Proposition 11.4.1. 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 $G: [a, b] \to E_{2}$.
Let $[\cdot]_{H}$ be a continuous seminorm on $H$, then there exists continuous seminorms $[\cdot]_{1}$ on $E_{1}$ and $[\cdot]_{2}$ on $E_{2}$ such that for any $f \in RS([a, b], G)$,
\[\braks{\int_a^bf dG}_{H} \le \sup_{x \in [a, b]}[f]_{1} \cdot [g]_{\text{var}, 2}\]
Proof. By Proposition 9.2.2, there exists continuous seminorms $[\cdot]_{1}$ on $E_{1}$ and $[\cdot]_{2}$ on $E_{2}$ such that $[xy]_{H} \le [x]_{1}[y]_{2}$ for all $(x, y) \in E_{1} \times E_{2}$.
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})]_{1}[G(x_{j}) - G(x_{j - 1})]_{2} \\&\le \sup_{x \in [a, b]}[f]_{1} \cdot V_{2, P}(G) \le \sup_{x \in [a, b]}[f]_{1} \cdot [g]_{\text{var}, 2}\end{align*}
$\square$