Separation by a Closed Hyperplane

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

Let $X$ be a real normed space and let $\Omega_1,\Omega_2\subset X$ be nonempty.

We say that $\Omega_1$ and $\Omega_2$ can be separated by a closed hyperplane if there exists a nonzero functional $x^\ast \in X^\ast$ (see dual space) such that

\langle x^\ast ,x\rangle \le \langle x^\ast ,y\rangle \quad\text{whenever }x\in\Omega_1,\ y\in\Omega_2.

Here "closed hyperplane" emphasizes that $x^\ast$ is continuous, so each level set $\{x\mid \langle x^\ast ,x\rangle=\alpha\}$ is closed; see continuity via closed level sets.