Let 
be a topological vector space whose continuous
dual 
may or may not separate points (i.e., may or may not be
T2). The weak-* (pronounced "weak
star") topology on 
is defined to be the 
-topology on 
, i.e., the coarsest topology
(the topology with the fewest open sets) under which
every element 
corresponds to a continuous map on 
. The weak-* topology is sometimes called the ultraweak topology
or the 
-weak
topology.
The fundamental observation to the above-stated definition is that every element 
induces a linear functional 
on 
. In particular, 
of the form
 |
for every element 
,
and because of this, one can view the space 
as a collection of linear functionals on 
and hence can define the 
-topology thereon.
Immediately following from the above is the fact 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 
is often called the 
-topology on 
and/or the weak topology on 
induced by 
, whereby it follows that the above-stated definition corresponds
to the special case of 
for 
a topological vector space.
The weak-* topology is fundamental throughout functional analysis, playing a fundamental role in a number of important theorems including the Banach-Alaoglu theorem.
