Bessel's inequality

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In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space with respect to an orthonormal sequence.

Let H be a Hilbert space, and suppose that e_1, e_2, ... is an orthonormal sequence in H. Then, for any x in H one has

\sum_{k=1}^{\infty}\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2

where 〈•,•〉 denotes the inner product in the Hilbert space H. If we define the infinite sum

x' = \sum_{k=1}^{\infty}\left\langle x,e_k\right\rangle e_k,

consisting of 'infinite sum' of vector resolute x in direction e_k, Bessel's inequality tells us that this series converges. One can think of it that there exists x' \in H which can be described in terms of potential basis e_1, e_2, ....

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  x' with  x).

Bessel's inequality follows from the identity:

0 \le \left\| x - \sum_{k=1}^n \langle x, e_k \rangle e_k\right\|^2 = \|x\|^2 - 2 \sum_{k=1}^n |\langle x, e_k \rangle |^2 + \sum_{k=1}^n | \langle x, e_k \rangle |^2 = \|x\|^2 - \sum_{k=1}^n | \langle x, e_k \rangle |^2,

which holds for any natural n.

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