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's fourteenth problem
gave the following counterexample to Hilbert's problem. The field "k" is a field containing 48 elements "a", ...,"a", for "i"=1, 2, 3 that are algebraically independent over the prime field. The ring "R" is the polynomial ring "k"["x"...,"x", "t"...,"t"] in 32 variables. The vector space "V" is a 13-dimensional vector space over "k" consisting of all vectors ("b"...,"b") in "k" orthogonal to each of the three vectors ("a", ...,"a") for "i"=1, 2, 3. The vector space "V" is a 13-dimensional commutative unipotent algebraic group under addition, and its elements act on "R" by fixing all elements "t" and taking "x" to "x" + "b""t". Then the ring of elements of "R" invariant under the action of the group "V" is not a finitely generated "k"-algebra.