View source for Riesz-Fischer theorem

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

<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>

(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

<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>

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.

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

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



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.V. Kadison,   J.R. Ringrose,   "Fundamentals of the theory of operator algebras" , '''1''' , Acad. Press  (1983)</TD></TR></table>