Exercise 13, § 3

a) Let E, F be two Fréchet spaces, G a separated locally convex space, 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 endowed with the weak topologies $\sigma(E', E)$, $\sigma(F', F)$ and $\sigma(G', G)$. Prove that, under these conditions, $u$ is a continuous mapping from $E' \times F'$ into $G'$ when $E'$, $F'$ and $G'$ are endowed with the strong topologies. (If, for $z \in G$, one sets $\langle z, u(x', y') \rangle = \langle v_z(x'), y' \rangle$, where $v_z(x') \in F$, first prove that when $z$ runs through a bounded set C of G, the set of the $v_z$ is equicontinuous when $E'$ is endowed with the strong topology and F with the initial topology; for this one will use IV, p. 51, Exer. 19, d). Then prove that there exists in $E'$ a neighbourhood V' of 0 for the strong topology such that when $z$ runs through C, the union of the sets $v_z(V')$ is bounded in F; use for this III, p. 48, Exer. 5.) b) Give an example showing that the conclusion of a) fails when one assumes that E is a Fréchet space and F a strict inductive limit of Fréchet spaces (III, p. 48, Exer. 3).