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Strict Separation by a Closed Hyperplane

Strict separation means there is a positive gap between the two sets under a continuous functional.
Strict Separation by a Closed Hyperplane

Let XX be a real and let Ω1,Ω2⊂X\Omega_1,\Omega_2\subset X be nonempty.

We say that Ω1\Omega_1 and Ω2\Omega_2 can be strictly separated by a closed hyperplane if there exist x∗∈X∗x^\ast \in X^\ast (see ) and real numbers α<β\alpha<\beta such that

⟨x∗,x⟩≤α<β≤⟨x∗,y⟩for all x∈Ω1, y∈Ω2. \langle x^\ast ,x\rangle \le \alpha < \beta \le \langle x^\ast ,y\rangle \quad\text{for all }x\in\Omega_1,\ y\in\Omega_2.

Equivalently, strict separation holds iff there exists x∗∈X∗x^\ast \in X^\ast with

sup⁡x∈Ω1⟨x∗,x⟩<inf⁡y∈Ω2⟨x∗,y⟩. \sup_{x\in\Omega_1}\langle x^\ast ,x\rangle < \inf_{y\in\Omega_2}\langle x^\ast ,y\rangle.

This notion strengthens by requiring a strict gap.