Added the Bochner integral.

src/fa/norm/normed.tex

@@ -86,3 +86,17 @@
 
     so $\sup_{T \in \mathcal{T}}\norm{T}_{L(E; F)} \le 2n/r$.
 \end{proof}
+
+
+\begin{proposition}
+\label{proposition:dual-norm}
+    Let $E$ be a normed space, then for any $x \in E$, 
+    \[
+    \norm{x}_E = \sup_{\substack{\phi \in E^* \\ \norm{\phi}_{E^*} = 1}}\dpn{x, \phi}{E}
+    \]
+\end{proposition}
+\begin{proof}
+    For any $\phi \in E^*$ with $\norm{\phi}_{E^*} = 1$, $\dpn{x, \phi}{E} \le \norm{x}_E \cdot \norm{\phi}_{E^*} = \norm{x}_E$. On the other hand, by the \hyperref[Hahn-Banach Theorem]{proposition:hahn-banach-utility}, there exists $x \in E^*$ such that $\dpn{x, \phi}{E} = \norm{x}_E$ and $\norm{\phi}_{E^*} \le 1$. 
+\end{proof}
+
+