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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.
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| 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]].
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| ==Statement of the lemma==
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| Let (''H'', 〈 , 〉) be a [[real number|real]] Hilbert space, and let ''U'' be an [[open set|open neighbourhood]] of 0 in ''H''. Let ''f'' : ''U'' → '''R''' be a (''k'' + 2)-times continuously [[differentiable function]] with ''k'' ≥ 1, i.e. ''f'' ∈ ''C''<sup>''k''+2</sup>(''U''; '''R'''). Assume that ''f''(0) = 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
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| :<math>H \ni x \mapsto \mathrm{D}^{2} f(0) ( x, - ) \in H^{*}. \, </math> | |
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| Then there exists a subneighbourhood ''V'' of 0 in ''U'', a [[diffeomorphism]] ''φ'' : ''V'' → ''V'' that is ''C''<sup>''k''</sup> with ''C''<sup>''k''</sup> inverse, and an [[invertible function|invertible]] [[symmetric operator]] ''A'' : ''H'' → ''H'', such that
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| :<math>f(x) = \langle A \varphi(x), \varphi(x) \rangle</math>
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| for all ''x'' ∈ ''V''.
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| ==Corollary==
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| Let ''f'' : ''U'' → '''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'' : ''V'' → ''V'' and an orthogonal decomposition
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| :<math>H = G \oplus G^{\perp},</math>
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| such that, if one writes
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| :<math>\psi (x) = y + z \mbox{ with } y \in G, z \in G^{\perp},</math>
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| then
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| :<math>f (\psi(x)) = \langle y, y \rangle - \langle z, z \rangle</math>
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| for all ''x'' ∈ ''V''.
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| ==References==
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| * {{cite book | last=Lang | first=Serge | title=Differential manifolds | publisher=Addison–Wesley Publishing Co., Inc. | location=Reading, Mass.–London–Don Mills, Ont. | year=1972 }}
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| {{DEFAULTSORT:Morse-Palais lemma}}
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| [[Category:Calculus of variations]]
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| [[Category:Hilbert space]]
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| [[Category:Lemmas]]
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