The extreme points of the unit ball of the James space $J$ and its dual spaces

We provide a new proof of S. Bellenot’s characterization of the extreme points of the unit ball $B_J$ of the James quasi-reflexive space $J$. We also provide an explicit description of the norm of $J^{**}$ which yields an analogous characterization for the extreme points of $B_{J^{**}}$. In the last part of the paper we describe the set of all extreme points of $B_{J^*}$ and its norm closure. It is remarkable that the descriptions of the extreme points of $B_J$ and $B_{J^*}$ are closely connected.