From e4da295fd916df68136d616a51d1c988b46ef48a Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Mon, 4 May 2026 16:04:09 -0400
Subject: [PATCH] Added introduction to polars.

---
 src/fa/duality/definitions.tex | 26 +++++++++++++++++++++++--
 src/fa/duality/index.tex       |  1 +
 src/fa/duality/polar.tex       | 35 ++++++++++++++++++++++++++++++++++
 3 files changed, 60 insertions(+), 2 deletions(-)
 create mode 100644 src/fa/duality/polar.tex

diff --git a/src/fa/duality/definitions.tex b/src/fa/duality/definitions.tex
index 936598b..f67f9b0 100644
--- a/src/fa/duality/definitions.tex
+++ b/src/fa/duality/definitions.tex
@@ -44,7 +44,29 @@
     
 \end{proof}
 
-
-
+\begin{lemma}
+\label{lemma:duality-dense}
+    Let $K \in \RC$ and $\dpn{E, F}{\lambda}$ be a duality over $K$, then for any subspace $F_0 \subset F$, the following are equivalent:
+    \begin{enumerate}
+        \item $\dpn{E, F_0}{\lambda}$ is a duality. 
+        \item For any $y_0 \in F$, $\seqf{x_j} \subset E$, and $\eps > 0$, there exists $y \in F_0$ such that for each $1 \le j \le n$, $\dpn{x_j, y}{\lambda} = \dpn{x_j, y_0}{\lambda}$.
+        \item $F_0$ is $\sigma(F, E)$-dense in $F$.
+    \end{enumerate}
+\end{lemma}
+\begin{proof}
+    (1) $\Rightarrow$ (2): Let $E_0 = \text{span}\bracs{x_j|1 \le j \le n}$, and
+    \[
+    \phi: E_0 \to K \quad x \mapsto \dpn{x, y_0}{\lambda}
+    \]
+    
+    Since $\dpn{E, F_0}{\lambda}$ is a duality, there exists $\seqf{y_j} \subset F_0$ such that for all $x \in E_0$, 
+    \[
+    |\dpn{x, \phi}{E_0}| \le \sum_{j = 1}^n |\dpn{x, y_j}{\lambda}|
+    \]
+
+    Hence $\phi \in L(E_0, \sigma(E_0, F_0); K)$. By the \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach}, there exists $\Phi \in L(E, \sigma(E, F_0); K)$ such that $\Phi|_{E_0} = \phi$. By \autoref{lemma:duality-dual}, there exists $y \in F_0$ such that $\dpn{x_j, y}{\lambda} = \dpn{x_j, y_0}{\lambda}$ for all $1 \le j \le n$. 
+
+    (3) $\Rightarrow$ (1): Let $x \in E$ such that $\dpn{x, y}{\lambda} = 0$ for all $y \in F_0$, then since $F_0$ is $\sigma(F, E)$-dense in $F$, $\dpn{x, y}{\lambda} = 0$ for all $y \in F$. Hence $x = 0$. 
+\end{proof}
 
 
diff --git a/src/fa/duality/index.tex b/src/fa/duality/index.tex
index edfa3f2..13b2f04 100644
--- a/src/fa/duality/index.tex
+++ b/src/fa/duality/index.tex
@@ -2,5 +2,6 @@
 \label{chap:duality}
 
 \input{./definitions.tex}
+\input{./polar.tex}
 
 
diff --git a/src/fa/duality/polar.tex b/src/fa/duality/polar.tex
new file mode 100644
index 0000000..fbfe1c9
--- /dev/null
+++ b/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}
+
+
+