Norm (mathematics): Difference between revisions

{{shortShort description|Length in a vector space}}

{{aboutAbout|normsthe ofconcept in [[normed vector 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 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''' or '''length''' of the vector. This norm can be defined as the [[square root]] of the [[inner product]] of a vector with itself.

The term '''pseudonorm''' has been used for several related meanings. It may be a synonym of "seminorm".<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 pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "<math>\,\leq\,</math>" in the homogeneity axiom.<ref>{{Cite web |title=Pseudo-norm - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Pseudo-norm |access-date=2022-05-12 |website=encyclopediaofmath.org}}</ref>{{Dubious|Encylopedia of Math definition of pseudonorm|date=April 2024}}

Some authors include non-negativity as part of the definition of "norm", although this is not necessary.

The relation "<math>p</math> is equivalent to <math>q</math>" is [[Reflexive relation|reflexive]], [[Symmetric relation|symmetric]] (<math>c q \leq p \leq C q</math> implies <math>\tfrac{1}{C} p \leq q \leq \tfrac{1}{c} p</math>), and [[Transitive relation|transitive]] and thus defines an [[equivalence relation]] on the set of all norms on <math>X.</math>

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

is a norm on the [[Dimension (vector space)|one-dimensional]] vector spaces formed by the [[real number|real]] or [[complex number]]s. The complex numbers form a [[dimension (vector space)|one-dimensional vector space]] over themselves and a two-dimensional vector space over the reals; the absolute value is a norm for these two structures.

This is the '''Euclidean norm''', which gives the ordinary distance from the origin to the point '''''X'''''—a consequence of the [[Pythagorean theorem]].

The Euclidean norm is by far the most commonly used norm on <math>\R^n,</math><ref name=":1" /> but there are other norms on this vector space as will be shown below.

However, all these norms are equivalent in the sense that they all define the same topology on finite-dimensional spaces.

The [[inner product]] of two vectors of a [[Euclidean vector space]] is the [[dot product]] of their [[coordinate vector]]s over an [[orthonormal basis]].

The Euclidean norm is also called the '''quadratic norm''', '''<math>L^2</math> norm''',<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Norm|url=https://mathworld.wolfram.com/Norm.html|access-date=2020-08-24|website=mathworld.wolfram.com|language=en}}</ref> '''<math>\ell^2</math> norm''', '''2-norm''', or '''square norm'''; see [[Lp space|<math>L^p</math> space]].

The Euclidean norm of a [[complex number]] is the [[Absolute value#Complex numbers|absolute value]] (also called the '''modulus''') of it, if the [[complex plane]] is identified with the [[Euclidean plane]] <math>\R^2.</math> This identification of the complex number <math>x + i y</math> as a vector in the Euclidean plane, makes the quantity <math display=inline>\sqrt{x^2 + y^2}</math> (as first suggested by Euler) the Euclidean norm associated with the complex number. For <math>z = x +iy</math>, the norm can also be written as <math>\sqrt{\bar z z}</math> where <math>\bar z</math> is the [[complex conjugate]] of <math>z\,.</math>

There are exactly four [[Hurwitz's theorem (composition algebras)|Euclidean Hurwitz algebra]]s over the [[real number]]s. These are the real numbers <math>\R,</math> the complex numbers <math>\Complex,</math> the [[quaternion]]s <math>\mathbb{H},</math> and lastly the [[octonion]]s <math>\mathbb{O},</math> where the dimensions of these spaces over the real numbers are <math>1, 2, 4, \text{ and } 8,</math> respectively.

The set of vectors whose 1-norm is a given constant forms the surface of a [[cross polytope]], ofwhich has dimension equivalentequal to thatthe dimension of the normvector space minus 1.

The Taxicab norm is also called the '''<math>\ell^1</math> norm'''. The distance derived from this norm is called the [[Manhattan distance]] or '''<math>\ell_1ell^1</math> distance'''.

Let <math>p \geq 1</math> be a real number.

The <math>p</math>-norm (also called <math>\ell_pell^p</math>-norm) of vector <math>\mathbf{x} = (x_1, \ldots, x_n)</math> is<ref name=":1" />

For <math>p = 2,</math> the <math>\|\,\cdot\,\|_2</math>-norm is even induced by a canonical [[inner product]] <math>\langle \,\cdot,\,\cdot\rangle,</math> meaning that <math display="inline">\|\mathbf{x}\|_2 = \sqrt{\langle \mathbf{x}, \mathbf{x} \rangle}</math> for all vectors <math>\mathbf{x}.</math> This inner product can be expressed in terms of the norm by using the [[polarization identity]].

while for the space <math>L^2(X, \mu)</math> associated with a [[measure (mathematics)|measure space]] <math>(X, \Sigma, \mu),</math> which consists of all [[square-integrable function]]s, this inner product is

This definition is still of some interest for <math>0 < p < 1,</math> but the resulting function does not define a norm,<ref>Except in <math>\R^1,</math> where it coincides with the Euclidean norm, and <math>\R^0,</math> where it is trivial.</ref> because it violates the [[triangle inequality]].

In probability and functional analysis, the zero norm induces a complete metric topology for the space of [[measurable function]]s and for the [[F-space]] of sequences with F–norm <math display="inline">(x_n) \mapsto \sum_n{2^{-n} x_n/(1+x_n)}.</math><ref name="RolewiczControl">{{Citation |title=Functional analysis and control theory: Linear systems |last=Rolewicz |first=Stefan |year=1987 |isbn=90-277-2186-6 |publisher=D. Reidel Publishing Co.; PWN—Polish Scientific Publishers |oclc=13064804 |edition=Translated from the Polish by Ewa Bednarczuk |series=Mathematics and its Applications (East European Series) |location=Dordrecht; Warsaw |volume=29 |pages=xvi,524 |mr=920371 |doi=10.1007/978-94-015-7758-8}}</ref>

In [[metric geometry]], the [[discrete metric]] takes the value one for distinct points and zero otherwise. When applied coordinate-wise to the elements of a vector space, the discrete distance defines the ''[[Hamming distance]]'', which is important in [[coding theory|coding]] and [[information theory]].

