View source for Strong topology

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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|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|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Köthe,   "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR></table>