The weak topology 
on a topological vector space 
is the coarsest topology
making every element of its continuous dual space 
continuous.
Here 
is the set of continuous linear functionals on 
. The weak topology is Hausdorff iff 
separates the points of 
. For a Fréchet
space, the original topology 
is often called the strong topology
to distinguish it from 
(Rudin 1991, p. 65).
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 of finite intersections
of sets of the form 
with 
and 
an open set in 
, including the empty intersection 
. 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 and 
its field of scalars (Rudin 1991, pp. 62-65).
