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.