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The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm". [1] A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "" in the homogeneity axiom.
Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces .
Norm map, a map from a pointset into the ordinals inducing a prewellordering; Norm group, a group in class field theory that is the image of the multiplicative group of a field; Norm function, a term in the study of Euclidean domains, sometimes used in place of "Euclidean function"
An example of such a space is the Fréchet space (), whose definition can be found in the article on spaces of test functions and distributions, because its topology is defined by a countable family of norms but it is not a normable space because there does not exist any norm ‖ ‖ on () such that the topology this norm induces is equal to .
Restricting this extended norm to the bounded functions (i.e., the functions with finite above extended norm) yields a (finite-valued) norm, called the uniform norm on . Note that the definition of uniform norm does not rely on any additional structure on the set X {\displaystyle X} , although in practice X {\displaystyle X} is often at least a ...
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The ...
In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. Formal definition [ edit ]