In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element <math>x</math> in a Hilbert space in respect to an orthonormal sequence. Let <math>H</math> be a Hilbert space, and suppose that <math>e_1, e_2, ...</math> is an orthonormal sequence in <math>H</math>. Then, for any <math>x</math> in <math>H</math> one has
- <math>\sum_{k=1}^{\infty}\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2 </math>
where <∙,∙> denotes the inner product in the Hilbert space <math>H</math>. If we define the infinite sum
- <math>x' = \sum_{k=1}^{\infty}\left\langle x,e_k\right\rangle e_k, </math>
Bessel's inequality tells us that this series converges. 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 <math> x'</math> with <math> x</math>). Bessel's inequality follows from the identity:
- <math>\| x - \sum_{k=1}^n \langle x, e_k \rangle e_k\|^2 = \|x\|^2 - 2 \operatorname{Re} \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</math>,
which holds for any <math>n</math>, excluding when <math>n</math> is less than -1 . This article incorporates material from Bessel inequality on PlanetMath, which is licensed under the GFDL.
External links
- Bessel's Inequality the article on Bessel's Inequality on MathWorld.
