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Transactions of the American Mathematical Society

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Separated sets and Auerbach systems in Banach spaces


Authors: Petr Hájek, Tomasz Kania and Tommaso Russo
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
Published electronically: August 5, 2020
MathSciNet review: 4155197
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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

Keywords: Banach space, Kottman’s theorem, the Elton–Odell theorem, unit sphere, separated set, reflexive space, Auerbach system, exposed point
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.
Article copyright: © Copyright 2020 American Mathematical Society