C¹ perturbations of a continuum of critical points

Given a real valued function having a nondegenerate compact manifold of critical points, some of these points survive under small C² perturbations. This is a well-known result in critical point theory. In 1986 Weinstein obtained the analogous conclusions when the perturbation is only C² and the ambient space is a finite dimensional manifold. In this work we present a complete proof for C¹ perturbations in infinite dimensional Hilbert spaces.