Added a characterisation of L^p.

src/fa/lc/convex.tex

@@ -7,6 +7,28 @@
     Let $E$ be a vector space over $K \in \RC$, then $A \subset E$ is \textbf{convex} if for any $x, y \in A$, $\bracs{\lambda x + (1 - \lambda) y| \lambda \in [0, 1]} \subset A$.
 \end{definition}
 
+\begin{definition}[Convex Hull]
+\label{definition:convex-hull}
+    Let $E$ be a vector space over $K \in \RC$ and $A \subset E$, then the set
+    \[
+    \text{Conv}(A) = \bracs{\sum_{j = 1}^n t_j x_j \bigg | \seqf{t_j} \subset [0, 1], \seqf{x_j} \subset E, \sum_{j = 1}^n t_j = 1 }
+    \]
+
+    is the \textbf{convex hull} of $A$. 
+\end{definition}
+
+\begin{definition}[Convex Circled Hull]
+\label{definition:convex-circled-hull}
+    Let $E$ be a vector space over $K \in \RC$ and $A \subset E$, then the set
+    \[
+    \Gamma(A) = \bracs{\sum_{j = 1}^n t_j x_j \bigg | \seqf{t_j} \subset K, \seqf{x_j} \subset E, \sum_{j = 1}^n |t_j| \le 1 }
+    \]
+
+    is the \textbf{convex circled hull} of $A$. 
+\end{definition}
+
+
+
 \begin{lemma}[{{\cite[II.1.1]{SchaeferWolff}}}]
 \label{lemma:convex-interior}
     Let $E$ be a TVS over $K \in \RC$, $A \subset E$ be convex, $x \in A^o$, and $y \in \ol{A}$, then
@@ -67,10 +89,6 @@
 
 
 
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 \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: