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 the origin in H. Let be a (k+2)-times continuously differentiable function with k \geq 1; that is, Assume that f(0) = 0 and that 0 is a non-degenerate critical point of f; that is, the second derivative D^2 f(0) defines an isomorphism of H with its continuous dual space H^* by Then there exists a subneighbourhood V of 0 in U, a diffeomorphism that is C^k with C^k inverse, and an invertible symmetric operator such that

Corollary

Let be such that 0 is a non-degenerate critical point. Then there exists a C^k-with-C^k-inverse diffeomorphism and an orthogonal decomposition such that, if one writes then