Theorem 5.3.8 (Change of Variables).label Let $[a,b]\suf\R$, $E,F,H$ be locally convex spaces over $K\in \curl{\R,\com}$, with $G\in C^{1}([a,b];F)$ and suppose there exists a continuous bilinear map $E\ti F\ni (x,y)\mto xy\in H$. For any bounded $f\in RS([a,b],G;E)$
\begin{align*}\integral{a}{b}f(t)G(dt)=\integral{a}{b}f(t)DG(t)dt\end{align*}
Proof. Let $\rho_{H}:H\to [0,\infty)$ be a continuous seminorm. Pick similarly $\rho_{E},\rho_{F}$ such that for any $x\in E$ and $y\in F$ we have
\begin{align*}\rho_{H}(xy)\leq\rho_{E}(x)\rho_{F}(y)\end{align*}
Since $f$ is bounded
\begin{align*}M\define \SUP{t\in[a,b]}\rho_{E}(f(t))<\infty\end{align*}
For any $(P=\curl{x_j}^{n}_{0},c=\curl{c_j}^{n}_{1})\in\scr{P}_{t}([a,b])$ such that $\s(P)\leq \de$ to be determined later
\begin{align*}\rho_{H}\parens{S(P,c,f,G)-S(P,c,fDG,\text{Id})}&=\rho_{H}\parens{\sums{n}{j=1}f(c_j)\parens{G(x_j)-G(x_{j-1})-(x_j-x_{j-1})DG(c_j)}}\\&\leq \sums{n}{j=1}\rho_{E}(f(c_{j}))\rho_{F}(G(x_{j})-G(x_{j-1})-(x_{j}-x_{j-1})DG(c_{j}))\\&\leq M\sums{n}{j=1}\rho_{F}(G(x_{j})-G(x_{j-1})-(x_{j}-x_{j-1})DG(c_{j}))\end{align*}
Fix $j$ and define
\begin{align*}H_{j}:[x_{j-1},x_{j}]\to F&&t\mto G(t)-G(c_{j})-(t-c_{j})DG(c_{j})\end{align*}
then $H_{j}\in C^{1}([x_{j-1},x_{j}],F)$ and by Mean Value Theorem 5.3.6
\begin{align*}\rho_{F}\parens{G(x_j)-G(x_{j-1})-(x_j-x_{j-1})DG(c_j)}&=\rho_{F}\parens{H_j(x_j)-H_j(x_{j-1})}\\&\leq \integral{x_{j-1}}{x_j}\rho_{F}(DH(t))dt\\&\leq (x_{j}-x_{j-1})\SUP{t\in [x_{j-1},x_j]}\rho_{F}(DH(t))\\&=(x_{j}-x_{j-1})\SUP{t\in [x_{j-1},x_j]}\rho_{F}(DG(t)-DG(c_{j}))\end{align*}
Observe $DG\in C([a,b];F)=UC([a,b];F)$ by Proposition 5.3.7 so $\fall\e>0,\exists\de>0$ such that after refining the partition if necessary we have
\begin{align*}\rho_{H}\parens{S(P,c,f,G)-S(P,c,fDG,\text{Id})}\leq M \sums{n}{j=1}(x_{j}-x_{j-1})\e=M (b-a)\e\end{align*}
as desired.$\square$