Corollary

The vector subspace $\mathscr{E}$ is generated by the characteristic functions of intervals.

Now let us consider the case where $E$ is a normed space over $\mathbf{R}$; it is then immediate that the characteristic function of an interval $J$ with endpoints $a, b$ ($a < b$) admits a primitive, namely the function equal to $a$ for $x \leq a$, to $x$ for $a \leq x \leq b$, and to $b$ for $x \geq b$. The cor. to prop. 2 thus shows that every step function with values in $E$ admits a primitive.

We can now apply the method set out in n' 2.