Corollary 1

— Let E and F be two Fréchet spaces, and G a locally convex space. Every separately continuous bilinear mapping from $E \times F$ into G is continuous.

In fact, every Fréchet space is barrelled (III, p. 25, corollary).

Let E and F be two locally convex spaces. We use $\mathscr{B}(E, F)$ to denote the space of continuous bilinear forms on $E \times F$, with the topology of uniform convergence on sets of the form $A \times B$, where A (resp. B) is bounded in E (resp. F). The formula

(for $x \in E, y \in F$ and $u \in \mathscr{B}(E, F)$) defines a continuous linear injective mapping $\phi$ from $\mathscr{B}(E, F)$ into $\mathscr{L}_b(E; F'_b)$.