Definition 5.3.1 (Riemann-Stieltjes Sum).label Let $[a,b]\suf\R$, $E,F,H$ be TVSs over $K\in \curl{\R,\com}$, $G:[a,b]\to F$ and suppose there exists a continuous bilinear map $E\ti F\ni (x,y)\mto xy\in H$. Let $f:[a,b]\to E$ and $(P=\curl{x_j}^{n}_{j=0},c=\curl{c_j}^{n}_{j=1})\in \scr{P}_{t}([a,b])$ then
\begin{align*}S(P,c,f,G)=\sums{n}{j=1}f(c_{j})\braks{G(x_j)-G(x_{j-1})}\end{align*}
is the Riemann-Stieltjes sum of $f$ with respect to $G$ and $(P,c)$.