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• # Artykuł - szczegóły

## Studia Mathematica

2003 | 159 | 3 | 403-415

## Reflexivity and approximate fixed points

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.

403-415

wydano
2003

### Twórcy

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