In functional and convex analysis, and related disciplines of mathematics, the polar set is a special convex set associated to any subset of a vector space , lying in the dual space The bipolar of a subset is the polar of but lies in (not ). 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 over the real or complex numbers ( and are often topological vector spaces (TVSs)). If is a vector space over the field then unless indicated otherwise, will usually, but not always, be some vector space of linear functionals on and the dual pairing will be the bilinear () defined by If is a topological vector space then the space will usually, but not always, be the continuous dual space of in which case the dual pairing will again be the evaluation map. Denote the closed ball of radius centered at the origin in the underlying scalar field of by Functional analytic definition Absolute polar Suppose that is a pairing. The polar or absolute polar of a subset of is the set: where denotes the image of the set under the map defined by If denotes the convex balanced hull of which by definition is the smallest convex and balanced subset of that contains then This is an affine shift of the geometric definition; it has the useful characterization that the functional-analytic polar of the unit ball (in ) is precisely the unit ball (in ). 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 (such as ambiguity) 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 (e.g. [Schaefer 1999]) define "polar" to