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Banach spaces with a weak cotype 2 property

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Abstract

We study the Banach spacesX with the following property: there is a numberδ in ]0,1[ such that for some constantC, any finite dimensional subspaceE ⊂X contains a subspaceF ⊂E with dimF≧δ dimE which isC-isomorphic to a Euclidean space. We show that if this holds for someδ in ]0,1[ then it also holds for allδ in ]0,1[ and we estimate the functionC=C(δ). We show that this property holds iff the “volume ratio” of the finite dimensional subspaces ofX are uniformly bounded. We also show that (althoughX can have this property without being of cotype 2)L 2(X) possesses this property iffX if of cotype 2. In the last part of the paper, we study theK-convex spaces which have a dual with the above property and we relate it to a certain extension property.

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Authors and Affiliations

  1. Tel Aviv University, Ramat Aviv, Israel

    Vitali D. Milman

  2. I.H.E.S. and Equipe d’Analyse, Université Paris VI, 4, Place Jussieu, 75230, Paris Cedex 05, France

    Vitali D. Milman & Gilles Pisier

Authors
  1. Vitali D. Milman
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  2. Gilles Pisier
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Milman, V.D., Pisier, G. Banach spaces with a weak cotype 2 property. Israel J. Math. 54, 139–158 (1986). https://doi.org/10.1007/BF02764939

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  • DOI: https://doi.org/10.1007/BF02764939

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