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