In the field of real or complex numbers, the distance of the discrete metric from zero is not homogeneous in the non-zero point; indeed, the distance from zero remains one as its non-zero argument approaches zero.

However, the discrete distance of a number from zero does satisfy the other properties of a norm, namely the triangle inequality and positive definiteness.

In [[signal processing]] and [[statistics]], [[David Donoho]] referred to the ''zero'' '''"'''''norm'''''"''' with quotation marks.

Following Donoho's notation, the zero "norm" of <math>x</math> is simply the number of non-zero coordinates of <math>x,</math> or the Hamming distance of the vector from zero.

When this "norm" is localized to a bounded set, it is the limit of <math>p</math>-norms as <math>p</math> approaches 0.

Of course, the zero "norm" is '''not''' truly a norm, because it is not [[homogeneous function#Positive homogeneity|positive homogeneous]].

Indeed, it is not even an F-norm in the sense described above, since it is discontinuous, jointly and severally, with respect to the scalar argument in scalar–vector multiplication and with respect to its vector argument.

The generalization of the above norms to an infinite number of components leads to [[Lp space|<math>\ell^p</math> and <math>L^p</math> spaces]] for <math>p \ge 1\,,</math> with norms

for complex-valued sequences and functions on <math>X \sube \R^n</math> respectively, which can be further generalized (see [[Haar measure]]). These norms are also valid in the limit as <math>p \rightarrow +\infty</math>, giving a [[supremum norm]], and are called <math>\ell^\infty</math> and <math>L^\infty\,.</math>

The concept of norm <math>N(z)</math> in [[composition algebra]]s does {{em|not}} share the usual properties of a norm assince it[[null mayvector]]s beare negative or zero for <math>z \neq 0allowed.</math> A composition algebra <math>(A, {}^*, N)</math> consists of an [[algebra over a field]] <math>A,</math> an [[involution (mathematics)|involution]] <math>{}^*,</math> and a [[quadratic form]] [[Degree of a field extension|<math>N(z) = z z^*</math>]] called the "norm".

The characteristic feature of composition algebras is the [[homomorphism]] property of <math>N</math>: for the product <math>w z</math> of two elements <math>w</math> and <math>z</math> of the composition algebra, its norm satisfies <math>N(wz) = N(w) N(z).</math> ForIn the case of [[division algebra]]s <math>\R,</math> <math>\Complex,</math> <math>\mathbb{H},</math> and '''<math>\mathbb{O'''}</math> the composition algebra norm is the square of the norm discussed above. In those cases the norm is a [[definite quadratic form]]. In otherthe composition[[split algebrasalgebra]]s the norm is an [[isotropic quadratic form]].

For any norm <math>p : X \to \R</math> on a vector space <math>X,</math> the [[reverse triangle inequality]] holds:

For the [[Lp space|<math>L^p</math> norms]], we have [[Hölder's inequality]]<ref name="GOLUB">{{cite book|last1=Golub|first1=Gene|title=Matrix Computations|last2=Van Loan|first2=Charles F.|publisher=The Johns Hopkins University Press|year=1996|isbn=0-8018-5413-X|edition=Third|location=Baltimore|page=53|author-link1=Gene_HGene H._Golub Golub}}</ref>

The concept of [[unit circle]] (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is a [[square (geometry)|square]], oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit [[circle]],; while for the infinity norm, it is aan differentaxis-aligned square. For any <math>p</math>-norm, it is a [[superellipse]] with congruent axes (see the accompanying illustration). Due to the definition of the norm, the unit circle must be [[convex set|convex]] and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, and <math>p \geq 1</math> for a <math>p</math>-norm).

In terms of the vector space, the seminorm defines a [[topology]] on the space, and this is a [[Hausdorff space|Hausdorff]] topology precisely when the seminorm can distinguish between distinct vectors, which is again equivalent to the seminorm being a norm. The topology thus defined (by either a norm or a seminorm) can be understood either in terms of sequences or open sets. A [[sequence]] of vectors <math>\{v_n\}</math> is said to [[modes of convergence|converge]] in norm to <math>v,</math> if <math>\left\|v_n - v\right\| \to 0</math> as <math>n \to \infty.</math> Equivalently, the topology consists of all sets that can be represented as a union of open [[ball (mathematics)|balls]]. If <math>(X, \|\cdot\|)</math> is a normed space then{{sfn|Narici|Beckenstein|2011|pp=107-113}}

For instance, if <math>p > r \geq 1</math> on <math>\Complex^n,</math> then<ref name="Relation between p-norms">{{cite web |url=https://math.stackexchange.com/questionsq/218046/relations-between-p-norms |title=Relation between p-norms|website=Mathematics Stack Exchange}}</ref>

That is,

* {{Khaleelulla Counterexamples in Topological Vector Spaces}} <!--{{sfn|Khaleelulla|1982|p=}}-->

* {{Schaefer Wolff Topological Vector Spaces|edition=2}} <!--{{sfn|Schaefer|1999|p=}}-->

* {{Trèves François Topological vector spaces, distributions and kernels}} <!--{{sfn|Trèves|2006|p=}}-->