Asymptotic Properties of Banach Spaces under Renormings

A classical problem in functional analysis has been to give a geometric characterization of reflexivity for a Banach space. The first result of this type was D.P. Milman’s [Mil] and B.J. Pettis’ [P] theorem that a uniformly convex space is reflexive. While perhaps considered elementary today it illustrated how a geometric property can be responsible for a topological property. Of course a Banach space can be reflexive without being uniformly convex, even under renormings, as shown by M.M. Day [D2]. The problem considered for years by functional analysts was whether there exists a weaker property of a geometric nature which is equivalent to reflexivity. In this paper we give an affirmative solution by demonstrating that such a property exists. The property was suggested in 1971 by V.D. Milman [Mi] (see also [DGZ], problem IV, p.177). We prove that a separable Banach space X is reflexive (if and) only if there exists an equivalent norm ||| · ||| on X so that whenever a sequence (xn) ⊆ X satisfies lim n lim m |||xn + xm||| = 2 lim n ||| xn|||