PlanetMath: operator norm

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operator norm

(Definition)

Let $A\colon {\mathsf V}\to{\mathsf W}$ be a linear map between normed vector spaces ${\mathsf V}$ and ${\mathsf W}$ . To each such map (operator) we can assign a non-negative number $\Vert A\Vert _{{\rm op}}$ defined by

$\displaystyle \Vert A\Vert _{{\rm op}} := \mathop{\sup_{{\mathbf v}\in{\mathsf ... ..._{{\mathbf v}\ne{\bf0}} \frac{\Vert A{\mathbf v}\Vert}{\Vert{\mathbf v}\Vert},$

where the supremum $\Vert A\Vert _{{\rm op}}$ could be finite or infinite. Equivalently, the above definition can be written as

$\displaystyle \Vert A\Vert _{\rm op}:= \mathop{\sup_{{\mathbf v}\in{\mathsf V}}... ...thbf v}\in{\mathsf V}}}_{0<\Vert{\mathbf v}\Vert\le1} \Vert A{\mathbf v}\Vert.$

By convention, if ${\mathsf V}$ is the zero vector space, any operator from ${\mathsf V}$ to ${\mathsf W}$ must be the zero operator and is assigned zero norm.

It turns out that $\Vert\cdot\Vert _{{\rm op}}$ satisfies all the properties of a norm and hence is called the operator norm (or the induced norm) of . The proof follows immediately from the definition:

positivity:: Since $\Vert A{\mathbf v}\Vert\ge 0$ , by definition $\Vert A\Vert _{\rm op}\ge 0$ . Also, $\Vert A{\mathbf v}\Vert = 0$ identically only if . Hence $\Vert A\Vert _{\rm op}= 0$ only if .
absolute homogeneity:: Since $\Vert\lambda A{\mathbf v}\Vert=\vert\lambda\vert \Vert A{\mathbf v}\Vert$ , by definition $\Vert\lambda A\Vert _{\rm op}= \vert\lambda\vert \Vert A\Vert _{\rm op}$ .
triangle inequality:: Since $\Vert(A+B){\mathbf v}\Vert=\Vert A{\mathbf v}+B{\mathbf v}\Vert\le \Vert A{\mathbf v}\Vert + \Vert B{\mathbf v}\Vert$ , by definition $\Vert A+B\Vert _{\rm op}\le \Vert A\Vert _{\rm op}+ \Vert B\Vert _{\rm op}$ .

If $\Vert A\Vert _{{\rm op}}$ is finite, we say that is a bounded. Otherwise, we say that is unbounded.

The space $L({\mathsf V},{\mathsf W})$ of bounded linear maps from ${\mathsf V}$ to ${\mathsf W}$ forms a vector space with $\Vert\cdot\Vert _{{\rm op}}$ as the natural norm.

Example

Suppose that ${\mathsf V}=({\mathbb{R}}^n,\Vert\cdot\Vert _p)$ and ${\mathsf W}=({\mathbb{R}}^n,\Vert\cdot\Vert _p)$ , where $\Vert\cdot\Vert _p$ is the vector p-norm. Then the operator norm $\Vert\cdot\Vert _{\rm op}= \Vert\cdot\Vert _p$ is the matrix p-norm.

"operator norm" is owned by CWoo. [ full author list (2) | owner history (1) ]

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See Also: vector norm, operator topologies

Other names:

induced norm

Also defines:

bounded linear map, unbounded linear map, bounded operator, unbounded operator

Attachments:: examples of bounded and unbounded operators (Example) by matte

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Cross-references: matrix p-norm, vector p-norm, vector space, properties, norm, zero operator, zero vector space, infinite, finite, supremum, number, operator, map, normed vector spaces, linear map
There are 32 references to this entry.

This is version 11 of operator norm, born on 2002-06-03, modified 2007-08-22.
Object id is 3018, canonical name is OperatorNorm.
Accessed 15978 times total.

Classification:

AMS MSC:	47L25 (Operator theory :: Linear spaces and algebras of operators :: Operator spaces )
	47A30 (Operator theory :: General theory of linear operators :: Norms )

Pending Errata and Addenda

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