Affine subspace/Translated linear subspace/Definition

Let ${\displaystyle {}V}$ be a vector space. An affine subspace of ${\displaystyle {}V}$ is (the empty set or) a subset of the form

${\displaystyle {}P+U={\left\{P+u\mid u\in U\right\}}\,,}$

where ${\displaystyle {}U\subseteq V}$ is a linear subspace and ${\displaystyle {}P\in V}$ is a vector.