Approximate Morse-Sard type results for non-separable Banach spaces

Abstract

For the Banach spaces E=c0(Γ),ℓp(Γ), where Γ is an arbitrary infinite set and 1<p<∞, we show that for every (non-zero) quotient F of E, every continuous function f:E→F can be uniformly approximated by smooth functions with no critical points, that is for every continuous function ε:E→(0,∞), there exists a Ck smooth function g:E→F such that g′(x) is surjective and ‖f(x)−g(x)‖≤ε(x) for all x∈E (k=∞ if E=c0(Γ) or p is an even integer, k=p−1 if p is an odd integer and k=[p] otherwise). Moreover, for a wide class of (not necessarily separable) infinite dimensional Banach spaces E and a suitable class of quotients F of E, every continuous function f:E→F can be uniformly approximated by Ck smooth functions with no critical points (k depending on the properties of the smoothness of the space E). In particular, for every Banach space E with Ck smooth partitions of unity (k∈N∪{∞}) and an infinite dimensional separable complemented subspace with a Ck smooth and LUR norm, we show that every continuous function f:E→Rn (n∈N) can be uniformly approximated by C1 smooth function with no critical points.