Reflexivity and approximate fixed points

Volume 159 / 2003

Eva Matoušková, Simeon Reich Studia Mathematica 159 (2003), 403-415 MSC: Primary 46B10, 47H10; Secondary 47H09. DOI: 10.4064/sm159-3-5


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.


  • Eva MatouškováMathematical Institute
    Czech Academy of Sciences
    Žitná 25
    CZ-11567 Praha, Czech Republic
  • Simeon ReichDepartment of Mathematics
    The Technion — Israel Institute of Technology
    32000 Haifa, Israel

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