Norm (mathematics)

This article is about the concept in normed spaces. For field theory, seeField norm.For ideals, seeIdeal norm.For commutative algebra, seeAbsolute value (algebra).For group theory, seeNorm (group).For norms in descriptive set theory, seeprewellordering.

Inmathematics,anormis afunctionfrom a real or complexvector spaceto the non-negative real numbers that behaves in certain ways like the distance from theorigin:itcommuteswith scaling, obeys a form of thetriangle inequality,and is zero only at the origin. In particular, theEuclidean distancein aEuclidean spaceis defined by a norm on the associatedEuclidean vector space,called theEuclidean norm,the2-norm,or, sometimes, themagnitudeorlengthof the vector. This norm can be defined as thesquare rootof theinner productof a vector with itself.

Aseminormsatisfies the first two properties of a norm, but may be zero for vectors other than the origin.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 termpseudonormhas been used for several related meanings. It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "$\,\leq \,$"in the homogeneity axiom. It can also refer to a norm that can take infinite values,or to certain functions parametrised by adirected set.

Definition

Given avector space$X$over asubfield$F$of the complex numbers$\mathbb {C},$anormon$X$is areal-valued function$p:X\to \mathbb {R}$with the following properties, where$|s|$denotes the usualabsolute valueof a scalar$s$:

1. Subadditivity/Triangle inequality:$p(x+y)\leq p(x)+p(y)$for all$x,y\in X.$
2. Absolute homogeneity:$p(sx)=|s|p(x)$for all$x\in X$and all scalars$s.$
3. Positive definiteness/positiveness/Point-separating:for all$x\in X,$if$p(x)=0$then$x=0.$
   • Because property (2.) implies$p(0)=0,$some authors replace property (3.) with the equivalent condition: for every$x\in X,$$p(x)=0$if and only if$x=0.$

Aseminormon$X$is a function$p:X\to \mathbb {R}$that has properties (1.) and (2.)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 if$p$is a norm (or more generally, a seminorm) then$p(0)=0$and that$p$also has the following property:

4. Non-negativity:$p(x)\geq 0$for all$x\in X.$

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 ";these definitions are not equivalent.

Equivalent norms

Suppose that$p$and$q$are two norms (or seminorms) on a vector space$X.$Then$p$and$q$are calledequivalent,if there exist two positive real constants$c$and$C$with$c>0$such that for every vector$x\in X,$ $$cq(x)\leq p(x)\leq Cq(x).$$ The relation "$p$is equivalent to$q$"isreflexive,symmetric($cq\leq p\leq Cq$implies${\tfrac {1}{C}}p\leq q\leq {\tfrac {1}{c}}p$), andtransitiveand thus defines anequivalence relationon the set of all norms on$X.$ The norms$p$and$q$are equivalent if and only if they induce the same topology on$X.$Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.

Notation

If a norm$p:X\to \mathbb {R}$is given on a vector space$X,$then the norm of a vector$z\in X$is usually denoted by enclosing it within double vertical lines:$\|z\|=p(z).$Such notation is also sometimes used if$p$is only a seminorm. For the length of a vector in Euclidean space (which is an example of a norm, asexplained below), the notation$|x|$with single vertical lines is also widespread.

Examples

Every (real or complex) vector space admits a norm: If$x_{\bullet }=\left(x_{i}\right)_{i\in I}$is aHamel basisfor a vector space$X$then the real-valued map that sends$x=\sum _{i\in I}s_{i}x_{i}\in X$(where all but finitely many of the scalars$s_{i}$are$0$) to$\sum _{i\in I}\left|s_{i}\right|$is a norm on$X.$There are also a large number of norms that exhibit additional properties that make them useful for specific problems.

Absolute-value norm

Theabsolute value $|x|$ is a norm on the vector space formed by therealorcomplex numbers.The complex numbers form aone-dimensional vector spaceover themselves and a two-dimensional vector space over the reals; the absolute value is a norm for these two structures.

Any norm$p$on a one-dimensional vector space$X$is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preservingisomorphismof vector spaces$f:\mathbb {F} \to X,$where$\mathbb {F}$is either$\mathbb {R}$or$\mathbb {C},$and norm-preserving means that$|x|=p(f(x)).$ This isomorphism is given by sending$1\in \mathbb {F}$to a vector of norm$1,$which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm.

