Dual Space
Every topological vector space has a continuous dual space—the set V* of all continuous linear functionals, i.e. continuous linear maps from the space into the base field K. A topology on the dual can be defined to be the coarsest topology such that the dual pairing V* × V → K 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 Banach space has a natural norm defined on it. However, it is very important in applications because of its compactness properties (see Banach–Alaoglu theorem).
Read more about this topic: Topological Vector Space
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