\indent A rigorous convergence theory for Galerkin methods for a model Helmholtz problem in $R^{d}, d=1,2,3,$ is presented. General conditions on the approximation properties of the approximation space are stated that ensure quasi-optimality of the method. As an application of the general theory, a full error analysis of the classical hp-version of the finite element method (hp-FEM) is presented where the dependence on the mesh width $h$, the approximation order $p$, and the wave number $k$ is given explicitly. In particular, it is shown that quasi-optimality is obtained under the conditions that $kh/p$ is sufficiently small and the polynomial degree $p$ is at least $O(log k)$. This result improves existing stability conditions substantially.