What is "local theory of Banach spaces"?

Albrecht Pietsch

Studia Mathematica (1999)

  • Volume: 135, Issue: 3, page 273-298
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Pietsch, Albrecht. "What is "local theory of Banach spaces"?." Studia Mathematica 135.3 (1999): 273-298. <http://eudml.org/doc/216655>.

@article{Pietsch1999,
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.},
author = {Pietsch, Albrecht},
journal = {Studia Mathematica},
keywords = {local theory of Banach spaces; local representability of operators},
language = {eng},
number = {3},
pages = {273-298},
title = {What is "local theory of Banach spaces"?},
url = {http://eudml.org/doc/216655},
volume = {135},
year = {1999},
}

TY - JOUR
AU - Pietsch, Albrecht
TI - What is "local theory of Banach spaces"?
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 3
SP - 273
EP - 298
AB - 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.
LA - eng
KW - local theory of Banach spaces; local representability of operators
UR - http://eudml.org/doc/216655
ER -

References

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  1. [bea 1] B. Beauzamy, Opérateurs uniformément convexifiants, Studia Math. 57 (1976), 103-139. Zbl0372.46016
  2. [bea 2] B. Beauzamy, Quelques propriétés des opérateurs uniformément convexifiants, ibid. 60 (1977), 211-222. Zbl0348.46017
  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 L p -spaces, Studia Math., to appear. Zbl0952.46008
  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. Zbl0412.46017
  9. [h-k] A. Hinrichs and T. Kaufhold, Distances of finite dimensional subspaces of L p -spaces, to appear. 
  10. [jam] C. James, Super-reflexive Banach spaces, Canad. J. Math. 24 (1972), 896-904. Zbl0222.46009
  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. Zbl0791.52003
  13. [p-r] A. Pełczyński and H. P. Rosenthal, Localization techniques in L p spaces, Studia Math. 52 (1975), 263-289. Zbl0297.46023
  14. [pie] A. Pietsch, Ultraprodukte von Operatoren in Banachräumen, Math. Nachr. 61 (1974), 123-132. Zbl0288.47037
  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. Zbl0919.46001
  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. Zbl0721.46004

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