Proposition 5.4.1 (Integrators Inequality).label Let $[a,b]\suf\R$, $E,F,H$ be locally convex spaces, $E\ti F\to H$ with $(x,y)\mto xy$ be a continuous bilinear map, $(P=\curl{x_j}_{0}^{n},c=\curl{c_j }_{1}^{n})\in\scr{P}_{t}([a,b])$ and $G:[a,b]\to F$. Let $\braks{\cd}_{H}$ be a continuous seminorm on $H$, then there exists continuous seminorms $\braks{\cd}_{E },\braks{\cd}_{F }$ on $E,F$ respectively such that
\begin{align*}\braks{S(P,c,f,G)}_{H }\leq \SUP{x\in[a,b]}\braks{f }_{E }\cd \braks{G }_{\text{var},F }\end{align*}
In particular for any $f\in RS([a,b],G)$
\begin{align*}\braks{\integral{a}{b}fdG}_{H}\leq \SUP{x\in[a,b]}\braks{f}_{E}\cd\braks{G}_{\text{var},F}\end{align*}
Proof. By Proposition 4.2.3, there exists continuous seminorms $\braks{\cd}_{E}$ on $E$ and $\braks{\cd}_{F }$ on $F$ such that $\braks{xy}_{H }\leq \braks{x }_{E }\braks{y }_{F }$ for all $(x,y)\in E\ti F$ then
\begin{align*}\braks{S(P,c,f,G)}_{H }&\leq \sums{n }{j=1 }\braks{f(c_j)\braks{G(x_j)-G(x_{j-1})}}_{H }\\&\leq \sums{n }{j=1 }\braks{f(c_j)}_{E}\braks{\braks{G(x_j)-G(x_{j-1})}}_{F}\\&\leq \SUP{x\in[a,b]}\braks{f}_{E}\cd V_{F ,P }\parens{G }\\&\leq \SUP{x\in[a,b]}\braks{f }_{E }\cd \braks{G }_{\text{var},F }\end{align*}
Since $(P,c)$ was arbitrary we are done.$\square$