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

• Jarno Talponen ^[1]
  1. [1] University of Eastern Finland

     University of Eastern Finland

     Kuopio, Finlandia

     
• Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 121, Nº 1, 2017, págs. 111-120
• Idioma: inglés
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• Resumen

  • In this note various geometric properties of a Banach space X are characterized by means of weaker corresponding geometric properties involving an ultrapower XU. The characterizations do not depend on the particular choice of the free ultrafilter U on N. For example, a point x∈SX is an MLUR point if and only if ȷ(x) (given by the canonical inclusion ȷ:X→XU) in BXU is an extreme point; a point x∈SX is LUR if and only if ȷ(x) is not contained in any non-degenerate line segment of SXU; a Banach space X is URED if and only if there are no x,y∈SXU, x≠y, with x−y∈ȷ(X).