On the approximate fixed point property in abstract spaces


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 ℓ 1-theorem for ℓ 1-sequences in metrizable case. The results are novel and generalize previous work obtained by the authors in Banach spaces. © 2011 Springer-Verlag.

Publication Title

Mathematische Zeitschrift