In mathematics, the Morse–Palais lemma is a result in the calculus of variations and theory of Hilbert spaces. Roughly speaking, it states that a smooth enough function near a critical point can be expressed as a quadratic form after a suitable change of coordinates.

The Morse–Palais lemma was originally proved in the finite-dimensional case by the American mathematician Marston Morse, using the Gram–Schmidt orthogonalization process. This result plays a crucial role in Morse theory. The generalization to Hilbert spaces is due to Richard Palais and Stephen Smale.

Statement of the lemma

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Let   be a real Hilbert space, and let   be an open neighbourhood of the origin in   Let   be a  -times continuously differentiable function with   that is,   Assume that   and that   is a non-degenerate critical point of   that is, the second derivative   defines an isomorphism of   with its continuous dual space   by  

Then there exists a subneighbourhood   of   in   a diffeomorphism   that is   with   inverse, and an invertible symmetric operator   such that  

Corollary

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Let   be   such that   is a non-degenerate critical point. Then there exists a  -with- -inverse diffeomorphism   and an orthogonal decomposition   such that, if one writes   then  

See also

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References

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  • Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison–Wesley Publishing Co., Inc.