Extremal structures in ultrapower spaces (Q972109)

This short note deals with extremal structures in ultrapowers of Banach spaces. We show that a unit vector \(x\in X\) is a strong extreme point if and only if for every ultrafilter \(U\), it is a strongly extreme point of the unit ball of the ultrapower space \((X)_U\). We also give an application of these ideas to approximation theory.