Israel Journal of Mathematics

, Volume 9, Issue 4, pp 488–506 | Cite as

On bases, finite dimensional decompositions and weaker structures in Banach spaces

  • W. B. Johnson
  • H. P. Rosenthal
  • M. Zippin


This is an investigation of the connections between bases and weaker structures in Banach spaces and their duals. It is proved, e.g., thatX has a basis ifX* does, and that ifX has a basis, thenX* has a basis provided thatX* is separable and satisfies Grothendieck’s approximation property; analogous results are obtained concerning π-structures and finite dimensional Schauder decompositions. The basic results are then applied to show that every separable p space has a basis.


Banach Space Approximation Property Natural Projection Separable Banach Space Invertible Operator 
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.


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  1. 1.
    M. M. Day,Normed Linear Spaces, Springer, 1958.Google Scholar
  2. 2.
    N. Dunford and J. T. Schwartz,Linear operators, Part I, New York, 1958.Google Scholar
  3. 3.
    A. Grothendieck,Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc.16 (1955).Google Scholar
  4. 4.
    W. B. Johnson,Finite dimensional Schauder decompositions in θλ and dual θλ spaces, Illinois J. Math. (to appear).Google Scholar
  5. 5.
    W. B. Johnson,On the existence of strongly series summable Markuschevich bases in Banach spaces (to appear).Google Scholar
  6. 6.
    S. Karlin,Bases in Banach spaces, Duke Math. J.15 (1948), 971–985.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    J. Lindenstrauss,Extension of compact operators, Mem. Amer. Math. Soc.48 (1964).Google Scholar
  8. 8.
    J. Lindenstrauss,On James’ paper “Separable conjugate spaces” (to appear).Google Scholar
  9. 9.
    J. Lindenstrauss and A. Pełczyński,Absolutely summing operators in ℒ p spaces and their applications, Studia Math.29 (1968), 275–326.MATHMathSciNetGoogle Scholar
  10. 10.
    J. Lindenstrauss and H. P. Rosenthal,The ℒ p spaces, Israel J. Math.7 (1969), 325–349.MATHMathSciNetGoogle Scholar
  11. 11.
    A. Pełczyński,Projections in certain Banach spaces, Studia Math.19 (1960), 209–228.MathSciNetMATHGoogle Scholar
  12. 12.
    A. Pełczyński,Universal bases, Studia Math.32 (1969), 247–268.MathSciNetMATHGoogle Scholar
  13. 13.
    A. Pełczyński and P. Wojtaszczyk,Banach spaces with finite dimensional expansions of identity and universal bases of finite dimensional subspaces, Studia Math. (to appear).Google Scholar
  14. 14.
    J. R. Retherford,Shrinking bases in Banach spaces, Amer. Math. Monthly73 (1966), 841–846.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A. E. Taylor,A geometric theorem and its application to biorthogonal systems, Bull. Amer. Math. Soc.53 (1947), 614–616.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1971

Authors and Affiliations

  • W. B. Johnson
    • 1
    • 2
  • H. P. Rosenthal
    • 1
    • 2
  • M. Zippin
    • 1
    • 2
  1. 1.University of HoustonHouston
  2. 2.University of CaliforniaBerkeley

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