Advertisement

Israel Journal of Mathematics

, Volume 107, Issue 1, pp 157–193 | Cite as

Convex unconditionality and summability of weakly null sequences

  • S. A. Argyros
  • S. Mercourakis
  • A. Tsarpalias
Article

Abstract

It is proved that every normalized weakly null sequence has a subsequence which is convexly unconditional. Further, a hierarchy of summability methods is introduced and with this we give a complete classification of the complexity of weakly null sequences.

Keywords

Banach Space Finite Subset Inductive Assumption Spreading Model Summability Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [An-Od] G. Androulakis and E. Odell,Distorting mixed Tsirelson spaces, preprint.Google Scholar
  2. [Al-Ar] D. Alspach and S. Argyros,Complexity of Weakly null sequences, Dissertations Mathematicae321 (1992), 1–44.MathSciNetGoogle Scholar
  3. [A-O] D. Alspach and E. Odell,Averaging Weakly Null sequences, Lecture Notes in Mathematics1332, Springer, Berlin, 1988.Google Scholar
  4. [A-D] S. Argyros and I. Deliyanni,Examples of asymptotic ℓ 1 Banach spaces, Transactions of the American Mathematical Society321 (1997), 973–995.CrossRefMathSciNetGoogle Scholar
  5. [B] J. Bourgain,Convergent sequences of continuous functions, Bulletin de la Société Mathématique de Belgique. Série B32 (1980), 235–249.zbMATHMathSciNetGoogle Scholar
  6. [Ell] E. Ellentuck,A new proof that analytic sets are Ramsey, Journal of Symbolic Logic39 (1974), 163–165.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [E] J. Elton, Thesis, Yale University.Google Scholar
  8. [E-M] P. Erdös and M. Magidor,A note on regular methods of summability and the Banach-Saks property, Proceedings of the American Mathematical Society59 (1976), 232–234.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [G-P] F. Galvin and K. Prikry,Borel sets and Ramsey's theorem, Journal of Symbolic Logic38 (1973), 193–198.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [G-M] W. T. Gowers and B. Maurey,The unconditional basic sequence problem, Journal of the American Mathematical Society6 (1993), 851–874.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [K] K. Stromberg,Introduction to Classical Real Analysis, Wadsworth, Belmont, CA, 1981.zbMATHGoogle Scholar
  12. [K-L] A. S. Kechris and A. Louveau,A classification of Baire class 1 functions, Transactions of the American Mathematical Society318 (1990), 209–236.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [M-R] B. Maurey and H. Rosenthal,Normalized weakly null sequences with no unconditional subsequence, Studia Mathematica61 (1977), 77–98.zbMATHMathSciNetGoogle Scholar
  14. [M-N] S. Mercourakis and S. Negrepontis,Banach spaces and topology II, inRecent Progress in General Topology (M. Husek and J. Vaan Mill, eds.), Elsevier, Amsterdam, 1992.Google Scholar
  15. [M] S. Mercourakis,On Cesaro summable sequences of continuous functions, Mathematica42 (1995), 87–104.zbMATHMathSciNetGoogle Scholar
  16. [N-W] C. St. J. A. Nash-Williams,On well quasi-ordering transfinite sequences, Proceedings of the Cambridge Philosophical Society61 (1965), 33–39.zbMATHMathSciNetCrossRefGoogle Scholar
  17. [O1] E. Odell,Applications of Ramsey theorems to Banach space theory, inNotes in Banach Spaces (H. E. Lacey, ed.), University of Texas Press, 1980, pp. 379–404.Google Scholar
  18. [O2] E. Odell,On Schreier unconditional sequences, Contemporary Mathematics144 (1993), 197–201.MathSciNetGoogle Scholar
  19. [O-TJ-W] E. Odell, N. Tomczak-Jaegermann and R. Wagner,Proximity to ℓ 1 and distortion in asymptotic ℓ 1 spaces preprint.Google Scholar
  20. [Sch] J. Schreier,Ein Gegenbeispiel zur Theorie der schwachen Konvergenz, Studia Mathematica2 (1930), 58–62.Google Scholar
  21. [Si] J. Silver,Every analytic set is Ramsey, Journal of Symbolic Logic35 (1970), 60–64.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [T] B. S. Tsirelson,Not every Banach space contains ℓ ν or c 0. Functional Analysis and its Applications8 (1974), 138–141.CrossRefGoogle Scholar

Copyright information

© The Magnes Press 1998

Authors and Affiliations

  • S. A. Argyros
    • 1
  • S. Mercourakis
    • 1
  • A. Tsarpalias
    • 1
  1. 1.Department of MathematicsUniversity of AthensAthensGreece

Personalised recommendations