In this paper, we present a convergence analysis for the space discretization of hyperbolic evolution problems in mixed form. The results of previous work on parabolic equations are extended to this situation, showing the relationships between the approximation of the underlying eigenvalue problem and the space discretization of the evolution problem. The theory is applied to the finite element approximation of the wave equation in mixed form and to the Maxwell equations. Some numerical results confirm the theory and make clear how a scheme that does not provide a spectrally correct discretization can perform badly when applied to the approximation of the evolution problem. Copyright © 2012 John Wiley & Sons, Ltd.