Spaces not containing ℓ1 have weak approximate fixed point property

2010/05/31 by Ondřej F. K. Kalenda · 1 citation
Computer Science · Mathematics · #Advanced Banach Space Theory #Fixed Point Theorems Analysis #Optimization and Variational Analysis #math.FA

paper · pdf · doi:10.1016/j.jmaa.2010.06.052

published as J. Math. Anal. Appl. 373 (2011), no. 1, 134-137 · 6 pages; the paper was reorganized a bit

arxiv created 2010/07/01 · openalex publication_date 2010/07/04 · arxiv updated 2011/03/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/31

Abstract

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→ 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 ℓ₁. As a byproduct we obtain a characterization of Banach spaces not containing ℓ₁ in terms of the weak topology.

Citations

• Pointwise Compact Sets of Baire-Measurable Functions
• A double-dual characterization of separable Banach spaces containingl 1
• A Characterization of Banach Spaces Containing l 1

Cited by

• On the approximate fixed point property in abstract spaces