Exercise 14, § 1

Let $E$ be a locally convex Hausdorff and quasi-complete space, $E'$ its dual. Show that on $E'_1$ the topology of compact convergence is the finest of the topologies compatible with the duality between $E$ and $E'$ and which induces the same topology as $\sigma(E', E)$ on every equicontinuous subset of $E'$ (\emph{cf.} IV, p. 48, exerc. 4).