A Proof of Series of A Product of Fourier Coefficients Converges Absolutely

Question

I am reading Rudin's RCA. In the section 4.26, Rudin says the Parseval theorem asserts $$\sum_{n=-\infty}^{\infty} \hat{f}(n)\overline{\hat{g}(n)}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)\overline{g(t)}dt$$ for all $f, g\in L^2(T)$.
Then Rudin says that the series on the left side of above equality converges absolutely.
How to justify this? If the case that $f=g$ holds, clearly the series converges absolutely.

Answer 1

Recall that $\ell^2$ is a Hilbert space, with inner product $\langle \hat{f}(n),\hat{g}(n)\rangle=\sum_n\hat{f}(n)\overline{\hat{g}(n)}.$ Using the Cauchy-Schwarz inequality, $$ \sum_n\left|\hat{f}(n)\overline{\hat{g}(n)}\right|=\langle |\hat{f}(n)|,|\hat{g}(n)|\rangle\leq\left(\sum\limits_n|\hat{f}(n)|^2\right)^{1/2}\left(\sum\limits_n|\hat{g}(n)|^2\right)^{1/2}<\infty,$$ since $f$ and $g$ are in $L^2(\mathbb{T}),$ so the sums of their Fourier series coefficients converge in $\ell_2.$