Uniform-to-proper duality of geometric properties of Banach spaces and their ultrapowers

Abstract

In this note various geometric properties of a Banach space $\mathrm{X} $ are characterized by means of weaker corresponding geometric properties involving an ultrapower $\mathrm{X} ^\mathcal {U}$. The characterizations do not depend on the particular choice of the free ultrafilter $\mathcal {U}$ on $\mathbb{N}$. For example, a point $x\in \mathbf{S} _\mathrm{X} $ is an MLUR point if and only if $\jmath (x)$ (given by the canonical inclusion $\jmath \colon \mathrm{X} \to \mathrm{X} ^\mathcal {U}$) in $\mathbb{B} _{\mathrm{X} ^\mathcal {U}}$ is an extreme point; a point $x\in \mathbf{S} _\mathrm{X} $ is LUR if and only if $\jmath (x)$ is not contained in any non-degenerate line segment of $\mathbf{S} _{\mathrm{X} ^\mathcal {U}}$; a Banach space $\mathrm{X} $ is URED if and only if there are no $x, y \in \mathbf{S} _{\mathrm{X} ^\mathcal {U}}$, $x \neq y$, with $x-y \in \jmath (\mathrm{X} )$.