PlanetMath: Bessel inequality

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Bessel inequality

(Theorem)

Let $\mathcal{H}$ be a Hilbert space, and suppose $e_1, e_2, \ldots \in \mathcal{H}$ is an orthonormal sequence. Then for any $x\in\mathcal{H}$ ,

$\displaystyle \sum_{k=1}^{\infty}\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2.$

Bessel's inequality immediately lets us define the sum

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

The inequality means that the series converges.

For a complete orthonormal series, we have Parseval's theorem, which replaces inequality with equality (and consequently with ).

"Bessel inequality" is owned by ariels.

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Attachments:: proof of Bessel inequality (Proof) by ariels

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Cross-references: equality, Parseval's theorem, complete, converges, series, inequality, sum, sequence, orthonormal, Hilbert space
There are 3 references to this entry.

This is version 2 of Bessel inequality, born on 2002-06-10, modified 2002-06-11.
Object id is 3089, canonical name is BesselInequality.
Accessed 8068 times total.

Classification:

AMS MSC:

46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology )

Pending Errata and Addenda

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