Ukrainian Mathematical Journal

, Volume 62, Issue 12, pp 1896–1905 | Cite as

Sard’s theorem for mappings between Fréchet manifolds

  • K. Eftekharinasab

We prove an infinite-dimensional version of Sard’s theorem for Fréchet manifolds. Let M (respectively, N) be a bounded Fréchet manifold with compatible metric d M (respectively, d N ) modeled on Fréchet spaces E (respectively, F) with standard metrics. Let f : M → N be an MC k -Lipschitz–Fredholm map with k > max{Ind f, 0}: Then the set of regular values of f is residual in N.


Vector Bundle Topological Vector Space Fredholm Operator Local Representative Linear Isomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    I. Kupka, “Counterexample to Morse–Sard theorem in the case of infinite-dimensional manifolds,” Proc. Amer. Math. Soc., 16, No. 5, 954–957 (1965).MathSciNetCrossRefGoogle Scholar
  2. 2.
    S. Smale, “An infinite dimensional version of Sard’s theorem,” Amer. J. Math., 87, No. 4, 861–866 (1965).MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    D. Keller, “Differential calculus in locally convex spaces,” Lect. Notes Math., 417 (1974).Google Scholar
  4. 4.
    O. Müller, “A metric approach to Fréchet geometry,” J. Geom. Phys., 58, No. 11, 1477–1500 (2008).MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    H. Glöckner, Implicit Functions from Topological Vector Spaces in the Presence of Metric Estimates, Preprint, Arxiv:math/6612673.-2006.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2011

Authors and Affiliations

  • K. Eftekharinasab
    • 1
  1. 1.Institute of MathematicsUkrainian National Academy of SciencesKievUkraine

Personalised recommendations