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 $f:U\to \mathbb {R}$ be a $(k+2)$-times continuously differentiable function with $k\geq 1;$ that is, $f\in C^{k+2}(U;\mathbb {R} ).$ 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 $f:U\to \mathbb {R}$ 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 $$f(\psi (x))=\langle y,y\rangle -\langle z,z\rangle \quad {\text{ for all }}x\in V.$$

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.