operator norm


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


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


By convention, if 𝖵 is the zero vector space, any operator from 𝖵 to 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:


Since A𝐯0, by definition Aop0. Also, A𝐯=0 identically only if A=0. Hence Aop=0 only if A=0.

Absolute homogeneity:

Since λA𝐯=|λ|A𝐯, by definition λAop=|λ|Aop.

Triangle inequality:

Since (A+B)𝐯=A𝐯+B𝐯A𝐯+B𝐯, by definition A+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.


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

Title operator norm
Canonical name OperatorNorm
Date of creation 2013-03-22 12:43:20
Last modified on 2013-03-22 12:43:20
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 15
Author asteroid (17536)
Entry type Definition
Classification msc 47L25
Classification msc 46A32
Classification msc 47A30
Synonym induced norm
Related topic VectorNorm
Related topic OperatorTopologies
Related topic HomomorphismsOfCAlgebrasAreContinuous
Related topic CAlgebra
Defines bounded linear map
Defines unbounded linear map
Defines bounded operator
Defines unbounded operator