Given an orthogonal function system $\phi_{n}(x)$ ($n=1,2,\dots$) with norm
$\left\vert \left\vert \phi _{n}\right\vert \right\vert =\sqrt{\left( \phi_{n}\cdot \overline{\phi }_{n}\right) }=\displaystyle\sqrt{\displaystyle\int_{a}^{b}\phi _{n}\left( x\right) \,\overline{\phi }_{n}\left( x\right) \;dx}$
and two functions $f(x)$ e $g(x)$ represented by the Fourier series
$f(x)\sim\displaystyle\sum_{n\ge 1} c_{n}\phi_{n}(x)$
$g(x)\sim\displaystyle\sum_{n\ge 1} d_{n}\phi_{n}(x)$
with coefficients $c_n,d_n$
$c_n=\dfrac{\left ( f\cdot \overline{\phi }_{n}\right)}{\left\vert \left\vert \phi _{n}\right\vert \right\vert ^{2}}=\dfrac{\displaystyle\int_{a}^{b}f\left( x\right)\overline{\phi }_{n}\left( x\right) \;dx}{\left\vert \left\vert \phi _{n}\right\vert \right\vert ^{2}}$
$d_n=\dfrac{\left ( f\cdot \overline{\phi }_{n}\right)}{\left\vert \left\vert \phi _{n}\right\vert \right\vert ^{2}}=\dfrac{\displaystyle\int_{a}^{b}g\left( x\right)\overline{\phi }_{n}\left( x\right) \;dx}{\left\vert \left\vert \phi _{n}\right\vert \right\vert ^{2}},$
and if
$\displaystyle\int_{a}^{b}\left\vert f(x)-\displaystyle\sum_{n=1}^{\infty}c_{n}\phi_{n}(x)\right\vert ^{2}dx=0$
and
$\displaystyle\int_{a}^{b}\left\vert g(x)-\displaystyle\sum_{n=1}^{\infty}d_{n}\phi_{n}(x)\right\vert ^{2}dx=0,$
then the following equality holds
$\displaystyle\int_{a}^{b}f(x)\overline{g}_{n}(x)\; dx=\displaystyle\sum_{n\ge 1} c_{n}\overline{d}_{n}||\phi_n||^2,$
a particular case of which, for $g(x)=f(x)$, is
$\displaystyle\int_{a}^{b}|f(x)|^2\; dx=\displaystyle\sum_{n\ge 1} |c_{n}|^2||\phi_n||^{2}.$