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Proximity to ℓ1and Distortion in Asymptotic L1Spaces

✍ Scribed by Edward Odell; Nicole Tomczak-Jaegermann; Roy Wagner

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
610 KB
Volume
150
Category
Article
ISSN
0022-1236

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✦ Synopsis

For an asymptotic l 1 space X with a basis (x i ) certain asymptotic l 1 constants, $ : (X) are defined for :<| 1 . $ : (X) measures the equivalence between all normalized block bases ( y i ) k i=1 of (x i ) which are S : -admissible with respect to (x i ) (S : is the : th-Schreier class of sets) and the unit vector basis of l k 1 . This leads to the concept of the delta spectrum of X, 2(X), which reflects the behavior of stabilized limits of $ : (X). The analogues of these constants under all renormings of X are also defined and studied. We investigate 2(X ) both in general and for spaces of bounded distortion. We also prove several results on distorting the classical Tsirelson's space T and its relatives.

1997 Academic Press

1. Introduction

The first non-trivial example of what is now called an asymptotic l 1 space was discovered by Tsirelson [26]. This space and its variations were extensively studied in many papers (see ). While the finite-dimensional asymptotic structure of these spaces is the same as that of l 1 , they do not article no. FU973106

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