2011/01/31 by Cleon S. Barroso, Ondřej F. K. Kalenda, Pei-Kee Lin +1
Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #Banach space #Bounded function #Combinatorics #Discrete mathematics #Fixed Point Theorems Analysis #Fixed-point property #Fixed-point theorem #Hausdorff space #Locally convex topological vector space #Mathematical analysis #Mathematics #Metrization theorem #Pure mathematics #Regular polygon #Separable space #Topological space #Topological vector space #Topology (electrical circuits) #Vector space #math.FA #math.GN #msc:46A03 #msc:47H10
paper · pdf · doi:10.1007/s00209-011-0915-6
published as Math. Z. 271 (2012), no. 3-4, 1271-1285 · 16 pages; the paper was slightly revised, some more explanations were added
arxiv created 2011/04/18 · openalex publication_date 2011/07/07 · arxiv updated 2012/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let X be a Hausdorff topological vector space, X^* its topological dual and Z a subset of X^*. In this paper, we establish some results concerning the σ(X,Z)-approximate fixed point property for bounded, closed convex subsets C of X. Three major situations are studied. First when Z is separable in the strong topology. Second when X is a metrizable locally convex space and Z=X^*, and third when X is not necessarily metrizable but admits a metrizable locally convex topology compatible with the duality. Our approach focuses on establishing the Fréchet-Urysohn property for certain sets with regarding the σ(X,Z)-topology. The support tools include the Brouwer's fixed point theorem and an analogous version of the classical Rosenthal's ℓ1-theorem for ℓ1-sequences in metrizable case. The results are novel and generalize previous work obtained by the authors in Banach spaces.