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.
Let formula_1 be a real Hilbert space, and let formula_2 be an open neighbourhood of the origin in formula_3 Let formula_4 be a formula_5-times continuously differentiable function with formula_6 that is, formula_7 Assume that formula_8 and that formula_9 is a non-degenerate critical point of formula_10 that is, the second derivative formula_11 defines an isomorphism of formula_12 with its continuous dual space formula_13 by
formula_14
Then there exists a subneighbourhood formula_15 of formula_9 in formula_17 a diffeomorphism formula_18 that is formula_19 with formula_19 inverse, and an invertible symmetric operator formula_21 such that
formula_22
Corollary.
Let formula_4 be formula_24 such that formula_9 is a non-degenerate critical point. Then there exists a formula_19-with-formula_19-inverse diffeomorphism formula_28 and an orthogonal decomposition
formula_29
such that, if one writes
formula_30
then
formula_31