Reflexivity and approximate fixed points

• Autorzy: Eva Matoušková , Simeon Reich
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 47H09: Contraction-type mappings, nonexpansive mappings, A -proper mappings, etc.
• 46B10: Duality and reflexivity
• 47H10: Fixed-point theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2003
• Tom: 159
• Numer: 3
• Strony: 403-415

Daty

wydano

2003

Twórcy

autor

Eva Matoušková

• Mathematical Institute, Czech Academy of Sciences, Žitná 25, CZ-11567 Praha, Czech Republic

autor

Simeon Reich

• Department of Mathematics, The Technion - Israel Institute of Technology, 32000 Haifa, Israel

Identyfikatory

DOI

10.4064/sm159-3-5