Abstract
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.
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C. S. Barroso’s research has been partially supported by the Brazilian-CNPq Grant. O. F. K. Kalenda supported in part by the grant GAAV IAA 100190901 and in part by the Research Project MSM 0021620839 from the Czech Ministry of Education. A very preliminary version of this work was presented at the 4th ENAMA (National Meeting on Mathematical Analysis and Applications) in Belém-PA, Brazil.
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Barroso, C.S., Kalenda, O.F.K. & Lin, PK. On the approximate fixed point property in abstract spaces. Math. Z. 271, 1271–1285 (2012). https://doi.org/10.1007/s00209-011-0915-6
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DOI: https://doi.org/10.1007/s00209-011-0915-6
Keywords
- Weak approximate fixed point property
- Metrizable locally convex space
- ℓ 1 sequence
- Fréchet–Urysohn space