TAILIEUCHUNG - Đề tài " The diameter of the isomorphism class of a Banach space "

Dedicated to the memory of V. I. Gurarii Abstract We prove that if X is a separable infinite dimensional Banach space then its isomorphism class has infinite diameter with respect to the Banach-Mazur distance. One step in the proof is to show that if X is elastic then X contains an isomorph of c0 . | Annals of Mathematics The diameter of the isomorphism class of a Banach space By W. B. Johnson and E. Odell Annals of Mathematics 162 2005 423 437 The diameter of the isomorphism class of a Banach space By W. B. Johnson and E. Odell Dedicated to the memory of V. I. Gurarii Abstract We prove that if X is a separable infinite dimensional Banach space then its isomorphism class has infinite diameter with respect to the Banach-Mazur distance. One step in the proof is to show that if X is elastic then X contains an isomorph of c0. We call X elastic if for some K oo for every Banach space Y which embeds into X the space Y is K-isomorphic to a subspace of X . We also prove that if X is a separable Banach space such that for some K o every isomorph of X is K-elastic then X is finite dimensional. 1. Introduction Given a Banach space X let D X be the diameter in the Banach-Mazur distance of the class of all Banach spaces which are isomorphic to X that is D X sup d X1 X2 X1 X2 are isomorphic toX where d X1 X2 is the infimum over all isomorphisms T from X1 onto X2 of TII T-1 . It is well known that if X is finite say N dimensional then cN D X N for some positive constant c which is independent of N. The upper bound is an immediate consequence of the classical result see . T-J p. 54 that d Y iff ỵfN for every N dimensional space Y. The lower bound is due to Gluskin G T-J p. 283 . It is natural to conjecture that D X must be infinite when X is infinite dimensional but this problem remains open. As far as we know this problem was first raised in print in the 1976 book of J. J. Schaffer S p. 99 . The problem was recently brought to the attention of the authors by V. I. Gurarii who Johnson was supported in part by NSF DMS-0200690 Texas Advanced Research Program 010366-163 and the Binational Science Foundation. Odell was supported in part by NSF DMS-0099366 and was a participant in the NSF supported Workshop in Linear Analysis and Probability at Texas A M University.

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