  Let Pi be a unitary representation of GL_2(Q_p), topologically of finite
length. We describe the sub-representation Pi^{an} made of its locally analytic
vectors, and its filtration by radius of analyticity, in terms of the phi-Gamma
module attached to Pi via the p-adic local Langlands correspondence, and we
deduce that the universal completion of Pi^{an} is Pi itself.
