The range of the gradient of a Lipschitz \(C^1\)-smooth bump in infinite dimensions
type: Item
instance of: scholarly article
title: The range of the gradient of a Lipschitz \(C^1\)-smooth bump in infinite dimensions
zbMATH Open document ID: 1010.46016
DOI: 10.1007/BF02784514
published in: Israel Journal of Mathematics
publication date: 2003-05-20T00:00:00Z
review text: A bump \(b:X\to\mathbb{R}\) is a continuous function \(b\) from a Banach space \(X\) to the reals which vanishes outside some bounded subset of \(X\). The gradient \(\nabla b\) is the Fréchet derivative, its values are in the adjoint space \(X^*\) and the range of \(\nabla b\) in \(X^*\) is denoted by \(R(\nabla b)\). The authors prove the following statement: Let \(X\) be an infinite dimensional Banach space with a Lipschitz \(C'\)-smooth bump. Let \(\Omega\subset X^*\) be an open connected set containing the origin and satisfying the following condition: There is a summable sequence \(a_0,a_1,\dots\) of position numbers such that every \(\eta\in \overline\Omega\) can be expressed as \(\lim_{i\to \infty} \xi_i\) for some sequence \(0=\xi_0,\xi_1,\xi_2\dots\) in \(\Omega\) such that \(\|\xi_{i+1}- \xi_i\|<a_i\), and the linear segment \(co\{\xi_i, \xi_{i+1}\}\) lies in \(\Omega\) for every \(i=1,2, \dots\). Then there exists a Lipschitz \(C'\)-smooth bump \(b:X\to[0,1]\) such that \(R(\nabla b)= \overline\Omega\).
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 49J50
zbMATH DE Number: 1874396
zbMATH Keywords: bump function
author: Jonathan M. Borwein
author: M. Fabian
author: P. D. Loewen
reviewed by: Ludvík Janoš
MaRDI profile type: Publication
cites work: James' theorem fails for starlike bodies
cites work: The failure of Rolle's theorem in infinite-dimensional Banach spaces
cites work: On the size of the sets of gradients of bump functions and starlike bodies on the Hilbert space
cites work: On smooth, nonlinear surjections of Banach spaces
cites work: The range of the gradient of a continuously differentiable bump
cites work: Q5538494
cites work: A note on norm attaining functionals
full work available at URL: https://doi.org/10.1007/bf02784514
OpenAlex ID: W2089900962
Recommended article: Range of the gradient of a smooth bump function in finite dimensions
Recommended article: The range of the gradient of a continuously differentiable bump
Recommended article: Q3545530
Recommended article: On the gradient set of Lipschitz maps
Recommended article: Q4526387
Recommended article: Properties of \(C^{1}\)-smooth functions whose gradient range has topological dimension 1
Recommended article: Uniform Lipschitz estimates in bumpy half-spaces
Recommended article: Q5391622
Recommended article: Gradient estimates for the 𝑝-Laplacian Lichnerowicz equation on smooth metric measure spaces
Recommended article: Non-Lipschitz functions with bounded gradient and related problems
description: scientific article; zbMATH DE number 1874396