Bessel's inequality

In mathematics, especially functional analysis, Bessel's inequality is a statement about the coefficients of an element ${\displaystyle x}$ in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828.^[1]

Let ${\displaystyle H}$ be a Hilbert space, and suppose that ${\displaystyle e_{1},e_{2},...}$ is an orthonormal sequence in ${\displaystyle H}$. Then, for any ${\displaystyle x}$ in ${\displaystyle H}$ one has

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

where ⟨·,·⟩ denotes the inner product in the Hilbert space ${\displaystyle H}$.^[2][3][4] If we define the infinite sum

${\displaystyle x'=\sum _{k=1}^{\infty }\left\langle x,e_{k}\right\rangle e_{k},}$

consisting of "infinite sum" of vector resolute ${\displaystyle x}$ in direction ${\displaystyle e_{k}}$, Bessel's inequality tells us that this series converges. One can think of it that there exists ${\displaystyle x'\in H}$ that can be described in terms of potential basis ${\displaystyle e_{1},e_{2},\dots }$.

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 ${\displaystyle x'}$ with ${\displaystyle x}$).

Bessel's inequality follows from the identity

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

which holds for any natural n.