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src/fa/rs/rs-bv.tex

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+\section{Riemann-Stieltjes Integrals and Functions of Bounded Variationo}
+\label{section:rs-bv}
+
+\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 $[\cdot]_H$ 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 $f \in RS([a, b], G)$,
+    \[
+    \braks{\int_a^bf dG}_H \le \sup_{x \in [a, b]}[f]_1 \cdot [g]_{\var, 2}
+    \]
+\end{proposition}
+\begin{proof}
+    By \ref{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 $(x, y) \in E_1 \times E_2$.
+
+    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}
+    \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 $G \in BV([a, b]; E_2)$.
+
+    For each continuous seminorm $\rho$ on $H$ and $f: [a, b] \to E$, define
+    \[
+    [f]_{u, \rho} = \sup_{x \in [a, b]}f(\rho)
+    \]
+    Let $\net{f} \subset RS([a, b], G)$ such that:
+    \begin{enumerate}
+        \item[(a)] $\rho(f_\alpha - f) \to 0$ for all continuous seminorm $\rho$ on $E_1$.
+        \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,
+    \begin{enumerate}
+        \item If $H$ is complete, then condition (a) may be omitted.
+        \item If $H$ is sequentially complete and $A = \nat$, then condition (b) may be omitted.
+    \end{enumerate}
+\end{proposition}
+\begin{proof}
+    Let $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_t([a, b])$, then
+    \begin{align*}
+        \rho\paren{S(P, c, f, G) - \lim_{\alpha \in A}\int_a^b f_\alpha dG} &\le \rho(S(P, c, f - f_\alpha, G)) \\
+        &+ \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 $\eps > 0$, 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 $\rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps/3$.
+    \end{enumerate}
+    Since $f_\alpha \in RS([a, b], G)$, there exists $P_0 \in \scp([a, b])$ such that if $P \ge P_0$,
+    \begin{enumerate}
+        \item[(3)] $\rho\paren{S(P, c, f_\alpha, G) - \int_a^b f_\alpha dG} < \eps/3$.
+    \end{enumerate}
+    Thus for any $(P, c) \in \scp_t([a, b])$ with $P \ge P_0$,
+    \[
+    \rho\paren{S(P, c, f, G) - \lim_{\alpha \in A}\int_a^b f_\alpha dG} < \eps
+    \]
+\end{proof}
+
+\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 $f \in C([a, b]; E_1)$, $G \in BV([a, b]; E_2)$, then
+    \begin{enumerate}
+        \item $f \in RS([a, b], G)$.
+        \item For any $\seq{(P_n, t_n)} \subset \scp_t([a, b])$ with $\sigma(P_n) \to 0$,
+        \[
+        \int_a^b fdG = \limv{n}S(P_n, t_n, f, G)
+        \]
+    \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 $(P = \seqfz{x_j}, c = \seqf{c_j}), (Q = \seqfz[m]{y_j}, d = \seqf[m]{d_j}) \in \scp_t([a, b])$ 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}
+    \end{align*}
+    Therefore for any two $(P, c), (Q, d) \in \scp_t([a, b])$,
+    \[
+    \rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q)) \end{array}}[f(x) - f(y)]_1 \cdot [G]_{\var, 2}
+    \]
+    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.
+
+    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}