Norm (mathematics) explained

In mathematics, a norm is a 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: it 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, the 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.^[1] 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.

The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm".^[2] A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "

" in the homogeneity axiom.^[3] It can also refer to a norm that can take infinite values,^[4] or to certain functions parametrised by a directed set.^[5]

Definition

over a subfield of the complex numbers a norm on is a real-valued function with the following properties, where denotes the usual absolute value of a scalar :^[6]

1. Subadditivity/Triangle inequality:

for all

1. Absolute homogeneity:

for all and all scalars

1. Positive definiteness/positiveness/: for all

if then

1. • Because property (2.) implies

some authors replace property (3.) with the equivalent condition: for every if and only if

A seminorm on

is a function that has properties (1.) and (2.)^[7] so that in particular, every norm is also a seminorm (and thus also a sublinear functional). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that if is a norm (or more generally, a seminorm) then and that also has the following property:

4. Non-negativity:

for all

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

Equivalent norms

Suppose that

and are two norms (or seminorms) on a vector space Then and are called equivalent, if there exist two positive real constants and with such that for every vector $$c\; q(x)\; \backslash leq\; p(x)\; \backslash leq\; C\; q(x).$$The relation " is equivalent to " is reflexive, symmetric ( implies ), and transitive and thus defines an equivalence relation on the set of all norms on The norms and are equivalent if and only if they induce the same topology on ^[8] Any two norms on a finite-dimensional space are equivalent but this does not extend to infinite-dimensional spaces.^[8]

Notation

If a norm

is given on a vector space then the norm of a vector is usually denoted by enclosing it within double vertical lines: Such notation is also sometimes used if is only a seminorm. For the length of a vector in Euclidean space (which is an example of a norm, as explained below), the notation with single vertical lines is also widespread.

Examples

Every (real or complex) vector space admits a norm: If

is a Hamel basis for a vector space then the real-valued map that sends (where all but finitely many of the scalars are ) to is a norm on There are also a large number of norms that exhibit additional properties that make them useful for specific problems.

Absolute-value norm

The absolute value

is a norm on the vector space formed by the real or complex numbers. The complex numbers form a 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.

Any norm

on a one-dimensional vector space is equivalent (up to scaling) to the absolute value norm, meaning that there is a norm-preserving isomorphism of vector spaces where is either or and norm-preserving means that This isomorphism is given by sending to a vector of norm which exists since such a vector is obtained by multiplying any non-zero vector by the inverse of its norm.

Euclidean norm

On the

-dimensional Euclidean space the intuitive notion of length of the vector is captured by the formula^[9] $$\backslash |\backslash boldsymbol\backslash |\_2\; :=\; \backslash sqrt.$$

This is the Euclidean norm, which gives the ordinary distance from the origin to the point X—a consequence of the Pythagorean theorem.This operation may also be referred to as "SRSS", which is an acronym for the square root of the sum of squares.^[10]

The Euclidean norm is by far the most commonly used norm on

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 vectors over an orthonormal basis.Hence, the Euclidean norm can be written in a coordinate-free way as$$\backslash |\backslash boldsymbol\backslash |\; :=\; \backslash sqrt.$$

The Euclidean norm is also called the quadratic norm,

norm,^[11] norm, 2-norm, or square norm; see

L^p

space.It defines a distance function called the Euclidean length, distance, or distance.

The set of vectors in

whose Euclidean norm is a given positive constant forms an

n

-sphere.

Euclidean norm of complex numbers

This identification of the complex number as a vector in the Euclidean plane, makes the quantity $\backslash sqrt$ (as first suggested by Euler) the Euclidean norm associated with the complex number. For , the norm can also be written as where is the complex conjugate of

Quaternions and octonions

See also: Quaternion and Octonion.

There are exactly four Euclidean Hurwitz algebras over the real numbers. These are the real numbers

the complex numbers the quaternions and lastly the octonions where the dimensions of these spaces over the real numbers are respectively.The canonical norms on and are their absolute value functions, as discussed previously.

The canonical norm on

of quaternions is defined by$$\backslash lVert\; q\; \backslash rVert\; =\; \backslash sqrt\; =\; \backslash sqrt\; =\; \backslash sqrt$$for every quaternion in This is the same as the Euclidean norm on considered as the vector space Similarly, the canonical norm on the octonions is just the Euclidean norm on

Finite-dimensional complex normed spaces

On an

-dimensional complex space the most common norm is$$\backslash |\backslash boldsymbol\backslash |\; :=\; \backslash sqrt\; =\; \backslash sqrt.$$

In this case, the norm can be expressed as the square root of the inner product of the vector and itself:$$\backslash |\backslash boldsymbol\backslash |\; :=\; \backslash sqrt,$$where

is represented as a column vector and denotes its conjugate transpose.

