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:$x\text{'}\; =\; sum\_\{k=1\}^\{infty\}leftlangle\; x,e\_k\; ight\; angle\; e\_k,$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 $x\text{'}$ 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.