Proposition 11.4.3 | Jerry's Digital Garden

Proposition 11.4.3. 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

$f \in RS([a, b], G)$.
For any $\seq{(P_n, t_n)}\subset \scp_{t}([a, b])$ with $\sigma(P_{n}) \to 0$,
\[\int_{a}^{b} fdG = \limv{n}S(P_{n}, t_{n}, f, G)\]

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]_{\text{var}, 2}\end{align*}

Therefore for any two $(P, c), (Q, d) \in \scp_{t}([a, b])$,

\[\rho(S(P, c, f, G) - S(Q, d, f, G)) \le 2 \cdot \sup_{\begin{array}{c}x, y \in [a, b] \\ |x - y| < \max(\sigma(P), \sigma(Q))\end{array}}[f(x) - f(y)]_{1} \cdot [G]_{\text{var}, 2}\]

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.

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 $\limv{n}S(P_{n}, t_{n}, f, G)$.$\square$