What is $\bigoplus_{(m,n)\in \mathbb{R}^2\times\mathbb{R}^2} \mathbb{R}\delta_{(m,n)}$? One way to define free vector space is to consider a set of functions. To be more concrete, if we let our field $\mathbb{F} = \mathbb{R}$, and our set be $\mathbb{R}^2\times\mathbb{R}^2$, then the element in the free vector space $C_\mathbb{R}(\mathbb{R}^2\times\mathbb{R}^2)$ would be functions $f:\mathbb{R}^2\times\mathbb{R}^2 \longrightarrow \mathbb{R}$ in form
$$f = \sum_{y\in \mathbb{R}^2\times\mathbb{R}^2} f(y)\delta_y$$
Yet, according to this notes, we can also formulate free vector space using direct sum. 
$$C_{\mathbb{R}}(\mathbb{R}^2\times\mathbb{R}^2) := \bigoplus_{(m,n)\in \mathbb{R}^2\times\mathbb{R}^2} \mathbb{R}\delta_{(m,n)}$$
I am having some trouble understanding the second notation since $\delta_{(m,n)}$ is a function. How do we take direct sum here? Are we taking direct sum of vector spaces? Does the following make sense (ignoring the fact that $\mathbb{R}^2\times\mathbb{R}^2$ is in fact uncountable)
$$\bigoplus_{(m,n)\in \mathbb{R}^2\times\mathbb{R}^2} \mathbb{R}\delta_{(m,n)}=\mathbb{R}\delta_{(v_1,w_1)} \oplus \mathbb{R}\delta_{(v_2,w_2)}\oplus\cdots$$
But what is the thing on the right hand side? Why is it equivalent to the first definition of free vector space?
 A: Here $\delta_{(m,n)}$ is considered as an element of the vector space $\mathbb{R}^{\mathbb{R}^2\times\mathbb{R}^2}$ of all functions $\mathbb{R}^2\times\mathbb{R}^2\to\mathbb{R}$ (with addition and scalar multiplication defined pointwise).  So $\mathbb{R}\delta_{(m,n)}$ is a one-dimensional subspace of $\mathbb{R}^{\mathbb{R}^2\times\mathbb{R}^2}$, consisting of all scalar-multiples of $\delta_{(m,n)}$.
Now, your description of $C_\mathbb{R}(\mathbb{R}^2\times\mathbb{R}^2)$ is also a subspace of $\mathbb{R}^{\mathbb{R}^2\times\mathbb{R}^2}$, and it contains $\mathbb{R}\delta_{(m,n)}$ for all $(m,n)$.  What is being asserted here is that in fact $C_\mathbb{R}(\mathbb{R}^2\times\mathbb{R}^2)$ is the internal direct sum of the subspaces $\mathbb{R}\delta_{(m,n)}$.  That means that each element $f\in C_\mathbb{R}(\mathbb{R}^2\times\mathbb{R}^2)$ can be written uniquely as a sum $\sum_{i=1}^n f_i$, where each $f_i$ is a nonzero element of a different $\mathbb{R}\delta_{(m,n)}$.  But indeed, this exactly what your description of $C_\mathbb{R}(\mathbb{R}^2\times\mathbb{R}^2)$ gives: any $f$ can be written as $\sum_y f(y)\delta_y$, where $y$ ranges over all points such that $f(y)\neq 0$, and $f(y)\delta_y$ is some nonzero element of $\mathbb{R}\delta_y$.  The uniqueness of this representation is just the fact that the vectors $\delta_{(m,n)}$ are linearly independent.
