As we all know that

If $\{\varphi_n\}$ is a sequence of characteristic functions of probability measures $\{ \mu_n \}$ on $\mathbb{R}$. And $\lim\varphi_n(t)$ exists for each $t\in \mathbb{R}$. Set $\varphi(t)=\lim\varphi_n(t)$, then if $\varphi$ is continuous at $t=0$, $\varphi$ is a characteristic function for some probability measure $\mu$, and $ \mu_n $ converge weakly to $\mu$.

I want to know whether the conclusion is valid for infinite Hilbert space case, e.g.

Assuming $\{ \mu_n \}$ is a sequence of probability measures in a separable Hilbert space $H$. $\{\varphi_n\}$ are the corresponding characteristic functions defined by $$ \varphi(\lambda) = \int_{H} e^{i\langle \lambda,x \rangle} \mu(\mathrm{d}x) ,\ \ \lambda\in H $$

$\varphi_n$ pointwisely converges to a continuous function $\varphi$.

Is $\varphi$ a characteristic function for some probability measure? If not, is there any counterexample?

What is the critical difference between finite and infinite dimension cases?

Are there any references?