Strong topology

A dual pair of vector spaces $ ( L, M) $ over a field $ k $ is a pair of vector spaces $ L $, $ M $ together with a non-degenerate bilinear form over $ k $,

$$ \phi : L \times M \rightarrow k. $$

I.e. $ \phi ( a _ {1} l _ {1} + a _ {2} l _ {2} , m)= a _ {1} \phi ( l _ {1} , m)+ a _ {2} \phi ( l _ {2} , m) $, $ \phi ( l, b _ {1} m _ {1} + b _ {2} m _ {2} ) = b _ {1} \phi ( l, m _ {1} )+ b _ {2} \phi ( l, m _ {2} ) $; $ \phi ( l, m)= 0 $ for all $ m \in M $ implies $ l= 0 $; $ \phi ( l, m)= 0 $ for all $ l \in L $ implies $ m= 0 $.

The weak topology on $ L $ defined by the dual pair $ ( L, M) $( given a topology on $ k $) is the weakest topology such that all the functionals $ \psi _ {m} : L \rightarrow k $, $ \psi _ {m} ( l) = \phi ( l, m) $, are continuous. More precisely, if $ k = \mathbf R $ or $ \mathbf C $ with the usual topology, this defines the weak topology on $ L $( and $ M $). If $ k $ is an arbitrary field with the discrete topology, this defines the so-called linear weak topology.

Let $ \mathfrak M $ be a collection of bounded subsets of $ L $( for the weak topology, i.e. every $ A \in \mathfrak M $ is weakly bounded, meaning that for every open subset $ U $ of $ 0 $ in the weak topology on $ L $ there is a $ \rho > 0 $ such that $ \rho A \subset U $). The topology $ \tau _ {\mathfrak M } $ on $ M $ is defined by the system of semi-norms $ \{ \rho _ {A} \} $, $ A \in \mathfrak M $, where $ \rho _ {A} ( x) = \sup _ {m \in A } | \phi ( m, x) | $( cf. Semi-norm). This topology is locally convex if and only if $ \cup \mathfrak M $ is a total set, i.e. it generates (in $ L $ as a vector space) all of $ L $. The topology $ \tau _ {\mathfrak M } $ is called the topology of uniform convergence on the sets of $ \mathfrak M $.

The finest topology on $ M $ which can be defined in terms of the dual pairs $ ( L, M) $ is the topology of uniform convergence on weakly bounded subsets of $ L $. This is the topology $ \tau _ {\mathfrak M } $ where $ \mathfrak M $ is the collection of all weakly bounded subsets of $ L $, and it is called the strong topology on $ M $, for brevity.

References

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