# 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

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

where 〈•,•〉 denotes the inner product in the Hilbert space $H$. If we define the infinite sum

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

consisting of 'infinite sum' of vector resolute $x$ in direction $e_k$, Bessel's inequality tells us that this series converges. One can think of it that there exists $x' \in H$ which can be described in terms of potential basis $e_1, e_2, ...$.

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:

$0 \le \left\| x - \sum_{k=1}^n \langle x, e_k \rangle e_k\right\|^2 = \|x\|^2 - 2 \sum_{k=1}^n |\langle x, e_k \rangle |^2 + \sum_{k=1}^n | \langle x, e_k \rangle |^2 = \|x\|^2 - \sum_{k=1}^n | \langle x, e_k \rangle |^2,$

which holds for any natural n.