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Grothendieck spaces: the landscape and perspectives

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Japanese Journal of Mathematics Aims and scope

Abstract

In 1973, Diestel published his seminal paper Grothendieck spaces and vector measures that drew a connection between Grothendieck spaces (Banach spaces for which weak- and weak*-sequential convergences in the dual space coincide) and vector measures. This connection was developed further in his book with J. Uhl Jr. Vector measures. Additionally, Diestel’s paper included a section with several open problems about the structural properties of Grothendieck spaces, and only half of them have been solved to this day.

The present paper aims at synthetically presenting the state of the art at subjectively selected corners of the theory of Banach spaces with the Grothendieck property, describing the main examples of spaces with this property, recording the solutions to Diestel’s problems, providing generalisations/extensions or new proofs of various results concerning Grothendieck spaces, and adding to the list further problems that we believe are of relevance and may reinvigorate a better-structured development of the theory.

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Acknowledgements

We thank the referee for a careful reading of the manuscript and for providing an interesting solution to Problem 8.

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Authors and Affiliations

  1. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cantabria, Avda. de los Castros s/n, 39071, Santander, Spain

    Manuel González

  2. Mathematical Institute, Czech Academy of Sciences, Žitná 25, 115 67, Praha 1, Czech Republic

    Tomasz Kania

  3. Institute of Mathematics and Computer Science, Jagiellonian University, Łojasiewicza 6, 30-348, Kraków, Poland

    Tomasz Kania

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  1. Manuel González
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Correspondence to Manuel González or Tomasz Kania.

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Communicated by: Yasuyuki Kawahigashi

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Research of M. González was partially supported by MICINN (Spain), Grant PID2019-103961GB-C22. T. Kania acknowledges with thanks funding received from SONATA 15 No. 2019/35/D/ST1/01734.

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González, M., Kania, T. Grothendieck spaces: the landscape and perspectives. Jpn. J. Math. 16, 247–313 (2021). https://doi.org/10.1007/s11537-021-2116-3

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