Bessel's inequality

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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.

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 which can be described in terms of potential basis .

For a complete orthonormal sequence (that is, for an orthonormal sequence which 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

External links

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This article incorporates material from Bessel inequality on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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