What does "Kuratowski's theorem" refer to in the context of dense linear subspaces being Borel?

Question

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.

Answer 1

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.

Answer 2

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".