Exercise 14, § 3

(a) Let E be a Fréchet space, E' its dual. Now prove that E', endowed with the topology of compact convergence or with a finer $\mathscr{S}$-topology, is complete ($cf.$ III, p. 23, Remark 1). If E is not reflexive, prove that E' is not infratoned for any of the $\mathscr{S}$-topologies finer than the topology of compact convergence and less fine than $\tau(E', E)$. (b) Let $(E_\alpha)_{\alpha \in A}$ be a family of Fréchet spaces, E a vector space, and for each $\alpha \in A$, let $h_\alpha$ be a linear mapping of $E_\alpha$ into E. We suppose that E, endowed with the finest locally convex topology making the $h_\alpha$ continuous (II, p. 29), is separated. Now 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).