Convergence in the mean of Fourier series 
I need to do some research on fourier series for my analysis class so I'm trying to find info (preferably a book or paper with the proof) on this:
"If $f$ is Riemann integrable on $[-l,l]$ then its fourier series converge in the mean to $f$ on $[-l,l]$"
Or in other words that "the fourier trigonometric system is complete over the Riemann integrable functions on $[-l,l]$"
I'm talking about this specific fourier series $$f \sim \frac{a_0}{2} + \sum_{n=1}^{\infty}{a_n\cos\frac{n\pi}{l}x + b_n\sin\frac{n\pi}{l}x}$$
Where $$a_n = \frac{1}{l}\int_{-l}^l{f(x)\cos\frac{n\pi}{l}x \, dx}$$
and $$b_n = \frac{1}{l}\int_{-l}^l{f(x)\sin\frac{n\pi}{l}x \, dx}$$
Any info you can give me on this would be appreciated.

Fourier Analysis an Introduction by Elias Stein and Rami Shakarchi.
