Copyright © 2010 T. Domínguez Benavides. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Assume that X is a Banach space such that its Szlenk index Sz(X) is less than or equal to the first infinite ordinal ω. We prove that X can be renormed in such a way that X with the resultant norm satisfies R(X)<2, where R(⋅) is the García-Falset coefficient. This leads us to prove that if X is a Banach space which can be continuously embedded in a Banach space Y with Sz(Y)≤ω, then, X can be renormed to satisfy the w-FPP. This result can be applied to Banach spaces which can be embedded in C(K), where K is a scattered compact topological space such that K(ω)=∅. Furthermore, for a Banach space (X,‖⋅‖), we consider a distance in the space 𝒫 of all norms in X which are equivalent to ‖⋅‖ (for which 𝒫 becomes a Baire space). If Sz(X)≤ω, we show that for almost all norms (in the sense of porosity) in 𝒫, X satisfies the w-FPP. For general reflexive spaces (independently of the Szlenk index), we prove another strong generic result in the sense of Baire category.