Separation by a Closed Hyperplane

Separation using a nonzero continuous functional in the dual space.

Separation by a Closed Hyperplane

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 .