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 separa...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2012-08, Vol.271 (3-4), p.1271-1285
Main Authors: Barroso, C. S., Kalenda, O. F. K., Lin, P.-K.
Format: Article
Language:English
Subjects:
Mathematics
Mathematics and Statistics
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Summary: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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-011-0915-6