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Tsirelson-like spaces and complexity of classes of Banach spaces

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas volume 112, pages 1101–1123 (2018)Cite this article

Abstract

Employing a construction of Tsirelson-like spaces due to Argyros and Deliyanni, we show that the class of all Banach spaces which are isomorphic to a subspace of \( c_{0} \) is a complete analytic set with respect to the Effros Borel structure of separable Banach spaces. Moreover, the classes of all separable spaces with the Schur property and of all separable spaces with the Dunford–Pettis property are \(\Pi ^{1}_{2} \)-complete.

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Acknowledgements

The author is grateful to the referees for numerous suggestions that helped to improve the manuscript.

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Authors and Affiliations

  1. Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75, Prague 8, Czech Republic

    Ondřej Kurka

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  1. Ondřej Kurka
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Correspondence to Ondřej Kurka.

Additional information

The research was supported by the grant GAČR 14-04892P. The author is a junior researcher in the University Centre for Mathematical Modelling, Applied Analysis and Computational Mathematics (MathMAC).

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Kurka, O. Tsirelson-like spaces and complexity of classes of Banach spaces. RACSAM 112, 1101–1123 (2018). https://doi.org/10.1007/s13398-017-0412-9

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  • DOI: https://doi.org/10.1007/s13398-017-0412-9

Keywords

  • Effros Borel structure
  • Complete analytic set
  • Tsirelson space
  • Banach space \( c_{0} \)
  • Schur property

Mathematics Subject Classification

  • Primary 46B25
  • 54H05
  • Secondary 46B03
  • 46B20