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.

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 $H$ .^[1]^[2]^[3] 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

0 \le \left\| x - \sum_{k=1}^n \langle x, e_k \rangle e_k\right\|^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,

which holds for any natural n.

Notes

↑ Saxe, Karen (2001-12-07). Beginning Functional Analysis. Springer Science & Business Media. p. 82. ISBN 9780387952246.
↑ Zorich, Vladimir A.; Cooke, R. (2004-01-22). Mathematical Analysis II. Springer Science & Business Media. pp. 508–509. ISBN 9783540406334.
↑ Vetterli, Martin; Kovačević, Jelena; Goyal, Vivek K. (2014-09-04). Foundations of Signal Processing. Cambridge University Press. p. 83. ISBN 9781139916578.

External links

Hazewinkel, Michiel, ed. (2001) [1994], "Bessel inequality", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
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.

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[1] Saxe, Karen (2001-12-07). Beginning Functional Analysis. Springer Science & Business Media. p. 82. ISBN 9780387952246.

[2] Zorich, Vladimir A.; Cooke, R. (2004-01-22). Mathematical Analysis II. Springer Science & Business Media. pp. 508–509. ISBN 9783540406334.

[3] Vetterli, Martin; Kovačević, Jelena; Goyal, Vivek K. (2014-09-04). Foundations of Signal Processing. Cambridge University Press. p. 83. ISBN 9781139916578.

Functional analysis (topics)
TVS types	Banach Banach Lattice Barrelled Bornological Brauner F-space Finite-dimensional Fréchet (tame) Hilbert (pre-Hilbert Polarization identity) LF-space Locally convex (Seminorms/Minkowski functionals) Mackey Montel Nuclear Normed (norm) Quasinormed Reflexive Riesz Smith Stereotype Strictly convex Webbed Topological tensor product (of Hilbert spaces)
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Operator theory	Banach algebras C-algebras Spectrum (C-algebra radius) Spectral theory (of ODEs Spectral theorem) Polar decomposition Singular value decomposition
Theorems	Banach–Alaoglu Banach–Mazur Banach–Saks Bessel's inequality Cauchy–Schwarz inequality Closed graph Closed range Eberlein–Šmulian Freudenthal spectral Gelfand–Mazur Gelfand–Naimark Goldstine Hahn–Banach (hyperplane separation) Kakutani fixed-point Krein–Milman Lomonosov's invariant subspace Mackey–Arens Mazur's lemma M. Riesz extension Riesz representation Open mapping Parseval's identity Schauder fixed-point
Analysis	Abstract Wiener space Bochner space Differentiation in Fréchet spaces Derivatives (Fréchet Gâteaux functional holomorphic) Integrals (Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak) Functional calculus (Borel continuous holomorphic) Inverse function theorem (Nash–Moser theorem) Measures (Lebesgue Projection-valued Vector) Weakly measurable function
Types of sets	Absolutely convex Absorbing Balanced Bounded Convex Convex cone (subset) Linear cone (subset) Radial Star-shaped Symmetric Zonotope
Subsets / set operations	Algebraic interior (core) Bounding points Convex hull Extreme point Interior Minkowski addition Polar

Bessel's inequality

See also

Notes

External links