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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 Gateaux differentiable norm includes the Hilbert space case and the formulae isdoi:10.2307/2000915 fatcat:ftcaxaprvfc6nccj7x5fmbib2u