Q:

Generalization of Lévy's continuity theorem for nuclear spaces

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.

A:

There is a partial result due to Boulicaut (1973), which states 

Theorem:  Let $E$ be a separable metrizable Hausdorff locally convex topological vector space. Then $E$ is nuclear if and only if for every sequence $\{\mu_n\}$ of tight probability measures, weak convergence to a tight probability measure $\mu$ is equivalent to the pointwise convergence of the characteristic functions of $\mu_n$ to the ch. f. of $\mu$. 

This makes characteristic function(al)s more useful than in separable Banach spaces, where they are only used for uniqueness but not weak convergence.

