In this paper we consider the Maxwell resolvent operator and its finite element approximation. In this framework it is natural the use of the edge element spaces and to impose the divergence constraint in a weak sense with the introduction of a Lagrange multiplier, following an idea by Kikuchi. We shall review some of the known properties for edge element approximations and prove some new result. In particular we shall prove a uniform convergence in the L2 norm for the sequence of discrete operators. These results, together with a general theory introduced by Brezzi, Rappaz and Raviart, allow an immediate proof of convergence for the finite element approximation of the time-harmonic Maxwell system.