A note on nonseparable Lipschitz-free spaces

Ramón J. Aliaga; Guillaume Grelier; Antonín Procházka

Commentationes Mathematicae Universitatis Carolinae (2024)

  • Issue: 1, page 31-44
  • ISSN: 0010-2628

Abstract

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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 ( 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.

How to cite

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Aliaga, Ramón J., Grelier, Guillaume, and Procházka, Antonín. "A note on nonseparable Lipschitz-free spaces." Commentationes Mathematicae Universitatis Carolinae (2024): 31-44. <http://eudml.org/doc/299953>.

@article{Aliaga2024,
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.},
author = {Aliaga, Ramón J., Grelier, Guillaume, Procházka, Antonín},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Lipschitz-free space; nonseparable Banach space; sequentially compact; Radon--Nikodým property},
language = {eng},
number = {1},
pages = {31-44},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A note on nonseparable Lipschitz-free spaces},
url = {http://eudml.org/doc/299953},
year = {2024},
}

TY - JOUR
AU - Aliaga, Ramón J.
AU - Grelier, Guillaume
AU - Procházka, Antonín
TI - A note on nonseparable Lipschitz-free spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2024
PB - Charles University in Prague, Faculty of Mathematics and Physics
IS - 1
SP - 31
EP - 44
AB - 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.
LA - eng
KW - Lipschitz-free space; nonseparable Banach space; sequentially compact; Radon--Nikodým property
UR - http://eudml.org/doc/299953
ER -

References

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