An approximation scheme for defining the Conley index of isolated critical points

In the present paper, we study isolated critical points of functionals dened on a real separable Hilbert space $H$ and satisfying the $H$-properness condition. We introduce the notion of Conley index of an isolated critical point and prove that it is homotopy invariant. The scheme suggested here for defining the Conley index is based on the application of finite-dimensional Conley index theory to finite-dimensional restrictions of the functional to be studied.