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
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.