A substitute for the Sard-Smale theorem in theC
1 case |
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Authors: | Jacobo Pejsachowicz Patrick J Rabier |
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Institution: | (1) Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy;(2) Department of Mathematics, University of Pittsburgh, 15260 Pittsburgh, PA, USA |
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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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Keywords: | |
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