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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

The Morse lemma on Banach spaces


Author: A. J. Tromba
Journal: Proc. Amer. Math. Soc. 34 (1972), 396-402
MSC: Primary 58E05; Secondary 58B10
DOI: https://doi.org/10.1090/S0002-9939-1972-0295395-9
MathSciNet review: 0295395
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Abstract: Let $ f:U \to R$ be a $ {C^3}$ map of an open subset U of a Banach space E. Let $ p \in U$ be a critical point of $ f(d{f_p} = 0)$. If E is a conjugate space $ (E = {F^ \ast })$ we define what it means for p to be nondegenerate. In this case there is a diffeomorphism $ \gamma $ of a neighborhood of p with a neighborhood of $ 0 \in E,\gamma (p) = 0$ with

$\displaystyle f \circ {\gamma ^{ - 1}}(x) = \frac{1}{2}{d^2}{f_p}(x,x) + f(p).$


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DOI: https://doi.org/10.1090/S0002-9939-1972-0295395-9
Article copyright: © Copyright 1972 American Mathematical Society