This exercise suggests an alternative proof of Plancherel's Theorem for $L^2(\mathbb{T})$. Given $f \in L^2(\mathbb{T})$, define $f^*(\theta) := \overline{f(-\theta)}$.
(a) Prove that for any trigonometric polynomial $f$, it follows from Fourier inversion that $$\Vert f \Vert_2^2 = (f * f^*)(0) = \sum_{-\infty}^\infty |\hat{f}(n)|^2.$$ (b) Use the above to prove that $f \mapsto \hat{f}$ is an isometry from $L^2(\mathbb{T})$ to $\ell^2(\mathbb{Z})$.
I have shown (a), but am having trouble with (b). I think that the density of trigonometric polynomials will help extend the above formula to all $f$.