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Fixed some typos.
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This commit is contained in:
Bokuan Li
2026-03-12 22:29:40 -04:00
parent 68ec8c38a7
commit 532bced3b7
5 changed files with 15 additions and 15 deletions
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2
.chktexrc
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@@ -1 +1 @@
CmdLine { -n24 -n9 -n17 -n25 }
CmdLine { -n24 -n9 -n17 -n25 -n3 }

2
document.tex
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@@ -9,7 +9,7 @@
\input{./src/fa/index}
\input{./src/measure/index}
\input{./src/dg/index}
\input{./src/process/index}
%\input{./src/process/index}
\bibliographystyle{alpha} % We choose the "plain" reference style
\bibliography{refs} % Entries are in the refs.bib file

2
src/cat/cat/universal.tex
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@@ -42,7 +42,7 @@
\begin{enumerate}
\item $P \in \obj{\catc}$.
\item For each $i \in I$, $\iota_i \in \mor{A_i P}$.
\item[\textbf{(U)}] For any pair $(C, \seqi{f})$ satisfying (1) and (2), there exists a unique $f \in \mor{P, C}$ such that the following diagram commutes
\item[(U)] For any pair $(C, \seqi{f})$ satisfying (1) and (2), there exists a unique $f \in \mor{P, C}$ such that the following diagram commutes
\[
\xymatrix{

14
src/fa/rs/bv.tex
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@@ -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}

10
src/fa/rs/rs-bv.tex
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@@ -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.