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Point derivations for Lipschitz functions andClarke's generalized derivative

Vladimir DemyanovDiethard Pallaschke — 1997

Applicationes Mathematicae

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 X and fixed v E the function f 0 ( x , v ) is continuous and sublinear in f L i p ( 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.

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