From f6c597687345fb06e5f5ecf305df82e837390d0f Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Mon, 25 May 2026 21:05:21 -0400
Subject: [PATCH] Fixed some typos.

---
 src/fa/rs/rs-bv.tex       | 4 ++--
 src/fa/tvs/complexify.tex | 2 +-
 2 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/fa/rs/rs-bv.tex b/src/fa/rs/rs-bv.tex
index 0a974d7..a7aceb8 100644
--- a/src/fa/rs/rs-bv.tex
+++ b/src/fa/rs/rs-bv.tex
@@ -123,7 +123,7 @@
     \end{align*}
 
 
-    Since $\alpha \in BV([a, b]; F)$, by \autoref{proposition:rs-bv-continuous}, for any $\seq{(P_n, c_n)} \subset \scp_t([a, b])$, 
+    Since $f \in C([a, b] \times [c, d]; E)$, $f$ is uniformly continuous by \autoref{proposition:uniform-continuous-compact}, and $\bracs{f(\cdot, t)|t \in [c, d]} \subset C([a, b]; E)$ is uniformly equicontinuous. As $\alpha \in BV([a, b]; F)$, by \autoref{proposition:rs-bv-continuous}, for any $\seq{(P_n, c_n)} \subset \scp_t([a, b])$, 
     \[
     \int_a^b \int_c^d f(s, t) \beta(dt) \alpha(ds) = \limv{n}S(P_n, c, g, \alpha)
     \]
@@ -133,7 +133,7 @@
     \limv{n}S(P_n, c_n, f(\cdot, t), \alpha) = \int_a^b f(s, t) \alpha(ds)
     \]
 
-    uniformly for all $t \in [c, d]$. Since $f \in C([a, b] \times [c, d]; E)$, $f$ is uniformly continuous by \autoref{proposition:uniform-continuous-compact}, and $\bracs{f(\cdot, t)|t \in [c, d]} \subset C([a, b]; E)$ is uniformly equicontinuous. As $\beta \in BV([c, d]; G)$, 
+    uniformly for all $t \in [c, d]$. Finally, given that $\beta \in BV([c, d]; G)$, 
     \[
     \int_c^d\int_a^b f(s, t) \alpha(ds) \beta(dt) = \limv{n}\int_c^d S(P_n, c_n, f(\cdot, t), \alpha) \beta(dt)
     \]
diff --git a/src/fa/tvs/complexify.tex b/src/fa/tvs/complexify.tex
index 877292d..e571628 100644
--- a/src/fa/tvs/complexify.tex
+++ b/src/fa/tvs/complexify.tex
@@ -14,7 +14,7 @@
         E \ar@{->}[u]^{\iota} \ar@{->}[ru]_{T} & 
         }
         \]
-        \item $\complex(E) = \iota(E) \oplus i\iota(E)$ as a vector space over $\real$. For each $z \in \complex(E)$ with $z = x + iy$, $x = \text{Re}(x)$ and $y = \text{Re}(y)$ are the \textbf{real} and \textbf{imaginary parts} of $z$. 
+        \item $\complex(E) = \iota(E) \oplus i\iota(E)$ as a vector space over $\real$. For each $z \in \complex(E)$ with $z = x + iy$, $x = \text{Re}(x)$ and $y = \text{Im}(y)$ are the \textbf{real} and \textbf{imaginary parts} of $z$. 
     \end{enumerate}
 
     The pair $(\complex(E), \iota)$ is the \textbf{complexification} of $E$, and