The content of Part I was the construction of the fundamental metric tensor; in Part II the conditions which make possible the probability interpretation for elements lying in the same coherent sector were discussed. The subject of this part (III) is the space of states as a whole. The existence of superselection rules allows the fundamental tensors formerly introduced as identity matrix in the subspace of the physical states to be given up. The mathematical form in which symmetry groups appear is shown by an example. Finally, a possible generalization for the dual element of an element is given. (auth)