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.
be a real Hilbert space, and let
be an open neighbourhood of the origin in
-times continuously differentiable function with
is a non-degenerate critical point of
that is, the second derivative
defines an isomorphism of
with its continuous dual space
Then there exists a subneighbourhood
a diffeomorphism
φ :
inverse, and an invertible symmetric operator
φ ( x ) , φ ( x ) ⟩
is a non-degenerate critical point.
Then there exists a
-inverse diffeomorphism
ψ :
and an orthogonal decomposition
such that, if one writes
ψ ( x ) = y + z
f ( ψ ( x ) ) = ⟨ y , y ⟩ − ⟨ z , z ⟩