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{{Short description|Length in a vector space}}
{{About|norms of [[normed vector space]]s|field theory|Field norm|ideals|Ideal norm|group theory|Norm (group)|norms in descriptive set theory|prewellordering}}
In [[mathematics]], a '''norm''' is a [[function (mathematics)|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 (mathematics)|origin]]: it [[Equivariant map|commutes]] with scaling, obeys a form of the [[triangle inequality]], and is zero only at the origin. In particular, the [[Euclidean distance]] in a [[Euclidean space]] is defined by a norm on the associated [[Euclidean vector space]], called the [[#Euclidean norm|Euclidean norm]], the [[#p-norm|2-norm]], or, sometimes, the '''magnitude''' of the vector. This norm can be defined as the [[square root]] of the [[inner product]] of a vector with itself
A [[seminorm]] satisfies the first two properties of a norm, but may be zero for vectors other than the origin.<ref name="Knapp">{{cite book|title=Basic Real Analysis|publisher=Birkhäuser|author=Knapp, A.W.|year=2005|page=[https://books.google.com/books?id=4ZZCAAAAQBAJ&pg=279] |isbn=978-0-817-63250-2}}</ref> A vector space with a specified norm is called a [[normed vector space]]. In a similar manner, a vector space with a seminorm is called a ''seminormed vector space''.
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