Definitions

# 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

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

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

$x\text{'} = sum_\left\{k=1\right\}^\left\{infty\right\}leftlangle x,e_krightrangle e_k,$
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 $x\text{'}$ with $x$).

Bessel's inequality follows from the identity:

$left| x - sum_\left\{k=1\right\}^n langle x, e_k rangle e_kright|^2 = |x|^2 - 2 sum_\left\{k=1\right\}^n |langle x, e_k rangle |^2 + sum_\left\{k=1\right\}^n | langle x, e_k rangle |^2 = |x|^2 - sum_\left\{k=1\right\}^n | langle x, e_k rangle |^2$,
which holds for any $n$, excluding when $n$ is less than 1.