What is the free abelian group of an infinite set?

Question

Given a set $S$, the free abelian group is defined as the set of formal sums of elements of $S$. Is this restricted to formal sums of a finite subset of elements of $S$? For example, is $S = \mathbb{R}$, then is $\sum_{z\in \mathbb{Z}}a_z z$ a legitimate element of the free abelian group on $\mathbb{R}$? What does the free abelian group on $\mathbb{R}$ even look like? What about the free abelian group on other infinite sets like $\mathbb{R}^2$, $\mathbb{Z}$, and the set of functions $f: \mathbb{R}\to\mathbb{R}$?