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You can find it in
Kadison and Ringrose (search Google books and then search inside for "weak Hilbert Schmidt"). Reducing the definition from multilinear to bilinear case and adapting a bit to the notation used in this article the definitions go something like
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for the completed tensor product. For the second issue you raise, the proof I know of this result uses Fubini's theorem, which I believe requires sigma-finite measures (does this imply separability of L^2?), although there are probably still better results.
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That's why I made the edits that I did, but any better fix would be fine with me. Anyway, the error is prehistoric; it was there when sillyrabbit copied it over in 2008 from another article. I'm surpirsed it hasn never been fixed.
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What happens I think is that any fixed element of the
Hilbertian tensor product belongs to the product of separable subspaces on both sides, so that the general case should follow from the proof in the separable case.
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I've put the definition of weakly
Hilbert Schmidt functional in this article, it has already bemoved from the Hilbert Schmidt operator article (probably because it was wrong). I hope that everything is correct now.
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In books they sometimes use a hat. And one cannot say that only the completed tensor product will appear in discussions: the space of simple functions is also important.
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of copies of the two-point space {-1, 1}, each copy equipped with the probability that gives mass 1/2 to each point. This is a probability but the
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I know what its trying to say, but, as a formula, the above just doesn't make sense. So, for example ... obviously, the intent is that
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2373:{\displaystyle {\begin{aligned}x_{1}\otimes x_{2}:H_{1}^{*}&\to H_{2}\\x^{*}&\mapsto x^{*}(x_{1})x_{2}\\\end{aligned}}}
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What guarantees the function defined in the text will be an inner product? I don't see why it must be positive definite.
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I forgot to answer one of your questions: you may define the product probability measure on the uncountable product
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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I find a little unsatisfactory that the completed tensor product is still denoted by
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981:{\displaystyle \phi _{v}=(u_{1},u_{2})\mapsto \langle L(u_{1},u_{2}),v\rangle }
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what does that mean? It's not defined here, nor at
Hilbert-Schmidt operator.--
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if it is a bounded bilinear functional (p. 127). A bounded linear mapping
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I am attempting to repair this problematic formula, which also occurs in
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I that's ok maybe somebody wants to put this into the article? (ezander)
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However it looks, it is wrong and also not what is written in the book.
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in the section "Examples and applications", giving the strange equation
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It turns out that the set of linear combinations is in fact dense in
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Also I am not sure that the restriction of separability is needed in
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mean? The intent seemed to be to use a \mapsto not a \to, so that
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First of all one has to make precise what Ο-field is taken on
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2064:{\displaystyle x_{1}\otimes x_{2}\mapsto x_{1}^{*}(-)x_{2}}
1788:{\displaystyle x_{1}\otimes x_{2}\mapsto x_{1}^{*}(-)x_{2}}
1987:{\displaystyle H_{1}\otimes H_{2}\to (H_{1}^{*}\to H_{2})}
255:{\displaystyle L^{2}(X)\otimes L^{2}(Y)=L^{2}(X\times Y).}
491:. What I'm a little unclear on is whether the products Ο
2130:{\displaystyle x_{1}\otimes x_{2}\in H_{1}\otimes H_{2}}
1517:{\displaystyle x^{*}\in H_{1}^{*}\to x^{*}(x_{1})x_{2}}
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but this doesn't make much sense either. By contrast,
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158:Notation for the Hilbertian tensor product
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484:{\displaystyle L^{2}(X)\otimes L^{2}(Y)}
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988:is a Hilbert-Schmidt functional and
335:{\displaystyle {\widehat {\otimes }}}
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95:This article is within the scope of
38:It is of interest to the following
1907:{\displaystyle H_{1}^{*}\to H_{2}}
14:
2400:Mid-priority mathematics articles
115:Knowledge:WikiProject Mathematics
2395:Start-Class mathematics articles
1795:does make sense, notationally.
613:{\displaystyle 1_{A}(x)1_{B}(y)}
118:Template:WikiProject Mathematics
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507:) span a dense subspace of L(
421:is an orthonormal basis of L(
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359:21:01, 30 November 2008 (UTC)
308:20:16, 30 November 2008 (UTC)
109:and see a list of open tasks.
