Abstrakt
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 sigma(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 Frechet-Urysohn property for certain sets with regarding the sigma(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.