Let $S^1 = \{ (x,y)\in \mathbb{R}^2 | x^2+y^2=1$. Let $z:\mathbb{R} \to S^1$ given by $z=(cos(\theta),sin(\theta))$. Define the map $z^* : C^0 (S^1) \to C^0 (\mathbb{R})$ by $z^* (f)= f \circ z$.
I am trying to prove the map $z^*$ is injective onto the subset of continuous periodic functions with period $2\pi$. I know how to prove surjectivity, however, I am not sure how to prove $z^*$ is injective and its image is a subset of the periodic functions with period of $2\pi$
Periodicity of the image of $z^*$ follows trivially from periodicity of $z$.
Let $z^*(f) = z^*(g)$.
Then, for all $x \in \mathbb{R}$, we have $f(z(x)) = g(z(x))$. But then clearly $f = g$, so $z^*$ is injective.