[en] The Generator Coordinate approach is used to construct a collective subspace of the full many-body Hilbert Space. The construction has a purely kinematical character, given the a priori (e.g., phenomenological) specification of a set of generator many-body states. It is based on the analysis of the properties of the overlape of the generator states and on the use of the standard spaces of square integrable functions of quantum mechanics. Some well konwn misbehaviours of the Generator Coordinate weight functions are clearly identified as of kinematical origin. A standard representation in the collective phase space is introduced which eliminates them. It is also indicated kow appropriate collective dynamical variables can be defined a posteriori. Hilbert-Schmidt overlap kernels and the Gaussian Overlap Approximation are treated as special examples