Factorization of weakly continuous differentiable mappings

Given real Banach spaces X and Y, let C _wbu¹(X, Y) be the space, introduced by R.M. Aron and J.B. Prolla, of C ¹ mappings from X into Y such that the mappings and their derivatives are weakly uniformly continuous on bounded sets. We show that f ∈ C _wbu¹(X, Y) if and only if f may be written in the form f = g ∘ S, where the intermediate space is normed, S is a precompact operator, and g is a Gateaux differentiable mapping with some additional properties.