 Weak-star separability of the entire dual

The continuous dual of a separable Banach space is separable in its weak-star topology, even when it is not norm separable. Banach-Alaoglu theorem and weak-star metrizability of the dual ball make the dual unit ball a separable compact metric space. The countable union of positive integer multiples of a countable dense subset of that ball is dense in the whole dual. This does not imply global weak-star metrizability.