Are these sequences total sequences?

Solution 1:

I think the key result you are missing is the Stone Weierstrass theorem.

$\boxed{1}$ Assume that $$\int_{-1}^1 z(t) t^n dt = 0.$$ Then, by linearity of the integral $$\int_{-1}^1 z(t)p(t) dt = 0$$ for all polynomial functions $p: [-1,1]\to \mathbb{C}$. By the Stone-Weierstrass theorem, these functions are uniformly dense in $[-1,1]$, so there is a sequence of polynomial functions $\{p_n\}_{n=1}^\infty$ such that $$ p_n \to \overline{z}$$ where the convergence is uniform. Hence, since we can switch an integral and a uniform limit, we get $$\int_{-1}^1 |z(t)|^2 dt = \int_{-1}^1 z(t)\overline{z}(t) dt = \lim_n \int_{-1}^1 z(t)p_n(t)dt = 0$$ from which it follows that $z=0$.

$\boxed{2}$ I leave this as an exercise for you after you understand the third subquestion, which I will now explain.

$\boxed{3}$ Proceed as follows:

(1) Use the Stone-Weierstrass theorem to prove that the span of $\{e^{int}\}_{n \in \mathbb{Z}}$ is uniformly dense in $C([-\pi, \pi])$. It may be useful to note that the relations $$e^{int}e^{imt}= e^{i(m+n)t}, \quad \overline{e^{int}}= e^{-int}$$ hold.

(2) Proceed as in exercise $\boxed{1}$.