In functional and convex analysis, and related disciplines of mathematics, the polar set of a vector space lying in the dual space The bipolar).

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 are often topological vector spaces (TVSs)).

will usually, but not always, be some vector space of linear functionals will be the bilinear () is a topological vector space will usually, but not always, be the continuous dual space in which case the dual pairing will again be the evaluation map.

Functional analytic definition

Absolute polar

is a pairing. The polar or absolute polar

denotes the image denotes the convex balanced hull which by definition is the smallest convex and balanced

This is an affine shift).

The prepolar or absolute prepolar

is also called the polar or absolute polar; 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

Real polar

The real polar and the real prepolar

As with the absolute prepolar, the real prepolar is usually called the real polar that is used in this article and in [Narici 2011]).

The real bipolar-closure of the convex hull

is balanced denotes the balanced hull

Competing definitions

almost always represents is standard notation.

We now briefly discuss how these various definitions relate to one another and when they are equivalent.

is a symmetric set

of unit length is a balanced set is not necessarily balanced.

Specialization for the canonical duality

Algebraic dual space

denote the algebraic dual space which is the set of all linear functionals becomes a Hausdorff complete locally convex topological vector space

denotes the convex balanced hull denote the Euclidean topology into a Hausdorff denotes the union of all closures (see this footnote then by the Banach–Alaoglu theorem is a weak-* compact

(that is, a vector subspace of the algebraic dual space) then the real-valued map

1. defined by

is a seminorm then by definition of the supremum defined above would not be real-valued and consequently, it would not be a seminorm.

Continuous dual space

is a topological vector space (TVS) with continuous dual space is the called the

denotes the closure

The Banach–Alaoglu theorem and this polar set is a compact subset is endowed with the weak-* topology (also known as the topology of pointwise convergence).

The bipolar theorem characterizes the bipolar of a subset of a topological vector space.

is endowed with its canonical dual norm.

Geometric definition for cones

The polar cone

; the dual relationship for a hyperplane yields that hyperplane's polar point.

Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article.

Properties

will be a pairing is the weak-* topology is the weak topology The term "polar" will refer to the polar.

• The (absolute) polar of a set is convex and balanced.

• is convex but is balanced.

• is closed under the weak-*-topology.

• is absorbing.

• is absorbing is absorbing

-closed convex hull

• -closed conic hull

).

's original topology.

Set relations

• is infinite.

• For a closed convex cone the polar cone

See also

Notes

References

Bibliography