请输入您要查询的字词:

 

单词 VectorNorm
释义

vector norm


A vector normMathworldPlanetmath on the real vector space V is a function f:V that satisfies the following properties:

f(x)=0x=0
f(x)0xV
f(x+y)f(x)+f(y)x,yV
f(αx)=|α|f(x)α,xV

Such a function is denoted as ||x||. Particular norms are distinguished by subscripts, suchas ||x||V, when referring to a norm in the space V. A unit vectorMathworldPlanetmath with respect to the norm |||| is a vector x satisfying||x||=1.

A vector norm on a complex vector space is defined similarly.

A common (and useful) example of a real norm is the Euclidean norm given by ||x||=(x12+x22++xn2)1/2 defined on V=n.Note, however, that there exists vector spacesMathworldPlanetmath which are metric, but upon which it is not possible to define a norm. If it possible, the space is called a normed vector space. Given a metric on the vector space, a necessary and sufficient condition for this space to be a normed space, is

d(x+a,y+a)=d(x,y)x,y,aV
d(αx,αy)=|α|d(x,y)x,yV,α

But given a norm, a metric can always be defined by the equation d(x,y)=||x-y||. Hence every normed space is a metric space.

Titlevector norm
Canonical nameVectorNorm
Date of creation2013-03-22 11:43:00
Last modified on2013-03-22 11:43:00
Ownermike (2826)
Last modified bymike (2826)
Numerical id30
Authormike (2826)
Entry typeDefinition
Classificationmsc 46B20
Classificationmsc 18-01
Classificationmsc 20H15
Classificationmsc 20B30
Related topicVector
Related topicMetric
Related topicNorm
Related topicVectorPnorm
Related topicNormedVectorSpace
Related topicMatrixNorm
Related topicMatrixPnorm
Related topicFrobeniusMatrixNorm
Related topicCauchySchwarzInequality
Related topicMetricSpace
Related topicVectorSpace
Related topicLpSpace
Related topicOperatorNorm
Related topicBoundedOperator
Related topicSemiNormMathworldPlanetmath
Related topicBanachSpace
Related topicHilbertSpace
Related topicUnitVector
Definesnormed vector space
DefinesEuclidean norm
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 17:21:38