The proximal normal formula in Banach space

Approximation by proximal normals to the Clarke generalized subdifferential for a distance function generated by a nonempty closed set and the normal cone to the set generated by the proximal normals are important tools in nonsmooth analysis. We give simple general versions of such formulae in infinite dimensional Banach spaces which satisfy different geometrical conditions. Our first class, of spaces with uniformly Gâteaux differentiable norm includes the Hilbert space case and the formulae is attained through dense subsets. Our second class, of reflexive Kadec smooth spaces is the most general for which such formulae can be obtained for all nonempty closed sets in the space. Our technique also allows us to establish the existence of solutions for a class of optimization problems substantially extending similar work of Ekeland and Lebourg.