Added the Mackey-Arens theorem.

src/fa/tvs/equicontinuous.tex

@@ -113,10 +113,10 @@
 % TODO: Replace this with a more general version involving polars in the future. 
 \begin{theorem}[Banach-Alaoglu]
 \label{theorem:alaoglu}
-    Let $E$ be a locally convex space over $K \in \RC$ and $\alg \subset E^*$ be equicontinuous, then $\alg$ is precompact with respect to $\sigma(E^*, E)$.
+    Let $E$ be a locally convex space over $K \in \RC$ and $\alg \subset E^*$ be equicontinuous, then $\alg$ is relatively compact with respect to $\sigma(E^*, E)$.
 \end{theorem}
 \begin{proof}
-    For each $x \in E$, $\alg(x) = \bracsn{\dpn{x, \phi}{E}|\phi \in \alg}$ is precompact by \autoref{proposition:equicontinuous-bounded}. By the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, 
+    For each $x \in E$, $\alg(x) = \bracsn{\dpn{x, \phi}{E}|\phi \in \alg}$ is relatively compact by \autoref{proposition:equicontinuous-bounded}. By the \hyperref[Arzelà-Ascoli Theorem]{theorem:arzela-ascoli}, 
     \begin{enumerate}
         \item[(C2)] The $\sigma(E^*, E)$-closure of $\alg$ in $\prod_{x \in E}K$ is equicontinuous. 
         \item[(C3)] The $\sigma(E^*, E)$-closure of $\alg$ in $\prod_{x \in E}K$ is compact.