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The Szlenk index and local l_1-indices

Dale Alspach·Robert Judd·Edward Odell
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Abstract

We introduce two new local l_1-indices of the same type as the Bourgain l_1 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 l_1. The l_1^+-weakly null index has the same form as the Bourgain l_1 index: if it is countable it must take values omega^alpha for some alpha<omega_1. The different l_1-indices are closely related and so knowing the Szlenk index of a Banach space helps us calculate its local l_1-index, via the l_1^+-weakly null index. We show that I(C(omega^{omega^alpha}))=omega^{1+alpha+1}.

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Dale Alspach, Robert Judd, Edward Odell. 2000-09-29. The Szlenk index and local l_1-indices. https://arxiv.org/abs/math/0009250

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