Journal of Symbolic Logic

The fan theorem and unique existence of maxima

Josef Berger, Douglas Bridges, and Peter Schuster

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Abstract

The existence and uniqueness of a maximum point for a continuous real—valued function on a metric space are investigated constructively. In particular, it is shown, in the spirit of reverse mathematics, that a natural unique existence theorem is equivalent to the fan theorem.

Article information

Source
J. Symbolic Logic, Volume 71, Issue 2 (2006), 713-720.

Dates
First available in Project Euclid: 2 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1146620167

Digital Object Identifier
doi:10.2178/jsl/1146620167

Mathematical Reviews number (MathSciNet)
MR2225902

Zentralblatt MATH identifier
1107.03064

Citation

Berger, Josef; Bridges, Douglas; Schuster, Peter. The fan theorem and unique existence of maxima. J. Symbolic Logic 71 (2006), no. 2, 713--720. doi:10.2178/jsl/1146620167. https://projecteuclid.org/euclid.jsl/1146620167


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References

  • P. Aczel and M. Rathjen, Notes on constructive set theory, Technical Report 40, Institut Mittag--Leffler, Royal Swedish Academy of Sciences,2001.
  • E. Bishop and D. Bridges, Constructive analysis, Grundlehren der Mathematischen Wissenschaften, vol. 279, Springer--Verlag, Heidelberg,1985.
  • D. Bridges, A constructive proximinality property of finite--dimensional linear spaces, Rocky Mountain Journal of Mathematics, vol. 11 (1981), no. 4, pp. 491--497.
  • --------, Recent progress in constructive approximation theory, The L.E.J. Brouwer Centenary Symposium (A.S. Troelstra and D. van Dalen, editors), North--Holland, Amsterdam,1982, pp. 41--50.
  • --------, Constructing local optima on a compact interval, preprint, Universität München,2003.
  • --------, Continuity and Lipschitz constants for continuous projections, preprint, University of Canterbury and Universität München,2003.
  • D. Bridges and F. Richman, Varieties of constructive mathematics, London Mathematical Society Lecture Notes, vol. 97, Cambridge University Press,1987.
  • M. Dummett, Elements of intuitionism, 2nd ed., Oxford Logic Guides, vol. 39, Clarendon Press, Oxford,2000.
  • H. Ishihara, Informal constructive reverse mathematics, Sūrikaisekikenkyūsho Kīkyūroko, vol. 1381(2004), pp. 108--117.
  • K-I Ko, Complexity theory of real functions, Birkhäuser, Boston--Basel--Berlin,1991.
  • U. Kohlenbach, Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin's proof for Chebycheff approximation, Annals of Pure and Applied Logic, vol. 64 (1993), pp. 27--94.
  • E. Specker, Nicht konstruktiv beweisbare Sätze der Analysis, Journal of Symbolic Logic, vol. 14 (1949), pp. 145--158.
  • K. Weihrauch, Computable analysis, Springer--Verlag, Heidelberg,2000.