Weak convergence can be tricky when dealing with infinite dimensional spaces. For example, the usual Levy's continuity theorem does not extend readily to separable Banach spaces.

Consider a (separable) Hilbert space $H$: we know that the sequence of $H$-valued random variables $Z^N$ converges in laws towards the random variable $Z$. We also know that the sequence of random processes $W^N \in C([0,T],H)$ (continuous functions from $[0,T]$ to $H$) converges in laws to a Brownian motion $W$.

**Question:** If we can show that for any $k,h_1, \ldots, h_n \in H$ and time $t_j$
$$E[e^{i(k,Z^N)}e^{\sum_{j=1}^N i(h_j,W_{t_j})}] \to E[e^{i(k,Z^N)}] \ E[e^{\sum_{j=1}^N i(h_j,W_{t_j})}],$$
is it enough to conclude that the sequence of couples $(Z^N,W^N)$ also converge in laws to $(Z,W)$, with $Z$ and $W$ independent ?