$X$ is a Banach space and it is separable, is there any simple counterexample to show the dual space $X^\ast$ is not separable?
2026-04-22 08:32:54.1776846774
an example to show separability of a Banach space does not imply separability of the dual space
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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).