# Riesz-Fischer theorem | π nosplit | β parent "Lebesgue integral of $L_{p}$ is complete but Riemann isn't" | 56, 4, 287

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:

$lim_{Nββ}β₯S_{N}fβfβ₯_{2}=0$