Proposition 12.15.3.label Let $E, F, G$ be TVSs over $K \in \RC$, and $\lambda: E \times F \to G$ be a separately continuous bilinear map. If one of the following holds:
- (B)
$E$ is Baire.
- (B’)
$E$ is barrelled and $G$ is locally convex.
then $\lambda$ is $B(E)$-hypocontinuous.
Proof. Since $\lambda$ is separately continuous, the mapping
\[E \to L(F; G) \quad x \mapsto \lambda(x, \cdot)\]
is continuous with respect to the strong operator topology on $L(F; G)$. As such, for each $B \subset E$ bounded, $\bracs{\lambda(x, \cdot)|x \in B}$ is bounded in $L_{s}(F; G)$. By the Banach-Steinhaus Theorem, $\bracs{\lambda(x, \cdot)|x \in B}$ is equicontinuous.$\square$
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