Exercise 1, § 2

Let E be a complete normed space over $\mathbf{R}$, F a topological space, J an interval of $\mathbf{R}$ not reducing to a single point; let $(t, \xi) \mapsto A(t, \xi)$ be a map from $J \times F$ into $\mathscr{L}(E)$, such that when $\xi$ tends to $\xi_0$, $A(t, \xi)$ tends uniformly to $A(t, \xi_0)$ in J. If $C(t, t_0, \xi)$ is the resolvent of the linear equation $d\mathbf{x}/dt = A(t, \xi).\mathbf{x}$ show that, for every compact interval $K \subset J$, $C(t, t_0, \xi)$ tends uniformly to $C(t, t_0, \xi_0)$ on $K \times K$ as $\xi$ tends to $\xi_0$.