Vector space span theorem I'm looking at ta theorem on linear algebra involving a spanning subsand I have a question regarding the definition:

Theorem: Let $S=\{v_1,v_2,\dots,v_r\}$ be a non-empty subset of a vector
  space $V$,  and the set $W=\{ \sum_{i=1}^{r} k_iv_i | k \in \mathbb{R}\}$
  as the set of all linear combinations of the vectors in $S$ is
  a subspace of $V$.
The set $W$ is the smallest subspace of $V$ that contains $S$, and is called the span of $S$ denoted by $span(S)$.

I'm a bit confused on what $W$ includes. Does it include a single sum of all the entries in $S$ multiplied by a scalar $k_i$ added together $W=\{(k_1 v_1 +k_2 v_2 + \dots k_i v_i)\}$? Does $k_i$ have to be a specific scalar for each $v_i$ in $S$? Or does it include more than one sums, if so with what scalars?

The span includes all sums of the form $k_1 v_1+k_2 v_2 +\cdots+k_r v_r$ (for all choices of $k_1,\dots, k_r \in\mathbb{R}$).
