Exercise 6, § 5

Let E, F and G be three topological vector spaces. A set H of bilinear mappings from $E \times F$ into G is said to be separately equicontinuous if for all $x \in E$, the set of linear mappings $u(x, .)$, where $u$ runs through H, is equicontinuous in $\mathscr{L}(F; G)$ and if for all $y \in F$, the set of linear mappings $u(., y)$, where $u$ runs through H, is equicontinuous in $\mathscr{L}(E; G)$. Suppose that F is metrizable, and that E is a Baire space ($cf.$ III, p. 43, exerc. 5 and V, p. 79, exerc. 15). Show that every separately equicontinuous set of bilinear mappings from $E \times F$ into G is equicontinuous ($cf.$ III, p. 42, exerc. 11).