  In this paper we give an explicit description of the universal unitary
completion of certain locally Q_p-analytic representations of GL_2(F), where F
is a finite extension of Q_p (this generalizes some results of Berger-Breuil
for F=Q_p). To this aim, we make use of certain Banach spaces of C^r functions
on O_F (for r a positive real number) introduced by the author.
