An operator norm is an object from the mathematical branch of functional analysis. These operator standards generalized idea to assign to an object a length on the amount of linear operators. Are the operators under consideration is continuous, then the operator norm is a real standard, otherwise the operator norm take the value infinity. The operator norm of a linear map between finite dimensional vector spaces is a natural matrix norm by choosing a base.
- 3.1 Natural matrix norms
- 3.2 The sequence space l2
- 3.3 norm of a (pseudo - ) differential operator
Let and be normed vector spaces and be a linear operator. Then the operator norm
With respect to the vector norms and
Defined. For this is equivalent to
The operator norm also met along with the characteristic standards for three properties definiteness, absolute homogeneity and triangle inequality, the following:
Are and linear operators, the respective operator norms are submultiplicative in addition to the usual norm properties. That is, it is
The operator norm of linear maps between finite dimensional vector spaces is always finite, since the unit sphere is a compact set. Thus, in the finite case, the operator norm is always a real standard. This is not always true for infinite-dimensional vector spaces. Operators whose norm infinity takes as a value are called unlimited. In areas with such unbounded operators, the operator norm is not strictly true standard. It can be shown that a linear operator between normed spaces if and only has a finite operator norm if it is limited, and thus steadily. In particular, this is the space of continuous linear operators on a normed vector space.
If it is complete, the operator space is complete. The space does not need to be completely in general.
Natural matrix norms
Since we can represent any linear operator between finite-dimensional vector spaces as a matrix, are special matrix norms, the natural or induced matrix norms, obvious examples of operator norms. The most important of these natural matrix norms are the following three.
- The column sum norm is induced by the sum Standard Standard:
- The spectral norm is induced by the Euclidean norm norm:
- The row sum norm is induced by the maximum norm norm:
However, not every matrix norm one operator norm. The total norm and the Frobenius norm, for example, no operator norms.
The sequence space l2
Let be a bounded sequence and thus an element of the sequence space, which is provided with the standard. Define now by a multiplication operator. Then for the corresponding operator norm
Norm of a ( pseudo-) differential operator
Be and be a bounded linear operator between Sobolev spaces. Such operators can be represented as a pseudo-differential operators. Under certain circumstances, particularly when the order of the Sobolev spaces is an integer, the pseudo-differential operators (weak) are differential operators. The space of (pseudo) differential operators can be fitted with an operator norm. Since the norm in the Sobolev space is given by the operator norm for the ( pseudo ) differential operators by