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In [[Mathematics|mathematics]], the '''Morse–Palais lemma''' is a result in the [[Calculus of variations|calculus of variations]] and theory of Hilbert spaces. Roughly speaking, it states that a smooth enough [[Function (mathematics)|function]] near a critical point can be expressed as a [[Quadratic form|quadratic form]] after a suitable change of coordinates.
The Morse–Palais lemma was originally proved in the finite-dimensional case by the {{wipe|United States}} [[Biography:Mathematician|mathematician]] [[Biography:Marston Morse|Marston Morse]], using the [[Gram–Schmidt process|Gram–Schmidt orthogonalization process]]. This result plays a crucial role in [[Morse theory]].  The generalization to Hilbert spaces is due to [[Biography:Richard Palais|Richard Palais]] and [[Biography:Stephen Smale|Stephen Smale]].

== Statement of the lemma ==

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

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

== Corollary ==

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

== See also ==

* {{annotated link|Fréchet derivative}}

== References ==

* {{cite book|last=Lang|first=Serge|title=Differential manifolds|publisher=Addison–Wesley Publishing Co., Inc.|location=Reading, Mass.&ndash;London&ndash;Don Mills, Ont.|year=1972}}

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[[Category:Calculus of variations]]

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