Euclidean norm

Further information:Euclidean normandEuclidean distance

On the$n$-dimensionalEuclidean space$\mathbb {R} ^{n},$the intuitive notion of length of the vector${\boldsymbol {x}}=\left(x_{1},x_{2},\ldots,x_{n}\right)$is captured by the formula $$\|{\boldsymbol {x}}\|_{2}:={\sqrt {x_{1}^{2}+\cdots +x_{n}^{2}}}.$$

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.

The Euclidean norm is by far the most commonly used norm on$\mathbb {R} ^{n},$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 productof two vectors of aEuclidean vector spaceis thedot productof theircoordinate vectorsover anorthonormal basis. Hence, the Euclidean norm can be written in a coordinate-free way as $$\|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.$$

The Euclidean norm is also called thequadratic norm,$L^{2}$norm,$\ell ^{2}$norm,2-norm,orsquare norm;see$L^{p}$space. It defines adistance functioncalled theEuclidean length,$L^{2}$distance,or$\ell ^{2}$distance.

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

Euclidean norm of complex numbers

See also:Dot product § Complex vectors

The Euclidean norm of acomplex numberis theabsolute value(also called themodulus) of it, if thecomplex planeis identified with theEuclidean plane$\mathbb {R} ^{2}.$This identification of the complex number$x+iy$as a vector in the Euclidean plane, makes the quantity${\textstyle {\sqrt {x^{2}+y^{2}}}}$(as first suggested by Euler) the Euclidean norm associated with the complex number. For$z=x+iy$,the norm can also be written as${\sqrt {{\bar {z}}z}}$where${\bar {z}}$is thecomplex conjugateof$z\,.$

Quaternions and octonions

See also:QuaternionandOctonion

There are exactly fourEuclidean Hurwitz algebrasover thereal numbers.These are the real numbers$\mathbb {R},$the complex numbers$\mathbb {C},$thequaternions$\mathbb {H},$and lastly theoctonions$\mathbb {O},$where the dimensions of these spaces over the real numbers are$1,2,4,{\text{ and }}8,$respectively. The canonical norms on$\mathbb {R}$and$\mathbb {C}$are theirabsolute valuefunctions, as discussed previously.

The canonical norm on$\mathbb {H}$ofquaternionsis defined by $$\lVert q\rVert ={\sqrt {\,qq^{*}~}}={\sqrt {\,q^{*}q~}}={\sqrt {\,a^{2}+b^{2}+c^{2}+d^{2}~}}$$ for every quaternion$q=a+b\,\mathbf {i} +c\,\mathbf {j} +d\,\mathbf {k}$in$\mathbb {H}.$This is the same as the Euclidean norm on$\mathbb {H}$considered as the vector space$\mathbb {R} ^{4}.$Similarly, the canonical norm on theoctonionsis just the Euclidean norm on$\mathbb {R} ^{8}.$

Finite-dimensional complex normed spaces

On an$n$-dimensionalcomplex space$\mathbb {C} ^{n},$the most common norm is $$\|{\boldsymbol {z}}\|:={\sqrt {\left|z_{1}\right|^{2}+\cdots +\left|z_{n}\right|^{2}}}={\sqrt {z_{1}{\bar {z}}_{1}+\cdots +z_{n}{\bar {z}}_{n}}}.$$

In this case, the norm can be expressed as thesquare rootof theinner productof the vector and itself: $$\|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}^{H}~{\boldsymbol {x}}}},$$ where${\boldsymbol {x}}$is represented as acolumn vector${\begin{bmatrix}x_{1}\;x_{2}\;\dots \;x_{n}\end{bmatrix}}^{\rm {T}}$and${\boldsymbol {x}}^{H}$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: $$\|{\boldsymbol {x}}\|:={\sqrt {{\boldsymbol {x}}\cdot {\boldsymbol {x}}}}.$$

Taxicab norm or Manhattan norm

Main article:Taxicab geometry

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

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 the$\ell ^{1}$norm.The distance derived from this norm is called theManhattan distanceor$\ell ^{1}$distance.

