Hyperplanes as Level Sets of Linear Functionals

In real vector spaces, Ω is a hyperplane iff Ω={x : f(x)=α} for some f≠0.

Hyperplanes as Level Sets of Linear Functionals

Let $X$ be a real vector space .

Proposition: A subset $\Omega\subset X$ is a hyperplane if and only if there exist a nonzero linear functional $f:X\to\mathbb{R}$ and a scalar $\alpha\in\mathbb{R}$ such that

\Omega=\{x\in X\mid f(x)=\alpha\}.

Context: One direction uses the decomposition of codimension-one subspaces (see codimension-one decomposition ). The other direction uses that $\ker f$ has codimension one (see codimension of kernels ).