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.