In mathematics, a self-adjoint operator on a finite-dimensional complex vector space "V" with inner product formula_1 (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map "A" (from "V" to itself) that is its own adjoint: formula_2 for all vectors v and w. If "V" is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of "A" is a Hermitian matrix, i.e., equal to its conjugate transpose "A". By the finite-dimensional spectral theorem, "V" has an orthonormal basis such that the matrix of "A" relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension.