Banach *-algebra

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 D²f(0) defines an isomorphism of H with its continuous dual space H^∗ by

$H\ni x\mapsto {\mathrm{D}}^{2}f(0)(x,-)\in H^{*}.$

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

$f(x)=⟨A\phi (x),\phi (x)⟩$

for all x ∈ V.

Corollary

Let f : U → R be C^k+2 such that 0 is a non-degenerate critical point. Then there exists a C^k-with-C^k-inverse diffeomorphism ψ : V → V and an orthogonal decomposition

$H=G\oplus G^{\perp },$

such that, if one writes

$\psi (x)=y+z\text{ with }y\in G,z\in G^{\perp },$

then

$f(\psi (x))=⟨y,y⟩-⟨z,z⟩$

for all x ∈ V.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534