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RETRACTED - Compact reduction in Lipschitz-free spaces

Part of: Normed linear spaces and Banach spaces; Banach lattices Spaces with richer structures

Published online by Cambridge University Press:  08 December 2021

Ramón J. Aliaga Open the ORCID record for Ramón J. Aliaga [Opens in a new window] ,
Camille Noûs ,
Colin Petitjean Open the ORCID record for Colin Petitjean [Opens in a new window]  and
Antonín Procházka Open the ORCID record for Antonín Procházka [Opens in a new window]
Ramón J. Aliaga
Affiliation:
Universitat Politècnica de València, Instituto Universitario de Matemática Pura y Aplicada, Camino de Vera S/N, 46022, Valencia, Spain (raalva@upvnet.upv.es)
Camille Noûs
Affiliation:
Laboratoire Cogitamus, 16, route de Gray, 25030, Besançon, France (camille.nous@cogitamus.fr)
Colin Petitjean
Affiliation:
LAMA, Univ Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS, F-77447, Marne-la-Vallée, France (colin.petitjean@univ-eiffel.fr)
Antonín Procházka
Affiliation:
Laboratoire de Mathématiques de Besançon, Université Bourgogne Franche-Comté, CNRS UMR-6623, 16, route de Gray, 25030, Besançon Cedex, France (antonin.prochazka@univ-fcomte.fr)
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Abstract

We prove a general principle satisfied by weakly precompact sets of Lipschitz-free spaces. By this principle, certain infinite dimensional phenomena in Lipschitz-free spaces over general metric spaces may be reduced to the same phenomena in free spaces over their compact subsets. As easy consequences we derive several new and some known results. The main new results are: $\mathcal {F}(X)$ is weakly sequentially complete for every superreflexive Banach space $X$, and $\mathcal {F}(M)$ has the Schur property and the approximation property for every scattered complete metric space $M$.

Keywords

Lipschitz-free spaceLipschitz functionLipschitz lifting propertySchur propertyapproximation propertyweak sequential completenessDunford–Pettis property

MSC classification

Primary: 46B20: Geometry and structure of normed linear spaces
Secondary: 54E50: Complete metric spaces
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 6 , December 2021 , pp. 1683 - 1699
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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