Advertisement

Set-Valued and Variational Analysis

, Volume 27, Issue 1, pp 223–240 | Cite as

A Pointwise Lipschitz Selection Theorem

  • Miek MesserschmidtEmail author
Article

Abstract

We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain. We apply our selection theorem to demonstrate a slight improvement to a well-known version of the classical Bartle-Graves Theorem: Any continuous linear surjection between infinite dimensional Banach spaces has a positively homogeneous continuous right inverse that is pointwise Lipschitz on a dense meager set of its domain. An example devised by Aharoni and Lindenstrauss shows that our pointwise Lipschitz selection theorem is in some sense optimal: It is impossible to improve our pointwise Lipschitz selection theorem to one that yields a selection that is pointwise Lipschitz on the whole of its domain in general.

Keywords

Selection theorem Pointwise Lipschitz map Bartle-Graves Theorem 

Mathematics Subject Classification (2010)

54C65 54C60 (primary) 46B99 (secondary) 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author would like to thank the MathOverflow community (Nate Eldredge in particular, for pointing out the example in Remark 3.5 to the author), and the anonymous referees of the paper for their constructive comments and suggestions.

Funding Information

The author’s research was funded by The Claude Leon Foundation.

References

  1. 1.
    Aharoni, I., Lindenstrauss, J.: Uniform equivalence between Banach spaces. Bull. Am. Math. Soc. 84(2), 281–283 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, 3rd edn. Springer, Berlin (2006)Google Scholar
  3. 3.
    Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis, vol. 1. American Mathematical Society, Providence (2000)zbMATHGoogle Scholar
  4. 4.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry. American Mathematical Society, Providence (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    de Jeu, M., Messerschmidt, M.: A strong open mapping theorem for surjections from cones onto Banach spaces. Adv. Math. 259, 43–66 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings, 2nd edn. Springer, New York (2014)Google Scholar
  7. 7.
    Dugundji, J.: Topology. Allyn and Bacon, Inc, Boston, London, Sydney (1978)zbMATHGoogle Scholar
  8. 8.
    Durand-Cartagena, E., Jaramillo, J.A.: Pointwise Lipschitz functions on metric spaces. J. Math. Anal. Appl. 363(2), 525–548 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Godefroy, G., Kalton, N.J.: Lipschitz-free Banach spaces. Studia Math 159(1), 121–141 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kalton, N.J.: The nonlinear geometry of Banach spaces. Rev. Mat. Comput. 21(1), 7–60 (2008)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Messerschmidt, M., Wortel, M.: The intrinsic metric on the unit sphere of a normed space, arXiv:1510.07442
  12. 12.
    Przeslawski, K., Yost, D.: Continuity properties of selectors and Michael’s theorem. Mich. Math. J. 36(1), 113–134 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Schäffear, J.J.: Inner diameter, perimeter, and girth of spheres. Math. Ann. 173, 59–79 (1967). addendum, ibid. 173 (1967), 79–82MathSciNetCrossRefGoogle Scholar
  14. 14.
    Stone, A.H.: Paracompactness and product spaces. Bull. Am. Math. Soc. 54, 977–982 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Tao, T.: An Introduction to Measure Theory, vol. 126. American Mathematical Society, Providence (2011)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of PretoriaPretoriaSouth Africa

Personalised recommendations