Separation by a Hyperplane

Let $X$ be a real vector space and let $\Omega_1,\Omega_2\subseteq X$ be nonempty.

The sets $\Omega_1$ and $\Omega_2$ can be separated by a hyperplane if there is a nonzero linear functional $f:X\to\mathbb R$ such that

Equivalently,

Any $\alpha$ between these two values gives a hyperplane $\{x\in X:f(x)=\alpha\}$ lying between the sets.