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 series and Hilbert polynomial
Thus formula_62 is an Artinian ring, which is a -vector space of dimension , and Jordan–Hölder theorem may be used for proving that formula_63 is the degree of the algebraic set . In fact, the multiplicity of a point is the number of occurrences of the corresponding maximal ideal in a composition series.