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. The inequality was derived by F.W. Bessel in 1828.

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 $$x' \in H$$ that can be described in terms of potential basis $$e_1, e_2, \dots$$.

For a complete orthonormal sequence (that is, for an orthonormal sequence that 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
 * $$\begin{align}

0 \leq \left\| x - \sum_{k=1}^n \langle x, e_k \rangle e_k\right\|^2 &= \|x\|^2 - 2 \sum_{k=1}^n \operatorname{Re} \langle x, \langle x, e_k \rangle e_k \rangle + \sum_{k=1}^n | \langle x, e_k \rangle |^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, \end{align}$$ which holds for any natural n.