Editing Topological vector space (section)

==Dual space==

{{Main|Algebraic dual space|Continuous dual space|Strong dual space}}

Every topological vector space has a [[continuous dual space]]&mdash;the set <math>X'</math> of all continuous linear functionals, that is, [[continuous linear map]]s from the space into the base field <math>\mathbb{K}.</math> A topology on the dual can be defined to be the coarsest topology such that the dual pairing each point evaluation <math>X' \to \mathbb{K}</math> is continuous. This turns the dual into a locally convex topological vector space. This topology is called the [[Weak topology|weak-* topology]].{{sfn|Rudin|1991|p=62-68 §3.8-3.14}} 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 <math>X</math> is a non-normable locally convex space, then the pairing map <math>X' \times X \to \mathbb{K}</math> is never continuous, no matter which vector space topology one chooses on <math>X'.</math> A topological vector space has a non-trivial continuous dual space if and only if it has a proper convex neighborhood of the origin.{{sfn|Narici|Beckenstein|2011|pp=177-220}}