Science Fair Project Encyclopedia
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 of a bounded linear operator:
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.
The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details