Proposition 7

Let E be a locally convex space and $\mu$ a promeasure on E. Suppose that $x'(\mu)$ is a Gaussian measure on $\mathbf{R}$ for every $x' \in E'$. Then $\mu$ is a Gaussian promeasure on E.

For every $x' \in E'$, let $Q(x')$ be the variance of the Gaussian measure $x'(\mu)$ on $\mathbf{R}$. One has $x'(\mu) = \gamma_{Q(x')}$, whence

by the definition of $\mathscr{F}\mu$ (No. 3, formula (2)). In other words, $\mathscr{F}\mu = e^{-Q/2}$, and it remains to prove that $Q$ is a positive quadratic form on $E'$.

For every closed linear subspace V of E with finite codimension, denote by $p_V$ the canonical mapping of E onto $E/V$, by $\mu_V$ the measure $p_V(\mu)$ on $E/V$, and set $Q_V = Q \circ {}^t p_V$. Since $E' = \bigcup_{V \in \mathscr{F}(E)} \operatorname{Im}({}^t p_V)$ and ${}^t p_V$ is injective, it suffices to prove that $Q_V$ is a positive quadratic form on $(E/V)'$. Let $u \in (E/V)'$ and $x' = {}^t p_V(u)$. We have

Prop. 6 then implies thus $Q_V$ is a positive quadratic form on $(E/V)'$.