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.
Hilbert space
Sobolev spaces, denoted by or , are Hilbert spaces. These are a special kind of function space in which differentiation may be performed, but that (unlike other Banach spaces such as the Hölder spaces) support the structure of an inner product. Because differentiation is permitted, Sobolev spaces are a convenient setting for the theory of partial differential equations. They also form the basis of the theory of direct methods in the calculus of variations.