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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 function|smooth]] enough [[function (mathematics)|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|American]] [[mathematician]] [[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 [[Richard Palais]] and [[Stephen Smale]].

==Statement of the lemma==

Let (''H'',&nbsp;〈&nbsp;,&nbsp;〉) be a [[real number|real]] Hilbert space, and let ''U'' be an [[open set|open neighbourhood]] of 0 in ''H''. Let ''f''&nbsp;:&nbsp;''U''&nbsp;→&nbsp;'''R''' be a (''k''&nbsp;+&nbsp;2)-times continuously [[differentiable function]] with ''k''&nbsp;≥&nbsp;1, i.e. ''f''&nbsp;∈&nbsp;''C''<sup>''k''+2</sup>(''U'';&nbsp;'''R'''). Assume that ''f''(0)&nbsp;=&nbsp;0 and that 0 is a non-degenerate [[critical point (mathematics)|critical point]] of ''f'', i.e. the second derivative D<sup>2</sup>''f''(0) defines an [[isomorphism]] of ''H'' with its [[continuous dual space]] ''H''<sup>∗</sup> by

:<math>H \ni x \mapsto \mathrm{D}^{2} f(0) ( x, - ) \in H^{*}. \, </math>

Then there exists a subneighbourhood ''V'' of 0 in ''U'', a [[diffeomorphism]] ''φ''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' that is ''C''<sup>''k''</sup> with ''C''<sup>''k''</sup> inverse, and an [[invertible function|invertible]] [[symmetric operator]] ''A''&nbsp;:&nbsp;''H''&nbsp;→&nbsp;''H'', such that

:<math>f(x) = \langle A \varphi(x), \varphi(x) \rangle</math>

for all ''x''&nbsp;∈&nbsp;''V''.

==Corollary==

Let ''f''&nbsp;:&nbsp;''U''&nbsp;→&nbsp;'''R''' be ''C''<sup>''k''+2</sup> such that 0 is a non-degenerate critical point. Then there exists a ''C''<sup>''k''</sup>-with-''C''<sup>''k''</sup>-inverse diffeomorphism ''ψ''&nbsp;:&nbsp;''V''&nbsp;→&nbsp;''V'' and an orthogonal decomposition

:<math>H = G \oplus G^{\perp},</math>

such that, if one writes

:<math>\psi (x) = y + z \mbox{ with } y \in G, z \in G^{\perp},</math>

then

:<math>f (\psi(x)) = \langle y, y \rangle - \langle z, z \rangle</math>

for all ''x''&nbsp;∈&nbsp;''V''.

==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]]
[[Category:Hilbert space]]
[[Category:Lemmas]]