Exercise 10, § 1

Let $f$ be a regulated function on $[a,\ b]$, taking values in $E$, let $g$ be a regulated function on $[a,\ c]$ ($c > b$), taking its values in $F$, and let $(\mathbf{x}, \mathbf{y}) \mapsto [\mathbf{x}.\mathbf{y}]$ be a continuous bilinear map from $E \times F$ into $G$ ($E, F, G$ being complete normed spaces). Show that

(reduce to the case where $f$ is a step function).