Proposition 2

Let E, F and G be three locally convex spaces. Suppose that E and F are metrizable and E is barrelled. Let H be a set of separately continuous bilinear mappings from E × F into G. Suppose that for every $x \in E$, the set of mappings $u(x, .)$ from F into G, where u runs through H, is equicontinuous. Then H is equicontinuous.

Let $U_n$ (resp. $V_n$) be a fundamental sequence of neighbourhoods of 0 in E (resp. F). If H is not equicontinuous, there exists a closed, convex, balanced neighbourhood W of 0 in G such that for all $n$, $H(U_n \times V_n)$ is not contained in W. There exists then a sequence of pairs $(x_n, y_n) \in U_n \times V_n$, and a sequence $(u_n)$ of elements of H, such that $u_n(x_n, y_n) \notin W$. Let $p$ be the gauge of W. For every $y \in F$ and every $u \in H$, the mapping $u(., y)$ from E into G is continuous, hence $p \circ u(., y)$ is a continuous semi-norm on E. On the other hand, for every $x \in E$, the set of mappings $u(x, .)$ for $u \in H$ is equicontinuous; since the sequence $(y_n)$ tends to 0, it is bounded, and the set of all $u(x, y_n)$, for $n \geq 0$ and $u \in H$, is bounded (III, p. 22, prop. 9). It follows from this that the function $p'(x) = \sup_{\substack{u \in H \\ n \geq 0}} p(u(x, y_n))$ is a lower semi-continuous semi-norm (finite) on E. Since E is barrelled, $p'$ is continuous (III, p. 24, corollary). Since $(x_n)$ tends to 0 in E, $p'(x_n)$ tends to 0, so that we have $p'(x_n) \leq 1$ if $n$ is large enough; but then $p(u_n(x_n, y_n)) \leq p'(x_n) \leq 1$, hence $u_n(x_n, y_n) \in W$, which contradicts the hypothesis on $u_n, x_n, y_n$.