Corollary 2

— Let $\mathscr{S}$ be a family of bounded subsets of E. The $\mathscr{S}$-topology on $\mathscr{L}(E; F)$ is the topology of uniform convergence on sets of the form $A \times B \subset E \times F'$, where A is in $\mathscr{S}$, and B belongs to the family of equicontinuous subsets of $F'$.

For every $u \in \mathscr{L}(E; F)$, every $A \in \mathscr{S}$ and every closed convex balanced neighbourhood V of 0 in F, the relation $u(A) \subset V$ is equivalent to « $j(u)(A \times V^\circ)$ is contained in the unit ball of K ».