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Embedding of k and a Theorem of Alon and Milman

  • Michel Talagrand
Conference paper
Part of the Operator Theory Advances and Applications book series (OT, volume 77)

Abstract

Consider normalized vectors (xi) i n in a Banach space X and set \( Av\left\{{\left\| {\sum\limits_{i = 1}^n {{\varepsilon_i}{x_i}}} \right\|;{\varepsilon_i} = \pm 1} \right\},\,{w_n} = \sup \left\{{\sum\limits_{i \leqslant n}^n {\left| {{x^*}\left({{x_i}} \right)} \right|;{x^*} \in {X^*},\left\| {{x^*}} \right\|} \leqslant 1} \right\} \). We prove that there exists a subset A of {1,…, n} of cardinality k at least n/32w n such that (x i)iA is 8M n isomorphic to the natural basis of k . This improves a result of Alon and Milman that obtained only A of cardinality \( {2^{ - 7}}\sqrt n /{M_n} \). The proof is much simpler than the orginal proof.

Keywords

Banach Space Limit Theorem Lower Estimate Conditional Expectation Natural Basis 
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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References

  1. [A-M]
    N. Alon, V. Milman, Embedding of k in finite dimensional Banach spaces, Israel J. Math 45 (1983), 365–380.MathSciNetCrossRefGoogle Scholar
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    E. Giné, J. Zinn, Some limit theorems for empirical processes, Ann. Probab. 12 (1984), 929–989.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [T1]
    M. Talagrand, Cotype and (q, l)-summing norm in a Banach space, Inventions Math. 110 (1992), 545–556.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [T2]
    M. Talagrand, Regularity of infinitely divisible processes, Ann. Probab. 21 (1993), 362–432.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 1995

Authors and Affiliations

  • Michel Talagrand
    • 1
    • 2
  1. 1.Equipe d Analyse-Tour 56 E.R.A. au C.N.R.S. no. 754Université Paris VIParis Cedex 05France
  2. 2.Department of MathematicsThe Ohio State UniversityColumbusUSA

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