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Norm (mathematics)

From Wikipedia, the free encyclopedia
(Redirected fromPseudonorm)
Length in a vector space
This article is about the concept in normed spaces. For other uses, seeNorm (disambiguation) § In mathematics.

Inmathematics, anorm is afunction from a real or complexvector space to the non-negative real numbers that behaves in certain ways like the distance from theorigin: itcommutes with scaling, obeys a form of thetriangle inequality, and zero is only at the origin. In particular, theEuclidean distance in aEuclidean space is defined by a norm on the associatedEuclidean vector space, called theEuclidean norm, the2-norm, or, sometimes, themagnitude orlength of the vector. This norm can be defined as thesquare root of theinner product of a vector with itself.

Aseminorm satisfies the first two properties of a norm but may be zero for vectors other than the origin.[1] A vector space with a specified norm is called anormed vector space. In a similar manner, a vector space with a seminorm is called aseminormed vector space.

The termpseudonorm has been used for several related meanings. It may be a synonym of "seminorm".[1] It can also refer to a norm that can take infinite values[2] or to certain functions parametrised by adirected set.[3]

Definition

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Given avector spaceX{\displaystyle X} over asubfieldF{\displaystyle F} of the complex numbersC,{\displaystyle \mathbb {C} ,} anorm onX{\displaystyle X} is areal-valued functionp:X→R{\displaystyle p:X\to \mathbb {R} } with the following properties, where|s|{\displaystyle |s|} denotes the usualabsolute value of a scalars{\displaystyle s}:[4]

  1. Subadditivity /Triangle inequality:
    p(x+y)≤p(x)+p(y){\displaystyle p(x+y)\leq p(x)+p(y)} for allx,y∈X.{\displaystyle x,y\in X.}
  2. Absolute homogeneity:
    p(sx)=|s|p(x){\displaystyle p(sx)=|s|p(x)} for allx∈X{\displaystyle x\in X} and all scalarss.{\displaystyle s.}
  3. Positive definiteness / Positiveness[5] /Point-separating:
    for allx∈X,{\displaystyle x\in X,} ifp(x)=0,{\displaystyle p(x)=0,} thenx=0.{\displaystyle x=0.}

Aseminorm onX{\displaystyle X} is a functionp:X→R{\displaystyle p:X\to \mathbb {R} } that has properties (1.) and (2.)[6] so that in particular, every norm is also a seminorm (and thus also asublinear functional). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that ifp{\displaystyle p} is a norm (or more generally, a seminorm), thenp(0)=0{\displaystyle p(0)=0} and thatp{\displaystyle p} also has the following property:

  2. Non-negativity:[5]p(x)≥0{\displaystyle p(x)\geq 0} for allx∈X.{\displaystyle x\in X.}

Some authors include non-negativity as part of the definition of "norm", although this is not necessary.Although this article defined "positive" to be a synonym of "positive definite", some authors instead define "positive" to be a synonym of "non-negative";[7] these definitions are not equivalent.

Equivalent norms

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Suppose thatp{\displaystyle p} andq{\displaystyle q} are two norms (or seminorms) on a vector spaceX.{\displaystyle X.} Thenp{\displaystyle p} andq{\displaystyle q} are calledequivalent, if there exist two positive real constantsc{\displaystyle c} andC{\displaystyle C} such that for every vectorx∈X,{\displaystyle x\in X,}cq(x)≤p(x)≤Cq(x).{\displaystyle cq(x)\leq p(x)\leq Cq(x).}The relation "p{\displaystyle p} is equivalent toq{\displaystyle q}" isreflexive,symmetric (cq≤p≤Cq{\displaystyle cq\leq p\leq Cq} implies1Cp≤q≤1cp{\displaystyle {\tfrac {1}{C}}p\leq q\leq {\tfrac {1}{c}}p}), andtransitive and thus defines anequivalence relation on the set of all norms onX.{\displaystyle X.}The normsp{\displaystyle p} andq{\displaystyle q} are equivalent if and only if they induce the same topology onX.{\displaystyle X.}[8] Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.[8]

Notation

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If a normp:X→R{\displaystyle p:X\to \mathbb {R} } is given on a vector spaceX,{\displaystyle X,} then the norm of a vectorz∈X{\displaystyle z\in X} is usually denoted by enclosing it within double vertical lines:‖z‖=p(z){\displaystyle \|z\|=p(z)}, as proposed byStefan Banach in his doctoral thesis from 1920. Such notation is also sometimes used ifp{\displaystyle p} is only a seminorm. For the length of a vector in Euclidean space (which is an example of a norm, asexplained below), the notation|x|{\displaystyle |x|} with single vertical lines is also widespread.

