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| In [[mathematics]], especially [[functional analysis]], '''Bessel's inequality''' is a statement about the coefficients of an element <math>x</math> in a [[Hilbert space]] with respect to an [[orthonormal]] [[sequence]].
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| Let <math>H</math> be a Hilbert space, and suppose that <math>e_1, e_2, ...</math> is an orthonormal sequence in <math>H</math>. Then, for any <math>x</math> in <math>H</math> one has
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| :<math>\sum_{k=1}^{\infty}\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2 </math>
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| where 〈•,•〉 denotes the [[inner product space|inner product]] in the Hilbert space <math>H</math>. If we define the infinite sum
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| :<math>x' = \sum_{k=1}^{\infty}\left\langle x,e_k\right\rangle e_k, </math>
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| consisting of 'infinite sum' of [[vector resolute]] <math>x</math> in direction <math>e_k</math>, Bessel's [[inequality (mathematics)|inequality]] tells us that this [[series (mathematics)|series]] [[Limit of a sequence|converges]]. One can think of it that there exists <math>x' \in H</math> which can be described in terms of potential basis <math>e_1, e_2, ...</math>.
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| For a complete orthonormal sequence (that is, for an orthonormal sequence which is a [[Orthonormal basis|basis]]), we have [[Parseval's identity]], which replaces the inequality with an equality (and consequently <math> x'</math> with <math> x</math>).
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| Bessel's inequality follows from the identity:
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| :<math>0 \le \left\| x - \sum_{k=1}^n \langle x, e_k \rangle e_k\right\|^2 = \|x\|^2 - 2 \sum_{k=1}^n |\langle x, e_k \rangle |^2 + \sum_{k=1}^n | \langle x, e_k \rangle |^2 = \|x\|^2 - \sum_{k=1}^n | \langle x, e_k \rangle |^2,</math>
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| which holds for any natural ''n''.
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| ==See also==
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| * [[Cauchy–Schwarz inequality]]
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| ==External links==
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| * {{springer|title=Bessel inequality|id=p/b015850}}
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| * [http://mathworld.wolfram.com/BesselsInequality.html Bessel's Inequality] the article on Bessel's Inequality on MathWorld.
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| {{PlanetMath attribution|title=Bessel inequality|id=3089}}
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| {{Functional Analysis}}
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| [[Category:Hilbert space]]
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| [[Category:Inequalities]]
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