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Difference between revisions of "Orthogonal basis"

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A system of pairwise orthogonal non-zero elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o0702801.png" /> of a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o0702802.png" />, such that any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o0702803.png" /> can be (uniquely) represented in the form of a norm-convergent series
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A system of pairwise orthogonal non-zero elements $e_1,\dots,e_n,\dots,$ of a Hilbert space $X$, such that any element $x\in X$ can be (uniquely) represented in the form of a norm-convergent series
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o0702804.png" /></td> </tr></table>
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$$x=\sum_ic_ie_i,$$
  
called the Fourier series of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o0702805.png" /> with respect to the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o0702806.png" />. The basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o0702807.png" /> is usually chosen such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o0702808.png" />, and is then called an orthonormal basis. In this case, the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o0702809.png" />, called the Fourier coefficients of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o07028010.png" /> relative to the orthonormal basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o07028011.png" />, take the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o07028012.png" />. A necessary and sufficient condition for an orthonormal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o07028013.png" /> to be a basis is the Parseval–Steklov equality
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called the Fourier series of the element $x$ with respect to the system $\{e_i\}$. The basis $\{e_i\}$ is usually chosen such that $\|e_i\|=1$, and is then called an orthonormal basis. In this case, the numbers $c_i$, called the Fourier coefficients of the element $x$ relative to the orthonormal basis $\{e_i\}$, take the form $c_i=(x,e_i)$. A necessary and sufficient condition for an orthonormal system $\{e_i\}$ to be a basis is the Parseval–Steklov equality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o07028014.png" /></td> </tr></table>
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$$\sum_i|(x,e_i)|^2=\|x\|^2,$$
  
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o07028015.png" />. A Hilbert space which has an orthonormal basis is separable and, conversely, in any separable Hilbert space an orthonormal basis exists. If an arbitrary system of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o07028016.png" /> is given such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o07028017.png" />, then in the case of a Hilbert space with a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o07028018.png" />, the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o07028019.png" /> converges in norm to an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o07028020.png" />. An isomorphism between any separable Hilbert space and the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070280/o07028021.png" /> is established in this way (Riesz–Fischer theorem).
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for any $x\in X$. A Hilbert space which has an orthonormal basis is separable and, conversely, in any separable Hilbert space an orthonormal basis exists. If an arbitrary system of numbers $\{c_i\}$ is given such that $\sum_i|c_i|^2<\infty$, then in the case of a Hilbert space with a basis $\{e_i\}$, the series $\sum_ic_ie_i$ converges in norm to an element $x\in X$. An isomorphism between any separable Hilbert space and the space $l_2$ is established in this way (Riesz–Fischer theorem).
  
 
====References====
 
====References====

Revision as of 08:45, 19 October 2014

A system of pairwise orthogonal non-zero elements $e_1,\dots,e_n,\dots,$ of a Hilbert space $X$, such that any element $x\in X$ can be (uniquely) represented in the form of a norm-convergent series

$$x=\sum_ic_ie_i,$$

called the Fourier series of the element $x$ with respect to the system $\{e_i\}$. The basis $\{e_i\}$ is usually chosen such that $\|e_i\|=1$, and is then called an orthonormal basis. In this case, the numbers $c_i$, called the Fourier coefficients of the element $x$ relative to the orthonormal basis $\{e_i\}$, take the form $c_i=(x,e_i)$. A necessary and sufficient condition for an orthonormal system $\{e_i\}$ to be a basis is the Parseval–Steklov equality

$$\sum_i|(x,e_i)|^2=\|x\|^2,$$

for any $x\in X$. A Hilbert space which has an orthonormal basis is separable and, conversely, in any separable Hilbert space an orthonormal basis exists. If an arbitrary system of numbers $\{c_i\}$ is given such that $\sum_i|c_i|^2<\infty$, then in the case of a Hilbert space with a basis $\{e_i\}$, the series $\sum_ic_ie_i$ converges in norm to an element $x\in X$. An isomorphism between any separable Hilbert space and the space $l_2$ is established in this way (Riesz–Fischer theorem).

References

[1] L.A. Lyusternik, V.I. Sobolev, "Elements of functional analysis" , Wiley & Hindustan Publ. Comp. (1974) (Translated from Russian)
[2] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Pitman (1981) (Translated from Russian)
[3] N.I. [N.I. Akhiezer] Achieser, I.M. [I.M. Glaz'man] Glasman, "Theorie der linearen Operatoren im Hilbert Raum" , Akademie Verlag (1958) (Translated from Russian)


Comments

References

[a1] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5
How to Cite This Entry:
Orthogonal basis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Orthogonal_basis&oldid=14572
This article was adapted from an original article by V.I. Sobolev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article