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  (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 mean "real polar" (rather than "absolute polar", as is done in this article) and use the notation  for it (rather than the notation  that is used in this article and in [Narici 2011]). 

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 (so the meaning of the notation  may vary from source to source). 
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 (or equivalently, if  and  are vector spaces over ) then  

If  is a symmetric set (that is,  or equivalently, ) then  where if in addition  is real-valued then 

If  and  are vector spaces over  (so that  is complex-valued) and if  (where note that this implies  and ), 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 (where this is equivalent to  for all unit length scalar ).
In particular, all definitions of the polar of  agree when  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 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

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 (TVS). 
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 (TVS). 
If  denotes the union of all closures  as  varies over all finite dimensional vector subspaces of  then  (see this footnote 
for an explanation). 
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  (that is, a vector subspace of the algebraic dual space of ) then the real-valued map 

  defined by   

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 (TVS) 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 (also known as the topology of pointwise convergence). 

If  satisfies  for all scalars  of unit length then one may replace the absolute value signs by  (the real part operator) 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  (or even any subset of the closed unit ball that contains the open unit ball) 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 (confusingly) 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 (absolute) 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 (i.e. -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 (taken with respect to ) of any 0-neighborhood base forms a fundamental family of equicontinuous subsets of  (i.e. given any bounded subset  of  there exists a neighborhood  of the origin in  such that ). 
 Conversely, if  is a locally convex TVS then the polars (taken with respect to ) 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

See also

Notes

References

Bibliography

  
  
  
  
  
  
  

Functional analysis
Linear functionals
Topological vector spaces