From 4e0efaf7f5e2bb5e652156bb56edff7b9cae3152 Mon Sep 17 00:00:00 2001
From: Bokuan Li <bokuan.li@mail.mcgill.ca>
Date: Fri, 1 May 2026 18:31:08 -0400
Subject: [PATCH] Adjusted formulation of a total variation statement.

---
 src/fa/lc/barrel.tex |  4 ++--
 src/fa/rs/bv.tex     | 12 +-----------
 2 files changed, 3 insertions(+), 13 deletions(-)

diff --git a/src/fa/lc/barrel.tex b/src/fa/lc/barrel.tex
index 7dbe5c4..f651928 100644
--- a/src/fa/lc/barrel.tex
+++ b/src/fa/lc/barrel.tex
@@ -23,7 +23,7 @@
     $(3) \Rightarrow (2)$: Let $D \subset E$ be a barrel and $\rho: E \to [0, \infty)$ be its gauge. By (4) of \autoref{definition:gauge}, $D = \bracs{\rho \le 1}$, so $\bracs{\rho > 1}$ is open, and $\rho$ is semicontinuous. By assumption, $\rho$ is continuous, so $D \in \cn_E(0)$ by (5) of \autoref{lemma:continuous-seminorm}. 
 \end{proof}
 
-\begin{summary}[Barreled Spaces]
+\begin{summary}
 \label{summary:barreled-space}
     The following types of locally convex spaces are barreled:
     \begin{enumerate}
@@ -61,7 +61,7 @@
 \label{proposition:barrel-limit}
     Let $\seqi{E}$ be locally convex spaces over $K \in \RC$, $E$ be a vector space over $K$, and $\seqi{T}$ such that $T_i \in \hom(E_i; E)$ for all $i \in I$, then the inductive locally convex topology on $E$ induced by $\seqi{T}$ is barreled.  
 \end{proposition}
-\begin{proof}[Proof, {{\cite[II.7.2]{SchaeferWolff}}}]
+\begin{proof}[Proof, {{\cite[II.7.2]{SchaeferWolff}}}. ]
     Let $D \subset E$ be a barrel, then for each $i \in I$, $T_i^{-1}(D) \subset E_i$ is also a barrel, and thus a neighbourhood of $0$ in $E_i$. By (5) of \autoref{definition:lc-inductive}, $D$ is a neighbourhood of $0$ in $E$, so $E$ is barreled. 
 \end{proof}
 
diff --git a/src/fa/rs/bv.tex b/src/fa/rs/bv.tex
index e6ca5db..7b8529c 100644
--- a/src/fa/rs/bv.tex
+++ b/src/fa/rs/bv.tex
@@ -65,7 +65,7 @@
     \begin{enumerate}
         \item $BV([a, b]; E)$ is a vector space.
         \item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}$ is a seminorm on $BV([a, b]; E)$.
-        \item For each continuous seminorm $\rho$ on $E$ and $M > 0$, $\bracs{[\cdot]_{\text{var}, \rho} \le M} \subset E^{[a, b]}$ is closed. 
+        \item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}: E^{[a, b]} \to [0, \infty]$ is lower semicontinuous. In particular, for any $M > 0$, $\bracs{[\cdot]_{\text{var}, \rho} \le M} \subset E^{[a, b]}$ is closed. 
         \item For any $f \in BV([a, b]; E)$ and continuous seminorm $\rho$ on $E$, $\sup_{x \in [a, b]}\rho(f(x)) \le \rho(f(a)) + [f]_{\text{var}, \rho}$.
     \end{enumerate}
     If $(E, \norm{\cdot}_E)$ is a normed vector space, then
@@ -74,16 +74,6 @@
     \end{enumerate}
 \end{definition}
 \begin{proof}[Proof {{\cite[Proposition X.1.1]{Lang}}}. ]
-    (3): Let $\rho$ be a continuous seminorm on $E$, $P = \seqf{x_j} \in \scp([a, b])$, and $f \in \overline{\bracs{[\cdot]_{\text{var}, \rho} \le M}}$. For any $\eps > 0$, there exists $g \in \bracs{[\cdot]_{\text{var}, \rho} \le M}$ such that $\rho(f(x_j) - g(x_j)) < \eps$ for each $1 \le j \le n$. In which case, 
-    \begin{align*}
-    V_{\rho, P}(f) &= \sum_{j = 1}^n \rho(f(x_j) - f(x_{j - 1})) \\
-    &\le 2n\eps + \sum_{j = 1}^n \rho(g(x_j) - g(x_{j - 1})) \\
-    &\le V_{\rho, P}(g) + 2n\eps \le M + 2n\eps
-    \end{align*}
-    
-
-    As the above holds for all $\eps > 0$, $V_{\rho, P}(f) \le M$. Since this holds for all $P \in \scp([a, b])$, $[f]_{\text{var}, \rho} \le M$.
-
     (5): For each $n \in \nat^+$, let
     \[
     D_n = \bracs{x \in [a, b]|\forall \eps > 0, \exists y \in (x - \eps, x + \eps): \norm{f(x) - f(y)}_E \ge 1/n}