Consider the Galerkin discretization of an abstract variational problem where the Hilbert space V is separable.

http://en.wikipedia.org/wiki/Galerki...stract_problem

Each of the subspaces Vn is generated by the first n terms of a sequence of elements of the separable Hilbert space V. This sequence is such that each of these subspaces will be dense in V. The problem is proving that there is a subsequence of the bounded sequence of discretized solutions , that converges weakly to the solution of the variational abstract problem , then proving that the same subsequence converges strongly and finally proving that the sequence of discretized solutions converges to the solution of the variational abstract problem .