Spaces not containing $\ell_1$ have weak aproximate fixed point property

Details Spaces not containing $\ell_1$ have weak aproximate fixed point property Spaces not containing $\ell_1$ have weak aproximate fixed point property

2010-05-07

arxiv.org/abs/1005.1218v2

J. Math. Anal. Appl. 373 (2011), no. 1, 134-137

Mathematics

Functional Analysis

6 pages; the paper was reorganized a bit

Scientific paper

10.1016/j.jmaa.2010.06.052

A nonempty closed convex bounded subset $C$ of a Banach space is said to have the weak approximate fixed point property if for every continuous map $f:C\to C$ there is a sequence $\{x_n\}$ in $C$ such that $x_n-f(x_n)$ converge weakly to 0. We prove in particular that $C$ has this property whenever it contains no sequence equivalent to the standard basis of $\ell_1$. As a byproduct we obtain a characterization of Banach spaces not containing $\ell_1$ in terms of the weak topology.

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