Let us consider the $n-$ dimensional sequence $(a_k)_{k\in\mathbb{Z}^n}$ defined to be a map $a:\mathbb{Z}^n \rightarrow \mathbb{C}$ such that $$a(k)=a_k.$$
Let $$ C_0(\mathbb{Z}^n)=\{(a_k)_{k\in \mathbb{Z}^n|\lim_{\|k\|\to \infty}a_k=0\} . $$ Is dual of $C_0(\mathbb{Z}^n)$ equal to $l^1(\mathbb{Z}^n)$ like the one dimensional case?
Show that by density of the finite sequences $$\|f\|_{\ell^1}=\infty \implies \sup_{a\in C^0,\|a\|_\infty \le 1} |f(a)|=\infty$$ which implies that there is a sequence $\|a_m\|\to 0,f(a_m)=1$ ie. $a\to f(a)$ is not continuous $C^0\to \Bbb{C}$.
Conversely if $f\in \ell^1$ then by density of the finite sequences $f(a)=\langle a,f\rangle$