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)


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.


Banach Space Limit Theorem Lower Estimate Conditional Expectation Natural Basis 
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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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