Exercise 20, § 1

Let $F$ be a Hausdorff locally convex space, $F'$ its dual, $\mathscr{G}'$ the set of subsets $A' \subset F'$ such that every sequence of points of $A'$ has a subsequence convergent for $\sigma(F', F)$. Show that, for a bounded subset $A$ of $F$, the following conditions are equivalent: $\alpha)$ $A$ is precompact for the $\mathscr{G}'$-topology; $\beta)$ every sequence convergent for $\sigma(F', F)$ converges uniformly on $A$. (Let $\mathscr{G}$ be the set formed by the finite subsets of $F$ and of $A$; observe that $\beta$) implies that every set $A' \in \mathscr{G}'$ is precompact for the $\mathscr{G}$-topology, using Exer. 6 of GT, II, §4. Then apply Exer. 4 of TVS, IV, §1.)