Functional analysis
Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the orthonormal basis. Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to formula_1. Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space. One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace. Many special cases of this invariant subspace problem have already been proven.
Projective Hilbert space
In mathematics and the foundations of quantum mechanics, the projective Hilbert space formula_1 of a complex Hilbert space formula_2 is the set of equivalence classes of vectors formula_3 in formula_2, with formula_5, for the relation formula_6 given by