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



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> *If $E$ is a separable Banach spaces, then there is a sequence $\{ \phi\_n \}$ in its dual $E^{\star}$ such that $$\|x\|=\sup\_n |\phi\_n(x)|$$*
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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.



The norm of a separable Banach space can be determined by countable continuous linear functionals?
For (1): Pick a countable dense set $x\_i\in E$, and for each $x\_i$ in that set, pick (by Hahn–Banach) a functional $\phi\_i$ of norm $1$ such that $\|x\_i\|=\phi\_i(x\_i)$.


For (2): duals of separable Banach spaces also have that property. For example, $L^{\infty}(\mathbb R)$ is the dual of $L^1(\mathbb R)$ and its norm is therefore determined by countably many functionals.