This formula is valid for any inner product space, including Euclidean and complex spaces. For complex spaces, the inner product is equivalent to the complex dot product. Hence the formula in this case can also be written using the following notation:$$\backslash |\backslash boldsymbol\backslash |\; :=\; \backslash sqrt.$$

Taxicab norm or Manhattan norm

See main article: Taxicab geometry.

$$\backslash |\backslash boldsymbol\backslash |\_1\; :=\; \backslash sum\_^n\; \backslash left|x\_i\backslash right|.$$The name relates to the distance a taxi has to drive in a rectangular street grid (like that of the New York borough of Manhattan) to get from the origin to the point

The set of vectors whose 1-norm is a given constant forms the surface of a cross polytope, which has dimension equal to the dimension of the vector space minus 1.The Taxicab norm is also called the

norm. The distance derived from this norm is called the Manhattan distance or distance.

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

In contrast,$$\backslash sum\_^n\; x\_i$$is not a norm because it may yield negative results.

p-norm

See main article: L^p space.

Let

be a real number.The -norm (also called -norm) of vector is$$\backslash |\backslash mathbf\backslash |\_p\; :=\; \backslash left(\backslash sum\_^n\; \backslash left|x\_i\backslash right|^p\backslash right)^.$$For we get the taxicab norm, for we get the Euclidean norm, and as approaches the -norm approaches the infinity norm or maximum norm:$$\backslash |\backslash mathbf\backslash |\_\backslash infty\; :=\; \backslash max\_i\; \backslash left|x\_i\backslash right|.$$The -norm is related to the generalized mean or power mean.

For

the -norm is even induced by a canonical inner product meaning that $\backslash |\backslash mathbf\backslash |\_2\; =\; \backslash sqrt$ for all vectors This inner product can be expressed in terms of the norm by using the polarization identity.On this inner product is the defined by$$\backslash langle\; \backslash left(x\_n\backslash right)\_,\; \backslash left(y\_n\backslash right)\_\; \backslash rangle\_\; ~=~\; \backslash sum\_n\; \backslash overline\; y\_n$$while for the space associated with a measure space which consists of all square-integrable functions, this inner product is$$\backslash langle\; f,\; g\; \backslash rangle\_\; =\; \backslash int\_X\; \backslash overline\; g(x)\backslash ,\; \backslash mathrm\; dx.$$

This definition is still of some interest for

but the resulting function does not define a norm,^[12] because it violates the triangle inequality.What is true for this case of even in the measurable analog, is that the corresponding class is a vector space, and it is also true that the function$$\backslash int\_X\; |f(x)\; -\; g(x)|^p\; ~\; \backslash mathrm\; d\; \backslash mu$$(without th root) defines a distance that makes into a complete metric topological vector space. These spaces are of great interest in functional analysis, probability theory and harmonic 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

-norm is given by$$\backslash frac\; \backslash |\backslash mathbf\backslash |\_p\; =\; \backslash frac\; .$$

The derivative with respect to

therefore, is$$\backslash frac\; =\backslash frac\; .$$where denotes Hadamard product and is used for absolute value of each component of the vector.

For the special case of

this becomes$$\backslash frac\; \backslash |\backslash mathbf\backslash |\_2\; =\; \backslash frac,$$or$$\backslash frac\; \backslash |\backslash mathbf\backslash |\_2\; =\; \backslash frac.$$

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

See main article: Maximum norm. If

is some vector such that then:$$\backslash |\backslash mathbf\backslash |\_\backslash infty\; :=\; \backslash max\; \backslash left(\backslash left|x\_1\backslash right|,\; \backslash ldots,\; \backslash left|x\_n\backslash right|\backslash right).$$

The set of vectors whose infinity norm is a given constant,

forms the surface of a hypercube with edge length

Zero norm

In probability and functional analysis, the zero norm induces a complete metric topology for the space of measurable functions and for the F-space of sequences with F–norm $(x\_n)\; \backslash mapsto\; \backslash sum\_n.$Here we mean by F-norm some real-valued function

on an F-space with distance such that The F-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 distance and discrete metric.

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 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.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.

In signal processing and statistics, David Donoho referred to the zero "norm" with quotation marks.Following Donoho's notation, the zero "norm" of

is simply the number of non-zero coordinates of or the Hamming distance of the vector from zero.When this "norm" is localized to a bounded set, it is the limit of -norms as approaches 0.Of course, the zero "norm" is not truly a norm, because it is not 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.Abusing terminology, some engineers omit Donoho's quotation marks and inappropriately call the number-of-non-zeros function the norm, echoing the notation for the Lebesgue space of measurable functions.

