ejpam-6613	406	9	,	,	PUNCT
ejpam-6613	406	10	69(1):223–246	69(1):223–246	PROPN
ejpam-6613	406	11	,	,	PUNCT
ejpam-6613	406	12	1959	1959	NUM
ejpam-6613	406	13	.	.	PUNCT
ejpam-6613	407	1	[	[	X
ejpam-6613	407	2	5	5	X
ejpam-6613	407	3	]	]	PUNCT
ejpam-6613	407	4	h.	h.	PROPN
ejpam-6613	407	5	nakajima	nakajima	PROPN
ejpam-6613	407	6	.	.	PUNCT
ejpam-6613	407	7	quiver	quiver	PROPN
ejpam-6613	407	8	varieties	variety	NOUN
ejpam-6613	407	9	and	and	CCONJ
ejpam-6613	407	10	tensor	tensor	NOUN
ejpam-6613	407	11	products	product	NOUN
ejpam-6613	407	12	.	.	PUNCT
ejpam-6613	408	1	inventiones	inventione	NOUN
ejpam-6613	408	2	mathematicae	mathematicae	PROPN
ejpam-6613	408	3	,	,	PUNCT
ejpam-6613	408	4	146:399–449	146:399–449	NUM
ejpam-6613	408	5	,	,	PUNCT
ejpam-6613	408	6	2001	2001	NUM
ejpam-6613	408	7	.	.	PUNCT
ejpam-6613	409	1	[	[	X
ejpam-6613	409	2	6	6	NUM
ejpam-6613	409	3	]	]	PUNCT
ejpam-6613	409	4	c.	c.	PROPN
ejpam-6613	409	5	a.	a.	PROPN
ejpam-6613	409	6	weibel	weibel	PROPN
ejpam-6613	409	7	.	.	PUNCT
ejpam-6613	410	1	an	an	DET
ejpam-6613	410	2	introduction	introduction	NOUN
ejpam-6613	410	3	to	to	ADP
ejpam-6613	410	4	homological	homological	ADJ
ejpam-6613	410	5	algebra	algebra	NOUN
ejpam-6613	410	6	.	.	PUNCT
ejpam-6613	411	1	number	number	NOUN
ejpam-6613	411	2	38	38	NUM
ejpam-6613	411	3	.	.	PUNCT
ejpam-6613	412	1	cambridge	cambridge	PROPN
ejpam-6613	412	2	university	university	PROPN
ejpam-6613	412	3	press	press	NOUN
ejpam-6613	412	4	,	,	PUNCT
ejpam-6613	412	5	1994	1994	NUM
ejpam-6613	412	6	.	.	PUNCT
ejpam-6613	413	1	[	[	X
ejpam-6613	413	2	7	7	X
ejpam-6613	413	3	]	]	PUNCT
ejpam-6613	413	4	j.	j.	PROPN
ejpam-6613	413	5	j.	j.	PROPN
ejpam-6613	413	6	rotman	rotman	PROPN
ejpam-6613	413	7	.	.	PUNCT
ejpam-6613	414	1	an	an	DET
ejpam-6613	414	2	introduction	introduction	NOUN
ejpam-6613	414	3	to	to	ADP
ejpam-6613	414	4	homological	homological	ADJ
ejpam-6613	414	5	algebra	algebra	NOUN
ejpam-6613	414	6	,	,	PUNCT
ejpam-6613	414	7	volume	volume	NOUN
ejpam-6613	414	8	2	2	NUM
ejpam-6613	414	9	.	.	PUNCT
ejpam-6613	414	10	springer	springer	NOUN
ejpam-6613	414	11	,	,	PUNCT
ejpam-6613	414	12	new	new	PROPN
ejpam-6613	414	13	york	york	PROPN
ejpam-6613	414	14	,	,	PUNCT
ejpam-6613	414	15	2009	2009	NUM
ejpam-6613	414	16	.	.	PUNCT
ejpam-6613	415	1	[	[	X
ejpam-6613	415	2	8	8	NUM
ejpam-6613	415	3	]	]	X
