Polar topology
Encyclopedia
In functional analysis
and related areas of mathematics
a polar topology, topology of
-convergence or topology of uniform convergence on the sets of 
is a method to define locally convex topologies on the vector space
s of a dual pair
.
and a family 
of sets in 
such that for all 
in 
the polar set 
is an absorbent subset of 
, the polar topology on 
is defined by a family of semi norms 
. For each 
in 
we define
.
The semi norm
is the gauge of the polar set 
.
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
and related areas of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
a polar topology, topology of
-convergence or topology of uniform convergence on the sets of is a method to define locally convex topologies on the vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s of a dual pair
Dual pair
In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form....
.
Definition
Given a dual pairDual pair
In functional analysis and related areas of mathematics a dual pair or dual system is a pair of vector spaces with an associated bilinear form....
and a family of sets in such that for all in the polar set is an absorbent subset of , the polar topology on is defined by a family of semi norms . For each in we define.The semi norm
is the gauge of the polar set .Examples
- a dual topologyDual topologyIn functional analysis and related areas of mathematics a dual topology is a locally convex topology on a dual pair, two vector spaces with a bilinear form defined on them, so that one vector space becomes the continuous dual of the other space....
is a polar topology (the converse is not necessarily true) - a locally convex topology is the polar topology defined by the family of equicontinuous sets of the dual spaceDual spaceIn mathematics, any vector space, V, has a corresponding dual vector space consisting of all linear functionals on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra...
, that is the sets of all continuous linear forms which are equicontinuous - Using the family of all finite sets in
we get the coarsest polar topology 
on 
. 
is identical to the weak topologyWeak topologyIn mathematics, weak topology is an alternative term for initial topology. The term is most commonly used for the initial topology of a topological vector space with respect to its continuous dual...
. - Using the family of all sets in
where the polar set is absorbent, we get the finest polar topology 
on 