Let $\mathbb{T}$ be the unit circle in the complex plane, that is $\mathbb{T}=\{z\in\mathbb{C}: |z| = 1\}$. I have read that there is a bijection between the class of complex-valued functions on $\mathbb{T}$ and the class of complex-valued $2\pi$-periodic functions on the real line $\mathbb{R}$. This bijection was explained like the following
A function $f$ on the unit circle induces a 2$\pi$-periodic function $g$ on the real line by $$ g(x) = f(e^{ix}), \;\;\; x\in\mathbb{R}. $$ This correspondence is clearly bijective.
I have tried to prove the result above myself but I am unfamiliar with the notion of a bijection between classes. If we are talking about the sets of functions on $\mathbb{T}$ and $\mathbb{R}$ then I understand. Is he just using sloppy wording? Does he mean to say that $f(e^{ix})$ is the bijection from the set of 2$\pi$-periodic, complex-valued functions on $\mathbb{R}$ to the set of functions on $\mathbb{T}$? If so, I need some help in showing that because I have not come up with anything myself.