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 ⟨, ⟩ over the real or complex numbers.
Functional analytic definition
Absolute polar
Suppose that is a pairing. The polar or absolute polar of a subset of is the set: where. This is an affine shift of the geometric definition; it has the useful characterization that the functional-analytic polar of the unit ball is precisely the unit ball. The prepolar or absolute prepolar of a subset of is the set: Very often, the prepolar of a subset of is also called the polar or absolute polar of and denoted by ; in practice, this reuse of notation and of the word "polar" rarely causes any issues and many authors do not even use the word "prepolar". The bipolar of a subset of, often denoted by, is the set ; that is,
Real polar
The real polar of a subset of is the set: and the real prepolar of a subset of is the set: As with the absolute prepolar, the real prepolar is usually called the real polar and is also denoted by. It's important to note that some authors define "polar" to mean "real polar" and use the notation for it. The real bipolar of a subset of, sometimes denoted by, is the set ; it is equal to the -closure of the convex hull of. For a subset of, is convex, -closed, and contains. In general, it is possible that but equality will hold if is balanced. Furthermore, A∘ = where bal denotes the balanced hull of Ar.
Competing definitions
The definition of the "polar" of a set is not universally agreed upon. Although this article defined "polar" to mean "absolute polar", some authors define "polar" to mean "real polar" and other authors use still other definitions. No matter how an author defines "polar", the notation almost always represents their choice of the definition. In particular, the polar of is sometimes defined as: where the notation is not standard notation. We now briefly discuss how these various definitions relate to one another and when they are equivalent. We always have and if is real-valued then. If is symmetric then where if in addition is real-valued then. If and are vector spaces over and if , then where if in addition A ⊆ A for all real then. Thus for all of these definitions of the polar set of to agree, it suffices that for all scalars of unit length . In particular, all definitions of the polar of agree when is a balanced set so that often, which of these competing definitions is used is immaterial. However, these difference in the definitions of the "polar" of a set do sometimes introduce subtle or important technical differences when is not necessarily balanced.
Specialization for the canonical duality
Suppose that is a topological vector space with continuous dual space. We consider the important special case where Y := and the brackets represent the canonical map. Thus is the canonical pairing. The polar of a subset A ⊆ X is: If satisfies sA ⊆ A for all scalars s of unit length then one may replace the absolute value signs by so that: The prepolar of a subset of is: If satisfies sB ⊆ B for all scalars s of unit length then one may replace the absolute value signs with so that: where B :=. The bipolar theorem characterizes the bipolar of a subset of a topological vector space. If is a normed space and S is the open or closed unit ball in then S∘ is the closed unit ball in the continuous dual space when is endowed with its canonical dual norm.
Geometric definition for cones
The polar cone of a convex coneA ⊆ X is the set This definition gives a duality on points and hyperplanes, writing the latter as the intersection of two oppositely-oriented half-spaces. The polar hyperplane of a point x ∈ X is the locus ; the dual relationship for a hyperplane yields that hyperplane's polar point. Some authors call a dual cone the polar cone; we will not follow that convention in this article.
Properties
Unless stated otherwise, we henceforth assume that is a pairing. Note that is the weak-* topology on while is the weak topology on. For any set
