From eed3a342e4d4b102a5ac78e5a48d36f97534a8c6 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Fri, 26 Jun 2026 12:59:13 -0400
Subject: [PATCH] Typo fixes.

---
 src/fa/duality/mackey.tex | 2 +-
 src/fa/duality/polar.tex  | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/fa/duality/mackey.tex b/src/fa/duality/mackey.tex
index be65a80..31648e5 100644
--- a/src/fa/duality/mackey.tex
+++ b/src/fa/duality/mackey.tex
@@ -52,7 +52,7 @@
     \bracsn{x \in E|\dpn{x, \psi}{E} \le \mu \forall \psi \in A} \subset \bracsn{x \in E|\dpn{x, \phi}{E} \le 1} = \bracs{\phi}^\circ
     \]
 
-    Since $\sigma$ is saturated, assume without loss of generality that $\mu = 1$ and that $A$ is convex, circled, and $\sigma(F, E)$-compact. In which case, let $A^\circ$ be the polar of $A$ with respect to $\dpn{E, E^*}{E}$, then
+    Since $\sigma$ is covering and saturated, assume without loss of generality that $\mu = 1$ and that $A$ is convex, circled, and $\sigma(F, E)$-compact with $0 \in A$. In which case, let $A^\circ$ be the polar of $A$ with respect to $\dpn{E, E^*}{E}$, then
     \[
     A^\circ = \bracsn{x \in E|\dpn{x, \psi}{E} \le 1 \forall \psi \in A} \subset \bracs{\phi}^\circ
     \]
diff --git a/src/fa/duality/polar.tex b/src/fa/duality/polar.tex
index bd81e90..85b0c05 100644
--- a/src/fa/duality/polar.tex
+++ b/src/fa/duality/polar.tex
@@ -113,7 +113,7 @@
 \begin{proof}[Proof, {{\cite[IV.1.5]{SchaeferWolff}}}. ]
     By \autoref{proposition:polar-properties}, $A^{\circ \circ}$ is a $\sigma(E, F)$-closed, convex set that contains $0$. Since $A^{\circ \circ} \supset A$, it is sufficient to show that $A^{\circ\circ} \subset \ol{\conv}(A \cup \bracs{0})$. 
 
-    Let $x_0 \in E \setminus \ol{\conv}(A \cup \bracs{0})$, then by the \hypreref[Hahn-Banach Theorem]{theorem:hahn-banach-geometric-2}, there exists $\phi: E \to \real$ such that:
+    Let $x_0 \in E \setminus \ol{\conv}(A \cup \bracs{0})$, then by the \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach-geometric-2}, there exists $\phi: E \to \real$ such that:
     \begin{enumerate}
         \item $\phi$ is $\sigma(E, F)$-continuous.
         \item $\phi(\ol{\conv}(A \cup \bracs{0})) \subset (-\infty, 1)$ and $\phi(x_0) > 1$.