Proposition 12.15.4.label Let $E, F, G$ be TVSs over $K \in \RC$, $\sigma \subset B(E)$ and $\tau \subset B(F)$ be ideals of bounded sets, and $\lambda: E \times F \to G$ be a bilinear mapping.

1. (1)

   If $\lambda$ is $\sigma$-hypocontinuous, then $\lambda$ is continuous on $S \times F$ for all $S \in \sigma$.

2. (2)

   If $\lambda$ is $(\sigma, \tau)$-hypocontinuous, then $\lambda$ is uniformly continuous on $S \times T$ for all $S \in \sigma$ and $T \in \tau$.