A characterization of smooth functions defined on a Banach space

A sufficient condition for a function defined on a Banach space to be ${C^k}$ is given. This enables us to characterize the ${C^\infty }$ 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 ${C^\infty }$ function from ${{\mathbf{R}}^m}$ into the domain of the function is ${C^\infty }$.