 A formal definition follows.The Hilbert space formula_67 is decomposed into three orthogonal subspaces

In the literature, these orthogonal subspaces are usually formulated in terms of the set formula_69 of orthogonal basis vectors for the Hilbert space formula_67.This set of basis vectors is divided up into subsets formula_71 and formula_72, such that

is the linear span of the basis vectors in the accept set.