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Difference between revisions of "Riesz-Fischer theorem"

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A theorem establishing a relationship between the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r0822401.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r0822402.png" />: If a system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r0822403.png" /> is orthonormal on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r0822404.png" /> (cf. [[Orthonormal system|Orthonormal system]]) and a sequence of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r0822405.png" /> is such that
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A theorem establishing a relationship between the spaces $l_2$ and $L_2[a,b]$: If a system of functions $\{\phi_n\}_{n=1}^\infty$ is orthonormal on the interval $[a,b]$ (cf. [[Orthonormal system|Orthonormal system]]) and a sequence of numbers $\{c_n\}_{n=1}^\infty$ is such that
  
<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/r/r082/r082240/r0822406.png" /></td> </tr></table>
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$$\sum_{n=1}^\infty c_n^2<\infty$$
  
(that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r0822407.png" />), then there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r0822408.png" /> for which
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(that is, $c_n\in l_2$), then there exists a function $f\in L_2[a,b]$ for which
  
<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/r/r082/r082240/r0822409.png" /></td> </tr></table>
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$$\int\limits_a^b|f(t)|^2dt=\sum_{n=1}^\infty c_n^2,\quad c_n=\int\limits_a^bf(t)\phi_n(t)dt.$$
  
Moreover, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r08224010.png" /> is unique as an element of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r08224011.png" />, i.e. up to its values on a set of Lebesgue measure zero. In particular, if the orthonormal system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r08224012.png" /> is closed (complete, cf. [[Complete system of functions|Complete system of functions]]) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r08224013.png" />, then, using the Riesz–Fischer theorem, one gets that the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r08224014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r082/r082240/r08224015.png" /> are isomorphic and isometric.
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Moreover, the function $f$ is unique as an element of the space $L_2[a,b]$, i.e. up to its values on a set of Lebesgue measure zero. In particular, if the orthonormal system $\{\phi_n\}$ is closed (complete, cf. [[Complete system of functions|Complete system of functions]]) in $L_2[a,b]$, then, using the Riesz–Fischer theorem, one gets that the spaces $l_2$ and $L_2[a,b]$ are isomorphic and isometric.
  
 
The theorem was proved independently by F. Riesz [[#References|[1]]] and E. Fischer [[#References|[2]]].
 
The theorem was proved independently by F. Riesz [[#References|[1]]] and E. Fischer [[#References|[2]]].

Revision as of 13:33, 9 August 2014

A theorem establishing a relationship between the spaces $l_2$ and $L_2[a,b]$: If a system of functions $\{\phi_n\}_{n=1}^\infty$ is orthonormal on the interval $[a,b]$ (cf. Orthonormal system) and a sequence of numbers $\{c_n\}_{n=1}^\infty$ is such that

$$\sum_{n=1}^\infty c_n^2<\infty$$

(that is, $c_n\in l_2$), then there exists a function $f\in L_2[a,b]$ for which

$$\int\limits_a^b|f(t)|^2dt=\sum_{n=1}^\infty c_n^2,\quad c_n=\int\limits_a^bf(t)\phi_n(t)dt.$$

Moreover, the function $f$ is unique as an element of the space $L_2[a,b]$, i.e. up to its values on a set of Lebesgue measure zero. In particular, if the orthonormal system $\{\phi_n\}$ is closed (complete, cf. Complete system of functions) in $L_2[a,b]$, then, using the Riesz–Fischer theorem, one gets that the spaces $l_2$ and $L_2[a,b]$ are isomorphic and isometric.

The theorem was proved independently by F. Riesz [1] and E. Fischer [2].

References

[1] F. Riesz, "Sur les systèmes orthogonaux de fonctions" C.R. Acad. Sci. Paris , 144 (1907) pp. 615–619
[2] E. Fischer, C.R. Acad. Sci. Paris , 144 (1907) pp. 1022–1024; 1148–1150
[3] I.P. Natanson, "Theory of functions of a real variable" , 1–2 , F. Ungar (1955–1961) (Translated from Russian)


Comments

References

[a1] R.V. Kadison, J.R. Ringrose, "Fundamentals of the theory of operator algebras" , 1 , Acad. Press (1983)
How to Cite This Entry:
Riesz-Fischer theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz-Fischer_theorem&oldid=32781
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article