In
functional and
convex analysis
Convex analysis is the branch of mathematics devoted to the study of properties of convex functions and convex sets, often with applications in convex optimization, convex minimization, a subdomain of optimization (mathematics), optimization theor ...
, and related disciplines of
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the polar set
is a special convex set associated to any subset
of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
lying in the
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by cons ...
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 space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
s (TVSs)).
If
is a vector space over the field
then unless indicated otherwise,
will usually, but not always, be some vector space of
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...
s on
and the dual pairing
will be the
bilinear () defined by
If
is a
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
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
In mathematics, a pairing is an ''R''- bilinear map from the Cartesian product of two ''R''- modules, where the underlying ring ''R'' is commutative.
Definition
Let ''R'' be a commutative ring with unit, and let ''M'', ''N'' and ''L'' be '' ...
.
The polar or absolute polar of a subset
of
is the set:
where
denotes the
image
An image or picture is a visual representation. An image can be Two-dimensional space, two-dimensional, such as a drawing, painting, or photograph, or Three-dimensional space, three-dimensional, such as a carving or sculpture. Images may be di ...
of the set
under the map
defined by
If
denotes the
convex balanced hull of
which by definition is the smallest
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
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.
chaefer 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
arici 2011.
The real bipolar of a subset
of
sometimes denoted by
is the set
;
it is equal to the
-closure of the
convex hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
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[
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
] (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
Algebraic dual space
If
is any vector space then let
denote the
algebraic dual space of
which is the set of all
linear functionals
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field (mat ...
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
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(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
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot ...
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
[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 ]
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
In mathematics, particularly in functional analysis, a seminorm is like a Norm (mathematics), norm but need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some Absorbing ...
on
If
then by definition of the
supremum
In mathematics, the infimum (abbreviated inf; : infima) of a subset S of a partially ordered set P is the greatest element in P that is less than or equal to each element of S, if such an element exists. If the infimum of S exists, it is unique, ...
,
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
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
(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
In functional analysis, the dual norm is a measure of size for a continuous function, continuous linear function defined on a normed vector space.
Definition
Let X be a normed vector space with norm \, \cdot\, and let X^* denote its continuous d ...
.
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
In mathematics, a pairing is an ''R''- bilinear map from the Cartesian product of two ''R''- modules, where the underlying ring ''R'' is commutative.
Definition
Let ''R'' be a commutative ring with unit, and let ''M'', ''N'' and ''L'' be '' ...
.
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
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
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
Functional analysis
Linear functionals
Topological vector spaces