ejpam-6613	415	4	d.	d.	PROPN
ejpam-6613	415	5	dey	dey	PROPN
ejpam-6613	415	6	.	.	PROPN
ejpam-6613	415	7	introduction	introduction	NOUN
ejpam-6613	415	8	to	to	PART
ejpam-6613	415	9	differential	differential	VERB
ejpam-6613	415	10	geometry	geometry	NOUN
ejpam-6613	415	11	with	with	ADP
ejpam-6613	415	12	tensor	tensor	NOUN
ejpam-6613	415	13	applications	application	NOUN
ejpam-6613	415	14	.	.	PUNCT
ejpam-6613	416	1	john	john	PROPN
ejpam-6613	416	2	wiley	wiley	PROPN
ejpam-6613	416	3	&	&	CCONJ
ejpam-6613	416	4	sons	son	NOUN
ejpam-6613	416	5	,	,	PUNCT
ejpam-6613	416	6	2022	2022	NUM
ejpam-6613	416	7	.	.	PUNCT
ejpam-6613	417	1	[	[	X
ejpam-6613	417	2	9	9	NUM
ejpam-6613	417	3	]	]	PUNCT
ejpam-6613	417	4	m.	m.	NOUN
ejpam-6613	417	5	nielsen	nielsen	PROPN
ejpam-6613	417	6	and	and	CCONJ
ejpam-6613	417	7	i.	i.	PROPN
ejpam-6613	417	8	chuang	chuang	PROPN
ejpam-6613	417	9	.	.	PUNCT
ejpam-6613	418	1	quantum	quantum	ADJ
ejpam-6613	418	2	computation	computation	NOUN
ejpam-6613	418	3	and	and	CCONJ
ejpam-6613	418	4	quantum	quantum	NOUN
ejpam-6613	418	5	information	information	NOUN
ejpam-6613	418	6	.	.	PUNCT
ejpam-6613	419	1	cambridge	cambridge	PROPN
ejpam-6613	419	2	university	university	PROPN
ejpam-6613	419	3	press	press	NOUN
ejpam-6613	419	4	,	,	PUNCT
ejpam-6613	419	5	2010	2010	NUM
ejpam-6613	419	6	.	.	PUNCT
ejpam-6613	420	1	[	[	X
ejpam-6613	420	2	10	10	NUM
ejpam-6613	420	3	]	]	PUNCT
ejpam-6613	420	4	m.	m.	NOUN
ejpam-6613	420	5	nakahara	nakahara	PROPN
ejpam-6613	420	6	.	.	PUNCT
ejpam-6613	421	1	geometry	geometry	NOUN
ejpam-6613	421	2	,	,	PUNCT
ejpam-6613	421	3	topology	topology	NOUN
ejpam-6613	421	4	and	and	CCONJ
ejpam-6613	421	5	physics	physics	PROPN
ejpam-6613	421	6	.	.	PUNCT
ejpam-6613	422	1	crc	crc	PROPN
ejpam-6613	422	2	press	press	PROPN
ejpam-6613	422	3	,	,	PUNCT
ejpam-6613	422	4	2018	2018	NUM
ejpam-6613	422	5	.	.	PUNCT
ejpam-6613	423	1	[	[	X
ejpam-6613	423	2	11	11	NUM
ejpam-6613	423	3	]	]	PUNCT
ejpam-6613	423	4	c.	c.	PROPN
ejpam-6613	423	5	s.	s.	PROPN
ejpam-6613	423	6	kubrusly	kubrusly	PROPN
ejpam-6613	423	7	.	.	PUNCT
ejpam-6613	424	1	a	a	DET
ejpam-6613	424	2	concise	concise	ADJ
ejpam-6613	424	3	introduction	introduction	NOUN
ejpam-6613	424	4	to	to	ADP
ejpam-6613	424	5	tensor	tensor	NOUN
ejpam-6613	424	6	product	product	NOUN
ejpam-6613	424	7	.	.	PUNCT
ejpam-6613	425	1	far	far	PROPN
