Completeness of Inner Product Spaces and G. Cattaneo

In the Eighties G. Cattaneo contributed to Hilbert space quantum mechanics models using the family of splitting subspaces of an inner product space in order to find completeness criteria. In this paper we show how that theory was developed in the last 25 years into a rich theory which has a close connection to quantum information. G. Cattaneo also posed an open problem which was solved only this year.