0
$\begingroup$

Let $X$ be a Banach space with the dual $X^\ast$. Let $\{x_n\}_{n=1}^{\infty}$ be a norm-bounded sequence in $X^\ast$, then can we claim that $x_n$ posses a weak$^\ast$-convergent subsequence by Banach-Alaoglu Theorem?

$\endgroup$
3
$\begingroup$

Yes, if $X$ is separable, no in general. See the Wikipedia page, e.g.

Beware that a compact space need not be sequentially compact. And IIRC, $X=\ell^\infty$ is an example where this happens for the dual ball.

$\endgroup$
  • 1
    $\begingroup$ Thanks. So in general we only can say it contains a weak$^* -$ convergent subnet. Right ? $\endgroup$ – Red shoes Jun 19 at 4:40
  • $\begingroup$ @Redshoes yes, a subnet is fine, that works in any compact space. $\endgroup$ – Henno Brandsma Jun 19 at 4:41

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.