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$X$ is a Banach space and it is separable, is there any simple counterexample to show the dual space $X^\ast$ is not separable?

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The dual of $\ell^1$ (clearly separable by finite sequences of rational numbers) is $\ell^\infty$ (clearly not separable, as you have $\mathfrak c$-many disjoint balls of radius $1/2$ around indicator functions).

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