Geodesic
A note on nonseparable Lipschitz-free spaces
Commentationes Mathematicae Universitatis Carolinae, Tome 65 (2024) no. 1, pp. 31-44
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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.
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.
DOI : 10.14712/1213-7243.2025.003
Classification : 46B20, 46B26, 46E15
Keywords: Lipschitz-free space; nonseparable Banach space; sequentially compact; Radon--Nikodým property 
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Aliaga, Ramón J.; Grelier, Guillaume; Procházka, Antonín. A note on nonseparable Lipschitz-free spaces. Commentationes Mathematicae Universitatis Carolinae, Tome 65 (2024) no. 1, pp. 31-44. doi: 10.14712/1213-7243.2025.003

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