Exercise 11, § 1

Let $E$ be a locally convex Hausdorff space, $E'$ its dual, $\mathscr{S}$ a covering of $E$ consisting of bounded subsets; we assign the $\mathscr{S}$-topology to $E'$. Show that, for the bilinear form $(x, x') \mapsto \langle x, x' \rangle$ to be continuous on $E \times E'$, it is necessary and sufficient that the topology of $E$ can be defined by a single norm, and that the $\mathscr{S}$-topology is the strong topology on $E'$ (\emph{cf.} III, p. 37, exerc. 2).