Reflexivity and approximate fixed points

Eva Matoušková ¹   ; Simeon Reich ²

¹ Mathematical Institute Czech Academy of Sciences Žitná 25 CZ-11567 Praha, Czech Republic
² Department of Mathematics The Technion — Israel Institute of Technology 32000 Haifa, Israel

Studia Mathematica, Tome 159 (2003) no. 3, pp. 403-415

Résumé

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.

Détail

DOI : 10.4064/sm159-3-5

Keywords: banach space reflexive only every bounded sequence contains norm attaining subsequence means contains subsequence which sup * limsup infty attained dual unit sphere * banach space reflexive only contains normalized sequence property every * there exists * limsup infty liminf infty combining result shafrir conclude every infinite dimensional banach space contains unbounded closed convex set which has approximate fixed point property nonexpansive mappings