Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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.