On denseness of a subspace of trigonometric polynomials in $L^1(\mathbb T).$

Question

$\mathbf {The \ Problem \ is}:$ Let, $e_n(\theta)=e^{in\theta}$ & $A=span\{e_n | n\geq 0\} \subset L^1(\mathbb T).$ Let $B=\bar A.$ Does $e_n \in B$ for any $n<0?$

$\mathbf {My \ approach}:$ The set of all trigonometric polynomials $\mathcal T$ is dense in $C(\mathbb T)$ under sup-norm(by Stone-Weierstrass) and if the answer to this question is positive, then $A$ becomes dense in $C(\mathbb T).$

But, is this right ? Any small hint ? Thanks in adv.

Answer 1

Hint: $\int e_n\overline {e_m}=0$ for $n <0$ and $ m\geq 0$. This implies $\int e_n\overline {f}=0$ for $n <0$ and $f \in B$. What happens if $f=e_n$?