The definition of a dcpo, or directed complete partial order, can be found here.
A real vector is formed with a basis set $B =\{ e_1, e_2 \ldots \}$, and the real numbers $\mathbb{R}$. A vector is a formal sum of the form $v = \Sigma_i a_i e_i$, for $a_i \in \mathbb{R}$. Let us define the "terms" as a span $\mathbb{R} \leftarrow Term \rightarrow \mathbb{R}$. Where the left arrow gives the real coefficient and the right arrow gives an integer, identifying the basis vector. This allows us to have a set of "terms", binding the basis vector to the coefficient. Two vectors may have some of the same terms. We can even say that, for $v,w$, $Term_v \subseteq Term_w$ which means that $w$ has all the terms of $v$ and perhaps some more (or no more)
Define the following ordering:
$v \le w$ if $Term_v \subseteq Term_w$
Does this define a dcpo? What kind of ordering is this? What are the compact elements of the dcpo?
There's something weird: can you have vectors with an infinite number of non-trivial coordinates (since you wrote "formal sum") in your set (let's call it $$)?
I assume it's not the case (as you tagged your question under "vector spaces"): in this case, your set is not a dcpo if your basis set is infinite: consider the set $$\left\{ \sum_{i≤n} e_i \mid n ∈ ℕ\right\}$$ it's a chain (hence a directed set), but it has no upper bound.
if your basis set is finite (let's denote by $n ∈ ℕ$ its cardinal), you can write every vector $v ≝ \sum\limits_{i=0}^n a_i e_i$ as the $n$-tuple $(a_i, \ldots, a_n) ∈ ℝ^n$. As it happens, since $$v ≝ \sum\limits_{i=0}^n a_i e_i ≤ w ≝ \sum\limits_{i=0}^n a_i' e_i \quad ⟺ \quad ∀i, \, a_i = 0 \, ∨ \; a_i = a_i'$$ your set $$ is a product of flat pointed dcpos (where $0 ≤ a$ for every $a ≠ 0$ and $a ≤ a$ for every $a$ (and you have no other order relations)), and, as such, is a dcpo (the category of dcpos is cartesian closed).
As for the compact elements in this case: one can show that every directed set is finite (of cardinal at most $n$), and thus contains its least upper bound. As a consequence, all non-zero vectors are compact.