Exercise 22, § 2

Let $F$ be a Hausdorff locally convex space, $\mathscr{G}$ the set of subsets of $F$ that are convex, balanced, and compact for $\sigma(F, F')$, and $\mathscr{G}'$ the set of subsets of $F'$ that are convex, equicontinuous, balanced, and compact for $\sigma(F', F'')$.

a) Suppose that every set in $\mathscr{G}$ is precompact for the $\mathscr{G}'$-topology. Show that every continuous linear mapping $u$ of $F$ into a quasi-complete, Hausdorff locally convex space $G$, that transforms the bounded subsets of $F$ into subsets relatively compact for $\sigma(G, G')$, transforms every set belonging to $\mathscr{G}$ into a subset of $G$ that is relatively compact for the original topology. (Use Exer. 4 of TVS, IV, §1 and Exer. 11 above.) Converse (consider the canonical mapping of $F$ into its completion for the $\mathscr{G}'$-topology).

b) Suppose that $F$ is a metrizable space, or a strict direct limit of metrizable spaces. Show that the condition of a) is satisfied if every Cauchy sequence for $\sigma(F, F')$ is a Cauchy sequence for the $\mathscr{G}'$-topology (make use of Šmulian’s theorem (TVS, IV, §5, Exer. 2 c))).

c) Suppose F is infra-barreled (TVS, III, §4, Exer. 7) and denote by $\mathscr{G}''$ the set of subsets of $F''$ that are convex, equicontinuous, balanced, and compact for $\sigma(F'', F''')$. Show that if every set in $\mathscr{G}'$ is precompact for the $\mathscr{G}''$-topology, then every set in $\mathscr{G}$ is precompact for the $\mathscr{G}'$-topology. (Use Exer. 4 of TVS, IV, §1, and observe that the original topology of F is induced by the strong topology on $F''$.)