@article{Talponen_2017, title={Uniform-to-proper duality of geometric properties of Banach spaces and their ultrapowers}, volume={121}, url={https://www.mscand.dk/article/view/26166}, DOI={10.7146/math.scand.a-26166}, abstractNote={<p>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 $x, y \in \mathbf{S} _{\mathrm{X} ^\mathcal {U }$, $x eq y$, with $x-y \in \jmath (\mathrm{X} )$.</p>}, number={1}, journal={MATHEMATICA SCANDINAVICA}, author={Talponen, Jarno}, year={2017}, month={Sep.}, pages={111–120} }