I'm aware that it's a trivial question, but I want to make sure that I'm understanding correctly what I'm studying, therefore I would like to ask you, can you tell me what does $R^D$ "say" in these three different cases? Does it always say the same thing or the meaning is different?
1) Let "G" be a finite-dimensional vector space of real functions in $R^D$.
2) $x \in R^D$
3) {$x_1, x_2, ..., x_m$} $\subseteq R^D$
These are all different, though you may have intended $2$ and $3$ to be the same if you've mixed up your vector and set notation.
For item $1$ the elements of $G$ are all functions. "in" is a little bit unclear, but I think most people would interpret it as meaning that each element $f\in G$ is a mapping $f:{\mathbb R}^D \rightarrow {\mathbb K}$ where ${\mathbb K}$ is either ${\mathbb R}$ or ${\mathbb C}$. It could however mean functions taking values in ${\mathbb R}^D$. Note that there aren't all that many (mathematically speaking) finite dimensional function spaces: ${\mathbb R}^n$ and $({\mathbb R}^n)^*$ for each $n$ are most of them (if you throw away enough topological structure you can find a few more, but they're not really interesting then).
For item $2$, ${\mathbb R}^D = {\mathbb R} \times {\mathbb R} \times \cdots {\mathbb R}$ is the cartesian product of $D$ copies of ${\mathbb R}$, and a typical element $x\in {\mathbb R}^D$ therefore can be represented as a vector of $D$ real numbers $(x_1, x_2, \ldots , x_D)$ As an example, consider the three orthogonal unit vectors in ${\mathbb R}^3: (1,0,0), (0,1,0)$ and $(0,0,1)$
For item $3$ you have a set of $m$ elements that is a subset of ${\mathbb R}^D$. As an example here, consider $\{x \in {\mathbb R}^3 | \|x\|\leq 1 \}$ where the norm is the $\max$ norm. Then this set contains all three vectors from the previous example, as well as many more.