Weak-star separability of the entire dual
= 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.