Fixed some typos.

src/fa/rs/bv.tex

@@ -11,7 +11,7 @@
 
     is the \textbf{variation} of $f$ with respect to $\rho$ and $P$. The supremum over all such partitions
     \[
-    [f]_{\var, \rho} = \sup_{P \in \scp([a, b])}V_{\rho, P}(f)
+    [f]_{\text{var}, \rho} = \sup_{P \in \scp([a, b])}V_{\rho, P}(f)
     \]
 
     is the \textbf{total variation} of $f$ on $[a, b]$ with respect to $\rho$.
@@ -21,19 +21,19 @@
 
 \begin{definition}[Bounded Variation, {{\cite[Proposition X.1.1]{Lang}}}]
 \label{definition:bounded-variation}
-    Let $E$ be a locally convex space, $\rho$ be a continuous seminorm on $E$, and $f: [a, b] \to E$. If $[f]_{\var, \rho} < \infty$, then $f$ is of \textbf{bounded variation} with respect to $\rho$.
+    Let $E$ be a locally convex space, $\rho$ be a continuous seminorm on $E$, and $f: [a, b] \to E$. If $[f]_{\text{var}, \rho} < \infty$, then $f$ is of \textbf{bounded variation} with respect to $\rho$.
 
     The space $BV([a, b]; E)$ is the set of functions $[a, b] \to E$ of bounded variation with respect to every continuous seminorm on $E$, and
     \begin{enumerate}
         \item $BV([a, b]; E)$ is a vector space.
-        \item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\var, \rho}$ is a seminorm on $BV([a, b]; E)$.
+        \item For each continuous seminorm $\rho$ on $E$, $[\cdot]_{\text{var}, \rho}$ is a seminorm on $BV([a, b]; E)$.
         \item Let $\fF$ be a filter on $BV([a, b]; E)$ and $f: [a, b] \to E$. If
         \begin{enumerate}
             \item $\pi_x(\fF) \to f(x)$ for all $x \in [a, b]$.
-            \item For every continuous seminorm $\rho$ on $E$, there exists $U \in \fF$ such that $\sup_{g \in U}[g]_{\var, \rho} = M_\rho < \infty$.
+            \item For every continuous seminorm $\rho$ on $E$, there exists $U \in \fF$ such that $\sup_{g \in U}[g]_{\text{var}, \rho} = M_\rho < \infty$.
         \end{enumerate}
-        then $f \in BV([a, b]; E)$ with $[f]_{\var, \rho} \le M_\rho$.
-        \item For any $f \in BV([a, b]; E)$ and continuous seminorm $\rho$ on $E$, $\sup_{x \in [a, b]}\rho(f(x)) \le \rho(f(a)) + [f]_{\var, \rho}$.
+        then $f \in BV([a, b]; E)$ with $[f]_{\text{var}, \rho} \le M_\rho$.
+        \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 space, then
     \begin{enumerate}
@@ -66,5 +66,5 @@
 
     Let $k \le N$ and suppose inductively that $E_k, I_k$ have been constructed. Let $x_k \in E_k$, then by (b), there exists $\eps > 0$ such that $[x_k - \eps, x_k + \eps] \subset I_k$ and $|E_k \setminus [x_k - \eps, x_k + \eps]| \ge N - k$. Let $y_k \in [x_k - \eps, x_k + \eps]$ such that $\norm{f(x_k) - f(y_k)} \ge 1/n$, $I_{k + 1} = I_k \setminus [x_k - \eps, x_k + \eps]$, and $E_{k+1} = E_k \setminus [x_k - \eps, x_k + \eps]$, then $I_k$ and $E_k$ satisfies (a) and (b).
 
