We consider the classical duality operators for convex objects suchas the polar of a convex set containing the origin, the dual norm,the Fenchel-transform of a convex function and the conjugate of aconvex cone. We give a new, sharper, unified treatment of the theoryof these operators, deriving generalized theorems of Hahn-Banach,Fenchel-Moreau and Dubovitsky-Milyutin for the conjugate of convexcones in not necessarily finite dimensional vector spaces and hencefor all the other duality operators of convex objects.