Szlenk and $w^\ast$-dentability indices of the Banach spaces $C([0,\alpha])$

TL;DR

This paper determines the Szlenk and $w^ ext{-} ext{dentability}$ indices of the Banach space of continuous functions on ordinal intervals, linking these indices to the ordinal's structure.

Contribution

It provides explicit calculations of Szlenk and $w^ ext{-} ext{dentability}$ indices for $C([0,eta])$ spaces based on ordinal analysis, a novel contribution in Banach space theory.

Findings

Szlenk index of $C([0,eta])$ is $oldsymbol{ extomega^{oldsymbol{ extgamma+1}}}$.

$w^ ext{-} ext{dentability}$ index of $C([0,eta])$ is $oldsymbol{ extomega^{1+ extgamma+1}}$.

Indices are explicitly linked to the ordinal structure of $eta$.

Abstract

Let $\alpha$ be an infinite ordinal and $\gamma$ the unique ordinal satisfying $\omega^{\omega^\gamma}\leq \alpha < \omega^{\omega^{\gamma+1}}$. We show that the Banach space $C([0,\,\alpha])$ of all continuous scalar-valued functions on the compact ordinal interval $[0,\,\alpha]$ has Szlenk index equal to $\omega^{\gamma+1}$ and $w^\ast$-dentability index equal to $\omega^{1+\gamma+1}$.