In functional and convex analysis, and related disciplines of mathematics, the polar set
is a special convex set associated to any subset
There are at least three competing definitions of the polar of a set, originating in projective geometry and convex analysis.
[1][citation needed] In each case, the definition describes a duality between certain subsets of a pairing of vector spaces
will be the bilinear evaluation (at a point) map defined by
in which case the dual pairing will again be the evaluation map.
centered at the origin in the underlying scalar field
which by definition is the smallest convex and balanced subset of
This is an affine shift of the geometric definition; it has the useful characterization that the functional-analytic polar of the unit ball (in
; 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".
[2] The definition of the "polar" of a set is not universally agreed upon.
No matter how an author defines "polar", the notation
almost always represents their choice of the definition (so the meaning of the notation
We now briefly discuss how these various definitions relate to one another and when they are equivalent.
is a balanced set (which is often, but not always, the case) so that often, which of these competing definitions is used is immaterial.
However, these differences in the definitions of the "polar" of a set
do sometimes introduce subtle or important technical differences when
under the topology of pointwise convergence so when
becomes a Hausdorff complete locally convex topological vector space (TVS).
into a Hausdorff topological vector space (TVS).
varies over all finite dimensional vector subspaces of
(that is, a vector subspace of the algebraic dual space of
Continuous dual space Suppose that
and this polar set is a compact subset of the continuous dual space
of unit length then one may replace the absolute value signs by
of unit length then one may replace the absolute value signs with
is the closed unit ball in the continuous dual space
This definition gives a duality on points and hyperplanes, writing the latter as the intersection of two oppositely-oriented half-spaces.
[3][citation needed] Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article.
The last two results explain why equicontinuous subsets of the continuous dual space play such a prominent role in the modern theory of functional analysis: because equicontinuous subsets encapsulate all information about the locally convex space