Let $X = C_0(\mathbb{R})$ and define $A:X \to X$ where $(Ax)(t) = a(t)x(t)$, $a\in C_0(\mathbb{R})$
$C_0(\mathbb{R})$ the space of continuous functions on $\mathbb{R}$ for which $\lim_{\left | t \right |\rightarrow \infty }x(t) = 0$ with metric $\left \| x \right \| = max_{t\in R}\left | x(t) \right |$
It was our class example since this is not easy example but we don't have time to solve it. So we should solve it by ourself. Pls, can anyone solve it with a little description? Or give some hints?
Thanks in advance.