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McShane-Whitney Pairs

  • Iosif Petrakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10307)

Abstract

We present a constructive version of the classical McShane-Whitney theorem on the extendability of real-valued Lipschitz functions defined on a subset of a metric space. Through the introduced notion of a McShane-Whitney pair we study some abstract properties of this extension theorem showing how the behavior of a Lipschitz function defined on the subspace of the pair affect its McShane-Whitney extensions on the space of the pair. As a consequence, a Lipschitz version of the theory around the Hahn-Banach theorem is formed. We work within Bishop’s informal system of constructive mathematics \(\mathrm {BISH}\).

References

  1. 1.
    Benyamini, Y., Lindenstrauss, J.: Geometric nonlinear functional analysis, vol. 1, American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI (2000)Google Scholar
  2. 2.
    Bishop, E.: Foundations of Constructive Analysis. McGraw-Hill, New York (1967)MATHGoogle Scholar
  3. 3.
    Bishop, E., Bridges, D.: Constructive Analysis, Grundlehren der mathematischen Wissenschaften. 279. Springer, Heidelberg (1985)Google Scholar
  4. 4.
    Brudnyi, A., Brudnyi, Y.: Methods of geometric analysis in extension and trace problems, Volume 1. Monographs in Mathematics, vol. 102. Birkhäuser/Springer, Basel (2012)CrossRefMATHGoogle Scholar
  5. 5.
    Bridges, D.S., Vîţă, L.S.: Techniques of Constructive Analysis. Universitext. Springer, New York (2006)CrossRefMATHGoogle Scholar
  6. 6.
    Bridges, D.S., Richman, F.: Varieties of Constructive Mathematics. Cambridge University Press, Cambridge (1987)CrossRefMATHGoogle Scholar
  7. 7.
    Ishihara, H.: On the constructive Hahn-Banach theorem. Bull. London. Math. Soc. 21, 79–81 (1989)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Julian, W., Philips, K.: Constructive bounded sequences and lipschitz functions. J. London Math. Soc. s2–31(3), 385–392 (1985)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kirszbraun, M.D.: Über die zusammenziehende und Lipschitzsche Transformationen. Fundam. Math. 22, 77–108 (1934)MATHGoogle Scholar
  10. 10.
    Loeb, I.: Lipschitz functions in constructive reverse mathematics. Logic J. IGPL 21(1), 28–43 (2013). (special issue on Non-Classical Mathematics)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Mandelkern, M.: Constructive continuity, Mem. Amer. Math. Soc. 277 (1983)Google Scholar
  12. 12.
    McShane, E.J.: Extension of range of functions. Bull. Amer. Math. Soc. 40(12), 837–842 (1934)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Petrakis, Iosif: A direct constructive proof of a stone-weierstrass theorem for metric spaces. In: Beckmann, Arnold, Bienvenu, Laurent, Jonoska, Nataša (eds.) CiE 2016. LNCS, vol. 9709, pp. 364–374. Springer, Cham (2016). doi: 10.1007/978-3-319-40189-8_37 Google Scholar
  14. 14.
    Tuominen, H.: Analysis in Metric Spaces, Lecture notes (2014)Google Scholar
  15. 15.
    Valentine, F.A.: On the extension of a vector function so as to preserve a Lipschitz condition. Bull. Amer. Math. Soc. 49(2), 100–108 (1943)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Valentine, F.A.: A Lipschitz condition preserving extension for a vector function. Amer. J. Math. 67, 83–93 (1945)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Weaver, N.: Lipschitz Algebras. World Scientific, Singapore (1999)CrossRefMATHGoogle Scholar
  18. 18.
    Wells, J.H., Williams, L.R.: Embeddings and Extensions in Analysis. Springer, Heidelberg (1975)CrossRefMATHGoogle Scholar
  19. 19.
    Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36(1), 63–89 (1934)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of MunichMunichGermany

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