# Strong topology (polar topology)

In functional analysis and related areas of mathematics the **strong topology** is the finest polar topology, the topology with the most open sets, on a dual pair. The coarsest polar topology is called weak topology.

## Definition

Let be a dual pair of vector spaces over the field of real ( ) or complex ( ) numbers. Let us denote by the system of all subsets bounded by elements of in the following sense:

Then the **strong topology**
on
is defined as the locally convex topology on
generated by the seminorms of the form

In the special case when
is a locally convex space, the **strong topology** on the (continuous) dual space
(i.e. on the space of all continuous linear functionals
) is defined as the strong topology
, and it coincides with the topology of uniform convergence on bounded sets in
, i.e. with the topology on
generated by the seminorms of the form

where
runs over the family of all bounded sets in
. The space
with this topology is called **strong dual space** of the space
and is denoted by
.

## Examples

- If is a normed vector space, then its (continuous) dual space with the strong topology coincides with the Banach dual space , i.e. with the space with the topology induced by the operator norm. Conversely -topology on is identical to the topology induced by the norm on .

## Properties

- If is a barrelled space, then its topology coincides with the strong topology on and with the Mackey topology on generated by the pairing .

## References

- Schaefer, Helmuth H. (1966).
*Topological vector spaces*. New York: The MacMillan Company. ISBN 0-387-98726-6.