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 United States 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 [math]\displaystyle{ (H, \langle \cdot ,\cdot \rangle) }[/math] be a real Hilbert space, and let [math]\displaystyle{ U }[/math] be an open neighbourhood of the origin in [math]\displaystyle{ H. }[/math] Let [math]\displaystyle{ f : U \to \R }[/math] be a [math]\displaystyle{ (k+2) }[/math]-times continuously differentiable function with [math]\displaystyle{ k \geq 1; }[/math] that is, [math]\displaystyle{ f \in C^{k+2}(U; \R). }[/math] Assume that [math]\displaystyle{ f(0) = 0 }[/math] and that [math]\displaystyle{ 0 }[/math] is a non-degenerate critical point of [math]\displaystyle{ f; }[/math] that is, the second derivative [math]\displaystyle{ D^2 f(0) }[/math] defines an isomorphism of [math]\displaystyle{ H }[/math] with its continuous dual space [math]\displaystyle{ H^* }[/math] by [math]\displaystyle{ H \ni x \mapsto \mathrm{D}^2 f(0) (x, -) \in H^*. }[/math]

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

Corollary

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

See also

• Fréchet derivative – Derivative defined on normed spaces

References

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