Bessel's inequality

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}leftvertleftlangle x,e_k ight angle ightvert^2 le leftVert x ightVert^2

where <∙,∙> denotes the inner product in the Hilbert space $$H. If we define the infinite sum:$$x' = sum_{k=1}^{infty}leftlangle x,e_k ight angle e_k, 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 $$x' with $$x).

Bessel's inequality follows from the identity::$$left| x - sum_{k=1}^n langle x, e_k angle e_k ight|^2 = |x|^2 - 2 sum_{k=1}^n |langle x, e_k angle |^2 + sum_{k=1}^n | langle x, e_k angle |^2 = |x|^2 - sum_{k=1}^n | langle x, e_k angle |^2,which holds for any $$n, excluding when $$n is less than 1.

External links

* [http://mathworld.wolfram.com/BesselsInequality.html Bessel's Inequality] the article on Bessel's Inequality on MathWorld.