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

2014/12/03 by Talponen, Jarno · 1 citation
#03H05 #46B10 #46B20 #46M07 #FOS: Mathematics #Functional Analysis (math.FA) #Logic (math.LO)

paper · doi:10.48550/arxiv.1412.1279

Abstract

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. For example, a point x∈ S_X is an MLUR point if and only if j(x) (given by the canonical inclusion j\colon X → X^U) in B_X^U is an extreme point; a point x∈ S_X is LUR if and only if j(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≠ y, with x-y ∈ j(X).

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