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In complex vector spaces, separation holds via the real part of a complex linear functional.
Let be a complex vector space and let be nonempty convex sets. Assume core(Ω₁) and .
Theorem: There exists a nonzero complex-linear functional on such that
Context: View as a real vector space and apply the real separation theorem to obtain a real linear functional . Then form a complex functional whose real part is .
A vector space over a field is a set equipped with two operations (functions) and such that for all and :
Vector spaces are the basic objects studied via linear maps and invariants such as the determinant and eigenvalues of operators.
Let be a vector space (typically over ). A set is convex if for all and all we have
Context. Equivalently, is convex iff it contains every line segment joining any two of its points. Convexity is the core geometric notion underlying convex analysis and optimization.
Let be a real vector space and let be nonempty convex sets. Assume core(Ω₁) and
Theorem: The sets and can be separated by a hyperplane.
Context: This strengthens the disjointness condition by requiring only that avoid the core of . The argument uses idempotence of core and the auxiliary separation lemma.