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.
Representation of a Lie superalgebra
A unitary representation of such a Lie algebra is a Z graded Hilbert space which is a representation of a Lie superalgebra as above together with the requirement that self-adjoint elements of the Lie superalgebra are represented by Hermitian transformations.