Let $X,Y$ be Banach spaces such that there exists an isomorphism between $X^*$ and $Y^*$. Then, can we claim that there exists an isomorphism between $X$ and $Y$?
2026-03-26 20:40:40.1774557640
If two Banach spaces have isomorphic duals, must they be isomorphic?
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No. One can be separable, the other non-separable yet they may have isometrically isomorphic duals. For the specific example take: $X=C[0,1]$ (which is separable) and $Y=C[0,1]\oplus_\infty c_0(\Gamma)$, where $\Gamma$ is any uncountable set of cardinality at most continuum (so that $Y$ is non-separable).