Uniform-to-proper duality of geometric properties of Banach spaces and their ultrapowers - Talponen, Jarno (MATHEMATICA SCANDINAVICA, 2017)
In this note various geometric properties of a Banach space $X $ are characterized by means of weaker corresponding geometric properties involving an ultrapower $X ^\mathcal {U}$. The characterizations do not depend on the particular choice of the free ultrafilter $\mathcal {U}$ on $\mathbb{N}$. For example, a point $x\in S _X $ is an MLUR point if and only if $\jmath (x)$ (given by the canonical inclusion $\jmath \colon X → X ^\mathcal {U}$) in $\mathbb{B} _{X ^\mathcal {U}}$ is an extreme point; a point $x\in S _X $ is LUR if and only if $\jmath (x)$ is not contained in any non-degenerate line segment of $S _{X ^\mathcal {U}}$; a Banach space $X $ is URED if and only if there are no $x, y \in S _{X ^\mathcal {U}}$, $x eq y$, with $x-y \in \jmath (X )$.