An equivalent theorem for Sobolev spaces in infinite dimensions

Question

There is a proposition which states:

Let $f\in W^{1}(U)$ be real valued and $h\in C^{1}(\mathbb{R})$ with $h'\in C_{b}(\mathbb{R})$. We then have $h\circ f\in W^{1}(U)$ and $$\partial_{j}(h\circ f)=(h'\circ f)\partial_{j}f$$ for $j\in\{1,...,d\}$.

Is there a similar theorem which works in the setting of $f\in W^{1,p}(X,\gamma)$? For example: for such an $f$ and $h\in C^{1}_{b}(X)$ we would have $h\circ f\in W^{1,p}(X,\gamma)$ and...etc.

The closest that I have found is Corollary 5.4.3 of Bogachev's 'Gaussian Measures' which states:

Let $f\in G^{p,n}(\gamma)$ and $\varphi\in C^{\infty}_{b}(\mathbb{R}^{1})$ (or, more generally $\varphi\in C^{1}_{b}(\mathbb{R}^{1})$). Then $\varphi\circ f\in G^{p,n}(\gamma)$....

But that's not quite what I need. I'm actually trying to prove that if $f\in W^{1,p}(X,\gamma)$ then so is $f^{+}$ and in the finite dimensional case (where the Sobolev space is endowed with the Lebesgue measure) this is proven using the first proposition that I stated.

Answer 1

The analogue to the top proposition is true, and it is irrelevant whether or not $X$ is a finite or infinite-dimensional Banach space. The only thing that matters is that you can construct a Borel measure $\mu$ on open $U \subset X$ and that the function $f$ in question is $\mu$-measurable. This is because:

1. The chain rule is valid for Frechet derivatives.
2. The construction of $L^p(\Omega, \mathbb{R})$ does not depend on the choice of measure space, so usual integration inequalities hold.

As long as we define $f$ to be in $W^1(U)$ if it is a $\mu$-measurable function such that

$$ \int_U |f(x)|^2 \, d\mu + \int_U \| Df(x) \|^2 \, d\mu < \infty, $$ where $\|Df(x)\|$ is the operator norm of the Frechet derivative of $f$ at $x$, the derivation of the proposition is independent of the dimension of $X$.