A characterization of smooth functions defined on a Banach space

Abstract

A sufficient condition for a function defined on a Banach space to be C k {C^k} is given. This enables us to characterize the C ∞ {C^\infty } functions from one Banach space into another Banach space as those functions that, for each positive integer m, have the property that the composition of the function with each C ∞ {C^\infty } function from R m {{\mathbf {R}}^m} into the domain of the function is C ∞ {C^\infty } .