Proposition 3

— Let E, F, G be three locally convex spaces, $\mathfrak{S}$ a family of bounded subsets of E. Let u be a separately continuous bilinear mapping from $E \times F$ into G. The following properties are equivalent :

a) For every neighbourhood W of 0 in G and every set $M \in \mathfrak{S}$, there exists a neighbourhood V of 0 in F such that $u(M \times V) \subset W$.

b) For every set $M \in \mathfrak{S}$, the image of M under the mapping $x \mapsto u(x, .)$ is an equicontinuous subset of $\mathscr{L}(F; G)$.

c) The mapping $y \mapsto u(., y)$ from F into $\mathscr{L}_{\mathfrak{S}}(E; G)$ is continuous.

a) expresses that $y \mapsto u(., y)$ is continuous at the point 0, on account of the definition of neighbourhoods of 0 in $\mathscr{L}_{\mathfrak{S}}(E; G)$ (III, p. 13); likewise a) expresses that the image of M under the mapping $x \mapsto u(x, .)$ is equicontinuous at the point 0 (III, p. 16).