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Ideal Convergence and Completeness of a Normed Space

Author

Listed:
  • Fernando León-Saavedra

    (Department of Mathematics, Faculty of Social Sciences and Communication, University of Cádiz, 11405 Jerez de la Frontera, Spain)

  • Francisco Javier Pérez-Fernández

    (Department of Mathematics, Faculty of Science, University of Cádiz, 1510 Puerto Real, Spain)

  • María del Pilar Romero de la Rosa

    (Department of Mathematics, CASEM, University of Cádiz, 11510 Puerto Real, Spain)

  • Antonio Sala

    (Department of Mathematics, Escuela Superior de Ingeniería, University of Cádiz, 11510 Puerto Real, Spain)

Abstract

We aim to unify several results which characterize when a series is weakly unconditionally Cauchy (wuc) in terms of the completeness of a convergence space associated to the wuc series. If, additionally, the space is completed for each wuc series, then the underlying space is complete. In the process the existing proofs are simplified and some unanswered questions are solved. This research line was originated in the PhD thesis of the second author. Since then, it has been possible to characterize the completeness of a normed spaces through different convergence subspaces (which are be defined using different kinds of convergence) associated to an unconditionally Cauchy sequence.

Suggested Citation

  • Fernando León-Saavedra & Francisco Javier Pérez-Fernández & María del Pilar Romero de la Rosa & Antonio Sala, 2019. "Ideal Convergence and Completeness of a Normed Space," Mathematics, MDPI, vol. 7(10), pages 1-11, September.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:10:p:897-:d:270557
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    Cited by:

    1. Fernando León-Saavedra & María del Pilar Romero de la Rosa & Antonio Sala, 2020. "Schur Lemma and Uniform Convergence of Series through Convergence Methods," Mathematics, MDPI, vol. 8(10), pages 1-11, October.

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