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

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

p-norm

Main article:L^pspace

Let$p\geq 1$be a real number. The$p$-norm (also called$\ell ^{p}$-norm) of vector$\mathbf {x} =(x_{1},\ldots,x_{n})$is $$\|\mathbf {x} \|_{p}:=\left(\sum _{i=1}^{n}\left|x_{i}\right|^{p}\right)^{1/p}.$$ For$p=1,$we get thetaxicab norm,for$p=2$we get theEuclidean norm,and as$p$approaches$\infty$the$p$-norm approaches theinfinity normormaximum norm: $$\|\mathbf {x} \|_{\infty }:=\max _{i}\left|x_{i}\right|.$$ The$p$-norm is related to thegeneralized meanor power mean.

For$p=2,$the$\|\,\cdot \,\|_{2}$-norm is even induced by a canonicalinner product$\langle \,\cdot,\,\cdot \rangle,$meaning that${\textstyle \|\mathbf {x} \|_{2}={\sqrt {\langle \mathbf {x},\mathbf {x} \rangle }}}$for all vectors$\mathbf {x}.$This inner product can be expressed in terms of the norm by using thepolarization identity. On$\ell ^{2},$this inner product is theEuclidean inner productdefined by $$\langle \left(x_{n}\right)_{n},\left(y_{n}\right)_{n}\rangle _{\ell ^{2}}~=~\sum _{n}{\overline {x_{n}}}y_{n}$$ while for the space$L^{2}(X,\mu )$associated with ameasure space$(X,\Sigma,\mu ),$which consists of allsquare-integrable functions,this inner product is $$\langle f,g\rangle _{L^{2}}=\int _{X}{\overline {f(x)}}g(x)\,\mathrm {d} x.$$

This definition is still of some interest for$0<p<1,$but the resulting function does not define a norm,because it violates thetriangle inequality. What is true for this case of$0<p<1,$even in the measurable analog, is that the corresponding$L^{p}$class is a vector space, and it is also true that the function $$\int _{X}|f(x)-g(x)|^{p}~\mathrm {d} \mu$$ (without$p$th root) defines a distance that makes$L^{p}(X)$into a complete metrictopological vector space.These spaces are of great interest infunctional analysis,probability theoryandharmonic 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 the$p$-norm is given by $${\frac {\partial }{\partial x_{k}}}\|\mathbf {x} \|_{p}={\frac {x_{k}\left|x_{k}\right|^{p-2}}{\|\mathbf {x} \|_{p}^{p-1}}}.$$

The derivative with respect to$x,$therefore, is $${\frac {\partial \|\mathbf {x} \|_{p}}{\partial \mathbf {x} }}={\frac {\mathbf {x} \circ |\mathbf {x} |^{p-2}}{\|\mathbf {x} \|_{p}^{p-1}}}.$$ where$\circ$denotesHadamard productand$|\cdot |$is used for absolute value of each component of the vector.

For the special case of$p=2,$this becomes $${\frac {\partial }{\partial x_{k}}}\|\mathbf {x} \|_{2}={\frac {x_{k}}{\|\mathbf {x} \|_{2}}},$$ or $${\frac {\partial }{\partial \mathbf {x} }}\|\mathbf {x} \|_{2}={\frac {\mathbf {x} }{\|\mathbf {x} \|_{2}}}.$$

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

$\|x\|_{\infty }=1$

Main article:Maximum norm

If$\mathbf {x}$is some vector such that$\mathbf {x} =(x_{1},x_{2},\ldots,x_{n}),$then: $$\|\mathbf {x} \|_{\infty }:=\max \left(\left|x_{1}\right|,\ldots,\left|x_{n}\right|\right).$$

The set of vectors whose infinity norm is a given constant,$c,$forms the surface of ahypercubewith edge length$2c.$

Zero norm

In probability and functional analysis, the zero norm induces a complete metric topology for the space ofmeasurable functionsand for theF-spaceof sequences with F–norm${\textstyle (x_{n})\mapsto \sum _{n}{2^{-n}x_{n}/(1+x_{n})}.}$ Here we mean byF-normsome real-valued function$\lVert \cdot \rVert$on an F-space with distance$d,$such that$\lVert x\rVert =d(x,0).$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

