  We bound the Gelfand-Kirillov dimension of unitary Banach space
representations of $p$-adic reductive groups, whose locally analytic vectors
afford an infinitesimal character. We use the bound to study Hecke eigenspaces
in completed cohomology of Shimura curves and $p$-adic Banach space
representations of the group of units of a quarternion algebra over $\mathbb
Q_p$ appearing in the $p$-adic Jacquet-Langlands correspondence, deducing
finiteness results in favourable cases.
