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 spaces of a dual pair.
A set is said to be bounded in with respect to , if for each element the set of values is bounded in :
This condition is equivalent to the requirement that the polar of the set in
is an absorbent set in , i.e.
Let now be a family of bounded sets in (with respect to ) with the following properties:
- each point of belongs to some set
- each two sets and are contained in some set :
- is closed under the operation of multiplication by scalars:
Then the seminorms of the form
define a Hausdorff locally convex topology on which is called the polar topology on generated by the family of sets . The sets
form a local base of this topology. A net of elements tends to an element in this topology if and only if
- if is the family of all bounded sets in then the polar topology on coincides with the strong topology,
- if is the family of all finite sets in then the polar topology on coincides with the weak topology,
- the topology of an arbitrary locally convex space can be described as the polar topology defined on by the family of all equicontinuous sets in the dual space .
- A.P.Robertson, W.Robertson (1964, III.2)
- In other words, iff and there is a neighbourhood of zero such that
- Robertson, A.P.; Robertson, W. (1964). Topological vector spaces. Cambridge University Press.