From ca5e81fdbc888f8c618150e51c7a8a4eca72cf92 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Sun, 3 May 2026 22:49:27 -0400
Subject: [PATCH] Started duality.

---
 src/fa/duality/definitions.tex | 50 ++++++++++++++++++++++++++++++++++
 src/fa/duality/index.tex       |  6 ++++
 src/fa/index.tex               |  1 +
 3 files changed, 57 insertions(+)
 create mode 100644 src/fa/duality/definitions.tex
 create mode 100644 src/fa/duality/index.tex

diff --git a/src/fa/duality/definitions.tex b/src/fa/duality/definitions.tex
new file mode 100644
index 0000000..865011f
--- /dev/null
+++ b/src/fa/duality/definitions.tex
@@ -0,0 +1,50 @@
+\section{Dual Systems}
+\label{section:dual-systems}
+
+\begin{definition}[Duality]
+\label{definition:duality}
+    Let $K$ be a field, $E, F$ be vector spaces over $K$, and $\lambda: E \times F \to K$ be a bilinear map, then the triple $(E, F, \lambda)$ is a \textbf{dual system/duality} over $K$ if
+    \begin{enumerate}[label=($S_{\arabic*}$)]
+        \item For any $x_0 \in E$, if $\lambda(x_0, y) = 0$ for all $y \in F$, then $x_0 = 0$. 
+        \item For any $y_0 \in E$, if $\lambda(x, y_0) = 0$ for all $x \in E$, then $y_0 = 0$.
+    \end{enumerate}
+
+    The mapping $\lamdba: E \times F \to K$ is the \textbf{canonical bilinear form} of the duality, denoted $(x, y) \mapsto \dpn{x, y}{\lambda}$, and the duality $(E, F, \lambda)$ is denoted $\dpn{E, F}{\lambda}$. 
+
+    In the context of a dual system, $E$ and $F$ are identified as subspaces of each others' algebraic duals. 
+\end{definition}
+
+\begin{definition}[Weak Topology]
+\label{definition:duality-weak-topology}
+    Let $K \in \RC$ and $\dpn{E, F}{\lambda}$ be a duality over $K$, then the weak topology generated by $F$, denoted $\sigma(E, F)$, is the \textbf{weak topology} of the duality on $E$. 
+\end{definition}
+
+\begin{lemma}
+\label{lemma:duality-dual}
+    Let $K \in \RC$ and $\dpn{E, F}{\lambda}$ be a duality over $K$, then the dual of $(E, \sigma(E, F))$ is $F$. In other words, for any $\phi \in L(E, \sigma(E, F); K)$, there exists a unique $y \in F$ such that $\dpn{x, \phi}{E} = \dpn{x, y}{\lambda}$ for all $x \in E$.  
+\end{lemma}
+\begin{proof}[Proof, {{\cite[IV.1.2]{SchaeferWolff}}}. ]
+    Since $\phi$ is continuous, there exists $\seqf{y_k} \subset F$ such that for all $x \in E$, 
+    \[
+    |\dpn{x, \phi}{\lambda}| \le \sum_{k = 1}^n |\dpn{x, y_k}{\lambda}|
+    \]
+
+    Assume without loss of generality that $\seqf{y_k}$ is linearly independent, then by the First Isomorphism Theorem, there exists $\Phi \in L(K^n; K)$ such that the following diagram commutes
+    \[
+    \xymatrix{
+    E \ar@{->}[rd]_{\phi} \ar@{->}[r]^{{(y_1, \cdots, y_n)}} & K^n \ar@{->}[d]^{\Phi} \\
+    & K
+    }
+    \]
+    
+    For each $1 \le k \le n$, let $e_k$ be the $k$-th standard basis vector in $K^n$, then for any $x \in E$,
+    \[
+    \dpn{x, \phi}{E} = \sum_{k = 1}^n \Phi(e_k) \dpn{x, y_k}{\lambda}
+    \]
+    
+\end{proof}
+
+
+
+
+
diff --git a/src/fa/duality/index.tex b/src/fa/duality/index.tex
new file mode 100644
index 0000000..edfa3f2
--- /dev/null
+++ b/src/fa/duality/index.tex
@@ -0,0 +1,6 @@
+\chapter{Duality}
+\label{chap:duality}
+
+\input{./definitions.tex}
+
+
diff --git a/src/fa/index.tex b/src/fa/index.tex
index b8c5ca5..adeace0 100644
--- a/src/fa/index.tex
+++ b/src/fa/index.tex
@@ -8,4 +8,5 @@
 \input{./rs/index.tex}
 \input{./lp/index.tex}
 \input{./order/index.tex}
+\input{./duality/index.tex}
 \input{./notation.tex}