Reflexivity and approximate fixed points
Volume 159 / 2003
Studia Mathematica 159 (2003), 403-415
MSC: Primary 46B10, 47H10; Secondary 47H09.
DOI: 10.4064/sm159-3-5
Abstract
A Banach space is reflexive if and only if every bounded sequence \{x_n\} in X contains a norm attaining subsequence. This means that it contains a subsequence \{x_{n_k}\} for which \sup_{f\in S_{X^*}}\limsup_{k\to \infty} f(x_{n_k}) is attained at some f in the dual unit sphere S_{X^*}. A Banach space X is not reflexive if and only if it contains a normalized sequence \{x_n\} with the property that for every f\in S_{X^*}, there exists g\in S_{X^*} such that \limsup_{n\to \infty}f(x_n)<\liminf_{n\to \infty}g(x_n). Combining this with a result of Shafrir, we conclude that every infinite-dimensional Banach space contains an unbounded closed convex set which has the approximate fixed point property for nonexpansive mappings.
