From ef91d9f91b6756beb8c2870917c3f41857b0c485 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Sun, 15 Mar 2026 23:04:46 -0400
Subject: [PATCH] Various typo fixes.

---
 src/dg/derivative/mvt.tex       | 4 ++--
 src/fa/lc/hahn-banach.tex       | 4 ++--
 src/fa/tvs/spaces-of-linear.tex | 2 +-
 3 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/dg/derivative/mvt.tex b/src/dg/derivative/mvt.tex
index caf2fef..2ee3e24 100644
--- a/src/dg/derivative/mvt.tex
+++ b/src/dg/derivative/mvt.tex
@@ -113,9 +113,9 @@
 \begin{proof}
     Let $g: [0, 1] \to F$ be defined by $g(t) = f((1 - t)x + ty)$. Since $f$ is Gateaux-differentiable, $g$ is differentiable by the chain rule \autoref{proposition:chain-rule-sets-conditions} with $Dg(t) = Df((1 - t)x + ty)(y - x)$, and continuous by \autoref{proposition:derivative-sets-real}.
 
-    By the \hyperref[Mean Value Theorem]{theorem:mean-value-theorem-line},
+    By the \hyperref[Mean Value Theorem]{theorem:mean-value-theorem-line}, $f(y) - f(x) = g(1) - g(0)$ is contained in
     \[
-    f(y) - f(x) = g(1) - g(0) \in \overline{\text{Conv}\bracs{Dg(t)|t \in [0, 1]}} = \overline{\text{Conv}\bracs{Df(z)(y - x)|z \in (x, y)}}
+    \overline{\text{Conv}\bracs{Dg(t)|t \in [0, 1]}} = \overline{\text{Conv}\bracs{Df(z)(y - x)|z \in (x, y)}}
     \]
 
 \end{proof}
diff --git a/src/fa/lc/hahn-banach.tex b/src/fa/lc/hahn-banach.tex
index f7a4491..f4a01f3 100644
--- a/src/fa/lc/hahn-banach.tex
+++ b/src/fa/lc/hahn-banach.tex
@@ -140,7 +140,7 @@
     \end{enumerate}
 \end{proposition}
 \begin{proof}
-    (1): Let $\rho_M: E \to [0, \infty)$ be the quotient of $\rho$ by $M$, then $\rho_M \le \rho$ is a continuous seminorm on $E$ by \autoref{definition:quotient-norm}. Let $\phi_0: Kx \to K$ be defined by $\lambda x \mapsto \lambda \rho_M(x)$. By the \hyperref[Hahn-Banach theorem]{theorem:hahn-banach}, there exists $\phi \in \hom{E; K}$ such that $\dpb{x, \phi}{E} = \rho_M(x)$ and $|\phi| \le \rho_M \le \rho$.
+    (1): Let $\rho_M: E \to [0, \infty)$ be the quotient of $\rho$ by $M$, then $\rho_M \le \rho$ is a continuous seminorm on $E$ by \autoref{definition:quotient-norm}. Let $\phi_0: Kx \to K$ be defined by $\lambda x \mapsto \lambda \rho_M(x)$. By the \hyperref[Hahn-Banach Theorem]{theorem:hahn-banach}, there exists $\phi \in \hom{E; K}$ such that $\dpb{x, \phi}{E} = \rho_M(x)$ and $|\phi| \le \rho_M \le \rho$.
 
     (2): By (1) applied to $M = \bracs{0}$.
 
@@ -149,7 +149,7 @@
 
 \begin{proposition}
 \label{proposition:seminorm-lsc}
-    Let $E$ be a locally convex space and $\rho: E \to [0, \infty)$ be a continuous seminorm, then $\rho: E_w \to [0, \infty)$ is lower semicontinuous and Borel measurable. 
+    Let $E$ be a locally convex space and $\rho: E \to [0, \infty)$ be a continuous seminorm, then $\rho: E_w \to [0, \infty)$ is lower semicontinuous and Borel measurable with respect to the weak topology.
 \end{proposition}
 \begin{proof}
     Let $x \in E$, then there exists $\phi_x \in E^*$ such that $\dpn{x, \phi_x}{E} = \rho(x)$ and $|\phi_x| \le \rho$. Thus
diff --git a/src/fa/tvs/spaces-of-linear.tex b/src/fa/tvs/spaces-of-linear.tex
index 5d05d01..027547f 100644
--- a/src/fa/tvs/spaces-of-linear.tex
+++ b/src/fa/tvs/spaces-of-linear.tex
@@ -188,7 +188,7 @@
 
     and
     \[
-    \psi_{x, \lambda}: F^E \to F \quad T \mapsto T(\lambda x) - Tx
+    \psi_{x, \lambda}: F^E \to F \quad T \mapsto T(\lambda x) - \lambda Tx
     \]
 
     are continuous with respect to the product topology. Since