I have some doubt about the proof of the following theorem using SWT (Rudin 8.15):
If $f$ is continuous (with period $2\pi$) and if $\epsilon >0$, then there is a trigonometric polynomial $P$ such that $\forall\ x \in \Bbb{R}$, $|P(x)-f(x)|< \epsilon$.
The proof is as follows:
If we identify $x$ and $x +2\pi$ (typo by Rudin?), we may regard the $2\pi$-periodic functions on $\Bbb{R}$ as functions on the unit circle $T$, by means of the mapping $x \mapsto e^{ix}$. The trigonometric polynomials form a self adjoint subalgebra $\mathscr{A}$, which separates points on $T$ and vanishes at no point of $T$. Since $T$ is compact, SWT tells us that $\mathscr{A}$ is dense in $C(T)$. This is exactly what the theorem asserts.
I have already verified that $\mathscr{A}$ is an subalgebra of $C(T)$ which is self adjoint, separates points, and vanishes at no point. However, I think the theorem asserts that $\mathscr{A}$ is dense in the set of (complex valued) continuous 2$\pi$ periodic functions $C_{per}$, not exactly $C(T)$. The proof looks intuitive to me but how can we regard the periodic functions on $\Bbb{R}$ as functions on $T$, rigorously?
Shall I attempt to show that the mapping $\phi: C_{per} \to C(T)$ defined by $\phi(f) = f \circ \arg$ is a homeomorphism so that the density can be taken to $C_{per}$?