Proposition 13.4.3.label 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
- (1)
$f \in RS([a, b], G)$.
- (2)
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 $[\cdot]_{E}$ 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
Therefore for any two $(P, c), (Q, d) \in \scp_{t}([a, b])$,
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 $\limv{n}S(P_{n}, t_{n}, f, G)$.$\square$