Bessel's inequality

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


  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.

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