Let 
be a topological
vector space whose continuous dual 
separates points (i.e., is T2).
The weak topology 
on 
is defined to be the coarsest/weakest
topology (that is, the topology with the fewest open
sets) under which each element of 
remains continuous on 
. To differentiate the topologies 
and 
, 
is sometimes referred to as the strong topology on 
.
Note that the weak topology is a special case of a more general concept. In particular, given a nonempty family 
of mappings from a set 
to a topological space 
, one can define a topology 
to be the collection of all unions
and finite intersections of sets of the form 
with 
and 
an open set in 
. The topology 
-often called the 
-topology on 
and/or the weak topology on 
induced by 
-is the coarsest topology in which every element 
is continuous on 
and so it follows that the above-stated definition corresponds
to the special case of 
for 
a topological vector space.
