On the semigroup of bounded C1-mappings

Abstract

Let E be a real Banach space. If f: E→E is (Fréchet-) differentiable at every point of E, the derivative of f at x is denoted by f'(x), which is an element of the Banach algebra ℒ=ℒ(E) of all linear continuous mappings of E into itself with the usual upper bound norm, and, if we put , we have .