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Limit sets in normed linear spaces

Volume 147 / 2017

Włodzimierz J. Charatonik, Alicja Samulewicz, Roman Wituła Colloquium Mathematicum 147 (2017), 35-42 MSC: Primary 54F15, 46B20; Secondary 40A05, 40A25. DOI: 10.4064/cm6868-5-2016 Published online: 21 November 2016

Abstract

The sets of all limit points of series with terms tending to 0 in normed linear spaces are characterized. An immediate conclusion is that a normed linear space $(X,\| \cdot \| )$ is infinite-dimensional if and only if there exists a series $\sum x_n$ of terms of $X$ with $x_n\to 0$ whose set of limit points contains exactly two different points of $X$. The last assertion could be extended to an arbitrary (greater than 1) finite number of points.

Authors

  • Włodzimierz J. CharatonikDepartment of Mathematics and Statistics
    Missouri University of Science and Technology
    Rolla, MO 65409, U.S.A.
    e-mail
  • Alicja SamulewiczInstitute of Mathematics
    Faculty of Applied Mathematics
    Silesian University of Technology
    Kaszubska 23
    44-101 Gliwice, Poland
    e-mail
  • Roman WitułaInstitute of Mathematics
    Faculty of Applied Mathematics
    Silesian University of Technology
    Kaszubska 23
    44-101 Gliwice, Poland
    e-mail

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