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 space
The adjoint of a densely defined unbounded operator is defined in essentially the same manner as for bounded operators. Self-adjoint unbounded operators play the role of the "observables" in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operators on the Hilbert space are:These correspond to the momentum and position observables, respectively. Note that neither nor is defined on all of , since in the case of the derivative need not exist, and in the case of the product function need not be square integrable. In both cases, the set of possible arguments form dense subspaces of .