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Advances in MathematicsVolume 392, 3 December 2021, Article number 108041

An Asplund space with norming Markuševič basis that is not weakly compactly generated(Article)(Open Access)

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  • aDepartment of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, Prague 6, 166 27, Czech Republic
  • bInstitute of Mathematics, Czech Academy of Sciences, Žitná 25, Prague 1, 115 67, Czech Republic
  • cDipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Via Cesare Saldini 50, Milano, 20133, Italy
  • dDepartment of Mathematics, University of Toronto, Toronto, ON M5S 2E4, Canada
  • eInstitut de Mathématiques de Jussieu, Paris, France
  • fMatematički Institut, SANU, Belgrade, Serbia

Abstract

We construct an Asplund Banach space X with a norming Markuševič basis such that X is not weakly compactly generated. This solves a long-standing open problem from the early nineties, originally due to Gilles Godefroy. En route to the proof, we construct a peculiar example of scattered compact space, that also solves a question due to Wiesław Kubiś and Arkady Leiderman. © 2021 Elsevier Inc.

Author keywords

Asplund Banach spaceNorming Markuševič basisSemi-Eberlein compact spaceWeakly compactly generated Banach spaceϱ-function

Funding details

Funding sponsor Funding number Acronym
Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro ApplicazioniINdAM/GNAMPA
Università degli Studi di MilanoUniMi
Istituto Nazionale di Alta Matematica "Francesco Severi"INdAM
CAAS CZ.02.1.01/0.0/0.0/16_019/0000778
Grantová Agentura České Republiky67985840,20-22230LGA ČR
Centre National de la Recherche ScientifiqueUMR7586CNRS

  • 1

    P. H?jek was supported in part by OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778. Research of T. Russo was supported by the GA?R project 20-22230L; RVO: 67985840 and by Gruppo Nazionale per l'Analisi Matematica, la Probabilit? e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy. J. Somaglia was supported by Universit? degli Studi di Milano, Research Support Plan 2019 and by Gruppo Nazionale per l'Analisi Matematica, la Probabilit? e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy. Research of S. Todor?evi? is partially supported by grants from NSERC (455916) and CNRS (UMR7586).

  • 2

    P. H\u00E1jek was supported in part by OPVVV CAAS CZ.02.1.01/0.0/0.0/16_019/0000778 . Research of T. Russo was supported by the GA\u010CR project 20-22230L ; RVO: 67985840 and by Gruppo Nazionale per l'Analisi Matematica, la Probabilit\u00E0 e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy. J. Somaglia was supported by Universit\u00E0 degli Studi di Milano , Research Support Plan 2019 and by Gruppo Nazionale per l'Analisi Matematica, la Probabilit\u00E0 e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy. Research of S. Todor\u010Devi\u0107 is partially supported by grants from NSERC ( 455916 ) and CNRS ( UMR7586 ).

  • ISSN: 00018708
  • Source Type: Journal
  • Original language: English
  • DOI: 10.1016/j.aim.2021.108041
  • Document Type: Article
  • Publisher: Academic Press Inc.

  Russo, T.; Institute of Mathematics, Czech Academy of Sciences, Žitná 25, Prague 1, Czech Republic;
© Copyright 2021 Elsevier B.V., All rights reserved.

Cited by 3 documents

de Bernardi, C.A. , Somaglia, J.
ROTUND GÂTEAUX SMOOTH NORMS WHICH ARE NOT LOCALLY UNIFORMLY ROTUND
(2024) Proceedings of the American Mathematical Society
Russo, T. , Somaglia, J.
Banach spaces of continuous functions without norming Markushevich bases
(2023) Mathematika
Correa, C. , Russo, T. , Somaglia, J.
Small semi-Eberlein compacta and inverse limits
(2021) Topology and its Applications
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