Smooth approximations without critical points of continuous mappings between Banach spaces, and diffeomorphic extractions of sets

2018/11/19 by Azagra, Daniel, Dobrowolski, Tadeusz, García-Bravo, Miguel
#Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1811.07587

Abstract

Let E, F be separable Hilbert spaces, and assume that E is infinite-dimensional. We show that 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 every x∈ E. We also provide a version of this result where E can be replaced with a Banach space from a large class (including all the classical spaces with smooth norms, such as c₀, ℓ_p or L^p, 1

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