Exercise 1, § 5

Let $E$, $F$ and $G$ be three locally convex spaces, $\mathscr{S}$ a cover of $E$ consisting of bounded sets. Show that if $u$ is a separately continuous bilinear mapping from $E \times F$ into $G$ such that for every set $M \in \mathscr{S}$ the restriction of $u$ to $M \times F$ is continuous, then $u$ is $\mathscr{S}$-hypocontinuous.

EXERCISES