Morse–Palais lemma

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 ${\displaystyle (H,⟨\cdot ,\cdot ⟩)}$ be a real Hilbert space, and let ${\displaystyle U}$ be an open neighbourhood of the origin in ${\displaystyle H.}$ Let ${\displaystyle f:U\to \mathbb{R}}$ be a ${\displaystyle (k+2)}$-times continuously differentiable function with ${\displaystyle k\ge 1;}$ that is, ${\displaystyle f\in C^{k+2}(U;\mathbb{R}).}$ Assume that ${\displaystyle f(0)=0}$ and that ${\displaystyle 0}$ is a non-degenerate critical point of ${\displaystyle f;}$ that is, the second derivative ${\displaystyle D^{2}f(0)}$ defines an isomorphism of ${\displaystyle H}$ with its continuous dual space ${\displaystyle H^{\ast }}$ by $${\displaystyle H\ni x\mapsto {\mathrm{D}}^{2}f(0)(x,-)\in H^{\ast }.}$$

Then there exists a subneighbourhood ${\displaystyle V}$ of ${\displaystyle 0}$ in ${\displaystyle U,}$ a diffeomorphism ${\displaystyle \phi :V\to V}$ that is ${\displaystyle C^k}$ with ${\displaystyle C^k}$ inverse, and an invertible symmetric operator ${\displaystyle A:H\to H,}$ such that $${\displaystyle f(x)=⟨A\phi (x),\phi (x)⟩\text{ for all }x\in V.}$$