Examples

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Every (real or complex) vector space admits a norm: Ifx∙=(xi)i∈I{\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}} is aHamel basis for a vector spaceX{\displaystyle X} then the real-valued map that sendsx=∑i∈Isixi∈X{\displaystyle x=\sum _{i\in I}s_{i}x_{i}\in X} (where all but finitely many of the scalarssi{\displaystyle s_{i}} are0{\displaystyle 0}) to∑i∈I|si|{\displaystyle \sum _{i\in I}\left|s_{i}\right|} is a norm onX.{\displaystyle X.}[9] There are also a large number of norms that exhibit additional properties that make them useful for specific problems.

Absolute-value norm

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"Absolute-value norm" redirects here. For the commutative algebra concept, seeAbsolute value (algebra).

Theabsolute value|x|{\displaystyle |x|}is a norm on the vector space formed by thereal orcomplex numbers. The complex numbers form aone-dimensional vector space over themselves and a two-dimensional vector space over the reals; the absolute value is a norm for these two structures.

Any normp{\displaystyle p} on a one-dimensional vector spaceX{\displaystyle X} is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preservingisomorphism of vector spacesf:F→X,{\displaystyle f:\mathbb {F} \to X,} whereF{\displaystyle \mathbb {F} } is eitherR{\displaystyle \mathbb {R} } orC,{\displaystyle \mathbb {C} ,} and norm-preserving means that|x|=p(f(x)).{\displaystyle |x|=p(f(x)).}This isomorphism is given by sending1∈F{\displaystyle 1\in \mathbb {F} } to a vector of norm1,{\displaystyle 1,} which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm.

Euclidean norm

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Further information:Euclidean norm andEuclidean distance

On then{\displaystyle n}-dimensionalEuclidean spaceRn,{\displaystyle \mathbb {R} ^{n},} the intuitive notion of length of the vectorx=(x1,x2,…,xn){\displaystyle {\boldsymbol {x}}=\left(x_{1},x_{2},\ldots ,x_{n}\right)} is captured by the formula[10]‖x‖2:=x12+⋯+xn2.{\displaystyle \|{\boldsymbol {x}}\|_{2}:={\sqrt {x_{1}^{2}+\cdots +x_{n}^{2}}}.}

This is theEuclidean norm, which gives the ordinary distance from the origin to the pointX—a consequence of thePythagorean theorem.This operation may also be referred to as "SRSS", which is an acronym for thesquareroot of thesum ofsquares.[11]

The Euclidean norm is by far the most commonly used norm onRn,{\displaystyle \mathbb {R} ^{n},}[10] 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.

Theinner product of two vectors of aEuclidean vector space is thedot product of theircoordinate vectors over anorthonormal basis.Hence, the Euclidean norm can be written in acoordinate-free way as‖x‖:=x⋅x.{\displaystyle \|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.}

The Euclidean norm is also called thequadratic norm,L2{\displaystyle L^{2}} norm,[12]ℓ2{\displaystyle \ell ^{2}} norm,2-norm, orsquare norm; seeLp{\displaystyle L^{p}} space.It defines adistance function called theEuclidean length,L2{\displaystyle L^{2}} distance, orℓ2{\displaystyle \ell ^{2}} distance.

The set of vectors inRn+1{\displaystyle \mathbb {R} ^{n+1}} whose Euclidean norm is a given positive constant forms ann{\displaystyle n}-sphere.

Euclidean norm of complex numbers

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See also:Dot product § Complex vectors

The Euclidean norm of acomplex number is theabsolute value (also called themodulus) of it, if thecomplex plane is identified with theEuclidean planeR2.{\displaystyle \mathbb {R} ^{2}.} This identification of the complex numberx+iy{\displaystyle x+iy} as a vector in the Euclidean plane, makes the quantityx2+y2{\textstyle {\sqrt {x^{2}+y^{2}}}} (as first suggested by Euler) the Euclidean norm associated with the complex number. Forz=x+iy{\displaystyle z=x+iy}, the norm can also be written asz¯z{\displaystyle {\sqrt {{\bar {z}}z}}} wherez¯{\displaystyle {\bar {z}}} is thecomplex conjugate ofz.{\displaystyle z\,.}

Quaternions and octonions

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See also:Quaternion andOctonion

There are exactly fourEuclidean Hurwitz algebras over thereal numbers. These are the real numbersR,{\displaystyle \mathbb {R} ,} the complex numbersC,{\displaystyle \mathbb {C} ,} thequaternionsH,{\displaystyle \mathbb {H} ,} and lastly theoctonionsO,{\displaystyle \mathbb {O} ,} where the dimensions of these spaces over the real numbers are1,2,4, and 8,{\displaystyle 1,2,4,{\text{ and }}8,} respectively.The canonical norms onR{\displaystyle \mathbb {R} } andC{\displaystyle \mathbb {C} } are theirabsolute value functions, as discussed previously.

