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arXiv · 1710.01638

Prescribed Szlenk index of separable Banch spaces

Ryan M. Causey·Gilles Lancien
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Abstract

In a previous work, the first named author described the set $\cal P$ of all values of the Szlenk indices of separable Banach spaces. We complete this result by showing that for any integer $n$ and any ordinal $\alpha$ in $\cal P$, there exists a separable Banach space $X$ such that the Szlenk of the dual of order $k$ of $X$ is equal to the first infinite ordinal $\omega$ for all $k$ in $\{0,..,n-1\}$ and equal to $\alpha$ for $k=n$. One of the ingredients is to show that the Lindenstrauss space and its dual both have a Szlenk index equal to $\omega$. We also show that any element of $\cal P$ can be realized as a Szlenk index of a reflexive Banach space with an unconditional basis.

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Ryan M. Causey, Gilles Lancien. 2017-10-04. Prescribed Szlenk index of separable Banch spaces. https://arxiv.org/abs/1710.01638

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