# Does $x_n$ posses a weak$^* -$ convergent sequence by banach-aluoghlo Theorem?

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?

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.
• Thanks. So in general we only can say it contains a weak$^* -$ convergent subnet. Right ? – Red shoes Jun 19 '19 at 4:40