Choice free fixed point property in separable Banach spaces

Gregoriades, Vassilios

doi:10.1090/S0002-9939-2015-12465-3

Choice free fixed point property in separable Banach spaces
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Proc. Amer. Math. Soc. 143 (2015), 2143-2157 Request permission

Abstract:

We show that the standard approach of minimal invariant sets, which applies Zorn’s Lemma and is used to prove fixed point theorems for non-expansive mappings in Banach spaces, can be applied without any reference to the full Axiom of Choice when the given Banach space is separable. Our method applies results from classical and effective descriptive set theory.

References

Spiros A. Argyros and Pandelis Dodos, Genericity and amalgamation of classes of Banach spaces, Adv. Math. 209 (2007), no. 2, 666–748. MR 2296312, DOI 10.1016/j.aim.2006.05.013
C. Bessaga and A. Pełczyński, A generalization of results of R. C. James concerning absolute bases in Banach spaces, Studia Math. 17 (1958), 165–174. MR 115071, DOI 10.4064/sm-17-2-165-174
Benoît Bossard, Codages des espaces de Banach séparables. Familles analytiques ou coanalytiques d’espaces de Banach, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 10, 1005–1010 (French, with English and French summaries). MR 1222962
J. Bourgain, D. H. Fremlin, and M. Talagrand, Pointwise compact sets of Baire-measurable functions, Amer. J. Math. 100 (1978), no. 4, 845–886. MR 509077, DOI 10.2307/2373913
Gabriel Debs, Effective properties in compact sets of Borel functions, Mathematika 34 (1987), no. 1, 64–68. MR 908840, DOI 10.1112/S0025579300013280
Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004, DOI 10.1007/978-1-4612-5200-9
Pandelis Dodos, Banach spaces and descriptive set theory: selected topics, Lecture Notes in Mathematics, vol. 1993, Springer-Verlag, Berlin, 2010. MR 2598479, DOI 10.1007/978-3-642-12153-1
Benno Fuchssteiner, Iterations and fixpoints, Pacific J. Math. 68 (1977), no. 1, 73–80. MR 513055, DOI 10.2140/pjm.1977.68.73
Kazimierz Goebel and W. A. Kirk, Some problems in metric fixed point theory, J. Fixed Point Theory Appl. 4 (2008), no. 1, 13–25. MR 2447958, DOI 10.1007/s11784-008-0076-3
Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258 (German). MR 190718, DOI 10.1002/mana.19650300312
V. Gregoriades, A dichotomy result for a pointwise summable sequence of operators, Ann. Pure Appl. Logic 160 (2009), no. 2, 154–162. MR 2541470, DOI 10.1016/j.apal.2009.02.003
Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597, DOI 10.1007/978-1-4612-4190-4
W. A. Kirk, An abstract fixed point theorem for nonexpansive mappings, Proc. Amer. Math. Soc. 82 (1981), no. 4, 640–642. MR 614894, DOI 10.1090/S0002-9939-1981-0614894-6
S. C. Kleene, Quantification of number-theoretic functions, Compositio Math. 14 (1959), 23–40. MR 103822
Motokiti Kondô, L’uniformisation des complémentaires analytiques, Proc. Imp. Acad. Tokyo 13 (1937), no. 8, 287–291 (French). MR 1568473
K. Kuratowski and C. Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13 (1965), 397–403 (English, with Russian summary). MR 188994
Yiannis N. Moschovakis, Uniformization in a playful universe, Bull. Amer. Math. Soc. 77 (1971), 731–736. MR 285390, DOI 10.1090/S0002-9904-1971-12789-1
Yiannis N. Moschovakis, Descriptive set theory, 2nd ed., Mathematical Surveys and Monographs, vol. 155, American Mathematical Society, Providence, RI, 2009. MR 2526093, DOI 10.1090/surv/155
A. Pełczyński, A proof of Eberlein-Šmulian theorem by an application of basic sequences, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 543–548. MR 172091
Stephen G. Simpson, Subsystems of second order arithmetic, 2nd ed., Perspectives in Logic, Cambridge University Press, Cambridge; Association for Symbolic Logic, Poughkeepsie, NY, 2009. MR 2517689, DOI 10.1017/CBO9780511581007
John von Neumann, On rings of operators. Reduction theory, Ann. of Math. (2) 50 (1949), 401–485. MR 29101, DOI 10.2307/1969463