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In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin.
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 .
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 ...
Operator norm, a map that assigns a length or size to any operator in a function space; Norm (abelian group), a map that assigns a length or size to any element of an abelian group; Field norm a map in algebraic number theory and Galois theory that generalizes the usual distance norm; Ideal norm, the ideal-theoretic generalization of the field norm
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. [1] A norm is a generalization of the intuitive notion of "length" in the physical world.
For a real number, the -norm or -norm of is defined by ‖ ‖ = (| | + | | + + | |) /. The absolute value bars can be dropped when p {\displaystyle p} is a rational number with an even numerator in its reduced form, and x {\displaystyle x} is drawn from the set of real numbers, or one of its subsets.
By definition, all Euclidean vectors have a magnitude (see above). However, a vector in an abstract vector space does not possess a magnitude. A vector space endowed with a norm, such as the Euclidean space, is called a normed vector space. [8] The norm of a vector v in a normed vector space can be considered to be the magnitude of v.