习题 7, § 5

设E、F和G是E的有界子集的三个拓扑向量空间、$\mathscr{S}$集合，以及从$E \times F$到G的单独连续双线性映射的H集合。 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. 类似地，对于 F 的有界子集的集合 $\mathfrak{T}$，我们可以计算 $\mathfrak{T}$-等次连续集合和 $(\mathscr{S}, \mathfrak{T})$-等次连续集合的定义概念。