Extraction of critical points of smooth functions on Banach spaces

Abstract

Let E be an infinite-dimensional separable Hilbert space. We show that for every C 1 function f :for every x ∈ E, f = ϕ outside U and ϕ has no critical points (Cϕ = ∅). This result can be generalized to the case where E = c0 or E = lp, 1 < p < ∞. In the case E = c0 it is also possible to get that ||Df (x) − Dϕ(x)|| ≤ ε(x) for every x ∈ E.