The weak-* topology (pronounced "weak star") on the continuous dual space 
of a topological vector space 
is the coarsest topology making
every evaluation map 
, for 
, continuous. It is also
called the topology of pointwise
convergence on 
.
Here 
consists of all continuous linear
functionals on 
.
The weak-* topology on 
is always T2,
because distinct linear functionals differ at
some 
.
This does not require 
to separate the points of 
(Rudin 1991, pp. 67-68).
The observation underlying this definition is that every element 
induces a linear
functional 
on 
.
In particular, 
is of the form
 |
for every element 
, and these evaluation linear
functionals, indexed by 
, define the weak-* topology on 
. Distinct points of 
need not induce distinct evaluation linear
functionals.
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
of finite intersections
of sets of the form 
with 
and 
an open set in 
, including the empty intersection 
(Rudin 1991, pp. 62-63). The topology 
is often called the 
-topology on 
, or the weak topology on
induced by 
.
The weak-* topology is the case 
, with 
the field of scalars of 
and 
.
The weak-* topology plays a fundamental role in functional analysis, including the Banach-Alaoglu theorem.
