Strong topology
Definition[edit]
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[edit]
 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 .