  In \cite[\S1.3]{Br2}, some unitary representations of ${\rm
GL}_2(\mathbf{Q}_p)$ on $p$-adic Banach spaces are associated to 2-dimensional
irreducible crystalline representations of ${\rm
Gal}(\bar{\mathbf{Q}}_p)/\mathbf{Q}_p)$. Some conjectures are formulated
concerning those Banach spaces (non triviality, topological irreducibility,
admissibility). We prove those conjectures by reinterpreting those Banach as
spaces of functions of a certain type on $\mathbf{Q}_p$, and then by using the
theory of $(\phi,\Gamma)$-modules associated to the corresponding crystalline
representations.
