Theorem 2 (Minlos)

— Let F be a locally convex space, $\mathscr{T}$ a locally convex topology on $F'$ intermediate to $\mathscr{T}_s$ and $\mathscr{T}_c$, and $\mu$ a promeasure on $F'_{\mathscr{T}}$. Assume that the Fourier transform $\Phi$ of $\mu$ is continuous on F for the Sazonov topology. Then $\mu$ is a measure on $F'_{\mathscr{T}}$.

Let $\varepsilon > 0$. Since $\Phi$ is continuous for the Sazonov topology on F, there exist two continuous positive quadratic forms Q and H on F such that $\operatorname{Tr}(Q/H) < +\infty$ and

for every $x \in F$ such that $Q(x) \leq 1$. By Prop. 8 of No. 8, $|\mathscr{R}\Phi(x)| \leq \Phi(0)$ for all $x \in F$, whence

for all $x \in F$.

Set $r = (12\Phi(0)\operatorname{Tr}(Q/H)\varepsilon^{-1})^{1/2}$ and denote by K the set of $x' \in F'_\mathscr{T}$ such that $\langle x, x' \rangle^2 \leq r^2 H(x)$ for all $x \in F$. Since $H^{1/2}$ is a continuous seminorm on $F$, the set K is equicontinuous and closed in $F'_\mathscr{T}$; it is therefore compact in $F'_\mathscr{T}$ by Ascoli’s theorem (GT, X, §2, No. 5, Cor. 1 of Th. 2).

Let V be a closed linear subspace of $F'_\mathscr{T}$ with finite codimension; then, V is the orthogonal of a finite-dimensional linear subspace T of F. Let $\mu_V$ be the measure on $T'$ that is the image of the promeasure $\mu$ on $F'_\mathscr{T}$ under the mapping $p_V$ that is the transpose of the canonical injection of T into F; its Fourier transform is the restriction of $\Phi$ to T. Finally, by the Hahn–Banach theorem (TVS, II, §3, No. 2, Cor. 1 of Th. 1), $p_V(K)$ is equal to the set $C_r$ of $x' \in T'$ such that $\langle x, x' \rangle^2 \leq r^2 H(x)$ for all $x \in T$. By the inequality (63), one can apply Prop. 10 of No. 9 to the measure $\mu_V$ on $T'$, on taking for q the restriction of $2\Phi(0)Q$ to T and for h that of H. Then $\operatorname{Tr}(q/h) \leq 2\Phi(0)\operatorname{Tr}Q/H)$, whence

Since $p_V$ defines, by passage to the quotient, an isomorphism of $F'_\mathscr{T}/V$ onto $T'$, Prop. 1 of No. 1 then shows that $\mu$ is a measure on $F'_\mathscr{T}$.

Q.E.D.