Theorem 4

— Let E be a locally convex space. The Fourier transformation is a bijection of the set of promeasures on E onto the set of functions of positive type on $E'$ whose restriction to every finite-dimensional subspace is continuous.

We know (No. 3, Prop. 3) that the Fourier transformation is injective. Let $\mu = (\mu_V)_{V \in \mathscr{F}(E)}$ be a promeasure on E and $\Phi$ its Fourier transform. Let T be a finite-dimensional subspace of $E'$ and let V be the orthogonal of T in E. One can identify T with the dual of $E/V$; the restriction $\Phi_T$ of $\Phi$ to T is the Fourier transform of the bounded measure $\mu_V$ on $E/V$. By Prop. 11, $\Phi_T$ is continuous and of positive type on T. Since T is arbitrary, it is clear that $\Phi$ is of positive type on $E'$.

Conversely, let $\Phi$ be a function of positive type on $E'$ whose restriction to every finite-dimensional subspace of $E'$ is continuous. For every $V \in \mathscr{F}(E)$, we identify the dual of $E/V$ with the orthogonal $V^\circ$ of V in $E'$; the restriction $\Phi_V$ of $\Phi$ to $V^\circ$ is continuous and of positive type and so, by Bochner's theorem, there exists a bounded (positive) measure $\mu_V$ on $E/V$ whose Fourier transform is $\Phi_V$. Let V and W in $\mathscr{F}(E)$ be such that $W \subset V$, and let $p_{VW}$ be the canonical mapping of $E/W$ onto $E/V$; with the identifications made, ${}^t p_{VW}$ is the injection of $V^\circ$ into $W^\circ$. By formula (4) of No. 3, we then have

(*) This question will be studied in a forthcoming chapter of the Book Théories spectrales. The reader may consult for this subject the book of L.H. LOOMIS, Abstract harmonic analysis, Van Nostrand, New York, 1953.

whence $pvw(\mu_w) = \mu_v$ by Prop. 3 of No. 3. Consequently, the family $\mu = (\mu_v)_{v \in \mathscr{F}(E)}$ is a promeasure on $E$; it is clear that $\Phi$ is the Fourier transform of $\mu$.