Bessel's inequality

In mathematics, 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>).

This article incorporates material from Bessel inequality on PlanetMath, which is licensed under the GFDL.