Proposition 13.4.1.label Let $[a, b] \subset \real$, $E, F, H$ be locally convex spaces, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, $G: [a, b] \to F$, and $f \in RS([a, b], G)$, then for any continuous seminorms $[\cdot]_{E}: E \to [0, \infty)$, $[\cdot]_{F}: F \to [0, \infty)$, and $[\cdot]_{H}: H \to [0, \infty)$ such that $[xy]_{H} \le [x]_{E}[y]_{F}$ for all $x \in E$ and $y \in F$,
\[\braks{\int_a^bf dG}_{H} \le \sup_{x \in [a, b]}[f]_{E} \cdot [g]_{\text{var}, F}\]
Proof. 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})]_{E}[G(x_{j}) - G(x_{j - 1})]_{F} \\&\le \sup_{x \in [a, b]}[f]_{E} \cdot V_{2, P}(G) \le \sup_{x \in [a, b]}[f]_{E} \cdot [g]_{\text{var}, F}\end{align*}
$\square$
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