Hahn-Banach

Using perturbation duality, we prove the Hahn–Banach theorem in Euclidean space. The theorem generalizes basic separation. The primal models a simple condition: for a convex set and an affine set, either the sets have a vector in common (optimum value zero) or they do not (optimum value infinity). For the latter condition, one obtains the Hahn–Banach separation theorem. For the former condition, one obtains the Hahn–Banach extension theorem. That the same primal/dual pair obtains both theorems reveals their intrinsic interconnection. If the two sets have no vector in common, one obtains the Hahn–Banach separation theorem: there exists a hyperplane containing the affine set such the convex set lies entirely on one side of the hyperplane. A value for the dual choice variable defines the hyperplane: select it so the value of the objective function is positive. If the two sets have a vector in common, then the optimum value is zero. For the optimum value of the dual to be zero is the defining condition for the Hahn–Banach extension theorem: there exists a linear function defined on a subspace such that a certain property holds. The theorem asserts that one can extend this linear function to the entire space, such that this same property holds. The perturbation variable for the primal defines the linear function on the subspace. Adding any solution to the primal to the perturbation variable extends the linear function to the entire space.