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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.
The standard absolute value on the integers. The standard absolute value on the complex numbers.; The p-adic absolute value on the rational numbers.; If R is the field of rational functions over a field F and () is a fixed irreducible polynomial over F, then the following defines an absolute value on R: for () in R define | | to be , where () = () and ((), ()) = = ((), ()).
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.
Other norms defined in terms of or include the weak , space norms (for <), the norm on Lebesgue space (,), and operator norms. Monotone sequences in S {\displaystyle S} that converge to sup S {\displaystyle \sup S} (or to inf S {\displaystyle \inf S} ) can also be used to help prove many of the formula given below, since addition and ...
For example, points (2, 0), (2, 1), and (2, 2) lie along the perimeter of a square and belong to the set of vectors whose sup norm is 2. In mathematical analysis, the uniform norm (or sup norm) assigns to real-or complex-valued bounded functions defined on a set the non-negative number
1 Formal definition. 2 Examples. ... In mathematics, the (field) norm is a particular mapping defined in field theory, ... For example, for ...
Any Euclidean norm is a Dedekind-Hasse norm; thus, (5) shows that a Euclidean domain is a PID. (4) compares to: (4) compares to: An integral domain is a UFD if and only if it is a GCD domain (i.e., a domain where every two elements have a greatest common divisor) satisfying the ascending chain condition on principal ideals.
Examples of inner products include the real and complex dot product; see the examples in inner product. Every inner product gives rise to a Euclidean l 2 {\displaystyle l_{2}} norm , called the canonical or induced norm , where the norm of a vector u {\displaystyle \mathbf {u} } is denoted and defined by