Proposition 4

Let $u$ be a $\mathscr{S}$-hypocontinuous bilinear mapping from $E \times F$ into $G$. For every set $M \in \mathscr{S}$, the restriction of $u$ to $M \times F$ is continuous, and $u(M \times Q)$ is bounded in $G$ for every bounded subset $Q$ of $F$.

The first assertion follows from cor. 3 of GT, X, § 2, No. 1. Let $W$ be a neighbourhood of 0 in $G$; there exists, by hypothesis, a neighbourhood $V$ of 0 in $F$ such that $u(M \times V) \subset W$. Since there exists $\lambda \neq 0$ such that $\lambda Q \subset V$, we have $\lambda u(M \times Q) = u(M \times \lambda Q) \subset W$, and this proves the second part of the proposition.