Consider $s_n(x) := e^{inx}$ for any $n \in \mathbb{Z}$, and define ${\cal A} := span\{s_n\}_{n \in \mathbb{Z}}$, I think there is a theorem says that ${\cal A}$ is dense in $(C([-\pi,\pi],\mathbb{C}),||\cdot||_{\infty})$
To prove this theorem, we need to invoke Stone-Weistrass Theorem, one thing I don't understand is that why ${\cal A}$ separates the points over $[-\pi,\pi]$? For any $f \in {\cal A}$ we must have $f(-\pi) = f(\pi)$ right?
Thanks for your help!
That's an excellent catch. Indeed, the trigonometric polynomials are dense in the space of periodic functions on an interval. Meaning that actually, trig polynomials are dense in the space $$ \left(C(S^1,\mathbb{C}),||\cdot||_\infty\right) $$