Editing Dual space (section)

{{Short description|In mathematics, vector space of linear forms}}
{{Use American English|date = March 2019}}

In [[mathematics]], any [[vector space]] ''<math>V</math>'' has a corresponding '''dual vector space''' (or just '''dual space''' for short) consisting of all [[linear form]]s on ''<math>V,</math>'' together with the vector space structure of [[pointwise]] addition and scalar multiplication by constants.

The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the {{em|algebraic dual space}}.
When defined for a [[topological vector space]], there is a subspace of the dual space, corresponding to [[continuous linear functional]]s, called the '''continuous dual space'''.

Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in [[tensor]] analysis with [[finite-dimensional]] vector spaces.
When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe [[Measure (mathematics)|measures]], [[Distribution (mathematics)|distributions]], and [[Hilbert space]]s. Consequently, the dual space is an important concept in [[functional analysis]].

Early terms for ''dual'' include ''polarer Raum'' [Hahn 1927], ''espace conjugué'', ''adjoint space'' [Alaoglu 1940], and ''transponierter Raum'' [Schauder 1930] and [Banach 1932]. The term ''dual'' is due to [[Nicolas Bourbaki|Bourbaki]] 1938.{{sfn | Narici|Beckenstein | 2011 | pp=225-273}}