The canonical norm onH{\displaystyle \mathbb {H} } ofquaternions is defined by‖q‖=qq∗ =q∗q =a2+b2+c2+d2 {\displaystyle \lVert q\rVert ={\sqrt {\,qq^{*}~}}={\sqrt {\,q^{*}q~}}={\sqrt {\,a^{2}+b^{2}+c^{2}+d^{2}~}}}for every quaternionq=a+bi+cj+dk{\displaystyle q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k} } inH.{\displaystyle \mathbb {H} .} This is the same as the Euclidean norm onH{\displaystyle \mathbb {H} } considered as the vector spaceR4.{\displaystyle \mathbb {R} ^{4}.} Similarly, the canonical norm on theoctonions is just the Euclidean norm onR8.{\displaystyle \mathbb {R} ^{8}.}

Finite-dimensional complex normed spaces

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On ann{\displaystyle n}-dimensionalcomplex spaceCn,{\displaystyle \mathbb {C} ^{n},} the most common norm is‖z‖:=|z1|2+⋯+|zn|2=z1z¯1+⋯+znz¯n.{\displaystyle \|{\boldsymbol {z}}\|:={\sqrt {\left|z_{1}\right|^{2}+\cdots +\left|z_{n}\right|^{2}}}={\sqrt {z_{1}{\bar {z}}_{1}+\cdots +z_{n}{\bar {z}}_{n}}}.}

In this case, the norm can be expressed as thesquare root of theinner product of the vector and itself:‖x‖:=xH x,{\displaystyle \|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}^{H}~{\boldsymbol {x}}}},}wherex{\displaystyle {\boldsymbol {x}}} is represented as acolumn vector[x1x2…xn]T{\displaystyle {\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{n}\end{bmatrix}}^{\rm {T}}} andxH{\displaystyle {\boldsymbol {x}}^{H}} denotes itsconjugate transpose.

This formula is valid for anyinner product space, including Euclidean and complex spaces. For complex spaces, the inner product is equivalent to thecomplex dot product. Hence the formula in this case can also be written using the following notation:‖x‖:=x⋅x.{\displaystyle \|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.}

Taxicab norm or Manhattan norm

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Main article:Taxicab geometry

‖x‖1:=∑i=1n|xi|.{\displaystyle \|{\boldsymbol {x}}\|_{1}:=\sum _{i=1}^{n}\left|x_{i}\right|.}The name relates to the distance a taxi has to drive in a rectangularstreet grid (like that of theNew York borough ofManhattan) to get from the origin to the pointx.{\displaystyle x.}

The set of vectors whose 1-norm is a given constant forms the surface of across polytope, which has dimension equal to the dimension of the vector space minus 1.The Taxicab norm is also called theℓ1{\displaystyle \ell ^{1}} norm. The distance derived from this norm is called theManhattan distance orℓ1{\displaystyle \ell ^{1}} distance.

The 1-norm is simply the sum of the absolute values of the columns.

In contrast,∑i=1nxi{\displaystyle \sum _{i=1}^{n}x_{i}}is not a norm because it may yield negative results.

p-norm

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Main article:Lp space

Letp≥1{\displaystyle p\geq 1} be a real number.Thep{\displaystyle p}-norm (also calledℓp{\displaystyle \ell ^{p}}-norm) of vectorx=(x1,…,xn){\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} is[10]‖x‖p:=(∑i=1n|xi|p)1/p.{\displaystyle \|\mathbf {x} \|_{p}:={\biggl (}\sum _{i=1}^{n}\left|x_{i}\right|^{p}{\biggr )}^{1/p}.}Forp=1,{\displaystyle p=1,} we get thetaxicab norm, forp=2{\displaystyle p=2} we get theEuclidean norm, and asp{\displaystyle p} approaches∞{\displaystyle \infty } thep{\displaystyle p}-norm approaches theinfinity norm ormaximum norm:‖x‖∞:=maxi|xi|.{\displaystyle \|\mathbf {x} \|_{\infty }:=\max _{i}\left|x_{i}\right|.}Thep{\displaystyle p}-norm is related to thegeneralized mean or power mean.

