In functional and convex analysis, and related disciplines of mathematics, the polar set A ∘ {\displaystyle A^{\circ }} is a special convex set associated to any subset A {\displaystyle A} of a vector space X , {\displaystyle X,} lying in the dual space X ′ . {\displaystyle X^{\prime }.} The bipolar of a subset is the polar of A ∘ , {\displaystyle A^{\circ },} but lies in X {\displaystyle X} (not X ′ ′ {\displaystyle X^{\prime \prime }} ).
Definitions There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis. In each case, the definition describes a duality between certain subsets of a pairing of vector spaces ⟨ X , Y ⟩ {\displaystyle \langle X,Y\rangle } over the real or complex numbers ( X {\displaystyle X} and Y {\displaystyle Y} are often topological vector spaces (TVSs)). If X {\displaystyle X} is a vector space over the field K {\displaystyle \mathbb {K} } then unless indicated otherwise, Y {\displaystyle Y} will usually, but not always, be some vector space of linear functionals on X {\displaystyle X} and the dual pairing ⟨ ⋅ , ⋅ ⟩ : X × Y → K {\displaystyle \langle \cdot ,\cdot \rangle :X\times Y\to \mathbb {K} } will be the bilinear evaluation (at a point) map defined by
⟨ x , f ⟩ := f ( x ) . {\displaystyle \langle x,f\rangle :=f(x).}
If X {\displaystyle X} is a topological vector space then the space Y {\displaystyle Y} will usually, but not always, be the continuous dual space of X , {\displaystyle X,} in which case the dual pairing will again be the evaluation map. Denote the closed ball of radius r ≥ 0 {\displaystyle r\geq 0} centered at the origin in the underlying scalar field K {\displaystyle \mathbb {K} } of X {\displaystyle X} by
B r := B r K := { s ∈ K : | s | ≤ r } . {\displaystyle B_{r}:=B_{r}^{\mathbb {K} }:=\{s\in \mathbb {K} :|s|\leq r\}.}
Functional analytic definition
Absolute polar Suppose that ⟨ X , Y ⟩ {\displaystyle \langle X,Y\rangle } is a pairing. The polar or absolute polar of a subset A {\displaystyle A} of X {\displaystyle X} is the set:
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