If and are linear operators, then the respective operator norms are sub-multiplicative in addition to the usual norm properties . That is, it applies
The operator norm of linear mappings between finite-dimensional vector spaces is always finite, since the unit sphere is a compact set . Thus, in the finite-dimensional case, the operator norm is always a real norm. This does not always apply to infinite-dimensional vector spaces. Operators whose norm takes infinite as their value are called unbounded . Strictly speaking, the operator norm is not a real norm on spaces with such unrestricted operators. One can show that a linear operator between normalized spaces has a finite operator norm if and only if it is bounded and thus continuous . In particular, this turns the space of continuous linear operators into a normalized vector space .
If is complete , the operator space is complete. The room doesn't need to be complete.
Since every linear operator between finite-dimensional vector spaces can be represented as a matrix , special matrix norms , the natural or induced matrix norms , are obvious examples of operator norms . The most important of these natural matrix norms are the following three.
Let be a bounded sequence and thus an element of the sequence space that is provided with the norm . Now define a multiplication operator by . Then applies to the corresponding operator norm
Norm of a (pseudo) differential operator
Let be and be a bounded linear operator between Sobolev spaces . Such operators can be represented as pseudo differential operators . Under certain circumstances, especially when the order of the Sobolev spaces is an integer, the pseudo differential operators are ( weak ) differential operators . The space of (pseudo) differential operators can be provided with an operator norm. Since the norm in Sobolev space is given by , the operator norm for the (pseudo) differential operators is by