Polar set

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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[edit]

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[edit]

Absolute polar[edit]

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[edit]

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[edit]

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 symmetric (i.e. 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 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[edit]

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[edit]

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[edit]

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[edit]

Notes[edit]

  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[edit]

  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[edit]