See also:Hamming distanceanddiscrete metric

Inmetric geometry,thediscrete metrictakes 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 incodingandinformation 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 processingandstatistics,David Donohoreferred to thezero"norm"with quotation marks. Following Donoho's notation, the zero "norm" of$x$is simply the number of non-zero coordinates of$x,$or the Hamming distance of the vector from zero. When this "norm" is localized to a bounded set, it is the limit of$p$-norms as$p$approaches 0. Of course, the zero "norm" isnottruly 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 engineersomit Donoho's quotation marks and inappropriately call the number-of-non-zeros function the$L^{0}$norm, echoing the notation for theLebesgue spaceofmeasurable functions.

Infinite dimensions

The generalization of the above norms to an infinite number of components leads to$\ell ^{p}$and$L^{p}$spacesfor$p\geq 1\,,$with norms

$$\|x\|_{p}={\bigg (}\sum _{i\in \mathbb {N} }\left|x_{i}\right|^{p}{\bigg )}^{1/p}{\text{ and }}\ \|f\|_{p,X}={\bigg (}\int _{X}|f(x)|^{p}~\mathrm {d} x{\bigg )}^{1/p}$$

for complex-valued sequences and functions on$X\subseteq \mathbb {R} ^{n}$respectively, which can be further generalized (seeHaar measure). These norms are also valid in the limit as$p\rightarrow +\infty$,giving asupremum norm,and are called$\ell ^{\infty }$and$L^{\infty }\,.$

Anyinner productinduces in a natural way the norm${\textstyle \|x\|:={\sqrt {\langle x,x\rangle }}.}$

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

Generally, these norms do not give the same topologies. For example, an infinite-dimensional$\ell ^{p}$space gives astrictly finer topologythan an infinite-dimensional$\ell ^{q}$space when$p<q\,.$

Composite norms

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

For any norm and anyinjectivelinear transformation$A$we can define a new norm of$x,$equal to $$\|Ax\|.$$ In 2D, with$A$a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. Each$A$applied to the taxicab norm, up to inversion and interchanging of axes, gives a different unit ball: aparallelogramof a particular shape, size, and orientation.

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

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

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

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

In abstract algebra

Main article:Field norm

Let$E$be afinite extensionof a field$k$ofinseparable degree$p^{\mu },$and let$k$have algebraic closure$K.$If the distinctembeddingsof$E$are$\left\{\sigma _{j}\right\}_{j},$then theGalois-theoretic normof an element$\alpha \in E$is the value${\textstyle \left(\prod _{j}{\sigma _{k}(\alpha )}\right)^{p^{\mu }}.}$As that function is homogeneous of degree$[E:k]$,the Galois-theoretic norm is not a norm in the sense of this article. However, the$[E:k]$-th root of the norm (assuming that concept makes sense) is a norm.

Composition algebras

The concept of norm$N(z)$incomposition algebrasdoesnotshare the usual properties of a norm sincenull vectorsare allowed. A composition algebra$(A,{}^{*},N)$consists of analgebra over a field$A,$aninvolution${}^{*},$and aquadratic form$N(z)=zz^{*}$called the "norm".

The characteristic feature of composition algebras is thehomomorphismproperty of$N$:for the product$wz$of two elements$w$and$z$of the composition algebra, its norm satisfies$N(wz)=N(w)N(z).$In the case ofdivision algebras$\mathbb {R},$$\mathbb {C},$$\mathbb {H},$and$\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 algebrasthe norm is anisotropic quadratic form.

Properties

For any norm$p:X\to \mathbb {R}$on a vector space$X,$thereverse triangle inequalityholds: $$p(x\pm y)\geq |p(x)-p(y)|{\text{ for all }}x,y\in X.$$ If$u:X\to Y$is a continuous linear map between normed spaces, then the norm of$u$and the norm of thetransposeof$u$are equal.

For the$L^{p}$norms,we haveHölder's inequality $$|\langle x,y\rangle |\leq \|x\|_{p}\|y\|_{q}\qquad {\frac {1}{p}}+{\frac {1}{q}}=1.$$ A special case of this is theCauchy–Schwarz inequality: $$\left|\langle x,y\rangle \right|\leq \|x\|_{2}\|y\|_{2}.$$

Illustrations ofunit circlesin different norms.

