Morse–Palais lemma

Let ${\displaystyle (H,\langle \cdot ,\cdot \rangle )}$ 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 ${\displaystyle k\geq 1;}$ 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 ${\displaystyle H^{*}}$ by

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

Then there exists a subneighbourhood ${\displaystyle V}$ of ${\displaystyle 0}$ in ${\displaystyle U,}$ a diffeomorphism ${\displaystyle \varphi :V\to V}$ that is ${\displaystyle C^{k}}$ with ${\displaystyle C^{k}}$ inverse, and an invertible symmetric operator ${\displaystyle A:H\to H,}$ such that

$${\displaystyle f(x)=\langle A\varphi (x),\varphi (x)\rangle \quad {\text{ for all }}x\in V.}$$

Let ${\displaystyle f:U\to \mathbb {R} }$ be ${\displaystyle f\in C^{k+2}}$ such that ${\displaystyle 0}$ is a non-degenerate critical point. Then there exists a ${\displaystyle C^{k}}$-with-${\displaystyle C^{k}}$-inverse diffeomorphism ${\displaystyle \psi :V\to V}$ and an orthogonal decomposition

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

such that, if one writes

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

then

$${\displaystyle f(\psi (x))=\langle y,y\rangle -\langle z,z\rangle \quad {\text{ for all }}x\in V.}$$

• Fréchet derivative – Derivative defined on normed spaces

• Lang, Serge (1972). Differential manifolds. Reading, Mass.–London–Don Mills, Ont.: Addison–Wesley Publishing Co., Inc.