  In this paper we introduce a class of Banach spaces of functions of class C^r
(where r is a positive real number) and the associated dual spaces of
distributions of order r, which turn out to be useful in p-adic Langlands
theory. We construct a Banach basis for these spaces and we give a criterion
for telling when a linear form on a space of locally Q_p-polynomial functions
extends to a distribution of order r. This generalises the classical results of
Amice-V\'elu and Vishik.
