@LeakyNun let $A$ be the span of $e^{2 \pi i nz}$, then we can't just say $A$ is dense in $C(S^1,\Bbb C)$ and $C(S^1,\Bbb C)$ is dense in $L^2(S^1)$ and hence $A$ is dense in $L^2(S^1)$ because there are two different topologies involved!
But this is not an actual problem because $\|f\|_{L^2(S^1)}^2 \leq \mathrm{vol}(S^1) \sup_{x \in S^1} |f(x)|^2$