Bessel's inequality

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In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element in a Hilbert space with respect to an orthonormal sequence.

Let be a Hilbert space, and suppose that is an orthonormal sequence in . Then, for any in one has

where 〈•,•〉 denotes the inner product in the Hilbert space .[1][2][3] If we define the infinite sum

consisting of "infinite sum" of vector resolute in direction , Bessel's inequality tells us that this series converges. One can think of it that there exists that can be described in terms of potential basis .

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 with ).

Bessel's inequality follows from the identity

which holds for any natural n.

See also[edit]


  1. ^ Saxe, Karen (2001-12-07). Beginning Functional Analysis. Springer Science & Business Media. p. 82. ISBN 9780387952246.
  2. ^ Zorich, Vladimir A.; Cooke, R. (2004-01-22). Mathematical Analysis II. Springer Science & Business Media. pp. 508–509. ISBN 9783540406334.
  3. ^ Vetterli, Martin; Kovačević, Jelena; Goyal, Vivek K. (2014-09-04). Foundations of Signal Processing. Cambridge University Press. p. 83. ISBN 9781139916578.

External links[edit]

This article incorporates material from Bessel inequality on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.