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Monatshefte für Mathematik

, Volume 182, Issue 1, pp 39–47 | Cite as

Nonseparable closed vector subspaces of separable topological vector spaces

  • Jerzy Ka̧kol
  • Arkady G. Leiderman
  • Sidney A. Morris
Article

Abstract

In 1983 P. Domański investigated the question: For which separable topological vector spaces E, does the separable space Open image in new window have a nonseparable closed vector subspace, where \(\hbox {c}\) is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space Open image in new window has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line \(\mathbb M\) the space of all continuous real-valued functions on \(\mathbb M\) endowed with the pointwise convergence topology, \(C_p(\mathbb M)\) contains a nonseparable closed vector subspace while \(C_p(\mathbb M)\) is separable.

Keywords

Locally convex topological vector space Separable topological space 

Mathematics Subject Classification

46A03 54D65 

Notes

Acknowledgments

The first mentioned author gratefully acknowledges the financial support he received from the Center for Advanced Studies in Mathematics of the Ben-Gurion University of the Negev during his visit May 5–12, 2015. The third mentioned author thanks Ben Gurion-University of the Negev for its hospitality during which much of the research for this paper was done.

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Copyright information

© Springer-Verlag Wien 2016

Authors and Affiliations

  • Jerzy Ka̧kol
    • 1
  • Arkady G. Leiderman
    • 2
  • Sidney A. Morris
    • 3
    • 4
  1. 1.Faculty of Mathematics and InformaticsA. Mickiewicz UniversityPoznańPoland
  2. 2.Department of MathematicsBen-Gurion University of the NegevBeer ShevaIsrael
  3. 3.Faculty of Science and TechnologyFederation University AustraliaBallaratAustralia
  4. 4.School of Engineering and Mathematical SciencesLa Trobe UniversityBundooraAustralia

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