9.7 Locally Convex Spaces of Linear Maps
Proposition 9.7.1. Let $T$ be a set, $E$ be a locally convex space defined by the seminorms $\seqi{[\cdot]}$, and $\mathfrak{S}\subset 2^{T}$ be an upward-directed family. For each $i \in I$ and $S \in \mathfrak{S}$, let
\[[\cdot]_{S, i}: E^{T} \to [0, \infty) \quad f \mapsto \sup_{x \in S}[f(x)]_{S, i}\]
then the $\mathfrak{S}$-uniform topology on $E^{T}$ is defined by the seminorms
\[\bracs{[\cdot]_{S, i}|S \in \mathfrak{S}, i \in I}\]
and hence locally convex.
Proof. By Proposition 6.1.3.$\square$