Some Results Associated with Bochner's Theorem

Let $\{ P_\alpha\}$ be a family of probability distributions in a separable Hilbert space (or more generally, in a space $X=Y^*$ conjugate to a countably-Hilbert space $Y$) and let $\{\chi_\alpha\}$ be the family of corresponding characteristic functionals. We investigate whether or not there exists a locally convex topology $\mathscr{T}$ with the following property:
The relative compactness of $\{{P_\alpha}\}$ is equivalent to uniform (with respect to $\alpha$) continuity of $\{\chi_\alpha\}$.
We prove that there is no such topology except for the case of the countably-Hilbert nuclear space $Y$.