On the Duality in the Theory of Smooth Manifolds

In this paper, we discuss an important and nontrivial theorem on evaluation homomorphisms. We state this theorem as a canonical duality between the family of all smooth mappings f ∈ Hom(M,M′) of a smooth real finite-dimensional manifold M into a similar manifold M′ and the family of homomorphisms φ of the algebra C^∞ (M′) of smooth scalar-valued functions on M′ into the analogous algebra C^∞ (M) on M, φ ∈ Hom (C^∞ (M′), C^∞ (M)). This formulation possesses the maximum natural generality and, at the same time, allows it to be used in applications in the standard canonical form.