Added relevant Hahn-Banach results.

src/fa/norm/normed.tex

@@ -1,7 +1,35 @@
-\section{Normed and Banach Spaces}
+\section{Norms}
 \label{section:normed-banach}
 
 
+\begin{definition}[Norm]
+\label{definition:norm}
+    Let $E$ be a vector space over $K \in \RC$ and $\norm{\cdot}_E: E \to [0, \infty)$, then $\norm{\cdot}_E$ is a \textbf{norm} if:
+    \begin{enumerate}
+        \item[(N)] For any $x \in E$, $\norm{x}_E = 0$ if and only if $x = 0$.
+        \item[(SN2)] For any $x \in E$ and $\lambda \in K$, $\norm{\lambda x}_E = \abs{\lambda} \norm{x}_E$.
+        \item[(SN3)] For any $x, y \in E$, $\norm{x + y}_E \le \norm{x}_E + \norm{y}_E$.
+    \end{enumerate}
+\end{definition}
+
+\begin{proposition}
+\label{proposition:norm-criterion}
+    Let $E$ be a separated TVS over $K \in \RC$, then the following are equivalent:
+    \begin{enumerate}
+        \item There exists $U \in \cn^o(0)$ bounded and convex.
+        \item There exists a norm $\norm{\cdot}_E: E \to [0, \infty)$ that induces the topology on $E$.
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    (1) $\Rightarrow$ (2): Using \ref{proposition:tvs-good-neighbourhood-base}, assume without loss of generality that $U$ is also circled. Let $V \in \cn^o(0)$, then there exists $\lambda \in K$ such that $\lambda V \supset U$ and $\abs{\lambda}^{-1}U \subset V$.
+
+    Let $\norm{\cdot}_E: E \to [0, \infty)$ be the gauge (\ref{definition:gauge}) of $U$. For each $r > 0$, let $B_E(0, r) = \bracs{x \in E|\norm{x}_E < r}$, then
+    \[
+    \bracs{\lambda U|\lambda > 0} = \bracs{B_E(0, r)|r > 0}
+    \]
+    is a fundamental system of neighbourhoods at $0$ for $E$. Therefore $\norm{\cdot}_E$ induces the topology on $E$.
+\end{proof}
+
 \begin{theorem}[Successive Approximation]
 \label{theorem:successive-approximation}
     Let $E, F$ be normed spaces, $T \in L(E; F)$, $C \ge 0$, and $\gamma \in (0, 1)$. If for all $y \in F$, there exists $x \in E$ such that: