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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.

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2 Answers 2

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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.

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  • $\begingroup$ A reference for this is e.g. "Functional Analysis and Semi-Groups" by Hille and Phillips, Theorem 2.8.5 $\endgroup$
    – Andrei Kh
    Commented Jan 15, 2020 at 21:15
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With respct to (2), we have the following result (Corolary 6.8 here):

A normed space have this property if and only if it is isometric to a subspace of $\ell_\infty$ .

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