Forp=2,{\displaystyle p=2,} the‖⋅‖2{\displaystyle \|\,\cdot \,\|_{2}}-norm is even induced by a canonicalinner product⟨⋅,⋅⟩,{\displaystyle \langle \,\cdot ,\,\cdot \rangle ,} meaning that‖x‖2=⟨x,x⟩{\textstyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x} ,\mathbf {x} \rangle }}} for all vectorsx.{\displaystyle \mathbf {x} .} This inner product can be expressed in terms of the norm by using thepolarization identity.Onℓ2,{\displaystyle \ell ^{2},} this inner product is theEuclidean inner product defined by⟨(xn)n,(yn)n⟩ℓ2 = ∑nxn¯yn{\displaystyle \langle \left(x_{n}\right)_{n},\left(y_{n}\right)_{n}\rangle _{\ell ^{2}}~=~\sum _{n}{\overline {x_{n}}}y_{n}}while for the spaceL2(X,μ){\displaystyle L^{2}(X,\mu )} associated with ameasure space(X,Σ,μ),{\displaystyle (X,\Sigma ,\mu ),} which consists of allsquare-integrable functions, this inner product is⟨f,g⟩L2=∫Xf(x)¯g(x)dx.{\displaystyle \langle f,g\rangle _{L^{2}}=\int _{X}{\overline {f(x)}}g(x)\,\mathrm {d} x.}

This definition is still of some interest for0<p<1,{\displaystyle 0<p<1,} but the resulting function does not define a norm,[13] because it violates thetriangle inequality.What is true for this case of0<p<1,{\displaystyle 0<p<1,} even in the measurable analog, is that the correspondingLp{\displaystyle L^{p}} class is a vector space, and it is also true that the function∫X|f(x)−g(x)|p dμ{\displaystyle \int _{X}|f(x)-g(x)|^{p}~\mathrm {d} \mu }(withoutp{\displaystyle p}th root) defines a distance that makesLp(X){\displaystyle L^{p}(X)} into a complete metrictopological vector space. These spaces are of great interest infunctional analysis,probability theory andharmonic analysis.However, aside from trivial cases, this topological vector space is not locally convex, and has no continuous non-zero linear forms. Thus the topological dual space contains only the zero functional.

The partial derivative of thep{\displaystyle p}-norm is given by∂∂xk‖x‖p=xk|xk|p−2‖x‖pp−1.{\displaystyle {\frac {\partial }{\partial x_{k}}}\|\mathbf {x} \|_{p}={\frac {x_{k}\left|x_{k}\right|^{p-2}}{\|\mathbf {x} \|_{p}^{p-1}}}.}

The derivative with respect tox,{\displaystyle x,} therefore, is∂‖x‖p∂x=(x∘|x|p−2‖x‖pp−1)⊤.{\displaystyle {\frac {\partial \|\mathbf {x} \|_{p}}{\partial \mathbf {x} }}=\left({\frac {\mathbf {x} \circ |\mathbf {x} |^{p-2}}{\|\mathbf {x} \|_{p}^{p-1}}}\right)^{\top }.}where∘{\displaystyle \circ } denotesHadamard product and|⋅|{\displaystyle |\cdot |} is used for absolute value of each component of the vector.

For the special case ofp=2,{\displaystyle p=2,} this becomes∂∂xk‖x‖2=xk‖x‖2,{\displaystyle {\frac {\partial }{\partial x_{k}}}\|\mathbf {x} \|_{2}={\frac {x_{k}}{\|\mathbf {x} \|_{2}}},}or∂∂x‖x‖2=(x‖x‖2)⊤.{\displaystyle {\frac {\partial }{\partial \mathbf {x} }}\|\mathbf {x} \|_{2}=\left({\frac {\mathbf {x} }{\|\mathbf {x} \|_{2}}}\right)^{\top }.}

Maximum norm (special case of: infinity norm, uniform norm, or supremum norm)

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‖x‖∞=1{\displaystyle \|x\|_{\infty }=1}
Main article:Maximum norm

Ifx{\displaystyle \mathbf {x} } is some vector such thatx=(x1,x2,…,xn),{\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{n}),} then:‖x‖∞:=max(|x1|,…,|xn|).{\displaystyle \|\mathbf {x} \|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots ,\left|x_{n}\right|\right).}

The set of vectors whose infinity norm is a given constant,c,{\displaystyle c,} forms the surface of ahypercube with edge length2c.{\displaystyle 2c.}

Energy norm

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The energy norm[14] of a vectorx=(x1,x2,…,xn)∈Rn{\displaystyle {\boldsymbol {x}}=\left(x_{1},x_{2},\ldots ,x_{n}\right)\in \mathbb {R} ^{n}} is defined in terms of asymmetricpositive definite matrixA∈Rn{\displaystyle A\in \mathbb {R} ^{n}} as

