Corollary 1

— Let $E_1$ and $E_2$ be two Hilbert spaces, $u$ a Hilbert–Schmidt mapping of $E_1$ into $E_2$, and $\mu$ a promeasure on $E_1$. Assume that the Fourier transform $\Phi$ of $\mu$ is continuous on $E_1$. Then the promeasure $\nu = u(\mu)$ is a measure on $E_2$ equipped with the weak topology.

With the identifications of $E_1$ and $E_2$ with their duals introduced in this No., the Fourier transform of $\nu$ is equal to $\Phi \circ u^*$, where $u^*$ is the adjoint of $u$. Now, $u^*$ is a Hilbert–Schmidt mapping of $E_2$ into $E_1$ (Annex, No. 2), and the quadratic form $y \mapsto \|u^*(y)\|^2$ on $E_2$ is therefore nuclear. If $(E_2)_\mathscr{S}$ denotes $E_2$ equipped with the Sazonov topology, $u^*$ is therefore a continuous linear mapping of $(E_2)_\mathscr{S}$ into $E_1$, and $\mathscr{F}_V = \Phi \circ u^*$ is continuous on $(E_2)_\mathscr{S}$; Theorem 3 then shows that $\nu$ is a measure on the space $E_2$ equipped with the weak topology.