Exercise 13, § 3

a) Let E, F be two Fréchet spaces, G a locally convex Hausdorff space and E', F', G' the duals of E, F, G respectively. Let $u$ be a bilinear mapping from $E' \times F'$ into $G'$, which is separately continuous (III, p. 28) when E', F', G' are assigned the weak topologies $\sigma(E', E)$, $\sigma(F', F)$ and $\sigma(G', G)$. Show that under these conditions, $u$ is a continuous mapping from $E' \times F'$ into $G'$ when E', F' and G' are assigned the strong topologies. (For $z \in G$, put $\langle z, u(x', y') \rangle = \langle v_z(x'), y' \rangle$ where $v_z(x') \in F$. First show that if $E'$ is assigned the strong topology and $F$ the initial topology, then the set of all $v_z$, as $z$ ranges over a bounded set $C$ in $G$, is equicontinuous; for this use IV, p. 51, exerc. 19, $d$). Next show that there exists a neighbourhood $V'$ of $0$ for the strong topology of $E'$ such that the union of the sets $v_z(V')$ as $z$ ranges over $C$ is bounded in $F$; for this use III, p. 47, exerc. 5.) b) Give an example to show that the conclusion of a) does not hold if we assume that $E$ is a Fréchet space and $F$ a strict inductive limit of Fréchet spaces (III, p. 47, exerc. 3).