Let $\ell^1:=\{f:\mathbb{N}\to\mathbb{C}\colon \|f\|_1:=\sum_{n=1}^\infty |f(n)|<\infty\}$ provided with $\|\cdot\|_1$. Show, that $S: \ell^1\to c_0'$ is an isometrical isomorphism.
First of all I am not sure on what $S$ is supposed to be, since there is missing a definition. Also $l^1\stackrel{?}{=}\ell^1$ might be a typo aswell. We defined $S_f: c_0\to\mathbb{C}$ by $S_f(g)=\sum_{n=1}^\infty g(n)f(n)$. But what is $S$? Also $c_0'$ notes the dual space of $c_0$.
Or do I have to define it myself, since I have to find a isomorphism of vectorspaces, which is an isometric?
Thanks in advance.