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operator norm
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(Definition)
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Let
 be a linear map between normed vector spaces
 and
 . To each such map (operator) $A$ we can assign a non-negative number $\|A\|_{\op}$ defined by
where the supremum $\|A\|_{\op}$ could be finite or infinite. Equivalently, the above definition can be written as
By convention, if
 is the zero vector space, any operator from
 to
 must be the zero operator and is assigned zero norm.
$\|A\|_{\op}$ is called the the operator norm (or the induced norm) of $A$ , for reasons that will be clear in the next section.
Definition - If $\|A\|_{\op}$ is finite, we say that $A$ is a bounded. Otherwise, we say that $A$ is unbounded.
It turns out that, for bounded operators, $\|\cdot\|_{\op}$ satisfies all the properties of a norm (hence the name operator norm). The proof follows immediately from the definition:
- Positivity:
- Since $\|A\v\|\ge 0$ , by definition $\|A\|_\op \ge 0$ . Also, $\|A\v\| = 0$ identically only if $A=0$ . Hence $\|A\|_\op = 0$ only if $A = 0$ .
- Absolute homogeneity:
- Since $\|\lambda A\v\|=|\lambda| \|A\v\|$ , by definition $\|\lambda A\|_\op = |\lambda| \|A\|_\op$ .
- Triangle inequality:
- Since $\|(A+B)\v\|=\|A\v+B\v\|\le \|A\v\| + \|B\v\|$ , by definition $\|A+B\|_\op \le \|A\|_\op + \|B\|_\op$ .
The set
 of bounded linear maps from
 to
 forms a vector space and $\|\cdot\|_{\op}$ defines a norm in it.
Suppose that
 and
 , where $\|\cdot\|_p$ is the vector p-norm. Then the operator norm $\|\cdot\|_\op = \|\cdot\|_p$ is the matrix p-norm.
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Cross-references: matrix p-norm, vector p-norm, vector space, proof, properties, clear, norm, zero operator, zero vector space, infinite, finite, supremum, number, operator, map, normed vector spaces, linear map
There are 54 references to this entry.
This is version 12 of operator norm, born on 2002-06-03, modified 2008-09-17.
Object id is 3018, canonical name is OperatorNorm.
Accessed 21515 times total.
Classification:
| AMS MSC: | 46A32 (Functional analysis :: Topological linear spaces and related structures :: Spaces of linear operators; topological tensor products; approximation properties) | | | 47L25 (Operator theory :: Linear spaces and algebras of operators :: Operator spaces ) | | | 47A30 (Operator theory :: General theory of linear operators :: Norms ) |
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Pending Errata and Addenda
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