Added the Singer representation theorem.

src/fa/norm/normed.tex

@@ -100,3 +100,13 @@
 \end{proof}
 
 
+% TODO: Replace this with a more general version involving polars in the future. 
+\begin{theorem}[Alaoglu's Theorem]
+\label{theorem:alaoglu}
+    Let $E$ be a normed vector space over $K \in \RC$, then $B^* = \bracsn{\phi \in E^*| \norm{x}_{E^*} \le 1}$ is compact in the weak*-topology. 
+\end{theorem}
+\begin{proof}
+    For each $x \in E$, $I_x = \bracsn{\dpn{x, \phi}{E}|\phi \in B^*}$ is compact. By \autoref{proposition:operator-space-completeness}, the closure of $B^*$ in $\prod_{x \in E}I_x$ is a subset of $\hom(E; K)$. Since $B^*$ is bounded, $I_x \subset \overline{B_K(0, 1)}$ for all $x \in E$, so the closure of $B^*$ is contained in $B^*$. By \autoref{theorem:tychonoff}, $\prod_{x \in E}I_x$ is compact. Therefore $B^*$ is compact with respect to the weak*-topology.
+\end{proof}
+
+