I am trying to show use Stone-Weierstrass Theorem to show that trig polynomials are dense in $L^2([0,2\pi])$, however the following things concerns me.
trig polynomails doesn't separate points since $f(0) = f(2\pi)$. My solution is to show that trig polynomial is dense in all $[0,2\pi - \epsilon]$, then let $\epsilon \rightarrow 0$, we could say that trig polynomials are dense almost everywhere in $[0,2\pi]$. (I doubt if this is correct)
Stone-Weierstrass theorem only provide density in $C(K)$ (all complex continuous functions on compact K). How could we extend this to the space of $L^2$?
I hope you could give me hints on these. Thank you!