An extension theorem and the C1 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 establishes the necessary and sufficient conditions for a C-1 equivalence to the standard form) and a C-1 splitting lemma in Hilbert space, for a C-2 singular germ. This improves the known version that only guarantees, under the same hypothesis, a C-0 equivalence. (C) Academie des sciences/Editions scientifiques et medicales Elsevier SAS.