Updated the separable dual proposition.

src/fa/duality/definitions.tex

@@ -14,6 +14,15 @@
     In the context of a dual system, $E$ and $F$ are identified as subspaces of each others' algebraic duals. 
 \end{definition}
 
+\begin{definition}[Norming Duality]
+\label{definition:norming-duality}
+    Let $K \in \RC$ and $\dpn{E, F}{\lambda}$ be a duality of normed vector spaces over $K$, then $\dpn{E, F}{\lambda}$ is \textbf{norming} if:
+    \begin{enumerate}
+        \item For each $x \in E$, $\norm{x}_E = \sup_{y \in F, \norm{y}_F \le 1}\dpn{x, y}{\lambda}$. 
+        \item For each $y \in F$, $\norm{y}_F = \sup_{x \in E, \norm{x}_E \le 1}\dpn{x, y}{\lambda}$. 
+    \end{enumerate}
+\end{definition}
+
 \begin{definition}[Weak Topology]
 \label{definition:duality-weak-topology}
     Let $K \in \RC$ and $\dpn{E, F}{\lambda}$ be a duality over $K$, then the weak topology generated by $F$, denoted $\sigma(E, F)$, is the \textbf{weak topology} of the duality on $E$.