This means that if the coefficients \( C_{p\lambda} \) belong to a unitary or orthogonal trasformation (using the Dirac bra-ket notation) $$ \vert p\rangle = \sum_{\lambda} C_{p\lambda}\vert\lambda\rangle, $$ orthogonality is preserved, that is \( \langle \alpha \vert \beta\rangle = \delta_{\alpha\beta} \) and \( \langle p \vert q\rangle = \delta_{pq} \).

This propertry is extremely useful when we build up a basis of many-body Stater determinant based states.

Note also that although a basis \( \vert \alpha\rangle \) contains an infinity of states, for practical calculations we have always to make some truncations.