Corollary 3

— Let E be a locally convex space. Every subset of E which is bounded for $\sigma(E, E')$ is bounded.

Let A be a subset of E. Saying that A is bounded for $\sigma(E, E')$ means that every continuous linear form on E is bounded on A ; Saying that A is bounded means that every continuous semi-norm on E is bounded on A. Let N be the closure of 0 in E and $\pi$ the canonical mapping from E onto E/N. The continuous linear forms on E are the mappings of the form $f \circ \pi$ with $f \in (E/N)'$ and we have an analogous characterization of continuous semi-norms on E. Replacing E by E/N and A by $\pi(A)$ we can thus limit ourselves to the case where E is Hausdorff.

Let $\mathfrak{S}$ be the set of equicontinuous subsets of $E'$; when $E'$ is assigned the topology $\sigma(E', E)$, $E$ can be identified with $(E')_{\mathfrak{S}}'$ (III, p. 19, cor. 1). Every closed equicontinuous subset of $E'$ is compact for $\sigma(E', E)$ (III, p. 17, cor. 2), hence complete for $\sigma(E', E)$. It is now enough to apply th. 2.