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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Separated sets and Auerbach systems in Banach spaces
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by Petr Hájek, Tomasz Kania and Tommaso Russo PDF
Trans. Amer. Math. Soc. 373 (2020), 6961-6998 Request permission

Abstract:

The paper elucidates the relationship between the density of a Banach space and possible sizes of Auerbach systems and well-separated subsets of its unit sphere. For example, it is proved that for a large enough space $X$, the unit sphere $S_X$ always contains an uncountable $(1+)$-separated subset. In order to achieve this, new results concerning the existence of large Auerbach systems are established, that happen to be sharp for the class of weakly Lindelöf determined (WLD) spaces. In fact, we offer the first consistent example of a non-separable WLD Banach space that contains no uncountable Auerbach system, as witnessed by a renorming of $c_0(\omega _1)$. Moreover, the following optimal results for the classes of, respectively, reflexive and super-reflexive spaces are established: the unit sphere of an infinite-dimensional reflexive space contains a symmetrically $(1+\varepsilon )$-separated subset of any regular cardinality not exceeding the density of $X$; should the space $X$ be super-reflexive, the unit sphere of $X$ contains such a subset of cardinality equal to the density of $X$. The said problem is studied for other classes of spaces too, including WLD spaces, RNP spaces, or strictly convex ones.
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Additional Information
  • Petr Hájek
  • Affiliation: Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Praha 6, Czech Republic
  • Email: hajek@math.cas.cz
  • Tomasz Kania
  • Affiliation: Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67 Praha 1, Czech Republic; and Institute of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
  • MR Author ID: 976766
  • ORCID: 0000-0002-2002-7230
  • Email: kania@math.cas.cz, tomasz.marcin.kania@gmail.com
  • Tommaso Russo
  • Affiliation: Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, 166 27 Praha 6, Czech Republic
  • MR Author ID: 1221908
  • ORCID: 0000-0003-3940-2771
  • Email: russotom@fel.cvut.cz
  • Received by editor(s): November 4, 2019
  • Published electronically: August 5, 2020
  • Additional Notes: The third-named author is the corresponding author.
    The research of the first-named author was supported in part by OPVVV CAAS CZ.02.1.01/0.0/0.0/16$\_$019/0000778.
    The second-named author acknowledges with thanks funding received from GAČR project 19-07129Y; RVO 67985840 (Czech Republic).
    The research of the third-named author was supported by the project International Mobility of Researchers in CTU CZ.02.2.69/0.0/0.0/16$\_$027/0008465 and by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of Istituto Nazionale di Alta Matematica (INdAM), Italy.
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 6961-6998
  • MSC (2010): Primary 46B20, 46B04; Secondary 46A35, 46B26
  • DOI: https://doi.org/10.1090/tran/8160
  • MathSciNet review: 4155197