Bessel's inequality
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In mathematics, Bessel's inequality is a statement about the coefficients of an element x in a Hilbert space in respect to an orthonormal sequence.
Let H be a Hilbert space, and suppose that e1,e2,... is an orthonormal sequence in H. Then, for any x in H one has
where <∙,∙> denotes the inner product in the Hilbert space H. If we define the infinite sum
Bessel's inequality tells us that this series converges.
For a complete orthonormal sequence (that is, for an orthonormal sequence which is a basis), we have Parseval's identity, which replaces the inequality with an equality (and consequently x' with x).
This article incorporates material from Bessel inequality on PlanetMath, which is licensed under the GFDL.
