Skip to main content
SpringerLink
Log in

On subseries convergent series and m-quasi-bases in topological linear spaces

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

We prove that the algebraic dimension of the linear span of sums of a subseries convergent series in a Hausdorff topological linear space is either finite or equals 2ℵ°. This result is applied to represent every infinite-dimensional metrizable linear spaee of cardinality 2ℵ° as the direct sum of a sequence of dense subspaces with a strange summability property. Moreover, we show that every infinite-dimensional separable metrizable linear space admits an m-quasi-basis which is not a quasi-basis.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bessaga, G., Peiczyński, A.: Selected topics in infinite-dimensional topology. Warsaw: Polish Scientific Publishers 1975

    Google Scholar 

  2. Christensen, J.P.R.: Compact convex sets and compact Choquet simpexes. Invent. Math. 19, 1–4 (1973)

    Google Scholar 

  3. Drewnowski, L.: Equivalence of Brooks-Jewett, Vitali-Hahn-Saks and Nikodym theorems. Bull. Acad. Polon. Sei. Sér. Sci. Math. Astronom. Phys. 20, 725–731 (1972)

    Google Scholar 

  4. — On minimally subspace-comparable F-spaees. J. Func. Anal. 26, 315–332 (1977)

    Google Scholar 

  5. — Labuda, I., Lipecki, Z.: Existence of quasi-bases for separable topological linear spaces. Arch. Math. (Basel), 37, 454–456 (1981)

    Google Scholar 

  6. Halmos, P.R.: A Hilbert space problem book. Toronto: Van Nostrand 1967

    Google Scholar 

  7. Klee, V.: On the borelian and projective types of linear subs-paces. Math. Seand. 6, 189–199 (1958)

    Google Scholar 

  8. Kliś, C.: An example of noncomplete normed (K)-space. Bull. Aead. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26, 415–420 (1978)

    Google Scholar 

  9. Labuda, I.: On the existence of non-trivial Saks sets and continuity of linear mappings acting on them. Ibid. 23, 885–890 (1975)

    Google Scholar 

  10. Mackey, G.W.: On infinite-dimensional linear spaces. Trans. Amer. Math. Soc. 57, 155–207 (1945)

    Google Scholar 

  11. Peck, N.T.: On nonlocally convex spaces. II. Math. Ann 178, 209–218 (1968)

    Google Scholar 

  12. Popoola, J.O., Tweddle, I.: On the dimension of a complete metrizable topological vector space. Canad. Math. Bull. 20, 271–272 (1977)

    Google Scholar 

  13. Rolewicz, S.: Metric linear spaces. Warsaw: Polish Scientific Publishers 1972

    Google Scholar 

  14. Singer, I.: Bases in Banach spaces. I. Berlin-Heidelberg-New York: Springer 1970

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences Poznan Branch, Mielzyriskiego 25/27, 61-725, Poznan, Poland

    Iwo Labuda

  2. Institute of Mathematics, Polish Academy of Sciences Wrocław Branch, Kopernika 18, 51-617, Wrocław, Poland

    Zbigniew Lipecki

Authors
  1. Iwo Labuda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zbigniew Lipecki
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Labuda, I., Lipecki, Z. On subseries convergent series and m-quasi-bases in topological linear spaces. Manuscripta Math 38, 87–98 (1982). https://doi.org/10.1007/BF01168388

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01168388

Keywords

Search

Navigation