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

Abstract

Let $\alpha$ be an infinite ordinal and $\gamma$ the unique ordinal satisfying ${\omega }^{{\omega }^{\gamma }}\le \alpha <{\omega }^{{\omega }^{\gamma +1}}$. 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 ${\omega }^{1+\gamma +1}$.