polaarsetest

Polaarsetest refers to polar sets in convex analysis and functional analysis. Given a subset A of a locally convex topological vector space X, its polar A° is defined as the set of all continuous linear functionals that are uniformly bounded by 1 on A. In a normed space, this means A° = { φ ∈ X' : sup_{x ∈ A} |φ(x)| ≤ 1 }, where X' denotes the continuous dual of X. The polar of a subset B of the dual space X' is similarly defined as B° = { x ∈ X : sup_{φ ∈ B} |φ(x)| ≤ 1 }.

Polars have several basic properties. If A ⊂ B, then B° ⊂ A°. Scaling behaves by (λA)° = (1/|λ|) A° for any nonzero scalar λ. The polar A° is always closed, convex, and balanced (or circled). The polar construction interacts with duality and helps describe boundedness and separation: a functional belongs to the polar of A precisely when it is controlled on A by the unit bound.

A central result is the bipolar theorem: in a locally convex space X, the bipolar A°° is the closed convex balanced hull of A (viewed inside the bidual X''). In practical terms, the double polar encodes a canonical way to recover a well-behaved hull of A from its action on functionals.

Polaarsetest play a key role in duality theory, optimization and separation theorems, and in the study of gauge and Minkowski functionals. In Estonian mathematical usage, polaarsetest denotes these polar sets and their properties within similar frameworks. See also polar, duality, convex analysis, and Minkowski functional.