dual space

Dual of a vector spaceMathworldPlanetmath; dual bases

Let V be a vector space over a field k. The dual of V, denoted by V, is the vector space of linear formsPlanetmathPlanetmath on V, i.e. linear mappings Vk. The operationsMathworldPlanetmath in V are defined pointwise:


for λK, vV and φ,ψV.

V is isomorphic to V if and only if the dimensionPlanetmathPlanetmathPlanetmathPlanetmath of V is finite. If not, then V has a larger (infiniteMathworldPlanetmath) dimension than V; in other words, the cardinal of any basis of V is strictly greater than the cardinal of any basis of V.

Even when V is finite-dimensional, there is no canonical or natural isomorphism VV. But on the other hand, a basis of V does define a basis of V, and moreover a bijection . For suppose ={b1,,bn}. For each i from 1 to n, define a mapping




It is easy to see that the βi are nonzero elements of V and are independent. Thus {β1,,βn} is a basis of V, called the dual basisMathworldPlanetmath of .

The dual of V is called the second dual or bidual of V. There is a very simple canonical injection VV, and it is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath if the dimension of V is finite. To see it, let x be any element of V and define a mapping x:Vk simply by


x is linear by definition, and it is readily verified that the mapping xx from V to V is linear and injectivePlanetmathPlanetmath.

If V is a topological vector space, the continuous dual V of V is the subspacePlanetmathPlanetmath of V consisting of the continuousPlanetmathPlanetmath linear forms.

A normed vector space V is said to be reflexiveMathworldPlanetmathPlanetmathPlanetmathPlanetmath if the natural embedding VV′′ is an isomorphism. For example, any finite dimensional space is reflexive, and any Hilbert spaceMathworldPlanetmath is reflexive by the Riesz representation theorem.


Linear forms are also known as linear functionalsMathworldPlanetmathPlanetmath.

Another way in which a linear mapping VV can arise is via a bilinear formPlanetmathPlanetmath


The notions of duality extend, in part, from vector spaces to modules, especially free modulesMathworldPlanetmathPlanetmath over commutative rings. A related notion is the duality in projective spaces.

Title dual spaceMathworldPlanetmathPlanetmath
Canonical name DualSpace
Date of creation 2013-03-22 12:16:52
Last modified on 2013-03-22 12:16:52
Owner Daume (40)
Last modified by Daume (40)
Numerical id 15
Author Daume (40)
Entry type Definition
Classification msc 15A99
Synonym algebraic dual
Synonym continuous dual
Synonym dual basis
Synonym reflexive
Synonym natural embedding
Synonym topological dual
Related topic DualHomomorphism
Related topic DoubleDualEmbedding
Related topic BanachSpace
Related topic Unimodular
Related topic LinearFunctional
Related topic BoundedLinearFunctionalsOnLpmu