Skip to main content
SpringerLink
Log in

The strong Schur property in Banach lattices

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

We prove that in the class of discrete Banach lattices the strong Schur property is equivalent to the disjoint strong Schur property (Theorem 3.1). Roughly speaking the strong Schur property holds iff an appropriate condition concerning sequences with positive pairwise disjoint terms is satisfied.

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. A. Ju. Abramovich, Some theorems on normed lattices, Vestnik Leningrad. Univ. Math. 4 (1977), 153–159.

  2. F. Albiac, N.J. Kalton, Topics in Banach space theory, Grad. Texts Math., vol. 233, Springer, (2006).

  3. C. Aliprantis, O. Burkinshaw, Locally solid Riesz spaces with applications to economics, Mathematical Surveys and Monographs, vol. 105, American Mathematical Society, (2003).

  4. J. Bourgain, H.P. Rosenthal, Martingales valued in certain subspaces of L1, Israel J. Math. 37 (1980), 54–75.

  5. G. Godefroy, N.J. Kalton, D. Li, On subspaces of L1 which embed into ℓ1, J. Reine Angew. Math., 471 (1996), 43–74.

  6. M. González, A. Martínez-Abejón, Ultrapowers of L1(μ) and the subsequence splitting principle, Israel J. Math., 122 (2001), 189–206.

  7. J. Hagler, C. Stegall, Banach spaces whose duals contain complemented subspaces isomorphic to C[0,1], J. Funct. Anal., 13 (1973), 233–251.

    Google Scholar 

  8. W.B. Johnson, E. Odell, Subspaces of Lpwhich embed into ℓ p , Comp. Math., 28 (1974), 37–49.

  9. N.J. Kalton, On subspaces of c0 and extension of operators into C(K)-spaces, Quart. J. Math., 52 (2001), 313–328.

  10. H. Knaust, E. Odell, On c0 sequences in Banach spaces, Israel J. Math., 67 (1989), 153–169.

    Google Scholar 

  11. E. Odell, On certain complemented subspaces of L1 with the strong Schur property, Longhorn Notes, (1983–1984), 177–184.

  12. L. Weis, Banach lattices with the subsequence splitting property, Proc. Amer. Math. Soc., 105 (1989), 87–96.

  13. W. Wnuk, Banach lattices with properties of the Schur type-a survey, Conf. Sem. Mat. Univ. Bari, 249 (1993), 1–25.

  14. W. Wnuk, Banach lattices with order continuous norms, Advanced Topics in Mathematics, Polish Scientific Publishers PWN, Warsaw, (1999).

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, A. Mickiewicz University, ul. Umultowska 87, 61-614, Poznań, Poland

    Witold Wnuk

Authors
  1. Witold Wnuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Witold Wnuk.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wnuk, W. The strong Schur property in Banach lattices. Positivity 13, 435–441 (2009). https://doi.org/10.1007/s11117-008-2252-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-008-2252-5

Mathematics Subject Classification (2000)

Keywords

Search

Navigation