How can I able to show that the following set is dense or not?

Question

How can I able to show that the following set is dense or not?
The set of trigonometric polynomials in the space of continuous functions on $[−\pi, \pi]$ which are $2\pi$-periodic (with the sup-norm topology).

Answer 1

The trigonometric polynomials are dense in $2\pi$-periodic functions. As pointed out in the comment, this can be easily done with techniques from Fourier analysis. But if you are willing to assume the theorem of Stone-Weierstrass, then it could be understood without using Fourier (also I vaguely remember Stone-Weierstrass can be established without heavy use of Fourier).

First note that the space of continuous $2\pi$-periodic functions on $[-\pi,\pi]$ is the same as the space of continuous functions on the unit circle, $C(\mathbb{S})$. Now since $\mathbb{S}$ is a compact subset of $\mathbb{C}$, and the inclusion function $\zeta$ is continuous and separates points on $\mathbb{S}$.

So by Stone-Weierstrass, the unital $C^*$-algebra generated by $\zeta$ (this space consisting of polynomials of $\zeta$ and $\bar{\zeta}$, and all the possible limiting points) is $C(\mathbb{S})$, so we know polynomials of $\zeta$ and its conjugate is dense in $C(\mathbb{S})$.

But these polynomials are exactly the trigonometric polynomials by Euler's formula.