Prescribed Szlenk index of separable Banach spaces
Volume 248 / 2019
Studia Mathematica 248 (2019), 109-127
MSC: 46B20.
DOI: 10.4064/sm171012-9-9
Published online: 4 March 2019
Abstract
In a previous work, the first named author described the set $\mathcal P$ 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 $\mathcal P$, there exists a separable Banach space $X$ such that the Szlenk index of the dual of order $k$ of $X$ is equal to the first infinite ordinal $\omega $ for all $k$ in $\{0,\ldots ,n-1\}$ and equal to $\alpha $ for $k=n$. One of the ingredients is to show that the Lindenstrauss space and its dual both have Szlenk index equal to $\omega $. We also show that any element of $\mathcal P$ can be realized as the Szlenk index of a reflexive Banach space with an unconditional basis.
