Homomorphisms between algebras of differentiable functions in infinite dimensions

Abstract

Let E and F be two real Banach spaces. For n = 0, 1, ...,1, let Cnw ub(E; F) be the space of n-times continuously differentiable functions f: E ! F such that, for each integer j _ n and each x 2 E, both the jth derivative mapping fj : E ! P(jE; F) and the polynomial fj(x) are weakly uniformly continuous on bounded subsets of E. This paper studies the characterization of the homomorphisms of the type A: Cnw ub(E;R) ! Cm wub(F;R) in terms of mappings g: F00 ! E00 which are differentiable when the biduals E00 and F00 are endowed with their bw_ topologies. The authors prove that every such homomorphism is automatically continuous when the spaces Cnw ub are given their natural topology.