Corollary 2

Let $x_0$ be a point of I, and for each function $f \in \mathscr{H}$ let $P(f)$ be the primitive of $f$ which vanishes at the point $x_0$; the map $f \mapsto P(f)$ of $\mathscr{H}$ into $\mathscr{F}_c(I; E)$ is a continuous linear mapping.

Cor. 1 to th. 1 allows us to establish the existence of primitives for certain categories of functions by the following procedure: if one knows that the functions belonging to a subset $\mathscr{A}$ of $\mathscr{F}_c(I; E)$ admit a primitive, so will the functions belonging to the closure in $\mathscr{F}_c(I; E)$ of the vector subspace generated by $\mathscr{A}$. We shall apply this method in the next subsection.