Bessel's inequality

In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828. Let be a Hilbert space, and suppose that is an orthonormal sequence in . Then, for any in one has where ⟨·,·⟩ denotes the inner product in the Hilbert space . If we define the infinite sum consisting of "infinite sum" of vector resolute in direction , Bessel's inequality tells us that this series converges. One can think of it that there exists that can be described in terms of potential basis .