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
The ring formula_51 has Krull dimension one, and is the ring of regular functions of a projective algebraic set formula_52 of dimension 0 consisting of a finite number of points, which may be multiple points. As formula_53 belongs to a regular sequence, none of these points belong to the hyperplane of equation formula_54 The complement of this hyperplane is an affine space that contains formula_55 This makes formula_52 an affine algebraic set, which has formula_57 as its ring of regular functions. The linear polynomial formula_58 is not a zero divisor in formula_59 and one has thus an exact sequence
which implies that 
Here we are using Hilbert series of filtered algebras, and the fact that the Hilbert series of a graded algebra is also its Hilbert series as filtered algebra.