View source for Riesz-Fischer theorem

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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

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

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

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

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|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]]].

====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>