Linear subspaces/Sum/Definition

For a ${\displaystyle {}K}$-vector space ${\displaystyle {}V}$ and a family of linear subspaces ${\displaystyle {}U_{1},\ldots ,U_{n}\subseteq V}$, we define the sum of these linear subspaces by

${\displaystyle {}U_{1}+\cdots +U_{n}={\left\{u_{1}+\cdots +u_{n}\mid u_{i}\in U_{i}\right\}}\,.}$