Let $x\in l^2(\mathbb{Z},\mathbb{C})$, we consider the fourier transform: $$F^*:l^2(\mathbb{Z},\mathbb{C})\rightarrow L^2([02\pi]\rightarrow\mathbb{C})$$ $$\mathcal{(F^*x)(y)}:=\sum_{n\in\mathcal{Z}}x_ne^{iny}$$
I need to prove that it's an isomorphism,
So first, is it well defined? if $||\sum_{n\in\mathcal{Z}}x_ne^{in(.)}||_{L^2}< \infty ?$