Timeline for Is the space of tempered distribution second countable?
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
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| Feb 10, 2014 at 21:10 | vote | accept | Thomas | ||
| Feb 10, 2014 at 20:25 | comment | added | Pedro Lauridsen Ribeiro | Hmm... perhaps I should update my answer with the above comments and clean them up a bit. | |
| Feb 10, 2014 at 20:21 | comment | added | Pedro Lauridsen Ribeiro | @Thomas: you can find it, for instance, in Gel'fand-Vilenkin's book, "Generalized Functions - Volume IV", Chapter IV, pp. 303ff. Once again, the precise relationship is connected to Jochen Wengenroth's answer: since $\mathscr{S}$ is separable, the $\sigma$-algebra generated by the Borel cylinder sets of $\mathscr{S}'$ contains the polar $A^\circ\subset\mathscr{S}'$ of every subset $A\subset\mathscr{S}$. This implies, together with the first countability of $\mathscr{S}$, that this $\sigma$-algebra includes all weak-* Borel sets - therefore, both $\sigma$-algebras coincide. | |
| Feb 10, 2014 at 19:57 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Small clarification
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| Feb 10, 2014 at 18:04 | comment | added | Thomas | @PedroLauridsenRibeiro: thank you very much for your very complete answer! If second countability is not the way do you have any reference to suggest to clarify the relationship between the borel and cylinder $\sigma$-field on $\mathcal S'$? | |
| Feb 10, 2014 at 17:28 | comment | added | Pedro Lauridsen Ribeiro | @Thomas: no, it's not quite the same thing. It simply means that a straightforward argument to get equality of both $\sigma$-algebras using second countability of the whole space is untenable (see, however, Jochen Wengenroth's answer for a more careful approach). The fact that cylinder set measures usually have very small supports has to do rather with the particular way they are constructed. | |
| Feb 10, 2014 at 17:23 | comment | added | Thomas | So in this case it means that the Borel $\sigma$-algebra and the cylinder $\sigma$-algebra are actually not the same? I found some references where Minlos-Bochner theorem is used to prove the existence of a gaussian measure on $\mathcal S'$. This measure is either defined on the cylinder or Borel $\sigma$-algebra. In this particular case the support of the measure is actually $L^2$ but what about more general applications of this theorem? | |
| Feb 10, 2014 at 15:51 | comment | added | alpha | @Pedro. The original articles of Sebastião e Silva are hard to come by but his spaces and results are described in the classic book by G. Köthe on topological vector spaces. | |
| Feb 10, 2014 at 15:49 | comment | added | Pedro Lauridsen Ribeiro | That, I didn't know... | |
| Feb 10, 2014 at 15:47 | comment | added | alpha | In fact, the property about continuity is even more miraculous---it applies to ANY function (not necessarily linear) with values in ANY topological space. | |
| Feb 10, 2014 at 15:44 | comment | added | Pedro Lauridsen Ribeiro | @alpha: I'm currently short of a reference on locally convex vector spaces to refresh my memory, but yes, you're actually right. It is actually even more general than that: any strict countable inductive limit of Fréchet-Schwartz spaces (that is, Fréchet spaces with a projective system such that the intertwining maps are compact), such as $\mathscr{D}$, $\mathscr{E}$ and $\mathscr{S}$, has this property. | |
| Feb 10, 2014 at 15:37 | comment | added | alpha | I think that it is a bit more subtle than that. It is because the space is a countable inductive of Banach spaces with compact intertwining mappings (in this case, they are even nuclear). The portuguese mathematician Sebastião e Silva proved this fact and many others about such spaces in the 50's---hence the name "Silva space". | |
| Feb 10, 2014 at 15:29 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Small typo corrected
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| Feb 10, 2014 at 15:26 | comment | added | Pedro Lauridsen Ribeiro | Thanks to the "magic" of linearity (and, of course, the fact that the codomain of functionals is a finite-dimensional locally convex vector space). | |
| Feb 10, 2014 at 15:23 | history | edited | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |
Added clarifications
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| Feb 10, 2014 at 15:19 | comment | added | Gerald Edgar | The interesting (and mysterious?) thing is that, even though it is not first-countable, continuity for linear functionals can be checked using convergent sequences only. | |
| Feb 10, 2014 at 15:11 | history | answered | Pedro Lauridsen Ribeiro | CC BY-SA 3.0 |