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.
Calkin algebra
In functional analysis, the Calkin algebra, named after John Wilson Calkin, is the quotient of "B"("H"), the ring of bounded linear operators on a separable infinite-dimensional Hilbert space "H", by the ideal "K"("H") of compact operators. Here the addition in "B"("H") is addition of operators and the multiplication in "B"("H") is composition of operators; it is easy to verify that these operations make "B"("H") into a ring. When scalar multiplication is also included, "B"("H") becomes in fact an algebra over the same field over which "H" is a Hilbert space.