Hahn-Banach Theorem
Statement (Real Version)
Statement (Complex Version)
Geometric Form (Separation Theorem)
Corollaries
Proof Technique (Zorn's Lemma)
The proof uses Zorn's lemma to extend the functional one-dimensional step at a time, maintaining the inequality at each step.
Applications
Dual Space Richness: The Hahn-Banach theorem guarantees the dual space is "large enough" to separate points and characterize the norm.
Supporting Hyperplanes: For any closed convex set and point , there exists a supporting hyperplane separating from .