Exercise 22, § 2

Let $F$ be a separated locally convex space, $\mathfrak{S}$ the set of convex, balanced and compact subsets of $F$ for $\sigma(F, F')$, $\mathfrak{S}'$ the set of convex, equicontinuous, balanced and compact subsets of $F'$ for $\sigma(F', F'')$.

a) Suppose that every subset of $\mathfrak{S}$ is precompact for the $\mathfrak{S}'$-topology. Prove that every continuous linear mapping $u$ from $F$ into a separated quasi-complete locally convex space $G$, which transforms the bounded subsets of $F$ into relatively compact subsets for $\sigma(G, G')$, transforms every subset belonging to $\mathfrak{S}$ into a relatively compact subset of $G$ for the initial topology. (Use Exercise 12 of Topological Vector Spaces, Chapter IV, § 1, and Exercise 11 above.) Converse (consider the canonical mapping of $F$ into its completion for the $\mathfrak{S}'$-topology).

b) Suppose that $F$ is a metrizable space or a strict inductive limit of metrizable spaces. Prove that the condition of a) is satisfied if every Cauchy sequence for $\sigma(F, F')$ is a Cauchy sequence for the $\mathfrak{S}'$-topology (use the Theorem of Šmulian (Topological Vector Spaces, Chapter IV, § 2, Exercise 13 c)).

c) Suppose that $F$ is infratonned (Topological Vector Spaces, Chapter III, § 2, Exercise 12) and denote by $\mathfrak{S}''$ the set of convex, equicontinuous, balanced and compact subsets of $F''$ for $\sigma(F'', F''')$. Prove that, if every subset of $\mathfrak{S}'$ is precompact for the $\mathfrak{S}''$-topology, then every subset of $\mathfrak{S}$ is precompact for the $\mathfrak{S}'$-topology. (Use Exercise 12 of Topological Vector Spaces, Chapter IV, § 1, and note that the initial topology of $F$ is induced by the strong topology on $F''$.)