Science Fair Project Encyclopedia

Science Fair Project Encyclopedia Contents Page

Bessel's inequality

In mathematics, Bessel's inequality is a statement about the coefficients of an element $x$ in a Hilbert space in 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

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

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 $x$ ).

Categories: Functional analysis

Last updated: 05-29-2005 03:19:45

Science Fair Project Encyclopedia Contents Page

10-26-2009 08:16:03
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details

All Science Fair Projects

Science Fair Project Encyclopedia for Schools!

Science Fair Project Encyclopedia

Bessel's inequality