Uniform-to-proper duality of geometric properties of Banach spaces and their ultrapowers
In this note various geometric properties of a Banach space X are characterized by means of weaker corresponding geometric properties involving an ultrapower X ^U. The characterizations do not depend on the particular choice of the free ultrafilter U on N. For example, a point xS _X is an MLUR point if and only if (x) (given by the canonical inclusion X → X ^U) in B _X ^U is an extreme point; a point xS _X is LUR if and only if (x) is not contained in any non-degenerate line segment of S _X ^U ; a Banach space X is URED if and only if there are no x, y S _X ^U , x eq y, with x-y (X ). , number=1, journal=MATHEMATICA SCANDINAVICA, author=Talponen, Jarno, year=2017, month=Sep., pages=111–120