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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Morse lemma on Banach spaces
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by A. J. Tromba
Proc. Amer. Math. Soc. 34 (1972), 396-402
DOI: https://doi.org/10.1090/S0002-9939-1972-0295395-9

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 \[ f \circ {\gamma ^{ - 1}}(x) = \frac {1}{2}{d^2}{f_p}(x,x) + f(p).\]
References
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Bibliographic Information
  • © Copyright 1972 American Mathematical Society
  • 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