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

$\sum _{k=1}^{\infty }\left\vert \left\langle x,e_{k}\right\rangle \right\vert ^{2}\leq \left\Vert x\right\Vert ^{2},$

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

${\begin{aligned}0\leq \left\|x-\sum _{k=1}^{n}\langle x,e_{k}\rangle e_{k}\right\|^{2}&=\|x\|^{2}-2\sum _{k=1}^{n}\operatorname {Re} \langle x,\langle x,e_{k}\rangle e_{k}\rangle +\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}\\&=\|x\|^{2}-2\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}+\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2}\\&=\|x\|^{2}-\sum _{k=1}^{n}|\langle x,e_{k}\rangle |^{2},\end{aligned}}$

which holds for any natural n.

See also

• Cauchy–Schwarz inequality
• Parseval's theorem

External links

• "Bessel inequality", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Bessel's Inequality the article on Bessel's Inequality on MathWorld.

This article incorporates material from Bessel inequality on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.