Corollary 1

The map which to every point $\mathbf{x}_0 \in E$ associates the continuous function $t \mapsto C(t, t_0).\mathbf{x}_0$, defined on J, is an isomorphism of the normed space E onto the vector space $\mathscr{I}$ of integrals of (4), endowed with the topology of compact convergence.

It is certainly a bijective linear map of E onto $\mathscr{I}$: now $C(t, t_0)$ is bounded on a compact set $K \subset J$, so $\| C(t, t_0).\mathbf{x}_0 \| \leq M \| \mathbf{x}_0 \|$ for any $t \in K$ and $\mathbf{x}_0 \in E$, which shows that this map is continuous; and since

it is clear that the inverse map is also continuous.