A measurable function defined on a closed interval is square integrable (and therefore in $L_{2}$) if and only if Fourier series converges in $L_{2}$ norm the function:

$lim_{N→∞}∥S_{N}f−f∥_{2}=0$

A measurable function defined on a closed interval is square integrable (and therefore in $L_{2}$) if and only if Fourier series converges in $L_{2}$ norm the function:

$lim_{N→∞}∥S_{N}f−f∥_{2}=0$