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.
Basil Hiley
In 2000, Brown and Hiley showed that the Schrödinger equation can be written in a purely algebraic form that is independent of any representation in a Hilbert space. This algebraic description is formulated in terms of two operator equations. The first of these (formulated in terms of the commutator) represents an alternative form of the quantum Liouville equation, which is well known to describe the conservation of probability, the second (formulated in terms of the anticommutator), which they dubbed the "quantum phase equation", describes the conservation of energy. This algebraic description in turn gives rise to descriptions in terms of multiple vector spaces, which Brown and Hiley call "shadow phase spaces" (adopting the term "shadow" from Michał Heller). These shadow phase space descriptions include the descriptions in terms of the "x"-space of the Bohm trajectory description, of the quantum phase space, and of the "p"-space. In the classical limit, the shadow phase spaces converge to one unique phase space. In their algebraic formulation of quantum mechanics the equation of motion takes on the same form as in the Heisenberg picture, except that the "bra" and "ket" in the bra–ket notation each stand for an element of the algebra and that the Heisenberg time evolution is an inner automorphism in the algebra.