Let $H$ be a real Hilbert Space. A stochastic process $W=\{W(h)\;;h\in H\}$ defined in a probability space $\Omega$ is an isonormal Gaussian process if $W$ is a centered Gaussian family of random variables such that $E(W(h)W(g))=\langle h,g\rangle$ for all $g,h\in H$. The mapping $h\to W(h)$ provides a linear isometry of $H$ onto a closed subspace of $L^2(\Omega)$.
I would like to show that if $h_n \to h$ in $H$ then $W(h_n) \to W(h)$ almost everywhere.
My attempts:
1) $h_n \to h$ so $W(h_n) \to W(h)$ in $L^2(\Omega)$. So we can obtain a subsequence $h_{n_k}$ such that $W(h_{n_k}) \to W(h)$ almost everywhere. Unfortunately, we have a subsequence.
2) I use a particular construction of $W$. If $(e_i)$ is an orthonormal basis of $H$ and $(\gamma_i)$ is a family of independant standard gaussian random variables then for any $h \in H$ the family $(\gamma_i\langle h,e_i\rangle)$ is summable in $L^2(\Omega)$ and $$ W(h) =\sum_{i \in I} \gamma_i\langle h,e_i\rangle $$ defines an isonormal process. The property seems true if $H$ is finite dimensional or separable (Is it really true ?). But it is unclear if $H$ is non-separable. Moreover, I find this argument strange since we use a specific construction of the isonormal process.