I'm cofused with the definition of the spaces $L^2(S^1)$ and $L^2[0,2\pi]$. I have read that we can consider $L^2(S^1)$ as the elements $f \in L^2[0,2\pi]$ such that $f(0)=f(2\pi)$ . But in my opinion the latter does not have sense because in $L^2[0,2\pi]$ we can't talk about point values.
Could you explain me what is the difference between $L^2(S^1)$ and $L^2[0,2\pi]$, please? I will appreciate if you can give me a reference related to this question.
$L^2([0,2\pi])$ is the completion of $C([0,2\pi])$ with respect to the $L^2$-norm. Now $C(S^1)$ can be viewed as the subspace of $C([0,2\pi])$ consisting of functions such that $f(0)=f(2\pi)$. Moreover, $L^2(S^1)$ is the completion of $C(S^1)$ with respect to the $L^2$-norm. This is the sense in which $L^2(S^1)$ can be viewed as a subspace of $L^2([0,2\pi])$.