\section{Riemann-Stieltjes Integrals and Functions of Bounded Variation}
\label{section:rs-bv}

\begin{proposition}
\label{proposition:rs-bound}
    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 $[\cdot]_E$ on $E$ and $[\cdot]_F$ on $F$ such that for any $f \in RS([a, b], G)$,
    \[
    \braks{\int_a^bf dG}_H \le \sup_{x \in [a, b]}[f]_E \cdot [g]_{\text{var}, F}
    \]

\end{proposition}
\begin{proof}
    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)]_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 $[a, b] \subset \real$, $E, F, H$ 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 $E$ 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 $E$, $[f_\alpha - f]_{u, \rho} \to 0$.
        \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,
    \begin{enumerate}
        \item If $H$ is complete, then condition (b) 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 $[\cdot]_E$ 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 $[f - f_\alpha]_E < \eps/(3[G]_{\text{var}, F})$.
        \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 $P \ge P_0$,
    \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 $[a, b] \subset \real$, $E, F$ 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 $f \in C([a, b]; E)$, $G \in BV([a, b]; F)$, then
    \begin{enumerate}
        \item $f \in RS([a, b], G)$.
        \item For any $\seq{(P_n, t_n)} \subset \scp_t([a, b])$ with $\sigma(P_n) \to 0$,
        \[
        \int_a^b fdG = \limv{n}S(P_n, t_n, f, G)
        \]

    \end{enumerate}
\end{proposition}
\begin{proof}
    Let $\rho$ be a continuous seminorm on $H$, and $[\cdot]_E$ 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)]_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 $(P, c), (Q, d) \in \scp_t([a, b])$,
    \[
    \rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c} x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q)) \end{array}}[f(x) - f(y)]_E \cdot [G]_{\text{var}, F}
    \]

    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}
