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
:$\sum_^\left\vert\left\langle x,e_k\right\rangle \right\vert^2 \le \left\Vert x\right\Vert^2,$

where ⟨·,·⟩ denotes the inner product in the Hilbert space $H$. If we define the infinite sum
:$x' = \sum_^\left\langle x,e_k\right\rangle e_k,$
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
:$\begin 0 \leq \left\, x - \sum_^n \langle x, e_k \rangle e_k\right\, ^2 &= \, x\, ^2 - 2 \sum_^n \operatorname \langle x, \langle x, e_k \rangle e_k \rangle + \sum_^n , \langle x, e_k \rangle , ^2 \\ &= \, x\, ^2 - 2 \sum_^n , \langle x, e_k \rangle , ^2 + \sum_^n , \langle x, e_k \rangle , ^2 \\ &= \, x\, ^2 - \sum_^n , \langle x, e_k \rangle , ^2, \end$
which holds for any natural ''n''.

See also

* Cauchy–Schwarz inequality
* Parseval's theorem

References

External links

*

Bessel's Inequality
the article on Bessel's Inequality on MathWorld.

{{Hilbert space

Hilbert space
Inequalities