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 $(H,⟨\cdot ,\cdot ⟩)$ 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 $k\ge 1;$ 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 $\phi :V\to V$ that is $C^k$ with $C^k$ inverse, and an invertible symmetric operator $A:H\to H,$ such that $$f(x)=⟨A\phi (x),\phi (x)⟩\text{ for all }x\in V.$$

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 $$\psi (x)=y+z\text{ with }y\in G,z\in G^{\perp },$$ then $$f(\psi (x))=⟨y,y⟩-⟨z,z⟩\text{ for all }x\in V.$$

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..