Polar set


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 .

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.
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 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, where denotes the balanced hull of

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 choice of the definition.
In particular, the polar of is sometimes defined as:
where the notation is standard notation.
We now briefly discuss how these various definitions relate to one another and when they are equivalent.
It is always the case that
and if is real-valued then
If is a symmetric set then where if in addition is real-valued then
If and are vector spaces over and if , then
where if in addition 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 differences 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

Algebraic dual space
If is any vector space then let denote the algebraic dual space of which is the set of all linear functionals on The vector space is always a closed subset of the space of all -valued functions on under the topology of pointwise convergence so when is endowed with the subspace topology, then becomes a Hausdorff complete locally convex topological vector space.
For any subset let
If are any subsets then and where denotes the convex balanced hull of
For any finite-dimensional vector subspace of let denote the Euclidean topology on which is the unique topology that makes into a Hausdorff topological vector space.
If denotes the union of all closures as varies over all finite dimensional vector subspaces of then .
If is an absorbing subset of then by the Banach–Alaoglu theorem, is a weak-* compact subset of
If is any non-empty subset of a vector space and if is any vector space of linear functionals on then the real-valued map
is a seminorm on If then by definition of the supremum, so that the map defined above would not be real-valued and consequently, it would not be a seminorm.
Continuous dual space
Suppose that is a topological vector space with continuous dual space
The important special case where and the brackets represent the canonical map:
is now considered.
The triple is the called the associated with
The polar of a subset with respect to this canonical pairing is:
For any subset where denotes the closure of in
The Banach–Alaoglu theorem states that if is a neighborhood of the origin in then and this polar set is a compact subset of the continuous dual space when is endowed with the weak-* topology.
If satisfies for all scalars of unit length then one may replace the absolute value signs by so that:
The prepolar of a subset of is:
If satisfies for all scalars of unit length then one may replace the absolute value signs with so that:
where
The bipolar theorem characterizes the bipolar of a subset of a topological vector space.
If is a normed space and is the open or closed unit ball in then 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 cone 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 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, will be a pairing.
The topology is the weak-* topology on while is the weak topology on
For any set denotes the real polar of and denotes the absolute polar of
The term "polar" will refer to the polar.
  • The polar of a set is convex and balanced.
  • The real polar of a subset of is convex but necessarily balanced; will be balanced if is balanced.
  • If for all scalars of unit length then
  • is closed in under the weak-*-topology on.
  • A subset of is weakly bounded if and only if is absorbing in.
  • For a dual pair where is a TVS and is its continuous dual space, if is bounded then is absorbing in If is locally convex and is absorbing in then is bounded in Moreover, a subset of is weakly bounded if and only if is absorbing in
  • The bipolar of a set is the -closed convex hull of that is the smallest -closed and convex set containing both and
  • * Similarly, the bidual cone of a cone is the -closed conic hull of
  • If is a base at the origin for a TVS then
  • If is a locally convex TVS then the polars of any 0-neighborhood base forms a fundamental family of equicontinuous subsets of .
  • * Conversely, if is a locally convex TVS then the polars of any fundamental family of equicontinuous subsets of form a neighborhood base of the origin in
  • Let be a TVS with a topology Then is a locally convex TVS topology if and only if is the topology of uniform convergence on the equicontinuous subsets of
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 's original topology.
Set relations
  • and
  • For all scalars and for all real and
  • However, for the real polar we have
  • For any finite collection of sets
  • If then and
  • * An immediate corollary is that ; equality necessarily holds when is finite and may fail to hold if is infinite.
  • and
  • If is a cone in then
  • If is a family of -closed subsets of containing then the real polar of is the closed convex hull of
  • If then
  • For a closed convex cone in a real vector space the polar cone is the polar of ; that is, where