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:

$lim_{N→∞}∥S_{N}f−f∥_{2}=0$

- Lebesgue integral of $L_{p}$ is complete but Riemann isn't | 91, 805, 11
- Lebesgue integral | 31, 1k, 15
- Calculus | 17, 7k, 159
- Mathematics | 17, 13k, 336
- Ciro Santilli's Homepage | 262, 182k, 3k

- Carleson's theorem | 140
- Fourier basis is complete for $L_{2}$ | 62, 246, 2
- Lebesgue integral of $L_{p}$ is complete but Riemann isn't | 91, 805, 11