This paper represents some basic mathematic theories for G^s spaces of functions that can be used for weakened weak (W²) formulations, upon which the smoothed finite element methods (S-FEMs) and the smoothed point interpolation methods (S-PIMs) are based for solving mechanics problems. We first introduce and prove properties of G^s spaces, such as the upper boundedness and convergence of the norms, which are in contrast with H¹ spaces. We then prove the equivalence of the G^snorms and its corresponding semi-norms. These mathematic theories are important and essential for the establishment oftheoretical frame and the development of relevant numerical approaches. Finally, we apply S-FEM modelsto solve some practical problems with large deformations.