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.
Geometric quantization
The next step in the process of geometric quantization is the choice of a polarization. A polarization is a choice at each point in formula_6 a Lagrangian subspace of the complexified tangent space of formula_6. The subspaces should form an integrable distribution, meaning that the commutator of two vector fields lying in the subspace at each point should also lie in the vector field at each point. The "quantum" (as opposed to prequantum) Hilbert space is the space of sections of formula_14 that are covariantly constant in the direction of the polarization. The idea is that in the quantum Hilbert space, the sections should be functions of only formula_27 variables on the formula_28-dimensional classical phase space.