and so does not belong to the kinematic Hilbert space formula_242. In these cases we take a dense subset formula_261 of formula_242 (intuitively this means either any point in formula_261 is either in formula_264 or arbitrarily close to a point in formula_242) with very good convergence properties and consider its dual space formula_266 (intuitively these map elements of formula_261 onto finite complex numbers in a linear manner), then formula_268 (as formula_266 contains distributional functions). The constraint operator is then implemented on this larger dual space, which contains distributional functions, under the adjoint action on the operator. One looks for solutions on this larger space. This comes at the price that the solutions must be given a new Hilbert space inner product with respect to which they are normalizable (see article on rigged Hilbert space). In this case we have a generalized projection operator on the new space of states. We cannot use the above formula for the new inner product as it diverges, instead the new inner product is given by the simply modification of the above,