Bessel's inequality

From HandWiki

In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element [math]\displaystyle{ x }[/math] in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828.[1] Let [math]\displaystyle{ H }[/math] be a Hilbert space, and suppose that [math]\displaystyle{ e_1, e_2, ... }[/math] is an orthonormal sequence in [math]\displaystyle{ H }[/math]. Then, for any [math]\displaystyle{ x }[/math] in [math]\displaystyle{ H }[/math] one has

[math]\displaystyle{ \sum_{k=1}^{\infty}\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2, }[/math]

where ⟨·,·⟩ denotes the inner product in the Hilbert space [math]\displaystyle{ H }[/math].[2][3][4] If we define the infinite sum

[math]\displaystyle{ x' = \sum_{k=1}^{\infty}\left\langle x,e_k\right\rangle e_k, }[/math]

consisting of "infinite sum" of vector resolute [math]\displaystyle{ x }[/math] in direction [math]\displaystyle{ e_k }[/math], Bessel's inequality tells us that this series converges. One can think of it that there exists [math]\displaystyle{ x' \in H }[/math] that can be described in terms of potential basis [math]\displaystyle{ e_1, e_2, \dots }[/math].

For a complete orthonormal sequence (that is, for an orthonormal sequence that is a basis), we have Parseval's identity, which replaces the inequality with an equality (and consequently [math]\displaystyle{ x' }[/math] with [math]\displaystyle{ x }[/math]).

Bessel's inequality follows from the identity

[math]\displaystyle{ \begin{align} 0 \leq \left\| x - \sum_{k=1}^n \langle x, e_k \rangle e_k\right\|^2 &= \|x\|^2 - 2 \sum_{k=1}^n \operatorname{Re} \langle x, \langle x, e_k \rangle e_k \rangle + \sum_{k=1}^n | \langle x, e_k \rangle |^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, \end{align} }[/math]

which holds for any natural n.

See also

References

  1. "Bessel inequality - Encyclopedia of Mathematics". https://www.encyclopediaofmath.org/index.php/Bessel_inequality. 
  2. Saxe, Karen (2001-12-07) (in en). Beginning Functional Analysis. Springer Science & Business Media. pp. 82. ISBN 9780387952246. https://books.google.com/books?id=QALoZC64ea0C. 
  3. Zorich, Vladimir A.; Cooke, R. (2004-01-22) (in en). Mathematical Analysis II. Springer Science & Business Media. pp. 508–509. ISBN 9783540406334. https://books.google.com/books?id=XF8W9W-eyrgC. 
  4. Vetterli, Martin; Kovačević, Jelena; Goyal, Vivek K. (2014-09-04) (in en). Foundations of Signal Processing. Cambridge University Press. pp. 83. ISBN 9781139916578. https://books.google.com/books?id=LBZEBAAAQBAJ. 

External links