Reflexivity and approximate fixed points
Eva Matoušková ; Simeon Reich
Studia Mathematica, Tome 157 (2003), p. 403-415 / Harvested from The Polish Digital Mathematics Library
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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 xnk for which supf∈SX*limsupk→∞f(xnk) is attained at some f in the dual unit sphere SX*. A Banach space X is not reflexive if and only if it contains a normalized sequence xₙ with the property that for every f∈SX*, there exists g∈SX* such that limsupn→∞f(xₙ)<liminfn→∞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.

Publié le : 2003-01-01
EUDML-ID : urn:eudml:doc:284385 
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Eva Matoušková; Simeon Reich. Reflexivity and approximate fixed points. Studia Mathematica, Tome 157 (2003) pp. 403-415. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm159-3-5/