ejpam-6613	425	2	east	east	PROPN
ejpam-6613	425	3	journal	journal	PROPN
ejpam-6613	425	4	of	of	ADP
ejpam-6613	425	5	mathematical	mathematical	ADJ
ejpam-6613	425	6	sciences	science	NOUN
ejpam-6613	425	7	,	,	PUNCT
ejpam-6613	425	8	22(2):137	22(2):137	NUM
ejpam-6613	425	9	,	,	PUNCT
ejpam-6613	425	10	2006	2006	NUM
ejpam-6613	425	11	.	.	PUNCT
ejpam-6613	426	1	[	[	X
ejpam-6613	426	2	12	12	NUM
ejpam-6613	426	3	]	]	PUNCT
ejpam-6613	426	4	k.	k.	PROPN
ejpam-6613	426	5	ferrer	ferrer	PROPN
ejpam-6613	426	6	,	,	PUNCT
ejpam-6613	426	7	o.	o.	PROPN
ejpam-6613	426	8	ferrer	ferrer	PROPN
ejpam-6613	426	9	,	,	PUNCT
ejpam-6613	426	10	and	and	CCONJ
ejpam-6613	426	11	a.	a.	PROPN
ejpam-6613	426	12	sierra	sierra	PROPN
ejpam-6613	426	13	.	.	PUNCT
ejpam-6613	427	1	tensor	tensor	NOUN
ejpam-6613	427	2	products	product	NOUN
ejpam-6613	427	3	of	of	ADP
ejpam-6613	427	4	spaces	space	NOUN
ejpam-6613	427	5	with	with	ADP
ejpam-6613	427	6	indefinite	indefinite	ADJ
ejpam-6613	427	7	metric	metric	ADJ
ejpam-6613	427	8	.	.	PUNCT
ejpam-6613	428	1	under	under	ADP
ejpam-6613	428	2	review	review	NOUN
ejpam-6613	428	3	,	,	PUNCT
ejpam-6613	428	4	2025	2025	NUM
ejpam-6613	428	5	.	.	PUNCT
ejpam-6613	429	1	[	[	X
ejpam-6613	429	2	13	13	NUM
ejpam-6613	429	3	]	]	PUNCT
ejpam-6613	429	4	s.	s.	PROPN
ejpam-6613	429	5	gähler	gähler	PROPN
ejpam-6613	429	6	.	.	PUNCT
ejpam-6613	430	1	lineare	lineare	ADJ
ejpam-6613	430	2	2	2	NUM
ejpam-6613	430	3	-	-	PUNCT
ejpam-6613	430	4	normierte	normierte	NOUN
ejpam-6613	430	5	räume	räume	PROPN
ejpam-6613	430	6	.	.	PUNCT
ejpam-6613	430	7	mathematische	mathematische	PROPN
ejpam-6613	430	8	nachrichten	nachrichten	PROPN
ejpam-6613	430	9	,	,	PUNCT
ejpam-6613	430	10	28(1	28(1	NOUN
ejpam-6613	430	11	-	-	PUNCT
ejpam-6613	430	12	2):1–43	2):1–43	NOUN
ejpam-6613	430	13	,	,	PUNCT
ejpam-6613	430	14	1964	1964	NUM
ejpam-6613	430	15	.	.	PUNCT
ejpam-6613	431	1	[	[	X
ejpam-6613	431	2	14	14	NUM
ejpam-6613	431	3	]	]	PUNCT
ejpam-6613	431	4	z.	z.	PROPN
ejpam-6613	431	5	lewandowska	lewandowska	PROPN
ejpam-6613	431	6	.	.	PUNCT
ejpam-6613	432	1	linear	linear	PROPN
ejpam-6613	432	2	operators	operator	NOUN
ejpam-6613	432	3	on	on	ADP
ejpam-6613	432	4	generalized	generalized	ADJ
ejpam-6613	432	5	2	2	NUM
ejpam-6613	432	6	-	-	PUNCT
ejpam-6613	432	7	normed	norme	VERB
ejpam-6613	432	8	spaces	space	NOUN
ejpam-6613	432	9	.	.	PUNCT
