How close are Galerkin eigenvectors to the best approximation available out of the trial subspace ? Under a variety of conditions the Galerkin method gives an approximate eigenvector that approaches asymptotically the projection of the exact eigenvector onto the trial subspace -- and this occurs more rapidly than the underlying rate of convergence of the approximate eigenvectors. Both orthogonal-Galerkin and Petrov-Galerkin methods are considered here with a special emphasis on nonselfadjoint problems. Consequences for the numerical treatment of elliptic PDEs discretized either with finite element methods or with spectral methods are discussed and an application to Krylov subspace methods for large scale matrix eigenvalue problems is presented. New lower bounds to the $sep$ of a pair of operators are developed as well.

39 pages