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.
be a Hilbert space, and suppose that
is an orthonormal sequence in
one has where ⟨·,·⟩ denotes the inner product in the Hilbert space
[2][3][4] If we define the infinite sum consisting of "infinite sum" of vector resolute
, Bessel's inequality tells us that this series converges.
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
Bessel's inequality follows from the identity which holds for any natural n. This article incorporates material from Bessel inequality on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.