Vector space/Spanned linear subspace/Definition

Linear span

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

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

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