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Kuratowski was a busy man who showed many results in topology and functional analysis, so when a writer says that some result follows from ''Kuratowski's theorem", it could apply to many different ones.

I am currently studying "Stochastic Partial Differential Equations: An Introduction", by Wei Liu and Michael Röckner and in Chapter 4, they introduce the Gelfand triple $\left(V,H,V^*\right)$ and sketch the situation where they work in.

For this Gelfand triple (or evolution triple), we have a reflexive Banach space $\left(V,\|\cdot\|_{V}\right)$ and a Hilbert space $\left(H,\langle\cdot,\cdot\rangle_{H}\right)$, such that $V \subset H$ and $V$ can be continuously and densely embedded in $H$. It also follows that $H^*$ can be densely embedded into $V^*$, by restricting the functionals on $H$ to $V$ (call this (isomorphic!) map $\rho$). As the Hilbert space $H$ and its dual are isomorphic by the Riesz representation map $\Phi$, we can identify $H$ and its image under the map $\rho\circ\Phi$; we will write $\bar{H}$ for $\rho(\Phi(H))$.

Then, they claim that by Kuratowski's theorem, we know that $$ V \in \mathcal{B}(H) \quad \text{ and } \quad \bar{H}\in\mathcal{B}\left(V^*\right) $$ but I don't know which theorem they refer to and that is my question. I've scoured the internet to find what they mean, but my attempts have not yet been fruitful, so I was wondering if any of you could help me out.

Ghostface
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    I might be inclined not so much to look for a theorem by Kuratowski, but for a book on PDEs written by authors who had more respect for their readers' time. – John Hughes Mar 09 '17 at 12:27
  • @NormalHuman: Excuse me, I forgot to include that $\mathcal{B}(H)$ is the Borel-$\sigma$-field of $H$. – Ghostface Mar 13 '17 at 11:48
  • @NormalHuman And I completely forgot to answer your question: the way this is set up is to show the existence of solutions to stochastic differential equations. They first assume the solutions are $V^*$-valued, but that they really are $\bar{H}$-valued, so can be translated back to being $H$-valued. That is essentially what it comes down to.

    If you think: what is this bloke rattling about, then I agree that my comment might not be particularly helpful.

     – Ghostface Mar 13 '17 at 11:59
  • @JohnHughes Any recommendations? – Ghostface Mar 13 '17 at 12:00
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    None at all -- this is completely outside my area of expertise. But it's a general rule: avoid authors who don't take your time seriously. – John Hughes Mar 13 '17 at 12:28
  • @JohnHughes Ah, that's a bummer. Thanks for your quick response anyway! – Ghostface Mar 13 '17 at 13:20
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    @johnhughes is certainly right about generally avoiding texts which send you on snipe hunts. With that said, it's slim pickings for stochastic PDE texts, but I prefer Prevot and Roeckner as an elementary text. – David Mar 13 '17 at 19:55

2 Answers2

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I think the authors refer to the following theorem:

Theorem. Suppose $X, Y$ are Polish spaces and $f : X \to Y$ is continuous. If $A \subset X$ is Borel and $f_{|A}$ is injective, then $f(A)$ is Borel.

It is a theorem of Lusin and Souslin, as stated in Classical Descriptive Set Theory by Alexander Kechris, Theorem 15.1 on page 89. Sometimes it's attributed to Lusin alone. I haven't been able to connect Kuratowski to this result.

Every separable Banach space is a Polish space, and the embeddings $V\subset H$ and $H^*\subset V^*$ are continuous injective maps. So one needs to assume that $H$ (equivalently, $V$) is separable, which I believe is a standard assumption when considering Gelfand triples.

  • Great find! Did it take long?

    (I indeed forgot to mention that these spaces were assumed to be separable.)

     – Ghostface Mar 14 '17 at 08:29
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As a complementary of the previous answer, Kuratowski proved a similar result to Souslin's for Borel measurable mappings (not only continuous mappings) in page 489 of his book "Topology: Volume I".

Liping Li
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