请输入您要查询的字词:

 

单词 DualSpace
释义

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:

(φ+ψ)(v)=φ(v)+ψ(v)
(λφ)(v)=λφ(v)

for λK, vV and φ,ψV.

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

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

βi:Vk

by

βi(kxkbk)=xi.

It is easy to see that the βi are nonzero elements of Vand are independent. Thus {β1,,βn} is a basis ofV, 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:Vksimply by

x(ϕ)=ϕ(x).

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

Dual of a topological vector spaceMathworldPlanetmath

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

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

Remarks

Linear forms are also known as linear functionalsMathworldPlanetmathPlanetmath.

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

V×Vk.

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

Titledual spaceMathworldPlanetmathPlanetmath
Canonical nameDualSpace
Date of creation2013-03-22 12:16:52
Last modified on2013-03-22 12:16:52
OwnerDaume (40)
Last modified byDaume (40)
Numerical id15
AuthorDaume (40)
Entry typeDefinition
Classificationmsc 15A99
Synonymalgebraic dual
Synonymcontinuous dual
Synonymdual basis
Synonymreflexive
Synonymnatural embedding
Synonymtopological dual
Related topicDualHomomorphism
Related topicDoubleDualEmbedding
Related topicBanachSpace
Related topicUnimodular
Related topicLinearFunctional
Related topicBoundedLinearFunctionalsOnLpmu
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 22:47:49