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.