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.
Contents
Statement of the lemma
Let
(
H
,
⟨
⋅
,
⋅
⟩
)
{\displaystyle (H,\langle \cdot ,\cdot \rangle )}
be a real Hilbert space, and let
U
{\displaystyle U}
be an open neighbourhood of the origin in
H
.
{\displaystyle H.}
Let
f
:
U
→
R
{\displaystyle f:U\to \mathbb {R} }
be a
(
k
+
2
)
{\displaystyle (k+2)}
-times continuously differentiable function with
k
≥
1
;
{\displaystyle k\geq 1;}
that is,
f
∈
C
k
+
2
(
U
;
R
)
.
{\displaystyle f\in C^{k+2}(U;\mathbb {R} ).}
Assume that
f
(
0
)
=
0
{\displaystyle f(0)=0}
and that
0
{\displaystyle 0}
is a non-degenerate critical point of
f
;
{\displaystyle f;}
that is, the second derivative
D
2
f
(
0
)
{\displaystyle D^{2}f(0)}
defines an isomorphism of
H
{\displaystyle H}
with its continuous dual space
H
∗
{\displaystyle H^{*}}
by
H
∋
x
↦
D
2
f
(
0
)
(
x
,
−
)
∈
H
∗
.
{\displaystyle H\ni x\mapsto \mathrm {D} ^{2}f(0)(x,-)\in H^{*}.}
Then there exists a subneighbourhood
V
{\displaystyle V}
of
0
{\displaystyle 0}
in
U
,
{\displaystyle U,}
a diffeomorphism
φ
:
V
→
V
{\displaystyle \varphi :V\to V}
that is
C
k
{\displaystyle C^{k}}
with
C
k
{\displaystyle C^{k}}
inverse, and an invertible symmetric operator
A
:
H
→
H
,
{\displaystyle A:H\to H,}
such that
f
(
x
)
=
⟨
A
φ
(
x
)
,
φ
(
x
)
⟩
for all
x
∈
V
.
{\displaystyle f(x)=\langle A\varphi (x),\varphi (x)\rangle \quad {\text{ for all }}x\in V.}
Corollary
Let
f
:
U
→
R
{\displaystyle f:U\to \mathbb {R} }
be
f
∈
C
k
+
2
{\displaystyle f\in C^{k+2}}
such that
0
{\displaystyle 0}
is a non-degenerate critical point. Then there exists a
C
k
{\displaystyle C^{k}}
-with-
C
k
{\displaystyle C^{k}}
-inverse diffeomorphism
ψ
:
V
→
V
{\displaystyle \psi :V\to V}