Recently I'm reading *Stochastic Equations in Infinite Dimensions*, a result is used many times. It is

If $E$ is a separable Banach spaces, then there is a sequence $\{ \phi_n \}$ in its dual $E^{\star}$ such that $$\|x\|=\sup_n |\phi_n(x)|$$

my question is

(1) How to prove it? Or where can I find the proof of it?

(2) Is there any other spaces that have this property? Where can I find related results?

thanks a lot.