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.
Uniformly hyperfinite algebra
One example of a UHF C*-algebra is the CAR algebra. It is defined as follows: let "H" be a separable complex Hilbert space "H" with orthonormal basis "f" and "L"("H") the bounded operators on "H", consider a linear map