First I'll give some context for my question. I'm learning about crossed products of dynamical systems involving $C^*$-algebras and I've just seen the definition of a covariant representation. I have no problem with this definition, but I'm working through an example to illustrate it and I'm unfamiliar with a piece of notation which I haven't managed to find, so I'd just like to clarify that it is what I think it is.
I'm attempting to define a covariant representation $(\pi,U)$ for the action of a group $G$ on $\ell^\infty(G):=\{$bounded functions $f:G\rightarrow\mathbb{C}$$\}$ given by $(g\cdot f)(h)=f(g^{-1}h)$. My notes say we will want to take $\pi$ to be a representation of $\ell^\infty(G)$ on $\ell^2(G)$. Unfortunately, I can't find the definition of the Hilbert space $\ell^2(G)$ anywhere. Here is my guess at its definition in analogy with my usual notion of $\ell^2$, and in light of the definition of $\ell^\infty(G)$ I do have:
$$\ell^2(G)=\Big\{ f:G\rightarrow\mathbb{C}:\sum\limits_{g\in G}|f(g)|^2<\infty\Big\}.$$
Is this the correct definition?
This is the correct definition. See, for example, page 7 of Isaac Goldbring's notes here.
One thing to note is that this is just the usual $L^2$ space on $G$ with respect to $G$'s Haar measure $\mu$, normalized so that $\mu({1}) = 1$ (although the particular normalization doesn't matter). Since $G$ is discrete, $\langle f, g \rangle = \int f\bar{g} d\mu = \sum_{\gamma \in G} f(\gamma)\bar{g}(\gamma)$. You can of course do a similar construction for other (locally compact Hausdorff) groups and their Haar measures.