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单词 NormedVectorSpace
释义

normed vector space


Let 𝔽 be a field which is either or . A over 𝔽 is a pair (V,) where V is a vector space over 𝔽 and :V is a function such that

  1. 1.

    v0 for all vV and v=0 if and only if v=0 in V (positive definiteness)

  2. 2.

    λv=|λ|v for all vV and all λ𝔽

  3. 3.

    v+wv+w for all v,wV (the triangle inequalityMathworldMathworldPlanetmath)

The function is called a norm on V.

Some properties of norms:

  1. 1.

    If W is a subspaceMathworldPlanetmath of V then W can be made into a normed spaceMathworldPlanetmath by simply restricting the norm on V to W. This is called the induced norm on W.

  2. 2.

    Any normed vector space (V,) is a metric space under the metric d:V×V given by d(u,v)=u-v. This is called the metric induced by the norm .

  3. 3.

    It follows that any normed space is a locally convex topological vector space, in the topologyMathworldPlanetmath induced by the metric defined above.

  4. 4.

    In this metric, the norm defines a continuous mapMathworldPlanetmath from V to - this is an easy consequence of the triangle inequality.

  5. 5.

    If (V,,) is an inner product spaceMathworldPlanetmath, then there is a natural induced norm given by v=v,v for all vV.

  6. 6.

    The norm is a convex function of its argument.

Titlenormed vector space
Canonical nameNormedVectorSpace
Date of creation2013-03-22 12:13:45
Last modified on2013-03-22 12:13:45
Ownerrspuzio (6075)
Last modified byrspuzio (6075)
Numerical id14
Authorrspuzio (6075)
Entry typeDefinition
Classificationmsc 46B99
Synonymnormed space
Synonymnormed linear space
Related topicCauchySchwarzInequality
Related topicVectorNorm
Related topicPseudometricSpace
Related topicMetricSpace
Related topicUnitVector
Related topicProofOfGramSchmidtOrthogonalizationProcedure
Related topicEveryNormedSpaceWithSchauderBasisIsSeparable
Related topicEveryNormedSpaceWithSchauderBasisIsSeparable2
Related topicFrobeniusProduct
Definesnorm
Definesmetric induced by a norm
Definesmetric induced by the norm
Definesinduced norm
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更新时间:2025/5/4 16:42:04