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Israel Journal of Mathematics

, Volume 20, Issue 3–4, pp 326–350 | Cite as

Martingales with values in uniformly convex spaces

  • Gilles Pisier
Article

Abstract

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ x(ɛ)/ɛ r whenɛ → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(ɛ)/ɛ q → ∞ whenɛ → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.

Keywords

Banach Space Convex Space Equivalent Norm Normed Linear Space Convex Norm 
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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Copyright information

© Hebrew University 1975

Authors and Affiliations

  • Gilles Pisier
    • 1
  1. 1.Centre de MathématiquesEcole PolytechniqueParisFrance

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