ejpam-6613	433	1	bulletin	bulletin	PROPN
ejpam-6613	433	2	mathématique	mathématique	PROPN
ejpam-6613	433	3	de	de	X
ejpam-6613	433	4	la	la	PROPN
ejpam-6613	433	5	société	société	PROPN
ejpam-6613	433	6	des	des	PROPN
ejpam-6613	433	7	sciences	sciences	PROPN
ejpam-6613	433	8	mathématiques	mathématiques	PROPN
ejpam-6613	433	9	de	de	PROPN
ejpam-6613	433	10	roumanie	roumanie	PROPN
ejpam-6613	433	11	,	,	PUNCT
ejpam-6613	433	12	pages	page	NOUN
ejpam-6613	433	13	353–368	353–368	NUM
ejpam-6613	433	14	,	,	PUNCT
ejpam-6613	433	15	1999	1999	NUM
ejpam-6613	433	16	.	.	PUNCT
ejpam-6613	434	1	m.	m.	PROPN
ejpam-6613	434	2	luis	luis	PROPN
ejpam-6613	434	3	,	,	PUNCT
ejpam-6613	434	4	f.	f.	PROPN
ejpam-6613	434	5	osmin	osmin	PROPN
ejpam-6613	434	6	,	,	PUNCT
ejpam-6613	434	7	s.	s.	PROPN
ejpam-6613	434	8	arley	arley	PROPN
ejpam-6613	434	9	/	/	SYM
ejpam-6613	434	10	eur	eur	PROPN
ejpam-6613	434	11	.	.	PUNCT
ejpam-6613	435	1	j.	j.	PROPN
ejpam-6613	435	2	pure	pure	PROPN
ejpam-6613	435	3	appl	appl	PROPN
ejpam-6613	435	4	.	.	PROPN
ejpam-6613	435	5	math	math	PROPN
ejpam-6613	435	6	,	,	PUNCT
ejpam-6613	435	7	18	18	NUM
ejpam-6613	435	8	(	(	PUNCT
ejpam-6613	435	9	4	4	NUM
ejpam-6613	435	10	)	)	PUNCT
ejpam-6613	435	11	(	(	PUNCT
ejpam-6613	435	12	2025	2025	NUM
ejpam-6613	435	13	)	)	PUNCT
ejpam-6613	435	14	,	,	PUNCT
ejpam-6613	435	15	6613	6613	NUM
ejpam-6613	435	16	17	17	NUM
ejpam-6613	435	17	of	of	ADP
ejpam-6613	435	18	17	17	NUM
ejpam-6613	435	19	[	[	SYM
ejpam-6613	435	20	15	15	NUM
ejpam-6613	435	21	]	]	X
ejpam-6613	435	22	c.	c.	PROPN
ejpam-6613	435	23	diminnie	diminnie	PROPN
ejpam-6613	435	24	,	,	PUNCT
ejpam-6613	435	25	s.	s.	PROPN
ejpam-6613	435	26	gähler	gähler	PROPN
ejpam-6613	435	27	,	,	PUNCT
ejpam-6613	435	28	and	and	CCONJ
ejpam-6613	435	29	a.	a.	NOUN
ejpam-6613	435	30	white	white	PROPN
ejpam-6613	435	31	.	.	PUNCT
ejpam-6613	436	1	strictly	strictly	ADV
ejpam-6613	436	2	convex	convex	VERB
ejpam-6613	436	3	linear	linear	ADJ
ejpam-6613	436	4	2	2	NUM
ejpam-6613	436	5	-	-	PUNCT
ejpam-6613	436	6	normed	norme	VERB
ejpam-6613	436	7	spaces	space	NOUN
ejpam-6613	436	8	.	.	PUNCT
ejpam-6613	437	1	mathematische	mathematische	PROPN
ejpam-6613	437	2	nachrichten	nachrichten	PROPN
ejpam-6613	437	3	,	,	PUNCT
ejpam-6613	437	4	59:319–324	59:319–324	PROPN
ejpam-6613	437	5	,	,	PUNCT
ejpam-6613	437	6	1974	1974	NUM
