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Bessel inequality
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(Theorem)
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Let $\Hilb$ be a Hilbert space, and suppose $e_1, e_2, \ldots \in \Hilb$ is an orthonormal sequence. Then for any $x\in\Hilb$ $$ \sum_{k=1}^{\infty}\size{\scalar{x}{e_k}}^2 \le \norm{x}^2. $$
Bessel's inequality immediately lets us define the sum $$ x' = \sum_{k=1}^{\infty}\scalar{x}{e_k}e_k. $$ The inequality means that the series converges.
For a complete orthonormal series, we have Parseval's theorem, which replaces inequality with equality (and consequently $x'$ with $x$ .
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"Bessel inequality" is owned 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 10205 times total.
Classification:
| AMS MSC: | 46C05 (Functional analysis :: Inner product spaces and their generalizations, Hilbert spaces :: Hilbert and pre-Hilbert spaces: geometry and topology ) |
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Pending Errata and Addenda
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