In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, any
vector space ''
'' has a corresponding dual vector space (or just dual space for short) consisting of all
linear form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
s on ''
'', 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 .
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''
ahn 1927 Ahn or AHN may refer to:
People
* Ahn (Korean surname), a Korean family name occasionally Romanized as ''An''
* Ahn Byeong-keun (born 1962, ), South Korean judoka
* Ahn Eak-tai (1906–1965, ), Korean composer and conductor
* Ahn Jung-hwan (born 19 ...
''espace conjugué'', ''adjoint space''
laoglu 1940 and ''transponierter Raum''
chauder 1930and
anach 1932 The term ''dual'' is due to Bourbaki 1938.
Algebraic dual space
Given any
vector space over a
field , the (algebraic) dual space
(alternatively denoted by
[ p. 19, §3.1] or
)
[For used in this way, see '' An Introduction to Manifolds'' ().
This notation is sometimes used when is reserved for some other meaning.
For instance, in the above text, is frequently used to denote the codifferential of '''', so that represents the pullback of the form .
uses to denote the algebraic dual of ''''. However, other authors use for the continuous dual, while reserving for the algebraic dual ().
] is defined as the set of all
linear maps ''
'' (
linear functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , th ...
s). Since linear maps are vector space
homomorphisms, the dual space may be denoted
.
[ p. 19, §3.1]
The dual space
itself becomes a vector space over ''
'' when equipped with an addition and scalar multiplication satisfying:
:
for all
, ''
'', and
.
Elements of the algebraic dual space
are sometimes called covectors or
one-forms.
The pairing of a functional ''
'' in the dual space
and an element ''
'' of ''
'' is sometimes denoted by a bracket: ''