The Morse lemma on Banach spaces

Abstract

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