Remark

— 1) Let $E$ be a Hilbert space, $E_s$ the space $E$ equipped with the weak topology, and $j$ the identity mapping of $E$ into $E_s$. One knows that $j$ defines a bijection of the set of promeasures on $E$ onto the corresponding set for $E_s$. Moreover, if $E$ is separable, it is a Polish space and $j$ defines a bijection of the set of bounded measures on $E$ onto the set of bounded measures on $E_s$ (§ 3, No. 3, Remark); it can be shown (the theorem of Phillips) that this theorem still holds if $E$ is not separable. Consequently, Theorem 3 furnishes criteria for a promeasure on $E$ to be a measure.

2) One can show (Annex, Exer. 7) that the Sazonov topology on a Hilbert space $E$ is defined by the semi-norms of the type $Q^{1/2}$ where $Q$ is a nuclear positive quadratic form on $E$.

12. Relations with functions of positive type