Non-separable Hilbert manifolds of continuous mappings

Let X,Y be separable metrizable spaces, where X is noncompact and Y is equipped with an admissible complete metric d. We show that the space C(X,Y) of continuous maps from X into Y equipped with the uniform topology is locally homeomorphic to the Hilbert space of weight 2ℵ0 if (1) (Y,d) is an ANRU, a uniform version of ANR and (2) the diameters of components of Y is bounded away from zero. The same conclusion holds for the subspace CB(X,Y) of bounded maps if Y is a connected complete Riemannian manifold.