Norm (mathematics)

In mathematics, a norm is a function from a real or

complex

vector space to the nonnegative 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 of a vector from the origin is a norm, called the Euclidean norm, or 2-norm, which may also be defined as the square root of the inner product of a vector with itself.

A pseudonorm or seminorm satisfies the first two properties of a norm, but may be zero for other vectors than the origin. 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''.

Definition

Given a vector space $X$ over a subfield of the complex numbers $\Complex,$ a norm on $X$ is a real-valued function $p : X \to \R$ with the following properties, where $, s,$ denotes the usual

absolute value

of a scalar $s$:

# Subadditivity/

Triangle inequality

: $p(x + y) \leq p(x) + p(y)$ for all $x, y \in X.$
# Absolute homogeneity: $p(s x) = \left, s\ p(x)$ for all $x \in X$ and all scalars $s.$
# Positive definiteness/: 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 $x = 0.$

A seminorm on $X$ is a function $p : X \to \R$ that has properties (1) and (2) 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 $p$ is a norm (or more generally, a seminorm) then $p(0) = 0$ and that $p$ also has the following property:

#

Nonnegativity: $p(x) \geq 0$ for all $x \in X.$

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

Equivalent norms

Suppose that and are two norms (or seminorms) on a vector space $X.$ Then and are called equivalent, if there exist two real constants and with such that for every vector $x \in X,$
$cq(x) \leq p(x)\leq Cq(x).$
The norms and 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 $p \colon X \to \R$ is given on a vector space , 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 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 $, x,$ with single vertical lines is also widespread.

In

LaTeX

and related markup languages, the double bar of norm notation is entered with the macro \, , which renders as $\, .$ The double vertical line used to denote parallel lines, parallel operator and parallel addition is entered with \parallel and is rendered as $\parallel.$ Although looking similar, these two macros must not be confused as \, denotes a

bracket

and \parallel denotes an operator. Therefore, their size and the spaces around them are not computed in the same way. Similarly, the single vertical bar is coded as , when used as a bracket, and as \mid when used as an operator.

In

Unicode

, the representation of the "double vertical line" character is . The "double vertical line" symbol should not be confused with the "parallel to" symbol, , which is intended to denote parallel lines and parallel operators. The double vertical line should also not be confused with , aimed to denote lateral clicks in linguistics.

The single vertical line , has a Unicode representation .

Examples

Every (real or complex) vector space admits a norm: If $x_ = \left(x_i\right)_$ is a Hamel basis for a vector space then the real-valued map that sends (where all but finitely many of the scalars are 0) 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

$\, x\, = , x,$
is a norm on the one-dimensional vector spaces formed by the real or

complex number

s.

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 $\mathbb$ is either $\R$ or and norm-preserving means that
This isomorphism is given by sending $1 \isin \mathbb$ to a vector of norm , which exists since such a vector is obtained by multiplying any nonzero vector by the inverse of its norm.

Euclidean norm

On the $n$-dimensional Euclidean space $\R^n,$ the intuitive notion of length of the vector $\boldsymbol = \left(x_1, x_2, \ldots, x_n\right)$ is captured by the formula
$\, \boldsymbol\, _2 := \sqrt.$

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.

The Euclidean norm is by far the most commonly used norm on $\R^n,$ 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.

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
$\, \boldsymbol\, := \sqrt.$

The Euclidean norm is also called the $L^2$ norm, $\ell^2$ norm, 2-norm, or square norm; see $L^p$ space.
It defines a

distance function

called the Euclidean length, $L^2$ distance, or $\ell^2$ distance.

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

Euclidean norm of complex numbers

The Euclidean norm of a

complex number

is the absolute value (also called the modulus) of it, if the complex plane is identified with the Euclidean plane $\R^2.$ This identification of the complex number $x + i y$ as a vector in the Euclidean plane, makes the quantity $\sqrt$ (as first suggested by Euler) the Euclidean norm associated with the complex number.

Quaternions and octonions

There are exactly four Euclidean Hurwitz algebras over the real numbers. These are the real numbers $\R,$ the complex numbers $\Complex,$ the

quaternion

s $\mathbb,$ and lastly the octonions $\mathbb,$ where the dimensions of these spaces over the real numbers are $1, 2, 4, \text 8,$ respectively.
The canonical norms on $\R$ and $\Complex$ are their

absolute value

functions, as discussed previously.

The canonical norm on $\mathbb$ of

quaternion

s is defined by
$\lVert q \rVert = \sqrt = \sqrt = \sqrt$
for every quaternion $q = a + b\,\mathbf i + c\,\mathbf j + d\,\mathbf k$ in $\mathbb.$ This is the same as the Euclidean norm on $\mathbb$ considered as the vector space $\R^4.$ Similarly, the canonical norm on the octonions is just the Euclidean norm on $\R^8.$

Finite-dimensional complex normed spaces

On an $n$-dimensional complex space $\Complex^n,$ the most common norm is
$\, \boldsymbol\, := \sqrt = \sqrt.$

In this case, the norm can be expressed as the

square root

of the inner product of the vector and itself:
$\, \boldsymbol\, := \sqrt,$
where $\boldsymbol$ is represented as a column vector \left(\left _1; x_2; \ldots, x_n\rightright), and $\boldsymbol^H$ 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:
$\, \boldsymbol\, := \sqrt.$

Taxicab norm or Manhattan norm

$\, \boldsymbol\, _1 := \sum_^n \left, x_i\.$
The name relates to the distance a taxi has to drive in a rectangular street grid 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 of dimension equivalent to that of the norm minus 1.
The Taxicab norm is also called the $\ell^1$ norm. The distance derived from this norm is called the

Manhattan distance

or ''ℓ''₁ distance.

