Morse–Palais lemma


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 be a real Hilbert space, and let U be an open neighbourhood of 0 in H. Let f : U → R be a -times continuously differentiable function with k ≥ 1, i.e. f ∈ Ck+2. Assume that f = 0 and that 0 is a non-degenerate critical point of f, i.e. the second derivative D2f defines an isomorphism of H with its continuous dual space H∗ by
Then there exists a subneighbourhood V of 0 in U, a diffeomorphism φ : V → V that is Ck with Ck inverse, and an invertible symmetric operator A : H → H, such that
for all x ∈ V.

Corollary

Let f : U → R be Ck+2 such that 0 is a non-degenerate critical point. Then there exists a Ck-with-Ck-inverse diffeomorphism ψ : V → V and an orthogonal decomposition
such that, if one writes
then
for all x ∈ V.