Proposition 13.4.2.label 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
Let $\net{f}\subset RS([a, b], G)$ such that:
- (a)
For each continuous seminorm $\rho$ on $E$, $[f_{\alpha} - f]_{u, \rho}\to 0$.
- (b)
$\lim_{\alpha \in A}\int_{a}^{b} f_{\alpha} dG$ exists.
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,
- (1)
If $H$ is complete, then condition (b) may be omitted.
- (2)
If $H$ is sequentially complete and $A = \nat^{+}$, then condition (b) may be omitted.
Proof. Let $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_{t}([a, b])$, then
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:
- (1)
$[f - f_{\alpha}]_{E} < \eps/(3[G]_{\text{var}, F})$.
- (2)
$\rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG}< \eps/3$.
Since $f_{\alpha} \in RS([a, b], G)$, there exists $P_{0} \in \scp([a, b])$ such that if $P \ge P_{0}$,
- (3)
$\rho\paren{S(P, c, f_\alpha, G) - \int_a^b f_\alpha dG}< \eps/3$.
Thus for any $(P, c) \in \scp_{t}([a, b])$ with $P \ge P_{0}$,
$\square$