# The Szlenk index and local ℓ1-indices

The Szlenk index and local ℓ1-indices We introduce two new local ℓ1-indices of the same type as the Bourgain ℓ1-index; the ℓ+ 1-index and the ℓ+ 1-weakly null index. We show that the ℓ+ 1-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain ℓ1. The ℓ+ 1-weakly null index has the same form as the Bourgain ℓ1-index: if it is countable it must take values ωα for some α<ω1. The different ℓ1-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its ℓ1-index, via the ℓ+ 1-weakly null index. We show that I(C(ωωα))=ω^1+α+1. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# The Szlenk index and local ℓ1-indices

, Volume 9 (1) – Sep 12, 2002
44 pages

/lp/springer_journal/the-szlenk-index-and-local-1-indices-DEA2mYAGRQ
Publisher
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-002-9781-0
Publisher site
See Article on Publisher Site

### Abstract

We introduce two new local ℓ1-indices of the same type as the Bourgain ℓ1-index; the ℓ+ 1-index and the ℓ+ 1-weakly null index. We show that the ℓ+ 1-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain ℓ1. The ℓ+ 1-weakly null index has the same form as the Bourgain ℓ1-index: if it is countable it must take values ωα for some α<ω1. The different ℓ1-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its ℓ1-index, via the ℓ+ 1-weakly null index. We show that I(C(ωωα))=ω^1+α+1.

### Journal

PositivitySpringer Journals

Published: Sep 12, 2002

### References

• Characterizations of reflexivity
3 James, R.C.

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