Dual space

In mathematics, every vector space

V

{\displaystyle V}

has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on

V

,

{\displaystyle V,}

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 algebraic dual space.

When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, 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 measures, distributions, and Hilbert spaces. 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 Bourbaki.

Algebraic dual space

Given any vector space

V

{\displaystyle V}

over a field

F

{\displaystyle F}

, the (algebraic) dual space

V

∗

{\displaystyle V^{*}}

(alternatively denoted by

V

∨

{\displaystyle V^{\lor }}

or

V

′

{\displaystyle V'}

) is defined as the set of all linear maps

φ

:

V

→

F

{\displaystyle \varphi :V\to F}

(linear functionals). Since linear maps are vector space homomorphisms, the dual space may be denoted

hom

⁡

(

V

,

F

)

{\displaystyle \hom(V,F)}

.