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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.
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
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 .
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 .
In mathematics, a normed algebra A is an algebra over a field which has a sub-multiplicative norm: , ‖ ‖ ‖ ‖ ‖ ‖. Some authors require it to have a multiplicative identity 1 A such that ║1 A ║ = 1.
Asymmetric norms differ from norms in that they need not satisfy the equality () = (). If the condition of positive definiteness is omitted, then p {\displaystyle p} is an asymmetric seminorm . A weaker condition than positive definiteness is non-degeneracy : that for x ≠ 0 , {\displaystyle x\neq 0,} at least one of the two numbers p ( x ...
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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