A Remark on Characteristic Functionals

For the case of separable Hilbert space $H$ two theorems are proved that give necessary and sufficient conditions for the functional $\chi(f)$, $f\in H$, to be the characteristic functional of some probability distribution of $H$.
In the first theorem the case $\int{\|x\|}^2\,dP<+\infty$ is investigated; the second one deals with the general case and the condition for it is given in the form of the continuity of $\chi(t)$ in the $\mathfrak J$-topology whose definition is contained in the paper.