Automatic Continuity and Weighted Composition Operators between Spaces of Vector-valued Differentiable Functions

Let E and F be Banach spaces. It is proved that if Ω and Ω are open subsets of R and R , respectively, and T is a linear biseparating map between two spaces of differentiable functions A(Ω, E) and A(Ω, F ), then p = q, n = m, and there exist a diffeomorphism h of class C from Ω onto Ω, and a map J : Ω → L(E, F ) of class s−C such that for every y ∈ Ω and every f ∈ A(Ω, E), (Tf)(y) = (Jy)(f(h(y))). In particular E and F are isomorphic as Banach spaces and, as a consequence, all linear biseparating maps are continuous for usual topologies in the spaces of differentiable functions.