The Morse Lemma on Banach Spaces

Let /: U-*R be a C3 map of an open subset U of a Banach space E. Letp £ U be a critical point of/ (df¡,=0). If £ is a conjugate space (E=F*) we define what it means for/» to be nondegenerate. In this case there is a diffeomorphism y of a neighborhood of p with a neighborhood of 0 E E, y(p)=0 with foy->(x) = id%(x,x)+f(p). In this paper we define a notion of nondegeneracy for critical points of smooth real valued functions defined on open subsets of a Banach space E which is a dual space, and we shall prove the Morse lemma with our definition. Earlier notions of nondegeneracy which implied the Morse lemma are stronger than ours and in particular had the unfortunate consequence of implying that £«¿£*. See Palais [3] and [4] for the case E= Hilbert space and EmE* respectively. Definition. Let £0 be a Banach space and let E=E* be the dual space of E0. Denote by ( , ):ExE0 the bilinear pairing of E0 and E given by (fx)=fix). In the paragraphs below we shall not be too careful about the order of the terms in < , ); e.g., (x,f) instead of (/, x) and no confusion should arise from this. A continuous linear map A e L(£, E0) is said to be symmetric if (y, Ax) = (x, Ay) for all x, y £ E. We note this by writing A e LsiE, E0). The proof of the following is standard. Lemma 1. If A e LS(E, E0) is injective then A(E) is dense in E0. Definition. Let ¡7c: £ be an open subset with/: U-+R C2-differentiable. We say that/ is weak (*) smooth if for each fixed / e U and he E the linear functional on E given by k~>d2ft(h, k) is continuous in the weak (*) topology of E induced by E0. If E is reflexive every C2-map on U is weak (*) smooth. Lemma 2. Let t e U be fixed with f: U-+R C2 and weak (*) smooth. Then d% induces a map AteLs(E,Eo) with k(At(h)) = (k, At(h))= d2f(h, k). Received by the editors February 16, 1971. AMS 1970 subject classifications. Primary 49F15, 58E05; Secondary 58C15. © American Mathematical Society 1972