What would be a basis for the free vector space $C(\mathbb{R}^2)$? I have been reading about free vector space, but I would like to see it on a concrete example. The definition I have is the following:

Let $X$ be a set. Define $C(X)=\{f:X\to \mathbb{F} \mid f^{-1}(\mathbb{F}-\{0\}) \text{ is finite} \}$. One can show that $C(X)$ is a vector space with underlying field $\mathbb{F}$.

In addition,

$C(X)$ also has a basis. Let $y \in X$. If we define $$\delta_y(x)=\begin{cases} 1,&\text{ if }x = y,
\\0,&\text{ if } x \ne y.\end{cases}$$
  Then it is easy to see that for any $f\in C(X)$, we have
  $$ f = \sum_{y\in X} f(y) \delta_y$$

These all make sense to me, except that I don't see what they look like! 
Let $X = \mathbb{R}^2$, $\mathbb{F} = \mathbb{R}$. What does $C(\mathbb{R}^2)$ look like? First, the dimension (or cardinality of basis) is uncountable. This is already quite terrifying. Second, I cannot even come up with a function $f:\mathbb{R}^2\to\mathbb{R}$ such that it has finite support. Can anyone help me write out a vector in $C(\mathbb{R}^2)$? 
Also, is $C(\mathbb{R}^2)$ isomorphic to (as vector spaces) any more familiar vector space? 
 A: Don't think about $\mathbb{R}^2=X$ in $C(\mathbb{R}^2)$ as having any of the structure one typically attributes to $\mathbb{R}^2$.  From the point of view of the free vector space, this is just a set with no additional important information except for its cardinality.  
Pick some arbitrary points $x_1,x_2,\ldots, x_n \in X$, and some arbitrary elements $\lambda_1,\ldots, \lambda_n \in \mathbb{R}$.  Then the function $f: X \to \mathbb{R}$ which sends $x_i$ to $\lambda_i$, and any other point $x$ to $0$, has finite support.  
In fact, all the elements in $C(X)$ look like this.
As you've noted, a basis is given by $\{\delta_x \mid x\in X\}$.  Let's just identify $x$ with $\delta_x$.  Then a function $f$ above can be written as $\lambda_1 x_1 + \lambda_2 x_2 + \cdots + \lambda_n x_n$.  This looks innocent enough.
The idea behind a free vector space is that we want to create a vector space which has $X$ (or at least something naturally associated with it) as a basis.  So we expect such a vector space to be the set of elements of the form above. 
The construction given is simply making this rigorous.
