Abstract: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.
Keywords: Lipschitz-free space; nonseparable Banach space; sequentially compact; Radon--Nikodým property
DOI: DOI 10.14712/1213-7243.2025.003
AMS Subject Classification: 46B20 46B26 46E15