Exercise 7, § 2

a) Let $t \mapsto A(t)$ be a map from an interval $J \subset \mathbf{R}$ into $\mathscr{L}(E)$, such that, for every $x \in E$, the map $t \mapsto A(t).x$ of $J$ into $E$ is continuous. Show that $t \mapsto \|A(t)\|$ is bounded on every compact interval $K \subset J$ (use Gen. Top., IX, p. 194, th. 2). Under these conditions show that, for every point $(t_0, x_0) \in J \times E$ the equation $d\mathbf{x}/dt = A(t).\mathbf{x} + \mathbf{b}(t)$ has one and only one solution defined on $J$ and equal to $\mathbf{x}_0$ at the point $t_0$. Further, if one denotes this solution by $\mathbf{u}(t, t_0, \mathbf{x}_0)$, the map $\mathbf{x}_0 \mapsto \mathbf{u}(t, t_0, \mathbf{x}_0)$ is a bijective bicontinuous linear map $C(t, t_0)$ of $E$ onto itself, which satisfies the relations (6) (IV, p. 179) and (7) (IV, p. 181); further, the map $(s, t) \mapsto C(s, t)$ is continuous from $J \times J$ into $\mathscr{L}(E)$.

b) Take for $E$ the space of sequences $\mathbf{x} = (x_n)_{n \in \mathbf{N}}$ of real numbers such that $\lim_{n \to \infty} x_n = 0$, endowed with the norm $\|\mathbf{x}\| = \sup |x_n|$, under which $E$ is complete. For each $t \in J = [0, 1]$, let $A(t)$ be the linear map of $E$ into itself such that

Show that $A(t)$ satisfies the conditions of a) but that the map $t \mapsto A(t)$ of $J$ into $\mathscr{L}(E)$ is not continuous at the point $t = 0$, and that the resolvent $C(t, t_0)$ of the equation $d\mathbf{x}/dt = A(t).\mathbf{x}$ is such that the map $t \mapsto C(t, t_0)$ of $J$ into $\mathscr{L}(E)$ is not differentiable at the point $t = 0$.