The Morse lemma on Banach spaces

Abstract:

Let $f:U \to R$ be a ${C^3}$ map of an open subset U of a Banach space E. Let $p \in U$ be a critical point of $f(d{f_p} = 0)$. If E is a conjugate space $(E = {F^ \ast })$ we define what it means for p to be nondegenerate. In this case there is a diffeomorphism $\gamma$ of a neighborhood of p with a neighborhood of $0 \in E,\gamma (p) = 0$ with \[ f \circ {\gamma ^{ - 1}}(x) = \frac {1}{2}{d^2}{f_p}(x,x) + f(p).\]