Strong topology

A dual pair of vector spaces over a field is a pair of vector spaces , together with a non-degenerate bilinear form over ,

I.e. , ; for all implies ; for all implies .

The weak topology on defined by the dual pair (given a topology on ) is the weakest topology such that all the functionals , , are continuous. More precisely, if or with the usual topology, this defines the weak topology on (and ). If is an arbitrary field with the discrete topology, this defines the so-called linear weak topology.

Let be a collection of bounded subsets of (for the weak topology, i.e. every is weakly bounded, meaning that for every open subset of in the weak topology on there is a such that ). The topology on is defined by the system of semi-norms , , where (cf. Semi-norm). This topology is locally convex if and only if is a total set, i.e. it generates (in as a vector space) all of . The topology is called the topology of uniform convergence on the sets of .

The finest topology on which can be defined in terms of the dual pairs is the topology of uniform convergence on weakly bounded subsets of . This is the topology where is the collection of all weakly bounded subsets of , and it is called the strong topology on , for brevity.

References

[a1]	G. Köthe, "Topological vector spaces" , 1 , Springer (1969)