From e68b9912404ee417493f87536d58efab4c9e0457 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Tue, 28 Apr 2026 14:20:27 -0400
Subject: [PATCH] Fixed typo in convex hull.

---
 src/fa/lc/convex.tex | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/fa/lc/convex.tex b/src/fa/lc/convex.tex
index 25cc1b3..934ba2b 100644
--- a/src/fa/lc/convex.tex
+++ b/src/fa/lc/convex.tex
@@ -184,7 +184,7 @@
     If the above holds, then $E$ is a \textbf{locally convex} space.
 \end{definition}
 \begin{proof}
-    $(1) \Rightarrow (2)$: Let $U \in \cn(0)$ be convex. By \autoref{proposition:tvs-good-neighbourhood-base}, there exists $V \in \cn(0)$ circled such that $V + V \subset U$. Let $W = \bracs{tx + (1 - t)y|x, y \in V}$ be the convex hull of $V$, then $W \subset U$ is convex and circled.
+    $(1) \Rightarrow (2)$: Let $U \in \cn(0)$ be convex. By \autoref{proposition:tvs-good-neighbourhood-base}, there exists $V \in \cn(0)$ circled such that $V + V \subset U$. Let $W = \text{Conv}(V)$ be the convex hull of $V$, then $W \subset U$ is convex and circled. 
 
     $(2) \Rightarrow (3)$: For each $V \in \cn(0)$ convex, circled, and radial, let $[\cdot]_V: E \to [0, \infty)$ be its gauge, then $[\cdot]_V$ is a seminorm. For each $x, y \in X$ and $r > 0$, $[x - y]_V < r$ if and only if $x - y \in rV$. Thus the uniformity induced by $[\cdot]_V$ corresponds to the uniformity generated by $\bracs{U_{rV}| r > 0}$, where $U_V = \bracs{(x, y) \in E|x - y \in V}$. Since this holds for all $U \in \cn(0)$, the topology of $E$ and the topology induced by $\bracs{[\cdot]_V| V \in \cn(0), \text{ convex, circled, radial}}$ coincide.