# Weak-$*$ sequential compactness of closed ball in bidual

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.

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

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 $$f_n(t) = \sin(nt)$$.