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Fixed some typos.
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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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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.