Theorem 2

Let $E_1$ and $E_2$ be two reflexive Fréchet spaces, and G a locally convex Hausdorff space. For $i = 1, 2$, let $F_i$ be the strong dual of $E_i$. Then every separately continuous bilinear mapping $u : F_1 \times F_2 \to G$ is continuous.

The space G is isomorphic to a subspace of a product of Banach spaces (II, p. 5, prop. 3). Therefore it is enough to prove the theorem under the additional hypothesis that G is a Banach space. But $F_1$ is barrelled and $F_2$ bornological (IV, p. 24, corollary), and $\mathscr{L}_b(F_2 ; G)$ is a Fréchet space (IV, p. 23, prop. 3). Let v denote the linear mapping from $F_1$ into $\mathscr{L}_b(F_2 , G)$ associated with u by the relation

Since $F_1$ is barrelled and u separately continuous, v is continuous (III, p. 31, prop. 6).

Since v is continuous, lemma 1 implies the existence of a neighbourhood $U_1$ of 0 in $F_1$ whose image under v is bounded in $\mathscr{L}_b(F_2 ; G)$. In other words, for every bounded subset $B_2$ in $F_2$, the set $u(U_1 \times B_2)$ is bounded in the Banach space G. Let $U_2$ be the set of all $x_2 \in F_2$ such that $\|u(x_1, x_2)\| \leqslant 1$ for all $x_1 \in U_1$. The set $U_2$ then absorbs every bounded subset; since $F_2$ is bornological, $U_2$ is a neighbourhood of 0 in $F_2$, and this proves that u is continuous.