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Concept
Dual space
Summary
In mathematics, any vector space has a corresponding dual vector space (or just dual space for short) consisting of all linear forms 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 [Hahn 1927], espace conjugué, adjoint space [Alaoglu 1940], and transponierter Raum [Schauder 1930] and [Banach 1932]. The term dual is due to Bourbaki 1938.
Given any vector space over a field , the (algebraic) dual space (alternatively denoted by or ) is defined as the set of all linear maps (linear functionals). Since linear maps are vector space homomorphisms, the dual space may be denoted .
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, one-forms, or linear forms.
The pairing of a functional in the dual space and an element of is sometimes denoted by a bracket:
or . This pairing defines a nondegenerate bilinear mapping called the natural pairing.
Dual basis
If is finite-dimensional, then has the same dimension as . Given a basis in , it is possible to construct a specific basis in , called the dual basis. This dual basis is a set of linear functionals on , defined by the relation
for any choice of coefficients .
Official source
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