ejpam-6613	437	7	.	.	PUNCT
ejpam-6613	438	1	[	[	X
ejpam-6613	438	2	16	16	NUM
ejpam-6613	438	3	]	]	X
ejpam-6613	438	4	c.	c.	PROPN
ejpam-6613	438	5	diminnie	diminnie	PROPN
ejpam-6613	438	6	,	,	PUNCT
ejpam-6613	438	7	s.	s.	PROPN
ejpam-6613	438	8	gähler	gähler	PROPN
ejpam-6613	438	9	,	,	PUNCT
ejpam-6613	438	10	and	and	CCONJ
ejpam-6613	438	11	a.	a.	NOUN
ejpam-6613	438	12	white	white	PROPN
ejpam-6613	438	13	.	.	PUNCT
ejpam-6613	439	1	remarks	remark	NOUN
ejpam-6613	439	2	on	on	ADP
ejpam-6613	439	3	generalization	generalization	NOUN
ejpam-6613	439	4	of	of	ADP
ejpam-6613	439	5	2	2	NUM
ejpam-6613	439	6	-	-	PUNCT
ejpam-6613	439	7	inner	inner	ADJ
ejpam-6613	439	8	products	product	NOUN
ejpam-6613	439	9	.	.	PUNCT
ejpam-6613	440	1	mathematische	mathematische	PROPN
ejpam-6613	440	2	nachrichten	nachrichten	PROPN
ejpam-6613	440	3	,	,	PUNCT
ejpam-6613	440	4	74:363–372	74:363–372	PROPN
ejpam-6613	440	5	,	,	PUNCT
ejpam-6613	440	6	1976	1976	NUM
ejpam-6613	440	7	.	.	PUNCT
ejpam-6613	441	1	[	[	X
ejpam-6613	441	2	17	17	NUM
ejpam-6613	441	3	]	]	X
ejpam-6613	441	4	c.	c.	PROPN
ejpam-6613	441	5	diminnie	diminnie	PROPN
ejpam-6613	441	6	,	,	PUNCT
ejpam-6613	441	7	s.	s.	PROPN
ejpam-6613	441	8	gähler	gähler	PROPN
ejpam-6613	441	9	,	,	PUNCT
ejpam-6613	441	10	and	and	CCONJ
ejpam-6613	441	11	a.	a.	NOUN
ejpam-6613	441	12	white	white	PROPN
ejpam-6613	441	13	.	.	PUNCT
ejpam-6613	442	1	2	2	NUM
ejpam-6613	442	2	-	-	PUNCT
ejpam-6613	442	3	inner	inner	ADJ
ejpam-6613	442	4	product	product	NOUN
ejpam-6613	442	5	spaces	space	VERB
ejpam-6613	442	6	.	.	PUNCT
ejpam-6613	443	1	part	part	PROPN
ejpam-6613	443	2	ii	ii	PROPN
ejpam-6613	443	3	.	.	PROPN
ejpam-6613	443	4	demonstratio	demonstratio	PROPN
ejpam-6613	443	5	mathematica	mathematica	PROPN
ejpam-6613	443	6	,	,	PUNCT
ejpam-6613	443	7	10:169–188	10:169–188	NUM
ejpam-6613	443	8	,	,	PUNCT
ejpam-6613	443	9	1977	1977	NUM
ejpam-6613	443	10	.	.	PUNCT
ejpam-6613	444	1	[	[	X
ejpam-6613	444	2	18	18	NUM
ejpam-6613	444	3	]	]	X
ejpam-6613	444	4	c.	c.	PROPN
ejpam-6613	444	5	diminnie	diminnie	PROPN
ejpam-6613	444	6	,	,	PUNCT
ejpam-6613	444	7	s.	s.	PROPN
ejpam-6613	444	8	gähler	gähler	PROPN
ejpam-6613	444	9	,	,	PUNCT
ejpam-6613	444	10	and	and	CCONJ
ejpam-6613	444	11	a.	a.	NOUN
ejpam-6613	444	12	white	white	PROPN
ejpam-6613	444	13	.	.	PUNCT
ejpam-6613	445	1	remarks	remark	NOUN
