Theorem 10.13.6.label Let $E, F, G$ be TVSs over $K \in \RC$ and $\alg$ be separately continuous bilinear maps from $E \times F$ to $G$. If one of the following holds:
- (B)
$E$ is Baire.
- (B’)
$E$ is barrelled and $G$ is locally convex.
and that
- (M)
$E$ and $F$ are both metrisable.
- (E)
For each $x \in E$, $\bracsn{\lambda(x, \cdot)|\lambda \in \alg}\subset L(F; G)$ is equicontinuous.
then $\alg$ is equicontinuous.
Proof, [III.5.1, SW99]. Let $\seq{(x_n, y_n)}\subset E \times F$ and $\seq{\lambda_n}\subset \alg$ such that $(x_{n}, y_{n}) \to 0$ as $n \to \infty$. Since $\seq{y_n}$ is convergent, for each $n \in \natp$ and $x \in E$, $\bracsn{\lambda_n(x, y_n)|n \in \natp}$ is bounded by (E) and Proposition 6.5.2. By (B) or (B’) and the Banach-Steinhaus Theorem, $\bracsn{\lambda_n(\cdot, y_n)|n \in \natp}$ is equicontinuous, and $\lambda_{n}(x_{n}, y_{n}) \to 0$ as $n \to \infty$ by Proposition 6.5.2. By (M) and Proposition 6.5.2, $\alg$ is equicontinuous at $0$, and hence equicontinuous by Lemma 10.13.5.$\square$