Download PDF - What is "local theory of Banach spaces"?All rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

What is "local theory of Banach spaces"?

Authors 1

Affiliations

  1. Mathematisches Institut, FSU Jena, D-07740 Jena, Germany

Abstract

Banach space theory splits into several subtheories. On the one hand, there are an isometric and an isomorphic part; on the other hand, we speak of global and local aspects. While the concepts of isometry and isomorphy are clear, everybody seems to have its own interpretation of what "local theory" means. In this essay we analyze this situation and propose rigorous definitions, which are based on new concepts of local representability of operators.

Bibliography

  1. [bea 1] B. Beauzamy, Opérateurs uniformément convexifiants, Studia Math. 57 (1976), 103-139.
  2. [bea 2] B. Beauzamy, Quelques propriétés des opérateurs uniformément convexifiants, ibid. 60 (1977), 211-222.
  3. [b-s] A. Brunel and L. Sucheston, On J-convexity and some ergodic super-properties of Banach spaces, Trans. Amer. Math. Soc. 204 (1975), 79-90.
  4. [d-k] D. Dacunha-Castelle et J. L. Krivine, Applications des ultraproduits à l'étude des espaces et des algèbres de Banach, Studia Math. 41 (1972), 315-344.
  5. [g-j] Y. Gordon and M. Junge, Volume ratios in Lp-spaces, Studia Math., to appear.
  6. [gro] A. Grothendieck, Sur certaines classes de suites dans les espaces de Banach, et le théorème de Dvoretzky-Rogers, Bol. Soc. Mat. São Paulo 8 (1956), 81-110.
  7. [hei 1] S. Heinrich, Finite representability and super-ideals of operators, Dissertationes Math. 172 (1980).
  8. [hei 2] S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72-104.
  9. [h-k] A. Hinrichs and T. Kaufhold, Distances of finite dimensional subspaces of Lp-spaces, to appear.
  10. [jam] C. James, Super-reflexive Banach spaces, Canad. J. Math. 24 (1972), 896-904.
  11. [kue] K.-D. Kürsten, On some problems of A. Pietsch, II, Teor. Funktsiĭ Funktsional. Anal. i Prilozhen. 29 (1978), 61-73 (in Russian).
  12. [l-m] J. Lindenstrauss and V. D. Milman, The local theory of normed spaces and its applications to convexity, in: Handbook of Convex Geometry, P. M. Gruber and J. M. Wills (eds.), Elsevier, 1993, 1150-1220.
  13. [p-r] A. Pełczyński and H. P. Rosenthal, Localization techniques in Lp spaces, Studia Math. 52 (1975), 263-289.
  14. [pie] A. Pietsch, Ultraprodukte von Operatoren in Banachräumen, Math. Nachr. 61 (1974), 123-132.
  15. [PIE] A. Pietsch, Operator Ideals, Deutscher Verlag Wiss., Berlin, 1978; North-Holland, Amsterdam, 1980.
  16. [P-W] A. Pietsch and J. Wenzel, Orthonormal Systems and Banach Space Geometry, Cambridge Univ. Press, 1998.
  17. [ste] J. Stern, Propriétés locales et ultrapuissances d'espaces de Banach, Sém. Maurey-Schwartz 1974/75, Exposés 7 et 8, École Polytechnique, Paris.
  18. [TOM] N. Tomczak-Jaegermann, Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Longman, Harlow, 1989.
Studia Mathematica
1999/135/3
Export to PBN, Opens in new tab
Pages:
273-298
Main language of publication
English
Received
1999-02-22
Published
1999
Exact and natural sciences