Remark

— Suppose the topology of $F$ is the finest locally convex topology on $F$ for which the linear mappings $h_\alpha : F_\alpha \to F$ are continuous (II, p. 27). Then condition c) of prop. 3 (III, p. 30) shows that if $E$ and $G$ are locally convex, then the bilinear mapping $u : E \times F \to G$ is $\mathscr{S}$-hypocontinuous if and only if each of the bilinear mappings

from $E \times F_\alpha$ into $G$ is $\mathscr{S}$-hypocontinuous.

Now suppose that $E$ is a locally convex space which is the strict inductive limit of an increasing sequence $(E_n)$ of closed vector subspaces of $E$ (II, p. 33); then every set $M \in \mathscr{S}$ is contained in one of the $E_n$ and is bounded in this subspace (III, p. 5, prop. 6). We denote by $\mathscr{S}_n$ the family of all subsets belonging to $\mathscr{S}$ contained in $E_n$.

Condition a) of prop. 3 (III, p. 30) shows that for a bilinear mapping $u : E \times F \to G$ to be $\mathfrak{S}$-hypocontinuous, it is necessary and sufficient that each of the restrictions $u_n : E_n \times F \to G$ of $u$ is $\mathfrak{S}_n$-hypocontinuous.