Extraction of critical points of smooth functions on Banach spaces

Let E be an infinite-dimensional separable Hilbert space. We show that for every C-1 function f : E -> R-d, every open set U with C-f = {x is an element of E : Df(x) is not surjective} subset of U and every continuous function epsilon -> E (0, infinity) there exists a C-1 mapping phi : E -> R-d such that parallel to f(x) -phi(x)parallel to <= epsilon(x) for every x is an element of E, f = phi outside U and phi has no critical points (C phi = empty set). This result can be generalized to the case where E = c(0) or E = l(p), 1 < p < infinity. In the case E = c(0) it is also possible to get that parallel to Df(x) - D phi(x)parallel to <=epsilon(x) for every x is an element of E. (C) 2019 Elsevier Inc. All rights reserved.