11.10 Locally Convex Spaces of Linear Maps

Proposition 11.10.1.label Let $T$ be a set, $\sigma \subset 2^{T}$ be an ideal, $E$ be a locally convex space over $K$, and $\cf \subset E^{T}$ be a subspace such that

1. (B)

   For each $f \in \cf$ and $S \in \sigma$, $f(S) \subset E$ is bounded.

For each $S \in \sigma$ and continuous seminorm $\rho: E \to [0, \infty)$, let

then the $\sigma$-uniform topology on $\cf$ is induced by seminorms of the form $\rho_{S}$, where $\rho$ is a continuous seminorm on $E$, and $S \in \sigma$. In which case, the $\sigma$-uniform topology on $\cf$ is locally convex.

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 $\sigma$.

Lemma 11.10.3.label Let $E$, $F$ be locally convex spaces over $K \in \RC$, $\sigma \subset 2^{E}$ be an ideal, and $\ol \sigma$ be its saturated hull, then the $\sigma$-uniformity and $\ol \sigma$-uniformity on $L(E; F)$ coincide.