Reflexivity and approximate fixed points
Volume 159 / 2003
Abstract
A Banach space $X$ 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.
