Abstract
IfF is a Fredholm mapping of indexΝ ∃ ℤ and classC max(Ν,0)+1 between separable Banach spaces, the Sard—Smale theorem yields the existence of arbitrarily small perturbations ofF having 0 as a regular value. The smoothness requirement cannot be weakened in the Sard—Smale theorem itself, at least whenΝ ≥ 0, but we prove that the approximation result remains valid irrespective of the indexΝ whenF is only of classC 1 and satisfies appropriate properness-like conditions. The separability of the spaces is not needed either. Everything carries over to the setting of Banach manifolds modeled on spaces with a norm of classC 1 away from the origin. We also show that in Banach spaces, theC 1 norm assumption can be dropped without major prejudice. The application to degree theory forC 1 Fredholm mappings of index 0 is developed in a separate paper.
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Pejsachowicz, J., Rabier, P.J. A substitute for the Sard-Smale theorem in theC 1 case. J. Anal. Math. 76, 265–288 (1998). https://doi.org/10.1007/BF02786938
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DOI: https://doi.org/10.1007/BF02786938