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 $$(H, \langle \cdot ,\cdot \rangle)$$ be a real Hilbert space, and let $$U$$ be an open neighbourhood of the origin in $$H.$$ Let $$f : U \to \R$$ be a $$(k+2)$$-times continuously differentiable function with $$k \geq 1;$$ that is, $$f \in C^{k+2}(U; \R).$$ 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 $$H \ni x \mapsto \mathrm{D}^2 f(0) (x, -) \in H^*.$$

Then there exists a subneighbourhood $$V$$ of $$0$$ in $$U,$$ a diffeomorphism $$\varphi : V \to V$$ that is $$C^k$$ with $$C^k$$ inverse, and an invertible symmetric operator $$A : H \to H,$$ such that $$f(x) = \langle A \varphi(x), \varphi(x) \rangle \quad \text{ for all } x \in V.$$

Corollary
Let $$f : U \to \R$$ be $$f \in C^{k+2}$$ such that $$0$$ is a non-degenerate critical point. Then there exists a $$C^k$$-with-$$C^k$$-inverse diffeomorphism $$\psi : V \to V$$ and an orthogonal decomposition $$H = G \oplus G^{\perp},$$ such that, if one writes $$\psi (x) = y + z \quad \mbox{ with } y \in G, z \in G^{\perp},$$ then $$f (\psi(x)) = \langle y, y \rangle - \langle z, z \rangle \quad \text{ for all } x \in V.$$