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 Mat. Zametki, 2007, Volume 82, Issue 5, Pages 729–735 (Mi mz4085)

Operations on Approximatively Compact Sets

I. A. Pyatyshev

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

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.

Keywords: Approximatively compact set, algebraic sum of sets, uniformly closed Banach space, Efimov–Stechkin space

DOI: https://doi.org/10.4213/mzm4085

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English version:
Mathematical Notes, 2007, 82:5, 653–659

Bibliographic databases:

UDC: 517.982.256
Revised: 16.04.2007

Citation: I. A. Pyatyshev, “Operations on Approximatively Compact Sets”, Mat. Zametki, 82:5 (2007), 729–735; Math. Notes, 82:5 (2007), 653–659

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz4085
• https://doi.org/10.4213/mzm4085
• http://mi.mathnet.ru/eng/mz/v82/i5/p729

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This publication is cited in the following articles:
1. De la Sen M., “Some Results on Fixed and Best Proximity Points of Precyclic Self-Mappings”, J. Appl. Math., 2013, 310106
2. De la Sen M., “Some Results on Fixed and Best Proximity Points of Multivalued Cyclic Self-Mappings with a Partial Order”, Abstract Appl. Anal., 2013, 968492
3. Luo Zh., Sun L., Zhang W., “a Remark on the Stability of Approximative Compactness”, J. Funct. space, 2016, 2734947
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