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Approximation by smooth functions with no critical points on separable Banach spaces

Azagra Rueda, Daniel and Jimenez Sevilla, Maria del Mar (2007) Approximation by smooth functions with no critical points on separable Banach spaces. Journal of Functional Analysis , 242 (1). pp. 1-36. ISSN 0022-1236

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Abstract

We characterize the class of separable Banach spaces X such that for every continuous function f : X -> Rand for every continuous function epsilon : X -> (0, +infinity) there exists a C-1 smooth function g: X -> R for which vertical bar f(x) - g(x)vertical bar <= epsilon(x) and g'(x) not equal 0 for all x is an element of X (that is, g has no critical points), as those infinite-dimensional Banach spaces X with separable dual X*. We also state sufficient conditions on a separable Banach space so that the function g can be taken to be of class C-p, for p = 1, 2,..., +infinity. In particular, we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces l(p)(N) and L-p(R-n). Some important consequences of the above results are (1) the existence of a non-linear Hahn-Banach theorem and the smooth approximation of closed sets, on the classes of spaces considered above; and (2) versions of all these results for a wide class of infinite-dimensional Banach manifolds.

Item Type:Article
Uncontrolled Keywords:Rolles theorem; Singular maps; Hilbert-space; Image size; Manifolds; Morse-Sard theorem; smooth bump functions; critical points; approximation by smooth functions; Sard functions
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:14770
Deposited On:17 Apr 2012 11:52
Last Modified:06 Feb 2014 10:08

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