Definition 11.3.1 (Riemann-Stieltjes Sum, [Section X.1, Lan93]). Let $[a, b] \subset \real$, $E_{1}, E_{2}, H$ be TVSs over $F \in \RC$, $E_{1} \times E_{2} \to H$ with $(x, y) \mapsto xy$ be a continuous bilinear map, and $G: [a, b] \to E_{2}$.
Let $f: [a, b] \to E_{1}$ and $(P = \seqfz{x_j}, c = \seqf{c_j}) \in \scp_{t}([a, b])$, then
\[S(P, c, f, G) = \sum_{j = 1}^{n} f(c_{j})[G(x_{j}) - G(x_{j - 1})]\]
is the Riemann-Stieltjes sum of $f$ with respect to $G$ and $(P, c)$.