1417:12:06, 8 December 2014 (UTC)
1346:13:00, 19 October 2011 (UTC)
731:14:20, 25 August 2011 (UTC)
712:14:09, 25 August 2010 (UTC)
433:is an orthonormal basis of
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1073:Hilbert-Schmidt functional
798:Hilbert-Schmidt functional
1579:{\displaystyle H_{1}^{*}}
565:) will span the products
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638:Yes, that makes sense.
141:project's priority scale
1268:{\displaystyle (f_{m})}
1235:{\displaystyle (e_{n})}
1055:{\displaystyle M\geq 0}
98:WikiProject Mathematics
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971:
968:
963:
959:
955:
950:
946:
942:
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936:
933:
930:
925:
921:
917:
912:
908:
904:
901:
896:
892:
871:
868:
865:
841:
838:
833:
829:
825:
820:
816:
812:
809:
785:
782:
777:
773:
769:
764:
760:
756:
753:
739:
738:
723:155.198.192.75
718:
715:
699:
698:Inner product?
696:
695:
694:
693:
692:
691:
690:
659:
658:
657:
656:
633:
632:
609:
606:
603:
598:
594:
590:
587:
584:
579:
575:
558:
550:
538:
537:
536:
535:
534:
533:
500:
492:
480:
477:
474:
469:
465:
461:
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455:
452:
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430:
426:
410:
394:
385:
383:
382:
381:
380:
362:
361:
328:
325:
288:
287:
263:
262:
251:
248:
245:
242:
239:
236:
231:
227:
223:
220:
217:
214:
209:
205:
201:
198:
195:
192:
187:
183:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
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26:
13:
10:
9:
6:
4:
3:
2:
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2357:
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2325:
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2304:
2300:
2294:
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2282:
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2274:
2269:
2265:
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2252:
2240:
2239:
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2236:
2235:
2234:
2230:
2226:
2220:
2202:
2198:
2189:
2185:
2178:
2173:
2169:
2162:
2154:
2149:
2145:
2122:
2118:
2114:
2109:
2105:
2101:
2096:
2092:
2088:
2083:
2079:
2056:
2052:
2045:
2037:
2032:
2028:
2019:
2015:
2011:
2006:
2002:
1976:
1972:
1963:
1958:
1954:
1942:
1938:
1934:
1929:
1925:
1915:
1899:
1895:
1886:
1881:
1877:
1854:
1850:
1841:
1837:
1828:
1824:
1815:
1810:
1806:
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1776:
1769:
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1752:
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1730:
1726:
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1690:
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1677:
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1664:
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1637:
1633:
1624:
1620:
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1607:
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1594:
1571:
1566:
1562:
1539:
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1479:
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1428:
1418:
1414:
1410:
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1351:
1350:
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1334:
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1257:
1253:
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1171:
1166:
1162:
1155:
1145:
1142:
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823:
818:
814:
810:
807:
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783:
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728:
724:
716:
714:
713:
709:
705:
697:
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685:
681:
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673:
669:
665:
664:
663:
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660:
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637:
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532:
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520:
514:
510:
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467:
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424:
420:
416:
408:
404:
400:
392:
391:
390:
389:
388:
387:
386:
379:
375:
371:
366:
365:
364:
363:
360:
356:
348:
326:
323:
312:
311:
310:
309:
305:
301:
297:
293:
286:
282:
278:
274:
271:
270:
269:
266:
249:
243:
240:
237:
229:
225:
221:
215:
207:
203:
199:
193:
185:
181:
173:
172:
171:
169:
165:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
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50:
45:
41:
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2221:
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1526:
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853:
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641:siββy rabbit
562:
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422:
418:
414:
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346:siββy rabbit
295:
291:
289:
284:
280:
277:(X Γ Y), if
276:
272:
267:
264:
167:
163:
161:
137:Mid-priority
136:
96:
62:Midβpriority
40:WikiProjects
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2389:Categories
2225:User:Linas
1360:User:Linas
744:A mapping
393:Well, if Ο
1062:(p. 131).
409:), and Ο
281:(X) and
139:on the
1994:which
36:scale.
796:is a
737:this:
2229:talk
2137:and
2071:for
1413:talk
1409:TSBM
1386:talk
1364:talk
1342:talk
1302:and
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727:talk
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684:talk
680:Bdmy
648:talk
626:talk
622:Bdmy
525:talk
374:talk
370:Bdmy
353:talk
304:talk
300:Bdmy
852:is
131:Mid
2391::
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2328:β¦
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2038:β
2025:β¦
2012:β
1969:β
1964:β
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42::
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