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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 $L^{2}[-\pi ,\pi]$

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    @Polymorpher I think you should post your comment as an answer to the question. Otherwise the users will be opening this "unanswered" question only to find there is nothing to do here.2012-06-09

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This question is already answered. Thanks to @Qiaochu Yuan.

Here $\mathcal{C}(S^{1})$ is merely space of functions defined on unit circle in $\mathbb{R}^{2}$ sense. It make sense to say this is equivalent to space of functions defined on $[-\pi,\pi]$ that have the same value on $\pi$ and $-\pi$.

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    Sorry. Now I have fixed it.2012-06-11