Reflexivity and approximate fixed points

Eva Matoušková, and Simeon Reich. "Reflexivity and approximate fixed points." Studia Mathematica 159.3 (2003): 403-415. <http://eudml.org/doc/284385>.

@article{EvaMatoušková2003,
abstract = {A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence $\{x_\{n_k\}\}$ for which $sup_\{f∈S_\{X*\}\} lim sup_\{k→∞\} f(x_\{n_k\})$ 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 xₙ with the property that for every $f ∈ S_\{X*\}$, there exists $g ∈ S_\{X*\}$ such that $lim sup_\{n→∞\}f(xₙ) < lim inf_\{n→∞\}g(xₙ)$. 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.},
author = {Eva Matoušková, Simeon Reich},
journal = {Studia Mathematica},
keywords = {reflexive Banach space; norm attaining sequence},
language = {eng},
number = {3},
pages = {403-415},
title = {Reflexivity and approximate fixed points},
url = {http://eudml.org/doc/284385},
volume = {159},
year = {2003},
}

TY - JOUR
AU - Eva Matoušková
AU - Simeon Reich
TI - Reflexivity and approximate fixed points
JO - Studia Mathematica
PY - 2003
VL - 159
IS - 3
SP - 403
EP - 415
AB - A Banach space X is reflexive if and only if every bounded sequence xₙ in X contains a norm attaining subsequence. This means that it contains a subsequence ${x_{n_k}}$ for which $sup_{f∈S_{X*}} lim sup_{k→∞} f(x_{n_k})$ 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 xₙ with the property that for every $f ∈ S_{X*}$, there exists $g ∈ S_{X*}$ such that $lim sup_{n→∞}f(xₙ) < lim inf_{n→∞}g(xₙ)$. 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.
LA - eng
KW - reflexive Banach space; norm attaining sequence
UR - http://eudml.org/doc/284385
ER -