Exercise 14, § 3

a) Let $E$ be a Fréchet space, $E'$ its dual. Show that $E'$, endowed with the topology of compact convergence or with a finer $\mathscr{S}$-topology, is complete ($cf.$ III, p. 22, Remark 1). If $E$ is not reflexive, show that $E'$ is not infrabarrelled for any $\mathscr{S}$-topology which is finer than the topology of compact convergence and coarser than $\tau(E', E)$. b) Let $(E_\alpha)_{\alpha \in A}$ be a family of Fréchet spaces, $E$ a vector space and for every $\alpha \in A$, let $h_\alpha$ be a linear mapping from $E_\alpha$ into $E$. Suppose that $E$, endowed with the finest locally convex topology for which the $h_\alpha$ are continuous (II, p. 27) is Hausdorff. Prove that the dual $E'$ of $E$, endowed with the topology of compact convergence or with any finer $\mathscr{S}$-topology, is complete ($cf.$ III, p. 20, th. 1).