Point derivations for Lipschitz functions andClarke's generalized derivative
Volume 24 / 1997
Applicationes Mathematicae 24 (1997), 465-474
DOI: 10.4064/am-24-4-465-474
Abstract
Clarke's generalized derivative $f^0(x,v)$ is studied as a function on the Banach algebra Lip(X,d) of bounded Lipschitz functions f defined on an open subset X of a normed vector space E. For fixed $x\in X$ and fixed $v\in E$ the function $f^0(x,v)$ is continuous and sublinear in $f\in Lip(X,d)$. It is shown that all linear functionals in the support set of this continuous sublinear function satisfy Leibniz's product rule and are thus point derivations. A characterization of the support set in terms of point derivations is given.
