What is $\mathcal{C}(S^{1})$ (Continuous function on a unit circle)? (Where $S^1$ denotes unit circle)
I saw this in a proof of showing Fourier Basis $S:=\{1,\sqrt{2}\cos{nx},\sqrt{2}\sin{nx}\}$ is an orthonormal basis of $L^{2}[-\pi ,\pi]$
The proof says $\mathcal{C}(S^{1})$ is equivalent to
$C^{*}[-\pi,\pi]=\{f\in C[-\pi,\pi]:f(-\pi)=f(\pi)\}$
then used the facts
S is dense in $\mathcal{C}(S^{1})$
$C^{*}[-\pi,\pi]$ is dense in $C[-\pi,\pi]$
and $C[-\pi,\pi]$ is dense in $L^{2}[-\pi ,\pi]$