ejpam-6613	445	2	on	on	ADP
ejpam-6613	445	3	strictly	strictly	ADV
ejpam-6613	445	4	convex	convex	ADJ
ejpam-6613	445	5	and	and	CCONJ
ejpam-6613	445	6	strictly	strictly	ADV
ejpam-6613	445	7	2	2	NUM
ejpam-6613	445	8	-	-	NUM
ejpam-6613	445	9	convex	convex	ADJ
ejpam-6613	445	10	2	2	NUM
ejpam-6613	445	11	-	-	PUNCT
ejpam-6613	445	12	normed	norme	VERB
ejpam-6613	445	13	spaces	space	NOUN
ejpam-6613	445	14	.	.	PUNCT
ejpam-6613	446	1	mathematische	mathematische	PROPN
ejpam-6613	446	2	nachrichten	nachrichten	PROPN
ejpam-6613	446	3	,	,	PUNCT
ejpam-6613	446	4	88:363–372	88:363–372	NUM
ejpam-6613	446	5	,	,	PUNCT
ejpam-6613	446	6	1979	1979	NUM
ejpam-6613	446	7	.	.	PUNCT
ejpam-6613	447	1	[	[	X
ejpam-6613	447	2	19	19	NUM
ejpam-6613	447	3	]	]	PUNCT
ejpam-6613	447	4	j.	j.	PROPN
ejpam-6613	447	5	hernández	hernández	PROPN
ejpam-6613	447	6	,	,	PUNCT
ejpam-6613	447	7	o.	o.	PROPN
ejpam-6613	447	8	ferrer	ferrer	PROPN
ejpam-6613	447	9	,	,	PUNCT
ejpam-6613	447	10	and	and	CCONJ
ejpam-6613	447	11	a.	a.	PROPN
ejpam-6613	447	12	sierra	sierra	PROPN
ejpam-6613	447	13	.	.	PUNCT
ejpam-6613	448	1	about	about	ADP
ejpam-6613	448	2	operators	operator	NOUN
ejpam-6613	448	3	on	on	ADP
ejpam-6613	448	4	generalized	generalized	ADJ
ejpam-6613	448	5	2	2	NUM
ejpam-6613	448	6	-	-	PUNCT
ejpam-6613	448	7	inner	inner	ADJ
ejpam-6613	448	8	spaces	space	NOUN
ejpam-6613	448	9	and	and	CCONJ
ejpam-6613	448	10	their	their	PRON
ejpam-6613	448	11	numerical	numerical	ADJ
ejpam-6613	448	12	range	range	NOUN
ejpam-6613	448	13	.	.	PUNCT
ejpam-6613	449	1	under	under	ADP
ejpam-6613	449	2	review	review	NOUN
ejpam-6613	449	3	,	,	PUNCT
ejpam-6613	449	4	2025	2025	NUM
ejpam-6613	449	5	.	.	PUNCT
ejpam-6613	450	1	[	[	X
ejpam-6613	450	2	20	20	NUM
ejpam-6613	450	3	]	]	X
ejpam-6613	450	4	y.	y.	PROPN
ejpam-6613	450	5	j.	j.	PROPN
ejpam-6613	450	6	cho	cho	PROPN
ejpam-6613	450	7	.	.	PUNCT
ejpam-6613	451	1	theory	theory	NOUN
ejpam-6613	451	2	of	of	ADP
ejpam-6613	451	3	2	2	NUM
ejpam-6613	451	4	-	-	PUNCT
ejpam-6613	451	5	inner	inner	ADJ
ejpam-6613	451	6	product	product	NOUN
ejpam-6613	451	7	spaces	space	VERB
ejpam-6613	451	8	.	.	PUNCT
ejpam-6613	452	1	nova	nova	PROPN
ejpam-6613	452	2	science	science	NOUN
ejpam-6613	452	3	publishers	publisher	NOUN
ejpam-6613	452	4	,	,	PUNCT
ejpam-6613	452	5	2001	2001	NUM
ejpam-6613	452	6	.	.	PUNCT
ejpam-6613	453	1	[	[	X