-    Therefore there exists pairs $\bracs{(x_k, y_k)|1 \le k \le N}$ such that $\norm{f(x_k) - f(y_k)}_E \ge 1/n$ for all $n$, and the smallest interval containing each $(x_k, y_k)$ are pairwise disjoint. Thus $[f]_{\var} \ge N/n$ for all $N \in \nat^+$, so $[f]_{\var} = \infty$.
+    Therefore there exists pairs $\bracs{(x_k, y_k)|1 \le k \le N}$ such that $\norm{f(x_k) - f(y_k)}_E \ge 1/n$ for all $n$, and the smallest interval containing each $(x_k, y_k)$ are pairwise disjoint. Thus $[f]_{\text{var}} \ge N/n$ for all $N \in \nat^+$, so $[f]_{\text{var}} = \infty$.
 \end{proof}

src/fa/rs/rs-bv.tex

@@ -7,7 +7,7 @@
 
     Let $[\cdot]_H$ be a continuous seminorm on $H$, then there exists continuous seminorms $[\cdot]_1$ on $E_1$ and $[\cdot]_2$ on $E_2$ such that for any $f \in RS([a, b], G)$,
     \[
-    \braks{\int_a^bf dG}_H \le \sup_{x \in [a, b]}[f]_1 \cdot [g]_{\var, 2}
+    \braks{\int_a^bf dG}_H \le \sup_{x \in [a, b]}[f]_1 \cdot [g]_{\text{var}, 2}
     \]
 
 \end{proposition}
@@ -17,7 +17,7 @@
     Let $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_t([a, b])$, then
     \begin{align*}
         [S(P, c, f, G)]_H &\le \sum_{j = 1}^n [f(c_j)[G(x_j) - G(x_{j - 1})]]_H \le \sum_{j = 1}^n [f(c_j)]_1[G(x_j) - G(x_{j - 1})]_2 \\
-        &\le \sup_{x \in [a, b]}[f]_1 \cdot V_{2, P}(G) \le \sup_{x \in [a, b]}[f]_1 \cdot [g]_{\var, 2}
+        &\le \sup_{x \in [a, b]}[f]_1 \cdot V_{2, P}(G) \le \sup_{x \in [a, b]}[f]_1 \cdot [g]_{\text{var}, 2}
     \end{align*}
 \end{proof}
 
@@ -52,7 +52,7 @@
 
     Let $\eps > 0$, then by assumption (a) and (b), there exists $\alpha \in A$ such that:
     \begin{enumerate}
-        \item $[f - f_\alpha]_1 < \eps/(3[G]_{\var, 2})$.
+        \item $[f - f_\alpha]_1 < \eps/(3[G]_{\text{var}, 2})$.
         \item $\rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps/3$.
     \end{enumerate}
     Since $f_\alpha \in RS([a, b], G)$, there exists $P_0 \in \scp([a, b])$ such that if $P \ge P_0$,
@@ -86,11 +86,11 @@
     Let $(P = \seqfz{x_j}, c = \seqf{c_j}), (Q = \seqfz[m]{y_j}, d = \seqf[m]{d_j}) \in \scp_t([a, b])$ with $Q \ge P$, then
     \begin{align*}
         \rho(S(P, c, f, G) - S(Q, d, f, G)) &\le \sum_{j = 1}^n \sum_{y_k \in [x_{j - 1}, x_j]}[f(c_j) - f(d_k)]_1[G(y_k) - G(y_{k - 1})]_2 \\
-        &\le \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \sigma(P) \end{array}}[f(x) - f(y)]_1 \cdot [G]_{\var, 2}
+        &\le \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \sigma(P) \end{array}}[f(x) - f(y)]_1 \cdot [G]_{\text{var}, 2}
     \end{align*}
     Therefore for any two $(P, c), (Q, d) \in \scp_t([a, b])$,
     \[
-    \rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q)) \end{array}}[f(x) - f(y)]_1 \cdot [G]_{\var, 2}
+    \rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q)) \end{array}}[f(x) - f(y)]_1 \cdot [G]_{\text{var}, 2}
     \]
 
     by passing through a common refinement. Since $f \in C([a, b]; E_1)$, this bound tends to $0$ as $\max(\sigma(P), \sigma(Q))$ tends to $0$, so $\angles{S(P, c, f, G)}_{(P, c) \in \scp_t([a, b])}$ is a Cauchy net.