Homomorphisms between algebras of differentiable functions in infinite dimensions (Q921465)

Let E and F be two real Banach spaces. For \(n=0,1,...,\infty\), let \(C^ n_{wub}(E;F)\) be the space of n-times continuously differentiable functions f: \(E\to F\) such that, for each integer \(j\leq n\) and each \(x\in E\), both the jth derivative mapping \(f^ j: E\to P(^ jE;F)\) and the polynomial \(f^ j(x)\) are weakly uniformly continuous on bounded subsets of E. This paper studies the characterization of the homomorphisms of the type A: \(C^ n_{wub}(E;R)\to C^ m_{wub}(F;R)\) in terms of mappings g: F\({}''\to E''\) which are differentiable when the biduals \(E''\) and \(F''\) are endowed with their \(bw^*\) topologies. The authors prove that every such homomorphism is automatically continuous when the spaces \(C^ n_{wub}\) are given their natural topology.