‖x‖A:=xT⋅A⋅x.{\displaystyle {\|{\boldsymbol {x}}\|}_{A}:={\sqrt {{\boldsymbol {x}}^{T}\cdot A\cdot {\boldsymbol {x}}}}.}

It is clear that ifA{\displaystyle A} is theidentity matrix, this norm corresponds to theEuclidean norm. IfA{\displaystyle A} is diagonal, this norm is also called aweighted norm. The energy norm is induced by the inner product given by⟨x,y⟩A:=xT⋅A⋅y{\displaystyle \langle {\boldsymbol {x}},{\boldsymbol {y}}\rangle _{A}:={\boldsymbol {x}}^{T}\cdot A\cdot {\boldsymbol {y}}} forx,y∈Rn{\displaystyle {\boldsymbol {x}},{\boldsymbol {y}}\in \mathbb {R} ^{n}}.

In general, the value of the norm is dependent on thespectrum ofA{\displaystyle A}: For a vectorx{\displaystyle {\boldsymbol {x}}} with a Euclidean norm of one, the value of‖x‖A{\displaystyle {\|{\boldsymbol {x}}\|}_{A}} is bounded from below and above by the smallest and largest absoluteeigenvalues ofA{\displaystyle A} respectively, where the bounds are achieved ifx{\displaystyle {\boldsymbol {x}}} coincides with the corresponding (normalized) eigenvectors. Based on the symmetricmatrix square rootA1/2{\displaystyle A^{1/2}}, the energy norm of a vector can be written in terms of the standard Euclidean norm as

‖x‖A=‖A1/2x‖2.{\displaystyle {\|{\boldsymbol {x}}\|}_{A}={\|A^{1/2}{\boldsymbol {x}}\|}_{2}.}

Zero norm

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In probability and functional analysis, the zero norm induces a complete metric topology for the space ofmeasurable functions and for theF-space of sequences with F–norm(xn)↦∑n2−nxn/(1+xn).{\textstyle (x_{n})\mapsto \sum _{n}{2^{-n}x_{n}/(1+x_{n})}.}[15]Here we mean byF-norm some real-valued function‖⋅‖{\displaystyle \lVert \cdot \rVert } on an F-space with distanced,{\displaystyle d,} such that‖x‖=d(x,0).{\displaystyle \lVert x\rVert =d(x,0).} TheF-norm described above is not a norm in the usual sense because it lacks the required homogeneity property.

Hamming distance of a vector from zero

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See also:Hamming distance anddiscrete metric

Inmetric geometry, thediscrete 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 theHamming distance, which is important incoding andinformation 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.When applied component-wise to vectors, the discrete distance from zero behaves like a non-homogeneous "norm", which counts the number of non-zero components in its vector argument; again, this non-homogeneous "norm" is discontinuous.

Insignal processing andstatistics,David Donoho referred to thezero"norm" with quotation marks.Following Donoho's notation, the zero "norm" ofx{\displaystyle x} is simply the number of non-zero coordinates ofx,{\displaystyle x,} or the Hamming distance of the vector from zero.When this "norm" is localized to a bounded set, it is the limit ofp{\displaystyle p}-norms asp{\displaystyle p} approaches 0.Of course, the zero "norm" isnot truly a norm, because it is notpositive 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.Abusing terminology, some engineers[who?] omit Donoho's quotation marks and inappropriately call the number-of-non-zeros function theL0{\displaystyle L^{0}} norm, echoing the notation for theLebesgue space ofmeasurable functions.

Infinite dimensions

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The generalization of the above norms to an infinite number of components leads toℓp{\displaystyle \ell ^{p}} andLp{\displaystyle L^{p}} spaces forp≥1,{\displaystyle p\geq 1\,,} with norms

‖x‖p=(∑i∈N|xi|p)1/p and  ‖f‖p,X=(∫X|f(x)|p dx)1/p{\displaystyle \|x\|_{p}={\bigg (}\sum _{i\in \mathbb {N} }\left|x_{i}\right|^{p}{\bigg )}^{1/p}{\text{ and }}\ \|f\|_{p,X}={\bigg (}\int _{X}|f(x)|^{p}~\mathrm {d} x{\bigg )}^{1/p}}

for complex-valued sequences and functions onX⊆Rn{\displaystyle X\subseteq \mathbb {R} ^{n}} respectively, which can be further generalized (seeHaar measure). These norms are also valid in the limit asp→+∞{\displaystyle p\rightarrow +\infty }, giving asupremum norm, and are calledℓ∞{\displaystyle \ell ^{\infty }} andL∞.{\displaystyle L^{\infty }\,.}

Anyinner product induces in a natural way the norm‖x‖:=⟨x,x⟩.{\textstyle \|x\|:={\sqrt {\langle x,x\rangle }}.}

Other examples of infinite-dimensional normed vector spaces can be found in theBanach space article.

