Added introduction to polars.

src/fa/duality/polar.tex

@@ -0,0 +1,35 @@
+\section{Polars}
+\label{section:polar}
+
+\begin{definition}[Real Polar]
+\label{definition:real-polar}
+    Let $\dpn{E, F}{\lambda}$ be a duality over $K \in \RC$ and $A \subset E$, then
+    \[
+    A^\circ = \bracsn{y \in F| \text{Re}\dpn{x, y}{\lambda} \le 1 \forall x \in A}
+    \]
+
+    is the \textbf{real polar} of $A$. 
+\end{definition}
+
+\begin{definition}[Absolute Polar]
+\label{definition:absolute-polar}
+    Let $\dpn{E, F}{\lambda}$ be a duality over $K \in \RC$ and $A \subset E$, then
+    \[
+    A^\square = \bracsn{y \in F|\ |\dpn{x, y}{\lambda}| \le 1 \forall x \in A}
+    \]
+
+    is the \textbf{absolute polar} of $A$. 
+\end{definition}
+
+\begin{proposition}
+\label{proposition:polar-properties}
+    Let $\dpn{E, F}{\lambda}$ be a duality over $K \in \RC$, then:
+    \begin{enumerate}
+        \item $\emptyset^\circ = \emptyset^\square = F$ and $F^\circ = F^\square = \bracs{0}$. 
+        \item For any $A, B \subset E$ and $\lambda \ne 0$, if $\lambda A \subset B$, then $B^\circ \subset \lambda^{-1}A^\circ$. 
+    \end{enumerate}
+    
+\end{proposition}
+
+
+