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Does separability follow from weak-* sequential separability of dual space?
$\omega^*$-separability of $l_\infty^*$.
Recently I read a Theorem stating, Let $X$ be a Banach space which is separable then every weakly compact subset is metrizable. I noticed that the separability is used in order to prove that $X^{*}$ is weakly* separable and in the rest of proof separability of $X$ is no longer used.
Motivated by that my question is, let $X$ be a Banach space if $X^*$ is $\mathrm{weakly}^*$ separable then does this imply that $X$ is separable, if not this would weaken the hypothesis.