Bipolar theorem

The bipolar theorem identifies the bipolar of a set in a paired vector space with its closed convex hull containing the origin, under the pairing’s appropriate topology.

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Dual pairing: A bilinear map between two vector spaces that lets elements of each space evaluate elements of the other. It supplies the evaluations used to define each polar.
Locally convex topological vector space: A topological vector space whose topology is generated by a family of seminorms. The standard theorem uses locally convex spaces and their continuous duals.
Hahn–Banach separation theorem: A theorem guaranteeing continuous linear functionals that separate suitable convex sets in locally convex spaces. Separation arguments establish that points outside the closed convex hull fail the bipolar inequalities.
Convex conjugate: The transform assigning a function the supremum of its affine linear underestimators. Set bipolarity is the geometric counterpart of recovering convex objects through dual transforms.
Polar set: The set of vectors whose pairing with every point of a given set satisfies a specified bound, commonly one. The theorem applies this operation twice to recover a hull.
Continuous dual space: The vector space of continuous linear functionals on a topological vector space. Its functionals form one side of the pairing in the usual formulation.
Fenchel–Moreau theorem: A theorem identifying a proper lower-semicontinuous convex function with its biconjugate. It extends the same dual-reconstruction pattern from sets to convex functions.
Fenchel duality: A framework relating an optimization problem to a dual problem built from convex conjugates. Polar constraints help express feasible sets and derive dual optimization formulations.
Convex hull: The smallest convex set containing a given set. Taking a bipolar enforces convexity, so nonconvex features disappear.
Weak topology: The coarsest topology making a specified family of linear functionals continuous. The closure in the conclusion depends on the topology selected by the dual pairing.