Bessel inequality

Let be a Hilbert spaceMathworldPlanetmath, and suppose e1,e2, is an orthonormal sequence. Then for any x,


Bessel’s inequalityMathworldPlanetmath immediately lets us define the sum


The inequality means that the series converges.

For a completePlanetmathPlanetmath orthonormal series, we have Parseval’s theorem, which replaces inequality with equality (and consequently x with x).

Title Bessel inequality
Canonical name BesselInequality
Date of creation 2013-03-22 12:46:38
Last modified on 2013-03-22 12:46:38
Owner ariels (338)
Last modified by ariels (338)
Numerical id 5
Author ariels (338)
Entry type Theorem
Classification msc 46C05