Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Choice free fixed point property in separable Banach spaces


Author: Vassilios Gregoriades
Journal: Proc. Amer. Math. Soc. 143 (2015), 2143-2157
MSC (2010): Primary 47H10, 03E15, 54H05, 54H25
DOI: https://doi.org/10.1090/S0002-9939-2015-12465-3
Published electronically: January 8, 2015
MathSciNet review: 3314122
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Spiros A. Argyros and Pandelis Dodos, Genericity and amalgamation of classes of Banach spaces, Adv. Math. 209 (2007), no. 2, 666-748. MR 2296312 (2008f:46014), https://doi.org/10.1016/j.aim.2006.05.013
  • [2] 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 0115071 (22 #5874)
  • [3] 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 (94g:46022)
  • [4] 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 (80b:54017), https://doi.org/10.2307/2373913
  • [5] Gabriel Debs, Effective properties in compact sets of Borel functions, Mathematika 34 (1987), no. 1, 64-68. MR 908840 (89b:03082), https://doi.org/10.1112/S0025579300013280
  • [6] Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004 (85i:46020)
  • [7] Pandelis Dodos, Banach spaces and descriptive set theory: selected topics, Lecture Notes in Mathematics, vol. 1993, Springer-Verlag, Berlin, 2010. MR 2598479 (2011j:46008)
  • [8] Benno Fuchssteiner, Iterations and fixpoints, Pacific J. Math. 68 (1977), no. 1, 73-80. MR 0513055 (58 #23795)
  • [9] 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 (2009h:54056), https://doi.org/10.1007/s11784-008-0076-3
  • [10] Dietrich Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965), 251-258 (German). MR 0190718 (32 #8129)
  • [11] V. Gregoriades, A dichotomy result for a pointwise summable sequence of operators, Ann. Pure Appl. Logic 160 (2009), no. 2, 154-162. MR 2541470 (2010k:03041), https://doi.org/10.1016/j.apal.2009.02.003
  • [12] Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513 (2004g:03071)
  • [13] Alexander S. Kechris, Classical descriptive set theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995. MR 1321597 (96e:03057)
  • [14] W. A. Kirk, An abstract fixed point theorem for nonexpansive mappings, Proc. Amer. Math. Soc. 82 (1981), no. 4, 640-642. MR 614894 (82f:54075), https://doi.org/10.2307/2043787
  • [15] S. C. Kleene, Quantification of number-theoretic functions, Compositio Math. 14 (1959), 23-40. MR 0103822 (21 #2586)
  • [16] Motokiti Kondô, L'uniformisation des complémentaires analytiques, Proc. Imp. Acad. 13 (1937), no. 8, 287-291 (French). MR 1568473
  • [17] 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 0188994 (32 #6421)
  • [18] Yiannis N. Moschovakis, Uniformization in a playful universe, Bull. Amer. Math. Soc. 77 (1971), 731-736. MR 0285390 (44 #2609)
  • [19] Yiannis N. Moschovakis, Descriptive set theory, 2nd ed., Mathematical Surveys and Monographs, vol. 155, American Mathematical Society, Providence, RI, 2009. MR 2526093 (2010f:03001)
  • [20] 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 0172091 (30:2317)
  • [21] Stephen G. Simpson, Subsystems of second order arithmetic, Second edition, Perspectives in Logic, Cambridge University Press, 2009. MR 2517689 (2010e:03073)
  • [22] John von Neumann, On rings of operators. Reduction theory, Ann. of Math. (2) 50 (1949), 401-485. MR 0029101 (10,548a)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47H10, 03E15, 54H05, 54H25

Retrieve articles in all journals with MSC (2010): 47H10, 03E15, 54H05, 54H25


Additional Information

Vassilios Gregoriades
Affiliation: Technische Universität Darmstadt, Fachbereich Mathematik, Arbeitsgruppe Logik, Schloßgartenstraße 7, 64289 Darmstadt, Germany
Email: gregoriades@mathematik.tu-darmstadt.de

DOI: https://doi.org/10.1090/S0002-9939-2015-12465-3
Keywords: Minimal invariant sets, non-expansive mappings, fixed point property, Axiom of Choice, effective descriptive set theory.
Received by editor(s): May 8, 2013
Received by editor(s) in revised form: November 5, 2013
Published electronically: January 8, 2015
Communicated by: Mirna Džamonja
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society