In mathematics, von Neumann's theorem is a result in the operator theory of linear operators on Hilbert spaces. Statement of the theorem Let and be Hilbert spaces, and let be an unbounded operator from into Suppose that is a closed operator and that is densely defined, that is, is dense in Let denote the adjoint of Then is also densely defined, and it is self-adjoint. That is, and the operators on the right- and left-hand sides have the same dense domain in References Operator theory Theorems in functional analysis