The Morse lemma on Banach spaces
HTML articles powered by AMS MathViewer
- by A. J. Tromba
- Proc. Amer. Math. Soc. 34 (1972), 396-402
- DOI: https://doi.org/10.1090/S0002-9939-1972-0295395-9
- PDF | Request permission
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
- Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
- Serge Lang, Introduction to differentiable manifolds, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155257
- Richard S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963), 299–340. MR 158410, DOI 10.1016/0040-9383(63)90013-2
- Richard S. Palais, The Morse lemma for Banach spaces, Bull. Amer. Math. Soc. 75 (1969), 968–971. MR 253378, DOI 10.1090/S0002-9904-1969-12318-9
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