Theorem 13.3.5 (Change of Variables).label Let $[a, b] \subset \real$, $E, F, H$ be locally convex spaces over $K \in \RC$, $E \times F \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G \in C^{1}([a, b]; F)$, then for any bounded $f \in RS([a, b], G; E)$,
Proof. Let $[\cdot]_{H}: H \to [0, \infty)$ be a continuous seminorm and $[\cdot]_{E}: E \to [0, \infty)$ and $[\cdot]_{F}: F \to [0, \infty)$ be continuous seminorms on $E$ and $F$, respectively, such that for any $x \in E$ and $y \in F$, $[xy]_{H} \le [x]_{E}[y]_{F}$.
Since $G \in C^{1}([a, b]; F)$, $DG \in UC([a, b]; F)$ by Proposition 6.4.5. Thus there exists $\delta > 0$ such that $[DG(x) - DG(y)]_{F} < \eps$ for all $x, y \in [a, b]$ with $|x - y| \le \delta$. Let $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_{t}([a, b])$ with $\sigma(P) \le \delta$, then by the Mean Value Theorem, for each $1 \le j \le n$,
so
and
$\square$
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