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.