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Operations on approximatively compact sets

Mathematical Notes volume 82, pages 653–659 (2007)Cite this article

Abstract

In the paper, the problem of preserving the property of approximative compactness under diverse operations is considered. In an arbitrary uniformly convex separable space, we construct an example of two approximatively compact sets whose intersection is not approximatively compact. An example of two linear approximatively compact sets for which the closure of their algebraic sum is not approximatively compact is constructed. In an arbitrary Banach space, we construct two nonlinear approximatively compact sets whose algebraic sum is closed but not approximatively compact. We also prove that any uniformly closed Banach space contains an approximatively compact cavity.

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Authors and Affiliations

  1. Moscow State University, Russia

    I. A. Pyatyshev

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  1. I. A. Pyatyshev
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Correspondence to I. A. Pyatyshev.

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Original Russian Text © I. A. Pyatyshev, 2007, published in Matematicheskie Zametki, 2007, Vol. 82, No. 5, pp. 729–735.

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Pyatyshev, I.A. Operations on approximatively compact sets. Math Notes 82, 653–659 (2007). https://doi.org/10.1134/S0001434607110089

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  • DOI: https://doi.org/10.1134/S0001434607110089

Key words

  • Approximatively compact set
  • algebraic sum of sets
  • uniformly closed Banach space
  • Efimov-Stechkin space
  • closed convex set
  • uniformly convex set