Replaced references to upward-directed families with ideals.

src/fa/lc/spaces-of-linear.tex

@@ -3,18 +3,23 @@
 
 \begin{proposition}
 \label{proposition:lc-spaces-linear-map}
-    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
+    Let $T$ be a set, $\sigma \subset 2^T$ be an ideal, and $E$ be a locally convex space over $K$. For each $S \in \sigma$ and continuous seminorm $\rho: E \to [0, \infty)$, let
     \[
-    [\cdot]_{S, i}: E^T \to [0, \infty) \quad f \mapsto \sup_{x \in S}[f(x)]_{S, i}
+    \rho_S: E^T \to [0, \infty] \quad f \mapsto \sup_{x \in S}\rho(f(x))
     \]
 
-    then the $\mathfrak{S}$-uniform topology on $E^T$ is defined by the seminorms
+    then the the $\sigma$-uniform topology on $E^T$ is defined by distances of the form
     \[
-    \bracs{[\cdot]_{S, i}|S \in \mathfrak{S}, i \in I}
+    d_{S, \rho}: E^T \times E^T \to [0, \infty] \quad (f, g) \mapsto \rho_S(f - g)
     \]
 
-    and hence locally convex.
+    In particular, if $\cf \subset E^T$ is a subspace such that
+    \begin{enumerate}
+        \item[(B)] For each $f \in \cf$ and $S \in \sigma$, $f(S) \subset E$ is bounded. 
+    \end{enumerate}
+
+    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. 
 \end{proposition}
 \begin{proof}
-    By \autoref{proposition:set-uniform-pseudometric}.
+    By \autoref{proposition:tvs-set-uniformity} and \autoref{proposition:set-uniform-pseudometric}.
 \end{proof}