In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828. Let H be a Hilbert space, and suppose that is an orthonormal sequence in H. Then, for any x in H one has where ⟨·,·⟩ denotes the inner product in the Hilbert space H. If we define the infinite sum consisting of "infinite sum" of vector resolute x in direction e_k, Bessel's inequality tells us that this series converges. One can think of it that there exists x' \in H that can be described in terms of potential basis. For a complete orthonormal sequence (that is, for an orthonormal sequence that is a basis), we have Parseval's identity, which replaces the inequality with an equality (and consequently x' with x). Bessel's inequality follows from the identity which holds for any natural n.