The Szlenk index and local l₁-indices

2000/09/29 by Dale Alspach, Alspach, Dale, Robert Judd +3
Mathematics · #46B #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B

paper · pdf · doi:10.48550/arxiv.math/0009250

LaTeX2e, 41 pages, to appear in Positivity

arxiv created 2000/09/29 · arxiv updated 2009/11/30

Abstract

We introduce two new local l₁-indices of the same type as the Bourgain l₁ index; the l₁⁺-index and the l₁⁺-weakly null index. We show that the l₁⁺-weakly null index of a Banach space X is the same as the Szlenk index of X, provided X does not contain l₁. The l₁⁺-weakly null index has the same form as the Bourgain l₁ index: if it is countable it must take values omega^alpha for some alpha<omega₁. The different l₁-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its local l₁-index, via the l₁⁺-weakly null index. We show that I(C(omega^omegaalpha))=omega^1+alpha+1.

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