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Separable determination of integrability and minimality of the Clarke subdifferential mapping. (English) Zbl 0937.49009

Frequently it is pointed out that the Clarke subdifferential of a Lipschitz function is too large to conclude about the structure of the associated function. In this manner, some years ago the authors presented large classes of functions (e.g., essentially smooth functions) for which their subdifferentials are well-behaved, i.e., they are \(D\)-representable (representable by Gâteaux-gradients which exist almost everywhere) and integrable.
In the present paper the authors show that the study of \(D\)-representability and integrability in Banach spaces can be reduced on the study of these properties in separable Banach spaces. The first main theorem points out that the Clarke subdifferential mapping \(\partial f\) of a Lipschitz function \(f\) is minimal (weak*) cusco iff there exists a rich family of closed separable subspaces \(Y\) such that \(\partial (f|_Y)\) is minimal (weak*) cusco. (We should remark that this minimality property is closely connected with \(D\)-representability.) In the second main theorem the authors show that \(\partial f\) is integrable iff there exists a rich family of closed separable subspaces \(Y\) such that \(\partial (f|_Y)\) is integrable.
The proofs of both theorems base on the nice result that for each Lipschitz function the family of those closed separable subspaces \(Y\) which ensure the reductions \[ \begin{alignedat}{2} f^0(x,y) &= (f|_Y)^0(x,y) &&\quad \forall x,y\in Y,\\ \partial f(x)|_{Y^*} &= \partial(f|_Y)(x) &&\quad \forall x\in Y \end{alignedat} \] is rich.

MSC:

49J52 Nonsmooth analysis
46G05 Derivatives of functions in infinite-dimensional spaces
49J50 Fréchet and Gateaux differentiability in optimization
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