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.
For a complete orthonormal sequence, we have Parseval's identity, which replaces the inequality with an equality.
Bessel's inequality follows from the identity
which holds for any natural n.
