I am interested in a generalization of the following finite-dimensional results in infinite dimensional vector-space with nuclear structure, especially for the cases of the spaces of distributions $\mathcal{D}'(\mathrm{R}^N)$ and $\mathcal{S}'(\mathrm{R}^N)$.

**Theorem 1:**
Let $X_n$, $n\in\mathbb{N}$ and $X$ be random variables in $\mathbb{R}^N$ and $\Phi_n$ and $\Phi$ their characteristic functions. Then,
$$\left( X_n \overset{\mathcal{L}}{\rightarrow} X \right) \Leftrightarrow \left( \forall \omega, \Phi_n(\omega) \rightarrow \Phi(\omega) \right).$$

**Theorem 2: (Lévy's continuity theorem)**
Again, the $X_n$'s are random variable and the $\Phi_n$'s are their characteristic functions. Assume that the limit $\lim \Phi_n(\omega)$ exists pointwise and is denoted by $\Phi(\omega)$. We then have the equivalence:

$(X_n)$ converges in law to some random variable $X$.

$\Phi$ is continuous at $0$.

Now I precise my question. Given a probability measure $\mu$ on $\mathcal{N}' = \mathcal{D}'(\mathrm{R}^N)$ or $\mathcal{S}'(\mathrm{R}^N)$, we can define its characteristic functional on $\mathcal{N}$ by $$\hat{\mu}(\varphi) = \int_{\mathcal{N}'} \mathrm{e}^{\mathrm{j} \langle u , \varphi \rangle} \mathrm{d}\mu (u).$$ This generalizes the concept characteristic function of a random variable (Bochner's theorem being generalized by Minlos's theorem). Is the following result true?

**Possible generalization of theorem 1:**
Let $\mu_n$, $n\in\mathbb{N}$ and $\mu$ be probability measures on $\mathcal{N}'$ and $\hat{\mu}_n$ and $\mu$ their characteristic functionals. Then,
$$\left( \mu_n \overset{\mathrm{weakly}}{\rightarrow} \mu \right) \Leftrightarrow \left( \forall \varphi, \hat{\mu}_n(\varphi) \rightarrow \mu(\varphi) \right).$$

Similarly, can we generalize the Lévy's continuity theorem?

Thanks for attention.