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Keywords:
Lipschitz-free space; nonseparable Banach space; sequentially compact; Radon--Nikodým property
Lipschitz-free space; nonseparable Banach space; sequentially compact; Radon--Nikodým property
Summary:
We prove that several classical Banach space properties are equivalent to separability for the class of Lipschitz-free spaces, including Corson's property ($\mathcal{C}$), Talponen's countable separation property, or being a Gâteaux differentiability space. On the other hand, we single out more general properties where this equivalence fails. In particular, the question whether the duals of nonseparable Lipschitz-free spaces have a weak$^*$ sequentially compact ball is undecidable in ZFC. Finally, we provide an example of a nonseparable dual Lipschitz-free space that fails the Radon--Nikodým property.
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