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Updated the dyadic rational numbers and RS integrals.
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Bokuan Li
2026-03-21 19:15:40 -04:00
parent d7d5db5f90
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src/fa/rs/rs-bv.tex
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@@ -3,36 +3,37 @@
\begin{proposition}
\label{proposition:rs-bound}
Let $[a, b] \subset \real$, $E_1, E_2, H$ be locally convex spaces, and $E_1 \times E_2 \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G: [a, b] \to E_2$.
Let $[a, b] \subset \real$, $E, F, H$ be locally convex spaces, and $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G: [a, b] \to F$.
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 $f \in RS([a, b], G)$,
Let $[\cdot]_H$ be a continuous seminorm on $H$, then there exists continuous seminorms $[\cdot]_E$ on $E$ and $[\cdot]_F$ on $F$ such that for any $f \in RS([a, b], G)$,
\[
\braks{\int_a^bf dG}_H \le \sup_{x \in [a, b]}[f]_1 \cdot [g]_{\text{var}, 2}
\braks{\int_a^bf dG}_H \le \sup_{x \in [a, b]}[f]_E \cdot [g]_{\text{var}, F}
\]
\end{proposition}
\begin{proof}
By \autoref{proposition:tvs-convex-multilinear}, there exists continuous seminorms $[\cdot]_1$ on $E_1$ and $[\cdot]_2$ on $E_2$ such that $[xy]_H \le [x]_1[y]_2$ for all $(x, y) \in E_1 \times E_2$.
By \autoref{proposition:tvs-convex-multilinear}, there exists continuous seminorms $[\cdot]_E$ on $E$ and $[\cdot]_F$ on $F$ such that $[xy]_H \le [x]_E[y]_F$ for all $(x, y) \in E \times F$.
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]_{\text{var}, 2}
[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)]_E[G(x_j) - G(x_{j - 1})]_F \\
&\le \sup_{x \in [a, b]}[f]_E \cdot V_{2, P}(G) \le \sup_{x \in [a, b]}[f]_E \cdot [g]_{\text{var}, F}
\end{align*}
\end{proof}
\begin{proposition}
\label{proposition:rs-complete}
Let $[a, b] \subset \real$, $E_1, E_2, H$ be locally convex spaces, and $E_1 \times E_2 \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G \in BV([a, b]; E_2)$.
Let $[a, b] \subset \real$, $E, F, H$ be locally convex spaces, and $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G \in BV([a, b]; F)$.
For each continuous seminorm $\rho$ on $E_1$ and $f: [a, b] \to E_1$, define
For each continuous seminorm $\rho$ on $E$ and $f: [a, b] \to E$, define
\[
[f]_{u, \rho} = \sup_{x \in [a, b]}\rho(f(x))
\]
Let $\net{f} \subset RS([a, b], G)$ such that:
\begin{enumerate}
\item[(a)] For each continuous seminorm $\rho$ on $E_1$, $[f_\alpha - f]_{u, \rho} \to 0$.
\item[(a)] For each continuous seminorm $\rho$ on $E$, $[f_\alpha - f]_{u, \rho} \to 0$.
\item[(b)] $\lim_{\alpha \in A}\int_a^b f_\alpha dG$ exists.
\end{enumerate}
then $f \in RS([a, b], G)$ and $\int_a^b f dG = \lim_{\alpha \in A}\int_a^b f_\alpha dG$. In particular,
@@ -48,11 +49,11 @@
&+ \rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG} \\
&+ \rho\paren{S(P, c, f_\alpha, G) - \int_a^b f_\alpha dG}
\end{align*}
Let $\rho$ be a continuous seminorm on $H$, and $[\cdot]_1$ and $[\cdot]_2$ be continuous seminorms on $E_1$ and $E_2$ such that $\rho(xy) \le [x]_1[y]_2$ for all $(x, y) \in E_1 \times E_2$.
Let $\rho$ be a continuous seminorm on $H$, and $[\cdot]_E$ and $[\cdot]_F$ be continuous seminorms on $E$ and $F$ such that $\rho(xy) \le [x]_E[y]_F$ for all $(x, y) \in E \times F$.
Let $\eps > 0$, then by assumption (a) and (b), there exists $\alpha \in A$ such that:
\begin{enumerate}
\item $[f - f_\alpha]_1 < \eps/(3[G]_{\text{var}, 2})$.
\item $[f - f_\alpha]_E < \eps/(3[G]_{\text{var}, F})$.
\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 $P \ge P_0$,
@@ -68,9 +69,9 @@
\begin{proposition}
\label{proposition:rs-bv-continuous}
Let $[a, b] \subset \real$, $E_1, E_2$ be locally convex spaces, $H$ be a sequentially complete locally convex space, and $E_1 \times E_2 \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map.
Let $[a, b] \subset \real$, $E, F$ be locally convex spaces, $H$ be a sequentially complete locally convex space, and $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map.
Let $f \in C([a, b]; E_1)$, $G \in BV([a, b]; E_2)$, then
Let $f \in C([a, b]; E)$, $G \in BV([a, b]; F)$, then
\begin{enumerate}
\item $f \in RS([a, b], G)$.
\item For any $\seq{(P_n, t_n)} \subset \scp_t([a, b])$ with $\sigma(P_n) \to 0$,
@@ -81,19 +82,20 @@
\end{enumerate}
\end{proposition}
\begin{proof}
Let $\rho$ be a continuous seminorm on $H$, and $[\cdot]_1$ and $[\cdot]_2$ be continuous seminorms on $E_1$ and $E_2$ such that $\rho(xy) \le [x]_1[y]_2$ for all $(x, y) \in E_1 \times E_2$.
Let $\rho$ be a continuous seminorm on $H$, and $[\cdot]_E$ and $[\cdot]_F$ be continuous seminorms on $E$ and $F$ such that $\rho(xy) \le [x]_E[y]_F$ for all $(x, y) \in E \times F$.
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]_{\text{var}, 2}
&\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)]_E[G(y_k) - G(y_{k - 1})]_F \\
&\le \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \sigma(P) \end{array}}[f(x) - f(y)]_E \cdot [G]_{\text{var}, F}
\end{align*}
Therefore for any two $(P, c), (Q, d) \in \scp_t([a, b])$,
\[
\rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q)) \end{array}}[f(x) - f(y)]_1 \cdot [G]_{\text{var}, 2}
\rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q)) \end{array}}[f(x) - f(y)]_E \cdot [G]_{\text{var}, F}
\]
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.
by passing through a common refinement. Since $f \in C([a, b]; E)$, 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.
In addition, for any $\seq{(P_n, t_n)}$ as in (2), $\limv{n}S(P_n, t_n, f, G)$ exists by sequential completeness. Since $\angles{S(P, c, f, G)}_{(P, c) \in \scp_t([a, b])}$ is Cauchy, the limit $\lim_{(P, c) \in \scp_t([a, b])}S(P, c, f, G)$ exists as well and is equal to $\limv{n}S(P_n, t_n, f, G)$.
\end{proof}