Added relevant Hahn-Banach results.

src/fa/lc/convex.tex

@@ -64,6 +64,8 @@
 
 
 
+
+
 \begin{definition}[Sublinear Functional]
 \label{definition:sublinear-functional}
     Let $E$ be a vector space over $K \in \RC$, then a \textbf{sublinear functional} is a mapping $\rho: E \to \real$ such that:
@@ -79,9 +81,9 @@
 \label{definition:seminorm}
     Let $E$ be a vector space over $K \in \RC$, then a \textbf{seminorm} on $E$ is a mapping $\rho: E \to [0, \infty)$ such that:
     \begin{enumerate}
-        \item $\rho(0) = 0$.
-        \item For any $x \in E$ and $\lambda \in K$, $\rho(\lambda x) = \abs{\lambda} \rho(x)$.
-        \item For any $x, y \in E$, $\rho(x + y) \le \rho(x) + \rho(y)$.
+        \item[(SN1)] $\rho(0) = 0$.
+        \item[(SN2)] For any $x \in E$ and $\lambda \in K$, $\rho(\lambda x) = \abs{\lambda} \rho(x)$.
+        \item[(SN3)] For any $x, y \in E$, $\rho(x + y) \le \rho(x) + \rho(y)$.
     \end{enumerate}
 \end{definition}
 
@@ -128,8 +130,8 @@
     \end{enumerate}
     In particular,
     \begin{enumerate}
-        \item If $A$ is convex, then $[\cdot]_A$ is a sublinear functional.
-        \item If $A$ is convex and circled, then $[\cdot]_A$ is a seminorm.
+        \item[(4)] If $A$ is convex, then $[\cdot]_A$ is a sublinear functional.
+        \item[(5)] If $A$ is convex and circled, then $[\cdot]_A$ is a seminorm.
     \end{enumerate}
 \end{definition}
 \begin{proof}