If $G$ is a topological group, how is $\ell^1(G)$ defined (is it necessary to require that $G$ be a topological group?)? From my understanding, it consists of all functions (continuous?) $f : G \to \Bbb{C}$ such that $\sum_{g \in G} |f(g)| < \infty$. But what does $\sum_{g \in G} |f(g)| < \infty$ mean? It should mean something like, "For every $\epsilon > 0$, there is some...(?)...such that $\sum_{g \in ?} |f(g)| < \epsilon$."
I guess I am equally interested in how one defines $\ell^p(G)$, which should be similar to the $p=1$ case.
$\ell^{1}(G)$ consists of all $f:G \to \mathbb C$ such that $\sup \{\sum_{\{g \in A\}} |f(g)|:A \subset G \, \text {finite}\} <\infty$. (The supremum is taken over all finite subsets $A$ of $G$). For $\ell^{p}$ just replace $|f(g)|$ by $|f(g)|^{p}$