MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 8 down vote accepted

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.

share|cite|improve this answer

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 $\ell_\infty$ .

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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