## 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.