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.
Sobolev space
The Sobolev space formula_96 is also denoted by formula_97 It is a Hilbert space, with an important subspace formula_98 defined to be the closure of the infinitely differentiable functions compactly supported in formula_1 in formula_97 The Sobolev norm defined above reduces here to