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.
KeywordsWeak approximate fixed point property Metrizable locally convex space ℓ1 sequence Fréchet–Urysohn space
Mathematics Subject Classification (2000)Primary 47H10 46A03
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