  In this paper we study the conforming Galerkin approximation of the problem:
find u $\in$ U such that a(u, v) = <L, v> for all v $\in$ V, where U and V are
Hilbert or Banach spaces, a is a continuous bilinear or sesquilinear form and L
$\in$ V' a given data. The approximate solution is sought in a finite
dimensional subspace of U, and test functions are taken in a finite dimensional
subspace of V. We provide a necessary and sufficient condition on the form a
for convergence of the Galerkin approximation, which is also equivalent to
convergence of the Galerkin approximation for the adjoint problem. We also
characterize the fact that U has a finite dimensional Schauder decomposition in
terms of properties related to the Galerkin approximation. In the case of
Hilbert spaces, we prove that the only bilinear or sesquilinear forms for which
any Galerkin approximation converges (this property is called the universal
Galerkin property) are the essentially coercive forms. In this case, a
generalization of the Aubin-Nitsche Theorem leads to optimal a priori estimates
in terms of regularity properties of the right-hand side L, as shown by several
applications. Finally, a section entitled "Supplement" provides some
consequences of our results for the approximation of saddle point problems.
