Bessel's inequality

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In mathematics, especially functional analysis, 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 $e 1, e 2,...$ is an orthonormal sequence in $H$ . Then, for any $x$ in $H$ one has

$\sum_{k=1}^{\infty}\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2$

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

$x' = \sum_{k=1}^{\infty}\left\langle x,e_k\right\rangle e_k,$

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:

$\| x - \sum_{k=1}^n \langle x, e_k \rangle \|^2 = \|x\|^2 - 2 \operatorname{Re} \sum_{k=1}^n |\langle x, e_k \rangle |^2 + \sum_{k=1}^n | \langle x, e_k \rangle |^2 = \|x\|^2 - \sum_{k=1}^n | \langle x, e_k \rangle |^2$ ,