The authors are interested in the approximation of the linear elasticity problem with the tangential-displacement normal-normal-stress finite element method. In this case, the tangential component of the displacement vector and the normal-normal component of the stress are sought in a re-formulation of the linear elasticity problem. Although it is well known that this method is convergent with the optimal order and robust with respect to shear and volume locking, still an analysis with respect to the natural norms of the arising spaces was missing. Here, the mathematical theory for the continuous problem is presented using the space \(\mathbf{H}(\mathbf{curl})\) for the displacement and the space \(\underline{\mathbf H}(\operatorname{div} \mathbf{div})\) for the stresses. The trace operators for the normal-normal stress are provided. The finite element problem is shown to be stable with respect to the \(\mathbf{H}(\mathbf{curl})\) norm and a discrete \(\mathbf{H}(\operatorname{div} \mathbf{div})\) norm. An a priori error estimate of optimal order with respect to these norms is shown.