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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 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'', 〈 , 〉) be a real Hilbert space, and let ''U'' be an 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''''k''+2(''U''; R). Assume that ''f''(0) = 0 and that 0 is a non-degenerate critical point of ''f'', i.e. the second derivative D2''f''(0) defines an isomorphism of ''H'' with its continuous dual space ''H''∗ by : Then there exists a subneighbourhood ''V'' of 0 in ''U'', a diffeomorphism ''φ'' : ''V'' → ''V'' that is ''C''''k'' with ''C''''k'' inverse, and an invertible symmetric operator ''A'' : ''H'' → ''H'', such that : for all ''x'' ∈ ''V''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Morse–Palais lemma」の詳細全文を読む スポンサード リンク
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