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Archiv der Mathematik

, Volume 108, Issue 1, pp 55–64 | Cite as

Ideals of subseries convergence and F-spaces

  • Lech Drewnowski
  • Iwo Labuda
Article

Abstract

Let X be an F-space and \({\boldsymbol x=(x_n)}\) be a sequence of vectors in X. Ideals \({\mathcal{C}(\boldsymbol x)}\) of subseries convergence are considered. In particular, we show that a characterization of the class of Banach spaces not containing c 0 obtained by using the ideals \({\mathcal{C}(\boldsymbol x)}\) breaks down in every Fréchet space not isomorphic to a Banach space. On the other hand, the result can be extended to some F-spaces via the definition of a new class of F-spaces satisfying a stronger version of the condition (O) of Orlicz. A theorem discriminating between the finite and infinite dimensional case is obtained about the family \({\mathcal{C}(X)}\) of all ideals associated with the F-space X.

Keywords

F-spaces Subseries convergence Unconditional convergence Ideal of sets \({F_{\sigma}}\) set \({F_{{\sigma}\delta}}\) set 

Mathematics Subject Classification

40A05 46A04 46A16 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceA. Mickiewicz UniversityPoznańPoland
  2. 2.Department of MathematicsThe University of MississippiOxfordUSA

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