Concerning differentiability properties of locally Lipschitz functions (Q2776678)

Let \(\psi\) be a locally Lipschitz function on a Banach space \(X\) and \(\partial\psi\) be the Clarke subdifferential mapping. If \(\psi\) is Gâteaux (Fréchet) differentiable on a dense subset \(D\subset X\) then for the sets \((CSC\psi')(x)\) given by NEWLINE\[NEWLINE(CSC\psi')(x)= \bigcap\{\overline{\text{co}}^{w^*}\psi'(V\cap D): V\text{ is an open neighborhood of }x\}NEWLINE\]NEWLINE it holds NEWLINE\[NEWLINE(CSC\psi')(x)\subset \partial\psi(x)\qquad\forall x\in X.NEWLINE\]NEWLINE In 1990, Preiss characterized Banach spaces (Asplund spaces and spaces with differentiable norm) in which equality holds, i.e., spaces for which the Clarke subdifferential can be formed in the classical manner by the Gâteaux (Fréchet) derivatives.NEWLINENEWLINENEWLINEThese results are generalized in the present paper. The author points out that this property can be extended to a Banach space \(Y\) if there exists a continuous linear mapping \(T: X\to Y\) with \(\overline{T(X)}= Y\).NEWLINENEWLINENEWLINEIn the second part of the paper it is shown that the locally Lipschitz function \(\psi\) is uniformly upper Dini subdifferentiable on a dense \(G_\delta\) subset of \(X\) if this function is assumed to be Fréchet differentiable on a dense subset. Hence, Asplund spaces can be characterized also by means of this property.NEWLINENEWLINEFor the entire collection see [Zbl 0974.00048].