Parallel Affine Set

Let $X$ be a vector space . An affine set $\Omega\subset X$ is said to be parallel to a linear subspace $L\subset X$ if

for some $\omega\in\Omega$.

By the translate characterization , every nonempty affine set is parallel to at least one subspace. The parallel subspace is unique and can be written explicitly as $\Omega-\Omega$; see the Ω−Ω proposition .

Examples:

• In $\mathbb{R}^n$, any affine subspace is parallel to its direction subspace (the translation of the affine set through the origin).