Let $C_c^\infty(\mathbb R^n)$ be the functions from $\mathbb R^n$ to $\mathbb R$ with compact support, further let $X$ be a separable Hilbert space with a fixed orthonormal basis $(e_n)_n$. Define the cylindrical functions:

\begin{align*} \text{Cyl}(X) := \{f : X \to &\mathbb R : \text{there exists } d \in \mathbb{N} \text{ and } \phi \in C_c^\infty(\mathbb R^n) \text{ such that }\\\ &f(x) = \phi(\langle x, e_1 \rangle, \ldots , \langle x, e_d \rangle ) \text{ for all } x \in X \} \end{align*}

Now, if $\langle . , . \rangle$ is the inner product on $X$ define $$\langle x, y \rangle_\omega := \sum_n \frac{1}{n^2} \langle x, e_n \rangle \langle e_n, y \rangle$$

Now, why is every $f \in \text{Cyl}(X)$ continuous with respect to $\langle . , . \rangle_\omega$? Sure, it is Lipschitz and continuous with respect to the weak topology (because it is with respect to the strong topology). Further I know that on bounded sets, the topology induced by $\langle . , . \rangle_\omega$ is the same as the weak topology. However, $f^{-1}(A)$ does not have to be bounded. What am I missing, I'm sure it is easy.

anyelements of $X$, not just elements of the chosen orthonormal basis. My class of functions seems to be strictly larger than yours... $\endgroup$Gradient Flows in Metric Spaces and in the Space of Probability Measuresby Ambrosio et al uses the definition I give (pg 113). Do you have a reference where I can find your version? $\endgroup$