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?
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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.
answered Jun 19 '19 at 4:37
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1$\begingroup$ Thanks. So in general we only can say it contains a weak$^* -$ convergent subnet. Right ? $\endgroup$ – Red shoes Jun 19 '19 at 4:40
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$\begingroup$ @Redshoes yes, a subnet is fine, that works in any compact space. $\endgroup$ – Henno Brandsma Jun 19 '19 at 4:41
