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.
Positivity – Springer Journals
Published: Sep 12, 2002
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