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Let $(X,\|\cdot\|)$ be a normed vector space and let $X^*,X^{**}$ denote its continuous and second continuous dual, each endowed with the usual norm. Let $B$ denote the closed unit ball of $X^{**}$.

Since $X^{**}$ is the dual of $X^*$, from the Banach-Alaoglu theorem, it is known that $B$ is compact in the weak-$*$ topology.

I want to know if $B$ is sequentially compact, or relatively sequentially compact (still in the weak-$*$ topology).


I know that the Eberlein-Šmulian theorem can be helpful when dealing with weak topologies, but here we're interested in the weak-$*$ topology.

I don't mind adding the assumption that $X$ is Banach. However, I don't want to add a reflexivity assumption on $X$.

I am not well-versed in functional analysis or weak topologies. Actually, my question comes from the theory of optimization of functions.

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  • $\begingroup$ do you mind adding separability assumptions? $\endgroup$ Commented Nov 14, 2021 at 11:10
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    $\begingroup$ Not true in general. If $X^{*}$ is separable then is is true. $\endgroup$ Commented Nov 14, 2021 at 11:44

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In the 2 examples below, the given sequences have no weak* convergent subsequences in the unit ball $B$ of the second dual.

  1. Take $X=\ell^1$ and let $(e_n)$ be the standard basis.
  2. Take $X=C([0,1])$ and let $(f_n)$ be the sequence of functions defined by $f_n(t) = \sin(nt)$.
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