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

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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   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 evaluation (at a point) map 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

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Absolute polar

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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

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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  [2] 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  [2]

For a subset   of     is convex,  -closed, and contains  [2] In general, it is possible that   but equality will hold if   is balanced. Furthermore,   where   denotes the balanced hull of  [2]

Competing definitions

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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 their 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 not 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[note 1] (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 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

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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[note 2] 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 canonical pairing 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

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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.[3][citation needed]

Some authors (confusingly) call a dual cone the polar cone; we will not follow that convention in this article.[4]

Properties

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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 absolute polar.

  • The (absolute) polar of a set is convex and balanced.[5]
  • The real polar   of a subset   of   is convex but not necessarily balanced;   will be balanced if   is balanced.[6]
  • If   for all scalars   of unit length then  
  •   is closed in   under the weak-*-topology on  .[3]
  • A subset   of   is weakly bounded (i.e.  -bounded) if and only if   is absorbing in  .[2]
  • For a dual pair   where   is a TVS and   is its continuous dual space, if   is bounded then   is absorbing in  [5] 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  [7]
  • If   is a base at the origin for a TVS   then  [8]
  • 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  ).[6]
    • 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  [6]
  • 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  [6]

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

  •  [6] and  
  • For all scalars     and for all real     and  
  •   However, for the real polar we have  [6]
  • 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  [5]
  • If   is a family of  -closed subsets of   containing   then the real polar of   is the closed convex hull of  [6]
  • If   then  [9]
  • For a closed convex cone   in a real vector space   the polar cone is the polar of  ; that is,   where  [1]

See also

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Notes

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  1. ^ Since for all of these completing definitions of the polar set   to agree, if   is real-valued then it suffices for   to be symmetric, while if   is complex-valued then it suffices that   for all real  
  2. ^ To prove that   let   If   is a finite-dimensional vector subspace of   then because   is continuous (as is true of all linear functionals on a finite-dimensional Hausdorff TVS), it follows from   and   being a closed set that   The union of all such sets is consequently also a subset of   which proves that   and so     In general, if   is any TVS-topology on   then  

References

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  1. ^ a b Aliprantis, C.D.; Border, K.C. (2007). Infinite Dimensional Analysis: A Hitchhiker's Guide (3 ed.). Springer. p. 215. doi:10.1007/3-540-29587-9. ISBN 978-3-540-32696-0.
  2. ^ a b c d e Narici & Beckenstein 2011, pp. 225–273.
  3. ^ a b Zălinescu, C. (2002). Convex Analysis in General Vector Spaces. River Edge, NJ: World Scientific. pp. 7–8. ISBN 978-9812380678.
  4. ^ Rockafellar, T.R. (1970). Convex Analysis. Princeton University. pp. 121-8. ISBN 978-0-691-01586-6.
  5. ^ a b c Trèves 2006, pp. 195–201.
  6. ^ a b c d e f g Schaefer & Wolff 1999, pp. 123–128.
  7. ^ Niculescu, C.P.; Persson, Lars-Erik (2018). Convex Functions and Their Applications. CMS Books in Mathematics. Cham, Switzerland: Springer. pp. 94–5, 134–5. doi:10.1007/978-3-319-78337-6. ISBN 978-3-319-78337-6.
  8. ^ Narici & Beckenstein 2011, p. 472.
  9. ^ Jarchow 1981, pp. 148–150.

Bibliography

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