In this paper, a variationally consistent contact formulation is considered and we provide an abstract framework for the a priori error analysis in the special case of frictionless contact and small deformations. Special emphasis is put on quadratic mortar finite element methods. It is shown that under quite weak assumptions on the Lagrange multiplier space $$mathcalO (h^t-1), 2 < t < frac52$$ , a priori results in the H 1-norm for the error in the displacement and in the H −1/2-norm for the error in the surface traction can be established provided that the solution is regular enough. We discuss several choices of Lagrange multipliers ranging from the standard lowest order conforming finite elements to locally defined biorthogonal basis functions. The crucial property for the analysis is that the basis functions have a local positive mean value. Numerical results are exemplarily presented for one particular choice of biorthogonal (i.e. dual) basis functions and also comprise the case of finite deformation contact.