Separation Theorems and Supporting Functionals in Normed Spaces: Structure, Duality, and Applications
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This paper presents a rigorous, application-oriented survey of separation theorems and supporting functionals for convex sets in normed and Banach spaces. Emphasizing geometric version of the Hahn-Banach theorem, this work develops clean conditions for strict and non-strict separation, existence of supporting hyperplanes, and links to dual cones and polar sets. We highlight the role of weak and weak-star topologies in separation and illustrate how these results undergird feasibility, sensitivity, and duality principles in convex optimization. Short, self-contained proofs and examples are provided to keep the exposition accessible while maintaining mathematical precision. The paper thereby complements classical treatments of linear functional extension by focusing on geometric separation mechanisms and their applied consequences, especially in linear programming, convex feasibility, and basic duality frameworks. This comprehensive investigation into separation theorems reveals fundamental geometric structures that underpin both theoretical functional analysis and practical optimization applications, providing a unified framework for understanding convex separation phenomena across diverse mathematical contexts.
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