Every topological vector space has a continuous dual space—the set of all continuous linear functionals, that is, continuous linear maps from the space into the base field A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation is continuous. This turns the dual into a locally convex topological vector space. This topology is called the weak-* topology. This may not be the only natural topology on the dual space; for instance, the dual of a normed space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see Banach–Alaoglu theorem). Caution: Whenever is a non-normable locally convex space, then the pairing map is never continuous, no matter which vector space topology one chooses on A topological vector space has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of the origin.
Every topological vector space has a continuous dual space—the set of all continuous linear functionals that is, continuous linear maps from the space into the base field A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation is continuous. This turns the dual into a locally convex topological vector space. This topology is called the weak-* topology. This may not be the only natural topology on the dual space; for instance, the dual of a normed space has a natural norm defined on it. However, it is very important in ap,plications because of its compactness properties (see Banach–Alaoglu theorem). Caution: Whenever is a non-normable locally convex space, then the pairing map is never continuous, no matter which vector space topology one chooses on A topological vector space has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of the origin.