I'm studying Real analysis, Folland section 8.4 page 257. I can't undersfand the below highlighted statement.
If $f\in L^1(\mathbb{T^n})$ and $\hat{f}$ $\in l^1(\mathbb{Z^n})$, then the Fourier series $\sum_k \hat{f}(x)e^{2\pi ik\cdot x}$ converges absolutely and uniformly to a function $g$. Since $l^1 \subset l^2$, it follows from Theorem 8.20 ($e^{2\pi ik\cdot x}$'s consist an orthonormal basis of $L^2(\mathbb{T^n})$) that $f \in L^2$ and that the series converges to $f$ in the $L^2$ norm.
From Theorem 8.20, I confirm that $g$ is in $L^2$. But, why $f$ should be in $L^2$?
$\hat{f}$ $\in l^1(\mathbb{Z^n})$ implies that $\hat{f}$ $\in l^2(\mathbb{Z^n})$ and this implies that the Fourier series converges in $L^2$ to $f$. Since it converges pointiwse to $g$ it follows that $f=g$ a.e..
[If the FS converges to $F$ then $F$ and $f$ are $L^{1}$ functions having the same Fourier coefficients, which implies that $F=f$ a.e. Hence the series converges to $f$ in $L^{2}$].