Proposition 11.4.2. 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 $E_{1}$ and $f: [a, b] \to E_{1}$, define
Let $\net{f}\subset RS([a, b], G)$ such that:
For each continuous seminorm $\rho$ on $E_{1}$, $[f_{\alpha} - f]_{u, \rho}\to 0$.
$\lim_{\alpha \in A}\int_{a}^{b} f_{\alpha} dG$ 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,
If $H$ is complete, then condition (b) may be omitted.
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 $[\cdot]_{1}$ 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:
$[f - f_{\alpha}]_{1} < \eps/(3[G]_{\text{var}, 2})$.
$\rho\paren{\int_a^b f_\alpha dG - \lim_{\alpha \in A}\int_a^b f_\alpha dG}< \eps/3$.
Since $f_{\alpha} \in RS([a, b], G)$, there exists $P_{0} \in \scp([a, b])$ such that if $P \ge P_{0}$,
$\rho\paren{S(P, c, f_\alpha, G) - \int_a^b f_\alpha dG}< \eps/3$.
Thus for any $(P, c) \in \scp_{t}([a, b])$ with $P \ge P_{0}$,