Lemma 1

Let E and F be two Hausdorff locally convex and quasi-complete spaces and u a continuous linear mapping from E into F. For ${}^t u$ to be a strict morphism from $F'_c$ into $E'_c$, it is necessary and sufficient that the image $u(E)$ of u be closed, and that every compact subset of $u(E)$ be the image under u of a compact subset of E.

By Mackey’s th. (IV, p. 2, th. 1) and the fact that on $E'$ (resp. F) the topology of compact convergence coincides with that of convex compact convergence (IV, p. 4), we can identity E (resp. F) with the dual $E'_c$ (resp. $F'_c$). Then $u$ is the transpose of ${}^t u$, and the equicontinuous subsets of E (resp. F) are the relatively compact sets. Lemma 1 then follows from prop. 2 (IV, p. 27), since $u(E)$ is closed in F if and only if it is closed for the weakened topology $\sigma(F, F')$ (IV, p. 4, prop. 2).