Generally, these norms do not give the same topologies. For example, an infinite-dimensionalℓp{\displaystyle \ell ^{p}} space gives astrictly finer topology than an infinite-dimensionalℓq{\displaystyle \ell ^{q}} space whenp<q.{\displaystyle p<q\,.}

Composite norms

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Other norms onRn{\displaystyle \mathbb {R} ^{n}} can be constructed by combining the above; for example‖x‖:=2|x1|+3|x2|2+max(|x3|,2|x4|)2{\displaystyle \|x\|:=2\left|x_{1}\right|+{\sqrt {3\left|x_{2}\right|^{2}+\max(\left|x_{3}\right|,2\left|x_{4}\right|)^{2}}}}is a norm onR4.{\displaystyle \mathbb {R} ^{4}.}

For any norm and anyinjectivelinear transformationA{\displaystyle A} we can define a new norm ofx,{\displaystyle x,} equal to‖Ax‖.{\displaystyle \|Ax\|.}In 2D, withA{\displaystyle A} a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. EachA{\displaystyle A} applied to the taxicab norm, up to inversion and interchanging of axes, gives a different unit ball: aparallelogram of a particular shape, size, and orientation.

In 3D, this is similar but different for the 1-norm (octahedrons) and the maximum norm (prisms with parallelogram base).

There are examples of norms that are not defined by "entrywise" formulas. For instance, theMinkowski functional of a centrally-symmetric convex body inRn{\displaystyle \mathbb {R} ^{n}} (centered at zero) defines a norm onRn{\displaystyle \mathbb {R} ^{n}} (see§ Classification of seminorms: absolutely convex absorbing sets below).

All the above formulas also yield norms onCn{\displaystyle \mathbb {C} ^{n}} without modification.

There are also norms on spaces of matrices (with real or complex entries), the so-calledmatrix norms.

In abstract algebra

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Main article:Field norm

LetE{\displaystyle E} be afinite extension of a fieldk{\displaystyle k} ofinseparable degreepμ,{\displaystyle p^{\mu },} and letk{\displaystyle k} have algebraic closureK.{\displaystyle K.} If the distinctembeddings ofE{\displaystyle E} are{σj}j,{\displaystyle \left\{\sigma _{j}\right\}_{j},} then theGalois-theoretic norm of an elementα∈E{\displaystyle \alpha \in E} is the value(∏jσk(α))pμ.{\textstyle \left(\prod _{j}{\sigma _{k}(\alpha )}\right)^{p^{\mu }}.} As that function is homogeneous of degree[E:k]{\displaystyle [E:k]}, the Galois-theoretic norm is not a norm in the sense of this article. However, the[E:k]{\displaystyle [E:k]}-th root of the norm (assuming that concept makes sense) is a norm.[16]

Composition algebras

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The concept of normN(z){\displaystyle N(z)} incomposition algebras doesnot share the usual properties of a norm sincenull vectors are allowed. A composition algebra(A,∗,N){\displaystyle (A,{}^{*},N)} consists of analgebra over a fieldA,{\displaystyle A,} aninvolution∗,{\displaystyle {}^{*},} and aquadratic formN(z)=zz∗{\displaystyle N(z)=zz^{*}} called the "norm".

The characteristic feature of composition algebras is thehomomorphism property ofN{\displaystyle N}: for the productwz{\displaystyle wz} of two elementsw{\displaystyle w} andz{\displaystyle z} of the composition algebra, its norm satisfiesN(wz)=N(w)N(z).{\displaystyle N(wz)=N(w)N(z).} In the case ofdivision algebrasR,{\displaystyle \mathbb {R} ,}C,{\displaystyle \mathbb {C} ,}H,{\displaystyle \mathbb {H} ,} andO{\displaystyle \mathbb {O} } the composition algebra norm is the square of the norm discussed above. In those cases the norm is adefinite quadratic form. In thesplit algebras the norm is anisotropic quadratic form.

