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.
CCR and CAR algebras
Let formula_16 be a Hilbert space. In the theory of operator algebras the CAR algebra is the unique C*-completion of the complex unital *-algebra generated by elements formula_37 subject to the relations
for any formula_28, formula_43.
When formula_16 is separable the CAR algebra is an AF algebra and in the special case formula_16 is infinite dimensional it is often written as formula_46.