Hyperplane

Let $X$ be a vector space . An affine set $\Omega\subset X$ is called a hyperplane if it has codimension one.

More precisely: if $\Omega\neq\emptyset$, it is parallel to a unique subspace $L=\Omega-\Omega$ (see Ω−Ω characterization ). Then $\Omega$ is a hyperplane iff $\operatorname{codim}(L)=1$.

In real vector spaces, hyperplanes are exactly level sets of nonzero linear functionals; see the level-set characterization .

Examples:

• In $\mathbb{R}^n$, the set $\{x\mid a^\top x=\alpha\}$ with $a\neq 0$ is a hyperplane.