Consider the following proof showing the density of the range of the fourier transfrom $\mathcal{F}: L^1[0,2\pi] \to c_0(\mathbb{Z})$
Let $\left(e^{(k)}\right)_{k \in \mathbb{Z}} \subset c_0(\mathbb{Z})$ be given by $e_n^{(k)}:=\delta_{n k}$, for every $k, n \in \mathbb{Z}$. Notice that $\left(e^{(k)}\right)_{k \in \mathbb{Z}}$ is dense in $c_0(\mathbb{Z})$. Moreover, we have $\left(e^{(k)}\right)_{k \in \mathbb{N}} \subset \mathcal{F}\left(L^1([0,2 \pi], \mathbb{C})\right)$, because $e^{(k)}=\mathcal{F}\left([0,2 \pi] \ni t \mapsto e^{i k t} \in \mathbb{C}\right)$ for every $k \in \mathbb{Z}$. Hence, we conclude that $\mathcal{F}\left(L^1([0,2 \pi], \mathbb{C})\right)$ is dense in $c_0(\mathbb{Z})$.
My problem is to understand how $\left(e^{(k)}\right)_{k \in \mathbb{Z}}$ is dense in $c_0(\mathbb{Z})$ as $c_0$ uses the infinity norm and for example choosing $(...0,0,2,0,0...) \in c_0(\mathbb{Z})$ This can't possibly be arbitrary close to any element $\left(e^{(k)}\right)$ right? Am I missing something? I begin to believe they may have meant the linear span $span(\left(e^{(k)}\right))$. This would indeed make sense to me.