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Relatively Uniform Convergence in Partially Ordered Vector Spaces Revisited

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Positivity and Noncommutative Analysis

Part of the book series: Trends in Mathematics ((TM))

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Abstract

We consider relatively uniform convergence of nets in a partially ordered vector space. We give an example of a set V  in a space X where adding the limits of nets in V  that converge in X does not produce the closure of V  in X. The closure of a set can be constructed by adding limits if that process is repeated by transfinite induction. We also consider closed sets and complete spaces and show that they coincide with sequentially closed sets and sequentially complete spaces, respectively.

To Ben de Pagter, on the occasion of his 65th birthday, with admiration and gratitude

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Author information

Authors and Affiliations

  1. Technische Universität Dresden, Fakultät Mathematik, Institut für Analysis, Dresden, Germany

    Anke Kalauch

  2. Leiden University, Mathematical Institute, Leiden, The Netherlands

    Onno van Gaans

Authors
  1. Anke Kalauch
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  2. Onno van Gaans
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Corresponding author

Correspondence to Onno van Gaans .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Mississippi, Oxford, MS, USA

    Gerard Buskes

  2. Mathematical Institute, Leiden University, Leiden, The Netherlands; Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, South Africa

    Marcel de Jeu

  3. College of Science and Engineering, Flinders University, Adelaide, Australia

    Peter Dodds

  4. Department of Mathematics, University of South Carolina, Columbia, SC, USA

    Anton Schep

  5. School of Mathematics and Statistics, University of New South Wales, Sydney, Australia

    Fedor Sukochev

  6. Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands

    Jan van Neerven

  7. Mathematical Sciences Research Centre, Queen’s University Belfast, Belfast, UK

    Anthony Wickstead

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Kalauch, A., Gaans, O.v. (2019). Relatively Uniform Convergence in Partially Ordered Vector Spaces Revisited. In: Buskes, G., et al. Positivity and Noncommutative Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10850-2_14

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