Bessel's inequality

In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element [math]\displaystyle{ x }[/math] in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828. Let [math]\displaystyle{ H }[/math] be a Hilbert space, and suppose that [math]\displaystyle{ e_1, e_2, ... }[/math] is an orthonormal sequence in [math]\displaystyle{ H }[/math]. Then, for any [math]\displaystyle{ x }[/math] in [math]\displaystyle{ H }[/math] one has

[math]\displaystyle{ \sum_{k=1}^{\infty}\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2, }[/math]

where ⟨·,·⟩ denotes the inner product in the Hilbert space [math]\displaystyle{ H }[/math]. If we define the infinite sum

[math]\displaystyle{ x' = \sum_{k=1}^{\infty}\left\langle x,e_k\right\rangle e_k, }[/math]

consisting of "infinite sum" of vector resolute [math]\displaystyle{ x }[/math] in direction [math]\displaystyle{ e_k }[/math], Bessel's inequality tells us that this series converges. One can think of it that there exists [math]\displaystyle{ x' \in H }[/math] that can be described in terms of potential basis [math]\displaystyle{ e_1, e_2, \dots }[/math].

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 [math]\displaystyle{ x' }[/math] with [math]\displaystyle{ x }[/math]).

Bessel's inequality follows from the identity

[math]\displaystyle{ \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} }[/math]

which holds for any natural n.

See also

• Cauchy–Schwarz inequality
• Parseval's theorem

External links

• Hazewinkel, Michiel, ed. (2001), "Bessel inequality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=p/b015850 
• Bessel's Inequality the article on Bessel's Inequality on MathWorld.