Schubert calculus and equivariant cohomology of grassmannians

We give a description of equivariant cohomology of grassmannians that places the theory into a general framework for cohomology theories of grassmannians. As a result we obtain a formalism for equivariant cohomology where the basic results of equivariant Schubert calculus, the basis theorem, Pieri's formula and Giambelli's formula can be obtained from the corresponding results of the general framework by a change of basis.In order to show that our formalism reflects the geometry of grassmannians we relate our theory to the treatment of equivariant cohomology of grassmannians by A. Knutson and T. Tao.