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.


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.

  • $\begingroup$ A reference for this is e.g. "Functional Analysis and Semi-Groups" by Hille and Phillips, Theorem 2.8.5 $\endgroup$ – Andrei Kh Jan 15 '20 at 21:15

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 $\ell_\infty$ .


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.