Unit group of group algebra can be identified with the space of function $G \rightarrow k?$

Question

Consider the group algebra $k[G].$ Supposedly, the unit group $k[G]$ can be identified as the space of functions $G$ to $k.$ I am a little confused by this. Suppose $\sum_{g \in G} a_gg$ is a unit. Then it has an inverse. That is, $\sum_{g \in G} \sum_{h \in G} a_{gh^{-1}}b_h g = 1.$ This means that $\sum_{h \in G}a_{gh^{-1}}b_h = 0$ for $g \neq e$ and 1 otherwise. However, how does this define a function from $G \rightarrow k?$ Furthermore, doesn't such a function define any such element of the group algebra as we can map $g \in G$ to its coefficient?