Exercise 7, § 5

Let E, F and G be three topological vector spaces, $\mathscr{S}$ a set of bounded subsets of E, and H a set of separately continuous bilinear mappings from $E \times F$ into G. The following properties are equivalent : $\alpha)$ For every neighbourhood W of 0 in G and every set $M \in \mathscr{S}$, there exists a neighbourhood V of 0 in F such that $u(M \times V) \subset W$ for all $u \in H$. $\beta)$ For every set $M \in \mathscr{S}$, the image of $H \times M$ under the mapping $(u, x) \mapsto u(x, .)$ is an equicontinuous subset of $\mathscr{L}(F; G)$. $\gamma)$ As $u$ runs through H, the set of mappings $y \mapsto u(., y)$ from F into $\mathscr{L}_{\mathscr{S}}(E; G)$ is equicontinuous. We then say that H is a $\mathscr{S}$-equihypocontinuous set of bilinear mappings (separately continuous) from $E \times F$ into G. Similarly for a set $\mathfrak{T}$ of bounded subsets of F, we define the notions of a $\mathfrak{T}$-equihypocontinuous set and a $(\mathscr{S}, \mathfrak{T})$-equihypocontinuous set.