Norm (mathematics): Difference between revisions

{{About|normsthe ofconcept [[normedin vectornormed space]]sspaces|field theory|Field norm|ideals|Ideal norm|commutative algebra|Absolute value (algebra)|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 ana [[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''' or '''length''' of the vector. This norm can be defined as the [[square root]] of the [[inner product]] of a vector with itself.

<math display="block">\|x\| = |x|</math>

is a norm on the [[Dimension (vector space)|one-dimensional]] vector space formed by the [[real number|real]] or [[complex number]]s. The (complex numbers, althoughform representeda in[[dimension two(vector dimensions,space)|one-dimensional canvector formspace]] over themselves and a two-dimensional vector space ofover dimensionthe one)reals; the absolute value is a norm for these two structures.