Properties

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For any normp:X→R{\displaystyle p:X\to \mathbb {R} } on a vector spaceX,{\displaystyle X,} thereverse triangle inequality holds:p(x±y)≥|p(x)−p(y)| for all x,y∈X.{\displaystyle p(x\pm y)\geq |p(x)-p(y)|{\text{ for all }}x,y\in X.}Ifu:X→Y{\displaystyle u:X\to Y} is a continuous linear map between normed spaces, then the norm ofu{\displaystyle u} and the norm of thetranspose ofu{\displaystyle u} are equal.[17]

For theLp{\displaystyle L^{p}} norms, we haveHölder's inequality[18]|⟨x,y⟩|≤‖x‖p‖y‖q1p+1q=1.{\displaystyle |\langle x,y\rangle |\leq \|x\|_{p}\|y\|_{q}\qquad {\frac {1}{p}}+{\frac {1}{q}}=1.}A special case of this is theCauchy–Schwarz inequality:[18]|⟨x,y⟩|≤‖x‖2‖y‖2.{\displaystyle \left|\langle x,y\rangle \right|\leq \|x\|_{2}\|y\|_{2}.}

Illustrations ofunit circles in different norms.

Every norm is aseminorm and thus satisfies allproperties of the latter. In turn, every seminorm is asublinear function and thus satisfies allproperties of the latter. In particular, every norm is aconvex function.

Equivalence

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The concept ofunit circle (the set of all vectors of norm 1) is different in different norms: for the 1-norm, the unit circle is asquare oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unitcircle; while for the infinity norm, it is an axis-aligned square. For anyp{\displaystyle p}-norm, it is asuperellipse with congruent axes (see the accompanying illustration). Due to the definition of the norm, the unit circle must beconvex and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, andp≥1{\displaystyle p\geq 1} for ap{\displaystyle p}-norm).

In terms of the vector space, the seminorm defines atopology on the space, and this is aHausdorff 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. Asequence of vectors{vn}{\displaystyle \{v_{n}\}} is said toconverge in norm tov,{\displaystyle v,} if‖vn−v‖→0{\displaystyle \left\|v_{n}-v\right\|\to 0} asn→∞.{\displaystyle n\to \infty .} Equivalently, the topology consists of all sets that can be represented as a union of openballs. If(X,‖⋅‖){\displaystyle (X,\|\cdot \|)} is a normed space then[19]‖x−y‖=‖x−z‖+‖z−y‖ for all x,y∈X and z∈[x,y].{\displaystyle \|x-y\|=\|x-z\|+\|z-y\|{\text{ for all }}x,y\in X{\text{ and }}z\in [x,y].}

Two norms‖⋅‖α{\displaystyle \|\cdot \|_{\alpha }} and‖⋅‖β{\displaystyle \|\cdot \|_{\beta }} on a vector spaceX{\displaystyle X} are calledequivalent if they induce the same topology,[8] which happens if and only if there exist positive real numbersC{\displaystyle C} andD{\displaystyle D} such that for allx∈X{\displaystyle x\in X}C‖x‖α≤‖x‖β≤D‖x‖α.{\displaystyle C\|x\|_{\alpha }\leq \|x\|_{\beta }\leq D\|x\|_{\alpha }.}For instance, ifp>r≥1{\displaystyle p>r\geq 1} onCn,{\displaystyle \mathbb {C} ^{n},} then[20]‖x‖p≤‖x‖r≤n(1/r−1/p)‖x‖p.{\displaystyle \|x\|_{p}\leq \|x\|_{r}\leq n^{(1/r-1/p)}\|x\|_{p}.}

In particular,‖x‖2≤‖x‖1≤n‖x‖2{\displaystyle \|x\|_{2}\leq \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}}‖x‖∞≤‖x‖2≤n‖x‖∞{\displaystyle \|x\|_{\infty }\leq \|x\|_{2}\leq {\sqrt {n}}\|x\|_{\infty }}‖x‖∞≤‖x‖1≤n‖x‖∞,{\displaystyle \|x\|_{\infty }\leq \|x\|_{1}\leq n\|x\|_{\infty },}That is,‖x‖∞≤‖x‖2≤‖x‖1≤n‖x‖2≤n‖x‖∞.{\displaystyle \|x\|_{\infty }\leq \|x\|_{2}\leq \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}\leq n\|x\|_{\infty }.}If the vector space is a finite-dimensional real or complex one, all norms are equivalent. On the other hand, in the case of infinite-dimensional vector spaces, not all norms are equivalent.

Equivalent norms define the same notions of continuity and convergence and for many purposes do not need to be distinguished. To be more precise the uniform structure defined by equivalent norms on the vector space isuniformly isomorphic.

