Definition 11.10.2 (Saturated Ideal).label Let $E$ be a locally convex space over $K \in \RC$ and $\sigma \subset 2^{E}$ be an ideal, then $\sigma$ is saturated if:

1. (1)

   For each $\lambda \in K$ and $S \in \sigma$, $\lambda S \in \sigma$.

2. (2)

   For each $S \in \sigma$, $\ol{\aconv}(S) \in \sigma$.

For any ideal $\sigma \subset 2^{E}$, the smallest saturated ideal $\ol \sigma$ containing it is the saturated hull of $\sigma$.