Proposition 7

Let $E, F, G$ be three locally convex spaces, $G$ being assumed Hausdorff; let $E_0$ (resp. $F_0$) be a dense vector subspace of $E$ (resp. $F$). Let $u$ be a separately continuous bilinear mapping from $E \times F$ into $G$.

1) If $u(E_0 \times F_0) = \{0\}$, then $u = 0$.

2) Let $\mathfrak{S}_0$ be a family of bounded subsets of $E_0$; if the restriction of $u$ to $E_0 \times F_0$ is $\mathfrak{S}_0$*-hypocontinuous then so is* $u$.

1) By hypothesis, for all $x \in E_0$, the continuous linear mapping $u(x, .)$ is null on $F_0$, hence on $F$: therefore for all $y \in F$, the continuous linear mapping $u(., y)$ is null on $E_0$, hence on $E$. This proves that $u = 0$.

2) For every closed neighbourhood $W$ of 0 in $G$ and for every set $M \in \mathfrak{S}_0$, there exists, by hypothesis, a neighbourhood $V$ of 0 in $F_0$ such that $u(M \times V) \subset W$. But $\overline{V}$ is a neighbourhood of 0 in $F$; for every $x \in M$, the relation $u(\{x\} \times V) \subset W$ implies that $u(\{x\} \times \overline{V}) \subset W$, since $u(x, .)$ is continuous and $W$ is closed; therefore $u(M \times \overline{V}) \subset W$, which proves that $u$ is $\mathfrak{S}_0$*-hypocontinuous.