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 C1 function f : E → Rd, every open set U with Cf := {x ∈ E : Df(x) is not surjective} ⊂ U and every continuous function ε : E → (0, ∞) there exists a C1 mapping ϕ : E → Rd such that ||f(x) −ϕ(x)|| ≤ ε(x)for every x ∈ E, f =ϕoutside U and ϕhas 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.