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 ).
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
Suppose that is a pairing.
The polar or absolute polar of a subset of is the set:
where denotes the 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,
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.
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In mathematics, a dual system, dual pair, or duality over a field is a triple consisting of two vector spaces and over and a non-degenerate bilinear map . Duality theory, the study of dual systems, is part of functional analysis. It is separate and distinct to Dual-system Theory in psychology. Pairings A or pair over a field is a triple which may also be denoted by consisting of two vector spaces and over (which this article assumes is the field either of real numbers or the complex numbers ).
In functional analysis and related areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated to include the set. A set that is not bounded is called unbounded. Bounded sets are a natural way to define locally convex polar topologies on the vector spaces in a dual pair, as the polar set of a bounded set is an absolutely convex and absorbing set. The concept was first introduced by John von Neumann and Andrey Kolmogorov in 1935.
In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. An operator between two normed spaces is a bounded linear operator if and only if it is a continuous linear operator. Continuous function (topology) and Discontinuous linear map Bounded operator Suppose that is a linear operator between two topological vector spaces (TVSs). The following are equivalent: is continuous.
This work studies linear elliptic problems under uncertainty. The major emphasis is on the deterministic treatment of such uncertainty. In particular, this work uses the Worst Scenario approach for the characterization of uncertainty on functional outputs ...
We use variational techniques to prove existence and nonexistence results for the following singular elliptic system: {div(vertical bar del u vertical bar(p-2)del u) = theta z(q)/u(1-0), u > 0 in Omega is an element of W-0(,1p) (Omega), -div(vertical bar d ...
In this paper we prove an existence result for the following singular elliptic system {z > 0 in Omega, z is an element of W-0(iota,p)(Omega) : -Delta(p)z = a(x)z(q-iota)u(theta) , u > 0 in Omega, u is an element of W-0(iota,p)(Omega) : -Delta(p)u = b(x)z(q ...