The weak topology 
on a topological vector space 
is the coarsest topology
making every element of its continuous
dual space 
continuous. It is Hausdorff iff 
separates the points
of 
. The original topology 
is sometimes called the strong topology
to distinguish it from 
.
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.
