Vector space/Spanned linear subspace/Definition

Let ${\displaystyle {}K}$ be a field, and let ${\displaystyle {}V}$ be a ${\displaystyle {}K}$-vector space. For a family ${\displaystyle {}v_{i}}$, ${\displaystyle {}i\in I}$, we set

${\displaystyle {}\langle v_{i},\,i\in I\rangle ={\left\{\sum _{i\in J}s_{i}v_{i}\mid s_{i}\in K,\,J\subseteq I{\text{ finite subset}}\right\}}\,,}$

and call this the linear span of the family, or the generated linear subspace.