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

Equivalence

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 asquareoriented 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 any$p$-norm, it is asuperellipsewith congruent axes (see the accompanying illustration). Due to the definition of the norm, the unit circle must beconvexand centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, and$p\geq 1$for a$p$-norm).

In terms of the vector space, the seminorm defines atopologyon the space, and this is aHausdorfftopology 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. Asequenceof vectors$\{v_{n}\}$is said toconvergein norm to$v,$if$\left\|v_{n}-v\right\|\to 0$as$n\to \infty.$Equivalently, the topology consists of all sets that can be represented as a union of openballs.If$(X,\|\cdot \|)$is a normed space then $\|x-y\|=\|x-z\|+\|z-y\|{\text{ for all }}x,y\in X{\text{ and }}z\in [x,y].$

Two norms$\|\cdot \|_{\alpha }$and$\|\cdot \|_{\beta }$on a vector space$X$are calledequivalentif they induce the same topology,which happens if and only if there exist positive real numbers$C$and$D$such that for all$x\in X$ $$C\|x\|_{\alpha }\leq \|x\|_{\beta }\leq D\|x\|_{\alpha }.$$ For instance, if$p>r\geq 1$on$\mathbb {C} ^{n},$then $$\|x\|_{p}\leq \|x\|_{r}\leq n^{(1/r-1/p)}\|x\|_{p}.$$

In particular, $$\|x\|_{2}\leq \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}$$ $$\|x\|_{\infty }\leq \|x\|_{2}\leq {\sqrt {n}}\|x\|_{\infty }$$ $$\|x\|_{\infty }\leq \|x\|_{1}\leq n\|x\|_{\infty },$$ That is, $$\|x\|_{\infty }\leq \|x\|_{2}\leq \|x\|_{1}\leq {\sqrt {n}}\|x\|_{2}\leq n\|x\|_{\infty }.$$ 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

Main article:Seminorm

All seminorms on a vector space$X$can be classified in terms ofabsolutely convexabsorbing subsets$A$of$X.$To each such subset corresponds a seminorm$p_{A}$called thegaugeof$A,$defined as $$p_{A}(x):=\inf\{r\in \mathbb {R}:r>0,x\in rA\}$$ where$\inf _{}$is theinfimum,with the property that $$\left\{x\in X:p_{A}(x)<1\right\}~\subseteq ~A~\subseteq ~\left\{x\in X:p_{A}(x)\leq 1\right\}.$$ Conversely:

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

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

norm case:

Suppose now that$(p)$contains a single$p:$since$(p)$isseparating,$p$is a norm, and$A=\{p<1\}$is its openunit ball.Then$A$is an absolutely convexboundedneighbourhood of 0, and$p=p_{A}$is continuous.

The converse is due toAndrey Kolmogorov:any locally convex and locally bounded topological vector space isnormable.Precisely:

If$X$is an absolutely convex bounded neighbourhood of 0, the gauge$g_{X}$(so that$X=\{g_{X}<1\}$is a norm.

See also

• Asymmetric norm– Generalization of the concept of a norm
• F-seminorm– A topological vector space whose topology can be defined by a metricPages displaying short descriptions of redirect targets
• Gowers norm
• Kadec norm– All infinite-dimensional, separable Banach spaces are homeomorphicPages displaying short descriptions of redirect targets
• Least-squares spectral analysis– Periodicity computation method
• Mahalanobis distance– Statistical distance measure
• Magnitude (mathematics)– Property determining comparison and ordering
• Matrix norm– Norm on a vector space of matrices
• Minkowski distance– Mathematical metric in normed vector space
• Minkowski functional– Function made from a set
• Operator norm– Measure of the "size" of linear operators
• Paranorm– A topological vector space whose topology can be defined by a metricPages displaying short descriptions of redirect targets
• Relation of norms and metrics– Mathematical space with a notion of distancePages displaying short descriptions of redirect targets
• Seminorm– nonnegative-real-valued function on a real or complex vector space that satisfies the triangle inequality and is absolutely homogenousPages displaying wikidata descriptions as a fallback
• Sublinear function– Type of function in linear algebra

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