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Operator norm

(Redirected from Norm topology)

In mathematics, the operator norm is a norm defined on the space of bounded operators between two normed spaces.

 Contents

Further analysis

More formally, the operator norm of a bounded linear operator L from V to W, where V and W are both normed real (or complex) vector spaces, is defined as the supremum of ||L(v)|| taken over all v in V of norm 1. This definition uses the property ||c.v|| = |c|.||v|| where c is a scalar, to restrict attention to v with ||v|| = 1. Geometrically we need (for real scalars) to look at one vector only on each ray out from the origin 0.

A linear operator is bounded (and hence continuous) precisely if it has a (finite) operator norm.

The operator norm indeed satisfies the conditions for being a norm, so the space of all bounded linear transformations from V to W is itself a normed vector space. It is complete if W is complete.

Equivalent definitions for the operator norm

One can show that the following equalities hold for the norm $\|L\|_{op}$ of a bounded linear operator:

$\|L\|_{op} = \sup\{\|Lv\|_W : v\in V \mbox{ with }\|v\|_V \le 1\}$
$= \sup\{\|Lv\|_W : v\in V \mbox{ with }\|v\|_V = 1\}$
$= \sup\left\{\frac{\|Lv\|_W}{\|v\|_V} : v\in V \mbox{ with }v\ne 0\right\}.$

Examples

An induced matrix norm is the operator norm of a matrix viewed as a linear transformation.

Computing the operator norm

Finding the norm of a given operator is often a difficult problem. Even for matrices it can be nontrivial. One way of finding the norm of a matrix A (with complex entries) is to compute A*A, which will be a non-negative hermitian operator, and then find its largest eigenvalue. This non-negative real number is then the square of the norm of A. However, it can be computationally difficult to find the largest eigenvalue of a given matrix.