Bessel's inequality

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

Let H be a Hilbert space, and suppose that e_1, e_2, ... is an orthonormal sequence in H. Then, for any x in H one has

sum_{k=1}^{infty}leftvertleftlangle x,e_krightrangle rightvert^2 le leftVert xrightVert^2

where <∙,∙> denotes the inner product in the Hilbert space H. If we define the infinite sum

x' = sum_{k=1}^{infty}leftlangle x,e_krightrangle e_k,
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).

Bessel's inequality follows from the identity:

left| x - sum_{k=1}^n langle x, e_k rangle e_kright|^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,
which holds for any n, excluding when n is less than 1.

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