请输入您要查询的字词:

 

单词 OperatorNorm
释义

operator norm


Definition

Let A:𝖵𝖶 be a linear map between normed vector spacesPlanetmathPlanetmath 𝖵 and𝖶. To each such map (operator) A we can assign a non-negative numberAop defined by

Aop:=sup𝐯𝖵A𝐯𝐯,

where the supremum Aop could be finite or infinite.Equivalently, the above definition can be written as

Aop:=sup𝐯𝖵A𝐯=sup𝐯𝖵A𝐯.

By convention, if 𝖵 is the zero vector space, any operator from 𝖵to 𝖶 must be the zero operator and is assigned zero norm.

Aop is called the the operator normMathworldPlanetmath (or the induced norm)of A, for reasons that will be clear in the next .

Operator norm is in fact a norm

Definition - If Aop is finite, we say that A is a. Otherwise, we say that A is .

It turns out that, for bounded operatorsMathworldPlanetmath, op satisfies all the properties of a norm(hence the name operator norm). The proof follows immediately from the definition:

Positivity:

Since A𝐯0, by definitionAop0. Also, A𝐯=0 identically only if A=0.Hence Aop=0 only if A=0.

Absolute homogeneity:

Since λA𝐯=|λ|A𝐯, by definitionλAop=|λ|Aop.

Triangle inequality:

Since (A+B)𝐯=A𝐯+B𝐯A𝐯+B𝐯, by definitionA+BopAop+Bop.

The set L(𝖵,𝖶) of bounded linear maps from 𝖵 to 𝖶 forms a vector spaceMathworldPlanetmath and op defines a norm in it.

Example

Suppose that 𝖵=(n,p) and 𝖶=(n,p), wherep is the vector p-norm. Then the operator normop=p is the matrix p-normMathworldPlanetmath.

Titleoperator norm
Canonical nameOperatorNorm
Date of creation2013-03-22 12:43:20
Last modified on2013-03-22 12:43:20
Ownerasteroid (17536)
Last modified byasteroid (17536)
Numerical id15
Authorasteroid (17536)
Entry typeDefinition
Classificationmsc 47L25
Classificationmsc 46A32
Classificationmsc 47A30
Synonyminduced norm
Related topicVectorNorm
Related topicOperatorTopologies
Related topicHomomorphismsOfCAlgebrasAreContinuous
Related topicCAlgebra
Definesbounded linear map
Definesunbounded linear map
Definesbounded operator
Definesunbounded operator
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 14:17:32