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 (that is, for an orthonormal sequence that is a basis), we have Parseval's identity, which replaces the inequality with an equality (and consequently with ).

Bessel's inequality follows from the identity

which holds for any natural n.

See also

• Cauchy–Schwarz inequality
• Parseval's theorem