Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. It's essentially just an infinitely dimensional vector space. ﻿A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used.
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.