Proposition 6

If $F$ is a barrelled space, every separately continuous bilinear mapping $u$ from $E \times F$ into a locally convex space $G$ is $\mathscr{S}$-hypocontinuous for every set $\mathscr{S}$ of bounded subsets of $E$.

In other words, the linear mapping $y \mapsto u(., y)$ from $F$ into $\mathscr{L}_b(E; G)$ is continuous.

It is enough (III, p. 30, prop. 3) to prove that the image of every bounded subset $M$ of $E$ under $x \mapsto u(x, .)$ is equicontinuous in $\mathscr{L}(F; G)$. But, by virtue of prop. 1 (III, p. 28) this image is a simply bounded subset of $\mathscr{L}(F; G)$, and since $F$ is barrelled, every simply bounded subset of $\mathscr{L}(F; G)$ is equicontinuous (III, p. 25, th. 1).