ejpam-6613	453	2	21	21	NUM
ejpam-6613	453	3	]	]	X
ejpam-6613	453	4	o.	o.	PROPN
ejpam-6613	453	5	ferrer	ferrer	PROPN
ejpam-6613	453	6	,	,	PUNCT
ejpam-6613	453	7	k.	k.	PROPN
ejpam-6613	453	8	ferrer	ferrer	PROPN
ejpam-6613	453	9	,	,	PUNCT
ejpam-6613	453	10	and	and	CCONJ
ejpam-6613	453	11	j.	j.	PROPN
ejpam-6613	453	12	cure	cure	PROPN
ejpam-6613	453	13	.	.	PUNCT
ejpam-6613	454	1	construction	construction	NOUN
ejpam-6613	454	2	of	of	ADP
ejpam-6613	454	3	spaces	space	NOUN
ejpam-6613	454	4	with	with	ADP
ejpam-6613	454	5	an	an	DET
ejpam-6613	454	6	indefinite	indefinite	ADJ
ejpam-6613	454	7	two	two	NUM
ejpam-6613	454	8	-	-	PUNCT
ejpam-6613	454	9	metric	metric	ADJ
ejpam-6613	454	10	and	and	CCONJ
ejpam-6613	454	11	applications	application	NOUN
ejpam-6613	454	12	,	,	PUNCT
ejpam-6613	454	13	2024	2024	NUM
ejpam-6613	454	14	.	.	PUNCT
ejpam-6613	455	1	preprint	preprint	NOUN
ejpam-6613	455	2	.	.	PUNCT
ejpam-6613	456	1	[	[	X
ejpam-6613	456	2	22	22	NUM
ejpam-6613	456	3	]	]	PUNCT
ejpam-6613	456	4	z.	z.	PROPN
ejpam-6613	456	5	lewandowska	lewandowska	PROPN
ejpam-6613	456	6	.	.	PUNCT
ejpam-6613	457	1	bounded	bound	VERB
ejpam-6613	457	2	2	2	NUM
ejpam-6613	457	3	-	-	PUNCT
ejpam-6613	457	4	linear	linear	NOUN
ejpam-6613	457	5	operators	operator	NOUN
ejpam-6613	457	6	on	on	ADP
ejpam-6613	457	7	2	2	NUM
ejpam-6613	457	8	-	-	PUNCT
ejpam-6613	457	9	normed	norme	VERB
ejpam-6613	457	10	sets	set	NOUN
ejpam-6613	457	11	.	.	PUNCT
ejpam-6613	458	1	glasnik	glasnik	PROPN
ejpam-6613	458	2	matematicki	matematicki	PROPN
ejpam-6613	458	3	,	,	PUNCT
ejpam-6613	458	4	39(2):301–312	39(2):301–312	PROPN
ejpam-6613	458	5	,	,	PUNCT
ejpam-6613	458	6	2004	2004	NUM
ejpam-6613	458	7	.	.	PUNCT
ejpam-6613	459	1	introduction	introduction	NOUN
ejpam-6613	459	2	preliminaries	preliminary	NOUN
ejpam-6613	459	3	main	main	ADJ
ejpam-6613	459	4	results	result	NOUN
ejpam-6613	459	5	2	2	NUM
ejpam-6613	459	6	-	-	PUNCT
ejpam-6613	459	7	tensor	tensor	NOUN
ejpam-6613	459	8	product	product	NOUN
ejpam-6613	459	9	algebraic	algebraic	ADJ
ejpam-6613	459	10	tensor	tensor	NOUN
ejpam-6613	459	11	product	product	NOUN
ejpam-6613	459	12	tensor	tensor	NOUN
ejpam-6613	459	13	product	product	NOUN
ejpam-6613	459	14	of	of	ADP
ejpam-6613	459	15	linear	linear	PROPN
ejpam-6613	459	16	operators	operator	NOUN
ejpam-6613	459	17	conclusions	conclusion	NOUN