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 metric
Let Ω be a convex open domain in a Euclidean space that does not contain a line. Given two distinct points "A" and "B" of Ω, let "X" and "Y" be the points at which the straight line "AB" intersects the boundary of Ω, where the order of the points is "X", "A", "B", "Y". Then the Hilbert distance "d"("A", "B") is the logarithm of the cross-ratio of this quadruple of points: