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:

$lim_{Nββ}β₯S_{N}fβfβ₯_{2}=0$

- Lebesgue integral of $L_{p}$ is complete but Riemann isn't | 89, 547, 7
- Lebesgue integral | 31, 851, 12
- Calculus | 11, 5k, 106
- Mathematics | 17, 10k, 260
- Ciro Santilli's Homepage | 238, 154k, 2k

- Carleson's theorem | 140
- Fourier basis is complete for $L_{2}$ | 60, 242, 2
- Lebesgue integral of $L_{p}$ is complete but Riemann isn't | 89, 547, 7