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

$\sum _{k=1}^{\mathrm{\infty }}{\left|⟨x,e_k⟩\right|}^2\le {\Vert x\Vert }^2,$

where ⟨·,·⟩ denotes the inner product in the Hilbert space $H$. If we define the infinite sum

$x^{\prime }=\sum _{k=1}^{\mathrm{\infty }}⟨x,e_k⟩e_k,$

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 $x^{\prime }\in H$ 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 $x^{\prime }$ with $x$).

Bessel's inequality follows from the identity

$\begin{array}{rl}0\le {\Vert x-\sum _{k=1}^n⟨x,e_k⟩e_k\Vert }^2& =\Vert x{\Vert }^2-2\sum _{k=1}^n\mathrm{Re}⟨x,⟨x,e_k⟩e_k⟩+\sum _{k=1}^n|⟨x,e_k⟩{|}^2\\ & =\Vert x{\Vert }^2-2\sum _{k=1}^n|⟨x,e_k⟩{|}^2+\sum _{k=1}^n|⟨x,e_k⟩{|}^2\\ & =\Vert x{\Vert }^2-\sum _{k=1}^n|⟨x,e_k⟩{|}^2,\end{array}$

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.