The range of the gradient of a Lipschitz \(C^1\)-smooth bump in infinite dimensions
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 \(R(\nabla b)= \overline\Omega\).
scientific article; zbMATH DE number 1874396