Infinite dimensions

The generalization of the above norms to an infinite number of components leads to

and

L^p

spaces for

with norms

$$\backslash |x\backslash |\_p\; =\; \backslash bigg(\backslash sum\_\; \backslash left|x\_i\backslash right|^p\backslash bigg)^\; \backslash text\backslash \; \backslash |f\backslash |\_\; =\; \backslash bigg(\backslash int\_X\; |f(x)|^p\; ~\; \backslash mathrm\; d\; x\backslash bigg)^$$

for complex-valued sequences and functions on

respectively, which can be further generalized (see Haar measure). These norms are also valid in the limit as , giving a supremum norm, and are called and

Any inner product induces in a natural way the norm $\backslash |x\backslash |\; :=\; \backslash sqrt.$

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

Generally, these norms do not give the same topologies. For example, an infinite-dimensional

space gives a strictly finer topology than an infinite-dimensional space when

Composite norms

Other norms on

can be constructed by combining the above; for example$$\backslash |x\backslash |\; :=\; 2\; \backslash left|x\_1\backslash right|\; +\; \backslash sqrt$$is a norm on we can define a new norm of equal to$$\backslash |A\; x\backslash |.$$In 2D, with a rotation by 45° and a suitable scaling, this changes the taxicab norm into the maximum norm. Each applied to the taxicab norm, up to inversion and interchanging of axes, gives a different unit ball: a parallelogram 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, the Minkowski functional of a centrally-symmetric convex body in

(centered at zero) defines a norm on (see below).

All the above formulas also yield norms on

without modification.

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

In abstract algebra

See main article: Field norm.

Let

be a finite extension of a field of inseparable degree and let have algebraic closure If the distinct embeddings of are then the Galois-theoretic norm of an element is the value $\backslash left(\backslash prod\_j\; \backslash right)^.$ 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 -th root of the norm (assuming that concept makes sense) is a norm.^[13]

Composition algebras

The concept of norm

in composition algebras does share the usual properties of a norm since null vectors are allowed. A composition algebra consists of an algebra over a field an involution and a quadratic form

N(z)=zz^*

called the "norm".

The characteristic feature of composition algebras is the homomorphism property of

: for the product of two elements and of the composition algebra, its norm satisfies In the case of division algebras and the composition algebra norm is the square of the norm discussed above. In those cases the norm is a definite quadratic form. In the split algebras the norm is an isotropic quadratic form.

Properties

For any norm

on a vector space the reverse triangle inequality holds:$$p(x\; \backslash pm\; y)\; \backslash geq\; |p(x)\; -\; p(y)|\; \backslash text\; x,\; y\; \backslash in\; X.$$If is a continuous linear map between normed spaces, then the norm of and the norm of the transpose of are equal.

For the

norms, we have Hölder's inequality^[14] $$|\backslash langle\; x,\; y\; \backslash rangle|\; \backslash leq\; \backslash |x\backslash |\_p\; \backslash |y\backslash |\_q\; \backslash qquad\; \backslash frac\; +\; \backslash frac\; =\; 1.$$A special case of this is the Cauchy–Schwarz inequality:$$\backslash left|\backslash langle\; x,\; y\; \backslash rangle\backslash right|\; \backslash leq\; \backslash |x\backslash |\_2\; \backslash |y\backslash |\_2.$$

Every norm is a seminorm and thus satisfies all properties of the latter. In turn, every seminorm is a sublinear function and thus satisfies all properties of the latter. In particular, every norm is a convex function.

Equivalence

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 oriented as a diamond; for the 2-norm (Euclidean norm), it is the well-known unit circle; while for the infinity norm, it is an axis-aligned square. For any

-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 and centrally symmetric (therefore, for example, the unit ball may be a rectangle but cannot be a triangle, and for a -norm).

In terms of the vector space, the seminorm defines a topology on the space, and this is a 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

is said to converge in norm to if as Equivalently, the topology consists of all sets that can be represented as a union of open balls. If is a normed space then

Two norms

and on a vector space are called if they induce the same topology,^[15] which happens if and only if there exist positive real numbers and such that for all $$C\; \backslash |x\backslash |\_\backslash alpha\; \backslash leq\; \backslash |x\backslash |\_\backslash beta\; \backslash leq\; D\; \backslash |x\backslash |\_\backslash alpha.$$For instance, if on then^[16] $$\backslash |x\backslash |\_p\; \backslash leq\; \backslash |x\backslash |\_r\; \backslash leq\; n^\; \backslash |x\backslash |\_p.$$

In particular,$$\backslash |x\backslash |\_2\; \backslash leq\; \backslash |x\backslash |\_1\; \backslash leq\; \backslash sqrt\; \backslash |x\backslash |\_2$$$$\backslash |x\backslash |\_\backslash infty\; \backslash leq\; \backslash |x\backslash |\_2\; \backslash leq\; \backslash sqrt\; \backslash |x\backslash |\_\backslash infty$$$$\backslash |x\backslash |\_\backslash infty\; \backslash leq\; \backslash |x\backslash |\_1\; \backslash leq\; n\; \backslash |x\backslash |\_\backslash infty,$$That is,$$\backslash |x\backslash |\_\backslash infty\; \backslash leq\; \backslash |x\backslash |\_2\; \backslash leq\; \backslash |x\backslash |\_1\; \backslash leq\; \backslash sqrt\; \backslash |x\backslash |\_2\; \backslash leq\; n\; \backslash |x\backslash |\_\backslash infty.$$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 is uniformly isomorphic.

Classification of seminorms: absolutely convex absorbing sets

See main article: Seminorm.

All seminorms on a vector space

can be classified in terms of absolutely convex absorbing subsets of To each such subset corresponds a seminorm called the gauge of defined as

inf