Differentiable mappings between weighted restricted products
  In this paper, we introduce restricted products for families of locally convex spaces and formulate criteria ensuring that mappings into such products are continuous or smooth. As a special case, can define restricted products of weighted function spaces and obtain results concerning continuity and differentiability properties of natural non-linear mappings between such spaces. These concepts and results are the basis for the study of weighted vector fields on Riemannian manifolds in a subsequent work (see [B. Walter, "Weighted diffeomorphism groups of Riemannian manifolds", arXiv: 1601.02834]), which serve as modelling spaces for suitable infinite-dimensional Lie groups of diffeomorphisms. 
Boris Walter
math.FA