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) 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) 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].


[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)



[a1] R.V. Kadison, J.R. Ringrose, "Fundamentals of the theory of operator algebras" , 1 , Acad. Press (1983)
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