Original Title: “Smooth approximations without critical points of continuous mappings between Banach spaces, and diffeomorphic extractions of sets”
Authors: Daniel Azagra (UCM), Tadeusz Dobrowolski (Pittsburgh State University) y Miguel García Bravo (UAM-ICMAT)
Source: Advances in Mathematics 354 (2019), 106756
Date of online publication: October 1, 2019
Link: https://doi.org/10.1016/j.aim.2019.106756
The Morse-Sard theorem asserts that if f : n→ m is sufficiently smooth –meaning that f is of class Ck, with k ≥ max{1, n − m + 1}–then its set of critical values is of measure zero. A good analogue of the Morse-Sard theorem does not hold in infinite dimensions. Indeed there are C∞ functions f : l2→ whose set of critical values contain intervals. However, in infinite dimensions one can still get the following Morse-Sard type approximation result, which generalizes many of the previous results that one can find in the literature in this context: Let E, F be separable Hilbert spaces, and assume that E is infinite-dimensional. Then, for every continuous mapping f : E → F and every continuous function ε : E→ (0, ∞) there exists a C∞mapping g : E → F such that ∥f(x) − g(x)∥≤ε(x) and Dg(x) : E → F is a surjective linear operator for everyx ∈ E.
In this article, Daniel Azagra, Tadeusz Dobrowolski, Miguel García Bravo prove this result and also provide a version of it, where E can be replaced with a Banach space from a large class –including all the classical spaces with smooth norms, such as c0, lp or Lp, 1 <p< ∞, and F can be taken to be any Banach space such that there exists a bounded linear operator from E onto F. In particular, for such E and F, every continuous mapping f : E → F can be uniformly approximated by smooth open mappings.
Part of the proof provides results of independent interest that improve some known theorems about diffeomorphic extractions of closed sets from infinite-dimensional Banach spaces or Hilbert manifolds. More precisely, it can be proved that if X a closed subset of E which is locally contained in the graph of a continuous function defined on a subspace of infinite codimension in E and taking values in its orthogonal complement, U is an open subset of E, and G an open cover of E, then there exists a C∞ diffeomorphism h of E \ X onto E \ (X \ U ) which is the identity on (E \ U ) \ X and is limited by G (meaning that for every x ∈ E \ X we may find a Gx ∈ G such that both x and h(x) are in Gx). This property ensures that h can be taken to be as close to the identity as we wish.
