Definition 12.15.2 (Hypocontinuity).label Let $E, F, G$ be TVSs over $K \in \RC$, $\sigma \subset 2^{E}$ be an ideal of bounded sets, and $\lambda: E \times F \to G$ be a separately continuous bilinear map, then the following are equivalent:

1. (1)

   For each $S \in \sigma$ and $V \in \cn_{G}(0)$, there exists $U \in \cn_{F}(0)$ such that $\lambda(S \times U) \subset V$.

2. (2)

   For each $S \in \sigma$, $\bracs{\lambda(x, \cdot)|x \in S}\subset F^{*}$ is equicontinuous.

If the above holds, then $\lambda$ is $\sigma$-hypocontinuous.

For any ideal $\tau \subset 2^{F}$ of bounded sets, $\lambda$ if $(\sigma, \tau)$-hypocontinuous if $\lambda$ is $\sigma$-hypocontinuous and $\tau$-hypocontinuous.