Classification of seminorms: absolutely convex absorbing sets

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Main article:Seminorm

All seminorms on a vector spaceX{\displaystyle X} can be classified in terms ofabsolutely convexabsorbing subsetsA{\displaystyle A} ofX.{\displaystyle X.} To each such subset corresponds a seminormpA{\displaystyle p_{A}} called thegauge ofA,{\displaystyle A,} defined aspA(x):=inf{r∈R:r>0,x∈rA}{\displaystyle p_{A}(x):=\inf\{r\in \mathbb {R} :r>0,x\in rA\}}whereinf{\displaystyle \inf _{}} is theinfimum, with the property that{x∈X:pA(x)<1} ⊆ A ⊆ {x∈X:pA(x)≤1}.{\displaystyle \left\{x\in X:p_{A}(x)<1\right\}~\subseteq ~A~\subseteq ~\left\{x\in X:p_{A}(x)\leq 1\right\}.}Conversely:

Anylocally convex topological vector space has alocal basis consisting of absolutely convex sets. A common method to construct such a basis is to use a family(p){\displaystyle (p)} of seminormsp{\displaystyle p} thatseparates points: the collection of all finite intersections of sets{p<1/n}{\displaystyle \{p<1/n\}} turns the space into alocally convex topological vector space so that every p iscontinuous.

Such a method is used to designweak and weak* topologies.

norm case:

Suppose now that(p){\displaystyle (p)} contains a singlep:{\displaystyle p:} since(p){\displaystyle (p)} isseparating,p{\displaystyle p} is a norm, andA={p<1}{\displaystyle A=\{p<1\}} is its openunit ball. ThenA{\displaystyle A} is an absolutely convexbounded neighbourhood of 0, andp=pA{\displaystyle p=p_{A}} is continuous.
The converse is due toAndrey Kolmogorov: any locally convex and locally bounded topological vector space isnormable. Precisely:
IfX{\displaystyle X} is an absolutely convex bounded neighbourhood of 0, the gaugegX{\displaystyle g_{X}} (so thatX={gX<1}{\displaystyle X=\{g_{X}<1\}} is a norm.

See also

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References

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  1. ^abKnapp, A.W. (2005).Basic Real Analysis. Birkhäuser. p. [1].ISBN 978-0-817-63250-2.
  2. ^"Pseudonorm".www.spektrum.de (in German). Retrieved2022-05-12.
  3. ^Hyers, D. H. (1939-09-01)."Pseudo-normed linear spaces and Abelian groups".Duke Mathematical Journal.5 (3).doi:10.1215/s0012-7094-39-00551-x.ISSN 0012-7094.
  4. ^Pugh, C.C. (2015).Real Mathematical Analysis. Springer. p. page 28.ISBN 978-3-319-17770-0.Prugovečki, E. (1981).Quantum Mechanics in Hilbert Space. p. page 20.
  5. ^abKubrusly 2011, p. 200.
  6. ^Rudin, W. (1991).Functional Analysis. p. 25.
  7. ^Narici & Beckenstein 2011, pp. 120–121.
  8. ^abcConrad, Keith."Equivalence of norms"(PDF).kconrad.math.uconn.edu. RetrievedSeptember 7, 2020.
  9. ^Wilansky 2013, pp. 20–21.
  10. ^abcWeisstein, Eric W."Vector Norm".mathworld.wolfram.com. Retrieved2020-08-24.
  11. ^Chopra, Anil (2012).Dynamics of Structures, 4th Ed. Prentice-Hall.ISBN 978-0-13-285803-8.
  12. ^Weisstein, Eric W."Norm".mathworld.wolfram.com. Retrieved2020-08-24.
  13. ^Except inR1,{\displaystyle \mathbb {R} ^{1},} where it coincides with the Euclidean norm, andR0,{\displaystyle \mathbb {R} ^{0},} where it is trivial.
  14. ^Saad, Yousef (2003),Iterative Methods for Sparse Linear Systems, p. 32,ISBN 978-0-89871-534-7
  15. ^Rolewicz, Stefan (1987),Functional analysis and control theory: Linear systems, Mathematics and its Applications (East European Series), vol. 29 (Translated from the Polish by Ewa Bednarczuk ed.), Dordrecht; Warsaw: D. Reidel Publishing Co.; PWN—Polish Scientific Publishers, pp. xvi, 524,doi:10.1007/978-94-015-7758-8,ISBN 90-277-2186-6,MR 0920371,OCLC 13064804
  16. ^Lang, Serge (2002) [1993].Algebra (Revised 3rd ed.). New York: Springer Verlag. p. 284.ISBN 0-387-95385-X.
  17. ^Trèves 2006, pp. 242–243.
  18. ^abGolub, Gene; Van Loan, Charles F. (1996).Matrix Computations (Third ed.). Baltimore: The Johns Hopkins University Press. p. 53.ISBN 0-8018-5413-X.
  19. ^Narici & Beckenstein 2011, pp. 107–113.
  20. ^"Relation between p-norms".Mathematics Stack Exchange.

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