Corollary

Let $E, F, G$ be three complete normed spaces over $\mathbf{R}$, and $(x, y) \mapsto x.y$ a continuous bilinear map from $E \times F$ into $G$. Let $\mathbf{f}, \mathbf{g}$ be two regulated functions on $I$, with values in $E$ and $F$ respectively. If $\mathbf{f}$ is bounded on $I$ and if the integral of g is absolutely convergent over I, then the integral of [f, g] is absolutely convergent.

Indeed, there exists a number h > 0 such that one has $\| [x.y] \| \leq h \| x \| . \| y \|$ identically (Gen. Top., IX, p. 173, th. 1); if one puts $k = \sup_{x \in I} \| f(x) \|$, then one has $\| [f(x).g(x)] \| \leq h k \| g(x) \|$ on I; the comparison principle now shows that the integral of [f.g] is absolutely convergent, and, from (2),