$\mathcal L^2$ convergence of the Fourier series of an $\mathcal L^1$ function?

Question

I'm reading Stein and Shakarchi's Fourier analysis. Here's a theorem from the book :

But how can we talk about the mean-square convergence of the fourier series of an integrable function? What if it's not square integrable? E.g. $f(x)=\frac{1}{\sqrt x}$ which is integrable but not square integrable. Even in the proof they have considered the 2-norm of $f$ without assuming $f$ to be square integrable. What am I missing?