Polarsetes

Polarsets, or polar sets, are a concept in functional analysis and convex geometry that describe a relationship between a subset of a vector space and the linear functionals acting on it.

Definition. Let X be a real or complex vector space and X′ its dual space of linear functionals. For a subset A ⊆ X, the polar A° is defined as A° = {f ∈ X′ : sup{ |f(x)| : x ∈ A } ≤ 1 }. In topological vector spaces, polar sets relate to boundedness of functionals on A and can be defined with respect to different bounds or topologies.

Key properties. Polar sets are always convex and balanced. They encode how large a family of functionals can act on A without exceeding a specified bound. The polar of the unit ball in a normed space X is exactly the unit ball in the dual space X′, highlighting the link between norms and dual functionals. The concept of polars also connects to the support function hA(y) = sup{x∈A} ⟨y,x⟩, since A° = {f ∈ X′ : sup{ |f(x)| : x ∈ A } ≤ 1 } can be viewed as a bound on the action of functionals on A via the support function.

Bipolar and finite dimensions. The bipolar A°° often equals the closed convex balanced hull of A under suitable topologies; in finite-dimensional spaces, A°° is the closed convex balanced hull of A. This property underpins duality and separation results in convex analysis.

Applications. Polar sets are used in duality theory, optimization, convex geometry, and the study of operator boundedness. They provide a bridge between a set and its certificates of bounded linear action.

See also. Polar set (mathematics), dual space, Hahn–Banach theorem, support function, Fenchel conjugate, bipolar theorem.