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 the "infinite sum" of the vector resolute ${\displaystyle x}$ in the directions ${\displaystyle e_{k}}$, 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.

In the theory of Fourier series, in the particular case of the Fourier orthonormal system, we get if ${\displaystyle f\colon \mathbb {R} \to \mathbb {C} }$ has period ${\displaystyle T}$,

${\displaystyle \sum _{k\in \mathbb {Z} }\left\vert \int _{0}^{T}e^{-2\pi ikt/T}f(t)\,\mathrm {d} t\right\vert ^{2}\leq T\int _{0}^{T}\vert f(t)\vert ^{2}\,\mathrm {d} t.}$

In the particular case where ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$, one has then

${\displaystyle \left\vert \int _{0}^{T}f(t)\,\mathrm {d} t\right\vert ^{2}+2\sum _{n=1}^{\infty }\left\vert \int _{0}^{T}\cos(2\pi kt/T)f(t)\,\mathrm {d} t\right\vert ^{2}+2\sum _{n=1}^{\infty }\left\vert \int _{0}^{T}\sin(2\pi kt/T)f(t)\,\mathrm {d} t\right\vert ^{2}\leq T\int _{0}^{T}\vert f(t)\vert ^{2}\,\mathrm {d} t.}$

More generally, if ${\displaystyle H}$ is a pre-Hilbert space and ${\displaystyle (e_{\alpha })_{\alpha \in A}}$ is an orthonormal system, then for every ${\displaystyle x\in H}$^[1]

${\displaystyle \sum _{\alpha \in A}|\langle x,e_{\alpha }\rangle |^{2}\leq \lVert x\rVert ^{2}}$

This is proved by noting that if ${\displaystyle F\subseteq A}$ is finite, then

${\displaystyle \sum _{\alpha \in F}|\langle x,e_{\alpha }\rangle |^{2}\leq \lVert x\rVert ^{2}}$

and then by definition of infinite sum

${\displaystyle \sum _{\alpha \in A}|\langle x,e_{\alpha }\rangle |^{2}=\sup {\Bigl \{}\sum _{\alpha \in F}|\langle x,e_{\alpha }\rangle |^{2}:F\subseteq A{\text{ is finite}}{\Bigr \}}\leq \lVert x\rVert ^{2}.}$

• Cauchy–Schwarz inequality
• Parseval's theorem

• "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.