I was reading one write-up written by my senior. In preliminaries, he set $\mathbb T$ as unit circle. Then he wrote following:
$L^2(\mathbb T)=\{f:\mathbb T \to \mathbb C :\frac{1}{2\pi}\int_{-\pi}^{\pi}\mid f(e^{it})\mid^{2}dt<\infty\}$. I am bit confused here. I know we can view function on unit circle as function on $\mathbb R$ which is $2\pi$ periodic. So, as $\mathbb T$ is unit circle, we can use parametrisation $z(t)=e^{it}$..$t\in[-\pi, \pi]$.Then what I was thinking is if we use definition of $L^2$, then
$L^2(\mathbb T)=\{f:\mathbb T \to \mathbb C:\frac{1}{2\pi}\int_{\mathbb T}\mid f(z)\mid^{2}dz< \infty\}$
$L^2(\mathbb T)=\{f:\mathbb T \to \mathbb C :\frac{1}{2\pi}\int_{-\pi}^{\pi}\mid f(e^{it})\mid^{2}ie^{it}dt< \infty\}$.
Is he missing factor $"ie^{it}"$ in his definition? Or I am wrong? Please help. Thanks.