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

In contrast,
$\sum_^n x_i$
is not a norm because it may yield negative results.

''p''-norm

Let be a real number.
The -norm (also called $\ell_p$-norm) of vector $\mathbf = (x_1, \ldots, x_n)$ is
$\left\, \mathbf\right\, _p := \left( \sum_^n \left, x_i\^p\right)^.$
For , we get the taxicab norm, for , we get the Euclidean norm, and as approaches $\infty$ the -norm approaches the infinity norm or maximum norm:
$\left\, \mathbf\right\, _\infty := \max_i \left, x_i\.$
The -norm is related to the generalized mean or power mean.

This definition is still of some interest for , but the resulting function does not define a norm, 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
$\int_X \left, f(x) - g(x)\^p ~ \mathrm d \mu$
(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
$\frac \, \mathbf\, _p = \frac .$

The derivative with respect to , therefore, is
$\frac =\frac .$
where denotes Hadamard product and $, \cdot,$ is used for absolute value of each component of the vector.

For the special case of , this becomes
$\frac \, \mathbf\, _2 = \frac,$
or
$\frac \left\, \mathbf\right\, _2 = \frac.$

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

If $\mathbf$ is some vector such that $\mathbf = (x_1, x_2, \ldots ,x_n),$ then:
$\, \mathbf\, _\infty := \max \left( \left, x_1\ , \ldots , \left, x_n\\right).$

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

hypercube

with edge length 2''c''.

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 $(x_n) \mapsto \sum_n.$
Here we mean by ''F-norm'' some real-valued function $\lVert \cdot \rVert$ 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

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-nonzeros function the ''L''⁰ 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 spaces, with norms

$\, x\, _p = \bigg( \sum_ \left, x_i\^p \bigg)^ \text\ \, f\, _ = \bigg( \int_X , f(x), ^p ~ \mathrm d x \bigg)^$

for complex-valued sequences and functions on $X \sube \R^n$ respectively, which can be further generalized (see Haar measure).

Any inner product induces in a natural way the norm

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

Composite norms

Other norms on $\R^n$ can be constructed by combining the above; for example
$\, x\, := 2 \left, x_1\ + \sqrt$
is a norm on

For any norm and any

injective

linear transformation we can define a new norm of , equal to
$\, A x\, .$
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 (

octahedron

s) 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 $\R^n$ (centered at zero) defines a norm on $\R^n$ (see below).

All the above formulas also yield norms on $\Complex^n$ without modification.

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

In abstract algebra

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 $\left(\prod_j \right)^.$ As that function is homogenous of degree , 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.

Composition algebras

The concept of norm $N(z)$ in composition algebras does ''not'' share the usual properties of a norm as it may be negative or zero for ''z'' ≠ 0. A composition algebra consists of an algebra over a field ''A'', an involution *, and a quadratic form $N(z) = zz^*,$ which is called the "norm".

The characteristic feature of composition algebras is the homomorphism property of ''N'': for the product ''wz'' of two elements ''w'' and ''z'' of the composition algebra, its norm satisfies $N(wz) = N(w)N(z).$ For $\R,$ $\Complex,$ $\mathbb,$ and O the composition algebra norm is the square of the norm discussed above. In those cases the norm is a definite quadratic form. In other composition algebras the norm is an isotropic quadratic form.

Properties

For any norm $p : X \to \R$ on a vector space $X,$ the reverse triangle inequality holds:
$p(x \pm y) \geq , p(x) - p(y), \text x, y \in X.$
If $u : X \to Y$ is a continuous linear map between normed spaces, then the norm of $u$ and the norm of the

transpose

of $u$ are equal.

For the ''L''^''p'' norms, we have Hölder's inequality
$, \langle x, y \rangle, \leq \, x\, _p \, y\, _q \qquad \frac + \frac = 1.$
A special case of this is the Cauchy–Schwarz inequality:
$\left, \langle x, y \rangle\ \leq \, x\, _2 \, y\, _2.$

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

, for the 2-norm (Euclidean norm), it is the well-known unit

circle

, while for the infinity norm, it is a different square. For any ''p''-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 $p \geq 1$ for a ''p''-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 $v,$ 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 open balls. If $(X, \, \cdot\, )$ is a normed space then
$\, x - y\, = \, x - z\, + \, z - y\, \text x, y \in X \text z \in$, y

Two norms $\, \cdot\, _\alpha$ and $\, \cdot\, _\beta$ on a vector space $X$ are called if 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 $\Complex^n,$ then
$\, x\, _p \leq \, x\, _r \leq n^ \, x\, _p.$

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

Classification of seminorms: absolutely convex absorbing sets

All seminorms on a vector space $X$ can be classified in terms of absolutely convex absorbing subsets ''A'' of $X.$ To each such subset corresponds a seminorm ''p_A'' called the

gauge

of ''A'', defined as
infimum