Knowlpedia

Separation by Closed Hyperplane Under an Interior Condition

If int(Ω1)≠∅ and int(Ω1)∩Ω2=∅, then Ω1 and Ω2 are separable by a continuous functional.

Let XX be a real and let Ω1,Ω2⊂X\Omega_1,\Omega_2\subset X be nonempty . Assume that ≠∅\neq\emptyset and

int⁡(Ω1)∩Ω2=∅.\operatorname{int}(\Omega_1)\cap\Omega_2=\emptyset.

Theorem: The sets Ω1\Omega_1 and Ω2\Omega_2 can be ; i.e., there exists x∗∈X∗∖{0}x^\ast \in X^\ast \setminus\{0\} such that

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

Context: In vector spaces, the analogous separation uses possibly discontinuous linear functionals (see ). The interior assumption ensures the separating functional is continuous, using .