An extension theorem and the C-1 splitting lemma in Hilbert space

We prove an extension theorem of Whitney type for a 1-jet on a Banach space Y of a certain kind to a C-1 function on X x Y, where X is a Hilbert space . With this result, we prove a C-1 parameterized Morse lemma (which establi shes the necessary and sufficient conditions for a C-1 equivalence to the s tandard form) and a C-1 splitting lemma in Hilbert space, for a C-2 singula r germ. This improves the known version that only guarantees, under the sam e hypothesis, a C-0 equivalence. (C) Academie des sciences/Editions scienti fiques et medicales Elsevier SAS.