Abstract: I will try to explain some ideas that enter in the proof of the following result, proved in joint work with Pierre Colmez: if $\Pi$ is a unitary admissible $p$-adic Banach space representation of $\mathrm{GL}_2(\mathbb{Q}_p)$ which is residually of finite length, then the universal unitary completion of the space of its locally analytic vectors is $\Pi$ itself. This crucially uses the p-adic Langlands correspondence and the theory of $\varphi-\Gamma$ modules.