# 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) $A$ we can assign a non-negative number $\|A\|_{{\rm op}}$ defined by

 $\|A\|_{{\rm op}}:=\mathop{\sup_{{\mathbf{v}}\in{\mathsf{V}}}}_{{\mathbf{v}}% \neq{\bf 0}}\frac{\|A{\mathbf{v}}\|}{\|{\mathbf{v}}\|},$

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

 $\|A\|_{\rm op}:=\mathop{\sup_{{\mathbf{v}}\in{\mathsf{V}}}}_{\|{\mathbf{v}}\|=% 1}\|A{\mathbf{v}}\|=\mathop{\sup_{{\mathbf{v}}\in{\mathsf{V}}}}_{0<\|{\mathbf{% v}}\|\leq 1}\|A{\mathbf{v}}\|.$

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.

$\|A\|_{{\rm op}}$ is called the the operator norm  (or the induced norm) of $A$, for reasons that will be clear in the next .

## Operator norm is in fact a norm

Definition - If $\|A\|_{{\rm op}}$ is finite, we say that $A$ is a . Otherwise, we say that $A$ is .

It turns out that, for bounded operators  , $\|\cdot\|_{{\rm op}}$ satisfies all the properties of a norm (hence the name operator norm). The proof follows immediately from the definition:

Positivity:

Since $\|A{\mathbf{v}}\|\geq 0$, by definition $\|A\|_{\rm op}\geq 0$. Also, $\|A{\mathbf{v}}\|=0$ identically only if $A=0$. Hence $\|A\|_{\rm op}=0$ only if $A=0$.

Absolute homogeneity:

Since $\|\lambda A{\mathbf{v}}\|=|\lambda|\|A{\mathbf{v}}\|$, by definition $\|\lambda A\|_{\rm op}=|\lambda|\|A\|_{\rm op}$.

Triangle inequality:

Since $\|(A+B){\mathbf{v}}\|=\|A{\mathbf{v}}+B{\mathbf{v}}\|\leq\|A{\mathbf{v}}\|+\|B% {\mathbf{v}}\|$, by definition $\|A+B\|_{\rm op}\leq\|A\|_{\rm op}+\|B\|_{\rm op}$.

The set $L({\mathsf{V}},{\mathsf{W}})$ of bounded linear maps from ${\mathsf{V}}$ to ${\mathsf{W}}$ forms a vector space  and $\|\cdot\|_{{\rm op}}$ defines a norm in it.

## Example

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

 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