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.
Crossed product
Suppose that "A" is a von Neumann algebra of operators acting on a Hilbert space "H" and "G" is a discrete group acting on "A". We let "K" be the Hilbert space of all square summable "H"-valued functions on "G". There is an action of "A" on "K"
given by 
for "k" in "K", "g", "h" in "G", and "a" in "A",
and there is an action of "G" on "K" given by
The crossed product formula_5 is the von Neumann algebra acting on "K" generated by the actions of "A" and "G" on "K". It does not depend (up to isomorphism) on the choice of the Hilbert space "H".