Theorem 2

Let $E$ be a locally convex Hausdorff space and $\hat{E}$ its completion; we identify $E$ and $\hat{E}$ with vector subspaces of ${E'}^*$. Then for an element $f \in {E'}^*$ to belong to $\hat{E}$, it is necessary and sufficient that the restriction of $f$ to every equicontinuous subset of $E'$ is continuous for the topology $\sigma(E', E)$.

The space $E$ can be identified with the topological dual of $E'$ when $E'$ is assigned the topology $\sigma(E', E)$ (II, p. 43, prop. 3); on the other hand, if $\mathscr{S}$ is the set of equicontinuous subsets of $E'$, the given topology on $E$ is the $\mathscr{S}$-topology (III, p. 19, cor. 1). Then it follows from III, p. 13, prop. 1, that the sets of $\mathscr{S}$ are bounded for $\sigma(E', E)$ (cf. later on, III, p. 22, prop. 9); in other words, $\mathscr{S}$ is an adapted and covering bornology for the topology $\sigma(E', E)$. Theorem 2 is then a consequence of th. 1 if we replace $E$ by $E'$ and $E'_\mathscr{S}$ by $E$.